484 HYDROMECHANICS [HYDRAULICS. VIII. STEADY FLOW OF WATER IN PIPES OF UNIFORM SECTION. 68. The ordinary theory of the flow of water in pipes, on which all practical formulae are based, assumes that the variation of velocity at different points of any cross section may be neglected. The water is considered as moving in plane layers, which are driven through the pipe against the frictional resistance, by the difference of pressure at or elevation of the ends of the pipe. If the motion is steady the velocity at each cross section remains the same from moment to moment, and if the cross sectional area is constant the velocity at all sections must be the same. Hence the motion is uniform. The most important resistance to the motion of the water is the surface friction of the pipe, and it is convenient to estimate this independently of some smaller resistances which will be accounted for presently. In any portion of a uniform pipe, excluding for the present the ends of the pipe, the water enters and leaves at the same velocity. For that portion therefore the work of the external forces and of the surface friction ; must be equal. /^. Let fig. 80 repre sent a very short portion of the pipe, of length dl, be tween cross sec tions at 2 and 2 + dz feet above any horizontal da tum line xx, the pressures at the cross sections being p and p + dp lb per square foot. Fur- Fig. 80. ther, let Q be the volume of flow or discharge of the pipe per second, n the area of a normal cross section, and x the perimeter of the pipe. The Q cubic feet, which flow through the space considered per second, weigh GQ lb, and fall through a height - dz feet. The work done by gravity is then -GQdz; a positive quantity if dz is negative, and vice versa. The resultant pressure parallel to the axis of the pipe is p - (p + dp) = - dp ft per square foot of the cross section. The work of this pressure on the volume Q is The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is x dl. The work expended in overcoming the frictional resistance per second is (see 64, eq. 3) or, since Q = flr, the negative sign being taken because the work is done against a resistance. Adding all these portions of work, and equating the result to zero, since the motion is uniform, Dividing by GQ, Integrating, - GQcfe - Qdp - CG *. Q dl = . n 2g
- + TT + ^n~ -|-*- constant (1).
Horizontal Fig. 81. 69. Let A and B (fig. 81) be any two sections of the pipes for which p, z, I have the values p lt r,, 7 U and p. 2 , z.,, L 2 respectively. or, if 1 2 - Z 1 = L, rearranging the terms, P 2 ~ . . . (2). Suppose pressure columns introduced at A and B. The water will rise in those columns to the heights ^ and ^ due to the G G pressures j; x and p. 2 at A and B. Hence the quantity represented in the figure by DE, the fall of level of the pressure columns, or virtual fall of the pipe. If there were no friction in the pipe, then by Bernoulli s equation there would be no fall of level of the pressure columns, the velocity being the same at A and B. Hence DE or h is the head lost in friction in the distance AB. The quantity = -i is termed the virtual slope A i * i . of the pipe or virtual fall per foot of length. It is sometimes termed very conveniently the relative fall. It will be denoted by the symbol i. The quantity which appears in many hydraulic equations is X called the hydraulic mean radius of the pipe. It will be denoted by m. Introducing these values, CV" II . I Q 2g-= m T = ml (3)> For pipes of circular section, and diameter d, n 4" d x ird 4 d h di h=( 4 L 4 mi. fV~ a ll al IA Then _ = _._ = _, (4): (4a); a z</ which shows that the head lost in friction is proportional to the head due to the velocity, and is found by multiplying that head by the coefficient . ct 70. Hydraulic Gradient or Line of Virtual Slojie.Jom CD. Since the head lost in friction is proportional to L, any intermediate pressure column between A and B will have its free surface on the line CD, and the vertical distance between CD and the pipe at any point measures the pressure, exclusive of atmospheric pressure, in the pipe at that point. If the pipe were laid along the line CD instead of AB, the water would flow at the same velocity by gravity without any change of pressure from section to section. Hence CD is termed the virtual slope or hydraulic gradient of the pipe. It is the line of free surface level for each point of the pipe. If an ordinary pipe, connecting reservoirs open to the air, rises at any point above the line of virtual slope, the pressure at that point is less than the atmospheric pressure transmitted through the pipe. At such a point there is a liability that air may be disengaged from the water, and the flow stopped or impeded by the accumulation of air. If the pipe rises more than 34 feet above the line of virtual slope, the pressure is negative. But as this is impossible, the con tinuity of the flow will be broken. If the pipe is not straight, the line of virtual slope becomes a curved line, but since in actual pipes the vertical alterations of level are generally small, compared with the length of the pipe, distances measured along the pipe are sensibly proportional to distances measured along the horizontal projection of the pipe. Hence the line of hydraulic gradient may be taken to be a straight line without error of practical importance. 71. Case of a Uniform Pipe connecting two Reservoirs, ichenallthc Resistances are taken into account. Let h (fig. 82) be the difference of level of the reservoirs, and r the velocity, in a pipe of length L and diameter d. The whole work done per second is virtually the
removal of Q cubic feet of water from the surface of the upper
