carnot engine and carnot cycle (definition and equation)

 carnot engine:-

Carnot engine is a perfect engine does not waste any heat . It it is defined with the help of carnot cycle

Carnot cycle :- carnot cycle is an ideal  cycle whose working is perfectly reversible.

According to carnot;efficiency of heat engine depend upon temperature of hot and cold reservior.

                       Efficiency = 1-Te

                                                Th

Working of carnot cycle

It contain source in temperature(Th) , sink(Tc) , insulator and a container.

It takes heat from source and throw in sink.

walls and Pistons are perfectly non conducting whereas  bottom is perfectly conducting .source and sink are at infinite heat capacity

Working  graph.

Work done in isothermal expansion

\[W_{1}=nR T_{h} ln \frac{v_{2}}{v_{1}} \\\ W_{3}=nR T_{h} ln \frac{v_{4}}{v_{3}}\rightarrow (1)...\]

Also work done in adiabatic process

\[W_{2}= \frac{1}{y-1}nR(T_{h}- T_c) \\\ W_{4}= \frac{1}{y-1}nR(T_{c}- t_{h})\]

Work done in adiabatic process is zero

\[W_{2}+ W_1=0\]

In thermodynamics expansion (b – c)

\[T_h V^{y-1}_2=T_c V^{y-1}_3 \rightarrow (2)...\]

In adiabatic compression (d – a)

\[T_c V^{y-1}_4 =T_h V^{y-1}_1 \rightarrow (3)...\]

Divide both equations 

\[ \frac{T_h V^{y-1}_2= T_c V^{y-1}_3}{T_h V^{y-1}_1=T_c V^{y-1}_4} \\\ ( \frac{ V_2}{V_1}) ^{y-1}=( \frac{ V_3}{V_4}) ^{y-1} \\\  \frac{ V_2 }{V_1}= \frac{ V_3}{V_4}  \]

Put the values of V4 and V3 in equation ( 1)

\[ W_3 = nR T_c ln( \frac{V_1}{V_2})^- \]

Total work done

\[W_1 +W_3\]

\[= nR T_h ln( \frac{V_2}{V_1})+nR T_c ln(\frac{V_1}{V_2})^- \]

\[= nR T_h ln( \frac{V_2}{V_1})(-nR T_c ln(\frac{V_2}{V_1})) \]

\[   = nR (T_h -T_c) ln( \frac{V_2}{V_1})\]

Efficiency

\[= \frac{W}{Q_1} = \eta =\frac{W}{W_1}\]

\[ \eta = \frac {nR (T_h -T_c) ln( \frac{V_2}{V_1}) }{nR (T_h ) ln( \frac{V_2}{V_1}) }\] 

\[ \eta = 1- \frac{T_c}{T_h}\]

click here  to see T-S diagrams of thermodynamic processes