585
Ind. Eng. Chem. Res. 1995,34, 585-590
Temperature Polarization Coefficients in Membrane Distillation Armando Veldzquez and Juan I. Mengual* Departamento de Fisica Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
Membrane distillation experiments have been carried out by using two Gelman PTFE porous hydrophobic membranes and various sodium chloride aqueous solutions. The influence of stirring rate on the experimental results has been discussed by means of the so-called temperature polarization coefficient. The influence of the remaining relevant parameters, such as solute concentration, mean temperature, and bulk temperature difference has been analyzed.
Introduction
As is well-known, membrane distillation usually refers to nonisothermal transport of water, in vapor phase, through the pores of a hydrophobic membrane that separates two aqueous solutions. Due to the liquidrejecting properties of the membrane material, the liquid water cannot enter the membrane pores unless a hydrostatic pressure greater than the so-called liquid entry pressure of water for the porous partition is applied. In the absence of such hydrostatic pressure, liquid-vapor interfaces are formed on both sides of the membrane pores due to surface tension forces. Under these conditions, if a temperature difference is applied, a water vapor pressure difference is created on both interfaces. Evaporation takes place at the hot interface, and after vapor is transported through the pores, condensation takes place at the cold interface. In this way, a water flux occurs through the membrane in the direction of hot to cold. Obviously, for membrane distillation to proceed, it is essential that the liquid water be excluded from the pores. In the literature there are several papers on this subject (see for example, Drioli and Wu (19851, Jonsson et al. (1985), Kimura et al. (1987), Peiia et al. (1993), Sarti et al. (19851, and Schofield et al. (1987)). Most of them refer t o pure water or t o different aqueous solutions, as well as to various hydrophobic membrane materials such as poly(tetrafluoroethy1ene) (PTFE), poly(viny1idene fluoride) (PVDF),or polypropylene (PP). In the present paper, two PTFE membranes were studied. Each membrane was placed separating two sodium chloride aqueous solutions of the same composition. An externally maintained temperature difference induces the corresponding transport of water through it, via membrane distillation. This transport of water has been measured for various values of mean temperature, temperature difference, solution concentration, and stirring rate. The dependence of the measured flux on the stirring rate has been interpreted in the framework of the temperature polarization model (see, for example, Schofield et al. (1987)), and the so-called temperature polarization coefficient (TPC) has been calculated in each case. The dependence of the TPC on the remaining variables quoted above has been analyzed. Theory The system to be studied consists of a porous hydrophobic membrane, held between two aqueous solutions of a nonvolatile component (both having the same initial solute concentration). A fixed bulk temperature difference, ATb, is maintained in both subsystems, and the
* To whom all correspondence should be addressed.
corresponding transmembrane water vapor pressure difference is created. Mass transfer by membrane distillation takes place due to the convective and/or diffusive transport of water vapor across the membrane pores, the driving force being the difference in vapor pressure. The mass transfer may be explained according to three different possibilities (see Schofield et al. (1987)): the Knudsen flux model, the Poiseuille flux model, and the diffusive model. In any case, the three models suggest a linear relationship between the volume flux and the water vapor pressure difference:
J=CM
(1)
where J is the volume flux per unit surface area of the membrane, AP is the transmembrane water vapor pressure difference, and C is a phenomenological coefficient valid for the system. It is worth mentioning that the water flux, from hot to cold, causes, as a consequence, a progressive increase of the solute concentration on the hot side and a progressive decrease on the cold side. In this way, a growth in time concentration difference appears that contributes negatively t o the transmembrane water vapor pressure difference. This phenomenon is called “osmotic distillation” (see Johnson et al. (1989) and Mengual et al. (1993)), and in this paper we are not interested in it. Consequently, the water fluxes will be considered only at the initial times, in such a way that the water vapor pressure difference is only due to nonisothermal effects and does not vary appreciably in time. That means that, in these initial times, the contribution of the osmotic distillation may be considered negligible. The vapor pressures at both liquid-vapor interfaces cannot be measured, but their difference may be expressed as a function of the corresponding transmembrane temperature difference, AT, by using standard thermodynamic procedures (Schofield et al., 1987). So, in a first approximation
J = BAT
(2)
where B is a new phenomenological coefficient called net nonisothermal coefficient. On the other hand, the literature on the subject and the experiments presented in this paper indicate that, in the cases of pure water and aqueous solutions, the volume flux depends on the stirring rate, w , the mean temperature, T , the bulk temperature difference, ATb, and the solute concentration, c. The dependence of J on w has been studied, in the case of pure water, by Schofield et al. (1987), Ortiz de Ziirate et al., (1990), etc., and the experimental results have been interpreted with reference to the concept of
Q888-5885/95/2634-Q585~Q9.QQfQ 0 1995 American Chemical Society
586 Ind. Eng. Chem. Res., Vol. 34,No. 2, 1995
“temperature polarization”. According to this idea, the measured fluxes are affected by the presence of unstirred liquid layers adjoining the membrane at both sides. In other words, the temperature difference on the two membrane surfaces, AT, is not the same as the one corresponding to the well-stirred bulk phases, ATb. Part of this externally applied temperature difference is dissipated through the unstirred liquid layers. This effect, which is called temperature polarization, leads t o lower transmembrane fluxes than expected. Thus, eq 2 may be rewritten
J = BAT = B‘ATb
(3)
where B‘ is the apparent or global nonisothermal coefficient. The quotient (ATIATb) = (B‘IB) is the temperature polarization coefficient and quantifies the effect of the unstirred layers on the transport phenomenon. This coefficient represents the fraction of the externally applied thermal driving force that contributes to the mass transfer. In other words, the TPC is a measure of the efficiency of heat transfer from the wellmixed bulk solution to the membrane-solution interface. It is worth noting that this concept of temperature polarization was proposed in papers dealing with pure water. The effect of a nonvolatile solute in the feed is to reduce the vapor pressure, to alter the fluid dynamics through density and viscosity and, finally, t o influence the heat transfer through thermal conductivity and heat capacity. This problem has been considered and discussed by Schofield et al. (1990). As a consequence of its definition, the TPC depends, obviously, on the stirring rate, and it may also depend on the remaining variables such as mean temperature, bulk temperature difference, and solute concentration. That means that, in the absence of any other evidence, each one of the TPC values should be only valid for the specific values of those variables for which the TPC was calculated. One aim of this paper is to experimentally check these dependences. Several methods have been proposed to evaluate the temperature polarization effects (see, for example, Schofield et al. (1987) and Ortiz de Zarate et al. (1990)). The most widely used consists of measuring the mass flux at various values of the stirring rate and extrapolating the data to an infinite stirring rate. If one considers the heat transfer, in steady state, through the composite system layer-membrane-layer, the following relationship is reached (Ortiz de Zarate et al., 1991):
dh AT = AT (4) bdh 2k 2BLd where d is the membrane thickness, K is the membrane thermal conductivity, L is the heat of vaporization of
+ +
water, and h is the film heat transfer for the layer (when this equation is written, the effects of both layers is assumed to be equal). In order t o obtain a relationship between the flux and the stirring rate, a hypothesis must be stated, referring t o the dependence of h on o. The dependence we propose is
+
h = h, a d (5) where ho and a are two parameters and y is a positive dimensionless exponent. This kind of dependence is suggested by the well-known correlation, obtained from the study of heat transfer through thermal boundary layers Nu = Rer (see Kreith and Black (1980)),with Nu the Nusselt number (proportional to the heat transfer coefficient) and Re the Reynolds number (proportional
to the stirring rate). In addition, it is worth saying that the appearance of the parameter ho in eq 5 takes into account that there is a heat transfer through the layer even in the absence of stirring. Equations 3-5 lead to
where J, and JO are the volume fluxes measured with stirring rate w and without stirring respectively and X and Yare complicated functions depending on B , ho, etc. According t o eq 6, the volume flux increases with the stirring rate, as indicated by the experiments. The flux value that would correspond to an infinite stirring rate, J,, that is, in the absence of polarization effects, may be obtained from the parameter X. Later on, this possibility will be discussed. On the other hand, if one considers again that the solute concentration in both subsystems does not vary appreciabily in the initial time interval considered, the dependence of the measured flux on solute concentration is expected t o be linear for relatively high solute concentrations (in this case the activity coefficient is practically a constant), but it is expected to deviate from linearity for low concentrations (in this case the activity coefficient depends on concentration). Otherwise, the dependence of the volume flux, J, on mean temperature, T , has been considered by several authors (see, for example, Drioli and Wu (1985) and Schofield et al. (1987)). In these papers, an Arrhenius type of dependencewas considered. Nevertheless, Ortiz de Zarate et al. (1991) proposed a more accurate model in which the following relationship is reached:
J = ( l / P ) f?Xp(-L/RT)ATb
(7) where L is the heat of vaporization of water and R is the gas constant. Finally, eq 3 predicts a linear dependence of the volume flux, J, on the temperature differences, AT or ATb. This linear behavior has been widely checked in the literature concerning pure water or dilute aqueous solutions. This fact indicates that the nonisothermal coefficients B and B’ do not depend on the corresponding temperature differences in the ranges studied.
Experimental Section
(a) Materials. Two commercial PTFE membranes were studied. Membrane 1was a Gelman TF-200 and membrane 2 was a TF-1000. These membranes are porous, hydrophobic, and industrially used in filtration processes. Both have irregular cavities going through the membrane matrix. Their principal characteristics, as specified by the manufacturer, are as follows: membrane 1,nominal pore radius 0.2 pm, thickness 178pm, and porosity 80%; membrane 2, nominal pore radius 1 pm, thickness 178 pm, and porosity 80%. The materials employed in the experiments were pure proanalysis grade sodium chloride, supplied by Prolabo, and pure water, distilled and deionized. (b) Apparatus. The experimental setup used (see Figure 1)was substantially similar t o that described previously (Ortiz de Zarate et al., 1991). The central part of the experimental device is a cell, which essentially consists of two equal cylindrical chambers having a volume of 300 mL 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 there were no leaks in the whole
Ind. Eng. Chem. Res., Vol. 34, No. 2,1995 587 Table 1. Volume Fluxes per Unit Surface Area d s ) as a Function of Solute Concentrati_on,Stirring Rate, and Bulk Temperature Difference (T= 35 "C) for Membrane 1 ATb w ("C) (rpm) 0.0 5
10
15
TB
TB
Figure 1. Experimental setup: M, membrane; S, magnetic stirrer; PM, propelling magnet; TB, thermostatic bath; R, reservoir; B, buret, T, thermometer.
assembly. The membrane surface area exposed to the m2. flow was 2.75 x 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 and compositions inside each chamber, the solutions were stirred by a chaindriven cell magnetic stirrer assembly. The temperatures were measured with platinum resistance thermometers placed near both sides of the membrane. Under these conditions, the temperature was constant within kO.1 "C. Flux measurements were carried out by collecting the liquid flowing out of the cold chamber into a buret. These measurements were made only in the beginning of the experiments (that is, when just 1 mL of water has been collected in the buret) in order to avoid the contribution of the concentration variations in both subsystems t o the phenomenon, as was explained in the Theory section. For the sake of security, the solute concentration in both semicells was measured, at the beginning and at the end of each experiment, by means of standard chemical titration (Mohr titration). The error in these measurements is k0.005 mom.
Results and Discussion In all the cases studied, the temperature-drivenfluxes took place from the hot to the cold chamber, as occurred in the case of pure water. In each case, the value of the nonisothermal volume flux was obtained by adjusting the initial experimental data (volume flowing into the cold chamber versus time) to a linear function by a least-squares procedure. As an example of the calculations carried out, the value of the slope (with its estimated standard deviation) is indicated, in a particular case, for membrane 1. This value is (5.96 f 0.01) x m3/s (the stirring rate was 200 rpm, the mean temperature was 50 "C, the solution concentration was 3.5 mom, and the bulk temperature difference was 15 K). The value of the correlation coefficient obtained was 0.999, in the most unfavorable case, for runs of at least 10 points. This confirmed that the assumptions leading to eqs 1-3 were correct within the ranges of measuring time, concentration, temperature, and stirring rate used.
20
25
30
0 75 150 200 250 300 0 75 150 200 250 300 0 75 150 200 250 300 0 75 150 200 250 300 0 75 150 200 250 300 0 75 150 200 250 300
0.16 0.48 0.62 0.67 0.71 0.74 0.35 0.94 1.23 1.34 1.45 1.51 0.57 1.36 1.76 1.93 2.05 2.18 0.81 1.83 2.42 2.73 2.83 2.96 1.09 2.31 3.04 3.34 3.56 3.77 1.36 2.76 3.72 4.15 4.41 4.64
concentration ( m o m ) 0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.13 0.42 0.56 0.60 0.64 0.67 0.31 0.84 1.11 1.23 1.32 1.40 0.53 1.28 1.70 1.85 1.97 2.08 0.78 1.70 2.29 2.44 2.69 2.81 1.03 2.19 2.81 3.18 3.40 3.59 1.30 2.52 3.34 3.73 4.12 4.33
0.13 0.41 0.52 0.58 0.63 0.67 0.30 0.82 1.07 1.18 1.25 1.34 0.50 1.23 1.61 1.79 1.91 1.98 0.74 1.60 2.17 2.36 2.57 2.71 0.98 2.08 2.73 3.05 3.30 3.50 1.25 2.48 3.26 3.64 3.96 4.21
0.12 0.37 0.50 0.55 0.60 0.63 0.28 0.78 1.04 1.14 1.21 1.28 0.46 1.16 1.55 1.71 1.84 1.93 0.67 1.53 2.08 2.28 2.48 2.63 0.93 1.97 2.61 2.91 3.16 3.36 1.18 2.30 3.09 3.50 3.83 3.97
0.10 0.36 0.47 0.53 0.58 0.60 0.26 0.74 0.98 1.08 1.16 1.22 0.45 1.09 1.47 1.62 1.75 1.83 0.65 1.47 2.01 2.19 2.38 2.51 0.90 1.86 2.39 2.79 3.03 3.12 1.16 2.21 2.97 3.35 3.58 3.87
0.10 0.34 0.46 0.50 0.54 0.57 0.24 0.70 0.94 1.05 1.12 1.17 0.42 1.05 1.41 1.58 1.69 1.77 0.61 1.43 1.90 2.09 2.29 2.40 0.84 1.78 2.35 2.66 2.88 3.00 1.10 2.12 2.87 3.25 3.51 3.71
0.09 0.34 0.43 0.49 0.52 0.55 0.21 0.67 0.90 1.01 1.09 1.14 0.36 1.00 1.37 1.52 1.64 1.73 0.55 1.36 1.82 2.03 2.21 2.33 0.78 1.68 2.29 2.55 2.79 2.92 1.03 2.02 2.73 3.11 3.34 3.58
0.08 0.29 0.41 0.45 0.49 0.52 0.19 0.63 0.86 0.95 1.02 1.07 0.33 0.94 1.27 1.41 1.53 1.62 0.53 1.28 1.71 1.89 2.09 2.17 0.74 1.61 2.17 2.39 2.63 2.79 0.96 1.96 2.63 2.95 3.20 3.42
0.06 0.27 0.39 0.44 0.47 0.51 0.18 0.59 0.81 0.90 0.97 1.02 0.32 0.85 1.25 1.35 1.45 1.56 0.50 1.20 1.63 1.80 1.96 2.08 0.69 1.45 2.08 2.29 2.51 2.71 0.89 1.87 2.52 2.77 3.07 3.29
In the case of membrane 1, the fluxes were determined in two sets of experiments. In the first set, the mean temperature was fured at 35 "C and the stirring rate, the solution concentration, and the bulk temperature difference were varied independently. The stirring rate values were 0,75,150,200,250, and 300 rpm. The concentration was varied from 0 to 4 mom, with increases of 0.5 mom. The bulk temperature difference was varied from 5 to 30 K, with increases of 5 K. In the second set, the bulk temperature difference was fured at 15 K and the stirring rate, the solution concentration, and the mean temperature were varied independently. The stirring rate and solution concentration values were chosen as above, and the mean temperature was varied from 25 to 50 "C, with increases of 5 K. The results corresponding to membrane 1 appear in Tables 1 and 2. In the case of membrane 2, the fluxes were determined in one set of experiments. The bulk temperature difference was fixed at 15 K and the stirring rate, the solution concentration, and the mean temperature were varied independently. The solution concentration values were chosen as above, and the mean temperature was varied from 25 to 50 "C, with increases of 5 K. The stirring rate was varied between 0 and 300 rpm, with increases of 50 rpm. The results corresponding to membrane 2 appear in Table 3. Some measurements were carried out in each case for each one of the membranes in order to check reproducibility and eliminate errors. The deviation of a given value from the mean value was 8%in the most unfavorable case, which confirms the accuracy of the measure-
588 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 Table 2. Volume Fluxes per Unit Surface Area m/s) as a Function of Solute Concentration, Stirring Rate, and Mean Temperature (ATb = 15 K) for Membrane 1 T w ("C) (rpm) 0.0 25
30
35
40
45
50
0 75 150 200 250 300 0 75 150 200 250 300 0 75 150 200 250 300 0 75 150 200 250 300 0 75 150 200 250 300 0 75 150 200 250 300
0.38 0.92 1.19 1.29 1.36 1.45 0.48 1.12 1.45 1.60 1.69 1.76 0.57 1.36 1.76 1.93 2.05 2.18 0.66 1.53 2.06 2.25 2.41 2.58 0.77 1.76 2.42 2.65 2.88 3.09 0.85 2.12 2.80 3.07 3.29 3.49
0.5 0.37 0.87 1.15 1.25 1.33 1.39 0.44 1.08 1.42 1.55 1.65 1.74 0.53 1.28 1.70 1.85 1.97 2.08 0.61 1.48 2.00 2.17 2.33 2.46 0.75 1.70 2.30 2.49 2.68 2.87 0.79 2.04 2.70 3.03 3.22 3.38
concentration (moVL) 1.0 1.5 2.0 2.5 3.0 0.34 0.85 1.10 1.23 1.31 1.35 0.42 1.02 1.37 1.48 1.59 1.68 0.50 1.23 1.61 1.79 1.91 1.98 0.56 1.40 1.91 2.08 2.25 2.37 0.66 1.60 2.21 2.44 2.61 2.79 0.72 1.89 2.54 2.81 3.05 3.22
0.33 0.80 1.05 1.18 1.26 1.31 0.40 0.97 1.28 1.42 1.54 1.60 0.46 1.16 1.55 1.71 1.84 1.93 0.54 1.31 1.81 1.98 2.13 2.25 0.59 1.52 2.12 2.33 2.52 2.69 0.67 1.76 2.48 2.64 2.91 3.10
0.31 0.77 1.00 1.12 1.20 1.27 0.37 0.91 1.26 1.35 1.47 1.55 0.45 1.09 1.47 1.62 1.75 1.83 0.50 1.27 1.73 1.92 2.04 2.18 0.56 1.43 2.01 2.19 2.37 2.55 0.64 1.70 2.35 2.56 2.78 3.00
0.28 0.72 0.98 1.09 1.15 1.20 0.35 0.87 1.19 1.31 1.40 1.49 0.42 1.05 1.41 1.58 1.69 1.77 0.45 1.20 1.64 1.83 1.97 2.11 0.53 1.37 1.91 2.08 2.28 2.44 0.60 1.57 2.19 2.43 2.66 2.81
0.28 0.66 0.89 0.99 1.07 1.13 0.32 0.82 1.11 1.23 1.34 1.41 0.36 1.00 1.37 1.52 1.64 1.73 0.43 1.12 1.57 1.74 1.88 2.01 0.48 1.31 1.82 2.02 2.19 2.34 0.56 1.52 2.15 2.35 2.58 2.74
3.5
4.0
0.26 0.64 0.87 0.96 1.05 1.10 0.28 0.79 1.06 1.18 1.28 1.35 0.33 0.94 1.27 1.41 1.53 1.62 0.37 1.06 1.50 1.66 1.82 1.94 0.42 1.23 1.71 1.91 2.06 2.18 0.48 1.33 1.97 2.17 2.39 2.57
0.24 0.60 0.83 0.93 1.01 1.05 0.27 0.73 1.00 1.13 1.22 1.30 0.32 0.85 1.25 1.35 1.45 1.56 0.35 0.98 1.39 1.56 1.71 1.83 0.38 1.14 1.60 1.76 1.97 2.07 0.42 1.24 1.86 2.05 2.27 2.45
w ("C) (rum) 0.0
concentration (mom)
F
ments. Each one of the fluxes appearing in Tables 1-3 corresponds to the mean value. Table 1shows that the nonisothermal flux increases, in an approximately linear way, with the bulk temperature difference in the studied range from 5 to 30 K (with the exception corresponding to the lowest value ATb = 5 K). This means that the global nonisothermal permeability, B', which is obtained from the slope of the straight line, is independent of the bulk temperature difference, ATb, in practically the whole range. Tables 2 and 3 show that there is an increase of the nonisothermal flux with the mean temperature. This fact was also reported previously by Schofield et al. (1987) and Ortiz de Zarate et al. (1991) for various systems (pure water or aqueous solutions-hydrophobic membranes). The type of dependence given by eq 7 was found to fit the experimental data well. Figures 2 and 3 show the results of the adjustment in some representative cases. A visual inspection of these figures suggests that the pairs of experimental data are adequately fitted. In addition, Tables 1-3 state that the volume flux decreases when the solute concentration increases, which is an expected result. Tables 1-3 show that the measured nonisothermal flux always increases with the stirring rate. This behavior has been reported previously in other papers for similar membranes and permeating fluids (Schofield et al., 1987; Ortiz de Zarate et al., 1990,1991). In order to quantify this effect, that is, to calculate the TPC values, we assume that if the system were stirred with infinite speed (o w), neither unstirred layer would exist nor, consequently, would temperature polarization effects exist. In other words, the apparent nonisother-
-
Table 3. Volume Fluxes per Unit Surface Area m/s) as a Function of Solute Concentration,Stirring Rate, and Mean Temperature (A!&, = 15 K) for Membrane 2
25
30
35
40
45
50
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 0.54 0.48 0.42 0.40 0.37 0.35 0.33 0.31 0.29 50 1.08 0.97 0.84 0.82 0.78 0.74 0.70 0.64 0.58 100 1.42 1.29 1.18 1.12 1.07 1.03 0.97 0.86 0.82 150 1.66 1.51 1.35 1.32 1.26 1.21 1.17 1.03 0.98 200 1.81 1.68 1.50 1.45 1.40 1.36 1.28 1.14 1.07 250 1.95 1.81 1.64 1.58 1.51 1.46 1.40 1.24 1.20 300 2.05 1.89 1.75 1.67 1.60 1.55 1.49 1.32 1.28 0 0.59 0.55 0.51 0.51 0.46 0.43 0.39 0.35 0.34 50 1.24 1.11 1.03 0.97 0.93 0.89 0.81 0.73 0.67 100 1.66 1.48 1.39 1.28 1.25 1.19 1.13 1.00 0.97 150 1.93 1.71 1.64 1.52 1.48 1.42 1.34 1.21 1.17 200 2.13 1.90 1.81 1.70 1.65 1.58 1.51 1.36 1.31 250 2.25 2.03 1.94 1.84 1.78 1.71 1.62 1.49 1.43 300 2.37 2.14 2.06 1.94 1.88 1.79 1.72 1.59 1.53 0 0.69 0.65 0.61 0.55 0.53 0.47 0.45 0.42 0.39 50 1.40 1.29 1.21 1.11 1.08 1.02 0.95 0.85 0.79 100 1.87 1.72 1.60 1.47 1.46 1.39 1.31 1.17 1.11 150 2.15 2.01 1.89 1.76 1.71 1.67 1.55 1.41 1.33 200 2.36 2.20 2.11 1.95 1.92 1.85 1.75 1.58 1.48 250 2.51 2.37 2.26 2.13 2.05 2.00 1.89 1.73 1.62 300 2.66 2.50 2.38 2.25 2.17 2.11 2.00 1.83 1.74 0 0.78 0.75 0.69 0.61 0.59 0.55 0.53 0.45 0.43 50 1.54 1.48 1.36 1.26 1.22 1.13 1.09 0.94 0.89 100 2.07 1.96 1.84 1.70 1.64 1.53 1.47 1.33 1.24 150 2.41 2.34 2.16 2.04 1.97 1.85 1.81 1.62 1.49 200 2.67 2.57 2.43 2.24 2.15 2.10 2.01 1.83 1.70 250 2.85 2.75 2.62 2.44 2.39 2.26 2.19 2.02 1.87 300 3.01 2.92 2.77 2.60 2.50 2.39 2.34 2.18 2.01 0 0.88 0.81 0.77 0.71 0.66 0.61 0.59 0.52 0.47 50 1.71 1.66 1.59 1.49 1.44 1.35 1.25 1.10 1.00 100 2.27 2.23 2.05 1.94 1.91 1.81 1.65 1.48 1.42 150 2.67 2.61 2.50 2.34 2.28 2.15 2.00 1.83 1.75 200 2.95 2.90 2.80 2.63 2.56 2.38 2.28 2.10 1.93 250 3.17 3.11 2.93 2.82 2.73 2.58 2.42 2.24 2.15 300 3.33 3.29 3.15 2.95 2.87 2.73 2.59 2.40 2.29 0 0.96 0.91 0.85 0.79 0.72 0.67 0.65 0.56 0.51 50 1.99 1.92 1.77 1.68 1.57 1.44 1.38 1.24 1.10 100 2.63 2.59 2.33 2.22 2.10 1.98 1.87 1.73 1.58 150 3.13 2.99 2.87 2.72 2.60 2.35 2.27 2.08 1.90 200 3.47 3.32 3.12 2.90 2.87 2.65 2.53 2.35 2.16 250 3.72 3.59 3.41 3.17 3.03 2.89 2.77 2.59 2.38 300 3.91 3.75 3.60 3.40 3.28 3.09 2.93 2.79 2.54
A
3.20 -
f
/
2.80 2.40 h
,
@
2.00 -
0
z 1.60 5
1.20 -
0.80 0.40
I
I
I
I
I
I
I
I
3
Figure 2. Volume flux per unit surface area versus mean temperature a t various concentrations (w = 200 rpm; ATb = 15 K)for membrane 1: c = (*) 0 , ( 0 )1,( A ) 2, ( 0 )3, and (+) 4 m o m .
mal coefficient that corresponds to an infinite stirring rate may be equalized to the net nonisothermal coefficient. This fact suggests the following calculation pro-
Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 589 3.60 3.20 2.5 2.80
-
Tw
2.0
2.40 h
2.00
'i
1.5
E
p 1.60
0
0,
2
v
5
1.20
5
1.0
0.80
0.5
L_
0.40 0.00 15
20
25
30
40
35
T
45
50
55
1 I
60
0
("C)
Figure 3. Volume flux per unit surface area versus mean temperature at various stirring rates (c = 1.5 M; ATb = 15 K) for 100, and (*) 50 membrane 2: w = (*) 300, (0)250, (A) 150, (0) rpm.
Figure 5. Volume flux per-unit surface area versus stirring rate at various concentrations (T= 35 "C; ATb = 20 K) for membrane 2: c = (*) 0, (0)1, (A) 2, (0)3, and (*) 4 m o m .
Table 4. Values of J , d s ) and y as a Function of @lute Concentration and Bulk Temperature Difference (T= 35 "C) for Membrane 1
3.0
i
concentration ( m o m )
ATb
2.5
("C) 5
0.0 1.04 0.93 0.98 2.11 y 0.92 J , 3.21 y 1.24 J , 3.64 y 1.04 J , 5.27 y 1.21 J , 6.75
y J, 10 y J,
2.0
15
h
I y1
Ei 1.5
20
0
z
25
v
5 1.0
30
0.5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.98 0.89 0.93 2.13 1.08 2.75 1.11 3.79 0.97 5.56 1.07 6.80
0.92 0.98 0.92 1.94 1.09 2.66 1.09 3.82 0.99 5.47 1.00 6.91
0.97 0.93 1.08 1.67 1.09 2.61 1.08 3.73 0.98 5.43 1.23 6.16
0.93 0.87 0.98 1.71 1.07 2.57 1.11 3.48 1.07 4.74 1.09 5.94
0.99 0.79 1.07 1.60 1.08 2.47 0.99 3.64 1.10 4.40 1.14 5.36
0.96 0.74 1.01 1.61 1.07 2.43 0.99 3.57 1.10 4.35 1.11 5.40
1.08 0.72 1.11 1.40 0.97 2.44 1.00 3.32 1.02 4.31 1.05 5.37
1.07 0.72 1.09 1.37 1.16 2.20 1.04 3.08 1.12 4.18 1.02 5.36
Table 5. Values of J , d s ) and y as a Function of Solute Concentration and Mean Temperature (hTb = 15 K)for Membrane 1
0.0 0
100
200
300
400
w (rpm)
T
("0
0.0
0.5
1.0
concentration ( m o m ) 1.5 2.0 2.5 3.0
3.5
4.0
y 0.96 1.07 1.11 1.11 1.03 1.09 1.09 1.09 1.09 J , 1.99 1.84 1.74 1.73 1.80 1.63 1.59 1.54 1.53
Figure 4. Volume flux per unit surface area versus stirring rate a t various mean temperatures (c = 2 mol&; ATb = 15 K) for membrane 1: = (*I 25, ( 0 )30, (A) 35, ( 0 )40, and (*) 45 "C.
25 30
y
cedure: The pairs of experimental data { w ; J,} may be fitted, according to eq 6, by using a three-parameter x2minimization method. As an example of the results obtained, Figures 4 and 5 show the adjustments in some representative cases. A visual inspection of these figures suggests that the fitting procedure seems to be adequate. Furthermore, the extrapolated values of the flux corresponding to an infinite stirring rate (no layers on the membrane surfaces), J,, were obtained from the parameters of the curves. These values, together with the y values, appear in Tables 4-6, which correspond to Tables 1-3, respectively. As a matter of fact, it is worth noticing that the y values are, in general, very close to unity. On the other hand, the extrapolated values of the fluxes may be used to obtain the corresponding dimensionless temperature polarization coefficients (with due reservations for the uncertainties inherent in all extrapolation processes). It is worth mentioning that this
35
J , 2.25 2.30 2.22 2.26 2.16 2.01 2.04 2.04 1.95 y 0.96 1.11 1.09 1.09 1.11 1.11 1.11 0.98 1.10 J , 3.07 2.68 2.63 2.60 2.47 2.41 2.34 2.45 2.19
40
y
1.10 1.06 1.09 1.03 1.07 1.10 1.04 0.96
1.04
1.06 1.12 1.12 1.13 1.13 1.02 1.10 1.08 1.07
J , 3.60 3.23 3.18 3.02 2.91 3.15 2.83 2.82 2.78 45
y
1.06 1.07 1.12 1.11 1.10 1.04 1.06 1.09 1.00
J , 4.55 3.99 3.81 3.74 3.59 3.62 3.39 3.01 3.22 50
y
0.97 1.09 0.99
1.07 1.02 1.11 1.10 1.11 1.11
J , 5.03 4.54 4.76 4.37 4.42 3.98 3.89 3.78 3.66
coefficient may be calculated for the different stirring rates appearing in Tables 1-3 under the corresponding experimental conditions. Table 7, which corresponds to membrane 1, shows the values of the TPC as a function of the bulk temperature difference and the solute concentration (for a mean temperature 35 "C). Table 8, which corresponds to membrane 1, shows the values of the TPC as a function of the mean temperature and the solute concentration (for a bulk temperature difference of 15 K). Finally, Table 9, which corresponds to membrane 2, shows the values of the TPC as a function of the mean temperature and the solute concentration
590 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 Table 6. Values of J , d s ) and y as a Function of Solute Concentration and Mean Temperature (ATb = 15 K) for Membrane 2 concentration ( m o m )
T ("C)
25
'J
30
J, y J,
35
y
40
J, y J,
45
y
J, 50
y
J,
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1.01 2.87 1.09 3.07 1.03 3.58 1.07 4.11 1.07 4.60 1.04 5.52
1.04 2.66 1.02 3.00 1.04 3.47 1.06 4.08 1.03 4.67 0.98 5.56
1.01 2.62 1.07 2.85 1.04 3.40 1.05 4.05 1.03 4.54 1.03 5.31
1.04 2.38 1.03 2.93 0.98 3.56 1.01 3.92 1.02 4.34 0.97 5.15
1.06 2.26 1.04 2.72 1.06 3.06 1.00 3.83 1.03 4.11 1.03 4.79
1.08 2.17 1.05 2.56 1.07 2.96 1.09 3.49 0.96 4.16 0.97 5.10
1.07 2.13 1.13 2.37 1.06 2.91 1.05 3.60 1.00 4.04 1.03 4.52
0.99 2.03 1.01 2.57 1.08 2.72 1.04 3.62 1.07 3.66 0.95 5.02
1.04 2.00 1.08 2.32 1.07 2.62 0.99 3.51 1.07 3.53 1.08 3.94
Table 7. TPC Values as a Function of Solute Concentration and Bulk Temperature Difference (o= 200 rpm; T = 35 "C) for Membrane 1 concentration ( m o m )
ATb ("C)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
5 10 15 20 25 30
0.72 0.64 0.60 0.75 0.63 0.61
0.67 0.58 0.67 0.64 0.57 0.55
0.59 0.61 0.67 0.62 0.56 0.53
0.59 0.68 0.66 0.61 0.54 0.56
0.61 0.63 0.63 0.63 0.59 0.56
0.63 0.66 0.64 0.59 0.60 0.61
0.66 0.63 0.63 0.57 0.59 0.58
0.63 0.68 0.58 0.57 0.55 0.55
0.61 0.66 0.61 0.58 0.55 0.52
Table 8. TPC Values as a Function of Solute Concentration and Mean Temperature (o= 200 rpm; ATb = 15 K) for Membrane 1 concentration (mol,&)
T ("C)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
25 30 35 40 45 50
0.65 0.71 0.63 0.63 0.58 0.61
0.68 0.67 0.69 0.67 0.62 0.67
0.71 0.67 0.68 0.65 0.64 0.59
0.68 0.63 0.66 0.66 0.62 0.60
0.62 0.63 0.66 0.66 0.61 0.58
0.66 0.65 0.66 0.58 0.58 0.61
0.62 0.60 0.65 0.61 0.60 0.60
0.62 0.58 0.58 0.59 0.63 0.57
0.61 0.60 0.62 0.56 0.55 0.56
Table 9. TPC Values as a Function of Solute Concentration and Mean Temperature (a= 200 rpm; hTb = 15 K) for Membrane 2 concentration (mob%)
T ("C)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
25 30 35 40 45 50
0.63 0.70 0.66 0.65 0.64 0.63
0.63 0.63 0.63 0.63 0.62 0.60
0.57 0.64 0.62 0.60 0.62 0.59
0.61 0.58 0.55 0.57 0.61 0.56
0.62 0.61 0.63 0.56 0.62 0.60
0.63 0.62 0.63 0.60 0.57 0.52
0.60 0.64 0.60 0.56 0.56 0.56
0.56 0.53 0.58 0.51 0.57 0.47
0.54 0.56 0.56 0.48 0.55
0.55
(for a bulk temperature difference of 15 K). In all cases the results refer t o an arbitrarily chosen stirring rate of 200 rpm, but equivalent tables might be obtained for each one of the stirring rates. Inspection of the data appearing in Tables 7-9 (and in equivalent tables for the other stirring rates) shows that the dependences of the TPC on temperature difference, solute concentration, and mean temperature are not clear. It is possible that a little decrease of the TPC with these variables exists, but this behavior is rather doubtful and it is more possible than there are not dependences a t all.
Conclusions (i) The measured nonisothermal fluxes increase with the stirring rate. This dependence has been quantified by using the concept of temperature polarization coefficient.
(ii) The TPC increases, as expected, with the stirring rate, but seems t o be virtually independent of the bulk temperature difference, the solute concentration, and the mean temperature in the studied range.
Acknowledgment This work has been financially supported by the Comunidad Aut6noma de Madrid. Nomenclature a = parameter B = net nonisothermal coefficient (m/sK) B' = apparent nonisothermal coefficient (m/s*K)
C = phenomenological coefficient (ndsbar) c = solute concentration (mol&) d = membrane thickness (m) h = film heat transfer (J/s*m2.K) ho = parameter J = volume flux per unit area ( n d s ) J,,= volume flux per unit area with stirring rate w ( d s )
JO = volume flux per unit area in absence of stirring ( d s ) J, = volume flux per unit area for infinite stirring rate (Ids)
k = membrane thermal conductivity (J/smK) L = heat of vaporization (J/mol) AP = pressure difference (bar) R = gas constant (J/mol.K) T = mean temperature ("C) AT = transmembrane temperature difference (K) ATb = bulk temperature difference (K) X = adjustment parameter Y = adjustment parameter w = stirring rate (rpm) y = positive dimensionless exponent
Literature Cited Drioli, E.; Wu, Y. Membrane Distillation: An Experimental Study. Desalination 1985,53,339-346. Johnson, R.A.; Valks, R. H.; LBfibvre, M. S. Osmotic Distillation. Aust. J . Biotechnol. 1989,3 (31, 206-211 Jonsson, A. S.; Wimmerstedt, R.; Harrison, A. C. Membrane Distillation: A Theoretical Study of Evaporation through Microporous Membranes. Sep. Sci. Technol. 1985,56,237-249. Kimura, S.; Nakao, S. I.; Shimatani, S. I. Transport Phenomena in Membrane Distillation. J . Membr. Sci. 1987,33,285-298. Kreith, F.; Black, W. Z. Basic Heat Transfer; Harper and Row: New York, 1980. Mengual, J . I.; Ortiz de Zarate, J. M.; Peiia, L.; Velazquez, A. Osmotic Distillation through Porous Hydrophobic Membranes. J . Membr. Sci. 1993,82, 129-140. Ortiz de Zarate, J. M.; Garcia-Lbpez, F.; Mengual, J. I. Temperature Polarization in Non-isothermal Mass Transport through Membranes. J . Chem. SOC.,Faraday Trans. 1990,86 (161, 2891-2896. Ortiz de Zarate, J . M.; Garcia-Lopez, F.; Mengual, J. I. Nonisothermal Water Transport through Membranes. J . Membr. Sci. 1991,56,181-194. Peiia, L.; Ortiz de Zarate, J. M.; Mengual, J. I. Steady States in Membrane Distillation: Influence of Membrane Wetting. J . Chem. SOC.,Faraday Trans. 1993,89(241,4333-4338. Sarti, G. C.; Gostoli, C.; Matulli, S. Low Energy Cost Desalination Processes using Hydrophobic Membranes. Desalination 1985, 56 (191, 227-286. Schofield,R. W.; Fane, A. G.; Fell, C. J. D. Heat and Mass Transfer in Membrane Distillation. J . Membr. Sci.1987,33,299-313. Schofield, R.W.; Fane, A. G.; Fell, C. J. D.; Macoun, R. Factors affecting Flux in Membrane Distillation. Desalination 1990, 77, 279-294.
Received for review May 20, 1994 Revised manuscript received October 3, 1994 Accepted October 18, 1994@ I39403232 Abstract published in Advance ACS Abstracts, December 15, 1994. @
