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Engineering of pervaporation systems: Exact and approximate expressions for the average flux during alcohol dehydration by single-pass pervaporation
Separation and Purification Technology 152 (2015) 160–163
Contents lists available at ScienceDirect
Separation and Purification Technology journal homepage: www.elsevier.com/locate/seppur
Engineering of pervaporation systems: Exact and approximate expressions for the average flux during alcohol dehydration by single-pass pervaporation Cilian Ó Súilleabháin a,b, Greg Foley a,⇑ a b
School of Biotechnology, Dublin City University, Dublin 9, Ireland School of Mechanical and Chemical Engineering, Cork Institute of Technology, Ireland
a r t i c l e
i n f o
Article history: Received 9 June 2015 Received in revised form 11 August 2015 Accepted 12 August 2015 Available online 13 August 2015 Keywords: Pervaporation Design Activation energy Exponential integral Average flux
a b s t r a c t Assuming a concentration-independent flux with an Arrhenius dependence on temperature, and using temperature- and composition-averaged physical properties, an exact analytical expression is derived for the average flux in an adiabatic, single-pass pervaporation module. This expression incorporates the Exponential Integral, a special function known to arise in the analysis and design of ultrafiltration systems. The range of feasible activation energies of permeation is established for IPA-water and ethanol–water systems. Within this range, and over the range of typical operating conditions, a simple approximation to the exact analytical expression is derived, namely, J av =Jf ¼ ðJ r =Jf Þ0:52 where Jf is the flux at the feed temperature and Jr is the flux at the retentate temperature. The 0.52 exponent is derived on the basis of minimising the maximum error over the range of conditions studied and the maximum error was found to be less than 1.2% in both systems. A value of n = 0.5 increases this maximum error to just under 2.5% for both systems but average errors in this instance remain below 0.5%. The approximate expression allows for the rapid estimation of pervaporation area for water–ethanol and water–IPA systems. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The most common application of pervaporation is the dehydration of aqueous azeotropes, in particular those of ethanol and isopropanol (IPA) [1]. Hot liquid feed contacts one side of a membrane and vapor permeates through a selective membrane. In industrial systems that operate adiabatically, the latent heat of vaporisation causes the liquid to cool as it passes through the module, leading to a reduction in the local flux. This complicates somewhat the underlying mathematics of these systems and, consequently, the development of a simple and accurate method for computing the average flux as a function of feed and retenate conditions would be an important step forward in the analysis and design of pervaporation systems [2,3]. Previous approaches to the modelling of pervaporation systems have focused on operating conditions, thermodynamic properties of the solution, and solute-membrane properties such as diffusivity. The complexity of these models is such that numerical solutions are often required [2–6]. ⇑ Corresponding author. E-mail address: [email protected] (G. Foley). http://dx.doi.org/10.1016/j.seppur.2015.08.021 1383-5866/Ó 2015 Elsevier B.V. All rights reserved.
While both Feng and Huang [7] and Baker et al. [8] recommend that pervaporation performance data be reported in terms of permeance, the reality remains that most data continues to be reported in terms of flux [1,8]. Thus, for the design engineer at least, it is important to formulate pervaporation design equations in terms of flux. For most systems, flux is a function of both permeability and the saturated vapour pressure of the liquid. Both of these parameters have Arrhenius-type temperature dependencies [7] and thus the flux is typically given by an expression of the form
J ¼ J 0 eEj =RT
ð1Þ
where J0 is the maximum flux through the membrane (i.e. the flux as T ? 1), Ej is the activation energy (a function of both the solution and the membrane) while R is the universal gas constant. By combining the appropriate mass and energy balances with Eq. (1), an expression is derived in this paper for the membrane area required to produce a certain mass flowrate of permeate. Based on this analysis, a simple approximate expression for the average flux is derived, thus providing a straightforward methodology for pervaporation system design. This requires knowledge of the flux at the feed and retentate conditions only. The analytical
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C. Ó Súilleabháin, G. Foley / Separation and Purification Technology 152 (2015) 160–163
solution and the approximate equation are both applicable for the full range of industrial operating conditions for ethanol and IPA dehydrations.
J av ¼
_p m ¼ m_ R 1 f A Jo
_p m
_f _ p =m 1m
ð13Þ
a
ebðb1Þ=s ds
Therefore 2. Model development Pervaporation flux is a function of solute concentration in many applications. However, there are some membranes for which flux is constant across a range of concentrations [9–12]; this work considers such systems. Hence, it is assumed throughout that the local flux is a function of the local temperature only and not the local concentration of the permeating solute. _ be the mass flowrate and let T be the temperature at a Let m point in the channel. The heat balance can then be written in differential form as
_ p TÞ ¼ Jhv dA dðmc
ð2Þ
where J is the local mass flux of permeate, cp is the local specific heat capacity, hv is the local enthalpy per unit mass of vapour and dA represents the differential area. The relevant mass balance is
_ ¼ JdA dm
ð3Þ
Dividing Eq. (2) by Eq. (3) gives
_ p cp T þ mc
dT ¼ hv _ dm
ð4Þ
Rearranging gives
Z
T
cp dT ¼ hv c p T
Tf
Z
_f m
_ dm _ m
_ m
ð5Þ
where the f subscript denotes the feed. Assuming constant or appropriately averaged physical properties, this can be integrated to give
_f m hv hv T¼ Tf _ cp cp m
ð6Þ
Combining Eqs. (1), (3) and (6) gives Ej h i _ m dA 1 R hcvp ðhcvp T f Þ m_f ¼ e _ dm Jo
ð7Þ
Therefore, the total area is found from computing the integral below:
A¼
1 Jo
Z
_f m _r m
_f m hv hv _ exp ðEj =RT f Þ 1 dm _ cp T f cp T f m
ð8Þ
To simplify this expression, we let
s¼
_ m _f m
ð9Þ
Therefore
A¼
_f m J0
Z
1
_ p =m _f 1m
a
ebðb1Þ=s ds
ð10Þ
where
a¼
Ej RT f
ð11Þ
hv cp T f
ð12Þ
and
b¼
From Eq. (10) it follows that the average flux is given by
_ p =m _ f Þea ðm J av ¼ R1 a Jf bðb1Þ=s ds _ e _
ð14Þ
1mp =mf
Carrying out the integration using a symbolic integrator (MathematicaÒ) we get 1 as aðb1Þ Z 1 bðbðs 1Þ þ 1Þebðs1Þþ1 aðb 1Þea=b Ei bðbðs1Þþ1Þ a bðb1Þ=s e ds ¼ 2 _ _ 1mp =mf b _f _ p =m 1m
ð15Þ
Therefore 2
J av b ea p ¼ aðb1Þ J f bea aðb 1Þea=b Ei aðb1Þ bð1 bpÞeað1pÞ 1bp þ aðb 1Þea=b Ei b
bð1bpÞ
ð16Þ where Ei is the Exponential Integral, a special function that has been shown to occur in the analysis of ultrafiltration systems [13], and p is defined by
p¼
_p m _f m
ð17Þ
The Exponential Integral can be computed readily with advanced computation packages but also from its series definition using spreadsheet software [13]. In the next section some numerical computations are presented with this equation and regression analysis is used to derive a simple and accurate approximation for the average flux. 3. Results and discussion While Eq. (16) poses no great computational challenges, it would be desirable to obtain a more user-friendly version of this equation, one that might be analogous to Cheryan’s well-known expression for the average flux in batch ultrafiltration [14]. Cheryan proposed that batch times can be accurately computed by taking the average flux to be the arithmetic mean of the initial and final fluxes, thus avoiding the need for numerical integration. The first task in this process is to establish a realistic range of values for the parameters a, b and p. Industrial systems normally operate with feed temperatures in the range 350–395 K [1]. The typical change in temperature, DT, i.e., (Tf Tr) is 15–30 K [1]. A permeate mass flowrate of 3–5% of the feed is typical for industrial systems [1]. Permeate typically contains at least 80% water and remains essentially fixed for many commercial systems [9,10,12,15–18]. Feed in industrial systems flows through a series of adiabatic pervaporation modules and in order to maintain the flux in each module at sufficiently high values, the retentate is re-heated to the initial feed temperature prior to entering subsequent modules. But over-frequent re-heating leads to complexity; the additional cost of piping and valves can be significant [1]. The optimum membrane area thus depends on the relative cost of membranes versus that of re-heating equipment. Studies have suggested that a fixed temperature drop be used for each stage [2] or that an equal membrane area be used for each stage [2,5]. It is proposed here that a Jr/Jf ratio of 0.4 be used and, on this basis, a maximum commercially feasible value of Ej can be computed. In comparison, Sosa and Espinosa used fixed temperature drops with Jr/Jf ratios of greater than 0.5 [5]. On the other hand, Bausa and Marquadt assumed that the average flux was 80% of the maximum flux: this implies a ratio P 0.65 [2].
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1.0
Building on this approach, one can write
E j Jr ¼ J o eRT r J o e RT f Jf Ej
0.9
Rearranging and applying the 0.4 criterion gives the following expression for the maximum value of Ej for industrial operation:
0.8
Emax ¼ j
R ln 0:4 ð1=T f 1=T r Þ
ð19Þ
Jav / Jf
ð18Þ
0.6
For a given value of Tf the retentate temperature in this equation is given by Eq. (6) as
hv hv 1 Tf 1p cp cp
Tr ¼
0.5
ð20Þ 0.4
which, for p = 0.03, becomes
hv hv 1 Tr ¼ Tf 1 0:03 cp cp
ð21Þ
Jr Jf
ð22Þ
0.7
0.8
0.9
1.0
0.9 0.8 0.7 Exact Approximate
0.6 0.5
!n ð23Þ
Table 2 shows the values of n obtained for both solutions using different ‘best-fit’ criteria. Now, using Eq. (23) the simplified design equation becomes
_ p Jf n m A¼ Jf Jr
0.6
1.0
These equations establish the maximum possible value of the activation energy to make pervaporation feasible. In summary, the procedure for computing the average flux over a range of operating conditions using Eq. (16) is outlined in Table 1. Figs. 1 and 2 are plots of Jav/Jf versus Jr/Jf for the full range of industrial operating conditions examined for ethanol and IPA dehydration. The following equation was proposed to approximate the simulated data.
J av ¼ Jf
0.5
Fig. 1. Jav/Jf versus Jr/Jf for water–ethanol. Solid curve denotes Eq. (23) with n = 0.52.
Jav / Jf
R ln 0:4 ¼ ð1=T f 1=ðT f 30ÞÞ
0.4
Jr / J f
Note that if this value for Tr is such that DT > 30 K, then
Emax j
Exact Approximate
0.7
ð24Þ
where it is at the discretion of the designer as to which value of n to use. In summary, the steps involved in calculating the ‘ideal’ membrane area required for a given separation using the approximate method are outlined in Table 3 below. The approximate approach outlined above is clearly simple and easy to use, and causes errors of less than 1.2% (based on the minimax criterion) for commercial operating conditions. Little data is required for the equation and such data is either readily available or easily determined. Of course the underlying mathematical model on which Eq. (16) is based is itself an approximation and some form of ‘efficiency factor’, not unlike the column efficiency that is sometimes employed in distillation design, will ultimately be needed to compute the real area when sizing industrial systems. Such an efficiency factor is likely to be a function of module design,
0.4 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Jr / J f Fig. 2. Jav/Jf versus Jr/Jf for water–IPA. Solid curve denotes Eq. (23) with n = 0.52.
Table 2 Best-fit values of n according to various criteria.
a
n
Max error (%)
Mean error (%)
Water–ethanol Least squares a Minimax n = 0.5
0.51 0.52 0.5
1.37 1.03 2.31
0.28 0.46 0.33
IPA-Water Least squares a Minimax n = 0.5
0.51 0.52 0.5
1.51 1.10 2.44
0.27 0.45 0.32
Minimisation of maximum error.
operating conditions and fluid properties. In that sense, the work outlined here is merely a first step towards the development of design equations that will give engineers the confidence to engage
Table 1 Algorithm for generating average flux ‘data’.
For a given value of Tf and feed composition, choose a value of p between 0.03 and 0.05 inclusive, and a permeate composition between 0.8 and 1.0 inclusive. Calculate the maximum value of Ej for these conditions. Choose a value of Ej less than or equal to this value. Evaluate the retenate composition by mass balance. Estimate a value of Tr and using this value, compute the weighted average values of hv and cp using appropriate data sources [19,20]. Iterate on Eq. (20) to get Tr. Compute Jr/Jf . Evaluate parameters a and b using Eqs. (11) and (12). Compute Jav/Jf using Eq. (16) Repeat for all combinations of the following parameters across the full range of industrial conditions: permeate-to-feed ratio, feed composition, permeate composition, feed temperature and Ej.
C. Ó Súilleabháin, G. Foley / Separation and Purification Technology 152 (2015) 160–163
163
Table 3 Algorithm for approximate calculation of membrane area.
For a given feed, choose a membrane for which the Arrhenius flux relationship and the permeate composition is known or can be determined by experiment. Choose a suitable feed temperature. Check that the Ej value is feasible using Eqs. (21) and (22). Choose a desired permeate/feed ratio. Determine the retentate composition by mass balance and Tr by iteration as described previously. Calculate Jr/Jf. Calculate the required membrane area using Eq. (24).
with pervaporation technology. Ongoing work involves applying the methodology outlined here to other dehydration processes, the expectation being that other systems will be characterised by different values of n in Eq. (23). In addition, innovative ways of operating pervaporation modules are being examined and new ways of selecting the best membrane for any given separation are being investigated. Further work will focus on the optimisation of multi-stage systems. There are two obvious areas where the current model can be further developed. The first is to go beyond the use of averaged physical properties and to incorporate the full temperature and concentration dependence of all properties into the analysis. The simplest way to do this is to work with the underlying differential equations rather than pursing an integral approach. In that case, numerical solutions will inevitably be needed. The second area would involve incorporating a composition-dependent flux term and again this would suggest a solution method based on numerical solution of the underlying differential equations. In addition to the modelling work, experimental work is required to determine the ‘efficiency factor’ mentioned above, and its dependence on operating conditions, membrane properties and fluid properties. 4. Conclusions This paper has described the first steps towards the development of easy-to-use engineering design equations for pervaporation. The general approach, which is based ultimately on an experimentally determined expression for the flux as a function of temperature, combined with mass and energy balances, is one that will be familiar to all chemical engineers. It is to be hoped that the familiarity of this approach will cause engineers to consider more seriously the use of pervaporation technology for not only alcohol dehydration processes but many other applications as well [21]. References [1] R.W. Baker, Membrane Technology and Applications, third ed., Wiley, West Sussex, UK, 2012. [2] J. Bausa, W. Marquardt, Shortcut design for hybrid membrane/distillation processes for the separation of non-ideal multicomponent mixtures, Ind. Eng. Chem. Res. 39 (2000) 1658–1672.
[3] J. Marriott, E. Sørensen, The optimal design of membrane systems, Chem. Eng. Sci. 58 (2003) 4991–5004. [4] N. Valentinyi, P. Mizsey, Comparison of pervaporation models with simulation of hybrid separation processes, Periodica Polytechnica Chem. Eng. 58 (1) (2014) 7–14. [5] M.A. Sosa, J. Espinosa, Feasibility analysis of isopropanol recovery by hybrid distillation/pervaporation process with the aid of conceptual models, Sep. Purif. Technol. 78 (2011) 237–244. [6] J.P.G. Villaleunga, Y. Cohen, Numerical model of non-isothermal pervaporation in a rectangular channel, J. Memb. Sci. 260 (2005) 119–130. [7] X. Feng, R.Y.M. Huang, Liquid separation by membrane pervaporation: a review, Ind. Eng. Chem. Res. 36 (1997) 1048–1066. [8] R.W. Baker, J.G. Wijmans, Y. Huang, Permeability, permeance and selectivity: a preferred way of reporting pervaporation performance data, J. Memb. Sci. 348 (2010) 346–352. [9] M. Pera-Titus, J. Llorens, J. Tejero, F. Cunill, Description of the pervaporation dehydration performance of A-type zeolite membranes: a modeling approach based on the Maxwell-Stefan theory, Catal. Today 118 (2006) 73–84. [10] S. Sommer, T. Melin, Influence of operation parameters on the separation of mixtures by pervaporation and vapor permeation with inorganic membranes. Part 1: Dehydration of solvents, Chem. Eng. Sci. 60 (2005) 4509–4523. [11] V. Van Hoof, C. Dotremont, A. Buekenhoudt, Performance of Mitsui NaA type zeolite membranes for the dehydration of organic solvents in comparison with commercial polymeric pervaporation membranes, Sep. Purif. Technol. 48 (2006) 304–309. [12] A.W. Verkerk, P. Van Male, M.A.G. Vorstman, J.T.F. Keurentjes, Description of dehydration performance of amorphous silica pervaporation membranes, J. Memb. Sci. 193 (2001) 227–238. [13] G. Foley, Three classic ultrafiltration problems solved with the exponential integral, Educ. Chem. Eng. 6 (2011) e90–e96. [14] M. Cheryan, Ultrafiltration and Microfiltration Handbook, CRC Press, Boca Raton, 1998. [15] T. Gallego-Lizon, E. Edwards, G. Lobiundo, L. Freitas dos Santos, Dehydration of water/t-butanol mixtures by pervaporation: comparative study of commercially available polymeric, microporous silica and zeolite membranes, J. Memb. Sci 197 (2002) 309–319. [16] K. Koch, A. Górak, Pevaporation of binary and ternary mixtures of acetone, isopropyl alcohol and water using polymeric membranes: experimental characterisation and modelling, Chem. Eng. Sci. 15 (2014) 95–114. [17] A. Urtiaga, E.D. Gorri, C. Casado, I. Ortiz, Pervaporative dehydration of industrial solvents using a zeolite NaA commercial membrane, Sep. Purif. Technol. 32 (2003) 207–213. [18] V. Van Hoof, L. Van den Abeele, C. Dotremont, A. Buekenhoudt, R. Leysen, Economic comparison between azeotropic distillation and different hybrid systems combining distillation with pervaporation for the dehydration of isopropanol, Sep. Purif. Technol. 37 (2004) 33–49. [19] C.L. Yaws, Chemical Properties Handbook, McGraw-Hill, New York, UK, 1999. [20] V. Majer, V. Svoboda, Enthalpies of Vaporization of Organics Compounds, IUPAC Chemical Data Series No. 32, Blackwell Scientific Publications, Oxford, 1985. [21] B. Van der Bruggen, P. Luis, Pervaporation as a tool in chemical engineering: a new era?, Curr Opin. Chem. Eng. 4 (2014) 47–53.
Contents lists available at ScienceDirect
Separation and Purification Technology journal homepage: www.elsevier.com/locate/seppur
Engineering of pervaporation systems: Exact and approximate expressions for the average flux during alcohol dehydration by single-pass pervaporation Cilian Ó Súilleabháin a,b, Greg Foley a,⇑ a b
School of Biotechnology, Dublin City University, Dublin 9, Ireland School of Mechanical and Chemical Engineering, Cork Institute of Technology, Ireland
a r t i c l e
i n f o
Article history: Received 9 June 2015 Received in revised form 11 August 2015 Accepted 12 August 2015 Available online 13 August 2015 Keywords: Pervaporation Design Activation energy Exponential integral Average flux
a b s t r a c t Assuming a concentration-independent flux with an Arrhenius dependence on temperature, and using temperature- and composition-averaged physical properties, an exact analytical expression is derived for the average flux in an adiabatic, single-pass pervaporation module. This expression incorporates the Exponential Integral, a special function known to arise in the analysis and design of ultrafiltration systems. The range of feasible activation energies of permeation is established for IPA-water and ethanol–water systems. Within this range, and over the range of typical operating conditions, a simple approximation to the exact analytical expression is derived, namely, J av =Jf ¼ ðJ r =Jf Þ0:52 where Jf is the flux at the feed temperature and Jr is the flux at the retentate temperature. The 0.52 exponent is derived on the basis of minimising the maximum error over the range of conditions studied and the maximum error was found to be less than 1.2% in both systems. A value of n = 0.5 increases this maximum error to just under 2.5% for both systems but average errors in this instance remain below 0.5%. The approximate expression allows for the rapid estimation of pervaporation area for water–ethanol and water–IPA systems. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The most common application of pervaporation is the dehydration of aqueous azeotropes, in particular those of ethanol and isopropanol (IPA) [1]. Hot liquid feed contacts one side of a membrane and vapor permeates through a selective membrane. In industrial systems that operate adiabatically, the latent heat of vaporisation causes the liquid to cool as it passes through the module, leading to a reduction in the local flux. This complicates somewhat the underlying mathematics of these systems and, consequently, the development of a simple and accurate method for computing the average flux as a function of feed and retenate conditions would be an important step forward in the analysis and design of pervaporation systems [2,3]. Previous approaches to the modelling of pervaporation systems have focused on operating conditions, thermodynamic properties of the solution, and solute-membrane properties such as diffusivity. The complexity of these models is such that numerical solutions are often required [2–6]. ⇑ Corresponding author. E-mail address: [email protected] (G. Foley). http://dx.doi.org/10.1016/j.seppur.2015.08.021 1383-5866/Ó 2015 Elsevier B.V. All rights reserved.
While both Feng and Huang [7] and Baker et al. [8] recommend that pervaporation performance data be reported in terms of permeance, the reality remains that most data continues to be reported in terms of flux [1,8]. Thus, for the design engineer at least, it is important to formulate pervaporation design equations in terms of flux. For most systems, flux is a function of both permeability and the saturated vapour pressure of the liquid. Both of these parameters have Arrhenius-type temperature dependencies [7] and thus the flux is typically given by an expression of the form
J ¼ J 0 eEj =RT
ð1Þ
where J0 is the maximum flux through the membrane (i.e. the flux as T ? 1), Ej is the activation energy (a function of both the solution and the membrane) while R is the universal gas constant. By combining the appropriate mass and energy balances with Eq. (1), an expression is derived in this paper for the membrane area required to produce a certain mass flowrate of permeate. Based on this analysis, a simple approximate expression for the average flux is derived, thus providing a straightforward methodology for pervaporation system design. This requires knowledge of the flux at the feed and retentate conditions only. The analytical
161
C. Ó Súilleabháin, G. Foley / Separation and Purification Technology 152 (2015) 160–163
solution and the approximate equation are both applicable for the full range of industrial operating conditions for ethanol and IPA dehydrations.
J av ¼
_p m ¼ m_ R 1 f A Jo
_p m
_f _ p =m 1m
ð13Þ
a
ebðb1Þ=s ds
Therefore 2. Model development Pervaporation flux is a function of solute concentration in many applications. However, there are some membranes for which flux is constant across a range of concentrations [9–12]; this work considers such systems. Hence, it is assumed throughout that the local flux is a function of the local temperature only and not the local concentration of the permeating solute. _ be the mass flowrate and let T be the temperature at a Let m point in the channel. The heat balance can then be written in differential form as
_ p TÞ ¼ Jhv dA dðmc
ð2Þ
where J is the local mass flux of permeate, cp is the local specific heat capacity, hv is the local enthalpy per unit mass of vapour and dA represents the differential area. The relevant mass balance is
_ ¼ JdA dm
ð3Þ
Dividing Eq. (2) by Eq. (3) gives
_ p cp T þ mc
dT ¼ hv _ dm
ð4Þ
Rearranging gives
Z
T
cp dT ¼ hv c p T
Tf
Z
_f m
_ dm _ m
_ m
ð5Þ
where the f subscript denotes the feed. Assuming constant or appropriately averaged physical properties, this can be integrated to give
_f m hv hv T¼ Tf _ cp cp m
ð6Þ
Combining Eqs. (1), (3) and (6) gives Ej h i _ m dA 1 R hcvp ðhcvp T f Þ m_f ¼ e _ dm Jo
ð7Þ
Therefore, the total area is found from computing the integral below:
A¼
1 Jo
Z
_f m _r m
_f m hv hv _ exp ðEj =RT f Þ 1 dm _ cp T f cp T f m
ð8Þ
To simplify this expression, we let
s¼
_ m _f m
ð9Þ
Therefore
A¼
_f m J0
Z
1
_ p =m _f 1m
a
ebðb1Þ=s ds
ð10Þ
where
a¼
Ej RT f
ð11Þ
hv cp T f
ð12Þ
and
b¼
From Eq. (10) it follows that the average flux is given by
_ p =m _ f Þea ðm J av ¼ R1 a Jf bðb1Þ=s ds _ e _
ð14Þ
1mp =mf
Carrying out the integration using a symbolic integrator (MathematicaÒ) we get 1 as aðb1Þ Z 1 bðbðs 1Þ þ 1Þebðs1Þþ1 aðb 1Þea=b Ei bðbðs1Þþ1Þ a bðb1Þ=s e ds ¼ 2 _ _ 1mp =mf b _f _ p =m 1m
ð15Þ
Therefore 2
J av b ea p ¼ aðb1Þ J f bea aðb 1Þea=b Ei aðb1Þ bð1 bpÞeað1pÞ 1bp þ aðb 1Þea=b Ei b
bð1bpÞ
ð16Þ where Ei is the Exponential Integral, a special function that has been shown to occur in the analysis of ultrafiltration systems [13], and p is defined by
p¼
_p m _f m
ð17Þ
The Exponential Integral can be computed readily with advanced computation packages but also from its series definition using spreadsheet software [13]. In the next section some numerical computations are presented with this equation and regression analysis is used to derive a simple and accurate approximation for the average flux. 3. Results and discussion While Eq. (16) poses no great computational challenges, it would be desirable to obtain a more user-friendly version of this equation, one that might be analogous to Cheryan’s well-known expression for the average flux in batch ultrafiltration [14]. Cheryan proposed that batch times can be accurately computed by taking the average flux to be the arithmetic mean of the initial and final fluxes, thus avoiding the need for numerical integration. The first task in this process is to establish a realistic range of values for the parameters a, b and p. Industrial systems normally operate with feed temperatures in the range 350–395 K [1]. The typical change in temperature, DT, i.e., (Tf Tr) is 15–30 K [1]. A permeate mass flowrate of 3–5% of the feed is typical for industrial systems [1]. Permeate typically contains at least 80% water and remains essentially fixed for many commercial systems [9,10,12,15–18]. Feed in industrial systems flows through a series of adiabatic pervaporation modules and in order to maintain the flux in each module at sufficiently high values, the retentate is re-heated to the initial feed temperature prior to entering subsequent modules. But over-frequent re-heating leads to complexity; the additional cost of piping and valves can be significant [1]. The optimum membrane area thus depends on the relative cost of membranes versus that of re-heating equipment. Studies have suggested that a fixed temperature drop be used for each stage [2] or that an equal membrane area be used for each stage [2,5]. It is proposed here that a Jr/Jf ratio of 0.4 be used and, on this basis, a maximum commercially feasible value of Ej can be computed. In comparison, Sosa and Espinosa used fixed temperature drops with Jr/Jf ratios of greater than 0.5 [5]. On the other hand, Bausa and Marquadt assumed that the average flux was 80% of the maximum flux: this implies a ratio P 0.65 [2].
162
C. Ó Súilleabháin, G. Foley / Separation and Purification Technology 152 (2015) 160–163
1.0
Building on this approach, one can write
E j Jr ¼ J o eRT r J o e RT f Jf Ej
0.9
Rearranging and applying the 0.4 criterion gives the following expression for the maximum value of Ej for industrial operation:
0.8
Emax ¼ j
R ln 0:4 ð1=T f 1=T r Þ
ð19Þ
Jav / Jf
ð18Þ
0.6
For a given value of Tf the retentate temperature in this equation is given by Eq. (6) as
hv hv 1 Tf 1p cp cp
Tr ¼
0.5
ð20Þ 0.4
which, for p = 0.03, becomes
hv hv 1 Tr ¼ Tf 1 0:03 cp cp
ð21Þ
Jr Jf
ð22Þ
0.7
0.8
0.9
1.0
0.9 0.8 0.7 Exact Approximate
0.6 0.5
!n ð23Þ
Table 2 shows the values of n obtained for both solutions using different ‘best-fit’ criteria. Now, using Eq. (23) the simplified design equation becomes
_ p Jf n m A¼ Jf Jr
0.6
1.0
These equations establish the maximum possible value of the activation energy to make pervaporation feasible. In summary, the procedure for computing the average flux over a range of operating conditions using Eq. (16) is outlined in Table 1. Figs. 1 and 2 are plots of Jav/Jf versus Jr/Jf for the full range of industrial operating conditions examined for ethanol and IPA dehydration. The following equation was proposed to approximate the simulated data.
J av ¼ Jf
0.5
Fig. 1. Jav/Jf versus Jr/Jf for water–ethanol. Solid curve denotes Eq. (23) with n = 0.52.
Jav / Jf
R ln 0:4 ¼ ð1=T f 1=ðT f 30ÞÞ
0.4
Jr / J f
Note that if this value for Tr is such that DT > 30 K, then
Emax j
Exact Approximate
0.7
ð24Þ
where it is at the discretion of the designer as to which value of n to use. In summary, the steps involved in calculating the ‘ideal’ membrane area required for a given separation using the approximate method are outlined in Table 3 below. The approximate approach outlined above is clearly simple and easy to use, and causes errors of less than 1.2% (based on the minimax criterion) for commercial operating conditions. Little data is required for the equation and such data is either readily available or easily determined. Of course the underlying mathematical model on which Eq. (16) is based is itself an approximation and some form of ‘efficiency factor’, not unlike the column efficiency that is sometimes employed in distillation design, will ultimately be needed to compute the real area when sizing industrial systems. Such an efficiency factor is likely to be a function of module design,
0.4 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Jr / J f Fig. 2. Jav/Jf versus Jr/Jf for water–IPA. Solid curve denotes Eq. (23) with n = 0.52.
Table 2 Best-fit values of n according to various criteria.
a
n
Max error (%)
Mean error (%)
Water–ethanol Least squares a Minimax n = 0.5
0.51 0.52 0.5
1.37 1.03 2.31
0.28 0.46 0.33
IPA-Water Least squares a Minimax n = 0.5
0.51 0.52 0.5
1.51 1.10 2.44
0.27 0.45 0.32
Minimisation of maximum error.
operating conditions and fluid properties. In that sense, the work outlined here is merely a first step towards the development of design equations that will give engineers the confidence to engage
Table 1 Algorithm for generating average flux ‘data’.
For a given value of Tf and feed composition, choose a value of p between 0.03 and 0.05 inclusive, and a permeate composition between 0.8 and 1.0 inclusive. Calculate the maximum value of Ej for these conditions. Choose a value of Ej less than or equal to this value. Evaluate the retenate composition by mass balance. Estimate a value of Tr and using this value, compute the weighted average values of hv and cp using appropriate data sources [19,20]. Iterate on Eq. (20) to get Tr. Compute Jr/Jf . Evaluate parameters a and b using Eqs. (11) and (12). Compute Jav/Jf using Eq. (16) Repeat for all combinations of the following parameters across the full range of industrial conditions: permeate-to-feed ratio, feed composition, permeate composition, feed temperature and Ej.
C. Ó Súilleabháin, G. Foley / Separation and Purification Technology 152 (2015) 160–163
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Table 3 Algorithm for approximate calculation of membrane area.
For a given feed, choose a membrane for which the Arrhenius flux relationship and the permeate composition is known or can be determined by experiment. Choose a suitable feed temperature. Check that the Ej value is feasible using Eqs. (21) and (22). Choose a desired permeate/feed ratio. Determine the retentate composition by mass balance and Tr by iteration as described previously. Calculate Jr/Jf. Calculate the required membrane area using Eq. (24).
with pervaporation technology. Ongoing work involves applying the methodology outlined here to other dehydration processes, the expectation being that other systems will be characterised by different values of n in Eq. (23). In addition, innovative ways of operating pervaporation modules are being examined and new ways of selecting the best membrane for any given separation are being investigated. Further work will focus on the optimisation of multi-stage systems. There are two obvious areas where the current model can be further developed. The first is to go beyond the use of averaged physical properties and to incorporate the full temperature and concentration dependence of all properties into the analysis. The simplest way to do this is to work with the underlying differential equations rather than pursing an integral approach. In that case, numerical solutions will inevitably be needed. The second area would involve incorporating a composition-dependent flux term and again this would suggest a solution method based on numerical solution of the underlying differential equations. In addition to the modelling work, experimental work is required to determine the ‘efficiency factor’ mentioned above, and its dependence on operating conditions, membrane properties and fluid properties. 4. Conclusions This paper has described the first steps towards the development of easy-to-use engineering design equations for pervaporation. The general approach, which is based ultimately on an experimentally determined expression for the flux as a function of temperature, combined with mass and energy balances, is one that will be familiar to all chemical engineers. It is to be hoped that the familiarity of this approach will cause engineers to consider more seriously the use of pervaporation technology for not only alcohol dehydration processes but many other applications as well [21]. References [1] R.W. Baker, Membrane Technology and Applications, third ed., Wiley, West Sussex, UK, 2012. [2] J. Bausa, W. Marquardt, Shortcut design for hybrid membrane/distillation processes for the separation of non-ideal multicomponent mixtures, Ind. Eng. Chem. Res. 39 (2000) 1658–1672.
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