Steady-state Heat Transfer Problems Research Articles

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>

Heat transfer comparison investigation of the permanent magnet synchronous motor for electric vehicles based on the BEM and the FEM

In the field of heat transfer in permanent magnet synchronous motors (PMSM) for electric vehicles, the boundary element method (BEM) has been applied for the first time to calculate the steady-state temperature of the PMSM with a spiral water-cooled system. In this investigation, the boundary-integration equation for the steady-state heat transfer problem of a water-cooled PMSM is first derived on the basis of thermodynamic theory, and the system of constant coefficient differential equations is obtained by discretizing its boundaries, while the temperature results obtained from the BEM are compared with the finite element method (FEM) results. Furthermore, the temperature distribution and heat transfer characteristics obtained from the FEM and BEM were verified twice using the PMSM prototype and test platform. The results show that the maximum relative error between the temperature calculation results of FEM and BEM is 1.97%, and the maximum relative error between the results of BEM and the test does not exceed 3%, which finally verifies the validity and accuracy of BEM in solving the heat transfer problems of water-cooled PMSM.

THERMAL ANALYSIS OF BI-DIRECTIONAL FUNCTIONALLY GRADED PLATES SUBJECTED TO HEAT GENERATION

The two-dimensional steady-state heat transfer problem is analyzed under the most general boundary conditions in a heterogeneous environment. It is assumed that the internal heat generation varies arbitrarily in two directions, while the thermal conductivity coefficient of the material is bi-directly (thickness-length) graded with the Mori-Tanaka model. The linear inhomogeneous partial differential equations produced under these conditions may not be solved by conventional methods. The governing equation of the system is numerically solved with high accuracy using the 2D pseudospectral Chebyshev method. Benchmark problems are used to check the accuracy of the method. It has been observed that the effect of volumetric heat generation on the temperature distribution is more than the heat transmission coefficient. Besides, this method is simple and effective and can be easily applied to such problems.

A New Nonlinear One-Dimensional Shape Function applied to Simulate a Steady State Heat Transfer Problem

Generally linear shape functions are considered for two noded elements as coefficients of higher order polynomial cannot be determined from two nodes. In the present work, attempts have been made to fit a non-linear trigonometric shape function for one dimensional two noded elements. As in case of nonlinearity, the dependent variable varies nonlinearly with the independent one, consideration of linear shape function may lead to erroneous results. Here, a one dimensional heat transfer problem is solved analytically, with the help of linear shape functions and with the help of newly developed nonlinear shape functions. Analysis shows that results obtained by considering nonlinear shape functions have quite good match with that by linear shape functions.

Determination of heat transfer representative element volume and three-dimensional thermal conductivity tensor of fractured rock masses

The natural fracture system exerts a significant influence on the physical and mechanical properties of rock masses. The heat transfer in fractured rock masses is much more complicated than that of intact rock due to the thermal contact resistance induced by natural fractures with varying size, aperture, roughness, and orientation. This work describes a method for determining the representative element volume (REV) of heat transfer and the 3D thermal conductivity tensor of fractured rock masses. The method is based on the element-wise averaging of temperature and heat flux, which can be obtained from a finite element solution to a steady state heat transfer problem with specific boundary conditions. A case study was conducted based on the geological data of a potential area for high level radioactive waste disposal in China. The results indicate that the size of heat transfer REV of the rock masses in the study area is approximately 18 m × 18 m × 18 m. 3D thermal conductivity tensor, the magnitudes and directions of the three principal thermal conductivities, and anisotropy of the fractured rock masses were obtained. Influences of main fracture parameters including density, disc diameter, and thermal contact resistance on the heat transfer characteristics were investigated. The research results in this work can provide a better understanding of the discontinuous heat transfer processes in fractured rock masses and their homogenization methods.

Semi-analytical solution to steady-state and transient heat transfer problem with variable conductivity properties of the domain

Introduction. Present paper studies the case of heat transfer problem where the conductivity property of the domain is not a constant value but varies depending on its temperature. In particular, a conductivity that linearly dependent onthe temperature is reviewed. A semi-analytical formulation is developed and proposed to solve steady-state and transient types of these kinds of heat transfer problems.
 Materials and methods. The domain is mathematically modeled by using finite element method where non-linearity property of the problem has been incorporated. For time-dependent (transient) case, the temporal solution has been achieved by analytical methods wherefore all necessary formulation has been derived. Non-linear equations have been solved by both Newton and Picard methods.
 Results. In order to proof check, the proposed calculations a wall with three layers has been calculated by using the proposed transient problem formulation. The derivative boundary conditions at the faces of the wall are given as the functions of time. Non-linearity has been solved by two different methods that gave the identical results. Having solved the non-linearity by these two methods, allowed to compare the efficiency of Newton method versus Picard method. Both methods reached the solution with the same number of iterations. The paper proposes the algorithms for solving the problems. The authors have used MatLab environment to implement those algorithms.
 Conclusions. Proposed formulation solves coupled heat transfer problems in multi-layered exterior walls. Multi-layered calculation ability allows to cover all non-homogeneous cases of the computation domain. The formulation solves the heat loss problems through the multi-layered walls with due and reliable accuracy.

ROLL-BOND HEAT EXCHANGER (RHE) DESIGN AND APPLICATION TO A CHOCOLATE COOLING TUNNEL - A NUMERICAL STUDY

The snack sector has a significant market today. It is expected that the snacks will look as promised on the packaging and that the product will have no deformations or defects. The entire area of liquid chocolate- covered snack products entering the chocolate cooling tunnel must be cooled homogeneously. If all surfaces of the chocolate are not sufficiently cooled, a problem of sticking to the conveyor belt can occur. This study investigates the performance of roll-bond heat exchangers under different operating conditions to solve the snack sector’s technical problems. For this purpose, a CAD model of a roll-bond heat exchanger is created. A steady-state heat transfer problem is established in the SolidWorks Flow Simulation environment, and the problem is solve numerically. The maximum, minimum, and average values of the fluid temperature, pressure difference, fluid velocity, and solid temperature distributions are calculated for different inlet velocities and temperatures. In addition, the standard deviations of the solid temperature values are calculated and evaluated. The results show that the roll-bond heat exchanger applies to the chocolate cooling tunnel and is much more efficient for heat transfer than the conventional design.

Analysis of Two-Dimensional Heat Transfer Problem Using the Boundary Integral Equation

In this paper, we examine the problem of two-dimensional heat equations with certain initial and boundary conditions being considered. In a two-dimensional heat transport problem, the boundary integral equation technique was applied. The problem is expressed by an integral equation using the fundamental solution in Green’s identity. In this study, we transform the boundary value problem for the steady-state heat transfer problem into a boundary integral equation and drive the solution of the two-dimensional heat transfer problem using the boundary integral equation for the mixed boundary value problem by using Green’s identity and fundamental solution.

An implicit high-order radial basis function-based differential quadrature-finite volume method on unstructured grids to simulate incompressible flows with heat transfer

A high-order implicit radial basis function-based differential quadrature-finite volume (IRBFDQ-FV) method is presented in this work to efficiently simulate incompressible flows with heat transfer on unstructured mesh. The velocity and temperature fields are solved by locally using the lattice Boltzmann flux solver and the high-order finite volume method. Specifically, the proposed highly accurate finite volume method utilizes a high-order Taylor polynomial to approximate the solution within every control cell. Spatial derivatives are the corresponding coefficients in the polynomial, and they are approximated by the meshless radial basis function-based differential quadrature (RBFDQ) method. The diffusive and convective fluxes at each cell interface are simultaneously evaluated through local reconstruction of lattice Boltzmann solution using D2Q9 lattice velocity model. To efficiently calculate the solution with high-order accuracy, an implicit time-marching method incorporating the lower-upper symmetric Gauss-Seidel (LU-SGS) and the explicit first stage, singly-diagonally implicit Runge-Kutta (ESDIRK) approaches is devised. The proposed method is comprehensively validated by a series of numerical experiments containing both steady-state and time-dependent heat transfer problems with/without curved boundaries at a wide variety of Rayleigh numbers and Grashof numbers. The obtained results demonstrate a high degree of accuracy and reliability of the proposed method for complex flows on unstructured mesh. In comparison with the classical second-order method, the proposed high-order method has better computational efficiency when comparable results are achieved.

The application of FEM-BEM coupling method for steady 2D heat transfer problems with multi-scale structure

This paper presents a finite element method and boundary element method (FEM-BEM) coupling scheme to study the steady-state 2D heat transfer problem for structures with multi-scale features. Based on the geometric information or material properties, the considered model is divided into FE sub-domain and BE sub-domain. The closed BE sub-domain in the mixed FEM-BEM model is defined as a finite super-element for which the effective stiffness matrix and effective fluxes are obtained by the self written BE code and hence can be assembled to the FE systems by the user subroutine (UEL) provided in ABAQUS. This method offers a number of key improvements compared with current analysis methods available for multi-scale structures. These improvements are: (i) the flexibility and convenience by using the powerful pre/post-processing in ABAQUS; (ii) the higher accuracy of the solution over the classical FEM; and (iii) the computational cost and time can be highly reduced by using the coupling method. Numerical results are compared with reference solutions and demonstrate the accuracy and effectiveness of the proposed approach.

Predicting subdifferential switching surface in a steady-state complex heat transfer problem using deep learning

Predicting subdifferential switching surface in a steady-state complex heat transfer problem using deep learning

A Linear Approach Suitable for a Class of Steady-State Heat Transfer Problems with Temperature-Dependent Thermal Conductivity

This work presents an useful tool for constructing the solution of steady-state heat transfer problems, with temperature-dependent thermal conductivity, by means of the solution of Poisson equations. Specifically, it will be presented a procedure for constructing the solution of a nonlinear second-order partial differential equation, subjected to Robin boundary conditions, by means of a sequence whose elements are obtained from the solution of very simple linear partial differential equations, also subjected to Robin boundary conditions. In addition, an a priori upper bound estimate for the solution is presented too. Some examples, involving temperature-dependent thermal conductivity, are presented, illustrating the use of numerical approximations.

Higher Order Multiscale Finite Element Method for Heat Transfer Modeling.

In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

Heat transfer from a particle in laminar flows of a variable thermal conductivity fluid

We revisit the classical problem of steady-state heat transfer from a single particle in a uniform laminar flow with the assumption that the thermal conductivity of the fluid changes linearly with the temperature. We use a combination of asymptotic and scaling analyses to derive approximate expressions for the dimensionless heat transfer coefficient, i.e., the Nusselt number Nu, of arbitrarily shaped particles. The results cover the entire range of the Peclet number Pe. We find that, for a constant temperature boundary condition and fixed geometry, the Nusselt number is essentially equal to the product of two terms, one of which is only a function of Pe while the other one is nearly independent of Pe and mainly depends on the proportionality constant of the conductivity-temperature relation. We also show that, in contrast, when a uniform heat flux is imposed on the surface of the particle, Nu can be estimated as a summation of a Pe-dependent piece and one that solely varies with the proportionality constant.

A Boundary Element Method for Steady-State Heat Transfer Problems Based on a Novel Type of Interpolation Elements

To improve the computational accuracy of the boundary element method (BEM) for solving the steady-state heat problems, a new method was studied with a new element interpolation method (called expansion element interpolation method). The novel expansion interpolation element was obtained through configuration of virtual nodes at the boundary of the traditional discontinuous element, to transform the original discontinuous element into a higher-order continuous element. The interpolation function constructed with the virtual node and the internal source node can accurately work in the continuous and discontinuous physical fields on the boundary, and the interpolation accuracy increases by 2 orders compared with that of the original discontinuous element. In addition, the boundary integral equation is only valid at the internal source nodes of the traditional discontinuous element and contains only the degrees of freedom of the source nodes, while the degrees of freedom of the virtual node can be eliminated through the relationship between the virtual node and the internal source node, so the size of the final linear system equations will not increase. This new interpolation element inherits the advantages of the traditional continuous and discontinuous elements and overcomes their disadvantages. The numerical results show that, the proposed method helps solve the steady-state heat transfer problem with high computational accuracy and convergence.

Adaptive Extended Isogeometric Analysis for Steady-State Heat Transfer in Heterogeneous Media

Steady-state heat transfer problems in heterogeneous solid are simulated by developing an adaptive extended isogeometric analysis (XIGA) method based on locally refined non-uniforms rational B-splines (LR NURBS). In the XIGA, the LR NURBS, which have a simple local refinement algorithm and good description ability for complex geometries, are employed to represent the geometry and discretize the field variables; and some special enrichment functions are introduced into the approximation of temperature field, thus the computational mesh is independent of the material interfaces, which are described with the level set method. Similar to the approximation of temperature field, a temperature gradient recovery technique for heterogeneous media is proposed, and based on the Zienkiewicz–Zhu recovery technique a posteriori error estimator is defined to automatically identify the locally refined regions. The convergence and performance properties of the developed method are verified by using three numerical examples. The numerical results show that (1) The convergence speed of the adaptive local refinement is faster than that of the uniform global refinement; (2) The convergence rate of the high-order basis functions is faster than that of the low-order basis functions; and (3) The existing inclusions change the local distributions of the temperature, and the extreme values of the temperature gradients take place around the inclusion interfaces.

A lattice Boltzmann model for the conjugate heat transfer

Conjugate heat transfer problems are of great importance in many applications, while the available lattice Boltzmann (LB) models have some limitations in the study of such problems with the heterogeneous media. In this paper, we develop a new LB model for the conjugate heat transfer problems where the evolution equation with a modified equilibrium distribution function and a source term is considered. Through the Chapman–Enksog analysis, the macroscopic temperature equation can be recovered correctly. In order to test the accuracy, efficiency and stability of this LB model, some simple but nontrivial benchmark problems, including the planar thermal Poiseuille flow with two immiscible fluids, the conjugate heat transfer between a stratified heterogeneous medium with variable thermophysical properties, the steady and unsteady-state planer thermal flows with the two immiscible fluids, the steady-state conjugate heat transfer problem with a vertical interface, and the natural convection in a square cavity filled with circular and square rods, are studied. It is found that the results obtained from the present LB model are in good agreement with the analytic solutions or some previous works. In addition, it is also demonstrated that the developed LB model has a second-order convergence rate in space, and is more stable than some previous LB models. Finally, it should be noted that in our work, there is no special treatment to the interface, and the continuity conditions at the interface can be satisfied automatically.

Isogeometric boundary element method for steady-state heat transfer with concentrated/surface heat sources

An isogeometric boundary element method (IGABEM) is proposed for solving the steady-state heat transfer problems with concentrated/surface internal heat sources. The isogeometric boundary element method (IGABEM) possesses the advantages of both the isogeometric analysis (IGA) and the boundary element method (BEM), the non-uniform rational B-spline (NURBS) basis functions used in the (computer-aided design) CAD system are flexible and stable in dealing with irregular boundaries, due to which the NURBS basis functions are applied to exactly reconstruct the boundary geometry of the analysis domain, and the physical variables are also approximated by the NURBS as the standard IGA does. Dirac delta function is introduced in the present IGABEM for dealing with the concentrated point heat sources, and a local coordinate system is proposed for the domain integration over line heat source based on the NURBS basis functions. As to the domain integrals caused by the surface heat sources, the radial integration method (RIM) is used to transform the domain integrals to boundary ones without any internal cells for the original version of IGABEM. Several numerical examples involving Dirichlet, Neumann and Robin boundary conditions are analyzed, and circular region with point/line heat source(sources), global and local surface heat sources within multiply connected regions are also considered. Verification of the proposed method has been gained by comparing the present results with those obtained by several other researchers, and solutions achieved by analytical expression together with the fine-meshed ANSYS model are also utilized for the purpose of comparison. Meanwhile, the computational efficiency and convergence of the present method is evaluated by comparing the IGABEM with the finite element method (FEM) as well as the traditional BEM. Good performance of the IGABEM is observed. Finally, heat transfer within a simplified printed circuit board (PCB) with both concentrated (line and point) and surface heat generations is studied, the temperature distribution is achieved, which verifies the applicability of the proposed technoque in practical engineering.

A new numerical technique for interval analysis of convection-diffusion heat transfer problems using LSE and optimization algorithm

This article devotes to the uncertain analysis of steady-state convection-diffusion heat transfer problems with interval input parameters in material properties, thermal source and boundary conditions. The optimization strategy is adopted to ensure a reliable bounds estimation when scales of interval width is larger, and a Galerkin system is derived to construct a Legendre Series Expansion (LSE) to surrogate FE solutions of deterministic problems, so as to reduce the computational expense in the optimization based bounds estimation with sufficient accuracy. A LSE-GS (global search), and a LSE-CM (combinatorial method) are presented to solve uncertain steady-state convection-diffusion heat transfer problems with multiple interval variables, and are extended to the fuzzy analysis. Various numerical examples are provided to verify the performance of the proposed method, and evidence the accuracy and effectiveness of the proposed methods for interval prediction of steady-state convection-diffusion heat transfer problems.

STEADY-STATE HEAT TRANSFER IN A PLATE WITH TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY (AN EXACT SOLUTION)

This work presents the solution of the steady-state heat transfer problem in a rectangular plate with an internal heat source in a context in which the thermal conductivity depends on the local temperature. This generalization of one of the most classical heat transfer problems is carried out with the aid of the Kirchhoff transformation and employs only well known tools, as the superposition of solutions and the Fourier series. The obtained results illustrate how the usual procedures may be extended for solving more realistic physical problems (since the thermal conductivity of any material is temperature-dependent). A general formula for evaluating the Kirchhoff transformation as well as its inverse is presented too. This work has a strong didactical contribution since such analytical solutions are not found in any classical heat transfer book. In addition, the main idea can be used in a lot of similar problems.