Glycerol acetylation considering competing dimerization

Article abstract of DOI:10.1002/kin.21282

Glycerol acetylation is a very interesting reaction for studies of consecutive kinetics. In this short communication, we present a pseudo-homogeneous model for the synthesis of triacetyl glycerol from the reaction of glycerol and acetic acid over strongly acidic Amberlyst-15 and Amberlyst-70 catalysts, considering a dimerization of diacetyl glycerol (DAG) into diglycerol tetraacetate as a parallel reaction and compare the results with a model without side reactions. The best fits were obtained for apparent zeroth-order dimerization and first-order consecutive reactions in the presence of acetic acid in excess and with removal of water. An adaptation was made for DAG. The proposed model shows that the formation of DAG is faster than the consumption of glycerol, which could be an explanation for the occurrence of DAG dimerization instead of other parallel reactions in acetylation.

Full text of DOI:10.1002/kin.21282

Received: 6 November 2018
DOI: 10.1002/kin.21282
Revised: 21 March 2019
Accepted: 11 April 2019
C O M M E N T
Glycerol acetylation considering competing dimerization
André V.-H. Soares1
Sumeet S. Kale2,3
Udo Armbruster2
Fabio B. Passos4
Shubhangi B. Umbarkar5
Mohan K. Dongare5,6
Andreas Martin2
1Instituto Federal do Rio de Janeiro, Duque
de Caxias, RJ, Brazil
Abstract
Glycerol acetylation is a very interesting reaction for studies of consecutive kinet-
ics. In this short communication, we present a pseudo-homogeneous model for the
synthesis of triacetyl glycerol from the reaction of glycerol and acetic acid over
strongly acidic Amberlyst-15 and Amberlyst-70 catalysts, considering a dimerization
of diacetyl glycerol (DAG) into diglycerol tetraacetate as a parallel reaction and com-
pare the results with a model without side reactions. The best fits were obtained for
apparent zeroth-order dimerization and first-order consecutive reactions in the pres-
ence of acetic acid in excess and with removal of water. An adaptation was made
for DAG. The proposed model shows that the formation of DAG is faster than the
consumption of glycerol, which could be an explanation for the occurrence of DAG
dimerization instead of other parallel reactions in acetylation.
2Leibniz-Institut für Katalyse e.V. an der
Universität Rostock, Rostock, Germany
3Institut National des Sciences Appliquées de
Toulouse, Laboratory of Physics and
Chemistry of Nano-Objects (LPCNO),
Toulouse, France
4Departamento Eng. Química e Petróleo,
Universidade Federal Fluminense, Niterói,
Brazil
5Catalysis Division, CSIR-National Chemical
Laboratory, Pune, India
6Mojj Engineering System Ltd., Pune, India
Correspondence
André V.-H. Soares, Instituto Federaldo Rio de
Janeiro, Av. Repúblicado Paraguai120, Duque
deCaxias, RJ, 25050-100, Brazil.
Email:andre.soares@ifrj.edu.br
K E Y W O R D S
apparent kinetics, diacetyl glycerol dimerization, glycerol acetylation
Fundinginformation
CoordenaçãodeAperfeiçoamentodePessoal
deNível Superior, Grant/AwardNumber:
99999.009893/2014-08;Bundesministerium
fürBildungund Forschung, Grant/Award
Number:IND11/045
chemicals. Monoacetyl glycerol (MAG, monoacetin), diacetyl
glycerol (DAG, diacetin), and triacetyl glycerol (TAG, tri-
1
INTRODUCTION
acetin) can be obtained from such esterification. Acetins have
applications in food and leather industry as plasticizers and
may also serve as solvent and fuel additives.8–11 However,
new processes for the transfer of lab results into pilot and
industrial scale, for example, for TAG, in large quantities
should rely on accurate kinetic parameters, therefore justify-
ing the efforts in describing the reaction through models that
carefully take parallel reactions into consideration, if they do
happen.
Biomass-derived chemicals have played a central role in
the debate surrounding the substitution of fossil-derived
resources. Phenol, acetic acid, and glycerol are some of
the resulting platform chemicals, serving also as model
compounds for studies in the interest of developing strong
bio-based chemical processes. In particular, surplus glyc-
erol is obtained from the biodiesel industry and its con-
version to valuable chemicals is under study in fundamen-
tal research1–3 but also in transfer of scientific results to
industrial scale.4–7 In this context, glycerol acetylation is an
important reaction from scientific and industrial viewpoints,
since it combines two biomass-derived compounds, glycerol
(G) and acetic acid (AA), for the production of value-added
Thus, we present herein a kinetic model for TAG synthe-
sis that stems from previously used models adjusted to the
actual problem of dimer formation as a side reaction, when
Amberlyst-15 and Amberlyst-70 are used as acidic catalysts,
Int J Chem Kinet. 2019;1–7.
wileyonlinelibrary.com/journal/kin
© 2019 Wiley Periodicals, Inc.
1

2
SOARES ET AL.
and the reaction is performed in conditions away from equilib-
rium. Owing to the use of entrainer toluene for water removal
via azeotropic mixture, the variation of the temperature was
not possible without changing important features of the
system.
only the parameters relative to water, G, AA, and TAG are
estimated by fitting the equilibrium data, leaving aside MAG
and DAG. Third, equilibrium data are obtained at 298 K, far
from the actual reaction temperatures. Fortunately, based on
the decrease in hydroxyl groups, it is reasonable to estimate
the values for the parameters for MAG and DAG adsorp-
tion between the ones obtained for glycerol and TAG, but
only for 298 K. It is also seen that depending on the activity
and amount of catalyst, mass transfer and equilibrium greatly
hamper the kinetic evaluation, since the selectivity toward
TAG is very low for the works using this approach.12–14
By setting the conditions of the reaction to avoid equi-
librium limitations in the formation of TAG, a straightfor-
ward pseudo-homogeneous consecutive kinetic model can
be advanced.15 This entails that apparent ka,1, ka,2, and ka,3
will suffice to describe the reaction behavior, leaving adsorp-
tion equilibria out of the model. Although it may first seem
there are too many approximations, the final model describes
the reaction dynamics for conditions away from equilibrium
consistently. Moreover, the model requires no estimation of
thermodynamic parameters, which could be cumbersome and
misleading, but only the apparent kinetic constants sufficient
for describing the concentrations of the products at a given
moment. In a recent work,16 we show that glycerol esterifica-
tion can be undertaken in conditions that shift equilibrium in
the direction of the formation of TAG, with total conversion
of glycerol, MAG, and DAG, and high TAG selectivity (94%).
To guarantee that the reaction is far from equilibrium, two
experimental conditions have to be assured: (1) water must be
stripped from the reaction medium by use of an adequate sol-
vent entrainer that does not interfere with the reaction; and (2)
acetic acid must be supplied in abundance. Thus, the apparent
kinetic rates ka,k can be expressed as a function of the con-
sumption of glycerol, MAG, and DAG. Through these consid-
erations, Zhou et al15 have neglected the backward reactions
and proposed the following set of differential equations for a
pseudo-homogeneous kinetic model for the reaction:
2
MODELS
2.1 Model A: Pseudo-homogeneous with no
side reactions
A straightforward consecutive model for the kinetics for the
production of TAG may be described considering the follow-
ing reactions:
ꢀ
→
1
G + AA ← MAG + H2O
(R1)
(R2)
(R3)
ꢀ
−1
ꢀ
→
2
MAG + AA ← DAG + H2O
ꢀ
−2
ꢀ
→
3
DAG + AA ← TAG + H2O
ꢀ
−3
where k , k , and k are the rate constants of the forward reac-
1
2
3
tions, and k−1, k−2, and k−3 are the constants for the backward
reactions.
This model was first proposed in detail by Gelosa et al12
and was applied with little modification by Hasabnis and
Mahajani13 and Hung et al.14 Gelosa et al12 presented a
Langmuir-Hinshelwood Hougen-Watson kinetic model based
on multicomponent adsorption equilibria, relying on parame-
ters estimated from equilibrium experiments. In their model,
the adsorbed concentration of the ith species, Γi, relates to its
liquid phase concentration, ci, in the following manner:
ꢂꢁΓ∞ꢁ ꢃꢁ
Γꢁ =
,
(1)
∑
ꢄ
1 + ꢅ=1 ꢂꢅꢃꢅ
ꢉꢃꢇ
ꢉꢊ
where Ki is the adsorption equilibrium constant, Γ∞ꢁ is the
pure-component saturation concentration for adsorption, and
N is the number of components in the mixture, aside from the
catalyst.
= −ꢀꢋ,1ꢃꢇ,
(3)
(4)
(5)
ꢉꢃMAG
ꢉꢊ
= ꢀꢋ,1ꢃꢇ − ꢀꢋ,2ꢃMAG
,
Thence, the reaction rates can be written as
(
)
(
)
ꢉꢃDAG
ꢉꢊ
ꢈ
ꢄ
ꢁ,1
= ꢀꢋ,2ꢃMAG − ꢀꢋ,3ꢃDAG
,
∏
ΓEQ,ꢁ
ꢂEΓQ,1
ꢆꢀ = ꢀꢀΓꢇΓAA 1 −
,
(2)
ꢁ=1
where ka,k are the apparent kinetic rates. The use of less
parameters simplifies the problem and gives robustness to the
model. Considering the initial condition that only glycerol is
where rk is the reaction rate of the kth reaction, kk is the respec-
tive kinetic constant, ꢂEΓQ,ꢀ is the thermodynamic equilib-
rium constant for each reaction, and vi,k is the stoichiometric
coefficient of the ith component in the kth reaction. Notwith-
standing its thoroughness, this approach offers some limita-
tions. First, an adsorption model has to be assumed. Second,
present with a concentration cG,0 at t , exact solutions to the
0
differential equations are the following:
−ꢀ
ꢊ
ꢋ,1
ꢃꢇ = ꢃ
ꢌ
,
(6)
ꢇ,0

SOARES ET AL.
3
model with n = 2, it should be noted that the dimerization in
the present model is a side reaction, with yields for DGTA
much lower than for TAG. By using n = 1, one considers
that the production of DGTA requires a first-order dependence
with DAG concentration, as happens to the other reactions in
the system. However, fits using n = 2 and n = 1 were not sat-
isfactory.
ꢀ
ꢋ,1ꢃꢇ,0
(
)
ꢊ
ꢌ−ꢀ ꢊ − ꢌ−ꢀ
,
(7)
ꢋ,1
ꢋ,2
ꢃMAG
=
ꢀ
ꢋ,2 − ꢀꢋ,1
(
)
ꢌ−ꢀ ꢊ − ꢌ−ꢀ ꢊ ꢌ−ꢀ ꢊ − ꢌ−ꢀ
ꢊ
ꢀ
ꢀ
ꢋ,1ꢀꢋ,2ꢃꢇ,0
ꢋ,2 − ꢀꢋ,1
ꢋ,2
ꢋ,3
1
3
ꢃDAG
=
−
.
ꢀ
ꢋ,3 − ꢀꢋ,1
ꢀꢋ,3 − ꢀꢋ,2
(8)
In conditions of reactant excess, the amount of adsorbed
reactants that occupy the surface of a heterogeneous catalyst
limits the reaction rate and may alter profoundly the intrin-
sic reaction order. A pseudo-zeroth order kinetic fit agrees
with such limitations for different cases.19,20 In the present
work, the parallel reaction for dimerization depends only on
the amount of adsorbed DAG, in an environment with excess
of AA, which reacts with DAG at a higher rate than DAG
reacts with itself. Therefore, the dimerization and other side
reactions should depend much less on the concentration of
any given compound, but rather on the available catalyst sites.
Assuming n = 0, Equations (11) and (12) then become
The expression for c
may be obtained by considering
TAG
the total mass balance to be
(
)
ꢃTAG = ꢃꢇ,0 − ꢃꢇ + ꢃMAG + ꢃDAG
.
(9)
Since the yield of a species i is defined as yi = ci/cG,0, the
molar balance can be rewritten more conveniently as
(
)
ꢃꢇ
ꢃꢇ,0
ꢍTAG = 1 −
+ ꢍMAG + ꢍDAG
.
(10)
ꢉꢃDAG
ꢉꢊ
2.2 Model B: Pseudo-homogeneous
+ ꢀꢋ,3ꢃDAG = ꢀꢋ,2ꢃMAG − ꢀꢋ,4
(13)
(14)
considering dimerization as side reaction
ꢉꢃDGTA
The previously presented model is very useful and clear, as
a reference. However, as we have previously reported for the
first time,16 diglycerol tetraacetate (DGTA) is formed through
etherification of DAG and its production competes with ester-
ification. In view of this, model A or any model that con-
siders only three consecutive reactions fails as a consistent
description of the system. Thence the model of consecutive
reactions for TAG production must be changed to accommo-
date this competition. Therefore, an extended model (B) is
presented by adding the side reaction R. 4 to the previous steps
(R1–R3):
= ꢀꢋ,4
,
ꢉꢊ
and the respective solutions are
(
)
ꢌ−ꢀ ꢊ − ꢌ−ꢀ ꢊ ꢌ−ꢀ ꢊ − ꢌ−ꢀ
ꢊ
ꢀ
ꢋ,1ꢀꢋ,2ꢃꢇ,0
ꢋ,2
ꢋ,3
1
3
ꢃDAG
=
−
ꢀ
ꢋ,2 − ꢀꢋ,1
ꢀ
ꢋ,3 − ꢀꢋ,1
ꢀꢋ,3 − ꢀꢋ,2
ꢀꢋ,4
ꢀꢋ,3
(
)
ꢌ−ꢀ ꢊ − 1 ,
(15)
ꢋ,3
+
ꢃDGTA = ꢀꢋ,4ꢊ.
(16)
ꢀ
ꢋ,4
→
2DAG
DGTA + H2O
(R4)
There is one inconvenience with Equation (15):
The equations corresponding to the reaction steps are the same
as Equations (3) and (4), plus the ones regarding the formation
of the dimer:
lim ꢃDAG = −(ꢀꢋ,4∕ꢀꢋ,3). This means that as time pro-
ꢊ→∞
gresses toward the end of the reaction, the concentration
of DAG will be negative until a constant value, which is
unacceptable. To overcome this problem, a correction term
may be introduced directly into Equation (15), to satisfy
ꢉꢃDAG
ꢉꢊ
= ꢀꢋ,2ꢃMAG − ꢀꢋ,4ꢃDAGꢎ − ꢀꢋ,3ꢃDAG
,
(11)
lim ꢃDAG = 0, and also cDAG(0) = 0. Since this problem
ꢊ→∞
ꢉꢃDGTA
= ꢀꢋ,4cꢎDAG
.
(12)
ꢋ,3
occurs only because of the term ꢀꢋ,4∕ꢀꢋ,3(ꢌ−ꢀ ꢊ − 1),
ꢉꢊ
the correction should be introduced here. A function like
ꢏ(ꢊ) = ꢊ∕(ꢊ + ꢋ) meets the aforementioned criteria, which
yields
Some difficulties arise from including Equations (11)
and (12) in this new model, because of the value for the con-
stant n. A typical dimerization reaction would be expected to
behave in a second-order kinetics,17,18 and reasonable values
for n should include or lie between 0 and 2. Therefore, for
n = 2, Equation (11) is transformed into a nonlinear differen-
tial equation, commonly known as the Riccati equation. Leav-
ing aside some mathematical complications that arise from a
(
)
ꢌ−ꢀ ꢊ − ꢌ−ꢀ ꢊ ꢌ−ꢀ ꢊ − ꢌ−ꢀ
ꢊ
ꢀ
ꢋ,1ꢀꢋ,2ꢃꢇ,0
ꢋ,2
ꢋ,3
1
3
ꢃDAG
=
−
ꢀ
ꢋ,2 − ꢀꢋ,1
ꢀ
ꢋ,3 − ꢀꢋ,1
ꢀꢋ,3 − ꢀꢋ,2
(
)
ꢀꢋ,4
ꢀꢋ,3
ꢋ
.
ꢊ
ꢌ−ꢀ
−
(17)
ꢋ,3
+
ꢊ + ꢋ

4
SOARES ET AL.
Constant a could be obtained from fits of the model, as
parallel dimerization reaction. Table 1 shows the values of the
apparent kinetic constants. A limitation on the experiments
impedes the assessment of the models for other temperatures,
since the reaction runs with toluene as an entrainer. This
means the reaction temperature stabilizes at 378 K, around
the boiling temperature of toluene (384 K).
done for ka,k. However, obtaining a from the model impacts
the accuracy of ka,k. Since a has the same dimension as t, that
is time, it could be interpreted as a scale factor, loosely related
to the order of magnitude for the consumption of DAG. Of
course, Equation (17) is not the solution for Equation (13).
Therefore, differentiation of Equation (17) renders the equa-
tion in substitution for Equation (13):
It should be clear by now that although model A does
describe MAG, DAG, and TAG concentrations very accu-
rately, model B is qualitatively different from model A
because it incorporates DGTA, an actual product that can-
not be ignored. Therefore, model A is not sufficient for the
description of the system. While model B has the advantage
of considering the DGTA yield, it describes poorly the yields
for MAG for both sets of data, as seen in Figures 1 and 2.
This could be attributed to the fact that the optimized func-
ꢉꢃDAG
+ ꢀꢋ,3ꢃDAG
ꢉꢊ
[
]
ꢋ
1
= ꢀꢋ,2ꢃMAG + ꢀ
− 1 . (18)
ꢋ,4 (ꢊ + ꢋ)
ꢀ
ꢋ,3 (ꢊ + ꢋ)
Once again, the mole balance may be used to calculate the
concentration of TAG and adapted to conveniently express its
yield, as done in Equation (10). The final expression arises
from combination of Equations (6), (7), (16), and (17), and
by assuming that two molecules of DAG are required to form
one dimer:
tion was y
and not yMAG. Another limit of both models is
the fact thTaAt Gthere is no distinction between the different iso-
mers of MAG and DAG that may be formed throughout the
reaction. MAG has two position isomers, as has DAG. Given
the position of the remaining hydroxyl groups in MAG and
DAG, one expects that some preferred isomer of each prod-
uct is more abundant due to the difference in reactivity, for
steric and polarity reasons. Further experimental work may
also shed light on these matters.
(
)
ꢃꢇ
ꢃꢇ,0
ꢍTAG = 1 −
+ ꢍMAG + ꢍDAG + 2ꢍDGTA
.
(19)
Some comments should be made regarding constant a.
From Equation (18), it can be seen that a is related to the
3
MATERIALS AND METHODS
time when c
peaks, and a corresponding value for c
,
DAG
MAG
The detailed experimental procedures and a discussion on
optimal product distribution are described elsewhere.16 In
general, experiments were conducted at 378 K in a glass
flask equipped with a Vigreux column, a modified Dean-Stark
apparatus with a thermometer, reflux condenser, and magnetic
stirrer at 1200 rpm. Typical amounts used: 10 g of glycerol and
varying AA: G ratios of 6:1 and 8:1 (mol:mol). The catalyst
amount used in the experiments was 500 mg, 60 g of toluene
was the entrainer used for water removal, and products were
analyzed by Gas Chromatography, using dodecane as internal
standard.
when dcDAG/dt = 0. Although it can be estimated from the
model, like any of the parameters ka,k, it is far more practical
to view it as part of the model, rather than a value that can
vary according to differing experimental conditions (such as
catalyst or AA concentration). Considering the dynamics of
DAG variation, constant a should not be much higher than
the time when c
peaks. Preliminary simulations using a
= 0.001, 0.01, 0.D1A, G1.0, and 10.0 showed no considerable dis-
agreement between model and results for a ≤ 1. Therefore, for
convenience, in this work a = 1, fixed for all data.
The quantitative difference between the fitted models and
the experimental results is subtle. For TAG, whose yield is the
target, very little is changed at the beginning of the reaction,
but there is an increasing gap toward the end, since model A
was developed considering no side reactions, whereas model
B agrees better with the actual experimental results. Model
B shows that the production of DAG is faster than the con-
sumption of glycerol (ka,2 > ka,1) for all AA: G ratios, which
is not expected from model A. This difference in the consecu-
tive reaction rate constants sheds light on why a side reaction
with DAG occurs, rather than with glycerol or MAG. For both
catalysts, dimerization is much slower than all of the reaction
steps. Moreover, the rate of dimerization decreases as the AA:
Nonlinear regression was done using the maximum likeli-
hood methodology through an algorithm specifically written
for that purpose with the software Scilab21 and was also cross-
checked with Polymath (student version) for least squares
nonlinear regression. For deeper understanding of the under-
lying mathematics of the maximum likelihood method and its
applications in kinetic models, the reader is referred to semi-
nal previous applications of this method.22–24
4
RESULTS AND DISCUSSION
2
G ratio increases. For AA: G = 6:1, model B fits showed R
Figure 1 shows liquid phase formation of MAG, DAG, TAG,
and DGTA for Amberlyst-15, and Figure 2 for Amberlyst-70,
using fits for reference model A and model B, containing the
adj
= 0.9972 for Amberlyst-15 and R 2 = 0.9970 for Amberlyst-
adj
70. For AA: G = 8:1 using Amberlyst-70, R 2 = 0.9934.
adj

SOARES ET AL.
5
F I G U R E 1 Yields for MAG, DAG, TAG, and DGTA considering experimental values and models A and B for Amberlyst-15 at a ratio of AA:
G = 6: 1 (T = 378 K; entrainer: toluene; t = 24 h; 0.5 g cat) [Color figure can be viewed at wileyonlinelibrary.com]
TA B L E 1 Apparent rate constants for Amberlyst-15 and Amberlyst-70 from models A and B
Amberlyst-15
Zhou et al15,a
2.668
Amberlyst-70
Model Ab
7.675
ka,k
ka,1
ka,2
ka,3
ka,4
ka,1
ka,2
ka,3
ka,4
Unit
AA: G (mol: mol)
Model Ab
2.784
1.525
0.206
–
Model Bb
1.209
Model Bb
1.147
−1
h
6: 1
6: 1
6: 1
6: 1
8: 1
8: 1
8: 1
8: 1
−1
h
1.211
3.703
1.104
16.302
0.259
−1
h
0.430
0.213
0.271
−1 −1
−4
−3
mol L
h
–
7.6 × 10
–
1.2 × 10
1.123
−1
h
20.175
0.826
−1
h
19.909
0.312
−1
h
0.352
−1 −1
−4
mol L
h
–
4.1 × 10
aExperimental conditions: 1100 rpm; T = 383 K; 2.645 g cat; 150 mL mixture (G+AA).
bExperimental conditions: 1200 rpm; T = 378 K; entrainer: 60 g toluene; 0.5 g cat; 10 g glycerol.
For Amberlyst-70, while model A shows increasing ka,1
with increasing AA ratio, model B does not follow this pat-
tern, but shows that ka,1 remains practically constant (≈
1.1 h−1) when increasing the AA ratio from 6:1 to 8:1.
It is instructive to note that in model A ka,2 decreases,
and ka,3 increases, as the concentration of AA increases.
Unless other processes take place, one would not expect
this from the structures of MAG and DAG and the relative
decreasing reactivity due to the fewer respective available
hydroxyl groups.
There is no appropriate comparison of values of ka,k with
the ones reported by Gelosa et al,12 replicated by Hasabnis
and Mahajani,13 because those authors use a ratio of mcat/m
= 15, which is overwhelmingly larger than the one used iGn
the experiments modeled herein (mcat/m = 0.05). On the
G
other hand, Zhou et al15 use model A, and their results are

6
SOARES ET AL.
F I G U R E 2 Yields for MAG, DAG, TAG, and DGTA considering experimental values and models A and B for Amberlyst-70 at a ratio of AA:
G = 6:1 (T = 378 K; entrainer: toluene; t = 24 h; 0.5 g cat) [Color figure can be viewed at wileyonlinelibrary.com]
remarkably consistent with this work (Table 1). Upon com-
parison, the role of water removal in the kinetics can be seen
in the fact that ka,k is slightly higher at 378 K with toluene
than at 383 K with no entrainer.
for the conditions that kinetically control the reaction there is
a nonnegligible amount of DAG that forms dimers.
ACKNOWLEDGMENTS
The Brazilian authors would like to thank CAPES for fund-
ing (Process 99999.009893/2014-08) and the other authors
gratefully acknowledge the financial support by DLR within
the frame of the BMBF-CSIR Cooperative Science Program
(project number IND 11/045).
5
CONCLUSIONS
The provided model was originally devised to assess the pro-
duction of TAG with a nonnegligible side reaction. For this
purpose, fits for the experimental curves using n = 1 and 2
for reaction order of DAG dimerization were unsatisfactory
in comparison to model A. However, our refined model built
upon pseudo-zeroth order as a side reaction for dimer forma-
tion responds better to the data. This agreement indicates that
it is more likely for an adsorbed DAG molecule to find an AA
molecule beside it than another DAG or MAG molecule. This
is likely a case in which the catalyst affects the intrinsic rate of
the dimerization reaction, provided that DGTA is limited by
the production of TAG. The fact that ka,2 > ka,1 > ka,3 > ka,4
in model B is counterintuitive. It reveals that although the for-
mation of DGTA occurs very slowly, it is the side product with
a higher yield than other side products, because the formation
of DAG is faster than the consumption of glycerol. Therefore,
ORCID
André V.-H. Soares
https://orcid.org/0000-0003-4576-7937
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How to cite this article: Soares AV, Kale SS,
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