_POWER OF LOCOMOTIVES AND TRAIN RESISTANCES._
CALCULATING POWER OF LOCOMOTIVES.
The capacity of engines is generally expressed in horse-power, which
is a measurable quantity; but, for several reasons, that method of
indicating power has not been usually applied to the locomotive. When
practical railroad men hear the size of cylinders, the diameter of
driving-wheels, and the boiler dimensions of a locomotive, mentioned,
they understand what kind of service the engine is adapted for, and
about the weight of train it can haul. As it has been found necessary
for designing and other purposes, to estimate, with some degree of
accuracy, the work a locomotive is capable of doing, it has become
usual to reckon the power of a locomotive by the tractive force it can
exert upon the rails.
PROPORTION OF ADHESION TO TRACTION.
Tractive force is the power which the pistons of the engine are capable
of exerting through the driving-wheels, to move the engine and train.
The efficiency of the engine’s traction is dependent upon the adhesion
of the wheels to the rails; for, where the adhesion is insufficient,
the pistons will slip the wheels, and no useful effect will result. To
prevent the wheels of ordinary American engines from slipping on dry
rails, the weight resting on the drivers must be about five times the
power exerted by the pistons to slip the wheels. To prevent slipping on
wet, unwashed rails, more than double the above weight would be needed.
In practice, locomotives are not provided with weight enough to prevent
the wheels from slipping on a greasy rail: the sand-boxes provide the
means of obtaining adhesion where the rails are in bad order. A common
practice is to place upon the drivers weight equal to about six times
the power exerted to slip the wheels, which leaves a small margin for
wet rails. Many locomotives have power sufficient to slip the wheels
on dry rails; but such engines generally have boilers too small for
the cylinders, or the distribution of weight on the drivers is badly
effected.
ESTIMATING TRACTIVE POWER.
The easiest way of calculating the tractive power of a locomotive is by
use of the following simple formula, first propounded by Pambour:--
_d^{2}Lp_
_T_ = ---------.
_D_
_d_ = the diameter of the cylinder in inches.
_L_ = the length of the stroke in inches.
_D_ = the diameter of driving-wheels in inches.
_p_ = the effective mean pressure on the piston in pounds per
square inch.
_T_ = the equivalent tractive force at the rails in pounds.
This, given in plain terms, reads: Square the diameter of piston in
inches; multiply by the length of stroke in inches; multiply by the
mean pressure of steam per square inch, and divide by the diameter of
the drivers in inches.
We will apply the calculations to the case of the standard Buchanan
passenger engine; the cylinders being 17 by 24 inches, the drivers 68
inches in diameter, and the effective pressure on pistons about 80
pounds. The problem is worked thus:--
17 inches diam. of cylinders.
17
----
289 = square of diameter.
24 inches stroke of piston.
----
6936
80 lbs. per square inch.
------
Diam. of drivers, 68)554880(8160
This gives 8,160 pounds as the pressure exerted to turn the drivers,
which may be accepted as a close approximation to the truth. In using
this formula, the mistake has frequently been made of taking the
quotient to represent the power developed by one cylinder, when, in
fact, it gives the power of both.
A method of calculating the locomotive traction that is a good deal
followed by our engineers, is to ascertain the foot-pounds of work the
engine is doing during each revolution of the drivers. By dividing the
total thus found by the circumference of the drivers in feet, the force
exerted through each foot that the engine moves is found. Taking the
same engine, the pistons 17 inches diameter, give 226.98 square inches
area. This is multiplied by the mean effective pressure of steam,
giving 226.98 × 80 = 18158.4 pounds pressing on each piston through the
whole stroke of four feet.
18158.4
2 pistons.
-------
36316.8
4 feet of stroke.
--------
145267.2
These figures show that 145267.2 foot-pounds are exerted during each
turn of the wheel, whose circumference is 17.8024; therefore, 145267 ÷
17.8024 = 8160, the number of pounds exerted through each foot moved.
There is still another way of figuring out tractive power by
calculating the rotational force and the leverage through which it is
applied to the locomotive wheels. The pressure of steam on the piston
is productive of two strains on the crank-axle. When the crank is
on the dead center, the pressure upon the crank-pin forces the axle
against the axle-box without causing any tendency to rotate: but, when
the crank-pin is on the quarter, the full leverage of the crank is
available for rotation; the leverage increasing gradually as the crank
gets away from the center. The mean effort upon the crank-pin during
one revolution of the crank is to the effort of the piston as .6366 is
to 1.
Now, to find the traction of the locomotive, we multiply the length
of our crank, which is 12 inches, by .6366, which gives 7.64 as the
length of the crank receiving constant pressure from the pistons.
The aggregate steam-pressure has already been found to be for each
piston, 18158.4 pounds. This is multiplied by 7.64; and the product is
divided by the radius of the wheel, which is 34 inches, giving as a
quotient 4080, being the power exerted on one side of the engine. That,
multiplied by 2, gives 8160 pounds,--the same as was found by the other
two methods.
HORSE-POWER OF LOCOMOTIVES.
When people wish to find the horse-power developed by a locomotive, the
indicator is generally employed, and the calculations made from the
diagram. Others figure out locomotive horse-power as they do that of
other engines. By this method, the engine whose traction we have been
investigating might have the horse-power calculated as follows:--
226.98 square inches piston area.
80 pounds mean pressure.
-------
18158.4 piston pressure.
4 feet piston travel each revolution.
-------
72633.6
2 cylinders.
--------
145267.2
145 revolutions per minute.
--------
20063944 ÷ 33000 = 608 horse-power when running
at a speed of 30 miles an hour.
No allowance has been made for frictional losses in any of these
calculations.
One horse-power is equivalent to the work performed in raising 33,000
pounds one foot high in one minute. One pound raised 33,000 feet
high, or 330 pounds raised 100 feet high, would amount to the same
thing. One horse-power is usually spoken of as 33,000 foot-pounds; and
engineers in this country always calculate work by foot-pounds,--that
is, so many pounds raised a certain number of feet. To indicate the
capacity of any prime motor, the foot-pounds of work it is capable of
raising in a given time must be stated. Although the work is often done
without any thing being raised vertically, the power represented would
be capable of raising the equivalent weight in the stated time.
FORMULAS OF TRAIN RESISTANCES.
The work which a locomotive performs in pulling a train is expended in
overcoming the resistances due to wheel-friction, gradients, curves,
and atmospheric or wind pressure. Formulas have been propounded for
calculating all train resistances, but they are utterly untrustworthy
for American railroad trains. The best known formula of this kind is
that given by D. K. Clark in his _Railway Machinery_. One calculation
will show how misleading its figures are when applied to American
railroad train resistances. Figured by the Clark formula, the total
resistance per ton of a passenger train running at a speed of fifty
miles an hour on a straight level track, is 22.6 pounds. By accurate
records with his dynagraph car, Professor P. H. Dudley found the total
resistances of an express train running at a speed of fifty-one miles
an hour, to be 11 pounds. The resistances are so much different under
different conditions, that nothing closer than a loose approximation
can be calculated of the work done by a locomotive, unless indicator or
dynamometer tests are made.
EXPERIMENTS OF TRAIN RESISTANCES ON THE ERIE RAILWAY.
In experiments made with a freight train on the Erie Railway in
1881, reported by Mr. F. M. Wilder to the Railway Master Mechanics’
Association, it was found that the total resistance on a level track
was from 3.25 to 4.5 pounds per ton at speeds under twenty miles an
hour. These figures will approximately represent the resistance due to
wheel and axial friction in summer; but this resistance will be higher
during cold weather, when the oil in the axle-boxes gets frozen. Track
in bad condition will also tend to increase the wheel resistance, and
improperly constructed trucks and wheels will entail the use of more
power to move the train. Where trucks are so defective that they do not
maintain the wheels revolving in parallel planes, the flanges of some
of the wheels will rub on the rail, increasing the resistance. Wheels
out of round; those having the axle out of center, however slight;
wheels of different size on the same axle; and numerous other car-truck
disorders,--all contribute their share in making a train pull hard.
CONDITIONS THAT INCREASE TRAIN RESISTANCES.
In a calm day the atmospheric resistance is very slight under a speed
of twenty miles an hour. To a fast train, atmospheric resistance
becomes an important obstruction. The atmosphere acts on the train
in various ways, that are hard to calculate with any degree of
accuracy,--head resistance to the locomotive, which is presumably equal
to the exposed area of the front of the engine and cab in square feet
multiplied by the air-pressure due to the speed; then, various parts of
the cars present surfaces that the air strikes against, and increases
the resistance; the raised and projecting roofs of passenger coaches
offer an ample area for the wind to hold the train by; and every
opening between the cars permits the wind to obstruct, to some extent,
each individual car. Where wind is blowing freely in a direction to
strike the train on the side, the resistance is greatly increased; the
retardation being due to the wind pushing the car sidewise, so that the
wheel-flanges rub against the rail, and also to the wind obtaining a
strong hold on the front of each car. In the case of a freight train,
the resistance is greatly increased when the doors of cars are left
open; for every car in that condition acts like a parachute to reduce
speed. Freight trains arranged with box cars and flat cars mixed,
obtain more than a fair share of obstruction from the atmosphere; for
every box car that has a space opened in front by a flat car, gets
nearly the full pressure of the wind in its front. It pays in coal to
incur some trouble and delay in putting box cars together. That also
enables the brakemen to get along the train more rapidly than where the
cars are mixed.
In the experiments already alluded to on the Erie Railway, it was
found, in the absence of wind, that the first car of a freight-train
produced atmospheric resistance equal to a surface of sixty-three feet,
multiplied by the air-pressure due to speed; and that each subsequent
car offered a resistance of twenty per cent of that due to the first
car.
RESISTANCE OF CURVES.
Curved track increases the resistance of trains in direct proportion
to the shortness of curvature. In European railways, the character of
the curves is nearly always denominated by the length of radius: in
this country, a railroad curve is described as of so many degrees. The
degree of a curve is determined by the angle subtended at its center by
a chord of 100 feet. To those who think of a curve by its radius, it
may be well to explain that a curve of one degree has a radius of 5,370
feet, and the radius of any curve can be ascertained by dividing these
figures by the number of degrees.
WORK DONE BY A LOCOMOTIVE PULLING A TRAIN.
To pull a train up an ascending gradient, the locomotive has to perform
work similar to the operation of a pile-driving engine in raising its
driving-block. The train is the block raised by the locomotive; and
the lift is not vertical, but up an inclined plane; yet the amount of
work done is reckoned in precisely the same way. When the engine of a
pile-driver raises a block weighing 1,000 pounds a distance of 30 feet,
the work done is 1,000 × 30 = 30,000 foot-pounds: when a locomotive
pulls a train weighing 1,000 tons over one mile rising 30 feet, the
engine performs 30,000 foot-tons of work in that distance by raising
the load alone. The total amount of work done will also include the
energy expended in overcoming wheel-friction and other ordinary train
resistances.
To find the tractive force which the engine must exert through each
foot of the mile traversed in pulling the train described, we must
divide the foot-pounds of work done, by the distance over which the
power was exerted. Thirty thousand foot-tons of work is 60,000,000
foot-pounds. To this we will add 5 pounds additional for every ton of
the train for every foot advanced to cover wheel and wind resistances,
making 86,400,000 foot-pounds of work that the engine has to perform in
hauling the train one mile. This, divided by the number of feet in a
mile, will give 16,363 pounds as the work the locomotive must perform
through each foot,--an effort which is entirely within the capacity of
many consolidation engines.
RECORD OF FAST EXPRESS TRAIN MADE BY PROFESSOR P. H. DUDLEY’S DYNAGRAPH
CAR.
Engineers interested in finding an approximation of the work done in
taking a fast train over a railroad, can make a close estimate by
studying out the figures given below by Professor Dudley. The table
shows the performance of an ordinary locomotive upon a train composed
of three eight-wheel and six twelve-wheel cars; weight 250 tons;
working-weight of engine and tender, 126,000 pounds; cylinders of
engine, 17 by 24 inches; diameter of drivers, 72 inches; weight on
drivers, 48,000 pounds; blowing-pressure on boiler, 135 pounds.
In starting the train, the locomotive would record a tension of 11,000
to 12,000 pounds for one or two hundred feet of distance. After hooking
up, the tension would decrease to about 2,800 or 3,000 pounds; and,
with this pull, the speed of 50 miles an hour was attained in the
fifth mile. As the speed increases, the resistance of the air against
the locomotive becomes greater, and more of its own power is required
to move itself. In starting a train, the working adhesion of the
steel-tired drivers on dry steel rails is about 33 per cent of the
weight upon them, and reduces as the speed increases.
_Tabulation of Part of a Trip of the Dynagraph Car on a Fast Express
Train._
KEY:
1 Numbers of Miles.
2 Time in Minutes and Seconds per Mile.
3 Speed in Miles per Hour.
4 Velocity of the Wind in Miles per Hour.
5 Approximate Grades.
6 Foot-Pounds of Work shown by Dynametrical Curve per Mile.
7 Foot-Pounds of Work per Minute expressed in Horse-Power.
8 Approximate calculated Foot-Pounds of Work required to move the
Locomotive in Horse-Power.
9 Sum of Columns 7 and 8.
===+======+=======+=====+================+============+=====+=====+====
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---+------+-------+-----+----------------+------------+-----+-----+----
1 | 2 54 | 20.68 | | Level | 24,116,233 | 252 | |
2 | 1 34 | 38.31 | 6.0 | Down 5 ft 3 in | 20,035,253 | 369 | 221 | 590
3 | 1 22 | 43.90 | 4.0 | Down 5 ft 3 in | 17,763,214 | 398 | 292 | 690
4 | 1 16 | 47.34 | 3.0 | Level | 15,904,273 | 383 | 418 | 791
5 | 1 11 | 50.70 | 4.5 | Level | 14,871,528 | 382 | 406 | 788
6 | 1 13 | 49.31 | 6.0 | Up 13 feet | 15,284,616 | 383 | 406 | 789
7 | 1 11 | 50.70 | 6.0 | Down 18 feet | 14,458,430 | 369 | 426 | 795
8 | 1 08 | 52.89 | 5.0 | Down 13 feet | 13,219,136 | 354 | 451 | 805
9 | 1 07 | 53.70 | 5.0 | Down 8 feet | 11,566,744 | 319 | 483 | 802
10 | 1 09 | 52.10 | 5.0 | Down 5 feet | 11,773,293 | 310 | 441 | 751
11 | 1 08 | 52.89 | 4.2 | Level | 11,773,293 | 316 | 447 | 763
12 | 1 09 | 52.10 | 5.2 | Down 8 feet | 12,806,038 | 337 | 456 | 793
13 | 1 10 | 51.43 | 6.0 | Level | 12,392,940 | 324 | 443 | 767
14 | 1 10 | 51.43 | 4.5 | Level | 12,806,038 | 339 | 426 | 765
15 | 1 10 | 51.43 | 4.0 | Level | 13,425,685 | 351 | 420 | 771
16 | 1 10 | 51.43 | 3.5 | Level | 13,299,136 | 345 | 415 | 760
17 | 1 08 | 52.89 | 3.0 | Level | 13,838,783 | 371 | 443 | 814
18 | 1 08 | 52.89 | 5.0 | Down 6 feet | 13,219,136 | 354 | 464 | 818
19 | 1 08 | 52.89 | 3.0 | Down 2 feet | 13,219,136 | 354 | 443 | 797
20 | 1 11 | 50.70 | 3.5 | Up 10 feet | 14,838,783 | 379 | 406 | 785
21 | 1 13 | 49.31 | 3.0 | Up 10 feet | 14,458,430 | 362 | 384 | 746
22 | 1 08 | 52.89 | 3.1 | Level | 12,392,940 | 332 | 443 | 775
23 | 1 07 | 53.70 | 3.1 | Down 10 feet | 12,186,391 | 333 | 462 | 797
===+======+=======+=====+================+============+=====+=====+====
The following calculations have been made, to indicate the trains that
various locomotives ought to pull.
_Weights of Train which Locomotives can haul at a Speed
of 20 Miles an Hour under Ordinary Conditions, in Tons
of 2,000 Pounds (not including the Weight of Engine and
Tender)._
=========================+=============================
| TYPE OF LOCOMOTIVE.
+---------+---------+---------
|Type “A.”|Type “B.”|Type “C.”
-------------------------+---------+---------+---------
_On straight track_: | | |
Level | 1,096 |1,664 | 2,226
Grade 20 feet per mile | 547 | 840½ | 1,128
” 40 ” ” | 350 | 545 | 734
” 60 ” ” | 249 | 390½ | 522
” 80 ” ” | 188 | 302 | 410
” 100 ” ” | 148 | 242 | 330
| | |
_On 5-degree curves_: | | |
Level | 921 |1,401½ | 1,876
Grade 20 feet per mile | 464 | 716 | 962
” 40 ” ” | 310 | 485 | 654
” 60 ” ” | 227 | 360½ | 488
” 80 ” ” | 173 | 279½ | 380
” 100 ” ” | 137 | 225½ | 308
| | |
_On 10-degree curves_: | | |
Level | 662 |1,013 | 1,358
Grade 20 feet per mile | 401 | 621½ | 836
” 40 ” ” | 278 | 477 | 590
” 60 ” ” | 207 | 330½ | 448
” 80 ” ” | 160 | 260 | 354
” 100 ” ” | 128 | 212 | 290
=========================+=========+=========+=========
Under the most favorable conditions, loads about fifty per cent
greater than those given above may be hauled.
The calculations are for three types of engine, designated in the
column titles as Type “A,” Type “B,” and Type “C;” these being as
follows:--
_Type A._--American locomotive, with four driving-wheels, and 12,000
pounds weight on each wheel, the total weight of engine being 36 tons.
_Type B._--Mogul or ten-wheeled locomotive, with six driving-wheels,
and 12,000 pounds weight on each wheel, the total weight of engine
being about 42 tons.
_Type C._--Consolidation locomotive, with eight driving-wheels, and
12,000 pounds weight on each wheel, the total weight of engine being
about 54 tons.