Tractive effort conversion to hp.

Anybody got any method or equations to transfer from tractive effort to horse power?

 

I'm not entirely certain that's possible. You can have locomotives of identical nominal tractive effort but of widely different horsepower. Take the 350hp 08, that has a nominal tractive effort of 35,000. This is almost identical to the nominal tractive effort of an A4 Pacific, a loco capable of over 2000hp.

 

That's because there is a missing variable - velocity.

 

Good point. So if the max or average or recommended speed could be taken into account. i.e. xxxhp at yy mph?
Or with diesels are the hp values taken at starting, max hp?

 

Force and power and dimensionally different, so as Spamcan says, you can't easily interconvert.

One way to think about it is that a force is needed to overcome a resistance. If you apply that force over a distance, you expend energy (in the older literature, they will often use the word "work" to mean energy). Power is the rate at which energy is expended (i.e. how much energy per unit time), which is why effectively velocity comes into the mix (because distance divided by time is velocity).

So imagine a piston of certain area that has steam at a certain pressure pushing against it. That piston will be subject to a certain force, (which will be related to the tractive effort - TE is a measure of force).

Ignoring the drop in pressure in the cylinder as a result of the expansion of steam (i.e. assuming a piston pushed back along its whole stroke by the full boiler pressure), a certain amount of work will be performed, i.e. energy will be expended, when the piston moves back along the piston. That is, the force x distance = energy.

Then to calculate the power, you need to know how quickly that piston stroke takes place. The faster the stroke, the shorter the time over which that energy is expended and therefore the greater the power. So Force x distance / time = power.

Obviously, the detailed measurement is quite technically complex to perform. But fundamentally, that is the reason why an A4 and an 08 shunter have broadly similar TE, but the A4 needs much greater power to be able to develop that TE at high speed [sup]*[/sup], whereas the 08 can generate a lot of TE, but only at low speeds.

[sup]*[/sup] In practice, the TE of the A4 falls of with speed - Mallard wasn't developing 35,000lbs at 126mph...

Tom

 

Isn't TE analogous to Torque?

With a motor vehicle, you can produce high torque at the wheels from a small capacity engine by use of a suitably low-geared drive train. And so you can with a steam loco, by a combination of long piston stroke (ie crank throw) and small driving wheel size. Hence we have industrial loco's with TE equivalent to a main line class 5 or 6, but a top speed of around 20mph. Speed is limited by boiler capacity and mechanical constraints such as piston speed and bearing wear/lubrication.

 

I agree that TE falls off with speed but I'd wager that at starting an A4 is still developing more hp that a Class 08 at starting although TE's would be largely similar.
Also if you rewheeled an 08 with smaller wheels, the TE would increase but the horsepower from the diesel generator would be the same for any given controller setting.

 

I didn't write anything about TE falling with speed although it does if all other variables are considered constant (they are not).

At any instant Power = Force x Velocity (force is the tractive effort).

To model this properly would lead us to partial differentials.

 

The other point to be considered is that all the energy used in the cylinders has to come from somewhere - by burning coal. So on paper you could design a locomotive with lots of big cylinders and a high boiler pressure that should in theory be powerful, but if - for example - you coupled a Merchant Navy front end to a Terrier grate, you couldn't burn coal fast enough to keep up with the demands from the cylinders.

So to generate power, you have to be able to burn coal at a certain rate.

A rough rule of thumb I have heard is that for a big loco, you can sustain (i.e. not mortgage the boiler) about 50hp per square foot of grate area. So a Merchant Navy (48.5sq ft) should allow a sustained power output of around 2400hp assuming a clean fire, good coal and a competent fireman. A West Country (38.5sq foot) could probably do about 1900hp sustained. Obviously that depends on a front end that can use the steam, and drafting arrangements that allow the coal to be burnt optimally.

Edit: Found the source for that figure - I overestimated a bit. They come from a study by Prof Tuplin in 1956. He gave figures of 49edhp per square foot for 5 minutes sustained, dropping to 36edhp for 60 minutes sustained. These are drawbar horsepower figures; the indicated horse power would be a bit higher. (ihp is what is developed at the cylinders; edhp is what is available to perform useful work pulling a train, after any internal losses in the locomotive).

Prof Tuplin also noted that:

Tom

 

Quite simply Horsepower = T.E (tonf) x speed (ft/s) x 4.073 or T.E.(lbf) x speed (mph) x 0.002667

 

But surely for that to work, you need to know the fall off of TE with speed, i.e. the TE in your calculation is the TE the loco is actually producing at the speed you happen to choose; not the nominal 85% starting TE.

As an example, if I put the nominal TE of an A4 (35,455lbs) at 126mph I get a power output of close to 12,000hp! I know Gresley is well regarded as a loco designer, but surely even Spamcan wouldn't believe the man was capable of designing a loco with that sort of power output!

Tom

 

A 12,000hp A4 would be really something. I'm sure HNG would have been able to design such a powerful machine but I think it would have been a bit bigger than an A4.
Have just been looking at a TE v speed graph of an electric loco and the fall off is quite marked once the thing gets into its stride.

 

Yeah, but would the frames stand up to the punishment? (Ducks and runs for cover...)

Tom

 

Monocoque design eliminating the need for frames.

 

Tuffer simply wanted to know the relationship between T.E. & horsepower. It was pointed out that speed has to be factored into this and I have answered the question quite simply to show the relationship between horsepower, tractive effort (force) and speed. My answer is correct. You, and others, have complicated both the question and the answer!

 

Yes, quite so. Although I haven't checked your unit conversion factors - I haven't worked in Imperial units for decades

However, keep in mind that this relationship is only correct for instantaneous velocity. I don't think it tells us anything very useful.

 

Quite so! One of the depressing things about steam locomotive engineers is that they seem unable to grow out of using Imperial Measurements.

 

I don't think it is depressing. I've used metric units for an awful lot longer than I care to admit but, as I can only think of one UK steam loco that has been built to metric standards (and I may be wrong on that), it is sensible to stick with the imperial units of their design and use. Injectors excepted, of course!

 

It is a fundamental building block in any design process. Yes, in that design process you have to take an awful lot more into account, notably efficiency, you can't get anywhere without that fundamental equation as a starting point.