235 zyxwvutsr Journal of M embrane Science, 69 (1992) 235- 250 Elsevier Science Publishers B.V.. Amsterdam Mass transfer in various hollow fiber geometries zyxwvutsrqponmlkjihgfed S.R. Wickramasinghe, Michael J. Semmens and E.L. Cussler Department of Chemical Engineering and M aterials Science, University of M innesota, M inneapolis, M N 55455 (USA) (Received July 1,199l; accepted in revised form December 17,199l) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON Abstract Mass transfer coefficients in commercial modules, including blood oxygenators, agree with literature correlations at high flows but are smaller at low flows. The smaller values at low flows probably result from channelling in the hollow fiber bundle. For the special case of flow within the fibers, the slight polydispersity of the hollow fibers causing this channelling can be used to predict deviations from the Ldvbque limit. These deviations can not be predicted from extensions to the L&&que analysis, or the analysis by Graetz. For the special case of flow outside the fibers, the mass transfer coefficients in commercial modules of various geometries are surprisingly similar, and fall below those of carefully handmade modules. These results can be used to develop still better membrane module designs. Keywords: membrane modules; mass transfer; blood oxygenators Introduction Hollow fiber membrane modules promise more rapid mass transfer than is commonly possible in conventional equipment. For example, mass transferred per equipment volume is about thirty times faster for gas absorption in hollow fibers than in packed towers [1,2]. Liquid extraction is six hundred times faster in fibers than in mixer settlers [3-71. This fast mass transfer in hollow fibers is due to their large surface area per volume, which is typically one hundred times bigger than in conventional equipment. However, the improved separations promised by hollow fibers will only be realized if the Correspondence to: E.L. Cussler, Dep. Chem. Eng. and Mater. Sci., University of Minnesota, 151 Amundson Hall, 421 Washington Ave., S.E., Minneapolis, MN 55455-0132, USA. 0376-7388/92/$05.00 larger area per volume is not compromised by a low overall mass transfer coefficient. In general, the mass transfer coefficient is a weighted average of the individual mass transfer coefficients in the feed, across the membrane, and in the permeate. In other words, the speed of the separations is controlled by the overall resistance to mass transfer; and this overall resistance is the sum of the mass transfer resistances in the feed, across the membrane, and in the permeate [ 81. This separation-controlling, overall mass transfer resistance has in the past been dominated by the resistance of the membrane. This was because the permeability of the membrane was low and because the membrane was thick. Extensive research both in industry and academia is changing this picture, producing ultrathin composite membranes with a much smaller mass transfer resistance. Parallel research on 0 1992 Elsevier Science Publishers B.V. All rights reserved. S.R. W ickramasinghe et al,/J. M embrane Sci. 69 (1992) 235- 250 zyxwvutsrq 236 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA microporous membranes has adjusted the pore size and membrane hydrophobicity, again yielding a much smaller mass transfer resistance. These reductions in membrane resistance are broadening the focus of membrane separations to include the resistances in the feed and permeate. This expanded focus has produced some design relations describing these resistances [ 5,9,10]. These design relations, commonly expressed as mass transfer correlations, allow development of better hollow fiber modules than those prototypes made when hollow fibers were first available. The mass transfer correlations often find support in corresponding heat transfer relations: the Sherwood and Nussult numbers will vary in the same way with Reynolds number, for example. More recent hollow fiber modules have explored new geometries where neither mass transfer correlations or heat transfer parallels exist. These new modules often outperform earlier designs, a tribute to their inventors’ intuition. Their development has been especially striking in the modules developed for blood oxygenation. These modules, the essential part of the heart-lung machines used in cardiac surgery, achieve extremely high mass transfer per unit volume. Such performance reduces the need for blood transfusions, and hence the risk of accidental infection from blood contaminated with HIV or hepatitis. The goals of this paper reflect both the substantial potential of hollow fiber modules and their accelerating development as blood oxygenators. These goals are focussed by three questions: (1) What mass transfer correlations are most accurate? (2) Which available hollow fiber geometries perform best? ( 3 ) Which new membrane geometries have the greatest potential? Clearly, the first question is easiest to answer, and the last is more speculative. The first question, the reliability of mass transfer correlations, has its basis in engineering science. There, a century of theoretical effort has produced sound theories for heat transfer which closely parallel mass transfer [ 11-131. These theories predict results in hollow fibers where they are applicable. In some important cases, they aren’t. For example, there is no theory for the effect of polydispersity of hollow fiber diameters. There is no theory for helically arranged hollow fibers. Both these situations are important practically. The second question, of which module geometry is best, will be seen to depend strongly on the final use of the module. For blood oxygenators, “best” means the most mass transferred per volume. For antibiotic extraction, “best” means the most mass transferred per dollar. We will explore how these different uses can influence module design. Finally, the third question asks how mass transfer operations are best accomplished. In the past, we accepted the fluid interfaces in packed towers or countercurrent extractors or distillation columns because we had no choice in the shape of the fluid-fluid interface. To be sure, we could get somewhat better mass transfer by replacing conventional packing with stacked, structured packing, but we still were constrained by loading and flooding. Now, we can use membranes to get the shape of interface which makes mass transfer best. We can have any shape of fluid-fluid interface which we want. But what do we want? We will start to answer these questions in this paper. We begin in the theory below to explore deviations of mass transfer in actual modules from the accepted theories of mass transfer. We then describe commercially available modules of various geometries, and report the mass transfer correlations which describe their performance. Finally, we discuss the relative per- 231 zyxwvutsr S.R. W ickramasingke et al./J. M embrane Sci. 69 (1992) 235- 250 formance of these modules, and suggest how This distribution function allows estimation of further improvements can be made. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC the various averages which appear in eqn. (1) : co Theory R= As mentioned above, we usually describe mass transfer in hollow fiber modules with an overall mass transfer coefficient, which is the reciprocal of an overall mass transfer resistance. This overall resistance is the sum of three individual resistances, that inside the fiber, that across the membrane, and that outside the tibers. Each individual resistance is in turn proportional to the reciprocal of an individual mass transfer coefficient. Often, one of the three individual coefficients will be much smaller than the other two, and hence dominates the overall mass transfer coefficient. In this section, we want to discuss the effect of polydisperse hollow fiber diameters on a dominant individual coefficient, and hence on the overall mass transfer coefficient. In particular, we choose the case where mass transfer in the fiber lumen dominates performance. We choose this case because it is easily described mathematically, though we recognize that other cases are also important. To explore this special case, we begin with the basic equation used to calculate the average mass transfer coefficient ( k) : I (3) rgdr 0 cc V=nl s Q= z~r2vgdr 0 (5) c-2 HAP = w I r4gdr 0 where 1is the module length, assumed constant for all fibers; Ap is the pressure drop through the fibers; and p is the feed viscosity. Note that eqn. (5) implies that the velocity u is given by the Hagen-Poiseuille law, i.e. that the flow is laminar. That will always be true here. Finally, the concentration (c) is given by (c) = nTr2ucgdr/Q (6) =i r4cgdr/ 4 r4gdr 0 (k)= $ln$ s 0 0 But the concentration in one fiber is given by in which R is the average fiber radius, Q is the average volumetric flow through a fiber lumen, V is the average volume of this lumen, co is the inlet solute concentration, and (c) is the average, “ cup-mixing” , concentration coming out. We now assume that the fiber radii are not all equal, but vary according to a distribution function g, defined so 03 gdr=l (4) zyxwvutsrqp r’gdr 0 (2) C -=e -2kl/ru (7) CO the analogue of eqn. (1) for a single fiber. When we combine eqns. (6) and (7 ), we obtain - cc> O” = Cl3 5 Co r4g&. r4e --6k~l~/&r~~&./ 0 (8) s 0 We must now evaluate these integrals. To do so, we assume that the distribution of radii is Gaussian: 238 S.R. Wickramasinghe et al./J. Menbrane Sci. 69 (1992) 235-250 zyxwvutsrqp uneven spacing is not as well known. Still, we can begin by assuming channels between the fi(9) bers which can be characterized by some apparwhere R, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA is an average radius and (Roe, ) 2 is the ent radius r. We can further assume that the variance of this distribution of radii. We now velocity within these channels is proportional can use this to show from eqns. (3) that R to r2, consistent both with the Blake-Kozeny equals R,. In this, we also assume that because equation and the Hagen-Poiseuille law. We can E,<< 1zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA the integration from zero to infinity is then parallel the analysis above to fined essentially that from minus infinity to infinity. Similarly, from eqn. (4)) (k) =k[l-(z,/m++:+...] V=dR; (l+$) (10) (15) and from eqn. (5 ) , Q= %[1+6r;+3$] (11) The average concentration given in eqn. (8) is more difficult to calculate analytically. As a convenient approximation, we expand the exponential in eqn. (8) as a power series in e,: e - 16kjdz/Apr3 = (1_6E~+...)e-16k~‘/2/A~R~ We then find cc> -= c0 Xe- lGk/@/ApR: (13) Finally, inserting this into eqn. (1)) expanding the logrithm as a power series, and combining with eqns. (10) and (ll), we find in which V is now the total module volume, Q is the total flow in the module, R, is the average channel radius, and E’ is the void fraction of fibers in the module. Details of this analysis are given elsewhere [ 141. While we recognize that eqn. (15) rests on imperfect assumptions, we are struck by the prediction that at low flow, the zyxwvutsr (12)average mass transfer coefficient (k) falls below k, that for a perfectly spaced fiber array. As a result, we expect that modules with uneven fiber spacing will show smaller mass transfer coefficients than modules with very exact spacing. We will discuss this point more fully in connection with the experiments, which are described next. Experimental (k)=k[l-(E +T}e:+...] (14) At low flow Q, the average mass transfer coefficient (k) will be less than the value k expected in one fiber. Thus, polydisperse hollow fibers produce uneven flows which in turn reduce the average mass transfer coefficient in the hollow fiber module. We can also analyse the effect of unequal flows outside the hollow fibers caused by unevenly spaced fibers. Such an analysis is more speculative than that given above because the The chemicals and procedure closely imitated those of earlier experiments, and so are described only very briefly [2,9,X]. Basically, water saturated with oxygen was pumped through the particular membrane module under study, using a FM1 Fluid Metering Inc. model RP-D high pressure liquid chromatography pump. Water saturated nitrogen under 10 psi flowed countercurrently to the liquid water. Oxygen concentrations in and out of the 239 zyxwvutsrq S.R. W ickramasinghe et al./J. M embrane Sci. 69 (1992) 235- 250 module were measured using a Orion model 9708-00 oxygen specific electrode. Nine different membrane modules were used in this work. All the modules used microporous polypropylene membranes; six used hollow fiber membranes and three used a crimped flat membrane. Three of the hollow fiber modules were of a shell-and-tube design, shown schematically in Fig. 1 (a). Two modules (HoechstCelanese model numbers 5010-8010 and 50108020, Charlotte, NC.) used fibers of 240 pm internal diameter, 30 pm wall thickness, 0.05 pm pore size, and 30% void fraction. The model 8010 contains 7500 fibers with an effective length of 18.4 cm; the model 8020 contains 12,500 fibers with an effective length of 24.8 cm. The third shell and tube module, of 1.0 m2 area, used a fabric made of these same hollow fibers as the warp, and with 26 pm nylon thread as a weft. This fabric was made in an attempt to reduce the channelling on the shell side of the module. Three of the modules used were commercial hollow fiber blood oxygenators, which differed (a) Flow Inside or Outside and Parallel primarily in the way in which the hollow fibers were arranged. The first used a helically wound bed, shown rchematically in Fig. l(b) (Medtronic “Maxima”, Anaheim, CA). This module contains 2800 fibers 48 cm long and 400 pm in diameter, with a wall thickness of 30 pm. The second uses a cylindrical bed of fibers, shown schematically in Fig. 1 (c) (Sarnes/3M model 16310, Ann Arbor, MI). This unit has 11,000 fibers, 10 cm long, with an internal diameter of 240 pm. The fiber in these units is also made by Hoechst-Celanese. The third blood oxygenator used a rectangular bed of hollow fibers, shown schematically in Fig. 1 (d) (Bard model William Harvey HF-5000, Billerica, MA; manufactured by Minntec, St. Paul, MN). This module contains 32,400 fibers, 13 cm long, with an internal diameter of 220 ,um and a wall thickness of 25 pm. These fibers, manufactured by Mitsubishi, give the same mass transfer performance when used under the same module geometry. However, modules with different geometry perform differently, as shown in the next section. (b) Flow Across a Helically Wound Bundle (e) Flow Along a Crimpled Flat Membrane Gas Out Water In Water Out 4- Wate In 4 Gas Out =+Gas Behind Membrane Water out (c) Flow Across a Cylindrical Bundle (d) Flow Across a Rectangular Bundle Gas out zyxwvutsrqponm Water Out ;t , Gas In +Gas Out ‘t’ Water In Fig. 1. Schematic drawings of the modules used. Three modules have the form in (a); one has the form in each of (b), (c), and (d); and three have the form in (e). Sources of the modules are given in the text. 240 S.R. Wickramasinghe et aL/J. Membrane Sci. 69 (1992) 235-250 zyxwvutsrq fined Sherwood numbers, defined as ( (iz) d/ The final three modules studied, which are D). The Sherwood numbers found from these also blood oxygenators, use a flat crimped experiments are plotted in Fig. 2 vs. Graetz membrane, shown schematically in Fig. 1 (e) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED numbers, defined as ( d2u/DZ). Data from four (Cobe Cardiovascular Inc. models EXCEL modules are shown in this figure. One set - the (050-123-000), ULTRA (150-120-000) and VP open circles - are from a module with very carePLUS (050-125-000)) Arvado, CO). These fully aligned and spaced hollow fibers [91.Two modules use flat Hoechst-Celanese memsets - the open and filled squares - are from branes, crimped to form blood channels shaped commercial modules with less carefully aligned like isosceles triangles with a base of 150 pm fibers which are much more tightly packed (Fig. and a side of 4 cm. These modules differ only la). The final data set - the open triangles in the number of channels, and hence in the are from a rectangular bed of hollow fibers (Fig. membrane area. Mass transfer results for all Id). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO these modules are given in the next section. The data for all four modules in Fig. 2 agree closely with each other. This agreement exists Results even though the data for the rectangular bed uses hollow fibers made by a different supplier In this paper, we report measurements of (Mitsubishi) than those for the three other mass transfer in six different types of memmodules (Hoechst-Celanese) . This agreement brane modules made by five different manufacsupports the contention that oxygen mass turers. In this section, we report the experitransfer is controlled by diffusion in the water, mental values, and emphasize differences and is unaffected by diffusion across the membetween the data. In the following section, we brane or on the shell side of the module [15]. discuss correlations inferred from these experMoreover, the data in Fig. 2 agree closely with iments, and contrast these with literature data the theoretical prediction of LQv$que [16], at wherever similarities exist. For convenience, we least at high Graetz numbers. Indeed, mass organize the report in this section under four transfer coefficients rarely agree with theory as geometries: exactly as in Fig. 2. (1) flow inside the hollow fibers; However, at low Graetz numbers, the Sher(2) flow outside and parallel to the hollow fibers; zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1OOL I (3) flow outside but across the hollow fibers; and (4 ) flow across crimped flat membranes. Results for modules with each of these geometries are discussed in detail below. Flowinside hallow fibers 0.1 In these experiments, water saturated with oxygen is pumped through the hollow fibers, and the space outside of the fibers is rapidly flushed with water-saturated nitrogen. Mass transfer coefficients calculated from the reduced oxygen concentrations are then used to 1 Graetz’N”mber 100 1000 zyxwvutsrqpon Fig. 2. Oxygen mass transfer out of water flowing within hollow fibers. Both the Sherwood and Graetz numbers are based on the internal fiber diameter d. (0 ) handmade, shell-and-tube module; ( 0,~) commercial shell-and-tube modules; and (A ) rectangular bed of hollow fibers. (-) LM que limit, (- - - - ) correction calculated from eqn. ( 14). S.R. W ickramasinghe et al./J. M embrane Sci. 69 (1992) 235- 250 241 zyxwvutsr wood numbers in Fig. 2 deviate from the theoretical prediction, even though they remain consistent with each other. Such deviations are expected, because LQvgque’s theory assumes that the oxygen concentration at the center of the fibers is unchanged [16]. Obviously, this will not be true for slow flows or long tubes, i.e. for low Graetz numbers. When the concept of Fig. 3. Oxygen mass transfer out of water flowing on the a thin diffusion layer close to the wall of the shell side of shell-and-tube modules. Both the Sherwood fiber is no longer valid, the Ldvgque solution is and Graetz numbers are based on the equivalent diameter expected to break down. Newman [17] pred.. (0,O) refer to the shell-and-tube module models 8010 sents an extension to the LQveque solution by (7500 fibers) and 8020 (12000 fibers) respectively. ( 0 ) refer to the module made with a hollow fiber fabric. (-,- considering the terms neglected by LQvgque. - - -) refer to the correlations of Yang [9] and Prasad [ 71, Naturally when the diffusion boundary layer respectively. reaches the center of the tube, Newman’s extension too is no longer valid. In these cases, the oxygen concentration in water flowing on one must use the more rigorous Graetz solution the shell side of modules like those in Fig. 1 (a). [ 18-201. Water-saturated nitrogen flows rapidly through Surprisingly, the deviations of the Sherwood each fiber’s lumen. Unlike the data for flow numbers from the theory are in the opposite within the fibers, the results for the three moddirection to the predicted improvements to the ules studied are different. Those for module LtMque solution. In other words, at low flows, # 8010, which has 7500 fibers, give mass transthe experimental results fall below the solid line fer coefficients about three times higher than in Fig. 2, but the Graetz solution lies above the those for module f8020, which has 12,500 filine. We believe that the deviations from theory bers. While the different number of fibers seems in Fig. 2 are caused by the slight polydispersity too small to cause such a large difference, the in the hollow fiber diameters, and not by other fibers in module # 8010 are potted in a dumblimitations of the LdvCque analysis. To test this bell shaped shell which might facilitate flow belief, we measured the diameters of individual around the fiber ends and reduce channelling. hollow fibers for module # 8020, and found that However, the data for module #8020 agree these diameters showed a standard deviation of closely with earlier correlations of similar mod5%. We then used eqn. (14) to estimate the ules containing 16,120, and 300 fibers [ 91. All change in (k). The result of this estimate, of these results show mass transfer coefficients shown as the dotted line in Fig. 2, is in reasonvarying almost linearly with velocity. In other able agreement with the data for all the modwords, all show Sherwood numbers proporules. Similar deviations have been observed by tional to Graetz numbers. others, for example Prasad and Sirkar [7,21] The data for the hollow fiber fabric module, and Zander et al. [ 221. Analogous deviations shown as squares in Fig. 3, seem more consishave also been observed in heat transfer tent with the result of module # 8020 than with [23,24]. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA those of module #8010. Interestingly, however, these data seem to show a smaller variaFlow outside and parallel to hollow fibers tion of Sherwood number with Graetz number. In the experiments, shown in Fig. 3, mass This smaller variation is consistent with one transfer coefficients are found from changes in earlier, careful study of mass transfer in this type of module [21] and with that expected performance is similar to that of crossflow heat from heat transfer in shell-and-tube heat exexchangers [ 91. changers [ 11,121. As it is easier to avoid chanThe close agreement of the Sherwood numnelling with a few dozen heat exchanger tubes bers in Fig. 4 suggests that the flow within the than with thousands of hollow fibers, we are modules must be similar, In other words, it sugtempted to attribute this smaller variation to gests that flow through a rectangular bed, flow reduced channelling. We have no quantitative through a helically wound bed, and flow through reason to do so now. Accordingly, we will ema cylindrical bundle all give similar boundary phasize module #8020 in the discussion later layers near the fiber surface, and hence similar in this paper. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA mass transfer coefficients. This seems inconsistent with experiments which claim that there is an optimum angle for winding helical modFlow outside and across hollow fibers ules [25]. The result certainly merits much more experimental attention, especially since similar modules with non-porous fibers are These experiments used the modules in Fig. strong candidates for gas separations. 1 (b zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ) , 1 (c ) and 1 (d) , and so involve somewhat We have also made preliminary experiments different flows. However, the Sherwood numfor flow across a mat made of the hollow fiber bers for these modules, shown in Fig. 4, agree fabric. We have not shown these data in Fig. 4 closely. All vary roughly linearly with Reynolds both because doing so crowds the figure and benumber at low Reynolds numbers, i.e. at low cause we plan to describe these results in much flow. All seem to approach a variation with the more detail in a later paper. Nevertheless, we 0.33 power of Reynolds number at higher flows. are sure that mass transfer coefficients for the This 0.33 power is consistent with the variahollow fiber fabric fall very close to those of the tions observed for flow perpendicular to 300 handbuilt module, and thus are dramatically carefully spaced fibers in a hand made module. higher at low flows than those for the commerHowever, the data for these modules seem to cially built modules. The implication is that the approach Sherwood numbers about half of hollow fiber fabric has more monodisperse voids those observed in the handmade module, whose than the commercial modules, and hence has less flow channeling. loo,,,.,,,, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Flow along flat, crimped microporous membranes Finally, we report mass transfer coefficients for crimped flat membranes made of the same membrane material as the hollow fibers used for the results in Figs. 2-4. The use of flat memFig. 4. Oxygen mass transfer out of water flowing across branes may seem out of place in work emphahollow fibers. The Sherwood and Reynolds numbers are sizing hollow fibers. However, the crimped, flat based on the outer fiber diameter. (A&,0) flow across membrane is basic to a commonly used blood fiber bundles that are rectangular, he&al, and cylindrical, oxygenator which is often described as the easrespectively (cf. Fig. 1) .zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (-) correlation based on carefully spaced, handmade modules [9]. iest to use clinically. As a result, we felt that it 243 zyxwvuts S.R. W ickramasinghe et al./J. M embrane Sci. 69 (1992) 235250 lated by plots of Sherwood number vs. Graetz or Reynolds number. In many cases, the correlations combine results for modules whose geometry would seem to vary significantly. But the very success of these correlations raises other questions. Two seem especially major: (1) How do the correlations obtained here J 1 I *c* compare with other, earlier efforts? 1 10 100 Graek Number zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (2 ) Which module geometries offer the fastest mass transfer? Fig. 5. Oxygen mass transfer across a crimped, flat memWe will begin to answer these questions in this brane. The Sherwood and Graetz numbers are defined as final section of this paper. (k)b/D and b’Q/DV. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (O,O,El,A) modules with membrane areas of 0.40,0.85,1.25, and 3.0 m*, respectively. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE 1 Correlations merited the same effort which we spent on the other units. In Table 1, we compare the mass transfer The data for the three flat membrane modcorrelations obtained here with those obtained ules studied agree closely with each other as earlier. Such comparisons should be made caushown in Fig. 5. Two points about this appartiously, for some aspects of correlations like ent agreement merit emphasis. First, Fig. 5 plots these have been much carefully studied than the Sherwood number vs. the Graetz number, others. We have tried to signal this caution in as Figs. 2 and 3 do; but these dimensionless the organization of the table. The first column groups are now differently defined. Both groups in the table gives the basic geometry studied. are now written in terms of the channel width The second column gives the range of flows, exb, rather than the fiber diameter d. Thus the pressed as a Graetz or a Reynolds number. The Sherwood number is written as ( ( It) b/D). The Graetz number is defined as (d2u/DZ) for flow Graetz number, now defined in terms of the toinside the fiber and as (d,2u/DZ) for flow outtal flow per module volume is written as ( b2Q/ side and parallel to the fibers. The Reynolds D V). The second point about the data in Fig. 5 number is defined as (du/v), except as indiis that the four modules reported were actually cated. For the flat membrane, the Graetz numthree separate units. One large module had a ber is defined as b2Q/D V. The third column in surface area of 3 m2; and an intermediate sized Table 1 gives the mass transfer coefficient vs. module had a surface area of 1.25 m2. The those variables which are actually altered in the smallest module, intended for pediatric use, experiments, and the fourth gives the dimencould be operated with an area of 0.4 m2. By sionless correlation inferred from this variausing different parts which use different memtion. The difference between these columns is branes, we could also operate the smallest modimportant. While both are consistent, the ule with an area of 0.85 m2. Thus the results in fourth often contains implicit assumptions. For Fig. 5 are for three modules, one of which was example, in the easiest case of fast flow inside operated in two different ways. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA the fibers, column (4) asserts that the mass Discussion The data in Figs. 2-5 show that mass transfer in these modules can be effectively corre- transfer coefficient (k) varies with the two thirds power of the diffusion coefficient D. We believe this assertion is correct, because it is consistent with theory and with experiments of TABLE 1 Mass transfer correlations for hollow fiber modules of varying geomet$ Flow geometry Flow range Experimental resultb Inferred correlation Literature correlation Remarks Flow inside fibers Gr>4 (k) =4.3x lo-sfu/lj”s Sk= 1.62G+ Sh= 1.62Gr”3 Grt4 (k) =1.5x 10-4(u/l) Theory and experiment don’t agree at low flows, apparently because of slight polydispersity in hollow fiber diameters (cf. eqn. 14). Flow outside and parallel to fibers” Gr<GO (k) Flow outside and across fibers Re> 2.5 (k)=8.1~10-‘u”.s Re<2.5 (k) Gr> 11 <k) =O.O025(Q/A)‘= Flow along a crimped flat membran&’ =2.5x10-% Sh=Sk&{T Sh=0.019Gr’.0 +7}t;+...] Sk=1,25 !& ( Gr<ll VL 0.93(V)1/3 > Sk = 0.39Re0?W’~s3 =~.OX~O-~U D The correlation obtained here, which agrees with the earlier result, can be written either vs. Graetz number or vs. Reynolds and Schmidt numbers. The values of (k) are less than those for well spaced fibers, but increase more with increasing velocity. Sh=6.0Gr0.35 The results at high flow agree closely with the literature correlation, but those at low flow do not. Sh= 1.25G? <k) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA =3.O(QlA) “Dimensionless groups are defined for the hollow fiber modules as follows: Sherwood numbers Sh= (k)d/D, Graetz number Gr=d%/DI; Reynolds number Re=du/v; Schmidt number SC= v/D. Note that d is the fiber diameter except as indicated. %nite are k: m/set; u: m/set; 1:m; Q: m3/sec; A: m2. ‘The characteristic length for this geometry is the equivalent diameter 4, equal to four times the cross section for flow divided by the wetted perimeter. dThe characteristic length for this geometry is the crimp length b; the Graets number is defined as b*Q/DV. S.R. W ickramasinghe et al./J. M embrane Sci. 69 (1992) 235- 250 others [6,9,10]; but we have not examined this assertion in the experiments reported in this paper. The fifth column in Table 1 lists correlations reported in earlier literature effects [7,%171. We now want to discuss each of the four geometries in Table 1 in more detail. For fast flow inside of the fibers, our results are consistent both with the theory of LQvQque,and with earlier experiments by us and others. Indeed, this correlation is so well established that the observed consistency seems more a justification of our experimental procedure than a new verification of this established result. In contrast, for slow flow inside the fibers, the observed mass transfer coefficients fall significantly below the accepted correlation, shown as the solid line in Fig. 2. As explained in the results section above, we believe that this is due to unequal diameters of the hollow fibers. Predictions based on this hypothesis, summarized by eqn. (14)) seem consistent with our experiments. (We urge caution in applying this theory quantitatively, for it is based on a Taylor series expansion. ) This result was for us unexpected, especially since it is in the opposite direction to other theoretical corrections for mass transfer out of a cylinder [ 17,201. It seems a disadvantage of hollow fiber modules. We can explain this effect in qualitative terms by imagining a module with only two hollow fibers of equal length, one of which has twice the diameter of the other. The big fiber will carry sixteen times the flow and have half the residence time as the small one [assuming, as does eqn. (5 ), that the pressure drop applied across both fibers is the same]. The big fiber will allow less mass transfer than expected from a correlation based on an average fiber diameter, equal to half the sum of the two diameters. Thus the apparent mass transfer coefficient calculated from eqn. (1) will be less than that theoretically estimated from the average fiber diameter. 245 zyxwvutsrq The results for flow outside of and parallel to the hollow fibers are much less conclusive, probably because the chance of channelling along the axis of the fiber bundle is so great. Our results do support the near-linear variation of mass transfer coefficient with fluid flow observed in some earlier studies [9,15,21]. Our results do not explicitly investigate the variation with void fraction because the modules studied use close-packed fibers [ 5 1. After reflection, we believe that correlating the results vs. Graetz number makes more sense than correlating them with some product of Reynolds and Schmidt numbers. This essentially presumes that the Sherwood number varies little with changes in viscosity, which will be true if the velocity profile is quickly established within the module. For example, the Sherwood number for mass transfer inside the fibers varies only with the Graetz number, and is independent of the viscosity. However, our belief that the Graetz number based correlation is preferable has not been experimentally scrutinized. The third geometry, involving flow outside of but perpendicular to a fiber bundle, is more interesting because it gives faster mass transfer. Not surprisingly, this geometry is that frequently chosen for blood oxygenators. The Sherwood numbers for the modules studied in this case are about half those of handmade modules with precisely spaced fibers [9]. At high Reynolds numbers, they may be approaching this handmade limit; but at low Reynolds numbers, they drop further, showing a near linear variation with Graetz number. This change is reflected in the correlations given in Table 1. At the same time, our preliminary experiments on hollow fiber fabric did agree with the values for the handbuilt modules at both high and low flow. We are pleasantly surprised that the same correlation works reasonably well for the three S.R. W ickramasinghe et al./J. M embrane Sci. 69 (1992) 235- 250 zyxwvutsrq 246 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA designsshowninFigs.l(b),l(c),andl(d).At the same time, we are disappointed that the Sherwood numbers at low flow lie below the correlations observed at high flow. We are unsure why this is so. One possible cause is the somewhat polydisperse channels between fibers, which cause uneven flows and lead to reduced mass transfer coefficients, as suggested by eqn. (15). This hypothesis is consistent with the hollow fiber fabric results which have more regular channels. However, we are much less confident that eqn. (15) is as useful as the result for flow inside the fibers, given by eqn. (14). Both equations assume that variations in the flow channels follow a Gaussian distribution. This is justified by measurements of internal fiber diameters, and hence is reasonable for eqn. (14). This Gaussian assumption is a speculation for gaps between fibers, and hence for eqn. (15). Moreover, a larger internal fiber diameter produces a larger flow which persists for the length of the module. A larger channel between fibers produces a larger flow for only one course of fiber; then this flow must find its way through new gaps in a new course of fibers. While polydisperse gaps between fibers seem a reasonable explanation for the decrease in module performance, we have not proved that this is the cause. The results for the crimped flat membranes also seem to show mass transfer correlations consistent with theoretical expectations at higher flows, dropping to lower values at low flow. In particular, the results in the triangular channels of these modules seem to give Sherwood numbers varying with the cube root of the Graetz number at Graetz numbers above eleven. This is consistent with the results expected for flow in a slit. The Sherwood numbers vary more linearly with Graetz number at lower flows. Reassuringly, the data for all modules, whose membrane area varies ten times, appear to fit the same correlation. Performance at equal flow per membrane area We now turn from the correlations inferred from this work to consider which module designs offer faster mass transfer. As we have discussed elsewhere, our considerations must include the choice of a basis for this comparison [ 26,271. In this paper, we give results for two choices: performance at constant flow per membrane area, and performance at constant flow per module volume. The former choice is better for those who want to use membrane modules for industrial separations. The latter is more appropriate for those designing blood oxygenators. We consider each choice below. To compare modules operated at equal flow per membrane area, we first make a mass balance on the module [8] to find the fraction removed 8 &I- -Cc) ,l_e-<WA/Q zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP (16) CO where A is the total area in the membrane module. This result is the general case of the mass balance given in eqn. (1) for flow inside hollow fibers, where the membrane area per fiber volume is (2/R). We want to maximize mass transfer, and hence the fraction transferred. Because the flow per area Q/A is constant, this means that we want to maximize the mass transfer coefficient (k) . Values of (iz) and 6’ for the specific module geometries used here are compared in Table 2. In this table, we have assumed a flow per area of 0.005 cm/set, characteristic of that used for absorption and extraction in membrane contactors. We have used the fiber size and module dimensions specific to the units studied here. We have chosen the physical properties appropriate for oxygen transfer from liquid water into nitrogen gas. Extensions to other module geometries and other chemical systems can be easily made using the correlations in Table 1. S.R. W ickramasinghe et al./J. M embrane Sci. 69 (1992) 235- 250 247 TABLE 2 Relative performance for different geometries with equal flow per membrane area. The flow per area of 0.005 cm/set is typical of that used in absorption or extraction. All physical properties assume oxygen dissolved in water being transferred across a microporous membrane into rapidly flowing, water saturated nitrogen Flow Module type geometry Inside fibers Outside fibers Parallel to flat membrane Membrane area Bed length Water flow (mx) (cm) (cma/sec) Percent k( zyxwvutsrqponmlkjihgfedcbaZY X 10~%m/sec) removed Shell and tube (Fig. la) 1.0 18 52 4.0 55 Shell and tube (Fig. la) Rectangular bundle (Fig. Id) 2.3 3.2 25 14 116 160 4.0 4.3 55 57 Cylindrical bundle (Fig. lc) Helical bundle (Fig. lb) Shell-and-tube (Fig. la) 1.8 2.0 2.9 10 12 25 90 100 146 8.5 9.8 0.3, 82 86 7 Rectangular bundle (Fig. Id) 3.7 14 185 Crimped membrane (Fig. le) Crimped membrane (Fig. le) 0.4 3.0 25 25 20 150 The results in Table 2 show that modules with water flowing outside of the fibers are usually more effective than modules with water inside the modules. Modules with a crimped, flat membrane perform between these two cases. Beyond these quick generalizations, there are curious subtilties. First, all the modules with water inside the fibers perform almost equally, whether the fibers are in a shell-and-tube or fiber bed geometry. Second, flow outside and across the fibers is at least ten times more effective than flow outside and parallel to the fibers. Third, crossflow modules perform best when the length of the fiber bed is greatest. Because the flow per area is fixed, a deep bed means a small cross-sectional area for flow, a high velocity, and hence a large mass transfer coefficient. The large coefficient in turn means a large fraction of the oxygen removed. Thus the results in Table 2 suggest that better performance will come from modules operated with flow across deep beds of hollow fibers. Such beds will have a higher pressure drop and hence a higher pumping cost. Based on other work [27], we expect that pumping costs will become important for beds with fiber diame- 19 6.3 6.3 98 12 72 zyxwvutsrqpon ters around 200 pm and a membrane cost of $10/m’-yr. Performance at equal flow per module volume The second basis for judging module performance, vs. equal flow per module volume, is more applicable to the design of blood oxygenators. This is because these oxygenators are used in heart surgery, where infection due to transfusions with contaminated blood is a major risk. Maximizing performance per module volume minimizes transfusions and hence risk. In analysing this case, we begin by rewriting eqn. (16)as: 8=1 _ -Cc) =l_e-<k>aviQ (171 CO where a is the membrane area per module volume. To maximize the fraction removed 8 at fixed flow per module volume (Q/V), we want to maximize the product (k) a. In contrast, in the earlier case, we wanted to maximize the mass transfer coefficient (Iz) . The results for equal flow per volume, shown in Table 3, exhibit many of the same character- S.R. Wickramasinghe et al./J. Membrane Sci. 69 (1992) 235-250 248 TABLE 3 Relative performance for different geometries with equal flow per module volume. The flow per volume of 1.0 see-l is typical of that in membrane oxygenators. AI1 physical properties assume oxygen dissolved in water being transferred across a microporous membrane into rapidly flowing, water saturated nitrogen Flow geometry Module type Module volume Module length Water flow (k) a (set-‘) (cm”/sec) (cm) (cm? Percent removed Inside fibers Shell and tube (Fig. la) Shell and tube (Fig. la) Rectangular bundle (Fig. Id) 62 140 180 18 25 14 62 140 180 0.71 0.71 0.80 51 51 55 Outside fibers Cylindrical bundle (Fig. lc ) Helical bundle (Fig. lb) Rectangular bundle (Fig. Id) Shell-and-tube (Fig. la) 150 150 165 240 10 12 14 25 150 150 165 240 1.53 1.81 4.0 0.08e 78 a3 98 7 Parallel to fiat Crimped membrane (Fig. le) membrane Crimped membrane (Fig. le) 70 450 25 25 70 450 0.57 0.63 44 41 istics of the results in Table 2. As before, crossflow modules are most effective, followed first by crimped membranes, and then by shell-andtube modules with flow inside the fibers. Shelland-tube modules with flow outside and parallel to the fibers are least effective under these conditions. The most effective modules among the crossflow devices are those with the greatest membrane area per volume a. A large value of a increases the fraction removed; it also implicitly increases the velocity past the fibers and hence the mass transfer coefficient. We urge caution in concluding that some blood oxygenators are better than others solely on the basis of the results in Table 3. Remember that these results are for oxygen being removed from water and not for oxygen diffusing into blood. Obviously, the choice of a blood oxygenator also depends on factors like clinical convenience and blood damage, factors which are not investigated here. Blood damage in particular may be increased by factors like high shear, factors which also increase mass transfer rates. We can draw more definite conclusions about the importance of membrane properties. In general, we expect the overall mass transfer coefficient K to be a function of the mass transfer coefficients across the membrane hMand in the liquid ( zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR k) : 1 -- 1 K=Hk,+(k) 1 (19) where H is a partition coefficient, the equilibrium concentration in the gas divided by that in the liquid. For oxygen in water, H is about twenty and kM is given by kM = D/6 (20) where D is the diffusion coefficient in the membrane and S is the membrane thickness. Again, for oxygen, D is about 0.05 cm2/sec and 6 is about 0.01 cm, so (l/Hk,) is about 0.001 set/ cm. In contrast, the largest value of (k) in Table 3 is about 0.02 cm/set, so (l/(k) ) is about fifty. As a result, K is dominated by (It), and independent of kM or 6. Phrased in other terms, module performance is independent of membrane properties. For blood rather than water, module performance is more complicated. The partition coefficient H now drops, and the mass transfer coefficient (k) can be accelerated by S.R. Wickramasinghe 249 zyxwvutsrq et al./J. Membrane Sci. 69 (1992) 235-250 List of symbols the oxygen-hemoglobin reaction. For the membrane properties to become important, HIzMmust be about equal to (k) [8]. For this membrane area per volume a to occur, the half life of the oxygen-hemoglobin blood channel size b reaction must be less than 10m6sec. concentration c zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The results in both Tables 2 and 3 are limited inlet concentration CO to the case of fast nitrogen flow on the gas side average concentration zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO of the membrane. Such fast flow insures that average fiber diameter: twice the average d the oxygen concentration in the nitrogen is alradius ways near zero, and the concentration differequivalent diameter: four times the cross ence responsible for mass transfer is kept near section for flow divided by the wetted its maximum possible value. Blood oxygenaperimeter tors are in fact operated under equivalent cong distribution function (eqn. 2) ditions: the air flow is kept high to maximize Graetz number Gr the concentration differences of oxygen and mass transfer coefficient k carbon dioxide, and hence maximize the mass (k) average mass transfer coefficient (eqn. 1) transfer of these species. module length 1 Modules used for chemical processing, inpressure P cluding those for absorption and extraction, will average water flow (eqn. 5) Q not be operated in this way. Instead, the two r radius fluids will usually flow countercurrently to each R,R, average radii (eqns 3 and 9 ) other. Such countercurrent contacting gives Re Reynolds number more complete separations than either concurSh Sherwood number rent flow or crossflow. Thus better membrane velocity u modules for chemical processing should try to V average volume (eqn. 4) include both local flow across the fibers, and countercurrent flow in the module itself. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF 60 standard deviation divided by the mean cc> Acknowledgements Bradley Reed (Hoechst-Celanese ), Marc Voorhees (Cobe Cardiovascular, Inc. ) James McCabe (Bard), Ron Leonard (Sarnes 3M), and Jean Pierson (Medtronic) provided the modules used in this work. We benefitted from discussions with Ravi Prasad (Hoechst-Celanese) , Marc Voorhees (Cobe Cardiovascular, Inc. ) and Wallace Jansen (Minntec ) . The work was largely supported by Hoechst-Celanese. 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