Heat Engine and efficiency

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Heat Engine, Carnot Cycle and Efficiency of Carnot Engine

Heat Engine

A heat engine is a device that converts a part of the heat supplied to it into useful mechanical work.

A heat engine must contain the following three essential parts:

1. Heat source
2. Mechanical arrangement and working substance
3. Heat sink

(i) Heat Source

A heat source is a heat reservoir maintained at a high temperature. It has a very large, ideally infinite, heat capacity so that its temperature remains constant even when a large amount of heat is continuously extracted from it.

(ii) Mechanical Arrangement and Working Substance

To convert heat into mechanical work, a hollow cylinder fitted with a movable piston is used as the mechanical arrangement. The working substance is placed inside the cylinder.

The working substance absorbs heat from the heat source and expands, thereby moving the piston and doing mechanical work.

The walls of the cylinder are thermally insulated from the surroundings, whereas its base is conducting so that heat can be transferred to or from the working substance when required.

An ideal gas is generally used as the theoretical working substance, whereas air or steam may be used in practical heat engines.

(iii) Heat Sink

A heat sink is a heat reservoir maintained at a temperature lower than that of the heat source. It also has a very large, ideally infinite, heat capacity, so that its temperature remains practically constant even when heat is rejected to it.

Fig. 2.2-1: Schematic representation of a heat engine

The working substance takes heat \(Q_1\) from the heat source at temperature \(T_1\). A part of this heat is converted into useful mechanical work \(W\), while the remaining heat \(Q_2\) is rejected to the heat sink at temperature \(T_2\).

\[ \boxed{Q_1 = W + Q_2} \]

After rejecting heat \(Q_2\), the working substance returns to its initial state. The complete sequence of processes is called one cycle, and the process is known as a cyclic process.

Efficiency of a Heat Engine

The efficiency of a heat engine is defined as the ratio of the useful work done by the engine to the heat received from the heat source.

\[ \boxed{ \eta = \frac{\text{Useful work done}} {\text{Heat received from source}} } \tag{1} \]

Therefore,

\[ \boxed{ \eta = \frac{W}{Q_1} } \tag{2} \]

For a complete cycle, the working substance returns to its initial state. Therefore, the change in its internal energy is zero:

\[ \Delta U = 0. \]

From the first law of thermodynamics,

\[ Q = \Delta U + W. \]

For the complete cycle,

\[ Q_1-Q_2 = 0+W. \]

Hence,

\[ \boxed{ W=Q_1-Q_2 } \tag{3} \]

Substituting Eq. (3) into Eq. (2),

\[ \eta = \frac{Q_1-Q_2}{Q_1}. \]

Therefore,

\[ \boxed{ \eta = 1-\frac{Q_2}{Q_1} } \tag{4} \]

The percentage efficiency is

\[ \boxed{ \eta(\%) = \left( 1-\frac{Q_2}{Q_1} \right)\times100 } \tag{5} \]

Important:

\(Q_1\) is the heat supplied by the source, \(Q_2\) is the heat rejected to the sink, and \(W=Q_1-Q_2\) is the useful work obtained during one complete cycle.

2.3 Carnot's Cycle and Carnot's Ideal Engine

In 1824, the French physicist Sadi Carnot proposed a theoretical heat engine to investigate the maximum possible efficiency of a heat engine.

A Carnot engine is an ideal heat engine in which all processes are reversible and no energy is lost due to friction or other irreversible effects.

A Carnot cycle is a cyclic process consisting of four reversible processes performed in a definite sequence, namely two isothermal processes and two adiabatic processes.

Main Parts of a Carnot Engine

(i) Heat Source

The heat source is a reservoir of effectively infinite heat capacity maintained at a high temperature \(T_1\) K. Its temperature remains constant even when heat is supplied to the working substance. Its upper surface is perfectly conducting.

(ii) Mechanical Arrangement and Working Substance

A hollow cylinder is used whose walls are perfectly insulating and whose base is perfectly conducting. A frictionless piston made of insulating material is fitted inside the cylinder.

An ideal gas is used as the working substance.

(iii) Heat Sink

The heat sink is a reservoir of effectively infinite heat capacity maintained at a lower temperature \(T_2\) K. It receives the rejected heat from the working substance while maintaining a constant temperature.

(iv) Insulating Stand

The stand is perfectly insulated. When the cylinder is placed on this stand, the working substance can expand or contract adiabatically without exchange of heat with the surroundings.

Working of Carnot Engine

According to Carnot, maximum work is obtained when the working substance is taken through a sequence of reversible processes and finally returned to its initial state.

The Carnot cycle consists of the following four stages:

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

Fig. 2.3-2: P–V diagram of the Carnot cycle

(i) First Process: Isothermal Expansion (A → B)

The cylinder is placed on the heat source. The temperature of the ideal gas becomes equal to the temperature \(T_1\) K of the heat source.

Let the initial pressure and volume of the gas at state \(A\) be \(P_A\) and \(V_A\), respectively.

The piston is allowed to move upward slowly. The gas expands isothermally at temperature \(T_1\). During this expansion, the gas absorbs heat \(Q_1\) from the heat source and reaches state \(B\), where its pressure and volume are \(P_B\) and \(V_B\).

For an ideal gas undergoing an isothermal process, \(\Delta U=0\). Therefore, from the first law of thermodynamics,

\[ Q_1=W_1. \tag{1} \]

The work done is

\[ W_1 = \int_{V_A}^{V_B}P\,dV. \]

For one mole of an ideal gas,

\[ PV=RT_1. \]

Hence,

\[ W_1 = RT_1 \int_{V_A}^{V_B}\frac{dV}{V}. \]

Therefore,

\[ \boxed{ Q_1=W_1 = RT_1 \ln\left(\frac{V_B}{V_A}\right) } \tag{2} \]

Since \[ \ln x=2.303\log_{10}x, \] we may also write

\[ \boxed{ Q_1 = 2.303RT_1 \log_{10} \left(\frac{V_B}{V_A}\right) }. \]

(ii) Second Process: Adiabatic Expansion (B → C)

The cylinder is removed from the heat source and placed on the insulating stand. The working substance is therefore completely thermally isolated from the surroundings.

The gas is allowed to expand slowly and adiabatically until its temperature falls from \(T_1\) to the temperature \(T_2\) of the heat sink.

The gas reaches state \(C\), where its pressure and volume are \(P_C\) and \(V_C\), respectively.

For an adiabatic process,

\[ \boxed{ PV^\gamma=\text{constant} } \tag{3} \]

where

\[ \boxed{ \gamma=\frac{C_P}{C_V} }. \]

Thus,

\[ P_BV_B^\gamma=P_CV_C^\gamma=K. \]

The work done during adiabatic expansion is

\[ W_2 = \int_{V_B}^{V_C}P\,dV. \]

Since \[ P=\frac{K}{V^\gamma}, \] we obtain

\[ W_2 = K \int_{V_B}^{V_C} V^{-\gamma}\,dV. \]

Therefore,

\[ W_2 = \frac{K}{1-\gamma} \left[ V_C^{1-\gamma} - V_B^{1-\gamma} \right]. \]

Hence,

\[ \boxed{ W_2 = \frac{P_BV_B-P_CV_C} {\gamma-1} } \]

For one mole of an ideal gas,

\[ P_BV_B=RT_1, \qquad P_CV_C=RT_2. \]

Therefore,

\[ \boxed{ W_2 = \frac{R(T_1-T_2)} {\gamma-1} }. \tag{4} \]

(iii) Third Process: Isothermal Compression (C → D)

The cylinder is removed from the insulating stand and placed on the heat sink. The gas is compressed slowly and isothermally at temperature \(T_2\) K.

The gas moves from state \(C\) to state \(D\). During compression, the gas rejects heat \(Q_2\) to the heat sink.

Since the process is isothermal for an ideal gas,

\[ \Delta U=0. \]

Hence the magnitude of heat rejected is equal to the magnitude of work done on the gas.

The work done by the gas is negative:

\[ W_3 = RT_2 \ln\left(\frac{V_D}{V_C}\right). \]

Since \(V_D

\[ \boxed{ W_3 = -RT_2 \ln\left(\frac{V_C}{V_D}\right) }. \tag{5} \]

Therefore, the heat rejected to the sink is

\[ \boxed{ Q_2 = RT_2 \ln\left(\frac{V_C}{V_D}\right) }. \tag{6} \]

(iv) Fourth Process: Adiabatic Compression (D → A)

The cylinder is again placed on the insulating stand. The gas is compressed slowly and adiabatically until its temperature rises from \(T_2\) to \(T_1\).

The gas returns from state \(D\) to the original state \(A\), completing the Carnot cycle.

The adiabatic relation is

\[ \boxed{ PV^\gamma=\text{constant} }. \tag{7} \]

The work done by the gas during this compression is negative. Its magnitude is

\[ \boxed{ W_4 = -\frac{R(T_1-T_2)} {\gamma-1} }. \tag{8} \]

Net Work Done in a Carnot Cycle

The work done by the gas during the first two processes is positive, whereas the work done by the gas during the last two processes is negative.

Therefore, the net work done in one complete Carnot cycle is

\[ W=W_1+W_2+W_3+W_4. \tag{9} \]

On substituting Eqs. (2), (4), (5), and (8),

\[ W = RT_1\ln\left(\frac{V_B}{V_A}\right) + \frac{R(T_1-T_2)}{\gamma-1} - RT_2\ln\left(\frac{V_C}{V_D}\right) - \frac{R(T_1-T_2)}{\gamma-1}. \]

The adiabatic terms cancel, giving

\[ W = RT_1\ln\left(\frac{V_B}{V_A}\right) - RT_2\ln\left(\frac{V_C}{V_D}\right). \]

Relation Between the Volumes in Carnot Cycle

For the adiabatic expansion \(B\rightarrow C\),

\[ T_1V_B^{\gamma-1} = T_2V_C^{\gamma-1}. \tag{10} \]

Therefore,

\[ \frac{V_C}{V_B} = \left( \frac{T_1}{T_2} \right)^{1/(\gamma-1)}. \tag{11} \]

Similarly, for the adiabatic compression \(D\rightarrow A\),

\[ T_2V_D^{\gamma-1} = T_1V_A^{\gamma-1}. \tag{12} \]

Hence,

\[ \frac{V_D}{V_A} = \left( \frac{T_1}{T_2} \right)^{1/(\gamma-1)}. \tag{13} \]

Comparing Eqs. (11) and (13),

\[ \frac{V_C}{V_B} = \frac{V_D}{V_A}. \]

Therefore,

\[ \boxed{ \frac{V_B}{V_A} = \frac{V_C}{V_D} }. \tag{14} \]

Using Eq. (14), the net work becomes

\[ W = R(T_1-T_2) \ln\left(\frac{V_B}{V_A}\right). \tag{15} \]

Thus,

\[ \boxed{ W = R(T_1-T_2) \ln\left(\frac{V_B}{V_A}\right) }. \]

This net work is equal to the heat converted into useful work:

\[ \boxed{ W=Q_1-Q_2 }. \tag{16} \]

Efficiency of Carnot Engine

The efficiency of a heat engine is

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

For the Carnot engine,

\[ Q_1 = RT_1 \ln\left(\frac{V_B}{V_A}\right). \]

Also,

\[ W = R(T_1-T_2) \ln\left(\frac{V_B}{V_A}\right). \]

Therefore,

\[ \eta = \frac{ R(T_1-T_2) \ln(V_B/V_A) }{ RT_1 \ln(V_B/V_A) }. \]

Cancelling the common terms,

\[ \boxed{ \eta = \frac{T_1-T_2}{T_1} } \tag{17} \]

Hence,

\[ \boxed{ \eta = 1-\frac{T_2}{T_1} }. \tag{18} \]

The percentage efficiency is therefore

\[ \boxed{ \eta(\%) = \left( 1-\frac{T_2}{T_1} \right)\times100 }. \tag{19} \]

Conclusions from the Efficiency of a Carnot Engine

1. The efficiency of a Carnot engine depends only on the temperatures of the heat source and heat sink and is independent of the nature of the working substance.
2. For a fixed source temperature \(T_1\), the efficiency increases as the sink temperature \(T_2\) decreases.
3. The efficiency increases when the temperature difference \(T_1-T_2\) increases.
4. From \[ \eta=1-\frac{T_2}{T_1}, \] 100% efficiency would require \(T_2=0\,\mathrm{K}\) or, as a limiting mathematical case, \(T_1\rightarrow\infty\). Such conditions cannot be achieved by a practical heat engine. Therefore, \[ \boxed{\eta<1}. \]
5. The efficiency of the Carnot engine is determined by the temperatures of the source and sink. The volume ratios in the Carnot cycle are related through the adiabatic processes, as shown by Eqs. (11)–(14).

Summary of the Carnot Cycle

Process	Path	Nature	Heat Transfer	Temperature
1	A → B	Isothermal expansion	Heat \(Q_1\) absorbed	\(T_1\)
2	B → C	Adiabatic expansion	\(Q=0\)	\(T_1\rightarrow T_2\)
3	C → D	Isothermal compression	Heat \(Q_2\) rejected	\(T_2\)
4	D → A	Adiabatic compression	\(Q=0\)	\(T_2\rightarrow T_1\)

Important Results of Carnot Engine

\[ \boxed{W=Q_1-Q_2} \]

\[ \boxed{ \eta = \frac{W}{Q_1} = 1-\frac{Q_2}{Q_1} } \]

\[ \boxed{ \eta_{\mathrm{Carnot}} = 1-\frac{T_2}{T_1} } \]

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