Volume flow through a one side closed capillary

[Bavier]

Hi everyone,

I'm doing a simulation and need some help.
A capillary which is closed on both ends with the length l (x=0 to x=l), with a radius R and the volume pi*R^2*l is dropped on a parachut at the time t=0 from a hight h above ground.
At t=0 the pressure inside the capillary is p_i0 (this should be smaller than the ambient pressure p_a at this particular hight). Also at t=0 one side of the capillary is opened (the other side will remain closed). Now due the pressure difference between ambient and inner pressure (p_a - p_i0) a volume flow of air flows in x-direction into the capillary. We can assume laminar flow of the compressible gas.
I tried to use Hagen-Poisuille (for compressible fluids) to calculate the volume flow through the capillary due to the pressure difference. However the pressure difference and thereby the volume flow are time and position dependent, i.e. that the ambient pressure is changing with hight (can be easily calculated with the barometric formula) and the inner pressure and the density of the air inside the capillary will also change. There will be a build up of a back pressure.
Like I said, my goal is to simulate this process, i.e. a time and positon dependent (x-value in the capillary) function of the volume flow.

It's really difficult and I'm not a physicist nor I'm an engineer.

Is using Hagen-Poisuille the right choice? The big problem that I have is that one side is closed.
Do you have any imputs?

Many thanks

 

[Bystander]

What's the I.D. for this capillary? What material? This looks like you're working on a technique for atmospheric sampling, and there are going to be a number of differentiations in composition between bulk and what ends up in the capillary depending upon diameter and surface properties.

 

[Bavier]

The I.D is 0.003m. The material can be neglected. Or we can assume that there is no adsorption of the air and also no permeation through the material and the material itself will be very smooth. There is of course diffusion within the air once the air is inside the capillary but at the moment we can neglect that.
Yes, you got it right. It's a kind of atmospheric sampling technique.

 

[Bystander]

The I.D is 0.003m.

I'm assuming you meant "mm." You might check data sheets on molecular sieves for pore behaviors. Terrel Hill's Thermodynamics of Small Systems (Statistical Thermo of Small Systems?) might also be of some use. Adsorption of air wasn't the problem that concerned me --- I was thinking you could easily wind up with capillaries full of water.

 

[Bavier]

Ok, maybe it's more like a pipe. The diameter is indeed 0.003m or 3mm. No, the water shouldn't be a problem. Since it's falling down, the temperature is increasing and there should be no condensation of the water. And it can only be used if there is no liquid water in the atmosphere (clouds). Or there is the possibility to use magnesiumperchlorat driers.

 

[Bystander]

Three mm --- that's a regular culvert. The timescale for equilibration of that with atm. has to be on the same order as sound speed divided by capillary length, and, no, adsorption is no problem in that case. Depending on capillary length and fall rate, you're more concerned with diffusion replacing your high altitude sample with whatever atmospheric composition you're passing through.

 

[Bavier]

The problem that I have is more about the formula. Like I said, I try now to make a simulation about this process. The problem that I don't know what I should use for the length. Since the pipe is on one side closed, the length is time dependent. Either if I use Hagen-Poisuille or this formula for the mass flow:
$M^*=\frac{dM}{dt} =\frac{\pi}{16} \frac{m}{kT} \frac{R^4}{L} \frac{1}{\eta} (p_i^2-p_a^2) ,$

wherby $m$: mass of an air molecule, $k$: Boltzmann-constant, $T$: Temperature, $R$: radius, $L$: pipe lenght, $\eta$: dynamic viscosity, $p_i$: inner pressure, $p_a$: ambient or outer pressure
The assume the flow through a pipe or capillary in a space. But in my case there is only the pipe and the air can't actually flow out at the end...

I tried to make it in an iterative way but it didn't work. I don't know how I should calculate the new $p_i$ after each step and also how to deal with the last part where the air can't flow anywhere. And like you said, it needs some time to get in an equilibrium which is actually never the case with the parameters I used. One goal of the simulation is also to play around with the parameters diameter and length. To see which fits best for different kind of purposes.

 

[Bavier]

Hi, I don't even know if I'm in the right forum here. Maybe the physics one is more appropriate...