a primary reason for reducing cutoff quickly is to minimize back pressure due to exhaust steam.
Assume it is an exponential decay, looking for the time-constant
greg - Philadelphia & Reading / Reading
You sure as hell won't find it that way; the residual heat in the steam makes it re-expand in volume as the pressure comes off, and this occurs throughout the mass of steam in the tract in addition to the pressure differential from the exhaust valve opening.
If you look at the continuous flow in a Steamotive plant, you will appreciate the shape and dimensions of that outsize exhaust plenum...
You tune the exhaust, modify the port unshrouding and opening, and get longer expansion from admission cutoff to exhaust. If you were wondering why we grouse about having different exhaust duration and timing relative to admission -- something a regular radial valve gear does with difficulty if at all -- thinking about this should be enlightening.
interested in pressure, not volume. doesn't the pressure decrease when allowed to expand?
i fyou can't answer how quickly the pressure decreases, can you explain how quickly it expands?
Ideal gas in a cylinder would expand out following a pressure gradient, and you would look at the flow resistance along the exhaust tract relative to pressure and mass flow to figure out what the resistance to free expansion would be. Note that much of the design of internal-combustion exhaust systems will give you formulae for this "assuming that the water of combustion stays a vapor".
Steam is different because its latent heat of condensation is so large. That means that you get nucleate condensation at lower pressure (which scotches getting much expansive work in LP cylinders of unsuperheated compounds) but the nuclei happily revaporize at exhaust release -- expanding spherically, so the exhaust volume chokes the tract rather than inducing flow down it. That is much of the 'back pressure' effect associated with high speed/short events, as the expanded mass of admitted steam all has to 'clear' during the time the exhaust valve is open. (The mass remaining in the cylinder when the valve is close to shrouding closed then contributes to compression, and thence to admission on the following return stroke in a DA, so it is not trivial to understand).
for those that might be interested ...
the plots show cylinder pressure (red), residual exhaust pressure (green) and zero (gray), for speeds of 1, 2, 4, 8 and 16 rev/sec which correspond to 5.5, 11, 21 and 43 mph for 60" drivers.
the plots show that at higher speeds, the residual exhaust pressure is relatively high for much of the cylinder cycle. At low speeds, the exhaust pressure has sufficient time to decrease such that the effective pressure, cyllinder pressure - exhaust pressure is relatively high over the entire cylinder cycle
the initial exhaust pressure depends on cutoff, so decreasing cutoff reduces the initial exhaust pressure and effective pressure.
added resulting sinusoidal force at the driver (cyan). The force plot shows that the back pressure from the exhaust has less effect at the beginning and end of each cylinder cycle.
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Interesting plot. What would be highly interesting here would be to see some indicator diagrams of exhaust pressure overlaid at the same cyclic rps and adjusted to the same scale dimensions.
I suspect you will find more than a little broadening of the curve as exhaust mass flow per second changes, particularly at high speed where the cutoff begins being lengthened again. I would repeat that a useful mathematical representation is likely to be an empirical set of curves for a particular 'instantiation', like all those irritating factors in Lawford Fry's boiler analysis, rather than a clean set of exponential decays.
Someone like Ed should find and post a couple of the accounts of the 'automatic cutoff control' developed in the early 1920s (I always suspected as part of the automatic-train-control scenario). That used back pressure as the control signal for adjusting cutoff.
you need to look at a Carnot cycle for a steam engine. Steam cut off during stroke varies puposely. keeping a train moving requires less steam then when starting BUT you are missing that the inlet valve and exhaust valve are both shut before the end of piston stroke allowing the piston to be cushioned as end of stroke pressure builds up. So pressure disipates during the piston stroke but builds again before the end of the stroke
ndbprrBUT you are missing that the inlet valve and exhaust valve are both shut before the end of piston stroke allowing the piston to be cushioned as end of stroke pressure builds up.
i'm aware of this (i.e. lead). it is a minor detail for my purposes
not being aware of back-pressure was a major oversight.
ndbprryou need to look at a Carnot cycle for a steam engine.
BUT you are missing that the inlet valve and exhaust valve are both shut before the end of piston stroke allowing the piston to be cushioned as end of stroke pressure builds up. So pressure dissipates during the piston stroke but builds again before the end of the stroke
The 'cushioning' is provided by the steam and vapor remaining in the cylinder after the (long-travel) valve has traversed the exhaust port and closed it off again... during the previous stroke to the one we're looking at exhaust expansion from. That compression not only 'cushions' piston reversal but keeps the port and passage pressure relatively high in the part of the return stroke up to the start of admission in the 'following' return stroke.
When we talk about the desirability of having different 'timing and duration' of admission and exhaust, we are referring to controlling the admission valve head's travel and speed differently from the exhaust valve's. That is not easily possible with a typical radial gear moving a piston valve spool forward and backward, no matter whether the valve gear has negative lead at starting or whatever.
All the OP is concerned with here is the fall in pressure during the time the volume of the cylinder near the end of its stroke is communicating through the open exhaust valve to the exhaust tract. This has nothing to do with what the admission valve did when the piston was starting that stroke, back at the other end of the cylinder.
Here are the events -- ASSuming you have an engine that is already producing some cyclic rpm, and measuring from FDC or BDC, not just the moment of steam admission. For convenience start by considering the piston is at FDC. There is residual steam at comparatively high pressure, via compression, ahead of this piston; this pressure is acting on the piston to drive it backward from 'just after FDC' but the pressure is falling (and temperature decreasing according to saturated relationship) as the piston starts accelerating backward.
At some point (which is in a typical DA the 'reverse' of the point of exhaust cutoff in the previous stroke) the admission valve starts to open. High-pressure steam now passes from the steam-chest volume into equilibrium in the cylinder; at low speed this is in fact reached and for the following portion of the stroke there is essentially full chest pressure on the piston, with mass flow of chest steam filling the increasing volume of the cylinder as the piston moves backward.
At some point the admission valve closes, and now instead of 'boiler pressure' sustaining piston thrust, you switch to expansive working, where a fixed mass of steam expands to continue pushing the piston down the cylinder. This is the part where superheat is so important: the combination of physical expansion and the extraction of work from piston thrust is causing some of the steam to change phase (which reduces volume and hence aggregate cylinder pressure). Some of this is 'nucleate condensation' (not wall or surface condensation) and it is an unavoidable issue while the steam is doing compressive work.
Note that the start of expansion-only working is not completely determined by valve cutoff (or 'riding cutoff' in older types of engine, e.g. Cuyahoga Cutoff in short-valve-travel engines with little fixed cutoff). There is some maximum percentage of time that admission to the cylinder is sustained per stroke, even when the valve gear is set to the maximum possible admission time. In some cases this is dramatically long -- the PRR I1s, for example, could not be operated below 50% cutoff. (That produces a number of other effects, for example the need to have slot ports for starting).
We see all kinds of discussion about getting high-pressure steam into the cylinder at high cyclic rpm, as the admission time especially at very short cutoff can get very short for a mass flow that has to start and stop). But there is comparatively less concern with what happens to the admitted steam at the end of the 'power stroke' -- a common idea being to get rid of it with lowest possible backpressure (hence the whole evolution of things like Kylchap or Lempor or Giesl front ends) forgetting that steam doesn't magically flow out of the cylinder with the same adroitness it was shoved in...
Now if you've been following along, we have gotten to the point the cylinder volume has exhausted down... where is the compression steam? The residual steam is still on the 'front side' of the piston, and the compression has to 'cushion' the back side. So how does the compression steam get there?
Remember back at the beginning, where I said there was some residual pressure in the cylinder? And in other discussions, that you can get the effect of prompt admission mass flow at high speed if the inlet-tract pressure is a good proportion or in fact equal to chest pressure at the moment admission begins? The residual 'equilibrium' steam that didn't go out the exhaust stays in the cylinder as the piston reverses and starts coming back... and it is compressed all the way back up the cylinder as the following double-action stroke takes place. You will recognize that there will be some efficient 'back pressure' that optimizes the amount of remaining mass to result in proper 'cushioning' and pressurization action back at the other end.
Do I sense a disturbance in the Force, voices crying out 'but that's not thermodynamically efficient!' The answer is yes... but the game here isn't thermodynamic maximum efficiency (or chasing Carnot efficiency) but getting the locomotive to make power at high cyclic rpm. It is comical to note the number of people who think a steam locomotive is given shorter and shorter cutoff until it is finally going as fast as it possibly can with 'just wisps of steam flicking in and out of the cylinders'. That is not what actually happens, and in fact you can recognize that it can't simply by looking at how aerodynamic resistance builds up with road speed -- you need more and more power to go really fast, and that's not going to come from anywhere but higher mass flow... which means longer effective admission. Interestingly, both the Mallard record speed and the highest test speed achieved in the N&W J 610 testing on PRR are within a couple of percent, in the low 40% cutoff range. This can be taken as the (empirically determined) range of expected cutoff that the valves, tracts, etc. will have to be capable of handling. In particular that is a lot of mass flow that has to get out of the cylinder when it has gotten done producing usable piston thrust...
if you didn't see this on MRH, we captured our algorithms in a Java app
What is the multiphysics simulation for modeling the valve, tract, and changes in steam quality for the duration of an exhaust event in this app?
You might have to brute-force it with numerical approximations for a range of loads and cutoff conditions -- perhaps quantized by % cutoff within the normal engine working range.
You will NOT get much, if anything, coherent out of using a simple exponential decay function, especially if you are using the result to calculate energy or entrainment characteristics in a model of a front end.
OvermodYou will NOT get much, if anything, coherent out of using a simple exponential decay function, especially if you are using the result to calculate energy or entrainment characteristics in a model of a front end.
how much error would there be?
.
so the error is of no concern for my needs
Look at an indicator card for your chosen cyclic and cutoff setting, and note the point where the exhaust begins to open at that cutoff. (This will automatically, if a bit unhelpfully empirically, give you the back-pressure effect on the exhaust release).
If you are lucky, the cylinder pressure and tract pressure will equalize before the valve fully closes to steam. That pressure will determine much of your compression characteristics -- for cushioning of the 'return' stroke, blowoff if excessive, etc.
You can then do curve fitting to get the best constants if you don't want a double-salient (or more complex) algorithm.
I suspect you might not need many of these fitted curves to be a 'library' for use in the app.
Of course I can't help but wonder if you could dust off Charlie Dockstader's programs and figure out what curves are needed when changing the various lengths of elements in those examples...
OvermodOf course I can't help but wonder if you could dust off Charlie Dockstader's programs and figure out what curves are needed when changing the various lengths of elements in those examples...