So models can and should play a central role in the development of large-scale SSF bioreactors. The remainder of this chapter gives an overview of the structure of mathematical models and the manner in which they are developed. The aim is not to teach those readers who do not have a background in modeling how to construct and solve models, but rather to increase their ability to interact with a modeling expert in the modeling process.

The structure of a model is presented in terms of a case study of a simple model of a well-mixed SSF bioreactor. Figure 12.3 shows the bioreactor, highlighting the various phenomena described by the model. Figure 12.4 shows the equations of this model, highlighting the fact that mathematical models of bioreactors contain two parts: the kinetic sub-model describes microbial growth kinetics, while the balance/transport sub-model describes transport phenomena and overall mass and energy balances. Work must be undertaken to generate data for both parts of the model.

Various symbols appear in these equations, representing different quantities. These quantities are of fundamentally different types, or, in other words, the various symbols represent a range of state variables, independent variables, operating variables and parameters. These are defined below.

State variables. These represent variable properties of the bioreactor, or the various phases within the bioreactor. For example, the state variables within the well-mixed SSF bioreactor model are the temperature of the substrate bed (T) and the amount of biomass in the bioreactor (X). They are called state variables because, together, the values for all these variables at a particular instant describe the state of the system at that instant. They are variables because their values vary as the independent variables change.

T = temperature of outlet air H = humidity of outlet air F = air flow rate (dry basis)

Microbial variables and parameters

X = biomass

/u = specific growth rate constant Xmax = maximum biomass YQ = yield of metabolic heat

Bed variables and parameters

T = bed temperature

Cpb = overall bed heat capacity

M = total bed mass

Thermodynamic constants

Cpar = heat capacity of dry air AHvap = heat of vaporization of water Cpvapor = heat capacity of water vapor

Associated with heat transfer through the bioreactor wall h = heat transfer coefficient A = area across which heat transfer takes place Tsurr = temperature of the surroundings

Fig. 12.3. A simple mathematical model for predicting the temperature within a well-mixed SSF bioreactor: The system modeled and the various variables and parameters involved in the model. Note that due to assumption of perfect mixing, the conditions within the bioreactor are equal to the outlet conditions

Independent variables. These represent variables that do not depend on the system and how it is operated. Rather the system depends on these variables. The independent variables that appear in models for SSF bioreactors are either time or space or both. In the current example it is assumed that the bioreactor is well mixed and therefore time is the only independent variable. In some cases the variations across space are significant while the variations in time occur only slowly. In this case, it might be appropriate to write the equations with space as the only independent variable, and the equation is referred to as a "pseudo-steadystate" equation. There are also bioreactors in which both the temporal and spatial variations are significant: the temperature at a specific position changes over time, and if the temperature is measured simultaneously at different locations within the substrate bed, the measured temperature varies with position. In this case both time and position appear as independent variables.

T = temperature of outlet air H = humidity of outlet air F = air flow rate (dry basis)

Bed variables and parameters

T = bed temperature

Cpb = overall bed heat capacity

M = total bed mass

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