SYSTEM DYNAMICS (ME 344)
Homework #9
TWO PAGES
Due THURSDAY 11/8/2012
1. Consider the system shown at right.
Assume the resistances R1 and R2
are constants, and the areas A1 and
A2 are constant so that the tank
capacitances are constant (C1 and
C2). Consider the inflow rate qmi to
be an input variable and the outflow
rate through the pipe with resistance
R2 to be an output variable.
The heights of liquid in the two tanks describe the state of the system. First, write
equations for the rate of change of height in each tank, in terms of the heights and the inflow
rate, as well as an equation for the outflow rate in terms of those variables. Then, write your
equations in standard state-space format ( x = Ax + Bu and y = Cx + Du ).
2. Consider the system shown at
right. Assume the resistances R1
and R2 are constants, and the areas
A1 and A2 are constant so that the
tank capacitances are constant (C1
and C2). Consider the inflow rate
qmi to be an input variable and the
height of liquid in the second tank,
h2, to be an output variable.
Write equations for the flow
through each pipe and the
changing height in each tank, and
use these to obtain the transfer
function relating input qmi to output h2, that is, derive H2 (s) / Qmi (s) .
3. A pressure vessel has a volume of 0.5 m3 and is filled with air at a nominal temperature of
300 K (27° C), near which the specific heats are approximately cp = 1.00 kJ/kgK and cv =
0.718 kJ/kgK. Calculate the pneumatic capacitance of the vessel
(a) for an isothermal process (constant C), and
(b) for an adiabatic process (strictly speaking, C will be a function of temperature, but
assume small temperature fluctuations near 300 K and simply calculate the value
corresponding to 300 K).
4. Consider the pneumatic pressure system shown below. Two rigid tanks, whose pneumatic
capacitances are C1 and C2 respectively, are connected to pipes with resistances R1 and R2 as
shown. The tanks have pressures p1 and p2 that are small deviations about some steady
reference pressure. A source of pressure pi (also measured relative to the same reference
pressure) is connected as shown. Derive the transfer function P2(s)/Pi(s), assuming an
isothermal process.