Heat Engines and Refrigerators: Principles and Applications (Chapter 21 Study Notes)

Heat Engines and Refrigerators

Introduction

This chapter explores the fundamental principles governing heat engines and refrigerators, focusing on the laws of thermodynamics, the concept of entropy, and the limits of efficiency for energy conversion devices. These concepts are central to understanding how energy is transformed and utilized in modern technology and industry.

Reversible and Irreversible Processes

Definitions and Examples

• Reversible Process: A process that can proceed in either direction without increasing the entropy of the universe. At the microscopic level, molecular collisions are typically reversible.

• Irreversible Process: A process that can only proceed in one direction, leading to an increase in entropy. Macroscopic events, such as a car crash, are irreversible.

• Spontaneous Heat Flow: Heat energy naturally flows from a hotter object to a colder one until thermal equilibrium is reached. The reverse does not occur spontaneously.

Example: When two gases at different temperatures are brought into contact, heat flows from the hotter to the colder gas until both reach the same temperature.

Entropy, Disorder, and Probability

Concept of Entropy

• Entropy (S): A measure of the disorder or randomness in a system. It quantifies the number of microscopic configurations (microstates) corresponding to a macroscopic state.

• Boltzmann's Entropy Formula:

$S = k_B \ln \Omega$

• Where $k_B$ is Boltzmann's constant and $\Omega$ is the number of microstates.

• Systems tend to evolve toward states with higher entropy (greater disorder and higher probability).

Example: A box of coins has lowest entropy when all coins are heads (only one arrangement), and highest entropy when half are heads and half are tails (many possible arrangements).

The Second Law of Thermodynamics

Formal and Informal Statements

• Formal Statement: The entropy of an isolated system never decreases; it increases until equilibrium is reached, or remains constant if already at equilibrium.

• Informal Statement #1: Heat energy spontaneously transfers from hot to cold, never from cold to hot.

• Informal Statement #2: The direction in which entropy increases defines the 'arrow of time.'

Implications: The second law explains why certain processes are irreversible and why energy transformations are never 100% efficient.

Thermodynamics: The Study of Energy Transformation

First and Second Laws

• First Law (Conservation of Energy): $\Delta E_{th} = Q + W$

• Second Law: Most macroscopic processes are irreversible; heat flows spontaneously from hot to cold.

Work, Heat, and Energy Reservoirs

Definitions

• Work Done by the System ($W_{by}$): The energy transferred from the system to its environment due to external forces.

• Energy Reservoir: An object so large that its temperature does not change when exchanging heat with a system. Hot and cold reservoirs are used in engines and refrigerators.

First Law in Terms of Work Done by the System:

$Q = \Delta E_{th} + W_{by}$

Heat Engines

Principles and Efficiency

• Heat Engine: A device that transforms heat energy into useful work via a cyclic process, requiring both a hot and a cold reservoir.

• Thermal Efficiency ($\eta$): The ratio of work output to heat input:

$\eta = \frac{W_{by}}{Q_H} = 1 - \frac{Q_C}{Q_H}$

• Where $Q_H$ is the heat absorbed from the hot reservoir and $Q_C$ is the heat expelled to the cold reservoir.

• Real engines have efficiencies much less than 100% due to unavoidable heat loss.

Example: Modern gasoline engines have maximum thermal efficiencies of about 25% to 30%.

Refrigerators and Coefficient of Performance

Principles and Performance

• Refrigerator: A device that uses external work to transfer heat from a cold reservoir to a hot reservoir (opposite of a heat engine).

• Coefficient of Performance (K): A measure of refrigerator efficiency:

$K = \frac{Q_C}{W_{in}}$

• Where $Q_C$ is the heat removed from the cold reservoir and $W_{in}$ is the work input.

• No refrigerator can have $K = \infty$; this would violate the second law.

Limits of Efficiency: Carnot Cycle

Carnot Engine and Maximum Efficiency

• Carnot Engine: An idealized, perfectly reversible engine operating between two temperatures using only isothermal and adiabatic processes.

• Carnot Efficiency ($\eta_{Carnot}$): The maximum possible efficiency for any engine operating between two reservoirs:

$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$

• Where $T_H$ and $T_C$ are the absolute temperatures (in Kelvin) of the hot and cold reservoirs, respectively.

• No real engine can exceed Carnot efficiency.

Carnot Coefficient of Performance for Refrigerators:

$K_{Carnot} = \frac{T_C}{T_H - T_C}$

Common Thermodynamic Cycles

Otto, Diesel, and Brayton Cycles

• Otto Cycle: Used in gasoline engines; involves isochoric and adiabatic processes.

• Diesel Cycle: Used in diesel engines; involves isobaric, isochoric, and adiabatic processes.

• Brayton Cycle: Used in jet engines and gas turbines; involves adiabatic compression and expansion, and isobaric heat addition and rejection.

Efficiency of Brayton Cycle: Increases with the pressure ratio, but levels off at high ratios.

Summary Table: Key Formulas and Concepts

Concept	Formula	Description
First Law of Thermodynamics	$\Delta E_{th} = Q + W$	Energy conservation in thermodynamic systems
Entropy (Boltzmann)	$S = k_B \ln \Omega$	Microscopic definition of entropy
Thermal Efficiency	$\eta = 1 - \frac{Q_C}{Q_H}$	Efficiency of a heat engine
Carnot Efficiency	$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$	Maximum possible efficiency
Coefficient of Performance (Refrigerator)	$K = \frac{Q_C}{W_{in}}$	Efficiency of a refrigerator
Carnot Coefficient of Performance	$K_{Carnot} = \frac{T_C}{T_H - T_C}$	Maximum possible refrigerator performance

Applications and Real-World Examples

• Power Plants: Steam turbines convert heat from burning fuel into electricity, but much energy is lost as waste heat due to thermodynamic limits.

• Automobile Engines: Only a fraction of the fuel's energy is converted to useful work; the rest is lost as heat.

• Refrigerators and Air Conditioners: Use work to transfer heat from cool interiors to warmer exteriors, limited by the second law.

Key Takeaways

• All real engines and refrigerators are limited by the laws of thermodynamics.

• Entropy provides a measure of disorder and determines the direction of spontaneous processes.

• The Carnot cycle sets the upper bound for efficiency; no real device can surpass it.