Document 6Mbqvyo8k3Za5RRaXYnZdwd4

72 CCHHAAPPTTEERR 44 1958; Guid^ , The curves in Fig.,4 may, be approximated very,closely, by the empiri(;a;l|| formula:1, '- r / c ioo*y1 J= 0.0055 1 + ( 20,000 \ - + Nr,) ' ";'m ^ >',41 '-Equation 8 is'applicable; to all liquids, and: 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 .. ' ` ioi PV Pi* - JlVt .Ill Pi* gd PM which may be arranged to give the loss in pressure, Pi L r : ffd pivi _ (H)l Fig. 5. Comparison or Velocity Profiles job 3 Different Reynolds Numbers but for Same Average Velocity Pressure Loss in Non-Circular Pipes . The formulas for friction loss in pipes are based on tfie use of pipes of*| circular cross-section. The same-formulas may be extended to non-|| circular sections, by suitable modification. In the basic formula, Equations 8, the internal diameter d is to be replaced by the hydraulic diameter! dH defined by the equation: ' -. If 4 X area of cross-section dH = ---------;------:------------:------ :------- :-- wetted perimeter of cross-section ,,,_J| (15)a SJ For example, in a rectangular duct, 1 ft by 2 ft, the cross-section areal is 2 sq ft, and the perimeter 6 ft. Then the hydraulic diameter will be3 dH = (4 x 2)/6 = F/3ft. ! In the case of a round pipe, 4 X vdV4 , dH =----- j " 7TU (16) In computing the Reynolds number, and from that the friction factor, the! hydraulic diameter is hot 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 xb where a < b, Nn,, for the purposes of calculating friction factors, is| Nr, = (o + 0.25dH).V>/p - Fluid Flow , 73 Table 1. Values of e fob Different Kinds of Pipe Type op Pipe e Smooth drawn tubing................................ Commercial steel or wrought iron.. Asphalted cast-iron.............................. Galvanized iron.................................. Cast-iron.............................................. Wood stave...................................... Concrete........................................... 0.00015 0.001 to 0.01 xiiis vaiue oi tv r. 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 1 dp 2-g dV* +-- p = 0 If, in addition, the flow is adiabatic, (18) pp-t -- Pun-* so that Equation 18 becomes (19) or by integration, I + Pil/* hdpl = 0 2g pj pi/* (20) i (V, - M + ji _ ^ H= 1 Pi i_ v -/ (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, fc, 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 for Compressible Fluids Compressible Fluid Helium and other monatomic gases. AAAmmirmma_no_r_dn^-iivao-utahcnei dr1uhd'iayiatdotromomgiceicngassuelfsid.e. .....................steam. Carbon dioxidei methane, natural gas, superheated steam, moist steam down to a quality of 97 percent.................... Sulfur dioxide, ethylene, acetylene............................................ Ratio k = Cp/c, 1.66 1.40 1.34 1.28 to 1.32 1.24 to 1.26