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Anachronist

Gold Member

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This seems simple, but I have some confusion. I'm trying to refine a numerical simulation to get away from my isothermal assumption, to account instead for isentropic temperature and pressure changes in pressure tank with an orifice.

Given an insulating air tank of volume ##V##, containing air at absolute pressure ##P_0## (Pascals) and temperature ##T_0## (degrees K), a mass of air ##\Delta m## (kg) is allowed to escape rapidly into the atmosphere at ambient pressure ##P_a## over a short interval ##\Delta t## (seconds). What is the tank's internal pressure and temperature ##P_1## and ##T_1## at the end of the interval?

Whether the air flow is choked or unchoked, or the orifice geometry, shouldn't matter; this is accounted for by the fact that we already know the mass flow rate.

I have this so far:

The initial density of air would be ##\rho_0 = \frac{P_0}{R T_0}##

The initial mass of air would be ##m_0 = \rho_0 V##

So after the change in mass, the new air density would be ##\rho_1 = \frac{m_0 - \Delta m}{V}##

...and that's kinda where I get stuck, getting a final temperature and pressure from there. The adiabatic relationship

$$\frac{T_1}{T_0} = \left( \frac{P_1}{P_0} \right)^{1-\frac{1}{\gamma}}$$

would need to be used, but I need to know if there's a relationship that connects density ratios to pressure ratios? I suspect it's something like this (I can't find a source yet):

$$\gamma \frac{\Delta \rho}{\rho} = \frac{\Delta P}{P}$$

Is there enough information to solve this problem?

Given an insulating air tank of volume ##V##, containing air at absolute pressure ##P_0## (Pascals) and temperature ##T_0## (degrees K), a mass of air ##\Delta m## (kg) is allowed to escape rapidly into the atmosphere at ambient pressure ##P_a## over a short interval ##\Delta t## (seconds). What is the tank's internal pressure and temperature ##P_1## and ##T_1## at the end of the interval?

Whether the air flow is choked or unchoked, or the orifice geometry, shouldn't matter; this is accounted for by the fact that we already know the mass flow rate.

I have this so far:

The initial density of air would be ##\rho_0 = \frac{P_0}{R T_0}##

The initial mass of air would be ##m_0 = \rho_0 V##

So after the change in mass, the new air density would be ##\rho_1 = \frac{m_0 - \Delta m}{V}##

...and that's kinda where I get stuck, getting a final temperature and pressure from there. The adiabatic relationship

$$\frac{T_1}{T_0} = \left( \frac{P_1}{P_0} \right)^{1-\frac{1}{\gamma}}$$

would need to be used, but I need to know if there's a relationship that connects density ratios to pressure ratios? I suspect it's something like this (I can't find a source yet):

$$\gamma \frac{\Delta \rho}{\rho} = \frac{\Delta P}{P}$$

Is there enough information to solve this problem?

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