ME-302 Thermodynamics - Jun 2025 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aDiscuss Clausius inequality and its role in defining the second law of thermodynamics.
   How to answer

   • State Clausius inequality: $\oint \frac{\delta Q}{T} \le 0$.
   • Explain cases: equal to zero for reversible cycle, less than zero for irreversible cycle, greater than zero is impossible.
   • Discuss its role as a criterion for reversibility and impossibility of processes.
   • Show how it leads to the definition of entropy as a property.
2. 7 Marks
   bThe specific heat capacity of the system during a certain process is given by $C_n = (0.4 + 0.004\text{ T})\text{ kJ/kg}^\circ\text{C}$. If the mass of the gas is $6\text{ kg}$ and its temperature changes from $25^\circ\text{C}$ to $125^\circ\text{C}$ find: i) Heat transferred ii) Mean specific heat of the gas
   How to answer

   • Formula: $Q = m \int_{T_1}^{T_2} (0.4 + 0.004 T) dT$.
   • Integrate: $\int_{25}^{125} (0.4 + 0.004 T) dT = [0.4T + 0.002 T^2]_{25}^{125} = (50 - 10) + 0.002(15625 - 625) = 40 + 30 = 70\text{ kJ/kg}$.
   • Calculate heat transferred: $Q = 6 \times 70 = 420\text{ kJ}$.
   • Calculate mean specific heat: $C_{mean} = \frac{70}{125 - 25} = 0.70\text{ kJ/kg}^\circ\text{C}$.
3. 7 Marks
   aDefine the Carnot cycle and its importance in thermodynamic efficiency. Also compare the efficiencies of Carnot, Otto, Diesel and Dual cycles.
   How to answer

   • Define Carnot cycle: four reversible processes (two isothermal, two isentropic).
   • Discuss significance: maximum possible efficiency between two given temperature limits (Carnot theorem).
   • Compare efficiencies under same compression ratio and heat input: $\eta_{Carnot} > \eta_{Otto} > \eta_{Dual} > \eta_{Diesel}$.
   • Compare under same maximum pressure and temperature: $\eta_{Diesel} > \eta_{Dual} > \eta_{Otto}$.
4. 7 Marks
   bA domestic food refrigerator maintains a temperature of $-12^\circ\text{C}$. The ambient air temperature is $35^\circ\text{C}$. If heat leaks into the freezer at the continuous rate of $2\text{ kJ/s}$ determine the least power necessary to pump this heat out continuously.
   How to answer

   • Convert temperatures to Kelvin: $T_L = 273.15 - 12 = 261.15\text{ K}$, $T_H = 273.15 + 35 = 308.15\text{ K}$.
   • Calculate maximum COP of the refrigerator: $COP_{max} = \frac{T_L}{T_H - T_L} = \frac{261.15}{308.15 - 261.15} = \frac{261.15}{47} \approx 5.556$.
   • Calculate least power required: $W_{min} = \frac{\dot{Q}_L}{COP_{max}} = \frac{2\text{ kW}}{5.556} \approx 0.360\text{ kW} = 360\text{ W}$.
5. 7 Marks
   aExplain the concept of pure substance and its phases in thermodynamics. Also discuss the formation of steam and its thermodynamic properties.
   How to answer

   • Define pure substance: homogeneous and invariable chemical composition throughout.
   • Explain different phases: solid, liquid, vapor, and phase equilibrium.
   • Describe steam formation process at constant pressure (subcooled liquid $\rightarrow$ saturated liquid $\rightarrow$ wet steam $\rightarrow$ dry saturated steam $\rightarrow$ superheated steam).
   • Sketch T-v or T-s diagram representing sensible heat, latent heat, and superheat.
6. 7 Marks
   bA rigid vessel of volume $0.86\text{ m}^3$ contains $1\text{ kg}$ of steam at a pressure of $2\text{ bar}$. Evaluate the specific volume, temperature, dryness fraction, internal energy, enthalpy and entropy of steam.
   How to answer

   • Calculate specific volume: $v = \frac{V}{m} = \frac{0.86}{1} = 0.86\text{ m}^3/\text{kg}$.
   • From steam tables at $p = 2\text{ bar}$ ($0.2\text{ MPa}$): $T_{sat} \approx 120.2^\circ\text{C}$, $v_f \approx 0.00106\text{ m}^3/\text{kg}$, $v_g \approx 0.8857\text{ m}^3/\text{kg}$. Since $v_f < v < v_g$, steam is wet, so temperature $T = T_{sat} \approx 120.2^\circ\text{C}$.
   • Determine dryness fraction: $x = \frac{v - v_f}{v_g - v_f} \approx \frac{0.86}{0.8857} \approx 0.971$.
   • Calculate enthalpy: $h = h_f + x h_{fg}$ using steam table values ($h_f \approx 504.7\text{ kJ/kg}, h_{fg} \approx 2201.9\text{ kJ/kg}$).
   • Calculate internal energy: $u = h - p v$ (with $p = 200\text{ kPa}$).
   • Calculate entropy: $s = s_f + x s_{fg}$ using steam table values ($s_f \approx 1.530\text{ kJ/kg}\cdot\text{K}, s_{fg} \approx 5.597\text{ kJ/kg}\cdot\text{K}$).
7. 7 Marks
   aDiscuss the PVT relationship in mixtures of ideal gases and its importance.
   How to answer

   • State Dalton's law of partial pressures: $P_m = \sum p_i$.
   • State Amagat's law of partial volumes: $V_m = \sum v_i$.
   • Derive the relationship: $\frac{p_i}{P_m} = \frac{v_i}{V_m} = y_i$ (mole fraction).
   • Discuss equation of state for mixture: $P_m V_m = n_m R_u T_m$ or $P_m V_m = m R_{mix} T$.
   • State its importance in combustion analysis, air conditioning, and chemical engineering processes.
8. 7 Marks
   bA reversible power cycle is used to drive a reversible heat pump cycle. The power cycle takes in $Q_1$ heat units at $T_1$ and rejects $Q_2$ at $T_2$. The heat pump abstracts $Q_4$ from the sink at $T_4$ and discharges $Q_3$ at $T_3$. Develop an expression for the ratio $Q_4/Q_1$ in terms of the four temperatures.
   How to answer

   • For reversible engine: Efficiency $\eta = \frac{W}{Q_1} = 1 - \frac{T_2}{T_1} = \frac{T_1 - T_2}{T_1} \implies W = Q_1 \frac{T_1 - T_2}{T_1}$.
   • For reversible heat pump: $W = Q_3 - Q_4$. For reversible cycle: $\frac{Q_3}{T_3} = \frac{Q_4}{T_4} \implies Q_3 = Q_4 \frac{T_3}{T_4}$.
   • Thus, $W = Q_4 \left(\frac{T_3 - T_4}{T_4}\right)$, or $\frac{Q_4}{W} = \frac{T_4}{T_3 - T_4}$.
   • Equating work output of engine to work input of heat pump: $Q_1 \frac{T_1 - T_2}{T_1} = Q_4 \frac{T_3 - T_4}{T_4}$.
   • Rearranging gives: $\frac{Q_4}{Q_1} = \frac{T_4(T_1 - T_2)}{T_1(T_3 - T_4)}$.
9. 7 Marks
   aDescribe the internal energy, enthalpy and specific heat of gas mixtures. Also explain the significance of Mollier charts in thermodynamic analysis.
   How to answer

   • Formulate internal energy of a gas mixture: $U_m = \sum m_i u_i$ and $u_m = \sum m_{f,i} u_i$.
   • Formulate enthalpy of a gas mixture: $H_m = \sum m_i h_i$ and $h_m = \sum m_{f,i} h_i$.
   • Formulate specific heats: $c_{p,m} = \sum m_{f,i} c_{p,i}$ and $c_{v,m} = \sum m_{f,i} c_{v,i}$.
   • Define Mollier chart: enthalpy-entropy ($h-s$) diagram.
   • Explain significance: simplifies steam turbine, nozzle, and throttling calculations directly giving isentropic enthalpy drops without steam tables.
10. 7 Marks
    bA sample of gasoline has a composition by mass of $85.5\%$ carbon and $14.5\%$ hydrogen. When the gasoline is burned in oxygen, it produces carbon dioxide and water. i) Write a balanced chemical equation for the combustion of gasoline (assume octane, $(\text{C}_8\text{H}_{18})$, as the representative compound). ii) If $500\text{ grams}$ of gasoline are burned, calculate the mass of oxygen required for complete combustion.
    How to answer

    • i) Balanced equation: $\text{C}_8\text{H}_{18} + 12.5\,\text{O}_2 \rightarrow 8\,\text{CO}_2 + 9\,\text{H}_2\text{O}$ (or $2\text{C}_8\text{H}_{18} + 25\text{O}_2 \rightarrow 16\text{CO}_2 + 18\text{H}_2\text{O}$).
    • ii) From elemental analysis per kg fuel: $\text{C} + \text{O}_2 \rightarrow \text{CO}_2 \implies 12\text{ g C needs } 32\text{ g } \text{O}_2$; $2\text{H}_2 + \text{O}_2 \rightarrow 2\text{H}_2\text{O} \implies 4\text{ g H needs } 32\text{ g } \text{O}_2$ ($1\text{ g H needs } 8\text{ g } \text{O}_2$).
    • Mass of oxygen per gram fuel: $0.855 \times \frac{32}{12} + 0.145 \times 8 = 2.28 + 1.16 = 3.44\text{ g O}_2\text{/g fuel}$.
    • (Alternatively via octane: $114\text{ g C}_8\text{H}_{18}$ requires $12.5 \times 32 = 400\text{ g O}_2 \implies 3.509\text{ g O}_2\text{/g}$).
    • For $500\text{ g}$ gasoline: $m_{O_2} = 500 \times 3.44 = 1720\text{ g}$ (or $500 \times \frac{400}{114} \approx 1754.4\text{ g}$ using pure octane).
11. 7 Marks
    aDiscuss the actual and theoretical combustion processes and their impact on engine performance.
    How to answer

    • Define theoretical (stoichiometric) combustion: exact theoretical amount of oxygen with complete conversion to $\text{CO}_2$ and $\text{H}_2\text{O}$.
    • Explain actual combustion: presence of excess or deficit air, incomplete combustion producing $\text{CO}$, unburned hydrocarbons, soot, and $\text{NO}_x$.
    • Discuss phenomena in actual engines: dissociation at high temperatures, variable specific heats, finite flame speed.
    • Detail the impact on performance: reduction in thermal efficiency, loss of power output, increased emissions, and fuel consumption.
12. 7 Marks
    bAnalyze the enthalpy of formation and reaction in thermodynamic systems.
    How to answer

    • Define standard enthalpy of formation ($\Delta h_f^\circ$): enthalpy change when a compound is formed from its stable elements at standard state ($25^\circ\text{C}$, $1\text{ atm}$).
    • Define enthalpy of reaction ($\Delta H_R$): difference between total enthalpy of products and total enthalpy of reactants.
    • State the relationship: $\Delta H_R^\circ = \sum N_p (\Delta h_f^\circ)_p - \sum N_r (\Delta h_f^\circ)_r$.
    • Analyze application in energy balance of open/closed reactive systems and flame temperature determination.
13. 7 Marks
    aDraw the T-s diagram for water showing the different regions i.e. wet region, dry saturated steam region, compressed liquid region etc.
    How to answer

    • Draw axes: Temperature ($T$) on y-axis and Entropy ($s$) on x-axis.
    • Sketch the vapor dome with the critical point at the apex.
    • Label saturated liquid line and saturated vapor line.
    • Clearly indicate regions: subcooled/compressed liquid region (left of saturated liquid line), two-phase wet mixture region (under the dome), superheated vapor region (right of saturated vapor line), and supercritical region (above critical point).
    • Draw typical constant pressure lines across all regions.
14. 7 Marks
    bDiscuss the measurement and significance of dryness fraction in steam engineering.
    How to answer

    • Define dryness fraction: $x = \frac{m_v}{m_v + m_l}$.
    • Discuss its significance: determines enthalpy, specific volume, and available work of steam; prevents blade erosion in steam turbines.
    • Explain measurement methods: separating calorimeter, throttling calorimeter, or combined separating and throttling calorimeter.
    • Outline operating principle and formula for the throttling calorimeter ($h_1 = h_2$ at constant enthalpy).
15. 7 Marks
    aA piston-cylinder device contains $0.5\text{ kg}$ of nitrogen gas at $250\text{ kPa}$ and $300\text{ K}$. The gas expands isothermally to a final pressure of $100\text{ kPa}$. Calculate the work done by the gas during this process.
    How to answer

    • Gas constant for nitrogen: $R = \frac{8.314}{28} \approx 0.2968\text{ kJ/kg}\cdot\text{K}$ (or $0.297\text{ kJ/kg}\cdot\text{K}$).
    • Isothermal work formula: $W = m R T \ln\left(\frac{P_1}{P_2}\right)$.
    • Substitute given values: $W = 0.5 \times 0.2968 \times 300 \times \ln\left(\frac{250}{100}\right)$.
    • Compute: $\ln(2.5) \approx 0.9163$.
    • Calculate result: $W = 0.5 \times 0.2968 \times 300 \times 0.9163 \approx 40.79\text{ kJ}$.
16. 7 Marks
    bWrite the expressions of efficiency of Otto and diesel cycles.
    How to answer

    • Expression for Otto cycle efficiency: $\eta_{Otto} = 1 - \frac{1}{r^{\gamma - 1}}$, defining $r$ (compression ratio) and $\gamma$ (ratio of specific heats).
    • Expression for Diesel cycle efficiency: $\eta_{Diesel} = 1 - \frac{1}{r^{\gamma - 1}} \left[ \frac{r_c^\gamma - 1}{\gamma (r_c - 1)} \right]$, defining $r_c$ (cutoff ratio).
    • Briefly explain the effect of compression ratio and cutoff ratio on both efficiencies.