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Pervaporation in hollow fibres:
Journal of Membrane Elsevier
Science
Science,
Publishers
PERVAPORATION EFFICIENCY
IN HOLLOW
-
Printed
GEnie Biologique,
J.C. MORA
and R. BES
FIBRES:
VIA THE FRACTAL
CONCEPT*
BP 45 F, 63170 AubiBres (France)
192, I.G.C. Chemin de la Loge, 31078
January
in The Netherlands
G. DJBLVEH
Laboratoire
(Received
321
321-331
Amsterdam
COEFFICIENT
R. PLUBPRASIT,
L.A. C.N.R.S.
21 (1984)
B.V.,
3, 1984;
accepted
in revised
Toulouse
Cedex (France)
form July 6, 1984)
Summary The complexity of the equations for the simultaneous transfer of momentum, heat and mass with a change in the physical state of a membrane means that only the use of an overall coefficient can account for the performance of an industrial hollow fibre pervaporator; this coefficient is expressed by means of the mathematical fractai concept.
1. Introduction Pervaporation is a gas-liquid separation method based on mass transfer through a membrane with a change in the physical state of the diffusing component. The application of this process on an industrial scale requires the use of a large transfer area contained in a small volume; the use of hollow fibres would therefore seem to be appropriate. The surface obtained in this way is not a traditional geometrical surface, but is rather a “volumetric” surface. The exchange area is much greater than the outside area calculated from the overall dimensions of the fibre bundle; this makes it quite different from a flat surface area (Fig. 1). For example, a bundle 0.25 m long with an outside diameter of 0.04 m (an outer area of the order of 0.01 m”) can have an internal surface area of the order of 1 m2, i.e., an area one hundred times greater than the outside area. The impossibility of integrating the equations based on traditional microscopic momentum, mass and energy balances show up the need for an overall empirical coefficient, an efficiency coefficient which could be determined experimentally. The problem encountered with hollow fibres is the same as in the case of catalytic chemical reactors. One of the major difficulties arises from the fact that the surface of a bundle of hollow fibres, like the surface *Paper
presented
0376-7388/84/$03.00
at IMTEC
‘83, November
8-10,
0 1984 Elsevier Science
1983,
Sydney,
Publishers
Australia.
B.V.
322
of a solid catalyst, is not a traditional one such as a plane or a sphere. The mathematical concept of the fractal seems to offer a convenient way of modelling such a surface. This paper shows how it is possible to find a new, convenient and pragmatic way of defining an efficiency coefficient by this means.
2R
aO=
ag >> 2flRZ Fig. 1. Specific
geometry
2. Mass transfer
of hollow-fibre
in a hollow-fibre
Z2
bundle.
bundle
Traditional modelling is based on a transfer law of the Fick type; the diffusion coefficient is generally obtained experimentally using a flat membrane. Experiments with isolated fibres are difficult to carry out with traditional methods, because of the very low value of the mass flux. So, for an industrial plant, an overall exchange coefficient must be used: Nj
= k,,Ao6
= A,J,
(1)
For a fibre, it is theoretically possible to take into account the effect of the cylindrical shape (Fig. 2b), but a correction factor must be introduced to account for modifications in the polymer structure which appear during spinning of the fibre (Fig. 2a). Therefore the mass flux now becomes: Nj’ = n, k,,Af6
= n,Af Jo = Af J,
(2)
In the case of a bundle, the flux lines are curved (Fig. 2c), menisci can appear, fibres can be deformed (Fig. 2d) and there are axial gradients in concentration, temperature and pressure (Fig. 2e). Even if it should be possible theoretically to devise a series of calculations which would lead to an overall efficiency coefficient, this is not feasible in practice and the transfer must be represented by an equation of the type: Nj” = vkoAp6
= vAP J,, = A, J
(3)
323
where Nj, Nj’, Nj” and 6 are the experimental results, and Ao, Af and A, the geometric area. The possibility in some cases of separating 77into two terms where that seems useful gives: (4)
77 = 77p77t
The coefficient q sums up, in a non-explicit manner, the particular geometrical form of the bundle, the changes in polymer structure, contact and polarization effects, the various gradients, etc. This coefficient is always determined experimentally.
POLYMER
i
i
ANj = k,a,r,
A
I
Fig. 2. Various
steps
in the calculation
F LENGTH
of mass transfer
in a hollow-fibre
bundle.
324
3. The fractal concept The fractal concept has been proposed to represent a particular sort of physical object [l-3] such as snow flakes (Fig. 3), river systems, powders [4,5], chaotic flows [6], lungs [7], turbulent fluctuations [8], transfer between phases [9,12] and electrochemical transfer [lO,ll]. This concept was first introduced by mathematicians such as Cantor. It was taken up by Mandelbrot [l] and adapted to the representation of objects or physical phenomena for use in numerical simulations.
Fig. 3. Fractal
model
of a snow flake.
The von Koch curve [3] provides a simple way for understanding this concept. A segment of a straight line undergoes a series of repetitive transformations, T (Fig. 4). Each of these operations consists in transforming a segment by homothethis with a ratio X, thus forming a broken line of N elements with N > l/h. In Fig. 4 we have N = 4 and X = l/3. By repeating this operation indefinitely, a final curve is obtained of infinite length and entirely located in a finite part of a plane. This curve has no tangent at any point. By setting up such a curve and using as the starting point an equilateral triangle, a mathematical model for a snow flake is obtained (Fig. 3). If the classical concept of geometrical dimension (Euclidean dimension) is generalized so as to give the fractal dimension, defined by the following relationship:
325 E3 -T ,I
-T ------,T
<
A
[==--p
N=4
N&
x = l/3
1
Fig. 4. Presentation
D of the fractal
‘; LINE
Log
concept
r-l II I’
using example
?
/I
”
IT
1
JL CURVE
SURFACE N =
l/x
l/x
D =
D=l
Fig. 5. Euclidean
and fractal
l/x’
1.26
dimensions.
z;z
of the von Koch
i
FRACTAL
=
l/h
N/Log
IT
T
_---_-__
=
x
A + l/N
N =
D=
l/x’
2
curve.
326
D=-
NXD
log N
(54
log l/X
=
(5b)
1
the von Koch curve is found to have a dimension D = log 4/lag 3 > 1, lying between 1 and 2. So this curve is more than a traditional line (dimension of 1) and less than a surface (dimension of 2). Indeed, if a straight line is divided up by transformations of the fractal type, they obey the relationship h = l/N, so D = 1. For a square we find N = l/h* so D = 2 (Fig. 5). 4. Fractal
model of a hollow-fibre
bundle
The model is based on the wetted perimeter of the fibres: N 27x. This perimeter is a tightly packed curve contained in an envelope (Fig. 6d) and can be compared to the von Koch curve. Each element can be related to the perimeter of a single fibre. The fractal concept can be adapted to this system in the following way:
(6)
by-‘\
c/A-.\
m / G!iGiF !\,
I I
\
REFERENCE
\
/
\I
D=l
D z1.74
Fig. 6. Application
-4
\
\
--
D =
D=
of the fractal
d/ /
1.66
2
concept
to a hollow-fibre
bundle.
327
Thus, the fractal dimension can be seen as a quantification of the fragmented character of the internal surface compared with the external surface of the bundle. This is illustrated in Fig. 6. For a single fibre r = R, therefore D = 1, and the surface is a simple cylinder (Fig. 6a). For fibres packed together, as their number, N, goes to infinity, the whole of the cross-section of the bundle becomes increasingly filled with fibre boundaries and, in the limiting case, the cross-section is entirely filled, so that D = 2 as for a flat surface. In the case of our experimental hollow-fibre bundle, D is about 1.75. Therefore, the transfer area of the bundle is more like a volume than a surface. This idea is implicitly accepted in chemical engineering: for a packed absorption column, the transfer coefficient is defined by reference to the surface area of the packing (D = l), but for a gas-solid catalytic reactor, the reaction rate is expressed with respect to unit volume (D = 2), even though the catalysis concerned is a surface phenomenon. It should be noted that the geometrical entropy [13] can be used to characterize the complexity of the wetted perimeter: length of internal curve length of outer envelope so = log 2XN This mathematical definition provides a justification for the choice of the outer radius, R, as the reference. 5. Efficiency coefficient The efficiency coefficient is given by eqn. (2): J=qJ,
(8)
The overall transfer taking place in the bundle results from the transfer through each fibre, so it is possible to consider this transfer as a fractal, the characteristic dimension of which is given by: N
mass transfer through one fibre, r Dt mass transfer through cylinder, R
= 1
(9)
This relationship is a comparison of the mass transfer through a real fibre in the bundle, with the transfer through a cylindrical fibre of large diameter (equivalent to a flat membrane). In terms of mass fluxes we have: N
so
(=-)
Dt= +)Dt (;)“” =
1
(10)
328
(11) By referring to eqn. (3), it can be seen that a distinction can be made between a transfer effect, vt, and a geometrical effect, np; the fractal analysis suggests the following correspondences: vp
+
Ot
‘+
geometrical
effect
+ D
transfer effect
+ D,
This analysis is probably not the only one possible. Equation (3) can be seen as a very special case an?, in fact, it is only based on pragmatic considerations with no strict theoretical support. 6. Comparison
with experimental
results
Experiments were performed on the separation by pervaporation of a binary water-ethanol mixture using commercial hollow-fibre bundles (0.20 m long, 0.04 m outside diameter, about 10,000 fibres of 200 pm diameter). The first results gave an efficiency coefficient, q, of less than 0.1 compared with a flat membrane of 10 cm2 area. The geometrical fractal dimension of the mass transfer is thus less than D, = 1.2. However, this result has no meaning without a study of the sensitivity of D to the parameters N, h and n. Tables 1 and 2 show that D is not very sensitive to X or N in the range of values used in our work. However, Table 3 shows that D, is quite sensitive to n. TABLE
1
Sensitivity N D
8,000 1.696
TABLE
l/180 1.774
TABLE
1
T
0.02
12,000 1.773
of D to h, N = 10,000 l/200 1.738
l/220 1.707
3
Sensitivity
Dt
10,000 1.738
2
Sensitivity h D
of D to N, h = l/200
of D, to q ; D = 1.738, 1.2 0.09
1.3 0.17
N = 10,000, 1.5 0.43
h = l/ZOO D 1
329
Table 4 gives experimental results for a flat membrane, a single fibre and a bundle. The overall efficiency coefficient for water (the major diffusing component) is always low and the bundle effect (J/J,) seems to be the more important. D, is very near unity, so the fractal reference can be taken with D, = 1 as follows: r
D-l
Jo = QJO = vtqJo
J = vf jj(
1
(12)
or D-l
J=v,J,
(
$
(13)
1
so qf (and 77,) are of the order of 1 instead of the order of 10m2 for qt (line 4 of Table 4) in the case of water (component j). TABLE
4
Experimental
results -Example x (vol.% 0
Flat membrane j k
J”
Isolated
of efficiency
coefficient
EtOH) 50
30
Pervaporation
and ultrafiltration
70
effects
160 z
110 25
Bundle 140
J
z
4.3 3.0
88 22 2.6 3.9
Bundle
j
JIJs=sp
0.87
k
Isolated fibre
0.039
0.12
Isolated fibre
2.7 2.9
0.59 0.005
0.13 0.0003
0.040 0.0003
0.024 0.14
0.010 0.0006
0.027 0.0007
0.0095 0.0010
0.030 0.18
0.045 0.59
0.071 0.78
j
0.33 0.18
j k by eqn. (12)
1.2 7.0
k Flat membrane
Defined
4.2 0.29
4.9 0.45
2.1 4.4
j k Flat membrane
qs
57 7.9
38 3.7
Bundle
Defined
114 21
98
47 7.5
JIJ, ‘17t
Qf
90
fiber
Js
J&Jo
80
43.5 : by eqn. (13)
Bundle: N = 7500, R = 1510.‘m, i = water, k = alcohol. Cuprophan, D = 1.78. J,, J,, J: 10m’mol-sec~‘-m~‘.
1.9
6.0
1.5 9.0
2.2 29.5
r = 100 X 10m”m. z = 0.17 m.
3.5 39
bad accuracy,
+
economic
0.5 0.03
1.3 0.03
flux too low
interest
0.5 0.05
+
330
Taking the composition x = 80% EtoH (column 5 of Table 4) the following comments are possible: (1) The low value of g (17= 2.4%) means that the flat membrane model is not adequate to describe a bundle. The name “efficiency coefficient” is therefore not correct: “adequation coefficient” of the model would be better. (2) The value 33% for qt may mean that the structure of the hollow-fibre membrane is not exactly the same as that of a flat membrane; however, compared to the value of 2.4%, this structure effect is not the more important (3) The number of fibers, IV, and the homothethis ratio, h, are the bestknown characteristics of the bundle: why not use them? (4) By reference to the fractal model (D = 1.75 in eqn. 11) and comparison with the experiments (77= 0.024), the corresponding value of Dt is about 1 (Table 3). (5) The efficiency coefficient qf = 1.2 (compared to 7) = 0.024) means that the fractal model is nearest to the experimental results and so may be a justification of this model. 7. Conclusion This paper offers a new way of taking into account the complexity of the pervaporation process in a hollow-fibre bundle. We consider the present work as a first attempt and this project will be continued, together with traditional modelling, to find out exactly what kind of phenomena are involved in the mass-transfer fractal dimension. Using traditional modelling based on heat, mass and momentum balances, it is possible to go from a diffusion coefficient to a local transfer coefficient, and from a local transfer coefficient to an overall coefficient for an isolated fibre. However, for fibre bundles, because of the complexity of the flow, only overall efficiency coefficients can be used. The approach which has been presented here offers a way of explicitly introducing the specific character of the transfer surface, so representing a step towards a theoretical expression for the efficiency coefficient. The mass-transfer fractal dimension lumps together all the other phenomena and the next step will be to analyze what effects are contained in this quantity. List of symbols
AD A
ex
A, Ao D Dt J JS
overall bundle area (m”) external area of the bundle (m’) lateral area of a fibre (m’) area of flat membrane (m’) fractal dimension of the bundle mass-transfer fractal dimension mass flux through the bundle (mol-sec-1-m-2) mass flux through a single fibre (mol-sec-‘-m-2)
mass flux through a flat membrane (mol-sec-‘-m-2) mass-transfer coefficient (J,, = ho 6) number of fibres molar flow of component j through a flat membrane (mol-set-‘) molar flow through an isolated fibre (mol-set-‘) molar flow through the bundle (mol-see-‘) fibre radius (m) radius of the bundle, cylindrical co-ordinate (m) geometrical entropy of the bundle fractal transformations composition (volume % ethanol) length of the bundle, dimension of a flat square membrane (m) homothethis ratio overall efficiency coefficient bundle efficiency coefficient efficiency coefficient, referred to fractal transfer efficiency coefficient (defined by eqn. 2) efficiency coefficient, referred to fractal and single fiber mass transfer driving force References
10 11
12
13
B. Mandelbrot, Les Objets Fractals, Flammarion, Paris, 1975. B. Mandelbrot, Fractals, W.H. Freeman, San Francisco, 1977. B. Mandelbrot, Les objets fractals, La Recherche, 85 (1978) 5. B.H. Kaye, Specification of ruggedness and/or texture of a fine particle profile by its fractal dimension, Powder Technol., 21(l) (1978) 1. A.G. Flook, Use of dilatation logic and quantiment to achieve fractal dimension characterisation of textured and structured profiles, Powder Technol., 21(2) (1978) 295. H. Mori, Fractal dimensions of chaotic flows of autonomous dissipative systems, Progr. Theor. Phys., 63(3) (1980) 1044. J. Lefevre, Teleogical optimization of a fractal tree model of the pulmonary vascular bed, Bull. Eur. Physiopath. Resp., 16( 1) (1980) 53. F. Stapleton, Turbulent fluctuations as fractal functionals of averages, Bull. Amer. Phys. Sot., 25(B) (1980) 1034. G. Djelveh and J.C. Mora, Heat and mass transfer surface between two phases by the concept of fractal object, paper presented at CHISA ‘81, Comm. C1.5, Prague, August 1981. A. Lemehaute and G. Grepy, Sur quelques propridtes de transferts electrochimiques en geombtrie fractale, C.R. Seances Acad. Sci. II, Met. Phys., 294( 14) (1982) 685. A. Lemehaute, A. Deguibert, M. Delaye and C. Filippi, Notes d’introduction de la cinetique des Bchanges d’bnergies et de mat&e sur les interfaces fractales, CR. Seances Acad. Sci., II, M&Z. Phys., 294(14) (1982) 835. G. Djelveh, P. Aptel, R.S. Bes and J.C. Mora, Modelisation de la pervaporation dans un faisceau de fibres creuses en tant qu’operation unitaire de genie chimique: Utilisation du concept d’objet fractal, European Workshop on Pervaporation, Nancy, 21-22 September 1982. Laboratoire de Mathematiques et d’Informatique, Universite de Bordeaux, Courrier C.N.R.S., 53 (1983) 5-9.
Science
Science,
Publishers
PERVAPORATION EFFICIENCY
IN HOLLOW
-
Printed
GEnie Biologique,
J.C. MORA
and R. BES
FIBRES:
VIA THE FRACTAL
CONCEPT*
BP 45 F, 63170 AubiBres (France)
192, I.G.C. Chemin de la Loge, 31078
January
in The Netherlands
G. DJBLVEH
Laboratoire
(Received
321
321-331
Amsterdam
COEFFICIENT
R. PLUBPRASIT,
L.A. C.N.R.S.
21 (1984)
B.V.,
3, 1984;
accepted
in revised
Toulouse
Cedex (France)
form July 6, 1984)
Summary The complexity of the equations for the simultaneous transfer of momentum, heat and mass with a change in the physical state of a membrane means that only the use of an overall coefficient can account for the performance of an industrial hollow fibre pervaporator; this coefficient is expressed by means of the mathematical fractai concept.
1. Introduction Pervaporation is a gas-liquid separation method based on mass transfer through a membrane with a change in the physical state of the diffusing component. The application of this process on an industrial scale requires the use of a large transfer area contained in a small volume; the use of hollow fibres would therefore seem to be appropriate. The surface obtained in this way is not a traditional geometrical surface, but is rather a “volumetric” surface. The exchange area is much greater than the outside area calculated from the overall dimensions of the fibre bundle; this makes it quite different from a flat surface area (Fig. 1). For example, a bundle 0.25 m long with an outside diameter of 0.04 m (an outer area of the order of 0.01 m”) can have an internal surface area of the order of 1 m2, i.e., an area one hundred times greater than the outside area. The impossibility of integrating the equations based on traditional microscopic momentum, mass and energy balances show up the need for an overall empirical coefficient, an efficiency coefficient which could be determined experimentally. The problem encountered with hollow fibres is the same as in the case of catalytic chemical reactors. One of the major difficulties arises from the fact that the surface of a bundle of hollow fibres, like the surface *Paper
presented
0376-7388/84/$03.00
at IMTEC
‘83, November
8-10,
0 1984 Elsevier Science
1983,
Sydney,
Publishers
Australia.
B.V.
322
of a solid catalyst, is not a traditional one such as a plane or a sphere. The mathematical concept of the fractal seems to offer a convenient way of modelling such a surface. This paper shows how it is possible to find a new, convenient and pragmatic way of defining an efficiency coefficient by this means.
2R
aO=
ag >> 2flRZ Fig. 1. Specific
geometry
2. Mass transfer
of hollow-fibre
in a hollow-fibre
Z2
bundle.
bundle
Traditional modelling is based on a transfer law of the Fick type; the diffusion coefficient is generally obtained experimentally using a flat membrane. Experiments with isolated fibres are difficult to carry out with traditional methods, because of the very low value of the mass flux. So, for an industrial plant, an overall exchange coefficient must be used: Nj
= k,,Ao6
= A,J,
(1)
For a fibre, it is theoretically possible to take into account the effect of the cylindrical shape (Fig. 2b), but a correction factor must be introduced to account for modifications in the polymer structure which appear during spinning of the fibre (Fig. 2a). Therefore the mass flux now becomes: Nj’ = n, k,,Af6
= n,Af Jo = Af J,
(2)
In the case of a bundle, the flux lines are curved (Fig. 2c), menisci can appear, fibres can be deformed (Fig. 2d) and there are axial gradients in concentration, temperature and pressure (Fig. 2e). Even if it should be possible theoretically to devise a series of calculations which would lead to an overall efficiency coefficient, this is not feasible in practice and the transfer must be represented by an equation of the type: Nj” = vkoAp6
= vAP J,, = A, J
(3)
323
where Nj, Nj’, Nj” and 6 are the experimental results, and Ao, Af and A, the geometric area. The possibility in some cases of separating 77into two terms where that seems useful gives: (4)
77 = 77p77t
The coefficient q sums up, in a non-explicit manner, the particular geometrical form of the bundle, the changes in polymer structure, contact and polarization effects, the various gradients, etc. This coefficient is always determined experimentally.
POLYMER
i
i
ANj = k,a,r,
A
I
Fig. 2. Various
steps
in the calculation
F LENGTH
of mass transfer
in a hollow-fibre
bundle.
324
3. The fractal concept The fractal concept has been proposed to represent a particular sort of physical object [l-3] such as snow flakes (Fig. 3), river systems, powders [4,5], chaotic flows [6], lungs [7], turbulent fluctuations [8], transfer between phases [9,12] and electrochemical transfer [lO,ll]. This concept was first introduced by mathematicians such as Cantor. It was taken up by Mandelbrot [l] and adapted to the representation of objects or physical phenomena for use in numerical simulations.
Fig. 3. Fractal
model
of a snow flake.
The von Koch curve [3] provides a simple way for understanding this concept. A segment of a straight line undergoes a series of repetitive transformations, T (Fig. 4). Each of these operations consists in transforming a segment by homothethis with a ratio X, thus forming a broken line of N elements with N > l/h. In Fig. 4 we have N = 4 and X = l/3. By repeating this operation indefinitely, a final curve is obtained of infinite length and entirely located in a finite part of a plane. This curve has no tangent at any point. By setting up such a curve and using as the starting point an equilateral triangle, a mathematical model for a snow flake is obtained (Fig. 3). If the classical concept of geometrical dimension (Euclidean dimension) is generalized so as to give the fractal dimension, defined by the following relationship:
325 E3 -T ,I
-T ------,T
<
A
[==--p
N=4
N&
x = l/3
1
Fig. 4. Presentation
D of the fractal
‘; LINE
Log
concept
r-l II I’
using example
?
/I
”
IT
1
JL CURVE
SURFACE N =
l/x
l/x
D =
D=l
Fig. 5. Euclidean
and fractal
l/x’
1.26
dimensions.
z;z
of the von Koch
i
FRACTAL
=
l/h
N/Log
IT
T
_---_-__
=
x
A + l/N
N =
D=
l/x’
2
curve.
326
D=-
NXD
log N
(54
log l/X
=
(5b)
1
the von Koch curve is found to have a dimension D = log 4/lag 3 > 1, lying between 1 and 2. So this curve is more than a traditional line (dimension of 1) and less than a surface (dimension of 2). Indeed, if a straight line is divided up by transformations of the fractal type, they obey the relationship h = l/N, so D = 1. For a square we find N = l/h* so D = 2 (Fig. 5). 4. Fractal
model of a hollow-fibre
bundle
The model is based on the wetted perimeter of the fibres: N 27x. This perimeter is a tightly packed curve contained in an envelope (Fig. 6d) and can be compared to the von Koch curve. Each element can be related to the perimeter of a single fibre. The fractal concept can be adapted to this system in the following way:
(6)
by-‘\
c/A-.\
m / G!iGiF !\,
I I
\
REFERENCE
\
/
\I
D=l
D z1.74
Fig. 6. Application
-4
\
\
--
D =
D=
of the fractal
d/ /
1.66
2
concept
to a hollow-fibre
bundle.
327
Thus, the fractal dimension can be seen as a quantification of the fragmented character of the internal surface compared with the external surface of the bundle. This is illustrated in Fig. 6. For a single fibre r = R, therefore D = 1, and the surface is a simple cylinder (Fig. 6a). For fibres packed together, as their number, N, goes to infinity, the whole of the cross-section of the bundle becomes increasingly filled with fibre boundaries and, in the limiting case, the cross-section is entirely filled, so that D = 2 as for a flat surface. In the case of our experimental hollow-fibre bundle, D is about 1.75. Therefore, the transfer area of the bundle is more like a volume than a surface. This idea is implicitly accepted in chemical engineering: for a packed absorption column, the transfer coefficient is defined by reference to the surface area of the packing (D = l), but for a gas-solid catalytic reactor, the reaction rate is expressed with respect to unit volume (D = 2), even though the catalysis concerned is a surface phenomenon. It should be noted that the geometrical entropy [13] can be used to characterize the complexity of the wetted perimeter: length of internal curve length of outer envelope so = log 2XN This mathematical definition provides a justification for the choice of the outer radius, R, as the reference. 5. Efficiency coefficient The efficiency coefficient is given by eqn. (2): J=qJ,
(8)
The overall transfer taking place in the bundle results from the transfer through each fibre, so it is possible to consider this transfer as a fractal, the characteristic dimension of which is given by: N
mass transfer through one fibre, r Dt mass transfer through cylinder, R
= 1
(9)
This relationship is a comparison of the mass transfer through a real fibre in the bundle, with the transfer through a cylindrical fibre of large diameter (equivalent to a flat membrane). In terms of mass fluxes we have: N
so
(=-)
Dt= +)Dt (;)“” =
1
(10)
328
(11) By referring to eqn. (3), it can be seen that a distinction can be made between a transfer effect, vt, and a geometrical effect, np; the fractal analysis suggests the following correspondences: vp
+
Ot
‘+
geometrical
effect
+ D
transfer effect
+ D,
This analysis is probably not the only one possible. Equation (3) can be seen as a very special case an?, in fact, it is only based on pragmatic considerations with no strict theoretical support. 6. Comparison
with experimental
results
Experiments were performed on the separation by pervaporation of a binary water-ethanol mixture using commercial hollow-fibre bundles (0.20 m long, 0.04 m outside diameter, about 10,000 fibres of 200 pm diameter). The first results gave an efficiency coefficient, q, of less than 0.1 compared with a flat membrane of 10 cm2 area. The geometrical fractal dimension of the mass transfer is thus less than D, = 1.2. However, this result has no meaning without a study of the sensitivity of D to the parameters N, h and n. Tables 1 and 2 show that D is not very sensitive to X or N in the range of values used in our work. However, Table 3 shows that D, is quite sensitive to n. TABLE
1
Sensitivity N D
8,000 1.696
TABLE
l/180 1.774
TABLE
1
T
0.02
12,000 1.773
of D to h, N = 10,000 l/200 1.738
l/220 1.707
3
Sensitivity
Dt
10,000 1.738
2
Sensitivity h D
of D to N, h = l/200
of D, to q ; D = 1.738, 1.2 0.09
1.3 0.17
N = 10,000, 1.5 0.43
h = l/ZOO D 1
329
Table 4 gives experimental results for a flat membrane, a single fibre and a bundle. The overall efficiency coefficient for water (the major diffusing component) is always low and the bundle effect (J/J,) seems to be the more important. D, is very near unity, so the fractal reference can be taken with D, = 1 as follows: r
D-l
Jo = QJO = vtqJo
J = vf jj(
1
(12)
or D-l
J=v,J,
(
$
(13)
1
so qf (and 77,) are of the order of 1 instead of the order of 10m2 for qt (line 4 of Table 4) in the case of water (component j). TABLE
4
Experimental
results -Example x (vol.% 0
Flat membrane j k
J”
Isolated
of efficiency
coefficient
EtOH) 50
30
Pervaporation
and ultrafiltration
70
effects
160 z
110 25
Bundle 140
J
z
4.3 3.0
88 22 2.6 3.9
Bundle
j
JIJs=sp
0.87
k
Isolated fibre
0.039
0.12
Isolated fibre
2.7 2.9
0.59 0.005
0.13 0.0003
0.040 0.0003
0.024 0.14
0.010 0.0006
0.027 0.0007
0.0095 0.0010
0.030 0.18
0.045 0.59
0.071 0.78
j
0.33 0.18
j k by eqn. (12)
1.2 7.0
k Flat membrane
Defined
4.2 0.29
4.9 0.45
2.1 4.4
j k Flat membrane
qs
57 7.9
38 3.7
Bundle
Defined
114 21
98
47 7.5
JIJ, ‘17t
Qf
90
fiber
Js
J&Jo
80
43.5 : by eqn. (13)
Bundle: N = 7500, R = 1510.‘m, i = water, k = alcohol. Cuprophan, D = 1.78. J,, J,, J: 10m’mol-sec~‘-m~‘.
1.9
6.0
1.5 9.0
2.2 29.5
r = 100 X 10m”m. z = 0.17 m.
3.5 39
bad accuracy,
+
economic
0.5 0.03
1.3 0.03
flux too low
interest
0.5 0.05
+
330
Taking the composition x = 80% EtoH (column 5 of Table 4) the following comments are possible: (1) The low value of g (17= 2.4%) means that the flat membrane model is not adequate to describe a bundle. The name “efficiency coefficient” is therefore not correct: “adequation coefficient” of the model would be better. (2) The value 33% for qt may mean that the structure of the hollow-fibre membrane is not exactly the same as that of a flat membrane; however, compared to the value of 2.4%, this structure effect is not the more important (3) The number of fibers, IV, and the homothethis ratio, h, are the bestknown characteristics of the bundle: why not use them? (4) By reference to the fractal model (D = 1.75 in eqn. 11) and comparison with the experiments (77= 0.024), the corresponding value of Dt is about 1 (Table 3). (5) The efficiency coefficient qf = 1.2 (compared to 7) = 0.024) means that the fractal model is nearest to the experimental results and so may be a justification of this model. 7. Conclusion This paper offers a new way of taking into account the complexity of the pervaporation process in a hollow-fibre bundle. We consider the present work as a first attempt and this project will be continued, together with traditional modelling, to find out exactly what kind of phenomena are involved in the mass-transfer fractal dimension. Using traditional modelling based on heat, mass and momentum balances, it is possible to go from a diffusion coefficient to a local transfer coefficient, and from a local transfer coefficient to an overall coefficient for an isolated fibre. However, for fibre bundles, because of the complexity of the flow, only overall efficiency coefficients can be used. The approach which has been presented here offers a way of explicitly introducing the specific character of the transfer surface, so representing a step towards a theoretical expression for the efficiency coefficient. The mass-transfer fractal dimension lumps together all the other phenomena and the next step will be to analyze what effects are contained in this quantity. List of symbols
AD A
ex
A, Ao D Dt J JS
overall bundle area (m”) external area of the bundle (m’) lateral area of a fibre (m’) area of flat membrane (m’) fractal dimension of the bundle mass-transfer fractal dimension mass flux through the bundle (mol-sec-1-m-2) mass flux through a single fibre (mol-sec-‘-m-2)
mass flux through a flat membrane (mol-sec-‘-m-2) mass-transfer coefficient (J,, = ho 6) number of fibres molar flow of component j through a flat membrane (mol-set-‘) molar flow through an isolated fibre (mol-set-‘) molar flow through the bundle (mol-see-‘) fibre radius (m) radius of the bundle, cylindrical co-ordinate (m) geometrical entropy of the bundle fractal transformations composition (volume % ethanol) length of the bundle, dimension of a flat square membrane (m) homothethis ratio overall efficiency coefficient bundle efficiency coefficient efficiency coefficient, referred to fractal transfer efficiency coefficient (defined by eqn. 2) efficiency coefficient, referred to fractal and single fiber mass transfer driving force References
10 11
12
13
B. Mandelbrot, Les Objets Fractals, Flammarion, Paris, 1975. B. Mandelbrot, Fractals, W.H. Freeman, San Francisco, 1977. B. Mandelbrot, Les objets fractals, La Recherche, 85 (1978) 5. B.H. Kaye, Specification of ruggedness and/or texture of a fine particle profile by its fractal dimension, Powder Technol., 21(l) (1978) 1. A.G. Flook, Use of dilatation logic and quantiment to achieve fractal dimension characterisation of textured and structured profiles, Powder Technol., 21(2) (1978) 295. H. Mori, Fractal dimensions of chaotic flows of autonomous dissipative systems, Progr. Theor. Phys., 63(3) (1980) 1044. J. Lefevre, Teleogical optimization of a fractal tree model of the pulmonary vascular bed, Bull. Eur. Physiopath. Resp., 16( 1) (1980) 53. F. Stapleton, Turbulent fluctuations as fractal functionals of averages, Bull. Amer. Phys. Sot., 25(B) (1980) 1034. G. Djelveh and J.C. Mora, Heat and mass transfer surface between two phases by the concept of fractal object, paper presented at CHISA ‘81, Comm. C1.5, Prague, August 1981. A. Lemehaute and G. Grepy, Sur quelques propridtes de transferts electrochimiques en geombtrie fractale, C.R. Seances Acad. Sci. II, Met. Phys., 294( 14) (1982) 685. A. Lemehaute, A. Deguibert, M. Delaye and C. Filippi, Notes d’introduction de la cinetique des Bchanges d’bnergies et de mat&e sur les interfaces fractales, CR. Seances Acad. Sci., II, M&Z. Phys., 294(14) (1982) 835. G. Djelveh, P. Aptel, R.S. Bes and J.C. Mora, Modelisation de la pervaporation dans un faisceau de fibres creuses en tant qu’operation unitaire de genie chimique: Utilisation du concept d’objet fractal, European Workshop on Pervaporation, Nancy, 21-22 September 1982. Laboratoire de Mathematiques et d’Informatique, Universite de Bordeaux, Courrier C.N.R.S., 53 (1983) 5-9.