# Air speed and Differential pressure

1. Jul 29, 2015

### MARECHAL

After a big mistake in this PF, I try to post my question here, hoping it is ok.
I'm looking for a more "simple" relation than Darcy's law.
I try to determinate what is the relation between air speed and differential pressure in a tube where there is a restrictor.
A kind of relation like V=kP, where V in m/s and P in Pa or mBar and k a constant.
I do not want to try experimentation to determine this relation, I'm looking for theoretical relation to calculate air speed with only a differential manometer with the + connected before the restrictor and the - after it.
In the Darcy's law, there is too much parameters I can't know.

2. Jul 29, 2015

### Staff: Mentor

This is a very poorly posed question. What is the approximate dimensions of the tube (e.g.,length and diameter), and what is the order of magnitude of the air volumetric or mass flow rate? For all we know, the tube is 10 km long, and has a diameter of 1 km. what is the approximate shape of the constriction?

3. Jul 30, 2015

### MARECHAL

Hello Sir,

Could you excuse my English too, I'm a French Man, not so strong in this language.
Yes it is a poor and simple question, but I was voluntary.
I can knowing the diameter of the tube, the min and the max air speed and the pressure before and after the restriction, in fact I came on PF to know if my question was judicious, if it was possible to know a formula more simple than the Darcy's law for my application.
I simply want to verify an experimentation in an air speed bench.
The internal diameter of the tube is D = 100 mm, air speed V is variable from 0,23 m/s to 3 m/s with the first restriction, with no restriction air speed start to 1,5 m/s until 8 m/s.
The bench is a kind of "ring" with two parts ( a high test vein and a low) and closed, a rotating vane anemometer is in the tube after the restriction where the flow is quasi laminar.
There is two fan in the ring I can adjust air speed with a very steady and adjusting voltage.
But my rotating vane anemometer is a mechanic system, I'm not sure my measurement could be in a good repeatability, I would prefer measuring differential pressure before and after restriction.
With no restriction of course, the problem is strong, and probably I would be constrain to put in the tube another kind of restrictor.
If you want I will make a drawing of this bench.

4. Jul 31, 2015

### Staff: Mentor

I must admit that I am struggling with your command of English. Don't feel badly. You know infinitely more English than I know of any other language.

Let me summarize my understanding. You have a tube 10 cm in diameter. The tube is in the shape of a torus. You have two fans situated diametrically across from one another blowing air in the same direction. Air speed is typically 1.5 to 8 m/s without a constriction in the torus, but is only 0.23 to 4 m/s with a constriction present. Presumably, the constriction is half-way between the fans.

Now I'm a little confused. Are you asking how to better measure the air velocity (presumably away from the constriction) than by using an anemometer? Or are you asking how to measure the pressure change across the constriction? Or, are you asking how you can calculate the gas velocity and the pressure change across the constriction without any measurement?

Chet

5. Aug 1, 2015

### MARECHAL

Chet,

Another thank to you.
Without schematic you understand the configuration of the bench so I'm not so bad in English, and rather glad to observe it :-)
Yes I would like to measure air speed in this bench with a delta P instrument, to verify my rotating vane anemometer "tell" the truth about air speed in the tube.
So I'm looking for a "conversion" between differential pressure P (Pa, mbar, hPa etc...) measured before and after the constriction and airspeed (m/s) measured with the rotating vane anemometer, if, of course, there is a relation...
I found on the web a more simple relation than Darcy's law, the relation is Q = (-k.S.(Pb - Pa)) / μ.L where (if I understand) Q is the flow, k permeability of the constriction, S the section of the constriction, (Pb-Pa) the differential pressure upstream and downstream the constriction μ the viscosity of air and L the length of the constriction.
But in this formula I have a unknown quantity : μ.
Of couse, on my bench I can measuring air speed, and differential pressure and determine a relation, but how to compare it with some theoretical law to verify my result?
This is the 'knot" of my problem.

6. Aug 2, 2015

### MARECHAL

Chet,

Excuse my precedent post :
I made a mistake here "But in this formula I have a unknown quantity : μ.", it is not μ but k.

7. Aug 2, 2015

Well part of the problem is that you are trying to apply an equation that does not apply to your physical situation. Darcy's law is for the flow through porous media, like water moving through sand or volcanic stone or something of that sort.

Now, when you are trying to measure airspeed, are you wanting to just figure out the average airspeed in the pipe at a given cross section or at some specific point in that cross section. The speed won't be uniform and will be parabolic with radial position in the pipe.

8. Aug 2, 2015

### MARECHAL

Hi, Sir

Yes Darcy'l Law is for porous media I noticed it, I had imagined the air across the restriction would have the same "behavior" than a liquide in the sand or something else.
The restriction in my system is only fabricate to have a minimum air speed to around 0,2 ms, the low speed of the fan was too important at the minimum voltage, the restriction is composed by a "sandwich" ; plastic straw to do a honeycomb weave in the pipe, a large fiber filter between the other honeycomb weave made again with plastic straw (I'm not sure of the translation of these technical word)
Yes I'm conscient I currently measuring the average of the air speed in the pipe due to the rotating anemometer (the section of the pipe is all integrated by the helical probe),
I would like measured differential pressure at the restriction and then deduce air peed and compare my results with the mechanical anemometer.
Do you think I must stop my investigation, is there any solution for me?
Yesterday I said "never problem, always solution", but I must admit it is not so evident.

9. Aug 2, 2015

I've been assuming that your "restriction" is actually a constriction, but to me, it now sounds like you actually have just a honeycomb of straws placed in your flow and it is simply some obstacle in the flow that induces a pressure drop. Is this more accurate to the situation? If so, then what velocity are you even trying to measure? The velocity will be the same on the upstream and downstream side of the straws due to mass conservation.

10. Aug 2, 2015

### Staff: Mentor

Yes. To add to what boneh3ad is asking, please give more details on the restriction. Does it fill the tube and not permit any bypass, or does it cover only part of the cross section? In your honeycomb, what are the details of the geometry (straw diameters, packing, length)? Maybe darcy's law is not such a bad approximation after all. If the geometry is a simple enough arrangement, then we can predict the value of k.

Chet

11. Aug 3, 2015

### MARECHAL

Ok Sirs,

Wonderful...
The restriction covering all internal section of the pipe or tube, it was made to decrease the velocity and create a steady and laminar flow from the fan to the rest of the ring bench in the pipe/tube section.
Yes, physically, I thought the restriction or constriction was a kind of media in the system, and, as it create upstream and downstream a different pressure, I thought I could measuring this difference to deduce air velocity of the rotating anemometer, and verify the truth air speed.
I must gave you a schematic, but I need to draw it on my computer.
Great thank's to you.

12. Aug 3, 2015

### MARECHAL

Dear Sirs,

Here the schematic of the bench :

#### Attached Files:

• ###### Delta P.pdf
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13. Aug 3, 2015

### Staff: Mentor

Please tell us more details about the geometry of the honeycomb. This is the only way we can have a chance of analyzing in advance the pressure drop - flow rate relationship for the honeycomb (i.e., the permeability k of the honeycomb). But note that, if the geometry is too complicated, you will have to do experiments to calibrate the permeability k. Then you can use that value for other tests. That's what we do when we characterize the flow properties of geological rocks.

Chet

14. Aug 4, 2015

### MrAnchovy

Why do you think that that the calculation of air speed from pressure difference measurements using a whole load of assumptions about the complicated fluid dynamics of the honeycomb (which is affected by a whole load of factors such as turbulence) will be any more accurate than that measured by the vane anemometer?

15. Aug 4, 2015

### MARECHAL

To Chet :

The honeycomb is cylindric, internal diameter : 100 mm is made by many plastic straws (for cocktails) with an internal diameter of 4 mm there are in the tube, the lengh of the two honeycomb with the large fiber filter is 26,5 mm, the large fiber filter is a white synthetic cotton (used for cleaned or filtering aquarium) compressed by the two honeycomb the thickness is 5 mm (average) ( I upload two photos if you want).
Yes probably I must experiment...

16. Aug 4, 2015

### MARECHAL

To Mister Anchovy :

Because the anemometer is a mechanical system, even if it is currently accurate, I'm not sure it was steady in time (dust and usury of bearing).
Because with my project I will have two "source" of measurements of air speed in the tube, I will know with more insurance results, I could compare two measurements, instead of one.

17. Aug 4, 2015

### Staff: Mentor

The permeability k of those honeycombs should be really easy to estimate. If the straws are laid out in an equilateral triangular array, what is the total cross sectional area that can be assigned to each straw?

If you neglect the flow air flow down the cusps between the straws, then you have a set of circular cylinders in parallel. Do you know the Hagen Poiseulle law for average flow velocity through a circular cylinder as a function of the pressure drop (in terms of the diameter and the viscosity of the fluid)?

Chet

18. Aug 4, 2015

### MARECHAL

Hi Chet,

Hoping I understand you :
Straws are laid out in an kind of hexagon for some of there, and cylinder for other, yes I can neglect the flow between the straws.
I do not be able to translate your question " total cross sectional area,..., assigned to each straws".
Yes I saw Poiseulle law but very rapidly, I must looking for it with more attention and come back here.
Do you think we saw the end of the tunnel :-) ? (tunnel for me).
I must thank you again for your patience.

19. Aug 4, 2015

### Staff: Mentor

The side of each equilateral triangle in the array is D, and each triangle contains half a cylinder of diameter D. So there are two triangles of side D per cylinder of diameter D. So, what is the ratio of the cylinder area to the area of the two triangles? (This is also the ratio of the open area of the cylinders to the total cross sectional area of the torus).

Chet

20. Aug 4, 2015

### MrAnchovy

You should research existing wind tunnel designs: those straws look too long to me and what's the point of having the fibre screen between 2 sets of straws?

21. Aug 4, 2015

### MARECHAL

To MrAnchovy:
Yes this bench is home made and currently existing.
Because of the too high air speed of the fan at it minimum voltage, I looked for having around 0,2 m/s at the minimum voltage of the fan, I had the idea to put this filter, this screen, between the two sets of straws only for this reason.
Yes probably the straws are too long, but it was difficult to cut them perfectly to another length, then I thought it was better for having the most laminar flow after this honeycomb...probably a mistake.

22. Aug 4, 2015

### MARECHAL

To Chet,

I need to read and re-read to understand your last post before answering you (if I can), I must tell you my level in geometry is very bad amplified by my precarious English vocabulary.

23. Aug 4, 2015

### MARECHAL

Chet,
If the nominal diameter of the straw is 4 mm, I can deduce the area of each equilateral triangular is : A = ((2.2). √3) / 4 so A = √3 = 1,732 mm2.
But after I don't know how to determine what you name by total cross sectional area. Could you suppose that each straws is composed with six equilateral triangular so the area should be 6.√3 mm2 ?

24. Aug 4, 2015

### Staff: Mentor

The side of the equilateral triangle joining the centers of three straws in contact within the triangular array is equal to the diameter of a straw D. This equilateral triangle contains just half a straw of diameter D. So there are two triangles per straw. The area of one triangle is $\frac{D^2}{4}\sqrt{3}$, so the area of two triangles is $\frac{D^2}{2}\sqrt{3}$. The flow area of a straw is $\pi\frac{D^2}{4}$. So the ratio of the flow area to the overall cross sectional area is $\frac{\pi}{2\sqrt{3}}=0.907$.

For laminar flow in a single tube, the average velocity down the tube is given by:

$$\bar{v}=\frac{D^2}{32\mu}\frac{Δp}{L}$$

The superficial velocity v (based on the total cross section) is related to the average velocity down the tube by:
$$v=0.907\bar{v}$$
So, in terms of the superficial velocity, we obtain:
$$v=0.907\frac{D^2}{32\mu}\frac{Δp}{L}$$
From this, it follows that the permeability of the honeycomb k is related to the diameter of a straw D by:
$$k=0.907\frac{D^2}{32}$$

Chet

25. Aug 5, 2015

Chet,