Appendix

Numerical Solution of the Kinetic Model

The numerical approximation for the solution of Eqs. (119–124) is calculated by a

finite difference scheme. After replacing the spatial derivations with difference

quotients, a system of ordinary differential equations for the concentration C at

discrete points is obtained.

The origin of the coordinate system at the chip center is located and the onedimensional

wood chip is divided into 2n slices with the width Dh = s/2n. Ci

denotes the concentration at height iDh; thus, Ci(t) = C(iDh,t). The derivation of a

smooth function can be approximated by a central difference quotient

df

dx _ x _ ≈ f _ x h _ _ f _ x _ h _

2 h

_ _126_

The difference quotient is applied consecutively in Eq. (119), with h= Dh/2 obtaining

the following difference equations

_ C

i _ t _ ≈ D

D h 2 _ Ci 1_ t _ _ 2 Ci _ t _ Ci _1_ t __ Rai i _ 1_ _____ n _ 1 _127_

To simplify expressions, it was assumed that D does not depend on the spatial

direction; the general case, however, can be solved using the same principle.

After approximating C2, C1, C0 with a quadratic polynomial and minding

[Eq. (121)], we obtain

C 0_ t _ ≈ 4

C 1_ t _ _

C 2_ t _ _128_

The same approximation for Cn, Cn – 1, Cn – 2 results in

∂ C _ s _2_ t _ ∂ z ≈ 1

2D h _3 Cn _ t _ _ 4 Cn _1_ t _ Cn _2_ t __ which, after combining with

Eqs. (122) and (124), yields

Cn _ t _ ≈ CBulk _ t _ _

D

k 2D h _3 Cn _ t _ _ 4 Cn _1_ t _ Cn _2_ t __ _129_

and

_ C

Bulk _ t _ ≈

VChip D

s VBulk D h _3 Cn _ t _ _ 4 Cn _1_ t _ Cn _2_ t __ _130_

Equations (127–130) define a system of differential algebraic equations (DAEs).

After elimination of C0(t) and Cn(t) by inserting Eq. (128) and Eq. (129) into Eqs.

228 4 Chemical Pulping Processes

(127) and (130), the DAEs simplify to a system of ordinary differential equations

(ODE) which can be solved by any standard numerical ODE solver that has good

stability properties, for example, an implicit Runge Kutta method. Euler’s – which

has excellent stability properties – is used in the sample code, and although a set

of linear equations must be solved for every time step, the method is very fast

because the system matrix is almost trigonal.

4.2.6

Process Chemistry of Kraft Cooking

Standard Batch Cooking Process

In standard batch cooking, the whole amount of chemicals required is charged

with the white liquor at the beginning of the cook. Certain amounts of black

liquor are introduced together with white liquor to increase the dry solids content

of the spent liquor prior to evaporation. The concept of conventional cooking

results in both a high concentration of effective alkali at the beginning of the cook

and a high concentration of dissolved solids towards the end of the cook which,

according to kinetic investigations, is disadvantageous with respect to delignification

efficiency and selectivity.

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