7 Ultrafiltration Rates and Rejection of Solutes by Cellulosic Hollow Fibers
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E. KLEIN, F. F. HOLLAND, and K. EBERLE Gulf South Research Institute, P.O. Box 26518, New Orleans, L A 70186 R. P. WENDT Gulf South Research Institute and Loyola University, New Orleans, L A 70118
A major o b j e c t i v e of fundamental s t u d i e s on h o l l o w - f i b e r h e m o f i l t e r s is t o c o r r e l a t e ultrafiltration r a t e s and s o l u t e clearances w i t h the operating v a r i a b l e s of the h e m o f i l t e r such as pressure, blood flow r a t e , and s o l u t e concentration i n the blood. The mathematical model f o r the process should be kept relatively simple t o facilitate day-to-day computations and a l l o w conceptual i n s i g h t s . The model developed f o r Cuprophan hollow f i b e r s in this study has two p a r t s : (1) intrinsic transport p r o p e r t i e s of the f i b e r s and (2) a fluid dynamic and thermodynamic d e s c r i p t i o n of the t e s t fluid (blood) w i t h i n the fibers. Transport P r o p e r t i e s . Important transmembrane t r a n s p o r t parameters of the fibers a r e Lp, the h y d r a u l i c c o n d u c t i v i t y ; Pm, the diffusive p e r m e a b i l i t y f o r a given s o l u t e ; a, the s o l u t e reflection coefficient; and R, the s o l u t e r e j e c t i o n . These coefficients appear in the f o l l o w i n g equations, which are assumed to be valid a t the steady s t a t e a t each p o s i t i o n Z along the fiber w a l l : J
= L (AP - a Air) J d-a)(C-C ) j = j (l-a)C _ s v w $ _^ 3). A t h i r d procedure was t r i e d , c a l l e d the "mid-point" approximation. This method leads to an a n a l y t i c form f o r the i n t e g r a l and a consequent q u a d r a t i c form f o r f ( R ) which can be solved a l g e b r a i c a l l y _ f o r R. According to the midp o i n t method, the average v a l u e , I , of the i n t e g r a l from 0 to E i s approximated by the i n t e g r a l evaluated a t E /2, ° a n d
a r e
i v e n
E
Qbs
v
L
c
E
I =
2
/C
1
3
ex [1.219(Pe E) / ]dE P
Z
=
£
1/J
m p
2
=
E exp(1.219(Pe E / 2 ) — [R+(l-R)exp(1.219(Pe E /2) 2
c
1 / 3
^
C
[R+(1-R)exp(l.219(Pe E) )](1-E)
_ - E I °
j
) (43)
1/3
)](l-E /2) c
Percentage e r r o r s i n t h i s approximation as a f u n c t i o n of E are i l l u s t r a t e d i n F i g u r e 4 f o r Pe = 1.0 and f o r R = 0.1 and 0.9.
In Synthetic Membranes: Volume II; Turbak, A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
KLEIN ET AL.
Cellulosic Hollow Fibers
85
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7.
Figure 4. Errors in I ("see Equation 42) incurred by using midpoint approximation at Pe = 1.0
In Synthetic Membranes: Volume II; Turbak, A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
86
SYNTHETIC M E M B R A N E S :
H F A N D U F USES
For E l e s s than 0.4 the e r r o r s a r e l e s s than about 2%, a t o l e r a b l e e r r o r f o r a n a l y s i s of most of our data. The r o o t R of the q u a d r a t i c f u n c t i o n f ( R ) , which i s obtained when E I i s subs t i t u t e d f o r the i n t e g r a l i n Equation 41, i s given by Equation 1A (Appendix A). We used t h i s equation f o r a l l of our c a l c u l a t i o n s i n c l u d i n g those w i t h E >0.4, s i n c e sample c a l c u l a t i o n s performed i t e r a t i v e l y and w i t h I evaluated by Simpson's r u l e , gave f i n a l values of R d i f f e r i n g by l e s s than 1% from those found by u s i n g the mid-point approximation f o r I . F i n d i n g R w i t h e i t h e r method r e q u i r e s a v a l u e f o r E F i g u r e 5 i s a graph of E versus Pe c a l c u l a t e d according to Equation 22. There was some concern about the s e n s i t i v i t y of the c a l c u l a t e d v a l u e f o r R to the choice of E . The asymptotic form f o r the boundary-layer modulus, Equation 57, was used f o r E>E even though that form i s t r u l y exact a t E>>E f o r a given w a l l P e c l e t number. L o c a l l y , the e x p o n e n t i a l form f o r C /C^ can be much l a r g e r than the asymptotic f u n c t i o n , (C / ^ ) A ' ^ the choice of E would seem t o be q u i t e c r i t i c a l . Sut i t i s the sum of the i n t e g r a l s along the e n t i r e f i b e r l e n g t h that helps determine R, not the l o c a l v a l u e s . To t e s t the s e n s i t i v i t y of R to the choice of E , the sum of i n t e g r a l s i n Equation 41 which help determine R was c a l c u l a t e d , c
c
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c
a t
E > E
C
c
A
N
S
s
J
o
A
[R+(l-R)(C /C )](l-E) w
+
[R+d-R)^/^)^]
b
c
c
a
r
(44)
e
Here (C /C, ) and ( / ) given by Equations 20 and 17, r e s p e c t i v e l y ; E i s a n asiumed v a l u e f o r E>_E , and E^ i s E f o r a given experiment. I n F i g u r e 6 i s a graph of the percentage e r r o r i n I i n c u r r e d by u s i n g E >E , [ I (E ) - I (E ) ] x 100/1 (E ) , versus E^, f o r R = 0.5, P =°0.5 I n d ^ = 8.3$0. Three values of E^ were used. A remarkabfe i n s e n s i t i v i t y t o the choice of EA^E i s shown f o r these c o n d i t i o n s , which a r e i n the midrange of values f o r most of the s a l i n e experiments. Thus, although a l l of the data was reduced by u s i n g Equation 41 w i t h E given by Equation 22, f o r most experiments the choice of 6 was not c r i t i c a l . Only f o r values of E >0.5 can the choice cause a p p r e c i a b l e e r r o r i n c a l c u l a t i o n s of R from R . I n f a c t , f o r experiments w i t h E 0.6 x 10~ cm/sec from a p l a t e a u when s a l i n e was the s o l v e n t , whereas i n serum the R versus J graphs were as p r e d i c t e d from the Spiegler-Kedem equation. The r e l a t i v e e f f e c t s of serum or BSA on R , and on r e s u l t a n t values f o r a and P^ f o r the two l a r g e s o l u t l s , are s m a l l . For cytochrome C i n serum and s a l i n e the d i f f e r e n c e between values i s at most 11%, and f o r myoglobin the d i f f e r e n c e i s about 4% (Table I I I ) . In summary, p r o t e i n i n the s o l v e n t r e s u l t s i n n e g l i g i b l y s m a l l i n c r e a s e s i n s o l u t e r e j e c t i o n by the c e l l u l o s i c membranes s t u d i e d i n t h i s work. F i n a l l y , we note the r a t h e r s m a l l d i f f e r e n c e s given i n Table I I between R c a l c u l a t e d according to u n i f o r m - w a l l - f l u x c
c
w
b
R = 1
v
m
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UF
In Synthetic Membranes: Volume II; Turbak, A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
In Synthetic Membranes: Volume II; Turbak, A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
3 .06
0 92
3 06
3 .06
Serum
Serum
Saline
II
II
II
3
4
5
3 06
3 06
Saline
0.71 + 0.10
0 55 + 0.30
0. 25 + 0. 05
0 81 + 0.60
0 11 + 0.04
V
0 10 + 0 03
0.73 + 0.03
V
0 09 + 0 03
0.75 + 0.05
-
14
Serum
-
13
0 92
Serum
-
-
V
12
0 06 + 0 08
0.03 + 0.02
0.09 + 0.03 0.82 + 0.02
0.03
0.07 + 0.03
± 0.84
0.85 + 0.05
0.04 + 0.02
0.76 + 0.04 0 07 + 0 04 0.66 + 0.06
0.02 + 0.02
1 73 + 4.28
0.02 + 0.02 0.03 0.77 + 0.03
0. 62 + 1. 27
0 86 + 0.69
0 10 + 0.03
0 70
Saline
IV
11
0.86
-
0.03 + 0.02
0 09 + 0 09
-
0. 31 + 0. 17
0.63 + 0.07
0.81 + 0.02
-
0.04 + 0.01
0.04 + 0.02
0.01 + 0.01
0 01 + 0 02
0.84 + 0.01
0.72 + 0.01
-
0 28 + 0.10
0.02 + 0.01
0.87 + 0.01
0 70 + 0.66
1 .40
Saline
IV
-
1 70 + 1.79
-
0 18 + 0.11
0 70
BAS (10%)
IV
9
-
-
-
1 30 + 1,26
0 24 + 0.16
10
(0.5%) 1 40
BAS
-
-
IV
0 04 + 0 02
8
0.89 + 0.05 0.91 + 0.03
0 04 + 0 02
0.83 + 0.04 0.82 + 0.03
-
-
-
3 06
-
0 92
Serum
Serum
III
III
7
0. 24 + 0. 03
0 90 + 0.60
0 14 + 0.02
0.04 + 0.01
0.88 + 0.02
0 03
0 05 + 0 01
0.03 + 0.03
0 06
0.89 + 0.03
4
0.83 + 0.03
0 04 + 0 01
K 10
0.85 + 0.02
0.82 + 0.01
-
-
-
-
0.05 + 0.01
Myoglobin Pm
0.84 + 0.05
a
-
A
-
10
-
X
0 04 + 0 01
CytochrotnePm
0.79 + 0.02
0
-
10*
-
X
-
Tnulin Pm
-
a
-
A
A
-
10
1.10 + 6.4
X
-
Pm
VALUES FOR o AND Pin MEASURED IN REJECTION EXPERIMENTS
0 10 + 0.39
a
B- 12
TABLE I I I .
6
0 .92
Serum
I
2
Serum
I
1
So Lvent
Bundle No.
System No.
Uo cm/s
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100
SYNTHETIC MEMBRANES:
HF
AND
UF
USES
theory, and c a l c u l a t e d from the R , according to Equations 8, 9 (with E = ET/2), and 14. Appreciable d i f f e r e n c e s occurr only for E >0.2. P r o t e i n U l t r a f i l t r a t i o n Experiments. In Table IV and i n Figures 16 through 19, t y p i c a l data obtained i n u l t r a f i l t r a t i o n experiments w i t h serum and BSA are shown. Values f o r TT i n Table IV were c a l c u l a t e d by using a rearrangement of Equation 5,
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7 = P - J /L v p
(50)
w i t h P = AP, the average transmembrane pressure and J the average f i l t r a t e v e l o c i t y . Graphs of TT versus C f o r serum and BSA (Figure 7) were used to c a l c u l a t e the average w a l l c o n c e n t r a t i o n , C , which produced the reverse-osmotic pressure and the consequent r e d u c t i o n of f i l t r a t e v e l o c i t y of p r o t e i n systems below that obtained i n s a l i n e at comparable pressures (cf dashed l i n e s i n Figures 16 through 18). For pressures a p p l i e d i n those systems, C never was l a r g e r than about 350 g/£ f o r serum or BSA. For BSA trie s o l u b i l i t y has been independently measured to be about 580 g/% (17), w e l l above the estimated w a l l concentrations i n our experiments. A l s o , the graphs of J versus P never reach a^ h o r i z o n t a l p l a t e a u r e g i o n where J becomes independent of P. Therefore, there can be no j u s t i f i c a t i o n f o r i n v o k i n g a _ g e l - l a y e r hypothesis to e x p l a i n the observed graphs of J versus P. Instead, as explained p r e v i o u s l y , a boundary l a y e r theory must be used to p r e d i c t C , TT, and hence J at the given o p e r a t i n g _ c o n d i t i o n s . The remarkable independence of the J versus P graphs from a x i a l v e l o c i t y and f i b e r l e n g t h , i l l u s t r a t e d i n Figures 14 through 17, suggests the e x i s t a n c e of an asymptotic boundary l a y e r r e g i o n of f u l l y developed flow. C a l c u l a t e d values f o r D, obtained by u s i n g the u n i f o r m - w a l l - f l u x are shown i n Tables V and VI. For serum, data p o i n t s f o r P >0.5 atm were excluded; f o r BSA, data p o i n t s at a l l pressures were included to c a l c u l a t e D. For serum (Table V) there i s a s m a l l trend to i n c r e a s e D w h i l e i n ^ r e a ^ i n g L and decreasing U , but an average v a l u e , D"= 1.41 x 10~ cm /sec, c o r r e l a t e s a l l o? the serum data f o r bundles I - I I I w e l l f o r P
