The Cybernetics of a Pulse Combustor

[WebPilot]

The Cybernetics of a Pulse Combustor

A while back, I obtained a copy of this book, "Cybernetics - Control and Communication in the Animal and the Machine" (Wiener 1948). Little did I know then, I would find an application for it, now.

If you're unsure what this science is go here:

http://www.scholarpedia.org/article/Cybernetics

Wiener himself emphasized the role of feedback mechanisms in the goal-oriented systems.
.....
Feedback control is an invisible thread in the history of technology, as studies on mechanical clocks, steam engines, aerodynamic and electronic devices show (Bernstein 2002). A device designed to measure and regulate the speed of a machine, called 'governor' was successfully built by James Watt. Interestingly, James Clerk Maxwell had a paper "On Governors" (Maxwell 1868), in which he anticipated some aspects of modern control engineering (Mayr 1971).

Surely, you've all wondered why a pulse combustor oscillates? Or wondered why when you first ignite it, its intensity increases or decreases until it seems to 'lock in' to some equilibrium running point?

There is a 'governor', or a 'feedback mechanism' at work in a pulse combustor and I think, I have discovered one. This 'invisible thread' makes my mathematical model 'lock in' to one of two stable limit cycles.

If there is interest, I shall continue ...



(c.t.)
2Mike-3a.gif

[Viv]

Hi Forrest

Your posts are always interesting even when its just about our other favorite topic cars and motorcycles

This book rings a bell with me but I don't recall ever reading it, it could be it was covered at college as we did control theory when I originally studied electronics, maybe this guys work came up for the maths section.

Viv

[WebPilot]

Hi Viv,

We've known each other for 8 years and if we lived closer to each other, we'd be "buds". Nice maserati.

It is not necessary for this discussion to get a copy of Wiener's book. I just mentioned it since it had a profound influence on me - as did Gleick's Chaos.

continuing with my discussion ...

It is possible to estimate how much energy is added and lost per cycle.



(c.t.)
cyber-1.gif

[Irvine.J]

Too many connections to the image! Please continue forest!

[WebPilot]

Hi Irvine,

I am very pleased to read your response.

continuing ...

Two things the reader must keep in mind:

1. delta EZ, or combustion, excites the vibration but
2. delta ED, gas (air/fuel) flow through the valves when the system (gauge) pressure is -tive , damps the vibration.

It's the yin and yang of pulse combustion.

[WebPilot]

(c.t.)
cyber-2.gif

This is a modification from the previous plot.

I have drawn 2 excitation curves (one red and one blue) and 1 damping curve. Notice that either of the excitations are above the damping - at least until they intersect with it. For the present discussion, please disregard the upper blue curve.

Suppose at some point in time the pressure amplitude is at a1. Notice from the graph, the damping energy, delta E_D, is less than that of the excitation, Delta E_Z. The pulse combustor pressure amplitude will continually rise as it cycles and will eventually reach the equilibrium point, S.

Now suppose at some other point in time the pressure amplitude is at a2. Notice from the graph, the damping energy, delta E_D, is greater than that of excitation, Delta E_Z. The pulse combustor pressure amplitude will continually diminish as it cycles and it too, will eventually reach the equilibrium point, S.


Point, S, is a stable equilibrium point. When the pulse combustor amplitude reaches this amplitude, the combustor is at a 'limit cycle'. Any deviation in pressure amplitude will return it to this point - at least, as long as there is fuel.

The energy curves being in this orientation allows their point of intersection to act as a 'governor' or 'feedback mechanism' - phenomena investigated by Norbert Wiener, James Watt and James Clerk Maxwell long ago.

>PS Norbert Wiener, mathematician, engineer and social philosopher, coined the word "cybernetics" from the Greek word meaning "steersman."

>PPS We all know of another governor - that on a 3-5 HP Briggs and Stratton or Tecumseh engine powering our lawn mowers or portable AC generators.

[WebPilot]

There is a physical 'entity' which I did not model at first, that does exactly as described in the previous. I now get stable limit cycles as predicted.

I shall now show you a series of progressions to the final result.

A little about my model:

• It has a little bit of fuel to start out with.
• If the flame goes out due to fuel starvation, it can 'restart' as soon as the fuel re-enters the chamber during the induction part of the cycle.
• I have to give it an initial pressure to start it - sound familiar?
• The following plots show where it is burning and where it is not while it cycles.

Here is my first phase diagram again. You will notice the burn rate is too high. It quickly 'burned up' what fuel it was initially allotted, and struggles to get more fuel into the chamber to keep it running.

It finally quits since its pressure amplitude is too small to 'suck in' anymore fuel.

[WebPilot]

This is still burning too much fuel per cycle, but there is much improvement over the last.

The simulation starts at center, spirals clockwise rapidly to the outside ring, begins to run out of fuel (note gray areas), orbit decreases until it runs out of simulation time.

[Mike Everman]

Faaaaaascinating, Forrrrest! Welcome back, BTW.

[WebPilot]

Hi Mike,

Thanks. It's good to be back. I need to document this while it is still somewhat fresh in my memory. I am already suffering from at times "what the frell was I thinking?"



Here is the first of three images of my 1st pc limit cycle. It still 'tickles' me to see her; it took me so long to write and troubleshoot the code in order to find it. This is my baby - sorry, just being a proud father.

Particulars:

• I gave it an initial 'kick' - starting pressure is less than final limit cycle pressure
• it used some of the initial starting fuel to reach the point where the valves open and it sucks in additional air/fuel mixture
• it reaches final limit cycle amplitude and stays there!
• note how well the mass fuel curves behave
• there are no gray lines in the upper plot - during the simulation it never runs out of fuel!

[Viv]

Hi Viv,

We've known each other for 8 years and if we lived closer to each other, we'd be "buds". Nice maserati.

Hi Forrest

Time flies and yes I think we would be friends after all this I will be planning a vacation down south of here for next year as the new project companies will be running autonomous by then and I wont have to spend so much of my time on them, there are a few people I would love to meet while in the US so we will make a list and plan an itinerary, I really would like to do a few custom car and bike shows while down there

The Maserati ? I am having a joke with the corporate management company I use on the projects about how many Maserati's I want, a black coupe sport for the weekends and a sensible silver Quattroporte for the weekdays.

Currently I am having trouble finding time to keep up with your threads and just keep coming up with new questions as you post the next installment, I love the implications for valve design though.

Viv

[WebPilot]

Hi Viv,

For sure, keep in touch. Bring that Maserati down.

Here in June, July, August there are big town 'car cruises' every year. In the 'burgh there is a a vintage Grand Prix in July. There is a big car/motorcycle racetrack nearby called BeaveRun. Their schedule of events is on the 'net. Then in mid spring there is the Pittsburgh air show and on Labor Day weekend there is the Cleveland air show. That's up the road from me, but I'd go. Pick and choose.

continuing with my discussion ...



Here is the second of three images of my 1st pc limit cycle. It still 'tickles' me to see this, too.

Particulars:

• I knew already what the final limit cycle pressure was, so I gave it an initial 'kick' at that pressure
• the oscillation remained at that amplitude
• it used little (if any) of the initial fuel/air and (i.e. the mean of the reactant mass line stays horizontal)
• look how well the mass burnt and mass inflow lines overlap each other!

[WebPilot]

Click here to see image in its entirety.

Last, but not least, here is the third and final of the three images of my 1st pc limit cycle. Yes, I'm still 'tickled'.

Particulars:
[*]I gave this one an initial 'kick' at a pressure greater than the final limit cycle. The orbit decayed directly to it.
[*]this 'sucked in' a little fuel as it started at a greater amplitude, but then settled down. This explains the 'rise' in the reactant mass line.
[*]the mass burnt and mass inflow lines are parallel.

In summary, I started this oscillator at amplitudes above, below and at the final limit cycle amplitude and it behaved admirably - just as it should and as described in my earlier diagram and discussion on 'feedback control'.

No brag ... just fact.

[WebPilot]

I need to clarify something I had posted here. It will have to wait until I return. Stay tuned.

Until I do, enjoy this. It does have something to do with my analysis.



Click here to view full image.

[WebPilot]

I have formulated an expression whose solution actually dictates the number of stable limit cycles (SLC) that exist for a given pulse combustor configuration. I found this to be quite a remarkable discovery.

The solution depends on a certain parameter. For further discussion, I shall refer to it as a 'knob'.

In the following plot, a stable limit cycle exists where the red and green curves cross. I have included on this plot the difference between these two curves only as an aid to numerical solution.

Upon inspection one will see that there are two intersections. Thus, there are two SLC's.