s ystem d ynamics m odeling in f luids e lectricity ,...

S

YSTEM

D

YNAMICS

M

ODELING

IN

F

LUIDS

, E

LECTRICITY

, H

EAT

,

AND

M

OTION

Examples, Practical Experience, and Philosophy

Hans U. FuchsDepartment of Physics

Zurich University of Applied Sciences at Winterthur8401 Winterthur, Switzerland

Invited Talk at the2006 GIREP Conference on Modeling in Physics and Physics EducationAMSTEL Institute of the University of Amsterdam, August 20-25, 2006

Hans U. Fuchs, ZHW 2006 2

Contents

Part 1

System Dynamics Modeling

How it is done, what kind of models it leads to, how simple it is, how easily it is integrated with experiments, and how it supports creativity

I want to show how easy and how practical it is to do SD modeling…

Part 2

Creativity and basic forms of thought

A theory of creativity in science, the gestalt of physical processes, imaginative structures and figurative thought, and analogical reasoning

I want to argue that SD modeling makes use of some fundamental human thought processes that correspond to how humans “see” nature…


Part 1

System Dynamics Modeling

How it is done, what kind of models it leads to, how simple it is, how easily it is integrated with experiments, and how it supports creativity


SD Modeling

—

Outside of physics

Undiscovered Gas ReservesDiscovery Usage

Reserve usage ratio

Price

roc price

Potential usage

Price ratio

Investment

~

Fraction invested

Revenue

Relative rur

Discovery cost

Relative rur

Initial usage

Growth factor

Desired rur

Initial undiscovered

Initialdiscovery

cost

Initial price

Initialdiscovery

cost

d

dtReserves Discovery Usage

Reserves

Dis

= −

( ) = …0

ccovery = …( )= …( )

f

Usage g


SD Modeling

A graphical approach to building models of dynamical systems by

combining the relations we perceive to hold in such systems. It makes use

of a very few structures which are projected onto virtually any type of

dynamical system and its processes, i.e., it makes strong use of analogical

reasoning.

Mathematically speaking, the models created are initial value problems of

spatially uniform elements.

There are several programs available that implement the SD approach.

Here, I shall use Stella as an example.


Two communicating oil containers

Flow

V 1 V 2

Flow

A 1 h 1h 2 A 2

Flow factor

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

0

50

100

150

0 100 200 300 400

Lev

el /

mm

Time / s


Two communicating water containers with outflow

0.00 75.00 150.00 225.00 3000.00

0.15

0.30

1

1

1

1

2

2

2

2

3 3

3

3

4

4

4

4

V 1

Flow 2

Flow factor 2V 2

Flow

A 1

p 1p 2

A 2

Flow factor

Level versus time

Flow =

IF ((P1-P2) > 0) THEN SQRT((P1-P2)/k1)

ELSE -SQRT(-(P1-P2)/k1)


Two gliders with magnets

0.00 0.75 1.50 2.250.30

0.60

0.90

1

1

1

1

2

22

3

33

4

4

4

p1 p2Ip magnet

v1 v2

m1m 2

x1

dx1 dt

x2

dx2 dt

delta x

Factor

Ip_magnet = Factor / delta_x^5

Position versus time


A model of a sheep’s aorta

V Aorta 1

delta P Ao 1

IV AV

LVP m

C

R AV

delta PR AV

V Aorta 2

IV Ao 1

C

delta P Ao 2

R Ao 1V Aorta 3

IV Ao 2

C

delta P Ao 3

R Ao 2 V Aorta 4

IV Ao 3

C

delta P Ao 4

R Ao 3 V Aorta 5

IV Ao 4

C

delta P Ao 5

R Ao 4

IV A

R Ao 5

1.10 1.55 2.00 2.450

8000

16000

1

1

1

122

2

2

33

3

3

Pressure versus time


A model of a sheep’s aorta with inductive effects

V Aorta

delta P Ao 1

C

V Aorta 2IV Ao 1

C

delta P Ao 2

dp L 1

R Ao

V Aorta 3IV Ao 2

C

R Ao IV Ao 3

IV 1

dIV dt 1L

dp L 2

IV 2

dIV dt 2

0.00 0.80 1-1e-04

1e-04

3e-04

1

1

2

2

Blod flow versus time


Two capacitors with battery

0.00 40.00 80.00 120.00-4.00

0.00

4.00

1

11 1

2

2 2 2

Q 1

U S

Q 2

IQ

C 1

U 1U 2

C 2

G

Voltage versus time

I_Q = G*(U_S+U_1-U_2)


Thermal equilibration

S 1 S 2S node

IS 2

Pi S

P diss

IS

K 1

T 1 T 2

K 2

GS

S 1 S 2

IS

K 1

T 1T 2

K 2

GS IS = GS*(T_1-T_2)

P_Diss = (T_1-T_2)*IS

Pi_S = P_diss/T_2


Thermal equilibration: Experiment

GS

S node

spec heat water

mass water 1

specific entropy 1

S water 1 S water 2

T water 1

IS

T ref

T water 2

T ref

IS 2

specific entropy 2

mass water 2

IS 1 loss IS 2 loss

T amb

T amb

GS loss GS loss

Pi S

Temperature versus time

0.00 375.00 750.00 1125.00 1500.00290

330

370

1

1

1 1

2

22 2

3

33 3

4

44 4


Peltier device: Principle

S 1 S TR 1 S 2

Q 1 Q 2

IS 2

IS 3

Pi S

IQ 1 IQ 2

UB

IQ 3

phi 1

RT 1 T 2

Entropy capcitance Entropy capcitance

Seebeck

IS TE

phi 2

U TE

R

IQ 2

R th

U PE

Charge capacitanceCharge capacitance

R ext

Peltier

U_TE = Seebeck*(T_2-T_1)

IS_TE = Peltier*IQ_2

Pi_S = P_diss/T_2

P_diss =

-IS_2*(T_2-T_1) + R*IQ_2^2


Thermal equilibration: Experiment

S 1 S TR 1 S 2

Q 1 Q 2

KS T

ST 1IS 2

IS 3

Pi S

IQ 1 IQ 2

UB

IQ 3

phi 1

RT 1 T 2

KSKS

Seebeck

IS TE

phi 2

U TE

R

IQ 2

ST 2

R th

U PE

C

C

KS T

Ta

GS TPE

S node 2

Ta

IS LT 1 IS LT 2

S node 1

R ext

GS TPE

GS T L

GS a

TT 1TT 2

GS a

IS 1

IS 11 IS 4

GS T L

Ta

Pi S 1

IS 41

Pi S 2

Ta

IS L 1 IS L 2

Peltier

Temperature versus time0.00 75.00 150.00 225.00 300.00290.00

300.00

310.00

1

1

1

1

2

2

2

23

3

3

3

4 4

4 4


Summary 1: Creating and using SD models

• We can construct models step by step by assembling a system made up of a few graphical symbols.

• The symbols represent basic elements of human reasoning (see Part 2).

• Complete models can be simulated.

• Simulation results can be compared to data take in the lab.

• Models can be extended, single relations can be changed, parameters can be adjusted… We move back and forth between modeling and experiment.

• SD modeling allows us to work on some fairly complex real life applications.

V 1

Flow 2

Flow factor 2V 2

Flow

A 1

p 1p 2

A 2

Flow factor


Summary 2: Structure of SD models in physics

Some quantities are stored, they can flow (and sometimes, they are produced). There are differences (of potentials) which lead to

• flows

• changes of flows

• production rates

Differences are produced by

• storage

• pumps (in a generalized sense)

When quantities flow through a difference, energy is released. The energy released is

• used to drive other processes (set up other differences)

• dissipated

• stored or transferred

V 1

Flow 2

Flow factor 2V 2

Flow

A 1

p 1p 2

A 2

Flow factor


Part 2

Creativity and basic forms of thought

A theory of creativity in science, the gestalt of physical processes, imaginative structures and figurative thought, and analogical reasoning


A model of creativity in the sciences

A bi-cycle represents the interaction between modeling and experiment. A cycle is a model for a type of cognitive tool.

Steps leading to the creation of hypotheses and the generation of good questions, can be represented as two additional cycles.

See Kieran Egan on cognitive tools.


The roots of creativity in the sciences

Creativity, in the sense of the generation of ideas and hypotheses, is rooted in

imaginative

structures

of human thought.

Imagination

(Vorstellungskraft: I. Kant), grows from the human capacity to

re-present

situations with the help of representational tools (mimesis, language, writing, drawing…).

Metaphor

is an example of this type of representation: One thing is represented in terms

of another.

Metaphor is the capacity, or cognitive tool, that enables people to see one thing in

terms of another

(Kieran Egan, 2005).

From metaphor grows the perception of (structural) similarity which is used in

analogical

reasoning

.

Let us see how humans conceptualize large classes of physical processes…


The gestalt of physical processes

Human perception of phenomena such as fluids, electricity, heat, motion

The concept of “heat,” for example, is abstracted by perception from the sum total of thermal experiences

in the form of a

gestalt

: An entity that encompasses more than the sum of its parts. While we do not

differentiate a gestalt of a collective of phenomena (such as electricity or heat) consciously, we do notice

aspects. The most fundamental

aspects

humans use to talk about such a

gestalt

are

Table 1: The gestalt of collectives of physical phenomena

A

SPECT

OF

GESTALT

M

ETAPHORIC

STRUCTURE

Intensity (quality)

Polarity such as light-dark, warm-cold, high-low, fast-slow, strong-weak. The concepts are structured metaphorically by the image schema of verticality (intensity as a level).

Quantity (substance)

Substance-like concepts are metaphorically structured in terms of fluid substances.

Force or power

Prototypical causation as the

gestalt

of direct manipulation.


Prototypical causation: The gestalt of direct manipulation

The

gestalt of direct manipulation

Lakoff (1987, p. 54), Lakoff and Johnson (1980, p. 70)

Aspects of the gestalt

1. There is an

agent

that does something.

2. There is a

patient

that undergoes a change to a new state.

3. Properties 1 and 2 constitute a single event; they overlap in time and space; the agent comes in contact with the patient.

4. Part of what the agent does (either the motion or the exercise of will) precedes the change in the patient.

5. The agent is the

energy source

; the patient is the

energy goal

; there is a

transfer of energy from the agent to patient

.

6. …


Evidence for the gestalt of physical processes 1

Journalism students judging the validity of linguistic expressions using heat and temperature

Persons are asked if they agree or disagree with certain expressions

• The temperature is high

• Today, the heat is high

• There is lots of heat in this room

• There is lots of temperature in this room

• Heat drives the engine

• Temperature drives the engine

a. Agreement (1) or disagreement (0) with expressions using heat and temperature. Expected results in paren-theses. Results of a questionnaire given to journalism students in Summer of 2004.

Tabelle 2: Agreement with classes of expressions

a

as substance as cause as level

Heat 0.67 (1) 0.77 (1) 0.14 (0)

Temperature 0.09 (0) 0.09 (0) 0.83 (1)



Sadi Carnot’s image of the waterfall and its comparison to the power of heat

Sadi Carnot (1796-1832)

Réflexions sur la puissance motrice du feu

D'après les notions établies jusqu'à présent, on peut comparer avec assez de justesse la

puissance motrice de la chaleur

à celle d'une

chute d'eau

[…]. La puissance motrice d'une chute d'eau dépend de sa hauteur et de la quantité du liquide; la puissance motrice de la chaleur dépend aussi de

la quantité de calorique

employé, et de ce qu'on pourrait nommer, de ce que nous appellerons en effet

la hauteur de sa chute

, c'est-à-dire de

la différence de température

des corps entre lesquels se fait l'échange du calorique.



The concept of heat in the Accademia del Cimento

The concept of heat of the members of the Accademia del Cimento: Saggi di naturali esperienze… (1667)

M. Wiser and S. Carey (1983): When Heat and Temperature were one.

“The Experimenters’ concept of heat had three aspects:

substance

(particles),

quality

(hotness), and

force

.”

A weakly differentiated gestalt

It seems that the Experimenters did not really distinguish between these aspects of the gestalt of heat.


The concept of heat in the Accademia del Cimento

The concept of heat of the members of the Accademia del Cimento: Saggi di naturali esperienze… (1667)

The description of thermal phenomena by the Experimenters demonstrates clearly the image corresponding to direct causa-tion: Hot or cold bodies are the sources of heat or cold. Heat or cold are emitted by the sources, and they influence other bod-ies. The Experimenters were interested in the “force” or “power” of heat (or of cold).

See M. Wiser and S. Carey (1983)



Visual expressions of the metaphors used to structure the aspects of the gestalt

Substance-based thinking(storage and flow/production)

Causative thinking / With feedback-thought

Causative thinking (interaction) / With feedback-thought

(Level differences)(Releasing energy)

Charge

Charge transport

Substance

Flow of substance

Momentum 1 Momentum 2

Momentum flow

Substance

Flow of substance

Momentum 1 Momentum 2

Momentum flow

VIV

PressureC

VIV

PressureC Power


Evidence 5: The gestalt of abstract concepts

Concepts such as

pain

or

love

are conceptualized in the same manner as physical concepts such as

heat or electricity. The have a similar gestalt with the same aspects of

quantity (substance) / intensity (quality) / force or power

Linguistic expressions

• The pain was to strong

• The level of pain went down

• More and more pain…

• The pain gained control of me

• The aspirin took away my headache

• The headache soured my mood

Entailments of the conceptualization

Two broken teeth means double the pain. More pain means higher intensity. More pain means the

pain is more powerful. Higher intensity leads to more power.


Entailments of the metaphoric structure of physical concepts

An example of entailments that can be brought into quantitative form

1.4 timesthe output

Double thewater current

1.4 timesthe height

Output

Double theoutput

Power =

Level difference · Current of substance

Carnot and analogies…


The structure of basic metaphors

Image schemata as primal source domain

Image schema is a recurring structure of, or within our cognitive processes, which establishes patterns of understanding and reasoning. It emerges from our bodily interactions, linguistic experience and historical context. [Image schemata are gestalts (M. Johnson, 1987, Chapter 5): The Body in the Mind).]

Examples: up/down, inside/outside, near/far, balance, object, process, path

Image schemata are projected metaphorically onto more abstract domains. (Metaphors are one-sided projections.)

Imageschema TARGET

Metaphoricprojection

SOURCE Domain TARGET Domain


Metaphors and analogical reasoning

Origin and meaning of analogies

When different domains of experience are structured metaphorically by the same source domains (such as by the same image schemata), these domains become comparable (they start to look similar).

This comparison can be applied in the construction of analogies. An analogy is a double-sided mapping (more or less symmetrical).

Domain 1

HEAT

Domain 2

Elektricity

Image schema 1

Vertical level

Image schema 2

Fluid substance

ANALOGY

METAPHOR


Three consequences…

…among many others…

1. Take seriously the metaphorical nature of graphical interfaces for human thought

processes.

The soul never thinks without an image. Aristotle, De Anima, Book III, Part 7

The Greeks […] never forgot that direct vision is the first and final source of wisdom.

Rudolf Arnheim, Visual Thinking, p. 12.

2. Make use of concepts that are “built into” humans…

… and you won’t have to worry so much “conceptual change.”

3. Make explicit use of analogies offered by the metaphoric nature of the human mind.

And I cherish more than anything else the Analogies, my most trustworthy masters.

Johannes Kepler (Optics, Quoted in Polya, 1973.)


LITERATURE

Accademia del Cimento (Magalotti, Lorenzo, 1667): Saggi di naturali esperienze fatte nell'Accademia del Cimento sotto la protezione del serenissimo principe Leopoldo di Toscana e descritte dal segretario di essa Accademia. Electronic Edition: Instituto e Museo di Storia della Scienza, Firenze, http://www.imss.fi.it/biblio/ebibdig.html

Arnheim R. (1969): Visual Thinking. University of California Press, Berkeley, CA.

Borer, Frommenwiler, Fuchs, Knoll, Kopacsy, Maurer, Schuetz, Studer (2005): Physik – ein systemdynamischer Zugang, 2. Auflage, h.e.p. verlag, Bern.

Carnot S. (1824): Réflexions sur la puissance motrice du feu. Édition critique avec Introduction et Commentaire, augmentée de documents d’archives et de divers manuscrits de Carnot, Paris : Librairie philosophique J. Vrin (1978). English: Reflections on the Motive Power of Fire. E. Mendoza (ed.), Peter Smith Publ., Gloucester, MA (1977). Deutsch: Betrachtungen über die bewegende Kraft des Feuers, in Ostwald, Willhelm, Ostwalds Klassiker der exakten Wissenschaften, Frankfurt am Main: Verlag Harri Deutsch (2003).

Egan K. (1988) Primary Understanding. Rutledge, NY.

Egan K. (1997): The Educated Mind. How Cognitive Tools Shape Our Understanding. The University of Chicago Press, Chicago.


Literature

Egan K. (2005): An Imaginative Approach to Teaching. Jossey-Bass, San Francisco.

Fuchs H. U. (1996): The Dynamics of Heat. Sprinnger-Verlag, New York.

Fuchs H. U. (2002): Modeling of Uniform Dynamical Systems. Orell Füssli, Zurich.

Gibbs R. W. (1994): The Poetics of Mind. Figurative Thought, Language, and Understanding. Cambridge University Press, UK.

Johnson M. (1987): The Body in the Mind. The Bodily Basis of Meaning, Imagination, and Reason. University of Chicago Press, Chicago.

Lakoff G. and Johnson M (1980): Metaphors We Live By. University of Chicago Press, Chicago (with a new Afterword, 2003).

Lakoff G. and Johnson M. (1999): Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought. Basic Books, New York.

Polya G. (1973): Mathematics and plausible reasoning. Volume 1. Princeton University Press, Princeton, NJ.

Wiser M. and S. Carey (1983): When Heat and Temperature were one, in D. Gentner and A. L. Stevens eds.: Mental Models, Lawrence Earlbaum Associates, Hillsdale, new Jersey, 1983 (pp. 267-297)

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