the Inertance valve

[WebPilot]

Dynamics of valve action due to pure fluid inertance

Fresh from the success of my dynamic modeling of a strip valve thread, I now turn my attention to the pfi valve. Yes, I am "on a roll".

Based on how a tube is used, it can be modeled via 3 differing dynamic fluid elements:

1. I, inertance
2. R, resistance
3. C, capacitance

Pulse combustors have been built in the past using non-linear resistances attached to a tube. By non-linear resistances I mean, more flow restriction in one direction than the other. This thread will use typical fluid resistances and identify the contribution of the other two elements for valve action.

An approximate answer to an engineering problem is often the best answer if the degree of approximation is known and a more precise answer is not really needed.

In the spirit of this, let's begin.

[WebPilot]

Here is my mad-scientist model of one tube and its

1. pure fluid resistance,
2. pure fluid inertance, and
3. A-type source, which is only a technical name for a device that can cause a sinusoidally varying pressure differential between the closed and open ends of the tube. The source has a 'knob' that can be turned to change the signal frequency.

[larry cottrill]

Now you're talking, Forrest! If I had the integrals, I could set it up on my 1960 vintage Heathkit Analog Computer. Ha. (I would need to get hold of a couple of 'tubes' of an entirely different kind to make this work. I'll bet they'd be expensive nowadays, too. Also, the machine only has nine op amps, which is pretty sparse.) On the analog computer, you really DO have knobs to change things. I used to do artillery trajectories, gradually changing the fluid medium from vacuum to syrup with just a slow turn of the knob. What fun! - ha, again.

Of course, in reality, I can't remember a thing about setting up integrals except for the simplest possible form, i.e. an input resistor and a feedback capacitor. But with a "network" of passive components in and about the feedback path, some very complex things (e.g. electric motors) can be accurately simulated. It's all in knowing how, of course -- and alas, there aren't many textbooks being written nowadays on analog computing.

One thing I don't get right offhand is the meaning of the + sign atop your "signal generator".

All wishful thinking / reminiscing aside, I really do like this kind of approach. Very cool.

L Cottrill

[WebPilot]

It's all in knowing how, of course -- and alas, there aren't many textbooks being written nowadays on analog computing.

One thing I don't get right offhand is the meaning of the + sign atop your "signal generator".

New books, are maybe non-existent. I have a couple, though of the old ones. As the new stuff replaces the old, people are throwing away the old. So the books, and the equipment could show up 'anywhere' if you are a 'rummager'. I picked up a Tektronix dual trace o'scope with manuals and its rack on wheels for $20US. It works.

The +tive sign is just a convention for the source; it shows which way the 'current' flows.

continuing ...

The above system is linear, so an analytical solution can be determined by hand. There is no need to program a numerical solution, neither digital nor analog. It took me about 2/3rds of a legal sheet of paper. I used what is known as the Laplace Transform method.

Naturally, I 'punched' that 'bad boy' equation into a spreadsheet for a 'quick look see' of its predictions.

The result is a function of six variables. The proper way is probably to use 'dimensional analysis'. However, here is the first of three plots to give the reader an idea of what goes on when one ratio is varied, and the others held constant.



Notice:

1. Again, here is the familiar 'bell-shaped' flow curve, that I showed you in my dynamic modeling of a strip valve thread.
2. Observe the ratio of resistance to inertance for this model.
3. Unfortunately, for this set of parameters, the flow continues out of the tube for the entire cycle. It never goes -tive therefore it never re-enters the tube.
4. This may be good for an 'out flow' tube but certainly not for 'in-flow'.

[WebPilot]

Notice for this ratio:

1. flow 'jumps' to a far greater amplitude than the previous one.
2. the flow is just beginning to go -tive at the very end of the cycle (cannot see this clearly due to scale used), so 'a bit' of re-entrant flow is beginning to occur
3. best suited for an 'outflow' tube.

[WebPilot]

And finally, for this ratio:

1. the out flow amplitude has been seriously attenuated.
2. the flow becomes -tive at end of cycle (see arrow), indicating flow reversal and re-entry.
3. approaching behavior best suited for an 'inflow' tube, naturally.
4. note in all cases, flow lags pressure by 90°.

[GRIM]

Hi Forrest

And finally, for this ratio:

I was horrified to read "Finally" , NOOOOO, there must be more

0.08 ,0.998, &18.0 are big steps ,

note in all cases, flow lags pressure by 90°.

Because you plotted it that way ?, or is there some cunning reason?

fascinating , please continue ,

[WebPilot]

Hi GRIM,

I am glad to read you are enjoying the thread.

Flow lags pressure by 90° or pressure leads flow by 90°.

This is a property of an RL circuit excited by a sinusoidal voltage in electricity/electronics. It's in textbooks or on the web and can be verified experimentally in the lab.

It also seems to be a property of a pulse combustor so, this is a plausible explanation.

Here is a comparison of the response of each of the three tubes taken over 3 cycles. In order to show more clearly the flow re-entrance, I've done a plot 'ZOOM in' .

[WebPilot]

The previous included the effect of a transient. If you look at the last plot, you may just detect an overall shifting down and to the right of the max/min amplitudes.

It's no BIGGIE. It just takes time in order for the system to settle down to a steady, but oscillating condition.

Let's look at this, next.

[WebPilot]

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This is what I mean by transient decay as the system response approaches the steady state oscillation.

[WebPilot]

For this group of tubes, after t=7.00 secs have elapsed,

1. tubes 2 and 3 have reached their steady state amplitudes (mean is 0).
2. tube 1 has not (mean flow is not 0).
3. source pressure included on plot to show differences in phases.

[WebPilot]

After almost a minute from starting, Q1 'settles down', too.

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Data for the 3 tubes I used.


view full image

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The previous two dynamic system elements are relatively easy to model for a slender tube. Since we are working with air, and it is compressible, fluidic capacitance turns out to be non-linear.

There's no need to fear ... Underdog is here!

PS I just saw the movie for the first time a couple of days ago. I just loved to see the 'doughnut' and 'jet trail' behind the 'flying beagle' as he accelerates across the sky.

I tried to draw just that over a month ago on my the Illustrated pulse jet in 3D thread. It's hard to get it 'just right'.