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Lecture_8_notes - Chemical Engineering 374 Fluid Mechanics...

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1 1 Chemical Engineering 374 Fluid Mechanics Fall 2011 Bernoulli Equation 2 dQ dt + dW s dt = d dt ρ ( u + 1 2 v 2 + gz ) V + ρ vA ( u + P ρ + 1 2 v 2 + gz ) out - [] in e mech e Accumulation Out In Generation Can rearrange to familiar (Accumulation) = (In) – (Out) + ( Generation ) Simplify Steady State Ws = 0 Q = 0 No friction (viscous effects) This and no Q give const. u • Incompressible à constant density P ρ + 1 2 v 2 + gz in = P ρ + 1 2 v 2 + gz out Δ P ρ + 1 2 v 2 + gz = 0 Or e mech is conserved
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2 Streamlines 3 4
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4 Consider streamlines, then mechanical energy on a streamline is constant. Can derive the Bernoulli equation by making the same set of assumptions and dot the momentum equation (force balance equation, not covered yet) with displacement along a streamline. Cengel and Boles give a simpler derivation in terms of Newton s Second Law (force balance), again along a streamline.
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