Study on simultaneous heat and mass transfer during the physical vapor transport of Hg₂Br₂ under ㎍ conditions

Abstract

A computational analysis has been carried out to get a thorough and full understanding on the effects of convective process parameters on double-diffusive convection during the growth of mercurous bromide ($Hg_2Br_2$) crystals on earth and under ${\mu}g$ conditions. The dimensional maximum magnitude of velocity vector, ${\mid}U{\mid}_{max}$ decreases much drasticlly near Ar = 1, and, then since Ar = 2, decreases. The ${\mu}g$ conditions less than $10^{-2}g$ make the effect of double-diffusion convection much reduced so that adequate advective-diffusion mass transfer could be obtained.

1. Introduction

During the past 50 years one area of interest in natural convection research has been the study in confined geometries-enclosures. Examples of the enclosures may be parallelograms, three-dimensional enclosures, spheres or concentric annuli. In particular, one of the important topics in material processing is the growth of crystals in a sealed chamber as discussed by Osrtarch [1]. Crystal species was transported inside a closed tube from a source solid material at high temperature region to the growing crystal at low temperature region. The driving potential for the transport of crystal species was a temperature difference from the source interface at the high temperature region to the crystal interface at the low temperature region, which the equilibrium composition of the gas mixture is moved between the two ends and, thus, generated a concentrated gradient. In gravitational fields the simultaneous occurrence of temperature and concentration gradients would lead to double diffusion convection [2]. Kim and his coworkers [3-29] have systematically performed numerical studies of natural convection in the vapor phase during physical vapor crystal growth. Duval [30] reported that there exist four flow structure regions. During the change from one region to another region, three distinct bifurcation phenomena occur. The flow field structure shifts from a unidirectional advective-diffusion flow to two convection rolls, subsequently to four convection rolls, and finally six convection rolls. Singh et al. [31, 32] and Amarasinghe et al. [33] have studied the mercurous halide materials have proved to be most promising materials in applications for acoustooptic materials and signal processing optics such as Bragg cells.

In order to control the final quality of crystal affected by convection fields, it is required to study double diffusion convection flow structure in the vapor phase. In the author’s previous results [34], the relations of aspect ratio and total molar flux was addressed and as a sequential study, in this paper the maximum magnitudes of velocity vector in the dimensional unit (cm/s) as a basis of convection intensity would be discussed with (1) aspect ratios, (2) Peclet numbers, (3) the driving potentials of the temperature differences between two ends in closed ampoules. A mixture of Hg₂Br₂ and argon is chosen as a systematic model both on earth and under μg environments, and a gravity vector of 10. Therefore, as a sequential study, in this paper one investigates numerically the characteristics of the double diffusion convection flow fields during the PVT processes of Hg₂Br₂ crystal growth.

2. Modeling

A steady state double diffusion convection in PVT crystal growth enclosure is considered with linear temperature profiles at sidewall boundary conditions, as plotted in Fig. 1. The detailed assumptions can be found in reference [34]. ux, uy symbolize the x- and y-component velocity along the x- and y-coordinates in the rectangular coordinate system (x, y), respectively and T, ωA, p stand for the temperature, mass fraction of species Hg₂Br₂ and pressure, respectively [34].

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

The superscript of * represents non-dimension state. The dependent variables appearing the governing equations are non-dimensionalized using the following groupings,

By applying the non-dimensional variables, the governing equations can be expressed in non-dimensional form:

The boundary conditions to the above equations (4) to (7) are given as follows:

The numerical investigation utilized the Semi-Implicit Method Pressure-Linked Equations Revised (SIMPLER) [35] iterative technique for the system of nonlinear, coupled governing partial differential equations. A 63 x 63 (x x y) and a 43 x 23 (x x y) grid system were used for Ar = 1 and Ar ≥ 2, respectively. The numerical verifications of one’s results can be found in references [3, 5, 6].

3. Results and Discussion

Double diffusion convection during the physical vapor transport is computationally investigated. When M_A ≠ M_B, the two molecular weights of Hg₂Br₂ and argon are different, solutally buoyancy driven convection is important compared with thermally buoyancy driven convection. This case arises when the temperature profile imposed on walls has little effect on the crystal growth rate. As pointed by Duval [30], the double diffusion convection flow fields can be characterized by 7 dimensionless numbers, i.e., solutal Grashof number (Gr_s), thermal Grashof number (Gr_t), aspect ratio (Ar), Prandtl number (Pr), Lewis number (Le), concentration number (C_v), and Peclet number (Pe). One’s interest is restricted on his study to investigate the relations of maximum magnitudes of velocity vector and double-diffusion convection parameters of aspect ratio, Peclet number and the driving force for the transport of crystal species.

Table 1 Typical thermo-physical properties (M_A = 560.988, M_B = 39.944)

Fig. 2 illustrates the effects of aspect ratio, Ar (L/H) on the dimensional maximum magnitude of velocity vector, |U|_max, for various aspect ratio, 1 ≤ Ar ≤ 10, based on ΔT = 30ºC (290ºC→260ºC), and 1 g0, one gravity vector, 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. As shown in Fig. 2, |U|_max diminishes first order exponentially with the aspect ratio, 1 ≤ Ar ≤ 10. This trend is an agreement with the previous results [34]. For the range of 1 ≤ Ar ≤ 2, the |U|_max diminishes rapidly, and, for 2 ≤ Ar ≤ 10, dwindles down slowly. In other words, the |U|_max decreases much sharply near Ar = 1, and, then since Ar = 2, decreases. With an increase in the aspect ratio, the side wall effects result in a decrease in the |U|_max, which is consistent with the previous results [30, 36]. In other words, the side wall effects enhance the viscous force which would alleviate the effects of the convection. As the horizontal narrow cavity approaches to Ar = 10, the |U|_max decreases due to the side wall effects. Therefore, the aspect ratios of the growth ampoule is proved one of important double-diffusive convection parameters. The |U|_max diagram versus the Peclet number is shown in Fig. 3. It is obvious that the relationship of the |U|_max versus the Peclet number is directly linear. The Peclet number is associated with the advections across the interfaces at the source and the crystal regions.

Fig. 2. The |U|_max as a function of the aspect ratio, Ar(L/H).

Fig. 3. The |U|_max as a function of the dimensionless Peclet number, Pe.

Fig. 4 shows the relation of ΔT and |U|_max is direct and linear for 10ºC ≤ ΔT ≤ 50ºC, where the temperature of the source region is fixed at 290ºC, with Ar =1, 1 g₀, P_B = 10 Torr, 1.8 × 10³ ≤ Gr_t ≤ 2.9 × 10³, 2.6 × 10⁴ ≤ Gr_s ≤ 4.9 × 10⁴. This relation illustrates that the temperature difference, as the driving force of physical vapor transport causes always thermally buoyancy driven convection through the density gradient associated with the gravity vector. Fig. 5 shows velocity vector, streamline, temperature, mass fraction profiles, based on Ar = 1, ΔT=20ºC (290ºC→270ºC), 1 g₀ and P_B = 10 Torr, |U|_max = 16.44 cm s⁻¹. As plotted in Fig. 5, there exists single convective roll in the vapor phase, and the flow pattern is asymmetrical against at x* = 0.5 and three-dimensional flow structure. Convection roll is positioned to the right crystal region wall and the upper wall. As shown in Fig. 5(d), the intervals of mass fraction exhibit close spacings, which indicates the diffusion-limited mass transfer.

Fig. 4. The |U|_max as a function of the temperature difference, ΔT (°C), Peclet number, Pe, based on Ar = 1, T_s = 290°C fixed, 1 g₀, P_B = 10 Torr, 1.8 × 10³ ≤ Gr_t ≤ 2.9 × 10³, 2.6 × 10⁴ ≤ Gr_s ≤ 4.9 × 10⁴.

Fig. 5. (a) Velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on Ar = 1, ΔT = 20°C (290°C→270°C), 1 g₀, P_B = 10 Torr, Pr = 0.99, Le = 0.18, C_v = 1.13, Pe = 2.12, Gr_t = 2.9 × 10³, Gr_s = 4.9 ×10⁴. |U|_max = 16.44 cm s⁻¹.

Fig. 6 shows the effects of gravity accelerations on |U|_max for 10⁻⁵ g₀ ≤ g_y ≤ 10 g₀, based on Ar = 1, ΔT = 50ºC (290ºC→240ºC), P_B = 10, Pr = 0.97, Le = 0.37, C_v = 1.03, Pe = 3.41. The solutally dominant convec-tion mode is predominant over the diffusion mode for 10⁻¹ g₀ ≤ g_y ≤ 10 g₀. The solutally dominant convection mode is transited into the diffusion mode at g_y = 0.1 g₀ and, since g_y = 0.1 g₀, down to g_y = 10⁻⁴ g₀, the diffusion becomes predominant. As seen in Fig. 6 |U|_max drop sharply for 10⁻¹ g₀ ≤ g_y ≤ 10 g₀. This indicates the mass transport is diffusion-dominated under gravity environments less than 0.1 g₀. One can see that the effect of solutally buoyancy driven convection is first important and then decreases rapidly and eventually the mode of transport becomes largely diffusion. Therefore, the parameter of the gravity vector is much important so that it could provide the researchers related to μg environments much motivations to perform numerical studies under μg environments.

Fig. 6. |U|_max as a function of gravity accelerations, 10⁻⁵ g₀ ≤ g_y ≤ 10 g₀, based on Ar = 1, ΔT=50°C (290°C→240°C), P_B = 10, Pr = 0.97, Le = 0.37, C_v = 1.03, Pe = 3.41.

Figs. 7 through 9 show velocity vector, streamline, temperature, mass fraction profiles, based on Ar = 1, ΔT=50ºC (290ºC→240ºC), P_B =10Torr, Pr=0.97, Le= 0.37, C_v = 1.03, Pe = 3.41 for three different gravities of 10⁻² g₀, 1 g₀ and 10 g₀. Figs. 7, 8 and 9 are related to 10⁻² g₀, 1 g₀ and 10 g₀, respectively. The corresponding |U|_max s are 7.7cm s⁻¹, 23.91 cm s⁻¹, 70.60cm s⁻¹, respectively. With increasing the magnitude of the gravity vector from 10⁻² g₀ to 1 g₀ by two order of magnitude, the corresponding |U|_max is increased by a factor of 3, whereas increasing the magnitude of the gravity vector from 1 g₀ to 10 g₀ by one order of magnitude, the corresponding |U|_max is increased by a factor of 2.

Fig. 7. (a) Velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on Ar = 1, ΔT=50°C (290°C→240°C), 10⁻² g₀, P_B = 10 Torr, Pr = 0.97, Le = 0.37, C_v = 1.03, Pe = 3.41, Gr_t = 1.8 × 10¹, Gr_s = 2.6 × 102. |U|_max = 7.7 cm s⁻¹.

Fig. 8. (a) Velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on Ar = 1, ΔT=50°C (290°C→240°C), 1 g₀, P_B = 10 Torr, Pr = 0.97, Le = 0.37, C_v = 1.03, Pe = 3.41, Gr_t = 1.8 × 10³, Gr_s = 2.6 × 10⁴. |U|_max = 23.91 cm s⁻¹.

Fig. 9. (a) Velocity vector profile, (b) streamline profile, (c) temperature profile, (d) mass fraction profile, based on Ar = 1, ΔT=50°C (290°C→240°C), 10 g₀, P_B = 10 Torr, Pr = 0.97, Le = 0.37, C_v = 1.03, Pe = 3.41, Gr_t = 1.8 × 10⁴, Gr_s = 2.6 × 10⁵. |U|_max = 70.60 cm s⁻¹.

4. Conclusions

It is concluded that the |U|_max diminishes first order exponentially with the aspect ratio, 1 ≤ Ar ≤ 10. For the range of 1 ≤ Ar ≤ 2, the |U|_max diminishes rapidly, and, for 2 ≤ Ar ≤ 10, dwindles down slowly. In other words, the |U|_max decreases much sharply near Ar = 1, and, then since Ar = 2, decreases. The μg conditions less than 10⁻² g make the effect of double-diffusion convection much reduced so that adequate advectivediffusion mass transfer could be obtained.