Pyrite Dissolutions Bioreactor

In this reactor, we used wild-type T. ferrooxidans kinetics to the bioreactor.

These are the mass balance of semi-batch bioreactor¹:

\[\begin{align*} \frac{d(XV)}{dt} & = V r_c(X,S) \\ \frac{d(PV)}{dt} & = V r_P(X,S) \\ \frac{d(SV)}{dt} & = F S_0 - \frac{1}{Y_{X/S}}V r_c(X,S) \end{align*}\]

The change of volume per time (hour) is defined as¹:

\[\frac{dV}{dt} = F(t)\]

The cell growth is expressed as¹:

\[\frac{dV}{dt} = F(t)\]

Monod model provides this equation to define cell specific growth rate as a function of substrate concentration¹:

\[ \mu(S) = \mu_{max}\frac{S}{K_S + S} \]

The rate of product formation is assumed to be a fraction of cell growth by-product:¹:

\[ r_P(X,S) = Y_{P/X}r_g(X,S) \]

Where \(Y_{P/X} \) is a product yield coefficient that serves as a ratio of product formation versus cell growth.¹:

\[ Y_{P/X} = \frac{\mbox{product mass}}{\mbox{new cells mass}} \]

The mass of new cells from cell growth is assumed to have a constant ratio with mass of substrate consumed, thus¹:

\[ Y_{X/S} = \frac{\mbox{new cells mass}}{\mbox{substrate consumed}} \]

In the semi-batch model, the volume is not constant. Thus, the dilution effect is affecting cell, product, and substrate concentrations. The differential equation for \(X\), \(P\), and \(S\) mass balance is extended according to chain rule¹:

\[ \begin{align*} \frac{d(XV)}{dt} & = V\frac{dX}{dt} + X\frac{dV}{dt} = V\frac{dX}{dt} + F(t)X \\ \frac{d(PV)}{dt} & = V\frac{dP}{dt} + P\frac{dV}{dt} = V\frac{dP}{dt} + F(t)P \\ \frac{d(SV)}{dt} & = V\frac{dS}{dt} + S\frac{dV}{dt} = V\frac{dS}{dt} + F(t)S \end{align*} \]

Rearranging this mass balance equation with the previous equations gives¹:

\[ \begin{align*} \frac{dX}{dt} & = - \frac{F(t)}{V}X + r_g(X,S) \\ \frac{dP}{dt} & = - \frac{F(t)}{V}P + r_P(X,S) \\ \frac{dS}{dt} & = \frac{F(t)}{V}(S_f - S) - \frac{1}{Y_{X/S}}r_g(X,S) \\ \frac{dV}{dt} & = F(t) \end{align*} \]

We used an assumption that states \(Fe^{3+}\) is a product that also acts as a competitive inhibitor for pyrite dissolution reaction. This assumption is based on the engineering cycle – design, build, test, and learn – that has been done on various possible inhibition schemes (competitive, uncompetitive, non-competitive) and various possible inhibition components ( \(Fe^{3+}\) and regulation by T. Ferrooxidans cells). The best scheme would be the one which gives the shortest reactor residence time.

The \(Fe^{3+}\) as competitive inhibitor assumption gives the following value for bioreactor-related variables mentioned in Table 2.

In this model, it is assumed that the operating conditions are maintained from the beginning until the end of the process batch. Then, the reaction was modelled via Jupyter Notebook.

These are the results of cell concentration (\( X \)), product concentration (\( P \)), substrate concentration (\( S \)), and total volume (\( V \)) in 4 hours batch time:

To determine the volume of the bioreactor, we implemented 20% overdesign to prevent reactor flooding. Thus, based on the final total volume of one batch:

\[ 25.2\ \mbox{Liter} \times 120\% = 30.24\ \mbox{Liter} \]