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Desalination by vacuum membrane distillation: sensitivity analysis
Separation and Purification Technology 33 (2003) 75 /87 www.elsevier.com/locate/seppur
Desalination by vacuum membrane distillation: sensitivity analysis Fawzi Banat *, Fahmi Abu Al-Rub, Khalid Bani-Melhem Department of Chemical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan Received 14 December 2001; received in revised form 19 December 2002; accepted 19 December 2002
Abstract In order to enhance the performance of the vacuum membrane distillation process in desalination, i.e. to get more flux, it is necessary to study the effect of operating parameters on the yield of distillate water. Simple techniques including the normalized dimensionless sensitivity factor and temperature polarization coefficient as well as the solution of the transport models were used for this purpose. The sensitivity of the mass flux to the process operating parameters including downstream pressure, feed temperature, feed flow rate, and membrane permeability were investigated. The mass flux of distillate water was highly sensitive to the feed temperature especially at high values of vacuum pressure. The mass flux was more sensitive to the vacuum pressure at low feed temperature levels than at the high ones. Since lowering the temperature polarization coefficient is essential to enhance the process performance considerable efforts should be directed toward maximizing the heat transfer coefficient through better module design. The predictions of the normalized dimensionless sensitivity analysis were in agreement with the results obtained from solving the transport model of the process. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Desalination; Vacuum membrane distillation; Sensitivity analysis
1. Introduction Vacuum membrane distillation, like any membrane distillation process, is a thermally driven process in which the convective mass transfer is the dominant mechanism for mass transfer. The driving force is maintained by applying vacuum at the downstream side to keep the pressure at this side below the equilibrium vapor pressure (see Fig. 1).
* Corresponding author. Fax: /962-2-709-5018. E-mail address: [email protected] (F. Banat).
The membrane in this process is a physical support for the vapor/liquid interface and does not affect the selectivity associated with the vapor/liquid equilibrium. In contrast, pervaporation process depends mainly on using a dense membrane, which alters the vapor /liquid equilibrium [1]. Vacuum membrane distillation has the potential to be one of the most common techniques used to separate dilute aqueous mixtures. Applications include the removal of ethanol from fermentation broth to prevent the inhibition of microorganisms used in fermentation [2] and for the removal of trace concentrations of VOC’s from water [1]. In
1383-5866/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S1383-5866(02)00221-6
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Fig. 1. Schematic drawing of vacuum membrane distillation process (1) feed side, (2) membrane, (3) vacuum side.
addition, VMD can be used when the feed contains non-volatile salt as in desalination. Depending on the type of application, the importance of the different design parameters in VMD is determined. For example, while in the removal of ethanol from fermentation broth, both the separation factor and the flux are the most important design parameters; the flux is the most important design parameter in the case where water contains non-volatile solutes. However, when water contains dissolved gases, maximizing the solute flux and minimizing the water flux are the most important design parameters. Compared with conventional separation techniques, VMD has many advantages: The vacuum can be applied only on a very limited volume of the membrane equipment and not on the entire equipment. VMD can be also operated at relatively low evaporation temperatures; typically below 50/60 8C. Thus, a low cost energy source is required to supply the heat for evaporation. Economically, VMD is found to be comparable [3,4] with respect to the separation costs of other membrane alternatives such as pervaporation [5,6]. Recently, VMD has become an active area of research by many. Most of the published work has focused on ethanol/water separation [2]. Other researchers studied the use of VMD in the removal of trace gases and VOC’s from water [1,7 /9]. To the author’s knowledge, the work of Bandini et al. [7] was the first and the only study to
investigate the effect of different variables on the VMD efficiency. They used the dimensionless sensitivity approach to study the sensitivity of the total flux in VMD to heat and mass transfer coefficients. They proposed a simple criterion that can be used to establish, without the need to solve the transport model, whether the total flux is controlled by heat or by mass transfer resistance. However, their study only focused on the sensitivity of mass flux to variations in the heat and mass transfer coefficients and did not tackle the sensitivity of mass flux to process operating parameters such as the vacuum pressure and the temperature of the feed stream. The objective of this study is therefore to extend the sensitivity analysis by Bandini et al. [7] to include the sensitivity of the mass flux to the controllable operating parameters of the VMD process such as the vacuum pressure, and feed temperature, again using pure water for demonstration. In addition to the dimensionless sensitivity approach, a temperature polarization coefficient-based analysis was performed to assess the effect of these parameters on the process performance. This approach requires solving the transport model of the process.
2. Theory The base-case that is considered in studying the relative importance of process parameters is the production of distillate water from water containing non-volatile solutes. Since the presence of nonvolatile matters such as salts in the feed solution only affects the absolute amount of produced flux but not the process sensitivity toward any process variable, salt-free water was considered as a feed solution. In fact, seawater of 35 000 ppm salinity has a vapor pressure which is approximately 1.84% lower than that of pure water [10]. Bandini et al. [7] made a similar assumption. 2.1. Mass and heat transfer in VMD As in any membrane distillation process, VMD is characterized by simultaneous heat and mass transfer. Mass transfer through the membrane process is associated with diffusive and convective
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transport of vapors through the microporous membrane. As is widely known, the mass transfer through a porous media depends mainly on the pore size of the membrane and the mean free molecular path of the transferring species. These two factors determine what mechanism of mass transfer will be dominant. In most of the membranes used in VMD, the pore size is usually smaller than the mean free path. Hence, the Knudsen mechanism is dominant [11]. Accordingly, the mass flux N is linearly related to the pressure difference across the membrane. pffiffiffiffiffiffi N Km M (PI PV ) (1)
The numerical value of u lies within the range [0 /1]. As u 0/0.0, TI 0/Tb, in this case the resistance in the liquid phase is negligible and the process is controlled by the resistance of the membrane. When u 0/1, TI 0/Tv so the resistance in the membrane is negligible and the process is controlled in the liquid phase. Thus, the polarization factor can be used as a tool in studying the process behavior.
where Km (s mol0.5/kg0.5 m) is the membrane permeability which depends on membrane characteristics and feed temperature. M is the molecular weight (kg/mol), PI is the interfacial partial pressure and PV is the downstream pressure. In the case of one volatile component in the feed, PI will be the vapor pressure of that component. The heat required for the evaporation at the interface is supplied by the heat flux through the liquid stream. The evaporation temperature at the interface is related to the feed bulk temperature in the liquid phase by the equation:
Before proceeding in studying the sensitivity of the VMD process to the process parameters, the following sensitivity factors are defined.
Nl h(Tb TI )
2.2. Temperature polarization The temperature polarization phenomenon occurs as a result of a temperature difference between the feed bulk temperature Tb, interfacial temperature TI, and the temperature in the vacuum side Tv. This temperature gradient is due to the heat flux through the liquid layer, which is needed to provide the required heat for evaporation at the membrane interface. The temperature variation across the membrane can be described by the temperature polarization factor that is defined as: Tb TI Tb Tv
1) the first order sensitivity factor of any model response R with respect to any of the model input parameters (Pi) is defined as [12]: s(R; Pi )
@R @Pi
(4)
2) the normalized dimensionless sensitivity factor is defined as [12]:
(2)
where h is the heat transfer coefficient in the liquid phase, and l is the latent heat of vaporization.
u
2.3. Sensitivity analysis
(3)
S(R; Pi)
@ln R @ln Pi
Pi @R R @Pi
s(R; Pi )
Pi R
(5)
where Pi is any parameter that may affect R . In the case of pure water, R represents the mass flux (N ) and Pi may be any one of the input parameters affecting N , i.e. Tb, PV, h and Km. As the parameters studied are not dimensionally homogeneous, normalized (dimensionless) parameters are used to better understand their physical significance. For the present case, the following normalized sensitivity factors are introduced. Expression for these sensitivity factors can be found using equation Eqs. (1) and (2). The final results are presented here while the detailed derivations are in the Appendix A. / The sensitivity factor of mass flux to the heat transfer coefficient; S (N , h):
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@ln N R1 S(N; h) @ln h 1 R1
(6)
s(N; Km )
pffiffiffiffiffiffiffi Km (PI PV )
1
1 R1
(18)
/ The sensitivity factor of mass flux to the permeability of the membrane; S (N , Km): S(N; Km )
@ln N @ln Km
1 1 R1
(7) 3. Results and discussion
/ The sensitivity factor of mass flux to the feed bulk temperature; S (N , Tb): S(N; Tb )
@ln N @ln Tb
R1 (1 R1 )R2
(8)
The sensitivity factor of mass flux to the vacuum pressure; S (N , PV): S(N; Pv )
@ln N @ln Pv
1 (1 R1 )R3
(9)
where: pffiffiffiffiffiffi M l dPI R1 Km h dTI R2 1 R3
PI PV
TI Tb
1
(10) (11) (12)
Using Clausius /Claperyon equation: dPI dTI
lMPI RTI2
(13)
and substituting Eq. (13) into Eq. (10), the expression for R1 can be written as: R1
M 3=2 Km l2 PI RTI2 h
(14)
The first order sensitivity factors of the above parameters are given by the following equations: h R1 s(N; Tb ) (15) l 1 R1 pffiffiffiffiffiffi 1 s(N; PV ) Km M (16) 1 R1 1 R1 s(N; h) (Tb TI ) (17) l 1 R1
In order to enhance the process performance, i.e. to get more flux, it is necessary to understand the sensitivity of the process to every input key parameter. In other words, what type of action should be undertaken to improve the flux for a given VMD system? To determine the relative importance of h , Km, Tb and PV on the mass flux, the relative sensitivity factors defined in Eqs. (6) / (9) are used. For the case of pure water in the feed, and at a fixed vacuum pressure, the permeate flux is controlled, as indicated by Eq. (1), by the membrane permeability and the vapor pressure of water at the membrane interface. The interfacial vapor pressure is a function of the interfacial temperature, so that the overall process seems to be controlled by two simultaneous processes; heat transfer through the liquid phase and mass transfer through the membrane. The parameters Tb and PV are responsible for establishing the driving force, therefore the sensitivity factors mentioned in Eqs. (8) and (9) are not only measuring the sensitivity of flux to the given parameters, but they are also measuring the sensitivity of flux to the driving forces associated by the given parameter. This behavior is explained in the coming discussion.
3.1. Sensitivity of the flux to the feed bulk temperature S (N , Tb) The flux and its sensitivity to the feed temperature are shown in Figs. 2 and 3, respectively. As shown in Fig. 2, increasing the feed bulk temperature results in an exponential increase in the pure water flux. This exponential behavior can be attributed to the exponential relationship between water vapor pressure and temperature.
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
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temperatures. This behavior would be obtained if the first order sensitivity analysis was considered (see Fig. 4). However, when considering the normalized sensitivity as in this work, the large change in first order sensitivity is divided by the large value of the flux, which results in small change in the normalized sensitivity at high Tb.
3.2. Sensitivity of the flux to the vacuum pressure S (N , PV)
Fig. 2. Effect of feed bulk temperature on mass flux.
Figs. 5 and 6 show, respectively, the absolute flux and its sensitivity to the vacuum pressure at different values of the liquid phase heat transfer coefficient. Fig. 5 shows that for a given heat transfer coefficient the flux decreases linearly with the increasing vacuum side pressure. This is due to the decrease in the driving force when vacuum pressure is increased. Fig. 5 shows also that for a given vacuum pressure, the flux of pure water increases by increasing the liquid phase heat transfer coefficient. This is obvious, since increasing the liquid phase heat transfer coefficient reduces the heat transfer resistance, consequently TI approaches Tb, and in doing so the mass transfer driving force is increased.
Fig. 3. Response of S (N , Tb) to the feed bulk temperature.
On the other hand, the sensitivity of the mass flux to the feed bulk temperature, as represented by the normalized sensitivity factor, decreases by increasing the feed bulk temperature. This is due to the fact that at lower feed temperature, the flux is small, so any small change in bulk temperature of the feed results in a large change in the normalized sensitivity. This interesting result seems contradictory to what is shown in Fig. 2 where the flux of water increases sharply at higher
Fig. 4. Response of the first order sensitivity to feed bulk temperature.
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Fig. 5. Effect of vacuum pressure on mass flux.
that there is no effect of the heat transfer coefficient on the S (N , PV) as all of the lines that represent different values of the h are concurring each other. The above results may be analyzed using R2 and R3 which may be considered as measures of the sensitivity of the relative driving force in both resistances with respect to the given parameter. When R2 0/0, TI 0/Tb, hence, the flux is highly sensitive to any small change in the membrane resistance. When R3 0/0, PI 0/PV, hence, the flux becomes highly sensitive to driving forces in the liquid phase and the vacuum side as well. Therefore, at lower temperatures and high vacuum pressure, the mass flux will be sensitive to both parameters PV and Tb. The positive value of S (N , Tb) in Fig. 3 indicates that increasing Tb increases the flux, while the negative value of S (N , PV) in Fig. 6 shows that increasing PV reduces the mass flux. 3.3. Sensitivity of transport coefficients The role of heat and mass transfer can be determined by considering the sensitivity factors of h and Km. According to Eqs. (6) and (7) the sensitivity factors of h and Km are related by the following equation: S(N; h)S(N; Km ) 1
Fig. 6. Response of S (N , PV) to vacuum pressure.
On the other hand, as shown in Fig. 6, the sensitivity of the flux increases as the vacuum pressure is increased. This occurs because an increase in the vacuum pressure decreases linearly the mass flux as given by Eq. (1). Accordingly, the normalized value of the mass flux becomes very sensitive to any small change in PV. Importantly also, the results obtained for different values of the heat transfer coefficient show
(19)
Eq. (19) is in an agreement with the work of Bandini et al. [7]. Since the production of pure water by VMD is characterized by two simultaneous resistances in series, heat and mass, one of these resistances may dominate over the other one. This can be determined by considering the sensitivity of mass flux to these coefficients. To determine the relative importance of h and Km on the mass flux, the relative sensitivity factors of the mass flux to these parameters are used. Fig. 7 shows the process sensitivity factors as a function of h for two bulk feed temperatures and for two different values of membrane permeability (Km / 2.0 /10 6 and Km /1.5 /10 5 s mol0.5/kg 0.5 m) [8]. It is clear that in the low range of heat transfer coefficient, the changes in h have the greatest relative influence on N , significantly more than
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Fig. 7. Response of S (N , h ) to liquid phase heat transfer coefficient.
Km, so that the process is controlled by the heat transfer resistance in the liquid film. In order to increase the flux, the heat transfer coefficient should be increased. Low heat transfer coefficients result in high temperature polarization. In this case, the difference between the bulk and the interfacial temperature increases and T1 approaches TV. If this occurs, the interfacial temperature is lowered and consequently the vapor pressure of pure water in the feed decreases resulting in a lower flux. Also, as shown in Fig. 7, as the feed temperature increases, the process becomes more sensitive to the heat transfer resistance in the liquid film. This can be attributed to the fact that the vapor pressure of water is more sensitive at high temperatures than at low ones. To elucidate this numerically, the vapor pressure of water at 25 8C is 23.76 mmHg and at 25.5 8C is 24.47 mmHg, while at 80 8C it is 355.1 mmHg and at 80.5 8C is 363 mmHg. A quick calculation shows that the temperature drop of 0.5 at 25.5 will lower the vapor pressure by a value of less than 1 mm Hg while at 80.5 8C, a temperature drop by 0.5 8C will lower the vapor pressure by about 8 mmHg. This means that careful consideration should be directed toward the heat transfer coefficient when operating at high temperatures. Lowering the
81
temperature polarization is essential to enhance the process performance. On the other hand, when the heat transfer coefficient is greater than about (1000 W/m2 K), the membrane permeability becomes more important as indicated by Eq. (19). Persistently increasing the heat transfer coefficient reduces the importance of the heat transfer resistance. Ultimately, the process becomes dominated by the membrane permeability (S(N , Km)0/1, S (N , h )0/ 0.0). However, if the membrane permeability is fixed, operating at high feed temperatures requires higher values of h than when operating at lower feed temperatures as shown in Fig. 7. This emphasizes the need to keep h as high as possible. Therefore, when (S (N , Km) 0/1.0), the process is completely controlled by the membrane permeability. To increase the permeate flux, the permeability of the membrane should be increased. This is more apparent at low temperatures than at high ones. Similarly, when (S (N , h)0/1), the liquid film heat transfer coefficient should be increased in order to increase the permeate flux, no matter how high the membrane permeability is. This discussion clearly shows that this process is a combined heat and mass transfer process. On the other hand, for the two membranes, the sensitivity factor decreases as the membrane permeability increases, and thus the process starts to be sensitive to the liquid phase heat transfer coefficient. This is clearly shown in Fig. 8 where the sensitivity factor S (N , Km) as a function of the membrane permeability is plotted. Physically, VMD will be heat transfer limited if the module design does not provide adequate heat transfer to the membrane surface. Conversely, the process will be mass transfer limited if the membrane permeability is too low. To improve the process performance, the membrane permeability and/or the module design should be improved. The membrane performance is mainly determined by the pore size and the membrane thickness. Higher fluxes require larger pore size and thinner membranes. The larger the pore size, the lower is the liquid entry pressure. The liquid entry pressure determines the repulsive properties of the membrane. If the feed pressure is higher than the liquid entry pressure then the liquid will flow continu-
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Fig. 8. Response of S (N , Km) to membrane permeability in VMD.
Fig. 9. Effect of feed temperature on VMD temperature polarization factor.
ously through the pores and no selective separation is achieved. In addition to that, the membrane should be thick enough to withstand the mechanical pressure in the process. As a result, there is little scope for improvement by developing special MD membranes. Attention would be better focused on designing VMD modules to provide high heat transfer coefficients. Practically, if VMD has been used for desalination, it will be operated at low pressures with a good membrane permeability to maximize the flux. Therefore, the process will be heat transfer limited. So maximizing the heat transfer coefficient by a good module design maximizes the mass flux.
Apparently, the polarization factor increases by increasing feed bulk temperature. When u 0/0.0, TI 0/Tb and the process are controlled by the membrane resistance. Actually it is important to keep u as small as possible. To overcome the increase in u that results from Tb increase, the heat transfer coefficient in the liquid layer bounding the membrane surface should be increased as well. This is consistent with the discussion aforementioned. The increase of the u factor with bulk temperature is expected since increasing the bulk temperature increases the mass flux that requires sufficient heat of vaporization. Providing such a heat increases the difference between the bulk temperature and the interfacial temperature, and in turn the polarization factor.
3.4. Temperature polarization factor Another methodology to study the dependency of flux on the heat transfer coefficient can be obtained by solving the transport model equations for the flux and temperature polarization. The concept of the temperature polarization factor will be used as a tool for evaluating the effect of the input parameters on maximizing the mass flux. The polarization coefficient as a function of feed bulk temperature is shown in Fig. 9. The model has been solved for three different values of vacuum pressure 500, 2000 and 5000 N/m2.
4. Conclusions Vacuum membrane distillation is a process that can be used for the production of distillate water from brackish or seawater. To priori know the parameters to which the process is sensitive, a sensitivity analysis study was carried out for this process. The main parameters considered in this study were vacuum pressure, membrane perme-
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
ability, feed temperature, and liquid side heat transfer coefficient (fluid flow). A simple approach known as the normalized dimensionless sensitivity factor technique along with the temperature polarization coefficient were used to determine the sensitivity of the process to each of the aforementioned parameters. The role of the heat transfer resistance in the liquid phase and the mass transfer resistance in the membrane matrix was also considered in this study. The process was determined to be sensitive to both the vacuum pressure and the feed temperature but was more sensitive to the feed temperature at high vacuum pressure levels and more sensitive to the vacuum pressure at low values of bulk feed temperatures. Improvement of membrane characteristics in terms of its permeability is important but more important is the improvement of module designs to provide high heat transfer coefficients. The temperature polarization factor-based analysis gave similar results as the dimensionless normalized approach. However, the later approach is handier to use than the former since it does not require solving the whole transport model.
Appendix A
dTI
lMPI
(A:3)
RTI2
1-The normalized sensitivity factor S (N , h ). S(N h)
@ln N @ln h
dN=N dh=h
dN h dh N
Then to get dN /dh , from Eq. (A.2) h N (Tb TI ) l
dh
1 h dTI (Tb TI ) l l dh
(A:5)
Now dTI/dh is needed. Equalize Eqs. (A.1) and (A.2) for N to get: pffiffiffiffiffiffi h (Tb TI ) Km M (PI PV ) l or: h lKm
pffiffiffiffiffiffi PI PV M Tb TI
dh lKm
pffiffiffiffiffiffi(Tb TI )dPI (PI PV )dTI M (Tb TI )2
pffiffiffiffiffiffi dh lKm M dTI (Tb TI )dPI =dTI (PI PV ) (Tb TI )2
dh dTI
lKm
pffiffiffiffiffiffi M
(dPI =dTI ) (PI PV )=(Tb TI )
(Tb TI )
from the Clausius /Claperyon Eq. (A.3):
and the Clausius /Claperyon equation:
dN
Tb T h dh I dh dTI l l l
or
Derivation of Eqs. (6) /(9). The derivation of Eqs. (6) /(9) is based on the permeate flux which can be written as: pffiffiffiffiffiffi N Km M (PI PV ) (A:1) Nl h(Tb TI ) (A:2) dPI
dN
83
(A:4)
dPI lMPI dTI RTI2 and from Eqs. (A.1) and (A.2): PI PV h pffiffiffiffiffiffi Tb TI lKm M Then: pffiffiffiffiffiffi(lMPI =RTI2 ) (h=lKm dh lKm M (Tb TI ) dTI
pffiffiffiffiffiffi M)
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or
2-The normalized sensitivity S (N , Km).
dh (l2 M 3=2 PI Km =RTI2 ) h dTI (Tb TI )
(A:6)
or: dTI dh
(Tb TI ) (l2 M 3=2 PI Km =RTI2 ) h
(A:7)
Now, substitute equations Eqs. (A.5) and (A.7) in Eq. (A.4) to get: S(N; h)
dln N dln h
dN=N dh=h
1 h dln TI (Tb TI ) l l dln h
dN h dh N
h N
From Eq. (A.1) h l N Tb TI then the term 1/l(Tb /TI)(h/N ) will be: 1 l (Tb TI ) 1 l (Tb TI ) and the other term: h (Tb TI ) h l (l2 M 3=2 PI Km =RTI2 ) h N
h (Tb TI ) l l (l2 M 3=2 PI Km =RTI2 ) h Tb TI
1 R1 1
where: R1
M 3=2 Km l2 PI RTI2 h
then: S(N; h) 1
1 R1 R1 1 R1 1
dN=N dN Km dKm =Km dKm N pffiffiffiffiffiffi From Eq. (A.1): N /Km/ M/(PI /PV). pffiffiffiffiffiffi pffiffiffiffiffiffiffi dN Km M (dPI )(PI PV ) M dKm S(N; Km )
@ln N
@ln Km
(A:8)
or pffiffiffiffiffiffi dPI pffiffiffiffiffiffi Km M (PI PV ) M dKm dKm dN
(A:9)
Now (dPI/dKm) is needed. Again, Equalize Eqs. (A.1) and (A.2) for N to get: h(T TI ) Km pffiffiffiffiffiffib l M (PI PV ) then h dKm pffiffiffiffiffiffi l M (PI PV )( dTI ) (Tb TI )dPI (PI PV )2 and dKm h pffiffiffiffiffiffi l M dPI (dTI =dPI ) ( (Tb TI )=(PI PV )) (PI PV ) from the Clausius /Claperyon equation, dPI dTI
lMPI RTI2
and from Eqs. (A.1) and (A.2): pffiffiffiffiffiffi Tb TI lKm M h PI PV then h pffiffiffiffiffiffi l M dPI pffiffiffiffiffiffi (RTI2 =lMPI ) ((lKm M )=h) (PI PV )
dKm
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
pffiffiffiffiffiffi l M h dKm PI PV p ffiffiffiffiffi ffi (RTI2 =lMPI ) ((lKm M )=h) dPI
dPI dKm
f g
(RTI2 h=M 3=2 l2 PI ) Km
or dPI 1 PI PV Km dKm RTI2 h=M 3=2 Km l2 PI (1) but RTI2 h M 3=2 Km l2 PI
Then:
R1 S(N; Km ) 1 1 R1
1 R1
then (A:10)
Now substitute Eqs. (A.9) and (A.10) into Eq. (A.8) to get: dln N dln Km
dN=N dKm =Km
1 (1 R1 )
:
3-The normalized sensitivity factor S (N , Tb).
1 PI PV : Km dKm (1=R1 ) (1) dPI
S(N; Km )
1 R1 (1=R1 ) (1) 1 R1
and the second part of Eq. (A.11) can be written as: pffiffiffiffiffiffi Km ((PI PV ) M ) N pffiffiffiffiffiffi 1 ((PI PV ) M ) pffiffiffiffiffiffi 1: M (PI PV )
PI PV
85
S(N; Tb )
pffiffiffiffiffiffi dPI pffiffiffiffiffiffi Km M (PI PV ) M dKm K m (A:11) N
the first part of the Eq. (A.11) can be written as:
pffiffiffiffiffiffi dPI Km Km M dKm N pffiffiffiffiffiffi 1 PI PV (Km M ) Km (1=R1 ) (1) 1 pffiffiffiffiffiffi M (PI PV )
dN Tb dTb N
(A:12)
h N (Tb TI ) l
dN Km dKm N
@ln N @ln Tb
dN
h l
dTb
h l
dTI
or: dN
h h dTI : dTb l l dTb
(A:13)
So dTI/dTb is needed. pffiffiffiffiffiffi Km M l (PI PV )TI Tb h
dTb
Km
pffiffiffiffiffiffi Ml dPI dTI h
Km
pffiffiffiffiffiffi M l dPI
or: dTb dTI
h
dTI
1
from the Clausius /Claperyon Eq. (A.3)
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F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
pffiffiffiffiffiffi dPI dN Km M 1 : dPV dPV
dPI lMPI dTI RTI2
So dPI/dPV is needed. From Eqs. (A.1) and (A.2): pffiffiffiffiffiffi pffiffiffiffiffiffi h (Tb TI ) Km M PI Km M PV l
then: dTb M 3=2 Km l2 PI 1 R1 1 dTI RTI2 h or: dTI dTb
or
1 1 R1
(A:14)
:
Substitute Eq. (A.14) in Eq. (A.13) to get: dN h h dTI h dT 1 I dTb l l dTb l dTb h 1 h R1 1 l 1 R1 l 1 R1
PV PI
h pffiffiffiffiffiffi (Tb TI ) lKm M
@ln N
S(N; Tb )
R1
Tb
1 R1 (Tb TI )
then dPV dPI
dPV dPI
1
h dTI pffiffiffiffiffiffi : lKm M dPI
dPI lMPI dTI RTI2
:
then dPV
Tb TI Tb
dPI
1
RTI2 h M 3=2 Km l2 PI
but
then: S(N; Tb )
RTI2 h
R1 : (1 R1 )R2
M 3=2 Km l2 PI
4-The normalized sensitivity factor S (N , PV). S(N; PV ) N Km
h pffiffiffiffiffiffi dTI lKm M
From the Clausius /Claperyon equation:
Let R2
and
dN Tb
@ln Tb dTb N h R1 lTb l 1 R1 h(Tb TI )
h pffiffiffiffiffiffi (Tb TI ) PV PI lKm M
then
@ln N dN PV @ln PV dPV N
pffiffiffiffiffiffi M (PI PV )
pffiffiffiffiffiffi dN Km M fdPI dPV g or
(A:16)
1 R1
then dPV
(A:15)
dPI
1 R1 R1
or: dPI dPV
R1 1 R1
:
(A:17)
Substitute Eqs. (A.16) and (A.17) into Eq. (A.15):
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
S(N; PV )
Km
@ln N dN PV @ln PV dPV N
pffiffiffiffiffiffi dPI PV M 1 dPV N
Km
pffiffiffiffiffiffi R1 P pffiffiffiffiffiffi V M 1 1 R1 Km M (PI PV ) 1
PV
1 R1 PI PV
:
Let R3
PI PV PV
then S(N; PV )
1 (1 R1 )R3
References [1] S. Bandini, A. Savedra, G. Sarti, Vacuum membrane distillation: experiments and modeling, AIChE J. 43 (1997) 398.
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[2] E. Hoffman, D. Pfenning, E. Philippsen, P. Schwahn, M. Seiber, D. Woermann, Evaporation of alcohol/water mixtures through hydrophobic porous membranes, J. Member. Sci. 34 (1987) 199. [3] G. Sarti, C. Gostoli, S. Bandini, Extraction of organic components from aqueous streams by vacuum membrane distillation, J. Member. Sci. 80 (1993) 21. [4] G. Sarti, C. Gostoli, Separation of liquid mixtures by membrane distillation, J. Member. Sci. 41 (1989) 211. [5] J. Wijmans, J. Kaschemetat, J. Davidson, W. Baker, Treatment of organic contaminated wastewater streams by pervaporation, Environ. Prog. 9 (1990) 262. [6] V. Lipski, P. Cote, The use of pervaporation for the removal of organic contaminants from water, Environ. Prog. 9 (1990) 254. [7] S. Bandini, C. Gostoli, G. Sarti, Separation efficiency in vacuum membrane distillation, J. Member. Sci. 73 (1992) 217. [8] S. Bandin, G. Sari, C. Gostoli, Vacuum membrane distillation: pervaporation through porous hydrophobic membranes, in: R. Bakish, (Ed.) Proceedings of Third International Conference on Pervaporation in Chemical Industry, Nancy, France, September 19 /22, 1988, p. 117. [9] F. Banat, J. Simandl, Removal of benzene traces from contaminated water by vacuum membrane distillation, Chem. Eng. Sci. 51 (1996) 1257. [10] K.S. Spiegler, Y. El-Sayed, A Desalination Primer, Balaban Desalination Publications, Rehovot, 1994. [11] F. Dullien, Porous Medi-Fluid Transport and Pore Structure, Academic Press, London, 1979. [12] A. Miguel, M. Jose, S. Fernanda, N. Rosa, S. Julila, Application of parametric sensitivity to batch process safety: theoretical and experimental studies, Chem. Eng. Technol. 19 (1996) 222.
Desalination by vacuum membrane distillation: sensitivity analysis Fawzi Banat *, Fahmi Abu Al-Rub, Khalid Bani-Melhem Department of Chemical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan Received 14 December 2001; received in revised form 19 December 2002; accepted 19 December 2002
Abstract In order to enhance the performance of the vacuum membrane distillation process in desalination, i.e. to get more flux, it is necessary to study the effect of operating parameters on the yield of distillate water. Simple techniques including the normalized dimensionless sensitivity factor and temperature polarization coefficient as well as the solution of the transport models were used for this purpose. The sensitivity of the mass flux to the process operating parameters including downstream pressure, feed temperature, feed flow rate, and membrane permeability were investigated. The mass flux of distillate water was highly sensitive to the feed temperature especially at high values of vacuum pressure. The mass flux was more sensitive to the vacuum pressure at low feed temperature levels than at the high ones. Since lowering the temperature polarization coefficient is essential to enhance the process performance considerable efforts should be directed toward maximizing the heat transfer coefficient through better module design. The predictions of the normalized dimensionless sensitivity analysis were in agreement with the results obtained from solving the transport model of the process. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Desalination; Vacuum membrane distillation; Sensitivity analysis
1. Introduction Vacuum membrane distillation, like any membrane distillation process, is a thermally driven process in which the convective mass transfer is the dominant mechanism for mass transfer. The driving force is maintained by applying vacuum at the downstream side to keep the pressure at this side below the equilibrium vapor pressure (see Fig. 1).
* Corresponding author. Fax: /962-2-709-5018. E-mail address: [email protected] (F. Banat).
The membrane in this process is a physical support for the vapor/liquid interface and does not affect the selectivity associated with the vapor/liquid equilibrium. In contrast, pervaporation process depends mainly on using a dense membrane, which alters the vapor /liquid equilibrium [1]. Vacuum membrane distillation has the potential to be one of the most common techniques used to separate dilute aqueous mixtures. Applications include the removal of ethanol from fermentation broth to prevent the inhibition of microorganisms used in fermentation [2] and for the removal of trace concentrations of VOC’s from water [1]. In
1383-5866/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S1383-5866(02)00221-6
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Fig. 1. Schematic drawing of vacuum membrane distillation process (1) feed side, (2) membrane, (3) vacuum side.
addition, VMD can be used when the feed contains non-volatile salt as in desalination. Depending on the type of application, the importance of the different design parameters in VMD is determined. For example, while in the removal of ethanol from fermentation broth, both the separation factor and the flux are the most important design parameters; the flux is the most important design parameter in the case where water contains non-volatile solutes. However, when water contains dissolved gases, maximizing the solute flux and minimizing the water flux are the most important design parameters. Compared with conventional separation techniques, VMD has many advantages: The vacuum can be applied only on a very limited volume of the membrane equipment and not on the entire equipment. VMD can be also operated at relatively low evaporation temperatures; typically below 50/60 8C. Thus, a low cost energy source is required to supply the heat for evaporation. Economically, VMD is found to be comparable [3,4] with respect to the separation costs of other membrane alternatives such as pervaporation [5,6]. Recently, VMD has become an active area of research by many. Most of the published work has focused on ethanol/water separation [2]. Other researchers studied the use of VMD in the removal of trace gases and VOC’s from water [1,7 /9]. To the author’s knowledge, the work of Bandini et al. [7] was the first and the only study to
investigate the effect of different variables on the VMD efficiency. They used the dimensionless sensitivity approach to study the sensitivity of the total flux in VMD to heat and mass transfer coefficients. They proposed a simple criterion that can be used to establish, without the need to solve the transport model, whether the total flux is controlled by heat or by mass transfer resistance. However, their study only focused on the sensitivity of mass flux to variations in the heat and mass transfer coefficients and did not tackle the sensitivity of mass flux to process operating parameters such as the vacuum pressure and the temperature of the feed stream. The objective of this study is therefore to extend the sensitivity analysis by Bandini et al. [7] to include the sensitivity of the mass flux to the controllable operating parameters of the VMD process such as the vacuum pressure, and feed temperature, again using pure water for demonstration. In addition to the dimensionless sensitivity approach, a temperature polarization coefficient-based analysis was performed to assess the effect of these parameters on the process performance. This approach requires solving the transport model of the process.
2. Theory The base-case that is considered in studying the relative importance of process parameters is the production of distillate water from water containing non-volatile solutes. Since the presence of nonvolatile matters such as salts in the feed solution only affects the absolute amount of produced flux but not the process sensitivity toward any process variable, salt-free water was considered as a feed solution. In fact, seawater of 35 000 ppm salinity has a vapor pressure which is approximately 1.84% lower than that of pure water [10]. Bandini et al. [7] made a similar assumption. 2.1. Mass and heat transfer in VMD As in any membrane distillation process, VMD is characterized by simultaneous heat and mass transfer. Mass transfer through the membrane process is associated with diffusive and convective
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
77
transport of vapors through the microporous membrane. As is widely known, the mass transfer through a porous media depends mainly on the pore size of the membrane and the mean free molecular path of the transferring species. These two factors determine what mechanism of mass transfer will be dominant. In most of the membranes used in VMD, the pore size is usually smaller than the mean free path. Hence, the Knudsen mechanism is dominant [11]. Accordingly, the mass flux N is linearly related to the pressure difference across the membrane. pffiffiffiffiffiffi N Km M (PI PV ) (1)
The numerical value of u lies within the range [0 /1]. As u 0/0.0, TI 0/Tb, in this case the resistance in the liquid phase is negligible and the process is controlled by the resistance of the membrane. When u 0/1, TI 0/Tv so the resistance in the membrane is negligible and the process is controlled in the liquid phase. Thus, the polarization factor can be used as a tool in studying the process behavior.
where Km (s mol0.5/kg0.5 m) is the membrane permeability which depends on membrane characteristics and feed temperature. M is the molecular weight (kg/mol), PI is the interfacial partial pressure and PV is the downstream pressure. In the case of one volatile component in the feed, PI will be the vapor pressure of that component. The heat required for the evaporation at the interface is supplied by the heat flux through the liquid stream. The evaporation temperature at the interface is related to the feed bulk temperature in the liquid phase by the equation:
Before proceeding in studying the sensitivity of the VMD process to the process parameters, the following sensitivity factors are defined.
Nl h(Tb TI )
2.2. Temperature polarization The temperature polarization phenomenon occurs as a result of a temperature difference between the feed bulk temperature Tb, interfacial temperature TI, and the temperature in the vacuum side Tv. This temperature gradient is due to the heat flux through the liquid layer, which is needed to provide the required heat for evaporation at the membrane interface. The temperature variation across the membrane can be described by the temperature polarization factor that is defined as: Tb TI Tb Tv
1) the first order sensitivity factor of any model response R with respect to any of the model input parameters (Pi) is defined as [12]: s(R; Pi )
@R @Pi
(4)
2) the normalized dimensionless sensitivity factor is defined as [12]:
(2)
where h is the heat transfer coefficient in the liquid phase, and l is the latent heat of vaporization.
u
2.3. Sensitivity analysis
(3)
S(R; Pi)
@ln R @ln Pi
Pi @R R @Pi
s(R; Pi )
Pi R
(5)
where Pi is any parameter that may affect R . In the case of pure water, R represents the mass flux (N ) and Pi may be any one of the input parameters affecting N , i.e. Tb, PV, h and Km. As the parameters studied are not dimensionally homogeneous, normalized (dimensionless) parameters are used to better understand their physical significance. For the present case, the following normalized sensitivity factors are introduced. Expression for these sensitivity factors can be found using equation Eqs. (1) and (2). The final results are presented here while the detailed derivations are in the Appendix A. / The sensitivity factor of mass flux to the heat transfer coefficient; S (N , h):
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F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
@ln N R1 S(N; h) @ln h 1 R1
(6)
s(N; Km )
pffiffiffiffiffiffiffi Km (PI PV )
1
1 R1
(18)
/ The sensitivity factor of mass flux to the permeability of the membrane; S (N , Km): S(N; Km )
@ln N @ln Km
1 1 R1
(7) 3. Results and discussion
/ The sensitivity factor of mass flux to the feed bulk temperature; S (N , Tb): S(N; Tb )
@ln N @ln Tb
R1 (1 R1 )R2
(8)
The sensitivity factor of mass flux to the vacuum pressure; S (N , PV): S(N; Pv )
@ln N @ln Pv
1 (1 R1 )R3
(9)
where: pffiffiffiffiffiffi M l dPI R1 Km h dTI R2 1 R3
PI PV
TI Tb
1
(10) (11) (12)
Using Clausius /Claperyon equation: dPI dTI
lMPI RTI2
(13)
and substituting Eq. (13) into Eq. (10), the expression for R1 can be written as: R1
M 3=2 Km l2 PI RTI2 h
(14)
The first order sensitivity factors of the above parameters are given by the following equations: h R1 s(N; Tb ) (15) l 1 R1 pffiffiffiffiffiffi 1 s(N; PV ) Km M (16) 1 R1 1 R1 s(N; h) (Tb TI ) (17) l 1 R1
In order to enhance the process performance, i.e. to get more flux, it is necessary to understand the sensitivity of the process to every input key parameter. In other words, what type of action should be undertaken to improve the flux for a given VMD system? To determine the relative importance of h , Km, Tb and PV on the mass flux, the relative sensitivity factors defined in Eqs. (6) / (9) are used. For the case of pure water in the feed, and at a fixed vacuum pressure, the permeate flux is controlled, as indicated by Eq. (1), by the membrane permeability and the vapor pressure of water at the membrane interface. The interfacial vapor pressure is a function of the interfacial temperature, so that the overall process seems to be controlled by two simultaneous processes; heat transfer through the liquid phase and mass transfer through the membrane. The parameters Tb and PV are responsible for establishing the driving force, therefore the sensitivity factors mentioned in Eqs. (8) and (9) are not only measuring the sensitivity of flux to the given parameters, but they are also measuring the sensitivity of flux to the driving forces associated by the given parameter. This behavior is explained in the coming discussion.
3.1. Sensitivity of the flux to the feed bulk temperature S (N , Tb) The flux and its sensitivity to the feed temperature are shown in Figs. 2 and 3, respectively. As shown in Fig. 2, increasing the feed bulk temperature results in an exponential increase in the pure water flux. This exponential behavior can be attributed to the exponential relationship between water vapor pressure and temperature.
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
79
temperatures. This behavior would be obtained if the first order sensitivity analysis was considered (see Fig. 4). However, when considering the normalized sensitivity as in this work, the large change in first order sensitivity is divided by the large value of the flux, which results in small change in the normalized sensitivity at high Tb.
3.2. Sensitivity of the flux to the vacuum pressure S (N , PV)
Fig. 2. Effect of feed bulk temperature on mass flux.
Figs. 5 and 6 show, respectively, the absolute flux and its sensitivity to the vacuum pressure at different values of the liquid phase heat transfer coefficient. Fig. 5 shows that for a given heat transfer coefficient the flux decreases linearly with the increasing vacuum side pressure. This is due to the decrease in the driving force when vacuum pressure is increased. Fig. 5 shows also that for a given vacuum pressure, the flux of pure water increases by increasing the liquid phase heat transfer coefficient. This is obvious, since increasing the liquid phase heat transfer coefficient reduces the heat transfer resistance, consequently TI approaches Tb, and in doing so the mass transfer driving force is increased.
Fig. 3. Response of S (N , Tb) to the feed bulk temperature.
On the other hand, the sensitivity of the mass flux to the feed bulk temperature, as represented by the normalized sensitivity factor, decreases by increasing the feed bulk temperature. This is due to the fact that at lower feed temperature, the flux is small, so any small change in bulk temperature of the feed results in a large change in the normalized sensitivity. This interesting result seems contradictory to what is shown in Fig. 2 where the flux of water increases sharply at higher
Fig. 4. Response of the first order sensitivity to feed bulk temperature.
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F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
Fig. 5. Effect of vacuum pressure on mass flux.
that there is no effect of the heat transfer coefficient on the S (N , PV) as all of the lines that represent different values of the h are concurring each other. The above results may be analyzed using R2 and R3 which may be considered as measures of the sensitivity of the relative driving force in both resistances with respect to the given parameter. When R2 0/0, TI 0/Tb, hence, the flux is highly sensitive to any small change in the membrane resistance. When R3 0/0, PI 0/PV, hence, the flux becomes highly sensitive to driving forces in the liquid phase and the vacuum side as well. Therefore, at lower temperatures and high vacuum pressure, the mass flux will be sensitive to both parameters PV and Tb. The positive value of S (N , Tb) in Fig. 3 indicates that increasing Tb increases the flux, while the negative value of S (N , PV) in Fig. 6 shows that increasing PV reduces the mass flux. 3.3. Sensitivity of transport coefficients The role of heat and mass transfer can be determined by considering the sensitivity factors of h and Km. According to Eqs. (6) and (7) the sensitivity factors of h and Km are related by the following equation: S(N; h)S(N; Km ) 1
Fig. 6. Response of S (N , PV) to vacuum pressure.
On the other hand, as shown in Fig. 6, the sensitivity of the flux increases as the vacuum pressure is increased. This occurs because an increase in the vacuum pressure decreases linearly the mass flux as given by Eq. (1). Accordingly, the normalized value of the mass flux becomes very sensitive to any small change in PV. Importantly also, the results obtained for different values of the heat transfer coefficient show
(19)
Eq. (19) is in an agreement with the work of Bandini et al. [7]. Since the production of pure water by VMD is characterized by two simultaneous resistances in series, heat and mass, one of these resistances may dominate over the other one. This can be determined by considering the sensitivity of mass flux to these coefficients. To determine the relative importance of h and Km on the mass flux, the relative sensitivity factors of the mass flux to these parameters are used. Fig. 7 shows the process sensitivity factors as a function of h for two bulk feed temperatures and for two different values of membrane permeability (Km / 2.0 /10 6 and Km /1.5 /10 5 s mol0.5/kg 0.5 m) [8]. It is clear that in the low range of heat transfer coefficient, the changes in h have the greatest relative influence on N , significantly more than
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
Fig. 7. Response of S (N , h ) to liquid phase heat transfer coefficient.
Km, so that the process is controlled by the heat transfer resistance in the liquid film. In order to increase the flux, the heat transfer coefficient should be increased. Low heat transfer coefficients result in high temperature polarization. In this case, the difference between the bulk and the interfacial temperature increases and T1 approaches TV. If this occurs, the interfacial temperature is lowered and consequently the vapor pressure of pure water in the feed decreases resulting in a lower flux. Also, as shown in Fig. 7, as the feed temperature increases, the process becomes more sensitive to the heat transfer resistance in the liquid film. This can be attributed to the fact that the vapor pressure of water is more sensitive at high temperatures than at low ones. To elucidate this numerically, the vapor pressure of water at 25 8C is 23.76 mmHg and at 25.5 8C is 24.47 mmHg, while at 80 8C it is 355.1 mmHg and at 80.5 8C is 363 mmHg. A quick calculation shows that the temperature drop of 0.5 at 25.5 will lower the vapor pressure by a value of less than 1 mm Hg while at 80.5 8C, a temperature drop by 0.5 8C will lower the vapor pressure by about 8 mmHg. This means that careful consideration should be directed toward the heat transfer coefficient when operating at high temperatures. Lowering the
81
temperature polarization is essential to enhance the process performance. On the other hand, when the heat transfer coefficient is greater than about (1000 W/m2 K), the membrane permeability becomes more important as indicated by Eq. (19). Persistently increasing the heat transfer coefficient reduces the importance of the heat transfer resistance. Ultimately, the process becomes dominated by the membrane permeability (S(N , Km)0/1, S (N , h )0/ 0.0). However, if the membrane permeability is fixed, operating at high feed temperatures requires higher values of h than when operating at lower feed temperatures as shown in Fig. 7. This emphasizes the need to keep h as high as possible. Therefore, when (S (N , Km) 0/1.0), the process is completely controlled by the membrane permeability. To increase the permeate flux, the permeability of the membrane should be increased. This is more apparent at low temperatures than at high ones. Similarly, when (S (N , h)0/1), the liquid film heat transfer coefficient should be increased in order to increase the permeate flux, no matter how high the membrane permeability is. This discussion clearly shows that this process is a combined heat and mass transfer process. On the other hand, for the two membranes, the sensitivity factor decreases as the membrane permeability increases, and thus the process starts to be sensitive to the liquid phase heat transfer coefficient. This is clearly shown in Fig. 8 where the sensitivity factor S (N , Km) as a function of the membrane permeability is plotted. Physically, VMD will be heat transfer limited if the module design does not provide adequate heat transfer to the membrane surface. Conversely, the process will be mass transfer limited if the membrane permeability is too low. To improve the process performance, the membrane permeability and/or the module design should be improved. The membrane performance is mainly determined by the pore size and the membrane thickness. Higher fluxes require larger pore size and thinner membranes. The larger the pore size, the lower is the liquid entry pressure. The liquid entry pressure determines the repulsive properties of the membrane. If the feed pressure is higher than the liquid entry pressure then the liquid will flow continu-
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Fig. 8. Response of S (N , Km) to membrane permeability in VMD.
Fig. 9. Effect of feed temperature on VMD temperature polarization factor.
ously through the pores and no selective separation is achieved. In addition to that, the membrane should be thick enough to withstand the mechanical pressure in the process. As a result, there is little scope for improvement by developing special MD membranes. Attention would be better focused on designing VMD modules to provide high heat transfer coefficients. Practically, if VMD has been used for desalination, it will be operated at low pressures with a good membrane permeability to maximize the flux. Therefore, the process will be heat transfer limited. So maximizing the heat transfer coefficient by a good module design maximizes the mass flux.
Apparently, the polarization factor increases by increasing feed bulk temperature. When u 0/0.0, TI 0/Tb and the process are controlled by the membrane resistance. Actually it is important to keep u as small as possible. To overcome the increase in u that results from Tb increase, the heat transfer coefficient in the liquid layer bounding the membrane surface should be increased as well. This is consistent with the discussion aforementioned. The increase of the u factor with bulk temperature is expected since increasing the bulk temperature increases the mass flux that requires sufficient heat of vaporization. Providing such a heat increases the difference between the bulk temperature and the interfacial temperature, and in turn the polarization factor.
3.4. Temperature polarization factor Another methodology to study the dependency of flux on the heat transfer coefficient can be obtained by solving the transport model equations for the flux and temperature polarization. The concept of the temperature polarization factor will be used as a tool for evaluating the effect of the input parameters on maximizing the mass flux. The polarization coefficient as a function of feed bulk temperature is shown in Fig. 9. The model has been solved for three different values of vacuum pressure 500, 2000 and 5000 N/m2.
4. Conclusions Vacuum membrane distillation is a process that can be used for the production of distillate water from brackish or seawater. To priori know the parameters to which the process is sensitive, a sensitivity analysis study was carried out for this process. The main parameters considered in this study were vacuum pressure, membrane perme-
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
ability, feed temperature, and liquid side heat transfer coefficient (fluid flow). A simple approach known as the normalized dimensionless sensitivity factor technique along with the temperature polarization coefficient were used to determine the sensitivity of the process to each of the aforementioned parameters. The role of the heat transfer resistance in the liquid phase and the mass transfer resistance in the membrane matrix was also considered in this study. The process was determined to be sensitive to both the vacuum pressure and the feed temperature but was more sensitive to the feed temperature at high vacuum pressure levels and more sensitive to the vacuum pressure at low values of bulk feed temperatures. Improvement of membrane characteristics in terms of its permeability is important but more important is the improvement of module designs to provide high heat transfer coefficients. The temperature polarization factor-based analysis gave similar results as the dimensionless normalized approach. However, the later approach is handier to use than the former since it does not require solving the whole transport model.
Appendix A
dTI
lMPI
(A:3)
RTI2
1-The normalized sensitivity factor S (N , h ). S(N h)
@ln N @ln h
dN=N dh=h
dN h dh N
Then to get dN /dh , from Eq. (A.2) h N (Tb TI ) l
dh
1 h dTI (Tb TI ) l l dh
(A:5)
Now dTI/dh is needed. Equalize Eqs. (A.1) and (A.2) for N to get: pffiffiffiffiffiffi h (Tb TI ) Km M (PI PV ) l or: h lKm
pffiffiffiffiffiffi PI PV M Tb TI
dh lKm
pffiffiffiffiffiffi(Tb TI )dPI (PI PV )dTI M (Tb TI )2
pffiffiffiffiffiffi dh lKm M dTI (Tb TI )dPI =dTI (PI PV ) (Tb TI )2
dh dTI
lKm
pffiffiffiffiffiffi M
(dPI =dTI ) (PI PV )=(Tb TI )
(Tb TI )
from the Clausius /Claperyon Eq. (A.3):
and the Clausius /Claperyon equation:
dN
Tb T h dh I dh dTI l l l
or
Derivation of Eqs. (6) /(9). The derivation of Eqs. (6) /(9) is based on the permeate flux which can be written as: pffiffiffiffiffiffi N Km M (PI PV ) (A:1) Nl h(Tb TI ) (A:2) dPI
dN
83
(A:4)
dPI lMPI dTI RTI2 and from Eqs. (A.1) and (A.2): PI PV h pffiffiffiffiffiffi Tb TI lKm M Then: pffiffiffiffiffiffi(lMPI =RTI2 ) (h=lKm dh lKm M (Tb TI ) dTI
pffiffiffiffiffiffi M)
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F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
or
2-The normalized sensitivity S (N , Km).
dh (l2 M 3=2 PI Km =RTI2 ) h dTI (Tb TI )
(A:6)
or: dTI dh
(Tb TI ) (l2 M 3=2 PI Km =RTI2 ) h
(A:7)
Now, substitute equations Eqs. (A.5) and (A.7) in Eq. (A.4) to get: S(N; h)
dln N dln h
dN=N dh=h
1 h dln TI (Tb TI ) l l dln h
dN h dh N
h N
From Eq. (A.1) h l N Tb TI then the term 1/l(Tb /TI)(h/N ) will be: 1 l (Tb TI ) 1 l (Tb TI ) and the other term: h (Tb TI ) h l (l2 M 3=2 PI Km =RTI2 ) h N
h (Tb TI ) l l (l2 M 3=2 PI Km =RTI2 ) h Tb TI
1 R1 1
where: R1
M 3=2 Km l2 PI RTI2 h
then: S(N; h) 1
1 R1 R1 1 R1 1
dN=N dN Km dKm =Km dKm N pffiffiffiffiffiffi From Eq. (A.1): N /Km/ M/(PI /PV). pffiffiffiffiffiffi pffiffiffiffiffiffiffi dN Km M (dPI )(PI PV ) M dKm S(N; Km )
@ln N
@ln Km
(A:8)
or pffiffiffiffiffiffi dPI pffiffiffiffiffiffi Km M (PI PV ) M dKm dKm dN
(A:9)
Now (dPI/dKm) is needed. Again, Equalize Eqs. (A.1) and (A.2) for N to get: h(T TI ) Km pffiffiffiffiffiffib l M (PI PV ) then h dKm pffiffiffiffiffiffi l M (PI PV )( dTI ) (Tb TI )dPI (PI PV )2 and dKm h pffiffiffiffiffiffi l M dPI (dTI =dPI ) ( (Tb TI )=(PI PV )) (PI PV ) from the Clausius /Claperyon equation, dPI dTI
lMPI RTI2
and from Eqs. (A.1) and (A.2): pffiffiffiffiffiffi Tb TI lKm M h PI PV then h pffiffiffiffiffiffi l M dPI pffiffiffiffiffiffi (RTI2 =lMPI ) ((lKm M )=h) (PI PV )
dKm
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
pffiffiffiffiffiffi l M h dKm PI PV p ffiffiffiffiffi ffi (RTI2 =lMPI ) ((lKm M )=h) dPI
dPI dKm
f g
(RTI2 h=M 3=2 l2 PI ) Km
or dPI 1 PI PV Km dKm RTI2 h=M 3=2 Km l2 PI (1) but RTI2 h M 3=2 Km l2 PI
Then:
R1 S(N; Km ) 1 1 R1
1 R1
then (A:10)
Now substitute Eqs. (A.9) and (A.10) into Eq. (A.8) to get: dln N dln Km
dN=N dKm =Km
1 (1 R1 )
:
3-The normalized sensitivity factor S (N , Tb).
1 PI PV : Km dKm (1=R1 ) (1) dPI
S(N; Km )
1 R1 (1=R1 ) (1) 1 R1
and the second part of Eq. (A.11) can be written as: pffiffiffiffiffiffi Km ((PI PV ) M ) N pffiffiffiffiffiffi 1 ((PI PV ) M ) pffiffiffiffiffiffi 1: M (PI PV )
PI PV
85
S(N; Tb )
pffiffiffiffiffiffi dPI pffiffiffiffiffiffi Km M (PI PV ) M dKm K m (A:11) N
the first part of the Eq. (A.11) can be written as:
pffiffiffiffiffiffi dPI Km Km M dKm N pffiffiffiffiffiffi 1 PI PV (Km M ) Km (1=R1 ) (1) 1 pffiffiffiffiffiffi M (PI PV )
dN Tb dTb N
(A:12)
h N (Tb TI ) l
dN Km dKm N
@ln N @ln Tb
dN
h l
dTb
h l
dTI
or: dN
h h dTI : dTb l l dTb
(A:13)
So dTI/dTb is needed. pffiffiffiffiffiffi Km M l (PI PV )TI Tb h
dTb
Km
pffiffiffiffiffiffi Ml dPI dTI h
Km
pffiffiffiffiffiffi M l dPI
or: dTb dTI
h
dTI
1
from the Clausius /Claperyon Eq. (A.3)
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F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
pffiffiffiffiffiffi dPI dN Km M 1 : dPV dPV
dPI lMPI dTI RTI2
So dPI/dPV is needed. From Eqs. (A.1) and (A.2): pffiffiffiffiffiffi pffiffiffiffiffiffi h (Tb TI ) Km M PI Km M PV l
then: dTb M 3=2 Km l2 PI 1 R1 1 dTI RTI2 h or: dTI dTb
or
1 1 R1
(A:14)
:
Substitute Eq. (A.14) in Eq. (A.13) to get: dN h h dTI h dT 1 I dTb l l dTb l dTb h 1 h R1 1 l 1 R1 l 1 R1
PV PI
h pffiffiffiffiffiffi (Tb TI ) lKm M
@ln N
S(N; Tb )
R1
Tb
1 R1 (Tb TI )
then dPV dPI
dPV dPI
1
h dTI pffiffiffiffiffiffi : lKm M dPI
dPI lMPI dTI RTI2
:
then dPV
Tb TI Tb
dPI
1
RTI2 h M 3=2 Km l2 PI
but
then: S(N; Tb )
RTI2 h
R1 : (1 R1 )R2
M 3=2 Km l2 PI
4-The normalized sensitivity factor S (N , PV). S(N; PV ) N Km
h pffiffiffiffiffiffi dTI lKm M
From the Clausius /Claperyon equation:
Let R2
and
dN Tb
@ln Tb dTb N h R1 lTb l 1 R1 h(Tb TI )
h pffiffiffiffiffiffi (Tb TI ) PV PI lKm M
then
@ln N dN PV @ln PV dPV N
pffiffiffiffiffiffi M (PI PV )
pffiffiffiffiffiffi dN Km M fdPI dPV g or
(A:16)
1 R1
then dPV
(A:15)
dPI
1 R1 R1
or: dPI dPV
R1 1 R1
:
(A:17)
Substitute Eqs. (A.16) and (A.17) into Eq. (A.15):
F. Banat et al. / Separation and Purification Technology 33 (2003) 75 /87
S(N; PV )
Km
@ln N dN PV @ln PV dPV N
pffiffiffiffiffiffi dPI PV M 1 dPV N
Km
pffiffiffiffiffiffi R1 P pffiffiffiffiffiffi V M 1 1 R1 Km M (PI PV ) 1
PV
1 R1 PI PV
:
Let R3
PI PV PV
then S(N; PV )
1 (1 R1 )R3
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87
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