ON THE IMPORTANCE OF A PROPER USE OF ENERGY …

ON THE IMPORTANCE OF A PROPER USE OF ENERGY FUNCTIONS IN TRANSPORT MODELSPeter BreedveldUniversity of [email protected]

• Context of justification (‘world of proofs’): model manipulation (mathematics, mathematical physics)

versus• Context of discovery/explanation: modeling as a

decision process, abstraction (engineering physics)

Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena

SETTING THE STAGE… VIEWPOINT

2

• Herein focus on reduction by– making well‐based initial modeling decisions– structured approach– variable categorization based on physical properties

– tools/notation that support modeling decisions– INSIGHT in physical behavior

• Mathematical result still open for further reduction!


SETTING THE STAGE… VIEWPOINT

3

• Ideal concepts (‘storage’, ‘transformation’, etc.): mental pictures

• Communication: depicting imaginary concepts into visible images – ideal ‘elements’

• Potential confusion of– ideal elements with tangible components– topological structure with spatial (geometrical)

structure(see Walter Lewin’s lecture about Faraday versus Kirchhoff:part 1: http://www.youtube.com/watch?v=eqjl‐qRy71w&NR=1,

part 2: http://www.youtube.com/watch?v=1bUWcy8HwpM&feature=related)


SETTING THE STAGE… TERMINOLOGY

4

• Physicists: dB/dt = 0 for Kirchhoff’s voltage law to hold (geometrical interpretation)

• Electrical engineers, graph theoreticians: d/dt is a voltage (topological interpretation of the quasistationary situation)

• i.o.w.: topological structure is NOT EQUAL TO geometrical structure (Lewin’s confusion)


EXAMPLE: FARADAY’S LAW

5

C SdE dl B dAdt

0

C SdE dl B dAdt

Sd d dn n BdAdt dt dt

• Ideal elements are conceptual ‘lumps’• Distributed system models treat configuration space

in a continuous waybutstill use conceptually ‘lumped’ concepts(!):– various balance equations (momentum, mass, etc.)– dissipation relations– etc.

• All conceptually separated…


‘LUMPED’ VERSUS ‘DISTRIBUTED’

6

• Basic physical concepts: e.g. quantities for which a conservation principle holds (momentum, charge, etc.)

• Physical interaction implies energy exchange == power (‘through’ conceptual ports)– even information exchange requires low power: back

effect is (made) negligible (e.g. sensor & amplifier)• Eulerian vs. Lagrangian coordinates (‘view point’)• Legendre transforms• Basics of (ir‐)reversible thermodynamics


PREREQUISITES

7

• separation between configuration and energy states: more insight • system boundary definitions may be mixed: Eulerian vs Lagrangian• concept of energy density (material of spatial) synonymous with

first degree homogeneous energy function of all possible extensive states in principle

• energy‐based modeling approach/notation:– automatically satisfies fundamental principles of physics when

all grammar rules are obeyed – systematic approach to the dynamic behavior of all properties

that may be convected in principle (e.g. momentum, electric charge, etc.);

• generalized thermodynamic framework of variables more general than Hamiltonian (generalized mechanical) framework:– some domains have no dual storage


‘PITCH’ OF POINTS TO MAKE

8

THERMODYNAMIC APPROACH

Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 9

Energy balance forthe total system

System boundary



‘Reservoirs’, sources


System boundary




Gibbs’ relation (Energy

‘balance’ of the reversible part)


System boundary







System boundary










System boundary






Irreversible thermodynamics of ‘forces’ and ‘fluxes’ (irreversible part)


System boundary








System boundary




System boundary









System boundary




• Difference is used to find the entropy production, power continuous structure not made explicit, except for instantaneous Carnot engines (1-junction for entropy flow)


System boundary




System boundary

• power continuous structure now made explicit

GJS

NETWORK THERMODYNAMIC APPROACH


• simplified picture, containing the same information…


GJS


Power continuous junction structure


GJS


Power continuous junction structure

Structure is conceptual !


GJS


Generalized Junction Structure (power

continuous)

Boundary conditions (sources)

Energy storage

Irreversible transformation (entropy

production)

BASIC MODEL STRUCTURE (GENERALIZED BOND GRAPH)


Anti-reciprocal part (generalized gyrator)

Weighted Junction Structure (reciprocal)






‘Dualizer’, rotation operator, cross product

Casimir extension of Onsager


Simple Junction Structure (generalized Kirchhoff

laws)

Weighted part (generalized transformer)





CONFIGURATION INFLUENCE ON ENERGETIC STRUCTURE• can ‘become’ an additional energy state: geometric

parameter in storage relation results in a force

– examples: LVDT, electret microphone, relay, (ideal) gas etc. (MP C)– cycles allow transduction– changes of causality correspond to Legendre transforms!! (relation to

dissipation)

• can modulate an energy relation– examples: crank‐slider mechanism, etc. (MTF)

• can switch a contact (or behavior)

,,

E q xF q x

x


ENERGY AS STARTING POINT• Energy

–– Homogeneous function of set of extensive state variables

• In (‘generalized’) mechanics : Hamiltonian–

• In thermodynamics of ‘simple’ systems: internal energy–– more species:

1,..., ,...i nE E q E q q q

1 1, ,..., ,... , ,..., ,...i k i kE H q p H q q q p p p

, ,E U q U V S N

1

1 1

, ,

, , ,..., ,...,

, , ,..., ,..., ,i m

i m

E U q U V S N

U V S N N N

U V S N N N N

• Energy and power: domain independent concepts• An energy function of a set of k conserved states q:• Result in a power:

• where is an equilibrium‐establishing variable or flow

• and is an

equilibrium‐determining variable or effort

• This distinction is lost in a Hamiltonian framework!


ENERGY‐BASED MODEL FORMULATION

30

1,... ,...i kE q q q

1 1

1 1

,... ,... ,... ,...k ki k i k i

i ii ii

dE q q q E q q q dqP e f

dt q dt

11

,... ,...,... ,... i k

i i ki

E q q qe q q q

q

ii

dqf

dt

• Note that a balance equation for a conserved, extensive state

• May now be written as


ENERGY‐BASED MODELING APPROACH

31

with = 1 depending on direction w.r.t. positive orientation

dq fdt

0f


GENERALIZED THERMODYNAMIC FRAMEWORK OF VARIABLES

q f t d

u td

tSS f d

tNN f d

f flow

eeffort generalized state

electric icurrent

uvoltage charge

magnetic uvoltage

icurrent magnetic flux linkage

thermal Ttemperature

fSentropy flow

entropy

chemical chemical potential

fNmolar flow

number of moles

q i t d


GENERALIZED THERMODYNAMIC FRAMEWORK OF VARIABLESf flow

eeffort generalized state

elastic/potential translation

vvelocity

Fforce displacement

kinetic translation

Fforce

vvelocity momentum

elastic/potential rotation

angular velocity

Ttorque angular displacement

kinetic rotation Ttorque

angular velocity

angularmomentum

elastic hydraulic

volume flow

ppressure volume

kinetic hydraulic

ppressure

volume flow

momentum of a flow tube

dt

b T t d

V t d

dp t

p F t d

x v t d

dq f t


MECHANICAL FRAMEWORK OF VARIABLES


SYNTHESIS OF MECHANICAL & THERMODYNAMIC FRAMEWORK

Split domains (therm.) and couple by SGY (mech.):

C-SGY-C

Only relaxation behavior: C-R

Oscillatory behavior (damped): C-I(-R)

One type of storageTwo types of storage

Thermodynamics:Mechanics:

GENERALIZED THERMODYNAMIC FRAMEWORK


UNRESOLVED ISSUE?

• What is analog to what?– Mass and coil or mass and capacitor?

or– Spring and capacitor or spring and coil?

• Long debates since mid thirties…


PHYSICAL MEANING OF THE SYMPLECTIC GYRATOR

• Electrical network (q,):–Only if quasi‐stationary (non‐radiating), Maxwell’s equations reduce to:

–Dualizing effect!

SGY0 0C C

magEe

dd

magft

dd

ut

mage i

dd

elecqft

elecf i

elecEeqelece u

dd

ut

mage i


PHYSICAL MEANING OF THE SYMPLECTIC GYRATOR

• Mechanical (x,p)–Only when in inertial frames, i.e. when Newton's 2nd law holds:

SGY0 0C C

potEex

dd

potxft

kinEep

dd

kinpft

dd

pFt

potf v

pote Fkine v

dd

pFt


‘SPECIAL’ STATESPosition/displacement• has a dual nature:

– energy state (related to a conservation or symmetry principle like all other states)– configuration state

Matter• convects all matter‐bound properties (not ‘available volume’!)• conjugate intensity depends on other intensities (Gibbs‐Duhem)• boundary criterionVolume• boundary criterionEntropy• can be ‘locally’ produced & is only ‘locally’ conserved (conceptual separation!)


SYSTEM (BOUNDARY) DEFINITION• Open (thermodynamic) systems• dN = 0 (dm=MdN=0) : ‘Lagrangian coordinates’, material boundary

criterion:N not a state

• dV = 0 (Adx=0): ‘Eulerian coordinates’ (control volume, spatial boundary criterion):

V not a state• Almost always mixed boundaries:

N and V remain states for system under study!• Tangible system: globally Lagrangian, locally Eulerian• Network of subsystems: globally Eulerian, locally Lagrangian (e.g.

network of elastic tubes)


EXAMPLE: FLUID SUPPLY SYSTEMCOMPONENT LEVEL

41

u iu

i

T

p

p

p

valve position

level

PWM

environmental influence

microprocessor/digital controller

"plant"

desired level

p

p

AD

containerAD

Demand

electric_motor

environment

fluid_lineH_bridgePID

power_supply

pump

Sensor

Setpoint valve


EXAMPLE: FLUID SUPPLY SYSTEMCONCEPTUAL ELEMENTS

42

level

u iu

i

T

p

p

valve position

PWM



"plant"

desired level

p

p

AD

1

g

C

AD

Demand

environment

GYH_bridge

I

1PID

power_supply

pump

R

Sensor

Setpoint valve0

level

u iu

i

T

p

p

valve position

PWM



"plant"

desired level

p

p

AD

1

g

C

AD

Demand

environment

GYH_bridge

I

I I

111PID

power_supply

pump

R

R R

Sensor

Setpoint valve0


EXAMPLE: FLUID SUPPLY SYSTEM

43


EXAMPLE: FLUID SUPPLY SYSTEM

44

level

p

u iu

i

T

p

p

valve position

PWM



"plant"

desired level

p

p

AD

AD

Demand

environment

H_bridgePID

power_supply

pump

Sensor

Setpoint valve

1

g

C

0

C CI I I

1 1 1

R R R

0 0GY

I I

11

R R

• Most dominant: power transduction• But also possible:

– Multiport energy storage (magnetic, kinetic, thermal), necessarily reversible (cycle process is always required!), linear decomposition results in a transducer and 1‐ports

– Separate (conceptual!): irreversible transduction, electrical resistance and mechanical friction

– etc.Berlin, May 19th, 2015Workshop on model order reduction of

transport-dominanted phenomena

EXAMPLE: ELECTRIC MOTOR

45

• Dominant: fluid resistance• 1) If relatively long and narrow: fluid inertia

– Conceptual structure: common flow, summation of pressure drops, not separated in space!

• 2) Compressibility:– Several segments (R,I,C)– Normal modes

• 3) Compressible fluid, convecting momentum and entropy (also charge, flux,…): conceptual structure

• 4) Elastic tube wall: mixed boundaries


EXAMPLE: FLUID LINE

46

• System boundary: globally Lagrangian (commonly at rest), locally Eulerian (commonly closed)

• Tangible subsystem boundaries (components): globally Eulerian (network w.r.t. system boundary), locally Lagrangian (components have fixed neighbors, exceptions require bookkeeping, while major system structure is maintained)

• Conceptual (abstract) subsystem boundaries: structure is based on distinction between basic dynamic behaviors and not based on material or spatial criteria


TRANSPORT IN ENGINEERING SYSTEMS

47


‘ENERGY DENSITY’ IMPLIES A FIRST DEGREE HOMOGENEOUS ENERGY

48

• Conserved energy has to be a homogeneous function of conserved states:

• Convection requires that addition of subsystems means interaction‐free addition of extensive states and addition of (extensive) energy:

• n=1 in principle (if all states are considered!)

1

1

1 1, , : ,1, ,1, ,

1 1, , : , ,1 , ,1 ,

n n

m

n n

V

V VE m V E E vm m m m m m m

mE m V E EV V V V V N

q q qq

q q qq

• Generalized Gibbs’ relation:

• First degree homogeneous energy implies zero degree constitutive relations (intensity!):

k‐1 independent intensities

• One‐port storage cannot exist in principle, unless constant states are considered parameters


FIRST DEGREE HOMOGENEOUS ENERGY

49

11 n

i i ii

EE q e qn q

q

0i i

i i i

E E Ee e

q q q

q q q

q q

1 1d d 0

k k

i i ii ii

Eq q eq


GENERALIZING GIBBS AND GIBBS‐DUHEM

50

1

1

1

1

(Generalized Gibbs)

0 d d d d d d (Generalized Gibbs-Duhem)

mtoti i

i ii

mtoti i

i ii

N qpu Ts pv v eN N N

N qps T v p v eN N N

1

1

1

1

(Gibbs)

0 (Gibbs-Duhem)

mtoti

ii

mtoti

ii

Nu Ts pv

N

NsdT vdp d d

N

Specific quantities are ‘weighting factors’


CONJUGATE FLOW RELATIONS

51

d dd d d d; ; d d d d d dd d d; 0!d d d

i i i i

convectedconvected convected

convected convected

q q N NN p p N Nt N t t N t t N tS S N Vt N t t

dd d d d; ; 0!d d d d d

i i

convected convectedconvected

N NS S N N Vt N t t N t t

Same specific quantities are ‘weighting factors’


PART OF A TUBE NETWORK

52

daxial V = 0(V not a convected property)daxial N 0

dradialN = 0dradial V 0(flexible tube, local changes)


globally Eulerian boundary (dV =0)

locally Lagrangian boundary (dN =0; e.g. elastic tube wall)

Total energy ‘flow’: dN dV dEe pv pdt dt dt

GENERIC PICTURE: MASS/MATERIAL FLOW

but the specific enthalpy h cannot serve as an equilibrium-determining variable


GENERIC PICTURE: SPLIT OUT FLOWS


GENERIC PICTURE: NO ‘SLIP’


GENERIC PICTURE: WITH ‘SLIP’ OF CONVECTED PROPERTIES


GENERIC PICTURE: MOMENTUM CONVECTION


SIMPLIFIED: MOMENTUM CONVECTION, RIGID TUBE WALL

51015 momentum

4

8 mass

11.5

2 p_momentum.e

-1

1

3p_mass.e

0

4 p_mass.f

0 2 4 6 8 10time {s}

-20

0

20 p_momentum.f

NON‐MINIMUM PHASE RESPONSE


Just qualitative,numbers haveno meaning!

NON‐MINIMUM PHASE RESPONSE


Linear System Pole-Zero Plot

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Re

-2

-1

0

1

2

0.10.20.30.40.50.6

0.7

0.8

0.9

• separation between configuration and energy states: more insight • system boundary definitions may be mixed: Eulerian vs Lagrangian• concept of energy density (material of spatial) synonymous with

first degree homogeneous energy function of all possible extensive states in principle

• energy‐based modeling approach:– automatically satisfies fundamental principles of physics when

all grammar rules are obeyed – systematic approach to the dynamic behavior of all properties

that may be convected in principle (e.g. momentum, electric charge, etc.);

• generalized thermodynamic framework of variables more general than Hamiltonian (generalized mechanical) framework:– some domains have no dual storage


CONCLUSIONS

61

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