1. Introduction
Supercritical CO
2 (SC-CO
2) extraction process is well known to be an efficient alternative to the use of toxic organic solvents. Indeed, it is a GRAS (Generally Recognized as Safe) solvent with tunable properties and selectivity. Thanks to its low critical temperature (31.06 °C), it can be used to extract heat-sensitive components. Furthermore, no separation step is needed since CO
2 is gaseous at ambient pressure and it can be recycled at industrial scale, enabling a clean and compact process. Supercritical CO
2 extraction is highly regarded in industry, with several industrial units already in use worldwide (China, France, Republic of Korea, etc.). Two examples of applications can be found in the following articles [
1,
2].
Experimental data on SC-CO
2 extraction from several biomass sources have been widely reported in the literature; a non-exhaustive list of studies can be found in the following references [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14]. Among the published works dealing with SC-CO
2 extraction, several studies dealing with the application of response surface methodology (RSM) for the optimization of supercritical CO
2 extraction of oil from several biomass were published. Some of these studies can be found in the following references [
15,
16,
17,
18,
19,
20]. Recently, to improve the understanding of the theoretical aspects of SC-CO
2 extraction, studies dealing with the modeling of extraction kinetics using a mass transfer model [
21,
22,
23,
24,
25] were published [
26,
27,
28,
29]. Scale-up studies have also been the subject of increasing interest in recent years [
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41]; these studies showed that the most pertinent criterion was the conservation of the CO
2/biomass mass ratio, found at the highest achievable yield, between the small and large scale. The establishment of extraction kinetics at various operating conditions is the only way to have access to the highest reachable yield.
Considering the significant amount of available data in the literature, one might think that it would be quite simple to select any published study about a targeted biomass to have access to the extraction kinetics and perform scale-up calculations for industrial purposes. However, this is not always possible as several drawbacks were identified:
- -
several studies dealing with RSM do not present the extraction kinetics;
- -
in some studies, only one extraction kinetic was presented. It is then not possible to know if this extraction kinetic corresponds to the best extraction condition (highest yield and fastest extraction kinetic);
- -
when one or several extraction kinetics were established for a selected biomass, the operating conditions proposed in the studies may be beyond the technical capabilities of the planned industrial-scale equipment;
- -
when the operating conditions can be applicable at industrial scale, some important information for scale-up calculations, such as the mass of introduced biomass, the CO2 mass flowrate, the particle size distribution, the CO2 flow direction (up flow or downflow), etc., was not reported.
Assuming that an article reports all the information required for precise scale-up calculation, the challenge would be to achieve a perfect match between the technical specificities of industrial-scale equipment and the operating conditions proposed in the article. In the case of a mismatch, the solution would be to carry out additional extraction experiments, which is time-consuming and costly. Another solution would be to be able to predict the SC-CO2 extraction kinetics from a biomass whatever the operating conditions, which is very challenging.
Mass transfer models [
23,
24] such as the Sovová’s broken and intact cell model (BICM) [
21], which consider physical parameters such as the extract solubility in SC-CO
2, transfer and/or diffusion coefficients, etc. can be considered for predicting extraction kinetics. Some parameters such as solute solubility in SC-CO
2 or diffusion coefficients may be calculated or approximated by using specified models or correlations [
42,
43,
44,
45]. However, parameters such as the highest achievable yield (
Cu), which depend on:
- -
the condition of pressure and temperature (as SC-CO2 has tunable solvent properties);
- -
the nature of the biomass (plant, flower, roots, kernels, etc.);
- -
the geographical growth location;
- -
and the grinding efficiency (particle size diameter).
The Sovová’s BICM has been used to predict the end (interaction and diffusion part) of the SC-CO
2 extraction kinetics from
Helichrysum italicum, Jatropha, and Argan kernels [
4,
5,
6] by applying adequate hypotheses. The highest reachable yield (
Cu) was known in the case of Jatropha and Argan kernels, and it was deduced by screening calculations in the case of
Helichrysum italicum. The methodology described in these articles aims to propose a possible shape of the extraction kinetics when the extraction experiments could not be conducted until the end. This methodology is then not fully predictive.
Del Valle et al. [
46] have considered the shrinking core model [
47] to propose a methodology to predict extraction kinetics. Their aim was to show that they were able to predict SC-CO
2 extraction kinetics of prepressed rapeseeds, olive husks, and flaked rosehip seeds at 40 °C and 300 bar. The experimental extraction kinetics were established at these operating conditions. Their methodology was based on the use of a set of several correlations for the calculations of the shrinking core model parameters: the film mass transfer coefficient, the diffusion coefficient, the effective diffusivity coefficient, and the solute solubility. These parameters were determined considering several steps calculations depending on the values of the dimensionless Reynolds, Biot, Sherwood, and Schmidt numbers but also by considering various types of correlations from the literature and a group contribution method. The methodology is also based on the mean particle size distribution and the surface structure of the biomass, considering various parameters such as the bed void fraction and the tortuosity. Nevertheless, it was necessary to verify the calculated value of the effective diffusivity coefficient with experimental data to be sure of the good prediction of the methodology. For better reliability, it was also necessary to consider this parameter as an adjustable parameter and add a correction factor: the microstructural correction factor to correctly describe the shape of the extraction kinetics. In addition, the highest reachable yield (
Cu) was not predicted; the values were taken from the experimental data used for establishing the extraction kinetics. Consequently, the methodology proposed by Del Valle [
46] could not be considered purely predictive, and the calculations were performed only at the experimental operating conditions of 300 bar and 40 °C. No predictions were performed outside this operating domain. Finally, their methodology requires integrating complex partial differential equations.
Reverchon et al. [
48] have used the BICM for the modeling and simulation of extraction kinetics from sunflower seeds, tomato seeds, coriander seeds and grape seeds. The experimental data were taken from the literature. The fraction of broken cells was not considered as an adjustable parameter; the authors used scanning electron microscopy (SEM) images to identify the broken cells and calculate this parameter. Unfortunately, the SEM image analysis was not explained in detail, making it impossible to understand how the identification of the broken and intact cells can be made on SEM images. As the surface structure of biomass is very different from one biomass to another, their methodology cannot be extrapolated to other studies. The authors have performed simulation tests considering a “generic” seed at 280 bar, 40 °C and 1.5 kg/h to evaluate the effects of particle size distribution (100, 500, and 2000 µm) on the extraction kinetics. They have considered a constant value of Cu; they took into consideration a set of parameters previously adjusted on experimental data, and they have considered three different particle diameters to perform calculations. Unfortunately, this methodology cannot be considered simulation calculations but extrapolation calculations. In addition, Reverchon et al.’s methodology requires integrating complex partial differential equations.
Cabeza et al. [
49] have proposed a methodology based on graphical extrapolations for the calculation of the mass transfer model parameters. Their methodology consisted of selecting the extraction kinetics of two biomass from the literature: sesame seeds and coffee grains. Fifteen SC-CO
2 extraction kinetics were considered for sesame seeds and nine for coffee grains. Various operating conditions of pressure (138–350 bar), temperature (40–80 °C), particle size distribution (450–1180 µm), and CO
2 volumetric flow rate (0.68–2 mL/min; the CO
2 mass flow rates are available in the works cited by Cabez et al. [
49]) were considered. These authors have performed modeling calculations on these data. In a second step, they have plotted the evolution of the model parameters depending on the variation of the operating conditions (e.g., the evolution of the mass transfer coefficient vs. the CO
2 flow rate or the particle size distribution, etc.) to establish correlations for the calculations of a model parameter at wanted operating conditions. Unfortunately, the authors do not give details for the number of experiments to perform or the number of correlations to establish for using their methodology. In addition, it also requires integrating complex partial differential equations.
The neural networks (NN) tool was also considered to describe and predict the extraction kinetics from
Nigella sativa L. [
50] and
Drimys angustifolia Miers [
51]. Despite NN being a powerful tool applied for prediction purposes, the main drawback of this method is that it requires not only a large amount of experimental data (more than 150 points) but also very specific knowledge and very specific mathematical skills. These works [
43,
44] show that the NN tool was able to describe the shape of the extraction kinetics obtained experimentally; no calculations were performed for an operating condition not investigated experimentally. Furthermore, no theoretical parameters such as solubility, diffusion coefficient, or partition coefficient can be calculated when NN calculations are applied.
In summary, in these studies [
46,
48,
49,
50,
51], several approaches were considered. When a mass transfer model was considered, only the method proposed by Cabeza et al. can be considered to be predictive, but no detail about how to apply the methodology was given. In the approach proposed by Del Valle et al. and Reverchon et al., the value of the highest reachable yield (
Cu) was not predicted; they have considered the experimental value of
Cu for their calculation procedure. In these three studies [
39,
41,
42], it was necessary to integrate complex partial differential equations. When the NN tool was considered, a huge amount of experimental data and very specific mathematical skills were needed. To our knowledge, the NN tool was used to describe experimental extraction kinetics; it was not used to plot extraction kinetics at operating conditions not performed experimentally. To our knowledge, no other studies dealing with the simulation of SC-CO
2 extraction kinetics were found in the literature.
Regardless of the chosen method (using a mass transfer model or NN tool), the prediction of extraction kinetics requires experimental data and especially the knowledge of the highest achievable yield (
Cu). The aim of this study is then to propose a new methodology for predicting SC-CO
2 extraction kinetics from biomass whatever the operating conditions of pressure and temperature inside an experimental operating domain. This new methodology, called predictive broken and intact cell model (PBCIM), aims to use the Sovová’s BICM [
21] analytical solutions combined with the Design of Experiments (DOE) tools such as RSM to predict SC-CO
2 extraction kinetics in a specific operating domain with defined operating conditions. RSM is a very powerful and useful method for predicting the variation of an output (called response) depending on various input parameters (called factors) in a specific operating domain [
52]. It requires a minimum of experiments, and its effectiveness is very well established. The main advantage of these two modeling tools (BICM and RSM) is that they do not require very advanced skills in mathematics, but only the knowledge of classical modeling methods.
2. Materials and Methods
2.1. Description of the Novel Methodology
The methodology chosen is called the predictive broken and intact cell model (PBICM). This methodology aims to use RSM to calculate the parameters of the Sovová’s BICM (response) to predict any extraction kinetics at operating conditions (factors) located inside the variation range of the factors studied in the design of experiments. For this purpose, the first step consisted of finding in the literature articles reporting extraction kinetics established according to a design reporting only the effects of pressure and temperature at a constant SC-CO
2 flow rate and mean particle size distribution with all the experimental data needed to perform the calculations: the mass of biomass introduced in the extraction autoclave (i.e., the extraction basket), the CO
2 mass flow rate, and the particle size distribution of the biomass. Hence, the following studies [
6,
27,
28,
53,
54] were selected to test this methodology. The type of biomass and the operating conditions (pressure (P), temperature (T), CO
2 mass flow rate (Q), mass of introduced biomass (N), mean particle size distribution (d
p), bed void fraction (ε), internal autoclave (i.e., basket) volume (mL), and the ratio of internal basket length (L) to internal basket diameter (D)) were reported in
Table 1. The next step consisted of modeling the extrapolated data with Sovová’s BICM equations described in
Section 2.2. Then, the parameters of the BICM were considered as answers in RSM to be able to predict their values whatever the operating condition of pressure and temperature located inside the domain of interest.
2.2. The Model of Broken and Intact Cells
The Sovová’s BICM [
21] is a very well-known and widely used model for describing SC-CO
2 extraction kinetics. This model was chosen because it describes very well the SC-CO
2 extraction kinetics obtained on a large panel of biomass: plants, seeds, kernels, roots, etc. A non-exhaustive list of studies (also containing additional information) can be found here [
4,
5,
6,
8,
27,
55,
56]. In brief, the model considers the biological structure of the biomass and accounts for the sudden reduction in extraction rate after the first extraction period that is observed during SC-CO
2 extractions from a sample biomass. The extraction can be of type A, B, C, or D. In deciding on extraction type, it is possible to refer to the aspect of the extraction curves reported in
Figure 1.
If the first part of the extraction curve consists of one straight section, the type is A or D. If it consists of two straight sections, it is of type B or C. It is worth noting that the curve shapes of type A and D and of type B and C are quite similar. Nevertheless, in the case of type A and B, the slope of the first straight section is related to the apparent solubility of the solute (ys) in SC-CO2. In the case of type C, the slope of the first straight section is close to 1, and for type D, the slope of the first straight section is lower than the apparent solute solubility in SC-CO2.
The shape of an SC-CO
2 extraction kinetic and thus the values of the parameters of the BICM equations depend on the nature of the biomass (oleaginous, plants, roots, kernels, etc.) and the type of flow in the extraction autoclave (plug flow or mixed cell). For instance, for microalgae or some Corsican plants, type A was found [
3,
4], while for Argan or Jatropha (oleaginous), type B was found [
5,
6]. The type of extraction can be identified only after performing the experiments and applying the BICM equations. In this article, only the equations for type A and B were selected due to the shapes of the extraction curves. It is worth noting that depending on the case study, type C and D can also be obtained [
21]. If type C or D are identified, the methodology developed here can be applied by considering the equations describing the shape of the extraction kinetics of type C and D [
21].
The model equations of type A extraction kinetic are given in Equations (1)–(4). Physically, ys is the apparent solute solubility of the extract in SC-CO
2, q
1 is the end of the solubility period (first extraction period),
r is the grinding efficiency,
ks is the solid phase mass transfer coefficient, and as is the specific surface between the regions of intact and broken cells. For type A, q
1 is defined as the intersection point between Equations (1) and (2). C
1 and C
2 are constant parameters allowing the calculation of
r (the fraction of broken cells) and
ks·
as (the product of the solid-phase mass transfer coefficient and the specific surface area between the regions of intact and broken cells) knowing the biomass bed porosity ε in the extraction autoclave.
Knowing the values of parameters ys, xu, C1, and C2, it is possible to plot the extraction kinetic of type A and calculate the values of r and ksas. These parameters (ys, xu, C1, and C2) were then considered as responses for the RSM.
The model equations describing type B extraction kinetic are given in Equations (5)–(12). Physically,
xt is the transition concentration,
K is the partition coefficient,
q1 is the end of the solubility period, and
qc is defined as the intersection point between Equations (7) and (8).
With the fluid phase density (density of SC-CO2) and the solid biomass density
Knowing the values of parameters ys, q1, xu, C1, C2, K, and xt, it is possible to plot the extraction kinetic of type B and calculate the values of r and ksas. It is worth noting that for type B, q1 (Equation (6)) was calculated using parameter r, and r was calculated using Equation (9) depending on the values of C1 and C2. However, parameter C1 was calculated using Equation (10) depending on the values of r and C2. Consequently, for the sake of simplicity, r was considered as a response in RSM, allowing the calculation of both parameters q1 and C1. Hence, the selected answers for type B extraction kinetic were ys, xu, C2, K, xt, and r.
The grinding efficiency (
r) represents the efficiency of the pretreatment (grinding and sieving); from a theoretical point of view, for a similar pre-treatment process, the fraction of broken cells is supposed to be constant. Nevertheless, as the pre-treatment cannot be homogeneous, some variations in the value of
r can be found. The fraction of broken cells was then considered as a parameter and not as a constant value. It is worth noting that the apparent solubilities (
ys) were modeled using the Chrastil model [
57] to verify the values predicted using the proposed methodology. A good agreement was found between these two methods (lower than 5%).
The modeling calculations were performed considering the insoluble quantities. However, the extraction kinetics were presented reporting the yield (mass of extracted solute in kg/mass of introduced biomass in kg), given by Equation (15), against the CO
2/biomass mass ratio (kg/kg), given by Equation (16), for more convenience.
The absolute average relative deviation (AARD), given in Equation (17), was calculated to estimate the accuracy of the predictions.
For all predictive calculations, a solid biomass density () of 1200 kg/m3 was considered.
2.3. Experimental Design
In the selected studies [
6,
27,
28,
53,
54], experimental designs were achieved in order to study the pressure and temperature parameters. The responses Y (Sovová’s BICM parameters) were modeled with a second-order polynomial model given in Equation (18). In order to calculate the coefficients, a central composite design with 9 experiments was performed. The design (in coded values) corresponding to these 9 experiments is given in
Table 2. The polynomial coefficients were calculated using Azurad software (Version 4.3.0, Azurad SAS, Marseille, France,
http://www.azurad.fr/logiciel-plans-experiences.php, accessed on 25 August 2024).
where
X1 and
X2 are coded values (−1, 0, 1) of pressure and temperature, respectively, and b
ij are the polynomial coefficients (dimensionless).
These 9 experiments were optimally selected using mathematical criteria (variance prediction functions) to calculate the Sovová’s BICM parameters in all the domain of interest by using Equation (18).
4. Conclusions
The predictive methodology developed in this study allows the prediction of SC-CO2 extraction kinetics at various operating conditions of pressure and temperature (at a constant CO2 flow rate and a constant mean particle size diameter) on various types of biomass inside an operating domain. Both extraction of type A and B can be predicted with satisfactory accuracy. It was also possible to predict the presence or absence of the retrograde solubility zone and also when it should end. The advantage of this methodology is that it does not need the use of complex tools (NN or integration of partial differential equations) and does not require a large number of experimental data (higher the 150 experimental points).
An experimental design of nine extraction kinetics (each extraction kinetic consist in five up to ten experimental points) given in
Table 2 (considering the variation of pressure and temperature at a set particle size distribution and CO
2 mass flow rate), and the analytical solution of the BICM proposed by Sovova (type A, B, C, or D) are needed to be able to use this predictive method. The use of this new innovative tool is powerful as it allows for a satisfactory prediction of an extraction kinetic inside the experimental operating domain without performing any additional extraction experiments. In addition, the calculations performed by this new method give access to all BICM parameters (also
Cu), which is not the case with the other published methods. The predicted BICM parameters were found to be consistent with the modeled parameters. In the worst case, it was possible to predict the parameters with the good order of magnitude, which is encouraging.
The aim of this manuscript is to encourage researchers to apply this methodology to various types of biomass considering the four BICM extraction types. To deepen the work proposed here, the next step will be to establish the RSM matrix in order to study the effects of various operating condition parameters (pressure, temperature, and SC-CO2 flowrate or pressure, temperature, and particle size distribution, etc.) to extend the applicability of the PBICM we have developed.