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Hollow fiber modules made with hollow fiber fabric
Journal of Membrane Science, 84 (1993) l-14 Elsevier Science Publishers B.V.. Amsterdam
Hollow fiber modules made with hollow fiber fabric S.R. Wickramasinghe, Michael J. Semmens and E.L. Cussler* Departments of Chemical Engineering and Materials Science, and of Civil and Mineral Engineering, Minnesota, Minneapolis, MN 55455 (USA)
University of
(Received November 3,1992; accepted in revised form April 27,1993)
Abstract We have measured the overall mass transfer coefficient of oxygen from water across a hollow fiber membrane into nitrogen. The membranes are microporous, so this overall mass transfer is controlled by the individual coefficient in the water. The membranes are incorporated into modules in which the fibers are carefully spaced, including modules made from hollow fiber fabric. At low flows, these modules give mass transfer which is up to ten times faster than that in commercial modules based on similar hollow fibers. This implies that fabric-based modules can give equivalent performance with less membrane area. The performance of the fabric-based modules approaches that of modules built by hand, one fiber at a time. It approaches that inferred for a single fiber from heat transfer correlations. Key words: concentration
polarization;
fiber membranes; microporous
Introduction This paper reports correlations for mass transfer across boundary layers adjacent to hollow fiber membranes. The correlations include results for a fabric woven of hollow fibers, which are dramatically better than most other membrane geometries. The results suggest routes to substantially superior designs of hollow fiber blood oxygenators. We believe that the results in this paper also offer clues to better module designs for other membrane separations. The success of all these separations depends on getting a large selective flux [ 11. The large flux is achieved with a large membrane area; this large area is the spur for developing hollow fibers. The selective flux +To whom correspondence should be addressed.
0376-7388/93/$06.00
and porous membranes; modules
most often means a selective overall mass transfer coefficient. This overall mass transfer coefficient depends both on the membrane itself and on the boundary layers adjacent to the membrane. To explore the consequences of this dependence, we consider three separate cases. First, we imagine a gas separation across thick non-porous membranes of high selectivity but modest permeability. The overall mass transfer coefficient in this case in dominated by the membrane’s permeability. For example, a membrane more permeable to hydrogen than nitrogen can separate these gases. A membrane more permeable to carbon dioxide than methane produces a permeate enriched in carbon dioxide. As a second case, we turn to blood oxygenation across non-selective microporous hydro-
0 1993 Elsevier Science Publishers B.V. All rights reserved.
2
phobic membranes [ 21. The overall mass transfer coefficient in this case is unaffected by the membrane’s permeability. Instead, it is dominated by diffusion in the boundary layers adjacent to the membrane. Hollow fiber blood oxygenation is selective because blood selectively absorbs oxygen. On this basis, we might conclude that studies of boundary layers have little relevance to separations with selective membranes. This is not true. To see why it is not true, we turn to the third special case. Our third special case is the hollow fiber drying of air [ 31. This case used thin non-porous membranes with a selectivity for water vapor over air of at least 10,000: 1. However, the performance of these modules gives a separation with a typical selectivity of perhaps 40: 1. The reason for this compromised selectivity comes from the overall mass transfer coefficient. For air, this overall coefficient is dominated by air’s low membrane permeability. For water vapor, the overall coefficient is controlled by boundary layer resistances. The membrane permeability of the water vapor is so high that it does not affect the water’s overall mass transfer coefficient. The selectivity of the process is given by the ratio of these overall coefficients, and not by the ratio of the membrane permeabilities. Many working on membrane separations may feel that their selective membranes are not affected by boundary layers. Those who work with thick films of low permeability are correct. Those who are trying to achieve higher fluxes of making their membranes thinner are heading towards the limit where hollow fiber membrane separations are affected by boundary layers. Thus to achieve still greater improvements in hollow fiber membrane separations, we need to reduce the resistance in these boundary layers. To do so, our group has studied commercially available hollow fiber blood oxygenators
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et al./J. Membrane Sci. 84 (1993) 1-14
[ 2 1. Our studies show that modules with greater membrane area per volume perform better than those with smaller area per volume. They show that the mass transfer coefficients for liquid flowing rapidly within the hollow fibers agree closely both with the predictions of fluid mechanics and with analogous experiments of heat transfer. All these results are expected and reassuring. But some of the results with commercial hollow fiber blood oxygenators are unexpected and disquieting. Two unexpected results occur when the liquid is slowly flowing across the hollow fibers. First, mass transfer resistances are similar for modules of different geometry. In particular, at equal flow over an equal membrane area, the mass transfer coefficients for a cylindrical bed of fibers, for a rectangular bed of fibers, and for an annular bed of fibers are the same within experimental error [ 21. This equivalence presumably reflects the same local boundary layers around each fiber. The second unexpected result from our studies of these commercial oxygenators is that the mass transfer coefficients, which agree with each other, fall significantly below those in modules which are carefully built by hand [ 41. The shortfall can be as much as a factor of ten. We might conclude that the hand-built modules, with each fiber painstakingly positioned, might be in error. However, because these test modules agree closely with theories and with heat transfer experiments, we believe the results are correct. Instead, we suspect that the commercial modules may be improved by better manufacture. Specifically, we believe that the performance of hollow fiber modules, including commercial blood oxygenators, may be improved by ensuring very uniform spacing between the hollow fibers. If our belief is correct, then the mass transfer coefficients for carefully spaced fibers in various geometries should be higher than those in existing commercial modules. They
S.R. Wickramasinghe et al./J. Membrane Sci. 84 (1993) I-14
should be the same as those in the hand-built modules constructed one fiber at a time. In this paper, we test this belief with experiments on the modules illustrated in Fig. 1. The first “axial” module, shown in Fig. 1 (a), is an annular bed of hollow fibers wound helically around a central filter core. Gas flows through the lumen of each fiber. Liquid enters into the porous central tube, is forced radially outwards across the fibers by the plug, and is then collected in the shell surrounding the fibers. Thus, this axial geometry imitates some which are
3
available commercially, though it may be more carefully assembled than is feasible under commercial conditions. The second module, which is not shown in Fig. 1, is much like the first, except that the central tube contains two plugs, and the space between the fibers and the shell is blocked by an O-ring. These plugs and O-ring divide the module into three nearly equal sections. Gas flows down the lumen of each fiber, just as in the first module. However, in the second module, the liquid enters the central tube, is de-
(a) Axial gasin
water out
(b) Fabric
(c) Vane
Fig. 1. Module geometries used in this work. The axially wound modules use hollow fibers wound helically around a central core. The fabric modules wind a hollow fiber fabric around a similar core. The vane module is an open ended box with a vane of hollow fiber fabric mounted diagonally inside the box.
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Fig. 2. The hollow fiber fabric. The fabric has a warp of 300 pm microporous polypropylene hollow fibers held at constant spacing by a weft of solid 2.5 pm polypropylene thread. The hollow fibers are not crimped, as shown in the inset.
fleeted by the first plug across the fibers into the shell, is then deflected by the O-ring back into the central tube, and is then deflected by the second plug so that it exits from the shell side. Thus the liquid crosses back and forth across the fiber bed three times. This module tries to achieve some of the crossflow necessary for a large mass transfer coefficient and some of the countercurrent contacting which is important for many efficient gas absorptions and
liquid extractions. As such, it is similar to commercial modules made for liquid-liquid extraction. The third module, shown in Fig. 1(b), is based on the knitted hollow fiber fabric shown in Fig. 2. Any fabric consists of the threads lengthwise on the loom, called the warp, and the cross-threads, called the weft or the woof. In our fabric, the warp consists of 300 pm microporous polypropylene hollow fibers, and the
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weft is made of a yarn of solid 2.5 pm solid polypropylene thread. The fabric enforces even spacing between the hollow fibers. We also tried weaving fabric with hollow fibers as the warp and the weft, but found that the fibers tended to crimp. We have very little crimping with the weft used here, as shown in the insert in Fig. 2. The module made with this fabric and shown in Fig. 1 (b) is very similar to the second module described above. The fabric, with the warp aligned axially, is wound around a porous central tube which contains four plugs as shown. The fabric-wrapped tube is placed in a shell which is blocked by three O-rings. The plugs and the O-rings divide the module into seven roughly equal sections. As before, gas flows through each fiber’s lumen. Again, liquid enters through the central tube, flows back and forth across the fabric, and exits via a port on the shell side. The fourth “vane” module, shown in Fig. 1 (c ), uses the hollow fiber fabric as a vane mounted diagonally in a rectangular box which has open ends. As before, gas flows through each hollow fiber lumen; the gas enters and leaves through tubular manifolds mounted along the diagonal as shown. The entire box is towed through liquid, so liquid flows both through and around the box. This type of module has occasionally been suggested as an artificial gill, either for a human diver or for a small non-nuclear submarine [ 5 1. This module’s purpose is to harvest dissolved oxygen for the diver’s breathing or for the submarine’s fuel cells. Whether this design is practical depends on the mass transfer achieved. We want to determine the mass transfer coefficients possible in each of these four modules. We will use these determinations to develop mass transfer correlations, and compare these to test our hypothesis that commercially available modules can be compromised by unevenly spaced hollow fibers. We can use established procedures to make these tests for the
5
two axially wound modules (Fig. la) and for the hollow fiber fabric module (Fig. lb). We cannot do so for the vane module in Fig. 1 (c) without first estimating the water velocities across the vane of fibers and around the module itself, We develop this velocity estimation in the theory section which follows. Theory In this section, we first want to review characteristics of the overall mass transfer coefficient, and then explore how this coefficient can be found for the vane module. The overall coefficient can be found from a mass balance on the oxygen in the water flowing in the module [ 61: O=
-Q $-k(c-e*)
(1)
where Q is the volumetric flow in the module, A is the total membrane area, and c and c* are the actual and equilibrium concentrations of oxygen in the water. In our experiments, the gas flow is present in excess, so the concentration c* is a constant. Hence eqn. (1) is easily integrated subject to the condition that the entering concentration is co: k=
(2)
An alternative form of this equation is also useful (3) where a is the membrane area per volume of the fiber bed, 1 is the length of the bed in the direction of flow, and u is the superficial velocity in the bed, i.e. the volumetric flow rate divided by the cross section for the flow. The overall mass transfer coefficient is in turn given by
6
S.R. Wickramasinghe
LL+s+k-kL
P
et al./J. Membrane Sci. 84 (1993) 1-14
1 kH
where kL is the mass transfer coefficient in the liquid flowing across the fibers, P is the permeability of the membrane of thickness 6, kc is the mass transfer coefficient in the gas, and H is Henry’s law constant, the ratio of gas to liquid concentration at equilibrium. Earlier experiments [2,4,7] conclusively show that for the air-water system studied here, the overall mass transfer coefficient is dominated by the coefficient in the liquid. It is not influenced by the membrane’s properties or by the excess gas flow. In other words, the term (l/it,) is much larger than the other two terms on the right hand side of eqn. (4). We can easily use eqns. (2 )-- (4 ) to find kL for the axial and fabric modules shown in Figs. 1 (a) and 1(b). We cannot easily use these equations for the vane module in Fig. 1 (c) because we do not measure the liquid phase concentrations and because we are unable to measure directly the water velocity across the fibers. We did not measure the liquid concentrations because the total liquid flow is much greater than the liquid flow actually passing through the module. This means that the change in the average water concentration is very small. Instead, we feed the vane module with slowly flowing water-saturated nitrogen and measure the oxygen concentration of the exiting nitrogen. We have derived slightly different but analogous expressions to eqns. ( 1)- (4 ) which are detailed elsewhere [8,9]. We use these to calculate kL for the vane module. The second problem, measurement of the water velocity across the vane modules fibers, is more serious. We are unable to measure this velocity directly at the low flows of interest here. We can measure the total flow approaching the module. To estimate the velocity across the fibers from this total flow, we must develop a theory. This theory begins with the drawing in
so
+
-
v,s,
Fig. 3. Details of vane module operation. The unit shown in Fig. 1 (c ) is suspended in a flume of water flow at velocity u,,. The velocity across the fibers is much less than u,, because of the form drag of the module.
Fig. 3, which defines the relevant velocities ui and cross sectional areas Si. From a mass balance, s,uo=s,ul
+ (So-S,)u,
(5)
The pressure drop Ap around the module equals that through the module:
-= 4J lug P
f(- 2
)
where p is the liquid’s density, f and f’ are the friction factors outside and inside of the vane module, and 1and Eare the length and the void fraction of the hollow fiber bed [ 10,111. While the quantity in square brackets in eqn. (6) presumes that the Ergun equation for packed beds holds for flow across the hollow fibers, we will largely use only the functional form of this equation, and not the specific numerical constants. We can easily combine eqns. (5) and (6) and solve for the velocity U,within the vane module:
so
b+l __ & ( a ,rJ
u1
-=UO
1
l+ (b+al)"-l
(7)
where a is a weighted ratio of friction factors: a=-
f’ f
so-s, 2_l
( > Sl
(8)
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et al./J. Membrane Sci. 84 (1993) 1-14
7
and b is a ratio of viscous drag to form drag:
TABLE 1
b
Geometrical parameters for the modules studied
15OVZ(1-@ =F-‘od2E3
(S,-S,)2 s,s;
(9)
Two limits of eqn. (7) are interesting. First, when b >> 1, Ill=
s;
(s*>u; t3
= 300 (l-c)2
u1=
S",
S;
(s,-s,)2~
s1
&O =[ #(So-s,)+Jfs,
1
;
1
0.35 1.05 2.45 0.060
25.9 25.9 16.7 48.8
0.81 0.81 0.87 0.63
“The areas are S, and S; respectively (cf. Fig. 3).
(11)
-
Depth of Bed area Bed void Cross sectional fiber bed per volume fraction area a (cm-‘) (cm) (cm’)
Simple axial 97 Bafiledaxial 32 Baffled fabric 13 Vane 100; 270”
-
fd2
Module description
u”
Since f andf’ will be almost equal in most practical cases, u1 will be about equal to uo. Taken together, eqns. (10) and (11) predict that the velocity u1 in the vane module will be proportional to the square of the incident velocity u. at low flow, but becomes linearly dependent on u. at high flow. We test these predictions and characteristics of all the modules with the experiments described in the following sections. Experimental The four modules tested in this work and shown schematically in Fig. 1 were all made of microporous polypropylene hollow fibers of 300 pm outside diameter, 25 ,umwall thickness, and 0.20 wall void fraction (Celgard@ Hoechst Celanese Corporation, Charlotte, NC). The key geometrical parameters of these four modules are given in Table 1. The first column describes the module, and the next four give appropriate dimensions.
Because details of module assembly are given elsewhere, only a synopsis is given here [ 21. The unbaffled and baffled axially wound modules were made by winding hollow fibers in groups of six around a central filter core (Internet, Minneapolis, MN) using a yarn winder (Leesona model 959 Charlotte, NC). In the baffled module two rubber stoppers were placed at regular intervals inside the central filter core. After the winding was complete, each fiber bundle was slipped into a second filter core and this second core was placed inside a glass shell. In the case of the baffled module, an O-ring was placed twothirds of the way down the length between the second filter core and the inner surface of the shell. As a result, the fluid flows across the fibers once in the unbaffled module and three times in the baffled module. The depth of the fiber bed in the unbaffled module was taken as the distance between the filter cores; the depth of the fiber bed in the baffled axial module is three times that of the unbaffled axial one. The modules were potted using two kinds of epoxy. The more viscous epoxy (3 M Scotch-Weld Brand 838 B/A TAN, St. Paul, MN) was used to hold the filter core in the center of the glass shell, while the less viscous epoxy (H.B. Fuller FE-5045 Brooklyn Center, MN) was used to insure an air-tight seal. Cylindrical screws made of acrylic were glued into both ends of the inner
a filter core. Again, both types of epoxy were used, the more viscous one to hold the screw in the center of the filter core, and the less viscous one to insure a tight seal. End caps to guide the fluid flows were then screwed into position. The end caps are removable and reusable. The hollow fiber fabric module was made with fabric containing 39 fibers per inch (Hoechst Celanese Corporation, Charlotte, NC ) . The fabric was wrapped around an identical central filter core which contained four rubber stoppers placed at regular intervals. As before, this bundle was then slipped into a second filter core, and the second core placed in a glass shell with three regularly spaced O-rings. Fluid entering the central filter core and leaving from the outer shell thus flows across the fiber bundle seven times. This module was potted like the axially wound modules; end caps were also similar; the total depth of the bed was taken as seven times the distance between the filter cores. The vane module, made of the same hollow fiber fabric, was built by placing two layers of the fabric between two polypropylene spacers. The spacers, made by the Nagle Company (Rochester, NY), were supplied by Cobe Cardiovascular (Lakewood, CO). Their purpose was to protect the membrane. In this case, though, we assumed the volume of the fiber bed was the volume of two layers of fabric rather than that between the spacers. The ends of the fibers were potted with epoxy (H-B. Fuller FE5045) into manifolds of 0.7 cm polycarbonate tubing which had been slit lengthwise. The manifolded fabric was then placed diagonally in a rectangular box 10 cm X 10 cm X 25 cm potted with the same epoxy. The box was an aluminum frame with the outside edges coated with a polypropylene mesh (Filmtec, Edina, MN). In addition, a 10 cm x 10 cm X 5 cm poly (methyl methacrylate) calming section was placed on the front of the box to guide the water into the box. The box was mounted with the total 30 cm length parallel to the flow in a flume 15cm wide
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by 39 cm high. The flume was over 6 m long, and was usually filled with water to a depth 25 to 33 cm, with an average depth of 29 cm. We tried to measure the velocities in the flume and in the vane module, but with mixed success. We found that the average velocity u,, in the flume measured with a V-notch gage agreed closely with that measured with an electromagnetic current meter (Marsh McBirney model 523, Gaithersburg, MD). At high flows, the velocity u1 within the vane module was about 30% lower than that in the flume, consistent with eqn. (11) . At the low flows where we wanted to study mass transfer, values of u1 seemed proportional to the square of uO,but the data were badly scattered. The low flow region is that studied in the mass transfer experiments. As a result, we choose to estimate u1using eqn. (10). Mass transfer of oxygen from water into nitrogen was measured in the axially wound and fabric modules in much the same way. Each module was fed with pure nitrogen ( > 99%; ~0.01% 02, Linde) and with water saturated with air. Oxygen concentrations in the inlet and outlet water streams were measured vs. water flow, nitrogen flow, and time, using a Orion Model 97-o-00 oxygen specific electrode. Mass transfer coefficients were calculated from these measurements using eqn. (2 ) or eqn. (3 ) . These coefficients were independent of nitrogen flow and reached steady state values within five minutes. For the vane module, the very large water flow makes measuring changes in oxygen concentration impractical. As a result, we measured small changes in the oxygen in the gas flow, using Microelectronics model MI-730 specific oxygen electrode (Londonderry, MA). The mass transfer coefficients were found by a parallel procedure described in detail elsewhere 191. Results This paper tests the hypothesis that better
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hollow fiber module performance requires better spacing of the hollow fibers. To test this hypothesis, we measure the mass transfer coefficients in four different modules and compare these coefficients with earlier measurements on a spectrum of module designs. We begin the comparison in this section for the axially wound, the hollow fiber fabric, and the vane modules. The mass transfer coefficients for the axially wound modules are shown as functions of velocity in Fig. 4. The coefficients are reported as Sherwood numbers (M/D), and the velocities are given as Reynolds numbers (&I/V). Because the liquid flow is outside of the fibers and because mass transfer in the liquid dominates the overall coefficient, we use the outside fiber diameter in these dimensionless groups. Because earlier correlations of mass and heat transfer in similar geometries use the superficial velocity, we use that definition in our calculations of Reynolds numbers. Data for both the unbaffled and baffled axial modules are shown in the figure: the open squares are for the unbaffled module, and the filled squares are for the module whose flow is baffled three times
Fig. 4. Mass transfer coefficients for axially wound modules. The unbaffled modules (open squares) are similar to those for baffled modules (filled squares). These coefficients are greater than those for commercial modules (triangles) but less than those of modules built one fiber at a time (the solid line).
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back and forth across the fibers. The data from the two modules are indistinguishable. As a result, the mass transfer coefficient is not significantly changed by the baffles. Figure 4 also compares our axially wound data with those of modules available commercially or assembled one fiber at a time. While the range of Reynolds numbers is slightly different, our axial data lie significantly above the data for commercial modules, shown as open triangles [ 21. These commercial module data include fibers wound axially, bunched cylindrically, and grouped rectangularly. We believe that our higher values reflect the difficulty in commercially assembling modules with very uniform fiber spacing. At the same time, our axially wound modules show mass transfer coefficients slightly below those for modules built one fiber at a time, which are summarized by the solid line in Fig. 4 [ 41. This solid line agrees closely with correlations for heat transfer across much larger but widely spaced tubes at similar Reynolds numbers. We were unable to measure mass transfer from a single hollow fiber because the volumetric flow through just one fiber is so small. We are reassured that the solid line in Fig. 4 also closely agrees with heat transfer results for a single tube. The results in Fig. 4 imply that carefully spacing axially wound fibers does improve module performance beyond that in commercial modules. However, the improvement is less than that expected for carefully spaced individual fibers, and requires such careful winding that module manufacture is slow. To get better results with less work, we turn to the hollow fiber fabric. Our measured mass transfer coefficients for a hollow fiber fabric module are plotted as open circles in Fig. 5. Again, these coefficients are plotted as Sherwood numbers based on the outer fiber diameter; again, the liquid velocities are plotted as Reynolds numbers based on the
10
S.R. Wickramasinghe
10
0.1Reynold:
100
Number Fig. 5. Mass transfer coefficients for hollow fiber fabric. These coefficients, shown as open circles, are above those of commercial modules, shown as triangles, but slightly below those of modules built one fiber at a time, shown as the solid line.
superficial velocity; again, our data are compared with commercial modules (the open triangles) and with modules built one fiber at a time (the solid line). Our fabric-based module data fall above the commercial module results. At low flows, the data in Fig. 5 mean that a fabric-based membrane oxygenator can achieve the same oxygen transfer as a commercial unit which has ten times more active membrane. These data present a strong argument for building modules with fabric. The fabric-based mass transfer coefficients fall slightly below those for modules built one fiber at a time, and slightly above those of the axially wound modules. This somewhat better performance is achieved in spite of the major difference in the work of making these modules. The fabric based module is easily - almost casually - made: the fabric is simply wound around the central filter core. The axially wound modules took us much more effort, though we believe that in time we could become more skilled in operating the winder. Building a module one fiber at a time takes weeks, even for a module containing only a few hundred fibers. But building a module with hollow fiber
et al./J. Membrane Sci. 84 (1993) l-14
fabric takes only a few minutes, and yet gives high mass transfer coefficients. The results in Fig. 5 also raise another question: what is the highest mass transfer coefficient possible at a given Reynolds number? After all, as we get better and better at carefully spacing the hollow fibers, we get higher and higher coefficients. Is there a maximum, and have we reached it? We suspect we have not. We can see no theoretical reason for a maximum either in theories of mass transfer or those of heat transfer. We suspect that assembling fabric modules so that the boundary layers from one course of fibers aid mixing on the next course may give still further gains. Finally, we turn to the mass transfer coefficients for the vane module. Our measured mass transfer coefficients for the vane module are plotted vs. velocity in Fig. 6. As before, the coefficients are given as Sherwood numbers based on the outer diameter of the fiber, and the velocities are expressed as Reynolds numbers. Overall Reynolds Number
‘O0U
11
1
l_dgl Reynolds 6mber Fig. 6. Mass transfer coefficients for a vane module. The coefficients for the module in Fig. 1 (c) can be correlated in terms of an overall Reynolds number based on the incident velocity u,, or in terms of an estimated local velocity in the fiber bed u;. The latter choice gives rough agreement with the correlation for modules built one fiber at a time (the solid line).
S.R. Wickramusingke et al./J. Membrane Sci. 84 (1993) l-14
Now, however, two different Reynolds numbers are possible. The first, overall Reynolds number, based on the velocity in the flume uo, is the more obvious, the more directly measured. The second local Reynolds number, based on the velocity u ‘1within the module and estimated from eqn. (lo), allows more direct comparison with other module results. In estimating u;, we have assumed the friction factor f is 0.5, typical of the values for flow around submerged objects. We have assumed the thickness of the hollow fiber bed is doubled from 0.06 to 0.12 cm by the 0.1 cm spacers located next to the membrane. These choices do not affect the variation of the Sherwood number with the local Reynolds number, but they do affect the constant relating these quantities. In other words, they do not change the slope of the data in Fig. 6, though they do change the position of the data. The mass transfer coefficients obtained for the vane module agree roughly with those obtained for the hand-built modules, summarized by the solid line in Fig. 6. This comparison is much less exact than those in Figs. 4 and 5 because of the significant approximations in using eqn. (10) to calculate the velocity and hence the local Reynolds number around the fiber. It is the analogue of this local Reynolds number which is used for the earlier correlations. The apparent consistency is reassuring, and is the starting point for the discussion which follows.
11
tractive, for these permit even fiber spacing and easy module assembly. These results do raise some unanswered questions. First, we want the most reliable correlation for mass transfer in these modules. Second, we want to know why this correlation differs from that found for commercially available modules. Third, we want to know which module design is best. These questions are discussed in the following paragraphs. The mass transfer correlations inferred from the various modules are summarized in Table 2. The first column is this table describes the module, the second gives the actual variations measured, and the third gives the range of flows used in these measurements. The fourth column gives the correlations inferred, where we have paralleled other studies of mass transfer in assuming a cube root dependence on the Schmidt number [ 4,121. The correlations in Table 2 are indistinguishable within the experimental error of our measurements. This equivalence, emphasized by the comparison of data in Fig. 7, results from our success in spacing the fibers evenly. After all, in each case, we are measuring the oxygen transferred from water flowing perpendicular to the hollow fiber. If the fibers are evenly spaced, the water velocity and the oxygen mass transfer should be similar. Thus we suggest as an average correlation
(12) Discussion The results given above show that carefully spaced hollow fibers give larger mass transfer coefficients than those in commercially assembled modules. At low flows, the coefficients can be ten times larger, suggesting that a module with such fibers will require only one tenth the membrane area as is currently used. Modules based on hollow fiber fabric seem especially at-
In taking this average, we have weighted each module design equally, even though the vane module data are less accurate. Quation (12) is our recommendation for future designs. We recognize that the Reynolds number variation in eqn. (12) is about half that we find for commercial modules. We believe that this difference reflects difficulties in spacing the fibers, and that we can express these difficulties in a variety of ways. For example, we can say
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TABLE 2 Correlations
of mass transfer
coefficients
in high performance
modules
Module geometry
Measured correlations”
Flow rangeb
Inferred correlationb*c Remarks
Modules built one fiber at a time
-
10-‘
Sh= 0.90Re0.40Sc0.33
This result is very close to the heat transfer correlation for a single tube
O.O3
Sh=0.49Re0.53Sc0.33
These modules were assembled by hand, slowly winding six fibers at a time
O.l
Sh=0.82Re0.4gSc
[41
Axially wound (Fig. la)
~=~.OX~O-~U~.~~
Fabric (Fig. lb)
a33 These modules are the most easily built
0.1~ Re < 10d
k=4.4x10-3(u’)0.461
Vane (Fig. lc)
Sh=0.80Re0?Sc0.33
The velocity used in the Reynolds number is that estimated from eqn. (10)
“Units are cm/set for both k and u. bathe Reynolds number shown is based on an outer fiber diameter of 0.030 cm, a kinematic viscosity for water of 0.01 cm”/ set, and a superficial velocity across the fibers. “The Schmidt number for oxygen transfer in water is taken as 480, implying a diffusion coefficient of 2.1 x 1O-5 cm2/sec. dThese values are based on the local Reynolds number (du;/u).
1001
0.01
I
10
a.1 Reynod
100
Number
Fig. 7. Comparing mass transfer coefficients for different geometries. The values for all the modules in Fig. 1 agree, but lie slightly below the results for modules built one fiber at a time (the solid line ) .
that unevenly spaced fibers cause dispersion, and blame any difference between eqn. ( 12)and the result for commercial modules on an empirically determined dispersion coefficient. Al-
ternatively, we can assert that the differences are due to a bed of fibers with polydisperse gaps between them [ 21. We then can interpret any differences in the correlations in terms of the standard deviation of the polydisperse gaps. This interpretation is successful in explaining both extraction and chromatography for liquids flowing within hollow fibers, rather than around the outside [ 13,141. We are uneasy about applying this idea here because it seems like curve fitting. As a third approach, we can assume that part of the flow bypasses the hollow fibers much as for the vane module in Fig. 3. We then can parallel the analysis in eqns. (5 ) - ( 11) in this paper to find the “real” velocity past the fiber. All these approaches seem speculative to us. We can reach more definite conclusions about which module design is “best”. To explore these conclusions, we must first decide what we mean
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et al./J. Membrane Sci. 84 (1993) 1-14
by “best”. Three choices seem key. First, we can define “best” as the most mass transferred per cost. We expect that this definition will be that most commonly used commercially. When the total cost is dominated by the capital cost of the membrane, the best module is the one with the greatest mass transfer coefficient, as shown in Fig. 7. But this figure shows that all designs have the same coefficients if the fibers are carefully spaced Thus there is no single best design on this basis. A second definition is that the “best” module gives the most mass transferred per module volume. This definition, commonly quoted for blood oxygenators, might also be used for separations where space is important, as on offshore oil platforms. This definition is equivalent to maximizing the mass transfer coefficient times the membrane area per volume. Which volume is important determines which design is best. If the important volume is that of the fiber bed, then the vane module is marginally better because its area per volume is slightly higher (cf. Table 1). If the important volume is that of the entire module, then the vane module is the poorest design because the fibers fill only a small fraction of the module itself (cf. Fig. lc). A third definition is that the “best” module gives the most mass transferred per power required to move the fluid. At low flows, the power is proportional to the square of the velocity across the fiber bed, at higher flows, it varies with the velocity cubed. Thus a small power means a very small velocity. The best mass transfer per power would probably come from an unbaffled module based on hollow fiber fabric, followed closely by an unbaffled axially wound module. Baffled modules of either type will have a higher mass transfer coefficient; but the higher mass transfer coefficient will be purchased by a higher velocity and hence a dramatically increased power. The vane module will be poorer than other designs, because most
of the flow will go around the module, not across the fibers. Thus in this paper, we present mass transfer correlations for oxygen transfer from water across hollow fiber membranes into excess nitrogen. The correlations show faster mass transfer when the follow fibers are evenly spaced. For example, the mass transfer in modules easily made from hollow fiber fabric approaches that expected for single hollow fibers. The results suggest how better blood oxygenators can be built now, and how other module designs can be improved in the future.
Acknowledgements This work was primarily supported by the Environmental Protection Agency (MTU 00569) and by the Hoechst Celanese Corporation, Separations Products Division. Other support came from the National Science Foundation (CTS 91-23837), from DARPA (9205112), and from General Mills. E.L. Cussler held the Amundson Professorship during a sabbatical leave at M.I.T., where he was graciously received.
List of symbols a : b c CO
C*
d D
membrane area per module volume ratio of fraction factors total membrane area ratio of viscous to form drag solute concentration in the liquid flowing out of the module solute concentration in the liquid flowing into the module hypothetical solute concentration in the liquid in equilibrium with the excess gas outer fiber diameter diffusion coefficient
S.R. Wickramasinghe
14
f,f H k kc, kL
1 P P
Q
Re Scl S, S; SC Sh u UO 4 u;
u2
6 E
V P
friction factors outside and inside the vane module Henry’s law coefficient overall mass transfer coefficient individual mass transfer coefficients in the gas and in the liquid, respectively length fiber bed in direction flow pressure membrane permeability volumetric flow Reynolds number, defined variously flume cross section vane module cross section fabric cross section Schmidt number Sherwood number superficial liquid velocity flume velocity velocity within the vane module local velocity across fibers of vane module velocity around the vane module membrane wall thickness void fraction of hollow fiber bed kinematic viscosity density
et al.jJ. Membrane
Sci. 84 (1993) l-14
References 1 2
3
4 5 6 7 8
9 10
11 12
13
14
Winston Ho and K.K. Sirkar, Membrane Handbook, Van Nostrand Reinhold, New York, NY, 1992. S.R. Wickramasinghe, M.J. Semmens and E.L. Cussler, Mass transfer in various hollow fiber geometries, J. Membrane Sci., 69 (1992) 235-250. K.L. Wang, S.H. McCray, D.D. Newbold and E.L. Cussler, Hollow fiber air drying, J. Membrane Sci., 72 (1992) 231-244. MC. Yang and E.L. Cussler, Designing hollow fiber contactors, AIChE J., 32 (1986) 1910-1916. M.C. Yang and E.L. Cussler, Artificial gills, J. Membrane Sci., 42 (1989) 273-284. E.L. Cussler, Diffusion, Cambridge University Press, New York, NY, 1984. R. Prasad and K.K. Sirkar, Hollow fiber solvent extraction, J. Membrane Sci., 50 (1990) 153-1’75. L. Dahuron, Designing liquid-liquid extractions in hollow fiber modules, Ph.D. Thesis, University of Minnesota, 1987. cf. also L. Dahuron and E.L. Cussler, Protein extractions with hollow fibers, AIChE J., 34 (1988)130-136. S.R. Wickramasinghe, The best hollow fiber module, Ph.D. thesis, University of Minnesota, 1993. R.B. Bird, W.E. Steward and E.M. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, NY, 1960. M.M. Denn, Process Fluid Mechanics, Prentice Hall, Englewood Cliffs, NJ, 1980. R. Prasad and K.K. Sirkar, Dispersion-free solvent extraction with microporous hollow-fiber modules, AIChE J., 34 (1988) 177-188. H.B. Ding and E.L. Cussler, Fractional extraction using hollow fibers with hydrogel filled walls, AIChE J., 37 (1991)855-862. D.K. Schisla, H.B. Ding, P.W. Carr and E.L. Cussler, Polydisperse tube diameters compromise multiple tubular chromatography, AIChE J., 39 (1993) 946-953.
Hollow fiber modules made with hollow fiber fabric S.R. Wickramasinghe, Michael J. Semmens and E.L. Cussler* Departments of Chemical Engineering and Materials Science, and of Civil and Mineral Engineering, Minnesota, Minneapolis, MN 55455 (USA)
University of
(Received November 3,1992; accepted in revised form April 27,1993)
Abstract We have measured the overall mass transfer coefficient of oxygen from water across a hollow fiber membrane into nitrogen. The membranes are microporous, so this overall mass transfer is controlled by the individual coefficient in the water. The membranes are incorporated into modules in which the fibers are carefully spaced, including modules made from hollow fiber fabric. At low flows, these modules give mass transfer which is up to ten times faster than that in commercial modules based on similar hollow fibers. This implies that fabric-based modules can give equivalent performance with less membrane area. The performance of the fabric-based modules approaches that of modules built by hand, one fiber at a time. It approaches that inferred for a single fiber from heat transfer correlations. Key words: concentration
polarization;
fiber membranes; microporous
Introduction This paper reports correlations for mass transfer across boundary layers adjacent to hollow fiber membranes. The correlations include results for a fabric woven of hollow fibers, which are dramatically better than most other membrane geometries. The results suggest routes to substantially superior designs of hollow fiber blood oxygenators. We believe that the results in this paper also offer clues to better module designs for other membrane separations. The success of all these separations depends on getting a large selective flux [ 11. The large flux is achieved with a large membrane area; this large area is the spur for developing hollow fibers. The selective flux +To whom correspondence should be addressed.
0376-7388/93/$06.00
and porous membranes; modules
most often means a selective overall mass transfer coefficient. This overall mass transfer coefficient depends both on the membrane itself and on the boundary layers adjacent to the membrane. To explore the consequences of this dependence, we consider three separate cases. First, we imagine a gas separation across thick non-porous membranes of high selectivity but modest permeability. The overall mass transfer coefficient in this case in dominated by the membrane’s permeability. For example, a membrane more permeable to hydrogen than nitrogen can separate these gases. A membrane more permeable to carbon dioxide than methane produces a permeate enriched in carbon dioxide. As a second case, we turn to blood oxygenation across non-selective microporous hydro-
0 1993 Elsevier Science Publishers B.V. All rights reserved.
2
phobic membranes [ 21. The overall mass transfer coefficient in this case is unaffected by the membrane’s permeability. Instead, it is dominated by diffusion in the boundary layers adjacent to the membrane. Hollow fiber blood oxygenation is selective because blood selectively absorbs oxygen. On this basis, we might conclude that studies of boundary layers have little relevance to separations with selective membranes. This is not true. To see why it is not true, we turn to the third special case. Our third special case is the hollow fiber drying of air [ 31. This case used thin non-porous membranes with a selectivity for water vapor over air of at least 10,000: 1. However, the performance of these modules gives a separation with a typical selectivity of perhaps 40: 1. The reason for this compromised selectivity comes from the overall mass transfer coefficient. For air, this overall coefficient is dominated by air’s low membrane permeability. For water vapor, the overall coefficient is controlled by boundary layer resistances. The membrane permeability of the water vapor is so high that it does not affect the water’s overall mass transfer coefficient. The selectivity of the process is given by the ratio of these overall coefficients, and not by the ratio of the membrane permeabilities. Many working on membrane separations may feel that their selective membranes are not affected by boundary layers. Those who work with thick films of low permeability are correct. Those who are trying to achieve higher fluxes of making their membranes thinner are heading towards the limit where hollow fiber membrane separations are affected by boundary layers. Thus to achieve still greater improvements in hollow fiber membrane separations, we need to reduce the resistance in these boundary layers. To do so, our group has studied commercially available hollow fiber blood oxygenators
S.R. Wickramasinghe
et al./J. Membrane Sci. 84 (1993) 1-14
[ 2 1. Our studies show that modules with greater membrane area per volume perform better than those with smaller area per volume. They show that the mass transfer coefficients for liquid flowing rapidly within the hollow fibers agree closely both with the predictions of fluid mechanics and with analogous experiments of heat transfer. All these results are expected and reassuring. But some of the results with commercial hollow fiber blood oxygenators are unexpected and disquieting. Two unexpected results occur when the liquid is slowly flowing across the hollow fibers. First, mass transfer resistances are similar for modules of different geometry. In particular, at equal flow over an equal membrane area, the mass transfer coefficients for a cylindrical bed of fibers, for a rectangular bed of fibers, and for an annular bed of fibers are the same within experimental error [ 21. This equivalence presumably reflects the same local boundary layers around each fiber. The second unexpected result from our studies of these commercial oxygenators is that the mass transfer coefficients, which agree with each other, fall significantly below those in modules which are carefully built by hand [ 41. The shortfall can be as much as a factor of ten. We might conclude that the hand-built modules, with each fiber painstakingly positioned, might be in error. However, because these test modules agree closely with theories and with heat transfer experiments, we believe the results are correct. Instead, we suspect that the commercial modules may be improved by better manufacture. Specifically, we believe that the performance of hollow fiber modules, including commercial blood oxygenators, may be improved by ensuring very uniform spacing between the hollow fibers. If our belief is correct, then the mass transfer coefficients for carefully spaced fibers in various geometries should be higher than those in existing commercial modules. They
S.R. Wickramasinghe et al./J. Membrane Sci. 84 (1993) I-14
should be the same as those in the hand-built modules constructed one fiber at a time. In this paper, we test this belief with experiments on the modules illustrated in Fig. 1. The first “axial” module, shown in Fig. 1 (a), is an annular bed of hollow fibers wound helically around a central filter core. Gas flows through the lumen of each fiber. Liquid enters into the porous central tube, is forced radially outwards across the fibers by the plug, and is then collected in the shell surrounding the fibers. Thus, this axial geometry imitates some which are
3
available commercially, though it may be more carefully assembled than is feasible under commercial conditions. The second module, which is not shown in Fig. 1, is much like the first, except that the central tube contains two plugs, and the space between the fibers and the shell is blocked by an O-ring. These plugs and O-ring divide the module into three nearly equal sections. Gas flows down the lumen of each fiber, just as in the first module. However, in the second module, the liquid enters the central tube, is de-
(a) Axial gasin
water out
(b) Fabric
(c) Vane
Fig. 1. Module geometries used in this work. The axially wound modules use hollow fibers wound helically around a central core. The fabric modules wind a hollow fiber fabric around a similar core. The vane module is an open ended box with a vane of hollow fiber fabric mounted diagonally inside the box.
S.R. Wickramasinghe
et al./J. Membrane
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Fig. 2. The hollow fiber fabric. The fabric has a warp of 300 pm microporous polypropylene hollow fibers held at constant spacing by a weft of solid 2.5 pm polypropylene thread. The hollow fibers are not crimped, as shown in the inset.
fleeted by the first plug across the fibers into the shell, is then deflected by the O-ring back into the central tube, and is then deflected by the second plug so that it exits from the shell side. Thus the liquid crosses back and forth across the fiber bed three times. This module tries to achieve some of the crossflow necessary for a large mass transfer coefficient and some of the countercurrent contacting which is important for many efficient gas absorptions and
liquid extractions. As such, it is similar to commercial modules made for liquid-liquid extraction. The third module, shown in Fig. 1(b), is based on the knitted hollow fiber fabric shown in Fig. 2. Any fabric consists of the threads lengthwise on the loom, called the warp, and the cross-threads, called the weft or the woof. In our fabric, the warp consists of 300 pm microporous polypropylene hollow fibers, and the
S.R. Wickramasinghe
et al./J. Membrane Sci. 84 (1993) 1-14
weft is made of a yarn of solid 2.5 pm solid polypropylene thread. The fabric enforces even spacing between the hollow fibers. We also tried weaving fabric with hollow fibers as the warp and the weft, but found that the fibers tended to crimp. We have very little crimping with the weft used here, as shown in the insert in Fig. 2. The module made with this fabric and shown in Fig. 1 (b) is very similar to the second module described above. The fabric, with the warp aligned axially, is wound around a porous central tube which contains four plugs as shown. The fabric-wrapped tube is placed in a shell which is blocked by three O-rings. The plugs and the O-rings divide the module into seven roughly equal sections. As before, gas flows through each fiber’s lumen. Again, liquid enters through the central tube, flows back and forth across the fabric, and exits via a port on the shell side. The fourth “vane” module, shown in Fig. 1 (c ), uses the hollow fiber fabric as a vane mounted diagonally in a rectangular box which has open ends. As before, gas flows through each hollow fiber lumen; the gas enters and leaves through tubular manifolds mounted along the diagonal as shown. The entire box is towed through liquid, so liquid flows both through and around the box. This type of module has occasionally been suggested as an artificial gill, either for a human diver or for a small non-nuclear submarine [ 5 1. This module’s purpose is to harvest dissolved oxygen for the diver’s breathing or for the submarine’s fuel cells. Whether this design is practical depends on the mass transfer achieved. We want to determine the mass transfer coefficients possible in each of these four modules. We will use these determinations to develop mass transfer correlations, and compare these to test our hypothesis that commercially available modules can be compromised by unevenly spaced hollow fibers. We can use established procedures to make these tests for the
5
two axially wound modules (Fig. la) and for the hollow fiber fabric module (Fig. lb). We cannot do so for the vane module in Fig. 1 (c) without first estimating the water velocities across the vane of fibers and around the module itself, We develop this velocity estimation in the theory section which follows. Theory In this section, we first want to review characteristics of the overall mass transfer coefficient, and then explore how this coefficient can be found for the vane module. The overall coefficient can be found from a mass balance on the oxygen in the water flowing in the module [ 61: O=
-Q $-k(c-e*)
(1)
where Q is the volumetric flow in the module, A is the total membrane area, and c and c* are the actual and equilibrium concentrations of oxygen in the water. In our experiments, the gas flow is present in excess, so the concentration c* is a constant. Hence eqn. (1) is easily integrated subject to the condition that the entering concentration is co: k=
(2)
An alternative form of this equation is also useful (3) where a is the membrane area per volume of the fiber bed, 1 is the length of the bed in the direction of flow, and u is the superficial velocity in the bed, i.e. the volumetric flow rate divided by the cross section for the flow. The overall mass transfer coefficient is in turn given by
6
S.R. Wickramasinghe
LL+s+k-kL
P
et al./J. Membrane Sci. 84 (1993) 1-14
1 kH
where kL is the mass transfer coefficient in the liquid flowing across the fibers, P is the permeability of the membrane of thickness 6, kc is the mass transfer coefficient in the gas, and H is Henry’s law constant, the ratio of gas to liquid concentration at equilibrium. Earlier experiments [2,4,7] conclusively show that for the air-water system studied here, the overall mass transfer coefficient is dominated by the coefficient in the liquid. It is not influenced by the membrane’s properties or by the excess gas flow. In other words, the term (l/it,) is much larger than the other two terms on the right hand side of eqn. (4). We can easily use eqns. (2 )-- (4 ) to find kL for the axial and fabric modules shown in Figs. 1 (a) and 1(b). We cannot easily use these equations for the vane module in Fig. 1 (c) because we do not measure the liquid phase concentrations and because we are unable to measure directly the water velocity across the fibers. We did not measure the liquid concentrations because the total liquid flow is much greater than the liquid flow actually passing through the module. This means that the change in the average water concentration is very small. Instead, we feed the vane module with slowly flowing water-saturated nitrogen and measure the oxygen concentration of the exiting nitrogen. We have derived slightly different but analogous expressions to eqns. ( 1)- (4 ) which are detailed elsewhere [8,9]. We use these to calculate kL for the vane module. The second problem, measurement of the water velocity across the vane modules fibers, is more serious. We are unable to measure this velocity directly at the low flows of interest here. We can measure the total flow approaching the module. To estimate the velocity across the fibers from this total flow, we must develop a theory. This theory begins with the drawing in
so
+
-
v,s,
Fig. 3. Details of vane module operation. The unit shown in Fig. 1 (c ) is suspended in a flume of water flow at velocity u,,. The velocity across the fibers is much less than u,, because of the form drag of the module.
Fig. 3, which defines the relevant velocities ui and cross sectional areas Si. From a mass balance, s,uo=s,ul
+ (So-S,)u,
(5)
The pressure drop Ap around the module equals that through the module:
-= 4J lug P
f(- 2
)
where p is the liquid’s density, f and f’ are the friction factors outside and inside of the vane module, and 1and Eare the length and the void fraction of the hollow fiber bed [ 10,111. While the quantity in square brackets in eqn. (6) presumes that the Ergun equation for packed beds holds for flow across the hollow fibers, we will largely use only the functional form of this equation, and not the specific numerical constants. We can easily combine eqns. (5) and (6) and solve for the velocity U,within the vane module:
so
b+l __ & ( a ,rJ
u1
-=UO
1
l+ (b+al)"-l
(7)
where a is a weighted ratio of friction factors: a=-
f’ f
so-s, 2_l
( > Sl
(8)
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et al./J. Membrane Sci. 84 (1993) 1-14
7
and b is a ratio of viscous drag to form drag:
TABLE 1
b
Geometrical parameters for the modules studied
15OVZ(1-@ =F-‘od2E3
(S,-S,)2 s,s;
(9)
Two limits of eqn. (7) are interesting. First, when b >> 1, Ill=
s;
(s*>u; t3
= 300 (l-c)2
u1=
S",
S;
(s,-s,)2~
s1
&O =[ #(So-s,)+Jfs,
1
;
1
0.35 1.05 2.45 0.060
25.9 25.9 16.7 48.8
0.81 0.81 0.87 0.63
“The areas are S, and S; respectively (cf. Fig. 3).
(11)
-
Depth of Bed area Bed void Cross sectional fiber bed per volume fraction area a (cm-‘) (cm) (cm’)
Simple axial 97 Bafiledaxial 32 Baffled fabric 13 Vane 100; 270”
-
fd2
Module description
u”
Since f andf’ will be almost equal in most practical cases, u1 will be about equal to uo. Taken together, eqns. (10) and (11) predict that the velocity u1 in the vane module will be proportional to the square of the incident velocity u. at low flow, but becomes linearly dependent on u. at high flow. We test these predictions and characteristics of all the modules with the experiments described in the following sections. Experimental The four modules tested in this work and shown schematically in Fig. 1 were all made of microporous polypropylene hollow fibers of 300 pm outside diameter, 25 ,umwall thickness, and 0.20 wall void fraction (Celgard@ Hoechst Celanese Corporation, Charlotte, NC). The key geometrical parameters of these four modules are given in Table 1. The first column describes the module, and the next four give appropriate dimensions.
Because details of module assembly are given elsewhere, only a synopsis is given here [ 21. The unbaffled and baffled axially wound modules were made by winding hollow fibers in groups of six around a central filter core (Internet, Minneapolis, MN) using a yarn winder (Leesona model 959 Charlotte, NC). In the baffled module two rubber stoppers were placed at regular intervals inside the central filter core. After the winding was complete, each fiber bundle was slipped into a second filter core and this second core was placed inside a glass shell. In the case of the baffled module, an O-ring was placed twothirds of the way down the length between the second filter core and the inner surface of the shell. As a result, the fluid flows across the fibers once in the unbaffled module and three times in the baffled module. The depth of the fiber bed in the unbaffled module was taken as the distance between the filter cores; the depth of the fiber bed in the baffled axial module is three times that of the unbaffled axial one. The modules were potted using two kinds of epoxy. The more viscous epoxy (3 M Scotch-Weld Brand 838 B/A TAN, St. Paul, MN) was used to hold the filter core in the center of the glass shell, while the less viscous epoxy (H.B. Fuller FE-5045 Brooklyn Center, MN) was used to insure an air-tight seal. Cylindrical screws made of acrylic were glued into both ends of the inner
a filter core. Again, both types of epoxy were used, the more viscous one to hold the screw in the center of the filter core, and the less viscous one to insure a tight seal. End caps to guide the fluid flows were then screwed into position. The end caps are removable and reusable. The hollow fiber fabric module was made with fabric containing 39 fibers per inch (Hoechst Celanese Corporation, Charlotte, NC ) . The fabric was wrapped around an identical central filter core which contained four rubber stoppers placed at regular intervals. As before, this bundle was then slipped into a second filter core, and the second core placed in a glass shell with three regularly spaced O-rings. Fluid entering the central filter core and leaving from the outer shell thus flows across the fiber bundle seven times. This module was potted like the axially wound modules; end caps were also similar; the total depth of the bed was taken as seven times the distance between the filter cores. The vane module, made of the same hollow fiber fabric, was built by placing two layers of the fabric between two polypropylene spacers. The spacers, made by the Nagle Company (Rochester, NY), were supplied by Cobe Cardiovascular (Lakewood, CO). Their purpose was to protect the membrane. In this case, though, we assumed the volume of the fiber bed was the volume of two layers of fabric rather than that between the spacers. The ends of the fibers were potted with epoxy (H-B. Fuller FE5045) into manifolds of 0.7 cm polycarbonate tubing which had been slit lengthwise. The manifolded fabric was then placed diagonally in a rectangular box 10 cm X 10 cm X 25 cm potted with the same epoxy. The box was an aluminum frame with the outside edges coated with a polypropylene mesh (Filmtec, Edina, MN). In addition, a 10 cm x 10 cm X 5 cm poly (methyl methacrylate) calming section was placed on the front of the box to guide the water into the box. The box was mounted with the total 30 cm length parallel to the flow in a flume 15cm wide
S.R. Wickramasinghe
et al./J. Membrane
Sci. 84 (1993) l-14
by 39 cm high. The flume was over 6 m long, and was usually filled with water to a depth 25 to 33 cm, with an average depth of 29 cm. We tried to measure the velocities in the flume and in the vane module, but with mixed success. We found that the average velocity u,, in the flume measured with a V-notch gage agreed closely with that measured with an electromagnetic current meter (Marsh McBirney model 523, Gaithersburg, MD). At high flows, the velocity u1 within the vane module was about 30% lower than that in the flume, consistent with eqn. (11) . At the low flows where we wanted to study mass transfer, values of u1 seemed proportional to the square of uO,but the data were badly scattered. The low flow region is that studied in the mass transfer experiments. As a result, we choose to estimate u1using eqn. (10). Mass transfer of oxygen from water into nitrogen was measured in the axially wound and fabric modules in much the same way. Each module was fed with pure nitrogen ( > 99%; ~0.01% 02, Linde) and with water saturated with air. Oxygen concentrations in the inlet and outlet water streams were measured vs. water flow, nitrogen flow, and time, using a Orion Model 97-o-00 oxygen specific electrode. Mass transfer coefficients were calculated from these measurements using eqn. (2 ) or eqn. (3 ) . These coefficients were independent of nitrogen flow and reached steady state values within five minutes. For the vane module, the very large water flow makes measuring changes in oxygen concentration impractical. As a result, we measured small changes in the oxygen in the gas flow, using Microelectronics model MI-730 specific oxygen electrode (Londonderry, MA). The mass transfer coefficients were found by a parallel procedure described in detail elsewhere 191. Results This paper tests the hypothesis that better
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et al./J. Membrane Sci. 84 (1993) 1-14
hollow fiber module performance requires better spacing of the hollow fibers. To test this hypothesis, we measure the mass transfer coefficients in four different modules and compare these coefficients with earlier measurements on a spectrum of module designs. We begin the comparison in this section for the axially wound, the hollow fiber fabric, and the vane modules. The mass transfer coefficients for the axially wound modules are shown as functions of velocity in Fig. 4. The coefficients are reported as Sherwood numbers (M/D), and the velocities are given as Reynolds numbers (&I/V). Because the liquid flow is outside of the fibers and because mass transfer in the liquid dominates the overall coefficient, we use the outside fiber diameter in these dimensionless groups. Because earlier correlations of mass and heat transfer in similar geometries use the superficial velocity, we use that definition in our calculations of Reynolds numbers. Data for both the unbaffled and baffled axial modules are shown in the figure: the open squares are for the unbaffled module, and the filled squares are for the module whose flow is baffled three times
Fig. 4. Mass transfer coefficients for axially wound modules. The unbaffled modules (open squares) are similar to those for baffled modules (filled squares). These coefficients are greater than those for commercial modules (triangles) but less than those of modules built one fiber at a time (the solid line).
9
back and forth across the fibers. The data from the two modules are indistinguishable. As a result, the mass transfer coefficient is not significantly changed by the baffles. Figure 4 also compares our axially wound data with those of modules available commercially or assembled one fiber at a time. While the range of Reynolds numbers is slightly different, our axial data lie significantly above the data for commercial modules, shown as open triangles [ 21. These commercial module data include fibers wound axially, bunched cylindrically, and grouped rectangularly. We believe that our higher values reflect the difficulty in commercially assembling modules with very uniform fiber spacing. At the same time, our axially wound modules show mass transfer coefficients slightly below those for modules built one fiber at a time, which are summarized by the solid line in Fig. 4 [ 41. This solid line agrees closely with correlations for heat transfer across much larger but widely spaced tubes at similar Reynolds numbers. We were unable to measure mass transfer from a single hollow fiber because the volumetric flow through just one fiber is so small. We are reassured that the solid line in Fig. 4 also closely agrees with heat transfer results for a single tube. The results in Fig. 4 imply that carefully spacing axially wound fibers does improve module performance beyond that in commercial modules. However, the improvement is less than that expected for carefully spaced individual fibers, and requires such careful winding that module manufacture is slow. To get better results with less work, we turn to the hollow fiber fabric. Our measured mass transfer coefficients for a hollow fiber fabric module are plotted as open circles in Fig. 5. Again, these coefficients are plotted as Sherwood numbers based on the outer fiber diameter; again, the liquid velocities are plotted as Reynolds numbers based on the
10
S.R. Wickramasinghe
10
0.1Reynold:
100
Number Fig. 5. Mass transfer coefficients for hollow fiber fabric. These coefficients, shown as open circles, are above those of commercial modules, shown as triangles, but slightly below those of modules built one fiber at a time, shown as the solid line.
superficial velocity; again, our data are compared with commercial modules (the open triangles) and with modules built one fiber at a time (the solid line). Our fabric-based module data fall above the commercial module results. At low flows, the data in Fig. 5 mean that a fabric-based membrane oxygenator can achieve the same oxygen transfer as a commercial unit which has ten times more active membrane. These data present a strong argument for building modules with fabric. The fabric-based mass transfer coefficients fall slightly below those for modules built one fiber at a time, and slightly above those of the axially wound modules. This somewhat better performance is achieved in spite of the major difference in the work of making these modules. The fabric based module is easily - almost casually - made: the fabric is simply wound around the central filter core. The axially wound modules took us much more effort, though we believe that in time we could become more skilled in operating the winder. Building a module one fiber at a time takes weeks, even for a module containing only a few hundred fibers. But building a module with hollow fiber
et al./J. Membrane Sci. 84 (1993) l-14
fabric takes only a few minutes, and yet gives high mass transfer coefficients. The results in Fig. 5 also raise another question: what is the highest mass transfer coefficient possible at a given Reynolds number? After all, as we get better and better at carefully spacing the hollow fibers, we get higher and higher coefficients. Is there a maximum, and have we reached it? We suspect we have not. We can see no theoretical reason for a maximum either in theories of mass transfer or those of heat transfer. We suspect that assembling fabric modules so that the boundary layers from one course of fibers aid mixing on the next course may give still further gains. Finally, we turn to the mass transfer coefficients for the vane module. Our measured mass transfer coefficients for the vane module are plotted vs. velocity in Fig. 6. As before, the coefficients are given as Sherwood numbers based on the outer diameter of the fiber, and the velocities are expressed as Reynolds numbers. Overall Reynolds Number
‘O0U
11
1
l_dgl Reynolds 6mber Fig. 6. Mass transfer coefficients for a vane module. The coefficients for the module in Fig. 1 (c) can be correlated in terms of an overall Reynolds number based on the incident velocity u,, or in terms of an estimated local velocity in the fiber bed u;. The latter choice gives rough agreement with the correlation for modules built one fiber at a time (the solid line).
S.R. Wickramusingke et al./J. Membrane Sci. 84 (1993) l-14
Now, however, two different Reynolds numbers are possible. The first, overall Reynolds number, based on the velocity in the flume uo, is the more obvious, the more directly measured. The second local Reynolds number, based on the velocity u ‘1within the module and estimated from eqn. (lo), allows more direct comparison with other module results. In estimating u;, we have assumed the friction factor f is 0.5, typical of the values for flow around submerged objects. We have assumed the thickness of the hollow fiber bed is doubled from 0.06 to 0.12 cm by the 0.1 cm spacers located next to the membrane. These choices do not affect the variation of the Sherwood number with the local Reynolds number, but they do affect the constant relating these quantities. In other words, they do not change the slope of the data in Fig. 6, though they do change the position of the data. The mass transfer coefficients obtained for the vane module agree roughly with those obtained for the hand-built modules, summarized by the solid line in Fig. 6. This comparison is much less exact than those in Figs. 4 and 5 because of the significant approximations in using eqn. (10) to calculate the velocity and hence the local Reynolds number around the fiber. It is the analogue of this local Reynolds number which is used for the earlier correlations. The apparent consistency is reassuring, and is the starting point for the discussion which follows.
11
tractive, for these permit even fiber spacing and easy module assembly. These results do raise some unanswered questions. First, we want the most reliable correlation for mass transfer in these modules. Second, we want to know why this correlation differs from that found for commercially available modules. Third, we want to know which module design is best. These questions are discussed in the following paragraphs. The mass transfer correlations inferred from the various modules are summarized in Table 2. The first column is this table describes the module, the second gives the actual variations measured, and the third gives the range of flows used in these measurements. The fourth column gives the correlations inferred, where we have paralleled other studies of mass transfer in assuming a cube root dependence on the Schmidt number [ 4,121. The correlations in Table 2 are indistinguishable within the experimental error of our measurements. This equivalence, emphasized by the comparison of data in Fig. 7, results from our success in spacing the fibers evenly. After all, in each case, we are measuring the oxygen transferred from water flowing perpendicular to the hollow fiber. If the fibers are evenly spaced, the water velocity and the oxygen mass transfer should be similar. Thus we suggest as an average correlation
(12) Discussion The results given above show that carefully spaced hollow fibers give larger mass transfer coefficients than those in commercially assembled modules. At low flows, the coefficients can be ten times larger, suggesting that a module with such fibers will require only one tenth the membrane area as is currently used. Modules based on hollow fiber fabric seem especially at-
In taking this average, we have weighted each module design equally, even though the vane module data are less accurate. Quation (12) is our recommendation for future designs. We recognize that the Reynolds number variation in eqn. (12) is about half that we find for commercial modules. We believe that this difference reflects difficulties in spacing the fibers, and that we can express these difficulties in a variety of ways. For example, we can say
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et al./J. Membrane
Sci. 84 (1993) 1-14
TABLE 2 Correlations
of mass transfer
coefficients
in high performance
modules
Module geometry
Measured correlations”
Flow rangeb
Inferred correlationb*c Remarks
Modules built one fiber at a time
-
10-‘
Sh= 0.90Re0.40Sc0.33
This result is very close to the heat transfer correlation for a single tube
O.O3
Sh=0.49Re0.53Sc0.33
These modules were assembled by hand, slowly winding six fibers at a time
O.l
Sh=0.82Re0.4gSc
[41
Axially wound (Fig. la)
~=~.OX~O-~U~.~~
Fabric (Fig. lb)
a33 These modules are the most easily built
0.1~ Re < 10d
k=4.4x10-3(u’)0.461
Vane (Fig. lc)
Sh=0.80Re0?Sc0.33
The velocity used in the Reynolds number is that estimated from eqn. (10)
“Units are cm/set for both k and u. bathe Reynolds number shown is based on an outer fiber diameter of 0.030 cm, a kinematic viscosity for water of 0.01 cm”/ set, and a superficial velocity across the fibers. “The Schmidt number for oxygen transfer in water is taken as 480, implying a diffusion coefficient of 2.1 x 1O-5 cm2/sec. dThese values are based on the local Reynolds number (du;/u).
1001
0.01
I
10
a.1 Reynod
100
Number
Fig. 7. Comparing mass transfer coefficients for different geometries. The values for all the modules in Fig. 1 agree, but lie slightly below the results for modules built one fiber at a time (the solid line ) .
that unevenly spaced fibers cause dispersion, and blame any difference between eqn. ( 12)and the result for commercial modules on an empirically determined dispersion coefficient. Al-
ternatively, we can assert that the differences are due to a bed of fibers with polydisperse gaps between them [ 21. We then can interpret any differences in the correlations in terms of the standard deviation of the polydisperse gaps. This interpretation is successful in explaining both extraction and chromatography for liquids flowing within hollow fibers, rather than around the outside [ 13,141. We are uneasy about applying this idea here because it seems like curve fitting. As a third approach, we can assume that part of the flow bypasses the hollow fibers much as for the vane module in Fig. 3. We then can parallel the analysis in eqns. (5 ) - ( 11) in this paper to find the “real” velocity past the fiber. All these approaches seem speculative to us. We can reach more definite conclusions about which module design is “best”. To explore these conclusions, we must first decide what we mean
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et al./J. Membrane Sci. 84 (1993) 1-14
by “best”. Three choices seem key. First, we can define “best” as the most mass transferred per cost. We expect that this definition will be that most commonly used commercially. When the total cost is dominated by the capital cost of the membrane, the best module is the one with the greatest mass transfer coefficient, as shown in Fig. 7. But this figure shows that all designs have the same coefficients if the fibers are carefully spaced Thus there is no single best design on this basis. A second definition is that the “best” module gives the most mass transferred per module volume. This definition, commonly quoted for blood oxygenators, might also be used for separations where space is important, as on offshore oil platforms. This definition is equivalent to maximizing the mass transfer coefficient times the membrane area per volume. Which volume is important determines which design is best. If the important volume is that of the fiber bed, then the vane module is marginally better because its area per volume is slightly higher (cf. Table 1). If the important volume is that of the entire module, then the vane module is the poorest design because the fibers fill only a small fraction of the module itself (cf. Fig. lc). A third definition is that the “best” module gives the most mass transferred per power required to move the fluid. At low flows, the power is proportional to the square of the velocity across the fiber bed, at higher flows, it varies with the velocity cubed. Thus a small power means a very small velocity. The best mass transfer per power would probably come from an unbaffled module based on hollow fiber fabric, followed closely by an unbaffled axially wound module. Baffled modules of either type will have a higher mass transfer coefficient; but the higher mass transfer coefficient will be purchased by a higher velocity and hence a dramatically increased power. The vane module will be poorer than other designs, because most
of the flow will go around the module, not across the fibers. Thus in this paper, we present mass transfer correlations for oxygen transfer from water across hollow fiber membranes into excess nitrogen. The correlations show faster mass transfer when the follow fibers are evenly spaced. For example, the mass transfer in modules easily made from hollow fiber fabric approaches that expected for single hollow fibers. The results suggest how better blood oxygenators can be built now, and how other module designs can be improved in the future.
Acknowledgements This work was primarily supported by the Environmental Protection Agency (MTU 00569) and by the Hoechst Celanese Corporation, Separations Products Division. Other support came from the National Science Foundation (CTS 91-23837), from DARPA (9205112), and from General Mills. E.L. Cussler held the Amundson Professorship during a sabbatical leave at M.I.T., where he was graciously received.
List of symbols a : b c CO
C*
d D
membrane area per module volume ratio of fraction factors total membrane area ratio of viscous to form drag solute concentration in the liquid flowing out of the module solute concentration in the liquid flowing into the module hypothetical solute concentration in the liquid in equilibrium with the excess gas outer fiber diameter diffusion coefficient
S.R. Wickramasinghe
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f,f H k kc, kL
1 P P
Q
Re Scl S, S; SC Sh u UO 4 u;
u2
6 E
V P
friction factors outside and inside the vane module Henry’s law coefficient overall mass transfer coefficient individual mass transfer coefficients in the gas and in the liquid, respectively length fiber bed in direction flow pressure membrane permeability volumetric flow Reynolds number, defined variously flume cross section vane module cross section fabric cross section Schmidt number Sherwood number superficial liquid velocity flume velocity velocity within the vane module local velocity across fibers of vane module velocity around the vane module membrane wall thickness void fraction of hollow fiber bed kinematic viscosity density
et al.jJ. Membrane
Sci. 84 (1993) l-14
References 1 2
3
4 5 6 7 8
9 10
11 12
13
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Winston Ho and K.K. Sirkar, Membrane Handbook, Van Nostrand Reinhold, New York, NY, 1992. S.R. Wickramasinghe, M.J. Semmens and E.L. Cussler, Mass transfer in various hollow fiber geometries, J. Membrane Sci., 69 (1992) 235-250. K.L. Wang, S.H. McCray, D.D. Newbold and E.L. Cussler, Hollow fiber air drying, J. Membrane Sci., 72 (1992) 231-244. MC. Yang and E.L. Cussler, Designing hollow fiber contactors, AIChE J., 32 (1986) 1910-1916. M.C. Yang and E.L. Cussler, Artificial gills, J. Membrane Sci., 42 (1989) 273-284. E.L. Cussler, Diffusion, Cambridge University Press, New York, NY, 1984. R. Prasad and K.K. Sirkar, Hollow fiber solvent extraction, J. Membrane Sci., 50 (1990) 153-1’75. L. Dahuron, Designing liquid-liquid extractions in hollow fiber modules, Ph.D. Thesis, University of Minnesota, 1987. cf. also L. Dahuron and E.L. Cussler, Protein extractions with hollow fibers, AIChE J., 34 (1988)130-136. S.R. Wickramasinghe, The best hollow fiber module, Ph.D. thesis, University of Minnesota, 1993. R.B. Bird, W.E. Steward and E.M. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, NY, 1960. M.M. Denn, Process Fluid Mechanics, Prentice Hall, Englewood Cliffs, NJ, 1980. R. Prasad and K.K. Sirkar, Dispersion-free solvent extraction with microporous hollow-fiber modules, AIChE J., 34 (1988) 177-188. H.B. Ding and E.L. Cussler, Fractional extraction using hollow fibers with hydrogel filled walls, AIChE J., 37 (1991)855-862. D.K. Schisla, H.B. Ding, P.W. Carr and E.L. Cussler, Polydisperse tube diameters compromise multiple tubular chromatography, AIChE J., 39 (1993) 946-953.