A method to evaluate the net membrane distillation coefficient

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:

[email protected]

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.


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



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



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.



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



(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



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


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


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-


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


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.



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



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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A method to evaluate the net membrane distillation coefficient