Calculating Tractive Effort for steam locomotives

[michael blunck]

Calculating Tractive Effort for steam locomotives


When modeling a set of train vehicles, an important information needed for the dynamics of a locomotive is "(max) tractive effort" (TE, resp max TE), which is determining the ability of a locomotive to pull up a train. This characteristic detail is somewhat mysterious, and is unfortunately missing from locomotive descriptions most of the time, be them derived from the internet or even from railway books.

The main problem with TE is that it cannot be derived automatically by TTDPatch, although there´s a "default value" for it, calculated from the weight of the locomotive. But this is only a very over-simplified model, taking into account only the maximum transferable force between driving wheels and rails for a fixed factor of friction (µ = 0.3). It neither takes into account the TE generated by the locomotive nor the special limits put in effect by the characteristics of the engine´s drive or its "adhesive weight". Moreover, special building schemes of steam locomotives, like Mallets, Fairlies, Kitson-Meyers, ... cannot be taken into account in that way.


Now, since a couple of months, I´m able to calculate correct TE from steam locomotive data detail, which is given e.g. on J.D.H. Smith´s steam locomotive tables.

This calculation is very accurate. I´m owning parts of the official technical manuals of the DRG (steam locomotives) which contain both calculated and measured TE values for german steam locomotives, and I have repeated that calculation with the needed technical data (cylinder dimensions, boiler pressure, ...). The outcome is very accurate (error < 1%), so it should be a feasible method to calculate TE for every steam locomotive.


So, what´s the benefit for .grf set authors?

I´ve already calculated some french steamers for Snail´s French Set and some serbian steamers for Wile´s Serbian Set and apparently, most of the data for TE (which had been derived from the internet before), have been shown to be wrong.

Moreover, the neat effect of the calculation is that every steam locomotive of every .grf set would get comparable numbers. No more "behemoths", which in real life shrink down to only moderate engines, and vice versa. Then, we have those american engines, where TE is given in pounds and, in addition to wrong numbers for TE at all, the conversion to kN introduces even more uncertainty.

In fact, we´re now able to calculate correct TE for every steam locomotive!


1. For those who like to do it yourself:

Tractive effort (TE) (or "tractive force") is in principal determined by boiler pressure, cylinder proportions, and size of driving wheels:

TE = D² * S * const * p / d

with D = cylinder diameter, S = piston stroke, p = boiler pressure, and d = driving wheel diameter. ("const" being a heuristic constant in the range 0.75 ... 0.85, depending on speed.)

Because of the nature of the limited friction (µ) between wheel and rail, TE is limited by the weight on the driving wheels ("adhesive weight", W_adh), i.e. non-driven and slipping driving wheels cannot transmit force. In addition, friction depends heavily on the condition of the rails: clean, dry or sanded rails increase friction, but grease, ice, mud, leaves, etc. will all cause the locomotive to slip before nominal TE is reached. Thus, to transmit TE as best as possible, the quotient W_adh / W_tot should be as large as possible, which would be best achieved by a locomotive having only driving axles (0-x-0 scheme, W_adh / W_tot = 1).

[This is an excerpt from http://www.tt-forums.net/viewtopic.php?p=573658#573658 ]


Plain steam

Let´s have an example (SNCF 141TD):

I´ll use a slightly different formula which the DRG used for "plain steam" in contrast to "compound steam" in its technical manuals:

TE = 0,8 * p * D² * S * z / 2d (for plain steam)

("z" being the number of cylinders)

According to J.D.H. Smith´s tables, numbers for the french 141TD are as follows:

d = 1420 mm -> 1.42 m (driving wheel diameter)
D = 510 mm -> 0.51m, D² = 0.2601 m² (cylinder area)
S = 660 mm (3) -> 0.66 m (cylinder stroke)
p = 16 atm -> 1621200.384 N/m² (boiler pressure)

Then,

TE = 0.8 * 1621200.384 N/m² * 0.2601 m² * 0.66 m * 3 / (1.42 m * 2) -> 235 kN

This is the "nominal maximum tractive effort".

I.e., for an adhesive weight of 75t and a friction coefficient µ between 0.25 (wet, slippery) and 0.33 (dry, clean) we get:

TE_max = 75 t * 9.81 m/s² * [0.25 ... 0.3 ... 0.33] -> 183 ... 220 ... 242 kN

[This is an excerpt from http://www.tt-forums.net/viewtopic.php?p=573863#573863]


Compound steam

Let´s have an example (MAVAG 601):

This one is a four-cylinder compound engine, hence:

TE = (2 * 0.85 * p * D_hp² * S) / (D_hp²/D_lp² +1) * d

d = 1440 mm -> 1.44 m (driving wheel diameter)
Dhp = 520 mm -> 0.52 m, Dhp² = 0.2704 m² (hp cylinder area)
Dlp = 850 mm -> 0.85 m, Dlp² = 0.7225 m² (lp cylinder area)
S = 660 mm -> 0.66 m (piston stroke)
p = 16 atm -> 1621200.384 N/m² (boiler pressure)

TE = (2 * 0.85 * 1621200.384 * 0.2704 * 0.66) / (0.2704 / 0.7225 +1) * 1.44
TE = (491854.04) / 1.9728 -> 249 kN

TE_max = 94.7 t * 9.81 m/s² * [0.25 ... 0.3 ... 0.33] -> [232 ... 278 ... 306] kN

[This is an excerpt from http://www.tt-forums.net/viewtopic.php?p=709546#p709546]



2. For those who don´t like to calculate TE yourself

Simply post the country and name resp. class number of the steamer in question here and I´ll do the calculation for you.

Please note that calculation of TE for diesels and electrics isn´t possible in this way.

regards
Michael

[Wile E. Coyote]

Michael, very useful things from you, as allways!

Please note that calculation of TE for diesels and electrics isn´t possible in this way.

Then, do you know how to calculate TE for them?

[michael blunck]

do you know how to calculate TE for [diesels & electrics]?

No, not really. OTOH, it shouldn´t be too hard to get real numbers for them, except for very old diesels, possibly.


BTW, I´ve taken a look on the NS steamers, and indeed, TE for them would be slightly higher than written on the NS web page (I think, most of the old numbers were by me too, Hyr?) due to the better calculation now (many of the NS steamers are 4-cylinder engines):

Code: Select all

type       old value new value
NS 2100    75.1kN    89kN
NS 3700    94kN      111kN
NS 3800    94kN      111kN
NS 3900    120.9kN   122kN
NS 4600    139.8kN   n/a
NS 4700    163.6kN   193kN
NS 5000II  95.1kN    112kN
NS 6200    104.9kN   123kN
NS 6300    144.4kN   170kN

regards
Michael

[Hyronymus]

Most values were by you, Michael but I was able to find a limited number in books or on the internet. Thanks for the recalculation .

[michael blunck]

BTW, what´s the difference between

Code: Select all

NS3700 2'C 1006kW 111kN 110km/h 72t 1910

and

Code: Select all

NS3800 2'C 1006kW 111kN 110km/h 72t 1920

except the latter´s streamlined cladding?

And if that´s the only difference, why doesn´t a streamline locomotive travel faster than the same engine without such cladding?

And what´s that NS 4600? I cannot find it on JDH Smith´s page which is very unusual?

regards
Michael

[Hyronymus]

You make some sharp observations about the 3800. It indeed is a streamlined 3700 and that says it all. The weight of the streamline outfit made it hard for the 3800's to effectively profit from the streamlined outfit. The Dutch didn't think it through entirely I guess. Or they realised early on that reduced travel time by streamlined trains would be minimal in a small country like The Netherlands. Did I mention they never constructed fitting coaches?

The 4600 series was a cross-over between the 3700 and 6200 series. They were constructed by Werkspoor N.V. in Amsterdam, no. 518-537 (20 engines).

[Dave]

Fascinating information, Michael. Another thing that can only increase the realism of the game!

[Snail]

I've read that, for compound steamers, the calculation is the same, but you only have to take into account the high pressure cylinders (or anyway the first ones to power the wheels). Is it accurate?

[michael blunck]

I've read that, for compound steamers, the calculation is the same, but you only have to take into account the high pressure cylinders (or anyway the first ones to power the wheels). Is it accurate?

Well, no. Both high-pressure (hp) and low-pressure (lp) cylinders count for TE. But let´s see.

In general, compound engines were designed in such a way that both hp and lp cylinders had equal force despite the drop in pressure. This was achieved by different cylinder areas and, hence, diameters. Usually, by assuming a loss of 60% pressure this would result to a ratio of 1 / 0.4 -> 2.5 for the cylinder areas, a value only slightly varying between engines / builders.

In general, TE for a compound engine would be

TE = (2 * 0.85 * p * D_hp² * S) / (r +1) * d

where r = D_hp²/D_lp² is the ratio of the area of the hp cylinder to that of the lp cylinder, i.e.:

TE = (2 * 0.85 * p * D_hp² * S) / (D_hp²/D_lp² +1) * d

In case D_hp = D_lp this results in the usual formula for plain steam:

-> TE = (2 * 0.85 * p * D² * S) / (D²/D² +1) * d

-> TE = ( 0.85 * p * D² * S) / d

Voilà. (see first post)


Problem is that many compound steam locomotives, especially mallets, could be started as "simple engines", i.e. with hp steam going to both the lp and hp cylinders. Otherwise they would not have been able to start a heavy train. E.g., in the UK, drivers were instructed not to switch into compound mode until a speed of 30mph was passed.

Maybe this custom is what you´re referencing to ("you only have to take into account the high pressure cylinders ")?

regards
Michael

[ostlandr]

Thanks for the excellent tech info!

For later American locomotives, especially those using the "super power" concept, weight on drivers (or rather, lack thereof) was a significant problem. For example, a 4-8-4 Northern type of the 1940s had a significant amount of it's weight on it's leading and trailing trucks. Despite it's huge potential horsepower, it didn't have significantly higher tractive effort than a USRA 2-10-0 of WW 1 vintage. Also, these locomotives (for marketing reasons) were fitted with large-diameter drivers which allowed them to reach 80 mph/130 kph. This was only warranted in locomotives that were regularly used at these speeds, which resulted in "racehorses being hitched to plows."
Trains magazine had an excellent article on this awhile back.

[Snail]

In general, TE for a compound engine would be

TE = (2 * 0.85 * p * D_hp² * S) / (r +1) * d

where r = D_hp²/D_lp² is the ratio of the area of the hp cylinder to that of the lp cylinder

I suppose S in that formula refers to high pressure cyls?

Problem is that many compound steam locomotives, especially mallets, could be started as "simple engines", i.e. with hp steam going to both the lp and hp cylinders. Otherwise they would not have been able to start a heavy train. E.g., in the UK, drivers were instructed not to switch into compound mode until a speed of 30mph was passed.

Maybe this custom is what you´re referencing to ("you only have to take into account the high pressure cylinders ")?

Well, I got this info from here, look down under the "Special Cases" label. And, my bad, it's not the high pressure cyls, but the low pressure ones. And it suggests to crank down the constant from .85 to .8. That's what I did to compute the TE of the French steamers...

[Raichase]

Being essentially a reference post means it won't be replied to very often, thus it may slip away from view, which would be a shame. I've stickied it to rectify this.

[michael blunck]

In general, TE for a compound engine would be

TE = (2 * 0.85 * p * D_hp² * S) / (r +1) * d

where r = D_hp²/D_lp² is the ratio of the area of the hp cylinder to that of the lp cylinder

I suppose S in that formula refers to high pressure cyls?

No, the variables are the same as in the first post:

p = pressure
S = stroke
d = driving wheel diameter
D_hp² = area of high-pressure cylinder
D_lp² = area of low-pressure cylinder


[Wiki article on tractive effort]

it suggests to crank down the constant from .85 to .8.

The constant 0.85 was the Association of American Railroads (AAR) standard for such calculations, [...] European designers used a constant of 0.6 instead of 0.85, [...]

Let´s see.

The german DRG used 0.85.

In Britain, the main-line railways generally used a constant of 0.85 [...]

Well, I´ve seen constants between 0.75 and 0.91 (ALCO). In fact, this is a factor to adjust "p" to the "mean effective pressure" (MEP), so understandingly, it varies from source to source. Feel free to experiment or try to get the original value the french railways used for their steamers. That would be interesting.

In the case of compound locomotives the tractive effort is calculated using the dimensions of the low-pressure cylinder(s) with a constant of 0.80 instead of 0.85.

Well, that´s just number shifting for another rule of thumb. Better use the formula given above.

Again, the validity seems doubtful because the actual starting tractive effort would depend upon which cylinder (high-pressure or low-pressure) happened to be in the starting position.

That´s more important. To keep things simple, I didn´t talk about individual compound designs like, e.g.,

- Smith (Midland type): 1hp and 2 lp cylinders

- deGlehn: 2 hp and 2 lp cylinders

- Vauclain: 2 hp and 2 lp cylinders superimposed

- Webb (LNWR): 2 hp and 1lp cylinders

- von Bories: 1 hp and 1 lp cylinder

almost all in a different arrangement.

That´s exactly why test runs were made to measure real TE.


You may read the Wikipedia article on compound steam locomotives, it´s quite interesting.

regards
Michael

[AndersI]

This is great!

I'd like to have the numbers for the Swedish Sb and R, as I've already drawn them. I'll get back to you when I have actually made any others (I do need to make an F too and was pondering a J).
(I think you already have all the important data for those, right?)

[michael blunck]

Calculation of max TE for some SJ steamers:

SJ littera "Sb" -> 100kN
SJ littera "R" -> 234kN
SJ littera "F" -> 96kN (compound engine)
SJ littera "J" -> 76kN

regards
Michael

[DanMacK]

Michael,

I've been using these extensively for some of my locos, and the numbers jive pretty good with realsitic specs.

One thing I do want to ask, do you have a specific formula for calculating Steam locomotive HP as well?

[michael blunck]

Michael,

Ah, sorry Dan. Didn´t notice your post.

[...] do you have a specific formula for calculating Steam locomotive HP as well?

No. Not really. You´ll need a pV-diagram of the locomotive in question (or an engine indicator (s.b.) to draw it yourself. And o/c, the loco in that case ).

Dobbie-McInnes.jpg (322.41 KiB) Viewed 2201 times

But even then, you´ll get only the theoretical mechanical work. And from that, net power can only be calculated to some degree, because the mechanical degree of efficiency would be unknown. (At least for you and me, possibly not for the engine manufacturer).

regards
Michael

[Snail]

Hi Michael,

Note: The formulas above are only valid for plain steam 2-cylinder engines, i.e., especially not for three- or four-cyclinder engines, Mallets or other special designs.

... so what would be the formula to calculate the TE for a Mallet engine? I've got a few here to calculate, I've got the data and I'd like to see the formula to come up with a plausible number.

[Nagyzee]

Check out the 34th page of the Serbian Train Set! Michael gave a formula for these types of steam locos there, and the TEs of two locos (a four cylinder and a two cylinder one) were calculated with it. So there are even examples to follow there.

[michael blunck]

... so what would be the formula to calculate the TE for a Mallet engine? I've got a few here to calculate, I've got the data and I'd like to see the formula to come up with a plausible number.

As Nagyzee already pointed out, there´s an example in the Serbian Railset thread.

In principle, a Mallet isn´t that different from an ordinary compound engine. Problem is that you don´t know the correct arrangement of cylinders in every case (check my post above).

However, it´d be highly interesting to calculate one of your Mallets and check against "real" TE (measured). I hope you do have at least one available with all stats known!

regards
Michael