Characterisation of three hydrophobic porous membranes
used in membrane distillation
Modelling and evaluation of their water vapour permeabilities
L. Mart´ınez
a,∗, F.J. Florido-D´ıaz
a, A. Hernández
b, P. Prádanos
baDepartamento de F´ısica Aplicada, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
bDepartamento 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 mea-surements, 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 con-tact 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. Mart´ınez).
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 configu-ration 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
Nomenclature
A factor defined in Eq. (11) (mol s/Kg m) B factor defined in Eq. (12) (mol s3/Kg2m) C water vapour permeability (mol/m2s Pa) d pore diameter (m)
DiK Knudsen diffusivity of species i (m2/s) Dij binary molecular diffusivity (Pa m2/s) Dij ordinary binary diffusion coefficient (m2/s) J molar flux (mol/m2s)
M molecular weight (kg/mol) n number of pores (m−2) N total number of pores (m−2) p pressure (Pa)
q tortuosity r pore radius (m) R gas constant (J/mol K) T temperature (K)
Greek letters
δ membrane thickness (m)
δeff 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
water vapour being transported by a Knudsen diffu-sion mechanism, employing membranes with a pore size lower than 0.2m. 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 0.2m. A
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 Knud-sen and molecular diffusion resistances for the vapour transport across the membrane have been explained in [8,9]. On the other hand, frequently, in the applica-tion of these models the membrane of void volumeε is considered as a collection of capillaries of diameter d and a tortuosity factor about 2, including the values given by the manufacturers forεand d in the model. As there are several methods of membrane character-isation, 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 dis-tillation 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 charac-teristics 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 dis-placement measurements. In the second step, the wa-ter vapour permeabilities have been evaluated. In both steps, the same general gas transport model has been applied.
2. Theory
2.1. The transport model
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 momen-tum transfer considerations. The flux relations are, of course, founded on transport laws for a single
capillary. According to mentioned transfer consid-erations, 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 di-rection 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 an-other species as a consequence of collisions between pairs of unlike molecules (molecular resistance); (c) by indirect transfer to the capillary walls via a se-quence of molecule–molecule collisions terminating in a molecule–wall collision (viscous resistance).
One may first consider three different simple situa-tions 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 esti-mated the rate of momentum transfer and hence deduced the corresponding flux relation as JiK
DiK = − 1
RT∇pi (1)
where JiK is the molar flux of species i and DiK the Knudsen diffusion coefficient,
DiK= 2 3r
8RT πMi
1/2
where r is the pore capillary, T the temperature, Mithe 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 con-stant pressure. A good exposition of the momen-tum 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
pjJiD−piJjD Dij = −
1
RT∇pi (2)
whereDij 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 es-tablishes as
JiV= − pir 2
8RTµ∇p (3)
whereµis the viscosity of the gas.
The real problem is to know the way these indi-vidual descriptions are coupled together to describe simultaneous transport by more than one mode or mechanism. Following with momentum–transfer ar-guments, Mason and Malinauskas [10] first consider that how Knudsen and molecular diffusive flows com-bine. 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 suffi-cient to balance the momentum transfer by wall colli-sions and by collicolli-sions with each of the other species in the mixture. So
−(∇pi)= −(∇pi)molecule−(∇pi)wall (4) where
−(∇pi)molecule=RTpjJiD−piJjD Dij
and
−(∇pi)wall=RTJiK 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
additive, and the currents (the fluxes JiK and JiD) are identical. In this way the relation
JiD DiK
+pjJiD−piJjD Dij = −
1
RT∇pi (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 the-ory, 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 equa-tions. This independence depends only on the fact that the various flows are proportional to gradients (linear laws), and that quantities of different tensorial char-acter do not couple in the linear approximation in isotropic systems (Curie’s theorem).
In this way and using an electrical analogy, Knud-sen 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 par-allel, where currents (fluxes) are additive (Fig. 1). So,
Ji =JiD+JiV (6)
where JiD and JiV are given by Eqs. (3) and (5), re-spectively.
Above equations have been obtained using simple momentum transfer arguments. Some of these ar-guments 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 indepen-dent of the preceding three modes of motion. This mechanism is considered negligible in MD mem-branes [1], as by definition of the MD phenomenon, molecule–membrane interaction is low and the sur-face 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 charac-terise porous membranes: electron microscopy; mer-cury intrusion; gas adsorption; thermoporometry; permporometry; bubble point; gas transport method; liquid–liquid displacement method; as well as other methods based on rejection performance using refer-ence molecules and particles. As indicated by Germic et al. [13], the characterisation technique should be chosen in such a way that the medium of charac-terisation and final application are similar. Since the final application is membrane distillation (that is wa-ter 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 increas-ing 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 ap-plied pressure difference across the sample, allowing its analysis to evaluate structural parameters of the
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
J (r)= − 1 RTqδ 2 3r 8RT πM
1/2 +pr¯ 2
8µ
p (7)
where qδis the effective pore length andp¯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 ap(i) is applied, both related by the Washburn equa-tion
r(i)= 2γp(i)cosθ (9)
whereγ is the surface tension of the air–liquid inter-face andθ is the contact angle between liquid phase and pore wall. For a fully wetting-liquid, cosθ =1. Taking into account Eq. (7), the molar flux J(j) through the membrane when ap(j) pressure difference is ap-plied can be expressed as
J (j) p(j)= π RTqδ j i=1 2 3r(i) 3n(i) 8RT πM
1/2
+r(i)4n(i)p(j)¯ 8µ
(10)
wherep(j)¯ is the average pressure in the membrane, when ap(j) is applied andδis the membrane thick-ness. If
A≡2 3 π RT 8RT πM
1/2
(11)
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
n(j) qδ =
(J (j)/p(j)) −j−1
i=1{[Ar(i)3+Br(i)4p(j)¯ ](n(i)/qδ)} Ar(j)3+Br(j)4p(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 pres-sure 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 versusp¯should be obtained in accordance with
J (k) p(k)= A m i=1
r(i)3n(i) qδ + B m i=1
r(i)4n(i) qδ
¯ p(k),
for allk > m (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), 1
qδ dn(j) dr(j) =
(n(j)/qδ)
r(j−1)−r(j) (15)
• The total number of pores, N, N
qδ =
m
i=1 n(i)
qδ (16)
• The superficial porosity of the membrane,εS, εS
qδ =
m
i=1
πr(i)2n(i)
qδ (17)
different to the void volume,ε,
• The average pore radius,
r =m
i=1
n(i)r(i)
n(i) (19)
In order to use these structural parameters in the equations of transport in MD, the determination of n(i)/qδis 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 appli-cations processing aqueous solutions of non-volatile solutes, and unless steps are taken to remove dis-solved air from the feed and permeate prior to pro-cessing, 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 ex-pression of the water vapour flux through a cylindri-cal pore of radius r can be obtained from Eqs. (3), (5) and (6)
J (r)= − 1 RT
1
(2/3)r(8RT/πMw)1/2+(pa/pDwa) −1
×∇pw (20)
where Mw is the water molecular weight, ∇pw the gradient of water vapour pressure, and the value of
pDwa(Pa/m2s) for water–air is given by [17] pDwa=4.46×10−6T2.334 (21) From the integration of Eq. (20), we obtain
J (r)= pDwa RT qδ¯ ln
pp
aDK(r)+pDwa pfaDK(r)+pD
wa
(22)
wherepaf andpapare the partial pressures of air in the feed and permeate ends of the membrane pores,T¯ the average temperature in the membrane, and
DK(r)= 2 3r
8RT¯ πMw
1/2
(23)
Taking into account that the membrane presents a pore size distribution, the water flux through the membrane can be calculated as
J=m
i=1 pDwa
RT¯ ln pp
aDK(r(i))+pDwa paDKf (r(i))+pD
wa
×πr(i)2n(i)
qδ (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
pf w−p
p w
(25)
wherepwf andppw are the partial pressures of water vapour in the feed and permeate ends of the pores. So, C can be calculated as
C=
m
i=1
pDwa/RT¯ pp
a−paf ln p
p
aDK(r(i))+pDwa pf
aDK(r(i))+pDwa
×πr(i)2n(i)
qδ (26)
If the pore radius is large enough, the Knudsen re-sistance in Eq. (20) is negligible and the expression for the flux in this molecular diffusion limit can be obtained as
Jmolecular=m
i=1
pDwa RT¯
ln p p a pf a
×πr(i)2n(i)
and the corresponding water vapour permeability, Cmolecular, calculated as
Cmolecular= pDwa RT¯
1 pa log
εs
qδ (28)
where pa log is the mean logarithmic of the partial pressure of air in the pore ends.
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 per-meate [18]. In this way, the air is no longer present to maintain a constant total pressure across the mem-brane. Also in VMD only trace amounts of air will exist within the membrane pores. In these cases con-sidering only water vapour within the pores, Eq. (14) gives for the membrane permeability
C=
2 3
π RT¯
8RT¯ πMw
1/2m
i=1
r(i)3n(i) qδ
+
π 8RT µ¯
m
i=1
r(i)4n(i) qδ
¯
pw (29)
whereµ 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 rela-tion 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
CK= 2 3
π RT¯
8RT¯
πM
1/2m
i=1
r(i)3n(i)
qδ (30)
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,
0.45 and 0.2m, respectively. Their porosity, as in-dicated by the manufacturer, is about 0.8. They are composite membranes formed by an actual porous PTFE layer with a thickness of about 60m on a PP screen support with big holes of about 500m in size.
3.2. Porometry
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 = 16×10−3Pa 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 sub-jected to increasing pressure applied by a compressed clean and dry air source at 313 K. In this type of ex-periment, 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 in-creasing 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 ex-amples 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
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 p¯ 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 experi-mental 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 influ-ences (dn(i)/dr(i),εs, 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/qδdn(i)/dr(i),εs/qδ and not n(i), dn(i)/dr(i) andεs. This is enough to predict water vapour per-meabilities in MD, that is the final application of the membranes studied.
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
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) εs/qδ(m−1) r(m)
TF1000 1.0 0.80 3.35 11100 0.325
TF450 0.45 0.80 5.10 8900 0.235
TF200 0.2 0.80 10.5 7900 0.155
pressure within the pores p (p=pa+pw) 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
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 Cmolecularis 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 differ-ence (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.
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 Cmolecularare 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 mea-surements of water flux and evaporation efficiency. The measurements were carried out for low trans-membrane temperature differences (lower than 10◦C) and average temperatures in the membrane ranging between 15 and 45◦C. In the treatment of the measure-ments, in order to evaluate C, this was assumed con-stant and the values obtained were(9.0±0.9)×10−4 and (6.6±0.6)×10−4mol/m2/s/Pa, for the TF450
and TF200 membranes, respectively. Also, the per-meability of TF1000 membrane was evaluated in another work [21] from similar MD measurements and a lightly different analysis. Values of C between 11 and 12×10−4mol/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 re-sults, we conclude that for these membranes with
pore sizes smaller than 0.6m, the Knudsen diffu-sion 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 ap-plication where the air within the membrane pores has been removed. The results on water vapour permeabil-ities obtained using Eqs. (29) and (30) are shown in Fig. 6, when feed and distillate are considered pure
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)).
water. They have been obtained considering a temper-ature 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 contri-bution 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. 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 distribu-tions. The values of the average pore size are slightly different from those given by the manu-facturer.
2. In the applications with stagnant air both molecular and Knudsen resistances are important, the molec-ular diffusion limited model resulting insufficient. 3. In the applications without stagnant air, both vis-cous 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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