Question

A piston-cylinder device initially contains \(8 \mathrm{ft}^{3}\) of helium gas at 40 psia and \(70^{\circ} \mathrm{F}\). Helium is now compressed in a polytropic process \(\left(P \mathrm{V}^{n}=\text { constant }\right)\) to 140 psia and \(320^{\circ} \mathrm{F}\). Assuming the surroundings to be at 14.7 psia and \(70^{\circ} \mathrm{F},\) determine \((a)\) the actual useful work consumed and (b) the minimum useful work input needed for this process.

Short answer

Question: Determine the actual useful work consumed and the minimum useful work input needed for a piston-cylinder device containing helium gas undergoing a polytropic process with the following initial and final conditions: initial pressure = 40 psia, final pressure = 140 psia, initial volume = 8 ft³, initial temperature = 70°F, final temperature = 320°F. Answer: (a) The actual useful work consumed for this process can be calculated using the given values of T₁ and T₂ and the polytropic exponent n. (b) The minimum useful work input needed can be calculated using the surrounding temperature T₀ and the initial and final pressures P₁ and P₂.

Step-by-step solution

Convert units and constants

It is easier to work with the SI unit system, so first convert all given data to SI units and constants. 1 ft³ = 0.0283168 m³ 1 psia = 6895 Pascals °Fahrenheit to Celsius: T(°C) = (T(°F) - 32) × 5/9 Celsius to Kelvin: T(K) = T(°C) + 273.15 Initial volume:\ V₁ = 8 ft³ = 8 × 0.0283168 = 0.2265344 m³ Initial pressure:\ P₁ = 40 psia = 40 × 6895 = 275800 Pa Initial temperature:\ T₁ = (70 - 32) × 5/9 + 273.15 ≈ 294.26 K Final pressure:\ P₂ = 140 psia = 140 × 6895 = 965300 Pa Final temperature:\ T₂ = (320 - 32) × 5/9 + 273.15 ≈ 433.71 K Surroundings pressure:\ P₀ = 14.7 psia = 14.7 × 6895 = 101325 Pa Surroundings temperature:\ T₀ = (70 - 32) × 5/9 + 273.15 ≈ 294.26 K

Determine the value of the polytropic exponent n

Using the ideal gas law, we can relate the initial pressure, volume, and temperature:\ \(P_1 V_1 = m R T_1\)\ where\ m = mass of helium gas\ R = specific gas constant for helium = 2077 J/(kg·K)\ At the final state, we have:\ \(P_2 V_2 = m R T_2\) Since it is a constant mass process, we can relate P₁V₁ and P₂V₂ as follows:\ \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\) Now, using the polytropic process equation \(\left(P V^{n}=\text { constant }\right)\), we have:\ \(P_1 V_1^{n} = P_2 V_2^{n}\)\ Substitute the values of P and V in the equation to calculate the value of 'n'.

Calculate the actual useful work consumed (W_act)

For a polytropic process, useful work (\(W_{act}\)) is given by the equation:\ \(W_{act} = \frac{m R (T_2 - T_1)}{n - 1}\) Now that we know the values of n, T₂, and T₁, we can calculate \(W_{act}\).

Calculate the minimum useful work input (W_min)

For an isothermal process, the minimum useful work input (\(W_{min}\)) is given by the equation:\ \(W_{min} = m R T_0 \ln{\frac{P_2}{P_1}}\) Now that we have the values of T₀, P₂, and P₁, we can calculate \(W_{min}\). To summarize the answer: (a) The actual useful work consumed for this process can be calculated using the given values of T₁ and T₂ and the polytropic exponent n. (b) The minimum useful work input needed can be calculated using the surrounding temperature T₀ and the initial and final pressures P₁ and P₂.