Three-dimensional Heat Transfer Problems Research Articles

A fast Bayesian parallel solution framework for large-scale parameter estimation of 3D inverse heat transfer problems

The inverse heat transfer problems (IHTP) have a wide range of applications in the engineering field. Bayesian methods using Markov Chain Monte Carlo (MCMC) have long been considered as a robust and effective method for solving inverse problems. However, the discretization of the problem domain by the spatio-temporal Galerkin skill, i.e., the finite element interpolation also includes the time dimension, making the scale of the unknown parameters extremely difficult for Bayesian calculations. In this paper, a fast Bayesian parallel sampling (FBPS) framework is proposed for large-scale parameter estimation of benchmark three-dimensional inverse heat transfer problems (3D-IHTP). The FBPS we developed achieves a parameter computation scale of 105 magnitude within minutes, through dimensionality reduction of the space-time dependent problem domain. The Hamiltonian Monte Carlo (HMC) sampler, which is proven to be more efficient for high-dimensional parameter estimation, is employed. Through several simulation tests of IHTP, it was confirmed that the solving efficiency of FBPS surpasses that of the traditional Bayesian strategy significantly. Finally, FBPS is successfully developed to estimate the unknown heat flux on the chip heat sink and pack interface effectively, given some simulated high resolution measurement data. The reliability and efficiency show that FBPS has the potential to support efficient prediction techniques for a class of IHTPs in engineering applications.

An inverse problem to estimate simultaneously the heat source strength for multiple integrated circuit chips on a printed circuit board

<abstract> <p>In this three-dimensional steady-state inverse heat transfer problem, we determine the magnitude of the spatially dependent volumetric heat source originating from multiple encapsulated chips mounted on a printed circuit board (PCB). Prior to the estimations, the functional form of the multiple heat sources is treated as unknown, leading to its classification as a function estimation challenge within the realm of inverse problems. The utilization of the conjugate gradient method (CGM) as an optimization tool is rooted in its distinct advantage of not requiring any a priori knowledge regarding the functional form of the unidentified quantities. Furthermore, the CGM empowers the simultaneous correction and estimation of multiple unknowns during each iteration, thereby ensuring the consistent possibility of precise estimates.</p> <p>To affirm the precision of the estimated heat source attributed to multiple chips, a series of numerical experiments were conducted. These experiments encompassed varying inlet air velocities and introduced measurement errors. Notably, the results revealed that meticulous measurements consistently yielded accurate heat generation assessments for the chips, regardless of the prevailing air velocity conditions. The findings underscored that the accuracy of chip heat generation estimates diminished as measurement errors escalated, predominantly due to the ill-posed nature inherent in the inverse problem.</p> </abstract>

Numerical investigation of heat transfer enhancement using (Fe3O4 and Ag-H2O) nanofluids in (converge-diverge) mini-channel heat sinks

The development of electronic devices in the last quarter of the last century brought about a revolution in the electronics industry. Continuous miniaturization and continued operation cause these devices to heat up, which leads to a decrease in their efficiency or damage to their components and that heat can reduce the efficiency of these devices and their life span in addition to the waste of energy, so providing an integrated and efficient cooling system has become an important part of the design of these devices. In this paper, a numerical study was carried out to increase the enhancement in the coefficient of heat transfer of a mini-channel heatsink. The new shape of a mini-channel (converge-diverge) was proposed and compared with the rectangular (straight) channel using pure water and (Fe3o4 and Ag-water) nanofluids as a coolant. In numerical analysis, the Comsol Multiphysics program was used to perform simulation and solved the three-dimensional fluid and heat transfer problem in a mini channel. A CFD module in Comsol Multiphysics is based upon the finite element method with a Galerkin approach to solving the partial differential equations governing. The base temperature, Nusselt Number, friction factor, and thermal resistance have been studied about channel arrangement, with Reynold’s number varieties from (200–1000) and Concentration volume ranges are used (0% – 0.075%). The minichannel heatsink is exposed to a constant heatflux (180 KW/m2) at the bottom. The numerical results indicate that nanofluid and (converge-diverge)mini-channel can increase the heat sink's hydrothermal performance and (Ag- H2O) nanofluid displays superior heat transfer performance than Fe3o4 nanofluid as enhancement in nusselt number reached 43.27% at volume concentration 0.075%.

XIGA: An eXtended IsoGeometric analysis approach for multi-material problems

Multi-material problems often exhibit complex geometries along with physical responses presenting large spatial gradients or discontinuities. In these cases, providing high-quality body-fitted finite element analysis meshes and obtaining accurate solutions remain challenging. Immersed boundary techniques provide elegant solutions for such problems. Enrichment methods alleviate the need for generating conforming analysis grids by capturing discontinuities within mesh elements. Additionally, increased accuracy of physical responses and geometry description can be achieved with higher-order approximation bases. In particular, using B-splines has become popular with the development of IsoGeometric Analysis. In this work, an eXtended IsoGeometric Analysis (XIGA) approach is proposed for multi-material problems. The computational domain geometry is described implicitly by level set functions. A novel generalized Heaviside enrichment strategy is employed to accommodate an arbitrary number of materials without artificially stiffening the physical response. Higher-order B-spline functions are used for both geometry representation and analysis. Boundary and interface conditions are enforced weakly via Nitsche’s method, and a new face-oriented ghost stabilization methodology is used to mitigate numerical instabilities arising from small material integration subdomains. Two- and three-dimensional heat transfer and elasticity problems are solved to validate the approach. Numerical studies provide insight into the ability to handle multiple materials considering sharp-edged and curved interfaces, as well as the impact of higher-order bases and stabilization on the solution accuracy and conditioning.

Boundary Condition Independent Thermal Network Modeling of High-Frequency Power Transformers

With increasing global power demands and the trend of miniaturization of power converters, electronic components are being pushed toward their thermal limits. This has accentuated the need for accurate and low-cost thermal models of electronic components. Lumped Parameter Thermal Network Modeling (LPTNM) is commonly used to evaluate the thermal performance of electronic components, but unlike semiconductors, transformer LPTNM is not standardized in the academic community. Due to this lack of standardization, most proposed transformer thermal models have not been evaluated for Boundary Condition Independence (BCI). This study presents a BCI thermal modeling approach for power transformers based on LPTNM. A thermal network model is developed by discretizing the transformer geometry into smaller volumes and evaluating the thermal resistances by approximating the three-dimensional heat transfer problem as a piecewise one-dimensional model. The network model/geometry discretization is iteratively improved for a set of different boundary conditions, resulting in a BCI model. The modeling approach is experimentally and numerically validated on a PQ40/30 transformer operating in a 3.3 kW switch mode power supply. The BCI transformer thermal model closely matches both numerical and experimental results with a fraction of the computational cost of the numerical model. The proposed BCI modeling approach provides a low-cost alternative to detailed thermal models and can predict transformer thermal performance for varied applications.

Efficient very high-order accurate polyhedral mesh finite volume scheme for 3D conjugate heat transfer problems in curved domains

The conjugate heat transfer problem is found in many engineering applications and comprises the heat transfer between solids and fluids in contact. The discretization method of these problems must guarantee that the interface conditions are properly satisfied, which is an additional difficulty when designing a numerical scheme with very high-order of convergence. The three-dimensional case with arbitrary curved domains is particularly challenging to treat, as the standard approaches to preserve the optimal convergence order become cumbersome with the use of curved meshes and non-linear transformations. Moreover, the implementation aspects of very high-order of convergence methods are critical for the three-dimensional case, as the computational cost of conventional algorithms often grows exponentially with the problem dimension. A very high-order accurate finite volume scheme with general polyhedral meshes is proposed in this work, which is able to solve three-dimensional conjugate heat transfer problems in arbitrary curved domains. The implementation of the proposed method is addressed and optimized calculations are derived to provide the same approximate solution at a significantly reduced computational cost. The provided set of numerical benchmarks, addressing several situations of the conjugate heat transfer problem, allows confirming that the optimal convergence order is effectively achieved. The results obtained also show that the proposed method overcomes the main shortcomings of the standard curved mesh approaches, while preserving the accuracy and the convergence order with the sole use of polyhedral meshes. A computational benchmark proves that substantial performance improvements are obtained from the proposed optimization, which represents a significant advance towards less demanding and more efficient numerical simulations.

Performance evaluation and optimization of flattened microchannel heat sinks for the electronic cooling application

The present study introduces a microchannel heat sink (MCHS) with a new geometry entitled as flattened, which is used to dissipate heat flux form high heat flux generation devices and evaluates its hydrothermal performance. Three-dimensional conjugate heat transfer problem with the assumptions of laminar and steady-state fluid flow has been solved numerically, based on finite volume method. Silicon and pure water with temperature-independent thermophysical properties form the solid part of the heat sink and the coolant, respectively. The performance of three geometric designs of flattened cross-sectional MCHS including a simple single-layer heat sink (Design-A), a heat sink with an adjustable horizontal separation plate (Design-B), and a double-layer MCHS with truncated upper channels (Design-C), is evaluated based on the changes of six effective parameters in the problem including the number of microchannels (N), wall thickness (Ww), thickness of the separation plate (δ), vertical position of the separation plate (H1), velocity ratio (VR), and the length ratio (LR), and under the thermal and hydrodynamic conditions of a uniform heat flux and four different pumping powers. Creating the lowest thermal resistance and the most uniform temperature distribution are the criteria for selecting the optimal designs. The results show that Design-C with the specifications of N = 72, Ww = 56 µm, δ = 25 µm, H1 = 300 µm, VR = 0.7, and LR = 0.7 has the best performance among the all cases by creating the thermal resistance of 0.1333 kW−1, which also indicates a $$7.9\%$$ performance improvement over similar studies.

Mathematical modeling and research of heat and moisture transfer processes in porous media

A mathematical model, numerical algorithm, solution, and results of the computational experiment on a computer are developed to predict the process of heat and moisture transfer in porous media, taking into account such factors as the internal heat and moisture release of a porous natural product on the example of raw cotton and its products of hulling, seeds of various crops. It also considers the effect of temperature and moisture content changes in the environment on the storage and drying of porous materials. In this study, the developed mathematical support of the object under research makes it possible to predict the change in temperature and moisture content at arbitrary points of the porous body and serves to prevent the loss of quality and spontaneous combustion of materials under solar radiation and to analyze and make managerial decisions. Based on the method of coordinate-wise splitting, a numerical algorithm for calculating three-dimensional heat and moisture transfer problems in areas of the parallelepiped type is presented. An implicit second-order difference scheme for calculating the required functions is presented. Based on the numerical calculations performed, it was established that moisture and heat transfer and their exchange with the environment occurs in the upper layers of the cotton pile; in the inner layers, there is an increase in temperature and moisture due to the respiration of raw cotton and its seeds, which depends on the degree of moisture of the raw cotton and their products of hulling.

Hybrid 1D + 2D Modelling for the Assessment of the Heat Transfer in the EU DEMO Water-Cooled Lithium-Lead Manifolds

The European demonstration fusion power reactor (EU DEMO) tokamak will be the first European fusion device to produce electricity and to include a breeding blanket (BB). In the framework of the design of the EU DEMO BB, the analysis of the heat transfer between the inlet and outlet manifold of the coolant is needed, to assess the actual cooling capability of the water entering the cooling channels, as well as the actual coolant outlet temperature from the machine. The complex, fully three-dimensional conjugate heat transfer problem is reduced here with a novel approach to a simpler one, decoupling the longitudinal and transverse scales for the heat transport by developing correlations for a conductive heat-transfer problem. While in the longitudinal direction a standard 1D model for the heat transport by fluid advection is adopted, a set of 2D finite elements analyses are run in the transverse direction, in order to lump the 2D heat conduction effects in suitable correlations. Such correlations are implemented in a 1D finite volume model with the 1D GEneral Tokamak THErmal-hydraulic Model (GETTHEM) code (Politecnico di Torino, Torino, Italy); the proposed approach thus reduces the 3D problem to a 1D one, allowing a parametric evaluation of the heat transfer in the entire blanket with a reduced computational cost. The deviation from nominal inlet and outlet temperature values, for the case of the Water-Cooled Lithium-Lead BB concept, is found to be always below 1.4 K and, in some cases, even to be beneficial. Consequently, the heat transfer among the manifolds at different temperatures can be safely (and conservatively) neglected.

Acceleration of high-order combined compact finite difference scheme for simulating three-dimensional flow and heat transfer problems in GPUs

In this article, the high-order upwinding combined compact difference scheme developed in a three-point grid stencil is applied to solve the incompressible Navier-Stokes (NS) and energy equations in three dimensions. The time integrator with symplectic property is employed to approximate the temporal derivative term in inviscid Euler equation so as to numerically retain the embedded Hamiltions and Casimir to get long-time accurate solutions. For the sake of computational efficiency in solving the three-dimensional NS equations, all the calculations will be accelerated using the hybrid CUDA and OpenAcc GPU programing models. The parallel speedup performance compared to the multicore of an Intel Xeon E5-2690V5 CPU is reported.

A novel method to detect hot spots and estimate strengths of discrete heat sources using liquid crystal thermography

This paper reports the results of an investigation to solve the inverse problem of estimating the strengths of different strip heat sources embedded in a flat plate under laminar steady-state forced convection and by way of this propose a novel method to detect hot spots from remote measurements. A Bayesian framework is adopted to infer the strength of the heat sources from thermochromic liquid crystal (TLC) temperature measurements. This framework consists of the forward model, the measured data, and the inverse model. The forward model simulates the conjugate three-dimensional heat transfer problem with the specified thermophysical properties, and the boundary conditions. The input data for the forward model is a combination of different heat source strengths, and the output is temperature data obtained at the bottom surface of the cork. The input-output data of the numerical simulations are used to build a proxy or surrogate (artificial neural network, ANN) that acts as a replacement for the actual forward model to increase the computational speed and decrease the computational time while solving the inverse problem. Calibrated thermochromic liquid crystal sheets are attached at the bottom surface of the cork for mapping the temperature data, so that the top surface where the convection takes place is undisturbed. In the inverse model, Bayesian statistics, along with the Gibbs sampling algorithm is adopted for analyzing the posterior distribution to estimate the mean, the maximum a posteriori and the standard deviation of the heat source strengths. Validation and robustness of the inverse methodology have been examined. The estimated heat source values are input to the forward model to determine the hot spot temperatures on the flat plate. This is a key spin-off from the present study, wherein based on temperature measurements at a convenient place, the hot spot in a geometry can be remotely ‘estimated.’ A comparison of the simulated and the measured values of the hot spot temperatures is reported for different flow Reynolds numbers.

Performance of Textured Metallized Teflon Coverings with High Diffuse Reflectance Coefficient in Low Earth Orbit on ISS

The data of mathematical modeling of the thermal state of a cutting tool under conditions of laser hardening are presented. The features of the implementation of the method of polyargument systems for solving the three-dimensional heat transfer problem with a localized movable laser heating source are considered. The data of numerical studies to establish patterns of the influence of the radius of the heating spot on the temperature regime of the hardened tool are presented. It is shown that varying this parameter can be used as an effective way of influencing the thermal state of a cutting tool during its hardening by a laser beam. The results of modeling the temperature conditions of the tool in a wide practically important range of changes in its sharpening angle are presented.

Update-grid reanalysis method based on NS-FEM for 3D heat transfer problems

Repeated modifications and iterative recalculations cost a great amount of computation time in the structural design process. Reanalysis is an efficient computational method that can ensure the accuracy of a solution in place of repeated full finite element analysis (FEA) and other numerical methods. In this study, an update-grid (UG) reanalysis (UGR) method is suggested to analyse three-dimensional (3D) heat transfer problems. Compared with other reanalysis methods, the suggested method easily establishes a mapping between initial meshes and modified meshes using the mapping relationship. Generally, a modified structure can be reanalysed by directly reusing transfer operators from the UR analysis. Therefore, the modified meshes can be solved by constructing a map from the initial meshes, even if the model is totally re-meshed. Moreover, considering the accuracy of the analysis, a node-based smoothed finite element method (NS-FEM) is used as the main solver. To evaluate the performance of the suggested method, several heat transfer problems are investigated. The results demonstrate that the UGR method is more accurate and efficient compared with the popular multi-grid (MG) preconditioned conjugate gradient (PCG) method.

NUMERICAL SIMULATION OF CONVECTIVE HEAT TRANSFER AND PRESSURE DROP IN TWO TYPES OF LONGITUDINALLY AND INTERNALLY FINNED TUBES

With two kinds of methods of boundary condition treatment, three-dimensional turbulent flow and heat transfer problems in two types of internally finned tubes with a blocked core-tube have been studied numerically by the realizable k−ε model. The numerical simulation results obtained from two calculation models were compared with experimental data. It was found that the simulation results obtained from the turbulent flow model are closer to the experimental values than those obtained from the laminar flow model. Meanwhile, it has been found that the critical Reynolds number for the flow that develops in internally finned tubes from a laminar flow to a turbulent one is much less than the Reynolds number for traditional bare tubes. The calculation results also indicate that the periodical ridges inside the finned tubes change the distribution of the inner flow field and temperature profile. Unlike straight tubes, in internally finned tubes, a secondary vortex flow emerges that plays a definitely destructive role for the flow boundary layer and increases the turbulent kinetic energy of the flow field. With the field synergy principle, a contrasting analysis of the intensified heat exchange mechanism for internally finned tubes and a bare annular tube was performed quantitatively. The results show that the field synergy degree of longitudinally ridged and internally finned tubes is better than that of bare annular tubes, which enhance heat transfer.

Anisotropic failure of Fourier theory in time-domain thermoreflectance experiments.

The applicability of Fourier's law to heat transfer problems relies on the assumption that heat carriers have mean free paths smaller than important length scales of the temperature profile. This assumption is not generally valid in nanoscale thermal transport problems where spacing between boundaries is small (<1 μm), and temperature gradients vary rapidly in space. Here we study the limits to Fourier theory for analysing three-dimensional heat transfer problems in systems with an interface. We characterize the relationship between the failure of Fourier theory, phonon mean free paths, important length scales of the temperature profile and interfacial-phonon scattering by time-domain thermoreflectance experiments on Si, Si0.99Ge0.01, boron-doped Si and MgO crystals. The failure of Fourier theory causes anisotropic thermal transport. In situations where Fourier theory fails, a simple radiative boundary condition on the heat diffusion equation cannot adequately describe interfacial thermal transport.

Three-Dimensional Thermal Analysis of Three-Phase Enclosed GIS Bus Bars

The ampacity of GIS bus bars is limited by the maximum operation temperature of the contacts. This paper employs the coupled eddy current, fluid and thermal finite-element method (FEM) to solve the three-dimensional (3-D) heat transfer problem in a three-phase enclosed GIS bus bar. The contact resistance is considered and described as a resistor between the conductor and the contact finger. In order to avoid convective boundary condition on the tank surface which is not easy to apply, the ambient air is introduced into the solution region. The temperature dependent thermal properties of SF 6 gas and air are considered. The temperatures calculated by proposed model are found to be in good agreement with the experimental data of a 126 kV three-phase enclosed bus bar prototype.

Numerical simulation of the freezing process in a moist soil

A numerical method is implemented for the solution of the unsteady three-dimensional heat transfer problem in moist soils with consideration of phase transitions in a region of subzero temperatures. The corresponding numerical model takes into account the physical nonlinearity of the soil. For the description of phase transitions in frozen soils, a model that consists of the four characteristic temperature zones where the basic thermodynamic soil parameters vary according to certain laws is used. The numerical simulation is carried out using a finite element method to discretize the boundary value problem in space and a finite difference scheme to discretize it in time. The obtained numerical solutions are compared with the analytical solutions to the Stefan problem.

A formalism for anisotropic heat transfer phenomena: Foundations and Green’s functions

A new complex-variable formalism for the analysis of three-dimensional (3D) steady-state heat transfer problems in homogeneous solids with general anisotropic behaviour is proposed in this paper. The derived method is based on the Radon transform, which is used in order to reduce the dimension of the problem to a two-dimensional (2D) Radon space where a solution can be easily handled via a complex-variable method. Subsequently, the 3D solution is obtained by applying the inverse Radon transform. Despite that the main goal of this work is to develop and illustrate the general methodology, the proposed formalism is further applied to derive new Green’s functions as application examples. Contributions include new forms for bimaterial and half-space Green’s functions for line, point, vortex and dipole heat sources. In particular Green’s functions due to heat vortex loop sources for infinite media, half-space and bimaterial systems are presented for the first time. The veracity and computability of the approach are demonstrated with some numerical examples.

A model-integrated localized collocation meshless method for large scale three-dimensional heat transfer problems

We present a Model Integrated Meshless Solver (MIMS) tailored to solve practical large-scale industrial problems. This is accomplished by developing a robust meshless technique as well as a comprehensive model generation procedure. By closely integrating the model generation process into the overall solution methodology, the presented techniques are able to fully exploit the strengths of the meshless approach to achieve levels of automation, stability, and accuracy currently unseen in the area of engineering analysis. Specifically, MIMS implements a blended meshless solution approach which utilizes a variety of shape functions to obtain a stable and accurate iteration process. This solution approach is then integrated with a newly developed, highly adaptive model generation process which employs a quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques are able to achieve directionally independent, automatic refinement of the underlying model, allowing the method to generate accurate solutions without the need for intermediate human involvement. In addition, by coupling the model generation with the solution process, the presented method is able to address the issue of ill-constructed geometric input such as small features, poorly formed faces, and other such pathologies often generated from solid models in the course of design and in the end to provide an intuitive, yet powerful approach to solving modern engineering analysis problems.

Effect of natural convection on conjugate heat transfer characteristics in liquid minichannel during phase change material melting

This article presents numerical and experimental simulation of three-dimensional conjugate heat transfer problem in mini-scaled thermal storage system. The conjugate problem includes melting process of phase change material in the presence of natural convection during laminar flow of heat transfer fluid through circular minichannel. The paraffin wax is used as a phase change material while the water is used as a heat transfer fluid. The main objective of this study is to investigate the effect of the phase change material natural convection during the melting process on the heat transfer fluid thermal characteristics as well as the impact of the natural convection on the melting performance itself. The thermal characteristics are represented by local Nusselt number ( Nu) and local surface temperature. The melting performance is evaluated by fusion time and liquid fraction profile. Two inlet temperatures and velocities of the heat transfer fluid are adopted to highlight the effect of the natural convection. Combination of the inlet temperatures and velocities of the heat transfer fluid forms four cases: case_1 (at [Formula: see text] = 353 °K, [Formula: see text] = 1 m/s), case_2 (at [Formula: see text] = 453 °K, [Formula: see text] = 1 m/s), case_3 (at [Formula: see text] = 353 °K, [Formula: see text] = 0.1 m/s), and case_4 (at [Formula: see text] = 453 °K, [Formula: see text] = 0.1 m/s). Experimental test rig was constructed to verify the computational results and good agreement between both results was achieved. The study shows that the heat transfer fluid encounters an erratic thermal behavior during the phase change material melting process. For example, the local surface temperature experiences dramatic increase and decrease at certain sections of the channel length. The magnitude of this temperature inconsistency interrelates closely to the strength of natural convection impact, and this can expose the minichannel (which has short length) to severe wall thermal stress. The local Nu experiences improvement in some section of the channel and at the same time it suffers from drastic deterioration in its value particularly at the channel end at which the convection current accommodates. The case with the lowest inlet velocity and the highest inlet temperature has the smallest fusion time at expense of the largest heat transfer fluid bulk temperature gradient before reaching the fusion time. The study is considered as a benchmark and helpful guidelines in the design of small-scaled thermal storage systems of phase change material.