Analogous Physical Systems BIOE 4200

Creating Mathematical Models Break down system into individual components or processes Need to model process outputs as function of inputs and process states Models can be obtained experimentally – Measure output using a range of inputs Models can be obtained theoretically – Derive equations using physical principles This lecture will focus on physical principles

Theoretical Modeling Many real physical systems can be modeled using a combination of ideal elements Processes represent “lumped” parameters – Idealized mechanical systems use point masses, springs, dampers – Idealized electrical circuits use resistors, capacitors, inductors Assume spatial (3-D) properties do not affect the results of your model Different types of idealized physical systems are governed by the same equations

Physical Variables Physical quantities can be categorized in two groups “Through” variables – Quantities that pass through ideal element – Value of through variable is same going into and coming out of ideal elements “Across” variables – Quantities are measured across ideal elements – Values do not make sense unless they are measured relative to a reference point

Physical Variables SystemThrough variableAcross variable TranslationForce (F)Velocity (v) Rotation Torque (  ) Angular velocity (  ) ElectricalCurrent (i)Voltage (V) Fluid Volumetric flow rate (Q) Pressure (P) ThermalHeat flow rate (q)Temperature (T)

Physical Variables Force is a through variable because of Newton’s 3 rd law – Pulling on one end of spring produces equal and opposite force on other end Velocity, pressure and temperature are across variables because they are relative – Pressure must be measured across two points – Temperature difference is relevant in heat transfer – Velocity is relative – Newtonian frame of reference

Ideal Elements Categorize ideal elements by how energy is transferred within the process Processes generally transfer energy from one source to another or convert energy from one form to another Energy dissipation – energy entering process is dissipated as heat loss Capacitive storage – energy entering the process is accumulated as velocity or charge Inductive storage – energy entering the process is stored as a force or electric field

Mathematical Relationships Energy Dissipation – Through ~ Across – Energy dissipated ~ Across 2 or Through 2 Capacitive Storage – Through ~ d(Across)/dt – Energy stored ~ Across 2 /2 Inductive Storage – Across ~ d(Through)/dt – Energy stored ~ Through 2 /2

Energy Dissipation SystemElementEquationEnergy Translation Damping b (friction) F = bvbv 2 Rotation Damping b (friction)  = b  b2b2 ElectricalResistance Ri = V/RV 2 /R Fluid Resistance R f (pipe) Q = P/R f P 2 /R f Thermal Resistance R t (insulation) q = T/R t T/R t

Capacitive Storage SystemElementEquationEnergy TranslationMass m F = m dv/dt (F = ma) mv 2 /2 RotationInertia J  = J d  /dtJ  2 /2 ElectricalCapacitor Ci = C dV/dtCV 2 /2 Fluid Fluid storage C f (balloon) Q = C f dP/dtC f P 2 /2 Thermal Thermal storage C t q = C t dT/dtCtTCtT

Inductive Storage SystemElementEquationEnergy Translation Linear spring k kv = dF/dt (F = kx) F 2 /2k Rotation Torsional spring k k  = d  /dt (  = k  )  2 /2k Electrical Inductor L (magnet/coil) V = L di/dtLi 2 /2 FluidFluid inertia IP = I dQ/dtIQ 2 /2 Thermal???