2.2.1. Proposed Kinetic Model
Based on the experimental data, a scheme of glycerol conversion is proposed (
Figure 2).
The assumptions underlying the kinetic modeling are given below:
(1) To construct a kinetic model of the reaction occurring in the volume, power law kinetic modeling was used. The reagent concentration exponents are based on the stoichiometry of the corresponding reactions. The LA and NaOH concentration exponents in the equation for the rate of consumption of lactic acid into acetic acid were calculated in the range from 0.1 to 1.5.
(2) The adsorption of hydroxyl ions on the Cu NPs surface is negligible and does not affect on the adsorption of substrates. The hydroxyl ion attacks the adsorbed substrate molecule from the volume in the course of the surface heterogeneous reactions.
(3) For the description of the surface reactions in the presence of Cu NPs catalyst, LH and ER type models were used. These models are commonly used realistic approach to derive the rate expression for heterogeneous reactions. The selection of models was based on the analysis of literature data on the kinetic modeling of the conversion of glycerol into PG [
37], LA [
38,
39] and other carboxylic acids [
40] in an alkaline media.
(3.1) To construct a kinetic model of the reaction
r1 occurring on the surface of a heterogeneous catalyst, Eley–Rideal type reaction mechanism was used. A glycerol molecule adsorbed on the catalyst surface is attacked by a hydroxyl ion from the reaction volume. The formed product (GA) desorbs from the surface (see
Section 2.2.2 for more details about this reaction pathway).
(3.2) For the surface reaction
r5, combined Eley–Rideal type and Langmuir–Hinshelwood type reaction mechanism was used. First, a hydroxyl ion attacks of a glycerol molecule on the surface with GA formation. Then, GA reacts with adsorbed hydrogen on the surface with PG formation (see
Section 2.2.2 for more details about this reaction pathway).
(3.3) For the surface reaction
r6, Langmuir–Hinshelwood type reaction mechanism was used. PG dehydrogenates with PA formation on the catalyst surface, and PA desorbs into the reaction volume (see
Section 2.2.2 for more details about this reaction pathway).
(4) The carboxylic acids are present in the system in the form of Na salts (carboxylic acid anion + Na cation) under strongly alkaline conditions. The adsorption of carboxylic anions on the Cu catalyst surface is negligible. Since the formation of LA and AA occurs in the volume, but not on the surface (reactions r1, r4, r6), their presence in the system can be neglected when describing the surface conversions.
(5) As shown previously (see
Table 1), the glycerol conversion does not occur in a given temperature range in the absence of sodium hydroxide both in the presence and absence of a copper catalyst. Thus, the NaOH concentration was included in all the kinetic equations.
(6) The adsorption of organic substrates is competitive.
(7) In situ formation of hydrogen occurs as a result of the formation of LA from both glycerol and PG (reactions
r1 and
r6), and its consumption occurs as a result of the hydrogenolysis of glycerol into PG (
r5). It is worth noting that hydrogen is produced much more than it is consumed by the reaction
r5, which leads to its excess in the reaction system. The hydrogen desorption rate from the catalyst surface increases and its solubility in the reaction volume decreases due to the high reaction temperature. As a result, the main amount of hydrogen moves to the gas phase (the hydrogen content in the gas phase in all the experiments exceeded 95% vol) (
Table S1 in the
Supplementary Information). Thus, the hydrogen pseudo equilibrium in the gas-liquid-catalyst surface system is created, i.e. the hydrogen content on the catalyst surface and in the liquid phase is almost unchanged. A similar pattern is observed for a homogeneous process. Thus, the hydrogen concentrations in the kinetic equations were included in the values of the effective reaction rate constants
k5,
ks2 and
ks3.
(8) The reaction mixture volume is constant.
2.2.2. Description of Proposed Model
Based on this mechanistic hypothesis, the reaction rate equations can be written as the sum of the homogeneous and heterogeneous-catalyzed ones, as follows:
1. r1: conversion of glycerol into LA.
This reaction rate is the sum of the homogeneous reaction rate in the volume and the heterogeneous reaction rate on the surface.
Reaction in the volume. The reaction includes the following steps [
20]: activation of glycerol molecule by HO
− with glycerolate–ion formation, its conversion into GA, further conversion of GA into unstable 2-hydroxypropenal, which is transformed into pyruvaldehide PA. Benzilic acid rearrangement reaction of PA into LA is the final step (
Figure 3).
Reaction on the surface. Eley–Rideal-type reaction mechanism. Under this mechanism, conversion of molecules comprises the following steps: equilibrium adsorption of glycerol molecules on the Cu surface, interaction of adsorbed glycerol molecules with hydroxyl ions which attack from solution, formation of intermediate reaction product (GA), its instantaneous desorption into the reaction volume, where instantaneous LA formation takes place under alkaline conditions according to the scheme:
Step 1. Adsorption of GLY on the surface vacant site (*):
Step 2. Activation of adsorbed glycerol molecule by HO
− with glycerolate–ion and hydrogen atom formation on the surface:
Step 3. Conversion of glycerolate–ion into GA on the catalyst surface:
Step 4. Desorption of GA from catalyst surface:
Step 5. Conversion of GA into PA through intermediate unstable 2-HP:
Step 6. Benzilic acid rearrangement of PA into LA (very fast):
Figure 4 presents a schematic conversion of glycerol into LA on the catalyst surface.
Thus, the generalized reaction rate equation is:
2.
r2: conversion of glycerol into diglycerol (
Figure 5).
The
Reaction in the volume. The reaction rate equation is:
3.
r3: conversion of diglycerol into glycerol (
Figure 5).
Reaction in the volume. The reaction rate equation is:
4.
r4: consumption of LA into AA in an alkaline media (
Figure 6).
Reaction in the volume. The reaction rate equation is:
5. r5: hydrogenolysis of glycerol into PG.
This reaction rate is the sum of the homogeneous reaction rate in the volume and the heterogeneous reaction rate on the surface.
Reaction in the volume. The reaction includes the following steps: activation of glycerol molecule by HO
− with glycerolate–ion formation, its conversion into GA, further conversion of GA into unstable 2–HP, which is transformed into PA, hydrogenation of 2–hydroxypropenal (2–HP) and PA with PG formation (
Figure 7).
Reaction on the surface. Combined Eley–Rideal (steps 1–4) and Langmuir–Hinshelwood (steps 5 and 6) mechanism.
Step 1. Adsorption of GLY on the surface vacant site (*):
Step 2. Activation of adsorbed glycerol molecule by HO
− with glycerolate–ion and hydrogen atom formation on the surface:
Step 3. Conversion of glycerolate–ion into GA on the catalyst surface (very fast):
Step 4. Elimination of water from GA molecule on the surface with unstable 2-HP formation:
Step 5. Hydrogenation of 2-HP on the surface:
Step 6. Desorption of PG from the surface into the volume:
Figure 8 presents a schematic conversion of glycerol into PG on the catalyst surface.
Thus, the reaction rate equation of PG formation from glycerol (competitive adsorption of GLY and PG) is:
6. r6: consumption of PG into LA.
Reaction on the surface. Langmuir–Hinshelwood mechanism.
Step 1: Adsorption of PG on the Cu surface vacant site (*):
Step 2: Dehydrogenation of PG into PA on the Cu surface:
Step 3: Desorption of PA from catalytic surface into reaction volume:
Step 4: Benzilic acid rearrangement of PA into LA (very fast in alkaline media):
A schematic representation of this mechanism is given in
Figure 9.
By analogy with the reaction rate equation
r5, the final equation (
ks3,—effective reaction rate constant of conversion PG into LA) can be written as:
7. r7: consumption of PG into others.
Reaction in the volume. The reaction rate equation is:
8. r8: consumption of glycerol into others.
Reaction in the volume. The reaction rate equation is:
All the reaction rate equations are placed in
Table 4 for clarity.
The concentration of NaOH was not measured during the experiments, but was determined by the difference between its initial concentration and the concentration of LA and AA. The formation of one mole of LA and AA required one mole of NaOH, respectively. The part of NaOH was consumed to absorb CO
2, but this fact was neglected and not considered in the kinetic model. Thus, the concentration of NaOH was calculated by the material balance proposed below:
The formation/consumption equations for each component are presented in
Table 5.
The analysis of the reaction mixture after ~100% glycerol conversion showed that the amount of H2 presented in the system does not contribute to further side reactions.
2.2.3. Criteria for Parameter Estimation
The following criteria were chosen to calculate the constants:
(A) Minimization of the residual sum of squares between the calculated and the experimental concentrations of the reaction mixture components for all experimental data.
(B) All the rate and adsorption constants must be positive; the reaction constants must increase with increasing the temperature, and the adsorption constants must decrease with increasing the temperature.
The residual sum of squares (RSS) between the calculated and experimental data was used to obtain the optimum kinetic parameters:
where
n refers to the considered experimental data set.
Mathematical processing and finding of the kinetic constants were performed in the MATLAB software (R2009a, Version 7.8.0.347, The MathWorks, Inc., Natick, MA, USA) using Runge–Kutta method (ode23s subroutine). Therefore, there were 82 parameters to be estimated, namely: 50 kinetic constants (7 constants for homogeneous and 3 constants for surface reactions for five temperature values (483, 493, 503, 513, and 518 K)), 10 adsorption constants of glycerol and PG on catalyst surface bGLY, bPG, 10 activation energies Eaj, 2 adsorption enthalpies, 10 concentration exponents for LA and AA in the reaction rate equation r4 optimized in the range from 0.1 to 1.5. The calculated concentration profiles were then compared with the experimental ones.
2.2.4. Determination of the Constants and Verification of the Model
The kinetic experiments were performed in the temperature range of 483–518 К, the glycerol concentration in solution was 0.27–2.06 mol·L−1, the initial molar ratio NaOH/glycerol was 0.25–3.0, the initial molar ratio glycerol/Cu was 0.26–160.0, the stirring speed was 1000 rpm, the reaction time was 0–480 min, the volume of the reaction mixture was 250 mL in all the experiments.
The calculated rate constants, adsorption constants and concentration exponents for LA and AA (
m,
n) in the reaction rate equation
r4 are presented in
Table 6.
The Arrhenius equation was used for the temperature dependence of reaction rates:
where
k0,i is the pre-exponential factor for the homogeneous reaction, mol min
−1;
k0,si is the pre-exponential factor for the surface reaction, mol·g
cat−1·min
−1;
Eai and
Ea,si is the activation energies for the homogeneous and surface reaction, respectively, J·mol
−1;
R is the universal gas constant; the gas constant value is 8.314 J·mol
−1·K
−1;
T is the temperature, K.
The van’t Hoff equation was used for the temperature dependence of adsorption constants:
where
bj is the adsorption constant of component
j on catalyst surface, L·mol
−1;
bo,j is the constant of integration, L·mol
−1; Δ
Hj is the adsorption enthalpy, J·mol
−1;
R is the universal gas constant; the gas constant value is 8.314 J·mol
−1·K
−1;
T is the temperature, K.
Figure 10 shows the ten Arrhenius plots used to calculate the activation energies and pre-exponential factors.
The results of the parameters calculation of the Arrhenius and van’t Hoff equations are presented in
Table 7. The parameters and their confidence intervals are determined at a significance level of 0.05 (95% confidence interval).
Comparison of the obtained values of activation energies with those described in the literature is shown in
Table 8.
Comparison of the calculated and experimental values of the component concentrations in the presence of NaOH without and with Cu NPs is shown in
Figure 11. The deviation of the calculated from the experimental values does not exceed 20%.
Figure 12 presents the experimental and calculated C/t-profiles of the glycerol conversion without and with Cu NPs catalyst.
Adequacy of the obtained catalytic conversion of glycerol into LA kinetic model was evaluated using the Fisher criterion (Fc) at a significance level of 0.05 (95% confidence interval). The Fc value must be lower than the Fcrit to consider the proposed model statistically significant.
The variance of optimization which shows the reproducibility of the results was 0.0024. The variance of adequacy which is equal to the sum of squares of deviations between the experimental and calculated values was 0.0009. Thereby, the experimental value of the Fisher’s criterion is equal 0.3864. Since the experimental value of the Fisher’s criterion is less than the critical value (2.5418), the obtained kinetic model adequately describes of the experimental data.