10. Linearizing √h

Nonlinear model:
Here the outflow is given by \[q_o=C\sqrt{h}\] From mass balance for a constant density system, \[q - C\sqrt{h} = A\frac{dh}{dt}\] Linearized model:
Here the outflow is taken as \[q_o=\frac{h}{R}\] In terms of deviation variables, i.e., \(Q=q-q_s\); and, \(H=h-h_s\), \[\frac{H(s)}{Q(s)} = \frac{R}{\tau s+1}\] where \[R = \frac{2\sqrt{h_s}}{C} \qquad \text{and} \qquad \tau = AR\]

\[\begin{aligned} q_o &= C\sqrt{h} \\ f(h) &= C\sqrt{h} \\ f(h) &\approx f(h_s) + \left(\frac{df(h)}{dh}\right)_{h_s}(h-h_s) \\ &\approx f(h_s) + \frac{C}{2\sqrt{h_s}}(h-h_s) \\ f(h) - f(h_s) &\approx \frac{C}{2\sqrt{h_s}}(h-h_s) \\ q_o - q_{os} &= \frac{C}{2\sqrt{h_s}}(h-h_s) \\ Q_o &= \frac{C}{2\sqrt{h_s}} H \end{aligned}\]