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J. of Supercritical Fluids 153 (2019) 104578

Contents lists available at ScienceDirect

The Journal of Supercritical Fluids

j our na l ho me page: www.elsev ier .com/ locate /supf lu

oupled extraction and crystal growth in supercritical solutions:odel and theory

han-Chao Hua,b, Xin-Rong Zhanga,b,∗

Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, ChinaBeijing Engineering Research Center of City Heat, Peking University, Beijing 100871, China

i g h l i g h t s

Conceptual model with integratedextraction and crystal growthdesigned.Double-diffusive convection identi-fied as the mechanism for solutetransport.Correlation equation for convectivetransport proposed via CFD simula-tions.Newly defined lower pseudocriticalendpoints are desirable operatingstates.

g r a p h i c a l a b s t r a c t

r t i c l e i n f o

rticle history:eceived 20 February 2019eceived in revised form 31 July 2019ccepted 31 July 2019vailable online 3 August 2019

eywords:

a b s t r a c t

For a supercritical solvent–solute system, as a critical end point (CEP) is approached, the solubilitybecomes sensitive to temperature variations. By virtue of this phenomenon, a conceptual model is pro-posed to combine extraction and crystal growth. CO2–naphthalene is chosen as the reference system.Theoretical optimizations suggest that the newly defined pseudo-CEPs are favorable reference states,especially those close to the lower CEP. In each near-CEP region, the performance is promoted whenthe CEP is approached. The temperature at the bottom wall should be higher than that at the top one

upercritical fluidsrystal growthxtractionouble-diffusive convectionransport phenomenon

to offset the stabilizing concentration gradient due to gravity and initiate double-diffusive convectionin the cooperative regime. Numerical simulations confirm that the performance of the current modeloperating at the optimized reference state is better than the previous experimental results in terms ofcrystal growth rates. The conceptual model provides an efficient configuration for coupled extraction andcrystal growth apparatuses.

© 2019 Elsevier B.V. All rights reserved.

∗ Corresponding author at: Department of Energy and Resources Engineering,ollege of Engineering, Peking University, Beijing 100871, China.

E-mail address: xrzhang@pku.edu.cn (X.-R. Zhang).

ttps://doi.org/10.1016/j.supflu.2019.104578896-8446/© 2019 Elsevier B.V. All rights reserved.

1. Introduction

A fluid is supercritical when its temperature and pressure areabove its critical point. The physical properties of supercriticalfluids (SCFs) are tunable with temperature and pressure, rang-

ing from liquid-like to gas-like properties. This behavior is themain technological reason for the various applications of SCFs [1].For example, extraction of natural compounds, such as caffeine,

2 Z.-C. Hu and X.-R. Zhang / J. of Supercritical Fluids 153 (2019) 104578

Nomenclature

AcronymsCP1 critical point of the solventCP2 critical point of the soluteAARD average absolute relative deviationCEP critical end pointCOO cooperative regimeDDC double-diffusive convectionEOS equation of stateFIN fingering regimeLCEP lower critical end pointOSC oscillatory regimeSCF supercritical fluidSLV solid–liquid–vapor equilibriumSTA stable regimeUCEP upper critical end point

Greek symbols˛ isothermal compressibility (Pa−1)

thermal expansion coefficient (K−1)� gravitational compressibility factor� viscosity (Pa s)� solubility efficiency� specific heat ratio∇T typical relative temperature gradient (K m−1)� concentration contraction coefficient� thermal conductivity (W m−1 K−1)∇c typical concentration gradient (m−1)∇ad adiabatic temperature gradient (K m−1)� density (kg m−3) supersaturation diffusivity ratio� indicator for the regime of DDC (degree)�c partial derivative of the solubility with respect to

concentration�p partial derivative of the solubility with respect to

pressure (Pa−1)�T partial derivative of the solubility with respect to

temperature (K−1)

Roman symbolsg gravitational acceleration vector (m s−2)u velocity vector (m s−1)m cross-boundary flow rate of the solute

(kg m−2 s−1 K−1)RaS solutal Rayleigh numberRaT thermal Rayleigh numberc concentration (mass fraction of the solute)cp specific heat at constant pressure (J kg−1 K−1)cv specific heat at constant volume (J kg−1 K−1)D diffusion coefficient (m2 s−1)d height (m)DT thermal diffusivity (m2 s−1)g gravitational acceleration (m s−2)K growth rate constant (mol m−2 s−1)k12 binary interaction parameterl12 binary interaction parameterM1 molar mass of the solvent (kg/mol)M2 molar mass of the solute (kg/mol)n1, n2 coefficients in the correlating equation

rT temperature gradient ratios solubility in mass fraction (maximum mass fraction

of the solute)T temperature (K)t time (s)u velocity in the x direction (m s−1)Vc growth rate of the crystal (m s−1)VT total volume (m3)VS molar volume of the solute (m3/mol)w velocity in the z direction (m s−1)x horizontal coordinate (m)ys solubility in mole fraction (maximum mole fraction

of the solute)z vertical coordinate (m)Nu Nusselt numberPr Prandtl numberSh Sherwood number

SuperscriptsT temperature

initial field

Subscriptsc concentrationd dynamicp pressurer reference states hydrostaticT temperatureth thermodynamic1 bottom wall2 top wallLCEP lower critical end pointS solute

p pressure (Pa)

UCEP upper critical end point

vitamins and active pharmaceutical ingredients, from materials isone of the main applications of SCFs [2]. There are also varioustechniques for the precipitation or crystallization of compoundsby SCFs, where SCFs can work either as solvents or antisolvents,including crystallization from supercritical solution, rapid expan-sion of a supercritical solution, gas antisolvent crystallization andsupercritical antisolvent precipitation [3].

There are two types of supercritical solvent–solute phase dia-grams [4], among which type II is more relevant than type II toextraction or crystallization using SCFs. Thus, the emphasis of thisstudy is placed on the type II phase diagram, as shown in Fig. 1.The solid curve at low temperature represents the vapor pressurecurve of the pure solvent, which ends at the critical point denotedby CP1. In this vicinity, there is a dotted-dashed curve standing forthe solid–liquid–vapor (SLV) equilibrium of the mixture due to thesolute dissolving into the solvent. This SLV curve ends at the lowercritical end point (LCEP), where the liquid and vapor phases becomeidentical in the presence of the solid phase [5]. The solid curves athigh temperature are the sublimation curve, the vapor pressurecurve (ending at CP2) and the melting curve of the pure solute [5],which intersect at the triple point. The other SLV curve appears athigh pressure due to the solvent dissolving into the liquid solute [5],which emanates from the triple point and terminates at the upper

critical end point (UCEP). The LCEP and UCEP are connected to CP1and CP2, respectively, by the dashed vapor–liquid critical curves ofthe mixture. The shaded region between the two CEPs, where anequilibrium is achieved between the solid and single-phase fluid, is

Z.-C. Hu and X.-R. Zhang / J. of Supercritical Fluids 153 (2019) 104578 3

Fig. 1. Type II phase diagram of the supercritical solvent–solute system in thetemperature–pressure plane. CP1: the critical point of the solvent; CP2: the criticalpoint of the solute; LCEP: lower critical end point; UCEP: upper critical end point;SLV: solid–liquid–vapor equilibrium.

Fig. 2. Schematic of the proposed model that combines extraction and crystalgtd

tMin

tcspapb

difampn

2

2

tbpa

srd

Fig. 3. Schematic of the physical model for mathematical modeling where x and z are

rowth processes. The situation plotted here is when the solubility increases withemperature and the top wall is hotter than the bottom wall or when the solubilityecreases with temperature and the bottom wall is hotter than the top wall.

he region where extraction and crystal growth would take place.ore importantly, the solubility of the solvent changes dramat-

cally with small variations in temperature and/or pressure [6,7]ear the two CEPs.

The question that naturally arises is whether one can combinehe two processes in one apparatus in order to take advantage of theritical behavior of the solubility. On the one hand, it is particularlyuitable for substances that require both processes in industrialroduction. On the other hand, such an apparatus may also oper-te in a single-purpose mode as an alternative method. This paperresents preliminary theoretical explorations regarding this ideaased on a conceptual model.

This paper is structured in the following way: Section 2 isevoted to the presentation of the model and its general mathemat-

cal descriptions, followed, in Section 3, by the specific formulationsor the CO2–naphthalene system. In Section 4, detailed theoreticalnalyses are presented to optimize the model from both ther-odynamic and hydrodynamic points of view. In Section 5, the

erformance of the optimized model is evaluated by means ofumerical simulations. The paper is summarized in Section 6.

. Problem description

.1. Physical model

The discussions of this paper are based on a conceptual modelhat combines extraction and crystal growth by virtue of the criticalehavior of the solubility. Fig. 2 shows the schematic of the pro-osed model, where a cavity with thermostats mounted at the topnd bottom walls is presented, while its lateral walls are adiabatic.

The operation of the model can be understood as follows: Theolution in the cavity is initially saturated and in thermal equilib-ium. Then, the thermostats are adjusted to achieve a temperatureifference between the two walls. Since the solubility depends on

the horizontal and vertical directions, respectively, g is the gravitational accelerationvector, d is the cavity’s height, T1 and T2 are the temperatures at the bottom and topwalls, respectively.

temperature, as the heat penetrates into the fluid, the regions nearthe walls are either unsaturated or supersaturated depending onthe behavior of the solubility to temperature change. The unsat-urated region provides the opportunity for extraction, while thesupersaturated region favors crystal growth. To combine extractionand crystal growth, the unsaturated region is filled with solute-enriched nutrients, and a crystal growth plate (with crystal seedson it) is inserted in the supersaturated region. In the transportregion, the solute transfer is accomplished by diffusion or naturalconvection.

2.2. Mathematical description

To study the conceptual model shown in Fig. 2, the emphasisof the mathematical modeling is placed on the transport region,where the extraction and crystal regions are regarded as twoidealized rigid interfaces subjected to desorption and adsorptionreactions, respectively.

Fig. 3 presents the simplified model, a square cavity of height dcontaining a supercritical binary mixture initially at saturation. Thehorizontal and vertical directions are denoted by x and z, respec-tively. The subscripts 1 and 2 are used to denote a physical propertyat z = 0 and d, respectively. The mixture is initially motionless, satu-rated, in thermal equilibrium and stratified under gravity g(0, − g).The reference state is defined as the initially average state, denotedby the subscript r. The lateral boundaries are adiabatic and imper-meable, and all of the four boundaries are no-slip. At t = 0 s, thetemperature T at the bottom and the top walls are adjusted fromTr to T1 and T2, respectively, resulting in a solubility difference. Thedesorption reaction due to extraction occurs at the horizontal wallof the higher solubility, while the adsorption reaction caused bycrystallization takes place at the other horizontal wall. Dependingon the value of the temperature difference, the physical process inthe cavity can be either pure diffusion or natural convection.

The governing equations are given by [8]

• conservation of mass:

∂�

∂t+ ∇ · (�u) = 0, (1)

• conservation of momentum:

∂�u∂t

+ ∇ · (�uu) = −∇pd + �∇2u + 13

�∇(∇ · u) + (� − �s)g, (2)

• conservation of energy:

∂�T

∂t+ ∇ · (�uT) = �

cp∇2T + ∇ad

g

∂pth

∂t− �s∇adw, (3)

• conservation of concentration:

∂�c + ∇ · (�uc) = D∇ · (�∇c), (4)

∂t

• equation of state (EOS):

� = �s + �r[˛(pth − pr) − ˇ(T − Tr) + �(c − cr)], (5)

4 upercritical Fluids 153 (2019) 104578

w�tpt

˛

aa

d

p

wtd

(ws

0

�

p

wa

p

wspi

tTs(sttu

2

d

N

Table 1Definitions and names of the governing parameters.

Symbols Definitions Names

RaT�r gˇ∇T d4

�DTThermal Rayleigh number

RaS�r g�∇c d4

�DTSolutal Rayleigh number

D/DT Diffusivity ratioPr �

�r DTPrandtl number

rT ∇ad/∇T Temperature gradient ratio

Z.-C. Hu and X.-R. Zhang / J. of S

here � is the density, t is the time, u(u, w) is the velocity vector, is the viscosity, � is the thermal conductivity, c is the concentra-ion of the solute (mass fraction), cp is the specific heat at constantressure, D is the diffusion coefficient, ∇ad = Trˇg/cp is the adiabaticemperature gradient, and

= 1�r

(∂�

∂p

)T,�

, = − 1�r

(∂�

∂T

)p,c

, � = 1�r

(∂�

∂c

)T,p

, (6)

re the isothermal compressibility, thermal expansion coefficientnd concentration contraction coefficient, respectively.

In the derivation of the above equations, the total pressure p wasecomposed into three parts:

= pth + ps + pd, (7)

here pth is the thermodynamic pressure representing the averagehermodynamic state (a function of t), ps is the hydrostatic pressureue to gravity (a function of z) obeying

dps

dz= −�sg, (8)

�s is the stratified initial density) and pd is the dynamic pressureorking to balance the inertial, viscous and body forces (time- and

pace-dependent).According to Eq. (8), by letting the mean �s and ps equal �r and

, respectively, �s and ps are obtained as

s = ��rexp(−�z/d)1 − exp(−�)

, (9)

s = �rgd

[exp(−�z/d)1 − exp(−�)

− 1�

], (10)

here � = ˛�rgd is a dimensionless parameter. pth is calculatedccording to the overall mass conservation:

th = pr +∫

VT[ˇ(T − Tr) − �(c − cr)]dVT +

∫VT

(� − �s)/�rdVT

˛VT, (11)

here VT is the total volume. In addition to the pressure decompo-ition in Eq. (7), through the low-Mach number approximation [9],d, as a high-order term, was omitted in Eqs. (3) and (5). In addition,n Eqs. (2), (3) and (5), ps has been substituted by �s through Eq. (8).

In this study, a linearized EOS is used, and all physical proper-ies are treated as constant values of those at the reference state.hese approximations are perfectly satisfactory when |T1 − T2| ismall. Even though the thermostats are adjusted in such a way thatT1 + T2)/2 = Tr to guarantee that the physical properties are repre-entative, there will be deviations if |T1 − T2| is too large. However,hey are still employed because they reduce the nonlinearities ofhe problem and facilitate theoretical analyses and numerical sim-lations.

.3. Initial and boundary conditions

According to the description in Section 2.2, the initial fields,enoted by bars, are governed by

u = w = pd = 0, T = Tr,

∇ �∇ c = 0,

p = pr +

∫VT

[( � − �s)/�r − �(c − cr)

]dVT

,

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪ (12)

th ˛VT

� = �s + �r [˛(pth − pr) + �(c − cr)] .

⎪⎪⎪⎪⎭ote that initially, c /= cr because of the nonuniformity of �.

� ˛�rgd Gravitational compressibility factor� cp/cv Specific heat ratio

Mathematically, the boundary conditions can be expressed as

x = 0 and d, u = w = 0,∂T

∂x= ∂c

∂x= 0,

z = 0, u = 0, w = − D

1 − c

∂c

∂z, T = T1,

z = d, u = 0, w = − D

1 − c

∂c

∂z, T = T2.

(13)

The nonzero w is a result of the partial permeability of the reac-tive walls (only the solute is allowed to go across) [10]. Accordingto previous discussions, the boundary conditions of c at the tworeactive walls can be generally expressed by

s1 − s2 ≥ 0, c1 : desorption, c2 : adsorption,

s1 − s2 < 0, c1 : adsorption, c2 : desorption,

}(14)

where s is the solubility in the unit of mass fraction. The adsorp-tion is caused by crystal growth while desorption is the resultof extraction. The exact expressions for these reactions are verycomplicated and rely strongly on the correlations reported inexperimental measurements. In the next section, the modeling ofthe CO2–naphthalene system will be presented, together with theformulations of Eq. (14).

2.4. Governing parameters

Under the current configuration, the natural convection isdriven by the gradients of T and c, and they generally have differ-ent rates of diffusion [11]. This kind of natural convection is calleddouble-diffusive convection (DDC).

The DDC of a compressible binary fluid is governed by sevenparameters [8]: thermal Rayleigh number RaT, solutal Rayleighnumber RaS, diffusivity ratio , Prandtl number Pr, temperaturegradient ratio rT, gravitational compressibility factor � and specificheat ratio � . The definitions of these parameters are summarizedin Table 1, where ∇T = (T1 − T2)/d − ∇ad is the typical relativetemperature gradient, ∇c = (c1 − c2)/d is the typical concentra-tion gradient, DT = �/(�rcp) is the thermal diffusivity and cv = cp −Trˇ2/(˛�r) is the specific heat at constant volume.

Physically, the two Rayleigh numbers measure the competitionbetween the driving force (buoyancy) and the drag force (diffusionand viscosity) in natural convection. In particular, RaT > 0 indicatesthat the buoyancy produced by T is positive, while RaT < 0 meansthat T depresses natural convection. In contrast, RaS < 0 suggeststhat c promotes convection, while RaS > 0 means that the effect ofc is negative. is defined as the ratio of mass diffusivity to ther-mal diffusivity. rT measures the relative importance of ∇ad to ∇T ,and � the density stratification caused by gravity. � indicates the

intensity of the piston effect (a special fast thermalization phe-nomenon), which mainly influences the route to the fully developedstate [12,13].

upercritical Fluids 153 (2019) 104578 5

ts

N

SS

S

Tct

3

araacaplcs

afhim

3

tdds

s

w(ye

3rdtab[ttTfkti

r

Z.-C. Hu and X.-R. Zhang / J. of S

Two response parameters are defined to quantitatively describehe convection. The intensity of convective heat transfer is mea-ured by the Nusselt number:

u = convective heat transferconductive heat transfer

= ∂T1/∂z or ∂T2/∂z

(T2 − T1)/d. (15)

imilarly, the performance of mass transfer is measured by theherwood number:

h = convective mass transferdiffusive mass transfer

=�1D1−c1

∂c1∂z

or �2D1−c2

∂c2∂z

�rD(c2 − c1)/d. (16)

he emphasis of this study is placed on the fully developed naturalonvection, which is featured by the statistical consistency betweenhe values of Sh and Nu calculated at the two reactive walls.

. Formulations for CO2–naphthalene system

A reference fluid system should be chosen to conduct detailednalyses. In this paper, the CO2–naphthalene system is taken as aeference system with a type II phase diagram (see Fig. 1). CO2,s a nontoxic, nonflammable, environmentally safe and widelyvailable solvent, is the most popular substance in industrial appli-ations. Moreover, the binary system has been studied extensively,nd there are sufficient data for the solubility [14–17], trans-ort properties [18–21] and crystal growth kinetics [22,23]. It is

egitimately supposed that due to the critical universality, the con-lusions based on one reference system could be extended to otherystems with the same type of phase diagram.

The task of this section is twofold. On the one hand, thedsorption and desorption reactions at the two reactive walls areormulated to obtain explicit expressions of Eq. (14). On the otherand, the thermodynamic and transport properties in the govern-

ng equations are calculated. Before dealing with the first task, theodeling of solubility is first introduced.

.1. Solubility

The model proposed by Skerget et al. [17] is implemented inhis study to estimate the solubility of naphthalene in CO2. Theetails are summarized in Appendix A. In their model, the solubility,enoted by ys, is expressed by the mole fraction of the solute in theaturated mixture. s can be calculated from ys by

= ysM1

(1 − ys)M1 + ysM2, (17)

here M1 and M2 are the molar masses of CO2 and naphthaleneM1 = 0.04401 kg/mol, M2 = 0.1282 kg/mol), respectively. Note thats and s are different representations of the solubility, and themphasis of this study is placed on the latter one.

For the CO2–naphthalene system, TLCEP = 307.65 K [24] is close to04.21 K, the critical temperature of CO2. McHugh and Paulaitis [14]eported that TUCEP is between 333.55 K and 338.05 K. Solubilityata are usually experimentally reported at different tempera-ures between TLCEP and TUCEP as functions of p. Many data arevailable for T = 308.15 K, 328.15 K and 333.55 K [14–17] scatteredetween TLCEP and TUCEP. To estimate the solubility, Skerget et al.17] fitted the parameters in Eq. (A.3), i.e., k12 and l12, at individualemperatures by minimizing the average absolute relative devia-ion (AARD) between the calculated ys and the experimental data.heir results are [17]: For T = 308.15 K, k12 = −0.0272, l12 = −0.1289;or T = 328.15 K, k12 = −0.0177, l12 = −0.1255; and for T = 333.55 K,12 = 0.0533, l12 = −0.0166. Symbolic computations are employed

o avoid lengthy algebraic calculations. The computations weremplemented based on the Symbolic Math Toolbox in MATLAB [25].

The variations in s against p are presented in Fig. 4 for latereference.

Fig. 4. Maximum mass fraction of the solute (naphthalene) in the supercritical sol-vent (CO2), i.e., the solubility s, plotted against pressure p at 308.15 K (solid curve),328.15 K (dash-dotted curve) and 333.55 K (dotted curve).

To facilitate theoretical analyses, similar to Eq. (5), s is linearizedwith respect to p, T, and c, leading to

s = sr + �p(pth − pr) + �p�rgd

[exp(−�z/d)1 − exp(−�)

− 1�

]+ �T (T − Tr) + �c(c − cr), (18)

where

�p =(

∂s

∂p

)T,c

, �T =(

∂s

∂T

)p,c

, �c =(

∂s

∂c

)T,p

, (19)

can be determined directly by virtue of the symbolic computations.Note that sr = cr because the reference state is saturated. In thederivation of Eq. (18), Eq. (7) was employed, where pd was omittedaccording to the conclusion from low-Mach number approximationand ps has been substituted by Eq. (10).

3.2. Adsorption and desorption reactions

As mentioned earlier, the extraction and crystal growth regionsin Fig. 2 are simplified into two rigid walls in Fig. 3 where desorp-tion (extraction) and adsorption (crystal growth) reactions occur.According to Eq. (14), the sign of (s1 − s2) determines the designa-tion of the types of reactions. After some computations based onEq. (18), (s1 − s2) is obtained as

s1 − s2 = �T (T1 − T2) + �p�rgd

1 − �c�, (20)

where

� = c1 − c2

s1 − s2(21)

is the solubility efficiency, defined as the ratio of actual concen-tration difference to its theoretically maximum value. Physically, �indicates that the system is either diffusion-limited (when � is closeto unity) or reaction-limited (when � is close to zero). Furthermore,our computations confirm that 0 < �c < 1 is satisfied at the threetemperatures for p values ranging from 5 MPa to 30 MPa. There-fore, the denominator is always positive, and the sign of (s1 − s2) isdetermined by the numerator.

The desorption reaction at the extraction wall depends on thespecific circumstance. Modeling a practical situation would losegenerality. Thus, the desorption reaction is assumed to be diffusion-limited. That is, local equilibrium is achieved at any time [26],namely,

desorption : c = s. (22)

Now, consider the adsorption reaction due to crystal growth. Atthe interface, the outflow solute crystallizes into a new layer of the

6 upercr

sm

wtVe

V

wtqi1

a

Fi

ippnfae

3

cS

datatccasUpo1dt

4

tdc

Z.-C. Hu and X.-R. Zhang / J. of S

olid solute of density �S with a building velocity of Vc [10]. Thisass balance relationship yields [10]

�D

1 − c

∣∣∣∣∂c

∂z

∣∣∣∣ = �SVc, (23)

here Vc is the growth rate of the crystal (always posi-ive), �S = M2/VS (VS is the molar volume of the solute, andS = 1.1 × 10−4 m3/mol for naphthalene). Vc is modeled by thexperimental correlation proposed by Tai and Cheng [22]:

c = KVS〈〉 = 100KVS

( 〈c〉1 − 〈c〉 − 〈s〉

1 − 〈s〉)

, (24)

here K is the growth rate constant, 〈〉 is the bulk supersatura-ion, 〈c〉 the bulk concentration and 〈s〉 the bulk solubility. The bulkuantities reflect the environment for crystal growth in their exper-

ments. In [22], the solubility is expressed in grams of the solute per00 g of solvent. That is why a unit conversion appears in Eq. (24).

Combining Eqs. (23) and (24) yields:

dsorption :�D

1 − c

∣∣∣∣∂c

∂z

∣∣∣∣ = 100KM2

( 〈c〉1 − 〈c〉 − 〈s〉

1 − 〈s〉)

. (25)

inally, using Eqs. (20), (22) and (25), Eq. (14) can be written explic-tly as

s1 − s2 ≥ 0, c1 = s1, − �2D

1 − c2

∂c2

∂z= 100KM2

(c2

1 − c2− s2

1 − s2

),

s1 − s2 < 0, c2 = s2,�1D

1 − c1

∂c1

∂z= 100KM2

(c1

1 − c1− s1

1 − s1

).

⎫⎪⎬⎪⎭ (26)

Tai and Cheng [22] measured the growth rate of naphthalenen supercritical CO2 at T = 318.15 K under various p values androvided K = 3.6 × 10−3 mol m−2 s−1. Since the real situation is com-licated and accurate experimental data at other temperatures areot available, K is treated as a constant throughout this study. In

act, it is valid as long as K does not vary dramatically with T and p,s confirmed by the experiments of Tai and Cheng [22] and Uchidat al. [23] for the CO2–naphthalene system.

.3. Thermodynamic and transport properties

Calculations of the thermodynamic properties, i.e., ˛, ˇ, � andp, are based on the Peng–Robinson EOS introduced in Appendix A.ome additional equations for cp are detailed in Appendix B.

In regard to the transport properties, because the experimentalata are scarcely available, � and � of the mixture are approximateds those of pure CO2 [26] at Tr and �r. The state-of-the-art correla-ions developed by Laesecke and Muzny [27] and Huber et al. [28]re employed to calculate � and �, respectively. In these correla-ions, is required in the calculations of the critical enhancementontributions. Therefore, of the mixture is used to ensure theonsistencies in the critical behavior of various properties. Such anpproximation is highly accurate when the state is close to LCEP,ince the solubility is substantially low; see Fig. 4. When close to theCEP, the approximation is still satisfactory. The results from theure-CO2 approximation are compared to the experimental dataf � at T = 328.15 K reported by Lamb et al. [20]; for p ranging from2 MPa to 100 MPa, the AARD is 6.17 %. D is obtained from the modeleveloped by Vaz et al. [29], where they report the AARD is 9.68 % forhe CO2–naphthalene system by comparing with 114 data points.

. Theory

So far, the conceptual model and its mathematical formula-ion have been elaborated. In this section, theoretical analyses areeveloped to optimize the model and provide detailed strategiesoncerning its operation.

itical Fluids 153 (2019) 104578

An objective function is a key component in an optimizationproblem. Here, it is natural to take the objective function as thecross-boundary flow rate of the solute, given by

m = �1D

1 − c1

∣∣∣∣∂c1

∂z

∣∣∣∣ or�2D

1 − c2

∣∣∣∣∂c2

∂z

∣∣∣∣ , (27)

i.e., m can be calculated at both walls. In the sense of the tem-poral average, m1 is equal to m2 in a fully developed convection.Combining Eqs. (16), (20) and (21), m can be further expressed by

m =∣∣∣∣�T (T1 − T2) + �p�rgd

�−1 − �c

∣∣∣∣ �rDShd

. (28)

Substituting Eq. (26) into Eq. (16), using the definition of � givenby Eq. (21) and neglecting high-order terms, one arrives at

� =(

�rDSh100KM2d

+ 1)−1

. (29)

Combining the above two equations leads to the final expression ofm

m =∣∣�T (T1 − T2) + �p�rgd

∣∣( 1100KM2

+ 1 − �c

Shd

�rD

)−1

. (30)

The optimization goal is to maximize m.The elements of Eq. (30) can be divided into six groups: (T1 − T2),

d, K, Sh, g, and physical properties at the reference state. Amongthese elements, K and g have been treated as constants in this study.Moreover, in this section, d = 0.01 m is fixed as a reference length.The effects of d on m will be separately discussed later.

After these simplifications, the independent factors in the cur-rent optimization problem are the reference state and (T1 − T2),while Sh is a response parameter of these two variables. Therefore,the task of this section can be briefly interpreted as exploring thebest reference state and the strategies of controlling T1 and T2 tomaximize m. Moreover, 0 < �c < 1 implies m is a monotonic increas-ing function of Sh. The optimization is performed in two steps.First, the thermodynamic optimization maximizes the sensitivityof m in terms of (T1 − T2). Second, the hydrodynamic optimizationmaximizes Sh for a given (T1 − T2).

4.1. Thermodynamic optimization

In thermodynamic optimization, the natural convection is omit-ted. By letting Sh = 1, the thermodynamic contribution of m,denoted by mth, is obtained as

mth ≡ m(Sh = 1)

=∣∣�T (T1 − T2) + �p�rgd

∣∣ [ 1100KM2

+ d(1 − �c)�rD

]−1

. (31)

The sensitivity of mth with respect to (T1 − T2) is measured by

mTth = dmth

d(T1 − T2)= |�T |

[1

100KM2+ d(1 − �c)

�rD

]−1

. (32)

Fig. 5 shows the variations in mTth with p at various temperatures.

In the low-pressure region, influenced by the LCEP, all three curvesdevelop local minimum values denoted by A, B and C. As T departsfrom TLCEP = 307.65 K, the locations of the minimum values movetoward the higher pressure and the slopes decay gradually. Mean-while, intriguingly, local peak values emerge in the high-pressureregion (D, E and F), which increase with T, along with the loca-

tions moving toward pUCEP. The trends described above indicate atransition from the LCEP-dominance to the UCEP-dominance.

At the local extreme values of mTth, mth is sensitive to the varia-

tion of (T1 − T2). Therefore, they are preferable reference states. On

Z.-C. Hu and X.-R. Zhang / J. of Supercritical Fluids 153 (2019) 104578 7

Fig. 5. Thermodynamic contribution of the cross-boundary flow rate of the solute,mT

th, given by Eq. (32), versus pressure p at 308.15 K (solid curve), 328.15 K (dash-

dotted curve) and 333.55 K (dashed curve) for the CO2–naphthalene system. Thelocal extreme values (denoted by A to F) are marked by circles, two of whichare enlarged in the two insets. The pressure at the upper critical end point,PP

ttp

sttasrdf

4

ffti

saiif

Table 3Dimensionless parameters at the reference states.

Label Pr � �

A 2.973 8.652 4.179 × 10−5 27.507B 0.724 2.501 5.662 × 10−6 5.065C 0.682 2.213 5.122 × 10−6 4.537

−7

STA : � = 90◦–180◦, (35)

TR

UCEP = 22.6 MPa, is obtained from [30], while that at the lower critical end point,LCEP = 7.5 MPa, is a rough estimation.

he basis of the concept of the pseudocritical point, it is conveniento call them pseudo-CEPs. According to Fig. 5, the performance of aseudo-CEP increases as its corresponding CEP is approached.

In fact, the pseudo-LCEPs and the pseudo-UCEPs can be repre-ented by two respective continuous curves in the p − T plane. Dueo the lack of data, they cannot be obtained. However, because mT

thends to infinity (i.e., |�T| tends to infinity) at both CEPs, it is reason-ble to argue that one can always obtain two equivalent referencetates that own the same mT

th in both near-LCEP and near-UCEPegions. Therefore, the thermodynamic optimization cannot pre-ict the preferable near-CEP region, which should be investigatedrom a hydrodynamic point of view.

.2. Hydrodynamic optimization

This section is dedicated to comparing the two near-CEP regionsrom a hydrodynamic point of view. Fig. 5 suggests there are sixavorable reference states, which can be divided into two groups:he pseudo-LCEPs, including A, B and C, and the pseudo-UCEPs,ncluding D, E and F.

Table 2 lists the physical properties, and Table 3 lists the dimen-ionless parameters at these points. In the near-LCEP region, i.e. A, Bnd C, s decreases with T (�T < 0), density is sensitive to the change

n T and c, and the dissolved naphthalene makes the fluid heav-er (� > 0). However, in the near-UCEP region, i.e., D, E and F, theseeatures are just the opposite. The preliminary conclusion is that

able 2eference states and corresponding properties.

Label Tr (K) pr (MPa) �r (kg/m3) sr

A 308.15 7.93 455.534 5.22B 328.15 11.7 526.066 2.88C 333.55 12.7 504.748 2.40D 308.15 26.1 935.964 5.16E 328.15 24.4 855.178 1.17F 333.55 24.3 893.198 2.09

Label cp (kJ/(kg K)) �T (K−1) �p (MPa−1)

A 22.445 −3.208 × 10−3 1.923 × 10−2

B 4.329 −7.323 × 10−4 1.027 × 10−2

C 3.862 −2.529 × 10−4 7.753 × 10−3

D 1.960 2.248 × 10−3 8.410 × 10−5

E 1.983 4.316 × 10−3 7.839 × 10−4

F 1.870 7.035 × 10−3 1.466 × 10−3

D 0.132 1.678 6.783 × 10 2.074E 0.179 1.554 7.917 × 10−7 2.055F 0.153 1.499 6.262 × 10−7 1.852

different CEP regions have distinct hydrodynamic behavior, whichfurther influences the performance.

4.2.1. The regime of convectionAs mentioned earlier, the natural convection in the cavity is cat-

egorized as DDC, where the two governing components are T andc. In fact, according to the roles of the two components and theirdiffusion rates, DDC can be divided into four regimes [11]:

• Stable Regime (STA): Convection is inhibited by both components.Sh = 1.

• Fingering Regime (FIN): Convection is promoted by the slow dif-fuser and is inhibited by the fast one. Sh ≥ 1.

• Oscillatory Regime (OSC): Convection is promoted by the fast dif-fuser and is inhibited by the slow one. Sh ≥ 1.

• Cooperative Regime (COO): Convection is promoted by both com-ponents. Sh ≥ 1.

It is fundamental to investigate the regime of convection priorto understanding the behavior of Sh. As discussed in Section 2.4,promotion or inhibition can be determined from the signs of RaTand RaS, while the relative size of diffusion rates can be inferredfrom .

To avoid lengthy descriptions, through heuristic derivations, thejudgment can be elegantly performed by the � − criterion:

� = arg(RaT + iRaS) + 180◦ max(RaTRaS, 0)RaTRaS

min( − 1, 0) − 1

, (33)

where arg() is a function that returns the argument of a com-plex number in the range of [0◦, 360◦]. Through this equation,the regimes of DDC are projected onto the phase space of �. Thecorrespondences between � and the regimes are

FIN : � = 0◦–90◦, 360◦–450◦, (34)

OSC : � = 180◦–270◦, (36)

COO : � = 270◦–360◦. (37)

(MPa−1) (K−1) �

1 × 10−3 9.352 × 10−1 1.729 × 10−1 33.7900 × 10−2 1.097 × 10−1 2.472 × 10−2 4.0900 × 10−2 1.034 × 10−1 2.171 × 10−2 3.5850 × 10−2 7.387 × 10−3 4.772 × 10−3 −0.4391 × 10−1 9.437 × 10−3 5.003 × 10−3 −0.2605 × 10−1 7.147 × 10−3 4.058 × 10−3 −0.268

�c � (Pa s) � (W m−2 K−1) D (mm2 s−1

0.723 3.121 × 10−5 0.081 2.354 × 10−2

0.526 3.534 × 10−5 0.061 1.945 × 10−2

0.415 3.366 × 10−5 0.058 2.054 × 10−2

0.075 9.700 × 10−5 0.113 8.153 × 10−3

0.284 7.703 × 10−5 0.098 1.036 × 10−2

0.797 8.491 × 10−5 0.106 9.695 × 10−3

8 Z.-C. Hu and X.-R. Zhang / J. of Supercritical Fluids 153 (2019) 104578

Fig. 6. Evolution of the regimes of double-diffusive convection with �T = |T1 − T2| in the phase diagram of the indicator � defined by Eq. (33) for the pseudocritical end pointsA to F in Table 2, where Sh = 1 is fixed. T1 and T2 are the temperatures at the lower and upper walls, respectively. STA: stable regime. FIN: fingering regime. OSC: oscillatoryregime. COO: cooperative regime. Solid curves: (T1 > T2), dashed curves: (T1 < T2).

Roi

R

web

tWS�

Moreover, the modeling of the CO2–naphthalene system allowsaS to be expressed as a function RaT. According to the definitionsf RaT and RaS in Table 1 and Eqs. (20), (21) and (29), the functions expressed as

aS =[

�T �ˇ−1RaT + (�T∇ad + �p�rg)�rg�d4

�DT

](

�rDSh100KM2d

+ 1 − �c

)−1

, (38)

here RaT is a function of (T1 − T2). Based on Eqs. (33) and (38), thevolutions of the regimes of convection varying with (T1 − T2) cane investigated for the reference states A to F in Table 2.

As indicated by Eq. (38), because 1 − �c > 0, Sh has no effect on

he sign of RaS. In other words, Sh does not alter the regime of DDC.

ithout loss of generality, the following discussions are begun withh = 1. Fig. 6 shows evolutions of the regimes of convection withT = |T1 − T2| at the pseudo-CEPs for Sh = 1. It is demonstrated that

the evolutions of the regimes are complex when �T is small. How-ever, the common feature is, when �T is large enough, all casesevolve into the STA if (T1 < T2) and the COO if (T1 > T2). Because �Tis anticipated to be higher than several mKs in actual applications,it is concluded that (T1 > T2) is preferable for both near-CEP regionsto take advantage of the convection instead of being motionless.

The influence of Sh > 1 can be understood by considering thelimiting case of Sh = Inf, which leads to RaS = 0 and � = 0◦. Therefore,because Sh > 1, the solid curves in the COO actually bend toward(but never go across) � = 0◦ as (T1 − T2) increases gradually. How-ever, the dashed curves in Fig. 6 remain unchanged since Sh = 1 inthe STA.

4.2.2. Evaluation of the Sherwood numberFig. 6 identifies that COO is the most relevant regime in actual

applications. Therefore, in terms of Sh, the two near-CEP regionsshould be compared in this regime.

Z.-C. Hu and X.-R. Zhang / J. of Supercritical Fluids 153 (2019) 104578 9

Fig. 7. Values of [RaT − (RaS/)] at six pseudocritical end points A to F of theCnt

ecsg

R

Tw

eaLTctLtt

ptpviaeprtitop

fistrip

5

tCtw

Fig. 8. Log–log plot of the Sherwood number Sh versus [RaT − (RaS/)], where RaT

O2–naphthalene system (see Fig. 5) for T1 − T2 = 10 mK. RaT is the thermal Rayleighumber, RaS is the solutal Rayleigh number, is the diffusivity ratio, T1 and T2 arehe temperatures at the bottom and top walls, respectively.

The level of Sh is proportional to the intensity of convection. Thestimation is relatively straightforward in the COO since both T and

promote the convection. According to the classical result of lineartability analysis, the criterion for the onset of DDC in the COO isiven by [11]

aT − (RaS/) > threshold. (39)

his equation suggests that the left-hand side, [RaT − (RaS/)], canork as an indicator for the intensity of DDC.

Fig. 7 presents the comparisons of [RaT − (RaS/)] at various ref-rence states for T1 − T2 = 10 mK, under which all reference statesre in COO. Comparing the values of [RaT − (RaS/)], for pseudo-CEPs, the ranking is A B > C, and for pseudo-UCEPs, F > E > D.herefore, in each CEP region, the intensity of DDC increases as theorresponding CEP is approached. However, comparisons betweenhe two regions suggest A B > C > F > E > D, namely, the pseudo-CEPs are much higher than the pseudo-UCEPs. Therefore, due tohe hydrodynamic advantages, the pseudo-LCEPs are preferable tohe pseudo-UCEPs.

The explanations for the hydrodynamic advantages of theseudo-LCEPs involve the critical behavior of physical proper-ies. For a pure fluid, it is well known that the variable physicalroperties in the critical region make it susceptible to natural con-ection [31]. For a binary fluid mixture, there are also anomaliesn physical properties. However, the extents of these anomaliesre generally weaker than those of pure fluids [32]. Nevertheless,xceptions occur for dilute mixtures, whose anomalies in physicalroperties are also strong [32]. Since the solubility at the LCEP iselatively lower than that at the UCEP (see Fig. 4), it is concludedhat the physical properties exhibit stronger anomalous behav-or near the LCEP than near the UCEP. Since the susceptibility inerms of buoyancy-driven convection is proportional to the extentf anomalies, pseudo-LCEPs are hydrodynamically preferable toseudo-UCEPs.

The theoretical optimization performed in this section identi-es that the pseudo-CEPs are preferable reference states. At thesetates, the best strategy to control T1 and T2 is (T1 > T2) so thathe DDC is cooperatively driven by T and c. In each near-CEPegion, the performance is promoted when the corresponding CEPs approached. In addition, the pseudo-LCEPs are superior to theseudo-UCEPs due to their hydrodynamic excellence.

. Numerical simulations

The optimization presented in the previous section suggests that

he state A is the best reference state among the available pseudo-EPs in Table 2. According to Fig. 6 (A), as (T1 − T2) grows from zero,he convection evolves into the STA, FIN, and COO in sequence, inhich the COO is the main concern. In this section, by taking state

is the thermal Rayleigh number, RaS is the solutal Rayleigh number and is thediffusivity ratio. Dots: data in Table 4 for the CO2–naphthalene system. Solid line:power law given by Eq. (40), which is fitted from cases 7 to 30 in Table 4.

A as the reference state, the actual performance of the system isevaluated through a series of numerical simulations for d = 0.01 mand RaT up to 1 × 107 in the COO. Then, the influences of d arestudied separately.

5.1. Numerical method

The governing equations along with the initial and boundaryconditions were solved numerically by the finite-volume method.The numerical schemes have second-order accuracy in space andfirst-order accuracy in time. The pressure–velocity coupling istreated by the block-coupled algorithm [33]. The time step is con-trolled by limiting the maximum Courant number to be lower than0.5. The numerical method has been elaborated in [34,35].

A nonuniform wall-refined grid with 256 × 256 grid points isused in this study. A grid-independence test has been performedby comparing the results of RaT = 1 ×107 from two refined grids:388 × 388 and 483 × 483. The values of Nu and Sh produced by thethree grids agree well, suggesting that the resolution of 256 × 256is sufficient.

5.2. Cooperative regime

According to Eq. (20), when (T1 − T2) surpasses−�p�rgd/�T = 0.2678 mK, s1 is smaller than s2, and correspondingly,the DDC enters COO. The setup of the cases and the main resultsare summarized in Table 4.

As (T1 − T2) increases, the response parameters, i.e., Nu and Sh,increase monotonically (see Table 4). Moreover, the form of con-vection changes progressively from steady state to periodic motionand further to chaotic state, i.e., in the direction of increasing non-linearity.

Fig. 8 presents Sh versus [RaT − (RaS/)] in the double logarith-mic coordinates. As discussed earlier, [RaT − (RaS/)] is regarded asa measurement of the intensity of DDC in the COO. The values ofSh for the chaotic cases, i.e., cases 7–30 in Table 4, are identified toobey the following power law (represented by a straight line on alog–log plot):

Sh = n1[RaT − (RaS/)]n2 , (40)

where n1 = 0.1123 and n2 = 0.2793 are fitted coefficients. Notethat n2 is close to 2/7 =0.2857, an exponent for Nu in theRayleigh–Bénard convection [36].

Note that since our mathematical modeling is based on the lin-earized EOS and constant properties, Eq. (40) is likely to deviate

from real situations when obvious variations in physical propertiesset in (when RaT is large enough). In fact, numerical simulationsincorporating real EOS and variable properties are extremely chal-lenging. Equation (40) can serve as a reference for real situations.

10 Z.-C. Hu and X.-R. Zhang / J. of Supercritical Fluids 153 (2019) 104578

Table 4Summary of the selected cases and results in the COO. For chaotic and periodic cases, time-averaged values of RaS, �, Nu, Sh and m are shown here.

Case no. (T1 − T2) (mK) RaT RaS Final state � Nu Sh m (kg/(m2 s)

1 0.2678 109000 0 Steady 0.943 1.4158 2.6132 02 0.2681 110000 −1851 Steady 0.942 1.4242 2.6536 8.624 × 10−12

3 0.2809 150000 −75525 Periodic 0.940 1.6211 2.7321 3.622 × 10−10

4 0.3001 210000 −178040 Periodic 0.927 1.9807 3.3931 1.060 × 10−9

5 0.3161 260000 −260210 Periodic 0.920 2.2689 3.7426 1.710 × 10−9

6 0.3321 310000 −341910 Periodic 0.916 2.3938 3.9476 2.369 × 10−9

7 0.3480 360000 −422720 Chaotic 0.913 2.7288 4.1076 3.048 × 10−9

8 0.3640 410000 −498010 Chaotic 0.909 3.0620 4.2979 3.757 × 10−9

9 0.3800 460000 −571900 Chaotic 0.903 3.4573 4.6482 4.666 × 10−9

10 0.3960 510000 −646930 Chaotic 0.899 3.6304 4.8147 5.468 × 10−9

11 0.4069 544000 −696469 Chaotic 0.898 3.8262 4.9509 6.053 × 10−9

12 0.5329 938000 −1251121 Chaotic 0.878 5.5735 5.9851 1.315 × 10−8

13 0.6589 1332000 −1787886 Chaotic 0.868 6.8768 6.5707 2.062 × 10−8

14 0.7849 1726000 −2294765 Chaotic 0.858 7.9307 7.1379 2.875 × 10−8

15 0.9109 2120000 −2812321 Chaotic 0.852 8.7330 7.4469 3.676 × 10−8

16 1.0369 2514000 −3313633 Chaotic 0.848 9.3988 7.6910 4.474 × 10−8

17 1.2889 3302000 −4273488 Chaotic 0.838 10.5478 8.3070 6.232 × 10−8

18 1.4150 3696000 −4791228 Chaotic 0.836 11.0505 8.4494 7.107 × 10−8

19 1.6670 4484000 −5745444 Chaotic 0.829 11.9622 8.8902 8.967 × 10−8

20 1.9190 5272000 −6664723 Chaotic 0.821 12.7637 9.3512 1.094 × 10−7

21 2.0450 5666000 −7099038 Chaotic 0.818 13.2059 9.5608 1.191 × 10−7

22 2.1710 6060000 −7515324 Chaotic 0.815 13.4426 9.7708 1.289 × 10−7

23 2.2970 6454000 −7997895 Chaotic 0.813 13.7742 9.8779 1.387 × 10−7

24 2.4230 6848000 −8495331 Chaotic 0.813 13.9954 9.9031 1.477 × 10−7

25 2.5490 7242000 −8890377 Chaotic 0.810 14.2959 10.1215 1.580 × 10−7

26 2.6750 7636000 −9255017 Chaotic 0.803 14.8627 10.5388 1.712 × 10−7

27 2.8010 8030000 −9796852 Chaotic 0.806 14.9027 10.3674 1.783 × 10−7

28 3.1791 9212000 −11119753 Cha −7

29 3.3051 9606000 −11467015 Cha30 3.4311 10000000 −11672524 Cha

Fig. 9. Cross-boundary flow rate of the solute calculated from Eq. (30), m, plottedagainst (T1 − T2) ranging from 3.48 × 10−4 K to 0.5 K. T1 and T2 are the temperaturesat the bottom and top walls, respectively. Solid curve: with double diffusive con-vds

(apWt

suastowtba

1−3×0.2793 0.1621

ection, namely, the Sherwood number Sh obeying Eq. (40). Dashed curve: withoutouble-diffusive convection, namely, Sh = 1. The physical properties are those oftate A in Table 2 for the CO2–naphthalene system.

Due to Eqs. (38) and (40), m can be calculated by Eq. (30) givenT1 − T2). The results are plotted in Fig. 9 for (T1 − T2) up to 0.5 K,long with those obtained by setting Sh = 1 for the purpose of com-arison. In both situations, m varies almost linearly with (T1 − T2).hen T1 − T2 = 0.5 K, m = 5.08 × 10−5 kg/(m2 s), corresponding to

he crystal growth rate of 4.98 × 10−8 m s−1.Uchida et al. [23] constructed a crystal growth chamber to mea-

ure the crystal growth rate of naphthalene in supercritical CO2nder constant T and p. In their experiments, supersaturation waschieved by cooling the continuously supplied and fresh saturatedolution. The cooling was maintained at 0.5 K, 1 K, or 2 K. However,he crystal growth rates were reported to be much lower than thene achieved by the present model for T1 − T2 = 0.5 K. Because there

as a magnetic stirrer in their experiments to externally improve

he mass transfer, the enhancement in the current model shoulde attributed to the optimized reference state, which guarantees

large solubility difference to promote the reaction. However, in

otic 0.799 15.7025 10.8346 2.115 × 10otic 0.798 15.8132 10.8907 2.192 × 10−7

otic 0.795 16.0681 11.1055 2.275 × 10−7

their experiments, the reference states were chosen to cover largeranges of p and T, so the solubility differences were not significant.

Furthermore, by comparing the two curves, a tenfold increasein m is achieved by the DDC. Although Sh is higher than 10 whenT1 − T2 > 3.43 mK (see Table 4), a better amplification does notappear due to the limitation of the crystal growth kinetics. There-fore, if somehow K can be increased, the amplification effect of DDCon m is anticipated to be improved. In a limiting case when K → inf(when the reaction is absolutely diffusion-limited), the amplifica-tion factor is exactly Sh, according to Eq. (30).

5.3. The effects of cavity height

In previous sections, the external factor d is treated as a con-stant of 0.01 m. Here T1 − T2 = 0.5 K is fixed and the effects of d onthe performance are investigated. The range of d considered in thissection is from 0.01 m to 1 m.

Fig. 10 (a) plots Sh as a function of [RaT − (RaS/)] for variouscavity heights, where a dramatic increase in the intensity of DDCis clearly seen. As d increases from 0.01 m to 1 m, [RaT − (RaS/)]amplifies almost six orders of magnitude. As a result, Sh shows anincrease of nearly fiftyfold. However, as shown in Fig. 10 (b), theenhanced convection does not bring about a sharp growth in m,which contrarily decreases gently with d.

This counterintuitive phenomenon can be interpreted by Eq.(30). In the COO of state A, m can be expressed by

m =[−�T (T1 − T2) − �p�rgd

](1

100KM2+ 1 − �c

�rD

d

Sh

)−1

. (41)

The numerator decreases with d (since �p > 0), while d iscoupled with Sh in the denominator. According to Eq. (40),

d/Sh ∝ d = d is an increasing function. Therefore, mdecreases monotonically with d, as shown in Fig. 10 (b). Theabove arguments suggest that the exponent in Eq. (40) is cru-cial to the behavior of m versus d. If the exponent is larger than

Z.-C. Hu and X.-R. Zhang / J. of Supercritical Fluids 153 (2019) 104578 11

F ermaT ow rafi ene sy

1tc

rs[nttoipr

6

bvC

fmpatawnpsaCtwTrm

auFa

C

ig. 10. (a) Sherwood number Sh plotted against [RaT − (RaS/)], where RaT is the thhe arrow indicates the direction of increasing cavity height d. (b) Cross-boundary flgures, the physical properties are those of state A in Table 2 for the CO2–naphthal

/3, the denominator also decreases with d. Therefore, the rela-ionship becomes rather complicated and depends on specificircumstances.

In this section, the performance of the model operating ateference state A was evaluated based on a series of numericalimulations of DDC in COO. A power-law relation between Sh andRaT − (RaS/)] was identified for the chaotic motions. The expo-ent in this power law influences the dependence of m on d. Whenhe exponent in this power law is smaller than 1/3, as is currentlyhe case, m decreases with d. Otherwise, the relationship dependsn specific situations. The DDC achieves an overall tenfold increasen m. The performance of the conceptual model operating at theoint A was confirmed to be better than the previous experimentalesults of crystal growth rates reported in [23].

. Conclusions

In this paper, a conceptual model is developed to com-ine extraction and crystal growth in a supercritical solution byirtue of the critical behavior of solubility near the CEPs. TheO2–naphthalene system is employed as a reference system.

Theoretical optimizations are performed to maximize the per-ormance, indicated by the cross-boundary flow rate of the solute˙ . Thermodynamic optimization identifies that the newly definedseudo-CEPs are preferable reference states. The temperaturet the bottom wall should be hotter than that at the top wallo take advantage of the convection cooperatively driven by Tnd c. In each near-CEP region, the performance is promotedhen the corresponding CEP is approached. Moreover, hydrody-amic optimization reveals that pseudo-LCEPs are preferable toseudo-UCEPs due to their hydrodynamic excellence. Numericalimulations based on the most favorable reference state among allvailable states are carried out to evaluate the performance in theOO. A power-law relation between Sh and [RaT − (RaS/)] is iden-ified. If the exponent of the power law is less than 1/3, m decreasesith d. Otherwise, the relationship depends on specific situations.

he performance of the current model operating at the optimizedeference state is confirmed to be better than the previous experi-ental results in terms of crystal growth rates.The model presented in this study features a simple structure

nd excellent performance, which provides a promising config-ration for coupled extraction and crystal growth apparatuses.urthermore, the methods and procedures reported here will serves a reference when applying this model to other systems.

onflicts of interest

The authors declare no conflicts of interest.

l Rayleigh number, RaS is the solutal Rayleigh number and is the diffusivity ratio.te of solute m calculated from Eq. (30), plotted against d from 0.01 m to 1 m. In bothstem and T1 − T2 = 0.5 K is fixed.

Acknowledgments

This work was supported by the Natural Science Foundation ofChina [grant number 51776002]; and the High-performance Com-puting Platform of Peking University.

Appendix A. Modeling of the solubility

As proposed in [17], when T and p are known, the solubility isgiven by

ys = psub

�pexp

[VS(p − psub)

RT

], (A.1)

where ys is the mole fraction of the solute in saturated solution (thesolubility in mole fraction), psub is the sublimation pressure of thesolid, VS is the molar volume of the solid, R = 8.314 J/(mol K) is theideal gas constant and � is the fugacity coefficient of the solute. psubof naphthalene is calculated by two coupled equations [17,37,38],which are combined and simplified into

ln psub = 37.7612 − 10890.1411T

, (A.2)

where the unit of T is K, and the unit of the calculated psub is Pa.VS = 1.1 × 10−4 m3/mol is approximated as a constant. � is calcu-lated from the Peng–Robinson EOS written for a mixture:

p = RT

V − b− a

V2 + 2bV − b2, (A.3)

where V is the molar volume, and a and b are functions of the molefraction of naphthalene (denoted by y):

a = (a1 + a2 − 2a12)y2 + 2(a12 − a1)y + a1, (A.4)

b = (b1 + b2 − 2b12)y2 + 2(b12 − b1)y + b1, (A.5)

with

ai = 0.45724R2T2

ci

pci

[1 + (0.37464 + 1.54226ωi − 0.26992ω2

i)

(1 −

√T

Tc i

)]2

,

(A.6)

bi = 0.07780RTci

pci, (A.7)

a12 = √a1a2(1 − k12), (A.8)

b12 = b1 + b2

2(1 − l12), (A.9)

where i = 1 and 2 represent CO2 and naphthalene, respectively. Theproperties required to calculate ai and bi are listed in Table A.5. The

12 Z.-C. Hu and X.-R. Zhang / J. of Supercr

Table A.5Critical properties, acentric factors and molar masses of CO2 and naphthalene.

Name i Tc K pc (MPa) ω M (kg/mol)

tee

w

a

b

r

Ap

l

c

wwa

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

CO2 1 304.21 7.38 0.225 0.04401Naphthalene 2 748.40 4.051 0.302 0.1282

wo adjustable parameters k12 and l12 are determined by fittingxperiment data of solubility [17]. For the Peng–Robinson EOS, thexpression for � is [39]

ln � = bN

b

(pV

RT− 1

)− ln

[p(V − b)

RT

]+ a

2√

2bRT

(aN

a− bN

b

)

ln

[V +

(1 −

√2)

b

V +(

1 +√

2)

b

], (A.10)

here

N = 2y(a2 − a12) + 2a12, (A.11)

N = (y2 − 2y)(2b12 − b1 − b2) + 2b12 − b1. (A.12)

Since � depends on y, iterations among Eqs. (A.1)–(A.10) areequired to determine ys.

ppendix B. Equations for the specific heat at constantressure

The specific heat at constant pressure, denoted by cp, is calcu-ated by [40]

cp1 = (3.259 + 1.356 × 10−3T + 1.502 × 10−5T2

−2.374 × 10−8T3 + 1.056 × 10−11T4)R; (B.1)

cp2 = (2.889 + 14.306 × 10−3T + 15.978 × 10−5T2

−23.930 × 10−8T3 + 10.173 × 10−11T4)R; (B.2)

p =(1 − y)cp1 + ycp2 + dHR/dT)p,�

(1 − y)M1 + yM2, (B.3)

here the unit of T is K, and cp1 and cp2 are in J/(mol K). In the frame-ork of Peng–Robinson EOS, the residual enthalpy HR is expressed

s

HR

RT= pV

RT− 1 + Tda/dT − a

2√

2bRTln

[V + (1 +

√2)b

V + (1 −√

2)b

]. (B.4)

eferences

[1] Z. Knez, M.K. Hrncic, M. Skerget, Particle formation and product formulationusing supercritical fluids, Annu. Rev. Chem. Biomol. Eng. 6 (2015) 379–407,http://dx.doi.org/10.1146/annurev-chembioeng-061114-123317.

[2] Z. Knez, E. Markocic, M. Leitgeb, M. Primozic, M.K. Hrncic, M. Skerget,Industrial applications of supercritical fluids: a review, Energy 77 (2014)235–243, http://dx.doi.org/10.1016/j.energy.2014.07.044.

[3] A. Bertucco, Precipitation and Crystallization Techniques, John Wiley & Sons,Ltd., 1999, pp. 108–126 (Chapter 6).

[4] B.C.-Y. Lu, D. Zhang, Solid-supercritical fluid phase equilibria, Pure Appl.Chem. 61 (1989) 1065–1074, http://dx.doi.org/10.1351/pac198961061065.

[5] G. Xu, A.M. Scurto, M. Castier, J.F. Brennecke, M.A. Stadtherr, Reliablecomputation of high-pressure solid–fluid equilibrium, Ind. Eng. Chem. Res. 39(2000) 1624–1636, http://dx.doi.org/10.1021/ie990653s.

[6] M. Gitterman, I. Procaccia, Quantitative theory of solubility of supercritical

fluids, J. Chem. Phys. 78 (1983) 2648–2654, http://dx.doi.org/10.1063/1.445023.

[7] B.-Y. Lu, D. Zhang, W. Sheng, Solubility enhancement in supercritical solvents,Pure Appl. Chem. 62 (1990) 2277–2285, http://dx.doi.org/10.1351/pac199062122277.

[

itical Fluids 153 (2019) 104578

[8] Z.-C. Hu, S.H. Davis, X.-R. Zhang, Onset of double-diffusive convection innear-critical gas mixtures, Phys. Rev. E 99 (2019) 033112, http://dx.doi.org/10.1103/PhysRevE.99.033112.

[9] S. Paulucci, Filtering of sound from the Navier–Stokes equations, Technicalreport, Sandia National Labs, 1982.

10] G. Westphal, F. Rosenberger, On diffusive–advective interfacial mass transfer,J. Cryst. Growth 43 (1978) 687–693, http://dx.doi.org/10.1016/0022-0248(78)90147-1.

11] J.S. Turner, Double-diffusive convection, in: J.S. Turner (Ed.), Buoyancy Effectsin Fluids, Cambridge University Press, Cambridge, 1973, pp. 251–287.

12] Z.-C. Hu, X.-R. Zhang, Piston effect induced by cross-boundary mass diffusionin a binary fluid mixture near its liquid–vapor critical point, Int. J. Heat MassTransf. 140 (2019) 691–704, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2019.06.022.

13] Y. Wei, Z.-C. Hu, B. Chong, J. Xu, The Rayleigh–Bénard convection innear-critical fluids: influences of the specific heat ratio, Numer. Heat Transf. A:Appl. 74 (2018) 931–947, http://dx.doi.org/10.1080/10407782.2018.1505095.

14] M. McHugh, M.E. Paulaitis, Solid solubilities of naphthalene and biphenyl insupercritical carbon dioxide, J. Chem. Eng. Data 25 (1980) 326–329, http://dx.doi.org/10.1021/je60087a018.

15] W.J. Schmitt, R.C. Reid, Solubility of monofunctional organic solids inchemically diverse supercritical fluids, J. Chem. Eng. Data 31 (1986) 204–212,http://dx.doi.org/10.1021/je00044a021.

16] M. Sauceau, J. Fages, J.-J. Letourneau, D. Richon, A novel apparatus for accuratemeasurements of solid solubilities in supercritical phases, Ind. Eng. Chem.Res. 39 (2000) 4609–4614, http://dx.doi.org/10.1021/ie000181d.

17] M. Skerget, Z. Novak-Pintaric, Z. Knez, Z. Kravanja, Estimation of solidsolubilities in supercritical carbon dioxide: Peng–Robinson adjustable binaryparameters in the near critical region, Fluid Phase Equilib. 203 (2002)111–132, http://dx.doi.org/10.1016/S0378-3812(02)00177-2.

18] H.H. Lauer, D. McManigill, R.D. Board, Mobile-phase transport properties ofliquefied gases in near critical and supercritical fluid chromatography, Anal.Chem. 55 (1983) 1370–1375, http://dx.doi.org/10.1021/ac00259a041.

19] P.R. Sassiat, P. Mourier, M.H. Caude, R.H. Rosset, Measurement of diffusioncoefficients in supercritical carbon dioxide and correlation with the equationof Wilke and Chang, Anal. Chem. 59 (1987) 1164–1170, http://dx.doi.org/10.1021/ac00135a020.

20] D.M. Lamb, S.T. Adamy, K.W. Woo, J. Jonas, Transport and relaxation ofnaphthalene in supercritical fluids, J. Phys. Chem. 93 (1989) 5002–5005,http://dx.doi.org/10.1021/j100349a065.

21] C.Y. Kong, K. Sone, T. Sako, T. Funazukuri, S. Kagei, Solubility determination oforganometallic complexes in supercritical carbon dioxide bychromatographic impulse response method, Fluid Phase Equilib. 302 (2011)347–353, http://dx.doi.org/10.1016/j.fluid.2010.09.034.

22] C.Y. Tai, C. Cheng, Growth of naphthalene crystals from supercritical CO2

solution, AIChE J. 41 (1995) 2227–2236, http://dx.doi.org/10.1002/aic.690411006.

23] H. Uchida, A. Manaka, M. Matsuoka, H. Takiyama, Growth phenomena ofsingle crystals of naphthalene in supercritical carbon dioxide, Cryst. GrowthDes. 4 (2004) 937–942, http://dx.doi.org/10.1021/cg034212u.

24] C. Nicolas, E. Neau, S. Meradji, I. Raspo, The Sanchez–Lacombe lattice fluidmodel for the modeling of solids in supercritical fluids, Fluid Phase Equilib.232 (2005) 219–229, http://dx.doi.org/10.1016/j.fluid.2005.03.015.

25] MATLAB Symbolic Math Toolbox, Version 8.1, The MathWorks, Natick, MA,USA, 2018.

26] I. Raspo, S. Meradji, B. Zappoli, Heterogeneous reaction induced by the pistoneffect in supercritical binary mixtures, Chem. Eng. Sci. 62 (2007) 4182–4192,http://dx.doi.org/10.1016/j.ces.2007.04.027.

27] A. Laesecke, C.D. Muzny, Reference correlation for the viscosity of carbondioxide, J. Phys. Chem. Ref. Data 46 (2017) 013107, http://dx.doi.org/10.1063/1.4977429.

28] M.L. Huber, E.A. Sykioti, M.J. Assael, R.A. Perkins, Reference correlation of thethermal conductivity of carbon dioxide from the triple point to 1100 K and upto 200 MPa, J. Phys. Chem. Ref. Data 45 (2016) 013102, http://dx.doi.org/10.1063/1.4940892.

29] R.V. Vaz, A.L. Magalhaes, C.M. Silva, Prediction of binary diffusion coefficientsin supercritical CO2 with improved behavior near the critical point, J.Supercrit. Fluid 91 (2014) 24–36, http://dx.doi.org/10.1016/j.supflu.2014.03.011.

30] D.M. Lamb, T.M. Barbara, J. Jonas, NMR study of solid naphthalene solubilitiesin supercritical carbon dioxide near the upper critical end point, J. Phys.Chem. 90 (1986) 4210–4215, http://dx.doi.org/10.1021/j100408a074.

31] S. Amiroudine, P. Bontoux, P. Larroudé, B. Gilly, B. Zappoli, Direct numericalsimulation of instabilities in a two-dimensional near-critical fluid layerheated from below, J. Fluid Mech. 442 (2001) 119–140, http://dx.doi.org/10.1017/S0022112001004967.

32] J.M.H.L. Sengers, E. Kiran, J.M.H.L. Sengers, Critical behavior of fluids: conceptsand applications, in: Supercritical Fluids: Fundamentals for Application,Springer Netherlands, Dordrecht, 1994, pp. 3–38.

33] M. Darwish, I. Sraj, F. Moukalled, A coupled finite volume solver for thesolution of incompressible flows on unstructured grids, J. Comput. Phys. 228

(2009) 180–201, http://dx.doi.org/10.1016/j.jcp.2008.08.027.

34] Z.-C. Hu, X.-R. Zhang, An improved decoupling algorithm for low Machnumber near-critical fluids, Comput. Fluids 145 (2017) 8–20, http://dx.doi.org/10.1016/j.compfluid.2016.12.009.

upercr

[

[

[

[Boca Raton, 1997.

[39] M. McHugh, V. Krukonis, Supercritical Fluid Extraction: Principles and

Z.-C. Hu and X.-R. Zhang / J. of S

35] Z.-C. Hu, X.-R. Zhang, Onset of convection in a near-critical binary fluidmixture driven by concentration gradient, J. Fluid Mech. 848 (2018)1098–1126, http://dx.doi.org/10.1017/jfm.2018.397.

36] L.P. Kadanoff, Turbulent heat flow: structures and scaling, Phys. Today 54(2001) 34–39, http://dx.doi.org/10.1063/1.1404847.

37] S. Sako, K. Ohgaki, T. Katayama, Solubilities of naphthalene and indole insupercritical fluids, J. Supercrit. Fluid 1 (1988) 1–6, http://dx.doi.org/10.1016/0896-8446(88)90003-4.

[

itical Fluids 153 (2019) 104578 13

38] E. Baum, Chemical Property Estimation: Theory and Application, CRC Press,

Practice, second ed., Elsevier, 1994.40] B.E. Poling, J.M. Prausnitz, J.P. O’Connell, The Properties of Gases and Liquids,

fifth ed., McGraw-Hill, New York, 2001.