Fermentation Technologies
Fermentation Reactors Mass Balance Analysis

A process design project necessarily almost always begins with the conduct of mass and energy balances of the process streams as they flow through the various process equipment. This accomplishes several things: most important of which is to ensure that the process complies with the fundamental Laws of Conservation of Mass and Energy, which can not be violated when operating in the Newtonian scale; the calculations also accomplish stipulating the feed and effluent specifications against which the actual design of each equipment would be performed, and also meet the needs for proper reporting of biohazards accounting as required by the authorities in ensuring better risk management. An alcohol fermentation process as such should also begin with such mass and energy balances; and a bioreactor within the process necessarily also must be subjected to these calculations as to be in all-round compliance.

The constituents of typical ethanol fermentation reactor stream besides the substrate consists of reactants for the anabolic reactions of the metabolic reactions required for cellular maintenance functions. Both the inlet stream and the outlet stream would have the same contents of the reactants of the anabolic reaction, though some materials in the outlet stream would have zero value and must be evaluated from the analysis of cellular growth kinetics.

Often the inlet stream quantities of some of these reactant substances are determined by the intended production volume that supports profitability of the operations. However, these calculations are generally iterative: The quantity of the microbe needed to support profitable operations may also be determined from the needed extent of conversion of the substrate.  Other values are calculated based on need: The supplements, minerals and organic factors are calculated based on the need of the microbes both at the start of the reaction and through the course of the reaction. In fact, the quantities needed during the reaction are actually reverse calculated from the end-results using the outlet stream concentration of the microbes. These calculations are performed by the methods shown in the Monod equation fermentation contextual analysis, which had as the primary object the use of the biochemical sciences to evaluate the prospective compliance with the laws of conservation of mass and energy.

The objective here then is to ascertain conservation of mass, by accounting  for the microbes-consumed quantities of the inlet-stream substances, when evaluating the outlet stream.

Batch Fermentation Reactor
The mass balance calculations for a Batch reactor is complicated only by the need to determine the number of new cell growths in order to calculate the in-broth balance of the supplements, minerals and organics.

         dX/dt = K_mXS/(K_e + S)                                  B.1

          dS/dt = kXS                                                    B.2

Given that the actual time dependent solution is not needed, all that is required is to divide B.1 by B.2 as to have the X variable differential at the top;

         dX/dS = (K_m/k)/(K + S)                          B.3

Now integrating the RHS from S = So to 0.05So  should give the final concentration of the microbes and then using the Embden-Meyerhoff pathway-based  techniques of materials evaluation, the end of reaction values of the remainder reactants are determined.

Continuous Flow Tank Fermentation Reactor
The inlet stream of this reactor in general will be sterile as the microbe placed in it at the start will continue to grow and be continuously stripped as well. There is at play a residence time during the microbes grow and are partially removed. Hence, the applicable equation is B.3 here denoted C.3

    dX/dS = (K_m/k)/(K + S)                             C.3

which is integrated in this case from S = So to S_τ and dX is simply X - Xo, yielding the result:

                    X - Xo = f(S_τ)                          C.4

where f(S_τ) is the result of the integration of the RHS of C.3 with respect to S. Then the Monod equation is solved as a function of S_τ by replacing X everywhere with (Xo + f(S_τ)) and then integrating:

                 (dS_τ/G(S_τ))                                C.5

and setting the result to Θ the residence time, where G(S_τ) is the result of inserting f(S_τ) into the Monod equation growth rate. The algebraic equation is then solved for the root S_τ from which the C.4 enables the evaluation of X and then the method explained and used for the Batch Reactor material balance is followed.

Tubular Flow Fermentation Reactor
The mass balance calculations for a Tubular Flow reactor is in most respects the same as for the Batch

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reactor, except that the time element is replaced with the axial distance coordinate element along which the broth flows, and provides the reaction time effect during which the new microbes grow whose count is needed to calculate the in-broth balance of the supplements, minerals and organics.

         dX/dz = K_mXS/(K_e + S)                                  T.1

          dS/dz = kXS                                                     T.2

Given that the actual time dependent solution is not needed all that is required is to divide on T.1 by T.2 as to have the X variable differential at the top;

         dX/dS = (K_m/k)/(K + S)                          T.3

The similarity of with B.3 once again suggests the method used for the Batch Reactor mass balance.

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