Modeling MultiElement Systems Using Bond Graphs

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Modeling Multi-Element SystemsUsing Bond Graphs
18.10.2001
Modeling Multi-Element Systems Using Bond Graphs
Jürgen Greifeneder François E. Cellier
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Contents
Jürgen Greifeneder Review on paper the main
aspects of paper 1 and 2 Pressure Cooker already
diskussed in 2, however, only in a really short
way, as the model is based on the multi-element
system theory also.

• Introduction
• Review
• Basics of Multi-Element Systems
• Mixture Properties
• Transport Phenomena
• Model of a Pressure Cooker
• Conclusions

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Introduction

• Describing a thermodynamical problem
  necessitates 3 variables.
• Separation in storage and dissipative elements.
• Storage elements calculate the potentials and
  therefore need to know about the matter, they are
  representing. Dissipative elements calculate
  flows and do not care, which matter they are
  dealing with (network theory).
• Elements do not know about each other.
• No quasi-stationary or flow-equilibrium
  assumptions were made.
• Contrary to earlier efforts in this field, this
  work delt with real, rather than pseudo bond
  graphs.

4
The C-field (storage element)
Jürgen Greifeneder 3 storage elements, but none
of them can calculate its potential on ist
own Bus- vs. Vektorbond

0
.
T
S
C
CF
C
C
g
q
.
p
M
0
0
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Basic dissipative Elements
Jürgen Greifeneder Unterscheidung zwischen
RF-Element und RF-Konzept !!! Hinweis, daß es
sich um Dichte und spezifische Entropie handelt

Conduction
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Traditional Thermodynamics

• Jürgen Greifeneder
• Advantages of n2-CF-Element
• no constraint equations
• topological model of a complex system would be
  simpler and more easy to understand
• Disadvantages
• The previously introduced structures would have
  to be extended
• ?Internal equations of the C-field would change
  in accordance with the composition of the mixture
• unnecessary complexity, especially in the case
  of simple systems
• Processes would be hidden, that the authors
  would like to make visible
• However, as this is the classical thermodynamical
  approach, one would certainly have done so also

One Temperature, one pressure and n partial
mass gt n2 equations.
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Multi-Element Mono-Phase Systems
Jürgen Greifeneder Each matter has its own
CF-Element. Each CF-Element is assumed to be a
direct neighbor of each other element The contact
surfaces between the different matters are
assumed to be infinitely large gt temperature and
pressure may equilibrate infinitely
fast. However, the corresponding transfer rates
cannot be chosen infinitely large and therefore,
the temperature and the pressure of the different
components can assume somewhat different values
(in the simulation). Although, if let alone, the
will equilibrate, eventually.

DVA CD
2
2
3
3
Ø
Ø
2
2
CF
3
DVA CD
DVA CD
3
2
2
Ø
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Ideal and Non-Ideal Mixtures
Jürgen Greifeneder Ideally mixed molecules are
distributed at random (prediction, which molecul
becomes a neighbor of which other molecules is
not possible

• In the process of mixing, additionally entropy
  will be created, which must be distributed among
  the participating components
• Distribution is a function of the molar
  fractions
• CF-Elements are not supposed to know about each
  other
• Þ only necessary information will be provided

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Ideal and Non-Ideal Mixtures
Jürgen Greifeneder Non-Ideal Mixtures Volume
will change also specific excess volume and
entropy of a non-ideal mixture are tabulated in
the literature

• In the process of mixing, additionally entropy
  will be created, which must be distributed among
  the participating components
• Distribution is a function of the molar
  fractions
• CF-Elements are not supposed to know about each
  other
• Þ only necessary information will be provided

CF
M1, V1, S1
M2, V2, S2
MI
x1, s1Ex, v1Ex
x2, s2Ex, v2Ex
1
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Entropy of Mixing
Jürgen Greifeneder Ideal Mixture Temperature
and pressure do not change. Free enthalpy does
change gt difference creates an entropy flow

1
T
T
.
.
S
S
1
1
p
p
1
CF
CF
q
q
12
11
1
1
g1 (T,p)
g1(T,p)
mix
1
.
.
M11
x11
M
M
1
1
Dg1
T
RS
.
CD DVA
.
MI
M
mix
DSid
1
1
1
T
x21
M21
T
.
.
S
S
2
2
p
p
1
CF
CF
q
q
22
21
2
2
g2 (T,p)
g2(T,p)
1
mix
.
.
M
M
2
2
Dg2
T
RS
.
.
M
mix
DSid
2
2
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Jürgen Greifeneder Non-ideal Mixtures or general
case (cold milk poured into hot
coffee) Differences also in the values of
temperature and pressure

An François eigentlich müßte ich hier erwähnen,
daß die drei (m)RS-Elemente einem RF-Element
entsprechen (vor allem, weil dies auf der
nächsten Folie verwendet wird). Allerdings habe
ich die interne RF-Struktur nie verwendet und
sehe dies auch nicht als erforderlich an. Daher
Was tun? Die nächste Folie rauslassen?
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Jürgen Greifeneder Pressure and Temperature may
adjust to their corresponding outside values gt
volume increases gt additionally Entropy will
be created. More volume and a higher Entropy
leads to a higher temperature (Mischungswärme)

CF
CD DVA
11
3
3
3
Ø
outside
3
3
Ø
3
Ø
RF
3
3
3
CD DVA
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Convection in Multi-Element Systems
Jürgen Greifeneder Vertical exchange as
discussed before Horizontal exchange coupled
RF-Elements. Only one of them is independent. The
flows of the others are fixed by the composition
of the emitting mixture.

CF
CF
11
21
RF DVA CD
3
3
3
3
3
DVA CD
3
Ø
Ø
DVA CD
3
3
3
3
RF DVA CD
3
3
3
vertical Exchange (mixture)
CD DVA
3
Ø
CD DVA
CF
Ø
CF
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horizontal Exchange (transport)
3
3
3
3
3
3
DVA CD
Ø
Ø
DVA CD
RF DVA CD
3
3
3
3
CF
CF
13
23
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Two-Element, Two-Phase, Two-Compartment
Convective System
Jürgen Greifeneder Top of figure Gas phase
bottom of figure fluid phase On the left hand
side one compartment on the right hand side the
other one Gas phase ideal gases gt no
MI-Elements necessaire. However, total volume is
needed to calculate the partial pressures, which
is needed for the condensation element Fluid
phase needs MI-Elements Evaporation and
Condensation are two independent processes!!

An François Wie genau muß ich auf diese
Abbildung eingehen?
Gas
Gas
CF
CF
CD DVA RF
21
22
3
3
3
3
3
DVA CD
3
Ø
Ø
DVA CD
3
RF DVA CD
3
3
3
3


3
3
3
Vges
Gas
3
Vges
Ø
Gas
3
CF
Ø
CF
DVA CD
DVA CD
11
12
3
3
CD Condensation/ Evaporation DVA
CD Condensation/ Evaporation DVA
CD Condensation/ Evaporation DVA
CD Condensation/ Evaporation DVA
phase- boundary
3
3
CD DVA RF
3
3
3
3
3
3
Fl.
3
Ø
3
CF
DVA CD
DVA CD
Fl.
Ø
3
3
11
CF
12
3
3
M11, T11, p 11
x21, DSE21, DVE21
M12, T12, p 12
3
3
Ø
DVA CD
x12, DSE12, DVE12
Ø
DVA CD
RF DVA CD
3
3
3
3
x21, DSE21, DVE21
x22, DSE22, DVE22
Fl.
Fl.
MI
CF
CF
MI
21
22
1
M21, T21, p 21
2
M22, T22, p 22
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Equilibration of Concentrations
Jürgen Greifeneder The concentrations in two
neighboring compartments may become different, as
each compartment can be connected to any other
compartment or an outside source. Internal
structure is RF-Element However, RF-Elements were
only provided with the state information of the
emitting CF-Element

CFi
CFi1
3
3
CD DVA KA
...
...
Ø
3
3
3
3
Ø
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Model of a Pressure Cooker
Jürgen Greifeneder Air is needed, to provide the
pressure cooker at room temperature with the
pressure of the environment. Having the same
volume, without the air, some water would have to
evaporate even at room temperature in order to
produce equilibrium pressure, which would be
considerably lower than 1 bar. Explain the used
components (animation)

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Model of a Pressure Cooker

Air in boundary layer
CD
air
RF Dp
CD
CD
DVA
water
CD
steam
RF Dp
KV
CD
Steam in boundary layer
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Temperature Graphs
Jürgen Greifeneder Show heating phase (almost
identical temperatures) Cooking phase Cooling
phase boundary layer cools down most rapidly,
the bulk follows somewhat slowly, and the water
cools down last.

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Pressure Graphs
Jürgen Greifeneder Pressure in bulk is
indistinguishable from that of the fluid Heating
phase no differences. Knee in the curve (roughly
at time 150s) represents the point where the
water begins to boil (380K 130 kPa). Cooling
phase pressure in boundary layer drops
temporarily below that of the bulk, because water
condensates more rapidly in the boundary layer
and because the two RF-elements cannot resupply
the boundary layer with air/steam frum the bulk
arbitrarily fast.

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Humidity Graphs
Jürgen Greifeneder Humidity partial pressure
of the steam / saturation pressure of
water Decrease of humidity during first heating
phase, as the saturation pressure located in
the denominator of the humidity has the same
gradient as the rising temperature. Small
differences can be seen, as the boundary layer
heats up a little faster than the bulk At time
150s the humidity starts climbing again, because
- just like the knee in the pressure trajectories
the water starts to boil and therefore steam is
being created by evaporation. Equilibrium state
reached at 32 Begin of cooling phase, the
temperature in the boundary layer drops down
rapidly, the corresponding humidity quickly
reaches 100 and dew starts to form on the cold
surface of the pressure cooker. Now, the two
gaseous phases are no longer identical in their
composition and therefore, diffusion occurs. The
(slower) cooling down of the bulk together with
the diffusion between the bulk and the boundary
layer pull the humidity of the bulk up, until it
reaches 100 and steam starts to condensate
directly via the phase boundary. The humidity
inside the pressure cooker will remain at 100
until the end of the simulation, as the only way
to lower the humidity would be, to raise the
temperature again or open the pressure cooker.

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Mass Fraction Graphs
Jürgen Greifeneder Mass fraction mass of steam
/ (mass of steam and air) Beginning No change.
After that, evaporation increases the mass
fraction until the equilibrium point of
approximately 23.6 When cooling starts, the
boundary layer cools down more rapidly than the
bulk. Also the pressure of the boundary layer
drops down more rapidly than that of the bulk.
However, the pressure euqilibrates much more
rapidly than the temperature. Thus, the pressure
in the bulk (and in the water!) decreases more
rapidly than the corresponding temperature, the
boiling point of the water decreases, and
consequently, additional water boils off. As a
consequence, the mass fraction of steamin the
bulk rises temporarily. However, the mass
fractions starts dropping again due to pressure
equilibration and diffusion. At time 1315 sec,
steam starts to condensate from the bulk, and
consequently, the mass fraction drops
sharply. The final equilibration of the two mass
fractions occurs primarily by means of diffusion.

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Conclusion

• The elements introduced suffice to model most
  thermodynamical problems
• Modeling each matter separately as a storage
  element and connecting them by means of
  dissipative elements (RF-concept) simplyfies the
  modeling task, offers insight into physical
  functioning of multi-element systems and leads to
  mathematical models that can be simulated in a
  numerically stable and highly accurate fashion.
• Models are still limited to systems without
  chemical reactions

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Danke
Thank you! Remerciement!
Ende
Jürgen Greifeneder François E. Cellier