108 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal Systemswhere capacitance, lb-ft^2 lbf
mass of gas in vessel, lb
gas pressure, lbfft^2
volume of vessel, ft^3
density, lbft^3
The capacitance of the pressure system depends on the type of expansion process
involved. The capacitance can be calculated by use of the ideal gas law. If the gas ex-
pansion process is polytropic and the change of state of the gas is between isothermal
and adiabatic, then
(4–10)
wheren=polytropic exponent.
For ideal gases,
or
where absolute pressure, lbfft^2
volume occupied by 1 mole of a gas, ft^3 lb-mole
universal gas constant, ft-lbflb-mole °R
absolute temperature, °R
specific volume of gas, ft^3 lb
molecular weight of gas per mole, lblb-mole
Thus
(4–11)
whereRgas=gas constant, ft-lbflb °R.
The polytropic exponent nis unity for isothermal expansion. For adiabatic expansion,
nis equal to the ratio of specific heats cpcv, where cpis the specific heat at constant pres-
sure and cvis the specific heat at constant volume. In many practical cases, the value of
nis approximately constant, and thus the capacitance may be considered constant.
The value of drdpis obtained from Equations (4–10) and (4–11). From
Equation (4–10) we have
or
Substituting Equation (4–11) into this last equation, we get
dr
dp
=
1
nRgas T
dr
dp
=
1
Knrn-^1
=
rn
pnrn-^1
=
r
pn
dp=Knrn-^1 dr
pv=
p
r
=
R
–
M
T=Rgas T
M=
v=
T=
R
–
=
v–=
p=
pv=
R
–
M
pv–=R T
–
T
pa
V
m
bn=
p
rn
=constant=K
r=
V=
p=
m=
C=
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