The Obstruction to the Navigation of Rivers Caused by the Piers of Bridges
The investigation, which was interrupted at the close of my last communication, as to the amount of power required for the ascent of a steamboat of specified dimen osions through a given draw, was based upon the supposi tion that, so far as vcrtical resistances were concerned, all the power of the engine was usefully employed in overcoming the resistances; that there was no loss of power between the point where it was derived from the engine, and the point where it was applied to overcome the
... d to overcome the resistances. In truth this is f Ilr otherwise. Let us look at the causes of loss of power in its translLission. Those that I will consider arise from three sources-the +it'ti"11 of the wheel, the obliquity with which the pad dle boards (in the case of the ordinary radial wheel) strike the water, and the slip of the wheel. Of thrse three causes of loss of power, the operation of the first two, in case of a steamboat ascending a draw, is similar to theh' operation in case of a steamboat moving in an unobstructed water·way. The operation of the slip is quite different. The first two may then be disposed of by assuming that, of the entire power given up by the engine, kth is consumed by them. As regards the introduction of specific values, I would state that my object is to show 11010 the values are to be used, not what values arc to be used; and while in each case the values given arc based upon the best data In my possession, yet I wish distinctly to disclaim for them great accuracy. The careful reader will readily distin guish between those values which arc liable to mria tion, and those which are not; for instance, when it is stated that, to lift a boat of specified weight up a remou of definite hight requires the expenditnre of a given amount of useful eff ect, there is no liability to error; but when we go fUl"ther, and state that to accomplish this useful effect requires an expendituTe of a particular amount of power at the engine, then we enter the region of uncertamty, the values varying with the medium through which the power is transmitted. In order to determine the loss of power occasioned by the slip of the wheel when in the draw, it will be neces sary to analyze the causes producing the slip. Take the case of a locomotive drawinga train of cars, at a definite uniform velocity, on a level dry track, where the ad 'hesion of the driving wheel to the rail is perfect, and · there is no slipping. The measure of the power expend-· ed, in going a specified distance, is the product of the · equivalent mean pressure on the piston, multiplied by · the distance passed over by the piston. The useful 'effect produced is the product of the constant tractive 'power required to draw such a train at such a velocit.y, 'multiplied by the actual distance passed over. Neglect-· ing the loss of power within the cngine itself by fri ction, · &c., these two products will be equal, and the power ex-· pended lYill equal the useful effect produced. Again, suppose the same train to maintain the same velocity , over another equal portion of oiled track, where the ad 'hesion is not perfect. The useful ef f ect produced wiII be 'the same as in the /il'st Cllse, for the oiling of the track may be regarded as having no effect on the tractive power required to move the train; but in order to main 'tain the same tractive power, on account of the slip of · the wheel, it must revolve faster than in the first case, which implies an increased velocity of piston, and ex penditure of power, without any increase of the useful effect prod uced. If, while the train advances three feet, the wheel rolls over four feet, the slip of the wheel will he one foot, or one-third of the velocity of the train: and 'as four pl\rts of power are required to produce three parts ()f useful effect, one· fourth of the power expended is is wasted, or we must add one-third to the useful effect, in order to obtain the power to be expended for its pro duction. Just so is it in the case of a steamboat. If the wheel revolved with a velocity equal to the velocity of the boat, the paddles would quietly enter the water with out striking against it; but as water is not a solid, but a yielding fluid, it is necessary that, in order to push the boat forward, the paddles must strike the water with an 'increased velocity. The increased velocity of the paddle wheel, requisite to convert the yielding water into a re· sisting medium, corresponds to the increased velocity of the driving wheel ofebe locomotive, rllquisite to maintaiu THE SCIENTIFIC AMERICAN. the necessary adhesion to the track. All power expend ed in overcoming the slip is lost in both cases. The amount of slip in the case of a steamboat will vary with different velocities, but for our purpose we may take the prpportions given above, in the illustration of the loco motive, viz., one-third of the velocity of the boat. This value is, however, based upon the supposition that the ,'elocity of the boat is uniform, and that the only resist ances are those due to the boat's own motion, at any �iven velocity in level water. Suppose, however, a steamboat has another boat in tow, then it is evident that to maintain any speed, the paddles must strike the wllter with more velocity than would be requisite for the same speed, if there was none tow-but the slip measures the velocity with which the paddles strike the water, aud consequently in such a case the slip will he materially increased. Hence, when ever any steamboat encounters a resistance greater than that which is due to the boat itself moving at its present "peed, the per·centage of slip will be increased. When a boat is increasing its speed, its resistance is measured, not merely by its present speed, but to this must be add ed the resistance due to the effort to increase the speed. When a steamboat is moving uniformly, at the rate of five miles per hour, the slip is less thaD of the same boat increasing its speed from four to six miles per hour, at the instant when it reaches five miles per hour. When a steamboat ascends a draw, it encoDnters greater resistance than that due to the velocity of the current added to its own velocity; hence, for the reasons just given, the relative loss of power by slip is materially increased. In "iew of what has been said, I have little doubt that in case of a steamboat ascending a draw, the loss of power from the three causes alluded to amounts to one-half of the whole power. In determin ing the power req nired to overcome the horizontal resistances, these losses were taken into ac count; but they were not laken into account in estim ating the power required to overcome the vertical resist ances. Hence, doubling the power previously given as useful efft!ct, we have, in the illustration of the last num ber, for the power required to overcome the vertical re sistanceswhen t = 60 seconds, t= 120 " t= 180 " and for their total power when t = 60 seconds, t= 120 " t = 180 162 Horse.power. 130 " 118 " 251 Horse-power. 176 " 153 " Estimating this in the erroneous manner indicated at the close of the preceding article, we should havewhen t = 60 seconds, 133 +66 = 199 Horse·powers t =120 " 75+34 = 109 " t=180 ·.... 59+22 = 81 .. The per-cenlll ges of error of this method would be in this case 20, 38, and 47. In order to contrast the result just obtained with an other, take the same steamboat already described, and exnmine its ascent through a draw, constructed precisely the same as the last, but where the original velocity of five and a half miles an hour was increased to six. Take the time of ascent at 120 seconds. Five and a half miles per hour increased to six is 8 feet per second in creased to 8.8. The corresponding hight of remou would be 0.23 feet. Here-W= 1,875,000 b = 0.23 t = 120 vo= (8+8.8)+2=8.4 I = 240 �:Vhichgives for the useful effect to be expended in over coming the vertical resistances P = TV b [1 + (tvO + I)] = 2,242,500 feet pounds. As this work is to be done in 120 seconds, we have P = 34 horse-power of useful effect, requirinl\' on account of losses 68 horse-power of expenditure. The relatIve horizontal velocity of the boat is VO + (1 + t)= 10.4 seconds, requiring for its ac complishment 104 horse-power. Hence the total power required will be 104 + 68 = 172 horse-power. In the preceding illustration, where t = 120, the power required was 176 horse-power, showing thllt, of two draws con structed precisely alike, it requireS'less power to carry a steamboat .p through one of them, where the velocity is six miles per hour than it does through the other, where the velocity is one mile less, or five miles per hour. The question naturally arises: If the eqnivalent mean power required to carry a steamboat up through the draw © 1860 SCIENTIFIC AMERICAN. INC. exceeds the maximum power of the engines, will it be impossible for such a boat to make the ascent? By no means. When a moving body meets any resistance tending to check its motion, the inertia of the bouy comes into play as an active power to aid in overcoming the resistance. When the velocity of a steamboat is di minished by the resistances encountered in ascending a draw, then the inertia of every pound's weight of the boat and cargo comes to the aid of the engine. The measure of the power stored Dp in the inertia of a bocly is (IV -1-64.4) v2• W being its weight in pounds, and v its velocity in feet per second. It. must be remembered, however, that this power is not available until the velo dty begins to be checked. To apply this to the case last given:-What is tht' least power of the engine of the steamboat, to allow it to make the ascent, and leave the dra w with an actual velocity of hal. a mile per hour? Half a mile per hour is 0.7 feet per second, which, added to the "elocity of the current above (8.) gives for t. he equivalent velocity with which the boat leaves the draw, 8.7 per second. Let x represent the number of horse-power required; !/ represent the greatest velocity which can he produced in still water by x; then!/ will be the relative velocity with which the boat approaches the draw. As the power required varies as the cube of the velocity, and as a velocity of 10 feet per second corre sponds to 93 horse-power, we have !/3 : 103 = x : 93, and x = (93 + 1000) !/3. The power (stored up in the inertia) with which the boat leaves the draw will be (1,875,000 + 64.4) + 8.72 = 2, 203, 710 fect Ibs. The power consumed in making the ascent will be 172 X 550 X 120 = 11,352,000 feet pounds. The power given out by the engine during the ascent will be xX 550X 120 =66,000x. The power (stored up in the inertia) with which the boat approaches the draw will be (1,875,000 -1-64.4) X !/2 = 29,115 !/2. The power present, plus the power developed during the ascent, must equal the power consumed during the ascent, plw; the power re maining after the ascent. Hence we have-29,115!/2 + 66,000 x = 1l,352,OCO + 2,203,710 = 13,555,710. 2!l, 115!/2 + 66,000 X (!l3+ IOOO)y3 = 13,555,710. Il, 705!/2 + 204Gy3 = 4,5IS,5iO. This equation may readily be solved by approximation; trying different values of!/, until one sufficientlyaccu rate is III Tived at. As!/ must evidently he greater than the velocity (8.7), with \vhich the boat leaves the draw, take at random !/ = 10. When !/=10 Il, 705 y2+2046y3=3,01 6,500 (to.1, mall) .. y=1l " "= 3,8!17,531 " "!/=11.5 " "= 4,395,190 " " !/=I1. 7 =4,605,420 (too larl1e) " FI1.6 " " =4,491l,4!J0 (too small) Hence as 1 L 6 is too small, and 11. 7 too large, the true ,'alue lies between them; but, as we only carry the value of the velocity to one place of decimals, 11.6 is nearer the true value than 11.7, hence we have-!/ = 11.6 x = (93 -1-1000) y3 = 145. Hence, mstead of 172 horse-power, we have 145 horse power, as the requisite capacity of the engine, to enable it to fulfill the given conditions_ In estimating the amount of power necessary to carry a steamboat up through a draw, we have considered only the power necessary to ascend the rell/Ou. The rell/OU is situated just at the head of the piers, consequently before a boat reaches the foot of the remou, or commences its ascent, it will have to encounter level water moving with the velocity V. The resistance off ered by this rapid current in the draw must be estimated accordin� to the method previously given for estimating hm'izontal resist ances, and the amount added to the power requireli for ascending the rell/ou. It is evident that the greater the I.mgth of the draw, the longer the time the boat will be exposed to the rapid current V; consequently the draw should be made as short as possible. In conclusion, I would state that the objects aimed at in the present investigation were twofold, lirst to obtain a simple and reliahle metnod of measuring the increase of velocity and hight of Tfmou, caused by the piers ot bridges; and secondly, and principally, having obtained these values, to indicate a method by which the oustruc tion to navigation resulting from the accelerated velocity of the current, and the piling-up of the waters, might he accurately measured. That the method here sUl!gested is faultless is not contended. I only claim that it is more accurate, full, and simple, than any other I have bee n able to find recorded.