Diffusion

In pulp washing operations, diffusion controls the exchange of dissolved substances

between the free-flowing liquor outside the fibers and the immobile liquor

inside the pulp fibers or locked in between fibers. Physically, diffusion is defined

as the net transport of molecules caused by their random thermal motion in an

attempt to equalize concentration differences.

The basic equation describing the mass transfer related to diffusion is Fick’s

First Law, which states that the flux of a diffusing substance is proportional to its

concentration gradient. In its one-dimensional form (see Fig. 5.5), Fick’s First

Law reads:

J _ _ D

∂ c

∂ x _8_

where J = flux of the diffusing substance (kg m–2 s–1); D = diffusion coefficient

(m2 s–1);and ∂ c /∂ x = incremental change of concentration with distance (kg m–4).

5 Pulp Washing

Fiber wall

J

Fig. 5.5 Simplified illustration of mass transfer by diffusion across the wall of a pulp fiber.

On a macroscopic scale, ∂ c /∂ x in our simplified system expresses the concentration

gradient of the diffusing substance across the fiber wall. A higher concentration

difference between inside and outside the fibers gives a higher flux of the diffusing

substance.

In pulp washing, the diffusion coefficient D is dependent upon the type of

wood furnish, the lignin content in the fiber wall, the pH and ionic strength of

the liquor, the temperature as well as the diffusing substance itself. The diffusion

coefficient decreases with higher lignin content in the fiber wall, whereas it

increases with rising temperature. Smaller molecules diffuse more easily, and

therefore have higher diffusion coefficients than larger molecules [1,14,15].

Diffusion takes time until the concentrations equal out. According to Fick’s Second

Law, the time rate of concentration change is again dependent upon the diffusion

coefficient:

∂ c

∂ t _ D

∂2 c

∂ x 2 _9_

Hence, the time-dependent equalization of concentrations is influenced by the

same factors as the diffusion flux. For example, the small sodium ion diffuses

very rapidly, and the equilibrium concentration inside and outside the fibers is

reached within seconds, whereas it may take hours to reach the equilibrium concentration

for the larger dissolved lignin molecules [15,16].

The time required for diffusion to occur must be provided by the practical

design of a pulp washing system. It is essential to allow dissolved substances to

pass from within the fibers to the surrounding liquor. Otherwise, the lack of time

for diffusion can substantially deteriorate the washing result, especially in cases of

large diffusing molecules and high initial concentration levels within the fibers,

such as after digesting or oxygen delignification.

5.2 Pulp Washing Theory

5.2.5

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