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Open Journal of Heat, Mass and Momentum Transfer
©Attribution 3.0 Unported (CC BY 3.0)
HMMT 2014, 2(3):58-69
DOI: 10.12966/hmmt.10.01.2014
Numerical Approach for Optimal Design of a Hollow Fiber
Dialyzer System
Y. Sano1,*, A. Horibe1, N. Haruki1, K. Nagase2 and A. Nakayama2,3
1
Graduate School of Natural Science and Technology, Okayama University3-1-1 Tsushima-naka, Kita-ku, Okayama, 700-8530 Japan
Dept. of Mechanical Engineering, Shizuoka University 3-5-1 Johoku, Naka-ku, Hamamatsu, 432-8561
3
School of Civil Engineering and Architecture, Wuhan Polytechnic University, Wuhan, Hubei 430023, China
2
*Corresponding author (Email: [email protected])
Abstract - A numerical approach for an optimal designing of a countercurrent hollow fiber dialyzer has been proposed by
utilizing the membrane transport model based on the porous media theory. The three-dimensional numerical computations have
been conducted to capture individual concentration fields, namely, blood, dialysate and membrane phases within a hollow fiber
dialyzer. Clearances for the Creatinine and Vitamin B12 obtained from present numerical simulations are compared against
available experimental date so as to examine the validity of the present method. Subsequently, the effects of housing of
countercurrent hollow fiber dialyzers on the mass transfer rate have been also investigated by utilizing several types of hollow
fiber dialyzers. Furthermore, a series of calculations have been carried out with variation of the number of hollow fibers so as to
find the effective hollow fiber dialyzer for given pumping power. The present numerical methods based on the membrane
transport model can be useful for optimization of hollow fiber dialyzer systems.
Keywords - Optimal design, Hollow fiber dialyzer system, Numerical simulation, Porous media theory
1. Introduction
Hollow fiber dialyzers are widely used in the therapy of hemodialysis, which is a method for removing waste products such as
Creatinine and Urea, as well as free water from the blood when the kidneys are in renal failure. This dialyzer utilizes a bundle of
hollow fibers of ultrafiltration membrane to remove metabolic endproducts from the human body. Mass diffusion and
untrafiltration processes through such membranes are most commonly described by Kedem-Katchalsky’s model [1], which
estimates the volume and solute flows of nonelectrolyte solutions across membranes. A number of analytical and numerical
models based on the Kedem-Katchalsky model have been reported in the literature [2-11]. However, in most of previous
investigations, the axial concentration distribution of the dialysis has been neglected. Ding et al. [12] accounted for both radial
and axial gradients of the blood and dialysate sides. In recent years, some numerical attempts have been made to simulate the
fluid and solute transport processes in fiber membrane systems (Shirazian, et al. [13]; Wang et al. [14]; Kumar and Upadhyay
[15]). However, it appears to be impossible to resolve the details of flow and concentration within the dialyzer containing
thousands of hollow fibers and dialysate compartment, even using a super-computer available today.
Under these situations, Sano and Nakayama [16] proposed a membrane transport model based on the volume averaging
theory [17-25] for the analysis of hollow fiber hemodialyzer systems. They provided three-dimensional numerical computations
[26] for the mass transfer phenomena within the three individual phases, namely the lumen, shell and the membrane phase in the
countercurrent hollow fiber dialyzer. Subsequently, this membrane transport model based on the porous media theory has been
extended to describe the concentration polarization phenomena associated with hollow fiber reverse osmosis desalination
systems [27].
In this paper, the membrane transport model introduced by Sano et al. [16] are examined to seek the effective hollow fiber
dialyzer in terms of housing of dialyzer and the number of hollow fibers. The three-dimensional numerical computations can
capture individual concentration fields within a hollow fiber dialyzer. We shall compare the numerical results against available
experimental data so as to examine the validity of the present numerical method based on the porous media approach.
Subsequently, the effects of housing of countercurrent hollow fiber dialyzers on the mass transfer rate will be investigated by
utilizing several types of housing. Furthermore, a series of calculations with variation of the number of hollow fibers reveal the
optimal conditions for the hollow fiber dialyzer for given pumping power.
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
59
2. Three-Concentration Model for Hollow Fiber Membrane Dialyzers
Figure 1 schematically shows a hollow fiber membrane countercurrent dialyzer, in which only outer fibers are shown for clarity.
In the porous media approach, we define individual velocities and species concentrations to the three phases, namely, the blood,
dialysate and membrane phases. Each phase is treated as a continuum filling the entire space of the dialyzer case. Thus, three
phases share the same space but with different volume fractions. Assigning the subscripts, b , d and m to the blood phase,
dialysate phase and membrane phase, respectively, the following relations can be found for the volume fractions  b ,  m  d , the
specific area of the blood compartment
ab
and that of the dialysate compartment a d :
b 

m  4
d

N
db
4
A
 2t m   d b
2
b

4
db
(1a)

b  4
2

 d  1  1  4

ad 
2
2
tm
db
tm
db

t 
1  m  b
 db 

t 
1  m   b
d b  

ab 
4 b
db
4 b
db

t
1  2 m
d
b

(1b)
(1c)
(2a)



(2b)
Note that N is the number of hollow fibers while A is the cross-sectional area of the dialyzer case. The membrane thickness,
the inner and outer diameters of the hollow fiber are indicated by
t m , db
and d d  d b  2t m , respectively.
Figure 1. Hollow fiber membrane countercurrent dialyzer
Sano and Nakayama [16] on the other hand introduced the local volume averaging theory[17-25] so as to describe complex
transport phenomena associated with three individual phases, namely, the shell and lumen side phases and membrane phase in
the countercurrent dialyzer system.They introduced the macroscopic governing equations, namely,the continuity, momentum
and species mass transfer equations forthe three individual phases as following:
60
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
For the blood (lumen)phase:
b
 b u j
  0
x j

 b c
b

t
 ui
b

t
b
 b u j
c
x j
 uj
b
b
ui

x j
b

x j

 p
(3)
b
xi

 2 ui
x j
b

2

b u j
Kb
b
(4)
ij

b
b

 c 
 D  c   D
 ab ht b c
b b dis jk
 b b x j
xk 


b
 c
m
  c
b
(5)
For the dialysate (shell) phase:
d
 d u j
 0
x j
u
 i
t
 d c
t
d

d
 d u j

d
c
d
 uj
d
x j
d
ui


x j
 p
 2 ui



d u j
2
xi
K d ij
x j
d

x j
(6)
d

d
d

 c 
 D  c   D
 ad ht d c
d d dis jk
 d d x j
xk 


d
(7)
d
 c
m
  c
m
(8)
For the membrane phase:
 m c
t
m


x j

m


 D  c   a h c b  c
m m jk
b tb

xk 

m
  c
b

 ad ht d c  c
d
m
  c
m
(9)
where
m


iR T
b
d
m
d 

(10)
u
n
dA

a
L
p

p


c  c 
b P
Abint j b j
M




Eqs. (3)–(5) describe the continuity, momentum and mass balance equations for the blood phase, while Eqs. (6)–(8) describe
these equations for the dialysate phase, respectively. In this study,these three- dimensional tonsorial set of the macroscopic
governing equations along with the mass balance Eq. (9) for the membrane phase are exploited for full three-dimensional
numerical calculations.
  ab J v 

1
V



As carrying out a volume averaging procedure, one decomposes a certain variable ~ into its intrinsic average 
f
and
spatial deviation ~ :
 
where the intrinsic average 
f
f
 ~
(11)

(12)
is defined as

f

1
Vf
Vf
dV .
The volume V f (f=b, p, m) is the volume space, which f phase in question (blood, dialysate or membrane) occupies within
the total local control volume V. As defined in Eq. (10),  is the ultrafiltration volume rate per unit volume, which is the product
of the specific area of the membrane surface ab and the total permeate volume flux J V through the membrane. As usual, the
membrane is characterized in terms of three parameters, namely, the hydraulic permeability LP , the solute permeability hm and

the reflection coefficient  . Moreover, the osmotic pressure is given by iR T
kmol K), M and T
m
m

M c
m
 c
d
, where i, R(=8341 J/kg
are the number of ions for ionized solutes (i.e. Vant Hoff factor), ideal gas constant, molecular weight of
solute and temperature, respectively. However, in a hemodialysis, it is a common practice to control the osmotic pressure
between the blood and dialysate (around 3000mOsm/l), such that the osmotic pressure can be negligible.
As carrying out three dimensionalnumerical calculations based on Eqs. (3)-(10), the permeability tensor is given by
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
 1

 K b xx

1
 0
K bij 

 0

K bxx 
0
1
K b yy
 b db 2
32
0
61

0 


0 

1 
K b zz 
(13)
 K byy  K bzz
(14)
The axial permeability component K bxx was estimated assuming laminar fully-developed flow in a tube, while the transverse
components K b  K b may virtually be set to zero. The axial dispersion diffusivity component may be given following
yy
zz
Nakayama et al. [28] as
D  u db
 b 
192  Db

b
Db disxx
The effective mass transfer coefficient
2

 >> D
b dis yy = Db diszz = 0


(15)
ht b for the case of negligible tortuosity within the membrane is defined as
 db  dd

db
1
1
2


ln 
ht b hb 2 p Dm  d b





  1  d b ln 1  t m
 hb 2 p Dm  d b





(16)
where Dm is the solute diffusion coefficient in water, the mass transfer coefficient for the lumen side is given by Leveque et al
[29].
Shb 
hb d b
13
 1.62Re b Scb db L 
Db
(17)
The axial permeability component within the dialysate phase may be evaluated using the hydraulic diameter concept as
K d xx
 d   d d b 2 

32   b d d 
2
(18)
while the transverse components may be estimated according to Kuwahara et al. [30] as
K d yy  K d zz 
 d 3d d 2
120 1   d 
(19)
The axial dispersion diffusivity component may be estimated using the hydraulic concept as
2
2
D  u   d d b  
(20)

 >> Dd dis = Dd dis
Dd disxx  d 
yy
zz
192  Dd   b d d  


The transverse dispersion diffusivity components Dd dis and Dd dis are usually as small as 1/20 of the axial dispersion
yy
zz
d
counterpart (Yang and Nakayama [31]). The effective mass transfer coefficient
ht b for the dialysate phase is defined as
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Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69

 d
dd
1
1
d


ln 
ht d hd 2 p Dm  d b  d d

2



  1  db
 hd 2 p Dm


t

1 2 m

t 
db
1  2 m  ln 

tm
db 

 1
db







(21)
where Fukuda et al. [32] introduced the mass transfer coefficient for the shell side as following.
Shd 
hd d e
13
0.5
 1.80 Re d Scd d e L 
Dd
(22)
These partial differential equations can be solved numerically for given initial and boundary conditions using a standard
numerical procedure.
3. Three-Dimensional Numerical Computations
Sano and Nakayama [26] introduced a three-dimensional numerical computation procedure based onSIMPLE algorithm [25] by
utilizing the membrane transport model [16].Since the continuity equations (3) and (6) contain  , it would not be
straightforward to formulate the pressure correction equation (i.e. discretized continuity equation). This problem can readily be
overcome, introducing the combined velocity vector:
ui   b ui
b
  d ui
d
(23)
such that
 uj
x j
Exploiting the combined velocity vector
0
(24)
u i , a usual discretization procedure can be used to formulate the pressure
correction equations from the discretized continuity equation and momentum equation.
A three-dimensional numerical model based on HospalFiltral 12 AN69HF is shown in Figure 2, in which the dialysate enters
from the tube connected vertically near the right end of the container and leaves from the tube connected vertically near its left
end.Geometric characteristics of the membrane module are provided by the manufacturer as follows:
L  20cm, N  8500, d b  220 μm, t m  45μm, A  11.94cm 2
while the hydraulic permeability of the membrane L p  5.63  10 11 m/s Pa are provided by Ding et al.[12].
Figure 2. Numerical model
(25)
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
63
In Figure 3, a cross-sectional velocity field in the dialysate phase is illustrated in terms of local velocity vectors, when the
dialysate and blood inlet volume flow rate are Qd in  500 ml / min and Qbin  200 ml / min . In this study, the dialysate inlet
pressure is given as boundary condition so as to achieve the total ultrafiltration rate Q f  Qbin  Qbout  20ml / min , which
commonly set in the hemodialysis. The dialysate fluid particles pass deep through the bundles of hollow fibers and change the
direction to flow horizontally, before making sharp turn to exit from the container. The dimensionless Creatinine concentration
fields for dialysate and blood phases c b, d c b on the vertical symmetrical plane are illustrated in Figure 4. The dimensionless
in
Creatinine concentration in the blood phase decreases as it flows from left to right, while the dialysate carries the waste in the
blood phase as it flows from right to left. The figures indicate that the waste concentration of blood going out from the right end
is quite non-uniform. The blood in lower layers away from the entrance of the dialysate is less purified than that in upper layers
close to it.
Figure 3. Velocity field in the dialysate phase
Dialysate phase
Blood phase
Figure 4. Concentration fields in dialysate and blood phases

The primary may interest in the clearance of the solute from blood to dialysate CL  Qbin cbin  Qbout cbout
c
bin
, which is
64
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
often used for the scale for the dialysis time, where c b is the bulk mean solute concentration.Figure 5 shows the pure diffusive
clearances CL0 for Creatinine and Vitamin B12 concentrations, namely the clearance in case of non-ultrafiltration in the
dialyzer,when the dialysate inlet volume flow rate and the total ultrafiltration rate are Qd in  500 ml / min and
Q f  0ml / min . The figure shows good agreement between the present numerical calculations and the experiment reported by
Jaffrin et al. [33]. In Figure 8, the clearances CL obtained from the calculations are plotted against the predicted ultrafiltration
volume flow rate Q f with the experimental data [33].These results indicate the validity of present numerical simulation
associated with countercurrent dialyzer systems. Thus, this method may be useful fordescribing complex transport process in
hemodialysis.
Figure 5. Pure diffusive clearance
Figure 6. Effect of ultrafiltration flow rate on the clearance enhancement
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
65
4. Optimal Design of Hollow Fiber Dialyzers
We shall investigate the effect of the housing on the efficiency of dialyzer by utilizing the several casings as shown in Figure
7.The geometric characteristics of the membrane are fixed as HospalFiltral 12 AN69HF. Creatinine clearances obtained from a
series of numerical simulations for each dialyzer casing are illustrated in Figure 8, where the dialysate and blood inlet volume
flow rate are Qd in  500 ml / min and Qbin  200 ml / min ,as well as the total ultrafiltration rate is Q f  20ml / min .As
can be seen from the Figure 8, the Creatinine clearance of the housing number 4 is the highest as compared with the others, in
which additional space is installedsurrounding the hollow fiber bundle at bothinlet and outlet of the dialysate.Furthermore, we
must evaluate whether homogeneous mass transfer is carried out in a dialyzer. In Figure 9, a meridian velocity field for the
dialysate phase within the housing number 4 is illustrated in terms of local velocity vectors. Dialysate permeates uniformly into
the hollow fiber bundle from the free space at both inlet and outlet in dialysate phase, the dialysate velocity field also become
uniform as compared with the original one as shown in Figure 3. Uniform flow of the dialysate phase may lead to the
homogeneous mass transfer in the dialyzer. Thus, the space surrounding the hollow fiber bundle at both inlet and outlet of the
dialysate is quite beneficial so as to achieve a high clearance value and the homogeneous mass transfer in dialyzer. In this study,
although the difference among four housings on the Creatinine clearance enhancement is small, we are confident that a series of
present numerical calculations for evaluating individual concentration fields, namely, blood, dialysate and membrane phases will
lead to optimization for the dialyzer.
Figure 7. Several types of housing of dialyzer
66
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
Figure 8. Effect of housings on the Creatinineclearance enhancement
Figure 9. Velocity field in the dialysate phase associated housing number 4
Subsequently, the effect ofgeometric configuration of hollow fibers on the clearance was investigated by changing the
number of hollow fibers for the given pumping power. A series of numerical simulations were carried out with the housing
number 4, in whichouter diameters of the hollow fiber was changed to 200μm from 220μm to estimatethe effect of diameters of
hollow fibers. The pumping power for blood and dialysate phases are defined as follows:
P.P.b ,d   u j
b ,d
 p
b ,d
(26)
dV
x j
The dialysate and blood inlet volume flow rates are to be determined so as to have a pumping power for the case of the
operating condition Qd in  500 ml / min , Qbin  200 ml / min and Q f  20 ml / min , respectively. The dialysate and blood
inlet volume flow rates obtained from numerical calculations for the case of several numbers of hollow fibers are tabulated in
Table 1. It can be seen from Table 1, the blood inlet volume flow rate increases with the number of hollow fibers.However the
blood inlet volume flow rateis less than the case of outer diameters of the hollow fiber 220μm.The effects of the number of
hollow fibers on the Creatinine clearance are illustrated in Figure 10. It is interesting to note that the optimum value of the
number of hollow fibers exists for a given pumping power, since specific surface area concerning mass transfer increases with
the number of hollow fibers, while dialysate inlet volume flow rate decreases as increasing of the number of hollow
fibers.Moreover Figure 10 shows that the effect of diameters of hollow fibers on the clearance is large due to the variance of the
blood inlet volume flow ratefor the given pumping power. It is found that presented numerical simulation may be useful for
optimization of hollow fiber dialyzer systems.
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
67
Figure 10. Effect of the number of hollow fibers on the Creatinineclearance enhancement
Table 1. Dialysate and blood inlet volume flow rate for given pumping power
N
5426
7234
9042
10851
12660
εd
0.7
0.6
0.5
0.4
0.3
Qbin[ml/min]
136
156
173
187
202
Qdin[ml/min]
1070
910
737
559
370
5. Conclusions
A general set of macroscopic governing equations for countercurrent hollow fiber dialyzers were examined to seek the effective
hollow fiber dialyzers. The effects of dialyzer housing configurations and number of hollow fibers were investigated in detail.
Available experimental data agree well with the present numerical estimates based on the membrane transport model. It has been
confirmed that uniform velocity filed of dialysate phase is essential to highly efficient housing of dialyzer in terms of a high
clearance value and the homogeneous mass transfer in dialyzer. Additional space surrounding the hollow fiber bundle at both
inlet and outlet of the dialysate is beneficial so as to achieve a high clearance value and homogeneous mass transfer in dialyzer. A
series of three-dimensional numerical calculations reveal that the optimum value of the number of hollow fibers exists for a given
pumping power. The presented numerical simulations may be useful for optimization of hollow fiber dialyzer systems.
6. Nomenclature
A
Aint
ab ,d
c
CL
CL0
d b,d
: cross-sectional area of the dialyzer case [m2]
: interface area between the fluid and membrane phases [m2]
specific surface area [1/m]
:
: solute concentration [kg/m3]
: clearance [m3/s]
: pure diffusive clearance [m3/s]
: inner and outer diameters of the hollow fiber[m]
Db ,d ,m
: solute diffusion coefficient[m2/s]
Ddis jk
: dispersion tensor [m2/s]
h
: mass transfer coefficient [m/s]
: effective mass transfer coefficient [m/s]
ht
i
JV
: Vant Hoff factor
: ultrafiltration velocity [m/s]
K b,d ,m
: permeability [m2]
L
: effective length of the dialyzer case[m]
: hydraulic permeability of membrane [m/s Pa]
Lp
M
ni
N
p
: molecular weight of the solute [−]
: unit vector pointing outward from the fluid side to membrane side [-]
:number of hollow fibers [-]
: pressure [Pa]
P.P.
Q
: pumping power [Pa]
: volume flow rate [m3/s]
Qf
: total ultrafiltration rate [m3/s]
R
: universal gas constant [J/kg kmol K]
: Re  u d  : Reynolds number[-]
Re
68
Open Journal of Heat, Mass and Momentum Transfer (2014) 58-69
Sc
: Sc   D : Schmidt number[-]
tm
T
ui
: membrane thickness [m]
: temperature
: velocity vector [m/s]
: combined velocity vector [m/s]
ui
: representative elementary volume [m3]
: initial total volume of blood [m3]
V
Vb
x, y, z : Cartesian coordinates [m]
Greek symbols
 b,d ,m : volume fraction [-]


: viscosity [Pa s]
: density [kg/ m3]
: reflection coefficient [−]
: perfusion rate [1/s]


Special symbols
~
:deviation from intrinsic average



:Darcian average
b,d ,m
:intrinsic average
Subscripts and superscripts
b: blood
d: dialysate
m: membrane
dis: dispersion
f: fluid
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