How to Calculate Pipe Oscillation Frequency with Conical Nozzle?

[Graham C. Williams]

Hi.
I'm new and hope you can point me in the right direction. I'm not too sure how to write expressions here. Have to do some tests. So I've attached a *.doc file that outlines the problem and what I'm looking for. I'm too sure how this all works so I might have to try again.

In short it's an expression to calculate the frequency of oscillation of a pipe with conical nozzle on the end.

Thanks in advance.
Graham.
P.S. I have a degree in Physics with Maths, but that was 25years ago and I'm more than a bit rusty now.

 

[Graham C. Williams]

Lets's see if this works.
Graham.

 

[Graham C. Williams]

We have Joy - The Doc File is below.

Hot Organ pipe resonance.

Background.
The general equation for the resonant frequencies of a straight pipe open at one end and closed at the other is approximately given by.

F= (2n-1) C/4(L+ Le ) (1)

For the straight pipe open at both ends we have.

F= nC /2 (L + Le ) (2)

: n = 1,2,3... C = velocity of sound at the reference temperature.
L = Length of pipe. Le = Total end correction

(Please I don't want to get into solutions of the wave equation unless it's absolutely necessary or can be easily done within a spreadsheet)

OK Let's add some heat.

If the whole pipe is hot, the speed of sound C in (1) and (2) can be corrected with the equation.

√(γa Ta / γb Tb ) (3)
: a is the ambient gas condition and
b is the hot gas condition
T is absolute temperature degrees K
γ is the ratio of specific heats and γa/ γb = 1

Where only part of the pipe is hot it would be better to correct the hot length of pipe to its equivalent length of cold pipe at the reference temperature Ta.
In this case (2) above would look like:

F = nC/ 2[ Lc + Le + (√(γa Ta / γb Tb )) * (Lh)]

: Lc is the cold portion of the pipe.
Le is the total end correction.
Lh is the hot length of pipe.

(Infact I would probably want to look at both types of pipe with the length corrected for ambient conditions, but this is an aside.)

That's the background. Now my problem - Can you help?

I would like to obtain expressions that will approximately predict the resonant frequency of either pipe type (acoustically open at both ends or acoustically open at one end only) when a conical nozzle is placed at the end(s).

Example 1.
A pipe of Length L and Dia. D is open at one end and closed at the other.
On the open end of this pipe is fixed a conical nozzle of half angle Θ and exit dia. d : D>d.
The question in this case would be.
What will be the Fundamental resonant frequency and the maximum allowed harmonic of this pipe under the following conditions?
1) With the whole pipe at ambient or some elevated temperature
2) With some fraction of the pipe at ambient and the rest at a higher temperature.

Example 2.
A pipe of Length L and Dia. D is open at both ends.
On one open end of this pipe is fixed a conical nozzle of half angle Θ'and exit dia d' and on the other open end is another conical nozzle of half angle Θ'' and exit dia. d''. The half angles are different as are the exit diameters.

The question is the same as in the example above.
What will be the Fundamental resonate frequency and the maximum allowed harmonic of this pipe under the following conditions?
1) With the whole pipe at ambient or some elevated temperature
2) With some fraction of the pipe at ambient and the rest at a higher temperature.



So, as I see it at the moment the expressions should take into account:
Temperature changes along the pipe. More correctly a temperature gradient.
The effect of the conical nozzle in terms of half angle and outlet / Inlet dia.
The effective cutting off of the upper harmonics as a result of the restricted exit diameter or the length to diameter ratio when the half angle is zero.
The gas velocity will have an effect upon this, Assume it's small.
Easily calculated in a spreadsheet (Excel).

I hope I've made things clear.
If anyone would like further Information/Explanation please contact me.
This equation is needed for an amateur project, not commercial, looking at burner oscillations in pipes.

Best Regards
Graham.

 

[sheldon]

Maybe this page will help you
http://www.physics.brown.edu/Studies/Demo/waves/waves.html

 

[Graham C. Williams]

Many Thanks.
Graham.