Document 85X1k2NwzpN4nrLJw3D6Zq3zZ

1957 Guide CHAPTER 4 68 Replacing v by its equal g/gcp (where p is density in pounds weight per cubic, foot) and rearranging. Equation 3 becomes -- dF* + - dp + dz + -- [J du + p dv -- J dq + dW] = 0 (4) 2g p 9 In the case of flow through a pipe, no outside work is performed so that dW = 0. Furthermore, J da + pdv = JT ds = J dq + JT ds' (5) where ddss' == tcohtaanlgcehainngeenintroepnytrodpuye. to internal irreversibility from turbulence and friction. Fluid Flow 69 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. 1 represents ideal conditions. It should also be noted that care must be taken in de termining the proper mean density. Accordingly, the Bernoulli equation is applied most conveniently to incompressible fluids for which density is constant. Pig. 1. Relation or Various Factobs in Bernoulli Equation Accordingly, Equation 4 may be written - dV1 + -- + dz + - JTds' = 0* (6) 29 p 9 In cases where there is no internal irreversibility, ds' = 0, and Equation 6 may be integrated to give 2g + p*+ 2g e* (7) where Pm is the proper mean density. This is commonly called the Bernoulli equation, named after the Swiss V2 * mathematician and physician who first propounded the theory. -- is | 7) ^ known as the velocity head, -- is the pressure head, and z is the elevation j head, all in feet of the fluidP; 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 pipe In the analysis of auboequeat portions oi this chapter the distinction between g and go will be omittedAnid ffoiD the dimensional Consistency the factor, q!Oct is not in general significant in fluid flow analysis*! Pressure Loss in Circular Pipes The pressure loss in circular pipes is customarily expressed by the formula: where flY* h, = 2gd (8) ht = the loss in head of the fluid under conditions of flow, in feet. 1 = the length of the pipe, in feet. F = 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. 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,