Pump. Piedmont. Here jets were made of 1, 2, 3, and 4 inches diameter; and the water received into cisterns most accurately formed of brick, and lined with stucco. It is the result of these experiments which we have taken for a measure of the deficiency. We may therefore consider the water as flowing through a hole of this contracted dimension, or substitute this for the real orifice in all calculations. For it is evident that if a mouth-piece (so to call it) were made, whose internal shape precisely tallied with the form which the jet assumes, and if this mouth-piece be applied to the orifice, the water will flow out without any obstruction. The vessel may therefore be considered as really having this mouth-piece. taches itself at the very entry of the pipe, and flows with a contracted jet. When the pipe is of this length, and the extremity is stopped with the finger, so that it begins to flow with a full mouth, no subsequent contraction is observed; but merely striking on the pipe with a key or the knuckle is generally sufficient to detach the water in an instant from the sides of the pipe, and reduce the efflux to 62 Nay, from this we derive a very important observation, that if, instead of allowing the water to flow through a hole of an inch area made in a thin plate, we make it flow through a hole in a thick plank, so formed that the external orifice shall have an inch area, but be widened internally, agreeably to the shape which nature forms, both the velocity and quantity will be that which the fundamental proposition determines. Michelotti measured with great care the form of the great jets of three and four inches diameter, and found that the bounding curve was an elongated trochoid. He then made a mouth-piece of this form for his jet of one inch, and another for his jet of two inches; and he found the discharges to be and; and he, with justice, ascribed the trifling deficiency which still remained, partly to friction and partly to his not having exactly suited his mouth-piece to the natural form. We imagine that this last circumstance was the sole cause: For, in the first place, the water in his experiments, before getting at his jet-holes, had to pass along a tube of eight inches diameter. Now a jet of four inches bears too great a proportion to this pipe; and its narrowness undoubtedly hindered the lateral columns from contributing to the efflux in their due proportion, and therefore rendered the jet less convergent. And, in the next place, there can be no doubt (and the observations of Daniel Bernoulli confirm it) but that this convergency begins within the vessel, and perhaps at a very considerable distance from the orifice. And we imagine, that if accurate observations could be made on the motion of the remote lateral particles within the vessel, and an internal mouth-piece were shaped according to the curve which is described by the remotest particle that we can observe, the efflux of water would almost perfectly tally with the theory. But indeed the coincidence is already sufficiently near for giving us very valuable information. We learn that the quantity of water which flows through a hole, in consequence of its own weight, or by the action of any force, may be increased one half by properly shaping the passage to this hole; for we see that it may be increased from 62 to near 99. But there is another modification of the efflux, which we confess our total incapacity to explain. If the water issues through a hole made in a plate whose thickness is about twice the diameter of the hole, or, to express it better, if it issues through a pipe whose length is about twice its diameter, the quantity discharged is nearly of what results from the proposition. If the pipe be longer than this, the quantity is diminished by friction, which increases as the length of the pipe increases. If the pipe be shorter, the water will not fill it, but de This effect is most unaccountable. It certainly arises from the mutual adhesion or attraction between the water and the sides of the pipe; but how this, acting at right angles to the motion, should produce an increase from 62 to 82, nearly, we cannot explain. It shows, however, the prodigious force of this attraction, which in the space of two or three inches is able to communicate a great velocity to a very great body of water. Indeed the experiments on capillary tubes show that the mutual attraction of the parts of water is some thousands of times greater than their weight. 54 100 We have only further to add, that every increase of pipe beyond two diameters is accompanied with a diminution of the discharge; but in what ratio this is diminished it is very difficult to determine. We shall only observe at present that the diminution is very great. A pipe of 2 inches diameter and 30 feet long has its discharge only of what it would be if only 4 inches long. If its length be 60 feet, its discharge will be no more than. A pipe of 1 inch diameter would have a discharge of and, in the same situation. Hence we may conclude that the discharge of a 4 inch pipe of 30 feet long will not exceed of what it would be if only 8 inches long. This will suffice for our present purposes; and the determination of the velocities and discharges in long conduits from pump machines must be referred to the article WATER-Works. At present we shall confine our attention to the pump itself, and to what will contribute to its improvement. 44 Before we can proceed to apply this fundamental proposition to our purpose, we must anticipate in a loose way a proposition of continual use in the construction of water-works. Pump. Let water be supposed stagnant in a vessel EFGH (fig. 37.), and let it be allowed to flow out by a cylin- Fig. 37. drical pipe HIKL, divided by any number of partitions B, C, D, &c. Whatever be the areas B, C, D, of these orifices, the velocity in the intermediate parts of the pipe will be the same; for as much passes through any one orifice in a second as passes through any other in the same time, or through any section of the intervening pipe. Let this velocity in the pipe be V, and let the area of the pipe be A. The velocity in the VA orifices B, C, D, must be VA, VA, YA, &c. Let g Pump. 84 To deter mine the motion of water, &c. 85 In the suckingpump, and all the three would require a head A2 V2 A' X 2g D1 Va A' A2 2g 2B I ; D-2. By this induction may easily be seen what head is necessary for producing the efflux through any number of orifices. Let the expence or quantity of water discharged in an unit of time (suppose a second) be expressed by the symbol Q. This is measured by the product of the velocity by the area of the orifice, and is therefore = VA, VA VA Q1 or x B, or x C, &c. and V1- ThereB C A fore we may compute the head of water (which we shall express by H) in reference to the quantity of water discharged, because this is generally the interesting Q1 circumstance. In this view we have H= X 2gA A' A' A1 + + B1 C D2 water necessary for producing the discharge increases in the proportion of the square of the quantity of water which is discharged. 2: which shows that the head of on These things being premised, it is an easy matter to determine the motion of water in a pump, and the quantity discharged, resulting from the action of any force the piston, or the force which must be applied to the piston in order to produce any required motion or quantity discharged. We have only to suppose that the force employed is the pressure of a column of water of the diameter of the working barrel; and this is over and above the force which is necessary for merely supporting the water at the height of the place of delivery. The motion of the water will be the same in both cases. Let us, first of all, consider a sucking-pump. The motion here depends on the pressure of the air, and will be the same as if the pump were lying horizontally, and communicated with a reservoir, in which is a head of water sufficient to overcome all the obstructions to the notion, and produce a velocity of efflux such as we desire. And here it must be noted that there is a limit. No velocity of the piston can make the water rise in the suction-pipe with a greater velocity than what would be produced by the pressure of a column of water 33 Pump feet high; that is, about 46 feet per second. Let the velocity of the piston be V, and the area of the working barrel be A. Then, if the water fills the barrel as fast as the piston is drawn up, the discharge during the rise of the piston, or the number of cubic feet of water per second, must be = V× A. This is always supposed, and we have already ascertained the circumstances which ensure this to happen. If, therefore, the water arrived with perfect freedom to the pi ston, the force necessary for giving it this velocity, or for discharging the quantity VX A in a second, would be equal to the weight of the pillar of water whose height is and base A. 2g It does not appear at first sight that the force necessary for producing this discharge has any thing to do with the obstructions to the ascent of the water into the pump, because this is produced by the pressure of the atmosphere, and it is the action of this pressure which is measured by the head of water necessary for produ cing the internal motion in the pump. But we must always recollect that the piston, before bringing up any water, and supporting it at a certain height, was pressed on both sides by the atmosphere. While the air supports the column below the piston, all the pressure expended in this support is abstracted from its pressure on the under part of the piston, while its upper part still supports the whole pressure. The atmosphere continues to press on the under surface of the piston, through the intermedium of the water in the suction-pipe, with the difference of these two forces. Now, while the piston is drawn up with the velocity V, more of the atmospheric pressure must be expended in causing the water to follow the piston; and it is only with the remainder of its whole pressure that it continues to press on the under surface of the piston. Therefore, in order that the piston may be raised with the velocity V, a force must be applied to it, over and above the force necessary for merely supporting the column of water, equal to that part of the atmospheric pressure thus employed; that is, equal to the weight of the head of water necessary for forcing the water up through the suction-pipe, and producing the velocity V in the working barrel. -1). + C2 2 I Therefore let В be the area of the mouth of the suction-pipe, and C the area of the fixed valve, and let the suction-pipe be of equal diameter with the working barrel. The head necessary for producing the velocity A' V on the working barrel is (A If 2g d express the density of water; that is, if d be the number of pounds in a cubic foot of water, then d Awill express the weight of a column whose base is A, B1 and height all being reckoned in feet. Therefore 2g the force which must be applied, when estimated in d AVA A1 pounds, will be p, = 2g 2g The first general observation to be made on what bas been said is, that the power which must be employed to produce the necessary motion, in opposition to all the obstacles, is in the proportion of the square of the velo Pump. city which we would produce, or the square of the quantity of water we would discharge. We have hitherto proceeded on the supposition, that there is no contraction of the jet in passing through these two orifices. This we know would be very far from the truth. We must therefore accommodate things to these circumstances, by diminishing B and C in the ratio of the contraction, and calling the diminished areas b and Ad V /A A' c; then we have p= 2g b2 - 62 What this diminution may be, depends on the form of the parts. If the fixed valve, and the entry into the pump, are simply holes in thin plates, then b%2B and c=% C. The entry is commonly widened or trumpet-shaped, which diminishes greatly the contraction but there are other obstacles in the way, arising from the strainer usually put round it to keep out filth. The valve may have its contraction greatly diminished also by its box being made bell-shaped internally; nay, even giving it a cylindrical box, in the manner of fig. 33. is better than no box at all, as in fig. 5.; for such a cylindrical box will have the unaccountable effect of the short tube, and make b=8% B, instead of 62 B. Thus we see that circumstances seemingly very trifling may produce great effects in the performance of a pump. We should have observed that the valve itself presents an obstacle which diminishes the motion, and requires an increase of power; and it would seem that in this respect the clack or butterfly valve is preferable to the button valve. 2 Example. Suppose the velocity of the piston to be 2 feet or 24 inches per second, and that the two contracted areas are each of the area of the which pump, is not much less than what obtains in ordinary pumps. V'A' We have 2g 36,75 inches, and the force which we must add to what will merely support the column is the weight of a pillar of water incumbent on the piston, and something more than three feet high. This would be a sensible 576 is greater than unity, which was the last term of the But this advantage of a smaller suction-pipe is in all (4 + 4 − 1) = 488 (25 + 25-1) taken as an example will require a head of water = portion of the whole force in raising water to small heights. We have supposed the suction-pipe to be of the same diameter with the working barrel; but it is usual to make it of smaller diameter, generally equal to the water way of the fixed valve. This makes a considerable 2g ba AV a change in the force necessary to be applied to the pi- C 13 feet and upwards. Besides, it must be observed Thus we have enabled the reader to ascertain the A1 A A1 + force necessary for producing any required discharge of Pump. Merely to support the column of water of this height B as nothing, and = 4, and so that = 4. and there is no contraction here; and therefore - A is also 16. Therefore 16, = 16. This is the height of a column of water whose base is d +(승) 2g 2gb × A-α This, we see again, is proportional to the square of the velocity of the piston in its descent, and has no relation to the height to which the water is raised. If the piston has a button valve, its surface is at least equal to a; and therefore the pressure is exerted on the water by the whole surface of the piston. In this case d V2 A3 we shall have p = considerably greater than 2g b before. We cannot ascertain this value with great precision, because it is extremely difficult, if possible, to determine the resistance in so complicated a case. But the formula is exact, if b can be given exactly; and we know within very moderate limits what it may amount to. In a pump of the very best construction, with a button valve, b cannot exceed one-half of A; A1 and therefore cannot be less than 8. In this In a good steam-engine pump A1 is also 16. And lastly, or M, 16+ 16 64 X 1,333, 2,309 feet, and the piston will move with the velocity of 2 feet 4 inches nearly. Its velocity will be less than this, on account both of the friction of the piston and the friction of the water in the suction-pipe. These two circumstances will probably reduce it to one foot eight inches; and it can hardly be less than this. We have taken no notice of the friction of the water in the working-barrel, or in the space above the piston; because it is in all cases quite insignificant. The longest pipes employed in our deep mines do not require more than a few inches of head to overcome it. But there is another circumstance which must not be This is very easily ascertained. Let the velocity of a as there will always be some contraction, let the dimi- V is about three feet per second, and is about J Рокор $5 We have hitherto been considering the sucking-pump and in the alone but the forcing pump is of more importance, forcingand apparently more difficult of investigation.-Here pump we have to overcome the obstructions in long pipes, with many bends, contractions, and other obstructions. But the consideration of what relates merely to the pump is abundantly simple. In most cases we have only to force the water into an air-vessel, in opposition to the elasticity of the air compressed in it, and to send it thither with a certain velocity, regulated by the quantity of water discharged in a given time. The elasticity of the air in the air-vessel propels it along the Main. We are not now speaking of the force necessary counterbalancing this pressure of the air in the air-vessel, which is equivalent to all the subsequent obstructions, but only of the force necessary for propelling the water out of the pump with the proper velocity. for We have in a manner determined this already. The piston is solid, and the water which it forces has to pass through a valve in the lateral pipe, and then to move in the direction of the main. The change of direction requires an addition of force to what is necessary for merely impelling the water through the valve. Its quantity is not easily determined by any theory, and it varies according to the abruptness of the turn. It appears from experiment, that when a pipe is bent to a right angle, without any curvature or rounding, the velocity is diminished about. This would augment the head of water about. This may be added to the contraction of the valve hole. Let c be its natural area, and whatever is the contraction competent to its form, increase it, and call the contracted area VA' Then this will a require a head of water = 2gc3. C. Pump. ly giving the velocity V to the water. whole is (+1); and the power p necessary for water is forced by the expansion of the confined air, Pump. This must be added to the head necessary for mereshould always be formed in this manner. For it is this 2g Therefore the which produces the motion during the returning part of the stroke in the pump constructed like fig. 13. N° 1. and during the whole stroke in N° 2. Neglecting this seemingly trifling circumstance will diminish the performance at least one-fifth. The construction of No 1. is the best, for it is hardly possible to make the passage of the other so free from the effects of contraction. The motion of the water during the returning stroke is very much contorted. 2g this purpose is dAV/A JAV2 (A2+1). 2g 2 g It cannot escape the observation of the reader, that in all these formulæ, expressing the height of the column of water which would produce the velocity V in the working barrel of the pump, the quantity which d AV multiplies the constant factor depends on the contracted passages which are in different parts of the pump, and increases in the duplicate proportion of the sum of those contractions. It is therefore of the utmost consequence to avoid all such, and to make the main which leads from the forcing-pump equal to the working barrel. If it be only of half the diameter, it has but one-fourth of the area, the velocity in the main is four times greater than that of the piston, and the force necessary for discharging the same quantity of water is 16 times greater. It is not, however, possible to avoid these contractions altogether, without making the main pipe wider than the barrel. For if only so wide, with an entry of the same size, the valve makes a considerable obstruction. Unskilful engineers endeavour to obviate this by making an enlargement in that part of the main which contains the valve. This is seen in fig. 14. at the valve L. If this be not done with great judgment, it will increase the obstructions. For if this enlargement is full of water, the water must move in the direction of its axis with a diminished velocity; and when it comes into the main, it must again be accelerated. In short, any abrupt enlargement which is to be afterwards contracted, does as much harm as a contraction, unless it 87 be so short that the water in the axis keeps its velocity Use of ex- till it reaches the contraction. Nothing would do more periments. service to an artist, who is not well founded in the theory of hydrodynamics, than to make a few simple and cheap experiments with a vessel like that of fig. 37. Let the horizontal pipe be about three inches diameter, and made in joints which can be added to each other. Let the joints be about six inches long, and the holes from one-fourth to a whole inch in diameter. Fill the vessel with water, and observe the time of its sinking three or four inches. Each joint should have a small hole in its upper side to let out the air; and when the water runs out by it, let it be stopped by a peg. He will see that the larger the pipe is in proportion to the orifices made in the partitions, the efflux is more diminished. We believe that no person would suspect this who has not considered the subject minutely. All angular enlargements, all boxes, into which the pipes from different working barrels, unite their water before it goes into a main, must therefore be avoided by an artist who would execute a good machine; and the different contractions which are unavoidable at the seats of valves and the perforations of pistons, &c. should be diminished by giving the parts a trumpetshape. In the air-vessels represented in fig. 13. this is of very great consequence. The throat O, through which the 2 88 water in There is one circumstance that we have not taken any Acceleranotice of, viz. the gradual acceleration of the motion of tion of the water in pumps. When a force is applied to the piston, motion of it does not in an instant communicate all the velocity pumps. which it acquires. It acts as gravity acts on heavy bodies; and if the resistances remained the same, it would produce, like gravity, an uniformly accelerated motion. But we have seen that the resistances (which are always measured by the force which just overcomes them) increase as the square of the velocity increases. They therefore quickly balance the action of the moving power, and the motion becomes uniform, in a time so short that we commit no error of any consequence by supposing it uniform from the beginning. It would have prodigiously embarrassed our investigations to have introduced this circumstance; and it is a matter of mere speculative curiosity: for most of our moving powers are unequal in their exertions, and these exertions are regulated by other laws. The pressure on a piston moved by a crank is as variable as its velocity, and in most cases is nearly in the inverse proportion of its velocity, as any mechanician will readily discover. The only case in which we could consider this matter with any degree of comprehensibility is that of a steamengine, or of a piston which forces by means of a weight lying on it. In both, the velocity becomes uniform in a very small fraction of a second." 89 of elemen tary books We have been very minute on this subject. For al- Deficiency though it is the only view of a pump which is of any importance, it is hardly ever understood even by profes- on this subsed engineers. And this is not peculiar to hydraulics, ject. but is seen in all the branches of practical mechanics. The elementary knowledge to be met with in such books as are generally perused by them, goes no farther than to state the forces which are in equilibrio by the intervention of a machine, or the proportion of the parts of a machine which will set two known forces in equilibrio. But when this equilibrium is destroyed by the superiority of one of the forces, the machine must move; and the only interesting question is, what will be the motion? Till this is answered with some precision, we have learned nothing of any importance. Few engineers are able to answer this question even in the simplest cases; and they cannot, from any confident science, say what will be the performance of an untried machine. They guess at it with a success proportioned to the multiplicity of their experience and their own sagacity. Yet this part of mechanics is as susceptible of accurate computation as the cases of equilibrium.-We therefore thought it our duty to point out the manner of proceeding so circumstantially, that every step should be plain and easy, and that conviction should always accompany our progress. This we think it has been in our power to do, by the very simple method of substituting a co lumn |