Calculation Of View Factors Research Articles

Contour integration for the exchange of radiant energy between diffuse rectangular geometries

View factor calculation is a key point in radiation heat transfer between surfaces. In thermal engineering it is required to compute the view factor between complex geometries, which include several parallel and inclined surfaces. Currently, no analytical expressions are available for this purpose, so the use of Summation Rules (SR) is required, however, when analyzing complex geometries, the handling of SR is extremely laborious. Therefore, this work aims to develop an equation for the calculation of the view factor in the radiative heat transfer of two arbitrary rectangular inclined or parallel surfaces, using the advantages provided by vector calculus. The exact solutions obtained with contour integration were compared with the values calculated via the Algebra Factor (AF) using the Hottel and Hamilton-Morgan Solution and with the results computed from numerical integration of the quadruple integral using Simpson's rule 1/3 with five intervals. The lack of similar precedents in the technical literature reinforces the practical and scientific nature of the proposed method of analysis.

A hybrid Monte-Carlo method applied to the calculation of view factors in cylindrical shells

A hybrid Monte-Carlo method (MC_H) is extended to calculate view factors for a system. The test case considered is a cylindrical shell. The MC_H has many advantages over the standard approach of using ray tracing (MC_RT); it is very efficient, view factor algebra can be used to reduce the number of view factors the MC_H is applied to, and the allocation of function evaluations for each view factor can be changed to account for the distance between surfaces and the surface areas. For the view factor system considered a typical speed-up of 48 compared to MC_RT is achieved, and for a hybrid quasi-Newton-method the speed-up is 480.

Contour integration for the view factor calculation between two rectangular surfaces

AbstractThe calculation of the view factor is an important consideration in radiative heat transfer analysis. Thermal engineering requires the view factor calculations between complex geometries containing parallel and inclined surfaces. There is currently no analytic formula available for this, so the summation rule (SR) must be used. However, when complex geometries are analyzed, processing SR becomes a very intensive task. Therefore, the present study aims to exploit vector calculus to develop a formula for calculating the view factors for radiative heat transfer from any two rectangular oblique or parallel surfaces. The exact solution obtained by means of the contour integration was compared with the numerical integration of the quadruple integral using the multiple Simpson's rule 1/3 with five intervals (MSR) and the summation rule via the Hamilton–Morgan solution (HMS). The lack of similar precedents in the literature reinforces the scientific and practical nature of the proposed method.

A finite element approach for nonlinear, transient heat conduction problems with convection, radiation or contact boundary conditions

Heat conduction problems with convection and/or radiation boundary conditions arise in nuclear and other engineering fields. This paper presents a finite element approach to solve transient, three-dimensional (3-D), nonlinear heat conduction problems using quadratic tetrahedral elements with area/volume coordinates and quadrature integrations. Boundary conditions modeled include surface convection, to-ambient radiation, and surface-to-surface radiation (using the radiosity matrix method, RMM). Instead of the commonly used hemi-cube method, L′Huilier’s theorem is used here to calculate view factors. Contact constraints are also imposed. Interface condition is set up for the interfaces of different material. The one- and two-step backward differentiation are used for time integration. Nonlinearities introduced by temperature-dependent material thermal properties and the Stefan–Boltzmann law for radiation are handled using the Picard and/or Newton–Raphson methods. The convergence rate of the numerical scheme is verified using an analytical solution obtained using the method of manufactured solutions (MMS). Several practical cases are simulated, including that of two spheres in contact with each other, and results are compared to those obtained using commercial codes.

Analytical method for calculating internal stray radiation from an IR spectrometer based on the view factor.

Optomechanical components such as the lens barrels and frames of IR spectrometers produce strong internal stray radiation, which reduces the instrument's SNR and dynamic range. An IR internal stray radiation calculation method based on an analytical model of the view factor is proposed. The mathematical model of the view factor calculation method of typical optomechanical components is established. For any IR optical systems, the internal stray radiation can be quickly and accurately calculated by adjusting the coordinate systems in the calculation method. Based on the proposed method, the internal stray radiation of a double-pass long-wave IR spectrometer was calculated. The calculation results are consistent with the simulation results. The RMS value of the relative error between the calculated value and the simulated value is around 11%. To verify the proposed method, an experiment was conducted to test the internal stray radiation of the long-wave IR spectrometer. The internal stray radiation test results agree with the calculated and simulated results, and the relative error between the test results and the calculation results is within 9%.

View factor computation and radiation energy analysis in baking oven with obstructions: Analytical and numerical method

AbstractIn a baking oven, radiation heat transfer is the most prevalent mode of heat exchange. In the present investigation, a new baking oven was designed with two hemi cylindrical domes (called reflectors) attached to the heating coils at the top and bottom of the baking oven. Firstly, view factors were computed by finite element and analytical methods with different bread dimensions, as well as with and without the reflectors. The network representative method and S2S method were employed to analyze the radiation energy transfer in the oven at different baking temperatures using the same bread sizes. The results obtained from numerical and analytical methods were fairly similar. The radiative heat energy absorbed by bread surface was higher with the reflector than without the reflector in the oven. As the number of obstructions increased, the radiative heat flux was reduced. The inclusion of reflector shield in oven affect the heat lux on the bread. The radiative heat flux received by bread was found to have increased on increasing the oven temperatures. Moreover, quality parameter such as the color of the bread with and without the reflectors were also examined. The oven reflector had a noticeable effect on the color of the bread (p < 0.05).Practical applicationsIn recent years, reducing energy consumption has become more important in the design of ovens. Although domestic electric ovens are a convenient way to bake quality goods, they have a very significant energy loss. Prior research has focused on CFD modeling for commercial ovens to assess the thermal behavior of the system and overall energy efficiency of baking. Baking ovens operate at higher temperatures, hence, heat transfer by radiation is highly essential. In reality, one or more surfaces in oven are obstructing the surfaces. To calculate view factors and achieve accurate radiative heat transfer in such cases, the shadow effect must be taken into account. To solve them precisely in this situation, numerical techniques are highly beneficial.

“An enhanced and optimized Monte Carlo method to calculate view factors in packed beds”

Comprehensive radiation studies on packed beds require the estimation of view factors between the surfaces, which often computationally expensive. This paper proposes a novel method based on Monte Carlo Ray Tracing for high-precision and low time computation processing of view factors in random assemblies of cylindrical packed beds. The Monte Carlo Ray Tracing method is improved by coupling it with Kowsary’s tangent sphere method under the parallel computation processing through Compute Unified Device Architecture, showing a reduction of the computational time in approximately 153 times less than a traditional Monte Carlo Ray Tracing method on the hardware used and obtaining a maximum relative error of 0.39% for configurations evaluated in the literature. Furthermore, a calculation methodology for view factors between particle–particle, particle–wall, and particle-lid under an original layer concept, is presented and applied on a set of randomly assembled monosized packed beds generated with the LIGGGHTS software, covering an average porosity range from 0.38 to 0.50 for the arrays. Finally, detailed analysis and discussion of the results are performed, allowing to find characteristic view factors for the particles according to their positions and correlations for view factors particle–wall and particle-lid which is defined as layer view factor method, considering error intervals for each interaction respectively, defining a view factor calculation methodology for packed beds within the studied porosity range.

Evaluating View Factors Using a Hybrid Monte-Carlo Method

AbstractThis paper demonstrates that the well-known method for calculating view factors, the Monte Carlo method, combined with ray tracing is not necessarily the most efficient strategy. The Monte Carlo method and quasi-Monte Carlo method combined with numerical integration, provided the surfaces in a configuration are not too close together, are more accurate for the same run-time than a ray tracing-based Monte Carlo method. The Monte Carlo method based on numerical integration is complementary to the Monte Carlo method based on ray tracing. When many rays are required to calculate an accurate view factor, few function evaluations in a numerical integration approach are necessary to achieve the same accuracy. Where the surfaces in a configuration are touching, the Monte Carlo method with numerical integration converges to the exact view factor very slowly due to a singularity in the view factor multi-integral. For these configurations, a hybrid Monte Carlo method and quasi-Monte Carlo method are demonstrated to be the stochastic methods of choice.

Radiation modelling of arbitrary two-dimensional surfaces using the surface-to-surface approach extended with a blocking algorithm

Calculating the radiation exchange between arbitrary surfaces using view factors is a common practice in the fields of industrial furnaces, climate modelling, solar power, thermal building design and so on. Here, a new model was developed to calculate the view factors of arbitrary two-dimensional geometries. This model is based on Hottel’s crossed strings method paired with an optimized algorithm to efficiently detect shadowing effects between the surfaces. The model’s accuracy and discretization dependency was tested against an analytical view factor calculation method using an example of two concentric, infinitely extended cylinders. The model can be applied to arbitrary linear discretized two-dimensional geometries. In particular, it guarantees the highest possible accuracy within the chosen discretization. The need for developing customized view factor equations of different two-dimensional geometries is therefore no longer necessary, since the convergence of the model with decreasing mesh size on analytical results can be demonstrated. Additionally, the performance and validity of the newly developed shadowing algorithm was tested against a common brute force approach and significant speedups were achieved. Furthermore, the additional application of the net radiation method for the calculation of the heat fluxes exchanged by radiation within those geometries is shown.

Calculation of View Factors for Building Simulations with an Open-Source Raytracing Tool

Longwave radiative heat transfer is a key determinant of energy consumption in buildings and view factor calculations are therefore required for the detailed simulation of heat transfer between buildings and their environment as well as for heat exchange within rooms. Typically, these calculations are either derived through analytical means or performed as a part of the simulation process. This paper describes the methodology for employing RADIANCE, a command-line open-source raytracing software, for performing view factor calculations. Since it was introduced in the late-1980s, RADIANCE has been almost exclusively employed as the back-end engine for lighting simulations. We discuss the theoretical basis for calculating view factors through Monte Carlo calculations with RADIANCE and propose a corresponding workflow. The results generated through RADIANCE are validated by comparing them with analytical solutions. The fundamental methodology proposed in this paper can be scaled up to calculate view factors for more complex, practical scenarios. Furthermore, the portability, multi-processing functionality and cross-platform compatibility offered by RADIANCE can also be employed in the calculation of view factors.

Simpler view factor calculations for mutually perpendicular rectangles

A foolproof formula and a simple-to-use approach are presented for the analytical calculation of view factor between two rectangles in mutually perpendicular planes. At present, the options available are: (a) a direct formula that is difficult to work with in general and unusable for particular cases; or (b) case-specific formulas that arise out of complicated view factor algebra manipulations. Tasked with such a view factor calculation and without case-specific view factor relations readily available, a user who is unable to derive them would be in a predicament. The proposed formula and the general calculation approach to be followed address these limitations. They are simple to adopt and the required calculation steps can be intuitively visualized by the user. As an added advantage, the proposed formula can be extended to derive other view factor relations for strips and rectangles in perpendicular planes. By all indications, some of these relations are unavailable in the literature at present.

The analysis of urban Sky View Factor calculation based on geometric form

Previous research has shown that there is a clear relationship between the urban morphology on one hand and the local solar irradiance and air temperature on the other hand. In order to describe the urban climatology and its spatial variations the sky view factor (SVF) plays a key role. This fraction of visible sky provides an indication of the street geometry and building density. Adding SVF as an evaluation element in urban planning be used to coordinate the spatial layout of the city, improve the built environment, and reasonably control the height and density of construction projects, which can effectively promote the refinement of urban planning and the construction of green cities.

An Enhanced and Optimized Monte Carlo Method to Calculate View Factors in Packed Beds

An Enhanced and Optimized Monte Carlo Method to Calculate View Factors in Packed Beds

A generic flame shape model and analytical models for geometric view factor calculation on the fire exposure surface

Fires have attracted special attention in recent years due to catastrophic accidents. Flame radiation is the dominant heat transfer mode in fires and it plays an important role in the structure damage. Therefore, for accurate theoretical assessment, it is critical to calculate the distribution of radiant heat flux on the fire exposure surface, which is closely related to the geometric view factor. Based on the previous experiment observations, a ‘back-to-back conical frustum’ is introduced to describe the visible flame shape of gas fire in cross wind. The calculation of geometric view factor on the ground level, a typical fire exposure surface, is deduced in detail. The expressions of horizontal view factor, vertical view factor and maximum view factor and the applicable scopes are given. For the same gas flow rate with different cross air velocities, the visible flame envelopes are described by the proposed shape model and traditional tilted cylinder model. By comparing the view factor predictions from the two models, the accuracy and applicability of the proposed flame shape model are discussed. The proposed model is validated using measurements from a reduced-scale tunnel fire experiment and a large-scale jet fire experiment from literature. The predicted radiant heat flux using the proposed flame shape model is in good agreement with the measured value, with an approximate 20% discrepancy.

A data-driven deep learning model of radiative heat transfer in dense granular systems

Radiative heat transfer has attracted increasing research interest in various dense granular systems, including large-scale nuclear pebble-bed reactors, fusion breeder blankets, packed beds, metal powder beds and granular lunar regolith simulants. The optimised ray tracing method (ORTM) with a Sobol quasi-random sequence is proposed to calculate the obstructed view factor between spheres, and it converges much faster than that with the traditional pseudorandom number sampling method. Based on discrete particle packing of nuclear pebble bed HTR-PM and random particle generation, a large obstructed view factor dataset of thermal radiation is generated by parallel CUDA computing for training a deep learning regression model to discuss the packing-property linkage for radiative heat transfer in the large-scale particle bed. Compared with other approaches, the deep residual neural network (ResNet) with the swish activation function applied in this paper is found to be the best neural network architecture for view factor prediction without vanishing gradients or model degradation problems during training. The testing error of the trained deep learning model is <1.0 × 10-4 for over 99.96% view-factor cases in nuclear pebble beds, and it has proven to be 7.3 × 105 times faster than the traditional numerical method in view-factor calculations. This makes it feasible to explore the local and macroscopic thermal radiation behaviours in large-scale pebble beds filled with up to 1.0 × 107 spheres.

Comparative Study on the Numerical Methods for View Factor Computation for Packed Pebble Beds: Back Propagation Neural Network Methods Versus Monte Carlo Methods

Abstract It's an unsolved problem to calculate the thermal radiation view factors among fuel pebbles as accurately and quickly as possible in the simulation of the temperature fields within the pebble-bed. In this study, a series of fully connected neural networks (FCNs) has been developed to realize the fast calculation of view factors. In order to verify the accuracy and effects of the networks, the neural networks are compared with the Monte Carlo (MC) algorithm. The results show that, in most cases, the relative errors of the FCN method can be controlled within 1.0%, and the prediction accurate probability is up to 99%. In comparisons of specific examples, the temperature errors of the FCN method and the MC method are less than 1 K within the range neural networks have covered. In addition, the time of neural networks for a single calculation is about 2–20 μs, which is even less than 10−4 of the time taken by the MC algorithm. In conclusion, neural networks can greatly improve computational efficiency while keeping the same accuracy as the MC algorithm, which makes real-time simulation of the temperature fields possible.

Simulation of a CubeSat with internal heat transfer using Finite Volume Method

Estimating temperature is essential both to design reliable CubeSats and to keep it under maximum operating efficiency. This paper presents the transient thermal simulation of a CubeSat 1U, where the heat transfer by conduction and radiation (external and internal) are solved considering a typical CubeSat mission launched from the International Space Station. The objective is to obtain the temperature field based on the Finite Volume Method (FVM), with inner heat transfer by radiation. The Gebhart method computes the internal heat exchange through successive reflections, and an obstruction model supports the calculation of view factors. Three boundary conditions for the inner side of the CubeSat are tested: without internal heat transfer by radiation, or zero emissivity (ϵ=0.0), with intermediate internal heat transfer by radiation (ϵ=0.5), and maximum internal heat transfer by radiation (ϵ=1.0). The results show a significant impact of the internal heat transfer by radiation in the temperature field of both inner and outer parts of the satellite, and therefore it should not be ignored. Good agreement of transient temperature is found between the FVM and another more straightforward formulation based on the Lumped Method, although the three-dimensional effects are significant and cannot be obtained with such a simplified model.

High Energy Rapid Modular Ensemble of Satellites Payload Thermal Analysis Using OpenFOAM

In a space mission design, the goal of the thermal control subsystem is to ensure that all components of the satellite stay within their operating temperature ranges. The problem becomes more critical if the mission involves nanosatellites with an astrophysics payload requiring a dedicated thermal design to keep it at low temperature, as in the case of the High Energy Rapid Modular Ensemble of Satellites mission. Therefore, a thermal analysis is performed using ESATAN, one of the most-used software packages among those suggested by the European Cooperation for Space Standardization. This gives a good overview of the temperature fields, but many geometrical simplifications must be introduced in the model and all thermal interfaces must be checked. In order to assess the effect of the latter, the ESATAN results are cross checked with those obtained using another software, OpenFOAM, in which the geometry is created directly from the computer-aided-design model of the satellite, resulting in a very accurate geometrical representation. By first comparing the view factor calculations against analytical test cases, some issues emerged. These are thoroughly analyzed and discussed, and it is finally shown that they do not affect the temperature field. The high level of detail of OpenFOAM highlighted interesting possible improvements in the thermal configuration of the satellite.

An Enhanced Monte Carlo Method to Calculate View Factors in Packed Beds and its Application Using Distance Approximation

An Enhanced Monte Carlo Method to Calculate View Factors in Packed Beds and its Application Using Distance Approximation

Accuracy of view factor calculations for digital terrain models of comets and asteroids

Context. Detailed shape and topographic models coupled with sophisticated thermal physics are critical elements to proper characterization of surfaces of small bodies in our solar system. Calculations of self-heating effects are especially important in the context of thermal evolution of non-convex surfaces, including craters, cracks, or openings between “rocks”. Aims. Our aim is to provide quantitative comparisons of multiple numerical methods for computing view factors for concave geometries and provide a more rigorous criteria for the validity of their application. Methods. We contrasted five methods of estimating the view factors. First, we studied specific geometries, including shared-edge facets for a reduced two-facet problem. Then, we applied these methods to the shape model of 67P/Churyumov-Gerasimenko. Nevertheless, the presented results are general and could be extended to shape models of other bodies as well. Results. The close loop transformation of the double area integration method for evaluating view factors of nearby or shared-edge facets is the most accurate, although computationally expensive. Two methods of facet subdivision we evaluate in this work provide reasonably accurate results for modest facet subdivision numbers, however, may result in a degraded performance for specific facet geometries. Increasing the number of subdivisions improves their accuracy, but also increases their computational burden. In practical applications, a trade-off between accuracy and computational speed has to be found, therefore, we propose a combined method based on a simple metric that incorporates a conditional application of various methods and an adaptive number of subdivisions. In our study case of a pit on 67P/CG, this method can reach average accuracy of 2–3% while being about an order of magnitude faster than the (most accurate) line integral method.