THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING CRYOGENIC HEAT TRANSFER John A Clark August, 1968 IP-823

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CRYOGENIC HEAT TRANSFER* by John A. Clark Chairman Department of Mechanical Engineering University of Michigan Ann Arbor * Prepared for Advances in Heat Transfer, Vol. V, edited by T. F. Irvine, Jr. and J. P. Hartnett, Academic Press, Inc. Reproduced by permission.

TABLE OF CONTENTS Page I. INTRODUCTION.. **.....**O..*o*...............*..e. o 1 II. CONDUCTION HEAT TRANSFER............................ 13 A. Conduction in Solids...................... 13 B. Low Temperature Insulation. 24 C. Interfacial Phenomena............................. 32 III. FORCED-CONVECTION PROC ESSES.............................. 47 A. Flows with Moderate Property Variation................ 48 1. Laminar Flow.......*...............*.....aI...* 49 2. Turbulent Flow.....................*..*oo......000 50 3. Transition Flow...................... 60 4. Flow Outside of Ducts...........*......*.......** 61 B. Flow with Large Property Variation............... 63 IV. NATURAL CONVECTION PROCESSES.......................... 77 V. PRESSURIZED-DISCHARGE PROCESSES FOR CRYOGENS.............. 83 VI. STRATIFICATION IN CRYOGENIC VESSELS....................... 101 VII. MULTI-PHASE PROCESSES........................ 121 A. Boiling Heat Transfer............................ 121 1. Pool Boiling............................*.. 123 a. Nucleate Boiling............v.........a.... 123 b. Maximum and Minimum Heat Flux.............e00 O 147 c. Transition Boiling........................... 163 do Film Boiling.................. 0.. 0. 0 0 * 164 2. Forced Convection Boiling........a................ 185 a. Sub-cooled, Nucleate Boiling..............*.. 185 b. Saturated (Film) Boiling...................... 189 c. Maximum Nucleate Boiling Heat Flux (Burnout). 194 d. Pressure-drop in Two-phase Flow.............. 199 3. Gravic and Agravic Effects on Boiling Heat Transfer 208 4. Injection Cooling......... e*.................*.. * 223 5. Frost Formation............................... 224 i

TABLE OF CONTENTS (CONTINUED) Page VIII. RADIATION................................ 233 IX. HELIUM II............. 0..... 245 NOMENCLATURE.... 285 REFERENCES................................................. 291 ii

I. INTRODUCTION The first half of this century has seen intensive progress in the physics of low temperatures. This has been followed by the natural development of engineering applications in design and research. One consequence of this has been the introduction of a new word -cryogenic- into the lexicon of technology. -This word means the production of cold or a process involving very low temperatures. It is a combination of two Greek words kryos and -gen, kryos meaning icy.cold and -gen indicating an act of production. Hence, kryos plus.gen becomes cryogen which, accordingly means a refrigerant. The suffix -ic is appended giving it the adjectival form cryogenic, although the word also may be used aS a noun as it frequently is. An -s is sometimes also added forming the word cryogenics, which implies a broad connotation for low'temperature phenomena. Thus, today cryogenics is used to describe the science and technology of low temperature phenomena that is, below -1500C (-238F), where marked changes occur in the physical properties of materials. The growth of cryogenic technology in recent years has been exceptionally rapid and has led one observer to predict that "cryogenics will be to the second half of the 20th century what high temperature processing was to the first".(l) By 1965 the cryogenics industry had developed into a $650million a year business with expectation that by 1970 it would surpass $1-billion in yearly sales.(2) Although much of the early impetus to this growth as well as its principal market has been the missile and space programs of the United States Governments much development is o - curing in private industry. Natural gas, mostly methane, can be liquified -1

-2and easily shipped in insulated containers by barge or ship for ultimate use great distances from its source. Liquified natural gas (LNG), for example, from North African and American fields is shipped to England(3,4) and for peak load service a 290,000 bbl LNG storage facility has been built in the Hackensack Meadows, New Jersey, by the Transcontinental Gas Pipeline Corporation.(4) The use of very low temperature in medicine has permitted quick freezing processes important to brain surgery and the preservation of living cells for biological use and research.(2'5'6'14) In a similar application many foods are frozen for storage and transportation and a technique known as "freeze-drying" has introduced a new concept in food preservation. At low temperatures many materials exhibit extremely low resistance to the flow of electricity. This is called "superconductivity" and at present 24 elements and numerous alloys have been found to possess this property. Cryoelectronics is a developing science devoted to the exploitation of "zero" resistance electrical components for magnetic devices, computer memories and circuits and other electrical apparatus. (References 2, 7, 8, 9). The cooling of surfaces by cryogens, usually liquid nitrogen, is used to produce very low pressures in high vacuum systems. Called cryopumping it is extensively employed in space simulation systems.(2,10' ) Liquid oxygen (LOX) is used by the steel industry in blast furnace operations to increase production and improve plant efficiency. It is produced in LOX plants adjacent to the mills some having a capacity in excess of 500 tons per day.(l8) Liquid hydrogen (LH2) and liquid oxygen are used in large quantities by the NASA, the AEC and the U.S. Air Force in doubtless the most widely publicized use of cryogens. Owing to its high specific impulse

-3of approximately 1000 lbf/lbm-sec, liquid hydrogen is chosen as the coolant/propellant for a solid core nuclear rocket.(l2'l3) The ROVER program at Los Alamos is set up to operate at a flow rate of liquid hydrogen of 100 lbm/sec at 1200 psi for 500 sec. Cryogenic rockets and weapon systems employ both liquid hydrogen and kerosene as fuels and LOX as the oxidizer. Although the specific impulse of these combinations is not as high as pure LH2inn the nuclear rocket their choice provides the distinct advantages of easy handling storability and transportation without deterioration. The early rocket of Prof. Goddard in 1926 used LOX as did the German V-2. Liquid hydrogen is now used in bubble chambers(15) and for industrial hydrogenation processing. The production of liquid hydrogen is growing in the U. S. with one plant producing about 65 tons per day with an expected U. S. consumption by 1966 to be 4000 tons per month.(16) The Linde Co. has designed and operated a 28,000 gal railroad tank car for the transcontinental shipment of LH2 safely and with negligible loss.(7) Liquid hydrogen also is commonly transported conveniently and safely interstate in large tank trucks. Some of the typical cryogenic fluids in common industrial use (18) are listed in the following table as given by Zenner. A few cryo-. A few cryogenic gases, such as CO and Neon, are omitted since they are presently of minor industrial importance as cryogenic gases. Scott(l9) lists approximately 35 fluids as cryogens. The transport properties of several cryogens as related to liquid water of 10OF and 1- atmosphere pressure are listed in Table II. The properties of water also are given for comparison.

-4TABLE I THERMODYNAMIC PROPERTIES OF COMMON CRYOGENS (p =1 atm) Boiling Liquid Heat of Vaporization Ft of Gas at Cryogenic Point Density BTU/Ft STP per Ft3 Substance OF Lbm/Ft3 BTU/Lbm | Liquid of Liquid Oxygen -297 71.3 91.6 6531 862 Nitrogen -320 50.4 85.6 4319 696 Argon -303 87.5 70.0 6126 846 Methane -259 26.5 219.0 5808 637 Flourine -307 94.3 74.0 6975 959 Hydrogen -423 4.4 193.0 849 844 Helium -452 7.8 8.8 69 754!1

TABLE II TRANSPORT PROPERTIES OF CRYOGENiIC LIQUIDS AT 1 ATMOSPHERE (Relative to liquid water at 10OF, except Prandtl number) Cryogenic T k'C V Liquid IR kw.I ~w a w — w i aw N1-39 0.221 0.815 0.488 1 0.220 0.263 1 0.556 1500) 2.20 0.425 02 / 162 0.239 f 1.150 0.4o6 0.287 0.240 0.51 22502.21 H2i 1 36.7 0.186 0.071 2.30 0.021 0.285 1.145 32 3) 1.170.667 He-I 1 7.6 0.040 0.126 1.15 0.0047 0.036 0.277 1.58(10j3) 0.613 0.0118 He-II 3.6 2(105) j_0.148 1.15 0.00117 0.0076 1.17(106) ---------- 3.05(1078)_ _ PROPERTIES OF LIQUID WATER AT I ATMOSPHERE A-ND lOOF T I kw i p, cpw Clw v, aw aC Pr OR BTU/HR-R-FTt BTU/LbmR Lbm/HRFT Ft2/HR Ft HR Dynes/Cm (m 0R:- BTU/H-R-F """" In/L~ / 1bKT /1~1 ____ 6o 4.- 1-3H20 -0 - j364 62 1.0 1.65 27.6(10-3) 5.88(103) 58.8 4.52 0.250 ~~~6o 0.364 - I -~~~~~~~~~~~~~~~~~~-9 *B is related to the "most unstable wave length", Xd by Xd 2jrit4 where B 1 d d V~~~~~~~ie-v

The use of cryogenic substances has introduced several unique problems in heat transfer. The handling and transport of these fluids at very low temperatures in the presence of an atmospheric ambient has necessitated the development of specialized insulating methods and design techniques. Because of these large temperature differences new insulation systems consisting of multiple layer aluminum foil separated by a low conducting glass fiber matrix in a vacuum have been developed. These insulations, sometimes called super insulation or multiple layer vacuum insulation, are very effective in reducing the rate of heat leakage into cryogenic systems. One type, which has as many as 150 layers of aluminum foil per inch, has a mean thermal conductivity at 0.05 microns of mercury of 1/700 of that of air.(24) Thermal conduction effects in the supporting structure of cryogenic containers require special attention in their design for minimum heat loss. Another consequence of the large temperature differences in the high probability for the existence of boiling and twophase flow in storage containers and transfer lines. This has required considerable research on these topics for cryogenic fluids. The condensation of water vapor, carbon dioxide and other gases on cryogenic cooled surfaces is another problem of importance in cryogenic systems. At low temperatures the physical properties of many substances are significantly temperature dependent. This has introduced the need to consider the question of variable properties in the analysis and calculation of heat transfer processes atcryogenic temperatures. Important cases in point are thermal conduction at low temperatures and convective phenomena in the region of the critical point, a circumstance fairly commonly encountered with cryogens. An example of this is the variation of the specific heat of hydrogen with both temperature and pressure, as shown in Figure 1.

9 6 k l 11: tHYDROGEN E a6 a C l l ll PCRIT * 187.7 PSIA SEADER, ET AL (27) PRESSURE, PSIOA iv-g50 11>|~~~180 I'S q,0,,,Dv/ I 140 4- - 0 0 4 1r0 t- 2ILI U) 20 50 100 200 500 K)O 2000 4000 TEMPERATURE, ORlgure i. Speciff: Heat at Constan,.t Pr'ssure of Para-Hydrogen.

-8A significant characteristic of most cryogens is that they behave as "classical" fluids. That is, their physical behavior follows the well established principles of mechanics and thermodynamics and they obey the laws of similarity. This means that the principles of similitude may be applied and that the scaling laws are valid. Thus, the fundamental concepts of fluid behavior based on the classical laws of physics may be expected and the formulation of the governing elevations for analysis may be carried out in the same manner as is so familiar in the case of non-cryogenic fluids. Accordingly, convective heat transfer correlations can be formulated in terms of such well-known dimensionless quantities as Nusselt number, Reynolds number, Prandtl number, Grashof number and length-to equivalent diameter ratio, among others. An important exception to the "classical" behavior which has become known at very low temperatures is the behavior of liquid helium (He-II) at temperatures below 2.19K and certain other "electron gases" in solids.(22) Under these circumstances the substances exhibit what is known as "superfluid" behavior, which among other characteristics includes an enormous increase in the heat conducting ability. This is a relatively unexplored field at present from the standpoint of transport phenomena although a few of the available results, principally regarding helium, will be cited later. At the present time the engineering applications of superfluids is limited but they promise to increase. Examples of "superfluid" properties of helium, silver and copper are shown in Figures 2 and 3 for the thermal conductivity and specific heat. The shape of the specific heat curve in Figure 3 resembles the Greek letter lambda which has given rise to the identification of this transition from normal to superfluid characteristic in helium as the "X-point".

-9108 He _- KEESOM (26) Ag He | / / 7>SCOTT (19} I0I H.rZ ki 0.014 BTU/HR FT-F AT 7,6 R 10 10 10 T, OR Figure 2. Superconductivity at Cryogenic (Below -1500C) Temperatures.

-103.0 2.5 -.o 2.0 - 1.0 -J'0-" - I -' - - 01.2 1.4 1.6 1. 2.0 2.2 2.4 2.6 2.6 3.O A MEASUREMENTS OF APRIL 21, 1932 O MEASUREMENTS OF APRIL 26, 1*32 o KEE$OM AND CLUSIUS Figure 3. Specific Heat for Helium I and II.

-11Several of the important and useful sources of reference for information on the subject of cryogenics, including discussions on heat transfer, will now be given. Each year the proceedings of the Cryogenic Engineering Conference held in the United States is published under the title Advances in Cryogenic Engineering, (Plenum Press). This publication is now (1967) in its twelfth volume and contains reviewed papers on all aspects of cryogenic engineering. The journal Cryogenics, published in England by Heywood and Co., Ltd., London since 1960, devotes its pages to papers on low temperature engineering and research. Probably the original book to broadly treat this subject is Cryogenic Engineering by R. B. Scott (1959), Reference (19). This has been followed by Technology and Uses of Liquid Hydrogen, edited by R. B. Scott, W. H., W. H. Denton and C. M. Nicholls (1964), Reference (16). Two additional volumes on low temperature technology are Applied Cryogenic Engineering, edited by R. W. Vance and W. M. Duke (1962), Reference (13) and Cryogenic Technology edited by R. W. Vance (1964), Reference (14). The specialized topic of superfluid behavior is discussed in Superfluid Physics by C. T. Lane (1962) Reference (22), as previously mentioned. The subject of materials behavior is presented by the ASTM in Behavior of Materials at Cryogenic Temperatures (1966), Reference (23), Symposium on Evaluation of Metallic Materials in Design For Low-Temperature Service, (1962), Reference (24) and Report on Physical Properties of Metals and Alloys From Cryogenic to Elevated Temperatures (1961), Reference (25). A valuable source for information on physical properties at cryogenic temperatures is the Cryogenic Data Center, Institute for Materials Research, Cryogenics Divisions (formerly the Cryogenics Engineering Laboratory) of the National Bureau of Standards in Boulder, Colorado. This Institute also publishes reports and papers on cryogenic

-12science and technology as well as frequent bulletins listing their available publications. The publications of the various commissions in the Proceedings of the International Congress of Regrigeration contains frequent papers on cryogenics. Other U. S. sources having occasional papers and reports on cryogenic heat transfer and related subjects are the Transactions of the American Society of Mechanical Engineers (ASME), especially its Journal of Heat Transfer (series C) and Journal of Basic Engineering (series D), the Journal of the American Institute of Chemical Engineers (AIChE) and the Journal of the American Institute of Aeronautics and Astronautics (AIAA). Reports and technical notes treating cryogenic processes also are issued periodically by the National Aeronautics and Space Administration (NASA). This chapter will summarize some of the available information relating specifically to heat transfer phenomena at low temperatures. The technical presentation will include conduction,insulation and interfacial processes, forced and natural convection processes, pressurizeddischarge processes, stratification phenomena in vessels, multi-phase processes including two-phase flow and frost formation, among others, radiation and transport phenomena in the superfluid helium II, Although most of the material to be presented may be applied to non-cryogenic behavior, the emphasis will be on applications valid for the cryogens. Where possible or available cryogenic data will be cited and the uniqueness of low temperature application high-lighted. Surveys of heat transfer results appropriate to low temperature may be found in References (13, 14, 16, and 19). The author has published an extensive discussion of the general subject in Reference (14). The present contribution will differ from that presentation in the sense of a specific emphasis on cryogenic heat transfer and the inclusion of the more recently available material.

II. CONDUCTION HEAT TRANSFER Probably the principal new problems associated with conduction heat transfer at cryogenic temperatures are those of variable thermal properties and low temperature insulation, as indicated in the Introduction. In addition, the principles of conduction heat transfer and diffusion phenomena have been recently applied to some important applications in cryogenics, namely, interfacial processes in single and multicomponent systems. These will be discussed later in this section. A. Conduction in Solids* The differential. equato-.)il governing tile steady and unrsteady' conduction of heat in isotropic subs'tbalnces having variable thermaal proeI.ti. ies and including thc i.nfluence of internal heat ge neratic. l o:i.s ra a k aT a k + kT ST _ 1 at Fp(p ax x |y a 6 d PS(x,Y zt) PCp This equation is written in cartesian coordinates for convenience only. The properties k, p, Cp are usually i-emperature dependent for a given substance. Typical low temperature variations in k and Cv (for solids Cv and Cp are essentially the same) are shown in Figures 4 and 5. Equation (1) is non-linear owing to the temperature dependency in k, p and Cp and accordingly is very difficult to solve. A simple transformation (28), however, can be used to aid in the solution. A new vari* The emphasis in this section will be on the influence of temperature dependent physical properties. Heat conduction analysis and calculation for uniform properties is well documented in References (29, 30, 31 and -13

500 400 300 200 Ag A SCOTT (19) 100 - Cu 80 B F 60 o 40 30208 O 6 C- CU - E i.c 3 os.~ 2- Cu IJJ 0.8 0,66 G 0.4 - 0.30.2 - Cu Al 0.1I 2 4 10 20 40 100 200 400 TEMPERATURE, "K Figure 4. Low Temperature Thermal Conductivity of Metals. A, silver 99.999Z pure; B, high purity copper; C, coalesced copper; D, copper, electrolytic tough pitch; E, aluminum single crystal; F, freemachining tellurium copper; G, aluminum, l00F; H, aluminum, 6063-T5; I, copper, phosphorus deoxidized; J, aluminum, 2024-T4; K, free-machining leaded brass.

-1525 -. 20 4. ~ (~ I ~~~SCOTT (19) 015........... I. I To convert these units to (BTU per lb-mole-deg R) divide the numberical values given in the figure by 4.184. Thus, the asyrmptotic value of 25 becomes its classical value of 3R, cal/gr-mole, which has exactly the same numerical as units in BTU/lb -mole-~ R. Sustase Sb stance _ u t Li 430o g 95 Na 160 Ga 125 K r100 In ^00 Be 980 T1 96 mg 320 Ti 350 Ca 230 Zr 28o br 170 Hf 213 A1 390 TH 245 Diamond 1850 Cr 440 Graphite 1500 MO 375 si 625 w 315 Ge 290 Mn 350 Sn (white) 165 a-Fe 430 Sn (gray) 240 y-Fe 320 Pb 86 Co 385 Sb 140 Ni 375 Bi 110 Ru 400 Ne 63 Rh 370 A 85 Pd 275 Cu 310 Pt 225 Ag 220 LiF 650 Au 180 NaC1 275 Zn 240 MgO 800 HA 213 Figure 5. Temperature Dependence of Specific Heat, Cv

able E is defined as T E = f k(T)dT, (2) TR where TR is an arbitrary reference temperature. Hence, we may formulate the space and time derivatives of T in Equation (1) as at k(T) at and aT a62E ax, /= 2 n etc. (4) With these transformations solutions to heat conduction problems are sought in terms of the function E. Since E is a unique function of T by Equation (2) the temperature distribution T(x,y,z,t) is determined from the function E(x,y,z,t). Thus, Equation (1) becomes E =a(T) + a + -2 + a(T) p"(x,y,z,t), (5) tx2 6y2 6z2 where a(T) is the variable thermal diffusivity, defined as k(T)/p(T)Cp(T), and is a function of temperature. Equation (5) is still non-linear since both a(T) and E are functions of temperature. However, it has been put into a more useful form than Equation (1). In case a(T) is a severe function of temperature it will be necessary to employ numerical methods to the integration of Equation (5), something which will be discussed later. For numerical formulations Equation (5) is a more convenient and useful representation than Equation (1). There are certain practical circumstances in which a(T) is much less variable with temperature than is k(T). This is a consequence of similar temperature variations of k(T) and cp(T) in some regions

-17such that their ratio remains essentially independent of temperature, For these conditions a(T) may be approximated by a constant value, designated as a*. Equation (5), for no internal heat generation,then becomes, -=E 6a*2E 62E 2 E1 -=a* I - +- + (6) at Lx2 ay2 az2j This result will be recognized as the classical linear diffusion equation for which a vast number of analytical solutions have been generated for various boundary conditions (29, 30). For a one-dimensional case equation (6) is written., E a* 62E (72 Further, for a convective heat transfer at x = 0 and x = 2L, Figure 6, with an ambient fluid at T., the typical convective boundary condition is (28), E(o_ = h [E(O,t) - E] (8) ax ko where, k(T,-TR) = Em. The initial condition would be written E(x,O) = Ei. Hence, we may write Equation (7) and its boundary condition Equation (8) in the following form, aE*(x*, Fo) d2E*(x*, Fo) and E*(O, IF0) Bi[E*(O,Fo)] (10) XA*

where E(x,t)-Eco hL E* =, Bi Ei-EW km (11) a* t x F0 -x L2 L in which L is a characteristic length. A comparison of Equations (9) and (10) for variable properties with the corresponding differential equation and boundary conditions for uniform properties shows them to be identical. Hence, for the same geometrical shape and initial condition their solutions also will be identical. This means that the various heat conduction charts formulated on the basis of uniform properties, such as the extensive tabulation of Schneider, (31) can be employed for the solution of a variable property problem in terms of the function E. Multidimensional problems also may be solved by the product method for those geometric shapes for which analytical solutions are available. The analytical solution to Equations (9) and (10) for a slab is given in Figure 6 in chart form as (1-E*) for variable properties of k, p, and cp for the case of constant thermal diffusivity a* Analytical solutions for steady-state heat conduction problems having variable properties may be obtained from solutions to Equation (5) with (aE/kt) set equal to zero. Thus, the governing equation is, +2E + 2E + a2E +p(xyz) =. (12) x2 5 2 z2 For a convective heat transfer to a wetting fluid at T, the boundary condition would be of the type given in Equation (8). In the absence

1.0K)O Lo ~0-00 200 100 40 20 0.8 0.63/ Nb~ ~ ~ ~~~~~~~o 4-~~~~~~~~~~ A-I OsH2 0.2 0.17 -- i=h kl I 0'w Y ~~~~~~~~", —''./ / /C Ia 0.c_ __0_i_ __ _ _ __ _ I I_ 0.001 0.005 0.01 0.05 OJ 0.5 I 5 I0 50 100 *t~ 0 (t Figure 6. Temperature Response of a Slab, 0 < x < 6 after sudden exposure to a uniformtemperature convective ambient T.0 at x = 0. arom Schneider. (31)

-20of internal heat generation, we have then a2E _ a2E a 2E _ (13) ax2 a2 Z This is the classical La Place equation well-known in field theory and in the study of diffusion phenomena. As may be noted the variable property problem in this case involves only variations in k(T) and not in the thermal diffusivity a(T) Analytical solutions to Equations (12) or (13) are rather broadly available in the standard literature(28'29'32) for a wide variety of problems. Such solutions also may be found by analogical techniques such as the analog field plotter. For these reasons they will not be reproduced here. The main difference in the present formulations from those currently available rests in the treatment of the function E, Equation (2), to determine the temperature distribution rather in the temperature itself. This is the principal distinction of the variable property problem. For those cases of unsteady heat conduction in which the property variation is very great and the constant thermal diffusivity approximation of Equation (6) cannot be made satisfactorily it will be necessary to solve Equation (5) by numerical methods using a digital computer. In this discussion we shall study the case without internal heat generation, ps(x,y,z,t) = 0 as this function even with spatial and time variations is a simple additive term in the numerical formulation which does not complicate the calculation in any special way. The most convenient from for a numerical calculation is to arrange the equations in an explicit fbrmulation. This enables a marching-type solution

-21and avoids the time-consuming iterative computer procedures of the implicittype formulations. The development to be given here will be for the twodimensional region shown in Figure 7 which has a convective heat transfer at the wetted boundary. For the interior point 0, using a square grid, the value of the function Eo at the (n+l) time interval may be computed in terms of the functions at the adjacent lattice points at the (nth) time interval (28) as, En -+ En + En + En + (M -4)En n+l 1 2 3 4 ~ o (14) o Mo where mo= (15) a(TO)At F'or variable propcrtbisi: a(T,) will vary with time and space and it will be necessary to up-date the calculation of Mo at the start of each new sequence of calculations. This may require adjustment in the time interval At. The numerical value of _M, is related to the stability of the calculation and will'be discussed below, For the surface poiJ,.t,' the value of the function Es at the (n+l) time interval is given('28) as L2E E1- 2NE' + M - (2N [-4) ]Es En+l J (16) Mls where Ms -- (17) a( Ts )t hAX Nsk (18)

-22-. L6 T 4 X X qPo 2 8 q8q h 7.Ax Ax AX INTERIOR POINT SURFACE POINT SQUARE GRID Ax, Ay THICKNESS b Figure 7. Finite-Difference Network for Typical a Interior and Surface Point Using a Square Grid!x = y.

-23 - 1 T k* f= T k(T)d'T (19) T T n Tn 00 s s and, n T T00 = J k(T)dT (20) T' TR The stability of the numerical calculation requires that, Ms > 2Ns + 4, (21) Mo > Ms, (22) for constant grid size. Because a(Ts) and to a lesser extent k* will vary during the calculation owing to changes in Ts, the value of Ms must be up-dated at each step to check the compliance with the stability criterion, Equation (21). Should changes in Ms become necessary it is probably best done by altering the size of the time interval, At rather than the grid size, ZLx Equations (14) and (16) will be recognized as "marching" type explicit formulations. Because of the restrictions on the time interval, At, this kind of formulation can require considerable computer time to complete a calculation. Its advantage, however, rests with its'explicit: form. Another type of "texplicit" formulation suitable for this kind of a problem and which is unconditionally stable without restriction on the size of /t is given in Reference (33). Owing to space limitations it will not be outlined here. The calculation is complete when the computed values of E are related to their corresponding values of T, using Equation (2). Steady heat conduction problems also usually require the use of numerical methods when the geometry of a region or the boundary conditions are not simple. The general formulation of the equations for the steady

-24case corresponding to the system in Figure 7 are given as follows(28) for the interior and surface points, E1 + E2 + E3 + E4 -4o - (23) and, 2E8 + E6 + E 2N E- [NI+2]Es - (24) Thle solution of Equations (23) and (24) is usually accomplished by tlh, "relaxation" method(29) or by iterative procedures on a d.igital compute]. The desired value of R0 and Rs, the residual;, i, s;ro but as a II:a't - tical matter they are reduced to as small a value (positive or negati v> —; as required by the demands of accuracy of the problem. The quantities Ns, E, and E are defined above. B. Low Temperature Insulation Since the end of World War II many new applications have been found for low temperature fluids. Probably the most significant of these in terms of quantity of liquid consumed is the use of liquid oxygen and hydrogen in rockets and other such applications in missiles and space exploration. A widening industrial use of these low temperature fluids can be identified, however, such as the use of oxygen in steel manufacture and nitrogen in food preservation. Associated with cryogenic application has been the important problem of insulation of the low temperature fluids from the ambient, both terrestrial and in space. This is actually an old problem as Sir James Dewar devoted considerable effort to the development of low temperature insulations at the turn of the century. The vacuum bottle, or thermos, can be credited to him Many of our modern techniques such as the use

-25of vacuum and low emissivity multiple-layer surfaces were recognized as important by Dewar. In this section we shall mention some of the principal features of insulations for cryogenic application. Kropschot(34'35'36) has published a thorough summary of this subject. The topic is discussed by Scott et al(16), Scott(l9) and found in the periodical Cryogenics and in the annual report of the Cryogenic Engineering Conference.(37) Owing to the effectiveness and the high performance requirements of these new insulants they are sometimes referred to as "super-insulations". The types of low temperature insulation may be divided into four categories (i) high vacuum; (ii) multiple-layer; (iii) powder; and (iv) rigid foam, Insulation systems may include combinations of these categories. High vacuum (less than 1 i of Hg) insulation is similar to that used in a thermos bottle. The transfer to heat is predominantly by radiation although there may be a significant contribution due to gaseous conduction if the vacuum is not sufficiently high. Insulation is achieved by maintaining as low a pressure as possible in the vacuum space which is enclosed by low emissivity surfaces, usually consisting of highly polished metallic coatings. Multiple-layer insulation is made of alternate layers of low conductivity fibers and thin, low emissivity metallic foil (usually aluminum) in high vacuum. As many as 150 layers of foil per inch is used giving as apparent thermal conductivity as low as 0.025 x 10-3 BTU/hr-ft-oF which is one of the lowest of any bulk cryogenic insulation developed to date. Figure 8 is a photograph of a typical multiple layer insulation without its vacuum jacket. The low vacuum reduces gas conduction

I I I 0 1 2 3 INCHES FIGURE 8. IULTI-LAYER VACUUM INSUlTION FOR CYROGENIC APPLICATION (COURTESY LINDE COMPANY).

2.0 SUPERINSULATION IO~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, 1.0 T = 5400 K T - 163~ K COLD 0.5 0 I.. 0.2 I- CS-5 ILI Z i 0.1 SUPERINSULATION *- 0.1 0.05 ~~~~~0.0 2....._ _ LINDE CO. (21) 0.05 0.1 0.5 1.0 5.0 10 50 100 ABSOLUTE PRESSURE, MICRONS OF MERCURY Figure 9. Performance of Multiple-layer Insulations as Functions of Vacuum.

0.5 _______ T*: 15000R (10400F) 0.2A 0.1 T*=9000 R_(4400OF) IL0.05 To 530OR (70OF) *0.02 0.01__ _ __ _ _ 0.005 Bel___ I 2 5 10 20 50~~ 100 2 00 50 10 0.002 In X 3c~~T Z-R Figure 10. Th Apaethra Codciiyo d Mulipe laye Inuaina aiu Budr eprtrs

-29500 200 /o.. / o 50 i__ m 120 I.T* =360R T to HOT SIDE Tx COLD SIDE LINDE Co. (2) 100 200 500 1000 2000 5000 10,000 Figure 11. The Apparent Thermal Conductivity of a Multiple-layrer Insulation at High Temperature.

103 r-. Note The words'nitrogen' and'helium' identify the t/ S E E l,, 1,/,, 1,.., l X-'-.. -300K0 114\0 1= 7610K.- -- -"=-'...=-..Z;i t roge n I10 PERLIT Ti -20o0 K PE'RL.IT E Helium. KROPSCHOT (34) 1 0' I0'4 I0'- 10" 10-' I 10 10' PRESSURE, mm Hg Figure 12. Thermal Conductivities of Evacuated Powders.

-31to a negligible amount and the multiple-layers of polished foil decreases the radiation contribution. The influence of pressure on the apparent thermal conductivity of a superinsulation reported by the Linde Co.(21) is shown in Figure 9. Values of the apparent thermal conductivity of the insulation for various boundary temperatures is given in Figure 10. The low values of this property are to be noted. The thermal conductivity of multiple-layer insulation is called "apparent" thermal conductivity because the mechanism of transfer is not purely diffusion. At low pressures where gaseous conduction is negligible heat is transferred primarily by radiation. For this reason the apparent thermal conductivity is a function of the boundary temperatures as shown in Figure 10. Multiplelayer insulation also can be used at high temperatures and some data on the performance of this material in this range of temperatures is given in Figure 11. Multiple-layer vacuum insulations also are discussed by Riede and Wang(39) and Paivanas, et al(40), The thermal conductivity of several powders commonly employed in vacuum insulation systems is shown in Figure 12 as a function of the pressure of the interstital gas. Below approximately 10-3 mm Hg the effect of gas pressure is negligible indicating that the principal mechanism of the transfer is radiation. Additional discussion of perlite as an insulant is given by Kropschot and Burgess.(41) Rigid foams which have found application in low temperature insulation are those which have a relatively closed cellular structure. Such a structure has the advantage of being inprevious to water vapor, an important characteristic in systems to be used in contact with the

-32atmosphere. Foams commonly used include polystyrene, epoxy, polyurethane, rubber and glass. Heat conduction through a foam is determined by convection and radiation within the cells and by conduction in the solid structure. Evacuation of a foam is effective in reducing its thermal conductivity although it still will be considerably higher than either multiple-layer or evacuated powder insulations. Data on the thermal conductivity of some selected foams presented by Kropschot(34) is given in Table III. The thermal properties of foams are also discussed by Haskins and Hertz(42) and Miller, et al.(43) Evacuation of most foams reduces the apparent thermal conductivity, indicating a partially open cellular structure. Data on this effect for polystyrene and epoxy forms, among others, are given by Kropschot,(34) as indicated in Table III. The opposite effect, diffusion of ambient gases into the cells of a foam, can cause an increase in its apparent thermal conductivity. This is especially significant in the case of the diffusion of hydrogen or helium. C. Interfacial Phenomena Interfacial transfer of heat and mass is intimately associated with both pressurization and stratification phenomena in cryogenic vessels containing coexistent liquid and vapor phases. It is, of course, the conditions at the liquid-vapor interface which couple the simultaneous transport processes in the liquid and gas phases. In a generalized sense knowledge of these interfacial phenomena is very incomplete. Yet, in terms of an idealized model, which is a reasonable representation of many practical circumstances, exact solutions for these transfer processes are known. The basis for most of these is the classical treatise of Carslaw and Jaeger.(30) In this section the subject of interfacial phenomena will parallel that given in Reference 44.

-33TABLE III THERMAL CONDUCTIVITY OF SOME SELECTED FOAMS* Foam Density Boundary Pressure Thermal Lbm/ft3 Temperatures Conductivity BTU/hr-ft-F Polystyrene 2.4 540-140~R 1 Atm O. 0191 2.9 540-140 ~R 1 Atm 0.0150 2.9 140-360R o10-5mm Hg. 0047 Epoxy 5.0 540-140o~R 1 Atm 0.0191 5.0 540-140~R 10-2mm Hg 0.0097 -5.0 540-1400R 4xl0-3mm Hg 0.0075 Polyurethane 5-8.8 540-140~R 1 Atm O. 0191 540-1400R 10-3mm Hg 0.oo69 Rubber 5 540-140~R 1 Atm 0,0208 Silica 10 540-140~R 1 Atm 0.0318 Glass 8.8 540-140~R 1 Atm 0.0202 *Data Taken from Kropschot(34)

-34Experience to-date points to three important generalizations (a) the interfacial temperature is essentially that of equilibrium (saturation) conditions corresponding to system pressure,* (b) during pressurized-discharge of a liquid from a vessel both condensation and evaporation of the cryogenic propellants at the interface are possible but usually are not significant factors, and (c) during self-pressurization of liquid containers interfacial evaporation occurs and the system pressure is governed by the vapor-pressure characteristics of the phases at the interfacial temperature. Schrage(45) presents the basic equations based on the statistical behavior of molecules from the kinetic theory applicable to condensation and vaporization phenomena. Balekjian and Katz(46) give experimental data on the depression of the liquid-vapor interface temperature below saturation temperature during the condensation of superheated vapors of Freon and water. Analytical and experimental investigations of liquid surface configurations for adiabatic processes in containers including the effects of low gravity, surface tension and draining are presented by Saad and Oliver(47) and Satterlee and Reynolds. (48) Knuth (49'54) in a series of two theore:tical papers solved the laminar transport equations governing interfacial growth for a single component system. The same problem was studied independently by Thomas and Morse(53) who presented both an exact solution and an approximate solution yielding an explicit expression for the interfacial mass transfer. The phase change of single component liquids and vapors in contact with various sub-strates is reported in Reference (0) and by ) and by Yang) Yang, et al. (5255) have *This has been the subject of direct measurement in many investigations in single component phases and is thought to be a reasonable assumption in multi-component systems although no known experimental confirmations of this have come to the author's attention. Departures from this are discussed later in this section.

-35applied the source theory to the solution of interfacial heat and mass transfer in multi-component phases producing approximate but simple formulations for the rates of phase growth and transient temperature and concentration distributions in liquid and vapor. An extension of the analytical work on single component systems to binary systems is given by Yang, et al. (56) where exact solutions to the simultaneous transient, heat and mass transfer between phases for a suddenly pressurized system are presented. Because this work is representative of the current analytical studies on interfacial phenomena it will be used as source material for much of the following discussion. Mass transfer by condensation or evaporation at a vapor-liquid interface depends on the relative rates of heat transfer from each phase at the interface. Should heat transfer from the vapor dominate that to the liquid, evaporation will occur at the interface; if the opposite is true, the vapor will condense; if the respective heat transfer rates are the same, neither evaporation nor condensation occurs and the interface remains stationary. These circumstances will exist generally. For physica. systems having convective action in both phases adjacent to the interface there is little known at present for predicting the interfacial transport of heat and mass.Clark, et al(54) treat the subject for cryogenic containers. The influence of liquid and vapor velocities in both laminar and turbulent motion on this process, the magnitude of molecular and eddy diffusion coefficients controlling simultaneous heat and mass transfer over an extended range of conditions and a model of the general mechanics for this process are largely unknown. However, significant progress has been made by adopting a simple, but reasonable model for study. One such model is shown in Figure 13 in

which a two-phase binary system initially in thermodynamic equilibrium (t < 0) is suddenly subjected to temperature, pressure and concentration transients (t = 0) in the vapor phase. The resulting transient, transport process (t > O) is governed by the following equations, assuming the origin of the X-coordinate to be at the initial location of the interface, thus causing the liquid to have a zero velocity, Vapor: T" (t aT",, (25) -— + u"(t),= - (25) 6t 6x 6x2 + u"(t) D"t (26) at ax ax2 Liquid: aT', 2T' (27) at ax2 ad' DI 2C' (28) 6t 6x2 with boundary and initial conditions as shown in Figure 13 and outlined in Reference (56). Conditions at the interface are taken to be those of thermodynamic equilibrium for t > 0. For uniform initial conditions the displacement of the interface X(t) from X = 0 is characterized by X(t) = 2\J7t,. (29) where X is the interfacial growth parameter and is positive for condensation and negative for evaporation. The interfacial and vapor velocities and interfacial mass flux are then, (56) dX(t) - (30) dt b t'

-37X X VAPOR P"'PO t VAPOR P.P" VAPOR P. P CD TSJ LIQUI\ LIQUID \ LIQUID t t tO t >O X X X P' P" VAPOR P pP t VAPOR Ps. VAPOR oilcb~~ I C" of' r O CO C C0 CsP'1c Ct \0 \\ D CD0. \ o-C O;D 0'~P co OD\D'P LIQUID LIQUID LIQUID t<O t O t >O Figure 13. Schematic Illustration of Temperature and Concentration Distributions in the Liquid and Vapor Regions at Several Different Times.

-38u"(t) = - P "(31) w dX(t)_ -A P' dt PT' t (32) The general solutions for T'(t) and T"(t) are Reference (56) T'(x,t) - T' L T," - To' 2 N/-67 T" - T"'(xt) = A"() erfc + A(p' P ) (34) T IOt" - 2 Wt' t (3j where 2 2 wAhee'(A) =, 6T A erfc(yTT) + oT e (7T e2 erfc(yTx) + aTe YT + erfc A) A"(%) T (36),,- k e - 6T A(1 + erf ) (36) e -2 erfc (7Tk) + T e 27T2(1 + erf \) and = hfg t( 5T (37) cpt (To"o - To') YT p" I a (38) (kpc%)" CT (kpc) (39) The interfacial temperature Ts at X(t) = 2j c'tt is given by T - T T - -O = A'(X) (1 + erf %) (40) T "- T' For a single component system k is determined from Equation (40) by specifying the interfacial temperature Ts, usually taken to be that of thermodynamic equilibrium. In a binary system it is necessary to

100 -I {_ EXPERIMENTAL DATA;-,~:3ATM, SAT. CF < ~ o ~ LN2 | 8..| (kpCp'/(kpco"l EVAPORATION L SATURATED CONDITNS 0,P 3ATM, o ID I I ATM 3 ATM H, a SAT H2 331 38.3 N2 5.06iO03 L50x 103 02 4.63[ IO3 1.94x 103 1H40 9.26IO0 2.59110 I01 102 103 104 0s (kpcp)'/(kpcp)" Figure 14. Interfacial Temperature for No Condensation Nor Evaporation, x= 0.

-40determine A by a coupling of Ts as expressed in Equation (40) with a similar expression for interfacial concentration Cs and C'" and the thermodynamic equilibrium data for the binary mixture. These results will not be given here but may be found in Reference (56). The vapor and liquid heat flux at the interface are expressed as (q/A)' = Ai1 e2 (41) k'(T," - Too) sl't and (q/A)" A" (1/ e' T2 (42) k"(TO" - To') 1&oe t A particular:ly useful case is that of \ = 0, which separates the conditions corresponding to evaporation and condensation. A simple formulation may be used as a criterion for judging in a particular instance which process may be expected to occur. The interfacial temperature for A = 0 is found from Equation (40) as T aT.Ts ~ Tx' =A'(O) To: -Toi w =0 1+ -T (43) 1 + (kpcp) (kpcp)" It may be shown that if (Ts-Too')/(To,"-TooT) is greater than A'(0), then X > 0 and condensation occurs, whereas if this temperature ratio is less than A'(O), then A < 0 and evaporation occurs. The usefulness of this criterion is that it provides a simple expression in terms of known system parameters and the thermal property ratio (kpcp)1/(kpcp) T. This expression is shown in Figure 14 along with

representative values of the thermal property ratio for 02, N2, lI2 and H20 at 1 and 3 atmospheres, saturated conditions. From this result it may be observed that interfacial evaporation may reasonably be expected in the pressurization of sub-cooled liquid hydrogen, whereas much larger temperature differences, Too"-Too', are required to cause evaporation at liquid nitrogen, oxygen or water interfaces. In these latter systems condensation may more often prevail. Experimental data for both liquid hydrogen systems,(58) where vaporization was reported, and liquid nitrogen systems,(59) where condensation was reported, are included in Figure 14. The relative positions of these data points on the figure confirm the prediction of Equation (43). This is further borne out by the growth rate parameter-gas temperature calculations shown in Figuresl5 and lb for tlhe pressurization o:' sub-cooled liquid hydrogen and nitrogen. Oxygen would behave similar to that of nitrogen in Figure 16. Experimental observations in both liquid hydrogen, li]quid oxygen and liquid nitrogen systems have indicated these same effects, (251) References (60) and (61). Aydelott reports evaporation from a liquid hydrogen interface in contact with its superheated vapor. The growth rate parameters for binary systems oxygen-nitrogen and helium-nitrogen systems are reported in References (55) and. (56). An important question relating to interfacial phenomena concerns the departure of the interfacial temperature from that corresponding to thermodynamic equilibrium. The difference between the equilibrium temperature T* and the interfacial temperature Ts may be shown to be closely approximated by the following expression(56) T* - Ts vfg T{3/2 2iR (kp/cp)' _ CF,/~ _ _ 1 -. t44) ii/bl 1 J her a;O t < t (:,

-42_ 0.04 - I- 0.03 - w:ZE a~~~ i P". ~'P15 PSIA ha 0.02 - CPaz46.5 PSIA z 0.01! 00 I I _ I l o 00 0 60 801 100 120 140 -. - \ CT" GAS TEMPERATURE, OR -0.01 2 I E I\ -0.03 - -0.04 - 0.05 s, INTERFACIAL INTERFACIAL - 0.06 CONDENSATION; EVAPORATION Figure 15. Gaseous Hydrogen-Liquid Hydrogen Interfacial Transfer for Non-Uniform Temperature and Interface Equilibrium.

-430.03 EXACT SOLUTION (56) \ —— SOLUTION BY THE SOURCE THEORY (55) 0.02 \ CONDENSATION (X>o) -400 -300 200 - tOO 200' T'~,,OF EVAPORATION O O (X<O) -0.02 Figure 16. Growth Rate Parameter of Liquid-Vapor Nitrogen System Pressurized from 1 atm. To 3 atm.

-44Here a1 is called the Knudsen condensation coefficient, an experimentally determined parameter discussed in References (45) and (46). For estimating purposes Ts on the right hand side of Equation (44) may be taken as T* with negligible error. At a pressure of 3 atm and for t in seconds, CF has the approximate value of 5.31x10l4 for hydrogen and 87.0x10l4 for oxygen. Hence, it may be seen that hydrogen interfaces respond considerably faster than oxygen interfaces for equivalent values of A/al. For t equal to 1 second (T*-Ts)(/lT1) is 0.000531~R for H2 and 0.0087~R for 02. Values of X are of the order 10-1 to 10-2. Thus, unless al is extremely small departures from thermodynamic equilibrium at the interface will be negligible except at vanishingly small times. Even thenthis departure will reduce rapidly at larger times. Olsen(62) has reported several degrees (F) superheat of a liquid hydrogen interface during pressurization. However, the superheat was observed to decay with time in a manner suggested by Equation (44). The transient condensation of a cryogenic vapor on an insulating substrate is reported in Reference (50). One of the problems associated with pressurization of cryogenic vessels is the high rate of initial condensation of the pressurant on the internal surfaces, including the liquid-vapor interface, and the consequent loss in pressure. The presence of an insulating material on these surfaces allows for an increased response time of surface temperature and a fairly rapid re-evaporation of a condensed liquid layer. An important consideration then concerns the identification of those thermal properties which may be used to select insulants for this purpose. It turns out(50) that a single property group, namely, k ~ cp, may be used to discriminate as to the suitability of insulants on whose surface a condensed layer of liquid will have

-45minimum residence time. Some typical values of k p cp are listed in Table IV. The low value of this property for styrofoam indicates the rapid surface temperature response which may be expected for this material. TABLE IV VALUES OF k p cp Substance kpcp)03, ( BTU/HR-F-Ft2)2HR Aluminum (70 ~F) 3 000 000. Water (Liquid, 70~F) 24,920. Nitrogen (Liquid, -320~F) 2,5310. Cork (1000F) 56. Fiberglas ('700F) 29. Styrofoam (40~') 9.

I

III. FORCED-CONVECTION PROCESSES In general, single phase forced convection heat transfer processes for cryogens may be described by the same scaling parameters as found useful for other substances, The exceptions to this rule include all transport phenomena in the region of the critical point. Here variations in the values of cp,, k, p and P with temperature and pressure are so great that correlation equations developed for "constant" property conditions are invalid. It is currently felt that heat transfer processes in the region of the critical point will be successfully described when the general problem of convective heat transfer in a system of severely variable properties in adequately solved. Some work has been accomplished along this line(64'65) for liquid hydrogen and will be discussed later. Another hypothesis tendered to explain near-critical point behavior is to ascribe it to a pseudo-boiling phenomena. (66) The frictional pressure drop for duct flow can be fairly reliably predicted using standard friction factor-Reynolds number correlations, except in the region of the critical point. The total pressure drop in many cases, however, is made up mostly of momentum changes resulting from density variations on heating or cooling, These can be computed using property data. (66) The most useful results to date for design purposes are the generalized empirical and semi-empirical correlations. These include correlations formulated from studies conducted on both cryogenic and non-cryogenic substances. (Reference 14, Chapter 5). Richards, Steward and Jacobs(67) have summarized some results for cryogenic substances, Data from this reference are given in Reference 14. Compact heat exchangers have been sized for cryogenic application(68'698-) using the -47

extensive tabulation of heat transfer and friction data of Kays and London.(70) For single phase flow the standard heat transfer and pressure-drop correlations have been employed for design.(69,71,81) Other heat exchanger design procedures for cryogenic application are reported by Bartlit and Williamson,(72) Hargisanand Stokes(73) and Kroeger.(82) Some of the results of Hargis and Stokes, who worked on Saturn I and Saturn V pressurization heat exchangers, will be discussed later. Single-phase convective heat transfer is characterized by the type of flow, laminar, transition of turbulent and the manner in which the flow is obtained, forced convection or natural convection. Heat transfer correlations depend on these characterizations as well as on the geometry of the flow, external, internal (duct flow) or separated. The principal scaling parameters are the Nusselt number Nu = hDe/k the Reynolds number Re = p V De/p., the Grashof number GR = De3p 2gpTo/L2, the Prandtl number Pr = cpp/k and the length.to diameter ratio L/De Where variable property effects are important these parameters are either evaluated at a particular temperature, such as the film temperature Tf = (Tw + Tb)/2, or an additional parameter consisting of (Tw/Tb) or (Vw/vb) may be included in the correlation. A. Flows With Moderate Property Variation Except for the superfluid condition most cryogens behave in a "classical" manner when their thermodynamic state is well removed from the critical state. For gases, this will always be true whenever an ideal gas equation of state describes the p, v, T relationship and the

-49transport properties (cp, i, k) vary only slightly with temperature. Under these circumstances the heat transfer correlations may be written for a moderate property variation corresponding to low to moderate differences in temperature between the surface and that of the fluid bulk. 1. Laminar Flow Laminar flow exists when disturbances decay when introduced into the flow. Generally, for flow inside of ducts this corresponds to a condition of a Reynolds number, based on diameter, less than approximately 2000. In this instance the Reynolds number is defined as pVD DeG Re (45) Here De is the equivalent diameter defined by 4 wetted flow area, Af 4 iAf (46) wetted perimeter, p P P which provides for consideration of ducts having non-circular cross section. Heat transfer data for laminar flow inside ducts having uniform wall temperature is correlated by the following two equations for h based on the arithmetic mean temperature difference. The subscript b indicates that physical properties are evaluated at mixed mean fluid temperature. (a) Re Pr/L- > 10 h18 |0De hD, 4_.8~ Af Re Pr 1/3 k~, b -.8.-

-50(b) Re Pr/L < 5 De hDe 1 Re Pr(48) k - 2 [L/D7 These equations are shown on Figure 17 taken from McAdams(74) and compared with some experimental data for air in circular ducts. The agreement is good at low values of Re'Pr/L- and satisfactory at higher values where De the data fall slightly above the curve for Equation (47). This probably was caused by a superposed effect of free convection. The parabolic velocity profile correlation, Equation (47) is shown for circular tubes for which Af/D_2 = -/4 2. Turbulent Flow Turbulent flow is found to exist when the flow Reynolds number is sufficiently large. For flow inside ducts a condition of fully turbulent flow occurs when the Reynolds number based on diameter, De, exceeds approximately 10,000. Between a Reynold number of 2000 and 10,000 the flow is in a transition state, being partly laminar and partly turbulent and characterized by some unsteadiness. Heat transfer data for turbulent flow inside ducts is correlated (+ 25%) by the following equation for 1 < Pr < 120 and 10,000 < Re < 500, 000, e= 0.023 (G) Prf 1l + (L ). (49) k ~ C\f / The subscript f in this expression indicates that the physical properties are to evaluated at the film temperature which is the arithmetic mean of the surface and mixed mean fluid temperature.

30 i i'. i _ i ii i i 20 " ya'l 7-IrRIPrIIl a == -_ w E I I - - _ _ _ __ EQ. 47 fOR CLb * s Z I I 2~ _ r _ _'SYMBOL D inchsl L/DL, 0i017 C5'ret' a Tn, e Dt' o8 1;Xti T L X I I | |ICADAMS ( 4) 0a4iinar Fowin _ __ - t 0,330 RePr X, Figure>17. Correlation of Heat Transfer Data for Laminar Flow in Ducts. o L~~,.. o0......,_ ~001.... 21-3 o 2 4 aa o. io 2 0 2 aC 2 32 D G L/b Figure 18. Correlation of Heat Transfer Data for Lamninar, Transition and Turbulent Flow in Ducts.

-52Another correlation which may be used to compute heat transfer coefficients in turbulent flow (+ 25%) and having the same restrictions as Equation(49), is written as follows: Ph pr./35() = 0.0205 (e) -0 1 +( (50) Except for the viscosity ratio correction factor,'w/7b, all physical properties are evaluated at the mixed mean bulk temperature, designated by the subscript b. A comparison of this expression with experimental data is shown in equivalent form in Figure 18. The influence of L/De as well as the general interrelationship of turbulent, transition and laminar heat transfer data also is shown. Thompson and Geery(75) studied heat transfer to liquid hydrogen at super critical pressures (680-1344 psia and inlet temperatures 57 to 860R) for turbulent flow in a 0.194 inch ID tube. Two regimes of flow were found characterized by the magnitude of the level of heat flux and wall-fluid temperature ratio. Except at low values of these quantities, the data were correlated (+ 30%) by Nub = 0.0217 Reb08 Prb.4 ) (51) as shown in Figure 19. Pressure drop was satisfactorily correlated by the conventional friction factor for Newtonian fluids, In low pressure regions where hydrogen has almost ideal gas behavior, Hendrick's, et al(66) found that local heat transfer data could be correlated by hxDe / 023 i(PfV De) (O cp) (52)04 kf -I

-53104 THOMPSON and GEERY (75) Nub=O0217 Re' Pr (Tw/T)"., - 10 LI I I I I, i,, i I, J 730 /A 08 To BTU/INO IEC REYNOLDS NUMBER, Rob PRESSURE INLET SYMBOL. (psa) TEMP ("R) PROPERTIES EVALUATED AT LOCAL 1340 86 BULK TEMPERATURES 1o 1270 74L/D 412 0 1210 77 A 730'9 0/A' 0.6 TO 6.0 BTU/IN" SEC v 730 82 0 730 55 o 720 60 c 680 57 Figure 19, Correlation of Forced-Convection Heat Transfer,

-54At higher pressures (12-50 atm) and for fluid temperatures in excess of 90~R, these authors find their local heat transfer data to correlate approximately as Equation (52), as shown in Figure 20. Hendricks, et al(66) summarize turbulent convective heat transfer correlations for hydrogen at pressures above and below the critical pressure (188 psia) in the range 1-100 atm, but not too close to the critical temperature (60~R)< This summary is given in Table V and includes some results for helium as well. A comparison of the various correlations in Table V is shown in Figure 21. Except for Equation (53) they all predict essentially the same magnitude of result over a large range of Tw/Tb. Equation (53) diverges severely for values of Tw/Tb greater than about 4. The range of actual experimental values of Tw/Tb is not great and since many modern power and propulsion systems require design at higher values of Tw/Tb additional data would be welcome. Hendricks, et al(66) recommend the use of Equation (57) at large ratios of Tw/Tb as this correlation is formulated in terms of local properties and consequently takes some account of property variation. At low pressures the pressure drop may be computed by the standard turbulent flow Equation (66). For pressures above the critical pressure and temperatures in excess of 90~R the fluid friction appears to be similar to that observed with other gases in pipes of varying roughness. In Reference 77 the isothermal fluid friction was comparable to results in the literature. However it was noticed that the nonisothermal friction factors were less than the isothermal values at the same bulk Reynolds number.

Tb>900R 2000 EQ(57), C.023 1000 Nuf Prf 0.4 =0 400 o HENDRICKS, ET AL (66) Q0(57), Ce.021 200 i0s lol Ref Figure 20. Correlation of Local Heat Transfer Data for Hydrogen p = 12 - 50 ATM HENDRIOKS, ET AL (66) 20 REF RE RE 79* REF 77 4-:REF,9 D.L/. ED2O > 2 BASED UPON EXPERIMENT,. - -- EXTRAPOLATED W I I I I I I iiii 2 4 6 810 20 40 6060100 T% / Tb Figure 21. Comparison of Convective Film Correlations Over a Range of TW/Tb

TABLE V (66) SUMMARY OF HEAT TRANSFER CORRELATIONS FOR REGIONS REMDTE FROM THE CRITICAL POINT. Entrance Max'TT bulk temp. range, Ref. Fluids Tb OR Correlation -0.3 0. 04 w,b(3 76 Hydrogen 2.8 360 to 560 Nub = 0.023Reb Prb (overall) 77 Hydrogen 0~~~~~~~~~~~~~I. 8 L 4(Tw ) 77 H I-Tydrogen 9 g 135 to 515 Nub o.o45Reb Prb T D Helium 11 (overall) lo.8 1/3(Tw -055 6 78 Hydrogen 4 88 to 130 Nub O.O2eb rb Tb 1+ Le I Tb ~~L/De (overall) 0. 04 L e ~, 79 Helium 4 1500 max Nuf = 0.0o4Ref08Prf (56) (overall) Unpublished hydrogen66 NU = CRef*8 Prf.4 C 0.021 (57) (overall)

-57In Reference 79 the friction coefficient with helium at high wall temperatures was observed to be predictable by the Karman-Nikuradse relation for turbulent flow: 4 log10(Re a) -0.4 (58) where f is the Fanning friction factor, defined as T f = w (59) pv2/2go and p = 4-f L- - p (6o) De / 2go McAdams(74) recommends a modified form of Equation (58) for flow in heated tubes having' Tw/Tb up to 2.5. The result is a reduction in the friction factor for the higher values of Tw/Tb, as was observed in Reference (77). For compressible flows, a significant contribution of momentum change to the total pressure drop must be anticipated. Hargis and Stokes(73) report on the design of heat exchangers for the Saturn I and Saturn V pressurization systems. These units consist of a helical tube coil through which the gaseous pressurant (H2, 02 or He) is pumped. The heat exchanger coil is heated by combustion gases from the rocket exhaust which flows over the outside of the coil. A comparison of test data and computer performance predictions for an oxygen heat exchanger is shown in Figure 22. The heat transfer correlation used for average tube side coefficient for oxygen which best fitted the data is Nub = 0,023 RebO'8 Prb (4 )Tw O 34 2 (61)

-58900 oo800 TEST DATA o COMPUTED, EQUATION (61) o |A COMPUTED, EQUATION (64) 700 L A n- 600 w 0 400 HARGIS and STOKES (73) 300 1.0 1.5 2.0 2.5 FLOWRATE w, lbs/sec Figure 22. Comparison of Test Data and Computed Performance Predictions for a Typical Oxygen Heat Exchanger.

-59where l1 and 42 are correction factors, given below. TABLE VI (a) OXYGEN CORRECTION FACTOR 41 FOR EQUATION (61) Tb Tw OR OR 6oo 1000 8o00 200 1.3 0.95 ----- 278 0.58 0.42 0.32 300 0.68 0.556 0.434 350 o.836 O.764 o.655 400 1.0 1.0 1.0 Over 400 1.0 1.0 1.0 (b) Correction Factor 02 For Equation (61) For L/De < 50: =, 1.48 (62) (L/De,)o' For L/De > 50:,2 = 1.oo (63) Equation (61) is shown in Figure 22 and agrees satisfactorily with the experimental data, As a demonstration of the need for property value correction Hargis and Stokes(73) include a comparison with a simplified correlation sometimes used in heat transfer correlations. This equation also shown in Figure 22 is Nub = 0.023 RebO'8 Prb~'4 (64)

-60This result compares much less favorably with the experimental data than does Equation (61). As may be observed Equation (61) for oxygen at temperatures above 400~R and L/De in excess of 50 is essentially the same correlation found to predict the behavior of supercritical hydrogen, Equation (51) and Equation (53). A discussion of the application of these various correlations to hydrogen cooled rocket nozzles is given by Benser and Graham.(83) The cooling of large masses by ducted supercritical helium is reported by Koln(84) and Koln, et al.(85) To maintain superconducting magnets at low temperature (40K) these investigators employed a forced circulation system of liquid helium at pressures above 100 atmospheres in small tubing integral with the object to be cooled. Performance data but not heat transfer data are reported. 3. Transition Flow Between the Reynolds numbers of 2000 to 10,000, the flow in a duct is in a transition condition. This is a flow region in which the characteristics of both laminar and turbulent flow co-exist. There also is a tendency for instability in the flow pattern. Very little is known about this flow regime and no really satisfactory method or correlation exists for computing its heat transfer coefficients. A residual L/De influence is observed which is greatest at the lower range of Reynolds numbers and gradually diminishes at higher Reynolds numbers. To obtain heat transfer coefficients in the transition region it is recommended that the data of Figure 18 be smoothed and the coefficients computed from the ordinate corresponding to the expected Reynolds number and the L/De

-61 - 4. Flow Outside of Ducts An important configuration encountered frequently is flow external and normal to the axis of a duct. Heat transfer data for the case of round ducts are correlated(86) by the expression =D0 Co.pVD~ n (65) the subscript f denotes the physical properties are evaluated at the film temperature. Both C and n are functions of the Reynolds and Prandtl numbers in accordance with Table VII below. TABLE VII p VD C C n Gases Liquids 1 - 4 0.330 0.891 0.989 Pr1/3 4 - 40 0.385 0.821 0.911 prl/3 40 - 4,000 0.466 0.615 0.683 Pr1/3 4,ooo - 40,000 0.618 10.174 0.193 Pr1/3 40,000 - 250,000 0.805 0o.0239 o.0266Prl/3 Experimental results for air are shown in Figure 23 and compared with the correlation, Equation (65). Data also are available for non-circular ducts and spheres and may be found in McAdams(74) and Knudsen and Katz. (86) The flow around a bluff object is of a variable nature. It starts as a laminar condition at the forward stagnation point, undergoes transition to a turbulent boundary layer and finally separates from the surface in the downstream regions producing a highly turbulent wake.

600 400 200 McADAMS(74) l l 100 40 ~ ~ ~ ~ ~~~~0 20 II; s IO 0.4 OLI 2 34 6810 2 34 6810 2 34 6 8 If 2 34 68 KP 2 34 68 04 2 34 6 810' 233 DOG Figure 23. -Heat Transfer Data for Air Flow'n Normal to a Single Cylinder.

-63For this reason the flow is characterized by the Reynolds number only without reference to a laminar or turbulent description. B. Flow With Large Property Variation The conditions corresponding to the limits of application of constant or variable property correlations - the ideal gaseous or liquid states, on one hand, and the critical or superfluid states on the other - are fairly easily identified. Their intermediate or transition states, where fluid property variation with temperature and/or pressure becomes severe, are much less easily defined. No simple critierion exists, One is best guided by the magnitude of the variation in such properties as k, cp and v in the range of temperatures expected in a design, McAdams (Reference 74) reports the data of Desmon and Sams(87) which indicates the influence of large temperature variations on heat transfer to air in turbulent flow. These results are shown in Figures 24 and 25 where the data are correlated in accordance with a modified form of Equations (64) and (52), respectively. In this instance the Reynolds number is evaluated at the bulk temperature. The corresponding property variations for air suggests a limiting criterion for the application of "moderate" property variation correlations. Hendricks, et al(66) present another guideline in the form of a dimensionless temperature ratio, originally suggested by E. R. G. Eckert. This ratio indicates the proximity of the fluid state to the critical state as measured by the temperature, Tm, at which the specific heat, cp, reaches its maximum. The criterion is given in Figure 26. The ordinate in this figure is the ratio of the experimentally

-64-. 64F CONVENTIONAL' Y~a0.02 X!2 F 11111 1-1 I I - MoADAMS (74) I t5 ___._a —._ A_ 4 = is _&i - 3 2 2000 27 548 6- 2579_ 3.17 54 5 _5- * 303 3.55 531:: 4' _! 4 2 3 4 56804 2 3 456 8 106 2 34 X DG4Ab Figure 24. Heat-Transer Date at InletAir Temperature of 531 to 5541R. Physical properties of air evaluated at bulk temperature.

-65McADAMMS (74) 0 0 - L SYMBOL T_ S Ts Tb 1l 1_ =- Io 981 1.60,>s=~~ 1 01 ~ lll 2.19 i101 i X- D 0pf 2.43 Figure 25. Corre>ation of HeatTransfer Data at Inlet-air Temperature of 531 to 533~R Using Modified Film Reynolds Number. 3.8 HENDRICKS, ET AL (66) O.30.303.55-!(.a104 E 2.6 A0 i0 -.12 -.08 o.04 0.04 08 ECKERT PARAMETER, TmiTb TwTb Figure 26. Comparrelatison of Local Experimental and Computed h Near Critical Bulk Temperature o Coputed h Based on Fim ipe Flow Correlatiods Number. TI;%

determined local heat transfer coefficient to that computed by Equation (57) (C = 0.023) for "moderate" property variation. As is shown, the ratio of the heat transfer coefficients increases sharpely as the fluid state approaches the critical state. The data shown are for liquid hydrogen and the bulk temperature where the largest coefficients were measured was approximately 90~R. An analysis of the velocity and temperature distributions, the wall shear stress and the heat transfer rate for the fully developed turbulent flow of supercritical hydrogen is given by Hess and Kunz. (64) Their analysis accounts for the influence of variable physical properties on the universal velocity and temperature profiles. Based on the method of Wiederecht and Sonnemann,(88) these authors have solved the momentum and energy transport equations for variation in the eddy diffusivities of heat and momentum as well as the thermodynamic and transport properties. The influence of these property variations on the flow is shown in Figures 27, 28 and 29. The universal velocity profile for constant properties is given in Figure 27 for Reynolds numbers of 50,000 and 500,000. The constant property results are compared with both velocity and temperature profile calculations for supercritical hydrogen in Figures 28 and 29. The kinematic viscosity ratio, vw/Vb, corresponding to the conditions of this calculation is 100, the inlet temperature and pressure is 67.2~R and 274 psia and the wall-bulk temperature ratio, Tw/Tb, is approximately 10. The influence of variable properties is most significant in the turbulent core.

-6730 LAMINAR BUFFER TURBULENT 2 SUBLAYE~ I 1REGION CORE * 0 w H ESS and KUNZ (64) 1 10 100 1000 10000 DIMENSIONLESS DISTANCE FROM WALL-Y+ FIGURE 27. UNIVERSAL VELOCITY PROFILE, 30 n U,, HESS and KUNZ (64) o 0 Q I 10 100 1000 DIMENSIONLESS DISTANCE FROM WALL -y FIGURE 28. THE EFFECT OF FLUID PROPERTY VARIATIONS ON THE DIMENSIONLESS VELOCITY PROFILE, g~~CI~~~ __ DIMENSONLES DSANCE PRo 2WALLI 0 I 111 I 01 II I I0 I00 I 000 FIGURE 29. THE EFFECT OF FLUID PROPERTY VARIATIONS ON THE DIMENSIONLESS TEMPERATURE PROFILE.

-68It was found necessary to relate the viscous damping constant for the eddy diffusivity of momentum to the kinematic viscosity ratio, Vw/Vb, in order to describe the heat transfer to supercritical hydrogen. The specific form of the functional relationship to do this was determined by selectively fitting the results of the calculation to experimental data. As a consequence, however, reasonable agreement was found between the analytical calculations and the experimental heat transfer data of Hendricks, et al. (66) The results are shown in Figure 30 where the ratio of the experimental Nusselt number to the computed from the theory is plotted against the ratio Vw/4 b, the principal governing parameter for this system. As may be seen the comparison is within + 20%, except for a few data in the range of vw/vb between 30 and 60. As a further test of their analysis Hess and Kunz computed the local Nusselt numbers and the wall temperature distribution along the heated length of a tube and compared their results with the experimental hydrogen data of Hendricks, et al.(66) A typical result is shown in Figure 31. Except in the entrance region (x/D < 30) the comparison of computed and experimental results is favorable. The lack of agreement for x/D less than 30 is attributed to an extended influence of variable properties on the thermal entrance region. The unusual temperature distribution, which has a peak, had previously been found only in boiling experiments. The region downstream from the peak with the decreasing wall temperature appears to be predicted well by the analytical results based on the hypothesis of variable fluid properties.

HSIS and KUNZ (04) u~ 0.DAAt O.Bl- p p —- DATA FROM REF 6 - o20 40 60 o 100 120 140 160 KINEMATIC VISCOSITY RATIO -/Vv: Figure 30. Comparison Between Experimental and Calculated Nueselt Numbers using the Proposed Analysis. ]00 Goo _ _. J _ _ _ O 400 -.-(1 - 200 2._. Lu5 1,0 0,5 ~ DO,35 IN z _ 0 10 20 30 40 50 60 7 LENGTH TO DIAMETER RATIO-X/D Figure 31, Experimental Data Exhibiting a Peak in the Tube Wall Temperature.

-70The analytic results obtained by Hess and Kunz, while providing some insight into the process, require extensive calculations using a digital computer in order to obtain useful numerical results. Hence, they are too complex for most practical design work. Because of this the following simpler formulation was obtained to fit the experimental data, Nuf = 0.0208 Ref8 Prf04 (1 + 0.01457 v ), (66) where all properties are evaluated at the film temperature, except that the bulk density is used to determine the velocity. Equation (66) is shown in relationship to the experimental data in Figure 32 as the ratio Nuf exp/Nuf calc The value of Nuf calc is computed from Equation (52) and the solid line is the ratio of Equations (66) and (52). The work of Hess and Kunz points to the solution of the near critical region problem as one of a judicious consideration of property variation. They have found that the kinematic viscosity ratio, vw/vb, is one of the principal governing parameters for that proper description of heat transfer in a region of having large property variation. Another explanation and correlation of near critical point heat transfer is the pseudofilm boiling hypothesis of Hendricks, et al,(66) This hypothesis is based on the similarity between the wall temperature distribution for heat transfer to supercritical hydrogen and that found in boiling experiments, as shown in Figure 31. It is thus postulated(66) that, "Instead of two-phases, there would be a continuum of densities between a light gas-like species and a heavy liquid-like species. Heavy, tightly packed clusters of molecules would migrate toward the wall and then would be broken up into smaller and lighter clusters and migrate

-71EQ.(66) F EQ.52 O _ _ — - - Using Equation (52 0 0: 1.0 DATA FROM REP 66 KINEMATIC VISCOSITY RATIO- V'w/V-D Figure 32. Comparison between Experimental and Calculated Nusselt Numbers Using Equation (52)o

-72toward the core of the fluid". The heavy specie was taken to be a fluid at the melting point and the light specie to be that or an ideal gas. Using the concept of a "vapor" quality, X2, to represent a mixture of these specie the mean bulk density is written 1 X2_ + 1-X2 (67) Pb Ppg,b )melt The analogy of supercritical hydrogen heat transfer with boiling is extended by introducing a modified form of the Martinelli twophase parameter X', as tt me lt pgf (68 XI~tX 2 r; i i0.1 I~pB~iijD~i (68) ttpgf \Pmelt This parameter is then used to correlate heat transfer data to hydrogen in the near critical region. The correlation is in the form of plot of of Nuf exp/Nuf m vsXt, as shown in Figure 33. Nuf is the exftmt exp perimental value of the Nusselt number and Nuf m is computed from P Vf D 0.4(69) Nufm= 0.023 AV (69 The fluid density p' is defined as fm 1 X2 l-X2 + (70) Pftl Ppg, f Pmelt and the average velocity in Equation (69) is computed on the basis of Rb, Equation (67).

-73CORRELATION BANO FOR'OILING HYOROGEN X2~ HENDRICKS, IT AL. (66).0X' -A I MARTINELLI TWO-PHASE PARAMETER, PSEUDoo-TWO PHASE, X?t Figure 33, Correlation of Local Near Critical Heat Transfer Data Using Modified Martinelli Parameter.

-74As may be seen in Figure 33 the data tend to form a correlation. The range of correlation for boiling hydrogen, to be discussed later in section VII, is also shown. The same data correlated by the pseudo-film boiling hypothesis were correlated by Hess and Kunz in Figure 32. In addition, Hess and Kunz replotted the data of Hendricks, et al,(66) and obtained the following equation which fitted the data + 20%, uf exp - 0.21 logeX + 0.549. (71) Nufm this relationship is given in Figure 34. Whether either the two correlating concepts is the "correct" one is unknown at the present. Each appears to represent the same data about equally well. In a sense both are "variable property" concepts. One gives recognition to the influence of property variation on the mechanics of the flow and the other employs an analogy with a boiling system which can be considered to have a variable mixture density. The correlation of Hess and Kunz(64) is probably a little easier to use and directs its attention more to the established principles of the mechanics of the flow.

E C' 0 NtIEIXI I/INIuOF S * bg =-'11 0.2111iI 1 MODIFIED MARTINELLI -Xtt FReference (o. Z 6 1 MODIFIED MARTINIELLI PARAMETER-X't Re fe rence (66).

IV. NATURAL CONVECTION PROCESSES Natural convection describes a type of convective heat transfer process in which fluid motion is a direct consequence of bouyant and viscous forces in the fluid. The only requirements for flow is that the fluid have a temperature gradient and be in a force field such as a gravity or centrifugal field. The coupling of the fluid motion with the temperature distribution gives rise to complexities in the mathematical analysis of such processes. However, empirical correlations have been obtained which confirm those theoretical solutions that are available. It has been found that for both laminar and turbulent flow next to a heated (or cooled) surface of various geometry a correlation (+ 25%) of the following form is obtained: hL [ f C I 2gfA \ 1 (72) - _.. - a (72) kf k f The term in the brackets is sometimes called the Rayleigh number and is thie product of tlhe Gralshof and:Btrand'tl numbers. The subscript f indicates that all physical properties are to be evaluated at the film temperature. For vertical plates and cy'linders the constant C and exponent n of Equation (72) are functions of -the Grashof-P:randtl number product, Gr. Pr, as shown in Table VIII below. TABLE VIII Gr. Pr C n Type of Flow 104 109 0.59 1/4 laminar 109 - 1012 0.13 1/3 turbulent -77

-78-... s expression has been found to confirm experimental measurements on liquid and gaseous nitrogen.(89'90) Partial comparison of this correlation with experimental results of liqiid helium and liquid hydrogen indicate qualitative agreement, except for helium data at approximately half an atmosphere pressure. (Reference 67, p. 14). Correlations of the form of Equation (72) are available for other geometry as well. Some of these are given by the following expressions:(74) Horizontal cylinders, 103 < Gr Pr < 109 hD Do3pf20-ATo c 1/4 0.53 ~ (73) k 2 This equation with the constant slightly less than 0.53 correlates natural convection data from a 1-inch sphere for subcooled liquid nitrogen at 3 and 5 atmospheres.(91) These data are shown in Figure 35. Heated horizontal square plates facing upward and cooled plates facing downward: Laminar flow, 105 < Gr*Pr < 2 x 107 hL =3pf2 gfATo cp j1 = 0.54... (74) kf kf Turbulent flow, 2 x 107 < Gr Pr < 3 x 1010 hL =0.4[L3pf2gfATO (C j/3 — = o.1 2 0 4~ (75) kf [ f k This expression has been confirmed experimentally for unbounded plates in force-fields resulting from system accelerations up to 21 times standard gravity.(92) Results from this investigation are given in Figure 36

-79PRESSURE SUBCOOUNG O P 3 ATM 15~F Nu O0.53 (GrPr)l"4 o P = 5 ATM 25~F P 3 ATM P 5 ATM I INCH DIAMETER SPHERE (a/g) I LIQUID NITROGEN 0 0 0 MERTE AND LEWIS (91) 10' LI I Ie I 10(T-;T), OF Figure 35. Heat Transfer in Natural Cor vecton with Liquid Nitrogen.

100 90 80 _ 70 / NuO 0.14 (GR PR) (TUBULENT) 60 _ 50- - - _ - - - - - _ _ _ _ 40 z 0 Nu 0.54 (GR Pt) /4(L AMINAAR z~~~~~~~~~~~ 30 ____ cx' 201 ____ - - -- _ _ _ LO"DIAMETER FLAT PLATE HEATING UPWARD L=0.9D I ot, 2 4 6 8 lol 2 4 6 GR PR FIGURE 35a* NATURAL CONVECTION OF LIQUID HYDROGEN ON A HORIZONTAL (260) FLAT PLATEs

SYTM ACCELERATI ON _M HEADFLOW SURFACE00~L 4' -oD0o(Gr Pr) 1 300 -_ Nu=&I14(Gr x INY5-McADAMS(74) -2-05-04 0 q/A= 4700 BTU/HR-FT2 " 01~~~~~~~~A, 9,440 Zx 10,870 OO_ AL.. 1 0,870, ~,, 24,450 I o/: 1.0' 2, 5.3 o,, 10.5 5 21, 4 6 8 10 2 3 4 6 8 I00' 2 -3 4 6 8 I011 Gr x Pr FIGURE 36, CORRELATION OF NATURAL CONVECTION DATA FOR SYSTEMS HAVING ACCELERATION NORMAL TO HEATED SURFACE.

-82for both a bounded and unbounded plate. The results for the bounded plate fall significantly above the correlation for the bounded plate, Equation (75), probably owing to increased eddy motion in the fluid. Similar results were found(93) for liquid nitrogen in the range a/g from 1 to 20, as shown in Figure ill. Natural convection data for liquid hydrogen on a horizontal flat plate is shown in Figure 35a. Agreement with Equations (74) and (75) is favorable.(260) Heated square plates facing downward or cooled plate facing upward, 3 x 105 < Gr.Pr < 3 x 1010 hL L30.27 L3pf2gfAT T (76) k0f 2 k f The characteristic dimension L in these correlations is the edge of the plate. It will be noted that for the laminar condition, exponent of 1/4, the length dependency is weak, L1/4, and for the turbulent condition, exponent 1/3, it is completely absent. For non-square geometry an estimate of a mean L is recommended for use in the foregoing expressions. During the orbital flight test of the Saturn I B, vehicle AS-203, the above correlations were found to be satisfactory for the prediction of heat transfer to liquid hydrogen and oxygen in a low gravity (a/g as low as 8x104) environment.(94'95)

V. PRESSURIZED-DISCHARGE PROCESSES FOR CRYOGENS During the past decade a great deal of research, engineering design effort and testing has been devoted to the problem of the pressurized-discharge of a cryogenic liquid from a container. Most of this effort has been directed at the optimization of the propellant tank design, determination of the pressurant requirements and the selection of the operating parameters for large rocket vehicles. Similar problems arise, however, in other applications such as the pressurized-transfer and storage of cryogens from vessels in ground installations. The discussion of the problems of pressurization included here will parallel that published by the author in Reference 44. Although the results are general they have been used almost exclusivly on cryogenic propellant systems for both ground and flight application. Considerations of propellant or storage tank design and performance characteristics and the interaction of these with other sub-systems requires an examination of several related processes. Of primary importance among these are: (a) pressurization, including the calculation of the transient temperature, velocity and concentration profiles in the gas space and the flow rate and quantity of pressurant; (b) liquid stratification, including the calculation of transient temperature, velocity and concentration distribution in the liquid; and (c) interfacial phenomena, including the study and prediction of mass and heat transfer rates across gas-liquid and gas-solid interfaces. Part (b) will follow this section as Section VI and Part (c) was discussed earlier in Section II, paragraph c. A typical flight propellant feed system to which these kinds of problems apply is shown schematically in Figure 37, taken from the paper of -83

(I:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4: Fiur 37. ~ s ~Iyica Propllan Feed S z fo Fligh Vehicle.

-85Pl.att,; et; alo (96). lThe liquid oxygen (LOX) tank is pressurized by a side st.ream of vaporized oxygen from the LOX pumps. The temperature, pressure and flow rate of gaseous oxygen (GOX) pressurant for the LOX tank is controlled by a heat exchange:r and pressure regulating system. During the LOX tank discht.arge pressurized O,.X at high.. temperature is introduced into the top of the tank through a di.ffulser, which is not:, shown, Heat is transferred betwe-en t.he LOX and the C(OX as well as between these fluids and the tank wall.s, Mass transfer in the form of vaporization or condensation may occur between the fluids. (See Figure 14). As a;result of these interactions temperature and concentration distributions are established in both the liquid and gas phases. These control. the process dynamics as will as the total pressurante gas consumption and liquid heating. The level. of pressure of the pressurant; is governed by the vehicle stru.ctu.ral and weight limi+t;ations and the suction bhead requirements for the turbo-pumps, The final residual mass of pressurant in -t.he tanks at t.he endc nf engi.ne irini.g:i.s fixed -by its pressure and mean t6emperature. It: is desired, of (ou:rse,'to keep this mass of pressurant at a m.in:imumr TI'emperat;lure st-r atificat;ion:in the liq.tlid, if severe, can cause reduced engine burning t:ime or loss of thr,'ust owing to pump cavitation resulting from the passage of warmer li.quid;from the region of the LOX-,O;X interface through -t.h.e p'nmp s 3Because of oscillations imparted to the vehicle during flight, baffles are mounted inside the tank to reduce the sloshing tendency of the propell.antso The effect of sloshing and splashing is to break up the layer of heated liquid at the interface. This results in rapid condensation of the pressurant and consequent, undesirable fluctuations in tank pressure and pressurant flow rateo

-86An indication of the importance of the pressurizing system to flight vehicle weight is shown in Figure 38, which is from a study by Nein and Thompson (60) The representation presented in this figure shows an approximate relationship between vehicle thrust and the mass/pressure ratio of the pressurant for several large rocket systems. For a Saturn V SlC Stage LOX tank pressure of about 22 psia at engine cut-off, for example, the mass of pressurant remaining in the LOX tank would be approximately 4500 pounds. Should the vehicle design be altered by reversing the relative positions of the LOX and fuel tanks a higher pressure would be required in the LOX to supply sufficient suction head for the turbo-pumps. An increase in tank pressure to 64 psia, for example, would result in a residual pressurant in the Saturn V SlC Stage of approximately 12,800 pounds. Figure 38 also shows the relationships between GOX, helium and LOX self-pressurization systems. Weight penalties are evident in the later two in comparison with a system using a heated GOX pressurant. For purposes of classification the analytical approaches to the calculation of the pressurant properties in a pressurized-discharge process will be divided into two classes, namely, (a) distributed systems and (b) lumped systems. A distributed system is one in which the temperature, composition, velocity or pressure are determined as functions of both space and time within the gas space. Such an analysis is the most general and usually provides the greatest amount of information concerning the process. The calculations are produced from rate-type (non-equilibrium) equations of mass, energy and momentum. Although for many purposes such analysis as these are the most desirable, they also are the most complex and often involve parameters and terms about which little is generally known'a priori. Analytical

MASS OF PRESSURANT p LB. GAS/PSIA 0 O~ I I -I 1 I 1 l l-1 1 1 — I I — m e;o_ m ~ ~ i CENTAUR AC-7'n i E ___ _ =....r.'n F ~aa z 0 IJUPITER, THOR' m SATURN lB w P D ) S IVB STAGE r' P' m I I IATLAS.P,-=. ~ ~.. __'__1-_10. 1 I STAGE W - ==_=__ SATURN I k% r C L X )S-I STAGE ffi CL 1 L I I I IL 411 \_ P~F _ ___ S-ICl STAGE

-88solutions of this kind in one-dimensional space and in time, which have the advantages of exactness and the disadvantages of certain idealizations, include the work of Arpaci, et al. (97, 98) and Clark, et al. (99). Arpaci (100) has presented an analysis of the gas space processes including axial heat transport, interfacial heat transfet and wall heat transfer. Recently numerical methods using the finite-difference approximation programmed on a digital computer have been employed to compute the gas space processes. Notable a among these is the paper by Epstein, et al. (101) and a paper by Roudebush (102). Owing to the considerable flexibility of the finite-difference approximation this method represents a fruitful approach to the calculation of pressurization phenomena for propellant tank design. A lumped-system is one in which only the mean properties of the gas space and tank wall are determined. Hence, such a property as the gas temperature is not found as a function of position within the gas space but rather the mean gas temperature is computed as a function of time only. Sometimes the calculation consists of determining over-all changes resulting from specified displacements of gas volume. These analyses are basically thermodynamic (equilibrium) in character and provide the minimum amount of useful information. Lumped-system studies have been reported by Burke, et al. (103), Moore, et al. (104), Bowersock and Reid (105), Bowersock, et al. (106), Humphrey (107), Gluck and Kline (61), Coxe and Tatom (108), Canty (109) and Momenthy (110). Most of these authors also report experimental data. Papers which are primary experimental in nature include the work of Van Wylen, et al. (111), Fenster, et al. (112), Oridn, et al. (113) and Nein and Head (114). For our present purposes emphasis will be placed on analysis or a distributed system.

-89In a series of two papers Arpaci, et al. (97, 98) have presented an analytical solution of the transient processes within the gas space of a pressurized-discharge system which predicts the response of both the gas, T(x,t), and the wall, Tw(x,t), as functions of space and time. This work appears to be the only analytical solution of a distributed character available. As such it possesses all the advantages of an exact solution, namely, it is continuous in space and time, it is subject to parameterization, it can be used to judge the completeness of an approximate solution and it can be employed with confidence to determine the influence of a single variable. Exact analytical solutions also are beset with inherent disadvantages as well. Among these are the limitations on the solution resulting from certain idealizations necessary to make the problem mathematically tractable. The analytical results are fairly complicated to use and require several tabulated or plotted functions to carry out numerical calculations. However, the idealizations are chosen to fit a great many practical systems and the results can be employed to evaluate experimental data and used in certain design and pre-design calculations. This method has been used by the NASA - Marshall Space Flight Center for preliminary design on the Saturn V, SlC stage and other designs. A unique feature of the results in Reference 98, which are an extension of those in Reference 97, is the allowance made in the computations of T(x, t) and T (x, t) for completely arbitrary and simultaneous time-dependent inputs of (1) inlet gas temperature, T (t), (2) ambient temperature, g T (t), and/or (3) ambient heat flux (q/A)o(t). The analytical model and the nature of the arbitrary time-dependent inputs which may be handled by this solution are shown in Figure 39. The idealizations imposed on the analysis are constant physical properties, including gas density, constant discharge rate, constant cross sectional area of container, constant pressure and

TYPES OF TRANSIENITS Tg(t) Tg(t) Tw(xt)- T(x,t) (q/A)o(t) ANDQR PRNESSURAT To (t) ~ t GAS MOVING — hatho INT:FACE AND SIMULTANEOUSLY LIUID I CONTAIJNER WALL ~ m TO (TO (tA)o)(t) 2 AND.OR 3 t 0 t ix (a) (b) Figure 39. Analytical Model and Time-Dependent Inputs for the Analysis of Arpaci, Et. A1.(98)

-91constant heat transfer coefficients.* The imposition of constant gas density is less severe than may first appear since the effects of this can be adequately circumvented by an iterative process of calculation described in Reference 97. In terms of dimensionless space and time variables, s and 5, and the dimensionless parameter A, the response of the pressurant gas and wall temperatures to these simultaneous, arbitrary, time-dependent inputs of the type 1, 2 and 3, Figure 39, are written, T(s,B,r) - Te = T( S6,,j) + T2(8,,r) + T (s,5) (77) and Tw(s,,) ) TQ T1w( S, +, ) + T (s, r)+ Tw (s 6) ( 78) 2 5 Expressions for T1...T3 and TWl... Tw are given (98) for arbitrary forms of the time-dependent inputs. Of particular interest is the linear time-dependent input disturbance of the form, D(bl)-D(o) D(6) = D(O) + ()((79) as shown in Figure 40. The various forms of the disturbance D(6) are summarized in Table IX. The temperatures T1...T and Tl T of Equations (77) and (78) for the linear disturbances in inlet pressurant gas temperature (D1), ambient temperature (D2) and ambient heat flux (D ) are given in terms of functions F1...F, f. f, G1... and gl. which are functions of s, b and B and defined and plotted in Reference 98. *An axial variation in gas temperature only is considered in this analysis and in others. Radial variations have been shown to be small in test models of Saturn I and 1/3 scale model of Saturn V LOX tanks (60) and in smaller containers (97).

-92DISTURBANCE INCREASING D(8). —.. D(O), INITIAL VALUE OF DISTURBANCE DISTURBANCE DECREASING LINEAR TIME-DEPENDENT DISTURBANCE FIGURE 40. LINEAR TINIE.DEPENDENT DISTURBANCE

-93TABLE DCIX SUMMARY OF LINEAR DISTURBANCE FUNCTIONS Case Symbol Type of Transient D(O) D(51)-D(O) Disturbance 1 D1 Inlet Pressurant Tg(O) - Tj Tg(bl)-Tg(O) Gas Temp. Tg(6) 2 D2 Ambient Temp. T0(O) - T T0(61)-T0(O) To(b) 61 3 D Ambient Heat Flux D(O) (q/A)o(O)po [(q/A)o(51)-(q/A)o(O)] p, | (q/A)0(O) hg p J IoO The importance of the linear time-dependent disturbance may be seen in Figure 41 where an arbitrary disturbance in Tg(6) is represented by a piece-wise fit of several linear combinations. In Reference 98 a method is outlined for computing the response of the pressurant gas and the wall temperatures for a typical disturbance as that shown in Figure 41. It is based on the superposition of solutions to linear systems. The influence of hg, the gas space heat transfer coefficient, was demonstrated in Reference 97 for an insulated container. The results are reproduced in Figure 42 showing the gas spaice mean temperature at the end of discharge as a function of hg for the experimental system of Reference 97. This indicates that above a certain value of hg the influences of this parameter is diminished. A knowledge of h is necessary to carry out calculations of T(x,t) g and Tw(x,t) using this method and most of the others. As may be seen from Figure 42 there is a range of hg in which its influence is quite significant.

ACTUAL ARBITRARY DISTURBANCE APPROXIMATE LINEAR FIT Tg( () - ~Tg(t8)- ~Tt, ( )'t ~ Tq (0) -T| 01~~~~I I I I 1 1 1 o 81 82 83 8 FIGURE 41. LINEAR PIECEWISE FIT TO AN ARBITRARY DISTURBANCE IN Tg(8) - T

ljo 08 06 0.2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0 4 8 12 16 20 24 28 32 34 BTU FIGURE 42. INFLUENCE OF hg ON SPATIAL MEAN GAS TEMPERATURE FOR THE CONTAINER GEOMETRY OF FIGURE 43

Hence it is very important to have reliable estimates of the gas space heat transfer coefficient. Various approaches have been employed to obtain numerical values for hg, none of them entirely satisfactory. In the experiments described in Reference 97 the theoretical curves for various values of h g were superimposed on the experimental results in order to find an approximate fit for the suitable value of h which describes the experiments. For that system (liquid N2 - gaseous N2) the suitable value of of 2 Btu/hr-ft -F adequately represented the data (see Figure 8, Reference 97). This is approximately the value one would obtain from natural convection heat transfer correlations and is reasonable in view of the low velocities expected in the gas space. In other systems, of course, different values of h must be g expected depending on the type of pressurant, its flow rate and diffuser-tank design and arrangement. The comparison of the theory (97) for a step-change in inlet gas temperature with the experimental results on a small insulated container (1 foot dia., 3 foot long) is given in Figure 43 for a LN2 - GN2 system and inlet gas temperatures from 200 to 1000 R. The same theory is compared with test data from Saturn I mock-up tanks 6.5 feet in dia. and 39 feet long in Figure 44 for various values of hg (60). In this case h of 10 Btu/hr-ft2-F represents the data very well. Experimental measurements in the same tank also indicated a heat transfer coefficient of about 10 Btu/hr-ft2-F for the GOX pressurant. In an attempt to avoid the limitations of the analytical solutions, Epstein, et al. (101) have presented a numerical method for calculation of the pressurization process. The computer program (101) has been modified by the NASA Marshall Space Flight Center to improve its correlation with test

0.8 Final Mean Density vs Inlet Gas Temperature 0.7 a ~~~~~~Adiabatic Case,~,., 68 ~ff3,t=IOe a Mobotic Cme~~~~~~Acp,=16.85 Btu/ft -OFJ 120 sec Lj_ 9hg=2 Btu/hr-ft- 1F, P=50 Psia THEORY EXPERIMENTSo 0@ im I \c ~ ~ ~~~~ ~(Ref.97 and go) E 0.5 Q, 1 0.4 WO.3~~~~~~~~~~ (I) __~ct) (q/A)O= 0 Z \ W 0.3 hg GNZ C3~~~~~ h0=O < 0.2 D= I Foot 0 L= 3 Feet 3r I LN W 0.1 r (P~~~~~ischarge <( 0 2 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 LL INLET GAS TEMPERATURETg,0R Figure 43 Comparison of Theory and Experiment for Pressurized-Discharge in Small Containers for Nitrogen as Pressurant.

H 0 LIQUD LEVEL -a t t=O 400..w 0uJBtu ~~~~~~~~~~~~~~h=2 7HR-'Ft:300 () (: (3 1 I II I h' 1 H 200-g z hg=15 cr~~r ~From Nein and Thompson (6C NASA-MSFC::3 I oA TEST U) 100 )3 130-10 a. 0 I0.10 20 3040 DISTANCE FROM TANK TOP,X (ft.) Figure44Comparison Between Experimental and Computed Pressurant Gas Temperature Gradient Facility 2, Oxygen as Pressurant, D = 6.5 Feet, L = 39 Feet

-99data on large tanks (60, 115). Comparison of these computer calculations with experimental data is given in Figure 45 which includes flight test data from Saturn SA-5. As can be seen a reasonable agreement is obtained for pressurant gas temperature distribution and pressurant flow rate. A distinct advantage of this computer program is that one output is the pressurant flow rate in terms of inputs such as measured propellant tank pressure and inlet temperature.

U) Ir PRESSURE U)m 0 60 w 50 PRESSUPANT 30 30 0 FLOWRATE LIL 0(0 U) RAG TEST DATA 10o COMPUTER(- - 15) Hgure45 CopsFrom Nein and Thompson (60) NASA-MSFC F 1t U) 0 50 100 150 (I.=,.~RANGE TIME, X Figure45 Comparison Between Experimental and Computed Pressurant Gas Flowrate Facility 1, SA - 5 Flight Test, Oxygen as Pressurant

VI. STRATIFICATION IN CRYOGENIC VESSELS Thermal stratification of a cryogenic liquid in a vessel results from external heat exchange and consequent non-equilibrium phenomena within the liquid. In both flight vehicles and storage containers the most dominant influence is heat transfer through the (vertical) side walls of the vessels. Other effects may be important as well, such as nuclear energy absorption within the liquid, heating from the bottom of the vessel and both heat and mass exchange between the liquid surface and the pressuranto The phenomena of thermal stratification is important to propellant tank design and operation as it influences the selection of venting devices, insulation, pumps and tank structure, among others. The pressure in the tank is directly related to the interface temperature which is established by the convective transport processes with the tank. In the case of liquid hydrogen a 1R increase in the liquid-vapor interface temperature represents approximately a 3 psi increase in tank pressure. As a result of stratification the pressure in cryogenic vessels has been found to be significantly greater than that corresponding to the vapor pressure at bulk (mixed) liquid temperature (58, 116, 133). This discussion of stratification will follow that presented by the author in Reference (44) and will include certain new information which has recently boecome available. Stratification is caused by the natural convective flow of heated liquid along the side-walls of a tank and into the upper regions near the liquid interface. Here it flows toward the center of the tank, dispersing and mixing, and causing a downward penetration of heated liquid, the depth of which increases with time. This depth is known as the thermal stratification layer. These processes of natural convective flow and mixing are extremely -101

-102complex and presently not well understood and.only imperfectly described. Anderson and Kolar (117) have shown that the stratification pattern is very dependent on whether the liquid heating is from the side-wall, bottom or by internal absorption of energy. Schlieren photographs indicate that side-wall heating produces the greatest amount of thermal stratification, Schwind and Vliet (118) have taken schlieren and shadowgraph photographs of the natural-convective processes resulting from side-wall heating at various levels of heat flux. A boundary layer type phenomenon was observed with vortex motion in the regions where the flow changes direction. Experimental measurements of liquid hydrogen stratification are presented by Barnett, et al. (119) on a full size propellant tank from the Saturn S-IV stage. As a result of these observations plus the application of physical intuition most analytical approaches to the prediction of stratification processes have started with the assumption of boundary layer flow along the wall. Essentially all analysis has been for side-wall heating only with uniform heat flux. Both laminar and turbulent layers have been studied. The model most commonly employed for these analyses is shown in Figure 46. Owing to a sidewall heat flux, (q/A)o,a boundary-layer forms at x = 0, growing in thickness 6 as it develops along the wall, intersecting with the bottom of the stratified layer, A(t), and finally mixing with the stratified layer itself of volume V(t). The unmixed bulk liquid is uniform at its initial temperature and the various analyses assume the heated stratified layer to be both mixed and unmixed at a higher temperature. While the boundary layer type of flow may seem reasonable, it should be pointed out that analysis based on this assumption usually proceeds from the application of boundary layer theory to transient internal flows taken from steady-flow results on a vertical flat plate in an infinite, uniform

-103PRESSURANT Ic ~ INLET PRESSURANT GAS LIQUID - VAPOR T SIDE-WALL ISTRATIFIED LIQUI HEAT FLUX i - - 0~ f LA0ER,>5VELOCITY DISTRIBUTION, -BOUNDARy LAYER,y~ X T, LIQUID DISCHARGE --— TO TURBO-PUMPS Figure 46. Typical Analytical Model for Liquid Stratification Analysis.

-105T - TB 2(H/R) ut (80) (q/A)0 (H/k) I Pr H(t) H. where, i?~fdyj (81) I= dn (81/) T(z) - TB (8~) Ts - TB According to assumption (5), 2 should have "similarity" characteristic-, i.*f. constant values for a given r, in which case its integral I ought to tL: constant also. Experimental measurements of centerline temperatures during heating are shown in Figure 47 for a single run and in Figure 48 for e..;'. runs. The requirement for a constant value of I is met reasonably well over a broad range of modified Grashof numbers and it does not appear to be significantly influenced by Prandtl number. The break in the function at Gr*H of about 101l may be a result of transition in the flow near the boundary. For constant sidewall heat flux the principal scaling parameter governing the process is a modified Rayleigh number, Ra*x, the product of a modified Grashof number, Gr*x, and the Prandtl number, Pr. Hence, Ra*x = Gr*X Pr = g 2(q/A)o X4 (p (84) X k02\k Boundary layer transition from laminar to turbulent flow occurs at a modified Reyleigh number of approximately 1011 (120). To carry out calculations using Equation (80) it is necessary to have information on A(t)/H. This is provided by an analysis of the boundary layer flow, and the results depend on whether the boundary layer is laminar or turbulent. The following result is based on the need

-1060 0 Z 0.4 WATER Pr 6.2 U) 0.6 0?100 SEC 0 0 200 SEC z l 300SEC 2 0 A 400SEC Z 0.8 H/R 2 232 1.0 0.2 0.4 0.6 0.8 1.0 T-Te DIMENSIONLESS TEMPERATURE RISEo rC Figure 47 Dimensionless Stratification Temperature Profiles (120).

1.0 Pr SYMBOL FLUID FROM REFERENCE (120) 1.3 - 6.7 0 WATER 3.9 - 4.8 o TRICHLOROETHYLENE 0.8 28 - 32 A 50% GLYCEROL 6.1 - 6.6 x METHYL ALCOHOL ii 16.3 - 8. v FREON 113 o 7.4 - 10 o 25% CALCIUM CHLORIDE x v 0.6 FILLED SYMBOLS DESIGNATE 0 TESTS WITH INITIAL VENTING o ~ O I Wo V 1- 0 o x z 0.4 0 0 0.2 9 I I 13 1 108 10 10 10 12 10 14 10 MODIFIED GRASHOF NUMBER GrH Figure 48 Energy Integral I as a Function of Modified Grashof Number for Prandtl Numbers Between 1.3 and 32 (with HIR = 2.3) (120)

-108to satisfy continuity requirements in the vessel. It is assumed that the rate of mass flow in the boundary layer at the plane A(t) is identical with the rate of change of mass within the stratified layer of volume V(t). Hence, I2~IcR I"O pu d~y H - nlt _ d[pV(t)], R2 d[pA(t)] (85) dt dt ( } dY) x = H - (t) dt dt If the density is assumed to be constant, then this becomes, (2R u dy Jx = H2 dA(t) (86) Equation (86) is integrated using velocity distributions for both laminar and turbulent boundary layers. For the case of turbulent flow the result is (120), L(t) H (i~tjH Gr*H (t) 1 - 1 + / H9 (t/92) ( )2/7 27 Ht i-1 0.092~ ~ Pr 2/3 1 + 0o.443 Pr2/3(87) This equation is compared with experimental data in Figure 49, where A(t) was determined from measurements of centerline temperature and time-zero corresponded with the instant the surface temperature started to increase. The general nature of the agreement with the shape and form of the data should be noted, although Equation (87) consistently predicts smaller stratification layers than those measured. It should be noted that in Equation (87) A(t) equals H only at infinite time, a circumstance which limits its application to small times following the start of transient, Equation (87) can be used in Equation (80) to compute surface temperature transients. It will be noted that the result in (87) is independent from that in (80), but the reverse is not

1.0 0 tOo 0 D 4 o0.8- EQUATION (87) I. 6 FOR: O U V 01 50% GLYCEROL- H20' 25 CC- H20 FROM REFERENCE (120) Z. / TRI CHLORETHYLENE t0 0 |4 n2METHYL ALCOHOL o FREON 113 UL 0.2 0 0 1 2 3 4 5 6 7 H 2/3 G/7 R IPr (0.443 Pr2/3 Figure 49 Comparison of Experimental Data for Stratified Layer Growth With Prediction by Equation (14) (with GrHA>4 x 12 and H1/ = 2.32)

-110true. The experimental results of surface temperature measurements are compared with Equation (80) in Reference (120) and a general agreement is found, as shown in Figure 50. The transient laminar natural convection processes in both the liquid and vapor in a partically filled container, including the influence of heat sources have been studied by Barakat and Clark (130), Barakat (131) and Clark, et al. (57). These investigations have involved the finite-difference representation and solution on a digital computer of the two-dimensional transient, transport equations for rectangular and cylindrical geometry. Work on a spherical geometry is currently underway (133). Application of the results of these studies is to the internal flow, temperature stratification and pressure rise in cryogenic propellant containers when subjected to various gravity fields, external heat flux or wall temperature disturbance. The isotherms and streamline pattern in a cylindrical container subjected to a sudden increase in wall heat flux of 2000 BTU/Hr - ft2 is shown in Figure 51 at a time of 60 seconds following the disturbance (130). The bottom is insulated and the liquid vapor interface is maintained at 80F. It is evident from these results that, while a boundary-layer type of phenomenom is suggested, the bottom corner and the liquid interface intersection introduce effects causing a significant departure from boundary layer flow. Near the liquid interface two-dimensional effects are evident,also. Two vortices, one near the wall and the second at the centerline are evident. Numerical calculations indicated a shifting towards a steady-state position approximately at the midpoint in the liquid space. A comparison of the theoretical and experimental results for water at lg in a cylindrical container is shown in Figure 32. The spatial mean values of the heat transfer coefficient and

-111i0 FROM REFERENCE (120) EQUATION (80) ) Jw ~L ~~~~~~FOR: _=t _31 t ",~50% GLYCEROL-H 0| O 10.~. 25 % CaCL -H 0 H2o _, o TRICHLORETHYLENE _ moo<1^8 METHYL ALCOHOL _t J aFREON 113 0 ~-04 1 I RIGHT HAND SIDE OF EQUATION (80) WITH 2 (a(e)/H) FROM EQUATION (87) Figure 50 comparison of Experimental Data for Surface Temperature Rise with Prediction by Equations(i)and (87) (with GrH Pr>4 x 1012 and HIR * 2. 32)

-112m.. i ~i 0, 37 (a) Isotherms (b) Streamlines FIGURE 51. COMPUTED VALUES OF ISOTHERMS AND STREAMLINES IN A CYLINDRICAL CONTAINER (q/A)o' 2000 BTU/HR FT TIME = 60 SEC.

-113LIQUID -VAPOR INTERFAC 30 20 CENTER LINE POSITION y-O 0 O - - -; *30 20 * y /2- IN. BELOW INTERFACE 10, r OFFSET: N7/-IN. 0 I.o y -I N. i,.0 6 10 - r w -IN. 20 Yu2IN. o Theory a10 r-' - IN. -Experiment j - 2 10 r 7Y " IN. ~ 20 - y - IN. 10 r"* IN. t0 20 y 6 /a -IN. 10 - r- IN. 0 40 80 120 160 200 240 Time, Seconds Fig. 52 Liquid temperature response in the cylindrical container,(I130) run 4. (q/A)o - 2000 tu/hr ft2, To. 80~F. FW D: H20; o/g 1.0

-114Nu/R 1/4 ratio were computed for the conditions in Figures 51 and 52. These results are shown in Figure 53 and correspond to a maximum Ra of approximately 6(109) based on height. Of interest is the rapid decrease in heat transfer coefficient and Nu following the start of the transient and their subsequent oscillation about a mean. The mean value about which'Nu/ 1/4 oscillates is approximately 0.54, which is close to the value 0.59, recommended for steady laminar, natural-convection from vertical surfaces (74). A similar oscillation in the velocity and temperature distributions has been reported by Barakat (131). One of the important consequences of thermal stratification in cryogenic propellant containers is the rate of pressure rise in the container resulting from the transfer of mass at the liquid-vapor interface. For a single component system the pressure in the vapor space corresponds very closely to the saturation pressure at the temperature of the interface. Thus, with stratification it is evident that container pressures can significantly exceed the pressure corresponding to saturation at the bulk liquid temperature. On the other hand, a sudden mixing of the fluid will result in rapid depressurization. These conditions have been studied by Huntley (133), among others. The computation of pressure-time histories in closed containers is given in Reference 57. This analysis is in the form of a numerical, finite-difference approximation to the transport equations in both liquid and gas phases with solution by a digital computer. A coupling of phase interaction at the interface is required in this analysis. Owing to the complexities of making detailed calculations at the interface, the limitations imposed by computer storage and uncertain-tie s concerning the conditions at incipient boiling an approximation was introduced in the analysis. This consisted of establishing an arbitrary value of wall temperature rise, tOTw, max., above the saturation

-115Z. 160 _ _ _ _ -2.0 w..140 ____... 1.8 LL~ LU. w 120 1r 0 O~~.. 1.6,TII1-III 00 _IA 80s 1.2 Ca Z 60 1 40 0.8 I- (NU RC, 1 w 20' _ II I''1 imi z.. 0#4 Z 00 40 so 120 160 200 240 3~r ~TIME, SEC. FIGURE 53, HEAT TRANSFER COEFFICIENT VS TIME,(130)

-116temperature, which the wall would not be permitted to exceed. The thermodynamic effect of this was introduced into the computer program by the evaporation of an appropriate amount of liquid directly into the vapor space, thus allowing for the superheat in the wall and the adjacent liquid. The influence of this arbitrary control is evident at heat flux as high as 180 BTTJ/He-Ft2o However, at a heat flux more closely corresponding to that or an insulated propellant container subject to solar radiation is space, 1 BrJ/Hr-Ft2, the influence of LTw, max is negligible. Under these conditions the actual wall temperature rise is less than 1~Fo Some typical temperature stratification patterns in both the liquid and vapor regions (57) are shown in Figure 54 for the self-pressurization of a closed cylindrical LOX container subjected to a uniform external heat flux of 180 BTU/Hr-Ft2 and grayity level of 10-5g. Except near the wall and liquid interface the liquid temperature rise is completely negligible but the relative stratification is quite large. The corresponding pressure rise and total mass of LOX evaporated is shown in Figure 55 for various values of ATwI max, The influence of heat flux on the pressure rise and total mass evaporated for ATw, max = 0~F is given in Figure 56. The pressure rise in the same LOX container for a heat flux of 1 BTU/Hr-Ft2 is very much lower even for an exposure time exceeding 3 hours, as shown in Figure 57. One of the principal problems of the finite-difference approximations used in (57, 130, 131) is that of numerical convergence and stability. This question is discussed in detail in (130) for the laminar transport equations and a simple stability criterion is given and its mathematical basis established. The numerical formulation is of an "explicit" form which reduces computer time to a practical minimum.

-U719.01 13.89 5.863 0 13.6 1 17.85 5.858 21 3 G I-D 11.51 5.14 - 11.33 TsiT -Tpt 2.42 TF SAT-To z5.6 8 F TS-To'T07.7I 1F.223x102 1.0 8.89XO a3 ~ 2.85 x IO' I.08x10 j5 I o06x10'3 2.73x10'2 7.7 xo169 3.72 x 2oF.07x10' T-To 4.66x10-T T-To T-To 6 MINUTES 12 MINUTES 18 MINUTES P ~ 17.46 PSIA Ps 20.40 PSIA Ps 22.74 PSIA LIQUID OXYGEN IN CYLINDRICAL TANK 5 FT DIA. x I0 FT LONG q/A - 180 BTU/HR- FT2 21x 31 GRID a/g =I0o33% INITIAL ULLAGE Po = 15 PSIA FIGURE 54. ISOTHERMS IN A CYLINDER LOX CONTAINER FOR ATw max2~F

-118~~2 ~~~~~~5 aF.^ Tw maox 25 24 IOF 23 221/,,10m 14 I I 21 - b. I CYLINDRICAL TANK 03 0 n,. / 5 FT. DIA. x I0 FT.Q 17 / 21 x 31 GRID 33 % INITIAL ULLAGE 1/6 q/A 180 BTU/HR-FT2:~'/ cl/g.10'~ 14 0 4 8 12 16 20 TIME - MINUTES - 9 7 I~F 2F FIU~ 5 PPSUERIEAN OT///SEVPRAE 3E~ ~ ~ NACLNRIA O OTIE

LIQUID OXYGEN IN CYLINDRICAL TANK 5 FT. DIA. x 10 FT. LONG 21x31 GRID a/g =10-O 30Q 33% INITIAL ULLAGE 3T-w max= OOF L 25 r q /Az 180 BTU/HR-FT2 w 20 aq/A IO BTU/HR- FT2 05 0 5 10 15 20 TIME-MINUTES il o 7J7 0: 6 = 2 0fiI p / q/AA 10 BTU/HR- FT2 TIME- MINUTES FIGURE 56. EFFECT OF HEAT FLUX ON PRESSURE RISE AND MASS EVAPORATED

-12034 32 / 30 28/ 26 4/ ATw max OOF 24 c, 22 x 20 18 / Q 16t F/ A//Tw max' 1 F and 2~F' 14 /: 12 Liquid Oxygen in Cylindrical Tank 5 ft. Dia. X 10 - 21 x 31 Grid 8 33 Initial Ullage Initial Pressure - 15 P SI -6 / / qq/ A 1. BTU/HR-Ft' 6 a/g Io'" L/D= 2 0 100 200 TIME - Minutes FIGURE 57, PRESSURE RISE IN A CYLINDRICAL LOX CONTAINER FOR AN EXTERNAL HEAT FLUX OF I.O BTU/HR-Ft'

VII. MULTI-PHASE PROCESSES A. Boiling Heat Transfer Boiling will exist at a surface when the surface temperature exceeds the saturation or bubble point temperature by a few degrees. This superheat depends upon the type of fluid and surface and the system pressure. It decreases to near zero as the pressure approaches the critical pressure. The word boiling is used to describe the process of vapor bubble generation within a liquid and almost always occurs at a solid surface in heat transfer systems. Two general types of boiling systems exist: pool boiling, a process similar to natural convection, and forced convection boiling. For each of these systems the fluid can be sub-cooled and thus have no net vapor generation or it can be saturated in which case a net vapor generation will occur. The boiling phenomenon itself is characterized by three regimes, shown in Figure 58, namely, nucleate boiling, transition boiling and film boiling. In addition, two other unique phenomena cre observed in boiling heat transfer. These are the conditions of maximum and minimum heat flux, also shown in Figure 58, which separate the transition boiling regime from those of nucleate and film boiling. For systems such as nuclear reactors and electronic equipment in which the heat flux (q/A) is an independent variable the point of maximum heat flux is of utmost importance. Should an attempt be made to increase the power level (and hence q/A) of such systems beyond that of the maximum heat flux corresponding to a given set of circumstances, the surfacefluid saturation temperature difference would increase to that of film boiling at this heat flux. As may be observed from Figure 58 this would result in heat transfer surface temperatures 1000 to 2000F above the fluid saturation temperature. For most fluid-surface combinations this would mean physical -121

I0 IEiii III I I i l POOL BOILING HEAT TRANSFER LIQUID NITROGEN (134) /jii - l MAXIMUM HEAT FLUX TRANSITION BOILING DATA OF BROMLEY 1/ -C 4 NUCLEATE BOILING I (REF. 135) / I0 A / 4m I i I I I I I 1 I I I I I r I I - - - - -FILM BOILING MINIMUM HEAT 31 I 2 3 4 5 6 7 8910 2 3 4 5 6 789102 2 3 4 5 6 7 8910 Tw- TsatF Figure 58. Typical pool boiling characteristic curve.

-123destruction of the surface. In cryogenic application this consequence is minimized owing to the low saturation temperatures but it still is to be avoided in most instances. Forced convection boiling has a similar character to that of pool boiling shown in Figure 5S, except that non-boiling region is usually more evident in the presentation of the data. Walters (156) has studied the forced convection boiling of liquid hydrogen in a 0.25 inch ID tube and some of his results are given in Figure 59. The literature of boiling heat transfer is very large. It will be possible in this section to describe only some of the significant results as related to cryogenic heat transfer. Excellent summaries of the subject have been prepared by Westwater (137), Balzhiser, et al. (158), Zuber and Fried (139), Richards, Steward and Jacobs (140), Giarratano and Smith (141, 142), Brentari, et al. (143), Seader, et al. (27), and Tong (144), for both cryogenic and non-cryogenic application. In general, the properties of pool boiling including nucleate, transition and film boiling and the maximum and minimum heat flux can be computed from available correlations and some knowledge of the fluid-surface characteristics. This is less true with forced convection boiling although some progress has been made and will be discussed later. Data on gravic and agravic effects will be given in a separate section. 1.- Pool Boilin a. Nucleate Boiling Nucleate boiling exists in the (q/A) - AT range from incipient

-124l'0,0'' 1' I I I l' DATA OF WALTERS (136) _oooo _ -__ _ _ ~ I|.7 AT.063 0,000 2.2 ATI,.0444 x < 51 R 2_0 0\ 60,000 50000 560 500 < Re < 60, - - - 40,000 30,___ -000 I I _ r JUNSTABL. 20,000 d R.. 21 NUCLATE STABL FILM BOL801 ING BOILING ___:2000 < M < 3 o, < <.0 3 s 3,000 / =-. 00 < R < 330, 00 4,000 _ -.5 I 2 3 4 5 10 20 30 40 50 100 200 300 TEMPERATURE DIFFERENCE, AT To, - Tb, DEG R Figure 59. Heat Flux vs. Temperature Difference.

-125boiling to the point of maximum heat flux. This type of boiling has probably received the greatest attention in terms of the total number of investigations owing to the relative simplicity of conducting measurements. Complete agreement is lacking among the results of the various investigators because of the important influence the kind and nature of the surface has on the process, among other things. This effect has yet to be adequately described. In addition to this variations in system geometry, method of taking data and uncertainities in measurement contribute to the general scatter of data. This general effect is seen in Figure 60, 61, 62 and 63, taken from Brentari and Smith (142), where the experimental data for nucleate and film boiling for Oxygen, Nitrogen, Hydrogen and Helium are shown. Except for the data of Lyon, et al. (145) the width in the band of the data reflects the spread of each investigator's measurements on a given system. The spread in the data of Lyon, et al. is a result of their study on a range of geometries, orientations and surfaces. For comparision, the correlation for nucleate boiling of Kutateladze (145, Equation 11.21, Equation 88) is included.* The results for the maximum heat flux, the minimum heat flux and film boiling will be discussed later. In general Kutateladze's equation represents the data reasonably well. It should be pointed out that this also will be true of several other correlating equations to be discussed below. Kutateladze's first correlation, originally derived for water and various organic liquids, is * Kutateladze (145) gives two equations (11.21 and 11.22, p..129, Reference 145.) His second equation is used by Seader, et al. (27) and Zuber and Fried (139) and given here as Equation (92). Each appears to give approximately the same results, although their (q/A) - AT relationship differ.

-126h ~// 5.25(10-4) (q/A)CP (gc) (() 2(a/2).125 ( gPg /0*7

(BTU/HR-Ft2)= 3180(WATTS /cM2), TATR= 1.8 (AT~K) 100oo 0 Experimental critical heat flux Correlation.06 c -.. Predicted critical heat flux 1< o/"0~ l0D 0.2 cm,..l I -0 E 1 Banchero,.oIal Ml kov t al. e, al. = 0.06 *o ~ ~~ D = 0.1-..' 3 D = 0.32 o1. I-; ___ ~.0 Correlation FIGUR~~o~~ P= I tm. Haselden _ __' (The points of minimum film E r n Petersd Film P oo lMikhail boiling are given by either the correlation of Lienhard & Wong(151) or of Zuber (162).) Hoge & Bnchero, Brickwedde t al. 0.01 0.10 1.0 10 100 1000 ~T, *K FIGURE 60 Experimental Nucleate and Film Pool Boiling of Oxygen at One Atmosphere Compared with the Predictive Correlations of Kutateladze and Breen and Westwater (142). See Refs. (145-174).

(BTU/HR- Ft)=3180 (WATTS/Cm2),aT OR 1.8 (T OK) Coo Frederkin9 D =.00136 cm. D =.00312 cm _____ t - ___ ortoA =.0 51C_" I I M~~~~erte' 8 o~orc O Experimental critical heat flux Kutteldze Predicted critical heat flux C elarKtion Roubioui Lyon! F ln'ro.,... ~,o - ~'""u/~1 t'~Z 10 E Cow lay, Hsu 8 Wstwote — N. __ cm _ (I, ~WeiHanson "' Richards 1.0 ____ 1 — H Johnson hs _-_/ Bradfield_ rl I I _ et ol M i khoil ar gv e th / Welli~'~~~[~ I lynn oo /t al. -D I.Ocm! Breen'& etaI1. Westwoatr Hoselden~.~ /[~, 7/[ Correlation (The points of minimum film 8 Peters eil 8~ boiling are given by eith r the correlation of Lienhard & Lapin et al. Wong(151) or of Zuber(162). ) 0.1 0.01 0.1 1.0 10 00 000 AT, OK FIGURE 61 Experimental Nucleate and Film Pool Boiling of Nitrogen at One Atmosphere Compared with the Pred'ctive Correlations of Kutateladze and Breen and Westwater (142). See Refs. (145-174).

(BTl/E-Ft)=3180 (WATTS/Cm2), AT R 1.8 (AT OK) l0 0 Experiment maximnmm heat flux utateladze — Os, - pedicted m|aximum heat flux Correlationt al. ulford, et ol. Mulford~... i 1.0 OM1.0~~ D _>Ql~ 1.0 cm. EExperalss ucleal~e S p~i~lm Pol Boiling o g aI I_ o — sBreen Ek Westwater( 0.1 x(The points of minimum film boiling Ti pbeand given by either the correlation ail Rof.ienh-ard & Wong (151) or of L Zuber (162).) 0.01 0.1 1.0 I0 I00 I 000 AT, OK FIGU]RE,62 Exper ntal Nucleate aMn F'iLm Pool Boiling of E4ydrogen at One Atmosphere Compared with the Predictive Correlations of Kutateladze and Breen azW Westwater (142). See Refs. (145-174).

(ITl/ - Ft>3180 (WATTS/Cm2), AT R 1.8 (AT OK) 1000 I redrng D =.o055 cm I'100 ~ ~ ~ ~ ~ ~S D.25 cm/!D.5K) cm Experimental maximum heat flux cm - 0 - Predicted mnximnu beat flux - Breen 8 Westwater 3)0 Correlation QO I`U tl=... ltO.04 cm., QOA E L t I I I I I I I I I E~astmon& IDatorsj. _ _.. A0~1_ i'"H,0 D- 0,1 cm. Reg'ion Paa a =1tIt o* P 1E2 6tm0-1 o{ri- - i'Ll(The points of minimum film boiling are Kutoeladie given by either the correlation of Correlati(5 /- -~ -ienbard & Wong (151) or of Zuber (162).) ~OI O00 001 EI 0.10 O 1.0 0 toJ 000 0,000 00,000 AT,'K FwGuRE 6 3 Experimental Nucleate and Film Pool Boiling of Helium Compared with the Predictive Correlations of Kutateladze and Breen and Westwater (li2). See Refs. (l15-.174).

-131From this result the (q/A) - AT relationship is, (q/A) = f(p) aT2'5 (88a) This equation has been computedby Brentari and Smith for a range of pressures as shown in Figures 64, 65, 66 and 67. for 02, N2, H2 and He. Since all data on which equation (88) is based were taken at normal gravity, the influence of g cannot explicitly be found from this result. It is probable the constant 3.25 (10-4) is valid only for g = 32.2 Ft/SEC2. This is also true for the correlations given in Table X. The presence of g in these correlations is primarily for dimensional considerations. Seader, et al. (27) have arranged several nucleate boiling correlations into the form, L* G* F(p)'Vq/A b (89) ( C G* which may be written in the form of a Reynolds Number, Stanton Number, Prandtl Number Correlation as Re F(p) ReB= - pa b (90) StB Pr The characteristic length, L*, mass velocity, G*, the property function, F(p), of the exponents a and b are given in Table X. It should be noted that a is the exponent on AT in the relation (q/A) = f(p) ATa. The value of a depends on the pressure and the nature of the surface. A comparison of -the various correlations in Table X is shown in Figures 69, 70, and 71 for H2, N2 and 02. They all are in fair agreement and generally fall within the range of the experimental data. The correlations of

(lTU//u-m Ft31, o (WATTS,/,C2, AT oR 1.8 (T oK) o Predicted maxmum heat flux. Eq. (10k) 10 - 0.01 0.10 t.0 10 100 1000 T, K Predictive Nucleate ad Film Pool Boilig Correlatios for ygstwatern (12) "p:,::~ ~ / /' 1: = i at 0tm. P =0.5~~1~/ I: (for fat plate and larg1e 1.07'//4r/-/I diametes, use D_>.0. ~jr j!~~/'by either the correlation of ienhard & 1Wong 0.01 0.10!.0 I0 I00 I000 FIGURE 64 Predictive Nucleate and Film Pool Boiling Correlations for Oxygen (142)

(BTU/Hr-Ft2) = 3180 (WATTS/cm2); AT 0R = 1.8( AT K ) io00 1 E <0* ASURED MAXIMUM HEAT FLUX.REF90) *EXPERIMENTAL DATA (90) J oX KUTA'ELADZE CORRELATION (146) / / Y //"/ / %~/w2 / p a 20 atm C E __ - =Io BREN & WESTWATER CORRELATION 0.01 0.10 I 0 I 0 I0 100 AT, KP FIGURE 65. PREDICTIVE NUCLEATE AND FILM POOL BOILING CORRELATIONS FOR NI TROGEN (42)

(BTU/HR-FT) 3180 (WATTS/cm~ PO (~ K 100 0 predicted maaxirr heat flux; P E Kurtateladze 0 -- (i46) ~~" Been & West'wate Correlation 16 a Oj C~~~~~~~~~~~~~~~~ ~ ~~orrelation (fO Tii I I I I I I l/ll/ln/lll I I L I ~~-V~~t~ P I atpl. fio fat` plate adnd large diameters, use D.>Z 1.0 cm.) (The point s of minimum film boili]3 are given by either the correlateo of Lienhard &e Wong (151l) Or Of Zuber (162) 0. I - 1.0 10OI 0.01 0.1 A, s K FIGUR 66~pREDITIM LEATEE AND FILM POOL BOILING CORRELATIONS142 FOR HYDROGENS 1

( BTU/Hr- Ftl) z 3180 (WATTS/cm2); AT ~R = 1.8(AT K) 1000 BREEN 8 WESTWATER CORRELATION (150) slt I atto. 100 1 1 1 I I I I D=-.OO1 cm L -PREDICTED _ I V y D=.002 cm MAXIMUM HEAT FLUX, Eq(104)/ D=.004 cm D=.006 cm 10 I 01 cm -.-liii-0~D=02 cm., /l W//} XrT+~O=.04 cm E 0 KUTATELADZE CORRELATION (146) = 0.1 cm - | O | | PI.P 0_ LLARE DIAMETERS,USE D21.O cm E 0.1P=1.0 (The points of minimum film boiling l0.01 -are given by either the correlation of Lienhard and Wong(151) or of Zuber,(162)) 0-0 001 0.001 0o.01 0.1 0 1.0 10 I00 I000 10,000 100,000 AT,'K FIGURE 67. PREDICTIVE NUCLEATE AND FILM BOILING CORRELATIONS FOR HELIUM.

II 0 O0 0 LLL~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-1 tf L MFfl 1-1 i 111,1 04 1TT H HI-II -t-F. T. I w.. S W It-t~~~lttIL —I II i-!1IIJI.[I I.0 z N [ |. W 0 l Il44 lt t g X g W- | tg RX ME; -'W~t0 —1Ng~gt. I~~~~~~~~~~~~~~~~~~~~~~ ~0~~~~~~~~~~~~~~~~~~~~~~~ I. I II- iiX T: w f,^: 4E-1TlTI TT~~~~~~~~~~-1- 1 tllHTTTT IT If Tf -lS I-Ft-tillTi T + k T T % s. W -41e1111 1 1 11111-111-1-fI1j++ -1-1 11 l-f~llitl0-I I IiiiitI- I~~~~~~~~~~ E, 4t TT g T g I StiL~~~~~~~~~~~~~~~liti~~~~tl~-t ~ ttk l dI 0 O0 I-4 434 ~~~~~~~~~~~'0 0 1-J H ii __ __ _ _ F11, I I I I IIJ-L I I H ~~~~~~~~T F.1 I I I _t r T __Rtmm 9t 1 I t —~tI Im1 ~~~~~~~~~~~~~~~~~~~~~~~~~oo ~~~~0,-I _____ __X_ TT~ ~~~~~~~~~~~1 fil I ___II__: _T _H ~ r l l Crrifr L.l ~ X l W X + i X ++1lllll 1 1 1 ll 1 1 | r~~~~~MI IIl* lFtiiARW LHfL-L RII, X I 3E e I f U X FFME-I I I I 1171 W. ruH F~f~l~ I I -H 0 u I E. X~r_ _ _ I _ uIIIII~ttiililifilT[HTI I 1 11111 lxliliilililulrlnilm ml II W IO q:SiE __f F_ -F|-11 ~tlllll~lII 1111 Il[H+XH0[FFI IIIA U~ ~ ~~~~~~~~~~O X y 111 llll 1"F4XQIr

to! I I I I I I I I I I I Note: NISHIKAWA AND YAMAOATA (182) PREDICTED HEAT FLUXES BELOW LIMITS OF THIS CURVE II I I I I I I II I I I... _._ DASHED LINES INDICATE RANlE OF EXPERIMENTAL DATA AT NEAR ATMOSPHERIC PREISURE - - -I -X -1 -O1 1.76 175 177 179 14 I 0 - r HYDROGEN 12.0 psla/ " I I I L/ i I I I,/,// i cIr IIITI I S/A I t # 1 17 t0"@''lRNO ZU lI / 4 ~1 Ref, No. AUTHOR 175 ROHSE NOWCgf Of 0015 -AL-L- TUPEIRH 7PORSTER AND OREIR SO LASOUNTZOV 176 PORSTER AND ZUSER /...17 MICHNKO 146 KUTATELADZE,Eq 11.22 - -" —- - -1t5 LEVY' III CHAN$ AND SNYDER 011 0.4 1.0 4.0 A O. 40. WALL SUPERHEAT, AT, R FIGURE 69. POOL NUCLEATE BOILING OF HYDROGEN AT 12.0 PSIA.27

-138J 179-175 180\ 176 146 7 181 182 177 NITROGEN / 15.2 PSIA I/ DI I II I IC R AG ir I /1 LL i.: t I I OF EXPERIMENTAL DATA AT ___ _ I /=(175) N EAR-ATMOSPHERNOWICs PRESSURE015 xldl __ /11 lll~n1177 1 IREF~17 FOSE&GE l! t- l! r/7(175) ROHSENOWC l'//, / D D(180) LABOUNTZOV -:: // / -/ -F EXPERE(181) CHANG a SNYDER (182) NISHIKAWA B YAMAGATA 10' 1.0 4.0 10 40 100 400 WALL SUPERHEAT, AT, OR Figure 70. Pool Nucleate Boiling of Nitrogen at 15.2 psia.(7)

IOs....'.../-01 1 I I I I179 / I / I __ OXYGEN 12.8 PSIA 178 N 14/,W t~/I ~ I!/ I r l s lDASHED LINES INDICATE RANGE / | _/1 I / |OF EXPERIMENTAL DATA AT, I 6 / / 1 NEAR-ATMOSPHERIC PRESSURE If m 10 "' io., 77. /?,/, ~//{! X /] =r fI (175) ROHSENOW, Ctf 0.015 -.76 / ~/r,,.t'182 (146) KUTATLADZE, EQ. 11.22 - 176 1/...2 WALLt. II 176) FORSTER a ZUBER Figure' Pl N t B (177) FORSTER a GREIF (178) LEVY (179) MICHENKO 180 /17/ / I (180) LABOUNTZOV /tf >L_|||(181) CHANG & SNYDER I _ (182) NISHIKAWA & YAMAGATA 1.0 4.0 I0 40 100 400 WALL SUPERHEAT, AT, OR Figure 71. Pool Nucleate Boiling of Oxygen at 12.8 psia.(27)

TABLE X SUMMARY OF POOL NUCLEATE BOILING EQUATIONS IN GENERALIZED STANTON NUMBER FORM (27) ReB = F(p) L* G* F(p) ReB or (St) a (pr)b IL T (q/A) L a (pr)b B A T CL G Reference |L* G* F(p) a b Eq. No. Rohsenow (175) | g a 11/2 |(qVA) 2.97 x 105 to 5.08 x 17 3 5.1 9 [. hf g (PL PVn Levy (178) CPL (PL - PV) Ts (q/A) PL 1P 4 hf ) from graph of Fig. 68 3 1 Zuber (176)1/2 (q/A) PL T -- 2 a(pv hfg PV (P - ) 1/2 Forster and ( PL P-P T ( Zuber (176) CPL f Forster and g(PV h hf PV 2 2 2 Greif (177) g 2 g 94

TABLE X (Continued) Reference L* G* F(p) a b Eq. No. Michenko (179) F1g2(q/A) PL"P 7/3 10 1.0 90 -P)h L 6.3 x 1]1/24 1g kPL fg PVO:c i P P Labountzov (180) CPL (PL - PV) Ts a (q/A) PL ReB < i02: 0.00391 2 4/3 hfg )2 J hfg ReB > 102: 0.00260 2.86 1.9 Chang and Snyder (181) CPL (PL - PV) Ts 0 (V/A) PL 57.2 x 10- 4 2.4 1 (PV hfg)2 J hfg Pv Nishikawa and Yasmagata (182) CPL (PL - pv) Ts (q/A) PL (PL - Pv) kL Ts g 1 3 1 (PV hf )2 J hf5 512 LM2 B pVhfg J gj /atm) 9 9 f9 Pv L 9 0 P h~, g M = 274.32 ft-1 B = 6.742 Btu/h f is a surface factor (See Ref. 182'

-142Rohsenow (175), Levy (178) and Michenico (179) were found to fit the nucleate boiling data for liquid nitrogen at one atmosphere from a polished copper sphere 1-inch in diameter (90). These data are shown in Figure 72. This fit was accomplished using a valve of Csf of 0.015 in Rohsenow's original equation. Zuber and Fried (139) report good agreement between the correlations of Rohsenow (175), Michenko (179), Forster and Zuber (176) and Labountzov (180) and experimental data of liquid hydrogen in nucleate boiling from a smooth flat surface, (148) in the pressure range from 0.8 to 5.1 ASM. In this comparison a value of 0.0147 was used for Csf in Rohsenow's correlation, essentially the same as used in Figures 69, 70, 71o Nucleate boiling heat transfer has been studied for hydrogen, neon, nitrogen and argon by Bewilogua, et al. (183) and for neon by Astruc, et al. (184). Each of these papers reports an hysteresis effect in the transition from natural convection to nucleate boiling, an effect ususally associated with systems in which contamination has been carefully avoided. A different nucleate boiling characteristic was obtained depending on whether the data were taken by increasing or decreasing the heat fluxo This is the same effect observed in the pool boiling of water at high pressures reported by Elrod, et al. (185). Experimental data of Astruc, et al. for neon are given in Figure 73, 74, and 75. The data in Figure 73 indicate a very steep nucleate boiling characteristic at low q/A and low pressures, possibly partly a result of the hysteresis effect. A comparison of these results at a pressure of 1 BAR (0o987 ATM.) is made with the measurements of Bewilogua, et al. and Lapin, et al. (186, 187) in Figure 74. Also shown is the nucleate boiling correlation of Kutateladze, Equation (88), indicating a moderate to poor agreement especially at heat flux above 1 watt/cm2. The hysteresis effect and the difference between the increasing-decreasing heat flux characteristic on nucleate boiling is shown clearly in Figure 75 for a pressure of 4 bars (3.948 ATM.).

-143wo~ F I I I I I I1I I IOI/I I-I250o 25sO/ 8 FOR LN2, o/g9 I: IDJ 6. A-(q/A)MIN PREDICTED BY BERENSON (112) -B- ATMIN PREDICTED BY BERENSON (112) - 4 - C - (q/A)MAX PREDICTED BY NOYES (195) D-RANGE OF A&TCRIT PREDICTED BY CHANG a SNYDER (181) 1/ A rU. I I I I I I A I I sI. O. I I _.__ = _z I I===z 104 __ -- _- = = - ----- -— l =I 2 -- -E/ (A _ AA 1 2 4 6 8 10 2 4 6 8 100 TW TSAT F Figure 72. Comparison of Results for Liquid Nitrogen at 1 ATM Reduced Manually and by Dijital Computer. a/g = 1, 1-inch Diameter Sphere. (9

(BTU/HR-FT2): 3180(W/CM2 ) AT,OR =1.8(AT,OC) 20 l0' o; I.1, ~M Atlr,; - A II d-1. -BAR - _.,.. / 0.5 _ __-4 P_2 BARS XP=:4BARS - - - + PI0 BARS _,;, I-? -r i- ~~ - P:18 BARS - OS ~ P:2OBARS o.~ B, i_ _._X- P-23BARS0.03-L — Nucleate Pool Boiling of Neon (Pt Wire, Diam. 0.01 5 cm) ( Pt Wire, Diam. 0.015 cm).

(BTU/HR-FT2): 3180(W/CMz) AT, ~R: 1.8 (AT, 0C) 10 I I I I " i BEWILOGUA (CAPILLARY) _PRESENT WORKi - I IWT (1)c BEWILOG O KUTATEET AL,ADZE TUE) CORRELAIlurl 7. E im TION 0,5 LAPIN t ET AL 0.1 0,. 0.5 1 5 10 aTIAT (C) Figure 74. Experimental Nucleate Pool Boiling of Neon at 1 Bar Compared With the Predictive Correlation of Kutateladze.

-146(BTU/HR-FT2):3180 (W/CM ) AT,OR: 1.8(AT,OC) * Ist POWER INCREASE + Ist DEC. 5 O 2nd " INC.'V 2nd " DEC. A 3rd " INC. I:-. - I -I I. I_ I 1 0.5 0.1 O.O3_I - v 0.05 0.2 0.5 1 5 ATSATTW-TSAT (OC) Figure 75. Hysteresis at Very low Heat Flux in Nucleate Pool Boiling of Liquid Neon. Absolute Pressure Four Bars.

-147Nucleate boiling data from an 0.811-inch diameter, 4-inch long gold plated cylinder at pressures from 1 ATM to 41.2 ATM (0.9 critical pressure)is reported by Sciance, et al. (188). Their nucleate and film boiling data are given in Figure 76. The correlations of Forster and Grief (177) and Madejski (189) were found unsatisfactory. However, a modified form of Rohsenow's correlation (175) in which pressure dependence was introduced was found to correlate the data for pressures below 0.7 the critical pressure. This equation is g/hfgPC 1 (cPLT ~/Tc'12.89 qAhfg1/2 5g(P' 1 ) 1 3.25(105) /T 1.18 This result is shown in Figure 77 in comparison with the methane data.- Equation (99) differs from Rohsenow's correlation by the inclusion of T/T and the exponents on the Prandtl number and cpO T/hfg. Lewis, et al. (91) have measured the influence of pressure and subcooling on nucleate boiling of N2 from a 1-inch diameter sphere. Their results for pressures of 1, 3 and 5 ATM and sub-cooling of 16, 22, 25F are shown in Figure 78. b. Maximum and Minimum Heat Flux The upper limit of heat flux for nucleate boiling and the lower limit of heat flux for film boiling, Figure 58, are each marked by unique physical states known as the maximum (or, first critical) heat flux and the minimum (or, second critical) heat flux. Each of these states is apparently characterized by a critical stability condition relating to the ability of liquid to maintain contact with the surface. Their physical states have been quite well defined by considerations of the stability of-the liquid-vapor interface. In the case

P/PC 10AS01 0~~~~~~~~002 0 0.60 H~~~~~~~~~~~~~~~~007 0'~~~~~~~~~G08 T OF~~~~~~~~~~~~~~~~~~~~~O W TSAT'" Figure 76. Nucleate and Film Boiling Data for Methane. (l88)

BEST FIT BASED ON Pr<0.7 STANDARD DEVIATION:-'0.124 * BASED ON LOG ORDINATE Pc:674 PSIA Tc 343.3 oR o 10~~~0 00 bI'. V e EQUATION (99),.I.1 V oh P/Pc, v ODO5 A 0.10r _ / v 0.15 0 0.20 * 0A0 00.50 0 0.70 A }: v e0 0.80 I ~ I.o. 0.01 CPL(TW TSAT T/TC' h. -- PR Figure 77. Nucleate Boiling Data for se85ne Compared with Proposed Correlation. ~1O)

10 C PATM 0 I SATURATED O 3 SATURATED O 3 SUBCOOLED 16OF A 5 SATURATED A 5 SUBCOOLED 22 0F 25 0F REFERENCE REFERENCE CURVE CURVE (90) SATURATED LIQUID (90) jC LEWIS, ET AL (91) ~~104 Cr I-INCH DIAMETER I-INCH DIAMETER SPHERE SPHERE SUBCOOLED LIQUID SATURATED LIQUID (o/g); (o/g):I 3 I L I I 1111 I 101 10 30 10 30 AT OF SAT' Figure 78. Effects of Pressure and Subcooling on Nucleate Boiling.

-151of the maximum heat flux the stability theory considers those conditions under which liquid no longer can adequately wet a heated surface. At a certain critical condition of bubble generation, the flow of vapor away from the surface has a destabilizing effect on the flow of liquid to the surface. This has been called the condition of Hemholtz instability. The minimum heat flux, on the other hand, represents a condition where, owing to interfacial stability of an overiding dense phase (for horizontal, upward facing surfaces, at least), the interface is disrupted and vapor alone no longer is in contact with the surface, but liquid comes in contact with it. This condition has been called one of Taylor instability and is described in terms of the critical wave length of an interfacial disturbance. As might be expected the heat flux from a surface behaves in an opposite manner in these liquid-deficient and vapor-deficient states: This effect is clearly evident from Figure 58 and is probably one of the most uniquely distinguishing characteristics of boiling. Kutateladze (190) was probably the first to point out that the critical heat flux conditions was a matter of hydrodynamic stability. However, Zuber (20)9 Chang (191) and Berenson (112) first applied the concepts of stability analysis to the process. Borishanskii (192) extended the work of Kutateladze to include the influence of viscosity, which usually is not particularly significant. Other work include that in References (149, 151, 193, 194, 196). A number of correlations for both maximum and minimum heat flux have appeared in the literature. Some of these can be represented by the functions (q)A) max. $ (100) hfgPv 1

-152and (q/A) min = (2) (101) hfgPvf 1/4 In most cases the dependence on acceleration is given as g. Because of this and the fact that Zuber (20) has shown on the basis of stability analysis that the constant in the correlations is a pure numeric (and not a function of the state of the fluid), each of the equations except that of Rohsenow and Griffith (194) is written with an acceleration dependence of (a/g) 1/4. This was not in the orginal equations. The value of g is that corresponding to gravitational acceleration on the earth's surface (32.2 ft/sec ), the condition under which the theories were based and essentially all the measurements were made. The various functions l! and 02 are listed in Tables XI and XII, taken from Seader, et al (27). It should be noted that both 0! and ~2 have the dimensions of a velocity. Several of the correlations in Table XI are compared with experimental maximum heat flux data for liquid nitrogen (90) in Table XIII. In this comparison the equations of Noyes (195) and Zuber (20) most closely represent the data on liquid nitrogen at 1 atm. The calculation from Noyes' equation is shown as line C in Figure 72. The computed result from the equation of Kutateladze, Equation (104), for (q/A)max is shown in Figures 60, 61, 62 and 63 in comparison with experimental data for 02, N2, H2 and He. The agreement is quite favorable. Since Zuber's and Kutateladze's equations are essentially identical the same agreement could be expected from Zuber's equation. Equation (104) of Kutateladze is also used as the upper limit to his nucleate boiling correlation as shown in Figures 64, 65, 66 and 67 for 02, N2, H2 and He.

-153TABLE XI SUMMARY OF POOL NUCLEATE BOILING MAXIMUM HEAT FLuX THEORIES (27) (q/A)max = ~1 hfg Pv Equation Reference tD [Length/Time] No. Rohsenow and PL - PV o.6 Griffith (194) 143) ft/hr 102 [ 1/4 1/2 1/4 Zuber ( 20) * g ] ) () 05* J \PL Q j/ Kutateladze (190) o.16 gg~ - )]o 1/4._L" P _, [a 990 2 L V)- 1/4 Borishanskii (192) [ )/4 1/405 [LX [' (o~-~)] 11/2/ 0.4 -[Lg(pL-PV), Noyes (195) 0.144 jol ] / P1 V 1( P ) (a) 1/ 106 Chang and Snyder 0 145[ 4 (L PV a 107 (181), L Pv Chgar ( KPL-PV)] 1/4( 1/4 K 0.098 vertical Chang (196) K [ ] 108 2 K = 0.13 horizontal i, -, i,,.l..,, Moissis and 0.18 o ( L Berenson (1 97) 1 + 2(p L) + * In a later discussion of Berenson's paper (112), Zuber gives a range for the constant in Equation (103) to be from 0.120 to 0.157.

TABLE XII SUMMARY OF POOL FIII BOILING MINIMUM HEAT FLUX THEORIES (27) (q/A)min hf (PV)f. 2 Equation Re ference l 2 [Length/Time] No. Zuber and 1/4 1/4 Tribus (195) K (), K1 = 0.099 to 0.131 110 i. Zuber and 1/4 o(L Tribus (195) a gg (alternateK 0.09 to 0.1441 IZuber (~20) I rra gr (PL-PV> i" (~" 1120 24 (PL+V )2 - Lienhard and 1 Berenson (112) K020 [ 990(PL PV) 113 Lienhard and 0 114 (1) 114 Wong (151) (PL + V) + (PL +l* g(PL - PV) + 2 -/4 g9a D2J * In a later discussion of Berenson's paper (112), Zuber gives a range for the constant in Equation (112) to be from 0.110 to 0.142.

-155TABLE XIII COMPARISON OF (q/A)max CORRELATIONS FOR SATURATED LIQUID NITROGEN (AT ATMOSPHERIC PRESSURE AND a/g = 1) AND EXPERIMENTAL DATA Investigator Equation No. (q/A),,, BTU/hr-ft2 Noyes (195) (106) 45,000 Chang and Snyder (181) (107) 56,500 Zuber (20) (103) 50,100 Kutateladze (190) (104) 61,000 Borishanskii (192) (105) 61,00ooo Chang (196) (108) 38,500-50,000 Moissis and Berenson (197) (109) 65,000 Merte and Clark (90) Experimental 47,000 + 1000 Chang and Synder (181) derived an expression for ATcrit by equating the maximum heat flux, Equation (107) to the product of the ATcrit and their correlation for h in nucleate boiling, resulting in: Tcr 00 721x 3(hfgpv) a[ggoa(ppv ) 1/4 ATcrit = x 1033 (a) (115) 1.450o P1/2k[CpTs (P- Pv) ] 2/AP7/5 The range in the constant arises from the assumption of from 25% to 50% coverage of the heater surface by the bubbles at the maximum heat flux, This range for liquid nitrogen is shown as lines D in Figure 72. The results bracket the experimental value of ATcrit' Maximum heat flux data for the cryogenic liquids 02, N2, H2, He and Ne taken by a large number of investigators on plates, cylinders and wires is shown in Figure 79. Comparison is made with Equation (104) but only fair to moderate agreement is found, Astruc, et al (184) found

-156. 100 60 40 20 ~-YO.09 XK 10 /~* NEON, ASTRUC ET AL 184).06 - - y NITROGEN, ROUBEAU1(147).04 NITROGEN, FLYNN ETAL(158) 66.4 NEXPERIMTROGEN, WTAEL a LACAZE (164).0 //2 HYDROGEN, CASTRUC ETAL (184) HYDROXGEN, LYROUBEAU (147)(198) oxvGEN, ANCEO L a LACAZE (165).01 - - OXYDROGEN, WEIL (166) 006 NITROGEN- HELIUM, LYON (198) 1.0 2 4 6 10 NITROGEN,20 40 60 100 200 40060047) NITROGEN, COWLEY ET AL (1546).04 13 NITROGEN, FLYNN ET AL (188) oNITROGEN, WE 8 ACAZ (164).021 (D1 I,8 HYDROGEN, CLASS ET AL (148) $ HYDROGEN, ROUBEAU (147) ~ HYDROGEN, WEIL a LACAZE (164).01 I G HYDROGEN, WEIL (166).00 &A HELIUM, LYON (199) 1.0 2 4 6 I0 20 40 60 I00 200 400 600 pvl Fluxes wl~tKh the Kutatele 2dze Mex~ ~orreletion,

-157that a best fit with their neon data was with a relation y = O.09Xk, whereas the remainder of the data in Figure 49, except for exceptionally low 02 and N2 points of Lyon, seemed to correlate best by y = 0.16Xk, the Kutateladze equation. Interestingly, the approximate mean curve through all the data would be that of Zuber (20), y = (7/24)Xk The maximum heat flux data of Science, et al (188) for the boiling of methane outside a 0.811 inch dia. tube over a wide range of pressures is shown in Figure 80 in comparison with Equation (106), the correlation of Noyes (195). An excellent order of agreement is found over the full range of pressures. Noyes' correlation and some cited by McAdams (74) for water and organic liquids are the only equations in which the thermal transport properties of the liquid are considered in the description of the process. The peaking of (q/A)max at P/Pc about 0.30 is typical. The influence of pressure, subcooling and a/g on the maximum heat flux is shown in Figure 81, from the results of Lewis, et al (91). The effect of both subcooling and pressure is to increase the magnitude of (q/A)ax. This observed influence of pressure is valid up to about 0.3 Pcrit., as shown in Figure 90. A comparison of Berenson's correlation, Equation (113), for the minimum heat flux is shown in Table XIV with experimental liquid nitrogen data at various accelerations (a/g) from 1 to 0.001 (90). The comparison is generally favorable and tends to confirm the (a/g)1/4 dependence predicted by the Equations (110 - 113), at least for an (a/g) above approximately 0.10. The calculation from Berenson's Equation is shown as line A in Figure 72.

-158150 NOYES CORRELATION, EQUATION (106) 0 0,00 LL - I I0 0 0.5 1.0 REDUCED PRESSURE /SIA Figure 80. Methane Burnout Het Flux Compare UREh the Noyes Correlationout He95 Equation. h the

~~~~~~~P=: ATA t - P=5 ATII \ | P=5 ATM a/g):= I SATURATED LIQUID LEWIS, ET AL (91) ATEDI P=5 ATM P:3 ATM a/g):0.1l7 SAT. SUB. wIv ~ L ~~P=3 ATM. I p SAT. IL /\ (o/g)<0.002 \~REFERENCE -rI~~~~~~ / ~~~~CURVE IREFERENCE P=l ATM CURyE (90) ATM m P=P I ATM (a/g)= _D. P. ATM SYMBOLS I-INCH DIAMETER SATURATED / (90) SPHERE LIQUID |/g 1.0 0.17 <0.002 SPHERE LIQUID do' I SATURATED v P=l ATM 3 O.OI<(a/g)<0.01<(a/g)<O.03 3 SATURATED 0 REE19 REF 19 3 SATURATED A — 3 SATURATED & | I-INCH DIAMETER SPHERE 5 SATURATED 13 O o. EFFECTS OF PRESSURE 8 SUBCOOL- b. EFFECT OF (o/g) 5 SUBCOOLED N (27 ) ING (27el') I I 11 II1 I 1 [I I I J1LLI 10 70 10 100 ATsAT OF Figure 81. Effects of Pressure, Subcooling, and (a/g) on (q/A),x.(91) (a/g on q/Amax'(1

TABLE XIV COMPARISON OF EXPERIMENTAL AND PREDICTED VALUES OF (q/A)min, ECJATION (113) Merte and Clark (90) a/g (q/A)min-predicted (q/A)min-experimental 1 2100 1700 - 2100.6 1850 1550.33 1590 1300 - 1400.20 1400 1300.03 5 875 7 870 - 1100.01 666 —.003 491 ~~.003 1~91~2 V180 - 530.001 374 - The correlations of Lienhard and Wong (151) and Zuber (20), Equations (114) and (112), respectively, for the minimum heat flux are given in Figures 60, 61, 62 and 63 for 02, N2, H2 and He in comparison with some experimental data. The comparisons are reasonable. The same correlations are used to establish the lower limit for flim boiling in Figures 64, 65, 66 and 67 for 02, N2, H2 and He The correlation of Lienhard and Wong is.an extension of the concept of Taylor instability introduced by Zuber and takes into account physical systems of small size. An interesting result is obtained for the ratio (q/A)max/(q/A)min using the correlations of Zuber, Equations (10) and (112). This ratio may be written,

-161(q/A)max i 1/2 (116)* (q/A)min VI Thus, with this simple result the maximum heat flux or the minimum heat flux may be estimated from a measurement of the other. Equation (116) suggests that (q/A)max and (q/A)min approach the same value of high pressures, a reasonable conclusion, confirmed by experience. An equation for ATmin was derived by Berenson (112) by equating the minimum heat flux, Equation (113), to he ATmin using his correlation (112) for film boiling, resulting in: ATmin = 0.127 f 7 1/3 gAc 1/2 ( -1/6 zSTmin = 0.127 Pvfhfg 9( p-pv) 1gof (117) This result for liquid nitrogen at a/g = 1 is also indicated on Figure 72 as line B. In this case the comparison is only fair. Lewis, et al (91) studied the influence of pressure and subcooling on (q/A)min for liquid nitrogen. Their measurements for pressures of 1, 3 and 5 ATM and an average sub-cooling of 29~F for N2 boiling on a 1-inch diameter sphere are given in Figure 82. Some data for the transition region are included for completeness. The effect of increase in both pressure and sub-cooling is to increase (q/A)min *Using the range of constants given in the footnotes to Equations (103) and (112'), this would be written (q/A)max =- (o0.845-l.436) ( |/ (116a) (q/A)min W / The results for N2 in (90) suggest the larger value of the constant best fits the data.

10- P, ATM o.EFFECT OF PRESSURE b. EFFECT OF SUBCOOLING -0 SATURATED I SATURATED P=5ATM I-INCH DIAMETER I-INCH DIAMETER SPHERE 03 SATURATED SPHERE (o/g):I _ 5 SATURATED P3ATM (/) P ATM SATURATED A 5 SUBCOOLED P=l ATM (RANGE 23~330; AVERAGE 290) IL D 10 m LEWIS, ET AL (91) SUB I0 10 P=5 ATM SAT P=3 ATM P=1 ATM REFERENCE REFRENCE CURVE CURVE P:I ATM 90) P:I ATM ____ ____ ____ ____ ___ I ISATURATED (90) 10 20 100 30 100 AT OF SAT' (91) Figure 82. Effects of Pressure and Subcooling on (A)iand Transition Boiling.

-163c. Transition Boiling The region of transition boiling, Figures 58 and 82 has not been studied very extensively nor is it described by any known correlation equations. It is a region intermediate between the liquiddeficient, maximum heat flux condition and the vapor-deficient, minimum heat flux condition. Because it represents a state transition from basically a liquid-solid, (q/A)max, heat transfer process to one in which the heat exchange is between the solid and vapor (q/A)min, the heat flux is drastically reduced as the process goes from the maximum to the minimum heat flux-condition. Only those physical systems having the surface temperature as the independent variable ever pass through the state of transition boiling. Systems which are controlled by establishing independent levels of (q/A), such as electrically, chemically or nuclear heated systems, make rapid transition from the points (q/A)max or (q/A)min to either film boiling or nucleate boiling. The data in Figures 28 and 82 were obtained by cooling a copper sphere from a film boiling condition to one of non-boiling. In this way the complete boiling characteristic is obtained. Presently, the only approach to the treatment of transition boiling is to make recourse to the basic (q/A) - AT data, where they are available. In the absence of data, it is suggested that the points (q/A)max - ATcrit and (q/A)min - ATmin be established using the correlations of the preceding section. The transition boiling region could then be estimated by fairing-in a suitable curve which would have an appropriate zero-slope at the maximum and minimum points. This will probably not produce data having a high degree of accuracy but could be used for preliminary estimates.

An interesting result in Figure 82 is the relative insensitivity of the transition region to pressure and sub-cooling. More data will be required before conclusions can be made on these effects, however. d. Film Boiling Film boiling can exist practically with cryogenic fluids because of their low saturation temperatures. In fact, it is possible to have a heat flux in film boiling greater than that of (q/A)max and still maintain sufficiently low surface temperatures to prevent melting. This may be seen especially fdr nitrogen in Figure 61 and helium in Figure 63. Because film boiling presents a reasonably well defined physical model (as compared with nucleate boiling) it has been treated analytically. Bromley (135) was one of the first to analyse this process and he modeled his study on the similarity to film condensation. Other such studies include those of Breen and Westwater (150), Frederking and Clark (200) Berenson (112), Chang (191) and Hsu and Westwater (152). In practical systems the total rate of heat transfer from a surface in film boiling will include an important component of radiation because of the high surface temperatures. This effect is discussed by Brentari, et al (143) who point out that if the surface temperature is below 400 to 4250K (260 to 3060F) the maximum error in neglecting radiation is less than 5% for film boiling of 02, N2, H2, and He. The error becomes approximately 50% for a doubling of the surface temperature. Bromley (135) correlated film boiling to liquid nitrogen on the outside of horizontal cylinder by hD- =0.62 D Pf( p-Pf)g hfg (1+0.4 CpfTsat (118) 0.62kf4 K+sat g(118) kf kft~fTsat hfg

-165For fluids of low latent heat (,ATSAT/hfg > 2) such as helium and hydrogen, Frederking (169) modified this result by the following -Do =0 5 i1/4(oPf g~fATsat. ( (119) ~ 0 522 (119) kf k 2 f For circumstances in which p~ \ pf, i.e., at pressures near the critical pressure, this result is very similar to that for single phase free convection from a horizontal cylinder, Equation (73). Bromley's result assumes the vapor film to be of negligible thickness compared with the cylinder diameter. For small wires this is no longer true. Frederking's results for the film boiling of helium I from small wires and the results of others on the liquids of nitrogen, oxygen and helium for cylinders ranging -in diameter from 0.0055 to 8.89 mm are correlated in terms of the Nusselt, Gaashof and Prandtl numbers in Figures83 and 84. At low values of the Grashof-Prandtl numbers, Figure 83, (10-7 < Gr'Pr < 10-1), Frederking's helium data are correlated by hD,3 2 Cp~ 0.11 hD= 2-5 ( (120) kf2 k The constant depends on the type of fluid. It is probably a function of the Prandtl number since the nitrogen data in Figure 83 fall slightly above those of helium for which the constant is equal to 2.5. It is probable the data in Figure 84 also are slightly spread owing a Prandtl number influence. More study is required to clarify this point. Merte and Clark (134) obtained liquid nitrogen film boiling data from a 1-inch dia sphere using the transient method, These data are given in Figure 58 where they are compared with Bromley's results for a 0.350

10, d x 10 mm DATA OF FREDERKING (169) x 5.5 Vl 13 oI0 He A 21.5 + 31.2 o 51 o3 13.4 10 N2 IV 31.2 0 51 OI 0.1 -7 -6 -5 -4 -3 -2 -I 0 LOG[ Db Pfg fATSAT cPI) Figure 83. Film Boiling Data for Helium and Nitrogen at Low Grashof-Prandtl Numbers from Cylinders.

I01 FREDERKING (169) 102 &A A d mm _~0 ~* 1.0 WEIL (166) N A1 8.89 BROMLEY (135) 0.635 1.01 BANCHERO, ( l'l 02 O 1.755 BARKER, e 3.22 BOLL (55) 19.05 -0 1.0 H2 WElL (166) -0 3.0 0.1 I _ - 0 1 2 3 4 5 6 7: p[ g.f 9 T AT cL Cl ] Figure 84. Film Boiling Data for Nitrogen, Oxygen and Hydrogen at High GrashofPrandtl Numbers From Cylinders.

-168inch cylinder. The agreement is good. These data have been correlated by Frederking and Clark (200) by hD0 = 0.14 Do3Pf(P -pf)g(1 + 0.5 c- Tsat) kf L lf2 k cpTsat (121) The results are given in Figure 85 for liquid nitrogen. The similarity of this result with that for turbulent single phase natural convection, Equation (72) or (75) should be noted. Equation (121) has also correlated the film boiling data of Banchero, et al (155). Lewis, et al (91) studied film boiling with liquid nitrogen from spheres of 1-inch, 1/2-inch and 1/4-inch diameter and from a 3-inch diameter flat disc. They investigated the influence of geometry, orientation and pressure on saturated film boiling and the effect of sub-cooling on film boiling. These data are given in Figures 86, 87, and 88 for pressures of 1, 3 and 5 ATM, average sub-cooling of 260F and at standard gravity. An effect of size is shown in Figure 86 is consistent with that predicted by Breen and Westwater in Figures 64 to 67. Both increases in pressure and sub-cooling result in increased heat flux in film boiling, as indicated in Figures 87 and 88. The data in reference (91) as well as that of Merte and Clark (90) are correlated over a range of pressures, sizes and a/g according to the parameters in Equation (121), as shown in Figure 89. The correlation of Frederking and Clark (200) is given for comparison. A reasonably valid correlation is obtained except for the data from the 1/4-inch diameter sphere. Also included are the freon 113 data of Pomerantz (201) taken on a 0.1875-inch dia. cylinder at (a/g) from 1 to 10. These data appear

-169MERTE S CLARK (134) 0 0 HSU & WESTWATER (152) -*> RUZICKA (174) /... — BROMLEY (135) ) EQ. (121)__ _ 1 _I___ 100 1000 ATSAT ST-AT'F Figure 85. Film Boiling H at Transfer Data for Nitrogen from a Sphere. 200

15( TEST SURFACE O I" DIA. SPHERE O 1/4" DIA. SPHERE - DISK - VERTICAL V DISK- HEATING UPWARD A DISK- HEATING DOWNWARD P=l ATM (o/g)= DISK-ALL ORIENTATIONS L. LEWIS, ETAL (91) 1/4" DIA. 3- 0 SPHERE -Q ~ ~ ~ ~ ~~~~~~ 0 Cr REFERENCE CURVE I" DIA. SPHERE (90) 0 1." 10 100 1000 TAT ~F SAT' Figure 86. Effects of Geometry and Orientation on Saturated Film Boiling. (91J

F INCH DIAMETER SPHERE o/gI Us P:5 ATM LEWIS, ET AL (91) P= 3 ATM Cr A ~~~REFERENCE CUIV P= I ATM 10 10~~~~~~~~~~~ 00 H0 SPA( Figure 87. Effect of Pressure on Saturated Film Boiling.(1)

los I-INCH DIAMETER SPHERE P=5 ATM (o/g)=l'J |I/ - \'-SUBCOOLED (RANGE 221- | / \ \ 330F; AVERAGE 26 OF) /SATURATED LEWIS, ET AL(91) In H / \ O / / REFERENCE CURVE P= I ATM (90) 1 10 100 00 AT o Sub,,coling on Film Boi Figure 88. Effect of Subcooling on Film Boiling.(91)

I0'' "' *'''',' * * I, I,, * J' I''I''' I I I', I I I LN2 DATA (SPHERE)(90,91) FREON 113 DATA P (ATM) I 3 (CYLINDER) PHERE PO*MERANTZ (201) ouI. ( 10) I El / o6 0 I J, o.-0 2 CORRELATION OF FREDERKING a CLARK (200) IH Y m 0I A L4 1 EQUATION (121) 1/2 O.0 5 X 1/4 LO 10D 7 + 0.171, o 10 V,o 3 _. o - -J IOC LO ATM CORRELATION OF BROMLEY (135), EQUATION (118) IOLz _o ~ c~Y ~ ~ - FOR LAMINAR FLOW': to _ x DATA SO MARKED WERE OBTAINED IN THE FILM BOILING REGION VERY CLOSE TO THE MINIMUM HEAT FLUX IO,,,,.....,I,,,.,,,,..;..,1,,,0,,,,2,,,, I o"'f 109I 1o 10o 2 Ra (Wf) )( L k )(h+0.5)() Figure 89. Comparison of Fractional Gravity Film Boiling Data for Liquid Nitrogen and Freon 113.(91)

-174to correlate well with Equation (121). Manson (202) found that her data for the film boiling in liquid nitrogen from a 3-inch dia. sphere fell about 7% below that predicted by Equation (121). The laminar film boiling correlation of Bromley is shown in Figure 89, indicating that transition to turbulent film boiling at Ra in the range 1707-108 The natural characteristic dimension for film boiling in systems of large physical dimension is probably the"critical" wavelength, d \ (112, 203), where =c 2 g(P~ /v ( g -1/ (122) and d = (X )3 (123) The flat plate data of Lewis, et al (91) for liquid nitrogen was correlated in terms of kc as shown in Figure 90. The best fit equation for what is believed to be turbulent film boiling is 1/2 Nu\ = 0.012 Ra (124) where, hAc Nu\ = kv kvf and Ra 2 k cp'sat + 5 (.(125) ~Ivf Ivf Also included in this figure is the laminar film boiling correlation of Berenson (112), which may be written h___ - 0.672 (126) F1

o10 TEST FLUID NITROGEN (91) WATER FREON-II SURFACES 3 DIA. DISK 18" F.R 1r 8 8 F.eP ORIENTVA- HORIZ. HORI ORIZ. HORIZIZ. TION VERT AT UP HEAT. DOWUP HEAT. UP HEAT UP P _10.16 __ I I 0.16 I 0 * a o ov 1 3 E | REF. (204) a 9 A EP A 0 EQUATION (126) cP I Allo' lIBERENSON (I112 - NuOk: 02 ( k:Z 102 -ONu Ra' Figure 90. Saturated Film Boiling Correlation for Flat Plates.(91)

-176where k3 (pi-p,)g hfg /4 F1 = ( v v T fg (127) and hfg = hfg (1 + 0.5 cvAT) (128) hfg The fact that the results fall above Berenson's equation is probably caused by turbulent flow in the vapor film. Although no criterion exists for the transition between laminar and turbulent flow, the results in Figure 90 suggests that is is approximately at Rak of 107 Of some interest in Figure 90 is the relative insensitivity of the data to orientation. The problem of representing cylindrical systems whose physical dimension is small compared with kC was treated by Breen and Westwater (150). Their correlation is 1/4 h kc kc = 0.59 + o.o069 (129) F2 D where, F2 = k3 Pv(PQ-Pv)ghfg2 ( a/ (130) k Lv AT and, 2 hfg2 = hfg 1 + 0.34 c (132) The functions F2 and F1 are virtually identical. Equation (129) was tested on several fluids, including the N2 data of Bromley (135), that of Banchero, et al (155) for 02 and the He data of Frederking (169).

-177The results are shown in Figure 91 and very good agreement is found. For systems in which Xc/Do <<< 1, Equation (129) becomes essentially the same as Berenson's result, Equation (126). Furthermore, under these circumstances h becomes independent of D. Equation (132) also is restricted to laminar flows. By comparing their results with the correlation of Bromley, Eauation (118), Breen and Westwater conclude that Bromley's equation is valid for 0.8 < Xc/Do < 8.0. The correlation of Breen and Westwater, Equation (129), was used to compute the film boiling curves in Figures 64 to 67. It also is shown in Figure 92 in compraison with the neon film boiling data of Astruc, et al (184) from small diameter wires. The agreement is quite favorable. Minimum heat flux points computed by the equations of Lienhard and Wong (151) and Zuber (162) are also shown. The film boiling data of Sciance, et al (188) for methane given in Figure 76 is correlated in Figure 93. The correlation of the data from an 0.811-inch dia. cylinder presented by these authors is F,~h~g I 0.276 Nu* 0.346 cpvaT(T/T) hg 7 (133) cpvAT (T/Tc)2 where, r g 1~ -1/2 -1/2 Nu* =a (q/A) 134) kvfAT g(P-r Pv) j ( Ra vf v c gfe-) o la(135) R\ 2= kvf2 k vf g g Equation (133) is shown in Figure 93 as the solid line,

BREEN AND WESTWATER (150) 10 14 Equation (129 N2 (135) 3 1 ow 3 U. ~~~Equation (129)' - Bromley) equation (118) (135) FIGU 0.3 G 3 10 30 100 300 Tc/Do FIGURE 91. GENERAL CORRELATION FOR FILM BOILING ON HORIZONTAL CYLINDERS, THE RANGE OF DIAMETERS IS FROM U.0022 TO 1.895 INN.

-17920 BREEN & WESTWAtER CORRELATION (150) - Dm0. Z CM D02. 2 CM — Dt l CM ~ONGO ~ J~YI~I 1 WIRE SYMBOLS PLATINUM DO.02 HI LEAD DvO.2 CM 0 LEAD D.OOS CM ORRE LATION 10 100 500 AT t(~C) Figure 92. Experimental Film Pool Boiling of Neon at 1 Bar Compared with the Predictive Corre ations of Breen and Westwate,(150) Leinhard and Wong, (151) and Zuber.(162)

p/pc 4 A 0.022 * 0-4 SCIANCE, ET AL (188) 2 0.05 0 0.5 2 A 0.10 0 0.6 EQUATION- (133) 1 0.15 0 0.7 EQUATION (136) 0 0.2 1 0.8 * *I0.3 0 0.9 0 IPc=674 PSIA Te:3440 T~w40R4 106 16'I 10' 10 106 10 Ra*8'aa~~~~~aaaa Tr2 Figure 93. Methane Film Boiling Data Compared with the Proposed Correlation.

100: "7: -! -I F 10 ~~~~~~~~~~~~~~~~~~~~~~~ii: H 0o T~~~~~~~~~~~~~~~~~~~ II 1.0 ~~~: i?; ii' 01 10 1; ~!10 100 O.9 0.5 0.1 i F I t ~ I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ * t i~~~~~~~~~~~~~~~~~~It+:=',x) PJ~ G R 94. R O N F F1 O ~~~IGR 9 LI RENOD NUB F ATR, ( I r j:F, OF- CH-EN, 0.1 1. 10 10 f ii ~~~~~~~~~~~~~~~~~~~~~9 05 0.1 f 1 T t I It I~~~~~~~~~~~ f rl f!X AIL X t X — ~~~ —h -I FGURE94. EYNODS NMBERFACTR* F OF HE.s(205

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-183Fdrced-Convection Boiling 5 16 go SS/5mm Pyrex pa 32 IbF/In.l obs 4 L/De ~ 48 a Va4.5 ft/sec 3 T -Tb 186- 72F _ 2 o V''8.tft/"sc TS - Tb 183 -122'F a L/De. 20 2 V ~ 15.1 ft /s ec Ts- Tb 197- 129'F.106 _ _ 9 _ 2 3L 9: I Pool Boiling 8.... sgo S5 ~7 F _ | ~Ip a 32 1 b/in.' abs 6 lTs-Tb 4 F TwTT /F Fig.~ ~~ _ 96P Focdcnetowac oin aaadpolbiigdt o *tin..-gg tub (38)

These same experimental data can be correlated in the manner of Breen and Westwater (150) by an equation similar to that of Berenson (112) and obtain an equally satisfactorily result. This may be done by fitting the data in Figure 93 by an equation in which the exponent in Equation (133) is changed from 0.276 to 0.250 and determining a new value for the constant. As a result the following equation is obtained h Ac1/4 -1/2 F1 = O. 745 Tc c F1 Equation (136) is shown by the dotted line in Figure 93. This result appears to correlate the data about equally well as Equation (133). Equation (136) has the same asymptotic form as that of Berenson, Equation (126). This is reasonable since the value of AC/Do for the system of Sciance., et al is small and the influence of size (Do ) should not be significant. The range of (T/Tc) for the data in Figure 93 is from 0.75 to 1.0. Thus, Equation (136) could be written, h %\c1/'4 = 1/2(17 =Ci(T/Tc)(37 F1 where -1/2 0.745< Cl(T/Tc < o.86o. This range of values is slightly higher than that of Berenson but with fair agreement considering the extended conditions and a different geometry of Sciance, et al. Furthermore, Equation (137) is in reasonable

-1852. Forced Convection Boiling a. Sub-cooled, Nucleate Boiling Nucleate boiling will occur in forced convection flow when the temperature of a heated surface exceeds the saturation temperature by a few degrees. When the bulk of the fluid is sub-cooled the bubbles which form at the surface condense in the liquid, either while still attached to the surface or after detaching and penetrating the liquid a short distance. The resulting flow phenomena in the region of the surface is enormously complex and not well understood or described at present. Owing to the action of the bubbles themselves the degree of liquid mixing or turbulence is significantly enhanced near the surface. As would be expected the heat transfer rate is significantly increased under these conditions. The process mechanics then appear to be governed by both bubble induced flows and the flow of the bulk liquid as influenced by the presence of the bubbles. Accordingly, most nucleate boiling heat transfer correlations in forced convection boiling attempt to account for these two simultaneous effects. At present there are very little forced convection, sub-cooled nucleate boiling data available for cryogenic liquids. The liquid hydrogen data of Walters in Figure 59 is one of the few examples. -Because of the limited amount of available cryogenic data., essentially all of the correlation methods to be discussed have been developed for noncryogenic substances. However., it may be expected that they also will apply to cryogenic liquids. Forced convection boiling with cryogenic flids isr' QY reviewedT =~7 by Seader, e t- al I(27) G~iarataoan Smith (I II)

-186The simplest method which has been proposed is to use the poo boiling equations alone for forced convection boiling. Zuber and Fried (139) in a review paper place the work of Kutateladze (146), Michenko (179) Gilmour (278), Labountzov (180) and Forster and Grief (177) in this category. This method, however, seems hazardous in view of the demonstrated variation between some of the forced convection and pool boiling data, especially with regard to the influence of sub-cooling. The data of Bergles and Rohsenow (38) for sub-cooled water, Figure 97, indicates a difference between these two kinds of boiling whereas that of Elrod, et al (185) for saturated boiling in pressurized water shows the data of nucleate poo boiling and forced convection boiling to merge at high heat flux. Rohsenow (38) and Clark and Rohsenow (63) have suggested that nucleate boiling data be predicted by a superposition of data for nonboiling forced convection and poo1 boiling. Thus, the heat flux in nucleate boiling, (q/A)N~, would be written, (q/A)NB = h(Tw - TI) + (/A)-p-, (138) where (q,/A )PB is the poo1 boiling heat flux corresponding to (Tw - T~ and h is computed from an appropriate non-boiling correlation, such,as hxDe oe.8 Pr. (52) -0.023Re Pr rf This method was employed on data for high pressure water at low velocities in tubfes (63~) w'here it was observed that at high (fully'developed nucleate

-1870~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 106 ___________ 8 7 6 54 38 -Pool Boiling 7 i6 go SS 2 6 p. 29 _bF/in, bs 5 0~~., Ts,-Tb 3IF 0 71 T. TAf. F Fig. 97 Influence of subcooling on pool boiling (38)

-188 - indicate that nucleate pool boiling and forced convection (fully developed) nucleate boiling are not the same. This suggests that the superposition method of Equation (138) should be used with discretion. A nucleate boiling correlation designed for saturated liquids has been proposed by Chen (205) and tested against data for water, methanol, cyclohexane, pentane, benzene and heptane. The average deviation of the data from Chens equation is + 11% for a vapor quality range from 1% to 70%. For flow in a vertical tube in the annular or annular-mist flow regime, Chen employed a weighted superposition hypothesis to account for the interaction between the vapor bubbles and the flowing liquid. Thus, the boiling heat flux is written (q/A)>= h(Tw - Tsat)F + (q/A)F S sat)F +(q/A)FZ (139 where h is computed from Equation (52), or equivalent, and (q/A) FZ is the nucleate pool boiling heat flux using the Forster-Zuber correlation, Equation (93). F is a two-phase correction factor given in Figure 94-i as a function of the parameter Xt- where lxO.9 P-2 0.1 (PvO05 (P1-a) X kV Pi S is a factor which accounts for the influence of the flow in suppressing the growth of the vapor bubbles. This factor is given in Figure 95, as a function of Re2F F'25 where Re, = DG(l-x) (14~11) 112

-189that the curves for forced convection surface (nucleate) boiling cannot be based on data for saturated pool boiling but must rather be based on actual forced-convection data." This results from the fact that the mechanics of the flow near a heated surface in saturated pool boiling is not similar to that in forced convection boiling, especially where sub-cooling exists, except perhaps at low velocities. Accordingly, the heat transfer rates also will be different. This, of course, complicates the prediction of forced-convection nucleate boiling data as it requires the measurement of these data. The difference between saturated nucleate pool boiling and forced convection nucleate boiling as well as the influence of sub-cooling on nucleate pool boiling are shown in Figures 96 and 97, taken from the paper of Bergles and Rohsenow (27). A method for the construction of the curve for the intermediate boiling region connecting single phase forced-convection and fully developed forcedI-convection nucleate boiling is given by IBergles and Rohsenow (277). This procedure, coupled with data for both the single-phase and boiling regions, then permits the construction of the complete (q/A) - At heat transfer characteristic curve. Since, the data given in (277) applies specifically to water., this construction method will not be reproduced here. b. Saturated (Film) Boiling Heat transfer to a flowing saturated liquid produces bubbles which do not condense but cause a net increase in the vapor fraction of the stream. This is called saturated boiling or, if the vapor

-190the large reduction in stream density which can occur. For flow inside of pipes and tubes the flow then accelerates, causing a momentum pressuredrop which in many instances becomes the predominent component of the total pressure-drop. Under these circumstances, the flow patterns in the stream are also very complex and difficult to define. These conditions contribute to the problem of correlating heat transfer data in this flow regime. Presently most of the available data is for liquid hydrogren. This includes the work of Hendricks, et al (66), Wright and Walters (78), Walters (136), Chen (205), Graham, et al (206), Chi (207,208) and Core, et al (209). Burke and Rawdon (210) and Laverty and Rohsenow (211) have studied nitrogen in film boiling. A summary of film boiling correlating methods is given in references (27), (141) and (143). Saturated and film boiling correlations have been formulated in terms of the Martine lli parameters, Xtt and the vapor mass fraction (quality) x, where, (C)0.9 ~110.1 Pv 0.5(10 Xtt __ ( )1(-) These correlations are written in terms of the ratio Nu,ex Nu, calc where, Nu exp. is the experimentally determined Nusselt number and Nu, calc. is a Nusselt number computed on the basis of a single phase correlation.

-191where, Nu,exp. = expx. D (143) kf, v o.8 o~ 4 Nu, calc,f,t.p. = 0.023 Refmtp Prf v (144) Refm t p= Pf,m,t.p. Vavg D (145) 1 Pf,m,t.p (x/pf) + [((146) Vavg = Pb A (147) 1 P1 ( (148) (X/pv) + [(lx/Q The correlation in the form of Equation (142) is shown in Figure 98 in comparison with experimental data. Although a general agreement is observed, quite a wide scatter exists in the data. Because the experiments were conducted on hydrogen alone, a correlation was sought (141) for the Nusselt number ratio of Equation (142) in terms of the local vapor quality x. The result is shown in Figure 99. There appears to be no significant improvement in the degree of correlation. However, some simplification is introduced since the determination of the ordinate x is easier than Xtt. An improvement in the correlation was obtained by Elerbrook, et al (212), by recognizing that the data in Figure 98 could be grouped into families of curves identified by constant "Boiling Number", Bo0 where, Bo=h G/ (149) hfg mix'

10 -.................. ~~Nuexp' _____- ___- =-NUexp exp(.0527-.416RnXtt-.008(inXtt)2) t-t~ ~ ~ Nu exp(. 1771I, - --......-4 —-.. Z~~~~~~~~~ I "`~~~~~~~I 0 00 ~ i~~~~~~~~~~~~~~~~~~~1 7 _ I )~~~~~~~~~~~~~~~~~~~~~~~~I Nuexp~~~~~~~~~~~~~~~~~~~...1 A0: Nu 0 -~ —~~ I' ~~- -- __ _ Hendricks et al (66) * Wright Ek Walters (78) A& Core et al (2 0 9.01.01.01.1 1.0 10 Xtt Figure 98 Two-phaseNusselt number ratio vs X (I14.001,01 1.0 ~.,

10 - i — I- i i t-+- _ _ _i~~~~~~~~~~c - i -i —C — -. —-- ~ - 9~8-4 —— 4- 4 — -... - - ___ _I ______ i I I*-. __ V4't "___ -..... -",. _ 0 Hendricks (66)- - ii_ —- -- - -4El Wright Oi Wolterst(78) A Core et al (209) -.01 _ -_1 -:'~]- - - I __ - - -' ~001 ~ 01 I 1.0 Figure 99 Two-phase Nusselt number ratio vs x(quality) (141)

-194where, w Gmix Ac * (150) The separation of the data by boiling number is shown in Figure 100. The boiling number may be interpreted physically as the ratio of the transverse mass velocity of the vapor formed at the wall, (q/A)/hfg, to the axial mass velocity of the mixture, Gmix. The boiling number is used to modify Equation (142) to give Nu, exp. -0.4 ~-Nu ca Bo0 = f(Xtt). (151) Nu, calc, f, t.p. This form of the correlation is shown in Figure 101 where it will be observed that a significant improvement in the correlation has been obtained. The modified Nusselt number ratio is shown in Figure 102 as a function of quality. In this case the degree of correlation also is improved. c. Maximum Nucleate Boiling Heat Flux (Burnout) Very little maximum nucleate boiling heat flux data are available for cryogens and no suitably reliable correlations are known to exist at present. The available data are those of Lewis, et al (213) for liquid nitrogen and liquid hydrogen. These data are compared with the following correlations of Lowdermilk, et al (214), originally derived for pressurized water in Figure 103. (q/A) 2L70; 1 < G < 150 (152) D02 (L/D)o.85 /)2 - 1400 G05 G 4 (153) _/) 0o2 0o; 150o< < lo0 (153) D (L/D).15 (L/D)

I0 Hendricks et 01(66) Avg. Boiling Number (1) 0.575x103 _ - I (2) 1. 51! i (3) 1.89 (4) 2. 10 (5) 2.75 (6) 3.23 (7) 4.09 (8) 4.55 (9) 5.89.01.001.01.1 10 10 100 xtt Figure 900 Two-phase Nusselt number ratio vs Xtt for only the data of Hendricks, et a1(66) illustrating separation of data according to boiling number (141)

i00_'D I I 1 1155 E~,I,IHT V 11, 100 - I Mm- A.,,lo " -: ___~~~~~~~~~~~~~"""""'.. __._.~,,_ _, 0 __~ _ —-......... \Nuco,__.f,,. I exp(2.35-.266 s nXtt -.0255(r tti Xtt) — ____~~~~~~~~~~~~~~~~~~ ___ I_ 0 Hendr icks et at (66) —--- _ _ la Wright EN Walters (78) --- -__ ACoretaI(209)- -- e —0 X 1.h.. ___ I~~~~~~~~~~~~ J j.;L~~~~~~~~~~~~~~~~~~~~~~.001.01. I I0 Xtt Figure 101 Two-phase Nusselt number ratio times boiling number factor vs X (141)

100 T 1 I0I - I I I I I I I I I I I ~ I z 10 el e e 80~~~ No~~; =exp(283 +.3173nx.000569( nx ) *Hoendricks eIt l (66) *Wright B Walters (78) iCoreLt aIj209) _ I 0T.,Ns n r i e i n r o.001.01.1 i E~igure 102 Two-phase Nusselt number ratio times boiling number factor vs x(quality)(141)

-198Liquid Test-section Exit Length-to- Tube pressure, pressure, diameter inside Ib/sq in. abs atm ratio diameter, in o Nitrogen } 50 -- 8 to 29 0.555 ~1050 Hydrogen 104 E C\\Water -- I 25to50.051to.188: t:.... _ _, ] l I _ _ _ =_ I _ _.___ IOL -I I II!0I 1I 1 1 lint==,_ c with _at-pr rn-rrePatin" nf T'.Irmilk et al, _,__ G/(L/D)2 liquids with water correlation of Lowdermilk, et al.(214)

-199As can be seen the correlations agree with the general trend of the data but predict values for (q/A)max which are greater by almost one order of magnitude than those observed for liquid nitrogen and hydrogen. Thus, Equations (152) and (153) should be used with discretion. d. Pressure Drop in Two-Phase Flow The total pressure drop in a flowing fluid is caused by the effects of viscous friction, the influence of body forces (such as gravity) and the momentum changes in the flows. For two phase systems subject to heat transfer at the wall, and thus to significant density variations, the latter effect, namely momentum changes, can be the principal factor in the pressure drop. In instances where this is true, the calculation of the pressure drop is greatly simplified providing the mixture densities can be determined. For exit qualities (xe) above 0.10-0.20, an assumption of a homogenous stream provides a reasonable estimate. An example of this is shown in Figure 104 where the total measured pressure drop across a 0.313-inch I.D. tube, 12-inches long is compared with that computed for simple momentum change. These results are those of Graham, et al (206) for the flow of liquid hydrogen. As can be observed, the two results are essentially identical, especially for the higher exit qualities. Because of the inherent large temperature difference between a cryogenic fluid and an atmospheric ambient the probability of boiling is great. The design of piping, transfer lines and associated pumping equipment depends on an estimate of two-phase flow phenomena. Two phase pressure drop is classified as adiabatic (without heat transfer) or diabatic (with heat transfer). Two-phase flow and pressure drop have

-200I I I I Run Wq b, inqXe (table I) lb/sec Btu/sec OR l 0 22-4 0.151 8.23 45.4 27 0 22-1.142 10.93 4' 38 Q 22-2.101 10.95 45.5 57 D 22-3.068 10.94,5.5 89 Experimental --- --- Analytical 80 Run 22-1 70 22-4 Cd 60 50 5,0 -.___ _ Ca ra~~~~~~~~~~~~~, 30 0O 2 4 6 8 10 12 Inlet Length, L, in. Exit Figure 104 Comparison of experimental and theoretical pressures along length of test section of 0.375-inch-outside-diameter Tnconel tube. (20 6)

-201been studied by Baker (215), Martinelli and Lockhart (216), Martinelli and Nelson (217), Leonhard and McMordie (218), Hatch and Jacobs (219), Lapin and Baver (220), Chenoweth and Martin (221) and Dukler, et al (222), among others. A simplified calculation method for total pressure drop for low quality flow of saturated freon-ll is given by Sugden, et al (223). Timmerhaus and Sugden (224) give a simplfied method for the calculation of the frictional pressure-drop for freon-ll, freon-12 and liquid hydrogen. Their method is similar to that proposed by McAdams, et al (225) and used by Davison, et al (226), who found the two-phase friction factor about 500 greater than smooth tube single phase friction.factor. Baker (215) has described several regimes of adiabatic twophase flow inside-pipes and ducts as, (a) bubble, (b) froth, (c) plug (vapor), (d) stratified (liquid-vapor), (e) wavy, (f) slug (liquid), (g) annular (liquid) and (h) mist (fog or homogeneous). These flow patterns were identified by Baker in terms of certain parameters of the flow as shown in Figure 105. This representation has become known as the "Baker Plot" and has been found to be reasonably valid for pressurized water systems. Lapin and Bauer (220) also have used it to identify the flow regimes for methane. The data points on Figure 105 are those of Leonhard and McMordie (218) for freon-12 and show a reasonable agreement with the flow regime boundaries. As the "Baker Plot" suggests the flow of two phases is a rather complex phenomenon. Several correlating methods for pressure drop exist in the literature and individual design groups frequently develop limited emperical forms valid for a specific range of application. Often these latter are modifications of existing correlations. In this

z 6III|STRATIF MiST FLOW I 4 ____o SLUG FLOW' r ASTR ATIFIED FLOW I I I I I I I I I 10 II a EXPERIMENTAL POINTS F-12(218) A - BUBBLE OR FROTH FLOW A T PLUG FLOW El - PLUG FLOW 10 OJ.2.4.6.8 1.0 2 4 6 8 0 20 40 6080100 2 4 6 8 103 2 4 6 8 0 G FIUE10.BKE LT O DABTCTW-HSEFO.(215) FIGURE 105, BAKER PLOT FOR ADIABATIC TWO-PHASE FLOW,

-203section it will be necessary to limit the description to but two and these will be those of Martinelli and his co-workers (216,217) for adiabatic and diabatic flow. These Martinelli correlations describe the essential features of the process and introduce the governing parameters. They also recognize a slip-flow between separated phases, something which is necessary for any realistic representation of the process. The Martinelli correlations have enjoyed a remarkable endurance over the years having been used with reasonable success on air-water mixtures at low pressures, on pressurized water systems and for cryogenic fluids. However, it should be mentioned that Lapin and Bauer (220) have found that an extrapulated form of the Chenoweth and Martin (221) correlation was most satisfactory for the calculation of frictional pressure drop for liquid nitrogen and methane. For the calculation of adiabatic turbulent two phase flow frictional pressure drop, Martinelli and Lockhart (216) present the following equations, (AT)TPF = 02 (154) where (ap) = frictional pressure drop per unit length in twoAL TPF phase flow = frictional pressure drop per unit length if liquid only flows in the pipe at rate w The function 0~ in Equation (154) is given in Figure 106 as a function of X, defined as x2 (a/ )g(155) (p//AL)g

-204where, = frictional pressure drop per unit length if gas only flows in the pipe at rate w The fractional pipe volume filled by liquid, RQ, and that of the gas, Rg, also is given in Figure 106. For diabatic flow, the effect of heat transfer is to change the relative vapor-liquid fraction along a pipe. In this case, Martinelli and Nelson (217) found the following equation to describe events at each axial location, (dpdL)TPF = (1 - x)2-n 2(p,x), (156) (dp/dL)o where (dp/dL)o is the frictional pressure gradient if only liquid were flowing at the total flow rate we + wg. The constant n is the exponent in the friction factor-Reynolds number relation, f = C/Re. For turbulent flow this was taken to be 0.25. The function ~0(p,x) is found (217) from Figure 106 corresponding to X which may be obtained from the following relation and is equivalent to that given in Equation (155), 2-n /2 n 2-n X =x2 = (p) (OL)2 (1 _(157) Pi 119g X Hence, for a condition in which the pressure level and the axial heat flux distribution (and thus x ) is specified, the two-phase frictional pressure drop is computed from, Ldp d H\/n\ 2 -n 2 PTPF = ( FdL~ = () (l-x) 0 (p,x) ~ dL o x 0 ALo (dx/dL) ('( o

-205If the axial heat flux is constant, dx/dL = xe/L, and Equation (158) becomes (noting L = AL), x (r) = 1 - x) 2- (p,x) d x(159) (AL TPF ALo xe 0 The single-phase pressure drop gradient, (Ap/AL)o is found from, (AP) 4 f OPV~ (160) Z,) 0 De 2go where, Wg + W v =w gW (161) F F o.o46 f = 0'O46 (162) Re 0.25 Re = DeVo (163) The momentum pressure-drop (217) is computed from, 2 2 2 (wg + w) (l - Xe) _ Xe p 1 (164) apm 2 P~AF g0o Re Rge Pg where RE and Rg correspond to the value of X, Equation (157) evaluated at x = xe. The total pressure drop would then be Ap = APTPF + ApM, the sum of Equations (158) and (164). This method for estimating two-phase frictional and momentum pressure drop been extensively employed in computing the performance of pressurized water boiling systems. Hatch and Jacobs (219) have used this method to correlate adiabatic and diabatic pressure drop for Freon-ll and hydrogen. Their data are shown in Figure 107, 108 and 109, where the parameter X was evaluated at the mean quality along the

-2061000 1.0 I00 0.1 1, I- 0.001 0,01 0.10 1.0 1I0 100 x Figure 106. Adiabatic two-phase flow pressure drop func I ~ I Ir I I I I m60 1 ~0 1 1 I 0.1 0.2 0.3 0.5 1.0 2 3 4 6 10 Figure 107. Comparison of two-phase (adiabatic) pres- elson. prediction. () Nelson predict ion. (21!9)

-207_00 60 _ _ 50 \ _ 40' I 0 r MARlTINELLI 4 - -= _, _ s, q- - AN0 NELSON 3' 0 3, r M i - S IN - Figure 108. Comparison of two-phase (diabatic) pressure drop data for boiling Fregn-ll with Martinelli-Nelson prediction.(21?) 3- -- I I-111 M —A —t ] - N. o. o. o. o.',.0 a = e 18, 4fx Figure 108. Comparison of two-phase (diabatic) pressure drop data for boillng Inl'~ ctin, I 1-1 1 -- 04.2.3.4 66,10 2 34 O lO Figure 109. Comparison ofi two-phase (diabatic)

-208test section. The Freon-ll data fell below the recommended curve of Martinelli - Nelson by about 20%. The authors attribute this partly to the existence of metastable liquid in the flow. The hydrogen data scatter considerably although the mean correlating line is also below the recommended curve. In both cases the experimental data closely parallel the Martinelli - Nelson correlation. As may be seen from these results two-phase pressure drop can exceed the single-phase pressure drop by several orders of magnitude for the same rates of flow. 3. Gravic and Agravic Effects on Boiling Heat Transfer In recent years the application of boiling heat transfer data for environments of varying gravity has become of importance. This is a result of the need for information on heat transfer for the design of both high acceleration systems such as booster rockets and re-entry vehicles as well as devices to operate in sub-terrestrial gravity environment as orbiting satellites and deep space probes. Results obtained from these studies will undoubtedly have application in other fields also, as well as providing an improved understanding of the process fundamentals. One important aspect of low-gravity studies to date has been the isolation of gravity or body-force controlled phenomena and an evaluation of their true influence on a process. Also, a lowgravity environment is effective in reducing natural convective motion thus enabling the effects of other forces to be determined. An approximate listing of the various gravity conditions to be expected for the operations of equipment is shown in Table XV. Essentially all of the low-gravity heat transfer data have been taken using drop-towers. Some data have been taken with a KC-135

-209TABLE XV APPROXIMATE RANGES OF a/g FOR EQUIPMENT OPERATION Source Application Order of Magnitude of a/i' 1. Centrifugal Rotations in Turbines 105 and compressors 2. Vehicle Acceleration Liftoff and Re-entry of 10 Rockets and Space Vehicles 3. Earth Gravity At Surface of Earth 1 4. Lunar Gravity At Surface of Moon 10-1 5. Aerodynamic Drag Drop Tower Experiments 10 to 10 6. Thrust from ullage Typical Thrust: 10 lbf 10-4 Control or From Radio-isotope Propulsion 7. Centrifugal (Space Vehicle Rotation)* a. To Maintain Vehicle Low Earth Orbit 10-6 Parallel to Earth High Earth Orbit 10-8 Surface b. Limit Cycle to Angular Velocity Due 10-8 Maintain Vehicle to Limit Cycle: Oriented Towards 0.05 deg/sec Sun or Star 8. Aerodynamic Drag* 100 mi Orbit 250 mi Orbit 10-8 400 mi Orbit 10-9 9. Solar Pressure* Deep Space Penetration; Low Absorptance Surface 10-10 Typical Vehicle: 100,000 lb class about 20 ft in diameter (227).

-210aircraft in a Keplerian trajectory and a few have been taken in a magnetic field using the principle of magnetic levitation. The drop-towers have so far proven to be the most convenient, reliable, accurate and least costly. Their great disadvantage, of course, is the relatively short period of test time permitted (1-4 sec.). Longer duration of test times may be expected when orbital laboratories are put into operation. High gravity heat transfer data have been obtained on centrifuge apparatus. Present capacity enables the attainment of gravity levels up to 1000 times earth gravity in completely instrumented test packages.* Nucleate and film pool boiling heat transfer data for liquid hydrogen for gravity levels (a/g) from 1 to 7 are reported by Graham, et al (228). Their results are shown in Figure 110. The influence of increased gravity forces is relatively insignificant in nucleate boiling but prominent in film boiling. Clark and Merte (93) and Merte and Clark (92) have studied the effect of gravity fields in the range 1 < a/g < 21 on pool nucleate boiling to saturated liquid nitrogen and water at atmospheric pressure. The results of this investigation are given in Figures 111 and 112. As in the case of the data for liquid hydrogen of Graham, et al (228) the influence of increased bouyant forces on nucleate boiling is not significant, at least in the range of bubble dominated boiling. The apparent influence of a/g on the low boiling heat flux in Figures 111 and 112 is attributed to the effect of single phase, natural convection. This is to be expected since single phase natural convection is uniquely related to bouyant forces. Thus, it is apparent that in the region of well-developed boiling it is the dynamic forces associated with the bubbles themselves which govern the process, the * 10-foot diameter centrifuge installed in the Heat Transfer Laboratory, Department of Mechanical Engineering, The University of Michigan, Ann Arbor, 1963.

-211-.2 NUCLEATE FILM BOILING BOILING.1.08.06 / z.4 RUN GRAVITY PRES- BULK.04 ga SURE, TEMPERATUREI gs PSIA R —,.. eP' 0 2972-2979 1 52.5 46,4 2 / {J 2983-3003 1 49.5 45,8 / A 3006-3024 7 52,3 X I I LLx I I.01 /.008.006.004.002 I I 2 4 6 8 10 20 40 60 80 100 200 400 TEMPERATURE DIFFERENCE, AT, R FIGURE 110. COMPARISON OF EFFECTS OF ACCELERATION ON NUCLEATE AND FILM BOILING FOR PARA-HYDROGEN,(228) AND FILM BOILING FOR PARA-HYDROGEN.(

-21240,000 30000 0 a/g I + a/g: 5 CO o/g ~10 20,000 - ~ a/og 15 a/o ~20 10,000 8,000 LIQUID NITROGEN 6,000 /// //INON- BOILING |A/ / /CONVECTION Nu / 0.0505 (Gr Pr)WO 1,000 800 1"" /// 400 2 3 4 5 6 7 8 9 10 15 20 Tw -TSAT, O F Figure 111. High-Gravity Nucleate Boiling Dat fo Liquid Nitrogen at Atmospheric Pressure.(1~93)

20 - - - = 90 70 _ 60.. EXPERIMENTAL DATA 40C FOR POOL BOILING TO / ESSENTIALLY SATURATED WATER m / o a/g S I'o 30 ~ 3 > l; tI *- /10 ~L 1{{1/ I @+,,,SI I-s o-..'.10 U'.- *' l.B S 20 & %tl t t ^- u 2C (229). -DATA OF COSTELLO a/gs I 10~/, 9 (" a -vs;o NON-BOILING CONVECTION 7 EUD 0. 0N: 85 (a P O Pft).0 10 15 20 25 30 40 50 60 708090100 Tw- Tat, Figure 112. Influence of system &cceleration rnormal to heating surface on convection and pool boiling. (92)

-214role of bouyant forces not being nearly as significant as previously believed. It is possible, of course, that at very high levels of a/g this condition will no longer prevail. This same effect will be seen in the low-gravity results to be discussed below in which the bouyant (body) forces have been essentially removed. High gravity data (a/g from 1 to 10) are reported by Pomerantz (201) for Freon-113. His film boiling results were given in Figure 89 and found to correlate reasonably well with Equation 121. Beckman and Merte (230) studied the process of bubble formation and growth photographically in water in the range of gravity (a/g) from 1 to 100. High speed motion pictures of the bubbles taken at 20,000 fps showed some remarkable dynamic effects. Maximum heat flux data of Ivey (231) for water are shown in Figure 113 for a/g up to 160. The 1/4-power influence of a/g on (q/A)max, predicted by Equations 103-109, is seen only to be approximated. Ivey found that an exponent on (a/g) of 0.273 would best fit his data. However, the maximum heat flux data of Merte and Clark(90) for liquid nitrogen at (a/g) ranging from 1.0 to about 0.10 shows a reasonably good agreement with the 1/4-power prediction. These data, taken from drop-tower measurements, are shown in Figure 114. For a/g less than about 0.10 the 1/4-power dependence of a/g seems no longer valid for calculations of the maximum heat flux in pool boiling. This effect is suggested in Figure 114. Lyon, et al (234) have made an extensive study of the influence of a/g on (q/A)ma in these very low ranges of acceleration using the unique technique of magnetic levitation. The results of these investigators is shown in Figure 115 for liquid oxygen where comparison is made with measurements of others for water, liquid nitrogen

z 2 (q/A)m, o/g I"' 0 — (q/A)x,ao/g. i..% is x -I li _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -/74 4Z %,o TVey (231) Ukaea I (,/)7 Usiski and Siegel (252) O( I 06 8 DIMENSIONLESS CENTRIFUGAL ACCELERATIONI AT TEST SECTION X 00 Ivey (231) Ukcaea I<(0/g)15 ~ U siskin and Siegel (232) O C((o/g) ~ I "' 40 60 so K )O 120 140 mo0 DIMENSIONLESS CENTRIFUGAL ACCELERATrON AT TEST SECTION RADIUS ( a/g ) Figure 113. Effect of system acceleration on maximum heat flux in pool boiling.

(q/A)mx by NOYES g- 6- REF. 195 I' iLSLOPE~4 1 0-2 2 3 4 5 6 7 89 0- 2 3 4 5 6 7 8 9 1 a/g Figure 114. Maximum Heat Flux Data for Liquid Nitrogen from 0 Sphere During Fractional Gravity. (90)

0.8, II 0.7,; u~ I I l 0.6 e!!, 0.5. / I -4.. EE 0.4 0o LYON ET AL234 370 G CHARGE 02 // 5V'V0 +(a/g) 0 LYON ET AL(23L) 170 G CHARGE 02. _ 0.3 s A LYON, ET AL(234) 170 G. CHANGE 02, USISKIN AND SIEGEL. DISTILLED WATER 0.2 L / I>- ON PLATINUM WIRE. 0.2 (90).MERTE AND CLARK.,90) NITROGEN ON COPPER SPHERE. 0.7 (233) 0.1 SHERLEY,(233) HYDROGEN ON LEAD FILM. o.0 -0.03 0.0 0.05 0.10 0.15 0.20 0.25 (o/g) VAPOR FIGURE 115. NORMALIZED MAX HEAT FLUX AT SMALL RELATIVE VAY9 ACCELERATIONS (CALCULATED FROM BULK LIQUID TEMPERATURES).

-218and liquid hydrogen. As may be observed, a considerable departure is found from the 1/4-power formula. At very low a/g the actual (q/A)max is significantly greater from that predicted by any of the Equations 103 to 109. At the present, the meaning of this is unclear. It seems probable that owing to the powerful dynamic influence of the bubbles a finite maximum heat flux will exist even at a condition of true zero-g. How the vapor will be removed from the surface under these conditions is unknown and doubtless would present a serious problem for sustained operation at true zero-g. However, at least for a short period following the initiation of zero-g it seems probable that (q/A)max would be finite and a significant fraction of that at normal earth gravity. An interesting feature of the work of Lyon, et al (234) is the control they could impose over both the magnitude and direction of the acceleration vector (body or bouyant force). Some of their results for a small negative acceleration also are shown in Figure 115. Under these circumstances the maximum heat flux rapidly approaches zero since the forces in the system tend to drive the vapor towards the surface. Gravity dependence on the minimum heat flux is also predicted in accordance with the 1/4-power relation in Equations 110 to 114, Table XII. A reasonable agreement with experiments using liquid nitrogen in the range a/g from 1.0 to 0.001 is shown in Table XIV. Low gravity heat transfer data of Merte and Clark (90, 93, 134), for liquid nitrogen in the nucleate and film boiling regions are given in Figure 116. The film boiling data for a/g of 1.0 are in good agreement with the results of others (135, 152, 174) for systems in which the ratio Xc/D is small. For the range of a/g from 1.0 to 0.20

105 - V o/g 0. OI, 1/2 INCH DIA. EX ao/g I, 1/2 INCH DIA. (q/A) max by Equation (106), a/g I * o/agO 1, I INCH DIA. o o/g - 0.01, I INCH DIA. 7/g=./ 0 HSU AND WESTWATER (152) U1"' a BROMLEY (135) ~ li*~ 0 RUZlIKa (174) _*~ A.~~A —-A EQUATION (121) o/gl * P 0 o/g:0.20 ** e o/g: 0.33 (q/A), Equation (106)o/g =.Ov Q0.0 max Ja/QvI.Vc *f06 Al ~9C~ZPB a Q/Q = 0.60 00 % IL /~~~~Ii * A o — cr 0~~~~~~~~~~~~~~~~ 4 0 * a 0 ~~ 0 \ /~~~~ ~,/'0 10 100 1000 ~TWISAT'~' OF Figure oiing eat Transfer to Liquid Nitrogen at Atmospheric Pressure.(90) 1~~~~0:::) 104500 ~~~~~~~~~~T-Tr " Figure~~~~~~~~ 01.Biigha Pase oLqi Ztgna topei rsue(O

-220these data correlate well with Equation 121 as shown in Figure 89. This indicates that the controlling force in film (pool) boiling is the body force. The nucleate boiling data in Figure 116 indicates a remarkable lack of sensitivity to the 100-fold reduction in gravity. A similar result is reported for liquid hydrogen by Shetley (233) as shown in Figure 117. Both of these sets of data are consistent with the observations of the influence of high-g on nucleate boiling which was indicated in Figures 110, 111 and 112. These results have led to the aforementioned conclusion that bouyant forces are not of prime significance in the process of nucleate boiling. The role of inertia forces resulting from bubble growth appears to have the principal influence in the process. These observations have suggested the formulation of a Froude number, Fr, criterion to indicate the relative importance of inertia and bouyant forces (235). Adelberg and Forster in a discussion of Reference (232) have formulated the Froude number as Fi _ 3R2R2 + R3R (165) Fr = (165) Fb R3(a/g)g This result can be evaluated further by introducing the following expression for bubble growth in a superheated liquid derived by Forster and Zuber (176). RR AT cpL pi'a ]2 (166)* fg Pv Equation (166) was derived for a uniformly superheated liquid. However, Zuber and Fried (139) report that it has been used as a first approximation to predict bubble growth rates in pool boiling. These authors also employ a factor of 2 on the left side of equation (166) which was not in the original formulation (176). For the present purposes either form would be adequate.

-221LIQUID HYDROGEN NUCLEATE BOILING DATA OF SHERLEY AND STEINLE AT ONE-g AND ZERO-g SOURCE: GENERAL DYNAMICS /ASTRONAUTICS ON J 8TATISTICAL ZERO-g CURVE STATISTICAL ONE-g CURVE ZERO-g OPEN VENT DATA KC-135 AIRCRAFT 10- -1 0 I 2 3 4 5 6 7 8 9 10 II.12 t- tat,F Figure 117. Nucleate boiling data for liquid hydrogen at standard and zero-gravity. (233)

-222Performing the indicated operationsin Equation (165), the bubble Froude number becomes Fr = R [ T Cp P (167) R(a/g)g hfg Pv The magnitude of the Froude number in Equation (167) indicates the relative importance of inertia forces and bouyant forces in the growth of a bubble. When the Froude number is of the order of 1 these two forces have approximately equal value. Should this ratio be greater than unity then the inertia forces may be expected to predominate. The opposite will be true for Froude numbers less than unity. Hence, Equation (167) may be employed as a criterion for a satisfactory level of "zero-gravity". As will be noted below, an adequate zero-g condition actually exists when a/g = 1 for nucleate boiling of cryogenic liquids and water. Calculations of the Froude number have been made for the liquids of nitrogen, oxygen and hydrogen (93, 235) and are given in Table XVI. Water is included for comparison. The data correspond to those conditions which may be reasonably expected near a heated surface in nucleate boiling where the dynamics of the bubbles have their greatest influence on the heat transfer. TABLE XVI FROUDE NUMBER FOR NUCLEATE BOILING (93) (P=15 psia, a/g=l, AT=160F, R=0.005 inch) Liquid Froude Number N2 452 02 546 H2 352 H20 13,900

-223As is evident from these calculations the importance of inertia forces far outweigh that of bouyant forces in nucleate boiling. For gravities less than a/g of 1 the effect becomes even more pronounced and it may be expected that at zero-gravity bubble inertia effects are infinitely more significant. Since it is believed that a micro-convection, or stirring effect of the bubbles, is the principal mechanism controlling the heat transfer rate in nucleate boiling it follows that this would be governed by a liquid flow pattern caused by the dynamics of bubble growth. In view of the results in Table XVI the process of nucleate boiling then appears to be gravity-insensitive, except possibly at very high a/g, the important mechanics being the inertial effects of the growing (and collapsing) bubble. In the case of sub-cooled nucleate boiling, inertial effects can be expected to be even greater since the bubble dynamics are more pronounced. Siegel (236) has presented a comprehensive review of the influence of reduced gravity on heat transfer in the previous volume of Advances in Heat Transfer. 4. Injection Cooling Another process which has found important application in cryogenic systems is the cooling of a cryogen by the injection of a noncondensible gas of low solubility. One significant application of this has been the injection of gaseous helium into the liquid-oxygen suction lines of large rocket boosters during pre-launch operations. This provides the necessary subcooling by the evaporation of the liquid into the helium bubbles to prevent pump cavitation and engine failure.at start-up (237, 258). This process has been analyzed in depth in recent

-224years. However, in the interest of brevity the available papers on the subject will be cited only. Larsen, et al (239) analyzed the process of gas injection including the effects of liquid evaporation, gas solubility, gas enthalpy flux, ambient heating and liquid displacement due to the presence of gas bubbles in the system. The analytical model is spacewise lumped and the results compare favorably with measurements on vertical columns of liquid oxygen cooled by the injection of gaseous helium and nitrogen. The production of solid argon by the injection of gaseous helium into liquid argon has been observed in experiments reported by Lytle and Stoner (240). The cooling of liquid hydrogen by helium gas injection has been studied by Schmidt (241). A detailed analytical investigation of the dynamics of single bubbles injected in a liquid is reported by Arpaci, et al (242). Reasonable agreement with the theory was obtained from high speed motion pictures of the dynamics of nitrogen and helium bubbles injected into water. 5. Frost Formation The problem of frost formation on cooled surfaces is not well defined although it is commonly encountered on cryogenic systems in an atmospheric ambient. Should cryogenic substances be stored in atmospheres other than the terrestrial other kinds of frost can be expected. The prediction of frost formation in atmospheric air is complicated by several factors. For surface temperatures below the temperature of liquid air the overriding layers of water and carbon dioxide frost have a liquid air substrate. This results in an instability of the growth of the frost causing it to coat a surface in patches. The

-225shearing action of gravity and aerodynamic drag forces cause a similar destruction of the frost layers. The mechanical structure of frost is largely unknown and its physical properties are not well tabulated. Frost formation from normal ambient air involves a complex process of simultaneous transient heat and mass transfer of at least two or three components (H20, C02, air) and two or three phases. A study of frost formation inside of tubes in forced convection is reported by Chen and Rohsenow (244). They attribute the increased heat transfer and pressure drop under conditions of frost to the roughness of the frost surface. Smith, et al (245) analyzed the problem of frost formation on cooled surfaces in forced convection and give results concerning the thermal conductivity of frost. They conclude that the thermal conductivity is a function of the conduction and diffusion paths through the frost. Barron and Han (243) investigated the formation of frost on vertical flat surfaces in laminar and turbulent natural convection. They observed that a simultaneous mass transfer of water vapor from the moist ambient air to the frost layer increased the rate of heat transfer. This effect was not large, however, being about 5% for a mass fraction of water vapor of 0.10. Their heat transfer data were correlated within + 20% by formulations in terms of Nusselt, Grashof, Prandtl and Schmidt numbers and parameters which relate to the effects of mass transfer. On the other hand the predicted mass transfer rates were about 10 times greater than those measured. This was attributed to the interference by frost particles in the boundary-layer to the diffusion of water vapor to the cold surface. The predominant component to the energy transport

-226was the convective component and the influence of thermal diffusion and diffusion thermo effect was found to be completely negligible. Barron and Han measured the thermal conductivity of the frost and found it to be related to its mean density and mean temperature. Some of their results are given in Figures 118 and 119. Studies of frost formation also have been reported by Holten (246) Van Gundy and Uglum (247), Arthur D. Little, Inc., (248), The National Bureau of Standards (249) and Loper and Heatherly (250). Holten (246) investigated the growth of frost on the outside of spherical aluminum containers in a free convection atmospheric ambient and found a definite dependence on the specific humidity. Some of his results are given in Figures 120 to 123. The rate of formation of frost was found to go through a maximum while its surface temperature increased. The fact that the latter did not reach 32~F after long times is attributed to the mechanical failure of the condensate which became excessive after 150 minutes. Thermal conductivity data for the frost is given in Figure 123 as a function of the average temperature. Similar data from reference (248) indicate k-values of frost to range from 0.02 BTU/HR-Ft-F at -275F to 0.035 BTU/HR-Ft-F at -200F, corresponding to frost thicknesses of 0.05 and 0.15 inches, respectively. Reference (248) also reports that only 20-40% of the water vapor that is predicted to diffuse to the cooled surface by mass transfer calculations is retained on the surface as frost, an observation similar to that of Barron and Han (243). Van Gundy and Uglum (247) conducted experiments on the mechanics of frost on liquid hydrogen cooled surfaces at 200K for the forced flow of moist air. They report an important influence of the condensation

-227Barron and Han (243) 0.10.08 -06 - * w 0.010.it04 _ _ - _ -J 0.015 =.02 t -300' -280' -260' -240' -220' -200' -180' MEAN FROST TEMPERATURE, OF Fig. 118. Mean apparent frost thermal conductivity as a function of the mean frost temperature, Tf 0.5(Tw + Ts). 0.07 m.05: A -,04 -.03.02 -r F.01 0 e: 0~U. 0 1.0 2.0 3.0 MEAN FROST DENSITY, Ibm/ft.? Fig. 119. Mean apparent frost thermal conductivity as a function of mean frost density

-228-5.24x10. _'2 0-~P HO TEN (246) 20 < 16 - I. 12 [ 0 / 0 8 s ar.' ~ <"S.H. a 30, 40 GRAINS/LB. O 4 [ U 40 30;8 cc: 0 30 60 90 120 150 180 210 240 TIME, minutes Figure 120. Rate of frost formation as a fUnction of time and specific humidity. -60. U. I-80 m, -100 0-120 0 TIME, minutes w -140 HOLTEN (246) -160 m 0 30 60 90 120 150 180 210 240 TIME, minutes function of time in natural convect~ion.

-22922xo 0 20 HOLTEN (246) H *70-SO w16.. z 1 2.4 8, O30 40 GRAINSbw 10 30 60 90 120 1 50 TIME. minutes Figure 122.-Frost thickness as a function of time and specific humidity. 2.41xl.I -. 2.00 - -- HOLTEN'(246) O tj Is 0-'" A0 o.r 250 -240 -230 -220 -210 -200 40 -160 -170 AVERAGE TEMPERIIATURE OF FROITF Figure 123. Apparent thermal conductivity of frost as a function of the average frost temperature.

-230of liquid air. The highest heat flux is obtained when the condensed air can flow off the surface. A low ambient temperature and low humidity was found to inhibit frost formation. Data for the steady frost thickness for this system are given in Figure 124 as a function of the absolute humidity and air velocity.

0.8 _____ K *6 0.6 Pol Id oO0.6 z 0.4 rro 0 -6 MPH L x - 13 MPH r,-21 MPH IL 0.21 x'e, VAN Y AND UGLUM (247) 00,Z (jUND~~000 00 50 K)0 150 200 250 300 350 ABSOLUTE HUMIDITY IN GRAINS PER POUND DRY AIR Figure 124. Influence of flow velocity and absolute humidity on front thickness.

VIII. RADIATION The principal problems of radiation heat transfer at cryogenic temperatures is the determination of the radiation properties of surfaces. Gaseous radiation is less of a problem since most of the substances which remain as gases at low temperatures are not significant radiators nor absorbers. Radiation enclosure calculation methods such as those of Hottel (74) and Gebhart (252) are valid under these conditions. However, low temperature systems have introduced a new consideration into the treatment of radiation heat transfer. This is the effect of condensed gases on the radiation properties of cold surfaces. Such condensate layers build up complex systems known as cryodeposits on the cold substrate. Much of the work to date has been for systems of H20 and C02, the common cryodeposits from a terrestrial ambient. In general this is an extension of the problem of frost and much yet remains to be learned about these systems. Knowledge of the radiation properties of the cryodeposits is important in connection with studies of cryopumping, simulation of the solar space environment and the storage of cryogenic fluids. Kropshot (13) gives some values of the total hemispherical emissivities for some common metals at low temperatures. His results are included in Table XVII. Other tabulations of the emissivities of various materials with different surface conditions as a function of temperature include the work of Fulk and Reynolds (253). The emissive properties of vacuum-deposited metallic coatings on polyester film for low temperature service has been investigated by Ruccia and Hinckley (254). These highly reflective surfaces are -233

-234TABLE XVII SELECTED MINIMUM TOTAL EMISSIVITIES* Surface Temp.,'K Surface 4 20 77 300 Copper 0.0050 0.008 0.018 Gold 0.01 0.02 Silver ooo0.0044 o.oo8 O 0.02 Aluminum 0.011 0.018 0.03 Magnesium 0.07 Chromium. 08. 08 Nickel 0.022 0.04 Rhodium I 0.078 Lead 0.012 0.036 0.05 Tin 0.012 0.013 0.05 Zinc 0.026 0.05 Brass 0.018 0.035 Stainless steel, 18-8 0.o48 0.08 50 Pb 50 Sn solder 0.032 Glass, paints, carbon >0.9 Silver plate on copper 0.013 0.017 Nickel plate on copper 0.027 0.033 These are actually absorptivities for radiation from a source at 3000K. Normal and hemispherical values are included indiscriminately. Data taken from Kropschot (13).

-235employed in multi-layer, vacuum insulation where, except for aluminum, savings in weight and cost as well as increases in strength are obtained by depositing copper, gold and silver on a polyester substrate. An important factor to be determined is the influence of thickness of' the metallic deposit on the radiative properties. Another is the effect of environmental conditions of temperature, humidity and contaminents on the stability and adhesion of the metal film. The results of Ruccia and Hinckley (254) for the emittance of metallic deposits of aluminum, gold, silver, silicon monoxide protective coating on silver, copper and silicon monoxide protective coating ori copper are given in Figure 125. The substrate consisted of 1/4-mil (0,00025 inch) DuPont, type A, polyester film. As the thickness of the deposited metal layer increases the emittance decreases until- an asymptote 0 o is reached at about 750-1000 A for silver and aluminum and 1500 A for gold. The lowest emittance is found for silver with copper, gold and aluminum following next in order of increasing emittance for any thickness. The protective coatings of SiO on silver and copper degrade their emittance by about 40%. The influence of environment is shown in Table XVIII. The tape test listed consisted of placing a strip of scotch tape over the film to determine the adhesive properties of metallic coating. Except for the 95% relative humidity environment the aluminum coating showed good stability. Gold appears to be less stable in all environments while silver and copper were the least stable, as may be seen in the table. Cunningham and Young (255) have studied the absorptance of a C02 cryodeposit on various substrates at 77~K (139~R). Their measurements

.04.03 6 Aluminum.02 W02 f~~~~~~~~~~~~~~~~~~~Gold__ LsWio 150 1v& SiO 75 tim.01_ _ _ _ _ _ _ _ 0 500 000 600 2000 Thickness of Metollized Coating- Angstrmns Fig. 125 Emittance of vacuum-metallized polyester film at 5530R for various metal coating materials and thicknesses. (254)

TABLE XVIII EMITTANCE OF VACUUM-METALLIZED POLYESTER FILM AT 553~R FOR VARIOUS METAL-COATINGS AND MATERIAL THICKNESSES (254) Sample Thick- ~ Tape Tape Tape Environment Film Source No. ness, A Start test, % 50 hr test, % 100 hr test, l Air atm., 45% rel. A1 ADL 48 790 0.021 0 0.021 0 0.0195 O humidity, 95~F Au ADL 35-2 783 0.015 0 0.0159 0 0.0148 0 huAgidt 9ADL 36 762 0.0133 0 0.0181 10.o016 10 SiO/Ag ADL 42 75/745 0.0160 0 0.0152 1 0.0165 O Cu ADL 58 675 0.013 0 0.0167 0 0.0174 O SiO/Cu ADL 52 75/761 0.0179 0 0.0178 0 0.0173 0 Al* NRC 305 376 0.01361 0 0.025 0 0.0291 O Au* Hastings 304-A 1000 0.025 O 0.025 0 0.0234 O Au* Nat.Met. 308 240 0.0214 0 0.0211 0 0.0235 O Air atm., 95% rel. Al ADL 49 862 0.0184 0 0.0225 0 0.0206 O humidity, 95~F Au ADL 35-1 940 0.0140 0 O. 0145 0 0.014 O Ag ADL 37 762 0.0111 0 o.o0144 20 0.0147 40 DO SiO/Ag ADL 43 75/745 0.0141 0 0.0199 20 0.0175 20 Cu ADL 59 675 0.0121 0 0.0437 0 0.0713 O SiO/Cu ADL 53 75/761 0.0174 0 0.0212 10 0.0254 10 Al NRC 321 435 0.0225 0 0.0229 0 0.243 o Au Hastings 304B 825 0.021 0 0.023 2 0.0225 30 Au Nat.Met. 322 212 0.0211 0 0.027 O 0.0271 O CO2 atm., 950F Al ADL 50 862 0.0203 0 0.0192 0 o.0184 0 Au ADL 34-1 1020 0.0152 0 0.0148 0 0.0187 0 SiO/Ag ADL 44 75/745 0.0150 0 0.0142 0 0.0207 O SiO/Cu ADL 54 75/761 0.0170 0 0.018 0.o0166 0 Au Hastings 333 953 0.0259 o 0.0273 0 0.0299 Au Hastings 330A 1840 0.0146 0 0.0146 0 0.0153 0 Salt atm., 95~F Al ADL 51 862 0.0191 0 0.0187 0 0.0200 0 Au ADL 33-2 455 0.0154 0 0.0153 0 0.0152 O SiO/Ag ADL 45 75/745 O.0198 0 0.0179 50 0.0165 100 SiO/Cu ADL 55 75/761 0.0228 0 0.0248 O 0.0255 O Au Hastings 336 1050 0.0228 0 0.0225 0 0.0202 40 Au Hastings 330B 2072 0.0127 0 0.0160o 0 0.0144 O *Purchased samples are coated on both sides. lQuestionable value; previous measurements indicate this value has a range of 0.023 to 0.027.

-238of the absorptance as a function of deposit thickness is shown in Figure 126. The results for the bare surface (t = 0) agree well with literature values for the black and polished surface and a weighted calculation based on area for the other surfaces, The influence of cryodeposit thickness is to change the absorptance significantly for small thickness but to have essentially no effect for thicknesses greater than about 0.8 mm. For substrates of low absorptivity the effect of the deposit is to increase the absorptivity whereas the opposite effect is observed for high absorptivity substrates. Even for relatively large thickness of CO2 the absorptivity of the cryodeposit on the various surfaces did not approach a common value. Apparently for a thickness in excess of about 0.8 mm the frost absorbs all radiation in the absorption band thus having no further influence on the radiating characteristics for greater thicknesses. The absorptance of water vapor and carbon dioxide cryodeposits at 770K (1390R) for both solar and 290~K (5220R) black body radiation is reported by Caren, et al (256). The substrates consisted of polished aluminum and aluminum coated with a black epoxy paint (cat-a-lac flat black). The absorptance of an H20 cryodeposit on these substrates is shown in Figure 127 as a function of deposit thickness for black radiation at 5220R. The influence of the H20 deposit is to increase the absorptance very significantly for the low absorptance substrate. The effect is much less for the painted surface. Agreement with data for H20 given in Reference 257 is good. The absorptance data for a CO2 deposit on the same substrates for black radiation at 522~R is given in Figure 128. Agreement with the results of Reference 257 is reasonably good but poor with the data of Cunningham'and Young (255).

o CO2 deposit on black wall 0 C02 deposit on 75 % black, 25 % copper wall o CO2 deposit on 25 % black, 75 % copper wall tA C02 deposit on well-polished copper wall 1.0 0.80.6'o 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Cryodeposit Thickness, t, mm Fig. 126 Effect of Cryodeposit Thickness and Wall Absorptivity on Complex Absorptance (255)

1.0 INITIAL a * 0.86 0.8 CAT-A-LAC BLACK ON / + POLISHED ALUMINUM ALUMINUM SUBSTRATE SUBSTRATE Z I OO E-0 0.6 mfI | / DOUGLAS DATA (257) > 0.4 L / X POLISHED CU SUBSTRATE 0.4 NOTES: 0.2 v DEPOSITING PRESSURE 4.6 x 104mm Hg DENSITY OF DEPOSITS 39 lb/ft3 I MIL 10-3 INCHES INITIAL a = 0.07 0.01 0.10 1.0 10 100 THICKNESS OF DEPOSIT (MILS) Fig.127 Absorptance of HaO cryodeposit for room temperature blackbody radiation.(2s6)

1.0 CAT-A-LAC B~LACK ON ALUMINUM SUBSTRATE 0.8 POLISHED ALUMINUM SUBSTRATE Un 0.6 DOUGLAS DATA (257) ~~t ~ I I ~~~ POIJSHED CUI SUBSTRATE ARO DATA (255) A PARSONS BLACK SUBSTRATE o..4 X- OWPOLISHED CU SUBSTRATE U ~ ~ ~ ~ ~ ~ ~ ~~~ 02 O 0.2 N —-----—' I ~OTE: DEPOSITING PRESSURE, 3.8 x 10 mm Hg DENSITY OF DEPOSIT, 91 LB/FT3 0 20 40 60 80 100 120 140 THICKNESS OF DEPOSIT (MILS) FIGURE 128. ABSORPTANCE OF CO2 CRYODEPOSIT FOR ROOM TEMPERATURE BLACKBODY RADIATION2

This variation is attributed to differences in the physical nature of the deposits. A solar source was simulated. by using filtered radiation from a Mercury-Xenon lamp. The absorptance of an H20 cryodeposit under these conditions is shown in Figure 129. In this case the absorptance decreases with deposit thickness for an essentially black substrate. An analysis of the thermal transport processes in a cryodeposit subject to radiation at the vacuum interface is given by McConnell (258). His results compare favorably with the experimental measurements of Caren, et al (256) for H20 on a reflecting substrate. Tien and Cravalho (259) survey recent advances in the study of the thermal radiation properties of solids at cryogenic temperatures. For radiative transport between solids they discuss the influence or non-gray surfaces (wave length and temperature dependent properties), non-equilibrium fields and the effect of small surface spacing, a factor of particular significance at cryogenic temperatures. This latter is important because of the concentration of low temperature thermal radiation at long wavelengths. Theoretical studies are needed to provide calculation procedures for the transport of radiation between surfaces whose spacing is of the same order or less than the wavelength at the maximum radiative heat flux.

1.0O INITIALam 0.96 0.8 s 0.6 hi C) Z LEGEND ~I- i I MERCURY-XENON LAMP O 0.4 MERCURY-XENON LAMP WITH CORNING FILTER NO. 1-69 U) 0O. NOTES: SUBSTRATE: CAT-A-LAC BLACK DEPOSITING PRESSURE 4.6 XIO-4mm Hg 00 0 20 40 60 80 100 120 CRYO-DEPOSIT THICKNESS (MILS) FIGURE 129. ABSORPTANCE OF H20 CRYODEPOSIT FOR SIMULATED SOLAR ENERGYi2

IX. HELIUM II Helium is unique among the various substances in that it has two known and distinctly different liquid phases. These are indicated in the phase diagram for helium in Figure 130. The liquid phase which exists in the temperature range from about 2.19~K to the critical point at 5.2~K behaves in a classical manner as do ordinary substances and the gaseous phase of helium. However, at temperature below approximately 2.190K liquid helium undergoes a remarkable transformation. Within a fraction of a degree below 2.19~K the heat conducting ability of the liquid increases in an astonishing manner by a factor of 107, as was indicated in Figure 2 and Table II. Also, its heat capacity increases by about a factor of 6 in this temperature region as was shown in Figure 3. Because of these characteristics this phase of the liquid is identified as helium II and sometimes called a "superfluid". The classical liquid phase is known as helium I. An interesting and potentially significant aspect of helium II from a technological view point is the fact that it remains as a liquid down to the lowest attainable temperatures and presumably to the absolute zero as well. This property will enable the fluid to be employed conveniently as a coolant or as a transfer medium for equipment and components designed to operate below 10K. At the present there are few such applications but they may be expected to increase in the future. Helium II may also be solidified if it is subjected to a sufficiently high pressure. The shape of the heat capacity curve has given the name "Lambda" to the line of transition between helium I and helium II. -245

-246CRITICAL POINT 1000 1000 X- LI NE -| LIQUID HELIUM I LIQUID HELIUM II 10c0 p | / SATURATION CURVE X — POINT 1j HELIUM GAS B C A 10 1 2 3 4 5 TEMPERATURE ( K) FIGURE 130. PHASE DIAGRAM FOR HELIUM.(261) FIGURE 130, PHASE DIAGRAM FOR HELIUM,

-247The properties of the "superfluid" helium II have long been of great interest to physicists. Much of the known results have been summarized by Lane(22) and Matheson(262) describes a symposium held on this substance consisting of 24 papers treating several aspects of the subject. Some of the extraordinary behavior of helium II, described by Lane, include its superfluidity, the mechano-caloric effect and thermo-mechanical effect, sometimes known also as the fountain effect. In order to explain these observed phenomena, helium II was postulated to be made up of two fluids. One fluid, known as the "superfluid", is without viscosity, entropy or heat capacity while the second fluid has all the properties of a normal substance. It is called the "normal fluid". These two fluids mix in all proportions with helium II consisting of all normal fluid at the X-point and all superfluid at the absolute zero. These assumptions are largely unproved but they have helped to "explain" certain observed behavior of He II. Obviously, this substance presents a challenge which requires a departure from classical concepts to obtain a quantitative description of its properties. One of these properties is its extremely great heat conducting ability. The definition of a "thermal conductivity" for He II, as was done in Figure 2, is principally a convenience for comparative purposes as He II does not follow the Fourier concept in the usual sense. If a thermal conductivity is computed for this substance it will be found to be highly temperature sensitive, be a function of the thermodynamic state and be influenced by the geometry and temperature gradient in its system.(263)5 Another factor in this is the fact that thermal transport ("conduction") in an apparently "stagnant" (zero net mass flow) He II is not a consequence of a strictly diffusive process.

-248Because of the presence of the superfluid within the mrass of He II powerful convective flows are established by the temperature differential. The large apparent thermal conductivity is a result of these internal flows. Nevertheless, there is useful value in formulating the thermal conduction characteristic in terms of an apparent thermal conductivity. In a presentation of a review of this subject Clement and Frederking(264) give the following expression for the apparent thermal conductivity of He II, 1/3 2/3 4/3 LT -2/3 kApp = C(T)T 2/ 4/ (T) 2/ (168) where ~n is the absolute viscosity of the normal fluid, p is the total fluid density, S is the liquid entropy per unit mass and C(T) is a function of temperature and shown in Figure 131. The combined property function, kAPP(o)w/ = c(T)T /3 p2/3 S4/3, (1r 9) is shown in Figure 132 and the corresponding values of kApp for various values of (AT/L) is given in Figure 133, both taken from Clement and Frederking.(265) The effectiveness of He II as a "conductor" may be seen from Figure 133. The maximum value of kApp shown is about 500 watts/cm-OK which may be compared with the thermal conductivity of room temperature copper, Figure 4, which is about 4 watts/cm-~K. For lower values of (aT/L) the kApp of He II is even greater. Of particular interest is maximum in the property group C(T)T /3 2/ S3 at approximately l.90K as shown in Figure 132 and the rapid reduction in

-24990 CHANNEL IA Z LHe n CROSS SECTION 0 I | KEESOM,SARIS,MEYER 80 0 CAPILLARY m 0.157 cm DIA'A CAPILLARY IC 0.0346 cm DIA O CAPILLARY ID 0.0346 cm DIA A KEESOM,DUYCKAERTS 0.00946 cm DIA 70 0o VINEN 0.24 cm x 0.645 cm * CHASE 0.08 cm DIA 60 50 40 30 20 0 100 1.0 1.2 1.4 1.6 1.8 2.0 2.2 T (eK) DIMENSIONLESS PARAMETER C(T) FOR SUPERCRITICAL HEAT TRANSPORT IN TUBES AT ZERO MASS FLOW(264) FIGURE 131

4/3 2/3 1/3 C(T)TS P 7n'5 10 1.2 14 16 1.8 20 TX22 T,~K FIGURE 132. PROPERTY FUNCTION C(T) TS4/3 p2/3 nn V TEMPERATURE (265). aS!

Kapp 600 AT/L; ~K/CM He 1 0005. 0.03 005400 w CMOK 200 - 1.2 14 16 1.8 20 Tx22 T.OK FIGURE 133. APPARENT THERMAL CONDUCTIVITY k OF He II app AS A FUNCTION OF LIQUID TEMPERATURE FOR CONSTANT TEMPERATURE GRADIENTS. (265)

-252kApp as the liquid temperature is either increased to the X-point or reduced toward the absolute zero. This is a consequence of the liquid becoming completely a normal fluid at the X-point with the thermal conductivity of liquid He I, and completely a superfluid, but with no heat conducting ability, at OOK. Because of the large magnitude of kApp He II can sustain only very small temperature differentials. Thus, even during large rates of thermal transport the temperature of He II appears uniform. There is, however, another effect which becomes of importance under these circumstances and has a primary influence on the transport of heat from a solid surface to He II. This is a boundary resistance known as the Kapitza. resistance after its discoverer.(266) At low heat flux rates to liquid He II Kapitza observed what appeared to be a temperature discontinuity between the surface and the liquid. It is possible this is an effect to be found with many other fluid-solid systems as well but owing to the large apparent conductivity of He II the temperature discontinuity is more evident in this case. The existence of this phenomenon is supported by theoretical considerations of Khalatnikov(267) who assumed a radiative transport and showed the Kapitza effect to depend on T3 and the acoustic and elastic properties of both the solid and the liquid. Theo temperature difference under these circumstances may be expressed in terms of a contact heat transfer coefficient (the reciprocal of a resistance) as ~~~~~~~~h = (il,(170) A limiting value to this coefficient, ho, corresponding to aT equal to zero, may be shown to be written(264)

-253ho = 4 aK T3 (171) and for larger AkT but under the restraint that AT < T, the coefficient h may be formulated in terms of ho as (264) h = f(A = 1+ + ()+ () (172) In Equation (171), aK is a parameter which depends on the particular properties of the solid and liquid. In most cases it is necessary to determine aK empirically as the theory of Khalatnikov, while correctly predicting a solid-fluid property dependence, does not predict its observed magnitude.(268) The temperature dependence is reasonably borne out by experiments with measured exponents ranging between 2.6 and 4~2. Thus, Equation (172) may be written h = 4 aK T f( T (173) and the corresponding heat flux, (q/A)K, in the range wherethe Kapitza resistance is controlling becomes (F/A)K = 4 aK T3 ATf(c) t (174) For a Pt - He II interface Kapitza's data(266) for AT << T may be expressed as(264) ho = o.o65 T3, (175)* The units are watts/cm -K and T is ~K.

-254and, (q/A) o 0.063 T3 d f(T) (176)* These general types of relationships have been confirmed in several experiments. Clement and Frederking(264) have studied the heat transfer from a silver surface to helium II in a short tube and for conditions of AT < T find that (q/A)KAg may be expressed as (q/A)Ag = 0.082 T3 AT f(), watts/cm2 (177) Their data are shown in Figure 134. Agreement between Equation (177) and the data, especially for low values of WT/T is very good. A correlation of the measured (q/A) and that computed by the T3-relation, Equation (177), is given in Figure 135. The comparison is favorable for low AT but a departure is observed at higher AT. It seems probable the reason for this is that at higher zT the surface temperature exceeds the X-point temperature and He II no longer is in contact with the solid, This would have the effect of introducing a thin, very low conducting film of helium I between the solid surface and the helium II in the liquid bulk. If this is the case then the measured (q/A) would fall below that predicted by the Kapitza-type theory, which is what is observed. At much larger AT a vapor film of He I would be expected to form at the surface. Under these circumstances it is possible that two films, one of He I vapor and the second of He I liquid, could co-exist in a helium II system in film boiling * The units are watts/cm2 and both T and T are K.

-255t,OK 2.0 / o 2.00 / x 1. 80 / o/ ~ X -o 1.70/ / 1.6 01.// o /// x 0 1.40 1.80 / / -N 2.00 / /I N ot /o 0/ xo/ E 1.2-'2 /1 / // 2.10+ /o /X / OK 1/7 / // 4 O~ ~ ~~~~~+ 0. 0/ 1.55 0.8 // / / 1.40 // - //,,,A'.. /',,,,,, /OX / // - L I I I I I I I I I A/ x./ -AT, K FIGURE 134. COMPARISON OF EXPERIMENTAL HEAT FLOW DENSITY q/A WITH SEMIEMPIRICAL CORRELATION ( 0.082 fAT T3. a. 0 0.2 0.4 0.6 0.8 1.0 1.2 IA 1.6 AT,eK FIGURE 135. RATIO (q//A) / (q/A) K,Ag. FOR DATA OF FIGURE 1314

-256for pressures above that of the X-point. In this circumstance the temperature discontinuity would probably be shifted to the liquid He I - liquid He II interface. Thus, for computing heat transfer rates to liquid He II a T3-relationship of the form of Equation (174) can be employed in which the constant aK must be determined for each solid-liquid combination. As a guideline this relationship may be used for IT up to the point at which the surface temperature exceeds the X-point temperature of He II. For conditions of higher surface temperature a He II system goes into other modes of heat transfer. These could involve some vapor formation (boiling) of a He I film or a film boiling condition of He I vapor in contact with a bulk liquid of He II. Because of its large heat transport characteristic bubbles or nucleate boiling have never been observed in helium II. As a guide to engineering calculations in these regions it is probably best to use the available experimental data, most of which is for natural convection processes from small, laboratory type surfaces. Some of these data will be discussed later. Further evidence of the T3-relationship for low (q/A) and (269) low AT is provided by Irey, et al. and shown here in Figure 136. The test section for these data was a horizontal cylinder of soft-glass, 1.88 mm in diameter and heat transfer was by natural convection from the outside surface of the cylinder to a He II bath.* The data for this combination appear to fit the relationship h 0.139 T3 very well. Clement and Frederking(264) have summarized the results of several investigators for heat transfer to liquid helium II from a number of In a later paper, Holdredge and McFadden(263) suggest that some of the data of Irey, et al. (269) are probably in error. These are data above the Kapitza range and below a AT of about 80~K and thus does not include those in Figure 136.

0 DATA OF IREY, ET al (269) h = 0.139 T3 E ~ / 0 0.5 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 TEMPERATURE,(K) FIGURE 136. VALUES OF h AT SMALL TEMPERATURE DIFFERENCES.(269)

-258different surfaces for conditions in which the Kapitza effect controls the process. These results are given in Figure 137 where ho is plotted against T3. A linear relationship indicates an agreement with Equation (171). As may be seen a reasonable validation of this type of relationship is borne out by most of the data. A low value of ho will produce the greater boundary AT for a given heat flux. A comparison of natural convection heat transfer data from a 1.79 mm diameter soft-lead glass tube to saturated and unsaturated ("sub-cooled") liquid helium II with the Kapitza resistance theory of Khalatnikov(267) is given by Madsen and McFadden(268) and shown in Figure 138. Good agreement is found for the low AT range but as the temperature difference is increased there apparently is an additional thermal resistance being created in the system. Some possibilities as to what this might be were discussed above although the exact nature of this resistance is presently unknown. As may be seen from the data in Figure 138 Khalatnikov's theory is valid to larger AT's for a subcooled condition than with a saturated liquid. Holdredge and McFadden (263) studied the Kapitza resistance in a saturated He II system with a test section similar to that in Reference 269 in which the influence of depth of immersion was examined. Their results in comparison with the theory of Khalatnikov are given in Figure 139. It is apparent that immersion depth is not an important factor in the Kapitza effect. Since the influence of depth is to establish the pressure level at the test section these results suggest that no vapor phase of helium I exists at the heated surface in these low AT regions. An effect of test section size is unobserved as well in this range of AT. The

Ag h0 O.065T3 D Pt KAPITZA (Pt LAYER) (q/A).~ (266) 1.0 F =1 CHALLIS I /AT\ I U Cu DRANSFELD (POLISHED) LT f J WILKS(2.70) jL O~~~ 0 A I Ni H:hoj 0.0142T'5 E 0.8 I! Ni WEY- YEN (ELECTROPOLI SHED) 7 I A~Ni:R:how 0.062T9 3\ VJ2(BEFORE POLISH) A Au JOHNSON AuC: howu(1/8)T 0O.6JONN V Cu LITTLE(272) CuC: hom 0.091T1 o I~~~~~~~~~~ rb. 0.4 0.2I 0 2 6POINT 0~~~41 0 2 4 8 10 T3, (OK)3 FIGURE 137. COMPARISON OF THE REDUCED HEAT TRANSFER COEFFICIENT WITH THE T -RELATIONSHIP FOR FACE-CENTERED CUBIC METALS.

10.0 y HALATNIKOVNS ThEORY N: 1 I ~ SAIURAJED BATH,o I I aUNSATURAND BAIH ILOL T =1A% T, =2.10 cn Pb 0.01 0.10 1.0 10.0 0.01 0.1'iL 1. lEMPERAlURE DIFFERECE A 0T EMPERATURE DiFEENEAT S0 FIGURE 138. VARIATION OF THE KAPITZA RESISTANCEe(268)

-2613 — DIAMETER IMMERSION (CM) (CM) 4A 0.245 4.2 x 0,245 11II. 0 2 O 0.145 " 4.2 o + 0.145 II1. SOLID LINE REPRESENTS ho / h FROM KHALATNIKOV THEORY 00 I~k )p+ 0 0 X -| A no,+ o+ 0 + Tb 1.96 K oL I1 + 0.01 0.1 1.0 TEMPERATURE DIFFERENCE,AT (K) FIG. 139 COMPARISON OF EXPERIMENTAL AND THEORETICAL KAPITZA RESISTANCES (263 )

-262experimental data seem to depart from the Khalatnikov theory at a AT of approximately 0.100K. Above the ALT corresponding to the range of the predominance of the Kapitza effect heat is transferred to liquid helium II by a transport mechanism which is largely unknown. It is called non-film boiling but does not appear to be similar to nucleate boiling. Data for this heat transfer process are given in Figures 140 to 145 taken from Holdredge and McFadden(263) and Madsen and McFadden. (268) In each case the test section is a soft-glass cylinder horizontally oriented in a bath of helium II. The data in Figures 140 and 141 are for a saturated liquid with various bulk temperatures, depths and test section size. In these data an influence of size is observed at all bath temperatures suggesting a controlling effect of bouyant forces. Depth, or pressure, does not appear to be significant until fairly large AT are obtained. This is thought to be the point for the onset of film boiling and the appearance at the surface of the gaseous phase of helium I. The surface temperature increases with heat flux very significantly at this point and the phenomenon of a maximum heat flux is observed. The effect of "subcooling" on the nonfilm boiling, the maximum heat flux and the transition to film boiling are shown in Figures 142 and 143 for a test section 1.79 mm in diameter.(268) The maximum heat flux is in the range 2 - 4 watts/cm2 which may be compared to a corresponding value of about 1 watt/cm2 for helium I as shown in Figure 63 for a pressure of 1/2 atmosphere. For a heat flux above a certain critical maximum the influence of the superfluid is destroyed, at leat locally, in the vicinity of the heated surface. This critical maximum heat flux is similar to the

-263I....,1 I.IIiI I I Tb 1.96 0. I 4_.. I 4 f Tb' 1.80,I DIAMETER DEPTH OF vMMERSIO 0O 0.145CM 11 CM A 0.245 11 X 0.245 4.2 + 0.145 4.2 0.01 0.1 1.0 5 TEMPERATURE DIFFERENCE, AT(K) FIG. 140 NON-FILM BOILING HEAT FLUX AS A FUNCTION OF TEMPERATURE DIFFERENCE (263)

-26410 0.1 I I 11111 I 1 111111 1 1 3./ DIAMETER DEPTH OF + 0.145 4.2 x 0.245 4.2 0010,01 1 — _.01 0.1 1 5 TEMPERATURE D-IFFERENCE, FAT(K) FIG. 141 NON-FILM BOILING HEAT FLUX AS A FUNCTION OF TEMPERATURE DIFFERENCE FOR A 2.1 K BATH (263)

10. x 0.10 C H'E AURAD, P 21.06H t -- A UNSATURAJED, Pb 775MMHG 0.01 0.01 0.10 1.0 10.0 100. lEMPERATURE DIFFERENCE, AT (K) FIGURE 142. BOILING CURVES FOR A BATH TEMPERATURE OF 1.96 K.(268)

10. 0.010 __ _ _ d~C~ TPTo SATIRATEDN, A 1.M IWt I I IA UNSATURATED, Pb75 MMHG 0.01 0.01 0.10 O 10.0 100. TEMPERATURE DIFFERENCE, AT (K) FIGURE 143. BOILING CURVES FOR A BATH TEMPERATURE OF 2.K.(2b8)

-26710 H RUN cm NO. X 24.2 I 0 15.2 I 13 226.6 3 5 _A 11.7 3 -V 3 3.8 2 AE34 00 _,Bg0 2 / % V / A\ E vV / O (q/A)MAX He r ~/ I I |(LYON: (199) — x HORIZONTAL v/0 t i SURFACE) 0.5 / /,,.( q/A)MA 0.148 C (T)TS 4/3,p 2/3n n 1/3 0.2 -/ K 0. 1 I I I I I I I I I 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 T,OK FIGURE 144, MAXIMUM HEAT FLUX AT VARIOUS DEPTHS OF IMMERSION VS, TEMPERATURE,

-268maximum nucleate boiling heat flux but its mechanism is probably quite different. Clement and Frederking(264) investigated this phenomenon for heat transfer to He II from a horizontal surface through a short tube. In this study the transport processes were of a more bounded type than those cited above in which a test surface was totally surrounded by liquid helium II. The low heat flux-low AT data for this bounded system were discussed above and given in Figures 134, 135 and 137. The measured values of (q/A)max are given in Figure 144 for various liquid depths H of the test surface below the He II bath surface. There appears to be very little effect of liquid depth (pressure) on (q/A)max. Owing to limitations of pump capacity the pressure was difficult to stabilize at the lower temperatures. In the vicinity of the X-point, however, the data are very accurate. The magnitudes of (q/A)mx are seen to increase from 0.10 watts/cm2 at the X-point to approximately 2 - 4 watts/cm2 at about 1.90K, the point or maximum apparent thermal conductivity for helium II. The results near the X-point coincide very (199) well with the low pressure maximum heat flux data of Lyon for helium I. Clement and Frederking have fitted a curve to their data which follows the general characteristic of the heat flux corresponding to the apparent thermal conductivity of helium II. The equation, which fits the data well near the X-point, is (q/)max 0.148 C(T)T Bn/3 p2/3 4/. (178) The influence of depth of immersion (pressure) on the temperature difference at the maximum heat flux is indicated to be completely

-269negligible in Figure 145. The temperature difference corresponding to the maximum heat flux is summarized in Figure 146 as a function of bath temperature. The accuracy is limited at the lower temperatures owing to restricted pumping capacities in the experimental apparatus. A lower bound to max is shown in Figure 146 which is that aT from Equation (174), the Kapitza relationship, at which the corresponding heat flux, (q/A)K, is equal to the maximum heat flux, (q/A)max, from Equation (178). There is quite a bit of scatter in the data. It should be noted that these ZT are about the same as those determined for the maximum heat flux in helium I as shown in Figure 63. Clement and Frederking (264) report (q/A)max values for helium II as high as 10 watts/cm2 but do not give the specific data. Reference is probably made to the small wire data such as that of Frederking. (275) Maximum heat flux data of the order of 10 watts/cm2 for a horizontal platinum wire 15 Ad diameter is reported by Rinderer and Haenseler(273) for liquid helium II at 1.40K. The magnitude of (q/A)max was found to depend on liquid height above the test surface. Lemieux and Leonard(274) investigated the maximum heat flux for a horizontal 76.2 p. diameter 90% platinum - 10% rhodium wire. Depths of immersion below the liquid-vapor interface ranged from 5 to 70 cm. In this way both the pressure at the heated surface and the heat flow path length were varied. A large influence of,depth of immersion was found. Figure 147 shows the maximum heat flux data for a depth of 30 cm as a function of liquid bath temperature. Near the X-point (q/A)max increases very rapidly with bulk temperature reaching a maximum at about 2.0~K. The maximum values are considerably greater

2 q =2.10 W/CM2 T =.880 "K0 0q /A)I I(Q /A) 0 5 1O 15 20 H, CM FIGURE 145. SOLID SURFACE EXCESS TEMPERATURE VS DEPTH OF IMMERSION IN THE VICINITY OF THE MAXIMUM HEAT FLUX. (264)

-2713.0 H, (cm) RUN, NO. x 24.2 I o0 15.2 I 2.5 ll 26.6 3 A 11.7 I I 2.0 I6 o~ I x o 1.5 0 OWN 1.0 A ATMAX For (q/A)K, t ( q/A) MA, 0.5 x z I I.I I I I 1,17 1.8 1.9 2.0 2.1 TX 2.2 T, (K) FIGURE 146, TEMPERATURE DIFFERENCES AT THE MAXIMUM HEAT FLUX AS A FUNCTION OF THE BATH TEMPERATURE,(

LEGEND SPEC. NO. LENGTH (CM) 20~ 6 1-9407 H | l7 1-8468 A a 1-9324 ) 1 1P8556 3 g I' ~15~ A 4 a U. 2 14 1 1-8 20 T -~K FIG 147.MAXIMUM HEAT FLUX AS A FUNCTION OF BATH TEMPERATURE AT A DEPTH OF IMMERSION OF 30 CM (274)

-273for this small diameter wire than for the larger unbounded cylinders (263,268) and the flat (bounded) plate. (264) The smoothed data taken as a function of depth are given in Figuxe 148. The influence of depth vanishes at the X-point. A comparison of the data of Lemieux and Leonard with those of others(263,275) as a function of depth is shown in Figure 149 for a bath temperature of 1.960K. Agreement (275) with the results of Frederking for a 50 p. diameter wire is good. However, for the cylinder (D = 0.245 cm) of Holdredge and McFadden(263) there is no agreement. Apparently, there is a very significant diameter or size influence on (q/A)max, perhaps similar to that observed in film boiling for small diameter cylinders by Breen and Westwater. (150) Lemieux and Leonard(274) report that on reduction of heat flux from a film boiling condition a minimum heat flux is reached, less than (q/A), at which the system no longer can support the vapor film. For lower heat flux the heat transport reverts to that of the superfluid. Some (q/A)min data are included in Figure 149. At heat flux rates in excess of (q/A)max a helium II system goes over into film boiling controlled by ordinary fluid and thermal phenomena. Under these conditions the superfluidity effects are destroyed a-,&the heated surface and are replaced with those circumstances which govern the natural. convection of a vapor layer in film boiling.(2"5) The liquid helium II is lifted off the surface but becomes an effective heat sink for the vapor. Since the greater thermal resistance resides in the vapor film the unusually great heat transport ability of the superfluid is no longer effective in promoting heat transfer.

25DEPTH Of (CM),OO 15 1'2 14 I-6 I-8 2"0 2-2 T- OK FIG. 148 MAXIMUM HEAT FLUX AS A FUNCTION OF BATH TEMPERATURE & DEPTH OF IMMERSION (274) _ ~ ~~~~~~~~~~~~~~~~~~~I I I i

24 w 0 I.~ X I * A (At LEGEND 8ZI |A + (q/A )MAX MCFADDEN 8 HOLDREDGE(263) 245 CM A * (q/A)MAX FREDERKING (275) 30 A @ (q/A )MAXLEMIEUX & LEONARD (274) 76-2 p A+ + + A (q/A )MAXLEMIEUX a LEONARD (274) 76-2J + 0 20 40 60 80 DEPTH OF IMMERSION - CM. FIG. 149 HEAT FLUX VERSUS DEPTH OF IMMERSION (T =1-96~K)

-276Rivers and McFadden(261) have analyzed the transport processes in laminar film boiling in liquid helium II for a flat plate and cylinder. Their results are given for both liquid and vapor films of helium I between the heated surface and the liquid helium II. The process is formulated in terms of the usual transport equations for laminar boundary layer flow but a temperature and velocity discontinuity is introduced at the film-helium II interface. The Nusselt number is determined in terms of the Grashof and Prandtl numbers and two parameters, the "interface enthalpy," Hi and the "interface heat flux," Qb The analytical results for the cylinder and flat plate are given in Figures 150 and 151 for films of liquid helium I (Pr = 0.4) and helium gas (Pr = 0.7). The following definitions are used Nu = -, (179) kf where D = r for a cylinder and L for a plate, Gr = g D3 pf (Pb-Pf) (180) 2 4f H. (181) where (Dh)i = enthalpy change at the He II interface, and in which (q/A)b is an "interface heat flux," a quantity required to compute Qb but about which little is presently known. From these

-277results Rivers and McFadden identify three regions. For Qb Gr-74< 0.01, the heat transfer follows ordinary film boiling characteristics in which the Nusselt number is a function of the Grashof and Prandtl numbers and, of course, Hi. In this region the convective processes dominate. Above Qb Gr1/4 of about 10, the process is described by the asympote Nu = Qb ~ Under these circumstances, the wall heat flux (q/A)w is identical with that at the interface, (q/A)b. Conductive mechanisms govern the phenomena in this region. For intermediate values of Qb Grl1/4 a transition region exists in which both convection and conduction effects coexist. Except in the "convection" region, Qb Gr1/4 < 0.01, it is necessary to determine (q/A)b in order to make calculations using the results in Figures 150 and 151. This is the key step in the calculation procedure for the transition region. The "interface heat flux," (q/A)b, is fixed by the state of the liquid helium II and is independent of the processes in the film. Accordingly, it has the role of a "heat sink" and enters the process as a boundary condition. Its magnitude is thought to depend on the bulk temperature of the liquid He II and the interfacial temperature discontinuity. However, its' priori determination has not yet been resolved nor have any quantitative formulations been established for its calculation. Fortunately, in the high iAT Range where Qb Grl/ is less than 0.01, (q/A)b is much less than (q/A)w, the wall heat flux, and thus it does not influence the process significantly. In this range, however, the classical film boiling correlations are not valid owing principally to the unusual temperature discontinuity and "heat sink," (q/A)b, boundary condition

-278at the helium II interface. There is presently a need to clarify the role of (q/A)b and to achieve an explicit expression for its calculation. Rivers and McFadden(261) report some experimental data of Holdredge for the case of film boiling from a horizontal cylinder in which the liquid helium II is separated from the heated surface by a film of helium I vapor. A comparison of the predicted and measured Nusselt numbers is shown in Figure 152 for primarily a "convective" condition in which (q/A)b has a subdued role. The comparison is generally favorable for the range of variables considered and the results tend to authenticate the boundary conditions established in the analysis. A study of isothermal, turbulent flow of liquid helium II at 1.50K in wide channels (0.003 to 0.015 inch) having an RMS rough(276) ness of about 0.0001 inch is reported by Frederking and Schweikle. Their results indicated that under these flow conditions He II behaved similar to ordinary fluids and could be correlated in terms of the usual similarity parameters. Pressure drop data correlated in terms of Reynolds number and friction factor are shown in Figure 153 in comparison with the Blasius formulation for turbulent flow. Although the general nature of the correlation is evident, the magnitude of the friction factor for a given Reynolds number is considerably greater than that predicted by the Blasius expression for smooth surfaces. This increase is significantly greater than would normally be expected for roughend surfaces in the experimental ranges of relative roughness (hydraulic radius/RMS roughness = 30 to 150) and Reynolds number.

10 FILM OF LIQ. HELIUM I, Pr = 0.4 OF HELIUM GAS Pr = 0.7 H NuGr4 100 Bonn — mm-mm~~ —;9 0,0 000 I Nu ObQ 0.01 0.1 10 QbGr' Fig. 150 Calculated results illustrating average heat transfer coefficient (Nu = I r/f) for a horizontal circular cylinder (261)

10 FILM OF LIQo HELIUM I, Pr 0.4 FI LM OF HELIUM GAS, Pr 0.7 H'i 100 Nu G'r110 0.01 Nu r Qb 0.01 0Gr 11111 QbGr Fig. 151 Calculated results illustrating average heat transfer coefficint (Nu lhiL/k) for a vertical lat plate (261)

100 x 0.245 CM DIAMETER 0.145 CM DIAMETER z -IJ Cl) tO U) z d o ~~~~~ x Irl a. I ~~~~~10 tOO PREDICTED NUSSELT NUMBER Fig. 152 A comparison of predicted values with experimental values of average Nusselt number for a horizontal circular cylinder (261)

-282105I II I I II'll TURBULENT FLOW I'~ F= 0.3164 Re BLASIUS Re 104._/~ CHANNEL WIDTH ~~- /c~~~3.003 INCH v.008 INCH o.010 INCH f t ^~~~~L.'015 INCH 103 a i,. 105 106 10o los f Re2 FIGURE 153, DIMENSIONLESS FLOW RATE (REYNOLDS NUMBER) VS DIMENSIONLESS PRESSURE (273) GRAD I ENT,

-283The increase in friction factor at Reynolds number as low as 103 as a consequence of roughness alone is quite unusual and may be a result of other factors. In Figure 153 the Reynolds number is defined as Re = pVD (183) ln where p is the total liquid density, ALn the absolute viscosity of the normal fluid and D the hydraulic diameter. The friction factor is defined from = f(b) 2P * (184) 2go Heat transfer data fdr zero net mass flow systems also were plotted using similarity parameters but less success was achieved. ACKNOWLEDGEMENT The author wishes to express his appreciation to the Industry Program of the College of Engineering, University of Michigan, for providing typing and drafting services on this manuscript and to Mr. Donald L. Danford, Assistant to the Program Director, for his cooperation.

NOMENCLATURE A area, ft2 or cm2 a acceleration, ft/sec2 a,a* thermal diffusivity, ft2/hr A' (X) see Equation (35) A" () see Equation (36) A flow area, ft2 f B function for Xc, see Equation (20) Bo boiling number, (q/A)/hfg Gmix Cp,Cv heat capacity, BTU/Lbm-~F, joules/mole-~K co concentration, Lbm/ft3 D',D" mass diffusivity, ft /hr De equivalent diameter, ft Do outside diameter, ft E k(T) integral, see Equation (2), BTU/hr-ft E value of E at T~, BTU/hr-ft Ei value of E at Ti,BTU/hr-ft E* defined by Equation (11) F two-phase flow correction factor, see Figure 94 F1 see Equation (127) F2 see Equation (130) F(p) see Table X Fo Fourier number, a*t/L2 f friction factor -285

-286NOMENCLATURE (CONT'D) G for Figure 105 only, vapor mass velocity, wv/A, Lbm/hr-ft2 G mass velocity, Lb./hr-ft2 G* see Table X Gmix Gmix see Equation (150) Gr Grashof number go conversion factor, 32.2 (Lbm/Lbf)(ft/sec2) g gravitional acceleration, ft/sec2 H liquid height, ft, Figure 46 Hi see Equation (181) h heat transfer coefficient, BTU/hr-ft -OF, watts/cm 2-K hx local heat transfer coefficient ho Kapitza effect heat transfer coefficient, Equation (171) hg gas space heat transfer coefficient g hfg latent heat, BTU/Lbm fg hl see Equation (128) fg h f2 see Equation (132) fg2 I see Equation (81) J conversion factor 778 ft-Lbf = 1 BTU k thermal conductivity, BTU/hr-ft-0F, watts/cm-~C k'j,k" thermal conductivity koo thermal conductivity at TX k* see Equation (19) kApp thermal conductivity of He-II, see Equation (168)

-287NOMENCLATURE (CONT'D) L characteristic length, ft L* see Table X L for Figure 105 only, liquid mass velocity, w2/A, Lbm/hr-ft2 M0,Ms stability modulus, Equations (15, 17) Ns biot modulus, Equation (18) Nu Nusselt number p pressure, psia, mm Hg p perimeter, ft pc critical pressure, psia p" volumetric heat generation, BTU/hr-ft3 Pr Prandtl number q heat flux, watts/cm2 (q/A) heat flux, BTU/hr-ft2 (q/A) maximum nucleate boiling heat flux max (q/A)min minimum film boiling heat flux (q/A)K Kapitza heat flux b see Equation (182) R residual R gas constant, (Lbf/Lbm)(ft/~R) Re Reynolds number Ra Rayleigh number, GrPr Rg see Figure 106 Ra see Figure 106

-28.8NOMENCLATURE (CONT'D) St Stanton number S entropy S two-phase flow correction factor, Figure 95 T,T',T" temperature R, ~K Tm temperature at maximum value of cp, ~R TOO ambient temperature, OR, OK T. initial temperature, ~R T interfacial temperature, ~R S Tsat saturation temperature, OR, OK Asat Tw - Tsat R, OK Tw wall temperature, ~R, OK T critical temperature ~R,:~K c t time, hours, seconds t+ universal temperature u,u',u" velocity, ft/sec u+ universal velocity vfg volume for vaporization, ft3/Lbm V velocity, ft/sec V average velocity w mass flow rate, Lbm/hr x coordinate, mass quality x* x/L Lx grid size X interface location

-289NOMENCLATURE (CONT'D) y coordinate y* y/L z coordinate z* z/L Subscripts b bulk f film g gas or vapor Q, liquid v vapor w wall 00 ambient Greek Symbols al',C" thermal diffusivity, ft2/hr absorptance expansion coefficient, R-1 5T see Equation (37) 6 disturbance parameter A(t) stratified layer thickness, ft YT see Equation (58) emittance 1 7-z/A(t) e Debye temperature, ~K, Figure 5

-290NOMENCLATURE (CONT' D) Xc critical wave length, ft, cm, Equation (122) Xd most dangerous wave length, ft, cm, Equation (123) X growth rate parameter, Equation (29) X for Figure 105 only [(pg/o.o75)(p/62.3)] II, viscosity, Lbm/hr-ft v kinematic viscosity, ft /hr p density Lbm/ft3, gr/cm3 ca surface tension, Lbf/ft, dynes/cm aT see Equation (39) 1 see Equation (100) and Table XI'D2 see Equation (101) and Table XII correction factor, Table VI 0p2 correction factor, Equations (62) and (63) correlation factor, see Figure 68 0? gsee Figure 106 see Equation (82) for Figure 105 only, (73/a)/[( Q(62.6)/p2)2 ] Xtt Martinelli parameter, Equation (68)

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