Document 2j9pMY85qGGMkENGEj8R0NnDR

40 CHAPTER 4 1960 Guide Fluid Flow 41 (Equation 7). In certain special cases, however, the exact equations for compresible flow may be stated. If flow occurs with no friction or other internal irreversibility Equation 6 to be incompressible. This is generally true in heating and ventilating air ducts. In terms of the Mach number Table 2___ Ratio of Specific Heat ot Constant Pressure to Specific Heat at Constant Volume for Compressible Fluids Fig, 5....Comparison of Velocity Profiles for 3 Different Reynolds Numbers but for Same Average Velocity 1 dp - dV* + - 0 2g p If, in addition, the flow is adiabatic, pp~-k ~ so that Equation 18 becomes l-Ay+^iZ-o 2q Pi pi'* CocaprvtsbU FkAf Ratio k " c,/c. (18) o* ^ 7* (28) oi> ~ 7i +V"'' 1.66 1.40 1.34 (19) and Carbon dioxide, methane, natural gas, super heated steam, moist steam down to a quality 1.28 to 1.32 p* (29) Sulfur dioxide, ethylene, acetylene................... 1.24 to 1.26 (20) pi 1 + L_i,,,. tiie pressure toss is less than 10 percent of the initial pressure. When the lass in bead is high, the formula to be used for gaag or by integration, The quantity, 80 that s /*r* Pi* ffdpiPi which may be arranged to give the loss in pressure, >]- (21) (13) This extension to compressible flow of Bernoulli's equation reduces to the more familiar form if the pressure change is small. j>- - J> (30) is called the stagnation pressure, and gives a measure of pres sure energy- For incompressible flow VS(P1 (38) If the initial velocity is sufficiently small, Mi* will be negligible J>1 - P. - Pi [l - a/X - LV fldpt*ij Tire ratio of specific heats, k, is used extensively in fluid (14) dynamics; values of k for various gases are given in Table 2. It is convenient in the analysis of compressible flow to in troduce the velocity of propagation of pressure impulses or, tcAere P* * P + 7T pV* " P + 7 (31) so that Pressure Loss in Non-Circular Pipes The formulas for friction loss in pipes are based on the use more familiarly, the sonic velocity, o. For perfect gases this is given by the equation: q-~rV (32) of pipes of circular cross-section. The same formulas may be extended to noncircular sections, by suitable modification. In the base formula, Equation 8, the internal diameter d is to be replaced by the hydraulic diameter dff defined by the equation : da 4 X area of cross-section wetted perimeter of cross-section (15) For example, in a rectangular duct, 1 ft by 2 ft, the crosssection area is 2 sq ft, and the perimeter 6 ft. Then the hy e* kgp/fi = kgRT Accordingly, Equation 21 may be written or, by rearramnggeemment, (22) and is the dynamic pressure. A total head- tube measures If this is computed and the figures are plotted, the curved stagnation pressure directly. line (partly solid and partly broken) of Fig. 6 is found. The From Equation 29 it follows that for frictionless, adiabatic flow (23) P* - Pi4 ""(33) maximum value of -- may be computed by differentiating Pi to with respect to pt and equating the result to sew. This This represents another extension of the Bernoulli equation operation produces the equation: to compressible flow. Friction will cause a loss in pressure (24) energy. e _ (-i-Vi v, \* + V (40) draulic diameter.will be dfi = (4 x 2)/Q *= 1H ft. In the case of a round pipe, 4 x rdV4 da d *d which permits the calculation of the ratio of pressures at entrance and exit of the steady flow device--pipe, orifice, or nozzle. From Equations 19 and 22 it follows that (16) Ideal Row through Nozzle or Orifice The majority of low-head measuring systems depend upon a correlation between pressure drop, area, and quantity of flow. The basic formulas may be stated on the assumption For air, with k *> 1.40, & 0.63. Actually, the broken part of the curve is not attained for 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 so that Of* T* \pi/ (25) that the flow is frictionless and adiabatic. Designating the main stream by station 1, and flow at some measuring re striction by station 2, the flow in pounds per second is .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, Mg., for the purposes of calculating fric k- 1 ov- V,*) +- a,1 -- ai` 0 2 (26) w piA\V% " P$.4,Vi or, in terms of Mach number, (34) tion factors, is and Mg. ~ (a + <h25d*)rp/p ' (17) k ~ 1 V This value of Ns, may be used in Equation 10 for laminar 21 +a,1 i i flow, and in Equation 12 or Fig. 4 for turbulent flow. The error in the approximation is somewhat greater for laminar i than for turbulent flow. Iu the former case, the relative error O)1 1 + k - 1.7,' 2 a.* (27) may be as much as 10 percent, while in the latter it almost The ratio of flow velocity to sonic velocity is known as the always is less than 3 percent. Mach number. u> =* AtVkflpun According to Equation 29 (35) ROW OF COMPRESSIBLE FLUIDS M - V/a ! In tiie flow of compressible fluids, the large density varia This parameter is particularly useful in compressible-flow i tions make it impracticable to use the Bernoulli equation. analysis. In general, if M 0.1 the flow may be raiHp.rri fig. 6.... Relation of Row of Gas to Pressure Drop in o Converging Tube J