on the importance of a proper use of energy …
TRANSCRIPT
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!
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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)
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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)
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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…
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
‘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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
‘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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 10
Energy balance forthe total system
‘Reservoirs’, sources
THERMODYNAMIC APPROACH
System boundary
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 11
Energy balance forthe total system
‘Reservoirs’, sources
Gibbs’ relation (Energy
‘balance’ of the reversible part)
THERMODYNAMIC APPROACH
System boundary
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 12
Energy balance forthe total system
‘Reservoirs’, sources
Gibbs’ relation (Energy
‘balance’ of the reversible part)
THERMODYNAMIC APPROACH
System boundary
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 13
Energy balance forthe total system
THERMODYNAMIC APPROACH
THERMODYNAMIC APPROACH
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 14
Energy balance forthe total system
‘Reservoirs’, sources
Gibbs’ relation (Energy
‘balance’ of the reversible part)
System boundary
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 15
Energy balance forthe total system
‘Reservoirs’, sources
Gibbs’ relation (Energy
‘balance’ of the reversible part)
Irreversible thermodynamics of ‘forces’ and ‘fluxes’ (irreversible part)
THERMODYNAMIC APPROACH
System boundary
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 16
Energy balance forthe total system
‘Reservoirs’, sources
Gibbs’ relation (Energy
‘balance’ of the reversible part)
Irreversible thermodynamics of ‘forces’ and ‘fluxes’ (irreversible part)
THERMODYNAMIC APPROACH
System boundary
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 17
Energy balance forthe total system
‘Reservoirs’, sources
System boundary
Gibbs’ relation (Energy
‘balance’ of the reversible part)
Irreversible thermodynamics of ‘forces’ and ‘fluxes’ (irreversible part)
THERMODYNAMIC APPROACH
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 18
Energy balance forthe total system
‘Reservoirs’, sources
THERMODYNAMIC APPROACH
System boundary
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 19
Energy balance forthe total system
‘Reservoirs’, sources
• Difference is used to find the entropy production, power continuous structure not made explicit, except for instantaneous Carnot engines (1-junction for entropy flow)
THERMODYNAMIC APPROACH
System boundary
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 20
Energy balance forthe total system
‘Reservoirs’, sources
System boundary
• power continuous structure now made explicit
GJS
NETWORK THERMODYNAMIC APPROACH
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 21
• simplified picture, containing the same information…
NETWORK THERMODYNAMIC APPROACH
GJS
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 22
Power continuous junction structure
NETWORK THERMODYNAMIC APPROACH
GJS
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 23
Power continuous junction structure
Structure is conceptual !
NETWORK THERMODYNAMIC APPROACH
GJS
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 24
Generalized Junction Structure (power
continuous)
Boundary conditions (sources)
Energy storage
Irreversible transformation (entropy
production)
BASIC MODEL STRUCTURE (GENERALIZED BOND GRAPH)
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 25
Anti-reciprocal part (generalized gyrator)
Weighted Junction Structure (reciprocal)
BASIC MODEL STRUCTURE (GENERALIZED BOND GRAPH)
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 26
Anti-reciprocal part (generalized gyrator)
Weighted Junction Structure (reciprocal)
BASIC MODEL STRUCTURE (GENERALIZED BOND GRAPH)
‘Dualizer’, rotation operator, cross product
Casimir extension of Onsager
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 27
Simple Junction Structure (generalized Kirchhoff
laws)
Weighted part (generalized transformer)
Anti-reciprocal part (generalized gyrator)
Weighted Junction Structure (reciprocal)
BASIC MODEL STRUCTURE (GENERALIZED BOND GRAPH)
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 28
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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 29
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!
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
ENERGY‐BASED MODELING APPROACH
31
with = 1 depending on direction w.r.t. positive orientation
dq fdt
0f
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 32
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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 33
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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 34
MECHANICAL FRAMEWORK OF VARIABLES
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 35
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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 36
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…
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 37
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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 38
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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 39
‘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!)
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 40
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)
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
EXAMPLE: FLUID SUPPLY SYSTEMCONCEPTUAL ELEMENTS
42
level
u iu
i
T
p
p
valve position
PWM
environmental influence
microprocessor/digital controller
"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
environmental influence
microprocessor/digital controller
"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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
EXAMPLE: FLUID SUPPLY SYSTEM
43
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
EXAMPLE: FLUID SUPPLY SYSTEM
44
level
p
u iu
i
T
p
p
valve position
PWM
environmental influence
microprocessor/digital controller
"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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
TRANSPORT IN ENGINEERING SYSTEMS
47
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
‘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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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’
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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’
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
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)
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 53
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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 54
GENERIC PICTURE: SPLIT OUT FLOWS
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 55
GENERIC PICTURE: NO ‘SLIP’
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 56
GENERIC PICTURE: WITH ‘SLIP’ OF CONVECTED PROPERTIES
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 57
GENERIC PICTURE: MOMENTUM CONVECTION
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 58
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
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 59
Just qualitative,numbers haveno meaning!
NON‐MINIMUM PHASE RESPONSE
Berlin, May 19th, 2015 Workshop on model order reduction of transport-dominanted phenomena 60
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
Berlin, May 19th, 2015Workshop on model order reduction of transport-dominanted phenomena
CONCLUSIONS
61