Bioreactors reactor equations

A model can be defined as a set of relationships between the variables of interest in the system being investigated. A set of relationships may be in the form of equations the variables depend on the use to which the model is applied. Therefore, mathematical equations based on mass and energy balances, transport phenomena, essential metabolic pathway, and physiology of the culture are employed to describe the reaction processes taking place in a bioreactor. These equations form a model that enables reactor outputs to be related to geometrical aspects and operating conditions of the system. 

This chapter describes the different types of batch and continuous bioreactors. The basic reactor concepts are described as well as the respective basic bioreactors design equations. The comparison of enzyme reactors is performed taking into account the enzyme kinetics. The modelhng and design of real reactors is discussed based on the several factors which influence their performance the immobilized biocatalyst kinetics, the external and internal mass transfer effects, the axial dispersion effects, and the operational stabihty of the immobilized biocatalyst. 

The calculation of heat transfer film coefficients in an air-lift bioreactor is more complex, as small reactors may operate under laminar flow conditions whereas large-scale vessels operate under turbulent flow conditions. It has been found that under laminar flow conditions, the fermentation broths show non-Newtonian behaviour, so the heat transfer coefficient can be evaluated with a modified form of the equation known as the Graetz-Leveque equation 9... 

Fermentation systems obey the same fundamental mass and energy balance relationships as do chemical reaction systems, but special difficulties arise in biological reactor modelling, owing to uncertainties in the kinetic rate expression and the reaction stoichiometry. In what follows, material balance equations are derived for the total mass, the mass of substrate and the cell mass for the case of the stirred tank bioreactor system (Dunn et ah, 2003). 

In the ideal plug-flow reactor (Figure 11.16) the continuous phase flows as a plug through the reactor i.e., there is no mixing or, in other words, no axial dispersion. Consequently, if a compound is consumed or produced, a concentration gradient will exist in the direction of flow. The mass balance is therefore first set up over an infinite small slice perpendicular to the direction of the flow with volume dV of the bioreactor. Assuming steady state and F =Fq=F, Equation (11.5) then is reduced to ... 

When the biochemical reactors are kinetically controlled, the batch bioreactors and the PFR are described by the same design equations (Equations (11.25) and (11.28)) and show a better performance than the CSTR in most cases, except for substrate inhibition kinetics. 

The activity of each compound ax in the reactor stage (R01) is calculated according to Equation 6 with the respective saturation pressure Ppsatx determined at the temperature of the bioreactor. 

Bioreactors that use enzymes but not microbial cells could be regarded as fermentors in the broadest sense. Although their modes of operation are similar to those of microbial fermentors, fed-batch operation is seldom practiced for enzyme reactors. The basic equations for batch and continuous reactors for... 

Although the yields and total growth rate are useful parameters, it is the state variables like the concentrations of biomass, substrate and product, and the culture parameters like the specific rates of growth, substrate uptake, etc., that provide a complete description of the bioreactor. One attempt in estimating these variables from R and the yields consisted of integrating the governing differential equation with known initial conditions and the measured values for R and the yields (9). For a batch reactor, for example, b was estimated by integrating... 

Knowing the concentrations in the absorber-outlet liquid stream C k. 0, the model of the bioreactor allows calculation of the concentrations in the reactor-outlet stream Ck =1. The model consists of mass-balance equations for FeEDTA complexes ... 

Membrane bioreactors have been modelled using approaches that have proven successful in the more conventional catalytic membrane reactor applications. The simplest membrane bioreactor system, as noted in Chapter 4, consists of two separate units, a bioreactor (typically a well-stirred batch reactor) coupled with an external hollow fiber or tubular or flat membrane module. These reactors have been modelled by coupling the classical equations of stirred tank reactors with the mathematical expressions describing membrane permeation. What makes this type of modelling unique is the complexity of the mecha-... 

A comprehensive model of a membrane bioreactor has been developed by Moueddeb et al [5.103] for a simple irreversible reaction A B. The goal of the model was to describe their experimental reactor system, which was described earlier in Chapter 4. The model equations were established by taking into account the effect of the biomass on the permeate flow rate in the annular volume. The mass balance equations for the substrate (A) and the product (B) in cylindrical coordinates, utilized by Moueddeb et al [5.103] are given as ... 

Usually, Nr -h 1 reactors will be required to absorb non-productive time (discharge, cleaning and filling of reactor). Solving the equation that represents enzyme inactivation under operation conditions (i.e. Eq. 5.76) and the equation that model conversion profiles within the biocatalyst bed in CPBR (Eq. 5.79), residual enzyme activity in each bioreactor after each time interval can be determined and feed flow-rate to each bioreactor during each interval calculated as ... 

These equations remain valid for bioreactors provided that one employs a suitable mathematical representation of the rate of disappearance of the substrate that is the limiting reagent. In Illustration 13.3 we employ an alternative form of the design equation to determine the holding time necessary to achieve a specified degree of conversion in a strictly batch bioreactor. This illustrative example also indicates how overall yield coefficients are employed as a vehicle for taking the stoichiometry of the reaction into account. Illustration 13.4 describes how one type of semibatch operation (the fed-batch mode) can be exploited to combine the potential advantages of batch and continuous flow operation of a stirred-tank reactor. 

Now we need to consider the equations that describe the operation of the bioreactor once the feed of fresh substrate to the reactor is initiated. For subsequent times, the volume of the suspension of the microorganism will be given by a relation of the form... 

As we saw in Section 8.3.2, one can develop equations describing the performance of an arbitrary CSTR in a cascade of CSTRs. We can also develop the corresponding equations for the nth bioreactor in an extended cascade of stirred tanks by conducting an analysis on the nth bioreactor. The feed stream exiting bioreactor n - 1 and entering reactor n has a volumetric flow rate V, a concentration of the hmiting substrate equal to s , and a biomass concentration. This stream enters well-stirred reactor n, which... 

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