1)One-Half Pound of Water Executes a Carnot Power Cycle. During the Isothermal Expansion

Problem set 7

1)One-half pound of water executes a Carnot power cycle. During the isothermal expansion, the water is heated at 600°F from a saturated liquid to a saturated vapor. The vapor then expands isentropically to a temperature of 90°F. The mixture is then compressed at constant temperature to a quality of 36.9%.

(a) Sketch this Carnot power cycle on a Ts diagram with the two phase region. Label each state and the temperature.
(b) Evaluate the heat and work for each process, in Btu.
(c) Evaluate the thermal efficiency.

W12 = 38.4 Btu,Q12 = 274.85 Btu

W23 = 200.2 Btu,Q23 = 0

W34 = − 142.61 Btu,Q34 = − 8.29 Btu

W41 = − 94.7 Btu,Q41 = 0

thermal efficiency = 48.1%

2)Carbon Dioxide (CO2) as an ideal gas executes a Carnot cycle while operating between thermal reservoirs at 450 and 100°F. The pressures at the initial and final states of the isothermal expansion are 400 and 200 lbf/in.2, respectively. The specific heat ratio is k = 1.24. Using the following formulas

as needed, determine
(a) the work and heat transfer for each of the four processes, in Btu/ lb.
(b) the thermal efficiency.

(c) the pressures at the initial and final states of the isothermal compression, in lbf/in.2

W12/m = 28.46 Btu,Q12/m = 28.46 Btu/m

W23/m = 65.8 Btu,Q23/m = 0

W34/m = − 17.52 Btu,Q34/m = − 17.52 Btu

W41/m = − 65.8 Btu,Q41/m = 0

thermal efficiency = 48.1%

P3 = 16.28 psia, P4 = 35.26 psia

3) Ocean temperature energy conversion (OTEC) power plants generate power by utilizing the naturally occurring decrease with depth of the temperature of ocean water. Near Florida, the ocean surface temperature is 300 K, while at a depth of 700 m the temperature is 280 K.
(a) Determine the maximum thermal efficiency for any power cycle operating between these temperatures.
(b) The thermal efficiency of existing OTEC plants is approximately 2 percent. Compare this with the result of part (a)

6.7%

4)At steady state, a heat pump provides 30,000 Btu/h to maintain a dwelling at 68°F on a day when the outside temperature is 35°F. The power input to the heat pump is 5 hp. If electricity costs 8 cents per kW · h, compare the actual operating cost with the minimum theoretical operating cost for each day of operation.

The actual operating cost: 7.16 $/day.
The minimum theoretical operating cost: 1.02$/day.

5)By supplying energy at an average rate of 22032 kJ/h, a heat pump maintains the temperature of a dwelling at 21°C. If electricity costs 8 cents per kW · h, determine the minimum theoretical operating cost for each day of operation if the heat pump receives energy by heat transfer from
(a) the outdoor air at −5°C.
(b) well water at 8ºC.

(a) $1.00/day(b) $0.50/day

6)A heat pump maintains a dwelling at temperature T when the outside temperature averages 5°C. The heat transfer rate through the walls and roof is 2000 kJ/h per degree of temperature
difference between the inside and outside. If electricity costs 8 cents per kW · h. Determine the minimum theoretical operating cost for each day of operation when T = 20ºC.

$0.82/day

7)A system executes a power cycle while receiving 1000 kJ by heat transfer at a temperature of 500 K and discharging energy by heat transfer at a temperature of 300 K. There are no
other heat transfers. Determine σcycle if the thermal efficiency is (a) 60%, (b) 40%, (c) 20%.

σcycle = − 0.667 kJ/K;0; 0.667 kJ/K

8)Determine the minimum theoretical power, in Btu/s, required at steady state by a refrigeration system to maintain a cryogenic sample at −195ºF in a laboratory at 70ºF, if energy
leaks by heat transfer to the sample from its surroundings at a rate of 0.054 Btu/s.

0.085 Btu/s

9)For each kW of power input to an ice maker at steady state, determine the maximum rate that ice can be produced, in kg/h, from liquid water at 0ºC. Assume that 333 kJ/kg of energy
must be removed by heat transfer to freeze water at 0ºC, and that the surroundings are at 20ºC.

147.6 kg/h