Mass transfer in various hollow fiber geometries

Journal of Membrane Science, 69 (1992) 235-250 Elsevier Science Publishers B.V.. Amsterdam

235

Mass transfer in various hollow fiber geometries

S.R. Wickramasinghe, Michael J. Semmens and E.L. Cussler Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 (USA)

(Received July 1,199l; accepted in revised form December 17,199l)

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 typi- cally one hundred times bigger than in conventional equipment.

However, the improved separations prom- ised 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 0 1992 Elsevier Science Publishers B.V. All rights reserved.

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 aca- demia is changing this picture, producing ultra- thin composite membranes with a much smaller mass transfer resistance. Parallel research on

236 S.R. Wickramasinghe et al,/J. Membrane Sci. 69 (1992) 235-250

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 ex- pressed 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 ex- plored new geometries where neither mass transfer correlations or heat transfer parallels exist. These new modules often outperform earlier designs, a tribute to their inventors’ in- tuition. Their development has been especially striking in the modules developed for blood ox- ygenation. 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 sub- stantial 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 sit- uations 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-

S.R. Wickramasingke et al./J. Membrane Sci. 69 (1992) 235-250 231

formance of these modules, and suggest how further improvements can be made.

Theory

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 ti- bers. 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) :

(k)= $ln$

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

s gdr=l

0

(2)

This distribution function allows estimation of the various averages which appear in eqn. (1) :

co

R= rgdr I

(3) 0

cc

V=nl r’gdr s

(4) 0

Q= z~r2vgdr

0 (5)

c-2

HAP = -

w I r4gdr

0

where 1 is 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

=i 4

(6)

r4cgdr/ r4gdr

0 0

But the concentration in one fiber is given by

C -=e

-2kl/ru

CO (7)

the analogue of eqn. (1) for a single fiber. When we combine eqns. (6) and (7 ), we obtain

cc> O” Co

- =

Cl3 5 r4e --6k~l~/&r~~&./

s

r4g&. (8)

0 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

uneven spacing is not as well known. Still, we (9) can begin by assuming channels between the fi-

bers which can be characterized by some appar- where R, is an average radius and (Roe, ) 2 is the variance of this distribution of radii. We now can use this to show from eqns. (3) that R equals R,. In this, we also assume that because E, << 1 the integration from zero to infinity is essentially that from minus infinity to infinity. Similarly, from eqn. (4))

V=dR; (l+$) (10)

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 ex- ponential in eqn. (8) as a power series in e,:

ent radius r. We can further assume that the velocity within these channels is proportional to r2, consistent both with the Blake-Kozeny equation and the Hagen-Poiseuille law. We can then parallel the analysis above to fined

(k) =k[l-(z,/m++:+...]

(15)

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

e - 16kjdz/Apr3 = (1_6E~+...)e-16k~‘/2/A~R~ (12)average mass transfer coefficient (k) falls be-

We then find low 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 con- nection with the experiments, which are described next.

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

(k)=k[l-(E +T}e:+...] (14)

Experimental

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 un- evenly spaced fibers. Such an analysis is more speculative than that given above because the

The chemicals and procedure closely imi- tated 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 chromato- graphy pump. Water saturated nitrogen under 10 psi flowed countercurrently to the liquid water. Oxygen concentrations in and out of the

S.R. Wickramasinghe et al./J. Membrane Sci. 69 (1992) 235-250 239

module were measured using a Orion model 97- 08-00 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 (Hoechst- Celanese model numbers 5010-8010 and 5010- 8020, 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

primarily in the way in which the hollow fibers were arranged. The first used a helically wound bed, shown rchematically in Fig. l(b) (Med- tronic “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 Wil- liam 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 Mit- subishi, 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.

(a) Flow Inside or Outside and Parallel (b) Flow Across a Helically Wound Bundle

Water Gas Out In

4 Wate Gas Out In

Water out

(c) Flow Across a Cylindrical Bundle

Water Out

(d) Flow Across a Rectangular Bundle

Gas In

;t ,

‘t’

+Gas Out

Water In

(e) Flow Along a Crimpled Flat Membrane

Water Out 4-

=+Gas Behind Membrane

Gas out

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

The final three modules studied, which are fined Sherwood numbers, defined as ( (iz) d/ also blood oxygenators, use a flat crimped D). The Sherwood numbers found from these membrane, shown schematically in Fig. 1 (e) experiments are plotted in Fig. 2 vs. Graetz (Cobe Cardiovascular Inc. models EXCEL numbers, defined as ( d2u/DZ). Data from four (050-123-000), ULTRA (150-120-000) and VP modules are shown in this figure. One set - the PLUS (050-125-000)) Arvado, CO). These open circles - are from a module with very care- modules use flat Hoechst-Celanese mem- fully aligned and spaced hollow fibers [ 91. Two branes, crimped to form blood channels shaped sets - the open and filled squares - are from

like isosceles triangles with a base of 150 pm commercial modules with less carefully aligned

and a side of 4 cm. These modules differ only fibers which are much more tightly packed (Fig.

in the number of channels, and hence in the la). The final data set - the open triangles -

membrane area. Mass transfer results for all are from a rectangular bed of hollow fibers (Fig.

these modules are given in the next section. Id).

Results

In this paper, we report measurements of mass transfer in six different types of membrane modules made by five different manufac- turers. In this section, we report the experimental values, and emphasize differences between the data. In the following section, we discuss correlations inferred from these experiments, and contrast these with literature data wherever similarities exist. For convenience, we organize the report in this section under four geometries: (1) flow inside the hollow fibers; (2) flow outside and parallel to the hollow

The data for all four modules in Fig. 2 agree closely with each other. This agreement exists even though the data for the rectangular bed uses hollow fibers made by a different supplier (Mitsubishi) than those for the three other modules (Hoechst-Celanese) . This agreement supports the contention that oxygen mass transfer is controlled by diffusion in the water, and is unaffected by diffusion across the membrane or on the shell side of the module [15]. Moreover, the data in Fig. 2 agree closely with the theoretical prediction of LQv$que [16], at least at high Graetz numbers. Indeed, mass transfer coefficients rarely agree with theory as exactly as in Fig. 2.

However, at low Graetz numbers, the Sher-

fibers; 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.

Flow inside hallow fibers

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

0.1 1 Graetz’N”mber 100 1000

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. (-) LMque limit, (- - - - ) correction calculated from eqn. ( 14).


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 a thin diffusion layer close to the wall of the fiber is no longer valid, the Ldvgque solution is expected to break down. Newman [17] pre- sents an extension to the LQveque solution by considering the terms neglected by LQvgque. Naturally when the diffusion boundary layer reaches the center of the tube, Newman’s extension too is no longer valid. In these cases, one must use the more rigorous Graetz solution [ 18-201.

Surprisingly, the deviations of the Sherwood numbers from the theory are in the opposite direction to the predicted improvements to the LtMque solution. In other words, at low flows, the experimental results fall below the solid line in Fig. 2, but the Graetz solution lies above the line. We believe that the deviations from theory in Fig. 2 are caused by the slight polydispersity in the hollow fiber diameters, and not by other limitations of the LdvCque analysis. To test this belief, we measured the diameters of individual hollow fibers for module # 8020, and found that these diameters showed a standard deviation of 5%. We then used eqn. (14) to estimate the change in (k). The result of this estimate, shown as the dotted line in Fig. 2, is in reasonable agreement with the data for all the modules. Similar deviations have been observed by others, for example Prasad and Sirkar [7,21] and Zander et al. [ 221. Analogous deviations have also been observed in heat transfer [23,24].

Flow outside and parallel to hollow fibers

In the experiments, shown in Fig. 3, mass transfer coefficients are found from changes in

Fig. 3. Oxygen mass transfer out of water flowing on the shell side of shell-and-tube modules. Both the Sherwood and Graetz numbers are based on the equivalent diameter d.. (0,O) refer to the shell-and-tube module models 8010 (7500 fibers) and 8020 (12000 fibers) respectively. ( 0 ) refer to the module made with a hollow fiber fabric. (-,- - - - -) refer to the correlations of Yang [9] and Prasad [ 71,

respectively.

the oxygen concentration in water flowing on the shell side of modules like those in Fig. 1 (a). Water-saturated nitrogen flows rapidly through each fiber’s lumen. Unlike the data for flow within the fibers, the results for the three modules studied are different. Those for module # 8010, which has 7500 fibers, give mass transfer coefficients about three times higher than those for module f8020, which has 12,500 fibers. While the different number of fibers seems too small to cause such a large difference, the fibers in module # 8010 are potted in a dumb- bell shaped shell which might facilitate flow around the fiber ends and reduce channelling. However, the data for module #8020 agree closely with earlier correlations of similar modules containing 16,120, and 300 fibers [ 91. All of these results show mass transfer coefficients varying almost linearly with velocity. In other words, all show Sherwood numbers proportional to Graetz numbers.

The data for the hollow fiber fabric module, shown as squares in Fig. 3, seem more consistent with the result of module # 8020 than with those of module #8010. Interestingly, however, these data seem to show a smaller variation of Sherwood number with Graetz number. This smaller variation is consistent with one earlier, careful study of mass transfer in this

type of module [21] and with that expected from heat transfer in shell-and-tube heat exchangers [ 11,121. As it is easier to avoid channelling with a few dozen heat exchanger tubes than with thousands of hollow fibers, we are tempted to attribute this smaller variation to reduced channelling. We have no quantitative reason to do so now. Accordingly, we will emphasize module #8020 in the discussion later in this paper.

Flow outside and across hollow fibers

These experiments used the modules in Fig. 1 (b ) , 1 (c ) and 1 (d) , and so involve somewhat different flows. However, the Sherwood numbers for these modules, shown in Fig. 4, agree closely. All vary roughly linearly with Reynolds number at low Reynolds numbers, i.e. at low flow. All seem to approach a variation with the 0.33 power of Reynolds number at higher flows. This 0.33 power is consistent with the variations observed for flow perpendicular to 300 carefully spaced fibers in a hand made module. However, the data for these modules seem to approach Sherwood numbers about half of those observed in the handmade module, whose

loo,,,.,,,,

Fig. 4. Oxygen mass transfer out of water flowing across hollow fibers. The Sherwood and Reynolds numbers are based on the outer fiber diameter. (A&,0) flow across fiber bundles that are rectangular, he&al, and cylindrical, respectively (cf. Fig. 1) . (-) correlation based on carefully spaced, handmade modules [9].

performance is similar to that of crossflow heat exchangers [ 91.

The close agreement of the Sherwood numbers in Fig. 4 suggests that the flow within the modules must be similar, In other words, it suggests that flow through a rectangular bed, flow through a helically wound bed, and flow through a cylindrical bundle all give similar boundary layers near the fiber surface, and hence similar mass transfer coefficients. This seems incon- sistent with experiments which claim that there is an optimum angle for winding helical modules [25]. The result certainly merits much more experimental attention, especially since similar modules with non-porous fibers are strong candidates for gas separations.

We have also made preliminary experiments for flow across a mat made of the hollow fiber fabric. We have not shown these data in Fig. 4 both because doing so crowds the figure and because we plan to describe these results in much more detail in a later paper. Nevertheless, we are sure that mass transfer coefficients for the hollow fiber fabric fall very close to those of the handbuilt module, and thus are dramatically higher at low flows than those for the commercially built modules. The implication is that the hollow fiber fabric has more monodisperse voids than the commercial modules, and hence has less flow channeling.

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 membranes may seem out of place in work empha- sizing hollow fibers. However, the crimped, flat membrane is basic to a commonly used blood oxygenator which is often described as the easiest to use clinically. As a result, we felt that it

S.R. Wickramasinghe et al./J. Membrane Sci. 69 (1992) 235250 243

1 I 1 *c* J 1 10 100

Graek Number

Fig. 5. Oxygen mass transfer across a crimped, flat membrane. The Sherwood and Graetz numbers are defined as (k)b/D and b’Q/DV. (O,O,El,A) modules with membrane areas of 0.40,0.85,1.25, and 3.0 m*, respectively.

merited the same effort which we spent on the other units.

The data for the three flat membrane modules studied agree closely with each other as shown in Fig. 5. Two points about this apparent agreement merit emphasis. First, Fig. 5 plots the Sherwood number vs. the Graetz number, as Figs. 2 and 3 do; but these dimensionless groups are now differently defined. Both groups are now written in terms of the channel width b, rather than the fiber diameter d. Thus the Sherwood number is written as ( ( It) b/D). The Graetz number, now defined in terms of the total flow per module volume is written as ( b2Q/ D V). The second point about the data in Fig. 5 is that the four modules reported were actually three separate units. One large module had a surface area of 3 m2; and an intermediate sized module had a surface area of 1.25 m2. The smallest module, intended for pediatric use, could be operated with an area of 0.4 m2. By using different parts which use different membranes, we could also operate the smallest module with an area of 0.85 m2. Thus the results in Fig. 5 are for three modules, one of which was operated in two different ways.

Discussion

The data in Figs. 2-5 show that mass transfer in these modules can be effectively corre-

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

compare with other, earlier efforts? (2 ) Which module geometries offer the fastest

mass transfer? We will begin to answer these questions in this final section of this paper.

Correlations

In Table 1, we compare the mass transfer correlations obtained here with those obtained earlier. Such comparisons should be made cau- tiously, for some aspects of correlations like these have been much carefully studied than others. We have tried to signal this caution in the organization of the table. The first column in the table gives the basic geometry studied. The second column gives the range of flows, ex- pressed as a Graetz or a Reynolds number. The Graetz number is defined as (d2u/DZ) for flow inside the fiber and as (d,2u/DZ) for flow outside and parallel to the fibers. The Reynolds number is defined as (du/v), except as indi- cated. For the flat membrane, the Graetz number is defined as b2Q/D V. The third column in Table 1 gives the mass transfer coefficient vs. those variables which are actually altered in the experiments, and the fourth gives the dimensionless correlation inferred from this variation. The difference between these columns is important. While both are consistent, the fourth often contains implicit assumptions. For example, in the easiest case of fast flow inside the fibers, column (4) asserts that the mass 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

TA

BL

E

1

Mas

s tr

ansf

er

corr

elat

ions

fo

r ho

llow

fib

er m

odul

es o

f va

ryin

g ge

omet

$

Flow

geo

met

ry

Flow

ins

ide

fibe

rs

Flow

ran

ge

Exp

erim

enta

l re

sultb

In

ferr

ed

corr

elat

ion

Lite

ratu

re

corr

elat

ion

Rem

arks

Gr>

4 (k

) =4

.3x

lo-s

fu/lj

”s

Sk=

1.62

G+

Sh=

1.

62G

r”3

The

ory

and

expe

rim

ent

don’

t ag

ree

at l

ow f

low

s, a

ppar

ently

be

caus

e G

rt4

(k)

=1.5

x 10

-4(u

/l)

Sh=

Sk&

{T

+7}

t;+

...]

of s

light

pol

ydis

pers

ity

in h

ollo

w f

iber

di

amet

ers

(cf.

eqn

. 14

).

Flow

out

side

an

d G

r<G

O

(k)

=2.5

x10-

%

Sh=0

.019

Gr’

.0

The

cor

rela

tion

obta

ined

he

re,

whi

ch

para

llel

to f

iber

s”

Sk=1

,25

!&

0.93

(V)1

/3

( >

VL

D

agre

es w

ith t

he e

arlie

r re

sult,

can

be

wri

tten

eith

er

vs.

Gra

etz

num

ber

or

vs.

Rey

nold

s an

d Sc

hmid

t nu

mbe

rs.

Flow

out

side

and

ac

ross

fib

ers

Re>

2.

5

Re<

2.5

(k)=

8.1~

10-‘

u”.s

(k)

=~.O

X~O

-~U

Sk =

0.3

9Re0

?W’~

s3

The

val

ues

of (

k)

are

less

tha

n th

ose

for

wel

l sp

aced

fib

ers,

but

in

crea

se

mor

e w

ith i

ncre

asin

g ve

loci

ty.

Flow

alo

ng a

cr

impe

d fl

at

mem

bran

&’

Gr>

11

<

k)

=O

.O02

5(Q

/A)‘

=

Gr<

ll <

k)

=3.

O(Q

lA)

Sh=6

.0G

r0.3

5

Sh=

1.

25G

?

The

res

ults

at

hig

h fl

ow a

gree

clo

sely

w

ith t

he l

itera

ture

co

rrel

atio

n,

but

thos

e at

low

flo

w d

o no

t.

“Dim

ensi

onle

ss

grou

ps

are

defi

ned

for

the

hollo

w f

iber

mod

ules

as

follo

ws:

She

rwoo

d nu

mbe

rs

Sh=

(k

)d/D

, G

raet

z nu

mbe

r G

r=d

%/

DI;

R

eyn

old

s nu

mbe

r R

e=d

u/

v;

Schm

idt

num

ber

SC=

v/D

. N

ote

that

d

is th

e fi

ber

diam

eter

ex

cept

as

indi

cate

d.

%ni

te

are

k: m

/set

; u:

m/s

et;

1: m

; Q

: m

3/se

c;

A:

m2.

‘T

he c

hara

cter

istic

le

ngth

fo

r th

is g

eom

etry

is

the

equ

ival

ent

diam

eter

4,

eq

ual

to f

our

times

the

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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 ge-

ometries 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 ver- ification 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, summa- rized by eqn. (14)) seem consistent with our experiments. (We urge caution in applying this theory quantitatively, for it is based on a Tay- lor series expansion. ) This result was for us un- expected, 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 resi- dence 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 theoreti- cally estimated from the average fiber diameter.

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 in- teresting because it gives faster mass transfer. Not surprisingly, this geometry is that fre- quently 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 ap- proaching this handmade limit; but at low Rey- nolds 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

246 S.R. Wickramasinghe et al./J. Membrane Sci. 69 (1992) 235-250

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 un- sure 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 specula- tion 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 Sher- wood 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. Re- assuringly, 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

CO

(16)

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.


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 mem-

brane into rapidly flowing, water saturated nitrogen

Flow Module type Membrane area Bed length Water flow k( X 10~%m/sec) Percent

geometry (mx) (cm) (cma/sec) removed

Inside fibers Shell and tube (Fig. la) 1.0 18 52 4.0 55

Shell and tube (Fig. la) 2.3 25 116 4.0 55

Rectangular bundle (Fig. Id) 3.2 14 160 4.3 57

Outside fibers Cylindrical bundle (Fig. lc) 1.8 10 90 8.5 82

Helical bundle (Fig. lb) 2.0 12 100 9.8 86

Shell-and-tube (Fig. la) 2.9 25 146 0.3, 7

Rectangular bundle (Fig. Id) 3.7 14 185 19 98

Parallel to flat Crimped membrane (Fig. le) 0.4 25 20 6.3 12

membrane Crimped membrane (Fig. le) 3.0 25 150 6.3 72

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. Be- cause 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-

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

CO (171

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-

248

TABLE 3

S.R. Wickramasinghe et al./J. Membrane Sci. 69 (1992) 235-250

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-‘) Percent removed (cm? (cm) (cm”/sec)

Inside fibers Shell and tube (Fig. la) 62 18 62 0.71 51 Shell and tube (Fig. la) 140 25 140 0.71 51 Rectangular bundle (Fig. Id) 180 14 180 0.80 55

Outside fibers Cylindrical bundle (Fig. lc ) 150 Helical bundle (Fig. lb) 150 Rectangular bundle (Fig. Id) 165 Shell-and-tube (Fig. la) 240

10 150 1.53 78 12 150 1.81 a3 14 165 4.0 98 25 240 0.08e 7

Parallel to fiat Crimped membrane (Fig. le) 70 25 membrane Crimped membrane (Fig. le) 450 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-and- tube modules with flow inside the fibers. Shell- and-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 im- plicitly 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. Remem- ber 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 hM and in the liquid ( k) :

1 1 1 -- K=Hk,+(k)

(19)

where H is a partition coefficient, the equilib- rium 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 Ta- ble 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


the oxygen-hemoglobin reaction. For the membrane properties to become important, HIzM must be about equal to (k) [8]. For this to occur, the half life of the oxygen-hemoglobin reaction must be less than 10m6 sec.

The results in both Tables 2 and 3 are limited to the case of fast nitrogen flow on the gas side of the membrane. Such fast flow insures that the oxygen concentration in the nitrogen is always near zero, and the concentration difference responsible for mass transfer is kept near its maximum possible value. Blood oxygenators are in fact operated under equivalent conditions: the air flow is kept high to maximize the concentration differences of oxygen and carbon dioxide, and hence maximize the mass transfer of these species.

Modules used for chemical processing, including those for absorption and extraction, will not be operated in this way. Instead, the two fluids will usually flow countercurrently to each other. Such countercurrent contacting gives more complete separations than either concur- rent flow or crossflow. Thus better membrane modules for chemical processing should try to include both local flow across the fibers, and countercurrent flow in the module itself.

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-Ce- lanese) , Marc Voorhees (Cobe Cardiovascular, Inc. ) and Wallace Jansen (Minntec ) . The work was largely supported by Hoechst-Celanese. Other support came from the National Science Foundation (grant CTS 89-12634), from Gen- eral Mills, and from the Center for Interfacial Engineering.

List of symbols

a

b c

CO

cc> d

g Gr

k

(k) 1

P

Q r

membrane area per volume blood channel size concentration inlet concentration average concentration average fiber diameter: twice the average radius equivalent diameter: four times the cross section for flow divided by the wetted perimeter distribution function (eqn. 2) Graetz number mass transfer coefficient average mass transfer coefficient (eqn. 1) module length pressure average water flow (eqn. 5) radius

R,R, average radii (eqns 3 and 9 ) Re Reynolds number Sh Sherwood number u velocity V average volume (eqn. 4)

60 standard deviation divided by the mean (eqn. 9)

; viscosity fraction of feed oxygen removed

References

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Mass transfer in various hollow fiber geometries

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