The figure below shows a liquid storage system consisting of two
interacting identical tanks. Tank 1 has one additional outlet by gravity, while
the outlet for tank 2 is through a pump. The two valves in the process are linear
and have an equal resistance (R = 2 hr/m²), the two tanks have equal cross-
sectional areas (A = 3 m²), and the process is isothermal. Initially, $q_1$ = 2 m³/hr,
the pump is shut off ($q_3$=0 m³/hr), and the tanks are at steady state.
Perform the following tasks:
a) Identify the state variables and inputs of this process.
b) Develop a transient model for this process.
c) Solve for the steady state values of the state variables.
d) Write the model in terms of deviation variables.
e) Obtain a Laplace domain representation of the model.
f) Obtain transfer functions relating $H_1(s)$ to both $Q_1(s)$ and $Q_3(s)$. And the
same for $H_2(s)$.
g) What are the values of the damping coefficient and time constant for
this second order system?
h) If $q_1$ is suddenly decreased to $q_1$ = 1.5 m³/hr (step change) while $q_3$ is kept
constant, use the final value theorem to claculate the new steady state.
i) If the pump is turned on to $q_3$ = 0.5 m³/hr while $q_1$ is kept at its original
value, use the final value theorem to claculate the new steady state.
j) Comment on the differences observed in (h) and (i).
If frequency analysis is performed on this system, sketch the
Normalized Amplitude Ratio Bode plot (vs. frequency) for the transfer
function relating $H_2(s)$ to $Q_1(s)$. Indicate all important features and
special points.
