PECULIARITIES OF THERMODYNAMIC SIMULATION WITH
THE METHOD OF BOUND AFFINITY.
B. Zilbergleyt,
System Dynamics Research Foundation, Chicago,
livent@ameritech.net
ABSTRACT. Thermodynamic simulation of the chemical and metallurgical systems is
the only method to predict composition of the systems in equilibrium. Regardless the
basic approach – Gibbs’ free energy minimization, maximization of the reaction
entropy, or solution of the equations for equilibrium constants  the conventional
strategy is always to find the most probable contents of chemical and phase species in
true thermodynamic equilibrium. Traditional simulation methods do not account for
interactions within the chemical system directly.
Recently introduced ideas in the thermodynamics of chemical systems account
explicitly the interactions between subsystems of complex systems, and lead to more
detailed and accurate evaluation of the systems’ composition in chemical equilibrium.
The basic equation of the theory at p, T=const in chemical equilibrium is
Gj*/RT – j j* (1j*) = 0,
where the reduced chaotic temperature j defines the intensity of the system’s
interaction with its environment, as well as its ability to resist to any shifts from
equilibrium due to that interaction. Solutions to logistic equation (1) constitute the
chemical system domain of states with 4 consecutive characteristic areas. For
thermodynamic simulation, 2 leftmost areas are of practical interest – area of
thermodynamic equilibrium (TdE) where the subsystem state totally matches traditional
thermodynamic equilibrium, and the area of open equilibrium (OpE) where the
subsystem state deviates from TdE but still dwells on the thermodynamic branch.
Regardless of the j value, the second term of the above equation equals to zero within
the TdE area due to j*=0; the logistic equation turns into traditional equation for
equilibrium constant, and traditional thermodynamic simulation is applicable within
this area. In the OpE area j*≠0, and the second term of the above equation should be
taken into account.
Thermodynamic simulation, based on the Method of Bound Affinity , (MBA), includes
a search for dependence of j value of the sub-system’s chemical reaction upon
deviation from thermodynamic equilibrium and the area limits as well as for the domain
characteristic points TdEj and OpEj for every subsystem at given n0kj, p, T and G0j.
When those parameters are found, the thermodynamic simulation consists of the joint
solution for the set of logistic equations (1) with the appropriate j values. Another
option is to minimize Gibbs’ free energy of the subsystems.
The application of MBA is exemplified by results for a complex system with a set of
subsystems.
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INTRODUCTION.
Practical applications of chemical thermodynamics are focused to a great extent on
thermodynamic simulation of the chemical systems. The goal of the simulation is to find the
most probable chemical and phase composition of the equilibrium system. Maximum
probability corresponds to thermodynamic equilibrium, the only state classical theory and based
on it methods can properly treat. The well-known Zeldovitch’s theorem [1], asserting a
uniqueness of the equilibrium state for isolated system, forms a cornerstone of the application,
thus prompting us to search that unique point in the space of states of the chemical system. In
the most of up to now developed simulation software [e.g., 2,3,4], the standard search usually
consists in finding out the global minimum of the whole system Gibbs’ free energy.
In the contrary to the commonly accepted approach, recently reported new theory of chemical
equilibrium as balance of the thermodynamic forces [5,6] is able to analyze wider scope of the
system states. The new theory places the sought states of chemical equilibrium in the domain of
states, specific for each chemical system. Depending upon the “strength” of the chemical
reaction in the system, that generally might be characterized by standard change of Gibbs’ free
energy, area of chemical equilibrium of a practical interest gets broken by two parts –namely
thermodynamic equilibrium and the area of the open equilibrium. Chemical equilibrium in the
first area totally matches classical thermodynamic equilibrium, the system considered an
isolated entity, and the simulation algorithm doesn’t change comparing to tradition. The
difference occurs when the system runs out of that area, ceases to be isolated, and new equation
members to account in the simulation become essential. Currently we call the method,
following from the theory the Method of Bound AffinityI.
This article is discussing some peculiarities of the new approach in a particular case of constant
pressure and temperature.
SOME FEATURES OF THE MBA.
One of the basic objects of the new theory is the domain of states of the chemical system - a set
of dependencies between the system’s deviation from true thermodynamic equilibrium and a
parameter characterizing system’s interaction with the environment, the reduced chaotic
temperature [6]. The name of this parameter reminds that its increase leads the chemical system
to bifurcations and chaos. At constant pressure and temperature, the system’s domain of states
is represented by bifurcation diagram, constituted by a set of solutions to the equation for the
system’s Gibbs’ free energy change, reduced by RT - a logistic equation with negative feed
back
ln[j(j,0)/j(j,j*)] j j*(1j*)=0.
(1)
*
In this equation j is the reduced chaotic temperature; j is the deviation of the system’s
chemical equilibrium from thermodynamic equilibrium in terms of reaction extent; it is positive
if the system state shifted towards initial reacting mixture, and negative for the opposite
direction; j(j,j*) is the mole fractions product of the j-reaction participants. Asterisk relates
the values to the state of chemical equilibrium. A set of solutions to build the domain of states
also can be obtained by minimization of the systems’ Gibbs’ free energy, reduced by RT
G j*= nkj*)  k/RT + nkj*) lnj(j, j*)  j (j*)2/2 = min.
(2)
It is easy to see that both of them turn to the appropriate classical equations if j*=0, that is in
true thermodynamic equilibrium of the isolated system.
____________________________________________________________________________
I
In some works we called it a Method of Chemical Dynamics. Currently it seems more
reasonable to return to the original name of the method, the Method of Bound Affinity (MBA)
[7].
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A typical domain of states for a simple reaction
PCl3+Cl2=PCl5.
(3)
is shown in Fig.1. The numbers at the curves define the thermodynamic equivalent of
transformation j [6], the mole amounts transformed on the reaction way to equilibrium.
1.0
j*
0.101
0.240
0.474
0.5
0.713
0.870
0.950
0.0
0
10
 TdEq
20
j
30
Fig.1. The part of the domain of states with positive deviations from TdE, reaction (3).
The curve from zero point of the reference frame up to the fork bifurcations presents
thermodynamic branch of evolution of the chemical system [8], and is apparently broken by 2
parts - the horizontal and ascending. The horizontal part is housing the states with zero
deviation from thermodynamic equilibrium; this is the TdE area, restricted by the TdE limit
value of j. The ascending parts before bifurcations are related to the open equilibria, or OpE
area. In the first bifurcation the thermodynamic branch decays, splitting by two new non-stable
branches. Further increase of the j value leads to the higher bifurcation period and eventually
to chaos.
Both equilibrium areas are typical for any direction of the system deviation from equilibrium,
while bifurcations were found only in cases of the shifts toward initial reacting mixtures as it
can be seen in Fig.2.
Initial reacting mixture, =0
1.0
j*
0.5
TDE
OPE
Bifurcations
0.0
0Line of TDE
-0.5
10
j
20
Exhausted reacting mixture,=max
Fig.2. Two-way bifurcation curve for j=0.713.
It is obvious from equations (1) and (2) that, depending upon the system states location in the
domain, we have either classical case of isolated equilibrium with j*=0 where we do not have
to take into account the j value in the calculations, or open equilibrium where we have to
account for j. The TdE limit defines the transition from classical, conventional simulation to
non-classical approach. Regarding the simulation problems, it occurs very important to know
the domain of states area limits to apply proper algorithms.
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AREA LIMITS AND REDUCED CHAOTIC TEMPERATURE.
The area limits may be found by direct computer simulation given initial composition and
thermodynamic parameters. At the same time the limits, TdE and OpE can be found analytically
with a good precision avoiding any simulation. Recall that equation (1) contains 2 functions,
logarithmic and parabolic. Both have at least one joint point at j*=0 (Fig.3) in the beginning of
the reference frame, providing for a trivial solution to equation (1) and retaining the system
within the TDE area. The curves may cross somewhere else at least one time more; in this case
the solution will differ from zero, and number of the roots will be more than one. There is no
intersection if
d()/dd[ln`*]/d
This condition leads to a universal formula to calculate TDE limit as
TDE =1+jkjn0kjkjj 

0
where n kj– initial amount and kj– stoichiometric coefficient of k-participant in j-system. We
offer the reader to check its derivation.

20
j*(1-j*)
10
=72
ln[((j,0)/(j,j*)]
0
-0.5
=10
0.0
0.5
j*
1.0
-10
F()
-20
Fic.3. The terms of equation (1) calculated for reaction (3), j=0.87 (T=348.15K).
Though the area with j*<0 is more complicated, formula (5) is still valid in cases when the
system gets exhausted by one or more of the reactants before the minimum of the logarithmic
term occurs. In case of reaction A+B=C with initial amounts of participants, corresponding to
1, 1, and 0 moles, formula (5) may be simplified as
 TDEjj
(6)
Fig.4 shows the comparison between values of TDE obtained by iterative process and the
calculated by formulae (5) and (6), reaction (3), in dependence on j.
As concerns to the OpE limit, it physically means the end of the thermodynamic branch
stability where the Liapunov exponent changes its value from negative to positive, and the
iterations start to diverge. If the logistic equation (1) is written in the form of
j(n+1)*=fjn*),
(7)
the OpE limit can be found as a point along the j axis where the | f`jn*)| value changes from
(-1) to (+1) [9]. As of now, we do not have ready formula for this limit.
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20
TdE
10
0
0.0
0.2
0.4
0.6
j
0.8
1.0
Fig.4. Calculated and simulated values TDE vs. j. Series о , Δ and □ represent results
calculated by equation (4), equation (5) and simulated correspondingly, reaction (3).
The dependence of the area limits upon standard change of Gibbs’ free energy for reaction (3)
is shown in Fig.5. Apparently, the “stronger” is the reaction in the system, the more extended
is its TdE area along the j axis; both limits, TdE and OpE get closer with the increase of
negative G0j merging at its extreme value.
30
jlim.
15
Bifurcations
OpE
TdE
0
-20
Gj st.
0
20
Fig.5. The area limits on the bifurcation diagram jlim. vs. G0j, kJ/m, reaction (3).
1.0
PbO-RO
j*
SrORO
0.5
CoO-RO
0.0
0
20
40 TDFext. 60
Fig.6. Shift vs. TDF in homological series of double oxides, reaction of the double oxides
with sulfur, HSC simulation.

To perform calculations in the OpE area one has to know the value of j. The
phenomenological theory offers several ways to find it. In the more general method, j can be
found directly from equation (1)
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j = ln[j(j, 0)/ j(j, j*)] / [j*(1j*)].
(8)

Varying the j and j* values, one can create the domain of states as a reference table
allowing for extraction of j during simulation, depending upon the first two parameters.
An alternative method consists in finding the equilibrium composition and the appropriate j
and j* values in the homological series of chemical reactions by varying the external TDF
[5]. This point by point method is illustrated by Fig.6. Standard change of Gibbs’ free energy
of the double oxide formation from oxides was accepted as the TDF.
Needless to say that prior to finding j one has to find j for the reaction in question at given
temperature. It can be done by any simulation method for thermodynamic equilibrium (at
j*=0).
EXAMPLES.
The Method of Bound Affinity obviously consists of 2 steps – mandatory solutions for
isolated subsystems precede solutions for the interacting subsystems.
Following example is a graphical image of a hypothetical chemical system with 5
subsystems. We tried to show possible different locations of the subsystems in their domains
of states in equilibrium. The internal pentagon connects the TdE limit values of the
subsystems, the external does the same for OpE limit values; the pentagon with the marks
connects conditionally simulated j values.
1
20
15
10
OpE limit
Simulated
 j values
5
2
TdE limit
5
0
4
3
Fig.7. A flat radar presentation of the sub-system domains of states, hypothetic chemical
system. Subsystem numbers are shown at the beams.
How different may be results of conventional and MBA thermodynamic simulation? To
answer this question, we used chemical systems with 2 subsystems where different reactions
between a couple of double oxides and hydrogen can run. Double oxides were taken to
reduce reaction activity of the basic oxides, in other words, to make the reaction less
“strong”, allowing usage of non-zero parabolic term in the OpE area in calculations. For
example, binding of CoO into CoO•TiO2 brings equilibrium reaction constant from 32.09 for
CoO+H2=Co+H2O to as low as 0.684 for CoO•TiO2+H2=Co+TiO2+H2O. Simulation was
carried using HSC and then MBA in mixtures of double oxides (Me1O•TiO2+Me2O•TiO2)
with hydrogen at T=973.16 and p=0.1MPa; results are shown in Table 1. We used following
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values of j, found by the point method for MeO and MeO•TiO2: reactions NiO and
NiO•TiO2 with hydrogen j=29.04, for CdO-reactions j=21.13, for CoO-reactions j=18.57.
Even if the used thermochemical data were far from perfect, the difference between results of
classical and MBA simulations is quite discernable.
Table 1.
Comparison between results, obtained by conventional (HSC) and MBA simulations.
Couple
NiO*TiO2
CdO*TiO2
HSC
0.126
0.322
MBA
0.164
0.276
Equilibrium contents, moles
Couple
HSC
MBA
NiO*TiO2
0.088
0.157
CoO*TiO2
0.488
0.389
Couple
CdO*TiO2
CoO*TiO2
HSC
0.208
0.437
MBA
0.258
0.380
CONCLUSION.
Described peculiarities of the Method of Bound Affinity at constant temperature and pressure
and ways to find area limits and running values of j, allow for application of the MBA to
thermodynamic simulation of chemical systems. Results by a classical simulation method and
the MBA are quite different; the new theory prompts us to consider the MBA results to be in
better correspondence with real chemical systems. Some results of this work were presented
in part earlier on [10].
REFERENCES.
[1] Zel’dovich, Ya. J. Phys. Chem.(USSR), 1938, 3, 385.
[2] Outokumpu HSC Chemistry; Outokumpu Research Oy, Finland,
[3]
[4]
[5]
[6]
www.outokumpu.fi/hsc/brochure.htm, 2000.
CEA: NASA Jet Propulsion Laboratory,
http://www.grc.nasa.gov/WWW/CEAWeb/index.html, 2002.
SolGasMix-PV, NESC9944,
http://www.nea.fr/abs/html/nesc9944.html (last update 1986).
Zilbergleyt B.: Equation of State of Chemical System: From True Equilibrium to
True Chaos, http://arXiv.org/abs/physics/0204074
Zilbergleyt B.: Equation of State of Chemical System: From True Equilibrium to
True Chaos, http://arXiv.org/abs/physics/0404082
Zilbergleyt B.: Russian Journal of Physical Chemistry, 52(10), 1978, p. 1447.
[7]
[8] Prigogin I.: The End of Certainty, New York, The Free Press, 1997.
[9] Beck C., Schlogl F.: Thermodynamics of Chaotic Systems, Cambridge, Cambridge University
Press, 1997.
[10] Zilbergleyt B., Zinigrad M.: Thermodynamic Simulation Of Complex Metallurgical and
Chemical Systems with The Method Of Chemical Dynamics, International Conference
“Modeling, Control, and Optimization in Ferrous and Non-Ferrous Industry”, Chicago, TMS,
2003.
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