A Catechism of the Steam Engine

by C. E. John Bourne

497. Q.--Does the expense of traction increase much with an increased speed?

A.--Yes; it increases very rapidly, partly from the undulation of the earth when a heavy train passes over it at a high velocity, but chiefly from the resistance of the atmosphere and blast pipe, which constitute the greatest of the impediments to motion at high speeds. At a speed of 30 miles an hour, the atmospheric resistance has been found in some cases to amount to about 12 lbs. a ton; and in side winds the resistance even exceeds this amount, partly in consequence of the additional friction caused from the flanges of the wheels being forced against the rails, and partly because the wind catches to a certain extent the front of every carriage, whereby the efficient breadth of each carriage, in giving motion to the air in the direction of the train, is very much increased. At a speed of 30 miles an hour, an engine evaporating 200 cubic feet of water in the hour, and therefore exerting about 200 horses power, will draw a load of 110 tons. Taking the friction of the train at 7-1/2 lbs. per ton, or 825 lbs. operating at the circumference of the driving wheel--which, with 5 ft. 6 in. wheels, and 18 in. stroke, is equivalent to 4,757 lbs. upon the piston--and taking the resistance of the blast pipe at 6 lbs. per square inch of the pistons, and the friction of the engine unloaded at 1 lb. per square inch, which, with pistons 12 in. in diameter, amount together to 1,582 lbs., and reckoning the increased friction of the engine due to the load at 1/7th of the load, as in some cases it has been found experimentally to be, though a much less proportion than this would probably be a nearer average, we have 7018.4 lbs. for the total load upon the pistons. At 30 miles an hour the speed of the pistons will be 457.8 feet per minute, and 7018.4 lbs. multiplied by 457.8 ft. per minute, are equal to 3213023.5 lbs. raised one foot high in the minute, which, divided by 33,000, gives 97.3 horses power as the power which would draw 110 tons upon a railway at a speed of 30 miles an hour, if there were no atmospheric resistance. The atmospheric resistance is at the rate of 12 lbs. a ton, with a load of 110 tons, equal to 1,320 lbs., moving at a speed of 30 miles an hour, which, when reduced, becomes 105.8 horses power, and this, added to 97.3, makes 203.1, instead of 200 horses power, as ascertained by a reference to the evaporative power of the boiler. This amount of atmospheric resistance, however, exceeds the average, and in some of the experiments for ascertaining the atmospheric resistance, a part of the resistance due to the curves and irregularities of the line has been counted as part of the atmospheric resistance.

498. Q.--Is the resistance per ton of the engine the same as the resistance per ton of the train?

A.--No; it is more, since the engine has not merely the resistance of the atmosphere and of the wheels to encounter, but the resistance of the machinery besides. According to Mr. Gooch's experiments upon a train weighing 100 tons, the resistance of the engine and tender at 13.1 miles per hour was found by the indicator to be 12.38 lbs.; the resistance per ton of the train, as ascertained by the dynamometer, was at the same speed 7.58 lbs., and the average resistance of locomotive and train was 9.04 lbs. At 20.2 miles per hour these resistances respectively became 19.0, 8.19, and 12.2 lbs. At 441 miles per hour the resistances became 34.0, 21.10, and 25.5 lbs., and at 57.4 miles an hour they became 35.5, 17.81, and 23.8 lbs.

499. Q.--Is it not maintained that the resistance of the atmosphere to the progress of railway trains increases as the square of the velocity?

A.--The atmospheric resistance, no doubt, increases as the square of the velocity, and the power, therefore, necessary to overcome it will increase as the cube of the velocity, since in doubling the speed four times, the power must be expended in overcoming the atmospheric resistance in half the time. At low speeds, the resistance does not increase very rapidly; but at high speeds, as the rapid increase in the atmospheric resistance causes the main resistance to be that arising from the atmosphere, the total resistance will vary nearly as the square of the velocity. Thus the resistance of a train, including locomotive and tender, will, at 15 miles an hour, be about 9.3 lbs. per ton; at 30 miles an hour it will be 13.2 lbs. per ton; and at 60 miles an hour, 29 lbs. per ton. If we suppose the same law of progression to continue up to 120 miles an hour, the resistance at that speed will be 92.2 lbs. per ton, and at 240 miles an hour the resistance will be 344.8 lbs. per ton. Thus, in doubling the speed from 60 to 120 miles per hour, the resistance does not fall much short of being increased fourfold, and the same remark applies to the increase of the speed from 120 to 240 miles an hour. These deductions and other deductions from Mr. Gooch's experiments on the resistance of railway trains, are fully discussed by Mr. Clark, in his Treatise on railway machinery, who gives the following rule for ascertaining the resistance of a train, supposing the line to be in good order, and free from curves:--To find the total resistance of the engine, tender, and train in pounds per ton, at any given speed. Square the speed in miles per hour; divide it by 171, and add 8 to the quotient. The result is the total resistance at the rails in lbs. per ton.

500.Q.--How comes it, that the resistance of fluids increases as the square of the velocity, instead of the velocity simply?

A.--Because the height necessary to generate the velocity with which the moving object strikes the fluid, or the fluid strikes the object, increases as the square of the velocity, and the resistance or the weight of a column of any fluid varies as the height. A falling body, as has been already explained, to have acquired twice the velocity, must have fallen through four times the height; the velocity generated by a column of any fluid is equal to that acquired by a body falling through the height of the column; and it is therefore clear, that the pressure due to any given velocity must be as the square of that velocity, the pressure being in every case as twice the altitude of the column. The work done, however, by a stream of air or other fluid in a given time, will vary as the cube of the velocity; for if the velocity of a stream of air be doubled, there will not only be four times the pressure exerted per square foot, but twice the quantity of air will be employed; and in windmills, accordingly, it is found, that the work done varies nearly as the cube of the velocity of the wind. If, however, the work done by a given quantity of air moving at different speeds be considered, it will vary as the squares of the speeds.

501. Q.--But in a case where there is no work done, and the resistance varies as the square of the speed, should not the power requisite to overcome that resistance vary as the square of the speed?

A.--It should if you consider the resistance over a given distance, and not the resistance during a given time. Supposing the resistance of a railway train to increase as the square of the speed, it would take four times the power, so far as atmospheric resistance is concerned, to accomplish a mile at the rate of 60 miles an hour, that it would take to accomplish a mile at 30 miles an hour; but in the former case there would be twice the number of miles accomplished in the same time, so that when the velocity of the train was doubled, we should require an engine that was capable of overcoming four times the resistance at twice the speed, or in other words, that was capable of exerting eight times the power, so far as regards the element of atmospheric resistance. We know by experience, however, that it is easier to attain high speeds on railways than in steam vessels, where the resistance does increase nearly as the square of the speed.