Dear Kartik Gehlot ,

This pdf may be helpful.

Thank you.

Dear Kartik Gehlot ,

This pdf may be helpful.

Thank you.

Dear Kartik Gehlot,

If the process follows Monod kinetics, the mass balance of the limiting substrate can be formulated for continuous stirred tank bioreactor as:

0 = FV*(CAF-CA)-1/Y*k*CA*CB/(KS+CA)*V

while for the biomass:

0 = FV*(-CB)+k*CA*CB/(KS+CA)*V

In the above equations, stationary conditions were assumed. FV denotes volumetric flow rate, V is the volume of liquid in the bioreactor, CAF - concentration of the limiting substrate (A) in the feeding stream. CA and CB denote substrate A and biomass B concentration in the liquid, respectively. k is maximum specific growth rate and KS is Monod (saturation) constant. Y is growth yield coefficient (kgB/kgA). It is assumed that the biomass is not supplied in the feeding stream (CBF=0). The model is transformed in order to introduce the dilution rate D. Then we have:

0 = D*(CAF-CA)-1/Y*k*CA*CB/(KS+CA)

0 = D*(-CB)+k*CA*CB/(KS+CA)

The system of equations can be simplified, by using the relationship RB=Y*RA (RB is the biomass growth rate and RA is the substrate A utilization rate).

0 = D*(CAF-CA)-1/Y*k*CA*CB/(KS+CA)

0 = D*(-CB)+k*CA*CB/(KS+CA)

In the biomass balance, CB is reduced (we can assume it is not equal zero to be specific):

0 = -D+k*CA/(KS+CA)

or

D = k*CA/(KS+CA)

In order to determine k and KS some transformations are required. Namely, the above equation can be presented as a formula of straight line. Inverting both sides gives:

1/D = (KS+CA)/(k*CA)

or

1/D = KS/k*(1/CA) + 1/k

The above equation is an equation of a straight line. We can write it in this way:

Y = A*X + B

where: Y=1/D, A=KS/k, X=1/CA and B=1/k

KS and k can be obtained by linear regression. To do so, You need to have results in the form

D1 : CA1

D2 : CA2

...

DN : CAN

Where CA1, CA2, ... CAN are stationary A concentrations obtained for corresponding dilution rates. After finding coefficients A and B using linear regression, k and KS are calculated using definitions of coefficients A and B. I hope I didn't make any mistake, as typically I use different symbols.

The doubt is in the kinetic model of the process. If this is alcoholic fermentation, then product inhibition should be taken into account. For such kinetic models, nonlinear regression should be used for the determination of kinetic parameters. I can recommend You the following paper:

Beyenal, Haluk, Suet Nee Chen, and Zbigniew Lewandowski. "The double substrate growth kinetics of Pseudomonas aeruginosa." Enzyme and Microbial Technology 32.1 (2003): 92-98.

In this paper, a different process is considered, however You can get a lot of information about determination of kinetic parameters.

In order to determine the specific growth rate, once again we will use the biomass balance:

D = k*CA/(KS+CA)

The expression at the right-hand side of the above equation is the specific growth rate, therefore it is equal to D (but only at stationary conditions).

About the determination of optimum value of the dilution rate. I have not do exactly such thing, but something similar. Actually, at the moment I am writing a paper about stationary properties of alcoholic fermentation in continuous stirred tank bioreactor.

It is not exactly clear what do You mean by an optimal dilution rate. You can recognize this as a value for which a maximum conversion of the substrate is obtained (alpha=(CAF-CA)/CAF). In order to find such a value of D it convenient to obtain steady-state branches for the model analysed. There are several software which can be used, for example DST for Matlab. Anyhow for Monod kinetic model it is possible to write a simple program which solves the model in loop (D is changed gradually). For more complex model it will not work due to possible multiplicity of steady states. After obtaining relationship ALFA(D) You can read the maximum value from the diagram or You can try to find numerically a maximum of the function ALFA(D).

You can find such relationships for example in my papers:

Skoneczny, Szymon, and Monika Cioch. "Modeling of continuous-flow bioreactors with a biofilm with the use of orthogonal collocation on finite elements." Chemical Engineering Communications (2018): 1-18.

or

Tabiś, Bolesław, Szymon Skoneczny, and Wojciech S. Stryjewski. "Nonlinear dynamics of a controlled stirred tank bioreactor with predator-prey relationship." Chemical and Process Engineering 35.3 (2014): 349-360.

or other papers concerning bifurcation of bioreactors or steady-state characteristics of bioreactors.

The optimal dilution rate can also be defined as a value for which maximum bioreactor productivity P is obtained. This quantity is defined as:

P = FV*(CAF-CA) [kg A/hr]

As it can be seen, P depends on volumetric flow rate, not on the dilution rate. You can use quantity

PV = 1/D*(CAF-CA) [kg A/(m3*hr)]

It can be called 'specific productivity'. Determination of optimal dilution rate is the same as the above presented, but You look for a maximum of PV(D).

I hope my answer will help.

Best regards

Szymon Skoneczny

It's simple if it's is a continous reactor then u=D which gives S=DKs/um-D ; and for calculating um plot ln(x/xo) vs t you will get a straight line and slope will be um and you know that final substrate concentration or steady state substrate concentration