2.3 Second Law of Thermodynamics

2.3 Second Law of Thermodynamics

1. Kelvin–Planck and Clausius Statements

Kelvin–Planck Statement

• Impossibility of a Perfect Heat Engine

• "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work."

• A heat engine must reject some heat to a low-temperature reservoir; it cannot convert all input heat into work.

$W_{net} = Q_H - Q_L > 0$

Where:

• $Q_H$ is heat input from high-temperature source

• $Q_L$ is heat rejected to low-temperature sink

• $W_{net}$ is net work output

Clausius Statement

• Impossibility of a Perfect Refrigerator

• "It is impossible for any device that operates on a cycle to transfer heat from a cooler body to a hotter body without any work input."

• Heat cannot flow spontaneously from a colder to a hotter body; work must be expended.

Equivalence of the Statements

• The Kelvin–Planck and Clausius statements are equivalent.

• Violation of one statement leads to violation of the other.

2. Entropy and Entropy Relations

Entropy ($S$)

• A measure of molecular disorder or randomness of a system.

• An extensive property (kJ/K).

• Specific entropy: $s = S/m$ (kJ/kg·K).

Definition

• For a reversible process: $dS = \left( \frac{\delta Q}{T} \right)_{rev}$

Entropy Change

• For any process (reversible or irreversible): $\Delta S = S_2 - S_1 \ge \int_1^2 \frac{\delta Q}{T}$

• The equality holds for reversible processes, inequality for irreversible processes.

Entropy Balance

• For a closed system: $\Delta S_{system} = S_2 - S_1 = \int_1^2 \frac{\delta Q}{T} + S_{gen}$ where $S_{gen} \ge 0$ is entropy generated due to irreversibilities.

• $S_{gen} = 0$ for reversible processes

• $S_{gen} > 0$ for irreversible processes

Entropy Relations for Ideal Gases

$\Delta s = s_2 - s_1 = c_p \ln\left(\frac{T_2}{T_1}\right) - R \ln\left(\frac{P_2}{P_1}\right)$

$\Delta s = s_2 - s_1 = c_v \ln\left(\frac{T_2}{T_1}\right) + R \ln\left(\frac{v_2}{v_1}\right)$

Principle of Increase of Entropy

• The entropy of an isolated system always increases or, in the limit of a reversible process, remains constant. $\Delta S_{isolated} \ge 0$

3. Isentropic Process and Efficiency

Isentropic Process

• A process during which the entropy remains constant.

• $s_2 = s_1$ or $\Delta s = 0$

• For reversible adiabatic processes: $Q = 0$ and $S_{gen} = 0$

Isentropic Relations for Ideal Gases (constant specific heats)

$Pv^k = \text{constant}$

$Tv^{k-1} = \text{constant}$

$\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(k-1)/k}$

where $k = c_p/c_v$

Isentropic Efficiency

• Measures how closely an actual device approaches the corresponding isentropic device.

Turbine Isentropic Efficiency

$\eta_t = \frac{W_{actual}}{W_{isentropic}} = \frac{h_1 - h_{2a}}{h_1 - h_{2s}}$

Compressor/Pump Isentropic Efficiency

$\eta_c = \frac{W_{isentropic}}{W_{actual}} = \frac{h_{2s} - h_1}{h_{2a} - h_1}$

Nozzle Isentropic Efficiency

$\eta_n = \frac{V_{2a}^2}{V_{2s}^2} = \frac{h_1 - h_{2a}}{h_1 - h_{2s}}$

4. Reversible and Irreversible Processes

Reversible Process

• A process that can be reversed without leaving any trace on the surroundings.

• System passes through a series of equilibrium states.

• Both system and surroundings can be returned to their initial states.

• Characteristics: Infinitely slow, no friction, no unrestrained expansion, no temperature gradients.

Irreversible Process

• A process that cannot be reversed without permanent changes to system or surroundings.

• Causes of irreversibility: Friction, unrestrained expansion, heat transfer through finite temperature difference, mixing of different substances, inelastic deformation, chemical reactions.

5. Heat Engines, Refrigerators, Heat Pumps

Heat Engine

• Converts heat into work.

• Operates between two thermal reservoirs ($T_H$ and $T_L$, where $T_H > T_L$).

Performance Parameters:

Thermal Efficiency: $\eta_{th} = \frac{W_{net}}{Q_H} = \frac{Q_H - Q_L}{Q_H} = 1 - \frac{Q_L}{Q_H}$

Refrigerator

• Transfers heat from low-temperature space to high-temperature space by consuming work.

Performance Parameters:

Coefficient of Performance (COP): $COP_R = \frac{Q_L}{W_{net}} = \frac{Q_L}{Q_H - Q_L}$

Heat Pump

• Transfers heat from low-temperature space to high-temperature space (heating application).

Performance Parameters:

Coefficient of Performance (COP): $COP_{HP} = \frac{Q_H}{W_{net}} = \frac{Q_H}{Q_H - Q_L}$

Relationship:

$COP_{HP} = COP_R + 1$

6. Carnot Cycle and Efficiency

Carnot Cycle

• A theoretical reversible cycle that provides the maximum possible efficiency between two temperature reservoirs.

• Consists of four reversible processes:

  1. Reversible Isothermal Expansion (at $T_H$)

  2. Reversible Adiabatic Expansion ($T_H \to T_L$)

  3. Reversible Isothermal Compression (at $T_L$)

  4. Reversible Adiabatic Compression ($T_L \to T_H$)

Carnot Principles

1. The efficiency of an irreversible heat engine is always less than the efficiency of a reversible one operating between the same two reservoirs.

2. All reversible heat engines operating between the same two reservoirs have the same efficiency.

Carnot Efficiency

• For a Carnot heat engine: $\eta_{Carnot} = 1 - \frac{T_L}{T_H}$ where temperatures are in Kelvin (absolute temperature).

Carnot COP

• For a Carnot refrigerator: $COP_{R,Carnot} = \frac{T_L}{T_H - T_L}$

• For a Carnot heat pump: $COP_{HP,Carnot} = \frac{T_H}{T_H - T_L}$

Important Notes

• Carnot efficiency depends only on the temperature levels, not on the working fluid.

• Actual devices have lower efficiencies than Carnot due to irreversibilities.

• Carnot cycle establishes the theoretical limit of performance for heat engines, refrigerators, and heat pumps.