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Heat Transfer in Laminar Flow in Vertical Concentric Annuli

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Heat Transfer in Laminar Flow in Vertical Concentric Annuli Literature Cited 4) Johnson, A.I, and Furter, W.F.: Can. J. Chem. Eng., 2, 78 (1960) 1) Ciparis, J. N. : "Data of Salt Effect in Vapor-Liquid Equili- 5) Perry, J. H.: "Chemical Engineer's Handbook", 4th. Ed. brium ", Lithuanian Agricultural Academy, Kaunas, U. S. S. R., McGraw-Hill, New York, 1963 1966 6) Smith, T.E. andBonner, R.F.: Ind. Eng. Chem., 41, 2867 2) HashitanL M., Hirata, M. and Hirose, Y. : Kagaku Kogaku, (1949) : 32, (No.2) 182 (1968) 7) Teshima, T., Hiyoshi, S., Matsuda, H., Monma, S. and 3) Hashitani, M. and Hirata, M.: J. Chem. Eng. Japan, 1, Iwabe, S.: J. Chem. Soc. Japan, Ind. Chem. Sect., 55, 801 (No. 2) 116 (1968) (1952) HEAT TRANSFER IN LAMINAR FLOW IN VERTICAL CONCENTRIC ANNULI NOBUO MITSUISHI, OSAMU MIYATAKE** AND MITSUGU TANAKA Dept. of Chem. Eng., Kyushu University, Fukuoka A theoretical analysis of heat transfer to Newtonian fluids in laminar flow in concentric annuli was developed under the conditions of constant temperature inner wall and insulated outer wall, taking into account the temperature dependency of viscosity and density. The arithmetic mean Nusselt number was obtained as a function of the Graetz number with K, flwl'fJLo and Grw/Rew as parameters. Furthermore, experimental data were obtained for two different diameter ratios K. It was found that theoretical predictions are in reasonably good accord with experimental data. Introduction (1) The temperature of the inner wall isuniform and the outer wall is insulated. The analytical solutions of problems involving heat (2) The laminar velocity profile is fully developed transfer to Newtonian fluids flowing in concentric an- at the inlet to the heat transfer section. nuli with constant temperature inner wall and insulated (3 ) Thermal conduction in the longitudinal direc- outer wall under fully developed laminar flow conditions tion is negligible. have been given in the form of an infinite series con- (4) Heat produced by viscous dissipation is neg- taining eigen values and eigen functions by R. Viskanta4) lected. and A. P. Hatton & A. Quarmby3). These theoretical (5) The fluid temperature is uniform at the inlet approaches have been developed under the assumptions to the heat transfer section. that the physical properties of the fluids are inde- (6) A steady state has been attained. pendent of temperature and natural convection effects Then the equation of motion of a flow in a vertical are negligible. annulus can be expressed as However for many industrial heat transfer problems 1 d ( du\ , dp , ,^ the change in physical properties due to heating or r dr\ drJ dz cooling must be considered. In the case of downwardflow, it suffices to substitute Hence the authors have extended the analysis to the -g for g. The following linear relations will be used case of heat transfer with temperature dependent viscos- to express the temperature dependency of the fluid ity and density. Experimental data obtained for two different diameter ratios k (=2n/2r0) were in reason- able agreement with the theoretical predictions. Theoretical Analysis The coordinates and geometry are shown in Fig. 1, and in the course of the analysis the following assump- tions are made: Received on October 30, 1968 Presented at the 33rd Annual Meeting of the Society of Chemical Engineers, Japan, at Kyoto, April 1968 Research Institute of Industrial Science, Kyushu Univer- sity Fig. I Diagram of the coordinate system VOL.2 1969NO.2 153 density and viscosity. ( 1 -K2) + P=toll -fcT- To)] (2) 4(1-kY Re*, /< = ft/[i + Kr- T.)] (3) For short contact times, temperature changes are [(/,+/O -(l --J-)(-/s +J.) restricted to a thin region near the inner wall. With- - 77-/c in such a thin region the velocity distribution may be represented by the following equation. + u = Y(du/dr\r=ri)iso(r - r<) (4) ft-(-^m1 where Gru Grw ff1 [1 - (1 -^/jO6>(£)] 4(1 -a:)2 ' Rew T = (du/dr\r=rt)/(du/dr\r=rt)uo (5) I (du/dr\r=ri)iso, which is the velocity gradient at the coil - 6(<o))da> - /. \d$\ inner wall, is obtained from the equation of the veloc- 1 X ity distribution for isothermal flow15 as follows. 2 7i+ 1- 2[-2*- (1 -*')/(*In*)] _0 r_ ,]] (du/dr\r=n)uo = [1+a:2+(l-yrO/ln/r] ' r0 =0 (15) Here Rewand Grw are defined as follows. Rew - DeUpJfJ-w (16) where Grw -7>o2g/3De3(T. - To)/^2 (17) /C = Vilro For short contact times, Eq. (4) is substituted into De will be defined by the following equation : the energy equation and the resulting expression is De=2(r0-n) (18) peraturesolved underto givethe conditionsthe followingof constantexpressioninnerfor walltempera-tem- ture distribution. r, which is defined by Eq. (5), can be rewritten using Eqs. (6) and (14) as (8) f , 1 Grw x 1+t^ 2(l-K2){l-lcy Rew L-h where [(/a +J<) -/,à"/, + (9) and <f>can be expressed as follows : (-75+76+/8-79+ 0 - <piso/Y (10) By defining Gz as Gz = WCp/i&z (ll) Cmaybe rewritten as X is determined as a function of GzT4 from Eq. (15) C = kz/pCPuro2 = tt(1 - a:2)/G^ (12) by trial and error. Using this X, with k, Pw/Po and Grw/ = [9tt(1 - K2)<f>uo/GzrYn (13) Rewas parameters, weare able to evaluate T as a func- tion of GzT, and establish the relation between Gz and Substituting Eqs. (2), (3) and (8) into the equation r. of motion and integrating the resulting equation, we get the following expression for the velocity distribu- For short contact times, the arithmetic meanNusselt tion. (See Appendix) number Numis given by the following equation. (1 -tc2)u , />og/3ro2(Tw - To) Num = JlmDe/k = qaveDe/k(TW ~ To) u = =3(1-*)/[r(D^cT] (20) [(7a+74) -(l-^L)(-75+/e) where ^9^C7 = C9tt(1 - K2)$i,o/GzLr']in (21) -/7-/9-ll- /«. Now,since the temperature distribution given by Eq. (8) is the result obtained by assuming that T is (![-'å + (i-£)*]| constant along the tube axis, Eq. (20) is also the result obtained for constant T. Since T varies along the tube rj- (> -£)«»]( - G)d$ axis, we must substitute a meanvalue of T into T of PogProKTv, - To) Eq. (20). Because it is not easy to evaluate the mean /*« value of T analytically, the authors choose the value of Ci - (i - ft.//B.)efe)] T at the mid-point of the heat transfer section as the J r/ro f [J w(l - 6(<o)}da) mean value of T. That is, in calculating Num from Eq. (14) (20), the value of T at 2Gzl is used as the mean value - j O>(1 - 0(o>))<*<»]<# ofr. where /'s are given by Eqs. (A4) through (A12). When the temperature dependency of physical pro- X is a value of the dimensionless radial coordinate perties is not taken into account, ^=1. Therefore Num where du/dr-0. It is determined from the following is given by the following equation : equation, obtained by inserting the boundary condition Num=3(1-k)/[r(j)y/^ucCL] (22) that velocity = 0 at inner wall into Eq. (14). 154 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN Gr, For small values of Gzl, Eq. (22) deviates from the The fluid is pressurized in a vessel and flows into a exact solutions3'0. The correction factor for Eq. (22) mixing chamber where the inlet temperature is meas- shown in Fig. 2 has been calculated by the authors ured. Then the fluid passes through an insulated based on the exact solutions. Consequently Num is entrance region of sufficient length to build-up a fully expressed as follows over the whole range of GzL. Num=3F(X-fc)I|r(4)VWu&l1 (23) If it is assumed that this F is applicable also to the case in which the change of physical properties is taken into account, Eq. (20) yields Num=3F0.-*)/[r(j)V9#r] (24) The results for Numcalculated from Eq. (24) are given in Figs.3 to 5*. Experimental Apparatus and Procedure Fig. 6 is a schematic diagram of the experimental apparatus. Fig. 2 Correction factor Fig. 3 Computed values of mean Nusselt number Fig. M Computed values of mean Nusselt number Detailed results are published in Memoirs of the Faculty of Engineering, Kyushu University, Vol, XXVIII, No. 3 VOL.2NO.2 969 155 Fig. 5 Computed values of mean Nusselt number Fig. 6 Schematic diagram of experimental apparatus developed velocity profile, flows into a heat transfer Experimental Results section where it is heated by a steam-heated inner wall, passes through another insulated annular space and into The results of experiment are shown in Figs. 7 and a mixing chamber. The inner wall temperature is 8. measured by four thermocouples equally spaced along In Fig. 7 the calculated values of Nusselt number the heat transfer section and the outlet temperature obtained by interpolating exact solutions3'0 for T=l is measured in the second mixing chamber. Flow rate are comparedwith experimental values. Disagreement is measured with a platform scale. between measured values and theoretical predictions Two kinds of heat transfer sections are used. One increases as Grw/Rew increases. section is a;=.0.354 (outside diameter of inner tube (2n) In Fig. 8 the values of Nusselt number calculated by =19.lmm, inside diameter of outer tube (2ro)=54.0 the authors from Eq. (24) are compared with the same mm,length of heat transfer section (L) =120cm, length experimental values. The experimental values agree of insulated entrance section (Le)=140cm) and the with theoretical predictions by the authors within about other £=0.746 (outside diameter of inner tube (2n)= 30%.
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