1 p 1 \u03c1 \u03b1 1 U 1 2 2 gy 1 p 2 \u03c1 \u03b1 2 U 2 2 2 gy 2 p U 2 2 gy represents the

1 p 1 ρ α 1 u 1 2 2 gy 1 p 2 ρ α 2 u 2 2 2 gy 2 p

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1 )= p 1 ρ + α 1 U 1 2 2 + gy 1 p 2 ρ + α 2 U 2 2 2 + gy 2 ( p / + U 2 /2+ gy ) represents the mechanical energy per unit mass at a flow cross section. ( u 2 - u 1 - Q / m ) is equal to the difference in mechanical energy per unit mass between sections 1 & 2. This term represents the conversion of mechanical energy at section (1) to unwanted thermal energy ( u 2 - u 1 ) and loss of energy via heat transfer (- Q / m ). So it is the total head loss . 2 2 2 2 2 1 2 1 1 1 t 2 2 gy U p gy U p h l 80
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h l t = h l + h l m h l - head loss due to frictional effects in fully developed flow in constant area conduits. h lm - minor losses due to entrances, fittings, area changes, etc. So, for a fully developed flow through a constant-area pipe, 2 1 2 1 y y g p p h l and if y 1 = y 2 , h l = p 1 p 2 ρ = p ρ For laminar flow, d p d x = p L = 128 μ Q π D 4 Q = U ( π D 2 /4) p = 32 L D μ U D h l = 32 L D μ U ρ D = L D U 2 2 64 μ ρ U D = 64 Re L D U 2 2 f = 64 Re h l = f L D U 2 2 81
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8.3. Turbulent flow In turbulent flow we cannot evaluate the pressure drop analytically. We must use experimental data and dimensional analysis. In fully developed turbulent flow, the pressure drop, p , due to friction in a horizontal constant-area pipe is known to depend on the pipe diameter, D , the pipe length, L , the pipe roughness, e , the average flow velocity, U , the fluid density, , and the fluid viscosity, . Therefore, p = p ( D , L , e , U , , ) Dimensional analysis, D e , D L , D U ρ μ U P 1 2 Δ h l = p / D e , D L , U h l Re 1 2 Experiments show that the non-dimensional head loss is directly proportional to L / D , hence: D e Re D L U h l , 2 2 or D e Re D L U h l , 2 2 2 1 Define the friction factor, f = 3 ( Re , e / D ) So, h l = f ( L / D ) ( U 2 / 2) f is determined experimentally. Moody Diagram .
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  • Fall '08
  • ZOHAR
  • Fluid Dynamics, Incompressible Flow, Turbulent Flow

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