Diffusion. All transfer of cooking chemicals into chips, and dissolved matter from the chips,

All transfer of cooking chemicals into chips, and dissolved matter from the chips,

will occur through diffusion only after complete penetration. Consequently, molecular

diffusion is a very important step in chemical pulping.

To understand the process of impregnation, much effort has been made to follow

the distribution of the active cooking chemicals within the void structure of

the wood by both experimental studies and theoretical considerations. McKibbins

contributed the first rather complete description of the diffusion of sodium ions

in kraft-cooked chips [34]. He measured the diffusivity of sodium ions by immersing

the cooked chips in distilled water, and compared the measured chip sodium

concentration as a function of time to those predicted by unsteady-state diffusion

theory.

Data were obtained for extraction in the transverse and longitudinal directions of

the wood for several temperatures and sample thicknesses.

Unsteady-state and unidirectional and isothermal diffusion in one dimension

may be described by Fick’s second law of diffusion according to Eq. (43):

∂ c

∂ t _ D

∂2 c

∂ x 2 _43_

It is assumed that diffusion occurs through a homogeneous material of constant

width or thickness, L, with an initial concentration ci, and that the solute leaves at

138 4 Chemical Pulping Processes

both faces which are maintained at a constant concentration, c0. There is no single

integral solution to this differential equation, but a variety of solutions have been

derived depending on the boundary and other conditions [35].

Equation (43) may be solved by applying the following initial and boundary conditions:

_ Initial conditions: t = 0, c = ci for all x

_ Boundary conditions: x = 0, L, c = c0 for t > 0

_ c0 concentration of sodium ions outside the chips

_ ci concentration of sodium ions inside the cooked chips

Considering these initial and boundary conditions yields Eq. (44):

Y _

c _ c 0

ci _ c 0 _

p2_

∞

n _1

_2 n _ 1_2 Exp _ __2 n _ 1_2p2 D _ t

L 2 _ _ _44_

where Y equals the average fraction of unextracted sodium ions.

The solution of this infinite series reduces to a single term for values of

(D· t · L–2) > 0.03, which is accomplished when Y < 0.6. In this very likely case,

Eq. (44) reduces to Eq. (45):

Y _

c _ c 0

ci _ c 0 _

p2 _ Exp _ _

p2 D _ t

L 2 _ _ _45_

According to Eq. (45), the diffusion coefficient can be determined from the slope

k of the linear correlation obtained by plotting the natural logarithm of Y against

time. The diffusion coefficient D can thus be calculated using Eq. (46):

D _

k _ L 2

p2 _46_

In case diffusion occurs in more than one direction, Eq. (43) must be expanded to

include the new coordinates. It has been shown that for certain geometries and

sets of boundary conditions, the solution for multidirectional diffusion is the

product of the solution for unidirectional diffusion for each of the coordinates

involved. Considering diffusion in the x, y, and z directions, the average concentration

of a rectangular parallelepiped will be equal to the product of the concentrations

obtained for each of these directions according to Eq. (47):

Yav _ Yx _ Yy _ Yz _47_

The diffusion coefficient, D, is determined by plotting the logarithm of the fractional

residual sodium content against the diffusion time. The values of the diffusion

coefficients are determined from the slope of the straight-line portion of

these curves (Fig. 4.8).

4.2 Kraft Pulping Processes 139

0 10 20 30

0,1

0,5

38 °C 71 °C

Fractional residual sodium, C/C

i

Diffusion time, min

Fig. 4.8 Residual sodium fractions versus extraction or

diffusion time for unidirectional longitudinal diffusion from

0.3175 cm-thick wood chips (according to McKibbins [34]).

The diffusion coefficients in longitudinal directions at 38 °C and 71 °C can be

determined by putting the calculated slopes determined from the natural logarithm

of Y against extraction or diffusion time in Fig. 4.8 into Eq. (46) according

to Eq. (48):

D 38_ C _ longitudinal _

6_95 _ 10_4 s _1 _ 3_175 _ 10_3 _ _2_ m 2

p2 _ 7_1 _ 10_10 m 2 _ s _1

D 71_ C _ longitudinal _

1_86 10_3 s _1 _ 3_175 _ 10_3 _ _2_ m 2

p2 _ 19_0 _ 10_10 m 2 _ s _1 _48_

Since the diffusion process is a rate phenomenon, D may be related to the temperature

by an Arrhenius-like relation. An associated activated energy, EA, is required

according to Glasstone, Laidler and Eyring to elevate the diffusing molecules to

that energy level sufficient to initiate molecular transport [36]. The diffusion coefficient

may thus be related to the temperature in the following manner [Eq. (49)]:

D _ A _ __ T _

_ Exp _

EA

R _ T _ _ _49_

A plot of the natural logarithm of the ratio D to the square root of the absolute

temperature against the reciprocal of the absolute temperature results in a

straight line with a slope dependent on the activation energy, EA. The experimen-

140 4 Chemical Pulping Processes

tal results obtained for the diffusion coefficients and activation energies for both

longitudinal and transverse direction are summarized in Tab. 4.11.

Tab. 4.11 Diffusion coefficients as a function of temperature and

activation energies according to [34](recalculated).

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