Question

An insulated \(260-\mathrm{ft}^{3}\) rigid tank contains air at 40 psia and \(180^{\circ} \mathrm{F}\). A valve connected to the tank is opened, and air is allowed to escape until the pressure inside drops to 20 psia. The air temperature during this process is maintained constant by an electric resistance heater placed in the tank. Determine \((a)\) the electrical work done during this process and \((b)\) the exergy destruction. Assume the surroundings to be at \(70^{\circ} \mathrm{F}\).

Short answer

Question: Calculate the electrical work done and exergy destruction during the process of an insulated tank filled with air losing pressure while being kept at a constant temperature. Answer: The electrical work done during the process is zero since there is no change in the internal energy of the air. The exergy destruction can be found by calculating the entropy change and using the equation Ex_dest=T0*(S_sys + S_sur).

Step-by-step solution

Calculate the amount of air remaining in the tank

Use the Ideal Gas Law to find the initial mass of air in the tank, then repeat for the final condition, since the temperature and volume are the same. The difference in mass is the mass of air that has been released. The equation for Ideal Gas Law is: P*V = m*R*T, where P is the pressure, V is the volume, m is mass, R is the specific gas constant and T is the absolute temperature.

Find the change in internal energy of air.

We can find this by multiplying the release mass of air with the change in specific enthalpy (h) and temperature. As it's a constant temperature process, effectively, Δh=0 and consequently, the total change in internal energy of air is zero.

Calculation of the electrical work done.

From the law of energy conservation, the electrical work done should equate to the change in internal energy of air in this case. As we found ΔU = 0, there's no energy change hence no electrical work needed.

Calculation of Exergy Destruction

To find exergy destruction, one needs to know the amount of entropy change from initial state to final state. As this problem exhibits an isothermal process, hence, the entropy change depends only on the pressure change at a given temperature, which can be found through S2-S1 = m*R*ln(P1/P2), where S is the entropy, P is the pressure. After computing the entropy change, exergy destruction can be found using the equation: Ex_dest=T0*(S_sys + S_sur) where T0 is the reference temperature and S_sys is the system entropy change, S_sur is the surroundings entropy change.