Analysis of steric hindrance reduction in pulsed protein ultrafiltration

Journal of Membrane Science, 85 (1993) 39-58 Elsevier Science Publishers B.V., Amsterdam

Analysis of steric hindrance ultrafiltration

39

reduction in pulsed protein

V.G.J. Rodgers* and Kathryn D. Miller Chemical and Biochemical Engineering Department, The University of Zowa, 125 C-B Iowa City, IA, 52242 (USA) (Received December 7,1992; accepted in revised form June 11,1993)

Abstract This study analyzes the effect of transmembrane pressure pulsing on transient at&c hindrance for BSA ultrafiltration in an unstirred batch cell using 100,000 MWCO cellulosic membranes. Studies were performed with fresh membranes and membranes preadsorbed with 0.1% IgG for 0.5,2.25 and 26 hours. The Kedem-Katchalsky model was used to determine the apparent sieving coefficients and estimate the wall concentrations. Hindered transport theory was used to estimate the resulting apparent pore size of the membranes for pulsed and nonpulsed cases. The apparent pore size was increased by 8% when pulsing was used in conjunction with fresh membranes. The intrinsic sieving coefficients were increased by 33% during pulsing. Pulsing also substantially increased the apparent pore size and the initial sieving coefficients for the IgG preadsorbed membrane cases. Based on the initial time data for the IgG preadsorbed membrane cases, it was determined that the presence of IgG on the membranes could affect the concentration of BSA in the permeate. It was also found that transmembrane pressure pulsing could partially eliminate this resistance to BSA permeability. Key words: ultrafiltration; transmembrane pressure pulsing; hindered transport; fouling; pore plugging; protein separation

Introduction Perhaps one of the most elusive elements of membrane transport resistances in protein ultrafiltration has been internal membrane fouling. While it has been established that concentration polarization resistance and external membrane fouling are the primary limitation to solvent flux [l-3], this can be controlled to some extent with increased shear rate [ 11. However, protein adsorption has been found to be the predominant factor in solute transport ‘To whom correspondence should be addressed.

0376-7388/93/$06.00

resistance [ 41. Increasing shear rate has not been effective in preventing internal membrane fouling or solute surface deposition. Recently [5,6], research on crossflow and batch cell ultrafiltration of a binary protein solution consisting of albumin (BSA, 69,000 Da) and y-globulin (IgG, 159,000 Da), separated with a 100,000 MWCO (molecular weight cutoff) membrane, showed that membrane fouling resistance could be reduced by the novel process of transmembrane pressure pulsing. Transmembrane pressure pulsing is a dynamic process in which the transmembrane pressure is oscillated between negative and positive val-

0 1993 Elsevier Science Publishers B.V. All rights reserved.

40

V.G.J. Rodgers and K.D. Miller/J.

ues at various frequencies and durations. This process has been able to improve solute transport by as much as two orders of magnitude in some crossflow studies [ 51. An analysis of the process implied that membrane fouling reduction was the dominant factor in improving the solute flux. However, it was also observed that during this process the maximum observed BSA concentration in the permeate was substantially less than that present when BSA is ultrafiltered without IgG. The observed phenomena may be caused by the adsorption of IgG in or near the pore that results in a reduced apparent pore size. However, studies have shown that pulsing significantly increases the concentration of IgG in binary protein ultrafiltration even when the concentration of BSA was only slightly increased [ 61. Using single solute solutions of BSA and IgG, it was found that the permeate flux for the pulsed cases was increased by an order of magnitude more than the nonpulsed cases going from 1.7 X 10m6m/set to 12 X 10V6 m/set. The concentration of BSA in the permeate during the pulsed case was near that observed for the nonpulsed case. This was expected since the permeate concentration was originally near the bulk concentration value. The single solute IgG study, however, showed an increase of six times in the permeate concentration. This implies that steric hindrance was reduced for the larger IgG molecule during pulsing. It is expected that transmembrane pressure pulsing improves solute flux by reducing protein adsorption and/or pore plugging in some pores. When a significant number of pores are cleared this results in an increase in intrinsic sieving coefficient and apparent pore size for the membrane (for a fixed porosity). Figure 1 schematically shows how this may be accomplished. It would be advantageous to quantitatively determine how much steric hindrance was reduced during the pulsing process to further

Membrane Sci. 85 (1993) 39-58

A PRIOR TO PULSING, PROTEINS ARE LODGED IN PORES Mcmhrane Section

Lodged Proteins Pores Blocked

Apparent Avg. Pore Size from Hindered Transpon Theory

B. AFTER PULSING, PORES CLEARED

Proteins Dislod

Apparent Avg. Pore Size from Hindered Transpon Theory

Fig. 1. Schematic diagram of proposed mechanism for reduction of steric hindrance due to transmemhrane pressure pulsing. In (A), the two left-most pores are blocked by lodged proteins and are excluded from any further transport of solute. Assuming the number of pores to be constant, the average pore size for solute transport is reduced. In (B), immediately after transmembrane pressure pulsing, the proteins in the two left-most pores have been dislodged allowing these pores now to transport solute. This effectively increases the average pore size for solute transport.

understand this separation problem. In addition, it would be advantageous to know the transient development of steric hindrance. The objective of this study is to use BSA in solution to analyze the potential transient reduction of steric hindrance in pulsed ultrafiltration with fresh membranes and membranes that were preadsorbed with IgG. A batch cell apparatus was used in this research. The effect of transmembrane pressure pulsing on the timedependent intrinsic sieving coefficients, wall concentration and average pore radii were also investigated. Osmotic pressure resistance model

V.G.J. Rodgers and K.D. Miller/J,

41

Membrane Sci. 85 (1993) 39-58

for flux and hindered transport theory were used in this analysis. The use of the batch cell ultrafiltration has some advantages over the crossflow cell in this analysis. It has previously been shown [ 71 that pulsing in the batch cell device did not effect the concentration polarization boundary layer. Using 30,000 MWCO membranes, the permeate flux was measured for the ultrafiltration of BSA. After pulsing at frequencies of 0.2,0.5, 2 and 5 Hz, it was concluded that pulsing had no significant impact on the permeate flux. However, transmembrane pressure pulsing with this device demonstrated a significant increase in the solute transport in semi-permeable ultrafiltration [ 61. Consequently, the effect of pulsing on internal fouling resistance can be isolated from potential polarization resistance modifications by using the batch cell.

ture that describe the flux dependency on the solution properties. Reviews of such models have recently been presented [ 3,8,9]. A model for flux dependency on solute concentration for unstirred batch cell ultrafiltration is the Kedem-Katchalsky model [ 10,111, J,=L,,(AP-aAx).

(3)

Here J,, is the velocity of the total volume flow, Lp is the hydraulic permeability of the membrane, AP is the hydrostatic pressure difference, a is the reflection coefficient and Ax is the osmotic pressure difference. This equation is not restricted to ideal dilute solutions in its present form [ 121. However, the value of the reflection coefficient, in terms of solute concentrations, is usually determined from an ideal dilute solution approximation to give [91, o=

Theoretical background

rep” eP’-l+r

(4)

where Pe is the Peclet number defined as For analysis of the effect of transmembrane pressure pulsing on membrane performance, it is appropriate to use the intrinsic sieving coefficient and the true retention coefficient which are

pe=

(l-dJv P

*

Here P is the solute diffusive permeability. When the true retention coefficient, r, is approximately 1, the reflection coefficient can be approximated by

L/w

frxr=l-2.

and r=l-Si

(2)

where C, is the permeate concentration and C, is the apparent solute concentration at the wall. The difficulty in evaluating the intrinsic sieving coefficient is that the wall concentration is often unknown. However, models have been developed to relate the solute wall concentrations and the solvent flux. Estimation of wall concentration

Several models exist throughout

the litera-

c C,

(6)

Thus eqn. (3) coupled with eqn. (6) results in a relationship between the solute wall concentration and the measured permeate flux provided a suitable representation of the osmotic pressure dependence on the solute concentrations can be found. The osmotic pressure for albumin is determined from a semi-empirical correlation developed by Vilker [ 131. The osmotic pressure is given by the expression a=$+xd

(7)

V.G. J. Rodgers and K.D. Miller/J.

42

where q.,

+(wp+A+;

+A2w;)

(8)

P

(9) where xr, is the osmotic pressure due to protein interaction, xd is the Donnan effect osmotic pressure, R is the ideal gas constant, T is the absolute temperature, w, is the mass concentration of the protein, Mp is the molecular weight of the protein, 2 is the macromolecular charge and m, is the salt molal concentration. The coefficients AI and A2 are adjusted to correct for pH and charge. Equations ( 3 ) , (6) - (9 ) can be used with experimentally determined values of hydraulic permeability, solvent flux and solute flux to iteratively estimate the solute wall concentration and, thus, the intrinsic sieving coefficients in the single solute transport studies. This model is valid for both the pulsed and the nonpulsed cases.

classical hydrodynamic theory has difficulty in obtaining relationships between pore radius and hydrodynamic parameters, and only series solutions for well defined geometries have been developed [ 15-171. However, recently it has been shown that these models, when modified, have had some success in predicting hydrodynamic parameters for asymmetric membrane transport of proteins [ 18,191 and dextran transport in asymmetric membranes [ 201. These studies provide the motivation for this research in applying classical hydrodynamics to quantitatively investigate the apparent average pore size and the relative variation of the average radius due to transmembrane pressure pulsing and the presence of preadsorbed IgG on the membranes. A complete review of classical hydrodynamics in hindered transport has been done by Deen [ 171 and the approach used in the study is described by Opong and Zydney [ 191. Therefore, only the final equations used in the analysis will be presented. The intrinsic sieving coefficient for steady-state transport of a solute through a pore is [ll]

(10)

Estimation of average pore size

One approach to determine the average pore size may be to analyze the hydraulic permeability of the membrane before and after pulsing to determine the significance of pore blockage. Although this has been an effective tool for analyzing the effect of adsorbed proteins on the average pore radius [ 41, it has been pointed out that the hydraulic permeability may not be as sensitive to relatively small reductions in radius as the solute transport [ 141. Transient values of hydraulic permeability would also be difficult to obtain. An alternate approach may be the use of available hindered solute transport theories. However, even with well defined geometries,

Membrane Sci. 85 (1993) 39-58

where $Jis the partitioning coefficient and Pe is the pore Peclet number which can be written as pe

KvL

=Tg7*

(11)

Here, Kc and Kd are the cross-sectionally averaged convective and diffusive hindrance coefficients, respectively. For cylindrical pores, with spherical solute particles and no interaction, it can be shown that the partitioning coefficient is [ 21,221 $= (l-2)2.

(12)

Models have been developed for specific geometries under different conditions, however, evidence has indicated that the centerline ap-

V.G.J. Rodgers and RD. Miller/J.

proximations produce reasonable hindrance coefficient estimates (Deen [ 171) . The model developed by Bungay and Brenner [ 231 will be used in this study, in keeping with the studies by Robertson and Zydney [ 181 and Opong and Zydney [ 191. In this model, eqn. (12) is used to represent the partition coefficient and the hydrodynamics coefficients can be expressed as Kdzv

671

(13)

nt

and (14) where Kt and KS are evaluated from the polynomial expansion in A as

(15) The coefficients are listed in Table 1. This model is accurate to within 2 10% of 1. Because the asymmetric membranes possess a pore size distribution, the solute to pore size ratio, A, can be considered an equivalent size TABLE 1 Coefficients for eqn. (15) [ 231 Subscript n 1 2 3 4 5 6 7

a, - 73160 77293150400 - 22.5083 -5.6117 - 0.3363 - 1.216 1.647

43

Membrane Sci. 85 (1993) 39-58

b, 7160 - 2227150 400 4.0180 - 3.9788 - 1.9215 4.392 5.006

ratio for the system to result in the obtained sieving coefficient [ 191. This parameter, however, is limited to the values between 0 and 1. Opong and Zydney [ 191 suggested that a more realistic value of this ratio for asymmetric membranes may-be obtained if the relationship for partition coefficient for a rigid spherical particle in a random arrangement of parallel planes is used [ 221, where +exp(

-2A*) .

(16)

In this expression, the ratio of the solute radius to pore radius, A*, is represented as A*=;

(17)

where t is the mean external length of the solute and s is the specific area (hydraulic radius) of the average pore. Note that A’ is not limited to values less than or equal to one, indicating that some transport is possible in this system for all particle sizes. These equations provide the framework used in this study to analyze the apparent average pore size from transient data. The procedure used to obtain an apparent average pore size consisted of first, using an iterative procedure to obtain an appropriate value of 3, to produce the sieving coefficient determined from the thermodynamic resistance model described above (eqns. 10-E). Then, the partitioning coefficient obtained in this analysis, together with the projected mean radius of the protein molecule, was used in eqns. (16) and (17) to obtain values for the average pore specific area. These values will be determined for every timesample to determine if any change is observed. Pseudo-steady state assumption The use of the hindered transport theory models presented requires the pseudo-steady state assumption to be valid during the batch cell and pulsed runs. Using an order of magni-

44

V.G.J. Rodgers and RD. Miller/J.

tude analysis, it was determined that the time constant for the batch process and the pulse intervals are sufficiently large to validate this assumption. The details of this are shown in the appendix. The pulse frequency is also sufficiently large to minimize any potential influence on the polarization boundary layer [ 61. Experimental

The experimental apparatus has been outlined elsewhere (Miller et al., [ 61) . In all cases 100,000 MWCO cellulosic membranes (Amicon Corp., Lexington, MA) were used as the separation medium. For the runs with preadsorbed IgG, a 0.1% solution of IgG was prepared and the membrane was placed in the solution to soak for either 0.5 hr, 2.25hr or 26 hr. Afterwards the membrane was placed in the cell and the experiments were performed. The pulse frequency was 0.2 Hz and the operating pressure was 70 kPa. The pulse amplitude was 70 kPa for all pulsed cases. Results and discussion Average error and repeatability

Repeated runs for the pulsed 2.25 hr and 26 hr preadsorbed IgG cases were performed to obtain an estimate of the average experimental error and repeatability for the sieving coefficient and apparent pore sizes. The average differences in the time averaged sieving coefficient were determined to be f. 5% and ? 3% for the 2.25 hr and 26 hr cases, respectively. The average differences between sieving coefficients at specific times were determined to be 59.9% and + 2.4% for the 2.25 hr and 26 hr cases, respectively. The average difference in the time averaged apparent pore specific areas were 2 4% and + 0.9%, respectively, for the 2.25 hr and 26 hr cases. Error values of + 3% and + 0.5% were determined for the average differ-

Membrane Sci. 85 (1993) 39-58

ence in the specific pore area at specified times, for the 2.25 hr and 26 hr cases, respectively. From these results the largest errors were used to represent the error in repeatability. These errors are the largest of either + 10% or kO.34~ 10m2 for sieving and the largest of either 24% or 20.35 A for the apparent pore specific area. The most conservative cases from these repeated runs are presented below. Permeate

flux

and concentration results

The permeate flux and concentration for the fresh membranes has already been discussed in detail elsewhere [6]. However, these results have been included in this discussion for comparison. Figure 2 shows the effect of transmembrane pressure pulsing on the permeate flux for the fresh membranes and the preadsorbed IgG cases. It is clear that the nonpulsed case for all the membranes resulted in a permeate flux which was significantly less than the pulsed case. For the preadsorbed membranes, the nonpulsed permeate fluxes after one hour were (1.9*O.14)X1O-6 m/set, (1.1&O.O3)X1O-6 m/set, and (0.9 + 0.03) x 10m6 m/set for the membranes preadsorbed in IgG for 0.5 hr, 2.25 hr and 26 hr, respectively. This is compared to values hour of pulsed after one (8.8?0.19)~10-6 m/set, (6.220.28)x10T6 m/set, and (4.8 + 0.22) x 10V6 m/set, respectively. It is also apparent that the permeate flux decreased as the time of adsorption was increased for all cases. This phenomena is most likely due to surface adsorption of IgG. It can be concluded that pulsing is not effective in completely eliminating fouling due to adsorption on the membrane surface. This is confirmed further when one compares the permeate flux without IgG adsorption to the case with only 0.5 hr of adsorption. In the case without preadsorbed IgG, the permeate flux was 12 x 10e6 m/set when pulsing, but, as stated above, this was reduced to

V.G.J. Rodgers and K.D. Miller/J.

Membrane Sci. 85 (1993) 39-58

45

Cl

Ohr,NoPuk

n

0lx.Pu1se

0

0.5 hr, No Pulse

.

0.5br.Pdse

A

2hr,NoPulse

.

2hr,Pulse

0

2bhr,NoMse

l

26iv.Pulse

0

0.5 br, pulsingstopped

v

2 hr.Pulsingstowed

*

26 hr,Pulsingstopped

0 0

1000

2000

3000

4000

5000 6000 Time (s)

7000

8000

9000

Kl

Fig. 2. Permeate flux versus ultrafiltration time for fresh and IgG preadsorbed membrane studies. Data are shown to represent the time in hours of preadsorbed IgG on the membrane, and whether the ultrafiltration process was operated with pulsing. The last three symbols in the legend represent the data that show how the permeate flux changed in the respective pulse case when pulsing was terminated. Average error in permeate tlux is f 0.3 X lo-’ mlsec.

8.8x lo-’ m/set after only 0.6 hr of exposure of the membrane to IgG. Analysis of the nonpulsed cases compare values of 1.7 X lob6 m/ set when no preadsorption was performed to (1.9 2 0.14) x 10m6m/set. These results are indistinguishable and do not provide any indication that additional fouling has taken place as a result of preadsorption. From a practical point of view, these results may also indicate that traditional backflushing, which may be on the order of minutes, cannot reduce membrane fouling in the same manner as transmembrane pressure pulsing. Figure 2 also shows the rapid decline in flux for each case when pulsing was turned off. In almost every case, the following sample had reduced to the nonpulsedpermeate flux level. This implies that, although permeate flux resistances are dominated by concentration polarization and external fouling, internal fouling can contribute to some flux reduction. A summary of BSA concentration in the permeate versus time is shown in Fig. 3. The fresh membrane shows a continuous difference between the permeate concentration for the

pulsed case and the nonpulsed case. The average permeate concentration for the fresh membrane studies was 8.4 & 0.78 g/L and 10.15 0.37 g/L for the nonpulsed and the pulsed cases, respectively. This results in an increase in permeate concentration of 20.2% due to pulsing. It is important to note that the pulse concentration is at the bulk concentration value, implying very little BSA rejection. For the preadsorbed cases, the initial concentration is lower than for the fresh membrane studies, with the membranes exposed to the IgG solution for 0.5 hr having the lowest initial BSA concentration. It is not clear why the increased exposure time of the membranes to the IgG solution resulted in slightly higher BSA concentrations. It is speculated that the increased soaking time, irrespective of the presence of IgG, may change the physical structure of the membrane due to slight but significant swelling on the thin selective skin that slightly affects the internal pore structure. Therefore, direct comparisons of studies across different exposure times should be made with caution. Since the fresh membranes had to be

V.G.J. Rodgers and K.D. Miller/J.

46

01 ,, 1. #//,. /I,m,j, 1000

2000

3000

$000 Time (s)

0

0 hr.No

n

0 hr, Pulse

o

0.5hr. ‘d<, Pulw

l

0.5hr, Pulse

A

2 hr. No Pulse

Pulse

A

2hr, Pulse

o

26 hr, No Pulse

l

26 hr, Pulse

I I

0

Membrane Sci. 85 (1993) 39-58

5000

I’

6000

/

7000

Xf 0

Fig. 3. Permeate BSA concentration versus ultrafiltration time for fresh and IgG preadsorbed membrane studies. Average error in concentration is + 0.1 X 10m3 kg/L.

soaked in distilled water for 0.5hr to remove the glycerin, it may be assumed that the affect of soaking time is negligible between the the fresh membrane studies and the studies for the membranes exposed to IgG for 0.5 hr. Comparing these two sets of studies, it is clear that during the initial phase of the ultrafiltration operation ( < 1500 set), the presence of IgG on the membrane significantly reduces the concentration of BSA in the permeate. The increase in BSA permeate concentration over time is probably due to the desorption of IgG from the membrane. However, one should note that BSA permeate concentration also increased in the nonpulsed case for the fresh membranes, although not as significantly. Recall that it was observed in previous research [ 2,5] that the presence of IgG in solution with BSA caused a significant reduction in the BSA permeate concentration (about 60% reduction for 1% BSA and 0.1% IgG solutions). The results of this study imply that IgG interaction with the membrane during binary protein ultrafiltration, may contribute to the reduced BSA concentration observed in those studies. One should note that, since the IgG present on

the membranes in this study resulted from establishing an equilibrium with a 0.1% solution of.IgG, this phenomenon may be even more severe in the binary protein case, where the presence of IgG at the membrane surface may be much larger than 0.1% and not likely to desorb. For large time, the concentration differences between the pulsed and the nonpulsed cases became nearly indistinguishable, implying significant desorption of the IgG from the membrane. The times required for permeate concentration to reach its near steady-state values are consistent with the times required for the nonpulsed fresh membrane to reach its near equilibrium value. The effect of stopping the pulse showed no change in the permeate concentration. Estimated

wall concentration

The wall concentrations for the single solute pulsed and nonpulsed studies were determined by applying the Kedem-Katchalsky model (eqns. 3, 6-9) to the measured flux and permeate concentrations. A Newton-Raphson model was developed that, upon an initial guess,

V.G.J. Rodgers and RD. Miller/J.

Membrane Sci. 85 (1993) 39-58

41

0

Ohr,NoPulse

A 2hr,NoPulse

0

1000

Fig. 4. Calculated wall concentration

2000

3000

4000 5000 Tie (s)

versus ultrafiltration

7000

6000

8000

time for studies using fresh and IgG preadsorbed

membranes.

TABLE 2 Summary of BSA transport Calculated values

No exposure Puke

Wall cont. (kg/L)

0.283 f 0.004

six 102

Pe

No pulse

A%

0.307 k 0.001

-6

2.7 0.26

11.4

10.6

f 0.10

Ik 0.24

41.1

Pulse

0.296 f 0.006

33

4.9 f 1.3

2.25 hr exposure No

8

739

0.306 rt 0.001

2.65

2.1

0.25

rt 0.59

f

0.10

A% are based on nonpulsed

studies (values time averaged)

10.6

9.9

I!I 0.25

If

36.4

13.8

k 5.9

f

Pulse

-3

0.297 + 0.003

f

3.1 f 0.15

2.8 k 0.25

11

10.9 f 0.23

10.7 f 0.23

2

cases. Repeatability

26

7

0.7 164

0.306

6.0

A%

Pulse

-3

0.298 + 0.006

0.001

6.9

3.2 +

3.5

No pulse 0.307

255

A%

-3

f 0.001 3.0

8

f

0.14

0.20

11.1

10.8

+ 0.12

Ii 0.12

19.1 f

f

error in s, f 0.35 A. Repeatability

converged to a solution for the wall concentration within a relative error in flux of k 5 x 10m3. Separate analysis was performed for all time

No pulse

24.5 f

7.4

26hr exposure

A%

Pulse

f

AI 6.6

membrane

30 mm exposure

3.6 f

s (A)

with IgG preadsorbed

4.5

3

324

f 6.5

1.3

error in Si, f 0.34 x 1O-2.

values for which permeate concentration and flux were available. Since & in eqn. (3 ) is reduced during mem-

48

V.G.J. Rodgers and RD. Miller/J.

brane fouling, the approximated value for it may affect the estimation for the wall concentration. A sensitivity analysis was performed to determine the significance of variation in hydraulic permeability on calculated sieving coefficients. It was found that an order of magnitude variation in hydraulic permeability (which reflects potential changes that could occur during ultrafiltration) resulted in only a 1.7% decrease in sieving coefficient. This translated to a 1% increase in the apparent pore size. Thus, these results imply that the more subtle changes in & due to membrane fouling are not likely to contribute significant error in the sieving coefficient and resulting apparent pore size. The calculated transient apparent wall concentrations are shown in Fig. 4. It can be seen that the pulsed case, with fresh membranes, resulted in the lowest calculated apparent wall concentration followed by the pulsed cases with increasing adsorption time of IgG. This is directly related to the increased permeate flux for the pulsed cases. However, the deviation in the estimated wall concentrations between pulsed and nonpulsed cases is approximately 6%. Table 2 summarizes the average wall concentrations for these studies.

Membrane Sci. 85 (1993) 39-58

Calculated sieving coefficients

The resulting sieving coefficients calculated for these runs are shown in Fig. 5. For nearly all cases, the sieving coefficient increased for the first 2000 set and then reached near steadystate values. The average sieving coefficient values for the fresh membrane studies were (3.6_+0.1)X10-2 and (2.750.26)x10-2 for the pulsed and the nonpulsed cases, respectively. The difference is outside of the experimentally determined average and represents an average increase in the sieving coefficient of 33% due to pulsing. Although the sieving coefficient did increase due to pulsing, it should be noted that the permeate concentration for both the pulsed and nonpulsed cases showed little rejection for albumin. The sieving coefficients for the preadsorbed IgG studies are also shown in Fig. 5. In these cases, the difference between pulsed and nonpulsed cases gradually reduced as the adsorption time of IgG was increased. The initial time values show a significant difference between the pulsed and the nonpulsed cases for each IgG exposure study with the maximum difference in the intrinsic sieving coefficient being ap-

-3.5 353.0

82.0 P 'c1.5 d

Fig. 5. Calculatedintrinsic sievingcoefficientversus ultrafiltrationtime for membranes.

0

0.5hr,NoF'ulse

A

2hr,NoPulse

o

26hr,NoPulse

studies using fresh and

IgG preadaorbed

V.G.J. Rodgers and K.D. Miller/J.

Membrane Sci. 85 (1993) 39-58

proximately 1.4 x lOA2for the cases which were preadsorbed with IgG for 0.5 hr. This represents an increase in sieving of 140% due to pulsing. Sieving for the complete run for this case was increased by an average of 26% due to pulsing. For the cases in which IgG was preadsorbed for 2.25 hr and 26 hr, the differences in sieving due to pulsing were not as dramatic, the maximum increase being about 30%. The average increase in sieving due to pulsing was 11% and 8%, respectively, which is on the same order as the experimental error. Table 2 summarizes these results. It appears from these results that transmembrane pressure pulsing can improve sieving even when the membrane is preadsorbed with IgG. However, since the highest observed sieving coefficients was during pulsing with fresh membranes, it may be concluded that some adsorbed proteins cannot be removed even during pulsing under these operating conditions. Apparent pore specific area

The apparent pore size was determined using the hydrodynamic theory for non interacting solid spheres in cylinders to approximate the partitioning coefficient (eqns. 10-15). This was accomplished by using a Regula-Falsi iteration procedure that converged on a solution for Iz when the value of the calculated sieving coefficient was obtained within an absolute error of + 1 x 10-4. The model was designed to accept only real-valued solutions between 0 and 1. The value of the partitioning coefficient obtained from these calculations was then used with eqns. (16) and (17) to determine the average apparent specific area of the pore. The projected mean diameter of albumin was determined to be 80.6 A [ 181. Figure 6 shows the time dependent apparent pore specific areas, s, for the fresh membrane runs. It is clear that the values for the pulsed case are significantly larger than those deter-

49

mined from the nonpulsed cases. The average increase in pore specific area due to pulsing was 0.8 A which is substantially larger than the error in repeatability of t 0.35 A. These average values of specific pore area were determined to be 10.620.24 A and 11.4+0.1 A, for the nonpulsed and pulsed cases, respectively. The average is determined from the mean for all time values calculated. This relatively constant value for calculated specific area during pulsing resulted even though the flux during the pulsed run reduced from an initial value of 2.3 x 10m5 m/s& to 1.2 x 10m5m/set within the first 2000 sec. The fact that these results are determined from data collected independently at different times throughout the runs gives considerable credibility to this method in providing some relative difference in the pulsed and the nonpulsed studies. The average apparent pore specific areas for the IgG preadsorbed membrane studies are shown in Fig. 7. As can be seen there are no significant distinctions between the pulsed cases and the nonpulsed cases after long time for any case. The short time trend is the same as that for the calculated sieving coefficients. The effect of IgG adsorption on the apparent specific pore area is difficult to determine from these results. It appears that the increase in adsorption time resulted in an increase in the apparent specific pore area, for the respective pulsed and nonpulsed cases. As stated earlier, this apparent increase may be due to swelling of the membrane semi-permeable thin skin when soaked for longer periods of time. However, the pulsing versus nonpulsing data can be analyzed for the same soaking time. This will eliminate any contribution from swelling in the analysis. Comparisons made between pulsed and nonpulsed cases for the same IgG exposure time show that the difference between the pulsed and the nonpulsed case reduced as the exposure time of the membrane to IgG was increased. It appears that desorption was taking

V.G.J. Rodgers and K.C. Miller/J.

50

Membrane Sci. 85 (1993) 39-58

9.0 8.5 ’ 8.0 / 0

1000

2000

3000

4000

5000

6000

7000

8000

Time (9

Fig. 6. Calculated average pore specific area versus ultrafiltration time for studies using fresh membranes.

A

8.5

:o

8.0;,, 0

2h1,NoF’ulse

I

I I,, moo

,,

1,. 2000

,’

I’,’ 3000

‘1” / ““! 4000 5000 Time (s)

‘~~‘i “‘I 6OOo 7000

1 8000

Fig. 7. Calculated average pore specific area versus ultrafiltration time for studies using IgG preadsorbed membranes.

place in that the pore specific area generally was increasing in the first 1500 sec. However, it again should be noted that this trend was also observed for the fresh membrane case without pulsing although the increase in pore specific area was not as intense. Comparing Figs. 6 and 7 we can see that it appears as though the nonpulsed membranes with the presence of IgG initially had a reduced specific pore area that increased significantly

during the ultrafiltration process to approach the value obtained for the fresh membrane runs after a long time. The calculated specific pore areas are most sensitive to permeate concentrations and the transient specific pore area generally follows the trend of the transient permeate concentration during the runs. The apparent increase in specific area for the membranes exposed to IgG for 0.5 hr follows the trend of the transient permeate concentration

V.G.J. Rodgers and K.D. Miller/J.

Membrane Sci. 85 (1993) 39-58

for BSA during these runs. During this period of time the permeate flux was decreasing. This phenomenon may be explained by the fact that the permeate flux is dominated by the boundary layer development above the membrane whose resistance increases over time [24]. Thus, the small transient changes in pore size are not enough to compensate for the increased external resistance. The concentration of BSA in the permeate is influenced by a small variation in the pore size. It is possible that the IgG is only weakly bound to the generally hydrophilic cellulosic structure of the membrane after an exposure of 0.5 hr and, thus, over a period of time, the IgG gradually desorbs from the membrane pores during ultrafiltration. These figures also imply that pulsing appears to increase the rate in which IgG is removed but the membrane does generally not return to the fresh membrane performance within the experimental error. Table 2 summarizes the apparent specific pore areas. The error determined for the repeatability experiments in transient specific area appears small enough to allow us to analyze the trend of specific area change over time and make comparisons confidently. To be complete however, it is important to analyze whether the error in the parameters used in the calculation could result in a larger potential error than observed. Error analysis of specific area calculation

A more thorough analysis of the approach used to determine the specific area is important before any final conclusions can be drawn. The primary models in determining these values include the relationship of the sieving coefficient (eqn. 10) with respect to the partitioning coefficient and the Peclet number as determined by eqn. ( 11)) and the relationship of the partitioning coefficient with respect to s (eqn. 17). The first part of this analysis is based on the as-

51

sumption that the solution is dilute and well modeled by spherical solute molecules in cylindrical pores. This is clearly not true for protein separation with asymmetric membranes but it may be argued that relative values in this analysis provide insight to the reduction of hindrance due to pulsing and the effect of IgG adsorption on hindrance. The value of the partitioning coefficient, the hydrodynamic coefficients and Pe are important in this analysis and these values are dependent on the value of A. Since il is the ratio of the spherical solute radius to the cylindrical pore radius, these results can be critical to the final solution. The following is a summary of the error analysis of these parameters in determining the sieving coefficient and apparent specific pore area. Reflection coefficient

The approximation used to determine the reflection coefficient could have some impact on the calculated sieving coefficient. The reflection coefficient used in this analysis is derived from a statistical mechanics model and assumed a dilute solution and that the retention coefficient was near 1. However, other approximations have been proposed for this parameter which are based on hydrodynamic theory. For convective dominant transport, when there is no long-range interaction potential between the solute and the pore wall, the convective reflection coefficient has been determined to be [ 17,25-271 af=l-@KC.

(18)

Another model for the reflection coefficient, the osmotic reflection coefficient, uses the flux model (eqn. 3) for very dilute solutions with neutral solutes, and has been shown to be [23,29 1

r&= (l-$)2.

(19)

52

V.G.J. Rodgers and K.D. Miller/J.

Membrane

Sci. 85 (1993) 39-58

TABLE 3 Comparison

of reflection

coefficients Used u= (l_Si)

Convective of= (l-@&I

A%

Osmotic a,= (l--$)2

A%

BSA No pulse Fresh membrane

0.973

0.973

0.0

0.955

- 1.8

BSA Pulse Fresh membrane

0.964

0.964

0.0

0.942

-2.3

BSA No pulse 0.5 hr exp.

0.979

0.979

0.0

0.965

-1.0

BSA Pulse 0.5 hr exp.

0.973

0.973

0.0

0.956

-1.7

BSA No pulse 2.25 hr exp.

0.972

0.972

0.0

0.954

- 1.9

BSA Pulse 2.25 hr exp.

0.969

0.970

0.1

0.950

-2.0

BSA No pulse 26 hr exp.

0.970

0.971

0.1

0.952

-1.9

BSA Pulse 26 hr exp.

0.968

0.968

0.0

0.948

-2.0

Study

A% values based on used reflection

coeffkient.

The values of the reflection coefficients from these models were determined and compared to the reflection coefficient used in this study. The tabulated results are shown in Table 3. The calculated reflection coefficients are within excellent agreement with the convective and osmotic reflection coefficients being within 2.3%. The difference is well within the error in the estimated sieving coefficient based on experimental calculations.

Hindrance coefficients In Table 4 it can be seen that the KC values are all within a narrow region for both the pulsed and the nonpulsed cases. The average variation in KC values for all runs was 0.7%. Pulsing, nor the presence of IgG did little to effect KC. Kd had an average variation during each run of 10.7%. Using average values of KC and Kd of 1.2and 6.77 x 10m3,respectively, for an average Peclet number, Pe, of 8, this corre-

V.G.J. Rodgers and K.D. Miller/J.

Membrane Sci. 85 (1993) 39-58

TABLE 4 Other hindrance parameters calculated in this study Study

#x10’

I

;1”

BSA No pulse Fresh membrane

2.26 iz 0.20

0.85 1.90 + + 0.006 0.09

1.19 6.20 f It 0.008 0.70

BSA 2.92 PIllSI? f Fresh membrane 0.09

0.83 k 0.003

1.77 k 0.01

1.22 !I 0.005

8.75 !I 0.30

BSA No pulse 0.5 hr exp.

1.76 f 0.50

0.87 2.04 f f 0.019 0.15

1.17 Ik 0.023

4.56 f 1.55

BSA P&e 0.5 hr exp.

2.22 f 0.20

0.85 1.91 f IL 0.007 0.05

1.19 + 0.008

6.33 f 0.59

BSA No pulse 2.25 hr exp.

2.35 k 0.19

0.85 + 0.006

1.88 f 0.04

1.20 6.58 + f 0.008 0.68

BSA Pulse 2.25 hr exp.

2.53 f 0.14

0.84 1.84 f f 0.006 0.03

1.20 7.29 + + 0.006 0.49

BSA No pulse 26 hr exp.

2.42 f 0.09

0.84 1.86 IL + 0.003 0.02

1.20 6.82 f ?I 0.004 0.40

BSA Pulse 26 hr exp.

2.65 f 0.11

0.84 1.82 k a 0.003 0.02

1.21 7.68 * f 0.005 0.41

KC

Kdx lo5

sponds to an error of 5 0.8. For an average partitioning coefficient of 2.4 x 10e3, these errors propagate to an error in the sieving coefficient of + 0.2 x low3 or 7%. This is on the same order as transient differences in sieving coefficient measured in each run which contributed less than 3% error in the apparent specific pore area. Table 4 shows a summary of the remaining calculated parameters used in this study. Partitioning coefficients

This analysis suggests that the dominant

53

factor in estimating the sieving coefficient in this study was the partitioning coefficient. Although the partitioning coefficient is also initially dependent on the value of ;2,the relationship for a random plane distribution, developed by Giddings et al. [ 221, is subsequently adopted for determining the apparent average pore specific area. These researchers indicated that this relationship, although superior to other relationships for the partitioning coefficient and the diameter ratios, was nevertheless, only an approximation. They also showed that variations in the aspect ratio of the solute could lead to deviations in the calculated partitioning coefficient. An appreciation of this effect requires additional studies with species having different aspect ratios than albumin. Studies associated with the transport of IgG, which has an aspect ratio of 5.3 as opposed to the 3 value for albumin, are forthcoming. The combined error in the calculated radii from all of the parameters is less than the average difference in the observed radii due to pulsed and nonpulsed cases. At this point it would be advantageous to compare how the hydraulic permeability analysis for pore radii compares to the estimated radii from hydrodynamic theory. Comparison of specific radius with hydraulic permeability determined radius

Initial hydraulic permeabilities studies were performed for the runs with preadsorbed IgG on the membranes. The resulting hydraulic permeabilities were determined to be 6.95O.36x1O-7 m-set-‘-kPa-’ and 7.1+ 0.33 x 10m7m-set-l-kPa-’ for the 2.25 hr preadsorbed case and the 26 hr preadsorbed case, respectively. This is compared to the average hydraulic permeability for the fresh membranes of 11.64 +0.79X 10m7 m-set-‘kPa-‘. These values are based on the nominal

54

V.G.J. Rodgers and K. D. Miller / J. Membrane Sci. 85 (1993) 39-58

cross-sectional area of the membrane of 14.52 cm2. From Darcy’s law the average pore size for straight cylindrical pores is related to the hydraulic permeability as .,“2 L -p-&L

(20)

where r is the average pore radius, L is the pore length, y is the pore area fraction and q is the solvent viscosity. Direct calculation of the average pore radius from this model requires a good approximation of the membrane void fraction and the average pore length [ 301. The pore length, which is also generally estimated, is on the order of the membrane thickness. Although the membrane thickness was averaged at 0.8 pm for the above calculations, the manufacturer indicated that the membrane thin skin thickness ranges from 0.1 to 1.5 ,um. This alone can contribute to a 4 fold range in the average pore radius. In addition, the pore area fraction has an r dependency which can be shown for cylindrical pores to be y=Enr2

(21)

where E is a constant associated with the nominal membrane area, and n is the number of pores available. In this model, if it is assumed that the number of pores is constant then, during the plugging process, the average radius must reduce to account for the reduction in pore area fraction. During severe plugging, in which some pores completely block water permeability, for a fixed n, these pores would contribute a pore radius of zero to the average. Using an r4 dependency, it appears that the average radius of the membrane was reduced by 12% when exposed to IgG for the 2.25 hr and 26 hr cases. For a void fraction of 0.8, the average radius, based on the hydraulic permeability data, is between 11 A and 42 A for the fresh membrane hydraulic permeability study.

The range is established to account for the uncertainty in the pore length. Taking the ratio of hydraulic permeabilities can result in determining a radial difference within 2%, based on propagation of error. Deviation in radius calculations using hydrodynamic theory during transient studies was found to be as low as 1.1% (see Table 2 ) . It is difficult to get initial pore size information from the hydrodynamic studies since the membrane pore size may have reduced substantially prior to the aquisition of the initial data point. However, using the pulsing data for the highest initial pore size, the average pore radii (twice the average specific area) were determined to be 23 A, 22.2 A, and 22.4 A for the fresh membrane case, the 2.25 hr exposure case and the 26 hr exposure case, respectively. This represents only a 3% change in the initial pore sizes and only an expected 12% difference in hydraulic permeabilities. The average radius values do fall within the range of the radii based on hydraulic permeability however. Thus the pore radii determined in this study can be assumed to be within a reasonable approximation although both the hydrodynamic theory based method and the hydraulic permeability method require significant approximations. Mochizuki and Zydney [20] used the hydrodynamic theory method presented in this manuscript and the hydraulic permeability method to compare estimated average pore radii for dextran transport through membranes. They found a factor of nearly two difference in the estimated pore radii using the two methods in some cases. To determine if the difference in radii between pulsed and nonpulsed studies could be determined, post-operative hydraulic permeability studies were performed for several separate studies for fresh membranes. The resulting hydraulic permeabilities, were all within the error of measurement. The difference in average pore radii based on hydrodynamic theory implies that the post-operative hydraulic

V.G.J. Rodgers and K.D. Miller/J.

Membrane Sci. 85 (1993) 39-58

permeability should differ by 25% to account for the radial change observed for the fresh membranes. However, since the average hydraulic permeability radii are based on the transport of solvent and the average hydrodynamic pore radii are based on solute transport, they can theoretically deviate. Conclusion Calculations of transient wall concentration and sieving coefficients along with apparent average pore size, based on hydrodynamic theory, were used to determine whether pulsing resulted in an apparent relative increase in the average pore area. It was concluded that pulsing appeared to increase the average pore radius available for transport by as much as 8% for fresh membranes. The sieving coefficient increased by approximately 33% during pulsing. This increase in average apparent pore specific area was a result of reducing the pore blockage and perhaps adsorption of proteins within the membrane pore or directly at the pore mouth. It was shown that this is not a result of reduced polarization resistance due to pulsing. Thus it can be concluded that pulsing does minimize the loss of pore size without damaging the membrane. This can have a significant effect on the current methods of ultrafiltration in many industrial and clinical settings. The preadsorbed IgG studies showed that BSA was potentially hindered by the presence of y-globulin on the membrane in the initial phase of the experiments. The desorption of IgG observed in these experiments is not not likely in binary protein studies [ 61. This, coupled with the fact that pulsing increased the transport of IgG, indicates that the observed BSA concentration reduction in the permeate in binary protein studies may be due to IgG-membrane interaction and solute-solute interaction between IgG and BSA. A further analysis of the

55

binary protein studies based on these methods is forthcoming. Acknowledgement The authors wish to thank A. L. Zydney for his valuable discussions. The authors would also like to thank the reviewers of this manuscript for their constructive comments and suggestions. The authors gratefully acknowledge the Amicon Corporation for their generous contributions of membranes. Financial support was provided by the GE Foundation, NIH Biomedical Seed Grant and The USDA (Grant No. A16-20-03). List of symbols membrane nominal surface area coefficient for eqn. (8) coefficient for eqn. (8) parameter in eqn (15) shown in Table 1 parameter in eqn. (15) shown in Table 1 radially averaged solute concentration in pore solute concentration solute concentration in the permeate solute concentration at the membrane wall solute concentration at the membrane wall species diffusivity constant in eqn. (21) velocity of total volume flow through the membrane convective hindrance coefficient in hindered transport model diffusive hindrance coefficient in hindered transport model polynomial expansion in 1 evaluated in eqns. (13) and (14) polynomial expansion in A evaluated in eqns. (14) and (14) pore length

V.G.J. Rodgers and K.D. Miller/J.

56 -

t

P

;

Pe R F

iii S

T t WP m,

V lJ * ; z

mean external length of solute hydraulic permeability molecular weight of the protein number of pores solute diffusive permeability Peclet number ideal gas constant intrinsic retention coefficient, 1 - Si pore radius intrinsic sieving coefficient, Cp/CW specific area of the average pore, volume/pore surface area absolute temperature time mass concentration of the protein salt concentration in solution bulk solution volume radially averaged velocity through pore solute velocity macromolecular charge axial direction in pore

Tl

r2

w 0

dimensionless time for bulk solution variation equation dimensionless time for convective transport through a pore dimensionless concentration frequency of pulsation

References 1 2

3 4

Greek letters 5

transmembrane pressure osmotic pressure drop across membrane solvent viscosity partitioning coefficient pore area fraction ratio of the solute radius to the pore radius solute radius to pore specific area, t/4s osmotic pressure osmotic pressure due to protein interaction osmotic pressure due to Donnan effect reflection coefficient convective reflection coefficient osmotic reflection coefficient characteristic time for bulk solution concentration variation characteristic time for transport through a pore characteristic time for ultrafiltration during pulsing

Membrane Sci. 85 (1993) 39-58

6

7

8

9

10

11

12

A.S. Michaels, New separation techniques for the CPI, Chem. Eng. Prog., 64 (12) (1968) 31. W.F. Blatt, A. Dravid, A.S. Michaels and L. Nelsen, Solute polarization and cake formation membrane ultrafiltration: causes, consequences and control techniques, in: J. Flinn (Ed.), Membrane Science and Technology, Plenum Press, New York, NY, 1970, p. 47. A. Jijnsson and G. Tragilrdh, Fundamental principles of ultrafiltration, Chem. Eng. Process, 27 (1990) 67. KC. Ingham, T.F. Busby, Y. Sahlestrom and F. Castino, Separation of macromolecules by ultrafiltration: influence of protein adsorption, protein-protein interaction, and concentration polarization, in: A.R. Cooper (Ed. ) , Polymer Science and Technology, Vol. 13, Ultrafiltration Membranes and Applications, Plenum Press, New York, NY, 1980, p. 141. V.G.J. Rodgers and R.E. Sparks, Reduction of membrane fouling in protein ultrafiltration, AIChE J., 37 (10) (1991) 1517. K.D. Miller, S. Weitzel and V.G.J. Rodgers, Reduction of membrane fouling in the presence of high polarization resistance, J. Membrane Sci., 76 (1992) 77. V.G.J. Rodgers, Transmembrane pressure pulsing in protein ultrafiltration, DSc Thesis, Washington University, 1989. G. Belfort, Membrane separation technology: an overview, in: H.R. Bungay and G. Belfort (Eds.), Advanced Biochemical Engineering, Wiley, New York, NY, 1987, pp. 239-297. E.A. Mason and H.K. Lonsdale, Statistical-mechanical theory of membrane transport, J. Membrane Sci., 51 (1990) 1. 0. Kedem and A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to non-electrolytes, Biochim. Biophys. Acta, 27 (1958) 229. K.S. Spiegler and 0. Kedem, Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes, Desalination, 1 (1966) 311. L.F. Del Castillo and E.A. Mason, Generalization of membrane reflection coefficients for nonideal, nonisothermal, multicomponent systems with external

V.G.J. Rodgers and K.D. Miller/J.

13 14 15

16

17 18

19

20

21

22

23

24

25

26 27

28

29

57

Membrane Sci. 85 (1993) 39-58

forces and viscous flow, J. Membrane Sci., 28 (1986) 229. V.L. Vilker, The ultrafiltration of biological macromolecules, PM Thesis, MIT, 1976. A.M. Brims and M.N. de Pinho, Mass transfer in ultrafiltration, J. Membrane Sci., 61 (1991) 49. J.L. Anderson and J.A. Quinn, Restricted transport in small pores: A model for steric exclusion and hindered particle motion, Biophys. J., 14 (1974) 130. H. Brenner and L.J. Gaydos, The constrained brownian movement of spherical particles in cylindrical pores of comparable radius, J. Colloid Interface Sci., 58 (1977) 313. W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J., 33 (1987) 1409. B.C. Robertson and A.L. Zydney, Hindered protein diffusion in asymmetric ultrafiltration membranes with highly constricted pores, J. Membrane Sci., 49 (1990) 287. W.S. Gpong and A.L. Zydney, Diffusive and convective protein transport through asymmetric membranes, AIChE J., 37 (1991) 1497. S. Mochixuki and A.L. Zydney, Dextran transport through asymmetric ultrafiltration membranes: comparison with hydrodynamic models, J. Membrane Sci., 68 (1992) 21. E. Renkin, Filtration, diffusion, and molecular sieving through porous cellulose membranes, J. Gen. Physiol., 38 (1954) 225. J.C. Giddings, E. Kucera, C.P. Russell and M.N. Myers, Statistical theory for the equilibrium distribution of rigid molecules in inert porous networks: exclusion chromatography, J. Phys. Chem., 72 (1968) 4397. P.M. Bungay and H. Brenner, The motion of a closely fitting sphere in a fluid filled tube, Int. J. Multiph. Flow, 1 (1973) 25. F.A. Williams, A nonlinear diffusion problem relevant to desalination by reverse osmosis, SIAM J. Appl. Math., 17 (1969) 59. C.P. Bean, The physics of porous membranes-neutral pores, in: G. Eiseman (Ed.), Membranes, Vol. 2, Marcel Dekker, New York, NY, 1972, p. 1. D.G. Levitt, General continuum analysis of transport through pores, Biophys. J., 15 (1975) 533. E.N. Lightfoot, J.B. Bassingthwaighte and E.F. Grabowski, Hydrodynamic models for diffusion in microporous membranes, Ann. Biomed. Eng., 4 (1976) 78. J.L. Anderson and D.M. Malone, Mechanism of osmotic flow in porous membranes, Biophys. J., 14 (1974) 957. J.L. Anderson, Configurational effect on the reflection coefficient for rigid solutes in capillary pores, J. Theor. Biol., 90 (1981) 405.

30

A.L. Zydney, Personal communications,

1992.

Appendix For the pseudo steady-state approximation to be valid, it is necessary that the characteristic time for the change in concentration in the bulk be much greater than the transport time in the membrane pore. Using order of magnitude analysis we can estimate these characteristic times. For the bulk solution the change in concentration, assuming constant density and diffusivity, can be expressed by vdc %=

-yA{cu*},,o

(Al)

where V is the bulk solution volume, u* is the solute velocity through the membrane evaluated at z = 0, A is the membrane cross-sectional area and y is the membrane void fraction. Now we define the dimensionless variables to be:

(AZ) Here 8, is the characteristic time for concentration variation in the bulk solution. Substituting these relationships into eqn. (Al) results in

dly-

-=

dr,

1

(A3)

resulting in the characteristic time becoming

The characteristic time for the transport of a radially averaged species through the pores can be obtained by writing the transient convective-diffusion equation through the pore as a,C+K,va,C=K~Da~C.

WI

Here C is the radially averaged species concentration, u is the radially averaged solvent veloc-

V.G.J. Rodgers and K.D.

58

ity, and D is the bulk solution diffusivity. Defining the additional dimensionless variables as ?j=;

r+

(A6) 2

and substituting these into eqn. (A5) results in (A7) where Pe is defined by eqn. ( 11) . In these studies Pe >> 1 during pulsing, implying that solute transport is dominated by convection. The convective characteristic time becomes

02+ c

and is valid for Pe on the order of one or greater. Using values from this study, and estimating u* to be on the order of the convective velocity, I&u, the characteristic time for the bulk solution concentration change, &, and the trans-

Miller1J.

Membrane Sci. 85 (1993) 39-58

port through the pore, S,, are 1 x lo6 set and 2 x 10-l set, respectively. Therefore the pseudo steady-state approximation is an excellent approximation. For the pulsed cases, it is expected that the transient fluxes develop immediately after pulsing. To approximate as steady state it is necessary that the pulse characteristic time is much larger than the characteristic time for the transport through the pore. The characteristic time for pulsing is defined as

W) where o is the frequency of pulsation. In this study the frequency was 0.2 Hz, and thus the characteristic time for pulsing is 5 sec. Since this is much larger than the convective characteristic time through the pore, the steadystate assumption is valid during the pulsed ultrafiltration cases.