- Email: [email protected]
Analysis of a hollow-fibre artificial kidney performing simultaneous dialysis and ultrafiltration
Chemical Engineering Science, Vol. 42. No. Printed in Great Britain.
I. pp. 133 -
142.
I987
000%2509/87 S3.00 + 0.00 Pergrmon Journats Ltd.
ANALYSIS OF A HOLLOW-FIBRE ARTIFICIAL PERFORMING SIMULTANEOUS DIALYSIS ULTRAFILTRATION
KIDNEY AND
M. ABBAS and V. P. TYAGI Department of Genie Biologique, Universite de Technologie de Compiegne, B.P. 233/60206 Cedex, France (Received
5 November
Compiegne
1985)
Abstract-The problem of mass transfer in a hollow-fibre artificial kidney performing simultaneous dialysis and ultrafiltration is formulated as a well-defined duct convective mass transfer problem. Such assumptions are made as are satisfied in test dialyser. An analytical solution for the concentration profile in the duct is obtained by using variable separation method. The necessary requirement that the solution must tend to pure dialysis case solution in the limit ultrafiltrationtendingto zero is fulfilled.Exact analytical expressions are derived for the eigenconstants of the solution. In using the solution, Sherwood numbers are emphasized as they provide useful guidelines for optimizations and design of the artificial kidney. Various types of Sherwood numbers are defined and studied.
1.
INTRODUCTION
The artificial kidney is one of the most important biomedical engineering devices. Achieving such characteristics with this device as efficiency and compactness are of great significance. In this direction, the mass transfer aspect of the device has been recognized to be given emphasis under research investigations. With regard to this aspect of the device theoretical studies are quite relevant and important. The main aim of a theoretical study is to help those who are concerned with the design and optimization of the device and to facilitate determining the various properties of the device. The theorization which has been made and is well recognized, and whose solution can possess the above-mentioned features may be stated as below. In any considered duct, the fluid flow is incompressible, Newtonian and laminar and is independent of concentration distribution. The velocity and concentration fields satisfy steady state conditions. The flow is fully developed at the mass transfer entrance section and in the onward region of mass transfer. The material properties are constant. Also constant are dialysate side mass transfer resistance and bulk concentration. The axial mass diffusion is negligible and the concentration distribution at mass transfer entrance section is constant. The mass flux to the inner surface of the duct wall is equal to the mass flux through the wall. The above-stated mass transfer problem for hollowfibre artificial kidney in the case of pure dialysis, was analysed by Davis and Parkinson (1970). Later Cooney et al. (1974) investigated the same problem in terms of the confluent hypergeometric function. The problem in the case of combined dialysis and ultrafiltration under the assumption of constant ultrafiltration velocity along the duct length, has been analysed
by Jagannathan and Shettigar (1977). Unfortunately, the analysis by Jagannathan and Shettigar carries some mistakes. In (Jagannathan and Shettigar, 1977) the expression for the axial velocity component is not correct which can be verified by going through the velocity field results in Yuan and Finkelstein (1956). Also the dimensionless axial coordinate is zero at the mass transfer entrance section as well as for zero ultrafiltration velocity. This means that eq. (2) of Jagannathan and Shettigar (1977) tells us that the pure dialysis case concentration is constant along the entire duct length, which is contrary to this fact. It is also found that if their analysis is subjected to the process of limit ultrafiltration tending to zero, the results of the case of pure dialysis are not deducible. In view of these observations, a fresh analysis is needed to be carried out for the problem. Furthermore as compared to Jagannathan and Shettigar (1977), there is a scope of carrying out some more work which is of considerable importance.
133
2. ANALYSIS
The basic equations in terms of dimensional quantities are as follows:
c=ci
-++
r
at
X=0
(2)
vwc = K,(c-cc,)+T,~& at dC
z_=O
at
r=O
r = R, x > 0
(3) (4)
M.
134
ABBAS
and V. P. TYAGI
where
-EE2r4+E3r6-E.#)
(IS)
and let parameter R, be called as ultrafiltration Reynolds number. Assume a solution to the eqs (13 j(16) of the form F = X(x)Y(r).
(6) and ci denotes the concentration at the mass transfer entrance section, v;. the ultrafiltration velocity, K, (= l/(R, + l/P,)) the wall mass transfer coefficient, R, the dialysate resistance, P,,, the membrane permeability, cD the bulk dialysate concentration, T, the membrane transmittance coefficient, U, the mean of U and G,,,(O)the value of xi,,,at X = 0. [Equations (5) and (6) are taken from Yuan and Finkelstein (1956) with some changes in notation.] With the following non-dimensionalization scheme
u=v
ii
u=m’ r,l-
(7)
VW
XD
F
(c -4)
=
(q -
where
With this product form for F, eqs (13), (15) and (16) are separable and the resulting relations are: $(I 1
(9)
CiB)
CiCShw Sh,
TR ) $1.
( 1-
&JR
= __
D
Y’=O
F=l
at
dF z+[Sh,-(l-TT,)J/]F
u = ;(I 1
t?F -=O &
-4E,&-+7,
x=0 =0
at
2 u=-f,Um r
x>O
r=O
r=O
1
(24) (25)
lj/xp4*‘-
(26)
A solution to eq. (23) subject to the boundary conditions (24) and (25) can be obtained by assuming g a&n “=Cl
(27)
(11)
alJ= 1
(13) (14)
at r=l,
r =
which satisfies eq. (25) readily and satisfies eq. (23) by having the coefficients as:
2
1)
at
x = (1 - 4E,
a2i+8
=
(15)
&
C{W+
(28)
12)1(1-P2)Ela2i+6
4
+ {Z/3* - (4i + s)+)E2QZi+
where parameter Sh, is called the wall Sherwood number, and 9 the ultrafiltration Peclet number, eqs (l)-(6) may be written in non-dimensional form as follows: Otrc
at
(22)
(10)
(12)
g++g=;g+$ug,(x>o,
= 0
where the prime has been used to denote differentiation with respect to argument and fi is an unknown parameter and is required to be determined. Equation (22) has a solution
Y = c&W
B=
where
-4E141/x)X’+~~X
Y’+[Sh,-(l-TT,)II/]Y=O
(8)
x = 2R*u,(O)
R’
(21)
+{(4i+4)IC/-3382}E3a2i+2 + {4b2 i=
(4i)ll/)E4a,;l;
-33, - 2, - 1, 0, 1, 2, .
.
(2%
where aa
= a_-4 = a_-6 = 0
(30)
and satisfies eq. (24) by demanding _zO C2n + Sh, - (I-
T’Wlaz. = 0.
(31)
(16)
= +(l-4E,I(/x) 1 (17)
Equation (31) is a transcendental equation in the unknown p and its roots, denoted by fi,,,, m = 1,2, . . . , may be called as eigenvalues. In terms of Pm, there are infinitely many functions X, and Y,,,, and the product functions X,Y,,,, m = 1,
135
Analysis of a hollow-fibreartificialkidney 2 . . , satisfy eqs (13), (15) and (16). Therefore the solution to the system of eqs (13b( 16) may be obtained by assuming F as a linear combination of the product functions X,Y, as follows:
2
F =
certain operations were performed with eq. (23) and eqs (24) and (25) were utilised. The following exact analytical solution for A, was obtained
-
A,,, = A,X,Y,
(32)
s
a;,
0
1,2,.
i =
i #j;
. . , j = 1,2,.
.
,
(33)
where P = j+(j)f)dr.
Performing this work for the evaluation of the eigenconstants A,,,, we get:
A,=
s0
m = 1, 2, . . . .
(35)
The solution for the local concentration c, is given by c = CiBt-Ci(1 -B)
g A,(1 m=1 [
-4E1$x)fl~‘4@
-B)
2
A,(1
In= 1
(37) The mixing-cup concentration is given as follows: C
M-
-
CiBf2Ci(l
--El)
5
A,(1
-4E14bxp4*
tit=1
According to eq. (35), the computation of the eigenconstants A, involves numerical integration. In fact, the earlier investigators (Jagannathan and Shettigar, 1977) also obtained the same type of equation for their eigenconstants and used a numerical method of integration to compute them. This way, the computation of eigenconstants becomes a substantial work. It is therefore of great significance to give an exact analytical solution for the eigenconstants. With reference to this a calculation was performed, where
(40)
= exp ( -
p’x)
(41)
numbers for
the fluid-wall
system
The mass transfer rate through the membrane over a small dialyser area dA, is given by drir= K,(c,
-44E,l(lxp4*
n=l,2,....
(39)
where ,V is to be regarded to be governed by eq. (31) with R, = t+G= 0. From the view point of artificial kidney design, the most significant and meaningful quantity is a mass appropriately nontransfer coefficient. An dimensionalized mass transfer coefficient is called as Sherwood number. We may define Sherwood numbers for the fluid-wall system as well as for the fluid medium as follows. Sherwood
This equation with r = 1 gives the wall concentration as follows: c, = CjBfC,(l
da,, = do;
Jim0x
exp (P) g Y, dr exp (P) g Y i dr
B= B ;
It may be realized that the computer time consumed by the above-given analytical solution for A, is negligible as compared to that consumed by a solution for A,,, which involves numerical integration. The results of the case of pure dialysis are obtained readily, by setting R. = Ic/ = 0 in the above results that have appeared after eq. (21); except in the results for X, for which a mathematical limit is to be evaluated. It is easily seen that
1 J‘ 9
1-
m = 1, 2, . . . )
where
1
exp (P) ;YiYj dr = 0;
TRW_)
n + Sh, - (1 - q+}P’&
III=1
where the coefficients A,, called eigenconstants, are determined by applying the initial condition (14) and making use of the orthogonality relations.
{Sh, - (I-
- ce) dA, + v&JR
d-4,.
(42)
The mass transfer ratecan also be defined in terms of individual mass transfer coefficients, based on the mixing-cup concentration and bulk dialysate concentration as: dti = h&,,,
- cD) dA, + h,(c,
- cD) dA,
(43)
or in terms of fluid-wall mass transfer coefficient as: drir= h,(c,
- cD) dA,
(4)
where hd is the ditTusive fluid-wall mass transfer coefficient, h, the ultrafiltration fluid-wall mass transfer coefficient and h, the fluid-wall mass transfer coefficient. Based on these three mass transfer coefficients, below we define three Sherwood numbers. Nsh, =
Shw
(cn’- CD)
(chf - CD)
NSh* = Nsh, +
Nsh..
(45)
(47)
M. ABBAS and V. P. TYAGI
136 Sherwood
numbers for
the fluid medium
The mass transfer rate to the membrane over a small dialyser area dA,, in view of eq. (3), is given by eq. (42). The mass transfer rate can also be defined in terms of individual mass transfer coefficients, based on the mixing-cup concentration and the wall concentration as: dti = ha’ (cM - c,) dA, + h :f,/‘(c, - c,) dA,
(48)
(4%
where ha’ is the diffusive fluid side mass transfer coefficient, hlf) the ultrafiltration fluid side mass transfer coefficient and ha) the fluid side mass transfer coefficient. Based on these fluid side mass transfer coefficients, below we define three Sherwood numbers. (cw - co)
Nit&)d = Sh,
(CM
-
G.)
)
N$‘=N;{‘+N~z’. e
(52)
s
d
It may be noted that all the six Sherwood numbers defined above are functions of axial distance, x and, therefore, they all are local Sherwood numbers. 3.
RESULTS
AND
membrane-solute systems are presented in the Table 1. For our theoretical study a test dialyser (number of hollow-fibres N, = 20000, radius R = 0.01 cm and
length L = 10 cm) is considered where fluid flow flux at mass transfer entrance section Q, is 200 ml/min. The value of v is taken to be 1.388 x lo-* cm*/s. A one-unit computer program was made to numerically compute, simultaneously, all the quantities (eigenvalues, eigenconstants, mixing-cup concentration, wall concentration, Sherwood numbers, etc.) of the case of combined dialysis and ultrafiltration as well as that of pure dialysis. The computation was performed on a CYBER system of Control Data Corporation installed at Tifr, Bombay. The program contained the numerical convergence criterion of lo-” for any infinite series. The computation was done in double precision. While going through the study of Jagannathan and Shettigar (1977), we noticed that their results for local concentration, mixing-cup concentration, etc., were not convincing and were not showing what was actually happening. Therefore in a separate paper (Abbas and Tyagi, 1985) we have compared the results of Jagannathan and Shettigar (1977) with the corresponding results based on the present analysis and presented some new results which are not found in their work. We have computed all six Sherwood numbers defined above, although in order to save space we will present the numerical results for three fluid-wall Sherwood numbers only. Previous investigators such as Popovich et al. (1971) and Jagannathan and
or in terms of fluid side mass transfer coefficient as: dti = h’/‘(c 0 M - cw) dA,
Recently Asahi has introduced a series of hollow-fibre artificial kidneys PAN 150, PAN 200 and PAN 250 with polyacrylonitrile (PAN) hollow-fibres (Sigdell, 1985). Only these two hollow-fibres are referred to here. The values of the membrane permeability P,, dialysate resistance R,, diffusivity D, etc. for the four
DISCUSSION
The total family of solutes requiring removal by the artificial kidney has not been defined as yet. Therefore, in a test dialyser, one studies for a selected range of low and intermediate molecular weight solutes. In this paper, we refer to solutes urea and vitamin B-12. Most of the hollow-fibre artificial kidneys in clinical use today are with cuprophan (CU) hollow-fibres.
Table 1. Values of wall Sherwood number Sh, referred to hollow-fibre radius R = 0.01 cm Urea Polyaerylonitrile Pin (cm/s) RD
(s/cm) KV (em/s) D (cm2/s) Sk Cuprophan Pin (em/s) %I @/em) K, (em/s) D
(cm’/@ Sh,
Vitamin B- 12
13.2 x lo-“ 0.0
13.2 x 1O-4
2.35 x lo-‘+ 360 8.947939
x 10-4
1.0 x 10-S 1.32
0.0
780
2.35 x 1O-4
1.985971 x 1O-4
0.23875 x 10 - ’ 0.8947939
11.5 x 10-h
0.984293 19
0.83182031
0.59 x 10-a
0.0
360
0.0
780
11.5 x 10-d
8.132956 x 10m4
0.59 x 10-4
0.564043 x 10-d
0.23875 x 1O-5
1.0 x 10-5 1.15
0.8132956
0.24712041
0.23624837
Analysis of a hollow-fibre artificialkidney
Shettigar (1977) have also given and discussed these Sherwood numbers only. Figures 1 and 2 illustrate the values of the diffusive fluid-wall Sherwodd number N,,. Figure 1 shows the variation of Nshd as a function of VW, whereas Fig. 2 shows the same as a function of X. In Fig. 1 the points of intersection of the curves with the axis of ordinates, represent the pure dialysis case fluid-wall Sherwood number. It may be noted that in simultaneous dialysis and ultrafiltration (SDU), diffusive and ultrafiltration processes are not independent of each other. Ultrafiltration influences diffusive processes and vice versa (Gupta and Jaffrin, 1984). In Fig. 1, it is seen that Nshl slightly increases with the increase in V,,. This can be explained as follows. In going from pure dialysis to SDU, flow field in the duct is influenced by two factors, namely: (1) walldirected radial fluid motion in the duct and (2) axial fluid motion reduction everywhere in the region x > 0. The effect of (1) is to increase wall concentration (cw) and mixing-cup concentration (cM) in the duct (Abbas
137
and Tyagi, 1985). On the other hand, the effect of (2) is to decrease concentration, because the lower the axial motion of fluid is then the lower the concentration is in the duct. The effect of (1) has its maximum intensity near x = 0, as the concentration differences near the wall are maximum here. On the other hand, the effect of (2) has its minimum intensity near x = 0, as axial fluid motion reduction is minimum here. However, the effect of (2) increases as x increases, because the axial fluid motion reduction increases as x increases. At such a distance as L = 10 cm, the effect of (2) overtakes and dominates that of (1). Thus at L = 10 cm, the given ultrafiltration has the effect of decreasing cM and slightly increasing c,. As a result c,/c, is increased and NSh, is, therefore, increased. In order to see the effect of dialysate resistance, we compare curve 2 with curve 3. We find that the effect of non-zero R, is to decrease NSh,. A reason for this may be as follows. In the presence of dialysate resistance, wall mass transfer coefficient K,, becomes smaller. Consequently, the wall Sherwood number is decreased
0.70 L -lOCM
0.69
-
=
0.96429319
ShwZ =
0.63142031
Sh
Wl
1) shw
0.66
TR
t Nshd 0.65
t ”
2n
m
2m
Ru -
0.6L -
3n
Ln
S?
3m
Lm
Sm
I
Y--t
0.63 -
3
1.256
2.s12
3.768
VtLT O-62
2)
5.024
6.28
=%I,,, =
0.94
Sh,,,
=
Sh,
TR
=
I.0
1’
Sh,
=
Sh,2,
TR
=
I-O
/HR)---
-
ms0.11634671
o-59. 0
0-l
o-2
O-3 yw(CM/HR)--
0-L
0 -!
Fig. 1. Diffusive fluid-wall Sherwood number in the cases of vitamin B-12 and polyacrylonitrile.
138
M.
ABBAS and V. P. TYAGI
Sh,
= 0.913L29319 1)
Y
=0-o,
R,=O.O
2)
t
W =O-17452, Ru=
0.30019
x to’
Nshd 3)
Y = 0.34904, Ru=0.60038
x lo4
0-65
0.02
0.06
O-10
O.lL
0.18
o-22
XFig. 2.
Diffusivefluid-wall Sherwoodnumbervs axialdistanceat T, polyacrylonitrile.
and NSh, is therefore decreased in view of its definition by eq. (45). The effects of membrane transmittance coefficient value less than unity are observed by comparing curve 1 with curve 2. It is seen that the effect of TR < 1 is to increase Nsh,. This may be understood as follows. By definition T, < 1 implies an accumulation of solute at and around r = 1. Due to this, the wall concentration increases significantly and there is also a small increase in cM. As a result the value of c,/c, increases and therefore Nsh, increases. In Fig. 2, three values of ultrafiltration velocity have been taken and they are: 0.0 cm/h (to refer the case of pure dialysis), 0.15 and 0.30 cm/h. The values of ultrafiltration Peclet number, ti and ultrafiltration Reynolds number, R, corresponding to the abovementioned values of VW are properly shown in the figure. We find that the diffusive fluid-wall Sherwood number, N,,,* decreases as axial distance, x increases. It tends to the value of wall Sherwood number, Sh, as x tends to zero. We also note that the variation of NSh, with x, for small values of x, is very fast and for large values of x it is very slow. At certain large X, the variation of NSL, becomes negligible and then at all the remaining higher x, NSh, remains constant. These variations of the diffusive fluid-wall Sherwood number with axial distance have already been explained by earlier pure dialysis studies. The variation of the Sherwood number with the ultrafiltration velocity parameter, shown in the figure, is new, which is more or less explained by the discussion above (preceding but one paragraph). The reason why this variation is more at large x is that the intensity of the effect of the axial fluid motion reduction is more there. Figures 3 and 4 illustrate the values of the ultrafiltration fluid-wall Sherwood number defined by eq. (46). Figure 3 shows the variation of NSh_ with respect to VW, whereas Fig. 4 shows the same with
= 0.94
in the casesof vitaminB- 12 and
respect to axial distance, x. In Fig. 3, it is seen that in all the cases, NSL. increases as V, increases, which is mainly because of the fact that NSh is directly proportional to Ic/.By comparing curve 1 w&h curve 2, we can see the effects of dialysate resistance. We find that the effect of non-zero R, is to slightly increase NSh_. This can be explained as follows. By taking a non-zero value of R,, the wall resistance to mass transfer becomes higher. As a consequence, the rate of extraction of solute by diffusive process is lowered, which in turn increases the concentration at interface points and at neighbourhood points in the duct. Thus, c, is increased. cM is also increased but not to that extent. Therefore, c,/cM is increased and Ns,,. is therefore increased. In Fig. 3, the effects of membrane transmittance coefficient less than unity are observed by observing curve 2 against curve 3. It is found that the effwt of TR -=c1 is to decrease NShU.This is mainly due to the fact that the convective mass flux through the membrane is directly proportional to TR by definition. In Fig. 4, all the curves refer to the case of urea. Curves 1 and 3 show the cases of zero dialysate resistance, whereas curves 2 and 4 show the cases of non-zero dialysate resistance. This figure shows that NShUdecreases as x increases. We observe that the variation of N,,,_against x continues only up to certain small x and then NShU becomes invariant for all remaining higher x. Figures 5 and 6 illustrate the values of fluid-wall Sherwood number, N,, Figure 5 shows the variation of NSh, as a function of0 v_,‘,., whereas, Fig. 6 shows the same as a function of axial coordinate, x. In Fig. 5, the left-hand side end points of the curves give pure dialysis case fluid-wall Sherwood numbers. While studying NShO,we may remember that this physical quantity is the sum of diffusive and ultrafiltration fluid-wall Sherwood numbers discussed above. For
139
Analysis of a hollow-fibre artificial kidney
”
2n
m
2m
Ru -
3n
4n
5n
irn
4-m
5n
UJ-
Sh
Wt
Wv2
1.256
o-5
t
2.512
3.768
5 -024
6-28C
V (LT/HR)-
1)
=
0.98429319
=
0.831S2031
Sh,
=
Sh,2,
m
=
I.0
Nsh, 0.4 2)
L=lOCM
3,
=
Sh,.,,,
TR
=
I-O
0.3 3)
0.2
0 -1
Shw
=
TR
=0.94
Shy,
t-l =
0 -20012808
ml =
0.
x IO
-4
11634671
0 V,(CM/HR)-
Fig. 3. Ultrafiltration fluid-wall Sherwood number in the cases of vitamin B-12 and polyacrylonitrile.
Wvl
=
l-15
Sh ,.,s2= 0.8132956
0.07 1)
0.06
2)
0.05 t Nsh”
Sh.,,
=
Sh,,
,
v
= O-04166,
Ru
=0.30019x1~4
Shw
-
Sh,,,2,
w
=
0.04166,
RU
=
0.30019
-4 x IO
o* OL 3)
r2 0.03
Ll
4)
0.02
=
Sh,,
w
=
0.08332,
RU
=
0.60038
Wv
=
Sh,,,.2,
w
=
O-08332,
%I
=
0-60038x
Shvu
> -4 x 10
164
0.01
0-c
o-04
0.12
0.28
0.20
O-36
x-r-
Fig. 4. Ultrafiltration fluid-wall Sherwood number vs axial distance in the cases of urea and cuprophan.
M.
140
l-2&
ABBAS
2.;12
and
P.
TYAGI
5.b2L
3:760 (LT~HR)
v
V.
6&
-
Ru -
n
2n
3n
Ln
5n
n-l
2m
3m
Im
5m
1
Sh,,
=O-98429319
ShwZ
= 0.83182031
v,-
L
=
IO CM I)
Sh,
= =
TR
2)
3)
Fig. 5. Fluid-wall
Sherwood number in the cases of vitamin
Sh,
=
1.32
TR
=
1.0
2)
Sh,
=
Shwl
TR
=
0.9L
Sh,
=
ShwZ
TR
=
I-O
n =
0.20012808
m =
0.1163~671
B-12
1)
3)
w
=
0.0,
R”
=
0.0
+ ‘’ =
v R,
2
9
,
,
x 10
-4
and polyacrylonitrile.
Ru
3
Sh,, I-O
o-04166,
=0.30019x
=
lo-&
0.08332,
=0.60038
x lO-4
1
*04
0.12
0.20
O-28
0.36
Y-
Fig. 6. Fluid-wall
Sherwood number vs axial distance in the cases of urea and polyacrylonitrile.
Analysis
of a hollow-fibre
particular values of the controlling parameters, a measure of the diffusive component of NSh_ is wall Sherwood number Sh, and that of the ultrafiltration component is ultrafiltration Peclet number +. The magnitudes of Sh, and @ will determine the effectiveness of the corresponding components. It is seen that in all the cases shown in Fig. 5, NSh_ increases as V, increases. A comparison of curves 1 and 3 reveals that the effect of non-zero dialysate resistance is to decrease N,,_. In other words the fluid-wall Sherwood number increases with an increase in the wall permeability. We can see the effects of T, -c 1 by observing curve 2 against curve 1. It is found that the effect of TR < 1 is to decrease NSIIO _ From this, we may conclude that for middle molecules polyacrylonitrile hollow-fibre is superior to cuprophan hollow-fibre, because the membrane transmittance coefficient T, of cuprophan for vitamin B-12 is significantly lower. Figure 6 refers to the solute urea and the PAN hollow-fibre. In Fig. 6, the curve corresponding to J/ = 0 represents the pure dialysis case fluid-wall Sherwood number. It is seen that NShOdecreases as x increases. Also from the figure it is clear that the region where the fluid-wall Sherwood number is changing rapidly is confined to a small inlet portion of the test dialyser. A physical explanation for this may be given as follows. Near x = 0, the mass flux is significantly higher, because the concentration differences near the wall are more. Due to this reason, we get large values for Nshonear the inlet of the dialyser. For small values of x, sotute concentration in the duct rapidly decreases in the axial direction. The rate of this decrease in the concentration is more near the wall as compared to the central region. Consequently, c, decreases faster than cM along the dialyser length. Therefore, c,/c, decreases in the main blood flow direction and, hence, NSh. is decreased with increase in x. Beyond certain value of x the mass flux is very low and the ratio of c, and cM becomes practically constant. This leads to a value of Ns4 which remains invariant for all remaining higher x. 4. CONCLUSIONS
A mass transfer problem simulating the blood flow mass transfer in a hollow-fibre artificial kidney performing simultaneous dialysis and ultrafiltration, has been considered theoretically. An exact analytical solution for the solute concentration distribution has been obtained by using variable separation method. The solution satisfies the necessary requirement that it should yield the solution of the pure dialysis case in the limit ultrafiltration tending to zero. Exact analytical expressions have been obtained for the eigenconstants of the solution, which implies a substantial reduction of numerical work and considerable saving of computer time. It has been found that the computer time consumed by one example of combined dialysis and ultrafiltration is nearly the half of that consumed by the corresponding example of pure dialysis. The paper has been devoted mainly to the study and of Sherwood numbers. Fluid-wall discussion
artificial
kidney
141
Sherwood numbers as well as fluid-side Sherwood numbers are discussed and presented graphically. The theoretical study predicts the following in terms of the fluid-wall Sherwood numbers.
(1) As compared to pure dialysis procedure, com-
(2)
bined dialysis and ultrafiltration procedure is of greater significance in the case of middle molecules as well as in the case of small molecules, The performance of the polyacrylonitrile hollowfibre is better than that of the cuprophan hollowfibre. NOTATION
A, B C 'D ci
‘M
CW D F
hd ha'
ho /,‘/I
h:: h”’ ”
Kw
eigenconstants introduced by eq. (32) parameter given by eq. (10) local concentration in the duct, gm/cm3 bulk dialysate concentration, gm/cm3 mass transfer entrance section concentration, gm/cm3 mixing-cup concentration, gm/cm3 wall concentration, gm/cm3 diffusion coefficient, cm’/s dimensionless form of c diffusive fluid-wall mass transfer coefficient, cm/s diffusive fluid-side mass transfer coefficient, cm/s fluid-wall mass transfer coefficient, cm/s fluid-side mass transfer coefficient, cm/s ultrafiltration fluid-wall mass transfer coefficient, cm/s ultrafiltration fluid-side mass transfer coefficient, cm/s wall mass transfer coefficient, l/(R,+ l/P,), cm/s diffusive fluid-wall Sherwood number diffusive fluid Sherwood number fluid-wall Sherwood number fluid Sherwood number ultrafiltration fluid-wall Sherwood number ultrafiltration fluid Sherwood number membrane permeability, cm/s flow rate at mass transfer entrance section in the duct, cm3/s radial coordinate, cm hollow-fibre radius, cm dialysate side mass transfer resistance, s/cm ultrafiltration Reynolds number wall Sherwood number membrane transmittance coefficient velocity component in Z-direction in the duct, cm/s mean of ri, cm/s velocity component in f-direction in the duct, cm/s ultrafiltration velocity, cm/s axial coordinate, cm eigenvalues, roots of eq. (31) kinematic viscosity, cm*/s ultrafiltration Peclet number
142
M. ABBAS and V. P. TYAGI REFERENCES
Abbas, M. and Tyagi, V. P., i985, Laminar mass transfer due to simultaneous dialysis and ultrafiltration in a circular duct dialyser. Proc. Int. Symp. Mathematical Modelling of Ecological, Environmental and Biological Systems, 27-30 August 1985, IIT, Kanpur, India. Cooney, D. 0.. Kim, S. S. and Davis, E. J., 1974, Analyses of mass transfer in hemodialyzers for laminar blood flow and homogeneous dialysate. Chem. Engng Sci. 29, 1731-1738. Davis, H. R. and Parkinson, G. V., 1970, Mass transfer from small capillaries with wall resistance in the laminar flow regime. Appl. Sci. Res. 22, 2Ck34. Gupta, B. B. and Jaffrin, M. Y., 1984, In vitro study of combined convection-diffusion mass transfer in he-
modialysers. lnt. J. Artif. Organ. 7, 263-268. Jagannathan, K. and Shettigar, U. R., 1977, Analysis of a tubular hemodialyzer--effect of ultrafiltration and dialysate concentration. Med. Biol. Engng Comput. 15, 50&511. Popovich, R. P., Christopher, T. G. and Babb, A. L., 1971,The effects of membrane diffusion and ultrafiltration properties on hemodialyzer design and performance. Chem. Engng Prog. Symp. Ser. 67, 105-115. Sigdell, J. E., 1985, New products for blood treatment. Artif. Organs 9, 207-208. Yuan, S. W. and Finkelstein, A. S., 1956, Laminar pipe flow with injection and suction through a porous wall. Trans. ASME 78, 719-724.
I. pp. 133 -
142.
I987
000%2509/87 S3.00 + 0.00 Pergrmon Journats Ltd.
ANALYSIS OF A HOLLOW-FIBRE ARTIFICIAL PERFORMING SIMULTANEOUS DIALYSIS ULTRAFILTRATION
KIDNEY AND
M. ABBAS and V. P. TYAGI Department of Genie Biologique, Universite de Technologie de Compiegne, B.P. 233/60206 Cedex, France (Received
5 November
Compiegne
1985)
Abstract-The problem of mass transfer in a hollow-fibre artificial kidney performing simultaneous dialysis and ultrafiltration is formulated as a well-defined duct convective mass transfer problem. Such assumptions are made as are satisfied in test dialyser. An analytical solution for the concentration profile in the duct is obtained by using variable separation method. The necessary requirement that the solution must tend to pure dialysis case solution in the limit ultrafiltrationtendingto zero is fulfilled.Exact analytical expressions are derived for the eigenconstants of the solution. In using the solution, Sherwood numbers are emphasized as they provide useful guidelines for optimizations and design of the artificial kidney. Various types of Sherwood numbers are defined and studied.
1.
INTRODUCTION
The artificial kidney is one of the most important biomedical engineering devices. Achieving such characteristics with this device as efficiency and compactness are of great significance. In this direction, the mass transfer aspect of the device has been recognized to be given emphasis under research investigations. With regard to this aspect of the device theoretical studies are quite relevant and important. The main aim of a theoretical study is to help those who are concerned with the design and optimization of the device and to facilitate determining the various properties of the device. The theorization which has been made and is well recognized, and whose solution can possess the above-mentioned features may be stated as below. In any considered duct, the fluid flow is incompressible, Newtonian and laminar and is independent of concentration distribution. The velocity and concentration fields satisfy steady state conditions. The flow is fully developed at the mass transfer entrance section and in the onward region of mass transfer. The material properties are constant. Also constant are dialysate side mass transfer resistance and bulk concentration. The axial mass diffusion is negligible and the concentration distribution at mass transfer entrance section is constant. The mass flux to the inner surface of the duct wall is equal to the mass flux through the wall. The above-stated mass transfer problem for hollowfibre artificial kidney in the case of pure dialysis, was analysed by Davis and Parkinson (1970). Later Cooney et al. (1974) investigated the same problem in terms of the confluent hypergeometric function. The problem in the case of combined dialysis and ultrafiltration under the assumption of constant ultrafiltration velocity along the duct length, has been analysed
by Jagannathan and Shettigar (1977). Unfortunately, the analysis by Jagannathan and Shettigar carries some mistakes. In (Jagannathan and Shettigar, 1977) the expression for the axial velocity component is not correct which can be verified by going through the velocity field results in Yuan and Finkelstein (1956). Also the dimensionless axial coordinate is zero at the mass transfer entrance section as well as for zero ultrafiltration velocity. This means that eq. (2) of Jagannathan and Shettigar (1977) tells us that the pure dialysis case concentration is constant along the entire duct length, which is contrary to this fact. It is also found that if their analysis is subjected to the process of limit ultrafiltration tending to zero, the results of the case of pure dialysis are not deducible. In view of these observations, a fresh analysis is needed to be carried out for the problem. Furthermore as compared to Jagannathan and Shettigar (1977), there is a scope of carrying out some more work which is of considerable importance.
133
2. ANALYSIS
The basic equations in terms of dimensional quantities are as follows:
c=ci
-++
r
at
X=0
(2)
vwc = K,(c-cc,)+T,~& at dC
z_=O
at
r=O
r = R, x > 0
(3) (4)
M.
134
ABBAS
and V. P. TYAGI
where
-EE2r4+E3r6-E.#)
(IS)
and let parameter R, be called as ultrafiltration Reynolds number. Assume a solution to the eqs (13 j(16) of the form F = X(x)Y(r).
(6) and ci denotes the concentration at the mass transfer entrance section, v;. the ultrafiltration velocity, K, (= l/(R, + l/P,)) the wall mass transfer coefficient, R, the dialysate resistance, P,,, the membrane permeability, cD the bulk dialysate concentration, T, the membrane transmittance coefficient, U, the mean of U and G,,,(O)the value of xi,,,at X = 0. [Equations (5) and (6) are taken from Yuan and Finkelstein (1956) with some changes in notation.] With the following non-dimensionalization scheme
u=v
ii
u=m’ r,l-
(7)
VW
XD
F
(c -4)
=
(q -
where
With this product form for F, eqs (13), (15) and (16) are separable and the resulting relations are: $(I 1
(9)
CiB)
CiCShw Sh,
TR ) $1.
( 1-
&JR
= __
D
Y’=O
F=l
at
dF z+[Sh,-(l-TT,)J/]F
u = ;(I 1
t?F -=O &
-4E,&-+7,
x=0 =0
at
2 u=-f,Um r
x>O
r=O
r=O
1
(24) (25)
lj/xp4*‘-
(26)
A solution to eq. (23) subject to the boundary conditions (24) and (25) can be obtained by assuming g a&n “=Cl
(27)
(11)
alJ= 1
(13) (14)
at r=l,
r =
which satisfies eq. (25) readily and satisfies eq. (23) by having the coefficients as:
2
1)
at
x = (1 - 4E,
a2i+8
=
(15)
&
C{W+
(28)
12)1(1-P2)Ela2i+6
4
+ {Z/3* - (4i + s)+)E2QZi+
where parameter Sh, is called the wall Sherwood number, and 9 the ultrafiltration Peclet number, eqs (l)-(6) may be written in non-dimensional form as follows: Otrc
at
(22)
(10)
(12)
g++g=;g+$ug,(x>o,
= 0
where the prime has been used to denote differentiation with respect to argument and fi is an unknown parameter and is required to be determined. Equation (22) has a solution
Y = c&W
B=
where
-4E141/x)X’+~~X
Y’+[Sh,-(l-TT,)II/]Y=O
(8)
x = 2R*u,(O)
R’
(21)
+{(4i+4)IC/-3382}E3a2i+2 + {4b2 i=
(4i)ll/)E4a,;l;
-33, - 2, - 1, 0, 1, 2, .
.
(2%
where aa
= a_-4 = a_-6 = 0
(30)
and satisfies eq. (24) by demanding _zO C2n + Sh, - (I-
T’Wlaz. = 0.
(31)
(16)
= +(l-4E,I(/x) 1 (17)
Equation (31) is a transcendental equation in the unknown p and its roots, denoted by fi,,,, m = 1,2, . . . , may be called as eigenvalues. In terms of Pm, there are infinitely many functions X, and Y,,,, and the product functions X,Y,,,, m = 1,
135
Analysis of a hollow-fibreartificialkidney 2 . . , satisfy eqs (13), (15) and (16). Therefore the solution to the system of eqs (13b( 16) may be obtained by assuming F as a linear combination of the product functions X,Y, as follows:
2
F =
certain operations were performed with eq. (23) and eqs (24) and (25) were utilised. The following exact analytical solution for A, was obtained
-
A,,, = A,X,Y,
(32)
s
a;,
0
1,2,.
i =
i #j;
. . , j = 1,2,.
.
,
(33)
where P = j+(j)f)dr.
Performing this work for the evaluation of the eigenconstants A,,,, we get:
A,=
s0
m = 1, 2, . . . .
(35)
The solution for the local concentration c, is given by c = CiBt-Ci(1 -B)
g A,(1 m=1 [
-4E1$x)fl~‘4@
-B)
2
A,(1
In= 1
(37) The mixing-cup concentration is given as follows: C
M-
-
CiBf2Ci(l
--El)
5
A,(1
-4E14bxp4*
tit=1
According to eq. (35), the computation of the eigenconstants A, involves numerical integration. In fact, the earlier investigators (Jagannathan and Shettigar, 1977) also obtained the same type of equation for their eigenconstants and used a numerical method of integration to compute them. This way, the computation of eigenconstants becomes a substantial work. It is therefore of great significance to give an exact analytical solution for the eigenconstants. With reference to this a calculation was performed, where
(40)
= exp ( -
p’x)
(41)
numbers for
the fluid-wall
system
The mass transfer rate through the membrane over a small dialyser area dA, is given by drir= K,(c,
-44E,l(lxp4*
n=l,2,....
(39)
where ,V is to be regarded to be governed by eq. (31) with R, = t+G= 0. From the view point of artificial kidney design, the most significant and meaningful quantity is a mass appropriately nontransfer coefficient. An dimensionalized mass transfer coefficient is called as Sherwood number. We may define Sherwood numbers for the fluid-wall system as well as for the fluid medium as follows. Sherwood
This equation with r = 1 gives the wall concentration as follows: c, = CjBfC,(l
da,, = do;
Jim0x
exp (P) g Y, dr exp (P) g Y i dr
B= B ;
It may be realized that the computer time consumed by the above-given analytical solution for A, is negligible as compared to that consumed by a solution for A,,, which involves numerical integration. The results of the case of pure dialysis are obtained readily, by setting R. = Ic/ = 0 in the above results that have appeared after eq. (21); except in the results for X, for which a mathematical limit is to be evaluated. It is easily seen that
1 J‘ 9
1-
m = 1, 2, . . . )
where
1
exp (P) ;YiYj dr = 0;
TRW_)
n + Sh, - (1 - q+}P’&
III=1
where the coefficients A,, called eigenconstants, are determined by applying the initial condition (14) and making use of the orthogonality relations.
{Sh, - (I-
- ce) dA, + v&JR
d-4,.
(42)
The mass transfer ratecan also be defined in terms of individual mass transfer coefficients, based on the mixing-cup concentration and bulk dialysate concentration as: dti = h&,,,
- cD) dA, + h,(c,
- cD) dA,
(43)
or in terms of fluid-wall mass transfer coefficient as: drir= h,(c,
- cD) dA,
(4)
where hd is the ditTusive fluid-wall mass transfer coefficient, h, the ultrafiltration fluid-wall mass transfer coefficient and h, the fluid-wall mass transfer coefficient. Based on these three mass transfer coefficients, below we define three Sherwood numbers. Nsh, =
Shw
(cn’- CD)
(chf - CD)
NSh* = Nsh, +
Nsh..
(45)
(47)
M. ABBAS and V. P. TYAGI
136 Sherwood
numbers for
the fluid medium
The mass transfer rate to the membrane over a small dialyser area dA,, in view of eq. (3), is given by eq. (42). The mass transfer rate can also be defined in terms of individual mass transfer coefficients, based on the mixing-cup concentration and the wall concentration as: dti = ha’ (cM - c,) dA, + h :f,/‘(c, - c,) dA,
(48)
(4%
where ha’ is the diffusive fluid side mass transfer coefficient, hlf) the ultrafiltration fluid side mass transfer coefficient and ha) the fluid side mass transfer coefficient. Based on these fluid side mass transfer coefficients, below we define three Sherwood numbers. (cw - co)
Nit&)d = Sh,
(CM
-
G.)
)
N$‘=N;{‘+N~z’. e
(52)
s
d
It may be noted that all the six Sherwood numbers defined above are functions of axial distance, x and, therefore, they all are local Sherwood numbers. 3.
RESULTS
AND
membrane-solute systems are presented in the Table 1. For our theoretical study a test dialyser (number of hollow-fibres N, = 20000, radius R = 0.01 cm and
length L = 10 cm) is considered where fluid flow flux at mass transfer entrance section Q, is 200 ml/min. The value of v is taken to be 1.388 x lo-* cm*/s. A one-unit computer program was made to numerically compute, simultaneously, all the quantities (eigenvalues, eigenconstants, mixing-cup concentration, wall concentration, Sherwood numbers, etc.) of the case of combined dialysis and ultrafiltration as well as that of pure dialysis. The computation was performed on a CYBER system of Control Data Corporation installed at Tifr, Bombay. The program contained the numerical convergence criterion of lo-” for any infinite series. The computation was done in double precision. While going through the study of Jagannathan and Shettigar (1977), we noticed that their results for local concentration, mixing-cup concentration, etc., were not convincing and were not showing what was actually happening. Therefore in a separate paper (Abbas and Tyagi, 1985) we have compared the results of Jagannathan and Shettigar (1977) with the corresponding results based on the present analysis and presented some new results which are not found in their work. We have computed all six Sherwood numbers defined above, although in order to save space we will present the numerical results for three fluid-wall Sherwood numbers only. Previous investigators such as Popovich et al. (1971) and Jagannathan and
or in terms of fluid side mass transfer coefficient as: dti = h’/‘(c 0 M - cw) dA,
Recently Asahi has introduced a series of hollow-fibre artificial kidneys PAN 150, PAN 200 and PAN 250 with polyacrylonitrile (PAN) hollow-fibres (Sigdell, 1985). Only these two hollow-fibres are referred to here. The values of the membrane permeability P,, dialysate resistance R,, diffusivity D, etc. for the four
DISCUSSION
The total family of solutes requiring removal by the artificial kidney has not been defined as yet. Therefore, in a test dialyser, one studies for a selected range of low and intermediate molecular weight solutes. In this paper, we refer to solutes urea and vitamin B-12. Most of the hollow-fibre artificial kidneys in clinical use today are with cuprophan (CU) hollow-fibres.
Table 1. Values of wall Sherwood number Sh, referred to hollow-fibre radius R = 0.01 cm Urea Polyaerylonitrile Pin (cm/s) RD
(s/cm) KV (em/s) D (cm2/s) Sk Cuprophan Pin (em/s) %I @/em) K, (em/s) D
(cm’/@ Sh,
Vitamin B- 12
13.2 x lo-“ 0.0
13.2 x 1O-4
2.35 x lo-‘+ 360 8.947939
x 10-4
1.0 x 10-S 1.32
0.0
780
2.35 x 1O-4
1.985971 x 1O-4
0.23875 x 10 - ’ 0.8947939
11.5 x 10-h
0.984293 19
0.83182031
0.59 x 10-a
0.0
360
0.0
780
11.5 x 10-d
8.132956 x 10m4
0.59 x 10-4
0.564043 x 10-d
0.23875 x 1O-5
1.0 x 10-5 1.15
0.8132956
0.24712041
0.23624837
Analysis of a hollow-fibre artificialkidney
Shettigar (1977) have also given and discussed these Sherwood numbers only. Figures 1 and 2 illustrate the values of the diffusive fluid-wall Sherwodd number N,,. Figure 1 shows the variation of Nshd as a function of VW, whereas Fig. 2 shows the same as a function of X. In Fig. 1 the points of intersection of the curves with the axis of ordinates, represent the pure dialysis case fluid-wall Sherwood number. It may be noted that in simultaneous dialysis and ultrafiltration (SDU), diffusive and ultrafiltration processes are not independent of each other. Ultrafiltration influences diffusive processes and vice versa (Gupta and Jaffrin, 1984). In Fig. 1, it is seen that Nshl slightly increases with the increase in V,,. This can be explained as follows. In going from pure dialysis to SDU, flow field in the duct is influenced by two factors, namely: (1) walldirected radial fluid motion in the duct and (2) axial fluid motion reduction everywhere in the region x > 0. The effect of (1) is to increase wall concentration (cw) and mixing-cup concentration (cM) in the duct (Abbas
137
and Tyagi, 1985). On the other hand, the effect of (2) is to decrease concentration, because the lower the axial motion of fluid is then the lower the concentration is in the duct. The effect of (1) has its maximum intensity near x = 0, as the concentration differences near the wall are maximum here. On the other hand, the effect of (2) has its minimum intensity near x = 0, as axial fluid motion reduction is minimum here. However, the effect of (2) increases as x increases, because the axial fluid motion reduction increases as x increases. At such a distance as L = 10 cm, the effect of (2) overtakes and dominates that of (1). Thus at L = 10 cm, the given ultrafiltration has the effect of decreasing cM and slightly increasing c,. As a result c,/c, is increased and NSh, is, therefore, increased. In order to see the effect of dialysate resistance, we compare curve 2 with curve 3. We find that the effect of non-zero R, is to decrease NSh,. A reason for this may be as follows. In the presence of dialysate resistance, wall mass transfer coefficient K,, becomes smaller. Consequently, the wall Sherwood number is decreased
0.70 L -lOCM
0.69
-
=
0.96429319
ShwZ =
0.63142031
Sh
Wl
1) shw
0.66
TR
t Nshd 0.65
t ”
2n
m
2m
Ru -
0.6L -
3n
Ln
S?
3m
Lm
Sm
I
Y--t
0.63 -
3
1.256
2.s12
3.768
VtLT O-62
2)
5.024
6.28
=%I,,, =
0.94
Sh,,,
=
Sh,
TR
=
I.0
1’
Sh,
=
Sh,2,
TR
=
I-O
/HR)---
-
ms0.11634671
o-59. 0
0-l
o-2
O-3 yw(CM/HR)--
0-L
0 -!
Fig. 1. Diffusive fluid-wall Sherwood number in the cases of vitamin B-12 and polyacrylonitrile.
138
M.
ABBAS and V. P. TYAGI
Sh,
= 0.913L29319 1)
Y
=0-o,
R,=O.O
2)
t
W =O-17452, Ru=
0.30019
x to’
Nshd 3)
Y = 0.34904, Ru=0.60038
x lo4
0-65
0.02
0.06
O-10
O.lL
0.18
o-22
XFig. 2.
Diffusivefluid-wall Sherwoodnumbervs axialdistanceat T, polyacrylonitrile.
and NSh, is therefore decreased in view of its definition by eq. (45). The effects of membrane transmittance coefficient value less than unity are observed by comparing curve 1 with curve 2. It is seen that the effect of TR < 1 is to increase Nsh,. This may be understood as follows. By definition T, < 1 implies an accumulation of solute at and around r = 1. Due to this, the wall concentration increases significantly and there is also a small increase in cM. As a result the value of c,/c, increases and therefore Nsh, increases. In Fig. 2, three values of ultrafiltration velocity have been taken and they are: 0.0 cm/h (to refer the case of pure dialysis), 0.15 and 0.30 cm/h. The values of ultrafiltration Peclet number, ti and ultrafiltration Reynolds number, R, corresponding to the abovementioned values of VW are properly shown in the figure. We find that the diffusive fluid-wall Sherwood number, N,,,* decreases as axial distance, x increases. It tends to the value of wall Sherwood number, Sh, as x tends to zero. We also note that the variation of NSh, with x, for small values of x, is very fast and for large values of x it is very slow. At certain large X, the variation of NSL, becomes negligible and then at all the remaining higher x, NSh, remains constant. These variations of the diffusive fluid-wall Sherwood number with axial distance have already been explained by earlier pure dialysis studies. The variation of the Sherwood number with the ultrafiltration velocity parameter, shown in the figure, is new, which is more or less explained by the discussion above (preceding but one paragraph). The reason why this variation is more at large x is that the intensity of the effect of the axial fluid motion reduction is more there. Figures 3 and 4 illustrate the values of the ultrafiltration fluid-wall Sherwood number defined by eq. (46). Figure 3 shows the variation of NSh_ with respect to VW, whereas Fig. 4 shows the same with
= 0.94
in the casesof vitaminB- 12 and
respect to axial distance, x. In Fig. 3, it is seen that in all the cases, NSL. increases as V, increases, which is mainly because of the fact that NSh is directly proportional to Ic/.By comparing curve 1 w&h curve 2, we can see the effects of dialysate resistance. We find that the effect of non-zero R, is to slightly increase NSh_. This can be explained as follows. By taking a non-zero value of R,, the wall resistance to mass transfer becomes higher. As a consequence, the rate of extraction of solute by diffusive process is lowered, which in turn increases the concentration at interface points and at neighbourhood points in the duct. Thus, c, is increased. cM is also increased but not to that extent. Therefore, c,/cM is increased and Ns,,. is therefore increased. In Fig. 3, the effects of membrane transmittance coefficient less than unity are observed by observing curve 2 against curve 3. It is found that the effwt of TR -=c1 is to decrease NShU.This is mainly due to the fact that the convective mass flux through the membrane is directly proportional to TR by definition. In Fig. 4, all the curves refer to the case of urea. Curves 1 and 3 show the cases of zero dialysate resistance, whereas curves 2 and 4 show the cases of non-zero dialysate resistance. This figure shows that NShUdecreases as x increases. We observe that the variation of N,,,_against x continues only up to certain small x and then NShU becomes invariant for all remaining higher x. Figures 5 and 6 illustrate the values of fluid-wall Sherwood number, N,, Figure 5 shows the variation of NSh, as a function of0 v_,‘,., whereas, Fig. 6 shows the same as a function of axial coordinate, x. In Fig. 5, the left-hand side end points of the curves give pure dialysis case fluid-wall Sherwood numbers. While studying NShO,we may remember that this physical quantity is the sum of diffusive and ultrafiltration fluid-wall Sherwood numbers discussed above. For
139
Analysis of a hollow-fibre artificial kidney
”
2n
m
2m
Ru -
3n
4n
5n
irn
4-m
5n
UJ-
Sh
Wt
Wv2
1.256
o-5
t
2.512
3.768
5 -024
6-28C
V (LT/HR)-
1)
=
0.98429319
=
0.831S2031
Sh,
=
Sh,2,
m
=
I.0
Nsh, 0.4 2)
L=lOCM
3,
=
Sh,.,,,
TR
=
I-O
0.3 3)
0.2
0 -1
Shw
=
TR
=0.94
Shy,
t-l =
0 -20012808
ml =
0.
x IO
-4
11634671
0 V,(CM/HR)-
Fig. 3. Ultrafiltration fluid-wall Sherwood number in the cases of vitamin B-12 and polyacrylonitrile.
Wvl
=
l-15
Sh ,.,s2= 0.8132956
0.07 1)
0.06
2)
0.05 t Nsh”
Sh.,,
=
Sh,,
,
v
= O-04166,
Ru
=0.30019x1~4
Shw
-
Sh,,,2,
w
=
0.04166,
RU
=
0.30019
-4 x IO
o* OL 3)
r2 0.03
Ll
4)
0.02
=
Sh,,
w
=
0.08332,
RU
=
0.60038
Wv
=
Sh,,,.2,
w
=
O-08332,
%I
=
0-60038x
Shvu
> -4 x 10
164
0.01
0-c
o-04
0.12
0.28
0.20
O-36
x-r-
Fig. 4. Ultrafiltration fluid-wall Sherwood number vs axial distance in the cases of urea and cuprophan.
M.
140
l-2&
ABBAS
2.;12
and
P.
TYAGI
5.b2L
3:760 (LT~HR)
v
V.
6&
-
Ru -
n
2n
3n
Ln
5n
n-l
2m
3m
Im
5m
1
Sh,,
=O-98429319
ShwZ
= 0.83182031
v,-
L
=
IO CM I)
Sh,
= =
TR
2)
3)
Fig. 5. Fluid-wall
Sherwood number in the cases of vitamin
Sh,
=
1.32
TR
=
1.0
2)
Sh,
=
Shwl
TR
=
0.9L
Sh,
=
ShwZ
TR
=
I-O
n =
0.20012808
m =
0.1163~671
B-12
1)
3)
w
=
0.0,
R”
=
0.0
+ ‘’ =
v R,
2
9
,
,
x 10
-4
and polyacrylonitrile.
Ru
3
Sh,, I-O
o-04166,
=0.30019x
=
lo-&
0.08332,
=0.60038
x lO-4
1
*04
0.12
0.20
O-28
0.36
Y-
Fig. 6. Fluid-wall
Sherwood number vs axial distance in the cases of urea and polyacrylonitrile.
Analysis
of a hollow-fibre
particular values of the controlling parameters, a measure of the diffusive component of NSh_ is wall Sherwood number Sh, and that of the ultrafiltration component is ultrafiltration Peclet number +. The magnitudes of Sh, and @ will determine the effectiveness of the corresponding components. It is seen that in all the cases shown in Fig. 5, NSh_ increases as V, increases. A comparison of curves 1 and 3 reveals that the effect of non-zero dialysate resistance is to decrease N,,_. In other words the fluid-wall Sherwood number increases with an increase in the wall permeability. We can see the effects of T, -c 1 by observing curve 2 against curve 1. It is found that the effect of TR < 1 is to decrease NSIIO _ From this, we may conclude that for middle molecules polyacrylonitrile hollow-fibre is superior to cuprophan hollow-fibre, because the membrane transmittance coefficient T, of cuprophan for vitamin B-12 is significantly lower. Figure 6 refers to the solute urea and the PAN hollow-fibre. In Fig. 6, the curve corresponding to J/ = 0 represents the pure dialysis case fluid-wall Sherwood number. It is seen that NShOdecreases as x increases. Also from the figure it is clear that the region where the fluid-wall Sherwood number is changing rapidly is confined to a small inlet portion of the test dialyser. A physical explanation for this may be given as follows. Near x = 0, the mass flux is significantly higher, because the concentration differences near the wall are more. Due to this reason, we get large values for Nshonear the inlet of the dialyser. For small values of x, sotute concentration in the duct rapidly decreases in the axial direction. The rate of this decrease in the concentration is more near the wall as compared to the central region. Consequently, c, decreases faster than cM along the dialyser length. Therefore, c,/c, decreases in the main blood flow direction and, hence, NSh. is decreased with increase in x. Beyond certain value of x the mass flux is very low and the ratio of c, and cM becomes practically constant. This leads to a value of Ns4 which remains invariant for all remaining higher x. 4. CONCLUSIONS
A mass transfer problem simulating the blood flow mass transfer in a hollow-fibre artificial kidney performing simultaneous dialysis and ultrafiltration, has been considered theoretically. An exact analytical solution for the solute concentration distribution has been obtained by using variable separation method. The solution satisfies the necessary requirement that it should yield the solution of the pure dialysis case in the limit ultrafiltration tending to zero. Exact analytical expressions have been obtained for the eigenconstants of the solution, which implies a substantial reduction of numerical work and considerable saving of computer time. It has been found that the computer time consumed by one example of combined dialysis and ultrafiltration is nearly the half of that consumed by the corresponding example of pure dialysis. The paper has been devoted mainly to the study and of Sherwood numbers. Fluid-wall discussion
artificial
kidney
141
Sherwood numbers as well as fluid-side Sherwood numbers are discussed and presented graphically. The theoretical study predicts the following in terms of the fluid-wall Sherwood numbers.
(1) As compared to pure dialysis procedure, com-
(2)
bined dialysis and ultrafiltration procedure is of greater significance in the case of middle molecules as well as in the case of small molecules, The performance of the polyacrylonitrile hollowfibre is better than that of the cuprophan hollowfibre. NOTATION
A, B C 'D ci
‘M
CW D F
hd ha'
ho /,‘/I
h:: h”’ ”
Kw
eigenconstants introduced by eq. (32) parameter given by eq. (10) local concentration in the duct, gm/cm3 bulk dialysate concentration, gm/cm3 mass transfer entrance section concentration, gm/cm3 mixing-cup concentration, gm/cm3 wall concentration, gm/cm3 diffusion coefficient, cm’/s dimensionless form of c diffusive fluid-wall mass transfer coefficient, cm/s diffusive fluid-side mass transfer coefficient, cm/s fluid-wall mass transfer coefficient, cm/s fluid-side mass transfer coefficient, cm/s ultrafiltration fluid-wall mass transfer coefficient, cm/s ultrafiltration fluid-side mass transfer coefficient, cm/s wall mass transfer coefficient, l/(R,+ l/P,), cm/s diffusive fluid-wall Sherwood number diffusive fluid Sherwood number fluid-wall Sherwood number fluid Sherwood number ultrafiltration fluid-wall Sherwood number ultrafiltration fluid Sherwood number membrane permeability, cm/s flow rate at mass transfer entrance section in the duct, cm3/s radial coordinate, cm hollow-fibre radius, cm dialysate side mass transfer resistance, s/cm ultrafiltration Reynolds number wall Sherwood number membrane transmittance coefficient velocity component in Z-direction in the duct, cm/s mean of ri, cm/s velocity component in f-direction in the duct, cm/s ultrafiltration velocity, cm/s axial coordinate, cm eigenvalues, roots of eq. (31) kinematic viscosity, cm*/s ultrafiltration Peclet number
142
M. ABBAS and V. P. TYAGI REFERENCES
Abbas, M. and Tyagi, V. P., i985, Laminar mass transfer due to simultaneous dialysis and ultrafiltration in a circular duct dialyser. Proc. Int. Symp. Mathematical Modelling of Ecological, Environmental and Biological Systems, 27-30 August 1985, IIT, Kanpur, India. Cooney, D. 0.. Kim, S. S. and Davis, E. J., 1974, Analyses of mass transfer in hemodialyzers for laminar blood flow and homogeneous dialysate. Chem. Engng Sci. 29, 1731-1738. Davis, H. R. and Parkinson, G. V., 1970, Mass transfer from small capillaries with wall resistance in the laminar flow regime. Appl. Sci. Res. 22, 2Ck34. Gupta, B. B. and Jaffrin, M. Y., 1984, In vitro study of combined convection-diffusion mass transfer in he-
modialysers. lnt. J. Artif. Organ. 7, 263-268. Jagannathan, K. and Shettigar, U. R., 1977, Analysis of a tubular hemodialyzer--effect of ultrafiltration and dialysate concentration. Med. Biol. Engng Comput. 15, 50&511. Popovich, R. P., Christopher, T. G. and Babb, A. L., 1971,The effects of membrane diffusion and ultrafiltration properties on hemodialyzer design and performance. Chem. Engng Prog. Symp. Ser. 67, 105-115. Sigdell, J. E., 1985, New products for blood treatment. Artif. Organs 9, 207-208. Yuan, S. W. and Finkelstein, A. S., 1956, Laminar pipe flow with injection and suction through a porous wall. Trans. ASME 78, 719-724.