Measurement of astaxanthin and squalene diffusivities in compressed liquid ethyl acetate by Taylor-Aris dispersion method

Separation and Purification Technology 234 (2020) 116046

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Measurement of astaxanthin and squalene diffusivities in compressed liquid ethyl acetate by Taylor-Aris dispersion method Bruno Zêzere, José M. Silva, Inês Portugal, José R.B. Gomes, Carlos M. Silva

T

⁎

Department of Chemistry, CICECO – Aveiro Institute of Materials, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal

A R T I C LE I N FO

A B S T R A C T

Keywords: Astaxanthin Squalene Diffusion coefficient Ethyl acetate Pressurized liquid

An increasing demand of consumers for natural bioactive compounds has boosted the interest of the scientific community for vegetables and fruits, with many efforts being made to identify new bioactive compounds and to develop sustainable extraction methods that preserve the compounds bioactivity. Such is the case of pressurized liquid extraction and supercritical fluid extraction using “green solvents” as supercritical CO2, ethanol, and ethyl acetate. The accurate design of these processes requires the knowledge of transport properties, namely, diffusion coefficients, D12 . Astaxanthin and squalene are two bioactive compounds whose D12 values were measured in compressed liquid ethyl acetate using a chromatographic technique. For astaxanthin the D12 values were in the range 0.817 × 10−5 − 1.223 × 10−5 cm2 s−1 (at 303.15–333.15 K and 1–100 bar) and for squalene they were in the range 1.052 × 10−5 − 1.822 × 10−5 cm2 s−1 (at 303.15–333.15 K and 1–150 bar). The influence of temperature, pressure and Stokes-Einstein coordinate was studied for both systems and explained with the most relevant theories of the literature.

1. Introduction

globulus leaves [17] and of stigmasterol from Eichhornia crassipes [18] using SC-CO2, pure or modified with ethanol. Experimental or accurately predicted D12 values of bioactive compounds in “green solvents” are both scarce and needed to implement PLE and SFE processes. In the particular case of multicomponent systems (i.e. mixtures of two or more solvents) the lack of experimental data and predictive equations for D12 is even more significant. As an alternative, the empirical mixing rules of Vignes [19] can be used to mix estimate the diffusion coefficients of mixtures, D12 , provided the D12 values of the pairs are known at the desired pressure and temperature conditions. Furthermore, if non-idealities prevail in the mixture the Maxwell-Stefan approach should be adopted [8,20]. Astaxanthin (C40 H52 O4 ) is a bioactive compound with relevant antioxidant properties. It has been reported to reduce the risk of diseases such as age-related macular degeneration and ischemia [21–25]. Astaxanthin can be found in many microorganisms and marine animals, such as shrimp, crayfish, salmon, trout, krill, microalgae or even in yeast. Due to its high concentration in marine animals, the residues of the fishing industry can be a profitable, reliable and sustainable source of astaxanthin [5,10,26]. A 3D representation of the chemical structure of astaxanthin is shown in Fig. 1 and some relevant properties can be found in Section 4.3. Squalene (C30 H50) is a terpenoid possessing antioxidant features beneficial to human health. For example, it has been shown to

Recently, the growing concern about human healthiness and the increasing demand of consumers for natural food and products, such as natural bioactive compounds, has raised the interest of both Industry and Academia for fruits and vegetables as a potential source [1–3]. Examples of such compounds are oleanolic acid, ursolic acid, astaxanthin, squalene and friedelin [4–7]. The calculation of the large-scale production of natural bioactive compounds requires the knowledge of specific transport properties, namely the diffusion coefficient (D12) . This property is critical for the design and optimization of industrial equipment and processes involving mass transfer phenomena [8,9] like pressurized liquid extraction (PLE) and supercritical fluid extraction (SFE) [7,10]. For example, PLE can be used to extract antibacterial biocompounds from Hancornia speciosa leaves using ethanol and ethyl acetate [11] and to extract lignans from Phyllanthus amarus using water, ethanol, and their mixtures [12]. In the case of SFE one may cite: the extraction of omega-3 rich oil from Ficus awkeotsang [13] and triglycerides from Aquilaria crassna seeds [14] using supercritical CO2 (SC-CO2); the extraction of watermelon oil from Citrullus lanatus (watermelon) seeds using SC-CO2, pure or modified with ethanol [15]; the extraction of tripterine from Tripterygium wilfordii using SC-CO2 modified with ethanol or ethyl acetate (EtOAc) [16]; the extraction of triterpenic acids from Eucalyptus ⁎

Corresponding author. E-mail address: [email protected] (C.M. Silva).

https://doi.org/10.1016/j.seppur.2019.116046 Received 14 June 2019; Received in revised form 28 August 2019; Accepted 7 September 2019 Available online 10 September 2019 1383-5866/ © 2019 Elsevier B.V. All rights reserved.

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Nomenclature

Apeak AARD Absmax C¯ C¯ (L, t ) CPB D D12 De DHB EtOAc H L LJ m Mi NAI NDP P PLE R0 Rc Re Rg S10 Sc SFE

t T TLSM u¯

area of the chromatographic peak average absolute relative deviation maximum absorbance of the peak average radial concentration of solute average concentration of solute at column outlet chromatographic peak broadening dispersion coefficient tracer diffusion coefficient Dean number, De = Re ζ Dymond-Hildebrand-Batchinski ethyl acetate theoretical plate’s high column length Lennard-Jones mass of a molecule molecular weight of the component i normalized absorbance intensity number of data points pressure pressurized liquid extraction column inner radius column coil radius Reynolds number universal gas constant symmetry factor at 10 % of high Schmidt number supercritical fluid extraction

time absolute temperature Tracer Liu-Silva-Macedo average linear velocity

Greek letters

εLJ,i

ζ μ1 λ ρ1 σLJ,i

Lennard-Jones energy parameter i (i = 1, 2) or mixture (i = 12) curvature ratio solvent viscosity wavelength solvent density Lennard-Jones molecular diameter of component i (i = 1, 2) or mixture (i = 12)

Subscripts 1 2 12 c

solvent solute solute-solvent pair critical property

Superscripts calc exp mix

calculated experimental mixture

Fig. 1. 3D representation of astaxanthin (C40H52O4) and squalene (C30H50). Atoms are represented as spheres with conventional color coding: hydrogen (white), carbon (black), oxygen (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

and olive oil) but concentrations are usually so small that extraction may be unfeasible. Some interesting sources are the plant Amaranthus [4,10], which is considered to have the highest concentration of squalene in the vegetal world (4.16 g squalene/kg seed), wine industry residues [29,32] and microalgae (even biomass residues from biodiesel production) [29,33]. A 3D representation of squalene chemical structure can be observed in Fig. 1, and some relevant properties can be found in Section 4.3. Ethyl acetate is a colorless liquid produced mainly to be used as a

effectively reduce serum cholesterol levels [27] and it exhibits a chemo preventive effect on colon cancer [28]. In industry, due to its hydrophobic properties it can be used to transport lipid soluble compounds [29]. It is also used in the cosmetic industry, since it reduces the possibility of allergies [30], and in food supplements [31]. Traditionally squalene was extracted from shark and whale liver oil but, in the last decades, international laws have limited shark and whale capture [29]. For this reason there is quest for squalene of vegetal origin. Squalene has been identified in many plant oils (e.g., sunflower, soybean, corn, 2

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solvent. It is considered a “green solvent” (environmental friendly) and it is approved for use in the food industry [34–36] and so it can be applied for the extraction of bioactive compounds. Diffusion data of bioactive compounds in ethyl acetate is very scarce, which highlights the pertinence and interest of this study. Hence, the diffusivities of astaxanthin and squalene in compressed liquid ethyl acetate at infinite dilution are determined by the chromatographic peak broadening (CPB) technique, and modeled by the important equations published in the literature.

[49,50]. Then the values of D12 are calculated (Eq. (2)) and the goodness of fitting is evaluated as follows: if ε < 1% the fitting is considered good; if 2% < ε < 3% the fitting is acceptable; finally, if ε > 3% the fitting is not acceptable [48,51]. The goodness of fit can also be evaluated by the empirical parameter known as symmetry factor, Si , where i denotes the height of the peak used in the calculation, which is typically 10 % (S10) . Usually, if S10 > 1.3 the peak is rejected [52,53]. For the curve fitting method to be applicable a set of restrictions must be met, namely: (i) laminar flow; (ii) negligible secondary flow effects inside the column, which means DeSc0.5<10 where De is the Dean number (De = Reζ −0.5 , where ζ = R c / R 0 is the ratio of tube coil radius, R c , to column radius, R 0 ) and Sc is the Schmidt number (Sc = μ1 /(ρ1 D12) where ρ1 and μ1 designate solvent density and viscosity, respectively) [54–56]; (iii) the concentration profiles must have an approximate Gaussian form, i.e., D /(Lu¯) < 0.01 [57]; (iv) temperature and pressure perturbations should be neglected, i.e., Lu¯/ D > 1000 [58].

2. Theoretical background 2.1. Chromatographic peak broadening (CPB) technique The CPB technique, also known as the Taylor dispersion method, is the most used experimental technique to measure diffusion coefficients [37,38]. This method is based on the fundamental work of Taylor [39–41] further developed by Aris [42]. Taylor focused his study on the dispersion of a solute in a laminar steady-state flow of a mobile phase through a tube of uniform diameter. Although not specifically thought to be used for the measurement of D12 [43], Giddings and Seager [44] used the Taylor method to measure D12 values of gases at low pressures. Later the technique was extended to dense gases by Balenovic et al. [45], to liquids by Ouano [46], and to supercritical fluids (SCF) by Swaid and Schneider [47]. In the CPB technique a sharp and small pulse of solute is injected into a carrier fluid flowing in laminar regime inside a capillary tube of inner radius R 0 . The pulse input will broaden due to the combined action of axial convection and molecular diffusion in the radial direction. The solute concentration, C¯ , measured at column outlet (z = L) is described as function of time, t , by [48]:

m ⎞ u¯ (L − ut ¯ )2 ⎤ C¯ (L, t ) = ⎛ exp ⎡− 2 ⎢ ⎥ πR Dt 4 πDt 4 0 ⎣ ⎦ ⎝ ⎠ ⎜

2.2. Modeling Eleven correlation and predictive models were tested with the experimental D12 values reported in this article. The assessed models were: the Dymond-Hildebrand-Batschinski (DHB) correlation [59–61], the DHB equation with temperature dependent VD parameter (VD = VD (T )) [62]; the predictive Tracer Lui-Silva-Macedo (TLSM) model and its derived correlations TLSMd and TLSMen [59,63,64]; the hydrodynamic predictive equations of Wilke-Chang [65,66] and TynCalus [66]; and four of the simple empirical and semi-empirical correlations of Magalhães et al. [80]. The models performance was assessed in terms of average absolute relative deviation, AARD, defined by:

⎟

(1)

AARD (%) =

where u¯ is the average linear velocity of the solvent, m the quantity of solute injected and D is defined by:

D = D12 +

100 NDP

NDP

∑ i=1

exp calc D12 − D12 exp D12

i

(4)

where NDP is the number of points.

R 02 u¯ 2 48D12

(2) 3. Materials and methods

The diffusion coefficient, D12 , can then be determined by two distinct ways, namely, using the theoretical plate height, H , and by curve fitting. For the latter, both D and u¯ (Eq. (1)) are optimized in order to minimize the root mean square error, ε , defined by:

3.1. Chemicals Astaxanthin (CAS number 472-61-7, purity of 99 wt.%) and squalene (CAS number 111-02-4, purity ≥98 (v/v) %) were purchased from Sigma-Aldrich. Ethyl acetate (CAS number 141-78-6) was purchased from CARLO ERBA Reagents S.A.S. (purity 99.99 wt.%). All chemicals were used without further purification.

1/2

t2 ⎛ ∫ (C exp (L,t ) − (C¯ (L,t ))2dt ⎞ ε = ⎜ t1 t2 ⎟ ∫t1 (C exp (L,t ))2dt ⎝ ⎠

(3)

where C exp is the experimental solute concentration and t1 and t2 are time values selected at 10 % of the maximum peak height, being t1 < t2

Fig. 2. Scheme of the experimental apparatus: (1) Ethyl acetate container, (2) syringe pump, (3) pre-heating column, (4) injector, (5) capillary column, (6) oven, (7) UV–vis detector, and (8) waste container [67]. 3

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Thus, for each solute several pulses were injected and the outlet peaks were recorded for different wavelengths in the respective range of detection, namely, 420–500 nm for astaxanthin and 250–320 nm for squalene. For all pulse inputs, the D12 values were calculated for each solute using the fitting curve method (see Section 2.1). The results presented in Fig. 3 for astaxanthin/EtOAc reveal a very small deviation of D12 and low fitting errors in the range 420–500 nm. The reproducibility of the results was tested by replicate injections and absorbance measurements at the wavelength with the lowest root mean square error, λ = 460 nm, using two solute concentrations: 0.20 mg mL−1 and 0.15 mg mL−1. For the lower concentration the D12 values exhibit smaller reproducibility hence future studies were performed with a solute concentration of 0.20 mg mL−1. Higher concentrations were not tested because the solubility of astaxanthin is very low [69] and its precipitation inside the diffusion column could occur and block the equipment. For the system squalene/EtOAc the reading wavelength was optimized by a similar process. The obtained results (not shown for simplicity) reveal that the linearity of the signal can be found between 275 and 300 nm with minimum error (ε ) at λ = 290 nm. Two concentrations of squalene were tested (0.43 g mL−1 and 0.64 g mL−1) with no significant differences in the D12 values. Further studies were performed with readings at 290 nm using the higher solute concentration simply because the signal was higher making it easier to analyze.

3.2. Experimental setup and procedure A scheme of the experimental apparatus [67] is presented in Fig. 2. The procedure involves pumping the solvent (EtOAc) from the reservoir (1) at constant flow rate using a syringe pumps (2) (Teledyne ISCO model 100 DM). The solvent is pre-heated in a stainless-steel column (3) placed inside a LSIS-B2 V/IC 22 oven (6) (Venticell, MMM Group) and fed to an open capillary column (5) (PEEK tubing, R 0 = 0.261 mm , L = 11.182 m , and R c = 0.150 m ) connected to a UV–vis detector (7) (UV Detector 2500, Knauer) set at a specific wavelength for each compound. After reaching steady-state conditions (i.e., constant pressure, temperature and baseline during 1–2 h after start-up), a small volume of solute (0.1 μL ) is injected in a short period (pulse input) using a C74H-1674 injector (4) from Valco Instruments Co. Inc. The outlet pressure is controlled by a Jasco BP-2080 back pressure regulator. The experimental conditions for the astaxanthin/EtOAc system were temperatures of 303.15, 313.15, 323.15 and 333.15 K, pressures of 1, 50 and 100 bar, and detector wavelength 460 nm (see Section 3.3). For the squalene/EtOAc system the assays were carried out at the same temperatures, pressures of 1, 75 and 150 bar and at a wavelength of 290 nm (see Section 3.3).

3.3. Setting UV–vis wavelength for optimum measurements 4. Experimental results-

The measurement of the diffusion coefficients relies on the analysis of the experimental absorbance profiles recorded at the exit of the diffusion column by the UV–visible detector. An optimal wavelength must be selected for each compound in order to provide the best response linearity with minimum experimental noise and error [48,68].

4.1. Astaxanthin in ethyl acetate results The applicability of the CPB method was assured in every assay

Fig. 3. Experimental results to assess the optimal detector wavelength (λ ) for astaxanthin/EtOAc system at 1 bar and 323.15 K: (a) Root mean square error, ε ; (b) Ratio of maximum absorbance to peak area (NAI = Absmax / Apeak ) ; (c) Preliminary D12 results for two different solute concentrations = 0.15 mg mL−1 and = 0.20 mg mL−1. 4

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Table 1 Experimental D12 values for astaxanthin in ethyl acetate and calculated density and viscosity of the solvent. Temperature (K)

Densityƚ

Viscosityǂ (cP)

Pressure (bar)

(g

303.15

1 50 100

0.888 0.892 0.897

0.399 0.422 0.444

0.842 ± 0.009 0.831 ± 0.005 0.817 ± 0.005

313.15

1 50 100

0.876 0.881 0.885

0.359 0.380 0.400

323.15

1 50 100

0.863 0.869 0.874

333.15

1 50 100

0.851 0.856 0.861

Table 2 Experimental D12 values for squalene in ethyl acetate and calculated density and viscosity of the solvent. Temperature (K)

Densityƚ

Viscosityǂ (cP)

Pressure (bar)

(g

303.15

1 75 150

0.888 0.895 0.901

0.399 0.433 0.464

1.171 ± 0.014 1.098 ± 0.008 1.052 ± 0.008

0.955 ± 0.011 0.950 ± 0.001 0.925 ± 0.002

313.15

1 75 150

0.876 0.883 0.9010

0.359 0.390 0.419

1.338 ± 0.008 1.265 ± 0.010 1.188 ± 0.015

0.325 0.344 0.363

1.102 ± 0.009 1.075 ± 0.003 1.034 ± 0.013

323.15

1 75 150

0.863 0.871 0.879

0.325 0.354 0.381

1.514 ± 0.023 1.419 ± 0.010 1.338 ± 0.008

0.295 0.314 0.332

1.223 ± 0.012 1.204 ± 0.011 1.169 ± 0.004

333.15

1 75 150

0.851 0.859 0.867

0.295 0.323 0.348

1.722 ± 0.020 1.603 ± 0.012 1.500 ± 0.005

cm−3 )

D12 ± ΔD12 (10−5

cm2

s−1)

ƚ

cm−3 )

D12 ± ΔD12 (10−5 cm2 s−1)

ƚ

Density calculated by the equations of Rackett [69] and Tait [70]. Viscosity values were taken from [72] or estimated at high pressure by the Lucas method [73].

Density calculated by the equations of Rackett [69] and Tait [70]. Viscosity values were taken from [72] or estimated at high pressure by the Lucas method [73].

Fig. 4. D12 values for astaxanthin/EtOAc against pressure at 303.15 K ( ), 313.15 K ( ), 323.15 K (●), and 333.15 K ( ).

Fig. 6. Experimental D12 values for squalene in ethyl acetate against pressure at 303.15 K ( ), 313.15 K ( ), 323.15 K (●), and 333.15 K ( ).

Fig. 5. Experimental D12 values for astaxanthin/EtOAc as function of StokesEinstein coordinate at 303.15 K ( ), 313.15 K ( ), 323.15 K (●), and 333.15 K ). ( ). The fitting of the data is represented by the line (

Fig. 7. D12 values for squalene/EtOAc as a function of Stokes-Einstein coordinate at 303.15 K ( ), 313.15 K ( ), 323.15 K (●) and 333.15 K ( ). The ) represents the data fitting. line (

since the previously mentioned restrictions (Section 2.1) were fulfilled, namely: (i) Laminar flow, with Re in the range 5.48–8.77; (ii) the Dean and Schmidt numbers were in the range of 0.229–0.366 and 289–670 respectively with DeSc0.5<10 ; (iii) the peaks were Gaussian ¯ ) < 0.01 with u¯ in the range 1.14–1.20 cm s−1); (iv) temperature (D /(uL

¯ / D was the range and pressure perturbations could be neglected since uL 1.07 × 108 − 1.61 × 108 . Goodness of fit was considered good or acceptable (ε ranged from 0.52 % to 2.10 %) and the empirical symmetry factor was nearly one, S10 ≅ 1. For these calculations the density of EtOAc was estimated by the equations of Rackett [70] and Tait [71],

ǂ

ǂ

5

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Table 3 Relevant properties of the chemical compounds used in this work. Compound

Mi (g mol−1)

Pc (bar)

Tc (K)

Vc (cm3 mol−1)

w

σLJ (Å)

εLJ/ kB (K)

Astaxanthin Squalene Ethyl acetate

596.84a 410.73a 88.11b

5.30c 7.03g 38.80b

1148.51c 716.50g 523.30b

1877.50c 1601.00c 286.00b

– – 0.361d

9.9803e 9.4641e 5.3148f

1004.6e 554.57e 405.03f

a

b c d e f g

Taken from safety data sheet; Taken from [70]. Estimated through the Joback’s method [70]. Taken from [66]. Estimated by Eqs. (8) and (9) of Ref. [64]. Taken from [59]. Taken from [79].

Table 4 Modelling results for D12 of astaxanthin and squalene in liquid ethyl acetate. Model

No. of parameters

Astaxanthin/EtOAc

Squalene/EtOAc

References

[59–61]

Parameters values DHB

DHB & VD (T )

2

3

TLSM TLSMd

0 1

TLSMen

1

Wilke-Chang Tyn-Calus

0 0

Magalhães et al., (Eq. (3) of ref. [80])

2

Magalhães et al., (Eq. (5) of ref. [80])

Magalhães et al., (Eq. (7) of ref. [80])

Magalhães et al., (Eq. (9) of ref. [80])

2

2

2

AARD (%)

1.70

0.69

BDHB (mol cm−1 s−1 K−0.5)

4.148 × 10−8

5.926 × 10−8

VD (cm3 mol−1)

8.716 × 101

8.775 × 101

AARD (%)

0.61

0.50

BDHB (mol cm−1 s−1 K−0.5)

2. 314 × 10−8

5.285 × 10−8

mVD (cm3 K mol−1)

− 1.182 × 10−1

− 1.958 × 10−2

b VD (cm3 mol−1)

1.140 × 102

9.245 × 101

AARD (%) AARD (%) k12,d (adm) AARD (%) k12,en (adm) AARD (%) AARD (%)

21.02 3.57

29.19 3.19

[59,63,64] [59,63,64]

1.078 × 10−1 3.13

1.502 × 10−1 2.54

[59,63,64]

4.101 × 10−1 13.11 53.56

6.370 × 10−1 8.62 20.64

[65,66] [66] [80]

AARD (%) aI (adm)

2.03

0.91

− 8.478 ×10−1

− 8.975 ×10−1

bI (adm)

− 1.813 × 101

− 1.748 × 101

AARD (%)

3.12

2.33

aII (cm2 cP s−1)

4.058 × 10−6

5.643 × 10−6

bII (cm2 s−1)

− 1.175 × 10−6

− 1.818 × 10−6

AARD (%)

aIII (cm5 g−1 K−1 s−1)

1.26 − 2.305 × 10−7

0.58 − 3.343 × 10−7

bIII (cm2 K−1 s−1)

2.332 × 10−7

3.354 × 10−7

AARD (%)

2.06

0.92

aIV (cm5 g−1 K−1 s−1)

4.823 × 10−9

4.263 × 10−9

bIV (cm2 K−1 s−1 cP)

9.868 × 10−9

1.420 × 10−8

[59–61]

[80]

[80]

[80]

EtOAc under isothermal and isobaric conditions are consistent with data reported in the literature for similar systems [43,62,77]. The hydrodynamic behavior of the system was studied adopting Stokes-Einstein coordinates. The plot of D12 versus T / μ1 (Fig. 5) exhibits a slightly linear relation with a positive origin intercept, 1.085 × 10−6 cm2 s−1. This indicates hydrodynamic behavior deviations analogous to those reported for aluminum acetylacetonate in ethanol [78] and gallic acid in ethanol [77] under similar conditions.

and viscosity values were taken from [72] or estimated at high pressure by the Lucas method [73]. The tracer diffusivities of astaxanthin in ethyl acetate varied between 0.817 × 10−5 and 1.223 × 10−5 cm2 s−1. The D12 values are reported in Table 1 and represented in Fig. 4 against pressure at fixed temperatures. The data presented in Table 1 and Fig. 4 reveal a strong effect of temperature and a smaller influence of pressure on D12 . The decrease of D12 as the pressure increases is in accordance with the free volume theory [59]. In fact, an increase of pressure leads to higher densities, which in practice results in a tighter packing of the solvent molecules, i.e., less free volume is available for the solute molecules to move. Besides, it increases the energetic barrier the molecules need to overcome in order to diffuse through the solvent [74,75]. Concerning the effect of temperature under isobaric conditions, the rise of D12 with temperature is explained not only by the decrease of density (which increases the free volume available) but also by the increase of the kinetic energy of the solute molecules [76]. Overall, the results recorded for astaxanthin/

4.2. Squalene in ethyl acetate results As in the previous section, the applicability of the method for the squalene/EtOAc system was ensured during every assay. In this case u¯ ranged from 1.16 to 1.21 cm s−1 and: (i) the flow was laminar, with Re in the range 5.87–9.00; (ii) the Dean and Schmidt numbers were in the range of 0.224–0.372 and 201–488, respectively with De Sc <10; (iii) ¯ < 0.01; and (iv) uL ¯ / D varied between the peaks were Gaussian, D / uL 1.67 × 108 and 2.39 × 108. The fitting was good (ε in the range 6

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Fig. 8. Calculated versus experimental values of D12 of: (a), (c) and (e) astaxanthin/EtOAc; (b), (d) and (f) squalene/EtOAc. Models: Wilke-Chang ( ), Magalhães et al. (Eq. (3) of [80]) ( ), TLSM ( ), TLSMd ( ), TLSMen (+), DHB (○) and DHB & VD (T ) (▴). For DHB and DHB & VD (T ) models (figures (e) and (f)), the temperatures are discriminated by color: 303.15 K (blue), 313.15 K (purple), 323.15 K (black), and 333.15 K (gray). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 5 Influence of the estimated critical constants on the calculated diffusivities of astaxanthin and squalene in liquid ethyl acetate. Values are the obtained average absolute relative deviations, AARD (%). Solute

Variation of the critical constant

TLSM

TLSMen

TLSMd

Wilke-Chang

Tyn-Calus

Squalene

No variationa 1.05Vc 0.95Vc

29.19 30.44 27.86

2.54 2.51 2.57

3.19 3.18 3.20

8.62 11.38 5.63

20.64 18.00 23.48

Astaxanthin

No variationa 1.05Vc 0.95Vc 1.05Tc 0.95Tc

21.02 22.37 19.58 22.05 19.94

3.13 3.06 3.17 3.13 3.13

3.57 3.56 3.58 3.60 3.54

13.11 9.69 18.12 No influence No influence

53.56 50.19 58.44 No influence No influence

a

In this case, the critical constants were the estimated ones, which are listed in Table 3, and for which the AARD values are those reported in Table 4.

7

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0.79–0.98 %) and the symmetry factor was S10 ≅ 1. D12 values for squalene/EtOAc were measured in the temperature range 303.15–333.15 K and pressure between 1 and 150 bar. The main results are presented in Table 2 and plotted in Fig. 6. The results for D12 of squalene/EtOAc exhibit the same trend as for the system astaxanthin/EtOAc regarding the influence of temperature, pressure and T / μ1. However, it should be noted that the influence of pressure is more pronounced for the squalene/EtOAc system with D12 decreasing up to 18 % when pressure goes from 1 bar to 150 bar. Although most liquids are assumed to be incompressible, the present results reveal the importance of pressure on diffusivity in liquid systems. A linear relation can be observed for D12 as function of StokesEinstein coordinate, T / μ1, represented in Fig. 7. The positive origin intercept, 9.933 × 10−7 cm2 s−1, although different from zero it is of the order of magnitude of the experimental uncertainty of D12 (see Table 2). Hence, it can be stated that the D12 for squalene/EtOAc follows the hydrodynamic behavior. Comparing both systems it is evident that D12 values for astaxanthin/EtOAc are lower than for squalene/EtOAc under the same conditions (i.e., P = 1 bar ). This is due to several reasons: (i) astaxanthin (C40 H52 O4 , 596.84 g mol−1) has a higher molecular weight than squalene (C30 H50 , 410.73 g mol−1) ; (ii) astaxanthin has two OH groups (see Fig. 1) enabling the formation of strong polar bonds with the solvent, hence the energy value for diffusion is higher than for squalene, which has no hydroxyl groups; (iii) the Lennard-Jones diameter, σLJ , of astaxanthin being higher than for squalene (see Table 3) implies that the former requires more energy to diffuse across the solvent. Furthermore, the previous explanation may also justify the more pronounced influence of temperature, pressure and viscosity on D12 values for the squalene/EtOAc system than for the astaxanthin/EtOAc system.

those fitted to thermodynamics data. As shown in Table 3, several critical properties were estimated to be used in the TLSM, TLSMen, TLSMd, Wilke-Chang, and Tyn-Calus models. Hence, it is important to assess the influence of the estimated critical constants on the calculated diffusivities of both systems. This study was performed whenever an experimental critical constant was unavailable, and variations of +5 % and −5 % around the estimated value were analyzed. Accordingly, the following cases were considered: the influence of the critical volume of squalene, and the impact of the critical temperature and volume of astaxanthine. Taking into account the calculated results shown in Table 5, one may conclude that such ± 5 % variations have a small impact on the TLSM, TLSMen, and TLSMd models, for which the computed AARD jumps lied between −1.44 % and +1.35 %. In the case of Wilke-Chang equation, the AARD differences were −3.43 % and 5.01 %, while for the Tyn-Calus model they were −3.37 % and 4.88 %. In conclusion, the family of TLSM models is more robust than the Wilke-Chang and Tyn-Calus equations, although none of them magnifies the propagation of the critical constants uncertainty. A data grouping effect can be clearly observed in Fig. 5 and slightly in Fig. 7, and this background effect is also translated to the models (see Fig. 8). For the DHB model this behavior can be corrected by expressing the minimum volume required for diffusion, VD, as a linear function of temperature (VD = m VD T + b VD) . As evidenced in Fig. 8 (e) (where the effect is more pronounced) the DHB & VD (T ) equation was able to disassemble this grouping effect providing lower AARD values (0.61 %). Furthermore, by observing all subplots of Fig. 8, one may say that the grouping effect is more pronounced for the astaxanthin/EtOAc system.

4.3. Modeling

The transport of the bioactive compounds astaxanthin and squalene in compressed liquid ethyl acetate (EtOAc) was studied using the chromatographic peak broadening (CPB) method. Diffusion coefficients, D12 , were measured in the range 303.15–333.15 K with pressures ranging from 1 to 100 bar, for astaxanthin, and from 1 to 150 bar, for squalene. For astaxanthin/EtOAc, D12 values were in the range 0.817 × 10−5 − 1.223 × 10−5 cm2 s−1 and in the range 1.052 × 10−5 − 1.822 × 10−5 cm2 s−1for squalene/EtOAc. The effects of temperature, pressure and Stokes-Einstein coordinate (T / μ1) were analyzed for both systems. In both systems the temperature imposes the most important influence upon D12 . From 303.15 to 333.15 K, D12 increases 44 % and 49 % in average in the case of astaxanthin/EtOAc and squalene/EtOAc, respectively. Pressure exhibits a much lower effect, namely in the case of astaxanthin/EtOAc. The hydrodynamic behavior of astaxanthin/EtOAc showed a non-negligible deviation from the hydrodynamic behavior. Several predictive and correlative equations from the literature were tested. Overall, for both systems the best results were achieved with the DHB & VD (T ) equation (AARD between 0.50 % and 0.61 %), the correlations of Magalhães et al. (AARD between 0.58 % and 3.12 %) and the 1-parameter TLSMd and TLSMen correlations (AARD between 2.54 % and 3.57 %).

5. Conclusions

The diffusion coefficients of astaxanthin and squalene in EtOAc were modeled by the eleven predictive/correlation models mentioned in Section 2.2, and the properties required for the calculations are compiled in Table 3: Mi , molecular weight; Pc , critical pressure; Tc , critical temperature; Vc , critical molar volume; w , acentric factor; σLJ , Lennard-Jones molecular diameter; εLJ/ kB , Lennard-Jones energy parameter. The calculated deviations (AARD values) and optimized parameters are listed together in Table 4. For some selected models, a graphical representation of calculated versus experimental D12 values for astaxanthin/EtOAc and squalene/EtOAc is presented in Fig. 8. The AARD results (Table 4) and the calculated vs experimental plots (Fig. 8) reveal good fitting for almost all models with AARD values in the range 0.61–53.56 % for astaxanthin/EtOAc and 0.50–29.19 % for squalene/EtOAc. Overall, the best results were obtained by the Dymond-Hildebrand-Batschinski (DHB) equation, especially when the minimum volume required for diffusion is expressed as a function of temperature (AARD values between 0.50 % and 1.70 %), and by the correlations of Magalhães et al. (AARD between 0.58 % and 3.12 %). The predictive Wilke-Chang equation provided reasonable results for both systems (AARD equal to 13.11 % and 8.62 %) exhibiting better performance than the Tyn-Calus equation (AARD values of 53.56 % and 20.64 %) and the Tracer Liu-Silva-Macedo (TLSM) model (AARD of 21.02 % and 29.19 %). The results for the predictive Tracer Liu-Silva-Macedo (TLSM) model were significantly improved by the TLSMd and TLSMen correlations that incorporate a binary interaction parameter, k12,d (TLSMd) and k12,en (TLSMen), in the respective mixing rules [67,77]. The AARD values dropped one order of magnitude, from 21.02 % to 3.57 % (TLSMd) and 3.13 % (TLSMen) in the case of astaxanthin/EtOAc and from 29.19 % to 3.19 % (TLSMd) and 2.54 % (TLSMen) for the system squalene/EtOAc. With respect to the Lennard-Jones parameters, it is worth noting that values obtained from transport properties are generally different from

Acknowledgements This work was developed in the scope of the project CICECO-Aveiro Institute of Materials (Ref. FCT UID/CTM/50011/2019), financed by National Funds through the FCT/MEC and when applicable co-financed by FEDER under the PT2020 Partnership Agreement. Bruno Zêzere thanks FCT for PhD grant SFRH/BD/137751/2018. References [1] M. Ivanovi, M.E. Alañón, Enhanced and green extraction of bioactive compounds

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