12
F.A. Perdomo et al. / Journal of Molecular Liquids 185 (2013) 8–12
a,b,i,j
based on the free energy minimization, and modeling the multi-
component system with the SAFT-VR approach for SW associating
chain molecules. Good agreement was found between the theo-
retical predicted points and the experimental measurements for dif-
ferent values of the reaction ratio R. Chemical equilibrium manifolds
were obtained varying the relation between reactants from low to
high values of acid lauric and for several temperatures at constant pres-
sure. In this way, the theoretical optimal value for R for maximum yield
in the ester production can be determined, a very valuable estimator in
the design of an esterification reactor. In previous work [7] we found
that the SAFT-VR modeling of FAME can be used to determine the opti-
mum molar fractions of the FAME compounds to maximize the calorific
efficiency of a biodisel ternary blend. In this article we have extended
this approach by modeling the reactive equilibrium involved in the pro-
duction of FAME compounds, that can be used in the optimization of
biodiesel production. The method is not restricted to SW potentials as
molecular interactions; the SAFT-VR approach was derived for a general
type of model potentials, either discrete or continuous, including the
Lennard–Jones case [8], and more robust SAFT approaches, like SAFT-γ
with Mie potentials [18], can be incorporated into the theoretical frame-
work presented here.
where Ka,b,i,j is the volume available for bonding, f
function of the a–b site-site interaction ϕ
tact value of the SW radial distribution. Since
is the Mayer
and gij (σij) is the con-
a,b,i,j
SW
!
a;b;i;j
a;b;i;j
−ϕ
f
¼ exp
−1
ð25Þ
kT
then for the case of SW anisotropic sites we have that
!
a;b;i;j
a;b;i;j
ε
f
¼ exp
−1
ð26Þ
kT
The fraction of molecules not bonded at a given site in the ternary
mixture is given by
−1
−1
X1 ¼ ð1 þ ρx X Δ þ ρx X Δ þ ρx X Δ þ 2ρx X Δ
Þ
1
1
11
2
2
12
3
3
13
4
4
14
ð27Þ
ð28Þ
X2 ¼ ð1 þ ρx X Δ þ ρx X Δ þ ρx X Δ þ 2ρx X Δ24Þ
1
1
21
2
2
22
3
3
23
4
4
−1
1
X3 ¼ ð1 þ ρx X Δ þ ρx X Δ þ ρx X Δ þ 2ρx X Δ
Þ
1
1
31
2
2
32
3
3
33
4
4
34
−
X4 ¼ ð1 þ ρx X Δ þ ρx X Δ þ ρx X Δ þ 2ρx X Δ44Þ
1
1
41
2
2
42
3
3
43
4
4
Acknowledgments
where subscripts 1–4 represent lauric acid, methyl alcohol, methyl
laureate and water respectively, and Δij characterizes the association
between a specie i and a molecule of specie j.
In this way,
We acknowledge the financial support from CONACYT through
grant 61418 and a PhD scholarship (FAP). The authors wish to thank
Eduardo Buenrostro-González (Instituto Mexicano del Petróleo, México)
for his very helpful comments and suggestions.
!
ASSOC
q
ASSOC
μ
∂a
ASSOC
¼
a
þ N
ð29Þ
Appendix A
kT
∂Nq
Ni;T;V
According to Eqs. (17) and (20), the chemical potential for the q
component of the mixture can also be given as the sum of different
energetic contributions,
where
!
"
#
ASSOC
q
si
n
si
μ
X
Xa;q sq
X
X
∂Xa;i
∂Nq
1
Xa;i
1
2
¼
lnXa;q
−
þ
þ
xi
−
ð30Þ
IDEAL
q
MONO
q
CHAIN
q
ASSOC
q
kT
2
2
μq
μ
μ
μ
μ
a¼1
i¼1
a¼1
¼
þ
þ
þ
ð21Þ
kT
kT
kT
kT
kT
Explicit expressions for the different terms in the right side of this
equation can be obtained in a straightforward way, following references
References
ASSOC
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[
9,15]. We give in this Appendix A more detailed account of μ
to its relevance for the systems studied here.
The Helmholtz free energy contribution due to association is given
q
, due
[
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[
[
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by [9]
"
#
ꢂ
ꢃ
ASSOC
n
si
X
X
Xa;i
[6] F.A. Perdomo, A. Gil-Villegas, Fluid Phase Equilibria 293 (2010) 182–189.
[7] F.A. Perdomo, A. Gil-Villegas, Fluid Phase Equilibria 306 (2011) 124–128.
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Journal of Chemical Physics 106 (1997) 4168–4186.
A
si
2
¼
xi
lnXa;i
−
þ
ð22Þ
NkT
2
i¼1
a¼1
[
9] A. Galindo, L.A. Davies, A. Gil-Villegas, G. Jackson, Molecular Physics 93 (1998)
41–252.
where s
i
is the number of associating sites on molecule i, the first sum
sites a on a molecule of spe-
2
is over species i and the second over all s
i
[
[
10] J. Beare-Rogers, A. Dieffenbacher, J.V. Holm, Pure and Applied Chemistry 73
(2001) 685–744.
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(1997) 5274–5543.
cies i. The term Xa,i is defined as the fraction of molecules i, in mix-
tures with other components, not bonded at site a and is given by
the mass action equation:
[
[
0
1
−1
sj
n
XX
[
14] J. Nocedal, S.J. Wright, Numerical Optimization, second edition Springer series,
@
A
Xa;i
¼
1 þ
ρx Xb;jΔa;b;i;j
ð23Þ
j
2000.
j¼1 b¼1
[
[
15] C. McCabe, A. Gil-Villegas, G. Jackson, Chemical Physics Letters 303 (1999) 27–36.
16] G.N.I. Clark, A.J. Haslam, A. Galindo, G. Jackson, Molecular Physics 104 (2006)
3
561–3581.
17] M.C. dos Ramos, H. Docherty, F.J. Blas, A. Galindo, Fluid Phase Equilibria 276
2009) 116–126.
j j
where ∑ indicates a summation over all sites on molecule j, a , b ,
j
bj
[
[
c
j
,…; ∑ indicates a summation over all components, and
(
18] C. Avendaño, T. Lafitte, A. Galindo, C.S. Adjiman, G. Jackson, E. Müller, Journal of
Physical Chemistry B 115 (2011) 11154–11169.
ꢄ
ꢅ
a;b;i;j a;b;i;j SW
Δa;b;i;j ¼ K
f
g
σij
ð24Þ
ij