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PHGN341: Thermal Physics Exam II - April 12, 2013 NAME: 1. (30) A nuclear power plant generates 3 GW (3 × 109 W) of electricity by operating on a Rankine cycle (steam engine) between {T1 = 20 C,P1 = .023 bar} and {T3 = 600 C, P3 = 300 bar}. (The steam tables are attached.) 2 (a) What is the enthalpy of 1 kg of water after the turbine power step (3 → 4)? 3 Pressure (Steam) (Water) 1 (Water + Steam) 4 Volume (b) Suppose the plant operates at an efficiency of 48%, what is the minimum mass of water that must pass through the generating plant turbines per second to generate 3 GW electric power? (c) The waste heat is dumped into a nearby river which has a flow rate of 2 × 105 kg per second. Upstream from the plant the temperature of the river water is 20 C. What is the average 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. (20) Estimate the temperature of the melting point of ice at 200 bar pressure. (Data: Latent heat: L = 0.333 MJ/kg, upon freezing 1000 cm3 of water (1 kg) expands to 1091 cm3 .) 3. (20) A methane (CH4 ) fuel cell operates by combining oxygen and methane to form water (liquid) carbon dioxide at standard pressure and temperature with the transfer of 4 electrons per molecule of water formed: CH4 + 2O2 → 2H2 O + CO2 . Table 1: Thermodynamic properties per mole of substance. Substance ∆f H (kJ) ∆f G (kJ) S (J/K) CP (J/K) H2 O (l) -285.83 -237.13 69.91 75.29 O2 (g) 0.0 0.0 205.14 29.38 CH4 (g) -74.81 -50.72 186.26 35.31 CO2 (g) -393.51 -394.36 213.74 37.11 a. Calculate the voltage of a methane fuel cell? b. How much heat is delivered to the environment per mole of water produced? 2 4. (30) Consider an ideal gas consisting of N indistinguishable (massless) neutrinos at temperature, T , confined to an area, A. The neutrino’s kinetic energy is given by the ultrarelativistic relation: E = pc where c is the speed of light. A 2 (Recall that the particle-in-a-box result for the density of states in 2 dimensions is d2 n = (2πh̄) 2 d p.) (a) Calculate the partition function for a single neutrino confined to area, A. (b) Calculate the average energy of one neutrino in two dimensions. (c) What is the partition function for the N-neutrino gas in terms of the single neutrino partition function? (d) Use your result from part (c) to calculate the 2-dimensional analog of the “pressure” (force per unit length along the boundary of the confining area) for the neutrino gas. 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 1 kg of saturated water/steam. P 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 C 2875 7.834 2866 7.312 2828 6.694 300 C 3074 8.216 3069 7.702 3051 7.123 2994 6.539 400 C 3278 8.544 3275 8.033 3264 7.465 3231 6.921 3097 6.212 2151 4.473 Table 2. Thermodynamic properties for 1 kg of superheated steam. 4 500 C 3488 8.834 3486 8.325 3479 7.762 3457 7.234 3374 6.597 3081 5.791 600 C 3705 9.098 3703 8.589 3698 8.029 3682 7.509 3625 6.903 3444 6.233