- Email: [email protected]
Predictive modelling of nanofiltration: membrane specification and process optimisation
DESALINATION Desalination 147 (2002) 197-203
ELSEVIER
www.elsevier.com/locate/desal
Predictive modelling of nanofiltration: membrane specification and process optimisation W. Richard Bowen*, Julian S. Welfoot Centre for Complex Fluids Processing, Department of Chemical and Biological Process Engineering, Universi@ of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK Tel. & Fax +44 (I 792) 295842; email: r. [email protected]. uk
Received 1 February 2002; accepted 15 February 2002
Abstract An important challenge for nanofiltration processes is the development of predictive models that convey a fundamental understanding and simple quantification of the governing phenomena in a way that has the potential for industrial application. The paper reviews one such approach, including: the formulation of a mathematically consistent description of rejection and flux at nanotiltration pores, the inclusion in calculations of pore size distributions with pore size dependent physical properties, the linearisation of these models to facilitate rapid calculations, and the use of such calculations to specify membranes for technically demanding separations. The present achievements and future challenges for predictive modelling of nanofiltration processes are assessed. Keywords:
Nanofiltration; Modelling; Prediction; Pore size distribution
1. Introduction There is an increasing demand for membrane nanofiltration (NF) processes on a world-wide basis. Important applications occur in the pharmaceutical industry, in drinking water treatment and in environmental protection. The specification and optimisation of such processes require the development of good predictive models. *Corresponding author. Presented at the International Julj 7-12, 2002.
Congress on Membranes
However, at a fundamental level, NF is a very Therefore, an important complex process. challenge is to develop models that convey a fundamental understanding and simple quantification of the governing phenomena in a way that has the potential for industrial application. The purpose of this paper is to review the development of one such approach. The review will outline: (1) the formulation of a mathematically consistent description of rejection and and Membrane
Processes
001 l-9164/02/$- See front matter 0 2002 Elsevier Science B.V. All rights reserved PII: SO0 I I-9 164(02)00534-9
(ICOW,
Toulouse, France,
198
W.R. Bowen, J.S. Werfoot /Desalination
147 (2002) 197-203
flux at NF pores, (2) the extension of this description to allow for a pore size distribution with pore size dependent physical properties, (3) the development and validation of a linearised solution of the model to facilitate engineering calculations, (4) an example of the use of model calculations to specify membranes for a challenging separation processes.
The most general description of partitioning between the feed/permeate solutions and the membrane includes steric, electrostatic and dielectric exclusion effects:
2. Mathematically rejection and flux
J$ =miexp(
consistent
description
of
The events controlling the performance ofNF membranes are taking place at a length scale of the order of a nanometre. This is a length scale at which macroscopic descriptions of hydrodynamics and rejection are beginning to break down - it is a length scale not much greater than atomic dimensions. Even so, all practically useful descriptions of NF to date have been based on a macroscopic approach to the two fundamental features of NF operation - partitioning between the feed/permeate solutions and transport through the membrane [ 11. The usual starting point for such calculations has been the extended Nernst-Planck equation:
j,=
d(lrqj) c,D. -c,D,~~---$V~~-D,~~
--
Z,CiD.
RT
dx
It is usually assumed that the first two terms on the right-hand side can be neglected, which can be shown to be reasonable. For uncharged solutes the fourth term on the right-hand side is also zero. For ions it is usual to obtain expressions for the potential gradient and concentration gradient:
I$g [&,c-
YI)‘i-‘ip]
W
_=
dx
‘9
-2 RT j=l F
(2)
q2ci
(Kc-Y)@
R=&+ 5 cxco= Wj
=(
Kc-Y
Pe'
I$c.V I
exP[ z)
(4)
For ions, the normal solution method is a fourthorder Runge-Kutta algorithm used for the simultaneous step-wise integration of concentration and potential gradients. Such an approach uses conditions of zero current and pore electroneutrality. For uncharged solutes in which partitioning is considered to arise only from steric effects, it is possible to derive an explicit expression for rejection:
(1) lp FdQ+K.
g*CD]
(5)
l-[l-(~C-Y)@]exp(-Pe’) andcxzAx = Wp
1Y2 ’ AP,
(6)
8’1Dp
Analysis shows that, when rejection is considered in terms of the dynamically calculated effective pressure driving force (AP,) [l], rejection of uncharged solutes depends only on the effective membrane pore radius (r,,) and rejection of charged solutes depends on rp, the effective membrane charge density (X,) and the value of A w,. Analyses of this kind have been successful in describing the rejection of uncharged solutes, ions and other charged solutes. These successes
W.R. Bowen, J.S. Welfoot/Desalination
suggest that macroscopic descriptions are useful, even if not strictly applicable. They have also been valuable for the description of some processes of specific industrial relevance, such as diafiltration of dye solutions. However, in order to identify how further progress might be made, it is worth considering some of the assumptions in these approaches: 1. The membranes have been considered as consisting of uniform, straight, cylindrical pores with solute transport corrected for hindered convection and diffusion. In reality, membranes generally contain a distribution of irregular pores. The corrections for hindered transport are based on spherical solutes and do not take solute charge into account. 2. Steric partitioning may be simply calculated. Molecular dynamic simulations suggest that ion solvation does not change greatly for rp> 0.35 nm [2]. Calculation ofelectrostatic partitioning is usually based on a Donnan expression that requires a knowledge of X, In general, this depends on membrane chemistry and specific adsorption of ions. It is not possible to predict&, so experimental isotherm data must be collected. There is less agreement on the calculation of AW, The major effect is probably due to a change in water dielectric constant (&,) due to confinement in the pore giving rise to a Born solvation energy difference [l]. Combining the different partitioning mechanisms as in Eq. (4) assumes that they act independently. 3. Activity coefficients in the membrane and bulk solution are usually assumed to be equal to unity. Activity coefficients in bulk solution could be calculated using fundamental or semi-empirical approaches. However, calculation in the pore is more questionable due to the unequal numbers of cations and anions and the effects of the charged pore walls. 4. The assumption of radially uniform potential has been shown to have good validity for NF pores. However, a variation of surface charge in an axial direction is possible. Again, specific ion
147 (2002) 197-203
199
adsorption may modify the effective membrane pore radius. These are only some of the limitations. Indeed, closer examination of the problem can only lead to agreement with the view expressed by Paul Anderson: “I have yet to see any problem, however complicated, which when looked at in the right way, did not become still more complicated”. Even so, the solution of engineering problems cannot wait for an exact scientific description of the process. Some further model developments on the basis already presented can be helpful. 3. Inclusion of pore size distributions pore size dependent physical properties
with
The inclusion of a pore size distribution in calculations can make model calculations more realistic. For ultrafiltration, the log-normal distribution has been most widely used [3]:
(7)
where
However, atomic force microscopy (AFM) imaging suggests that it would be better to use a truncated distribution [4]:
Truncation also gives better agreement with experimental data, as the large pore size “tail” of
200
W.R. Bowen, J.S. Welfoot / Desalination 147 (2002) 197-203
the distribution may otherwise dominate both rejection and flux. Truncation also greatly decreases computation time. AFM cannot directly give distributions for pores of nanometre dimensions due to convolution of the tip and pore. However, a useful approach is to use the shape of the distribution obtained by AFM in conjunction with experimental rejection data for an unchanged solute to calculate a corrected distribution [4]. Measurements indicate that r,,,,=:2r*. As mentioned previously for the case of dielectric constant, the physical properties of the solvent in NF pores are affected by the confinement. Experimental evidence suggests that the first layer of solvent molecules around the pore wall will have an increased viscosity and a reduced dielectric constant compared to the bulk values. Averaging of viscosity and dielectic constant on a geometric basis allows estimation of the effective pore solvent properties [4]:
Experimental evaluation of cP on this basis from salt rejection at a membrane isoelectric point (where only steric and electrostatic partitioning remain) is in reasonable agreement with molecular dynamic simulations [5]. The expression for 17assumes that the ordered layer has a viscosity ten times that of bulk water [6], the best available value but one which requires more detailed examination. Such effects are much greater than electroviscous effects for pores of nanometre dimensions. In practice, the effect on rejection of variation of q with pore radius is not very great if considered in terms of effective pressure driving force (though very significant if considered in terms of membrane flux). Introduction of a pore size distribution gives better agreement with experimental salt rejection data at NF membranes compared to calculations assuming uniform pores, especially at high presssure, the region of industrial interest. 4. Linearised
1=1+18 rl*
cp =
d
-9
0
80-2(80:*)[
calcu-
d
Calculations of membrane performance can be very demanding in computer time, especially as membrane specification and process optimisation require interative calculations. Finite difference linearisation of the pore concentration gradient, shown here for a binary electrolyte [7]:
rP
ii
model for engineering
lations
2
;I
+ [80-e* j( ;)’
(‘)
(10)
where dci AC, -‘--= AX dx
cl (Ax) -ci(O) Ax
ct U-0+ ci (Ax) and
cl,av
=
2
(11)
201
W.R. Bowen, J.S. Werfoot / Desalination 147 (2002) 197-203
greatly simplifies the solution of the threeparameter model for electrolyte rejection through removal of the requirement for numerical integration of the extended Nernst-Planck equation. Comparison with pilot plant data and with the results of the full model, including investigation of pore concentration profiles, shows the assumption of linearity to be valid over a wide range of NF conditions. The linearised model may be extended and shown to give good agreement with experimental data for multielectrolyte mixtures. Overall, the linearised model is a powerful tool for characterisation of NF membranes and the subsequent prediction of separation performance. The computational demands are modest in terms of time and complexity. Linearisation is especially useful for the calculation of electrolyte rejections including a pore size distribution. 5. Membrane
specification
and process opti-
misation
The models developed may be used to specify membranes and to ident@ the optimum operating conditions for industrially important separations. This is especially useful for technically demanding separations, such as the fractionation of molecules close in molecular weight. For very demanding calculations the models may also be useful in guiding the development of membranes for a particular purpose. As an example, in the pharmaceutical industry there may be a requirement to recover and purify a product of molecular weight 800 Da from an impurity of molecular weight 300 Da. Due to stringent specification of such products and their high value, a purity of >99% with a recovery of >95% may be required. For the purposes of illustrative calculation, it may be assumed that both product and impurity are uncharged and present in the ratio 1O:l by weight. A typical processing option would be to first concentrate the process liquid, say by a factor of 10 at AP, =
Table 1 Productpurity and product recovery with membranes of differing pore size distributions Membrane specification
Product purity, wt%
Product recovery, wt%
U* = 0.25r*
95
95.2
99 99.9
76.4
95 99
98.0 93.3
99.9
87.1
U* = O.lOr*
55.3
Note: Membranes specified with identical hydraulic permeabilities, equivalent to a uniform pore membrane just reaching 100% product retention. Calculation carried out using Eqs. (5)-(g).
4.0 MPa, followed by diafiltration to wash away the impurity, at say AP, = 1.OMPa (effectively in constant flux operation). AFM studies show that most commercial membranes have a pore size distribution such that U* = 0.25r* [8]. Results of example calculations are shown in Table 1. It may be seen that is is not possible to come close to the product specification with such commercial membranes. Also shown in the table are results for a hypothetical membrane of narrower pore size distribution, CJ*= 0.1 Or*. Use of such a membrane would give a product very close to the specification required. Investment in the development of such improved membranes can thus be technically justified [9]. 6. Conclusions The paper has described the development of a continuum approach to NF processes that has the potential for industrial application. Within such a framework useful calculations can be made. Indeed, given the many known limitations of such models, they are surprisingly successful. This is in part due to the use of experimental parameters in the calculations which may implicitly compensate for the shortcomings.
202
W.R. Bowen, J.S. Welfoot / Desalination
Model calculations are already valuable in predicting the separation of uncharged solutes, though it is essential that a pore size distribution is included if fractionation is to be quantitatively assessed. A significant weakness of the present approach for ions and other charged solutes is the requirement to experimentally determine the effective membrane charge density from separation data. Isotherms can be used to describe ion adsorption, though they are not always successful for multicomponent systems. Independent measurements quantifying membrane charge density or potential are difficult to apply successfully to process predictions. The inclusion of a dielectric exclusion term is essential for ions and other charged solutes, but further work is needed to determine the most appropriate means for its calculation. There is a need to learn more about the properties of solvents and solutes in confined volumes. Molecular dynamic calculations of properties such as dielectric constant and ion transport are useful. However, they need to be verified experimentally if they are to be used with confidence. It is likely that new theoretical and experimental approaches will need to be developed if substantial progress is to be made. In this context, membrane technologists need to be aware of advances being made in the biosciences, such as those describing solvent and ion transport in protein transport channels. Finally, there is the possibility that advances in understanding the physics of solutions and increases in computing power will allow the application of molecular level descriptions ofNF - but this is likely to be many years in the future.
7. Symbols a, b
-
Stokes radius of solute or ion i, m Parameter defined by Eq. (7), dimensionless
147 (2002) 197-203
‘i
-
Ci
-
Cf
-
cp Ci,, ci.w
-
d
-
DP Di,p L fk
-
F
-
j,
Concentration of ion i within pore, mol me3 Ionic bulk solution concentration, mol me3 Bulk feed concentration, mol rnM3 Bulk permeate concentration, mol mm3 Bulk permeate concentration of ion i, mol m-3 Wall concentration of ion i, mol mm3 Thickness of the oriented solvent layer, m Solute pore diffusion coefficient, m2 s-’ Pore diffusion coefficient of ion i, m2 s-’ Electronic charge, 1.602~ 10-l’ C Theoretical probability density function, m-’ Truncated probability density function, m-’ Faraday constant, 96487 C mall’ Flux of ion i (pore area basis), mol mm2 S
K,
-
Ki,c
-
k
Pe r r max rP
-
r*
R(r) R R T V K,V, L Y,Y, -
-I
Boltzmann constant, 1.381~10-~~ J K-’ Uncharged solute hindrance factor for convection, dimensionless Hindrance factor for convection of ion i, dimensionless Modified Peclet number, dimensionless Pore radius, m Upper limit of the truncated pore size distribution, m Effective pore radius, m Mean pore radius, m Porewise rejection, dimensionless Overall rejection, dimensionless Universal gas constant, 8.314 J mol-’ K-’ Absolute temperature, K Solvent velocity, m s-l Partial molar volume, m3 mol-’ Axial position within the pore, m Effective charge density, mol mm3 pimensionless function, dimensionless
W.R. Bowen, J.S. Weljoot /Desalination =I
-
147 (2002) 197-203
203
Acknowledgements
Valence of ion i, dimensionless
Greek
AP - Applied pressure, N m-* AP, - Effective pressure driving force, N mm2
We thank the UK Engineering and Physical Sciences Research Council and the UK Biotechnology and Biological Sciences Research Council for funding this work. We thank Avecia for a CASE studentship for JSW.
[AP~=AP-ATC]
AIT - Dynamic osmotic pressure difference
References VI
Aw, -
Dielectric exclusion (Born energy), J
energy
barrier
PI [31
A$D - Donnan potential, V Eb
-
% &*
-
?I
-
YI r’: rl
-
Bulk dielectric constant, dimensionless Pore dielectric constant, dimensionless Dielectric constant of oriented water layer, dimensionless Vacuum permittivity, 8.854~ lo-‘* J-’ C* m-’ Activity coefficient within pore Activity coefficient in bulk solution
-
Solvent viscosity within pore, N s mm2 Bulk solvent viscosity, N s m-* Ratio of ionic or uncharged solute
@
-
radius/ pore radius, dimensionless Uncharged solute steric partition coefficient, dimensionless [= (1 -a)*]
Qi
-
0*
-
Steric partition coefficient of ion i, dimensionless [= (1 -a)*] Distribution standard deviation, m
6
-
Potential within the pore, V
z”
141
151
WI
[71
181
[91
W.R. Bowen and J.S. Welfoot, Modelling the performance of membrane nanofiltration - critical assessment and model development, Chem. Eng. Sci, in press. R.M. Lynden-Bell and J.C. Rasaiah, Mobility and solvation of ions in channels, J. Chem. Phys., 105 (1996) 92669280. A.L. Zydney, P. Aimar, M. Meireles, J.M. Pimbley and G. Belfort, Use of the log-normal probability density function to analyze membrane pore size distributions: functional forms and discrepancies, J. Membr. Sci., 91 (1994) 293-298. W.R. Bowen and J.S. Welfoot, Modelling the performance of membrane nanofiltration - pore size distribution effects, Chem. Eng. Sci, in press. S. Senapati and A. Chandra, Dielectric constant of water in a nanocavity, J. Chem. Phys. B, 105 (2001) 5106-5109. G. Belfort, J. Scherig and D.O. Seevers, Nuclear magnetic resonance relaxation studies of adsorbed water on porous glass of varying sizes, J. Colloid Interface Sci., 47 (1974) 106-l 16. W.R. Bowen and J.S. Welfoot, A linearised model for nanofiltration: development and assessment, AIChE J., in press. W.R. Bowen and T.A. Doneva, Atomic force microscopy characterisation of ultrafiltration membranes: correspondence between surface pore dimensions and molecular weight cut-off, Surf. Interface Anal., 29 (2000) 544-547. W.R. Bowen and J.S. Welfoot, full paper in preparation.
ELSEVIER
www.elsevier.com/locate/desal
Predictive modelling of nanofiltration: membrane specification and process optimisation W. Richard Bowen*, Julian S. Welfoot Centre for Complex Fluids Processing, Department of Chemical and Biological Process Engineering, Universi@ of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK Tel. & Fax +44 (I 792) 295842; email: r. [email protected]. uk
Received 1 February 2002; accepted 15 February 2002
Abstract An important challenge for nanofiltration processes is the development of predictive models that convey a fundamental understanding and simple quantification of the governing phenomena in a way that has the potential for industrial application. The paper reviews one such approach, including: the formulation of a mathematically consistent description of rejection and flux at nanotiltration pores, the inclusion in calculations of pore size distributions with pore size dependent physical properties, the linearisation of these models to facilitate rapid calculations, and the use of such calculations to specify membranes for technically demanding separations. The present achievements and future challenges for predictive modelling of nanofiltration processes are assessed. Keywords:
Nanofiltration; Modelling; Prediction; Pore size distribution
1. Introduction There is an increasing demand for membrane nanofiltration (NF) processes on a world-wide basis. Important applications occur in the pharmaceutical industry, in drinking water treatment and in environmental protection. The specification and optimisation of such processes require the development of good predictive models. *Corresponding author. Presented at the International Julj 7-12, 2002.
Congress on Membranes
However, at a fundamental level, NF is a very Therefore, an important complex process. challenge is to develop models that convey a fundamental understanding and simple quantification of the governing phenomena in a way that has the potential for industrial application. The purpose of this paper is to review the development of one such approach. The review will outline: (1) the formulation of a mathematically consistent description of rejection and and Membrane
Processes
001 l-9164/02/$- See front matter 0 2002 Elsevier Science B.V. All rights reserved PII: SO0 I I-9 164(02)00534-9
(ICOW,
Toulouse, France,
198
W.R. Bowen, J.S. Werfoot /Desalination
147 (2002) 197-203
flux at NF pores, (2) the extension of this description to allow for a pore size distribution with pore size dependent physical properties, (3) the development and validation of a linearised solution of the model to facilitate engineering calculations, (4) an example of the use of model calculations to specify membranes for a challenging separation processes.
The most general description of partitioning between the feed/permeate solutions and the membrane includes steric, electrostatic and dielectric exclusion effects:
2. Mathematically rejection and flux
J$ =miexp(
consistent
description
of
The events controlling the performance ofNF membranes are taking place at a length scale of the order of a nanometre. This is a length scale at which macroscopic descriptions of hydrodynamics and rejection are beginning to break down - it is a length scale not much greater than atomic dimensions. Even so, all practically useful descriptions of NF to date have been based on a macroscopic approach to the two fundamental features of NF operation - partitioning between the feed/permeate solutions and transport through the membrane [ 11. The usual starting point for such calculations has been the extended Nernst-Planck equation:
j,=
d(lrqj) c,D. -c,D,~~---$V~~-D,~~
--
Z,CiD.
RT
dx
It is usually assumed that the first two terms on the right-hand side can be neglected, which can be shown to be reasonable. For uncharged solutes the fourth term on the right-hand side is also zero. For ions it is usual to obtain expressions for the potential gradient and concentration gradient:
I$g [&,c-
YI)‘i-‘ip]
W
_=
dx
‘9
-2 RT j=l F
(2)
q2ci
(Kc-Y)@
R=&+ 5 cxco= Wj
=(
Kc-Y
Pe'
I$c.V I
exP[ z)
(4)
For ions, the normal solution method is a fourthorder Runge-Kutta algorithm used for the simultaneous step-wise integration of concentration and potential gradients. Such an approach uses conditions of zero current and pore electroneutrality. For uncharged solutes in which partitioning is considered to arise only from steric effects, it is possible to derive an explicit expression for rejection:
(1) lp FdQ+K.
g*CD]
(5)
l-[l-(~C-Y)@]exp(-Pe’) andcxzAx = Wp
1Y2 ’ AP,
(6)
8’1Dp
Analysis shows that, when rejection is considered in terms of the dynamically calculated effective pressure driving force (AP,) [l], rejection of uncharged solutes depends only on the effective membrane pore radius (r,,) and rejection of charged solutes depends on rp, the effective membrane charge density (X,) and the value of A w,. Analyses of this kind have been successful in describing the rejection of uncharged solutes, ions and other charged solutes. These successes
W.R. Bowen, J.S. Welfoot/Desalination
suggest that macroscopic descriptions are useful, even if not strictly applicable. They have also been valuable for the description of some processes of specific industrial relevance, such as diafiltration of dye solutions. However, in order to identify how further progress might be made, it is worth considering some of the assumptions in these approaches: 1. The membranes have been considered as consisting of uniform, straight, cylindrical pores with solute transport corrected for hindered convection and diffusion. In reality, membranes generally contain a distribution of irregular pores. The corrections for hindered transport are based on spherical solutes and do not take solute charge into account. 2. Steric partitioning may be simply calculated. Molecular dynamic simulations suggest that ion solvation does not change greatly for rp> 0.35 nm [2]. Calculation ofelectrostatic partitioning is usually based on a Donnan expression that requires a knowledge of X, In general, this depends on membrane chemistry and specific adsorption of ions. It is not possible to predict&, so experimental isotherm data must be collected. There is less agreement on the calculation of AW, The major effect is probably due to a change in water dielectric constant (&,) due to confinement in the pore giving rise to a Born solvation energy difference [l]. Combining the different partitioning mechanisms as in Eq. (4) assumes that they act independently. 3. Activity coefficients in the membrane and bulk solution are usually assumed to be equal to unity. Activity coefficients in bulk solution could be calculated using fundamental or semi-empirical approaches. However, calculation in the pore is more questionable due to the unequal numbers of cations and anions and the effects of the charged pore walls. 4. The assumption of radially uniform potential has been shown to have good validity for NF pores. However, a variation of surface charge in an axial direction is possible. Again, specific ion
147 (2002) 197-203
199
adsorption may modify the effective membrane pore radius. These are only some of the limitations. Indeed, closer examination of the problem can only lead to agreement with the view expressed by Paul Anderson: “I have yet to see any problem, however complicated, which when looked at in the right way, did not become still more complicated”. Even so, the solution of engineering problems cannot wait for an exact scientific description of the process. Some further model developments on the basis already presented can be helpful. 3. Inclusion of pore size distributions pore size dependent physical properties
with
The inclusion of a pore size distribution in calculations can make model calculations more realistic. For ultrafiltration, the log-normal distribution has been most widely used [3]:
(7)
where
However, atomic force microscopy (AFM) imaging suggests that it would be better to use a truncated distribution [4]:
Truncation also gives better agreement with experimental data, as the large pore size “tail” of
200
W.R. Bowen, J.S. Welfoot / Desalination 147 (2002) 197-203
the distribution may otherwise dominate both rejection and flux. Truncation also greatly decreases computation time. AFM cannot directly give distributions for pores of nanometre dimensions due to convolution of the tip and pore. However, a useful approach is to use the shape of the distribution obtained by AFM in conjunction with experimental rejection data for an unchanged solute to calculate a corrected distribution [4]. Measurements indicate that r,,,,=:2r*. As mentioned previously for the case of dielectric constant, the physical properties of the solvent in NF pores are affected by the confinement. Experimental evidence suggests that the first layer of solvent molecules around the pore wall will have an increased viscosity and a reduced dielectric constant compared to the bulk values. Averaging of viscosity and dielectic constant on a geometric basis allows estimation of the effective pore solvent properties [4]:
Experimental evaluation of cP on this basis from salt rejection at a membrane isoelectric point (where only steric and electrostatic partitioning remain) is in reasonable agreement with molecular dynamic simulations [5]. The expression for 17assumes that the ordered layer has a viscosity ten times that of bulk water [6], the best available value but one which requires more detailed examination. Such effects are much greater than electroviscous effects for pores of nanometre dimensions. In practice, the effect on rejection of variation of q with pore radius is not very great if considered in terms of effective pressure driving force (though very significant if considered in terms of membrane flux). Introduction of a pore size distribution gives better agreement with experimental salt rejection data at NF membranes compared to calculations assuming uniform pores, especially at high presssure, the region of industrial interest. 4. Linearised
1=1+18 rl*
cp =
d
-9
0
80-2(80:*)[
calcu-
d
Calculations of membrane performance can be very demanding in computer time, especially as membrane specification and process optimisation require interative calculations. Finite difference linearisation of the pore concentration gradient, shown here for a binary electrolyte [7]:
rP
ii
model for engineering
lations
2
;I
+ [80-e* j( ;)’
(‘)
(10)
where dci AC, -‘--= AX dx
cl (Ax) -ci(O) Ax
ct U-0+ ci (Ax) and
cl,av
=
2
(11)
201
W.R. Bowen, J.S. Werfoot / Desalination 147 (2002) 197-203
greatly simplifies the solution of the threeparameter model for electrolyte rejection through removal of the requirement for numerical integration of the extended Nernst-Planck equation. Comparison with pilot plant data and with the results of the full model, including investigation of pore concentration profiles, shows the assumption of linearity to be valid over a wide range of NF conditions. The linearised model may be extended and shown to give good agreement with experimental data for multielectrolyte mixtures. Overall, the linearised model is a powerful tool for characterisation of NF membranes and the subsequent prediction of separation performance. The computational demands are modest in terms of time and complexity. Linearisation is especially useful for the calculation of electrolyte rejections including a pore size distribution. 5. Membrane
specification
and process opti-
misation
The models developed may be used to specify membranes and to ident@ the optimum operating conditions for industrially important separations. This is especially useful for technically demanding separations, such as the fractionation of molecules close in molecular weight. For very demanding calculations the models may also be useful in guiding the development of membranes for a particular purpose. As an example, in the pharmaceutical industry there may be a requirement to recover and purify a product of molecular weight 800 Da from an impurity of molecular weight 300 Da. Due to stringent specification of such products and their high value, a purity of >99% with a recovery of >95% may be required. For the purposes of illustrative calculation, it may be assumed that both product and impurity are uncharged and present in the ratio 1O:l by weight. A typical processing option would be to first concentrate the process liquid, say by a factor of 10 at AP, =
Table 1 Productpurity and product recovery with membranes of differing pore size distributions Membrane specification
Product purity, wt%
Product recovery, wt%
U* = 0.25r*
95
95.2
99 99.9
76.4
95 99
98.0 93.3
99.9
87.1
U* = O.lOr*
55.3
Note: Membranes specified with identical hydraulic permeabilities, equivalent to a uniform pore membrane just reaching 100% product retention. Calculation carried out using Eqs. (5)-(g).
4.0 MPa, followed by diafiltration to wash away the impurity, at say AP, = 1.OMPa (effectively in constant flux operation). AFM studies show that most commercial membranes have a pore size distribution such that U* = 0.25r* [8]. Results of example calculations are shown in Table 1. It may be seen that is is not possible to come close to the product specification with such commercial membranes. Also shown in the table are results for a hypothetical membrane of narrower pore size distribution, CJ*= 0.1 Or*. Use of such a membrane would give a product very close to the specification required. Investment in the development of such improved membranes can thus be technically justified [9]. 6. Conclusions The paper has described the development of a continuum approach to NF processes that has the potential for industrial application. Within such a framework useful calculations can be made. Indeed, given the many known limitations of such models, they are surprisingly successful. This is in part due to the use of experimental parameters in the calculations which may implicitly compensate for the shortcomings.
202
W.R. Bowen, J.S. Welfoot / Desalination
Model calculations are already valuable in predicting the separation of uncharged solutes, though it is essential that a pore size distribution is included if fractionation is to be quantitatively assessed. A significant weakness of the present approach for ions and other charged solutes is the requirement to experimentally determine the effective membrane charge density from separation data. Isotherms can be used to describe ion adsorption, though they are not always successful for multicomponent systems. Independent measurements quantifying membrane charge density or potential are difficult to apply successfully to process predictions. The inclusion of a dielectric exclusion term is essential for ions and other charged solutes, but further work is needed to determine the most appropriate means for its calculation. There is a need to learn more about the properties of solvents and solutes in confined volumes. Molecular dynamic calculations of properties such as dielectric constant and ion transport are useful. However, they need to be verified experimentally if they are to be used with confidence. It is likely that new theoretical and experimental approaches will need to be developed if substantial progress is to be made. In this context, membrane technologists need to be aware of advances being made in the biosciences, such as those describing solvent and ion transport in protein transport channels. Finally, there is the possibility that advances in understanding the physics of solutions and increases in computing power will allow the application of molecular level descriptions ofNF - but this is likely to be many years in the future.
7. Symbols a, b
-
Stokes radius of solute or ion i, m Parameter defined by Eq. (7), dimensionless
147 (2002) 197-203
‘i
-
Ci
-
Cf
-
cp Ci,, ci.w
-
d
-
DP Di,p L fk
-
F
-
j,
Concentration of ion i within pore, mol me3 Ionic bulk solution concentration, mol me3 Bulk feed concentration, mol rnM3 Bulk permeate concentration, mol mm3 Bulk permeate concentration of ion i, mol m-3 Wall concentration of ion i, mol mm3 Thickness of the oriented solvent layer, m Solute pore diffusion coefficient, m2 s-’ Pore diffusion coefficient of ion i, m2 s-’ Electronic charge, 1.602~ 10-l’ C Theoretical probability density function, m-’ Truncated probability density function, m-’ Faraday constant, 96487 C mall’ Flux of ion i (pore area basis), mol mm2 S
K,
-
Ki,c
-
k
Pe r r max rP
-
r*
R(r) R R T V K,V, L Y,Y, -
-I
Boltzmann constant, 1.381~10-~~ J K-’ Uncharged solute hindrance factor for convection, dimensionless Hindrance factor for convection of ion i, dimensionless Modified Peclet number, dimensionless Pore radius, m Upper limit of the truncated pore size distribution, m Effective pore radius, m Mean pore radius, m Porewise rejection, dimensionless Overall rejection, dimensionless Universal gas constant, 8.314 J mol-’ K-’ Absolute temperature, K Solvent velocity, m s-l Partial molar volume, m3 mol-’ Axial position within the pore, m Effective charge density, mol mm3 pimensionless function, dimensionless
W.R. Bowen, J.S. Weljoot /Desalination =I
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147 (2002) 197-203
203
Acknowledgements
Valence of ion i, dimensionless
Greek
AP - Applied pressure, N m-* AP, - Effective pressure driving force, N mm2
We thank the UK Engineering and Physical Sciences Research Council and the UK Biotechnology and Biological Sciences Research Council for funding this work. We thank Avecia for a CASE studentship for JSW.
[AP~=AP-ATC]
AIT - Dynamic osmotic pressure difference
References VI
Aw, -
Dielectric exclusion (Born energy), J
energy
barrier
PI [31
A$D - Donnan potential, V Eb
-
% &*
-
?I
-
YI r’: rl
-
Bulk dielectric constant, dimensionless Pore dielectric constant, dimensionless Dielectric constant of oriented water layer, dimensionless Vacuum permittivity, 8.854~ lo-‘* J-’ C* m-’ Activity coefficient within pore Activity coefficient in bulk solution
-
Solvent viscosity within pore, N s mm2 Bulk solvent viscosity, N s m-* Ratio of ionic or uncharged solute
@
-
radius/ pore radius, dimensionless Uncharged solute steric partition coefficient, dimensionless [= (1 -a)*]
Qi
-
0*
-
Steric partition coefficient of ion i, dimensionless [= (1 -a)*] Distribution standard deviation, m
6
-
Potential within the pore, V
z”
141
151
WI
[71
181
[91
W.R. Bowen and J.S. Welfoot, Modelling the performance of membrane nanofiltration - critical assessment and model development, Chem. Eng. Sci, in press. R.M. Lynden-Bell and J.C. Rasaiah, Mobility and solvation of ions in channels, J. Chem. Phys., 105 (1996) 92669280. A.L. Zydney, P. Aimar, M. Meireles, J.M. Pimbley and G. Belfort, Use of the log-normal probability density function to analyze membrane pore size distributions: functional forms and discrepancies, J. Membr. Sci., 91 (1994) 293-298. W.R. Bowen and J.S. Welfoot, Modelling the performance of membrane nanofiltration - pore size distribution effects, Chem. Eng. Sci, in press. S. Senapati and A. Chandra, Dielectric constant of water in a nanocavity, J. Chem. Phys. B, 105 (2001) 5106-5109. G. Belfort, J. Scherig and D.O. Seevers, Nuclear magnetic resonance relaxation studies of adsorbed water on porous glass of varying sizes, J. Colloid Interface Sci., 47 (1974) 106-l 16. W.R. Bowen and J.S. Welfoot, A linearised model for nanofiltration: development and assessment, AIChE J., in press. W.R. Bowen and T.A. Doneva, Atomic force microscopy characterisation of ultrafiltration membranes: correspondence between surface pore dimensions and molecular weight cut-off, Surf. Interface Anal., 29 (2000) 544-547. W.R. Bowen and J.S. Welfoot, full paper in preparation.