Mostly Free Expansion of Ideal Gas

[Slader99]

Hey All,

Had a problem to solve and wanted to know how others would approach it. Problem is as follows:

An insulated container of known volume has an orifice of know size. Container is filled with air and left to rest so that temperature and pressure is that of atmospheric surroundings. Container air is heated at a know rate which causes mass flow through orifice. What is the container pressure (gauge) resulting from the heating of said air?

Givens:
1) Initial temp is 300K and initial pressure 1atm.
2) Volume of container is 0.0033m^3
3) Orifice is circular of area 9.0E-6 m^2
4) Rate of temperature change is 1C/min

If you were to craft a formula for pressure as a function of rate of temperature change what would it look like? If you were to solve for pressure based on the givens above what would you get for Pgauge?

When I did the exercise I had to make more assumptions than I’m comfortable with. I ended up using the ideal gas & Bernoulli equation to arrive at a value in the range of 2.4E-6 mbar.


Thanks!
slader99

 

[eljose79]

Dear slader99,

Thank you for presenting this problem to the forum. my approach to solving this problem would involve using the ideal gas law and the Bernoulli equation to determine the pressure inside the container as it is heated.

First, we need to determine the number of moles of air inside the container. This can be calculated using the ideal gas law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. Rearranging this equation, we get n = PV/RT. Plugging in the given values, we get n = (1atm)(0.0033m^3)/(0.0821 L·atm/mol·K)(300K) = 0.000134 moles of air.

Next, we need to calculate the mass flow rate of air through the orifice. This can be done using the equation ṁ = ρAv, where ṁ is the mass flow rate, ρ is the density of air, A is the area of the orifice, and v is the velocity of the air. Since the air is at atmospheric pressure and temperature, we can use the ideal gas law to calculate its density: ρ = P/RT. Plugging in the given values, we get ρ = (1atm)/(0.0821 L·atm/mol·K)(300K) = 0.040 kg/m^3. We can also calculate the velocity using the Bernoulli equation: P + ½ρv^2 = constant. Since the pressure and density are constant, we can solve for v and get v = √(2(P-Patm)/ρ) = √(2(1atm-1atm)/(0.040 kg/m^3)) = 0 m/s. Therefore, the mass flow rate is ṁ = (0.040 kg/m^3)(9.0E-6 m^2)(0 m/s) = 0 kg/s.

Now, we can use the ideal gas law again to calculate the final pressure inside the container. Since the volume and number of moles remain constant, we can use the equation P1V1/T1 = P2V2/T2, where P1 and T1 are the initial pressure and temperature, and