# Monod-kinetic: General question on implementation

Hello everyone,

I am modeling a adsorption column for dissolved organic carbon (DOC) removal in a wastewater treatment plant. Since DOC removal is not only caused by adsorption but also by biodegradation/bioconversion i was wondering if there is a possibility to model biodegradation by a „REACTION MODEL“ or futhermore by the monod-equation. Here are the equations based on literature (see below).

The reaction term R_{T} is given by:

R_{T} = \frac{\varepsilon_B}{Y_C} (\mu_{max} \frac{c}{K_s + c}) c_{BM}

Where c_{BM} is the biomass concentration expressed by:

\frac{dc_{BM}}{dt} = \mu_{max} \frac{c}{K_s + c} c_{BM} - k_{decay} c_{BM}

And c is the concentration of the degraded substrate expressed by:

\frac{dc}{dt} = -\frac{1}{Y_c} (\mu_{max} \frac{c}{K_s + c}) c_{BM}

With \varepsilon_B the bedporosity, \mu_{max} the maximum growth rate of biomass, K_s the half saturation constant, k_{decay} the decay constant of the biomass. Y_c is the ratio of biomass production and substrate consumption which describes the change of substrate concentration over time.
Regarding the equations the biomass growth is dependend on substrate availability and vice versa the substrat degradation depends on the present biomass at a given time.

I was wondering if there is a option to implement a monod-equation in CADET and how much effort/knowledge is required to do so?
Am i right that such a model would require a additional „concentration“ in CADET to expresses the biomass growth?

Literature: Worch, Eckhard (2012). Adsorption technology in water treatment: Fundamentals, processes and modeling. Berlin: de gruyter. ISBN: 978-3-11-024022-1

@p.jans What you could do is to split up biomass growth into micro kinetics:

Since the Monod model is analogous to the Michaelis-Menten kinetics, this would look something like:

\ce{E + S <=>[k_f][k_r] ES ->[k_{cat}] P + E}

\frac{d[E]}{dt} = - k_f [E][S] + k_r [ES] + k_{cat}[ES]
\frac{d[S]}{dt} = - k_f [E][S] + k_r [ES]
\frac{d[ES]}{dt} = k_f [E][S] - k_r [ES] - k_{cat}[ES]
\frac{d[P]}{dt} = k_{cat}[ES]

The equation \mu = \mu_{max} \frac{S}{K_s + S} is non-linear in s and hence cannot be directly described by mass action law. However, it can be decomposed into micro kinetics that follows mass action law. Actually all macro kinetics are derived this way, also the more complicated ones with multiple substrates, activators and inhibitors.

So, in CADET we have tow options: 1) use the micro kinetic form in the mass action law formalism that is already implemented or 2) implement new non-linear reaction models for enzyme kinetics.

For option 1, K_s = \frac{k_r+k_{cat}}{k_f} and \mu_{max} = k_{cat} E_0 with E_0 = E + ES. This allows to calculate \mu_{max}, K_s and E_0 from k_f, k_r and k_{cat} and vice versa. The open degree of freedom can be closed, by setting k_f to a very large value that approximates the rapid equilibrium assumption for E and ES which is made by the Michaelis-Menten model. (Once implemented in CADET we can use rapid-equilibrium here.)

The Monod model assumes the whole cell as one catalyst. The substrate consumption can easily modeled as described above.

This looks great. Good to hear that there is a way to describe such a model.

Until now i am not sure if a “First order rate law” or “monod kinetic” describes my model best and i first need futher study on the rate limiting process (biomass growth or substrat availability) in the adsorption column to make my decision clear which one to use.
I will get back to this topic soon.
Thank you very much for your help.

Best regards
Pit