Document 9JLE0maOLx7pqzGk91KpVyxvR

38 CHAPTER 4 1959 Guide (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 - iv> + ^ _ 0 If, in addition, the flow is adiabatic, fig. 5 .... Comparison of Velocity Profiles for 3 Different Reynolds Numbers but for Same Average Velocity the pressure Joss is less than 10 percent of the initial pressure. When the loss in head is high, the formula to be used for gases is ^ Pi4 P<1 p,t>, which may be arranged to give the loss in pressure. Pi - P* - Pi |"l - A/1 - LV ffdpifi J (13) M 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 d* defined by the equation: dg -- 4----X----a-r-e. a of cro_ss-section wetted perimeter of cross-section (13) pp~* = pur* so that Equation 18 becomes (is) i iV + ^ % - 0 2g pi p1'* (20) or by integration, 5`"-TO+ ffie[(=)f?-i]-o > 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. It is convenient in the analysis of compressible flow to in troduce the velocity of propagation of pressure impulses or, more familiarly, the sonic velocity, a. For perfect gases this is given by the equation: a* - kgp/p = kgRT Accordingly, Equation 21 may be written H(Vt-V,<)+ [)V - ij _ 0 (22) (23) or, by rearrangement, 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 draulic diameter will be da = (4 x 2)/6 = 1>$ ft. In tiie case of a round pipe, 4 X rd*/4 dtf *d 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 fric tion factors, is AT* - (o + 0.25dB)VP/p (17) This value of Na, may be used in Equation 10 for laminar flow, is 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. which permits the calculation of tiie ratio of pressures at entrance and exit of the steady flow device--pipe, orifice, or nosale. From Equations 19 22 it follows that so that (25) Vt*) -+- oj* -- <n* 0 (26) and I Vt* (27) The ratio of flow velocity to sonic velocity is known as the Mach number, ROW OF COMPRESSIBLE RUIDS In the flow of compressible fluids, the large density variations make it impracticable to use the Bernoulli equation, M-v/a This parameter is particularly useful in compressible flow analysis. In general, if M 0.1 the flow may be considered Fluid-Flow' to be incompressible. This is generally true in heating and ventilating air ducts. In terms of the Mach number a** Tt di* Ti (28) 39 Table 2___ Ratio of Specific Heat at Constant Pressure to Specific Heat at Constant Volume for Compressible Fluids Compmsibt* Fluid Ratio k " c/c 1.66 Carbon dioxide, methane, natural gas, super heated steam, moist steam down to a quality Sulfur dioxide, ethylene, acetylene.................... 1.34 1.28 to 1.32 1.24 to L26 The quantity, 0+4^ (30) is called the stagnation pressure, and gives a measure of pres sure energy. For incompressible flow where p + -- pV* p + q ig q = j- rV< (31) (32) and is the dynamic pressure. A total head tube, measures stagnation pressure directly. From Equation 29 it follows that for frictionless, adiabatic flow pf - p* (33) This represents another extension of the Bernoulli equation to compressible flow. Friction will cause a loss in pressure energy. Ideal Flow 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 that the flow is frictioDless arfd 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 w - ptAiV, '= PtAVt (34) or, in terms of Mach number, to = Atilt Vkfpun (35) According to Equation 29 so that A If the initial velocity is sufficiently small, Mi* will be negligible so that - VSteF[teF-J - If this is computed and the figures are plotted, the eurved line (partly solid and partly broken) of Fig. 6 is found. The maximum value of ^ may be computed by differentiating Pi w with respect to Pi and equating the result to zero. This operation produces the equation: Pt (40) Pi For air, with k = 1.40, -- = 0.53. 7h Actually, the broken part of the curve is not attained for Fig. 6___ Relation of Flow of Gas to Pressure Drop in o Converging Tube