The cellulose predictions are well within experimental error, as shown in
Fig. 20.
Particularly encouraging is
that the cellulose fractional yield at
time zero, 1.042, which was fitted using the reaction kinetics study (southern
pine) data, works quite we
TIME, HOURS
Fig.
18.
Reaction kinetics study CU/4 lignin vs.
Series CV simulated two runs from Aurell and Hartler. 5 3
time.
The study was one
of the first to report carbohydrate data along with the usual lignin and yield
data.
Estimating chip size was a
-45-
The method of lines reduces the set of nonlinear partial differential
equations to a stiff set of nonlinear ordinary differential equations, which is
solved using DGEAR from the IMSL library. 5 2
For example a 3 x 3 x 3 grid re-
places the system of
-42-
able to condense to lignin or carbohydrate; models with a condensation rate dependence of cfw_[lignin] x [dissolved lignin] were inferior to models with a condensation rate dependence of [dissolved lignin].
Reaction schemes which include
explicit con
-43-
numerical solution is a polynomial function of the distance between the grid
points (grid spacing).
For problems with known analytical solutions, the numeri-
cal solution rapidly approaches the analytical solution as the grid spacing decreases.
For n
-44-
accuracy is desired;
the number of points required in general is the sum of the
order of accuracy required plus the order of the derivative estimated.
Even
though only five points in each direction are shown in shadow, six values are
available in the
-41-
where L is lignin and H is hemicellulose.
The transition between initial and
bulk phases (Ht) was fitted along with the other parameters in the glucomannan
and xylan models above.
and 0.868 for xylan.
The fitted transition values are 0.752 for glucom
-40Pulp viscosity 18 was best modeled as a single pathway reaction driven by NaOH:
where Up [=] cp.
Equation (57) is consistent with the experimental observation
of Kubes et al.16 that plots of 1/Up vs. time are linear.
The constant 1.268
represents pulp
The activation energy for xylan peeling due to NaSH was fixed to a value expected
for diffusion controlled reactions after attempts to fit the activation energy
resulted in values approaching zero.
The stopping reaction was modeled as two parallel reactio
-38-
The three carbohydrate fractions (cellulose, glucomannan, and xylan) were best
modeled by the reaction network shown in Fig. 15.
NaOH
N
D
The carbohydrate reaction network can be summarized as
where C is cellulose, glucomannan, or xylan, N is the nat
-36-
Each model represents a mechanistically plausible reaction scheme.
For
instance, carbohydrate models accounting for just the peeling reaction were considered as well as models accounting for simultaneous peeling, stopping, and
cleavage.
The reaction
where N is native lignin, R is residual lignin,
dissolved solids, and t [=] hr.
D is dissolved lignin,
DS is
The bulk delignification phase rate constant
kbulk was calculated as the sum of three parallel reactions, dependent on NaOH,
NaSH, and AQ, respect
-33-
Model discrimination criterion D increases our knowledge as to which model
is correct.
D is essentially the sum of squares of the differences in model
predictions between all unique pairs of models [weighted by functions of relative probability (n) a
-34The models were fitted using NONLINWOOD, a nonlinear regression program.5 0
I wrote a program named MDPE (Model Discrimination/Parameter Estimation) to
A sample data deck, sample output, and the source code listing for
optimize C.
MDPE are presented in
-32-
to approximate cfw_heat capacity x density of the wood-liquor mixture, giving an
expression for heat generation:
RT = -13.345 (NaOH]/at
(42)
where RT [=] K/hr and a[NaOH]/at [=]M/hr.
REACTION KINETICS STUDY
A reaction kinetics study was undertaken to
-31-
Thermal Diffusion
Unlike chemical diffusion, thermal diffusion in wood is isotropic.
This is
because the thermal diffusivity of wood is nearly identical to the thermal diffusivity of water.4 4
sivity (DT).
This allows the use of tabulated data for wa
Diffusion Rates in Wood
Diffusion in wood was modeled as the product of bulk liquor diffusion and
ECCSA.
ECCSA vs. pH at 100% yield for spruce 5
Two sets of data were used:
(Fig. 1), and ECCSA vs. yield at pH 13.2 for pine 5 (Fig. 2).
Gustafson 3 2 fit
th
-30-
Equations (30) to (32) are for yields between 100 and 65%; Eq. (33) to (35) are
for yields below 65%.
The regions in the yield - [NaOH] plane where ECCSA is
known are summarized in Fig. 9.
Combined formulas for ECCSA (longitudinal, radial, and tangen
-28-
where R is the gas constant, F is the Faraday, n+ is the cation valence, n-is
the anion valence, 1+ is the cationic limiting conductance, and 1- is
anionic limiting conductance.
the
The diffusivity of AQ (0.650 x 10- 9 m2 /s) was
estimated from the W
-27-
PARAMETERS AND THEIR SOURCES
Bulk Liquor Diffusion Coefficients
The diffusion rates of all species are calculated from the Stokes-Einstein
equation 3 4 for the diffusion of a sphere in a continuous solvent:
where kB is Boltzmann's constant, Us is sol
-26-
where u is the concentration of diffusing species, D is
diffusivity, R is reac-
tion rate, and x, y, and z are mutually orthogonal axes within the chip.
The
axes are arbitrarily aligned with the longitudinal, radial, and tangential planes
in the wood
-23-
RESULTS AND DISCUSSION
APPROACH
A dynamic model of the kraft pulping process was developed in three stages.
The first two stages were conducted in parallel.
Preliminary numerical analysis
work was done to identify a satisfactory numerical solution me
-24-
necessitated either by continued ignorance of certain aspects of the kraft process or by insufficient computational resources.
The assumptions and the
rationale for the assumptions are given below.
1.
The chip is completely filled with liquor or wate
-25-
formed during the chipping process.
The chip model would prob-
ably significantly underpredict the chemical concentrations at
the center of a heavily fissured chip.
3.
The chip is a homogeneous anisotropic gel under pulping conditions.
This assumptio
-21-
Anthraquinone is an effective catalyst with limited applicability due to its
high cost.
Including AQ would facilitate efforts to find optimal conditions for
its use.
I believe there is a need for a more fundamental model of the kraft pulping
process
-22-
THESIS OBJECTIVES
Since the introduction of the kraft pulping process, knowledge pertaining to
the effects of diffusion limitation of pulping rates on pulp properties has been
restricted mostly to large scale effects - increased rejects, reduced yiel