This is a modified excerpt from a recent report prepared by IETek for a study of the overpressure protection safety of batch (Kraft and Sulfite) digesters. The following is provided as an example of technologies with which IETek can help its customers. For more information please contact IETek.
PROCESS MODELING BASICS for BATCH DIGESTERS
(a brief introduction)
©
IETek 2002
DIGESTER
MODELING and PROCESS ANALYSIS
The
model developed for this project was based on the fundamental principles of
material and energy balances and the laws of equilibrium thermodynamics.
Throughout the report theoretical and technical jargon are kept to a minimum
level while emphasizing reasoning for basic underlying issues in a style
hopefully understandable by all potential readers.
The
focus of modeling is to predict vessel pressure as a function of all other
process related variables. Due to the very nature of batch processes in general,
the model is transient or dynamic in nature. To the
extent possible the model is designed to look and behave like the real process
so that it can provide predictive and interaction capabilities that relate the
influence of key variables to overpressure prediction. Figures 7 and 8 provide
two complimentary summary descriptions of the model.
The primary assumptions are:
(a) thermodynamic equilibrium exists between liquid and vapor phases, (b) ideal
gas law and Dalton’s law of additive partial pressures are applicable, (c)
compressible vapor phase is the combination of chip pore volume not occupied by
free liquor and the freeboard space, (d) chip compaction is determined by
established correlations, (e) for heat of reaction considerations cooking rates
can be estimated by H-factor based correlations, (f) steam enthalpy and fluid
properties are given by established steam tables, (g) all direct steam
introduced into the digester condense, transfer heat and equilibrate with its
environment, (h) approximations of constant physical properties like specific
heats are suitable for the purposes of this project, (i) relative liquid density
variation with temperature is governed by data provided for saturated water, (j)
volume of mixing is zero or negligible.
Ideal
gas law is given by the traditional equation
(1)
where
p = pressure [kPa], V = volume [m3], n = number of moles, R = ideal
gas constant = 8.314425 kPa m3/(0K kmol), T = temperature
[0K].
Another
form of the ideal gas law that is more useful for modeling purposes is
(2)
where
r
= density [kg/m3] and Mw = molecular weight [kg/kmol].
At
any given time there are three possible phases in the digester: chips, liquor
and vapor. Total vapor is the combination of freeboard volume and the pore
volume in chips that are not already occupied by free liquor. Dynamic material
balances are required for each phase. Considering the fixed digester volume as
the reference, formulations of the material balances for each phase are
It
is worth stating here that SO2 equilibrium between liquid and vapor
phases are governed by the experimental data provided by Perry’s 7th
ed Chemical Engineering Handbook (1997). Water vapor/liquid equilibrium follows
saturation values available in any steam tables. For this work, Perry’s
handbook and Himmelblau (1974) are used. For both SO2 and water
relationships tabulated values are numerically interpolated in the model to
estimate present conditions for a given temperature.
Dynamic
heat transfer equations are also formulated in a similar pattern to material
balances. Due to thermal and phase equilibrium assumptions there are only two
enthalpy balance equations that are pertinent for the model, one for the vessel
walls and the other for the complete contents of the digester.
(4)
Block diagram representation of modeling approach.
Thermal
capacity is defined as the total heat content or Σ[Mass*Cp*(T-Tref)],
where Cp is heat capacity, Tref is reference temperature for enthalpy
calculation and Σ [..] represents the additive contributions of each phase
present at any particular time. Enthalpy content of vapor is a function of
temperature and pressure and significantly different than liquid. Due to this
difference, there is an evaporative cooling effect as vapors escape the digester
(venting) and as additional vapor is produced to make up the difference in vapor
phase. Material and energy balance equations combined with equilibrium relations
help us capture and quantify this important phenomenon in the model. As it
should be obvious, when air and vapor mixture in a gas phase is vented
proportionate volumetric amounts of gases are released. Although total air
content in the vapor phase decreases, the vapor deficiency is quickly
compensated through evaporation from the liquid phase. An important challenge of
the modeling equations and the algorithm is to maintain these delicate balances
between material and energy transfers while keeping thermodynamic equilibria
satisfied.
Chip characterization is an
important aspect of modeling. Wet chips are composed of solid fiber, pore
volume, bound water that swells the fiber matrix and adsorb into fibers, free
liquor occupying space in the pores, and vapor and air in the remainder pore
volume in equilibrium with the liquid phase. As chips undergo digestion
reactions fibers lose mass. When chips are contained within liquor, free liquor
diffuses into pores and eventually fully occupies chip pore volume. The model
keeps track of the fraction of chip pores that are filled, or conversely
available for vapor holdup, at any time during the cook cycle. Naturally, vapor
space within chip pores behaves similarly to freeboard in terms of accommodating
a compressible gas mixture in equilibrium with the liquid it is in contact with.
Chip
compaction under pressure is given by Harkonen (1987) correlation as
(5)
where
h
is the fractional volume of space between chips compared to total space
occupied, and called the compaction. Kappa # is the traditional measure of the
extent of reaction that signifies the fraction of lignin remaining in the pulp.
In this work, the initial (loose or stationary) compaction for each mill
application was experimentally determined from chip samples collected and
substituted for the 0.644 term of the correlation. Specific numbers used for
each mill are different and listed separately in Appendix 6. Other
experimentally determined pertinent chip characterizations like porosity and
bulk density are also listed in the appendices. In eq.(5) P is the pressure felt
by chips during compaction in kPa units.
H-factor
is a commonly used measure to describe the extent of cook. K.E. Vroom (1957)
proposed it as
(6)
where
t is time in hours and T is in oK.
Dalton’s
law of additive partial pressures provides the necessary basis for computing
total pressure in a known volume of known mass quantities at a given
temperature. Partial pressures are computed either from the ideal gas law or
from equilibrium relations. The thermodynamics of water liquid-vapor equilibrium
as provided by steam tables are for pure conditions. For real cases as we have
it in batch digesters, where additional components besides water are present,
Rault’s law provides some guidance about the expected change in equilibrium
partial pressure of water vapor. It is called the lowering or the depression of
vapor pressure by non-volatile solutes. The non-volatile solute in the liquor of
a digester is the combined agglomerate of dissolved solids that are produced by
digestion reactions. The Rault’s law states that
(7)
where
the reduction in vapor pressure is proportionally adjusted by the mole fraction
of the solute, xs. The computations from Eq. (7) can be converted to
the equivalent of elevation of boiling point in the presence of a solute, which
for water results in approximately 0.5130C/molal solute. These
theoretical results would have been directly applicable to digester modeling if
we knew exactly what the dissolved solids are. The dissolved solids are the
ingredients that make black liquor when concentrated and have no known simple
chemical characterization. Therefore, for practical reasons we use Equation (7)
in a generic form as
(8)
where
Φ is simply a correction factor for pure component vapor pressures.
Considering the theoretical guidance of Equation (7), Φ should be a time
dependent correction as the solute concentrations increase during the cook
cycle. For practical reasons and in the absence of
appropriate high fidelity and high frequency (laboratory) thermodynamic data,
the model simply uses a constant value of Φ ~ 0.925, which agrees rather
well with the experimental evidence.
For additional information or questions please contact
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