Pseudo First-principle Models

More elaborate models consider different reaction conditions such as hydroxide

ion and hydrogen sulfide ion concentrations, temperature, and classification into

different pulping stages or the different reactivities of wood components [7,15,

27–35]. Some models even consider chip dimensions, ion strength, dissolved solids

and pulping additives such as anthraquinone or polysulfide [31,36–39]. The

more comprehensive and partly mechanistic models use either the concept of

parallel or consecutive reactions of the wood components [7,12]. The former is

often referred to as the Purdue model, derived at Purdue University, USA [40],

while the latter was developed primarily at University of Washington, USA, with

Gustafson as the prominent representative [28].

Both models, however, can be traced back to the work of Wilder and Daleski

[36], Kleinert [41], LeMon and Teder [27] and Olm and Tistad [35]. The Purdue

model, which was developed by Smith and subsequently modified by Christensen,

includes an in-depth description of the main wood components as high- and lowreactive

lignin (HR-L, LR-L), cellulose (C), the hemicellulose components galactoglucomannan

(GGM) and arabinoxylan (AX), dissolved solids, and extractives.

The latter are assumed to react very quickly, while the digester is being filled with

cooking liquor. Hydroxide ions and hydrogen sulfide ions are considered to be

active species of the cooking liquor.

A plot of the non-lignin constituents removed during pulping against lignin dissolved

indicates the initial, bulk and residual delignification pattern for both kraft

and soda processes (see Fig. 4.22). The shape of diagram is almost independent of

the operating conditions, type of wood or equipment. The kinetics of dissolution

of the five wood components is expressed as a set of first-order ordinary differential

equations according to the general equation:

dXj

dt _ k ′_ OH __ k ″_ OH __ a _ HS __ b _ ___ Xj _ Xj _0_ _94_

where X is the concentration of the wood components j, Xj, 0 is the unreactive portion

of the reactant j, k′ and k” are rate constants as a function of temperature

using the Arrhenius equation.

Further, mass transfer equations were provided to calculate differences in the

concentration of caustic and hydrogen sulfide between the free and entrapped

liquors [42]. The model also consists of mass and energy balances formulated as

first-order ordinary differential equations that are solved in 50 compartments

along the digester [43]. The Purdue model considers only the initial and bulk

198 4 Chemical Pulping Processes

stages. Recently, Lindgren and Lindstrom reported that the amount of lignin in

the pulp at the transition from bulk to residual delignification is decreased by

increasing the [OH– ]concentration in the final stage and the [HS– ]concentration

in the bulk phase. They also concluded that the residual lignin is not created during

the main part of the cook but exists already at the beginning of the bulk phase.

Moreover, the amount of residual lignin is also decreased by reducing the ionic

strength in the bulk phase [33]. The corresponding kinetic model can be assigned

as a development of the Purdue model because it also uses the concept of parallel

reactions. Lindgren and Lindstrom (and later also Andersson) pointed out that the

existence of unreactive wood components is questionable because the amount of

residual phase lignin can be lowered using reinforced conditions [44]. A comprehensive

model based on the concept of three consecutive reaction steps was developed

by Gustafson et al. [28]. This concept, however, suffers from the abrupt transitions

between the three phases, initial, bulk and residual. The wood components

are divided into two components, lignin and carbohydrates. The separation of carbohydrates

into cellulose and hemicellulose was accomplished by Pu et al. [45].

The rate expression for the initial period is taken from Olm and Tistad [35], for

the bulk delignification from LeMon and Teder [27]and the form of the residual

delignification originates from Norden and Teder [46]. The rate expressions [Eqs.

(95–97)]are:

Initial stage:

dL

dt _ 36_2*__ T _* Exp __4810_ T _* L _95_

Bulk stage:

dL

dt _ k 1_ OH _ k 2_ OH _ 0_5 S _ 0_4 _ _ L _96_

Residual stage:

dL

dt _ Exp 19_64 _

T _ __ OH _ 0_7* L _97_

Further, the calculation of carbohydrates is simplified by using a functional relationship

between the time derivative of carbohydrates and the time derivative of

lignin. Thus, the carbohydrate degradation is always a ratio of the lignin degradation,

which is an inadmissible assumption following the results from Lindgren [47].

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