Analogous Physical Systems

1. Analogous Physical Systems BIOE 4200

2. 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

3. 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

4. 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

5. Physical Variables

6. Physical Variables • Force is a through variable because of Newton’s 3rd 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

7. 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

8. Mathematical Relationships • Energy Dissipation • Through ~ Across • Energy dissipated ~ Across2 or Through2 • Capacitive Storage • Through ~ d(Across)/dt • Energy stored ~ Across2/2 • Inductive Storage • Across ~ d(Through)/dt • Energy stored ~ Through2/2

9. Energy Dissipation

10. Capacitive Storage

11. Inductive Storage