Question

Two carnot engines a and b are operated in series. The first one a receives heat at t1 = 600k and rejects to a reservoir at temperature t2. The second engine b receives heat rejected by the first engine and in turn, rejects to a heat reservoir at t3 = 400k. Calculate the temperature t2 if the work outputs of the two engines are equal

Answer 1

Work done by first engine W =  Q1 – Q2

Work done by 2nd engine W = Q2 – Q3

Thus,

Q1 – Q2 = Q2 – Q3

(As work output of the two engines are equal)

Dividing both sides by Q1

1 – Q2/Q1  = Q2/Q1 – Q3/Q1

=> 1-T/T1 = Q2/ Q1(1- Q3/ Q2)

=> 1-T/T1 = Q2/ Q1(1- T3/ T)

=>  1-T/T1 = T/T1 (1- T3/ T)

=> T1/T – 1 = 1 - T3/ T

=> T1/T + T3/ T = 2

=> 1/T (T1 + T3) = 2

=>T = (T1 + T3)/2

=> T = (600+400)/2 = 500 K

Cheers!!