Document 9JeVd460LJ3p38aq1vE6qyjGe

68 CHAPTER 4 1952 Guide Replacing v by its equal g/gj> (where p is density in pounds weight per cubic foot) and rearranging, Equation 3 becomes . -- dV1 + ~ dp + dz + -- [J du + pdv -- J dq + dW] = 0 2? P g (4) In the case of flow through a pipe, ho outside work is performed so that dW = 0. Furthermore, ' y ' -Jdu ^ p dv = JTds = J dq + JT da' , ' (5) where - ........ ds = total change in entropy. ds' = change in entropy due to internal irreversibility from turbulence and friction.................... . i:. ' , ' Fluid Flow 69 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,' arid that Fig. 1 represents ideal conditions. It should also be noted? that care must be taken in determining the proper mean density. Accordingly, the Bernoulli equation is applied most conveniently to incompressible fluids for which density is constant. ........ ........ : , :-r Fig. 1. Relation of Various Factors in Bernoulli Equation Accordingly, Equation 4 may be written ............ - dV> + -- + dz 4- - JTds'. = 0* 2g p g In cases where.there is-no internal irreversibility, ds',may be integrated to give (6) 0, and'Equation 6 -- H------- + zi = -- -i-------------b Zj 2g . Pm .. . .2g Pm (7) wherep. is the proper mean density. This is commonly called' the Bernoulli equation, named after the Swiss 71 mathematician and physician. who first propounded the theory. ^ is known as the velocity head, - is the pressure head, and z is the elevation P head, all in feet of the fluid; the.total head, ht is the sum of the other three heads. Fig. 1 shows diagrammatically the relation of the various factors. The pressure at point 2 is lower than at point 1 because, of the elevation of point 2 over point 1, and the velocity at point 2 is lower than at point 1 because of the larger pipe diameter at point 2. If the In the analysis of subsequent portions of this chapter the distinction between g and ge will be omitted. Aeide from the dimensional consistency the factor, g/ge, is not in general significant in fluid now analysis. . * " UT in 11 filYlA I I 1: Viscosity to Temperature of Air Pressure Loss in Circular Pipes The pressure loss in circular pipes is customarily expressed by the formula: . flV* fit = ---------- 2gd 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. g = the acceleration due to gravity = 32.174 ft per (second) (second), d = the internal diameter of the pipe, in feet. / = a dimensionless friction coefficient. (8) The' formula is generally known by the name of Darcy of Fanning, though it seems to have been originated.by d'Aubisson de Voisins.in 'i834. The factor / is a function of the Reynolds number, n (9)