Continuous System Modeling

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ThermoBondLib A New Modelica Library for
Modeling Convective Flows
François E. Cellier ETH Zürich, Switzerland
Jürgen Greifeneder University of Kaiserslautern,
Germany
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Properties of Bond Graphs

• Bond graphs represent the power flowing through a
  physical system.
• Since every physical system must observe the laws
  of energy conservation, all such systems can be
  represented topologically by means of the power
  flows between neighboring energy storages.
• In most physical systems, power can be expressed
  as a product of two adjugate variables, an effort
  (e) and a flow (f).

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Properties of Bond Graphs II
Representation of a bond

• Since a bond references two variables, we need
  two equations to evaluate them.
• In all systems, the effort and flow variables are
  evaluated at opposite ends of the bond.
• The side that evaluates the flow variable is
  often marked with a small vertical bar, the
  causality stroke.

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Effort Flow Generalized Momentum Generalized Position
e f p q
Electrical Circuits Voltage u (V) Current i (A) Magnetic Flux ? (Vsec) Charge q (Asec)
Translational Systems Force F (N) Velocity v (m / sec) Momentum M (Nsec) Position x (m)
Rotational Systems Torque T (Nm) Angular Velocity ? (rad / sec) Torsion T (Nmsec) Angle ? (rad)
Hydraulic Systems Pressure p (N / m2) Volume Flow q (m3 / sec) Pressure Momentum G (Nsec / m2) Volume V (m3)
Chemical Systems Chem. Potential ? (J / mol) Molar Flow ? (mol/sec) - Number of Moles n (mol)
Thermodynamic Systems Temperature T (K) Entropy Flow S (W / K) - Entropy S (J / K )
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Example Bond Graph of Electrical Circuit
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Example Bond Graph of Electrical Circuit II
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Example Bond Graph of Electrical Circuit III
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Example Bond Graph of Electrical Circuit IV
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Convective Flows

• When mass moves macroscopically from one place to
  another, it always carries its volume and its
  heat along. These are inseparably properties of
  the material representing the mass.
• Consequently, a single bond no longer suffices to
  describe convective flows.
• Each convective flow is described by two
  independent variables, e.g. temperature and
  pressure, or temperature and volume, and
  therefore, we require at least two parallel bonds.

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Convective Flows II

• Since the internal energy of material has three
  components
• we chose to represent the convective flow by
  three parallel bonds.

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Convective Flows III
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Thermo-bond Connectors
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Heat Dissipation
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Heat Dissipation II
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Volume Work
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Capacitive Fields
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Capacitive Fields II
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Capacitive Fields III
?
p TRM/V
pV TRM
T T0exp((ss0 - R(ln(v)-ln(v0 )))/cv)
?
T/T0 exp((ss0 - R(ln(v/v0 )))/cv)
?
ln(T/T0 ) (ss0 - Rln(v/v0 ))/cv
?
cvln(T/T0 ) ss0 - Rln(v/v0 )
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Capacitive Fields IV
g T(cp s)
?
h cpT
g h - Ts
for ideal gases
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The Pressure Cooker
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The Pressure Cooker II
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(No Transcript)
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Simulation of Pressure Cooker

• We are now ready to compile and simulate the
  model.

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Simulation Results
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Simulation Results II
Heating is sufficiently slow that the temperature
values of the different media are essentially
indistinguishable. The heat exchangers have a
smaller time constant than the heating. During
the cooling phase, the picture is very different.
When cold water is poured over the pressure
cooker, air and steam in the small boundary layer
cool down almost instantly. Air and steam in the
bulk cool down more slowly, and the liquid water
cools down last.
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Simulation Results III
The pressure values are essentially
indistinguishable throughout the
simulation. During the heating phase, the
pressures rise first due to rising temperature.
After about 150 seconds, the liquid water begins
to boil, after which the pressure rises faster,
because more steam is produced (water vapor
occupies more space at the same temperature than
liquid water). The difference between boundary
layer and bulk pressure values in the cooling
phase is a numerical artifact.
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Simulation Results IV
The relative humidity decreases at first, because
the saturation pressure rises with temperature,
i.e., more humidity can be stored at higher
temperatures. As boiling begins, the humidity
rises sharply, since additional vapor is
produced. In the cooling phase, the humidity
quickly goes into saturation, and stays there,
because the only way to ever get out of
saturation again would be by reheating the water.
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Simulation Results V
The mass fraction defines the percentage of water
vapor contained in the air/steam mixture. Until
the water begins to boil, the mass fraction is
constant. It then rises rapidly until it reaches
a new equilibrium, where evaporation and
condensation balance out. During the cooling
phase, the boundary layer cools down quickly, and
can no longer hold the water vapor contained.
Some falls out as water, whereas other steam gets
pushed into the bulk, temporarily increasing the
mass fraction there even further.
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The Air Balloon

• We got a problem. Whereas the air balloon
  operates under conditions of constant pressure
  (isobaric conditions), the gas bottle operates
  under con-ditions of constant volume (isochoric
  conditions).
• Our air model so far is an isobaric model.

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The Air Balloon II

• We measure the volumetric flow leaving the gas
  bottle and generate a volumetric flow of equal
  size in the modulated flow source. The energy
  for that flow comes out of the thermal domain
  (the gas bottle cools down.

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The Air Balloon III
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The Air Balloon Simulation Results
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The Water Loop
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The Water Loop Simulation Results
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Conclusions

• Modeling convective flows correctly using the
  bond graph approach to modeling, i.e., taking
  into account volumetric flows, mass flows, and
  heat flows, requires a new class of bonds, called
  thermo-bonds.
• A new bond graph library was introduced that
  operates on this new class of vector bonds.
• At the top level, the user may frequently not
  notice any black bonds or black component
  models. The entire model seems to be located at
  the higher, more abstract thermo-bond graph
  layer.
• Yet internally, the red thermo-bond graphs are
  being resolved into the black regular bond
  graphs.

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Conclusions II

• The new approach to dealing with mass flows
  offers a compact and fairly intuitive vehicle for
  describing convective flows in an
  object-oriented, physically correct manner.
• Model wrapping techniques shall be introduced at
  a later time to offer a yet more intuitive user
  interface.
• The capacitive fields describe the properties of
  fluids. As of now, the only fluids that have
  been described are air, water, and water vapor.
• In the future, more capacitive fields shall be
  added to the library, e.g. for the description of
  different classes of industrial oils as well as
  different types of glycols.

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References I

• Greifeneder, J. and F.E. Cellier (2001),
  Modeling convective flows using bond graphs,
  Proc. ICBGM01, Intl. Conference on Bond Graph
  Modeling and Simulation, Phoenix, Arizona, pp.
  276 284.
• Greifeneder, J. and F.E. Cellier (2001),
  Modeling multi-phase systems using bond graphs,
  Proc. ICBGM01, Intl. Conference on Bond Graph
  Modeling and Simulation, Phoenix, Arizona, pp.
  285 291.
• Greifeneder, J. and F.E. Cellier (2001),
  Modeling multi-element systems using bond
  graphs, Proc. ESS01, European Simulation
  Symposium, Marseille, France, pp. 758 766.

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References II

• Greifeneder, J. (2001), Modellierung
  thermodynamischer Phänomene mittels Bondgraphen,
  Diploma Project, Institut für Systemdynamik und
  Regelungstechnik, University of Stuttgart,
  Germany.
• Cellier, F.E. and A. Nebot (2005), The Modelica
  Bond Graph Library, Proc. 4th Intl. Modelica
  Conference, Hamburg, Germany, Vol.1, pp. 57-65.
• Zimmer, D. and F.E. Cellier (2006), The Modelica
  Multi-bond Graph Library, Proc. 5th Intl.
  Modelica Conference, Vienna, Austria, Vol.2, pp.
  559-568.