Thermal efficiency of a heat engine operating between two thermal reservoirs of 700 C and 100°C is 40% of the Carnot efficiency. Heat input the engine is 12 MJ. a) What is the thermal efficiency of the heat engine? b) Calculate the work output of the engine c) What amount of heat is transferred to the low temperature reservoir?
Answers
Answer:
Two Carnot engines A and B are operated in series. The first one A receives heat at 800 K and rejects to a reservoir at temperature T. The second engine B receives the heat rejected by the first engine and in turn rejects to a heat reservoir at 300 K. Calculate the temperature T for the following situations:
(a) The outputs of the two engines are equal and
(b) The efficiencies of two engines are equal
Explanation:
Given:
T
1
=800K
T
3
=400K
To find:
Temperature T
(i) when output of two engines are equal
(ii) when efficiencies of two engines are equal.
Let the output of both engines be W.
Let the engine A take Q
1
heat as input at temperature T
1
and gives out heat Q
2
at temperature T the second engine B receive Q
2
as input and give out Q
3
at temperature T
2
to the sink.
Work done by engine, A W=Q
1
–Q
2
Work done by engine, B W=Q
2
–Q
3
Thus,
Q
1
–Q
2
=Q
2
–Q
3
Dividing both sides by Q
1
,
1–
Q
1
Q
2
=
Q
1
Q
2
–
Q
1
Q
3
⟹1−
T
1
T
=
Q
1
Q
2
(1−
Q
2
Q
3
)
⟹1−
T
1
T
=
Q
1
Q
2
(1−
T
T
3
)
⟹1−
T
1
T
=
T
1
T
(1−
T
T
3
)
⟹
T
T
1
–1=1−
T
T
3
⟹
T
T
1
+
T
T
3
=2
⟹
T
1
(T1+T3)=2
⟹T=
2
(T
1
+T
3
)
⟹T=
2
(900+400)
=650K
is the Temperature when output of the engines are equal.
Let the efficiency of both engines be η. Now considering both engines efficiency are equal. This gives,
1−
T
1
T
=1−
T
T
3
⟹
T
1
T
=
T
T
3
⟹T
2
=T
1
×T
3
⟹T
2
=800×400=320000
T=565.68K is the temperature when efficiencies of the both the engines are equal.