Mass Transfer Characteristics in a Channel with Symmetric Wavy Wall for Steady Flow

MASS TRANSFER CHARACTERISTICS IN A CHANNEL WITH SYMMETRIC WAVY WALL FOR STEADY FLOW

TATSUO NISHIMURA, Yoshiji OHORI, Yoshihiko KAJIMOTO AND YUJI KAWAMURA Department of Chemical Engineering, Hiroshima University, Higashi-Hiroshima 724

Key Words: Mass Transfer, WavyChannel, Leveque Theory, Electrochemical Method, Laminar Flow, Turbulent Flow

Mass transfer characteristics in a channel with symmetric wavy wall were investigated by the Leveque theory and the electrochemical method. The channel used has a geometry similar to that of the Oxford membraneblood oxygenator. The flow regime covered ranged from laminar to turbulent flow. The local Sherwood number distributions indicate that the reversed-flow region significantly differs from the forward-flow region in masstransfer characteristics. For laminar flow, masstransfer enhancementof the wavy channel is scarcely expected as compared with the corresponding straight channel, but is found to be remarkable for turbulent flow.

Introduction Sparrow et al.11} presented numerical solutions of a fully developed mass transfer at low Schmidt numbers The channel or tube with a periodically converging- in a tube with triangular wavy wall for laminar flow. diverging cross section is one of several devices em- Experiments by the naphthalene sublimation tech- ployed for enhancing the heat and mass transfer nique also verified the numerical predictions of the efficiency of processes having high Peclet numbers, Sherwood number. They found that this tube is not such as in compact heat exchangers, electrodialysis an attractive enhancementdevice for mass transfer and membraneblood oxygenators. under laminar flow as compared with the correspond- There have been several studies of heat and mass ing tube with straight wall. Furthermore, Mendeset transfer in such a tube or channel. Chowet al2) first al.6) used this kind of tube for turbulent mass transfer presented the effects of constriction of tube or channel and recognized a large enhancement in contrast to the with sinusoidal wavy wall on heat trasnfer at the case of laminar flow. entrance region in a fully laminar flow by a per- As mentioned above, most heat and mass transfer turbation method. The solutions are limited to high studies are limited to the laminar flow regime. Also, Prandtl number at low Reynolds numbers, and the the relationship between fluid flow and heat or mass length of wall constriction is large compared to the transfer is not well understood for lack of detained meanradius of tube or the meanheight of channel. flow characteristics. We9)previously investigated flow They indicated that the thermal entry length changes characteristics such as flow pattern, pressure drop and very little from the corresponding tube or channel wall shear stress in a channel with symmetric sinu- with straight wall. Saito et al.10) also studied laminar soidal wavy wall, which has a geometry similar to heat transfer at low Prandtl numbers for the same that of the Oxford membrane oxygenator of geometry by a finite difference method, and they Bellhouse et al.l) The flow regime covered ranged obtained results similar to those of Chowet al. from a fully developed laminar flow to turbulent flow. Fedkiw et al?A) analyzed mass transfer at high In this study we analytically and experimentally Peclet numbers for creeping flow in sinusoidal wavy obtained the mass transfer coefficients at high Peclet tubes as a model of packed beds. The effective numbers for laminar and turbulent flow in the same Sherwood numbers depended upon two ratios of channel as that used in the previous study,9) and the three geometric parameters: the wave length, mean mass transfer mechanismwas interpreted on the basis radius and wave amplitude. of the flow results. Furthermore, mass transfer performance comparisons between the wavy channel Received April 20, 1985. Correspondence concerning this article should be addressed to T. Nishimura. Y. Ohori is now with Mitsui Petrochemical Co., Ltd., Koga-gun used in this study and the corresponding channel 740. Y. Kajimoto is now with Tokuyama Soda Co., Ltd., Tokuyama 745. with straigth wall were carried out to assess the effec-

550 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN tiveness of the wavychannel as a mass transfer enhancement device.

1. Analysis The mass transfer coefficients for laminar flow may be obtained as the result of numerical solution of the Fig. 1. Geometry of wavy channel. convection-diffusion equation in a similar wayto that described in the previous study.9) However,for high Schmidt number the concentration boundary layer formed on the wall is so thin that even the finest element which is possible to compute is not sufficient to detect the concentration profiles. So the Leveque theory is used to calculate the mass transfer coefficients. This approach is similar to that taken by Fedkiw et al.3) Figure 1 shows a segment of the channel used in this study. The diffusion of streamwise direction is assumed to be negligible, and the velocity profile is taken to be linear near the wall. The local Sherwood number is given as Shx=Hjr{4I^J2j(9®^H^ (1) Fig. 2. Mass transfer models.

The Leveque theory is not valid when flow separation occurs. In the previous flow study, the channel used has the onset of separation at Re=\5. So this theory is limited to Reynolds numbers less than 15. In this Reynolds number range, the concentration boundary layer on the wall is developed in the streamwise direction as shown in model 1 of Fig. 2. However, since mixing eddies becomes a dominant flow structure in the turbulent flow regime (Re > 350), Fig. 3. Details of test section and positions of electrodes. as observed in the previous study, the boundary layer might be destroyed by the eddies. The bound- separated by a distance of 28mm. This is the same ary layer is periodically developed from the reattach- geometry as that for the above analysis. ment point in each channel per one wavelength as The Sherwood numbers were obtained by measur- shown in model 2 of Fig. 2. That is, the average ing the diffusional carrent electrochemically.7) Four Sherwoodnumber which is calculated for a stream- cathodes consisting of nickel-plated brass {L\k= 1, 2, wise length equal to the wave length (hereafter called 3 and 4) were used to determine the effect of the a cycle), has the same value for all cycles. length of the mass transfer section on the cycle- Fromthe above discussion the calculation in this average Sherwoodnumber. These cathodes were lo- study is limited two extreme cases, i.e., low and high cated from the 6th to 9th wave in the lower wavy plate Reynolds numbers because at intermediate Reynolds as shown in the upper part of Fig. 3, where the fluid numbers prediction of the Sherwood numbers is very flow is the fully developed one. The local Sherwood difficult by the Leveque theory and numerical analysis numbers were determined by point cathodes of at the present time. Wall velocity gradients for this 0.9mmdiameter at nine chordwise locations along channel are given in the previous flow study.9) the surface per cycle as shownin the lower part ofFig. 3. These cathodes were imbedded in but isolated 2. Experimental Apparatus and Procedure electrically from the main cathode. The upper wavy The experimental apparatus is the same as that plate, made of nickel-plated brass, was used for the used in the previous flow study.9) The test section anode. The area ratio of anode to cathode was from consists of a pair of sinusoidal wavy plates placed 10.4 to 41.6. symmetrically about the flow axis, with a mean gap of The electrolyte used contained 0.01 N potassium 13mmas shown in Fig. 3. The aspect ratio of the ferri-ferro cyanide and 1.0 N sodium hydroxide. The cross section W/HaYis 15.38. Each wavy plate has an applied electric potential was held constant at amplitude-to-length ratio 2a/X of 0.25 and ten crests 800mV, which provided the diffusional controlled

VOL 18 NO. 6 1985 551 condition. The Sherwood number is related to the at Reynolds numbers higher than 1000, and the difFusional current by assumption of a mixing region between successive cycles is confirmed. Sh = idHJzFCbA@ (2) 3.2 Local Sherwood number The physical properties of the electrolyte presented by Figure 5 shows the local Sherwood number distri- Mackley5) were used in these experiments. butions at low Reynolds numbers, calculated from Measurementswere performed in the Reynolds num- model 1. The Sherwood numbers are oscillatory in ber range 100 to 10,000. However, experiments at nature with decreasing amplitides as the fluid moves Reynolds numbers less than 100 could not be per- awayfrom the entrance of the mass transfer section. formed due to limitations on the operating condition The maximumand minimumSherwoodnumbers per of the experimental apparatus. cycle are located on the minimum and maximum cross sections of this channel respectively, and these 3. Results and Discussion positions correspond to the maximumand minimum 3.1 Average Sherwood number points of the wall shear stress indicated in the pre- Figure 4 shows the relationship between the average vious study.9) The difference between the maximum Sherwood number and the Reynolds number for and minimum Sherwood numbers increases with the different values of length of the mass transfer section. Reynolds number. The solid lines indicate the results calculated from the Figure 6 shows the Sherwood number distribution analysis prescribed above, and the dotted lines are per cycle in the fully developed region at Re= 1830, interpolations between solid lines of model 1 and calculated from model2 as an example. The max- model 2. At low Reynolds numbers, the Sherwood imumand minimumSherwood numbers are located number of model 1 As proportional to the 1/3 ex- on the reattachment and separation points respec- ponent of the Reynolds number and to the -1/3 tively, which is of course predicted from the assum- exponent of the length of the mass transfer section. ptions of model 2 and is contrary to the case in the The same results have been obtained for straight low Reynolds number range as shown in Fig. 5. This channels and wavy tubes studies by Fedkiw et al?) On distribution is similar to that in in-line tube banks the other hand, at high Reynolds numbers the presented by Nishimura et al.8) Symbols in this figure Sherwood number of model 2 is independent of the represent the experimental results. The agreement length of the mass transfer section, which is of course between analysis and experiment is fairly good. included in the assumptions of model 2, and also the The relationship between the local Sherwood num- effect of the Reynolds number is larger than that at ber in the fully developed region and the Reynolds low Reynolds numbers. number is shown in Fig. 7. The Sherwood number at Symbols also shown in Fig. 4 represent the exper- the minimumcross section, where the flow near the imental results by the electrochemical method. The wall is a forward flow, increases monotonously with effect of the mass transfer length slightly appears at the Reynolds number. While at the maximumcross Reynolds numbers less than 350, which is the tran- section, where the flow near the wall is a reversed sition point from laminar to turbulent flow as observ- flow, the variation of the Sherwood number is not ed in the previous study, but not at higher Reynolds monotonous; a significant increment occurs in the numbers. Thus the Sherwood number for even L\X = 1 Reynolds number range 350 to 1000, which cor- can be regarded as the cycle-average fully developed responds to the flow transition indicated in the pre- value in the turbulent flow regime. The Sherwood vious study. Thus it is deduced that the unsteady number at Reynolds numbers higher than 1000 is vortex motion in the turbulent flow regime observed proportional to the 0.6 exponent of the Reynolds in the previous study remarkably affects the mass number, which is low when compared to the 0.8 transfer in the reversed-flow region rather than in the exponent for straight channels in the turbulent flow forward-flow region as well as the behavior of wall regime. This is not unexpected, because the wavy shear stress as shown in the previous study. channel used contains large zones of recirculation not present in straight channels. Such a flow situation 4. Mass Transfer Enhancement has also been observed for tube banks. Nishimura et To examine the enhancement provided by the al.8) presented the correlation of Sherwood number channel with wavy wall, the present average for in-line tube banks, which involves the 0.63 ex- Sherwood numbers were compared with those for ponent of the Reynolds number. Thus it is deduced the channel with straight wall. The condition of com- that the mass transfer in the wavychannel used has a parison is that the height of the straight channel is certain similarity to that in tube banks. equal to the average height of the wavy channel, and Fromthe comparison between analysis and experi- also that the flow rate and fluid properties are ment, model 2 can predict well the Sherwood number equal, i.e., equal Reynolds number. So the incre-

552 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN Fig. 4. Average Sherwood number versus Reynolds number.

Fig. 7. Local Sherwoodnumber versus Reynolds number.

The Sherwood numbers for the straight channel Fig. 5. Local Sherwood number distributions at low Reynolds numbers. needed for the aforementioned performance appraisal are calculated by the Leveque theory as well as those in the wavy channel. The wall shear stresses necessary for the calculation are obtained from the correlations of friction factor conventionally used. However, for the turbulent flow regime (ite>2000), the local Sherwood number for larger values of the mass transfer length considered mayapproach the fully developed values. Therefore, the average Sherwood numbers at Reynolds numbers higher than 2000 are calculated from the local Sherwood numbers at the entrance and fully-developed regions. The following correlations are used for the straight channel: Re<2000 , Sh = 1.HRe1/3Sc1/3(^/L)1/3 (3) Re > 2000 , for entrance region Shx=0.137Re7/12Sc1/3(Xlx)1/3 (4) Fig. 6. Comparison of calculated and measured local Sherwood number distributions. for fully-developed region Shx = 0.0202Re°-8Sc1'3 (5) ment in the mass transfer surface for the wavy channel as compared with that for the straight Figure 8 shows the local Sherwood number distri- channel (1.14 times) is neglected in this study. butions for the wavyand straight channels at Re=10

VOL 18 NO. 6 1985 553 Fig. 9. Comparison of average Sherwood numbers for

wavvanHstraicrlit nhnnnpiQwavyand straight channels. Fig. 8. Comparison of local Sherwood numbers for wavy and straight channels at Re= \0.

as an example. The minimumcross section of the wavy channel enhances mass transfer, but its maximum cross section decreases mass transfer. The overall effects of the minimum and the maximum cross sections upon the average Sherwood number cancel each other out and thus the average Sherwood numberfor the wavychannel is almost equal to that for the straight channel. This result is similar to that of Chow et al.,2) Saito et al.10) and Sparrow et ai11] Accordingly, it may be deduced that the channel or tube with a periodically converging-diverging cross section is not an effective enhancement device at low Fig. 10. ShJShs versus Reynolds number. Reynolds numbers. Figure 9 shows a comparison of the average remarkable for turbulent flow. Sherwood number at LjX = 2. Enhancement is expect- 2) The cycle-average Sherwood number for ed from about Re=350, which is the flow transition laminar flow is proportional to the 1/3 exponent of point for the wavy, channel. The Sherwood number the Reynolds number and to the -1/3 exponent of ratio of the wavychannel to the straight channel is the mass transfer length. For turbulent flow, the plotted against the Reynolds number in Fig. 10. The Sherwood number is proportional to about the 0.6 Sherwood number ratio increases with Reynolds exponent of the Reynolds number and the effect of number up to 2000, which is the transition point of mass transfer length is small. the straight channel, but at higher Reynolds numbers 3) The increment of the local Sherwood numbers its ratio decreases gradually, because both wavy and with Reynolds number for the flow transitional re- straight channels in this Reynolds number range gime is remarkable at the maximumcross section (Re >2000) show tubulent flow behavior. The trend of rather than at the minimumcross section of the wavy enhancement is more significant with increasing value channel. of the mass transfer length. This is effective for the The results obtained here correspond to a specific enhancement device. geometry, but give at least qualitative information about the mass transfer characteristics of other wavy Conclusions channels which contain large zones of recirculation. Westudied analytically and experimentally mass In the future, the effect of geometric parameters on transfer characteristics at high Peclet numbers for mass transfer enhancement will be considered. laminar and turbulent flow in a wavychannel having a geometry similar to that of the Oxford membrane Acknowledgment oxygenator of Bellhouse et al. The authors acknowledge with thanks the assistance of Messrs. 1) For laminar flow, mass transfer enhancement T. Yoshino and A. Tarumoto in the experiments. This work was of the wavychannel is scarcely expected as compared supported by the Science Research Foundation of the Ministry of with the corresponding straight channel, but becomes Education, Science and Culture, Japan (Grant No. 5870092). 554 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN Nomenclature X = wavelength [m]

A v = kinematic viscosity [m2/s] [m2]

a area of the mass transfer electrode waveamplitude [m] Cb bulk concentration [kg-mol/m3] Literature Cited 2) molecular diffusivity [m2/s] 1) Bellhouse, B. J., F. H. Bellhouse, C. M. Curl, T. I. MacMillan, F Faraday's constant [C/kg-mol] A. J. Gunning, E. H. Spratt, S. B. Macmurray and J. M. average spacing between wavy walls [m] Nelems: Trans. Am. Soc. Artif. Int. Organs., 19, 72 (1973). h diffusional current [A] 2) Chow, J. C. F. and K. Soda: Trans. ASME, J. of Heat k mass transfer coefficient [m/s] Transfer, 95, 352 (1973). L mass transfer length [m] 3) Fedkiw, P. and J. Newman: AIChEJ., 25, 1077 (1979). Q volumetric flow rate [m3/s] 4) Fedkiw, P. and J. Newman: AIChE J., 23, 255 (1977). Re Reynolds number ( = Havuav/v) 5) Mackley, N. V.: Ph. D. Thesis, Univ. ofAston in Birmingham critical Reynolds number for straight channel (1973). Recs ( =2000) critical Reynolds number for wavy channel [-] 6) Mendes, P. S. and E. M. Sparrow: Trans. ASME, J. of Heat (=350) Transfer, 106, 55 (1984). Recw Schmidt number [-] 7) Mizushina, T.: Advances in Heat Transfer, 7, 87 Academic Sc [-] Press (1977). Sh cycle-average Sherwood number (=kHav/@) [-] 8) Nishimura, T. and Y. Kawamura: Kagaku Kogaku Shs Sherwoodnumberfor straight channel [-] Ronbunshu, 7, 469 (1981); Heat Transfer Japanese Research, Shw Sherwoodnumberfor wavychannel 10, 82 (1981). Shx local Sherwood number 9) Nishimura, T., Y. Ohori and Y. Kawamura: /. Chem. Eng. Sx local wall velocity gradient Japan, 17, 466 (1984). "av velocity based on i/av (=Q/WHav) 10) Saito, T. and Y. Ito: Preprints of 18th National Heat Transfer w width of wavy wall Symposium of Japan, 277 (1981). x distance of streamwise direction ll) Sparrow, E. M. and.A. T. Prata: Numerical Heat Transfer, 6, 441 (1983). z number of electrons exchanged

EQUATION OF STATE BASED ON A GROUP CONTRIBUTION MODEL APPLICABLE TO VAPOR AND LIQUID PHASES AND ITS APPLICATION TO /i-ALKANE SYSTEMS

Shigeki TAKISHIMA and Shozaburo SAITO Department of Chemical Engineering, Tohoku University, Sendai 980

Key Words : Equation of State, Group Contribution Model, Hole Theory, Nonrandom Distribution Model, Vapor-Liquid Equilibrium, Normal Alkane Anequation of state is derived from a group contribution modelwhich takes into account the effect of local molecular distribution on potential energy in terms of the non-random two-liquid (NRTL) model. Model parameters for methyl and methylene groups are determined from the data reduction of saturated properties of n- alkanes. It is shownthat the equation of state correlates well the saturated properties of «-alkanes and successfully predicts the vapor-liquid equilibria of /i-alkane binary mixtures.

molecules of which only the molecular structure may Introduction be known. In particular, it is believed that the de- Group contribution models constitute a method for velopment of group contribution models will make predicting thermodynamic quantities from the prop- prediction possible for polymer systems as well as for erties of the commonstructural units of molecules. systems of molecules for which critical point data are The development of group contribution models will not available. be of advantage to the prediction of the properties of Manythermodynamic quantities can be calculated from an equation of state. There are two ways of Received June 7, 1985. Correspondence concerning this article should be addressed to S.Saito. applying the group contribution models, in relation to

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