CROSS ANALYSIS OF DYNAMIC MODELLING
OF AN ECONOMICAL ENERGY SYSTEM
AND A DISSIPATIVE PHYSICAL SYSTEM
J.C. MORA”, RS, BES@)
(I) Asian Institute of Technology
GPO Box 2754, Bangkok 10501, THAILAND
(2) ENSIGCICNRS, Toulouse, FRANCE
ABSTRACT
Actual Dissipative Systems are not workable without energy supply and energy dissipation. Among these
systems are economical systems as well physical systems such as fluidized bed. A System Dynamics
Analysis of a simple macro-economical system related to the energy sector of a country shows causal
loops including energy supplies, This analysis can be applied to a fluidized bed. This system is a
classical engineering dynamic system. Due to its complexity it has been highly investigated on the basis
of time averaged regimes but a few studies exist about the true dynamical aspect. This is mainly due to
the fact that the basic equations are complex. Several length and time scales exist simultaneously, The
theoretical equations should comprise the classical hydrodynamic equations for fluid containing solid
particles in suspension. The complete set of theoretical equations is for the time being not available.
From analogies with the economical systems it is possible to identify causal loops which are not usually
considered in the classical modelling of a fluidized bed, i.e interaction with energy supply. Introducing
this phenomena and using a very simple equation it is possible to show that space structuration can occur.
Then by using a simplified non-linear hydrodynamic equation a chaotic complex behavior, in agreement
with experiments, can be simulated. The analysis of the model shows that one of the origin of the
difficulties to use the models lies in the existence of several time and space scales. This kind of analysis
is helpful to understand fluctuant behavior of many other systems including economical systems.
THE PROBLEM
#
Many mathematical tools used for economical sciences or human sciences come from exact sciences such
as thermodynamics,
Little has been done, up to now, in order to apply tools specifically developed for economical systems,
ie. “index”, "indicators", "causal loop" to classical physical systems.
This paper shows how a "System Dynamics" analysis applied to a macroeconomical system can help to
improve the modelling of pure engineering system such as a fluidized bed.
ECONOMICAL MODEL
The simplified economical model is based on macroanalysis of the energy sector of a country.
The energy supplies include: importation, and domestic resources,
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Energy is consumed for:
~ domestic transportation,
- intemational exchanges,
- organization of the energy sector,
- intemal development,
~ well being and goods production.
Uncertainty on energy supply needs to partly control energy importation. This can be done through:
- investment in oil producer countries,
- control of oil or gas transportation.
These two conditions are directly dependent on the development of the country as well as its growth rate.
Internal energy consumption is directly linked to the level of development. This energy consumption can
be considered by this fact as an “indicator"or "index" of the economical activity of the country for the
purpose of the present analysis. ¢
‘Two countries defined as above will have exchanges. These exchanges can be purely economically or
energy related, i.e. energy or equipment related to energy.
Exchange capabilities of one country are directly dependant on its internals development. A poor country
cannot “invest” in many other countries while a rich country can do that. These exchanges can be related
to the “energy indicator” as defined above.
Regarding the intemational transportation a developed country is able to afford to get airline companies,
this sector consumes energy but at the same time gives a larger geographical influence, i.e. space
structuration.
Finally a causal type diagram can be summarized by the figure 1.
The energy consumption is in fact dissipated into heat: engines, heating, so that the energy consumption
is fully comparable to the entropy source of classical physical system.
‘Without energy these systems are not workable: energy is compulsory but of course not sufficient.
Regarding the figure 1, world energy resources are limited so that there is a competition in order to
control external energy supplies. Two oil importer countries will have limited possibilities of investment a
and joint ventures in an oil exporter sountry. A too high energy demand from one country could be taken S
out of the possibility for the other.
FLUIDIZED BED
A fluidized bed is composed of solid particles (sand, catalyst, etc.) suspended into an up flow of fluid
(gas or liquid), The drag forces counterbalance the gravity force (Figure 3).
The controlling parameter is the fluid flow. At low flow rate the particle, bed is fixed, then it becomes
fluidized ie. particles are suspended in the flow. Bubbles and channels appear at higher flow rate.
Finally the particles are transported at very high flow rate.
Fluidization is characterized by the pressure drop. This is the main characteristic of fluidization. The
other characteristic is the void fraction, this increases up to one when transportation occurs,
The two parameters fluctuate around average values. The void fraction can be related to the various
regimes: fixed bed, bubbling, channelling.
At least four length scales are to be simultaneously considered:
- the molecular scale for the fluid flow,
- the grains scale,
~ the bubbles and channels scales,
~ the equipment scale.
Writting classical fluid mechanics equations is difficult, even with simplified basic equation. Another
difficulty lies on the definition of the border conditions, i.e. fluid distribution, bed height, particles shapes,
etc.
For the time being modelling cannot be disconnected from a pure phenomenological analysis.
Most of the experimental results and equations use time and space averaged values but very few can be
found for time and space fluctuations,
CAUSAL DIAGRAM
The fluidized bed is considered as series of cells. Each of these cells are characterized by dissipated
energy. The regimes: fixed bed, fluidized, bubbling etc ... depends on this energy. One cell interacts
with two other cells according to viscosity like interactions. It has been already shown how dissipated
energy per unit of time and volume can be consider as an “activity index". Bes (1982),
Figure 2 shows the causal diagramme. This diagramme comprises influences on gas feed, which is
normally not considered.
Comparison with the macroeconomical system is as follows:
- In both cases energy dissipation is compulsory, without energy the two systems are not
workable.
- Energy dissipated into heat, through particles shocks and viscosity, should be compared to
wasted energy.
- The two systems develop an internal organization which is directly linked to energy
consumption,
- Energy dissipation is connected to the development of length scales. Domestic and international
transportation for the macroeconomical system, bubbles and channels for the fluidized bed.
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- A self organization occurs in both systems. The use of a fraction of energy for self
organizations permits to decrease energy intensity compared to a non-organized system.
Fluidized bed compared to a fixed bed dissipated less energy for the same fluid flow rate. This behavior
can be related to a Curie-Prigogine Thermodynamic principle.
Following the analogy with the macroeconomical system, the fluidized bed should not be considered
independently from the fluid supply. Energy supply is not hexogeneous.
BASIC MODELLING
The basic equation for fluidizations are a steady state equation, i.e. force balance.
a) Equipment scale
Pressure force - gravity force = 0 Y qd)
b) Particle scale
Drag force gravity force = 0 @
which define the minimum fluidization velocity.
Most of the phenomenological equations are derived from steady state balance.
However as fluctuations obviously occur this equation should be the consequence of a non-steady state
equation,
The momentum balance is a way to find one of these equations,
The simplest equation is:
dv/dt = kV,- Kv - mg @)
For minimum of fluidization a true steady state exists: i.e. the particles are at rest so that v = 0 and dv/dt
=0.
Finally the equation becomes
dv/dt = kV - Ky @
where V is the excess gas velocity over the minimum of fluidization,
This equation is however unable to represent any fluctuations or space structuration.
The fluidized bed is considered as series of cells with feedback on fluid feed. When the fluid flow
increases in a given cell, it decreases in the other consecutive cells. Two consecutive cells could be with
the following conditions,
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Energy Consumption
Self Organization in C=
I
rf
Energy Saving!———} Country ——
yA Country B
Domestic Production Transport
_t i
Oil Gas producers
Energy Consumption
Self Organization Cc ces
[. Bubbles
= cana |
Cell B
Minimum Energy Transport
Consumption . j
| Energy Supply (fluid) |
Figure 1: Economical system
Figure 2: Fluidized bed
VatVar = Vo
dv jdt = kV,-Ky,
Avg 4/dt = VV.) -K Vaet
These equations are compatible with the two following steady states.
Vv, =V, Vatt = Vo
State 1 State 2
Ver = 0 : Vest = 0 8)
These conditions could represent series of alternate channeling zones (V,, = V,) and fixed bed zones (V,,
=). This is the case, for fluidization of sawdust.
‘This kind of representation is of course very simplified but however better than the initial equation as at
least one space organization can be represented.
V,, OF Vn41 represents the velocity of the particles or clusters of particles, Along time the averaged value
should be zero with fluctuations around zero.
The dissipative effect can be seen as resulting from acceleration phases, by the gas, followed by
dissipation of kinetic energy by shocks. V, is linked to the energy supplied to the fluid, the following
general equation has been proposed.
daydt = Ka, - ka, (2)
where a,, a, are the square roots of the dissipated energy per unit of time and volume of fluidized phase.
In agreement with a generalized non-steady state reaction - diffusion equation. a, has been proposed to
be a “dissipative indicator" or a "dissipative index".
The two following equations have the same steady state,
da dt = (k’a, - k a,) (10)
da,/at = fla, )(k’a,- k a,) an
but the second one is non-linear and as such could have much more complex behavior especially when
this one is used under its time discretized form.
This equation can be put under a logistic form.
Xx(t+1) = g(x)(1-x) (12)
The simplest case g(x) = 4 ux is the most commonly used to illustrate a chaotic numerical process. Hao
Bai Lin (1990)
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Working only with steady state or time and length averaged values makes the complex behavior hidden
from equation point of view in spite of an obviously fluctuant system.
These general considerations open ways to consider discretized equations, or cellular automata, Gabsi
(1990), as actual models instead of considering only differential equations to be true models.
MODELLING
A more complex model can be considered using analogy with hydrodynamic Navier_Stokes equation
using viscosity. The equation is of the following type:
av av
> HAP HUES (13)
The adaptation to a fluidized bed will be, CHEHBOUNI (1985)
av ay 4
2 -coftryond| a4)
In order to consider that the particles are not moving out of the bed, on time averaged basis, only time
fluctuating solutions should be considered,
Finally the equation using the dissipative indicator is:
aa. de as)
= sofa pi
A simple discretized equation is:
race} = K afA,-al + Da@,yy+ 1° 2a) (16)
The simple cases, D = 0
OF &41 = 84 = A and K = 0 will give a logistic like function.
Coming back to the basic equations in steady state conditions. k and j1 depends on the void fraction, this
latter depending on the fluid flow rate which depends itself on the energy supplied to the pump or
compressor.
MODEL ANALYSIS
A basic equation can be written under the following way.
x(t+1) = x[K A,(1-x) + Dt 41° 2X] an
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Iterative numerical processes permit to calculate "x" profiles versus the rank of cells.
Due to the non linearity complex behavior occurs with the following typical cases:
- At low value of A, there is no fluctuations when A increases fluctuations appear.
- The system is highly dependant on initial conditions, difference of 10°, destabilized completely
the system which can pass from a time converging process to a chaotic process.
“A,” tepresent the energy supplied to the system, and has been taken so that
ZA, = NA,
Figure (4) shows example values of "x” versus the length position at different time. This emphasizes the
structurations and the fact that for some external constraints, A”, two consecutive cells are rigidly linked.
When "A" increases the general trend of one cell, is kept, but with fluctuations, compared to the neighbor
cells.
CONCLUSION #
An economical system and a physical system could have similar causal diagramme. From this analogy,
it is possible to improve, at least qualitatively the model of the physical system in order to better fit with
actual behavior.
The use of “index” of "indicator" permits to get an overview about the trend of the systems. The use
of the simplified physical model permits to emphasizes, space and time chaotic structuration which is in
good qualitative agreement with actual behavior. .
As a consequence the fluidized bed model could lead to some help to understand complex economical
behavior which are not necessarily originated in complex equations,
a [es > - 462 -
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2.992 pe
Seta gn ntereNatg
Pada |
o.seones |
Figure 4 : Four examples of dynamic simulation
Houzontal axis: cells rank
Vertical axis : dissipative index
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NOTATIONS
a dissipative index, time t
ay dissipative index, volume n, time t
ay dissipative index, averaged value
At) dissipative index, time t+1
A dissipative index, fluid
D coefficient (dimension defined by the equation)
f(a,) mathematical function of a,
g gravity constant, (m/s/s)
g(@) function of a
a(x) function of x
kk coefficient (dimension defined by the equation)
K Coefficient (dimension defined by the equation)
m mass (m)
n rank of a cell, or a volume
N total number of cells
Pp Pressure (P,)
t time (s)
v velocity, particle (m/s)
v velocity, gas or liquid (m/s)
x variable
z length position (m)
rm viscosity (m2/s)
z gas (or fluid)
a rank of a cell
a constant or averaged value
REFERENCES
BES R.S., MORA J.C. (1982). Modelisation Entropigue et Simulation du Comportement d’une Systeme
Dissipatif Complexe, Proceed. Int. AMSE Conf. Modelling and Simulation, 51, 216-220 Paris,
Mesnard.
CHEHBOUNI F., GABSI S., BES R.S., MORA J.C. (1985). Modelisation des Fluctuations dans un Lit
Fluidisé par une Processus d’Interaction _ Dissipative, Toulouse, Sept. 18-20, Proceed itme
Journées Europe Fluidization:
GABSI S., CHEHBOURI F., BES R.S., MORA J.C. (1990). Simulation of the Fluctuant Behavior of
a Fluidized Bed by Means of Cellular Automatic. CHISA 90, Prague, Czekosolvakia.
HAO BAI LIN (1990), Chaos, New York, The Viking Press.
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