Randall D. Knight — Heat Engines and Refrigerators: Problem 73

The gasoline engine in your car can be modeled as the Otto cycle shown in FIGURE CP21.73. A fuel-air mixture is sprayed into the cylinder at point 1, where the piston is at its farthest distance from the spark plug. This mixture is compressed as the piston moves toward the spark plug during the adiabatic compression stroke. The spark plug fires at point 2, releasing heat energy that had been stored in the gasoline. The fuel burns so quickly that the piston doesn't have time to move, so the heating is an isochoric process. The hot, high-pressure gas then pushes the piston outward during the power stroke. Finally, an exhaust valve opens to allow the gas temperature and pressure to drop back to their initial values before starting the cycle over again. (a) Analyze the Otto cycle and show that the work done per cycle is \( W_{\text{out}} = \frac{nR}{1 - \gamma} (T_2 - T_1 + T_4 - T_3) \) (b) Use the adiabatic connection between \( T_1 \) and \( T_2 \) and also between \( T_3 \) and \( T_4 \) to show that the thermal efficiency of the Otto cycle is \( \eta = 1 - \frac{1}{r^{(\gamma-1)}} \) where \( r = V_{\text{max}}/V_{\text{min}} \) is the engine's compression ratio. (c) Graph \( \eta \) versus \( r \) out to \( r = 30 \) for a diatomic gas.