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A reversible heat engine rejects 50 percent of the heat supplied during a cycle of operation. If this engine is reversed and operates as a heat pump, then what is its coefficient of performance?
- a)1.0
- b)1.5
- c)2.0
- d)2.5
Correct answer is option 'C'. Can you explain this answer?
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A reversible heat engine rejects 50 percent of the heat supplied durin...
If the engine is reversed and operated as the Heat Pump.
Then COP coefficient of performance
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A reversible heat engine rejects 50 percent of the heat supplied durin...
Given: Reversible heat engine rejects 50% of the heat supplied during a cycle of operation.
To find: Coefficient of performance (COP) when the engine operates as a heat pump.
Solution:
1. Coefficient of performance for a heat engine
The coefficient of performance for a heat engine is defined as the ratio of the work output to the heat input. It is given by:
COP_he = W/Q_h
where, W = work output, Q_h = heat input
For a reversible heat engine, the efficiency is given by:
η = 1 - Q_c/Q_h
where, Q_c = heat rejected
Since the engine rejects 50% of the heat supplied, we have:
Q_c = 0.5Q_h
Substituting this in the efficiency equation, we get:
η = 1 - 0.5 = 0.5
Therefore, the work output is:
W = ηQ_h = 0.5Q_h
Substituting this in the COP equation, we get:
COP_he = 0.5
2. Coefficient of performance for a heat pump
The coefficient of performance for a heat pump is defined as the ratio of the heat output to the work input. It is given by:
COP_hp = Q_h/W
When the reversible heat engine operates as a heat pump, the direction of the heat and work transfers is reversed. Therefore, the work input and heat output are the same as the work output and heat input of the engine, respectively. Hence, we have:
W = 0.5Q_h
Q_h = Q_c + Q_out
Since the engine is operating as a heat pump, the heat output is greater than the heat input, i.e., Q_out > Q_h. Therefore, we have:
Q_h = Q_c + Q_out > Q_c
Substituting W = 0.5Q_h in the COP equation for a heat pump, we get:
COP_hp = Q_h/W = Q_h/(0.5Q_h) = 2.0
Therefore, the coefficient of performance of the reversible heat engine when it operates as a heat pump is 2.0.
To find: Coefficient of performance (COP) when the engine operates as a heat pump.
Solution:
1. Coefficient of performance for a heat engine
The coefficient of performance for a heat engine is defined as the ratio of the work output to the heat input. It is given by:
COP_he = W/Q_h
where, W = work output, Q_h = heat input
For a reversible heat engine, the efficiency is given by:
η = 1 - Q_c/Q_h
where, Q_c = heat rejected
Since the engine rejects 50% of the heat supplied, we have:
Q_c = 0.5Q_h
Substituting this in the efficiency equation, we get:
η = 1 - 0.5 = 0.5
Therefore, the work output is:
W = ηQ_h = 0.5Q_h
Substituting this in the COP equation, we get:
COP_he = 0.5
2. Coefficient of performance for a heat pump
The coefficient of performance for a heat pump is defined as the ratio of the heat output to the work input. It is given by:
COP_hp = Q_h/W
When the reversible heat engine operates as a heat pump, the direction of the heat and work transfers is reversed. Therefore, the work input and heat output are the same as the work output and heat input of the engine, respectively. Hence, we have:
W = 0.5Q_h
Q_h = Q_c + Q_out
Since the engine is operating as a heat pump, the heat output is greater than the heat input, i.e., Q_out > Q_h. Therefore, we have:
Q_h = Q_c + Q_out > Q_c
Substituting W = 0.5Q_h in the COP equation for a heat pump, we get:
COP_hp = Q_h/W = Q_h/(0.5Q_h) = 2.0
Therefore, the coefficient of performance of the reversible heat engine when it operates as a heat pump is 2.0.
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A reversible heat engine rejects 50 percent of the heat supplied during a cycle of operation. If this engine is reversed and operates as a heat pump, then what is its coefficient of performance?a)1.0b)1.5c)2.0d)2.5Correct answer is option 'C'. Can you explain this answer?
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