AN ANALYTICAL SOLUTION OF THE CONVECTIVE DRYING OF A MULTICOMPONENT LIQUID FILM

15th International Drying Symposium (IDS 2006) Budapest, Hungary, 20-23 August 2006

AN ANALYTICAL SOLUTION OF THE CONVECTIVE DRYING OF A MULTICOMPONENT LIQUID FILM

R. Gamero

, A. Picado

, F. Luna

and J. Martínez

1Faculty of Chemical Engineering, National University of Engineering (UNI) PO Box 5595, Managua, Nicaragua

Tel.:+505 2701523, E-mail: rafaelg@ket.kth.se

2Café Soluble S.A.

PO Box 429, Managua, Nicaragua

Tel.:+505 2331122 x199, E-mail: fluna@cafesoluble.com

3Dept. of Chemical Engineering and Technology, Royal Institute of Technology (KTH) S-10044 Stockholm, Sweden

Tel.:+468 7906570, E-mail: jmc@ket.kth.se

Abstract: Analytical solutions of the diffusion and conduction equations applied to liquid- side-controlled convective drying of a multicomponent liquid film are developed.

Assuming constant physical properties of the liquid, the equations describing interactive mass transfer are decoupled by a similarity transformation and solved simultaneously with conduction equation by the method of variable separation. Variations of physical properties along the process trajectory are taken into account by a piecewise application of the solution in time intervals with averaged coefficients from previous time steps. Despite simplifications, the analytical solution gives a good insight into the selectivity of the drying process and is computationally fast.

Keywords: evaporation, mass transfer, multicomponent diffusion, selectivity, ternary mixture

INTRODUCTION

The drying of varnish layers, pharmaceuticals, coated laminates, magnetic storage media and juice concentrates are some of the industrial applications where the drying of multicomponent liquid films is important. When the moisture consists of a multicomponent mixture the drying process has a great influence on the quality of the product. For instance, water should be removed from liquid foods after drying whereas the aroma compounds should be retained, toxic components should be removed from pharmaceuticals and coated surfaces should retain compounds that make them smooth and mechanically stable. The fulfilment of these requirements depends greatly on the conditions of drying.

During drying of a liquid film by convection, evaporation is governed by several mechanisms:

transport in the liquid phase, in the gas phase or by equilibrium. The conditions required for different controlling steps to prevail are discussed by Schlünder (1982) in connection to the isothermal evaporation of a binary mixture. Drying intensity determines the controlling step. At moderate gas velocities and temperatures the process is likely to be

controlled by the gas-side mass and heat transfer or equilibrium. Under intensive drying, the resistance to mass and heat transfer in the liquid phase becomes important. Riede and Schlünder (1990) extend the results to the effects of gas preloading and the presence of a third component of negligible volatility. For gas-phase-controlled drying, Luna and Martínez (1999) show that a deep understanding of the process can be obtained by a stability analysis of the ordinary differential equations that describe the dynamical system. Liquid-side-controlled drying of multicomponent mixtures has been analysed by Pakowski (1994). When liquid-side resistance cannot be neglected, the complexity of convective drying of multicomponent liquid mixtures is such that only an approximate analysis or a numerical solution of the equations that describe the process is possible. This approach has been used to study the drying of polymeric films by Guerrier et al. (1998). The main difficulties to obtain particular analytical solutions are the complex interactions of transport mechanisms and phase equilibrium as well as the strong dependence of the drying process on composition, temperature, and the contact mode between the phases. Recently, Luna et al. (2005) has developed

an analytical solution of the multicomponent diffusion equation for isothermal drying of a liquid film assuming constant physical properties.

The purpose of this study is to extend this solution to the non-isothermal case. The solution is applied to the drying of the liquid mixtures: ethanol- methylethylketone-toluene and water-2-propanol- glycerol, the last being a mixture with a component of negligible volatility.

THEORY Diffusion in liquid phase

Diffusion of n species in a single phase is described by the differential equations of continuity for all n components of the mixture. Applied to mass transfer in a liquid film it yields:

∂(C_Lx_i)

∂t + ∇ ⋅ (G_tx_i) = − ∇ ⋅ J_i (1) where i = 1, 2,…n, xi is the molar fraction of compound i in the liquid, Ji is the diffusion flux of component i, CL is the total concentration and Gt the total molar flux. The molar flux Gi, with respect to a stationary coordinate frame of reference, are related to the diffusion fluxes by:

G_i= J_i+ x_iG_t (2) Since ∑x_i = 1 and ∑J_i = 0, only n-1 equations (1) are independent and an additional condition, specific for a given diffusion problem, is required in order to determine the molar fluxes. This extra piece of information is frequently represented by an interrelationship between the molar fluxes. For example, if the liquid mixture contains a component of negligible volatility or if evaporation occurs in presence of a non-condensing gas the process is regarded as diffusion through a stagnant component in the respective phase.

In addition to initial and boundary conditions, the solution of the set of equations (1) requires a relationship between diffusion fluxes and mole fraction gradients in the system. One of these relationships is provided by a generalization of Fick’s law for binary diffusion. For one-dimensional diffusion, it is expressed in compact matrix notation as follows:

J = − C_LD∂x

∂z (3)

where D is the matrix of multicomponent diffusion coefficients. The column vectors of diffusion fluxes and composition gradients have the dimension n-1 to match the dimensions of matrix D, which is of order n-1 × n-1. This expresses the fact that the nth component does not diffuse independently. In non- ideal mixtures, the matrix of multicomponent diffusion coefficients is defined as:

D = B⁻¹Γ (4)

The matrix B, which can be regarded as a kinetic contribution to the multicomponent diffusion coefficients, has the elements:

∑

≠= +

=

n

i k 1

k ik

k in

ii i D

x D

B x (5)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

−

=

≠

in i ij

ij D

1 D x 1 ) j i (

B (6)

where i, j = 1, 2,…n-1 and D_ij are the Maxwell- Stefan diffusion coefficients. The elements of the matrix of thermodynamic factors, Γ, are given by:

j i j ij i

ij lnx

ln x x

∂ γ + ∂ δ

=

Γ (7)

where γ_i is the activity coefficient of compound i and δ_i,j is the Kronecker delta (δ_i,j = 1 for i = j and δ_i,j = 0 for i ≠ j). For ideal solutions, the matrix of thermodynamic factors reduces to the identity matrix.

Combining equations (1) and (3) in one dimension:

∂(C_Lx)

∂t = ∂

∂z C_LD∂x

∂z

⎛

⎝ ⎜ ⎞

⎠ ⎟ − ∂(G_tx)

∂z (8)

The total molar flux Gt vanishes in equimolar counter-diffusion. The second term of the right hand side can also be neglected when diffusion is the dominant mechanism for mass transfer.

Drying of a liquid film

The drying of a liquid film into an inert gas is schematically described in Fig. 1. To simulate the evaporation of a multicomponent liquid film requires the solution of equations (8) together with an energy balance and respective initial and boundary conditions.

T x, yδ

q

Gg

Tδ g

g,u T ,

δ0

y

Fig. 1. Schematic of drying of a liquid film into an inert gas

In addition, a total balance is necessary to determine the changes of film thickness.

Governing equations

If diffusion is the main contribution to mass transfer and, if average and constant values of CL and D are assumed, equations (8) reduce to the following:

∂x

∂t = D∂²x

∂z² (9)

If conduction is the only mechanism for heat transfer in the liquid the corresponding equation to describe changes of temperature is the conduction equation:

∂T

∂t = D_h ∂²T

∂z² (10)

These equations represent a system of coupled partial differential equations. If evaporation and convection heat occurs only at the surface of the film and the initial composition as well as temperature of the liquid are given functions of z, the initial and boundary conditions are:

At t = 0 and 0 ≤ z ≤ δ,

x = x₀{z}, T = T₀{z} (11) At z = 0 and t > 0,

z 0 , T z=0 =

∂

∂

∂

∂x (12)

At z = δ and t > 0,

g g

1 n , g L

) T (T z h k T C z

TG λ x G

D

−

−

∂ =

− ∂

∂ =

− ∂

δ

− (13)

where λ is a column vector of heat of vaporisation.

The subscript g and superscript T denotes gas phase and transposition respectively. The subscript n-1 in the column vector of evaporation fluxes in gas phase indicates that only n-1 of the fluxes are considered to match the dimension of the independent diffusion fluxes at the liquid side.

Evaporation rates at the liquid surface

If diffusional interactions in gas phase are included evaporation fluxes may be written as:

G^g = K{y_δ− y_∞} (14) where the matrix K is the matrix product βEk in which β embodies an extra relationship between the fluxes to calculate molar fluxes from diffusion fluxes, E is a matrix of correction factors to account for the finite mass transfer rate and k is a matrix of mass transfer coefficients at zero mass transfer rate.

The columns vectors yδ and y∞ are the molar fractions of the vapours at the gas-liquid interface and the bulk of the gas respectively (Taylor and Krishna, 1993). These methods were applied to drying by Martínez and Setterwall (1991). The analytical solution of equation (9) requires K to be a diagonal matrix. In order to maintain the description of diffusional interactions, effective mass transfer coefficients that produce the same evaporation fluxes

at the free liquid surface that equation (14) can be defined:

G^g= K_eff{y_δ− y_∞} (15) where Keff is a diagonal matrix with the following elements:

K_i,i,eff = G_i,g{y_i,δ− y_i,∞}⁻¹ (16) where i = 1, 2,…n. If the molar fractions in gas phase are considered in equilibrium with the liquid at the interface, then at z = δ:

y_δ = 1

P_t P⁰γ x_n= K_γx_n (17) is obtained, with Pt being the total pressure. P⁰ and γ are diagonal matrices containing the saturated vapour pressures of the pure liquids, and activity coefficients respectively. The subscript n indicates that the vector x contains the molar fractions of the n components of the liquid mixture.

Moving boundary

If the changes of film thickness are not neglected the invariance of the fluxes at the moving phase boundary can be expressed as:

dt }d C C

{ _{L n g}

g

L δ

yδ x

G

G = + − (18)

Under non-isothermal conditions, the following total mass balance gives the rate of change of film thickness:

dδ

dt = − 1

C _L G_g+ δ ∂C _L

∂x

⎛

⎝ ⎜ ⎞

⎠ ⎟

Tdx dt+∂C _L

∂T

∂T

∂t

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ (19)

where the scalar Gg is the total evaporation flux. The other terms on the right hand side vanishes if the total concentration is constant. Introducing the expression for the fluxes in gas phase in equation (18):

• − ∞

=Ξ x K y

G_{L n eff} (20)

is obtained, with

dt }d C C

{ _{l g}

eff

δ

γ

γ I K

K K

Ξ^•= + − (21)

At the interface liquid side, mass transport is molecular. The boundary condition can be rewritten:

b

L z

C D x=Ξx+y

∂

− ∂ (22)

with

y^b= − {K_effy_∞}_n−1 (23) where the square matrix Ξ is of order n-1 and consists of the first n-1 column and rows of the original matrix Ξ^•. To maintain the dimensional consistency the last element of the vectors is

disregarded and, to indicate this, the vectors are denoted by the subscript n-1.

Analytical solution

The solution of the equation (9) was reported by Luna et al. (2005):

1 b m 1

0 m 0

1 m

2m 1 ˆ 2 2m 2 2m 1

) cos(

d ) ˆ ( ) cos(

e ) )(

( 2

y Ξ ν u

ν

ξ ξ ν ξ ν P Ξ

x ^Dν

−

∞

=

τ

−

−

−

−

⎭ ζ

⎬⎫

⎩⎨

⎧ ζ ζ ζ

×

+ + +

=

∫

∑

(24)

where uˆ₀ =P⁻¹(Ξx₀+y_b) and P is the modal matrix whose columns are the eigenvectors of the adimensional matrix:

∞

−

= − 1

~ 1

Ξ D Ξ

Ξ D

D Ξ (25)

and

Ξ Ξ D

D

ξ Ξ ¹

1 L ˆ

C

−

∞

= −δ

(26)

where Dˆ is a diagonal matrix with the eigenvalues of D~

.

Equation (24) provides the mole fractions of n-1 components in the liquid. The mole fraction of the nth component is calculated taking advantage of:

x_n = 1− x_j

j=1

∑

The solution of the equation (10) is similar given the dimensionless temperature:

) cos(

e d ) ( T ) cos(

a) a )(

a ( 2

m , h 2 h

m , Dˆ h 1

0 h,m 0

1 m

1 2 2,m 2 h 2,m h

ζ

⎭ ν

⎬⎫

⎩⎨

⎧ ν ζ ζ ζ

×

+ + ν + ν

= Θ

τ ν

−

∞

=

−

∫

∑

(28)

where

a ) T T )(

b T (

T _g Θ− ^g − ^δ

−

= (29)

with

{ }

) T h(T b a k and a h

g g

− δ

−

= δ

= λ^TG

(30) The eigenvalues in equations (24) and (28) are defined implicitly by:

tanν_m = ξν⁻¹_m and tanν^h,m= aν⁻¹_h,m (31)

To preserve the formalism of matrix product, the integral in equation (24) is a diagonal matrix that contains the value of the integral.

Even though the solution is only valid for constant physical properties the variation of coefficients for the whole process can be taken into account by a piecewise application of the analytical solution along the process trajectory. That is, by performing the solution in successive steps where the final conditions of the previous step are used to calculate the coefficients and, as initial condition of the next step. The changes of film thickness can reliably be calculated step by step by solving simultaneously equation (19) with a proper initial condition.

RESULTS AND DISCUSSION

Calculations were performed with two liquid mixtures: water-2-propanol-glycerol and ethanol- methylethylketone-toluene. The evaporation fluxes were calculated according to equation (14) using an algorithm reported by Taylor (1982) with diffusion through stationary air as a bootstrap relationship. The matrix of corrections factors was evaluated using the linearised theory. Mass transfer coefficients at zero- mass transfer rate were computed from correlations for a gas stream flowing parallel to a flat surface with binary diffusion coefficients predicted by the method of Fuller (Reid et al., 1987). The physical properties of pure components and mixtures were evaluated at the average composition between the bulk gas phase and the interface using methods described by Reid et al. (1987). Activity coefficients were calculated according to the Wilson equation with parameters from Gmehling and Onken (1982). Antoine method was used for computing the vapour pressure of pure liquids. For determining the Maxwell-Stefan diffusion coefficients in liquid phase the method of Bandrowski and Kubaczka (1982) was used. In this method, the matrix of the multicomponent diffusion coefficients is calculated by applying an empirical exponent to the matrix of thermodynamics factors:

D = B⁻¹Γ^η (32) The following values of exponent η were used for the examined systems: 1 for water-2-propanol-glycerol and 0.1 for ethanol-methylethylketone-toluene.

Convergence of the solution

According to previous work of Luna et al. (2005) with the isothermal case it suffices with around 50 eigenvalues to achieve a satisfactory accuracy. The higher the concentration of the volatile components the lesser the number of eigenvalues required. At a given composition, the component of lowest molar fraction controls the convergence. Drying intensity also influences the optimal number of eigenvalues.

The convergence is faster at lower drying rates.

Fig. 2. Effects of the number of time steps on mean composition (a, b, c) and mean liquid temperature (d). Mixture ethanol–MEK–toluene. ug = 5 m/s, Tg = 353 K, y∞ = [0 0 0]^T, T = 323 K, x0 = [0.6 0.2]^T, l = 5

10^-3 m, δ₀ = 1 mm.

In order to elucidate how many times the analytical solution should be evaluated along a process trajectory to obtain a proper description of the evaporation process, calculations were performed with an increasing number of time steps. The calculations were carried out for the same total time divided into an increasing number logarithmically spaced time intervals, except at the very beginning of the evaporation. (See Fig. 2). As revealed by Fig. 2d, the convergence of liquid temperature requires more time steps. In the case of compositions the solutions do not differ appreciably for more than 100 time steps used in the calculations.

Liquid-side kinetic separation factor

A parameter that can be regarded as representative for the resistance of the phase is the liquid-side kinetic separation factor (Luna et al., 2005):

K_L = exp − δ_p

C_L D_∞ G_k,L

k=1

∑

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟ (33)

where δ_p is the penetration depth, i.e. the distance from the surface where concentration gradients vanish. If mass transfer within the gas phase is the controlling step, the separation factor approaches unity. If the resistance is located in the liquid phase the separation factor approaches zero.

Comparison with experimental results

The results of the calculations for the ternary mixture ethanol–methylethylketone–toluene at two different initial conditions are compared with experimental data reported by Martínez (1990) in Figs. 3 and 4.

Considerations on mass and heat transfer coefficients also reported in that work were taken into account for the calculations.

Considering the approximations the agreement is fairly good. The main reason for the deviation is probably related to the rapid evaporation process and the existence of convective mass transport in liquid phase.

The changes of the liquid separation factor shown in Fig. 3 imply a shift of the controlling mechanism from liquid side to gas side during the drying process. Note that the abscissas in Fig. 3 and 4 are proportional to the total number of moles. Since evaporation is considerably faster at the beginning of the process, the period of time during which evaporation is controlled by a combination of mechanisms is longer than the one that may be perceived in Fig. 3 and 4.

Drying of liquid mixture containing a component of negligible volatility

Fig. 5 shows the changes of composition and temperature, during the evaporation of a liquid film containing a non-volatile. The liquid mixture water–

2-propanol–glycerol is a strongly non-ideal system.

The high viscosity of glycerol solutions is connected to low diffusion coefficients and the resistance against mass transfer within the liquid can be considerable. The surface of the liquid film is rapidly depleted from volatile compounds with a consequent decrease of their molar fractions whereas the solution is enriched with glycerol. Since glycerol has a negligible volatility the liquid film will consist only of glycerol at the final stage and the diffusion coefficients will decrease during the evaporation process. Two factors will counteract the increase of resistance due to the increase of glycerol concentration; the increase of liquid temperature (see Fig. 5b) that will reduces solution viscosity and the reduction of the film thickness. It becomes apparent

by studying the variations of the liquid-side kinetics separation factor in Fig. 5a that is not zero during most of the drying process.

CONCLUSIONS

Solutions to the diffusion and conduction equations applied to the drying of a multicomponent liquid film have been presented. The equations for mass transfer were decoupled by a similarity transformation and solved by the method of variable separation. The solution is applied to the drying of ternary mixtures, one of them containing a component of negligible volatility.

(a) ml0 = 11.9 g, T = 328 K, x10 = x20 = 0.33 (b) ml0 = 12.6 g, T = 317 K, x10 = 0.6, x20 = 0.1 Fig. 3. Liquid mean composition versus relative number of moles. Comparison between experiments and calculations (denoted by lines). ug = 1.3 m/s, Tg = 334 K, y∞ = [0 0 0], l = 3 10^-2 m, δ0 = 3 mm, ms = 145 g

(a) ml0 = 11.9 g, T0 = 328 K, x10 = x20 = 0.33 (b) ml0 = 12.6 g, T0 = 317 K, x10 = 0.6, x20 = 0.1 Fig. 4. Liquid temperature versus time. Comparison between experiments and calculations at the same

conditions as Fig. 3.

To consider the changes of physical properties along the process trajectory, the calculations were carried out by a stepwise application of the analytical solution with non-uniform initial conditions and the total balance to determine the changes in film thickness. Due to the assumption of constant physical properties in each time step it is not expected that the analytical model is able to predict a drying process where physical properties changes drastically along the film.

The solution agrees qualitatively well with experimental results for the evaporation of the ternary mixture ethanol–methylethylketone–toluene.

Calculations show that irrespective of the initial drying conditions it is difficult to obtain an evaporation process that is completely controlled by the liquid-side mass transfer. This is due to two factors: first, the evaporation rates decrease rapidly as the liquid surface is depleted of the more volatile components at the initial stages of evaporation;

second, the film thickness decreases as evaporation progresses and the resistance in the liquid side is reduced due to the shorter diffusion path.

In the presence of a non-volatile liquid resistance is also reduced by the increase of liquid temperature that counteracts the effect of non-volatile concentration and solution viscosity on diffusion coefficients.

The analytical solution presented in this study may be a useful tool for exploring the drying process and choosing appropriate drying conditions. It can be also used to extrapolate composition gradients in time and accelerate the convergence of numerical solution when more rigorous models are solved numerically.

ACKNOWLEDGEMENT

The authors gratefully acknowledge the financial support provided by the Swedish International Development Agency (SIDA/SAREC) for this work.

NOMENCLATURE a Dimensionless variable defined

by equation (30) –

B Matrix with elements defined by equations (5) and (6)

s m^-2 b Dimensionless variable defined

by equation (30) –

C Concentration kmol m^-3

D Matrix of multicomponent diffusion coefficients

m² s-¹ Dh Thermal diffusivity m² s^-1

D Generalised Maxwell-Stefan diffusion coefficients

m² s^-1 G Molar flux referred to a

stationary coordinate reference frame

kmol m^-2 s^-1

G Column vector of fluxes kmol m^-2 s^-1 h Heat transfer coefficient W m^-2 K^-1

I Identity matrix –

J Diffusion flux kmol m^-2 s^-1 J Column vector of diffusion

fluxes

kmol m^-2 s^-1 k Matrix of mass transfer

coefficients at zero mass transfer rate

kmol m^-2 s^-1

k Thermal conductivity W m^-1 K^-1 KL Liquid-side kinetic separation

factor –

K Matrix product βEk kmol m^-2 s^-1 Keff Diagonal matrix defined by

equation (16) kmol m^-2 s^-1

(a) (b) Fig. 5. Liquid mean composition versus relative number of moles and liquid mean temperature versus time.

Mixture water–2-propanol–glycerol. ug = 5 m/s, Tg = 353 K, y∞ = [0 0 0]^T, T = 323 K, x0 = [0.6 0.2]^T, l = 5 10^-3 m, δ0 = 1 mm.

Kγ Diagonal matrix defined by

equation (17) kmol m^-2 s^-1

l Sample length m

m Mass kg

n Number of condensable

components –

P Pressure Pa

P⁰ Diagonal matrix of saturated vapour pressures

Pa P Modal matrix, equation (24) –

q Heat flux kJ m^-2 s^-1

T Temperature K

t Time S

u Column vector defined by yb

x Ξ u= +

ug Gas velocity m s^-1

x Molar fraction in liquid phase x Column vector of molar

fractions in liquid phase kmol kmol^-1 y Molar fraction in gas phase kmol kmol^-1 y Column vector of molar

fractions in gas phase kmol kmol^-1 yb Column vector defined by

equation (23)

kmol kmol^-1

z Space dimension M

Greek symbols

β Bootstrap matrix –

γ Activity coefficient – Γ Matrix of thermodynamic

factors –

δ Film thickness M

δ_ι,ϕ Kronecker delta –

δp Penetration depth M

ν Eigenvalues –

ν Matrix of eigenvalues

λ Vector of heat of vaporisation kJ kmol^-1 Ε Matrix of correction factors – Ξ^• Matrix defined by equation

(21)

kmol m^-2 s^-1 Ξ Matrix reduced from matrix Ξ^• kmol m^-2 s^-1 Θ Dimensionless temperature –

ξ Matrix defined by equation (26)

– ζ Dimensionless thickness –

Subscripts Superscripts 0 Initial quantity 0 saturation

eff Effective value T Matrix transposition

g Gas η Exponent in

equation (32)

L Liquid

m Index

s Solid Overscripts

t Total ~ Dimensionles

s quantity

δ Interface ^ Transformed

quantity

∞ Gas bulk – Mean values

REFERENCES

Bandrowski, J. and A. Kubaczka (1982), On the prediction of diffusivities in multicomponent liquid systems. Chem. Eng. Sci., Vol. 37, pp.

1309-1313.

Gmehling, J. and U. Onken (1982), Vapor-liquid equilibrium data collections. DECHEMA, Chemistry data Series, I/1a, I/2a.

Guerrier, B., C. Bouchard, C. Allain and C. Bénard (1998), Drying kinetics of polymer films. AIChE J, Vol. 44, pp. 791-798.

Luna, F., K.E. Birgersson and J. Martínez (2005), Diffusion equation applied to isothermal drying of a multicomponent liquid film, Drying Technology, Vol. 23, pp. 1953-1975.

Luna, F. and J. Martínez (1999), Stability analysis in multicomponent drying of homogeneous liquid mixtures. Chem. Eng. Sci., Vol. 54, pp. 5823- 5837.

Martínez, J. and F. Settertwall (1991), Convective drying of solids wetted with multicomponent liquid mixtures. Chem. Eng. Sci., Vol. 46, pp.

2235-2252.

Martínez, J. (1990), Convective drying of solids wetted with multicomponent solvent mixtures.

PhD thesis, Royal Institute of Technology, Stockholm.

Pakowski, Z. (1994), Drying of solids containing multicomponent mixture: recent developments.

Proceedings of the 10th International Drying Symposium, Gold Coast, Vol. A, pp. 27-38.

Reid, R.C., J.M. Praustnitz and B.E. Poling (1987), The properties of gases and liquids. 4th Ed., Mc Graw-Hill, New York.

Riede, T. and E.U. Schlünder (1990), Selective evaporation of a ternary mixture containing one non-volatile component with regard to drying process. Chem. Eng. & Proc., Vol. 28, pp. 151- 163.

Schlünder, E.U. (1982), Progress towards understanding the drying of materials wetted by binary mixtures. Proceedings of the 3rd International Drying Symposium. Birmingham, Vol. 2, pp. 315-325.

Taylor, R. (1982), Film models for multicomponent mass transfer: computational method II: the linearized theory. Computer & Chem. Engng., Vol. 6, pp. 69-75.

Taylor, R. and R. Krishna (1993), Multicomponent mass transfer. John Wiley & Sons, New York.