Consider the following diagram of a empty tank being filled with a liquid
Considering constant and equal denstity, it's total mass balance is
$$\frac{dV_1}{dt}=q_e-q_s$$
But I don't know how to model $V(h)$
I do know that below $h_0$
$$V=\frac{h \pi r^2}{3}$$
where
$$r(h)=\frac{R_t}{h_0} h$$
And that above $h_0$
$$V=h \pi R_t^2 + \frac{h \pi r^2}{3}$$
But how do I consider bot volumes in a single expresion of $V(h)$ considering their respective conditions of $h \le h_o$ and $h>h_0$ respectively?