Hans U. Fuchs Center for Applied Mathematics and Physics ...

Q

UANTITY

, I

NTENSITY

,

AND

P

OWER

OF

H

EAT

A Direct Approach to a Dynamical Theory of Heat

Hans U. FuchsCenter for Applied Mathematics and Physics

Zurich University of Applied Sciences at Winterthur8401 Winterthur, Switzerland

Talk at the Università degli studi di Modena e Reggio Emilia 12 Novembre 2010

Hans U. Fuchs, ZHAW 2010 2

G

OAL

…

In this talk, I wish to discuss how (

1

) a

dynamical theory of heat

can be constructed; for this I use the Second Edition of

The Dynamics of Heat

(Springer, New York 2010). At the same time I hope to clarify (

2

) its

conceptual roots

and show (

3

)

how it can be taught

.

Note: To make goals (2) and (3) clear, I will use a

form of language

throughout that reflects the

figurative

nature of

human thought

. This should give you a feeling for the

form of conceptualization

I use when teaching the physics of dynamical systems.

Covers of DoH:


Contents

Part 1 Dynamical Fluid and Electric Phenomena

Part 2 A Generalized Approach to Energy in Physical Processes

Part 3 Figurative Structure of Macroscopic Physics

Part 4 Quantity, Intensity, and Power of Heat

Part 5 Introducing a Theory of Dynamical Thermal Processes

Part 6 Uniform Models of Viscous Fluids


Part 1

D

YNAMICAL

F

LUID

AND

E

LECTRIC

P

HENOMENA

As an example, an exercise in modeling of electric windkessel circuits and the mammalian systemic blood flow system is presented.

We will see that the models make use of

only a small handful of concepts

which have

analogous structures

in fluids and electricity.


E

LECTRIC

W

INDKESSEL

AND

M

AMMALIAN

S

YSTEMIC

B

LOOD

F

LOW

: E

XPERIMENTS

BBBBBBBBBBB

B

B

B

B

BBB

B

B

B

B

BBBBBBBBBBBBBB

B

B

B

B

B

BB

B

B

B

B

B

BBBBBBBBBBBBBBBBBBBBBBBBBBB

B

B

B

B

B

B

BB

B

B

B

B

B

BBBBBBBBBBBBBBBBBBBBBB

B

B

B

B

B

BBBBB

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B

B

B

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B

B

B

B

B

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B

B

B

B

B

B

BBBBBBBBBBBBBBBBBBB

B

B

B

B

JJJJJJJJJJJJJ

J

J

J

J

JJJJJJJJJJJJJJJJJJJJJ

JJJ

JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ

J

J

J

J

J

JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ

JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ

J

J

J

J

J

JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ

-1

0

1

2

3

4

5

6

10 15 20

Vol

tage

/ V

Time / s50

0

40

80

120

160

-100

0

100

200

300

0 0.2 0.4 0.6 0.8 1Pr

essu

re /

mm

Hg

Flow

/ m

l/s

Time / s


E

LECTRIC

W

INDKESSEL

AND

M

AMMALIAN

S

YSTEMIC

B

LOOD

F

LOW

: M

ODELS

Leftventricle

Aorta

Aorticvalve

Arteries andcapillaries

Charge

IQ 2

UC

IQ 1

R2

Capacitance

R1

UD

UR 2UR 1

USXXX

X

XXXX

X

XXXXXXX

X

X

XXXXXX

X

XXXXX

XX

XXXXX

-1

0

1

2

3

4

5

6

0 5 10 15 20

Vol

tage

/ V

Time / s

Volume

IV 2

ΔpC

IV 1

R2

Capacitance

R1

Δp V

ΔpR 2ΔpR 1

Δp LV

0

4000

8000

12000

16000

0 0.5 1 1.5 2

Pres

sure

/ Pa

Time / s

dV

dtI I V V

p p p p

p p

p V C

I p R p I p R

V V

R LV C V

R C

C

V R R V R

= − ( ) =

= − −

=

=

= >( ) =

1 2 0

1

2

1 1 1 1 2 2 2

0

0

,

,

∆ ∆ ∆ ∆

∆ ∆

∆

∆ ∆ ∆


S

TRUCTURE

OF

D

YNAMICAL

M

ODELS

Basic quantities to build hydraulic models with:

Volume, volume current, pressure and pressure difference

Law of balance

Loop rule

Constitutive relation 1: Capacitive law

Constitutive relation 2: Flow law

dV

dtI I V V

p p p p

p p

p V C

I p R p I p R

V V

R LV C V

R C

C

V R R V R

= − ( ) =

= − −

=

=

= >( ) =

1 2 0

1

2

1 1 1 1 2 2 2

0

0

,

,

∆ ∆ ∆ ∆

∆ ∆

∆

∆ ∆ ∆

Pot

enti

al

PositionA B C D E F AG H

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

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

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

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

0E+0

1E-5

2E-5

3E-5

4E-5

5E-5

6E-5

0 1000 2000 3000

I V /

m3 /

s

∆p / Pa

V

P

Flux

FluxFlux

SystemSystem content

System boundary

Surroundings


Part 2

A G

ENERALIZED

A

PPROACH

TO

E

NERGY

IN

P

HYSICAL

P

ROCESSES

Here is a generalized approach to energy that owes much to Sadi Carnot’s and J. W. Gibb’s thermodynamics. Energy is conceived as the same quantity in all physical processes—there are no “forms of energy.”

Carnot’s image of

waterfalls

used as an analogy for heat engines can be generalized to all processes. It suggests that we should think of

energy

being

released

(or used) in physical processes…


W

ATERFALLS

AND

THE

P

OWER

OF

P

ROCESSES

Power = Level Difference · Flow

IV

P21 P1

IQ

ϕ2ϕ

Releasingenergy

Usingenergy

Pump

Electriccharge Fluid

Energy

1.4 timethe output

Double thecurrent

1.4 times the height

Output

Double theoutput


E

NERGY

T

RANSPORT

AND

E

NERGY

S

TORAGE

Transport

Storage

E

FFICIENCY

Energy Energy EnergyEnergy

IV

1ϕ2ϕ

Turbine

IQ

P2P1

Generator

Heat

Electric Heater

IL

ω 1 ω 2

Energy current = Level · Flow

This relation holds only for con-ductive transports (not for radi-ation or convection).

IV

1ϕ2ϕ

Pump

IQ

P2P1

Energy

V

E

PressureVessel

V

E

PressureVessel

Turbine andGenerator

1ϕ 2ϕP1 P2

IQIV

Energy Energy Energy

IV

P2P1

IQ

1ϕ2ϕ

PUMP

Heat

Balance of energy:

dE

dtI IE E= + +…1 2


Part 3

F

IGURATIVE

S

TRUCTURES

OF

M

ACROSCOPIC

P

HYSICS

The human mind is

embodied

, it functions by creating

schemas.

Our thinking is figurative. Literal Language and thought are the exception (if they exist at all)…

The most important structures of the human mind are image schemas, gestalts of rich phenomena, metaphors, and larger narrative units (such as stories).

The concepts of macroscopic physics can be understood in terms of the Force Dynamic Gestalt structured with the help of metaphoric projection of a few simple schemas.


FROM PERCEPTUAL GESTALTS TO REASONING

Gestalts: The act of perception is one of abstraction, products of perception are abstractions (gestalts, patterns, configurations) where the whole is simpler than the sum of its parts.

The power of the human mind is in part due to our ability to differentiate the gestalts of rich phenomena, i.e., to elaborate on the aspects of gestalts by using schemas (which are themselves gestalts) and their metaphoric projections.

In this way we fill the phenomena with embodied meaning. The schemas applied are simple yet they have internal structure and this structure can be made use of in reasoning.

A particular gestalt or schema to understand rich phenomena is what I call the Force Dynamic Gestalt. (Fuchs, 2010b)


RICH EXPERIENTIAL GESTALTS IN PHYSICS: FDG OF ELECTRICITY

The basic perceptual function of Figure-Ground Reversal applies to physical phenomena as well. Example: a. Charge is draining from this element; b. This body is on a higher electric level. This may well be the root of the concept of quantity of electricity (if we take the notion of polarity as given).

Schemas Linguistic expressions

Quantity (amount)Container, Path

The capacitor stores a lot of electricityElectricity (charge) flows through this connector

PolarityIntensity, Level of…Level difference

This body is strongly charged (weakly charged)The capacitor has been charged to a high levelThe difference of potentials is called voltage

Force or power Agent and patient, causation

Electricity drives this pumpElectricity is used to…

Tension The difference of intensities creates an electric tension

Balance Voltages of the capacitors have finally equilibrated

Letting, hindering The resistor obstructs the flow of charge


FDG: SCHEMAS, METAPHORIC PROJECTIONS, AND ANALOGICAL REASONING

If we use the same schemas and metaphoric projections to understand different phenomena,

these phenomena attain a degree of similarity in our mind. This can be used in a form of

analogical reasoning (analogy as structure mapping, Gentner, 1983).

ELECTRICITY

Vertical level

Fluid substance

METAPHOR

Causation(force, power,

energy…)

etc.

Resistance

Balance

HEAT

ANALOGY


Part 4

QUANTITY, INTENSITY, AND POWER OF HEAT — FROM THE ACCADEMIA DEL CIMENTO TO SADI CARNOT AND BEYOND

Early scientific accounts of the phenomenon of heat (and cold) clearly demonstrate that we employ the Force Dynamic Gestalt of heat to understand thermal processes.

Sadi Carnot achieved a clear differentiation of the three main aspects of the FDG of heat: quantity, intensity, and power.

The accounts assume that we can create a dynamical theory of heat, not just one of the statics of heat.


EVIDENCE FOR THE GESTALT OF THERMAL PROCESSES

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 FORCE DYNAMIC GESTALT OF HEAT

Sadi Carnot (1796-1832)

Réflexions sur la puissance motrice du feu (1824)

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.


Part 5

INTRODUCING A THEORY OF DYNAMICAL THERMAL PROCESSES

The following points, taken from Sadi Carnot and from a direct analogy with fluid and electric processes, lead to a dynamical theory of heat (Fuchs, 2010a):

Properties of a quantity of heat (entropy): storage, flow, and production, balance of entropy

Temperature as the thermal level and temperature differences as thermal driving force

The power of heat (entropy) in analogy to the power of a waterfall


HEATING COLD WATER IN A WARM ROOM (DOH, CHAPTER 4.6)

Law of balance of entropy:

Constitutive relations:

h S

Spec entropy

Spec heat water

P mixer S water

T water

IS

Mass water

Pi S mixer

T amb

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

[[[[[[[[[[[[[[[[[[[[[[[[[[XXXXXXXXXX

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

270

280

290

300

310

0E+0 4E+4 8E+4

Tem

pera

ture

/ K

Time / s

T_amb

T_water

A

dS

dtIw

S S= + Π

1ϕ2ϕ

IMMERSION HEATER

T

IW

IS

ΠS

PelIW

IQ

Pdiss

T T s c

I Ah T T

T

ref

S S a

S diss

= ( )= − −( )

=

exp

Π P


PELTIER DEVICE (DOH, CHAPTER 4.7)

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

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

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

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

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

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

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

20

25

30

35

0 2500 5000 7500 10000

Tem

pera

ture

/ C

Time / s

Heat

T21 T1

IQ

ϕ2ϕ

Peltier Pump I I G T T

U T T R I

S Q S

i Q

= − −( )= −( ) +

=

α

ε

α ε

2 1

2 1

Thermalcapacitor 1

Thermalcapacitor 2

Electriccapacitor 1

Electriccapacitor 2

IS

IQ

ΠS

+

UTE

+

T1 T2

IS,TE

2ϕ1ϕ

S 1 Node S 2

Q 1 Q 2

IS 2

IS 3

Pi S

IQ 1 IQ 2

U B

IQ 3

phi 1

R i

T 1 T 2

KK

SC

IS TE

phi 2

U TE

R i

IQ 2

R S

U PE

CC

IS 4IS 1

R ext

PC


OPTIMIZING THERMAL SOLAR COLLECTORS (DOH, CHAPTER 9.4)

Cold airInsulation

AbsorberHot air

IS,air in

IS,air outIS,loss

IS,rad Tair

Tamb

Environment

CollectorTair in

Π S1Π S2

Π S3

Π S4

Π S5

IS,aa

Tabs

Tair

1.840

1.850

1.860

1.870

1.880

250

300

350

400

450

0 4 8 12 16

Ent

ropy

pro

duct

ion

rate

/ W

/K

Ene

rgy

curr

ents

/ W

Length / m

ΠS

0

0

4

3

1 4

3

1

1

3 4 5

1

5

1 2 3 4 5

= − − +

= − + + +

=( )

= −⎛

⎝⎜

⎞

⎠⎟

=

…

= + + + +

I I I

I I

IA

T

T TI

I

T

S rad S aa S loss S

S aa S conv S S S

S radsun

Sabs sun

E abs

SE pump

air

S S S S S S

, . ,

, ,

,

,

,

Π

Π Π Π

Π

Π

Π Π Π Π Π Π

τα G


Part 6

UNIFORM MODELS OF VISCOUS FLUIDS

Here, a spatially uniform model of the thermo-fluidics of a viscous fluid is presented. It follows the model of viscous, heat-conducting fluids in continuum physics (Müller, 1985).

It shows how time appears explicitly in our evolution equations: we get initial value problems for the processes undergone by a fluid.


THERMODYNAMICS OF A SINGLE VISCOUS FLUID (DOH, CHAPTER 10.1)

ASSUMPTIONS

CONSEQUENCES

˙

˙, ,

S I

E I I

S S

E comp E th

= +

= +

Π P V T V P V T aV

S S V T V E E V T V P P V T V

I I

I PV

E

E th S

E comp

, , ˙ , ˙

, , ˙ , , , ˙ , , , ˙ ,

˙,

,

( ) = ( ) +

= ( ) = ( ) = ( ) …

∝

= −

I T I

a

TV

I V Ta

TV

E T S P V

E th S

S

S V V

E

,

˙

˙ ˙ ˙

˙ ˙ ˙

=

= −

= + +

= −

Π

Λ Κ

2

2

Application of these re-sults to particular exam-ples leads to initial value problems (evolu-tion equations for temper-ature and volume).


A Final Word…

ON THE FIGURATIVE CONCEPTUALIZATION IN TRADITIONAL THERMODYNAMICS

Fundamentally, Clausius and Kelvin use the same imaginative figures to conceptualize of thermal processes: quantity of heat, temperature, and power of heat.

The theory of the Force Dynamic Gestalt shows that traditional thermodynamics uses a messed up version of the FDG of thermal processes: quantity and power of heat are mixed up.

Naturally, the concept of quantity of heat as different from its power comes back to haunt us in the form of entropy… (Fuchs, 1986)


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.firenze.it/biblio/esaggi.html

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).

Fuchs H. U. (1986): A surrealistic tale of electricity. Am.J.Phys. 54, 907-909.

Fuchs H. U. (2010a): The Dynamics of Heat. A Unified Approach to Thermodynamics and Heat Transfer. Springer, New York. (First edition: 1996.)

Fuchs H. U. (2010b): Force Dynamic Gestalt, Metaphor, and Scientific Thought. Innovazione nella didattica delle scienze nella scuola primaria: al crocevia fra discipline scientifiche e umanistiche. Università degli studi di Modena e Reggio Emilia, 12-13 Novembre 2010.

Gentner D. (1983): Structure Mapping: A Theoretical Framework for Analogy. Cognitive Science 7, 155-170.

Müller I. (1985): Thermodynamics. Pitman, Boston MA.

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

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