Using an expression and an equation to get an ODE to describe something.

Question

I have an expression and an equation, that I need to use to show that ODE describes something.

Let me put it into context

I have an expression for the Rate at Anti-Freeze flows $\mathcal{IN}$ and $\mathcal{OUT}$ of an compartment:

$$\dfrac{dA}{dt}=\bigg[C_{in}Q_{in}-\bigg(\frac{AQ_{out}}{V_{tot}}\bigg)\bigg]$$

and I also have an equation to show concentration $$C =\frac{A}{V_{tot}}$$

and I need to use these two, to show the ODE describing Concentration of Anti-Freeze in the compartment, $C(t)$, is

$$[V_o\space+ (Q_{in}-Q_{out})t ]\dfrac{dC}{dt}\space+ Q_{in}C =Q_{in}C_{in}$$

Any help would be grateful. Any questions about the problem I am happy to answer.

Answer 1

You should perhaps check out the definitions of volumetric flow rate (Q), if you are not familiar with it. Also it seems like there is an assumption on the $Q_{in}$ and $Q_{out}$ missing - they are constant (?). That will make the following sort out nicely:

$$ Q_{tot} = Q_{in} - Q_{out} = \dot{V}_{tot} \\ \Rightarrow V_{tot}= V_0 + t(Q_{in} - Q_{out}) $$ (simply integrate $V_{tot}$, which is simple when the VFR's are constant)

Let's rewrite the compartment concentration to $C V_{tot} = A$. If you differentiate this (product rule), and inserting the above, you get

$$ \dot{C} V_{tot} + C \dot{V}_{tot} = \dot{A} \Rightarrow \dot{C}(V_0 + t(Q_{in} - Q_{out})) + C (Q_{in} - Q_{out}) = \dot{A} = C_{in} Q_{in} - C Q_{out} $$ (last equality is from the first differential equation you had). Then simply move one term to the other side and you are there.