Kinetics of Cellulose Chain Scissions

The kinetics of carbohydrate degradation, observed as a viscosity drop or cellulose

chains scissions, have been studied to a much smaller extent than the kinetics of

delignification. The rate expression describing the cellulose chain scissions, derived

in Section 4.2.5.2.1, Chapter Carbohydrate degradation, can be used to calculate

the average degree of polymerization parallel to model the extent of delignification

and carbohydrate degradation. Equation (90) describes the cellulose chain

scissions as a function of the most important reaction conditions, temperature,

effective alkali concentration and time. Due to lack of experimental data, there is

only one expression for the whole pulping process [see Eq. (90)].

DPn _ t _

DPn _0 _ __ 4_35 _ 1015 _ Exp _

_ T ___ OH _ 1_77_ t _90_

For simplicity, DPv can be used instead of DPn. DPv is calculated from intrinsic

viscosity according to SCAN-CM-15:88; assuming a DPn,0 ≅ DPV = 5080 = Intrinsic

viscosity (IV) = (50800.76) · 2.28 = 1495 ml g–1 for softwood pulps (see Fig. 4.26), a

reaction time of 120 min at 160 °C (isothermal conditions), a constant [OH– ]of

0.9 mol L–1, and a chain scission [OH– ]of, CS, 1.10. 10–4/AHG can be calculated

which corresponds to a DPV of 1/(CS + (DPn,0) DPv = 1/(CS + DPn,0

–1) = 3258 =

IV = 1065 mL g–1.

4.2.5.3.4 Concentration of Cooking Chemicals, [OH– ] and [HS– ] [28]

Experimental evidence indicates that diffusion in porous wood material may limit

the pulping rates during the early stages of kraft pulping [28,35]. The transport of

chemicals and dissolved solids between the wood chips and the bulk liquor

includes diffusion processes [42]. The diffusivities of the wood components are

zero as they are bound in the wood. The diffusion of the degradation products is

not considered as the pulping reactions are assumed to be irreversible. Because of

the very low consumption rate of hydrogen sulfide throughout the cook, an equal

distribution between solid and liquid phases can be expected. The diffusivity of

sodium hydroxide, however, must be estimated. As the diffusion process is a rate

phenomenon, the diffusion coefficient may be related to temperature by an Arrhenius-

type relationship [42]. McKibbins measured diffusivity by immersing cooked

chips in distilled water and comparing the measured chip sodium concentration

dependency on time to that predicted by non-stationary diffusion theory [42]. Diffusion

was considered to be proportional to the concentration difference of the

respective alkali in the entrapped and free liquor. Hence, for a set of different temperatures,

the following expression was derived [Eq. (114)].

D _ 3_4*10_2 _ __ _ T

_ Exp _

2452_4

T _ _ _114_

where D is the diffusion coefficient (in cm2 min–1) and T was temperature (in K).

4.2 Kraft Pulping Processes 217

Because McKibbins measured cooked chips at a given level of pH and cooking

intensity, D had to be corrected with respect to these variables. This approach was

made in the 1960s by Hartler [84], and earlier by Hagglund [85] and Backstrom

[86]. These authors identified relationships between pH and the so-called effective

capillary cross-sectional area (ECCSA), and between the pulp yield and ECCSA.

ECCSA is a measure of the diffusion area in the wood chips. Thus, Eq. (114) is

corrected with respect to pH and lignin content with the ECCSA data published

by Hartler [84]. Benko reported that the diffusion of sodium hydroxide is approximately

12-fold faster than that of lignin fragments due to the higher molecular

weight of the latter [87]. The corrected value of D is expressed in Eq. (115):

D _ 5_7*10_2 _ __ T _

_ Exp _

2452_4

T _ __ _2_0 _ a L 0_13 __ OH _ 0_550_58 _ __115_

where aL is the mass fraction of lignin.

The constant, 0.057, is calculated by demanding the D-values from Eqs. (114)

and (115) to be equal at selected conditions: T = 170 °C, aL = 0.03 and [OH]=

0.38 mol L–1.

Neglecting transverse diffusion (across the fibers), one-dimensional wood chips

with thickness, s, can be considered. At the beginning of the cook (t = t0), it is

assumed that the average [OH– ]in the bound liquor (chip phase) depends on the

wood density, qdc, and the chip moisture according to the following equation:

_ OH _ bound _

Vbl _ MCd

Vbl _ ___ OH _ free _116_

where MCd [l kg–1 dry wood]= MCw/(1 – MCw), which is the average [OH– ]resulting

from diluting the penetrating liquor with chip moisture.

The volume of bound liquor, Vbl, can be estimated considering the following

simple relationship between the dry wood density, qdc, and the density of wood

substance, qw according to Eq. (117):

VV _ Vwc _ Vws _

mwc

_ dc _

mwc

_ wc _

_ dc _

1_53 _117_

where VV is the void volume, Vwc the chip volume, Vws the volume of wood substance,

mwc the mass of wood substance, and qw the density of wood substance,

which is approximately 1.53 and constant for all practical purposes, wood species

constant and even pure cellulosic material.

The degree of penetration, P, determines the volume of bound liquor, Vbl, using

Eq. (116):

Vbl _ P _ VV _118_

P is nondimensional.

In case of full impregnation, Vbl equals VV.

218 4 Chemical Pulping Processes

Example:

Given: P = 1; qdc = 0.45 kg L–1; MCw = 0.45 L kg–1; [OH]free = 1 mol L–1.

Result: VV = Vbl = 1/0.45 – 1/1.52 = 1.56 L kg–1; MCd = 0.45/(1 – 0.45) = 0.82 L kg–1

[OH]bound = {(1.56 – 0.82)/1.56}. 1 mol L–1 = 0.47 mol L–1.

During the course of the heating-up period, the pulping reactions start. The

generated degradation products neutralize alkali as they diffuse out of the chips.

Alkali is transported from the bulk phase to the boundary layer, and then diffuses

into the chip to replace the sodium hydroxide consumed by the degradation products.

Diffusion of alkali within the porous chip structure obeys Fick’s second law

of diffusion. The one-dimensional wood chip is divided into 2n slices with the

width, Dh = s/2n. For calculation of the gradient of the [OH]within the chip, a

value of 20 was chosen for n (see Fig. 4.32).

Fick’s second law of diffusion for a one-dimensional chip is expressed as:

∂ C

∂ t _ z _ t _ _

∂

∂ z _ D

∂ C

∂ z _ z _ t __ _ Ra for t _ 0_ 0 ≤ z ≤ s _2 _119_

where C is the alkali concentration (mol L–1), D the coefficient of diffusion

(cm2 min–1) and Ra the reaction rate of [OH], (mol L–1. min–1).

The initial concentration is given as

C _ z _ 0_ _ C 0_ z _ for 0 ≤ z ≤ s _2 _120_

constant under assumption in Eq. (114).

-20 -10 0 10 20

S/2

[OH-]170°C

[OH-]125°C

[OH-]80°C

chip centre boundary

layer

boundary

layer

-S/2

Äh

Fig. 4.32 Schematic of a one-dimensional chip

model; s = chip thickness; 2n slices with width

Dh = s/2n; the gradient of bound [OH– ]across

the chip has been calculated for three cooking

stages at: the start of the cook at 80 °C (solid

curve); after reaching 125 °C in the heat-up time

(broken curve); and at the start of the cooking

phase at 170 °C (dotted curve).

4.2 Kraft Pulping Processes 219

In the chip center, the plane of symmetry, no concentration gradient occurs at t

> 0, as expressed in Eq. (121):

∂ C

∂ z _0_ t _ _ 0 t _ 0 _121

The mass flow rate at the solid–liquid interface is considered as proportional to

the concentration difference of the free and entrapped liquor, and must equal the

mass flow rate which is expressed by Fick’s first law of diffusion according to

Eq. (122):

D

∂ C

∂ z _

s

_ t _ _ k _ Cbulk _ t _ _ C _

s

_ t __ _122_

where k is the mass transfer coefficient.

As it is assumed that the bulk phase is homogeneous and well-stirred, the hydroxide

ion concentration in the free liquor has no concentration gradient and is

described as Cbulk. According to the conservation of mass, the mass of free liquor

decreases by the mass passing through the boundary layer. Thus, the balance on

the bulk phase gives the following equation:

dCbulk Vbulk

dt _ t _ _ _ AC D

∂ C

∂ z _

s

_ t _ _123_

Vbulk is assumed to be the constant volume of the free liquor, and AC the surface

area where the mass transfer takes place. The total area is double the circular

(chip) area, AC = 2r2p. The disk (chip) volume calculates to Vchip= r2ps, so that with

AC = 2Vchip/s, Eq. (123) leads to:

dCBulk

dt _ t _ _ _

2 VChip

sVBulk

D

∂ C

∂ z _

s

_ t _ t _ 0 _124_

The system is fully described by Eqs. (119–124). A method for the numerical

solution of the differential equations to determine the [OH– ]across the chip thickness

is described in the Appendix.

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