VELOCITY AND TEMPERATURE DISTRIBUTIONS IN VERTICAL CONCENTRIC ANNULUS WITH COMBINED FREE AND FORCED LAMINAR CONVECTION

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VELOCITY AND TEMPERATURE DISTRIBUTIONS IN VERTICAL CONCENTRIC ANNULUS WITH COMBINED FREE AND FORCED LAMINAR CONVECTION Tamotsu HANZAWA,Akihisa SAKO and Kunio KATO Department of Chemical Engineering,

1 VELOCITY AND TEMPERATURE DISTRIBUTIONS IN VERTICAL CONCENTRIC ANNULUS WITH COMBINED FREE AND FORCED LAMINAR CONVECTION Tamotsu HANZAWA,Akihisa SAKO and Kunio KATO Department of Chemical Engineering, GunmaUniversity, Kiryu 376 Key Words: Heat Transfer, Vertical Annulus, Free-Forced Convection, Combined Laminar Flow, Numerical Analysis Velocity and temperature distributions and heat transfer coefficient with combined free and forced laminar convection were investigated for annular flow between two vertical concentric cylinders, where a part of the inner cylinder was heated. By visualization of the flow pattern with a tracer, the fundamental equations of flow and heat transfer were derived and solved numerically. The velocity and temperature distributions in the annulus and the heat transfer coefficient on the heated wall were calculated with air. The range of Grashof number was x 107, that of Reynolds number , radius ratio and aspect ratio (cylinder length divided by annular gap) The experimental temperature distributions and the heat transfer coefficientwerein close agreement with the calculated values. Introduction Whenfluid flows in the laminar flow range through a vertical concentric annulus of which the inner wall is partially heated, the density of fluid near the heated wall is decreased by the temperature rise and free convection is generated. The free convection combined with laminar forced convection causes higher heat transfer rates. Many theoretical approaches have been taken to the combined forced-free laminar heat transfer in a vertical annulus.1'2'4~8) However, these theoretical investigations were mostly restricted to the low Rayleigh number range (Ra < ) and the whole inner wall was heated. In the barrel-type epitaxial reactor used in the semiconductor manufacturing industry, by contrast, the Rayleigh number is higher than 106 and the inner wall was partially heated. The lack of theoretical approach to the problem of combined forced-free laminar annular flow for the higher range of Rayleigh number and the practical importance of this problem in the field of epitaxial reactor motivated this work. In this study, velocity and temperature distributions and rate of heat transfer with combined free and forced laminar convection are investigated for the annular flow path between two concentric vertical cylinders, of which the inner cylinder is partially heated. By visualization of the flow pattern with a Received June 28, Correspondence concerning this article should be addressed to T. Hanzawa. A. Sako is now with Kao Co., Ltd., Wakayama 641. tracer, the fundamental equations for flow and heat transfer are derived and solved numerically. Velocity and temperature distributions and the heat transfer coefficient are calculated. 1. Fundamental Equations and Numerical Calculations According to the observation of flow pattern (TiCl4 tracer) in a preliminary experiment, it seems reasonable to analyze the transport phenomenaby considering two-dimensional circulating flow in the vertical cross section. Let us consider the heat transfer phenomenon with combined free and forced laminar convection in an axisymmetric flow in an annulus, where a part of the inner tube is heated and the other surface is kept at a constant temperature, as shownin Fig. 1. If it is assumed that the physical properties of the fluid are constant and independent of temperature, that the buoyancy is directly proportional to the temperature difference and that there is no component of velocity in the circumferential direction, the dimensionless fundamental equations and boundaryconditions are obtained as follows: 1 d dv (1) JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

2 R=R0, 0<ZsJZL; U=V=T=O Z=0, RS<R<RO; U=T=O v-yi 2VV U-k2) -ir-rxr+li^jwrx) [ln(l//c) where (5) v r0 ro dv Z=ZL, RS<R<RO; U=-=-=0 du Fig. 1. Coordinate system for a vertical annular enclosure. Table 1. Grid size and number of iterations AR AZ Number of iterations dt dt 1f1 3fndT\ d2t] UM+VIz=VrU^{RJR)+m (4) (boundary conditions) R =R [ <z<za, ZB<Z^ZL; U=V=T=0 S'\ZA^Z^ZB; U=V=0, T=l The dimensionless stream function ij/ and vorticity C were introduced to Eqs. (l)-(5) and each equation was written in a finite difference form by the "upwind method", and they were then solved numerically by a relaxation method. The optimum values of the relaxative condition have been found by trial and error for a given system of grid points. Representative values of grid size and iteration number for good convergence are given in Table 1. The calculations were performed over the ranges lte=25-800, Gr=0-3 x 107, Pr=0.7, th= k, V= K, L=6, 12, 24cm, 0y,ro)= (7.0,ll.15), (3.05,8.5)cm, z^=9cm and #=30, 33cm. 2. Calculation Results 2.1 Velocity profile Figure 2 shows the streamlines obtained numerically for the cases where the Reynolds number is 25, 400 and 800, respectively. Since the forced convection Fig. 2. Streamline with Re=25, 400 and 800 for fh=473k, DJL=0.692, rs/ro=0.628 and Pr=0.7. VOL 19 NO

3 Fig. 3. Streamline with Gr=O and 2.7x 106 for Re=400, DJL=0.692, rs/r0=0.628 and Pr=0.1. Fig. 5. Effect of Gr on calculated axial velocity profile. Fig. 4. Effect of Re on calculated axial velocity profile. is smaller than the free convection at Re=25and 400, a back flow arises near the outer tube wall in the exit region of the test section. Such a back flow was observed by the flow pattern of tracer in the preliminary experiment, too. The effect of Reynolds number on the streamlines was observed at other values of Gr, DJL and rs/ro. Figure 3 shows the streamlines for the cases where the Grashof number is 0 and 2.74 x 106, respectively. From Figs. 3 and 2, when Grashof number is increased the axial velocity near the heated surface becomes high by the large free convection. Since there 98 is no free convection at Gr= 0, the streamlines become parallel to the axial coordinate. Onthe other hand, at Gr=2.1A x 106 the streamlines are drawn toward the heated surface and at Gr= 1.1 x 107 back flow arises in the upper portion of the test section. Figure 4 shows the axial velocity profiles at the upper edge of the heated surface, with Reynolds number as a parameter. The axial velocity takes its maximumnear the heated surface by the effect of free convection. The effect of free convection upon the velocity profile is very large. Since the free convection at Re=25 is larger than the forced convection, a vortex with back flow arises. Figure 5 shows the axial velocity profiles at the upper edge of the heated surface with Grashof num- JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

4 Fig. 6. Effect of z on calculated axial velocity profile. Fig. 7. Temperature distributions with Re=25 and 400 for /H=473K, De/L=0.692, rs/ro=0.62s and Pr=0.1. ber as a parameter. From Fig. 5, it is found that the axial velocity near the heated surface increases with increasing Grashof number. Figure 6 shows the axial velocity profiles at various distances from the bottom of the test section at Re=25. From Fig. 6, it is found that the axial velocity profile is strongly changed with axial distance z by the free convection. 2.2 Temperature distribution Figure 7 shows the isotherms obtained numerically for the cases where the Reynolds number is 25 and 400, respectively, and this figure can be connected with the streamlines shown in Fig. 2. When the Reynolds number is increased, the temperature gradient near the heated surface becomes steep. Figure 8 shows the isotherms for the cases where the Grashof number is 0 and 2.74 x 106, respectively, VOL 19 NO Fig. 8. Temperature distributions with Gr=0 and 2.7 x 106 for Re=400, DJL=0.692, rs/ro=0.628 and Pr=0.7. and the conditions of this figure correspond to those of Fig. 3. From Fig. 8, when the Grashof number is increased the temperature gradient near the heated surface becomessteep. Figure 9 shows the isotherms for the cases where L is 6 cm and 24cm, respectively. From Fig. 9, the effect of L on the temperature distribution is smaller than that on the streamlines. 3. Experimental Apparatus and Procedure To check the calculated results, the temperature distributions in a vertical annulus were measured. Figure 10 shows a schematic diagram of the experimental apparatus. Stainless steel tubes (rs = 3.03 and 7.0cm) were used as the inner tube of the annulus and Pyrex glass tubes (ro=8.5, 10.5 and ll.2cm) were used as the outer tube. The length of the test section was 21cm, 30cm or 33cm, the length being changed according to that of the heating section. The annulus had an entrance region (60cm) and an exit region (50cm), and the total length was about 140cm. To adjust the gas flow, a distributor with proper pressure drop wasused at the entrance of the annulus. The heating section was located in the center of the test section and its length was varied from 6cm to 24cm. The heating section consisted of the stainless steel tube, in which Kanthal A wires of0.6mm0 were put as heaters on two or three separate areas and were heated individually.to obtain isothermal condition. The upper and lower ends of the heating section were separated from the nonheating portion of the inner tube by adiabatic plates (Silica Board-1 made in Isolite Co., Ltd. (1=0.0558W/m-K)) to prevent heat transfer by conduction from the heating section. For measurementof the wall temperature in the 99

5 Fig. 9. Temperature distributions with De/L= 1.38 and (L=6 and 24cm) for th=473k, rs/r0= 0.628andPr=0.7. radial direction by moving the slider to measure the local temperature in the annulus. The flow rate of air was in the range of Re=20to 800. The experimental procedure was almost the same as that described in the previous paper.3) The average Nusselt number from the heated surface of the inner tube may be obtained from temperature gradients near the heated surface, as follows: Fig. 10. Experimental apparatus. test section, CAthermocouples were placed at seven points in the axial direction on the outside wall of the inner tube and at three points in the axial direction on the inside wall of the outer tube as shown in Fig. 10. To measure correctly the radial temperature distribution in the annulus, gas temperature was measured by use of a sheathed thermocouple probe of very small diameter (0.20 mni(/>) and the radial spacing for measurement was 1.0mmin the boundary layer and 10.0mm in the ambient fluid. Regulated air was fed into the annulus and then the sheathed thermocouple probe was traversed in the 100 or from the heat balance between subsequent two planes perpendicular to the vertical axis at a distance of AZ. In this experiment, the velocity distributions in the test section were not measured. WhenRe was small, vortex flow with back flow was observed in the preliminary experiment and the axial velocity became negative. Therefore, in this study, the average Nusselt number was obtained by the temperature gradient. In the case of large Re numberthe values calculated by the heat balance roughly agreed with those obtained from the temperature gradient. 4. Experimental Results and Comparison with Calculation 4.1 Temperature distribution Figures ll and 12 show the experimental temperature distributions in the cross section at the middle point of the heated surface with Re or Gr as a parameter. In Figs. ll and 12, the lines show the JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

6 Fig. ll. Comparison of calculated temperature distributions with experimental ones in the crosssectional direction for Gr=13 x 106 and Re=25, 100 and 400. Fig. 12. Comparison of calculated temperature distributions with experimental ones in the crosssectional direction for Re=100 and Gr=53x 106, 1.2x 107 and 2.8x 107. theoretical temperature distributions, which were obtained by the method mentioned in Section 1 under the same conditions as those of this experiment. From Figs. ll and 12, the calculated values are in sufficiently good agreement with the experimental ones. WhenRe or Gr is increased, both the experimental and calculated radial temperature gradients near the heated surface become steep. 4.2 Nusselt number Figure 13 shows the relationship between Nu and Re with Gr as a parameter. In this figure, the marked points and each line show the experimental values and the calculated ones, respectively. In Fig. 13, the solid line and the broken line show the calculated values for the case without free convection. The broken line represents the calculated values in the case where the axial thermal conduction is not considered (d2 T/dZ2 = 0). Figure 14 shows the relationship between Nu and Gr with Re as a parameter. From Fig. 14, both the experimental and the calculated Nu increase with increasing Gr. As is seen from Figs. 13 and 14, the calculation shows fairly good agreement with the experimental data. It is suggested that relatively correct heat transfer coefficients can be estimated by this numerical calculation. Figure 15 shows a comparison of the experimental Nu with the calculated Nu. From Figs. 13 and 14, the VOL 19 NO Fig. 13. Comparison of calculated Nu with experimental values for Gr=2.5x 106, 4.9x 106 and 9.0x 106. Fig. 14. Comparison of calculated Nu with experimental values for Re=25, 100 and 400. effect of Re, Gr and DJL on the calculated Nu approximately agreed with the effect of Re, Gr and DJL on the experimental Nu. From these facts, the 101

7 Fig. 16. Average Nu compared with some numerical results. Fig. 15. Comparison of calculated Nu with experimental values. heat transfer coefficient from the isothermally heated inner tube in the vertical concentric annulus can be estimated by this numerical calculation. The fact that data showed good agreement with the theoretical predictions mayindicate the validity of the assumptions used for the theoretical approach developed in the preceding section. 5. Discussion The effect offree convection on Nu is quite large in the range of small Reynolds number, as shown in Fig. 13, and becomes small with increasing Reynolds number. This fact may be explained as follows; when the Grashof number is large, the axial velocity caused by free convection is quite large in the range of small Reynolds number. Nu becomes large even if the Reynolds number is small. For this reason, the exponent for the Reynolds number in the empirical equation for the Nusselt number is very small. The effect of axial thermal conduction of fluid on heat transfer is increased with decreasing Reynolds number, as shown by broken line in Fig. 13. In the low Reynolds number range, the effect of axial thermal conduction on the total heat transfer rate cannot be neglected. As the Reynolds number becomes large, the effect of thermal conduction on the heat transfer rate may be negligibly small. However, in the range of present operating conditions, the axial thermal conduction term in Eq. (4) should be considered in this numerical calculation. Figure 16 shows the calculated results for the Nusselt number for a vertical heated surface in a flow system similar to the system used by the authors. In Fig. 16, there are some discrepancies in slope between previous investigations and the present one. These discrepancies maybe due to differences of the operating conditions in the system, such as length of heated surface or ro/rs ratio. 102 C onclusion Velocity and temperature distributions were investigated for the case where a part of the inner tube wall of a vertical concentric annulus was isothermally heated and gas flowed upward through the annulus in forced laminar convection. The fundamental equations derived under certain numerically to analyze the assumptions were solved heat transfer in the annulus. The following results were obtained. (1) Streamlines near the heated wall were drawn toward the heated surface except at large Reynolds number. A vortex with back flow arose in the upper portion of the system at Gr= 106. (2) Axial velocity near the heated surface increased with increasing Grashof number and changed with axial distance. (3) The radial the vicinity of temperature gradient was steep in the heated surface. The temperature distribution in the radial direction was affected strongly by Re, Gr, DJL and rs/ro. (4) The calculated streamlines and temperature distributions agreed closely with the visualized streamlines and the measured temperature distributions. (5) There was reasonable agreement between the average Nusselt numbercalculated from the radial temperature gradient over the heated surface and the experimental values. Acknowledgment The authors wish to thank Mr. Sakujiro Ishizuka of Shin-Etsu Chemical Co., Ltd. and Mr. Nobuo Kato of Baika Boshoku Ltd. for their assistance with the experimental work. Nomenclature Cp = specific heat of fluid [J/g - K] De = hydraulic diameter of annulus, 2(r0 - rs) [cm] Gr g = Grashof number, = gravitational D\ à" gfi(th acceleration - tw)/v2 [cm/s2] H h - height of test section = average heat transfer coefficient [cm] [W/cm2à" K] L = length of heating section [cm] Nu = average Nusselt number, h - DJX P = dimensionless pressure, p -D^/pv2 Pr = Prandtl number, Cp - /i/x p = pressure [Pa] JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

8 Ra Re Ro Rs R' T t th tw U u V v Z Zl z c AR AZ X dimensionless radial distance, r/de Rayleigh number, Grà" Pr Reynolds number, De à"v/v dimensionless rold. inner radius of outer tube, dimensionless outer radius of inner tube, rs/de dimensionless radial distance, p ia ( Subscripts ) A B M (r-rs)/(ro - rs) O radial distance [cm] S inner radius of outer tube [cm] outer radius of inner tube [cm] dimensionless temperature, (t- tw)/{th- tw) temperature [K] temperature of heating surface [K] temperature of cooling surface [K] dimensionless velocity in indirection, De à" u/v velocity in r-direction [cm/s] dimensionless velocity in Z-direction, De à" v/v velocity in z-direction dimensionless axial distance, z/de dimensionless height, H/De axial distance volumetric coefficient of expansion dimensionless vorticity dimensionless grid size in ^-direction dimensionless grid size in Z-direction thermal conductivity of fluid viscosity of fluid [cm/s] [K"1] [W/cnj - K] [Pa s] ( Superscript) kinematic viscosity of fluid density of fluid dimensionless stream function lower end of heating section upper end of heating section middle point of heating section outer tube of annulus inner tube of annulus =average Literature Cited 1) El-Shaarawi, M. A. I. and A. Sarhan: /. Heat Transfer, 102, 617 (1980). 2) El-Shaarawi, M. A. I. and A. Sarhan: Ind. Eng. Chem. Fundamentals, 20, 388 (1981). 3) Hanzawa, T., A. Sako, H. Endo, M. Kagawa, T. Sunaga and K. Kato: /. Chem. Eng. Japan, 19, 78 (1986). 4) Maitra, D. and K. S. Raju: /. Heat Transfer, 97, 135 (1975). 5) Rokerya, M. S. and M. Iqbal: Int. J. Heat andmass Transfer, 14, 491 (1971). 6) Sherwin, K.: British Chem. Eng., 13, 569 (1968). 7) Sherwin, K. and J. D. Wallis: Fourth Int. Heat Transfer Conference, Paris, IV, Paper No. NC3.9 (1970). 8) Shumway, R. W. and D. M. McEligot: Nuclear Sci. Eng., 46, 394 (1971). VOL 19 NO

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