# Porta's Derivation of Lempor Theory

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• #5175
Chris Corney
Participant

Martin, I’m probably not the best person to discuss this with, because I am agreeing with everything you say!. I guess the angle comes from interpretation (or otherwise!) of Porta’s Lempor paper.
My background is as an electrical engineer, but I’m happy to pick up any snippets of wisdom that I can on this subject. I have my doubts about the turbulent entrainment theory, which seems to suggest the more turbulence you produce the better the entrainment will be.
Regarding the geometry I’ve been toying with idea of the nozzles angled to produce a hyperboloid steam jet, possibly feeding into a hyperboloid chimney.
https://en.wikipedia.org/wiki/Hyperboloid_structure
How would you make one? It’s possible to produce a casting from a polystyrene pattern, and you can cut polystyrene using a hot wire, set to the appropriate angle to give a hyperboloid.
If you then wanted to improve the surface finish you could electrochemically machine the surface using an electrode made from wires arranged in a hyperboloid.
I once found a patent from the 1950’s describing a hyperboloid ejector, but the internet link has now disappeared. If anyone is interested in making one I’d be happy to help with the mathematics.
In your experience have you come across anything like this?
Regards
Chris

#5176
Martin Johnson
Participant

Hello Chris,
Hyperboloids – an interesting concept. As to manufacture- for chimney or nozzles the internal form woud these days be cut on a CNC lathe, feed in the co-ordinates and there you are. For those of us working without CNC you carefully mark out or otherwise cut a form tool of the correct form and use that in a lathe.

I take it you also mean the nozzles would be arranged pointing in a “helix”. Producing a nozzle block with seatings for such nozzles is also reasonably easy to manufacture – really no more difficult than the splaying out nozzles of the S160 exhaust shown on this site.

Benefits – they would be quite subtle I think, but you do get more path length of the steam jet for a given height before entering the mixing chamber; so it helps to solve that eternal fight against vertical height to keep within loading gauge.

Martin

#5177
Chris Corney
Participant

Hello Martin,
No, a helix is the shape of a coil spring. When steam leaves the nozzle it will travel in a straight line.
An example of a hyperboloid is the traditional concrete power station cooling tower, where the steel reinforcing bars travel in a straight line from the base to the top of the tower, inclined at the appropriate angle. Other examples are in the pictures in the Wikipedia page above.
Regards
Chris

#5178
Chris Corney
Participant

Martin,

Further to the post above, I was thinking that the convergent-divergent shape of a hyperboloid might be of advantage when designing an ejector.
A chimney would have to be machined internally, and I have been advised that the typical reach of a profile follower attachment is about 125mm.
The exhaust on the S160 was designed before I joined the group, and I understand that the nozzle angle was carefully considered. The exhaust certainly performs succesfully.
Regards
Chris

#5179
Martin Johnson
Participant

Hi chris,
Shape of Mixing Chamber – I am currently reading a paper on air ejectors by LJ Kastner & JR Spooner on air ejector design. They make the point that you can design (and calculate) on the assumption of constant pressure mixing (which gives a reducing / expanding chamber) or constant area (parallel chamber) or indeed any combination of the two. They make no conclusions about which might give better efficiency. The paper can be found here:

https://journals.sagepub.com/doi/abs/10.1243/PIME_PROC_1950_162_021_02

It is interesting in other respects:
Convergent – divergent driving nozzles were not found to be best for transonic flow in the driving nozzle.
The theory advanced takes account of increasing entropy in the driving and mixing sections (which Koopman & Porta do not)
It confirms that a diffuser taper of 5 to 10 degrees included is the right range – even with the highly disturbed entry flows.

Chimney Machining – What size chimney are we talking about? If full size, I would suggest casting would be the way to go for complex shapes. That being the case, the problem resolves into making the corresponding male shape which is easy. For model work, boring bars are best if kept to a length / diameter ratio of 5 to 6 max. Even in my wee shop, my biggest boring bar is 25 diameter, giving about 6″ of reach, but remember that reach goes up with the size of hole being tackled.

Martin

#5180
Chris Corney
Participant

Hi Martin,

There was a paper recently which Jos Koopmans mentioned on the National Preservation forum (I’ll try to find it later). From this I drew the conclusions that a convergent mixing chamber could be shorter than a parallel type, with similar performance. This would be useful for railway applications.
I’m just trying to interpret your second point with respect to the principle of de Laval nozzles, which have been around for eons. I’m thinking that if transonic flow occurred in a de Laval nozzle it would probably just create shocks.
For transonic flow in diffusers, Prof. Eames at Nottingham University has done some work on CRMC theory (constant rate of momentum change), with one or two papers availableon the internet; you may find these interesting.
While various arrangements of ejectors using conical and parallel sections are proposed, I’m wondering if a smooth curve would be better. Hence my suggestion of a hyperboloid. CRMC theory also proposes a curve, although it is a different curve and potentially more difficult to manufacture than a hyperboloid.
For chimney machining we are thinking of miniature up to standard gauges, although I’m thinking that a de Laval nozzle with a CRMC divergent section could theoretically have some advantages with pulsating flows.
We discussed castings with complex curves in the group a while ago, and someone asked how you would go about improving the surface finish of a casting.
There are a number of miniature locomotives running with saturated boilers and exhausts with de Laval nozzles, and I can’t help wondering how these would operate with wet steam. If there is a sharp drop in pressure in the nozzle, does this just cause an appreciable amount of condensation?
Chris

#5181
Chris Corney
Participant

Martin,

The paper mentioned in the post above is “Optimum control of diffuser shapes for non-uniform flow” by G.P.Bentham, I.J.Hewitt, C.P. Please and P.A.D. Bird.

Chris

#5280
Martin Johnson
Participant

I have found the answer to my original question and it is this:
Porta’s formula is derived from Strahl’s which is derived from conservation of energy.
The more usual formulation (such as ESDU) is derived from a consideration of momentum conservation.
Prof. Bill Hall points out that in a system such as a loco front end, the losses are so great that working from energy conservation means any minor errors in determining the energy input less the energy lost in “shock losses” means major errors in the remainder (which is what we are interested in). That is why Hall (and others) favour the momentum approach.

It does beg the questions – what is the mathematical difference between the two approaches? Which gives the “safer” design? And is the difference significant for the typical quantities experienced in a loco front end?
I am not aware that anybody has worked through those questions.

Martin

#5282
Chris Corney
Participant

Hi Martin,
I hope I’ve understood your question correctly, let me know if I’ve missed the point.
My take on this is that it case of “horses for courses”
Conservation of momentum is one of the fundamental laws of physics. To change the momentum of a body it is necessary to apply a force to it. Every force has an equal and opposite reaction, so the body to which the reaction force applies will experience an equal and opposite change in momentum. The total momentum of the system remains constant.
So if you are mixing fluids with different velocities, the overall momentum will remain constant. however the kinetic energy of the mixture will be less that the combined KE of the constituent parts. Therefore it makes sense to use the momentum equation.
If a fluid passes through a nozzle of diffuser, the walls of the nozzle/diffuser will apply a force to the fluid and the momentum of the fluid will change. However, if the nozzle/diffuser is reasonably efficient, the energy of the fluid will remain reasonable constant, based on Bernoulli’s equation, and in this case it makes more sense to use conservation of energy.

Chris

#5293
Martin Johnson
Participant

In the case of a loco front end, the efficiency is actually miniscule when viewed as a pump. So any attempt at using energy as the main consideration means energy in is a large number – probably known. Losses are a large number but can be estimated (more or less!). The output is a small number, being the energy in minus the losses. However, any error in the other two terms will cause huge errors in the output.

Typical numbers for a Black 5, expressed in megapascals – 340 energy in. 30 out, hence losses in the order of 310.

One has a similar problem with the momentum approach, but are not faced with trying to estimate loss terms for sharp changes in velocity (Particularly Terms 2 & 3 of equation 1 in Porta’s 1974 paper). Here, momentum expressed in megapascals – 210 momentum in, about 30 out.

As far as I can see, the energy approach will give a smaller blast pipe orifice just running some typical numbers, but if you work it all back the difference between approaches is quite small (about 5% on diameter for the Black 5 example). Hence the reason why the Strahl / Porta approach “works”.

Martin

#5294
Chris Corney
Participant

My usual approach with a very inefficient device such as this is to try and identify the losses and then see if there is anything that can be done to reduce them. Obviously it’s necessary to stay within the loading gauge and – these days – limitations imposed by the heritage appearance.
Presumably the kinetic energy of the gases leaving the chimney is included in the losses, but there is not much that can be done about this.
Other losses that spring to mind are
– mixing losses due to steam and smokebox gases at different velocities.
– supersonic shock losses
– internal friction.
As you can tell from previous discussions I’ve been trying to understand how a Kylchap works. The double Kylchap successfully draughts the boiler but with a much larger orifice area, and lower back pressure than the single chimney it replaced, so there must be something beneficial going on there!

Chris

#5295
Martin Johnson
Participant

Stop press.
I have just found a mathematical error in my comparison of momentum and energy approaches. (Missed a divisor of 2 in one of the energy terms!).

What I now have is exact correlation between the momentum approach and the energy approach as set out by Porta. (Which he does hint at in his 1974 paper, but gives no proof.) At present this is limited to an analysis of a parallel chimney, since my aim was to investigate differences in approach, and tapered chimneys complicate matters dramatically.

So, it doesn’t matter which way you go, it comes to the same answer.

Martin
(One happy bunny)

#5296
Chris Corney
Participant

Hi Martin,
That’s good news.
I’m wondering if it would be worth writing up your findings for future reference by other members of the group.
Regards
Chris

#5303
Martin Johnson
Participant

I’m wondering if it would be worth writing up your findings for future reference by other members of the group.

I’m working on it Chris, but these things grind awful slow when you have a steam engine to build.

Martin

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