Document wDJpeYMK0wn8BKyxxDO0NmYK3

72 CHAPTER 4 1952 Guide Theicurves in Fig. 4 may be approximated very closely by'the empirical formula:1 v : .. ' ' (12) *' Equation ;8 is'apjflicable to all 'liquids; and to - gases whfeh the pressure loss is less thatflO percent of the-initial pressure.-'1 .When' the loss in.hesid .is high, the formula to be used for gases is ' '`.'.v' .Pt` - iv . Pi* , gdpit/i , ... (13) which may be an1! to give the loss in pressure^'' pi - pi=pi /r flVS - gd pah (14) Fiq. 5. Comparison of Velocity Profiles for 3 Different Reynolds . ; Numbebs-but fob' Same Average Velocity Pressure Loss in Non-Circular Pipes The formulas for. friction loss in piipes.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 d,, defined by the equation: i- 4 X area of cross-section uh = ~----------------------------- ----------------:-- 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 1V4 J (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, Nb, for the purposes of calculating friction factors, is IVr. = (o-f 0.25dH)7p/M . . (17) Fluid Row 73 Table 1.,YappJ5S. OF,.e f,or: Different Kinds of Pipe . . , . iypE of Pipe' ,e Asphalted cast-iron............................... ....... Galvanized iron.............................................................. Wood stave............. .. _,.......... __________ :................. V-;......' Concrete.............................:.................... 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 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 ttie'Iattef 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 - dV* + -- = 0 2g p If, in addition, the flow is adiabatic, Pp * = pun * so that Equation 18 becomes or by integration, -1 dV* + P--iltk dp '= 0 2g ' pi p,rt (18) (19) (20) i(y?_; 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,'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 Pressdre to Specific ; Heat at Constant Volume fob Compressible Fluids COMPBESSIBLE Fl.UlD Ratio k Cp/c, Helimi! and other monatomic gases.......................................... Air and other diatomic gases.................................................... Ammonia and hydrogen sulfide.................................................. Carbon dioxide, methane, natural gas, superheated steam, moist steam down to a quality of 97 percent...................... sulfur dioxide, ethylene, acetylene. .-.................................... 1.66 1.40 1.34 1.28 to 1.32 1.24 to 1.26