# Bioreactor washout dilution rate

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## Homework Statement

The production of a product P from a particular gram negative bacteria follows the Monod growth law
$$-r_{S} = \frac {\mu_{max}C_{S}C_{C}}{K_{M}+C_{S}}$$
with $\mu_{max} = 1 \hspace{0.05 in} hr^{-1}$, $K_{M} = 0.25 \hspace{0.05 in} g/L$, and $Y_{C/S} = 0.5 \hspace{0.05 in} g/g$

The reaction is now to be carried out in a CSTR with $C_{S0}$ = 20 g/L and $C_{C0}$ = 0 g/L. What is the dilution rate at which washout occurs?

## The Attempt at a Solution

I do a mass balance on the cell being produced
$$V \frac {dC_{C}}{dt} = -\nu C_{S} + (r_{g} - r_{d})V$$
Assume the death rate is negligible
$$\frac {dC_{C}}{dt} = -DC_{S} + r_{g}$$
$$DC_{S} = r_{g}$$
Now this is where I run into a problem. I am not sure how how to express $r_{g}$. I get confused because 3 different variables are used, $r_{S}$, $r_{C}$, and $r_{g}$ in these bioreactor problems. I know S is the substrate and C is cells, but what is g? Is g and C the same thing? Anyways, I thought $r_{g} = -r_{S}$, but it turns out from the solution manual that $r_{g} = -Y_{C/S}r_{S}$, but I thought that $r_{C} = -Y_{C/S}r_{S}$.

This is part (c) in the problem, but part (a) states
The reaction is to be carried out in a batch reactor with the initial cell concentration of $C_{C0}$ = 0.1 g/L and substrate concentration of $C_{S0}$ = 20 g/L
$C_{C} = C_{C0} + Y_{C/S}(C_{S0} - C_{S})$
Plot $-r_{S}$, $-r_{C}$, $C_{S}$, and $C_{C}$, as a function of time.
I know $r_{C} = -Y_{C/S}r_{S}$, but I don't know the difference between $r_{g}$ and $r_{C}$. They use different variables in the same problem, and I am uncertain if they mean the same thing or mean something different.

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NascentOxygen
Staff Emeritus
Just took the final for this class, so it's moot now! but in this case, $r_{C}$ and $r_{g}$ are the same, because $r_{C} = r_{g} - r_{d}$, where g is cell generation, and d is cell death. We are assuming cell death is negligible.