Thermodynamics (ME-302) - Unit 5 Short Notes

UNIT 5: Second Law of Thermodynamics & Advanced Topics

1. Second Law of Thermodynamics

Statements

• Kelvin-Planck Statement: It is impossible to construct a device that, operating in a cycle, will produce no effect other than the extraction of heat from a single reservoir and the performance of an equivalent amount of work.

  Implication: A heat engine must have at least two thermal reservoirs. 100% conversion of heat to work is impossible.

• Clausius Statement: It is impossible to construct a device that, operating in a cycle, will produce no effect other than the transfer of heat from a cooler body to a hotter body.

  Equivalence: Violation of one statement implies violation of the other. They are two sides of the same law.

Clausius Inequality

For any cyclic process, the cyclic integral of $$\displaystyle \frac{\delta Q}{T} $$ is less than or equal to zero.

$$ \oint \frac{\delta Q}{T} \leq 0 $$

• Equality ($$\displaystyle = $$): Holds for a reversible cycle.

• Inequality ($$\displaystyle < $$): Holds for an irreversible cycle.

• Significance: It provides a mathematical criterion to determine if a cycle is reversible or irreversible. It is the mathematical expression of the second law.

• Role in Defining Entropy: For a reversible process between states 1 and 2, $$\displaystyle dS = \frac{\delta Q_{rev}}{T} $$. The Clausius inequality proves that entropy $S$ is a property (state function).

Carnot Cycle

A theoretical reversible cycle consisting of four processes:

1. Reversible Isothermal heat addition ($$\displaystyle T_H $$)

2. Reversible Adiabatic expansion

3. Reversible Isothermal heat rejection ($$\displaystyle T_L $$)

4. Reversible Adiabatic compression

Efficiency Derivation:

For reversible processes, $$\displaystyle \frac{Q_H}{T_H} = \frac{Q_L}{T_L} $$ (from Clausius inequality for reversible cycle).

$$ \eta_{Carnot} = 1 - \frac{Q_L}{Q_H} = 1 - \frac{T_L}{T_H} $$

Importance: Sets the maximum possible efficiency for any heat engine operating between two given temperature reservoirs $$\displaystyle T_H $$ and $$\displaystyle T_L $$. All real engines have lower efficiency.

Heat Engines, Refrigerators, and Heat Pumps

Relationship: $$\displaystyle COP_{HP} = COP_R + 1 $$

Combined Cycles (Reversible Engine + Reversible Refrigerator)

A reversible power cycle (engine) drives a reversible refrigeration cycle.

• Engine: $$\displaystyle Q_1 $$ at $$\displaystyle T_1 $$, $$\displaystyle Q_2 $$ at $$\displaystyle T_2 $$, Work $$\displaystyle W = Q_1 - Q_2 $$

• Refrigerator: $$\displaystyle Q_4 $$ from $$\displaystyle T_4 $$, $$\displaystyle Q_3 $$ at $$\displaystyle T_3 $$, Work $$\displaystyle W = Q_3 - Q_4 $$

• Since $W$ is same: $$\displaystyle Q_1 - Q_2 = Q_3 - Q_4 $$

• For reversible cycles: $$\displaystyle \frac{Q_1}{T_1} = \frac{Q_2}{T_2} $$ and $$\displaystyle \frac{Q_3}{T_3} = \frac{Q_4}{T_4} $$

• Derivation of $$\displaystyle \frac{Q_2}{Q_1} $$:

  From $$\displaystyle Q_1 - Q_2 = Q_3 - Q_4 $$ and substituting $$\displaystyle Q_3 = Q_4 \frac{T_3}{T_4} $$:

  $$\displaystyle Q_1 - Q_2 = Q_4 \left( \frac{T_3}{T_4} - 1 \right) $$

  Also, $$\displaystyle \frac{Q_2}{Q_1} = \frac{T_2}{T_1} \Rightarrow Q_2 = Q_1 \frac{T_2}{T_1} $$

  Solving simultaneously yields:

  \boxed{\frac{Q_2}{Q_1} = \frac{T_2 (T_3 - T_4)}{T_1 T_3 - T_2 T_4}}

Availability (Exergy)

• Definition: The maximum useful work obtainable as a system comes to equilibrium with a dead state (environment at $$\displaystyle T_0, P_0 $$).

• Availability Function (for a closed system):

$$ \psi = (U - U_0) + P_0(V - V_0) - T_0(S - S_0) $$

For a flowing fluid, **Flow Exergy** or **Flow Availability** is:

$$ \psi = (H - H_0) - T_0(S - S_0) $$

where $$\displaystyle H_0, S_0 $$ are enthalpy and entropy at dead state.

• Is it a Property? Yes. Exergy is a property because it is defined in terms of state properties ($U, H, S, V$) and the fixed dead state properties.

• Significance: Quantifies the "quality" or "usefulness" of energy. It measures the maximum theoretical work potential.

• Maximum Work Concept: The reversible work done by a system as it reaches equilibrium with the environment equals the decrease in exergy of the system.

$$ W_{max, net} = \Delta \psi_{system} \quad (\text{for a closed system interacting with environment}) $$

2. Pure Substances and Phase Change

Definition and Characteristics

A pure substance has a fixed chemical composition throughout. It can exist in multiple phases (solid, liquid, vapor) but the chemical identity is uniform (e.g., pure water, pure nitrogen). A mixture of gases like air is often treated as a pure substance for thermodynamic analysis.

Phase Diagrams

• P-V Diagram: Shows saturation dome. Left curve: saturated liquid line. Right curve: saturated vapor line. Inside dome: wet region (liquid-vapor mixture).

• T-s Diagram: Similar dome. Shows constant temperature lines (isotherms) within the dome are horizontal.

• P-T Diagram: Shows triple point (unique P,T where all three phases coexist) and critical point (beyond which liquid and vapor are indistinguishable).

Triple Point: Unique state where solid, liquid, and vapor coexist in equilibrium.

Critical Point: State where the properties of saturated liquid and saturated vapor become identical.

Steam Formation (Constant Pressure Heating)

Heating liquid water at constant pressure (e.g., in a boiler):

1. Compressed Liquid: Subcooled liquid (T < Tsat at given P).

2. Saturated Liquid (f=0): Starts boiling at Tsat.

3. Wet Steam (0 < f < 1): Liquid and vapor mixture. Temperature and pressure are dependent (Tsat at P).

4. Saturated Vapor (f=1): Last drop of liquid evaporates.

5. Superheated Steam (f=1, T > Tsat): Vapor phase only.

Graphical Representation:

• T-v Diagram: Horizontal line during phase change (constant T).

• P-v Diagram: Vertical line during phase change? No. Pressure is constant during the process, so it's a horizontal line on a P-v diagram.

Steam Tables and Mollier Chart

• Steam Tables: Tabulate properties ($v, h, s, u$) of water/steam at saturation and superheat. Usage: Find properties given P & T (or f). Interpolation often needed.

• Mollier Chart (h-s diagram): Graphical representation of steam properties. Significance: For isentropic processes (turbines, nozzles), follow constant entropy line to read $$\displaystyle h_1, h_2 $$ and find work $$\displaystyle W = h_1 - h_2 $$. Isobars diverge because $$\displaystyle (\frac{\partial h}{\partial s})_P = T $$, and T increases with s at higher pressures.

Dryness Fraction (Quality)

• Definition: Mass fraction of vapor in a saturated liquid-vapor mixture.

$$ x = \frac{m_{vapor}}{m_{total}}, \quad 0 \leq x \leq 1 $$

• Measurement:

  1. Separating Calorimeter: Physically separates liquid and vapor, measures masses.

  2. Throttling Calorimeter: Steam throttled to low pressure, measures T & P after throttling. Assuming $$\displaystyle h_{before} = h_{after} $$ (isenthalpic), use superheated steam tables to find $x$.

• Engineering Significance: Determines the volume fraction (vapor occupies most volume) and is crucial for calculating properties in the wet region.

Thermodynamic Properties in Different Regions

3. Ideal Gases and Gas Mixtures

Ideal Gases (Single Component)

• Equation of State: $$\displaystyle Pv = RT $$ (specific) or $$\displaystyle PV = mRT $$.

• Specific Heats:

  • $$\displaystyle C_v = \left( \frac{\partial u}{\partial T} \right)_v $$

  • $$\displaystyle C_p = \left( \frac{\partial h}{\partial T} \right)_p $$

  • Relationship: $$\displaystyle C_p - C_v = R $$

• Temperature-Dependent $$\displaystyle C_p $$: e.g., $$\displaystyle C_p = a + bT $$ (kJ/kg·K).

  • Mean Specific Heat $$\displaystyle \bar{C_p} $$ over $$\displaystyle T_1 $$ to $$\displaystyle T_2 $$:

$$ \bar{C_p} = \frac{1}{T_2 - T_1} \int_{T_1}^{T_2} C_p \, dT $$

    Heat transferred: $$\displaystyle Q = m \int_{T_1}^{T_2} C_p \, dT = m \bar{C_p} (T_2 - T_1) $$.

• Thermodynamic Processes (Closed System, Ideal Gas):

  • Isothermal ($$\displaystyle T=const $$): $$\displaystyle W = Q = mRT \ln\left(\frac{V_2}{V_1}\right) $$

  • Adiabatic ($$\displaystyle Q=0, reversible $$): $$\displaystyle PV^\gamma = const $$, $$\displaystyle T V^{\gamma-1}=const $$, $$\displaystyle T^{\gamma} P^{1-\gamma}=const $$ <br> Work: $$\displaystyle W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} = m C_v (T_1 - T_2) $$

  • Polytropic ($$\displaystyle PV^n = const $$): Work: $$\displaystyle W = \frac{P_1 V_1 - P_2 V_2}{n - 1} $$ (n ≠ 1)

  • Free Expansion (Joule Expansion): $$\displaystyle W=0, Q=0, \Delta U=0 $$ (ideal gas), $$\displaystyle \Delta T=0 $$.

• First Law for Closed System: $$\displaystyle Q = \Delta U + W $$

Ideal Gas Mixtures

• Dalton's Law of Partial Pressures: Total pressure equals sum of partial pressures. $$\displaystyle P = \sum P_i $$, where $$\displaystyle P_i = n_i R T / V $$.

• Amagat's Law of Partial Volumes: Total volume equals sum of partial volumes. $$\displaystyle V = \sum V_i $$, where $$\displaystyle V_i = n_i R T / P $$.

• P-V-T Relationships: Use mass-weighted or mole-weighted averages.

  • Gas Constant for Mixture: $$\displaystyle R_{mix} = \sum Y_i R_i $$ (mass basis) or $$\displaystyle R_{mix} = \sum x_i R_i $$ (mole basis).

  • Molecular Weight: $$\displaystyle M_{mix} = \sum Y_i M_i $$ or $$\displaystyle M_{mix} = \frac{1}{\sum (y_i / M_i)} $$.

• Properties of Mixtures:

  • Internal Energy & Enthalpy: Extensive, additive. $$\displaystyle U = \sum U_i $$, $$\displaystyle H = \sum H_i $$. Per unit mass: $$\displaystyle u = \sum Y_i u_i(T) $$, $$\displaystyle h = \sum Y_i h_i(T) $$. Depend only on T for ideal gases.

  • Entropy: $$\displaystyle S = \sum S_i $$. For mixing, entropy of mixing is positive.

  • Specific Heats: $$\displaystyle C_v = \sum Y_i C_{vi}(T) $$, $$\displaystyle C_p = \sum Y_i C_{pi}(T) $$.

• Adiabatic Saturation Process:

  • Condition: Air-vapor mixture adiabatically saturated by liquid water spray. Constant enthalpy process ($$\displaystyle h_{in} = h_{out} $$).

  • Significance: Basis for psychrometric charts (h-s diagram for air-water vapor). Used in cooling towers and humidification processes.

4. Thermodynamic Cycles

Air-Standard Assumptions

1. Working fluid is air, behaves as an ideal gas with constant specific heats.

2. All processes are internally reversible.

3. Heat addition and rejection are with external heat reservoirs.

4. No friction, pressure losses, or heat losses to surroundings.

5. Closed cycle (same air mass recirculates).

Power Cycles

Cycle Comparison: For same compression ratio $r$, $$\displaystyle \eta_{Carnot} > \eta_{Otto} > \eta_{Diesel} > \eta_{Dual} $$ (at typical $\rho$). Diesel efficiency decreases as $\rho$ increases.

Refrigeration Cycles

• Carnot Refrigeration Cycle: Reverse Carnot cycle.

$$ COP_{R, Carnot} = \frac{T_L}{T_H - T_L} $$

• Heat Pump: Same cycle, different objective.

$$ COP_{HP, Carnot} = \frac{T_H}{T_H - T_L} $$

Graphical Representation

All cycles must be drawn on P-V and T-s diagrams. Key: Direction of processes (clockwise = power cycle, counter-clockwise = refrigeration cycle).

5. Combustion Thermodynamics

Fuels

• Primary: Natural gas, coal, crude oil (found in nature).

• Secondary (Derived): Gasoline, diesel, LPG, kerosene (derived from primary).

• Analysis:

  • Ultimate Analysis: % by mass of C, H, O, N, S, Ash, Moisture. Used for stoichiometry.

  • Proximate Analysis: % of Moisture, Volatile Matter, Fixed Carbon, Ash. Used for combustion behavior.

Stoichiometry

1. Balanced Equation (Complete Combustion): Fuel + $a$ (Air) $$\displaystyle \rightarrow $$ Products.

   • $$\displaystyle C_xH_y + a (O_2 + 3.76 N_2) \rightarrow x CO_2 + \frac{y}{2} H_2O + 3.76a N_2 $$

   • Theoretical $$\displaystyle a_{th} $$ from oxygen balance: $$\displaystyle 2x + \frac{y}{4} = 2a_{th} $$

2. Theoretical Air: Exact amount of air for complete combustion ($$\displaystyle a = a_{th} $$).

3. Excess Air: $$\displaystyle \text{% Excess Air} = \frac{a - a_{th}}{a_{th}} \times 100 $$. Used to ensure complete combustion.

4. Air-Fuel Ratio (AFR): $$\displaystyle AFR = \frac{m_{air}}{m_{fuel}} = \frac{28.97 \times a}{M_{fuel}} $$.

Enthalpy of Formation & Reaction

• Enthalpy of Formation ($$\displaystyle \Delta H_f $$): Enthalpy change when 1 mole of compound is formed from its elements in their standard states at specified T (usually 298 K). Elements in standard state have $$\displaystyle \Delta H_f^0 = 0 $$.

• Enthalpy of Reaction ($$\displaystyle \Delta H_{Rxn} $$): Enthalpy change for the reaction as written.

  • At Standard Conditions ($$\displaystyle T_0 $$): $$\displaystyle \Delta H_{Rxn}^0 = \sum n_p \Delta H_{f,p}^0 - \sum n_r \Delta H_{f,r}^0 $$ (Hess's Law).

  • At Temperature T: $$\displaystyle \Delta H_{Rxn}(T) = \Delta H_{Rxn}^0 + \int_{T_0}^T \Delta C_p \, dT $$.

First Law Analysis of Reacting Systems

For a steady-flow combustor (neglecting KE/PE changes):

$$ \dot{m}_{fuel} h_{fuel} + \dot{m}_{air} h_{air} = \dot{m}_{prod} h_{prod} + \dot{Q} - \dot{W} $$

Often $\dot{Q} \approx 0$, $\dot{W} \approx 0$ (adiabatic, no shaft work).

• Heating Values:

  • Higher Heating Value (HHV): Heat released when products are cooled to initial temperature (T0) and water vapor condenses.

  • Lower Heating Value (LHV): Heat released when products are cooled to T0 but water remains as vapor. $$\displaystyle LHV = HHV - m_{H_2} \cdot h_{fg} $$ (latent heat of water in products).

Adiabatic Flame Temperature

• Definition: Temperature of combustion products when no heat is lost to surroundings ($$\displaystyle Q=0 $$).

• Calculation: From energy balance: $$\displaystyle H_{reactants}(T_0) = H_{products}(T_{ad}) $$. Solve iteratively for $$\displaystyle T_{ad} $$ using product enthalpies (which depend on T). With dissociation, product composition also depends on T.

Actual vs. Theoretical Combustion

• Theoretical (Ideal): Perfect mixing, complete combustion, no heat loss, products at $$\displaystyle T_{ad} $$.

• Actual: Incomplete combustion (CO, soot), heat loss to walls, pressure drop, dissociation at high T, excess air.

• Impact: Lower efficiency (less useful work), pollutants (CO, NOx, soot), lower $$\displaystyle T_{ad} $$ than theoretical.

Combustion Products

• Volumetric Composition: From product moles (using excess air) and total moles at given P,T.

• Dew Point: Temperature at which water vapor in products begins to condense at constant pressure. Important for corrosion in exhaust systems. Found from partial pressure of water vapor and steam tables.

6. Advanced Topics and Special Processes

Third Law of Thermodynamics

• Statement (Nernst Heat Theorem): The entropy of a pure crystalline substance is zero at absolute zero temperature.

• Implications:

  1. Provides a reference point for absolute entropy calculations.

  2. Unattainability Principle: Absolute zero temperature cannot be reached by any finite number of processes.

Limitations of the First Law

• Cannot determine the direction of a process (e.g., heat flows from hot to cold, not reverse).

• Cannot quantify the quality or degradation of energy (e.g., work is "high-quality", heat at low T is "low-quality").

• Does not set an upper limit on cycle efficiency (Carnot efficiency comes from Second Law).

Real Gases

• P-V-T Surface: More complex than ideal gas. Shows a critical point and a saturation dome similar to pure substances.

• Deviations from Ideal Behavior: Significant at high pressures and low temperatures. Caused by:

  1. Finite molecular volume (repulsive forces).

  2. Intermolecular attractive forces.

• Equations of State: Van der Waals ($$\displaystyle (P + a/v^2)(v - b) = RT $$), Redlich-Kwong, etc.

Temperature Scales and Measurement

• New Scales: Based on thermodynamic principles (e.g., ideal gas thermometer).

• Constant Volume Gas Thermometer: Most accurate for defining thermodynamic temperature. Pressure of gas at constant volume is proportional to T. Used to realize the Kelvin scale.

• Constant Pressure Gas Thermometer: Volume proportional to T. Less accurate due to gas non-idealities and bulb expansion.

Special Processes

• Free Expansion (Joule Expansion): Gas expands into an evacuated space with no resistance, adiabatic ($$\displaystyle Q=0 $$), no work done ($$\displaystyle W=0 $$). For ideal gas: $$\displaystyle \Delta U = 0 $$, $$\displaystyle \Delta T = 0 $$. For real gas: $\Delta T \neq 0$ (Joule-Thomson effect).

• Polytropic Process ($$\displaystyle PV^n = const $$): Generalizes many processes:

  • $$\displaystyle n=0 \rightarrow $$ Constant Pressure

  • $$\displaystyle n=1 \rightarrow $$ Isothermal

  • $$\displaystyle n=\gamma \rightarrow $$ Isentropic (Adiabatic reversible)

  • $$\displaystyle n=\infty \rightarrow $$ Constant Volume

  • Work: $$\displaystyle W = \frac{P_1 V_1 - P_2 V_2}{n - 1} $$ (for $n \neq 1$).

• Adiabatic Saturation: (See Gas Mixtures). Constant enthalpy process for air-vapor mixture.