Alessandro Colotti

Dominique Monnet

Alexandre Goldsztejn

Franck Plestan

New convergence conditions for the super twisting algorithm based on a Lyapunov approach

Keywords: Finite time stability, super twisting control, nonsmooth Lyapunov function

A simple proof is proposed that extends Seeber and Horn's recent sufficient conditions (Automatica 2017) for the finite time convergence of the super twisting algorithm to a larger class of perturbations. The new convergence conditions are compared to state of the art conditions, and turn out to accept sensibly smaller gains.

Introduction

Lyapunov functions is one key approach for proving the finite time stability of sliding mode controllers [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via lyapunov function designing[END_REF][START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF][START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF]. They often offer some insights: e.g., the twisting controller can be assimilated to a nonsmooth harmonic oscillator with dissipation that enjoys an energy related Lyapunov function (see, e.g., Example 3 of [2,p.63]). The Lyapunov approach is easier when no or specifically structured uncertainties are involved. Several works have used a Lyapunov analysis on systems with no or special uncertainty and extended it to reach more general uncertainties, e.g., [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via lyapunov function designing[END_REF][START_REF] Seeber | A novel method to estimate the reaching time of the super-twisting algorithm[END_REF]. The convergence conditions [START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF] for the super twisting algorithm (STA) with perturbation given in [START_REF] Castillo | Supertwisting algorithm in presence of time and state dependent perturbations[END_REF] is used to obtain new convergence conditions for the more generally perturbed STA [START_REF] Cortes | Discontinuous dynamical systems[END_REF].

Seeber and Horn investigated in [START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF] the finite time stability of the STA

ẋ1 (t) = -k 1 φ(x 1 (t)) + x 2 (t) (1a) ẋ2 (t) = -k 2 sign x 1 (t) + δ(t), (1b)
where φ(x) = |x| Email addresses: alessandro.colotti@ls2n.fr (Alessandro Colotti), dominique.monnet@ec-nantes.fr (Dominique Monnet), alexandre.goldsztejn@ls2n.fr (Alexandre Goldsztejn), franck.plestan@ec-nantes.fr (Franck Plestan).

in the rest of the paper, trajectories are defined in the sense of Filippov [START_REF] Aleksej | Differential Equations with Discontinuous Righthand Sides: Control Systems[END_REF]. Finite time stability of the STA (1) means converging to x 1 (t) = 0 and x 2 (t) = 0 in finite time. They provided the following sufficient conditions for finite time stability:

Theorem 1 ([9]) The system (1) is finite time stable if k 2 > L and k 2 1 > k 2 + L.
We build on this result and derive new finite time stability sufficient conditions for a larger class of perturbations (modeled with functions g(t), h(t) and d(t)), that commonly appear when modeling physical systems controlled by STA:

ẋ1 (t) = h(t) + g(t) -k 1 φ(x 1 (t)) + x 2 (t) (2a) ẋ2 (t) = -k 2 sign x 1 (t) + d(t). (2b)
We assume that the perturbations g(t

| γ(t)| ≤ M . If | ġ(t)| ≤ G and | ḣ(t)| ≤ H, then M = H/g + G /g 2 is a valid bound for | γ(t)|.
The integral correction x 2 (t) does not converge to zero as (1) does, therefore the finite time stability of the STA (2) is understood here as only x 1 (t) converges to zero in finite time, we say (2) is x 1 -finite time stable. Our main result is the following theorem:

Theorem 2 The system (2) is x 1 -finite time stable if k 2 > M + d M and g k 2 1 > k 2 + M + d M .
A slightly more restrictive condition is compactly stated as

1 2 g k 2 1 > k 2 > M + d M .
The perturbations g(t), h(t) and d(t) wider the scope of [START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF]. Both g(t), h(t) allow tackling more practical situations, where the system is affine with respect to the input and presents uncertain functions g(t) and h(t).

The perturbation d(t) has no real practical usage but is kept here for the sake of generality. The differentiability with bounded derivative is typically assumed for the investigation of the finite time convergence of the STA, e.g., in [START_REF] Shtessel | Sliding mode control and observation[END_REF][START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via lyapunov function designing[END_REF]. In [START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF] Theorem 3 and in [START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF] Section 3, g(t) = 1 but h(t) is not assumed to be bounded or differentiable, and finite time convergence is proven only under some technical additional assumption entailing that h(t) vanishes at the origin 1 . Theorem 2 could be extended to handle the perturbation η(t) |x 1 | used in [START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF] but details are not provided here for the sake of simplicity and clarity 2 . Dealing with unbounded uncertainties that are allowed to grow linearly with x 1 , typically satisfying |h(t)| ≤ α 1 +α 2 |x 1 |, increases the practical scope of the investigation. Up to the knowledge of the authors, it requires modifications of the STA, e.g., generalizing the command law [START_REF] Castillo | Supertwisting algorithm in presence of time and state dependent perturbations[END_REF]. Other results on output feedback sliding mode controllers also achieve handling unbounded perturbations, e.g., [START_REF] Hugo | Global adaptive hosm differentiators via monitoring functions and hybrid state-norm observers for output feedback[END_REF].

In Section 3 two consecutive changes of variables will transform ( 2) into ( 1) with a time-varying integral gain k 2 (t). The extension of stability properties to time variable gains is not trivial (see, e.g., Section 6 in [START_REF] Moreno | Lyapunov Approach for Analysis and Design of Second Order Sliding Mode Algorithms[END_REF]). The extension of Theorem 1 to time variable integral gain k 2 (t) is performed in Section 2. Finally, the conditions of Theorem 2 is compared to the classical conditions [START_REF] Shtessel | Sliding mode control and observation[END_REF][START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via lyapunov function designing[END_REF] in Section 4.

Extension to a time-varying integral gain

We consider here that the integral gain k 2 in system ( 1) is a time-varying gain k 2 (t), and we generalize Seeber and Horn's sufficient conditions to accept a time-varying integral gain k 2 (t). Its proof is an extension of Seeber and Horn's original proof: their argument is based on the (time-invariant) Lyapunov function,

V (x) =          2 x 2 2 + 3α 2 k 2 1 x 1 -x 2 if x ∈ Σ 1 2 x 2 2 -3α 2 k 2 1 x 1 + x 2 if x ∈ Σ 2 3 |x 2 | otherwise, (3)
1 The condition |h(t)| ≤ η(t) |x1| is used in [START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF] and the con-

dition |h(t)| ≤ η(t) |x1| + x 2 1 in [5]
. Both allow unbounded |h(t)| but entail h(t) converges to zero when x1(t) does. 2 The constraint in Theorem 2 would then be g k 2 1 > k2 + M + dM + ηM , where ηM would be a bound on η(t).

where Σ

1 = x 1 ≥ 0 ∩ x 2 ≤ αk 1 √ x 1 and Σ 2 = x 1 ≤ 0 ∩ x 2 ≥ αk 1 √ -x 1 .
The argument readily extends to a variable integral gain since k 2 does not influence the Lyapunov function, preserving the negativeness of the Lie derivative where it exists. However, their proof cannot be extended to time-varying gain k 1 (t) because (3) depends on k 1 .

We deal with the states where the locally Lipschitz Lyapunov function is not differentiable by using the nonsmooth version of Lyapunov's stability theorem (see, e.g., [START_REF] Cortes | Discontinuous dynamical systems[END_REF]). Let Lf V (x) denote the set-valued Lie derivative of V w.r.t. f evaluated at x, where f represents the system's vector field. Intuitively, at the points where V (x) does not exist Lf V (x) is the convex hull of the limit points of neighbor Lie derivatives. We recall that, in analogy with Lyapunov's stability theorem, one needs to investigate the negativeness of max Lf V (x) to assess the system's stability (see, e.g., [START_REF] Cortes | Discontinuous dynamical systems[END_REF] for details). Before stating Theorem 1, we introduce Lemma 1 which, when f is continuous, allows to assess the negativeness of Lf V (x) by considering only the values of V (x) where V (x) is differentiable (the proof of Lemma 1 is given in Appendix A).

Lemma 1

Consider an open subset N of the state space. We assume that f (t, x) is continuous w.r.t. x for all x ∈ N and all t, and that V (x) be locally Lipschitz. Let r(x) be a real valued function continuous on N . If, for all t, V (t, x) = ∇V (x) T f (t, x) ≤ r(x) on every point where V is differentiable on N , then, for all t, max Lf V (t, x) ≤ r(x) on N .

Remark 1 Lemma 1 allows for any function r(x) to be used, in practice r(x) = -or r(x) = -V (x) α , 0 < α < 1, can be used to assess finite time stability.

The following theorem extends Seeber and Horn's theorem to a time-varying integral gain. The assumption that k 2 (t) is strictly greater than |δ(t)| can be stated in two natural ways: first by using a lower bound k 2 ≤ k 2 (t) and enforcing k 2 ≥ L as in Theorem 1. Second, by enforcing the inequality k 2 (t) ≥ |δ(t)| + at each time, with an arbitrary > 0. The first condition is sensibly stronger than the second, and leads to a more restrictive finite time stability condition than Theorem 2 (with an additional factor g/g). This is why the second statement is used in the following time-varying integral gain version of Theorem 1: Proof. The same Lyapunov function (3) is used here, except for the choice of the parameter α < 1 that needs to account for in the statement: here one can choose α such that

Theorem 3 The system (1)
(αk 1 ) 2 ≥ k 2 (t) + |δ(t)| + 1 2 (4)
Wherever the Lyapunov function is differentiable, the upper bound on the Lie derivative computed in Eq. ( 14) in [START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF] still holds when k 2 varies in time, meaning that

V ≤ 3 max k 2 (t) -δ(t) -(αk 1 ) 2 , 1 2 α(α -1)k 2 1 . (5)
As in the original proof, Eq. ( 4) entails k 2 (t) -δ(t) -(αk 1 ) 2 ≤ -1 2 and 0 < α < 1 entails α(α -1) < 0, showing that the maximum of them is strictly negative. Now, we need to investigate the states where V is not differentiable. It is well known that no sliding motion can occur along the set S 1 = {x| x 1 = 0, x 2 = 0}, whether k 2 is fixed or time-varying. The other set where

V is not differentiable is S 2 = {x| x 2 = αk 1 |x 1 |, x 1 = 0}. Since system (1) is continuous on N := R 2 \S 1 , we can apply Lemma 1 with r(x) = 3 max -1 2 , 1 2 α(α -1)k 2 1
to show that the same strictly negative upper bound that holds when V is differentiable also holds on the whole set N , which includes the set S 2 . 2

3 Extension to practical uncertainties

Centered system

The STA (2) aims x 1 (t) = 0 and ẋ1 (t) = 0, the integral correction x 2 (t) being foreseen not to converge. Indeed, the value of x 2 (t) enforced at steady state can be computed with respect to uncertainties: if x 1 (t) = 0 and ẋ1 (t) = 0 then x 2 (t) = -h(t)/g(t). Classically, by defining the system y 1 (t) = x 1 (t) and y 2 (t) = x 2 (t) + h(t)/g(t), we obtain a system that is expected to converge to zero. It satisfies the following ODE:

ẏ1 (t) = g(t) -k 1 φ(y 1 (t)) + y 2 (t) (6a) ẏ2 (t) = -k 2 sign y 1 (t) + d(t) + γ(t), (6b)
with

γ(t) = d dt h(t) g(t) = ḣ(t) g(t) - ġ(t) h(t) g(t) 2 . ( 7
)
The change of variable x → y is a diffeomorphism for all times, therefore Theorem 2 of [3, p.99] proves that Filippov solutions x(t) map to Filippov solutions y(t) and vice-versa. Since y 1 (t) = x 1 (t) the y 1 -finite time stability of the ( 6) is equivalent to the x 1 -finite time stability of (2).

Time shifted system

We introduce a rescaling of time z(t) = y(τ (t)), that is going to move the impact of the perturbation g(t) from (6a) to (6b), so as to match the perturbation structure of (1). We define τ (t) by the following ODE:

τ (t) = 1 g(τ (t)) , (8)
with τ (0) = 0. Since g(t) ∈ [g, g] is bounded away from zero by g > 0 and is differentiable with bounded derivative ġ(t) ∈ [-G, G], the function 1/g(τ ) is continuously differentiable with bounded derivative. As a consequence the ODE (8) has a unique solution, which is globally defined in R. Furthermore, from [START_REF] Seeber | A novel method to estimate the reaching time of the super-twisting algorithm[END_REF] and the bounds on g(t) we obtain 1/g ≤ τ (t) ≤ 1/g, which proves that τ (t) is strictly increasing and satisfies,

t g ≤ τ (t) ≤ t g . (9)
This makes τ (t) a valid rescale of time. The time-rescaled system z(t) = y(τ (t)) satisfies the following relation:

ż(t) = τ (t) ẏ(τ (t)) = 1 g(τ (t)) ẏ(τ (t)), (10)
obtained using the chain rule and (8). Finally using ( 6) we obtain the ODE governing z(t):

ż1 (t) = -k 1 φ(z 1 (t)) + z 2 (t) (11a) ż2 (t) = - k 2 g(t) sign z 1 (t) + d(t) + γ(t) g(t) , (11b)
where g(t) = g(τ (t)), γ(t) = γ(τ (t)) and d(t) = d(τ (t)).

The system (11) is in the scope of Theorem 3 with

k 2 (t) = k 2 g(t) and δ(t) = d(t) + γ(t) g(t) . (12)
Its z 1 -finite time stability is equivalent to y 1 -finite time stability of (6) because of [START_REF] Seeber | Stability proof for a wellestablished super-twisting parameter setting[END_REF]. The bounds

g(t) ∈ [g, g], γ(t) ∈ [-M, M ] and d(t) ∈ [-d M , d M ]
hold for their time-shifted counterparts g(t), γ(t) and d(t).

Proof of Theorem 2

Suppose the conditions

k 2 > M + d M and g k 2 1 > k 2 + M +d M of Theorem 2 hold. Since γ(t) ∈ [-M, M ], g(t) > g and d(t) ∈ [-d M , d M ], there exists > 0 such that both k 2 ≥ |γ(t) + d(t)| + and g(t) k 2 1 ≥ k 2 + |γ(t) + d(t)
| + hold for all times. Dividing both inequalities by g(t) < g, we prove that

k 2 (t) ≥ |δ(t)| + g and k 2 1 ≥ k 2 (t) + |δ(t)| + g , (13)
where k 2 (t) and δ(t) are defined in (12). This allows applying Theorem 3 to the STA (11) with (12), hence proving its z 1 -finite time stability.

Comparison to state-of-the-art conditions

Moreno [START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF] handles the case g(t) = 1 and h(t) bounded with no differentiability assumption, but does not prove x 1 -finite time stability in this case (he does handle successfully more structured perturbations proportional to |x 1 | + x 2 1 ). Castillo, Fridman and Moreno [START_REF] Castillo | Supertwisting algorithm in presence of time and state dependent perturbations[END_REF] handle more general perturbations than here, with a generalized STA (GSTA). However, the finite time stability conditions for the GSTA do not apply to the STA 3 .

In order to handle the same differentiable perturbations g(t) and h(t) as in (2), Shtessel, Edwards, Fridman and Levant [START_REF] Shtessel | Sliding mode control and observation[END_REF] have modified the STA by adding a switching law to keep u bounded in [-U m , U m ]. They proved some sufficient conditions on k 1 , k 2 and U m for the x 1 -finite time stability, see [10, p.156]. After some algebraic manipulations 4 , one can prove that these conditions imply

k 2 > M and g k 2 1 > 2 g g 2 (k 2 + M ), ( 14
)
where M = H/g + G /g 2 uses the bounds defined in Section 1. The first bound on k 2 is the same as in Theorem 2. However, the second bound on k 2 1 is seen to be similar but more restrictive that the one in Theorem 2 because of the factor 2(g/g) 2 . In particular, if the interval [g, g] is large and/or g is close to zero then (g/g) 2 is sensibly greater than 1 and the conditions of [START_REF] Shtessel | Sliding mode control and observation[END_REF] become sensibly more restrictive than Theorem 2. In addition to allowing for sensibly lower gains, the new condition of Theorem 2 does not require the switching law to keep u ∈ [-U m , U m ], which simplifies the usage of the STA.

Polyakov and Poznyak in [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via lyapunov function designing[END_REF] proved that the conditions 32H < k 2 1 < 8(k 2 -H) entails the x 1 -finite time convergence of the STA (2) in the restricted case g(t) = 1. In this restricted case, M = H holds and our conditions become 2H < k 2 + H < k 2 1 . None of these two conditions imply the other: gain values are accepted by Polyakov and Poznyak's condition and rejected by Theorem 2 if and only if 32H < k 2 1 ≤ k 2 + H (the black boundary area in Figure 1). Theorem 2 accepts pairs of small gains greater but arbitrarily close to the limit pair of gains k 1 = √ 2H and k 2 = H (see the lower-left point 3 The STA is obtained from the GSTA in [START_REF] Castillo | Supertwisting algorithm in presence of time and state dependent perturbations[END_REF] by setting β to zero, but this is incompatible with the conditions of Theorem 1 [1, p. 5 Eq. ( 15)] because they involve a factor 1/β. For strictly positive β, their conditions also contain a factor 1/(km -kM ), which is not applicable in the case where g(t) is constant, i.e., km = kM . 4 Among other basic manipulations, setting q and Um to the best cases q ∈ {0, 1} and Um = C in order to weaken the conditions, and using the fact that 0 < gk2 -C ≤ gk2 + C with C = H + G /g = g M being the smallest upper bound to | ḣ(t) + ġ(t) u(t)| subject to |u(t)| ≤ Um that can be obtained using bounds on the perturbations given in Section 1. in Figure 1), while pairs of gains accepted by Polyakov and Poznyak must satisfy k 1 ≥ √ 32H and k 2 ≥ 5H (see the upper-right point in Figure 1), so the new conditions of Theorem 2 accept sensibly lower gains, the greater H the more sensitive the improvement. Both conditions are therefore seen to be complementary. In particular, they prove together that the STA ( 2) is x 1 -finite time stable for all positive gains in the case where g(t) = 1 and ḣ(t) = 0, which was not proved previously up to the authors knowledge. The new condition of Theorem 2 are valuable because they allow for sensibly smaller gains.

1 2

 1 sign x, subject to the absolutely continuous bounded perturbation δ(t) ∈ [-L, L]. Here and Corresponding author A. Colotti.

Figure 1 .

 1 Figure 1. Gains (k1, k2) satisfying the conditions of Theorem 2 in the textured area, and satisfying the conditions of Polyakov and Poznyak [6] in the gray area. The graphic displays the numerical value H = 1.5.

  with time-varying integral gain k 2 (t) is finite time stable if there exists > 0 such that k 2 (t) ≥ |δ(t)| + and k 2 1 ≥ k 2 (t) + |δ(t)| + holds for all times.

Acknowledgements

This work was partially supported by Région Pays de la Loire as part of the O2GRACE project and by the French Agence National de la Recherche through the grants ANR-16-CE33-0024 and ANR-18-CE33-0011.

A Proof of Lemma 1

Let t be an arbitrary time instant and let Ω V be the set of non-differentiability points of V inside N . On N \Ω V , Lf V (t, x) = { V (t, x)}, so the lemma is trivially true. For

x ∈ Ω V , by the continuity of f we have that Lf V (t, x) consists of the convex combinations of limit points of V (t, x i ) at neighboring points x i where V is differentiable (this is a direct consequence of the definition of Lf V and a few of its basic properties which can be found, e.g., in [START_REF] Cortes | Discontinuous dynamical systems[END_REF]). Since Lf V (t, x) is a compact interval of real numbers, its max is a vertex of the one dimensional convex hull, meaning that there exists a converging sequence x i → x such that V (t, x i ) → max Lf V (t, x). But V (t, x i ) ≤ r(x i ) for every point in the sequence, meaning that, by continuity of r, we have max Lf V (t, x) ≤ r(x), which holds for all times since t is arbitrary. 2


Laura Reyes

Clémence Nikitine

Léa Vilcocq

Pascal Fongarland

Autocatalyzed and heterogeneously catalyzed esterification kinetics of glycolic acid with ethanol

The chemical equilibrium and reaction kinetics in homogeneously and heterogeneously catalyzed systems were investigated for the reversible esterification reaction of glycolic acid with ethanol. In a first approach, the thermodynamic equilibrium of the reaction was established for temperatures between 50°C and 75°C. To differentiate catalytic from homogeneous contribution, the esterification was first studied without catalyst and data obtained were then used to model the autocatalyzed-homogeneous reaction kinetics. The impact of different commercial solid catalysts such as zeolites, oxides and ion-exchange resins over the esterification reaction was also investigated. For the latter, the amount of total acid sites was quantified in order to correlate this parameter to the reaction rate and glycolic acid conversion. Finally, the effects of catalyst loading, temperature, presence of water and recyclability were examined for the most efficient catalyst, Amberlyst 70. Moreover, a kinetic adjustment of the experimental results was carried out by fitting the data to a pseudo-homogeneous model and comparing to adsorption-based models.

Introduction

The esterification of carboxylic acids has been widely studied as it is a highly appealing reaction for the chemical industry. Beyond glycolic acid production, derived esters also find direct applications. The industrial application of glycolates is wide, ranging from solvents, intermediates for the production of fine chemicals, biodegradable polymers, additives for fuels, beauty products and pharmaceuticals. [1][2][3] The ester production processes such as hydrogenation or partial oxidation generally require catalysts based on precious metals, high temperature and/or pressure. 4 Generally, the production of esters derived from glycolic, catalyzed or not, are petroleum-based. 5 Nowadays, with the transition of the industry to long-term and more sustainable processes, the use of renewable materials as feedstock has been encouraged. [START_REF] Ramaswamy | Separation and Purification Technologies in Biorefineries[END_REF] Therefore, the development of more efficient, greener and practical ester production processes remains a significant challenge in the modern chemical industry. Under this framework it has even been suggested that esters of carboxylic acids and hydroxy acids with short carbon chain can be recovered industrially from biomass wastes such as black liquor. [START_REF] Reyes | [END_REF] At large scale, esterification reactions are usually catalyzed by strong mineral acids (H2SO4, HCl, and H3PO4). Still, these homogeneous catalysts have been related to problems such as waste toxicity, corrosion and malfunctions in the industrial set ups. 8 Therefore, there is a need for heterogeneous acid catalysts capable to compete in activity with the traditional homogenous ones. Moreover, with the shift of the chemical industry to greener processes the use of heterogeneous catalyst is privileged. 9,10 The employment of heterogeneous catalysts represents some benefits as providing an ease in the separation of products and catalyst, avoiding laborious separations and recovery processes. 10 Among the most reviewed solid catalysts for the esterification of carboxylic acids are polymer resins, [11][12][13][14][15][16] zeolites [17][18][19][20] and oxides. [21][22][23] The esterification of glycolic acid, as well as other carboxylic acids, can be catalyzed by the acid itself. Autocatalysis can be due to undissociated acid molecules of the acid as well as the proton resulting from its dissociation. 24 In order to better describe the kinetics of these reactions, a term that accounts for the autocatalysis phenomenon was added to the homogeneous model traditionally employed. 24 For glycolic acid, the use of this so-called homogeneous-autocatalyzed model was not found in the literature. However, it has already been described for short chain carboxylic acid esterification reactions, primarily in studies of acetic and lactic acids. 13,[24][25][26][27] Studies on the heterogeneously catalyzed esterification of glycolic acid mainly report the use of ion exchange resins, 4,28,29 heteropolyacids, 30 and super acids (TiO2/SO4 2-, SO4 2-/ZrO2, SO4 2-/ZrO2/La 3+ ). [31][32][33] The use of enzymatic catalyst has been also investigated for this reaction. 3 Concerning the alcohol involved in the reaction, research has mainly been focused short chain alcohols (C1-C4) as the case for most of others carboxylic acids esterification. 4,28,[30][31][32]34,35 Glycolic acid esterification with fatty alcohols ranging from C8-C16 has also been studied. 3 It is important to highlight that even though the employment of an acid catalyst can increase the rate of esterification, it does not determine its end point due to the fact that this type of reaction is governed by thermodynamic equilibrium. 36 Usually, the conception of separation and recovery process requires, among other things, information on the reaction kinetics to correctly model the reaction-separation unit. 37 Several studies have been addressed on the reaction kinetic models in presence of solid catalysts. The reported studies focus on not only pseudo homogeneous models but also adsorption models such as Langmuir-Hinshelwood and Eley-Rideal. 14,[38][39][40][41][42] This study focuses on the esterification reaction of glycolic acid with ethanol as a reaction model (Scheme 1). Preliminary tests were performed in order to determine the nature of the autocatalytic activity and to obtain the optimum operational parameters, which were later employed in the catalytic screening. A variety of zeolites, resins and oxides were selected as heterogeneous acid catalysts for the catalytic test. Following the screening study, for the best performing catalyst, the influences of water, temperature and catalyst loading were investigated. Finally, kinetic parameters were obtained by fitting the experimental data to the most commonly employed kinetic models.

Experimental Procedure

The study on the autocatalytic esterification of glycolic acid with ethanol, as well as the heterogeneous catalytic tests, were carried out in a stainless steel autoclave (120 mL) operating in batch mode. The temperature inside the reactor was controlled within ±0.5 °C. Solid catalysts were suspended in the reaction mixture by mechanical stirring. A rubber seal allowed sampling by means of a syringe and needle, thus granting to work in a closed system. All reactants were weighed and then loaded into the reactor, when working with heterogeneous catalysts, the latter was charged directly with the reactant solution as the system worked in batch mode. The zero time for a given run was established as the time at which the reaction medium reached the set temperature.

Analytical Methods

Each sample was filtered with hydrophobic syringe filters (0.45 µm), diluted (1g sample/10 g ACN) in a pre-cooled vial containing acetonitrile (ACN) and kept in the freezer until analysis. The analysis of the compounds (glycolic acid, ethanol, and ethyl glycolate) was carried out by HPLC Shimadzu with a Phenomenex Kinetex XB-C18 column (250x4.6 mm), equipped with a refractive index and a UV detector. The mobile phase employed was acidified water (0.0025 N H2SO4 and acetonitrile in a 90/10 (v/v %) proportion. Glycolic acid and glycolate ester are detectable in UV at a wavelength of 210 nm whereas ethanol was analyzed by refractive index. Solutions of known concentrations were used as external standards. The correlation of the concentration to the peaks area allowed to determine the concentration of each compound (ESI).The concentrations obtained by HPLC analysis were used to calculate the evolution of the molar concentration with reaction time. From the data obtained, selectivity and yield were calculated using the equation shown below.

𝐴𝑐𝑖𝑑 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 (%) =

* 100 (1)

𝑌𝑖𝑒𝑙𝑑 [%] = * 100 (2)
Where 𝑛 and 𝑛 represent the number moles of acid at t=0 and at the corresponding sampling time respectively, and 𝑛 corresponds to the number of moles of ethyl glycolate produced. Due to the lack of an instrument capable of determining the amount of water such as Karl Fischer, the mass balance could not be determined. However, to corroborate the reliability of the analysis method, a carbon balance was formulated as illustrated by equation 3. Additionally, it has been shown that the experiments are repeatable with a margin of error of ±5% as illustrated in the ESI.

𝐶𝑎𝑟𝑏𝑜𝑛 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 (%) =

Titration of Acid Catalytic Sites

The potentiometric titration of the solid catalyst was carried out with a compact titrator Mettler Toledo G20S. The protocol followed for this measurement was the same for all the catalysts, 250 mg of fresh catalyst, previously dried at 100° C for one hour, was suspended by stirring in 50 mL of a 0.1 M NaCl solution and 0.1 M NaOH solution was used as titrant. The Gran plot method was applied to determine equivalence points corresponding to strong, medium, weak and total acidities, as described by K. Yu et al,. 43

Results and discussion

Autocatalyzed reaction

The first steps aimed to optimize the parameters for the catalytic study on the esterification of glycolic acid with ethanol as well as to get an insight into the autocatalyzed nature of this reaction. The influences of temperature, acid/alcohol ratio and the presence of water in the initial reaction mixture were investigated in the absence of an external catalyst. Table 1 summarizes the experiments carried out for the parametric study. In studies on the esterification reactions of glycolic acid, the formation of by-products has been reported, with glycolide being the most common one. Furthermore, the carbon balance determined using equation 3, meant the formation of by-products could be ruled out.

Influence of the temperature

The temperature was varied between 50-75°C. Tests at higher reaction temperatures were excluded due to possible degradation of the glycolic acid, and to avoid ethanol stripping.

In addition, reactions below 50°C slowed down to the point where hardly any conversion was observed. Figure 1 compares the glycolic acid conversion at four different temperatures. Sun et al. 45 demonstrated that an esterification reaction with ethanol working close to its boiling point of the ethanol (78°C at atmospheric pressure) can lead to a higher conversion. However, past this point the conversion declines because the vaporized ethanol filling the head space of the reactor is no longer available for the reaction. 45 Since the initial reaction rate is directly correlated to the temperature, it is logical to observe that when the temperature decreases, the reaction slows down.

It was also noted that even at 75 °C, equilibrium was not obtained after 6 hours of reaction.

Influence of the Acid/Alcohol ratio

The stoichiometry of the glycolic acid esterification requires an equimolar ratio between the alcohol and the acid. However, due to the low solubility of glycolic acid in ethanol at molar equivalences of 1 to 3, the selected molar ratios for the study of the impact of the acid to ethanol ratio were 1/3, 1/4 and 1/6. Figure 2 compares these three molar ratios for the glycolic acid conversion. At first glance, the ratio between the two reactants has little influence on the reaction speed within the first hour of reaction for the ratios 1/3 and 1/4. However, regarding the more diluted system a clear trend is marked. At stronger acid concentrations the reaction is favored while in dilute systems the reaction rate decreases.

The initial reaction rate was determined for all experiments by plotting the glycolic acid concentration against time and performing a linear regression of the data on the early part of the curve. Figure 3 illustrates the influence of the acid/alcohol molar ratio on the initial reaction rate, where a direct relationship between the initial reaction rate and the acid concentration was found. This trend can be explained by the nature of the reaction. It is known that esterification reactions can be catalyzed by acids, hence a higher concentration of acid should favor the reaction due to a higher concentration of protons in the medium. This behavior has already been described in literature, as for example in the kinetics study of the esterification of acetic acid with isobutanol by M.R Altıokka and A. Çıtak. 39 The authors concluded that in these reactions, the relationship between the initial reaction rate and the alcohol concentration is linear at low alcohol concentrations, and the two variables become of each other at high alcohol concentration.

Influence of the water content

The ffect of adding water to the initial reaction medium was studied. Water was added to the initial mixture with a water/ethanol molar ratio of 1 to 3, which corresponded approximately to 10 % wt. Figure 4 illustrates the impact of this on the reaction rate and conversion. The addition of water to the initial reaction medium strongly affects the reaction by decreasing the conversion and slowing down the reaction rate.

As the reaction has not reached the equilibrium, the inhibitory effect of the water presence in the departure solution may be attributed to dilution of the system. As previously mentioned, a high concentration of acid in the solution favors the conversion, therefore it is as expected that a drop in the glycolic acid conversion is observed due to the dilution of the catalyst, glycolic acid itself, in the reaction medium. After the preliminary study of the operational parameters, it was concluded that the optimal temperature for the reaction is 70°C. Even though at 75°C slightly better conversion was achieved, working so close to the boiling point of the ethanol might cause ethanol volatilization from the liquid phase reaction. Additionally, it was found that there is a direct linear correlation between the acid concentration in the medium and the reaction rate. Finally, it was also determined that presence of water in the initial reaction mixture has a great influence on the reaction whereby the rate is slowed down significantly.

Kinetic model Thermodynamic equilibrium

Little information is found in literature on the thermodynamic equilibrium in the esterification of glycolic acid with ethanol. In the absence of a literature reference value, the concentration equilibrium constant was calculated using the equilibrium compositions determined in experiments on a homogeneous system, catalyzed by H2SO4, at different temperatures (ESI).

To account for the non-ideal mixing behavior of the bulk liquid phase, the activity of the components(𝐾 ) was used instead of the concentration of the components (Kx). Considering that molecules with different sizes and molecular interactions are found in the solution, the mixture cannot be considered as an ideal system and so a model capable of representing these nonidealities is required. In strongly non-ideal systems at low pressure (P < 10 bar) it is standard procedure to use an activity coefficient model to describe behavior far from ideality, such as UNIFAC, UNIQUAC or NRTL. [START_REF] De Hemptinne | Select Thermodynamic Models for Process Simulation -A Practical Guide using a Three Steps Methodology[END_REF] Correlative models such as NRTL or UNIQUAC are preferred if the experimental parameters are known. Otherwise, in the absence of experimental data, as is the case here, UNIFAC can be employed as a predictive model. [START_REF] Gmehling | [END_REF] For a non-ideal liquid phase reaction, the chemical equilibrium can be described by the expression:

𝐾 (𝑇) = ∏ 𝛾 𝑥 (4)
In which 𝑣 represents the stoichiometric coefficient, 𝛾 the activity coefficient and 𝑥 the molar fraction at the equilibrium for each component 𝑖 in the mixture. This can be translated as the product of the ratio of the molar fractions of the products over the reactants, by the ratio of the activity coefficients of products over reagents, for a reaction in general. The equation below represents this expression fitted to the studied reaction.

𝐾 = 𝐾 𝐾 = ( ) ( ) (5)
In which 𝑥 , 𝑥 and 𝛾 , 𝛾 represent the products (ethyl glycolate and water) concentrations and activity coefficients calculated at equilibrium, and 𝑥 , 𝑥 and 𝛾 , 𝛾 represent the reactants (glycolic acid and ethanol) concentrations and activity coefficients at equilibrium. The activity coefficients of glycolic acid, ethanol, ethyl glycolate and water were estimated by using the UNIFAC model, using the equilibrium compositions for a temperature range of 50°C-75°C. The obtained values were used to plot a Van't Hoff diagram, shown in Figure 5, from which it was possible to extract the information on the equilibrium constant as a function of temperature. Equation 6represents the correlation obtained from the regression of the experimental data on activity coefficients. 𝐿𝑛 𝐾 = 2.7 -

The Van't Hoff diagram shows that the slopes of the curves present a similar trend. Still, the Keq and Kx values differ by a factor around 1.5-1.6, confirming that the behavior of the system is far from ideal within this range of temperatures. Moreover, using the regression extracted from the Van't Hoff plot, it is also possible to estimate the values of the entropy and enthalpy of the chemical reaction as it is shown by the correlation:

𝐿𝑛 𝐾 = - ∆ ° + ∆ ° (7)
Where, ∆ 𝐻° and ∆ 𝑆° represent the standard enthalpy and entropy of the reaction and 𝑅 the universal gas constant.

The calculated values for the enthalpy and entropy of reaction were 4.3 kJ.mol -1 and -22.5 J.mol -1 .K -1 respectively. The reaction enthalpy value as well as the behavior of the correlation curve of the equilibrium constant with temperature suggest that the reaction can be considered slightly endothermic. Usually, esterification is considered as an athermic or even slightly endothermic reaction. 48 Regarding the esterification of glycolic acid with methanol, values for the reaction enthalpy of approximately 15 kJ.mol -1 have been reported in studies which also used UNIFAC to correct the non-ideality of the system. 4,28 Additionally, in the kinetic study of the esterification of glycolic acid with propanol, butanol and octanol, J. Aparicio 35 reported values of 30.73 kJ.mol -1 , 23.76 kJ.mol -1 and 16.36 kJ.mol -1 respectively. However, in this case a different thermodynamic model was employed (NRTL). Thus, predictive and correlative thermodynamic models presents similar trends agree with the slightly endothermic values for the enthalpy of reaction in the esterification of glycolic acid. Despite the enthalpy value obtained in this study being low, similar observation have been reported for the esterification of other carboxylic acids with ethanol when comparing to references or theoretical values. 15,40,49 C. Pereira et al., 15 found a reaction enthalpy of around 4.28 kJ.mol -1 in the esterification of lactic acid with ethanol. Moreover, A. Orjuela 40,49 found that the slope for the equilibrium constant of succinic acid esterification with ethanol to be close to zero, showing little dependence on the temperature and so assuming heat of reaction as zero.

Parameter fitting

The experimental results of the parametrical study were used to form composition versus time profiles that were fitted to the kinetic equation. Usually, esterification reactions studied without the presence of a heterogeneous catalyst are modeled as homogeneous reactions whether or not they are catalyzed by another compound other than the main acid. This is represented by the equation below:

𝑟 = 𝑘 𝑎 𝑎 - (8)
In which 𝑘 stands for: 𝑘 𝑒𝑥𝑝 -, where 𝑘 and 𝐸 represent the pre-exponential factor and the activation energy and R the universal gas constant, 𝑎 , 𝑎 , 𝑎 , 𝑎 represents the activity of the reagents (ethanol and glycolic acid) and products (ethyl glycolate and water) respectively and 𝐾 accounts for the equilibrium constant. However, it is known that the esterification of carboxylic acids can be catalyzed by undissociated molecules of the acid and/or by the proton resulting from its dissociation. 24 Pöpken et al. 25 developed a kinetic model for this type of reaction, in which the autocatalytic factor is taken into account.

𝑘 𝑎 𝑎 𝑎 - (9)
Where 𝑎 , 𝑎 , 𝑎 and 𝑎 are the activities for the carboxylic acid, alcohol, ester and water respectively. The application of this model to short chain carboxylic acid esterification reactions has been already studied mainly focused in acetic and lactic acids. 13,[24][25][26][27] The difference between this model and the traditional homogeneous model is based on the addition of the term 𝑎 , which accounts for the activity of the protons dissociated from the main acid (glycolic acid). 25 As mentioned above, this serves as a catalyst both for the straightforward reaction of the esterification as well as for the hydrolysis of the ester formed. The alpha (𝑎) parameter can take two values depending on whether the reaction occurs via a dissociated acid or not. If it is assumed that catalysis occurs via an undissociated acid, giving rise to a trimolecular reaction where two acid molecules are present, with one acting as a reactant and the other as a catalyst, the value for this constant becomes 1. 25 Alternatively, if the reaction occurs via the dissociation of the acid, this parameter takes the value of 0.5 25 The employment of the ordinary differential equations solver ode45 in MATLAB ® enabled the numerical integration of the mass balance equations in a batch reactor that describes the kinetic model using the ordinary differential equations by the fourth-order Runge-Kutta method. Parameter estimation for the different models was carried out by the minimization of the sum of residual squares (SRS) between the calculated and experimental values as presented in equation 10.

𝑆𝑅𝑆 = ∑ ∑ 𝑥 -𝑥 (10)
With 𝑁𝑐 as number of compounds, 𝑁𝑠 as number of samples per experiments, 𝑛 as the number of experiments and 𝑥 , 𝑥 as the experimental and predicted molar concentrations. For the estimation of the two parameters (activation energy and pre-exponential factor), the concentration profiles of the 7 experiments in Table 1, which accounts for 12 individual concentrations of glycolic acid, ethanol and ester were employed, thus guaranteeing sufficient independent data to perform the error minimization and parameter regression. In order to differentiate between the three models, the experimental data was fitted to each one of the models in Table 2. Additionally, for the optimized parameters the absolute error and mean average relative residual were using equations 11 and 12. The results of the parameter estimation are displayed in Table 3.

𝐸𝑟𝑟 = ∑ ∑ 𝑥 -𝑥 (11)
𝑀𝐴𝑅𝑅 (%) = 100 * * ∑ ∑ (12)
Moreover, a statistical analysis was performed to estimate the correlation coefficients between pairs of determined parameters and their corresponding p-values (95 % confidence intervals). The optimization was performed several times starting from different initial values to avoid local minima. The correlation coefficients were estimated from the covariance calculated with the Jacobian Matrix.

Considering the estimation of the activation energy, the choice of model has little impact on the value regressed, as all the models are in agreement with values between 46-48 kJ.mol -1 . This is not the case for the value of the pre-exponential factor, as it is logical, due to the structure of the kinetic model. Little attention has been paid to the autocatalytic element of esterification with carboxylic acids; most of the reported literature is focused on studies of the kinetics of the reaction catalyzed by external sources different to the acid itself. Moreover, for the esterification of glycolic acid with ethanol no reference was found. Still, the estimated value is in agreement with the energy of activation reported in the literature for short chain carboxylic acids such as acetic acid and also for alpha hydroxy carboxylic acids, such as lactic, acid ranging between 42-63 kJ.mol -1 . 13,[24][25][26][27] As can be seen in Table 3 the experimental kinetics correspond better to the models that take into account the catalytic properties of the glycolic acid itself. Moreover, they are better reproduced when alpha takes the value 1 rather than 0.5, suggesting that the reaction is more likely catalyzed by molecular glycolic acid. However, due to the small difference between both autocatalytic models, it is difficult to differentiate between them, leading to the alternative where the acid is partially dissociated. It could also be considered as a valid and consistent hypothesis supported by the corresponding low pKa value for this acid (3.83). Similarly for the case of lactic acid (pKa 3.8), the obtained error values for the two homogeneousautocatalytic models lead to a narrow deviation between them, making difficult to distinguish. P. Delgado et al. 13 reported that for the esterification of lactic acid with ethanol, the autocatalysis occurs via dissociated molecules, whereas M. T. Sanz et al. 26 reported that the autocatalysis is more likely due to the interaction of the molecular acid in the reaction of lactic acid with methanol.

A parity plot describing this adjustment for the molecular catalyzed reaction is shown in Figure 6A. Calculated and predicted concentrations are illustrated in Figure 6B, where it is also possible to appreciate that the concentration profiles correspond with good precision to those obtained through experiments.

(A)

Heterogeneous catalyzed reaction

Catalysts screening

For this study a series of heterogeneous acid catalysts were selected to be tested in the glycolic acid esterification with ethanol. Following the results of the initial parametric study the temperature and acid/alcohol ratio were fixed at 70°C and 1/4 respectively. External diffusion limitations were discarded for all catalysts and for the best performant one, the kinetic regime was ensured (see Table 4 and Figure 16).

Zeolites

Zeolites can be employed in a broad number of applications such as molecular sieving, purification of effluents by adsorption, ionic exchange and catalysis. 50 However in the context of this investigation the zeolites were selected for their qualities as acid catalysts required for esterification reactions. Three different commercial zeolite structures were studied: Faujasite (FAU), Penstasil (MFI) and Mordenite (MOR). In addition to the diversity of the structures, different Si/Al ratios were tested. Figure 7 compares the progress of glycolic acid conversion between different commercial zeolites. In general, zeolite structure seems to have an impact on the conversion of glycolic acid. Comparing the FAU, MOR and MFI (40:1, 10:1 and 40:1, respectively) with acidity in a similar range, the FAU structure presented the highest glycolic acid conversion. Mordenite and MFI (10:1 and 40:1, respectively) showed a similar behavior, with MFI displaying a slightly better catalytic performance. Furthermore, for two FAU structures with different Si/Al ratio and acidity values, the resulting conversion differs. In this case, it was observed that the lower the Si/Al ratio, lower the catalytic activity and thus glycolic acid conversion. It is important to mention that the FAU 5.1/1 zeolite presented a lower performance compared to the uncatalyzed reaction under the same operating conditions. A drop of approximately 7 % in conversion compared to the benchmark suggests an inhibitory rather than a reaction-promoting behavior by this zeolite, suggesting that not only the structure but also the alumina content has an impact on the catalytic activity of zeolites in the esterification reaction of glycolic acid. The differences in the performances of the different structures can be explained by the relationship between the average pore diameter and the yield. Maheria et al. 51 stated that bulkier acid molecules can suffer from diffusional and geometrical limitations with respect to the pore size and geometry affect significantly the levulinic acid conversion. 51 Furthermore Prinsen et al., 52 suggested that hydrophobicity/hydrophilicity and pore size éare also relevant factors in esterification reactions rather that only the acidity. 52 The influence of the structure, Si/Al ratio and the acidity on the catalytic properties of zeolites is not straightforward. The rate and efficiency of proton transfer in zeolites, as discussed previously, depends on a variety of factors linked to the ease of the molecules to diffuse or transport through narrow confined cavities and closed channels in the zeolites to the acid sites, and the concentration of active sites and their enclosed geometry.

Glycolic acid conversion [%] Time [min]

Ion-exchange resins

The choice of the resins tested was based on their chemical composition and morphological structure properly adapted to the model reaction. These polymeric materials present different morphological characteristics such as surface area, average pore diameter and even physical shape, but also differ in catalytic properties as the concentration of acids sites. The latter one is directly related to the availability of acid sites for either per volume or mass of catalyst. Figure 8 compares the evolution of glycolic acid conversion for different commercial ion exchange resins as catalysts. All ion exchange resins tested presented a significant catalytic activity in the esterification of glycolic acid. The Amberlyst series performed better than to Dowex and Nafion polymers. When using Amberlyst 70 the equilibrium was reached faster than with Amberlyst 15 and 36, which presented similar reaction times to reach equilibrium. These experimental results are in agreement with the work described by Orjuela et al. 40 which stated that the catalytic activity of Amberlyst 70 may over perform Amberlyst 15 in esterification reactions even though if the former one exhibits lower pore volume and surface area. It was also found that Amberlyst 70 is a more effective catalyst in the esterification reaction than Amberlyst 36, consistent with the results obtained by Hoşgün & Çitak 53 in the kinetic study of propionic acid with isobutyl alcohol. Despite the fact that Amberlyst catalyst and Dowex have different surface areas and pore diameters, the four catalyst presented similar final conversions. However, the faster reaction rate presented by Amberlyst 70 may be related to the fact that this resin has a lower degree of polymerization in comparison to the other macro reticular polymers, which could play a role in the accessibility and transport of reactive molecules to the active sites. Moreover, the performance of the Nafion catalyst being below the average presented by the other resins tested is probably linked to its low ionic exchange capacity.

Titanium oxide

Along with zeolites and resins, mixed and metal oxides are the most commonly employed solid acids. 54 Figure 9 shows the conversion of glycolic acid with and without the presence of TiO2 as catalysts. When comparing TiO2 the reaction with the benchmark (autocatalyzed reaction) at same operational conditions, it is clear that the reaction is promoted by the acidic properties of the titanium oxide. Titanium oxide displayed a significant activity in the reaction with a conversion value around 57 % at the end of 6 hours, approximately 20 % higher than that of the benchmark reaction. Moreover, the titanium oxide presented a better performance as an acid catalyst in the glycolic acid esterification with ethanol than all the zeolites tested in this study.

Comparative study and selection of catalysts

To further establish the parameters that favor the production of ethyl glycolate, the acid-base titration of all the commercial catalysts employed in this study was carried out. This allowed to differentiate and quantify the different acid sites: strong, medium and weak, that were present in each solid. The results of the titration are presented in Figure 10.The quantification of acid sites by potentiometric titration is a straightforward procedure that consists of the direct titration of an aqueous suspension of the sample of powdered solid with a dilute base to a neutral endpoint. 55 The disadvantage of this technique is related to the imprecise representation of the phenomena occurring on the solid surface during its use as an acid catalyst, which might be radically different in a water suspension. Water can also drastically alter the structural properties of the solid, causing decantation, structural collapse and swelling. In general, all the catalysts tested in the catalytic screening showed catalytic activity in the esterification reaction. Resins presented the highest conversion, followed by titanium oxide and finally the group of zeolites. The common property of these catalysts is their acidic characteristics. Figure 11 displays the relation between the initial reaction rate and total acid site concentration of the catalyst.

The illustration gives a clear trend among the different families of solid catalysts used in this study. It also suggests that the primary influencing factor of the heterogeneously catalyzed glycolic acid esterification with ethanol is the total amount of acid sites. However, it is necessary to emphasize that this is valid as long as these sites are accessible to the reagents. The reactive molecules contrast in molecular sizes, where glycolic acid is a much more bulky molecule than ethanol. Catalysts such as resins and oxides have a larger pores, making the active sites more accessible. On the contrary, catalytic materials such as zeolites have smaller pore sizes in addition to the complex structures of channels and cages in which bulkier molecules can be trapped before arriving to the active site, thus decreasing their catalytic performance. Even if zeolites have a larger surface area and in some cases a notably higher concentration of active sites than oxides, their low accessibility may be the cause of their poor performance. The comparison between resins and oxides can also fit in this hypothesis. In this case, the Amberlyst series of resins present superior values in pore size and concentration of active sites, which makes them more efficient. Furthermore, the characterization suggests that not only the total amount but also the strength of the acid sites can be an influencing parameter in the reaction, indicated by the fact the best performing catalysts were ion exchange resins and, that only strong acid sites were found in these solids. This allows us to hypothesize that these might be the sites that have the biggest influence on the esterification of glycolic acid with ethanol.

Parametric study with Amberlyst 70

To continue the study the effects of temperature, catalyst loading, presence of water and reusability have been examined in detail for the glycolic acid esterification with ethanol in presence of Amberlyst 70 as a result of being the most efficient of all catalyst tested in the reaction.

Influence of the temperature

The influence of the reaction temperature on the esterification of glycolic acid was studied in the temperature range 50-70°C keeping the other reaction parameters constant. Figure 12 illustrates the effect of the three different temperatures. It has been observed that the glycolic acid conversion at a set reaction time increases with the increase of reaction temperature. The reaction at 50 °C did not reach constant conversion even in the last hour of reaction, while the reaction at 70 °C required only half this time to reach this point. The glycolic acid conversion in the reaction catalyzed by Amberlyst 70 increases from 83 to 88 % by changing the reaction temperature from 50 to 70°C at the end of 6 hours of reaction. In contrast an increase in the temperature had a much greater impact the autocatalyzed reaction, with 16% difference between the tests carried out at 50 and 70°C in the same period of time (Figure 1). The low variation in the final conversion achieved for the reactions in the presence of Amberlyst at different temperatures is explained by the fact that the tests either reached or got close to thermodynamic equilibrium. Influence of the water content

The effect of initial water amount on the reaction rate and conversion of glycolic acid was investigated by adding a molar ratio of 1 mol water to 3 mol of ethanol, corresponding to around 10 wt% of water, to the departure solution. The comparison of impact of the water addition over the autocatalyzed and heterogeneously catalyzed reaction is displayed in Figure 13. In the two scenarios, with or without catalyst, the presence of water in the initial solution has an inhibitory effect. These results agree with those found by Nguyen et al., 56 in the study of the esterification of lactic acid in dilute media, where it was found that resins and even the uncatalyzed reactions suffer from decrease in conversion due to the presence of water. In this case, the presence of water in the starting reaction medium shows a similar trend for the reaction catalyzed by the polymeric and the autocatalyzed reaction as the initial reaction rate decreased in both cases. However, the conversion was more affected at the end for the heterogeneously catalyzed reaction, as it decreased by around 15 % compared to 8 % for the autocatalyzed reaction after 6 hours of reaction. This may be due to the fact that when working with catalyzed reactions in presence of water in the reaction medium not just one but two negative effects may arise: the catalyst may be deactivated due to a fast interaction between the water and the catalyst, and the reaction itself may be affected. Since this is a reversible and equilibrium reaction, the presence of a product from the beginning of the reaction could shift the equilibrium towards the reactants, and therefore a lower final conversion is achieved. Moreover, the intentional addition of water can also cause a dilution effect, decreasing the reactant as well as the catalyst concentration in the medium, thus resulting in lower conversion.

Influence of catalyst loading and reusability

Heterogeneous catalyst greatly influence the reactions by providing active sites for the reaction to take place in. An increase in the catalyst loading increases the number of available catalytic sites. In order to test the effect of the catalyst loading, the reaction was tested in the presence of three different mass percentages of Amberlyst 70 :1 % , 2 % and 3 % (Figure 14). As the catalyst amount increases the glycolic acid conversion rate increases. Moreover, in the plot of the initial reaction rate against catalyst amount a linear correlation between the two variables was observed. The recyclability of the heterogeneous catalysts is an important factor in their economical applicability. 57 Heterogeneous catalysts may suffer from phenomena such as leaching decreasing their activity. The resins can be deactivated by SO3H groups leaching from the matrix. 56 For the recycling tests, the catalyst was recovered by filtration from the reaction medium and washed with ethanol, then dried in an oven overnight at 60°C. Higher temperatures of drying were avoided as resins suffer from gradual desulfurization when subjected to temperatures higher than 120°C. Amberlyst 70 was used over 3 consecutive runs, for a total reaction time of 18 hours of reaction. No significant variations were observed in the performance of the resin (ESI). The initial reaction rates presented in the reaction with the fresh and recycled catalysts are presented in Figure 15. Amberlyst 70 showed promise to be recyclable compared to fresh catalysts with little variation in initial rate possibly due to product or reagent residues resulting from the previous reaction blocking the active sites, or partial loss of acidity and/or strength of the active sites when exposed to the reaction medium.

Mass transfer limitations

Before performing rigorous kinetic modeling, an analysis of diffusion limitations should be conducted, in which the two sensitive variables are the transport of the reactants from within the fluid to the surface and their transport from the catalyst surface to the active sites. All this is to guarantee that the reaction is being developed in a kinetic regime. The intraparticle mass transfer limitations were estimated for each experience by the Weisz-Prater criterion. 58 Table 4 displays the parameters employed for the evaluation of mass transfer resistances for the Weisz-Prater criterion. Values significantly inferior to the unit (0.05-0.08) were obtained for the Weisz-Prater criterion, thus t can be stated that the internal mass transfer resistance has minimal impact in the glycolic acid esterification on presence of Amberlyst 70. Moreover, tests varying the stirring rate at similar operational conditions demonstrated that external diffusional limitations do not impact the heterogeneously catalyzed reaction (Figure 16). This in agreement with several authors that have stated that for reactions catalyzed by Amberlyst type resins these kind of resistances are usually negligible. 40,56,[62][63][64][65]

Kinetic parameter fitting

It is important to highlight that even if earlier in this study the catalytic component from glycolic acid was taken into account in the parametric adjustment, this contribution becomes negligible when compared to the catalytic effect of the ion exchange resin due to the important amount of acid sites provided by the solid catalyst. Catalytic tests at different initial acid concentrations allowed to corroborate this hypothesis (ESI). Therefore, only conventional heterogeneous kinetic models were considered for fitting the experimental data.

Several studies have been undertaken on the kinetic models for the esterification reactions in the presence of a solid catalyst.

The reported studies focus on Langmuir-Hinshelwood, Eley-Rideal and pseudo homogeneous type models. 14, [38][39][40][41][42] Pseudo-homogeneous (PH) model assumes that the reactants reach the active surface on the catalyst easily thus reacting as in a homogeneous system. The Eley-Rideal (ER) model assumes that only one of the two reactants adsorbs onto the active site, forms the adsorbed species and then reacts with the other reactant forming the products, only one of the resultants is generated as an adsorbed state on the surface of the solid. On the contrary, Langmuir-Hinshelwood model (LH) states that the all the species generated as adsorbed species on active sites. These species are product of the interaction, in the same way, of two absorbed species. The kinetic models adapted for the esterification reaction of glycolic acid with ethanol in a heterogeneous system are listed in Table 5, where 𝐾 ,𝐾 , 𝐾 and 𝐾 stand for the adsorption constants of the reactants (ethanol and glycolic acid) and products (ethyl glycolate and water) in the corresponding order. The experimental data obtained for Amberlyst 70 was fitted to the three kinetic models, as has been already described in the autocatalytic section concerning the software, solver of equations and minimization algorithm.

Here, there is a maximum of o total parameters to be estimated, however, the number of independent data was kept sufficient as the algorithm was fed with the concentration profiles of the ). No reference value was found in the literature, however the obtained activation energy is in agreement with the reported one for glycolic acid esterification with a solid catalyst and linear chain alcohols C1-C8 (34-59 kJ.mol -1 ). 4,34,35 one hand J. Aparicio 35 reported values of 52 kJ.mol -1 , 56 kJ.mol -1 and 34 kJ.mol -1 for the activation energy of propanol, butanol and octanol respectively obtained employing ER and LH models. On the other hand C. Mutschler 34 and S. Yang et al. 4 demonstrated that the PH model is able to correctly describe the profiles for the esterification of glycolic acid with butanol (48 kJ.mol -1 ) and methanol (59 kJ.mol -1 ) respectively. The narrow difference in the error criteria may suggest that both adsorption and PH model are able to describe the reaction kinetics, with ER being the less suitable to correctly represent the experimental data (higher SRS and ErrAbs values). For the ER model, two scenarios were considered: either the ester and the acid or the ester and the alcohol are present in adsorbed form on the solid. On the grounds that these were the quantified species, the best fit for the optimized parameters was the glycolic acid and ester couple where the ester is slightly more strongly adsorbed onto the catalyst. Higher adsorption constants were obtained for the LH model, however, in agreement with the trend found for ER, glycolic acid and ethyl glycolate were the most strongly absorbed species. Taking into account the fact that the values of the adsorption constants obtained are not negligible, it can be established that the model that describes the experiments more strictly is the LH model. Moreover, for the error criteria it might be inferred that the best fitting for the experimental data is the LH model. Despite the MARR value and being the simplest of the considered kinetic models, the results obtained with the PH model may also be considered as accurate for the he glycolic acid esterification with ethanol. As we perform an optimization on values close to zero, low molar concentrations, and normalizing the value, the error naturally increases and so as the PH model has only 2 adjustable parameters, in contrast to 6 and 8 for the ER and LH models, the MARR calculated is higher. As previously discussed, it had been demonstrated that PH model can represent with precision the kinetics of esterification in the presence of ion exchange resins. 29,36,37,43 A graphical representation of the adjustment of the LH model to the prediction of the concentration profiles for one of the experiments carried out is in Figure 17 where is illustrated that all predicted values agree with experimental data within an error of 15 %.

(A) (B)

Conclusions

In this study, the kinetics of the esterification of glycolic acid with ethanol homogeneously autocatalyzed and heterogeneously were investigated. In the parametrical study, the reaction was found to be favored by an increase in temperature and acid molar fraction, whereas the dilution of the initial mixture with water had an inhibitory effect. Under optimized conditions (Acid/Alcohol 1/4 [mol/mol], 70°C) up to 38% conversion of glycolic acid was achieved in a medium with no catalyst other than the acid itself. Additionally, it was established that the reaction is more likely to be catalyzed by glycolic acid molecules than via the solvated protons. In order to determine the relationship between the equilibrium constant and the temperature, the correction of non-idealities was performed using the UNIFAC group contribution method. This led to the conclusion that this reaction is slightly endothermic with an activation energy of 47-48 kJ.mol -1 . Among the reviewed solid catalysts, resins stand out as being the best performant in the esterification of glycolic acid with ethanol followed by the oxide and finally the zeolites the less performing (2-12% increase in conversion). Concerning heterogeneous catalysis, in the presence of Amberlyst 70 (2 w% Acid/Alcohol 1/4 [mol/mol], 70°C), equilibrium conversion (about 90%) was reached within 2 to 3 hours of the reaction. It was established that among the most sensitive parameters of the heterogeneously catalyzed reaction are the accessibility, quantity and strength of acid sites. Further studies carried out in the presence of Amberlyst 70 revealed a direct correlation between the catalyst loading and initial reaction rate. Moreover, a similar behavior was observed when temperature was increased compared to the autocatalyzed system. Contrary, when water is added to the initial solution the glycolic acid conversion is more affected when working in the presence of the ion exchange resin. Consecutive tests, evaluation of the Weizs Prater criterion and test at different stirring rates suggests that Amberlyst 70 could be recycled for several runs and that the kinetics of the glycolic acid esterification with ethanol is affected to minimum extent by internal or external mass transfer. Three kinetic models were considered to fit the experimental data obtained of the heterogeneously catalyzed reaction with Amberlyst70. The results obtained from the optimized parameters suggest that LH model is able to more accurately describe the kinetics of the esterification reaction of glycolic acid with ethanol. Still the results of the PH model were also found to be accurate when a simpler mathematical form is sought.

Scheme 1

 1 Scheme 1 Esterification reaction of glycolic acid with ethanol

Figure 1 .

 1 Figure 1. Temperature influence in the esterification reaction. Conditions: Acid/Alcohol 1/4 [mol/mol], 1400 rpm. (•) 75°C ; (■) 70°C; (▲) 60°C; ( ) 50°C.

Figure 2 .

 2 Figure 2. Acid/Alcohols molar ratio influence in the esterification reaction. Conditions: 60°C, 1400 rpm. (•) 1/3; (■) 1/4; (▲) 1/6.

Figure 4 .

 4 Figure 4. Influence of the water presence in the initial mixture Conditions: Acid/Alcohol 1/4 [mol/mol], 60°C, 1400 rpm. (•) No water added; (■) Water/Ethanol 1/3 [mol/mol]

Figure 5 .

 5 Figure 5. Van't Hoff diagram for glycolic acid esterification ethanol regressed from experimental data (■) Molar concentration and activity; ( ) Molar concentration.

Figure 7 .

 7 Figure 7. Influence of the zeolite structure and Si/Al ration in the esterification reaction. Conditions: Acid/Alcohol 1/4 [mol/mol] , 1400 rpm, 70°C, Catalyst 2 wt%. Zeolites (•) FAU 40/1 ; (■) MFI 40/1; (▲) MOR 10/1; ( ) MFI11.5/1;( ) FAU 2.5/1.

Figure 8 .

 8 Figure 8. Influence of the ion exchange resin in the esterification reaction. Conditions: Acid/Alcohol 1/4 [mol/mol], 1400 rpm, 70°C, Catalyst 2 wt%. Ion exchange resin: (•) Amberlyst 70 ; (■) Amberlyst 15; ( ) Amberlyst 36; ( ▲) Dowex 50XW8-100;( ) Nafion NR50.

Figure 9 .

 9 Figure 9. Comparison between benchmark (non-catalyzed at same operational

55

 55

  water is one of the reaction products, these phenomena would occur albeit to a lesser extent, over the course of the reaction. Despite this, the results obtained during this study are accurate approximations of the values described in the literature for the solid catalysts studied, with a variation of 15 % inherent to this type of technique. With the exception of FAU 2.5 in which no strong sites were found, zeolites were shown to have all three types of acid sites. Titanium oxide presented similar values regarding weak and strong sites, with fewer medium acid sites. Contrary to the characterization of the other two types of solids, the ion exchange resins did not show medium or weak acid sites, thus the total number of acid sites represents only strong acid sites.

Figure 11 .

 11 Figure 11. Influence of the total acid sites concentration over the initial reaction rate. Conditions: Acid/Alcohol 1/4 [mol/mol], 1400 rpm, 70°C, Catalyst 2 wt%. (•) Ion exchange resin; (■) Oxide; (▲) Zeolite.

Figure 10 .

 10 Figure 10. Characterization by titration of different acid sites present in the commercial catalyst

Figure 12 .

 12 Figure 12. Influence of the temperature over the reaction catalyzed by Amberlyst 70. Conditions: Acid/Alcohol 1/4 [mol/mol], 1400 rpm, Catalyst 2 wt%. (•) 70°C; (■) 60°C; (▲) 50°C

Figure 13 .

 13 Figure 13. Influence of water in the departure solution over glycolic acid conversion. Conditions: Acid/Alcohol 1/4 [mol/mol], 70°C, Catalyst 2 wt%. (■)Without water; ( ) Water 10 wt%.

Figure 14 .

 14 Figure 14. Influence of the catalyst amount in the reaction. Conditions: Acid/Alcohol 1/4 [mol/mol], 1400 rpm, 70°C.

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 26

Figure 15 .

 15 Figure 15. Influence of the reuse of Amberlyst 70. Conditions Acid/Alcohol 1/4 [mol/mol], 70°C, Catalyst 2 wt%.

Figure 16 .

 16 Figure 16. Influence of the stirring rate over the reaction catalyzed by Amberlyst 70.

Figure 17 .

 17 Figure 17. (A) Parity plot of the heterogeneously catalyzed glycolic acid esterification with ethanol by Amberlyst 70 fitted to LH (B) Experimental and predicted concentration profiles of heterogeneously catalyzed glycolic acid esterification with ethanol by Amberlyst70 fitted to LH. Conditions: Acid/Alcohol 1/4 [mol/mol], 70°C, 1400 rpm, Catalyst 2 wt%. Mole fractions: Experimental (•) Ethanol; (■) Glycolic acid; (▲) Ethyl glycolate; (---) ±15%; dotted lines: predicted profile concentrations.

Table 1 .

 1 Summary of the parametric study experiments

		Parametric study
	Temperature [°C]	Acid/Alcohol [mole ratio]	Water/Alcohol [mole ratio]
	50	1/4	-
	60	1/4	-
	70	1/4	-
	75	1/4	-
	60	1/3	-
	60	1/6	-
	60	1/4	1:3

Table 2 .

 2 Summary of autocatalytic kinetic models

Table 3 .

 3 Optimized parameters with 95% confidence intervals for the different kinetic models

	Model

Figure 6. (A) Parity plot of the autocatalyzed glycolic acid esterification with ethanol fitted to the molecular autocatalyzed model (B) Experimental and predicted molar concentrations of autocatalyzed glycolic acid esterification with ethanol fitted to the molecular autocatalyzed model. Conditions: Acid/Alcohol ratio 1/4 [mol/mol], 60°C, 1400 rpm; Mole fractions: Experimental (•) Ethanol; (■) Glycolic acid; (▲) Ethyl glycolate; (---) ±15%; dotted lines: predicted profile concentrations.

Table 4 .

 4 Parameters employed for the evaluation of mass transfer resistances.

	Conditions: Acid/Alcohol 1/4 [mol/mol], 70°C, Catalyst 2 wt%.
	Parameter	value
	Catalyst loading by mass of reacting media (𝑤 )
	[kgcat. kgsln -1 ]	0.020
	Density of the reacting media (⍴ ) [kg. m -3 ]	789
	Density of the catalyst (⍴ ) [kg. m -3 ]	1000 40,59
	Association factor Wilke -Chang model (ethanol)	1.50 60
	Glycolic acid molar volume at normal boiling point
	[cm 3 .mol -1	75.80*
	Observed reaction rate for the heterogeneously
	catalyzed reaction (𝑟 ) [kmol. kgcat -1 s -1 ]	5.27E-05
	Diffusion coefficient for glycolic acid (𝐷 ) [m 2 . s -1 ]	2.73E-09
	Effective diffusivity of glycolic (𝐷 ) [m 2 . s -1 ]	9.00E-10
	Weisz-Prater criterion	0.08
	*Calculated by Group/Atom contribution for Le bas method 61

Table 5 .

 5 Kinetic laws developed for the different kinetic models

  8 experiments each one accounting for 12 individual concentrations of the three quantified species. The estimated parameters are shown in table 6. The optimization was initialized from different values to avoid optimization in local minimum points of the function surface. Adsorption based models shown big difference concerning the pre-exponential factor. The LH model presented the highest value, whereas ER presented rather a value in the same range of PH. Considering the activation energies, the optimized values were quite similar for adsorption based models and PH (42-43 kJ.mol -1

Table 6 .

 6 Optimized parameters with 95% confidence intervals for the different heterogeneous kinetic models

	Model

Acknowledgements

Laura Reyes thanks the Ecole Doctorale de Chimie de Lyon for her PhD grant. This work was supported by the LABEX iMUST of the University of Lyon (ANR-10-LABX-0064), created within the program « Investissements d'Avenir » set up by the french government and managed by the French National Research Agency (ANR).

Author contributions

Conflicts of interest

There are no conflicts to declare.

Notes and references ‡ Footnotes rela ng to the main text should appear here. These might include comments relevant not central to the matter under discussion, limited experimental and spectral data, and crystallographic data.