What Level of Detail Should Be Used to Describe the Growth Kinetics

A crucial decision that must be made is whether to try to describe the spatial distribution of the system components at the microscale within the kinetic part of a model of an SSF bioreactor. The two key questions are:

1. Growth should be described as depending on which factors?

2. Should the spatial distribution of the biomass at the microscale be described?

13.2.1 Growth Should Be Treated as Depending on Which Factors?

Ideally, in a model describing the kinetics of microbial growth, the growth rate should be described as depending on those environmental factors that are important in influencing it. The problem in trying to meet this ideal in a model of an SSF bioreactor can be illustrated by comparing the implications, for both SSF and SLF, of a decision to include nutrient concentrations as one of the factors that determine the growth rate.

In SLF it is typically reasonable to assume that the fermentation broth is well mixed and therefore that the nutrient concentration is uniform throughout the broth. It is then a simple matter to use the Monod equation to describe the specific growth rate as a function of the nutrient concentration (Fig. 13.1(a)). In turn, it is also a simple matter to describe how the nutrient concentration changes during growth. In many cases such a simple model describes the growth curve quite well.

The situation is quite different in SSF. Mass transfer inside the substrate particle is limited to diffusion and, as a result of consumption of nutrients by the microorganism, concentration gradients will arise within the substrate (Fig. 2.8). As shown in Fig. 13.1(b), the Monod equation could be used to describe how growth depends on nutrient concentration, but it is not at all a simple matter to describe the nutrient concentration experienced by the microorganism. For example, even if a soluble nutrient is used, it is necessary to use equations that describe the diffusion of the nutrient within the substrate particle. If a polymeric nutrient is used, extra equations will be necessary in order to describe the processes of enzyme release, diffusion, and action. The problem is that the diffusion equations are written in terms of changes in both time and space, or, in other words, they are partial differential equations. Partial differential equations are significantly more difficult to solve than ordinary differential equations. As will be seen later, partial differential equations will typically arise in the modeling of the macroscale transport processes. A model with partial differential equations at both the microscale and mac-roscale would be highly complex and difficult to solve, requiring long solution times, as much as hours or days.

Actually, the situation is more complex still. SSF substrates typically involve a range of different carbon and energy sources and other nutrients, and these might be used sequentially or in parallel in a complex manner.

Besides the problems with the complexity introduced into the model, there is the extra problem of validating the model. In SLF it is a simple matter to withdraw a sample from the bioreactor and determine the nutrient and biomass concentrations. Plots of experimentally measured biomass and nutrient concentrations against time can then be compared with the model predictions. In SSF it is not a simple matter to determine experimentally the nutrient concentration experienced by the microorganism. Certainly it is not valid simply to homogenize a sample within a volume of water and then determine the nutrient concentration. This gives an average nutrient concentration that says nothing about the nutrient concentration gradients within the substrate particle. It would be necessary to use an analytical method that was capable of giving nutrient concentration as a function of position.

Given these modeling and experimental difficulties, if the aim is to develop a fast-solving model, the growth rate should not be expressed as a function of the in-tra-particle concentration of any component, neither of nutrients, nor of O2, nor of protons (pH). Typically the growth kinetic equation would be empirical and the parameters of the equation would be described as functions of one or more of biomass concentration, temperature, and water activity.

dX

MmaxS x

dt

Ks +S

dS

1 dX

dt

Since the liquid is well mixed, each variable represents the value at all points within the fermentation broth

S and X are functions of the radial position r

*

dX dt

Mraax S r ' Ks +S|

^Xr r

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