Theoretical study of the transport processes occurring during the evaporation step in asymmetric membrane casting

Journal ofMembrane Science, 29 (1986) 11-36 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

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THEORETICAL STUDY OF THE TRANSPORT PROCESSES OCCURRING DURING THE EVAPORATION STEP IN ASYMMETRIC MEMBRANE CASTING

WILLIAM B. KRANTZ, RODERICK J. RAY*, ROBERT L. SAN1 and KEVIN J. GLEASON

Department of Chemical Engineering, University of Colorado, Boulder, (U.S.A.)

(Received August 28,1984; accepted in revised form March 19,1986)

Summary

CO 80309

A mathematical model is developed for the evaporation step in asymmetric membrane casting which allows for the convective transport induced by local film shrinkage because of both solvent loss and the excess volume of mixing effect. Realistic boundary conditions account for the gas phase mass transfer characteristics. Predictions for the cellulose acetate/acetone system are presented for the instantaneous concentration profiles, film thickness, and time required for the interface to skin. The results are presented in the form of dimensionless correlations which allow predicting the behavior during evaporative casting for generalized operating conditions. The predictions are compared with limited data for cellulose acetate/acetone. The model suggests that once the polymer/solvent system has been selected, the most influential process parameter is the gas phase mass transfer coefficient and that inadequate control of this parameter may account for the variabil- ity in membranes cast in similar devices operated under ostensibly the same conditions.

Introduction

Relatively high flux, permselective asymmetric membranes consisting of a very thin (< 100 nm), relatively dense skin supported by a thick (- 100 pm) porous substructural layer, were developed in 1962 by Loeb and Sourirajan [ 11. The Loeb-Sourirajan or Manjikian procedure for preparing solvent-cast asymmetric membranes consists of the following four steps: (1)

(2)

(3)

A thin film of a solution of polymer, solvent, and possibly other addi- tives is cast on a smooth solid surface. Rapid solvent loss from the film is induced by contact with a gas phase or precipitation bath consisting of a nonsolvent (for the polymer) which causes interface solidification or “skinning” of the film by a relatively dense polymer layer. Subsequent or continued immersion of the cast film in a precipitation

*Present address: Bend Research, Inc., 64550 Research Road, Bend, OR 97701-8599, U.S.A.

0376-7388/86/$03.50 0 1986 Elsevier Science Publishers B.V.

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bath results in a sharply decreasing solvent loss rate thereby creating the porous substructure.

(4) The resulting cast film may be densified by thermal annealing in order to impart its rejection properties.

Although it is generally agreed that solvent-cast asymmetric membranes are formed via a phase-inversion process [Z] , the specific mechanisms involved in the above preparation procedure are not fully understood. This paper presents the first results of a comprehensive study to develop predictive models for the mechanisms involved in the Loeb-Sourirajan casting process. As such, it focuses on describing the evaporation step which can be used for some systems (such as cellulose acetate/acetone solutions) in lieu of immediate immersion into a precipitation bath in order to create the thin dense skin. Modeling the evaporation step is thought to be an appropriate starting point for this study for several reasons. First, it is not generally agreed how important the evaporation procedure is in the formation of skin- type membranes. Second, prior attempts [3,4] at modeling the evaporation step have not led to reliable estimates of the skin thickness or to predicted solvent evaporation rates in agreement with experimental observations. Third, modeling the evaporation step is the logical precursor to modeling the more complicated solvent bath precipitation step. Finally, modeling the evaporation step permits assessing the magnitude of the concentration gradient at the film interface which is thought to influence the small-scale morpho- logical structure via a mechanism recently described by the authors [5].

Related studies

Anderson and Ullman [3] were the first to recognize that the marked decrease in the diffusion coefficient associated with decreased solvent concentration plays a significant role in forming the skinned layer. They developed a predictive model for the concentration profiles in evaporative casting of a binary solution subject to the limiting assumptions of an infinitely deep casting solution thickness, negligible film shrinkage, and a specified constant concentration at the casting solution/gas phase interface. These limiting assumptions precluded obtaining accurate estimates of the concentration profiles and skin thickness. Indeed, for thinner initial casting solution film thick- nesses, lower temperatures or poor gas phase mass transfer, the solvent concentration at the impermeable lower surface can be affected. Both the solvent loss as well as the effect of the large excess volume of mixing cause significant local film shrinkage which in turn superimposes convective transport on the solvent diffusion. Finally, the instantaneous solvent concentration at the casting solution/gas phase interface is a function of the gas phase mass transfer characteristics.

Several of these limiting assumptions were addressed in a subsequent paper by Castellari and Ottani [4] who considered a finite casting solution depth, allowed for uniform film thinning due to solvent loss, and utilized an

13

empirical flux relationship as a boundary condition thereby allowing for the interface concentration to decrease with time. Despite these improvements, Cast.ellari and Ottani’s model predicts that excessively long times are required for skinning. The principal limitations in this model appear to be ne- glecting the convection effects due to the local volume contraction and util- izing a somewhat unrealistic flux boundary condition at the free surface. Furthermore, Castellari and Ottani as well as Anderson and Ullman base their model predictions on data for the self diffusion coefficient rather than the binary or mutual diffusion coefficient. This oversight introduces considerable error into the predictions of these models since, for example, the self diffusion coefficient can be more than two orders of magnitude larger than the binary diffusion coefficient in the concentration range of interest in the evaporative casting of cellulose acetate/acetone solutions.

Relatively few experimental studies of evaporative casting have appeared in the literature. Kunst and Sourirajan [6] studied evaporative casting of a ternary system. The only study of evaporative casting of a binary system which can be used to test the models of interest here is that of Ataka and Sasaki [7] who carried out a gravimetric analysis of the evaporative solvent loss from cellulose acetate/acetone casting solutions. Unfortunately these investigators provide no details on their gas phase transport conditions, thus precluding determining any gas phase mass transfer coefficients for use in the models.

This brief review indicates that a more refined model of the evaporative casting step in asymmetric membrane casting is needed. This model must account for convective contributions to the mass transfer flux because of local film thinning arising both from direct solvent loss as well as excess volume of mixing contraction effects. In addition, a realistic boundary condition must be used at the casting solution/gas phase interface which accounts in some way for the gas phase mass transfer characteristics. Finally, quantitative predictions must employ appropriate data for the binary or mutual diffusion coefficient.

Theoretical development

System of interest The system of interest here, shown in Fig. 1, consists of a horizontally in-

finite liquid layer of depth L(t), composed of a nonvolatile polymer and a volatile solvent whose mass concentration is p A. This liquid layer rests upon an impermeable solid surface (such as a sheet of glass) and is initially assumed to be at rest. The vertical spatial coordinate, y, is defined positive upward, with y = 0 being the solid surface. The liquid/gas interface, which is moving downwards as the solvent evaporates, is at y = L(t).

At time zero, a liquid/gas interface is suddenly created by exposing a film of casting solution to the gas phase. The solvent (initially at concentration pO) begins to evaporate into the gas phase at a rate proportional to the over-

14

Gas Phase

Liquid Phase

‘Solid/Liquid Interface

Fig. 1. Schematic of thinning, two-component liquid layer undergoing evaporation of the volatile component.

all gas phase mass transfer coefficient, h,. This lowers the solvent concentration at the liquid interface and causes diffusion of solvent to the interface from below, thus setting up the concentration profile shown in Fig. 1. The flux of solvent upward is described by Fick’s law of diffusion which includes both a diffusive flux and a convection term; the latter term arises because there is a net mass-average velocity since solvent is being evaporated from the film. A complicating factor is that the film thickness is changing both because of solvent loss as well as the excess volume of mixing effect. That is, as the solvent concentration decreases, the total mass density of the solution changes according to an appropriate equation of state. This results in a more rapid thinning rate if the solution density increases with decreased solvent concentration, as it does for most polymer/solvent systems. Hence, the mass-average velocity, which contributes to the convection term in Fick’s law of diffusion, arises because of the combined effects of solvent mass loss and density change. Note that the excess volume of mixing effect is highly localized near the liquid/gas interface where the solvent concentration change is most significant. A further complicating factor is that the diffusion coefficient will decrease markedly near the interface because of solvent loss. This implies that the concentration gradients will be very steep within a thin region near the interface. Note that since the flux of the nonvolatile polymer is zero at the liquid/gas interface, Fick’s law of diffusion implies that the diffusive flux of polymer downward at the interface is exactly balanced by the convective flux of polymer upward.

The model developed here accounts for all of the aforementioned effects and results in a set of equations that allow the determination of the component concentration profiles at any desired time after exposure to the gas phase up to the initiation of skinning. As the solvent concentration decreases in time, the solution viscosity increases markedly. In the model developed here, the upper interface of the casting solution is assumed to be an unde-

15

formable rigid solid when the solvent concentration reaches some critical minimum value. This critical minimum value does not necessarily correspond to a well-defined gelation point; certainly it does not in the important case of cellulose acetate/acetone, which does not display any miscibility gap over the full concentration range of interest here. The model developed here then describes the transport in the evaporative casting process up to the point at which solid-like behavior occurs at the upper interface.

For many polymer/solvent systems, the solvent diffusion coefficient can vary over six or more orders of magnitude [3] , over the full range of solvent concentrations. For example, the concentration dependence of the binary or mutual diffusion coefficient D for the cellulose acetate/acetone system can be represented by an appropriate modification of the equation used by Anderson and Ullman [3] and others [4] for the self-diffusion coefficient:

where Do is the diffusion coefficient of the volatile solvent (acetone) in the polymer (cellulose acetate) at a mass concentration of the solvent, pA = 0, A and B are empirical parameters, and 01 is the ratio of the binary to self diffu-

’ sion coefficient given by Crank [ 81

aGA ti = (PAIRT) - aPA

in which T is the absolute temperature, R is the gas constant and GA is the chemical potential of the solvent. The latter can be obtained as a function of concentration using Flory-Huggins theory [9], whose applicability will be justified later. In general, the ratio of the binary to self diffusion coefficient can be fit to the following functional form for use in eqn. (1):

(3)

where c,, c2, c3, c4 and c5 are empirical constants specific to the polymer/ solvent system of interest.

The density of most polymer/solvent solutions is a function of concentration and often can be described by a linear function of the solvent mass fraction, WA [lo]. For simplicity, the polymer density here will be assumed to be a linear function of the solvent mass concentration, pA, given by:

PT =Pp +KzPA (4)

where p T is the nonconstant liquid phase mass density, pp is the mass density of the pure polymer, and K2 is an empirically determined constant that can be either positive or negative. If indeed the solution density is a linear function of the solvent mass fraction, then eqn. (4) will be a reasonable ap- proximation for many polymer systems.

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Derivation of one-dimensional diffusion equation for a 1ocaEly thinning film One-dimensional unsteady-state Fickian diffusion of the solvent in a sys-

tem such as shown in Fig. 1 is described by

awA 1 apA - -PA" =- ay at (5)

where u is the dimensional mass-average velocity and t is time. To put this equation into a more useful form, it is necessary to find an expression for the mass-average velocity, u. This expression is obtained by the integration of the overall continuity equation as follows:

I y ah4 “3 dy=-j- ay dy 0 0

(6)

After substitution of eqns. (4) and (5) into eqn. (6) and simplification, the equation for the mass-average velocity is found to be

D& awA u= -PT--

ay PP

(7)

Equation (7) is then substituted into eqn. (5) to yield

(8)

Equation (8) is the one-dimensional diffusion equation for a liquid layer undergoing local thinning due to solvent loss. Note that this equation is highly nonlinear due to the concentration-dependent binary diffusion coefficient, D, defined by eqn. (1).

Derivation‘of initial and boundary conditions The onedimensional diffusion equation, eqn. (8), is a partial differential

equation (PDE), second-order in y and first-order in t. Solution of this equation thus requires two boundary conditions - a condition at the bottom of the liquid layer and a condition at the upper, moving boundary - as well as an initial condition.

The condition imposed at the lower boundary is zero flux of any component through the impermeable solid. This condition is described by

awA nA =- pTD-

ay +pAu=Oaty=O (9)

where nA is the mass flux of the solvent with respect to a stationary coordinate system. After substitution of eqn. (4), eqn. (9) can be simplified to

apA - =Oaty=O. ay

(10)

The upper boundary condition must take the moving upper interface into account. The first step is a mass balance on the solvent over the entire film. This yields

d L(t)

X, s PA dY = -k,(C, - C, ) (11)

where the left-hand side is the loss of the solvent from the liquid film per unit time and the right-hand side is the flux of the solvent into the gas phase. In eqn. (ll), k, represents the convective mass transfer coefficient which is assumed here to be constant; C, represents the instantaneous gas phase concentration of the solvent in equilibrium with a given concentration of the solvent at the surface that will be determined below; and C, is the concentration of the solvent in the bulk gas phase, i.e., at Y S L(t).

In order to put eqn. (11) into a more convenient form, an expression for the change in liquid film depth, L(t), with time is needed. This expression is obtained through an overall mass balance on the liquid layer, given by

c=(t) -=u-

k,(C, - C, 1 dt PT -

(12)

Combination of eqns. (11) and (12), substitution of eqn. (7) for the mass- average velocity, and considerable simplification yield the upper boundary condition

PTD ~WA

(w.4 - - = k,(C, - C, ) at Y = L(t).

1) aY (13)

Equations (8), (11) and (13) require an explicit equation for the instantaneous liquid-layer depth, L(t). This expression is derived by substituting eqn. (13) into eqn. (12) in order to eliminate the term k,(C, - C, ). Substi- tution of eqn. (7) to eliminate the mass-average velocity and simplification then yields

L(t) = Lo + St D 0

(14)

Equation (14) indicates, as expected, that transport of solvent out of the liquid film results in a decrease in the depth of the film.

The final unknown in this set of equations is the instantaneous gas phase concentration of the solvent, C,,, that appears in the upper boundary condition, eqn. (13). This concentration will be estimated through the use of the Ideal Gas Law and the Lewis and Randall fugacity rule [ll] :

(15)

18

where P is the total pressure of the system, yA is the gas phase mole fraction of the solvent, Pi is the vapor pressure of the solvent, 3tA is the liquid phase mole fraction of the solvent, and ?A is the activity coefficient of the solvent in the liquid phase. Due to the large molecular weight of polymers and the strong intermolecular interactions of many polymer/solvent systems, the nonideality of these solutions must be taken into account.

A model for polymer/solvent solutions was derived by Flory and Huggins [9] . Their model assumes that the solution is athermal - that is, the heat of mixing is zero. (However, this theory does not assume a zero volume of mixing, and thus is appropriate for this analysis.) Flory and Huggins modeled the polymer molecule as a chain consisting of a large number of mobile segments, each having the same size as a solvent molecule. These assumptions are questionable for many membrane polymer/solvent systems; nevertheless, this model yields results that are qualitatively in agreement with the ob- served behavior of the system of interest here.

Under these assumptions, the activity coefficient, yA, is determined from the relation

lny. =ln l- l- [ ( ;)%]+(+%I (16)

where m is the number of segments in a polymer molecule (the number of monomers) and Qp, is the volume fraction of the polymer molecule and is given by

ap = CA +mcp

(17)

where cp is the molar concentration of the polymer and CA is the molar concentration of the solvent.

With this expression for the activity coefficient, along with eqn. (15) and the Ideal Gas Law, the gas-phase concentration of the volatile component is given by

where M,A is the molecular weight of the solvent, R is the universal gas constant, and T is the absolute temperature.

After substituting eqn. (18) into the upper boundary condition, eqn. (13), and assuming the concentration of the volatile component in the bulk gas phase is zero, (i.e., C, = 0), the upper boundary condition becomes

PTD awA kc~I,MwAYAXA - -= -1 ay RT

at y = L(t). wA

(19)

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Finally, the initial condition is determined from the physics of the problem studied here. The liquid film is assumed to be at some initial concentration of the volatile component, pO, when it is suddenly exposed to the gas phase. The initial condition is therefore

,0* =po att=Oforally. (20)

Summary of assumptions The assumptions that were made in order to arrive at the model equations

are summarized below: l The film is assumed to shrink only in the y-direction l The analysis assumes binary polymer/solvent solutions l The diffusive mass transfer is assumed to be Fickian l The entire evaporation process is assumed to be isothermal l The overall solution density is assumed to be linearly dependent on the

mass concentration l The binary diffusion coefficient is assumed to be concentration-dependent

and of the form given by eqn. (1) l The gas phase is assumed to be ideal l The polymer/solvent mixing is assumed to be athermal such that the sol-

vent activity coefficient is described by Flory-Huggins theory l The gas phase mass transfer characteristics are assumed to be described by

a lumped parameter approach with a constant mass transfer coefficient l The bulk gas phase is assumed to contain no volatile component, that is

c, = 0 8 The equations describe the behavior of the liquid film only to the point of

solidification or skinning

Solution methodology

This system of equations cannot be solved analytically because it is highly nonlinear. A numerical solution technique thus was employed. Because eqn. (8) is a partial differential equation, an algorithm called PDEONE [12] was employed that numerically recast this equation and the boundary conditions into a set of ordinary differential equations (ODES). This representation then was solved with the aid of another numerical package called GEAR [13]. These solutions resulted in concentration profile predictions at any desired time for a given set of conditions. In the first part of this section, a solution to the set of dimensional equations (eqns. 8, 10, 13, 19 and 20) will be presented for clarity. Then, this set of dimensional equations will be nondimensionalized and solved by the same numerical methods, thus yielding a more useful set of generalized plots that can be applied to a broad class of polymer/solvent systems and evaporative casting conditions.

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Solution of dimensional equations When solving this set of dimensional equations numerically, it is necessary

to specify completely the system of interest. In this section, the results of a solution describing the evaporation step of a specific cellulose acetate/acetone system are presented. These parameters describe a freshly cast cellulose acetate/acetone film of the type studied by Ataka and Sasaki [7]. The parameters describing this system are given in Table 1. The values of A and B given in Table 1 were obtained from fitting the data of Park [14] and Anderson and Ullman [3] for the self-diffusion coefficient. Flory-Huggins theory was used in order to evaluate the chemical potential term in eqn. (2) and thereby obtain the values of cI through cs in Table 1 which permit determination of the binary diffusion coefficient from measurements of the self diffusion coefficient via eqn. (3).

Figure 2 is a plot of the liquid phase acetone concentration at the casting solution/gas phase interface as a function of time. As indicated in Table 1, solidification or skinning was assumed to occur for this system at an acetone mass concentration of pA = 0.177 g/cm3, the value used by Castellari and Ottani [4]. This plot indicates that the casting solution skins approximately 3.6 set after the film is cast for the conditions defined in Table 1.

Figure 3 shows a plot of the slope of the acetone concentration profile at the casting solution/gas phase interface versus time. This plot indicates

TABLE 1

Parameters describing cellulose acetate/acetone cast film

Parameters Value

Initial acetone concentration Initial film thickness Diffusion coefficient parameters

Acetone vapor pressure Acetone molecular weight Solution density parameter Cellulose acetate density Cellulose acetate molecular weight (for activity

coefficient determination) Number of monomers

PO = 0.703 g/cm3 L, = 100 pm D, = 2.25 x lo-‘2 cmz/sec A = 0.0512 B = 0.0064 g/cm3 Pi = 0.326 atm M = 58 g/m01 Kzw~-0.6711

PP = 1.32 g/cm”

M WP = 30,000 g/m01

(for activity coefficient determination) Mass transfer coefficient Concentration of acetone in the bulk air Assumed temperature Acetone concentration at skinning Constants appearing in eqn. (3)

m = 202 k, = 0.78 cm/set c, = 0 T= 300K pA = 0.177 g/cm3 c, = -5.19 cm3/g c, = 14.13 cm6/g’ c, = -25.72 cm9/g3 c, = 27.56 cm13/g4 c, = -12.70 cm’5/gS

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Surface Shinning

0 0 1 2 3 4

Time, t (see)

Fig. 2. Surface concentration versus time for a freshly cast, evaporating cellulose acetate/ acetone film. The initial layer depth is L, = 100 ~.rrn; the depth at the initiation of skin-

ning is 80 pm.

% -2ooo- ‘9 *

-3000 -

Surface Skinning

-40001 0

I 1 2 3

+, Time, t ised

J 4

Fig. 3. Slope of the acetone concentration profile at the gas/liquid interface of the cast film versus time for cellulose acetate/acetone. The initial layer depth is 100 hrn; the depth at the initiation of skinning is 80 pm.

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that the concentration profile is relatively flat for more than 3 of the 3.6 set that it takes for the surface to solidify or skin. Only during the last 0.6 set does the slope of the concentration profile become steep. This result has been shown to have important implications for a recent hypothesis advanced to explain the surface morphology of asymmetric membranes [ 51. The concentration profile in the casting solution at the instant when the interface just solidifies or skins, 3.6 set after casting in this case, is shown in Fig. 4. Again note the steepness of the concentration profile at the liquid/gas interface.

“O-

/ 0 I I I

0 2 4 6 6L0,) Distance from the Solid/Liquid Interface, y X lo3 km)

Fig. 4. Acetone concentration profile at the initiation of skinning. The initial depth is 100 pm; the depth at the initiation of skinning is 80 pm.

Nondimensionalization and solution of dimensionless equations As was mentioned above, the disadvantage of a numerical solution relative

to an analytical solution is that it results in numbers and not a functional relationship between the variables of interest and the parameters of the problem. A numerical solution to the dimensional set of equations necessitates using the computer to generate the predictions for each new set of conditions of interest. This is inconvenient, costly, and unwieldy. A technique known as ‘scaling’ [15] was applied to the set of equations in order to obtain the minimum parametric representation and the most useful form of the dimensionless variables and parameters appropriate to this problem. This

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scaling technique then permits presenting the results of the numerical solution in the form of generalized plots interrelating the dimensionless variables and parameters of the problem. These generalized plots can be determined using the results of relatively few numerical runs. The resulting generalized plots permit determining the skinning time, cast film thickness at the inception of skinning, concentration profiles, and other useful quantities for a broad class of polymer/solvent systems and evaporative casting conditions without the need to rerun the numerical solution scheme for the specific conditions in question.

By means of the scaling technique, scale factors were chosen for nondimensionalization of the variables so as to isolate those parameters that are independent of the chemical nature of the polymer/solvent system. That is, many of the parameters of this system assume fixed values once the compo- nents of the system have been chosen. These are parameters such as A, B, cl--c5, pp, and K, introduced in equations (l), (3), and (4). However, other parameters can still be varied - e.g., the mass transfer coefficient, k,, and the initial solvent concentration pO. It is advantageous, then, to isolate these nonconstant parameters into as few dimensionless groups as possible for the purpose of generating universal plots.

The variables of this problem, t, y, and p A) are nondimensionalized using the scale factors determined by the scaling technique [15]. The resulting dimensionless variables are :

tDo t* = - L2,

*= Y Y Lo

(21)

(22)

and

pz =!?. (23) PO

Finally, the method yields eleven dimensionless groups made up of parameters from the system of interest. These groups are as follows:

N, = (W?JKALO )!W’P,&)

Nz ‘PO/B N3 =A N4 =m

N, =K, N6 =ppIB N7 =c,B (24)

N8 = czB2 Ng = c3B3

Nlo = caB4 N,l = c5B5

where R is the universal gas constant; T is the system temperature; MwA is the molecular weight of the solvent; and D,, is the diffusion coefficient of a solvent molecule in a matrix of polymer, i.e., at infinite dilution.

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The advantage of this analysis is evident from the nature of these eleven dimensionless groups. Once a specific polymer/solvent system is specified, the last nine dimensionless groups (N3-N11) assume fixed values. Therefore, the quantities of interest for any polymer/solvent system (such as the layer depth of skinning or the shape of the concentration profiles) are func- tions of only the two remaining dimensionless groups, N, and Nz.

The parameter N, can be thought of as a dimensionless mass transfer coefficient. For example, as the velocity of the gas phase relative to the liquid phase increases, the dimensionless mass transfer coefficient, k,, increases, thus increasing the parameter N1. Likewise, the parameter Nz corresponds to a dimensionless initial solvent concentration. Finally, the dimensionless groups N3 through N,, correspond to the physical characteristics of the specific polymer/solvent system of interest. The values of these groups for the groups for the cellulose acetate/acetone system considered in Fig. 5 through 7 are given in Table 2.

Perhaps the primary quantity of importance in this analysis is the time required for initiating skinning of the interface of a given polymer/solvent system for any set of casting conditions. In Fig. 5, the dimensionless time at which the surface skins is plotted versus the dimensionless group N,. Note that this plot is for a specific value of the dimensionless group N,; that is, a specific initial concentration of the volatile solvent. The data points in Fig. 5 are for the skinning initiation times inferred from the experiments of Ataka and Sasaki [7] and will be discussed in a subsequent section.

N, _ kc pa’ MWALO RTppDo

x 10-S

Fig. 5. Dimensionless time for the initiation of skinning versus the dimensionless group N, for cellulose acetate/acetone. The value of the dimensionless group N, corresponds to an initial acetone concentration of 0.703 g/cm3. Data from Ataka and Sasaki [7].

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A quantity of interest during the preparation of cast polymeric membranes is the thickness of the film at the instant the liquid surface begins to solidify or skin. This quantity is denoted by L(t,) and is obtained by evalu- ating eqn. (14) at t = t,. Figure 6 shows a plot of L(t,), nondimensionalized with respect to the initial depth LO, as a function of the dimensionless group N1. In Fig. 5, curves representing the dimensionless depth at skinning versus the dimensionless group N, are presented for three values of the dimensionless group N, for the cellulose acetate/acetone system.

Finally, Fig. 7 shows the volatile component concentration profiles at the instant that the surface region skins for different sets of parameters. Note that the abscissa in Fig. 7 is the distance measured from the solid/liquid interface nondimensionalized so that the upper liquid film surface is always at unity despite the fact that the dimensional thickness differs for each value of N1.

1.2

l.C

0.6

i”

3 0.6

Ii

0.4

0.2

N2 = 62.5

77

-J 0 5 10 15 20 25 30

N, X 1O-5

Fig. 6. Dimensionless liquid layer thickness at the initiation of skinning versus the dimensionless group N, for varying values of the dimensionless group N, for cellulose acetate/acetone. Values of N, of 62.5, 93.8, and 110 correspond to initial acetone mass concentrations of 0.40 g/cm3, 0.60 g/cm’, and 0.73 g/cm3, respectively.

TABLE 2

Dimensionless groups for the cellulose acetate/acetone system

N, = 5.12 x 1o-2 N, = 2.02 x lo2 N, = -6.71 x lo-’

N, = 2.06 x lo2 N, = -3.32 x lo-= N, = 5.79 x 1O-4 N, = -6.74 x 1O-6 N,, = 4.62 x lo-* N,, = -1.36 x lo-‘*

26

0.8-

PA ‘PO

0.6 -

0.4-

N2=110

0 I I I

0 0.2 0.4 0.6 0.0 1.0

y/L(ts)

Fig. 7. Dimensionless volatile component concentration versus dimensionless distance from the solid surface for varying values of the dimensionless group N, for cellulose acetate/acetone. The value of the dimensionless group N, corresponds to an initial acetone concentration of 0.703 g/cm3. Note that y/L(t,) = 0 corresponds to the solid/ liquid interface and y/L(&) = 1 corresponds to the liquid/gas interface.

Discussion of results

Use of the dimensionless plots In order to illustrate the utility of expressing the results of this numerical

solution to the evaporative casting problem in dimensionless form, a simple example will be used to demonstrate the use of Figs. 5, 6 and 7. Since these figures were prepared for the cellulose acetate/acetone system, we necessarily must again consider this system whose dimensionless parameters N,-N,, are given in Table 2. Specification of the casting conditions given in Table 1 then implies that the variable dimensionless groups N, and N, are given by N, = 20 X lo5 and Nz = 110. The latter value when used with Fig. 5 implies that tJl,/Li = 0.8 X 10m7. Hence, the dimensional skinning time is found to be 3.6 sec. These values of N1 and N2 when used with Fig. 6 indicate that L(t)/L, = 0.80. Hence, the dimensional film thickness at skinning is 80 pm. Finally, these values of N1 and N, when used in Fig. 7 permit determination of the acetone concentration at any location in the film at the instant of surface skinning. For example, at y/L(&) = 0.8, PA /pO = 0.76, thus indicating that ,‘,A = 0.53 g/cm3.

The utility of these generalized plots then should be obvious. Figures 5, 6

27

and 7 provide a complete description of the evaporative casting of the cellulose acetate/acetone system and provide far more information than the numerical solution results presented in dimensional form such as those shown in Figs. 2-4. Similar generalized dimensionless plots of course could be developed for other polymer/solvent systems of interest in the preparation of asymmetric membranes.

Implications of modeling results In addition to permitting the determination of skinning time;instantane-

ous film thickness, solvent concentration, and other quantities of interest in evaporative casting, the generalized plots in Fig. 5, 6 and 7 provide insight into the transport processes occurring during the evaporation step of a freshly cast polymer/solvent film. As such, they indicate possible steps that can be taken to optimize the properties of asymmetric membranes prepared via an evaporative casting step. First, the time between casting of a film and skinning of the surface has important implications in the study of membrane formation. Figure 5 provides this information for the cellulose acetate/ acetone system, as well as insight into the factors affecting the time to the inception of skinning. As the mass transfer coefficient, k,, increases and thus the parameter N, increases (for example, due to an increase in the gas phase velocity relative to the film), the time to initiate surface skinning decreases. Figure 5 suggests, however, that there are diminishing returns. That is, at larger values of the parameter N,, a substantial increase in the magnitude of the parameter N1 produces almost no effect on the dimensionless time to initiate skinning. This suggests that the concentration gradient becomes steep enough to provide an adequate driving force to transport solvent to the surface region and slow the lowering of the liquid phase concentration of the solvent at the surface. Figure 5 also indicates that at low values of the parameter N,, a small increase in the mass transfer coefficient (and thus in the parameter N,) has a very large effect on the time to initiate skinning. This suggests that the flow characteristics in the gas phase should be carefully controlled during the evaporation step in membrane formation.

Another trend suggested by Fig. 5 is the effect of the initial casting solution thickness. Doubling the initial film thickness doubles the value of the parameter N1. This will result in a decrease in the dimensionless time required to initiate skinning for any region of the curve, However, the dimensional time for skinning will increase with increasing initial liquid film thickness.

Figure 6 is a plot of the dimensionless thickness of the liquid layer when the surface region begins to solidify versus the parameter N1 for three values of the parameter NZ. This plot indicates that as the mass transfer coefficient, k,, is increased (and thus the value of N1 increases), the film thickness at skinning also increases. The reason for this effect is clear. As the mass transfer coefficient increases, the solvent near the bottom of the layer has less time to diffuse toward the upper surface and transfer to the gas phase.

28

Therefore, there is more total solvent in the system at surface skinning; thus, the overall thickness is greater because less film shrinkage occurs.

The curvature at low values of the parameter N, shown in Fig. 6 is the result of an interesting effect. In this region, an increase in N1 increases the final thickness more than the same percentage increase does at high values of N1. The reason for this can be seen from the dimensionless concentration profiles shown in Fig. 7. As the latter figure indicates, the entire layer tends to be near the solidification concentration when the surface region skins for low values of N,. This implies that the diffusion coefficient is low every- where; hence, the mass transfer flux is low as well. Increasing the mass transfer coefficient, k,, at low values of the parameter N, then causes an en- hanced effect because the surface region is not supplied with solvent as easily at low concentrations. At higher values of N1, the bulk of the liquid is at a higher solvent concentration; hence, the solvent transport in the bulk of the liquid film is more rapid and the effect of N, on final thickness is linear. The final thickness discussed here is the thickness when the surface of the film just skins. The model cannot describe the behavior of the film after the surface skins since the transport characteristics and boundary conditions change markedly from those assumed in the present model.

The effect of varying the initial solvent concentration, pO, can also be determined from Fig. 6. As the initial solvent concentration is raised and thereby N2 is increased, the thickness of the liquid layer at surface skinning is lowered. This is as expected, since an increased initial solvent concentration corresponds to less polymer at skinning and thus a thinner film.

The dimensionless concentration profiles presented in Fig. 7 indicate how an increase in the value of the parameter N1 affects the solvent concentration profile for the cellulose acetate/acetone system. The trends shown in Fig. 7 are as expected, given the results already discussed in this section. An increase in N1 yields a steeper solvent concentration profile at the surface. As N1 decreases, the solvent has more time to diffuse to the surface and transfer into the gas phase; hence, smaller values of N1 imply smaller concentrations at the wall.

Comparison of results with previous models The evaporative casting models available for comparison are those of

Anderson and Ullman [3] and Castellari and Ottani [4]. However, a comparison between the model developed here and these prior models is not particularly meaningful since both of these prior models based their predictions on data for the self diffusion coefficient of acetone in cellulose acetate rather than the binary or mutual diffusion coefficient which was employed in the present model. Since the difference between the two diffusion coefficients can exceed two orders of magnitude, the effect of using an in- correct transport coefficient in these earlier models will mask the effects of the other refinements accounted for in the present model. Nonetheless, some general comparisons can be made.

As was mentioned earlier, the pioneering evaporative casting model of Anderson and IJllman [3] assumed that the casting solution was effectively infinitely thick. Figure 7 indicates that this assumption is invalid if the dimensionless group N1 is of order unity. Low values of the group N1 correspond to small values of the mass transfer coefficient, k,, and the initial cast film thickness, Lo, or to lower temperatures at which the vapor pressure, P6, is reduced.

29

Recall that the principal improvements in the new evaporative casting model over that developed by Castellari and Ottani are incorporation of local film thinning induced bulk transport, use of the binary rather than the self diffusion coefficient, and a lumped parameter description of the gas phase mass transfer. The net effects of local rather than uniform film shrinkage combined with use of the smaller binary rather than self diffusion coefficient are to steepen the concentration gradient near the casting solution/gas interface, decrease the time required for skinning, and increase the film thickness at the inception of skinning. These effects follow from the fact that both of these model refinements result in a decreased ability of the bulk casting solution to supply the interfacial region with solvent.

Comparison of results with experimental studies The definitive test of a model of course is to compare its predictions with

experimental studies. As indicated in the review of related studies, the only quantitative study of the evaporative casting of binary polymer/solvent solutions is that of Ataka and Sasaki [7] who used a gravimetric technique to study the cellulose acetate/acetone system for five initial cast film thick- nesses at two different temperatures; the initial solvent to polymer ratio in all their experiments was 2.9:l.O. Since these investigators do not provide any information on the gas phase mass transfer characteristics for their experiments, it is impossible to make any truly quantitative test of the model developed in this paper using their data. However, an estimate of the mass transfer coefficient for Ataka and Sasaki’s experiments will be made here to provide a basis for at least a qualitative comparison with theory.

The physical situation which probably best describes the gas phase mass transfer in the experiment of Ataka and Sasaki is that of free convection induced by the evaporation of a more dense gas (acetone) into a less dense gas (air) from a horizontal flat plate of infinite dimensions. This free convection situation is analogous to the free convection induced when a cold horizontal flat plate is exposed to a warmer overlying ambient gas phase. The heat transfer coefficient for the latter situation for the special case of a square flat plate is given by a correlation in McAdams [ 161. By invoking the analogy between heat and mass transfer, one then can cast this correlation into the following form applicable to free convection mass transfer from a square flat plate :

kcw -= 0.27 (Gr SC)” 4

(25)

30

where 3 X lo5 < fined by

Gr- W3&?W,

P2

Gr SC < 3 X lOlo, in which Gr is the Grasshof number de-

(26)

and SC is the Schmidt number defined by

&r-L. fJPg

(27)

W is the plate width, g is the gravitational acceleration, c is the concentration coefficient of volumetric expansion, ~1 is the gas phase shear viscosity, and AyA is the difference between the gas phase mole fraction at the liquid/gas interface and in the ambient gas phase.

The applicable properties of the cellulose acetate/acetone system given in Table 1 along with the parameters given in Table 3 were used in estimating the mass transfer coefficients at the temperatures of 16°C and 28°C used in the experiments of Ataka and Sasaki. The activity coefficients, YA, were evaluated using eqns. (16) and (17). The values reported in Table 3 represent the mean values determined as the average of the value at t = 0 and the value at t = t,. The gas phase acetone mole fractions, yA , were determined from eqn. (15). Again the values of yA reported in Table 3 represent mean values. The physical properties pg , {, and I-( were determined for the mean acetone concentration averaged between the liquid/gas interface and the ambient gas phase, whose acetone concentration was assumed to be zero. Since the gas phase contained more acetone at the higher temperature, the density value reported for 28°C is higher than that for 16°C and the viscosity value is lower. The plate width of 11.8 cm reported in Table 3 corresponds to the

TABLE 3

Parameters used in analyzing Ataka and Sasaki’s [ 7 ] data

Parameter Temperature

16°C 28°C

TA

Pk YA

PI

h P W SC Gr

kc

0.804 0.804 0.203 bar 0.343 bar 0.163 0.276 1.35 X 10m3 g/cm3 1.36 x 10-3g/cm3 1.69 x 10m4 g/( cm-see) 1.66 X 10v4g/(cm-set) 0.119 cm2/sec 0.126 cm2/sec

-0.884 -0.842 11.8 cm 11.8 cm

1.05 0.97 1.48 x lo7 2.51 x 10’ 0.17 cm/set 0.20 cm/set

31

width of a square plate having the same area as that of the 7 cm by 20 cm plates used by Ataka and Sasaki. These parameters then were used in eqn. (25) in order to obtain the following estimates of the gas phase mass transfer coefficients for the experiments of Ataka and Sasaki: k, = 0.17 cm/set at 16” C and k, = 0.20 cm/set at 28” C.

One particularly useful quantity for testing the model predictions is the time required for the initiation of skinning or solidification. By plotting the skinning initiation time in the dimensionless form used in preparing Fig. 5, it should be possible to collapse Ataka and Sasaki’s data for both temperatures onto a single line. The skinning initiation time is obtained from Ataka and Sasaki’s plots of solvent-to-polymer weight fraction versus time as that point at which the slope exhibits a well-defined decrease. This decrease is assumed here to indicate that the surface has skinned, thereby offering a greatly increased resistance to solvent evaporation. The values of the mass transfer coefficient obtained above then were used to determine the dimensionless skinning initiation time tz and dimensionless gas phase mass transfer coefficient N1 for each film thickness for the two data sets. The correspond- ing values of the initial film thickness Lo, skinning initiation time t,, dimensionless skinning initiation time tz and dimensionless group N1 are summarized in Table 4. The data for 16°C are plotted as the circles in Fig. 5 and the data for 28°C are plotted as the triangles. The general trend in the data agrees with that predicted by the model developed here, although one cannot claim quantitative accuracy. In particular, it appears that the model properly accounts for the temperature dependence in the experiments of Ataka and Sasaki since the data for 16°C and 28°C generally overlap.

One must keep in mind that there are many uncertainties in analyzing the data of Ataka and Sasaki. In particular, determination of the time required for the initiation of skinning is difficult, especially for thicker cast

TABLE 4

Skinning initiation time as a function of initial cast film thickness inferred from the data of Ataka and Sasaki [ 71

Lo (ml h (see) t,* x 10’ N, x lO+

28” C data: 100 150 200 300 500

16’C data: 100 36.3 8.2 2.8 150 62.0 6.2 4.3 200 91.7 5.2 5.7 300 127 3.2 8.5 500 271 2.4 14.2

19.2 35.8 48.0 60.0

139

4.3 5.4 3.6 8.2 2.7 10.9 1.5 16.3 1.3 27.1

32

films. Moreover, Ataka and Sasaki did not use square flat plates for which the correlation described by eqn. (25) is strictly applicable. Added to the uncertainties are the many assumptions invoked in the model development such as the use of Flory-Huggins theory to estimate the activity coefficients. Hence, in view of these uncertainties, the agreement between experiment and theory seen in Fig. 5 is encouraging. In fact, when one considers that the model development here is based on first principles without invoking any curve-fitting constants, the agreement shown in Fig. 5 is rather re- markable! One might conclude from this comparison between this model and the experiments of Ataka and Sasaki that the principal effect of changing the temperature in evaporative casting, is to change the volatile solvent vapor pressure which appears in the dimensionless group N,. Moreover, it would appear that the gas phase mass transfer in the experiments of At&a and Sasaki is described by free convection mass transfer induced by the evaporation of a more dense gas into an ambient less dense gas. The evaporation of the solvent creates a shroud of more dense gas above the flat plate which rolls off into the less dense ambient gas phase at the edges of the plate.

Further modeling considerations Until definitive experiments are carried out on the evaporative casting of

a binary polymer/solvent system, it is impossible to assess the accuracy of the new model developed here. This model development has pointed out em- phatically that any such experiments must pay careful attention to defining the gas phase mass transfer conditions since the mass transfer coefficient, k,, plays a very significant role in determining the evaporative casting process. The influence of the gas phase mass transfer coefficient appears to have been underestimated in prior studies. Indeed, subtle variations in this parameter because of changes in the gas phase hydrodynamics may well explain why casting apparatuses of the same design produce different cast membranes under ostensibly identical casting conditions. It is possible that manipulating the gas phase mass transfer coefficient can be used to advantage in the manu- facture of membranes by controlling the gas phase hydrodynamics in order to obtain optimal skin and substructure properties. Clearly, increasing the gas phase mass transfer coefficient will decrease the skinning time, thereby increasing the residual amount of solvent remaining at the inception of surface skinning. A probable consequence of increasing the mass transfer coefficient then might be to densify the skin since the solidification will proceed more slowly subsequent to the initiation of skinning.

It is worth commenting on the potentially limiting assumptions of the model developed here. Most significant among these are the assumptions of an isothermal evaporative casting process, the assumption of a diffusive transport process aided only by the one-dimensional bulk convection arising from film thinning, and the assumption of a constant gas phase mass transfer coefficient. Recent unpublished experiments of the authors using infrared

33

thermographic imaging to measure the temperature of the instantaneous casting solution/gas phase interface indicate that the interface temperature drops significantly on a very short time scale, apparently due to the inability of the bulk phase to supply the heat of vaporization to the interfacial region. The data in Fig. 5 suggest that evaporative cooling effects may have been significant in the experiments of Ataka and Sasaki since the theory is seen to predict conservative skinning initiation times for the thicker films. Since more solvent must be evaporated in order to skin thicker films, evaporative cooling will be more significant for thicker films. It may be necessary then to refine the present model to incorporate coupling with the unsteady-state energy equation in order to account for evaporative cooling.

Recently the authors have published a paper [ 51, which indicates that the steep gradients characteristic of short time-scale processes, such as evaporative casting, can generate spontaneous interfacially induced free convection in the bulk solution. If such effects are of importance in evaporative casting, then the model equations must necessarily account for the additional transport due to this three-dimensional free convection. The net effect of any such free convection would be to increase the skinning time and to decrease the cast film thickness at skinning.

The assumption that the gas phase mass transfer coefficient is constant is expedient since little is known about these transfer coefficients under high gradient unsteady-state conditions. The possibility exists that h, is a mono- tonically decreasing function of time due to the progressive decrease in the gas phase concentration gradients.

The authors currently are conducting experimental studies of evaporative casting of the cellulose acetate/acetone system using infrared thermographic imaging to measure the instantaneous free surface temperature in order to provide a definitive test of the evaporative casting models discussed in this paper. The authors are also extending this model to the evaporative casting of polymer/duplex solvent systems and to casting directly into a liquid precipitation bath. These studies will be discussed in a future paper.

Conclusions

l A new evaporative casting model for binary systems has been developed which incorporates local film thinning and a lumped parameter description of the gas phase transport.

l A method has been developed for casting the results of this numerical solution into the form of generalized correlations for the concentration profiles, skinning time and instantaneous film thickness which are independent of the process parameters once the polymer/solvent system has been specified.

l The model predicts that once the polymer/solvent system has been selected, the most influential process parameter is the gas phase mass transfer coefficient.

34

l The model suggests that the gas phase mass transfer in the experiments of Ataka and Sasaki [7] corresponded to free convection induced by the evaporation of acetone into less dense ambient air and that the principal effect of temperature in these experiments was to change the vapor pressure of the evaporating solvent.

l Experimental studies of evaporative casting employing controlled or known gas phase hydrodynamics are needed in order to provide a definitive test of the mathematical model.

l Interface cooling because of rapid solvent evaporation and interfacially induced free convection in the bulk casting solution may be important effects to include in further model development.

Acknowledgements

The authors gratefully acknowledge financial support through the Na- tional Science Foundation Grant No. CPE-8121841, the U.S. Department of Energy Grant No. DE-FG48-81801074, and a University of Colorado Dean’s Fellowship. This work was greatly aided through helpful discussions with Professor B.U. Felderhof, Professor P.G. Glugla, Professor R.K. Jain, Dr. R-E. Kesting, Dr. H.K. Lonsdale, Professor C. Majdarelli, Professor C.A. Smolders, and Professor H. Strathmann.

List of symbols

AB

CA

CP Cl, cz, c3

c4r c5

%

c, D

ig 0

g GA Gr

kc K2 Lo L(t)

Parameters appearing in the solvent diffusion coefficient defined in eqn. (1) Molar concentration of the solvent Molar concentration of the polymer Empirical parameters appearing in eqn. (3) for the ratio of the binary to self diffusion coefficient Gas phase mass concentration of the volatile component in equilibrium with liquid phase concentration at the interface Ambient gas phase mass concentration Binary diffusion coefficient of the solvent in the liquid defined by eqn. (1) Binary diffusion coefficient of the volatile solvent in the gas Diffusion coefficient of the solvent in the liquid at infinite dilution Gravitational acceleration Chemical potential of the solvent in solution Grasshof number defined by eqn. (26) Gas phase mass transfer coefficient defined by eqn. (11) Empirical constant appearing in eqn. (4) Initial depth of the casting solution layer Depth of casting solution at time t

L(ts)

k.4 nA

N2 NJ, Ne, NS Ncs, NT, Ns Nq,Nlo, NII P

PiI R SC t t* T u

wA

W

xA

Y

Y” YA ff

YA

5 P

PA

d

pg PO

PT

@P

References

36

Depth of casting solution at the initiation of skinning Number of monomers in a polymer molecule Molecular weight of the volatile solvent Mass flux of the solvent with respect to a stationary coordinate system Dimensionless gas phase mass transfer coefficient defined by eqn. (24) Dimensionless skinning initiation time defined by eqn. (24) Dimensionless groups defined by eqn. (24)

Total pressure in gas phase Vapor pressure of the volatile solvent Universal gas constant Schmidt number defined by eqn. (27) Dimensional time Dimensionless time defined by eqn. (21) Absolute temperature Mass-average fluid velocity in y direction Mass fraction of the volatile solvent Flat plate width Liquid phase mole fraction of the volatile solvent Vertical coordinate measured from the solid surface Dimensionless vertical coordinate defined by eqn. (22) Gas phase mole fraction of the volatile solvent Ratio of the binary to self diffusion coefficient determined by eqn. (2) Liquid phase activity coefficient of the volatile solvent defined by eqn. (15) Concentration volume expansion coefficient of the gas phase Shear viscosity of the gas phase Mass concentration of the volatile solvent in the liquid phase Dimensionless mass concentration of the volatile solvent in the liquid phase defined by eqn. (23) Mass density of the gas phase Initial mass concentration of the volatile solvent in the liquid phase Mass density of the liquid phase determined by eqn. (4) Volume fraction of the polymer in the liquid phase defined by eqn. (17)

1 S. Loeb and S. Sourirajan, Adv. Chem. Ser., 38 (1962) 117. 2 R.E. Kesting, Synthetic Polymeric Membranes, McGraw-Hill, New York, NY, 1971.

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3 J.E. Anderson and R. Ullman, Mathematical analysis of factors influencing the skin thickness of asymmetric reverse osmosis membranes, J. Appl. Phys., 44 (10) (1973) 4303.

4 C. Castellari and S. Ottani, Preparation of reverse osmosis membranes. A numerical analysis of asymmetric membrane formation by solvent evaporation from cellulose acetate casting solutions, J. Membrane Sci., 9 (1981) 29.

5 R.J. Ray, W.B. Krantz and R.L. Sani, A model for the initiation and regularity of large-scale fingers in asymmetric membranes, J. Membrane Sci., 23 (1985) 155.

6 B. Kunst and S. Sourirajan, Evaporation rate and equilibrium phase separation data in relation to casting conditions and performance of porous cellulose acetate reverse osmosis membranes, J. Appl. Polym. Sci., 14 (1970) 1983.

7 M. Ataka and K. Sasaki, Gravimetric analysis of membrane casting. 1. Cellulose acetate-acetone binary casting solutions, J. Membrane Sci., 11 (1982) 11.

8 J. Crank, The Mathematics of Diffusion, 2nd edn., Clarendon Press, Oxford, 1979. 9 J.M. Prausnitz, Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall,

Englewood Cliffs, NJ, 1969. 10 W.E. Stevens, C.S. Dunn and C.A. Petty, Surface tension induced cavitation in poly-

meric membranes during gelation, paper presented at the 73rd AIChE Annual Meet- ing, Chicago, IL, 1980.

11 J.M. Smith and H.C. van Ness, Introduction to Chemical Engineering Thermody- namics, 3rd edn., McGraw-Hill, New York, NY, 1975.

12 R.F. Sincovec and N.K. Madsen, Software for nonlinear partial differential equations, ACM Trans., Math. Softw., 1( 3) (1975) 232.

13 A.C. Hindmarsh, GEAR, ordinary differential equation system solver, CID-30001 Rev. 3, Lawrence Livermore Laboratory, P.O. Box 808, Livermore, CA, 1974.

14 G.S. Park, Radioactive studies of diffusion in polymer systems. Part 3 - Sorption and self-diffusion in the acetone + cellulose acetate system, Trans. Faraday Sot., 57 (1961) 2314.

15 W.B. Krantz, Scaling initial and boundary value problems as a teaching tool for a course in transport phenomena, Chem. Eng. Educ., 4( 3) (1970) 145.

16 W.H. McAdams, Heat Transmission, 3rd edn., McGraw-Hill, New York, NY, 1954.

Documents

Theoretical study of the transport processes occurring during the evaporation step in asymmetric membrane casting