New Approach to Modeling Supercritical CO2 Extraction of Cuticular

Article pubs.acs.org/IECR

New Approach to Modeling Supercritical CO2 Extraction of Cuticular Waxes: Interplay between Solubility and Kinetics Helena Sovova*,† and Roumiana P. Stateva‡ †

Institute of Chemical Process Fundamentals of the Czech Academy of Sciences, v. v. i., Rozvojova 135, 16502 Prague, Czech Republic ‡ Institute of Chemical Engineering, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria ABSTRACT: This paper presents a new approach to modeling the kinetics of a supercritical fluid extraction of a mixture of solids. Experimental data on pure long-chain n-alkanes solubility in supercritical CO2 (scCO2) was reviewed, and selected data was correlated applying a semiempirical density-based model and an equation of state. On the example of extraction of peppermint cuticular wax (Roy et al. J. Chem. Technol. Biotechnol. 1996, 67, 21), the solubility of representative n-alkanes in ternary and quaternary systems with scCO2 was predicted using the thermodynamic modeling framework advocated. The results obtained were implemented in mass balance equations of an extraction model. The agreement between the changes in extract composition observed experimentally and those predicted by the new approach developed is very good.

1. INTRODUCTION Extraction with supercritical CO2 (scCO2) is particularly suitable for obtaining lipophilic natural substances for application in food, pharmaceutical, and cosmetic industries because of mild operational temperatures, high solvent selectivity, and its easy removal from the extract. The scCO2 selectivity is enabled by pressure and temperature dependence on its solvation power. Components of essential oils, mostly volatile terpene hydrocarbons and oxygenated terpenes, are among the best soluble substances in scCO2. Their extraction from plants with scCO2 is, however, always accompanied by the extraction of less soluble, nonvolatile cuticular wax.1 Waxes cover the aerial parts of plants, serve as a permeability barrier for water and other molecules, and prevent contamination of plant tissues with dirt and microorganisms. In the first approximation, the extractions of essential oil and of cuticular wax are parallel and noninteracting processes.2 We found the omnipresence of cuticular waxes in scCO2 extracts as a good reason to examine their solubility in the supercritical solvent and the kinetics of the extraction process. Moreover, cuticular waxes themselves can be target compounds because they are used as raw materials in the production of cosmetics, detergents, soaps, drugs, and other products.3 The initial concentration of cuticular wax in CO2 flowing out of an extractor is similar to the solubility of their major components, long-chain n-alkanes, while the concentration of volatile oil is much lower than its solubility in CO2.2,4,5 To explain this phenomenon, Reverchon hypothesized different mass transfer mechanisms for the two compound families: while cuticular waxes are easily extracted from the plant surface, volatile oil has to overcome internal mass transfer resistance.1 Goto et al.6 assumed that both cuticular wax and volatile oil are extracted from solid surface but the volatile oil affinity to plant matrix is much stronger than that of cuticular wax. They modeled the equilibrium between the solid and supercritical phases using the BET isotherm with different equilibrium © 2015 American Chemical Society

constants for volatile oil (KBET = 15) and wax (KBET = 1). To reduce the number of adjusted BET parameters, the maximum amount of solute adsorbed as monolayer was assumed to be equal to the initial amount of solute in plant. However, according to the BET isotherm, the initial equilibrium fluid phase concentration of wax would not exceed one-half of its solubility, in contrast to the concentrations similar to the solubility observed in scCO2 extraction. To the best of our knowledge, the paper of Goto et al.6 is the last attempt to model and predict the kinetics of SFE of cuticular waxes. The solubility of long-chain n-alkanes in CO2 sharply decreases with decreasing temperature in contrast to that of terpenoids, which are well soluble in liquid CO2. To separate the extracted waxes from the essential oil components, a twostage separation procedure is applied. The extract solution in the dense CO2 flows to the first separator maintained at 0 °C or below this temperature where cuticular waxes precipitate; volatile substances are then separated from gaseous CO2 in the second separator at higher temperature and lower pressure.1,7 According to Reverchon and colleagues,2,8 odd-numbered C25−C35 n-alkanes represent the main components of cuticular waxes in CO2 extracts from vegetable matrices like herbs, flowers, and roots. Similarly, Roy et al.4 determined the composition of waxes in CO2 extracts of peppermint leaves as a mixture of C25−C35 alkanes with C31 and C33 n-alkanes as major components. Kubátová et al.9 analyzed the waxes in CO2 extracts of peppermint and savory leaves as C27−C35 n-alkanes with C29 and C31 n-alkanes as the major components. Nonacosane was the major component (39.3%) of the wax extracted with scCO2 from St. John’s wort.10 On the other hand, the composition of wax extracted from wheat straw was much more complex: wax esters, β-diketones, alkanes, Received: Revised: Accepted: Published: 4861

February 23, 2015 April 1, 2015 April 15, 2015 April 15, 2015 DOI: 10.1021/acs.iecr.5b00741 Ind. Eng. Chem. Res. 2015, 54, 4861−4870

Article

Industrial & Engineering Chemistry Research octacosanol, sterols, and fatty acids.11 Nevertheless, the alkanes comprised C29−C33 alkanes with n-C31 being the major component. A comparison of plant leaf waxes composition was carried out by Bi et al.12 for 26 sorts of plants of Amaranthaceae, Verbenaceae, Myrtaceae, and other plant families grown in China. C31 n-alkane was the most abundant in 19 plant sorts, C33 n-alkane in four sorts, C27 and C29 nalkanes, each, in two sorts, and C35 n-alkane in one sort. In this work a new approach to modeling supercritical extraction of cuticular waxes from vegetal matrices is introduced and demonstrated on the example of scCO2 extraction of long chain n-alkanes from peppermint leaves. We calculate not only the overall yield of waxes but also the evolution of extract composition in the course of extraction and compare the results obtained with the data Roy et al.,4 undoubtedly the most comprehensive data on supercritical fluid extraction and fractionation of cuticular waxes published until present. For this purpose we examine the change in solubility of model wax components in ternary and quaternary systems relative to their binary solubility and in dependence on the mixture composition, which is changing during the extraction.

Table 1. Conditions of Measurement of Long-Chain nAlkanes Solubility in scCO2 nalkane C24 C24 C24 C24 C24 C25 C25 C26 C26 C27 C28 C28 C28 C28 C28 C28 C28 C28 C29 C29 C30 C32 C32 C32 C33 C36 C36 C36

2. LITERATURE ON SOLUBILITY OF n-ALKANES IN scCO2 The literature on long-chain n-alkanes in scCO2 is relatively rich; hence, we decided to use the experimental data published as a basis for the modeling and solubility correlation applying a thermodynamic model and a semiempirical equation. The highpressure solubility data available were obtained using different measurement techniques as dynamic method, synthetic method, equilibrium method with spectrometric vapor phase analysis, and chromatographic method (Table 1).13−21 The solubility data measured for a particular n-alkane at a particular temperature can be directly compared, as shown in Figure 1 for the most reported alkane, n-octadecanone, at the two most frequent temperatures. The chromatographic method applied by Chandler et al.13 gives somewhat higher values in the upper part of the pressure range than the dynamic method, but otherwise the agreement with the data of McHugh et al.14 and Reverchon et al.,15 measured at 318.2 K, is good. The solubility values reported by Yau and Tsai16 are higher than the others (they are not shown in Figure 1b because they exceed 0.0034 kg·kg−1 CO2), and most probably they should not be taken into account. The data on the pressure dependence of solubility measured near 308 K, however, split into two groups (Figure 1b). According to McHugh et al.14 and Kalaga and Trebble,17 the solubility of n-C28 is almost independent of pressure, while according to Chandler et al.13 and Reverchon et al.15 the solubility increases with increasing pressure, similarly to the pressure dependence observed at higher temperature. Furuya and Teja,18 who measured at 310 and 313 K, reported an almost negligible pressure dependence of n-C24, n-C25, and nC26 solubility in CO2, in contrast to n-C27 and n-C29 solubility for which the pressure dependence was evident. Both Chandler et al.13 and Reverchon et al.15 found a pronounced pressure dependence of solubility for all n-alkanes and temperatures examined. Melting point depression of n-C28 under pressurized CO2 was measured by McHugh et al.14 The melting point 62 °C, at normal pressure, was decreasing with increasing pressure until 14.1 MPa where the minimum melting temperature 52.5 °C was reached, and at higher pressures it slowly increased. In

a

temp/K 308.2 310 308.2, 313.2, 318.2 343a a 329.7 , 339.2a, 348.1a, 357.1a 308.2 313 313 343a 313 307.8, 318.5, 323.3, 325.1 308.2, 318.2 308.2, 313.2, 318.2 308.2 308.2, 313.2, 318.2 338.5a, 348a, 357a, 366.4a 343a 323.7a, 343.2a, 362.5a 308.2, 317.2 313 308.2, 318.2 308.2, 318.2, 328.2 318.2, 328.2, 338.2 343a 308.2, 318.2, 328.2, 338.2a 318.2, 328.2, 338.2 318.2, 333.2 344a, 348.5a

pressure/ MPa 10−24 13−46 5−21 18−26 17−30 10−24 15−36 14−41 16−26 10−52 12−28 10−24 8−28 10−21 5−21 14−29 16−24 20−57 10−24 12−18 9−25 12−24 5−21 18−31 12−24 3-21 16−24 20−29

method

ref

chromatographic synthetic dynamic spectrometric synthetic

13 18 16 20 21

chromatographic synthetic synthetic spectrometric synthetic dynamic, synthetic chromatographic dynamic dynamic dynamic synthetic spectrometric spectrometric chromatographic synthetic dynamic chromatographic dynamic spectrometric chromatographic dynamic chromatographic synthetic

13 18 18 20 18 14 13 15 17 16 21 20 19 13 18 15 13 16 20 13 16 13 21

n-Alkane is in liquid state.

Figure 1. Literature data on n-octadecane solubility in CO2 (mass fractions) at different temperatures: (a) 318.5 K (ref 14) and 318.2 K (all other data); (b) 307.9 K (ref 14) and 308.2 K (all other data). For Chrastil correlation at 318.2 and 308.2 K (solid lines), see eq 29.

Table 1, the solubility measurements carried out with liquid alkanes are marked with an a. In the present work, however, only the solid−scCO2 equilibrium was of interest, and thus the data published in refs 19−21 were omitted. Furthermore, taking into consideration that the solubility data published by Chandler et al.13 covers the highest number of n-alkanes in the widest range (C24, C25, C28, C29, C32, C33, and C36), we have chosen it as the basis for our calculations. 4862

DOI: 10.1021/acs.iecr.5b00741 Ind. Eng. Chem. Res. 2015, 54, 4861−4870

Article

Industrial & Engineering Chemistry Research

3. NEW APPROACH TO MODELING THE KINETICS OF scCO2 EXTRACTION OF CUTICULAR WAXES Up until the present, there have been several studies published on the kinetics of parallel extraction of volatile oils and cuticular waxes, among them the pioneering work of Reverchon et al.2 who characterized CO2 extracts of basil, marjoram, and rosemary leaves extracted at 40 °C and 10 MPa as essential oil and cuticular waxes. In order to distinguish the essential oil extracted from a product of steam distillation or hydrodistillation, we denote the former as volatile oil. The extract was precipitated in two separators in series; waxes were completely recovered in the first separator at 8 MPa and 0 °C and the volatile oil in the second separator. The initial extraction rate of the waxes was by order of magnitude lower than that of the volatile oil, and thus, the extraction of volatile oil was practically complete after (50−100 min, while waxes showed an approximately constant extraction rate until the extraction run was finished. The extracts were analyzed using gas chromatography and, in the case of waxes, also supercritical chromatography. The white solid material precipitated in the first separator consisted mainly of n-alkanes C25−C35. The mixtures of n-alkanes were characterized by the mean molecular weight from 408 to 430, which is in the range C29−C31. Reverchon et al.2 derived a model for the extraction controlled by internal and external mass transfer resistances, and, although they applied it to both the volatile oil and waxes, they were aware that cuticular waxes are located in a narrow layer on the particle surface. Thus, waxes overcome only the external mass transfer resistance, which is usually much smaller than the internal resistance. Roy et al.4 extracted volatile oil and cuticular waxes from peppermint leaves at 313 K and 10−30 MPa and fractionated the extract collected in a tube into seven or eight fractions for each run by changing the tubes at definite time intervals. The fractions were analyzed by gas chromatography at different conditions for volatile oil and waxes. The waxes were composed of C25−C33 alkanes; the concentration of C32 and C33 alkanes in the mixture was observed to increase with the progress of the extraction at the expense of shorter-chain alkanes. The increase of longer-chain alkanes percentage in the mixture was also observed with increasing extraction pressure. The extraction of both volatile oil and waxes was proved to be controlled by phase equilibrium because the extraction yields were dependent on the amount of solvent passed through the extractor and independent of the solvent flow rate. The initial wax concentration in the solution flowing out of the extractor was by 23−55% lower than the solubility of n-C30 alkane reported by Reverchon et al.15 These results on the kinetics of volatile oil and cuticular wax extraction from peppermint leaves were modeled by Goto et al.,6 who derived a versatile model based on the implementation of the BET isotherm. The model allows to account for both extraction controlled by solute adsorption on the surface of vegetable matrix and simple leaching where solute dissolves from inert matrix. The dimensionless form of the BET isotherm is xs =

KBETxp

(1 − xp)[1 + (KBET − 1)xp] w xs = s wt

,

xp =

where ws and wt denote the solute content in the solid phase and its maximum content corresponding to monolayer adsorption, respectively; w+ is the equilibrium content of solute in the fluid phase at the solid−fluid interface, wsat is the solubility, and KBET is the adsorption equilibrium constant indicating the strength of interaction between solute and solid. The other features of a simplified form of the model were the following: porous particles with equilibrium established on the pore wall, linear driving force, and no concentration gradient in the extraction bed. To reduce the number of adjustable model parameters, it was assumed that the initial content of the solute in the plant equals monolayer saturation: xs = 1 for t = 0.6 The solubility of waxes was set equal to the solubility of triacontane, and the only adjusted parameter KBET was found to be equal to 1 for cuticular wax. The assumption made in our work is that the cuticular waxes extracted are located on the solid particles’ surface. Thus, the mass transfer rate expressed in mass of solute per time and mass of CO2 in the void of extraction bed is j=

w+ − w , tf

tf =

ε k f a0

(2)

where w (kg·kg−1 CO2) is the content of solute in bulk fluid phase, ε is the free void of the extraction bed, and kfa0 is the volumetric mass transfer coefficient. The BET isotherm, eq 1, is replaced in the model by the semiempirical relationship proposed by del Valle and Urrego.22 Its dimensionless form is xp =

xs xsb ⎛ x ⎞ ⎜⎜1 − s ⎟⎟ + Keq Keq ⎠ 1 + xsb ⎝

(3)

where the meanings of xp and xs are same as in eq 1. The exponent b ≫ 1 in eq 3 controls the steepness of transition between xp = xs/Keq and xp = 1, which takes place near xs = 1. The value of b was set equal to 7 in our calculations. The flexibility of eq 3, compared to the BET isotherm, according to which the fluid phase concentration only very slowly approaches its asymptotic value 1 with increasing xs, is illustrated in Figure 2. In the dimensional form of the del Valle−Urrego equation,22 the transition between dissolution of free solute and desorption of solute interacting with matrix occurs in the vicinity of transition concentration wt (kg·kg−1

w+ , wsat Figure 2. Effect of solute−matrix interaction according to the BET isotherm (for KBET = 1) and the dimensionless del Valle− Urrego equation22 (for Keq equal to 2 and 10).

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solid solutes solubilities in a multicomponent system with a supercritical solvent. The calculation of a solid solute solubility in a binary mixture with a supercritical component is presented in great detail in many books and papers.23 The number of papers devoted to ternary and multicomponent mixtures, however, is limited see, for example, refs 24−32. In our work we correlate the solubility of several heavy linear n-alkanes, model representatives of cuticular waxes, in scCO2. Although waxes are structurally among the simplest hydrocarbons, their solubility does not follow ideal solution behavior.33 Hence, to correctly model the solubility of waxes in scCO2, the TMF applied should incorporate robust and efficient methods for estimating the missing properties of the heavy n-alkanes, reflect the nonideality of the systems under consideration by adopting appropriate thermodynamic models, and guarantee steady convergence of the algorithms and numerical techniques for solving the equilibrium relations. In what follows, we will apply the dense gas approach as the thermodynamic model to correlate the solubility of waxes in scCO2.34 It treats the supercritical fluid (SCF) phase as a gas and uses an equation of state (EoS) directly in solid−fluid equilibrium calculations by introducing a solid-phase fugacity function defined in terms of a fluid-phase reference state. McHugh et al.34 suggested using the solid vapor pressure as the reference fugacity of the solid. The dense gas approach begins with equating the fugacities of the solute in the solid phase and in the supercritical fluid phase

plant). Desorption is characterized by linear equilibrium relationship w+ = Kws with partition coefficient K (kg plant· kg−1 CO2): w+ = Kws +

wsb wtb

+

wsb

(wsat − Kws),

wsat wtKeq

K=

(4)

The mass balance equations for a differential bed without concentration gradient are dw w + = j, dt tr

dws = −q′trj , dt

e = q′

∫0

t

w dt

(5)

with initial conditions w = w0 ,

ws = ws0 ,

ws0 + q′trw0 = wsu ,

e=0

for (6)

t=0 −1

−1

where t (min) is the extraction time, q′ (kg·min ·kg plant) is the specific flow rate, tr (min) is the residence time, e (kg· kg−1plant) is the extraction yield, and wsu (kg·kg−1plant) is the content of solute in the plant loaded in the extractor. For a multicomponent extraction, the model equations were rewritten for the ith component dws, i

dwi w + i = ji , dt tr

dt

= −q′trji ,

ei = q′

∫0

t

wi dt (7)

with initial conditions wi = w0i , for

ws, i = ws0, i ,

ws0, i + q′trw0i = wsu, i ,

ei = 0

f isolid = fiF

(8)

t=0

where is the fugacity of the solute in the solid phase and f iF is fugacity of the solute in the supercritical phase. We begin by a brief outline of the thermodynamics of binary solid−SCF (gas) equilibria, followed by the multicomponent solid−SCF (gas) equilibria. 4.1. Pure Solid. The application of the equi-fugacity condition (eq 10) to the simple case of binary solid−SCF (gas) equilibria requires expressions for the fugacity of the solute (denoted by subscript 2) in the solid and in the supercritical phase, respectively. The fugacity of the solute in the solid phase is

and with the specific mass transfer rate ji =

wi+ − wi , tf

wi+ = K iws, i + wsum =

∑ ws,i

b wsum b wtb + wsum

(10)

f isolid

(wsat, i − K iws, i), (9)

Thus, two equilibrium parameters are characteristic of each of the mixture components: solubility and partition coefficient. The transition concentration is compared with the sum of concentrations of all components. As the solubility of cuticular waxes in scCO2 is by orders of magnitude lower than the solubility of volatile oils, their extraction is much slower. When the extraction of volatile oil is almost finished, the content of wax in the plant is usually higher than the transition concentration, and thus its extraction kinetics is not affected by solute−matrix interaction. Moreover, the mass transfer resistance in the supercritical solvent is very small, and the wax extraction is therefore controlled by phase equilibrium. Thus, to model the coextraction of cuticular wax with essential oil, phase equilibrium of wax components + scCO2 should be studied which calls for a robust thermodynamic framework allowing reliable calculation of wax solubility in the supercritical solvent.

f 2solid = P2sφ2sexp

∫P

P

s 2

v2solid dP RT

(11)

where PS2(T) is the sublimation (vapor) pressure of the pure solid, φS2 is the fugacity coefficient at sublimation pressure, and vsolid is the molar volume of the solid, all at temperature T. 2 The fugacity of the solute in the supercritical phase is f2F = y2 Pφ2F

(12)

φF2

where is the fugacity coefficient and y2 the solubility (mole fraction) of the solute in the supercritical phase and P is the system’s pressure. For phase equilibrium between a high-boiling compound and a supercritical fluid, the following assumptions are usually introduced: (1) the solid solute remains pure since the size and shape of solute and solvent molecules are ordinarily sufficiently different and hence solid solutions do not form; (2) the molar volume of the solid solute is independent of pressure; (3) the saturated vapor of the solid solute−vapor (pure) system behaves as an ideal gas. Hence, taking into consideration that

4. THERMODYNAMIC MODELING FRAMEWORK The thermodynamic modeling framework (TMF) applied in our study allows calculation of a single solid solute solubility in a binary mixture with a supercritical solvent and of multiple 4864


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In this work, we use eq 18 to calculate the solubility of the ith solute in the supercritical phase, yi, which, as in the case of the simple binary system, is a function of visolid, PiS, and φFi . The latter is calculated applying a thermodynamic model. 4.3. Thermodynamic Model. In this work we apply as thermodynamic model the Soave−Redlich−Kwong cubic EoS (SRK EoS). If the SRK cubic EoS is written in terms of compressibility Z

the saturation pressure of the solute at the system temperature is (usually) much lower than one bar, the fugacity coefficient of the pure solid can be accepted to be unity. Applying the above assumptions, eq 11 can be rewritten as follows: ⎛ v solid(P − P S(T )) ⎞ 2 ⎟⎟ f 2solid = P2sexp⎜⎜ 2 RT ⎝ ⎠

(13)

Z3 − Z2 + (A − B − B2 )Z − AB = 0

Then, the mole fraction of the solid component in the supercritical phase, i.e., its solubility in the SCF, can be expressed as

y2 =

P2s

⎡ v solid(P − P s) ⎤ exp⎢⎣ 2 RT 2 ⎥⎦

P

φ2F

then the expression for the fugacity coefficient of the ith solid solute in the supercritical phase φFi is ⎛ bi Pb ⎞ ⎟ + (Z − 1) ln φi = −ln⎜z − ⎝ RT ⎠ b N ⎡ ⎤ bi ⎥ a ⎢1 0.5 0.5 (2ai ∑ xjaj (1 − kij)) − − b ⎥⎦ bRT ⎢⎣ a j=1

(14)

In the above, the most important variable is the fugacity coefficient of the solid solute in the supercritical fluid phase, φF2 . It is always far removed from unity and can produce very large enhancement factors. Therefore, the pure solute solubility is a function of its fusion properties and fugacity coefficient in the supercritical phase, and the system temperature and pressure. The fugacity coefficient φF2 is calculated by a thermodynamic model. 4.2. Multiple Solid Solutes. For the more complex case of multicomponent solid−SCF (gas) equilibria, following Prausnitz et al.,23 an assumption that the fugacity of a component in a mixture is proportional to its mole fraction, known as the Lewis fugacity rule, can be introduced in order to calculate the fugacity of the ith solute in the solid phase. Namely f isolid = sifi

⎛ b⎞ ln⎜1 + ⎟ ⎝ V⎠

bi = 0.08664 b=

=

⎝

− PiS(T )) ⎞ ⎟⎟ RT ⎠

N

a=

(16)

∑ ∑ xixj(aiaj)0.5 (1 − kij) (24)

aP (RT )2

and

⎛ bii + bjj ⎞ bij = ⎜ ⎟ ⎝ 2 ⎠

⎡ v solid(P − P S) ⎤ exp⎣⎢ i RT i ⎦⎥

B=

bP RT

(25)

(26)

The application of the SRK EoS as the thermodynamic model to calculate the solubilities of the n-alkanes in scCO2 requires information about their critical temperatures, pressure, and acentric factors. 4.4. Pure Component Properties. As highlighted in the above paragraphs, in order to calculate the solubility of solid solutes in scCO2, applying the TMF outlined, two groups of pure component properties are required: fusion and critical properties. For heavy compounds, like long-chain n-alkanes, there are either very limited or completely missing property data. Hence, all properties required that are not available have to be estimated.

(18)

PiS Pφi F

and

In this study we use the one-parameter-per-pair (1PWDW) version of the van der Waals one fluid mixing rule, and hence the expression for the cross-co-volume parameter is, respectively:

It should be noted that in many cases (see, for example, ref 24) the above relation can be further simplified taking into consideration that the exponential correction in eq 18 is often small and negligible and hence the solubility can be expressed as

yi = si

(23)

N

i=1 j=1

A=

(22)

(RTci)2 , Pci

mi = 0.48 + 1.574ωi − 0.176ωi2

(17)

φi F

aci = 0.42748

αi0.5 = 1 + mi(1 − Tr0.5 i )

The mole fraction of the ith solid component in the supercritical phase, i.e., its solubility in the SCF at the temperature and pressure of interest, is then PS yi = si i P

and ⎛ bi + bj ⎞ ⎟(1 − lij) ⎝ 2 ⎠

j

ai = aciαi ,

The fugacity of the ith solute in the supercritical phase is fiF = yP φi F i

RTci Pci

∑ ∑ xixj⎜ i

(15)

⎛ v solid(P siPiSexp⎜⎜ i

(21)

where

where si is the mole fraction of the ith solute in the system and f i is the fugacity of the pure solute. The fugacity of the pure solid solute f i is given by eq 11, and introducing the expression in eq 14 and applying the above assumptions, the fugacity of the ith solid solute in the mixture is then f isolid

(20)

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Figure 3. Composition of peppermint cuticular wax according to Roy et al.4 and its simplifications.

For the purpose of modeling, we represent the complex wax composition by two- and three-component mixtures. In the first case, the mass fractions of n-C25, n-C27, n-C28, n-C29, n-C30, and n-C31 were added and represented by n-C31; n-C32, *C32, and nC33 were represented by n-C33. In the second case, the mass fractions of four alkanes n-C25, n-C27, n-C28, and n-C29 were represented by n-C29, those of three alkanes n-C30, n-C31, and n-C32 were represented by n-C31, and those of *C32 and n-C33 were represented by n-C33. Figure 3 illustrates the reduction of the number of mixture components for cuticular wax extracted at 30 MPa and 313 K, the most severe extraction conditions applied by Roy et al.4 As the extraction yield was maximal and the extraction selectivity was minimal at these conditions, we make the completely justified assumption that this composition was the closest to the composition of cuticular waxes in peppermint leaves. 5.2. Solubility of a Pure n-Alkane in scCO2. To correlate the solubility of a pure long chain n-alkane in scCO2, a semiempirical model and an EoS were applied in our study. 5.2.1. Semiempirical Density-Based Equations Model. The solubility of n-alkanes with n carbon atoms was correlated with temperature T (K) and CO2 density d (kg·m−3) using the Chrastil equation:39

In our study, the sublimation pressures of the solid n-alkanes are estimated applying the Clapeyron relation, integrated from the triple-point temperature Tt and pressure Pt and assuming a negligible dependence of the sublimation enthalpy ΔH° with respect to temperature is applied. In rare cases, the triple-point conditions (the temperature and pressure at the triple point) for a solute are known experimentally.35 However, for almost all heavy compounds there is little difference, usually less than 0.1 K, between the triple-point temperature and the normal melting temperature. Hence, in the Clapeyron relation instead of the triple-point temperature the normal melting temperature of the solid solute can be used. The solid molar volume of the components of interest in our study is estimated applying the method of Bondi,36 while their critical parameters are estimated applying the method of Wakeham et al.37 With regard to the latter, although, as pointed out previously, there are no experimental data available, still there should be an assessment made whether the values estimated are reasonable. In view of this, the generalized semitheoretical expression advocated in ref 38 is applied to evaluate the reliability of the critical properties estimated, according to Tc/pc = 9.0673 + 0.43309(Q w1.3 + Q w1.95)

⎛a ⎞ wsat = d k − 1exp⎜ + b⎟, ⎝T ⎠

(27)

where Tc is in Kelvin and Pc is in bar. The dimensionless parameter Qw is a measure of the van der Waals molecular surface area and is calculated as the sum of the group area parameters, Qk

Qw =

∑ νkQ k k

b = 1.4n + 46.86,

a = 1.89n2 − 233n − 10794, k = 0.23n

for

24 ≤ n ≤ 36 (29)

The parameters of Chrastil equation were determined minimizing the average absolute relative deviation (AARD) of solubility calculated from 77 data points of Chandler’s data. The AARD value was 5.38%. Four points measured for n-C33 at the highest temperature, 338.2 K, were omitted because they were by 38−64% lower than the calculated solubility. The reason to discard those points is also that the chromatographic method used by Chandler et al.13 was applicable to solid solutes only. They estimated the melting point depression for longchain n-alkanes in dense CO2 to be around 10 K but did not take into account that the n-C33 melting point at normal pressure is 344−346 K. Hence, the experimental temperature was above n-C33 estimated melting point in scCO2. 5.2.2. Equation of State. The application of the SRK cubic EoS to correlate solubility of a solid solute requires knowledge of its fusion and critical properties. For the three long-chain n-alkanes, chosen in this work to represent cuticular waxes, melting temperature is the only property measured experimentally and available; in addition, the enthalpy of sublimation of n-nonacosane is also known.

(28)

where vk is the number of times group k appears in the molecule. The group area parameters Qk are available in the UNIFAC tables.

5. RESULTS AND DISCUSSION 5.1. Simplified Composition of Peppermint Waxes. Roy et al.4 determined the composition (% w/w) of peppermint cuticular waxes in CO2 extracts as a mixture of nalkanes from n-C 25 to n-C 33 and a branched alkane, ethyltriacontane (*C32). The major components of all extract samples were n-C31 and n-C33. The percentage of less soluble alkanes of higher molecular weight in the extracted wax increased with increasing extraction pressure at the expense of more soluble alkanes. As the extracts were fractionated into seven or eight fractions for each run, the increase of less soluble wax components percentage can be observed also with the progress of extraction. 4866


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Industrial & Engineering Chemistry Research Values for the sublimation enthalpies of n-C31 and n-C33 are not available experimentally and are taken from the literature.40 For each of the three n-alkanes, the triple-point pressure is calculated from the EoS. The critical parameters of the three n-alkanes were estimated and reported in several studies; refs 41 and 42 are just two such examples. To access the reliability of our estimations, we apply eq 27. The ratio of the n-alkanes’ values estimated by us (given in bold) gives a very good approximation to the theoretically calculated Tc/pc ratio (K bar−1), namely, 124.52 versus 125.14, 138.07 versus 139.87, and 153.61 versus 155.45, for the nnonacosane, n-hentriacontane, and n-tritriacontane, respectively. For comparison, the critical parameter values estimated and reported in ref 41 produce the following Tc/pc ratios: 140.58 and 180.62 for the n-nonacosane and n-tritriacontane, respectively. The values of all properties of the three n-alkanes required for the modeling, either experimentally measured or estimated in this study, are listed in Table 2.

Figure 4. Solubility of n-C29 in scCO2 (mole fractions). Comparison with experimental data13 at T = 308 and 318 K. Solid lines correlations, applying the SRK EoS with kij = (0.185 and 0.191), respectively. Symbolsexperimental data, ref 13.

Table 2. n-Nonacosane, n-Hentriacontane, and nTritriacontane Pure Component Properties pc/MPa Tc/K p/MPa (triple point) VS/m3 mol−1 ΔHs/J mol−1 Tm/K Pssubl/MPa at T = 308.2 K Pssubl/MPa at T = 313 K Pssubl/MPa at T = 318.2 K Pssubl/MPa at T = 328.2 K a

n-C29 H60

n-C31H64

n-C33H68

0.673a 838a 1.32 × 10−8 4.553 × 10−4 220000b 336.85b 8.81 × 10−12

0.612a 845a 7.23 × 10−9 4.860 × 10−4 261100c 341.1b 3.84 × 10−13

0.554a 851a 4.43 × 10−9 5.167 × 10−4 274200c 344.3c 5.84 × 10−14

3.47 × 10−11 1.31 × 10−10

1.95 × 10−12 9.44 × 10−12

3.22 × 10−13 1.69 × 10−12 3.97 × 10−11

Estimated in this study. bExperimental. cFrom ref 40.

To correlate the solubilites of n-C29 and n-C33 in scCO2, the unlike-pair interaction parameters, kij, were estimated by minimizing the absolute average deviation between the experimental data of ref 13 and calculated solubility (in mole fractions) of the two n-alkanes at the three temperatures of interest applying a standard optimization procedure. The agreement between the experimentally measured and EoS correlated solubilities of n-C29 and n-C33 in scCO2 is shown in Figures 4 and 5, respectively. The AARD is 1.84%. 5.3. Solubility of Solid n-Alkanes in Ternary and Quaternary Systems with scCO2 at T = 313 K. The semiempirical Chrastil equation was derived for binary systems solute + scCO2, and it cannot be applied to calculate the solubility of two or more solids in the solvent. In view of the above, we model the multiple solid−scCO2 equilibria applying the algorithm outlined in section 4.2. For the systems n-C31 + nC33 + CO2 and n-C29 + n-C31 + nC33 + CO2, respectively, the interaction parameters at T = 313 K were estimated taking into consideration the weak dependence of the binary parameters for n-alkanes + CO2 on temperature, a fact that has been reported by other investigators.41,43 The solubilities calculated are shown on Figures 6 and 7, respectively.

Figure 5. Solubility of n-C33 in scCO2 (mole fractions). Comparison with experimental data13 at T = 308.2, 318.2, and 328.2 K. Solid linescorrelations, applying the SRK EoS with kij = 0.0751, 0.0853, 0.0962, respectively. Symbolsexperimental data, ref 13.

5.4. Kinetics of scCO2 Extraction of n-Alkanes from Peppermint Leaves. In order to calculate the yield and composition of peppermint wax extract at extraction conditions 313 K and 17.6 MPa, applied by Roy et al.,4 eqs 7 and 9 were integrated. The model parameters tr = 1745 s and q′ = 0.00274 s−1 were calculated from CO2 density 816 kg·m−3 according to NIST44 and the data given by Roy et al.4 and Goto et al.:6 feed 0.023 kg, flow rate 6.3 × 10−5 kg·s−1, bed void fraction 0.827, specific interfacial area a0 = 2250 m−1, and plant density 816 kg· m−3. The mass transfer coefficient, kf = 4.2 × 10−5 m·s−1, was calculated according to Catchpole and King.45 Its value is in a good agreement with kf = 5.5 × 10−5 m·s−1 estimated6 for conditions 313 K and 13.6 MPa, moreover taking into consideration that the latter should be higher than the mass transfer coefficient at 17.6 MPa. The characteristic time of external mass transfer tf = 8.7 s (eq 2) is much shorter than the residence time tr, indicating independence of extraction yield on 4867


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Second, continuous functions wsat,i(z1), i = 1, 2 for the solubility (in mass fractions) were fitted to the results wsat,1 = −0.000277z12 + 0.000749z1 − 0.00002134

(31)

wsat,2 = −0.003208z13 + 0.00475z12 − 0.002625z1 + 0.0007176

(32)

as shown in Figure 8.

Figure 6. Solubilities (in mass fractions) of n-C31 and n-C33 in the ternary system n-C31 + n-C33 + CO2 at T = 313 K, with the solid phase composition shown in Figure 3, calculated applying the SRK EoS with k13 = 0.134, k23 = 0.0797, respectively.

Figure 8. Solubility of n-C31 and n-C33 in scCO2 at 313 K and 17.6 MPa in dependence on the mixture composition.

At each integration step eqs 31 and 32 were applied in eq 9. The exponent b in eq 9 was set equal to 7. The experimental and calculated extraction curves are compared in Figure 9. The extraction rates in the initial stage were determined solely by the solubility wsat,1 and wsat,2. The only model parameters to be adjusted by matching the calculated extraction curves to experimental ones were the characteristics of solute−matrix interaction, which affected only later stages of extraction. Their values were wt = 0.0065 kg·kg−1

Figure 7. Solubility (in mass fractions) of n-C29, n-C31, and n-C33 in the quaternary system (n-C29 + n-C31 + n-C33 + CO2) at T = 313 K, with the solid phase composition shown in Figure 3, calculated applying the SRK EoS with k14 = 0.188, k24 = 0.134, k34 = 0.0797, respectively.

mass transfer resistance in these experiments, proved also experimentally.4 The wax composition was reduced to two main components, n-C31 and n-C33, as described in section 5.1. The initial content of n-alkanes in peppermint leaves was then wsu,1 = 0.00917 kg· kg−1 and wsu,2 = 0.00583 kg·kg−1, where 1 stands for n-C31 and 2 stands for n-C33. The simulated composition of cuticular wax was expressed in terms of mass fraction ws,1 z1 = ws,1 + ws,2 (30) The solubility of both components in scCO2 depends on z1, which decreases in the course of extraction because the more soluble component 1 is extracted preferentially. First, the solubility of the two solutes was calculated as described in section 4.2. for five values of z1 in the range 0.23−0.61, where the upper limit corresponds to the initial wax composition.

Figure 9. Experimental data4 and calculated extraction curves for peppermint culticular wax represented by n-alkanes C31 and C33. Extraction conditions: 313 K, 17.6 MPa, negligible mass transfer resistance. 4868


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Industrial & Engineering Chemistry Research and K1 = K2 = 0.010 kg·kg−1. The evolution of wax composition in extract fractions is shown in Figure 10.

Thus, the kinetics of cuticular wax extraction could be estimated only approximately from the solubility of its major n-alkane in scCO2, very probably because cuticular waxes comprise a number of other compounds, not only long chain nalkanes. Still, the changes, in the course of extraction, of the simplified composition of n-alkanes in extract are predicted from the solubility of n-alkanes mixture in scCO2 quite well.

6. CONCLUSIONS The strategy developed and applied in this paper to model supercritical fluid extraction of cuticular waxes consists of several subsequent steps: (i) simplification of the composition of extracted mixture; (ii) calculation of mixture components’ solubility in scCO2 at discrete points within a range covering all compositions from initial composition of the mixture in the plant to mixture’s composition enriched by the less soluble component(s), as expected in the course of extraction, using the TMF; (iii) fitting continuous functions of solid phase composition to the above solubilities; (iv) implementing the solubility functions in mass balance equations of a model for extraction and integrating the equations. Needless to say, mathematical modeling of the dependence of CO2 extract composition on extraction time and/or solvent consumption is very challenging because scCO2 extraction is mostly applied on natural products. The latter usually consist of a large number of substances exhibiting different solubility, and these differences can be utilized to control the content of target compounds in the extract by properly chosen extraction conditions. It is only fair to say, however, that the conditions are adjusted rather by experience than based on reliable information since mathematical models able to provide it are missing. In view of this, it is our belief that the new approach suggested and applied in this paper is a step in the right direction and will contribute to the development of mathematical models for supercritical fluid extraction describing not only the total extraction yield but also the extract composition and its evolution in the course of extraction.

Figure 10. n-Alkanes mass ratio in successively sampled extract fractions. Comparison of experimental data4 with kinetic model where the transition from free dissolution to desorption occurs after the first four fractions have been sampled. Extraction conditions: 313 K, 17.6 MPa, negligible mass transfer resistance.

5.5. Is Extraction Rate of Cuticular Waxes Predictable? As cuticular waxes overcome only a small mass transfer resistance from plant surface to bulk fluid, a saturated solution flows out of the extractor under common extraction conditions. Thus, it might seem to be straightforward to estimate the rate of cuticular wax extraction from plant using its solubility calculated from eq 29 where the number of carbons of the most abundant long-chain n-alkane is substituted for n. However, comparison with experimental data shows that such approach is too simplified. Though the most abundant alkane in cuticular peppermint wax was n-C31,4 the saturated concentration of wax extracted with scCO2 corresponds better to the solubility of nC33 (Figure 11a).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; phone: +420 220 390 234. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Reverchon, E. Supercritical fluid extraction and fractionation of essential oils and related products. J. Supercrit. Fluids 1997, 10, 1−37. (2) Reverchon, E.; Donsi, G.; Sesti Osséo, L. Modeling of supercritical fluid extraction from herbaceous matrices. Ind. Eng. Chem. Res. 1993, 32, 2721−2726. (3) Lee, S. B.; Suh, M. C. Recent advances in cuticular wax biosynthesis and its regulation in Arabidopsis. Mol. Plant 2013, 6, 246− 249. (4) Roy, B. C.; Goto, M.; Kodama, A.; Hirose, T. Supercritical CO2 extraction of essential oils and cuticular waxes from peppermint leaves. J. Chem. Technol. Biotechnol. 1996, 67, 21−26. (5) Gaspar, F. Extraction of essential oils and cuticular waxes with compressed CO2: Effect of extraction pressure and temperature. Ind. Eng. Chem. Res. 2002, 41, 2497−2503. (6) Goto, M.; Roy, B. C.; Kodama, A.; Hirose, T. Modeling supercritical fluid extraction process involving solute−solid interaction. J. Chem. Eng. Jpn. 1998, 31, 171−177.

Figure 11. Comparison of the concentration of cuticular waxes at the extractor exit (symbols) with the solubility of n-alkanes in scCO2 according to eq 29 (solid lines): (a) peppermint wax4 and solubility of n-C31 and n-C33; (b) oregano wax5 and solubility of n-C27.

Another comparison of the solubility of n-alkanes and cuticular waxes can be carried out on the basis of extraction experiments of Gaspar5 who determined the load of oregano wax in scCO2 in the earliest stage of extraction at T = 310 and 320 K (Figure 11b). The composition of oregano cuticular wax was not specified. In this particular case, the exit wax concentration corresponded best to the solubility of pure nC27 in scCO2; yet, the former was less dependent on the extraction pressure than the solubility of the pure n-alkane. 4869


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New Approach to Modeling Supercritical CO2 Extraction of Cuticular

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