PHGN341: Thermal Physics
Exam II - April 13, 2012
NAME:
1. (30) A 2000 Megawatt (electrical) nuclear-powered generating plant operates on a Rankine cycle steam engine between
(T1 = 20 C,P1 = .023 bar) and (T3 = 500 C, P3 = 300 bar). The steam tables are attached.
2
(a) Based on the calculated efficiency, what is the minimum mass of water that
must pass through the generating plant turbines per second? Show all work.
3
Pressure
(Steam)
(Water)
1
(Water + Steam)
4
Volume
(b) The waste heat is dumped into a nearby river which has a flow rate of 4 × 104 kg per second. Upstream from the
plant the temperature of the river water is 20 C. What is the temperature of the river water downstream from the
plant? The heat capacity of water at constant pressure is cp = 4.18 kJ/(kg-K).
1
2. (30)Aluminum silicate (Al2 SiO5 ) can exist in three crystalline forms: kyanite, andalusite, and sillimanite. The thermodynamic properties for one mole of these forms are given in the table below.
Substance
Kyanite
Andalusite
Sillimanite
∆f H (kJ)
-2594.29
-2590.27
-2587.76
∆f G (kJ)
-2443.88
-2442.66
-2440.99
S (J/K)
83.81
93.22
96.11
V (cm3 )
44.09
51.53
49.90
(a) The temperature of the earth increases by about 20 C per kilometer of depth while the pressure increases by about
32 MPa per kilometer of depth. Assume that the ground level temperature is 25 C (298 K) and the pressure is 1 bar
(105 Pa). Write down the equations for the temperature and pressure as a function of depth, z.
(b) What is the stable form of aluminum silicate at a depth of 50 km? Show all work and give your reasoning.
2
3. (40)(a) Consider an ideal gas consisting of N indistinguishable (massless) neutrinos at temperature, T . Recall the
particle-in-a-box result for the density of states in momentum space: d3 n = hV3 d3 p. Use this to calculateRthe partition
function for a single neutrino by integrating the Boltzmann factor times the density of states, i.e. Z =
d3 n e−βEp ,
where the kinetic energy is given by the ultrarelativistic relation: Ep = pc where c is the speed of light.
(b) Calculate the average energy of one neutrino. Comment on your result.
(c) What is the partition function for the N-neutrino gas? (Use the Sterling’s approximation result, N ! ' N N e−N .)
(d) From the Helmholtz free energy, F = −kT ln Z, calculate the pressure for the neutrino gas. Comment on your
result.
3
Steam Tables
T (◦ C)
0
10
20
30
50
100
P (bar)
0.006
0.012
0.023
0.042
0.123
1.013
Hwater (kJ)
0
42
84
126
209
419
Hsteam (kJ)
2501
2520
2538
2556
2592
2676
Swater (kJ/K)
0
0.151
0.297
0.437
0.704
1.307
Ssteam (kJ/K)
9.156
8.901
8.667
8.453
8.076
7.355
Table 1. Thermodynamic properties of saturated water/steam for 1 kg of material.
P (bar)
1.0
3.0
10
30
100
300
Property
H (kJ)
S(kJ/K)
H (kJ)
S(kJ/K)
H (kJ)
S(kJ/K)
H (kJ)
S(kJ/K)
H (kJ)
S(kJ/K)
H (kJ)
S(kJ/K)
200
2875
7.834
2866
7.312
2828
6.694
300
3074
8.216
3069
7.702
3051
7.123
2994
6.539
400
3278
8.544
3275
8.033
3264
7.465
3231
6.921
3097
6.212
2151
4.473
500
3488
8.834
3486
8.325
3479
7.762
3457
7.234
3374
6.597
3081
5.791
Table 2. Thermodynamic properties of superheated steam for 1 kg of material.
4
600
3705
9.098
3703
8.589
3698
8.029
3682
7.509
3625
6.903
3444
6.233