Question

Using EES (or other) software, study the effect of variable specific heats on the thermal efficiency of the ideal Otto cycle using air as the working fluid. At the beginning of the compression process, air is at \(100 \mathrm{kPa}\) and \(300 \mathrm{K}\). Determine the percentage of error involved in using constant specific heat values at room temperature for the following combinations of compression ratios and maximum cycle temperatures: \(r=6,8,10,12\) and \(T_{\max }=1000,1500,2000,2500 \mathrm{K}\)

Short answer

Solution: To determine the percentage of error, calculate the thermal efficiency of the Otto cycle considering both variable and constant specific heats. Then, use the formula for error to find the percentage of error between the two efficiencies: \( \text{error} = \frac{\eta_{var} - \eta_{const}}{\eta_{const}} \times 100\% \). Perform these calculations for each combination of compression ratios and maximum cycle temperatures.

Step-by-step solution

Calculate the thermal efficiency considering variable specific heats

To do this, use the formula for the thermal efficiency of the Otto cycle with variable specific heats: \(\eta_{var} = 1 - \frac{T_2 - T_1}{(T_3 - T_4) (\gamma} - 1)\), where \(T_1 = 300\,K\), \(T_2\), \(T_3\), and \(T_4\) are the temperatures at points 2, 3, and 4 of the Otto cycle, respectively, and \(\gamma\) is the ratio of specific heats, which varies with temperature. For each combination of compression ratio r and maximum cycle temperature \(T_{\max}\), we need to find the temperatures \(T_2\), \(T_3\), and \(T_4\) using the temperature ratios \(T_2/T_1 = (r)^{\gamma - 1}\), \(T_3/T_2 = (T_{\max}/T_2)\), and \(T_4/T_3 = (1/r)^{\gamma - 1}\). The value of \(\gamma\) for each combination must be obtained from the software.

Calculate the thermal efficiency considering constant specific heats

Use the formula for the thermal efficiency of the Otto cycle with constant specific heats: \(\eta_{const} = 1 - \frac{1}{r^{\gamma} - 1}\), where \(\gamma\) is the ratio of specific heats at room temperature (use the value provided by the software). For each combination of compression ratio and maximum cycle temperature, calculate the constant specific heat thermal efficiency using the given \(\gamma\).

Calculate the percentage of error

For each combination of compression ratio and maximum cycle temperature, find the percentage of error between the variable and constant specific heat efficiencies using the formula: \(\text{error} = \frac{\eta_{var} - \eta_{const}}{\eta_{const}} \times 100\%\) After performing the above steps, we can determine the percentage of error involved in using constant specific heat values at room temperature for each given combination of compression ratios and maximum cycle temperatures.