A method to evaluate the net membrane distillation coef®cient
Luis PenÄa, M. Paz Godino, Juan I. Mengual*
Department of Applied Physics I, University Complutense of Madrid, 28040 Madrid, Spain
Received 22 September 1997; received in revised form 5 January 1998; accepted 7 January 1998
Abstract
A method is proposed that permits to evaluate the net membrane distillation coef®cient. The method, which completes and
is based on a model previously reported, emphasizes the in¯uence of the unstirred boundary layers on the membrane
distillation processes. The searched coef®cient is obtained from a set of measurements of thermally driven ¯uxes, osmotically
driven ¯uxes, and thermally and osmotically driven ¯uxes. The theory has been applied to the results obtained with two porous
hydrophobic membranes, in different experimental conditions. The experiments show that the net coef®cient decreases with
the mean temperature and permits to quantify the temperature polarization effects. In addition, the heat of vaporization of
water may be estimated. # 1998 Elsevier Science B.V.
Keywords: Membrane distillation; Net coef®cient; Porous hydrophobic membranes
1. Introduction
Temperature-driven transport of water in the vapor
phase through porous hydrophobic partitions has been
known since the late 1960s. The process was called
`̀ thermal membrane distillation'' [1±7]. Later on, in
the 1980s, it was observed that the same membranes
could be used in a concentration-driven process
termed `̀ osmotic membrane distillation'' [8,9]. In
all these cases the membrane material is water repel-
lent, so liquid water cannot enter the pores unless a
hydrostatic pressure, exceeding the so-called `̀ Liquid
Entry Pressure of Water (LEPw)'', is applied [4,10]. In
absence of such pressure difference, a liquid±vapor
interface is formed on either side of the membrane
pores. It is worth noticing that the driving force is,
same in both the cases, a water vapor pressure differ-
ence. Consequently, the distinction between `̀ thermal
membrane distillation'' and `̀ osmotic membrane dis-
tillation'' is rather arti®cial although the physical
origin of the pressure difference is not the same. In
[11], a model was proposed that considers the mem-
brane distillation as an operation in which the pressure
difference is due, in the most general case, to the
simultaneous actions of temperature and concentra-
tion gradients.
In the literature, the membrane distillation pro-
cesses are usually described by assuming a linear
relationship that relates the mass ¯ux and the water
vapor pressure difference, through the so-called net
membrane distillation coef®cient. The calculation of
this coef®cient is very dif®cult because the pressure
difference is often a complex function of the tempera-
Journal of Membrane Science 143 (1998) 219±233
*Corresponding author. Fax: 34 1 3945191; e-mail:
0376-7388/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved.
P I I S 0 3 7 6 - 7 3 8 8 ( 9 8 ) 0 0 0 1 4 - 3
tures and compositions at the membrane surfaces
(which are not directly measurable), not the bulk
conditions (which are easily known quantities). This
problem is closely related with the presence, at both
sides of the membrane, of unstirred boundary layers,
whose effects must be taken into account. In order to
overcome this dif®culty, simultaneous solution of the
heat and mass transfer equations must be carried out.
This implies the acceptance of several assumptions
concerning the membrane distillation process as well
as the solution properties, or the solving of the equa-
tions via iteration, preferably with the aid of a com-
puter [12]. Another possibility exists in which the
membrane distillation ¯uxes are measured at different
stirring rates, and the value corresponding to an in®-
nite stirring rate, which means in absence of boundary
layers, is obtained [9,13]. This method has got the
inconveniences inherent in all extrapolation processes.
In what follows, we propose a method, based on the
model proposed in [11], that permits to calculate the
net membrane distillation coef®cient from a set of
measurements of temperature-driven ¯uxes, concen-
tration-driven ¯uxes and temperature-and-concentra-
tion-driven ¯uxes. In addition, the quantitative effects
of the layers may be evaluated. The method has been
applied to the experimental results obtained with two
membranes.
2. Theoretical approach
The de®ning phenomenon `̀ direct contact mem-
brane distillation'' is relatively simple: a hydrophobic
microporous membrane is maintained between two
chambers. Each one of the chambers contains pure
water or an aqueous solution of a non-volatile com-
ponent. The liquids in the chambers are stirred with a
common engine whose rotation speed is controlled.
The temperature and the solute concentration in the
chambers may be varied independently. Three meth-
ods may be employed to impose a water vapour
pressure difference across the membrane to drive ¯ux:
(1), to establish a transmembrane temperature differ-
ence, �T; (2), to establish a transmembrane concen-
tration difference, �c; or (3), to establish both
differences simultaneously.
In any case, the water evaporates from one of the
liquid±vapor interfaces, diffuses and/or convects
across the membrane and is condensed on the opposite
liquid±vapor interface. This water ¯ux may be
explained in the framework of different transport
models: the Knudsen model, the Poiseuille model,
the transition Knudsen±Poiseuille models and the
diffusive model. The correct choice of the most ade-
quate of them requires some knowledge of the mem-
brane morphology, including porosity, tortuosity, pore
radius, etc. This problem has been extensively dis-
cussed by Scho®eld et al. [6,14]. In most cases, the
models suggest that the volume ¯ux per unit surface
area of the membrane, J, may be written as a linear
function of the transmembrane water vapor pressure
difference, �P:
J � A ��P (1)
where A is the net membrane distillation coef®cient,
which may depend on mean temperature (through
mean vapor pressure, vapor viscosity, etc.), membrane
structure, etc.
The water vapor pressure at each interface, P,
depends on the absolute temperature, T, and on the
solute concentration, c. This dependence may be
expressed, in a good approximation, by means of
the value corresponding to pure water, P�(T), and
the water activity, a(c,T):
P�c; T� � a�c; T� � P� �T� (2)
where the upper index `̀ �'' means pure water. As is
well known, the relation between the vapor pressure of
pure water and the absolute temperature may be
described very accurately by an exponential depen-
dence of the form:
P� �T� / exp ÿ L
R � T� �
(3)
where L is the heat of vaporization of water and R is
the gas constant.
The problem of relating the unsuitable magnitude
�P, in Eq. (1), with more appropriate transmembrane
quantities �T and �c was discussed in [11] and the
method proposed there will be followed in the present
paper. As a ®rst step, a series expansion of the
exponential term in Eq. (3) is carried out and the
following equation is reached:
J � A � P� �T� ��a� P
� �T� � a � L
R � T2��T
� �(4)
220 L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233
In this equation, T and a are the transmembrane
mean values of the temperature and the water
activity.
It is necessary to take into account the existence of
boundary layers adjoining the membrane at both sides
because of molecular diffusion across them is often a
very limiting factor for the membrane distillation mass
transfer. The presence of the layers implies that the
transmembrane temperature difference, �T, is always
smaller than the one existing between the bulk phases,
�Tb�ThÿTc (Th and Tc being the temperatures of the
hot and the cold bulk phases, respectively). The
phenomenon is called `̀ temperature polarization'',
and is quanti®ed by using the `̀ temperature polariza-
tion coef®cient, �'', de®ned by: ���T/�Tb. This
coef®cient represents the fraction of the externally
applied thermal driving force that contributes to the
mass transfer [6,13]. In the same way, the phenom-
enon of `̀ concentration polarization'' exists, which
gives rise to a difference between the solute concen-
tration at the interface membrane-unstirred layer, cm,
and in the corresponding bulk phase, cb [9].
The contribution of the boundary layers to the MD
mass transfer has been studied in [6,11,13]. According
to the model developed in [11], Eq. (4) may be
rewritten in the form:
J � 1
A� cb
�� d�Po
dc
����Cb
" #� �Po
b ÿL ��
2 � R � T2
� ��Pob ��Tb� � L ��
R � T2� P� �T� ��Tb (5)
where the mass transfer coef®cient has been de®ned as
usual, ���/D, D being the ordinary diffusion coef®-
cient and � the layer thickness. In this equation, �Pob
represents the part of the bulk pressure difference due
to the concentration difference. The value of this
pressure difference may be easily related to the cor-
responding bulk concentration difference, �cb, by
means of thermodynamic tables. It is worth pointing
out that three contributions to the ¯ux appear in
Eq. (5). In what follows, the `̀ osmotic'' contribution
to the ¯ux will be considered always positive. The
existence of a coupling term, proportional to the
Fig. 1. Schematic representation of the experimental setup.
L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233 221
product (�Pob ��Tb), states that the ¯ux cannot be
obtained as a simple addition of two terms propor-
tional to �Pob and �Tb respectively.
This last equation may be simpli®ed in the limits
when only one of the externally imposed temperature
or concentration differences is acting on the system.
For instance, if one imposes the condition �Tb � 0,
one has:
Jo � 1
A� cb
�� d�Po
dc
����cb
" #ÿ1
��Pob (6)
where the upper index `̀ o'' in J means that the ¯ux, in
this case, is due to a concentration difference. It is
worth pointing out that the concentration-driven water
transfer through the membrane causes an energy ¯ux
in the same direction. As a consequence, a transmem-
brane temperature difference is created, which gives
rise to a decrease in the water vapor pressure differ-
ence. This effect has been discussed by Mengual et al.
[9] and its quantitative in¯uence in the present case
will be considered later in Section 4.
On the contrary, if one imposes the condition
�Pob � 0; one has:
Jt � A � L ��R � T2
� P� �T� ��Tb (7)
where the upper index `̀ t'' in J means that the ¯ux, in
this case, is due only to a temperature difference.
Eq. (5) may be rewritten with the help of Eqs. (6)
and (7):
J
Jo� 1ÿ Jt � 1
2 � A � P� �T� ÿ1
A ��Pob
� �(8)
This equation permits to relate the ¯ux obtained in a
Fig. 2. Flux J versus �cb at different �Tb values. Membrane TF-200; T�308C.
222 L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233
general combined experiment, J, with the `̀ pure''
¯uxes, Jt and Jo, through the coef®cient A. In addition,
Eq. (8) states that the values of J/Jo and Jt are linearly
related through the term in brackets, which depends on
mean temperature. This fact suggests a calculation
procedure for the net coef®cient A. This possibility
will be developed later, in Section 4.
3. Experimental
3.1. Materials
Two commercial membranes have been studied: the
TF-200 and the TF-1000, both supplied by Gelman.
These membranes are made of PTFE (polytetra¯uor-
ethylene), supported by a polypropylene net. Their
principal characteristics, as speci®ed by the manufac-
turer, are as follows:
� TF-200: pore size 0.2 mm; thickness 178 mm;
empty volume fraction 80%.
� TF-1000: pore size 1.0 mm; thickness 178 mm;
empty volume fraction 80%.
The liquids employed in the experiments were water
and aqueous solutions of sodium chloride. Pure pro-
analysis grade chemicals and pure water (deionized
and distilled) were used.
3.2. Apparatuses and experimental method
The experimental device employed, which corre-
sponds to the embodiment termed `̀ direct contact
membrane distillation'', was essentially similar to
those described previously in [9,11] (see Fig. 1). Its
Fig. 3. Flux J versus �cb at different �Tb values. Membrane TF-200; T�358C.
L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233 223
central part is a cell, which basically consists of
two equal cylindrical chambers having a length of
20.5 cm and made of stainless steel. The membrane
was ®xed between the chambers by means of a PVC
holder. Three viton O-rings were employed to ensure
there were no leaks in the whole assembly. The
membrane surface area exposed to the ¯ow was
2.75�10ÿ3 m2.
The temperature requirements were set by connect-
ing each chamber, through the corresponding water
jacket, to a different thermostat. In order to improve
the uniformity of temperatures and concentrations
inside each chamber, the liquids were stirred by a
chain-driven cell magnetic stirrer assembly. Tempera-
tures were measured with platinum resistance thermo-
meters placed near both sides of the membrane. Under
these conditions, the temperature was constant within
�0.18C. The solute concentration was measured by
means of standard chemical titration (Mohr titration).
The maximum error in these measurements is
�0.05 mol lÿ1. The value of the volume ¯ux was
obtained, in each case, by adjusting the experimental
data (volume ¯owing into the corresponding chamber
versus time) to a linear function by a least squares
procedure.
4. Results and discussion
For reasons of simplicity, in what follows the
considered operation will be denoted TMD (thermal
membrane distillation) or OMD (osmotic mem-
brane distillation) when the water vapor pressure
difference is obtained by means of an externally
imposed temperature difference or an externally
imposed concentration difference, respectively. When
Fig. 4. Flux J versus �cb at different �Tb values. Membrane TF-200; T�408C.
224 L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233
the two differences exist simultaneously, the operation
will be denoted CMD (combined membrane distilla-
tion).
The water ¯ux through both membranes was mea-
sured, in different experimental conditions, with a
constant stirring rate of 200 rpm. The parameters
mean temperature, bulk temperature difference and
solute concentration were varied independently. The
stirring rate was not varied in the present paper,
because its in¯uence on this kind of systems has been
deeply studied in some previous papers (see, for
instance, Refs. [11] or [15]). The mean temperature
was varied between 308C and 508C, with steps of 5 K.
The solute concentrations were: 0, 2, 3, 4 and
5 mol lÿ1. The bulk temperature differences were 0,
�4, �6, �8 and �10 K, (the signs � and ÿ mean
proactive or counteractive with the concentration dif-
ference). The results are shown in Figs. 2±11, corre-
sponding to the membranes TF-200 and TF-1000,
respectively. It is worth saying that each one of the
data appearing in Figs. 2±11 is affected by its corre-
sponding error value, which has not been represented
for reasons of simplicity. The quantitative importance
of the errors may be appreciated by the following
illustrative example: the value of the ¯ux, with its
estimated standard deviation, is: (1.50�0.03)�10ÿ6 m sÿ1, for the membrane TF-200; mean tem-
perature 358C; solute concentration 3 mol lÿ1 and
bulk temperature difference �6 K.
The purposes of this set of measurements are: (1)
to study the in¯uence of the varying parameters on
the ¯ux value, (2) to calculate the net membrane
distillation coef®cient, A, as a function of the mean
temperature, and (3) to quantify the temperature
polarization coef®cient, �, in different experimental
conditions.
Fig. 5. Flux J versus �cb at different �Tb values. Membrane TF-200; T�458C.
L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233 225
In what follows, we are going to discuss the results,
as well as to check the validity of the equations
obtained in Section 2.
In Figs. 2±11, the ¯ux values corresponding to the
solute concentration 0 mol lÿ1 refer to the TMD
¯uxes, Jt, described by Eq. (7). In the same way,
the ¯ux values corresponding to the bulk temperature
difference of 0 K refer to the OMD ¯uxes, Jo,
described by Eq. (6). Finally, the remaining values
correspond to the CMD ¯uxes, J, described by Eq. (5).
As it was already said in Section 2, in the experi-
ments carried out under externally imposed isothermal
bulk conditions, an energy transfer takes place through
the membrane and a transmembrane temperature dif-
ference is created. The numerical value of this effect
may be roughly estimated, by means of the same
procedure employed in [9]. If one assumes that the
energy transfer takes place by a convection mechan-
ism, the temperature difference is smaller than 0.6 K,
for the membrane TF-200, in the most unfavourable
case, and smaller than 0.7 K, for the membrane TF-
1000, in the same conditions. In both cases the TMD
¯uxes originated are much smaller [9,11,13] than the
OMD ¯uxes reported in the present paper and, con-
sequently, the in¯uence of these temperature differ-
ences may be considered negligible.
In order to obtain the value of the parameter A, for
each one of the membranes and mean temperatures,
the following procedure is used: The set of ¯ux data J,
Jo and Jt, corresponding to a given membrane and a
given mean temperature, is prepared as a set of pairs
{J/Jo;Jt}, and ®tted to the straight line suggested in
Eq. (8), by means of a least squares method. The slope
of the straight line depends on mean temperature
Fig. 6. Flux J versus �cb at different �Tb values. Membrane TF-200; T�508C.
226 L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233
(through coef®cient A and pressure P� �T��. A subse-
quent straightforward calculation, in which the values
of P� �T� and �Po
b are obtained from Tables, permits to
get the searched A value from this slope.
In Fig. 12 the A values corresponding to both
membranes are plotted versus the mean temperature.
In both cases, it may be observed that the net mem-
brane distillation coef®cient decreases with the mean
temperature. This trend suggests that the transport
mechanism responsible for the vapor ¯ux through
the membrane pores is Knudsen type (see Scho®eld
et al. [6]).
On the other hand, Eq. (7) relates the temperature
polarization coef®cient, �, with the TMD ¯ux, the
coef®cient A, and other known quantities, such as the
mean temperature, the bulk temperature difference or
the heat of vaporization, L (obtained from Tables). The
values of Jt may be obtained from Figs. 2±11 and the A
values from Fig. 12. With this in mind, the coef®cient
� has been evaluated, for both membranes, at different
values of the mean temperature and the bulk tempera-
ture difference. The results are shown in Table 1.
A visual inspection of the data appearing in Table 1
suggests that the coef®cient seems to be virtually
independent of both mean temperature and bulk tem-
perature difference. This trend is similar to the one
previously reported in [13,15] for this kind of systems.
In order to be more rigorous, the pairs of data f�; Tghave been adjusted to a horizontal line, by using a �2-
minimization procedure. From this analysis the prob-
ability of independence (zero slope) was calculated in
all cases. The results of this analysis is: the probability
of independence of � with T runs from 96% to 99%,
for the membrane TF-200, and from 91% to 97%, for
Fig. 7. Flux J versus �cb at different �Tb values. Membrane TF-1000; T�308C.
L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233 227
the membrane TF-1000, which reasonably permits us
to state that the coef®cient � may be considered
independent on mean temperature. Otherwise, the
reported � values run between 0.49 and 0.59, for
the membrane TF-200, and between 0.40 and 0.49,
for the membrane TF-1000. These values fall within
Fig. 8. Flux J versus �cb at different �Tb values. Membrane TF-1000; T�358C.
Table 1
Temperature polarization coefficient, �, at different temperature conditions
�Tb (K) T�308C T�358C T�408C T�458C T�508C
Membrane TF-200
10 0.51 0.59 0.55 0.58 0.51
8 0.54 0.58 0.56 0.58 0.51
6 0.54 0.58 0.55 0.58 0.49
4 0.54 0.59 0.55 0.59 0.49
Membrane TF-1000
10 0.40 0.45 0.43 0.44 0.49
8 0.42 0.44 0.44 0.44 0.49
6 0.40 0.46 0.44 0.46 0.48
4 0.40 0.43 0.44 0.44 0.44
228 L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233
the range of 0.4 to 0.7, which is recommended for well
designed membrane distillation systems [12].
Finally, it is worth saying that our model permits to
get estimated values of the heat of vaporization of
water, L. In the literature [9,13,16±19] this problem
has been treated by taking into account the depen-
dence of the `̀ pure'' TMD and OMD ¯uxes on mean
absolute temperature. In these papers it is implicitly
assumed that the net membrane distillation coef®cient
does not depend on temperature and it is calculated an
apparent activation energy for the process. It is worth
saying that the values so obtained are always lower
than the expected value of L. On the contrary, our
model suggests a somewhat different calculation pro-
Fig. 9. Flux J versus �cb at different �Tb values. Membrane TF-1000; T�408C.
Table 2
Estimated values of the heat of vaporization of water
Membranes L (kJ molÿ1)
TMD Results OMD Results Ref. [17] Ref. [13] Ref. [16] Ref. [19] Ref. [9]
TF-200 42 48 30 25 20 26 30
TF-1000 43 41 19 17 22 26 33
L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233 229
cedure, in which the dependence of coef®cient A on T
is taken into account. With this in mind, the pairs of
values Jt; T are ®tted, according to Eqs. (7) and (3), by
using a �2-minimization procedure, and the same
procedure is used with the data fJo; Tg, and the
Eqs. (6) and (3). The results obtained appear in
Table 2. In addition, the table shows the results
obtained by different authors. For example, TMD
¯uxes through membranes of this type have been
measured, at different mean temperatures, by Pagliuca
et al. [16], Ortiz de ZaÂrate et al. [13,17], Scho®eld et
al. [18] and VaÂzquez±GonzaÂlez and MartõÂnez [19].
Similarly, OMD ¯uxes, for these membranes and
temperatures, have been measured by Mengual et
al. [9]. The values in the table must be compared with
the measured values of the heat of vaporization. In
thermodynamics tables, it may be seen that L varies
between 44 and 43 kJ molÿ1, in the considered tem-
perature range from 308C to 508C. Obviously, our
method provides better values than the former one,
which may be considered as an additional test in
support of our model.
5. Conclusions
A previously presented model, which permits to
study membrane distillation processes, is completed.
The model emphasizes the relevance of the boundary
layers on the phenomena.
The model permits to obtain the net membrane
distillation coef®cient from a set of membrane dis-
tillation ¯uxes, measured in different temperature and
concentration conditions.
Fig. 10. Flux J versus �cb at different �Tb values. Membrane TF-1000; T�458C.
230 L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233
The obtained coef®cient decreases with mean tem-
perature, which suggests that a Knudsen type mechan-
ism is the responsible for the water ¯ux.
The previous results permit to estimate the tem-
perature polarization coef®cient and the heat of vapor-
ization of water. This last value is closer to the
experimentally determined one than the obtained with
other calculation procedures.
6. List of symbols
a activity
A net membrane distillation coefficient
(m2 s kgÿ1)
c molar concentration (mol lÿ1)
D diffusion coefficient (m2 sÿ1)
J volume flux (m3 mÿ2 sÿ1)
L heat of vaporization of water (J molÿ1)
P pressure (Pa)
R gas constant (J molÿ1 Kÿ1)
T temperature (K)
6.1. Greek letters
� layer thickness (m)
� mass transfer coefficient (m sÿ1)
� temperature polarization coefficient
6.2. Subscripts
� pure
o osmotic
t thermal
Fig. 11. Flux J versus �cb at different �Tb values. Membrane TF-1000; T�508C.
L. PenÄa et al. / Journal of Membrane Science 143 (1998) 219±233 231
6.3. Superscripts
b at the bulk phase
m at the membrane surface
h hot phase
c cold phase
Acknowledgements
Economical support from the CICYT is acknowl-
edged.
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