Journal of the Korean Crystal Growth and Crystal Technology (한국결정성장학회지)

The Korea Association of Crystal Growth (한국결정성장학회)
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Heat transfer study of double diffusive natural convection in a two-dimensional enclosure at different aspect ratios and thermal Grashof number during the physical vapor transport of mercurous bromide (Hg2Br2): Part I. Heat transfer

  • Ha, Sung Ho (Department of Chemical Engineering, Hannam University) ;
  • Kim, Geug Tae (Department of Chemical Engineering, Hannam University)
  • Received : 2022.01.18
  • Accepted : 2022.01.27
  • Published : 2022.02.28
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Abstract

A computational study of combined thermal and solutal convection (double diffusive convection) in a sealed crystal growth reactor is presented, based on a two-dimensional numerical analysis of the nonlinear and strongly coupled partial differential equations and their associated boundary conditions. The average Nusselt numbers for the source regions are greater than those at the crystal regions for 9.73 × 103 ≤ Grt ≤ 6.22 × 105. The average Nusselt numbers for the source regions varies linearly and increases directly with the thermal Grashof number form 9.73 × 103 ≤ Grt ≤ 6.22 × 105 for aspect ratio, Ar (transport length-to-width) = 1 and 2. Additionally, the average Nusselt numbers for the crystal regions at Ar = 1 are much greater than those at Ar = 2. Also, the occurrence of one unicellular flow structure is caused by both the thermal and solutal convection, which is inherent during the physical vapor transport of Hg2Br2. When the aspect ratio of the enclosure increases, the fluid movement is hindered and results in the decrease of thermal buoyancy force.

Keywords

1. Introduction

Recently, there has been drawn much attention in the applications of acousto-optic materials and signal processing optics for thallium arsenic selenide (Tl3AsSe3), and mercurous chloride (Hg2Cl2) and mercurous bromide (Hg2Br2), which leads to extensive research works [1-10]. One of the important topics in material processing is the combined thermal and solutal convection, i.e., double diffusive convection which occurs inherently due to the driving forces of temperature and solutal gradients on earth. Yang and Zhao [11] investigated a two-dimensional computational study of double-diffusive convection in rectangular enclosures with various geometries and Liu et al. [12] carried out numerically the importance of the Dufour and Soret effects utilizing multiplerelaxation-time lattice Boltzmann model in double diffusive convection systems. Chakkingal et al. [13] examined the flow regimes in a cubical enclosure with adiabatic cylindrical obstacle, and Meften [14] performed the study of double diffusive convection from a point of view of conditional stability. More recently, Hamimid et al. [15] studied on the importance of limit of the buoyancy effect for double-diffusive convection and Chauhan et al. [16] reported the influence of viscous dissipation on thermo-solutal convection in a square enclosure. Kim et al. [17-20] have performed computational study of combined thermal and solutal convection of mercurous chloride and mercurous bromide vapors in the sealed crystal growth reactors. Weaver and Viskanta [21] investigated diffusive-advection convection with the out of mass flux at the hot walls and the in of mass flux at the cold walls in two-dimensional rectangular enclosures.

In this study, a computational study of combined thermal and solutal convection (double diffusive convection) in a sealed crystal growth reactor is presented, based on a two-dimensional numerical analysis of the nonlinear and strongly coupled partial differential equations and their associated boundary conditions with a compressible model of ideal gas law. A computational study of velocity vectors and streamlines, isotherms and iso-mass concentrations is performed for \(\Delta\)T = 20ºC, the temperature difference between the source (350ºC) and crystal regions (330ºC), the partial pressures of component argon of 20 Torr.

2. Analysis

We consider the PVT (physical vapor transport) crys-tal growth in a two-dimensional enclosure, as shown in Fig. 1. The two-dimensional configuration is a square enclosure of aspect ratio, Ar = 1 and an enclosure of aspect ratio, Ar = 2, with one vertical surface maintained at high temperature, referred as a source region, and with the opposite vertical wall kept at low temperature, referred as a crystal region. Hg2Br2 (species A) at the source region is transported into at crystal region. Argon (species B) as an inert carrier gas, insoluble in Hg2Br2 (species A) exists in the vapor phase within the enclosure. Analysis is achieved for the mathematical model for double diffusive natural convection system based on the following assumptions: 
• The fluid is assumed be Newtonian and incompressible.
• The fluid is assumed to be laminar.
• The flow is assumed to be two-dimensional and independent of time.
• The fluid is assumed to behave as a continuum and to be no slip at the walls.
• The thermo-physical properties of the fluid except the density variation in the buoyancy term are considered to be constant. The density depends on the perfect gas law.
• Radiation, viscous dissipation effects are assumed to be negligible.

Fig.1.jpg 이미지

Fig. 1. System schematic descriptions and coordinates for twodimensional numerical simulations of a PVT crystal growth enclosure of major component Hg2Br2 (A) and impurity argon (B).

2.1. Dimensionless form of equations

The dimensionless quantities are:

x* = x/H, y* = y/H, u = vx/U, v = vy/U, p* = p/\(\rho \)0U2, T* = (T-Tc)/(Th-Tc), \(\omega \)*A,= (\(\omega \)A-\(\omega \)A,c)/(\(\omega _{A,s}\),-\(\omega _{A,c}\),), U = \(\alpha \)/H,

where x, y are rectangular coordinate along the transport length and the height of the enclosure, respectively, and p is the pressure. The U is the characteristic velocity and H is the height of the enclosure. The dimensionless parameters (Prandtl number, Pr; Lewis number, Le; Grashof number, Gr) in the following governing equations can be found in reference [22]. The \(\beta \) is the coefficient of thermal volume expansion. The mass concentration \(\omega _{A}\) is defined as \(\omega _{A}\)=\(\frac{x_{A}M_{A}}{x_{A}M_{A}+x_{B}M_{B}}\), x is mole fraction of component A, M is the number-mean molecular weight of mixture.

By applying the dimensionless variables, the governing equations can be expressed by non-dimensionless form:

수식1.jpg 이미지

The associated boundary conditions are given as follows:

수식2.jpg 이미지

The heat transfer of the diffusive-convection in the enclosure can be expressed in terms of the local Nuy, from the definition of dimensionless Nusselt number,Nuy = hH/k and the average convective Nusselt Nu =\(-\int Nu_{y}\)(y*)dy*. When Qd illustrates the increased heat transfer at the wall due to double-diffusive convection and Qa = \(\rho \)*uT* is the dimensionless energy flux due to advection, the total energy transferred at the wall is expressed by Nu = Qd + Qa [21].

2.2. Method of solution

The numerical investigation utilized the Semi-Implicit Method Pressure-Linked Equations Revised (SIMPLER) [23] iterative technique for the two-dimensional case for heat transfer Grashof numbers ranged from 9.73 × 103 to 6.22 × 105.

3. Results and Discussion

When the molecular weight of Hg2Br2 (MA = 560.988 g/gmol) is not equal to the molecular weight of argon (MB= 39.944 g/gmol), i.e., MA\(\neq \)MB, the transport phe-nomena in the vapor phase during the physical vapor transport (PVT) are much complicated because of both thermal and solutal buoyancy driven natural convection and the mass flux of Hg2Br2 (A) in the source and crystal interfaces. Table 1 lists up the average Nusselt numbers for the heat transfer study of double diffusive natural convection in a two-dimensional enclosure at different aspect ratios and thermal Grashof number during the physical vapor transport of mercurous bromide (Hg2Br2). Table 2 shows in details that the average Nusselt numbers at the source and crystal interfaces due to the diffusive-convection and the advection, corresponding to Table 1. The typical process parameter used in this study can be found in reference [22].

Table 1
The average Nusselt numbers for the heat transfer study of double diffusive natural convection in a two-dimensional enclosure at different aspect ratios and thermal Grashof numbers during the physical vapor transport of mercurous bromide (Hg2Br2)

Table1.jpg 이미지

Table 2
The average Nusselt numbers at the source and crystal interfaces due to the diffusive-convection and the advection, corresponding to Table 1

Table2.jpg 이미지

Figure 2 shows the average Nusselt numbers for the crystal regions, Nuc as a function of the thermal Grashof number (Grt) at two different aspect ratios, Ar = 1 and 2. As shown in Fig. 2, Nuc varies linearly and increases directly with the thermal Grashof number form 9.73×103\(\leq \)Grt\(\leq \)6.22×105 for Ar (transport length-towidth, L/H) = 1 and 2. Additionally, the average Nusselt numbers for the crystal regions at Ar = 1 are much  greater than those at Ar = 2. Figure 3 shows the average Nusselt numbers for the source regions, Nus, as a function of the thermal Grashof number at two different aspect ratios, Ar = 1 and 2. Like the case of the average Nusselt numbers for the crystal regions, Nuc, the average Nusselt numbers for the source regions, Nus varies linearly and increases directly with the thermal Grashof number form 9.73×103\(\leq \)Grt\(\leq \)6.22×105 for Ar (transport length-to-width, L/H) = 1 and 2. Note that the average Nusselt numbers for the source regions at Ar = 1 are greater than those at Ar = 2 for 9.73×103\(\leq \)Grt\(\leq \)7.78 ×104, but that the average Nusselt numbers for the source regions at Ar = 2 is greater than those at Ar = 1 for 2.62 × 105\(\leq \)Grt\(\leq \)6.62×105. At this time, the reason of this finding is not clear considering the comparisons of maximum magnitudes of velocity vectors.

Fig.2.jpg 이미지

Fig. 2. The average Nusselt numbers for the crystal regions as a function of the thermal Grashof number at two different aspect ratios, Ar = 1 and 2.

Fig.3.jpg 이미지

Fig. 3. The average Nusselt numbers for the source regions as a function of the thermal Grashof number at two different aspect ratios, Ar = 1 and 2.

Figure 4 shows the maximum magnitudes of velocity vector, |U|max as a function of the thermal Grashof number at two different aspect ratios, Ar = 1 and 2. The maximum magnitudes of velocity vector, |U|max increases directly and linearly with the thermal Grashof number form 9.73×103\(\leq \)Grt\(\leq \)6.22×105 for Ar (transport lengthto-width, L/H) = 1 and 2. The |U|max at Ar = 1 are much greater than those at Ar = 2 by a factor of 3. This finding reflects the effect of walls along the transport length. The thermal Grashof number for the case C4 (|U|max =19.6 cm/sec), is greater one order of magnitude than that for the case C2, but, from the comparisons of the maximum magnitudes of the velocity vector, |U|max, indicative of the contribution of thermal convection, it is concluded that the maximum magnitudes of the velocity vector, |U|max is increased with the thermal Grashof number by 50 %. The detailed flow characteristics of cases C2 and C4 are presented in Figs. 6 and 7, respectively.

Fig.4.jpg 이미지

Fig. 4. The maximum magnitudes of velocity vector, |U|max as a function of the thermal Grashof number at two different aspect ratios, Ar = 1 and 2.

Figure 5 shows the average Nusselt numbers for both the crystal and the source regions as a function of the thermal Grashof number at the aspect ratio of 2, Ar = 2. As shown in Fig. 5, it is found that the average Nusselt numbers for the source regions are greater than those at the crystal regions for 9.73×103\(\leq \)Grt\(\leq \)6.22×105.

Fig.5.jpg 이미지

Fig. 5. The average Nusselt numbers for both the crystal and the source regions as a function of the thermal Grashof number at the aspect ratio of 2, Ar = 2.

Figure 6 shows the velocity vector fields (|U|max = 13.4 cm/sec), streamlines (\(\psi\)min = 0, \(\psi\)max = 124.4, \(\delta \psi \) = 10), isotherms (\(\delta\)T* = 0.1), iso-mass concentrations (\(\delta \omega \)* = 0.1), for the heat transfer case, Table 1, case C2 with thermal Grashof number (Grt) =7.78×104, aspect ratio = 1. As shown in Fig. 6, one convection cell appears and seems asymmetrical against the mid-transport length of the enclosure (x* = 0.5). Figure 7 is concerned with the heat transfer case, Table 1, case C4 with thermal Grashof number (Grt) = 6.22×105, aspect ratio = 1. As shown in Figs. 6 and 7, the temperature and mass concentration gradients in the vertical walls near the crystal regions are greater than those in the horizontal walls near the crystal regions. Figures 6 and 7 indicate the presence of velocity boundary layers along the crystal regions.

Fig.6.jpg 이미지

Fig. 6. (a) velocity vector fields (|U|max = 13.4 cm/s), (b) streamlines (\(\psi\)min=0, \(\psi\)max=124.4, \(\delta \psi \)=10), (c) isotherms (\(\delta\)T* = 0.1), (d) iso-mass concentrations (\(\delta \omega \)* = 0.1), for the heat transfer case, Table 1, case C2 with thermal Grashof number (Grt) = 7.78 × 104, aspect ratio (Ar) = 1.

Fig.7.jpg 이미지

Fig. 7. (a) velocity vector fields (|U|max = 19.6 cm/s), (b) streamlines (\(\psi\)min=-1.3, \(\psi\)max = 217.5, \(\delta \psi \)=10), (c) isotherms (\(\delta\)T* = 0.1), (d) iso-mass concentrations (\(\delta \omega \)*=0.1), for the heat transfer case, Table 1, case C4 with thermal Grashof number (Grt) = 6.22 × 105, aspect ratio (Ar) = 1.

Figures 8 and 9 show the velocity vector fields, streamlines, isotherms, iso-mass concentrations, for the heat transfer case, Table 1, cases C6 and C8, aspect ratio = 2, respectively. Figures 8 and 9 illustrate the presence of velocity boundary layers along the bottom walls and crystal regions. The effect of the aspect ratio may be illustrated by comparing the flow structure characteristics of the velocity vector fields, streamlines, isotherms, iso-mass concentrations, i.e., (1) the case of Figs. 6 and 8 for the fixed thermal Grashof number of 7.78×104, (2) the case of Figs. 7 and 9 for the fixed thermal Grashof number of 6.22×105. As shown in Figs. 6 through 9, the occurrence of one unicellular flow structure is caused by both the thermal and solutal convection, which is inherent during the physical vapor transport of Hg2Br2. Comparisons of one case of Figs. 6 and 8, and the other case of Figs. 7 and 9 show that the convective flow structure is stabilized and the maximum magnitude of velocity vector decreases with increasing the aspect ratio, which is related to the wall effects. In other words, Increasing the aspect ratio of the enclosure results in the hindrance of the fluid movement and the reduction of thermal buoyancy force. 

Fig.8.jpg 이미지

Fig. 8. (a) velocity vector fields (|U|max = 4.57 cm/s), (b) streamlines (\(\psi\)min = 0, \(\psi\)max = 124.4, \(\delta \psi \) = 10), (c) isotherms (\(\delta\)T* = 0.1), (d) iso-mass concentrations (\(\delta \omega \)*=0.1), for the heat transfer case, Table 1, case C6 with thermal Grashof number (Grt) = 7.78 × 104, aspect ratio (Ar) = 2.

Fig.9.jpg 이미지

Fig. 9. (a) velocity vector fields (|U|max = 6.72 cm/s), (b) streamlines (\(\psi\)min = 0, \(\psi\)max = 124.4, \(\delta \psi \)=10), (c) isotherms (\(\delta\)T* = 0.1), (d) iso-mass concentrations (\(\delta \omega \)* = 0.1), for the heat transfer case, Table 1, case C8 with thermal Grashof number (Grt) = 6.62 × 105, aspect ratio (Ar) = 2.

4. Conclusions

In conclusions, the average Nusselt numbers for the source regions are greater than those at the crystal regions for 9.73×103\(\leq \)Grt\(\leq \) 6.22 × 105. The average Nusselt numbers for the source regions varies linearly and increases directly with the thermal Grashof number form 9.73×103\(\leq \)Grt\(\leq \) 6.22 × 105 for Ar (transport lengthto-width, L/H) = 1 and 2. Additionally, the average Nusselt numbers for the crystal regions at Ar = 1 are much greater than those at Ar = 2. Also, the occurrence of one unicellular flow structure is caused by both the thermal and solutal convection, which is inherent during the physical vapor transport of Hg2Br2. When the aspect ratio of the enclosure increases, the movement of fluid is retarded and results in the decrease of thermal buoy-ancy force.

Acknowledgement

This work was financially supported by the 2019 Hannam University Research Fund under Grant No. 2019A114 (April 1, 2019 through March 31, 2020).

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