4128
Ind. Eng. Chem. Res. 1998, 37, 4128-4135
Effects of Polarization on Mass Transport through Hydrophobic Porous Membranes L. Martı´nez-Dı´ez and M. I. Va´ zquez-Gonza´ lez* Departamento de Fı´sica Aplicada, Facultad de Ciencias, Universidad de Ma´ laga, 29071 Ma´ laga, Spain
The aim of this study was to analyze the transport of water vapor by membrane distillation through a polytetrafluoroethylene flat membrane. Experiments were carried out for different salt concentrations in the feed, average temperatures, and recirculation rates at membrane sides. The influence of these operating conditions on water-vapor flux is discussed, taking into account mass and heat transfers within the membrane and the adjoining liquid phases. The polarization layers formed on either side of the membrane prove to be important in the interpretation of experimental results. Introduction In membrane distillation a hydrophobic membrane separates two aqueous liquids at different temperatures. In the present work we have a salt solution at an elevated temperature on one side of the membrane and pure water at a lower temperature on the other side. Both fluids do not penetrate the microporous membrane since a membrane material that is not wetted by aqueous phases has been selected (hydrophobic property). The imposed temperature difference results in a vapor pressure difference through the membrane. So the water evaporates on the warm side and then flows from the warm membrane side to the cold membrane side where it condenses. In the final analysis it is the vapor pressure difference that determines the mass transport. In fact, as indicated by Lawson and Lloyd (1997) in a recent review on membrane distillation, different transport models have been used in order to describe the water vapor transport through the membrane (Knudsen model, Poiseuille model, diffusive model, etc.). All the models suggest that the mass flux J may be written
J ) C(pm1 - pm2)
(1)
where C is the membrane permeability depending on different parameters of the membrane (pore size, porosity, thickness, temperature range, and gas pressure inside the pores). In eq 1, pm1 and pm2 are the vapor pressures at the warm and cold membrane surfaces, respectively. They are a function of the temperature and concentration at these membrane surfaces. Since the membrane distillation process relies on phase changes to perform mass transport, the latent heat of vaporization must be transferred to and from the feed and permeate vapor-liquid interfaces. As a consequence, thermal gradients are created in the liquid boundary layers of the membrane (Figure 1) and the bulk temperatures Tb1 and Tb2 are different from temperatures Tm1 and Tm2 in the liquid-vapor interfaces. The concept of temperature polarization defined as * To whom correspondence should be addressed. E-mail:
[email protected].
Figure 1. Temperature and salt concentration profiles in membrane distillation. Temperature and concentration polarizations are shown.
τ)
Tm1 - Tm2 Tb1 - Tb2
(2)
was proposed in order to quantify the fraction of the externally applied thermal driving force that contributes to the mass transfer. This concept of temperature polarization was introduced in papers dealing with pure water and several methods have been proposed for its evaluation (see, for example, Schofield et al. (1987) and Ortiz de Za´rate et al. (1990)), referring to it in the literature with normal values of τ within the range of 0.4-0.7 that show the importance of considering this phenomenon in order to interpret the flux results. When solutions of nonvolatile solutes are used as feed, the concentration polarization in the solution side of the membrane has to be considered together with the temperature polarization and cm1 is different from cb1. The concentration polarization is a major cause of flux reduction in other membrane processes such as ultrafiltration, but the combined effect of high-transfer coefficients and low-to-moderate fluxes reduce its effect in membrane distillation, as reported in the literature (Schofield et al. (1990); Calabro and Drioli (1996)). So, in the present work we deal with the following points: 1. Taking into account that the real driving force for transport is the vapor pressure difference, we introduce the coefficient f
S0888-5885(97)00911-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/10/1998
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4129
f)
pm1 - pm2 pb1 - pb2
(3)
in order to measure the reduction of the imposed force (pb1 - pb2) for transport as a consequence of the existence of both temperature and concentration polarizations. This f coefficient coincides aproximately with τ (see further on) when we work with water, but when we work with solutions of nonvolatile solutes as feed, they differ (although the polarization with the concentration will be of little importance) as p depends on c and T. In these cases f (and not τ) explains the real reduction of the generalized force and therefore of the flux. 2. The method proposed here for evaluating f is an extension of the method proposed by Schofield et al. (1987) for evaluating τ when water is used as the feed. The modified method takes into account the presence of solute in the feed. 3. The f dependence with experimental parameters, solution concentration, recirculation flow rate, and temperature has been analyzed. Experimental Section Experimental tests were performed using flat polytetrafluoroethylene (PTFE) membranes manufactured by Gelman Instrument Co. as TF 200 (80% void fraction, 60-µm thickness, 0.2-µm nominal pore size). The membrane cell is a tangential filtration cell manufactured by Millipore Corp. as Minitan-S, and it is schematically shown in Figure 2. In this membrane cell a flat sheet membrane separates the distilland and distillate liquid phases. In all experimental runs the membrane was maintained in a horizontal position. Both hot (distilland, lower) and cold (distillate, upper) liquid phases are formed by nine prismatic channels (provided by each silicone sheet) of approximately 0.45 × 7.0 × 55.0 mm, giving an effective membrane area of 34.6 × 10-4 m2. The hot (water or salt solutions) and cold (initially pure water) liquids are preheated in each corresponding thermostatic bath and are then pumped tangentially on both sides of the membrane in countercurrent directions (Figure 3). Recirculation flow rate, temperatures, conductivities, and pressures are monitored; they are constant in each experiment. The temperatures of the bulk liquid phases are measured at the entrance (Tb1-in, Tb2-in) and exit (Tb1-out, Tb2-out) of the membrane cell. Average values of the temperatures Tb1, Tb2, and Tb were calculated as
Tb1 )
Tb1-in + Tb1-out 2
(4)
Tb2 )
Tb2-in + Tb2-out 2
(5)
Tb1 + Tb2 2
(6)
Tb )
The temperature difference ∆Tb between the bulk phases at both sides of the membrane was evaluated as
∆Tb ) Tb1 - Tb2 As mentioned in the Introduction, due to the existence of temperature gradients in the liquid phases adjacent
Figure 2. Membrane holder showing the entrance (1) and exit (2) of cold water, the entrance (3) and exit (4) of hot solution, the silicone separators (5) and (7), and the membrane (6).
to the membrane, this ∆Tb is higher than the actual temperature difference between both sides of the membrane. Therefore, we indicate the following in order to differentiate them:
∆Tm ) Tm1 - Tm2p where Tm1 and Tm2 are the average temperatures at the hot and cold membrane sides, respectively. The J distillate flux is measured by timing and weighing of the water which flows out of the capillary attached to the top of the cold reservoir. The final composition of the distillate liquid was measured, and a total salt rejection was always observed. In the present work a series of experiments has been performed. A temperature difference was maintained between the thermostatic baths for each experiment, and the corresponding mass flux through the membrane was measured for different recirculation flow rates and solute concentrations in the hot semicell. Experiments were carried out for different temperatures. Theory In membrane distillation both mass and heat transfer occur through the hydrophobic membrane. The mass transport can be described by eq 1 which, as indicated by Sarti et al. (1985), can be written
J ) C[(p* m1 - p* m2)(1 - xm) - p* m(xm1 - xm2)]
(7)
where the symbol * refers to the vapor pressure of pure water at a determinate temperature, x refers to the mole fraction of dissolved species, and xm1 and xm2 (zero in all experiments) are the x values at the membrane surfaces. Finally, xm is defined as the mean value of xm1 and xm2.
4130 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
Figure 3. Experimental device that includes membrane cell (MC), membrane (M), thermocouples at entrance (1-in, 2-in) and exit (1-out, 2-out) of membrane cell, pressure transducers (3), pumps (4), flowmeters (5), conductivity meters (6), reservoirs (7), and collector of distillate water (8).
The interfacial mole fraction can be calculated from the interfacial concentration, which is given by the film model for concentration polarization:
cm1 ) cb1 exp(J/Fk)
(8)
where F is the liquid-phase density and k is the solute mass-transfer coefficient. Given the characteristics of our system (see below), the heat transfer to and from the membrane surface is controlled by forced convection and the Graetz or Le´veque solutions (Porter (1972), Holman (1989)) for laminar flow channels,
(
)
dh Nu ) 1.62 RePr L
0.33
(9)
h1(Tb1 - Tm1) )
modified for mass transfer,
(
)
dh Sh ) 1.62 ReSc L
0.33
(10)
may be used to evaluate the mass-transfer coefficient:
( )
k ) 1.62
vD2 dhL
0.33
(11)
Here, Nu, Pr, Sh, Re, and Sc are the Nusselt, Prandtl, Sherwood, Reynolds, and Schmidt numbers, respectively. L, dh, and v are the length, hydraulic diameter, and fluid velocity in the channels of the membrane cell, respectively. Finally, D is the diffusion coefficient of solute in water. Equation 7 can be written
J)C
(dp* dT )
Tm
[(Tm1 - Tm2) - ∆Tth](1 - xm) (12)
where Tm is the mean value of Tm1 and Tm2 and where we have made the approximation
p*m1 - p* m2 dp* ) Tm1 - Tm2 dT Tm
( )
(13)
which as indicated by Schofield et al. (1987) is convenient if (Tm1 - Tm2) is little. In eq 12 ∆Tth is related to the reduction in vapor pressure caused by dissolved species in the distilland and is given by
∆Tth )
RT2 xm1 - xm2 Mλ 1 - xm
where R is the gas constant, M the molecular weight of water, and λ their latent heat of vaporization. The ∆Tth value is indeed a threshold value since for ∆Tm larger than ∆Tth a positive driving force is obtained, with fluxes toward the cold side, while for ∆Tm lower than ∆Tth we have a negative driving force, giving rise to fluxes directed toward the warm membrane side. Since, as opposed to the temperatures Tb1 and Tb2, the temperatures Tm1 and Tm2 are difficult to measure, Tb1 and Tb2 are, as a rule, inserted in eq 12. To do this, we must introduce the heat-transfer coefficients (h1, h2) in the liquid films near the membrane and the heat transfer by conduction (km) through the membrane. In this way, for the stationary thermal flux across the membrane system we can write
(14)
km (T - Tm2) + Jλ ) h2(Tm2 - Tb2) δ m1 (15)
where δ is the membrane thickness and km can be calculated for the porous membrane as
km ) kg + (1 - )ks
(16)
with kg and ks being the thermal conductivities of the gas phase and the solid phase respectively and the membrane porosity. These equations show that the process is characterized by simultaneous mass- and heat-transfer resistances through the membrane and the external liquid phases (Schofield et al. (1990)). Thus, the overall process rate appears to be controlled by the heat and mass transfer through both the membrane and liquid phases. We will discuss the relative role played by both membrane and liquid phases by means of the polarization coefficient f, defined in eq 3. This f coefficient reflects the overall effect of the temperature and concentration polarization in the liquid layers adjacent to the membrane. For instance, in the case of a very fast heat and mass transfer through the liquid boundary layers, the Tm1 temperature approaches the bulk temperature Tb1 and Tm2 the Tb2. On the other hand, the cm1 concentration approaches the cb1 concentration. As a consequence, pm1 approaches the pb1 and pm2 the pb2, resulting in f f 1. In this case the process rate is completely controlled by the membrane and the polarization is negligible. Otherwise, for low heat- and mass-transfer values through the liquid boundary layers and a large C value, the pm1 approaches the pm2 and f f 0. In this case the
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4131
process rate is completely controlled by the liquid phases and polarization is very important. To calculate f, we can deduce from eqs 12 and 15
{ [
]}
km 1 dp* C ) λ(1 - xm) + h dT δ 1 dp* (Tb1 - Tb2) + C λ∆Tth(1 - xm) (17) h dT
(Tm1 - Tm2) 1 +
where
1 1 1 + h1 h2
h)
(18)
is the overall film heat-transfer coefficient for the liquid boundary layers of the membrane. From eqs 12 and 17
[
]
1 1 + ) λJ Tb1 - Tb2 dp* - ∆Tth (1 - xm)Cλ dT 1 km 1+ h δ 1 1 km 1+ h h δ (19) Tb1 - Tb2 - ∆Tth 1 km 1+ h δ
( )
(
)
that allow us to calculate h and C, and therefore f. When water is the distilland, eqs 17 and 19 can be written
Tm1 - Tm2 )
1 1+
(T - Tb2) H b1 h
(20)
and
km Tb1 - Tb2 1 + δh 1 + ) Jλ dp* h Cλ dT
(21)
where
H ) Cλ
dp* km + dT δ
(22)
is the effective heat transfer coefficient for the membrane. In this case eq 20 allows the calculation of the temperature polarization coefficient eq 2, as
τ)
1 1+
H h
(dp* dT )
Tm
p* b1 - p* b2 Tb1 - Tb2
(24)
are good when (Tb1 - Tb2) is little. Results and Discussion As previously mentioned, experiments were carried out for fixed temperatures in the membrane module. In all the experiments ∆Tb ) 10 °C, while the Tb1 average temperatures varied from 20 to 50 °C at increments of about 7 °C, and that of Tb2 from 10 to 40 °C, also at increments of about 7 °C. In each experiment the liquid recirculation rate Q was the same (except for small fluctuations, always under 5%) on both sides of the membrane. Different experiments were carried out applying the recirculation rates of 7, 11, and 15 × 10-6 m3/s (that is, a linear velocity down the channel of about 0.25, 0.39, and 0.53 m/s, respectively). The transmembrane pressure gradient was minimum for all the experiments while the absolute pressure in the membrane varied from 1.05 × 105 Pa for the lower recirculation rate to 1.3 × 105 Pa for the higher recirculation rate. Four different experiments were carried out with water and sodium chloride solutions of 0.55, 1.15, and 1.67 M, respectively, as a warm liquid. The temperature of the feed and distillate streams were measured at the entrances and exits of the module. For all experiments, the values of (Tb1-in - Tb1-out) and (Tb2-out - Tb2-in) do not differ to more than a 0.1 °C accuracy of the temperature measurements. From this result we conclude that the heat loss to the environment was negligible, as was expected due to the plastic material of the membrane module and the insulating material with which it was covered. However, values of (Tb1-in - Tb1-out) lower than 1.5 °C have been obtained. For all experiments a water flux from the hot phase to the cold phase was always observed. Conductivity measurements carried out on the permeate side (Figure 3) were always inferior to within 10 µs/cm after several hours of functioning. Considering the concentration of the feed solution, the volume of the distillate and the reservoir volume, retention was more than 99.99%. In the first place we have estimated the type of regime (laminar or turbulent) and if the influence of the free convection is important. So, we have calculated the numbers of Reynolds, Prandtl, and Grashof. The Reynolds number for the fluid in the channels of the membrane module varied from 150 to 800, depending on the experimental conditions. However, the GrashofPrandtl (Rayleigh) number product was evaluated using the equivalent hydraulic diameter dh as the characteristic dimension, and a temperature difference between the membrane surface and the bulk solution was typically 2 °C. We obtain for the product
(23)
In experiments where the water is the distilland, there is no concentration polarization. As a consequence, the temperature polarization coefficient τ obtained by eq 23 should approximately coincide with the polarization coefficient f obtained by eq 3, as the approximated equations (13) and (24)
)
GrPr )
gβF2cp 3 d (T - Tb) µk1 h m
(25)
values ranging from 7 to 58. Having in mind the analysis of Metais and Eckert (1964), we can conclude that the effects of free convection are negligible. So, a laminar regime of forced convection can be expected for flow through the channels of a membrane module.
4132 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
Figure 7. ∆Tb/Jλ vs 1/(dp*/dT) corresponding to the flux results obtained when the recirculation rate was 15 × 10-6 m3/s. Symbols same as those for Figure 4. Figure 4. Water flux through the membrane vs feed temperature for a recirculation rate of 7 × 10-6 m3/s. The different symbols correspond to different hot feeds: ), water; 4, cb1 ) 0.55 mol L-1; ×, cb1 ) 1.15 mol L-1; O, cb1 ) 1.67 mol L-1. In the experiment ∆Tb ) 10 °C.
Figure 5. Water flux through the membrane vs feed temperature for a recirculation rate of 11 × 10-6 m3/s. In the experiment ∆Tb ) 10 °C. Symbols same as those for Figure 4.
Figure 6. Water flux through the membrane vs feed temperature for a recirculation rate of 15 × 10-6 m3/s. In the experiment ∆Tb ) 10 °C. Symbols same as those for Figure 4.
In Figures 4-6 the distillate fluxes for the three recirculation rates are displayed as a function of the imposed temperatures when different salt concentra-
tions are used. We can see how the distillate flux monotonically increases with the recirculation rate. The flux also increases when the temperature in the membrane increases. Finally, we can conclude that for the same Tb and Q the distillate fluxes are lower than when the concentration of the salt solutions were higher. To interpret these flux results and calculate the polarization coefficients, analyses of eqs 21 and 19 for the experimental water flux corresponding to the same recirculation rate and salt concentration but different temperatures have been carried out following a method similar to that of Schofield et al. (1987). So, for each J we evaluated (dp*/dT) for the corresponding Tb temperature. Immediately after, we made plots of (1/Jλ) versus 1/(dp*/dT). Representative plots are shown in Figure 7. All the plots have a correlation coefficient >0.99. This linearity of the data provides support for the mentioned form of eqs 21 and 19 for average temperatures from 15 to 45 °C. This linearity is because in the second member of these equations only the term (dp*/dT) changes significantly with the temperature in the temperature interval indicated. To obtain h and C from the intercept and the slope of the above-mentioned plots, we evaluated km from eq 16 using the value of kg ) 0.027 W m-1 K-1 and ks) 0.22 W m-1 K-1 obtained from the literature (Perry (1963); Speraty (1989)). In the same way we evaluated ∆Tth and xm using eqs 8, 11, and 14. In all the experiments xm2 ) 0. The little variation of ∆Tth and xm with the temperature (in the interval 15-45 °C) have negligible effects to evaluate h and C. In the same way h and C values so obtained are temperature average values. Values of h and C obtained as mentioned above are shown in Table 1 as a function of the recirculation rate Q and of the concentration cb1. The results show that as Q increases, h increases. This tendency is in accordance with the model prediction of the convective heat-transfer theory for a laminar flow in the flat-sheet configuration. Besides, inferior differences of 16% are observed between the estimated h values and the predicted values by eq 9. However, a little decrease of C with Q is observed as a consequence of the increase of the total pressure in the membrane pores as Q increases. As indicated above, the experimental C and h values allow us to interpret the flux results and also quantify the polarization coefficient. Figure 8 shows the τ results obtained when water is
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4133 Table 1. h and C Values Obtained from Equations 21 and 19 Q × 106 (m3 s-1)
h (W m-2 K-1)
C × 107 (kg m-2 s-1 Pa-1)
water
7 11 15
1800 ( 90 2050 ( 80 2690 ( 140
14.9 ( 0.5 14.8 ( 0.4 13.7 ( 0.4
cb1 ) 0.55
7 11 15
1770 ( 80 2150 ( 120 2655 ( 190
14.5 ( 0.4 13.7 ( 0.4 13.2 ( 0.4
cb1 ) 1.15
7 11 15
1570 ( 70 1870 ( 140 2150 ( 80
15.6 ( 0.5 14.8 ( 0.3 14.5 ( 0.3
cb1 ) 1.67
7 11 15
1610 ( 50 1890 ( 50 2119 ( 130
14.9 ( 0.5 14.4 ( 0.7 14.7 ( 0.5
Figure 10. Polarization coefficient vs feed temperature when the feed is a solution 0.55 M. Symbols same as those for Figure 8.
Figure 8. Temperature polarization coefficient vs feed temperature when the hot feed is water. The different symbols correspond to different recirculation rates. 4, 7 × 10-6 m3/s; ×, 11 × 10-6 m3/s; ), 15 × 10-6 m3/s. Figure 11. Polarization coefficient vs feed temperature when the feed is a solution 1.15 M. Symbols same as those for Figure 8.
Figure 9. Polarization coefficient vs feed temperature when the feed is water. Symbols same as those for Figure 8.
the feed in the experiment. Here, τ is calculated from eqs 22 and 23. Figures 9-12 show the f values obtained for the different experiments. These f values are calculated as indicated in the following paragraph. From the C values obtained, eq 1 allows for the calculation of (pm1 - pm2) for each experiment. However, from the bibliography of Robinson and Stokes (1959) we have calculated the vapor pressures pb1 and pb2 corresponding to the temperature and salt concentrations
Figure 12. Polarization coefficient vs feed temperature when the feed is a solution 1.67 M. Symbols same as those for Figure 8.
that the bulk phases have in each experiment. Last, eq 3 allows for the calculation of the corresponding f value. From the analysis of the f values obtained as indicated we conclude the following: 1. As was hoped, the f and τ results obtained when the feed is pure water are very similar; the small differences are due to experimental error. These values show that approximately 50% of the applied driving
4134 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
force is not effective. In this case, the cause is the temperature polarization. 2. The f values obtained of about 0.5 indicate that there is no clear predominant role of either the membrane or the liquid layers adjacent to the membrane in the transport control (Schofield et al. (1990)). This value of 0.5 shows an important polarization. Most of it is due to temperature polarization because concentration polarization is very low. In fact, eq 8 gives concentration polarization values cm1/cb1 up to 1.04, and as a consequence, the corresponding vapor pressures are very similar. 3. The f values increase as the recirculation rate increases in agreement with the fact that the recirculation rate alters the fluid dynamics and influences the heat and mass transfer in the boundary liquid layers. When Q increases, the increase of h and k has as an outcome the decrease of the temperature and concentration polarizations. 4. The f values decrease slightly when the solution concentration increases. This is due to the change of the vapor pressure with the concentration and to the little increase of the temperature polarization. In fact, the results in Table 1 show how h decreases slowly with the concentration. 5. The f values decrease as the temperature increases. This is because the larger mass fluxes obtained as the temperature increases involve more important heat fluxes through the liquid phases, so that the resistance offered by a heat transfer in the liquid phases increases with relation to the resistance to a transmembrane mass transport. 6. The observed increase of the distillate flux with the recirculation rate corresponds to the obtained increase of the heat transfer. However, we know that the distillate flux increases with the temperature because (dp/dT) also increases, but (dp/dT) increases faster with temperature than the distillate fluxes. This is because, as we said in point 5, f decreases with the temperature and as a consequence the fluxes do not increase proportionally to (dp/dT) (Martı´nez-Dı´ez et al. (1998)). Finally, the decrease of the flux as cb1 increases is due to the decrease of the vapor pressure and to the increase of polarization. 7. As a consequence from the above-mentioned data we conclude that any attempt to explain any membrane distillation results must take into account the significant influence of polarization. List of Symbols c ) concentration, mol L-1 C ) membrane distillation coefficient, kg m-2 s-1 Pa-1 cp ) heat capacity, J kg-1 K-1 D ) diffusion coefficient of solute, m2 s-1 dh ) equivalent hydraulic diameter, m f ) polarization coefficient g ) acceleration of gravity, m s-2 Gr ) Grashof number ) (gβF2/µ2)dh3(Tm - Tb) H ) membrane heat-transfer coefficient, W m-2 K-1 h ) film heat-transfer coefficient, W m-2 K-1 J ) mass flux through membrane, kg m-2 s-1 k ) solute mass-transfer coefficient, m s-1 kg ) thermal conductivity of the air, W m-1 K-1 kl ) thermal conductivity of the liquid, W m-1 K-1 km ) thermal conductivity of the membrane, W m-1 K-1 ks ) thermal conductivity of the solid phase of the membrane,W m-1 K-1
L ) channel length, m M ) molecular weight, kg mol-1 Nu ) Nusselt number ) hdh/kl p ) vapor pressure, Pa p* ) vapor pressure of pure water, Pa pb ) mean vapor pressure for the bulk solution, Pa pm ) mean vapor pressure at membrane surface, Pa Pr ) Prandtl number ) cpµ/kl Q ) liquid recirculation rate, m3 s-1 R ) gas constant, J mol-1 K-1 Ra ) Rayleigh number ) GrPr Reynolds number ) vdhF/µ Schmidt number ) µ/(DF) Sherwood number ) kdh/D T ) temperature, K Tb ) mean temperature in the bulk phases, K Tm ) mean temperature at membrane surface, K ∆Tb ) mean temperature difference between the bulk phases, K ∆Tm ) mean temperature difference between the membrane surfaces, K x ) mole fraction of dissolved species v ) fluid velocity down the channel, m s-1 Greek Symbols β ) volume coefficient of expansion, K-1 δ ) membrane thickness, m ) porosity λ ) latent heat of vaporization, J kg-1 F ) density, kg m-3 τ ) temperature polarization coefficient µ ) viscosity, Pa s Subscripts in, out ) defined in Figure 4 1 ) hot side 2 ) cold side m ) membrane surface b ) bulk face
Literature Cited Calabro, V.; Drioli, E. Polarization Phenomena in Integrated Reverse Osmosis and Membrane Distillation for Seawater Desalination and Wastewater Treatment. Desalination 1996, 108, 81. Holman, J. P. Heat Transfer; McGraw-Hill: New York, 1989. Lawson, R. W.; Lloyd, D. R. Membrane Distillation. J. Membr. Sci. 1997, 124, 1. Martı´nez-Dı´ez, L.; Va´zquez-Gonza´lez M. I.; Florido-Dı´az, F. J. Temperature Polarization Coefficients in Membrane Distillation. Sep. Sci. Technol. 1998, 33 (6), 787. Metais, B.; Eckert, E. R. G. Forced, mixed and free convection regimes. J. Heat Transfer 1964, 86 (C), 295. Ortiz de Za´rate, J. M.; Garcı´a-Lo´pez, F.; Mengual, J. I. Temperature Polarization in non-Isothermal Mass Transport through Membranes. J. Chem. Soc., Faraday Trans. 1990, 86 (16), 2891. Perry, J. H. Chemical Engineers Handbook, 4th ed.; McGrawHill: New York, 1963. Porter, M. C. Concentration Polarization with Membrane Ultrafiltration. Ind. Eng. Chem. Prod. Res. Dev. 1972, 11, 234. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworth and Co.: London, 1959. Sarti, G. C.; Gostoli, C.; Matulli, S. Low Energy Cost Desalination Processes Using Hydrophobic Membranes. Desalination 1985, 56, 277. Schofield, R. W.; Fane, A. G.; Fell, C. J. D. Heat and Mass Transfer in Membrane Distillation. J. Membr. Sci. 1987, 33, 299. Schofield, R. W.; Fane A. G.; Fell, C. J. D. Gas and Vapour Transport through Microporous Membranes. II. Membrane Distillation. J. Membr. Sci. 1990, 53, 173.
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4135 Schofield, R. W.; Fane, A. G.; Fell, C. J. D.; Macoun, R. Factors Affecting Flux in Membrane Distillation. Desalination 1990, 77, 279.
Received for review December 15, 1997 Revised manuscript received July 13, 1998 Accepted July 23, 1998
Speraty, C. A. Physical Constants of Fluoropolymers. In Polymer Handbook, 3rd ed.; Wiley: New York, 1989.
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