A New Empirical Model for Calculation the Effective Diffusion

Article pubs.acs.org/IECR

A New Empirical Model for Calculation the Effective Diffusion Coefficient for Solid−Liquid Extraction from Plants Chavdar Chilev, Velichka Koleva,* and Evgeni Simeonov University of Chemical Technology and Metallurgy 8 Kl. Ohridski, 1756 Sofia, Bulgaria ABSTRACT: The current work is concerned with obtaining a new empirical model for the time-variable effective diffusion coefficient (Deff) by extraction from plant materials. It has been applied for describing a solid liquid extraction from Cotinus coggygria and compared with experimentally obtained extraction kinetics. The experiments have been performed by changing the liquid−solid ratio (ξ = 0.01, 0.02, and 0.03 m3 kg−1), temperature (20, 40, and 50 °C), and extraction solvents (water, 50% ethanol in water, and 70% ethanol in water). A nonlinear seven-parametric model for Deff is obtained in order to represent the influence of those parameters on the extraction process and Deff, respectively. The parameters of the model are obtained by a nonlinear regression of Deff values calculated using the method of regular regime. All experimental conditions have been taken into consideration. A very good coincidence between experimental and empirically obtained data has been found. This is a proof for the ability of the function to describe an extraction process. It can be used to estimate and control the process.

1. INTRODUCTION The solid−liquid extraction from plant materials is a process whose products are widely used in pharmaceutics, cosmetics, and the food industry, and it has also application to environmental purposes. The solid−liquid extraction from Cotinus coggygria is actually due to the possible application of extracts in the medical and pharmaceutical industry. Cotinus coggygria contains mainly flavonoids, tannins, oils, and potassium salts, and it has antiseptical, anti-inflammatory, and styptic properties. This provokes numerous scientific studies focused on the process kinetics, the diffusivities, the yield of extraction, or the process design. The modeling is a powerful tool for optimization of the equipment, simulation, design, and control, thus allowing theoretical description of the process and evaluation of the mass-transfer coefficients. Defining the effective diffusion coefficient (Deff) is critical for the mathematical modeling of the solid−liquid extraction and the design of equipment for plant material extraction. Deff quantitatively represents all physical and physiochemical processes in the solid phase. Its value is a function of the solid material and extractable physical properties, concentration, temperature. However, it is not influenced by the conditions on the phase boundary.1 The effective diffusion coefficient is a main kinetic parameter for the extraction from plant material, since this process is usually limited by the internal diffusion.2−4 ⎛ε⎞ Deff = D⎜ ⎟ ⎝σ ⎠

Many studies of the parameter Deff show that its value can decrease or remain constant during the solid−liquid extraction process. By assuming that Deff is a constant, it can be obtained by solving the one-dimensional Fick’s law for the three “classical” shapes of solid phase: plate, sphere, and cylinder.15−17 However, Deff usually varies during the process. Different ways of describing this can be found in the literature, such as viewing Deff as an exponential function of the extractable concentration in the solid phase18 or solid concentration and changing the internal porosity of the solid material during the process.4 There are currently plenty of investigations on the possibility of theoretically describing solid−liquid extraction, and Deff estimation especially. They describe the difficulties by obtaining the analytical and numerical solutions, and especially their adequacy by designing the extraction process in production scale.2−4,15 The use of empirical models is an alternative. They have very limited possible applications, since they do not show the physical meaning of the complicated nonstationary diffusion process in solid phase and do not summarize a certain class of phenomena.19 With this empirical model, we would like to present a relatively easy and adequate solution for the engineering of plant extraction, which can be implemented in the control devices for real-time management of the extraction process. The objective of the current work is to investigate the different mathematical models that have been used to describe the variation of Deff as influenced by extraction process conditions such as time, temperature, liquid−solid ratio, and solvent. Furthermore, based on that knowledge, our intention

(1)

where D is the diffusivity of the solute in the solvent, ε the internal porosity of the solid, and σ the tortuosity of the pores. Thus, Deff is proportional to the free diffusivity, which depends on several structural factors, such as porosity, geometry of capillaries, ratio between extractable molecule size and pore size, etc.5−14 © 2014 American Chemical Society

Received: Revised: Accepted: Published: 6288

July 30, 2013 February 25, 2014 March 24, 2014 March 24, 2014 dx.doi.org/10.1021/ie402473r | Ind. Eng. Chem. Res. 2014, 53, 6288−6296

Industrial & Engineering Chemistry Research

Article

was to create a model that would most accurately describe the impact of all factors on Deff.

C0 − C 2 1 = − C0 − Cm 1+β

n=1

4(υ + 1) μn 2 + 4(υ + 1)2 β(1 + β)

⎛ μ2 Deff τ ⎞ × exp⎜⎜ − n 2 ⎟⎟ R ⎠ ⎝

2. MATERIALS AND METHODS 2.1. Plant Material. Leaves from Cotinus coggygria were used as a solid phase by the kinetic experiments. The plant has application in the fields of medicine (antiseptic, inflammatory, and styptic) and coloring. 2.2. Extraction Design. The kinetic experiments were performed in a stirred vessel. The ground raw material, with the suitable particle size, was placed in a reactor and ethanol−water solutions in different ratios were poured on top of the material. The processing temperature was kept constant and the mixture was stirred continuously. To ensure that internal diffusion was limited, the angular velocity of the mixer was controlled. After certain time of extraction, samples from the liquid were taken, filtrated through pleated filter paper (to ensure that any residual solids are removed), and prepared for analysis. 2.3. Experimental Conditions. The kinetic study was carried out by periodical extraction from Cotinus coggygria in a stirred vessel. The processing temperature was 20, 40, or 50 °C. In order to eliminate the external mass-transfer resistance, the velocity of the stirrer was maintained at n = 5 s−1. The experiments were performed using three different liquid−solid ratios: ξ = 0.01 m3 kg−1; ξ = 0.02 m3 kg−1, and ξ = 0.03 m3 kg−1. Ethanol−water mixtures of 50% ethanol in water, 70% ethanol in water, and pure water were used as process solvents. Ten grams (10 g) of the solid phase was used for each experiment and was mixed with defined solvent volume according to the desired liquid− solid ratio. The effective volume of the stirred vessel was 0.5 L. The extracts were filtered through filter paper (Boeco Germany, grade 6). The concentration of tannins in the liquid phase (C1) after the extraction was measured. Each point of the kinetic curve was established, based on the average value of three independent experiments. Other process parameters included the density of the solid phase (ρsol = 961 kg m−3), the characteristic solid particle size (R = 1.5 × 10−4 m), and the initial tannins concentration in the solid phase (C0 = 353.66 kg m−3). 2.4. Analytical Method for the Extract Analysis. The concentration of tannins in the extracts (liquid phase) was measured by the modified Loewental method.20,21 It is based on the titration with KMnO4. Two milliliters (2 mL) of the filtrated extract were added to 750 mL of distilled water containing 25 mL of indicator indigocarmin-sulfonic acid. A standard solution of KMnO4 (0.02 mol/ L = 0.1 N, Titrisol, Merck) was used for the titration. The concentration of the extracted valuable compounds is calculated using the volume of KMnO4 used for the oxidation process. 2.5. Kinetic Study, Diffusion Coefficients, and Modeling. Two approaches were considered by calculation Deff by nonstationary mass transfer: the methods of regular regime and standard function. They are both based on the comparison between experimental data by nonstationary mass transfer and the analytical/numerical data for solids in the three “classical” shapes. The proposed methods consist of analysis of the kinetic behavior during a periodical solid−liquid extraction considering the following diffusion model:

⎡ ∂ 2C ∂C 2(x , τ ) t ∂C 2 ⎤ ⎥ = Deff ⎢ 22 + ∂τ X ∂X ⎦ ⎣ ∂X

∞

∑

(3)

where C0 is the initial concentration in the solid phase; Cm = C1i by periodical processes, where C1i the initial concentration in the liquid phase, C̅ 2 the average concentration in the solid phase; Deff is the effective diffusion coefficient in pores of the solid phase; R is the size of the solid particle; τ is the time; β = C1eq/(C0 − C1eq), where C1eq is the equilibrium concentration in the liquid phase; μi represents the roots of the characteristic equation; and ν = −1/2 for the plate-shaped solid phase. For a Fourier number of Fo > 0.1 (Fo = Deffτ/R), we can narrow the first term of eq 3 to C0 − C 2 4(υ + 1) 1 = − 2 C0 − Cm 1+β μ1 + 4(υ + 1)2 β(1 + β) ⎛ μ 2 Deff τ ⎞ × exp⎜⎜ − 1 2 ⎟⎟ R ⎠ ⎝

(4)

and, by comparison of this equation with the standard function (eq 5),

C0 − C 2 = A Φ − BΦ exp(− HΦτ ) C0 − Cm

Φ*(τ ) =

(5)

It is obvious that the coefficients A, B, and H are respectively equal to16

AΦ = BΦ =

HΦ =

1 1+β μ12

(6)

4(t + 1) + 4(t + 1)2 β(1 + β)

(7)

μ12 Deff R2

(8)

by A ≅ 1 (β = 0), the coefficient of the effective diffusion Deff is

Deff =

HΦR2BΦ 2(t + 1)

(9)

where t = 0 for the plate-shaped solid phase. 2.5.2. Method Using the Regular Regime. This method is based on a comparison between the experimentally obtained data by a nonconstant mass transfer from the solid into the liquid phase with the analytical solutions obtained under the same conditions of mass transfer. It is known that, at the beginning of this regime, Deff = constant and for eq 2 for the plate-shaped solid phase by τ = 0,C2 = C2 = constant, and C1i = 0, we can obtain C2 = C0

∞

∑ 1

⎡⎛ ⎤ π2 ⎞ 8 2 Deff τ ⎢ ⎥ − − exp (2 1) n ⎜ ⎟ 4⎠ π 2(2n − 1)2 R2 ⎦ ⎣⎝

(10)

where C1 is the concentration in the liquid phase and C1i represents the initial concentration in the liquid phase. For Fo > 0.1, we can narrow eq 10 to its first term:

(2)

where C2 is the concentration in the solid phase. There are two methods for determination of Deff based on the combination between analytical solutions of eq 2 and experimentally obtained C2 = f(τ) for the boundary case Bi → ∞; β → 0;3,16 here, Bi is the Biot number (Bi = kR/Deff) and k is the external coefficient of mass transfer (given in units of m/s). 2.5.1. Method Using the Standard Function. This method is based on the comparison between the analytical solution and the standard function. Equation 2, in combination with boundary conditions, gives the following results for the three classical forms of the solid phase:3,16

⎛ D τ⎞ C̅ 2 = B1 exp⎜ − μ eff2 ⎟ ⎝ C0 R ⎠

(11)

For the investigated system Bi → ∞;B1 = 8/π2; μ1 = π/2,1 eq 11 becomes

⎛C ⎞ ⎛D τ⎞ log⎜ 2 ⎟ = log B1 − 0.434μ12 ⎜ eff2 ⎟ ⎝ R ⎠ ⎝ C0 ⎠ 6289

(12)

dx.doi.org/10.1021/ie402473r | Ind. Eng. Chem. Res. 2014, 53, 6288−6296


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Table 1. Experimental Data for Extracted Tannin Concentration in the Liquid Phase under Different Process Conditions C1 (kg m−3)

C1 (kg m−3)

τ (s)

T = 20 °C

T = 40 °C

T = 50 °C

600 900 1200 1800 3600 7200

4.3649 4.9884 6.2355 6.4434 7.6905 8.5219

7.27 7.69 8.52 9.14 9.77 10.39

7.4826 8.9376 8.9376 10.1847 10.1847 10.8082

ξ = 0.01 m kg 3

14.3417 14.5495 16.7319 19.1222 19.5379 19.5379

−1

C1 (kg m−3)

ξ = 0.03 m kg 3

4.3649 4.9884 5.8198 6.4434 6.2355 6.8591

−1

H2O 3.1178 4.1570 4.7806 5.8198 5.8198

70% EtOH/H2O 7.2748 7.5865 8.9376 9.5611 10.6004

3. RESULTS AND DISCUSSION 3.1. Kinetic Results. The experimental data for target compounds extraction under different process conditions are presented in Table 1. This table presents the experimental results obtained under different working conditions for carrying out kinetic experiments. The experimentally received data for the extraction kinetics for the investigated solid−liquid system can be described with an acceptable accuracy by eq 13: C1 = A − B exp( − Hτ )

(13)

The values of A, B, and H are estimated by nonlinear regression of experimental data and presented in Table 2 for different temperatures, solid−liquid ratios, and different extraction solvents. Table 2. Coefficients A, B, and H in the Model Equation (eq 13) A

B

H

R

Figure 1. Mathematical modeling of experimental results by different process temperatures using eq 13.

2

For Different Temperatures T = 20 °C 8.089 7.958 0.00113 9.842 9.754 0.001872 T = 40 °C; ξ = 0.02 m3 kg−1; 50% ethanol in water T = 50 °C 10.41 10.38 0.002025 For Different Solid/Liquid Ratios ξ = 0.01 m3 kg−1 19.43 19.28 0.001843 ξ = 0.03 m3 kg−1 6.597 6.591 0.001724 For Different Extraction Solvents water 5.962 5.955 0.000924 70% ethanol/water 9.819 9.738 0.001898

0.9828 0.9863 0.993 0.9872 0.9918 0.9955 0.9786

Every experimentally obtained kinetic curve (presented by eq 13) contains, in a hidden way, all the factors influencing the complicated unstationary diffusion process of extraction. The standard function is obtained using Laplace transformations of eq 13. Equation 12 is used to describe the regular regime and the values for C2 are obtained by eq 13 for each extraction time. Figure 1 presents the experimental data and their mathematical description using eq 13 at different process temperatures, ξ = 0.02 m3 kg−1, and with 50% ethanol in water as the extraction solvent mixture. There is a very good correlation between the experimental and the mathematical model data. By increasing the process temperature the concentration of extractables increases for the same period of extraction. At 50 °C, the equilibrium concentration of tannins in the liquid phase is C1eq = 10.39 kg m−3. Figure 2 shows a comparison between the kinetics using different liquid−solid ratios at 40 °C and 50% ethanol in water as the liquid phase. The valuable compounds concentration in

Figure 2. Mathematical modeling of experimental results by different liquid−solid ratios using eq 13.

the solvent phase clearly has the highest value for ξ = 0.01 m3 kg−1 by equal other conditions. Figure 3 shows the tannins concentration in the liquid phase by applying different solvents/solvent combinations in the extraction process; t = 40 °C and ξ = 0.02 m3 kg−1. It has its lowest value by extraction with water, and it reaches equilibrium at C1eq = 9.83 kg m−3, using a 50% ethanol in water solution, and C1eq = 9.94 kg m−3 using a 70% ethanol in water solution. 3.2. Presenting the Kinetics via the Method of Standard Function. The standard function (eq 5) is 6290



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solvent concentration) influencing Deff, those factors have been studied separately. The values of Deff using the method of regular regime are used as a base, and since they are obtained directly from the experimental data, they are called “experimental results”. The value of Deff changes the most with time, i.e., the parameter time has the bigger impact on Deff. Therefore, using mathematical analysis and the choice of model function, the variable of time (τ) is taken as the primary influence and the influence of the other parameters is always considered with respect to the time. By choosing the right model function to describe the influence of parameters on Deff, many equations were tested. Only those giving the best accuracy are presented in the current article. 3.4.1. Extraction Time Influencing Deff. The main extraction kinetic parameter for internal diffusion regime is Deff of the target compounds in the pores of the solid phase. It was obtained using the methods of regular regime and standard function. The first method allows one to determine the change of Deff in time until reaching a constant value, i.e., regular regime. For the method of standard function, a constant Deff value is used. Both methods are compared for their ability to accurately describe an extraction process and match the experimentally obtained data. Figure 4 presents the obtained

Figure 3. Mathematical modeling of experimental results by different solvents/solvent mixtures using eq 13.

calculated based on the kinetic equation (eq 13). In order to use this method for presenting the extraction kinetics, it is necessary that the extraction process reaches equilibrium and an internal diffusion regime (Bi > 30)18 is present. The coefficients of effective diffusion (Deff) are estimated using the method of standard function by eq 8, and the results are presented in Table 3. Table 3. Coefficients of Effective Diffusion Calculated Using the Standard Function Deff (× 10−11 m2 s−1)

parameter

For Different Temperatures T = 20 °C T = 40 °C; ξ = 0.02 m3 kg−1; 50% ethanol/water T = 50 °C For Different Liquid/Solid Ratios ξ = 0.01 m3 kg−1 ξ = 0.03 m3 kg−1 For Different Solutions water 70% ethanol in water

1.1937 2.0301 2.2051 1.9460 1.9017 1.0269 2.0598

Figure 4. Experimental results for Deff and its modeling by eq 14.

3.3. Presenting the Kinetics via the Method of Regular Regime. The method of regular regime (eq 12) was used to calculate Deff based on all values of concentration in Table 1. The results are listed in Table 4. 3.4. Parameters Influencing the Value of the Coefficient of Effective Diffusion (Deff): Discussion. To obtain a multidimensional nonlinear function representing all different factors (time, temperature, liquid−solid ratio, and

values by the method of regular regime by extraction of tannins from Cotinus coggygria at a temperature of T = 40 °C, a liquid− solid ratio of ξ = 0.02 m3 kg−1, and 50% ethanol in water as a solvent mixture. They are called Deff‑exp, since they were calculated directly from the experimental data for tannins concentration in the liquid phase.

Table 4. Coefficients of Effective Diffusion Calculated Using the Method of Regular Regime Deff (× 10−10 m2 s−1)

Deff (× 10−10 m2 s−1)

τ (s)

T = 20 °C

T = 40 °C

T = 50 °C

600 900 1200 1800 3600 7200

0.2105 0.1489 0.1265 0.0862 0.0495 0.0274

0.2827 0.1978 0.1644 0.1194 0.0659 0.0370

0.2896 0.2320 0.1740 0.1419 0.0710 0.0406

ξ = 0.01 m kg 3

0.2796 0.1886 0.1612 0.1273 0.0659 0.0329 6291

−1

Deff (× 10−10 m2 s−1)

ξ = 0.03 m kg 3

0.2613 0.1931 0.1691 0.1295 0.0616 0.0362

−1

H2O 0.1251 0.1032 0.0730 0.0404 0.0202

70% EtOH/H2O 0.2829 0.1954 0.1160 0.0637 0.0387



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The function Deff = f(t) is monotonically increasing. It is presented in Figure 6 for τ = 1200 s and data from Figure 5.

Deff changes exponentially with the time. Similar behavior was observed by all studied cases (different process conditions). Thus, the function of Deff, relative to the time, can be described with an equation such as Deff = a exp( − bτ ) +c exp( − dτ )

(14)

This is a four-parameter model, where the parameters (a, b, c, d) can be obtained by nonlinear regression of the experimental data. This modeling is presented in Figure 4 and a = 2.63 × 10−11, b = −0.001416, c = 9.818 × 10−12, d = −0.0001964; R2 = 0.9977. There is a very good correlation between the experimental and calculated data. 3.4.2. Process Temperature Influencing Deff. Figure 5 presents the values of Deff by extraction from Cotinus coggygria

Figure 6. Deff experimental data approximation using eqs 15 and 16.

The nonlinearity of the function is clear. Other approximationssecond-grade-polynomial and power functionalso were performed. The results also are presented in Figure 6. Deff = p1 T 2 + p2 t + p3

(15)

Deff = aT b

(16)

The obtained coefficients are as follows: p1 = −3.117 × 10−15; p2 = 3.765 × 10−13; p3 = 6.367 × 10−12; a = 4.255 × 10−12; b = 0.363. The graphical interface “cftool” in the software program Matlab was used. It is clear from the graph that the polynomial function described by eq 15 is more accurate (R2 = 1) in the description, compared to eq 16 (R2 = 0.993). However, from an experimental point of view, both methods are accurate enough to describe the data in the interval from 20 °C to 50 °C. To define Deff as a function of the process temperature and time of extraction (Deff = f(T,τ)) a combination of eqs 1,2, and 3 was used:

Figure 5. Experimental values of Deff at different process temperatures.

at different process temperatures, with all other conditions kept equal. The values are calculated by both methodsregular regime and standard function. Please note that Deff calculated by the method of standard function (marked as Dst on the graph) has only one value, i.e., it is independent of time. Therefore, Dst is represented by bold symbols in Figure 5. It is placed in such a way (within the time frame) that its value can be compared with that obtained via the regular regime method. For example, the value of Deff calculated by the standard function at 20 °C is marked with a bold plus sign (“+”) on the graph for time τ = 1200 s, in order to compare it with the value from the regular regime method (marked with a regular plus sign (“+”) on the graph) for that time of extraction, with all other conditions kept equal. The same rule applies for other studied process temperatures (see Figures 8 and 11, presented later in this work). The processing temperature influences the coefficient of effective diffusion in the following way: on one hand, by changing the temperature, the free path of the molecules in the liquid phase also changes. That is a direct influence of the temperature on Deff. On the other hand, by increasing the temperature, the solubility of the target compounds in the liquid phase also increases. This leads to intensifying the extraction process. Hence, Deff should increase by increasing process temperature. This fact is confirmed by the experimental results (presented in Figure 2) obtained by the methods of regular regime and standard function.

Deff = (aT 2 + bT + c)[exp( − dτ ) +f exp( − gτ )]

(17)

Deff = (aT b)[exp( − cτ ) +d exp( − fτ )]

(18)

In eqs 17 and 18, the variables are the temperature (T) and the time (τ); a, b, c, d, f, and g are coefficients of the model functions. Figure 7 shows the experimental points from Figure 5 and their modeling by eqs 17 and 18. A good alignment between the experimental and calculated values is present. The nonlinear regression is calculated using the “nlinfit” function in MatLab. To optimize the fitting process results, it is very important to state an appropriate initial approximation variation interval of the parameters. Therefore, the results from the one-dimensional fitting of eqs 14, 15, and 16 are used as initial approximations for the parameters in eqs 17 and 18 (see Table 5). The surfaces obtained using eqs 17 and 18 are basically the same. In terms of accuracy, both equations are applicable. However, by using nonlinear regression, it is preferable to use 6292



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Figure 7. Experimentally obtained values for Deff and their mathematical modeling using eqs 17 and 18.

Figure 8. Experimental values of Deff by different liquid−solid ratios.

equations with smallest possible number of parameters in order to limit the complexity. Therefore, eq 18 is easier to be applied for experimental data description because it is more practical. 3.4.3. Liquid−Solid Ratios Influencing Deff. The liquid− solid ratio in an extraction process influences indirectly Deff by affecting the kinetic of the experiment. In order to study this effect, a series of experiments were performed using three different liquid−solid ratios: ξ = 0.01 m3 kg−1, ξ = 0.02 m3 kg−1, and ξ = 0.03 m3 kg−1. The other process conditions were kept constant: solid particle size, R = 1.5 × 10−4 m; temperature, T = 40 °C; and 50% ethanol in water as solvent. Figure 8 presents the values of Deff obtained by the methods of regular regime and standard function (Dst) at different liquid−solid ratios. There is a good alignment between the data calculated by the two different methods. A comparison between Figures 8 and 5 leads to the conclusion that the influence of the liquid−solid ratio on Deff is much less than that of the processing temperature. Figure 9 shows the variation of Deff, depending on the liquid−solid ratio at different extraction times (600, 900, 1200, 1800, 3600, and 7200 s). Deff clearly varies differently over time for the different ξ values. For example, Deff decreases by increasing ξ at τ = 600 s and vice versa at τ = 1200 s. At a time of τ = 1800 s, the initial Deff value decreases as ξ increases from 0.01 to 0.02 (i.e., the first and second points at this time).

Figure 9. Variation in the values of Deff by different liquid−solid ratios at different extraction times.

When ξ decreases from 0.02 to 0.03 (i.e., the second and third points at the time τ = 1800 s), Deff increases. In all of these cases, Deff = f(ξ) is a nonlinear function, depending on the

Table 5. Values Obtained for the Parameters of eqs 17 and 18 parameter

initial approximation

a b c d f g

−3 × 10−15 4 × 10−13 6 × 10−12 −1.6 × 10−3 10 × 10−12 −2 × 10−4

a b c d f

4.241 × 10−12 0.3644 −1.595 × 10−3 9.409 × 10−12 −1.816 × 10−4

result Equation 17 −9 × 10−15 1.314 × 10−12 2.087 × 10−12 −1.162 × 10−3 8.047 × 10−12 −1.092 × 10−4 Equation 18 3.805 × 10−12 0.6288 −1.145 × 10−3 7.805 × 10−12 −1.043 × 10−4 6293

variation interval −3.3 × 10−16 3.69 × 10−13 2.602 × 10−12 −1.329 × 10−3 1.211 × 10−12 −4.278 × 10−5

−1.4 × 10−14 2.998 × 10−12 3.019 × 10−11 −6.952 × 10−3 1,488 × 10−11 −2.611 × 10−4

7.98 × 10−13 0.3135 −7.042 × 10−4 1.247 × 10−12 −4.442 × 10−5

8.408 × 10−12 0.9441 −1.585 × 10−3 1.436 × 10−11 −2.53 × 10−4



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liquid−solid ratio, and the following polynomial (similar to eq 15) was chosen for its best description: Deff = p1 ξ 2 + p2 ξ + p3

(19)

Based on eq 19, a two-dimensional nonlinear function, Deff = f(ξ,τ), was developed: Deff = (aξ 2 + bξ + c)[exp( − dτ ) +f exp( − gτ )]

(20)

where the variables are the liquid−solid ratio ξ and time τ; a, b, c, d, f, and g are coefficients of the model. The experimental points from Figure 9 and their modeling (using eq 20) are shown in Figure 10. There is a good comparison between the experimental and calculated results. The nonlinear regression is obtained by the “nlinfit” in MatLab software.

Figure 11. Experimental values of Deff for different extraction solvents/ solvent mixtures.

Figure 10. Experimental results of Deff and their modeling using eq 20.

3.4.4. Extraction Solvent Influencing Deff. The solvent/ solvent mixture used for extraction directly influences Deff. The type of solvent affects the mobility of the liquid molecules and free diffusion (Poiseuille diffusion). However, it has an indirect impact on the extraction kinetics and Deff. Three different solvent/solvent combinations were used in this study: pure water, 50% ethanol in water, and 70% ethanol in water. In order to investigate their influence on the extraction, the other process conditions were kept constant: T = 40 °C; solid particle size, R = 1.5 × 10−4 m; and ξ = 0.02 m3 kg−1. The values of Deff calculated by the methods of regular regime and standard function (see Tables 3 and 4). There is a good correlation between the values of Deff obtained by both methods (regular regime and standard function) for different solvent/solvent combination (see Figure 11). Deff increases in value by adding ethanol to pure water (i.e., ethanol improves the kinetics of extraction). However, its concentration in the interval of 50%−70% has very limited impact on Deff. The values of Deff for different extraction solvents at 900 s are shown in Figure 12. A polynomial is used for the function approximation. Deff = p1 c 2 + p2 c + p3

Figure 12. Approximation of experimental results for Deff using eq 21.

To calculate Deff by eq 21, a minimum value of c = 0.00001 is used. This assumption was done in order to avoid unreal values for Deff in the case of pure water. A similar relation between Deff and extraction solvent was observed for different extraction times. Equation 21 was used as a base for a two-dimensional nonlinear function to describe Deff, depending on the extraction solvent and time (Deff = f(c,τ)). Deff = (ac 2 + bc + c)[exp( − dτ ) +f exp( − gτ )]

(22)

where the variables are the ethanol concentration in the extraction solvent c and time τ; a, b, c, d, f, and g are coefficients of the model. 3.4.5. Combined Impact of All Studied Parameters on Deff. The impact of different parameters on Deff has been studied separately in points 1−4. Generally, for each solvent(s)− extractable(s) system, Deff changes over time and is dependent on the processing temperature, liquid−solid ratio, and type of solvent (ethanol concentration, in this case).

(21)

Deff = f (T , ξ , c , τ )

where c is the concentration of ethanol in water. 6294

(23)



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where T is the temperature (°C), ξ the liquid−solid ratio (m3 kg−1), c the ethanol concentration (m3(EthOH) m−3(EthOH +H2O)), and τ the time (s). The results obtained above are used to explicitly define the function given as eq 23. Deff changes the most with the time. Therefore, eq 14 is taken as a base. To include other process parameters in the function, the constant a is presented as a = g(t,ξ,c): Deff = f [g (T , ξ , c) exp( − bτ )+c exp( − dτ )]

(24)

The process temperature is the next parameter of importance that influences Deff. Equation 16 is used to explicit the function g = g(t,ξ,c). Equation 24 transforms to Deff = f [q(ξ , c)T b exp( − bτ ) +c exp( − dτ )]

(25)

Equation 25 is similar to eq 18, only the parameter a in eq 18 is presented as a function of two variables: a = q(ξ,c). A polynomial is used to explicitly define q = q(ξ,c): q(ξ , c) = a(ξc)2 + bξc + d

Figure 13. Experimental results for Deff by different temperatures and their modeling by eq 27.

(26)

By combining eqs 25 and 26 into eq 23, we obtain the following: Deff = f [(a(ξc)2 + bξc + d)T f exp( − mτ )+n exp( − pτ )] (27)

Equation 27 is a seven-parameter model and its parameters a, b, d, f, m, n, and p can be calculated via nonlinear regression of the values for Deff obtained by the method of the regular regime for all process conditions. Table 6 summarizes the values of the parameters of eq 27. Table 6. Parameter Values for eq 27 parameter

value

a b d f m n p

2.009 × 10−8 4.127 × 10−10 1.185 × 10−12 0.6261 −1.113 × 10−3 7.7789 x10−12 −1.121 × 10−4

Figure 14. Experimental results for Deff for different liquid−solid ratios and their modeling by eq 27.

Figure 13 presents the results for Deff for different process temperatures. Dexp stands for the data of Deff from Table 4, obtained by the method of regular regime, and Dcal are determined by eq 27. Good alignment of the results is present. Figure 14 summarizes the data for Deff for different liquid− solid ratios from Table 4 and model data (eq 27). There is a good correlation between the experimental and model calculated values. Figure 15 shows the values for Deff for different concentration of ethanol in the extraction solvent from Table 4 and model data (eq 27). There is a good correlation between the experimental and model calculated numbers. Considering the results presented in Figures 13, 14, and 15, we can conclude that the suggested mathematical model (eq 27) describes the behavior of Deff in the investigated range of process parameters (t, ξ, and c) very well. This function can be used successfully for modeling kinetics by extraction from plant material using water−alcoholic solutions as a solvent system. The obtained model is empirical. It has no thermodynamical explanation. The base of its creation is the numerical analysis of

Figure 15. Experimental results for Deff by different extraction solvents and their modeling by eq 27. 6295



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the experimental data, their accurate description, and simultaneous modeling of all studied process parameters.

REFERENCES

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4. CONCLUSIONS The kinetics by extraction from Cotinus coggygria was obtained. The influence of the temperature, liquid−solid ratio, and extraction solvent composition on the speed of the extraction process is quantitatively presented. The results show that the intensity of the process increases by increasing the temperature and ethanol concentration in the extracting mixture. The coefficient of effective diffusion is calculated based on the experimental extraction kinetics. The methods of regular regime and standard function were used. There is a very good correlation between the values obtained by both methods. This fact confirms the appropriate use of the methods and the quality of the experimental results. A simple nonlinear function for determining Deff for different extraction conditions was found. The model is compared with experimental values for Deff (regular regime method), and it was verified that the suggested mathematical description can be used for process modeling by extraction from plant materials. Furthermore, the model can be easily applied for numerical solution of extraction kinetics and electronic devices for extraction control. In this way, the development of the model is justified, from a practical point of view.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +35928163296. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE A = coefficient defined by eq 13 AΦ = coefficient defined by eq 5 B = coefficient defined by eq 13 BΦ = coefficient defined by eq 5 Bi = Biot number; Bi = kR/Deff C2 = concentration in the solid phase, kg m−3 C2 = average concentration in the solid phase, kg m−3 C1eq = equilibrium concentration in the liquid phase, kg m−3 C1i = initial concentration in the liquid phase, kg m−3 C1 = concentration in the liquid phase, kg m−3 C0 = initial concentration in the solid phase, kg m−3 Deff = effective diffusion coefficient, m2 s−1 D = diffusivity of the solute in the solvent, m2 s−1 H = coefficient defined by eq 13 HΦ = coefficient defined by eq 5 k = external coefficient of mass transfer, m s−1 n = rnumber of evolutions in the stirred vessel, s−1 R = particle characteristic size, m T = temperature, °C t = shape factor (t = 0 for a plate), by eq 9 x = coordinate, m

Greek Letters

ε = internal porosity of the solid, m3 m−3 Φ* = standard function μi = roots of the characteristic equation ρsol = density of the solid phase, kg m−3 τ = time, s ν = shape factor (ν = −1/2 for plate) by eq 3 ξ = liquid−solid ratio, m3 kg−1 6296


A New Empirical Model for Calculation the Effective Diffusion

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