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Uniflow design theory

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Anatol

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Hello Everyone
I have been thinking a lot about uniflow engines and I have some questions. In the literature, it is argued that uniflow engines derive efficiency from the fact that the exhaust exits at BCD and so does not cool the cylinder head/inlet ports.
But - whatever 'exhaust' is not exhausted close to BDC is recompressed, and by the time the piston is at TDC, the pressure is/can be greater than supply steam pressure, which means that when the inlet valve opens, compressed exhaust steam is pushed back down the inlet passage, until piston has dropped far enough to draw the exhaust back in to the cylinder and then be followed by fresh steam. this is clearly inefficient in two ways - recompressing exhaust takes energy, and the delay in getting fresh stream also seems undesirable.
Is my understanding correct?
I know some Uniflow engines have/had relief valves in the cylinder head - this would allow the remaining compressed exhaust gas to be exhausted (but only after it had been compressed and at the cost of cooling the head (?).
What kind of valves were these - passive spring loaded or cam actuated...?
I've seen some double acting single cylinder designs - are these preferable to single acting types?
thanks for any input!
 

Anatol

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Thanks Dave - I've been to the wikipedia page, and other sources.

"The steam remaining within the cylinder after the exhaust ports are closed is trapped, and this trapped steam is compressed by the returning piston. This is thermodynamically desirable as it preheats the hot end of the cylinder before the admission of steam. However, the risk of excessive compression often results in small auxiliary exhaust ports being included at the cylinder heads."

"This is thermodynamically desirable" I'm dubious about this, the devil, as always, in in the details, and I'm not a thermodynamics grad. In any case, "the risk of excessive compression" is not just an explosion risk, its also - it seems also highly ineffficient, but the addition of " small auxiliary exhaust ports" seems to admit defeat.

"To gain the maximum potential work from the engine a high reciprocation rate is required, typically 80% faster than a double-acting counterflow type engine. This causes the opening times of the inlet valves to be very short, putting great strain on a delicate mechanical part" This is also a concern. Though IC poppet valves are now highly developed.
 

Asm109

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Your first concern about recompressed pressure being higher than incoming steam is baseless. It would be a violation of the second law of thermo for it to happen.
Hot High pressure steam is admitted to cylinder. Steam is expanded, doing work and cools down. Now cool steam is vented at bottom. Now a small portion of the cool steam is trapped and compressed, heating up and requiring some work ( inefficiency) on it. You started with a small volume of hot high pressure steam. Through some away and recompressed, you cannot get back up to where you started.
 

Charles Lamont

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I have just had a look at that Wikipedia article. It needs work. The parts you are quoting contain no references. Some of it is just wrong.

The inlet valve gear of a uniflow engine is no different from any other trip gear, except that it is often working a double-beat poppet valve, which will likely be lighter and subject to less friction than a comparable Corliss valve. It is true that while any engine is likely to be thermodynamically more efficient at a high piston speed, trip gear is not suitable for very high speeds.

Compared with a simple slide-valve engine running at the same speed, a uniflow engine needs a bigger cylinder swept volume to produce the same power output, but it will do so using less steam. It will be more efficient.

The relief valves may be primarily for clearing condensate during warm-up - to avoid a hydraulic lock.

Asm109 - Nice try but no cigar. It is possible to have excessive compression. Typically a uniflow's inlet valve will close at say 5% of piston stroke, but the clearance volume the exhaust steam is compressed into at dead centre is far less. This can be accommodated by designing a sufficiently large clearance volume. However, providing auxilliary exhaust valves allows a smaller quantity of exhuast steam to be compressed into a smaller clearance volume, with less negative work done on it. This improves thermodynamic efficiency at the expense of greater mechanical complexity.
 

Anatol

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Hi Asm
with respect, I am not persuaded. Forget the steam for a minute. Imagine cylinder at BDC, full of air at 1 atmosphere pressure. Now compress that volume, from, say 10 cu in to 1 cu in, then to 1/10th cu in, at some point you'll get a pressure higher than supply steam pressure, regardless of temperature.
That recompression is doing work, taking energy from the engine output. Of course, compressing poppet valve return springs is also doing work and taking energy. I don't know the relative scale of the losses.
 

Anatol

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Charles Lamont - I'd drafted the revised reply but did not realise I'd omitted to send it :(
Thanks for your reply.
"double-beat poppet valve" - Can you elaborate? I looked at the wikipedia page but it was not that helpful.
"Compared with a simple slide-valve engine running at the same speed, a uniflow engine needs a bigger cylinder swept volume to produce the same power output, but it will do so using less steam. It will be more efficient."

thankyou, very informative, (cylinder swept volume = (roughly) stroke?)

"The relief valves may be primarily for clearing condensate during warm-up - to avoid a hydraulic lock."
so are you saying that the energy lost to recomperession is negligible?

"Typically a uniflow's inlet valve will close at say 5% of piston stroke"
wpiudl it open before TDC? If so, would the recompressed exhaust effectively prevent inflow of supply steam till piston moved down?

"This can be accommodated by designing a sufficiently large clearance volume"
this being space in cylinder head where piston does not go? Hoe large would a "sufficiently large clearance volume" be, as percentage of total cylinder volume? 5%? 10%?

thanks!
 

Charles Lamont

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Anatol

You are right that the compression part of the cycle requires work, but that work brings the exhaust steam up to a state closer to that in the steam chest, so that when the inlet valve there is no sudden inrush of steam., which would be wasteful.

'Convincing' you qualitatively may be difficult. You are unlikely to grasp this stuff properly without some study of thermodynamics.

Cylinder swept volume is the area of the piston times the stroke.

No, I am saying I think the relief valves you say you have seen probably don't serve the purpose you think they do.

Yes the inlet valve starts to open before dead center, to ensure that it is sufficiently widely open to admit the necessary amount of steam freely without loss through "wire drawing" when the piston starts to gather speed.

Size of clearance volume, I don't know off the top of my head, and I need to get some breakfast, but nearer 5%..

Haven't we already been through all this at: Designing steam engines ?
 

Richard Hed

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Hello Everyone
I have been thinking a lot about uniflow engines and I have some questions. In the literature, it is argued that uniflow engines derive efficiency from the fact that the exhaust exits at BCD and so does not cool the cylinder head/inlet ports.
But - whatever 'exhaust' is not exhausted close to BDC is recompressed, and by the time the piston is at TDC, the pressure is/can be greater than supply steam pressure, which means that when the inlet valve opens, compressed exhaust steam is pushed back down the inlet passage, until piston has dropped far enough to draw the exhaust back in to the cylinder and then be followed by fresh steam. this is clearly inefficient in two ways - recompressing exhaust takes energy, and the delay in getting fresh stream also seems undesirable.
Is my understanding correct?
I know some Uniflow engines have/had relief valves in the cylinder head - this would allow the remaining compressed exhaust gas to be exhausted (but only after it had been compressed and at the cost of cooling the head (?).
What kind of valves were these - passive spring loaded or cam actuated...?
I've seen some double acting single cylinder designs - are these preferable to single acting types?
thanks for any input!
that is not possible. To have a return pressure greater than the steam pressure would be going against the law of conservation of energy. (caveats with that of course.) If you couldn't exhaust at least 90% of the used steam, my guess, is that any machine would not work at all or at least very poorly. However, examine the exhaust pports of other types of engines, particularly the slide valve, on so many of the little engines: the steam enters the slide and the valve moves in such a way that it goes into the cylinder body thru a tiny orifice almost at the center of the part! (This is really a poor design.) then the steam follows a pathway to the end of that body to the inside top (or bottom) of the actual cylinder where the piston resides. The steam pushes the piston. When the piston reaches near or at BDC, all that steam, that is, the same amount that went in weight wise, has to return thru the same pathway that allowed it in. Now that pathway has not enlarged but the volume of steam is now about 10 or more times as large under lower pressure--how does it escape quickly enough to not cause a similar problem to what you are talking about? Well, inside the valve, the escape port is a bit bigger, which helps, and the time for the valve to be open is a bit longer.

The steam manages to get out thru the same passageway it came in! This causes serious problems many to do with the size and shape of the tunnel system in these little engines: turbulence is the bane of moving any fluid quickly and with out loss of velocity thru friction. The least amount of friction caused by any sort of conveyance is a round, polished surface. The reason? a circle has the least "surface" for any given "volume" and I don't have to explain why to polish. So imagine a square thru hole for a moment, a square is the shape that has four sides and has the least amount of "surface" for it's volume, approximating a circle (that is better than a rectangular form). Now when that form is a tube, the corners get terrible turbulence, sukking up a lot of energy and slowing down the fluid.(This is particularly important in foundry work in the green sand.) It is, however, easier to make a round hole thru metal than a square one. (I had some square drill bits but they all broke, dang!) but there is more to it than that. There are size constraints and possibly the need to have a minimum "volume" which means one has to squish that round hole down to an oval. Well, ovals are harder to drill than square holes are. (I was hoping some clever manufacturer would come up with oval drills.) So what happens is a series of round holes are drilled in beside each other till you get an approximation of the size you needs. YOu can chip out the remaining peices or mill them. So after a "square hole" the next best thing is a pentagon hole which is really next to impossible to do, and then a hexagon. The hexagon may not be as difficult, ultimately as the square. But this should be noticed: That even-sided holes are far easier to do than odd number of sides. (I mean look, guys, can you even build a hole with ONE side? how would you do a triangle?) The point is that as you get more and more sides, you approach a circle which is really an infinite sided, closed geometric design.

The next feature that causes turbulence is: Corners. Every corner is going to slow your fluid down. Engineers typically have tools to round the corners of piping and whatnot for large projects, the larger the radius, the less turbulence. But worse than turbulence (or maybe the grand caliph of turbulence) is the "bounce". This is when a fluid is let directly into a flat, perpendicular to the flow, surface. it strikes that surface ded on and bounces back blocking the fluid from coming in. This sets up a vibrational type of turbulence. (Imagine a theatre pakt full of popkorn popping people when the theatre catches fire--the people jam the exits. If even ONE person stands picking his pickel nosed protuberance just outside the doors of the theatre, somebody dies! the same at soccer games in Brazil when the fans get in fights and then rush the gates.) Anyway, perpendicular hole/interfaces are the worst of all. MUCH better is to have a 45deg. angled ramp which would reflect the incoming fluid at 90 deg. to the march of the flow. You can approach the circular method again over a ramp, but in this case, the best shape is not EXACTLY circular, it is a hyperbola or maybe parabola. Well, try to make THAT with what we amateurs have! A ball ended mill end would be just fine and the amount of difference is like the tiny difference between Einsteins theory of gravitation and Newtons theory for a ball dropt 10 feet from the surface of the earth.

With all that said, the little steam engines that use this little valving system are VERY poorly designed. So, along comes the Corliss which overcame some of these problems and created others. (I just made that up.) Later in the 1880's or later (does anybody know the date of inception for the uni-flo?) the uniflo idea was hit upon. This is where there IS NO VALVE! They are not spring loader or cams--they are HOLES in the cylinder wall that allow the steam to exhaust when the piston passes the holes. Clearly, there is going to be VERY LITTLE turbulence in exhausting! There are no square or round tubes or corners to let the steam out, it is simply holes!

My understanding of the problem about why these uniflos never really took off is that the port exhaust causes some kind of trouble with the piston and rings as they pass.

******************************************************************************************
(Well now, with that grand tome, I expect some literary criticism: Did the plot work out? Was the hero killed properly? Was it logically connected from opening to end? Were the sentences perked up with lively witicisms? How were the subtle, wry jokes percieved? Will I get the Nobel Prize for literature like Bob Dylan? Please feel free to write a positive review.)
 

Richard Hed

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Anatol

You are right that the compression part of the cycle requires work, but that work brings the exhaust steam up to a state closer to that in the steam chest, so that when the inlet valve there is no sudden inrush of steam., which would be wasteful.

'Convincing' you qualitatively may be difficult. You are unlikely to grasp this stuff properly without some study of thermodynamics.

Cylinder swept volume is the area of the piston times the stroke.

No, I am saying I think the relief valves you say you have seen probably don't serve the purpose you think they do.

Yes the inlet valve starts to open before dead center, to ensure that it is sufficiently widely open to admit the necessary amount of steam freely without loss through "wire drawing" when the piston starts to gather speed.

Size of clearance volume, I don't know off the top of my head, and I need to get some breakfast, but nearer 5%..

Haven't we already been through all this at: Designing steam engines ?
Well thanx for showing me that "designing steam engines" forum. My big problem with this particular discussion is that nowhere do I see exactly how much of the steam is exhausted? This is important. If 99% of steam is exhaussted, the tiny 1% left is virtually meaningless. If 95% is exhausted, then the work of re-compression would be small but still significant. If 90% then it is VERY significant. What are the statistics? Where is the cutoff to significance?
 

Anatol

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Charles
thankyou for your note

You are right that the compression part of the cycle requires work, but that work brings the exhaust steam up to a state closer to that in the steam chest, so that when the inlet valve there is no sudden inrush of steam., which would be wasteful. 'Convincing' you qualitatively may be difficult. You are unlikely to grasp this stuff properly without some study of thermodynamics.
I am no applied physicist, but I do grasp some of the basics of thermodynamics. And I am ready to accept recommendations from those who know more! This all very interesting.

To paraphrase - the recompression of exhaust gas takes energy but that's ok because - even though the recompression uses energy that can only be partially scavenged on the next power stroke - the advantage is that it prevents "sudden inrush of steam".
I'm prepared to accept this if you explain why a sudden inrush of steam is bad.
So the standard counterflow is inefficient because of the "sudden inrush of steam" phenomenon? Is this essentially the same argument as "the inlet end of the cylinder remains hot because exhaust happens at the other end"?

"I am saying I think the relief valves you say you have seen probably don't serve the purpose you think they do."

have you seen the experimental uniflow engine of Dan Gelbart? (easy to find on youtube). His relief valve certainly functions to exhaust all exhaust.

"Yes the inlet valve starts to open before dead center, to ensure that it is sufficiently widely open to admit the necessary amount of steam freely without loss through "wire drawing" when the piston starts to gather speed."

I was surprised to se you say previously that the inlet closes at about 5% of power stroke, because, even if the opening was symmetrical (5% before TDC) that would only be 10% of power stroke, thats less than I would have expected.

"Haven't we already been through all this at: Designing steam engines ?"

I went back to that thread, you're right there's lots of good info there, ! ( I had to stop partaking in the forum about that time for personal reasons - and haven't been back till now. I guess I'd forgotten some of the details we'd discussed, but that discussion didn't mention uniflow, mostly wobblers.)
thanks
A.
 
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Steamchick

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Hi, re "sudden inrush of steam". Our erstwhile and esteemed Victorian predecessors did not have precise "Einstein's theories", but their livelyhoods depended upon steam so they managed it empirically, and pretty effectively. So to my point. When steam expands very rapidly, it is effectively adiabatic. Hence the energy within the steam powers the expansion. If suitably superheated, then only some or all the superheat energy powers the expansion (gas laws for adiabatic expansion hold true), but with insufficient superheat, the steam will then power the expansion by condensing some steam to water. Hence you need to refer to steam tables or otherwise combine gas laws and condensation calculations to mathematically predict what is happening. The Victorian engineers also understood the effect of shock waves and constant pressure change causing zones of higher and lower pressure, as they experienced the effect of the condensation causing "wet" steam when and where they didn't expect it to occur. In James Watts' era, this wasn't understood, but Corliss and his generation were designing for improved flow and pressure efficiency, based on a lot of real data from designs with problems, and some better designs, clever ideas and experiment to see what actually bore useful results. Finding the Design Textbooks of those eras is a bit difficult, as they have mostly become the latest theoretical driven text books that we have for use today. And so we fail to appreciate why they made certain design decisions. (Our judgemental position is not the same as theirs).
E.g. "wire-drawing" is a readily understood expression, but is really comparing something incompressible like water, steel, copper, nylon, with what they are really needing too understand: a 2-state (steam and water) compressible gas, that changes state from gas to steam or vice-versa in a non-linear fashion. But the phrase helped their less-educated brethren to understand flow and what they termed "back-pressure" to describe frictional losses of flow in passages. They could also easily compare work done in wire drawing with the change of cross-sectional area, and imagine a similar effect on steam in valve passages. Vis-a-vis speed of wire before and after drawing.

Hence they found it advantageous to have some back-pressure at the end of stroke to avoid a "rapid inrush" which caused condensate to form - which then wetted the metal surfaces and applied the steam oil carried in the steam to cylinder ends, not the bores, where they wanted it. Also condensate collecting at cylinder ends needed spring pressure relief valves to bleed out the condensate to avoid hydraulic lock at the end of stroke. So they learned from experience, not theory, that this mis-named wire-drawing was better avoided by larger passages and some back pressure in the cylinder, maybe at some thermodynamic expense. But their studies made engines more reliable, which saved maintenance costs and lost earnings in return. So we should try and find some original texts to understand these "antiquated" designs, not necessarily simply apply modern theory.
I can write more drivel and rubbish ideas if you wish.
I am enjoying this fascinating thread.
More comment please.
K
 

Cogsy

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Well, you wanted some critique so I can provide a start.

that is not possible. To have a return pressure greater than the steam pressure would be going against the law of conservation of energy. (caveats with that of course.)
Not true - even if the cylinder dropped to atmospheric at BDC the pressure developed by compression could be very high, depending on the compression ratio of the cylinder. I know we don't normally consider a compression ratio for a steam engine but if you have a piston moving up a sealed bore you are going to get compression. Now the energy required to compress this volume of gas (one cylinders' worth) to 21 PSI might need, say, one cylinders' worth of steam at 5 PSI, but if you're bringing in 5 cylinders' worth of steam at 20 PSI then you have plenty of energy to get the work done with no thermodynamic laws broken at all.
 

Richard Hed

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Hi, re "sudden inrush of steam". Our erstwhile and esteemed Victorian predecessors did not have precise "Einstein's theories", but their livelyhoods depended upon steam so they managed it empirically, and pretty effectively. So to my point. When steam expands very rapidly, it is effectively adiabatic. Hence the energy within the steam powers the expansion. If suitably superheated, then only some or all the superheat energy powers the expansion (gas laws for adiabatic expansion hold true), but with insufficient superheat, the steam will then power the expansion by condensing some steam to water. Hence you need to refer to steam tables or otherwise combine gas laws and condensation calculations to mathematically predict what is happening. The Victorian engineers also understood the effect of shock waves and constant pressure change causing zones of higher and lower pressure, as they experienced the effect of the condensation causing "wet" steam when and where they didn't expect it to occur. In James Watts' era, this wasn't understood, but Corliss and his generation were designing for improved flow and pressure efficiency, based on a lot of real data from designs with problems, and some better designs, clever ideas and experiment to see what actually bore useful results. Finding the Design Textbooks of those eras is a bit difficult, as they have mostly become the latest theoretical driven text books that we have for use today. And so we fail to appreciate why they made certain design decisions. (Our judgemental position is not the same as theirs).
E.g. "wire-drawing" is a readily understood expression, but is really comparing something incompressible like water, steel, copper, nylon, with what they are really needing too understand: a 2-state (steam and water) compressible gas, that changes state from gas to steam or vice-versa in a non-linear fashion. But the phrase helped their less-educated brethren to understand flow and what they termed "back-pressure" to describe frictional losses of flow in passages. They could also easily compare work done in wire drawing with the change of cross-sectional area, and imagine a similar effect on steam in valve passages. Vis-a-vis speed of wire before and after drawing.

Hence they found it advantageous to have some back-pressure at the end of stroke to avoid a "rapid inrush" which caused condensate to form - which then wetted the metal surfaces and applied the steam oil carried in the steam to cylinder ends, not the bores, where they wanted it. Also condensate collecting at cylinder ends needed spring pressure relief valves to bleed out the condensate to avoid hydraulic lock at the end of stroke. So they learned from experience, not theory, that this mis-named wire-drawing was better avoided by larger passages and some back pressure in the cylinder, maybe at some thermodynamic expense. But their studies made engines more reliable, which saved maintenance costs and lost earnings in return. So we should try and find some original texts to understand these "antiquated" designs, not necessarily simply apply modern theory.
I can write more drivel and rubbish ideas if you wish.
I am enjoying this fascinating thread.
More comment please.
K
More rubbish, please.
 

Richard Hed

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Well, you wanted some critique so I can provide a start.



Not true - even if the cylinder dropped to atmospheric at BDC the pressure developed by compression could be very high, depending on the compression ratio of the cylinder. I know we don't normally consider a compression ratio for a steam engine but if you have a piston moving up a sealed bore you are going to get compression. Now the energy required to compress this volume of gas (one cylinders' worth) to 21 PSI might need, say, one cylinders' worth of steam at 5 PSI, but if you're bringing in 5 cylinders' worth of steam at 20 PSI then you have plenty of energy to get the work done with no thermodynamic laws broken at all.
Thankyew, however, I did point out (caveats with that plate of ice cream and strawberries.) Well, did you like the plot?
 
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Charles Lamont

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This discussion about losses is a matter of the second law of thermodynamics.
I suggest an interlude for a brief explanation (bear with the intruduction):
 

Richard Hed

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I am still wondering, however, in all this discussion, just how much exhausted steam DOES get exhausted thru the exhaust port. It certainly is relevant to designing an engine well (something the uniflo should do exceptionally well>). I too am trying to design a uniflo with up to 5 HP that powers something like a generator or aacts exactly like the old 5hps one would find on a farm--made to grind corn, power a saw blade, maybe grind grains, and leap tall buildings, and of course generate electricity and pump water. (I hide the secret fact that I am a --god forbid - a , a ... a prepper! waiting for the world to fall apart so I, I, Cornelicus Heddicus will become DICTATOR of the world, or OZ, which ever comes first, all because I, Cornelicus Heddicus, am the only one in the world to have the forsight to build this 5 HP engine. Unfortunately it is a hollow crown sinc there will be no-one left to lord it over.)
Hope you don't tell anyone that.

Seriously, on various engines, how is it determined how much used steam is exhausted. Are there tables somewhere? Formulas? Empiric tests?
 

Charles Lamont

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Going back to Anatol's OP, the efficiency of a uniflow engine exhausting to atmosphere will likely benefit from auxilliary exhaust valves (controlled by valve gear).
A condensing uniflow engine exhausting into a good vacuum doesn't need them.

Without auxilliary exhaust valves, a non-condensing engine needs a large clearance volume to avoid over-compression.

Relief valves (which are different from exhaust valves) are needed particularly in the condensing case because they can have a smaller clearance volume of 1.5 to 2 percent of the swept volume without over-compression, but should the vacuum be lost the compression pressure could get dangerously high without relief valves.

I could not make out anything Dan Gelbart said over the racket the engine was making, especially as he appeared to have his back to the microphone.

Any sudden process is bad in thermodynamics as it is 'irreversible'. This is what the Second Law is all about. As Michael Flanders sings, "That's entropy, man."

Richard's long ramble about flow resistance in tortuous passages is generally on the right track, but I would suggest that in the case of a normal slide valve cylinder the thermal losses due to passing the hot steam and cool exhaust alternately through the same duct are considerably greater than the flow resistance, but that is no more than a well-educated guess. The Reynolds Number of the steam flow in model engine passages is much lower than in full size, mitigating the effect of the multiple holes and sharp corners.

As to the suggested problems with piston rings passing over ports, I am not convinced. Most main-line steam locomotives built after about 1925 had piston valves in which rings pass over ports, and in far more trying conditions than found in stationary engines, particularly as a result of pumping air through the cylinders when coasting.

Steamchick, in the 1909 edition of his famous 'Heat Engines' W Ripper* says, "The laws which govern the condensation of steam in the cylinder are not at present fully understood."

* William Ripper - Wikipedia
 
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Charles Lamont

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I am still wondering, however, in all this discussion, just how much exhausted steam DOES get exhausted thru the exhaust port. It certainly is relevant to designing an engine well (something the uniflo should do exceptionally well>). I too am trying to design a uniflo with up to 5 HP that powers something like a generator or aacts exactly like the old 5hps one would find on a farm--made to grind corn, power a saw blade, maybe grind grains, and leap tall buildings, and of course generate electricity and pump water. (I hide the secret fact that I am a --god forbid - a , a ... a prepper! waiting for the world to fall apart so I, I, Cornelicus Heddicus will become DICTATOR of the world, or OZ, which ever comes first, all because I, Cornelicus Heddicus, am the only one in the world to have the forsight to build this 5 HP engine. Unfortunately it is a hollow crown sinc there will be no-one left to lord it over.)
Hope you don't tell anyone that.

Seriously, on various engines, how is it determined how much used steam is exhausted. Are there tables somewhere? Formulas? Empiric tests?
It depends on the back pressure. An atmospheric engine will exhaust down to a bit above an atmosphere. In a steam locomotive, where the exhaust blast is used to draw the fire, the back-pressure is a bit higher. A condensing engine cylinder is more thoroughly evacuated by the condenser vacuum. So, as the exhaust valve closes, you will have trapped in the cylinder whatever volume that represents of more-or-less dry saturated steam at whatever the back pressure is. Steam Tables will allow you to calculate the mass. Equally, if you know the inlet pressure, and cut-off, and the amount any superheat, you can calculate the mass of steam you are starting with (neglecting condensation). The difference is the amount exhausted. I suggest you look into cylinder P-V or 'indicator' diagrams as a starting point.

For another way of thinking about the quantity of steam exhausted, I am tempted to quote Tom Lehrer: "Life is like a sewer ... you get out of it what you put in".
 
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