1.5 kg of air is initially at 200 kPa and 30oC in an expandable container. The air is heated so that it expands following the relationship PV1.2 = constant. The container also holds a fan which adds a total of 50 kJ of work to the system during the process. The pressure at the end of the process is 100 kPa.

Question

(a) Determine the final volume of the air, the final temperature, and the amount of heat transfer during the process.
(b) Develop a computer model for this problem, and plot the final volume, final temperature, and the amount of heat transfer during the process for final pressures ranging between 50 kPa and 190 kPa.
Given: Air; m = 1.5 kg; P1 = 200 kPa; T1 = 30oC = 303 K; Wrs = -50 kJ; P2 = 100 kPa; PV1.2 = constant.
Assume: ?KE = ?PE = 0. Air is an ideal gas (R = 0.287 kJ/kg-K).
What will be an ideal response?

Answer 1

Solution: (a) The First Law for a closed system reduces to

Q – W = m (u2 – u1)

The final volume is found from PV1.2 = constant: P1V11.2 = P2V21.2

V1 = mRT1/P1 = 0.652 m3

V2 = 1.16 m3

T2 = P2V2/mR = 269 K

For a polytropic process with n = 1.2, the moving boundary work is

Wmb=P2V2?P1V1/1-n = 72 kJ

The net work is W = Wmb + Wrs = 22 kJ

The temperature change is relatively small, so assume that the air has constant specific heats: cv = 0.718 kJ/kg-K: u2 – u1 = cv(T2 – T1)

Therefore, from the First Law, Q = -14.6 kJ



(b) Using the above approach, the following plots can be made for the given final pressure range:





Note: The energy addition from the fan impacts the needed heat transfer. Initially, heat must be removed as the pressure drops from 200 kPa, but eventually the conditions have changed enough that heat must be added. There is insufficient energy added by the work from the fan at that point.