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Dead-endfiltrationexperimentsonmodel
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Desalination 177 (2005) 303–315
Dead-end filtration experiments on model dispersions:
comparison of VFM data and the Kozeny–Carman model
E. Braunsa*, D. Teunckensb, C. Dotremonta, E. Van Hoofa, W. Doyena, D. Vanheckeb
a
Vito (Flemish Institute for Technological Research), Process Technology, Boeretang 200, B-2400, Mol, Belgium
Tel. +32 (14) 33 6913; Fax +32 (14) 32 65 86; email: [email protected]
b
De Nayer Institute, Jan De Nayerlaan 5, B-2860, Sint-Katelijne-Waver, Belgium
Received 26 August 2004; accepted 20 December 2004
Abstract
Permeate flux decline is a major item when considering pressure driven membrane filtration. The decline is a
result of the deposition of (dispersed) feed materials as a layer on the membrane surface. This means loss of
production of permeate or, when applying a higher compensating pressure, loss of energy. A number of
characterization techniques such as SDI (Silt Density Index) and MFI (Modified Fouling Index) are available to
study such effects by using a dead-end filtration set-up. A more recent technique, VFM (Vito Fouling Measurement),
is described in [1,2]. VFM is a pragmatic characterization method which presents dead end flux decline results as
a graph or as a table formatted “multi value index”. To the author’s opinion it is very difficult to squeeze the
complex permeate flux decline behaviour of a real feed into a one number “model” such as SDI or MFI without
losing crucial information or even missing a crucial zone within the complete set of data. In this paper it is
demonstrated that, even for a simple situation of a model dispersion of ceramic powders in water, it is improbable
to achieve a correct mathematical description of the hydraulic conditions during cake formation on a membrane
surface. Therefore, a universally applicable mathematical model which predicts in an accurate way the flux decline,
for a real feed with a complex composition, seems impracticable. This argues in favour of the experimental approach,
such as the VFM.
Keywords: Membrane; Permeate; Flux; Decline; Fouling; VFM; Model; Dispersion; Carman; Kozeny
1. Introduction
Permeate flux “modelling” can be approached
mathematically through a number of formats [3,4].
*Corresponding author.
A number of flux decline measurement techniques
has also evolved [1,2] (see also author’s note at
the end of this paper). The resistance in series
model (also used in the MFI) is often applied in
trying to describe the permeate flux deterioration:
0011-9164/05/$– See front matter © 2005 Elsevier B.V. All rights reserved
304
E. Brauns et al. / Desalination 177 (2005) 303–315
⎛ A ⋅ ∆P ⎞ 1
d V ⎛ A ⋅ ∆P ⎞
1
=⎜
=⎜
⎟⋅
⎟⋅
dT ⎝ η ⎠ ( Rm + R f ) ⎝ η ⎠ Rtot (1)
with V — permeate volume, m³; t — time, s; ∆P
— transmembrane pressure drop, Pa; η —
absolute viscosity, kg/m.s; R m — membrane
resistance, m–1; Rf — all additional resistance from
the layer on the membrane surface, m–1; Rtot —
total hydraulic resistance, m–1; A — membrane
surface area, m².
When using the resistance in series model, the
assumption is made that the additional hydraulic
resistance towards permeate flux can simply be
added to the original hydraulic resistance of the
membrane.
By integration of Eq. (1) one obtains:
t
V
η⋅ R f (V )
η ⋅ Rm
⋅ dV + ∫
⋅ dV
∆P ⋅ A
∆P ⋅ A
0
0
V
V
η ⋅ Rm
η
=
⋅V +
⋅ R f (V ) ⋅ dV
∆P ⋅ A
∆P ⋅ A ∫0
tests on dispersions of non-compressible
“model” fine powders in water.
• theoretical Rf(V) values as calculated from the
Kozeny–Carman equation [4,5,8,9]
The objective of the comparison was to highlight eventual differences between measured
values by VFM and calculated theoretical values,
as a result of the uncertainties on the parameter
values within the theoretical models. Other theoretical models than Kozeny–Carman could have
been compared with, but in this paper only the
Kozeny–Carman approach is used. Dead-end
filtration tests on non-compressible particles lead
to a build-up of those particles on the membrane
surface and a cake-filtration situation is obtained.
2. Permeate flux decline measurement method
VFM
∫ dt = ∫
0
• VFM data as obtained from dead-end filtration
(2)
In order to be able to predict the decreasing
trend of the permeate flux it is necessary to know
exactly the function Rf(V) in a way Eq. (2) can be
solved. When introducing a specific Rf(V) in
Eq. (2) to obtain a model after integration it is
clear that experimental data should fit the assumed
Rf(V). If this is not the case it should be concluded
that the assumed Rf(V) is inaccurate or even
oversimplified.
Since Rf(V) is very complex in practice and is
related to numerous possible interactions between
constituents in the feed and the membrane itself,
it is very difficult to define an appropriate Rf(V)
in most real cases. However, it is feasible to measure in a pragmatic way the complex Rf(V) by the
VFM method as explained in [1]. It is then possible
to compare such VFM data with calculated results
from a specific theoretical Rf(V) model. In this
paper such a comparison is made between:
The experimental set-up as shown in Fig. 1 is
the same as described in [1]. It consists of a computerised and automated gravimetric measurement
of the permeate vs. time. The permeate is obtained
from a dead-end filtration cell which is fed from
an air pressurised stainless steel tank, containing
the feed. The permeate is collected in a recipient
on an electronic balance and the permeate mass
data vs. time is sampled by the computer (sampling
frequency is of the order of magnitude of a few
seconds). The mass is converted to volume in order
to obtain the permeate volume (V) vs. time (t) experimental data. In this way a very large number of
(ti, Vi) data pairs are obtained. As explained in [1]
it is feasible to use the discrete (ti, Vi) pairs to estimate the derivative dV/dt (thus permeate volume
flow) for each ti by applying a regression technique on the data pairs around each (ti, Vi). At this
time the regression method has even been improved by having a self optimizing search for the
ultimate regression around each (ti, Vi). Therefore
the number of regressions even has increased
when compared to the technique described in [1].
E. Brauns et al. / Desalination 177 (2005) 303–315
305
Fig. 1. Experimental setup.
For t = 0 the value of Rm can be estimated from
Eq. (1) since dV/dt at t = 0 could also be estimated
from the (ti, Vi) data by regression. Moreover it is
also possible to estimate for each (ti, Vi) the value
of Rf and Rtot from Eq. (1) since the value of dV/dt
was estimated by the regression technique.
In [1,2] it has also been indicated that a specific
representation of the flux decline data can be obtained in a graphical format by plotting the ratio
volume/surface area vs. the ratio Rtot/Rm. In an
analogous way a plot of the ratio volume/surface
area vs. the ratio Rf/Rm can be constructed, if this
would be more relevant in an application but the
Rtot/Rm approach was selected as preferential for
the moment. A VFM plot has a universal readability and informs rapidly on the permeate flux
decline characteristics of a feed. A horizontal flat
curve is very negative since it indicates a very
rapid decline of flux; so the association of a low,
horizontal curve with a low flux is obvious. A
vertical steep curve indicates that a large amount
of feed can pass the filter without having an important effect on the Rtot value; so the association
of a steep curve with a high flux is also obvious.
Moreover, the curvatures of a VFM plot could
reveal information on (stage like) phenomena of
permeate flux decline. In this respect the phenomenon of a possible compression of a layer on
the membrane surface can be mentioned. Compression of such a layer will occur when the hydraulic pressure gradient within the layer induces
a shear stress which equals the shear stress limit
for plastic deformation of the layer. The VFM data
could also be reported as a table, thus introducing
a multi value index covering the complete permeate flux decline phenomena between t = 0 and
306
E. Brauns et al. / Desalination 177 (2005) 303–315
some specified Rtot/Rm value, corresponding to a
flux decline limit as defined by the user. Such a
multi value permeate flux decline index, represented in a table, has also a universal readability
in the same way as the VFM plot itself. In such a
table, the parameter time can also be incorporated
to give additional information, as illustrated in [2].
The VFM method was used here to evaluate
the Rtot,i and Rf,i values for each (ti, Vi) in order to
compare these with the corresponding values as
calculated from the Kozeny–Carman equation.
would be the case if the ceramic particles are
piled up one on one another in an ideal way
into a rigid, non-deformable and stationary
stack. The non-deformability of such a stack
can not be proved to be true but, for modelling
purposes, this assumption is the only way to
avoid an increased complexity of modelling
the stack regarding e.g. a static and nondynamic porosity etc.
• the homogeneity of the deposited layer
• the possibility to calculate the thickness of the
deposited layer
3. Model dispersions and hydraulic resistance
model
Ideal model dispersions for the envisaged
experiments would have consisted of perfect
spherical and monodispers ceramic particles. Such
ideal shaped particles are however not representative for real filtration situations and therefore
it was decided to use irregularly shaped particles,
in order to have a better idea of the correlation
between the theoretical model approach and the
VFM results. Scanning electron microscope
(SEM) images of the silicon carbide (SiC)
powders which were used in this respect are shown
in Figs. 2–4. Their characteristics, as obtained
from the manufacturer (HC Starck) or measured
at Vito, are shown in Table 1. The theoretical
density (thus the density of non-porous material)
of SiC is 3.21 g/cm³. From the SEM images the
3.1. Model dispersion
For a comparison between the measured Rtot,i
and Rf,i values and the corresponding theoretical
model values it was decided to perform experiments with model dispersions of non-compressible ceramic powders in water. By using extremely fine ceramic powders of known particle size
and by dispersing those in water up to a specific
concentration, such dispersions can be used as a
feed in the VFM set-up. In the dead-end filtration
set-up of the VFM the permeate volume can be
related at a specific time ti to the amount of ceramic
particles deposited on the filter surface, since the
dispersion concentration is known. When assuming
an incompressible layer of deposited ceramic
particles, it is also assumed that the cake layer is
deposited homogeneously over the membrane
surface and that the thickness of the deposited
layer can be estimated from the:
• volume of permeate, passed through the
membrane
• dispersion concentration
• packing characteristics of the dispersed
powders.
With respect to the theoretical modelling it thus
should be noticed that, up to now, there are already
three assumptions being made regarding:
• the incompressibility of the cake layer: this
Table 1
Characteristics of silicon carbide powder
Type
1
BET Specific surface, m²/g
BET Specific surface, m²/g2
Particle size, µm, at fraction
90% smaller than1a
Particle size, µm, at fraction
50% smaller than1a
Particle size, µm at fraction
10% smaller than1a
1
2
UF05
UF10 UF15
4–6
4.73
4.4
9–11
9.06
1.8
1.4
0.7
0.55
0.3
0.2
0.1
14–16
15.1
1.0
As specified by H.C.Starck (a = laser diffraction)
As measured at Vito
E. Brauns et al. / Desalination 177 (2005) 303–315
Fig. 2. Silicon carbide powder UF05.
Fig. 3. Silicon carbide powder UF10.
Fig. 4. Silicon carbide powder UF15.
307
308
E. Brauns et al. / Desalination 177 (2005) 303–315
difference in particle size between the three
powder types is obvious. This is also clear from
the specific surface value which expresses the
surface area per gram of powder. The UF15 powder
has the highest specific surface area value, indicating a very fine particle size distribution. The
UF SiC powders are produced by milling SiC to
very fine powders. A higher milling degree produces finer powders in a way the three different
types can be produced by applying specific milling
and classification (“sieving”) parameter values.
In order to obtain a good dispersion of the SiC
powders in water, a number of different dispersion
methods were evaluated by placing a mixture
recipient either in:
• an ultrasonic dispersion vessel. Ultrasonic
energy is very efficient in breaking up agglomerates.
• a tumbling mixer (Turbula) which executes a
complex rotation and shaking movement. The
treatment in a Turbula is typical 15 min. To
enhance the breaking up of agglomerates a
small amount of small aluminium oxide beads
were added. Such beads can implement shear
forces on SiC agglomerates in a way the individual particles are set free in the dispersion.
The beads can easily be sieved out of the dispersion after mixing.
• a laboratory shaker (Gerhardt). The recipient
is moved during about 24 h on a small platform, having a horizontal circular movement .
• a mechanical vibratory mixer (Grantham). The
recipient is shaken at a considerable frequency
but at a rather low amplitude (a few millimetres).
The quality of a dispersion was measured
through the VFM itself. The results are illustrated
in Fig. 5. The VFM curve indicated with “Unmixed”
is the result of dispersing the SiC powder without
any further dispersion action than slight and short
manual stirring. The other VFM curves, obtained
from the dispersions produced by the four different
dispersion methods, are closely grouped together
and are situated much lower then the “unmixed”
curve. Since it is to be expected that there are still
agglomerates remaining in the “unmixed” dispersion it is also expected that such agglomerates
may contribute to larger pores in the cake, being
deposited on the membrane surface, in a way the
hydraulic resistance of the cake from the “unmixed” dispersion is much less than the well
treated dispersions. The experimental results in
Fig. 5 thus prove that the dispersion techniques
are successful in this case and that the VFM
method effectively is able to detect such effects.
Some VFM results are also shown in Table 2
illustrating the alternative table format of the VFM
Fig. 5. Effect of dispersion method on VFM curve.
309
E. Brauns et al. / Desalination 177 (2005) 303–315
Table 2
VFM results for some dispersion techniques on SiC powder
Rtot/Rm
Unmixed
Laboratory shaker
Turbula
Mechanical vibration
V/A, m³/m²
1
2
4
6
8
10
0
0
0
0
1.3
0.193
0.205
0.224
3.3
0.469
0.527
0.585
5.4
0.777
0.855
0.948
7.5
1.08
1.20
1.31
1.38
1.53
1.66
data presentation. As explained in [1,2] the VFM
is intended to be a filtration characterisation
method representing the fingerprint of a feed
permeate flux decline potential as a graph or a
multi value table (V/A vs. Rtot/Rm).
Since the ultrasonic treatment of a dispersion
is very straightforward, this technique was selected to prepare the dispersions for the experiments.
Nevertheless it was decided to add a dispersion
promoting agent to the SiC powder since such an
agent enhances further the deagglomeration of
possible remaining agglomerates.
The effect of the addition of 17 µL TMAH per
gram SiC powder is shown in Fig. 6. Information
on the use of TMAH as a dispersant for SiC carbide powder can be found in [6]. The VFM curve
for the dispersant based dispersion of SiC powder
clearly shows an increase in hydraulic resistance
when compared to the VFM curve of the dis-
persion without dispersant. When using a dispersant, this observation is believed to result from
a better dispersion and therefore a better packing
of the individual SiC powder particles in the cake
on the membrane surface. It was decided therefore
to proceed with the VFM experiments by using
ultrasonic as the dispersion method and by using
TMAH as a dispersant.
3.2. Kozeny–Carman model approach and cake
layer resistance
In order to investigate the correlation between
the actual VFM data and existing theoretical
models the Kozeny–Carman model was selected
[5]. The former assumption was also maintained
stating that the cake resistance Rf is a hydraulic
resistance in series with the hydraulic resistance
Rm of the membrane. Moreover, the assumption
Fig. 6. Effect of the addition of vs. a dispersant
to the SiC powder.
310
E. Brauns et al. / Desalination 177 (2005) 303–315
was made that Rm is a constant during the experiments. Such simplifications and assumptions even
emphasize the complexity of the theoretical
modelling of permeate flux decline since, when
Rm would be a function of the permeate volume V,
an additional model would be needed to represent
Rm(V) which would make the whole approach
extremely difficult. Therefore, in this publication,
the Rm value as obtained from the VFM measurement, was considered to be a constant during the
experiment. This assumption then enables to
restrict the modelling to calculate Rf(V) values
corresponding to the cake layer, consisting of
deposited SiC powder particles.
With respect to the hydraulic resistance Rf of
the cake layer of deposited SiC powder particles
it can be noted that the flux Jf in the cake is proportional to the pressure drop ∆Pf (Pa) over the cake
and inversely proportional to the dynamic (=absolute) viscosity η (Pa.s):
Jf =
1 ∆Pf
⋅
Rf
η
(3)
According to the Kozeny–Carman equation the
flux within the porous stack of particles can be
modelled as [7]:
Jf =
∆Pf 1
1 1
ε3
⋅ 2⋅
⋅
⋅
2
kc S0 (1 − ε )
η H
(4)
kc is the Kozeny constant [8,9]; S0 is the specific
surface of the powder on a volume basis, m²/m³;
ε is the porosity (void fraction) of the cake; H is
the height (thickness), m, of the cake.
By combining Eqs. (3) and (4) one obtains:
R f = kc ⋅ S
2
0
(1 − ε )
⋅
ε3
2
⋅H
(5)
The value for the hydraulic resistance Rf of the
SiC powder cake on the membrane therefore can
be calculated in principal from Eq. (5). There are
Table 3
Values for the Kozeny constant as reported in the literature
Researcher Material
Value of kc
Uchikoshi
Uchikoshi
Uchikoshi
Sobue
5.8–7.4
3.8
4.1–5.6
6.2
Zirconia powder suspension A
Zirconia powder suspension B
Alumina powder
Silicon powder
four parameters kc, S0, ε and H which need to be
evaluated in order to calculate Rf. These parameters are discussed first.
The value of Kozeny constant kc is often set to
5, according to the experimental work of Carman.
In the literature however the value for kc is reported
to deviate from 5. Examples, as given by Uchikoshi
[7], are indicated in Table 3.
As a result it is obvious that kc is not to be considered as a constant, simply having a value of 5.
As indicated in [7] the BET method allows to
measure the specific surface SBET (m²/g) of a
powder. The BET measurement determines the
amount of gas needed to obtain a monomolecular
layer (mostly nitrogen) on the powder surface.
Since the occupied area by one gas molecule is
known, it is then also possible to calculate SBET
for the powder. The value of S0 can be easily calculated from SBET by multiplying SBET by the
theoretical density of the powder material [7].
Having a theoretical density of 3210 kg/m³ for
SiC the value of S0 for the SiC powder therefore
is:
S 0 = 3200 × 1000 × S BET
(6)
The value of the porosity ε of the porous SiC
cake on the membrane surface is another important parameter value. The measurement of ε for a
dry powder (compact) is well known in the
powder metallurgy or technical ceramic industry.
These methods comprise the measurement of
apparent density, tap density and green density
(of compacted powder) but can however not be
E. Brauns et al. / Desalination 177 (2005) 303–315
applied to the wet cake on the membrane surface.
The thickness H of the cake can be calculated
from the permeate volume V, the SiC powder
dispersion concentration cSiC, kg/m³, the membrane surface area, m², and the apparent density
ρapp, kg/m³, of the cake. According to the assumptions, as already discussed, the value for H can be
calculated from:
H=
V ⋅ cSiC
V ⋅ cSiC
=
ρapp ⋅ S m 3210 ⋅ (1 − ε ) ⋅ S m
(7)
When combining Eqs. (5) and (7) one obtains
for the SiC powder dispersions :
R f = kc ⋅ S02 ⋅
(1 − ε )
ε
3
⋅
V ⋅ cSiC
3210 ⋅ S m
(8)
As explained in 2, the VFM method is based
on a sampling procedure which produces a large
number of (ti, Vi) data pairs. When applying the
VFM calculations on these data pairs, the
experimental Rf,i values are obtained for each data
pair. When applying Eq. (8) for each value Vi of a
data pair (ti, Vi) it is obvious that the theoretical
Kozeny–Carman model value for Rf can be calculated and compared with the experimental Rf
value. Since V, cSiC and Sm are known while S0 can
be obtained from a BET measurement, the crucial
parameters within Eq. (8) are kc and ε.
4. Experimental VFM results and comparison
with Kozeny–Carman model
The SiC powder dispersions were produced
as explained already by ultrasonic treatment. The
dispersion concentration of the three types of powders was 0.050 kg/m³. VFM experiments were performed in the set-up, as shown in Fig. 1, by using
47 mm diameter type Millipore filters of 0.45 µm
pore size. The dead-end VFM filtration experiments were done at a pressure of 207 kPa. For
each sampled data pair (ti, Vi) the experimental
311
VFM value of Rf was determined. By using Eq. (8)
the theoretical value of R f according to the
Kozeny–Carman model could be calculated also.
Since there are hundreds of sampled data-pairs it
is impractical to present the values of the experimental and Kozeny–Carman values for Rf in a
table format. The results are however graphically
shown in Figs. 7–9. In those figures the VFM related (experimental) values for Rf are represented
by index A.
The theoretical Rf values as calculated from
Eq. (8) were obtained as follows:
• in paragraph 3, it was explained that the
parameter kc is crucial to calculate Eq. (8). It
was also explained that in the literature it is
shown that the Kozeny “constant” kc in practice
not always equals to the value of 5. As a result
of the uncertainty with respect to the value of
kc it was decided to do model simulations
through Eq. (8) by using values for kc in the
range –20% and +20%, thus 4 to 6 as discrete
values kc = 4, kc = 5 and kc = 6. In this way the
effect of the kc value uncertainty on the Rf
values results could be visualised and quantified.
• the porosity parameter ε is also crucial in
Eq. (8). Since no evaluation method was available to determine the actual porosity fraction
within the SiC powder cake, building up
dynamically during the dead end filtration on
the membrane surface, it was decided to do
model simulations over a selected and relevant
porosity value region. Specific information on
porosity values related to slip and gel casting
of SiC can be found in the literature, e.g. [6,10].
During slip casting a dispersion of ceramic
powder particles is in contact with a porous
medium by which the water is extracted from
the dispersion. On the porous medium a cake
layer is building up and since the porous medium
has a specific shape, ceramic components are
produced in this way. The thus formed parts
are then further treated and sintered in order
to obtain a solid component. The slip casting
method therefore is analogous to filtration
312
E. Brauns et al. / Desalination 177 (2005) 303–315
through a membrane. During pressure slip
casting the applied pressure increases the
removal of water in order to increase production speed. This can be compared to the
increase of permeate flux in membrane filtration by increasing the feed pressure. From the
casting data in the literature it can be concluded
that the porosity ε that can be expected for the
SiC cake on the membrane surface is rather
low (<0.4) since the relative densities of slip
casting components are reported to be above
60% (this means ε < 0.40) and even up to 74%
(ε = 0.26). In order to have additional information on the topic of porosity it was decided
to perform some pragmatic experiments with
the SiC powders UF05 and UF15. Table 4
shows the theoretical amounts of water and SiC
powder that would be needed to produce a
volume of 1 cm³ of undeformable cake, having
a perfect rigid stacking of powder particles and
having a specific porosity value. When actually
mixing these amounts of water (with TMAH
addition) to the SiC powder it was observed
that in the case of UF05 powder the obtained
mixture for ε = 0.60 still behaved like a viscous
fluid but not resulted in a rigid cake. This was
also true for ε = 0.55 while for ε = 0.50 the
mixture obtained a paste like character, but
with a viscosity still allowing the paste to be
deformed by hand. These limited experiments
on UF05 and UF15 powders confirmed the
findings in the literature of having low (at least
lower than 0.50) values for ε for a rigid cake
of SiC particles.
• to investigate the likelihood of a good fit of
the Kozeny–Carman approach through Eq. (8),
a number of calculations were performed using
the value combinations for kc and ε as illustrated in Table 5.
The results of the calculations are demonstrated in Figs. 7–9. The index A was used for the
Rf graph vs. permeate volume V as measured by
the VFM method. It is to be remarked that the
VFM related curves with index A all show a linear
Rf vs. V curve for UF05, UF10 and UF15 (Table 6).
Table 5
Kozeny–Carman model simulations by different combinations for values of kc and ε
Index B
kc
ε
5
0.40
C
D
E
F
G
H
5
0.45
5
0.50
5
0.55
5
0.60
4
0.50
6
0.50
Table 6
Linear regressions on the A-indexed graphs from VFM
results
A-graphs
(VFM)
Slope of the linear
regression line
Rf vs. V
Linear regression
correlation
coefficient
UF05
UF10
UF15
2.39×1013
4.81×1013
15.8×1013
0.99989
0.99956
0.99837
Table 4
Theoretical amounts of water and SiC powder for a theoretical porosity value
Theoretical case of 1 cm³ of rigid, undeformable cake
Theoretical porosity ε
Water theoretically needed, g
SiC powder theoretically needed, g
Total mass of theoretical volume of 1 cm³, g
“0.50”
0.500
1.605
2.105
“0.55”
0.550
1.445
1.995
“0.60”
0.600
1.284
1.884
Deformable
—
Liquid
—
Liquid
Deformable
Actual behaviour of UF05 and UF15 SiC powder mixtures with water
UF05/water mixture
UF15/water mixture
E. Brauns et al. / Desalination 177 (2005) 303–315
Fig. 7. Rf vs. permeate volume for UF05.
Fig. 8. Rf vs. permeate volume for UF10.
Fig. 9. Rf vs. permeate volume for UF15.
313
314
E. Brauns et al. / Desalination 177 (2005) 303–315
This linearity proves the expected cake behaviour
of the solid and undeformable SiC particles during
their build up on the membrane surface as an incompressible layer. It is also to be noticed in Table 6
that the coarsest powder (UF05) effectively shows
the highest permeability, thus the lowest Rf value,
as expected.
4.1. Discussion of the results
• Regarding the correlation between the VFM
result for Rf (graph A) and the Kozeny–Carman
based Rf it is to be noticed in Fig. 7 that for
index B (ε = 0.40 and kc = 5) the Rf graph B is
positioned much higher than graph A. This
means that for a “normal” kc = 5 and a relevant
ε = 0.40 value the Kozeny–Carman model
approach deviates largely from the measurement. For a permeate volume of 0.0035 m³ the
Rf as measured by the VFM equals to 8.39×
1010 m–1 whereas Rf as calculated from Eq. (8)
equals to 42.7×1010 m–1 .
• Moreover, the value of ε is expected to be
lower than at least 0.50 but, in contradiction, a
matching fit between model and experiment
can only be obtained by increasing the value
of up to a value of 0.60 (index F). Only in that
case the graphs A and F coincide. Since a value
of 0.60 for ε is very unlikely, as based on literature data and some experiments as reported in
Table 4, this shows that the Kozeny–Carman
model approach with kc = 5 is not suitable in
this case.
• When changing kc to the values, as reported in
literature, of 4 (index G) and 6 (index H) and
assuming a porosity of 0.50 it can be remarked
that the curve G is positioned somewhat lower
than curve D (also with a porosity of 0.50 but
with kc = 5). The curve H is positioned somewhat higher than curve D. In order to force the
curves G and H down towards to the position
of curve A it was again necessary to use a value
of 0.60 for the porosity. Since a value of ε =
0.60 is unlikely this is again an indication that
the Kozeny–Carman based model approach is
not well adapted in this case.
• Figs. 8 and 9 show the same trends for UF10
and UF15 respectively. So the same conclusions are valid for UF10 and UF15: to have a
good fit between the Kozeny–Carman based
model and the VFM experimental result for Rf
vs. permeate volume V, the model Eq. (8)
seems to need parameters values for kc and ε
which are in contradiction with the physical
reality, even in the “simple” case of a rigid and
incompressible cake of only SiC particles.
From these observations the VFM method
appears to be a practical method to measure the
permeate flux decline characteristics of a feed.
Even in the case of the theoretical modelling of a
simple cake behaviour of SiC particles on a membrane there is no universally applicable value for
the Kozeny constant and the model approach
needs experimental deduced parameter values to
obtain a good fit with the observed experimental
values. Since these parameter values have a weak
physical correlation it can be merely stated that
such parameters values are “correcting” parameter
values in order to obtain a good fit with the experimental results. In the case of a more complex feed
with different constituents, present at the same
time, giving rise to a non-rigid and compressible
layer, the model approach will certainly obtain a
mathematical curve-fitting character. In that case
the VFM is a measurement technique that characterises permeate flux decline and even is able to
produce data which are needed by those who
prefer a specific mathematical curve fitting model
and who can use for that matter the VFM data to
determine the model parameters values.
5. Conclusions
Experimental results from VFM measurements
on “model” dispersions of SiC powders show that
a Kozeny–Carman approach to model the hydraulic resistance Rf vs. permeate volume is founded
E. Brauns et al. / Desalination 177 (2005) 303–315
on two crucial parameters: the Kozeny constant
and the cake porosity. The Kozeny “constant” kc
however is reported in the literature to show values
in reality which differ from the value of 5. In this
paper the effect of a scatter of +20% and –20%
on this value was investigated. In literature even
larger value dispersions are mentioned. The
investigated spread of 40% on the kc value invokes
a first important constraint on the model approach.
Regarding the porosity of the cake it was shown
that physically relevant values fail in order to
obtain a good fit with the experimental results.
This imposes even a larger constraint on the model
approach. It is therefore believed that the theoretical model needs additional correction coefficients for obtaining a good fit. As a result, the VFM
method seems to be a valid candidate as well as a
pragmatic characterization method for the determination of basic permeate flux decline data of a
feed as well as a method to produce the data
needed for modelling the layer resistance Rf(V)
by curve fitting.
Acknowledgment
The authors would like to thank Rabah Mouazer
at Vito for discussions on the cake porosity and
the experiments performed on mixtures of SiC
powders in order to determine the characteristics
of the mixtures as mentioned in Table 4. Rabah
Mouazer is researching gel casting of ceramic
materials.
Authors note
In the proof of reference [1] a type-setting error
by the publisher was overlooked, only in the middle
part of Eq. (7) p. 35 with respect to the special
case of the MFI. The correct notation of Eq. (7) is:
t
∫ dt =
0
V
η ⋅ Rm
η
I ⋅R
⋅V +
⋅∫
⋅ V ⋅ dV
∆P ⋅ A
∆P ⋅ A 0 A
V
η ⋅ Rm
η⋅ I ⋅ R
=
⋅V +
⋅
V ⋅ dV
∆P ⋅ A
∆P ⋅ A2 ∫0
315
(7)
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