Document 7QdYVZpvG54qaB6jg5Ge8pbE

74 CHAPTER 4 1952 Guide It is convenient in the analysis of compressible flow to introduce the velocity of propagation of pressure impulses or, more familiarly, the sonic velocity, a. For perfect gases this is given by the equation: o' = kgp/p = kgRTl (22) Accordingly, Equation 21 may be written iivt - y,) + 0 (23) or, by rearrangement, k-- Or (24) 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 ^ = 2* = fP'Yr a,' T, \pj (25) so that k - 1 (W -Vfi+'af 2 a,' 0 and (26) 1 + k - 2 1 VV d,1 a,' t ^ I T,' 1 + 2 a,*. (27) The ratio of flow velocity to sonic velocity is known as the Mach number, M = V/o This parameter is particularly useful in compressible flow analysis. In i general, if M g 0.1 the flow may be considered to be incompressible. This is generally true in heating and ventilating air ducts. " In terms of the Mach number M T, Ol* ' 7\ 1 + k--- 1 1 . + k--- 1 W and k--1 1 + ~Y~ M,* i*-> P: Pi 1+ (29) f. The quantity, p = p (30) Fluid Flow 75 is called,the stagnation pressure, and gives a measure of pressure energy. For incompressible flow " . : tcfere . Tfi = p+~pyt=p+..g .. .. (3i) ' 9 = ^-p?V . .. (32) and is the dynamic pressure. A. total head; tube measures stagnation pres sure directly. .. . : From Equation-29 it follows that for frictionless, adiabatic-flow i , p. = p. \ ' (33) This represents another extension of the Bernoulli equation to compressible flow. Friction will cause a loss in pressure energy. ' ; !,r . > Ideal Flow through Nozzle or Orifice The majority of low 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 frictionless and adiabatic. Designating the main stream by station 1, and flow at some measuring restriction by station 2, the flow in pounds per sepond is ' to = piA iVi = p?ArVz (34) or, in terms of Mach number, to = AzMz y/kgpzPt According to Equation 29 1 + k- 1 i+SrM*' (35) (36) from which " ..Mt. : (37) so that to -- At k-1 1+-- W /?? [(S)-]i1 --. Afi'/Afj* t (38) ff the initial velocity is sufficiently small,`Mi1 will be negligible so that If this is computed and. the figures are plotted, the curved1 line (partly akk- ~ -