(a) The power output of a six-cylinder, four-stroke CI engine is absorbed by a hydraulic dynamometer for which the law is P = WN/20000, where P is the power in kW, W is the brake load in newton and N is the engine speed in r.p.m. The following observ

Question

(a) The power output of a six-cylinder, four-stroke CI engine is absorbed by a hydraulic dynamometer for which the law is P = WN/20000, where P is the power in kW, W is the brake load in newton and N is the engine speed in r.p.m. The following observations are made during a test on the engine: Bore = 100 mm; Stroke = 110 mm; Brake load = 540 N; Engine speed = 2500 r.p.m.; C/H ratio of the fuel (by mass) = 83/17; Ambient pressure = 1·0 bar; Ambient temperature = 27 °C; Time taken for 100 cc of fuel consumption = 18 s; Fuel density = 780 kg/m³; Calorific value of the fuel = 45 MJ/kg; Mass flow rate of atmospheric air consumed by the engine = 5·301126 kg/min. Calculate the bmep, bsfc, brake thermal efficiency, volumetric efficiency and the percentage of excess air. Given, R_air = 0·287 kJ/kg-K. (20 marks)

(b) (i) Draw typical velocity triangles of a stage of a reaction turbine, clearly showing the various velocities. (ii) Derive an expression to show that the optimum value of ρ, the blade-to-steam speed ratio for a Parsons reaction turbine is given by ρ = cos α, where α is the inlet angle of the fixed blades. (iii) Also, show that the maximum efficiency of the Parsons reaction turbine is given by η_maximum = (2 cos² α)/(1 + cos² α). (iv) Draw the velocity triangles of the Parsons reaction turbine operating at ρ_optimum. (20 marks)

(c) Explain briefly the various methods of air-conditioning duct design. (10 marks)

UPSC Answer Check · Accepted Answer

Calculate all engine performance parameters in part (a) using systematic thermodynamic formulas, ensuring unit consistency throughout. For part (b), derive the optimum blade speed ratio and maximum efficiency expressions using velocity triangle analysis, with clear diagrams at ρ_optimum. For part (c), enumerate duct design methods with brief explanations. Allocate approximately 40% time to (a), 40% to (b), and 20% to (c) based on marks distribution.
- Part (a): Brake power P = WN/20000 = 67.5 kW; bmep = (60×P×1000)/(L×A×n×N/2) = 7.82 bar; bsfc = ṁ_f/P = 0.208 kg/kWh; brake thermal efficiency = P/(ṁ_f×CV) = 38.46%; volumetric efficiency = (ṁ_a×R×T_a)/(P_a×V_s×N/2×n) = 85.2%; stoichiometric air-fuel ratio from C/H=83/17 gives excess air ≈ 25%
- Part (b)(i): Velocity triangles showing inlet/outlet absolute velocities (V₁, V₂), relative velocities (V_{r1}, V_{r2}), blade velocities (u), and angles (α, β, φ, ψ) for 50% reaction stage
- Part (b)(ii): Derivation using symmetrical triangles (V_{r1}=V₂, V₁=V_{r2}), power output expression, and dP/du = 0 leading to ρ_opt = u/V₁ = cos α
- Part (b)(iii): Substitution of ρ_opt into efficiency expression η = 2ρ(cos α - ρ)/(1 - 2ρ cos α + ρ²) yielding η_max = 2cos²α/(1+cos²α)
- Part (b)(iv): Velocity triangles at ρ_opt showing α = β₂ and φ = ψ with V_{r2} perpendicular to blade or specific geometric relationships
- Part (c): Equal friction method, velocity reduction method, static regain method, and T-method (total pressure method) with brief principles of each

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	For (a): correctly applies four-stroke cycle factor (N/2), uses dynamometer law properly, distinguishes between brake and indicated quantities, applies correct stoichiometry for diesel fuel; for (b): recognizes 50% reaction symmetry, correctly identifies that α = β₂ and φ = ψ; for (c): correctly distinguishes all four duct design methods and their applicability	Minor errors in cycle factor or confusion between mass and volume flow rates in (a); recognizes reaction turbine but misses symmetry conditions in (b); lists methods in (c) but conflates principles	Treats engine as two-stroke or uses wrong cycle factor; confuses impulse and reaction turbine principles; cannot identify more than two duct design methods
Numerical accuracy	20%	10	All five parameters in (a) computed correctly: bmep ≈ 7.82 bar, bsfc ≈ 0.208 kg/kWh, η_bth ≈ 38.5%, η_vol ≈ 85%, excess air ≈ 25%; consistent use of SI units with proper conversions (mm→m, cc→m³, min→s, bar→Pa where needed)	Three to four correct values with minor arithmetic slips (e.g., power of 10 error, missing 60 factor in bmep); mostly correct unit handling	Two or fewer correct values; major errors like using stroke in mm directly, confusing kg/min with kg/s, or order-of-magnitude errors in final answers
Diagram quality	20%	10	For (b)(i): Two complete velocity triangles (inlet/outlet) with all velocities labelled (V₁, V₂, u, V_{r1}, V_{r2}, V_{w1}, V_{w2}, V_{f1}, V_{f2}), angles clearly marked (α, β₁, φ, ψ), consistent scale and direction arrows; for (b)(iv): distinct triangles at ρ_opt showing the special geometric condition	Triangles drawn but some velocities or angles missing, or inconsistent scale between inlet and outlet; ρ_opt triangles not clearly distinguished	Single triangle only, or triangles without velocity components; no diagrams for (b)(iv); arrows missing or contradictory flow directions
Step-by-step derivation	20%	10	For (b)(ii): Starts with power per unit mass flow P = u(V_{w1}+V_{w2}), expresses whirl velocities in terms of u and angles, applies 50% reaction symmetry, differentiates with respect to u or ρ, shows dη/dρ = 0 condition clearly; for (b)(iii): substitutes ρ_opt back into efficiency expression with algebraic simplification shown	Jumps to final expressions with key steps missing; or uses energy equation without showing velocity triangle relationships; correct final answer but derivation unclear	No derivation shown—only final expressions stated; or fundamental errors in applying Euler turbine equation or differentiation
Practical interpretation	20%	10	For (a): comments on typical diesel engine values (bmep ~7-10 bar, η_bth ~35-42%), compares with automotive diesel standards, notes significance of excess air for complete combustion and smoke limitation; for (c): indicates when each duct method is preferred (equal friction for uniform pressure, static regain for long ducts, velocity reduction for variable loads)	Brief mention that values are reasonable for diesel engine; generic statement about duct design without specific application context	No interpretation of numerical results; no indication of which duct method suits which application; treats all parts as purely theoretical exercises

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