Subtopic: Thermodynamic principles of I.C. engines
Question Statement:
Air enters into an engine cylinder at a pressure of 100 kPa and temperature of 300 K in the suction stroke of an ideal Otto cycle. The cylinder clearance volume is 600 cm³ and compression ratio is 8:1. The expansion stroke of the cycle is polytropic in nature. The pressure at the beginning of the expansion stroke is 10 MPa and the temperature at the end of the expansion stroke is 1800 K. The polytropic exponent of the expansion stroke is ________.
Official Answer Key:
1.30 to 1.39
Step-by-Step Solution & Explanation
Volumes: V_c = 600 cm³; r = (V_c + V_s)/V_c = 8 ⇒ V₁ = 8 × 600 = 4800 cm³, V₂ = V₃ = 600 cm³.
State 3 (start of expansion): P₃ = 10 MPa = 10000 kPa, V₃ = 600 cm³.
Using PV/T = constant between state 1 and state 3 (same mass of air):
T₃ = T₁ × (P₃V₃)/(P₁V₁) = 300 × (10000 × 600)/(100 × 4800) = 300 × 12.5 = 3750 K.
Polytropic expansion 3 → 4 with V₄ = 4800 cm³, T₄ = 1800 K:
T₃/T₄ = (V₄/V₃)^{n−1} ⇒ 3750/1800 = 8^{n−1} ⇒ 2.0833 = 8^{n−1}
n − 1 = ln(2.0833)/ln 8 = 0.7340/2.0794 = 0.3530 ⇒ n = 1.353 — within the official range 1.30–1.39.
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