Section 4–3 / Pneumatic Systems 109
The capacitance Cis then obtained as
(4–12)
The capacitance of a given vessel is constant if the temperature stays constant. (In many
practical cases, the polytropic exponent nis approximately 1.0~1.2 for gases in unin-
sulated metal vessels.)
Pressure Systems. Consider the system shown in Figure 4–4(a). If we assume
only small deviations in the variables from their respective steady-state values, then this
system may be considered linear.
Let us define
gas pressure in the vessel at steady state (before changes in pressure have
occurred), lbfft^2
pi=small change in inflow gas pressure, lbfft^2
po=small change in gas pressure in the vessel, lbfft^2
V=volume of the vessel, ft^3
m=mass of gas in the vessel, lb
q=gas flow rate, lbsec
r=density of gas, lb/ft^3
For small values of piandpo, the resistance Rgiven by Equation (4–8) becomes constant
and may be written as
The capacitance Cis given by Equation (4–9), or
Since the pressure change dpotimes the capacitance Cis equal to the gas added to the
vessel during dtseconds, we obtain
or
which can be written as
Ifpiandpoare considered the input and output, respectively, then the transfer function
of the system is
whereRChas the dimension of time and is the time constant of the system.
Po(s)
Pi(s)
=
1
RCs+ 1
RC
dpo
dt
+po=pi
C
dpo
dt
=
pi-po
R
Cdpo=q dt
C=
dm
dp
R=
pi-po
q
P
–
=
C=
V
nRgas T
