Chapter 6

The resistance to bodies moving through the water increases as the square of the velocity; and the power, or coal, necessary to produce speed varies or increases as the cube of the velocity. This is a law founded in nature, and verified by facts and universal experience. Its enunciation is at first startling to those who have not reflected on the subject, and who as a general thing suppose that, if a vessel will run 8 miles per hour on a given quantity of coal, she ought to run 16 miles per hour on double that quantity. I think that it may be safely asserted that in all cases of high speed, and ordinary dynamic or working efficiency in the ship, the resistance increases more rapidly than as the squares. The _rationale_ of the law is this: the power necessary to overcome the resistance of the water at the vessel's bow and the friction increases as the square; again, the power necessary to overcome the natural inertia of the vessel and set it in motion, increases this again as the square of the velocity, and the two together constitute the aggregate resistance which makes it necessary that the power for increasing a vessel's speed shall increase as the cube of the velocity. But whatever the _rationale_, the law itself is an admitted fact by all theoretical engineers, and is proven in practice by all steamships. In evidence of this, I will give the following opinions.

In his treatise on "The Marine Engine," Mr. Robert Murray, who is a member of the Board of Trade in Southampton, England, says in speaking of the "Natural law regulating the speed of a steamer," page 104: "These results chiefly depend upon the natural law that _the power expended in propelling a steamship through the water varies as the cube of the velocity_. This law is modified by the retarding effect of the _increased resisting surface_, consequent upon the weight of the engines and fuel, so that the horse power increases in a somewhat higher ratio than that named." It must be understood that when he speaks of power, horse power, etc., it is simply another form of representing the quantity of coal burned; as the power is in the direct ratio of the quantity of fuel.

Bourne, the great Scotch writer upon the Screw Propeller, in his large volume published by Longmans, London, page 145, says, in concluding a sentence on the expensiveness of vessels: "Since it is known that the resistance of vessels increases more rapidly than the square of the velocity in the case of considerable speeds."

Again, at page 236, on "the resistance of bodies moving through the water," he says: "In the case of very sharp vessels, the resistance appears to increase nearly as the square of the velocity, but in case of vessels of the ordinary amount of sharpness the resistance increases more rapidly than the square of the velocity."

Again, on page 231, in speaking of the folly of a company attempting to run steamers sufficiently rapidly for the mails at the price paid for them, he says: "At the same time an increased rate of speed has to be maintained, which is, of course, tantamount to a further reduction of the payment. In fact, their position upon the Red Sea line is now this, that they would be better without the mails than with them, as the mere expense of the increased quantity of fuel necessary to realize the increased speed which they have undertaken to maintain, will swallow up the whole of the Government subvention. _To increase the speed of a vessel from 8 to 10 knots it is necessary that the engine power should be doubled._" This work of Mr. Bourne is now the standard of authority on the subject of which he treats, the world over.

Again, Mr. James R. Napier, of London, known as one of the largest and most skilled engine-builders in Great Britain, in the discussion of the dynamic efficiency of steamships in the proceedings of the "British Association" in 1856, page 436, says: "_The power in similar vessels, I here take for granted, at present varies as the cube of the velocity._" The power simply represents the coal; in fact, it is the coal.

Mr. Charles Atherton, the able and distinguished Chief Engineer of Her Majesty's Royal Dock Yard, at Woolwich, has published a volume, called "Steamship Capability," a smaller volume on "Marine Engine Classification," and several elaborate papers for the British Association, the Society of Arts, London, the Association of Civil Engineers, and the Artisans' Journal, for the purpose of properly exposing the high cost of steam freight transport as based on the law above noticed, and the ruinous expense of running certain classes of vessels of an inferior dynamic efficiency. When but a few weeks since in London, I asked the Editor of the "Artisan," if any engineer in England disputed the laws relative to power, on which Mr. Atherton based his arguments. He replied that he had never heard of one who did. I asked Mr. Atherton myself, if in the case of the newest and most improved steamers, with the best possible models for speed, he had ever found any defect in the law of, the resistance as the squares, and the power as the cubes of the velocity. He replied that he had not; and that he regarded the law as founded in nature, and had everywhere seen it verified in practice in the many experiments which it was his duty to conduct with steam vessels in and out of the Royal Navy. I think, therefore, that with all of these high authorities, the doctrine will be admitted as a law of power and speed, and consequently of the consumption of coal and the high cost of running steamers at mail speeds.

It is not my purpose here to discuss this law, or treat generally or specially of the theory of steam navigation. It will suffice that I point out clearly its existence and the prominent methods of its application only, as these are necessary to the general deduction which I propose making, that rapid steamships can not support themselves on their own receipts. The general reader can pass over these formulae to p. 69, and look at their results.

I. TO FIND THE CONSUMPTION OF FUEL NECESSARY TO INCREASE THE SPEED OF A STEAMER.

Suppose that a steamer running eight miles per hour consumes forty tons of coal per day: how much coal will she consume per day at nine miles per hour? The calculation is as follows:

8^3 : 9^3 :: 40 : required consumption, which is, 56.95 tons. Here the speed has increased 12-1/2 per cent., while the quantity of fuel consumed increased 42-1/2 per cent.

Suppose, again, that we wish to increase the speed from 8 to 10, and from 8 to 16 miles per hour. The formula stands the same, thus:

Miles. Miles. Tons Coal. Tons Coal. 8^3 : 10^3 :: 40 : _x_, = 78.1 8^3 : 16^3 :: 40 : _x_, = 320.

II. TO FIND THE SPEED CORRESPONDING TO A DIMINISHED CONSUMPTION OF FUEL.

Murray has given some convenient formulae, which I will here adopt. Suppose a vessel of 500 horse power run 12 knots per hour on 40 tons coal per day: what will be the speed if she burn only 30 tons per day? Thus:

40 : 30 :: 12^3 : V^3 (or cube of the required velocity,) Or, reduced, 4 : 3 :: 1728 : V^3, Equation, 3 x 1728 = 5184 = 4V^3, Or, 5184/4 = Cube root of 1296 = 10.902 knots = V, required velocity.

Thus, we reduce the quantity of coal one fourth, but the speed is reduced but little above one twelfth.

III. RELATION BETWEEN THE CONSUMPTION OF FUEL, AND THE LENGTH AND VELOCITY OF VOYAGE.

The consumption of fuel on two or more given voyages will vary as the square of the velocity multiplied into the distance travelled. Thus, during a voyage of 1200 miles, average speed 10 knots, the consumption of coal is 150 tons: we wish to know the consumption for 1800 miles at 8 knots. Thus:

150 tons : C required Consumption :: 10^2 knots x 1200 miles : 8^2, Knots x 1800 miles. Then, C x 100 x 1200 = 150 x 64 x 1800,* Or, C x 120,000 = 17,280,000 Reduced to C = 1728/12 = 144 tons consumption.

Suppose, again, that we wish to know the rate of speed for 1800 miles, if the coals used be the same as on another voyage of 1200 miles, with 150 tons coal, and ten knots speed:

We substitute former consumption, 150 tons for C, as in the equation above, marked *, and V^2 (square of the required velocity) for 64, and have,

150 x 100 x 1200 = 150 x V^2 x 1800, Or, 120,000 = 1800V^2, Reduced, 1200/18 = V^2, And V = square root of 66.66 = 8.15 knots.

From the foregoing easily intelligible formulae we can ascertain with approximate certainty the large quantity of coal necessary to increase speed, the large saving of coal in reducing speed, as well as the means of accommodating the fuel to the voyage, or the voyage to the fuel. It is not necessary here to study very closely the economy of fuel, as this is a question affecting the transport of freight alone. When the mails are to be transported, economy of fuel is not the object desired, but speed; and, consequently, we must submit to extravagance of fuel. This large expenditure of coal is not necessary in the case of freights, as they may be transported slowly, and, consequently, cheaply. But one of the principal reasons for rapid transport of the mails is that they may largely anticipate freights in their time of arrival, and consequently control their movements.

I recently had an excellent opportunity of testing the large quantity of fuel saved on a slight reduction of the speed, and give it as illustrative of the law advanced. We were on the United States Mail steamer "Fulton," Captain Wotton, and running at 13 miles per hour. Some of the tubes became unfit for use in one of the boilers, and the fires were extinguished and the steam and water drawn off from this boiler, leaving the other one, of the same size, to propel the ship. An intelligent gentleman who happened to know that we were using only one boiler, and consequently, but half the power, remarked to me that it was very strange that the ship was still going about eleven miles per hour, without any sail. He said: "It is strange, sir; two boilers of equal size drove us thirteen miles per hour; and here now but one boiler drives us nearly eleven miles, or nearly as fast; when common-sense teaches that the one boiler would drive us only six and a half miles per hour. How is that?" I then explained to him very clearly the natural law relative to power and speed, (_See Rule II., page 68_,) which he at once comprehended and admitted, but with the remark: "Indeed, sir, I would have testified that she ought with one boiler to have gone at only half the speed; or that going at six miles with one boiler, she would go twelve with two."

As it will be interesting to the general reader to examine the details of the increased consumption of fuel at increased rates of speed, I present the following elaborate table recently prepared by Mr. Atherton for his new edition of "Steamship Capability," according to the formula above noticed, and the performance of the best type of vessel in the Royal Navy, the steamer "Rattler." Mr. A. found a higher efficiency in this vessel per horse power than any other in the Navy, and consequently based the consumption of coal in the table on the assumption that the mail and passenger vessels generally should be of as good contractive type as "Rattler." I shall present also another table showing a much larger consumption of fuel by an inferior type of vessel. I use these tables because they are thoroughly correct, and quite as perfect as any that I could construct on the same formula; and because they carry with them the weight of probably the highest authority in Great Britain.

COAL TABLE: No. I.

_Displacement,[B] Speed, and Fuel consumed per Day, for Mail, Passenger, and Freight Steamers, whose locomotive performance is equal to that of the best class of ocean steam vessels; assuming the consumption of fuel to be 4-1/2 lbs. per indicated horse power per hour, equal to 33,000 lbs. raised one foot in one minute. The quantity consumed is expressed in tons per day of 24 hours._

[B] Displacement refers to the number of cubic feet of water displaced by the hull; allowing thirty-five cubic feet to the ton.

KEY: A: SHIP'S DISPLACEMENT.

+----+----+----+----+----+----+----+----+----+----+----+----+----+----+---- | SPEED PER HOUR.--NAUTICAL MILES. A +----+----+----+----+----+----+----+----+----+----+----+----+----+----+---- | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 -----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+---- TONS.|TONS|TONS|TONS|TONS|TONS|TONS|TONS|TONS|TONS|TONS|TONS|TONS|TONS|TONS|TONS -----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+---- 100|1.04|1.65|2.47|3.51|4.82|6.41|8.32|10.6|13.2|16.3|19.7|23.7|28.1|33.0|38.5 125|1.20|1.92|2.86|4.07|5.59|7.44|9.66|12.3|15.3|18.9|22.9|27.5|32.6|38.3|44.7 150|1.36|2.16|3.23|4.60|6.31|8.40|10.9|13.9|17.3|21.3|25.9|31.0|36.8|43.3|50.5 175|1.51|2.40|3.58|5.10|7.00|9.31|12.1|15.4|19.2|23.6|28.7|34.4|40.8|48.0|56.0 200|1.65|2.62|3.91|5.57|7.65|10.2|13.2|16.8|21.0|25.8|31.3|37.6|44.6|52.4|61.2 | | | | | | | | | | | | | | | 250|1.92|3.04|4.54|6.47|8.87|11.8|15.3|19.5|24.3|29.9|36.3|43.6|51.7|60.9|71.0 300|2.25|3.44|5.13|7.30|10.0|13.3|17.3|22.0|27.5|33.8|41.0|49.2|58.4|68.7|80.1 350|2.40|3.81|5.68|8.09|11.1|14.8|19.2|24.4|30.5|37.5|45.5|54.5|64.7|76.2|88.8 400|2.62|4.16|6.21|8.85|12.1|16.2|21.0|26.7|33.3|41.0|49.7|59.6|70.8|83.3|97.1 450|2.84|4.50|6.72|9.57|13.1|17.5|22.7|28.8|36.0|44.3|53.8|64.5|76.6|90.1|105 | | | | | | | | | | | | | | | 500|3.04|4.83|7.21|10.3|14.1|18.7|24.3|30.9|38.6|47.5|57.7|69.2|82.1|96.6|113 600|3.43|5.46|8.14|11.6|15.9|21.2|27.5|34.9|43.6|53.7|65.1|78.1|92.8|109 |127 700|3.81|6.05|9.02|12.8|17.6|23.5|30.4|38.7|48.4|59.5|72.2|86.6|103 |121 |141 800|4.16|6.61|9.87|14.0|19.3|25.6|33.3|42.3|52.9|65.0|78.9|94.6|112 |132 |154 900|4.50|7.15|10.7|15.2|20.8|27.7|36.0|45.8|57.2|70.4|85.4|102 |122 |143 |167 | | | | | | | | | | | | | | | 1000|4.83|7.67|11.4|16.3|22.4|29.8|38.6|49.1|61.3|75.5|91.6|110 |130 |153 |179 1250|5.60|8.90|13.3|18.9|26.0|34.5|44.8|57.0|71.2|87.6|106 |127 |151 |178 |208 1500|6.33|10.0|15.0|21.4|29.3|39.0|50.6|64.4|80.4|98.9|120 |144 |171 |201 |234 1750|7.01|11.1|16.6|23.7|32.5|43.2|56.1|71.3|89.1|110 |133 |159 |189 |223 |260 2000|7.66|12.2|18.2|25.9|35.5|47.3|61.3|77.9|97.4|120 |145 |174 |207 |243 |284 | | | | | | | | | | | | | | | 2500|8.89|14.1|21.1|30.0|41.2|54.8|71.2|90.5|113 |139 |169 |202 |240 |283 |329 3000|10.0|16.0|23.8|33.9|46.5|61.9|80.4|102 |128 |157 |191 |228 |271 |319 |372 3500|11.1|17.7|26.1|37.6|51.5|68.6|89.0|113 |141 |174 |211 |253 |301 |354 |412 4000|12.2|19.3|28.8|41.1|56.3|75.0|97.3|124 |155 |190 |231 |277 |329 |386 |451 5000|14.1|22.4|33.5|47.7|65.4|87.0|113 |144 |179 |221 |268 |321 |381 |448 |523 | | | | | | | | | | | | | | | 6000|15.9|25.3|37.8|53.8|73.8|98.3|128 |162 |203 |249 |302 |363 |431 |506 |591 7000|17.7|28.1|41.9|59.6|81.8|109 |141 |180 |224 |276 |335 |402 |477 |501 |654 8000|19.3|30.7|45.8|65.2|89.4|119 |155 |196 |245 |302 |366 |439 |522 |613 |715 9000|20.9|33.2|49.5|70.5|96.7|129 |167 |215 |265 |327 |396 |475 |564 |663 |774 10000|22.4|35.6|53.1|75.6|104 |138 |179 |228 |285 |350 |425 |510 |605 |712 |830 | | | | | | | | | | | | | | | 12500|26.0|41.3|61.7|87.8|120 |160 |208 |265 |330 |406 |493 |592 |702 |826 |963 15000|29.4|46.6|69.6|99.1|136 |181 |235 |299 |373 |459 |557 |668 |793 |933 |1088 20000|35.6|56.5|84.4|120 |165 |219 |285 |362 |452 |556 |675 |809 |961 |1130|1318 25000|41.3|65.6|97.9|139 |191 |254 |330 |420 |525 |645 |783 |939 |1115|1311|1529 30000|46.6|74.0|111 |157 |216 |287 |373 |474 |592 |728 |884 |1060|1258|1480|1727 -----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----

By the inspection of this table we can see in condensed form the coal-cost of any speed as high as twenty miles per hour, and for any size of vessel from one hundred tons to thirty thousand tons. Let us find in the left hand column a vessel of 2,500 tons displacement. Pursuing the line along to the right we find in the second column 8.89 tons of coal, which a steamer of this displacement would burn in 24 hours, if running, as indicated at the head of the column, 6 Nautical miles per hour.

In the next column, under the head of 7 Nautical miles per hour, we find that she would burn in one day 14.1 tons; or one and a half times as much coal to gain one sixth more speed:

Again, at 8 miles per hour she burns 21.1 tons; nearly three times as much as at six miles:

At 9 miles she burns 30 tons: above twice as much as at 7, and nearly four times as much as at 6, although the speed is but half doubled:

At 10 miles per hour she burns 41.2 tons; about twice as much as at 8 miles, although the speed is increased only one fourth. At 10 she burns 34 per cent. more than at 9, although the increase of speed is only eleven per cent. (_See pages 67 and 68_):

At 11 miles per hour she will burn 54.8 or 55 tons; nearly three times as much as at 8 miles per hour, and six times as much as at 6 miles per hour:

At 12 miles per hour she will burn 71.2; about thirty per cent. more than at eleven miles per hour, although gaining but 9 per cent. in speed; nearly twice as much as at ten miles per hour, three and a half times as much as at 8, five times as much as at 7, and above eight times as much as at 6 miles per hour. It is here seen that to double the speed the consumption of fuel has increased eight-fold, which verifies my statements hitherto made on this subject. We have already seen that to gain two miles of speed on any stated speed, it was necessary to double the quantity of fuel used.

At 13 miles per hour she burns 90.5 tons. This is burning two and a fourth times as much coal as if she ran only 10 miles per hour. Now, at this speed, the steamer will reach Southampton or Liverpool in 10 days and 6 hours, which is equivalent to 10 days and 12 hours burning fuel, allowing six hours for heating and starting, and which would make an aggregate consumption of 950 tons of coal for the passage of this steamer of 2,500 displacement or probably 3,000 tons register.

At 14 miles per hour she burns 113 tons. This is nearly three times as much as 10 miles per hour. At this speed the steamer would reach Southampton or Liverpool in 9 days, 12 hours, and 30 minutes, supposing the distance to be 3,200 miles from New-York, or say 9 days 18-1/2 hours coal-burning time, and would consume an aggregate of 1,104-1/2 tons. As this is but little above the distance from New-York to Southampton, and under that from Panama to California, and about the tonnage of the steamers running, the time being within eleven days generally, it will be seen how large is the cost of running the steamers of the Pacific Mail Steamship Company, those on the European routes, and also those between New-York and Aspinwall. As the route of the Havre and Bremen steamers is much longer, they are compelled to run slightly slower, or they would be filled up with their own fuel and power. Taking a Collins steamer of 3,000 tons, which we find in the line below, and we see that in running 14 miles per hour as they have frequently done, the consumption would be 128 tons per day, or 1,252 tons for the passage. And yet, one of those steamers could make 12 miles per hour on 80.4 tons per day, or at 11 miles per hour on 61.9, or less than half that used at 14. But pursuing this table we see that,

At 15 miles per hour she would burn 139 tons, or three and a half times as much as at 10 miles.

At 16 miles per hour she would burn 169 tons, or precisely eight times as much as at 8 miles per hour. Here again doubling the speed is found to be an enormous expense.

At 17 miles per hour she burns 202 tons per day.

At 18 miles per hour the consumption is 240 tons per day.

At 19 miles per hour she burns 283 tons coal per day; and

At 20 miles per hour she burns 329 tons per day. At 20 miles per hour she would run 480 miles per day, a thing as yet wholly unheard of, and would consume on the voyage of 6 days and 16 hours, say 6 days and 22 hours, 2,276 tons of coal. It would be clearly impossible for her to carry her own fuel; as the immense boiler and engine power necessary to secure this speed would of itself fill a ship of this size, to say nothing of the fuel which also would nearly fill it. Then, we may never expect any such ship to attain any such speed as seventeen, eighteen, or twenty miles per hour on so long a voyage without recoaling.

Seeing thus the enormous increase in the consumption of fuel for a moderate increase in the speed, we are enabled the better to appreciate the large expense incurred in running ocean steamers sufficiently rapidly for successful mail and passenger purposes. We will further pursue these inquiries by examining in this table the consumption for vessels of 6,000 tons, which would make the displacement of the ship nearly 5,000 tons, such as the "Adriatic," the "Vanderbilt," and the "Niagara." It appears that at 8 miles per hour they would consume 33 tons per day; at 10 miles, 65 tons; at 12 miles, 113 tons; at 13 miles, 144 tons; at 14 miles, 179 tons; at 15 miles, 221 tons; and at 16 miles, 268 tons per day. This is supposing this speed to be maintained on an average across the ocean, in all kinds of weather, which this size of steamer could not do without more engine and boiler power than any of them have. With such additional power the ships noticed would have scarcely any available room for freight or any thing else. One thing is very clear from this table, that when steamers run at very moderately slow rates of speed, their consumption of fuel is very small; and that when they leave this low freighting speed, for that of the necessarily rapid mails and passengers, the consumption increases to an extent and with a rapidity that would seem almost incredible at first view.

COAL TABLE: No. II.

_The following coal table is constructed in all respects as the preceding, but for a lower type of vessels, or those whose coefficient of Dynamic performance is inferior to that upon which the previous table is estimated. As a consequence, this style of vessel requires more fuel._

KEY: A: SHIP'S DISPLACEMENT.