Question

Investigate the effect of maximum cycle temperature on the net work per unit mass of air for air-standard Otto cycles with compression ratios of 5,8, and $11.$ At the beginning of the compression process, $p_1=1$ bar and $T_1=295\mathrm{K}$. Let the maximum temperature in each case vary from 1000 to $2200\mathrm{K}$.

Short answer

Calculate net work per unit mass using the Otto cycle formula and plot results for different compression ratios and maximum temperatures.

Step-by-step solution

- Define Given Values

Identify and list all the given values.- Compression ratios: 5, 8, 11- Initial pressure: $p_1=1$ bar- Initial temperature: $T_1=295$ K- Maximum temperature range: $T_3=1000$ K to $T_3=2200$ K

- Understand the Otto Cycle

The Otto cycle consists of four processes: an isentropic compression, a constant-volume heat addition, an isentropic expansion, and a constant-volume heat rejection. The net work per unit mass of air can be determined from these processes.

- Apply Isentropic Relations

Use the isentropic relations to determine various state properties. For the compression process, $$T_2=T_1(r_c{)}^{(\frac{\gamma -1}{\gamma })}$$ where $r_c$ is the compression ratio and $\gamma$ is the specific heat ratio (assume $\gamma =1.4$).

- Calculate State 2 Properties

For each compression ratio, calculate the temperature after the isentropic compression (state 2). For r_c = 5, 8, and 11: $$T_{25}=295\times 5^{(\frac{1.4-1}{1.4})}=526\mathrm{K}$$$$T_{28}=295\times 8^{(\frac{1.4-1}{1.4})}=643\mathrm{K}$$$$T_{211}=295\times {11}^{(\frac{1.4-1}{1.4})}=740\mathrm{K}$$

- Apply the Heat Addition Process

For each case, let the maximum temperature $T_3$ vary from 1000 K to 2200 K. Use $Q_{in}=c_v(T_3-T_2)$ to calculate the heat added, where $c_v$ is the specific heat at constant volume.

- Apply Isentropic Expansion

Determine the temperature after the isentropic expansion (state 4) using the relation: $$T_4=T_3(r_c{)}^{(\frac{1-\gamma }{\gamma })}$$

- Calculate State 4 Properties

Calculate $T_4$ for each $T_3$ and compression ratio. For example, for $T_3=10002200\mathrm{K}$ at $r_c=5$:$$T_{45,1000}=1000\times 5^{(\frac{1-1.4}{1.4})}=561\mathrm{K}$$$$T_{45,2200}=2200\times 5^{(\frac{1-1.4}{1.4})}=1235\mathrm{K}$$

- Calculate Net Work Per Unit Mass

The net work per unit mass is given by: $$W_{net}=Q_{in}-Q_{out}{Q}_{out}=c_v(T_4-T_1)$$. Calculate $W_{net}$ for each compression ratio and maximum temperature combination.

- Plot Results

Plot the net work per unit mass against the maximum temperature for each compression ratio (5, 8, and 11) to visualize the effect of maximum cycle temperature.

Key concepts

Otto cycle

The Otto cycle is a fundamental concept in thermodynamics, particularly for internal combustion engines. It's a theoretical model that helps understand how engines convert fuel into work. The cycle consists of four key processes:
1. **Isentropic Compression:** The air-fuel mixture is compressed, increasing its pressure and temperature, but no heat is exchanged with the surroundings.
2. **Constant-Volume Heat Addition:** The compressed air-fuel mixture ignites. This process increases the temperature and pressure at constant volume.
3. **Isentropic Expansion:** The high-pressure, high-temperature gas expands, doing work on the piston, similar to the compression process but in reverse.
4. **Constant-Volume Heat Rejection:** The heat is expelled at constant volume. Understanding these processes helps in analyzing the performance and efficiency of an engine. Higher maximum cycle temperatures can lead to more efficient engines, but they also place more stress on engine components. Each process impacts the net work produced by the engine.

Compression Ratio

The compression ratio is crucial for understanding the efficiency of the Otto cycle. It's defined as the ratio of the total cylinder volume when the piston is at the bottom of the stroke (largest volume) to when the piston is at the top of the stroke (smallest volume). Mathematically, it is represented as:
$\text{Compression Ratio}=\frac{V_1}{V_2}$ where $V_1$ is the initial volume and $V_2$ is the final compressed volume. Higher compression ratios lead to greater efficiency because they allow for more work to be extracted from the same amount of fuel. However, an increase in compression ratio also raises the temperature and pressure inside the cylinder, which can cause engine knocking—a condition where fuel pre-ignites before the spark plug fires, potentially damaging the engine.

Isentropic Compression

Isentropic compression is a reversible adiabatic process where the entropy remains constant. For the Otto cycle, it occurs when the air-fuel mixture is compressed in the cylinder. The temperature and pressure of the gas increase without any heat exchange with the environment. The relationship between temperature and pressure during isentropic compression is crucial for calculating the states of the system. It's given by:
$$T_2=T_1\times r_c^{(\frac{\text{γ}-1}{\text{γ}})}$$ where $T_1$ and $T_2$ are the initial and final temperatures, $r_c$ is the compression ratio, and $\gamma$ (gamma) is the specific heat ratio, typically around 1.4 for air. This step helps calculate the temperature of the air-fuel mixture after the compression stage, which is essential for further calculations in the cycle.

Net Work Calculation

Calculating the net work per unit mass is essential for assessing an engine's performance. In the Otto cycle, the net work is the difference between the heat added during the combustion phase and the heat rejected during the exhaust phase. The net work can be calculated as:
$$W_{\text{net}}=Q_{\text{in}}-Q_{\text{out}}$$ where $Q_{\text{in}}=c_v\times (T_3-T_2)$ and $Q_{\text{out}}=c_v\times (T_4-T_1)$, with $T_3$ and $T_4$ being the temperatures after combustion and expansion, respectively, and $c_v$ being the specific heat at constant volume. This calculation involves determining the temperatures at different stages of the cycle, particularly after compression ($T_2$) and expansion ($T_4$). By varying the maximum cycle temperature ($T_3$), one can analyze its effect on the net work, providing insights into optimizing engine performance.