The influence of the diffusional boundary layer on dye sorption from finite baths

J N Etters

Textile Sciences, Unioersity of Georgia, Athens, GA 30602. USA

A non-rigorous technique for approximating the effect of the diffusional boundary layer on the sorption of dye from finite baths is proposed. Dimensionless halftimes of sorption aregiven for various fractional equilibrium bath exhaustion and boundary layer effect conditions. A t higher fractional equilibrium exhaustions of the dyebath, the boundary layer is shown to have a greater influence on half-time of sorption.

INTRODUCTION Isothermal sorption of dyes by textile fibres from well stirred media of limited volume is a process first described mathe- matically by Wilson (11. The functional form of Wilson's equation is given by Eqn 1:

Even when the first three assumptions listed above are found to hold reasonably well, the applicability of Wilson's equation still breaks down when sorption of dye occurs under conditions of insufficient dyebath flow or agitation. Such conditions are common in practice and lead to the development of a concentration gradient between dye in the dyebath and dye at the fibre surface. This concentration gradient is referred to as a diffusional boundary layer. As the 'thickness' of this imaginary layer increases with decreasing

Mt - = f ( ~ f t / r ~ , E") M,

where M, is the concentration of dye in the fibre at a given time t , M, is the concentration of dye in the fibre at equilibrium, Df is the concentration-independent diffusion coefficient of dye in the fibre, r is the fibre radius and E, is the equilibrium dyebath exhaustion. The computational form of Wilson's equation is given in Eqn 2:

in which a equals (1 - €,)/Em, and the qn terms are roots of the transcendental Eqn 3:

with J o and J1 being respectively zero- and first-order Bessel functions.

For the strict applicability of Wilson's equation to real textile sorption processes the following assumptions must hold: 1. The fibre is a morphologically stable, homogeneous

endless cylinder 2. The diffusion coefficient is a constant concentration-

independent quantity 3. The sorption isotherm is linear 4. No significant boundary layer influence exists.

Only rarely in real textile sorption systems are all these assumptions valid, thereby enabling Wilson's equation to be used. Nevertheless, the equation serves as a solid foundation for studies involving sorption processes, although such studies at times may result in nothing more than descriptions of deviations from the ideal of Fickian formalism.

dyebath agitation, the speed of sorption (or desorption) of dye by the fibre decreases [2]. Newman was the first to provide a mathematical solution to the effect of the diffusion- al boundary layer on mass transport (31. The equivalent equation of Crank is, however, notationally less encumbered [4]. Both equations may be stated in functional form by Eqn 4:

where Mt/M, and Q t / r 2 are as given previously, and L is a dimensionless parameter given by Eqn 5 [5]:

(5)

where D, and Df are the diffusion coefficients of the dye in the medium and the fibre respectively, K is the distribution coefficient of dye between the dyeing medium and the fibre surface, r is the fibre radius and S, is the diffusional boundary layer. The computational form of Crank's equation is given by Eqn 6:

where the Pn terms are the roots of the transcendental Eqn 7:

in which J O and J1 again are zero- and first-order Bessel functions.

The first three of the four previously listed assumptions that are necessary for the applicability of Eqn 2, are also

114 JSDC Volume 107 March 1991

necessary for the applicability of Eqn 6. However, just as it is necessary that L = = (zero boundary layer) for Eqn 2 to be applicable, so E , = 0 (infinite bath) is a prerequisite before Eqn 6 can be used. As far we know, no ready-made computational solution exists for the functional relationship given by Eqn 8:

Over ten years ago Crank suggested that a solution to Eqn 8 can be determined by the use of finite difference or finite element computational techniques [6]. However, during the past ten years there appears to have been no published attempt to solve the problem by the use of Crank’s suggested method.

The purpose of the present work is to propose a new non- rigorous method by which mass transfer can be approximat- ed for finite bath systems in which a diffusional boundary layer exists. We believe that the new technique, which is based on physico-chemical relationships found in transitional kinetics [7], can be useful in the analysis of commercial dyeing systems. In such systems dye often is taken up by the fibre from finite baths having less than ideal flow velocities through the textile substrate.

TRANSITIONAL KINETICS A sorption system can be considered to be governed by transitional kinetics if the sorption process changes from infinite bath to finite bath kinetics during the course of dyeing. In the case of infinite bath kinetics the concentration of dye at the fibre surface does not change during dyeing. However, with finite bath kinetics the concentration of dye at the fibre surface continuously decreases throughout dye- ing until equilibrium uptake of dye by the fibre is achieved. McGregor and Etters have explored transitional kinetics for disperse dyeing systems [7], in which sorption can be controlled by either infinite, finite, or transitional kinetics, depending on the values of certain parameters. As previous- ly shown [7], infinite bath kinetics will occur if the value of Co is as specified in Eqn 9:

( 9 )

where C, is the apparent initial concentration of both particulate and dissolved dye, Csat is the saturation solubility of dye in the dyebath, and a is as previously given.

Finite kinetics will occur if the value of Co is as specified in Eqn 10:

and transitional kinetics will occur if the value of Co is as specified in Eqn 11:

Transitional systems are characterised by having two different fractional dye uptake values at a given time: a fractional fibre saturation value fsat and a fractional equilib- rium uptake value f-. These values are equivalent to M , / M , in Crank’s and Wilson’s equations respectively. Transition

from infinite bath kinetics to finite bath kinetics will occur at specific value of fsat and f,. Fractional saturation uptake at transition is defined by Eqn 12:

and fractional equilibrium uptake at transition is related to this value at transition by (Eqn 13):

f m = (Csat / a CoE=)fsat (13)

At transition the relationship defined by Eqn 14 also exists:

When Eqn 14 is substituted into Eqn 12, the ratio term in both equations is eliminated and a more general equation relating fsat to fm at transition results (Eqn 15):

Eqn 15 relates fsat to f, for all fractional uptake values at transition for the condition, 0 < E, < 1. Although the equation strictly holds for transitional disperse dyeing sys- tems, it will be shown that the equation also can be adopted for use in a more general fashion to aid in determining the effect of the diffusional boundary layer on dye uptake from finite baths.

COMPUTATIONAL PROCEDURE In the case of a finite dyebath in which no diffusional boundary layer exists (L = m), values for the dimensionless parameter Dft/r2 may be determined for given values of M t / M m , at a given value of E,, by the use of iterative solu- tions to Wilson’s equation. If the value of M,/M, is 0.5, the resulting value of Dft/r2 is referred to as a dimensionless half-time. (The term ‘dimensionless time’ is based on the fact that the dimensions of each term in Dft/r2 cancel, yet only the time t is not a constant.) Such half-times are at a mini- mum when L = =, but increase as L decreases. The question to be answered is: How much does the dimensionless half- time increase for a given value of L? The answer to this question is facilitated by the use of Eqn 15.

Suppose one wishes to compute the dimensionless half- time for a finite bath system in which €, is equal to a value x between 0 and 1, and L is equal to a value y less than infinity. Such a dimensionless parameter can be designated by Eqn 16:

[.Df t/r2], (€,=x. L = y ) = xo

where the superscript 03 indicates a finite bath system. The first step is to compute the value of Dft/r2 corresponding to M J M , = 0.5, €, = x, and L = m by iteration, using Wilson’s equation. The resulting parameter can be designated by Eqn 17:

[ Df t / r ‘1, = x1 ( E , = x . L = - )

The next step is to determine by the use of Eqn 15 the value of fsat that corresponds to f, = 0.5. Crank’s equation is then used to calculate by iteration the value of Dft/r2

JSDC Volume 107 March 1991 115

TABLE 1

Dimensionless half-times as a function of E, and L

E, L = c a L = 200 L = 50 L = 12.5 L = 3.125

0.995 0.990 0.980 0.960 0.920 0.840 0.680 0.360 0.000

3.6745 x 1.4609 x 5.7742 x 2.2557 x 8.6183 x 3.1596 x 1.0789 x 3.3029 x 6.3058 x

2.0484 x 4.9798 x 1.3036 x 3.7219 x 1.1497 x 3.7037 x 1.1761 x 3.4632 x 6.5157 x

6.4195 x 1.4144 x 3.2296 x

1.9636 x 5.2813 x 1.4625 x 3.9394 x 10-2 7.1416 x

7.7335 10-4

2.2665 x

1.0128 x 2.1999 x 4.8837 x 1.1074 x 2.5432 x 5.7790 x 9.5879 x

4.7534 10-4 8.5047 x 1.7390 x

7.4019 x 1.5357 x lo-* 3.1704 x 6.4221 x 1.2525 x lo-’ 1.8735 x lo-’

3.5775 10-3

corresponding to fsat, E , = 0 , L = y, designated by Eqn 18:

where the superscript ‘sat’ indicates an infinite bath system. Crank’s equation is then used for one final iterative calcula- tion to determine the value of Dft/r2 corresponding to the previously determined value of fsat, but for E , = 0 and L = -, designated by Eqn 19:

[Df t / r2Isa t ( E _ = O , L=-) = x3

Using the subscripted values of x in Eqns 16-19 for no- tational convenience, one can show that the finite bath dimensionless half-time, corrected for boundary layer effect, is given by Eqn 20:

xo = x1 + (x2 -x3) (20)

The technique given above has been used to calculate dimensionless half-times for various values of E, and L. These values are given in Table 1 and are believed to be accurate to at least four significant figures. Table 1 reveals that the half-times are more sensitive to boundary layer influence at higher values of E,. Non-uniform flow velocities are the norm in commercial dyeing equipment. Therefore level dyeings in these dyebaths are likely to be more difficult to achieve at high equilibrium bath exhaustion than at low equilibrium exhaustion.

The proposed technique may be used to calculate Dft/r2 as a function of E, and L for values of M t / M , other than 0.5. In fact, a three-dimensional matrix can be constructed in

which the three axes are E,, L or 1/L and Mt/M,, with the response variable being Dft / r2 . After such a matrix has been constructed, the matrix can serve as a basis for either three- dimensional interpolation or for the formulation of analytical approximations. Such approximations will permit M J M , to be estimated directly from a knowledge of D$/r2, E,, and L. Computations that involve the proposed method can be facilitated by the use of Urbanik’s recent contributions to diffusion mathematics IS].

CONCLUSION In the absence of a ready-made equation that permits the estimation of dye sorption as a function of both equilibrium bath exhaustion and boundary layer effect, the non-rigorous technique given here is proposed as an interim solution. Perhaps other, more meticulous, mathematical methods will be suggested in the future that may offer improvements to the proposed technique. But until such methods become available, it is believed that the present approach can fill an analytical void and thereby serve to assist progress in the analysis of sorption processes.

REFERENCES 1. 2.

3. 4.

5.

6. J Crank, Personal communication. 7.

8.

A H Wilson, Phil. Mag.. 39 (1948) 48. V G Levich, Physico-chemical hydrodynamics (Englewocd Cliffs: Prentice-Hall, 1962). A B Newman, Trans. Amer. Inst. Chem. Eng., 27 (1931) 203. J Crank, The mathematics of diffusion (Oxford: Clarendon Press, 1975). RMcGregor, Diffusion andsorption in fibresandfilms(NewYork: Academic Press, 1974).

R McGregor and J N Etters, Text. Chem. Colorist, 11 (9) (1979) 202/59. A Urbanik, Text. Chem. Colorist, 21 (6) (1989) 33.

Erratum In the January 1991 issue of the Journal (p. 33), it should have been mentioned that the Society is now affiliated to the South African Dyers and Finishers Association. The editor apologises for any embarrassment caused by this omission.

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