THE UNIVERSITY OF MICHIGAN
INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING
THE EFFECT OF AXIAL TURBULENCE PROMOTERS ON
HEAT TRANSFER AND PRESSURE DROP INSIDE A TUBE
Lawrence B. Evans
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy in the
University of Michigan
Department of Chemical and Metallurgical Engineering
1962
June, 1962
IP-570

ACKNOWLEDGMENTS
The author wishes to express his appreciation to the following
individuals and organizations for their contributions.
Professor Stuart W. Churchill, chairman of the committee, for
his interest, encouragement, and for his wholehearted and prompt cooperation on every occasion.
Professor John A. Clark for his many suggestions concerning the
experimental technique and for his generously allowing the complete use
of the facilities of the Heat Transfer and Thermodynamics Laboratory of
the Department of Mechanical Engineering.
Dr. Robert R. White (Atlantic Refining Company) for initially
suggesting the topic and serving as chairman of the committee during the
first part of the investigation.
The other members of the doctoral committee for their advice,
encouragement, and willingness to be of help at any time.
Professor Julius T. Banchero for serving on the committee during
the early phases of the work.
The staff of the Department of Chemical and Metallurgical Engineering and the staff of the Heat Transfer and Thermodynamics Laboratory
of the Department of Mechanical Engineering for their assistance during
all phases of the work.
The Industry Program of the College of Engineering for the preparation of the dissertation.
The Procter and Gamble Company for their fellowship during the
academic year 1956-57.
The Continental Oil Company for their fellowship during the
academic year 1957-58.
The University of Michigan for its fellowship during the academic year 1958-59
ii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS oo...o......................... ii
LIST OF TABLES o o o o 0 o o 0 O o o o. o oo o o... o o.o o o. o vi
LIST OF FIGURES..............viiO
I:TJ OF FI*.G~U.RESJ o o o n n a o o o, a a a o o o o a o o o o o o o o o < o o o o o o o non o o a o a o o e D o o n o a Vi l
3INSL,~OTRDCTDION^o eon. ~oo 9 o o...o,o,o oo o Qo, o,o,0 o,, o o o o o. o... o o, o l
NTREOUCTICL CONSIDERATIONS AND P REIOUS WO RK O. 0O.. O 5 o o o o
THEORETICAL CONSIDERATIONS AND PREVIOUS WORKo....... c e-o.. eo ne 5
Mathematical Statement of the General Problem.............. 5
Dimensional Analys is.a o. e.. e........... 6 6
Pressure Drop of Turbulence Promoters Based on Drag of a
Single Bluff Body o. o o o o o o o.. e o............ o 7
Review of Previous Work o 0.... oo oe o o........ 9
EXPERIMENTAL APPARATUS AND PROCEDURE.. o o. o o.. o.... o. e o 26
Description of the Equipment............................. 26
Description of Turbulence Promoters o oo......o.......oo.. 533
Description of the Procedure............................ 40
Method of Calculating Heat Transfer Coefficients............ 44
Method of Calculating Friction Factors 0.. oo.........e...... 47
EXPERIMENTAL RESULTS AND DISCUSSION OF RESULTS o o o.... o........ 50
Empty Tube e o o o o o o o o o o o Q o. o c o o 0 50
Solid Rod in the Center of the Tube o................o......... 55
Disks Evenly Spaced and Centered in the Tube.............. 65
Streamline Shapes Evenly Spaced and Centered in the Tubeo o. 88
Reliability of the Data for all Geometries0...eeooo..o.... 102
ECONOMICS.........eoo o o o o eooooo................. 1.... 11
Mathematical Model of a Heat Exchanger o..... o o o o o o o o 1 11
Factors Which Affect the Economics ooo..o................... 114
Procedure for Designing the Optimum Heat Exchanger o......... 116
Example Design of a Typical Heat Exchanger.................. 123
Conclusions Regarding the Economics of Using Turbulence
Promoters oo.0 ooooooo eeo.................................0 130
SUMMARY AND CONCLUSIONS.o.e.o o..e o0 o 0 0 o o o e..0.. o.. o.oe o. 139
BIBLIOGRAPHYo.. eo. O o. o 0 o.oo.o o.... 0 0 143
iv

TABLE OF CONTENTS (continued)
Page
APPENDICES....................................................
APPENDIX A - DESCRIPTION OF COMPUTER TECHNIQUES FOR PROCESSING DATA. o OOO.. o O..... OaDo.OQOo no O.oo 146
APPENDIX B - DERIVATIONS.............. o................... 168
APPENDIX C - ORIGINAL AND PROCESSED DATA...oo............. 202
APPENDIX D - CALCULATIONS REQUIRED FOR EXAMPLE HEAT
EXCHANGER DESIGN.................. o........ o 217
V

LIST OF TABLES
Table Page
I FUNCTIONS F(d), G(d), AND H(d) USED IN EQUIVALENT
FRICTION FACTOR CORRELATIONS FOR ANNULI.................. 41
II POSITION OF THERMOCOUPLES ON TEST SECTION RELATIVE
TO OTHER ITEMS....... o..........o o o o o o o o 32
III BLUFF-BODY TURBULENCE PROMOTER COMBINATIONS USED IN
THIS INVESTIGATION o...... o ooe o.. o.... 39
IV THERMOCOUPLE EMF vs. DEGREES CENTIGRADE FOR COPPERCONSTANTAN THERMOCOUPLES14.o..o......... oo ooo ooooo. 147
V FIRST THREE ROOTS Xi OF EQUATION (B-72) AND ASSOCIATED
VALUES OF Si FOR b/a 1 o100, 1o2418, AND 1.5000 o.....00 187
VI THE FUNCTION A(Xn, r/a) FOR b/a = 1 100, o12418, AND
1 500 AS A FUNCTION OF (r - a)/(b - a) oo....eo.......... 1.88
VII SUMMARY OF EXPERIMENTAL CONDITIONS.......o...... e..o 205
VIII CONSTANTS C(s,d) AND n(sd) USED IN FRICTION FACTOR
CORRELATION EQUATION (64) FOR INDIVIDUAL PROMOTER
COMBINATIONS..o o o o o o ooo... o..... o... o...... o.. 206
IX CONSTANS C(sd) AND n(s,d) USED IN CORRELATION
EQUATION (65) OF EFFECTIVE DRAG COEFFICIENT FOR
INDIVIDUAL PROMOTER COMBINATIONS...........o.......... 207
X CONSTANTS C(sd) AND n(s'd) USED IN CORRELATION
EQUATION (67) OF MEAN HEAT TRANSFER COEFFICIENTS FOR
INDIVIDUAL PROMOTER COMBINATIONS o o o oo.... o o o.o o o. 208
XI PRESSURE DROP RESULTS................................... 209
210
211
XII INTEGRATED RESULTS FROM HEAT TRANSFER EASUREMENTS..... 212
XIII LOCAL HEAT TRANSFER MEASUREMENTS o o o o..o o oo o o. O O O 213
214
215
216
vi

LIST OF FIGURES
Figure Page
1 Cross-Section of Tube with Bluff Body Turbulence
Promoter Inserted to Illustrate the Difference
Between Disk and Streamline Shape o. o o o.............. 4
2 Comparison of Friction Factor Correlations for
Annuli, Ratio of;AP Predicted by Correlations to AP
Calculated Using Blasius Equation and Hydraulic Radius
vso Diameter Ratio d, for Re = 1,000, 20,000 and
40,000 O O O o o oo oo o o o ooo o..o. a o o oo o o oo o o o.o o o 16
3 Photograph of Overall View of Equipment.e...o........... 27
4 Photograph of Closeup of Test Section o...o............ 27
5 Photograph of Thermocouple Switches and Recording
Potentiometer o.o......o.oooooo.oooooooooooo. o. eoo.ooooe 27
6 Schematic Diagram of the Apparatus..o.....o.. o........oo 28
7 Diagram of the -Test Section o.........oo.... o....o..... 30
8 Diagram of Disk and Streamline Shape Showing Relative
Dimensions and Method of Mounting..................o.. 36
9 Photograph of Individual Turbulence Promoter Shapes
Usedo o.oo........................... 37
10 Photograph of a String of Disks and a String of
Streamline Shapes o o o o o o o o. o o. o o o.. o o o o o37
11 Friction Factor for the Empty Tube as a Function of
Reynolds Number, f vso Re................................ 51
12 Sample Values of the Local Heat Transfer Coefficients
for the Empty Tube as a Function of Longitudinal
Position, h(z) vso z...O.........O...OOooooOOeo o....... 52
13 Nusselt Number for the Er pty Tube as a Function of
Reynolds Number, Nu/(Prl/3(t/-w)0~o14) vso Re............ 54
14 Friction Factor for the Tube with a Solid Rod in the
Center as a Function of Reynolds Number, f vso Re,
with Parameters of d..........oooo...oo..o.............. 56
15 Friction Factor for the Tube with a Solid Rod in the
Center as a Function of Reynolds Number, Based on
Equivalent Diameters, f* vso Re* o o o....... o............ 57
vii

LIST OF FIGURES (continued)
Figure Page
16 Sample Values of the Local Heat Transfer Coefficient
for a Tube with a Threaded Rod in the Center as a
Function of Longitudinal Position, h(z) vso z, for
d = 0 * 0 e o o e.. o o e.... o...... o...... 59
17 Sample Values of the Local Heat Transfer Coefficient
for a Tube with a Solid. Rod in the Center as a
Function of Longitudinal Position, h(z) vso z, for
d = 006250. e. o. o...OO....... o... eO.e............ 60
18 Sample Values of the Local Heat Transfer Coefficient
for a Tube with aSolid Rod in the Center as a Function
of Longitudinal Position, h(z) vs. z3 for d = 0.750.... 61
19 Nusselt Numbers for the Tube with a Solid Rod in the
Center as a Function of Reynolds Number, Nu vs. Re,
with Parameters of d,....ooooo o...o.e...oo....o.....oo 63
20 Nusselt Number for the Tube with a Solid Rod in the
Center as a Function of Reynolds Number Based on
Equivalent Diameters, Nu*/Pr1/3[1/pwy]014 vs. Re*..... 64
21 Friction Factor,for Disks as a Function of Reynolds
Number, f vs. Re, for s = 2 with Parameters of d....... 66
22 Friction Factor for Disks as a Function of Reynolds
Number, f vso Re, for s= 4 with Parameters of d.,..... 67
23 Friction Factor for Disks as a Function of Reynolds
Number, f vso Re, for s = 8 with Parameters of do..... 68
24 Friction Factor for Disks as a Function of Reynolds
Number, f vso Re, for s = 12 with Parameters of doo.... 69
25 Effective Drag Coefficient for Disks as a Function
of Reynolds Number, fD vso Re, for s = 12, 8, 4
and 2 with Parameters of dooo...oo,,.............. 71
26 Effective Drag Coefficient for Disks as a Function
of Free Area, fD vso Af, for s = 12, 8, 4, and 2
with Parameters of Re..,............................ 72
27 Effective Drag Coefficient for Disks as a Function
of Spacing, fD vs. s, for d = 0.625, 0.750, and
0.875 and Any Reynolds Number o.......,..o oooo..,..o. 73
viii

LIST OF FIGURES (continued)
Figure Page
28 Test of Effective Drag Coefficient Cosrrelation, fD
Values Predicted by Correlation vs. Values Measured
Experimentally for Disks and Streamline Shapes...... o oo 75
29 Sample Values of Local Heat Transfer Coefficient
Ratio for Disks as a Function of Longitudinal Position
from Disk with Reynolds Number Approximately 10,000,
h/hO vs. x, for d = 0.625, 0750, and 0.875..........o.. 77
30 Overall Mean Heat Transfer Coefficient Ratio for Disks
as a Function of Reynolds Number^ hm/hO vSo Re, for
s = 12, 8, 4, and 2 with Parameters of do o e o.... o o o o 80
31 Overall Mean Heat Transfer Coefficient Ratio for Disks
as a Function of Free Area, h/h0 VSo Af, for s = 12, 8,
4, and 2 with Parameters of Re........... o o o............ 82
32 Overall Mean Heat Transfer Coefficient Ratio for Disks
as a Function of Spacing, hm/ho vs. s9 for Re = 10,000,
20,000, and 40o,000 with Parameters of d............... 83
33 Test of Heat Transfer Correlation, hm/h Ratios Predicted by Correlation vs. Values Measured Experimentally
for Disks and Streamline Shapes...........oo.............. 85
34 Friction Factor for Streamline Shapes as a Function of
Reynolds Numbery f vs. Re, for s - 4 with Parameters of
d Friction.Factor for Streamline Shapes as a Function of
35 Friction. Factor for Streamline Shapes as a Function of
Reynolds Number, f vs. Re, for s = 8 with Parameters of
36 Friction Factor for Streamline Shapes as a Function of
Reynolds Number, f vs. Re, for s = 12 with Parameters of
da..... 0 0 0o a 0 0 o 0 a o o 0 a o 0 a. o. a 0 O0. O o. 0 o.. a 0.. 0. a.. a a 0 0 o. c e o 0 91
37 Effective Drag Coefficient for Streamline Shapes as a
Function of Reynolds Number, fD vso Re, for s = 12, 8,
and 4 with Parameters of d....................... 93.
38 Effective Drag Coefficient for Streamline Shapes as a
Function of Free Area, fD vso Af, for s = 12, 8, and 4
with Parameters of Re.....,...,.,,.,.,oo.O............. 94
ix

LIST OF FIGURES (continued)
Figure Page
39 Effective Drag Coefficient for Streamline Shapes
as a Function of Spacing, fD vs. s, for d = 00625,
0,750, and 0.875 with Parameters of Re............... 95
40 Sample Values of Local Heat Transfer Coefficient Ratio
for Streamline Shapes as a Function of Longitudinal
Position from Shape with Reynolds Number Approximately
10,000, h/ho,.vS,x, for d.= 0.625, 0O750, and 0.875.... 97
41 Overall Mean Heat Transfer Coefficient Ratio for
Streamline Shapes as a Function of Reynolds Number,
hm/ho vs. Re, for s = 12, 8, 4, and 0 with Parameters
of d o. e o..... o o e. o o e o o o o o.. o o o o o o 98
42 Overall Mean Heat Transfer Coefficient Ratio for
Streamline Shapes as a Function of Free Area,
h/h0 vs. Af, for s = 12, 8, 4, and 0 with Parameters
of Re e o o O O O Oa e o o...o o o o oo.o oo oe o o 100
43 Overall Mean Heat Transfer Coefficient Ratio for
Streamline Shapes as a Function of Spacing, h /h0
vso s, for Re = 10,0000 20,000, and 40o000 wiTh
Parameters of d....................................... 101
44 Frequency Distribution of Heat Balance Errors.......... 107
45 Frequency Distribution of Difference Between Local
Angular AT and Mean ZT for all Three Angles for
Streamline Shapes................. e......... o.... 109
46 Frequency Distribution of Difference Between Local
Angular AT and Mean ZT for all Three Angles for Disks... 109
47 Illustration of Heat Exchanger Costs for a Given
Geometry and Tube Diameter as a Function of Nusselt
Number............ o.... oooooooooooooooooo..o..o oo.oo. 121
48 Fixed Cost, Pumping Cost, and Total Cost per BTU
as a Function of Nusselt Number for Example Heat
Exchanger Design Using an Empty Tube Geometry,
Parameters of d. o.................. o o............ 126
49 Fixed Cost, Pumping Cost, and Total Cost per BTU
as a Function of Nusselt Number for Example Heat
Exchanger Design Using Turbulence Promoters with
D = 0o50 5inch o.....oeoo.o.o ooOO.O*.. oe.....*..e.. 129
x

LIST OF FIGURES (continued)
Figure Page
50 Ratio of Pumping Cost for Disks to Pumping Cost for
Empty Tube as a Function of Nusselt Number, (E/Q)/
(E/Q)ET vs. Nu for s = 12, 8, 4, and 2 with Parameters
of d.................................. o o...o 131
51 Ratio of Pumping Cost for Streamline Shapes and Solid
Rod to Pumping Cost for Empty Tube as a Function of
Nusselt Number, (E/Q)/(E/Q)ET vs Nu for s = 12, 8, 4,
and 0 with Parameters of d.......................... 132
52 Flow Rate as a Function of Rotameter Reading, GPM vso
Reading, for Rotameters 1, 2, 3, and 4................... 149
53 Sample Raw Data Sheet for Pressure Drop Measurements..... 152
54 Listing of Data Cards for Pressure Drop Measurements
of Sample Problem....................................... 154
55 Computer Analysis of Pressure Drop Data for Sample
Problem....................................o............ 155
56 Raw Data Sheet for Heat Transfer Measurements.,.......... 157
57 Listing of Data Cards for Heat Transfer Measurements
of Sample Problem. o.....o..o.. o...o................o...... 159
58 Computer Analysis of Heat Transfer Data for Sample
Problem, Pages I, II, III, and IV.... o................. 160
161
59 Boundary Conditions for Solution of Conduction Equation
Considering Axial Conduction into the Non-Heat-Generating
Portion of the Tube....... o........................... 179
60 Dimensionless Form of Solution for Axial Conduction into
the Non-Heat-Generating Portion of the Tube, Showing
that the End Effect is Negligible....................... 186
61 Damping Function Defined by Equation (B-80) as a
Function of the Parameter na/S,6(na/S, b/a) vs. na/S,
for b/a 4o.2 o........................................ 190
62 Sample Hypothetical Values of Fluid Temperature and
Inside Wall Temperature as a Function of Longitudinal
Position, Tf(y) and Ta(y) vs. y for Rapidly Varying
Inside Wall Temperature o........... o.......oo............ 192
xi

LIST OF FIGURES (continued)
Figure Page
63 Effect of Neglecting Damping of Outside Wall
Temperature When Inside Wall Temperature is Rapidly
Varying with Longitudinal Position................. 192
64 Frequency Distribution of Error Introduced by AZAR
Recorder. o..................................... 197
xii

NOMENCLATURE
a Inside radius of tube, ft.
a Dimensionless inside radius of tube (2a/L) used only in Part 4
of Appendix B
a * a3 Arbitrary constants used in Part 4 of Appendix B and as coefficients of polynomial in Equation(A-17I
A Total heat transfer surface inside the tubes in a heat exchanger,
ft2
A Rate of heat generation per unit volume of tube wall, BTU/hr-ft3
A0 Expression defined by Equation (53),i.e., rate of heat generation in the tube wall at 0 deg. F, BTU/hr-ft3
A2 Term defined by Equation (A-13) used in calculating ATgeneration
A3 Term defined by Equation (A-14) used in calculating ATgeneration
A4 Term defined by Equation (A-15) used in caclulating q(z)
Af Fraction free area at point of maximum radius of promoter, dimensionless
An Coefficient in orthogonal function expansion of g(r) used only
in Part 4 of Appendix B
A.tube Cross-sectional area of the tube, ft2
A Projected area of the promoter, ft2
p
b Outside radius of tube, ft.
b Dimensionless outside radius of tube (2b/L) used only in Part 4
of Appendix B
blb2 Arbitrary constants used in Part 4 of Appendix B
B1 ~~B3 Expressions defined by Equations (112), (113), and (114) used
to obtain E/Q as a function of Nusselt number from correlations
B Coefficient in Fourier series expansion of hypothetical periodic
~n inside wall temperature, deg. F
c Heat capacity of fluid, BTU/lbm - deg. F
xiii

NOMENCLATURE (continued)
Ci' oC Various constants used in correlations, dimensionless
C(s,d) Constant used in correlations of friction factor, effective drag
coefficient, and mean heat transfer coefficient ratio for individual promoter combinations, dimensionless
CE Cost per unit pumping energy, dollars/BTU
CF Fixed cost proportionality constant (fixed cost = CFAm), dollars/
ft2m
CW Cost of working fluid in heat transfer equipment, dollars/lbm
d Ratio of diameter of bluff-body turbulence promoter (at point of
maximum diameter) to diameter of tube, dimensionless
D Inside diameter of tube, ft.
D* Equivalent tube diameter used in correlations for an annulus, ft.
Dn Coefficient in orthogonal function expansion defined by Equation
(B-61) used only in Part 4 of Appendix B
Dp Diameter of bluff-body promoter at point of maximum diameter, ft.
E Energy expended in pumping, BTU/hr
E Electrical potential, volts
E/Q Ratio of pumping energy to total rate of heat transfer for a
specified heat exchanger with specified inside tube geometry
(function of Nusselt number), dimensionless
(E/Q)ET Ratio of pumping energy to total rate of heat transfer for a
specified heat exchanger using an empty tube geometry (function
of Nusselt number), dimensionless
f Friction factor based on inside tube diameter and superficial
velocity as defined by Equation (22), dimensionless
f* Friction factor based on equivalent diameter D* and true mean
velocity in tube, dimensionless
f Friction factor calculated for an empty tube as a function of
mass flow rate using Equation (15), dimensionless
fD Effective drag coefficient for a single bluff body, dimensionless
xiv

NOMENCLATURE (continued)
F(d) Function used for interpolating graphical equivalent friction
factor correlation of Lohrenz and Kurata
g(C),g(y) Any arbitrary function
gc Gravitational constant, lbm - ft/lbf - sec2
G(d) Function used in friction factor correlation of Meter and Bird,
dimensionless
h Local inside heat transfer coefficient including resistance of
tube wall, BTU/hr - deg. F - ft2
h' Effective outside heat transfer coefficient including resistance
of tube wall, BTU/hr - deg. F - ft2
ho Mean inside heat transfer coefficient for empty tube as calculated
from Sieder-Tate equation, BTU/hr - deg. F - ft2
h Convection coefficient from Fiberglas insulation to air,
BTU/hr - deg. F - ft2
hm Mean inside heat transfer coefficient, BTU/hr - deg. F - ft2
hm/ho Ratio of mean heat transfer coefficient for any inside tube
geometry to coefficient for empty tube at same mass flow rate
h(z)est Estimate of local heat transfer coefficient for empty tube geometry or tube with solid rod which considers change in physical
properties of fluid. Defined by Equation (63), BTU/hr - deg. F -
ft2
H(d) Function used in friction factor correlation of Meter and Bird,
dimensionless
I Electric current, amps
IO(x) Modified Bessel function of the first kind and order zero
I1(x) Modified Bessel function of the first kind and order one
J Electric current density in tube wall, amps/ft2
Jo(X) Bessel function of the first kind and order zero
J (x) Bessel function of the first kind and order one
Jc Conversion factor (777.5 ft - lbf/BTU)
xv

NOMENCLATURE (continued)
k Thermal conductivity of fluid, BTU/hr - deg. F - ft
K Thermal conductivity of tube wall, BTU/hr - deg. F - ft
Ko Thermal conductivity of tube wall at 0 deg. F, BTU/hr - deg. F -
ft
Kb Thermal conductivity of tube wall evaluated at the outside wall
temperature, BTU/hr - deg. F - ft
K f Thermal conductivity of Fiberglas insulation, BTU/hr - deg. F -
fg ft
Kmica Thermal conductivity of mica insulation, BTU/hr - dego F - ft
K0(x) Modified Bessel function of the second kind and order zero
Kl(x) Modified Bessel function of the second kind and order one
L Length cf.heated portion of tube, ft
L Distance between pressure taps for experimental apparatus, ft
P
m Slope of steady temperature increase in fluid and inside wall
temperature (m = Q/WcL), deg. F/ft
m Exponent of inside surface area such that fixed heat exchanger
cost is proportional to Am, dimensionless
n(s,d) Exponent used in correlations of friction factor, effective drag
coefficient, and mean heat transfer coefficient ratio for individual promoter combinations, dimensionless
nl, n2 Exponents used in various correlations, dimensionless
n Number of individual bluff bodies contained in string of turbulence
promoters, dimensionless
Ntube Number of tubes in parallel in a shell and tube heat exchanger,
dimensionless
Nu Nusselt number, dimensionless
Nu* Nusselt number based on equivalent diameter, dimensionless
WNoptimum Value of Nusselt number for which the total cost of a heat exchanger is lowest, dimensionless
xvi

NOMENCLATURE (continued)
p Exponent of Nusselt number defined by Equation (111) used to
obtain (E/Q) as a function of Nu from correlations, dimensionless
P Local fluid pressure, lbf/ft2
AP Longitudinal pressure drop between pressure taps, lbf/ft2
Pform Pressure drop due to form drag of individual bluff body which
would be present if there were no drag on the tube wall, lbf/ft
APtotal Total pressure drop caused by both form drag and drag of the
tube wall, lbf/ft2
APt Pressure drop occurring along length (npS) occupied by string
of turbulence promoters (estimated by Equation (61)), lbf/ft
APET Pressure drop due to drag on the tube wall that would be present
without turbulence promoters, lbf/ft2
Pr Prandtl number, dimensionless
Prm Overall mean value of Prandtl number obtained by integrating
local Prandtl number Prz over length of tube, dimensionless
Prz Prandtl number of fluid flowing in tube evaluated at local mixedmean fluid temperature, dimensionless
q Local rate of heat transfer per unit area, BTU/hr - ft2
qL Heat loss per unit area of outside tube wall through insulation,
BTU/hr - ft2
Q. Total rate of heat transfer, BTU/hr
Qin Total rate of heat input to tube by generation in tube wall as
calculated using Equation (70), BTU/hr
Qout Total rate of heat removal by water as calculated using Equation
(71), BTU/hr
r Radial distance from center of tube, ft.
r Dimensionless radial distance (2r/L) used only in Part 4 of
Appendix B
rins Outside radius of Fiberglas insulation, ft.
R Electrical resistance, ohms
xvii

NOMENCLATURE (continued)
Rp Radius of turbulence promoter, ft.
Re Reynolds number, dimensionless
Rem Overall mean value of Reynolds number obtained by integrating
local Reynolds number Re over length of tube, dimensionless
Rez Reynolds number of fluid flowing in tube evaluated at local
mixed-mean fluid temperature, dimensionless
Re* Reynolds number based on equivalent diamter and true mean velocity,
dimensionless'F^^,r-,'O Dummy variables used only in Part 4 of Appendix B
s Ratio of spacing between promoters to tube diameter, dimensionless
S Spacing between turbulence promoters, fto
tmica Thickness of mica insulation, ft.
T Temperature, dego F
T Dimensionless temperature (4KT/L2Ao) used only in Part 4 of Appendix B
TI Solution in Region I (the heat-generating zone) of the conduction
equation which accounts for axial conduction into the non4heatgenerating portion of the tube, dimensionless
TII Solution in Region II (the non-heat-generating zone) of the
conduction equation which accounts for axial conduction into the
non-heat-generating portion of the tube, dimensionless
Ta Temperature at inside tube wall, deg. F
Tamb Ambient air temperature, dego F
Tb Temperature at outside radius of tube, deg. F
Tinlet Mixed-mean temperature of the fluid at Z = 0, the:inlet of the
test section (actually measured at the inlet to the equipment),
deg. F
Tf Mixed-mean temperature of the fluid, deg. F
xviii

NOMENCLATURE (continued)
Toutlet Mixed-mean temperature of the fluid at Z = L, the outlet of the
test section (actually measured at the outlet of the equipment),
deg. F
Ttc Temperature measured by outside wall thermocouple at outside
edge of mica sheet, deg. F
Twall Temperature at the inside tube wall, deg. F
ATgeneration Temperature difference between inside and outside wall caused
by internal generation of heat, deg. F
ATm Mean temperature difference which provides driving force for
heat transfer, deg. F
AT Maximum difference between inside wall temperature and fluid
temperature for hypothetical, periodic inside wall temperature
distribution, deg. F
mAT Mean difference between inside wall temperature and fluid tempermean
ature for hypothetical, periodic inside wall temperature distribution, deg. F
ATmin Minimum difference between inside wall temperature and fluid
temperature for hypothetical, periodic inside wall temperature
distribution, deg. F
ATperiodic Hypothetical periodic inside wall temperature, deg. F
AT'periodic Damped component of hypothetical periodic inside wall temperature which would be measured at outside wall, deg. F
u True mean velocity of fluid, ft/sec
U Superficial mean velocity of fluid, ft/sec
Umax Mean fluid velocity at point of minimum free area, ft/sec
Uoa Overall heat transfer coefficient defined by Equation (74),
BTU/hr - deg. F - ft2
dV Infinitesimal volume of tube wall, ft3
V,V1,V2 Dummy variables used only in Part 4 of Appendix B, dimensionless
Vr Local fluid velocity in the radial direction, ft/hr
Vz Local fluid velocity in the longitudinal direction, ft/hr
xix

NOMENCLATURE (continued)
W Mass flow rate of fluid, lbm/hr
x Ratio of distance from point of maximum diameter of promoter to
diameter of tube, dimensionless
X Distance from first point of maximum radius of turbulence promoter, ft.
y Distance from center of heated section of tube, ft.
y Dimensionless axial distance from center of heated section (2y/L),
used only in Part 4 of Appendix B.Y1Y2 Two arbitrary longitudinal positions along tube, between which
the hypothetical periodic inside wall temperature is expanded in
a Fourier series, ft.
YO(x) Bessel function of the second kind and order zero
Yl(x) Bessel function of the second kind and order one
t+3TO Dummy variables used only in Part 4 of Appendix B
z Ratio of longitudinal distance from beginning of heating to diameter of tube, dimensionless
Z Longitudinal distance from beginning of heating, ft.
ZO Reference value of longitudinal position at which temperature,
pressure, and velocities are known in statement of theoretical
problem, ft.
~a Ratio of equivalent diameter to inside diameter of tube (used
in correlations for an annulus), dimensionless
3B Rate of change of thermal conductivity of tube wall with temperature, 1/deg. F
Y Rate of change of electrical resistivity of tube wall with temperature, l/deg, F
6(na/S,b/a) Damping function defined by Equation (B-80), dimensionless
Ratio of heat flux at inside wall to heat flux at outside wall
of an annulus (in the limiting case of two parallel plates)
used in theoretical analysis of Barrow, dimensionless
xx

NOMENCLATURE (continued)
Xn Eigenvalues defined by Equation (B-72), dimensionless
Xn Dummy Eigenvalue defined by Equation (B-49) and used only in
Part 4 of Appendix B
A(Xnr/a) Set of orthogonal functions associated with eigenvalue Xn and
defined by Equation (B-53), dimensionless
Dynamic viscosity of fluid, lbm/ft - hr`w Viscosity of fluid in tube evaluated at the inside wall temperature, lbm/ft - hr
(.L/uw)m Overall mean value of viscosity ratio obtained by integrating
local viscosity ratio (j/t) z over length of tube, dimensionless
( L/U^)z Viscosity ratio of fluid inside tube evaluated at local mixedmean fluid temperature, dimensionless
In Parameter associated with eigenvalue Xn and defined by Equation
(B-73)
p Density of fluid, lbm/ft3
p Electrical resistivity of tube wall, ohm-ft.
p Mean electrical resistivity of tube wall, ohm-ft.
pO Electrical resistivity of tube wall at 0 deg. F, ohm-ft
0 (1 + 7yb)/(l + PTb)
0(d) Function used in friction factor correlation for annuli by Meter
and Bird, dimensionless
xxi

INTRODUCTION
In recent years the problem of improving the rate of heat
transfer to a fluid flowing in a tube has been a subject of increasing importance. The advent of the nuclear reactor with its large
heat flux requirements has demanded the development of improved technique for obtaining high heat transfer rates. The rapid expansion of the
chemical process industry has made it necessary to improve the performance of existing equipment for exchanging heat in order to keep plants
already built from becoming o bsoleteo The exploration of space has
created requirements for heat transfer equipment which must both be
compact and must consume a minimum of power for operation.
One technique for improving the rate of heat transfer is to
insert devices commonly referred to as "turbulence promoters" inside
the tube to disturb the flow, enhance the mixing, and thus reduce the
resistance to heat transfer. Many types of devices have been suggested
for this service including such things as roughening elements attached
to the tube wall, twisted metal strips to impart a swirling motion to
the fluid, bluff objects in the center of the tube, and packing material.
Unfortunately, any insert which improves the rate of heat
transfer by enhancing the mixing also increases the pressure drop, and
more energy is required to pump the fluid through the tubeo Therefore,
in order to determine the economic feasibility of turbulence promoters
in a given heat transfer application it is necessary to be able to predict quantitatively the rate of heat transfer and the pressure drop as a
function of flow rate, tube diameter, geometry of the turbulence promoters,

-2physical properties of the fluid, and temperature boundary condition at
the tube wall.
At the present time the information concerning the effect of
turbulence promoters is of insufficient scope and reliability for design
purposes. One of the main difficulties in studying turbulence promotion
is the large number of possible geometrical configurations. The only promoters that have been systematically varied are twisted metal strips and
roughening elements attached to the tube wall.
In particular, little data exist for the effect of bluff objects
inside a tube on the pressure drop and the rate of heat transfer from the
tube wall to the fluid. There is a particular scarcity of data for local
heat transfer coefficients and for fluids other than air. Apparently no
previous investigation has considered the effect of streamlining a bluffbody turbulence promoter.
The purpose of this investigation was, therefore, to obtain
experimental values'of the pressure drop and local rates of heat transfer
for bluff-body turbulence promoters, including streamline shapes, in
water.
Because of the large number of independent variables, those
which were expected to be less significant were fixed. Thus, only one
fluid (water); only one tube diameter (one inch inside diameter); and one
boundary condition (constant wall heat flux) were utilized. This reduced
the problem to a consideration of the geometry of the promoter (shape,
diameter, spacing) and flow rate of the fluid. To further simplify the

-3consideration of geometry, only promoters centered in the tube, symmetric
about the axis and at'.uniform spacing were, studied.
Thus, the variables considered in this investigation were: flow
rate of the fluid, shape of the promoter, ratio of promoter diameter to
inside tube diameter, and the ratio of promoter spacing to inside tube
diameter. The effect of promoter shape was studied by considering two
dissimilar shapes: disks and a streamline body as shown in Figure 1.
These shapes represent the two extremes of no streamlining and a great
deal of streamlining. Data were also obtained for a solid rod in the
center of the tube in an attempt to find out what happens in the limiting
case of promoters at zero spacing. Data were obtained with the empty tube
for comparison.
An attempt was made to obtain data of sufficient scope and accuracy to enable it to be used for the following purposes:
1. Determine whether bluff-body turbulence promoters can be
used economically to improve the rate of heat transfer to a fluid flowing
in a tube.
2. Determine whether there is an optimum geometry for the promoter
3. Obtain correlations to permit the design of turbulence promoters for the most promising geometries.

-4- q,q- q — ),
SITOPDE VIEW SIDE TP VIEW
4- q q —>
Promoter Inserted to Il lustrate the Difference
Promoter Inserted to Illustrate the Difference

THEORETICAL CONSIDERATIONS AND PREVIOUS WORK
Mathematical Statement of the General Problem
The mathematical statement of the general problem studied in
this investigation is as follows:
A fluid is flowing under steady-state conditions in a cylinder
of diameter D in which there are a series of obstructions, symmetric about
the axis, evenly spaced at distance S apart. The obstructions as shown
below are defined by the equation
Rp = Rp(X) (1)
where Rp is the radius of the promoter and X is the distance from the
first point of maximum radius of the promoter.
Point of Maximum Radius
of Promote
S, Z
-5

Boundary conditions are specified by giving the velocity of the
fluid in the longitudinal and radial directions, temperatures, and pressures
at some reference point, ZO, i.e. Vz(r,Zo), Vr(r,Zo), T(r,Zo), and
P(r,Zo) are specified, where Z is the longitudinal distance down the
cylinder. In addition the wall temperature distribution Twall(Z) (or
using the same notation, T(D/2,Z)) must be given for Z greater than ZO.
The problem is to determine the velocities, temperature, and
pressure atany point: Vz(r,Z), Vr(r,Z), T(r,Z), and P(r,Z). This, in
turn, will either enable the rate of heat transfer and pressure drop to
be calculated or will eliminate the need for such information.
The solution of the Navier-Stokes equations together with the
energy equation subject to the boundary conditions just described should —
in theory —provide the-answer. Unfortunately, there is no known, general
solution to these equations for turbulent flow in even an empty tube —
much less one in which bluff objects have been introduced. The complex,
little-understood nature of turbulent flow makes this approach impossible
at the present time.
Dimensional Analysis
In dimensionless form the rate of heat transfer can be expressed
in terms of the Nusselt number
Nu - hD/k (2)
where the heat transfer coefficient h is defined as
h = )
- wall - Tf

-7The overall pressure drop can be expressed in terms of the
following dimensionless ratio called the friction factor
f gc D (4)
2p U2
For a given set of boundary conditions, a given shape of bluff
body, and a fluid whose physical properties can be assumed to be independent of temperature, the specification of the rate of heat transfer and
pressure drop should be a function of the following dimensionless variables:
1. Dimensionless flow rate, Re =DpU/I
2. Prandtl number, Pr = cP/k
3. Ratio of promoter diameter to tube diameter, d = Dp/D
4. Ratio of promoter spacing to tube diameter, s = S/D
Pressure Drop of Turbulence Promoters Based on Drag
of a Single Bluff Body
One approach to the problem of predicting the pressure drop for
a combination of bluff body turbulence promoters is to assume that the
pressure drop for a series of bluff body promoters is composed of two
parts: 1) drag on the tube wall which would be present even if there
were no bluff body, and 2) form drag of the bluff body.
The pressure drop due to drag on the tube that would be present
in a length S of the empty tube containing a single bluff body is
2fo p U2 S
ET gc D
2f0 p U2 s
-, (6)
Qc

The pressure drop due to the form drag of the body may be expressed
AJDP Umaxa p (y)
-LPform (7)
2 gc Atube
fD U2 d2
Uf p 2d2 (8)
2. (i d
O (1 d")
where A = projected area of the promoter
Atube cross-sectional area of the tube
Umx = mean superficial velocity at point of minimum
free area
fD effective drag coefficient
The total pressure drop is
APtotal= -f (9)
gc
where f = friction factor for turbulence promoters based on
the inside tube diameter and the mean velocity in
the empty tube
Since \Ptotal PET form (10)
then 2 2 f0 p U2 s f2 p U d21)
then.2 f P -(II)
gc gc 2 gc (-d2)2
-f 1- -1 L (12)
The -preceding analysis is quite simplified. One would not
expect "fD above to have the same value as the drag coefficient measured
in an infinite fluid of uniform velocity. Furthermore, fD should be a
function of spacing s, since as the promoters get closer and closer

-9together, their wakes will start interfering with each other and the drag
coefficient should decrease. The coefficient fD should also be a function of the diameter ratio d since it involves or includes a wall effect.
For these reasons the coefficient fD will, henceforth, be referred to as
an "effective drag coefficient".
If the fraction free area Af is defined as
Af = (1-d2) (13)
then a trial correlation for the rate of momentum transfer might be made
by plotting the effective drag coefficient
2
4 Af s r (,
f=4 A 0 (14)
versus s, d, and Reynolds number.
Review of Previous Work
Empty Tube —the Limiting Case of Infinite Promoter Spacing or Zero
Promoter Diameter
When the spacing becomes infinite or the diameter of the promoter goes to zero, the geometry reduces to the case of an empty tube.
This case has been the subject of a very large number of theoretical and
experimental investigations which are well summarized by McAdams(28) and
Knudsen and Katz(22)
A generally accepted empirical correlation for the friction
factor in a smooth tube is that given by Nikuradse(33
-_ 4.0 log1o (Re f/) - 0.40 (15)
~~~o~n)- o.o(f

10This is the quantity used throughout this dissertation when referring to
the friction factor f0 for the empty tube.
In addition, correlations for the friction factor in an empty
tube which are useful because of their explicit nature include that of
Blasius
f = 00o79 Re~ 025 (16)
and that of Colburn(lO) based upon the j-factor
f - o0o46 Re- ~20 (17)
The generally accepted empirical correlation for the Nusselt
number is that given by Sieder and Tate(38)
Nu C= ReO8 prl/ (/ )001.4. (18)
valid for Re > 10 000 and Pr > 0O70o
Sieder and'Tate give a value of 0o027 for C2, Bird, Stewart, and Lightfoot (4 suggest 0o026, McAdams(28) recommends 0.023, and Drexel and
McAdams(14) correlated data for air with a constant of 0.021i
Tube with a Solid Rod in the Center —the Limiting Case of Promoters
at Zero Spacing
When the promoter spacing goes to zero, the geometry reduces to
the case of flow in an annuluso Data for an annulus are generally correlated with Nusselt numbers, friction factors, and Reynolds numbers
defined on the basis of the velocity in the annulus and some equivalent
diameter, This leads to confusion when compared with results for an
empty tube or a tube with a set of turbulence promoters whose spacing is
not equal to zero. In this later case data are generally correlated on
the basis of the superficial velocity in the empty tube and the inside

-11tube diameter. In the interest of clarity the following conventions will
be adopted for this presentation.
1. The terms Re, f, Nu, and D will refer to values evaluated
using the inside tube diameter D and the superficial velocity U based
on flow in the empty tube. In other words,
U w (19)
p it D2
Re = D p U (20)
4 w
4- w (21)
p D ^D
g D
2 p U2 _
g r 2p3D5 (22)
32 W2
Nu = D (2)
k
2. The terms Re*, f*, Nu*, and D* will refer to values evaluated using an equivalent diameter D* and the mean velocity u in the
annulus. In other words,
44w
u= 4 (23)
p t D2 (l-d2)
0 D* /D (24)
Re* = D* p u (25)
4 D* W (26)
D2 (-d )(26)

-12Re (27)
(27)
(1-d2)
f p t (1-d2) D4D (8)
(28)
32 wP
= (1-d2)2 f (29)
Nu* -h D* (o)
NU* (30)
= a Nu (31)
Various correlations have been proposed for pressure drop in an
annulus. Knudsen and Katz(22) recommend
f* = 0.076 Re*'0*25 (32)
with a = 1 - d (2a)
(12)
Davis(12) gives
* = 0.055 (l-d)-0-10 Re*-020 ()
with a = - d (33a)
(42)
Whalker, Whan, and Rothfus(42) present
f* = 0.079 Re*O0'25 (34)
= 1 + (1-d) (34a)
l in d
Lohrenz and Kurata (26) have recently presented a graphical
correlation of f* vs. Re* based on
= + d2 + (-d2) (35)
l n d
Their definition of equivalent diameter (i.e. c) has the advantage that,
when plotted on logarithmic coordinates, all friction factors for laminar
flow fall on the same straight line defined by
*.= 16 (36)
Re*

and the critical Reynolds number where flow deviates widely from laminar
behavior occurs at Re* = 2300. For fully developed turbulent flow, however, (Re* > 10,000) this correlation yields a family of lines, each
corresponding to a different diameter ratio d. Lohrenz and. Kurata present
lines only for the limiting values of d = 0 and d = loO and one intermediate value corresponding to d = 0.33o In order to compare their
graphical correlation quantitatively with other correlations for annuli,
it is necessary to have an equation for interpolation. The following is
suggested for the equivalent Reynolds number range 10,000 < Re* < 4o0,00o
f* 0~079 F(d) Re*-0'25 (37)
where F(d) is a tabulated function of d given in Table I.
(29)
Meter and Bird9) propose on semi-theoretical grounds the
following equation
1 - G(d) logo10(d)Re* - f* - H(d) (38)
with a d (38a)
and
O(d) 1 1 + d2 (l-d) (38b)
(l-d)2 ln d
where G(d) and H(d) are complicated functions of d whose values
are tabulated in Table Io
Because of the different definitions of equivalent diameter
used by different investigators, it is difficult to compare directly the
various correlations of f* vSo Re*o This difficulty can be eliminated,
however, by converting all of the equivalent friction factor correlations
of the form f* vs. Re* to the type ordinarily used in empty tubes,

-14TABLE I
FUNCTIONS F(d), G(d), AND H(d) USED IN EQUIVALENT FRICTION FACTOR
CORRELATIONS FOR ANNULI
d F(d) G(d) H(d)
OO o.0918 4~ooo o04oo
0.05 0.918 30747 0o293
0o1 0 0o910 35736 00239
o015 0O903 3.738 0.208
0.20 0881 35746 O.186
0.30 0o853 3o771 0o154
o40 0o,830 3801o 0.131
0o 5 0 817 3 833 0 o lll
o60 o o807 3o866 0.093
0.70 0,798 3.900 0o076
0o80 0.779 3 933 0 o60
0.90 0.771 30967 0o046
1.00 0.762 4oo00 0o031

-15f vs. Re, by using the relations
Re* ~ Re (27)
fif = (39)
(1-d)2 2a
In other words, for a given mass flow rate of fluid in the
annulus the Reynolds number may be calculated using
Re -4 W (21)
p J D
Then, using any particular correlation for which a is specified, Re* may
be calculated, f* may be obtained as a function of Re* from the correlation, and the value converted to f for comparison with other correlations.
This was done for each of the five correlations, using various
values of diameter ratio d and Reynolds number Re. In addition, the
friction factor was calculated using the Blasius equation (16) and the
hydraulic radius. In effect, this is simply one more correlation of the
form
f* - 0.079 Re*-0~25 (40)
with a = - d (40a)
The comparison is shown in Figure 2. To eliminate the wide
variation of f with d at a given Reynolds number, the ratio of the
friction factor f obtained using each of the five correlations to that
obtained using Equation (40) is plotted. This is the same as plotting
the ratio of the pressure drop (for flow through the annulus at a given

000'0 pue'000OOO'OOO'OT = aH JoJ'p oTH aGq-GaulWeT'SA snfTPa oTTIn'eJpXH pus uOT fnbg s enTSBgI
SUTsn paT PeITnoD V Jo OT0s'BH'TTfnUUV OJ
SUOT;B^aJO0.o;'sej UOTI;OTJl JO osTduoj zduO a9n~o
P''OIJV.l H13.L3WVl
~1~ 1~1~1 ~ 801
6I0
ZlFVI OGNV N3SOnN)I - - _ _...~' ~n ~ ~,~ol, 01 1
^^ Ol OOdONe
110 QN, H I__ IVI
/L
P..lI' 63
~ ~ ~ ~ ~ ~~ ~ 6'0
Z.LVI ONV N3SOnNa I c -
w.b.- ") em-<
~ ~- I~~ ~1 CI
-' - __ 1*, 0000*0^ ^ ip
~ j ~ 1 ~ 1 ~ ~ 1 ~1 ~ 8'0 m
o
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 6'0 a)
_ _)I_ CIVNV N3San.x _ __ >
p00001 -71, 4 i /.J VI

-17mass flow rate) predicted by each correlation to that predicted by
Equation (40). It can be seen that the correlation of Walker, Whan, and
Rothfus(42) agrees very well with that of Meter and Bird(29). The correlation of Lohrenz and Kurata(26) predicts pressure drops about ten per
cent lower than that of Walker, Whan, and Rothfus, but this could well be
due to the difficulty in interpolating their plot for different values of
d. Considering the reliability of the data, any of the three correlations is probably acceptable.
Lohrenz and Kurata's choice of equivalent diameter has the
advantage of correlating all diameter ratios with a single curve in the
laminar region and of predicting a single value of the equivalent Reynolds number Re* for transition from laminar flow, but it has the disadvantage of producing separate lines for each diameter ratio in the
turbulent region and the correlation suffers from lack of an analytic
expression for calculating the friction factor. Walker, Whan, and
Rothfus' correlation on the other hand, has the advantage of producing
a single curve for all diameter ratios in the fully developed turbulent
region, but of producing separate lines in the laminar region and separate transition equivalent Reynolds numbers Re* for each different
value of d.
Much less has been done on the study of heat transfer to a
fluid flowing in an annulus than on pressure drop. This is particularly
true for heat transfer to the outside wall of the annulus.
Barrow(2) made a theoretical study of convection heat transfer
coefficients for turbulent flow between parallel plates in which the

-i8heat flux at each wall was arbitrary. The assumptions were: 1) constant
eddy diffusivity of heat over the channel, and 2) the thermal diffusivity
was negligible compared with the eddy diffusivity.
He defined q as the rate of heat transfer at one plate and
cq as the rate of heat transfer at the other plate and obtained the following expressions for the Nusselt number
Nu - 01986 Re/8 Pr (41)
5.03 (2 + R) Re1/8 + 9774 [Pr -(2 + l)]
at the plate where the rate of heat transfer is q, and
Nu 01- 986 Re7/8 Pr (42)
5.03 (2 + 1/i) Re1/8 + 9.74 [Pr - (2 + l/q)]
at the plate where the rate of heat transfer is Tqo
For T = -1, corresponding to symmetrical heating or cooling
(i.e. equal rate of heat transfer at each wall), the Nusselt numbers are
the same and agree well w'ith
Nu* - 0,023 Re*8 Pr (43)
where c = 1 - de
However, with r- = 0 (corresponding to heat transfer at one wall
only, which is the case of interest in this study) the Nusselt number
predicted by Equation (41) is about 30 per cent lower than the value
given by Equation (43) at a Re* 1= O,000oard 40 percent lower at Re* = 40,000.
Only one set of data for heat transfer to the outside wall of
an annulus could be found in the literature: that of Monrad and Pelton(30)
for a solid rod in the center of a tube with a diameter ratio d of
0o541e On the basis of their data they propose

-19NuX = 0.023 Re*08 Prn (44)
with a= 1 -d (44a)
and n = 0.3 for cooling
n = 0.4 for heating
Turbulence Promoters
In this section a review of previous experimental studies of
different types of turbulence promoters will be presented in chronolog(23)
ical order. The work done by Koch(2) which will be described in its
place is the only extensive work done on bluff-body turbulence promoters.
Because of the extensive work done with rough tubes, and because
roughening elements are usually placed in a different class from other
types of turbulence promoters, no attempt is made to review the experiments on the effect of roughness of the tube wall. A complete summary
of previous work done in this field, however, is provided by Nunner(34).
Royds' work^) reported in 1921 is apparently the earliest.
His equipment consisted of a horizontal, double-tube heat exchanger. Hot
air on the inside was cooled by water on the outside. The inside diameter
of the inside tube was 2-5/8 in. The length of the test section was
about 7 ft; no hydrodynamic entry length was provided.
The turbulence promoters (or retarders as he called them) for
which results were presented consisted of twisted metal strips of different pitch. The strips were 1-15/16 in. by 0.10 in. Apparently they
rested on the bottom of the tube.
The experimental data were not presented, but overall heat
transfer coefficients were plotted. In the pressure drops reported, the
change in kinetic energy due to the change in temperature of the air was
not taken into account.

-20Royds' general conclusion was that his retarders increased the
rate of heat transfer, but were slightly less efficient when the energy
required for pumping is compared with that for plain tubes.
Colburn and King(-9) presented their results in 1931. Their
test section was a 3 ft. horizontal length of 2-5/8 in. steel tube through
which hot air was passed. The air was cooled by water flowing through a
1/4 inch copper coil soldered around the tube.
Ten turbulence promoters of the following designs were tested:
2 large twisted steel strips
3 small twisted steel strips
2 copper wire spirals
1 propeller-shaped brass baffle
1 set of copper wire spirals
Although it is not clear how the turbulence promoters were supported in
the tube, it is most likely that they rested on the bottom of the tube.
Data were obtained and overall heat transfer coefficients
determined for a Reynolds number range of about 3,000 to 10,000. No
hydrodynamic entry length was provided; the air entered the test section
directly from a mixing chamber,
The general conclusions were: 1) heat transfer coefficients
can be materially increased by the use of baffles, etc. 2) values for
almost any type of packing of baffle lie on the same curve of heat transfer vso pressure drop, so that if the pressure drop for a new baffle is
known, the heat transfer coefficient can be estimated.
Nagaoka and Wantanabe(31) also published their results in 19310
The equipment consisted of a horizontal, double-tube heat exchanger in
which water on the inside was heated by hot transformer oil on the outside.

-21The inside tube had an inside diameter of 1.06 in. and a length of
5.37 ft. A hydrodynamic entry length of 3.28 ft. was provided.
Twenty-two different turbulence promoters were tested. Each
consisted of wires of various shapes wound in spiral coils such that the
outside diameter of the coil was the same as the inside diameter of the
tube. Approximately five flow rates were tried with each promoter, covering a range of Reynolds number of 4,000 to 20,000. Overall heat transfer coefficients were given.
Colburn(ll) reviewed in 1942 the data and results of both his
earlier work with King and the work of Nagaoka and Wantanabe.
Seigel reported his work(39) (which primarily concerned applications to air-conditioning and refrigerating coils) in 1946. Data were
taken for heating water in a 5/8 inch outside diameter horizontal copper
tube. The water was heated by passing an electric current through the
tube.
Three types of promoters were tested:
1. twisted copper strips 0.02 inches thick
2. spiral wire spring
3. 3/8 inch copper tube with sealed ends
The test section was 10 fto long with no hydrodynamic entry
length. No data on the pressure drop were reported; overall heat transfer coefficients were plotted as a function of water flow ra-te in GPM.
The general conclusion of Seigel was that spiral springs gave
the biggest increase in heat transfer and the pressure drop was least
when the distance between turns was largest,

-22Measurements made by Evans and Sarjant( 5) for cooling of high
temperature gases were reported in 1951. Air, heated electrically, was
cooled by water flowing through copper tubes coiled around the tube.
Turbulence promoters tested were:
1o a two inch solid rod centered in the tube for
one set of data and resting on the bottom for
another
2o twisted metal strips 2-1/2 inch by 3/32 inch
(pitches of 1/7, 1/9, 1/12, and 1/14)
The solid rods did not extend for the complete length of the
test section. This uncertainty in the geometry as well as the complications of high temperature makes it very difficult to compare their results
with the results of others.
(27)
Margolis( obtained data in 1957 using two different fluids:
water and air; his data are presented and discussed by Kreith and Margolis(24'25) The equipment consisted of a horizontal, single-pass heat
exchanger in which the fluid was heated in the tube by condensing steam
on the outside. Two inside diameters of tubes were tested (0o53 in. and
1.12 in.).
The test section was 42 inches long with no hydrodynamic entry
length between the mixing chamber and the test section. Overall heat
transfer coefficients were obtained for nine different turbulence promoters
consisting of the following:
3 twisted strips of metal
6 wire coils
About seven flow rates were tried for each promoter covering a range of
Reynolds number from 1,000 to 10,000.

-23Some of the conclusions were: 1) the rate of heat transfer per
unit area in vortex flow is considerably larger than in axial flow through
straight tubes and ducts; 2) a qualitative difference exists between the
results for air and water which is difficult to explain; 3) the effect
is due largely to the centrifugal force which is present.
Koch(23) published in 1958 data for turbulence promoters using
(34)
the same equipment which Nunner used to study the effect of roughness.
The equipment consisted of a horizontal tube in which air was heated by
condensing steam on the outside of the tube. The test section was 3.21 ft.
long and the tube had an inside diameter of 1.97 in. A hydrodynamic entry
length of 8.19 ft. was provided.
Overall heat transfer coefficients were determined for the following types of turbulence promoters:
14 orifices
10 disks
4 propeller devices
2 rings
3 twisted strips of metal
4 types of packing material
In addition, some velocity distributions and wall shear stress
measurements were made for a few orifices,
The general conclusion was that twisted metal strips, orifices,
and propellers were most effective. The use of turbulence promoters is
economical in some Cases.
In 1960 Gambill, Bundy, and Wansbrough(l7'18) reported the
results of a comprehensive study of the effect of twisted metal strips on
heat transfer and pressure drop to water flowing in a tube. Data were
taken for non-boiling heat transfer, boiling heat transfer, and burnout.

~24
A series of inside tube diameters ranging from O136 to 0.249 inches awas
studied over a Reynolds number range of 10,000 to 200,000 No special
entry length was provided for the twisted tapeso
In 1962 Gambill and Bundy(19) presented additional heat transfer results obtained using the same equipment as for their previous work,
but with ethylene glycol as the working fluid~ An overall correlation
was obtained for predicting heat transfer rates and friction factors for
any tube containing twisted metal strips using any fluid with Prandtl
number greater than that of airo
Miscellaneous Related Studies
This section presents a review of several investigations which,
although they did not study directly the effect of turbulence promoters
on, the rate of heat transfer and pressure drop to a fluid flowing in a
tube, have results which pertain in part to the problemo
Kemeny and Cyphers 21) presented experimental data on the rate
of heat transfer' and pressure drop from water to the inside tube of an,
annulus with surface spoilers. The surface spoilers consisted of both
semi-circular protrusions and depressions wound helically around the tubeO
A very good presentation of the economics of artificially increasing the
heat transfer at the expense of pressure drop is giveno
Sundstrom and Churchill(40) and Zartman and Churc:hill(43) as
part of separate studies of heat transfer from gas flames in a cylindrical burner measured local rates of heat transfer from hot air to cold
water flowing outside the tubeo The flow was disturbed by the presence
of a single, disk-shape bluff-body flame holder centered in the burner o

-25Sundstrom and Churchill's tube was one inch in inside diameter, the flame
holder provided 48 per cent free area and the Reynolds number varied from
5,000 to 20,000. Zartman and Churchill's tube was five inches in inside
diameter; data were taken for several holders with free areas less than
10 per cent. The Reynolds number was at 14,000 for each case.
Boelter, Young, and Iverson(6) determined local heat transfer
rates to air in the entrance region of a tube for a wide variety of hydrodynamic entrances, some of which might be classed as turbulence promoters.
Faruqui and Knudsen(16) measured heat transfer rates in short
tubes in which the flow upstream had been obstructed by orifices.

EXPERIMENTAL APPARATUS AND PROCEDURE
The apparatus was designed to obtain accurate local convective
heat transfer coefficients and values of the overall pressure drop for
water flowing in a tubeO The equipment was constructed in a manner to
allow any arbitrary devices to be easily inserted in the center of the
tube. A vertical direction of flow was chosen to provide symmetry about
the axis
Description of the Equipment
The essential parts of the equipment are: 1) water supply and
metering system, 2) electrically heated test section, 3) thermocouples
and accessories, and 4) manometer assembly.
Figure 3 is a photograph of the overall view of the equipment
showing the rotameters, manometers, and control panel Figure 4 is a
closeup of the test section. Figure 5 is a closeup of the thermocouple
switches and recording potentiometero Each part of the equipment will
be described in turn.
Water Supply and Metering System
The best illustration of the water supply and metering system
is given by Figure 6, the schematic diagram of the apparatuso Water
enters the, apparatus from the city water lines and is metered by one of.
four rotameterso It then flows through the vertical test section from
top to bottom and is discharged into the city drain lines The flow
rate and gage pressure are regulated by valves at the inlet and outlet
of the equipment.
-26~

Overall View of I VX}!~!iiSziiii!11i~ii!- i0Stf i0i''t'i
Equipment 0'#ff " PIPsi i i::
Se c tion.
Figure 5. Photograph of
Thermocouple Switches
and Recording Potentiometer...........~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-.~.............~~~~~~~~~~~~~~~~~~~~~~~~~;ii -ii~i

PLASTIC
PIPE
PIP Y~7 ~INLET WATER
INLET
THERMOCOUPLE
MIXING BED
PLASTIC
TUBE
MR~~~~~~~~~~~~~~~~~~~~~~~R
VOLT co
VMETER,
+
D.C. TO DRAIN
GENERATOR.0
I —
SHUN^f
MANOMETER ASSEMBLY
TO DRAIN
IXING BED
OUTLET THERMOCOUPLE
Figure 6. Schematic Diagram of the Apparatus.

-29The rotameters were calibrated by measuring the time required
for a given weight of water (weighed on scales) to pass through the equipment at several flow rates. Calibration curves and their equations are
given in Appendix A.
The inlet and outlet water temperatures are measured with thermocouples immediately before and after the test section. In each case, prior
to having its temperature measured, the water flows through a packed bed
2 inches x 10 inches filled with 3/8 inch Raschig rings to insure good
mixing.
Electrically Heated Test Section
Figure 7 is a diagram of the test section which consists of a
stainless steel tube 119.88 inches in length with an inside diameter of
1.005 inches and an outside diameter of 1.240 inches. The first 49.69
inches serve as a hydrodynamic calming section. The next 64.34 inches are
the heating section. The final 5.85 inches act as an outlet section. The
test section is connected to the water supply and metering system by two
1-1/4 inch tube connectors.
Heat is generated electrically in the heating section by passing an electric current through the tube wall. The current is generated
by a 12 volt, 3000 amp d.c. generator whose output can be varied from 4
to 36 kilowatts. The electrical terminals at the extremes of the heating section are copper bus bars 4 inches wide and 3/4 inches thick. Holes
were drilled in the center of the bus bars approximately two inches from
each end and the steel tube was silver soldered to the bus bars. As can
best be seen from the detailed diagram in Figure 7 the bus bars were

-30Li..
3g TOP I ^(^TOP
BRASS~ VIE
SPIDER
NUT
)!!!1~- i ~a~/-IWASHER
Z SIDE -F,INSULATED
t }IIS VIEW TEFLON BUSHING
TO MANOMETER
-- \ PRESSURE 1
J - II TAP i
<''>.II i
I SI DETAILED DIAGRAM
_ y ~ I,.I II OF BRASS SPIDER
3ES~ - llli (^ —-^~TUBE~ iHOLES
5,'!lBOLTS I
i II I
1'1 -X "x*^SILVER
SOLDERED
Io~' III
U' II "1 THERMOCOUPLES
iII BOLT
_o ii
I I0
SILVER
i 7 DETAILED DIAGRAM
5 ji OF BUS BAR CONNECTION
i THEPRES
-' I'"SURE
TOMETER',1{1
TUBE
~SWNOTE: FIBERGLAS INSULATION
Figure 7. Di~agram of the Test' Sec+~io~~,

-31bent at right angles and provided with bolts so that the test section
could be easily disconnected from the generator.
The heating section is insulated electrically from the water
supply and metering system by a one foot length of 1-1/4 inch plastic pipe
which serves as the final section of the inlet water pipe.
The entire test section from the measurement of the inlet water
temperature to the measurement of the outlet water temperature is thermally
insulated with Fiberglas insulation 1-1/2 inches thick.
The electric current through the heating section is measured by
noting the emf (in millivolts) across a 5000 amp shunt. This emf is
measured with a Leeds and Northrup 8662 precision portable potentiometer.
The voltage across the heating section is measured with a d.c. voltmeter.
Thermocouples and Accessories
The outside wall temperature of the heating section is measured
at 22 axial stations at one angular position with copper-constantan
thermocouples. In addition at three of the 22 axial stations temperatures
are measured at two other angular positions, each 120 degrees apart. A
list of the locations of the thermocouples relative to the other items
comprising the test section is given in Table II.
The thermocouples are held in place by an epoxy resin and are
insulated electrically from the heating section by a thin (0.002 inch)
sheet of mica.
The thermocouple emfs may be measured one of two ways: First,
by the 8662 potentiometer; or second, by a Leeds and Northrup 20 point
AZAR Speedomax recording potentiometer with arbitrary zero and arbitrary

-32TABLE II
POSITION OF THERMOCOUPLES ON TEST SECTION
RELATIVE TO OTHER ITEMS
Longitudinal Position Angular Position
Item (Tube Diameters) (Degrees)
Top of Tube -49.44
Pressure Tap -12o35
Bus Bar 0.00
(Beginning of Heating)
Thermocouple 1R 1.49 0
Thermocouple 2R o41 0
Thermocouple 3R 9.42 0
Thermocouple 4R 13.40 0
Thermocouple 5R 17.41 0
Thermocouple 6R 22.17 0
Thermocouple 7R 25 34 0
Thermocouple 8R 29 32 0
Thermocouple 9R 33 30 0
Thermocouple 20L 340 0
Thermocouple 19L 35.30 0
Thermocouple 18L 36.29 0
Thermocouple lOR 37.28 0
Thermocouple 17L 38.28 0
Thermocouple 16L 39.28 0
Thermocouple 15L 40o28 0
Thermocouple 11RP 41.20 0
Thermocouple 12R 45.27 0
Thermocouple 13R 49.22 0
Thermocouple 14R 53.26 0
Thermocouple 15R 57028 0
Thermocouple 16R 62.47 0
Thermocouple 14L 34.30 120
Thermocouple 13L 36.29 120
Thermocouple 12L 38.28 120
Thermocouple 1L 34030 240
Thermocouple 10L 36.29 240
Thermocouple 9L 38.28 240
Bus Bar 64.02
(End. of Heating)
Pressure Tap 65 67
Bottom of Tube 69084

-33range. By use of 20 double-pole-double-throw knife switches, the AZAR
recorder is able to record up to 40 thermocouple emfs.
The thermocouples were calibrated prior to use using a constant
temperature oil bath and precision thermometers calibrated by the U. S.
Bureau of Standards over the range from 40 deg. F to 250 deg. F. The
calibration agreed with that appearing in the International Critical
Tables(32).
Manometer Assembly
Pressure taps are located 12.41 inches before and 1.66 inches
after the heating section making a total distance between taps of 78.41
inches. The holes are 1/16 inch in diameter; connections to the manometer
assembly are made with l/4 inch copper tubing silver soldered to the
outside of the test section. The manometer assembly is insulated electrically from the heating section by using a 12 inch piece of plastic tubing
for the connection in place of the copper tubing at one point in the
system.
The manometers consist of two vertical, 100 inch, single-tube,
King manometers. The first contains mercury as the indicating fluid;
the second used an organic oil of specific gravity 1.750 (referred to as
Purple Fluid). A pressure gage is connected to measure the gage pressure
at either the inlet or outlet to the heating section. With the apparatus
pressure drops from 0.02 to 60 psi could be measured with reasonable
accuracy.
Description of the Turbulence Promoters
Easy insertion of arbitrary devices in the center of the tube
required the following features: 1) that it be easy to disconnect the

-34,
test section from the rest of the equipment; and, 2) that there be a
relatively simple method of centering the devices in the tube.
In order to disconnect the test section, the following steps
were necessary:
A. All water to the system was turned off and the drain valve
opened to drain water from the apparatus.
B. The tube connector at the top of the stainless steel tube
was disconnected,
C. Both pressure tap connections were disconnected.
Do Four bolts were removed from the bus bar connectors at the
top and bottom of the heating section.
E. The test section was swung down into a horizontal position
on the floor where the bottom tube connector was disconnected and the
test section moved awayo A swivel joint was provided at the floor just
below the test section so that the test section could be rotated down.
This whole procedure could be done in less than 30 minutes,
Reassembly of the equipment also required approximately 30 minutes and
consisted of doing the opposite of the above steps in reverse order.
Provision was made for centering a 1/8 inch rod in the tube and
keeping it in tension. Thus, any arbitrary device could be mounted in
the tube so long as it could be attached to the 1/8 inch rodo
Two brass spiders were provided at each end of the stainless
steel tube as shown in the detailed diagram of Figure 7. The 1/8 inch
rod (usually made of brass) was threaded at each end and could be put in
tension in the center of the tube by passing each end through the spider

-35and tightening the nuts on the end. A teflon bushing in the spider and
use of rubber washers made certain that the rod was insulated electrically
from the heating section.
Since the 1/8 inch rod was mounted vertically, centered at each
end of the tube, and held in tension, any device centered on the rod was
centered in the tube when the rod was in place. Each of the three types
of devices: disks, streamline shapes, and solid rods will now be described
and the method of mounting them on the rod will be explained.
Disks
Disks were prepared having diameters of 5/8, 3/4, and 7/8 inches
and a thickness of 1/8 inch. They were machined from nylon rod and a hole
slightly larger than 1/8 inch was drilled in the center of each. The disks
were held in place on the centering rod with two aluminum collars, 1/4 inch
in diameter, before and after each disk. The collars were held in place
with a small set screw. A diagram of the disk and method of mounting it
on the rod is shown in Figure 8.
Streamline Shapes
The shapes, while resembling teardrops, were actually composed
of a cone tangent to a sphere. This is illustrated in Figure 8 with the
important relative dimensions noted.
The shapes were machined from nylon rod and had diameters (at
the point of maximum diameter) of 5/8, 3/4, and 7/8 inches. A hole slightly
larger than 1/8 inch was drilled in the center of each shape so that it
could be mounted on the rod. Each shape was held in place on the rod by
a small set screw. As an added precaution to insure that the devices

-36TOP VIEW - ( TOP VIEW
CENTERING
PINS
~ - 1/4" 1
Iu
I I
I I
II
0
SIDE VIEW SIDE VIEW
J 3 Y ^ DISK STREAMLINE
\ ^ ^S \ ^/II ^ SHAPE
h ~I _
II
Figure 8. Diagram of Disk and Streamline Shape
Showing Relative Dimensions and Method
of Mounting.

-37Figure 9. Photograph of Individual Turbulence Promroter Shapes Used.
Figure 10. Photograph of a String of Disks and a String
of Streamline Shapes.

-38were centered, three aluminum pins (3/32 inches in diameter) were installed at the point of maximum diameter of the shape. The pins were
120 degrees apart, extending far enough so that when the shape was mounted
on the rod in the tube, the pins just touched the tube wall, holding the
device in place.
Figure 9 is a photograph of the six individual bluff-body turbulence promoters (i.e. disks of three diameters and streamline shapes of
three diameters) utilized in this investigation. One turbulence promoter
combination, by-definition, consists of a string of individual bluff bodies
all of the same shape and diameter spaced an equal distance apart. An
illustration of a combination of disks and a combination of streamline
shapes is shown in the photograph of Figure 10
The 21 different bluff-body turbulence promoter combinations
used in this investigation are itemized in Table III. An attempt was made
to choose the number of promoters such that the-total length of the string
of promoters was 48 inches (which is almost exactly the same as 48 tube
diameters). For various reasons, however, as noted in the footnotes to
Table III it was necessary to use a shorter string of promoters in some
cases. The variation in the length of the string of promoters used in
the experimental measurements in no way impairs the usefulness or generality of the data. Since local heat transfer coefficients were measured,
the total length of the string is immat-eal; it is necessary only to
obtain data near one representative promoter or group of promoters. As
will be shown later, the variation in length of the string of promoters
was corrected for in the calculation of friction factors.

-39TABLE III
BLUFF-BODY TURBULENCE PROMOTER COMBINATIONS USED
IN THIS INVESTIGATION
Shape d s np Length of String
Disk 0.625 12 4 48
Disk 0.750 12 4 48
Disk 0.875 12 4 48
Disk 0.625 8 6 48
Disk 0.750 8 6 48
Disk 0.875 8 6 48
Disk 0.625 4 12 48
Disk 0.750 4 12 48
Disk 0.875 4 8 32*
Disk 0.625 2 11 22**
Disk 0.750 2 11 22**
Disk 0.875 2 8 16*
Streamline Shape 0.625 12 4 48
Streamline Shape 0.750 12 4 48
Streamline Shape 0.875 12 4 48
Streamline Shape 0.625.8 6 48
Streamline Shape 0.750 8 6 48
Streamline Shape 0.875 8 6 48
Streamline Shape 0.625 4 6 24**
Streamline Shape 0.750 4 6 24**
Streamline Shape 0.875 4 6 24**
* Total length of string of promoters shortened because of
excessive pressure drop which would result, reducing maximum
flow rate which could be obtained.
** Total length of string of promoters shortened to save the
expense and effort of fabricating a large number of shapes.

-4oSolid Rods
The first type of solid rod tested was simply the 1/8 inch
brass rod used to mount the disks and streamline shapes.
The second type of solid rod tried was a 1/4 inch threaded
brass rod. The rod was mounted (not in tension) using a special spider
at the top only and centered with three teflon centering spiders, each
held in place with nuts threaded on the rod, These tests were originally
made with the idea of using a threaded rod to mount the disks and streamline shapes. Although this proved impractical, the data fall in the same
class as the other data for solid rods and seem worthy of presentation.
The third type of solid rod tested consisted of 5/8 and 3/4
inch brass rods. At each end of the rods 1/8 inch holes were drilled in
the center and threaded so that studs could be screwed into the rod and
used with the brass spiders to center the rod at each end of the tube.
At two points along the rod (dividing the rod approximately into thirds)
two 1/8 inch holes were drilled perpendicular to the axis of the rod,
each of the two holes being 1/2 inch apart and perpendicular to each other.
Nylon rod segments 1/8 inch in diameter (each one inch long) were inserted
through the holes so that they protruded the same distance on each side
of the rod. The nylon rod segments were held in place by small set screws
perpendicular to each rodo Thus, when the brass rod was in the tube, it
was centered at two points in addition to the ends.
Description of the Procedure
The procedure used to obtain heat transfer and pressure drop
data will be described for a typical situation.

-41The first step (assuming that the equipment was still set up
from a previous run) was to remove the test section from the rest of the
equipment as described in the description of the turbulence promoters.
The centering rod on which the turbulence promoter combination from the
previous run was mounted was removed and, if a different diameter or
shape device was to be tested, the old devices were removed from the
centering rod and new ones installed. If the same shape and diameter
devices were to be tested again, but at a different spacing, they were
simply loosened. In either case, the devices were moved to the desired
position and the necessary set screws were tightened to hold the objects
in place on the rod.
The distance from the end of the rod to each object was then
measured and recorded on a data sheet. The centering rod was inserted
in the spiders at each end of the tube, the nuts on the end of the rod
were tightened, and the distance from the top of'the tube to the end of
the centering rod was measur.ed and recorded. Thus, the distance from
the top of the tube to each turbulence promoter could be determined from
the measurements.
The test section was then reconnected to the water supply and
metering system; the electrical terminals were connected; the pressure
taps were connected and the water was turned on to the system. The water
was always discharged directly to the drain (bypassing the test section)
at maximum flow rate for about 15 minutes to remove any foreign material
that may have accumulated in the pipes.

_42The manometer lines were bled to remove any air and the equipment was ready for taking data. Pressure drop data were usually taken
first, by varying the flow rate and recording the rotameter reading, the
manometer scale being used, the manometer reading, and the temperature.
The flow rate was controlled by a valve at the outlet of the equipment so
that back pressure was always maintained on the test section, thus, assuring that the tube was full of water,
The procedure for taking heat transfer data was more involved.
The first step was to adjust the flow rate of the water through the test
section to the desired value. The d.c. generator was then started and
the voltage set so as to give the desired rate of heat input,
The temperatures throughout the system were observed (either
using the 8662 potentiometer or the AZAR Speedomax recording potentiometer to record the thermocouple emfs) at intervals of about five minutes until steady-state conditions were attained (ioeo until the temperatures did not change). For the first run of a session about two hours were
usually required for the system to come to steady state because it took
a long time for the generator and electrical leads to heat up to their
equilibrium temperatures. For subsequent runs less than 15 minutes were
usually required to achieve a steady state.
When steady state was achieved, the zero and range of the AZAR
recorder were adjusted so that the minimum wall temperature of the tube
was at zero on the recorder and the maximum wall temperature was at about
90 per cent of full scale. Two 8662 portable potentiometers were set so

-43that they provided reference emfs which recorded at about 0 and 80 per
cent of full scale on the AZAR recorder.
At this point the data taking commenced. The following items
were recorded on a data sheet: 1) the rotameter number and reading,
2) the manometer scale and manometer reading, 3) the voltmeter scale
and voltage across the test section, 4) the number of millivolts across
the 5000 amp shunt (i.e. the current through the test section), 5) the
inlet water thermocouple reading, 6) the outlet water thermocouple reading, 7) the value of the two reference emfs supplied by the 8662 potentiometer, and 8) one or more randomly selected wall temperatures. Items 4,
5, 6, and 8 were obtained using one of the 8662 potentiometers.
The AZAR recorder was then started with all the knife switches
set in the same position. As soon as 20 points had been recorded, twelve
of the knife switches (for the twelve channels which had two thermocouples
attached to them) were thrown to give additional temperature readings.
After the next 20 points (eight of which were the same) had been recorded,
the knife switches were returned to their original positions. This was
repeated four times so that the 32 different emfs were recorded four
times.
At this time the original 8 items of data (i.e. rotameter readings, manometer readings, etc.) were measured again and recorded, This
terminated a run and the generator was turned off, the water rate adjusted
to a new value and the whole procedure repeated for a new run.
Generally, five flow rates were used (or, in other words, five
heat transfer runs were made) for each turbulence promoter combination.

-44Data were almost always taken in the evening to minimize any variation
in the water pressure caused by other users in the building turning water
on and off. A complete set of pressure drop data and heat transfer data
at five flow rates could usually be obtained in one evening.
Method of Calculating Heat Transfer Coefficients
The local convective heat transfer coefficient is defined as
h(z) (Z) (45)
Th) (z) - T (z)
wall f
where z = longitudinal distance from beginning of heating (tube diameters)
h(z) = local heat transfer coefficient (BTU/hr - deg F - ft2)
q(z) = local rate of heat transfer per unit area (BTU/hr - ft2)
Twall(z) temperature of the inside tube wall (deg F)
Tf(z) = mixed mean temperature of the fluid (deg F)
The general procedure for obtaining these local heat transfer
coefficients from the experimental measurements is as follows:
1. q(z) is determined by measuring the electric current and
knowing the electrical resistance of the tube.
2. Tf(z) is obtained from an energy balance. The heat added
to the water is integrated from the beginning of the tube to z.
3o Twall(Z) is obtained by measuring the outside wall temperature and calculating the inside wall temperature from solutions of the
conduction equation.
When the electric current passes through the tube wall, generating heat, a temperature gradient is developed so that the heat will
flow to the inside wallo The differential equation (usually referred

-45to as the conduction equation which describes this process is
r K + K aT +A= (46)
r br Kr Y
where T = temperature (deg F)
r = radius (ft)
y = longitudinal distance (ft)
A = rate of heat generation per unit volume of tube wall
(BTU/hr - ft3)
K = thermal conductivity of the tube wall (BTU/hr - deg F - ft)
The rate of heat generation A as derived in Appendix B is
2- 2
A 3.41276 p- 2 (47)
Po (1 + 7T)t2 (b2 - a2)
where p = electrical resistivity of the tube wall, assumed to be a
linear function of temperature (ohm-ft)
P= PO (1 + yT) (48)
- (b2 _- 2)
and - o (b2 (49)
b
b r dr
a l+7yT(r)
The symbol p for electrical restivity will always be written with a
bar over it to distinguish it from the fluid density p
Equation (46) which is a non-linear partial differential equation can be simplified considerably and reduced to a non-linear ordinary
differential equation by considering that axial conduction is not important. Axial conduction, however, can arise and influence the solution
of (46) in one of two ways: 1) by conduction of heat into the nonheat-generating portion of the tube; and, 2) by presence of a nonconstant axial temperature gradient at the inside wall. Each of these

_46possibilities is considered in Appendix B and shown to produce a negligible effect.
Therefore, neglecting axial conduction (T= constant) and letting
K= K (1 + T) (50)
then the conduction equation becomes
d T I dT 3.41276 I -Pm_
d2T 1 dT +.41276 (51)
dr2 r dr 1 + PT
dr2 r dr 1 + PT Ldr KO o (1-+ yT)(1 + PT)(b2 a2)2
with the boundary conditions
TI(b) = 0 (52a),T(a) = T (52b)
a
2- 2
Define' A = 3.41276 IPm (53)
2 (b2 2)2
PO i (b - a)
The solution as suggested by Clark(8) is
A (b2 2)
T(b) - T(a) - 2 n b - a (5)
2 Kb - 2 J
2 Kb 6(1 + 7Tb)(l +T b)
where, of course, T(a) is the value Twall which is required and T(b) is
the outside wall temperature which is measured.
The difference between T(b) and T(a) caused by generation of
heat in the tube wall is often referred to as ATgeneration.
The rate of heat transfer to the fluid at the inside wall is
found by1 -b
(z) - A r dr (
1/a

-475.41276 p(ol + 7Tb(z))I (56)
(56)
2i2 (b2 - a2) a
The mean fluid temperature is given by
z
2t- a/ q(Z) dZ
Tf Ti +. (57)
f inlet W
or, since q(Z) is very nearly constant
Tf Tinlet + (Touet - Tinlet) (58)
Thus, in practice, Equation (56) was used to obtain q(z);
Equation (54) was used to obtain T11 and Equation (58) was used to
obtain Tf(Z). All data processing was performed on the IBM 704. A
detailed description of the computational method and the numerical values
of the constants in the preceding equations is given in Appendix Ao
Method of Calculating Friction Factors
The friction factor is defined as
f g (4)
2 p U2 L p
x2 D5
ge "2 p D5Lp (22)
32 W2 L
4 w
since U 4 (19)
TpD
with L = distance between pressure taps (ft)
p
AP = pressure drop (lbf/ft2)
As explained previously, the use of the superficial velocity
U and the inside diameter of the tube D allows the same definition of
f to apply for the empty tube, the tube with turbulence promoters, and
the tube with a solid rod in the center

-48It is tacitly assumed, however, that the geometry of the tube
is the same for the entire distance Lp between pressure taps. This is
certainly true for the empty tube and for the tube with a solid rod in
the center. But for the turbulence promoters the string of promoters
occupied only part of the distance between pressure taps. This is illustrated below.
1.I nP~Lp
npS
1,1 _________I I
/p
tp
The desired friction factor for the tube with turbulence
promoters is defined as
f g- D =_ (59)
2pU2
2 p IT Tn S
g 2p D5 - p (6)
(60)
32 W2 np S
This friction factor takes into account only the pressure loss due to
drag of the bluff-bodies and drag of the tube wall for the length of
tube occupied by the string of uniformly spaced turbulence promoters.
Unfortunately Ptp could not be measured, but instead only
the overall pressure drop AP was obtained. An attempt at correcting

-49this difficulty, however, was made by estimating the pressure drop due
to the tube wall between the pressure taps and string of turbulence promoters and subtracting it from tLP to obtain APtp. Thus,
32 W2 f0
tp = 32 5 (L-np S) (61)
gc i pD
where f0 is the friction factor for empty tubes as calculated using
Nikuradse's correlation equation (15) for smooth tubes. This, of course,
neglects the pressure loss due to drag on the 1/8 inch rod (on which the
promoters were mounted) in the length of tube for which there were no
promoters. As will be shown later, the difference between the friction
factor for a smooth tube and that for a tube with a 1/8 inch rod in the
center is very small. -For almbst all geometries, the difference between
Stp and AP is a very small per cent of P. Substituting Equation (61)
into Equation (59) an expression for the friction factor for bluff-body
turbulence promoters which- do not occupy the complete distance between
pressure taps is obtained,
f c fo ( - l (62)
Equation (22) was used to calculate friction factors for the
Equation (22) was used to calculate friction factors for the
empty tube and the tube with a solid rod in the center and Equation (62)
was used to calculate friction factors for a string of turbulence promoters. A description of the computer program used to process the
pressure drop data as well as specific values of the constants in the
preceding equations is given in Appendix A.

EXPERIMENTAL RESULTS AND DISCUSSION OF RESULTS
Empty Tube
Pressure Drop
Data were obtained for the empty tube primarily for use as a
check on the reliability of the experimental apparatus and procedure.
Friction factors are plotted versus Reynolds number in Figure 11. They
agree well with the accepted correlation for smooth tubes, Equation (15),
of Nikuradse(33).
Heat Transfer
Sample values of the local heat transfer coefficient h(z)
are presented in Figure 12 as a function of z for the empty tube to
illustrate the local variation. The integrated mean value for each case
is indicated by the solid line. A definite thermal entrance region is
observed extending from 5 to 15 diameters from the beginning of heating.
In some cases the heat transfer coefficient increases gradually throughout the heated section. Both of these effects were noted by
Hartnett(20) in his experimental study of the entrance region. Although
Hartnett could not account for the gradual increase in the heat transfer
coefficient along the tube, the most logical explanation seems to be the
favorable change in the physical properties of the water (particularly
the viscosity) as it is heated. An estimate of the local heat transfer
coefficient, taking into account the change in physical properties of
the fluid is given by
h(z) t = (hm) ez rz (z /)z (63)
o.8 1/53,,o0.14
Rem Pm (/0w)m
-50

0.020 A......
0.015 ~~ - ~~-~~ -
NIKURADSE'S CORRELATION FOR SMOOTH TUBES
0.010 - EQUATION (15) -
-? 00.009 0 _ _ _
W~-' 0.008 --
0.004
0.002 4 Lo
1010
4W
Re
Figure 11. Friction Factor for the Emnty Tube as a Function of
Reynolds Number, f vs. Re.

-521500
RUN: R-I-E
Re 4W 45,500
1400
1300
cN 1100
ILU.
I -) N: RUN: P-12
0
I Re = 31,000
I1000
900 ~ ~RUN:R-I-B
^ r~~~~~~~~.Re= 24,100
N: 800
Ui.
w
0
pr: I hm
w
^ 700... sh ( z )est FROM EQU. (63)
RUN: P-11
m 600 ^ ~ ~~ ~Re 14,.900
700
400 -______________ RUN:R-1-H
JI~ I I I ~~~Re = 8,800
300
0'
0 10 20 30 40 50 60 70
DISTANCE FROM BEGINNING OF HEATINGz (TUBE DIAMETERS)
Figure 12. Sample Values of Local Heat Transfer Coefficients
for the Empty Tube as a Function of Longitudinal
Position, h(z) vs. z.

-53The subscript z indicates that the quantity is a local value
evaluated at the local mixed-mean fluid temperature Tf(z). The subscript
m indicates that the quantity is a mean value and has been obtained by
integrating the local value over the entire length of the heated section
(with the exception of the first fifteen tube diameters). The term hm
is the integrated mean value of the measured local heat transfer coefficients h(z).
The longitudinal distribution of the heat transfer coefficient
due to change in the physical properties of the fluid as it was heated
was estimated using Equation (63) and plotted as a dashed line in Figure
12. It can be seen that Equation (63) satisfactorily explains the gradual
increase in local heat transfer coefficient with increasing fluid temperature. The solid line in Figure 12 is the value of hm.
It should be noted that in order to use Equation (63) it is
necessary to know the value of hm which must be obtained from the data.
The form of Equation (63) is such that when h(z)est is integrated, the
resulting mean value is forced to have a value of h.
In some cases the local heat transfer coefficients "scatter"
slightly. This scatter appears to be random and is probably due to
errors in reading thermocouples.
Nusselt numbers based on the overall integrated mean heat transfer coefficients are presented in Figure 13 versus Reynolds number. The
values have been divided by Prl/3(p/w) to reduce the effect of physical property variation. It is seen that the results agree well with
(3 8)
the Sieder-Tate equation in the region of Reynolds number greater

-54200
70~~
100 - ~~
80 ~
570 ~ ~ ~ ~ ~~- ~~ ~ ~ ~~__
30 ~~ ofe-no -~~-r
20 - ~~~ - ~ 4~~~P
2x10" 10 lo'
Figure 13. Nusselt Number for the Empty Tube as a Function
of Reynolds Number, = y vs. Re.
pr,3 (oL/')0.i4

-55than 10,000 for which the Sieder-Tate equation is valid. As would be
expected, in the Reynolds number range Re < 10,000 corresponding to the
transition region of flow for heat transfer the data points fall below
a line extrapolating Equation (18).
Solid Rod in the Center of the Tube
Pressure Drop
Friction factors defined by Equation (22) are plotted in
Figure 14 as a function of the Reynolds number (4W/i g D). This method
emphasizes. the variation in pressure drop with geometry at the same mass
flow rate. The rod with d = 0.75 gives a pressure drop over 20 times
that of the empty tube, while the rod with d = 0.125 increases the
pressure drop by only about fifteen per cent.
In Figure 15 the results are compared with the equivalent friction
factor correlation for annuli recommended by Lohrenz and Kurata(26) In
this case the friction factors and Reynolds numbers are based on an equivalent diameter and the mean velocity of the fluid in the annulus. In
other words,
f*- (1d2)2 a f (29)
Re* = Re (27)
(1-d)
a 1 + d2 + (b 1) (35)
The data fall between the lines corresponding to the limiting
cases of d = 0 and d = 1 and are well within the reliability of the
correlation as stated by Lohrenz and Kurata(26) The effect of roughness

-560.20..
0"..20'C.. d =-0.750
0.10
0.08
0.06
o~o~-~-o 3. =~.~.d=0.625
0.04
L~J 0.03
" QN 0.02
d =0.250
r C 0.010C)^,0 lt4 (THREADED ROD)
ii 0.010 _ _ __ _ __ _
0.008 -T~ T ~
0.006 d = 0.125
EMPTY TUBE
EQU. (15)
0.004 -~~~
0.003
0.002
4x103 o10 10 4x0lo
4W
Re: /.=TD
Figure 14. Friction Factor for the Tube with a Solid Rod in the
Center as a Function of Reynolds Number, f vs. Re,
with Parameters of d.

-57I0
0 0 -d =0.125(SMOOTH ROD)
o 0.030 -~ * -d =0.250(THREADED ROD)
0 E ~l E -d =0.625 (SMOOTH ROD)
A -d 0.750 (SMOOTH ROD)
ic 0.020 _____ ____ _______~~ _______ _ SOLID LINES ARE GRAPHICAL
a-^~ u~~~uwJ ~CORRELATION RECOMMENDED BY
r..0 LOHRENZ AND KURATA
- -0.010 - -....e D aGIEBEQ.
*b 0.008 ~- — _~ ~~~__d =0.0
0.004......
0.006 -- 0 - -
0.003
0.002
3x10 a 10w
Re* idZ,o D 0,GIVEN BY EQU.(35)
Figure 15. Friction Factor for the Tube with a Solid Rod in the
Center as a Function of Reynolds Number Based on
Equivalent Diameters, f* vs. Re*.

-58on the results for the threaded rod is quite noticeable in this method
of presentation.
Heat Transfer
Sample values of the local heat transfer coefficients are plotted
in Figures 16, 17, and 18 for the solid rods. For each of these rods
there were at least two centering supports: one located before the heated
section and one located somewhere within the heated section, Some of the
local values of h are not considered representative because they are in
the thermal entrance region or because they were increased due to disturbances in the flow near a centering support. These points are indicated
by dashed circles and were not used in computing the mean value indicated
by the solid line.
Some of the same features are noted in the plots of h(z) vs. z
that were noted in the results for the empty tube —particularly the increase
in h as the fluid is heated.
In many cases the effect is accentuated because the change in
physical properties may cause a variation in the local Reynolds number
such that the flow at the outlet of the heated section is in the fully
developed region of flow (Re* > 10,000) while the local Reynolds number
at the inlet is in the transition region (2300 < Re* < 10,000).
Equation (63) can not predict the effects of a change in
regime of flow caused by a change in the physical properties of the
fluid. The reason is that Nu* is roughly proportional to Re* in the
transition region of flow, while it is proportional to Re* 8 in fully
developed turbulent flow. Accordingly, it is noticed that the biggest

-591500 1
1400
RUN:R-2-E
Re= l = 32,200
1300 >I I I I- ~~~
9I- 00
1200'm: _ ~
CX. ~Ph( z )st FROM EQU. (63)
i 900
3M El RUN:R-2-G r J
200.- ~ r.-'^ 700 ~I ____ ~- DASHED POINTS NEAR CENTERING.J SUPPORTS OR ENTRY REGION
USED IN MNOT USED IN MEAN
0 10 20 30 40 50 60 70
Position, h(z) vs. z, for d 00
300 ~ RUN:R-2-1. Re= 5,300 r200 \___________________;C
0 10 20 30 40 50 60 70
DISTANCE FROM BEGINNING OF HEATING, z (TUBE DIAMETERS)
Figure 16. Sample Values of the Local Heat Transfer Coefficient for a Tube with a Threaded Rod in
the Center as a Function of Longitudinal
Position, h(z) vs. z, for d ~0.250~

-602300
2200 r - ^
DASHED POINTS NEAR CENTERING
2100 SUPPORTS OR ENTRY REGION
NOT USED IN MEAN
2000
0
1900 RUN:R-28-B ~
Re= 4W 44,000
1800
1700
1600 ~~
1500 hm
- 1400 -- U.- h ()st FROM EQU.(63)
1400 est; 1300
z
w
1200 RUN: R-28 C L
W, Re 18,800
0 1100 1 5,13- 1
0r1000
O I ECIUE I~_
i' 900
w 800
700.Z,'-'
o 700 U~ RUN:R-28-D,.,
-J Re = 11,100 - 0
600 _______
500 -.
400
400 RUN: R-28-E r
-REJECT ANCUSED IN MEAN BEI REJN USED IN MEAN ~I REJECTR
200
0 10 20 30 405060 70
DISTANCE FROM BEGINNING OF HEATING, z (TUBE DIAMETERS)
Figur e Values of the Local Heat Transfer Coefficient for a Tube with a Solid Rod in the
Center as a Function of Longitudinal Position,
h(z) vs. z, for d = 0.625.

3400
RUN R-29-A 0
3300 Re4 W 44,20
Re -C274400
3200 ~ ~~
2600 RUN- R-29- - -
3R100 ~~a349300
2500 -_: _I ___
12400
s _ SUR —-h(z)e FROM EATION _
2300 3
170o
21600 UN - R -29-C r-,
0 - -Re 342,300 E | t,' _' i'"
O 1500 20 3 40 5 6 -
o 9009
4 00 ~~~ Re~2 6100 __
1300
wA" 5 DASHED POINTS NEAR CENTERINGS
D ______SUPPORTSORENTRY REGION NOT
700 _
RUN-R-29-E' I'
400 Re' 6,400 R~^.~0~
200 ~~
<:"REJEECT- DNEAUSED IN MEANREJEC USED IN MEAN REJECT --
010 20 30 40 50 60 70 80
DISTANCE FROM BEGINNING OF HEATING, z (TUBE DIAMETERS)
Figure 18. Sample Values of the Local Heat Transfer Coefficient for
a Tube with a Solid Rod in the Center as a Function of
Longitudinal Position, h(z) vs. z, for d ~ 0.750.

-62failure of the dashed lines to explain the increase in heat transfer
coefficient as the fluid is heated occurs at the smaller values of Reynolds number. (Note: The transition region of flow, 2,300 < Re* < 10,000,
corresponds to 5,000 < Re < 21,190 for d = 0.750 and corresponds to
3,000 < Re < 14,700 for d = 0.125^)
The variation in local heat transfer coefficient caused by
changes in the physical properties, however, produces effects which are
scarcely noticeable when the local coefficients are integrated to obtain
mean values.
The dependence of Nusselt number on Reynolds number for the
solid rod geometry is indicated in Figure 19 for different values of
the diameter ratio d. For the solid rod with d = 0.125 the Nusselt
number is actually less than (or equal to) that for the empty tube at
the same Reynolds number, while for the rod with d = 0.750 it is over
two times the value for the empty tube at Reynolds numbers greater than
20,000. The effect of the transition region of flow at low Reynolds
numbers is quite evident.
A correlation making use of equivalent diameters is shown in
Figure 20. In this plot the equivalent Nusselt numbers (divided by
Prl/5 (u/u)0.l4)based on the integrated heat transfer coefficient for
a solid rod and the equivalent diameter suggested by Lohrenz and Kurata
is plotted versus the equivalent Reynolds number Re*. Several observations may be noted: 1) The data are consistent with the data of Monrad
and Pelton. 2) The line which best fits the data is about ten per cent
below the line for the Sieder-Tate equation, indicating that a value of

-631000- --
600
a d 0.750 (SMOOTH ROD _
E- d =0.625 (SMOOTH ROD)_ _
400 = ~ — d =0.250 (THREADED ROD) __ _
0 d =0.125 (SMOOTH ROD)
300
2103 1
//
20
4- /W
Re 4
Figure 19. Nusselt Numbers for the Tube with a Solid Rod in the
Center as a Function of Reynolds Number, Nu vs. Re,
with Parameters of d.
Re/=
30~~~~~~~~~~~f T
Fiue1.Nse-cNmesfrth/uewt oi.Rd n-h
Cetra[aFnto-oieni/ ubr N s e
wit Paaeesod.

-64500 ~,
_ __ _____ 0o d =0.125(SMOOTH ROD)
400~ d =0.250(THREADED ROD) -
o — 300 E] d= 0.625 (SMOOTH ROD)
3> ^ d 0.750(SMOOTH ROD) -
" X d =0.541 (DATA OF MONRAD AND PELTON)' 200
> 200 -— ~~ ~ ~ __ _ _ _ _ ~~ - ~- _-_t I00 -. 80
60 0.027 Re*
50
a.. 40
V 30
30I /LINE DRAWN THROUGH
],^^ DATA OTHER THAN
"', ~ 20 - THAT FOR 1/4 INCH
-- ~. THREADED ROD./
10
8
~ -d 4W
Re:I- a, GIVEN BY EQU.(35)
Figure 20. Nusselt Number for the Tube with a Solid Rod in the
Center as a Function of Reynolds Number Based on
Equivalent Diameters, Nu*/(Prl/3[p1/]4] OI) vs. Re*.

0.024 should be used for the constant C2 in the correlation. This trend
is suggested by the theoretical analysis of Barrow(2). 3) The effect of
roughness of the rod on the equivalent Nusselt number Nu* is much less
than the effect on the equivalent friction factor f* with the points
for the threaded rod of diameter ratio d = 0.25 falling only slightly
above the line drawn through points for the smooth tube.
Disks Evenly Spaced and Centered in the Tube
Pressure Drop
Friction factors for flow around the disks in the center of the
tube are presented in Figures 21, 22, 23, and 24 for s = 2, 4, 8, and 12
with the diameter ratio d as a parameter. Since these friction factors
are not based on any equivalent diameter, they indicate directly the
effect of the disk geometry on the pressure drop. It is seen that for
disks at a spacing of 2 tube diameters with a diameter ratio of 0.875
the pressure drop is over 300 times that for an empty tube. As would be
expected, except for small diameter ratios and large spacing, the friction
factors are practically independent of Reynolds number.
For a particular geometry (i.e. a given value of d and s) the
friction factor can be represented by an equation of the form
100 f = C(s,d) Ren(s,d) (64)
The constants C(s,d) and n(s,d) evaluated by the method of least
squares are listed as a function of s and d in Table VIII in Appendix
C. The lines drawn with the data for each geometry in Figures 21, 22, 23,
and 24 are the lines given by the above equation.

-6640
d =0.875
2.0 ~ ~~~
0.8
0.4 _.. _="
x 0.3 I 2 1
W4
0.08
0.023
4xd1 10 10i2x1
4W
Re D
Figure 21. Friction Factor for Disks as a Function
of Reynolds Number, f vs. Re, for s = 2
with Parameters of d.

-674.0 _ _
3.0 —20 --
d =0.875
1.0 -- -
0.8 -- ___
0.6
0.4
00 0 -. Fin n__r m F0. = =.=3, d: =0.2750
10.30 - -- 11
w
0.2
0.10i-o~ )_._-Q -.k ~. 0,o_^ l- d =0.625
006
0.034 ---
0.04
0.03 ~ ---- ~~ ~
4x10^ 10 10- 2xlO
4W
Re / Lr
Figure 22. Friction Factor for Disks as a Function
of Reynolds Number, f vs. Re, for s = 4
with Parameters of d.

~p. Jo sJa9q auTB.Txc[ q31 T
9 - s JOJ'a ~SA p j. aquin splouX@aG J
uoOTounI's s ssT- oj O oC oBj[ c o o, uoTq. oTJJ aO lnS,
Mo.
OIXZ 01 01 OIXbt
tOOO
_ _ _ _ _ - _~ ooo..00'0..... I ~ I~ III)'n3-03 OO'o
~3Bni AldW3
- ~-_~ __~- ~900'0
8000
_~0'0
OSL-'O P |E3>; |
~ --- ~ ~ ~ * *~ -- - - ~ ~ 90g0
I ~ ~ " - ~ ~ ~ ~ -0')~- ~ ~800
oG'o= p 1.. 1 L0 __C.C o aE o
0v'0
090
~ 92.8'0 ^ ^^ ^ 08'0
-89

-691.00. 1.
0.80
0.60.
0.140.0. -..
0.30 _ - ~~ -
0.010 Ry= _be _ = - v e o s 1
0.08
0.006 t
0.004 EQU. " =.6(15)2 -5
C 0.003~
0.0028 -.
4x1o0 104 10 a2xl
4W
Re==
Figure 24. Friction Factor for Disks as a Function
of Reynolds Number, f vs. Re, for s = 12
with Parameters of d.

-70In Figure 25 a trial correlation is attempted, based upon
characterizing the pressure drop by an effective drag coefficient (defined
by Equation (14)) for a single bluff body rather than by a friction factor
of the type used for smooth tubes. The effective drag coefficient fD
is plotted versus Reynolds number for s = 12, 8, 4, and 2 with diameter
ratio as a parameter. These plots bring the curves for different diameter
ratios close together, particularly when it is noted that the ordinates
are arithmetic rather than logarithmic.
As with the friction factors, for a particular geometry the
effective drag coefficient can be represented by an equation of the form
100 fD = C(s,d) Ren(s,d) (65)
The constants C(s,d) and n(s,d) evaluated by the method of least
squares are listed as a function of s and d in Table IX in Appendix Co
The lines drawn with the data for each geometry in Figure 25 are the
lines given by the above equation.
The effective drag coefficients are cross-plotted versus free
area in Figure 26 for four different spacings with Reynolds number as a
parameter. They are cross-plotted versus spacing in Figure 27 at three
different diameter ratios with Reynolds number as a parameter. Again, in
both of these plots the ordinate is in arithmetic coordinates. As is
expected, the drag coefficient decreases as the disks get closer together
(and their wakes start interfering with each other). The slight variation of fD with free area may well be the result of scatter in the data
from which the cross-plots were prepared rather than a real, though second
order effect.

250,, lllt1 1 250
s 12
200d 0.7501 d =0.875
A.~~~~~~~~~~~~~~il.
3 150 ~-/ d=.625 150
-LlU^ Q i I I I I I i I I I I I I I 1, I I i I I I \'"- d =0.750
100 100.625 10 - ~
o0 DRAG. COEFFICIENT 0
FOR SINGLE DISK IN
50 - AN INFINITE FLUID -, - --, 50
0 O
3 4 6 8 10 20 30 40 60 100 3 4 6 8 10 20 30 40 60 100
R e/1000= (4W/,LrD ) x 15 Re/1000 = (4 W/TD) x 10
250 250 -I
s =8 s 2
200 d 0 750___ _ __ 3 200
d =0.875
R 100 ~~ x 100 ~ -c-, ~~ __i
50 150
34 6810 20 3040 60 100 34 6810 20 3040 60 100
Re/1000 (4W/ITWD) x 10 Re/1000=(4W/$7TD) x l0e
Figure 25. Effective Drag Coefficient for Disks as a Function of
Reynolds Number, fD vs. Re, for ss = 12, 8, 4 andi 2
with Parameters of d.

s 1 2 s=4
200 ~ s=Re=10,000 200
g40 Hl,^ | ~-41 1 +Re 40,000
160- 160 Re
Re 40,000
z z
0 0
I- 120 -120
4 4
80 80
0
8 40 0 40
0 0I
0.20 0.30 0.40 0.60 0.80 1.0 0.20 0.30 0.40 0.60 0.80 1.0
FRACTION FREE AREA, Af FRACTION FREE AREA, Af
o~f Free Area, fD vs. Af, for s ^ 125 8> 4 and 2 ~If
witbParamet20ers ofRe.
200-~ 2001 s:~ *
Re = 40,000 s =8 s=2
V, I 60 Re= 10,000,___.~~ 160~~ Re = 40,000
2 20 o 1 120.~
D ^ ^Re 1 0,000
w 80~~~ ag 80-t
Q0~" O
o0 0 ~~_4________0_
0.2 0.3 0.4 0.6 0.8 1.0 0.2 0.3 0.4 0.6 0.8 1.0
FRACTION FREE AREA, Af FRACTION FREE AREA, Af
Figure 26. Effective Drag Coefficient for Disks as a Function
of Free Area, fD vs. Af, for s w 12, 8, 4, and 2
withl Parameters of Re.

-73d =0.625
200 ~~~
240
too
160
0
w /
200
o
040
C
0 2 4 6 8 10 12
SPACING s (TUBE DIAMETERS)
240 v d_ = 0._75d 750 __
200
z 160
= 120
80 /
0 /
o /
0 2 4 6 8 10 12
SPACING s (TUBE DIAMETERS)
d -- 0.875
240
- 200
z
2 160
o 120,
$ 80
o
0 40
0 2 4 6 8 10 12
SPACING s (TUBE DIAMETERS)
Figure 27. Effective Drag Coefficient for Disks as a Function of
Spacing, fD vs. s, for d = 0.625, 0.750, and 0.875 and
any Reynolds Number.

-74The generally accepted value of the drag coefficient for a
disk in an infinite fluid with a uniform flow is independent of Reynolds
number when the flow is turbulent and has a value of 1.10. This line is
drawn in Figure 25 for s = 12 to indicate that the measured coefficients
are at least in the expected range. One might expect the effective drag
coefficients for disks in a tube to approach lolO as the diameter ratio
becomes small and the spacing large. In this case, however, the effect
of the solid rod on which the disks are mounted and the effect of the
non-uniform flow (i.e., the fluid velocity in a tube is a logarithmic
function of radius) would have to be considered.
The cross-plots indicate that the effective drag coefficients
for disks are relatively independent of free area and Reynolds number.
Therefore, a generalized correlation for disks of the form
C s
100 fD- JJ (66)
C2 + s
is suggested. Best values of the constants for the above, generalized
correlation were obtained from the data of this investigation and found
to be
Ci = 156
C2 = 0o78
An indication of the validity of the correlation is given in Figure 28
where fD predicted by Equation (66) is plotted versus the value
measured experimentally. All of the data were correlated with an average
deviation of 6.6 per cent.
The only similar type data available in the literature for
comparison are those taken by Koch (25) Koch presented plots of

-75220
20C0II + I 0%'/
DISKS
z 80___
o 180
w C - DATA OF KOCH
0 4i 80 __0.4 < d < 0.8/ 2
40
0 0.62<s 5 3.92
160C 2 -
/ /
o / SEE LEGEND
w 1 00/
u 0/ d 0.625 0.750 0.875
w 80
I I r I $ 0 2 0
a60 ~ ~7 ~ ~A A~~ X
80
40
0 8 El B B
2. ~~' 12 0 Q
0 40 80 120 160 200 240 280
100 fD (MEASURED EXPERIMENTALLY)
1 00
0I.// /
o STREAMLINE SHAPES / -0%
_ 80
i.0 60
w
I- // /
0 40
o /
w
a- //'' 20 /SEE LEGEND ABOVE
0 FOR MEANING OF
0 y/ DATA POIN rs
00 20 40 60 80 100 120 140
100 fD( MEASURED EXPERIMENTALLY)
Figure 28. Test of Effective Drag Coefficient Correlation, fD values
Predicted by Correlation vs. Values Measured Experimentally
for Disks and Streamline Shapes.

-76experimentally measured friction factors using air as the fluid as a
function of Reynolds number for seven combinations of s and d. The
spacing s ranged from 0.62 to 3092 and the diameter ratio d was
between 0.4 and 0o8. For Reynolds numbers between 10,000 and 40,000
the agreement of Koch's data with Equation (66) is indicated in Figure
28. It can be seen that, in general, Koch's experimental results are
approximately 50 to 100 per cent greater than would be predicted by
Equation (66). Possible explanations for this discrepancy are: 1) Koch's
disks were centered in the horizontal tube with various centering supports
which probably contributed to the total pressure drop, while the disks
used in this experiment were mounted in a vertical tube and only centered
at each end of the test section; 2) the size of Koch's centering rod
which is not specified may have been such that his rod contributed more
drag than the centering rod used in this investigation, and 3) any
slight error in the measurement of the flow rate of the fluid or the
dimensions of the tube and promoters are greatly magnified in the calculation of effective drag coefficients.
A better test of the correlation would be obtained if data
at larger spacings were available for comparison. Unfortunately, however, the largest promoter spacings for which Koch reports pressure drop
results is a spacing of 3~92 tube diameters and this corresponds to a
diameter ratio. of 0o40 where the influence of the centering rod is large.
Heat Transfer
Local values of the heat transfer coefficient ratio h/h0 are
plotted in Figure 29 versus distance from the disk x in tube diameters

-774.5
4.0 - -s =12; Re11,600 _
SOLID POINTS FOR LAST
PROMOTER IN STRING -s 8; Re 10,700
3.5 ~-s = 4; Re = 9,200
3.0-~- ~ ~ d=0.625
I.0
E..
/////,/////// ///://///?///'/// /////,/// /// // /
0.5.
-O /77,'77.77/ /77777777 /77 7//'/7/,'///////77//77/0 1 2 3 4 5 6 7 8 9 10 11 12 13
DISTANCE FROM PROMOTER, x ( TUBE DIAMETERS)
4.0 _ ___ ___ __ __ __ ___ -s 12 Re =12,300_
SOLID POINTS FOR LAST E1 -s 8: Re =11,200
3.00 _ _ _ _ _
2.5 o- a __ _____
2.0 ~-~
1.5
1.0
//x/J//J/////////// //////,,^////// /// ///. " ///' //,. /
0.5 - - -
o 7 /777//////77/ 77 /// 7///7 77////// ///. // 7 //
0 I 2 3 4 5 6 7 8 9 10 11 12 13
DISTANCE FROM PROMOTER, x (TUBE DIAMETERS)
4.5 - _ 0 -s 12; Re =12, 100
SOLID POINTS FOR LAST E -s - 8; Re = 10,900
PR MOTER IN STRING
4.0 -a -i of L t P-sio Ds 4; Re =10,700
0.o. d=.875
3.0- 0. 2.5 -- ____ _
2.0 -_ _____ ___ ___ — _____ __
1.5
1.0 —-~_______ __
0.5 - - --
771 /77//777,/ // ///,//,//,,7 /////// ////'// 777
0 ~ —-
0 1 2 3 4 5 6 7 8 9 10 11 12 13
DISTANCE FROM PROMOTER, x (TUBE DIAMETERS)
Figure 29. Sample Values of Local Heat Transfer Coefficient Ratio for Disks
as a Function of Longitudinal Position from Disk with Reynolds
Number Approximately 10,000, h/h0 vs. x, for d = 0.625, 0.750,
and 0.875.

-78for three different diameter ratios. The coefficient h refers to the
local heat transfer coefficient. The quantity ho refers to the value
calculated for the empty tube using the Sieder-Tate equation and the same
mass flow rate. A diagram of the inside of the tube is shown in each
plot to indicate the relative dimensions of the disk and the position at
which the' local heat transfer coefficient is presented.
In some cases for values of x nearly equal to the spacing s
the corresponding local value of h/ho was influenced more by the presence of the next promoter than by the decaying effect of the previous
promoter. In these cases the value h/h0 was plotted ahead of the
promoter instead of behind or, in other words, the distance from the
promoter was considered negative and interpreted as the distance to the
next promotero
For each case illustrated in Figure 29 the set of experimental
results corresponding to the Reynolds number nearest 10,000 was selected.
The following may be noted from the figure:
1o In general, the points fall on the same smooth curve regardless of spacing downstream from the disk. This is expected, since the
effect of constricting the flow overpowers the effect of the previous
history of the fluid. The biggest separation of curves for different
spacings occurs for the diameter ratio 0~625 which is the case in which
the fluid flow is least constricted.
2. The point of maximum heat transfer coefficient appears to
be shifted downstream about two tube diameters from the disk in the case
of the disk with the largest diameter ratio. This may be explained by

-79two factors: a) the point of maximum velocity of the:fluid does not
occur at the point of minimum free area, but is shifted downstream due
to a "vena-contracta" effect; b) axial conduction along the tube wall
causes the step change in heat transfer coefficient which might be expected to be damped, somewhat. (This last effect is considered in more
detail in Part 4 of Appendix B.)
3. The heat transfer coefficient approaches the heat transfer
coefficient for the empty tube about 12 to 15 diameters downstream from
the disk.
The integrated mean heat transfer coefficients were calculated
by fitting a first, second, and third order polynomial by the method of
least squares to the local coefficients as a function of x, the dimensionless distance in tube diameters from the promoter. The polynomial was
then integrated analytically to obtain the mean value. In almost every
case the mean results for all three polynomials agreed very well (usually
to within one per cent). The values for the first order polynomials were
chosen for consistency. Data points in the vicinity of the first and
last promoter in a string were not used in calculating the mean value
in order to eliminate any entrance or exit effec't.
The mean coefficients (in the form hm/h0 where h0 is the
heat transfer coefficient for the empty tube as calculated by the SiederTate equation at the same mass flow rate) are presented in Figure 30
versus Reynolds number for each spacing with d as a parameter.
For a particular geometry the heat transfer coefficient ratios
fall roughly on a line given by

liii! d'0875 A
3.5~ s=12_ -~ _ 3.5 -- _. -
SOLID DATA POINTS
FOR CHECK RUNS
3.0 - ~ ~ ~ -3.0 - - 2
0
2.0 d~~ ~ ~ - ~ - 2.0
S =4
I.5 -'('-0I.5
I I I.I D P r I r 1.0 I I
4 6 8 10 20 40 60 4 10 20 40 60
Re/1000 =(4W//.1rD)x10- Re/1000=(4W/LtrD)xIOI
3.5,~' [ 1 1 11, 1 1 1 3.5
s: 8.2
3.0 3.0 - ~
2.5 2.5 - - -_
1. ~~______-~ ~ ~ ~___________I _ ~~__________ ~~~________
4 6810 204 60 46810 20 0 560.Re/ (4W/D)xO Re/1000 (4W/.D)xO
I.0 1.0
SOLID DATA POINTS
FOR CHECK RUNS
4 6 8 I0 20 40 60 4 6 8 I0 20 40 60
Re/ I000 (4W/~Tr D )x1 0 -3 Re/1000; (4 W/I~ 7rD) x 10'5
Figure 30. Overall Mean Heat Transfer Coefficient Ratio for Disks as a Function of
Reynolds Number, hm/h0 vs. Re, for s = 12, 8, 4j and 2 with Parameters of d.

hm/ho= C(s,d) Ren(Sd) (67)
The constants C(s,d) and. l'n(s,d) evaluated by the method of least
squares are given as a function of s and d in Table X of Appendix C.
The lines drawn with the data for each geometry in Figure 30 are the
lines given by the above equation.
It is not surprising that the ratio hm/ho decreases as the
Reynolds number increases, since at high Reynolds numbers the flow is
extremely turbulent and the added disturbance produced by the disks
does not produce as great an effect.
The results are cross-plotted versus logarithm of the free area
in Figure 31 for s = 12, 8, 4, and 2 with Reynolds number as a parameter.
It is seen that the cross-plotted values of the hm/h0 ratio produce an
almost straight line approaching unity as the free area approaches one.
The data are cross-plotted versus spacing in Figure 32 for Reynolds numbers of 10,000, 20,000, and 40,000 with diameter ratio as a parameter.
There is a spacing at which the maximum heat transfer coefficient occurs
before the coefficient starts decreasing (as the disk spacing is reduced) to
the value for a solid rod in the center of the tube.
A generalized correlation of the heat transfer coefficient ratio
as a function of diameter ratio:. d, spacing s, and Reynolds number of
the following form was suggested by the appearance of the cross-plotso
hm n C 1
m-= 1 + C1 e (-ln Af) I- (68)
Bet v s of 1 0' s for te a c
Best values of the constants for the above, generalized correlation were

4.0 - ~ 11 11- -- 4.0 ~\ 1 1 1
s:12 s =4
3.5- 35
3.0 -~~~ 3.0
0Re =10,000
0 0
as Re-20000 r Re= 10,000
Re z 20,000
2.5 2.5 ~ Re=20,000
Re =40,000 Res40,000 ~2060
2.0 -~ " ^ ^^ ~'' - ~~ ^ Y -
1.5 ~~' L5
1.0 ~~~~~~~~~~1.0023 046 08 1
Q2 03 04 0.6 Q8 1 02 0 04 0.6 Q8 1.0
FRACTION FREE AREA,Af FRACTION FREE AREA, Af
4.0 4.0 i - - --
s:8 s: = 2
I ~~~~~~~~~~~~~~~~~~~~~~~R)
3.5 ~ — 3. - _ _~~ 3.5
Re:10,000
30 ___ / —Re20,000 3.0 Re~ I0,000- -~
0 Re:40,000 Re 20,000
<.c^< Re=40,000
1 2.5 - 25 -
i.5' ^ ie _____ _____ _____ ____ < __1.5 -N
02 03 Q4 0.6 0.8 80 ^ 02 0.3 04 Q6 08.0
FRACTION FREE AREAAf FRACTION FREE AREA, Af
Figure 31. Overall Mean Heat Transfer Coefficient Ratio for Disks as a Function of
Free Area, h/ho vs. Af, for s = 12, 8, 4, and 2 with Parameters of Re.

-834.5 ~' "1Re = 10,000
4.0
d _ 0.875
355
d 0.750
0 3.0
2Re 20,000
0.625 _
1.5
1. 0 0___4__'0 2 4 6 8 10 12
SPACING, s (TUBE DIAMETERS)
4.5
Re 20,000
4.0
&5 3 L ~ — d = 0.875
10 L I I II0.750 1
2.5
1'.5 ~s.625 ~
0 2 4 6 8 10 12
SPACING, s (TUBE DIAMETERS)
4.5, f Re 40,000
4.0Parameters of
d =0.875
2.0
1.0
s, for Re 10,000, 20,000, and. 40,000 with
SPrameters of dI

-84obtained from the experimental data of this investigation and found
to be
CG = 3.28
C2 = 0.15
C3 = 1.7
C4 = 1.9
nl =-0,14
As an indication of the validity of the correlation hm/h predicted by the correlation equation (68) is plotted in Figure 33 versus
the value measured experimentally for each combination of d, s, and Re,
All of the data are correlated with an average deviation of 5.6 per
cent. The type of plot given by Figure 33 tends to emphasize the points
which deviate from the correlation, since the many points which are correctly predicted overlap one another.
There are several sources of heat transfer data for disks to
use for comparison. First, there is the work by Koch(23) in which experimental values of the Nusselt number are plotted as a function of Reynolds
number using air as the fluid, for six different disk combinations of
the same type as that used in this investigation. The diameter ratio
varied between 0.4 and 0.8 and the spacing was between 0.62 and 2.80.
As indicated in Figure 33 the agreement is good with all of Koch's data
predicted with an average deviation of 9,8 per cent by Equation (68).
A second source of data is that of Sundstrom and Churchill( )
The major portion of their experimental work concerned heat transfer
from gas flames to a cylindrical tube one inch in inside diameter. As
part of their investigation, however, non-burning, local rates of heat

-855.0 DISKS/ /
+I /
Z
2 ~ KOCH'S RESULTS / /
< 4.0 ~ SUNDSTROM & CHURCHILLSj /~
w RESULTS -O/
w 2.0 2..0_
E 1.0
4 A_____ A
/ _ 0
0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
id //
hm/ho MEASURED EXPERIMENTALLY
5 1.0
STREAMLINE SHAPES
>3.0 / A~
/o
a.
0 1.0 2.0 3.0 4.0 5.0 6.0 70
hm/ho MEASURED EXPERIMENTALLY
=
FiguE 1.0r Correlation, Ratios
SEE LEGEND ABOVE
Figure 33. Test of Heat Transfer Correlatonues Measuredatios
Experimentally for Disks and Streamline Shapes.

-86transfer from hot air to cold water flowing outside the tube were measured.
The flow was disturbed by the presence of a single disk-shape, bluff-body
flame holder centered in the burner. The flame holder provided 48 per
cent free area and the Reynolds number varied from 5,000 to 20,000.
For a Reynolds number of 10,000 Sundstrom and Churchill noted a
maximum value of hm/hO of 2.6 occuring about 1-1/2 tube diameters downstream from the flame holder. Strictly speaking, the results of their
work are not comparable with the results of this investigation and with
Koch's work, since they had only one disk rather than a series of them.
The effect of spacing can be simulated, though, by considering the
integrated average heat transfer coefficient from the flame holder to a
distance s tube diameters downstream as corresponding to the same
results that would be obtained if a series of promoters were spaced a
distance s tube diameters apart. Using this technique, results of their
experiments may be compared with the predictions of Equation (68). It is
found that, in general, their measurements were approximately ten per
cent below the values predicted.
(4j)
A third source of data is the work of Zartman and Churchill(3)
Their measurements were made with the same type of apparatus and with
practically the same intent as those of Sundstrom and Churchill except
that they used a tube of five inch inside diameter, only one Reynolds
number (14,000) and disk-shape, bluff-body flame holders which provided
free areas of 9 per cent and 5 per cent. A maximum value of hm/h0 of
4.5 was found to occur at the flame holder. Values predicted by Equation (68) for these very low values of Af (which represent quite an

-87extrapolation of the equation) were almost two times the values observed
experimentally by Zartman and Churchill. This indicates that the correlation can not be used with confidence for values of Af much smaller
than 0.234 (i.e., the smallest Af for which data were obtained in this
investigation)
Both the results of Koch and of Sundstrom and Churchill in
which the Reynolds number was varied confirm the trend of decreasing
hm/ho with increasing Reynolds number for any given geometry.
Koch presented a graphical correlation of hm/hO as a function
of d and s for use in the Reynolds number range 10,000 to 40,000.
His correlation was based upon results for ten combinations of s and d
(of which data for only seven are given in his paper) with the diameter
ratio ranging from 0.40 to 0.80 and the spacing from 0.62 to 7.82 tube
diameters. This correlation seems to be in error for very close spacing, in that the predicted value of hm/h0 keeps increasing with closer
spacing and there is no spacing which produces a maximum value. For a
free area of 0.50 and a spacing of 1.05 a value of hm/h0 of 4.2 is predicted by his graph. This does not seem reasonable, since for a free
area of 0.48 Sundstrom and Churchill found the maximum value of the
local coefficient ratio h/h to be 2.6. It would seem that at close
spacings the highest value of the mean heat transfer coefficient would
not be greater than the maximum value of the local coefficient for a
single disk and, in fact, would probably start to get smaller and approach
the value for a solid rod.
The probable explanation for the difficulty with Koch's corre-.
lation at close spacings is that it was based on too few combinations of

-88d and s. In any experiment of this nature the high heat transfer coefficients are the most difficult to measure, since they correspond to the
smallest temperature differences. Thus, with only one or two sets of
data (corresponding to close spacings and high values of hm/ho) in
error the whole correlation would be easily biased.
Even though Koch's correlation does not agree with Equation (68)
at close spacings, the bulk of his data do.
Streamline Shapes Evenly Spaced and Centered in the Tube
Results for the streamline shapes have been plotted in exactly
the same manner as the results for the disks. In other words, for every
plot of results for the streamline shapes there has already been presented
a corresponding plot for the disks. Therefore, attention will be called
only to the dissimilarities between the results for streamline shapes
and the results for disks.
Pressure Drop
Friction factors are plotted versus Reynolds number in Figures
34, 35, and 36 for s = 4, 8, and 12 with d as a parameter. It can be
seen that the pressure drop is approximately a fourth that observed for
the disks. Unlike the results for the disks, the friction factors for
the streamline shapes decrease as they do for a smooth tube with increasing Reynolds number, even for the large diameter shapes,
The lines drawn with the data in Figures 34-36 are given by
the equation
100 f = C(s,d) Ren(s,d) (64)

-890.80
~5 —z_.z~rZ ~d = 0.875
0.40
0.20
0.1 0 ~ --— ~~_-~ —'I -.d=0.750
40 30
0. 08
NI~L r D
0.04
of Reynolds Numberjf vs. Be, for s = 4 with Param-':eters of d0625
0.01
0.008
~ ~ —" ~- - ______ ___ __ -EMPTY TUBE
~ ~- ~'-;E^^ ___ ^ C Pu. (15)
0.004
0.002 -
4x10O0' 10 iO1
4W
Re =~
Figure 34. Friction Factor for Streamline Shapes as a Function
of Reynolds Number, f vs. Re, for s = 4 with Parameters of d.

-900.80 -—..
0.40....
d -=0.875
0.20
0.10
0.08 —.
- - o. 0^ d:0-7 50o __ ~ -
0 0.04.
0. 625
0.02
0.01
0.008.. —~~-EMPTY TUBEEQU. (15)..
0.004
0.002..~.....
4x10 10 10 80
4 W
Re =
pL I D
Figure 35. Friction Factor for Streamline Shapes as a
Function of Reynolds Number, f vSo Re, for
s = 8 with Parameters of d.

-910.80
0.40 - --
AA~
A A
0.20 ~ -- ~ ^A^ ^~- d 0.875
0.10
0.08
E l
1wiha ed O,0.625
0.0 I__
0.008
0.004 -EMPTY TUBE -
EOU. (15)
0.002 1~ ~
4x10 10 l0o 10
4W
Re =.
/t. D
Figure 36. Friction Factor for Streamline Shapes as a
Function of Reynolds Number, f vs. Re, for
s ~ 12 with Parameters of d.

-92The constants C(s,d) and n(s,d) are listed as a function of s and
d in Table VIII of Appendix C.
The effective drag coefficients are shown as a function of
Reynolds number for s = 12, 8, and 4 in Figure 37 with d as a parameter.
They are cross-plotted versus free area in Figure 38 and versus spacing
in Figure 39. The lines drawn with the data in Figure 37 are given by
the equation
100 f = C(s,d) Ren(s'd) (65)
The constants C(s,d) and n(s,d) are listed in Table IX of Appendix C
as a function of s and d.
Figure 37 indicates that the effective drag coefficient for
streamline shapes has a slight dependence on Reynolds number. Therefore,
a generalized correlation for streamline shapes is suggested of the form
C1s Re (
~00 fD -1 + 1000 (69)
Best values of the constants were obtained from the data of this investigation and found to be
C = 117
C2 = 16
n =-0.12
An indication of the validity of the correlation is given in Figure 28
where values of fD predicted by Equation (69) are plotted versus
values measured experimentally. All of the data are correlated with an
average deviation of 7.95 per cento There apparently are no similar
type data available in the literature for comparison.

125 ~ ^ - 125
s=12 s =4
100 100 8 10 200 100 10 20 40 60 1OO
c 0.875
d 0.750' 0 75
oo o_
25 ~ 25i
0 s
100 d X~d0.625 Figure 37. Effective Drag Coeff icient for Stream_ l1 t3C 1l l l lil line Shapes as a Function of Reynolds
4 with Parameters of d.
0*- 50
3 4 6 8 10 20 40 60 100
Re/1000: (4W//LUTD)x 10
Re/1000 = 4W//rDKOx ID$

100..100.......
s =12 s =4
80 80
60 = 60
gI /^^^~Re =40,000 w
Re=20,000 - --- 40
0 R
o ~10,000 ---
o 0/ Re 10,000
20 20
0 0
0.2 0.3 0.4 0.6 0.8 1.0 0.2 0.3 0.4 0.6 0.8 1.0
FRACTION FREE AREA, Af FRACTION FREE AREA, Af
100...
s=8
80
O l | _ _ _ | l l l l | FFigure 38. Effective Drag Coefficient for
t 0 I I I I I |' Streamline Shapes as a Function
j 60.l*...... of Free Area, f vs Af, for
CYog /t 1/ 1~ s = 12, 8, and with Parameters
of Re.
o 40 Reo=2,00
0 Re= I 0,000
20
0.2 0.3 0.4 0.6 0.8 1.0
FRACTION FREE AREA, Af

-95140
_ 100
w 80
120 _ _
o_^.^^^^^^\\^~. ~Re= 0,000
40 ~~^^^~"~^ ^~ ~~ Re=20,000.
20 //,//~~~ __ _Re =40,000
20
40
0 2 4 6 8 10 12
SPACING, s(TUBE DIAMETERS)
140 i
120 ___ d =0.750
1 00
1 80 -~~~
80
40 ~_~_\\ Re= 10,000
206^~ Re= 20,000
20 ~ ^ ~~ ~ ~ ~~^ — Re =40,000
0 2 4 6 8 10 12
SPACING, s (TUBE DIAMETERS)
140. f
120 -' d = 0.875' 100
S 80
o 60
O_... f, f~'"- Re = I0,000
4o ___" __ __ _\- Re 20,000
4y 0. =Re= 40,000
20
0 2 4 6 810 12
SPACING, s (TUBE DIAMETERS)
Figure 39. Effective Drag Coefficient for Streamline Shapes
as a Function of Spacing, fD vs. s, for d = 0.625,
0.750, and 0.875 with Parameters of Re.

-96Heat Transfer
The local heat transfer coefficient for Reynolds numbers of
approximately 10,000 are plotted for diameter ratios of 0.625, 0.750,
and 0.875 in Figure 40. The shape and position of the streamline bodies
are illustrated with a diagram just as for the disks. Some interesting
features of these curves are:
1. The points for all three diameters (including even d = 0.625)
fall on approximately the same smooth curve. This is to be expected,
since the streamline shapes should produce a smaller wake than the disks
and, hence, should interfere less with each other.
2. The point of maximum heat transfer is shifted downstream
from the point of maximum velocity just as for the disks (although
because of the scatter in the data at this point, it is difficult to tell
exactly where the maximum occurs). There is reason to believe that some
separation occurs in flow around the streamline shape just as it does
around the disk, thus producing a vena-contracta, Even though there may
be separation in the flow around the streamline shape, one would still
expect the pressure drop to be considerably less than for a disk. The
reason is that the tail of the streamline shape physically occupies
volume where wasted turbulent motion takes place in the wake of a disk.
3. The heat transfer coefficients approach those of an empty
tube within a shorter distance for the streamline shapes than for the
disks.
Values of hm/ho are plotted versus Reynolds number in Figure
41 for s = 12, 8, 4, and 0 (for the solid rod), with d as a parameter.

-974.0 - - 1-s=12; Res 9,800
0 - o s8; Re 10,600
3.5 d=0.625 --- s;4, Re =9,800~0
2.5
2.0
1.0
0.5
Q 05 2 3 45 6 7 8 9 -' 11 12
DISTANCE FROM PROMOTER (TUBE DIAMETERS)
4,5
(SOLID DATA POINTS ARE AFTER
40 LAST PROMOTER IN STRING) -,812 Re 12300~3 5D~ X ~~ - 4 -A -:4 R sQ12 Re 12,300
3.05 -
20- o-rr- ---
1.0 - -
0.5
U I 2 3 4 5 6 7 8 9 e10 II,2
DISTANCE FROM PROMOTER, x (TUBE DIAMETERS)
4 3.5 _ ~ _ 0 - a - s 8; R- 11,- 0
3.5 &'-s-4- d 875;ReIOO
0 0 - (SOUD DATA POINTS ARE AFTER
DISTANC FOLAST P ROMOTER IN STRINGS)
Figure 40. Sample Values of Local Heat Transfer Coefficient Ratio for
Strearlnine Shapes as a Function of Longitudinal Position from
Shape with Reynolds Number Approximately 10,000, l/h0 vs. x,
for d = 0.625, 0.750, and Q.875.
1.5
o 1 2 3 4 5 6 7 8 9 10 1 A A,

40 ~ 1 40 H:4
s:12 -d= 0.875
3.5 3.5
AUDIBLE VIBRATION OBSERVED
FOR THIS DATA
3.0 3.0
0 0
2.5 2.5
c. ~< IIIIII ~~~.=0.875 d 0.750
2.0 -'2.0
d 0.625
d = 0.750
1.5 __i~_ I ir4_ _ 1.5' - -
-d = 0.625
~~I0I)~~46 0 1.03 4 6 8 10 20 40 60 100 3 4 6 8 10 20 40 60 100
Re/I000 (4W/,lrD) x 16' Re/1000 (4W/,lrD)x I5)
40 3.0
3.5.. s:=8 2.5' ~ — s:O-
3.0 ~ ~ — ~ ~_ — 2.0 ~- /-d = 0.750
2.5 1.5 /
E E'~~d ^0. 6 2 5
2.0 - I- C- _ -;:
~^^[^^3^^^~~~ 3-Q.':. d~0.125
d 0.625
1.5 _~
SOLID POINTS ARE FOR
THREADED ROD. d= 0.250 I
3 4 6 8 10 20 40 60 100 3 4 6 810 20 40 60 100
Re/1000 (4W//LTD) xlCe Re/1000: (4W/LrD) x i0$
Figure 41, Overall Mean Heat Transfer Coefficient Ratio for Streamline Shapes as
a Function of Reynolds Number, hm/h0 vs. Re, for s - 12, 8, 4, and 0
with Parameters of d.

-99The lines drawn with the data are given by the equation
hm/h = C(s,d) Ren:(sd) (67)
with the constants C(s,d) and n(s,d) given in Table X of Appendix C
as a function of s and d.
The data are cross-plotted versus free area in Figure 42 for
s = 12, 8, 4, and 0 with Reynolds number as a parameter. They are crossplotted versus s in Figure 43 for Reynolds numbers of 10,000, 20,000,
and 40,000 with diameter ratio as a parameter.
The following observations should be noted: As the data for
the largest diameter ratio (d = 0.875) and the closest spacing (s = 4)
were taken, noticeable vibration of the shapes against the tube wall was
observed from the sound produced. At the completion of the run, when
the string of promoters was removed, it was found that the centering
pins had been badly damaged. The only other run for which audible vibration was heard was that for d = 0.875 and s = 8, but in this instance
no damage to the centering pins took place.
The vibration phenomenon was probably caused by pressure disturbances created by the periodic shedding of vortices from the streamline shape and very likely depends in addition upon the natural frequency
of the centering rod,
Approximately the same behavior is shown as for the disks with
the heat transfer coefficients about 20 to 50 per cent lower for the
streamline shapes than for the disks. A much smoother transition to
the case of a solid rod in the center of the tube occurs for the streamline shapes as the spacing is decreased than for the diskso

3.5 I 3.5s =12' 8 4
30 - 30
-Re =10,000 2.5
2.5 R/__~ ~ 2.5
2 / -Re =20,000 2 Re=100000
E Re2.0 R40,000 - Res20,000 __
~^~~.~Re^ ~"~~" ^~^~R=40P000
~~1.5,,'t~ I I -~1.5
0.2 0.3 0.4 0.6 0.8 1.0 0.2 0.3 0.4 0.6 08 10
FRACTION FREE AREA,Af FRACTION FREE AREA,Af
3~3.5 - o3
s', I I =8. s =;0 \ / — EQU. (133)
8:8 8:0 Y
3:0 Re = 10,000 - 30
Re =20,000
2.5 Re ~~4oo 25
0 "
Re= 20,000
-~^"1.5 1 I " ~ e 10 00 -1.5
Re= I
1.0
Q2 03 0.4 Q6 0.8 1.0 02 03 0.4 Q6 0.8 10
FRACTION FREE AREA, Af FRACTION FREE AREA, Af
Figure 42. Overall Mean Heat Transfer Coefficient Ratio for Streamline Shapes as a
Function of Free Area, hm/h0 vs. Af, for s = 12, 8, 4, and 0 with
Parameters of Re.

-101 -
4.0
3.5._
^^^ ~ y~d 0.875
3.0
2.5 - d 0.750
2.0
1.5
0 ^ S 2 ~d~;0,625
0 2 4 6 8 10 12
SPACING,s (TUBE DIAMETERS)
4.0
Re 20,000
3.5
3.0d:0.875 _
3.0,J~ __'__
2.5
&.~Is 0.750
2.0
1.5
1.0
0 2 4 6 8 10 12
SPACING,s(TUBE DIAMETERS)
4.0
Re =40,000
3.5
3.0
E 2.5
2D _____~~ d= 0.750
2.0 ~~_. ___.__
1.5
Zd 0.625
1.0
0 2 4 6 8 10 12
SPACING, s(TUBE DIAMETERS)
Figure 43. Overall Mean Heat Transfer Coefficient Ratio for
Streamline Shapes as a Function of Spacing, h /h vs.
s, for Re = 1 20,000, and 40,000 with m 0
Parameters of d.

-102A generalized correlation of the heat transfer coefficient
ratio for streamline shapes as a function of diameter ratio d, spacing s,
and Reynolds number was obtained of the following form
m = + C1 (-n A) Re (7)
h0 10, 000 1 + C2 s
Best values of the constants were obtained from the data of this investigation and found to be
C = 2o04
C2 = 014
nl 0 1ll
As an indication of the validity of the correlation, values of
hm/ho predicted by the correlation are plotted versus values measured
experimentally in Figure 335 All of the data are correlated with an
average deviation of 703 per cent. The biggest deviation from the correlation occurs at close spacing and large diameter ratio, particularly for
d = 0.875 and s = 4. Since this is the set of results for which noticeable
vibration of the shapes occurred, the high experimental values are probably due to this cause. Thus, the correlation given by Equation (70) is
probably best applicable only to streamline shapes in which there is no
vibration
There are apparently no data available in the literature for
comparison.
Reliability of the Data for All Geometries
In order to effectively utilize the experimental results just
presented, some judgment concerning the reliability of the data must be

-103made by the user. For this purpose a critical discussion of the errors,
reproducibilities, etc. of the data will be given,
Pressure Drop
The relative error (i.e. the precision) of the friction factors
may be estimated from the form of the defining equation used in this investigation. (62)
- gc 2 p D5 -AP IP -L
f 52LID rP 0 11P_ (62)
32 WL np S. p
It should be noted that when the product npS is equal to L (i.e., the
string of promoters occupies the complete distance between pressure taps)
Equation (62) reduces to Equation (22). Equation (22) was used for the
empty tube and solid rod geometries.
The density of the water and the dimensions of the experimental
equipment were known quite accurately. The last term in brackets in the
expression above contributes from ten per cent to less than 0.3 per cent
of the value of the friction factor.''Thus, almost all of
the uncertainty in the experimental friction factor arises from errors in
measuring the pressure drop and the mass flow rateo
The pressure drop could be measured to within about 0.10 inches
of indicating fluid, so that the relative error depends upon the magnitude of the pressure drop. At a Reynolds number of 10,000 the pressure
drop for the empty tube was about 2 inches of indicating fluid; at a
Reynolds number of 50,000 the pressure drop was about 20 inches; and,
for some turbulence promoting geometries the pressure drop was 100 inches
of indicating fluid. Thus, the error in measuring the pressure drop
contributed an error ranging from 0.1 to 5 per cent, but usually less than
2 per cent,

The mass flow rate, determined from the rotameter readings,
may be considered as accurate to within, one per cent, which when added
to the error just discussed (doubled since W2 is required) produces an
estimate of 2 to 7 per cent for the precision of the friction factor
results. The excellent agreement of the experimental friction factors
with accepted correlations for empty tubes and annuli confirms the estimate
of the precision of the data.
The reproducibility of the data, however, requires a separate
consideration and depends almost entirely upon the ability to reproduce
the exact geometry. In particular, for the types of systems studied in
this investigation, reproducing the geometry involves centering the devices
(or measuring the degree of eccentricity). It is estimated that the variation in the per cent eccentricity of the solid rods and streamline shapes
(which had centering supports) was within about 20 per cent, while the per
cent eccentricity of the disks (which had no centering supports) was probably within about 40 per cento The per cent eccentricity is defined as
the per cent of annulus width by which the inner element (at its maximum
diameter) is eccentric.
A theoretical analysis by Deisler and Taylor(1) for a solid
rod in the center of a tube with d 0.286 indicates that the friction
factor for an eccentric annulus is about ten per cent lower than the
friction factor for a concentric annulus when the per cent eccentricity
is 60 per cent and is about 25 per cent lower when the eccentricity is
100 per cent. This effect of eccentricity (difficult to control experimentally) probably accounts for part of the difficulty encountered in
the past by various investigators in obtaining a good, generalized friction
factor correlation for annuli.

-105It is difficult to estimate the effect of eccentricity on the
pressure drop results for bluff bodies. It is likely, however, that
part of the dependence of the effective drag coefficient on free area
observed in the cross-plots of Figures 26 and 38 is due to slightly
different eccentricities.
A check on the reprclcib.lity of the results was obtained for
two strings of disks with d = 0.750 and s = 12 and 8 in which the
strings were prepared and inserted completely independently in runs made
a month and a year respectively from the original measurements. In both
cases the check results differed by about ten per cent from the original
results.
Therefore, in summary, the following may be stated:
1. The precision of the pressure drop data probably varies
from about 2 to 7 per cent with most of the results corresponding to the
lower figure.
2. The reproducibility of the pressure drop data is probably
within about 15 per cent with most of the problems of reproducibility
consisting of difficulty in controlling the exact degree of eccentricity.
The irreproducibility appears as scatter in the overall correlations.
Heat Transfer
The precision of the local heat transfer coefficients may be
estimated from the form of the defining equation
h(z) - q) (45)
Twa (z)- Tf(z)

-106Errors may be introduced from two sources: 1) errors in measuring the experimental variables: Tb(z), Tinlet, Toutlet, and I; 2) errors propagated in the calculation of q(z) and ATgeneration due to
uncertainties in the parameters of the experimental apparatus: a, b,
po K0' Y, and A. A statistical analysis of the propagation of errors
given in Appendix B indicates that q(z) is accurate to within + 5 per
cent and the temperature difference between fluid and inside wall is
accurate to within about + 2 dego Fo
The percentage error in the local heat transfer coefficient,
however, depends upon the magnitude of the total difference between fluid
and inside wall. This temperature difference ranged from 12 to 100 deg. F
with the large majority of the runs at a mean temperature difference of
about 25 deg. F. The estimated precision of the heat transfer coefficients,
therefore, may be taken as being about 8 per cent.
There are two important tests of the precision of the data which
confirm the preceding analysis: 1) the agreement of the overall Nusselt
numbers measured for an empty tube with those predicted by the generally
accepted correlations; and 2) the agreement of the heat balances.
The first (agreement with accepted correlations) has already
been demonstrated in Figure 13
The percentage error in the heat balances is shown in the frequency plot of Figure 44 where the per cent error is defined as follows,
L
Qin 2 q(Z)dZ (70)
0
gout = W c(Toutlet - Tinlet) (71)
(&in - &out) x 100
Per Cent Error - (in out) x 10072)
0o5(Qin + Qout)

-10720
18
16
14 -
> 12
I0
8
6
4
2
9 -8 -7 -6 -5 -4 -3- - 0 2 3 4 5
PER CENT ERROR IN HEAT BALANCE
Figure 44. Frequency Distribution of Heat Balance Errors.

-1o8It is seen that this percentage error generally falls between
plus one per cent and minus five per cent. Since the temperature rise of
the water was usually between 5 deg. F and 20 deg. F, an error of one per
cent in the heat balance corresponds to an error in measuring the difference between inlet and outlet water temperature of 0.05 to 0.20 deg. F.
Thus, the heat balance error seems to be within acceptable limits.
The reproducibility of the local heat transfer coefficients
is subject to the same difficulty in centering the devices as was the
reproducibility of the friction factors. For the heat transfer results,
however, a better estimate is available, since measurements were made at
three different angular positions for three different axial positions
for each run. The angular variation is indicated in Figures 45 and 46
where the frequency of the difference between the local angular AT and
the average AT for all three angular positions is plotted versus the
difference. It is seen that the difference was generally less than
- 2 deg. F which means that any irreproducibility in the local heat transfer coefficient caused by inexact axial symmetry was less than 10 to 15
per cent.
A test of the reproducibility of the local heat transfer
coefficients for disks and streamline shapes is given by the plots of
Figures 29 and 40 since the data for different values of s were taken
as part of completely separate runs. In fact, the. local data near each
of the 4 to 12 bluff bodies used in each run may be thought of as data
from a different experiment since any eccentricity of the bodies probably varied somewhat from promoter to promoter along the tube. The

-10990
AVERAGE AT FOR ALL
80 RUNS WAS 32~F
70
>- 60
z
LC
D 50
0
u. 40 -
30
20
10IO
-7 -6 -5 -4 -3 -2 -I 0 1 2 3 4 5 6
DIFFERENCE BETWEEN AT AT GIVEN ANGLE AND MEAN AT FOR ALL
THREE ANGLES
Figure 45. Frequency Distribution of Difference Between Local
Angular AT and Mean AT for All Three Angles for
Streamline Shapes.
240
210 AVERAGE AT FOR ALL
RUNS WAS 16 OF
180
s150
z
1 120
90
60
30
-7 -6 -5 -4 -3 -2 -I 0 1 2 3 4 5 6
DIFFERENCE BETWEEN AT AT GIVEN ANGLE AND MEAN AT FOR ALL
THREE ANGLES
Figure 46. Frequency Distribution of Difference Between Local
Angular aT and Mean AT for All Three Angles for Disks.

-110effect of inexact axial symmetry probably accounts for some of the
scatter of the local heat transfer data, but the fact that all the data
fall roughly on the same curve indicates that the local data are reproducible.
The reproducibility of the mean heat transfer coefficients
should be considerably better than that for the local heat transfer coefficients, since the data are subjected to an averaging process. This is
indicated by the results of check runs made for disks at d = 0.750 and
s = 12 and 8 just as for the friction factors. As shown in Figure 30 the
mean heat transfer coefficients for the check runs (which were taken one
month and one year respectively from the original runs) fall on the same
smooth curves as the results for the original runso This confirms the
good reproducibility of the mean transfer coefficients.
In summary the following may be stated:
1i The precision of the heat transfer coefficients is probably
within, 8 per cent.
2. The reproducibility of the local heat transfer coefficients
is within about 10 to 15 per cent and the reproducibility of the mean
heat transfer coefficients is considerably better than thato

ECONOMICS
One of the primary goals of this investigation is to determine
whether turbulence promoters can be used economically to improve the rate
of heat transfer to a fluid flowing in a tube and whether there exists an
optimum type of promoter. Any general economic study of this nature, however, is difficult because of the many different types of equipment in
which the transfer of heat and momentum to a fluid flowing in a tube is
an important part.
Mathematical Model of a Heat Exchanger
Throughout the consideration of the economics of turbulence
promoters it will be assumed, for convenience in explanation, that heat
is being transferred to a cool fluid inside the tube and, thus, the temperature of the fluid is being increased. The conclusions reached are
equally applicable to the situation where heat is transferred from the
fluid and the temperature of the fluid is lowered.
A schematic diagram of a piece of equipment for transferring
heat to a fluid flowing in a tube is shown below
Source of Heat, Q''Iw
~:~^,)'', i Working
- JL C D, \ Fluid
L T
Source of
Pumping Energy
- ~ ~

-112The essential characteristics of the apparatus with typical
examples are:
1. Source of heat, Q
a hot fluid flowing outside the tube
b. condensing vapor
c. electrical generation of heat
d. chemical reaction
e. nuclear reaction
2. A single tube or group of tubes in parallel of specified
inside geometry, length L, and inside diameter D. The
number of tubes is denoted by Ntube.
3. A working fluid (with physical properties p, k, c, p) which
flows through the tube at mass flow rate, W
a. water
b. air
c. liquid metal
4. A source of pumping energy, E
a. pump
-bo head due to height of reservoir
5. A temperature driving force to cause heat to flow to the
tube(s), ZTm
The equations which relate the variables (with typical units)
are
Q = total rate of heat transfer (BTU/hr)
Uoa A A Tm (73)
where Uoa = overall heat transfer coefficient (BTU/hr-deg F-ft2)
~~~~~~~~~~1 ~~~(74)
l/h + l/h
Nu (k/D) (75)
1 + h/h'
Nu (k/D) (76)
Nu
1 + -
1 h'D/k
A = total heat transfer surface based on inside tube diameter
(ft2)

-113= Ntube D L (77)
E = pumping energy required (BTU/hr)
-W A P
J P (78)
c
AP = overall pressure drop (lbf/ft2)
2 f L p U2 (7
(79)
gc D
The Nusselt number Nu and friction factor f are generally
empirical functions of the inside diameter D, mass flow rate W, physical properties of the fluid (c, k,, p) and the geometry of the inside
of the tube
Nu = function of (D, W, A, p, c, k, geometry) (80a)
f = function of (D, W, l, p, geometry),, (81a)
In some cases, of course, Nu and f may depend on still
other factors such as AITM L, pressure, etc. For many geometries
(including that of the empty tube) the forms of the above functions are
Nu = C2 Re r (80)
Nu C Re prnl/ (80)
-n
f = C1 Re (81)
where Re = 4 W (82)
K D Ntube
Pr = c -H (83)
The term h is the mean, inside, convective heat transfer coefficient, while the term hi is an effective coefficient which takes into
account the resistance of the rest of the heat exchanger to the transfer
of heat,

-ll4When h' is very large compared to h, then almost all of the
resistance to heat transfer is provided by the flow inside the tube and
Uoa h (84)
or, in other words, the value of the mean inside heat transfer coefficient
is controlling.
On the other hand, when h' is very small compared with h, then
Uoa h' (85)
and the rate of heat transfer will be independent of conditions inside
the tube.
Factors Which Affect the Economics
In order to determine the economic desirability of using various
turbulence promoting devices it is necessary to consider in some detail
the factors which affect the economics of heat exchangers in general.
Thus, a large part of the discussion which follows will be equally applicable to the design of conventional heat transfer equipment.
In order to design the optimum heat exchanger the designer must
strike a proper balance between fixed and operating costs. This immediately introduces complications, since there will be both fixed and operating costs associated with the part of the exchanger outside the tube.
In order to avoid these complications the term "cost" will be interpreted
to mean "the cost which is influenced by the selection of the geometry of
the inside of the tube,"
In addition the following assumptions will be made:
1. Q is specified

-1152 W is specified
3. ATm is specified
4. h1 is specified
The problem, then, is to make the best selection of D, L, Ntube'
and inside tube geometry. Throughout this presentation the term L will
mean the effective length. Thus, if multiple tube passes are used, the
effective length L is the actual tube length times the number of tube
passes.
The fixed costs for a given unit of time are usually expressed
as a percentage of the initial investment and include such items as depreciation and maintenance. The amount of the initial investment depends
upon the size of the exchanger, the type, materials of construction, etc.
As a first approximation, however, the fixed cost is given by
Fixed Cost = CF Am (86)
where the coefficient CF (in typical units of dollars/ft2m-hr) depends
upon the materials of construction and type of exchanger; the exponent m
of the area is often taken as 0.60o
The operating costs consist mainly of the cost of pumping energy
and the actual cost of the fluid. If the fluid is some product, its cost
is probably considered negligible, but if it is a utility (for example
cooling water), its cost may be significant.
The cost of pumping is given by
Pumping Cost = CE E (87)
where CE is the cost of pumping energy (in typical units of dollars/BTU).
The cost of the fluid is given by
Fluid Cost = Cw W (88)
w~~~~~~~~~(8

-116where CW is the cost of the fluid (in typical units of dollars/lbm).
The total operating cost is given by
Operating Cost = CE E + CW W (89)
and the total cost is given by
Total Cost = CF Am + CE E + CW W (90)
Each of the above costs per unit heat transfer is given by
Fixed Cost/BTU = CF Am/Q (91)
Pumping Cost/BTU = CE E/Q (92)
Fluid Cost/BTU = CW W/Q (93)
Operating Cost/BTU = CE E/Q + CW W/Q (94)
Total Cost/BTU = CF Am/Q + CE E/Q + CW W/Q (95)
The fluid cost is being carried throughout this analysis, even
though both W and Q are assumed to be specified and, hence, the fluid
cost is independent of the variables remaining to be selected. This is
done to simplify a consideration later of removing the restrictions requiring that both W and Q be known. On the basis of the assumptions
made thus far, the relevant cost is given by
Relevant Cost = CF Am + CE E (96)
and Relevant Cost/BTU = CF Am/Q + CE E/Q (97)
Procedure for Designing the Optimum Heat Exchanger
One procedure for minimizing the relevant cost/BTU is as follows:
1. Obtain an expression for Am/Q and E/Q for any particular
geometry in terms of the variables D, L, and Ntbeo

-1172. Select the particular set of variables which produces the
minimum relevant cost/BTU as calculated by Equation (97).
As a starting point it will be assumed that the inside tube
geometry is specified, for example as an empty tube. This establishes
the specific form of the functional relationships (80a) and (81a).
Next, a reasonable value of the inside tube diameter is selected
on a fairly arbitrary basis.
The remaining variables to be selected are Nt and L. Only
tube
one of these, however, may be chosen independently. Once Ntube is specified the fluid velocity and Reynolds number in the tube are also determined. This enables the Nusselt number to be calculated from the empirical
correlation for the given geometry which, in turn, sets the value of Uoa,
required area, and hence the length L.
For purposes of comparing different geometries it will be more
useful to select a desired (or optimum) Nusselt number and calculate the
required number of tubes (and the length) rather than specify the number
of tubes and calculate the resulting Nusselt number. The derivation of
Am/Q and E/Q in terms of D, Nu, and the specified data is given below.
m
Am -J " 1 (98)
Q Q ] m
since
A. 1 (99)
Q UoaA Tm
~then ATm (D/k ATm)m [1 + Nu/(h'D/k)]m
then - ~- (100)
Q m 1-m
Nu Q

-118It can be seen that Am/Q and, thus, the fixed cost depend only
on Nu, D, and the parameters which were assumed specified. The fixed cost
is independent of the empirical correlation for friction factor and Nusselt
number inside the tube and, thus, does not depend upon the choice of inside
geometry.
When Nu is small compared with h'D/k, then Am/Q produces a
straight line with slope -m when plotted versus Nu on logarithmic
coordinates. This is the case when the value of the inside heat transfer
coefficient is the limiting factor in the rate of heat transfer. On the
other hand, when Nu is large compared with h'D/k then Am/Q is independent of the Nusselt number inside the tube and is given by
Am (D/k Tm) (101)
Q (h'D/k)m Ql-m
r 11
= ~1 j 1 (102)
h' AkT Q i m
An expression for E/Q will now be derived.
E W A P (78)
Jc p
since P = 32 f L (103)
gc 2 p D5 N2
g D tube
kt vDN Re
and W = D tube (104)
4
f Re3 L p3 J N
then E =tube (105)
2 Jc g P D5
and E = f Re3 l + huj Pr 2 (106)
Q 2 Nu h'D 2 D2
Jc gc P D c gTm

-119Since, for any given tube geometry
Nu = function of Re
f = function of Re
then expressions for the friction. factor and Reynolds number in terms of
the Nusselt number may be substituted into Equation (106). For example,
for the empty tube geometry (or any other geometry for which Equations (80)
and (81) are valid)
f C1 Re- (81)
n2p/ (80)
Nu = C2 Re Pr (80)
Thus, Re - 1 /n (107)
and f = C1 Nu n2 +
Q 2 ru c2 Prl/3 L /kJ LJ g p D c
(109)
The expression for E/Q is of the general form
C u
^ ^"1- 3-n) /
E - 1 N nu] N(110)
a Dependent only on empir- ( )
The exps2 sn fra correlationE i for particu
C lar geometry. Independent of
2C^ etce
2 k geometry. Dependent mainly
2 on tube diameter, physical
B s properties of the fluid, (114)
3 J- 2DcT nm,1etc
JC g D etc.
B2 = ~ Independent of inside tube (113)~~^~

-120When B2Nu is much smaller than unity (i.e. inside heat transfer
coefficient controlling) then E/Q vs. Nu on logarithmic coordinates
yields a straight line with slope p-l. Likewise, when B2Nu is much
larger than one (rate of heat transfer independent of Nu inside the tube)
then E/Q also plots as a straight line with a slope of p.
Substitution of Equations (100) and (110) into the cost equation
(97) gives the cost per BTU in terms of Q, Tmj D, Nu, and the physical
properties of the fluid. The first three of these items (total rate of
heat transfer Q, mean temperature difference ATm' and equivalent outside heat transfer coefficient h') were assumed specified. Thusa relation for the cost per BTU has been obtained as a function of the inside
tube diameter D, the Nusselt number Nu in the tube, and the specified
constant parameters.
Since the fixed cost decreases with Nusselt number and the pumping cost increases, it is evident that there will be an optimum value of
Nu at which the total cost is a minimum. The various costs are illustrated
in Figure 47. The case where the inside heat transfer is limiting (io.e
h' = co and, therefore, B2 = 0) is of particular interest.
In this case
Total Cost CF (D/kAm)" CE B1 B3 Nu1
BTU Num Ql-m prP/3-1
The optimum Nusselt number can be found by setting the derivative of this expression to zero and solving for Nu to obtain
Nuoptim m CF (D/kumTm)m p +m-l
(optimum C (116)B p
(p-l) CE B1 Bg pr(1-p/) Q

-1212 A.
10 -LIMIT AS
/ h'/h —
B Nu >>I
*CF(D/k^Tm) ( + 82 Nu) /
FIXED COST= N /
10 //
TOTAL COST —D "^^ OPTIMUM Nu7 /
w
LIMIT AS
iw h'/h — co
(i.e. B2 Nu O)' l
l-M / -MIT AS h/h - 0
o B2Nu >>I
PUMPING COST /
p-i
Pr 1
/ /\-LIMIT AS h'/h oo
I/ / (i.e. B Nu -O)
82 Th D /
/
20 1 103 1
NUSSELT NUMBER, LOGARITHMIC COORDINATES
Figure 47. Illustration of Heat Exchanger Costs for a Given
Geometry and Tube Diameter as a Function of
Nusselt Number.

-122This is not valid, of course, for non-zero values of B2 As
the effective outside heat transfer coefficient becomes more important,
the optimum Nusselt number will be less than the value given by Equation
(116).
From an examination of Equation (116) and the definition of B3
it can be seen that
2 + m
NUoptimum a [D] P-i- 1 (117a)
1 -m
a [AT] P. + m - 1 (117b)
1 -m
a [Q] pi + m. (117c)
For common values of'p and m (i.e. p = 3.5, m = 0.6)
Noptimum a D( 8a)
0.129
a a6 (118b)
m
0 Qo29 (118c)
Thus, for example, a change of diameter by a factor of two
changes the optimum Nusselt number by a factor of approximately 1.8.
On the other hand, a change in ATm or Q by a factor of 10 changes the
optimum Nusselt number by only about + 35 per cent. The same proportionality is true, but to a lesser extent, for non-zero values of B2.
Once the desired (or optimum) Nusselt number is selected, the
remaining design parameters Ntube and L may be calculated as follows.
From Equation (82), valid for any geometry, the number of tubes is given
by
Ntube n- D e)
^ ^ p. J D R e

-123and, specifically for a geometry for which Equations (80) and (81) are
valid c 1/
N 4W rC2 rl/31 (120)
Ntube - it D Nu
The tube length can be obtained by substituting Equations (77) and
(76) into Equation (73) and rearranging
L [1 + Nu/(h'D/k)] (121)
Nu Ntube k Tm
and, specifically for a geometry for which Equations (80) and (81) are
valid
L Q D [1 + Nu/(h'D/k)]Nu(l/n2) )
L = (122}
Z4 W k Sm (C2 Prl/3)1/n2
Example Design of a Typical Heat Exchanger
The relative importance of the variables on the design of the
optimum heat exchanger can best be shown by an example problem. This
problem is to design an optimum heat exchanger for a particular application in which actual numerical values of the parameters are specified.
The quantitative results of the solution of the problem will,
naturally, be applicable to some other arbitrary heat exchanger design
only insofar as the costs and specified variables are the same as for
this example. Throughout this presentation, however, the qualitative
effect of assuming different costs and different values of the specified
variables will be considered in some detail.
The statement of the problem is as follows: A condenser is to
be designed in which some vapor condenses on the outside of the tube and
water flows inside the tube. The amount of vapor to be condensed is such

-124that the required rate of heat transfer by the condenser is 10,000,000
BTU/hr. The equivalent outside heat transfer coefficient (including the
effect of any extended surface) is 3000 BTU/hr-deg. F - ft2
The mass flow rate of the water is 250,000 lbm/hr and the mean
temperature difference is 100 dego Fo The specified variables, then, are
W = 250,000 lbm/hr
Q = 10,000,000 BTU/hr
ATm = 100 dego F
hl = 3000 BTU/hr-deg F - ft2
Values of the physical properties of water will be assumed
constant as specified below
c = 1 BTU/lbm-dego F
p = 624 lbm/ft3
k = 0e353 BTU/hr-dego F - ft
= 2o42 lbm/ft - hr
Pr - 6855
The following costs will be assumed
lo The initial investment in dollars is given by
Initial investment = 100 A0~6 (123)
where the area A is measured in ft2o It will be assumed that the depreciation per year is ten per cent of the initial investment, the maintenance cost per year is also ten per cent of the initial investment and
the number of operating hours per year is 87600 Thus,
Fixed Cost (0o20)(100) AOo6 (124)
8760

-125and CF = 2.28 x 10-3 (dollars/ft1'2 hr) (125)
with m = 0o6 (126)
2. The cost of electric power will be taken as $0.01 per kilowatt hour. Based on a mechanical efficiency of 60 per cent for the pump
0,~01 (127)
CE (0o60)(3412.76)
= 4,88 x 10-7 dollars/BTU
Design calculations are presented in detail in Appendix D with
only the results summarized here.
For purposes of comparison and illustration of the problems
involved, designs will be considered initially using an empty tube with
three different values of the inside diameter: 0.25, 0.50, and 1.0 inch.
Since the purpose of this example problem is to illustrate the relationship between the variables, rather than to build a specific heat exchanger, no attempt will be made to choose values of the tube diameter,
number of tubes, tube length, etc. corresponding to dimensions of heat
exchanger components available commercially.
The fixed cost, pumping cost, and total cost for each diameter
tube are plotted as a function of Nusselt number in Figure 48.
It can be seen that for each tube diameter there is a particular Nusselt number for which the total cost is a minimum. The "flattness"
of the minimum appears exaggerated because the costs are plotted on logarithmic coordinates. Nonetheless, any value of the Nusselt number.within
+ 25 per cent of the optimum value would probably be acceptable. It can
also be seen that the cost of designing for a Nusselt number considerably

-126-7
EMPTY TUBE GEOMETRY
__ _ __.. _~ ~~1 ^ -~J /~ I~
FIXED COST
0
-9
71_______ ___ +_f - -PUMPING COST
/=0.25 =O-50 D=1
-10
10-I0 1 -
20o 2
20 10I 103 104
NUSSELT NUMBER, Nu.
Figure 48. Fixed Cost, Pumping Cost, and Total Cost Per BTU as a
Function of Nusselt Number for Example Heat Exchanger
Design Using an Empty Tube Geometry. Parameters of d.

-127greater than the optimum is much higher than the cost of designing for
one that is too small.
As the tube diameter is increased the curve of fixed cost vs.
Nu is raised, while the curve of pumping cost vs. Nu is lowered which
results in a shift of the optimum Nusselt number toward higher values of
Nu for larger tube diameters.
Despite the fact that the optimum Nusselt number is quite different for different diameters, the total cost at the optimum is almost
identical.
The optimum exchanger designs are summarized below.
Diameter 0.25 inch 0.50 inch 1.00 inch
Optimum Nu 175 330 600
x 1-7 -.0715 x 7 -7
Total Cost 0.031 x 10 0.0315 x 10 0.0321 x 10
(dollars/BTU)
Number of Tubes 241 55 13
Length (ft) 4.2 96. 21.4
Next, two designs will be considered using two types of turbulence promoters in a tube of diameter 0.50 inch for comparison with the
design for the empty tube. The two geometries (selected arbitrarily for
purposes of illustration) are
Geometry I Disks; d = 0.625; s = 4
Geometry II Disks; d = 0.875; s = 8
It should be noted that for any of the geometries studied in
the experimental portion of this investigation

-128100 f = C(s,d) Ren(s'd) (65)
hm/hO = C(s,d) Ren(sd) (67)
with the appropriate functions C(s,d) and n(s,d) -- different for each
geometry and not the same for calculating f as for calculating hm/h0 --
tabulated in Tables VIII and X of Appendix C. Since
h (0.027) k Re 8 prl/3 (128)
0 D
The constants C1, C2, n1 and n2 may be readily obtained.
The fixed cost, pumping cost, and total cost for each geometry
are plotted as a function of Nusselt number in Figure 49. For comparison,
the curve for the empty tube is also included. It can be seen that for
the same tube diameter the fixed cost is the same for all three geometries.
The curve of pumping cost vs. Nusselt number, however, is higher for both
turbulence promoters than for the empty tube. The curve for geometry II
is higher than the curve for geometry I.
The optimum exchanger designs are summarized below.
Empty Tube Geometry I Geometry II
disks disks
s=4 s=8
d = 0.625 d 0.875
Optimum Nu 330 300 245
Total Cost 0.0315 x 10-7 0.032 x 10-7 0.035 x 10-7
(dollars/BTU)
Number of Tubes 55 143 159
Length (ft).9.6 3.96 3.86
- _ _ _ _ __ _ _ __ _ _ _ _ _ _ _ _ _ _

-129D 0.50 IN_ / = = ______
GEOMETRY I DISKS s4, d 0.625 /4 - /
GEOMETRY I DISKS 8 8, d 0.875 1/
10.... ~ ~ I I II~_~_ _ _ I 1
~E -' TOTAL COST //
I --- - ~ -—, r;:,.-~_I I _ _
0
-10
I0I
10
20 102 3 10
NUSSELT NUMBER, Nu.
Figure 49. Fixed Cost, Pumping Cost, and Total Cost Per BTU
as a Function of Nusselt Number for Example Heat
Exchanger Design Using Turbulence Promoters with
D 0.50 inch.
~D;=0.50 inch.

-130It can be seen that in each case the turbulence promoting geometry produces a higher total cost than the empty tube, but for all geometries the costs are within ten per cento This example illustrates an
important conclusion: The only way a turbulence promoting geometry can
produce a lower total cost is for its curve of E/Q vs, Nusselt number
to be below that for an empty tube.
Even though the E/Q curve for the turbulence promoter does lie
considerably above that for the empty tube, if an exchanger is properly
designed (in the optimum manner) for the promoter, its total cost may be
only slightly greater than that for the optimum exchanger designed using
empty tubes. This is well illustrated by the example problem~ The
effect of geometries with high E/Q vs. Nu curves is to lower the optimum
Nusselt number.
Conclusions Regarding the Economics of Using Turbulence Promoters
It was illustrated in the solution of the example problem that
the total cost of a heat exchanger designed in an optimum manner to use
turbulence promoters may be only slightly greater than the total cost of
an exchanger designed in an optimum manner using an empty tubeo Nonetheless, it was shown that the only way one inside tube geometry can produce
a lower total cost than another is for its curve of E/Q vSo Nu to be
lower This suggests that a valuable measure of any proposed turbulence
promoting scheme for improving the rate of heat transfer in a tube is
the ratio of E/Q for the turbulence promoter to (E/Q)ET for the empty
tube as a function of Nusselt number This ratio is plotted as a function
of Nu in Figure 50 for disks and in Figure 51 for streamline shapes and
the solid rodo

-131DISKS S 12 DISKS S= 8
~ w
wI 4-. I W
>=O ~EMPTY TUBE < EMPTY TUBE 0
Iz'1,I TIKS_ _ I SKS/2
__ _ ___ _ _.I.il
0.1 0.1 ~~~
10 100 I10 00 100 I000
NUSSELT NUMBER,N NuUSSELT NUMBR,Nu
101010 10101 0
___ ___ __KS 84 _____ ____ DISKS S~2
NUSSELT NUMBER, Nu~ Ir-'V I, N N
Figure 50. Ratio of Pumping Cost for Disks to Pumping CoSt for' WI Ip
Me as a TUBFun n EMPTY NUBE
vs10 u for s 100 12,000 108, 4, and 100 1000
NUSSELT NUMBER. Nu NUSSELT NUMB~R,NU
Figure.5~ Ratio of Pumping Cost for Disks to Pumping Cost for
Empty Tube as a Function of Nusselt Number, (E/Q)/(E/Q)ET
vs* Nu for s ~ 12, 8, 4, and 2 with Parameters of d.

-13210 - r 1 I I 10 I 1STREAMLINE SHAPES S =12 STREAMLINE SHAPES S =8
Wa It =$ =ej
EMPTY'" d= 0.625 w6dZ
EMPTY TUBE
NZ. EMPTY TUBE
N.
IiJ ~ ~ ~ ~ ~ ~ ~ ~ L ~ ~ ~ ~ ~ ~ ~ ~
IOJ IO ~ ~ I O IO~0.1
10 100 1000 10 100 1000
NUSSELT NUMBER, Nu NUSSELT NUMBER,Nu
I V __ __ _ _ _ SOLID ROD S0 __
STREAMLINE SHAPES S =4
d~.50' d:0.60
W l I I I I I I I IIII
0.80.J I!. I ~.~ I
10 100 1000 10 100 1000
NUSSELT NUMBER,Nu NUSSELT NUMBER, Nu
Figure 51. Ratio of Pumping Cost for Streamline Shapes and Solid
Rod to Pumping Cost for Empty Tube as a Function of
Nusselt Number, (E/Q)/(E/Q)E vs. Nu for s = 12, 8, 4,
and 0 with Parameters of do

-133It can be seen that all of the promoter combinations except the
solid rods become less efficient at higher Nusselt numbers. For the disks
at spacings greater than 8 tube diameters and streamline shapes at spacings of 12 tube diameters, the most economical design is the one with the
smallest diameter and'tle.east economical is the one with the greatest diameter.
At closer spacings, particularly for the streamline shapes, the trend is
reversed and the most economical promoter is the one with the largest diameter. This is also true for the solid rod in the center. In other words,
it appears that systems which behave similar to a solid rod are best with
large diameter ratios, while systems which behave like individual bluff
bodies are best with small diameter ratios. It is also noticed that both
the disks and streamline shapes tend to become more efficient at closer
spacings, but in the case of the disks a spacing of 4 diameters is more
efficient than either 2 or 8 diameters, indicating an optimum spacing of
about 4 tube diameters for disks.
One important observation to be made from the example problem
is that even though the total cost is almost the same for the two exchangers using the turbulence promoters and the one using empty tubes,
the "shape" of the exchangers is quite different. Both of the optimally
designed exchangers using turbulence promoters were much shorter and required more tubes than the optimally designed exchanger using empty tubes.
This is a general characteristic of well-designed exchangers employing
turbulence promoters
Effect of Variables Assumed Specified
In the economic study it was assumed that four variables were
specified by the process: W, Q, AT, and h'o The following comments are

-134intended for the cases where these variables are not specified, but are subject to optimizationo
As was shown earlier, changing Em has little effect on the
value of the optimum Nusselt number for any type geometry. The total
cost, however, (excluding the cost of fluid) per unit of heat transfer is
less for higher values of STm This is obvious, since increasing Tm
increases the rate of heat transfer without any increase in area or pumping power required0 Of courses in order to obtain high values of Tm it
is usually necessary to provide large mass flow rates W of the fluid, so
that the optimum value of ATm is obtained by balancing the increased
cost of the fluid against the decreased fixed and pumping costs per unit:
of heat transfer. If the-Nusselt number in the tube is known (and, hence,
the value of Uoa) charts prepared by Colburn and presented by Perry(35)
are available for quickly obtaining the optimum value of AT based on
the relative cost of fluid and fixed cost per unit area. Thus, an approximate estimate of the best AT should be adequate for selecting the optimum Nusselt number for any geometry. On the basis of this optimum Nusselt
number, Colburn's charts can be used to obtain a more exact estimate of
the optimum value of AT ~ This may be continued, if necessary, until an
optimum set of conditions is obtained, but usually the first assumption
will be adequate.
Changing the mass flow rate W has no effect on the optimum
geometry or Nusselt number if it does not affect AT o For a constant
temperature difference, increasing the mass flow rate simply reduces the
length and increases the number of tubes (keeping the Nusselt number and

-135required area the same). If changing W also changes.AT then the comm
ments in the preceding paragraph are applicable.
The outside heat transfer coefficient h' is dependent upon
conditions external to the geometry of the inside of the tube, and hence,
must be assumed specified. Also the rate of heat transfer Q will almost
always be specified.
It should be noted that anything which tends to result in a large
heat exchanger reduces the fixed cost relative to the pumping cost per
unit heat transfer. This, in turn, favors operating at lower Nusselt numbers where turbulence promoters are most effective. Thus, high values of
Q, low values of h', and low values of ATm all tend to favor the use
of turbulence promoters.
Use of Turbulence Promoters in Design of New Exchangers
On the basis of the preceding economic study, including the
results of the example problem, some recommendations will be made concerning the applicability of including turbulence promoters in the design of
new heat transfer equipment.
From Figures 50 and 51 it can be seen that the pumping power
required per unit heat transfer for the types of turbulence promoters corsidered in this investigation ranges from about 80 to 500 per cent of that
required using an empty tube. Since the promoters are almost always most
effective at low Nusselt numbers, it would appear that their greatest
promise is in designs where pumping cost is very high compared with the
fixed cost, since in these cases the optimum Nusselt number is usually
low.

-136What is probably more important than the savings in cost by
using promoters is the added flexibility in design which may be afforded,
usually with only a slight increase in the total cost of operation. Their
use adds one more independent variable that the designer may have in specifying the exchanger. Even after the fluid velocity in the tubes has been
set, it is still possible to independently raise the heat transfer coefficient by a factor frcn less than 10 per cent to almost 400 per cent. It
has been shown that, properly designed, this may impose only a slight
additional costo
The use of turbulence promoters is essentially equivalent to
designing a longer exchanger with fewer tubes (or, in other words, to
adding more tube passes). This suggests that whenever a study indicates
that the best designed exchanger would require more tube passes than it
is feasible to build, then turbulence promoters should certainly be considered.
In certain cases a design may be encountered where one tube pass
will not produce a high enough heat transfer coefficient, but two tube
passes causes an excessive pressure drop, produces a heat transfer coefficient greater than that required, and reduces the effective value of ATem
In this situation a better solution might be to install promoters for at
least part of the length of the exchanger.
The use of turbulence promoters also has the advantage that the
local heat transfer coefficient can be controlled. This may be useful when
the local rate of heat transfer varies considerably along the length of
the tube as, for example, in a chemical reactor~

-137Use of Turbulence Promoters in Improving Existing Exchangers
The considerations involved in improving the performance of an
existing exchanger are simpler in some respects and more complicated in
others. The problem is simpler in that, since the exchanger is already
built, the number of tubes, diameter, length, etc. are set and there are
a minimum of variables which are free to be changed and, hence, fewer
which need to be considered. On the other hand, the problem is more complicated, since when one variable imbedded within a process is changed,
often all the others are affected in a manner which is hard to predict and
for which the economics are not known.
The rate of heat transfer by an exchanger is given by
Q = Uoa A Tm (73)
In an existing exchanger the area A is fixed, so there are only two means
for improving the rate of heat transfer: increase the overall coefficient
Uoa or increase the temperature difference ATm. Now, increasing ATm
involves changing the process external to the heat exchanger, for example
by increasing the mass flow rate of the fluid through the exchanger or
changing certain process temperatures. Since the advisability of this
approach depends strongly upon the specific situation it will not be considered further.
If it is desired to increase Uoa without changing the mass
flow rate, there are essentially two alternatives: 1) increase the fluid
velocity by increasing the number of tube passes, or 2) install some sort
of turbulence promoting devices. In many cases increasing the number of
tube passes will not be feasible, particularly when the exchanger is a
multiple pass exchanger to begin with,

-138It is in this case that the use of turbulence promoters provides
a quick, simple method of improving the performance of an existing heat
exchanger. The data and correlations of the experimental phase of this
investigation provide a guide to the selection of the best type of promoter for a given application. The use of promoters has the big advantage
that changes to the process in which the exchanger operates are localized
as much as possibleo
Streamlined vs. Non-streamlined Promoters
There appears to be little advantage (or disadvantage) in using
streamline shapes in preference to diskso Since the cost of fabricating
streamline shapes is greater than the cost of fabricating disks and since
possible problems with vibration may occur with their use, the use of disks
is recommended over the use of streamline shapes for most applications.

SUMMARY AND CONCLUSIONS
The main objectives of this investigation were twofold: 1) to
obtain generalized correlations for predicting the rate of heat transfer
and the pressure drop for a fluid flowing in a tube in which bluff-body
turbulence promoters were mounted axially, and 2) to determine whether
bluff-body turbulence promoters can be used economically in equipment for
transferring heato
The variables chosen for investigation were the shape of the
bluff-body turbulence promoter, the ratio of the diameter of the promoter
to the inside diameter of the tube, the spacing between promoters, flow
rate of the fluid, and physical properties of the fluids
Two different shapes were investigated which represent extremes
in the degree of streamlining, The first was a disk, the second was a
combination of a hemisphere and cone to form a teardrop-like shape. In
addition, data were obtained for. a solid rod centered in the tube which
corresponds to a string of bluff-bodies at zero spacing0
A variety of diameter ratios, spacings and flow rates were tested
with one fluids water0 All results are presented in the form of dimensionless ratios to extend the applicability of the results0
Two different techniques were employed to characterize the
pressure drop0 The first was to define a friction factor similar to that
used for smooth tubes as given by Equation (22)o The second was to define
an effective drag coefficient for each bluff-body similar to that used
for immersed objects in an infinite fluid with uniform flowo The effective
drag coefficient is defined by Equation (14)o
-1^9

-140The rate of heat transfer was characterized by a ratio h /h
where hm is the mean overall heat transfer coefficient for a tube with
turbulence promoters and h0 is the mean overall heat transfer coefficient
(given by the Sieder-Tate equation) for an empty tube and the same mass flow
rate. When presented in this form, it is presumed that the results of this
investigation using water are applicable to any other fluid for which the
Sieder-Tate equation is valid.
Separate correlations for the friction factor, effective drag
coefficient, and heat transfer coefficient ratio in terms of Reynolds number
were obtained for each combination of shape, spacing, and promoter diameter.
In addition, generalized correlations were obtained for each shape to predict the effective drag coefficient and heat transfer coefficient ratio in
terms of spacing, promoter diameter, and Reynolds number. The generalized
correlations correlate the data of this investigation with an average deviation of less than 10 per cent.
A comparison was made of existing correlations for the friction
factor based upon the use of an equivalent diameter for the annuli, and
it was found that recent correlations of Lohrenz and Kurata(26), Meter and
Bird(29), and Walker, Whan, and Rothfus(42) give comparable results. Accordingly, a correlation for predicting the rate of heat transfer to the outside wall of an annulus was developed using the equivalent diameter suggested
by Lohrenz and Kurata.
A comprehensive study of the economics of using turbulence promoters in heat transfer equipment was made and a general procedure was
outlined for evaluating the effectiveness of any turbulence promoting scheme.
This procedure was illustrated with an example problem.

-141The main conclusions resulting from this study may be summarized
as follows.
1. The pressure drop caused.by..a':, fluid.. flowing in a tube
in which axial bluff-body turbulence promoters have been inserted can be
better described by an effective drag coefficient based on the drag of a
single bluff body than by a friction factor of the type normally used for
smooth tubes.
2. For diameterratics between 0.625 and 0.875 the effective drag
coefficients for disks axially centered in the tube are correlated by
100 f 156 S (129)
0D 1 + 0.78 s
3. For diameter ratios between 0.625 and 0.875 the effective
drag coefficients for streamline shapes of the shape shown in Figure 8,
axially centered in the tube, are correlated by
F -0.12
100 f 117 s Re (130)
D 1 + 1.6 s 10 000
4. The mean heat transfer coefficient ratios for disks axially
centered in the tube with diameter ratio between 0.625 and 0.875 are correlated as follows.
h +.28 Af) -0.14
h - 1 -.28 (-ln A,) (131)
h 10,000
1 1.7
X 1 + 0.15 s 11.9 + s4
5. The mean heat transfer coefficient ratios for streamline
shapes of the shape shown in Figure 8, axially centered in the tube with
diameter ratio between 0.625 and 0.875 are correlated as

-142h R-O..11
-= 1 + 2.04 (-in Af) F Re 1 (132)
ho Lio,oooJ 1i + 0.14 sJ
6. The rate of heat transfer to the outside wall of an annulus
is best correlated in terms of an "equivalent Nusselt number" which is a
function of the "equivalent Reynolds number." The recommended correlation
is
Nu* = 0.024 Re*08 Prl/3 (/kw)0.14 (133)
where Nu* and Re* are based on the equivalent diameter recommended by
Lohrenz and Kurata(26) and defined by Equation (35).
7. The most important parameter required for an economic evaluation of turbulence promoters in a given heat transfer application is the
dimensionless ratio of pumping energy required to total rate of heat transfer, E/Q,as a function of the Nusselt number produced. This parameter may
be obtained from experimental measurements by use of Equation (106).
8. When designed for use at low Nusselt numbers, the use of
certain types of turbulence promoters as indicated by Figures 50 and 51
can produce a more economical heat exchanger than the use of empty tubes,
9. By careful design the use of an inefficient turbulence promoter can be used to advantage in giving additional flexibility in designing a new heat exchanger or in improving an old one. Although the total
cost of using an inefficient turbulence promoter will always be higher than
that for an empty tube, it may be only slightly higher.

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16. Faruqui, A. A. and Knudsen, J. G., "Rates of Heat Transfer from
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17. Gambill, W. R., Bundy, Ro D., and Wansbrough, R. W., "Heat Transfer,
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APPENDIX A
DESCRIPTION OF COMPUTER TECHNIQUES FOR PROCESSING DATA
In this section the method of processing pressure drop and
heat transfer data will be discussed, Some of the problems of converting the raw measurements into usable engineering units and evaluating
the physical properties of water were common to both types of data, so
they will be explained first.
Because of the tremendous amount of data generated by the
experiments, all data processing was performed on the IBM 704 computer.
Most of the procedures for converting the raw data and evaluating
physical properties were actually incorporated into general-purpose
computer subroutines and used by the main data processing program,
Conversion of Raw Measurements into Engineering Units
and Evaluation of Physical Properties
Thermocouple Readings
The copper-constantan thermocouples used were calibrated
against precision thermometers certified by the National Bureau of
Standards in a constant temperature oil bath over the range 50 deg
F to 220 deg F and found to agree with tabulated values in the
International Critical Tables. (32) The range of values which is
of interest is reproduced in Table IVo All thermocouple emfs were
converted to deg F using linear interpolation on the table and converting from deg C.to deg F with
Deg F = 1l8 Deg C.+ 3,2 (A-l)
-146

-147TABLE IV
THERMOCOUPLE EMF VS, DEGREES CENTIGRADE FOR COPPERCONSTANTAN THERMOCOUPLES
EMF TEMP M EP EMF TEMPMP
(mv) (Deg C) (mv) (Deg C) (mv) (Deg C)
0 00 0,00 1, 50 370 38 3, 00 72.08
0,10 2.59 1160 39-77 3.10 74,31
0o20 5.16 1,70 42, 15 3.20 76.,54
0 30 7.72 1, 80 44 51 3.30 78.76
o 40 10,27 1,90 46,86 3.40 80.97
0,50 12.80 2, 00 49.20 3.50 83517
0o.60 15.32 2,10 51.53 3. 60 85.537
0,70 17,83 2-20 53. 85 370 87,56
o,80 20*32 2 30 56.16 3.80 89.74
0 90 22. 80 2, 40 58,46 3.90 91.91
1,00 25,27 2 50 60,76 4,00 94.07
1,10 27.72 2, 60 3,0o4 4,10 96,23
1,20 30,15 2,70 65.31 4,20 98,38
1, 30 32 57 2. 80 67 58 430 100, 52
1.40 34.98 2,90 69.83 4,40 102,66

~18Rotameter Readings
Four rotameters (numbered 1 to 4) were available covering
different flow rates. These rotameters had assorted scales, depending
upon their original use. Each, however, was calibrated prior to use
by measuring the time required for a given weight of water to fill a
tank while the water was flowing at a constant rate. ach of the
rotameters was found to have a linear scale; the calibration curves
are presented in Figure 52. The rotameter number, range in gpm, range
in Reynolds number for water flowing in the test section at 60 deg F
and equations for calculating the flow rate in gpm from the rotameter
reading are given below,
Equation for
Rotameter Range of Re Calculating
Number Range of GPM at 60 deg F Flow Rate
1 01l to 0o7 340 to 2000 + 0,02+ 0454x Reading
2 0 15 to 2.3 1400 to 6200 -0 395+ 001095xReading
3 o08 to 3o8 2300 to 11,000 -0X075+ 0o3265xReading
4 2ol to 18 6000 to 50,000 -0 100+ Oo299x Reading
About 80 per cent of the data were taken using rot.ameter 4, 20 per
cent using rotameter 2, and less than one per cent using rotameters 1 and 3o
Pressure Drop
All pressure drops were measured using the single-tube King manometers with either mercury or purple indicating fluid of specific gravity
1.750. The equations for converting the pressure drop to lbf/in2 (psi)
are

-1491.0 I 2.0.- -
0.8 Sr 1.6
0.6 ~ ~ - 1.2 ~
0. - I - /0. 0P
0.4 -~~ ~ / ~ ~~, 0.8
0.2 _ - 04
0 —.~ 0 —
0 0.4 0.8 1.2 1.6 0 8 16 24
ROTAMETER NO. I READING ROTAMETER NO.2 READING
5 - X 20 i-~'
4 12 1 6 20 40 60
3 ~~- 12 -
Reading for Rotameters 1 2. 3 and 4
0 4 8 12 16 20 40 60
ROTAMETER NO.3 READING ROTAMETER NO.4 READING
Figure 52. Flow Rate as a Function of Rotameter Reading, GPM vs.
Reading, for Rotameters 1, 2. 3, and 4.

-150= tinches mercury (A-2a)
Z~p (A-2a)
psi 2.20392
Pinches purple fluid (A-2b)
psi 36,90533
Viscosity of Water
The viscosity of water at any temperature T was calculated
using Binghama's( formula as given by Perry(35) This formula is
= =~.. 1 (A-3)
2.1482 [T -8.435 + I8078 4+ (T - 8435)] - 120
where the viscosity u is given in centipoiseso
Thermal Conductivity of Water
The thermal conductivity of water as a function of the temperature T was obtained by fitting a second order polynomial to tabulated
values obtained by Timrot and Vargaftik(41) and presented by Perry35)
These values are
Temperature (Deg F) k (BTU/hr-deg F-ft)
32 0.343
100 O.363
200 0 393
The polynomial is
k = 00343 + 2.941 x 10o4 (T- 32) + 350O14 x 10-8 (T-32)(T-100) (A-4)

-151Density and Heat Capacity of Water
The density of water over the range of fluid temperatures
encountered was taken as
p = 6243 lbm/ft3
The heat capacity was taken as
c = 1.0 BTU/lb -deg F
Processing of Pressure Drop Data
The method of processing data on the IBM 704 can best be
described by considering a sample set of data. Thus, we will examine
the pressure drop data from run A-18 taken for streamline shapes with
d = 0,750 and s = 8.
The raw data sheet is shown in Figure 53. Each line on the
data sheet corresponds to one IBM card. The computer program was
written so that items not repeated on a line were assumed to be the
same as on the preceding line, The term labeled PSCALE signifies
which indicating fluid was being used, If it was 0.0 then the
purple fluid was used; if it was 1.0 then mercury was used. The
term labeled RATIO is (l/w ).w for all isothermal runs this was
1*000, A listing (i,e,, printed copy) of the punched cards for this
set of data is shown in Figure 54, An "L" in column 71 of the cards
signified the end of the set of remarks or the end of the set of data,
A blank card also signified the end of the set of data if the "L" was
not punched in column 71,

-152DATA SHEET FOR PRESSURE DROP MEASUREMENTS
RUN NUMBER DATE NP S
Fi-\8 l _-9-c \ 6. _ _ _ __ _
I 15,M A 36 co
3/% tircH1 T- "'E o,PS AT C ICK -
j1 ~4. r_.. L
___I___S^ ________ 22 _______________ _____-______
OBS. O. TEMP. NROTA ROTA P-IN P-OUT P-SCALE RATIO. ___ 041. ^.... 0.... 0...ooa.....
i 1' 1 1. |4..I. I.._
____.- 49,0
___________. ________
__ _ _ _____o __10_
____________ ______o 1_____________. _':i._
_ _ __ 2I 2 _3 __:3 az 5T 64 7_9
Figure 53. Sample Raw Data Sheet for Pressure Drop Measurements.

-153The data processing will be illustrated with the fourth
from the last data value on the sheet, The raw measurements were:
1 Thermocouple reading of inlet water temperature:
0 462 millivolts,
2* Rotameter number 4 reading: 45,4
35 Pressure drop: 87,8 inches purple fluid (sp. gr. 1,750)o
Converting these to engineering units as previously described,
1. Inlet water temperature: 53.3 deg F
24 Flow Rate: 13,47 gpm
3, Pressure drop: 2*379 psi
The Reynolds number was calculated using
Re = 4 (21)
4 x (62.43/7.481) x 60 x gpm
2,42 x i x 3,14159 x (1.005/12)
3145.5 gpm (A-5)
ktM~~~~~(A-5)
For the specific example, Re = 34,1315
The friction factor was calculated using
r -= w - -1 (62)
32 S
29075 (-APpsi) F78,41 78. 41 (A-6)
f = 2.05(gy ) [-.f [7, 1] (A-6)
(gpm)2 Ln S J LnS
For this set of data, the friction factor for the empty tube was calculated using Nikuradse's equation

-154A-18 1-9-61 6* 8,
3/4 INCH DEARDROPS AT 8 INCH SPACING L.462 2, 14,4 0 95 0o 001 0oo001 1 0
19,8 2 1
21* 0 3,0
24*6 3 45
4, 11,4 6*55
15,0 11,6
19,4 18,15
22 5 24, 2
26,0 31, 4
30,0 40.1
33.5 49,o
37,6 59.8
4o06 7005
45^4 87,8
51,0 6,8 1.
54 6 7.7
61 5 9.2 L
Figure 54, Listing of Data Cards for Pressure Drop
Measurements of Sample Problenj,
f ~ ~ ~: ~ (A-7)
= (1.73718 in ( f Re) - 040)2
0
for Re = 34,131, fo = 000570 and the experimental friction factor for
the above set of data is 0o05862. (Note: In the above formula fo was
obtained by iterating until (fO)assumed agreed with (fO)calculated to
three significant figures.)
The printed output frotf the computer containing the analysis
of the pressure drop data just illustrated is shown in Figure 55, The
particular data point which was illustrated is the point numbered 14
under: the column headed OBSV, In addition to the Reynolds number and
experimental friction factor, empirical estimates for the empty tube
were calculated using various other approximations in addition to

~ —---------— ~~_ —_ —__ —— _-_ —-__-_ —_ —__ —----— _@liUJL~J^J^E-~KP~SII^E 0 —^-DFSeaIR —-_ —--------------------------------------------— _ —R UIN NiL iiEJER A- I'8
DfiT TOKi:EN 1-9-61
-_-______,____-)-S,.^OiiE__3.i^_6________
DACiO~TAOPTARTIlL FCESSED DTPROCESE
1 53.3 " 2 ~ 1~4. 40" 1.13 ~ 0.35~ 0. 00 ""~ll-rTcHES PURPLE ~ 2.57 1.0O00'. —-~'_ -_-___-____^___2__~____ _ _____-__ —---— _ —-— ___-____ —-— _-_-_ —-— _
3 53.3 2 21.0r0 1.90 3.00G 0. 00 IHICHES P-URPLE ~ 8. 1R3 1.000
______ ^ _______ 53.3 ______ ______ 24. 60 _______ 2.30 _______. t'45 ________ 0.00 ___ IHCHES PURF-LE _________.35 ______ 1.000
~5 53.33 4 INCH11.40 3.31 TE.5ROP.00 ICHES PURPLE 17.5 INCH POCIi1.000
~ _~ —^___, ____ — ~ -_ 3._3_ __4__~ ___J50____:'_ 1~0 0_0___ICESPR'E ___..^^_ _JpO..
------------------------ - -----------------------------------------— ~ —---— l~l~~ —--------------------— 0
PART IALLY FPOOCESSED OATA
~5V.3 TE ~P 4 ~ 1RO.4OT ~.o INLET P OUTLET P PRCHESURE SCPLE DELTA7 P Y ~ 000.'-DC F CPSIA*-100)
1-. ~ _-5 3 _ _4 37 1_?.40 ____'~1 - - 24.SD _ 0._p0___ _IlNpHES URPLE_ __ _ 7^Z __ _ __OL - 0
53.3 4 21~.00 1.7'4 0.00 INCHES PURPLE 5.08 1.000
103 53.3 -4 30.00 8.87 40.10 0.00 INCHES PURPLE 108.5 1.000
~ n ~ 5T3~.3~-4.~ ^72~0 3. ~ Q5 ~ 0 i UU INCHES PURPLE 1 32. 771.000
-- 5 53.3 14 11-0 3 - 3 6 59.80 U'0.0 INCHES PURPLE 162.3 1.000
-— 14 53.3 4 45.40 13.47 87.0 0.00 INCHES PURPLE 237.90 1.000
53.3 ~~~~~~~~ ~ ~~~~~~~~. U 81500 iNCHiES PURPLE 4D.I 1.000
53.3 5 U4 nj INCHES PURPLE G.51.000
- ~ - ~~~~~~~~~~~~~~~~~~~~~~~~~~.G _ D 0 INC:HES PURPLE 8. 1.000
~~ "~ ~^ -- - ----- 3.^~ ~~ "~4 ~~ ~~ ~T 7 ^- -- ~^T T ~ ~' ~~^~s-~ ~' ~"~T~-o ~ "~'~'TC E~''TE TrF7- -- - ~3T:8: ~r" "~"~^ ooo~ ~~
10 __ 53.3 4 _______ O U.2 7.70 0.00 INCHES PERCURVPLE 134.33 1.000
~37 ~ 4 351^ 43~ ~~0 UI U 7INCHES UMERCPLRV 417.32 1.000
X 1000 0 100 ~HH PUPE1. 1000, 10 ~
-~-JL —-_ —— ~- - -— 2 -9 —-7.. S___. ___ ~ 2 _ _ J. ____J _
4 4.1 7.58 0.65 3..388 0.5.951 0.985 8.257 8.060 9.313 8.3EE 0 1.8.0 007
U —---— C. c'U I —-- fT1nHEZ TERrFVr,`7 — 7h ------- -~,TT r
_53..3 4__54_ I G. 27' 7 —7U i3-n I0 8 CHE MERCURY 3193iC1.000
14 "5.823 7U 2U U0.904 0. 220 ".'~12: INCHES M4ER.1CURY 4"73.61 1.00052
i ____ 5 ______ 8.381 7. 180 0I.826 0i.834 0.755 079 0.819.8E.696 8.608 9.'507 9. 099 8.766 _____
---- ------ F -— F-=a- - L-i,3 - -- f: LE H - L F E1 L —-- -----------------
6'' 11.107 1000 0.77liL0 I 0.71 4! 0. 5 - 9 C,
2.8 9 4 6 6 7. 026940 0.6.'QC.6~ff"'5 1 UZ -----— 9~9.'5 — ~ 9.517 10.133 ~ 107.470 7.859
1 —------- ~ —_LL~_IIS___~^II.- ----— __
10 22.468 6.160 "0.645' 07646 "7:.625" 0.6 07631 "~.~546 97 53T "9.934 9.871" 9".765
_____ U _____ 25. 119 6.0C24i 0.628 0.6G29,D 0.60 0.609 0.614 9.59.9 9.584 9.934 9.889 9.816 _____
~ 2 284.491 2 82 71 0. 09~0.61 1a 0.592 ~0.537~0.597~,.550~.532..825~3.309 8.758
13 3!3~43~5 5:383 ii538 i L`Ci5 f 53G 385C1 3.Z-1 6 10 841 0. 1 10.65
14 34.1131 f.l 0i0 0 "0.58 1 "570 ~59 " 57: 1i. 10 0578 101.827 10.9314 10.28407
4 41.0823 5.57 ~ 0.904 07557" "075 ~ "07I546" 1 1.736.1504.8 10.831 9".884 "10.913
-' 8.381 710'. 67~ 7l O_9 S G':D0.'. 008 9.50O7 9. 099 8. 760
11.107 7.2C3 0 1 074 _1 1'07 10.202'.U. +-46.325 5E.591C 0.538 0.542 0.537 C0.528i 053 1382 1322 1420 10.591 1D0.528'' _1 1
Figure 55. Compu-ter Analysis of Pressure Drop Data for Sample Problem*
1948 - U'7 0.GG9 G, 7!_.3 C,' C 0G5 7 9.G1e 1.101 1034.5
10 2 2. 4 6 I G 0 0.6-45 0. G. 4 G 01..17,.~~ -R'51'9 1094 9. 271 9' 765
13 30. 4,"EIG 5.ii nI 0 C)1R nF.l 10F 054, 1.1 r "'.304 ~.11,10284
15 3=. 377 G.033 0. 5 0 -. - ii5 i li791 Il 0.9-c
Figure —-- 55 Cmptr nlyi of G10 Presue ro Dat for 4Sample2 Problem -----

-156Nikuradse's, The friction factor calculated using the Blasius
equation is labeled F-BLAS; the friction factor calculated using
Colburn's j-factor equation is labeled F-J; the friction factor
calculated using Nikuradse's equation is labeled F-NIK, Other
estimates of the empty-tube friction factor are labeled F-DREW
and F-MOODY,
The ratio of the friction factor obtained experimentally
for each turbulence promoter combination to the friction factor for
the empty tube using each of the empirical approximations was calculated and printed for each run,
Processing of Heat Transfer Data
The method of" processing heat transfer data will be
illustrated using a set of data for the same turbulence promoter
combination (streamline shapes with d = 0 750 and s = 8) as was used
to illustrate the method of processing the pressure drop data, The
particular run, R-19-B, corresponds to only one flow rate,
The raw data sheet is shown in Figure 56, As in the case
of the pressure drop data, each line on the data sheet corresponds
to one IBM card. The first line contains the run number and date,*
The next few lines are for remarks; the last data card containing
remarks has an "L" punched in column 71 to indicate that it is the
last card containing remarks. The next set of cards (the last of
which also has an "L" in column 71) are readings of rotameter number,
rotameter reading, voltmeter scale, voltage, millivolts across 5000
amp shunt, ambient thermocouple reading, inlet water thermocouple

-157DATA SHEET FOR HEAT TRANSFER MEASUREMENTS
RUN NUMBER DATEI~
5eAcIuG.. - PScLE PuJiPL_ L
SP4c~tvc, Ps~CILE - PORLE |L|
NROTA ROTA VM VOLTS AMPSMV EAMB EIN EO`UT PIN POUT
- 4-. ^. 9. ^\4 loco o', o.^ 3.' 0,'ao FT A3 _.__ Jco. o.Si.3-, 1 7L
9L lOL iL 2L 13 1L 5L 16L 7 1 8L
EEi TC
REF EMF'- ~o a0,,w1__ ____ _
1 R R 5R 67R 9R 1OR
o^ SS 06 0.Q o o1w1 o o^7 ~519.0 1 0( A %o 0-6O o.a o
TR I 12R R 14R 15R 16R 17R 18R 19R 2OR
_o._66 0o, 0 o-7o 0 0.o.1 0o t 0o.* O.S' o(-oO o0i'0
9L 10L L1L 1L 13L 14L 15L 16L 17L 18L
o.14 o- 0660 %6 o 0.0S o-t10 0.5T o.605^ o.5S 0,39 0.1S1 ~9L -20L
IR 2R R 4R 5R R 7R SR 9R LOR
o.;- 0.69 0/ o 04 0Qsj 0. l 04,1 0o..a 0. a -l.I
11R 12R 5R 4R 5R 6R 17R 18R 9R OR
0 10o.C o60, 01 0 ooo 0.%5 0|.9 0.1- 0 I. 0o.0t)1
R 2R R 4R 5R 6R 7R 8R 9R 1OR
O5,61 O o9 o OrI i 0 0 a Q O o ~3(a19 o4 AI1 0.660 0- -
1IR 12R L3R 14R 5R 6R 17R 8R L9R 2OR
06.0 1o.3a7 0l 0.361 -T o) 01~ 0 -39 I o.oo %r o.l
1TOL 11L 12L 3L ~1L 15L L6L L7L.8L
o^^T O.C>\ c ^ io.3C^ Q^ o0, ^ o1sV 610-(o ~ 0.3< O ^10 1-6 o
-1 I,ij. AI
oa+~\ ~,,.. 1.,1.........
1R 2R!3R 4R 5R 6R 7R 8R 9R 10R
0.S~l 1 10 ki 0.S60j ~ O5~ O 0m' 01 5lO:I o -a
11R 12R 13R I14R 15R 16R 17R L8R 19R 20R
LO.661 0 0 1 -!0,0 o.3,0 o. 0 oo s.' O.. 0 3. o.o.1. _..YOL 11L 12L 13L 14L 15L 16L 17L 18L
0. 4A O 60 ol. O 3.306 IO\lo 0 - 051 06 0C.~ 0I2OL S sI
oa^ o 0.____1 _______ _ ______1_
t g I-, 29 36 3 0 0' 6 -q- 71
Figure 56. Raw Data Sheet for Heat Transfer Measurements.

-158reading, outlet water thermocouple reading, inlet water pressure,
and outlet water pressure. The next line contains the identification numbers of reference thermocouples (used in converting AZAR
readings to emfs as will be described) followed on the following
line by reference thermocouple emfs.
The remaining 16 lines (or cards) on the data sheet are
for readings taken from the chart of the AZAR recorder, Four complete
sets of 32 AZAR readings are used for each experimental run. A listing of the punched cards of this run is shown in Figure 57. The
printed computer output (requiring four pages) which is the complete
analysis of this set of data is shown in Figure 58 for pages I, II,
III, and IV.
The first step by the computer program consists of reading
the data. Where there are more than one set of values for one item
of data, the average and standard deviation of each were calculated,
Values of the input data corresponding to all of the raw observations
were printed on page I. of the computer analysis along with the mean
values and standard deviation for each particular item, A quick
visual check of the standard deviations of each item served to
eliminate errors in punching cards. Each of the AZAR readings was
converted to thermocouple emf and identified with its longitudinal
and angular position on the tube, The conversion to emf was made
using the following formulas.
EMF2 - EMF1
RANGE = (A-8)
AZAR2 - AZAR1
ZERO = AZAR1 - EMF1/RANGE (A-9)
emfi = RANGE (azari - ZERO) (A-10)

-159R-19-B 1-9-61
3/4 INCH TEARDROPS AT 8 INCH SPACING* PACALE = PURPLE L
4. 42.4 2. 9*2 21.83 1.05.300.552 73.5 0.
42.4 9.23 21.81.300 *556 73,6 L
19, 20. 12.
1*40 2.20 1*675 L
0.559 0*691 0*717 0*111 0.567 0*192 0.609 0*242 0.658 0*250
0.661 0.319 0.710 0.360 0*718 0.815 0.390 0.154 0.020 0.860
0.424 0*260 0.626 0*305 0.165 0.578 0*605 0.528 0.394 0 158
0.245 0.638
0,558 0*689 0.718 0.118 0*568 0*195 0611 0.240 0*655 0.250
0,661 0.318 0.708 0.360 0*720 0*815 0.395 0.155 0.014 0.861
0,422 0.259 0.630 0*308 0.170 0.579 0*605 0.527 0.395 0 155
0,248 0.640
0.560 0.690 0.720 0.118 0.561 0.200 0.612 0.241 0.660 0.248
0,660 0.321 0.710 0.361 0*715 0.815 0,397 0.155 0,015 0*861
0.425 0,261 0.627 0.309 0.169 0.578 0,609 0.526 0*395 0.156
0,249 0*641
0,561 0,690 0.720 0*114 0.565 0*190 0.608 0.245 0,659 0.250
0,664 0.308: 0.710 0*360 0.719 0.815 0,3'92 0.155 0,021 0.861
0.425 0.260 0.629 0.306 0.170 0.579 0.606 0.526 0.396 0.158
0,249 0.642 L
Figure 57. Listing of Data Cards for Heat Transfer Measurements of
Sample Problem.

-160- — ______________________ ___ANALYSIS OF HEAT TRANSFER DATAF
RUN NUMBER R-19-8 - PAGE I
DATR -TAKEN 1-9-61
DATA PROCCESSED 1-20-61
REM.RKS -
3/4 INCH TEARROR PS AT 8 INCH SPACII G. PFRCRLE= PURPLE
7 —7 —-----— DATA OTHER THAN RECORDER THERMOCOUPLE RERDI-NGS-l-... —------....v... -----— VT P' —-- fO-T —' ------ El --— T -- -; —-^ — --—' —----------------— N lRTdf A -------- gM A -----— S rrfa100 Eb~ C P
4. 42.400 2. -9.200 21.830 1.050 0.300 0.552 73.500 6 0.S00.__ 42.400 0. 9.230 21.:10 0.000 0.300 0.556 73.600 0.000
MERAN VALUE 4. 42.400 2. 9.215 21.920 1.050 0.300 0.554 73.550 0.000
STANDARD DEVIATION 0. 0.000 O. 0.015 0.010 0.000 0.000 0.002 0.051 0.000
CALIBRATION READINGS FOR RECORDER THERMOCOUPLES. —--—.._________-_ —______ ______ ____
TC NUNBER 19 20 12
1.400 2.200 1.675 -
MERA VALUE 1.400 2.200 1.675
STDO. DEV. 0.000 0.000 0.000...__...,,_____________________________TOC__OUPL _____________ ______....,________________________
THERMOCOUPLE READINGS, MEAN. VLUE., AND STANDARD DEVIATIONS
IR 2R 3R +R 5R GR 7R SR 9R 20L 19L 18L OR 17L 16L 15L 11R 12R 13R 14R
0.559 0.691 0.717 0.111 0.567 0.192 0.609 0.242 0.658 0.638 0.2q5 0.158 0.250 0.394 0.528 0.605 0.661 0.319 0.710 0.360
0.560 0.690 0.720 0.118 0.561 0.200 0.612 0.241 0.660 0.641 0.249 0.156 0.248 0.395 0.526 0.609 0.660 0.321 0.710 0.361
0.561 0.690 0.720 0.114 0.565 0.190 0.608 0.245 0.659 0.642 0.249 0.158 0.250 0.396 0.526.606 0.664 0.308 0.710 0.360
-0.559 0.690 0.719 0.115 0.565 0.194 0.610 0.242..t4T.2f8 0.157.95 0 5 666 0
--— o o —-- o — ----, —oo —. —-r --------------------------------------------------— o —-0 —----------- ------------—. —THERMOCOUPLE READINGS. MEAN VALUES, AND STANDARD DEVIATIONS CCONTINUED)
i.A 5'R.. -... - ---- K —-- 16-4L 1I —-— t' -.. -7L —— 11 —-— R —-1-''R 20R
0.718 0.815 0.578 0.165 0.305 0.626 0.260 0.424 0.390 0.154 0.020 0.860
O. 7T -0 — 8-ft C15 o75TE-1T7O-b.08-T30;b —ff. 25w —~.:2;-f.S -59 T6b~- - 4- -------------------------------------------------------------
0.715 0.815 0.i57 0.169 0.309 0.627 0.261 0.425 0.397 0.155 0.015 0.861
— 0.719 0815.0.579 0.170 0.-3O9 *.629 0.260 0.425 0.392 0.155 0.021 0.861
0.... 0'8,;15:t.F]'~'Oj~-~' 0.578 0.168 0.307 0.6.28 0.260 0.424 0,393 0.1SS 0.017 0.861
- -- - O. -O i s - o-; l 6 d. 3d-7-'o 2 — 0 -— 4-. —---- -- -- -0. —67 —------------------------------------------------------------
- _ E4EI —-_ o 6 42 —------— _ —-------------- - ------ - ---- ----— _ —-- - - - - -__ -==._ _v =8RAW PATB-FOQ-RVg -1PAGE T9-8 T
ROTAMETER 4 READING = 42.40 PAGE I
VOLTMETER READING 9.21 CMETER SCALE = 2)
OUTLET WATER THERMOCOUPLE 0.554 MILLIVOLTS
RMB T AI = RSR JH9EM~p4QC= QU~tM Q1J2 _ ~___=__ _ — __ —-----— ====-=_=======-==-=========_===
MILLIVUOLTS ACROSS 5000 AMP SHUNT 21.820; —-' ---- - - - - - -..._...__,=_............................... I IM E R=S ___===
1R 1.49. 1914 0.0539
2R 5,41 2. -3 0.61-
3R 9.42 2.065 0.0624
5R 17.41 1.920 0.0685''-="="= -"= —=g='=C —- "-'-' —-— i-rTr~"~= —-== "T3G 5r~"~"~""=oZF —- --— ===-========== = —"" — ------------
7R8 25.34 1.962 0.0558
9t_ 33.30 2.008 0.0389
~2OL 4. 173O 1 H69, O1WfirD== = ====== =====C=============
19L 35,30 1.61 0.0155
1OR 37.29 1.62O 0.015s
-~ I7fL 38.25 1 t. Y~s' """ " — ""~"~"s~758 0-.01- 1 —-. —5 -(02.9sD- - ------ - ----
ltL 36.a 1. es3 oot s1
18R 41.20 2.011 0.0390
-~ — r i4H... sj.='==' i "'"====== ======^m ==3=
13R 49. 22 2. 057 0,0624
R158 57.28 2,g O. 0719
14L 34.30 1.932 0.Q00Q
12L 38.28 1.675 0. 0000
IOL 36.29 1.630 0.0000.......... -- —..-. -.V —-~.. —-. —-- ----------- - - -—.
17R 0.00 1.757 0.0000
_ - "- TR - --- _ 8-..__0o = __0_o- ----- _ ------— = —---- ---- ---
19R 0.00 1.400 0.0000o..-..NL — ET'T PRESSURE'3- ENK —3'F0'Ri0WiUT-T
OUTLET PRESSURE = 0.000 INCHES OF KING. PURPLE FLUID
$, -
Figure 58. Computer Analysis of Heat Transfer Data for Sample Problem,
Pages I, II, III, and IV.

__________ —----- -- ---- --- -- SUMMARY OF-PROCESSED DATA FOR RUN R-19-B _ --------- — _
PAGE III
FLOW RATE = 12.58 GPM.P I
INLET WATER TEMPERATURE = 45.89 DEGREES F.
OUTLET iWTER TEFMERTURE = 57.49 DEGREES F.
AMBIENT AIR TEMPERATURE = 79.69 DEGREES F.
ELECTRIC CURRENT = 2182.0 AMPS
OVERALL VOLTAGE DROP = 9.31 VOLTS
CHANNEL POSITION ANGULAR DISTANCE DISTANCE DELTA T H RE/1000 THEORETICAL H/HTM
(DIAMETERS) POSITION TO FROM (DEG. F) CBTU/HR- H
(DEGREES) PROMOTER PROMOTER DEG-SO FT) CSAME RS H)
IR 1.49 0.0 9,20 0.00 43.5 1155.3 28.433 948.5 1.184
- 2R 5.41 0. 0 5. 28.00 48.0 1050.4 28.767 -6:~...-:07~
3R 9.42 0.0 1.27 0.00 48.4 1042.1 29.110 966.2 1.068
4R 13.40 0.0 5.25 2.71 24.2 2053.0 29.452 932.9 2.103
5R 17.41 0.0 1.24 6.72 40.8 1230.5 29.798 965.1 1.261
6, 22.17 0.0 4.44 3.52 25.3 1967.8 30.211 946.1 TT- 2.0
7R 25.34 0.0 1.27 6.69 41.2 1221.4 30.487 975.9 1.251
-~~ —---------- ---- -— g~ —- --- -7- ---------
8R 29.32 0.0 5.25 2.71 25.7...1942.4 --— 0.-3- 5 9-5-.-' 6. —--- T..9.
9R 33.30 0.0 1.27 6.69 41.7 1208.9 31.184 986.9 1.239
20L 34.30 0.0 0.27 7.69 40.8 1234.7 31.273 986.8 1.265
19L 35.30 0.0 7.23 0.73 24.8 2009.1 31.361 962.2 2.058
f8C —------— 36. -xs —------— 0.-0 —-------------------— r 7-2 —------- 21.-6 —------------- 2...... ~E1..."36.27..0-.0 —— ~o. 6.24 1.72 ~21.6-3....- ~-..3].-48 - 957.T..? -
10R 37.28 0.0 5.25 2.71 24.5 2032.7 31.536 964.3 2.083
_ —----------—._. —-_ — Tg^.7 2 ~ —— ^~ —5 ----- 4. ------ f —------ ------- --------— ~ —
17L 38.2T 0. 4..5 3.71-..2 - 15-.. T.-6:4.f-...T-.
16L 39.28 0.0 3.25 4.71 35.3 1420.7 31.712 984.7 1.456
t5L 40.28 0.0 2.25 ~ 5.71 38.3 1312.2 31.801 99.7 1.344
11R 41.20 0.0 1.33 6.63 40.4 1247.7 31.883 995.1 1.278
12R 45-.7 0.0 5.~22 ".-Y25.81"W"35.- 32745 " 97.7 f-.6T
13R 49.22 0.0 1.27 6.69 40.8 1235;2 32.598 1006.1 1.265
-4R8 5 —------— 5-3. -6 --------—. 6 55'0 — -~ —--- -----— 7 — 26 --- ~' —-- 291 —-----— 9TT-.T —---— ~32. ~ —------ 7.5 --------- ---------
15R 57.28 0.0 0.00 6.79 39.7 1270.3 33.322 1014.7 1.302
1GR 62.47 0.0 0.00 11'.98 42.6 1186.5 33.792' 102.S8 1.21t6
14L 34.30 120.0 0.27 7.69 38.3 1312.2 31.273 982.9 1.344
13. L- 362..69. 1 — 2-0.0- 6..24..- 1.72 2~1. ~9 -2. —6 —.6- 44- 958.-.32. —-----
12L 36.28 120.0 4.25 3.71 26.7 1871.5 31.624 969.2 1.917
11 L..34 -.... 0. -..269.?..-T 073 15....3...T. ------— 997. --— 1
10L 36.29 240.0 6.24 1.72 25.1 1984.3 31.448 964.0 2.033
9L 38.26 240.0 4.25.71 31.4 1594.9 31.624 977.0 1.634
~ EA~ VA~LUES 1877? 8 1.08 - 77
"""NT —------- -CrL —HT-I — -------------------------------- - --- ------ --- ---------------------------------
INTEGRATED LOCAL HEAT INPUT _ 70.796 Mll BTU/HR
OVERALL HEAT INPUT = 69.328 MM BTU/HR
HEAT REMOVED BY WRTER = 73.012 MM ETU/HR
HEATl LOSS ON BASIS OF ITTEGRATED~LOCAL INPUT = -2.216 tI,: STU/Hr
____PER_ CENTLOSS =__ -3. 04
HEAT LOSS ON BASIS OF OVERALL INPUT = -3.684 MM BTU/HR
PER CENT LOSS = -5.05
MERN VR TO THE.14 = 1.0699 MEAN PR TO THE 1/3 = 2.0683 MEAN PHVSICAL PROPERTIES FACTOR = 9.92154
PAGE IV... ~~_ g-~BADGITINAL PROCESSED OD'TA FOR RUN R-19_-B
CHANNEL ___ TC) ______TB WA______TER TEP.
CMMBTU'/.R) CDEG. F) CDEG. F CDEG. F)
IR 50.240 89.7 116.9 46.2
2R 50.392 94.8 122.2 46.9
3R 50.425 96.0 123.3 47.6
4R 49.743 72.6 99.8 48.3
5R 50.247 89.9 117.2 49.0
6R 49.82:0 75.2 102.5 49.9
7R 50.299 91.7 119.0 50.5
8R 49.86S 76.9 104.1 51.2
9R 50.355 93.6 120.9 51.9
20L 50.334 92.9 120.2 52.1
- 1 -,-9Lo 49.- 75 77.1 104.4 52.3
18L 49.785 74.0 101.3 52.5
- OR 4-;877 77.2 104.4 52.6
17L 50.049 83.1 110.3 52.8
16L 50.202 88.3 115.6 7 53.0
15L 50.295 91.5 118.8 53.2
11R 50.3599 93;. 121.05 —-. T-.-..- 7 —-5- 4
12R 49.957 79.9 107.2 54.1
3R —------- 1 50.4 -------- --------- 5 — - ~5.5 2.9 -54 —------
14R 50.008 81.7 108.9 55.5
15R 50.424 96.0 123.3 5. 3
16R 50.536 99.8 127.1 57.2
----------— 4L ---- ------ -0-4 ~ T —------------. —--
13L 49.796 74.4 101.7 52.5
- ~. C --------. ~ "- 4 —----- 4.-4- 797T5 —- 1 —9 — -TO2... —-- "".8 - "711L 50.320 92.4 119.7 52.1
10L 49.890 77.6 104.9 52.5
9L 50.083 84.2 111.5 52.8
MEAN VALUES 50..1& 87.7 115.0 51.7
---------------------------------— _ —-g -r-e —5n —
Figure 58, (Continued)

-162where EMF2 = upper reference emf
AZAR2 = AZAR reading for channel recording upper reference emf
EMF1 = lower reference emf
AZAR1 = AZAR reading for channel recording lower reference emf
azar = arbitary AZAR reading to be converted to emf
emfi = emf corresponding to azari
For example, in the data of Run R-19-B
2,200 - 1 400
RANGE 081 - 0.9478 millivolts
0,861 - 0*017
ZERO = 0 017 - 1,400/0.9478 = -1,460
Thus, for channel 9R, for example, where the mean value of the
four AZAR readings is 0.658 the emf is given by
emf 0.9478 (0658 + 1^460)
=- 2.008 millivolts
All of the mean values of each item of raw data were printed on
Page II of the computer analysis. The raw data were converted into
engineering units and presented on Pages III and IV- At each axial
and angular position on the tube the following items are presented on
Page III of the analysis.
III-1. Channel number
III-24 Distance from beginning of heating
III-3, Angular position on the tube
III-4* Distance to downstream promoter (tube diameters)
III-5. Distance from upstream promoter (tube diameters)
III-6, Wall temperature - fluid temperature (deg F)

-163111-7. h(z), local experimental heat transfer coefficient
(BTU/hr-deg F-ft2)
III-8o Re,, local Reynolds number calculated using local
fluid temperature
III-9. Empirical estimate of h calculated using SiederTate equation based on local values of the physical
properties (BTU/hr-deg F-ft2)
III-10O h(z)/h0 where h(z) is given by item III-7 and h0 is
the overall integrated value of III-8.
On Page IV of the computer analysis the following items are presented,
IV-lo Channel number
IV-2. Local rate of heat flux (BTU/hr-ft2), i.e., q(z)
IV-3 Inside wall temperature (deg F)
IV-4o Outside wall temperature (deg F)
IV-5o Water temperature (deg F)
Mean values of III-, III-8, III-9, IV-2, IV-3, IV-4, and IV-5
were obtained by multiplying each local value by the fraction of the total
tube length it represents (listed on Page II of the computer analysis as
WEIGHT FACTOR) and summing. In addition, local values of ([/Mw) and Pr
were integrated in this manner to obtain mean values.
The overall heat input is obtained by
L
Qin = 2J qj(Z) dZ (70)
The heat removed by the water was obtained by
Qout W c (Toutlet - Tinlet) (71)
and, thus, the heat losses were calculated.

-164The application of Equations (54), (56), (58), and (45)
to calculate Twall(z), q(z) Tf(z), and h(z) can best be illustrated
by an example. Consider the data for channel 9R. This thermocouple
is located 33.30 tube diameters from the beginning of heating; it is
at the 0 degree angular position (i, eo where the large majority of
the thermocouples were located). This thermocouple is located lo27
diameters upstream from one promoter and 6.69 diameters downstream
from another promoter. The outside tube temperature which was measured
was 120.9 deg F.
According to Equation (54) the inside tube temperature is
given by an equation of the form
Twall(Z) T (z) — A2(0 12) - ( (A-ll)
And, according to Equation (56) the local rate of heat flux is given by
an equation of the form
q(z) = A4(l + yTb) I2 (A-12)
where
3,41276 2 b2 b ( -a)b2A PO -~~^ ^ ~ab In - - (A-13)
2 2 2 (b a2 K2 Ia 2 L
2 2 (b2_ a2)2 K0
A5 = L2 62 P2 XJ (3 2 (b a) (A-14)
3.41276 po
A4 = (A-15)
2C (b2 - a2) a
and
X 1 + Tb(A-16)
1+ Tb

-1655The electrical resistivity data p and y for type 304 stainless steel were obtained from the Allegheny Ludlum Steel Corporation(l)
Thermal conductivity data K0 and P were obtained from Shelton(37)5
The values of the parameters in the above constants are
a = 0,5025 inches
b = 0,624 inches
p 2,265 x 10-6 ohm-ft
0
K0 = 85 BTU/hr-deg F-ft
y = 0,00062 (deg F)-1
= 0 04000517 (deg F)-1
From which the numerical values of the constants are
A2 = 5-6259 x 10 (deg F)/amps2
A = 9,8891 x 10-15 (deg F)/amps
A4 - 94,8389 x 10-4 BTU/hr-ft2-amps2
For the inside wall temperature of 120,9 deg F
= 1 + 0,00062 x 120,9
1 + 0,000517 x 120 9
= 1 0117
Since the electric current was 2182 amps, then
I2 4 8176 x 10 amps2
and

-166(98891 x o-15)(8178 x 106)2
Twa-11 = 120.9 -(5.6259)(4.8178) (9 889 x 6 )(4.878 x o
Twl - ^ ^ ) (.o 62 )(1.o749)
120*9 -27,10 - 0,20
= 93.6 deg F
q(z) - (9,8398 x 10-3)(2.0749)(2,182 x 103)2
= 50,355 BTU/hr-ft2
Tf = 45.89 + 3330 (57749 - 45,89)
64.02
= 51,9 deg F
h(z) = 50,3555
93.6 - 51,9
= 1208,9 BTU/hr-deg F-ft2
The mean value hm/h0 of the ratio of the local heat transfer
coefficient to the overall value for the empty tube h(z)/ho was obtained
by fitting a polynomial to the local values as a function of the distance
(in tube diameters) downstream from the nearest promoter. For this
purpose, values downstream from the first and last promoter were not
considered, The following are those which were used for the data of
run R-19-B.
Diameter from Beginning Distance from h(z)/h0
of heating, z Nearest Promoter, x
22,17 3*52 2.016
25*34 6.69 1,251
29 32 2.71 1.999
33.30 6.69 1.239
34.30 7.69 1.265
35530 0,73 2,058
36.29 1, 72 2.367
37.28 2*71 2,083
38,38 3.71 1.696
39.28 4.71 1,456
40. 28 5*71 1. 344
41.20 6,63 1.278
45.27 2.74 1*983
49.22 6,69 1.265

-167The polynomials which were fitted were of the form
h(x)/ = a + a x + a2 x2 + a x3 (A-17)
where the coefficients were obtained by the method of least squares,
Values of the coefficients for a first, second, and third order polynomial
and the above data are shown below with the integrated mean heat transfer
coefficient for each case,
Order a0 a1 a2 a3 hE/ho
1 2.433 -0,1719 1 745
2 2,414 -0,1605 -0,001311 1 744
3 1 935 0 3824 -0,1594 0,1292 1.718
The value hm/h0 = 1.745 is the value which was used,

APPENDIX B
DERIVATIONS
Derivation of the Conduction Equation
The solution of the conduction equation subject to the
correct boundary conditions is essential to the successful measurement
of local heat transfer coefficients by the experimental technique used
in this investigation, Because of the importance of this equation, it
will be derived in this section; the effect of the various simplifying
assumptions made in the solution will be shown to be negligible.
A diagram of the longitudinal cross-section of the tube -wall,
insulated at the outside, is shown below to illustrate the nomenclature.
Heat generation is in the shaded area, In this analysis the longitudinal distance from the center of the heated section will be denoted by
the symbol y, Thus, the heating section begins at y = -L/2 and
extends to y = +L/2, The inside radius is a; the outside radius is b.
r
/// HEAT GNERA TION X//
\.y = -L/2 Ly= -L/2
F-ay +L/2
Flow of Flux
1-r =0
-y O +y
-168

-169When an electric current passes through the tube wall, heat
is generated, This causes a temperature gradient to be developed so
that the heat will flow to the inside wall,
The differential equation (i.e, the conduction equation)
which describes this process is well known and can be derived as follows,
An annular element of the tube bounded by cylindrical surfaces at radius
r and r + dr and bounded by planes perpendicular to the axis at y
and y + dy is considered as shown below,
dy
Expressions for the.heat entering and leaving the element
across each plane can be.-.written as
aT
Input at r -K(27t r dy) ar (B-la)
dr
Output at r + dr -K(2J r dy) -K (2g r dy) ] dr (B-lb)
r~ r /r r
Input at y -K(2J r dr) a - (B-lc)
6T. _T (B-ld)
Output at y + dy -K(2g r dr) [K (2Y r dr) T] dy (B-id)
by by ay
These expressions are then substituted into the formula
Input + Generation - Output = 0 (B-2)

-170which is simply the first law of thermodynamics for a steady-state
system. The resulting differential equation is
1Fa rK IFK lT + A =O0 (46)
r Or L Or J by L y
where T = temperature (deg F)
r = radius (ft)
y = longitudinal distance from center of
heated section (ft)
A = rate of heat generation per unit volume of tube
wall (BTU/hr-ft3)
K = thermal conductivity of the tube wall (BTU/hr-deg F-ft),
The next step is to derive an expression for the rate of heat
generation A in terms of the electric current and the physical properties
of the tube. The rate of heat generation per unit volume dV of the tube
wall is given by
A 41276 (dI) R
dV
where.d = electric current passing through the infinitesimal
volume dV (amps)
R = electrical resistance of the volume (ohms),
However, the volume of any annular segment of the tube of
length dy is
dV - 2r r dr dy (B-4)
The current flowing through the element is
dI = 2t r J(r) dr (B-5)

-171where J(r) = current density at r (amps/ft2)0
The electrical resistance of the element is
R =- y (B-6)
2i r dr
where p = electrical resistivity (ohm-ft), The resistivity can
usually be expressed as a linear function of temperature so that
= Po(l+ yT) (48)
with << 1.
Substituting the above expressions into Equation (B-3) the
following is obtained,
A = 3.41276 J (r) p (1 + XT) (B-7)
By assuming that the voltage gradient dE/dy is independent of radius,
then
J(r) - dE/dy (B-8)
po (l+yT)
I Pom (B-9)
P (1 + 7T) (b - a2)
where Pm is an average value of the resistivity determined using the
condition
b
I = 2m S; J.(r)dr (B-10)
a
b
1= I pm.2m r dr
I = a.. I P r~- (B-11)
(1 + yT) ~ (b - a2)
a o

-172b
2 I Pm r dr
Thus, p (b2 - a2) ~ 1+ ~T(r) (B-12)
p ( -a I + 7T(r)
a
or (b2 a2
or P (49)
a 1 + yT(r)
and A,476 1)
p (1 + yT) (b2 - a2)
Equation (46) and the expressions defined by (47) and (49)
represent a complete derivation of the conduction equation for heat generation in an infinite hollow cylinder, This is a non-linear partial
differential equation, The boundary condition at the two cylindrical
surfaces r = a and r = b are
- (b,y) = 0 insulated surface (B-13)
8r
T (a,y) = g(y) arbitrary function of (B-14)
y depending upon the
inside heat transfer
coefficient
The boundary conditions at any given pair of planes perpendicular
to the axis of the tube (which are necessary in order to completely specify
the problem) are more complicated since, in reality, the hollow cylinder is
not infinite, but is composed of a finite length section in which heat is
generated. This section is bounded on each end by a section in which no
heat is generated and into which there will be some conduction of heat,
However, the problem may be treated in the following manner:

-1731. Assume that axial conduction may be neglected except within
a certain, short distance from each end of the section of the tube in
which heat is generated, (This implies that aT/yy is constant. )
2. Show that with the assumption of step 1 the problem is
reduced to that of solving a non-linear, ordinary differential equation,
Solve this equation and show that the effect of variation of physical
properties of the tube wall with temperature is small, although computable. (This is done in Part 2 of this appendix)
3~ Since the non-linear nature of the differential equation
can be shown to be of second-order importance, solve the linear form of
the partial differential equation which allows for heat conduction into
the non-heat-generating portion of the tube and show that, indeed, axial
conduction into this portion of the tube can be neglected within one
tube diameter from each end of the heated section, (This is done in
Part 3 of this appendix,)
4. Solve the linear form of the differential equation for an
infinite hollow cylinder with an arbitrary temperature distribution at
the inside wall. Thus, determine the effect of axial conduction caused
by a fluctuating inside wall temperature distribution and show that this
effect is negligible, (This is done in Part 4 of this appendix )
Effect of Temperature Dependence of Physical Properties of the Tube
Assume that the thermal conductivity and electrical resistivity
of the stainless steel tube are linear functions of temperature so that
K =-K0 (1 + BT) (50)

-174and p = p (1 + T) (48)
In order to make Equation (46) solvable, neglect axial conduction (TT/4y'= constant) so that the equation becomes an ordinary,
but non-linear differential equation.
d2T + 1 dT + D L + 3 41276 I Pm-
dr2 r.dr (1 + T) Lrl KO~ (1 + yT)(1 + 2T) (b -;a )
(51)
with the boundary conditions
dT (b) = 0 (52a)
dr
T(a) = T(a) (52b)
simplify by letting
3*41276 i2 -2
A0 -O - 2 2 (53)
PO (b -a )
and the solution as suggested by Clark(8) is
An 2 b3\ (b2 - a2)l
T(b) - T(a) = b b2 in
2 Kb- n 2 j
+ Tb (b-a)L (54)
[ 2 Kb_ L6 (1 + yTb)(l + PTO)where the subscript b indicates that the quantity is evaluated at a
temperature T(b). Notice that the first part of the solution is simply
the solution to the linear form of the differential equation, while the
last part is a "correction" to take care of the effect of the temperature
dependence of K and p,

-175The temperature difference T(b) - T(a) caused by the
generation of heat in the tube wall is referred to as Ageneration
As was shown in Appendix A for the particular dimensions
and physical properties of the tube used in the experimental portion
of this investigation, Equation (54) is of the following form.
T(b)- T(a) = 5.62591 ] + 9.89 x l L1000 (1 + yTb)(l + Tb)
(B-15)
where
1 +- 7 (A.l6)
1 + Tb
Since, for Tb < 250 deg F
1.0 < (1 + Tb) < 1,20
1,0 < (1 + Tb) < 1.20
10o < < 1*03
The ratio of the "correction term" to the non-corrected temperature
difference is approximately
Temperature Dependence Correction 9.8891 10-3
x 10
Total Temperature Difference 5.6259 1000
F3 12
1.75 x 10-3 [ 10
Thus, at the maximum value of electric current (I = 3000 amps)
the correction is approximately 1,5 per cent. This correction was made
in calculation of all inside wall temperatures from the measurements of
the outside wall temperatures. However, the effect seems small enough
so that any conclusions (for example, on the importance of axial conduction)

-176made on the basis of solutions to the linear partial differential
equations should be valid.
Effect of Axial Conduction into the
Non-Heat-Generating Portion of the Tube
Axial conduction can arise in one of two ways: 1) through
conduction into the non-heat-generating portion of the tube; and 2)
through presence of a nonconstant axial temperature gradient at the
inside wall. In this section of the appendix'-th first way will be
cons-i&ered; in Part 3 of this appendix the second way will be considered.
For this analysis consider the linear form of Equation (46)0
a2T +1 aT +aT +oA = (B-16)
r2 rr y y K
A solution to this equation is required for a hollow cylinder
of inside radius a and outside radius b, where (A/K) is independent of
temperature and heat is generated in a finite section of the tube between
y = -L/2 and y = +L/2.
Thus, for -L/2 < y < +L/2 A = A0
-o < y < -L/2) (B-17)
+L/2 < y < + A= 0
The boundary conditions are
T (a,y) = 0 (B-18a)
- (b,y) = 0 (B-18b)
or
aT (0,r) = 0 (B-i8c)
by
aT (+,r)'= O (B-18d)
ay

-177It is obvious that the solution will be symmetric about
the center of the heated section (y = O) so that it is only necessary
to obtain a solution for 0 < y < oo
Equation (B-16) may be placed in dimensionless form by
defining the following dimensionless variables.
y 2y/L (B-19)
r = 2r/L (B-20)
a = 2a/L (B-21)
b = 2b/L (B-22)
T 4T/(L Ao) (B-23)
The resulting dimensionless equation is
+ + A 0 (B-24)
r or y
where for
0 < y< A =l
(B-25)
1 < y < oo A= O
with the boundary conditions
T(ay) = 0 (B-26a)
( y) = 0 (B-26b)
ai (o,0) = 0 (B-26c)
i (cor) - 0 (B-26d)
oy
Henceforth, the notation Ty and T- will denote aT/8y.and aT/=r,
re spectively,

-178Divide the cylinder into two regions, I and II (the heatgenerating and non-heat generating regions), as shown in Figure 59.
Then set T(r~,l) = g(r) at the boundary between I and II, where g(r)
is some unknown, arbitrary function,
In Region I
aT +1 a T + + 1 (B-27)
-2 6 m2
br r arb ay2
with the boundary conditions
^A (by) = 0 (B-28a)
T (ay) = (B-28b)
T (ro) = 0 (B-28c)
Ty (rl). - g(r) (B-28d)
In Region II
a +T 1 + - a0 T 0 (B-29)
ar2 r ar ay2
with the boundary conditions
T? (,) = 0 (B-35a)
T (1,3) = O (B-30b)
T, (ro) = 0 (B-30c)
^T (r,1) - g(r) (B-30d)
Consider, first, only the solution for region I.
Let (B-31)
Let I = V - (2 - a2)/ (B-l)
Then
-2- = o (B-52)
3r2 r or oy

-179s T/ar = / = o
r b
bT/y= g(r) y =.o
r = a
T B =1 T = o0
y=0 y=l y==
Region I Region II
r =
Figure 59. Boundary Conditions for Solution of Conduction Equation
Considering Axial Conduction into the Non-Heat-Generating
Portion of the Tube.

-180subject to the boundary conditions
^V (b,) = /2 (B-33a)
5 (^,a) = o (B-33b)
7? (^,o) = 0 (B-33c)
Vs (,il) = g(r) (B-33d)
This can be solved by the method of separation of variables.
Assume V = r(r) () (B-34)
T2 d238 d"R A2M2
Then r d2 + r - r = 0 (B-35)
d. di?
d~ + X y = 0(B-36)
dy2
For X positive
q+ = al Io (Xre) + a2 K0(?)) (B-37)
+) = bl sin ( ) + b2 cos (,y) (B-38).2
For X negative
- = al J0 (r) - a2 YO(B-39)
~_ = b1 sinh ( x) + b2 cosh ( ) (B-40)
2
For X zero
o = al in r + a (B-41)
= bl.+ b2 (B-42)
Attempt to find two solutions, V1 and V2 such that
= V1 + V2 (B-43)

-181with V1 satisfying the conditions
V (bY) = /2 VY (0) = 0 (B-44a)
to
V (ay) = 0 V (r,1) = 1 (B-44d)
y
and with V2 satifying
VA (by) = 0 V (i,0) = 0 (B-45a)
to
V (,y) =0 V. (Fr1) = g(r) (B-45d)
y
M2 AA
The solutions for X = 0 will satisfy the requirements for V1 if
bl =1 a1 = 2 /2
(B-46)
b2 a0 = n na
or, in other words
A2 u /IVI (B-47)
1 = (b /2) ln (/a) (B
The solutions for X negative will satisfy the requirements
for V2 if
b 0 l a, Yo(. a)
b1 al = O(n) (B-48)
A -\ A\ A
b2 = n sinh kn 2 0(Xna)
2 n n
In order to satisfy the remaining boundary condition, Xn will be chosen
such that,a ~ ~k ~'~n~) ~ ~ ~i b = 0 (B-49)
Yo (kna) J1 (n) - J0 (na) Y1 (nb) = (BFor convenience it will be useful to define a new parameter
X = X a (B-50)
n n

-182so that the relation above is replaced by
Y (X) (X b/) - J (Xn) Y ( b/) 0 (B-51)
On 1 n 0 n 1 n
Thus,
All~~~ ~cosh X,~/'
V2 Y o(n r/) - JO (Xn) Yo (Xnr/W) n/a (B-52)
Xnsinh Xn/a
At this point it will be useful to define the function
A(knr/ = o(~) o= Yo( n) J (r/a) - JO(n) Yo (r/) (B-53)
It can be shown that the A(Xkn,/a) form an orthogonal set of functions
so that the arbitrary function g(r') may be expanded in the following
manner
00
g(r) = a A A(n'(/) ) (B-54)
n-=
Therefore, let
n=l _ n sinh X /a
n1al n
and
00
cosh Xn y~/~
V = (b /2) ln (r/a) + n An A(Xn'r /) (</a) sinh, (-6)
n=l
Thus,
00
- ( 2) + nA(Xn,/E) cosh /a (B-57)
T 1 n +I n=l ( nn/) sinh li (B-57)
The only remaining problem is the selection of the A.

-183The solution in Region II may be written by inspection as
= A(x /$) eP (An / ) (B-58)
n=l n
n
Now, equate the temperatures at y = 1I
T(1) = T(1) (B-59)
and
00'2 r2 1 2 ^2 V\2 / cosh X,/' _
b In - (r - a ) + A A(kn'/~) cash xn/a
^ (X,/a-) sinh Xn/
-- An A(Xr/)
n=l
= _- A (B-60)
n-lLet
~ oo~~~~~~0
b /2 in (r/a) - (r - a)/4 = Dn A( (B-61)
n=l
where the Dn are to be determined from Sturm-Liouiville theory,
Then
A cosh' /-'
D ^ An cos..n/ A (B-62)
n n/a sinh Xn/a X/
A/V
The solution for An becomes
A = - (Xn/a) sinh )n/a exp (- Xn/) Dn (B-63)
The expansion for Dn which satisfies Equation (B-61) is
D,r 2 _ (B-622
nD 2 21 -(B-62)
Dn =- 2 t2 1)
n n

where
n = Yo(Xn)/Yl(x/) = Jo(X)/Jl(X/a) (B-63)
Thus
A - (sinh Xn/a) exp (-Xn/a) 2T a
A I ~)- ^(B-6.4)
-Xn(n - 1)
Substituting into Equations (B-57) and (B-58)
2 4
00
2 2 1 e xp['n (y - 1)] + exp [- )]]A( )
n-i a aY " n
n=i n n
(B-67)
and
00
na ^ [I- exp (-2X/a)]
T >[ - [1 -2~n~ ) — exp [-(Xn/a)(y 1 i)]A(r/a)
n2l kn [ n - 11
(B-68)
Replacing the dimensionless variables by the physical variables, the
following solution is obtained. It may be observed that the solution
(B-67) is valid for y < 0 and is symmetric about y = 0,
For jyJ < L/2' 2AO l b 2 [ ()/r) 1
T(r,y) A ()n ( _ (-L) - 2J
2K a a 2 a
00
+ 2(2l) Lexpn (y- L/2)] + exp[a (y + L/2)]A(Xn r/a)}
n=l n n
(B-69)

-185for y > L/2
ArO a
2Kn
x A(Xn,r/a) (B-70)
where A(Xn,r/a) = Y0(n) Jo(nr/a) - J(Xn) Yo(nr/a) (B-71)
with the Xn satisfying
Y0 (n) J1 ( b/a) J0 (in) Y1 (X b/a) (B-72)
and
Jo(Xn) Yo(Xn)
n ^() - (n)(B-73)
J1(Xnb/a) Yl(Xnb/a)
The first three roots of Equation (B-72) were determined for
values of b/a equal to 1,100, 1.2418, and 1,5000. The results are presented in Table V. In Table VI the corresponding values of the function
A(Xn,r/a) are tabulated,
It should be noted that the first part of the solution (B-69)
outside the summation is simply the solution to the linear, ordinary
differential equation, The part inside the summation accounts for
axial conduction into the portion of the tube where there is no heat
generation,
The dimensionless form of the solution (at r = b) is plotted
in Figure 60, It is apparent that the "end effect" extends for a distance of less than one or two diameters from the beginning or end of

-1861.0
0.8
g6
_ 0
I- 0.4
0.2
-Q8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
DISTANCE FROM END OF HEATED SECTION, Y -L/2 (TUBE DIAMETERS)
Figure 60. Dimensionless Form of Solution for Axial Conduction
into the Non-Heat-Generating Portion of the Tube,
Showing that the End Effect is Negligible.

-187the heating section and outside this region axial conduction into
the non-heat-generating portion of the tube is negligible.
TABLE V
FIRST THREE ROOTS Xi OF EQUATION (B-72) AND
ASSOCIATED VALUES OF i FOR b/a = 1.100, 1.2418,
AND 1.5000.
b/a = 1,1000 b/a = 1,2418 b/a = 1,5000
X1 15 4061 6,2146 2,8899
2 47.024 19. 3979 9.3448
3 78. 4804 32. 4268 15.6602
1.0479 1,1092 1.2054
2 -1.0487 -1, 1138 -1.2228
U3 1.0488 1,1142 1.2240
Effect of Axial Variation of Inside Wall Temperature Gradient
It was shown in Part 2 of this appendix that longitudinal
conduction into the non-heat-generating portion of the tube can be
neglected except for a short distance within one diameter of each end
of the heated section, In this part of the appendix, it will be shown
that the effect of longitudinal conduction caused by any axial variation in the inside wall temperature gradient is also negligible.
For this analysis a solution will be developed for the
general case where T(a,y) is not zero, but some arbitrary function
g(y). This solution will then be interpreted in terms of the parameters
of the experimental apparatus used in this investigation,

-188TABLE VI
THE FUNCTION A(X,r/a) FOR b/a = 1,100, 1.2418,
AND 1,500 AS AFUNCTION OF (r- a)/(b - a)
r -a b/a = 1,1000 b/a= 1,2418
b-a
A1 A2 A3 A1 A2 A3
000 0 0000 0, O 0,.00 0,00000 0,00000 0,00000 0,00000
0.100 -0 00o63 -000610 -0, 00570 -0.01515 -0,91466 -0,01370
0,200 -0,0124 -0.01083 -oo00803 -0.02962 -0, 02585 -0 1917
0,300 -0o01 -0-,01317.-0,00566 -0o.04309 -o,03126 -0,01345
0,400 -0,0234 -0.01264 -0,0.0002 -0.05530 -0.02987 -0.00009
0,500 -0, 0281 -0,00939, 00558 -0,06601 -0,02214 0 01303
0,600 -0,0320 -0, 00414 0.00788 -0, 07502 -0.00983 0,.01834
0, 700 -0.0352 0 00196 0,0055 7 -0o,08217 0. 00431 0.01295
0 800 - 0375 0.00757 0. oo000oo4 -0, 08734 0, 01725 0,.00018
0o 900 -0 0389 0.01150 -0, 00547 -, g09046 0, 02624 -0,01244
1 000 -0,0394 0,01291 -0:.00773 -0,09150 0 02944 -0,01761
r - a b/a = 1.5000
b-a A A A
b - a 1 - 2 3
0.000 0,00000 0,00000 0 00000
0,100 3 -0,0095 -002994 -0,02798
0.200 -0.05984 -0,05222 -0,03874
0 300 -0, 08621 -0.06256 -0 02696
0,.400 -0,1097 -0-05936 -0,00031
o 5 00 -0.13001 -0.04378 o* 02543
o 600 -0,14685 -0o01963 o, 03564
o 700 -.0.16006 0,00774 0.*02509
0,800 -0.16951 0,03248 0, o00058
0g900 -0,17515, 04951 -0,02340
1,000 -0,.17702 0.05553 -0, 033-17
If a solution can be obtained to
+2 + 1 T 6 = 0 (B-74)
ar2 r ar ay
(ie,, the linear equation for no heat generation in the tube wall)
subject to the boundary conditions
(by) = 0 (B-75a)
T(ay) r g(y) (B-7
T(a,y) - g(y) (B_75b)

-189and if the solution is added to the solution to (B-16) which
accounts for the effect of heat generation, then (because the
partial differential equations are linear) one will still have
a solution to (B-16) but with the boundary condition
T(a,y) = 0 (B-18a)
replaced by
T(a,y) = g(y) (B-76)
Represent g(y) by the sum of a linear term plus a Fourier
series expansion between the points yl and y2.
00
g(y) = T(a,y1) + m(y-yl) + Bn cos nt (y2 Y)j (B-77)
n=l (Y2-Y
Then simplify the writing of g(y) by replacing y2 y with S and from
Carslaw and Jaeger(7) the solution is
T(r,y) = T(a,yl) + m(y - Y1)
00
+' [ (Y -Y1)] I0(ncr/S)K1(nb/S)+ K0(n~tr/S)I1(ncb/S)
+; n En L S IO(nica/S)K1(nitb/S)+ KO(nTa/S) I(nJtb/S)
n=l
(B-78)
In particular
00
T(b,y) = T(a,y)+ m(y-y) + 5(na/S,b/a) cos [nc (y-yl)/S]
n=l
(B-79)
where o(na/S,b/a) is a "daamping function" equal to
5(na/Sb/a) I0(n cr/S)Kl(nmb/S) + K0(nrr/S)Il(nrb/S) (B-80)
I0(n~ta/S)Kl(ncb/S) + K0(n1a/S )I(ntb/S)
This function is plotted in Figure 61 for b/a = 1.24.

-1901.0
0.8
0.6
A I I I \b/a =1.24
0.4
0.2
0 1 2 3 4 5
na/s
Figure 61. Damping Function Defined by Equation (B-80)
as a Function of the Parameter na/S,
5(na/S, b/a) vs. na/S, for b/a = 1.24.

-191Of course ATgeneration must be added to the above in order
to account for the temperature difference between inside and outside
tube wall due to internal generation of heat, The general solution
will now be interpreted in light of the specific experimental conditions encountered in this investigation~
For water being heated at a constant rate as it flows in
an empty tube with constant heat transfer coefficient the inside wall
temperature distribution is linear, showing only the steady rise in
temperature caused by heating of the water. From Equation (B-79) it
is seen that this linear temperature change is transmitted through
the tube with only the magnitude of the temperature changed by a
constant amount ATgeeration along the length of the tube as would
be given by Equation (54)~ This is also true for a tube with a solid
rod in the center, or for any other geometry for which h(z) is constant,
However, when turbulence promoters are inserted at uniform
spacing, causing the inside heat transfer coefficient to vary periodically, then the inside temperature will also vary periodically and
this variation will be superimposed on the linear increase in the
temperature caused by the heating of the watero Thus, a typical
(although extreme) example of an inside wall temperature distribution
is shown in Figure 62~
The equation describing the inside wall temperature as a
function of y is
T(y) = Tf(-L/2) + m(y - L/2) + me + eriodic (B-8)

-192I3 E
w
-y 0 +
LONGITUDINAL POSITION, y
Figure 62. Sample Hypothetical Values of Fluid Temperature
and Inside Wall Temperature as a Function of
Longitudinal Position, Tf(y) and Ta(y) vs y, for
Rapidly Varying Inside Wall Temperature.
I'Q
OUTSIDE WALL TEMPERATURE
FOR NO DAMPING
090E 0 -
-0.85 DAMPED VALUE AT
-090 l ~ ~ ~7OUT'SIDE WALL ~ ~
0.895
-0.3 -0.2 -0.1 0 0.10 0.20
LONGITUDINAL POSITION, y
Figure 63. Effect of Neglecting Damping of Outside Wall
Temperature When Inside Wall Temperature is
Rapidly Varying with Longitudinal Position.

-193The fluctuating component of the temperature difference ATperiodic
in this example is defined for 0 < y < S by
ATperiodic - ATmean = [ATmax - ATmean][S - 1] (B-82)
The Fourier series expansion of the last term is
00
ATperiodic - Amean Ak cos (2kl ( B3)
ATmax -ATmean k=O S
00
= 8 X ( 1~ cosF(2k l>yl (B-84)
2 (2k + 1)2 cs i
k=O
The temperature distribution at the outside wall according to
Equation (B-79) is given by
T(y) Tf(-L/2) + m(y +L/2) + Tmean + ATperiodic +ATgeneration (B85)
where Tgeneration is the temperature increase from r = a to r = b
caused by heat generation alone. If the solution of the linear ordinary
differential equation is used, this term is
A~ _F_ b2 In (b.) — (b2 2\7
A~T =. -^ [^ Ib n ( (b - a ) (B-86)
generation 2K 2 j
The term ATerio dic is the "damped" fluctuating component of
the temperature difference
AT' = AT' 5(na/S,b/a) (B-87)
periodic periodic
For the example chosen (S = s = 2 tube diameters, S/a = 4, b/a = 1o24)
the first 11 coefficients of the Fourier series expansion are listed

-194below, with the argument of the damping function, and with the value
of the damping function to be used in Equation (B-80),
k Ak (2k + 1)/4 5[(2k+ 1)/4, 1.24]
0 8,106 x 0o-l 0,25 0o9822
1 9.006 x 10-2 0.75 08568
2 3.,242 x 10-2 1 025. 06731
3 1.654 x 10-2 1 75 o 4956
4 1.001 x 10-2 2,25 0.3425
5 6,696 x 10-3 2, 75 0 2465
6 4,796 x 10-3 3. 25 0, 1711
7 3.603 x 10-3 3 75, 1184
8 2,805 x 10-3 4,25 0,0818
9 2,245 x 10-3 4,75 0,0565
10 1,838 x 10-3 525 0,0391
Values of the term (ZT - AT m )/(ma - ZTa ) are
periodic mean max mean
listed below, In the first column the values are for the true function
given by Equation (B-82), In the second column the values are for the
Fourier series expansion valid at the inside wall (or at the outside
wall if there is no damping), In the third column are values for the
damped Fourier series valid at the outside wall, The "true values"
(which would be predicted neglecting damping) and the damped value predicted are plotted in Figure 63,
y True Value Approximation by Damped Value at
Fourier Series Outside Wall
0.00 -1, 0000 -0,9816 -0,9102
0,20 -0..8000 -0.8009 -007926
0,40 -o6000ooo -o,6013 -0o5995
o,.60o -4000 -4010 -o 4000
0 80 — 0 2000 — 0,2005 -0 2000
1,.00 0,0000 0,0000 0.0000
1,20 0. 2000 0, 2005 0,2000
1i 40 04ooo000 4010 4oo000
1,60 0O6000 0,6013 0*5995
1.,80 o,8000 0,8009 0, 7926
2,.00 1,0000 0,9816 0,9102

-195It can be seen that the damping causes an error of approximately 7 per cent of [ATmax - ATmean] at the acute "peak" of the
fluctuating temperature distribution. At all the other points tabulated
above, however, there is a negligible difference between the damped and
undamped temperature distribution, Since, 1) this is a very extreme
example emphasizing damping; 2) the error at the worst point is only
7 per cent of [TmaxT - ATmean] and [ATmax - Tmean] is usually only
a small part of the total temperature difference between fluid and wall;
3) the error is compensated for in obtaining the integrated mean heat
transfer coefficient, because it reduces the temperature difference at
one point, but increases it at another. The conclusion is that axial
conduction caused by variation in the inside wall temperature may be
neglected in processing the experimental data of this investigation.
Estimate of Accuracy
In the preceding section it was shown that, in theory, if
the outside tube temperature were known exactly as a function of longitudinal position and if the physical properties of the tube were known
exactly, then the inside temperature of the tube (and, in turn, the
heat transfer coefficient) could be determined exactly. In this
section a discussion is given of the errors introduced by the practical
limitation that the outside wall temperature and the physical properties of the tube are not known exactly,
Accuracy of the AZAR Recorder
The emf produced by the thermocouples on the outside tube
wall were measured by a Leeds and Northrup Speedomax AZAR recorder~

-196An average of four readings (each recorded at intervals of about
30 seconds) was taken to reduce any random errors by the recorder1
For each run in addition to the two known reference emfs introduced
into the AZAR recorder for use in calibration, at least one of the
tube wall thermocouple emfs was measured separately using the 8662
portable precision potentiometer. The particular thermocouple to
be used was selected randomly for each run and was used as a check
on the accuracy of the AZAR recorder,
A frequency distribution of the difference between the emf
obtained from the AZAR recorder and the check value obtained by the
8662 potentiometer is shown in Figure 646 It can be seen that the
error is roughly normally distributed with a mean of zero and was
usually less than + 15 microvolts. This corresponds to an error of
about + 0,70 deg F,
Effect of Temperature Difference Across Mica Insulation
A second source of error in measuring the outside wall
temperature is the fact that the thermocouples were separated from
the outside tube wall by a thin (0,002 inch) sheet of mica to provide
electrical insulation, An exaggerated diagram of the outside wall
thermocouple is shown below;
The following nomenclature will be used to estimate the
temperature difference across the sheet of mica,
let b = outside radius of tube (0,612 inches)
rins = outside radius of insulation (15 inches)
tmica = thickness of mica (0o002 inches)

~19730
28
26
24
22
20
18
16
z
l 14
10
8
6
4
2
-25 -20 -1 -10 -5 0 5 10 15 20 25 30
ERROR INTRODUCED BY AZAR RECORDER, MICROVOLTS
Figure 64. Frequency Distribution of Error
Introduced by AZAR Recorder.

-198Tb = temperature at outside radius of tube
Tins = temperature at rins
Tamb = ambient air temperature
Tt = temperature measured by thermocouple at outside
edge of mica sheet
qL= heat loss per unit area measured at the outside
radius of the tube
Kmica = thermal conductivity of mica (0o25 BTU/hr-deg F-ft)
Kfg = thermal conductivity of Fiberglas (0.05 BTU/hr-deg F-ft)
fg
hc = convection coefficient for heat transfer by natural
convection from Fiberglas to airtube wall
Fiberglas
mica ~ ~air
thermocouple
a b r.
ins
Using simplified correlations recommended, by McAdams(28) for
the convection coefficient from vertical cylinders to air, the largest
value obtainable for temperature differences less than 140 deg F at
distances greater than 1 foot from the bottom of the tube is
1 BTU/hr-deg F-ft2. The following equations may be written

-199Tb Tamb = tmica + b n rins/b+ 1 (B88)
LL i'fg (B-88)
L nKmica Kfg hc rins/b
F 0,002 0612 In 2,52 1 1 (B-89)
L12 x 025 12 x 0 05 x 2,52j
= 0,667 x 10-3 + 0,945 + 0.397
= 134
The ratio
Tb - Ttc = 0,667 x 10-3
Tb - Tab 1.34
b4 amb^~~ 14 ~(B-90)
= 498 x 10
or, in other words, the error due to reading Ttc instead of Tb is
approximately 0,05 p'er cent of the difference between Tb and Tamb.
Since the outside wall temperature was always less than 100 deg F
greater than the ambient air temperature, the error caused by the
temperature difference across the mica insulation was less than 0.05
deg F.
Propagation of Errors in Calculating Local Heat Transfer Coefficients
The two important calculated quantities (ioeo, dependent
variables) which must be evaluated from the physical properties of
the tube and the experimental measurements (ie,, the independent
variables) in order to obtain the local heat transfer coefficient
are q(z) and ATgeneration The error in these calculated quantites
which is propagated by the defining Equations (56) and (54) will be
estimated in this section,

-200In symbolic form
q(z) = function of (ab,pOKOyTbI) (B-91)
Ageneration = function of (abop KO,IYYTb I) (B-92)
A propagation constant for eachof.thie independent variables (ab,,,,etc )
applicable to each of the above relations will be defined, The propagation constant is the percentage change in dependent variable oaueed
by a change of one per cent in the value of the independent variable
(with the other independent variables held constant), These constants
were evaluated by numerical partial differentiation of the defining
expressions' They are listed below,
PropagatiJn
Independent Propagation Constant. for
Variable Constant for q(z) ZTgeneration
a -1,2 +2, 8
b -0o8 -6 1
Y +0,09 +0,09
3_B~~ ~-0,06 0
p+ 1,0 +10 +.0
Ko +1 0 0
Tb +0, 01 +0,09
I +2 0 +22 0
The estimated maximum error associated with each independent
variable is listed below,
Independent
Variable Absolute Error Per Cent Error
a o 0,002, in 0,o 4%
b 0.002 in, 0, 4%
~' Error is. Relative 5,0%
_f Error is Relative 5^ 00
p0 Error is.Relative 5.0%
K Error is Relative' 5,. 0%
I~ 5 amps 0,.5
Tb 1 deg F 0o670

-201The maximum percentage error associated with q(z) and
Tgeneration is the sum of the absolute values of the products of
the relative error of each of the independent variables and its
propagation constant. This is the error that would occur if each
error were in the proper direction so that it exerted its maximum
effect. In practice, these errors tend to compensate and a better,
statistical estimate of the relative error of the.ependent variable
is given by the square root of the sum of the squares of the products
of the relative errors and propagation constants. On the basis of the
statistical procedure, q(z) is accurate to within + 5 per cent and
Tgeneration is accurate to within + 7 per cento
The temperature difference between inside wall and fluid
(Ta - Tf) is actually required in the calculation of the heat transfer
coefficiento The error associated in this term is given by the sum of
the errors (or statistically by the square root of the sum of the
squares of the errors) in Tb, ATgeneration d and Tf
The error in determining Tb (caused by error introduced by
the temperature difference across the mica insulation and by the AZAR
recorder) is less than 1 deg Fo
The error in determining Tf is less than 0~05 deg Fo
The error in calculating ATgeneration was shown to be less
than 7 per cent, Since ATgeneration was almost always less than 25
deg F, the absolute error in it was less than 1475 deg Fb
Thus the error in (Ta - Tf) on a statistical basis may be
considered less than + 2~08 deg F4

APPENDIX C
ORIGINAL AND PROCESSED DATA
In the course of this investigation over 15,000 local heat transfer coefficients and over 750 values of the overall pressure drop were
measured. In order to present all of the original data a volume almost as
large as this dissertation would be required. An attempt will be made,
however, to present the data in a semi-processed form with enough information so that the values of the original measured variables can be "backcalculated."
Table VII is a summary of the experimental conditions used. It
lists the geometry, diameter ratio, spacing, number of promoters (for
bluff-body promoters), heat transfer run number, number of flow rates for
which heat transfer data were obtained, and the pressure drop run number.
This table serves as a guide to the location of the datao In order to
find the original data corresponding to any particular geometry, the first
step is to locate the appropriate run number from Table VII.
Tables VIII, IX, and X present the constants C(s,d) and n(s,d)
for use in the Equations (64), (65), and (67) to predict the friction
factor f, the effective drag coefficient fD, or the heat transfer coefficient ratio hm/h0 as a function of Reynolds number for a given geometry. In addition, values of the respective variables calculated by the
appropriate equation are tabulated at various values of the Reynolds number. These are the values used in preparing the cross-plots of Figures
26, 27, 31, 32, 38, 39, 42, and 43.
Table XI presents the isothermal friction factor and effective
drag coefficient (where applicable) as a function of Reynolds number for
-202

-203each pressure drop run. The water temperature for each run was always
greater than 40 deg. F and less than 70 deg. F, depending upon the particular time of year when the run was madeo The exact value of the fluid temperature for each run is listed (as the inlet temperature) with the heat
transfer data in Table XII.
Thus, the mass flow rate can be "back-calculated" using Equation
(21). Since the number of promoters np is given in Table III, the pressure drop may be calculated using Equation (62) or (22)~ The inches of
purple indicating fluid or mercury may then be calculated using Equation
(A-2a) or (A-2b). Most of the pressure drops were measured with the
purple fluid, unless this was greater than 100 inches in which case mercury
was used.
Table XII presents the integrated results of the heat transfer
measurements. For each geometry, diameter ratio, and spacing, the overall
(i.e. mean) value of Reynolds number Rem, heat flux q, heat transfer
coefficient h for the empty tube calculated using the Sieder-Tate Equa0
tion (18), heat transfer coefficient ratio hm/ho0 the factor (,/w)~l4 x
Prl/3 (k/D), and the inlet water temperature are listed.
Table XIII presents the local heat transfer coefficient ratios
h/h0 at each longitudinal position on the tube. In addition, the distance
from the nearest promoter x (where appropriate) is giveno For longitudinal
stations upstream from the first promoter, the distance from the promoter
is listed arbitrarily as zero0
From Table XII the heat flux q and inlet water temperature
Tinlet may be obtained The fluid temperature at any longitudinal distance z

20o4in tube diameters from the beginning of heating may be calculated from
D2
Tf(z) = T 2inlet+ (C-i)
inlet W
From Table XII the mean value of the heat transfer coefficient hg for
the empty tube is listed. Thus, using the value h/h0 at each longitudinal position, the actual value of the heat transfer coefficient may be
obtained at each longitudinal position. Since h and q are known, the
inside wall temperature may be determined from
Twall(Z) Tf(z) + q/h (C-2)
In a similar fashion, almost all of the measured experimental variables
can be recovered from the data presented in this appendix.
For longitudinal positions marked with an asterisk (*) the data
were for an angular position of 120 degrees and for the longitudinal positions marked with a plus (+) the data were for an angular position of 240
degrees.
All of the information in Tables VIII to XIII was obtained from
punched cards used and prepared by the computer at various stages in the
data processing. Thus, the number of significant figures does not indicate
the estimated precision of the quantity printed. A detailed discussion of
the precision of the data is given on pages 10l-110O

-205TALE VII
SUMMARY OF EXPERIMENTAL CONDITIONS
Geometry d s n Heat Transfer F* Pressure Drop
P Run Numbers Run Number
EMPTY TUBE 0.000 P-1 TO P-14 14 P-1
EMPTY TUBE 0.000 -- R-1-A TO R-1-I 9 P-2
SOLID ROD IN CENTER OF TUBE 0.125 0.0 - R-3-A TO R-3-E 5 A-2
THREADED SOLID ROD IN CENTER OF TUBE 0.250 0,0 - R-2-A TO R-2-I 9 A-1
SOLID ROD IN CENTER OF TUBE 0625 0.0 - R-28-A TO R-28-E 5 A-27
SOLID ROD IN CENTER OF TUBE 0.750 0.0 -- R-29-A TO R-29-E 5 A-28
DISKS 0.625 12.0 4 R-13-A TO R-13-E 5 A-11
DISKS.0,625 8,0 6 R-12-B TO R-12-E 4 A-12
DISKS 0.625 4,0 12 R-14-A TO R-14-E 5 A-13
DISKS 0.625 2,0 11 R-24-A TO R-24-E 5 A-23
DISKS 0.750 12.0 4 R-5-A4B.C.AND E 4 A-4
DISKS 0.750 12.0 4 R-6-D TO R-6-E 2 A-5
DISKS 0.750 8.0 6 R-7-A TO R-7-E 5 A-6
DISKS 0.750 8.0 6 R-27-A TO R-27-E 5 A-26
DISKS.0750 4.0 12 R-8-A TO R-8-E 5 A-7
DISKS 0.750 2.0 11 R-26-A TO R-26-E 5 A-25
DISKS 0875 12.0 4 R-9-A TO R-9-E 5 A-8
DISKS 0.875 8,0 6 R-O1-A TO R-10-E 5 A-9
DISKS 0.875 4,0 8 R-11-A TO R'll-E 5 A-10
DISKS 0.875 2.0 8 R-25-A TO R-25-E 5 A-24
STREAMLINE SHAPES 0.625 12.0 4 R-15-A TO R-15-E 5 A-14
STREAMLINE SHAPES 0.625 8.0 6 R-16-A TO R-16-E 5 A-15
STREAMLINE SHAPES 0.625 4.0 6 R-17-A TO R-17-E 5 A-16
STREAMLINE SHAPES 0.750 12.0 4 R-18-A R-R-18-E 5 A-17
STREAMLINE SHAPES 0,750 8,0 6 R-19-A TO R-19-E 5 A-18
STREAMLINE SHAPES 0.750 4.0 6 R-20-A TO R-20-E 5 A-19
STREAMLINE SHAPES 0.875 12*0 4 R-21-A TO R-21-E 5 A-20
STREAMLINE SHAPES 0.875 8.0 6 R-22-A TO R-22-E 5 A-21
STREAMLINE SHAPES 0.8-75 4.0 6 R-23-A TO R-23-E 5 A-22
* Number of flow rates for which heat transfer data were obtained

-206TABLE VIII
CONSTANT C(s,d) AND n(s,d) USED IN FRICTION FACTOR
CORRELATION EQUATION (64) FOR INDIVIDUAL PROMOTER COMBINATIONS
100lxf at various values of Re/1000
Re/1000
Geometry AP Run s d C(s,d) n(s,d) 5 10 20 30 40 50
SOLID ROD A-1 0.0 0.250 6.6240 -0*1255 2*274 2.085 1,911 1.816 1.752 1.704
SOLID ROD A-2 0.0 0.125 7.5207 -0.2307 1,053 0*898 0o765 0.697 0.652 0*619
SOLID ROD A-27 0.0 0.625 19.190 -0*1271 6.738 6.169 5.649 5.365 5.172 5.028
SOLID ROD A-28 0.0 0.750 51.594 -0.1217 18.34 16*85 15.49 14.74 14.23 13*86
DISK A-11 12.0 0.625 8.1420 -0.0620 4.910 4.600 4.180 4.110 4.070 3.999
DISK A-12 8.0 0.625 7.4316 -0.0173 6.417 6.341 6.265 6.221 6.190 6.167
DISK A-13 4.0 0.625 6.7241 0.0425 9,657 9.945 10.24 10.42 10.55 10.65
DISK A-23 2.0 0.625 8.3359 0.0731 15.52 16.33 17.18 17.70 18.07 18.37
DISK A-4 12.0 0.750 64.253 -0.1612 16*27 14.55 13.01 12.09 11.64 11.22
DISK A-5 12.0 0.750 15.320 -0.0100 14.08 13.98 13.89 13.83 13.79 13.76
DISK A-6 8.0 0.750 17.833 0.0092 19.30 19.42 19.55 19.62 19.67 19.71
DISK A-26 8.0 0.750 14.255 0.0162 16.37 16.55 16.74 16.85 16.93 16.99
DISK A-7 4.0 0.750 25.298 0.0220 30.53 30.99 31.48 31.76 31.96 32.11
DISK A-25 2.0 0.750 8.5280 0.1740 37.53 42.35 47.80 51.30 53.94 56.08
DISK A-8 12.0 0.875 56.884 -0.0084 52.97 52.67 52*36 52.18 52.06 51.96
DISK A-9 8.0 0.875 72*954 0.0066 77.20 77.55 77*91 78.11 78.26 78.38
DISK A-10 4.0 0.875 101.94 0.0435 147.7 152.2 156.8 159.6 161.7 163.2
DISK A-24 2.0 0.875 126.48 0.0670 223.9 234*5 245.7 252.5 257.4 261.3
STREAMLINE A-14 12.0 0.625 9.2961 -0.1579 2.423 2.172 1.947 1l826 1.745 1.685
STREAMLINE A-15 8.0 0.625 15*734 -0.1754 3.533 3.128 2.770 2.580 2.453 2.359
STREAMLINE A-16 4.0 0.625 20*934 -0.1657 5.105 4.551 4.057 3.793 3.616 3.486
STREAMLINE A-17 12.0 0.750 17*041 -0.1310 5*583 5*099 4.657 4.417 4.254 4.131
STREAMLINE A-18 8.0 0.750 28*835 -0.1520 7,902 7.111 0,400 6.017 5.759 5.567
STREAMLINE A-19 4.0 01750 37.155 -0.1264 12.66 11.60 10.63 10.09 9.736 9.465
STREAMLINE A-20 12,0 0.875 70.880 -0.1190 25*73 23.69 21.81 20.78 20*08 19*56
STREAMLINE A-21 8.0 0.875 123.05 -0.1355 38.81 35.33 32.16 30,44 29.28 28.41
STREAMLINE A-22 4.0 0.875 190.68 -0.1230 66.87 61.41 56.39 53.64 51.78 0=38

-207TABLE IX
CONSTANTS C(s,d) AND n(s,d) USED IN CORRELATION EQUATION (65)
OF EFFECTIVE DRAG COEFFICIENT FOR INDIVIDUAL PROMOTER COMBINATIONS
100XfD at various values of Re/1000
Re/1000
Geometry AP Run s d C(s,d) n(s,d) 5 10 20 30 40 50
DISK A-11 12. -0.62-5222.07 -0.0280 174.9 171*5 168.2 166.3 165.0 164.0
DISK A-12 8.0 0.625 170.60 -0.0020 167.8 167.6 167,3 167.2 167l1 167*0
DISK A-13 4.0 0.625 103.49 0.0357 140.3 143.8 147.4 149.6 151.1 152.3
DISK A-23 2.0 0.625 38.280 0.1192 105.7 114.8 124.7 130.9 135.4 139.1
DISK A-4 12.0 0.750 244.42 -0.0265 195.0 191.5 188*0 186.0 184.6 183.5
DISK A-5 12.0 0*750 241,49 -0.0100 221.8 220*2 218,7 217.8 217.2 216.2
DISK A-6 8.0 0.750 203.13 0.0017 206.0 206.3 206.5 206.6 206*7 206.8
DISK A-26 8.0 0.750 132.00 0.02'88 168.8 172.2 175*7 177.8 179*3 180.4
DISK A-7 4.0 0*750 132.58 0.0248 163.8 166.6 169.5 171.? 172.4 173.4
DISK A-25 2*0 0.750 30.520 0.1458 105*7 116*9 129.3 137.2 143.1 147.8
DISK A-8 12.0 0.875 204.09 -0.0121 184*0 182.5 180.9 180.0 179.4 179.0
DISK A-9 8.0 0.875 161-70 0.0095 175.4 176.5 177.7 178.4 178*9 179.2
DISK A-10 4.0 0.875 137.02 0.0277 173.6 176.9 180.4 182.4 183.9 185.0
DISK A-24 2.0 0.875 87.250 0.0478 131.2 135.6 140.2 142.9 144.9 146.9
STREAMLINE A-14 12.0 04625 150.67 -0.0949 67.15 62.88 58.87 56.66 55.13 53.98
STREAMLINE A-15 8.0 0.625 199.67 -0.1153 74.80 69.05 63.75 60.84 58.85 57t36
STREAMLINE A-16 4.0 0.625 199.04 -0.1361 62.42 56.80 51.69 48.91 47.03 45.62
STREAMLINE A-17 12.0 0.750 173.02 -0.0985 74.75 69.81 65.20 62.65 60.90 59.58
STREAMLINE A-18 8*0 0*750 263.32 -0.1449 76.60 69.28?o.66 59,08 56*67 54.86
STREAMLINE A-19 4.0 0.750 160.71 -0.1096 63.20 58.58 54*30 51*94 50*33 49*11
STREAMLINE A-20 12.0 0.875 248*56 -0.1236 86*72 79*60 73*07 69*49 67*07 65*24
STREAMLINE A-21 8.0 0*875 262*70 -0*1302 86*62 79*15 72*31 68.59 66*07 64*18
STREAMLINE A-22 4*0 0*875 198.72 -0.1145 74.93 69.21 63*93 61*04 59.06 57.57

-208TABLE X
CONSTANT C(s,d) AND n(s,d) USED IN CORRELATION EQUATION (67) OF MEAN HEAT
TRANSFER COEFFICIENTS FOR INDIVIDUAL PROMOTER COMBINATIONS
h/h0 at various values of Re/1000
Re/1000
Geometry HT Run s d C(s,d) n(s,d) 5 10 20 30 40 50
DISK R-13 12*0 0.625 2.7532 -0.0605 1.645 1l57? 1.512 1.476 1.456 1.431
DISK R-12 8.0 0.625 3.9618 -0.0792 20017 1.909 1.808 1.750 1.711 1681
DISK R-14 4.0 0.625 6.3460 -0.1059 2.575 2.392 2.223 2.130 2.066 2.018
DISK R-24 2.0 0.625 3*6284 -0.0534 2.302 2.218 2.137 2.092 2.060 2.035
DISK R-5+6 12.0 0.750 7.0341 -0.1355 2.218 2.019 1.838 1.740 1.673 1.624
DISK R-7+27 8.0 0.750 6.0226 -0*1008 2.553 2*380 2.219 2.131 2,070 2.023
DISK R-8 4.0 0.750 9.5482 -0.1267 3.246 2.973 2.723 2.587 2.494 2.424
DISK R-26 2.0 0.750 6.0535 -0.0803 3.054 2.889 2.732 2.645 2.584 2.558
DISK R-9 12.0 0.875 5.5118 -0.0849 2.675 2.522 2.378 2.298 2.242 2.200
DISK R-10 8.0 0.875 8.4104 -0.1112 3.262 3.020 2.796 2.672 2.588 2.524
DISK R-11 4.0 0.875 5.7542 -0.0479 3.825 3.700 3.598 3.510 3.462 3.426
DISK R-25 2.0 0.875 4.8231 -0.0144 4.265 4.223 4.181 4.156 4.13:9 4.126
STREAMLINE R-15 12.0 0.625 1.2934 -0.0053 1.237 1.232 1.228 1.225 1.223 1.222
STREAMLINE R-16 8.0 0.625 2.1630 -0.0487 1.429 1.382 1.336 1.310 1.292 1.278
STREAMLINE R-17 4.0 0.625 4.0624 -0.0983 1.759 1.643 1.535 1.475 1.434 1.403
STREAMLINE R-18 12.0 0.750 2.1214 -0.032:8 1.604 1.568 1.533 1.513 1.499 1.4"8
STREAMLINE R-19 8.0 0.750 5.4733'-0.1094 2.156 1.998 1,853 1.?773 1.718 1.676
STREAMLINE R-20 4.0 0.750 4.3002 -0.0781 2.211 2*099 1,983 1.922 1.'879 1.847
STREAMLINE R-21 12.0 0.875 4.6661 -0.0873 2.218 2.088 1*96'5 1.896 1.849 1.814
STREAMLINE R-22 8.0 0*.75 8*2288 -0.1208 2.940 2.704 2.486 2.367 2.287 2.226
STREAMLINE R-23 4.0 0.875 3.8897-0.0122-3.506 3.477 3.448 3.431..419 3.409

-209TABLE XI
PRESSURE DROP RESULTS
Run P-1 Run P-2 Run A-1 Run A-2
Empty Tube Empty Tube Threaded Rod Plain Rod
d = 0.250 d = 0.125
Re/1000 0 oxfRe/1000 10oxf Re/1000 100Xf Re/1000 1000 x
3.765 0.697 2,349 0.778 2.895 1711 3.133 1.007
3.645 0.697 2.099 0.979 1,919 2.007 4.646 0.919
5.545 0.662 3.495 0.881 3.829 2.078 5,829 0.876
5.639 0.582 4*831 0.838 4.997 2.182 8.599 0.904
6.495 0.673 5.233 0.790 7.307 2.137 10.624 0.922
9.751 0.675 5.446 0.848 11.731 2.075 14.748 0.809
8.106 0.723 6.918 0.761 15.152 1.996 18.498 0.782
13.194 0.681 8.953 0.789 20.624 1.899 22.998 0.740
17.295 0.648 12.001 0,708 25.785 1.855 28.248 0.708
21.563 0.615 16.157 0.659 32.750 1.792 33.872 0.672
25.663 0.604 21.961 0.621 37.724 1.755 39.872 0.653
32.777 0.578 28.336 0.569 43.622 0*639
40.309 0.540 32.705 0.568 44.747 0.636
45.330 0.530 35.899 0.555
49.933 0.513 38.479 0.545
41.650 0.538
Run A-4 Run A-5 Run A-6 Run A-7
Disks Disks Disks Disks
d = 0.750 s = 12.0 d = 0.750 s = 12.0 d = 0.750 s = 8.0 d =0.750 s = 8.0
Re/1000 1OOxf lOOxf Re/ 10i00 lOO xfD Re/1000 10Oxf 0lxfD Re/1000 OOxf lOOxf
2.635 12.675 188.509 2.812 14.534 219.240'0978 37.469 212.909 2.109 32.057 167.910
3.906 12.562 188.753 3.817 14.244 216,113 2.500 20.872 214.724 3.825 30.906 162.756
4.930 13.003 197.036 4.990 14.893 227.959 3.988 19.819 204 923 5 145 32.080 169 605
6.605 12.828 195.400 5.660 14.097 215.501 5.442 19.315 200.375 6.695 30.993 164.049
7.764 17.983 280.214 6.465 14.733 226.430 6.189 19.666 204.552 7.890 28.809 152.366
13.270 12.392 190.680 7^001 14.461 222.298 7.187 19.874 207.207 9.073 30.807 163.410
18.098 11.752 181.109 9.007' 14 597 225 441 8.233 18.594 193.605 11.531 32.731 174.150
23.858 12.065 186.929 10.871 13.908 214.819 9.548 20.818 218.168 15.629 32.687 174.214
28.941 11.809 183.203 14.133 14199 220.382 10.487 18.453 192.624 19.544 32.411 172.914
35.294 12.000 186.761 16.928 14*143 219*980 11.802 18*769 196,318 24.161 32.000 170.855
41.647 11821 184.181 20.003 12.274 189.899 12.835 20.123 211.235 30.163 32.159 171.893
45.882 12.069 188.425 24.383 13.718 213.981 15.653 20.296 213.507 33.395 32.258 172.508
52.658 11.774 183.870 25.315 13.540 211.164 17.626 19.321 203.110 34.780 28.696 153*144
28.483 13 337 208 124 19.504 19.184 201.798 36.627 31.819 170.183
33.090 13.457 210.420 21.383 19.123 201.292 41.706 31.679 169*510
36.750 14.063 220.543 25.140 19.204 202.439 44.015 32.337 173*129
40.867 14 031 220 240 27.957 20.027 211,568 50.940 31.988 171.323
43.612 13.769 216 091 34.626 19.505 206.201 53.249 32.439 173.806
44.984 13.721 215.368 38.195 19.618 207.570 56.019 32.623 174*839
45.899 13.646 214.183 43.831 19.916 211.001
47.272 13.932 218.912 47.118 19,875 210 650
48.644 13.908 218.575 49'.936 20.307 215.428
50.016 14.349 225,831 53.693 19.000 201.287
51.846 13.935 219.137 55.196 19.989 212.091
53.219 13.994 2201.50 56.511 19.655 208.483
55 049 13.859 218.008
Run A-8 Run A-9 Run A-10 Run A-1
Disks Disks Disks Disks
0.875 sd = 0.875 = 0.875 s = 8.0 d = 0.875 s = 4.0 d = 0.625 s = 12.0
R2 10.3 1 0X7 f D p 100Xf Xf00 Re/1000 l0OXf 10OXfD Re/1000 lOOXf lO0XfD
2,232 64.759 218.908 1.87'8 72063 162.551 1.369 199.451 227*350 3.616 4.677 i66.453
3.689 55.585 187.905 3.627 81 954 185.799 2.316 164.706 187*720 4.768 5.020 185.776
5.146 55.931 189.423 5 046 80.643 183.007 3*164 152.679 174.039 5.656 4.874 181*185
6.409 56.162 190.412 6.563 77.839 176.723 3.948 161.492 184.235 6*643 4*718 175*887
9.045 54.661 185.513 6.761 76.132 172,820 5.384 151.972 173.406 7 874 4 536 169.383
10.380 50.608171.653 9.244 77.231 175.505 6.331 152.421 173,969 9.671 4.599 174.282
12.248 52.106 176.923 10*153 74.336 168.903 6.788 156,280 178.418 14*701 4*394 168.594
16.164 52.893 179.806 13.063 80,140 182.340 7.814 144.377 164.791 17,126 46326 166.693
20.613 53*495 182.017 16.064 79.900 181.874 9.240 150.990 172*425 21.259 4 274 165.914
23.282 51*010 173.523 21.520 78.362 178.452 10*578 152.266 173*921 24.672 4.228 164.844
28 1764 52511 178.788 24.521 78.509 178.835 12*271 157.316 179*752 28*894 4.151 162.313
30.845 52*989 1 80478 27.431 77.503 176 563 14 233 159 271 182.027 34 553 4.106 161.419
34.404 52 848 180.042 31*250 77.602 176,832 15.748 157,738 180.288 38.326 4.006 157.468
37.252 53.559 182.527 34.705 78 745 179 488 20.384 156.746 179.198 42.369 4.089 161.828
41.345 46.435 158.038 37.252 77 500 176 651 25.554 161.048 184.176 44.255 4.132 164.033
48.196 53.893 183*786 39 343 78 847 179 760 28.764 159.254 182.136 47.309 4.271 170.741
49,976 53.159 181.273 44.435 78 402 178 772 31.081 160.076 183.092 50.812 42-66 170.895
46.891 77.175 175.970 34.469 160.494 183.588 55.843 4.230 169.745
37.054 159.712 18-2701
38.926 159.254 182.182
37.054 159 712 182.701

-210TABLE XI (CONT'D)
Run A-12 Run A-13 Run A-14 Run A-15
Disks Disks Stre amline Shapes Streamline Shapes
d = 0.625 s = 8.0 d = 0.625 s 40 d 0.625 s = 12.0 d = 0:625 s = 8.0
Re/1000 10OXf lOOXf Re/1000 lOOxf 10xfD Re/100I0 laOxf lOOxf le/1000 lOOxf 100x
3,706 6.692 172.495 3.278 10.259 139.916 1 163 3.685 100.379 2.503 3.328 66 194
5,116 6.729 176.416 4.666 10.676 147.863 2,692 1 8'95 35,0&6 4.586 3.432 75.221
5.944 6.428 168.446 5.738 10.456 145.353 3.857 2.331 60.301 5.205 3.648 82.837
6.403 6.332 166.085 6.558 10.126 140.837 4.973 2 343 64 146 5 880 3.612 82.700
7.336 6.212 163.416 7 980 9.708 135.173 6.471 2 316 65 995 6.890 3.218 71.892
8.256 6.288 166.538 8.841 9.876 138.069 7,399 2,239 63 926 11.271 3.106 71 709
9,763 6.172 164,098 12.372 10.279 145.228 11044 2 114 62,116 12.424 3.035 70.118
11.771 6.298 169.066 14.956 10 109' 143.169 14.026 2,090 63,079 15 498 2.872 66.379
13 696 6.267 168.985 17.367 10.054 142.723 15 020 2.039 61. 308 17.804 2.810 65i209
16.207 7.629 211.322 19.521 10.263 146.193 15.550 2.047 61.950 20.759 2.708 62.858
21.312 6.078 165.499 20.382 10.181 145.050 18.599 1.976 60.085 24.104 2.688 62,944
26.082 6.003 164.138 25.980 10 230. 146.353 21.316 1 947 59.752 26 413 2.613 61.070
26.919 6.088 166.862 30.286 10.501 150.805 23.237 1.911 58.711 29,599 2.586 60.738
30.852 5.986 164.340 34.851 11.270 162.788 25.623 1.876 57.775 32.386 2.533 59.500
35.706 5.929 163.199 41.568 10.209 146. 992 29. 864 1.874 58.675 34 377 2.529 59 620
40.309 6.198 171.853 43.205 10.386 149.756 31.919 1.793 55.395 36.767 2.516 59.491
43.405 6.648 185.820 46.047 10,710 154.802 34.238 1.783 55.366 40.350 2.428 57.173
47.422 6.121 170.113 48.114 10.711 154.896 36 226 1 759 54 608 41 545 2.453 58.044
50.686 6'110 170.016 50.525 10.526 152.170 38;877 1,720 53.241 43.536 2.438 57.763
52.937 10.584 153.134 40 865 1 728 53.891 46.323 2408 57 078.
48.713 2,353 55.587
Run A-16 Run A-18
StreamlineShapes Streamline Shapes Streamline Shapes Run A-19
d = 0.625 s =.o d 0750 s = 12.0 d = 0.750 s = 8.0 Strealine Shapes
~~Re/ 1000 l00xf l0o R150 lx box Re/1000 looXf 100Xf d = 0.750 s = 4.0
Re/1000 100Xf Xf l lx "100XfD Re100 100f 100XfD
- DxRe/ 1000 lOOXf lOOxf
2.986 4.073 45.360 3.003 4.441 54.752 2.994 8.054 75.832 2.700 12.477 61.805
3.943 4.489 53.024 4.393 5.210 69.230 4.491 7.981 76.394 4.754 13.100 66.153
4,713 4.944 60.721 4.940 5.370 72.373 4.824 10.042 99.051 5,456 13.194 66.866
6.160 4.932 61.604 7.571 5.187 71.119 5.823 7.833 75.538 75869 11.797 59.742
8.391 4.640 58.257 10.622 5.061 70.243 8.381 7.186 69.427 8.902 12.035 61.185
10.941 4.493 56.862 11.300 5.227 73.154 11.107 7.287 71.163 13.330 11.430 58.327
13.299 4.457 56.881 15.029 4.921 69.020 14.440 6.743 65.776 17.609 10.869 55.538
16.677 4,225 53.960 17,063 4 665 65 195 16 788 6..662 65.179 20.856 10.475 53.541
20.310 3.953 50.313 19.368 46649 65.274 19.438 6.449 63.122 22,995 10.200 52.126
21 776 3 964 50 646 20.996 4 744 67 034 22.468 6 160 60.222 27.349 9.850 50.359
23.050 3.909 49.941 24.861 4 376 61 442 25.119 6.024 58.923 29.636 10.104 51.803
24 644 3.823 48.785 27.573 4 425 62 487 28 224 5.821 56.895 32.440 9.848 50.476
27.703 3.931 50.685 31.302 4.137 58.071 30,496 5.889 57 752 35 760 9.753 50.029
30.061' 3,884 50.144 3k4353 4.350 61.753 34.131 5.862 57.623 39.228 10.004 51.460
32.611 3.786 48.822 37.404 4.398 62.718 38.373 6.033 59,651 42.179 9.832 50.573
34.523 3.647 46.823 39.777 4.343 61.947 41.099 5.956 58.906 44.762 9.788 50.373
38.347 3.641 46.939 40.794 4.314 61.525 46.325 5.590 55.080
Run A-20
Run A-21 Run A-22
Streamline Shapes Run A-23
tShapes Streamline Shapes Streamline Shapes Run A-23
R 0.870 lO lO f d = 0.875 s = 8.0 d = 0.875 s = 4.0 0s
/looo 100Xf 100xfD Re/00 100xf 100XRf Re/1000 100Xf lOO1Xf 00xf 62 s 2.0
2.356 28,457 937959 Re1000 lo~xf lo0xf
24.16,4 37.967 84,904 2.729 68,001 76.775 3.129 13.473 94-85
4.457 27,414 91,079 129134 94
54225 27.029 891903 5.178 38.803 86,962 3.857 69.731 78,890 4.024 14.405 101.969
65343 247.02 89.130 5.808 39,340 88,263 4.634 66.875 75.672 5 291 15.713 112.496
96212 24.414 81*359 9.628 36.380 81,734 5,135 67,409 76,317 6.830 14.695 105.240
11.801 23 064 876 882 11.499 35440 79657 6.887 63.987 72.473 8.866 17.087 123,879
13 620 2230343 76.21 13.071 30,925 69,347 8.803 64.283 72.877 11.308 16.514 119.900
162559 22.340603 74 1 15,615 32.793 73,709 10.718 59.871 67.859 12.326 15.732 114.079
216059 22.060 743635 17.711 34.436 77,530 13.797 57.436 65 118 15.515 16.313 118.815
24.489 22711 274.01 21.827 32.185 72.439 17.834 57.380 65,105 17.687 17.889 130.970
247489 210272 71835 26.093 31.342 70.565 19.681 56.493 64.105 22.504 17.089 125,177
30.565 210005 708324 30.283 31.131 70.128 23.581 55.000 62,424 24.336 16.729 122.529
32.614 200558 68815 32,529 29,873 67,262 26.113 54.556 61.931 26.286 17.496 128.447
3.1439 20. 55 68.57 0 35.298 29.440 66,293' 27481 5670 6207017540 128900
3~5.439_______________40823 51439 58.422 39208 18265 3706

-211TABLE XI (CONT'D)
Run A-24 Run A-25 Run A-26
Disks Disks Disks
d 0.875 s= 2.0 d= 0.750 s 2.0 d 0.750 s =.0
Re/ O 1000 f lOOX10XfD00 lOOXf 1OXf Re/1000 100 Xf 10XfD
2*866 232.988 133.096 1.360 60.395 160.579 2.375 14.917 149.676
3.798 225.507 128.854 2*813 41*244 109.251 3.684 17.018 174.164
4.515 229,885 131.396 3.713 40.778 108.227 4.727 17.474 179.917
5.112 228.556 130.653 4*383 42*296 112.493 6*400 16*550 170.712
6.307 224.014 128.077 4*798 42*254 112.447 7*195 17.022 176.155
8.134 220.539 126.116 7.602 40.870 108.993 9.380 16*862 175.051
10.548 239,.141 136.825 10,055 41.037 109.612 11*102 16.629 172.887
12.049 236.767 135.477 14.033 42.705 114.326 15.340 16*273 169.663
13.158 244.368 139*849 18*209 51*979 139*694 18*121 16.399 171.342
14.790 250.647 143,466 21.325 50.720 136.336 20*703 16*852 176.508
17.400 244.516 139.963 21.948 47.098 126.488 21.630 16*990 178.085
20.010 245.860 140,747 24.491 48.357 129*960 23.749 16.658 174.625
24.577 2480895 142.507 25.860 48*244 129*674 26*266 16.621 174.383
27.187 247.591 141.767 28.142 50.688 136.360 28.583 16.958 178.183
31.429 248 666 142.396 29.511 52.389 141 009 30.239 16.967 178.367
33*357 53*776 144.830 32.887 16*932 178.110
35.639 53.991 145.439 35.536 16.967 178.602
40.529 50.990 137.314 38.847 17.089 180.055
39.906 16.841 177.392
Run A-27 Run A-28
Plain Solid Rod Plain Solid Rod
d = 0.625 d = 0.750
Re/1000 100Xf Re/1000 lOOXf
1.383 15.119 3.275 22.734
3.590 7.302 4.342 19.496
4.717 6.102 4.755 18.399
6.195 6.413 5.119 18.459
8.311 6.447 9.321 15.100
10.362 6.205 11.375 17.226
14.977 5.908 13.430 16.543
16.259 5.780 16.014 16.549
18.182 5.608 18.665 16.182
20.746 5.519 20.852 15.648
22.412 5.384 22.641 15.548
24.720 5.333 23.966 15.568
27.668 5,503 25.292 15.297
29.271 5.448 26.948 15.258
31.194 5.389 29.997 14.907
33.437 5.260 32.846 14.415
35.040 5.234 34.238 14.096
38.565 5.421 38.877 13.206

-212TABLE XII
INTEGRATED RESULTS FROM HEAT TRANSFER MEASUREMENTS
Run Geometry d s ^ s R U h0 hm/h ~ Tin Run Geometry d s Re h hh0 Tin{
Rn em r1000 1000 1000
P-i EMPTY TUBE 0.000 -- 49.854 9.276 1302.0 1.008 8.416 59.2 R-15-A STREAMLINE 0.62512.0 45.253 83.617 1351.9 1.224 9.454 45.3
P-2 EMPTY TUBE 0.000 -- 56.618 103.713 1518.1 11038 8.876 59.7 R-15-B STREAMLINE 0.625 12.0 34.088 80.290 1082.4 1.224 9.504 45.1
P-3 EMPTY TUBE 0.000 -- 51.059 37.575 1362.8 0.998 8.644 58.9 R-15-C STREAMLINE 0.625 12.0 22.419 60.199 770.3 1.226 9.464 45.4
P-4 EMPTY TUBE 0000 -- 10.235 35.520 381.5 0.967 8.803 60.0 R-15-D STREAMLINE 0.625 12.0 9.844 38.361 390.6 1.230 9.313 46.8
P-B EMPTY TUBE 0.000 -- 49.938 39.494 1355.1 0.980 8.749 56.9 -15-E STREAMLINE 0.625 12.0 5.685 15.731 248.7 1.238 9.163 48.0
P-6 EMPTY TUBE 0.000 -- 23.886 39.432 760.3 0.962 8.863 57.8 R-16-A STREAMLINE 0.625 8.0 46.186 80.803 1342.9 1.297 9.238 48.7
P-7 EMPTY TUBE 0.000 -- 37.604 39.337 1082e4 0.975 8.770 578 R-16-B STREAMLINE 0.625 8.0 33.811 58.650 1048.2 1.290 9.255 47.6
P-B EMPTY TUBE 0.000 -- 7.052 10.912 280.6 0.914 8.679 59.2 9-16-C STREAMLINE 0.625 8.0 22.363 51.222 750.6 1.319 9.234 48.5
P-9 EMPTY TUBE 0.000 -- 2.819 9.517 131.0 1.245 8.485 59.9 9-16-D STREAMLINE 0.625 8.0 10.565 30.066 406.3 1.386 9.118 49.6
P-10 EMPTY TUBE 0.000 -- 6.763 8.396 268.0 0.888 8.570 60.8 R-16-E STREAMLINE 0.625 8.0 6.304 13.434 263.0 1.413 8.910 51.7
P-ll EMPTY TUBE 0.000 -- 14.857 30.903 547.3 0.935 9.332 48.2 9-17-A STREAMLINE 0.625 4.0 45.709 70.169 1336.0 1.440 9.265 47.7
P-12 EMPTY TUBE 0.000 -- 30.951 50.187 984.2 0.951 9.324 48.5 R-17-B STREAMLINE 0.625 4.0 33.758 57.388 1046.5 1.451 9.251 48.1
P-13 EMPTY TUBE 0.000 - 43.707 49.907 1305.1 0.947 9378 45.8 9-17-C STREAMLINE 0.625 4.0 21.688 41.707 729.9 1.490 9.196 49.2
P-14 EMPTY TUBE 0000 -- 43.689 52.765 1303.3 1.060 9.367 45.0 9-17-D STREAMLINE 0.625 4.0 9.758 24.701 381.7 1.650 9.126 50.1
R-1-A EMPTY TUBE 0.000 -- 21.829 8.334 725.3 0.964 9.078 44.8 R -17-E STREAMLINE 0.625 4.0 5.869 11.355 253.6 1.744 9.093 49.2
R-1-B EMPTY TUBE 0.000 -- 24.085 41.402 818.0 0.974 9.472 45.0 9-18-A STREAMLINE 0.750 12.0 43.232 59.314 1276.1 1.516 9.251 45.6
R-1-C EMPTY TUBE 0.000 -- 27.085 93.400 918.7 0.964 9.722 45.1 9-18-B STREAMLINE 0.750 12.0 32.102 45.603 1001.3 1.494 9.211 46.2
R-1-D EMPTY TUBE 0.000 -- 43.983 8.701 1227.7 0.952 8.772 50O4 9-18-C STREAMLINE 0.750 12.0 21.221 36.093 716.5 1.524 9.182 46.7
R-1-E EMPTY TUBE 0.000 -- 45.532 47.221 1316.5 1.008 9.153 49.1 9-18-0 STREAMLINE 0.750 12.0 9.338 23.259 368.6 1.557 9.126 47.6
R-1-F EMPTY TUBE 0.000 -- 46.075 95.956 1386.4 1.017 9.560 46.0 R-18-E STREAMLINE 0.750 12.0 5.725 12.963 245.7 1.614 8.991 49.0
R-1-G EMPTY TUBE 0.000 -- 10.179 42.003 4130 0.903 9.588 45.4 9-19-A STREAMLINE 0.750 8.0 43.892 53.706 1284.7 1.680 9.200 45.5
R-1-H EMPTY TUBE 0000 -- 8.501 10.685 346.6 0.887 9.228 46.3 9-19-B STREAMLINE 0.750 8.0 31.087 50.185 976.1 1.745 9.215 45.9
R-1-I EMPTY TUBE 0.000 -- 5.622 7.593 245.3 0.848 9.091 49.2 9-19-C STREAMLINE 0.750 8.0 21.470 45.695 721.8 1.859 9.171 46.9
R-2-A SOLID ROD 0.250 0.0 40.907 10.502 1185.7 1.152 8.978 45.6 9-19-0 STREAMLINE 0.750 8.0 10.424 26.618 397.6 2.102 9.015 48.0
R-2-B SOLID ROD 0.250 0.0 42.370 42.079 1257.9 1.120 9.264 45.2 9-19-E STREAMLINE 0.750 8.0 6.154 13.198 258.4 2.021 8.925 49.1
R-2-C SOLID ROD 0.250 0.0 45.245 94.776 1353.8 1.214 9.472 44.9 9-20-A STREAMLINE 0.750 4.0 47.222 68.788 1353.7 1.845 9.147 49.2
R-2-0 SOLID ROD 0.250 0.0 27.027 9.364 853.8 1.113 9.007 44.5 9-20-B STREAMLINE 0.750 4.0 32.559 50.606 1005.1 1.899 9.145 48.9
R-2-E SOLID ROD 0.250 0.0 32.239 94.536 10329 1.162 9.495 45.9 9-20-C STREAMLINE 0.750 4.0 20.862 45.749 706.9 2.001 9.195 48.6
R-2-F SOLID ROD 0.250 0.0 16.477 9.217 574.2 1.074 9.000 46.8 9-20-0 STREAMLINE 0.750 4.0 10.683 26.098 410.4 2.105 9.123 49.2
R-2-G SOLID ROD 0.250 0.0 19.966 66.561 696.8 1.068 9.411 47.4 R-20-E STREAMLINE 0.750 4.0 5.981 12.576 255.3 2.156 9.023 50.1
R-2-H SOLID ROD 0.250 0.0 8.765 13.330 352.2 0.911 9.154 4892 9-21-A STREAMLINE 0.875 12.0 43.288 63.426 1269.8 1.837 9.197 45.8.R-2-I SOLID ROD 0.250 0.0 5.270 7.702 232.5 0.778 9.080 49.8 R-21-B STREAMLINE 0.875 12.0 32.777 48.950 1012.3 1.880 9.160 45.9
R-3-A SOLID ROD 0.125 0.0 49.745 36.026 1355.4 0.996 8.778 55.5 9-21-C STREAMLINE 0.875 12.0 20.209 31.325 681.6 1.948 9.082 46.7
R-3-B SOLID ROD 0.125 0.0 36.556 29.280 1068.7 0.963 8.856 54.0 9-21-0 STREAMLINE 0.875 12.0 10.223 23.464 390.8 2.129 8.995 48.2
R-3-C SOLID ROD 0.125 0.0 23.372 23.848 747.4 0.967 8859 54.5 R-21-E STREAMLINE 0.875 12.0 5.718 13.436 242.1 2.166 8.874 49.4
R-3-0 SOLID RO9 0.125 0.0 9.202 8.956 349.2 0.832 8.725 56.8 9-22-A STREAMLINE 0.875 8.0 42.932 56.576 1252.3 2.203 9.129 45.9
R-3-E SOLID ROD 0.125 0.0 6.326 6.753 258.1 0.754 8.702 58.2 9-22-B STREAMLINE 0.875 8.0 30.639 49.393 952.6 2.358 9.099 46.4
R-5-A DISKS 0.750 12.0 50.518 28.866 1337.7 1.598 8.556 57.4 9-22-C STREAMLINE 0.875 8.0 21.574 45.161 714.2 2.530 9.041 47.3
R-5-B DISKS 0.750 12.0 36.797 27.442 1045.9 1.683 8.622 56.0 9-22-0 STREAMLINE 0.875 8.0 11.329 25.823 420.6 2.768 8.918 48.4
R-B-C DISKS 0.750 12.0 25.889 27.853 772.0 1.790 8.434 61.3. -22-E..STREAMLINE 0.875 8.0 5.776 12.925 24'.9 2.792 8.790 50.1
R-3-E DISKS 0.750 12.0 7.007 7.774 267.6 2.072 8.319 62.5 -23-A STREAMLINE 0.875 4.0 46*995 67.522 1319.8 3.453 8.954 51.3
R-6-0 DISKS 0.750 12.0 12.317 15.853 415.1 2.058 8.216 66.2 9-23-B STREAMLINE 0.875 4.0 32.127 47.401 987.3 3.390 9.080 48.2
R-6-E DISKS 0.750 12.0 6.910 11.342 260.3 2.095 8.188 66.6 9-23-C STREAMLINE 0.875 4.0 21*488 46.467 719.5 3.454 9.141 47.7
R-7-A DISKS 0.750 8.0 55.763 35.447 1382.6 2.020 8.172 67.5 9-23-0 STREAMLINE 0.875 4.0 11.019 28.048.419.0 3.426 9.093 48.2
R-7-B DISKS 0.750 8.0 42.072 28.819 1106.8 2.008 8.197 66.8 R-23-E STREAMLINE 0.875 4.0 5.677 13.612 242.3 3.538 8.941 49.9
R-7-C DISKS 0.750 8.0 26.609 23.501 766.2 2.186 8.187 67.0 9-24-A DISKS 0.625 2.0 45.003 69.438 1343.8 2.052 9.438 42.7
R-7-0 DISKS 0.750 8.0 11.170 12.067 379.9 2.428 8.131 67.5 9-24-B DISKS 0.625 2.0 29.534 47.125 952.3 2.139 9.368 43.3
R-7-E DISKS 0.750 8.0 7.681 8.668 280.0 2.461 8.08668.7 9-24-C DISKS 0.625 2.0 17.632 36.518 629.1 2.106 9.359 44.1
R-8-A DISKS 0.750 4.0 54.345 38.632 1375.9 2.326 8.302 63.5 9-24-0 DISKS 0.625 2.0 10.302 26.808 405.5 2.159 9.286 45.5
R-8-B DISKS 0.750 4.0 42.285 31.139 1123.3 2.518 8.285 63.5 R-24-E DISKS 0.625 2.0 5.725 13.481 250.1 2.341 9.157 47.1
R-8-C DISKS 0.750 4.0 26.744 24.177 776.9 2.681 8.268 63.7 9-25-A DISKS 0.875 2.0 33.612 66.924 1073.1 4.177 9.526 41.5
R-8-D DISKS 0.750 4.0 10.750 13.061 371.6 2.984 8.201 64.5. -25-B DISKS 0.875 2.0 26.854 47.103 886.2 4.215 9.410 42.7
R-8-E DISKS 0.750 4.0 7.089 8.877 265.0 3.045 8.163 64.9 9-25-C DISKS 0.875 2.0 18.163 40.272 641.4 4.087 9.324 44.6
R-9-A DISKS 0.875 12.0 52.216 35.269 1321.0 2.170 8.230 65.7 9-25-D DISKS 0.875 2.0 10.107 25.924 395.3 4.199 9.193 47.2
R-9-B DISKS 0.875 12.0 40.626 28.177 1078.9 2.253 8.217 66.0 R-25-E DISKS 0.875 2.0 6.019 13.221 257.7 4.301 9.065 48.9
R-9-C DISKS 0.875.12.0 27.195 25.230 783.0 2,.311 8.223 66.0 9-26-A DISKS 0.750 2.0 43.890 62.657 1290.2 2.520 9,244 45.5
R-9-D DISKS 0.875 12.0 12.068 13.536 406.7 2.545 8.183 66.2 9-26-8 DISKS 0.750 2.0 32.324 51.167 1002.0 2.629 9.172 46.8
R-9-E DISKS 0.875 12.0 7.181 9.000 267.6 2.547 8.156 66.3 9-26-C DISKS 0.750 2.0 18.592 47.713 643.0 2.846 9.0182 47.8
R-10-A DISKS 0.875 8.0 48.655 35.721 1249.0 2.485 8.234 65.2 9-26-D DISKS 0.750 2.0 11.048 27.998 420.0 2.850 9.096 48.6
R-O10-B DISKS 0.875 8.0 37.349 30.612 1007.9 2.630 8.211 65.6 R-26-E DISKS 0.750 2.0 6.158 13.397 259.8 2.970 8.972 50.1
R-10-C DISKS 0.875 8.0 26.487 25.011 766.2 2.755 8.217 65.3 9-27-A DISKS 0.750 8.0 41.630 55.134 1234.5 2.093 9.223 44.1
R-10-D DISKS 0.875 8.0 10.944 13.391 375.1 2.994 8.162 65.8 R-27-B DISKS 0.750 8.0 31.151 60.673 987.5 2.060 9.312 43.5
R-10-E DISKS 0.875 8.0 7.087 9.084 2643 3.114 8.143 65.8 9-27-C DISKS 0.750 8.0 18.330 43.113 645.3 2.270 9.307 42.8
R-11-A DISKS 0.875 4.0 42.300 31.636 1130.0 3.404 8.332 62.5 9-27-0 DISKS 06750 8.0 10.275 26.823 402.0 2.337 9.218 43.1
R-ll-B DISKS 0.875 4.0 30.804 27.522 877.3 3,505 8.338 62.3 R-27-E DISKS 0.750 8.0 5.408 13.028 236.9 2.488 9.075 44.5
R-11-C DISKS 0.875 4.0 21.200 22.149 651.0 3.686 8.344 61.9 9-28-A SOLID ROD 0.625 0.0 43.995 80.223 1291.2 1.718 9.235 44.6
R-11-D DISKS 0.875 4.0 10.707 13.140 375.8 3.607 8.323 65.2 R-28-B SOLID ROD 0.625 0.0 32.903 69.404 1022.5 1.664 9.231 44.8
R-11-E DISKS 0.875 4.0 7.035 9.100. 267.3 3.781 8.282 62.8 9-28-C SOLID ROD 0.625 0.0 18.813 52.291 654.0 1.539 9.246 45.1
R-12-B DISKS 0.625 8.0 40.145 34.037 1095.0 1.677 8.420 62.1 9-28-D SOLID ROD 0.625 0.0 11.100 37.947 432.0 1.321 9.333 44.1
R-12-C DISKS 0.625 8.0 23.812 23.890 719.5 1.814 8.403 62.2 R-28-E SOLID ROD 0.625 0.0 5.918 15.391 265.3 0.825 9.462 44.9
R-12-D DISKS 0.625 8.0 10.678 13.666 374.5 1.962 8.310 64.0 9-29-A SOLID ROD 0.750 0.0 44.201 85.710 1271.4 2.470 9.061 45.2
R-12-E DISKS 0.625 8.0 7.110 9.272 269.4 1.904 8.278 64.6 R-29-B SOLID ROD 0.750 0.0 34.287 75.702 1036.2 2.379 9.052 45.4
R-13-A DISKS 0.625 12.0 55.363 36.550 1401.5 1.425 8.332 64.6 R -29-C SOLID ROD 0.750 0.0 21.335 54.031 703.7 2.248 8.992 46.4
R-13-B DISKS 0.625 12.0 40.521 30.052 1095.8 1.450 8.362 63.8 R-29-D SOLID ROD 0. 7.0 12.098 39.175 444.4 1.963 8.958 47.2
R-13-C DISKS 0.625 12.0 25.811 23.622 765.4 1.478 8.380 63.6.R-29-E SOLID ROD 0.750 0.0 6.415 22.352 269.2 1.287 9.025 48.8
R-13-0 DISKS 0.625 12.0 11.593 13.021 401.0 1.577 8.331 64.1. ______________________________
R-13-E DISKS 0.625 12.0 7.144 8.815 271.1 1.603 8.298 64.5
R-14-A DISKS 0.625 4.0 53.538 40.735 1387.9 10979 8.476 59.9 o L// l0 14 il/ak5 D
9-14-B DISKS 0.625 4.0 37.688 32.553 1045.5 2.130 8.456 59.9 J Integrated mean value of (/) Pr (k/)
R-14-C DISKS 0.625 4.0 24.775 26.210 747.1 2.131 8.453 60.1 q is in units of (BTU/hr-ft2)
R-14-D DISKS 0.625 4.0 9.242 13.465 337.1 2.491 8.397 60.6 T is the inlet tr temperature in eg F
R-14-E DISKS 0.625 4.0 6.826 10.136 263.0 2.433 8350 61.2 ins the inlet water temperature in deg F

TABLE XIII
LOCAL HEAT TRANSFER MEASUREMENTS
EMPTY.TUSIBE
THREADEPD ROO IN CENTER D~ 0*290
RUN P-i P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-IO P-lI P-12
RUN R-2-A R-2-B R-2-C R-2-D R-2-E R-2-F R-2-G R-2-H R-2-1
Z H/HO H/O H/HO H/HO /HO H/O H/O H/O H/HO H/HO /"HO /HO z H/HO H/O H/HO H/O H/HO H/HO H/HO H/HO H/HO
1.49 1105 11203 1.074 1160 1.071 1099 1.083 1.178 0.851 1.155 1.118 1.06 1149 1265 1226 1277 115 24 1146 1234 1.163 21180
5.41 1.061 1.044 1.029 0.950 1.015 0 977 0.994 0.972 1076 0.911 0.957 0.972 5341 1201 1.141 1.191 1.136.s113 1.078 1.021 0.939 0.861
9.42 1.034 1.040 1.021 0.909 1.004 0.962 0.991 0.924 1.267 0.889 0.932 0 962 9.42 1.153 1.119 1183 1.130 1.109 1.054 0.995 0.846 0.764
13.40 1.021 1.030 1i007 0.908 0.990 0.953 0.980 0.909 1.254 I0.81 0 922 0 952 13.40 1.187 1.111 1.190 1.119 1.120 1.080 1*007 0*870 0.733
17.41 1.021 1.030 0.999 0.912 0.984 0.952 0.978 *904 -1 152 0.878 0.919 0.952 17.41 1.168 1.110 1.200 1123 1.138 1.080 1.033 0.884 0.736
22.17 1 031 1.040 1.005 0.932 0.985 0.958 0.978 0901 1.162 0.877 0926 0.957 22.17 1.162 1.117 i.222 1.121 1.164 1.086 1.064 0.901 0.753
25.34 1.035 1.015 1.007 0 944 0.990 0.959 0.982 0.901 1.179 0.881 0.930 0.959 25.34 1.165 1.123 1.221 1.143 1.172 1.097 1.a076 0.911 0.768
29.32 1.028 1.052 1.008 0.957 0.988 0.964 0.985 0.909 1.217 0. 891 0.938 0.959 29.32 1.219 1.142 1.249 11.73 1.199 1.135 1 107 0.937 0 799
33.30 1.022 1.053 1.013 0.976 0 993 0.972 0.990 0.921 1.244 08o 6 0.942 0.965 33.30 1.358 1.418 1.531 1.400 1.542 1444 1.531 1.518 1 487
37.28 1.009 1.047 0.997 0987 0.982 0.971 0.979 0.922 1.258 0.892 0.943 0.953 37.28 1.176 1.164 1.274 1.197 1.259 1.162 1.209 1.149 1.101
41.20 0o 991 1.063 1.006 1.005 0.991 0.980 0.986 0.932 1.285 0.898 0.952 0.961 41.20 1.130 1.136 1.243 1.141 1.201 1*073 1.121 0.985 0.907
45.27 0.973 1.036 0.985 1.004 0.966 0.964 0.964 0929 1.294 0.892 0.945 0.943 45.27 1.138 1.123 1.231 1.093 1s189 1.071 1.103 0.950 0 829
49.22 0.985 1.052 0.995 1.026 0.978 0.976 0.971 0.941 1.337 0.904 0.956 0.957 49.22 1.140 1.135 1.245 1.103 1.207 107l1 1.128 0 950 0.819
53.26 0 985 1.049 0.985 1.032 0.967 0.967 0.966 0.924 1.330 0.902 0.955 0.949 53.26 1.127 1.127 1.227 1.102 1.193 1.061 1.122 0.945 0815
57.28 0.998 1.071 0 999 1.053 0.978 0.986 0.980 0.942 1.367 0 915 0.971 0 962 57.28 11.33 1.133 1.238 1.091 1.211 1.073 1.145 0.962 0.830
62.47 0.971 1.108 1.023 1.098 1.000 1.014 1.003 0.981 1.447 0.931 1.001 0.994 62.47 1.125 1.185 1.324 1066 1.296 1.064 1.224 1.006 0.878 j y
H
EMPTY TUBE PLAIN ROD IN CENTER D a 0.125 DISKS S *~ 120 D * 0.750 U)
RUN P-13 P-14 R-1-A R-I-B R-1-C R-I-D R-I1-E R-I-F R-1-G R-I-H R-1-I RUN R-3-A R-3-B R-3-C R-3-0 R-3-E R-5- A 8 C 0 E
Z H/HO H/HO HWHO /HO H/HO H/O H/HO H/HO H/H H/HO H/O z WHO H/HO H/O H/O H/HO X H/NO H/HO H/HO __ H/HO'149 1.035 1.135 1.072 1.&129 1.116 1.065 1.107 1 115 1.142 1#092'l131 ~ 1.49 1.076 1I097 1.117 1.092' 1078 0.00 1.192 1 141 1.208 1.197
541 0.984 1.076 1.009 0 998 0.959 1.013 1.044 1.031 0.922 0.943 0.932 5.41 1.022 0.993 0 971 0.897 0.836 0.00 1.107 1.054 1.047 0.894
9 42 o.974 1.066 0.988 0.977 0.937 0.994 1.025 1.021 0.866 0.901 0.876 9.42 1.013 0.979 0.965 0.859 0.791 0.00 1.081 1025 1.056 0.59
13.40 0.962 1.055 0.988 0.970 0.936 0.994 1.019 1.013 0.847 0.886 -0850 13.40 1.002 0.955 0 957 0.834 0.749 0.00 1.047 1.001 1.010 0842
17.41 o.957 1.054 0.974 0.974 0.941 0.971 1.015 1.013 0.855 0.883 0.843 17.41 0.995 0.949 0.951 0.826 0.738 0.00 2.313 2.002 2.049 2.284
22.17 0.955 1.060 0.969 0 975 0.953 0.953 1.014 1.021 0.866 0.879 0.841 22.17 1.008 0959 0.965 0.835 0.742 3.77 1.674 1 841 1958 221
25.34 0.956 1.066 0.974 0.978 0.958 0.963 1.015 1.023 0.878 0.885 0845 25.34 1.008 0.961 0.966 0835 0.744 6.94 1.331 1 435 1.510 1.635
29.32 o.959 1.066 0.984 0.981 0.965 0.976 1.018 1.023 0.891 0.880 0.850 29 32 1.008 0.969 0.974 0.833 0.745 0.96 2.483 2.237 2.35 2.746
33.30 0.959 1.070 0.974 0.993 0.984 0.954 1.027 1.041 0.915 0.892 0.850 33.30 1o.005 0.966 0970 0.828 0.751 4.94 1.613 1.852 1.946 2.402
37.28 0.951 1.065 0t964 0.979 0.978 0.956 1.011 1.025 0.921 0.890 0.857 37.28 0.990 0.961 0.965 0.825 0.752 8.92 1.273 1.392 1.441 1598
41.20 0.950 1.066 0.955 0.990 0.996 0.937 1.019 1.040 0.945 0.896 0.856 41.20 1.002 0.972 0.976 0.837 0.766 0.84 2.503 2.266 2 358 2.640
45.27 0.929 1.057 0.936 0.965 0.972 0.913 0.984 1.002 0.939 0.885 0.848 45.27 0.978 0.959 0.963 0.828 0.758 4.91 1503 1.696 1875 2 326
49.22 0.944 1.071 0.945 0.976 0.987 0.924 0.995 1014 0.960 0.900.860 49.22 0.989 0.970 0.978 0.842 0770 8.86 1232 1.372 1.465 1603
53.26 0.936 1.067 0.960 0.982 1.000 0.942 1.003 1.026 0.973 0.901 0.858 53.26 0.99S 0980 0.984 0.852 0.785 0.96 3.040 2.877 3.087 3.281
57.28 0.947 1.074 0.954 0,994 1.016 0.941 1.014 1.038 0.992 0.910 0.864 57.28 0.999 0.9 0.992 0.854 0.789 4.98 1.406 1.384 1.503 1620
62.47 0.975 1.106 0.909 1.019 1.055 0.911 1.042 1.077 1.034 0.907 0.876 62.47 1.041 1014 1.020 0.884 0813 10.17 1.188 1.223 1.317 1.256
Note: This table was prepared directly from IBM cards,: making it necessary to use all
upper case letters. The term H/HO in the Table refers to h/ho; the symbol D in
the table refers to d; and the symbol S. in the table refers to s.

TABLE XIII (COOT'D)
DISKS S = 12.0 D = 0.750 DISKS S = 8.0 D = 0.750 DISKS S 8.0 D = 0.875 DISKS S = 4.0 D = 0.875
RUN R-6-DR-6-E R-7- A B C D E RUN R-1O A B. C D E R-11- A B C D E
Z X H/HO H/HO X H/HO H/HO H/HO H/HO H/HO z X H/HO H/HO H/HO H/HO H/HO X H/HO H/HO H/HO H/HO H/HO
1.49 000 1177 175 0.00 1.130 1.117 1.145 1.134 1125 1.9~00011521163116311501128 0.00 1123 1.141 1.151 11091114
5.41 0.00 0.961 0.906 0.00 1.053 1.030 1.016 0.944 0.915 5.41 0.00 1.074 1.070 1.043 0.948 0.913 0.00 1.033 1.030 0.998 0.921 0.882
9.42 0.00 0.939 0.864 0.00 1.055 1.026 1.016 0.929 0.893 9.42 0.00 1.080 1.071 1.042 0.927 0.880 0.00 1.030 1 023 0.988 -0.894 0.845
13.40 0.16 2.037 2.191 0.78 2.083 2.251 2.458 2.587 2.653 13.40 1.90 2.878 3.030 3.038 3.027 3 147 0.00 1.017 1.011 0.979 0.882 0.827
17.41 4.17 2.495 2.599 4.79 1.667 1.594 1.708 2.006 2.112 17.41 5.91 2.040 2.196 2.396 2.684 2.888 0.00 2.163 2.284 2.422 2.562 2.763
22.17 8.93 1.687 1.749 1.59 2.472 2.643 2.848 2.932 2.950 22.17 2.71 3.601 3.742 3.703 3.663 3.764 0.72 4.539 4.489 4.518 4.192 4.317
25.34 0.16 2.524 2.657 4.76 1.680 19617 1.724 1.927 1.987 25.34 5.88 1.964 2.018 2.167 2s 441 2.504 3.89 3.052 3.257 3.532 3.578 3.680
29.32 4.14 2.660 2.683 0.78 2.300 2.472 2.749 2.957 3.057 29.32 1.90 2.800 3.073 3.183 3.299 3 484 3.89 3.851 4.028 4.446 4.608 4.849
33.30 8.12 1.724 1.753 4.76 1.718 1.662 1.786 1.990 2.054 33.30 5.88 2.109 2.1522 2276 2.573 2.655 3.89 3.449 3.501 3.753 3.836 4.147
34.30 9.12 1.555 1.577 5.76 1.601 1 571 1,.654 1.727 1.775 34.30 6.88 1.902 1.936 1.927 2.180 2.251 0.91 2.976 3.007 3.097 2.918 3.037
35.30 10.12 1.459 1.464 6.76 2.456 2.502 2.761 3103 3.107 35.50 7.88 2.593 2.622 2.848 3.237 3.432 1.91 3.358 3.497 3.682 3.525 3.697
36.29 11.11 1.316 1.306 7.75 2.264 2.213 2.325 2.545 2.610 36.29 0.91 2.797 2.986 3.030 2.957 3.037 2.90 3.162 3.335 3.564 3.645 3.792
37.28 0.16 2.773 2 810 0.78 2.341 2.405 2.742 3.010 3.013 37.28 1.90 2.742 3.014 3.198 3.328 3.476 3.89 3.548 3.570 3 822 4.092 4.448
38.28 1.16 2.287 2.400 1.78 2.267 2.289 2.536 2.925 2.884 38.28 2.90 2.644 2.951 3.207 3.524 3.563 0.91 3.071 3.164 3.295 3.1843311
39.28 2.16 2.836 2.898 2.78 2.138 2.054 2.286 2.640 2.668 39.28 3.90 2.474 2.674 2.920 3.303 3.373 1.91 3.238 3.408 3.670 3.597 3.754
40.28 3.16 2.694 2.742 3.78 1.978 1.872 2.031 2.389 2.407 40.28 4.,0 2.291 2.367 2.545 2.928 3.043 2 91 2.906 3.055 3.261 3.283 3.475
41.20 4.08 2.553 2.558 4.70 1.898 1.812 1.943 2.167 2.238 41.20 5.82 2.162 2.204 2.238 2.570 2.718 3.83 3.324 3.516 3.778 3.861 4.183
45.27 8.15 1.680 1.735 0.81 1.967 1.899 2.152 2.613 2.618 45.27 1.93 2.353 2.479 2.693 3.121 3.346 3.92 3 590 3.953 4.415 4.601 5.021
49.22 0.16 1.873 1.908 4.76 1.405 1.384 1.460 1.523 1.525 49.22 5.88 1.928 2.060 2.088 2.368 2 52.8 3.89 2.224 2.394 2.631 2.686 2.822
53.26 4.20 1.812 1.950 0.84 1.884 1.858 1.996 2.014 1.988 53.26 1.96 2.176 2.297 2.383 3.052 3.214 7.93 1.456 1.507 1.566 1.539 1.559
57.28 8.22 1.420 1.405 4.86 1.295 1.282 1.359 1.373 1.372 57.28 5.98 1.734 1.766 1.777 1.869 1.885 1195 1.233 1.263 1.262 1.218 1.221
62.47 13.41 1.195 1.152 10.05 1.275 1.220 1.254 1.229 1.205 62.47 11.17 1.329 1.327 1.309 1.320 1.368 17.14 1.206 1.213 1.187 1.094 1.071
34.30* 9.12 1.618 1.661 5.76 1.765 1.670 1.775 1.926 1.955 34.30* 6.88 1.996 2.029 2.055 2.343 2.414 0.91 5 161 5.038 5.086 4.570 4.667
36.29* 11.11 1.401 1.431 7.75 3.280 3.217 3.262 3.232 3.234 36.29* 0.91 5.218 5.343 4.910 4.742 4.699 2.90 3.549 3,654 3.880 3.801 3.976
39.28* 2.16 2.608 2.665 2.78 2.572 2.684 2.980 3.173 3.153 39.28* 3.90 3.405 3.773 3.788 4.010 4.015 1.91 5.819 5.711 5.875 5.369 5.492
34.30+ 9.12 1.486 1.552 5.76 1.524 1.520 1.633 1.735 1.826 34.30+ 6.88 1.760 1.820 1.835 2.101 2.229 0.91 4.361 4.434 4.595 4.232 4.340 R
36.29+ 11.11 1 323 1.364 7.75 2.374 2.124 1.969 2.143 2.220 36.29+ 0.91 4.344 4.520 4.332 4.307 4.324 2.90 3.016 3.209 3.440 3.464 3.641 H
39.28+ 2.16 1.761 1.863 2.78 2.443 2.508 2.622 2.937 2.890 39.28+ 3.90 3.090 3.461 3.607 3 869 3.926 1.91 4.769 4.880 5.160 4.878 5.093
DISKS S = 4.0 D = 0.750 DISKS S = 12.0 D = 0.875 DISKS S = 8.0 D = 0.625 DISKS S = 12.0 D = 0.625
RUN R-8- A B C D E R-9- A a C D E RUN R-12- B C D E R-13- A B C D
z X H/HO H/HO H/HO H/HO H/HO X H/HO H/HO H/HO H/HO H/HO Z X _ H/HO H/HO H/HO H/HO X H/HO H/HO H/HO H/HO H/HO
1.49 0.00 1.142 1.140 1.152 1.162 1.ll 0 000 1.154 1151 1.154 1.19 I1L22 1~ 49 0.00 1.118 1.141 1.147 1.082 0.00 1.124.111-4 1.126 1.121 1.108
5.41 0.00 1.073 1056 1.032 0.965 0.906 0.00 1.083 1.060 1.033 0.973 0.898 5.41 0.00 1.041 1.004 0.940 0.869 0.00 1.049 1.023 1.005 0.948 0.891
9.42 0.00 1.506 1.475 1.570 1.815 1.911 0.00 1.070 1.048 1.024 0.948 0.861 9.42 0.00 1.032 1.004 0.920 0.842 0.00 1.049 1,026 1.003 0.930 0.658
13.40 3.82 2.486 2.685 2.825 3.376 3.526 0.00 1.783 1.837 1.827 2.109 2.202 13.40 1.90 1.944 2.152 2.259 2.164 0.00 1.348 1.358 1.375 1.5721639
17.41 3.85 2.432 2.682 2.868 3.262 3.290 3.79 2.665 2.838 1.749 3.339 3.315 17.41 5.91 1 442 1.469 1.585 1.564 3.17 1.523 1.523 1.606 1.842 1.857
22.17 0.59 3.038 3.078 3.025 3.079 3.105 8.55 1.791 1.708 2.929 1.882 1.934 22.17 2.77 1.963 2.184 2.233 2.110 7.93 1.528 1.445 1.430 1.464 1.475
25.34 3.76 2.170 2.460 2.605 2.912 2.976 11.72 1.975 2.059 2.118 2.600 2.718 25.34 5.94 1.420 1.456 1.514 1.488 11.10 1.489 1.472 1.487 1.675 1.738
29.32 3.82 2.358 2.658 2.816 3.262 3.376 3.76 2.656 2.830 2.921 3.386 3.369 29.32 1.90 2.081 2.405 2.530 2.379 3.14 1.507 1.530 1.521 1.782 1.839
33.30 3.82 2.427 2.762 2.909 3.290 3.401 7.74 1.727 1.625 1.739 19182030 3330 588 142114801548 1.481 7.12 1.265 1.255 1.300 1.3381340
34.30 0.84 2.383 2.665 2.734 2.989 3.027 8.74 1.540 1.499 1.565 1.619 1.689 34.30 6.88 1.367 1.408 1.464 1.385 8.12 1.240 1.215 1.2661 1299 1286
35.30 1.84 2.277 2.559 2.728 3.039 2.996 9.74 1.461 1.430 1.467 1.504 1.498 35.30 7.88 19959 2.285 2.533 2.433 9.12 1.265 1.220 1.263 1.274 1.255
36.29 2.83 2.070 2.168 2.353 2.620 2.598 10.73 1.364 1.353 1.354 1.397 1.377 36.29 0.91 1.930 2.1172 2343 2.307 10.11 1.204 1.163 1.204 1.212 1.187
37.28 3.82 2.411 2.788 3,0O29 3.488 3.628 11.72 2.222 2.331 2.400 2.834 2.912 37.28 1.90 1.924 2.179 2.401 2.280 11.10 1.451 1.447 1.509 1.586 1.598
38.28 0.91 2.169 2.337 2.502 2.819 2.863 0.78 2.319 2.443 2.338 2.486 2.454 38.28 2.90 1.702 1.782 2.055 2.007 0.16 2.023 2.071 2.249 2.1682150
39.28 1.91 2.168 2.352 2.614 2.905 2.909 1.78 2.706 2.895 2.934 3.227 3.153 39.28 3.90 1.539 1.599 1.761 1.748 1.16 1.563 1.765 1.606 1.972 2.089
40.28 2.91 2.014 2.065 2.280 2.533 2.542 2.78 2.831 3.052 3.168 3.511 3.438 40.28 4.90 1.451 1.521 1.638 1.583 2.16 1.489 1.605 1.556 1.811 1.918
41.20 3.83 2.294 2.533 2.744 3.089 3.235 3.70 2.819 3.062 3.127 3 511 3.455 41.20 5.82 1.452 1.509 1.585 1.509 3.08 1.478 1.531 1.527 1.654 1.757
45.27 0.93 2.247 2.461 2.798 3.281 3.531 7977 1.638 1.611 1.724 1.836 1904 45.27 1.99 1.720 1.821 2.093 2.212 7.15 1.197 1.179 1.231 1.290 1.289
49.22 4.88 1.927 1.919 2.155 2.547 2.677 11.72 1.795 1.737 1.891 2.066 2.146 49.22 5.94 1.283 1.329 1.394 1.336 11.10 1.359 1.334 1.432 1.436 1.414
53.26 3.95 1.820 1.841 1.960 2.290 2.414 3.82 2.063 1.959 2.169 2.607 2.031 57.28 5.98 1.185 1.191 1.216 1.194 7.22 1.128 1.186 1.109 1.1161141
57.28 3.99 1375 1.409 1.448 1.505 1.509 7.84 1.628 1 609 1.696 1.751 1.739 53.26 1.96 1.644 1.714 1 780 1.642 3.20 1.210 1.350 1.213 1.264 1.281
62.47 9.18 1.248 1.248 1.234 1.215 1.206 13.03 1.319 1.281 1.278 1.257 1.217 62.47 11.17 1.180 1.156 1.134.1.085 12.41 le171' 1.152 1.115 1.080 1.054
34.30* 0.84 3.603 3.629 3.466 3.436 3.421 8.74 1.654 1.572 1.644 1.686 1.755 34.30* 6.88 1.479 1.490 1 541 1.487 8.12 1.323 1.293 1.333 1.356 1.345
36.29* 2.83 2.298 2.546 2.596 2.771 2.785 10.73 1.537 1.476 1.488 1.476 1.455 36.29* 0.91 1.908 1.831 1.869 1.874 10.11 1.317 1.273 1.287 1.289 1.260
39.28* 1.91 3.531 3.641 3.445 3.423 3.430 1.78 4.999 4.426 4.102 4.097 3.811 39.28* 3.90 1.918 2.060 2.240 2.134 1.16 1.721 1.786 1.619 1.855 1.858
34.30+ 0.84 2.658 2.828 2.698 2.641 2.646 8.74 1.458 1.442 1.492 1.548 1.589 34.30+ 6.88 1.357 1.409 1.481 1.410 8.12 1.219 1 222 1.242 1.285 1.281
36.29+ 2.83 2.073 2.168 2.334 2.555 2.577 10.73 1.347 1.324 1.353 1.405 1.403 36.29+ 0.91 1.533 1.410 1.455 1.400 10.11 1.183 1 161 1.177 1.224 1.218
39.28+ 1.91 44444 2.792 2.657 2.601 2.586 1.78 3643 3.416 3.246 3.141 2.978 39.28+ 3.90 1.762 1.778 1.881 1.794 1.16 1.196 1.220 1.145 1.2631300

TABLE XIII (CONT'D)
STREAMLINE S = 12.0 D = 0.750 STREAMLINE S = 8.0 D = 0.750
RUN R-14- A B C' D E R-15- A B C D E RUN R-18- A B C D E R-19- A B C D E
Z X H/HO H/H O H/HO H/ HO H/HO X H/HO H/HO H/HO H/HO H/HO Z X H/HO H/ /HO H/HO H/HO X H/HO H/HO H/HO H/HO H/HO
1,49 0.00 1.123 1.203 1.140 1.210 1.132 0.00 1,151 1,150 1161 1.140 1,105 1,49 0.00 1.193 1.175 1.183 1.165 1.139 0.00 1.192 1.184 1.213 1.281 1.144
5.41 0.00 1.050 1.137 1.006 1.005 0.900 0.00 1.064 1.026 0.992 0.912 0.843 5,41 0.00 1.114 1.077 1.033 0.946 0.883 0o00 1.122 1.076 1.051 1.054 0 911
9,42 0O00 1.080 1.167 1.056 1.042 0.935 0.00 1.074 1.039 0 993 0.868 0.812 9,42 0.00 1.105 1.062 1.014 0.879 0.826 0.00 1.115 1,008 1.032 0.997 0.849
13.40 3.76 1.747 1.988 1.986 2.273 2.135 0.28 1.109 1.115 1.122 1.056 1.017 13.40 0.47 1.851 1.809 1.866 1.724 1t644 2.71 2 075 2 103 2.297 2.547 2.523
17.41 3.79 2.005 2.290 2.386 2.816 2.725 4.29 1.170 l1165 1.176 1.195 1.242 17.41 4.48 1.400 1.421 1.486 1.554 1.635 6.72 1.264 1.261 1.293 1.439 1.376
22.17 0.59 2.577 2.550 2.408 2.525 2.400 9,05 1.295 1.254 1.195 1.100 1.054 22.17 9.24 1.170 1.141 1.121 1.088 1.117 3.52 1.886 2.016 2.212 2.597 2.484
25.34 3.76 1.731 1.917 2.150 2.544 2.448 0.28 1.184 1.196 1.208 1.187 1.164 25.34 0.47 2.000 1.915 1.951 1.905 1.949 6.69 1.256 1.251 1.293 1.426 1.325
29,32 3.76 1.881 2.198 2.471 2.947 2.890 4.26 1.204 1.205 1.220 1.271 1.316 29.32 4.45 1.446 1.455 1.506 1.607. 1.682 2.71 1.869 1.990 2.141 2.417 2 448
33.30 3.76 1.842 2.025 2.314 2.769 2.734 8.24 1.128 1.101 1.073 1.030 1.011 33.30 8.43 1.195 1.161 1.141 1.110 1.126 6.69 1.248 1.239 1.268 1.396 1.310
34.30 0.78 2.199 2.538 2.373 2.723 2.641 9.24 1.109 1.082 1.049 0.992 0.962 34.30 9.43 1.174 1.137 1.102 1.055 1.057 7.69 1.274 1.265 1.294 1.424 1.339
35.30 1l78 1.914 2.070 2.107 2.462 2.372 10.24 1.147 1.118 1.076 1.004 0.955 35.30 10.43 1.190 1.145 1.104 1.039 1.029 0.73 1.955 2.058 2.155 2.370 2.263
36.29 2.77 1.691 1.727 1.786 2.170 2.086 11.23 1.135 1,116 1.082 1.014 0.968 36.29 11.42 1.253 1.220 1.190 1.126 1.124 1.72 2.304 2.367 2.576 2.769 2.637
37.28 3.76 1.868 2.048 2.333 2.840 2.745 0.28 1.384 1.384 1.389 1.376 1.336 37.28 0.47 2.072 1.953 1.968 1.993 2.028 2.71 1.941 2.083 2.256 2.530 2.515
38,28 0.78 1.996 2.286 2.248 2,690 2.691 1.28 1.660 1.727 1.816 1.909 1.994 38.28 1.47 2.287 2.309 2.467 2.509 2.604 3.71 1.635 1.696 1.827 2.155 2.097
39.28 1.78 1.779 1.860 1.859 2.287 2.290 2.28 1.356 1.404 1.471 1.601 1.725 471142114561541 786 682
39.28 2.47 1.901 1.921 2.088 2.315 2.550 4.71 1.421 1.456 1.541 1.786 1.682
40.28 2378 1.620 1.646 1.657 12994 12962 3428 1.262 1.293 1.339218 441 1.527 40.28.3.47 1.632 1.677 1.761 1.967 2.108 5.71 1.328 1.344 1.400 1.598 1.464
41520 3.70 1.806 1.870 1.976 2.435 2.332.207 1.258 1.268 16288 10352 1.403 41.20 4.39 1.477 1.500 1.573 1.707 1.799 6.63 1.275 1.278 1.321 1.487 1.390
45927 3.796 1677 17829 1.915 21434 2.455 8027 1*107 1.092 14079 14061 1.043 45.27 8.46 1.173 1.148 1.137 1.138 1.168 2.74 1.871 1.983 2.139 2.529 2.467
49,22 3*76 1*655 1*756 1*758 2*139 2o170 0#28 1#395 1*415 1*441 1*401 1#305 49.22 0.47 1.871 1.828 1.8133 1761 1.735 6.69 1.265 1.265 1.301 1.478 1.368
53.26 3.82 1.609 1.707 1.674 1.905 1.877 4.32 1.170 1.180 1.207 1.291 1.327 53.26 4.51 1.457 1.474 1.541 1.658 1.718 2.77 1.891 1.961 2.109 2.476 2.342
57.28 3.86 1.343 1.385 1.283 1.416 1.358 8.34 1.122 1.107 1.092 1.081 1.049 5728 85312321206120312051229 67913011302134415271403
62.47 9.05 1.237 1.239 1.144 1.175 1.092 13.53 1.144 1.131 1.113 1.096 1.017 62.47 13.72 1.238 1.184 1.146 1.078 1.055 11.98 1.251 1.216 1.209 1.271 1.103
34.30* 0.78 2.419 2.507 2.25:6 2.391 2.305 9.24 1.196 1.158 1.111 1037 1.004 34.30* 9.43 1.253 1.203 1.162 1.099 1.098 7.69 1.370 1.344 1.358 1.471 1.352
36.29* 2.77 1.884 1,939 1.995 2.303 2.242 11.23 1.266 1.240 1.193 1.109 1.060 36429* 114214111367133612591248 172231623262448 503 2397
39.28* 1.78 2.314 2*481 2.285 2.423 2.320 1.28 1.764 1791 1831 836 1852 3629* 1142 1411 1367 1.336 1.259 1.248 1.72 2.316 2.326 2.448 2.503 2.397
3430+ 0.782 1914 1921 1 2761 1884 1815 9.24 1.122 19108 1073 1018 975 39.28* 1.47 2.344 2.323 2.349 2.193 2.286 3.71 1.866 1.917 2.010 2.282 2.160
34.30+ 0e78 1,914 1,921 le761 1,884 l815 9,24 1,122 1,108 1,073 1.018 0,975 " \ \ \^, n
36.29+ 2.77 1.719 1.719 1.789 2.104 2.016 11.23 1.137 1.130 1.109 1.063 1.017 34.30+ 9.43 1.181 1.144 1.131 1.106 1.110
36io29, 2,77 1 -719 1 -719 I -fQ ->inA ) n& 11it ii-a7 i -3.f i io ie\Lti f\i-f 34.30+ 9.43 1.181 1.144 1.131 1.106 1.110 7.69 1,280 1.280 1.313 1.463 1.35 8!
789 104 016 23 137 130 1093629+ 11.42 1.282 1.269 1.269 1.246 1.249 1.72 1.917 2,033 2.153 2.403 2.373 o
39.28+ 1.78 1.884 1.886 1.729 1.849 1.782 1.28 1.516 1.560 1.619 1.676 1.704. 39.28+ 1.47 1.963 1.933 2.051 2.162 2.312 3.71 1.588 1.634 1.730 1.973 1.905_
STREAMLINE S = 8.0 D = 0.625 STREAMLINE S = 4.0 D = 0.625 STREAMLINE S = 4.0 D = 0.750 STREAMLINE S = 12.0 D = 0.625 1
RUN R-16- A B C D E R-17- A B C D E RUN R-20- A B C D E R-21- A B C D E
Z X H/HO H/HO H/HO H/HO H/HO X H/HO H/HO H/HO H/HO H/HO Z X H/HO H/HO H/HO H/HO H/HO x H/HO H/HO H/HO H/HO H/HO
1.49 0.00 1.158 1.148 1.158 1.168 1,109 0.00 1,184 1.169 1.172 1.169 1.155 1.49 0.00 1.155 1.154 1.165 1.159 1.095 0.00 1.187 1.176 1.178 1.213 1.141
5.41 0.00 1.077 1.043 1001 0.945 0.868 0.00 1.090 1l047 1.003 0.938 0.859 5.41 0.00 1.072 1.043 0.992 0.945 0.860 0.00 1.110 1.074 1.037 0.989 0.882
9,42 0.00 1.063 1.029 0.978 0.884 0.812 0.00 1.073 1,029 0e974 0.867 0.791 9.42 0.00 1.060 1.025 0.967 0.882 0.798 0.00 1.100 1.061 1.013 0.927 0.818
13.40 2.40 1.368 1.379 1.430 1.543 1.643 0.00 le061 10e21 0.968 0.844 0.752 13.40 0.00 1.046 1.016 0.959 0.864 0.761 0.53 2.400 2.470 2.470 2.357 1.935
17.41 6.41 1.110 1.090 1.075 1.073 1.064 0.00 1.056 1l017 0.967 0.849 0.734 17.41 0.00 1.046 1.015 0.964 0.872 0.761 4.54 1.860 1,877 2.014 2.132 2.387
22.17 3.20 1.398 1.420 1.467 1.586 1.694 0,00 1.047 1.013 0.964 0.849 0.728 22.17 0,00 1.041 1.011 0.966 0.878 0.765 9.30 1,335 1.333 1.324 1.388 1.466
25.34 6.37 1.146 1.126 1.114 1.116 1.108 0.00 l1076 1.042 0.995 0.888 0.761 25.34 0.00 1.127 1.108 1.075 0.992 0.872 0.53 2.706 2.856 2.852 2.915 2.693
29.32 2.39 1.465 1.483 1.541 1.672 1.749 3.14 1.318 1,322 1.346 1.462 1.532 29.32 3.45 1.687 1.743 1.873 2.017 2.091 4.51 1.692 1.742 1.854 2.172 2.361
33,30 6.37 1l153 1.133 1.121 1.129 1l118 3.20 1.277 1.271 1.294 1.409 1.539 33.30 3.45 1.596 1 655 1.772 1.928 1.988 8.49 1 275 1.259 1.261 1.385 1.477
34.30 7.37 1.163 1.14'1 1.133 1.137 1.135 0.29 1.336 1.351 1.409 1.561 1.693 34.30 0.47 2.198 2.192 2.183 2.177 2.129 9.49 1.233 1.203 1.196 1.285 1.356
35.30 0.35 1e322 1.263 1.302 1.322 1.304 1.29 1.7'69 1.781 1.854 2.000 2.079 35.30 1.47 2.119 2.184 2.322 2.357 2.417 10.49 1.239 1,206 1.181 1.254 1.287
36.29 1.34 1.686 1.700 1.776 1.929 2.010 2.28 1.507 1.517 1.576 1.732 1.843 36.29 2.46 1.727 1.801 1.931 2.063 2.093 11.48 1.331 1,316 1.307 1.382 1.430
37.28 2.33 1.403 1.417 1 466 1,596 1l635 3.27 1.367 1.372 1.411 1.532 1e618 37.28 3.45 1.593 1.651 1.786 1.936 1l997 0.53 2.978 3.093 3.066 3.188 2.927
38.28 3.33 1.255 1.256 1.299 1.383 1.437 0.32 1.323 1.351 1.327 1.604 1.713 38.28 0,47 1.961 1.941 1.974 2.037 2,095 1.53 2.781 2.723 2.815 3.114 2.870
39.28 4.33 1.197 1.186 1.206 1.253 1.276 1.32 1.686 1.695 1.744 1.934 1.961 39.28 1.47 2.028 2.104 2.246 2.358 2.460 2.53 2.495 2.617 2.921 3.415 3.566
40.28 5.33 1.174 1.154 1.163 1.185 1l188 2.32 1.444 1,459 1.507 1l686 1.776 40.28 2,47 1.722 1.786 1.925 2.084 2.163 3.53 2.112 2.241 2 517 2.843 3.037
41.20 6.25 1.149 1.130 1.125 1.139 1.124 3.24 1.341 1.344 1.377 1.499 1.590 41.20 3.39 1.566 1.616 1.747 1.901 1.988 4,45 1l841 1.994 2.148 2.343 2.561
45.27 2.36 1.388 1.413 1.485 1.632 1.683 3.36 1l313 1.324 1.362 1.478 1.562 45.27 3.48 1.424 1.626 1.768 1.967 2.034 8.52 1 307 1.326 1.350 1.438 1.504
49.22 6.31 1.159 1.142 1.14-4 1.165 1l149 3.39 1.313 1.315 1.340 1 446 1.507 49.22 3.45 1.462 1.498 1l587 1.698 1.673 0.53 2.981 3.059 3.075 2.944 3.020
53.26 2.39 1.420 1.437 1.499 1.629 1,701 7.43 1.135 1.111 1.088 1.096 1.075 53.26 7.49 1.194 1.174 1.180 1.190 1.168 4.57 1.751 1.841 2.123 2.348 2.499
57,28 6.41 1.191 1.165 1.153 1.157 1.126 11.45 1.121 1,091 1.053 1.007 0.927 57.28 11.51 1.159 1.120 1102 1.054 0.993 8.59 1.301 1.300 1.374 1.522 1.568
62.47 11.60 1.190 1.145 1.109 1.048 0.951 16.64 1.190 1.130 1.088 1.020 0.902 62.47 16.70 1.198 1.144 1.120 1.041 0.938 13.78 1.246 1.198 1.152 1.172 l1138
34.30* 7.37 1.262 1.217 1.198 1l184 1.171 0.29 1.497 le509 1.559 1.705 1.845 34*30* 0.47 2.068 1.992 1.981 1.981 1.960 9e49 1.309 1.276 1.258 1.326 1.430
36.29* 1.34 1.871 1.853 1.911 2.013 2.063 2.28 1.665 1.650 1.674 1.800 1.873 36.29* 2.46 1.920 1.944 2.024 2.094 2.080 11.48 1.511 1.496 1.463 1.555 1.606
39,28* 3.33 1.424 1.403 1.436 1.508 1.536 0.32 1.606 1.620 1.675 1.835 1.923 39.28* 0.47 2.138 2.044 2.025 1.988 1.989 1.53 3.374 3.752 3.926 4.133 4.038
34.30+ 7,37 1.185 1.154 1l146 1.130 1 105 0.29 1.302 1l327 1.378 1.529 1.629 34.,30+ 0.47 1.984 1.907 1.844 1.797 1 773 9.49 1.222 1.201 1.201 201.287 1388
36,29+ 1.34 1.525 1,532 1,597 1.710 1.723 2.28 1.413 1.422 1.457 1.612 1.719 36.29+ 2.46 1.632 1.687 1.814 1.964 2.026 11.48 1.391 1.393 1.407 1.499 1.578
39,28+ 3.33 1.265 1.253 l1280 1.335 1385 0.32 1.342 1.364 1.419 1.580 1.675 39.28+ 0.47 2.032 1.978 1.964 1.939 1.873 1.53 3.001 3.277 3.542 4.012 3.895

TABLE XIII (CONT'D)
STREAMLINE S = 8.0 D = 0,625 STREAMLINE S = 4.0 0 = 0.625 DISKS S = 2.0 D = 0750 D ISKS S = 8.0 D = 0.750
RUN R-22- A B C D E R-23- A B C D E RUN R-26- A B C D E R-27- A B C D E
Z X H/HO H/HO H/HO H/HO H/HO X H/HO H/HO H/HO H/HO H/HO Z X H/HO H/HO H/HO H/HO H/HO X H/H H/HO H/H H/HO H/HO
L 1,~ ~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-4'9 0,00 1,185 1,*19627-.t 116 00 08 1,120 1,204 1,194
1.49 0.00 1 185 17 1203 1208 1136 OOU 1.177 1187 1.195 1171 1115 1 49 000 1152 1162 IT 1 177 1 116 O00 1180 1182 1204 1194 1127
5,41 0.00 1.105 1.080 1.034 0.989 0.882 0.00 1.089 1.068 1.017 0.942 0.842 5.41 0.00 1.065 1.034 0.957 0.941 0.851 0.00 1.088 1.043 1.006 0.964 0.856
9.42 0.00 1.099 1.072 1.017 0.931 0.821 0.00 1.076 1.049 0.993 0.882 0.785 9.42 0.00 1.052 1.017 0 "926 0.878 0.789 0.00 1.074 1.026 0.973 0.8950.791
13.40 2.33 2.703 2.878 2.975 3.469 3.340 0.00 1.061 1.031 0.980 0.863 0 755 13.40 0.00 1.042 0.999 0.924 0.854 0.756 0O00 2 104 2.106 2.230 2.394 2.653
17.41 6.34 1.610 1.786 1.843 1.830 le913 0.00 1.062 1.025 0.984 0.868 0.756 17.41 0.00 1042 0.998 0.939 0.862 0.751 3.98 2.006 2.083 2.237 2.467 2.570
22.17 3.14 2.659 2.788 2.936 3.298 3.385 0.00 1.056 1.023 1.003 0.870 0.755 22.17 0.00 1.037 0.990 0.939 0.861 0.751 0.78 2.916 2.980 3.232 3.122 3.393
2.5,34 6.31 1.538 1.665 1.819 1.911 1.979 0.00 1.171 1.139 1.147 1.125 1.186 25.34 0.00 1.061 1.016 0.969 0.891 0.779 3.95 1.955 1l883 2.188 2.316 2.418
29.32 2.33 2.730 2.838 2.978 3.482 3.663 3.33 3.102 3.190 3,463 3.756 4.598 29.32 0.09 3.521 4.244 3.923 3.506 3.698 7.93 2.787 2.587 3.133 3.247 3.307
33.30 6.31 1.454 1.596 1.713 1.813 1.805 3.33 2.893 2,892 3'006 3.377 3.902 33.30 0.09 3.122 4.105 4.217 4.114 4.007 3.95 1.911 1.819 2.089 2.216 2.366
34.30 7.31 1.582 1.708 1.814 1.937 1.988 0.35 3.993 3.870 3.612 2.511 2.193 34.30 1 09 2.751 2.795 3.089 2.794 2.883 4.95 1.799 1.623 1.856 2.006 2.147
35.30 0.35 3.491 3.696 3.659 3.638 3.142 1.35 4.135 4.024 4 151 3.839 3.657 35,30 0.10 2.240 2.346 2.691 2.484 2.551 5.95 1.707 1.560 1.700 1.892 1.962
36.29 1.34 3.420 3.524 3.329 3.626 3.593 2.34 3.483 3.480 3.517 4.028 4.175 36.29 1.09 2.133 2.246 2.465 2.767 2.943 6.94 1.590 1.409 1.563 1.748 1793
37,28 2.33 2.607 2.797 3.045 3.435 3.506 3.33 2 922 2.867 3.109 3.419 3e850 37.28 0.09 3.253 3.676 4.240 4.053 4.150 7.93 2.864 2.487 2.729 3,176 2.808
38.28 3.33 2.137 2.358 2.800 3.189 3.372 0.35 3.876 3.759 3.420 2.849 2.694 38,28 1.09 2.377 2.433 2.586 2.627 2.821 0.97 2.417 2.470 2.595 2-770 3.193
39.28 4.33 1 787 2.026 2.350 2.600 2.662 1.35 3.743 3.711 3.888 4.044 3.724 39.28 0.10 3.080 3.806 4.077 4.202 4.058 1.97 2.031 2.255 2.500 2.395 2.630
40.28 5.33 1.616 1.801 2.037 2.201 2.273 2.35 3.320 3.306 3.461 3.877 4.257 40.28 1,10 2.613 2.648 2.853 2.863 2.923 2.97 1.924 29028 2.162 2-078 2.244
41.20 6.25 1.500 1.659 1.852 2.006 2.021 3.27 2 916 2.851 3.142 3.464 3.874 41.20 0.03 2.508 2.748 3.129 3.461 3.585 3.89 1.816 1.833 1.911 1.909 2.004
45.27 2.36 2.496 2.654 3.024 3.358 3.459 3.36 2.693 2.616 2.916 3-012 3.597 45.27 0.12 3.028 3.561 3.784 3.891 3.926 7.96 1.990 1.977 2.129 2.388 2.696
49.22 6.31 1.404 1.480 1.666 1.824 1.831 3.33 2.591 2.526 2.711 2.760 3 156 49.22 2.08 1.992 2.214 2.264 2.288 2.344 3.95 1.550 1.512 1.673 1.509 1.577
53.26 2.39 2.654 2.904 3.273 3.664 3.823 7.37 1.484 1.488 1.567 1.639 1.732 53.26 6.12 1.324 1.371 1.390 1.437 1.409 0.03 1.934 1.909 1.964 2.175 2.347
57.28 6.41 1.470 1.577 1.674 1.781 1.783 11.39 1.271 1.240 1.252 1.255 1.288 57.28 10.14 1.198 1.185 1 166 1.172 1.116 4.05 1.422 1.540 1.554 1.547 1.515
62.47 11.60 1.253 1.267 1.266 1.243 1.185 16.58 1.274 1.210 1.188 1.115 1095 62.47 15.33 1.213 1.186 1.141 1.081 0.985 9.24 1.304 1.277 1.241 1.246 1216
34.30* 7.31 1.584 1.697 1.792 1.889 1.930 0.35 4.278 4.084 4.018 3.832 3.937 34.30* 1.09 3.507 3.930.3748 3.432 3.265 4.95 1.974 1.838 2.096 2.162 2.309
36.29* 1.34 3.663 3.564 3.245 3.369 3.259 2.34 3.858 3.698 3.774 4.036 4.175 36.29* 1.09 2.554 2.743 3.030 3.256 3.292 6.94 1.754 1.541 1.714 1.893 1.926
39.28* 3.33 2.433 2.698 2.985 3.298 3.393 0.35 4.460 4.110 3.964 3.937 3.805 39.28* 1.09 3.172 3.568 3.613 3.349 3.168 0.97 3.419 39219 3.311 3.328 3.475
34.30+ 7.31 1.500 1.646 1.800 1.950 2.032 0.35 3.330 3.247 3.184 3.247 3.314 34.30+ 1.09 2.869 3.182 3.301 2.891 2.724 4.95 1.823 1.685 1.941 2.007 2118
36.29+ 1.34 3.067 3.206 3.188 3.548 3.582 2.34 2.902 2.938 3.186 3.462 3.560 36.29+ 1.09 2.180 2.394 2.681 2.790 2.874 6.94 1.542 1.401 1.606 1.770 1.845 13
39.28+ 3.33 2.171 2.443 2.840 3.162 3.221 0.35 3.431 3.288 3.225 3.460 3.613 39.28+ 1.09 2.834 3.030 3.232 2.767 2.960 0.97 2.524 2.715 2.697 2.725 2.606 H
DISKS S = 2.0 D = 0.625 DISKS S = 2.0 D = 0.875 PLAIN ROD IN CENTER D = 0.625 PLAIN ROD IN CENTER D = 0.750 I
RUN R-24- A B C D E R-25- A B C D E RUN R-28- A B C D E R-29- A B C D E
Z X H/HO H/HO H/HO H/HO H/HO X H/HO H/HO H/HO H/HO H/HO Z H/HO H/HO H/HO H/HO H/HO H/HO H/H H/HO H/HO H/HO
1.49 0.00 1.157 1.182 1.158 1.170 1.115 0.00 1.197 1.190 1.179 1.171 1.108 1.49 1.832 1.778 1.710 1.596 1.389 2.470 2.394 2.282 2.100 1,763
5.41 0.00 1.068 1.062 0.974 0.937 0.842 0.00 1.077 1.054 0.998 0.943 0.861 5.41 1 754 1.662 1.480 1.277 0.940 2.407 2.305 2.110 1.716 1.196
9,42 0.00 1.062 1.041 0.943 0.868 0.781 0.00 1.062 1 037 0.968 0.878 0.800 9,42 1.734 1.649 1.455 1.179 0.810 2.381 2.288 2.108 1.697 1.036
13.40 0.00 1.081 1.074 0.993 0.901 0.807 0.00 1.050 1,024 0.956 0.845 0.752 13.40 1.723 1.648 1.483 1.171 0.723 2.394 2.308 2.150 1.788 1.029
17.41 0.00 1. O055 1.033 0.950 0.897 0.785 0.00 1.052 1,025 0.961 0.851 0.743 17.41 1.707 1.642 1.487 1.227 0.665 2.350 2,280 2.142 1.803 1.117
2217 000 1.035 1.015 0.924 0.852 0.765 O.'O 1.049 1.021 0.983 0.850 0.741 22.17 1.700 1.640 1.492 1.242 0.626 2.452 2,377 2.239 1.917 1.211
2534 0C00 1.056 1.036 0.957 0.862 0.764 0.00 1.067 1.037 0.978 0.872 0.765 25.34 1.737 1.680 1.535 1.286 0.624 2.504 2.433 2.289 1.980 1.244
29,32 1.96 2.431 2.730 2.928 2.919 3.041 O.OU 1,068 1,040 0.981.0.880 0.780 29 32 2.202 2.164 2.114 1.965 1.258 2 728 2,643 2.470 2.231 1.470
33,30 1.96 2.381 2.562 2.662 2.782 2,710 1.65 4.041 4.481 4.729 5.304 5.584 33.30 1.840 1.805 1.727 1.642 1.356 2.604 2.527 2.395 2.285 1926
34*30 0.97 2.082 2.145 2.118 2.165 2.420 0.66 4.392 4.367 4.024 3.827 3.849 34,30 1.816 1.737 1.652 1.558 1.301 2.542 2,462 2.309 2.124 1.803
35,30 1.97 2.383 2.650 2.649 2.778 2.736 1.66 3.660 3.760 4.258 4.975 4.811 35.30 1.828 1.862 1.623 1.490 1.273 2.559 2.485 2.301 2.061 1.706
36.29 0.97 2.116 2.191 2.136 2.159 2.377 0.66 4.794 4.743 4.174 4.014 4.068 36.29 1.799 1.707 1.586 1.449 1.217 2.529 2,466 2.308 2.017 1.605
37,28 1.96 2.243 2. 530 2.527 2.659 2.701 1.65 3.719 3.630 4.084 5.066 5.177 37,28 1.785 1.718 1.575 1.411 1.168 2.536 2484 2.311 2019 1535
38.28 0.97 1.995 2.092 2.060 2.128 2.308 0.66 4.404 4.591 4.080 3.968 4.013 38.28 1.723 1.631 1.515 1.345 1.108 2.417 2.370 2.252 1.943 1.438
39,28 1.97 2.204 2.383 2 366 2.432 2.616 1.66 3.834 3.957 4.756 5.240 5.551 39.28 1.729 1.634 1,525 1.340 1.161 2.436 2.393 2.273 1.965 1.398
40.28 0.98 2.003 2.100 2.070 2.136 2.211 0.67 4.307 4.375 3.831 3.838 3.902 40.28 1.739 1.651 1.547 1.353 1 075 - 2.487 2.442 2.315 1.999 1.383
41.20 1.90 2.035 2.258 2.309 2.450 2.697 1.59 3.327 3.272 3.843 4.028 4.326 41.20 1.716 1.655 1.536 1 33-3.1041 2.461 2.409 2.282 1.997 1.351
45.27 1.99 2.041 2.222 2.157 2.280 2.396 1.68 3.516 3.578 4.104 4.500 4.994 45.27 1 698 1.666 1 564 1 373 0.936 2 502 2.387 2.265 2.013 1.266
49.22 1.96 1.678 1.771 1.714 1.76.6 1.726 3.64 1.801 1.788 1.870 2.071 2.140 49.22 1.738 1.704 1.607 1.433 0.899 2.555 2,434 2.314 2.091 1.351
53.26 6.00 1.266 1.267 1.234 1.249 1.218 7.68 1.300 1.270 1.257 1.297 1.294 53.26 1.760 1.734 1.643 1.481 0.899 2.556 2.433 2.324 2.116 1.510
57.28 10.02 1.185 1.158 1.112 1.100 1.038 11 70 1.193 1.153 1.123 1.081 1.038 57,28 1.810 1.781 1.688 1.533 0.943 2.639 2.503 2.385 2.182 1.642
62,47 15.21 1.216 1.177 1.108 1.057 0,968 16.89 1.224 1.169 1.135 1.040 0.935 62.47 1.964 1.915 1.812 1.648 1.049 2 820 2,741 2.584 2.383 1.805
34.30* 0.97 2.443 2.512 2.487 2.380 2.372 0.66 6.067 5.648 5.250 4.980 4.867 34.30* 2.116 2.005 1.892 1.662 1.086 3.048 2.931 2.674 2.345 1.904
36.29 — 0.97 2.330 2.429 2.392 2.378 2.402 0.66 5.91.7 5.673 5.622 5.207 5.091 36.29* 2.095 1.965 1.835 1.638 1.228 3.026 2.905 2.648 2.250 1.707
39.28* C.97 2.377 2.500 2.434 2.353 2.366 0.66 6.194 5.749 5.702 5.334 5.057 38.28* 2.077 1.945 1.811 1.611. 1.297 3 031 2.919 2.676 2.269 1.613
34.30+ 0.97 2 178 2 164 2 107 2 118 2.042 0.66 4 397 4. 563 4.504 4.473 4.216 34.30+ 1.816 1.742 1.694 1.610 1.351 2.422 2.381 2260 2038 1667
36.29+ 0.97 2.141 2.208 2.155 2.162 2.141 0.66 4518 4697 5.09 1469 1229 2440 2401 2.275 19991572
39.28+ G097 2.132 2.220 44444 2.151 2.119 0.66 4.461 4.658 4.959 5.077 5.057 38.28+ 1.767 1.699 1.600 1.432 1 159 2.488 2.450 2 334 2 055 1.518
__-__ ___._ _____ __ 4

APPENDIX D
CALCULATIONS REQUIRED FOR EXAMPLE HEAT EXCHANGER DESIGN
The specified data were
W 250,000 lbm/hr
Q = 10,000,000 BTU/hr
AT = 100 deg. F
h' 3000 BTU/hr - deg. F - ft
CF 2~28 x 10-3 dollars/ftl* - hr
CE = 4.88 x 10-7 dollars/BTU
c = 1 BTU/lm- deg. F
p -62.4 1bm/ft3
k - 0 355 BTU/hr - dego F - ft
= 2.42 lbm/ft - hr
Unless otherwise specified, all calculations will be made for a
tube diameter of one incho Values of calculated quantities corresponding
to other diameters may be obtained by ratios,
Pr =- c_ 1 x 42 68555 (82)
k 0o353
pr1/3 - 108995 (D-l)
Factors Needed to Calculate Fixed Cost
D 1 I (D-2)
k ATm 12 x 0.353 x 100
- 2~3607 x 103 (BTU/hr ft2)
Lk Dm = 2.6535 x 10"2 (BTU/hr - ft2)~ (D-3)
-217

-218B2 ~ (h' D/k)-1 (113)
= 0.353 x 12 (D-4)
3000 x 1
= 1.4120 x 10-3
1-rn 04 (D-5)
Q"m _ (10,000,000)~ 4
630.96 (BTU/hr) 4-5)
CF (D/k ATm)m [1 + B2 Nu]
Fixed Cost - m (D-6)
Qlm Num
Fixed Cost = (2.28 x 10-3)(2.6535 x 10 2)(1 + 1.4120 x 10-3Nu) 0' 6 (7)
(630.96) Nu0'
For D = 1 Fixed Cost 0.98855 x 10 7 (1 + 1.4120 x 10 Nu) (D-8)
For D Fixed Cost (D-8)
0.6
Nu
-7 o. 0.6
For D = 0.5 Fixed Cost = 0.63256 x 10 (1 + 28240 x 10 Nu) (D-9)
Nu
0.6
For D 0.25 Fixed Cost = 0.41731 x 10-7 (1 + 56480 x 103 Nu)0 (Nu
The preceeding three equations were used to plot the fixed cost
curves in Figure 48. The second equation was also used to plot the fixed
cost in Figure 49.
The limiting fixed cost for B2Nu > > 1 was calculated as follows
CF
CF
Limiting Fixed Cost 1-
(Q3 ( ~ ~AT) (D-ll)
m
(2.28 x 10o3) (D-12)
(630.96) (3000 x 100).6

-219= 1.8692 x 10-9 (dollars/BTU)
Factors Needed to Calculate Pumping Cost for Empty Tube Geometry
f = C Re - (81)
Nu = C2 Ren2 Pr1/3 (80)
C 0.079; 0.27; n = 0.250; n = 0.80
_3 - n 53 - 0.25
n2 0.80(i)
C1 0.079 4
B1 = = 0.97453 x 104 (112)
2 (C2)P 2 (0.027)3
2
B = - 2 (114)
3 J g 2 p D c
JC gc PLTm
=... _ (2.42/3600)2 (D-13)
(777.5) (322)(62.4)2 (1/12)2 ()(100oo)
= 6.675 x 1015
pr(P/3-1) = (6,8555) 14583 = 13241 (D-14)
CE B1 Nup- (1 + B2 Nu) B3
Pumping Cost = (D-15)
pr (P/~r1)
Pumping Cost = (4.88 x 10-7) (0.97453 x 104) Nu2'4375
(1 + 1.4120 x 10-3 Nu) x 6.675 x 10-15 (D-16)
lo3241
For D = 1.0 Pumping Cost = 2.5275 x 10-17Nu24375(1 + 1.4120 x 10"- Nu)(D-17)

-220For D = 0.5 Pumping Cost = 9.5900 x 10-17 Nu24375 (1 + 2.8240 x 10-3 Nu) (D-18)
For D = 0.25 Pumping Cost = 38.360 x 10-17Nu2" 4375 (1 + 5.6480 x 10-3 Nu) (D-19)
The preceding equations were used to plot the pumping cost curves
in Figure 48; the second equation was used to plot the pumping cost for the
empty tube in Figure 49.
Factors Needed to Calculate Design Parameters from Optimum
Nusselt Numbers
4 W (C2 prl/3) l/n2
Ntube =/ (120)
x i D Nul/n2
(4) (250,000) (0.027 x 1.8995)1(D-2
(D-20)
(2.42)(3.14159)(D/12) Nu 25
0.38522 x 105 (
(D-21)
D Nu1'25
0.38522 x 105
For D =0.5 Ntube 0= (5 125 (D-223)
For D = 0.25 Ntube =0.5)(175)122 x 25 241 (D-24)
L =. QZ D [1 + Nu/(h'D/k)] Nu(/n2) (122)
4 W k Tm (C2 prl/) /n2
L= (107)(2.42)(D/12) [E + Nu/(0,353/3000 D) ] Nu0.25 (D-25)
(4) (250,000) (0.353) () (0.027 x 1.8995)1,25
= 2.3408 D (1 + 1,4120 x 10 -3 Nu) Nu025 (D-26)

-221For D = 1 L = 2.3408 x 1 x (1 + 1.4120 x 10-3 x 600) x (600)0.25 (D-27)
= 21.4 ft.
For D = 0.5 L = 9.6. ft. (D-28)
For D = 0.250 L =.K2 ft. (D-29)
Factors Needed to Evaluate Turbulence Promoting Geometries
All factors are for D = 0.50 inch
Pumping Cost
Geometry I Disks d = 0.625 s = 4
C1 = 0.06724; C2 = 0.17134; n1 -0.0425; n2 = 0.6941
p 3 + 0.0425 - 4.3834 (D-30)
0.6941
B1 0.0 4 0.76717 x 102 (D-31)
2 (0.17134) 4.
pr(p/3-1) = (6.8555)0'46113 = 2.4295 (D-32)
Geometry II Disks d =0.875 s = 8
C1 - 0.72954; C 0.22708; n = -0.0066; n = 0.6888
1 2 1 2
3 + o.oo66
P 30o.6888 4.65 (D-33)
B 0.72954 = 2.362 x 102 (D-4)
2 (0.22708)43650
Pr(P/3-1) = (6.8555)0'55 = 2.4699 (D-35)
For Geometry I
Pumping Cost = 4.1144 x 1019 Nu33834 (1 + 2.8240 x 10-3 Nu) (D-36)
This equation was used to plot the pumping cost curve in Figure 49.

-222Ntube 5.175 x 10 (D-37)
Nu1.4407
5.175 x 105 142.6 (D-38)
(300>'.4407
170a) l 44407
L = 0.17424 (1 + 2.8240 x 10-3 Nu) Nu04407 (D-39)
= 0.17424 (1 + 2.8240 x 10-3 x 300)(300)0.4407 (D-40)
= 3.958 ft.
For Geometry II
Pumping Cost = 12.461 x 10-19 Nu3 650(1 + 2.8240 x 10-3 Nu) (D-41)
This equation was used to plot the pumping cost curve in Figure 49.
4.7025 x 10
N - (D-42)
tube Nu.4518
4.7025 x 105
= 158.8 (D-43)
(245).4518
L = 0.19172 (1 + 2.8240 x 10-3 Nu) Nu04518 (D-44)
= 0.19172 (1 + 2.8240 x 10-3 x 245) (245)~'4518 (D-45)
= 3.859 ft.

UNIVERSITY OF MICHIGAN
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