462 HYDROMECHANICS [HYDRAULICS, (2); that is, the normal velocities are inversely as the areas of the cross sections. This is true of the mean velocities, if at each section the velocity of the stream varies. In a river of varying slope the velocity varies with the slope. It is easy therefore to see that in parts of large cross section the slope is smaller than in parts of .small cross section. If we conceive a space in a liquid bounded by normal sections at AJ, A 2 and between A 1; A 2 by stream lines (tig. 18), then, as there is no How across the stream lines, as in a stream with rigid boundaries. In the case of compressible fluids the variation of volume due to the difference of pressure at the two sections must be taken into account. If the motion is steady the weight of fluid between two cross sections of a stream must remain constant. Hence the weight flowing in must be the same as the weight flowing out. Let p 1} p. 2 be the pressures, v lt v. 2 the velocities, G lt G 2 the weight per cubic foot of fluid, at cross sections of a stream of areas A L , A 2 . The volumes of inflow and outflow are A]?^ and A 2 t> 2 , and, if the weights of these are the same, and hence, from (5a) 9, if the temperature is constant,
- 1
III. PHENOMENA OF THE DISCHARGE OF LIQUIDS FROM ORIFICES AS ASCERTAINABLE BY EXPERIMENTS. 15. When a liquid issues vertically from a small orifice, it forms a jet which rises nearly to the level of the free surface of the liquid in the vessel from which it flows. The difference of level h r (fig. 19) is so small that it may be at once suspected to be due either to air resistance on the surface of the jet or to the viscosity of the liquid or to fric tion against the sides of the orifice. Neglect ing for the moment this small quantity, we may infer, from the elevation of the jet, that each molecule on leaving the orifice possessed the velocity required to lift it against gravity to the height h. From or- tlinary dynamics, the relation between the velocity and height of projection is given by Fig. 19. t he equation v=V2(jh (1). As this velocity is nearly reached in the flow from well-formed orifices, it is sometimes called the theoretical velocity of discharge. This relation was first obtained by Torricelli. If the orifice is of a suitable conoidal form, the water issues in filaments normal to the plane of the orifice. Let o> be the area of the orifice, then the discharge per second must be, fromeq. (1), Q = art 1 u V2//h nearly (2). This is often termed the theoretical discharge. Use of tJie term Head in Hydraulics. The term head is an old millwright s term, and meant primarily the height through which a mass of water -descended in actuating a hydraulic machine. Since the water in fig. 19 descends through a height h to the orifice, we may say there are h feet of head above the orifice. Still more gene rally any mass of liquid h feet above a horizontal plane may be said to have h feet of elevation head relatively to that datum plane. Further, since the pressure y at the orifice which produces outflow is connected with h by the relation _ = ft, the quantity may be G =" G termed the pressure head at the orifice. Lastly, the velocity v is connected with h by the relation - = h, so that ^- may be termed the head due to the velocity v. 16. Coefficients of Velocity and Resistance. As the actual velocity of discharge differs from /2gh by a small quantity, let the actual velocity __ i* f. /o//^ ( c -** where c v is a coefficient to be determined by experiment, called the coefficient of velocity. This coefficient is found to be tolerably con stant for different heads with well-formed simple orifices, and it very often has the value 97. The difference between the velocity of discharge and the velocity due to the head may be reckoned in another way. The total height h causing outflow consists of two parts, one part h e expended in producing the velocity of outflow, another h r in overcoming the resistances due to viscosity and friction. Let h r = cji e , where c r is a coefficient determined by experiment, and called the coefficient of resistance of the orifice. It is tolerably constant for different heads with well-formed orifices. Then (4). V
h The relation between c v and c r for any orifice is easily found : (5). (5a). Thus if c, = 97, then c r = 0628. That is, for such nn orifice about 6 j per cent, of the head is expended in overcoming frictional resistances to flow. Coefficient of Contraction Sharp-edged Orifices in Plane Surfaces. When a jet issues from an aperture in a vessel, it may either spring clear from the inner edge of the orifice as at a or b (fig. 20), or it may adhere to the sides of the orifice as at c. The former con- dition will be found if the orifice is bevelled outwards as at a, so as to be sharp edged, and it will also occur generally for a prismatic aperture like b, provided the thickness of the vessel round the aper ture is less than the diameter of the jot. But if the thickness is greater the condition shown at c will occur. When the discharge takes place as at a or l>, the section of the jet is smaller than the section of the orifice. This is due to the formation of the jet from filaments converging to the orifice in all directions inside the vessel. The inertia of the filaments opposes siitiden change of direction of motion at the edge of the orifice, and the convergence continues for a distance of about half the diameter of the orifice beyond it. Let o> be the area of the orifice, and r c co the area of the jet at the point where convergence ceases ; then c c is a coefficient to be determined experimentallv for each^kind of orifice, called the coefficient of contraction. When the orifice is a sharp-edged orifice in a plane surface, the value of c r is on the average 64, or the section of the jet is very nearly five-eighths of
the area of the orifice.
