Old spec 1.2 example, two tank hydraulic system (ENGR5001)

Old spec 1.2 example, two tank hydraulic systemTutorialOld spec ENGR2022:115 min

ENGR202 (old spec) Control 1.2 Mechanistic Models, two tank hydraulic system (slides 14-17; generalised form in 1.3 slides 24-25)

Two tanks are connected in cascade (see figure). Tank 1 (area $A_1$ , head $h_1$ ) is filled at flow rate $Q_1$ and drains through an orifice at flow rate $Q_2$ into tank 2 (area $A_2$ , head $h_2$ ), which drains through its own orifice at flow rate $Q_3$ . Assume, in addition to the single tank assumptions, that each outflow is proportional to its own head: $Q_2 = K_1h_1$ and $Q_3 = K_2h_2$ .

Two tanks in cascade: Q_1 into tank 1 (area A_1, head h_1), Q_2 from tank 1 into tank 2 (area A_2, head h_2), Q_3 out of tank 2.

Formulas you may need

Volume balance $A\dfrac{dh}{dt} = Q_{in} - Q_{out}$ ; linear orifice $Q = Kh$ (learn this)
Generalised second order form $\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2 x = Ku(t)$ (learn this)

(a)
Derive a first order model for each tank.
(b)
Eliminate $Q_2$ to obtain a single model relating $Q_3$ to $Q_1$ .
(c)
Put the model into generalised second order form, giving $\omega_n$ , $\zeta$ and the steady state gain. Show that the system can never oscillate.