- Email: [email protected]
Further comments on the diffusion equations for membrane formation
Journal of Membrane Science, 43 (1989) 319-323 Elsevier Science Publishers B.V.. Amsterdam - Printed in The Netherlands
319
Short Communication
FURTHER COMMENTS ON THE DIFFUSION EQUATIONS FOR MEMBRANE FORMATION
A.J. MCHUGH* Department
of Chemical Engineering, University of Illinois, Urbana, IL 61801 (U.S.A.)
and L. YILMAZ Department of Chemical Engineering, Middle East Technical University, Ankara (Turkey) (Received December 29,1987; accepted in revised form September 13,1988)
In a recent paper in this journal, Reuvers, van den Berg, and Smolders [l] presented a derivation of the ternary diffusion equations for mass transfer during the immersion precipitation step of the phase inversion process. A transformation of variables was used to justify the resultant transient equations which had been given without derivation in an earlier paper by Cohen, Tanny and Prager [ 21. The claim, as stated in the paper by Cohen et al., is that by formulating the diffusion equations in terms of the volume, m, of polymer between the membrane-bath interface and the point of observation, “the motion of the polymer never has to explicitely treated” [ 2, p. 4801. Reuvers et al. also referred to a derivation of ours [3] in which we took issue with the equations of Cohen et al. and the claim that the above-mentioned transformation automatically removes the convected flux term accounting for the polymer motion. In our derivation we used the transformation variables as defined by Cohen et al. [ 2, p. 4801 and also explicitely written out by Wijmans, Altena and Smolders [4] (their eqn. 1) in their ad-hoc derivation. The sole purpose of our note was to demonstrate that the transformation equations and definitions as giuerzby both sets of authors were inconsistent and lead to an incorrect omission of the convected flux term. In their derivation, Reuvers et al. use an explicitly time- and position-dependent expression for the transformation variable, m (as opposed to both Cohen et al. and Wijmans et al. who expressed m as only position dependent), along with what they consider to be the appropriate transformation equations to go from position-time space to m,t-space. From this they conclude that the continuity equations of Cohen et al. are correct and that our criticism is therefore unfounded. Our purpose here is to show that the derivation of Reuvers et *To whom correspondence should be addressed.
0376-7388/89/$03.50
0 1989 Elsevier Science Publishers B.V.
320
al. is not rigorously correct and when properly worked still very clearly leads to the presence of a bulk flow term. The issue at hand is important since proper formulation of the diffusion equations lies at the heart of membrane formation modeling and the conclusions one draws from such modeling are necessarily limited by the assumptions inherent in the starting equations. Equation derivation We start with the one-dimensional form of the ternary diffusion equations as given by us [ 3, eqns. (11) ] and also explicitely written (using a slightly different nomenclature) by Reuvers et al. (eqns. (8)) [ 1 ] )
In these equations subscripts refer to nonsolvent (1 ), solvent (2 ) , or polymer (3 ) , qbiare the volume fractions and Ji is the volume flux of component i relative to the polymer velocity, i.e., Ji= & ( ui- z+). The position coordinate, z, is measured with respect to an arbitrary, laboratory-fixed origin, and locates a point in the polymer-solvent-nonsolvent region of the cast film, as indicated in Fig. 1. During the course of the counter diffusion process, the membrane-bath interface may move from its initial position, zo, to some arbitrary position, z = 1(t ), which varies with time in some arbitrary but unspecified fashion. The precise time dependence can, in general, only be determined through simultaneous solution of the diffusion equations and associated boundary and initial conditions. Since one would normally expect the membrane film to shrink as the solvent-nonsolvent exchange takes place, the interface position in Fig. 1 is indicated accordingly. However the mathematical transformation is independent of the direction of the interface motion. We should also point out that, in
z=o
coagulation -_----
bath
,--__-------___I-
cast
film
initial
-both-film
interface
position
interface
at time,
t
support
Fig. 1. Sketch of coordinates used to define mass transfer equations. Note that z=O represents the origin for the fixed coordinate and z= 1(t) denotes the position of the moving interface which is initially at zO.
321
their paper, Cohen et al. use Z(t) as the position of a propagating diffusion layer inside the cast film while referring to the interface position as “surface”. In our opinion, errors in the transformation to @-space have resulted from confusions in the mathematical definitions of m and the correspondingly required mathematics. Cohen et al. specifically state that m is the volume of polymer (per unit of area) between the moving interface and the particular point of observation in the film. However, their equation for m [ 2, p. 4801 does not contain an explicit time dependence, nor does that given by Wijmans et al. [ 4, eqn. (1) 1. It was those equations, as written, that were used in our derivation of the missing polymer flow term [ 3, eqn. (12 ) 1. On the other hand, Reuvers et al., while writing m as a function of both position and time, misleadingly use the origin of the fixed laboratory coordinates as the lower limit on their defining integral (their eqn. 9). Using the definition of m as stated by Cohen et al., one must have the following (see Fig. 1)
where c denotes an arbitrary position in the membrane rule for the required derivatives in eqn. (1) , one has
film. Applying the chain
(3)
t/,=~/,<
(4)
From eqn. (2 )
where @3m=G3(Z( t) ,t) is the volume fraction of polymer at the moving interface and ZL,= dl/dt is the membrane-bath interface velocity. The continuity equation for the polymer in z&space [ 3, eqn. (7b) ] is
(6) Substitution
of eqn. (6) in eqn. (5 ) gives the following
where u 3m= u3 (Z( t) ,t) is the polymer velocity at the film-bath interface. Unless one arbitrarily assumes that the film interface velocity will be precisely that of the polymer, the second bracketed term in eqn. (7 ) will not be zero.
322
From eqn. (2 ) one also has
dm -=
a2 t
(8)
$3
Therefore, upon substitution of eqns. (3 ), (4)) (7)) and (8) into eqn. ( 11, the following set of diffusion equations results
api
0
$3 p=--at
a&
( >
dJi
$43
(~3m---U,) am
@3mp
i= 1,2
(9)
am
In these equations it is understood that the dependent variables, & and & are functions of m and t, while the term (uym- ZL~)&~ is a function of time only. The form of the convected term in these equations is similar to that derived in our earlier equations (i.e. [ 3, eqns. (12) ] ), however, its interpretation here is different in that it represents the bulk flow due to the difference between the polymer velocity at the interface, ugrn,and that of the interface itself, u,. For the reasons we have mentioned, this term will not, in general, be zero and must therefore be determined through the coupled boundary conditions for the component material balances at the moving interface [ 51. One of the transformation equations used by Reuvers et al., namely their eqn. (ll), is in error since it improperly equates the polymer velocity to the derivative of the spatial coordinate at constant m, i.e., u3 = (&/dt) 1m (in our nomenclature). As we pointed out earlier [3], the proper definition is u3= (&/at) 1c where subscript, <, refers to the material coordinates of the polymer molecules. The proper relationship for ug in m,t-space can be derived by straightforward application of the chain rule. Using the fact that m= m (z,t), in combination with eqns. (7) and (8)) one has
From this discussion we conclude that in order for the diffusion equations as initially presented by Cohen et al., and modified by Reuvers et al., to be correct, one must indeed make a restrictive assumption regarding the nature of the polymer motion during diffusion.
References 1
2
A.J. Reuvers, J.W.A. van den Berg and C.A. Smolders, Formation of membranes by means of immersion precipitation. Part I. A model to describe mass transfer during immersion precipitation, J. Membrane Sci., 34 (1987) 45. C. Cohen, G.B. Tanny and S. Prager, Diffusion-controlled formation of porous structures in ternary polymer systems, J. Polym. Sci., Polym. Phys. Ed., 17 (1979)
477.
323 3 A.J. McHugh and L. Yilmaz, The diffusion equations for polymer membrane formation in ternary systems, J. Polym. Sci., Polym. Phys. Ed., 23 (1985) 1271. 4 J.G. Wijmans, F.W. Altena and C.A. Smolders, Diffusion during the immersion precipitation process, J. Polym. Sci., Polym. Phys. Ed., 22 (1984) 519. 5 K.J. Ruschak and C.A. Miller, Spontaneous emulsification in ternary systems, Ind. Eng. Chem., Fundam., 11 (1972) 534.
319
Short Communication
FURTHER COMMENTS ON THE DIFFUSION EQUATIONS FOR MEMBRANE FORMATION
A.J. MCHUGH* Department
of Chemical Engineering, University of Illinois, Urbana, IL 61801 (U.S.A.)
and L. YILMAZ Department of Chemical Engineering, Middle East Technical University, Ankara (Turkey) (Received December 29,1987; accepted in revised form September 13,1988)
In a recent paper in this journal, Reuvers, van den Berg, and Smolders [l] presented a derivation of the ternary diffusion equations for mass transfer during the immersion precipitation step of the phase inversion process. A transformation of variables was used to justify the resultant transient equations which had been given without derivation in an earlier paper by Cohen, Tanny and Prager [ 21. The claim, as stated in the paper by Cohen et al., is that by formulating the diffusion equations in terms of the volume, m, of polymer between the membrane-bath interface and the point of observation, “the motion of the polymer never has to explicitely treated” [ 2, p. 4801. Reuvers et al. also referred to a derivation of ours [3] in which we took issue with the equations of Cohen et al. and the claim that the above-mentioned transformation automatically removes the convected flux term accounting for the polymer motion. In our derivation we used the transformation variables as defined by Cohen et al. [ 2, p. 4801 and also explicitely written out by Wijmans, Altena and Smolders [4] (their eqn. 1) in their ad-hoc derivation. The sole purpose of our note was to demonstrate that the transformation equations and definitions as giuerzby both sets of authors were inconsistent and lead to an incorrect omission of the convected flux term. In their derivation, Reuvers et al. use an explicitly time- and position-dependent expression for the transformation variable, m (as opposed to both Cohen et al. and Wijmans et al. who expressed m as only position dependent), along with what they consider to be the appropriate transformation equations to go from position-time space to m,t-space. From this they conclude that the continuity equations of Cohen et al. are correct and that our criticism is therefore unfounded. Our purpose here is to show that the derivation of Reuvers et *To whom correspondence should be addressed.
0376-7388/89/$03.50
0 1989 Elsevier Science Publishers B.V.
320
al. is not rigorously correct and when properly worked still very clearly leads to the presence of a bulk flow term. The issue at hand is important since proper formulation of the diffusion equations lies at the heart of membrane formation modeling and the conclusions one draws from such modeling are necessarily limited by the assumptions inherent in the starting equations. Equation derivation We start with the one-dimensional form of the ternary diffusion equations as given by us [ 3, eqns. (11) ] and also explicitely written (using a slightly different nomenclature) by Reuvers et al. (eqns. (8)) [ 1 ] )
In these equations subscripts refer to nonsolvent (1 ), solvent (2 ) , or polymer (3 ) , qbiare the volume fractions and Ji is the volume flux of component i relative to the polymer velocity, i.e., Ji= & ( ui- z+). The position coordinate, z, is measured with respect to an arbitrary, laboratory-fixed origin, and locates a point in the polymer-solvent-nonsolvent region of the cast film, as indicated in Fig. 1. During the course of the counter diffusion process, the membrane-bath interface may move from its initial position, zo, to some arbitrary position, z = 1(t ), which varies with time in some arbitrary but unspecified fashion. The precise time dependence can, in general, only be determined through simultaneous solution of the diffusion equations and associated boundary and initial conditions. Since one would normally expect the membrane film to shrink as the solvent-nonsolvent exchange takes place, the interface position in Fig. 1 is indicated accordingly. However the mathematical transformation is independent of the direction of the interface motion. We should also point out that, in
z=o
coagulation -_----
bath
,--__-------___I-
cast
film
initial
-both-film
interface
position
interface
at time,
t
support
Fig. 1. Sketch of coordinates used to define mass transfer equations. Note that z=O represents the origin for the fixed coordinate and z= 1(t) denotes the position of the moving interface which is initially at zO.
321
their paper, Cohen et al. use Z(t) as the position of a propagating diffusion layer inside the cast film while referring to the interface position as “surface”. In our opinion, errors in the transformation to @-space have resulted from confusions in the mathematical definitions of m and the correspondingly required mathematics. Cohen et al. specifically state that m is the volume of polymer (per unit of area) between the moving interface and the particular point of observation in the film. However, their equation for m [ 2, p. 4801 does not contain an explicit time dependence, nor does that given by Wijmans et al. [ 4, eqn. (1) 1. It was those equations, as written, that were used in our derivation of the missing polymer flow term [ 3, eqn. (12 ) 1. On the other hand, Reuvers et al., while writing m as a function of both position and time, misleadingly use the origin of the fixed laboratory coordinates as the lower limit on their defining integral (their eqn. 9). Using the definition of m as stated by Cohen et al., one must have the following (see Fig. 1)
where c denotes an arbitrary position in the membrane rule for the required derivatives in eqn. (1) , one has
film. Applying the chain
(3)
t/,=~/,<
(4)
From eqn. (2 )
where @3m=G3(Z( t) ,t) is the volume fraction of polymer at the moving interface and ZL,= dl/dt is the membrane-bath interface velocity. The continuity equation for the polymer in z&space [ 3, eqn. (7b) ] is
(6) Substitution
of eqn. (6) in eqn. (5 ) gives the following
where u 3m= u3 (Z( t) ,t) is the polymer velocity at the film-bath interface. Unless one arbitrarily assumes that the film interface velocity will be precisely that of the polymer, the second bracketed term in eqn. (7 ) will not be zero.
322
From eqn. (2 ) one also has
dm -=
a2 t
(8)
$3
Therefore, upon substitution of eqns. (3 ), (4)) (7)) and (8) into eqn. ( 11, the following set of diffusion equations results
api
0
$3 p=--at
a&
( >
dJi
$43
(~3m---U,) am
@3mp
i= 1,2
(9)
am
In these equations it is understood that the dependent variables, & and & are functions of m and t, while the term (uym- ZL~)&~ is a function of time only. The form of the convected term in these equations is similar to that derived in our earlier equations (i.e. [ 3, eqns. (12) ] ), however, its interpretation here is different in that it represents the bulk flow due to the difference between the polymer velocity at the interface, ugrn,and that of the interface itself, u,. For the reasons we have mentioned, this term will not, in general, be zero and must therefore be determined through the coupled boundary conditions for the component material balances at the moving interface [ 51. One of the transformation equations used by Reuvers et al., namely their eqn. (ll), is in error since it improperly equates the polymer velocity to the derivative of the spatial coordinate at constant m, i.e., u3 = (&/dt) 1m (in our nomenclature). As we pointed out earlier [3], the proper definition is u3= (&/at) 1c where subscript, <, refers to the material coordinates of the polymer molecules. The proper relationship for ug in m,t-space can be derived by straightforward application of the chain rule. Using the fact that m= m (z,t), in combination with eqns. (7) and (8)) one has
From this discussion we conclude that in order for the diffusion equations as initially presented by Cohen et al., and modified by Reuvers et al., to be correct, one must indeed make a restrictive assumption regarding the nature of the polymer motion during diffusion.
References 1
2
A.J. Reuvers, J.W.A. van den Berg and C.A. Smolders, Formation of membranes by means of immersion precipitation. Part I. A model to describe mass transfer during immersion precipitation, J. Membrane Sci., 34 (1987) 45. C. Cohen, G.B. Tanny and S. Prager, Diffusion-controlled formation of porous structures in ternary polymer systems, J. Polym. Sci., Polym. Phys. Ed., 17 (1979)
477.
323 3 A.J. McHugh and L. Yilmaz, The diffusion equations for polymer membrane formation in ternary systems, J. Polym. Sci., Polym. Phys. Ed., 23 (1985) 1271. 4 J.G. Wijmans, F.W. Altena and C.A. Smolders, Diffusion during the immersion precipitation process, J. Polym. Sci., Polym. Phys. Ed., 22 (1984) 519. 5 K.J. Ruschak and C.A. Miller, Spontaneous emulsification in ternary systems, Ind. Eng. Chem., Fundam., 11 (1972) 534.