V. Using Analogy in Modeling

Use of Metaphor and Analogy

one of the most powerful intellectual methods for developing understanding

Generalized Variabes

abstractions of physical variables that allow similar functional relations between variables to bo constructed when dealing with physically different systems

What do generalized variables relations allow?

1. many different kids of systems to be analyzed according to a common set of procedures

2. allow analogies to be made such that well-developed methods in one-domain (e.g. engineering/mechanics) can be applied to solve problems in another domain (e.g physiology/circuits)

Effort Variable

a quantity that results from expenditure of energy, if unbalanced, causes something to move; requires a reference or a comparison between different locations

ex. force (mechanics), hydraulic pressure (fluid), voltage (electrical)

Generalized Flow, f (Flow variable)

the quantity that changes its location with time as a result of differences in the effort variable

ex. velocity (from force), volumetric flow (from hydraulic pressure), current (from voltage)

Derived Variables

can be derivatived from effort and flow; generalized dispalcement and momentum

Generalized Displacement

take flow variable and integrate

Generalized Momentum

integrate effort variable

Generalized Variables for Different Physical Systems Table Picture

Flow is always opposed by what?

some impedance

Hindrance of Flow

is by the impedence is overcome by effort

Generalized Impedance

the ratio of effort to flow

Impedance

generally not a constant; an intrinsic property of the system that defines the amount of effort needed to achieve a given flow; combines all constitutive relations of importance

What causes flow through an impedance?

the expenditure of energy always accompanies the application of effort

Power

the rate at which energy is applied

Generated Power

the work performed in applying the power to create a given

Generalized System Elements

can be defined by the manner in which effort is related to flow

Generalized Resistance Element, R

system element possesses a resiatance property if the effort is proportional to flow

Relationships for Generalized Resistance Elements

resistance increases w/ distance but decreases w/ SA and diffusion coefficient

resistance decreases w/ increasing kinetic rate constant and vessel volume

Generalized Potential Energy Storage Element

element stores PE when effort applied to system increases as a result of an increase in displacement

C - generalized capacitance

impedance is not a simple scale factor (Z* = e/f = (1/c * integral fdt) = h/c)

energy put into potential energy storage element is recoverable and equals the work performed on the element (w = integral (efdt) = integral (C dedt dt) = Ce²/2 = h²/2C

typical ex. capacitar and a spring

Generalized PE Stroage Element Relationships

Generalized Kinetic Energy Storag Element

an element stores KE when the effort applied to the system increases as a result of an increase in the rate of change of flow (e = L df.dt → f = 1/L integral e dt = 1/L P (L - generalized inertance???)

impedance is not a simple scale factor (Z* = e / (1/L * integral e dt))

energy put into a KE storage element is recoverable and equals the work performed on the element (w = integral ef dt = integral L df/dt f dt = L integral f df = (1/2) LF²

ex. resistor and mass (not found in chemical and thermal systems)

Balancing Techniques for Building Models of Systems with Interacting Passive Elements

need these to build dynamic eqns for systems of multiple passive elements

Balancing Laws

displacement balance yields generalized node law

energy balance yields generalized loop law

sign convention: flow is in the clockwise direction around the loop and the + pole of effort is at flow entry for all elements but the last

Passive Element Combinatiosn

series + parallel resistance elements

series + parallel PE storage elements

resistance + PE storage elements

series combination of all 3 elements

parallel connection of all 3 elements

system dynamic behavior depends on arrangement amongst elements as much as the dynamic behavior of the elements themselves

System dynamic behavior depends on what?

on the arrangement amongst elements as much as the dynamic behavior of the elements themselves

Series + Parallel Resistance Elements

Series + Parallel PE Storage Elements

Resistance + PE Energy Storage Elements

Series Combindation of all 3 of our Elements

Parallel Combination of the 3 Elements