6.1 Draining of a Tank
209
Tasks
Build a mathematical model of fluid flow into the tank based on the principle of
equivalence of mathematical models. Derive the differential equation for the liquid
level in the tank. Show the relationship between the height h of the liquid and the
input flow rate q i .
Formulation of Problem 2
Consider the reservoir from the previous example with the base area A = 1 m
2 . Let
the input stream be constant for the first 150 s, after which it increases three times:
q i =
f, t < 150 s
3 f, t ≥ 150 s
where f is the given parameter. When performing the simulation, we set f =
0.15 m
3
/s. And finally, we will limit the flow at the outlet of the tank to the minimum
V min = 0 m
3
/s and maximum V min = 0.5 m
3
/s.
Tasks
Controlling the output stream with the help of a proportional-integral controller (PI
controller) with an integrated gain K = 0.1 m
2
/s and a time constant T = 10 s, it is
required to maintain the tank filling level at a height h ref = 0.25 m.
Replace the PI controller with the PID controller and compare the resulting models. You need the value of the time constant of the differential gain, take it equal to
T d = 5 s.
Formulation of Problem 3
Consider a system consisting of three identical reservoirs, with a fluid flow controlled
by three PI regulators. Suppose that the level h ref = 0.2 m is required to be maintained
in the first and third tanks, and the level increase to h ref = 0.4 m is permissible in
the second tank.
Tasks
Build a hierarchical model of the system using the components created in the previous
tasks.
Modeling and computational Experiment 1
We construct a mathematical model of fluid intake into the tank based on the principle of equivalence of hydraulic systems and electrical circuits. The main elements
of hydraulic systems are hydraulic resistance, hydraulic capacity, and inertia (see
Fig. 6.3).
These elements are similar to their electrical equivalents—resistance, capacitance,
and inductance. The electric current in this approach is equivalent to the fluid flow
209
Tasks
Build a mathematical model of fluid flow into the tank based on the principle of
equivalence of mathematical models. Derive the differential equation for the liquid
level in the tank. Show the relationship between the height h of the liquid and the
input flow rate q i .
Formulation of Problem 2
Consider the reservoir from the previous example with the base area A = 1 m
2 . Let
the input stream be constant for the first 150 s, after which it increases three times:
q i =
f, t < 150 s
3 f, t ≥ 150 s
where f is the given parameter. When performing the simulation, we set f =
0.15 m
3
/s. And finally, we will limit the flow at the outlet of the tank to the minimum
V min = 0 m
3
/s and maximum V min = 0.5 m
3
/s.
Tasks
Controlling the output stream with the help of a proportional-integral controller (PI
controller) with an integrated gain K = 0.1 m
2
/s and a time constant T = 10 s, it is
required to maintain the tank filling level at a height h ref = 0.25 m.
Replace the PI controller with the PID controller and compare the resulting models. You need the value of the time constant of the differential gain, take it equal to
T d = 5 s.
Formulation of Problem 3
Consider a system consisting of three identical reservoirs, with a fluid flow controlled
by three PI regulators. Suppose that the level h ref = 0.2 m is required to be maintained
in the first and third tanks, and the level increase to h ref = 0.4 m is permissible in
the second tank.
Tasks
Build a hierarchical model of the system using the components created in the previous
tasks.
Modeling and computational Experiment 1
We construct a mathematical model of fluid intake into the tank based on the principle of equivalence of hydraulic systems and electrical circuits. The main elements
of hydraulic systems are hydraulic resistance, hydraulic capacity, and inertia (see
Fig. 6.3).
These elements are similar to their electrical equivalents—resistance, capacitance,
and inductance. The electric current in this approach is equivalent to the fluid flow