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18th European Symposium on Computer Aided Process Engineering – ESCAPE 18
Bertrand Braunschweig and Xavier Joulia (Editors)
© 2008 Elsevier B.V./Ltd. All rights reserved.
Thermochemical Multi-phase Models Applying the
Constrained Gibbs Energy Method
Risto Pajarre, Peter Blomberg, Pertti Koukkari
VTT, Process Chemistry, P.O. Box 1000 FIN-02044 VTT, Finland; [email protected]
Abstract
Computation of chemical equilibria in multiphase systems by Gibbs free energy
minimization under constraints set by the material balance has increasing interest in
many application fields, including materials technology, metallurgy and chemical
engineering. The results are utilised in multi-phase equilibrium studies or as parts of
equilibrium-based process simulation. Yet, there exist a number of practical problems
where the chemical system is influenced by other constraining factors such as surface
energy or electrochemical charge transport. For such systems, an extended Gibbs energy
method has been applied. In the new method, the potential energy is introduced to the
Gibbs energy calculation as a Legendre transformed work term divided into substance
specific contributions. The additional constraint potential is then represented by a
supplementary undetermined Lagrange multiplier.
In addition, upper bounds on the amounts of products can be set, which then limit the
maximum extents of selected spontaneous chemical reactions in terms of affinity. The
range of Gibbs energy calculations can then be extended to new intricate systems.
Example models based on free energy minimisation have been made e.g. for surface and
interfacial systems, where the surface, interfacial or adsorbed atomic or molecular
layers are modeled as separate phases. In an analogous fashion the partitioning effect of
a semi-permeable membrane in a two-compartment aqueous system can be modeled. In
such system the large ions, not permeable through the membrane, cause an uneven
charge distribution of ionic species between the two compartments. In this case, the
electrochemical potential difference between the two aqueous phases becomes
calculated for the multi-component system. The calculated results are consistent with
the Donnan equilibrium theory; however the multi-phase system may also include the
gas phase and several precipitating phases, which extends the applicability of the new
method. Finally, similar constraints can also be set to extents of reaction advancements,
allowing usage of Gibbs energy calculations in dynamic reaction rate controlled
systems.
Keywords: Gibbs energy calculation, multi-phase chemistry, surface tension, ion
exchange, reaction rate
1. Theory
When additional energy or work terms affect the Gibbs free energy, [G = G(T,p,nk)] it is
customary to transform the total differential of the Gibbs function to read as follows
(Alberty 2001):
N
dG   SdT  Vdp    k dnk 
k 1
 F dx
i
i
i
(1)
R. Pajarre et al.
2
where S denotes the entropy and V the volume of the system, μk = μk(T, p, nk) is the
chemical potential of the species (k), T is temperature, p is pressure and nk refers to
mole amounts of chemical substances. The last term (Fidxi) refers to additional
generalised work effects.
The Gibbs energy minimisation requires the use of the chemical potentials μk (partial
molar Gibbs energies) of the constituents of the system. If additional work terms are not
included, the multi-phase chemical equilibrium is solved with the mass balance
constraints as the necessary subsidiary conditions. Yet, in many physical circumstances
the potentials appear constrained also by other typically work-related factors. Then,
immaterial constraints for chemical potentials can be applied together with the mass
balance relations in a multi-phase Gibbs energy minimisation software. The additional
conditions arise from a work or energy effect or they are due to a pre-defined affinity or
extent of reaction. In the Gibbs energy minimisation performed by the Lagrange method
the new constraints are set with additional Lagrangian multipliers (Koukkari, Pajarre
2006a). They assume the same mathematical form as the mass balance conditions, but
with a zero molar mass for the new components. The equations governing the chemical
potentials and their respective constraints in the multi-phase system become as follows:
k 
N
a
k 1
kj
l
a
j
(k  1,2,...N )
(2)
nk   j  0 ( j 1,2, l )
(3)
j 1
kj
where N is number of different chemical constituents as formed from l system
components by means of the conservation matrix akj. The πj’s represent the chemical
potentials of the components in the system (Koukkari, Pajarre 2006a; Perez-Cisneros et
al. 2007) and equation (3) includes in addition to the mass balance constraints the
necessary immaterial conditions, such as electroneutrality, conservation of surface area,
metastability of a given constituent etc.. Equations (2) and (3) can then be used to solve
for the unknowns (nk, πj). For equilibrium problems, the Gibbs energy minimisation
with given input amounts and the system temperature and pressure is sufficient to solve
for the constraint potential and system composition. In non-equilibrium systems, an
external algorithm of e.g. reaction kinetics can furthermore be necessary to set the
extent of change (Koukkari, Pajarre 2006b). In Table 1 some examples of the
constraints and their respective conjugate potentials have been listed.
Thermochemical Multi-phase Models Applying the Constrained Gibbs Energy Method
Table 1. Examples of constraint potentials in the Gibbs free energy method
System
Constraint
Conjugate potential
Examples
Chemical
equilibrium
systems
Mass balance
Chemical potential of system
components
Multi-phase
chemical
equilibria
N
a
k 1
Systems with
surface/interface
equilibria
kj
nk  b j
Surface area
Ns
 Ak nks  A
j
G  k

b j a kj
Surface energy per unit area
(A0)
 area 
k 1
Electrochemical
multi-phase
systems
Charge neutrality
N
 z k nk  0
Electric potential energy per
mole of charge
 q  
k 1
Rate-controlled
complex
systems
Reacted amount
from the extent of
reaction
N
a
k 1
kr
nk  bri 
G
  A0
barea
G
 F  
bq 
Affinity of chemical reactions
 r  
i
G G

  Affi
bri  i
Phase diagrams
Interface
compositions
Nanoparticle
phase diagrams
Membrane
systems
fibre and pulp
suspensions
Metastable
phases or
constituents;
Non-equilibrium
chemical
reactors
akj = element of conservation matrix ; N = nr. of constituents. ; bj = molar amount of
component jth component, b(X) = molar amount of component ‘X’ (b(area) = A/A0)
πi = Chemical potential of ith component ; π(X) = Chemical potential of component ‘X’
A = total surface area; Ak = molar surface area ;  = surface tension; A0 = constant with
dimensions of m2/mol (numerical value arbitrary)
zk = charge number of species ; F = Faraday const. ;  = electric potential difference
between phases
Affi = Affinity of ith reaction
2. Example cases
2.1. Surface energy of mixtures
Surface energy can appear as an additional term in the Gibbs energy function of a multicomponent system. Assuming a flat, monomolecular surface layer where each
constituent of the surface phase occupies a characteristic molar surface area Ak, the
Gibbs energy can be written as:
3
R. Pajarre et al.
4
G    kb nkb    ks nks  Ak nks
k
k
(4)
k
where supersrcipts b and s refer to bulk and surface respectively. At equilibrium, the
total surface area (A) is constant and the immaterial constraint corresponding equation
(3) can be written as presented in Table 1. (Pajarre et al. 2006). With this and material
balance constraints formulated in such a way, that a minor amount of the surface phase
will appear, the surface tension can be solved by min(G) calculation as a conjugate
potential, together with the chemical potentials of the bulk and surface phases.. The
molar surface areas and surface energies of pure substances are needed for the standard
chemical potentials of the surface constituents:
 k0, s   k0  Ak k
(5)
It follows that the composition and surface tension of a mixture can be solved by Gibbs
energy minimisation when the initial amounts, the molar surface areas and surface
tensions of pure substances and the excess energy functions are given as input data. In
Fig. 1 the method has been applied to a binary water-ethanol system.
Surface tension (N/m)
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0
0.2
0.4
0.6
0.8
1
x(C2H5OH) bulk
Figure 1. Surface tension of a water-ethanol system.
2.2. Donnan potential and salt solubilities in pulp suspensions
The concept of ion-exchanging (Donnan) equilibrium has been conventionally used to
characterize electrolyte interactions with cellulose fibres (Towers, Scallan 1996).
Because of its simplicity it is an attractive approach to modelling of fibre-ion
interactions. The aqueous solvent in the pulp system is divided between two subvolumes: The volume of water surrounding the fibres and the volume of water inside
the fibre walls. The amount of water enclosed by the fibre walls depends on the type of
pulp and to some extent the chemical composition of the suspension liquid, typical
values being around 1-1.5 l/kg of dry weight. This value can be measured and is then
regarded as constant. Pulp contains fiber-bound acidic groups that can dissociate giving
the fibre an anionic charge. The amount of these groups, as well as their dissociation
constants can be quantified with standard potentiometric methods, and thus, the
respective G-values for Gibbs energy calculations are accessible.
A basic Donnan model however, is insufficient to deal several practical problems that
deal with dissolution and precipitation equilibria. These may involve carbonaceous and
other salts as well as the effect of gaseous pH-agents, such as carbon dioxide and
chlorine dioxide. Thence, a multi-phase approach based on the Gibbs free energy
Thermochemical Multi-phase Models Applying the Constrained Gibbs Energy Method
minimisation was developed, where the Donnan theory for ion exchange between two
aqueous phases was incorporated to a multi-phase Gibbs’ian system. (Koukkari et al.
2002). The necessary constraint is electroneutrality in both aqueous phases, and the
electrochemical potential difference becomes solved formally as another Lagrange
coefficient as presented in Table 1. This approach also enables the use of a more
elaborate ionic interaction models, such as that of Pitzer, instead of the ideal solution
assumption. Model curve in Fig. 2 has been calculated assuming just one acidic group
(pKa=4) as reported by Towers and Scallan.
Donnan potential, V .
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01
0
2
4
6
8
10
12
pH
Ca
Mg
Mn
model
Figure 2. The Donnan potential has been calculated as a function of pH and compared to
measured values of the concentration ratios of cationic species Ca2+, Mg2+ and Mn2+ in a pulp
suspension. The open diamond at highest pH value is for Manganese corrected for calculated
precipitation of solid hydroxide.
2.3. Affinity of a rate constrained reaction
Analogously with the equilibrium work terms it is possible to constrain the chemical
affinity of selected reactions of a multicomponent system. Thus the Gibbs energy
minimisation can be done for systems where one encounters metastable species or nonequilibrium constituents whose molar amounts are controlled by reaction rates. The
constraining by reaction rates is most practically performed by determining the extent of
a given reaction (r) from the experimental rate data (Koukkari and Niemelä 1997;
Perez-Cisneros et al. 1997). As this can be done incrementally for any reaction with
known reaction kinetics, the extent of reaction can be ‘frozen’ for any instant of time
while the min(G) procedure is performed for the whole system, as presented in Table 1.
The amount of the constraining component remains conserved during the minimisation,
while otherwise the system may reach equilibrium. In terms of affinities of the reactions
one may conclude that the affinity of fast (equilibrium) reactions becomes zero, while
the affinity of the constrained reaction results with a positive (non-zero) value until the
global equilibrium is reached (Fig. 3). Performing these calculations assumes local
chemical equilibrium to be valid for the homogeneous (well-stirred) system (or in
system that can be divided to such parts) and that the chemical relaxation rate is slow
compared with other relaxation times, including local heat exchange. The model results
can be validated by measured laboratory or process data.
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R. Pajarre et al.
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Gibbs free energy [kJ]
-1930
-1940
-1950
-1960
140
1
0.8
0.6
0.4
0.2
0
120
100
0
5
-1970
10 15 20 25
Time [ms]
Gibbs energy
-1980
Affinity
80
60
40
Affinity [kJ/mol]
Extent of reaction
-1920
-1990
20
-2000
-2010
0
0
5
10
15
20
25
Time [ms]
Figure 3. The reaction rate controlled oxidation reaction of TiCl 4 in the Ti-O2-Cl2 -system at 1360
C, showing the monotonically decreasing total Gibbs free energy and the affinity of the oxidation
reaction on the same time scale.
3. Conclusion
The constrained Gibbs energy applies to modeling of such systems as the
electrochemical Donnan equilibria in aqueous systems containing semi-permeable
interfaces, phase composition in surface-energy controlled systems and in systems with
affinities controlled by chemical reaction kinetics. The new method has been
successfully applied in calculating distribution coefficients for metal cations together
with pH-values in pulp suspensions, in calculation of surface tension of various
mixtures and in thermochemical process modelling involving chemical reaction rates. It
can be used in assessment of thermodynamic properties, service and design
applications, process scale-up, study of new chemical concepts and search for new,
more economic process alternatives.
References
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thermochemistry, Comput. Chem. Eng., 21(3), 245 – 253
P. Koukkari, R. Pajarre, 2006, Calculation of constrained equilibria by Gibbs energy
minimisation, Calphad, 30, 18-26.
P. Koukkari R. Pajarre, 2006, Introducing mechanistic kinetics to the Lagrangian Gibbs energy
calculation, Comput. Chem. Eng., 30(6-7), 1189-1196.
P. Koukkari, R. Pajarre, H. Pakarinen, 2002, Modelling of the ion exchange in pulp suspensions
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