"375 is the number. It's theoretical in the same sense that 2 plus 2 is theoretically 4."
Except for very large values of 2.
Calculating the power of a locomotive is not too difficult but calculating train resistance is more complicated. In the days of steam, engine ratings were usually given in adjusted tons, not actual tons, and engines had different ratings depending on grade, temperatures, or class of train. Take the New Haven for example using its 1927 engine rating book. Tonnage ratings were given for a "normal" train of cars weighing 50 tons. However, the tonnage had to be equated for varying resistance of cars of different weights, that is, a loaded car or an empty had different resistance. If the normal rating was 2500 tons in 50 cars the equated tonnage would be increased by 12 tons for each car below the normal number, that is the rating would increase to 2512 because loaded cars offer less resistance. But to complicate matters, the rating was increased by only 8 tons if the route was one designated a heavily graded route. Also, the rating was reduced for temperatures below 32 degrees or reduced more below 19 degrees due to greater bearing resistance. Moreover, the "normal" rating was for slow freights; fast freights were given lower tonnage ratings designated "B", "C", "D" or "E". So if you asked what a New Haven R-1 4-8-2 could haul according to its rating, it would depend on the mix of empties and loads, the time of year, the specific route and whether it was a drag, symbol, or fast freight.
nhrandCalculating the power of a locomotive is not too difficult
timz nhrand The calculation is easy, long as you don't worry about whether the answer is right.
nhrand
The calculation is easy, long as you don't worry about whether the answer is right.
This belongs in Bartlett's.
(The sad thing is that it doesn't matter if the calculation is easy or hard as long as you don't have to worry whether the answer is right. And there are plenty of people who go through the numbers knowing that the assumptions likely exceed the meaningful-error range of the model or even the data. At least with pro-formas there's the excuse that people expect to see them before they move on.)
Determining the tractive force produced by the cylinders of a steam locomotive is fairly simple if you accept some assumptions and know specifications. The major assumption is the mean effective steam pressure over the stroke which is usually assumed to be 85% of the boiler pressure at starting but could be different so the calculation is never certain. At speed the mean effectve pressure is almost a guess unless you are using a test indicator which was infrequent and most locomotives were never tested with an indicator. To estimate what a locomotive will actually pull you need to know the tractive force at the steam tender's drawbar. That is you have to know how much of the power of the cylinders is being used to move the locomotive and tender. The resistance of the many bearings of a large locomotive can be very substantial and making assumptions about the resistance can be way off. The bottom line is that what you need to know is well known but what you do know is often insufficient. Nevertheless, you have to make assumptions and a good estimate is often good enough even if not precise.
The thing with Google is that search results are never the same twice.
"Steam locomotive mechanical resistance" until recently turned up the Web site of a man in Austrailia who offered detail mathematical formulas for determining what needs to be subtracted from "indicated" tractive effort (the calculated value of the direct effect of steam acting on the cylinders) to obtain tractive effort at the wheel rim and in turn tractive effort at the drawbar. These formulas took into account all of the sources of friction -- of the pistons sliding in their cylinders, the piston rod sliding in the gland that contains the steam pressure, the crosshead, the crank pins and rods, along with the valve gear and valve. These sources of resistance are before you even consider the Davis formula resistance for the locomotive as a wheeled vehicle on steel rails along with its aerodynamic resistance.
This person was interested in quantifying the sources of mechanical resistance to answer the question you are discussing -- what is the true tractive effort at the drawbar and how does that relate to the tractive effort calculated from steam pressure acting on the cylinder faces?
The interesting this is that the resistance related to steam producing tractive effort at the wheel rim varies with load and speed, but the resistance does not drop off in strict proportion to reduction in load. So at light load (short cutoff working), the mechanical efficiency diminishes from the value at heavy load (long cutoff). Given all of the other source of inefficiency in a steam locomotive, the mechanical resistance does not appear to have been studied in the depth of this one source, but it is a major contributor to losses.
If you think about the low friction of the steel wheel rolling on a steel rail, the resistance of the locomotive when you take its piston-valve-and-rod mechanism into account is multiples of that, more on the level of a radial tire.
This site
http://5at.co.uk/index.php/definitions/terrms-and-definitions/resistance.html
gives some indication of the larger friction of a locomotive in relation to an unpowered by pistons and rods train car but without accounting for the variation in locomotive friction with tractive effort or any resistance value during "drifting."
If GM "killed the electric car", what am I doing standing next to an EV-1, a half a block from the WSOR tracks?
Old issues of Baldwin Locomotives provide some interesting numbers. For example, the January 1932 issue had an article describing dynamometer car tests on a Lehigh Valley 4-8-4 5100 built in early 1931. Included was a table listing engine friction (pounds of resistance) from 54 test cards taken at various locations. The resistance of the engine ranged from 2753 to 5634 lbs. A chart showing this in terms of horsepower showed, for example, that when the cylinders of the 4-8-4 were producing around 3,800 horsepower near 35 miles per hour, horsepower at the drawbar of the tender fell to about 3,500 horsepower due to the consumption of power to move the locomotive. The engine was new so it was said that the mechanical eficiency would probably be increased as the engine becomes more thoroughly worked in.
For design calculations, Baldwin considered the mechanical resistance of a locomotive to be 25 pounds per ton of weight on drivers at all speeds; the resistance of tender and engine trucks was taken from a table in which resistance was determined by speed and weight. For example, for design purposes it was assumed that an 85 ton tender moving at 30 mph would use about 365 lbs of tractive effort.
The July 1932 issue is a favorite of mine because it compared three eight-coupled locomotives of roughly equal starting tractive effort and showed how their different boiler capacities resulted in dramatic differences at speed. The locomotives were a Reading 2-8-0, Frisco 2-8-2 and a Lehigh Valley 4-8-4. At 5 mph each produced about 65,000 lbs of rated tractive force and about 60,000 at the tender drawbar. Using the resistance of 70-ton cars, Baldwin calculated that each could move about 95 cars at 5 mph. However, at 25 mph, the 2-8-0 could only handle 39 cars, the 2-8-2 62 cars and the 4-8-4 69 cars. At 50 mph the 2-8-0 was down to 12 cars, the 2-8-2 was down to 23 and the 4-8-4 down to 27. Although, the Frisco 2-8-2 had more weight on drivers than the 4-8-4, the Northern could maintain a higher mean effective pressure in the cylinders at higher speeds because of the larger boiler. The 4-8-4 could produce about 70,000 lbs of steam per hour, nearly 10,000 lbs more than the 2-8-2. The Reading 2-8-0 was a great engine with a slow coal train and had nearly the weight on drivers as the 4-8-4 but produced only only about 44,000 lbs. of steam an hour and consequently when steam consumption rose at the higher piston speeds, the massive 2-8-0 was no match for the 4-8-4.
nhrandAt 5 mph each produced about 65,000 lbs of rated tractive force and about 60,000 at the tender drawbar. Using the resistance of 70-ton cars, Baldwin calculated that each could move about 95 cars at 5 mph.
timz nhrand At 5 mph each produced about 65,000 lbs of rated tractive force and about 60,000 at the tender drawbar. Using the resistance of 70-ton cars, Baldwin calculated that each could move about 95 cars at 5 mph. Baldwin thought it took 60000 lb drawbar pull to move 95 70-ton cars at 5 mph? On level track? Did they say where they got that?
nhrand At 5 mph each produced about 65,000 lbs of rated tractive force and about 60,000 at the tender drawbar. Using the resistance of 70-ton cars, Baldwin calculated that each could move about 95 cars at 5 mph.
Baldwin thought it took 60000 lb drawbar pull to move 95 70-ton cars at 5 mph? On level track? Did they say where they got that?
Tim,
The most direct way was to test it in the real world. Empirical evidence.
The other way, and this addresses the original question as well, is the "Davis Formula", which simplified is: R=1.3+29/W+0.045V. R is resitance in pounds per ton, W is axle load in tons, and V is speed in miles per hour. There is also a complicated wind resistance component which is negligible at low speeds used to figure tonnage ratings. To the formula you must add Grade resistance at 20#/ton/%, and curve resistance at 1 pound/ton per degree of curve.
Lets run Baldwin's numbers and ASSUME that the 60,000# is at the drawbar. Also assume these are 70 ton gross, nominal 50 ton, cars which seems reasonable for 1900-1940.
R=1.3+29/70/4+.045*5. Be sure to do the 70/4 first to get 17.5.
R=1.3+1.66+.225=3.185#/ton
Now back calculate to check 95*70 = 6650 tons which goes into 60,000 9.02 times which implies resistance of 9.02 pounds per ton vs. 3.185 calculated. We did not account for TE required to accellerate that mass to 5 MPH. The Davis equation assumes virtually flawless track. Bad track both increases rolling resistnace and decreases actual TE. The other issue is adhesion of the loco.
Calculated TE for steam locos generally gives no thought to adhesion. Most steam engines had a "factor of adhestion", typically of about 4, which meant that the weight on the drivers was 4 times the rated TE, which was figured at the wheel rim. The inverse of factor of adhesion is coefficent of friction, which under perfect conditions at low speed is at best 25%. Wet rail will reduce the cof to about 18%, and sand will add about 7%.
The other big deal is that diesel TE drops off as speed increases. Steam does too but at different rates than diesels and for different designs of steam engines drop of differently. The calculations for speed do include air resistance which becomes ever more important as speed increases.
The point is that the answer to the question "What can this engine pull" depends almost entirely on grade, curves, and speed. Without specifying these, the question is unanswerable.
As a practical matter the railroads are always running empirical tests in the sense that every train is a test. They know how to assign power to accomplish the purpose; be it dragging coal up Cranberry Grade, which is a TE game at low speed, or running UPS trailers across the transcon, which is a Horsepower per ton (HPPT)game at high speed.
Mac McCulloch
This reply is to answer how Baldwin computed the train resistance in the article I previously cited which compared a 2-8-0, 2-8-2 and 4-8-4. Baldwin used the formula: R=1.3+(106+2V/W+1)+0.001 V squared. For example, this showed that 70-ton cars had a resistance of 6 pounds per ton on straight level track at 40 miles per hour. I failed to mention that in the example I cited, Baldwin calculated resistance on a 0.3 per cent ascending grade. Therefore, Baldwin added an amount for the grade for the various speeds used in their comparison. On the grade total resistance at 5mph was given as 9.1 per ton, at 20 mph 10, at 40 mph 11.8, and at 50 mph 13 per ton. For the 5 mph comparison, they calculated for the Frisco 2-8-2 a tractive force at rim of drivers of 66,200 lbs with 60,400 lbs pull back of tender. They then used the tender drawbar pull of 9.1 per ton resistance on the 0.3 per cent grade resulting in a hauling capacity of 6,640 tons or 95 70-ton cars. At 40 mph, Baldwin calculated the 2-8-2's drawbar pull was down to 27,700 lbs and its hauling capacity 2,350 tons or 34 70-ton cars.
Hope this helps with the Baldwin example -- Edward J. Ozog
Our community is FREE to join. To participate you must either login or register for an account.