Exit velocity of gas through two orifices

[Albo]

If a gas flowing through a tube would meet two orifices with different diameters in the same cross section, how much of the gas would flow through each orifice and what would the gas velocity be in dependence of orifice diameter?

 

[berkeman]

Welcome to PF.

If a gas flowing through a tube would meet two orifices with different diameters in the same cross section, how much of the gas would flow through each orifice and what would the gas velocity be in dependence of orifice diameter?

Do you mean 2 orifices in a flat cylindrical plate sealing the end of a tube? Can you show us a sketch? (use "Attach files" below the Edit window to upload a PDF or JPEG image)

Also, is this question for schoolwork? What do you think will happen to the flow through the 2 orifices?

 

[Albo]

Hi! Thank you so much for the quick answer.

No the question isnt school related, i'm trying to develop a flow straightener element and trying to figure out the effect of combining different orifice sizes to the velocity profile and therefore the laminarity (if thats a word :D).

Yes you understood my question correctly! I'm thinking that the amount of gas flowing through each orifice is proportional to their cross sectional area. If i understand correctly tho the as the gas velocity through the orifice is antiproportional to the area the gas velocity through both orifices would be the same which suprised me. Just wanted to double check that i didnt mess something up

 

[erobz]

Hi! Thank you so much for the quick answer.

No the question isnt school related, i'm trying to develop a flow straightener element and trying to figure out the effect of combining different orifice sizes to the velocity profile and therefore the laminarity (if thats a word :D).

Yes you understood my question correctly! I'm thinking that the amount of gas flowing through each orifice is proportional to their cross sectional area. If i understand correctly tho the as the gas velocity through the orifice is antiproportional to the area the gas velocity through both orifices would be the same which suprised me. Just wanted to double check that i didnt mess something upView attachment 333791

The head loss through each orifice would be the same as the other (as the orifices are in parallel). What system of equations did you solve to arrive at your conclusion?

EDIT: I assume you use:

$$ h_{l_1} = h_{l_2} $$

$$ h_l \approx \frac{ (V_o - V_u)^2}{2g} $$

Where $V_o$ is the orifice velocity, and $V_u$ is the velocity of the flow upstream.

$$ \frac{ (V_{o_1} - V_u)^2}{2g} =\frac{ (V_{o_2} - V_u)^2}{2g} $$

$$ \implies V_{o_1} = V_{o_2} $$

I wouldn't think it to be extremely precise, but perhaps its fine.

 

[Chestermiller]

The distribution of the flow between the two orifices depends on the locations of the two orifices relative to one another and relative to the tube wall. This is a pretty complicated fluid mechanics problem, even for laminar flow.

 

[Albo]

The head loss through each orifice would be the same as the other (as the orifices are in parallel).

How do you know that? Sorry if the answer to this question is straight forward but it isnt to me. Ofc the pressure at the outlet is the same for both orifices but wouldnt the pressure drop in the center of the orifice be different because its size dependent? Maybe im just confused.

Regarding your derivation: no i went much simpler and just assumed the orifices to be tubes (which they obviously are) with crossection A_1 and A_2. Then i assumed the volume flow through each tube is proportional to the area
-->
q1 = Q * A_1 / (A_1+A_2)
q2 = Q * A_2 / (A_1+A_2)

The velocity through a tube is then just the volume flow divided by the cross section
-->
v1 = q1/A_1
v2 = q2/A_2

This would always lead to
--> v1 = v2

Of course these assumptions are not true since im not accounting for pressure drop and since (like Chestermiller) said it depents strongly on orifice placement and relative position to the tube and so on... Im just looking for some qualitative feeling for it

 

[Albo]

The distribution of the flow between the two orifices depends on the locations of the two orifices relative to one another and relative to the tube wall. This is a pretty complicated fluid mechanics problem, even for laminar flow.

Yes that is true. For a real answer i would probably need some CFD simulations. But i dont want to go too much into detail with it and just find out experimentally if it works the way i want it too.

Basically my question originates from the problem: Which leads to a flow which is 'longer' 'stable' (laminar for a larger distance)?

(simplified ofc in top down view)

Version 1

or
Version 2

 

[erobz]

How do you know that?

The approximate head loss for flow through an orifice? I looked it up in my textbook.

Or that the flows divide by head loss all other things equal?

Sorry if the answer to this question is straight forward but it isnt to me. Ofc the pressure at the outlet is the same for both orifices but wouldnt the pressure drop in the center of the orifice be different because its size dependent? Maybe im just confused.

I think it could ( and perhaps is by @Chestermiller ) be argued that it's an oversimplification and that $V_{o_1} \neq V_{o_2}$.

 

[Albo]

Or that the flows divide by head loss all other things equal?

Yeah this one. Which textbook are you referring to? id like to read it myself :)

 

[erobz]

Yeah this one. Which textbook are you referring to? id like to read it myself :)

It's an "oldish" textbook now. Engineering Fluid Mechanics; Crowe, Elger, Williams, Roberson; 9th edition.

 

[erobz]

Yeah this one.

Imagine you have a large pipe that splits into two of equal diameter. One of those pipes is 100 m long and the other is 1000 m long, they both eject to atmosphere without changing elevation. Your result says the flow divides equally (same areas), does that result seem reasonable to you?

 

[Albo]

Imagine you have a large pipe that splits into two of equal diameter. One of those pipes is 100 m long and the other is 1000 m long, they both eject to atmosphere without changing elevation. Your result says the flow divides equally (same areas), does that result seem reasonable to you?

Interesting question and maybe this is where im confused. The pressure difference would be the same but the friction in the tube would slow down the gas in the longer tube more leading to a lower mass output, correct?

 

[erobz]

Interesting question and maybe this is where im confused. The pressure difference would be the same but the friction in the tube would slow down the gas in the longer tube more leading to a lower mass output, correct?

Correct, there is more frictional work done on the flow in the longer tube. That equates to waste heat generation, which either warms the flow, or escapes via heat transfer to the environment. That extra heat is taken from the kinetic energy of the flows.

If you assume uniform flow properties over the cross section, you can write Work-Energy between the junction and each of the end points, the result is:

$$ \overbrace{\frac{V_1^2}{2g}}^{\text{kinetic head branch 1}} + \overbrace{\sum_{j \to 1} h_l}^{ \text{waste heat branch 1} }= \overbrace{\frac{V_2^2}{2g}}^{\text{kinetic head branch 2}} + \overbrace{\sum_{j \to 2} h_l}^{ \text{waste heat branch 2} } $$

The head loss in a branch can be found via Darcy-Weisbach:

$$ h_l = \frac{f L}{D} \frac{V^2}{2g} $$

Then you would use continuity ( incompressible flow ) to solve for the velocities (assuming the flow is slow enough to neglect effects compressibility - since you are considering "air" )

It should go without saying that there are obviously more refined flow models, but the idea should be fundamental.