Question

Explain Carnot Engine!?

Answer 1

A Carnot heat engine is a hypothetical engine that operates on the reversible Carnot cycle. The basic model for this engine was developed by Nicolas Léonard Sadi Carnot in 1824. The Carnot engine model was graphically expanded upon by Benoît Paul Émile Clapeyron in 1834 and mathematically elaborated upon by Rudolf Clausius in 1857 from which the concept of entropy emerged.Every single thermodynamic system exists in a particular state. A thermodynamic cycle occurs when a system is taken through a series of different states, and finally returned to its initial state. In the process of going through this cycle, the system may perform work on its surroundings, thereby acting as a heat engine.A heat engine acts by transferring energy from a warm region to a cool region of space and, in the process, converting some of that energy to mechanical work. The cycle may also be reversed. The system may be worked upon by an external force, and in the process, it can transfer thermal energy from a cooler system to a warmer one, thereby acting as a refrigerator or heat pump rather than a heat engine.In the adjacent diagram, from Carnot's 1824 work, Reflections on the Motive Power of Fire, there are "two bodies A and B, kept each at a constant temperature, that of A being higher than that of B. These two bodies to which we can give, or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. We will call the first the furnace and the second the refrigerator.” Carnot then explains how we can obtain motive power, i.e.., “work”, by carrying a certain quantity of heat from body A to body B.

Answer 2

$\mathcal{Carnot\:\:Cycle}$

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

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:

$\textsc{1.\:Isothermal\:Expansion}$

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

$\textsc{2.\:Adiabatic\:Expansion}$

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

$\textsc{3.\:Isothermal\:Compression}$

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.

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

=-area CDLNC

$\textsc{4.\:Adiabatic\:Compression}$

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.