lecture 17 - Thermal and Fluids Engineering I Lecture 17...

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Thermal and Fluids Engineering I Lecture 17 Page 1 Lecture 17 – Turbulent Flow Reynolds Number For fully-developed laminar flow in a horizontal pipe: 2 8 m L P R µ ∆= V The pressure drop, P , depends on , , ,and . m LR V In turbulent flow, P also depends on these four parameters, but the relationship is more complex. To reduce the number of variables, we non-dimensionalize. 2 2 83 2 2 mm LL P D D µµ = ⎛⎞ ⎜⎟ ⎝⎠ VV To find a representative pressure, consider Bernoulli’s equation along a streamline in stagnation flow: 22 11 2 2 12 PP gz gz ρρ ++ = 2 2 2 P P += V
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Thermal and Fluids Engineering I Lecture 17 Page 2 2 1 12 2 P P ρ += V static pressure + dynamic pressure = stagnation pressure We now use dynamic pressure to define a non-dimensional pressure as: * 2 1 2 m P P ∆= V Also define a non-dimensional length * L L D = () * *2 2 32 1 2 m m DL P D µ ⎛⎞ ⎜⎟ ⎝⎠ V V * * 64 m P LD = V Define the Reynolds number as char L Re = V where L char is a characteristic length for the geometry under consideration. In a tube flow, L char is D . * * 64 D P LR e = laminar, fully developed This is a non-dimensional pressure drop per unit length of pipe. It depends only on the Reynolds number. By experiment in pipe flow:
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Thermal and Fluids Engineering I Lecture 17 Page 3 Re < 2100 laminar 2100 < Re < 4000 transitional 4000 < Re turbulent Friction Factor Define the Darcy friction factor as * * P f L = Since * 2 1 2 m P P ρ ∆= V and * L L D = 2 1 2 m P f L D = ⎛⎞ ⎜⎟ ⎝⎠ V 2 2 m L Pf D V laminar or turbulent
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lecture 17 - Thermal and Fluids Engineering I Lecture 17...

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