Document zz4B07Dv48Kw336Rrk5ojw1R3

76 CHAPTER 4 1950 Guide Replacing v by its equal g/g*p (where p is density in pounds weight per cubic foot) and rearranging, Equation 3 becomes ' cubic -dV* 2g + -dp p + dz + -- [Jdu g + pdn - Jdq + dW\ = 0 (4) In the case of flow through a pipe, no outside work is performed dW = 0. Furthermore, so that where J du + p do = JT d* *= J dq + JT ds1 (5) da total change in entropy. dt' = change in entropy due to interna! irreversibility from turbulence and friction. Fluid Flow 77 pipe diameter were the same throughout, the velocity, and consequently the velocity head, would be the same at both points, but the higher elevation at point 2 would still be responsible for a loss in pressure. The utility of the equation is evident, though it should be remembered that in it the effects of friction and turbulence are neglected, and that Fig. I --ideal conditions. It should also be noted that care must be J-dV1 + -- + dz + - JT ds' = 0* 2g t> g In cases where there is no internal irreversibility, ds' may be integrated to crlvc* (6) 0, and Equation 6 2g 2(7 Pa where p,, is the proper mean density i`he- --..uunut cv^uAiiiuu, --named -alter t*h(. a* mathematician and physician who first propounded the theory. known as the velocity head, - is the pressure head, and z is the elevatio head, all in feet of the fluid;P the total head, ht is the sum of the othe three heads. Fig. 1 shows diagrammatically the relation of the variou factors. The pressure at point 2 is lower than at point 1 because o the elevation of point 2 over point 1, and the velocity at point 2 is lowe than at point 1 because of the larger pipe diameter.at point 2. If th< * In the analysis of subsequent portions of this chapter the distinction between 9 and g9 will be omitted Adds from dimensional consistency the factor, 9(9: is not in c*neral significant in fluid flow analysis. Fio. 2. Relation of Kinematic Viscosity to Temperature of Air taken in determining the proper mean density. Accordingly, the Bernoulli equation is applied most conveniently to incompressible fluids for which density is constant. Pressure Loss in Circular Pipes The pressure loss in circular pipes is customarily expressed by the formula: fiv* A 2gd (8) where hi * the loss in head of the fluid under conditions of flow, in feet. I - the length of the pipe, in feet. V = the velocity, in feet per second. 9 the acceleration due to gravity = 32.174 ft per (second) (second), d = the internal diameter of the pipe, in feet. / = a dimensionless friction coefficient. The formula is generally known by the name of Darcy or Fanning, though it seems to have been originated by d'Aubisson de Voisins in 1834. The factor / is a function of the Reynolds number, Ne.m d Vp p (9)