Non-isothermal water transport through membranes

Non-isothermal water transport through membranes

Journal ofMembrane Science, 56 (1991) 181-194 Elsevier Science Publishers B.V., Amsterdam 181 Non-isothermal water transport through membranes J.M. ...

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Journal ofMembrane Science, 56 (1991) 181-194 Elsevier Science Publishers B.V., Amsterdam


Non-isothermal water transport through membranes J.M. Or&Z&ate,

F. Garcia Lopez and J.I. Mengual*


de Fisica Aplicada

28040 Madrid,


I (Termologia),


de Fisica,



(Received April 17,199O; accepted in revised form August 27,199O)

Abstract Non-isothermal mass transport through membranes has been studied. A model which considers a heat transport by convection through the unstirred layers has been developed. This leads to a simple dependence of the transport coefficient on stirring rate. The model has been applied to the experimental results obtained from five commercial membranes using water as permeant fluid. The experiments were carried out under a temperature difference of 5 K. The mean temperature and the stirring rate were changed independently. The mean temperature was changed between 32.5” C and 57.5 oC with steps of 5K. The stirring rate was changed between 0 and 250 or 360 rpm. porous membranes; membrane distillation; poly(tetrafluoroethylene) poly (vinylidene fluoride) membrane; non-isothermal mass transport



Introduction In the last years considerable effort has been devoted to theoretical and experimental studies of non-isothermal mass transport through membranes. Pioneer papers [l-7] have treated the relation between non-isothermal mass transport and the electrical nature of solutions and membranes. Other papers on the subject explained that non-isothermal mass transport exists only with suitably dense membranes [8,9]. In such cases the names of thermal osmosis or thermoosmosis was used. However, in several contributions [ 10-141, it has been shown that non-isothermal mass transport can also occur through porous walls. This observation was explained by using new views and ideas. In this respect it is worth quoting the contributions [ 15-171, where non-isothermal water transport through hydrophobic porous membranes is thought to be due to “membrane distillation” or “capillary evaporation”. Other authors [l&19] explained the non-isothermal transport by supposing that a liquid phase wets the membrane. Furthermore, there are qualitative and quantitative discrepancies between the results obtained by different authors, even when they refer to similar systems. Most of these discrepancies have been successfully explained by Bellucci [ 201 by using the concept of “temperature po*To whom all correspondence should be addressed. 0376-7388/91/$03.50

0 1991-

Elsevier Science Publishers B.V.


larization” introduced by Vink and Chisthi [ 211. The temperature polarization refers to the loss of driving force brought about by temperature gradients in the fluids adjoining the membrane. As is well known non-isothermal mass transport is influenced by the mean temperature and the temperature difference as well as by the presence of “unstirred liquid layers” on the membrane surfaces. In most of the previous papers, it was supposed that the increase of the transport coefficient with mean temperature could be expressed by a classical Arrhenius function. Nevertheless in Ref. [22] a different dependence was found by developing the “membrane distillation” model. Moreover, in Ref. [ 231, a “differential transport coefficient” was introduced in order to obtain a measured transport coefficient closely related to the continuous or local one. Still there is the problem of the unstirred liquid layers. In this respect it is worth quoting that it was empirically observed that this effect is strongly correlated with the existence of stirring in the bulk phases at both sides of the membrane [ 20,21,24]. In the present paper it is supposed that a convective type heat transfer takes place through the layers adjoining the membrane at both sides. This leads to a simple dependence of the transport coefficient on the stirring rate. The experimental results obtained from five membranes have been treated with this method and the result may be considered satisfactory. Theory Non-isothermal transport of fluids through membranes has generally been studied with the theory of Thermodynamics of Irreversible Processes [ 25,261. The system to be studied consists of a membrane which is held between two pure liquid subsystems I and II. Let us suppose that a difference in temperature, AT, may be created between both subsystems (T being the temperature of the cold subsystem and T + dT the temperature of the hot one). This difference induces the corresponding gradient in temperature across the membrane. If we assume that AT is small enough, the following linear relationship applies [25]: J=g




where J is the flow of material in moles of fluid passing per unit time through the membrane of cross section q, d is the membrane thickness, and B is the integral, global, or average phenomenological coefficient the so called “global non-isothermal permeability”. The coefficient B depends on the following parameters; (a) the nature of the membrane, (b) the nature of the fluid, (c) the average temperature of the system, and (d) (to a lesser extent) the average pressure of the system. A relationship similar to eqn. (1) can be written to include a difference in hydrostatic pressure [ 251.


In the Thermodynamics of Irreversible Processes [ 251 the membrane is usually considered as a “black box”; all flows and driving forces are referred to compartments I and II, while the membrane appears only as a barrier which is capable of maintaining a temperature difference; there is however, a problem concerning the relation between the non-isothermal permeability and the difference of temperature. As is well known, the transport coefficient is a mean value averaged over the range between the temperatures T and T+ AT of the two compartments. The precise form of the average is dictated by the temperature profile and hence by unknown properties of the membrane. In order to relate the experimental value B (T,T+AT) with a value corresponding to the mean temperature, we have followed the method proposed in Refs. [ 23,271. The experiments have been carried out across a “differential” temperature difference (the word “differential” means that this is the minimum temperature difference for which the experiments can be carried out ) . There is a new important factor that complicates the interpretation of the experimental results in the case of non-isothermal transmembrane flows. It is the presence of so-called “unstirred liquid layers” adjoining the membrane at both sides. The previous equation is based on the implicit assumption that the liquids on both sides of the membrane are so well-stirred that the temperatures at the membrane faces are the same as the bulk temperatures. It is however quite impossible to stir any liquid right up to the membrane-liquid interface and, with the usual rates of stirring used in our experiments, there is an effective unstirred layer. As a consequence the difference of the two surface temperatures, AT, is not the difference of the bulk temperatures AT,,. In fact, a part of this externally applied temperature difference is dissipated in the unstirred liquid layers in the vicinity of the membrane surfaces. This leads to lower transmembrane fluxes than expected. The effect has been called “temperature polarization” [20,21], and masks the real magnitude of the non-isothermal effects. Thus eqn. (1) could be written: J+AT+



where B’ is the measured global non-isothermal permeability which is different from the real global non-isothermal permeability B. In the experiments, the membrane was placed between two symmetrical chambers. A horizontal flow-pipe capillary was attached to each chamber. Equation (2 ) describes the phenomenon of non-isothermal matter transport and can be solved in the following way. If we assume that the thermodynamic force AT, is constant and the global phenomenological coefficient B’ is independent of the state variables, eqn. (2 ) becomes dV dt



where t is the time and dV/dt is the volume flow of liquid in the horizontal capillaries; c is a kinetic constant, which is related to the global measured permeability, and has the value MqB’dTJdp. M andp, are the molar mass and the liquid density respectively. Integration of eqn. (3) gives a straight line whose slope is equal to c. Several methods have been proposed to eliminate or at least to reduce the effect of temperature polarization. The one proposed by Vink and Chisthi [ 211 is based on the fact that the thickness of the liquid layer decreases with stirring rate. Thus, by performing measurements at different stirring rates and extrapolating the data to an infinite stirring rate, the effect of temperature polarization can be estimated. The work of Vink and Chisthi [21] only refers to evolution towards steady states in which gradients of hydrostatic pressure and temperature (opposed to one another) are present simultaneously. In their paper they assure that it was found empirically that the real value of nonisothermal pressure difference could be obtained by linear extrapolation of l/ (dP - A’, ) against l/u, where u is the rate of stirring (revolutions per minute), dP is the measured non-isothermal pressure difference for this rate of stirring, and dP, is the measured value of non-isothermal pressure difference without stirring. In their opinion one may assume that the values obtained by extrapolation correspond to the true temperature gradient in the membrane (with due reservations for the uncertainties inherent in all extrapolation processes). In what follows we suppose that there is a heat transfer through both unstirred layers. The heat transfer is due to a convective type mechanism. This leads to a linear dependence of 1/ (B’ - Bb ) on l/u, similar to the one proposed by Vink and Chisthi (Bb is the measured values of B’ without stirring). It should be remarked that, following the reasoning of Refs. [ 23,271, the B’ value obtained by extrapolation, Bb, can be identified with the true global non-isothermal permeability B. In what follows the theory of heat transfer within the membrane and adjoining fluids will be developed in order to study the temperature polarization, Let us assume that in compartments I and II, the bulk temperatures are Tbl and Tb2respectively. At the membrane-liquid interfaces the temperatures are T, and T2 respectively. In steady state for the heat transfer, we have: h,(T,,-T,)=H(T,-T,)=h,(T,-T,,)


where h, and h, are the film heat transfer coefficients for the liquid in compartments I and II, and H is the effective heat transfer coefficient for the membrane-permeant system. Written in this form, eqn. (4) may be used in the cases of thermoosmosis and “membrane distillation” (see Refs. [ 201 and [ 281 respectively). Figure 1 shows the temperature profile corresponding to eqn. (4) and the situation of both layers with respect to the membrane. If we assume that the bulk temperature difference, LIT, = T,,, - Tb2 is not too



Fig. 1. Scheme of the membrane with the adjoining liquid layers.

large and the stirring rates are equal on both sides of the membrane, then h, E h, = h and Tbl - T, E Tz - Tb2. Thus using eqn. (4) we obtain: AT=AT,,p

h (5)


Or, similarly, by using eqn. (3) we get: B’ =I?-




It is a matter of fact that the effect of temperature polarization decreases with stirring rate. This fact can easily be taken into account by assuming that the film heat transfer coefficient of the layer increases with stirring rate: h=h,,+au


where ho is the film heat transfer coefficient when there is no stirring and (Yis a parameter to be determined. By using eqns. (6) and (7), simple algebra leads to: 1


B’ -Bb



where X_h,+2H -2HB and




(hl +2w2 2HBa


There is another problem concerning the relation between the non-isothermal coefficient and the mean temperature. Most papers on the subject show that there is an increase of the non-isothermal permeability with mean temperature. This trend may be expressed by a classical Arrhenius function [ 201. Nevertheless a new type of dependence was proposed [ 221. The work was based on the “membrane distillation” model. As is well known liquid water cannot penetrate into hydrophobic pores, except under suitable pressure. Within each pore evaporation occurs at the hot side and condensation on the cold side. In this case the water flow through the membrane is proportional to the difference of vapour pressures at both sides of the membrane: Jx [P,(~+dT/2)-P,(~-dT/2)] If we assume that the vapor is an ideal gas, the Clapeyron equation may be used. If we also consider that (AT)< (2T) and (LdT) < (2RP2), L being the heat of evaporation of water, we finally get: B’=AF-*

exp( -L/R5?)


where A is a proportionality constant. Experimental Materials Four PTFE (Teflon) and one PVDF (Polyvinylidene fluoride) membrane have been studied. The membranes are hydrophobic and industrially used in filtration processes. All of them are grossly porous partitions, with irregular cavities going through the membrane thickness. Their principal characteristics (as specified by the manufacturer) are indicated in Table 1. The liquid employed in the experiments was pure water (de-ionized and double-distilled). The water was degassed by twenty minutes of vigorous boiling before each experiment. Apparatus The apparatus used in this research and the methodology employed are both nearly identical to the ones described in previous papers [ 24,271. The central part of the experimental device is a cell which essentially consists of two equal stainless steel cylindrical chambers having a length of 20.5 cm. The membrane was fixed between the chambers with the help of a PVC holder. Three viton Orings were employed to ensure the absence of leaks in the whole assembly. Each chamber was connected to a horizontal flow-pipe capillary. Both capillaries were placed at the same level in order to ensure the absence of pressure differ-

187 TABLE 1 Membranes employed in the present study and their principal manufacturer Membrane

TF-1000 Gelman TF-450 Gelman TF-200 Gelman FHLP Millipore GVHP Millipore


PTFE supported on polypropylene net PTFE supported on polypropylene net PTFE supported on polypropylene net PTFE supported on polyethylene PVDF unsupported


as specified by the

Pore radius” (pm)

Thickness (flm )

















void voiumeb


“Pore radius is defined with reference to smallest particle size which the membrane is able to retain. bFractional void volume is defined as ratio of inner void volume to total membrane volume. This quantity is not necessarily equal to the ratio pore surface/total membrane surface.

ences across the membrane. Each chamber with its corresponding capillary could be filled separately by means of a three-way stopcock connecting the chamber with a liquid reservoir located at a higher level. The membrane surface area exposed to the flow was q= 2.75 x 10m3m2. The temperature requirements were set by connecting each chamber, through the corresponding water jacket, to a different thermostat. In order to ensure the uniformity of temperatures in each chamber the water was stirred by a chaindrive cell magnetic stirrer assembly. Each set consisting of chamber, capillary and liquid reservoir was placed in a large air bath which provided a temperature controlled environmental housing for the non-isothermal experiments. Temperatures were measured with platinum resistance thermometers placed near both sides of the membrane. Under these conditions, the temperature errors were + 0.1 ‘C. Results and discussion Experimentally we have found that the measurements could be carried about across a minimum temperature difference of 5K. For smaller values (say 2K or 3K) the results are doubtful and the experimental errors become very important [ 231. Therefore 5K has been chosen as the temperature difference corresponding to the differential temperature gradient. All the flows were observed from the warmer to the cooler compartment. The value of the non-isothermal water flux was obtained, in each case, by


adjusting the experimental data (position of the meniscus versus time) to a linear function by a least squares method. As an example of the calculations carried out we quote the values of slopes (with their estimated standard deviations) in a particular case for each membrane: l

TF-1000: (3.33 ? 0.05) x 10e3 ml/set



(2.19 ? 0.02) x 10v3 ml/set



(2.65 ? 0.02) x lop3 ml/set



(2.64 + 0.08) x 10e3 ml/set



(1.07 2 0.05) x lop3 ml/set

In all these cases the mean temperature was 42.5’ C. The value of the correlation coefficient obtained in the most unfavourable case was 0.999 for runs of at least 15 points. This confirmed that the assumptions leading to eqn. (3) were correct within the range of temperature used. Determination of the kinetic constant c for non-isothermal flows, according to eqns. (2) and (3)) was made in two sets of experiments for each membrane. The temperature difference between both compartments was fixed at 5K. The stirring rate and the mean temperature were varied independently. In the first set, the stirring rate values were 0,50,100,150,200, and 250 rpm for the membranes TF-450 and GVHP; and 0,50,100,150,200,250,300, and 360 rpm for the membranes TF-200, TF-1000 and FHLP (in all cases the mean temperature was 42.5” C). The purpose of this set was to determine the influence of the stirring rate on the phenomenon. The results corresponding to this set are shown in Table 2. In the second set the mean temperature values were 32.5, 37.5, 42.5, 47.5, 52.5, and 57.5”C, and the stirring rate was 200 rpm for the membranes TF-450 and GVHP, and 300 rpm for the membranes TF-1000, TF200, and FHLP. The purpose of this set was to determine the influence of the mean temperature on the phenomenon. The results corresponding to this set are shown in Table 3. Several measurements were carried out in each case in order to check reproducibility and eliminate errors. The deviation of a given value from the mean value was 6% in the most unfavourable case, which confirms the accuracy of the measurements. Each one of the B’ values appearing in Tables 2 and 3 corresponds to the mean value. In all cases the dependence of B’ on u confirms that the unstirred liquid layers have not been eliminated. This result agrees with those of [ 24,271. Table 2 shows that the measured global non-isothermal permeability increases with stirring rate. As discussed above, eqn. (8)) a linear dependence of l/ (B’ -B& ) on l/u is expected. This linear dependence may be extrapolated to infinite stirring rate in order to obtain the intercept. As discussed above, this intercept may be identified with l/ (B-B;) (B being the value of B’ at infinite stirring rate). Nevertheless and in order to obtain the extrapolated value, an alternative method has been employed. It consists of fitting the values of u/ (B’ - Bb ) versus U.In this case a linear dependence is expected:



Now, the value of Bb, is obtained from the slope. The reason to work in this way is that the errors of extrapolation are lower in this case. In order to test the validity of the last statement, we display in Fig. 2 the experimental data (B’, u) for all the membranes. These data have been adjusted by a least squares method to eqns. (8) and (12). The results of both adjustments are displayed in Fig. 2 too. A visual inspection of the curves suggests that the experimental data are better adjusted by eqn. (12). There is a reason that explains this fact. In eqn. (8) the values of l/u are irregularly spread along the n-axis and consequently the values of B’ measured for the lower stirring rates are given more weight. TABLE 2 Differential non-isothermal permeability ( x lo6 mol-m-‘-see -‘-K--l) for the different values of stirring rate. The row corresponding to v = co has been obtained by extrapolation Membrane



TF-1000 T=42.5”C

TF-450 T=45.0°C

TF-200 T=42.5”C

FHLP T’=42.5”C


50 100 150 200 250 300 360

0.40 f 0.02 0.95 + 0.03 1.47 f 0.05 1.83+0.02 2.04 k 0.04 2.25 f 0.04 2.39 k 0.03 2.68 k 0.05

0.41 f 0.04 0.89 + 0.01 1.26f0.01 1.47 * 0.03 1.62 + 0.03 1.78f0.02 -

0.34 f 0.01 0.81 k 0.03 1.27 + 0.02 1.52 k 0.01 1.62 + 0.04 1.74kO.02 1.91 k 0.01 2.01+ 0.03

0.27 k 0.04 0.58 + 0.05 0.87 + 0.04 1.06+0.03 1.20+0.08 1.30 ?I 0.06 1.39 * 0.04 1.46 + 0.06

0.10 !I 0.01 0.23 k 0.02 0.40 + 0.03 0.515 0.02 0.54 + 0.04 0.62 + 0.02


4.5 f 0.3


3.0 + 0.2

2.37 kO.13

1.7 * 0.7



TABLE 3 Differential non-isothermal permeability ( x lo6 mol-m-‘-set-I-K-‘) mean temperature. The stirring rate is indicated in each case

for the different values of

Membrane v(rpm)







TF- 1000 TF-450 TF-200 FHLP GVHP

3.OlkO.02 2.51kO.03 2.88kO.01 2.03kO.02 0.53kO.01

2.71f0.02 2.26kO.04 2.64f0.07 1.84kO.02 0.47LO.01

2.61+0.01 1.76kO.01 2.23kO.02 1.65f0.04 0.39kO.01

2.39kO.03 1.57kO.01 1.91kO.01 1.39kO.04 0.34f0.02

2.23kO.07 1.33LO.01 1.671kO.02 1.23kO.07 0.29kO.01

1.98kO.01 1.16kO.02 1.39TO.02 1.04f0.02 0.23+0.01

300 200 300 300 200

Fig. 2. Variation

of non-isothermal


( * 1 TF-200; (0 ) FHLP and ( q ) GVHP.

with stirring rate.

( l ) TF-1000;

( + ) TF-450


In order to be more rigourous, the fitness of both adjustments has been checked by a X2-method. Table 4 shows the x2 values at the minima calculated by eqns. (8) and (12) for the membranes. In all these cases the x2 values are lower at the minimum calculated by eqn. (12). This fact suggests that the adjustment corresponding to eqn. (12) is better than the one corresponding to eqn. (8). On the other hand, when we compare Table 4 with the x2-Tables we observe that the results are as follows: for the membranes TF-1000, TF-200, FHLP and GVHP we have a confidence degree about the unfitness of eqn. (8) of, at least, 90%. This value is 80% for the membrane TF-450. Finally, Table 3 shows that there is an increase of measured global nonisothermal permeability with mean temperature. This trend may be expressed by a classical Arrhenius function [ 22,231: B’ =B” exp ( -E/RF), where E is the apparent activation energy for matter transport under a temperature gradient, and T is the mean absolute temperature. Alternatively we may use eqn. (11) to relate B’ with ii. In order to check which of the models is the most appropriate in our case, the experimental data (B’, pi) have been adjusted (by a least squares method) to the functions: l

1nB’ =x-y/p;



(Arrhenius model)

=x’ -y’/F;

(eqn. (11) model)

The results appear in Table 5. In the second and third columns appear the values of the apparent activation energies and of the correlation coefficients obtained in the Arrhenius model. In the fourth and fifth columns appear the values of L and of the correlation coefficients obtained from eqn. (11). In all cases the correlation coefficients obtained from eqn. (11) are a little better than those obtained from the Arrhenius model. Nevertheless the difference is very small and does not seem significant. In Ref. [ 191 the apparent activation energies for similar PTFE membranes were calculated. The values varied between 2.0 x lo4 J/mol and 3.6 X lo4 J/mol (the mean temperature was varied between 20’ C and 70 oC ) . On the other side our values may be compared with those found in the literature for other types TABLE 4 Values of x2 obtained at the minima calculated by eqns. (8) and (12). The B’ values are those of Table 2 Membrane


for eqn. (8)

TF-1000 TF-450 TF-200 FHLP GVHP

21.2 3.33 23.2 15.5 13.3

&in for eqn. (12) 8.12 1.11 8.75 3.33 3.33



Variation of differential temperature: Correlation Membrane

non-isothermal permeability ( X 10” mol-m-‘-set-‘-Km’) coefficients and parameters of both models




with mean


EX lo4 J-mol-’


LX lo4 J-mol-’


TF-1000 TF-450

1.33 * 0.08 2.671!10.14

0.993 0.994

1.86zkO.01 3.2OkO.15



2.49 I! 0.09

0.998 0.997

3.01 z!I0.08


2.27 LO.08 2.77kO.11


2.80 k 0.08 3.30 k 0.09

0.996 0.999 0.998 0.998

of membranes. For instance, Mengual et al. [24] found E values of 1.9x lo* J/ mol for two cellulose acetate membranes. Rastogi and Singh [9] obtained an E value of 3.43 x lo4 J/mol for a cellophane 600 membrane. Dariel and Kedem [8] found an E value of 2.80 x lo4 J/mol for another cellulose acetate membrane. Bellucci et al. [12], obtained a value of 2.01~ lo4 J/mol for an AP20 Millipore membrane. On the other hand the increase of non-isothermal permeability with mean temperature agrees with the results of Refs. [8,9,19,20,24], but disagrees with those of Refs. [5,7]. List of symbols A

proportionality constant (mol-K-m- ‘-sec- ’ ) global non-isothermal permeability (mol-m-‘-sec-‘-K-l) measured global non-isothermal permeability (mol-mkinetic constant (m3-set-‘) membrane thickness (m) E apparent activation energy (J-mol- ’ ) h film heat transfer coefficient (W-m-2-K-‘) H effective heat transfer coefficient (W-m-2-K-‘) J flow of matter (mol-K- ’ ) L heat of evaporation (J-mol-’ ) M molar mass (kg-mol-’ ) PV vapour pressure (Pa) cross section of the membrane (m”) i gas constant (J-mol-‘-K-l) T temperature (K) t time (set) V volume (m”) stirring rate (rpm) adjustment parameter (m-set-K-mol’)

B B’



‘-sec- l-K_ ’ )


X’ Y Y’ a c P

adjustment parameter adjustment parameter adjustment parameter adjustment parameter fractional void volume density ( kg-me3)

(m-K-mol-’ ) (m-K-mol-’ ) (m-set-K-mol-’ (J-m-‘-K-’ )


References 1 2 3 4 5 6 7 8 9 10 11

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17 18 19

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