Double-diffusive convection affected by conductive and insulating side walls during physical vapor transport of Hg₂Br₂

Abstract

In last few decades, although thermal and/or solutal buoyancy-driven recirculating flows in a closed ampoule have been intensively studies as a model problem, there exist interesting total molar flux of Hg₂Br₂ that have been unreported in the literature. It is concluded that the total molar flux of Hg₂Br₂(A) increases linearly and directly as the temperature difference regions in the range of 10℃ ≤ ΔT ≤ 50°, 3.5 × 10³ ≤ Gr_t ≤ 4.08 × 10³, 4.94 × 10⁴ ≤ Gr_s ≤ 6.87 × 10⁴. For the range of 10 Torr ≤ P_B ≤ 150 Torr, the total molar flux of Hg₂Br₂(A) decays second order exponentially as the partial pressure of component B (argon as an impurity), P_B increases. From the view point of energy transport, the fewer the partial pressure of component B (argon), P_B is, the more the energy transport is achieved.

1. Introduction

During the past few decades one area of interest in double diffusion has been the study of the physical vapor transport (PVT) processes in a sealed chamber. In recent years, the problems of Cattaneo-Christov double diffusion have been studied for Williamson nanomaterials slip flow subject to porous medium [1], and bi-directional stretched nanofluid flow with Cattaneo-Christov double diffusion [2]. Muhammad et al. [3] addressed Darcy-Forchheimer flow over an exponentially stretching curved surface with Cattaneo-Christov double diffusion. Asha and Sunitha [4] reported thermal radiation and hall effects on peristaltic blood flow with double diffusion in the presence of nanoparticles.

Mercurous halide materials are well known as the most promising materials in applications for acoustooptic materials and signal processing optics, for example, Bragg cells. Singh and his group investigated systematically the growth and characterization and development of large single crystals of Hg₂Br₂ [5-11]. Many reports of Hg₂Br₂ could be found in references [12-16]. Kim and his coworkers [17-19] have performed twodimensional numerical studies of double diffusion convection in the vapor phase during physical vapor crystal growth. Duval [20] published that four flow structure regions appear during the physical vapor transport of mercurous chloride crystal growth.

Our numerical simulations are motivated by the desire to study the influences of hybrid thermal boundary conditions on the convective flow because the final quality of crystal is affected by convection fields. In this paper, the effects of the temperature differences between the source and crystal, the Peclet number, Pe, and the partial pressure of component B (argon as inert gas), P_B on the total molar flux of Hg₂Br₂ and the maximum magnitudes of velocity vector, |U|_max in the dimensional unit (cm/sec) shall be addressed.

2. Numerical Simulations

Consider steady state thermal and solutal buoyancy driven recirculating flows of Hg₂Br₂(A)-argon (B) with thermo-physical properties listed in Table 1, in PVT crystal growth enclosure for hybrid thermal boundary conditions with linear temperature profiles, i.e., conductive walls, and insulating walls, shown in Fig. 1, accompanied by a 42 × 22 (x × y) grid system. The detailed assumptions and nomenclature can be found in reference [17]. Also, the dimensionless parameters of Prandtl, Lewis, Peclet, Grashof, concentration, aspect ratio are described in reference [20].

Table 1. Thermo-physical properties of Hg₂Br₂(A)-argon (B) (MA = 560.988, M_B = 39.944) at \(\Delta\)T = 50ºC, P_B = 10 Torr

Kinematic viscosity  Thermal diffusivity  Binary diffusivity  Coefficient of thermal volume expansion  Density of mixture  Prandtl number  Lewis number  Peclet number  Concentration number  Thermal Grashof number  Solutal Grashof number

0.44 cm2/sec  0.45 cm2/sec2   1.28 cm2/sec2  0.0017 (1/ºC)  0.000599 g/cm3  0.97  0.35  3.65  1.02  3.5 × 103  4.94 × 104

 

Fig. 1. System schematic and coordinates for numerical simulation of PVT crystal growth reactor of Hg₂Br₂(A)-argon (B).

In non-dimensional form, continuity, Navier-Stokes momenta, energy transport, and mass transport are governed by:

The boundary conditions corresponding to the above equations (1) to (4) are given as follows:

On the walls

(0 < x*< 1, y*= 0 and 1):

(0 < x*< 3, y*= 0 and 1):

(3 < x*< 4, y*= 0 and 1):

On the source (x*=0,  0 < y*< 1 ):

\(\nu (0,y^{*})=0,\)
 

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

\(\omega _{A}^{*}(0,y^{*})=1.\)

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

\(\nu (L/H,y^{*})=0,\)

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

\(\omega _{A}^{*}(L/H,y^{*})=0.\)

The code verification of one’s results can be found in reference [17], for the Semi-Implicit Method Pressure-Linked Equations Revised (SIMPLER) [21].

3. Results and Discussion

When M_A \(\neq \)M_B, the two molecular weights of Hg₂Br₂ and argon are different, i.e., M_A = 560.988 g/gmol, M_B =39.944 g/gmol, solutally buoyancy driven convection is much important compared with thermally buoyancy driven convection, but solutal and/or thermal convection are coupled and the effects of thermal buoyancy convection cannot be neglected during the physical vapor transport of Hg₂Br₂ in the vapor phase. Therefore, our interest is restricted on our studies to investigate the relations of the driving force, the temperature difference, \(\Delta \)T, the maximum magnitudes of velocity vector, |U|_max and the Peclet number for the transport of crystal species. 

As shown in Fig. 2, it is clear that the total molar flux of Hg₂Br₂(A) increases linearly and directly as the temperature difference in the range of 10ºC \(\leq\)\(\Delta \)T \(\leq\) 50ºC, 3.5 × 10³\(\leq\)Gr_t\(\leq\)4.08×10³, 4.94×10⁴\(\leq\)Gr_s\(\leq\)6.87×10⁴. For \(\Delta \)T= 10ºC, the corresponding thermal (Gr_t) and solutal (Gr_s) Grashof number is 4.08× 103and 6.87 × 104, respectively; for \(\Delta \)T=50ºC, the corresponding thermal (Gr_t) and solutal (Gr_s) Grashof number is 3.5×10³ and 4.94×10⁴, respectively. With increasing the temperature difference, the corresponding thermal (Gr_t) and solutal(Gr_s) Grashof number reversely decrease, which reflected the variations in the density of the mixture of Hg₂Br₂ and argon. In other words, for \(\Delta \)T=10ºC, the kinematic viscosity is 0.18 cm²/sec; for \(\Delta \)T = 50ºC, the kinematic viscosity is 0.44 cm²/sec. The system considered in Fig.1 is Ar (aspect ratio, transport length-to-width) = 4, T_s (source temperature)= 300ºC, P_B (partial pressure of component B, argon) = 10 Torr, on earth. For \(\Delta \)T=10ºC, the thermal diffusivity and binary mass diffusivity is 0.18, and 1.28 cm²/sec; for \(\Delta \)T=50ºC, the thermal diffusivity and binary mass diffusivity is 0.45, and 1.28 cm²/sec. For the range of 10ºC\(\leq\)\(\Delta \)T\(\leq\)30ºC, the total molar flux of Hg₂Br₂(A) increases sharply with increasing the temperature difference, whereas for the range of 30ºC\(\leq\)\(\Delta \)T\(\leq\)50ºC, the total molar flux of Hg₂Br₂(A) increases relatively gradually. As mentioned before, this difference is likely be due to the variations in the density of the mixture of Hg₂Br₂ and argon.

Fig. 2. The total molar flux of Hg₂Br₂(A) as a function of the temperature difference, \(\Delta \)T (ºC), based on aspect ratio = 4, T_s =300ºC, PB = 10Torr, 3.5 × 10³\(\leq\)Gr_t \(\leq\)4.08 × 10³, 4.94 × 10⁴\(\leq\)Gr_s \(\leq\)6.87 × 10⁴, on earth.

Figure 3 illustrates the effects of Peclet number, Pe on as the temperature difference, \(\Delta \)T in the range of 10ºC\(\leq\)\(\Delta \)T\(\leq\)50ºC, 3.5 × 10³\(\leq\)Gr_t \(\leq\)4.08 × 10³, 4.94 × 10⁴\(\leq\)Gr_s \(\leq\)6.87 × 10⁴. The Peclet number, Pe increases linearly and directly with the temperature differences. Figure 4 shows the relation of the maximum magnitude of velocity vector, |U|max and Peclet number, Pe for 10ºC\(\leq\)\(\Delta \)T\(\leq\)50ºC, aspect ratio = 4, T_s = 300ºC, P_B = 10 Torr, on earth. This relation illustrates that the |U|_max increases linearly with the Peclet number, Pe. Note that the Peclet number is at the source and the crystal regions. 

Fig. 3. The Peclet number, Pe as a function of the temperature difference, \(\Delta \)T (ºC), based on aspect ratio = 4, T_s = 300ºC, P_B =10Torr, 3.5 × 10³\(\leq\)Gr_t \(\leq\)4.08×10³, 4.94 × 10⁴\(\leq\)Gr_s \(\leq\)6.87 × 10⁴, on earth.

Fig. 4. The |U|_max as a function of the dimensionless Peclet number, Pe, based on aspect ratio = 4, T_s = 300ºC, P_B = 10 Torr, 3.5 ×103\(\leq\)Gr_t\(\leq\)4.08 × 10³, 4.94×10⁴\(\leq\)Gr_s \(\leq\)6.87 × 10⁴, on earth.

Figure 5 shows the profiles of velocity vector, streamline, temperature, mass concentration, based on aspect ratio = 4, \(\Delta \)T=50ºC (300ºC→250ºC), PB =10Torr, |U|_^max =1.87 cm/sec, on earth. As plotted in Fig. 5, there exists small one convective cell in the vapor phase, and the flow structure is asymmetrical against at y*= 0.5 and three-dimensional flow structure. For the flow regions along the transport length at the bottom region, i.e., 0\(\leq\)y*\(\leq\)0.5, the one-dimensional Stefan flows appear. Temperature profile shown in Fig. 5(c) is related to the hybrid thermal boundary conditions; for 0\(\leq\)x*\(\leq\)1, conductive walls and for 1\(\leq\)x*\(\leq\)3, insulating walls, for 3\(\leq\)x*\(\leq\)4, conductive walls. Close spacings of mass concentration shown in Fig. 5(d) exhibits the mechanism of the diffusion-limited mass transfer. 

Fig. 5. (a) Velocity vector, (b) streamline, (c) temperature, (d) mass concentration profile, based on aspect ratio = 4, \(\Delta \)T = 50ºC(300ºC→250ºC), P_B = 10Torr, Peclet number = 3.65, thermal Grashof number (Gr_t) = 3.5 × 103, solutal Grashof number (Gr_s) =4.94 × 104, Prandtl number = 0.97, Lewis number = 0.35, concentration parameter = 1.02, total pressure = 108 Torr, |U|_max = 1.87 cm/sec, on earth.

Figure 6 shows the effects of the partial pressure of component B (argon), PB, on the total molar flux of Hg₂Br₂(A), for 10 Torr\(\leq\)P_B\(\leq\)150 Torr, based on aspect ratio = 4, \(\Delta \)T=50ºC, T_s = 300ºC, P_B = 10 Torr, 1.8 × 10³\(\leq\)Gr_t \(\leq\)2.9×10³, 2.6×10⁴\(\leq\)Gr_s \(\leq\)4.9×10⁴, on earth. For the range of 10 Torr\(\leq\)P_B\(\leq\)150 Torr, the total molar flux of Hg₂Br₂(A) decays second order exponentially with the partial pressure of component B (argon), P_B. Figure 7 shows the influences of the partial pressure of component B (argon), P_B, on the Peclet number, Pe, for 10Torr\(\leq\)PB\(\leq\)150 Torr, corresponding to Fig. 6. 

Fig. 6. The total molar flux of Hg₂Br₂(A) as the partial pressure of component B, PB, based on aspect ratio = 4, \(\Delta \)T = 50ºC, T_s = 300ºC, P_B = 10Torr, 1.8 × 10³\(\leq\)Gr_t \(\leq\)2.9×10³, 2.6×10⁴\(\leq\)Gr_s \(\leq\)4.9 × 10⁴, on earth.

Fig. 7. The Peclet number, Pe as the partial pressure of component B, PB, \(\Delta \)T (ºC), based on aspect ratio = 4, T_s = 300ºC, P_B = 10Torr, 3.5×10³\(\leq\)Gr_t \(\leq\)4.6×10³, 4.9×10⁴\(\leq\)Gr_s \(\leq\)5.6×10⁴, on earth.

Figure 8 shows the profiles of velocity vector, streamline, temperature, mass concentration, based on aspect ratio = 4, \(\Delta \)T=50ºC (300ºC→250ºC), PB =60Torr, |U|_max =0.99 cm/sec, on earth. As shown in Fig. 8, one convective roll is present in the vapor phase. In a comparison of Fig. 8(c) temperature with Fig. 5(c) temperature, from the view point of energy transport, the fewer the partial pressure of component B (argon), P_B is, the more the energy transport is achieved.

Fig. 8. (a) Velocity vector, (b) streamline, (c) temperature, (d) mass concentration profile, based on aspect ratio = 4, \(\Delta \)T=50ºC (300ºC→250ºC), P_B = 60Torr, Peclet number=2.19, thermal Grashof number (Gr_t) = 3.8×10³, solutal Grashof number (Gr_s)=5.07×10⁴, Prandtl number=0.92, Lewis number=0.51, concentration parameter=1.12, total pressure=158 Torr, |U|_max = 0.99 cm/sec, on earth.

4. Conclusions

It is concluded that the total molar flux of Hg₂Br₂(A) increases linearly and directly as the temperature difference, \(\Delta \)T in the range of 10ºC\(\leq\)\(\Delta \)T\(\leq\)50ºC, 3.5×10³\(\leq\)Gr_t \(\leq\)4.08 × 10³, 4.94×10⁴\(\leq\)Gr_s \(\leq\)6.87×10⁴. For the range of 10 Torr\(\leq\)P_B\(\leq\)150 Torr, the total molar flux of Hg₂Br₂(A) decays second order exponentially with the partial pressure of component B (argon), P_B. From the view point of energy transport, the fewer the partial pressure of component B (argon), P_B i s, the m ore the energy transport is achieved.