# Boundary Conditions for Simulations of Fluid Flow and Temperature Field during Ammonothermal Crystal Growth—A Machine-Learning Assisted Study of Autoclave Wall Temperature Distribution

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Simulations of Temperature Distribution and Fluid Flow

_{e}, i.e., the sum of all radiation fluxes traversing the volume (regardless of direction and wavelength) is defined by Equation (3):

_{3}is the radiosity temperature in Kelvin, which is a separate variable in Phoenics. At the boundaries of transparent medium and solids, as well as within solids, the radiosity temperature is equal to the first phase temperature T

_{1}. The radiant flux at such boundaries, however, depends not only on the gradient of T

_{3}in the medium near the wall but also on the emissivity of the wall. The differential equation for T

_{3}is as follows (Equation (4)):

_{3}represents the thermal conductivity pertaining to the radiosity temperature T

_{3}described by Equation (5):

_{gap}represents the gap between nearby solid walls. The source term ${\dot{Q}}_{T3}$ is related to the first phase temperature T

_{1}and the radiosity temperature T

_{3}by Equation (6):

_{4}Cl-NH

_{3}-system [26] and that it is unknown whether the fluid remains clear to infrared radiation under process conditions.

^{3}and using linear extrapolation of NIST [24] data on viscosity to 550 °C (i.e., a dynamic viscosity of about 4.01 × 10

^{−5}Pa∙s), the Reynolds number R is about 2.02 × 10

^{3}for 0.02 m/s to 2.02 × 10

^{4}for 0.2 m/s. Therefore, a transition from laminar to turbulent flow is expected to occur depending on the location inside the cavity. Prantl and Rayleigh numbers are about 1.04 and 2.51 × 10

^{9}, respectively, which strongly suggests turbulent flow (for the given Prantl number, a transition to turbulent flow can be expected at Rayleigh numbers around 2.5 × 10

^{4}[29]). However, there are doubts as to how precisely extrapolated viscosity of pure ammonia matches the viscosity of the actual, solute-containing fluid [30]. A higher viscosity, as suspected based on in-situ x-ray monitoring of the diffusion of Ga-containing solutes [30], could push the Reynolds and Rayleigh numbers back to the laminar range, especially for small autoclave diameters. It may be worth noting that a significant increase of viscosity due to mineralizer addition is assumed for hydrothermal using KOH mineralizer [31]. Masuda et al. identify this increase of viscosity due to mineralizer addition as the most significant deviation from pure solvent properties [31]. Though not verified by measurements under process conditions, this assumption is based on a viscosity increase of up to five times at 44 wt% at room temperature [31]. Masuda et al. expect the results of their study be applicable also to ammonothermal growth of GaN [31]. In conclusion, both the experimental ammonothermal observation in [30] as well as the knowledge on mineralizer-containing hydrothermal solutions at ambient conditions support suspecting a non-negligibly increased viscosity. However, quantitative knowledge is missing, possible differences between the effects of different mineralizers remain unknown, and a possible influence of Ga-containing solutes beyond the effect of the mineralizer itself remains unclear.

^{+}, is defined as described in Equation (7):

_{molecular}the absolute viscosity of the fluid in laminar motion.

^{+}is defined as u∙(ρ/τ)

^{1/2}. The two dimensionless quantities y

^{+}and u

^{+}are related by a differentiable formula known as Spalding’s law of the Wall (Equation (8)).

^{+}and the local Reynolds number R. This dimensionless viscosity is used in the momentum equations, thus LVEL model is a form of zero-equation turbulence model. The dimensionless effective viscosity ν

^{+}is described as follows (Equation (9)):

^{+}and y

^{+}equals the local Reynolds number R, it can be computed using Equation (10) for every location:

^{+}is computed by an iterative Newton–Raphson procedure. Consequently, the dimensionless effective viscosity ν

^{+}and the effective viscosity can also be computed for every point in the flow, which permits accounting for the effects of turbulences.

_{1}/A and $\dot{Q}$

_{2}/A. Moreover, in the case of type A cases, there is an additional energy equation as stated above in equation 4 for the radiosity temperature T

_{3}.

^{5}Pa everywhere outside the ammonia-filled cavity and p = 1.0 × 10

^{8}Pa inside the ammonia-filled cavity, velocity components v = 1.0 × 10

^{–10}m/s and w = 1.0 × 10

^{−10}m/s in all fluid-filled regions, T

_{1}= T

_{3}= 20 °C.

^{5}Pa everywhere outside the ammonia-filled cavity and p = 1.0 × 10

^{8}Pa inside the ammonia-filled cavity; velocity components v = 1.0 × 10

^{−10}m/s and w = 1.0 × 10

^{−10}m/s in all fluid-filled regions; T

_{1}= 20 °C within the autoclave head (335 mm ≤ Z ≤ 370 mm). At the wall sections corresponding to the position of the heaters, the set temperatures were used as initial conditions in type B and C cases, i.e.: For 175 mm ≤ Z ≤ 335 mm and Y = −35 mm (region of top heater): ${T}_{1}={T}_{TC,1}$; for 0 mm ≤ Z ≤ 137.5 mm and Y = −35 mm (region of bottom heater): ${T}_{1}={T}_{TC,2}$. Outside the mentioned locations, the initial temperature for cases B and C was T

_{1}= 500 °C.

_{z}and v

_{y}as well as for the scalar variable LTLS and T3. The use of Conjugate-Gradient-Residual Solver is known to be often advantageous for pressure P1 (p) when modeling buoyancy-driven flows with complex geometry [39]. The same holds for TEM1 (temperature T

_{1}), which denotes the first-phase temperature in Phoenics, in complex conjugate heat transfer cases [39]. The variable LTLS is an auxiliary variable needed for the calculation of the distance to the nearest solid wall and the gap width between two nearest solid walls [23]. T3 represents the radiosity temperature used by IMMERSOL radiation model [23]. For TEM1, CGRS solver with AMG preconditioner was used. For the density ρ (or DEN1 in Phoenics), the STONE solver (the default solver in Phoenics that is based on Stone’s Strongly Implicit method [39]) was used.

^{−2}

_{.}For case A3 as an example, the normalized residuals were 4.371 × 10

^{−2}% for P1, 1.564 × 10

^{−1}% for V1, 3.021 × 10

^{−1}% for W1, 1.887 × 10

^{−2}% for LTLS, 2.503 × 10

^{−2}% for TEM1, 3.098 × 10

^{−5}% for DEN1, and 1.340 × 10

^{−3}% for T3, respectively. A grid sensitivity test was conducted by doubling the number of cells in each region of the cartesian grid. However, direct comparison proved to be difficult. With the decreased cell sizes (beyond the previously determined well-converging, approximately optimal grid), the residuals increased, seemingly because convergence deteriorated. The unusual effect that a finer (intermediate) grid size can sometimes produce worse results is mentioned in Phoenics Encyclopedia [23] in an evaluation that compares LVEL to other turbulence models, albeit no explanation is given for this behavior. Assuming that the cause is the deterioration of convergence, it would be necessary to re-optimize convergence-promoting measures such as relaxation times for each variable in order to present a space grid sensitivity study that does not suffer from convergence issues (but would again not represent a straightforward comparison). We have therefore optimized discretization in space rather for obtaining convergence than for achieving full grid-independence of the solutions. In conclusion, minor deviations due to grid dependence may exist in those cases where solutions with different geometry had to be compared. To minimize this effect, the space grids were designed for type A as well as for type B and C cases with all solids in place. The simulations without internal solids were then performed using the same grids as with internal solids.

#### 2.2. Machine Learning for Adjusting Input Parameters of Simulations

_{1}and $\dot{Q}$

_{2}, for the top and bottom heaters, respectively. Other quantities will carry subscripts 1 and 2 for the top and bottom zones as well. A combination of parametric runs and a machine learning model was used to accelerate the process of determining values for $\dot{Q}$

_{1}and $\dot{Q}$

_{2}. The idea behind this is that for an otherwise given model, it may be possible to find a sufficiently accurate description of the relationship between heater powers and temperatures at the thermocouple locations that is much simpler and computationally less expensive than a simulation of heat transfer and fluid flow. Machine learning models using simulation results as input data have already been applied for modelling flow velocity and supersaturation in SiC solution growth, aiming at more efficient optimization of growth parameters [42]. This shows that fluid flow can be modelled relatively accurately using machine learning algorithms, suggesting that our approach for power tuning may also be feasible. Also, a machine learning model may be able to capture the effects of changes to the physical model studied by re-training on a relatively small number of simulations run with these variations as additional features. A flowchart visualizing the workflow of the integrated approach using both physics-based simulations and machine learning is depicted in Figure 2. The initial values for $\dot{Q}$

_{1}and $\dot{Q}$

_{2}need to be guessed well enough to produce a converging simulation while limiting temperatures to a physically reasonable range (20 °C to 3000 °C were used). Therefore, it can be necessary to do (or start and cancel if not converging due to exceeding the upper temperature limit) a few simulations initially. Once a rough estimate of the maximum power had been found, an evenly spaced grid of values was used to obtain more initial training data by conducting parametric runs. New values for $\dot{Q}$

_{1}and $\dot{Q}$

_{2}were chosen based on the predictions of the machine learning models. For this purpose, the models were used to predict temperatures T

_{TC,1}and T

_{TC,2}for 10,000 different power settings, which were created by generating two sets of random numbers and scaling them to the range of powers that had initially been found to be suitable for keeping temperatures in a reasonable range. From those predictions, 10 to 20 sets of power settings were chosen by searching for those that yield T

_{TC,1}and T

_{TC,2}closest to T

_{set,1}and T

_{set,2}. These were then used in the next round of simulations (for technical reasons, the number of simulations that can be conducted within one parametric run depends on the number of digits of the parameters, hence the number had to be decreased when approaching the target values).

_{TC,1}and T

_{TC,2}), MultiOutputRegressor was used when using models that do not natively support multiple outputs (random forest regressor). The number of features can be as low as two ($\dot{Q}$

_{1}and $\dot{Q}$

_{2}) as long as the physics-based simulation considers only one model that does not vary except for the heater power settings. The model was initially trained on a subset of data with heater powers being the only two features, and subsequently re-trained as features had to be added and corresponding data became available.

## 3. Results and Discussion

#### 3.1. Results and Performance of Machine Learning Algorithms

_{TC,1}and T

_{TC,2}, whereas this is less clear for nutrient porosity. The number of seeds and open/space ratio do not show a clearly significant impact on the temperatures T

_{TC,1}and T

_{TC,2}, thus we conclude that they have either no or only a weak influence on wall temperatures for a given combination of heater powers. It is interesting that the power of the bottom heater is recognized as a more important feature than the power of the top heater. This is a further indication of the importance of convective heat transfer, as convective heat transfer will cause heat transfer from the bottom to the top. This also fits well with a study by Li et al. [18] on industry scale hydrothermal autoclaves that views the bottom heater as the main heater and even assumes an outgoing heat flux for the autoclave wall in the region of the top heater.

_{TC,1}= 550 °C and T

_{TC,2}= 650 °C), which is probably due to a higher density of training data in that region. This higher density of data probably also is the reason that the accuracy of the models appears to be somewhat better in this core region of interest, thus, the errors and accuracies calculated for the entire data space underestimate the usefulness of the models. However, due to the high accuracy that would be desirable for the core region of interest both models were found to be most useful for speeding up the process of finding approximately suitable power settings. For finetuning (decreasing deviations from set temperatures from about 5 K to less than 1 K, manual adaptation of power settings was therefore sometimes found to be the easier path.

_{TC,1}and T

_{TC,2}but a combination of temperatures can occur as a result of more than one combination of power settings. It was therefore more practical to use powers as features and make a large number of predictions for random power settings and select those that happened to come close to the desired temperatures for further consideration.

#### 3.2. Results of Simulations That Include the Entire Growth Setup

_{TC,1}and T

_{TC,2}for all evaluated type A cases deviate by only 0.3 K on average (maximum deviation: 0.9 K) and are thus deemed fully comparable for the purpose of this study. Specific values are denoted in Figure 8.

#### 3.3. Accuracy Losses with Conventional Boundary Condition Definitions

#### 3.4. Refined Boundary Condition Definition Methods

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | |

DG | dataset group |

NN | neural network |

RFR | random forest regressor |

Variables and physical quantities | |

c_{p} | specific heat capacity |

E | Young’s modulus |

k | thermal conductivity |

p | pressure (representing first phase pressure P1 in Phoenics) |

t | time |

T_{1} | temperature (representing first phase temperature TEM1 in Phoenics) |

T_{3} | radiosity temperature (representing variable T3 in Phoenics) |

T_{set,1} | set temperature at control thermocouple location of zone 1 (top heater zone) |

T_{set,2} | set temperature at control thermocouple location of zone 2 (bottom heater zone) |

T_{TC,1} | temperature at control thermocouple location of zone 1 (top heater zone) |

T_{TC,2} | temperature at control thermocouple location of zone 2 (bottom heater zone) |

α | linear thermal expansion coefficient |

α_{v} | volumetric thermal expansion coefficient |

β | compressibility |

ε | emissivity |

λ_{3} | thermal conductivity pertaining to the radiosity temperature T_{3} |

ν | kinematic viscosity |

ρ | density |

$\dot{Q}$ | energy source term |

$\dot{Q}$_{1} | power of heater zone 1 (rate of heat supply at top heater) |

$\dot{Q}$_{2} | power of heater zone 2 (rate of heat supply at bottom heater) |

M | mass of fluid in a mesh cell |

m | mass-inflow rate |

T_{set} | set temperature for heater control |

y^{+} | dimensionless distance from the wall |

u | velocity |

v | velocity component in Y-direction |

w | velocity component in Z-direction |

u^{+} | auxiliary variable defined as u∙(ρ/τ)1/2 |

v^{+} | dimensionless effective viscosity |

L | distance from the nearest wall |

VEL | local velocity |

dX | distance between mesh cell centers |

A | cell-face area |

H | enthalpy |

J_{e} | radiosity |

${\epsilon}_{1}^{\prime}$ | emissivity per unit length |

${s}_{1}^{\prime}$ | scattering coefficient per unit length |

W_{gap} | gap between nearby solid walls |

τ | shear stress in the fluid |

µ | dynamic viscosity |

µ_{molecular} | the absolute (dynamic) viscosity of the fluid in laminar motion |

R | local Reynolds number |

Re | Reynolds number |

K | Permeability of the porous medium |

F | Forchheimer coefficient |

D | binary parameter to combine equations for free flow and porous media flow |

${d}_{p}$ | particle diameter of the porous medium |

Subscripts | |

− | neighboring mesh cell in negative direction |

+ | neighboring mesh cell in positive direction |

abs | absolute (referring to absolute temperature in Kelvin) |

eff | effective |

ref | reference |

t | current time step |

t−1 | previous time step |

TC | thermocouple/location of control temperature measurement |

T3 | pertaining to heat radiation/radiosity temperature T3 |

Operators | |

≊ | equal or almost equal |

≇ | neither exactly nor approximately equal |

Constants | |

C | Karman constant (equal to 0.417) |

σ | Stefan–Boltzmann constant |

g | gravity |

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**Figure 1.**Geometry and materials used in simulation of temperature distribution and fluid flow. (

**a**) Domain of type A cases, with fluid regions indicated by color (blue: Air, red: Supercritical NH

_{3}). The distance between bottom domain boundary and bottom of the furnace is 15 mm, (

**b**) Close-up of geometry showing the solid materials. The red areas correspond to the two heaters. (

**c**) Simulation domain, geometry, and materials for cases of type B and C, containing only the autoclave and its interior. X, Y, and Z represent the positive directions of the cartesian coordinate system used to describe the geometry in the simulation. Gravity acts in the negative Z direction.

**Figure 3.**Overview of the distribution of data with different power settings in the final dataset. Darker color corresponds to a higher amount of datapoints in the respective region, as also indicated by the bar plots.

**Figure 4.**Overview of generated simulation data. Data are split by different features in the subplots as indicated by the legends. Darker regions indicate overlap of datapoint symbols. Cyan symbols represent cases with internal solids, black datapoints indicate cases without solids inside the autoclave.

**Figure 5.**Results using the complete dataset at the end of this study. (

**a**) Overview of simulated data together with those predicted by the random forest regressor and neural network, (

**b**) close-up of the temperature region of interest, (

**c**) feature importances obtained from the random forest regressor model (error bars represent inter-trees variability).

**Figure 6.**Exemplarily comparison of prediction data from random forest regressor and neural network to test data. (

**a**) Bottom temperature T

_{TC,2}as a function of top power $\dot{Q}$

_{1}, (

**b**) bottom temperature T

_{TC,2}as a function of bottom power $\dot{Q}$

_{2}, (

**c**) splitting of the data by the third feature, namely flow model.

**Figure 7.**Overview of temperature distribution in the entire domain for cases A1, A2, and A3, respectively. A brief description of the characteristics of each case is given in the insets at the top.

**Figure 8.**Overview of velocity distribution in the entire domain for cases A1, A2, and A3, respectively. For better visibility, thick arrows were added as guides to the eye.

**Figure 9.**Close-ups showing the temperature distribution in the walls and interior of the autoclave (left subfigure of each case) and velocity distribution inside the autoclave (right subfigure of each case). For better visibility, thick arrows were added as guides to the eye.

**Figure 10.**Temperature distributions at outer autoclave walls determined by simulations of the complete setup including the furnace. (

**a**) Vertical autoclave wall. (

**b**) Horizontal autoclave walls (horizontal position of 0 corresponds to symmetry axis of autoclave). For the top autoclave wall, a position in the autoclave head corresponding to the thickness of the bottom autoclave wall was chosen, as this mimics a suitable choice for simulations that do not include the furnace.

**Figure 11.**Comparison of internal temperatures (

**a**,

**c**) and flow velocities (

**b**,

**d**) depending on the method of boundary condition definition along the centerline of the autoclave (

**a**,

**b**) and in the vicinity of the inner autoclave wall (

**c**,

**d**). Regions with zero velocity originate from fully blocking solids.

**Figure 12.**Comparison of internal temperatures (

**a**,

**c**) and flow velocities (

**b**,

**d**) depending on the method of boundary condition definition in radial direction at the center of the nutrient (

**a**,

**b**) and at the center of the middle seed (

**c**,

**d**). Regions with zero velocity originate from fully blocking solids.

**Figure 13.**Temperature distributions (top) and flow fields (bottom) for case type

**A**(including furnace and surrounding),

**B**(heater-long fixed temperature), and type

**C**(i.e., using the outer wall temperature distribution extracted from the corresponding furnace-including simulation A3). The plots of the velocity fields have been changed in aspect ratio to improve readability of the figure, and the figures are rotated with respect to gravity (bottom of the autoclave is on the left side of the figure). For better visibility, thick arrows were added as guides to the eye.

Material | Description |
---|---|

Air (1) | At 20 °C, 1 atm, treated as incompressible |

NH_{3} (2) | At 426.6 °C, 100 MPa, treated as incompressible |

Steel (3) | At 27 °C, C = 1% |

Ni-Cr superalloy (4) | At 600 °C |

Active part of resistive heater (5) | At 600 °C |

Ni-Cr superalloy (6) | At 538 °C |

Insulation (7) | |

Baffle (8) | |

GaN (bulk) (9) | Wurtzite GaN |

**Table 2.**Material properties of fluids in simulation of temperature distribution and fluid flow. Data for air (at ambient temperature) were taken from Phoenics inbuilt database, data for supercritical ammonia at 426.9 °C (the upper end of the temperature range for which data are available) were obtained from NIST database [24].

Gas/Fluid | ρ/(kg/m^{3}) | ν/(m^{2}/s) | c_{p}/(J/(kg∙K)) | k/(W/(m∙K) | α/(1/K) | β/(m^{2}/N) |
---|---|---|---|---|---|---|

Air (1) | 1.189 | 1.544 × 10^{−5} | 1005.0 | 0.02580 | 3.410 × 10^{−3} | 0.0 |

NH_{3,sc} (2) | 233.950 | 1.570 × 10^{−7} | 4216.5 | 0.16224 | 2.631 × 10^{−3} | 0.0 |

Solid | ρ/(kg/m^{3}) | c_{p}/(J/(kg∙K)) | k/(W/(m∙K) | (1/E)/(m^{2}/N) | ε/(1) |
---|---|---|---|---|---|

Steel (3) | 7800.0 | 473.00 | 43.00 | 0.50 × 10^{−11} | 0.2 |

Ni-Cr superalloy (4) | 8260.0 | 533.00 | 20.60 | 6.00 × 10^{−6} | 0.8 |

Active part of resistive heater (5) | 7150.0 | 750.00 | 20.00 | 5.88 × 10^{−6} | 1.0 |

Ni-Cr superalloy (6) | 8249.0 | 452.00 | 18.90 | 5.30 × 10^{−6} | 0.8 |

Insulation (7) | 12.0 | 840.00 | 0.04 | 1.00 × 10^{−9} | 0.2 |

Baffle (8) | 10,220.0 | 251.00 | 138.00 | 3.03 × 10^{−6} | 1.0 |

GaN (bulk) (9) | 6150.0 | 518.41 | 100.13 | 3.39 × 10^{−6} | 1.0 |

Label | Geometry Type | Flow | Interior Solids | Boundary Condition |
---|---|---|---|---|

A1 | With furnace | Laminar | None | Heater power |

A2 | With furnace | Turbulent | None | Heater power |

A3 | With furnace | Turbulent | Nutrient, baffle and seeds | Heater power |

B | Autoclave only | Turbulent | Nutrient, baffle and seeds | Heater-long fixed T |

C | Autoclave only | Turbulent | Nutrient, baffle and seeds | Fixed outer wall T distribution from A3 |

**Table 5.**Groups of datasets used for model evaluation. For the group that varies in the number of features (DG2), more detailed information is given in brackets in the following format: (Number of simulation datasets, number of features varying within these datasets).

Dataset Group | Varying Parameter | Parameter Values |
---|---|---|

DG1 | Number of simulation data sets | 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 137 |

DG2 | Number and types of features varying within dataset | No internal solids (54/3), no internal solids but only turbulent flow (59/3), all features but only laminar flow (50/5), all features but only turbulent flow (85/5) |

**Table 6.**Summary of typical data characterizing the accuracy and time-efficiency of random forest regressor and neural network models in this study. The term average indicates that values are an average of those for T

_{TC,1}and T

_{TC,2}. The values in the table represent averages obtained using 19 datasets as described in Table 5.

Model Type | Mean Absolute Error (Average)/K | Mean Accuracy (Average)/% | Hyperparameter Optimization Time/min | Training Time (Best Estimator)/s |
---|---|---|---|---|

RFR | 39.21 ± 19.89 | 92.61 ± 7.08 | 0.18 ± 0.34 | 1.19 ± 0.27 |

NN | 14.94 ± 11.95 | 96.55 ± 3.44 | 11.28 ± 36.91 | 17.10 ± 19.50 |

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## Share and Cite

**MDPI and ACS Style**

Schimmel, S.; Tomida, D.; Saito, M.; Bao, Q.; Ishiguro, T.; Honda, Y.; Chichibu, S.; Amano, H.
Boundary Conditions for Simulations of Fluid Flow and Temperature Field during Ammonothermal Crystal Growth—A Machine-Learning Assisted Study of Autoclave Wall Temperature Distribution. *Crystals* **2021**, *11*, 254.
https://doi.org/10.3390/cryst11030254

**AMA Style**

Schimmel S, Tomida D, Saito M, Bao Q, Ishiguro T, Honda Y, Chichibu S, Amano H.
Boundary Conditions for Simulations of Fluid Flow and Temperature Field during Ammonothermal Crystal Growth—A Machine-Learning Assisted Study of Autoclave Wall Temperature Distribution. *Crystals*. 2021; 11(3):254.
https://doi.org/10.3390/cryst11030254

**Chicago/Turabian Style**

Schimmel, Saskia, Daisuke Tomida, Makoto Saito, Quanxi Bao, Toru Ishiguro, Yoshio Honda, Shigefusa Chichibu, and Hiroshi Amano.
2021. "Boundary Conditions for Simulations of Fluid Flow and Temperature Field during Ammonothermal Crystal Growth—A Machine-Learning Assisted Study of Autoclave Wall Temperature Distribution" *Crystals* 11, no. 3: 254.
https://doi.org/10.3390/cryst11030254