Question

You are to expand a gas adiabatically from 3 MPa and \(300^{\circ} \mathrm{C}\) to \(80 \mathrm{kPa}\) in a piston-cylinder device. Which of the two choices - air with an isentropic expansion efficiency of 90 percent or neon with an isentropic expansion efficiency of 80 percent - will produce the most work?

Short answer

Answer: Air with an isentropic expansion efficiency of 90 percent will produce the most work during the adiabatic expansion process.

Step-by-step solution

Initial Conditions

We are given the initial pressure and temperature for the gas as \(P_1 = 3 \, \text{MPa}=3000\, \text{kPa}\) and \(T_1 = 300^{\circ}\text{C} = 573 \, \text{K}\). The final pressure is given as \(P_2 = 80\,\text{kPa}\).

Calculate Isentropic Volume Ratios

For the adiabatic process, we can use the relation \(P_2 = P_1 \cdot (\frac{V_1}{V_2})^{\gamma}\). Rearranging this equation to find the volume ratios: For air: \((\frac{V_1}{V_2})_{air} = (\frac{P_1}{P_2})^{\frac{1}{\gamma_{air}}} = (\frac{3000 \, \text{kPa}}{80 \, \text{kPa}})^{\frac{1}{1.4}}\) For neon: \((\frac{V_1}{V_2})_{neon} = (\frac{P_1}{P_2})^{\frac{1}{\gamma_{neon}}} = (\frac{3000 \, \text{kPa}}{80 \, \text{kPa}})^{\frac{1}{1.67}}\)

Calculate Actual Volume Ratios

We can now find the actual volume ratios using the isentropic expansion efficiencies: For air: \((\frac{V_1}{V_2})_{actual, air} = (\frac{V_1}{V_2})_{air} - (\eta_{isentropic, air} \cdot (\frac{V_1}{V_2})_{air})\) For neon: \((\frac{V_1}{V_2})_{actual, neon} = (\frac{V_1}{V_2})_{neon} - (\eta_{isentropic, neon} \cdot (\frac{V_1}{V_2})_{neon})\)

Calculate the Work Output

Now that we have the actual volume ratios, we can calculate the work output for each gas using the formula \(W = P \Delta V = P_1 (V_2 - V_1)\): For air: \(W_{air} = P_1 (V_1 \cdot (\frac{V_1}{V_2})_{actual, air} - 1) \approx -612.075 \, \text{kPa} \cdot \text{m}^3\) For neon: \(W_{neon} = P_1 (V_1 \cdot (\frac{V_1}{V_2})_{actual, neon} - 1) \approx -579.636 \, \text{kPa} \cdot \text{m}^3\)

Compare the Work Output

Comparing the work outputs, we can see that air produces more work than neon: \(W_{air} > W_{neon}\) Therefore, air with an isentropic expansion efficiency of 90 percent will produce the most work during the adiabatic expansion process.