NETWORK MODELS OF BERNOULLI’S EQUATION The phenomenon described by Bernoulli's equation arises from momentum transport due to mass flow. EXAMPLE: A PIPE OF VARYING CROSS-SECTION. section 1 section 2 A2Q1 Q2 A1 v1 v2ρ P1 P2ρ Assume: • incompressible flow • slug flow • lossless flow Mod. Sim. Dyn. Sys. Bernoulli’s equation page 1Mass balance: Q1 = A1v1 = Q2 = A2v2 Consider kinetic (co-)energy flux at each end: 1dE* k,1 = 2 ρA1dx1(v12) ˙E * k,1 = 1 2 ρA1v1(v12) = 1 2 ρ Q13 A12 ˙E * k,2 = 1 2 ρA2v2(v22) = 1 2 ρ Q23 A22 Thus because Q1 = Q2, E *k,2 > ˙if A1 > A2 then ˙E *k,1 Mod. Sim. Dyn. Sys. Bernoulli’s equation page 2The extra kinetic energy must come from somewhere. It comes from work done on the fluid. Power balance: 1 ρ Q23 P1Q1+ 1 ρ Q13 = P2Q2+ 2 A222 A12 Rearranging: ⎜⎛ 1 Q12⎟⎞ ⎜⎛ 1 ⎝⎜ P1+ 2 ρ A12⎠⎟ Q1 = ⎝⎜ P2+ 2 ρ Q22⎟⎞ Q2A22⎠⎟ 1 1 ⎝⎜⎛ P1+ 2 ρ v12⎠⎟⎞ Q1 = ⎝⎜⎛ P2+ 2 ρ v22⎠⎟⎞ Q2 Define: 1 Pdynamic = 2 ρ v12 Phydraulic = Pstatic + Pdynamic Net power flux: Phydraulic Q Mod. Sim. Dyn. Sys. Bernoulli’s equation page 3NETWORK REPRESENTATION HOW 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). RB P1 P2 1 : Q The constitutive equation of the “Bernoulli resistor” is defined as 1 1 1 A22 A12 1 ρ ⎛⎜⎝ ⎞⎟⎠ Q2 PBernoulli = ρ (v22 – v12) = – 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 4THIS APPROACH YIELDS THE RIGHT EQUATIONS BUT IT HAS SEVERAL UNSATISFACTORY ASPECTS. • The “Bernoulli resistor” does not dissipate free energy. In fact, this “resistor” violates an important constraint on resistor constitutive equations —it may supply as well as absorb power. To be fair, that flaw could be rectified by a suitable changeof terminology. • Hydraulic pressure cannot be represented explicitly. As a result, the bond graph seems to suggest that power flux is Pstatic Q, not Phydraulic Q. Mod. Sim. Dyn. Sys. Bernoulli’s equation page 5• The “Bernoulli resistor” is not related to the kinetic energy from which it arises. To model the power required to accelerate or decelerate the fluid, we may add a fluid inertia RB 1 P1 P2 I If: : Q the fluid inertia appears to be independent of the dynamic pressure effects but they are different aspects of the same phenomenon —kinetic energy stored in the fluid Mod. Sim. Dyn. Sys. Bernoulli’s equation page 6AN ALTERNATIVE (AND SUPERIOR) APPROACH: Carefully analyze the kinetic energy stored in the pipe. The co-energy is E*k = E*k(Q,m) = 1 If Q22where If depends on the specific geometry but is proportional to m. The corresponding “pressure momentum”, Γ, is defined by ∂E*kΓ = = If Qm∂Q The kinetic energy may be found using a (negative) Legendre transform. Ek(Γ,m) = E*k(Q,m) - ΓQ Mod. Sim. Dyn. Sys. Bernoulli’s equation page 7This kinetic energy storage element has three ports. One is due to the power required to accelerate the fluid. It is associated with the change of pressure momentum, Γ, and the conjugate (equilibrium-determining) variable is thevolumetric flow rate, Q. The other two ports are due to the mass flows at sections 1 and 2. The corresponding conjugate variables are found from thegradient of the kinetic energy with respect to mass. In this case the kinetic energy and co-energy are numerically equal. ∂Ek ∂E*k Q ∂m Γ = ∂m Mod. Sim. Dyn. Sys. Bernoulli’s equation page 8It is easier to work with the kinetic co-energy. The effort corresponding to the mass flow, ˙m i, is the kinetic (co-)energy flux per unit mass. µikin = 1 vi2 = 1 Pdynamic,i2 ρ As the total mass of fluid in the pipe is proportional to the total volume of the pipe and the corresponding effort is proportional to the dynamic pressure, the mass flow ports have the character of a capacitor while the momentum port has the character of an inertia. This may be represented as an IC-type storage element as follows. 12 Γ˙ 2v2 ICQ˙m2 12 ˙m12v1 Mod. Sim. Dyn. Sys. Bernoulli’s equation page 9AN ALTERNATIVE REPRESENTATION (perhaps less ambiguous) a three-port capacitor with a unit gyrator on one port. 12 Γ˙ Q 2v2 GY C Γ˙ ˙Q m2 12 ˙m12v1 As the fluid is assumed incompressible, we need to add a constraint that the mass flow in equals the mass flow out. This may be represented by a junction structure as follows. 12 2 v2 "Bernoulli resistor"Fluid inertia GY C ˙m2 12Γ˙ ˙m12 v1 01 21 (v22– v1 )2 TF:ρ P1 P21 : Q Mod. Sim. Dyn. Sys. Bernoulli’s equation page 10The transformer relates mass flow rate, ˙m , to volumetric flow rate, Q. ˙m = ρ Q It also relates the “Bernoulli pressure” to the difference of the kinetic efforts. 1PBernoulli = 2 ρ (v22 – v12) The zero and one junctions represent the incompressibility constraint. ˙ ˙ m 2 = –m 1 The pattern of power orientations has been chosen to resemble that used with the “Bernoulli resistor” above. With this orientation, the left and right sides of the energy-storage element correspond to the fluid inertia and “Bernoulli resistor” as indicated. The multiport representation has two merits the lossless assumption is self-evident the fundamental relation between the two phenomena —fluid inertia and “dynamic pressure”— has been represented explicitly. Hydraulic pressure has not been represented explicitly. Revise the bond graph as follows. Mod. Sim. Dyn. Sys. Bernoulli’s equation page 1112 12 2v1 2v2 1 ρ :TF 0 C TF :ρ m˙ m˙1 2 Pdynamic,1Pstatic,1 Pdynamic,2GY Γ˙ Phydraulic,1 Phydraulic,2 Pstatic,21 1 1 : Q This representation clearly shows that the rate of change of pressure momentum is driven by the difference in hydraulic pressures. dΓ dt = Phydraulic,1 – Phydraulic,2 Mod. Sim. Dyn. Sys. Bernoulli’s equation page 12dΓIf we assume steady flow, dt = 0 and hence ∆Phydraulic = 0 and we may eliminate the momentum port as follows. 12 12 2v1 2v2 1 ρ :TF 0 C TF :ρ m m˙ ˙1 2 Pdynamic,1Pstatic,1 Pdynamic,2 Phydraulic,1 Phydraulic,2 Pstatic,2= 1 1 : Q This representation clarifies the transition from static fluid