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The solution of Eqs. (68) and (69) is calculated numerically by a finite difference
scheme. The origin of the coordinate system is located at the cylinder center. For
the radial concentration profile [Eq. (68)], the interval [–a, a] is divided into 2n
pieces of equal size Dh= a/n, for the axial concentration profile [Eq. (69)]the interval
[–Z/2, Z/2]is divided into 2n pieces of size Dh = Z/2n. Ci denotes the concentration
at 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
_ _70_
4.2 Kraft Pulping Processes 163
To obtain an approximation for a second-order derivation, the second-order derivation
is replaced by a central difference quotient of first-order derivations, after
which the first-order derivations are replaced by central difference quotients.
The resulting difference equations for Eq. (68) are
_ C
i _ t _≈ Dr
D h 2 __1
2 i _ Ci 1_ t _ _ 2 Ci _ t _ Ci _1_ t ___1 _
2 i __ i _ 1_ _____ n _ 1 _71_
and
_ C
i _ t _≈ Dz
D h 2 _ Ci 1_ t _ _ 2 Ci _ t _ Ci _1_ t __ i _ 1_ _____ n _ 1 _72_
for Eq. (67).
The condition that C is finite at r = 0 in Eq. (68) implies ∂ C
∂ r _0_ t _ _ 0 for t > 0
and symmetry of problem Eq. (69) implies ∂ C
∂ z _0_ t _ _ 0 for t > 0. After approximation
of C2, C1,C0 with a quadratic polynomial this conditions transform into:
C 0_ t _≈ 4
C 1_ t _ _
C 2_ t _ _73_
After inserting Eq. (73) into Eqs. (71) and (72) respectively, a system of ordinary
differential equations (ODE) is obtained which can be solved by any standard
numerical ODE solver with good stability properties.
Euler’s implicit method is used in the sample code. Only a set of linear equations
with a tridiagonal system matrix is solved each time step.
4.2.4
Chemistry of Kraft Cooking
Antje Potthast
Lignin Reactions
In pulping operations the lignin macromolecule must be degraded and solubilized
to a major extent. Inter-lignin linkages are cleaved and the fragments dissolved
in the pulping liquor. The reactivity of different lignin moieties towards
pulping chemicals and pulping conditions is highly dependent on the chemical
structure. An understanding of the major reactions of lignin moieties has been
established by experiments applying low molecular-weight model compounds featuring
lignin substructures, assisted by modern analytical techniques. The reactivity
of lignin subunits differs most notably depending on whether the phenolic
units are etherified, or not. In general, the reactivity of phenolic moieties is signif-
164 4 Chemical Pulping Processes
icantly enhanced over nonphenolic lignin units. In the following sections, the
reactions of the more reactive phenolic units under alkaline pulping conditions
will be addressed.
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