Thermodynamics Fundamentals and Applications

Posted on Sep 11, 2024 in Physics

Thermodynamics Fundamentals

a. Zeroth Law of Thermodynamics and its Significance

The Zeroth Law of Thermodynamics states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. In other words:

“If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A must also be in thermal equilibrium with C.”

Significance:

• Establishes the concept of temperature and the basis for measuring it.
• Allows for the construction of thermometers and temperature scales.
• Provides a foundation for defining thermal equilibrium, which is crucial for understanding and applying the laws of thermodynamics.

b. Clausius Inequality

Statement: The Clausius inequality states that for any cyclic process in a thermodynamic system:

∮ δQ/T ≤ 0

where:

• ∮ denotes a closed path in a cyclic process,
• δQ is the differential amount of heat added to the system, and
• T is the temperature at the boundary where heat transfer occurs.

Proof Outline:

• Consider a cyclic process where the system returns to its initial state.
• Apply the Kelvin-Planck statement (no cyclic process can convert all heat into work without other effects) to derive that the inequality holds for all possible heat transfer processes within a cycle, including reversible and irreversible processes.

c. Joule-Thomson Coefficient and its Significance

The Joule-Thomson coefficient (μ) describes how the temperature of a gas changes when it expands or is compressed at constant enthalpy (no heat exchange with surroundings). It is defined as:

μ = (∂T/∂P)_H

Significance:

• Determines whether a gas will cool or heat up during expansion or compression under constant enthalpy conditions.
• Essential in the design and operation of refrigeration and liquefaction processes (such as in gas liquefaction plants).
• Helps in understanding the behavior of real gases, especially when dealing with non-ideal gas behavior.

d. P-V & T-S Diagrams for Stirling and Ericsson Cycles

Stirling Cycle (P-V Diagram):

• The Stirling cycle consists of two isothermal processes and two constant-volume processes.
• During isothermal compression and expansion, the gas exchanges heat with the surroundings, resulting in a PV diagram that resembles a pair of isothermal curves connected by constant-volume lines.

Ericsson Cycle (T-S Diagram):

• The Ericsson cycle is a theoretical thermodynamic cycle used in heat engines.
• It consists of two isothermal processes and two isentropic processes.
• The T-S diagram for the Ericsson cycle typically shows two vertical lines (isothermal processes) and two horizontal lines (isentropic processes).

e. Definitions

(a) Dryness Fraction

The dryness fraction (x) is a measure of the quality of a saturated steam mixture. It represents the proportion of the mass of the steam that is in the vapor phase compared to the total mass of the mixture. It is defined as:

x = mass of vapor / total mass of mixture

The dryness fraction ranges from 0 (completely liquid) to 1 (completely vapor).

(b) Critical Point

The critical point is the end point of the phase equilibrium curve between the liquid and vapor phases. At the critical point, the properties of the liquid and vapor phases become indistinguishable. The temperature, pressure, and specific volume at the critical point are known as the critical temperature (T_c), critical pressure (P_c), and critical volume (V_c) respectively.

(c) Triple Point

The triple point is the unique condition in which all three phases of a substance (solid, liquid, and gas) coexist in thermodynamic equilibrium. The triple point of water, for example, occurs at a temperature of 0.01°C and a pressure of 611.657 pascals.

(d) Degree of Superheat

The degree of superheat is the amount by which the temperature of a superheated vapor exceeds the saturation temperature (the temperature at which vaporization occurs) at a given pressure. It is calculated as:

Degree of Superheat = T_superheated – T_saturation

where:

• T_superheated is the temperature of the superheated vapor and
• T_saturation is the saturation temperature at the same pressure.

f. Sonic Velocity and Mach Number

Sonic Velocity (Speed of Sound)

The speed of sound, also known as sonic velocity, is the speed at which a small disturbance or sound wave propagates through a medium. In a gas, this speed depends on the properties of the gas, such as temperature and molecular composition. The speed of sound in a gas is given by the formula:

c = speed of sound
γ = adiabatic index (ratio of specific heats)
R = universal gas constant
T = absolute temperature of the gas
M = molar mass of the gas

In dry air at 20°C (68°F), the speed of sound is approximately 343 meters per second (m/s) or 1235 kilometers per hour (km/h).

Mach Number

The Mach number (M) is a dimensionless quantity that represents the ratio of the speed of an object moving through a fluid to the speed of sound in that fluid. It is used to characterize the flow regime around an object, such as an aircraft, and is defined as:

M = v/c

v = velocity of the object relative to the fluid
c = speed of sound in the fluid

The Mach number helps to classify flow regimes as follows:

• Subsonic: M < 1 (Object is moving slower than the speed of sound)
• Transonic: M ≈ 1 (Object is moving at about the speed of sound, typically 0.8 < M < 1.2)
• Supersonic: 1 < M < 5 (Object is moving faster than the speed of sound)
• Hypersonic: M > 5 (Object is moving much faster than the speed of sound)

Significance of Sonic Velocity and Mach Number:

Sonic Velocity (Speed of Sound):

• Acoustics: The speed of sound is fundamental in acoustics for understanding sound propagation.
• Meteorology: It affects how sound travels through the atmosphere and can influence weather phenomena.
• Aerospace Engineering: The speed of sound in air affects aircraft design, particularly for supersonic and hypersonic vehicles.

Mach Number:

• Aerodynamics: The Mach number is crucial in determining the aerodynamic characteristics of an aircraft. Different flight regimes (subsonic, transonic, supersonic, hypersonic) have vastly different aerodynamic behaviors.
• Compressible Flow: The Mach number helps in analyzing compressible flows where density changes significantly, such as in jet engines and nozzles.
• Shock Waves: At supersonic speeds, the Mach number determines the strength and behavior of shock waves formed in front of the moving object.
• Performance: Aircraft performance, including lift, drag, and engine efficiency, varies with the Mach number, guiding design and operational strategies.

g. Effect of Varying Back Pressure on Nozzle Performance

In a nozzle, the back pressure (Pb) is the pressure at the exit of the nozzle. The performance of the nozzle is significantly affected by changes in this back pressure:

• Choking Condition: If the back pressure equals or exceeds the critical pressure ratio (for ideal gases, Pb/Pa = (2/(γ + 1))^(γ/(γ-1))), the flow through the nozzle becomes choked.
• Subsonic Flow: For back pressures lower than the critical pressure ratio, increasing the back pressure generally decreases the mass flow rate. This is because higher back pressures reduce the pressure difference driving the flow through the nozzle.
• Supersonic Flow: In supersonic flow regimes, an increase in back pressure can sometimes increase the mass flow rate if it is below the critical pressure ratio. This is because higher back pressures can increase the pressure difference.

Thermodynamic Cycles

Reheat Rankine Cycle

The Reheat Rankine cycle is an improvement of the basic Rankine cycle, designed to enhance the efficiency and performance of steam power plants. The key modification in the Reheat Rankine cycle is the reheat process, which involves reheating the steam between the high-pressure and low-pressure turbines.

Process Description:

1. Boiler (1-2): Water is heated at constant pressure until it becomes high-pressure steam.
2. High-Pressure Turbine (2-3): The high-pressure steam expands in the high-pressure turbine, doing work and generating power.
3. Reheater (3-4): The partially expanded steam is sent back to the boiler to be reheated to a higher temperature at constant pressure.
4. Low-Pressure Turbine (4-5): The reheated steam expands again in the low-pressure turbine, producing additional work.
5. Condenser (5-6): The steam is condensed into water at constant pressure and low temperature.
6. Pump (6-1): The condensed water is pumped back to the boiler, completing the cycle.

Advantages of the Reheat Rankine Cycle:

• Improved Efficiency: By reheating the steam, the average temperature at which heat is added to the cycle is increased, thus improving the thermal efficiency.
• Reduced Moisture Content: Reheating reduces the moisture content of the steam at the final stages of expansion in the turbine, reducing turbine blade erosion and improving reliability and maintenance intervals.
• Higher Work Output: Reheating allows for additional expansion in the low-pressure turbine, leading to a higher total work output from the cycle.

Applications:

The Reheat Rankine cycle is commonly used in large-scale thermal power plants, including coal-fired, nuclear, and some gas-fired power plants, to improve efficiency and performance while reducing mechanical stresses on equipment.

Thermodynamic Concepts

a) Zeroth Law of Thermodynamics

Statement: The Zeroth Law of Thermodynamics states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

Significance:

• Foundation of Temperature: It establishes the concept of temperature and allows for the definition of a temperature scale.
• Thermodynamic Equilibrium: It defines what it means for two systems to be at the same temperature, which is crucial for applying the other laws of thermodynamics.
• Measurement: It forms the basis for constructing thermometers and measuring temperature accurately.

b) Difference between Heat Engine, Refrigerator, and Heat Pump

Heat Engine

• Purpose: Converts thermal energy (heat) into mechanical work.
• Process: Typically operates in a cyclic manner (e.g., Rankine cycle for steam engines, Otto cycle for gasoline engines).
• Example: Steam turbines, internal combustion engines.

Refrigerator

• Purpose: Transfers heat from a cold reservoir (inside the refrigerator) to a hot reservoir (outside environment).
• Process: Uses mechanical work to move heat against its natural direction (from cold to hot).
• Example: Household refrigerators, air conditioners.

Heat Pump

• Purpose: Similar to a refrigerator but reverses the direction of heat transfer, moving heat from a cold reservoir (outside environment) to a hot reservoir (inside the building).
• Process: Requires mechanical work input to achieve heat transfer against the natural gradient.
• Example: Heat pumps used for heating buildings in winter.

c) Mollier’s Diagram

A Mollier diagram, also known as an enthalpy-entropy (h-s) chart, is a graphical representation used in thermodynamics to visualize the properties of steam and other substances. It is particularly useful for engineers and scientists working with thermodynamic cycles, such as those in power plants and refrigeration systems.

Key Features:

• Axes: The horizontal axis represents entropy (s), typically in units of kJ/(kg·K). The vertical axis represents enthalpy (h), typically in units of kJ/kg.
• Isobars: Lines of constant pressure (isobars) are plotted on the diagram, helping users determine how enthalpy and entropy change with pressure.
• Isotherms: Lines of constant temperature (isotherms) are also plotted, indicating the relationship between enthalpy and entropy at different temperatures.
• Saturation Lines: The diagram includes lines for saturated liquid and saturated vapor, which meet at the critical point. These lines separate the liquid, vapor, and mixed-phase regions.
• Quality Lines: Lines of constant quality (dryness fraction) can be found in the mixed-phase region, indicating the ratio of vapor to liquid in the mixture.

Applications:

• Steam Turbines: Engineers use the Mollier diagram to analyze and design steam turbine stages by plotting the expansion process of steam through the turbine.
• Refrigeration Cycles: It helps in visualizing refrigeration cycles, such as the Rankine and refrigeration cycles, by showing how refrigerants change state through different components.
• Heat Exchangers: The diagram aids in understanding the heat exchange processes by showing enthalpy changes at constant pressure.

Advantages:

• Ease of Use: The Mollier diagram provides a straightforward way to visualize complex thermodynamic processes without extensive calculations.
• Quick Analysis: Engineers can quickly determine the state of steam or other substances and the efficiency of thermodynamic cycles.

d) Maxwell’s Relations

Maxwell’s relations are a set of equations in thermodynamics derived from the second law of thermodynamics and the definition of the thermodynamic potentials (Helmholtz free energy, Gibbs free energy, internal energy, and enthalpy). These relations provide a way to express the changes in thermodynamic variables and can be used to derive various thermodynamic properties.

There are four Maxwell relations, each derived from one of the thermodynamic potentials:

Significance:

• Simplification: Maxwell’s relations simplify the calculation of changes in entropy, volume, pressure, and temperature by linking these variables through partial derivatives.
• Thermodynamic Properties: They help in deriving expressions for thermodynamic properties such as specific heats, compressibilities, and expansion coefficients.
• Interrelations: These relations highlight the interdependence of different thermodynamic variables and are essential for the study of thermodynamic systems in equilibrium.

e) Intensive and Extensive Properties

Intensive Properties

• Definition: Properties that do not depend on the amount of matter or the size of the system. They remain constant regardless of the quantity of substance present.
• Examples: Temperature, pressure, density, specific heat, specific volume.
• Characteristics: Intensive properties are used to characterize the state of the system and are uniform throughout a homogeneous system.

Extensive Properties

• Definition: Properties that depend on the amount of matter or the size of the system. They change when the size or quantity of the substance changes.
• Examples: Mass, volume, internal energy, enthalpy, entropy.
• Characteristics: Extensive properties are additive for subsystems. For example, the total volume of a system is the sum of the volumes of its parts.

f) Thermodynamic Systems

A thermodynamic system is a defined quantity of matter or a region in space under study, separated from its surroundings by boundaries. The type of system dictates how it interacts with its surroundings.

Types of Thermodynamic Systems:

• Open System:
  • Definition: A system that can exchange both energy (heat and work) and matter with its surroundings.
  • Example: An open container of boiling water.
• Closed System:
  • Definition: A system that can exchange energy (heat and work) but not matter with its surroundings.
  • Example: A sealed, heated gas cylinder.
• Isolated System:
  • Definition: A system that cannot exchange energy or matter with its surroundings.
  • Example: An insulated thermos flask.

g) Principle of Increase of Entropy

The principle of increase of entropy is a fundamental concept in the second law of thermodynamics. It states that for any spontaneous process, the total entropy of the system and its surroundings always increases. Entropy can remain constant in a reversible process but can never decrease.

Key Points:

• Isolated Systems: In an isolated system, the entropy either increases or remains constant; it never decreases.
• Direction of Processes: This principle explains the natural direction of processes, indicating that natural processes tend to move towards a state of maximum entropy or disorder.
• Thermodynamic Equilibrium: When a system reaches maximum entropy, it is said to be in thermodynamic equilibrium.

h) P-V and T-S Diagrams for Atkinson and Lenoir Cycles

Atkinson Cycle

• P-V Diagram: The Atkinson cycle is characterized by a longer expansion stroke than the compression stroke. It involves adiabatic compression, isochoric (constant volume) heat addition, adiabatic expansion, and isochoric heat rejection.
• T-S Diagram: The T-S diagram for the Atkinson cycle shows the respective entropy changes during isochoric and adiabatic processes.

Lenoir Cycle

• P-V Diagram: The Lenoir cycle involves isochoric heat addition, isentropic expansion, and isobaric heat rejection.
• T-S Diagram: The T-S diagram for the Lenoir cycle shows entropy changes during isochoric heat addition and isobaric heat rejection.

i) Definitions

(a) Availability (Exergy)

• Definition: The maximum useful work that can be obtained from a system as it comes into equilibrium with its surroundings. It measures the quality or usefulness of energy.
• Significance: Availability quantifies the potential of a system to perform work, considering both the system and environmental conditions.

(b) Unavailability

• Definition: The part of energy that cannot be converted into work due to irreversibilities and entropy production. It is the complement of availability.
• Significance: Unavailability represents energy losses due to inefficiencies in a process.

(c) Dead State

• Definition: The state of a system in which it is in thermodynamic equilibrium with its surroundings. At this state, the system has no potential to perform useful work.
• Significance: The dead state is used as a reference to measure availability or exergy.

(d) Joule-Thomson Coefficient

• Definition: The Joule-Thomson coefficient (μ) is a measure of the temperature change of a real gas or liquid when it is forced through a valve or porous plug while maintaining constant enthalpy.
• Significance: The Joule-Thomson coefficient indicates whether a gas will cool or heat up during expansion at constant enthalpy. The sign and magnitude of μ depend on the specific gas and its conditions of temperature and pressure.

j) Internal Energy as a Property of a System

Internal energy (U) is a state function that depends only on the current state of the system (its temperature, pressure, and composition), and not on the path taken to reach that state. This can be shown through the following reasoning:

In a closed system undergoing a process, the change in internal energy (ΔU) is given by the heat added to the system (Q) minus the work done by the system (W): ΔU = Q – W.

Since Q and W depend only on the initial and final states, ΔU also depends only on the initial and final states.

Therefore, internal energy is a property of the system because its value is uniquely determined by the system’s state (temperature, pressure, and composition).

k) Definitions

• Wet Steam: Steam that contains both vapor and liquid phases in equilibrium.
• Superheated Steam: Steam that is at a temperature higher than its saturation temperature corresponding to its pressure, and exists in a gaseous state only.
• Dryness Fraction: The ratio of the mass of vapor to the total mass of the mixture (vapor + liquid) in a wet steam mixture.
• Saturation Temperature: The temperature at which a substance (e.g., water) changes phase from liquid to vapor (or vice versa) at a given pressure.

l) Cut-off Ratio and Assumptions of Air Standard Cycle

• Cut-off Ratio: It is the ratio of the volume of the cylinder at the point of cut-off (when the supply of heat is cut off) to the volume at the beginning of the compression stroke in an internal combustion engine. A higher cut-off ratio typically leads to increased efficiency in the engine.
• Assumptions of Air Standard Cycle:
  • Perfect Gas: The working fluid (air) is treated as an ideal gas throughout the cycle.
  • Constant Specific Heats: Specific heats (cp and cv) of the working fluid are assumed constant.
  • No Friction: All processes are assumed to be frictionless.
  • Quasi-Equilibrium: All processes are assumed to occur in quasi-equilibrium.

m) Joule-Thomson Coefficient, Inversion Point, and Inversion Curve

• Joule-Thomson Coefficient: It describes how the temperature of a gas changes when it expands or is compressed under constant enthalpy (no heat exchange with the surroundings).
• Inversion Point: The temperature and pressure conditions where the Joule-Thomson coefficient is zero, indicating no temperature change upon expansion or compression.
• Inversion Curve: A curve on a temperature-pressure diagram that shows the conditions (temperature and pressure) at which the Joule-Thomson coefficient changes sign (from positive to negative or vice versa).

n) Effect of Variation in Back Pressure on C-D Nozzle Performance

The back pressure (Pb) is the pressure at the exit of a nozzle. It affects the performance of the nozzle in the following ways:

• Subsonic Flow (Pb < P_critical): Increasing back pressure decreases mass flow rate and exit velocity. The nozzle is not choked.
• Choked Flow (Pb = P_critical): At critical pressure ratio, further increases in back pressure do not increase mass flow rate; the flow remains choked.
• Supersonic Flow (Pb > P_critical): For supersonic flow, increasing back pressure can increase mass flow rate as it increases the pressure difference across the nozzle exit.