Question

Air is expanded in a polytropic process with \(n=\) 1.2 from \(1 \mathrm{MPa}\) and \(400^{\circ} \mathrm{C}\) to \(110 \mathrm{kPa}\) in a piston-cylinder device. Determine the final temperature of the air.

Short answer

Answer: The final temperature of the air after the polytropic expansion process is approximately -48.73°C.

Step-by-step solution

Convert pressure measurements to the same units

First, we need to convert the initial pressure from \(MPa\) to \(kPa\) for consistency: \(p_1 = 1000 \,\text{kPa}\) The final pressure is: \(p_2 = 110 \,\text{kPa}\)

Use the polytropic relation to find the final volume ratio

Let's work with the volume ratio, which we'll denote as \(v_r = \frac{V_2}{V_1}\): \(p_1V_1^n = p_2V_2^n \Rightarrow \frac{p_2}{p_1} = \frac{V_1^n}{V_2^n} \Rightarrow v_r = \left(\frac{p_1}{p_2}\right)^{\frac{1}{n}}\) Plug in the values for \(p_1\), \(p_2\), and \(n\) to find the volume ratio: \(v_r = \left(\frac{1000}{110}\right)^{\frac{1}{1.2}} \approx 2.586\)

Use the ideal gas equation

At both initial and final states, we can write the ideal gas equation as: \(p_1V_1 = mR{T_1}\) (Initial state) \(p_2V_2 = mR{T_2}\) (Final state) Divide the two equations for the initial and final states: \(\frac{p_1V_1}{p_2V_2} = \frac{T_1}{T_2}\) And also divide the volume terms to arrive the ratio: \(\frac{p_1}{p_2} = \frac{T_1}{T_2} \cdot \frac{V_1}{V_2}\)

Find the final temperature

Now we can plug in the values and ratios we obtained from the previous steps to solve for the final temperature: \(\frac{1000}{110} = \frac{T_1}{T_2} \cdot \frac{V_1}{V_2} \Rightarrow \frac{1000}{110} = \frac{T_1}{T_2} \cdot v_r\) First, we need to convert the initial temperature from Celsius to Kelvin: \(T_1 = 400^{\circ} \mathrm{C} + 273.15 = 673.15\, \text{K}\) Now plug in the values for \(T_1\) and \(v_r\): \(\frac{1000}{110} = \frac{673.15}{T_2} \cdot 2.586\) Solve for the final temperature: \(T_2 = \frac{673.15 \cdot 2.586}{\frac{1000}{110}} \approx 224.42\, \text{K}\) Finally, convert the final temperature back to Celsius: \(T_2 = 224.42 - 273.15 = -48.73^{\circ} \mathrm{C}\)

Final Answer

The final temperature of the air after the polytropic expansion process is approximately \(-48.73^{\circ} \mathrm{C}\).