Osmotic distillation through porous hydrophobic membranes

Journal of Membrane hence, 82 (1993) 129-140 Elsevrer Science Publishers B V , Amsterdam

129

Osmotic distillation through porous hydrophobic membranes

Juan I. Mengual*, Jose M. Ortiz de Z&rate, Luis Peiia and Armando Vela’zquez Departamento de Fikca Apltcada I (Termologia), Facultad de Fikca, Unwerstdad Complutense, 28040 Madrid (Spawn)

(Received December 17,1992, accepted m revrsed form March 4,1993)

Abstract

Osmotrc drstrllatron through porous hydrophobic membranes has been studied theoretmally and experimentally. In partmular the influence of the unstarred drffiion boundary layer has been determmed A model has been developed which descrrbes the contrrbutrons of solutron concentratron and strrrmg rate The influence of these and other relevant parameters 1s discussed

Key words osmotm drstillatron, membrane drstrllatron, porous hydrophobic membranes

Introduction

Non-isothermal transport of water through microporous hydrophobic partitions has being studied since the mid 1960s. The process was called “membrane distillation” [l-8]. Later, in the 19808, it was suggested that the same kind of membranes could be applied to a novel process termed “osmotic distillation” [g-12]. In some aspects both phenomena may be considered to be closely related, although there are some remarkable differences. In both cases, it is absolutely necessary to maintain a water vapour pressure difference across the membrane pores, in order to get the thermodynamic force causing the transport process, which is a difference in water chemical potential. However, the physical origin of that vapour pressure difference is quite different. In the case of membrane distillation, a temperature difference induces a

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corresponding difference in vapour pressure at both ends of the membrane pores, whereas, in the case of osmotic distillation, this difference is due to a difference in the composition of the bulk phases adjoining the membrane at both sides.

Some references to this new phenomenon, osmotic distillation, may be found in papers concerning membrane distillation experiments carried out with different aqueous solutions [5,13-171. There it was reported that, when the temperature difference was maintained constant, the membrane distillation flux de- creased upon increasing solute concentration difference. In experiments with the highest concentration hfferences, water fluxes were observed even from the cold to the hot side. This behaviour is consistent with the expected decrease in the water vapour pressure at the end of the pores, and therefore in the total driving force through the membrane. In Refs. [ 5,7-131 the effect of the concentration difference was

130 J I Mengual et al /J Membrane SCL 82 (1993) 129-140

considered only as a factor affecting the membrane distillation flux, not as a new phenomenon and, therefore, it did not receive a special denomination. Nevertheless, in Refs. [g-12 J water fluxes were observed, in isothermal conditions, when concentration differences were maintained between both sides of the membrane. These water fluxes originated from an apparent osmotic pressure, which gave reason to the name “osmotic dstillation”.

The scarce literature on osmotic distillation refers mainly to its practical aspects such as the concentration of various liquids (milk, fruit juices, etc. ). The goal of this paper is rather different from that of Refs. [g-12]. Our attention focusses on the physical-chemical aspects of osmotic distillation, instead of their practical applications. A model has been developed, which permits the study, both theoretically and experimentally, of the influence of concentration polarization on the transport process With this in mind, the influence on the phenomenon of some significant parameters such as solution concentration, stirring rate, and mean temperature is analyzed. The experiments have been carried out with five commercial porous hydrophobic membranes and using pure water and aqueous solutions of sodium chloride.

Theory

The system to be studied consists of a porous hydrophobic membrane that is held between two well stirred liquid subsystems, I and II. Subsystem I is pure water and subsystem II is an aqueous solution of sodium chloride, both maintained at the same temperature. The pres- ence of a non-volatile solute at subsystem II re- duces the water vapour pressure of this subsystem, and, consequently, a water vapour pressure difference between both sides of the membrane pores appears. Mass transfer is expected to oc- cur by convective and/or diffusive transport of water vapour across the membrane pores, the

driving force being the difference in vapour pressure.

The volume flux may be explained by three different models: the Knudsen model, the Polseuille model and the molecular diffusion model. The problem of deciding which of these options is applicable has been discussed in Refs [ 11,171. Each of the models has its own hmi- tations and, consequently, can only be used under certam conditions. For instance, the use of the Knudsen or Polseuille models involves considerations about the relative size of the mean free molecular path of the water vapour, as compared with the pore radius, while the use of the diffusive model requires that the partial pressure of air trapped within the pores is not too low. In any case, the three models suggest a linear dependence of volume flux on vapour pressure dfference:

J=CAP,,, (1)

where J IS the volume flux per unit surface area of the membrane, A?‘,,, is the water vapour pressure difference between both ends of the membrane pores, and C is a phenomenological coefficient valid for the system. In what follows coefficient C will be called osmotic distillation coefficient. Obviously the range of validity of the above equation must be determined by the experiments.

Equation (1) cannot be used, in this form, to study the transport process, due to the existence of concentration polarization which affects the flux values in the case of multlcom- ponent systems. The concentration polarization, m the present case, originates from the existence of an unstirred diffusional boundary layer adjoining the membrane on the solution side. As a consequence, a part of the externally maintained concentration difference is dissi- pated in that solution layer. In this way the concentration at the interphase membrane- unstirred layer, c,, is different from that of the well-stirred bulk phase, cb (see Fig. 1). This

J Z Mengual et al jJ Membrane Set 82 (1993) 129-140 131

C

-i-+Jc(x+dx) I

cb

BULK

I I I I

0 X x t dx 5

Fig 1 Schematic chagram of the boundary layer

leads to lower transmembrane fluxes than expected. Thus eqn. ( 1) could be rewritten:

J=CAP,,,=C’AP, (2)

where APb is the water vapour pressure difference corresponding to the bulk concentration and to pure water, and C’ is the measured or apparent osmotic distillation coefficient. As will be seen later, the experiments confirmed that the measured volume flux and, consequently, the apparent coefficient C’ depend on (a) the membrane, (b) the concentration of the solution, (c) the sturmg rate, and (d) the temperature. Therefore, it would be very interesting to find a relationship between both coefficients C and C’ by means of experimental parameters of the system (solution concentration, stirring rate, etc. ).

In what follows we will analyze the negative contribution of the unstirred solution layer to the transport process. The simplest model of a concentration unstirred layer is the Nernst film

model [ 18,191. Following this, the unstirred layer is a thin liquid film, which separates the membrane surface from the well-stirred bulk phase. Eddy diffusion is assumed to be negligible, thus limiting mass transport within the film to diffusion and convection mechanisms. In this way water and solute transport through the film take place perpemhcularly to the membrane surface. This situation is illustrated in Fig. 1, where a differential slab, within the unstirred layer, has been considered. Let c(x) be the solution concentration at a distance x from the membrane surface, and D the &ffusion coefficient. The mcoming solute fluxes, across imaginary planes located at x and x+ dx, are Jc(x) and -D(dc/dr),+, respectively. &ml- larly, the outcoming solute fluxes, across the same planes, are -D(dc/d.r). and Jc(x+dx) respectively. Once the steady state has been es- tablished, the solute flux is determined by the followmg differential equation [ 20,211, which is valid as a first approximation:

132 J I Mengual et al /J Membrane Set 82 (1993) 129-140

d2c J dc -- ---= d.X2 Ddx O

The general solution of eqn. (3) is:

c(x)=A+Bexp

(3)

where A and B are constants that may be determined with the help of the following boundary conditions, that refer to the membrane surface (x=0): c=c,, and D(dc/&),= Jc, This last relationship is due to the fact that the solute is non-volatile.

Taking into account the above boundary conditions, eqn. (4) yields: B = c, and A = 0. In order to obtain a relationship between the known bulk concentration, cb, and the un- known concentration at the membrane surface, c,, we assume that the solution concentration at the interphase unstirred layer/bulk phase (~=a) is just cb. In this case:

cb = C, eXp (J/k) (5)

In eqn. (5) the mass transfer coefficient for the unstirred layer, k=D/S, has been intro- duced as usual [ 19,221. This last equation may be considered equivalent to the so-called “fun- damental equation for concentration polarization”, that appears in papers on microfiltra- tion, ultrafiltration, and reverse osmosis. It is worth quoting that, in those cases, the volume flux leads to situations where c, > cb, just the opposite of the present case, where c, < cb (see Fig. 1).

As pointed out previously, we are looking for a relationship between the osmotic destillation coefficients C and C’. At this point, once the concentration profile in the layer has been determined, it is necessary to make some approx- imations. Let us assume that the flux is much lower than the mass transfer coefficient for the unstirred layer, which implies that c,,, and cb are not very different (see eqn. 5). In this case the

following first order approximation may be used:

cc, -cb) ulk

(6)

where all the symbols have their usual mean- ing. The fitness of the assumption leading to eqn. (6) will be determined by the experiments.

Assuming again that J<< k, and using the first order of the exponential term in a Taylor expansion, eqns. (l), (5), and (6) give:

(7)

Equation (7) provides the searched relationship between coefficients C and C’ C’ =C!k/ (k+ [dAP/dclbu&cb). In this expression, it may be seen that coefficient C’ depends on: (i ) the bulk concentration (through the factor cb), (ii) the nature of the solution (through the factor [d.AR/dclbUlk), (iii) the stirring rate (through the factor k), and (iv) the membrane (through the coefficient C) .

On the other hand, the experiments show that the measured flux and, therefore, the apparent osmotic distillation coefficient increase with the stirring rate. Equation (7) suggests that the dependence must be searched through the mass transfer coefficient for the unstirred layer. In order to evaluate this dependence, let us intro- duce the following hypothesis relating the mass transfer coefficient and the stirring rate, o:

k=a+bd (8)

where a and b are parameters, and y is a positive dimensionless number to be determined. This hypothesis is suggested by the well-known correlation, obtained from the study of diffusion boundary layer problems: ShccReY [ 221, Sh being the Sherwood number (proportional to the mass transfer coefficient), and Re the Rey- nolds number (proportional to the stirring

J I Menguul et al/J Membrane Set 82 (1993) 129-140 133

rate). Although this correlation between Sher- wood and Reynolds numbers is derived for crossflow situations, it should be pointed out that it was successfully applied by Malone and Anderson [22] for a diaphragm cell whose ge- ometry and stirring assembly were similar to those employed in the present paper. In addition, it is worth quoting that the appearance of the parameter a in eqn. (8) takes into account that a volume flux has been observed in the absence of stirring. This means a non-null mass transfer coefficient for w = 0. As above, the fitness of the hypothesis will be confirmed by the experiments.

Equations (7) and (8) lead to:

1 -=x+;y Jo,-Jo

(9)

where J, is the volume flux, measured at stirring rate o, and Jo is the volume flux without stirring. X and Y are complicated functions de- pending on a, b, and known quantities such as temperature, solution concentration, etc. It is worth mentioning that the relation between flux and stirring rate proposed in eqn. (9) is similar to those applied successfully in related problems of temperature polarization in thermoos- mosis [23] and in membrane distillation [ 24,251. The flux value that would correspond to an infinite stirring rate, J,, that is in absence of concentration polarization effects, may be obtained from the parameter X. Conse- quently, the true osmotic distillation coefficient may be obtained as: C = (Jo + l/X) /A&,.

As a summary it may be stated that eqn. (7) and the hypothesis proposed in eqn. (8) pro- vide a method to study the influence of the concentration polarization on the transport phenomenon. With this method the dependence of flux on solution concentration and stirring rate can be analyzed by means of eqns. (7) and (9) respectively.

On the other hand, in order to explain the observed dependence of the volume flux on

temperature, it is necessary to make some thermodynamic considerations. As is well known, the water vapour pressure in subsystem I (pure side) increases with absolute temperature. This trend is usually described by using an Arrhen- ius type of dependence. The water vapour pressure m subsystem II (solution side) may be written as the vapour pressure corresponding to pure water multiplied by the water activity in the solution. In standard thermodynamic tables it may be seen that, in the studied concentration range, the activity may be considered to be practically non-dependent on temperature. Therefore, the dependence of water vapour pressure on absolute temperature may be assumed to be similar for both subsystems. That means that an Arrhenius type of dependence between the volume flux, J, and the absolute temperature, T, is expected to be adequate:

J=J* exp( -L/RT) (10)

where J* is a parameter, L is the heat of vaporization of water, and R is the gas constant.

Experimental

(a) Materrals

Five commercial membranes, water repel- lent and industrially used in filtration processes, have been studied. The first is a Milh- pore PVDF (polyvinylidene) membrane, termed GVHP, the second is a Millipore PTFE (polytetrafluorethylene) membrane, termed FHLP, and the three other are Gelman PTFE membranes, termed TF-1000, TF-450, and TF- 200. All these membranes are grossly porous partitions, with irregular cavities going through the membrane thickness. Their principal char- acteristics, as specified by the manufacturers, are: l Millipore GVHP: nominal pore radius=0.2 pm; thickness = 125 pm; porosity = 70%. l Millipore FHLP: nominal pore radius= 0.2

134

pm; thickness = 175 pm; porosity = 70%. l Gelman TF-1ooO: nominal pore radius = 1 pm; thickness = 178 pm; porosity = 80%. l Gelman TF-450: nominal pore radius=O.45 pm; thickness = 178 pm; porosity = 80%. l Gelman TF-200: nominal pore radius=0.2 pm; thickness = 178 p; porosity = 80%.

The materials employed in the experiments were pure water (bi-distilled and de-ionized) and pure pro-analysis grade sodium chloride supplied by Probus.

(b) Apparatus

The experimental setup used was essentially similar to that described previously [ 23-251. Figure 2 illustrates the whole experimental assembly. The central part of the experimental device is a cell, which essentially consists of two equal cylindrical chambers having a length of 20.5 cm and made of stainless steel. The membrane was fixed between the chambers by means of a PVC holder. Three Viton O-rings were employed to ensure the absence of leaks in the whole assembly. The membrane surface area exposed to the flow was 27.5 cm’.

The temperature requirements were ob-

J I Mengual et al /J Membrane SCL 82 (1993) 129-140

tamed by connecting each chamber, through the corresponding water Jacket, to the same thermostat. In order to ensure uniformity of temperatures and concentrations m each chamber, the solutions were stirred by a chain-driven cell magnetic stirrer assembly. Temperatures were measured with platinum resistance thermom- eters, placed near both sides of the membrane. Under these conditions, it was checked that the temperature in each chamber was constant within + 0 1 ‘C.

Chamber II was filled with sodium chlonde aqueous solution and chamber I with pure water. The solution chamber was connected to a pipette and the other to a water reservoir. Both pipette and reservoir were placed at the same level in order to ensure the absence of hydro- static pressure differences across the membrane. At the beginning of each experiment, a variable osmotic distillation flux was observed. After a short time this flux became stabilized and was then measured. Solution concentration was measured at the end of each experiment by taking a sample from the chamber. The procedure was a standard chemical titration for the chloride ion (Mohr titration). The error of these measurements is ? 0.02 mol/l. The mea-

TO THERMOSTAT

FROM THERMOSTAT

Fig 2 Expenmental assembly (M) membrane, (H) membrane bolder, (MS) magnetic stirrer, (PM) propelling magnet, (T) thermometer, (TJ) thermostatizedjacket

J I Mengual et al /J Membrane SCL 82 (1993) 129-140 135

surements confirm that the variation in the solution concentration may be considered negligible. This result is not unexpected, taking into account that the volume flowing into the chamber is much smaller than the chamber volume itself.

Results and discussion

The value of the volume flux was obtained, in each case, by adjusting the experimental data (volume flowing into the chamber II versus time ) to a linear function by a X2-minimization procedure. As an example of the calculations carried out, we quote the values of the slopes (with their estimated standard deviations), for each membrane, in the case of solution concentration 3 M, stirring rate 200 rpm and temperature 40 ’ C: l Millipore GVHP, (7.97 2 0.03) x 10-l’ m3/ sec. l Millipore FHLP, (12.56 + 0.04) x lo-” m”/ sec. l Gelman TF-1000, (19.96 20.03) x 10-l’ m”/ sec. l Gelman TF-450, (10.73 5 0.05) x lo-” m”/ sec. l Gelman TF-200, (13.1420.04) x lo-” m”/ sec.

Three sets of experiments were carried out for each membrane. In the first set the chosen stirring rate was 200 rpm and the temperature was fixed at 40 o C; the solution concentration was varied between the values 0.5; 1; 1.5; 2; 2.5; 3; 4; and 5 M. The purpose of this set was to determine the influence of the concentration on the phenomenon. In the second set, the temperature was 40’ C and the solution concentration was 3 M; the stirring rate was varied between the values 0; 50; 100; 150; 200; 250; 300; and 350 rpm The purpose of this set was to determine the influence of the stirring rate on the phenomenon. In the third set, the sturmg rate was fixed at 200 rpm and the solution con-

centration was chosen 3 A$ the temperature was varied between 10°C and 6O”C, with increases of 10°C. The purpose of this set was to deter- mme the influence of the temperature on the phenomenon.

The results corresponding to the three sets of measurements appear in Tables 1 and 2 and m Fig. 3 respectively. Several measurements were carried out in each case, in order to check reproducibility and eliminate errors. The de- viation of a given value from the mean value was 5% in the most unfavourable case, which confirms the accuracy of the measurements. Each one of the fluxes appearing in Tables 1 and 2 and m Fig 3 corresponds to the mean value.

Table 1 shows that the measured flux increases with the solution concentration over the range studied (0.5 M to 5 M). As stated m the Theory section, the dependence of fluxes on solution concentration may be described by means of eqn. (7)) which includes, znter &a, the values of AP,., and [ dAP/dc Ibulk. These values were obtained from standard thermodynamic Ta- bles [ 261, by means of a quadratic interpola- tion of the data. In order to confirm the dependence proposed m eqn. (7)) the experimental points {J,cb} were adjusted by using a two-parameter (k and C) non-linear X2-munmization procedure. The results are shown in Fig. 4. A visual inspection of this figure confirms that the fitting is adequate. In addition, the procedure allows us to obtain numerical values for k and C. These values are displayed in the two last columns of Table 1. As a matter of fact, it is worth noting that the values of k are similar (in a range of 2 15% from the mean value) for all studed membranes, which is not an unexpected result if one takes into account that k is a property of the boundary layer and not of the membrane used. In addition, according to Ta- ble 1, the values of k are about one order of magnitude higher than those of J. This fact

136

TABLE 1

J I Mengual et al /J Membrane Scr 82 (1993) 129-140

Volume fluxes per unit area ( x 10s m-set-‘), at various molar concentrations The last columns show the values of parameters k and C

Membrane Molar concentration (M) kx10’ cx 10” (m-sec- ’ ) (m3-set-‘-N-l)

05 1 15 2 25 3 4 5

GVHP 0 462 0 855 131 176 2 28 2 90 4 10 5 14 68f15 345fO07 FHLP 0 78 123 2 04 2 94 3 50 4 57 5 94 7 51 54+8 553f013 TF-1000 0 898 2 07 2 97 3 94 491 6 17 8 21 10 78 63f8 811f014 TF-450 0051 135 205 280 3 54 3 90 5 89 759 64f7 538+020 TF-200 0 767 143 2 18 3 10 3 78 4 78 609 836 73+14 578fO22

TABLE 2

Volume fluxes per umt area ( x 10s m-set-I), at vanous stmng rates The last columns show the values of parameters J,, y, and C

Membrane Stirrmg rate (rpm ) J,xlO’ Y CXlO” (m-set-‘) (m3-set-‘-N-l)

0 50 100 150 200 250 300 350

GVHP 223 254 274 281 290 293 295 301 316fO08 1 llf0 17 3 64+009 FHLP 3.19 371 405 430 457 465 481 489 59 f04 102+017 66 f05 TF-1000 408 471 536 574 617 631 655 679 804+05 119f014 92 +06 TF-450 2 31 2 89 3 26 3 54 3 90 3 98 406 4 14 5 OO+O 20 109fOlO 575f023 TF-200 338 400 4.39 455 478 491 499 502 564fO25 102+013 65 f03

confirms that the starting assumption leading to eqns. (6) and (7) was adequate.

Table 2 shows that the volume flux increases, as expected, with the stirring rate. The experimental results {J;o} have been fitted to eqn. (9) by means of a non-linear X2-minimization method. The results corresponding to the fitting procedure are displayed in Fig. 5. A visual inspection of this figure confms that the fitting is adequate. Extrapolation of these curves to o+oo gives the values of J,. In addition, a new value for coefficient C may be obtained as the quotient between J, and A&. The values of J, (obtained by extrapolation), y (obtained from the fitting procedure), and C appear in the three last columns of Table 2. Two com- ments may be made in this respect: First, the effect of concentration polarization is very im-

portant (the numerical values of the fluxes may be increased by up to 40-50% when comparing the results corresponding to absence of stirring with those obtained by extrapolation to an infinite stirring rate), and, second, the values of y are similar for all the membranes, which is an expected result if one takes into account that y is a property of the boundary layer and not of the membrane.

The values of coefficient C reported in Ta- bles 1 and 2 have been obtained by different methods, and may be compared with one another for the same membrane. A visual inspection shows (taking into account the errors in- herent in all extrapolation processes) that the agreement is acceptable, the largest discrep- ancy being 22%.

Finally Fig. 3 shows that the measured vol-

J I Mengual et al /J Membrane See 82 (1993) 129-140

12 -I

o/ I , I I I I 1 / 1 I ’ 1 0 10 20 30 40 50 60

Temperature (“C)

Fig 3 Dependence of volume flux per unit area with tem- Fig 5 Dependence of volume flux per unit area with stlr- perature. (0) GVHP, (0) FHLP, (0 ) TF-1000, ( * ) TF- rmg rate ( l ) GVHP, (0) FHLP, (0) TF-1000, (* ) TF- 450, (A ) TF-200 The contmuous lmes are the fit curves 450, ( A ) TF’-200 The contmuous lines are the fit curves of the expenmental points to eqn (10) of the experimental pomts to eqn (9 )

4

Fig 4 Dependence of volume flux per unit area with molar concentration (0 ) GVHP, (0) FHLP, ( q ) TF-1000, ( * ) TF-450, (A ) TF-200 The contmuous lmes are the fit curves of the experimental pomta to eqn (7)

ume flux increases with the temperature. This fact was also reported previously [ 121 for other systems (fruit juices and SR 72 PTFE membrane modules), but only with two tempera-

,,I ,I ,(,I,8 ,I ,r 0 50 100 150 200 250 300 350

Stm-mg rate (rpm)

4oc

137

)

tures (29’ C and 40’ C ) being considered. The classical Arrhenius type of dependence, given by eqn. (lo), was found to fit well the experimental data, as can be seen in Fig. 3. A visual inspection of this figure suggests that the experimental points are fitted almost equally well for all the membranes On the other hand, the heat of vaporization of water may be obtained from the exponent of the exponential term. The values obtained are (3.48 2 0.10) x lo4 J/mol; (2.9150.17) x lo4 J/mol; (3.2950.10) x lo4 J/mol; (3.0750.09) x lo4 J/mol; and (3.05 2 0.12) x lo4 J/mol, for membranes Mil- lipore GVHP, Mlllipore FHLP, Gelman TF- 1000, Gelman TF-450, and Gelman TF-200 respectively. These values may be compared with the 4.323 x lo4 J/mol, that has been obtained from the data of water vapour pressures reported in the literature [26] (at 40°C). The accordance may be considered acceptable.

There is still another factor to be taken into account. The vapour flux takes place through evaporation on one side of each one of the membrane pores and condensation at the op-

138

posite side. This results in a decrease of the temperature at the pure water side, and an increase of the temperature at the solution side. Hence, a temperature difference between both sides IS created, that gives rise to a decrease m the original water vapour pressure difference. This problem has been considered elsewhere [ 11,121. In the present case, a rough estimation of the numerical influence of this factor may be made by assuming that there is a heat transfer, through the membrane, due to the created temperature difference (see Appendix). If one assumes that the heat transfer takes place by a convection mechanism, the temperature difference may be estimated as N 0.8 K, for the largest flux. On the other hand, if one assumes that the heat transfer is due to a conduction mechanism, the temperature difference may be estimated as -0.5 K, for the same flux. In both cases the membrane distillation fluxes originated are much smaller [ 241 than the osmotic distillation fluxes reported in the present paper, and, consequently, the influence of the created temperature difference may be considered negligible.

Conclusions

(1) The osmotic distillation process has been studied with five porous hydrophobic membranes. The influence of some relevant parameters, such as solution concentration, stirring rate and temperature, has been analyzed. It has been observed that the fluxes increase with solution concentration, sturing rate and temperature.

(2 ) A model has been proposed that permits to study, theoretically and experimentally, the influence of the diffusion boundary layer on the osmotic distillation. The model permits to obtain two relationships showing the explicit dependence of volume flux on bulk concentration and on stirring rate. The agreement between the model predictions and the experimental re-

J I Mengual et al /J Membrane See 82 (1993) 129-140

sults may be considered satisfactory (3) Some characteristic parameters of the

boundary layer have been calculated, such as the mass transfer coefficient, k, and the dimensionless number y.

(4 ) The osmotic distillation coefficient, which is a characteristic parameter of the membrane, has been calculated by two methods, with an acceptable agreement with one another.

Acknowledgments

This work has been financially supported by the Comunidad Autonoma de Madrid (242/92 ) .

List of symbols

Ti b

B

c

C

C’

D J J* k L P R Re Sh T

Gc Y

parameter (m/set ) parameter (mol/l) parameter (m) parameter (mol/l) molar concentration (mol/l) osmotic distillation coefficient ( m3/N- set ) global osmotic distillation coefficient ( m3/N-sec ) diffusion coefficient (m’/sec) volume flux per unit area (m/set) parameter (m/set) mass transfer coefficient (m/set )

heat of vaporization of water (J/mol) pressure (Pa) gas constant (J/mol-K) Reynolds number Sherwood number absolute temperature (K ) spatial coordinate (m) parameter (set/m) parameter (l/m)

Greek letters

6 layer thickness (m)

JZ Mengual et al/J Membrane Scz 82 (1993) 129-140

Y dimensionless number 0 stirring rate (l/set)

Subscripts

b at the bulk phase

zp at the membrane surface m absence of stirring

w at stirring rate Lr) 0 extrapolated to infinite stirring rate

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140

Appendix

Estimation of the created temperature difference:

(a) Conductwn mecharusm*

JL=KSAT/G

where S is the surface of the membrane exposed to flow, 27.5 cm2, 6 is the thickness of the layer, obtained as &D/k. Values of L, K (ther-

J I Mengual et al /J Membrane SCL 82 (1993) 129-140

ma1 conductivity of water), and D are obtained from tables.

(b) Convection mechanzsm

JL=hSAT

Values of h (film heat transfer coefficient) are obtained from the data of Ref. [23], h= 1,900 W_m-2_R-1.

In both cases AT represents the half of the created temperature difference, due to the processes of evaporation and condensation of water at the membrane faces.

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Osmotic distillation through porous hydrophobic membranes