bernoulli_resist

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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/27/2012 for the course MECHANICAL 2.141 taught by Professor Nevillehogan during the Fall '06 term at MIT.

Page1 / 15

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online