bernoulli_resist

# bernoulli_resist - NETWORK MODELS OF BERNOULLI'S EQUATION...

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NETWORK MODELS OF BERNOULLI’S EQUATION The phenomenon described by Bernoulli's equation arises from momentum transport due to mass flow. E XAMPLE : A PIPE OF VARYING CROSS - SECTION . section 1 section 2 A 2 Q 1 Q 2 A 1 v 1 v 2 ρ P 1 P 2 ρ Assume: incompressible flow slug flow lossless flow Mod. Sim. Dyn. Sys. Bernoulli’s equation page 1

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Mass balance: Q 1 = A 1 v 1 = Q 2 = A 2 v 2 Consider kinetic (co-)energy flux at each end: 1 dE * k,1 = 2 ρ A 1 dx 1 (v 1 2 ) ˙ E * k,1 = 1 2 ρ A 1 v 1 (v 1 2 ) = 1 2 ρ Q 1 3 A 1 2 ˙ E * k,2 = 1 2 ρ A 2 v 2 (v 2 2 ) = 1 2 ρ Q 2 3 A 2 2 Thus because Q 1 = Q 2 , E * k,2 > ˙ if A 1 > A 2 then ˙ E * k,1 Mod. Sim. Dyn. Sys. Bernoulli’s equation page 2
The extra kinetic energy must come from somewhere. It comes from work done on the fluid. Power balance: 1 ρ Q 2 3 P 1 Q 1 + 1 ρ Q 1 3 = P 2 Q 2 + 2 A 2 2 2 A 1 2 Rearranging: 1 Q 1 2 1 P 1 + 2 ρ A 1 2 Q 1 = P 2 + 2 ρ Q 2 2 Q 2 A 2 2 1 1 P 1 + 2 ρ v 1 2 Q 1 = P 2 + 2 ρ v 2 2 Q 2 Define: 1 P dynamic = 2 ρ v 1 2 P hydraulic = P static + P dynamic Net power flux: P hydraulic Q Mod. Sim. Dyn. Sys. Bernoulli’s equation page 3

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NETWORK REPRESENTATION H OW DO YOU DEPICT THIS PHENOMENON IN A NETWORK ( MODEL ? One possibility is to define a “Bernoulli resistor” (see Karnopp, D. C. (1972) “Bond Graph Models for Fluid Dynamic Systems.” ASME J. Dyn. Sys. Meas. & Cont. pp. 222-229; Karnopp, D. C, Margolis, D. L. & Rosenberg, R. C. (1990) System Dynamics: A Unified Approach, 2nd. Ed. Wiley Interscience). R B P 1 P 2 1 : Q The constitutive equation of the “Bernoulli resistor” is defined as 1 1 1 A 2 2 A 1 2 1 ρ Q 2 P Bernoulli = ρ (v 2 2 – v 1 2 ) = 2 2 This element is called a “resistor” because it relates a pressure drop to a flow rate. Mod. Sim. Dyn. Sys. Bernoulli’s equation page 4
T HIS APPROACH YIELDS THE RIGHT EQUATIONS BUT IT HAS SEVERAL UNSATISFACTORY ASPECTS .

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