ME 200 - HOMEWORK #6 – FALL 2014
DUE: Friday, October 31, 2014
Please note that you must show appropriate sketches for each problem.
6.1
(a) An inventor claims to have developed a refrigeration unit with a COP of 8.5 which maintains
the refrigerated space at -10oC while operating in a room where the temperature is 25oC. Can this
claim be correct?
(b) Consider an engine in outer space which operates on the Carnot cycle. The only way in which
heat can be transferred from the engine is by radiation. The rate at which the heat is radiated is
proportional to the fourth power of the absolute temperature and to the area of the radiating
surface. Show that for a given power output and a given TH, the area of the radiator will be a
minimum when TL/TH = ¾.
6.2
A power cycle receives 1120 Btu by heat transfer from a reservoir at 1080oF and discharges
energy by heat transfer to a reservoir at 280oF. The thermal efficiency of the cycle is 68% of that
for a reversible power cycle operating between the same reservoirs. For the actual cycle,
determine the thermal efficiency and energy discharged to the cold reservoir, in Btu.
6.3
The thermal efficiency of reversible power cycle operating between hot and cold reservoirs is
38%. Evaluate the COP of:
(a) a reversible refrigeration cycle operating between the same two reservoirs.
(b) a reversible heat pump cycle operating between the same two reservoirs.
6.4
A refrigeration cycle maintains a clean room at 68oF by removing energy entering the room by
heat transfer at the rate of 0.21 Btu/s. The cycle rejects energy by heat transfer to the outdoors
where the temperature is 80oF. Assume steady state operation.
(a) If the rate at which the cycle rejects energy by heat transfer to the outdoors is 0.24 Btu/s,
determine the power required, in Btu/s.
(b) Determine the power required for a reversible refrigeration cycle operating between cold and
hot reservoirs at 68oF and 80oF, respectively, and the corresponding rate at which energy is
rejected by heat transfer to the outdoors, each in Btu/s.
6.5
The interior of a building is maintained at 23oC by a heat pump cycle operating at steady state
while receiving thermal energy from well water at 12oC. The thermal energy discharged to the
building is at a rate of 116,250 kJ/h. Over a period of 21 days, a meter reads electricity
consumption of 2080 kW.h to run the heat pump. Determine:
(a) the amount of thermal energy that the heat pump receives over the 21-day period from the
well water , in kJ.
(b) the heat pump’s coefficient of performance.
(c) the coefficient of performance of a reversible heat pump cycle operating between hot and cold
reservoirs at 23oC and 12oC.
6.6
A gas within a piston-cylinder assembly executes a Carnot power cycle during which the
isothermal expansion occurs at TH = 720 K and the isothermal compression occurs at TC = 340 K.
Determine:
(a) the thermal efficiency
(b) the percent change in thermal efficiency if TH increases by 18% while TC remains the same.
(c) the percent change in thermal efficiency if TC decreases by 18% while TH remains the same.
(d) the percent change in thermal efficiency if TH increases by 18% and TC decreases by 18%.
6.7
A thermodynamic cycle operates between hot and cold reservoirs at 1120 K and 540 K,
respectively, while receiving energy via heat transfer from the hot reservoir at a rate of 1620 kW.
The cycle discharges energy heat transfer to the cold reservoir, and develops a power at a rate of
(a) 1000 kW, (b) 750 kW, (c) 0 kW. For each case, use the appropriate equation of a time-rate
basis to determine whether the cycle operates reversibly, operates irreversibly, or is impossible.
6.8
A heat engine operates between reservoirs at 1000 and 300 K. Some of the power output from the
heat engine drives a refrigerator which operates between 250 and 300 K. The heat-transfer input
at 1000 K to the heat engine is 1000 kJ, and the heat transfer rejected from the refrigerator is 2700
kJ.
(a) If the heat engine and refrigerator are both internally reversible devices, find the percentage of
the heat-engine work output used to drive the refrigerator.
(b) If the COP of the refrigerator is 80% of the value for an internally reversible device operating
between the given temperatures, determine the actual work (kJ) required to operate the
refrigerator.
(c) In (b), what is the heat transfer rate from the heat engine to the cold reservoir assuming the
thermal efficiency of the heat engine is 35%
6.9
An irreversible heat pump is designed to remove heat at 7oC from the atmosphere and supply
43,200 kJ/h to an industrial process at 420 K. The COP of the actual heat pump is 60% of that of
an internally reversible heat pump operating between the same two temperatures. The heat pump
is driven, through a transmission, by the output of a heat engine which receives heat transfer from
a reservoir at 1050 K and rejects heat to the same industrial process at 420 K. The thermal
efficiency of the heat engine is 75% of that of an internally reversible heat engine operating
between the same two temperatures. The transmission which delivers the power to the heat pump
from the heat engine has an efficiency of 80%. Determine
(a) the power output required for the heat pump (kW),
(b) the rate of heat input to the heat engine (kW), and
(c) the heat transfer rate from the heat engine to the industrial process (kW).