Document LJXyxNoRB8N11NmMdQOqRLkmz

72, CHAPTER 4 1956 Guidei The curves in Fig. 4 may be approximated very closely by the empirical formula:1 /- 0.0055T1^1 +(/20,000-e + --10*\Jw* Equation 8 is applicable to all liquids, ahd to gases when the pressure loss is less than 10 percent of the initial pressure. When the loss in. head is high, the formula to be used for gases is Pt - p.1 jlVf Pi* . gd pin (13) which may be arranged to give the loss in pressure, ''-"-"[`-'I/'-.tS] (14) Fig. 5. Comparison of Velocity Profiles hob 3 Different Reynolds . Numbers but fob Same Average Velocity ^ Pressure loss in Non-Circular Pipes The. formulas for friction loss in pipes are based on the use of pipes of circular cross-section. The same formulas may be extended to non circular sections, by suitable modification. > In the basic formula, Equation 8,. the internal diameter d is to be replaced by the hydraulic diameter dH defined by the equation: ^______ 4 X area of cross-section wetted perimeter of cross-section For example, in a rectangular duct, 1 ft by 2 ft, the cross-section area is 2 sq ft, and the perimeter 6 ft. Then the hydraulic diameter will be dH = (4 x 2)/6 = l1/, ft. In the case of a round pipe, 4 x *d*/4 - ----- ;--: -- d . (16) In computing the Reynolds number, and from that the friction factor, the hydraulic diameter, is not to be used; A better approximate procedure is to replace the length in the Reynolds number by the shortest dimension plus one-fourth of the hydraulic diameter. Thus, in a duct of dimension a x b where a < b, Nr,, for the purposes, of calculating; friction factors, is ; - Nr, = (o 4- 02Sdu)Vp/ii : (17) Fluid-Flow. 73. Table 1. Values of e fob Diffebent Kinds of Pipe Type of Pipe Smooth drawn tubing....... Commercial steel or wrought iron Asphalted cast-iron........................ Galvanised iron............................ Cast-iron........................,,................ Wood stave.................. ................. Concrete............................ ............ Riveted steel.................. ............. 6.000005 0.00015 0.0004 0.0005 . : 0.00085 0.0006 to 0.003 0.001 to 0.01 0.003 to 0.03 This value of Nr, may be used in Equation 10 for laminar flow, and in Equation 12 or Fig. 4 for turbulent flow. The error in the approximation is somewhat greater for laminar than for turbulent flow; In the. former case, the relative error may be as much as 10 percent, while in the latter it almost always is less than 3 percent. FLOW OF COMPRESSIBLE FLUIDS In the flow of compressible . fluids, the large density variations make impracticable the use of the Bernoulli equation, (Equation 7). In certain special cases, however, the exact equations for. compressible flow may: be stated. If flow occurs with no friction or other internal irreversibility, Equation 6 becomes ' -2g dV* + --p = 0 If, in addition, the flow is adiabatic, . P/T* = pu>T* so that Equation 18 becomes _1 dy + VE\^l_k d-Ev ' = o 2g or by integration, (18) (19) (20) (21) This extension to compressible flow of Bernoulli's equation reduces to the more familiar form if the pressure change is small. The ratio of specific heats, fe, is used extensively in fluid dynamics; values of k for various gases are given in Table 2. Table 2. Ratio of Specific Heat at Constant Pressure to Specific , Heat at Constant Volume fob Compressible Fluids Compressible Fluid Ratio b - Cp/cv Carbon dioxide." methane, natural gas, superheated steam, 1.66 1.40 1.34 1.28;tolv32 1 1.24 to 1.26