Question

37. What is Carnot engine?
Explain the construction and various operations for Carnot's heat engine working with two
temperatures. A reversible heat engine operates with an efficiency of 50%.
If during each cycle it rejects 150 cal to a reservoir of heat at 30°C, then what is the temperature of the other reservoir?​

Answer 1

Part-1

$\mathcal{CARNOT\:\:CYCLE}$

Carnot Engine is an ideal heat engine which is based on Carnot's reversible cycle.

Construction:

1) Source of heat. The source is maintained at a fixed higher temperature from which the working substance draws heat.

2) Sink of heat. The sink is maintained at a fixed lower temperature to which any amount of heat can be rejected by the working substance.

3) Working substance. A perfect gas acts the working substance.

The Carnot cycle consists of the following four stages:

1. Isothermal expansion
2. Adiabatic expansion
3. Isothermal expansion
4. Adiabatic compression

$\implies$Consider one gram mole of an ideal gas enclosed in the cylinder. Let $\sf{V_1,\:P_1,\:T_1}$ be the initial volume, pressure and temperature of the gas. The initial state of the gas is represented by the point A on P-V diagram. Now, the four processes are:

$\mathbb{1.\:ISOTHERMAL\:EXPANSION}$

Since the expansion is happening isothermally, therefore, temperature of the gas remains constant. This operation is represented by the isothermal curve AB. Let the amount of heat energy absorbed in the process be $\sf{Q_1\:and\:W_1}$ be the corresponding amount of work done by the gas in expanding isothermally from $\sf{A\left(V_1,\:P_1\right)}$ to $\sf{B\left(V_2,\:P_2\right)}$

$\sf{\therefore\:Q_1=W_1=\displaystyle\int\limits_{V_1}^{V_2}P\,dV=RT_1\log_e\dfrac{V_2}{V_1}}$

=area ABMKA

$\mathbb{2.\:ADIABATIC\:EXPANSION}$

The gas is allowed to expand further from $\sf{B\left(V_2,\:P_2\right)\:to\:C\left(V_3,\:P_3\right)}$ Temperature of gas falls to $\sf{T_2}$, the expansion is adiabatic and is represented by the adiabatic curve BC. Let $\sf{W_2}$ be the work done by the gas in expanding adiabatically.

$\sf{\therefore\:W_2=\displaystyle\int\limits_{V_2}^{V_3}P\,dV=\dfrac{R\left(T_2-T_1\right)}{1-\gamma}}$

=area BCNMB

$\mathbb{3\:ISOTHERMAL\:COMPRESSION}$

The gas is compressed until its pressure is $\sf{P_4}$ and volume is $\sf{V_4}$. This process is isothermal and is represented by the isothermal curve CD. Let $\sf{Q_2}$ be the amount of heat energy rejected to the sink and $\sf{W_3}$ be the amount of work done on the gas in compressing it isothermally.

$\sf{\therefore\:Q_2=W_3=\displaystyle\:\int\limits_{V_3}^{V_4}-P\,dV=-RT_2\log_e\dfrac{V_3}{V_4}}\\\sf{=RT_2\log_e\dfrac{V_3}{V_4}}$

=-area CDLNC

$\mathbb{4.\:ADIABATIC\:COMPRESSION}$

The gas is compressed to its initial volume $\sf{V_1}$ and pressure $\sf{P_1}$. Let $\sf{W_4}$ be the work done on the gas in compressing it adiabatically.

$\sf{\therefore\:W_4=\displaystyle\int\limits_{V_4}^{V_1}-P\,dV=\dfrac{-R\left(T_2-T_1\right)}{\left(1-\gamma\right)}}$

=-areaDAKLD

Work done by the engine per cycle,

Total work done by the gas=$\sf{W_1+W_2}$

Total work done on the gas=$\sf{W_3+W_4}$

Net work done by the gas in a complete cycle, $\sf{W=W_1+W_2-\left(W_3+W_4\right)}$

$\sf{W_2=W_4}$, in magnitude,

$\sf{\therefore\:W=W_1-W_3}$

$\boxed{\boxed{\sf{W=Q_1-Q_2}}}$

In terms of area,

W= area ABMKA + area BCNMB - area CDLNC - area DAKLD

$\boxed{\boxed{\sf{W=area\:ABCDA}}}$

Hence, in Carnot heat engine, net work done by the gas per cycle is numerically equal to the area of the loop representing the cycle.

Part-2

Efficiency of heat engine =$\sf{\eta=50\%=0.5}$

$\sf{Let\:T_1=temperature\:of\:source(To\:be\:found)}\\\sf{and\:T_2=temperature\:of\:sink=30^\circ}$

$\sf{we\:know\:that\eta=1-\dfrac{T_2}{T_1}}\\\sf{0.5=1-\dfrac{30}{T_1}}\\\sf{0.5=\dfrac{30}{T_1}}\\\sf{T_1=\dfrac{30}{0.5}=60^\circ\:C}$

The temperature of other reservoir is 60°.