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Drying of solvent-borne polymeric coatings: I. Modeling the drying process
XIIRFACf
COATINGS
/iZ/OLOtt
ELSEVIER
Surface and Coatings Technology99 (1998) 253-256
Drying of solvent-borne polymeric coatings: I. Modeling the drying process R . S a u r e a, G . R . W a g n e r b.,, E . - U . S c h l t i n d e r b
Krupp-Uhde GmbH. Friedrich Uhde-Str. 15. D-44141 Dormmnd, Germatzy b hrs'timtffir Thermisc/lc K,@dzrenstectmik, Unicersiti~t Karlsruhe (TH), D-76128 Karlsruhe, Germany Received 9 June 1997; accepted 23 September 1997
Abstract Solvent evaporation of a polymeric coating is modeled as a transient one-dimensional diffusion process in a slab shrinking in the direction perpendicular to the coating surface. The grid is selected in a way that there is always the same mass of the nonvolatile component between two grid lines. Kick's law of diffusion applies to this system with a shrinking coordinate when the diffusion coefficient is chosen as a function of solvent content. The resulting differential equation is solved numerically using finite differences. The boundary condition on the surface of the coating is given by the gas side mass transfer together with the assumption of thermodynamic equilibrium. The model allows the prediction of drying times or the determination of diffusion coefficients from drying experiments. © 1998 Elsevier Science S.A.
Ke,vwordsv Concentration-dependent diffusion coefficient: Drying model; Fickian diffusion; Polymeric coatings
1. Introduction Polymeric coatings, e.g. magnetic tapes or pressuresensitive adhesive tapes, are produced on a very large scale. The material to be applied is dissolved in organic solvents and subsequently dried in an oven with forced air convection. The gas flow is usually countercurrent or impinging. Since the drying rate can decrease by factor of 1000 in the region of low solvent content, knowledge of the exact drying time is essential for the design of the dryer. In the first part of this series, a model for the calculation of drying rates of polymeric, solvent-based coatings will be presented. In Part II, the theory will be proved by experiments and the effect of the experimental variation of different parameters will be discussed. The third part will report on coatings containing solvent mixtures.
2. Theory The mass transfer from the liquid coating to the ambient atmosphere can be divided into three successive transport steps: * Corresponding author. 0257-8972/98/$19.00 © 1998 ElsevierScience B.V. All rights reserved. PH S0257-8972(97) 00564- 1
(1) diffusion of liquid solvent from the inside of the coating to its surface; (2) evaporation of solvent on the surface of the coating; (3) transport of solvent vapor from the coating surface into the bulk of the atmosphere.
2.1. Diffusion of solvent ffom inside, the coating to its
SlOfuge When describing the liquid side mass transfer the following assumptions were made: ( I ) there is no temperature gradient within the coating; (2) the mass transfer is perpendicular to the coating surface, there is no diffusion parallel to the surface; (3) Internal stresses do not affect the mass transfer. (4) microconvection forced by concentration gradients or by gradients of the surface tension does not occur; (5) there is no chemical reaction like polymerization that influences the diffusion process. For a binary mixture, the diffusive fluxes with respect to the total volume-average velocity: =
+ P2 I/2 tt2
(I)
254
R. Saure et al. I Surface and Coatings Technology 99 (1998) 253-256
=
Eq. (6). This results in:
are given by j~' = plv (lt 1 -- HV ).
(2)
j~' = pV(u~ -u~),
(3)
where 1 denotes the solvent and 2 the polymer, with the specific partial volume Vi = ( 0 ~ i )
(4) T,p,t~'Ij~ i
and the density p[ = --.
(5)
V
J~P_
jv
(13)
We can rewrite Fick's law in an analogous way: jp = - D e Op~
(14)
with a different diffusion coefficient D e and the polymerfixed (shrinking) coordinate f. The density in this reference frame is now defined with the volume of the polymer V> i.e.'
Mi Mi Xi P~ = Vz - M2~z - (12"
(t5)
Neglecting volume changes of mixing results in: p~ p~ ÷pv ¢2 =1
(6)
and choosing the reference velocity Uv yields
The relationship between the different diffusio~ coefficients can easily be derived by using the definitions for the fluxes in Eqs. (8) and (14) together with the relationship between the fluxes in Eq. (13), resulting in:
JlV P1 "~J~' ~2 = P [ (/1(b/t--HV) "}-P} V2(l/2 - - " v )
1
= ~71gel "t-p2V V2~t2--1¢ * V =0;
(7)
i.e. there is no volume change in a control volume. Introducing Fick's law: (8)
and the definition of a rnoisture content on a dry basis
E __2-= p2VI22,
(9)
together with Eq. (6) yields the following expression of the continuity equation 1
~X1 = ~
(1 + VIlP2"Xj) 2 Ot
1+
VI/P2_,YI) 2 5~z ' (10)
In the polymer-fixed reference frame, the reference velocity is set equal to the velocity of the nonvolatile component. So there is no flux of the nonvolatile component across the boundary of a control volume. We therefore obtain for the fluxes: ,P V Jl = Pl (/41 -- H2)
(16)
{17)
Eq. (16) can be simplified further to D e = L~,,,. t [P2~ p:)2 = D " ~ I
P[
= .~_; X2 i 1
P2
( 11 )
(t8)
with the volume fraction of the polymer 4)2. This relationship is also indicated by Crank [1]. A mass balance for component 1 for an arbitrarily chosen distance d~ yields: Op1e Oj~ dL" 'P e _ dr, (19) -. =Jl,~ -J1,¢ + d; c~t Of or
0p P
3
De,
.
(20)
Eq. (20) has the familiar form of Fick's second law, but, of course, with different variables. With Eqs. (15) and (18), we obtain a form with more customary variables:
<
and •P
De Opv Oz l D v 3PV 0z @~ pV ("2 az
Since the two coordinates are related to each other by:
jv = _ D v @v 0z
X1
(p~' 1~2)2
V
J2 ----P2 (Zt2 -- U2) = 0.
(12)
A relationship between the fluxes of one component in different reference frames can be obtained by eliminating u~ from Eqs. (2) and ( 11 ), substituting the velocity difference with the help of Eq. (3) and finely using
0--t- - Of
0~ )'
2.2. Initial and boun&try conditions
(21) £
Before a polymeric coating is dried in an oven, it js cast on the substrate. Since the time between the casting
R. Saute et a[. / Sul:[ace and Coatings Tec/lno/ogy 99 (1998) 253-256
and the drying process is short compared with the total drying time, the initial condition that the concentration of the solvent is equally distributed within the film at t = 0 is justified: Xl(f=O,
(22)
0 _ ~ _ ~ ~max) = X1,0 .
Since the solvent cannot permeate the substrate, the boundary condition at the bottom of the film is J~;=o=0<=~ ?XI?~. ~ = o = 0 ' t > - 0 '
(23)
The boundary condition on the surface of the coating is related to the gas side mass transfer by: J~
1 DV q~ -~X 1 ; =;,~.×, t > 0 , = ffTt = -- -~-~-
(24)
and we finally obtain -c~[ -
= --//"1 ~%_ ~DVq52 .
.
(25)
255
For only a single solvent evaporating from the coating, ~' becomes unity. The mass transfer coefficient fi~ depends on the geometry of the dryer and the gas velocity. For the widely used slot nozzle dryers, fi, may be calculated from Ref. [6] together with Lewis' law. For the experimental set-up used in this study, fi~ was calculated on the basis of the theory of boundary layers for slabs [7]. In the case of a drying atmosphere with only traces of solvent and high gas velocities, compared with the solvent evaporation, f~,~. approaches zero.
3. Calculations Eq. (21) was solved numerically by means of finite differences. Since the diffusion coefficient is not known a priori, it was fitted to experimental results at different temperatures. An implicit differencing scheme, the Crank-Nicholson method, was applied. The algorithm was programmed on a 60486-PC. It turned out to be sufficient to have 20 grid lines to describe the concentration profile within a 50 btm PVAc coating. A detailed
2.3. Et,aporation o/'the so/t,ent on the su@lce oj'the coating T = 40 °C m
The activity a; of a solvent is a strong function of the solvent composition on the coating surface. For the calculation of the gas-side mass transfer rate and thus for the boundary condition, the solvent content on the coating surface is needed. Models like those suggested by Flow [2] and by Huggins [3] or the UNIFAC-FV model [4] can be used for calculating the activity of the solvent as a function of its concentration in the coating, Therefore, the mole fi'action of the solvent at the gasliquid interface )/Ph can be calculated as follows:
a,(X.
~7~ ph =
T)p'[
( 26 )
P
2.4. Transport oJ'so/cent t,aporJi'om the coating stoface into the bull< o/'the am~osphere The vapor flux fl'om a liquid surface to an inert atmosphere can be calculated fl'om the Stefan-Maxweli equation for the diffusion in dilute gases [5]: P
alS(T). t: i
=
Z "s/Sb" J
-
7g
io8 I i
?
D
I ~
PVAc = 49 g / m a
u = 0 4 3 m/s
2
~> 1,5
Experiment Calculation N u m b e r s : T i m e [s]
~12
t_
i "~,6
=_ o
o5
i" 1 ~4% ?&50
200©
0
50
i00 Tune/[s]
150
200
Fig. 1. Calculated drying curve of a poly(vinyI acetate) coating on an aluminium plate at 40 :C.
2
>
~
,-~
_
~..... ~
"-..
\
"
0.5
Parameter Time /[s]
1
O
(27)
2
05 "
~._
.-&
P
The molar evaporation rate t; i
G4
I;i --f'i,~c
n'z¢=2Qe ~--~T,,,~l& In
&/~1;
'W <
~--
~'~is defined by:
0
80 ' ~ 2 8 24 20 200 "~'+-~- " ,
20
40
60
. " "-
80
100
120
140
160
180
Absolute Co-ordinate / [gm]
(28)
Fig. 2. Calculated concentration profiles in the total volume specific coordinate system.
256
R. Saute el al. / Szoface and Coatings Technology 99 (1998) 253-256
jl M~ 37/i rili P :¢ p~
Parameter: Time / [s]
I
~
~.
2
-----20-16 .... - - _
"~ 1 - 2 ° ° ) 0 ) 0
-
:
~2~4 2
:
~
04 06 Dmaensionless Grid Co,ordinate
0.8
"t
Iooo<.
02
I
Fig. 3. Calculated concentration profiles in the shrinking coordinate system.
description of the calculation algorithm can be found elsewhere [8]. An example for a calculated drying curve of a coating containing methanol and the calculated moisture profiles both in volume-fixed and in shrinking coordinates is given in Figs. 1-3. A more comprehensive comparison of the results of the calculation and the experimental data including effects of the experimental variation of several parameters will be given in part II of this series.
4. Conclusions
The mass transfer in a polymeric coating containing a pure organic solvent can be modeled as a transient diffusion process with the gas-side mass transfer and the sorptive equilibrium as boundary condition on the surface. The resulting differential equation is of the Fickian type with a concentration-dependent diffusion coefficient. The shrinking effect can be taken into consideration by using a polymer-fixed reference fi'ame.
Acknowledgement
We wish to thank Holger Niemann for his participation in this work during his studies at the University of Karlsruhe. We also wish to thank the Deutsche Forschungsgemeinschaft ( D F G ) for financial support.
Notation
ai D
activity [Pa/Pa] diffusion coefficient [m2/s]
_
flux [kg/(m 2 s)] mass [kg] molar mass [kg/mol] mass flux per square unit [kg/m 2 S] pressure [Pa] vapor pressure [Pal universal gas constant [kJ/(mol K)] t'i molar evaporation rate [-] T absolute temperature [K] t time [s] u~ velocity [m/s] . . . . V total volume [m 3] l?; specific partial volume [m3/kg] 2",- solvent content [kg solvent/kg polymer] y~ mole fraction [-] z coordinate perpendicular to coating surface in Volume-specific coordinate system [m]
Greek letters fii mean mass transfer coefficient [m/s] ~b~ volume fraction [-1 = Pi partial density [kg/m 3] coordinate perpendicular to coating surface in shrinking coordinate system [m]
Subscripts i 1 2
component i solvent polymer
Superscripts V total volmne related reference frame P polymer volume related reference frame Ph phase
References [ 1] J. Crank, The Mathematics of Diffusion, 2nd ed., Clarendon Press, Oxford, 1975. [2] P.J. Flory, J. Chem. Phys. 9 (I94t) 660. [3] M.L. Huggin§~ J. Chem. Phys. 9 (1941) 440. [4] T. Oishi, J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev. 17 (3) (1978) 333. [5] E.-U. Schltinder, Einftihrung in die Stoffabertragung, Vieweg, Wiesbaden, 1996. [6] H. Martin, E.-U, Schltinder, Chem. lng. Techn. 45 (t973) 290. [7] H.D. Baehr, K. Stephan, W~.rme-und Stofffibertragung, Springer, Berlin, 1994, p. 330. [8] R. Saure, Zur Trocknung von [6sungsmittelfeuchten Polymerfilmen, VDI Fortschr.-Ber. VDI Reihe 3 no. 407, VDI Verlag, D~isseldorf, 1995.
COATINGS
/iZ/OLOtt
ELSEVIER
Surface and Coatings Technology99 (1998) 253-256
Drying of solvent-borne polymeric coatings: I. Modeling the drying process R . S a u r e a, G . R . W a g n e r b.,, E . - U . S c h l t i n d e r b
Krupp-Uhde GmbH. Friedrich Uhde-Str. 15. D-44141 Dormmnd, Germatzy b hrs'timtffir Thermisc/lc K,@dzrenstectmik, Unicersiti~t Karlsruhe (TH), D-76128 Karlsruhe, Germany Received 9 June 1997; accepted 23 September 1997
Abstract Solvent evaporation of a polymeric coating is modeled as a transient one-dimensional diffusion process in a slab shrinking in the direction perpendicular to the coating surface. The grid is selected in a way that there is always the same mass of the nonvolatile component between two grid lines. Kick's law of diffusion applies to this system with a shrinking coordinate when the diffusion coefficient is chosen as a function of solvent content. The resulting differential equation is solved numerically using finite differences. The boundary condition on the surface of the coating is given by the gas side mass transfer together with the assumption of thermodynamic equilibrium. The model allows the prediction of drying times or the determination of diffusion coefficients from drying experiments. © 1998 Elsevier Science S.A.
Ke,vwordsv Concentration-dependent diffusion coefficient: Drying model; Fickian diffusion; Polymeric coatings
1. Introduction Polymeric coatings, e.g. magnetic tapes or pressuresensitive adhesive tapes, are produced on a very large scale. The material to be applied is dissolved in organic solvents and subsequently dried in an oven with forced air convection. The gas flow is usually countercurrent or impinging. Since the drying rate can decrease by factor of 1000 in the region of low solvent content, knowledge of the exact drying time is essential for the design of the dryer. In the first part of this series, a model for the calculation of drying rates of polymeric, solvent-based coatings will be presented. In Part II, the theory will be proved by experiments and the effect of the experimental variation of different parameters will be discussed. The third part will report on coatings containing solvent mixtures.
2. Theory The mass transfer from the liquid coating to the ambient atmosphere can be divided into three successive transport steps: * Corresponding author. 0257-8972/98/$19.00 © 1998 ElsevierScience B.V. All rights reserved. PH S0257-8972(97) 00564- 1
(1) diffusion of liquid solvent from the inside of the coating to its surface; (2) evaporation of solvent on the surface of the coating; (3) transport of solvent vapor from the coating surface into the bulk of the atmosphere.
2.1. Diffusion of solvent ffom inside, the coating to its
SlOfuge When describing the liquid side mass transfer the following assumptions were made: ( I ) there is no temperature gradient within the coating; (2) the mass transfer is perpendicular to the coating surface, there is no diffusion parallel to the surface; (3) Internal stresses do not affect the mass transfer. (4) microconvection forced by concentration gradients or by gradients of the surface tension does not occur; (5) there is no chemical reaction like polymerization that influences the diffusion process. For a binary mixture, the diffusive fluxes with respect to the total volume-average velocity: =
+ P2 I/2 tt2
(I)
254
R. Saure et al. I Surface and Coatings Technology 99 (1998) 253-256
=
Eq. (6). This results in:
are given by j~' = plv (lt 1 -- HV ).
(2)
j~' = pV(u~ -u~),
(3)
where 1 denotes the solvent and 2 the polymer, with the specific partial volume Vi = ( 0 ~ i )
(4) T,p,t~'Ij~ i
and the density p[ = --.
(5)
V
J~P_
jv
(13)
We can rewrite Fick's law in an analogous way: jp = - D e Op~
(14)
with a different diffusion coefficient D e and the polymerfixed (shrinking) coordinate f. The density in this reference frame is now defined with the volume of the polymer V> i.e.'
Mi Mi Xi P~ = Vz - M2~z - (12"
(t5)
Neglecting volume changes of mixing results in: p~ p~ ÷pv ¢2 =1
(6)
and choosing the reference velocity Uv yields
The relationship between the different diffusio~ coefficients can easily be derived by using the definitions for the fluxes in Eqs. (8) and (14) together with the relationship between the fluxes in Eq. (13), resulting in:
JlV P1 "~J~' ~2 = P [ (/1(b/t--HV) "}-P} V2(l/2 - - " v )
1
= ~71gel "t-p2V V2~t2--1¢ * V =0;
(7)
i.e. there is no volume change in a control volume. Introducing Fick's law: (8)
and the definition of a rnoisture content on a dry basis
E __2-= p2VI22,
(9)
together with Eq. (6) yields the following expression of the continuity equation 1
~X1 = ~
(1 + VIlP2"Xj) 2 Ot
1+
VI/P2_,YI) 2 5~z ' (10)
In the polymer-fixed reference frame, the reference velocity is set equal to the velocity of the nonvolatile component. So there is no flux of the nonvolatile component across the boundary of a control volume. We therefore obtain for the fluxes: ,P V Jl = Pl (/41 -- H2)
(16)
{17)
Eq. (16) can be simplified further to D e = L~,,,. t [P2~ p:)2 = D " ~ I
P[
= .~_; X2 i 1
P2
( 11 )
(t8)
with the volume fraction of the polymer 4)2. This relationship is also indicated by Crank [1]. A mass balance for component 1 for an arbitrarily chosen distance d~ yields: Op1e Oj~ dL" 'P e _ dr, (19) -. =Jl,~ -J1,¢ + d; c~t Of or
0p P
3
De,
.
(20)
Eq. (20) has the familiar form of Fick's second law, but, of course, with different variables. With Eqs. (15) and (18), we obtain a form with more customary variables:
<
and •P
De Opv Oz l D v 3PV 0z @~ pV ("2 az
Since the two coordinates are related to each other by:
jv = _ D v @v 0z
X1
(p~' 1~2)2
V
J2 ----P2 (Zt2 -- U2) = 0.
(12)
A relationship between the fluxes of one component in different reference frames can be obtained by eliminating u~ from Eqs. (2) and ( 11 ), substituting the velocity difference with the help of Eq. (3) and finely using
0--t- - Of
0~ )'
2.2. Initial and boun&try conditions
(21) £
Before a polymeric coating is dried in an oven, it js cast on the substrate. Since the time between the casting
R. Saute et a[. / Sul:[ace and Coatings Tec/lno/ogy 99 (1998) 253-256
and the drying process is short compared with the total drying time, the initial condition that the concentration of the solvent is equally distributed within the film at t = 0 is justified: Xl(f=O,
(22)
0 _ ~ _ ~ ~max) = X1,0 .
Since the solvent cannot permeate the substrate, the boundary condition at the bottom of the film is J~;=o=0<=~ ?XI?~. ~ = o = 0 ' t > - 0 '
(23)
The boundary condition on the surface of the coating is related to the gas side mass transfer by: J~
1 DV q~ -~X 1 ; =;,~.×, t > 0 , = ffTt = -- -~-~-
(24)
and we finally obtain -c~[ -
= --//"1 ~%_ ~DVq52 .
.
(25)
255
For only a single solvent evaporating from the coating, ~' becomes unity. The mass transfer coefficient fi~ depends on the geometry of the dryer and the gas velocity. For the widely used slot nozzle dryers, fi, may be calculated from Ref. [6] together with Lewis' law. For the experimental set-up used in this study, fi~ was calculated on the basis of the theory of boundary layers for slabs [7]. In the case of a drying atmosphere with only traces of solvent and high gas velocities, compared with the solvent evaporation, f~,~. approaches zero.
3. Calculations Eq. (21) was solved numerically by means of finite differences. Since the diffusion coefficient is not known a priori, it was fitted to experimental results at different temperatures. An implicit differencing scheme, the Crank-Nicholson method, was applied. The algorithm was programmed on a 60486-PC. It turned out to be sufficient to have 20 grid lines to describe the concentration profile within a 50 btm PVAc coating. A detailed
2.3. Et,aporation o/'the so/t,ent on the su@lce oj'the coating T = 40 °C m
The activity a; of a solvent is a strong function of the solvent composition on the coating surface. For the calculation of the gas-side mass transfer rate and thus for the boundary condition, the solvent content on the coating surface is needed. Models like those suggested by Flow [2] and by Huggins [3] or the UNIFAC-FV model [4] can be used for calculating the activity of the solvent as a function of its concentration in the coating, Therefore, the mole fi'action of the solvent at the gasliquid interface )/Ph can be calculated as follows:
a,(X.
~7~ ph =
T)p'[
( 26 )
P
2.4. Transport oJ'so/cent t,aporJi'om the coating stoface into the bull< o/'the am~osphere The vapor flux fl'om a liquid surface to an inert atmosphere can be calculated fl'om the Stefan-Maxweli equation for the diffusion in dilute gases [5]: P
alS(T). t: i
=
Z "s/Sb" J
-
7g
io8 I i
?
D
I ~
PVAc = 49 g / m a
u = 0 4 3 m/s
2
~> 1,5
Experiment Calculation N u m b e r s : T i m e [s]
~12
t_
i "~,6
=_ o
o5
i" 1 ~4% ?&50
200©
0
50
i00 Tune/[s]
150
200
Fig. 1. Calculated drying curve of a poly(vinyI acetate) coating on an aluminium plate at 40 :C.
2
>
~
,-~
_
~..... ~
"-..
\
"
0.5
Parameter Time /[s]
1
O
(27)
2
05 "
~._
.-&
P
The molar evaporation rate t; i
G4
I;i --f'i,~c
n'z¢=2Qe ~--~T,,,~l& In
&/~1;
'W <
~--
~'~is defined by:
0
80 ' ~ 2 8 24 20 200 "~'+-~- " ,
20
40
60
. " "-
80
100
120
140
160
180
Absolute Co-ordinate / [gm]
(28)
Fig. 2. Calculated concentration profiles in the total volume specific coordinate system.
256
R. Saute el al. / Szoface and Coatings Technology 99 (1998) 253-256
jl M~ 37/i rili P :¢ p~
Parameter: Time / [s]
I
~
~.
2
-----20-16 .... - - _
"~ 1 - 2 ° ° ) 0 ) 0
-
:
~2~4 2
:
~
04 06 Dmaensionless Grid Co,ordinate
0.8
"t
Iooo<.
02
I
Fig. 3. Calculated concentration profiles in the shrinking coordinate system.
description of the calculation algorithm can be found elsewhere [8]. An example for a calculated drying curve of a coating containing methanol and the calculated moisture profiles both in volume-fixed and in shrinking coordinates is given in Figs. 1-3. A more comprehensive comparison of the results of the calculation and the experimental data including effects of the experimental variation of several parameters will be given in part II of this series.
4. Conclusions
The mass transfer in a polymeric coating containing a pure organic solvent can be modeled as a transient diffusion process with the gas-side mass transfer and the sorptive equilibrium as boundary condition on the surface. The resulting differential equation is of the Fickian type with a concentration-dependent diffusion coefficient. The shrinking effect can be taken into consideration by using a polymer-fixed reference fi'ame.
Acknowledgement
We wish to thank Holger Niemann for his participation in this work during his studies at the University of Karlsruhe. We also wish to thank the Deutsche Forschungsgemeinschaft ( D F G ) for financial support.
Notation
ai D
activity [Pa/Pa] diffusion coefficient [m2/s]
_
flux [kg/(m 2 s)] mass [kg] molar mass [kg/mol] mass flux per square unit [kg/m 2 S] pressure [Pa] vapor pressure [Pal universal gas constant [kJ/(mol K)] t'i molar evaporation rate [-] T absolute temperature [K] t time [s] u~ velocity [m/s] . . . . V total volume [m 3] l?; specific partial volume [m3/kg] 2",- solvent content [kg solvent/kg polymer] y~ mole fraction [-] z coordinate perpendicular to coating surface in Volume-specific coordinate system [m]
Greek letters fii mean mass transfer coefficient [m/s] ~b~ volume fraction [-1 = Pi partial density [kg/m 3] coordinate perpendicular to coating surface in shrinking coordinate system [m]
Subscripts i 1 2
component i solvent polymer
Superscripts V total volmne related reference frame P polymer volume related reference frame Ph phase
References [ 1] J. Crank, The Mathematics of Diffusion, 2nd ed., Clarendon Press, Oxford, 1975. [2] P.J. Flory, J. Chem. Phys. 9 (I94t) 660. [3] M.L. Huggin§~ J. Chem. Phys. 9 (1941) 440. [4] T. Oishi, J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev. 17 (3) (1978) 333. [5] E.-U. Schltinder, Einftihrung in die Stoffabertragung, Vieweg, Wiesbaden, 1996. [6] H. Martin, E.-U, Schltinder, Chem. lng. Techn. 45 (t973) 290. [7] H.D. Baehr, K. Stephan, W~.rme-und Stofffibertragung, Springer, Berlin, 1994, p. 330. [8] R. Saure, Zur Trocknung von [6sungsmittelfeuchten Polymerfilmen, VDI Fortschr.-Ber. VDI Reihe 3 no. 407, VDI Verlag, D~isseldorf, 1995.