fluidmech

In order to calculate the frictional losses we use

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Unformatted text preview: RESSURE Consider a fluid flowing with mean velocity um. If the kinetic energy of the fluid is converted into flow or fluid energy, the pressure would increase. The pressure rise due to this conversion is called the dynamic pressure. KE = ½ mum2 Q is the volume flow rate and ρ = m/Q Flow Energy = p Q Equating ½ mum2 = p Q p = mu2/2Q = ½ ρ um2 3.1.2 WALL SHEAR STRESS τo The wall shear stress is the shear stress in the layer of fluid next to the wall of the pipe. Fig.3.1 ⎛ du ⎞ The shear stress in the layer next to the wall is τ o = µ ⎜ ⎟ ⎜ dy ⎝ ⎠ wall The shear force resisting flow is Fs = τ oπLD The resulting pressure drop produces a force of F p = Equating forces gives τ o = D∆p 4L ∆pπD 2 4 © D.J.DUNN 23 3.1.3 Cf = FRICTION COEFFICIENT for LAMINAR FLOW Wall Shear Stress 2D∆p = Dynamic Pressure 4Lρu 2 m From Poiseuille’s equation ∆p = 3.1.4 ⎛ 2D 32µLu m Hence C f = ⎜ 2 ⎜ 4Lρu 2 D m ⎝ ⎞⎛ 32µLu ⎞ 16µ 16 ⎟⎜ ⎟= 2 = 2 ⎟⎝ D ⎠ ρu D R e m ⎠ DARCY FORMULA This formula is mainly used for calculating the pressure loss in a pipe due to turbulent flow but it can be used for laminar flow also. Turbulent flow in pipes occurs when the Reynolds Number exceeds 2500 but this is not a clear point so 3000 is used to be sure. In order to calculate the frictional losses we use the concept of friction coefficient symbol Cf. This was defined as follows. Cf = Wall Shear Stress 2D∆p = Dynamic Pressure 4Lρu 2 m Rearranging equation to make ∆p the subject ∆p = 4C f Lρu 2 m 2D This is often expressed as a friction head hf ∆p 4C f Lu 2 m = hf = 2gD ρg This is the Darcy formula. In the case of laminar flow, Darcy's and Poiseuille's equations must give the same result so equating them gives 4C f Lu 2 32µLu m m = 2gD ρgD 2 16µ 16 Cf = = ρu m D R e This is the same result as before for laminar flow. Turbulent flow may be safely assumed in pipes when the Reynolds’ Number exceeds 3000. In order to calculate the frictional losses we use the concept of friction coefficient symbol Cf. Note that in older textbooks Cf was written as f but now the symbol f represents 4Cf. 3.1.5 FLUID RESISTANCE Fluid resistance is an alternative approach to solving problems involving losses. The above equations may be expressed in terms of flow rate Q by substituting u = Q/A hf = 4C f Lu 2 4C f LQ 2 m = 2gD 2gDA 2 Substituting A =πD2/4 we get the following. hf = 32C f LQ 2 = RQ 2 25 gπ D R is the fluid resistance or restriction. R = 32C f L gπ 2 D 5 © D.J.DUNN 24 If we want pressure loss instead of head loss the equations are as follows. p f = ρgh f = 32ρC f LQ 2 = RQ 2 25 πD R is the fluid resistance or restriction. R = 32 ρC f L π 2 D5 It should be noted that R contains the friction coefficient and this is a variable with velocity and surface roughness so R should be used with care. 3.2 MOODY DIAGRAM AND RELATIVE SURFACE ROUGHNESS In general the friction head is some function of um such that hf = φumn. Clearly for l...
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This document was uploaded on 02/07/2014.

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