The additional cost of one foot of rise and fall is not altogether independent of the rate of grade. We can, however, divide grades into three groups, within which we may say that the cost of a foot of rise and fall is practically uniform. In the first class are grades which may be operated without changing the work of the engine, and which have practically no other effect than a harmless fluctuation of the velocity, but a grade which belongs to this class, when considering a fast passenger-train, will belong to another class when considering a slow and heavy freight-train, and, since it is the slow and heavy freight-trains which must be chiefly considered, a grade will usually be classified with respect to them. The limit of class A therefore depends on the maximum allowable speed, and also depends on the length of the grade and the depth of the sag. If it is permissible to operate all trains through the sag without making any change in the handling of the engine, without changing the throttle-valve or the position of the links, and especially without the use of brakes, then the effect of the sag on the operation of trains is possibly zero, and the sag or hump has no financial importance. The above conditions assume that the engine is working uniformly throughout, that all potential energy which is lost on the steeper grade is regained on the lighter grade. In the case of a sag the change in potential energy merely comes in reversed order. If train resistance and tractive effort were actually independent of velocity, the assumption that class A has no influence on train expenses would be almost theoretically precise. The operation of momentum grades has been considered in Chapter XI (Momentum Grades. 124. Velocity Head). In classifying a sag or a hump we must therefore consider whether trains which run over or through it may be operated without changing any operating conditions. We will very often discover that passenger-trains may be so operated, while freight-trains will need to be handled differently. Therefore such a hump or sag will be classified as belonging to class A for passenger-trains, and to class B or possibly class C for freight-trains.

The next classification (B) applies to grades on which a change of operating conditions becomes necessary. On the down grade it becomes necessary to partially, if not entirely, close the throttle, in order that the velocity shall not become too great. On the other hand, the up grade is so steep that the engine must work considerably harder, consume more coal, and perhaps operate with a longer cut-off and therefore less economy than is possible on the lighter grades. As long as brakes are not used there is no actual loss of energy, except that steam will probably be wasted by being blown through the safety-valve while the train is running down the grade with the throttle closed. The chief disadvantage is due to the uneconomical working of the train on the heavier up grade. On the down grade the losses in fuel consumption, due to radiation, etc., will become a much larger percentage than usual of the useful work actually obtained from the engine.

The third class (C) includes humps which are so high and sags which are so deep that brakes must be applied on the down grade in order to prevent excessive velocity. The loss by the application of brakes is very heavy. The brakes require power for their application which is a considerable tax on the locomotive. The use of brakes causes wear of the brake-shoes and of the wheel-tires, which hastens the deterioration of the rolling-stock. Their use destroys the kinetic or potential energy which had previously been created, while the tax on the locomotive on the corresponding ascending grade is very great.

It may be seen that the classification of these humps and sags is more a matter of the total height of the hump or the depth of the sag rather than the rate of grade. The sag or hump which has a comparatively steep grade on both sides may be almost harmless if the grades are short, or (which amounts to the same thing) if the height or depth is small. On the other hand, a comparatively light grade may become of importance because it is so deep. It is not usual, however, that a sag or hump could be classified in class C if the grade is very light, since it requires a considerable grade to cause a train to attain a dangerous velocity when the steam is turned off. Excessively long sags or humps are practically outside of the line of problems which may thus be considered.

186. Effect On Operating Expenses

As in the previous chapter we may at once throw out a large proportion of the items of expense of an average train-mile. In "maintenance of way and structures." Items 2 to 6 will be variously affected according to the classification of grades. The other items are evidently unaffected.