Desalination by vacuum membrane distillation: sensitivity analysis

Transcription

1 Separation and Purification Tecnology 33 (2003) 75/87 Desalination by vacuum membrane distillation: sensitivity analysis Fawzi Banat *, Fami Abu Al-Rub, Kalid Bani-Melem Department of Cemical Engineering, Jordan University of Science and Tecnology, P.O. Box 3030, Irbid 220, Jordan Received 4 December 200; received in revised form 9 December 2002; accepted 9 December 2002 Abstract In order to enance te performance of te vacuum membrane distillation process in desalination, i.e. to get more flux, it is necessary to study te effect of operating parameters on te yield of distillate water. Simple tecniques including te normalized dimensionless sensitivity factor and temperature polarization coefficient as well as te solution of te transport models were used for tis purpose. Te sensitivity of te mass flux to te process operating parameters including downstream pressure, feed temperature, feed flow rate, and membrane permeability were investigated. Te mass flux of distillate water was igly sensitive to te feed temperature especially at ig values of vacuum pressure. Te mass flux was more sensitive to te vacuum pressure at low feed temperature levels tan at te ig ones. Since lowering te temperature polarization coefficient is essential to enance te process performance considerable efforts sould be directed toward maximizing te eat transfer coefficient troug better module design. Te predictions of te normalized dimensionless sensitivity analysis were in agreement wit te results obtained from solving te transport model of te process. # 2003 Elsevier Science B.V. All rigts reserved. Keywords: Desalination; Vacuum membrane distillation; Sensitivity analysis. Introduction Vacuum membrane distillation, like any membrane distillation process, is a termally driven process in wic te convective mass transfer is te dominant mecanism for mass transfer. Te driving force is maintained by applying vacuum at te downstream side to keep te pressure at tis side below te equilibrium vapor pressure (see Fig. ). * Corresponding autor. Fax: / address: banatf@just.edu.jo (F. Banat). Te membrane in tis process is a pysical support for te vapor/liquid interface and does not affect te selectivity associated wit te vapor/liquid equilibrium. In contrast, pervaporation process depends mainly on using a dense membrane, wic alters te vapor /liquid equilibrium []. Vacuum membrane distillation as te potential to be one of te most common tecniques used to separate dilute aqueous mixtures. Applications include te removal of etanol from fermentation brot to prevent te inibition of microorganisms used in fermentation [2] and for te removal of trace concentrations of VOC s from water []. In /03/$ - see front matter # 2003 Elsevier Science B.V. All rigts reserved. doi:0.06/s (02)0022-6

2 76 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 Fig.. Scematic drawing of vacuum membrane distillation process () feed side, (2) membrane, (3) vacuum side. addition, VMD can be used wen te feed contains non-volatile salt as in desalination. Depending on te type of application, te importance of te different design parameters in VMD is determined. For example, wile in te removal of etanol from fermentation brot, bot te separation factor and te flux are te most important design parameters; te flux is te most important design parameter in te case were water contains non-volatile solutes. However, wen water contains dissolved gases, maximizing te solute flux and minimizing te water flux are te most important design parameters. Compared wit conventional separation tecniques, VMD as many advantages: Te vacuum can be applied only on a very limited volume of te membrane equipment and not on te entire equipment. VMD can be also operated at relatively low evaporation temperatures; typically below 50/60 8C. Tus, a low cost energy source is required to supply te eat for evaporation. Economically, VMD is found to be comparable [3,4] wit respect to te separation costs of oter membrane alternatives suc as pervaporation [5,6]. Recently, VMD as become an active area of researc by many. Most of te publised work as focused on etanol/water separation [2]. Oter researcers studied te use of VMD in te removal of trace gases and VOC s from water [,7/9]. To te autor s knowledge, te work of Bandini et al. [7] was te first and te only study to investigate te effect of different variables on te VMD efficiency. Tey used te dimensionless sensitivity approac to study te sensitivity of te total flux in VMD to eat and mass transfer coefficients. Tey proposed a simple criterion tat can be used to establis, witout te need to solve te transport model, weter te total flux is controlled by eat or by mass transfer resistance. However, teir study only focused on te sensitivity of mass flux to variations in te eat and mass transfer coefficients and did not tackle te sensitivity of mass flux to process operating parameters suc as te vacuum pressure and te temperature of te feed stream. Te objective of tis study is terefore to extend te sensitivity analysis by Bandini et al. [7] to include te sensitivity of te mass flux to te controllable operating parameters of te VMD process suc as te vacuum pressure, and feed temperature, again using pure water for demonstration. In addition to te dimensionless sensitivity approac, a temperature polarization coefficient-based analysis was performed to assess te effect of tese parameters on te process performance. Tis approac requires solving te transport model of te process. 2. Teory Te base-case tat is considered in studying te relative importance of process parameters is te production of distillate water from water containing non-volatile solutes. Since te presence of nonvolatile matters suc as salts in te feed solution only affects te absolute amount of produced flux but not te process sensitivity toward any process variable, salt-free water was considered as a feed solution. In fact, seawater of ppm salinity as a vapor pressure wic is approximately.84% lower tan tat of pure water [0]. Bandini et al. [7] made a similar assumption. 2.. Mass and eat transfer in VMD As in any membrane distillation process, VMD is caracterized by simultaneous eat and mass transfer. Mass transfer troug te membrane process is associated wit diffusive and convective

3 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 77 transport of vapors troug te microporous membrane. As is widely known, te mass transfer troug a porous media depends mainly on te pore size of te membrane and te mean free molecular pat of te transferring species. Tese two factors determine wat mecanism of mass transfer will be dominant. In most of te membranes used in VMD, te pore size is usually smaller tan te mean free pat. Hence, te Knudsen mecanism is dominant []. Accordingly, te mass flux N is linearly related to te pressure difference across te membrane. N M (PI P V ) () were (s mol 0.5 /kg 0.5 m) is te membrane permeability wic depends on membrane caracteristics and feed temperature. M is te molecular weigt (kg/mol), P I is te interfacial partial pressure and P V is te downstream pressure. In te case of one volatile component in te feed, P I will be te vapor pressure of tat component. Te eat required for te evaporation at te interface is supplied by te eat flux troug te liquid stream. Te evaporation temperature at te interface is related to te feed bulk temperature in te liquid pase by te equation: Nl(T b T I ) (2) were is te eat transfer coefficient in te liquid pase, and l is te latent eat of vaporization Temperature polarization Te temperature polarization penomenon occurs as a result of a temperature difference between te feed bulk temperature T b, interfacial temperature T I, and te temperature in te vacuum side T v. Tis temperature gradient is due to te eat flux troug te liquid layer, wic is needed to provide te required eat for evaporation at te membrane interface. Te temperature variation across te membrane can be described by te temperature polarization factor tat is defined as: u T b T I T b T v (3) Te numerical value of u lies witin te range [0/]. As u 0/0.0, T I 0/T b, in tis case te resistance in te liquid pase is negligible and te process is controlled by te resistance of te membrane. Wen u 0/, T I 0/T v so te resistance in te membrane is negligible and te process is controlled in te liquid pase. Tus, te polarization factor can be used as a tool in studying te process beavior Sensitivity analysis Before proceeding in studying te sensitivity of te VMD process to te process parameters, te following sensitivity factors are defined. ) te first order sensitivity factor of any model response R wit respect to any of te model input parameters (P i ) is defined as [2]: s(r; P i (4) 2) te normalized dimensionless sensitivity factor is defined as [2]: S(R; R P s(r; P i ) P P i i R (5) were P i is any parameter tat may affect R. In te case of pure water, R represents te mass flux (N) and P i may be any one of te input parameters affecting N, i.e. T b, P V, and. As te parameters studied are not dimensionally omogeneous, normalized (dimensionless) parameters are used to better understand teir pysical significance. For te present case, te following normalized sensitivity factors are introduced. Expression for tese sensitivity factors can be found using equation Eqs. () and (2). Te final results are presented ere wile te detailed derivations are in te Appendix A. / Te sensitivity factor of mass flux to te eat transfer coefficient; S(N, ):

4 78 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 S(N; R R (6) p s(n; ) ffiffiffiffiffiffi (PI P V ) R (8) / Te sensitivity factor of mass flux to te permeability of te membrane; S(N, ): S(N; N R / Te sensitivity factor of mass flux to te feed bulk temperature; S(N, T b ): S(N; T b N R T b ( R )R 2 Te sensitivity factor of mass flux to te vacuum pressure; S(N, P V ): S(N; P v N P v ( R )R 3 were: M l R (0) R 2 T I T b () R 3 P I (2) P V Using Clausius /Claperyon equation: lmp I (3) RTI 2 and substituting Eq. (3) into Eq. (0), te expression for R can be written as: R M3=2 l 2 P I RTI 2 (4) Te first order sensitivity factors of te above parameters are given by te following equations: s(n; T b ) R (5) l R s(n; P V ) M (6) R s(n; ) l (T b T I ) R (7) R 3. Results and discussion In order to enance te process performance, i.e. to get more flux, it is necessary to understand te sensitivity of te process to every input key parameter. In oter words, wat type of action sould be undertaken to improve te flux for a given VMD system? To determine te relative importance of,, T b and P V on te mass flux, te relative sensitivity factors defined in Eqs. (6) / (9) are used. For te case of pure water in te feed, and at a fixed vacuum pressure, te permeate flux is controlled, as indicated by Eq. (), by te membrane permeability and te vapor pressure of water at te membrane interface. Te interfacial vapor pressure is a function of te interfacial temperature, so tat te overall process seems to be controlled by two simultaneous processes; eat transfer troug te liquid pase and mass transfer troug te membrane. Te parameters T b and P V are responsible for establising te driving force, terefore te sensitivity factors mentioned in Eqs. (8) and (9) are not only measuring te sensitivity of flux to te given parameters, but tey are also measuring te sensitivity of flux to te driving forces associated by te given parameter. Tis beavior is explained in te coming discussion. 3.. Sensitivity of te flux to te feed bulk temperature S(N, T b ) Te flux and its sensitivity to te feed temperature are sown in Figs. 2 and 3, respectively. As sown in Fig. 2, increasing te feed bulk temperature results in an exponential increase in te pure water flux. Tis exponential beavior can be attributed to te exponential relationsip between water vapor pressure and temperature.

5 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 79 temperatures. Tis beavior would be obtained if te first order sensitivity analysis was considered (see Fig. 4). However, wen considering te normalized sensitivity as in tis work, te large cange in first order sensitivity is divided by te large value of te flux, wic results in small cange in te normalized sensitivity at ig T b Sensitivity of te flux to te vacuum pressure S(N, P V ) Fig. 2. Effect of feed bulk temperature on mass flux. Figs. 5 and 6 sow, respectively, te absolute flux and its sensitivity to te vacuum pressure at different values of te liquid pase eat transfer coefficient. Fig. 5 sows tat for a given eat transfer coefficient te flux decreases linearly wit te increasing vacuum side pressure. Tis is due to te decrease in te driving force wen vacuum pressure is increased. Fig. 5 sows also tat for a given vacuum pressure, te flux of pure water increases by increasing te liquid pase eat transfer coefficient. Tis is obvious, since increasing te liquid pase eat transfer coefficient reduces te eat transfer resistance, consequently T I approaces T b, and in doing so te mass transfer driving force is increased. Fig. 3. Response of S (N, T b ) to te feed bulk temperature. On te oter and, te sensitivity of te mass flux to te feed bulk temperature, as represented by te normalized sensitivity factor, decreases by increasing te feed bulk temperature. Tis is due to te fact tat at lower feed temperature, te flux is small, so any small cange in bulk temperature of te feed results in a large cange in te normalized sensitivity. Tis interesting result seems contradictory to wat is sown in Fig. 2 were te flux of water increases sarply at iger Fig. 4. Response of te first order sensitivity to feed bulk temperature.

6 80 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 Fig. 5. Effect of vacuum pressure on mass flux. tat tere is no effect of te eat transfer coefficient on te S(N, P V ) as all of te lines tat represent different values of te are concurring eac oter. Te above results may be analyzed using R 2 and R 3 wic may be considered as measures of te sensitivity of te relative driving force in bot resistances wit respect to te given parameter. Wen R 2 0/0, T I 0/T b, ence, te flux is igly sensitive to any small cange in te membrane resistance. Wen R 3 0/0, P I 0/P V, ence, te flux becomes igly sensitive to driving forces in te liquid pase and te vacuum side as well. Terefore, at lower temperatures and ig vacuum pressure, te mass flux will be sensitive to bot parameters P V and T b. Te positive value of S(N, T b )infig. 3 indicates tat increasing T b increases te flux, wile te negative value of S(N, P V )in Fig. 6 sows tat increasing P V reduces te mass flux Sensitivity of transport coefficients Fig. 6. Response of S(N, P V )tovacuum pressure. On te oter and, as sown in Fig. 6, te sensitivity of te flux increases as te vacuum pressure is increased. Tis occurs because an increase in te vacuum pressure decreases linearly te mass flux as given by Eq. (). Accordingly, te normalized value of te mass flux becomes very sensitive to any small cange in P V. Importantly also, te results obtained for different values of te eat transfer coefficient sow Te role of eat and mass transfer can be determined by considering te sensitivity factors of and. According to Eqs. (6) and (7) te sensitivity factors of and are related by te following equation: S(N; )S(N; ) (9) Eq. (9) is in an agreement wit te work of Bandini et al. [7]. Since te production of pure water by VMD is caracterized by two simultaneous resistances in series, eat and mass, one of tese resistances may dominate over te oter one. Tis can be determined by considering te sensitivity of mass flux to tese coefficients. To determine te relative importance of and on te mass flux, te relative sensitivity factors of te mass flux to tese parameters are used. Fig. 7 sows te process sensitivity factors as a function of for two bulk feed temperatures and for two different values of membrane permeability ( / 2.0/0 6 and /.5/0 5 s mol 0.5 /kg 0.5 m) [8]. It is clear tat in te low range of eat transfer coefficient, te canges in ave te greatest relative influence on N, significantly more tan

7 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 8 Fig. 7. Response of S(N, ) to liquid pase eat transfer coefficient., so tat te process is controlled by te eat transfer resistance in te liquid film. In order to increase te flux, te eat transfer coefficient sould be increased. Low eat transfer coefficients result in ig temperature polarization. In tis case, te difference between te bulk and te interfacial temperature increases and T approaces T V. If tis occurs, te interfacial temperature is lowered and consequently te vapor pressure of pure water in te feed decreases resulting in a lower flux. Also, as sown in Fig. 7, as te feed temperature increases, te process becomes more sensitive to te eat transfer resistance in te liquid film. Tis can be attributed to te fact tat te vapor pressure of water is more sensitive at ig temperatures tan at low ones. To elucidate tis numerically, te vapor pressure of water at 25 8C is mmhg and at C is mmhg, wile at 80 8C it is 355. mmhg and at C is 363 mmhg. A quick calculation sows tat te temperature drop of 0.5 at 25.5 will lower te vapor pressure by a value of less tan mm Hg wile at C, a temperature drop by 0.5 8C will lower te vapor pressure by about 8 mmhg. Tis means tat careful consideration sould be directed toward te eat transfer coefficient wen operating at ig temperatures. Lowering te temperature polarization is essential to enance te process performance. On te oter and, wen te eat transfer coefficient is greater tan about (000 W/m 2 K), te membrane permeability becomes more important as indicated by Eq. (9). Persistently increasing te eat transfer coefficient reduces te importance of te eat transfer resistance. Ultimately, te process becomes dominated by te membrane permeability (S(N, )0/, S(N, )0/ 0.0). However, if te membrane permeability is fixed, operating at ig feed temperatures requires iger values of tan wen operating at lower feed temperatures as sown in Fig. 7. Tis empasizes te need to keep as ig as possible. Terefore, wen (S(N, )0/.0), te process is completely controlled by te membrane permeability. To increase te permeate flux, te permeability of te membrane sould be increased. Tis is more apparent at low temperatures tan at ig ones. Similarly, wen (S(N, )0/), te liquid film eat transfer coefficient sould be increased in order to increase te permeate flux, no matter ow ig te membrane permeability is. Tis discussion clearly sows tat tis process is a combined eat and mass transfer process. On te oter and, for te two membranes, te sensitivity factor decreases as te membrane permeability increases, and tus te process starts to be sensitive to te liquid pase eat transfer coefficient. Tis is clearly sown in Fig. 8 were te sensitivity factor S(N, ) as a function of te membrane permeability is plotted. Pysically, VMD will be eat transfer limited if te module design does not provide adequate eat transfer to te membrane surface. Conversely, te process will be mass transfer limited if te membrane permeability is too low. To improve te process performance, te membrane permeability and/or te module design sould be improved. Te membrane performance is mainly determined by te pore size and te membrane tickness. Higer fluxes require larger pore size and tinner membranes. Te larger te pore size, te lower is te liquid entry pressure. Te liquid entry pressure determines te repulsive properties of te membrane. If te feed pressure is iger tan te liquid entry pressure ten te liquid will flow continu-

8 82 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 Fig. 8. Response of S(N, ) to membrane permeability in VMD. ously troug te pores and no selective separation is acieved. In addition to tat, te membrane sould be tick enoug to witstand te mecanical pressure in te process. As a result, tere is little scope for improvement by developing special MD membranes. Attention would be better focused on designing VMD modules to provide ig eat transfer coefficients. Practically, if VMD as been used for desalination, it will be operated at low pressures wit a good membrane permeability to maximize te flux. Terefore, te process will be eat transfer limited. So maximizing te eat transfer coefficient by a good module design maximizes te mass flux Temperature polarization factor Anoter metodology to study te dependency of flux on te eat transfer coefficient can be obtained by solving te transport model equations for te flux and temperature polarization. Te concept of te temperature polarization factor will be used as a tool for evaluating te effect of te input parameters on maximizing te mass flux. Te polarization coefficient as a function of feed bulk temperature is sown in Fig. 9. Te model as been solved for tree different values of vacuum pressure 500, 2000 and 5000 N/m 2. Fig. 9. Effect of feed temperature on VMD temperature polarization factor. Apparently, te polarization factor increases by increasing feed bulk temperature. Wen u 0/0.0, T I 0/T b and te process are controlled by te membrane resistance. Actually it is important to keep u as small as possible. To overcome te increase in u tat results from T b increase, te eat transfer coefficient in te liquid layer bounding te membrane surface sould be increased as well. Tis is consistent wit te discussion aforementioned. Te increase of te u factor wit bulk temperature is expected since increasing te bulk temperature increases te mass flux tat requires sufficient eat of vaporization. Providing suc a eat increases te difference between te bulk temperature and te interfacial temperature, and in turn te polarization factor. 4. Conclusions Vacuum membrane distillation is a process tat can be used for te production of distillate water from brackis or seawater. To priori know te parameters to wic te process is sensitive, a sensitivity analysis study was carried out for tis process. Te main parameters considered in tis study were vacuum pressure, membrane perme-

9 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 83 ability, feed temperature, and liquid side eat transfer coefficient (fluid flow). A simple approac known as te normalized dimensionless sensitivity factor tecnique along wit te temperature polarization coefficient were used to determine te sensitivity of te process to eac of te aforementioned parameters. Te role of te eat transfer resistance in te liquid pase and te mass transfer resistance in te membrane matrix was also considered in tis study. Te process was determined to be sensitive to bot te vacuum pressure and te feed temperature but was more sensitive to te feed temperature at ig vacuum pressure levels and more sensitive to te vacuum pressure at low values of bulk feed temperatures. Improvement of membrane caracteristics in terms of its permeability is important but more important is te improvement of module designs to provide ig eat transfer coefficients. Te temperature polarization factor-based analysis gave similar results as te dimensionless normalized approac. However, te later approac is andier to use tan te former since it does not require solving te wole transport model. Appendix A Derivation of Eqs. (6) /(9). Te derivation of Eqs. (6)/(9) is based on te permeate flux wic can be written as: N M (PI P V ) (A:) Nl(T b T I ) (A:2) and te Clausius /Claperyon equation: lmp I (A:3) RTI 2 -Te normalized sensitivity factor S(N, ). S(N dn=n d= dn (A:4) d N Ten to get dn/d, from Eq. (A.2) N l (T b T I ) dn T b l d TI l d l dn d l (T b T I ) (A:5) l d Now /d is needed. Equalize Eqs. (A.) and (A.2) for N to get: l (T b T I )K m M (PI P V ) or: P l M I P V T b T I (T dl M b T I ) (P I P V ) (T b T I ) 2 d l M (Tb T I ) = (P I P V ) (T b T I ) 2 or d l M (dpi =dt I ) (P I P V )=(T b T I ) (T b T I ) from te Clausius /Claperyon Eq. (A.3): lmp I RT 2 I and from Eqs. (A.) and (A.2): P I P V T b T I l M Ten: d (lmp l M I =RTI 2) (=lk m M ) (T b T I )

10 84 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 or d (l2 M 3=2 P I =RTI 2) (A:6) (T b T I ) or: d (T b T I ) (l 2 M 3=2 P I =RTI 2) (A:7) Now, substitute equations Eqs. (A.5) and (A.7) in Eq. (A.4) to get: S(N; ) dln N dln dn=n d= dn d N l (T b T I ) l From Eq. (A.) N l T b T I dln T I dln N ten te term /l(t b /T I )(/N) will be: l (T b T I ) l (T b T I ) and te oter term: (T b T I ) l (l 2 M 3=2 P I =RTI 2) N were: (T b T I ) l l (l 2 M 3=2 P I =RTI 2) T b T I R R M3=2 l 2 P I RTI 2 ten: R S(N; ) R R 2-Te normalized sensitivity S(N, ). S(N; N dn=n dn d = d N From Eq. (A.): N/ / M/(P I /P V ). ffiffi dn M (dpi )(P I P V ) M dkm or dn dp M I (P I P V ) M (A:9) d d Now ( /d ) is needed. Again, Equalize Eqs. (A.) and (A.2) for N to get: (T b T I ) l M (PI P V ) ten d p ffiffiffiffiffi l M (PI P V )( ) (T b T I ) (P I P V ) 2 and d l M (dti =dp I ) ((T b T I )=(P I P V )) (P I P V ) from te Clausius /Claperyon equation, lmp I RT 2 I and from Eqs. (A.) and (A.2): T b T I lk m M P I P V ten d l M (RT 2 I =lmp I ) ((l M )=) (P I P V )

11 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 85 l M d P I P V (RTI 2=lMP I ) ((l M )=) P fg I P V d (RTI 2=M3=2 l 2 P I ) or d but RTI 2 M 3=2 l 2 P I R ten d P I P V RT 2 I =M3=2 l 2 P I () PI P V (=R ) () : (A:0) Now substitute Eqs. (A.9) and (A.0) into Eq. (A.8) to get: (=R ) () R R and te second part of Eq. (A.) can be written as: Km ((P I P V ) M ) N ((P I P V ) M ) : M (PI P V ) Ten: R S(N; ) R ( R ) : 3-Te normalized sensitivity factor S(N, T b ). S(N; T b N dn T b T b dt b N N l (T b T I ) S(N; ) dln N dln dn=n d = dn d N dn l dt b l dp M I (P I P V ) M d Km (A:) N te first part of te Eq. (A.) can be written as: dp M I Km d N ( M ) PI P V (=R ) () M (PI P V ) or: dn dt b l : l dt b So /dt b is needed. T b K m M l (P I P V )T I dt b K m M l or: dt b K m M l from te Clausius /Claperyon Eq. (A.3) (A:3)

12 86 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 lmp I RT 2 I ten: dt b M3=2 l 2 P I RTI 2 R or: : (A:4) dt b R Substitute Eq. (A.4) in Eq. (A.3) to get: dn dt b l dt I l dt b l dt b R l R l R ten S(N; T b N dn T T b dt b N R lt b l R (T b T I ) Let R 2 T b T I T b ten: R R T b (T b T I ) : S(N; T b ) R : ( R )R 2 4-Te normalized sensitivity factor S(N, P V ). S(N; P V N dn P V P V dp V N N dn or M (PI P V ) M fdpi dp V g dn dp M I : (A:6) dp V dp V So /dp V is needed. From Eqs. (A.) and (A.2): M PV l (T b T I )K m M PI or P V P I (T b T I ) l M P V P I (T b T I ) l M ten dp V l M and dp V : l M From te Clausius /Claperyon equation: lmp I RT 2 I ten dp V RT I 2 M 3=2 l 2 P I but RTI 2 M 3=2 l 2 P I R ten dp V R R or: R : (A:7) dp V R Substitute Eqs. (A.6) and (A.7) into Eq. (A.5):

13 F. Banat et al. / Separation and Purification Tecnology 33 (2003) 75/87 87 S(N; P V P V dn dp V P V N dp M I dp V R Let PV N R M R PV P I P V : R 3 P I P V P V ten S(N; P V ) ( R )R 3 References P V M (PI P V ) [] S. Bandini, A. Savedra, G. Sarti, Vacuum membrane distillation: experiments and modeling, AICE J. 43 (997) 398. [2] E. Hoffman, D. Pfenning, E. Pilippsen, P. Scwan, M. Seiber, D. Woermann, Evaporation of alcool/water mixtures troug ydropobic porous membranes, J. Member. Sci. 34 (987) 99. [3] G. Sarti, C. Gostoli, S. Bandini, Extraction of organic components from aqueous streams by vacuum membrane distillation, J. Member. Sci. 80 (993) 2. [4] G. Sarti, C. Gostoli, Separation of liquid mixtures by membrane distillation, J. Member. Sci. 4 (989) 2. [5] J. Wijmans, J. Kascemetat, J. Davidson, W. Baker, Treatment of organic contaminated wastewater streams by pervaporation, Environ. Prog. 9 (990) 262. [6] V. Lipski, P. Cote, Te use of pervaporation for te removal of organic contaminants from water, Environ. Prog. 9 (990) 254. [7] S. Bandini, C. Gostoli, G. Sarti, Separation efficiency in vacuum membrane distillation, J. Member. Sci. 73 (992) 27. [8] S. Bandin, G. Sari, C. Gostoli, Vacuum membrane distillation: pervaporation troug porous ydropobic membranes, in: R. Bakis, (Ed.) Proceedings of Tird International Conference on Pervaporation in Cemical Industry, Nancy, France, September 9/22, 988, p. 7. [9] F. Banat, J. Simandl, Removal of benzene traces from contaminated water by vacuum membrane distillation, Cem. Eng. Sci. 5 (996) 257. [0] K.S. Spiegler, Y. El-Sayed, A Desalination Primer, Balaban Desalination Publications, Reovot, 994. [] F. Dullien, Porous Medi-Fluid Transport and Pore Structure, Academic Press, London, 979. [2] A. Miguel, M. Jose, S. Fernanda, N. Rosa, S. Julila, Application of parametric sensitivity to batc process safety: teoretical and experimental studies, Cem. Eng. Tecnol. 9 (996) 222.