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Bernoulli equation - BERNOULLIEQUATION (nomassaccumulation...

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BERNOULLI EQUATION First Law for one input and one output stream (no mass accumulation) { } { } 2 2 2 1 2 2 1 2 1 2 2 2 ˆ ˆ ( ) S v v v d U M mgh dV MH MH M mg h h Q W P dt dt + + = + + + + ± ± ± ± ± For a steady state (open system) we have { } 2 2 0 v d U M mgh dt + + = and 0 dV dt = Then, we write { } 2 2 1 2 1 2 1 2 2 2 ˆ ˆ 0 ( ) ( ) S v v M H H M mg h h Q W = + + + + ± ± ± ± We now consider a differential device , so that 2 1 ˆ ˆ ˆ ( ) H H dH = and so on. Thus, 2 2 ˆ S v W Q dH d mgdh M M δ δ = − + + ± ± ± ± We use H=U+PV to get: 2 2 ˆ ˆ ( ) S v W Q dU d PV d mgdh M M δ δ + = − + + ± ± ± ± We now replace ˆ ˆ ˆ dU TdS pdV = and expand ˆ ˆ ˆ ( ) d PV PdV VdP = + to get: 2 2 ˆ ˆ ˆ ˆ S v W Q TdS PdV PdV VdP d mgdh M M δ δ + + = − + + ± ± ± ± . Thus: 2 2 ˆ ˆ S v W Q TdS VdP d mgdh M M δ δ + = − + + ± ± ± ± .
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