C. D. PURDON, M. AM. SOC. C. E. (by letter).--Some years ago the writer,
in making studies for grade revision, found that the tractive power of a
locomotive up grade becomes less as the length of the grade increases,
and in some unknown proportion. This was a practical confirmation of the
saying of locomotive engineers, that the engine "got tired" on long
grades. On a well-known Western railroad, with which the writer is
familiar, experiments were made for the purpose of rating its
locomotives. The locomotives were first divided into classes according
to their tractive power, this being calculated by the usual rule, with
factors of size of cylinders, boiler pressure, and diameter of drivers,
also by taking one-fourth of the weight on the drivers, and using the
lesser of the two results as the tractive power.
Locomotives of different classes, and hauling known loads, were run over
a freight division, the cars being weighed for the purpose; thus the
maximum load which could be handled over a division, or different parts
of a division, was ascertained, and this proportion of tonnage to
tractive power was used in rating all classes.
Of course, this method was not mathematically accurate, as the condition
of track, the weather, and the personal equation of the locomotive
engineers all had an effect, but, later, when correcting the rating by
tests with dynamometers, it was found that the results were fairly
practical.
There were three hills where the rate of grade was the same as the rest
of the division, but where the length was much in excess of other grades
of the same rate.
Designating these hills as _A_, _B_, and _C_, the lengths are,
respectively, 2.44, 3.57, and 4.41 miles. There were no other grades of
the same rate exceeding 1 mile.
In one class of freight engines, 10-wheel Brooks, the weight of the
engine was 197,900 lb.; tender, 132,800 lb.; weight on drivers, 142,600
lb.; boiler pressure, 200 lb.; and tractive power of cylinders, 33,300
lb.
On Hill _A_ these engines are rated at 865 tons, as compared with 945 on
other parts of the division. As the engine weighs 165 tons and the
caboose 15 tons, 180 tons should be added, making the figures, 1,045 and
1,125 tons. Thus the length of the grade, 2.44 miles, makes the tractive
power on it 92% of that on shorter grades.
On Hill _B_, the rating, adding 180 tons as above, is 1,160 and 1,230
tons, respectively, giving 94% for 3.57 miles.
On Hill _C_, the rating, with 180 tons added, is 1,130 and 1,230 tons,
making 92% for 4.41 miles.
Taking the same basis as the author, namely, 4.7 lb. per ton, rate of
grade × 20, and weight on drivers, gives:
Hill _A_, 18.078%, remainder of division, 19.462%
Hill _B_, 20.068%, " " " 21.279%
Hill _C_, 19.549%, " " " 21.279%
It will be noted that the author uses the weight on the drivers as the
criterion, but the tractive power is not directly as the weight on the
drivers, some engines being over-cylindered, or under-cylindered; in the
class of engines above mentioned the tractive power is 23.35% of the
weight on the drivers.
The writer made a study of several dynamometer tests on Hill _C_. There
is a grade of the same rate, about 1 mile long, near this hill, and a
station near its foot, but there is sufficient level grade between this
station and the foot of the hill to get a good start.
All the engines of the above class, loaded for Hill _C_, gained speed on
the 1-mile grade, but began to fall below the theoretical speed at a
point about 2-1/4 miles from the foot of the hill. This condition
occurred when the trains stopped at the station and also when they
passed it at a rate of some 16 or 18 miles per hour, the speed becoming
less and less as the top of the hill was approached.
The writer concludes that the author might stretch his opinion as to
using heavier rates of grade on shorter hills than 10 miles, and indeed
his diagram seems to intimate as much, and that, for economical
operation, the maximum rate of grade should be reduced after a length of
about 2 miles has been reached, and more and more in proportion to the
length of the hill, in order that the same rating could be applied all
over a division.
This conclusion might be modified by local conditions, such as an
important town where cars might be added to or taken from the train.
While it does not seem practicable to the writer to calculate what the
reduction of rate of grade should be, a consensus of results of
operation on different lengths of grade might give sufficient data to
reach some conclusion on the matter.
The American Railway Engineering and Maintenance of Way Association has
a Committee on "Railway Economics," which is studying such matters, but
so far as the writer knows it has not given this question any
consideration.
The writer hopes that the author will follow up this subject, and that
other members will join, as a full discussion will no doubt bring some
results on a question which seems to be highly important.
JOHN C. TRAUTWINE, JR., ASSOC. AM. SOC. C. E. (by letter).--In his
collection of data, Mr. Randolph includes two ancient cases taken from
the earliest editions (1872-1883) of Trautwine's "Civil Engineer's
Pocket-Book," referring to performances on the Mahanoy and Broad
Mountain Railroad (now the Frackville Branch of the Reading) and on the
Pennsylvania Railroad, respectively.
In the private notes of John C. Trautwine, Sr., these two cases are
recorded as follows:
"On the Mahanoy & Broad Mtn. R. R., _tank_ Engines of 35 tons, _all
on 8 drivers_, draw 40 _empty_ coal cars weighing 100 tons, _up_ a
continuous grade of 175 ft. per mile for 3-1/2 miles; & around
curves of 450, 500, 600 ft. &c. rad., at 8 miles an hour. (1864)
This is equal to 77-14/100 tons for a 27-ton engine." (Vol. III, p.
176.)
"On the Penn Central 95 ft. grades for 9-3/4 miles, a 29-ton engine
all on 8 drivers takes 125 tons of freight and 112 tons of engine,
tender, & cars, in all 237 tons,[C] and a passenger engine takes up
3 cars at 24 miles an hour (large 8 wheels). When more than 3, an
auxiliary engine."
It will be seen that Mr. Randolph is well within bounds in ascribing to
the Mahanoy and Broad Mountain case (his No. 10) a date "certainly prior
to 1882," the date being given, in the notes, as 1864; while another
entry just below it, for the Pennsylvania Railroad case, is dated 1860.
It also seems, as stated by Mr. Randolph, quite probable that the
frictional resistance (6 lb. per 2,000 lb.) assumed by him in the
calculation is far below the actual for this Case 10. The small, empty,
four-wheel cars weighed only 4,400 lb. each. Furthermore, the "tons," in
the Trautwine reports of these experiments, were tons of 2,240 lb. On
the other hand, the maximum curvature was 12° 45' (not 14°, as given by
the author), and the engine was a tank locomotive, whereas the author
has credited it with a 25-ton tender.
After making all corrections, it will be found that, in order to bring
the point, for this Case 10, up to the author's curve, instead of his 6
lb. per 2,000 lb., a frictional resistance of 66 lb. per 2,000 lb. would
be required, a resistance just equal to the gravity resistance on the
3.3% grade, making a total resistance of 132 lb. per 2,000 lb.
While this 66 lb. per ton is very high, it is perhaps not too high for
the known conditions, as above described. For modern rolling stock, Mr.
A. K. Shurtleff gives the formula:[D]
Frictional resistance, on tangent, }
in pounds per 2,000 pounds } = 1 + 90 ÷ C,
where _C_ = weight of car and load, in tons of 2,000 lb. This would
give, for 4,400-lb. (2.2-ton) cars, a frictional resistance of 42 lb.
per 2,000 lb.; and, on the usual assumption of 0.8 lb. per 2,000 lb. for
each degree of curvature, the 12.75° curves of this line would give 10
lb. per ton additional, making a total of 52 lb. per 2,000 lb. over and
above grade resistance, under modern conditions.
In the 9th to 17th editions of Trautwine (1885-1900), these early
accounts were superseded by numerous later instances, including some of
those quoted by the author.
In the 18th and 19th editions (1902-1909) are given data respecting
performances on the Catawissa Branch of the Reading (Shamokin Division)
in 1898-1901. These give the maximum and minimum loads hauled up a
nearly continuous grade of 31.47 ft. per mile (0.59%) from Catawissa to
Lofty (34.03 miles) by engines of different classes, with different
helpers and without helpers.
Table 2 (in which the writer follows the author in assuming frictional
resistance at 4.7 lb. per 2,000 lb.) shows the cases giving the maximum
and minimum values of the quantity represented by the ordinates in the
author's diagram, namely, "Traction, in percentage of weight on
drivers."
It will be seen that the maximum percentage (16.1) is practically
identical with that found by the author (16) for grade lengths exceeding
17 miles.
Near the middle of the 34-mile distance there is a stretch of 1.51
miles, on which the average grade is only 5.93 ft. per mile (0.112%),
and this stretch divides the remaining distance into two practically
continuous grades, 19.39 and 13.13 miles long, respectively; but, as the
same loads are hauled over these two portions by the same engines, the
results are virtually identical, the maxima furnishing two more points
closely coinciding with the author's diagram.