The conservation equation for the culture can be stated as
with p and Q denoting the density and volumetric effluent rate, respectively, of the culture and pp and Qp the density and volumetric flow rate, respectively, of the sterile feed (usually liquid). Q is trivial in batch and fed-batch operations, while Qp is trivial in a batch operation. The mass balance in Eq. 2.7 is simplified via a customary assumption that pp and p are not significantly different. This assumption is reasonable since the densities of nutrient medium and biomass are not substantially different. The simplified form of Eq. 2.7 is
The culture is comprised of the biotic phase (cell mass) and the abiotic (extracellular) phase. Let the concentration of biomass (X, cell mass) be denoted as Cx (C'x = X, the notation popular in the biochemical engineering literature is X) and the density of biomass as pb. It is then not difficult to deduce that the volume fractions of the biotic and abiotic phases in the culture are Cx/pb and (1 - Cx/pb), respectively. The volumes of biotic and abiotic phases (Vj, and Va, respectively) and the volumetric flow rates of these phases in the bioreactor effluent (in the case of continuous culture, Qb and Qa, respectively) can then be expressed as
pb Pb pb Pb
The volume fraction of the biotic phase is commonly considered to be negligible. While this consideration is valid in cultures that are dilute in biomass
(Cx Pt), in cultures that are concentrated in biomass, neglecting the volume fraction of the biotic phase may lead to certain pitfalls.
Before proceeding further, some comments are in order regarding bases for different species. For a particular specie, the basis for choice is based on how the specie is monitored. Thus, the biomass concentration is expressed on the basis of unit culture volume, concentrations of species present in the abiotic phase are expressed on the basis of unit abiotic phase volume, and concentrations of intracellular species are usually expressed on the basis of unit biomass amount. For rate processes occurring entirely in the abiotic phase, the basis is the volume of the abiotic phase (Va), while for rate processes occurring in the biotic phase (metabolic reactions) and at the interface between the abiotic and biotic phases (such as species transport), the basis is the amount of the biotic phase. On a larger scale, a single cell is viewed also as a catalyst (hence the name biocatalyst), or in a stricter sense, an autocatalyst, since resource utilization and generation of end products of cellular metabolism are promoted by the cell. The rates of proliferation/replication of a living species and other processes associated with it (utilization of resources and synthesis of end products) are as a result proportional to the amount of the living species.
Approaches to representation of the biotic phase according to the number of components (species) used for such representation and whether or not the biotic phase is viewed as a heterogeneous collection of discrete cells have been succinctly classified by Fredrickson and Tsuchiya . Representations which view each cell as a multicomponent mixture are referred to as structured representations, while those which view the biotic phase as a single component (like any specie in the abiotic phase) are termed unstructured representations. An unsegregated representation is based on use of average cellular properties and does not account for cell-to-cell heterogeneity. A segregated representation, on the other hand, involves description of behavior of discrete, heterogeneous cells suspended in the culture and thereby accounts for cell-to-cell variations. The segregated-structured representation is most suitable for a bioreactor. In order to have tractable representations of biotic phase, it is often assumed that the cell-to-cell variations do not substantially influence the kinetic processes in the biotic phase. The segregated representation can then be reduced to an unsegregated representation based on average cell properties. The discussion in this chapter is limited to the latter perspective. With this in mind, the conservation equation for the biotic phase can be stated as
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