Conduction Heat Transfer Equation Research Articles

Estimating Non-Linear Heat Flux in Continuous Billet Casting Process With Limited Sensors

In a continuous billet caster, the heat transfer from the pool of molten steel to the cooling water via the copper mold depends non-linearly upon several casting parameters. A reliable and accurate methodology based on the principle of inverse heat transfer technique has been proposed in this work to obtain the nonlinear heat flux of a continuous billet casting mold. Differential evolution optimization technique has been used, in conjunction with the 2-D and 3-D steady state conduction heat transfer equations to develop the complete algorithm. It has been observed that both the 2-D and 3-D models are able to accurately predict the heat flux for different numbers and locations of temperature sensors. While the 2-D model estimates the 1-D heat flux profile varying along casting direction within a very short time period, the 3-D model offers the advantage of estimating a more accurate 2-D heat flux profile varying along both the mold length and width. The 2-D model has also been used to estimate the heat flux for an industrial continuous casting mold using measured plant data. It is observed that the present methodology accurately estimates the boundary heat flux in the continuous billet caster mold.

The Two-Dimensional Conduction Heat Transfer Equation on a Square Plate: Explicit vs. Crank-Nicolson Method in MS Excel Spreadsheet

In this paper, the two-dimensional conduction heat transfer equation on a square plate is analyzed using a finite difference method. We have developed both the forward time-centered space (FTCS) and Crank-Nicolson (CN) finite difference schemes for the two-dimensional heat equation, employing Taylor series. Subsequently, these schemes were employed to solve the governing equations. The primary objective of this study is to compare the efficiency of the two methods in solving the conduction heat transfer equation. This was accomplished by implementing Spreadsheet Excel instructions. The results are presented, highlighting a comparison between the exact and approximate solutions. Furthermore, to demonstrate the convergence of the numerical schemes, we estimated the error between the actual and approximate solutions for a specific numerical problem and presented the results graphically. The data utilized in this research included the thermal conductivity of the medium of the square plate concerning width, grid, and compliance with initial and boundary conditions. The findings indicate that the Crank-Nicolson method is more accurate than the forward time-centered space method, as it approaches the exact solution more effectively. Furthermore, this study confirms that the solution and simulation of the heat transfer equation on a square plate can be accurately performed using an Excel spreadsheet as well as other numerical software.

Validation of a Theoretical Model for Laser Welding Thermal Field by Multi-Physics Numerical Simulation

Theoretical models based on solutions of the conduction heat transfer equation have been widely proposed to calculate the thermal fields generated during laser welding, revealing simplification benefits and limitations in the accuracy of the results. In previous papers, the authors have introduced a parameterized analytical model based on the configuration of a virtual system of multiple mobile heat sources that simulates the effects of an actual keyhole welding mode by setting the system parameters so as to fit the calculated contours of the fusion zone in the weld cross-section of the experimental one. Even though a basic validation was already carried out by experimental detection, in order to further strengthen the model validity, this article deals with an extensive comparison between the results obtained by a multi-physics numerical simulation, performed by a commercial CFD software, and a theoretical one. The two different approaches were applied to the laser beam welding of butt-positioned AISI 304L steel plates. The investigation was focused on the effects of the keyhole on the main morphological features of the melt pool and fusion zone, and on the thermal fields obtained by the two models. The intrinsic differences between the two approaches, and how they are reflected in the corresponding results, were discussed. Satisfactory results were obtained by comparing the thermal fields, with a substantial convergence of the results, so as to validate the analytical model, assess the accuracy of its results, and define its application limits.

Simulation of Heat Storage and Heat Regeneration in Phase Change Material

The present study explores numerically the energy storage and energy regeneration during Melting and Solidification processes in Phase Change Materials (PCM) used in Latent Heat Thermal Energy Storage (LHTES) systems. Transient two-dimensional (2-D) conduction heat transfer equations with phase change have been solved utilizing the Explicit Finite Difference Method (FDM) and Grid Generation technique. A Fortran computer program was built to solve the problem. The study included four different Paraffin's. The effects of container geometrical shape, which included cylindrical and square sections of the same volume and heat transfer area, the container volume or mass of PCM, variation of mass flow rate of heat transfer fluid (HTF), and temperatures difference between PCM and HTF were all investigated. Results showed that the PCMs in a cylindrical container melt and solidify quicker than the square container. The increase in mass flow rate and/or temperature difference decreases the time required for complete phase change. Paraffin's solidify quicker than they melt and store more energy than they release

Comparison of heat transfer enhancement between open and closed cell porous metal structures for solid–liquid phase change

PurposeThe purpose of this study is to analyze heat transfer for solid–liquid phase change in two inclined cavities assisted with open cell and closed cell porous structures for enhancement of heat transfer and compare them.Design/methodology/approachThe heat transfer analysis is done numerically. The set of conservation equations for mass, momentum and energy for phase change material (PCM) and conduction heat transfer equation for metal frame are solved. Furthermore, temperature and solid–liquid fraction distributions for a cavity filled only with PCM are also obtained for comparison. The porosity is 0.9 for both porous structures. Rayleigh number and inclination angle change from 1 to 108, and from −90° to 90°, respectively.FindingsThe present study reveals that the use of closed cell structures not only can make phase change faster than open cell structure (except for Ra = 108 and = 90°) but also provide more stable process. The use of a closed cell porous structure in a cavity with PCM can reduce melting period up to 55% more than a cavity with an open cell porous structure. The rate of this additional enhancement depends on Rayleigh number and inclination angle.Originality/valueTo the best of the authors’ knowledge, this is the first time that the comparison between closed cell and open cell porous structures for heat transfer enhancement in a solid/liquid phase change process is reported. Authors believe that the present study will lead more attentions on the use of closed cell porous structures.

A novel temperature prediction method without using energy equation based on physics-informed neural network (PINN): A case study on plate- circular/square pin-fin heat sinks

This study introduces a new physics-informed neural networks (PINN)-based prediction method to determine the temperature pattern of fluid and fins when flow passes over plate-circular/ plate-square pin fin heat sinks (PCPFHS / PSPFHS). The proposed method is based on calculating the velocity pattern on the fins' surface. For this target, a training algorithm based on the feed-forward neural network (FNN) (110 layers of learnable weights and 16 neurons in each layer), a nonlinear activation function (rectified linear unit, "ReLU") and the Adam method for optimization is used. The training algorithm is fed by transient large eddy simulation (LES) results at every 0.01 s time step for the total physical time of 100 s. According to the input parameter type, the training:validation ratio is varied between 70:30 and 90:10 in order to keep the coefficient of determination (R2) at its maximum. The automatic differentiation employed the forward accumulation approach to reduce calculation costs, while the transient training matrix fed the neural network. The adaptive gradient (AdaGrad) method is also to improve convergence process and its speed up. Based on the developed calculation tools, the temperature pattern for the flow and over the fins are calculated according to the energy balance on the fin surface and the transient pattern of velocity predicted by PINN. After careful validation with experimental data and sensitivity analysis on the number of neurons and layers, the thermal behavior of PCPFHS and PSPFHS are determined using the conduction heat transfer equation inside the fins via the finite element method and by assuming a heat balance between the fins' surface and airflow. As a result of the proposed method, it is possible to reduce the number of equations in the calculation process of these parameters. According to the results, it is found that PSPFHS has an average Nusselt number which is 9.63% greater than the one in PCPFHS. However, compared to PCPFHS, PSPFHS shows a -17.78% reduced vorticity ratio at Re = 4,865. The results indicated that, for long calculation times (for instance, at 2,000 s physical time), the PINN method reduces calculation costs up to 35% compared to the technique that directly solves the energy conservation in the whole domain.

Volume penalization method for solving coupled radiative-conductive heat transfer problems in complex geometries

A novel immersed boundary method for simulating coupled radiative-conductive heat transfer based on a volume penalization method is proposed and verified. A smooth source/sink term is introduced to the governing equations for radiative and conductive heat transfer in order to take into account the absorption, reflection and emission at a solid surface. The proposed algorithm is successfully implemented into an open-source CFD solver, OpenFOAM with adaptive mesh refinement to guarantee sufficiently fine grid resolution near solid surfaces. Benchmark problems of pure radiation and combined conduction-radiation are considered for verifying the proposed method. The present simulation results under various absorption and emission conditions show good agreement with analytical solutions as well as existing literature data within around 5% errors. In addition to its superior performance in terms of accuracy, the present approach does not require special treatments around the solid surface with arbitrary complex geometry to impose boundary conditions for radiative heat transfer, and thereby wide applicability in scientific and engineering problems.

Use of the Alternating Directions Method in Cylindrical Coordinates, Applied to the Two-Dimension Unsteady Conduction Heat Transfer

Abstract Finding analytical solutions for the unsteady conduction heat transfer equation is generally very difficult due to the intrinsic complexity of the problem. In most cases, the equation has no analytical solutions. A numerical approach that surmounts these difficulties consists of the finite difference technique. The alternate directions finite-difference technique especially suits to the two-dimension heat conduction. The paper considers a hollow cylinder consisting of a conduction heat transfer medium, and the two-direction unsteady heat propagation within this medium is studied by means of the alternate directions method. The approach simulates the heat conduction process during a charge-discharge cycle of a single borehole heat storage.

Coupled neutronic core and subchannel analysis of nanofluids in VVER-1000 type reactor

Abstract This study is aimed to perform the coupled thermal-hydraulic/neutronic analysis of nanofluids as the coolant in the hot fuel assembly of VVER-1000 reactor core. Water-based nanofluid containing various volume fractions of Al2O3 nanoparticle is analyzed. WIMS and CITATION codes are used for neutronic simulation of the reactor core, calculating neutron flux and thermal power distribution. In the thermal-hydraulic modeling, the porous media approach is used to analyze the thermal behavior of the reactor core and the subchannel analysis is used to calculate the hottest fuel assembly thermal-hydraulic parameters. The derived conservation equations for coolant and conduction heat transfer equation for fuel and clad are discretized by Finite volume method and solved numerically using visual FORTRAN program. Finally the analysis results for nanofluids and pure water are compared together. The achieved results show that at low concentration (0.1 percent volume fraction) alumina is the optimum nanoparticles for normal reactor operation.

Modeling Thermal Behavior of Mass Concrete Structures at Early Age

In this study, a three-dimensional (3D) mathematical model based on conduction heat transfer in a solid is developed to predict the temperature profile of large concrete elements during the hydration process at early ages. Four different concrete mix designs are tested in the laboratory and on site. Heat of hydration and activation energy of each concrete mix is determined in the laboratory and temperature dependent thermo-physical parameters are considered in the simulations. The transient 3D conduction heat transfer equation with a heat generation source term is discretized by using finite volume method. The mathematical model is validated with temperature-time histories measured from four 1.8 m cube concrete blocks that are cast on site. Results obtained from the 3D model show that variable thermal properties are considered to accurately predict the temperature gradients. According to the analysis, the numerical model demonstrates that it can produce accurate predictions for temperature analysis within the 1.8 m large concrete elements.

Panel flutter analysis of cracked functionally graded plates in yawed supersonic flow with thermal effects

In the present paper, an isogeometric analysis formulation along with the Nitsche technique is implemented based on the first order shear deformation plate theory to investigate the aero-thermo-elastic panel flutter behavior of a functionally graded (FG) plate containing cracks in the supersonic flow field. A FG material is considered with temperature-dependent mechanical properties distributed in the plate thickness according to the power law. The temperature distribution in the plate thickness is calculated through solution of the one-dimensional steady-state conductive heat transfer equation while the surface temperatures are kept fixed. The physical contact and the crack growth phenomena are overlooked in the present research. The panel flutter behavior is investigated under combination of crack orientation, crack length, thermal conditions, flow direction, FG volume fraction index and boundary conditions. The accuracy and quality of the present isogeometric formulation is shown in comparison with those available in literature.

Coarse-grained discrete element method of particle behavior and heat transfer in a rotary kiln

Recent improvements in computational capabilities have made numerical simulation of the particle behavior and heat transfer in a rotary kiln reactor possible via the discrete element method (DEM). However, even with current computational capabilities, it is still difficult to simulate particle behavior and heat transfer in a rotary kiln reactor on the manufacturing scale because an enormous number of particles would be required. To address this issue, here, we propose for the first time a coarse-graining method for the Hertz–Mindlin contact force, the rolling friction, and conductive heat transfer between particles called the modified coarse-grained method for granular shear flow (M-CGSF). In M-CGSF, the scaling laws and governing equations for motion and conductive heat transfer of coarse-grained particles were derived according to matching centroid motions, elastic and rotational energy, and the overall heat transfer rate between the cluster of original particles and coarse-grained particles. Verification tests showed that coarse-graining of the particle dynamics, particle mixing, and particle heating behaviors in a rotary kiln were successfully performed by applying M-CGSF, demonstrating the effectiveness of M-CGSF for coarse-grained DEM simulations with conductive heat transfer in a rotary kiln reactor.

A parallelizable mesh-free approach for simulation of heat conduction in ferromagnetic particulate materials under magnetic field effect

A novel mesh-free approach has been developed to solve the conduction heat transfer equations in a particulate bed domain of ferromagnetic powder with and without magnetic field effect. The proposed method is based on a probabilistic approach to defining various configurations of energy transfer paths around an arbitrary particle called “local packed element”, and results are compared with experimental data. Furthermore, a Monte-Carlo approach is used to calculate the magnetic field of a permanent magnet with an arbitrary shape on any arbitrary point around it, and the outcome is also compared against measurements. In both cases, the proposed mesh-free method shows excellent agreement with available empirical results. The proposed method is then used to model the effect of an external magnetic field in the conductive heat transfer in a particulate bed of ferromagnetic powder. It is concluded that the presence of a magnetic field increases thermal diffusion in ferromagnetic powder bed domains by increasing thermal conductivity.

Three-dimensional analysis model of electric heating fabrics considering the skin metabolism

In low temperature environment, electric heating clothing can provide extra heat for human body through built-in heat source, so it has better thermal insulation effect. The thermal analysis is the initial step for electric heating clothing design. The current thermal analysis of electric heating textiles focuses on the fabric itself instead of the effect of skin tissue metabolism and heat production. In order to improve the accuracy of skin surface temperature prediction, the biological heat transfer need be modeled to analyze the internal temperature distribution of the heating suit system. In this paper, a three-dimensional (3D) thermal analysis model of electric heating clothing combined with human skin tissue is established. Firstly, the coupling analysis of Fourier heat conduction and Pennes biological heat transfer equation is carried out. Then the reliability of the 3D thermal analysis model is verified by finite element analysis (FEA). The results show that the fitting error between the three-dimensional model analysis data and FEA simulation data is 5°C, which proves that the model can accurately predict the system temperature. Finally, we make further research about the effects of ambient temperature, clothing layer thickness, and input power on the maximum skin surface temperature. This study provides theoretical foundation for the design of wearable thermal management fabric.

Thermal-hydraulic analysis of Al2O3 nanofluid as a coolant in Canadian supercritical water reactor by porous media approach

In this study, thermal-hydraulic analysis of Canadian Supercritical Water Reactor (Canadian SCWR) with nanofluid coolant has been done by porous media approach. In the applied analysis the derived conservation equations of mass, momentum and energy for coolant and conduction heat transfer equation for fuel and clad were discretised by finite volume method and solved numerically using C# program. Nanofluid coolant with various concentration of Al2O3 (Alumina) nanoparticle (up to 40% mass fraction) was selected for this study. Since in Canadian SCWR coolant and moderator are separated, higher mass fraction of nanofluid can be applied in this concept. The achieved results show that by increasing Al2O3 mass fraction, coolant heat transfer coefficient and average coolant temperature grow up in the middle of the core and pressure drop will be raised. Average clad and fuel temperature in the core was calculated. The results show that by using Alumina nanofluid with 40% nanoparticle mass fraction, maximum clad wall and fuel temperatures decrease about 20 °C and 30 °C respectively.

On fine particles synthesis using three-zone reactor

This paper reports a numerical study on thermal and mass dispersion in carbon combustion synthesis of oxides (CCSO) in porous media. A volume averaging of the microscopic conservation equations over an elementary volume is applied to derive the macroscopic conservation equations for convective and conductive heat and mass transfer in a porous media consisting of gas components mixture and solid particles of reactant species. The governing system of macroequations is the conservation of momentum, mass and energy for multicomponent gas-solid media using the heat and mass exchange between gas and solid phases. The model developed in dimensionless variables and similarity parameters is applied to numerical simulation of barium titanate micron particles synthesis in three-zone reactor using the thermal and mass dispersion. The results are in satisfactory agreement with experimental measurements. The dispersion strongly influences on the thermal front propagation as well as on the barium titanate synthesis rate. The three-zone reactor is more efficient compared to the flow-reactor.

A The Analysis of Energy Efficiency and Emission Rate of a Cookware coated with Enamel Material

Metal cookware has been used many people for its better thermal conductivity. However this material also has some weaknesses for its reaction with acidic food which will contaminate the food. Metal-base material coated with enamel is one of the solutions. The hard and non-reactive surface has cause this material has been widely used for processing food. Lower thermal conductivity is one of the problems of enamel cookware. The experimental analysis has been conducted to investigate energy efficiency, emission rate and thermal conductivity. The efficiency is calculated by comparing the useful energy (work) with the input energy. The emission rate is calculated by multiply energy consumption with emission factor. Meanwhile the thermal conductivity is calculated by using simply conduction heat transfer equation. Based on the analysis, enamel cookware has much lower thermal conductivity than aluminum cookware. However this physical property does not much affects the efficiency. The most dominant factors which should considered are type of fuel and technology of stove which consume higher energy inefficiently and emits much higher hazardous gas

TiB2–SiC-based ceramics as alternative efficient micro heat exchangers

Ceramics as functional materials durable at corrosive media or high temperatures are suitable candidates for manufacturing heat exchangers. In the present work, a series of numerical simulations was carried out to investigate the heat transfer in micro heat exchanger made by TiB 2 –SiC and TiB 2 –SiC doped with 2 wt% carbon fiber. Experimental thermal diffusivity data of materials were obtained by laser flash method and used for numerical simulations. This article demonstrates the feasibility of better heat transfer and performance for ceramic heat exchangers. Conduction heat transfer equation in solid walls of heat exchanger along with fluid flow governing equation was used for solid and fluid domains. The governing equations were discretized by the finite element method and solved by COMSOL Multiphysics software. The obtained results showed that the thermal conductivity of both TiB 2 -based composites are approximately the same and small amount of carbon fiber had no considerable effect on the thermal properties of the composites. For the ceramic heat exchanger, a higher heat transfer with 15.5% enhancement occurs using TiB 2 composites compared to utilization of Al 2 O 3 .

Analytical solution of steady state heat conduction equations in irregular domains with various BCs by use of Schwarz-Christoffel conformal mapping

Undoubtedly, analytical solutions of conduction heat transfer problems play an important role in formation of both qualitative and quantitative understanding of the heat transfer phenomenon and other phenomena related to it. Analytical methods for solving this class of problems, especially in irregular domains, contain much complexity, making the solution process extremely difficult. This paper demonstrates, in a general manner, the use of conformal mapping, specifically the Schwarz-Christoffel (SC) mapping, to solve the steady-state conduction heat transfer equations (namely Laplace and Poisson) and to obtain the analytical exact solution. The exact temperature distribution can then be obtained for irregular domains (i.e. domains with irregular boundaries) for homogeneous and non-homogeneous conditions. To get to this goal and to clarify the method more, for a specific case, namely a triangular plate with a right angle on the origin and constant-temperature boundary conditions, the steady-state heat conduction equation has been analytically solved using the conformal mapping method. In the analytical solution process, the importance and capabilities of complex variables and their applications, especially the SC mapping, is highlighted. Finally, in order to validate the exact solution, the problem has been solved using the finite-element software COMSOL Multi-physics 5.0 and the results have been compared to the analytical solution. Good agreement between two solutions has confirmed the accuracy of the analytical solution.

Two-dimensional transient heat conduction in multi-layered composite media with temperature dependent thermal diffusivity using floating random walk Monte-Carlo method

A novel mesh-free Monte-Carlo method for two-dimensional transient heat conduction in composite media with temperature dependent thermal properties is presented. The proposed approach is based on expressing the solution of the transient conductive heat transfer equation, in domains with temperature-dependent material properties, as a combination of two solutions: Bessel functions and integrals of peripheral temperature. The proposed approach is used to solve transient conduction in composite layered materials with temperature dependent thermal diffusivity. Results are compared against others obtained using a conventional finite element approach. Experimental results for heat transfer in a nonhomogeneous domain (composite layered material) are presented to demonstrate the performance of the proposed approach.