The usual method of rating locomotives at present is that of tonnage. That is to say, a locomotive is rated to handle a train, weighing a certain number of tons, over a division. This is preferred to a given number of loaded or empty cars because of the indefinite variation in the weights of the loads and the cars themselves.
In the determination of a-locomotive rating there are several factors to be considered, namely, the power of the locomotive, adhesion to the rail, resistance of the train including the normal resistance on a level, and that due to grades and curves, value of momentum, effect of empty cars, and the effect of the weather and seasons.
The power of a locomotive and its adhesion to the rails has already been considered. From the formula given, the tractive power can be calculated very closely from data already at hand.
There are three methods in use for obtaining the proper tonnage rating. First, a practical method which consists in trying out each class of engine on each critical or controlling part of the division and continuing the trials until the limit is reached. Second, a more rapid and satisfactory method is to determine the theoretical rating. Third, the most satisfactory method is, first, to determine the theoretical rating and then to check the results by actual trials.
The value of the momentum of a train is a very important element in the determination of the tonnage rating of locomotives on most railroads. In mountainous regions, with long heavy grades, there is little opportunity to take advantage of momentum, while on undulating roads, it may be utilized to the greatest advantage. An approach to a grade at a high velocity when it can be reduced in ascending the same, enables the engine to handle greater loads than would otherwise be possible without such assistance. Hence, stops, crossings, curves, water tanks, etc., will interfere with the make-up of a train if so located as to prevent the use of momentum. It is necessary, therefore, to keep all these points in mind when figuring the rating of a locomotive for handling trains over an undulating division.
The ordinary method of allowing for momentum is to deduct the velocity head from the total ascent and consider the grade easier by that amount.
For example: Suppose that a one per cent grade 5,000 feet long is so situated that trains could approach it at a high speed. The total rise of the grade would be 50 feet but 15 feet of that amount could be overcome by the energy of the train, leaving 35 feet that the train must be raised or lifted by the engine. The grade in which the rise is 35 feet in 5,000 would be a 0.7 per cent grade, so that if the engine could exert sufficient force to overcome the train resistance and that due to a 0.7 per cent grade, the train could be lifted the remainder of the height by its kinetic energy. In this case, the 5,000 feet of one per cent grade could be replaced by a grade of 0.7 per cent 5,000 feet long, and the effect on the load hauled by the engine Would be the same if in the latter case the energy of the train were not taken into account. Since the height to which the kinetic energy raises the train is independent of the length of the grade, its effect becomes far less when the grades are long than when short. Thus, for a one per cent grade 1,000 feet long, the total rise being only 10 feet, the kinetic energy would be more than sufficient to raise the weight of the train up the entire grade leaving only the frictional resistance to be overcome by the engine; whereas if the grade were 50,000 feet in length, or a total rise of 500 feet, the energy of the train would only reduce this rise about 15 feet, leaving a rise of 485 feet or the equivalent of a 0.99 per cent grade to be overcome by the engine, a reduction not worth considering.
It is thus seen that the length of a grade exerts a great influence on the value of the momentum.
Within ordinary limits, the following formula gives very accurate results
where
T = number of tons including engine, which can be hauled over a grade with velocities of V and v
d = diameter of cylinder in inches
L = length of stroke in inches
p1, = mean effective pressure in pounds per square inch
D = diameter of driver in inches.
R' = resistance in pounds per ton on a level track due to friction, air curves, and velocity, which may be taken at 8 pounds per ton
a = grade in feet per mile
l = length of grade in feet
V = velocity in miles per hour at foot of grade
v = velocity in miles per hour at top of grade
Thus, with an engine having cylinders 17 inches in diameter, a stroke of 24 inches, driving wheels 62 inches in diameter, and running at a velocity of 30 miles per hour, the formula gave a rating of 738 tons. On actual tests, it was possible to handle 734 tons with a speed of 10 miles an hour at the top of the grade.
The effect of empty cars is to reduce the total tonnage of the train below what could be handled if they were all loaded. The resistance of empty cars when on a straight and level track varies from 30 to 50 per cent more per ton of weight than loaded cars.
In using the formula given above, loaded cars are assumed. For empty cars, 40 per cent should be added. That is to say, if a train is composed of empty and loaded cars and is found to have a ceptain resistance, 40 per cent should be added to the portion of resistance due to the empty cars.
There is considerable difference of opinion regarding the allowance which should be made for the conditions of weather, etc. The following is a fair allowance which has been found to give satisfactory results in practice: Seven per cent reduction for frosty or wet rails; fifteen per cent reduction for from freezing to zero temperature; and twenty per cent reduction for from zero to twenty degrees below. The use of pushing or helping engines over the most difficult grades of an undulating track will increase the train load and thus reduce the cost of transportation.
