2.1 Thermodynamics Basics

2.1 Thermodynamics Basics

1. System and Surroundings

1.1 Definitions

1. System:

   • The specific region or quantity of matter being studied.

   • Example: Gas inside a piston-cylinder assembly.

2. Surroundings:

   • Everything external to the system that can interact with it.

   • Example: The atmosphere outside the cylinder, the piston itself.

3. Boundary:

   • The real or imaginary surface separating the system from its surroundings.

   • It can be fixed or movable, real or imaginary.

1.2 Types of Systems

1. Closed System (Control Mass):

   • Mass: Cannot cross the boundary.

   • Energy: Can cross the boundary as heat or work.

   • Example: A sealed, rigid container being heated.

2. Open System (Control Volume):

   • Mass: Can flow in and out across the boundary.

   • Energy: Can cross with mass flow, as heat, and as work.

   • Example: A turbine, compressor, or nozzle.

3. Isolated System:

   • Mass: Cannot cross the boundary.

   • Energy: Cannot cross the boundary.

   • Example: A perfectly insulated thermos flask (idealized).

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2. Temperature and Thermodynamic Properties

2.1 Temperature

1. Definition: A measure of the average kinetic energy of molecules in a substance. It indicates the "hotness" or "coldness."

2. Scales:

   • Absolute Scales: Independent of material properties.

     • Kelvin (K): SI unit. $T(K) = T(°C) + 273.15$

     • Rankine (R): $T(R) = T(°F) + 459.67$

   • Relative Scales:

     • Celsius (°C): $T(°C) = T(K) - 273.15$

     • Fahrenheit (°F): $T(°F) = 1.8T(°C) + 32$

2.2 Thermodynamic Properties

1. Definition: Macroscopic characteristics of a system that can be measured or calculated.

2. Classification:

   • Intensive Properties: Independent of the mass of the system.

     • Examples: Temperature ($T$), Pressure ($P$), Density ($\rho$), Specific Volume ($v$).

   • Extensive Properties: Depend on the mass or size of the system.

     • Examples: Mass ($m$), Volume ($V$), Total Energy ($E$), Enthalpy ($H$).

   • Specific Property: An extensive property per unit mass. It becomes intensive.

     • Example: Specific volume $v = V/m$.

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3. State and Path Functions

3.1 State Function (Point Function)

1. Definition: A property whose value depends only on the current state of the system, not on the path taken to reach that state.

2. Mathematical Implication: Has an exact differential. Change is calculated as: $\Delta X = X_2 - X_1$

3. Examples:

   • Pressure ($P$), Temperature ($T$), Volume ($V$).

   • Internal Energy ($U$), Enthalpy ($H$), Entropy ($S$).

3.2 Path Function

1. Definition: A quantity whose value depends on the path followed during a process, not just the initial and final states.

2. Mathematical Implication: Has an inexact differential. The amount transferred is denoted by $\delta X$ or $X$ (not $\Delta X$).

3. Examples:

   • Work ($W$): $W = \int_{1}^{2} P \, dV$. Value depends on the $P-V$ path.

   • Heat Transfer ($Q$): Depends on the process path.

3.3 Key Difference

• For a cycle (initial state = final state):

  • Net change in any State Function = 0. ($\oint dU = 0, \oint dH = 0$)

  • Net Work and Heat Transfer are generally NOT zero. ($\oint \delta W \neq 0, \oint \delta Q \neq 0$)

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4. Thermodynamic Equilibrium

4.1 Definition

A system is in thermodynamic equilibrium if it satisfies all three of the following conditions simultaneously:

1. Mechanical Equilibrium: No unbalanced forces. Pressure is uniform throughout the system.

2. Thermal Equilibrium: No temperature gradients. Temperature is uniform throughout.

3. Chemical Equilibrium: No chemical reactions or diffusion of species. Composition is uniform and constant.

4.2 The State Postulate

• For a simple, compressible substance (the most common type), the state is completely defined by two independent intensive properties.

• Example: Knowing $P$ and $T$ is enough to determine $v$, $u$, $h$, etc., for a pure substance.

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5. Zeroth Law of Thermodynamics

5.1 Statement

If two bodies ($A$ and $B$) are each in thermal equilibrium with a third body ($C$), then they are in thermal equilibrium with each other.

5.2 Significance

1. Foundation of Temperature Measurement: It justifies the use of thermometers.

2. Transitive Property: It establishes that thermal equilibrium is a transitive relation (if $A = C$ and $B = C$, then $A = B$).

3. Allows Temperature Scales: It makes it meaningful to say "Body A has the same temperature as Body B."

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6. Ideal Gas and Gas Equations

6.1 Ideal Gas Definition

A hypothetical gas that obeys the Ideal Gas Equation of State exactly. It assumes:

• Molecules are point masses with no volume.

• No intermolecular forces except during perfectly elastic collisions.

6.2 Ideal Gas Equation of State

$P v = R T$ $P V = m R T = n \bar{R} T$

• $P$ = Absolute pressure (Pa)

• $v$ = Specific volume (m³/kg)

• $V$ = Total volume (m³)

• $T$ = Absolute temperature (K)

• $R$ = Specific gas constant (J/kg·K). $R = \bar{R} / M$, where $M$ is molar mass.

• $\bar{R}$ = Universal gas constant = 8.314 kJ/kmol·K

• $n$ = Number of moles

• $m$ = Mass (kg)

6.3 When to Use Ideal Gas Model?

• Good approximation when the gas is at low pressure and high temperature relative to its critical point.

• Commonly used for air, oxygen, nitrogen, etc., at atmospheric conditions.

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7. Specific Volume and Quality

7.1 Specific Volume ($v$)

• Definition: Volume per unit mass. $v = V/m$ (m³/kg).

• For a Pure Substance: It is an intensive property that varies with $P$ and $T$.

7.2 Quality ($x$) for Two-Phase Mixtures

• Definition: The mass fraction of vapor in a saturated liquid-vapor mixture. $x = \frac{m_g}{m_f + m_g}$ where $m_g$ = mass of vapor, $m_f$ = mass of liquid.

• Range: $0 \le x \le 1$

  • $x = 0$: Saturated liquid (no vapor).

  • $x = 1$: Saturated vapor (no liquid).

  • $0 < x < 1$: Wet mixture (liquid + vapor).

• Quality is undefined for single-phase regions (superheated vapor or subcooled liquid).

7.3 Using Quality to Find Properties

For a two-phase mixture, any extensive property ($y$) can be found using the lever rule: $y = y_f + x (y_g - y_f)$ where:

• $y$ = Property of the mixture (e.g., $v$, $u$, $h$, $s$).

• $y_f$ = Property of saturated liquid (from tables at given $P$ or $T$).

• $y_g$ = Property of saturated vapor (from tables at given $P$ or $T$).

• $(y_g - y_f)$ = Property change during vaporization (e.g., $h_{fg}$ = enthalpy of vaporization).

• Specific Volume Example: $v = v_f + x (v_g - v_f)$

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8. Two-Phase Systems (Pure Substances)

8.1 Phases of a Pure Substance

1. Solid Phase: Molecules are fixed in a lattice.

2. Liquid Phase: Molecules are close but can move freely.

3. Vapor/Gas Phase: Molecules are far apart and move randomly.

8.2 Phase-Change Processes (at constant pressure)

• Melting/Fusion: Solid → Liquid.

• Vaporization/Boiling: Liquid → Vapor.

• Sublimation: Solid → Vapor (without passing through liquid phase).

8.3 Property Behavior During Phase Change

• During a phase change at constant pressure and temperature:

  • Temperature remains constant (saturation temperature $T_{sat}$).

  • Pressure remains constant (saturation pressure $P_{sat}$).

  • Internal energy, enthalpy, and entropy increase.

  • Volume usually increases (especially during vaporization).

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9. Property Tables and Charts

9.1 Purpose

To provide accurate thermodynamic property data ($P, T, v, u, h, s$) for substances, especially where equations of state are complex (e.g., water/steam, refrigerants).

9.2 Common Table Formats

1. Saturation (Temperature) Table:

   • Lists properties for saturated liquid (f) and saturated vapor (g) as a function of temperature.

   • Columns: $T_{sat}$, $P_{sat}$, $v_f$, $v_g$, $u_f$, $u_g$, $h_f$, $h_g$, $s_f$, $s_g$.

2. Saturation (Pressure) Table:

   • Lists properties for saturated liquid and vapor as a function of pressure.

   • Columns: $P_{sat}$, $T_{sat}$, $v_f$, $v_g$, etc.

3. Superheated Vapor Table:

   • Lists properties for vapor at temperatures above the saturation temperature for a given pressure.

   • Entries: For a fixed pressure, properties are given at various temperatures.

4. Compressed Liquid Table:

   • Lists properties for liquid at pressures above the saturation pressure for a given temperature.

   • Often approximated using saturated liquid data at the same temperature (since liquid properties are weakly dependent on pressure).

9.3 How to Use the Tables

1. Identify the State:

   • Compare given $P$ and $T$ with $P_{sat}$ at that $T$ (or $T_{sat}$ at that $P$).

2. Determine the Phase Region:

   • Given (P,T):

     • If $T < T_{sat}(P)$ → Compressed Liquid.

     • If $T = T_{sat}(P)$ → Saturated Mixture (find quality $x$ if needed).

     • If $T > T_{sat}(P)$ → Superheated Vapor.

   • Given (P,v):

     • If $v < v_f(P)$ → Compressed Liquid.

     • If $v_f(P) < v < v_g(P)$ → Saturated Mixture. Calculate $x = (v - v_f)/(v_g - v_f)$.

     • If $v > v_g(P)$ → Superheated Vapor.

3. Extract Data:

   • Single Phase (Compressed Liquid, Superheated Vapor): Read directly from the appropriate table.

   • Two Phase (Saturated Mixture): Use $y = y_f + x(y_g - y_f)$ with data from the saturation table.

9.4 Property Diagrams (Charts)

• T-v Diagram: Shows temperature vs. specific volume. Clearly displays saturated liquid-vapor dome and critical point.

• P-v Diagram: Shows pressure vs. specific volume. Essential for analyzing work in thermodynamic cycles.

• h-s Diagram (Mollier Chart): Shows enthalpy vs. entropy. Useful for analyzing turbines and compressors.

• P-h Diagram: Shows pressure vs. enthalpy. Widely used in refrigeration cycle analysis.