Document a4kzK5Nwd396Q1bga0vDKD00X

78 CHAPTER 4....... 1948 Guide For instance, in a square duct, 1 ft on a side, handling air, the hydraulic radius is J4 or 0.25. If the same duct is handling water, flowing 9 in. deep, the hydraulic radius is 0.75/2.5 or 0.30. Note in this latter case that the wetted perimeter does not include the distance across the free surface. In the case of a round pipe Rb = = -|-or rf = 4J?a (12) Substituting Equation 12 in Equation 6, A'Re 4RnVe and in the flow Equation 5, flV' ht 8gRn and finally in the compressible fluid flow Equation 10, (13) (14) pi -- Pi flVi 4g Rb PiVi (15) Equations 13, 14, anfl 15 may be used to compute the flow in pipes and ducts of non-circular section and in any type of conduit not flowing full. They should not be used when the flow is laminar. FLOW OF COMPRESSIBLE FLUIDS The energy equation for the flow of compressible fluids, as represented by the gases, is derived from Equation 1. Assuming that no heat is transferred to the fluid, that no work is done, and that there is no differ ence in elevation. Equation 1 becomes -TjJ + Jui + plVl = + Jut + piV, (16) or, after rearrangement, Vi* - V,* 2g = PlVl -- plVl + j(ui -- a,) (17) Since internal energy is dependent only on temperature, where ui -- a, -- cv{Ti -- Ti) (18) Cv = the specific heat of the gas at constant volume. Ti and Tj = the absolute temperatures in Fahrenheit degrees at points 1 and 2, re spectively. Substituting Equation 18 in Equation 17, Now V,' - V* 2* -- piVi -- piVi 4- Jcv(Ti -- Ti) Cy -- R Ak -1) (19) (20) 1' { . ' "' i Fluid Flout ._____________ !. where . ' ..... _ ________ ,____ 79 ' R = the gas constant in the expression. pv = RT ; (21) k = the ratio of the specific heat at constant pressure to the specific heat at constant volume. , This ratio, 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 Heat at Constant Volume for Compressible Fluids Compressible Fluid Carbon dioxide, methane, natural gas, superheated steam, moist steam down to a quality of 97 per cent-------------------- kRatio = cp/<v 1.66 1.40 1.34 1.28 to 1.32 1.24 to 1.26 Substituting Equations 20 and 21 in Equation 19, the ehergy equation becomes v.* -- v,1 h - -2g-- - z-Erj CP** ~ ft*) (22) While this is a convenient form of equation, it does not include all the necessary specifications. If the steady flow process is frictionless ^and reversible, - (-S-)* - B3> By introducing this relation in Equation 22 it is possible to reduce that equation to: This form of the equation is applicable not only to flow in pipes, blit also to flow through orifices and nozzles. A significant factor in the flow of compressible fluids is the velocity of sound, Fso, which for present purposes will be considered as the velocity at which sound will travel in the fluid at its density at the first or inlet section of the flow system being considered. The velocity of sound is expressed as V,, _ Jl*. (25) This formula may be developed rationally and agrees perfectly with experimental results. Substituting Equation 25 in Equation 24: