An insulated 8-m3 rigid tank contains air at 600 kPa and 400 K. A valve connected to the tank is now opened and air is allowed to escape until the pressure inside the tank drops to 200 kPa. The air temperature is maintained constant by an electric heater during the process.
1-Calculate the amount of discharged air in kg.
2-Calculate the power consumed by the electric heater in watt.
3-Explain how the enthalpy of discharged air is calculated if its temperature changes during the process.
Tables for specific energy/enthalpy of air as an ideal gas are given for 100 KPa and 25 C in book.
ideal gas law - PV = mR(air)T
v = V/m
for a general control volume, dE/dt = Qcv(per sec) - Wcv(per sec) + Σmi(inflow per sec)*(hi + .5(Vi^2) + gZi) - Σme(outflow per sec)*(he + .5*(Ve^2) + gZe)
power = work per second = Wcv(per sec) [there's no way to insert a dot directly above a letter, to represent work per second)
first law of thermo - ΔU = Q - W
The Attempt at a Solution
For part 1 -
I assume it's an ideal gas because no critical constants for air are given to me and I don't know if it's accepted to go outside the book and given info here, so I can't use a Z factor chart. And I'm not given masses or specific volumes anyway, so this is a must as far as I can tell.
I rearrange the ideal gas law to get PV/RT = m, and solve for the initial and final mass inside the box.
Mi = 41.81 Kg
Mf = 13.94 Kg
While I'm here, I use the specific internal energy and enthalpy at 400k (u/h doesn't vary with P much) to find initial and final U and H for the contents of the box.
Taking the differences gives me
ΔU = 7984.47 KJ
ΔH = 11184.23 KJ
M(discharged) = difference between initial and final mass in box = 27.87 Kg
For part 2 -
Firstly, this is where I'm stuck, but i'll go ahead and state what I tried.
Since I know that specific internal energy doesn't really vary with pressure, I assume it to be constant since the temperature is held constant by the generator. So by that logic, and considering the box is insulated (thus Q = 0), I say that the change in internal energy of the box is due to a loss of air mass.
So I guess this is where I'm stuck perhaps both conceptually and mathematically. For a general control volume I have the giant formula above, but since heat transfer = 0 due to insulation, no air flows IN, and I'm assuming kinetic/potential energy are negligible (no velocities /areas / heights given) , I whittle it down and simplify it to
dEcv/dt = - Wcv(per sec) - Σ me(outflow per sec) * (he) = -Wcv(per sec) - He(enthalpy outflow per sec)
Now, I wasn't given any time frame for this to happen during, so the easy way of figuring out flow rates isn't possible here. To try and take at least some step forward I simplify further and say that if not accounting for actual rates and instead just looking at total change, (basically just multiplying everything by whatever unknown total time this process takes)
ΔEcv = -W - He ?
And I'm assuming that the change in energy is simply ΔU, essentially? Since I'm not accounting for any kinetic or potential energy? Do I need to keep in mind that the air in the box does boundary work when it goes from high pressure to the outside low pressure?
I'm pretty sure what I want is the work per second, because that would be equivalent to the energy supplied by the generator in order to keep temperature constant. Is this the correct interpretation?
And if so, how exactly do I get to it if I have no time frame? I'm not given any velocities or areas so I'm unsure how else to establish a mass flow rate, or any flow "rate" for that matter. Do I just need to assume it all happens in 1 second or something, make some kind of arbitrary assumption or something?
Basically, how do I need to visualize this, and where are my errors in interpretation or conceptualization? Any and all help is greatly appreciated, thanks.