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(
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
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%. 31.8mm, inside diameter of outer tube (2ro)=42.6 These figures also give the experimental data for mm,length of heat transfer section (L)=100cm, laminar heat transfer of F. G. Carpenter et alP length of insulated entrance section (Le) =93cm). The working fluids used were water, CMC(carboxy Conclusions methyl cellulose) aqueous solutions and HEC(hydroxy ethyl cellulose) aqueous solutions. Although these fluids The authors obtained an analytical solution of heat display non-Newtonian behavior at high shear rates, transfer to Newtonian fluids flowing through a con- after violent mixing they show no non-Newtonian be- centric annulus with constant temperature inner wall havior at the shear rates employed in this work. and insulated outer wall under fully developed laminar
156 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN Fig. 7 Comparison of experimental data with theoretical Fig. 8 Comparison of experimental data with theoretical predictions for 7*- I predictions flow conditions, taking the temperature dependency of 2pQgproHTw-To) [fT ^ 7N / fiw\ viscosity and density into account. (1 -k2)Mw L V Po ) X (-h+76) -77-79-(l--^)(/8-/9)l Furthermore, experimental data were obtained with the two kinds of heat transfer sections of different diameter ratios tc (-2ri/2r0) and it was found that where, theoretical predictions are in reasonably good accord 7iU,^)=2 Z (f-4-VWz with experimental data. hU, x, Gzr) = 2\\[e^)U- -tpid&x Jk Jx \ g / Appendix n rx-j n h(fc) = \ X\JK J1CJK~\ codwd^dl Derivation of Velocity Distribution h{tc,Gzr) = \\[\?a)6{
Nomenclature X X Q){1-&{(D))d(D b = coefficient in viscosity-temperature function [°C~1] Cp = specific heat of fluid > [kcal-kg"1- ^"1] -©(I-0(o)))^te J* J De = equivalent diameter of annulus [m] .F = correction factor [-] Grw = Grashof number =Po2g^DeHTw~T0) //V2 [-] =Kt+«)-iAi-t^r?+(i-^>] G^ = Graetz number =WC,,/£* [-]
VOL.2NO.2 1969 157 GzL Graetz number = WCp/kL [-] = dimensionless temperature = (Tw- T)/ (Tw-To) g acceleration of gravity [m-hr" 2] [-] hm [kcal-m-2-hr- 1-°C-1] = diameter ratio =2rt/2r0 I's meamheat transfer coefficient - dimcnsicnlcss radial coordinate where du/dr=0 [-] k denned by Eqs. (A4)~(A12) [-] = fluid viscosity [kg-rrT^ hr" 1] L thermal conductivity of fluid [kcal-m^-hr-1-0^1] =inlet fluid viscosity [kg-m^-hr"1] Le Num = length of heat transfer section = fluid viscosity evaluated at wall temperature P =length of insulated entrance section qave = mean Nusselt number [kg-m^-hr"1] Rew [kgà"m dimensionless radial coordinate -rlr0 [-] = fluid pressure p
r [m] [-]hr"2] = average heat flux [kcalhr" 1] à" m fluid density [kg - irf3] = Reynolds number =Deupo/^w [-] r% - radial coordinate Po inlet fluid density [kg-nT 3] To = outside radius of inner tube [m] inverse dimensionless velocity gradient at the inner T =inside radius of outer tube [m] wall [-] To [m] kso = inverse dimensionless velocity gradient at the inner Tw = fluid temperature [°C] u =inlet fluid temperature wall for isothermal flow [-] = inner wall temperature [°C] u = fluid velocity [°C] W = mean fluid velocity [m-hr" 1] Literature cited = massflow rate [m-hr" 1] = axial coordinate [kg-hr- 1] = coefficient of thermal expansion 1) Bird, R. B., Stewart, W. E. and Light foot, E. N. : Transport Cm] Phenomena, pp. 51~54, John Wiley and Sons, Inc., NewYork = gammafunction =0.89298 (1960) = defined by Eq. (5) 2) Carpenter, F. G., Colburn, A. P., Schcenborn, E. M. and r å £å = kz/pCpUTo2 [-] = kL/pCpUTo2 Wurster, A. : Trans. Am. Inst. Chem. Engrs., 42, 165 (1946) = defined by Eq. (9) [-] 3) Hatton, A. P. and Quarmby, A. : Int. J. Heat Mass Trans- å [-] fer, 5, 973 (1962) [-] 4) Viskanta, R. : Argonne National Laboratory, Argonne, Illi- [-] nois, Report ANL-6441 (1961)
KINETICS OF THE HYDROGENATBONOF MESITYL OXIDE s ANALYSIS OF THE REACTION MECHANISM BY NONLINEAR LEAST SQUARES METHOD*
KENJI HASHIMOTO, KEIICHI TSUTO**, KAZUHISA MIYAMOTO***, NOBORU HASHIMOTO*, NORIHIRO GOTOn, TOSHIO TADA, AND SHINJI NAGATA Dept. of Chem. Eng., Kyoto University, Kyoto, Japan
A systematic approach to determining the kinetics of multiple reactions catalyzed by solid catalyst is demonstrated for the hydrogenation of mesityl oxide. First, obviously inadequate models were eliminated by a check of the initial rates, and secondly the precise parameters were evaluated by a nonlinear least squares method. It was found that this reaction is a consecutive reaction consisting of two steps in both of which reaction occurs between the dissociated hydrogen atoms adsorbed on the surface of the catalyst and the adsorbed molecules of the organic substances.
Introduction numberof parameters in the Hougen-Watsonrate equa- tions is large compared with experimental data points, Recently the determination of parameters in kinetic and the correlations amongthe estimated parameters are equations has been carried out by a nonlinear least usually high. In addition, for multiple reactions the squares method2'4>6). A large number of rate equations reaction scheme must be determined simultaneously. On for vapor-phase catalytic reactions have been successfully the other hand, the numberof iterations for convergence, represented by models of the Hougen-Watson type. In the converged values of parameters and the sum of this case, however, the estimation of parameters by the squares of residuals are strongly dependent on the nonlinear least squares method, is difficult, because the starting values of parameters5\ For these reasons, prior to applying the technique of nonlinear estimation it is Received on September 30, 1968 desirable to select several possible reaction models and Central Reseach Lab., Kao Soap Co., Ltd. Dept. of Pharmaceutical Engineering Chemistry, Osaka to estimate approximate values of unknownparameters. Univ. The purpose of this paper is to demonstrate the Japan Gasoline Co., Ltd. method of determining the mechanism of complicated Niihama Technical College. catalytic reactions. The reaction studied is the hydro-
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