> Questions about tractive effort, etc. of steam locos.
Well, this is an interesting subject. I am currently writing a book
about it. It will be 300-400 pages, dealing with energy efficiency,
power production and tractive effort characteristics. But, to
summarise the tractive effort characteristics….
Steam is a direct drive mechanism, the tractive force is coupled in a
fixed manner and essentially without intervening linkages to the
pressure in the cylinders. The relationship between the pressure in
the cylinders and the tractive effort at the wheel is conditioned by
the geometry of the link, varies roughly as the sine of the angle of
rotation of the wheel (providing the connecting rod is "sufficiently
long") and is the sum total of that contributed by the different
cylinders (which, in a 2-cylinder engine are 90 degrees out of phase).
Superimposed on this rhythmical variability is that introduced by the
varying pressure within the cylinder. At slow speeds, the pressure in
the cylinder is set primarily by the boiler pressure and how the
"cut-off" is set. The pressure at the start of the stroke is high,
near to boiler pressure, and remains at that level until cut-off.
Thereafter it falls along an adiabatic/isentropic path according to a
law essentially given by P*(V^x)= k, where x is about 1.33 and k is a
constant. V is the volume, directly proportional to the excursion of
the piston. At a certain point, the exhaust port is opened and the
pressure falls quickly to a lower value, perhaps 1.2 atmospheres. It
remains generally at this level while the steam is expelled during the
return stroke, until the exhaust closes and the remaining steam is
compressed along another adiabatic line (but with x=1.1 approx). The
consequence is that the pressure follows a complex cycle and is
mirrored on the other side of the piston, 180 degrees out of phase.
The driving force is the difference between the two. This difference
in pressure can be averaged over the entire cycle and is called the
"mean effective pressure" (m.e.p.). mep rises as the cut-off rises,
because the cylinder spends more of its time at the higher starting
pressure and falls adiabatically a smaller amount during expansion.
The relationship between cut-off (the "gear" of a steam loco) and
pressure is a roughly hyperbolic one. At maximum cut-off, usually in
the 70 to 80% range for locos with conventional valve gears, the mep
in the cylinder will be 85% of the boiler pressure. The overall
effect is that the force exerted on the rail follows a form that I
would describe as roughly shaped by a bishop's mitre. The ratio of
the peak of this curve to its average value is called the "coefficient
of fluctuation", which is a measure of how "slippery" the locomotive
will be. At any one point, the sum tractive effort of the coupled
wheels will be the result of that contributed by each of the
cylinders. This tends to raise the overall level and squash down the
peakiness of the "bishops mitre"….. higher, less variable effort. At
higher speeds, however, two things happen. First, the cut-off is
shortened by the driver and secondly the admission pressure falls
below that of the boiler. This is called throttling, it occurs
because of "friction" in the (relatively) narrow pipes and valves, and
is an isenthalpic process that essentially drops both the pressure and
the ability of the steam to liberate energy for a given expansion. At
very high speeds, say 400 rpm, the mep can be less than half the
theoretical value because of this effect. From this, it will be
realised that the tractive effort varies over a single wheel rotation
and also falls off with speed. At high speeds, inertial forces in the
system considerably distort the "mitre", although they do not alter
the average level. Locomotives were rated by their tractive effort,
which was the AVERAGE theoretical tractive effort over one wheel
revolution for a given mep, usually 85% of boiler pressure. There is
a standard formula for this, essentially TE= .85*d^2Pl/D, where
d=cylinder diameter, P=mep, l = cylinder stroke and D = wheel
diameter. This formula is relatively accurate in predicting mean
tractive effort at low speed. If a further factor, the "diagram
factor", varying with speed, is taken into account, this formula
predicts the tractive effort at higher speeds too.
Now, we can answer the question about speed tractive effort curves.
Tractive effort is constant for a steam loco at low speeds, up to a
certain speed, determined by the point at which the gear begins to be
wound back or the effect of throttling begins to be become pronounced.
This constant TE is, to some extent, also determined by whether it is
high enough to slip the wheels despite the weight pressing down on
them. Usually steel on steel has a coefficient of friction of about
0.3, so the rule of thumb was to not design a loco where the rated
tractive effort was more than 30% of the adhesive weight. As speed
rises, the tractive effort falls away, first at a high rate, then at a
slower rate, assymptoting toward zero. By the top rated speed of the
locomotive, it could well be down to 15% of the rated value.
Wheel slip, as mentioned, occurs when the peak tractive effort (the
top of the "bishops mitre") exceeds the product: (coefficient of
friction) * (adhesive weight). The adhesive weight depends upon the
mass of the loco, how many wheels it presses down on and how much of
the mass rests on the driving, versus the other wheels (a bit
tautological, this is). The tractive effort, however, doesn't depend
upon the number of wheels…. One could theoretically get the same
tractive effort out of a 2-2-0 as a 4-14-4, provided you had machinery
that wouldn't snap under the strain! But the adhesive weight
determines whether this is possible without slipping. Two cylinder
locos are for MOST of their wheel rotation, more prone to slip than 3
or 4-cylinder engines (higher coefficient of fluctuation). Naturally,
the chance of slipping is higher at low speeds where mep is higher and
throttling is not prominent. The coefficient of friction of
steel-on-steel also affects the propensity to slip. Damp, greasy
rails might have a coefficient as low as 0.15; well-sanded, clean ones
as high as 0.5.
As for controlling a wheel slip when it occurs, the usual practice is
to close the regulator, called a throttle in the U.S.. This throttles
the pressure down (as in throttling, described above), and reduces the
mep, so that the TE falls below the adhesion limit. The regulator is
opened again when the slip is controlled. To my knowledge no-one ever
developed an automatic system for this, it was always
human-controlled.
"High tech" steam locos are mostly oxymorons, at least as far as
tractive effort goes. The high tech S.African loco probably refers to
a loco with a modified "producer gas" combustion bed. The French,
especially, high-teched their locos by reducing steam friction
(throttling) and thereby increasing power and efficiency. The
Argentinians were still doing this sort of thing in the late 1960's.
For a couple of high tech locos, one a spoof, one probably not, see
this newsgroup for April 1st and Railway Digest for November 1994.
These proposals surface from time to time and were in fact the reason
for my writing the book.
Does this help?
Geoff Lambert