Polytropic Process
During actual expansion and compression processes of gases, pressure and
volume are often related by PVnC, where nand Care constants. A
process of this kind is called a polytropic process(Fig. 4 –9). Below we
develop a general expression for the work done during a polytropic process.
The pressure for a polytropic process can be expressed as
(4 –8)Substituting this relation into Eq. 4 –2, we obtain
(4 –9)since. For an ideal gas (PVmRT), this equation can
also be written as
(4 –10)For the special case of n1 the boundary work becomes
For an ideal gas this result is equivalent to the isothermal process discussed
in the previous example.
Wb
21P dV
21CV (^1) dVPV lnaV^2
V 1
b
Wb
mR 1 T 2 T 12
1 n
¬¬n 1 ¬¬ 1 kJ 2
CP 1 V 1 nP 2 V 2 n
Wb
2
1
P^ dV
2
1
CV
n¬dVC V^2
n (^1) V
1
n 1
n 1
P 2 V 2 P 1 V 1
1 n
PCVn
Chapter 4 | 171
PVn = const.
2
1
P
V
GAS
P 1
P 2
V 1 V 2
PVn = C = const.
P 1 V 1 n= P 2 V 2 n
FIGURE 4 –9
Schematic and P-Vdiagram for a
polytropic process.
EXAMPLE 4 –4 Expansion of a Gas against a Spring
A piston–cylinder device contains 0.05 m^3 of a gas initially at 200 kPa. At
this state, a linear spring that has a spring constant of 150 kN/m is touching
the piston but exerting no force on it. Now heat is transferred to the gas,
causing the piston to rise and to compress the spring until the volume inside
the cylinder doubles. If the cross-sectional area of the piston is 0.25 m^2 ,
determine (a) the final pressure inside the cylinder, (b) the total work done by
Use actual data from the experiment
shown here to find the polytropic
exponentfor expanding air. See
end-of-chapter problem 4 –174.
© Ronald Mullisen
EXPERIMENT