3.1 Thermodynamic State Postulate, Properties & P-V-T Equations of State

3.1 Thermodynamic State Postulate, Properties & P-V-T Equations of State

Thermodynamics is the branch of physical science that governs energy transformation, heat, work, and the fundamental properties of matter. For candidate mechanical engineers preparing for the Philippine Regulation Commission (PRC) Mechanical Engineering Licensure Examination (MELE), mastering property relations, phase diagrams, and equations of state is essential for solving complex thermal power plant, refrigeration, and HVAC engineering problems.

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1. Thermodynamic Definitions & The State Postulate

A thermodynamic system is defined as a quantity of matter or a region in space chosen for study. The mass or region outside the system is called the surroundings, and the real or imaginary surface that separates the system from its surroundings is the boundary.

Thermodynamic properties are classified into two broad categories:

• Intensive Properties: Independent of the mass or size of the system. Examples include Temperature ($T$), Pressure ($P$), Density ($ ho$), Specific Volume ($v$), Specific Enthalpy ($h$), and Specific Internal Energy ($u$).
• Extensive Properties: Depend directly on the total mass or extent of the system. Examples include Total Mass ($m$), Total Volume ($V$), Total Internal Energy ($U$), Total Enthalpy ($H$), and Total Entropy ($S$).

Specific Properties: Extensive properties expressed per unit mass are intensive properties (e.g., $v = V/m$, $u = U/m$, $h = H/m$).

The Zeroth Law of Thermodynamics

The Zeroth Law of Thermodynamics states that if two bodies are in thermal equilibrium with a third body, they are also in thermal equilibrium with each other. This fundamental law serves as the physical basis for temperature measurement; it validates that two bodies have the same temperature if they are in thermal equilibrium without needing physical contact.

The State Postulate

To completely specify the thermodynamic state of a system, a specific number of properties must be fixed. The State Postulate for a simple compressible system states:

$\text{The state of a simple compressible system is completely specified by two independent, intensive properties.}$

A simple compressible system is defined as a system in the absence of electrical, magnetic, gravitational, motion, and surface tension effects. For instance, if pressure ($P$) and specific volume ($v$) of superheated steam are known, all other intensive properties such as temperature ($T$), specific internal energy ($u$), and specific enthalpy ($h$) are fixed.

$\text{State Specification}: \quad T = T(P, v), \quad u = u(P, v), \quad h = h(P, v)$

Note: Two properties are independent if one property can be varied while the other is held constant. Inside a two-phase saturation dome (e.g., boiling water), temperature and pressure are dependent ($P = P_{\text{sat}}(T)$); thus, temperature and pressure alone cannot fix the state of a two-phase mixture.

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2. Pure Substances & Phase-Change Phenomena

A pure substance has a fixed chemical composition throughout. Water, nitrogen, helium, and carbon dioxide are pure substances. A mixture of two or more phases of a pure substance (such as liquid water and water vapor) is also a pure substance, provided the chemical composition of all phases remains identical.

Phase-Change Definitions

During a constant-pressure heating process of a pure substance (e.g., water at $1\text{ atm} = 101.325\text{ kPa}$):

1. Compressed (or Subcooled) Liquid: A liquid that is not on the verge of vaporizing ($T < T_{\text{sat}}$ at given $P$).
2. Saturated Liquid: A liquid that is on the verge of vaporizing ($T = T_{\text{sat}}$, subscript $f$).
3. Saturated Liquid-Vapor Mixture: The two-phase state where liquid and vapor co-exist in equilibrium ($T = T_{\text{sat}}, P = P_{\text{sat}}$).
4. Saturated Vapor: A vapor that is on the verge of condensing ($T = T_{\text{sat}}$, subscript $g$).
5. Superheated Vapor: A vapor that is not on the verge of condensing ($T > T_{\text{sat}}$ at given $P$).

Vapor Quality ($x$)

In a saturated liquid-vapor mixture, the mass fraction of vapor is defined as the vapor quality ($x$):

$x = \frac{m_{\text{vapor}}}{m_{\text{total}}} = \frac{m_g}{m_f + m_g}$

Quality is meaningful only for saturated mixtures ($0 \le x \le 1$). For saturated liquid, $x = 0$; for saturated vapor, $x = 1$.

Using quality, any specific property $y$ ($v, u, h, s$) of a two-phase mixture is computed as:

$y = y_f + x y_{fg} = (1 - x) y_f + x y_g$

where $y_{fg} = y_g - y_f$ is the property difference between saturated vapor and saturated liquid.

Critical Point & Phase Diagrams

• Critical Point: The state at which the saturated liquid and saturated vapor states are identical. Above the critical pressure ($P_{\text{cr}}$) and critical temperature ($T_{\text{cr}}$), there is no distinct phase-change process.
• Triple Point: The state at which all three phases (solid, liquid, vapor) coexist in equilibrium. For water, $T_{\text{triple}} = 0.01^\circ\text{C}$ ($273.16\text{ K}$) and $P_{\text{triple}} = 0.6117\text{ kPa}$.

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3. Ideal Gas Equation of State

An equation of state is any mathematical relationship among pressure, temperature, and specific volume. The simplest and most widely used equation of state is the Ideal Gas Law:

$P v = R T \quad \implies \quad P V = m R T$

where:

• $P$ = Absolute pressure (kPa or N/m²)
• $V$ = Total volume (m³)
• $v$ = Specific volume (m³/kg)
• $m$ = Mass of gas (kg)
• $T$ = Absolute temperature (Kelvin, $\text{K} = ^\circ\text{C} + 273.15$)
• $R$ = Specific gas constant (kJ/kg·K)

The specific gas constant $R$ is related to the Universal Gas Constant ($\bar{R}$) and the molar mass ($M$) of the gas:

$R = \frac{\bar{R}}{M}, \quad \text{where } \bar{R} = 8.31447 \text{ kJ/(kmol}\cdot\text{K)} = 8.31447 \text{ kPa}\cdot\text{m}^3\text{/(kmol}\cdot\text{K)}$

$\text{Molar form of Ideal Gas Law}: \quad P V = N \bar{R} T$

where $N = m/M$ is the number of kmoles of the gas.

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4. Real Gas Behavior & Real Gas Equations of State

Real gases deviate significantly from ideal gas behavior at high pressures ($P \approx P_{\text{cr}}$) and low temperatures ($T \approx T_{\text{cr}}$) because intermolecular forces and finite molecular volumes can no longer be neglected.

Compressibility Factor ($Z$)

The deviation from ideal behavior is accounted for by the Compressibility Factor ($Z$):

$Z = \frac{P v}{R T} = \frac{v_{\text{actual}}}{v_{\text{ideal}}}$

For an ideal gas, $Z = 1$ under all conditions. For real gases, $Z$ can be greater than or less than 1 depending on pressure and temperature.

Principle of Corresponding States

The principle states that the compressibility factor $Z$ for all gases is approximately the same at the same Reduced Temperature ($T_r$) and Reduced Pressure ($P_r$):

$T_r = \frac{T}{T_{\text{cr}}}, \quad P_r = \frac{P}{P_{\text{cr}}}$

To avoid iterations when $v$ is unknown, a Pseudo-Reduced Specific Volume ($v'_r$) is used:

$v'_r = \frac{v_{\text{actual}}}{R T_{\text{cr}} / P_{\text{cr}}}$

van der Waals Equation of State

Proposed in 1873, the van der Waals equation improves upon the ideal gas law by adding two empirical constants: $a$ to account for intermolecular attractive forces, and $b$ to account for the volume occupied by gas molecules:

$\left(P + \frac{a}{v^2}\right)(v - b) = R T$

The constants $a$ and $b$ are determined from the critical-point criteria (where $(\partial P / \partial v)_T = 0$ and $(\partial^2 P / \partial v^2)_T = 0$):

$a = \frac{27 R^2 T_{\text{cr}}^2}{64 P_{\text{cr}}}, \quad b = \frac{R T_{\text{cr}}}{8 P_{\text{cr}}}$

Beattie-Bridgeman Equation of State

The Beattie-Bridgeman equation (1928) is a high-accuracy empirical equation based on five experimentally determined constants:

$P = \frac{\bar{R} T}{\bar{v}^2}\left(1 - \frac{c}{\bar{v} T^3}\right)(\bar{v} + B) - \frac{A}{\bar{v}^2}$

where $\bar{v}$ is molar specific volume (m³/kmol), and:

$A = A_0 \left(1 - \frac{a}{\bar{v}}\right), \quad B = B_0 \left(1 - \frac{b}{\bar{v}}\right)$

The constants $A_0, B_0, a, b,$ and $c$ are published tabulated values for individual gases.

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5. Step-by-Step Worked Numerical Calculation

Problem Statement

A rigid closed pressure vessel with a volume of $V = 0.50\text{ m}^3$ contains $m = 10.0\text{ kg}$ of Refrigerant-134a at a uniform temperature of $T = -10.0^\circ\text{C}$.

Given saturation properties for R-134a at $T = -10.0^\circ\text{C}$:

• $P_{\text{sat}} = 201.73\text{ kPa}$
• $v_f = 0.0007534\text{ m}^3\text{/kg}$
• $v_g = 0.099590\text{ m}^3\text{/kg}$
• $h_f = 38.43\text{ kJ/kg}$
• $h_{fg} = 206.20\text{ kJ/kg}$
• $u_f = 38.28\text{ kJ/kg}$
• $u_{fg} = 186.72\text{ kJ/kg}$

Calculate:

1. The specific volume of R-134a in the tank.
2. The thermodynamic state phase and vapor quality ($x$).
3. The pressure ($P$) inside the tank.
4. The total enthalpy ($H$) of R-134a in the tank in kilojoules (kJ).

Step-by-Step Solution

Step 1: Compute specific volume ($v$) $v = \frac{V}{m} = \frac{0.50\text{ m}^3}{10.0\text{ kg}} = 0.0500\text{ m}^3\text{/kg}$

Step 2: Determine state phase and vapor quality ($x$) Compare $v$ with $v_f$ and $v_g$ at $T = -10.0^\circ\text{C}$: $v_f = 0.0007534\text{ m}^3\text{/kg} < v = 0.0500\text{ m}^3\text{/kg} < v_g = 0.099590\text{ m}^3\text{/kg}$

Since $v_f < v < v_g$, the refrigerant is in the Saturated Liquid-Vapor Mixture Region.

Calculate quality $x$: $v = v_f + x(v_g - v_f) = v_f + x v_{fg}$ $x = \frac{v - v_f}{v_g - v_f} = \frac{0.0500 - 0.0007534}{0.099590 - 0.0007534} = \frac{0.0492466}{0.0988366} = 0.49826 \quad (49.83\%)$

Step 3: Determine tank pressure ($P$) For a saturated liquid-vapor mixture, the pressure equals the saturation pressure at the given temperature: $P = P_{\text{sat}} = 201.73\text{ kPa}$

Step 4: Compute specific enthalpy ($h$) and total enthalpy ($H$) $h = h_f + x h_{fg} = 38.43 + (0.49826 \times 206.20) = 38.43 + 102.74 = 141.17\text{ kJ/kg}$

Total enthalpy $H$: $H = m h = 10.0\text{ kg} \times 141.17\text{ kJ/kg} = 1411.7\text{ kJ}$

Final Summary of Results:

• Specific volume $v = 0.0500\text{ m}^3\text{/kg}$
• State: Saturated mixture with quality $x = 0.4983$ (49.83% vapor mass fraction)
• Tank Pressure $P = 201.73\text{ kPa}$
• Total Enthalpy $H = 1411.7\text{ kJ}$