Solution– Heat, Internal Energy and the First Law
1 A mass of 0.1 kg of water fills a piston-cylinder apparatus with an initial volume of V1 =
0.01 m3. The water then undergoes a three step path where the initial and final states
are the same:
a. an expansion to a volume of V2 = 0.02 m3 following the path equation P = a + b/V
where a = 3 MPa and b = –0.02 MPa·m3.
b. a constant volume decrease in pressure P3 = P1, the initial pressure.
c. a constant pressure decrease in volume to V4 = V1, the initial volume.
Find the heat transfer for (i) part a of the path and (ii) the entire path.
We can use the path equation to find the work for part a as pathPdV.
1
2
12 ln
2
1V
V
bVVadV
V
b
aPdVW
V
Vapath
a
kJMJmMPa
m
m
mMPammMPa
V
V
bVVaWa
137.16016137.0016137.0
01.0
02.0
ln02.001.002.03ln
3
3
3
333
1
2
12
The heat transfer along this path can be found from the first law: Qa = Ua + Wa = (u2 – u1) + Wa.
We can find the internal energy, u1 at the initial state v1 = V1/m = (0.01 m3)/(0.1 kg) = 0.1 m3/kg
and P1 = a + b/V1 = (3 MPa) + (-0.02 MPa·m3) / (0.01 m3) = 1 MPa. From the saturation table at
P1 = 1 MPa we see that this specific volume is between vf and vg, so we must be in the mixed
region. We can find the internal energy from the quality:
5115.0
1933.0
001127.01.0
)1(
)1(
3
33
1
11
1
kg
m
kg
m
kg
m
MPaPv
MPaPvv
x
fg
f
kg
kJ
kg
kJ
kg
kJ
MPaPuxMPaPuu fgf2.16938.1821)5115.0(37.761)1()1( 1111
Since the tables do not have internal energy, we have to compute it from the enthalpy. At the
final state, P2 = a + b/V2 = 3 MPa + (–0.02 MPa·m3)/(0.2 m3) = 2 MPa, and v2 = V2/m = (0.02
m3)/(0.1 kg) = 0.2 m3/kg. At this pressure and specific volume we see that the temperature is
very close to 600oC. If we used interpolation we would find the final enthalpy as follows:
kg
kJ
kg
m
kg
m
kg
m
kg
m
kg
kJ
kg
kJ
kg
kJ
h5.369319960.02.0
19960.021144.0
7.36897.3802
7.3689 33
33
2
We can now find the internal energy at the end of the first path, u2.
kg
kJ
mMPa
kJ
kg
m
MPa
kg
kJ
vPhu 5.329310002.0
2
7.3689
3
3
2222
We can now find the heat transfer for step a from the first law.
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