Document G6ZqpZQXZXqyKrXMq77ed4jkN

38 CHAPTER 4 1960 Guide the Swiss maf>pmtiVtan and physicist who first propounded V* P the theory, -- is known as the velocity head, - is the pres sure head, and s is the elevation head, all in feet of the fluid; the total head, A* is the sum of the other three heads. Pig. 1 shows diagrammatieally 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 ai point 2 is lower than at point 1 because of the larger pipe diameter at point 2. If the 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 hi pressure. The utility of the equation is evident, though it hnM be re membered 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 determining the proper mean density. Accordingly, the Bernoulliequation is applied most conveniently to incompressible fluids for which density is constant Pressure Loss in Orculor Pipes The pressure loss in circular pipes is customarily expressed by the formula: where ./ 'JO. (8) hf *=* the loss in head of the Sind under conditions of Sow, feet. l -- the length of the pipe, feet. V -- the velocity, feet per second. g * the acceleration due to gravity * 32.174 ft per (sec ond) (second). d "* the internal diameter of the pipe, feet. / -- a dimensionless friction coefficient. The formula is generally known by the name of Darcy or Fanning. Rg. 3 .... Relation of Kinematic Viscosity to Temperature of Water The factor / is a function of the Reynolds number, AVP V*.- p (9) Ns, ** Reynolds number. p the density, pounds per eubic foot. ft * the absolute viscosity, pounds per foot-second. Both / and the Reynolds number are dimenmnnWa To aid in computing the Reynolds number, values of the kine matic viscosity, are shown as a function of temperature for air.in fig. 2, and for water in Fig. 3. fig. 4 shows the relation between / and the Reynolds number, adapted from a review by Moody.1 The straight line sloping downward at the left of the chart supplies the values of / for laminar flow determined by the formula: 64 / N* (10) With laminar flow, the velocity profile is a parabola, having tiie formula: V phf (f* - I/) (11) r = the radios of the pipe, feet. L =* distance perpendicularly from the axis of the pipe, feet. Accordingly, the maximum velocity occurs at the center of the pipe and is twice the average velocity; the average velocity is found when' L = 0.707 r. It is worth noting that roughness of the pipe wall has no effect on the loss in head for laminar flow. Between values of the Reynolds number of 2000 and 4000, 1 Superior number* refer to Use reference* *t tb* end of chapter. Fluid Row there is an unstable region where the flow changes from l<unjnft.r to turbulent, or vice versa. The actual value is im possible of prediction for any conditions of flow, though in general it may be said that the prevailing type of flow per sists into the unstable region; however, once a change starts, it proceeds very rapidly. When the flow is turbulent, the velocity profile is essentially parabolic over four-fifths of the pipe diameter, but near the pipe walls, the effect of friction becomes evident, and in the boundary layer at the pipe wall the flow is laminar. Fig. 5 compares tire velocity profiles for three different Reynolds numbers, but for the same average velocity. The lower curve in the turbulent region in Fig. 4 represents the relation of / to the Reynolds number for smooth pipe, such as drawn brass tubing or glass tubing. The effect of roughness on /, an effect which is considerable in turbulent flow, is open to some conjecture; artificially roughened pipes, . for instance, give results at variance with actual tests. The corves above the smooth pipe curve of Fig. 4 represent a summary of tests on rough pipe, each of them identified by a value of e/d, with e signifying the absolute roughness in feet. Values of e for different pipes are given in Table 1. To find the friction loss for any pipe, follow the curve with 39 Table 1.... Values of e for Different Kinds of Pipe Typo of pipo o Commercial steel or wrought iron.............. 0.000005 0.00015 0.0004 0.0005 0.00085 0.0006 to 0.003 0.001 to 0.01 0.003 to 0.03 the proper value of e/d, to the pertinent value of Nb. , and from this point proceed horizontally to left margin to find the value of / for use in Equation 8. The curves in Fig. 4 may be approximated very closely by the empirical formula:1 S - 0.0055 |\ + ^20,000 | + "J (12) Equation 8 is applicable to all liquids, and to gases when Hof*> The ftraigbt iim of hft thowt votee* of Friction Factor tor fastnor flow. KeprMod bf pormiutoa from ASMS TrswoctioB*. Fig. 4.... Relation Between Friction Factor and Reynolds Number