1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. 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
$$ \frac {dC_{C}}{dt} = -DC_{S} + r_{g} $$
Assume steady state
$$ DC_{S} = r_{g} $$
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

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.

I think this would be better placed in the Biology, Chemistry & Earth Homework forum. Left to your own devices, a web search for a similar worked example is what you need.

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.