HEAT TRANSFER THROUGH GASES CONTAINED BETWEEN TWO VERTICAL CYLINDERS AT DIFFERENT TEMPERATURES by William H. Lipkea and George S. Springer Fluid Dynamics Laboratory Department -.of Mechanical Engineering College of Engineering The University of Michigan Ann Arbor, Michigan This work was supported by the National Science Foundation under grant No. GK-1745, and administered through the Office of Research Administration, The University of Michigan.

HEAT TRANSFER THROUGH GASES CONTAINED BETWEEN TWO VERTICAL CYLINDERS AT DIFFERENT TEMPERATURES William H. Lipkea and George S. Springer Fluid Dynamics Laboratory, Department of Mechanical Engineering The University of Michigan, Ann Arbor, Michigan 48104 Heat transfer through gases contained between vertical concentric cylinders was investigated when the radius of the inner cylinder is small compared to the radius of the outer cylinder. Experiments were performed in a modified hot wire type thermal conductivity cell in which end effects could be evaluated in addition to the overall heat transfer between the cylinders. Measurements were made with argon, helium, and neon in the pressure range 0.1-620mm Hg, for the temperature differences between the cylinders of 10,50 and 1000C, and for the length to the outer diameter ratios between 3.3 and 6.7. The results show that below certain Rayleigh numbers and length to diameter ratios heat is transferred from the hot to cold boundary by conduction only. End effects contribute only in the corner regions. The distance to which these end effects penetrate was determined experimentally, and compared to analytical results obtained from a simple heat balance in the corner region. Relationships, based on the experimental data, were also obtained for the average heat transfer both in the corner regions and over the entire length of the cylinders. It is shown that these results may be used to estimate the error introduced in thermal conductivity and thermal accommodation coefficient measurements by neglecting end effects.

2 INTRODUCTION A knowledge of heat transfer through gases contained between concentric cylinders at different temperatures is required in many problems of practical interest. It is needed, for instance, in the experimental determination of thermal conductivities and thermal accommodation coefficients of gases. Heat transfer between concentric cylinders has been measured in connection with such experiments. To determine thermal conductivities or thermal accommodation coefficients not the total heat transfer, but only the heat conducted through the gas must be known. Thus, considerable efforts have been made in these experiments to minimize the effects of natural convection with the result that in most of the available data heat transfer due to convection is negligible. This investigation was undertaken to study heat transfer by both conduction and convection between two vertical concentric cylinders. Radiative transfer will not be considered. In most cases it is not negligible but is independent of the conductive and convective heat transfer and once the experimental conditions have been specified it can be either measured or calculated with reasonable accuracy. In addition to studying the overall heat transfer between the cylinders, special ** attention is given to the evaluation of the end effects. End effects become of particular importance in thermal conductivity and thermal accommodation coefficient measurements when the temperature difference between the inner and outer cylinders is large; a condition that has been encountered recently, in attempts to extend the use of the concentric cylinder geometry to higher temperatures [3 - 5]. This investigation was motivated by the problems arising due to end effects in such experiments. The results will be presented in general form, however, so as to * Good summaries of these experiments may be found in references 11 and [2] ** It is noted here that the end effects can be caused predominantly by convection in the gas but are not limited to convection effects only.

3 be applicable to other problems of similar geometry. EXPERIMENTAL Here we shall be concerned with the following problem: a gas is contained between two vertical concentric cylinders of radii a, and b, and length S. The radius of the outer cylinder, b, is very large compared to a (b/a 71) and the length S is large compared to b(S/b? 1). The temperatures of the inner and outer cylinders are Ta and Tb respectively. The ratio (Ta-Tb)/Ta is not necessarily small compared to one and thus the nronerties of the gas cannot be assumed to be constant. The two horizontal. surfaces which bound the gas layer are impervious to heat and mass flow. The aim aas to construct an apparatus which approximates well the above conditions and also allow the variation of the significant parameters over a wide range. Ther experiments were nerformed in a modified hot wire typr)e apparatus 6J in which the gas, the pressure, the temperature difference between the cylinders, and the lengths of the cylinders could be varied independently. The radii a and b were fixed. The rest section, shown schematically in Fig. 1, consists of a thin tungsten filament su-nnorted axially in a vertical Pyrex tube. Three sections of the filament, differing in length, were isolated by potential leads similar in concept to those described in refs. 7-9. The three sections were included in order to evaluate the influences of wire lengths on the end effects. The effective length of the outer cylinder (distance S in Figures 1,5) about each of these sections can be varied by moving teflon discs inside the tube. These moving disc are mounted on a 3/16 inch diameter stainless steel rod which has a left and right threaded portion, designed so that the distance between the disc can be altered by turing the:rod. The tihreaded rod and also the center Fil ament are supported by two stationary discs instal led near each end of the tube and connecte.d

4 by two 1/8 inch diameter supporting rods. The two supporting rods and the threaded rod are placed 1200 apart very close to the walls of the Pyrex tube. It is understood that the positions and sizes of these rods might affect the heat transfer. To test this, thermal conductivities of gases were measured with the apparatus. The results of these experiments [63 indicate that in the range of present experimental conditions the rods have negligible effects on the heat transfer. The data was obtained by the following experimental procedure. At each pressure and at each temperature difference between filament and tube, the position of the movable discs was varied ranging from a distance equal to the length of the filament section (S H, see Fig. 5) to the largest,ossible distance between the discs. For each position of the discs (i. e., for each S value) the power input to the filament section under consideration (A-A, B-B, or C-C, Fig. 1) was measured both in the presence of the gas, Qt, and in vacuum, Qv ( lxlO -7mm l-g). It was found that the % values were independent of the position of the discs, as long as the distance between the discs was larger than the distance between the potential leads being used. The total power input to the filament may be expressed as Qk is the heat conducted by the gas from the filament, Qe is the heat transferred from the filament to the gas due to end effects, Qr is the heat loss from the filament by radiation and Q0 is the heat conducted along the filament to the supports. From eq. (1) the sum QH 0 Qk + 0e is QF = - Qv (2) In evaluating QH from eq. (2) it is implicitly assumed that Q is the sum of the radiation and support losses (Qv = Qr + Qf) and that it is independent of

5 pressure. Data was taken after steady state conditions were reached. The temperature of the outer cylinder was maintained constant by immersing it in an oil bath. The temperature of this bath was held at 34.50 + 0.020 C during the tests. In the experiments that follow the inner surface of the test tube was presumed to have been at this same temperature, denoted as Tb. The average temperature of the filament, Ta, was determined by measuring its resistance [ 10. Recognizing that the temperature distribution along the filament is not entirely uniform, the term filament temperature, as used here, refers to the average temperature corresponding to the measured resistance of the appropriate filament section. The temperatures of the teflon discs were not recorded. The test tube was connected to the vacuum pumps and the gas supply tanks through a glass vacuum system. Pressures were measured with a McLeod gauge, a U-tube mercury manometer and an ionization gauge. The electrical measurements were made with a high precision d. c. Millivolt Standard, together with an optical galvanometer and suitable standard resistors. Test gases of higest quality ("Airco" in Pyrex) were used throughout the experiments, but no other efforts were made to obtain clean surfaces. The experimental data reported in the following were obtained with argon, neon, and helium for temperature differences of 10, 50 and 100~C and in the pressure range 0.1 - 620mm Hg. PENETRATION DEPTH Under certain conditions, in the center part of the cylinders the heat transfer will be by conduction only; i.e., at a distance from the ends the heat transfer through the gas is described by the Fourier equation ~11]. First, the conditions will be determined under wlhich such a "conduction regimel" exists. The experiments of Eckert and Carlson t113 on natural convection in air between two vertical plates show that local heat transfer conditions are

6 different in directly opposite corners (two lower, or two upper corners) and are similar in diagonially opposite ones. The two different types of corners are denoted as starting and departure corners. The distance from the end where the ends effects become negligible and the conduction regime begins is referred to as the penetration depth Zp. Although the penetration depths in P the starting and departure corners may be somewhat different [ 11, here no distinction will be made between them. In order to find the penetration depth we apply a heat balance in the corner region [ 11, 12] Z, 2 a c=), TS~z)d Td (3) 0 o Qc is the enthalpy carried by convection through a horizontal plane at Zp, and q and qd are the heat fluxes per unit length in the starting and departure corners (Fig. 2). Equation (3) can be applied only when the end effects do not penetrate beyond the center plane of the cylinders (Zp C H/2). It will be assumed p now a) that qs varies as Z 4 L13] b) that at Z = Zp, qs is equal to the conductive heat transfer per unit length, qk, and c) that qd is independent of the position and is equal to qk. With these assumptions eq. (3) may be integrated to yield (4) The parameters Qc and qk are given by the equations -J - 4, X L 7T wrd.Cd (5a)

dT LA,=[2Tia dcr lr- (5b) w(r) is the axial velocity and all other symbols are defined in the nomenclature. Equation (5a) is based on the assumption that in the conduction regime (Z = Z ) the flow is laminar and fully developed. In order to evaluate Qc and qk, the temperature and velocity distributions must be known at Z = Z. In the conduction regime heat is conducted in the radial direction only and thus, at Z = Zp, the conservation equations for mass, momentum and energy are dr w= O r (6a) r dr (r/ d (6b) i d (, — ii\r dr dr (6c) C/= e<t+C) (6d) C is a small correction to the density. It was introduced so that the continuity and momentum equations could be satisfied simultaneously without kInowledge of the detailed flow field in the corners L31. In the calculations that follow, at any point C was always less than 0.1 and generally was about 0.01. The average density has been defined as b a2 ( r -- (7) = __ - 9

8 In eq. (6b) the approximation has been made that density differences due to temperature differences are only of importance in producing differences in the buoyancy force. The equation of state for gases then has the form [121 Cr-)-'r)= T'=Ln (8) The boundary conditions corresponding to eqs. (6a - 6c) are (9) r'- b w —<=o T- -— b The temperature difference between the cylinders may be large (Tb/Ta~>l) and hence k and,/C cannot be taken to be constants. Equations (6-9) were integrated numerically using a high speed digital computer. Solutions for eqs. (4-9) were obtained for a wide range of conditions, namely for three monatomic gases (A, Ne, He) and four diatomic gases (C02, 02' N2, and H2), four temperature differences (zT = 10, 50, 100, 5000 C), four pressures (400, 600, 760 mm Hg) and five radius ratios (b/a = 10, 25, 50, 100, 250, using a = 0.5, 0.05, 0.005, and 0.0005 cm). The gas properties used in calculations were taken from refs. [14, 151. The computed numerical results were transformed into the following dimensionless groups: penetration depth 2b axial velocity W = W (10) Rayleigh number Ra r=,Q_",,T (? ~p /,~ T k =T

9 It was found that for the entire range of parameters employed in the solutions both w* and Z* can be correlated extremely well with the Rayleigh number defined above. Calculated axial velocity profiles are shown in Fig. 3 for various Rayleigh numbers. These profiles change very little for different radius ratios as long as the radius ratio, b/a, is larger than about 10. The calculated penetration depths are shown in Fig. 4. The results of the calculations are represented by a solid line and can be well approximated for b/a ) 10 by the relation: 44o~ (11) It will be shown presently that eq. (11) is applicable only for Ra > 4400. The above expression is expected to be a reasonable approximation for Z* only as long as the conductive and convective heat transfers are of the same order of magnitude in the corner region. Then the convective heat transfer is small compared to the heat conduction then dimensional considerations [37 suggest that the penetration depth becomes constant, its value being of the order of the difference between the radii (b-a). Penetration depths were measured in argon, helium and neon with the apparatus described in the previous section. The penetration depth was determined by measuring the heat transfer from a certain section of length H of the filament (AA, BB, or CC, Fig. 1) while varying the actual length S, of the cylinders (Figs. 1,5). A typical result indicating the variation of the heat transfer with S is shown in Fig. 5. In principle, the penetration depth is reached when there is no further change in the measured heat transfer with a change in the distance S. Here, however, the penetration depth was taken to correspond to the conditions,

10 that for a one cm change in S the heat transfer changes less than 0.1% compared to its maximum value. The experimentally determined penetration depths are shown in Fig. 4. In certain cases the condition given above could not be reached and the penetration depth was obtained by extrapolation of the data. These test points are indicated by a short line attached to them. In Fig. 4, penetration depths are also shown deduced from Gregory and Marshall's experiments in 02, N2, and C02 L16, 17. Gregory and Marshall used two test tubes of different lengths in gaseous connection to measure thermal conductivities. The penetration depths in their experiments can be evaluated from their carefully detailed data. For Rayleigh numbers greater than about 4400 the data agrees fairly well with the analytical result (eq. 11), provided that Z p H/2. At lower Rayleigh p numbers the penetration depth becomes a constant and equal to the diameter of the outer tube Z~P = I C Ks ~f 440o ) (12) The results in Fig. 4 also indicate that, as expected, the penetration depth is independent of the cylinder length as long as the end effects do not extend beyond the center plane of the cylinders. In the experiments the pressure was reduced to such low levels that rarefaction effects became important. The highest Knudsen number (based on the diameter of the filament) in the experiments was about 10. At this value nearly free molecule conditions exist in the gas L181. It is interesting to note, however, that even at such a high degree of rarefaction the penetration depths remain equal to 2b. It is noted here also that in the present experiments the Knudsen numbers

11 corresponding to Ra = 4400 were about 0.01. At these Knudsen numbers the temperature jump effects are negligible, eqs. (6,9) are valid, and consequently eq. (11) may be used in estimating the penetration depth. The conditions can now be established under which a conduction regime exists when end effects do not penetrate beyond the center plane of the cylinders, i.e. when Z < 2H. Using eqs. (11,12) the limiting conditions for the conduction regime were determined and are shown in Fig. 6. In this figure the Rayleigh number is based on the radius b so that the results for the concentric cylinder geometry can be compared with the results obtained by Eckert and Carlson l111 and by Batchelor L12I for vertical parallel plates. As can be seen the Rayleigh numbers limiting the conduction regime as given by the present results are similar to the values presented by Eckert and Carlson, and are less than the values given by Batchelor. However, the slope of the line bounding the conduction regime is identical to the one given by Batchelor. Heat Transfer in the Conduction Regime The conductive and convective heat transfer between the two cylinders will be now evaluated based on heat transfer measurements made in argon, neon, and helium. In these measurements the moveable discswere positioned at either one of the potential leads (AA, BB, or CC) resulting in length to diameter ratios H/D = 3.3, 5, 6.7 Corner Region Without distinguishing between the starting and departure corners an average Nusselt number is defined in the corner region f LL a(13) h (13)

12 D is the diameter of the outer tube and h is the overall heat transfer coefficient C. in the corner region given by the equation (Z p H/2) P, 7Fl'Toa-2F, (. H- 2Z) Ad o z) FIpt2 (14) Equations (13) and (14) may be rearranged to yield LN,- i_ + 2 ez 2 (2 5 a 2 iso04) 2IT " lk o>,T o w b/(15) For the present data, eq. (15) can be applied only in the Rayleigh number range indicated. At Ra. 25 rarefaction effects become significant, k is not a constant and the heat conducted from the filament, Qk' cannot be calculated from the Fourier equation. At Ra > 2x10 the condition Z < H/2 is not valid p anymore. Corner Nusselt numbers calculated from eq. (14) are shown in Fig. 7. In calculating Nuc, average values for the thermal conductivity were used corresponding to the temperatures (Ta + Tb) /2, and Z was computed from eqs. (11) or (12). For QH the measured heat transfer values were substituted. The results indicate that in the appropriate range of Rayleigh numbers Nu is c nearly a constant, at 0.355.* In the present experiments log (b/a) - 5.86, and Nu can be expressed as...-= 0 013 (16) *Note that the average Nusselt number remains constant at least up to Rcla lxlO (Fig. 7) even though Z > H/2 p

13 Average heat transfer. An average heat transfer coefficient for the heat transfer between the inner and outer cylinders including the corner regions may be defined by the equation (17) Using eqs. (14,17) one obtains for the average Nusselt number _ _ 2- 4 _l 4 \IN2 Nt4 _) (18) Substituting eqs. (11,12 and 16) into eq. (18) we get a2-1 No,- -- - + o.,z D (2' 2x c44o o) iOE bl14 t(19) and... 5, q c?,0 Z, A (o.4.o. <, 2,,14) WJ 4- R'lot ~~(20) The above results (eq. 19,20) may be used to estimate the errors arising in thermal conductivity measurements due to the neglecting of end effects. For example, in typical hot wire type thermal conductivity cells H/D = 20 and b/a = 200. Then, for Ra = 8800, by including the end effects the Nusselt number is 0.379, while neglecting end effects it is 0.377. Thus, in this case end effects may introduce about a one percent error into the measurements.

14 Finally, it is noted here that at Ra 1 x 105 steady state conditions cannot be reached and significant fluctuations in heat transfer are observed. It has yet to be determined whether these fluctuations are due to turbulence or to instabilities such as observed by Elder [ 19 and by Vest [20] between parallel plates using fluids with high Prandtl numbers. ACKNOWLEDGMENTS This work was supported by the National Science Foundation under grant No. GK-17 45.

REFERENCES (1) H. Y. Wachman, "The Thermal Accommodation Coefficient; A Critical Survey," ARS Journal, 32, 2 (1962) (2) N.- V. Tsederberg, "Thermal Conductivity of Gases and Liquids," The MIT Press, Cambridge, Massachusetts, (1965) pp 1-65 (3) N. C. Blais and J. B. Mann, "Thermal Conductivity of Helium and Hydrogen at High Temperatures," J. Chem. Phys. 32, 1459, (1960) (4) M. P. Saksena and S. C. Saxena, "Measurement of Thermal Conductivity of Gases using Thermal Diffusion Columns," Phys. Fluids, 9, 1595 (1966) (5) D. V. Roach and L. B. Thomas, "Determination of the Thermal Accommodation Coefficient of Gases on Clean Surfaces at Temperatures Above 3000 K by the Temperature Jump Method," Rarefied Gas Dynamics (ed. C. L. Brundin) Academic Press, New York, N. Y. (1967), 1, pp 163 (6) G. S. Springer and R. H. Ulbrich, "Modified Hot Wire Type Thermal Conductivity Cell," Rev. Sci. Instrum. 38, 938 (1967) (7) W. J. Taylor and H. L. Johnston, "An Improved Hot Wire Cell for Accurate Measurements of Thermal Conductivities of Gases Over a Wide Temperature Range," J. Chem. Phys.,14, 219 (1946) (8) L. B. Thomas and E. B. Schoefield, "Thermal Accommodation Coefficient of Helium on a Bare Tungsten Surface," J.Chem. Phys., 23, 861 (1955) (9) A. Dybbs and G. S. Springer, "Heat Conduction Experiments In Rarefied Gases Between Concentric Cylinders," Phys. Fluids, 8, 1946 (1965) (10) L. B. Thomas and R. E. Brown, "The Accommodation Coefficients of Gases on Platinum as a Function of Pressure," J. Chem. Phys., 18, 1367 (1950) (11) E. R. G. Eckert and W. O. Carlson, "Natural Convection in an Air Layer Enclosed Between Two Vertical Plates With Different Temperatures," Intl. J. Heat and Mass Transfer, 2, 106 (1961) (12) G. K. Batchelor, "Heat Transfer by Free Convection Across A Closed Cavity Between Vertical Boundaries at Different Temperatures," Quart. Appl. Math, 12, 209 (1954) (13) E. M. Sparrow and J. D. Gregg, "Laminar Free Convection from Vertical Plates with Uniform Heat Flux," Trans.ASME 78, 1824 (1956) (14) J. Hilsenrath et. al. "Tables of Thermal Properties of Gases," Natl. Bureau of Standards, Circular 564 (1955) (15) J. Hilsenrath and Y. S. Touloukian, "The Viscosity, Thermal Conductivity and Prandtl Number for Air, 02 N2, NO, H2, H20 He and A," Trans.ASME, 76, 967 (1954) (16) H. Gregory and S. Marshall, "The Thermal Conductivity of Carbon Dioxide," Proc. Roy. Soc. A114, 354, (1927)

16 (17) H. Gregory and S. Marshall, "The Thermal Conductivites of Oxygen and Nitrogen," Proc. Roy. Soc. A118, 594 (1928) (18). G. S. Springer and R. Ratonyi, "Heat Conduction From Circular Cylinders in Rarefied Gases," J. Heat Transfer, 87, 493 (1965) (19) J. W. Elder, "Laminar Free Convection In a Vertical Slot," J. Fluid Mech. 23, 77, (1965) (20) C. M. Vest, "Stability of Natural Convection In a Vertical Slot" Ph. D. Thesis, University of Michigan (1967)

17 NONEMCLATURE a radius of inner cylinder (cm) b radius of outer cylinder (cm) c specific heat (watts/gm-0C) D diameter of outer cylinder (cm) g gravitational acceleration (cm/sec2) h heat transfer coefficient (watts/cm- C) H distance between potential leads (cm) k thermal conductivity (watts/cm-~C/cm)q heat flow per unit length (watts/cm) Q heat flow (watts) r radial coordinate (cm) S actual length of cylinders (cm) T temperature (OC) aLT temperature difference = T -T (0C) w axial velocity (cm/sec) Z distance from corner in axial direction (cm) Zp penetration depth (cm) density (gm/cm3) corrected density = P(1+C) (gm/cm3) mean free path (cm) viscosity (gm/cm-sec) C correction to density Nu Nusselt number = hk/D

18 Pr Prandtl number T= 3 Ra Rayleigh number Gr Pr - c'Q T () /Tt w* axial velocity = t= Z* penetration depth = Z /2b p p Indices a evaluated at a b evaluated at b c convective d departure corner e due to end effects f along the filament H based on height, H, when S=H k conductive r radiation s starting corner t total v in vacuum indicates average value

19 FIGURE CAPTIONS Fig. 1 Schematic of test section Fig. 2 Sketch of lower end of tube indicating the corner region. Fig. 3 Calculated axial velociy distributions (a = 0.005 cm) Fig. 4 Penetration depth Fig. 5 Data, showing the variation of heat transfer from length H of the filament due to changes in the distance S. Fig. 6 Limits of the conduction regime. (Rayleigh number is based on radius b) Fig. 7 Average Nusselt numbers in the corner region

______ ___-_Iron Bar (Rotated by TO 7 <_ X 3Magnet) Vacuum Stationary Disc (Teflon) Spring Support Left Hand Thread ___ t =(SMovable Disc ( Tef Ion) Potential Leads o. (Dia 0.0040) o~ Al II /) ----— ~Main Filament o Tungsten oW mae o jr (Dia = 0.008534) o Outer Tube gOII(~ (Pyrex) A Ii 2- Supporting Rod (0.317 Dia) Right Hand __~ ~Thread (0.476 Dia) Movable Disc (Teflon) ___________Stationary Disc (Teflon) -H I. D. 2.90 All Dims. in cm O.D. 3.15

b a TT I (r) I I I IQs Qd NW. p I z I I z r Fig. 2

D-q D-J NOI.ISOd IVIClVI 0'1 8'0 9'0 too Z''0 I-' 0 001 I'0 0001 \o 0 -'\0 r oa g'O oo001: =o/q T\j, 6'0 0* 0'1 -", I *\,..'.. _ OOOg \./ Z ~El0~

2 C 0.8,, t, It R a to *~o i N 10 ~8 |H/D- 3.3; 5; 6.7 I: A 0 ~ * I-6 - HE 0 QmJ NE A -- 4 02 V 0z GN2 REGORY a MARSHALL 02 (H/ D 21 ) C02 KNUDSEN NUMBER RANGE ANASIS L 2 X Z |Kn= — = 0.00110 Zp CL/ J J J I I1 1 JJ I i iI I!i I I Il I I i -0 I0 o2 0 3 4 105 RAYLEIGH NUMBER Ra Fig. 4

0 o "~ -0-&~ Q~ QARGON 242 P18 mmHg x- T0=135.20~c IH o Tb= 34.500c a 2.2~~41 H=140.035 cm 2"""" uu 240 - - z 239 - I ~ 238 I w 237 236 t-S = H I I I I I I I I II I 10 12 14 16 18 20 22 24 26 DISTANCE S (cm) Fig. 5

10 10 10 10 10 100 H / b 10 CONDUCTION REGIME 5~ 4 r ~~~~~~~~~~VERTICAL PL)TS / ~~~ECKERT ar CARL'O 191 BATCHEELOR ( 19" 4 1~~~ / 10 10- 10 1.0 1 RAYLEIGH NUMBER (Ra)b Fig - 6

O H/~HD ~'3.3 5 6,7 zi 0.5 ARGON H1ELIUM t3 cr IrlA &Q C) CD NHON A O0 aD& AjL ~3 O~~~~~~~~~il~~0 0 0\bo 0 3~~~ LJ U ~~0.5 3 L t RAY LE1GG NUMBER: RU vIjIs