Thermodynamics (ME-302) - Unit 2 Short Notes

UNIT 2: THERMODYNAMICS - SHORT NOTES (RGPV FOCUS)

1. FUNDAMENTAL LAWS & CONCEPTS

First Law of Thermodynamics (Closed System)

• Statement: Energy cannot be created or destroyed, only transformed. For a closed system undergoing a cycle, net heat transfer equals net work transfer.

$$Q_{net} = W_{net}$$

• For a Process (Non-Flow): The change in internal energy ($\Delta U$) equals net heat added minus net work done by the system.

$$\boxed{Q = \Delta U + W}$$

where $W$ is work done **by** the system.

• Energy as a Property: Internal energy ($U$) and enthalpy ($$\displaystyle H = U + pV $$) are state functions (properties).

• Steady Flow Energy Equation (SFEE): For a control volume (open system) in steady state:

$$\dot{Q} - \dot{W} = \dot{m}(h_2 - h_1 + \frac{V_2^2 - V_1^2}{2} + g(z_2 - z_1))$$

Often simplified for turbines/compressors: $$\displaystyle \dot{W} = \dot{m}(h_1 - h_2) $$ (turbine) or $$\displaystyle \dot{W} = \dot{m}(h_2 - h_1) $$ (compressor).

• Limitations of First Law: It does not indicate the direction of a process or the quality of energy (e.g., why heat doesn't spontaneously flow from cold to hot). It sets no limits on efficiency.

Second Law of Thermodynamics

• Kelvin-Planck Statement: It is impossible to construct a device that, operating in a cycle, will produce no other effect than the extraction of heat from a single reservoir and the performance of an equivalent amount of work. (No 100% efficient heat engine).

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

• Equivalence: If one statement is violated, the other can be shown to be violated. They are equivalent.

• Clausius Inequality: For any thermodynamic cycle,

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

Equality holds for a **reversible cycle**. The inequality is a mathematical statement of the Second Law.

• Defining Entropy: Clausius inequality leads to the definition of entropy ($S$) as a property. For a reversible process, $$\displaystyle dS = \frac{\delta Q_{rev}}{T} $$. For any process, $$\displaystyle dS \geq \frac{\delta Q}{T} $$.

  [!TIP] Exam Focus: Be prepared to derive/justify that $$\displaystyle \oint \frac{\delta Q}{T} \leq 0 $$ implies entropy is a property.

Third Law of Thermodynamics

• Statement (Nernst's Theorem): The entropy of a perfect crystalline substance is zero at absolute zero temperature (0 K).

• Significance:

  1. Provides a reference point for calculating absolute entropy values (from $$\displaystyle S = \int_0^T \frac{C_p}{T} dT $$).

  2. States that absolute zero is unattainable in a finite number of steps.

  3. As $$\displaystyle T \rightarrow 0 $$, the heat capacities of all substances approach zero.

2. AVAILABILITY (EXERGY) & IRREVERSIBILITY

• Availability (Exergy) Function: The maximum useful work obtainable as a system comes to equilibrium with a specified reference environment (dead state: $$\displaystyle T_0, p_0 $$).

  For a closed system:

$$\boxed{A = (U - U_0) + p_0(V - V_0) - T_0(S - S_0)}$$

Often simplified if reference state properties are zero: $$\displaystyle A = U + p_0V - T_0S $$.

• Is Availability a Property? Yes. It is defined in terms of state properties ($U, V, S$) and constant environmental parameters ($$\displaystyle T_0, p_0 $$). Its change depends only on initial and final states.

• Maximum Work: For a system interacting only with the environment, the maximum work is the decrease in availability: $$\displaystyle W_{max} = -\Delta A $$.

• Irreversibility (I): The loss of useful work due to irreversibilities.

$$\boxed{I = T_0 \Delta S_{gen}}$$

where $$\displaystyle \Delta S_{gen} > 0 $$ for an irreversible process.

• Second Law Efficiency (η_II): Ratio of actual work output to maximum possible work output (or actual COP to reversible COP).

$$\eta_{II} = \frac{\text{Actual Performance}}{\text{Reversible (Maximum) Performance}}$$

• Solved Problem Example (Steel Blocks): Available energy of a block cooling from $T$ to $$\displaystyle T_0 $$:

$$A = m C_v \left[ (T - T_0) - T_0 \ln\left(\frac{T}{T_0}\right) \right]$$

> [!TIP] **Common Pitfall:** Do not confuse availability with energy. Availability is *destroyed* by irreversibility; total energy is conserved (First Law).

3. PROPERTIES OF PURE SUBSTANCES (WATER & STEAM)

Key Concepts & Diagrams

• Pure Substance: A homogeneous material with a fixed chemical composition (e.g., water, nitrogen). Can exist in different phases (solid, liquid, vapor).

• P-V Diagram: Shows saturation curve separating liquid/vapor regions. Triple point (all 3 phases coexist) and critical point (liquid-vapor distinction vanishes).

• T-s Diagram: Shows constant pressure lines diverge in wet region, converge in superheated region. Saturation curve is vertical in wet region.

• P-V-T Surface: 3D surface for a real substance. The "dome" encloses the two-phase region.

  DiagramSEARCH: "p-v-t surface water steam 3d diagram"

Steam Formation & Properties

1. Compressed Liquid: $$\displaystyle T < T_{sat} $$ at given $P$. Properties ≈ saturated liquid ($f$).

2. Saturated Liquid: State $f$ on saturation line. $$\displaystyle v = v_f $$, $$\displaystyle h = h_f $$, $$\displaystyle s = s_f $$.

3. Wet Vapor (Mixture): Inside the dome. Quality $x$ (dryness fraction) is mass of vapor / total mass.

$$\boxed{h = h_f + x h_{fg}} \quad \boxed{s = s_f + x s_{fg}} \quad \boxed{v = v_f + x v_{fg}}$$

4. Saturated Vapor: State $g$ on saturation line. $$\displaystyle x = 1 $$.

5. Superheated Vapor: $$\displaystyle T > T_{sat} $$ at given $P$. Properties found from superheated tables or Mollier chart.

• Sensible Heat: Heat added to change temperature without phase change ($\Delta h$ in compressed liquid or superheated vapor regions).

• Latent Heat: Heat added during phase change at constant $T$ and $P$ ($$\displaystyle h_{fg} $$ for evaporation, $$\displaystyle h_{fg} $$ for condensation).

Steam Tables & Mollier Chart

• Steam Tables: Provide $$\displaystyle v_f, v_g, h_f, h_g, s_f, s_g $$ at saturation, and $h, s$ for superheated steam at given $P,T$. Crucial for numerical problems.

• Mollier Chart (h-s): Graphical representation. Useful for steam expansion processes (turbines, throttling).

  • Isobars diverge in wet region (constant $P$ lines spread apart as entropy increases) because $$\displaystyle dh = T ds + v dp $$. At constant $P$, $$\displaystyle dh = T ds $$, and $T$ increases with $s$ in wet region.

  DiagramSEARCH: "Mollier diagram steam isobar divergence"

Dryness Fraction (x)

• Definition: $$\displaystyle x = \frac{m_{vapor}}{m_{total}} $$ (0 ≤ x ≤ 1). For saturated liquid, x=0; for saturated vapor, x=1.

• Entropy of Evaporation: $$\displaystyle s_{fg} = s_g - s_f $$. The increase in entropy when 1 kg of saturated liquid evaporates at constant T and P.

• Measurement:

  1. Throttling Calorimeter: Steam throttled to low pressure, measures $$\displaystyle T_2 $$ (≈ saturation temp at low $P$). From $$\displaystyle h_1 = h_2 $$ (isenthalpic), find $x$ at initial state using $$\displaystyle h_2 = h_f @ P_2 + x h_{fg} @ P_2 $$.

  2. Separating Calorimeter: Steam passed through a separator; condensed water and vapor masses measured directly. $$\displaystyle x = \frac{m_{vapor}}{m_{vapor} + m_{liquid}} $$.

  DiagramCANVAS: "Sketch of throttling calorimeter: steam inlet, throttling valve, outlet pipe, thermometer measuring T2 at low pressure"

4. IDEAL & REAL GAS BEHAVIOR; GAS MIXTURES

Ideal Gas

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

• Boyle's Law: $P \propto 1/V$ at constant $T$.

• Charles' Law: $V \propto T$ at constant $P$.

• Specific Heats: $$\displaystyle C_p $$, $$\displaystyle C_v $$. $$\displaystyle \gamma = C_p/C_v $$. For monatomic gas, $$\displaystyle \gamma = 1.67 $$; diatomic, $\gamma \approx 1.4$.

• Mean Specific Heat: For $$\displaystyle C_p = a + bT $$,

$$C_{p,mean} = \frac{1}{T_2 - T_1} \int_{T_1}^{T_2} (a + bT) dT = a + \frac{b(T_1 + T_2)}{2}$$

• Polytropic Process: $$\displaystyle PV^n = C $$. Work done:

$$W = \frac{P_2V_2 - P_1V_1}{1 - n} \quad (\text{for } n \neq 1)$$

Heat transfer: $$\displaystyle Q = \Delta U + W = m C_v (T_2 - T_1) + W $$.

Gas Mixtures (Ideal)

• Dalton's Law: Total pressure equals sum of partial pressures. $$\displaystyle P = \sum P_i $$, where $$\displaystyle P_i = \frac{m_i R_i T}{V} $$.

• Amagat's Law: Total volume equals sum of partial volumes at total $P$ and $T$. $$\displaystyle V = \sum V_i $$.

• Mixture Properties (per unit mass of mixture):

$$u_{mix} = \sum y_i u_i \quad h_{mix} = \sum y_i h_i \quad C_{v,mix} = \sum y_i C_{v,i} \quad C_{p,mix} = \sum y_i C_{p,i}$$

where $$\displaystyle y_i = m_i/m $$ is mass fraction.

> [!TIP] **Key Point:** For ideal gas mixtures, internal energy and enthalpy are functions of temperature only, and the mixture's specific heat is the mass-weighted average of components' specific heats.

Real Gases

• Compressibility Factor (Z): $$\displaystyle \boxed{PV = Z m R T} $$. $$\displaystyle Z = 1 $$ for ideal gas. $$\displaystyle Z < 1 $$ indicates attractive forces dominate; $$\displaystyle Z > 1 $$ indicates repulsive forces dominate.

• P-V-T Surface: For real gases, the $P-V-T$ surface does not have a sharp "kink" at the saturation dome like pure substances. Van der Waals equation is a common EOS: $$\displaystyle (P + a/v^2)(v - b) = RT $$.

5. SECOND LAW APPLICATIONS: HEAT ENGINES, REFRIGERATORS & HEAT PUMPS

• Thermal Efficiency (η) of Heat Engine:

$$\eta = \frac{W_{net}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}$$

• COP of Refrigerator/Heat Pump:

$$\text{COP}_{ref} = \frac{Q_{L}}{W_{net}} \quad \text{COP}_{hp} = \frac{Q_{H}}{W_{net}}$$

Note: $$\displaystyle \text{COP}_{hp} = \text{COP}_{ref} + 1 $$.

• Reversible vs. Irreversible Cycles: Reversible cycles have maximum efficiency/COP. Irreversible cycles have lower performance due to friction, unrestrained expansion, etc.

• Carnot Cycle: Reversible cycle consisting of two isothermal and two adiabatic processes.

  • Efficiency:

$$\boxed{\eta_{Carnot} = 1 - \frac{T_L}{T_H}} \quad (T \text{ in Kelvin})$$

*   **Importance:** Sets the **upper limit** for any heat engine operating between $$\displaystyle T_H $$ and $$\displaystyle T_L $$. All reversible engines between same reservoirs have same efficiency.

• Comparison of Air-Standard Efficiencies (for same compression ratio $r$):

$$\eta_{Otto} > \eta_{Dual} > \eta_{Diesel}$$

(Derivations in Unit 6).

• Combined Cycles (Reversible Engine driving Reversible Refrigerator):

  • Engine: $$\displaystyle \frac{Q_2}{Q_1} = \frac{T_2}{T_1} $$

  • Refrigerator: $$\displaystyle \frac{Q_4}{Q_3} = \frac{T_4}{T_3} $$

  • Net work: $$\displaystyle W_{net} = Q_1 - Q_2 = Q_3 - Q_4 $$

  • Derive $$\displaystyle \frac{Q_2}{Q_1} $$:

    From energy balance: $$\displaystyle Q_3 = Q_2 + W_{net} = Q_2 + (Q_1 - Q_2) = Q_1 $$.

    So, $$\displaystyle Q_3 = Q_1 $$.

    Then, $$\displaystyle \frac{Q_2}{Q_1} = \frac{Q_2}{Q_3} = \frac{T_2}{T_1} $$ (from engine) and also $$\displaystyle \frac{Q_2}{Q_1} = \frac{Q_2}{Q_3} = \frac{T_2}{T_3} $$ (from refrigerator: $$\displaystyle Q_4/Q_3 = T_4/T_3 \Rightarrow Q_2/Q_3 = (Q_3 - Q_4)/Q_3 = 1 - T_4/T_3 $$). Equating gives relation between temperatures.

    Final Expression: $$\displaystyle \frac{Q_2}{Q_1} = \frac{T_2}{T_1} = \frac{T_2}{T_3} $$ (implies $$\displaystyle T_1 = T_3 $$ for consistency in typical setup).

  [!TIP] Exam Trap: In combined cycle problems, carefully identify which reservoir temperatures are connected. Often $$\displaystyle T_2 = T_3 $$ (engine reject = refrigerator absorb).

6. AIR-STANDARD CYCLES (IDEAL IC ENGINES)

Assumptions: Air as ideal gas with constant $$\displaystyle C_p, C_v $$; closed system; reversible processes; no friction, no heat loss.

Otto Cycle (Constant Volume Heat Addition)

• Processes: 1-2: Isentropic compression; 2-3: Constant volume heat addition; 3-4: Isentropic expansion; 4-1: Constant volume heat rejection.

• Efficiency Derivation:

  From isentropic relations: $$\displaystyle T_2 = T_1 r^{\gamma-1} $$, $$\displaystyle T_3 = T_2 \frac{q_{in}}{C_v T_2} = T_2 + \frac{q_{in}}{C_v} $$, $$\displaystyle T_4 = T_3 / r^{\gamma-1} $$.

$$\eta = 1 - \frac{Q_{out}}{Q_{in}} = 1 - \frac{C_v(T_4 - T_1)}{C_v(T_3 - T_2)} = 1 - \frac{T_1(r^{\gamma-1} - 1)}{T_3 - T_2}$$

Substituting and simplifying:

$$\boxed{\eta_{Otto} = 1 - \frac{1}{r^{\gamma-1}}}$$

> 
DiagramSEARCH: "Otto cycle pv ts diagram"

Diesel Cycle (Constant Pressure Heat Addition)

• Processes: 1-2: Isentropic compression; 2-3: Constant pressure heat addition; 3-4: Isentropic expansion; 4-1: Constant volume heat rejection.

• Cut-off Ratio: $$\displaystyle \rho = \frac{V_3}{V_2} $$.

• Efficiency Derivation:

  $$\displaystyle T_2 = T_1 r^{\gamma-1} $$, $$\displaystyle T_3 = T_2 \rho $$, $$\displaystyle T_4 = T_3 r^{-\gamma} \rho^{\gamma} $$.

$$\eta = 1 - \frac{C_v(T_4 - T_1)}{C_p(T_3 - T_2)} = 1 - \frac{1}{r^{\gamma-1}} \left[ \frac{\rho^{\gamma} - 1}{\gamma(\rho - 1)} \right]$$

$$\boxed{\eta_{Diesel} = 1 - \frac{1}{r^{\gamma-1}} \cdot \frac{\rho^{\gamma} - 1}{\gamma(\rho - 1)}}$$

> 
DiagramSEARCH: "Diesel cycle pv ts diagram"

Dual Cycle (Mixed Heat Addition)

• Part constant volume, part constant pressure heat addition.

• Efficiency lies between Otto and Diesel for same $r$.

7. COMBUSTION & FUELS

Fuels & Heating Value

• Primary Fuels: Natural gas, coal, crude oil (used directly).

• Secondary Fuels: Derived from primary (e.g., gasoline, diesel, LPG).

• Heating Value: Heat released when fuel burns completely and products cool to initial temperature.

  • HHV (Higher/ Gross): Products include liquid water (condensed).

  • LHV (Lower/ Net): Products include water vapor (typical for engines). $$\displaystyle LHV = HHV - m_{H_2} \cdot h_{fg} $$ (latent heat of vaporization of hydrogen in fuel).

Stoichiometry (Theoretical Combustion)

1. Ultimate Analysis: Mass % of C, H₂, O₂, N₂, S, moisture, ash.

2. Balanced Equation (e.g., for C₈H₁₈):

$$\mathrm{C_8H_{18}} + a(O_2 + 3.76N_2) \rightarrow bCO_2 + cH_2O + dN_2$$

Balance C, H, O to find $a$ (theoretical O₂ moles).

3. Theoretical Air: $$\displaystyle m_{air,th} = \frac{32a}{0.21} \cdot \frac{M_{fuel}}{100} $$ if analysis is % by mass.

4. Excess Air: % excess = $$\displaystyle \frac{m_{air,actual} - m_{air,th}}{m_{air,th}} \times 100 $$. More excess air → more $$\displaystyle N_2 $$ and $$\displaystyle O_2 $$ in products, lower flame temperature.

5. Volumetric Composition of Dry Flue Gas:

$$\%CO_2 = \frac{b}{b + d + (a - b/2 - c/2)} \times 100 \quad (\text{on dry basis})$$

where $(a - b/2 - c/2)$ is excess $$\displaystyle O_2 $$ moles.

Enthalpy of Reaction & Formation

• Enthalpy of Formation (ΔH_f°): Enthalpy change when 1 mole of compound is formed from its elements in their standard states at specified $$\displaystyle T_0 $$ (usually 298 K, 1 atm). Elements in standard state have $$\displaystyle \Delta H_f° = 0 $$.

• Enthalpy of Reaction (ΔH_rxn): Enthalpy change when reaction occurs as written.

$$\boxed{\Delta H_{rxn} = \sum \nu_i \Delta H_{f°}(\text{products}) - \sum \nu_i \Delta H_{f°}(\text{reactants})}$$

where $$\displaystyle \nu_i $$ are stoichiometric coefficients (positive for products, negative for reactants).

• First Law for Reacting Systems (Steady Flow, Constant P):

$$\dot{Q} - \dot{W}_{shaft} = \dot{m}_{products} h_{products} - \dot{m}_{reactants} h_{reactants}$$

Often $$\displaystyle \dot{W}_{shaft} = 0 $$ for combustion chamber.

Adiabatic Flame Temperature (AFT)

• Definition: Temperature of products when combustion occurs adiabatically (no heat loss) and at constant pressure, with complete combustion and no dissociation.

• Calculation: Energy balance: $$\displaystyle H_{reactants} = H_{products} $$.

  Use steam tables for $$\displaystyle h_{products}(T_{AFT}) $$. Solve iteratively for $$\displaystyle T_{AFT} $$.

  [!TIP] Key: Assume complete combustion, write product composition (including excess $$\displaystyle O_2 $$, $$\displaystyle N_2 $$), set sum of enthalpies of products (as function of T) equal to sum of enthalpies of reactants (at standard $$\displaystyle T_0 $$), solve for T.

8. SPECIFIC TOPICS FROM PAST PAPERS (REVISION & SYNTHESIS)

Temperature Scales & Measurement

• New Scale Conversion: If $$\displaystyle T_N = a + b T_C $$, find $a,b$ from fixed points (e.g., ice point, steam point).

• Constant Volume Gas Thermometer: Most accurate for defining temperature scale. Based on $P \propto T$ at constant $V$ for ideal gas. Used to realize ITS-90.

Work & Heat in Special Processes

• Free Expansion (Joule Expansion): Gas expands into vacuum. $$\displaystyle W = 0 $$, $$\displaystyle Q = 0 $$ (adiabatic), so $$\displaystyle \Delta U = 0 $$. For ideal gas, $$\displaystyle \Delta T = 0 $$. For real gas, temperature change indicates intermolecular forces.

• Work in Filling Flexible Container (Balloon): $$\displaystyle W = \int P_{gas} dV $$. For balloon expanding against constant atmospheric pressure $$\displaystyle P_0 $$, $$\displaystyle W = P_0 \Delta V $$ if process is quasi-static.

• Work in Cyclic Process: Net work = area enclosed by cycle on P-V diagram. Clockwise cycle: net work output.

Key Definitions (One-Liners)

• Heat Engine: Device that converts heat into work cyclically.

• Refrigerator: Device that removes heat from a cold reservoir and rejects to a hot reservoir using work input.

• Heat Pump: Refrigerator with the hot reservoir as the desired output.

• Cycle: A sequence of processes returning the system to its initial state.

• Air Standard Efficiency: Efficiency of an ideal cycle assuming air as ideal gas with constant specific heats and reversible processes.

• Triple Point: Unique $P,T$ where solid, liquid, and vapor coexist in equilibrium.

Short Note Topics (Past Paper Favorites)

• Mollier Diagram (h-s): Graphical chart for steam/refrigerants. Isobars diverge in wet region because $$\displaystyle (\partial h/\partial s)_P = T $$, and $T$ increases with $s$ in the two-phase region.

• Limitations of First Law: Does not predict direction of process, does not quantify energy degradation, cannot account for irreversibilities.

• Basic Concept of Third Law: $$\displaystyle S \rightarrow 0 $$ as $$\displaystyle T \rightarrow 0 $$ for perfect crystal. Allows absolute entropy calculation; absolute zero unattainable.

• Actual vs. Theoretical Combustion: Theoretical assumes perfect mixing, complete combustion, no heat loss. Actual has incomplete combustion, dissociation, heat loss, pressure drop, excess air → lower efficiency, lower AFT, different product composition.

• Brayton Cycle (Gas Turbine): 1-2: Isentropic compression; 2-3: Constant pressure heat addition; 3-4: Isentropic expansion; 4-1: Constant pressure heat rejection. Efficiency: $$\displaystyle \eta = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}} $$, where $$\displaystyle r_p = P_2/P_1 $$ (pressure ratio).

• Boyle's & Charles' Laws: Boyle: $P \propto 1/V$ at const $T$. Charles: $V \propto T$ at const $P$. Combined: $$\displaystyle PV/T = constant $$ for fixed mass of gas.

• Kelvin-Planck Statement: (See Section 1 above).