Kinetics of green solid-liquid extraction of useful compounds from plant materials: kinetics coefficients and modeling

1 Introduction

Solid-liquid extraction is a process which gives products that are widely used in the pharmaceutical, cosmetics, tobacco, and food industries and also applied for environmental purposes. The latter provoked numerous scientific studies focused on the process kinetics [1], [2], [3], yield of extraction [2], [4], [5], [6], [7], and process design [8], [9].

The mathematical modeling is a powerful tool for equipment optimization, simulation, design, and control allowing theoretical description of the process and evaluation of the mass transfer coefficients [3], [10], [11], [12]. The possibility to extract maximum valuable compounds in short time is of significant interest for the practice.

In scientific literature, the processes of solid-liquid extraction of biologically active substances from various natural materials are described and simulated by several mathematical models. Some of the extensively applied theoretical models: of Crank [13], the film theory and unsteady state diffusion model [14], are derived from Fick’s second law of diffusion [6], [15]. Model equations that are based on a second-order rate law are used for the modeling of conventional and nonconventional extraction processes [16]. Commonly applied empirical models were proposed also by So and Macdonald, Patricelli, Peleg, Perez and applied for modeling of rapeseed extraction, oil extraction and wild sunflower extraction, respectively [17], [18]. The kinetics equations of Ponomaryov and a model based on nonstationary diffusion through the plant material were applied for describing the kinetics of total extractives and inositol hexaphosphate from sesame seeds [19]. Resinoid extraction from Hypericum perforatum L. aerial parts was successfully modeled by the parabolic diffusion model, power law model, Weibull’s and Elovich’s equations [20]. The effect of particle size and solid-liquid ratio on the kinetics of green solid-liquid extraction of andrographolide from Andrographispaniculata (AP) were investigated. The second-order rate law showed a better agreement with the experimental data. It was established that the extraction rate coefficient significantly increased with a decrease in particle size and an increase in solid-liquid ratio [21]. So and Macdonald’s model was suitable to describe the kinetics of the three stages (fast initial washing period, fast diffusion, and slow diffusion) of caffeine and total catechins extraction from green tea at different pressures [17]. Two mathematical models were developed by Winitsorn et al. [22] for solid-liquid extraction from tamarind seed coat and green tea. When comparing results of neglecting and considering external mass transfer resistance, both models displayed satisfactory applicability for prediction of the extraction yield.

In order to set up an experimental installation the diffusion type and the factors that are of main importance for the process have to be known. There are three different approaches for that purpose: analytical solution for the three classic shapes of the solid phase, numerical solutions based on simplifications, and experimental data description applying the method of standard or characteristic functions [9], [23].

The extraction of valuable components from plant raw material is almost ever limited by the mass transfer inside the pores of the solid phase. Each experimental kinetic curve (dependence) represents in a hidden way all the factors influencing the diffusion process velocity like: polydispersion, anisotropy, solid particles form, and liquid phase concentration change. These factors are given quantitatively by the effective diffusion coefficient (D_eff) [3], [12], [23].

Such a kinetic curve can be represented by an equation of the type:

where A, B, and H are constants determined numerically on the basis of the experimental data that have specific physical meaning, compared with the analytical solutions of the process of solid phase extraction [12]. Quantitatively, these factors are reported by the effective diffusion coefficient (D_eff). The exact calculation of D_eff is important for the process engineering. The combination of the experimental and process analytical data is used for the calculation of D_eff using the method of Regular regime [12], [23].

The aim of this work is to study experimentally the effect of agitation rate on the diffusion mode during the extraction of valuable target components from plant raw materials, as well as to determine the values of the kinetics coefficients (effective diffusivity, D_eff and partial mass transfer coefficient, k) by the Regular regime method.

2 Materials and methods

3 Kinetics study, diffusion coefficients, and modeling

4 Results and discussion

4.1 Kinetics study

4.1.1 Effect of agitation rate on the rate limiting stage of the extraction process

Series of experiments with both studied systems at various agitation rates, extraction temperature t=20°C and solvent-water, were conducted.

System I Nicotiana tabacum L. leaves-water: Five experiments at agitation rates n=1, 1.5, 2, 3; and 4/s, solid/liquid ratio ξ=0.04 m³/kg and extraction time τ=900 s were conducted. The derived relationship between the concentration of the tobacco concrete in the liquid phase and the agitation rate is graphically displayed in Figure 1.

Figure 1:

Dependence of tobacco concrete liquid phase concentration on the agitation rate at t=20°C; ζ=0.04 m³/kg and solvent water.

System II Geranium sanguineum L. roots-water: Figure 2 presents the dependence of the liquid phase concentration of the extractable components on the agitation rate. The plot was obtained by the data from five experiments conducted at agitation rates n=0.5, 1, 1.5, 2, and 4/s, solid/liquid ratio ξ=0.03 m³/kg and extraction time τ= 3600 s.

Figure 2:

Dependence of Geranium sanguineum L. liquid phase concentration on the agitation rate at t=20°C; ζ=0.03 m³/kg and solvent water.

The plots on Figures 1 and 2 displayed that at agitation rates n>1.5–2 s⁻¹, the process was limited by external diffusion, i.e. the liquid concentration of the extractable components remains constant at these conditions. Consequently, the external mass transfer resistance was eliminated.

4.1.2 Kinetics of periodical batch extraction

The kinetics of periodical batch extraction from the plant raw materials – tobacco (Nicotiana tabacum L.) and red geranium (Geranium sanguineum L.) in a stirred tank was investigated at the following working conditions:

• for extraction from tobacco leaves – extraction temperature t=20°C, solvent water, solid/liquid ratio ξ=0.05 m³/kg and agitation rate n=3 s⁻¹;

• for extraction from red geranium roots – extraction temperature t=20°C, solvent water, solid/liquid ratio ξ=0.04 m³/kg and agitation rate n=4 s⁻¹.

The kinetics at constant technological parameters (solvent, solid/liquid ratio, and temperature) could be described by an equation of type (1), where A, B, and H are constants determined numerically on the basis of the experimental data. The values of these constants are presented in Table 2.

Table 2:

Values of the constants A, B, and H in Eq. (1).

Plant raw material	Working conditions	A	B	H×103
Tobacco leaves	t=20°C; solvent – water;	6.443	6.234	0.549
	ξ=0.05 m3/kg; n=4 s−1
Red geranium roots	t=20°C; solvent – water;	3.883	3.705	0.254
	ξ=0.04 m3/kg; n=3 s−1

The experimental kinetic curves are presented in Figure 3 for System I and in Figure 4 for System II.

Figure 3:

Dependence of concentration on time at t=20°C; ζ=0.05 m³/kg and solvent water for System I.

Figure 4:

Dependence of concentration on time at t=20°C; ζ=0.04 m³/kg and solvent water for System II.

4.2 Numerical solution and determination of the diffusion coefficients by the Regular regime method

The programming environment Matlab 8.0.1 was used for modeling of the obtained experimental data. For accurate modeling of an extraction process, the effective diffusivity, in the equation describing solid phase mass transfer, has to be approximated with an appropriate function dependent on time.

The results in the present study were obtained by applying Eq. (10) and the solid phase form was accounted:

• for tobacco leaves – R=0.75×10⁻³ m; B₁=8/π²; μ₁=π/2; ρ_s=930 kg/m³; ε=0.7 m³/m³;

• for red geranium roots – R=0.5×10⁻³ m; B₁=6/π²; μ₁=π; ρ_s=1100 kg/m³; ε=0.4 m³/m³.

A four-parameter model for the effective diffusivity was suggested:

(11)Deff=a⋅ebτ+c⋅edτ

The parameters there in can be determined by linear regression of the experimental data.

The applicability of the model was analyzed based on the experimental data and model plots presented in Figures 5 and 6 for System I and System II, respectively.

Figure 5:

Experimental data and model curve of D_eff for System I at t=20°C and ξ=0.05 m³/kg.

Figure 6:

Experimental data and model curve of D_eff for System II at t=20°C and ξ=0.04 m³/kg.

The results obtained display that the value of D_eff decreases with time to a final value. The authors suppose that the reduction observed is related to the fact that with the progress of the process, extraction of the substance from pores that were previously unavailable or too hard to be reached, takes place. These are usually micropores, in which the mass transfer is limited and the general mass transfer as a whole decreases, i.e. the value of D_eff decreases. The intensity of the mass transfer in the boundary layer around the particle depends on the partial mass transfer coefficient k. This coefficient mainly depends on the nature of the solvent and the extractable component and its value is in the range of 10⁻⁶–10⁻⁴ m/s. Depending on the values of k and the effective diffusivity (respectively Biot number) the rate limiting stage – external or intraparticle diffusion, could be determined. The values of the partial mass transfer coefficient, k, were derived by the Regular regime method and presented in Table 3 for both studied systems.

Table 3:

Values of the constants A, B, H, Bi number, and the partial mass transfer coefficient k.

System	N	A	B	H ×103	Bi	k×106
	s−1	–	–	–	–	m/s
Tobacco-water	3	6.443	6.324	0.549	61.63	1.796
Geranium-water	4	3.883	3.705	0.254	72.31	0.192

The values of the constants A, B, and H are obtained at time τ=900 s for the system tobacco-water and at τ=3600 s for the system geranium-water.

5 Conclusions

The kinetics of extraction of valuable target compounds from tobacco leaves (Nicotiana tabacum L.) and red geranium roots (Geranium sanguineum L.) with solvent water at periodic conditions in a batch mode was experimentally investigated. The effect of agitation rate on the regime of the extraction process was determined by means of the calculated Biot number Bi 61.63 for System I and Bi 72.31 for System II. The experimentally obtained kinetics data were presented by the Regular regime method for both studied systems. The partial mass transfer coefficients: k=1.796×10⁻⁶ m/s for System I, and k=0.192×10⁻⁶ m/s for System II, were calculated. An empirical four-parameter model for variable effective diffusivity, D_eff, was suggested. The empirical solutions that were obtained in the recent study display its satisfactory applicability to the experimental kinetics data of extraction of valuable target compounds from plant raw materials.