Bernoulli equation

Bernoulli equation - BERNOULLIEQUATION

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BERNOULLI EQUATION First Law for one input and one output stream (no mass accumulation) { } {} 2 22 12 2 1 2 ˆˆ () S v vv dU M m g h dV MH MH M mg h h Q W P dt dt ⎡⎤ ++ ⎢⎥ ⎣⎦ =− + + + + ± ±±± ± For a steady state (open system) we have { } 2 2 0 v m g h dt = and 0 dV dt = Then, we write { } 1 2 0( ) S M HH M m g h h Q W + + + + ± ±± ± We now consider a differential device , so that 21 ˆ d H −= and so on. Thus, 2 2 ˆ S v W Q dH d mgdh M M δ + + ± ± We use H=U+PV to get: 2 2 S v W Q dU d PV d mgdh M M += + + ± ± We now replace ˆ dU TdS pdV and expand ˆ dPV PdV VdP =+ to get: 2 2 ˆ ˆ S v W Q TdS PdV PdV VdP d mgdh M M −++ = + + ± ± . Thus: 2 2 ˆ ˆ S v W Q TdS VdP d mgdh M M + + ± ± .
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Now, the entropy balance is: 12 ˆˆ gen dS Q M SM S S dt T ⎡⎤ =−+ + ⎣⎦ ± ± ±± But steady state requires 0 dS dt = , which gives the differential form: ˆ gen TS Q TdS M M δ =+ ± ± Then ˆ gen TS Q TdS M M =− ± ± We replace this in the equation above: 2 2 ˆ S gen v W TS TdS VdP d mgdh TdS M M += +− + ±
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This note was uploaded on 08/31/2011 for the course CHE 3473 taught by Professor Staff during the Spring '08 term at Oklahoma State.

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Bernoulli equation - BERNOULLIEQUATION

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