Document gEvr6pBVk2zn8DoebD0Mer2n9

78 CHAPTER 4 1951 The curves in Fig. 4 may be approximated very closely by the empirical :1 formula:1 / = D055 [l + (20.000 l + (12) 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 J Pi vf flVS P.* = gd pith (13) which may be rearranged to give the loss in pressure, Pi - P> IlVx' 1 ffd pita J (14) Fig. 5. Comparison of Velocitt Profiles for 3 Different Reynolds . N umbers but for 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 noncircular 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 dH -------------------:-----;--------------------------- wetted perimeter of cross-section . ,_ (15) 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 dH 4 X f/4 , (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 1.25 times the shortest dimension. Thus, in a duct of dimension a x b where a < b, Nb*, for the purposes of calculating friction factors, is N'bu ~ 1.25a Vp/p (17) (Fluid Plow- ' " .. . . .79 ' ' 'Tablelt; Values of e 'for Different ' Kind's of Pipe - Ttpbo Pipk . Smooth drawn tubing........ .... Commercial steel or wrought 1 Asphalted cast-iron. Galvanized iron-----Cast-iron..... ........... Wood stave.............. Concrete.. :........... Riveted steel........... 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 This value of A*r may be used in Equation io 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-dyi + d-p =0 (is> If, in addition, the flow is adiabatic, Pp~* = P1P1' . that Equation 18 becomes pi1** dp ~ dV> + -- 3; = 0 2g (19) (20) or by integration, -H-- (W - Vi*) + Pi k 2g"` fc - 1 Piv , (21) Ihis 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, k, 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 Air nnws-r*NT VOLUME FOR COMPRESSIBLE FLUIDS __ _ Compressible Fluid , _- ______ . Carbon dioxide,, methane, natural gas; superheated steam, Sulfur dioxide, ethylene, acetylene............................. Ratio k -- cp/c 1.66 1.40 1.34 1.28 to 1.32 1.24 to 1.26