The Otto-cycle engine in a Mercedes-Benz SLK230 has a compression ratio of 8.8. (a) What is the ideal efficiency of the engine? Use g = 1.40.

Verified step by step guidance

1. Identify the given values: the compression ratio (r) is 8.8 and the specific heat ratio (\(\gamma\)) is 1.40.
2. Recall the formula for the ideal efficiency (\(\eta\)) of an Otto cycle engine, which is \(\eta = 1 - \frac{1}{r^{(\gamma - 1)}}\).
3. Substitute the given values into the formula: \(\eta = 1 - \frac{1}{8.8^{(1.40 - 1)}}\).
4. Calculate the value inside the parentheses, which is \(\gamma - 1 = 1.40 - 1 = 0.40\).
5. Raise the compression ratio to the power of 0.40, then compute the efficiency using the formula.

Key Concepts

Otto Cycle

The Otto cycle is a thermodynamic cycle that describes the functioning of a gasoline engine. It consists of two adiabatic processes and two isochoric processes, which together convert fuel into mechanical work. The cycle is characterized by its compression and expansion strokes, where the air-fuel mixture is compressed and ignited, leading to power generation.

Compression Ratio

The compression ratio is the ratio of the maximum to minimum volume in the combustion chamber of an engine. It is a critical factor in determining the efficiency and power output of an engine; higher compression ratios generally lead to greater efficiency, as they allow for more complete combustion of the fuel. In the context of the Otto cycle, it directly influences the ideal thermal efficiency.

Thermal Efficiency

Thermal efficiency is a measure of how well an engine converts the heat from fuel into work. For the Otto cycle, the ideal thermal efficiency can be calculated using the formula: η = 1 - (1 / r^(γ-1)), where r is the compression ratio and γ (gamma) is the specific heat ratio of the working fluid. This efficiency indicates the maximum possible performance of the engine under ideal conditions.