In a mathematical model of a bioreactor, it will be necessary to write terms in the mass and energy balance equations to describe the changes that the microorganism causes in its local environment since, in turn, the changes in the local environment affect the rate of growth of the microorganism (Fig. 17.1). The mass and energy balance equations and the terms in them that are related to transport phenomena will be presented in later chapters. The current chapter concerns itself with those terms within these balance equations that are related to growth and maintenance metabolism by the microorganism.
Various effects that we may wish to include in a bioreactor model are:
• the liberation of waste metabolic heat;
• the consumption of substrate (i.e., overall residual substrate) or particular nutrients;
• the consumption of O2 and production of CO2;
• the production of water;
• the formation of products.
Typically the aim will be to link each of these growth-related activities with the kinetic equation. Note that it is not necessary for the kinetic equation to be expressed in terms of dry biomass. If the appropriate parameters are used, the growth-related activities can be related to a kinetic equation expressed in terms of a biomass component or activity.
The equations in this chapter are written in absolute terms, that is, the symbol X represents the total mass of biomass within the bioreactor (kg). The equations can also be expressed in terms of the "absolute biomass concentration" by dividing the entire equation by the initial mass of dry solids within the bioreactor (Do, kg). Once this is done, the ratio XDo can be replaced by the symbol CXA (the meanings of these symbols are explained in Chap. 15). In this case, the corresponding rate term will have the units of kg or mol per kg-IDS (initial dry solids).
O2 (or CO2) balance has terms for
• O2 (or CO2) entering and leaving bioreactor in air flow
• consumption (or production) by microorganism
May be important for process monitoring
Energy balance has terms for
• energy entering and leaving bioreactor in air flow
• heat removal by conduction/convection/evaporation
• liberation of waste metabolic heat
Water balance has terms for water entering and leaving bioreactor in air flow consumption in polymer hydrolysis production by respiration evaporation
Balanc equatio for the e/tran ns wr biore;i'
port itten ctor these balances determine the local aw and temperature growth rate depends f on the resulting aw and } V temperature of the J substrate release of products consumption of substrate
• calculation of overall nutrient levels
• overall changes in bed mass, for predicting:
(1) relative biomass concentrations (2) bed shrinkage
Fig. 17.1. The importance of describing changes in the environment caused by growth and maintenance activities. These changes in turn affect growth (see also Fig. 12.4)
In this chapter all rates (given the symbol "r" and an appropriate subscript, and with units of kg or mol per hour) are calculated as positive numbers, regardless of whether they represent components that are produced or consumed. When the time comes to incorporate these terms within balance equations, it will be necessary to subtract the term if it is for a component that is consumed and to add the term if it is for a component that is produced. Also note that stoichiometric coefficients within these equations can be written in two ways. For example, to express the stoichiometric relationship between A and B, we can use YAB (kg-A kg-B-1) or YBA (kg-B kg-A-1). These can be easily interchanged by remembering that one is the reciprocal of the other (i.e., YAB = 1/YBA).
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