Conduction Heat Transfer Problems Research Articles

Estimation of Multiple Parameters in Semitransparent Mediums Based on an Improved Grey Wolf Optimization Algorithm

This work investigates the inverse coupled radiation–conduction problem for estimating thermophysical parameters and source terms by an improved grey wolf optimization (GWO). The transient coupled radiation–conduction heat transfer problem in participating slab media is solved by the finite volume method. The radiative intensities on both boundaries are adopted as known measurement information in the inverse model. To overcome the disadvantages of the original GWO algorithm, an improved grey wolf algorithm (IGWO) is developed by introducing the weight strategy and nonlinear factors. Three benchmark functions are adopted to prove that the IGWO has a faster convergence speed and higher estimation accuracy than the original one. The IGWO is applied to inverse the thermophysical parameters and source terms based on the coupled radiation–conduction model; the results indicate that the IGWO is accurate and effective for estimating refractive index, absorption coefficient, and source terms.

Refining the diffusive compressible Euler model

We revisit the molecular arguments underpinning the diffusive compressible Euler model (proposed in Svärd (2018)), and conclude that a heat diffusive term is not accounted for in the energy equation of the original model. The new heat diffusive term complements the diffusion of the conserved variables already present in the model. By studying a diffusive and conductive heat transfer problem, we quantify the new term in the updated model and evaluate it in the context sound attenuation. Although the new diffusive term is important for certain problems, we also demonstrate that it has a negligible effect on the aerodynamic validation cases that the original model has already successfully passed. Furthermore, the updated system is compatible with the weak well-posedness that has previously been established for the original system.

Enhanced surrogate modelling of heat conduction problems using physics-informed neural network framework

Solving partial differential equations (PDEs) using deep-learning techniques provides opportunities for surrogate models that require no labelled data, e.g., CFD results, from the domain interior other than the boundary and initial conditions. We propose a new ansatz of the solution incorporated with a physics-informed neural network (PINN) for solving PDEs to impose the boundary conditions (BCs) with hard constraints. This ansatz comprises three subnetworks: a boundary function, a distance function, and a deep neural network (DNN). The new model performance is assessed thoroughly in terms of convergence speed and accuracy. To this end, we apply the PINN models to conduction heat transfer problems with different geometries and BCs. The results of 1D, 2D and 3D problems are compared with conventional numerical methods and analytical results. The results reveal that the neural networks (NNs) model with the proposed ansatz outperforms counterpart PINN models in the literature and leads to faster convergence with better accuracy, especially for higher dimensions, i.e., three-dimensional case studies.

Monte Carlo solution of heat conductivity problems with quadratic nonlinearity on the boundary of domain

In this work, we study the approach connected with branching Markov processes. Branching Markov random processes are created to solve boundary-value problems (BVR) with polynomial nonlinearities. The realization of these processes creates so called “trees”. Unbiased estimators of the solution of the nonlinear problem are constructed on these random trees. We also calculated in parallel way the variance (statistical error) of the constructed unbiased estimators. We offer computational algorithms for solving some nonlinear diffusion problems which frequently appears in engineering problems, particularly in heat conductivity. The case of a quadratic nonlinearity for the boundary BVP is considered in detail. Algorithms, for diffusion BVPs with nonlinear boundary conditions, differ from proposed for linear diffusion BVPs algorithms early and we compare their efficiencies. The described algorithms applied to the computational problem of thermal engineering in the presence of a nonlinear boundary condition. Also, this problem is solved as a boundary value problem of conductive heat transfer.

Nonstationary radiative–conductive heat transfer problem in an absolutely black body with semitransparent inclusions

We consider a nonstationary initial‐boundary value problem governing a radiative–conductive heat transfer in an absolutely black body with semitransparent inclusions. To describe the radiative transfer, the integro‐differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We proved at the first time the unique solvability of this problem. In addition, we proved the stability of solutions with respect to data, established a comparison theorem and results on improving the properties of solutions with an increasing on the summability of the data.

Discontinuous finite element method for transient conductive heat transfer in layered media with thermal contact resistance

Research on accurately solving the transient conductive heat transfer problem in layered media has both scientific and engineering significance. Numerical algorithms up to date usually suffer from additional dubious treatments when dealing with the temperature discontinuity induced by the thermal contact resistance (TCR) between the inner interface. In this paper, we extend the discontinuous finite element method (DFEM) to solve the problem of transient conductive heat transfer in a layered slab with TCR interfaces. The correctness of the DFEM for handling the TCR-induced temperature discontinuity is verified at first. Then we utilize the DFEM to solve the transient conductive heat transfer problem in a three-layered medium subjected to Dirichlet, Neumann, and Robin boundary conditions, respectively. The time-dependent temperature along the slab and the temperature discontinuity across the TCR interfaces are captured and analyzed. Results indicate the present DFEM can naturally capture the TCR interface-induced temperature discontinuities accurately. The time-dependent temperature and temperature discontinuities across the TCR interfaces presented in this work can serve as benchmarks for later research on conductive heat transfer in layered media.

Diffusive mass transfer from a Janus sphere

A highly accurate approximate solution is developed for the problem of diffusive mass transfer from a two-sided sphere. The problem is primarily motivated by the Marangoni-driven motion of a chemically active spherical surfer along a water-air interface, but it is also related to the self-diffusiophoresis of catalytic colloids, as well as to the conduction heat transfer and electrostatics problems involving partially insulated spheres.

Finite Difference Method Applied in Two-Dimensional Heat Conduction Problem in the Permanent Regime in Rectangular Coordinates

The solution of many conduction heat transfer problems is found by two-dimensional simplification using the analytical method since different points has different initial temperatures. The temperature at each point of a given element can be analyzed through the Heat Equation that, in some cases, converges to analytical solutions without precision and is far from the real. However, with the application of the Finite Difference Method (FDM), it is possible to solve it numerically in a relatively fast way, providing satisfactory results for the most varied boundary conditions and diverse geometries, characteristics of heat transfer problems by conduction. This study solved two problems inside a plate with and without heat generation involved in temperature distribution. Algorithms were built with the aid of the Matlab programming language, and applied to obtain a numerical solution using the FDM numerical method. The computational and analytical solutions were then compared. Under certain conditions of the parameters involved in the phenomenon of each problem, the numerical method was very efficient for presenting errors less than or equal to 0.003 and 0.03, respectively, for cases without and with heat generation.

On the Objective Function for Topology Optimization of Heat Sinks

In topology optimization, the optimal layout of the structure is obtained through an iterative update of the geometry. Since this update procedure utilizes the gradient of the objective function, it is important to select an appropriate objective function in order to obtain a meaningful geometry. From this point of view, the mathematical definition and physical meaning of the objective functions for topology optimization of heat sinks are reviewed in this article. For this, studies from conductive heat transfer problems to recent computational fluid dynamics (CFD)-based conjugate heat transfer problems are covered. In most of the studies, the objective functions are defined in order to minimize the thermal resistance of the heat sink or to minimize the maximum temperature of the heat source. In place of thermal resistance, thermal compliance is commonly used as the objective function. However, it is worth noting that minimization of the thermal compliance may lead to considerable deviation of the design from that based on the minimum thermal resistance. This is because the thermal compliance is linearly proportional to the average temperature of the heat source in the computational domain, and the average temperature of the heat source is usually different from the maximum temperature of the heat source. In heat conduction problems, the difference between the maximum temperature and the average temperature becomes large when the thermal conductivity of the solid material becomes low. In convection problems, the difference becomes large when the streamwise fluid temperature variation becomes large due to the low heat capacity rate of the fluid.

Investigation of transient coupled conduction and radiation heat transfer in the linearly anisotropic scattering cylindrical medium by spectral collocation method

In this paper, a spectral collocation method (SCM) is developed to solve transient conduction and radiation heat transfer problems in infinitely long annular cylinders with participating medium. The participating medium is an absorbing, emitting and linearly anisotropic scattering medium. To avoid the complex calculation of ray tracing, the radiative transfer equation and the transient heat conduction equation are expressed in eulerian coordinates. In the solving process, for the radiative transfer equation, both spatial and angular domains are discretized by the SCM; for the transient heat conduction equation, the spatial domain is discretized by the SCM, and the time domain is discretized by the fully implicit Euler scheme. The spatial derivative term, angular derivative term and scattering source integral term are approximated by the high order spectral collocation scheme. Compared with available data in references, the SCM can achieve very good accuracy even using few nodes. Thus, the SCM is a very effective method to achieve high order accuracy for transient coupled conduction and radiation heat transfer in an anisotropic scattering cylindrical medium. Moreover, the effects of conduction-radiation parameter, optical thickness, and wall emissivity on the transient temperature distribution and energy flow rate have also been comprehensively investigated.

Nonstationary Radiative–Conductive Heat Transfer Problem in a Semitransparent Body with Absolutely Black Inclusions

The paper is devoted to a nonstationary initial–boundary value problem governing complex heat exchange in a convex semitransparent body containing several absolutely black inclusions. The existence and uniqueness of a weak solution to this problem are proven herein. In addition, the stability of solutions with respect to data, a comparison theorem and the results of improving the properties of solutions with an increase in the summability of the data were established. All results are global in terms of time and data.

Heating and Convective Cooling of Two-Dimensional Solids Using Diffusion and Cattaneo–Vernotte Models

Thermal diffusion and wave phenomena in a two-dimensional solid subjected to convective cooling and an internal exponential heat source are investigated. Both diffusion and Cattaneo–Vernotte (C–V) heat conduction models are solved using the superposition principle in conjunction with the solution structure theorems. For comparison purposes, both models are solved to demonstrate the flexibility of the technique and to show differences between the results. With cooling at the exposed surface, peak temperatures occur in the interior of the medium instead of at exposed surfaces. Peak temperature predictions from the C–V model are higher than those from a conventional diffusion model, due to the thermal lagging phenomenon. Because of the inherent thermal relaxation characteristic in the C–V model, energy transport takes the form of wave propagation rather than instantaneous energy transport at a nonphysical infinite speed as seen in the diffusion model. The study demonstrates the flexibility of the solution structure theorems in solving conduction heat transfer problems in two-dimensional solids with complex boundary conditions and heat loads.

Unique solvability of a stationary radiative–conductive heat transfer problem in a semitransparent body with absolutely black inclusions

We consider a stationary boundary value problem describing a radiative–conductive heat transfer in a semitransparent body with absolutely black inclusions. To describe the radiative transfer, the integro-differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We proved at the first time the unique solvability of this problem. Besides, we proved the comparison theorems and established the results on improving the properties of solutions with increasing exponents of data summability.

A new optimum numerical method for analysis of nonlinear conductive heat transfer problems

In this research, a new optimum numerical method is introduced to analyze nonlinear conductive heat transfer problems. The governing equation and the boundary conditions are combined using the penalty method to define a proper cost function. Furthermore, a Moving Least Squares (MLS) approach is applied to approximate the derivatives in the introduced nonlinear objective function. The objective function is minimized using particle swarm optimization (PSO) to find un-known nodal values. Obtained numerical results are compared with those reported in literature. The comparisons clearly illustrate the successfulness of the introduced strategy to solve the nonlinear heat transfer and its superiority in comparison with other well-known schemes.

Effects of transient processes for thermal simulations of the Central European Basin

Abstract. Transient processes play a major role in geophysical applications. In this paper, we quantify the significant influence arising from transient processes for conductive heat transfer problems for sedimentary basin systems. We demonstrate how the thermal properties are affected when changing the system from a stationary to a non-stationary (transient) state and what impact time-dependent boundary conditions (as derived from paleoclimate information) have on the system's overall response. Furthermore, we emphasize the importance of the time-stepping approach adopted to numerically solve for the transient case and the overall simulation duration since both factors exert a direct influence on the sensitivities of the thermal properties. We employ global sensitivity analyses to quantify not only the impact arising from the thermal properties but also their parameter correlations. Furthermore, we showcase how the results of such sensitivity analysis can be used to gain further insights into the complex Central European Basin System in central and northern Europe. This computationally very demanding workflow becomes feasible through the construction of high-precision surrogate models based on the reduced basis (RB) method.

Estimation of distributed heat flux parameters in localized heating processes

In this work a numerical-experimental methodology is presented based on a combined finite element model and experimental tests, for the determination of the heat flux distribution in oxy-acetylene localized heating processes commonly employed in the shipbuilding industry. The Levenberg-Marquardt algorithm is employed in the inverse conduction heat transfer problem using the results of experimental tests in thin flat circular steel plates with different thicknesses subjected to heat flux applied over the heated area, considered constant from the beginning to the end of heating, on the centre of the plates, followed by natural cooling. The transient temperatures in different points of the plate are measured by five thermocouples, affixed opposite to the oxy-acetylene flame. Additionally, the thermal analysis is performed using a non-linear two-dimensional axisymmetric finite element model to describe the heating and cooling processes, and the calculated and measured temperatures are compared. A statistical analysis is performed to demonstrate the stability of the methodology and the reliability of the results. Average residuals from the discrete in blocks and continuous Gaussian heat flux distribution models are compared. Good agreement is achieved with the use of the inverse problem methodology.

An efficient inverse approach for reconstructing time- and space-dependent heat flux of participating medium* *Project supported by the Natural Science Foundation of Chongqing (CSTC, Grant No. 2019JCYJ-MSXMX0441).

The decentralized fuzzy inference method (DFIM) is employed as an optimization technique to reconstruct time- and space-dependent heat flux of two-dimensional (2D) participating medium. The forward coupled radiative and conductive heat transfer problem is solved by a combination of finite volume method and discrete ordinate method. The reconstruction task is formulated as an inverse problem, and the DFIM is used to reconstruct the unknown heat flux. No prior information on the heat flux distribution is required for the inverse analysis. All retrieval results illustrate that the time- and space-dependent heat flux of participating medium can be exactly recovered by the DFIM. The present method is proved to be more efficient and accurate than other optimization techniques. The effects of heat flux form, initial guess, medium property, and measurement error on reconstruction results are investigated. Simulated results indicate that the DFIM is robust to reconstruct different kinds of heat fluxes even with noisy data.

Coupled optical and thermal analyses of a new type of solar water heaters using parabolic trough reflectors

A new type of solar water heating system using an array of parabolic trough collectors (PTCs) was investigated. A coupled simulation technique was used to solve the complex radiation, convection, and conduction heat transfer problem inside the system. The realistic non-uniform heat flux at the walls of the receiver pipe was obtained by optical analysis and was used simultaneously in thermal modeling. A comprehensive model by considering thermal losses under the non-uniformity of heat flux was proposed and verified with experimental data. In the experiments, the water with different inlet temperatures and flow rates was tested. The obtained results demonstrate that the location of the receiver pipe relative to the PTCs significantly affects the system's thermal efficiency. For the system under study, the best position of the absorber pipe was found at the d/f ratio of 0.8. At the optimal position of absorber pipe, the average daily system's thermal efficiency is about 70%, and for an absorber pipe located at the focal line of the collectors, it is less than 60%. It can be concluded that the thermal efficiency of the current system is higher than that of conventional systems under the same conditions.

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.

Discrete and asymptotic approximations for one stationary radiative–conductive heat transfer problem

AbstractSpecial discrete and asymptotic approximations are proposed for the boundary value problem describing a stationary radiative–conductive heat transfer in a system of absolutely black heat-conducting rods of circular cross-section. Results of numerical experiments are presented to confirm the efficiency of proposed approximations.