Question 432: Thermodynamics & KTG

Question

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other
engine to produce work) between temperature, T1
 and T2
. The temperature of the hot reservoir of
the first engine is T1
 and the temperature of the cold reservoir of the second engine is T2
. T is
temperature of the sink of first engine which is also the source for the second which is also the
source for the second engine. How is T related to T1
 and T2
. If both engines perform equal amount
of work?

Accepted Answer

Q
H
: Heat input to 1
st
engine
Q
L
: Heat rejected from 1
st
engine
Q'
L
: Heat rejected from 2
nd
engine
T : Lower temperature of first engine
Work done by 1
st
engine = work done by 2
nd
engine
Q
H
Ã¢ÂÂ Q
L
= Q
L
Ã¢ÂÂ Q'
L
$$ \Rightarrow $$ 2Q
L
= Q
H
+ Q'
L
$$ \Rightarrow $$ 2 = $${{{Q_H}} \over {{Q_L}}} + {{Q{'_L}} \over {{Q_L}}}$$
$$ \Rightarrow $$ 2 = $${{{T_1}} \over T} + {{{T_2}} \over T}$$
$$ \Rightarrow $$ $$T = {{{T_1} + {T_2}} \over 2}$$

Answer

$$T = {{2{T_1}{T_2}} \over {{T_1} + {T_2}}}$$

Answer

$$T = \sqrt {{T_1}{T_2}} $$

Answer

$$T = {{{T_1} + {T_2}} \over 2}$$

Answer

T = 0

Thermodynamics & KTG

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