What is the step response of a filling chamber with changing air pressure?

[LoopInt]

Homework Statement

This is a experimental problem that I am trying to model:
I have an air source with constant pressure P connected with a small chamber (cylinder). The chamber has initial internal pressure P_0 and as time t goes, the pressure in the chamber raises to P.
So, find the step response of P_chamber(t).

Homework Equations

Thats a great question itself, hehe. These are the ones I tried:

P+ρU^2+ρgh = const
PV=mRT
dm/dt = ρU A

The Attempt at a Solution

P_chamber = (m(t)RT)/V

m varies over time, V is constant, R is constant and T is a problem. (don't know if I can keep it constant, but I did)
I assume the tank is very large compared with the chamber, so U for the tank is 0. I also neglected the hydrostatic term as ρ and h are very small.

So,
P=P_0+P_chamber+ρU^2

well, I'm stuck at this point. Can't reach a diff eq.

Experimentally the step response looks like P_chamber(t) = P(1-e^(-t/B)) ; where B is a time constant
So it should be a first order diff. eq.

Thanks in advance!

 

[Chestermiller]

You're using the wrong equations, and you're missing an equation. You need to model the flow through the valve. Let Δp represent the pressure drop across the valve, and let r represent the resistance to flow through the valve, such that

dm/dt = Δp/r
From the ideal gas law,

\frac{dp}{dt}=\frac{RT}{MV}\frac{dm}{dt}=\frac{RT}{MV}\frac{P-p}{r}

This will give you the functionality you observe experimentally.

 

[LoopInt]

Thanks!
This is indeed the response I observed. However I have some further questions about this model.
1st: Differentiating both sides from the ideal gas gives:
dp/dt=RT/V * dm/dt , so Whats the M you put in the equation and where it comes from?

2nd: How can I calculate the resistance "r" given the geometry of the pipe, and/or valve.

 

[Chestermiller]

Thanks!
This is indeed the response I observed. However I have some further questions about this model.
1st: Differentiating both sides from the ideal gas gives:
dp/dt=RT/V * dm/dt , so Whats the M you put in the equation and where it comes from?

M is the molecular weight of the gas.

2nd: How can I calculate the resistance "r" given the geometry of the pipe, and/or valve.

This is a complicated thing to calculate from scratch, since it depends on the internal geometry of the value and on the fluid dynamics taking place in the valve. But, what you can do is back out the value of r from your experimental data. Given the initial conditions, what is the solution to your differential equation for p as a function of time t (assuming r is constant)?

Chet

 

[LoopInt]

Ok, so I found that:

p(t)=C1e^(-RTt/(MrV))+P
p(0)=0
So,
C1-P=0 ; C1=P
p(t)=P(e^(-RTt)/(MrV)+1)

And I will plot and estimate r from the data.

But that M still Bugs me. I'm using the ideal gas law like this
P=mRT/V
where m is the mass in kg, R the air constant T temp and V volume.
so I get
dp/dt=RT/V * dm/dt

 

[Chestermiller]

Ok, so I found that:

p(t)=C1e^(-RTt/(MrV))+P
p(0)=0
So,
C1-P=0 ; C1=P
p(t)=P(e^(-RTt)/(MrV)+1)

That's not what I get. And how come you took p(0) = 0? Was the cylinder totally evacuated to begin with? I get:
\frac{P-p(t)}{P-p(0)}=e^{-\frac{RTt}{MVr}}
or, equivalently:

\ln{\left(\frac{P-p(t)}{P-p(0)}\right)}=-\frac{RTt}{MVr}

And I will plot and estimate r from the data.

I suggest plotting (P-p(t)) vs t on a semi-log plot. The semi-log scale will be used for (P-p(t)). You should get a straight line, and should be able to determine the desired parameters from the slope and intercept.

But that M still Bugs me. I'm using the ideal gas law like this
P=mRT/V
where m is the mass in kg, R the air constant T temp and V volume.
so I get
dp/dt=RT/V * dm/dt

OK. I was taking R as the molar gas constant. So your R is my R/M.

Chet

 

[LoopInt]

I was using p(0) as 0, because it was gauge pressure. I realized I should rather use the absolute pressure.

Just out of curiosity:

Will this be still valid for large delta P? I found the Hagen-Poiseuille equation, but it only works for low mach numbers, so maybe this one will too.
If so, how can I calculate the mass flow rate for a large delta P? How to know if for a given tube diameter and the pressure difference, there will be sonic flow?

The NASA site gives a formula to calculate the mass flow rate, but It depends on the mach number. There must be a way of calculating the mach number from a pressure difference and the pipe diameter, but I'm not finding a clear answer.

 

[Chestermiller]

I was using p(0) as 0, because it was gauge pressure. I realized I should rather use the absolute pressure.

Just out of curiosity:

Will this be still valid for large delta P? I found the Hagen-Poiseuille equation, but it only works for low mach numbers, so maybe this one will too.
If so, how can I calculate the mass flow rate for a large delta P? How to know if for a given tube diameter and the pressure difference, there will be sonic flow?

The NASA site gives a formula to calculate the mass flow rate, but It depends on the mach number. There must be a way of calculating the mach number from a pressure difference and the pipe diameter, but I'm not finding a clear answer.

Even if the starting pressure in your chamber is much lower than your source pressure (how do they compare?), most of the time for the pressure in the chamber to rise to P occurs then the pressures are not very different. During that part of the filling, the mach number will probably be low. In any event, using Hagen-Poiseuille won't be appropriate because the flow is going to be turbulent. So you need to use the turbulent flow equations for the pipe; this can include taking into account the compressibility of gas. But, the valve can be a problem (because the internal geometry is complicated and can be varied), and you may have to measure the valve characteristics experimentally. You need to get the pressure drop-flow rate relationship for the valve, if, in practice, the pressure drop for the valve is significant compared to the pressure drop in the pipe.

Chet

 

[LoopInt]

Thanks for the reply.

Even if the starting pressure in your chamber is much lower than your source pressure (how do they compare?)

I have a tank which is 4-8 bar connected to a pressure regulator at 2.5 bar. There is a pipe connecting the pressure regulator to the valve, and the valve to the chamber at atmospheric pressure, initially.

most of the time for the pressure in the chamber to rise to P occurs then the pressures are not very different. During that part of the filling, the mach number will probably be low.

So that means I can work with incompressible flow equations? (Those of MACH 0.3 and below?)

So you need to use the turbulent flow equations for the pipe

Any suggestion? There seems to be a lot of them.

You need to get the pressure drop-flow rate relationship for the valve

I can measure the pressure drop using 2 pressure sensors that I have, but how will I know the mass flow rate?

Also, there is something interesting happen when the chamber is filling. The mass flow rate is almost constant. That is why I thought of mach numbers, because if there is a throat somewhere, the mass flow rate will be limited, right?
And so $$\dot{m}=M*a*\rho*A$$
Where M = 1
a is the speed of sound
rho the air density
and A the area of the "throat"

 

[Chestermiller]

Thanks for the reply.

I have a tank which is 4-8 bar connected to a pressure regulator at 2.5 bar. There is a pipe connecting the pressure regulator to the valve, and the valve to the chamber at atmospheric pressure, initially.

So that means I can work with incompressible flow equations? (Those of MACH 0.3 and below?)

So, the real pressure drop in the pipe is only 2.5 bars. I would use the compressible flow equations, but, if the flow rate is low, assuming that the flow is isothermal might be a pretty good approximation.

Any suggestion? There seems to be a lot of them.

See Chapter 6 of Transport Phenomena by Bird, Stewart, and Lightfoot, particularly Examples 6.2.1 and 6.2.2.

I can measure the pressure drop using 2 pressure sensors that I have, but how will I know the mass flow rate?

Are the two pressure sensors located in the pipe? If so, then this is great. What are typical readings of the two pressure sensors? How long is the pipe, and what is its diameter?

Also, there is something interesting happen when the chamber is filling. The mass flow rate is almost constant. That is why I thought of mach numbers, because if there is a throat somewhere, the mass flow rate will be limited, right?
And so $$\dot{m}=M*a*\rho*A$$
Where M = 1
a is the speed of sound
rho the air density
and A the area of the "throat"

I don't think so. I would start out by assuming that the flow is not close to sonic. How does the resistance of the valve compare with the resistance of the pipe when the valve is opened? Is the resistance of the valve negligible? What is the reading on the pressure regulator when the valve is closed? It's not clear how this relates. If you really do have turbulent flow in the pipe, the mass flow rate of air will not be a linear function of the pressure drop. You will have to modify your pressure balance equation and its solution for the case of turbulent flow.

Chet