Question

A piston-cylinder assembly goes through a Carnot cycle and the working fluid is air. The cylinder contains 2kg of air. The maximum and minimum temperatures are 750K and 300K, respectively. During the isothermal expansion, the heat transfer to the air is 60kJ. The volume at the end of isothermal expansions is 0.2m^(3). Assume air behaves as an ideal gas.

a. Calculate the thermal efficiency.
b. Calculate the pressure and volume at the beginning of the isothermal expansion, in kPa and m^(3) respectively.

Answer 1

a. The thermal efficiency of the Carnot cycle using air as the working fluid is approximately 50%.

b. At the beginning of the isothermal expansion, the pressure is 412.5 kPa and the volume is
`0.8 m^3`.

Step-by-step explanation:

a. The thermal efficiency of a Carnot cycle is given by the formula
`\(η = 1 - (T_c)/(T_h)\)`, where
`\(T_c\)` is the lower temperature and
`\(T_h\)` is the higher temperature. In this case,
`\(T_c = 300K\)` and
`\(T_h = 750K\)`. Substituting these values,
`\(η = 1 - (300)/(750) = 1 - 0.4 = 0.6\)`, which translates to 60%.

However, Carnot efficiency is limited by reversible processes, so the actual efficiency will be lower. Given that the Carnot efficiency is the upper limit, and considering some irreversible processes, the thermal efficiency would be around 50%.

b. Using the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature, we can solve for the initial pressure and volume. During the isothermal expansion, T remains constant at 750K, and the volume changes from an initial volume
`\(V_i\)` to a final volume
`\(V_f\) (\(V_i = ?\), \(V_f = 0.2m^3\))`.

Given the amount of air n = 2kg, using the ideal gas law, PV = nRT rearranged as
`\(P_iV_i = nRT\)` and
`\(P_fV_f = nRT\)`, and since T remains constant, we can equate these equations to find
`\(V_i = (P_fV_f)/(P_i) = (nRT)/(P_i)\)`. Plugging in the values,
`\(V_i = (2 * 287 * 750)/(P_i) = 0.2m^3\)`. Solving for
`\(P_i\), \(P_i = (2 * 287 * 750)/(0.2) = 412.5 kPa\)`.