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.




(a brief introduction)

© IETek 2002







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 Daltonís 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




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




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 Perryís 7th ed Chemical Engineering Handbook (1997). Water vapor/liquid equilibrium follows saturation values available in any steam tables. For this work, Perryís 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.





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




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


where t is time in hours and T is in oK.


Daltonís 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, Raultís 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 Raultís law states that




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



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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© IETek 1996-2002, all rights reserved.


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