PNG 550
Reactive Transport in the Subsurface

5.3 Monod Rate Laws

PrintPrint

Dual Monod kinetics

The kinetics of microbe-mediated reactions are often described by Monod rate laws in the following form:

\begin{equation}R=\mu_{\max } C_{C_{5} H_{7} O_{2} N} \frac{C_{D}}{K_{m, D}+C_{D}} \frac{C_{A}}{K_{m, A}+C_{A}}\end{equation}

Here $\mu$ is the rate constant (mol/s/microbe cell), $C_{C_{5} H_{7} O_{2} N}$ is the concentration of microbe cells (cells/m3), Cd and Ca are the concentrations of electron donor and acceptor of the reaction (mol/m3). The Km,D and Km,A are the half-saturation coefficients of the electron donor and acceptors (mol/m3), respectively. These coefficients are the concentrations at which half of the maximum rates are reached for the electron donor and acceptor, respectively. If the electron donor is not limiting, it means that $C_{D} ? K_{m, D}$, so that the term $\frac{C_{D}}{K_{m, D}+C_{D}}$  is essentially 1.

Dual Monod kinetics with inhibition term

In order to represent the biogeochemical redox ladder, we will need to introduce an additional term in the dual Monod rate law, the inhibition term. A Monod rate law with an inhibition term looks as follows:

\begin{equation}R=\mu_{\max } C_{C_{5} H_{7} O_{2} N} \frac{C_{D}}{K_{m, D}+C_{D}} \frac{C_{A}}{K_{m, A}+C_{A}} \frac{K_{\mathrm{I}, H}}{K_{\mathrm{I}, H}+C_{H}}\end{equation}



Here the KI,H is the inhibition coefficient for the inhibiting chemical H. In contrast to the Monod terms, the inhibition terms become 1 (not inhibiting) only when KI,H?CH.

As an example, in a system where oxygen and nitrate coexist, which is very common in agricultural soils, aerobic oxidation will occur first before denitrification occurs. The sequence of that can be represented by the following:

\begin{equation}R_{O_{2}}=\mu_{\max , O_{2}} C_{C_{5} H_{7} O_{2} N\left(O_{2}\right)} \frac{C_{D}}{K_{m, D}+C_{D}} \frac{C_{O_{2}}}{K_{m, O_{2}}+C_{O_{2}}}\end{equation}


\begin{equation}R_{N O_{3}}=\mu_{\max , N O_{3}} C_{C_{5} H_{7} O_{2} N\left(N O_{3}\right)} \frac{C_{D}}{K_{m, D}+C_{D}} \frac{C_{N O_{3}}}{K_{m, N O_{3}}+C_{N O_{3}}} \frac{K_{\mathrm{I}, O_{2}}}{K_{\mathrm{I}, O_{2}}+C_{O_{2}}}\end{equation}



These two rate laws, with proper parameterization, will ensure that denitrification reaction occurs only when O2 is depleted to a certain extent that the term $\frac{K_{I, O_{2}}}{K_{I, O_{2}}+C_{O_{2}}}$ approaches 1.0. If other electron acceptors that are lower than nitrate in the redox ladder also occur, then multiple inhibition terms are needed. For example, if there is also iron oxide in the system, we will need the following for the iron reduction rate law:

\begin{equation}R_{F e(O H)_{3}}=\mu_{\max , F e} C_{C_{5} H_{7} O_{2} N, F e} \frac{C_{D}}{K_{m, D}+C_{D}} \frac{C_{F e(O H)_{3}}}{K_{m, F e(O H)_{3}}+C_{F e(O H)_{3}}} \frac{K_{\mathrm{I}, O_{2}}}{K_{\mathrm{I}, O_{2}}+C_{O_{2}}} \frac{K_{\mathrm{I}, N O_{3}}}{K_{\mathrm{I}, N O_{3}}+C_{N O_{3}}}\end{equation}



Where the additional litrate inhibition term will allow iron reduction to occur at sufficiently significant rates only when nitrate level is low compared to the inhibition constant value.