We apply our techniques of dimensional analysis to non-dimesionalize the chemostat model equations.
We will work with the chemostat model derived in notes 7:
\[ \begin{align} \frac{dN}{dt} &= G(C)N - \frac{F}{V}N \\ \frac{dC}{dt} &= -\alpha G(C)N - \frac{F}{V}C + \frac{F}{V}C_{0} \end{align} \]
Observe the following, in terms of unit dimensions, we must have
\(\left[G(C) \right] = \frac{1}{T}\),
\(\left[\frac{F}{V} \right] = \frac{1}{T}\),
\([\alpha] = \frac{[C]}{[N]}\), and
\([C_{0}] = [C]\)
Consider the case where we choose a Michaelis-Menten rate law for \(G(C)\) as in notes 8. That is, take
\(G(C) = \frac{K_{\text{max}}C}{k_{n} + C}\)
Note that this requires
\([K_{\text{max}}] = \frac{1}{T}\), and
\([k_{n}] = [C]\)
Thus, our model equations are
\[ \begin{align} \frac{dN}{dt} &= \frac{K_{\text{max}}C}{k_{n} + C}N - \frac{F}{V}N \\ \frac{dC}{dt} &= -\alpha \frac{K_{\text{max}}C}{k_{n} + C} N - \frac{F}{V}C + \frac{F}{V}C_{0} \end{align} \] and these are the equations we will non-dimensionalize.
We will carry out a generic non-dimensionalization as described in notes 10. Let \(N^{\ast}\) and \(C^{\ast}\) be characteristic scales for concentrations and let \(t^{\ast}\) be a characteristic time scale. Then, define \(x=\frac{N}{N^{\ast}}\), \(y=\frac{C}{C^{\ast}}\), and \(\tau=\frac{t}{t^{\ast}}\). Explicit expressions for the scales \(N^{\ast}\), \(C^{\ast}\), and \(t^{\ast}\) will be determined later.
By the chain rule we have
\[ \begin{align} \frac{dx}{d\tau} &= \frac{dx}{dt}\frac{dt}{d\tau} = t^{\ast}\frac{d}{dt}\frac{N}{N^{\ast}}\\ \frac{dy}{d\tau} &= \frac{dy}{dt}\frac{dt}{d\tau} = t^{\ast}\frac{d}{dt}\frac{C}{C^{\ast}} \end{align} \] which implies that
\[ \begin{align} \frac{dx}{d\tau} &= \frac{t^{\ast}}{N^{\ast}}\left( \frac{K_{\text{max}}C}{k_{n} + C}N - \frac{F}{V}N\right)\\ \frac{dy}{d\tau} &= \frac{t^{\ast}}{C^{\ast}}\left(-\alpha \frac{K_{\text{max}}C}{k_{n} + C} N - \frac{F}{V}C + \frac{F}{V}C_{0} \right) \end{align} \] or equivalently
\[ \begin{align} \frac{dx}{d\tau} &= t^{\ast} \frac{K_{\text{max}}C^{\ast}y}{k_{n} + C^{\ast}y} x - t^{\ast} \frac{F}{V} x \\ \frac{dy}{d\tau} &= -\alpha \frac{t^{\ast}}{C^{\ast}} \frac{K_{\text{max}}C^{\ast}y}{k_{n} + C^{\ast}y} N^{\ast}x - t^{\ast}\frac{F}{V}y + \frac{t^{\ast}}{C^{\ast}} \frac{F}{V}C_{0} \end{align} \]
Now we need to choose expressions for the scale factors \(N^{\ast}\), \(C^{\ast}\), and \(t^{\ast}\). Based on our earlier dimensional analysis, it is reasonable to choose
\(t^{\ast}=\frac{V}{F}\), and
\(C^{\ast}=k_{n}\).
Doing so leads to
\[ \begin{align} \frac{dx}{d\tau} &= \frac{V}{F} \frac{K_{\text{max}}k_{n}y}{k_{n} + k_{n}y} x - \frac{V}{F} \frac{F}{V} x \\ \frac{dy}{d\tau} &= -\alpha \frac{\frac{V}{F}}{k_{n}} \frac{K_{\text{max}}k_{n}y}{k_{n} + k_{n}y} N^{\ast}x - \frac{V}{F}\frac{F}{V}y + \frac{\frac{V}{F}}{k_{n}} \frac{F}{V}C_{0} \end{align} \]
or equivalently,
\[ \begin{align} \frac{dx}{d\tau} &= \frac{V K_{\text{max}}}{F} \frac{y}{1 + y} x - x \\ \frac{dy}{d\tau} &= - \frac{\alpha V K_{\text{max}}}{F k_{n}} N^{\ast} \frac{y}{1 + y} x - y + \frac{C_{0}}{k_{n}} \end{align} \] It remains to choose an expression for \(N^{\ast}\). To do this, we examine the unit dimensions for \(\frac{\alpha V K_{\text{max}}}{F k_{n}}\). We have
\[ \left[\frac{\alpha V K_{\text{max}}}{F k_{n}} \right] = \left[\alpha \right]\left[\frac{V}{F} \right]\left[\frac{K_{\text{max}}}{k_{n}} \right] = \frac{[C]}{[N]} T \frac{\frac{1}{T}}{[C]} = \frac{1}{[N]} \]
Thus, we may take
\[ N^{\ast} = \frac{F k_{n}}{\alpha V K_{\text{max}}} \]
which gives
\[ \begin{align} \frac{dx}{d\tau} &= a_{1} \frac{y}{1 + y} x - x \\ \frac{dy}{d\tau} &= - \frac{y}{1 + y} x - y + a_{2} \end{align} \]
where \(a_{1} = \frac{V K_{\text{max}}}{F}\) and \(a_{2} = \frac{C_{0}}{k_{n}}\) are dimensionless quantities.
A non-dimensional version of the chemostat model equations is
\[ \begin{align} \frac{dx}{d\tau} &= a_{1} \frac{y}{1 + y} x - x \\ \frac{dy}{d\tau} &= - \frac{y}{1 + y} x - y + a_{2} \end{align} \]
with \(a_{1} = \frac{V K_{\text{max}}}{F}\) and \(a_{2} = \frac{C_{0}}{k_{n}}\). Notice that we have reduced a model expression involving six parameters to an equivalent expression with only two parameters.