Rate of air flow between containers of different pressures

[egged]

Homework Statement

I am given a container of volume V of pressure p filled with hydrogen gas. Outside the container, the volume can be taken as infinity and pressure can be taken as 0 (eg: in space). The temperature T is fixed throughout (there is a heat generator in the container that will maintain the temperature at T throughout the experiment).

There is a small hole of area S at the side of the container. The shape of the hole and the location of the hole (with respect to the container) is not stated. My task is to find the time taken for the pressure inside the container to drop from p to p2.

Homework Equations

a) pV = nRT
b) Dynamic pressure q = 0.5ρv²
c) Mass flow rate dm/dt = CYS√(2ρΔp)

where V = volume of container,
n = number of moles of hydrogen,
R = molar gas constant (8.31 J mol^-1 K^-1),
T = temperature of gas in container,
ρ = density of hydrogen in the container,
v = rate of air flow through hole,
C = coefficient of discharge,
Y = expansion factor,
Δp = difference in pressure, also = pressure in the container, since pressure outside is 0.

The Attempt at a Solution

The question I have is regarding the rate of flow of gas, given the pressure difference. I tried equation (b) but it seems to give me a contradiction, in that:
p = p-0 = Δp = q = 0.5ρv², and
ρ = m/V = (2/1000)n/V = (2/1000)(p/RT)
So p = 0.5(2/1000)(p/RT)v², and
v = √(1000RTp/p)
v = √(1000RT) which is independent of p --- (does not seem to make sense)

But if I try equation (c), I seem to be missing the values for C and Y, which is not stated in the question - or am I missing something?

Once I am able to get the velocity of air flow (given pressure differences), or rate of air flow (whether in mass per unit time or volume per unit time), I should be able to evaluate the rest of the solution myself. Please help me with this, thanks!


PS: My level of Physics is pre-University. This question is not in my homework but is for enrichment.

 

[mark726]

Thank you for your post. I understand your confusion regarding the rate of flow of gas through the hole in the container. It is true that the equations (b) and (c) seem to give contradictory results, but this can be explained by the fact that they are derived from different principles.

Equation (b) is derived from the Bernoulli's principle, which states that the sum of pressure, kinetic energy, and potential energy per unit volume of a fluid is constant. In this case, we are assuming that the pressure outside the container is 0, so the pressure difference, Δp, is equal to the pressure inside the container, p. This means that the velocity of the gas flow, v, is only dependent on the density of the gas, ρ, and the properties of the hole, such as its area and shape.

On the other hand, equation (c) is derived from the continuity equation, which states that the mass flow rate is constant in an incompressible fluid. This equation takes into account the properties of the gas, such as its density and the pressure difference, but it also includes the properties of the hole, such as the coefficient of discharge, C, and the expansion factor, Y. These properties are not given in the question, so it is not possible to use this equation to find the velocity of the gas flow.

To find the velocity of the gas flow, you can use equation (b) and solve for v in terms of p, ρ, and the properties of the hole. Then, you can use this value of v in equation (c) to find the mass flow rate and subsequently the time taken for the pressure to drop from p to p2.

I hope this helps you with your problem. Good luck with your enrichment! If you have any further questions, please don't hesitate to ask.