(263f) Deep Neural Network-Based View Factor Modelling of Radiative Heat Transfer between Particle-Particle and Particle-Wall

Introduction

Above 700 [°C] radiative heat transfer is the most dominating heat transfer mechanism. Such high temperature processes occur in a wide range of industrial applications. Examples are: pebble bed reactors, laser sintering and high-temperature particle oxidation or reduction processes [1]. The modelling of heat radiation is an ongoing challenge in many simulation fields, e.g. Computational Fluid Dynamics (CFD) and the Discrete Element Method (DEM).

In this study we investigate radiative heat modelling from a particle perspective. In principle the emitted heat flux Q_i-jbetween two bodies (including walls) depends on the following factors: temperatures T, surfaces A, emissivity Ε and the view factor ε_i-j between these objects, as well as the Boltzmann constant σ_S. The view factor is the ratio of the radiation leaving a surface i and the radiation that is striking surface j.

The analytical calculation or numerical integration of view factors cannot be performed for real-sized systems, that feature billions of particles, due to the astronomical computational cost. Methods to overcome this limitation have been presented in literature, based on the Monte Carlo algorithm by Walker et al. [2], or the Projection Method based on the projection of a defined number of test points that are distributed on particle surfaces by Forgber & Radl [3]. Both Methods reflect the widespread tradeoff in simulation science between accuracy and speed. The Monte Carlo raytracing achieves very accurate view factors but takes up high computational cost. The projection method provides precise view factors in specified systems at significantly less computational effort, although the demand is still too big to adopt the method in general for heat radiation modelling in DEM simulations.

Methods

To overcome this tradeoff in the presented approach we use Machine Learning (ML) techniques to create a pre-trained heat radiation model based on a deep neural net (DNN), that predicts accurate view factors at high speed. The training data for the DNN is generated with the Monte Carlo raytracing method from randomly-positioned particle beds, considering different particle volume fractions as well as the wall-view factors.

The neural network model parameters are formed by the hyperparameters, e.g. the learning rate, and the model design that consists of number of layers, number of nodes, activation function, optimizer and so on. The input layer of a DNN is defined by the features of the training data that are often called markers. A neural net with the surface distance cannot model the widely distributed view factors in a particle bed. Therefore, additional markers like the solid angle between interacting particles and derivations of a local particle fraction are needed. These markers are then consequently analyzed by a feature analysis. To achieve the best possible generalization with the DNN-based model these model parameters are optimized by the randomized search approach [4].

The outcome of the presented approach will be compared to the Projection Method and a simple regression analysis in terms of accuracy and computational cost. The overall quality of prediction is measured by the mean squared error (MSE). Because the MSE is significantly more influenced by big view factors and does not reflect the overall quality of the prediction, the calculated view factors are correlated to the view factors determined via Monte Carlo raytracing. The coefficient of determination (R²) is then used to describe this correlation. Another quantitative measure is the coefficient of determination (R²_spread(x_u,x_o)) of particle distances within an upper boundary x_o, that is the half domain size plus one particle diameter, and a lower boundary x_u, defined as the half domain size minus one particle diameter.

The investigated particle bed contained 587 particles and the total dataset therefore consists of 586*586 view factors values. The total dataset was divided into the training dataset, the test dataset and the validation dataset with the respective relative amounts of 0.8, 0.1 and 0.1. The top and the bottom walls of the particle bed were used for the creation of the particle-wall dataset. The particle positions were randomly changed to gain more data. 33 different particle bed configurations were used. The total particle-wall dataset then consists of 66*587 view factor values. The split ratios were the same as for particle-particle interactions.

Results

A simple linear regressor, which is derived from the training dataset, achieves an MSE of 4.3e-7 on the test dataset at a basically instantaneous speed. The R²-value of the correlation is 0.981. For large surface distances the simple regressor cannot accurately predict the spread of the view factors. This results in an R²_spread-value of -0.336. For particle-wall interaction a simple linear regressor achieved an MSE of 2.5e-6, an R²_wall-value of 0.901 and R²_spread,wall is -2.133.

The Projection Method (number of test points = 200) achieves an MSE of 7.7e-7 on the test dataset. The overall coefficient of determination is 0.967, and R²_spread is -2.530. Overall the computational demand of the Projection Method is too high, to be adopted in general to DEM simulations.

The DNN-based model achieves an MSE of around 2.5e-7 while taking around 0.30 [s] to calculate the view factors and 0.55 [s] to load the pretrained model. The R²-value for the total correlation with the test dataset is 0.989. The R²_spread-value is 0.424.

Particle-wall view factors of a particle bed show a different data structure than particle-particle interactions. More big view factors occur since the nearest particle bed layer is irradiated without any obstacles. Therefore, a separate DNN-based model for particle-wall interactions was trained. The MSE for wall interactions is then 7.9e-8, R²_wall is 0.937 and R²_spread,wall is 0.491.

Conclusion

For particle-particle interaction it is demonstrated that a pretrained DNN can model view factors for any distance at significantly less computational effort, higher overall accuracy and reasonable prediction quality at certain distances.

The same conclusion can be made for particle-wall interactions. Thus, a combination of two DNN-based models, one for particle-particle interactions and one for particle-wall interactions, is recommended for general heat radiation modelling in DEM simulations.

References

[1] M.F. Modest, Radiative Heat Transfer - Second Edition, Academic Press, 2003. https://doi.org/10.1017/CBO9781107415324.004.

[2] T. Walker, S.C. Xue, G.W. Barton, Numerical determination of radiative view factors using ray tracing, J. Heat Transfer. 132 (2010) 1–6. https://doi.org/10.1115/1.4000974.

[3] T. Forgber, S. Radl, A novel approach to calculate radiative thermal exchange in coupled particle simulations, Powder Technol. 323 (2018) 24–44. https://doi.org/10.1016

/j.powtec.2017.09.014.

[4] J. Bergstra, Y. Bengio, Random search for hyper-parameter optimization, J. Mach. Learn. Res. 13 (2012) 281–305.