Question

How does the thermal efficiency of an ideal Otto cycle change with the compression ratio of the engine and the specific heat ratio of the working fluid?

Short answer

Answer: The thermal efficiency of the Otto cycle increases with an increase in both the compression ratio and specific heat ratio. Higher compression ratios and specific heat ratios lead to more efficient engines. However, there is a practical limit to the compression ratio due to engine design constraints and the risk of knocking.

Step-by-step solution

Understand the Otto Cycle Processes

The Otto cycle consists of four processes: 1. Adiabatic compression (1-2): The working fluid is compressed adiabatically which means there is no heat transfer during the compression. This process increases the temperature and pressure of the fluid. 2. Isochoric heat addition (2-3): The volume of the working fluid remains constant, and heat is added to the fluid. This process further increases the temperature and pressure of the fluid. 3. Adiabatic expansion (3-4): The working fluid expands adiabatically which means there is no heat transfer during the expansion. This process decreases the temperature and pressure of the fluid. 4. Isochoric heat rejection (4-1): The volume of the working fluid remains constant, and heat is extracted from the fluid. This process decreases the temperature and pressure of the fluid to the initial state before repeating the cycle.

Definition of Thermal Efficiency of Otto Cycle

The thermal efficiency (η) of the Otto cycle is defined as the ratio of the net work output (W_net) to the heat input (Q_in). Mathematically, this can be written as: η = \frac{W_{net}}{Q_{in}}

Calculating the Work Output and Heat Input

To find the efficiency, we need to calculate the work output (W_net) and heat input (Q_in). For the Otto cycle, the work output can be calculated as the difference between work done during the adiabatic expansion (W_expansion) and the adiabatic compression (W_compression): W_{net} = W_{expansion} - W_{compression} And the heat input can be calculated as the heat added during the isochoric heat addition process: Q_{in} = Q_{addition}

Express Efficiency in terms of Compression Ratio and Specific Heat Ratio

The compression ratio (r) is defined as the ratio of the initial volume (V_1) to the final volume (V_2) during the adiabatic compression process: r = \frac{V_1}{V_2} The specific heat ratio (k) is the ratio of the specific heat at constant pressure (C_p) to the specific heat at constant volume (C_v): k = \frac{C_p}{C_v} Using these parameters and the ideal gas law, we can express the thermal efficiency of the Otto cycle as: η = 1 - \frac{1}{r^{k-1}}

Analyzing how Efficiency Changes with Compression Ratio and Specific Heat Ratio

From the derived expression of efficiency: η = 1 - \frac{1}{r^{k-1}} It can be concluded that the thermal efficiency of the Otto cycle increases with an increase in the compression ratio (r) and specific heat ratio (k). This implies that higher compression ratios and higher specific heat ratios lead to more efficient engines. However, there is a practical limit to the compression ratio due to engine design constraints and the risk of knocking.