How much time to pressure balance two air chambers

[adam walker]

A smooth flexible tube 1mm in diameter and 1 foot in length holds air at normal / atmospheric pressure (14.7psi).

the tube is open at one end. the other end is connected to a tank by an open valve.

the tube is peristaltically compressed so all the air in the tube is forced into the tank.

the tank is dimensioned so that the air in the tank is pressurised to twice normal pressure (29.4psi)

the valve to the tank is now closed trapping the air in the tank .

the tube is allowed to naturally reflates to atmospheric pressure.

the valve to the tank is now opened:

Clearly the pressure differential between the tank and the tube results in air within the tank venting into the tube and out the open end - until the pressure in the tube and the tank stabilise at normal pressure.

MY QUESTION

Clearly that process is fast. But how fast?

I would like to know how much time that takes.

Can anyone tell me?

Thanks

 

[A.T.]

https://en.wikipedia.org/wiki/Speed_of_sound

 

[sophiecentaur]

Hi Adam and welcome to PF.
Could you clear up some questions about the experimental details? Is the tank pressurised just by the volume of air that was originally in the tube? that would mean its volume is similar to the tube volume. Does the temperature of the tank return to room temperature after it's been pressurised? For the sake of the experiment, is the tube 'rigid' or are you allowing it to be 'rubber' (implied by the peristaltic method for emptying the tube into the tank)?
The problem is not uncommon and it has various different facets. The time taken for an initial disturbance to make itself felt at the other end of the tube will be related to the 'speed of sound' in the air (as in post #2) but there are other factors. The pressurised tank will be discharging its excess pressure through the tube and the drag of the air will provide a back pressure. The situation would be analogous to a Capacitor discharging through a resistor with an exponential decay where.
V_{at time t} = V_start exp(-t/RC)i
The basic maths is the same and t's a simpler situation.
The time constant (time for V to fall to 1/e) of the start V will be RC. You now want an equivalent quantity for R and C. R, the resistance of the tube, will depend on the radius and the length and the equivalent to C will be the the pressure increase for a given volume of added air.
I will leave that with you to see if it makes sense so far.

 

[adam walker]

Hi Adam and welcome to PF.
Could you clear up some questions about the experimental details? Is the tank pressurised just by the volume of air that was originally in the tube? that would mean its volume is similar to the tube volume. Does the temperature of the tank return to room temperature after it's been pressurised? For the sake of the experiment, is the tube 'rigid' or are you allowing it to be 'rubber' (implied by the peristaltic method for emptying the tube into the tank)?
The problem is not uncommon and it has various different facets. The time taken for an initial disturbance to make itself felt at the other end of the tube will be related to the 'speed of sound' in the air (as in post #2) but there are other factors. The pressurised tank will be discharging its excess pressure through the tube and the drag of the air will provide a back pressure. The situation would be analogous to a Capacitor discharging through a resistor with an exponential decay where.
V_{at time t} = V_start exp(-t/RC)i
The basic maths is the same and t's a simpler situation.
The time constant (time for V to fall to 1/e) of the start V will be RC. You now want an equivalent quantity for R and C. R, the resistance of the tube, will depend on the radius and the length and the equivalent to C will be the the pressure increase for a given volume of added air.
I will leave that with you to see if it makes sense so far.

Thank you for your reply. I am very poor with equations and so perhaps it would help if I reframed the question in simpler terms:NOTE: This is a question where all the elements are ideal; there is no need to consider real world issues e.g. temperature / flow resistance etc.

A smooth tube 1mm in diameter and 1 metre in length holds air at normal / atmospheric pressure (14.7psi).
the tube is open at one end to the atmosphere. The other end of the tube is connected to a rigid tank via a valve which is closed.
The valve is 1mm in diameter.
The tank volume is half that of the tube.
The air pressure in the tank is twice that of the tube (29.4psi)
the valve to the tank is now opened:
(assume that this opening is instantaneous)
Because the tank holds air at twice atmospheric pressure one half of this air will vent into the tube.
The air already in the tube will then be displaced into the atmosphere.
This air movement will last for as long as it takes for the air pressure in the tank and the tube to settle at 14.7psi.
My assumption is this process would take only fractions of a second.
My question is: can this time be calculated.
(in the abstract – ie ignoring tube friction etc.)

And if so - what is the answer?

Thanks again for your time.
Adam

 

[davidwinth]

Here is your old one...