02 - First Order Systems

Mass balance: \[\begin{aligned} \left(\begin{array}{c}\text{Rate of} \\ \text{mass flow in} \end{array}\right) \ - \ \left(\begin{array}{c}\text{Rate of} \\ \text{mass flow out} \end{array}\right) &= \left(\begin{array}{c}\text{Rate of accumulation} \\ \text{of mass in tank} \end{array}\right) \\ \rho q(t) - \rho q_o(t) &= \frac{d(\rho Ah)}{dt}\\ \text{Taking $\rho$ and $A$ as constant,}\\ q(t) - q_o(t) &= A \frac{dh}{dt} \\ q - q_o &=A\frac{dh}{dt} \end{aligned}\] The flow rate \(q_o\) depends on the liquid level \(h\) and resistance \(R\), as below: \[q_o = \frac{h}{R}\] Therefore, \[\begin{aligned} q-\frac{h}{R} &= A\frac{dh}{dt} \end{aligned}\]