Analysis of Transport in a Pressure-Driven Membrane Separation

9 Analysis of Transport in a Pressure-Driven Membrane Separation Process Downloaded by PENNSYLVANIA STATE UNIV on July 1, 2012 | http://pubs.acs.org Publication Date: January 1, 1985 | doi: 10.1021/bk-1985-0281.ch009

STEPHEN W. THIEL, DOUGLAS R. LLOYD, and J. M. DICKSON1 Department of Chemical Engineering, University of Texas at Austin, Austin, TX 78712

The analysis of transport through a membrane in a pressure-driven membrane separation process need not include assumptions about the microstructure of the membrane. An analysis of transport is presented which interprets phenomenological relations in terms of the strengths of physicochemical interactions in the solute-solvent-membrane material system. No assumptions about the membrane microstructure are made. The results of this analysis are compared with the results of an analysis of transport through finely porous membranes.

Transport through a membrane is often interpreted in terms of an assumed microstructure for the membrane. Dense, homogeneous membranes are often described using a solution-diffusion analysis (1-3), while membranes with large pores are described by simple Fickian diffusion superimposed on laminar flow (2). Merten (2) suggested that most reverse osmosis and many ultrafiltration membranes are intermediate in structure and behavior between the extremes of dense membranes and large-pored membranes. Merten called these membranes finely porous. Mason, Wendt, and Bresler noted that a simple pore flow model can be used to correlate membrane performance over a wide range of conditions, including conditions which violate many of the physical assumptions of the model (4). They suggested that the success of the pore flow model can be explained by dimensional analysis. Consequently, the microstructure of the membrane plays a relatively minor role in determining the form of the mathematical relationships that govern membrane permeation. That is, the microstructure of the membrane does not determine the form of the transport equations,

1

Current address: Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 0097-6156/85/0281-O113$06.00/0 © 1985 American Chemical Society

In Reverse Osmosis and Ultrafiltration; Sourirajan, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

Downloaded by PENNSYLVANIA STATE UNIV on July 1, 2012 | http://pubs.acs.org Publication Date: January 1, 1985 | doi: 10.1021/bk-1985-0281.ch009

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REVERSE OSMOSIS AND ULTRAFILTRATION

only the values of the parameters in those equations. Even though structure and transport are related, membrane structure is not always a reliable key to transport modelling and, conversely, transport behavior is not always a reliable indication of membrane structure. One approach to transport modelling without assumptions about the microstructure of the membrane uses phenomenological relations. These relations correlate fluxes with thermodynamic forces through coefficients which are not necessarily tied to any given physical assumptions. Many investigators have interpreted the phenomenological coefficients in terms of friction (5-8). This friction is a resistance to motion due to attraction between molecules. For example, Jonsson and Boesen (9) derived a frictional model that is formally similar to the phenomenologicallybased Spiegler-Kedem model (10). This paper considers the transport of dilute binary solutions through membranes under the influence of pressure and concentration gradients. Transport is analyzed using phenomenological relations; no assumptions are made about the microstructure of the membrane. The phenomenological coefficients and external forces are interpreted in terms of intermolecular forces. This analysis is compared with the analysis of finely porous membranes in light of physicochemical interactions in the solute-solvent-membrane material system. Preliminary Assumptions and Definitions The mathematical analysis that follows applies to a steady-state pressure-driven membrane separation process separating a dilute, binary, isothermal, non-reacting, liquid solution. The membrane is flat. The analysis considers only transport within the membrane; film theory can be used to account for mass transfer effects in the adjacent liquid (11). The system is assumed to be uniform in directions parallel to the surface of the membrane; all transport occurs in the z-direction, normal to the surface of the membrane. The coordinate z increases from the retentate to the permeate. The membrane phase is chosen to include only the solute and solvent within the polymer matrix. The effects of the membrane material on permeation are accounted for as external forces acting on the solute and solvent. The effective thickness of the membrane phase, T, includes the effects of tortuosity in the transport path. The permeating components are partitioned between the membrane phase and the adjacent liquid phase. This equilibrium is described by the equation X

AM " K A X A

(1)

The quantity X. is the solute mole fraction in the membrane phase, X is the solute mole fraction in the liquid phase, and K. is a partition coefficient. The partition coefficient on the retentate side of the membrane, K A ? , can be different from the partition coefficient on the permeate side of the membrane, K ~.


9.

THIEL ET AL.

Pressure-Driven

Membrane

Separation

General Transport Considerations Irreversible thermodynamics (12-14) provides a general description of nonequilibrium systems, and so describes processes which advance at a measurably rapid rate. Irreversible thermodynamics suggests that for an n-component system the natural choices for fluxes, J., and forces, Y., are


and

where the molar-average velocity, V, is defined as

Throughout this paper, C is molar density, X. is the mole fraction of component i, v, is the velocity of component i relative to stationary coordinates, F. is the external force acting on component i, and y. is the chemical potential of component i. The flux J is the diffusive molar flux of component i relative to the molar-average velocity. In typical pressure-driven membrane separation processes, the diffusive fluxes cannot be measured; experimental results instead provide values for the fluxes relative to stationary coordinates, N . Because the membrane is stationary, stationary coordinates are defined relative to the membrane. The fluxes N are defined as

This definition can be rewritten as

Introducing the total molar flux, N , Equation 6 becomes

Equation 7 expresses the flux of a component relative to stationary coordinates as the sum of two fluxes: the flux due to bulk motion and the flux due to diffusion. The flux due to bulk motion, X.N , represents molar-average motion; the inclusion of this term does not imply viscous flow. Assume that the fluxes and forces are linearly related through the phenomenological relations


115

116



where the L , are phenomenological coefficients which relate the flux of component i to the force on component k. DeGroot (12) showed that not all of the L , are independent, and that in a binary system,

Equation 9 is a generalization of the observation that mass diffusion in a binary system is characterized by one diffusivity. Inserting Equation 9 into Equation 8 provides, for a binary system, the result

The phenomenological coefficient L can be interpreted in terms of intermolecular forces (2):

where the coefficient $ represents the resistance to mutual motion of the solute and solvent; . is the reciprocal of the mobility of the solute-solvent system. liquation 11 can be inserted into Equation 10 to provide the relationship

Equation 12 is central to the analysis of solute transport. The external force, F., represents the effect of chemical and physical interactions between the membrane material and component i on the transport of component i. This external force is assumed to be proportional to the difference between the velocity of component i and the velocity of the membrane material (15). Since the membrane material is stationary, F can be expressed as

where C is the molar density of the membrane phase. Although the values of the coefficients $. and R are determined by the physicochemical interactions in the solute-solvent-membrane system, the introduction of these coefficients does not require the introduction of a specific microstructure for the membrane (8).


9. THIEL ET AL.

Pressure-Driven Membrane Separation

Solute Transport


A relationship between separation and flux can be derived starting with Equation 12. The total force acting on the solute, Y.-Y , can be divided into an external force term and a chemical potential term using Equation 3:

The external force term can be expanded using Equation 13 to obtain the result

For dilute solutions, X Equation 15 then becomes

and XL~ are approximately unity.

The chemical potential term can be rewritten using the Gibbs-Duhem equation for the free energy of the system. For an inert, isothermal, binary system, this equation is

where V is the molar volume of the system, and p is the pressure of the system. Equation 17 can be rearranged to yield the result

The differential of the solute chemical potential is given by

where V is the partial molar volume of the solute. Since X^ is, by assumption, approximately unity, substituting Equation 19 into Equation 18 provides the result

Merten (2) noted that the contribution of pressure to chemical potential, V.dp, is often negligible compared to the contribution of composition to chemical potential, RTd(£n X ) . Similarly, the analysis presented here assumes that (V - Vjclp is negligible compared to RTd(&n X . ) . Making this approximation, combining


117


118

Equations 12, 14, 16, and 20 provides the differential equation


Combining Equation 21 with Equation 7 yields, after rearrangement,

where the b. are functions defined by the equation

The quantities b , b , C , and $ are assumed to be constant across the membrane. Assuming equilibrium at the membrane-liquid interfaces using Equation 1, Equation 22 can be integrated across the membrane and rearranged to yield the result

Equation 24 indicates that the behavior of the system depends on the relative magnitudes of the solute-membrane, solvent-membrane, and solute-solvent interactions. Solvent Transport Since the permeating solution is dilute, the solvent is treated as a pure component. The effects that result from the presence of solute are accounted for as external forces. Consider first the transport of a pure component. For pure component B, the phenomenological relations expressed in Equation 8 reduce to the single equation

However, by definition,

Since V=v for pure component B, the diffusive flux of component B, J.,, is zero. Therefore, the force Y_ must also be zero:

or, more directly,


9. THIEL ET AL.

119



According to Equation 28, the chemical potential gradient is balanced by external forces at steady-state. The net force acting on the solvent in a pressure-driven membrane separation process includes not only the force due to the motion of the solvent, but also a force due to the motion of the solute. Consequently, Equation 28 can be expanded to provide the result

where ' is a coefficient relating the external force acting on the solvent to the velocity of the solute. Equation 29 can be rearranged to yield the equation

Assuming constant values for C , , and ', Equation 30 can be integrated across the membrane to provide the result

Assuming equilibrium at both membrane-liquid interfaces, Au R =Au B . Since the chemical potential difference of the solvent across the membrane can be written as

Equation 31 can be rewritten as

Although Equation 33 is formally similar to the equation the for total flux through finely porous membranes (2^9), the derivation of Equation 33 uses no assumptions about the microstructure of the membrane. Comparison with the Analysis of Transport Through a Finely Porous Membrane The analysis of transport through finely porous membranes yields the result


(2)


120


where e is the surface porosity of the membrane, and the K ' are partition coefficients for equilibrium between free solution and the pore fluid. Equation 34 is similar to Equation 24 derived above. This resemblance is not surprising, since the two analyses share several key assumptions. First, both analyses describe binary diffusion, since both choose a membrane phase which excludes the membrane material. Second, both analyses express the results of intermolecular interactions in terms of partitioning and forces opposing motion. Finally, both analyses include external forces and chemical potential in the thermodynamic force terms. There is one major difference between this analysis and the analysis of a finely porous membrane: this analysis includes the force acting on the solvent, Y , in the phenomenological equation governing solute transport. The chemical potential term in the equation for solute transport is unaffected by the inclusion of the force Y because the contribution of pressure to chemical potential is negligible and because the mole fraction of solvent is nearly unity. The inclusion of Y does, however, lead to the introduction of the function b , thereby influencing the physical interpretation of the parameters in the solute transport equation. The presence of the term b-. indicates a role for solvent-membrane material interactions that the analysis of the finely porous model does not suggest. This additional role for solvent-membrane material interactions can be clarified by considering the separation achieved as the flux becomes infinitely great. This limiting separation provides a theoretical limit on the performance of the system, a limit which has been observed in laboratory studies (9,16,17). Equation 34 gives the limiting separation as b /KA ', while Equation 24 gives the limiting separation as (b./bg)/*^. T n e quantities b /K ' and ^ b A ^ B ^ K A 2 n a v e different relationships to the physicochemical interactions in the solute-solvent-membrane material system. Consider first the quantity b A / K ' , obtained from the analysis of a finely porous membrane. The ninetion b. is a result of the strength of solute-membrane material interactions relative to the strength of solute-solvent interactions, and so increases with increasing solute-membrane material affinity. The partition coefficient K. ' is a result of the strength of solute-membrane material interactions relative to the strength of solvent-membrane material interactions, and so increases with increasing solutemembrane material affinity but decreases with increasing solventmembrane material affinity. Consequently, b A / K A ' must increase as solvent-membrane material affinity increases relative to solutesolvent affinity. One cannot determine whether b /K ' increases or decreases with solute-membrane material affinity without more information about b and K . ' . Now consider tne quantity (b./b )/KA9. The terms b A and K A „ vary as before. The function b fs a measure of the strength or solvent-membrane material interactions relative to the strength of solute-solvent interactions, and so increases with increasing



9. THIEL ET AL.


121

solvent-membrane material affinity. Both the ratio b A /b R and the coefficient K.? increase as the strength of solute-membrane material interactions increases relative to the strength of solvent-membrane material interactions. Consequently, one cannot determine how the ratio ( D A / D «)/K A9 varies with the physicochemical properties of the solute-solvent-membrane material system without more information about the quantities b , b , and K. ? . Based on an analysis of the finely porous model, one would expect the limiting separation to increase as solvent-membrane material affinity increases relative to solute-solvent affinity. The analysis presented in this paper shows that the limiting separation does not necessarily increase as the affinity between the solvent and the membrane material increases. Conclusions This paper has presented an analysis of transport through membranes which uses no assumptions about the microstructure of the membrane. Since physicochemical interactions among the solute, solvent, and membrane material determine the strength of intermolecular forces, these interactions determine the values of the coefficients which relate separation to flux: ., , and K. .. The values of these coefficients are closely related to both tne chemical nature of the system and to the physical properties — the structure — of the system. Consequently, this work represents only a partial solution to the problem of transport through membranes. A complete understanding of the permeation process calls not only for an understanding of the transport equations and their fundamental basis, but also for an understanding of the relationship between physicochemical properties and the forces acting in the solute-solvent-membrane material system. Nomenclature A

Pure water permeability [kmol/m -s-kPa]

b.

A function representing the relative magnitudes of forces due to solute-membrane interactions and solute-solvent interactions [dimensionless]

C

Molar density [kmol/m3]

C.. M

Molar density of membrane phase [kmol/m ]

C.__ iM

Molar density of component i in the membrane phase [kmol/m ]

F.

External force acting on component i [(kg-m/s2)/kmol]

J

Molar diffusive flux of component i relative to the molar-average velocity [kmol/m2-s]

KA. Aj

Partition coefficient for solute at location j [dimensionless] f

K J

Partition coefficient for solute at location j for a porous membrane [dimensionless] In Reverse Osmosis and Ultrafiltration; Sourirajan, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.


122


L

Phenomenological coefficient for a binary system [(kmol/m2-s)/((kg-m/s2)/kmol)]

L ,

Phenomenological coefficient relating the force acting on component k to the flux of component i [(kmol/m2-s)/((kg-m/s2)/kmol)]

n

Number of components [dimensionless]

N

Molar flux of component i relative to stationary coordinates [kmol/m2-s]

p

Pressure [kPa]

p

Pressure at location j [kPa]

R

Gas law constant [J/kmol-K]

T

Thermodynamic temperature [K]

v.

Velocity of component i [m/s]

V

Molar-average velocity [m/s]

V

Molar volume of system [m3/kmol]

V.

Partial molar volume of component i [m3/kmol]

X..

Mole fraction of component i at location j [dimensionless]

Y.

Thermodynamic force acting on component i [(kg-m/s2)/kmol]

z

Position within the membrane phase [m]

e

Surface porosity of the membrane [dimensionless]

u.

Chemical potential of component i [J/kmol]

y.

Chemical potential of component i in the membrane phase [J/kmol]

m

ir.

Osmotic pressure at location i [kPa]

"

Coefficient related to the diffusivity of the system A — B [((kg-m/s2)/kmol)/(m/s)]

•

Coefficient relating the solute velocity relative to the membrane to the external force acting on the solvent [((kg-m/s2)/kmol)/(m/s)]

Coefficient relating the velocity of component i relative to the membrane to the external force acting on component i [((kg-m/s2)/kmol)/(m/s)]

T

Effective thickness of the membrane phase [m]


9. THIEL ET AL.



Subscripts: A

Solute

B

Solvent

T

Total solution (solute and solvent)

2

Retentate immediately adjacent to the membrane

3

Permeate

Acknowledgments The authors are grateful for the support provided by the National Science Foundation (Separation Processes Program, grant number CPE-8312671), and also for the support provided by the Separations Research Program, Department of Chemical Engineering, and Graduate School of the University of Texas at Austin. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Lonsdale, H. K.; Merten, U.; Riley, R. L. J. Appl. Polymer Sci. 1965, 9, 1341-1362. Merten, U. In "Desalination by Reverse Osmosis"; U. Merten, Ed.; MIT Press: Cambridge, 1966; p. 15-54. Burghoff, H. G.; Lee, K. L.; Pusch, W. J. Appl. Polym. Sci. 1980, 25, 323-347. Mason, E. A.; Wendt, R. P.; Bresler, E. H. J. Membr. Sci. 1980, 6, 283-298. Laity, R. W. J. Phys. Chem. 1959, 63, 80-83. Smit, J.A.M.; Eijsermans, J. C.; Staverman, A. J. J. Phys. Chem. 1975, 79, 2168-2175. Schmitt, A.; Craig, J. B. J. Phys. Chem. 1977, 81, 1338-1342. Lorimer, J. W. J. Chem. Soc., Faraday Trans. II 1978, 74, 75-83. Jonsson, G.; Boesen, C. E. Desalination 1975, 17, 145-165. Spiegler, K. S.; Kedem, O. Desalination 1966, 1, 311-326. Kimura, S.; Sourirajan, S. AIChE J. 1967, 13, 497-503. DeGroot, S. R. "Thermodynamics of Irreversible Processes"; North-Holland Publishing Company: Amsterdam, 1958. DeGroot, S. R.; Mazur, P. "Non-Equilibrium Thermodynamics"; North-Holland Publishing Company: Amsterdam, 1962. Prigogine, I. "Introduction to Thermodynamics of Irreversible Processes", 3rd ed.; Interscience Publishers: New York, 1967. Spiegler, K. S. Trans. Faraday Soc. 1958, 54, 1408-1428. Jonsson, G. Desalination 1978, 24, 19-37. Baker, R. W.; Eirich, F. R.; Strathmann, H. J. Phys. Chem. 1972, 76, 238-242.

RECEIVED February 22, 1985


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Analysis of Transport in a Pressure-Driven Membrane Separation

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