Question

Draw the

p - V

diagram and

T - S

diagram of a carnot cycle and explain these. What is the efficiency of a carnot engine.

Accepted Answer

P-V Diagram

The working substance in carnot engine undergoes into a cyclic process in four steps:

(I) isothermal expansion:

$W_{1} = 2.303 n r T_{1} \log (\frac{V_{2}}{V_{1}}) . . . . . . . . . . (1) W_{1} = A r e a u n d e r c u r v e A B$

(II) Adiabatic Expansion:

$W_{2} = \frac{n R (T_{1} - T_{2})}{r - 1} . . . . . . . . . (2) W_{2} = A r e a u n d e r c u r v e B C$

(III) Isothermal compression:

$W_{3} = 2.3.3 n R T_{2} \log (\frac{V_{3}}{V_{4}}) W_{3} = A r e a u n d e r c u r v e C D$

(IV) Adiabatic Compression:

$W_{4} = \frac{n R (T_{1} - T_{2})}{r - 1} W_{4} = A r e a u n d e r c u r v e A D$

Net work done in cyclic process:

$W = W_{1} + W_{2} - W_{3} - W_{4}$

$W = W_{1} - W_{3}$

$W = Q_{1} - Q_{3} (W_{1} = Q_{1}, W_{2} = Q_{2}, W_{3} = Q_{3} a n d W_{4} = Q_{4})$

$W = 2.303 n R T_{1} \log (\frac{T_{2}}{T_{1}}) - 2.303 n R T_{1} \log (\frac{T_{3}}{T_{4}})$

T-S Diagram

Available energy per cycle:

$Q_{1} - Q_{2} = (2 T_{1} \times S_{1} + T_{1} \times \frac{1}{2} \times 2 S_{1}) - 2 T_{1} S_{1}$

$= T_{1} S_{1}$

$Q = 3 T_{1} S_{1}$

$E f f i c i e n c y o f t h e e n g i n e (η) = \frac{Q_{1} - Q_{2}}{Q_{1}}$

$= \frac{T_{1} S_{1}}{3 T_{1} S_{1}} = \frac{1}{3}$

$E f f i c i e n c y o f C a r n o t e n g i n e i s g i v e n b y η = 1 - \frac{T_{2}}{T_{1}}$

Draw the $p - V$ diagram and $T - S$ diagram of a carnot cycle and explain these. What is the efficiency of a carnot engine.

Important Questions on Thermodynamics