Question

1: Efficient Engineer
An engineer is given the task to extract the maximum work out of two bodies A and B whose
heat capacities are CA and CB respectively. They are kept at temperatures TA and TB ( in
Kelvin scale). The engineer first tries to put these two bodies in contact.
(a) What would be the amount of work extracted during this process?
The engineer then tries to design an engine to get the best output ( work ) from this situation.
(b) What in your opinion would be the best method to achieve this goal?
(c) Find the value of the parameter for this ”engine” that will guarantee that the work is
extracted in the method is indeed maximum.

Answer 1

As per the question,

The temperature of the body A is $T_A$ and temperature of the body B be $T_B$ heat capacities of A and B are, $C_A$ and $C_B$ ,

Let heat transfer to the hot reservoir. $dQ_1=C_AdT_A$

and the heat transfer to cold reservoir $dQ_A=C_BdT_B$

a) When we will make the contact to two bodies, then heat will get transfer from one body to another body. so sum of the change in entropy will be zero.

So, $dW=dQ_1- dQ_2=C_AT_A-C_BT_B$

b) No heat should loss in the environment.

The body should have less surface area, it means it should have in sphere.

c) Now for the maximum work

$\dfrac{dQ_1}{T_A}+\dfrac{dQ_2}{T_B}=0$

so we can write it as

$C_A\dfrac{dT_A}{T_A}+C_B\dfrac{dT_B}{T_B}=0$

$\Rightarrow C_A \int \dfrac{dT_A}{T_A}+C_B\int\dfrac{dT_B}{T_B}=0$

$\Rightarrow C_A \ln T_A+C_B\ln T_B=\ln(T_AT_B)$

now taking the differentiation of both side

$\Rightarrow C_A d\ln T_A+C_B d\ln T_B=d \ln(T_AT_B)$

$\Rightarrow d\ln T_A^{C_A} + d\ln T_B^{C_B}=d \ln(T_AT_B)$

$\Rightarrow d\ln{ T_A^{C_A}}{ T_B^{C_B} }=d \ln(T_AT_B)$

if $T_A=T_B=T_f$

So, we can conclude that $T_f=\dfrac{}{}$ $\sqrt[C_A C_B]{T_AT_B}$

hence work =$W=C_A(T_A−T_f)−C_B(T_f−T_B)$

$W=C_A T_A-C_BT_B-C_AT_f-C_BT_f$

$W=C_AT_A-C_BT_B-C_A\sqrt[C_A C_B]{T_AT_B}-C_B\sqrt[C_A C_B]{T_AT_B}$