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Linear approximations for the description of solute flux through permselective membranes
journalof M:xsE ELSEVIER
JoumalofMembraneScience95
(1994) 179-184
Linear approximations for the description of solute flux through permselective membranes Jacek Waniewski Institute ofBiocybernetics and Biomedical Engineering, Polish Academy of Science, Trojdena 4, PL-02-109 Warsaw,Poland
Received 20 January 1994; accepted in revised form 10 May 1994
Abstract Solute flux in the combined diffusive and convective transport through a homogeneous permselective membrane is a non-linear function of volume flux. A set of linear approximations for this function, which are based on the application of the weighted mean value of boundary solute concentrations for the description of the mean intramembrane concentration, is reviewed. The choice of a particular approximation depends on the range of the Peclet number involved in the investigated problem. Furthermore, it is shown that the original Kedem-Katchalsky formalism, which used a logarithmic mean for the mean intramembrane solute concentration, is a particular case of the general non-linear formalism for zero net solute flux with non-vanishing diffusive and convective fluxes. Thus, the Kedem-Katchalsky logarithmic mean concentration may be used as a linear approximation for the case of diffusive transport in the opposite direction to convective transport. Linear approximations of solute flux equations are also useful for simple phenomenological descriptions of complex membrane systems, which otherwise would need sophisticated mathematical models. Keywords: Irreversible thermodynamics; Membrane transport; Diffusion; Convective transport
1.Introduction
Js=P(c,-cR)+(l-o)Jvc~
The contemporary theoretical description of the combined diffusive and convective transport through a permselective membrane was originated by Staverman [ I] and Kedem and Katchalsky [ 21. In their papers the phenomenological coefficients, described usually as diffusive permeability, P, and the Staverman reflection coefficient, a, were introduced according to the general rules of linear irreversible thermodynamics, and the so-called Kedem-Katchalsky “practical equation” for solute flux, Js, through a discontinuous system was proposed [ 2 ] : 0376-7388/94/$07.00
(1) where Jv is the volume flux, cL and CRare the solute concentrations at the left and the right side of the membrane, respectively, see Fig. 1, and CM is the mean intramembrane concentration. The following logarithmic mean value, c,,, of c, and CRwas proposed for CM[ 2 ] : CL
CR
(2)
and the arithmetic mean CL +cR Car --
0 1994 Elsevier Science B.V. All rights reserved
SsDlO376-7388(94)00110-K
-CR
“b=h CL-In 2
(3)
180
J. Waniewski/Journal of Membrane Science 95 (1994) 179-184
of both sides of Eq. (4) from 0 to 6 (Fig. 1), assuming steady-state conditions (i.e., Js and Jv independent of x) and a homogeneous membrane (i.e., Pd and o independent of x), yields Eq. (1) for Js with
Membrane
cM=(l-f)CL+fiR
(5)
where %$exp(iv)
Fig. 1. Schematic drawing of a concentration the permselective membrane.
profile inside
as an approximation of cl,, for (CL-cR)/cL < 1, i.e. CRClose to CL,Was suggested for practical applications [ 2 1. Next, Katchalsky and Curran provided an explanation of formula (2) by analogy with the thermodynamic description of osmotic pressure in the case of an ideal permselective membrane [ 3 1. However, their derivation should be valid only for conditions close to the equilibrium state, i.e., for Jv%O and Jsx 0, for which the definition of osmotic pressure is applicable [ 41. Therefore q, should be considered as an approximate value of the true mean intramembrane concentration CMin Eq. ( 1), if both Jv and Js are small. In spite of many applications of q,, Eq. (2)) many researchers questioned this approach [ 5-101. Moreover, it was recognized that the above set of equations led to paradoxical values of concentrations (negative or infinite) for the two-membrane system [ 10 1. The correct description of fluxes has been provided using the following local equation for Js within the membrane [ 561: J,=-P$+(l--a)J,c
where c=c(x) is the solute concentration in an aqueous solution that would be at equilibrium with the membrane in an imaginary inlinitesima1 slit perpendicular to the x-direction at point x in the membrane [ 61 and Pd is the local diffusive permeability of the membrane. Integration
- 1
where ;1= ( 1-a) /P, P= Pd/B, and 6 is the membrane thickness [ 4- 12 ].I 1Jv 1 is often called the Peclet number, Pe [ 4-12 1. This approach was already used in the 1930s for the description of the mixed diffusive and convective flow of atoms in a gas medium by Hertz [ 13 ] as well as for the description of the solute flow through completely permeable membranes, i.e., with o=O, by Manegold and Soft [ 141. Eq. (4) for membranes with o> 0 was formulated by Patlak et al. [ 5 1, and later re-discovered by several researchers, see, e.g. [ 6,15 1. In the literature on physiological applications of the membrane transport theory the local approach, Eqs. (4)(6)) is often referred to as the Hertzian equation or the Patlak equation [ 161, whereas in the literature on the technological applications the paper by Spiegler and Kedem [ 6 ] is the main reference [ 41. Eqs. (4)-( 6 ) may also be derived from the microscopic description of the hindered transport of solutes in liquid-filled pores of molecular dimensions, the hindered transport theory, and applied for a special kind of porous membranes [ 16,171. A great advantage of this approach is that the transport parameters, such as P and a, may be expressed by microscopic parameters, such as the size, shape and electrical charge of the solutes and pores. However, the practical applications of the hindered transport theory are relatively limited (in fact to track-etched membranes only [ 16,17 ] ), in contrast to the general thermodynamic approach. Another attempt to find a relationship between the microscopic description of the membrane and the solute structures is the fiber matrix theory [ 16 1. The “global” description of Js, Eqs. ( 1 ), (5 )
181
J. Waniewski /Journal of Membrane Science 95 (1994) 179-l 84
and (6)) yielded by the local approach, Eq. (4)) is non-linear in Jv. This non-linearity was widely discussed and even the applicability of linear irreversible thermodynamics for the description of the combined convective and diffusive transport was questioned [ 731. It is worth to note that the local Eq. (4) can be derived from linear irreversible thermodynamics and in fact it describes the linear dependence of flows and thermodynamic forces, according to the main postulate of this theory. However, the coefficient ( 1 - a)c in Eq. (4) depends on the thermodynamic state of the system (i.e., on the concentration of the solute). This phenomenon, although non-contradictory to the basic assumptions of linear irreversible thermodynamics, leads to non-linearity in the global description. However, linear approximations have been proposed for practical applications [ 151. The original logarithmic Kedem-Katchalsky formula for cM,Eq. (2 ), is not currently in use in the membrane science, nevertheless it is still quoted in some textbooks and review papers [4,16,18]. The problem of the relation of the logarithmic formula for CMto the local approach as well as the problem of the correctness of the Kedem-Katchalsky derivation of q, have never been fully elucidated. Some researchers stated that the logarithmic expression for CM,Eq. (2) “had nothing to do with thermodynamics. It was an arbitrary choice” [ 191. Recently, Tanimura et al. analyzed Eq. ( 1) using a frictional model and noted that the logarithmic mean, Eq. (2), may be used in Eq. ( 1) assuming that P is modified to fit to Eq. (1) with CMdescribed by Eq. (5) [ 201. The modification of P was attributed to the deviation of the solution inside the membrane from the equilibrium with an imaginary free solution [ 201. The objective of the current paper is to show that the original Kedem-Katchalsky approach was correct within the implicit assumptions involved. The misunderstanding of these assumptions was a cause of the later discussion concerning the Kedem-Katchalsky formalism. We also show that the logarithmic mean concentration is a specific case of the local approach for Js=O, which condition, however, does not necessarily
represent the equilibrium in the system. Furthermore, we discuss various approximations of the general formula ( 5 ) , which have been proposed for practical applications [ 15,2 1,221. 2. Generalized Kedem-Katchalsky logarithmic mean concentration Eq. (4) can be considered as a differential equation for variable c. At steady state, when Js and Jv are independent of x, the solution of Eq. (4)) assuming that Wv # 0, is c(x) =y+ (cL -Y)e”uvls
(7) where y= Js/ [ ( 1 -a) Jv]. Note, that the solution of Eq. (4) for Wv = 0 may be obtained from Eq. (7 ) using the limit Wv-+O, which should be considered as two separate cases: (1) (l-a)Jv-+Owiththelimitc(x)=~~-Jsx/(P&, and (2) P+oc, with the limit c(x) =c~ The calculation of the average solute concentration in the membrane CM=(l/G)_f$(x)dx, using Eq. (7), yields CL-CR Cl, =Y+
(8)
ln(cL-y)-ln(cR-y)
Eq. (8) is a generalization of the original KedemKatchalsky formula (2 ) . In general, concentration profiles expressed by Eq. (7 ) depend on both Js and Jv and therefore the average concentration cl,, Eq. (8), depends also on Js and Jv. In the case of Js=O, i.e., y=O, the concentration profile, described by Eq. ( 7 ) , is exponential and Eq. (8) simplifies to Eq. (2). However, this does not mean that stationary states of volume and mass transports are assumed. In fact, it follows from Eq. (4) that a nonvanishing flux Jv may exist in the direction opposite to the x direction but the convective transport is balanced by the diffusive transport. For practical purposes the description of CM with Eqs. (5 ) and (6) instead of Eq. (8 ) is perhaps more useful. Both descriptions are, however, equivalent. Namely, from Eq. (7) one has for x=6 cR=y+ (cL-r)eti” and Eqs. (8) and (9) yield Eq. (5).
(9)
J. Waniewski /Journal of Membrane Science 95 (1994) 179- 184
182
cR/cL=0.05 (Fig. 2) and c,/c,=OS (Fig. 3). For practical applications two methods of linearization can be utilized. The first one consists of an approximation of a curve by its tangent lines or asymptotes. In this manner three important linear approximations of the function Js versus Jv can be obtained [l&12,15]: cM=cX, Eq. (3), which defines a tangent line at the point WV= 0 and may be used as an approximation for 1WV 1cK 1, see Figs. 2 and 3, as well as cM= cL for tiv > 1 and cM= CRfor tiv CK- 1, which define asymptotic lines for WV+ + 03, respectively [ 15 1. Therefore the following expressions can be used as approximations for Js:
3. Linear approximations The solute flux, Js, described by Eqs. ( 1) , ( 5 ) and (6), depends on the volume flux, Jv, in a non-linear way. Js as function of Jv is presented in Figs. 2 and 3 using non-dimensional variables T=Js/P/cL and c=Wv; i.e., graphs of the solutions of equation [= 1-CR/CL + r[ 1-f( 0 + f(c)cR/cL], withf(c) described by Eq. (6), and
cL+cL
for 1WV 1-=K1
(10)
J~=P(cL-cR)+(~-~)J~cL
for Wv >> 1 -2-i -4
, -3
-2
-1
0
1
2
3
4
Js=P(cL
for&*
~Jv Fig. 2. Normalized solute flux, J,/P/c,, versus A.&, for cR/ cL=0.05. (p) Js described by Eqs. (l), (5) and (6); (- - -) Js described by Eq. (13) with F=OS; (- - -) Js described by Eq. (13) with F=0.33; (...) Js described by the original Kedem-Katchalsky formalism, Eqs. ( 1) and (2).
-CR) +
(11) (1 -ff)JvcR
-1
(12)
Another type of linearization uses chords of non-linear curves. For practical applications the most interesting question concerns linearization in intervals between Wv= 0 and an assumed (maximal) value, AJ,,,,,, of Wv. This can be achieved assuming that f=F, where F is a constant equal tof(Wv,,,,), so that (cf. [ 151) JS,F=P(cL-+)+(l-CT)JV
[(l-F)cL+FcR] (13)
-4
-3
-2
-1
0
1
2
3
4
hJV
Fig. 3. Normalized solute flux, J,/P/c,, versus A.& for c,/ cL=0.5 (-) JsdescribedbyEqs. (l), (5)and (6); (---) Js described by Eq. ( 13) with F=0.5; (- - -) Js described by Eq. ( 13 ) with F=0.33; ( ... ) Js described by the original Kedem-Katchalsky formalism, Eqs. ( 1) and (2).
is equal to Js at WV= 0 and W,=Wv,,,,, Figs. 2 and 3. By definition JS,F is a linear function of Jv, cL and CR.In particular, F= l/3 was proposed as a convenient linearization in the range 0
J. Waniewski /Journal of MembraneScience 95 (1994) 179-184
(14)
i.e., exactly the original Kedem-Katchalsky formula, Eq. (2). Thus, the Kedem-Katchalsky formula describes the chord of the curve Js versus Jv drawn across the points
[Jv=(~l(l-a))ln(c,lc~),Jsl=O and
183
- l.O
[Jv=O, Js=P(c,_-cR)]
cf. Figs. 2 and 3. To evaluate the adequacy of these approximations a factor CDdefined as J,=(l-@)J,,
(15)
with
(16)
should be analyzed [ 15 1. In the case of Wv>O, @ is maximal if CL/ ( CL -CR) = 1, i.e., if CR= 0. The eVahatiOU Of maximal possible error, i.e., 0 for CR=0, for some of the approximate linear equations is presented in Fig. 4 as a function of Wv. The relative maximal error is within the limits + 12% for l.O-
0.5.-__--
-_~~_l_====_~_~_l-------;:;~~~~,~_,~~_~_~.
-l.O-4
' I -3
. I' -2
I -1
0
'..., *... X\. *.**.... -.._ *.. 'k. I. 'A. . .. 1 3 4 1 2
WV
Fig. 4. Relative error, @, of linear approximations sus&.JsdescribedbyEq.(13)with(---)F=0.5;(---) F=0.33; (-.-.-) Fc0.66; (...) @= kO.1.
of Js ver-
4. Conclusions The Staverman-Kedem-Katchalsky “practical” approach to the theoretical description of solute transport through a permselective membrane has originated applications of irreversible thermodynamics in the membrane sciences. As limitations for the proposed equation for solute flux were found, the equation was replaced with the correct one which was derived using the local application of the same thermodynamic description as that formulated by Staverman, Kedem and Katchalsky [ l-3 1. However, it has not been recognized that the original Kedem-Katchalsky equation is a particular case of the general description and that it may be used as a linear approximation for the description of transmembrane solute flux if convective flux is opposite to diffusive flux. Another linear approximation for the description of solute flux, Eq. (3), was already proposed by Kedem and Katchalsky [ 2 1, and is likely to be the most widely used simplified description of the solute flux. This approximation is valid if diffusive transport is prevailing over convective transport. On the other hand, during prevailing convective transport, the mean intramembrane concentration is approximated by the concentration in the convective flow entering the membrane. The intermediate cases of combined diffusive and convective transport were discussed by Villarroel et al. [ 15 1. The weighted mean concentration, ( 1 - F)cR + FcL, was proposed for the approximation of the mean intra-
J. Waniewski/Journal of Membrane Science 95 (1994) 179-184
184
membrane concentration. Values of 0 I FI 0.5 may be used for J,2 0, whereas values of 0.5 I FI 1 provide approximate descriptions for Jv S 0, assuming Js 2 0 for both cases (Fig. 4). The linear approximations have been derived for homogeneous membranes and are adequate for specific conditions of the membrane transport (i.e., the actual value of the Peclet number). However, membranes used in practical applications or found in biological systems are often far from being homogeneous. Nevertheless, simple and practical descriptions of such complex membrane systems are useful in engineering, physiology and medicine. For this purpose, the linear approximations of thermodynamic description of the membrane transport may be considered as a phenomenological approach validated mainly by its simplicity and usefulness. Recent examples of such studies may be found in the evaluation of combined diffusive and convective solute transport in hemodialyzers [ 2 1 ] and in peritoneal dialysis [ 22 1. In both cases linear approximations of the solute flux provided an accurate description of the measured quantities, although the reasons for such approximations did not follow from the description of the transport through the homogeneous membrane.
References [ 1] A.J. Staverman, The theory of measurement of osmotic pressure, Ret, Trav. Chim., 70 ( 195 1) 344-352.
[ 210. Kedem and A. Katchalsky, Thermodynamic
analysis of the permeability of biological membranes to nonelectrolytes, B&him. Biophys. Acta, 27 (1958) 229246. [ 3]A. Katchalsky and P.F. Curran, Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, 1965. [ 4 ] E.A. Mason and H.K. Lonsdale, Statistical-mechanical theory of membrane transport, J. Membrane Sci., 51 (1990) l-81. [ 51 C.S. Patlak, D.A. Goldstein and J.F. Hoffman, The flow of solute and solvent across a two-membrane system, J. Theor. Biol., 5 ( 1963) 426-442. [ 6lK.S. Spiegler and 0. Kedem, Thermodynamics of hypertiltration (reverse osmosis): criteria for efficient membranes, Desalination, 1 ( 1966) 31 l-326.
[ 71 E.H. Bresler and R.P. Wendt, Diffusive and convective flow across membranes: irreversible thermodynamic approach, Science, 163 (1969) 944-945. [ 81 E.H. Bresler and R.P. Wendt, Replay to mouring, Science, 166 (1969) 1438. [9]R.I. Sha’li, G.T. Rich, D.C. Mikulecky and A.K. Solomon, Determination of urea permeability in red cells by minimum method, J. Gen. Phys., 55 (1970) 427-450. [ 1OlS.R. Thomas and DC. Mikulecky, Transcapillary solute exchange, Microvasc. Res., 15 ( 1978) 207-220. [ 111 L. Axel, Flow limits of Kedem-Katchalsky equations for fluid flux, Bull. Math. Biol., 38 (1976) 671-677. [ 121 D.C. Mickulecky, A simple network thermodynamic method for series - parallel coupled flows. II. The nonlinear theory, with applications to coupled solute and volume flow in a series membrane, J. Theor. Biol., 69 (1977) 51 l-541. [ 13 ] G. Hertz, Ein neues Verfahren zur Trennung von Gasgemischen durch Diffusion, Physik. Z., 23 (1922) 433434. [ 14]E. Manegold and K. Soft, Uber Kapillarsysteme. XIV. Die Dynamik osmotisher Zellen), Kolloid Z., 59 ( 1932) 179-195. [ 15 ] F. Villarroel, E. Klein and F. Holland, Solute flux in hemodialysis and hemoliltration membranes, Trans. Am. Sot. Artif. Intern. Organs, 23 (1977) 225-233. [ 161 F.-R.E. Curry, Mechanics and thermodynamics of transcapillary exchange, in E.M. Renkin and CC. Michel (Eds. ), Handbook of Physiology - Cardiovascular System IV, American Physiological Society, Bethesda, 1984, pp. 309-374. [ 171W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J, 33 (1987) 1409-1425. [ 181 C. Crone and 0. Christensen, Transcapillary transport of small solutes and water, in AC. Guyton and D.V. Young (Eds. ) , Cardiovascular Physiology III, Intemational Review of Physiology, Vol. 18, University Park Press, Baltimore, 1979, pp. 149-2 13. [ 19lM.A. Katz and E.H. Bresler, Osmosis, in N.C. Staub and A.E. Taylor (Eds.), Edema, Raven Press, New York, NY, 1984, pp. 39-60. [ 201s. Tanimura, S. Nakao and S. Kimura, Transport equation for a membrane based on a frictional model, J. Membrane Sci., 84 (1993) 79-91. [ 2 1 ] J. Waniewski, P. Lucjanek and A. Werynski, Alternative descriptions of combined diffusive and convective mass transport in hemodialyzer, Artif. Organs, 17 ( 1993) 37. [ 221J. Waniewski, A. Werynski, 0. Heimbtlrger and B. Lindholm, Simple membrane models for peritoneal dialysis: evaluation of diffusive and convective solute transport, ASAIO J., 38 (1992) 788-796.
JoumalofMembraneScience95
(1994) 179-184
Linear approximations for the description of solute flux through permselective membranes Jacek Waniewski Institute ofBiocybernetics and Biomedical Engineering, Polish Academy of Science, Trojdena 4, PL-02-109 Warsaw,Poland
Received 20 January 1994; accepted in revised form 10 May 1994
Abstract Solute flux in the combined diffusive and convective transport through a homogeneous permselective membrane is a non-linear function of volume flux. A set of linear approximations for this function, which are based on the application of the weighted mean value of boundary solute concentrations for the description of the mean intramembrane concentration, is reviewed. The choice of a particular approximation depends on the range of the Peclet number involved in the investigated problem. Furthermore, it is shown that the original Kedem-Katchalsky formalism, which used a logarithmic mean for the mean intramembrane solute concentration, is a particular case of the general non-linear formalism for zero net solute flux with non-vanishing diffusive and convective fluxes. Thus, the Kedem-Katchalsky logarithmic mean concentration may be used as a linear approximation for the case of diffusive transport in the opposite direction to convective transport. Linear approximations of solute flux equations are also useful for simple phenomenological descriptions of complex membrane systems, which otherwise would need sophisticated mathematical models. Keywords: Irreversible thermodynamics; Membrane transport; Diffusion; Convective transport
1.Introduction
Js=P(c,-cR)+(l-o)Jvc~
The contemporary theoretical description of the combined diffusive and convective transport through a permselective membrane was originated by Staverman [ I] and Kedem and Katchalsky [ 21. In their papers the phenomenological coefficients, described usually as diffusive permeability, P, and the Staverman reflection coefficient, a, were introduced according to the general rules of linear irreversible thermodynamics, and the so-called Kedem-Katchalsky “practical equation” for solute flux, Js, through a discontinuous system was proposed [ 2 ] : 0376-7388/94/$07.00
(1) where Jv is the volume flux, cL and CRare the solute concentrations at the left and the right side of the membrane, respectively, see Fig. 1, and CM is the mean intramembrane concentration. The following logarithmic mean value, c,,, of c, and CRwas proposed for CM[ 2 ] : CL
CR
(2)
and the arithmetic mean CL +cR Car --
0 1994 Elsevier Science B.V. All rights reserved
SsDlO376-7388(94)00110-K
-CR
“b=h CL-In 2
(3)
180
J. Waniewski/Journal of Membrane Science 95 (1994) 179-184
of both sides of Eq. (4) from 0 to 6 (Fig. 1), assuming steady-state conditions (i.e., Js and Jv independent of x) and a homogeneous membrane (i.e., Pd and o independent of x), yields Eq. (1) for Js with
Membrane
cM=(l-f)CL+fiR
(5)
where %$exp(iv)
Fig. 1. Schematic drawing of a concentration the permselective membrane.
profile inside
as an approximation of cl,, for (CL-cR)/cL < 1, i.e. CRClose to CL,Was suggested for practical applications [ 2 1. Next, Katchalsky and Curran provided an explanation of formula (2) by analogy with the thermodynamic description of osmotic pressure in the case of an ideal permselective membrane [ 3 1. However, their derivation should be valid only for conditions close to the equilibrium state, i.e., for Jv%O and Jsx 0, for which the definition of osmotic pressure is applicable [ 41. Therefore q, should be considered as an approximate value of the true mean intramembrane concentration CMin Eq. ( 1), if both Jv and Js are small. In spite of many applications of q,, Eq. (2)) many researchers questioned this approach [ 5-101. Moreover, it was recognized that the above set of equations led to paradoxical values of concentrations (negative or infinite) for the two-membrane system [ 10 1. The correct description of fluxes has been provided using the following local equation for Js within the membrane [ 561: J,=-P$+(l--a)J,c
where c=c(x) is the solute concentration in an aqueous solution that would be at equilibrium with the membrane in an imaginary inlinitesima1 slit perpendicular to the x-direction at point x in the membrane [ 61 and Pd is the local diffusive permeability of the membrane. Integration
- 1
where ;1= ( 1-a) /P, P= Pd/B, and 6 is the membrane thickness [ 4- 12 ].I 1Jv 1 is often called the Peclet number, Pe [ 4-12 1. This approach was already used in the 1930s for the description of the mixed diffusive and convective flow of atoms in a gas medium by Hertz [ 13 ] as well as for the description of the solute flow through completely permeable membranes, i.e., with o=O, by Manegold and Soft [ 141. Eq. (4) for membranes with o> 0 was formulated by Patlak et al. [ 5 1, and later re-discovered by several researchers, see, e.g. [ 6,15 1. In the literature on physiological applications of the membrane transport theory the local approach, Eqs. (4)(6)) is often referred to as the Hertzian equation or the Patlak equation [ 161, whereas in the literature on the technological applications the paper by Spiegler and Kedem [ 6 ] is the main reference [ 41. Eqs. (4)-( 6 ) may also be derived from the microscopic description of the hindered transport of solutes in liquid-filled pores of molecular dimensions, the hindered transport theory, and applied for a special kind of porous membranes [ 16,171. A great advantage of this approach is that the transport parameters, such as P and a, may be expressed by microscopic parameters, such as the size, shape and electrical charge of the solutes and pores. However, the practical applications of the hindered transport theory are relatively limited (in fact to track-etched membranes only [ 16,17 ] ), in contrast to the general thermodynamic approach. Another attempt to find a relationship between the microscopic description of the membrane and the solute structures is the fiber matrix theory [ 16 1. The “global” description of Js, Eqs. ( 1 ), (5 )
181
J. Waniewski /Journal of Membrane Science 95 (1994) 179-l 84
and (6)) yielded by the local approach, Eq. (4)) is non-linear in Jv. This non-linearity was widely discussed and even the applicability of linear irreversible thermodynamics for the description of the combined convective and diffusive transport was questioned [ 731. It is worth to note that the local Eq. (4) can be derived from linear irreversible thermodynamics and in fact it describes the linear dependence of flows and thermodynamic forces, according to the main postulate of this theory. However, the coefficient ( 1 - a)c in Eq. (4) depends on the thermodynamic state of the system (i.e., on the concentration of the solute). This phenomenon, although non-contradictory to the basic assumptions of linear irreversible thermodynamics, leads to non-linearity in the global description. However, linear approximations have been proposed for practical applications [ 151. The original logarithmic Kedem-Katchalsky formula for cM,Eq. (2 ), is not currently in use in the membrane science, nevertheless it is still quoted in some textbooks and review papers [4,16,18]. The problem of the relation of the logarithmic formula for CMto the local approach as well as the problem of the correctness of the Kedem-Katchalsky derivation of q, have never been fully elucidated. Some researchers stated that the logarithmic expression for CM,Eq. (2) “had nothing to do with thermodynamics. It was an arbitrary choice” [ 191. Recently, Tanimura et al. analyzed Eq. ( 1) using a frictional model and noted that the logarithmic mean, Eq. (2), may be used in Eq. ( 1) assuming that P is modified to fit to Eq. (1) with CMdescribed by Eq. (5) [ 201. The modification of P was attributed to the deviation of the solution inside the membrane from the equilibrium with an imaginary free solution [ 201. The objective of the current paper is to show that the original Kedem-Katchalsky approach was correct within the implicit assumptions involved. The misunderstanding of these assumptions was a cause of the later discussion concerning the Kedem-Katchalsky formalism. We also show that the logarithmic mean concentration is a specific case of the local approach for Js=O, which condition, however, does not necessarily
represent the equilibrium in the system. Furthermore, we discuss various approximations of the general formula ( 5 ) , which have been proposed for practical applications [ 15,2 1,221. 2. Generalized Kedem-Katchalsky logarithmic mean concentration Eq. (4) can be considered as a differential equation for variable c. At steady state, when Js and Jv are independent of x, the solution of Eq. (4)) assuming that Wv # 0, is c(x) =y+ (cL -Y)e”uvls
(7) where y= Js/ [ ( 1 -a) Jv]. Note, that the solution of Eq. (4) for Wv = 0 may be obtained from Eq. (7 ) using the limit Wv-+O, which should be considered as two separate cases: (1) (l-a)Jv-+Owiththelimitc(x)=~~-Jsx/(P&, and (2) P+oc, with the limit c(x) =c~ The calculation of the average solute concentration in the membrane CM=(l/G)_f$(x)dx, using Eq. (7), yields CL-CR Cl, =Y+
(8)
ln(cL-y)-ln(cR-y)
Eq. (8) is a generalization of the original KedemKatchalsky formula (2 ) . In general, concentration profiles expressed by Eq. (7 ) depend on both Js and Jv and therefore the average concentration cl,, Eq. (8), depends also on Js and Jv. In the case of Js=O, i.e., y=O, the concentration profile, described by Eq. ( 7 ) , is exponential and Eq. (8) simplifies to Eq. (2). However, this does not mean that stationary states of volume and mass transports are assumed. In fact, it follows from Eq. (4) that a nonvanishing flux Jv may exist in the direction opposite to the x direction but the convective transport is balanced by the diffusive transport. For practical purposes the description of CM with Eqs. (5 ) and (6) instead of Eq. (8 ) is perhaps more useful. Both descriptions are, however, equivalent. Namely, from Eq. (7) one has for x=6 cR=y+ (cL-r)eti” and Eqs. (8) and (9) yield Eq. (5).
(9)
J. Waniewski /Journal of Membrane Science 95 (1994) 179- 184
182
cR/cL=0.05 (Fig. 2) and c,/c,=OS (Fig. 3). For practical applications two methods of linearization can be utilized. The first one consists of an approximation of a curve by its tangent lines or asymptotes. In this manner three important linear approximations of the function Js versus Jv can be obtained [l&12,15]: cM=cX, Eq. (3), which defines a tangent line at the point WV= 0 and may be used as an approximation for 1WV 1cK 1, see Figs. 2 and 3, as well as cM= cL for tiv > 1 and cM= CRfor tiv CK- 1, which define asymptotic lines for WV+ + 03, respectively [ 15 1. Therefore the following expressions can be used as approximations for Js:
3. Linear approximations The solute flux, Js, described by Eqs. ( 1) , ( 5 ) and (6), depends on the volume flux, Jv, in a non-linear way. Js as function of Jv is presented in Figs. 2 and 3 using non-dimensional variables T=Js/P/cL and c=Wv; i.e., graphs of the solutions of equation [= 1-CR/CL + r[ 1-f( 0 + f(c)cR/cL], withf(c) described by Eq. (6), and
cL+cL
for 1WV 1-=K1
(10)
J~=P(cL-cR)+(~-~)J~cL
for Wv >> 1 -2-i -4
, -3
-2
-1
0
1
2
3
4
Js=P(cL
for&*
~Jv Fig. 2. Normalized solute flux, J,/P/c,, versus A.&, for cR/ cL=0.05. (p) Js described by Eqs. (l), (5) and (6); (- - -) Js described by Eq. (13) with F=OS; (- - -) Js described by Eq. (13) with F=0.33; (...) Js described by the original Kedem-Katchalsky formalism, Eqs. ( 1) and (2).
-CR) +
(11) (1 -ff)JvcR
-1
(12)
Another type of linearization uses chords of non-linear curves. For practical applications the most interesting question concerns linearization in intervals between Wv= 0 and an assumed (maximal) value, AJ,,,,,, of Wv. This can be achieved assuming that f=F, where F is a constant equal tof(Wv,,,,), so that (cf. [ 151) JS,F=P(cL-+)+(l-CT)JV
[(l-F)cL+FcR] (13)
-4
-3
-2
-1
0
1
2
3
4
hJV
Fig. 3. Normalized solute flux, J,/P/c,, versus A.& for c,/ cL=0.5 (-) JsdescribedbyEqs. (l), (5)and (6); (---) Js described by Eq. ( 13) with F=0.5; (- - -) Js described by Eq. ( 13 ) with F=0.33; ( ... ) Js described by the original Kedem-Katchalsky formalism, Eqs. ( 1) and (2).
is equal to Js at WV= 0 and W,=Wv,,,,, Figs. 2 and 3. By definition JS,F is a linear function of Jv, cL and CR.In particular, F= l/3 was proposed as a convenient linearization in the range 0
J. Waniewski /Journal of MembraneScience 95 (1994) 179-184
(14)
i.e., exactly the original Kedem-Katchalsky formula, Eq. (2). Thus, the Kedem-Katchalsky formula describes the chord of the curve Js versus Jv drawn across the points
[Jv=(~l(l-a))ln(c,lc~),Jsl=O and
183
- l.O
[Jv=O, Js=P(c,_-cR)]
cf. Figs. 2 and 3. To evaluate the adequacy of these approximations a factor CDdefined as J,=(l-@)J,,
(15)
with
(16)
should be analyzed [ 15 1. In the case of Wv>O, @ is maximal if CL/ ( CL -CR) = 1, i.e., if CR= 0. The eVahatiOU Of maximal possible error, i.e., 0 for CR=0, for some of the approximate linear equations is presented in Fig. 4 as a function of Wv. The relative maximal error is within the limits + 12% for l.O-
0.5.-__--
-_~~_l_====_~_~_l-------;:;~~~~,~_,~~_~_~.
-l.O-4
' I -3
. I' -2
I -1
0
'..., *... X\. *.**.... -.._ *.. 'k. I. 'A. . .. 1 3 4 1 2
WV
Fig. 4. Relative error, @, of linear approximations sus&.JsdescribedbyEq.(13)with(---)F=0.5;(---) F=0.33; (-.-.-) Fc0.66; (...) @= kO.1.
of Js ver-
4. Conclusions The Staverman-Kedem-Katchalsky “practical” approach to the theoretical description of solute transport through a permselective membrane has originated applications of irreversible thermodynamics in the membrane sciences. As limitations for the proposed equation for solute flux were found, the equation was replaced with the correct one which was derived using the local application of the same thermodynamic description as that formulated by Staverman, Kedem and Katchalsky [ l-3 1. However, it has not been recognized that the original Kedem-Katchalsky equation is a particular case of the general description and that it may be used as a linear approximation for the description of transmembrane solute flux if convective flux is opposite to diffusive flux. Another linear approximation for the description of solute flux, Eq. (3), was already proposed by Kedem and Katchalsky [ 2 1, and is likely to be the most widely used simplified description of the solute flux. This approximation is valid if diffusive transport is prevailing over convective transport. On the other hand, during prevailing convective transport, the mean intramembrane concentration is approximated by the concentration in the convective flow entering the membrane. The intermediate cases of combined diffusive and convective transport were discussed by Villarroel et al. [ 15 1. The weighted mean concentration, ( 1 - F)cR + FcL, was proposed for the approximation of the mean intra-
J. Waniewski/Journal of Membrane Science 95 (1994) 179-184
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membrane concentration. Values of 0 I FI 0.5 may be used for J,2 0, whereas values of 0.5 I FI 1 provide approximate descriptions for Jv S 0, assuming Js 2 0 for both cases (Fig. 4). The linear approximations have been derived for homogeneous membranes and are adequate for specific conditions of the membrane transport (i.e., the actual value of the Peclet number). However, membranes used in practical applications or found in biological systems are often far from being homogeneous. Nevertheless, simple and practical descriptions of such complex membrane systems are useful in engineering, physiology and medicine. For this purpose, the linear approximations of thermodynamic description of the membrane transport may be considered as a phenomenological approach validated mainly by its simplicity and usefulness. Recent examples of such studies may be found in the evaluation of combined diffusive and convective solute transport in hemodialyzers [ 2 1 ] and in peritoneal dialysis [ 22 1. In both cases linear approximations of the solute flux provided an accurate description of the measured quantities, although the reasons for such approximations did not follow from the description of the transport through the homogeneous membrane.
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[ 210. Kedem and A. Katchalsky, Thermodynamic
analysis of the permeability of biological membranes to nonelectrolytes, B&him. Biophys. Acta, 27 (1958) 229246. [ 3]A. Katchalsky and P.F. Curran, Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, 1965. [ 4 ] E.A. Mason and H.K. Lonsdale, Statistical-mechanical theory of membrane transport, J. Membrane Sci., 51 (1990) l-81. [ 51 C.S. Patlak, D.A. Goldstein and J.F. Hoffman, The flow of solute and solvent across a two-membrane system, J. Theor. Biol., 5 ( 1963) 426-442. [ 6lK.S. Spiegler and 0. Kedem, Thermodynamics of hypertiltration (reverse osmosis): criteria for efficient membranes, Desalination, 1 ( 1966) 31 l-326.
[ 71 E.H. Bresler and R.P. Wendt, Diffusive and convective flow across membranes: irreversible thermodynamic approach, Science, 163 (1969) 944-945. [ 81 E.H. Bresler and R.P. Wendt, Replay to mouring, Science, 166 (1969) 1438. [9]R.I. Sha’li, G.T. Rich, D.C. Mikulecky and A.K. Solomon, Determination of urea permeability in red cells by minimum method, J. Gen. Phys., 55 (1970) 427-450. [ 1OlS.R. Thomas and DC. Mikulecky, Transcapillary solute exchange, Microvasc. Res., 15 ( 1978) 207-220. [ 111 L. Axel, Flow limits of Kedem-Katchalsky equations for fluid flux, Bull. Math. Biol., 38 (1976) 671-677. [ 121 D.C. Mickulecky, A simple network thermodynamic method for series - parallel coupled flows. II. The nonlinear theory, with applications to coupled solute and volume flow in a series membrane, J. Theor. Biol., 69 (1977) 51 l-541. [ 13 ] G. Hertz, Ein neues Verfahren zur Trennung von Gasgemischen durch Diffusion, Physik. Z., 23 (1922) 433434. [ 14]E. Manegold and K. Soft, Uber Kapillarsysteme. XIV. Die Dynamik osmotisher Zellen), Kolloid Z., 59 ( 1932) 179-195. [ 15 ] F. Villarroel, E. Klein and F. Holland, Solute flux in hemodialysis and hemoliltration membranes, Trans. Am. Sot. Artif. Intern. Organs, 23 (1977) 225-233. [ 161 F.-R.E. Curry, Mechanics and thermodynamics of transcapillary exchange, in E.M. Renkin and CC. Michel (Eds. ), Handbook of Physiology - Cardiovascular System IV, American Physiological Society, Bethesda, 1984, pp. 309-374. [ 171W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J, 33 (1987) 1409-1425. [ 181 C. Crone and 0. Christensen, Transcapillary transport of small solutes and water, in AC. Guyton and D.V. Young (Eds. ) , Cardiovascular Physiology III, Intemational Review of Physiology, Vol. 18, University Park Press, Baltimore, 1979, pp. 149-2 13. [ 19lM.A. Katz and E.H. Bresler, Osmosis, in N.C. Staub and A.E. Taylor (Eds.), Edema, Raven Press, New York, NY, 1984, pp. 39-60. [ 201s. Tanimura, S. Nakao and S. Kimura, Transport equation for a membrane based on a frictional model, J. Membrane Sci., 84 (1993) 79-91. [ 2 1 ] J. Waniewski, P. Lucjanek and A. Werynski, Alternative descriptions of combined diffusive and convective mass transport in hemodialyzer, Artif. Organs, 17 ( 1993) 37. [ 221J. Waniewski, A. Werynski, 0. Heimbtlrger and B. Lindholm, Simple membrane models for peritoneal dialysis: evaluation of diffusive and convective solute transport, ASAIO J., 38 (1992) 788-796.