Pressure differential and air flow

[kibestar]

I'm having a hard time understanding air flow caused by pressure differentials. Given the following contextual image, crafted by myself:

Bare with me here, these are two interconnected boxes open to the Earth's atmosphere. The orange arrow represents an air pump, such as a vacuum cleaner's electric motor, which fills this connection's space completely. When off, atmospheric pressure is observed at every point. Now turn the pump on. I can clearly predict based on common sense that an air flow will be generated, following this path: atmosphere->box 1->pump->box 2->atmosphere. However, I realize I don't understand much of what's happening. I would suppose box 1 would experience a sudden drop of pressure, while a rise of pressure would occur in box 2, dragging air in and out... But how does this happen? The initial location of the air flow is the atmosphere, and so is its final location. The pressure differential between the atmosphere and the atmosphere is 0... Summarizing into questions:

1- How is their an air flow if the net pressure differential is 0?
2- Is the pressure drop in box 1 the same in every single point of its interior? Same goes for box 2.
3- If so, is it exactly the same or just a very good approximation?
4- If it is exactly the same, why can't the atmosphere be considered an extension of the box and also suffer the pressure drop/increase? (Supposing one of the boxes is now sealed, leading to question 5)
5- What happens with respect to air flow and pressure changes as the opening areas on each box is decreased/increased? What if one of them or both of them are sealed?

I'm very confused, and I suspect this might be some basic mechanics that I'm just not following... In such case, I apologize.

Thanks in advance!

 

[UltrafastPED]

Pressure wants to equalize ... thus winds move from high pressure regions to low pressure regions.

What produces low and high pressure regions? Differences in surface temperatures for one. Thus clouds, day and night, etc.

Your vacuum cleaner has a low pressure region (input) and a high pressure region (output), with the pump doing the work. So stuff moves in at the inlet, and stuff flows out into the bag. When you turn the pump off, everything quickly falls back to normal.

 

[kibestar]

OK, thank you, but what about all of my other questions?

 

[Chestermiller]

When you turn on the pump, the pressure in chamber 2 will begin to rise, and the pressure in chamber 1 will begin to fall. The pump is sending air from chamber 1 into chamber 2. The pressure in each of the two chambers will not be uniform. Over the bulk of each of the two chambers the pressure will be uniform, but, in chamber 2, close to the exit, the pressure will be decreasing toward atmospheric pressure, and in chamber 1 the pressure will be increasing toward the entrance (atmospheric). The volumetric flow out of the exit hole in chamber 2 can be described in terms of the pressure difference between the bulk of chamber 2 and the atmospheric pressure at the exit, using the Bernoulli's equation. Similarly for the volumetric flow into chamber 1. The accumulation of gas in chamber 2 and the buildup of pressure in chamber 2 can be described using the ideal gas law, in conjunction with a material balance on the chamber. Similarly for the accumulation of gas in chamber 1. Eventually the system will reach steady state, and the pressures within the two chambers will no longer be changing with time.

Chet

 

[kibestar]

That was a very informative answer, thank you.

What is a good approximation to consider the "bulk" of each chamber? And with regard to the exit holes, a modification on their area would produce what change in air flow (I'm assuming the pressure drop/increase would remain the same, since the pump is the same?)?
Also, is the absolute value of the pressure drop equal to that of the increase, or is this dependent on the chamber openings?

Thanks.

 

[Chestermiller]

That was a very informative answer, thank you.

What is a good approximation to consider the "bulk" of each chamber?

Everything more than a few exit hole diameters away from the exit hole.

And with regard to the exit holes, a modification on their area would produce what change in air flow (I'm assuming the pressure drop/increase would remain the same, since the pump is the same?)?

What does Bernoulli's equation tell you?

Also, is the absolute value of the pressure drop equal to that of the increase, or is this dependent on the chamber openings?

I don't quite understand this question. Which pressure drop and which increase are you referring to?

Chet

 

[kibestar]

I'm sorry, I don't have knowledge of Bernoulli's equation (just graduated high school). Will look it up.
With respect to the pressure drop/increase, I'm referring to the ones caused by the air pump in each plenum.

Thanks, once more.

 

[Chestermiller]

I'm sorry, I don't have knowledge of Bernoulli's equation (just graduated high school). Will look it up.
With respect to the pressure drop/increase, I'm referring to the ones caused by the air pump in each plenum.

Thanks, once more.

If you increase the hole diameter in the high pressure chamber, then the pressure in the chamber will be less. If you increase the hole diameter in the low pressure chamber, then the pressure in the chamber will become higher. As a high school student, you don't have enough background to analyze this problem in detail quantitatively.

Regarding the pump. There is an equation describing the so called "pump characteristic." This relates the three parameters: pump rotational speed, pressure increase, and volumetric flow rate. You need to know this relationship from experimental measurements to mathematically model the problem. If the material being processed is compressible like air, there are also other parameters involved (such as the gas density coming into the low pressure end of the pump). The pump characteristic is often provided at the time of purchase by the pump manufacturer.

Chet

 

[kibestar]

Chet, I looked up Bernoulli's equation, and now I have some new questions!

The accumulation of gas in chamber 2 and the buildup of pressure in chamber 2 can be described using the ideal gas law

So I'd have to use the compressible flow equation for this situation? And, knowing the pressure in the bulk region, I wouldn't actually need to take into account the characteristics of the air pump in use, since I'd be solving a system of equations for two speed variables, one at a point contained on the cross section of the entrance or exit hole, and another for a point in the bulk region. Correct?

Also, would there be any conservative forces to consider in this situation?

Thanks, you've been very helpful.

 

[Chestermiller]

Chet, I looked up Bernoulli's equation, and now I have some new questions!


So I'd have to use the compressible flow equation for this situation? And, knowing the pressure in the bulk region, I wouldn't actually need to take into account the characteristics of the air pump in use, since I'd be solving a system of equations for two speed variables, one at a point contained on the cross section of the entrance or exit hole, and another for a point in the bulk region. Correct?

No. Even when the system reaches steady state, you would have to match the mass flow rate of gas out the exit hole with the mass flow rate into the chamber from the pump. Also, the velocity in the bulk region would typically be negligible.

Also, would their be any conservative forces to consider in this situation?

If you're thinking about gravity, that would usually be negligible in this type of situation.

Chet

 

[kibestar]

No. Even when the system reaches steady state, you would have to match the mass flow rate of gas out the exit hole with the mass flow rate into the chamber from the pump. Also, the velocity in the bulk region would typically be negligible.

If you're thinking about gravity, that would usually be negligible in this type of situation.

Chet

Is your "no" directed towards which statement? Use of compressible flow equation, non-need of taking into consideration the pump specifications, system of equations or everything?

 

[Chestermiller]

Is your "no" directed towards which statement?

It referred to non-need of taking into consideration the pump specifications.

Chet

 

[kibestar]

But why isn't the system of equations enough to find air flow? Two equations with two variables... What am I missing?

 

[Chestermiller]

But why isn't the system of equations enough to find air flow? Two equations with two variables... What am I missing?

I don't know. I'll send you a private message with the formulation of the model equations.

Chet