Assume a cylinder of gas with known pressure (P), temperature (T) and constant volume (V). This cylinder contains specific moles of air (n). Suddenly, a hole with diameter of (d) is formed on the body of cylinder. How long does it take to cylinder's pressure reaches atmosphere pressure?
Dear Hamidreza,
Do you think the answer would depend on whether the cylinder is horizontal or vertical and also if the temperature will stay constant or not?
Ramin Moghadasi
Hi Ramin
Not sure about your first question but about temperature, it is not significant and can be ignored.
The answer to your question (under certain assumptions [1]) can be found in [2]. Note that the unspecified constant Mw introduced in [2] is somewhat odd. I propose the following changes to the notation in [2].
Instead of n, I would use M(t), the total mass of the gas in the container (here a cylinder) at time t. Assuming the gas in the container to be mono-atomic, with mass per atom equal to ma, for the total number N(t) of atoms in the container at time t one has
(1) N(t) = M(t)/ma.
For the mass density at time t one has
(2) ρ(t) = M(t)/V,
where V is the volume of the container (which is assumed to be a constant, independent of t). Note that this mass density ρ(t) is encountered in the Bernoulli equation
(3) v(t) = (2 (P(t) - Pat)/ρ(t))½
relating the gas velocity v(t) at time t to the pressure P(t) of gas in the container and the atmospheric pressure Pat prevailing outside the container. The equation for the conservation of mass can therefore be written as
(4) dM(t)/dt = -A v(t)M(t)/V ≡ -A v(t)ρ(t),
where A denotes the area of the hole in the container. For the ideal gas in equilibrium one has (see Huang) PV = NkT, where P is pressure, k the universal gas constant (also denoted by R) and T the absolute temperature. Assuming the gas in the container to be ideal and in quasi-static equilibrium throughout the entire process of gas discharge, for the pressure P(t) in the container at time t one has
(5) P(t) = kT N(t)/V = kT M(t)/(maV) ≡ kT ρ(t).
The rest amounts to repeating the algebra in [2]. Note that the solution presented in [2] (which I have verified to be correct) depends on one undetermined constant c1, which is to be fixed by equating P(t=0) with the pressure of the gas in the container before the release of the gas from it has been initiated.
Lastly, there is a considerable body of literature on the subject matter, some of which related to vacuum technology. Under certain assumptions, the fluid-dynamics problems can be mapped onto problems of electronic circuits. For instance, the problem at hand is analogous to the dynamics of the discharge of a capacitor, with the charge stored in the capacitor representing the pressure difference, the charge flow representing the mass flow, etc. Consult e.g. [3], in particular the contribution by R. Gordon Livesey (Chapter 2 in [3]).
____
[1] The assumptions are (1) that the equation of state for the ideal gas* applies to the gas in the tank (*See p. 5 in Statistical Mechanics, 2nd edition, by K Huang), (2) the gas is mono-atomic, (3) the gas in the container is in quasi-static equilibrium throughout the entire process of gas discharge, and (4) the Bernoulli equation** applies to the flow of gas out of the tank - inviscid and steady flow (**See Sec. 3.4, p. 49, in Fluid Mechanics, by Potter and Wiggert). These assumptions can all be relaxed, however at a considerable analytic and computational cost.
[2] https://physics.stackexchange.com/questions/408932/pressurized-tank-emptying-over-time-through-a-given-area
[3] JM Lafferty, editor, Foundations of Vacuum Science and Technology (John Wiley & Sons, New York, 1998).
Behnam Farid . Your reply was pretty useful. Thanks for your time Sir.
Hamidreza Asaadian : My pleasure. Incidentally, you could make the calculations more accurate by using a more accurate equation of state than that of an ideal gas. Think of the van der Waals equation of state, and more systematically, the virial expansion [1] -- taking account of the second virial coefficient is trivial (see in particular Sec. 7.6 of [1]).
[1] K Huang, Statistical Mechanics, 2nd edition (John Wiley & Sons, New York, 1987).
Behnam Farid I needed that calculations to model the pressure fluctuation in two-phase hilly terrain pipeline. I think using simplified model is more practical to predict this phenomena. By the way I will have a look over that book. Thanks again for your precious comments.
Hamidreza Asaadian : You are welcome. Incidentally, for a 'hilly terrain', you need to take account of the influence of the variation in the gravitational energy (pressure varies with height). For details, consult any good text book on fluid mechanics (you may begin with that by Potter and Wiggert, which I have cited in Ref. [1] of my first comment on this page). In any case, you should always test the results of your calculations against some experimental observations on a small-scale model before relaying on your computational results on large scale.
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