Journal of Membrane Science 203 (2002) 15–27
Characterisation of three hydrophobic porous membranes
used in membrane distillation
Modelling and evaluation of their water vapour permeabilities
L. Martı́nez a,∗ , F.J. Florido-Dı́az a , A. Hernández b , P. Prádanos b
a
b
Departamento de Fı́sica Aplicada, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
Departamento de Termodinámica y Fisica Aplicada, Facultad de Ciencias, Universidad de Valladolid, 47071 Valladolid, Spain
Received 17 July 2001; received in revised form 25 October 2001; accepted 26 October 2001
Abstract
Pore size distributions have been obtained for three hydrophobic porous membranes from air–liquid displacement measurements, assuming the common model of cylindrical capillaries for the membrane and using flux equations which includes
both diffusive and viscous mechanisms for transport in the gas phase in pores. The pore size distribution so obtained and the
same flux equations are used in order to predict the water vapour permeability through the membranes, characterised when
employed in different membrane distillation configurations and operating conditions. In membrane distillation applications,
where no viscous flux exists, the role of both Knudsen and molecular diffusion resistances is analysed. In membrane distillation
applications which include diffusive and viscous transport, the contribution of both mechanisms is analysed. © 2002 Elsevier
Science B.V. All rights reserved.
Keywords: Membrane distillation; Membrane characterisation; Capillary model; Porous membrane; Pore size
1. Introduction
An application of hydrophobic porous membranes
is membrane distillation (MD). In this application,
a heated aqueous feed solution is brought into contact with one side of the hydrophobic membrane.
The hydrophobic nature of the membrane prevents
penetration of the aqueous solution into the pores,
resulting in a vapour–liquid interface at each pore
entrance. Different methods [1] may be employed
to impose a vapour pressure difference across the
membrane to drive the flux: (a) the permeate side of
the membrane may consist of a condensing liquid
∗ Corresponding author. Tel.: +34-95-2131924;
fax: +34-95-2132000.
E-mail address: [email protected] (L. Martı́nez).
in direct contact with the membrane (DCMD); (b) a
condensing surface separated from the membrane by
an air gap (AGMD); (c) a sweeping gas (SGMD); or
(d) a vacuum (VMD). Regardless of the MD configuration used, water evaporates from the liquid–vapour
interface on the feed side of the membrane, diffuses
and/or convects across the membrane pores, and is
either condensed or removed from the membrane
module as vapour on the permeate side.
Modelling of mass (vapour) transfer within the
membrane pores has received the most interest from
MD investigators. Several MD models are available
in the literature, each considering one or more of
the following mass transfer mechanisms across the
membrane: viscous flow; Knudsen and molecular
diffusion. Sarti et al. [2] and Sarti and co-workers
[3] modelled their VMD experiments considering the
0376-7388/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 6 - 7 3 8 8 ( 0 1 ) 0 0 7 1 9 - 0
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L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
Nomenclature
A
B
C
d
Di K
Dij
Dij
J
M
n
N
p
q
r
R
T
factor defined in Eq. (11) (mol s/Kg m)
factor defined in Eq. (12) (mol s3 /Kg2 m)
water vapour permeability (mol/m2 s Pa)
pore diameter (m)
Knudsen diffusivity of species i (m2 /s)
binary molecular diffusivity (Pa m2 /s)
ordinary binary diffusion coefficient (m2 /s)
molar flux (mol/m2 s)
molecular weight (kg/mol)
number of pores (m−2 )
total number of pores (m−2 )
pressure (Pa)
tortuosity
pore radius (m)
gas constant (J/mol K)
temperature (K)
Greek letters
δ
membrane thickness (m)
δ eff length of the membrane pores (m)
ε
void volume of the membrane
εS
superficial porostity
γ
surface tension (N/m)
µ
viscosity (Pa s)
θ
contact angle
Subscripts
a
air
D
diffusion
i, j counter of gas mixture components
K
Knudsen flow
V
viscous flow
w
water
Superscripts
f
feed
p
permeate
model which accounts for both Knudsen and viscous
mass transport across the membrane was used by
Lawson and Lloyd [7] to predict the performance of
VMD. The experimental results assuming both Knudsen and molecular diffusion resistances for the vapour
transport across the membrane have been explained
in [8,9]. On the other hand, frequently, in the application of these models the membrane of void volume ε
is considered as a collection of capillaries of diameter
d and a tortuosity factor about 2, including the values
given by the manufacturers for ε and d in the model.
As there are several methods of membrane characterisation, the d value provided by the manufacturer can
be unsuitable in the gas transport model. In the same
way, value 2 for the tortuosity factor is approximated.
Summing up, when we examine the membrane distillation literature, we observe that: (a) there are few
characterisation works of the membranes used; and
(b) different transport models are used. With regard to
the second question, it is necessary to have in mind
that frequently in MD applications, the pore size and
the mean free path of the permeating molecules are
comparable. For this reason, no consideration of all
transport mechanisms can be a very simplified model.
The aim of this work was to predict the water vapour
permeability of a MD membrane using a gas transport
model applicable in the entire range of pore sizes and
mean free paths. Knowledge of the structural characteristics of the membrane is necessary to apply this
model. So, in the first step, we have obtained the pore
size distribution of the membranes from air–liquid displacement measurements. In the second step, the water vapour permeabilities have been evaluated. In both
steps, the same general gas transport model has been
applied.
2. Theory
2.1. The transport model
water vapour being transported by a Knudsen diffusion mechanism, employing membranes with a pore
size lower than 0.2 ␮m. A molecular diffusion model
in the study of their DCMD and AGMD experiments
have been taken into account in [4–6], employing
membranes with a pore size higher than 0.2 ␮m. A
If we have a porous media or a porous membrane
filled by a gas mixture and a pressure gradient exists
through the membrane, a form for the flux relations
can be obtained, modelling the porous medium as a
bundle of cylindrical capillaries and using momentum transfer considerations. The flux relations are,
of course, founded on transport laws for a single
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
capillary. According to mentioned transfer considerations, encounters between molecules or between
a molecule and the capillary walls are accompanied
by momentum transfer. As a result, there are three
mechanisms [10,11] by which a given species of a
gas mixture may lose momentum in the motion direction through a capillary, (a) by a direct transfer
to the capillary walls as a result of molecule–wall
collisions (Knudsen resistance); (b) by transfer to another species as a consequence of collisions between
pairs of unlike molecules (molecular resistance); (c)
by indirect transfer to the capillary walls via a sequence of molecule–molecule collisions terminating
in a molecule–wall collision (viscous resistance).
One may first consider three different simple situations in which each of the three separate mechanisms
in turn operates alone. For each of these situations, the
relations between fluxes and pressure gradients can be
found without too much difficulty.
(a) When the capillary diameter is small compared
with the mean free path lengths in the gas mixture,
the partial pressure gradient of species i, ∇pi , is
balanced by momentum transfer to the wall by the
first mechanism above mentioned. Knudsen estimated the rate of momentum transfer and hence
deduced the corresponding flux relation as
JiK
1
= − ∇pi
DiK
RT
(1)
where Ji K is the molar flux of species i and Di K the
Knudsen diffusion coefficient,
2
8RT 1/2
DiK = r
3
πMi
where r is the pore capillary, T the temperature, Mi the
molecular weight of species i and R the gas constant.
(b) It is also possible to give a detailed derivation of
the transport laws at the opposite limit, where the
capillary diameter is very large compared with the
mean free path lengths in the gas mixture at constant pressure. A good exposition of the momentum transfer arguments can be found in the book
of Present [12]. For a binary mixture of species i
and j, the flux relation is obtained as
pj JiD − pi Jj D
1
= − ∇pi
RT
Dij
(2)
17
where Dij denotes the mutual diffusion coefficient of
the two species
Dij = pDij
where p is the total pressure and Dij the ordinary
diffusion coefficient.
(c) When the capillary diameter is large compared
with mean free path lengths and a pure substance
is present in the capillary we have the classical
Poiseuille problem. The viscous flow relation establishes as
JiV = −
pi r 2
∇p
8RTµ
(3)
where µ is the viscosity of the gas.
The real problem is to know the way these individual descriptions are coupled together to describe
simultaneous transport by more than one mode or
mechanism. Following with momentum–transfer arguments, Mason and Malinauskas [10] first consider
that how Knudsen and molecular diffusive flows combine. They consider a binary mixture at uniform total
pressure, but where gradients of the partial pressure of
the species are present. According to Newton’s second
law of motion, if species i is not accelerated on the
average, then the average momentum transferred by
collisions with the walls and with other species must
be balanced by the gradient of the partial pressure of
the species, ∇pi . This force can be considered to be
made up of separated contributions, each just sufficient to balance the momentum transfer by wall collisions and by collisions with each of the other species
in the mixture. So
−(∇pi ) = −(∇pi )molecule − (∇pi )wall
(4)
where
−(∇pi )molecule = RT
pj JiD − pi Jj D
Dij
and
−(∇pi )wall = RT
JiK
DiK
For better comprehension of this combination,
Mason and Malinaukas [10] suggest thinking in an
electrical analogous circuit comprising two resistors
in series where voltage drops (pressure gradients) are
18
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
additive, and the currents (the fluxes Ji K and Ji D ) are
identical. In this way the relation
pj JiD − pi Jj D
JiD
1
+
= − ∇pi
DiK
Dij
RT
(5)
is obtained to describe the diffusion of component i
(a similar equation would be obtained for species j) of
a binary mixture at uniform total pressure throughout
the entire pressure range between the Knudsen or free
molecule limit and the molecular or continuum limit.
If a gradient of total pressure exists, the viscous
fluxes must be added to the diffusive flux. The reason
for this simple additivity follows from the kinetic theory, in that there are no viscous terms in the diffusion
equations and no diffusion terms in the viscous-flow
equations, the two are entirely independent in the sense
that there are no direct coupling terms in these equations. This independence depends only on the fact that
the various flows are proportional to gradients (linear
laws), and that quantities of different tensorial character do not couple in the linear approximation in
isotropic systems (Curie’s theorem).
In this way and using an electrical analogy, Knudsen and molecular diffusion resistances are combined
like resistors in series, where voltage drops (pressure
gradients) are additive, and the resultant flow is then
combined with the viscous flow like resistors in parallel, where currents (fluxes) are additive (Fig. 1). So,
Ji = JiD + JiV
(6)
where Ji D and Ji V are given by Eqs. (3) and (5), respectively.
Above equations have been obtained using simple
momentum transfer arguments. Some of these arguments are based largely on plausibility rather on
detailed theory. Nevertheless, the same equations are
Fig. 1. Mass transfer resistances for transport of species i through
the porous membrane.
obtained using the dusty gas model, that is a more
theoretically sound way to regard diffusion through
porous media, based on the well developed kinetic
theory. The dusty gas model equations in the form
given by Eqs. (3), (5) and (6) are considered a first
good approximation to the gas transport of binary
mixtures in a cylindrical capillary.
The dusty gas model also includes a pathway for
surface diffusion in which molecules move along
a solid surface in an adsorbed layer. In the dusty
gas model, this motion is assumed to be independent of the preceding three modes of motion. This
mechanism is considered negligible in MD membranes [1], as by definition of the MD phenomenon,
molecule–membrane interaction is low and the surface diffusion area in these membranes is relatively
small compared to the pore area.
2.2. Analysis of data on gas–liquid displacement
Different methods have been used to characterise porous membranes: electron microscopy; mercury intrusion; gas adsorption; thermoporometry;
permporometry; bubble point; gas transport method;
liquid–liquid displacement method; as well as other
methods based on rejection performance using reference molecules and particles. As indicated by Germic
et al. [13], the characterisation technique should be
chosen in such a way that the medium of characterisation and final application are similar. Since the
final application is membrane distillation (that is water vapour transport), we have chosen as medium
of characterisation the gas transport method. This
method has been frequently used for estimation of
mean pore size and pore size distributions of many
commercial membranes and has reached the status of
the recommended standard [14,15]. In this method,
the membrane sample under test is thoroughly wetted
with a low surface tension, low viscosity and low
volatility liquid and placed in a holder. An increasing
inert gas pressure (air was used) is applied on one
side of the sample, at the other side, the pressure is
kept constant (atmospheric pressure). The increasing gas pressure difference causes progressive liquid
emptying of smaller and smaller pores. The gas flux
across the sample is recorded as a function of the applied pressure difference across the sample, allowing
its analysis to evaluate structural parameters of the
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
membrane. However, the theoretical basis used for the
analysis of these accumulated data in the automated
commercial equipment (as the Coulter porometer
used here) has been criticised by some authors [16],
because it does not take into account the nature of
gas flow in pores. In this work, we have taken it
into account applying the model above developed, as
indicated in the following sections.
According to Eqs. (3), (5) and (6), the molar flux
of a simple gas (in the present analysis the air has
been asssumed a simple gas) through a long, circular
capillary of radius r and tortuosity q is
1
2
8RT 1/2 p̄r 2
J (r) = −
r
+
p
(7)
RTqδ 3
πM
8µ
where qδ is the effective pore length and p̄ the average
pressure in the pore.
In the porous membrane, pores of different sizes
are present all of which contribute to transport. We
approximate the pore size distribution function by a
discrete distribution with m classes. The ith class, i =
1, 2, . . . , m, have a width
r(i) = r(i − 1) − r(i)
(8)
where r(i) is the radius of the smallest dry pore when
a p(i) is applied, both related by the Washburn equation
2γ cos θ
r(i) =
(9)
p(i)
where γ is the surface tension of the air–liquid interface and θ is the contact angle between liquid phase
and pore wall. For a fully wetting-liquid, cos θ = 1.
Taking into account Eq. (7), the molar flux J(j) through
the membrane when a p(j) pressure difference is applied can be expressed as
j
J (j )
π 2
8RT 1/2
3
=
r(i) n(i)
p(j ) RTqδ
3
πM
i=1
r(i)4 n(i)p̄(j )
(10)
+
8µ
where p̄(j ) is the average pressure in the membrane,
when a p(j) is applied and δ is the membrane thickness. If
2 π 8RT 1/2
A≡
(11)
3 RT πM
19
and
B≡
π
8RTµ
(12)
thus, the number n(j) of pores with radius between
r(j − 1) and r(j) can be expressed as
(J (j )/p(j ))
j −1
− i=1 {[Ar(i)3 +Br(i)4 p̄(j )](n(i)/qδ)}
n(j )
=
qδ
Ar(j )3 + Br(j )4 p̄(j )
(13)
In this way, the pore number of the m classes can
be calculated, the last class m, corresponding to the
smallest membrane pores, with radius r(m) and a pressure difference applied p(m). When this pressure
difference is applied, the membrane is completely
dry and if higher pressure differences are applied, a
linear behaviour of J/p versus p̄ should be obtained
in accordance with
m
m
J (k)
3 n(i)
4 n(i)
= A r(i)
+ B r(i)
p̄(k),
p(k)
qδ
qδ
i=1
for all k > m
i=1
(14)
where all the figures between brackets are constant,
once we have fixed the type of membrane, the gas and
the work temperature. The knowledge of n(j)/qδ by
means of Eq. (13) allows us to calculate the following
cases.
• The pore size distribution, dn(j)/dr(j),
(n(j )/qδ)
1 dn(j )
=
qδ dr(j )
r(j − 1) − r(j )
(15)
• The total number of pores, N,
m
n(i)
N
=
qδ
qδ
(16)
i=1
• The superficial porosity of the membrane, ε S ,
m
n(i)
εS
π r(i)2
=
qδ
qδ
(17)
i=1
different to the void volume, ε,
ε = εS q
(18)
20
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
• The average pore radius,
r =
m
n(i)r(i)
i=1
n(i)
pDwa (Pa/m2 s) for water–air is given by [17]
(19)
pDwa = 4.46 × 10−6 T 2.334
(21)
From the integration of Eq. (20), we obtain
p
In order to use these structural parameters in the
equations of transport in MD, the determination of
n(i)/qδ is enough, not necessarily being the estimation
of q.
2.3. Evaluation of water vapour permeability
in MD
Although Eqs. (3), (5) and (6) were derived for
isothermal flux, they have been successfully applied
to non-isothermal systems, via the inclusion of terms
for thermal diffusion and thermal transpiration. It has
been shown [11] that these terms are negligible in the
MD operating regime, and the average temperature in
the membrane is used in place of T in the transport
model equations. Using this result, we are going to
evaluate the membrane permeability in two different
situations, when air is present in the pores and when
air is not present in the pores.
2.3.1. MD through stagnant air within the pores
In atmospheric pressure DCMD and AGMD applications processing aqueous solutions of non-volatile
solutes, and unless steps are taken to remove dissolved air from the feed and permeate prior to processing, the dissolved air acts just as a volatile solute,
exerting a partial pressure at the feed and permeate
vapour–liquid interfaces. But, the solubility of air in
water is so low that the mass transfer resistance in the
liquid boundary layers adjacent to the membrane can
completely impede air flux. So the air can be treated
as a stagnant film. Considering the air flux to be 0,
and the total pressure a constant, the following expression of the water vapour flux through a cylindrical pore of radius r can be obtained from Eqs. (3), (5)
and (6)
−1
1
1
J (r) = −
+(p
/pD
)
a
wa
RT (2/3)r(8RT/π Mw )1/2
×∇pw
(20)
where Mw is the water molecular weight, ∇pw the
gradient of water vapour pressure, and the value of
J (r) =
pa DK (r) + pDwa
pDwa
ln f
R T̄ qδ pa DK (r) + pDwa
(22)
p
where paf and pa are the partial pressures of air in the
feed and permeate ends of the membrane pores, T̄ the
average temperature in the membrane, and
1/2
8R T̄
2
DK (r) = r
(23)
3
π Mw
Taking into account that the membrane presents a pore
size distribution, the water flux through the membrane
can be calculated as
p
m
pDwa
pa DK (r(i)) + pDwa
ln
J=
paf DK (r(i)) + pDwa
R T̄
i=1
n(i)
qδ
×π r(i)2
(24)
The water vapour permeability of the membrane C, is
defined as the relation between the water flux and the
driving force for the mass transport
C=
J
f
pw
(25)
p
− pw
p
f and p are the partial pressures of water
where pw
w
vapour in the feed and permeate ends of the pores. So,
C can be calculated as
p
m
pDwa /R T̄
pa DK (r(i)) + pDwa
C=
ln
p
paf DK (r(i)) + pDwa
pa − paf
i=1
×π r(i)2
n(i)
qδ
(26)
If the pore radius is large enough, the Knudsen resistance in Eq. (20) is negligible and the expression
for the flux in this molecular diffusion limit can be
obtained as
p
m pa
pDwa
ln
Jmolecular =
paf
R T̄
i=1
×π r(i)2
n(i)
qδ
(27)
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
and the corresponding water vapour permeability,
Cmolecular , calculated as
Cmolecular =
pDwa 1 εs
R T̄ pa log qδ
(28)
where pa log is the mean logarithmic of the partial
pressure of air in the pore ends.
21
0.45 and 0.2 ␮m, respectively. Their porosity, as indicated by the manufacturer, is about 0.8. They are
composite membranes formed by an actual porous
PTFE layer with a thickness of about 60 ␮m on a
PP screen support with big holes of about 500 ␮m
in size.
3.2. Porometry
2.3.2. MD without stagnant air within the pores
One way to increase the membrane permeability in
DCMD is to remove the stagnant air from within the
membrane, for example by degassing the feed and permeate [18]. In this way, the air is no longer present
to maintain a constant total pressure across the membrane. Also in VMD only trace amounts of air will
exist within the membrane pores. In these cases considering only water vapour within the pores, Eq. (14)
gives for the membrane permeability
1/2 m
2 π
8R T̄
r(i)3 n(i)
C=
3 R T̄ πMw
qδ
i=1
m
π r(i)4 n(i)
+
(29)
p̄w
qδ
8R T̄ µ
i=1
where µ is the viscosity of water vapour. The total
permeability is the result of the addition of Knudsen
and viscous contributions.
If the membrane has relatively small pores in relation to the mean free path of water (that depends on
1/p), the viscous contribution to flux is negligible and
the expression for the permeability in this Knudsen
limit is
1/2 m
r(i)3 n(i)
2 π
8R T̄
CK =
(30)
3 R T̄ πM
qδ
i=1
that will be considered later.
3. Experimental
3.1. Membranes
We have studied three commercial hydrophobic
membranes manufactured by Gelman Instrument
and marketed as TF1000, TF450 and TF200. Their
pore sizes as given by the manufacturer are 1.0,
A Coulter Porometer II manufactured by Coulter
Electronics Ltd. was used [19]. For the analysis, the
membrane sample was first thoroughly wetted with a
liquid (Coulter Porofilm) of low surface tension (γ =
16 × 10−3 Pa m), low vapour pressure (3 mmHg at
298 K), and low reactivity, which was assumed to fill
all the pores given that it has a zero contact angle with
virtually all materials. The wetted sample was subjected to increasing pressure applied by a compressed
clean and dry air source at 313 K. In this type of experiment, as the pressure of air increases, it will reach
a point where it can overcome the surface tension of
the liquid in the largest pores and will push the liquid
out. According to the Washburn equation, further increasing of the pressure allows the air to flow through
smaller pores. This was in fact observed when the air
flux across the sample and the applied pressure were
monitored as liquid was being expelled. The ordinary
orientation for the membrane was chosen, with the air
flowing from the separation layer to the support. As
there is a big relation between the size of the holes of
the membrane PP support and the PTFE layer pores,
the pressure drop through the support was considered
negligible.
4. Results and discussion
4.1. Results on morphological characterisation
Different samples of each membrane have been
analysed in the Coulter porometer. Representative examples of the results obtained on air flow are shown
in Fig. 2. The corresponding pore size distributions
obtained from Eqs. (13) and (15) are shown in Fig. 3.
It is seen that these distributions are approximately
Gaussian. The smallest membrane pores of radius r(m)
considered for each membrane were those for which
the right side of Eq. (13) was a negative number. It
22
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
Fig. 2. Pressure and temperature normalised air fluxes vs. pressure difference applied, as obtained in the Coulter porometer for the three
membranes studied: (a) TF1000; (b) TF450; and (c) TF200.
was checked that the J/p versus p̄ plot was linear
for p applied higher than the corresponding to the
first negative value, in accordance with Eq. (14). Thus
indicating that the negative values are due to experimental error. In any case, flowmeter sensibility does
not allow the detection of the contribution of lower
pores to the air flow, if they were present.
The average pore radius evaluated from Eq. (19)
are shown in Table 1. Some differences between these
values and the ones given by the manufacturer have
been found. Table 1 also shows other characteristics
of the tested membranes. We can see that the pore
number n(i) and also the magnitudes in which it influences (dn(i)/dr(i), ε s , N) are calculated as a relation to
the tortuosity-thickness product qδ. So, the analysis of
porometer data made here allows the determination of
n(i)/qδ, 1/qδ dn(i)/dr(i), εs /qδ and not n(i), dn(i)/dr(i)
and ε s . This is enough to predict water vapour permeabilities in MD, that is the final application of the
membranes studied.
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
23
Fig. 3. Pore size distributions corresponding to the porometer results shown in Fig. 2.
4.2. Predictions on water vapour permeability
4.2.1. MD through stagnant air within the pores
The permeability C has been evaluated according
to Eq. (26) for a DCMD application, where the total
pressure within the pores p (p = p a + p w ) is assumed
1 atm. The feed is assumed to be water, and water is
also considered in the distillate side of the membrane.
Two types of situations are considered. In the first
one, the temperature at the cold ends of the pores is
Table 1
Structural parameters as indicated by the manufacturer and as obtained from the analysis of porometer results
Membrane
Nominal pore
size (␮m)
Nominal void
N/qδ (1016 × m−3 )
ε s /qδ (m−1 )
r (␮m)
TF1000
TF450
TF200
1.0
0.45
0.2
0.80
0.80
0.80
3.35
5.10
10.5
11100
8900
7900
0.325
0.235
0.155
24
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
Fig. 4. Water vapour permeability vs. average temperature in a MD application where the total pressure within the pores is assumed 1 bar
and the temperature at the pore outlets is 20 ◦ C. The temperature at the pore inlets is ranging between 30 and 80 ◦ C. C is the permeability
as given by the model (Eq. (26)), and Cmolecular is the permeability corresponding to the molecular diffusion limit (Eq. (28)).
assumed as 20 ◦ C, and the temperature at the hot ends
of the pores is ranging between 30 and 80 ◦ C. The
results obtained are shown in Fig. 4. The results for
a second situation where a little temperature difference (4 ◦ C) through the membrane exists are shown
in Fig. 5. Here the temperature at the pore inlets
is considered to be ranging between 22 and 82 ◦ C,
while the corresponding temperature at pore outlets
is considered to be ranging between 18 and 78 ◦ C.
In both situations, it is observed how C increases
as the temperature increases. This increase of C is
a consequence of the molecular resistance decrease
due to a lower pressure of the stagnant air within the
pores as the temperature increases. So, for the same
average temperature, C becomes bigger in the first
situation analysed than in the second one. Moreover,
we can see that when the temperature changes in a
wide range C cannot be considered a constant.
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
25
Fig. 5. Water vapour permeability vs. average temperature in a MD application where the total pressure within the pores is assumed 1 bar
and the temperature difference through the membrane is 4 ◦ C. C and Cmolecular are labels as in Fig. 4.
In a previous work [20], we proposed a method
to evaluate the water vapour permeability of the
membrane in a DCMD application, from MD measurements of water flux and evaporation efficiency.
The measurements were carried out for low transmembrane temperature differences (lower than 10 ◦ C)
and average temperatures in the membrane ranging
between 15 and 45 ◦ C. In the treatment of the measurements, in order to evaluate C, this was assumed constant and the values obtained were (9.0 ± 0.9) × 10−4
and (6.6 ± 0.6) × 10−4 mol/m2 /s/Pa, for the TF450
and TF200 membranes, respectively. Also, the permeability of TF1000 membrane was evaluated in
another work [21] from similar MD measurements
and a lightly different analysis. Values of C between
11 and 12 × 10−4 mol/m2 /s/Pa were estimated. As can
be seen, the accordance with the results here obtained
is good.
In Figs. 4 and 5, together with the C values, the
corresponding results for Cmolecular obtained from
Eq. (28) are shown. By comparing the obtained results, we conclude that for these membranes with
26
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
pore sizes smaller than 0.6 ␮m, the Knudsen diffusion resistance is important. This resistance is greater
when the smaller pore size is considered.
4.2.2. MD without stagnant air within the pores
Distillate flux has been calculated for a DCMD application where the air within the membrane pores has
been removed. The results on water vapour permeabilities obtained using Eqs. (29) and (30) are shown in
Fig. 6, when feed and distillate are considered pure
water. They have been obtained considering a temperature of 20 ◦ C at the cold entrances of the pores and
temperatures ranging between 30 and 80 ◦ C at the hot
ends of the pores. The ratio f of the viscous to the
Knudsen contributions is calculated and the results are
shown in Fig. 7. We can see that the viscous contribution is negligible for the membranes studied at low
water vapour pressures. However, for membranes with
larger pores, the viscous contribution can range up to
25% of the Knudsen contribution.
Fig. 6. Water vapour permeability vs. average pressure of water in the membrane pores for a MD application where the air within the
membrane pores has been removed. They have been obtained considering a temperature of 20 ◦ C at the cold entrances of the pores and
temperatures ranging between 30 and 80 ◦ C at the hot ends of the pores. C is the permeability as given by the model (Eq. (29)), and CK
is the permeability corresponding to the Knudsen limit (Eq. (30)).
L. Martı́nez et al. / Journal of Membrane Science 203 (2002) 15–27
Fig. 7. Viscous to Knudsen contribution ratio vs. average pressure
of water vapour in the membrane pores for the same conditions
specified in Fig. 6.
5. Conclusions
From the analysis carried out, we can conclude that
for the membranes studied, frequently used in MD
applications,
1. The membranes have narrow pore size distributions. The values of the average pore size are
slightly different from those given by the manufacturer.
2. In the applications with stagnant air both molecular
and Knudsen resistances are important, the molecular diffusion limited model resulting insufficient.
3. In the applications without stagnant air, both viscous and Knudsen contributions are important in
general. Nevertheless, the viscous contribution is
little for the membranes with smaller pores at low
water vapour pressures (low temperatures).
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