## Bke

Calculation of bed properties

Cpb = (spa(Cpa +fX) + (1-s)psCps)/pb at z = 0 T = Tn

Initial values at t = 0 T = To 0 <z <H at t = 0 X = Xo 0 <z <H

Fig. 24.1. Summary of the model used in the first packed-bed case study. The subscripts for the parameters p, k, and Cp are "s" for substrate particle, "a" for air, and "b" for the weighted average calculated for the bed

24.2.1 Synopsis of the Mathematical Model and its Solution

The model is based on the work of Sangsurasak (Sangsurasak and Mitchell 1995, 1998). It is almost identical to the version used by Mitchell et al. (1999), although the equation describing the effect of temperature on growth has been changed. The full model is not reproduced here; Fig. 24.1 summarizes its main features. Growth occurs according to the logistic equation, where the specific growth rate constant is affected by the temperature of the solid in a manner identical to that described by Eq. (22.1) (see Sect. 22.2). The energy balance takes into account axial convection, conduction, and evaporation and the production of metabolic heat. These two differential equations allow the biomass and temperature to be predicted as functions of time and space. The fact that the model is so simple means that it has many implicit assumptions and simplifications. Amongst these, some of the most important are:

• Growth depends only on biomass density and temperature. The bed does not dry out sufficiently during the fermentation to limit the growth;

• The bed is treated as a single pseudo-homogeneous phase that has the average properties of the gas and solid phases;

• Biomass does not move in space;

• Growth does not affect the void fraction;

• The substrate bed properties do not change with temperature or during consumption of substrate and production of biomass;

• Flow phenomena arising from increased pressure drop are not important;

• The air is always in thermal and moisture equilibrium with the solid (i.e., as the air heats up as it passes through the bed, water evaporates from the solid to maintain saturation of the air).

Table 24.1. gives the values of parameters, initial values of state variables and values of operating variables used in the base-case simulation. Note that some parameters are considered as constants even though they are not truly constant. For example the heat capacity of the air will change as the air passes through the bed, due to the increase in water content.

The mathematical model summarized in Fig. 24.1 contains partial differential equations. This model is solved by application of orthogonal collocation to convert each partial differential equation into a set of ordinary differential equations, which can then be integrated numerically. The principles of orthogonal collocation are beyond the scope of this book.

Note that the differential term dHsaJdT in the energy balance is given by Eq. (19.20) (see Sect. 19.4.1). It has been used to replace the constant value of 0.00246 kg-H2O kg-dry-air °C-1 used by Mitchell et al. (1999).

Table 24.1. Values of the parameters and variables used for the base case simulation

Symbol Significance

Base case value and unitsa'

Design and operating variables (can be varied in the input file for the model)

Table 24.1. Values of the parameters and variables used for the base case simulation

Design and operating variables (can be varied in the input file for the model)