1. Introduction

The subject of natural convection in enclosures has been widely recognized and extensive research has been performed. Most available studies have focused on natural convection with thermal gradients in enclosures, which appears in buildings. Some applications have been extended to nuclear reactor systems, materials processing, natural ventilation in buildings, and solar energy. Recently, research has been extended to the transport in porous media [1,2], and to double-diffusive natural convection [3,4]. However, we shall refer to double-diffusive convection relevant to the study of physical vapor transport (PVT) in the ampoules of vapor crystal growth. In general, PVT is well known as a crystal growth process, which has many advantages over conventional crystal process like a melt-growth because it can be performed at low temperatures and provides relatively high interfacial morphological stability [5].

Duval and coworkers [6-18] have computationally attempted to understand the influence of convective parameters on natural convective flow in vapor crystal growth enclosures, e.g., physical vapor transport (PVT) system. They reported that unsteady flow is shown to be correlated with a degradation of crystal quality as quantified by light scattering pattern measurements. An unsteady flow becomes steady (diffusive-advection) in a microgravity environment of 10^-3 g as predicted by the diffusion-limited model. Duval [12] addressed that there are four regions which distinct flow field structure occur. During the transition from one region to another, three distinct bifurcation events occur. The flow field structure transits from a unidirectional advective-diffusion flow to two cells, subsequently to four cells, and finally six cells. Singh et al. [19, 20] have characterized mercurous bromide crystals grown by the physical vapor transport (PVT) method in closed ampoules within two-zone and multizone crystal growth furnaces. They have reported the materials exhibit an exciting material for acousto-optic tunable filters (AOTF) and Bragg cells suitable for optical signal processing. Mercurous bromide crystals showed a strong cleavage plane in the orientation, and its velocity of the acoustic wave was observed to be 273 m/s. More recently, Amarasinghe et al. [21] grew single crystals of mercurous halides by the physical vapor transport (PVT) method and characterized the orientation and the crystalline quality of the grown crystals using high resolution x-ray diffraction technique. The grown mercurous iodide (Hg₂I₂) crystals are 48 mm in diameter, 400-600 grams in weight and 70-80 mm in height. Mercurous halides are proved to be promising materials in applications for acousto-optic tunable filters (AOTF’s) for 8~12 µm wavelength region.

With regard to the physical vapor transport method, Shi et al. [22] prepared 6H-type silicon carbide whiskers and studied both the growth of SiC whiskers and the mechanism of SiC nucleation, and found that there exist two major factors: the sublimation of raw materials, and the transport of the vapor species. Fanton et al. [23] published the results that the addition of hydrogen related to the SiC crystals grown by the PVT techniques had nothing with the level of boron contamination in all samples. Paorici et al. [24] carried out experiments of PVT techniques for urotropine to prove one-dimensional diffusion model to get qualitative results.

In so far as crystal quality is affected by the convection field, it is important to understand double-diffusive convective flow structure in the vapor phase. In particular, much study of double-diffusive convection is needed as a basis on the ground for future space-flight experiments in South Korea. This motivation causes us to investigate effects of double-diffusive natural convection on PVT for a mixture of Hg₂Br₂ and argon on earth and under microgravity environments. A mixture of Hg₂Br₂ and argon is chosen as a systematic model study for further study into microgravity environments. In this study we investigate computationally the characteristics of the double-diffusive convective flow structures in the PVT processes of Hg₂Br₂ crystal growth.

2. System and Mathematical Formulation

We consider a steady state double diffusive natural convection in PVT crystal growth enclosure caused by the driving force of the vapor crystal growth, e.g., ΔT, the source and the crystal regions, T_s-T_c along with conducting linear temperature profile at wall boundary conditions, as shown schematically in Fig. 1. The mass fluxes of sublimation-condensation at the interfaces, i.e., the finite normal velocities at the interfaces, are expressed by Stefan flow deduced from the one-dimensional diffusion-limited model [25]. The gravitational force vector is aligned in the negative y-direction. The thermophysical properties of the fluid in the vapor phase are assumed to be constant. With regard to the density dependency of the buoyancy term in the momentum equation, the density is calculated through the ideal gas equation of state and is dependent on temperature and pressure. The u_x and u_y denote the velocity components along the x- and y-coordinates in the Cartesian coordinate system (x, y), and T, ω_A, p denote the temperature, mass fraction of species A (Hg₂Br₂) and pressure, respectively, where the superscript of * denotes the dimensionless variable [11,12,13].

Fig. 1. Schematic and coordinates for modeling and simulation of PVT crystal growth reactor of Hg₂Br₂(A) and argon (B).

The dimensionless variables are scaled as follows:

\(x^{*}=\frac{x}{H}, y^{*}=\frac{y}{H}\),             (1)

\(\mathrm{u}=\frac{\mathrm{u}_{x}}{\mathrm{U}_{c}}, \mathrm{v}=\frac{\mathrm{u}_{y}}{\mathrm{U}_{c}}, \mathrm{p}=\frac{\mathrm{p}}{\rho_{c} \mathrm{U}_{c}^{2}}\),             (2)

\(\mathrm{T}^{*}=\frac{\mathrm{T}-\mathrm{T}_{c}}{\mathrm{T}_{s}-\mathrm{T}_{c}}, \omega_{A}^{*}=\frac{\omega_{A}-\omega_{A, c}}{\omega_{A, s}-\omega_{A, c}}\),             (3)

The non-dimensional governing equations in the Cartesian coordinate system (x, y) are given by:

\(\nabla^{*} \bullet \mathrm{V}=0\),             (4)

\(\overrightarrow{\mathrm{V}} \cdot \nabla^{*} \overrightarrow{\mathrm{V}}=-\nabla^{*} \mathrm{p}^{\prime}+\operatorname{Pr} \nabla^{* 2} \overrightarrow{\mathrm{V}}-\frac{\mathrm{Gr} \cdot \mathrm{Pr}^{2}}{\mathrm{AR}^{3}} \cdot \frac{\left(1-\rho^{*}\right)}{\beta \Delta \mathrm{T}}\),             (5)

\(\overrightarrow{\mathrm{V}} \cdot \nabla^{*} \mathrm{T}^{*}=\nabla^{* 2} \mathrm{T}^{*}\),             (6)

\(\overrightarrow{\mathrm{V}} \bullet \nabla^{*} \omega_{A}^{*}=\frac{1}{\mathrm{Le}} \nabla^{\prime 2} \omega_{A}^{*}\).             (7)

The non-dimensional boundary conditions are

On the walls (0 < x* < L/H, y* = 0 and 1):

\(u\left(x^{*}, 0\right)=u\left(x^{*}, 1\right)=v\left(x^{*} 0\right)=v\left(x^{*}, 1\right)=0\)

\(\frac{\partial \omega_{A}^{*}\left(\mathrm{x}^{*}, 0\right)}{\partial \mathrm{y}^{*}}=\frac{\partial \omega_{A}^{*}\left(\mathrm{x}^{*}, 1\right)}{\partial \mathrm{y}^{*}}=0\),

\(\mathrm{T}^{*}\left(\mathrm{x}^{*}, 0\right)=-\frac{1}{\mathrm{AR}} \cdot \mathrm{x}^{*}+1\)             (8)

On the source (x* = 0, 0 < y* < 1):

\(u\left(0, y^{*}\right)=-\frac{1}{L e} \frac{1}{(C v-1)} \frac{\partial \omega_{A}^{*}\left(0, y^{*}\right)}{\partial y^{*}}\),

\(v\left(0, y^{*}\right)=0\),

\(\mathrm{T}^{*}\left(0, \mathrm{y}^{*}\right)=1\),

\(\omega_{A}\left(0, y^{*}\right)=1\).             (9)

On the crystal (x* = L/H, 0 < y* < 1):

\(u\left(L / H, y^{*}\right)=-\frac{1}{L e} \frac{1}{(C v)} \frac{\partial \omega_{A}^{*}\left(L / H, y^{*}\right)}{\partial x^{*}}\)

\(v\left(L / H, y^{*}\right)=0\),

\(\mathrm{T}^{*}\left(\mathrm{L} / \mathrm{H}, \mathrm{y}^{*}\right)=0\),

\(\omega_{A}^{*}\left(\mathrm{L} / \mathrm{H}, \mathrm{y}^{*}\right)=0\).             (10)

where

\(\text { aspect ratio, } \mathrm{AR}=\mathrm{L} / \mathrm{H}\)             (11a)

\(\text { concentration number, } \mathrm{C}_{v}=\left(1-\omega_{A, c}\right) / \Delta \omega\)             (11b)

\(Lewis number, Le =\kappa / D_{A B}\)             (11c)

\(Prandtl number, \mathrm{Pr}=v / \mathrm{k}\)             (11d)

\(\text { thermal Grashof number, Gr }=\mathrm{g} \beta \Delta \mathrm{TH}^{3} / \mathrm{v}^{2}\)             (11e)

The Peclet number, Pe is defined as

\(\mathrm{Pe}=\frac{\mathrm{U}_{a d v} \mathrm{L}}{\mathrm{D}_{A B}}\)             (11f)

which represents the ratio of advective to diffusive flux. The U_adv is the streaming velocity based on the sublimation and condensation advective-diffusive flux. When the advective-diffusive flux is dominant, the flow field is stabilized [10]. Pe may be rewritten from one-dimensional model in the form [26],

\(\mathrm{Pe}=\ln \frac{\mathrm{O}_{B}(\mathrm{L})}{\omega_{B}(0)}=\ln \frac{\mathrm{p}_{B}(\mathrm{L})}{\mathrm{p}_{B}(0)}\)             (11f)

In the dimensionless parameters, which appear in the governing equations, the physicochemical properties of the gas mixture are obtained from a gas kinetic theory using Chapman-Enskog’s formulas [27], due to lack of available data: the viscosity of and thermal conductivity of argon, the mixture’s viscosity and thermal conductivity, the mass diffusivity for binary mixtures. The collision integral functions for estimations of viscosities of Hg₂Br and argon, diffusivity for binary mixture of Hg₂Br and argon can be found in reference [28]. The method of Wilke is used for estimation of the viscosity of Hg₂Br and argon [28]. Because the total vapor pressure is assumed to be constant during the physical vapor transport of Hg₂Br in the vapor phase, the mass fraction of species A (Hg₂Br) is defined as

\(\omega_{A}=\frac{1}{1+\left(\mathrm{P}_{T} / \mathrm{p}_{A}-1\right) \mathrm{M}_{B} / \mathrm{M}_{A}}\).             (12)

The vapor pressure [29] p_A of Hg₂Br₂ (in the unit of Pascal) can be evaluated from the following formula as a function of temperature:

\(\mathrm{p}_{A}=\mathrm{e}^{(a-b / T)}\),             (13)

in which a = 9.956, b = 4563. P_T denotes the total pressure, and the partial pressures for A(B) are denoted by p_A (p_B). Invoking of the Boussinesq approximation of the density [11] yields.

\(\rho=\bar{\rho}(1-\beta \Delta \mathrm{T}+\tau \Delta \omega)\),

where β is the coefficient of thermal volume expansion, and γ is the coefficient of solutal volume expansion,

\(\beta=-\frac{1}{\rho}\left(\frac{\partial \rho}{\partial \mathrm{T}}\right)_{\omega_{A}}, \tau=\frac{1}{\rho}\left(\frac{\partial \rho}{\partial \omega}\right)_{T}\).

The ratio of thermal and solutal buoyancy forces [30] is defined as

\(\mathrm{N}=\frac{-\tau \Delta \omega}{\beta \Delta \mathrm{T}}\).             (14)

To solve the discretization equations for the system of nonlinear, coupled governing partial differential equations, the semi-implicit method pressure-linked equations revised (SIMPLER) algorithm proposed by Patankar [31] with a power law scheme was used. A 43 × 23 (x × y) and a 63 43 × 23 (x × y) grid system were used for AR = 1 and AR ≥ 2, respectively. The iteration procedure was continued until the following convergence criterion was satisfied:

\(\left|\frac{\phi_{j+1}-\phi_{j}}{\phi_{j}}\right|<10^{-5}, \text {for all } \phi\),

where φ represents any dimensionless dependent variable being computed, e.g., u,v,T* and ω*_A, and j refers to the value of φ at the jth iteration level. The numerical verifications of our results have been done in references [13,16,17].

3. Results and Discussion

For the system at hand, we are interested in double-diffusive convection physical vapor transport with the presence of impurity, argon. When the M_A≠M_B is modeled, i.e., the molecular weight of component A is not equal to that of component B, the solutal convection is important, in particular when the imposed temperature profile has little effect on the total molar flux. The double-diffuisve convection flow structure can be quantified by the seven dimensionless governing parameters of aspect ratio (AR, L/H), concentration number (Cv, (1-ω_A,c)/Δω), Lewis number (Le, κ/D_AB), Peclet number (P_e, U_advL/ D_AB), Prandtl number (Pr, n/k), thermal Grashof number (Gr_t, gβΔTH³ / ν²), and solutal Grashof number (Gr_s, gγΔωH³/ν²) [11].

The excess pressure of the impurity component affects the binary diffusion coefficient, which determines the advective-diffusion flux at the interfaces. However, we restrict our study to investigating how a double-diffusive convection flow will affect the molar flux and its distribution across an interface in the crystal region. Thus, the effects of the parameters of excess pressure of the impurity, argon and the molecular weight, total pressure on the double diffusion natural convection are not examined in this study. Throughout the computational works in this study, the source temperature, T_s = 290 ^oC and width H = 2 cm are fixed, and the characteristic length H is introduced in the calculations of thermal and solutal Grashof numbers. Numerical results are presented in the form of velocity vectors, streamlines, and iso-mass fraction contours, and

Fig. 2 shows the effects of aspect ratio, AR (L/H) on the total molar flux of Hg₂Br₂ in terms of moles cm^-2s^-1 for various aspect ratio, 1 ≤ AR ≤ 10, based on ΔT = 30 ^oC (290 ^oC→260 ^oC), and 1 g, one gravity acceleration (where 1 g = 980.665 cm s^-2), P_B = 10 Torr, Pr = 0.99, Le = 0.15, C_v = 1.31, Pe = 1.4, Gr_t = 2.26 × 10³ , Gr_s = 3.87 × 10⁴ . Note that the governing dimensionless parameters of Prandtl, Lewis, concentration, Peclet, and Grashof numbers are obtained for the mixture of Hg₂Br₂ and argon, in Fig. 2 through Fig. 11. As plotted in Fig. 2, the total molar fluxes decay first-order exponentially with the aspect ratio, 1 ≤ AR ≤ 10. For the range of 1 ≤ AR ≤ 2, the molar fluxes drop significantly, and, for 2 ≤ AR ≤ 10, decrease slowly. So, the total molar fluxes decrease very sharply near AR = 1, and, then since AR = 2, decrease. With increasing the aspect ratio, the wall effects result in a decrease in the molar flux, which is consistent with the previous results [12,32]. It is found that the total mass flux occurs at the square cavity, and as we approach the horizontal narrow cavity AR = 10, the total mass flux decreases, which is reflected in the double diffusion convective flow fields, as addressed later, in Figs. 3 through 5. Therefore, the double-diffusive convection can be operated by altering the aspect ratio of the growth ampoule.

Fig. 2. Effects of aspect ratio (AR, L/H) on the total molar flux (moles cm^-2s^-1) of Hg₂Br₂, based on ΔT=30 ℃ (290 ℃→260 ℃), 1 g, P_B = 10 Torr, Pr = 0.99, Le = 0.23, C_v = 1.07, Pe = 2.64, Gr_t= 2.8 × 10³, Gr_s = 4.4 × 10⁴.

Figs. 3 through 5 show that (a) velocity vector, (b) streamline, (c) temperature, and (d) mass fraction profile for three cases of AR = 1, 2, and 8, ΔT = 30 ^oC (290 ^oC→260 ^oC), and 1 g. As depicted in Figs. 3 and 4, the single double diffusive convection unicellular flow structure occurs in the vapor phase and the convection flows toward the crystal interface in the lower half of the enclosure, which results in providing this part of the crystal interface with vapor supersaturated in component of Hg₂Br₂. The counter-clockwise single cell indicates the major role of the double-diffusive convection, because thermal convection alone, with the left part of the enclosure warmer than the right part, would result in an oppositely rotating cell, and thus, in augmented total molar flux in the upper half of the crystal interface [33]. The unicellular convective flow is asymmetrical along both the x-direction and the y-direction, which shows the dominating convection flow field. The flow field looks like the advective-diffusion mode along the bottom side wall because of the side wall effects damping the convection. Figs. 3(c) and 4(c) illustrate changes in which isotherms are positioned in the main part of the enclosure and convection near the crystal interface is transited into conduction. Moreover, it is observed from iso-mass fractions in 3(d) and 4(d) that the diffusion-limited mechanism is a critical factor in the crystal growth in the crystal interface because the intervals between isomass fractions near the crystal interface is tightly narrow so that the convection is switched over diffusion.

Fig. 3. (a) velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on ΔT = 30 ℃ (290 ℃→260 ℃), 1 g, P_B = 10 Torr, Pr = 0.99, Le = 0.23, C_v = 1.07, Pe = 2.64, Gr_t = 2.8 × 10³, Gr_s = 4.4 × 10⁴. The dimensional maximum magnitude of velocity vector is 18.03 cm s^-1.

Fig. 4. (a) velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on ΔT = 30 ℃ (290 ℃→260 ℃), 1 g, P_B = 10 Torr, Pr = 0.99, Le = 0.23, C_v = 1.07, Pe = 2.64, Gr_t = 2.8 × 10³, Gr_s = 4.4 × 10⁴. The dimensional maximum magnitude of velocity vector is 5.28 cm s^-1.

Fig. 5. (a) velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on ΔT=30 ℃ (290 ℃→260 ℃), 1 g, P_B = 10 Torr, Pr = 0.99, Le = 0.23, C_v = 1.07, Pe = 2.64, Gr_t = 2.8 × 10³, Gr_s = 4.4 × 10⁴. The dimensional maximum magnitude of velocity vector is 0.63 cm s^-1.

With increasing the aspect ratio of the horizontal enclosure from AR = 1 and 2 to AR = 8, as illustrated in Figs. 3 through 5, the convection field transits from the convection to the advective-diffusion mode and, the velocity and streamline fields become stable and exhibit the advective-dffusion dominated flow. It is due to fact that the wall effects enhance the viscous force which would alleviate the effects of the convection. The velocity vector, Fig. 5(a) and streamline profile, Fig. 5(b), show the parabolic advective-diffusion profile. The isotherms in Fig. 5(c) and the iso-mass fraction in Fig. 5(d) are well equally placed, supporting the advective-diffusion model.

Fig. 6 shows the relationship between total molar flux (moles cm^-2s^-1) of Hg₂Br₂ and the temperature difference between the source and the crystal regions, 10 ^oC ≤ ΔT ≤ 50 ^oC, based on AR = 1, T_s = 290 ^oC fixed, 1 g, P_B = 10 Torr, 1.8 × 10³ ≤ Grt≤ 2.9 × 10³, 2.6 × 10⁴ ≤ Gr_s ≤ 4.9 × 10⁴ . As shown in Fig. 6, the total molar flux of Hg₂Br₂ for 10 ^oC ≤ ΔT ≤ 50 ^oC, has a direct and linearly relationship with the temperature difference, ΔT. For 10 ^oC ≤ ΔT ≤ 30 ^oC, the total molar flux is significantly dependent on the temperature difference, while for 30 ^oC ≤ ΔT ≤ 50 ^oC, it varies slightly with the temperature difference. The temperature gradient for 10 ^oC ≤ ΔT ≤ 30 ^oC is, 8.5 × 10^-6 moles cm^-2s^-1K^-1 and for 10 ^oC ≤ ΔT ≤ 50 ^oC, 1.7 × 10^-6 moles cm^-2s^-1 K^-1. Therefore, the temperature gradient for 10 ^oC ≤ ΔT ≤ 30 ^oC is greater than for 30 ^oC ≤ ΔT ≤ 50 ^oC by a factor of 5. With increasing the driving force of PVT crystal growth, i.e., DT, the temperature difference between the source and the crystal, an increase of total molar fluxes is strongly reflected for the low temperature differences, 10 ^oC ≤ ΔT ≤ 30 ^oC, because of thermally buoyancy-driven convection. On the other hand, the effects of thermally buoyancy-driven convection are weakened for 30 ^oC ≤ ΔT ≤ 50 ^oC. As observed clearly in Figs. 8(a), 8(b), and 8(c), there are little changes in streamlines and isotherms and iso-mass fractions. In Fig. 6, the relation between the temperature difference and Peclet number is found to be direct and linear. Note that the Peclet number is related to thermodynamic variables [26].

Fig. 6. Relationship of total molar flux of Hg₂Br₂ in terms of moles cm^-2s^-1 and the temperature difference, ΔT (℃), Peclet number, Pe, based on AR = 1, T_s = 290 ℃ fixed, 1 g, P_B = 10 Torr, 1.8 × 10³ ≤ Gr_t ≤ 2.9 × 10³, 2.6 × 10⁴ ≤ Gr_s ≤ 4.9 × 10⁴.

Fig. 7 shows the relationship between total molar flux (moles cm^-2s^-1) of Hg₂Br₂ and the Peclet number, corresponding to Fig.6. Also, there is a direct and linear relationship between the total molar flux of Hg₂Br₂ and the Peclet number. Peclet number is used instead of the thermal Grashof number, because with increasing the temperature difference between the source and the crystal, at the source temperature of 290 ^oC fixed, the inverse value of the square corresponding kinematic viscosity, 1/ν² increases so that the thermal Grashof number is not considered as a quantity parameter in this study. The study of thermal properties on thermal convection would remain for further research in the future. The gradient of total mass flux to Peclet number is 1.02 moles cm^-2s^-1. Also, for considering the buoyancy parameter which is defined in Equation (14), N = -17 at ΔT = 10 ^oC, and the opposing buoyancy-induced boundary flows are expected.

Fig. 7. Total molar flux of Hg₂Br₂ as a function of the dimensionless Peclet number, Pe, corresponding to Fig. 6, based on AR = 1, T_s = 290 ℃ fixed, 1 g, P_B = 10 Torr, 1.8 × 10³ ≤ Gr_t ≤ 2.9 × 10³, 2.6 × 10⁴ ≤ Gr_s ≤ 4.9 × 10⁴.

Figs. 8(a), 8(b) and 8(c) show iso-streamlines, and isotherms, and iso-mass fractions, for four different temperature differences, AR = 1, 1 g, and PB = 10 Torr. As shown in Fig. 8(a), there occurs a single unicellular convective roll for the process conditions, and the convective roll exhibits asymmetrical against the center lines of x* = 0.5, and y* = 0.5, and the cell is in counter-clockwise, refer to Fig. 3(a). As mentioned, there is little variation in the structure and streamline value, temperature, and mass fraction for two cases of (c) ΔT = 40 ^oC and (d) ΔT = 50 ^oC, through the Figs. 8(a) through (c).

Fig. 8. (A) Streamline profile: (a) ΔT = 10 ℃, (b) ΔT = 20 ℃, (c) ΔT = 40 ℃, (d) ΔT = 50 ℃, based on AR = 1, 1 g and PB = 10 Torr. (B) Temperature profile: (a) ΔT = 10 ℃, (b) ΔT = 20 ℃, (c) ΔT = 40 ℃, (d) ΔT = 50 ℃, based on AR = 1, 1 g and PB = 10 Torr.

Fig. 8. (continued) (C) Mass fraction profile: (a) ΔT = 10 ℃, (b) ΔT = 20 ℃, (c) ΔT = 40 ℃, (d) ΔT = 50 ℃, based on AR = 1, 1 g and PB = 10 Torr.

Fig. 9 shows the total molar flux of Hg₂Br₂ in terms of moles cm^-2s^-1 as a function of gravity acceleration, 10^-5 g ≤ g_y ≤ 1 g, based on AR = 1, ΔT = 50 ^oC (290 ^oC→240 ^oC), and PB = 10. The double diffusive convection is dominated over the advection-diffusion for 10^-2 g ≤ g_y ≤ 1 g. The convection mode shifts into diffusion down to g_y = 10^-2 g and, from g_y = 10^-2g, down to g_y = 10^-5g, the diffusion becomes predominant. As seen in Fig. 9, the total molar fluxes drop sharply for 10^-2g ≤ g_y ≤ 1 g. This indicates the mass transport is diffusion-dominated under the microgravity environments less than 10^-2g. This tendency is approximately consistent with previous results [34,35]. Duval et al. [15] presented that an unsteady flow becomes diffusiveadvective flow under microgravity environments of gravity levels less than 10^-3g, and Nadarajah et al. [33] suggested 10^-1g an adequate level for diffusive transport. We can see that the effect of double diffusive convection is first important and then decreases rapidly and eventually the mode of transport becomes purely diffusion. Since the gravitational parameter is very important, a thorough understanding of double diffusive convection would be required for future spaceflight experiments in South Korea. Fig. 10 shows (a) velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on AR = 1, ΔT = 50 ^oC (290 ^oC→240 ^oC), 10^-3g and PB = 10. The dimensional maximum magnitude of velocity vector of 7.63 cm s^-1 is obtained. The velocity vector and streamline profiles in Figs. 10(a) and (b) exhibit a typical advective-diffusion pattern. Energy is transported through conduction near the crystal interface because the tight fine intervals of isotherms appear in the neighborhood of the crystal region, while the vapor of Hg₂Br₂ is transported through the diffusion mechanism in the neighborhood of the crystal interface due to the tight fine intervals of iso-mass fractions. Not shown here, there are little changes in streamlines, isotherms and iso-mass fraction contours for 10^-2g and 10^-3g cases.

Fig. 9. Total molar flux of Hg₂Br₂ in terms of moles cm^-2s^-1 as a function of gravity acceleration, 10^-5 g ≤ g_y ≤ 1 g, based on AR = 1, ΔT=50 ℃ (290 ℃→240 ℃), and P_B = 10, Pr = 0.97, Le = 0.37, C_v = 1.03, Pe = 3.41.

Fig. 10. (a) velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on AR = 1, ΔT=50℃ (290 ℃→240 ℃), 10^-3 g, P_B = 10, Pr = 0.97, Le = 0.37, C_v = 1.03, Pe = 3.41. The dimensional maximum magnitude of velocity vector is 7.63 cm s^-1.

To examine the effect of gravity accelerations on the interfacial distributions of total molar flux (moles cm^-2s^-1) of Hg₂Br₂, interfacial distributions of molar flux (moles cm^-2s^-1) of Hg₂Br₂ for four gravity accelerations (1 g, 10^-1g, 10^-2g, and 10^-3g, where 1 g = 980.665 cm s^-2), based on AR = 1, ΔT = 50 ^oC (290 ^oC→240 ^oC), and PB = 10 are presented in Fig. 11. We can see from Fig. 11 that double diffusive convection enhances the overall mass transport of component A, Hg₂Br₂ and consequently also the total mass flux, but at the expense of resulting crystal uniformity in the total mass flux. The effects of double diffusive convection are reflected through the density gradient and binary diffusivity coefficient. The total molar flux along the interfacial position at the crystal for 1 g and 10^-1g has asymmetrical profiles at the center position y = 1.0 cm, which indicates three-dimensional convective flow and the presence of single unicellular roll. For 10^-2g, and 10^-3g, the total mass flux profiles look to be symmetrical at the center position y = 1.0 cm, which implies the advective-diffusion convection in the enclosure is predominant over the double diffusion convection. The convection is found to be directly and intimately related to the non-uniformity of crystal quality desired.

Fig. 11. Interfacial distributions of total molar flux (moles cm^-2s^-1) of Hg₂Br₂ for four gravity accelerations (1 g, 10^-1 g, 10^-2 g, and 10^-3 g, where 1 g = 980.665 cm s^-2), based on AR = 1, ΔT = 50 ℃ (290 ℃→240 ℃), P_B = 10, Pr = 0.97, Le = 0.37, C_v = 1.03, Pe = 3.41.

 

4. Conclusions

It is concluded that the total molar fluxes decay first-order exponentially with the aspect ratio, 1 ≤ AR ≤ 10. With increasing the aspect ratio of the horizontal enclosure from AR = 1 up to AR = 10, the convection field changes from the convection to the advective-diffusion mode and the flow structures become stable. Microgravity environments less than 10^-2g make the effect of double-diffusive convection much reduced, so that adequate advective-diffusion transport can be obtained. Double diffusive natural convection flow structures are single unicellular roll and three-dimensional flow pattern. The total molar flux of Hg₂Br₂ for 10 ^oC ≤ ΔT ≤ 50 ^oC, has a direct and linear relationship with the temperature difference, ΔT. Boundary flows are expected because of the opposing buoyancy effects (the buoyancy parameter N < 0) for 10 ^oC ≤ ΔT ≤ 50 ^oC.

Acknowledgment

This work was financially supported by 2018 Hannam University Research Fund (Project Number: 2018A009).

Nomenclature

A : component A, Hg₂Br₂

B : component B, argon

D_AB : diffusivity of A and B

g : standard gravitational acceleration constant, 980.665 cms^-2

H : height (cm)

L : transport length, width (cm)

M : molecular weight of component A, B

P : pressure

P_T : total pressure

T : temperature

ΔT : temperature difference between source and crystal, T_s-T_c

Δω : mass fraction difference between source and crystal, ω_A,s-ω_A,c

x : x-coordinate

y : y-coordinate

u : dimensionless x-component velocity

U_adv : characteristic velocity based on the diffusive-advective flux

U_c : characteristic velocity, k/H

|U|_max : dimensional maximum magnitude of velocity vector

v : dimensionless y-component velocity

u_x : x-component velocity

u_y : y-component velocity

V : velocity vector

Subscripts and Superscripts

A : component A, Hg₂Br₂

Adv : advection

B : component B, Ar

c : crystal

s : source

T : total vapor pressure

* : dimensionless

Greek Letters

β : coefficient of thermal volume expansion

γ : coefficient of solutal volume expansion

κ : thermal diffusivity

ν : kinematic viscosity

∇* : (∂/∂x* )+(∂/∂y* )

∇*² : (∂² /∂x*²)+(∂² /∂y*²)

φ : variable (u, v, T*, ω*_A)

ω : mass fraction meaning dimensionless mass concentration

ρ : density of fluid with component A and B