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CS 367: Model-Based Reasoning Lecture 16 (03/14/2002) Gautam Biswas 7/17/2015 1 Today’s Lecture Last Lectures: Bond Graphs and Causality State Space Equations from Bond Graphs Today’s Lecture 7/17/2015 More Complex Examples 20-SIM 2 Electrical Circuits: Example 2 Try this one: 7/17/2015 3 Mechanical Model: Example 2 Try this one: 7/17/2015 4 State Equations Linear x A. x B.u x1 a11 . x1 .... a1n x n b11u1 .... b1m um x 2 a 21. x1 .... a 2 n x n b21u1 .... b2 m um .. . x n a n1. x1 .... a nn x n bn1u1 .... bnm um Nonlinear x ( x, u) x 1 1 ( x1 ,...., x n , u1 ,....,um ) x 2 2 ( x1 ,...., x n , u1 ,....,um ) 7/17/2015 .. . x n n ( x1 ,...., x n , u1 ,....,um ) 5 Causality for basic multiports Note that a lot of the causal considerations are based on 7/17/2015 algebraic relations 6 Causality Assignment: Example 3 Try this one: 7/17/2015 7 Equation Generation: Example 2 7/17/2015 8 Extending Modeling to other domains Fluid Systems e(t) – Pressure, P(t) f(t) – Volume flow rate, Q(t) Momentum, p = e.dt = Pp, integral of pressure Displacement, q = Q.dt = V, volume of flow Power, P(t).Q(t) Energy (kinetic): Q(t).dPp Energy (potential): P(t).dV Fluid Port: a place where we can define an average pressure, P and a volume flow rate, Q Examples of ports: (i) end of a pipe or tube (ii) threaded hole in a hydraulic pump 7/17/2015 9 Fluid Ports Flow through ports transfers energy P – force/unit area Q – volume flow rate P.Q = power = force . displacement / time Moving fluid also has kinetic energy But it can be ignored if 1 Q 2 P ( ) 2 A 7/17/2015 10 Fluid Domain: Resistor P1 Q Q1, P1 Q2, P2 1 P 1 Q2 2 P3 Q3 l R Pipe, Porous plug or a Constriction P 3 R.Q3 Q1 Q 2 Q3 flow P 3 P1 P 2 : pressure drop How do we compute R? For thin, long tubes with incompressible and laminar flow 128. .l ; :viscosity,l : length, d : diameter .d 4 For turbulent flow : R 3 4 P3 at .Q3 Q3 ; at :experimental constant Re 7/17/2015 Resistance relationship changes for orifices, etc. for turbulent flow 4. .Q ; Re 2000 laminar .d. above 4000 definitely turbulent 11 Fluid Capacitor C P1 Q1 Q3 Q1 Q2 P1, P2, P3 For rigid pipe P3 Q3 P 0 Q2 2 P1 P2 P3 density Q3 Q1 Q2 h height P3 .g.h Note : P3 .g.P3 A g gravitational const. V3 A ;C V3 volume C .g A area V .r .l C3 0 0 ; r0 radius B B l length; B bulk modulus (gives us pressure change when liquid is compressed) 2 7/17/2015 12 Fluid Inertia P1 Q1 area, A I P2 Q2 P1 Q1 P3 Q3 P3 P1 P2 P 1 Q2 2 Q1 Q2 Q3 Linear System I 3 .Q3 PP3 momentum net force on slug of liquid P1. A P2 . A ; fluid velocity QA3 Mass of slug . A.l F P1. A P2 . A m.a . A.l . QA3 As pressure difference .l .l P3 P1 P2 .Q3 ; I 3 fluid inertia increases, flow accelerates A A quicker for larger pipes Interesting point : A I 3 7/17/2015 13 Fluid effort sources 7/17/2015 14 Fluid flow sources at rate Q(t) 7/17/2015 15 Modeling a fluid line Simplest case: just a lumped resistance Distributed Model (break pipe into shorter segments) 7/17/2015 16 7/17/2015 17 Schematic of Secondary Sodium Cooling Loop sodium flow stopping valve (normally opened) R2 Vin main motor secondary sodium pump I IH X intermediate heat exchanger f11 e33 R1 e22 evaporator sodium overflow tank GY C E V COFC R4 R5 7/17/2015 e19 feed water loop sodium flow stopping valve (normally opened) e14 super heater CSH feed water loop R3 18 Nuclear reactor Heat Transport from reactor core to turbine – two loops primary and secondary loops. We model the secondary loop – use liquid sodium to transport heat from primary loop to evaporator and super heater where steam is produced to drive the turbine Primary components: intermediate heat exchanger, pump and motor system, super heater, evaporator, pipes, valves, and overflow tank. 7/17/2015 19 Building Bond Graph Model Step 1: Focus on energy and mass balance in fluid domain + mechanical characteristics of main motor & pump Assume motor is an AC synchronous motor; don’t model electric field effects – assume it is present as soon as motor is turned on. (dynamic electrical effects of motor not modeled) 7/17/2015 20