The air supply to a 2000 ft3 office has been shut off overnight to conserve utilities, and the room temperature has dropped to 40oF. In the morning, a worker resets the thermostat to 70oF, and 200 ft3/min of air at 120oF begins to flow into the office through a heating duct. The air is well-mixed within the room and an equal mass flow of air at room temperature is withdrawn through a return duct. The air pressure is essentially 1 atm everywhere. Ignoring heat exchange with the surroundings, as well as any changes in kinetic or potential energies, estimate how long it takes for the room temperature to reach 70oF. Plot room temperature as a function of time. Assume g = 1.4 for the air.
HINTS :
This is a transient or unsteady process because helium enters the system (the balloon). Assume the He behaves as an ideal gas, but check to see if this is a good assumption. Use the IG EOS to determine the initial mass of He in the balloon. After you determine V2, calculating Wb is easy ! Then, simultaneously solve two equations in two unknowns. The equations are: the IG EOS applied to the final state of the balloon and the transient form of the 1st Law applied to this process. The two unknowns are: T2, mHe,2 .
The catch is that we must determine values for U1, U2 and Hin. These are NOT ΔU's and ΔH's but real U's and H's. In order to do this (just like the steam tables) we MUST choose a reference state. A reference state is a T, P and phase at which YOU choose to make EITHER U or H zero kJ/kg. I want you to use a reference state of U = 0 for He gas at 22 oC and 100 kPa. The P doesn't actually matter because He is treated as an IG in this problem so U and H are not functions of P anyway.
Once you have a ref state, use a Hypothetical Process Path from the ref state to states 1, 2 and inlet to evaluate U1, U2 and Hin using the IG EOS and CV and CP given in the problem.
For He, use: CP = 5.1926 kJ/kg-K and CV = 3.1156 kJ/kg-K.
11 comments:
dr B,
may i know what does gamma represent?
I'm pretty stuck in general on this one. How about an overall hint from Dr. B?
I know the specific heat capacity is a constant and I have a simplified equation for the first law that includes Troom. My problem is that I don't know how to incorporate time into the equation. The mass flow rate includes time but is that where it is supposed to come in?? Something seems fishy to me in my first law simplification. Little help?
Y:
Gamma is the heat capacity artio. Gamma = Cp / Cv.
Can we please post questions to question 9?
Sparticus:
Anything you ask for, sir or madame, after all the help you have given others here.
The key here is that Tin, Cp, Cv and Tref are all constants and that mdot(in) = mdot(out)=mdot.
Transient 1st Law:
dEsys/dt = min * Hin - mout Hout
But min = mout = m and Esys = msys * Usys
H = Cp*(T-Tref) and U = Cv*(T-Tref)
Pick any Tref, b/c it cancels anyway !
Assume Cp and Cv and definitely Tref are constants.
When you separate variables, you should get an ordinary differential equation that looks alot like...
dTsys/(Tin-Tsys) = {some constnat stuff} * dt
Integrate and solve for t for part (a) and use Tsys = 70 degF. In part (b), solve for Tsys as a function of t.
I hope this does the trick !
Sparticus 9:08:
You asked at just the right time b/c I am still online !
Time comes in through the accumulation term on the left side of the 1st Law: d(Esys)/dt. Here, Esys = msys*Usys=msys*Cv*(Tsys-Tref).
This simplifies to:
msys*Cv*d(Tsys)/dt because msys, Cv and Traf are all constants.
Next, separate variables and integrate !
Best of luck !
Perhaps I have done the equations wrong, but I found that mdot cancels, leaving the equation as Cv(Tsys-Tref) = Cp(Tin-Tsys), which leaves me without something to integrate.
I figure maybe it should be Cv(dTsys/dt-Tref) = Cp(Tin-Tsys), but that still leaves me with a Tref, which should have canceled.
Questioning:
Yes, you did make an error. I think you cancelled msystem with mdot. Msystem is the mass of air in the room and mdot is the mass flow rate of air into and.or out the room. They are constants, but they are not equal.
I calculated msys using the average room temperature of 55 degC and the ideal gas EOS. I calculated mdot using the IG EOS and T = 120 degF = 579.67 degR.
Did you mean 55F for room temp instead of 55C?
Anon 7:53:
Doh !
Yes, I meant 55 degF.
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