Question

Show that the thermal efficiency of the Carnot cycle in terms of the isentropic compression ratio 𝑟𝑘 is given by 𝑒 = 1 − (1 /(𝑟𝑘) 𝑘−1). 

Answer 1

thermal efficiency:

$𝑒 = 1 − \frac{1 }{𝑟_𝑘^{𝑘−1}}$

$e=\frac{work\ done}{heat\ supplied}$

for Isentropic Compression 4-1:

$e=1-\frac{T_3}{T_2}$

$\frac{T_3}{T_2}=(\frac{V_2}{V_3})^{k-1}=(\frac{1}{r_k})^{k-1}$

$𝑒 = 1 − \frac{1 }{𝑟_𝑘^{𝑘−1}}$