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 per- formance 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 en- hancement 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 coef- ficients. 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 stream- wise 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 sepa- ration 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.