timz Paul Milenkovic could we consider the Niagara to be operated at 7 lbs water/lb coal, 13 lbs water/hp-hr, 4000 hp, less than half peak evaporation at 52,000 lb water/hr, 7400 lb coal/hr. "Could we consider" a 4-8-4 producing 4000 hp on 7400 lb/hr of coal? You mean, could it do that? Sounds unlikely, of course. Are you considering something else?
Paul Milenkovic could we consider the Niagara to be operated at 7 lbs water/lb coal, 13 lbs water/hp-hr, 4000 hp, less than half peak evaporation at 52,000 lb water/hr, 7400 lb coal/hr.
"Could we consider" a 4-8-4 producing 4000 hp on 7400 lb/hr of coal?
You mean, could it do that? Sounds unlikely, of course. Are you considering something else?
Whereas I am offering speculation, I am indeed suggesting that a Niagara could be run that efficiently at part-power, and I am invoking Tom Morrison (2018) The American Steam Locomotive in the 20th Century as a source that this could have been a practice in the late steam era.
There is a narrative that especially the French and to some extent the Germans and the British were building much more efficient steam locomotives than the ham-fisted Americans. Perhaps some of the shade thrown on late-era US designs is in the way figures are quoted?
If GM "killed the electric car", what am I doing standing next to an EV-1, a half a block from the WSOR tracks?
I suggest separating the auxiliary loads needed to 'run the boiler' or steam generator from those used to operate the engine -- with one special category (draft induction) kept separate from both. There is also a different set of 'auxiliaries' in play for starting (cylinder cocks open) and certain modes of drifting, when you get to operation.
I used the 'idling' consumption (in the absence of a Direct Steam type system, or electrical pressure maintaining as on rebuilt 8055) and this involves things like continuous blowdown that you might not know to account for (personally, I don't like current continuous blowdown as an option and plan to use different approaches that recover at least some proportion of the water mass and heat)
Problem is that many of the auxiliaries were designed to run efficiently at or near peak cylinder efficiency mass flow, or to suit the anticipated demand of a forced boiler, and are inefficient at idle. Therefore some of the heat-balance data are nonlinear and change nonproportionally.
oltmannd Paul Milenkovic Whereas I haven't accounted for the steam consumption of auxiliaries (air pump, turbogenerator, whistle, stoker engine, stoker steam jets) where auxiliaries are removed from the diesel ratings, diesels are commonly rated "at the input to the main generator" before losses in the electric drive come into play. Not quite. Diesels are rated at horspower into the main generator to be used for traction (at AAR standard conditions - 60 degree F air and fuel, 28.86" barometer). Power for auxiliaries is already taken out. (fans, aux gen, TM blower, etc.). Builders will usually provide RRs with info stating BHP, THP, and NTHP (BHP is Engine brake HP, THP is HP for traction into main generator, and NTHP is net HP out of main generator head to TMs).
Paul Milenkovic Whereas I haven't accounted for the steam consumption of auxiliaries (air pump, turbogenerator, whistle, stoker engine, stoker steam jets) where auxiliaries are removed from the diesel ratings, diesels are commonly rated "at the input to the main generator" before losses in the electric drive come into play.
Not quite. Diesels are rated at horspower into the main generator to be used for traction (at AAR standard conditions - 60 degree F air and fuel, 28.86" barometer). Power for auxiliaries is already taken out. (fans, aux gen, TM blower, etc.). Builders will usually provide RRs with info stating BHP, THP, and NTHP (BHP is Engine brake HP, THP is HP for traction into main generator, and NTHP is net HP out of main generator head to TMs).
I think that is pretty much what I am saying. The 3000 HP rating of an SD40-2 is "before the traction generator" but "after the auxiliary loads."
I was contrasting this with my figures for a steam locomotive, also "before the mechanical rod transmission of power" but without taking deductions for auxiliary loads.
Paul MilenkovicWhereas I haven't accounted for the steam consumption of auxiliaries (air pump, turbogenerator, whistle, stoker engine, stoker steam jets) where auxiliaries are removed from the diesel ratings, diesels are commonly rated "at the input to the main generator" before losses in the electric drive come into play.
-Don (Random stuff, mostly about trains - what else? http://blerfblog.blogspot.com/)
Paul Milenkoviccould we consider the Niagara to be operated at 7 lbs water/lb coal, 13 lbs water/hp-hr, 4000 hp, less than half peak evaporation at 52,000 lb water/hr, 7400 lb coal/hr.
The Niagara has been observed to do even better than that.
Bill Tuplin ("The Steam Locomotive, Its Form and Function) was something of a low-pressure enthusiast, and seems to seldom have passed up an opportunity to note that increased maintenance of high-pressure boilers may have outweighed the thermodynamic advantages of higher throttle pressure (if adequate superheat is provided). It was therefore probably delightful to him to observe a Niagara, on what I recall being characterized as a way freight (probably break-in after service at Harmon) doing the job of a 2-8-0 on a 2-8-0's budget of fuel and water. The fireman was reportedly using sliding-pressure firing, probably to about 180psi, and of course there was lavish superheat available from shortly after opening the throttle. So things scaled still lower with that boiler.
There was a concern, though, with the steam separation in the 'domeless' boiler, particularly at the expected steam mass flow associated with higher firing rate. The Timken rods were extremely intolerant of priming carryover, and probably excessive peak compression, and one set of repairs of the resulting damage would pay for many, many tons of coal and pans worth of water.
NYC was also one of the roads to replace whistle use with 'pneuphonic' horns, recognizing the heat and mass loss involved (much greater than that expended in the air compressors for a horn blown on main-reservoir pressure). I'll grant you that the whistle had it all esthetically ... but particularly when aggressive water treatment has to be in use for alloy boilers, the mass loss and water-rate complication involved in even a little crossing-signal blowing adds up fairly dramatically.
There is, as far as I know, no good test result on use of the Hancock Turbo-Inspirator by extending its shaft to drive a dynamo, and then using that power to run some of the auxliiaries that did not 'scale' well to be driven directly by steam. Experiments on this are anticipated for 5550 when built out to the point the boiler can be steamed. While the Turbo-Inspirator was more designed to be light than particularly effective (the one fitted on C&O 614 was removed and its 'platform' actually used for a hose reel at one point!) I think it can be modified to do several jobs well if expanded to make use of a larger turbine running at more constant speed with adjustable load.
I am just speculating here, but could we consider the Niagara to be operated at 7 lbs water/lb coal, 13 lbs water/hp-hr, 4000 hp, less than half peak evaporation at 52,000 lb water/hr, 7400 lb coal/hr.
This firing rate is 74 lb/sq ft, almost a third of the firing rate for max hp, consistent with the higher specific evaporation rate I assume above. By not "forcing" this locomotive, 4000 indicated HP is still quite a bit, and your coal rate is now 1.85 lb/hp-hr or 9.7%, putting in the range of figures quoted for the more efficient British steam locomotives?
This isn't from changing a thing about this excellent locomotive apart from how you choose to brag about its performance?
timz Paul Milenkovic where the 6% thermal efficiency for modern steam in US practice comes from If you mean how is the figure calculated -- no argument about that, is there? Convert the BTUs of the coal burned to foot-pounds, and compare to the foot-pounds in the cylinder, or at the drawbar.
Paul Milenkovic where the 6% thermal efficiency for modern steam in US practice comes from
If you mean how is the figure calculated -- no argument about that, is there? Convert the BTUs of the coal burned to foot-pounds, and compare to the foot-pounds in the cylinder, or at the drawbar.
Yes, what you say is well-known, but what I am asking, where does a figure of 3 lb coal/hp-hr come from? The case of Alfred Bruce's "mystery 4-8-4" (surprise, an ALCo product and the legendary Niagara), I speculate it could come from 16.7 lb steam/hp-hr, a figure I suggest comes from achieving peak hp by operating at high speed that lowers condensation loss but at a generous cutoff using a lot of steam, and from raising 5.5 lb steam/lb coal, which occurs at a high, inefficient firing rate. The coal consumption may have been considerably lower if the Niagara was not operated at peak horsepower, which author Tom Morrison hints may have been railroad practice.
Overmod gives a synopsis of late-steam era efforts and technology to improve the notably poor thermal efficiency of the Stephensonian-pattern steam locomotive along with what may have worked and what may have not been worth the trouble.
As to why consider thermal efficiency at all, especially in a time and a place when coal was cheap and plentiful (although the post-WW-II labor actions to give miners their due changed the equation, hastening Dieselization it is said), Wardale explains that the sheer quantity of coal placed a capacity burden on the railroads consuming in. If you could use coal more efficiently, that would free up a lot of track capacity, a lesser-noted advantage to the thriftier diesels. Even today, the cost of transporting coal from mine to the power plants and other remaining users is said to exceed to the cost of digging it out of the ground, especially for the bulkier, low-BTU Powder River Basin coal that is extracted from open-pit mines with special-purpose jumbo-sized excavating equipment.
Apart from the discussed improvements to steam, I am posing the question as to what it even means if someone says "this locomotive is 8% thermally efficient" or "that locomotive required 3 lb coal/hp-hr."
Consider the efficiency of producing hp from superheated steam as claimed by annotations on cylinder indicator diagrams on pp 258-259 of Wardale's "Red Devil." This locomotive is claimed to have a particularly low flow resistance steam circuit although its boiler pressure is quite moderate by late-steam standards (1464 kPa translates to 212 psi in more conventional, human-centric units). At 42 MPH, 52% cutoff, it used 15.8 lb steam/hp-hr. This is data Wardale reduced from an electronic indicator diagram that he admits isn't an exact match to data from a mechanical indicator diagram no showing such rounded corners of the trace, but still, this is not that far off from the 16.7 lb/hp-hr for putatively a Niagara at peak power.
This speed and power is equivalent to 34,100 lb of tractive effort, before correcting for friction in the cylinders, rods and valve gear. I posted somewhat recently that John Knowles posting on mechanical efficiency vanished from the Web, Wardale appears to have only a weak grasp on that quantity, but Ralph Johnson's book quotes, "20 lb friction per ton of adhesive weight." In contrast with the much lower Davis formula rolling resistance, this puts the "effective" rolling resistance at 1 part in 100 for friction in the actual steam engine plus, say, 1 part in 500 from the Davis formula into the level of friction of a radial tire on a modern passenger car. Keep in mind that a diesel engine or your car engine has considerable internal mechanical friction. Knowles went into more detail, but I will assume the 1 part in 100 fraction is a constant addition to the locomotive rolling resistance and does not diminish with reduced power, differing from Wardale quoting a constant 90% as the steam engine "mechanical efficiency."
For the Red Devil engine number 3450 having 83 tons of adhesive weight, this puts the tractive effort "at the wheel rim" at 34100 - 83 times 20 or 32400 lbs.
Overmod or someone else can correct me on this, but when Wardale states "cylinder steam flow" of 27500 kg/hr (60500 lb/hr -- a rip-roaring steam flow for a 3.5-foot "Cape" gauge 4-8-4), he is talking about pounds of steam past the valves, and this does not account for any recovery of energy and water with a feedwater heater -- I shall include the benefits of a feedwater heater in improving efficiency and reducing the "water rate" into the lbs water (into the cylinders)/lbs coal half of the efficiency equation. This also applies to with the possible use of a combustion air preheater that Overmod talks about, Porta included in his future plans and Wardale hints at, which also improves efficiency and recovers some additonal water.
Wardale's published indicator diagrams are a "tease" of what is possible with his modified locomotive because they are few in number and they sample speed and cutoff in haphazard fashion. He also has indicator diagrams for an unmodified 25NC locomotive for comparison, but he doesn't give the steam rate. I offer the following table derived from Wardale's data:
15.8 lb/hp, 42 MPH, 52% cutoff, 3850 hp, 32400 lb, 19.5% adhesion
12.8 lb/hp, 43 MPH, 24% cutoff, 3000 hp, 24500 lb, 14.8% adhesion
12.1 lb/hp, 53 MPH, 28% cutoff, 3650 hp, 25800 lb, 15.6% adhesion
10.5 lb/hp, 53 MPH, 18% cutoff, 3136 hp, 16800 lh, 10.1% adhesion
I ranked these by steam consumption, and even the somewhat slower speed is showing somewhat higher steam consumption at a lower cutoff, suggesting that superheat has not fully vanquished the condensation problem.
The hp is indicated "in the cylinders" whereas the tractive effort at adhesion factors are "wheel rim" using the Ralph Johnson of Baldwin figure for mechanical friction in the actual steam "engine."
By the way, Wardale makes a big deal that the "proper" place to measure hp, tractive effort and efficiency is "at the drawbar", essentially the wheel rim values after the deduction for the Davis-formula rolling resistance and aero drag. The steam locomotive, especially for high-power, large-firebox, high boiler pressure modern steam carries a lot of its weight off the drivers, and then there is the tender to drag around because of its appetite for coal and water, but I still regard the "in the cylinders" value as a more reasonable base for comparison. Whereas I haven't accounted for the steam consumption of auxiliaries (air pump, turbogenerator, whistle, stoker engine, stoker steam jets) where auxiliaries are removed from the diesel ratings, diesels are commonly rated "at the input to the main generator" before losses in the electric drive come into play.
At the 27,500 kg/hr (60500 lb/hr) steam flow for the top entry in the table, Wardale Fig. 117 on p 299 gives 59% boiler efficiency or 6.3 lb water/lb reference coal (South African coal was of lower btu), giving 2.5 lb (reference, high-BTU) coal/hp-hr or 7.2% efficiency. This, of course, assumed that his gas-producer combustion system (GPCS) is working properly, which is not a sure thing as he writes about.
At the 15,000 kg/hr (33000 lb/hr) steam flow for the bottom entry in the table, Wardale's boiler efficiency gives 68% or 7.3 lb water/hp-hr, giving 1.44 lb coal/hp-hr or 12.5% efficiency, which is consistent with the chart in Fig. 82 on p 267.
Same locomotive, big range in efficiency numbers depending on how it is operated, as they say your gas mileage may vary (YMMV) from the EPA gas mileage sticker on the car you just purchased. Again, the 2018 Tom Morrison books suggests that the New York Central may not have operated its Niagara at its full 6600 (indicated) hp capability requiring an enormous 110,000 lb/hr evaporation rate, but rather at some lower power levels that save coal.
Paul Milenkovicwhere the 6% thermal efficiency for modern steam in US practice comes from
The place to start is with the coal fuel, and that right there opens a sort of can-of-worms controversy. There are some who advise designing the locomotive to burn 'run-of-mine' coal, to minimize cost -- perhaps the most glaring example of this was the ACE3000, which would have used sophisticated modular 'coal pods' designed to be taken off the tender and fueled remotely ... likely on swap-body truck chassis ... at mines offering the cheapest price. (Which I think would be suicide on a sophisticated heavy 4-8-4 with obligate condensing and at least the premise of semi-automated firing... but that's just an opinion.) The AAR and other industry sources recommended (through the late 1940s, after which it increasingly ceased to matter) the opposite approach: coal mix of good ranks and ashing characteristics, properly sized to about 2" and then handled to minimize fines, and kept well-washed. I would note that some clean-coal approaches recommend co-firing with dolomite (to flux the glassing components in the ash and knock down some of the sulfur if the coal contains it, as many do) and this can be applied as a slurry to the lump coal at a convenient time during the sizing and washing.)
Thermodynamics often fails to consider the effects of having to deal with the post-combustion ash. Porta did some studies (regrettably mostly still untranslated from his somewhat idiomatic Spanish) including Detroit-style 'ashaveyor' systems to move ash to modular storage for longer run time and better ecological 'optics' -- but there are issues to be dealt with. I believe Wardale's concern with unburned fuel falling through the grates and 'quenching' in the ash is interesting, but it may be much less difficult on a North American engine with proper FireBar-style rocking grates and good secondary-air arrangements -- the difficulty being that with proper 'fuel design' there is no more place for 'recycling' the remnants in the firebox than there was for all the crackpot schemes of recycling unburned 'sparks' from various places in the combustion-gas path that Angus Sinclair (and the locomotive in my avatar) so derided.
That the flame in as much of the radiant section be luminous is confirmed from other material (notably a somewhat sarcastic article about 'college boys ordering zero smoke' from around 1910 that discussed the empirical observation that sometimes visible smoke needfully accompanied best performance) and at least to me it has clear implications for practical North American firing if you're interested in chasing a practical method of GPCS. At some point you will hit the same quandary that Tom Burlingame did: the sophistication required to burn coal effectively rapidly hits the limits of what even a good human fireman can watch and respond to, and the required 'computer assistance', while practical in many stationary plants, rapidly goes out of cost and reliability bounds when implemented on a typical reciprocating locomotive (cf. Wardale's sarcastic invocation of Bulleid concerning 'computerized valve gear' saying 'well, it wouldn't be a steam locomotive then').
There are also implications for the draft, which tie fairly intimately with some characteristics of the steam circuit and of water-rate concerns. I recognized the need for fairly early prediction and anticipation of load changes as early as the 1970s; in a system of GIS-enabled GPS pervasiveness (as in Carnegie-Mellon's proposal to enhance autonomous-vehicle performance) this becomes almost trivial to incorporate in a firing-assistance system. But what is important is the ability to get sustainable but rapid steaming out of a boiler that is (1) not forced beyond its economical generation range, and (2) is not subjected to careful management of differential thermal expansion (something that seems to have repeatedly bit Porta's designs on the butt in practice).
Lima's late response (in keeping with the fascinating but regrettable story about why the Alleghenies were so grossly overweight and the fact then so carefully covered up) was to design the rear firebox to enhance radiant uptake... at the cost of enormous additional structure, sometimes in exotic alloys that didn't hold up well in practice, necessitating an additional carrying axle and other structure to support even on an eight-coupled design. One is tempted to note, as for Franklin type C valve gear, that the success of this approach aside from Alleghenies can almost be open-and-shut determined from how many of these dream locomotives Lima actually sold. Considerations have been noted that boiler design is different for 'long life' than it was at the height of cheapest-possible commodity locomotive use, where it was deemed feasible to design for an unrebuilt service life as short as two years before major maintenance. Armchair designers and crayonista thermodynamicist fiends tend to forget things like that...
There was an interesting tradeoff developing in the 1940s regarding the choice of pressure. As will be determined in later analysis in these posts, a reciprocating locomotive can run with reasonable thermodynamic efficiency on a fairly wide range of throttle pressure -- subject more to careful 'steam-drying' superheat management than actual use of nominal pressure in the engine itself -- with the Scylla being the mechanical problems with higher pressure and the Charybdis being effective water rate and its issues. One has to design the firing system to suit a boiler system that optimizes the tradeoff -- occasionally by taking elements of both depending on circumstance. Which is why I bring it up now.
There is no point, and never was any point, in having tubes much shorter, or much longer, than the 'magic ratio' that gives about 20' in practice. (The actual 'best ratio' was supposed to be right around 406 to 408-and-a-fraction, but this is a bit like calculating the events in British Caprotti to be accurate to 3% cutoff when in practice you'd never use anything less that about 15% unless actively drifting.) It would be nice to have confirmed results on optimal flue length with Besler tubes, but very little application of those to conventional firetube boilers was conducted (the Stanley boiler, which used a multiplicity of small tubes for structural strength, and use of tapered monotubes neatly bracketing the use in small vehicle boilers, and maintenance of flues containing them being onerous when crap fuel and to-a-price servicing are "economic" concerns) but I suspect that some approximation of hydraulic depth for flues containing Beslers can be easily confirmed even with CFD.
Where the low-hanging fruit is more easily plucked is in the Rankine cycle, particularly nonlinear places like Holcroft-Anderson recompression or long-gas-path 'steaming' economization not pressure-equalized with the convection section's nominal pressure. A reasonable guiding adage is 'a pound's worth of heat conserved is a pound that doesn't need to be burned' -- but that is subject to a number of economic considerations outside that of pure efficiency.
Keep in mind that all the factors you're considering are not necessarily scaled correctly for modern power, and that a number of trends in late big-steam practice were counterproductive in actual practice. In my opinion it's best to start with first principles to be sure you've modeled the subsystems reasonably 'accurately' (which is a big problem in Fry's detail formulae, which quietly rely on empirical constants from early-'20s practice not good for later designs).
I presume you have downloaded and read Fry's book now that it's been released as a PDF. It is one of the must-read references if you actually want to design locomotives.
I have lost the patent reference to Snyder's preheaters, but you should look them up (and, arguably, use them). These use relatively spent exhaust steam to preheat the primary air entering between the ashpan and the grate, with the option of further condensing or pumping the condensate into the tender to increase the effective water rate. Reportedly C&O on test got a better than 10% improvement (although I disremember exactly on what) -- the point being that all this improvement was substantially free as far as cost or flexibility of operation are concerned.
Likewise, DNB in the water legs, although a serious problem, cannot be directly observed or quantified, but it can be addressed with better 'waterwall' circulation. Best is probably still a Lamont firebox, which runs about 6x the steam demand per hour through waterwall passages and then uses a cyclone arrangement to do mechanical steam separation with the circulation velocity. But you can get a long way toward this by using the Cunningham arrangement, which is a jet pump fed from downcoming areas in the convection boiler that feeds nozzles strategically placed in the water legs with fan diffusers above them. Cunningham did not (to my knowledge) subsequently optimize where the enhanced leg circulation goes once it reaches to the crown (note that it has to get back through the convection section too, at some point) so there is still some opportunity to visit undiscovered country of great potential beauty.
At present it looks as if the 'late' Koopmans characterization of what multiple nozzle front ends do is the best 'working' principle, but there are still things that were done in front ends that I think make great common sense but may not 'work as anticipated'. One of these is precisely how the combustion gas becomes velocity-matched and then entrained in the 'jet' coming from the exhaust nozzles. At one point Wardale thought that putting a hollow flow-smoothed duct at the 'mouth' of the exhaust petticoat was the best way to induce the flow from 'all the tubes and flues' to converge cleanly around the base of the stack and then rotate direction through what may be a large number of degrees to line up with the steam exhaust. It is surprising how little attention this gets.
It is easy to forget that the whole of the fire, except for a few shocks in the gas flow, is subatmospheric, and that oxygen only represents about a fifth of the available air. This has implications for generating the kind of luminous flame that maximizes practical radiant uptake by the direct heating surfaces over a reliably wide range of 'turndown'. It also gives you another handle on why forcing the boiler to the grate limit can be a poor practice.
Somewhere in here we should take up Besler tubes as well, as once you have arranged not to throw random sticky soot in the tubes and flues, these make thermodynamic sense in the convection section. You have to understand what they actually do, though, and I was lucky to have come across a chemical-industry reference from the early '80s that actually covered this (in a different context).
As a fun exercise: see whether you can find parameters for the PRR Q2 boiler (as modified for the 8000hp mechanical turbine) and see if you can 'improve' it to provide the vaporware 9000hp that PRR advertised (circa 1948) it would be able to make 'in competition' with first-generation diesels.
The difficulties with long-path heat recovery in economizing sections a la Franco-Crosti have been long solved: the first order of business is to fire only with good washed 2" coal of the proper rank and ash characteristics -- as promoted in publications like Railway Age up to the end of the '40s -- and the second order of business is to avoid sulfur without co-firing with dolomite; in any case we now have better coatings and alloys to deal with corrosion below the dew point of sulfuric acid/sulfur trioxide, and again the most critical characteristic of a large modern locomotive -- water rate -- can be economized nicely even before we get into the joy of recovering the latent heat liberated by the condensation of the water in the fuel exhaust (which, for example, the multipass Donlee TurboFire XL could do within stack-train clearances...) and, at least theoretically, use some of that mass for water-rate reduction as well.
Proper use of the steam is a whole subject in itself, but definitely one that bears on locomotive performance when we eventually get around to it.
Great post, Paul. I will leave it to you and others engineering pros to apply a derived formula to well-known locomotives in North America and Europe.
In evaluating claims of steam locomotive thermal efficiency under different conditions and circumstances, I choose to consider the efficiency in which coal is converted into superheated steam by the boilers separately from the efficiency in converting steam into indicated horsepower or "power in the cylinders." I will get to power "at the wheel rim" and "at the drawbar" later on.
Alfred Bruce (1952) The Steam Locomotive in America, p 142 Chart 15 gives curves "of a 484 steam locomotive." He is being coy: a boiler capable of evaporating 110,000 lbs steam/hr to produce a peak of 6600 indicated horse power ("in the cylinders) is no garden-variety Northern -- it is probably Kieffer's Niagara built by Bruce's employer ALCo. This stated evaporation and horsepower is a steam rate of 16.7 lb/hp-hr -- under some unstated conditions of speed and cutoff.
Bruce goes on for some pages talking about "Cole ratios" (an ALCo employee preceding Bruce by some decades) and grate area and how it really isn't grate area that matters but firebox volume, with 7 cu ft of firebox volume typically provided for every square-foot of grate area leading to Chart 16 on p 176. Here, the evaporation in pounds of water/pounds of coal, my second efficiency factor, is charted to be about 7.5 at the low end of firing rate to 5 at the high end. The low end corresponds to about 50 lb coal fired/sq-ft of grate and the high end reaches 200 lb coal/sq-ft of grate.
Now 50 lb coal/sq-ft of grate is a rather low rate, but Tom Morrison (2018) The American Steam Locomotive in the 20th Century hints that towards the end of steam, some railroads were aiming to "derate" their steam locomotives to fire them at this low rate to get better efficiency. 200 lb coal/sq-ft, on the other hand, is what one would fire during test to have bragging rights as the the peak horsepower of a particular locomotive.
In Chart 16 on p 176 of Bruce, the efficiency of converting coal to superheated steam is about 70 percent at the low end whereas it dips to 45 percent at the high end, with nearly half coal being blown out the stack as partially combusted fines when "forcing" a boiler to high output. 70 percent boiler efficiency seems rather low, but remember that you don't want the flue gases condensing and ruining the tubes -- something that was tried to that unfortunate effect in the Franco-Crosti system of heat recovery for feed-water heating. So a boiler starts out with about 80% efficiency in recovering heat from the flue gases. From graphs in Wardale (2013), The Red Devil and Other Tales of the Age of Steam, the graphs of "combustion efficiency" seem to extrapolate to no better than 90% at zero rate of combustion. This suggests that fully 10% of the carbon in the coal is swept up and thrown out with the ashpan discharge, where Wardale hints at an "afterburning ashpan" as some attempt to not waste that fuel fraction.
Let's start with a "reference coal" that I assign 14,000 BTU/lb as its "low heating value."
Standard Grade Coal - Heat Value (engineeringtoolbox.com)
Since no one attempts to extract heat from coal until the water generated by burning its volatile component is condensed -- the sulfur in coal makes that water quite corrosive -- I am using the "lower heating value" where that water remains vapor as it exits the stack. There is also a much larger spread between the low and high heating value of natural gas with a much higher hydrogen fraction, and this is what a "high efficiency" gas furnace does with its "secondary heat exchanger", but we won't attempt that with coal.
Indulge me to use easier-to-remember rounded conversion factors of 3400 BTU/kWHr, .75 kHWr/hp-hr, a coal-rate of 3 lb/hp-hr works about to
(3 lb coal) (14000 BTU/lb) (1 kW Hr/3400 BTU)/ (.75 kWHr) = 16.47 BTU heat in/BTU work out or a thermal efficiency of 1/16.47 = 6.07 % thermal efficiency. I shall round this to 6% thermal efficiency for a locomotive burning three pounds of a high BTU coal per horsepower-hour of mechanical output.
These numbers should be easy to remember. Burning 3 lb/hp-hour is 6% thermal efficiency which is considered characteristic of "late-era" steam, about 2 lb/hp-hour (claimed as peak efficiency for some locomotives under some conditions) works out to 9% efficiency and so on.
OK, now let's break out these numbers into the two factors. Back to Chart 15 in Bruce -- if you are firing at a high rate, you are getting maybe 5.5 lb steam/lb coal from Chart 16, with 16.7 lb steam/hp-hr in the notes to chart 15, or about 3 lb coal/hp-hr and the 6% rule-of-thumb thermal efficiency of late-generation steam in American practice. 3 lb coal/hp-hr times 6600 Hp and you are at 19,800 lb coal/hr, 101 sq-ft of grate, this brings us to 196 lb coal/hr/sq-ft grate, where these numbers "tie up" with the low boiler efficiency under peak hp conditions.
Let me leave it for discussion right now as to where the 6% thermal efficiency for modern steam in US practice comes from and consider later on how the lbs steam/lbs coal ratio (coal ratio? Cole ratio?) along with the lbs steam/hp-hr can vary under different designs and operating conditions.
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