Solution Of Transfer Equation Research Articles

Transient radiative-conductive heat transfer modeling in constructional semitransparent silica ceramics

The article deals with transient radiative – conductive heat transfer investigation in constructional silica ceramics with 10% of total porosity in 2D slab geometry. An empirical model, the so-called effective thermal conductivity (ETC) approximation, and rigorous radiative heat transfer (RCHT) problem solution are compared. Effective thermal conductivity coefficients were determined by both quasi-stationary and non-stationary sample heating and corresponding conductive heat transfer inverse problem solution. Rigorous RCHT calculations were based on radiative transfer equation solution. Spectral directional-hemispherical reflectances were used for scattering and absorption coefficients determination by an identification technique. Identification was based on direct problem solution by invariant embedding technique and parametric optimization. Mie theory and independent scattering approximation with regard to scatterers (pores) size distribution were used for mean scattering cosine estimation. It was shown that in the case of non-stationary temperature regime with rapid heating and cooling stages ETC models fail to predict through the thickness temperature distribution with absolute error less than 502013100 K. The absolute value of this error is strongly affected by the heating schedule. Moreover, calculations revealed several features of the heat flux through the unheated boundary that could not be taken into considerations by effective thermal conductivity approaches.

Inverse estimation of variable thermal parameters in a functionally graded annular fin using dragon fly optimization

ABSTRACTThis paper presents an inverse study of heat transfer of a conductive, convective and radiative annular fin made of a functionally graded material. Three major parameters such as conductive–convective parameter, conductive–radiative parameter and the parameter describing the variation of thermal conductivity are inversely estimated from a specified temperature field. The forward solution of temperature field is obtained from the closed form solution of nonlinear heat transfer equation using Homotopy perturbation method (HPM). A dragonfly algorithm that simulates the swarming behaviour of dragonflies, as analogous, is employed in finding out the inverse parameters. The temperature values of the forward solution are used as input data for the inverse analysis. The inverse parameters are then estimated iteratively by minimizing the objective function until the guessed temperature field approximately satisfies the preassigned temperature field of the forward solution. The inverse simulation following HPM-based forward solution converges faster than ordinary differential equation-based forward solution. The reconstructed temperature fields obtained from the various combination of inverse parameters give good agreement (∼1% error) with the desired temperature field. Thus, the presented inverse model provides an opportunity to the fin designer for selecting the several feasible combinations of thermal parameters suggesting the material design that result in a prescribed temperature field.

Splitting methods for the numerical solution of multi-component mass transfer problems

In a multi-component system the diffusion of a certain species is dictated not only by its own concentration gradient but also by the concentration gradient of the other species. In this case, the mathematical model is a system of strongly coupled second order elliptic/parabolic partial differential equations. In this paper, we adapt the splitting method for numerical solution of multi-component mass transfer equations, with emphasis on the linear ternary systems. We prove the positive definiteness assumptions for the discrete problem matrices which ensure the stability of the method. The numerical experiments performed confirmed the theoretical results, and the results obtained show good numerical performances.

Extended Irreversible Thermodynamics Versus Second Law Analysis of High-Order Dual-Phase-Lag Heat Transfer

This study introduces an analysis of high-order dual-phase-lag (DPL) heat transfer equation and its thermodynamic consistency. The frameworks of extended irreversible thermodynamics (EIT) and traditional second law are employed to investigate the compatibility of DPL model by evaluating the entropy production rates (EPR). Applying an analytical approach showed that both the first- and second-order approximations of the DPL model are compatible with the traditional second law of thermodynamics under certain circumstances. If the heat flux is the cause of temperature gradient in the medium (over diffused or flux precedence (FP) heat flow), the DPL model is compatible with the traditional second law without any constraints. Otherwise, when the temperature gradient is the cause of heat flux (gradient precedence (GP) heat flow), the conditions of stable solution of the DPL heat transfer equation should be considered to obtain compatible solution with the local equilibrium thermodynamics. Finally, an insight inspection has been presented to declare precisely the influence of high-order terms on the EPRs.

Finite Element Analysis to Determine Residual Stress in Electron Beam Welding of CuCrZr alloy Plates and Experimental Validation

The study on determining residual stresses in welding process through finite element (FE) analysis could receive much attention in the last decades. The prediction of residual stresses and temperature distribution is important for many industries, as it can ensure more efficient mechanical design. Electron beam welding (EBW) of copper chromium zirconium (CuCrZr) alloy was carried out in the bead-on-plate configuration. The temperature fields and stress distributions in the welded joint during the welding process were numerically simulated and experimentally verified. 3D finite element models were developed using the solution of heat transfer equations in Ansys mechanical APDL. Here, conical heat source was used for simulation. The temperature distribution was obtained, and stress analysis was carried out with the developed model using the temperature dependent material properties for CuCrZr.The temperature distribution was symmetric along the weld center, and the thermal stress also appeared to be in symmetric distribution. The predicted results were verified through real experiments, and a good agreement between them was achieved.

An Efficient Acceleration of Solving Heat and Mass Transfer Equations with the Second Kind Boundary Conditions in Capillary Porous Composite Cylinder Using Programmable Graphics Hardware

With the recent developments in computing technology, increased efforts have gone into simulation of various scientific methods and phenomenon in engineering fields. One such case is the simulation of heat and mass transfer in capillary porous media, which is becoming more and more important in analysing various scenarios in engineering applications. Analysing such heat and mass transfer phenomenon in a given environment requires us to simulate it. This entails simulation of coupled heat mass transfer equations. However, this process of numerical solution of heat and mass transfer equations is very much time consuming. Therefore, this paper aims at utilizing one of the acceleration techniques developed in the graphics community that exploits a graphics processing unit (GPU) which is applied to the numerical solutions of heat and mass transfer equations. The nVidia Compute Unified Device Architecture (CUDA) programming model caters a good method of applying parallel computing to program the graphical processing unit. This paper shows a good improvement in the performance while solving the heat and mass transfer equations for capillary porous composite cylinder with the second kind of boundary conditions numerically running on GPU. This heat and mass transfer simulation is implemented using CUDA platform on nVidia Quadro FX 4800 graphics card. Our experimental results depict the drastic performance improvement when GPU is used to perform heat and mass transfer simulation. GPU can significantly accelerate the performance with a maximum observed speedup of more than 7-fold times. Therefore, the GPU is a good approach to accelerate the heat and mass transfer simulation.

Исследование интегральных характеристик охлаждающей системы с термоэлектрическим модулем

Thermoelectric modules (TEM) are devices using which it is possible to implement either thermoelectric cooling due to the Peltier effect or electricity generation due to the Seebeck effect. In the first case, electric energy is directly converted into thermal energy, and in the second case, thermal energy is converted into electricity. Recent advances in the development of new thermoelectric nanostructured materials with characteristics significantly better than those found in bulk ones, prompted an increased interest in cooling and electricity generation through the use of thermoelectric devices. Special engineering design techniques are used in developing thermoelectric systems. However, such techniques only yield successful results for some simple designs. If the design of a thermoelectric device is non-trivial, some approximations have to be applied for the design, which seriously affects the accuracy of the result. The existing engineering methods for calculating the parameters and integral characteristics of TEMs involve a large number of simplifications and shortcomings, due to which the calculated integral characteristics of a TEM can differ significantly from their actual ones. In the field of thermoelectric devices, numerical simulation has to be used to solve various problems, in particular, for studying and evaluating promising designs of thermoelectric devices. The article presents the mathematical model of a cooling system equipped with TEMs. Numerical solution of the conjugate heat transfer equations describing the cooling system operation is obtained. It is shown that the TEM operating parameters are influenced by a number of factors that should preferably be taken into account in temperature distribution calculations. It is also shown that under real conditions the integral TEM operation parameters may differ considerably from those calculated according to the standard engineering procedures. Owing to its containing a rigorous body of mathematics, the developed model makes it possible to calculate the TEM parameters and integral characteristics under the real conditions of their operation.

Mathematical modeling of laser linear thermal effects on the anterior layer of the human eye

In this paper, mathematical analysis of thermal effects of excimer lasers on the anterior side of the human eye is presented, where linear effect of absorption by the human eye is considered. To this end, Argon Fluoride (ArF) and Holmium:Yttrium-Aluminum-Garent (Ho:YAG) lasers are utilized in this investigation. A three-dimensional model of the human eye with actual dimensions is employed and finite element method (FEM) is utilized to numerically solve the governing (Penne) heat transfer equation. The simulation results suggest the corneal temperature of 263°C and 83.4°C for ArF and Ho:YAG laser radiations, respectively, and show less heat penetration depth in comparison to the previous reports. Moreover, the heat transfer equation is solved semi-analytically in one-dimension. It is shown that the exploited simulation results are also consistent with those derived from the semi-analytical solution of the Penne heat transfer equation for both types of laser radiations.

Study on coal char ignition by radiant heat flux.

The study on coal char ignition by CO2-continuous laser was carried out. The coal char samples of T-grade bituminous coal and 2B-grade lignite were studied via CO2-laser ignition setup. Ignition delay times were determined at ambient condition in heat flux density range 90–200 W/cm2. The average ignition delay time value for lignite samples were 2 times lower while this difference is larger in high heat flux region and lower in low heat flux region. The kinetic constants for overall oxidation reaction were determined using analytic solution of simplified one-dimensional heat transfer equation with radiant heat transfer boundary condition. The activation energy for lignite char was found to be less than it is for bituminous coal char by approximately 20 %.

3D Thermal Analysis for CFETR Pre-Superheated Water-Cooled Blanket

Initial China Fusion Engineering Test Reactor (CFETR), blanket design, includes water-cooled ceramic breeder (WCCB) blanket operating in pre-superheated regime. This condition allows efficient cooling; however it requires accurate control and analysis to avoid zones with excessive heat flux. Analysis of the coolant flow and heat transfer in CFETR Pre-Superheated Blanket was performed using ANSYS CFX and included: 3D coolant flow analysis, external volumetric and surface heating effect, and two-phase wall boiling. ASIPP CAD Model imported directly into ANSYS Workbench Design Modeler as a STEP file. Fluid volume is created using Design Modeler Fill operation, and converting Inlet and Outlet surfaces. Meshing was performed using CFX method available within the framework of the ANSYS mesh generator. Application of tetrahedral elements for meshing of the internal regions allowed automatic mesh generation. Advanced sizing functions were used with automatic mesh inflation depending on wall proximity and curvature. Conjugated heat transfer analysis was performed including solution of heat transfer equations in solid and liquid parts, and solution of the flow equations in the liquid parts. Coolant flow in that was assumed turbulent and was resolved using Reynolds averaged Navier-Stokes equations with Shear Stress Transport turbulence model. RPI model for wall driven boiling is used. Inhomogeneous two-phase flow is resolved solving two sets of momentum and energy equations for liquid and steam. Results showed ability of the model to simulate two-phase boiling flow in complex configuration.

Differential inverse inelastic mean free paths and differential surface excitation probability in aluminium in the energy range of 0.5–120 keV

Electron energy loss and photoelectron spectra are obtained on the basis of transfer equation solution. The boundary problem is solved using invariant imbedding approach. The partial intensity method is used to calculate electron energy loss and photoelectron spectra. Differential inelastic electron scattering cross sections in bulk aluminium and in a surface layer are extracted by fitting procedure in an energy range 0.5–120 keV. The method of spectra computation is tested by comparison of results with the experimental data of five foreign scientific laboratories. The physical meaning of the surface-excitation parameter is discussed.

Finite integral transform-based analytical solutions of dual phase lag bio-heat transfer equation

• Analytical solution of the non-Fourier bio heat transfer equation. • Finite integral transform. • Time independent/dependent boundary conditions. • Coupling of the transient RTE and non-Fourier heat conduction models. • Thermal analysis of laser-irradiated biological tissue phantoms. A finite integral transform (FIT)-based analytical solution to the dual phase lag (DPL) bio-heat transfer equation has been developed. One of the potential applications of this analytical approach is in the field of photo-thermal therapy, wherein the interest lies in determining the thermal response of laser-irradiated biological samples. In order to demonstrate the applicability of the generalized analytical solutions, three problems have been formulated: (1) time independent boundary conditions (constant surface temperature heating), (2) time dependent boundary conditions (medium subjected to sinusoidal surface heating), and (3) biological tissue phantoms subjected to short-pulse laser irradiation. In the context of the case study involving biological tissue phantoms, the FIT-based analytical solutions of Fourier, as well as non-Fourier, heat conduction equations have been coupled with a numerical solution of the transient form of the radiative transfer equation (RTE) to determine the resultant temperature distribution. Performance of the FIT-based approach has been assessed by comparing the results of the present study with those reported in the literature. A comparison of DPL-based analytical solutions with those obtained using the conventional Fourier and hyperbolic heat conduction models has been presented. The relative influence of relaxation times associated with the temperature gradients ( τ T ) and heat flux ( τ q ) on the resultant thermal profiles has also been discussed. To the best of the knowledge of the authors, the present study is the first successful attempt at developing complete FIT-based analytical solution(s) of non-Fourier heat conduction equation(s), which have subsequently been coupled with numerical solutions of the transient form of the RTE. The work finds its importance in a range of areas such as material processing, photo-thermal therapy, etc.

Modeling of the Dynamics of Non-radiative Energy Transfer in Tm3+, Tb3+: LiYF4-Based Electronic Materials

The fluorescent decay of the levels 3H4 and 3F4 of Tm3+ in LiYF4 crystals doped with Tb3+ and Eu3+ was mathematically modeled to estimate the optimal doping levels to maximize the laser emission at 1.5 μm of the transition 3H4 → 3F4. The analysis is carried out both through the solution of the master energy transfer equations that govern the non-radiative processes of energy and Monte Carlo simulations. The analysis is improved with the experimental data of the fluorescence decay to include quadrupole and dipole interactions simultaneously. The importance of considering these interactions is that the optimal concentrations of impurities predicted for these luminescent systems are lower than those reported with the use of traditional models.

On the theory of ternary melt crystallization with a non-linear phase diagram

The present study is concerned with a theoretical analysis of unidirectional solidification process of ternary melts in the presence of a phase transition (mushy) layer. A new analytical solution of heat and mass transfer equations describing the steady-state crystallization scenario is found with allowance for a non-linear liquidus equation. The model under consideration takes into account the presence of two phase transition layers, namely, the primary and cotectic mushy regions. We demonstrate that the phase diagram nonlinearity leads to substantial changes of analytical solutions.

Application of differential transformation method (DTM) for heat and mass transfer in a porous channel

In the present paper a differential transformation method (DTM) is used to obtain the solution of momentum and heat transfer equations of non-Newtonian fluid flow in an axisymmetric channel with porous wall. The comparison between the results from the differential transformation method and numerical method are in well agreement which proofs the capability of this method for solving such problems. After this validity, results are investigated for the velocity and temperature for various values of Reynolds number, Prandtl number and power law index.

Numerical solution of the parabolic multicomponent convection–diffusion mass transfer equations by a splitting method

ABSTRACTThe splitting method used in a previous study for the numerical solution of mass transfer equations in ternary systems is generalized to mixtures with n-components. The diffusion coefficients are considered constant. Theoretical results about the stability of the method are presented, as well as numerical simulations for mixtures with n = 4, 5, and 6. The numerical experiments confirmed the theoretical results and show good numerical performances. Moreover, multicomponent diffusion effects without an imposed concentration gradient are investigated for mixtures with n = 4, 5, and 6 components.

Comparison of the earth's crust geothermal field in Northern Tien Shan region with seismic characteristics

The deep crustal temperatures of the Northern Tien Shan region are being calculated on the basis of direct measurements of the heat flow and empirical relation of geothermal parameters and seismic velocities. For this purpose the applied method of numerical solution of conductive heat transfer equation takes into account additional sources of heat release due to radioactive decay. Thermoelastic deformations, reaching 20-40% of the total lithostatic pressure at depths of 35-60 km, are determined. The criteria controlling the distribution of hypocenters of earthquakes have been determined in accordance with the morphology of the regions of excess temperatures and thermal stresses.

Analysis of mass transfer equations during solid-liquid extraction and its application for vanilla extraction kinetics modeling

Rigorous dimensionless analysis and analytical solution of mass transfer equation (punctual and space averaged) governing the solid-liquid extraction dynamic were performed. Analytical solutions result in continuous and analytical maps between solute mass fractions in extract with respect to time which were used both for solute diffusivity estimation and process modeling. In order to validate the models, experimental extraction at equilibrium of 1 cm long vanilla beans pieces were obtained at 0.1, 0.2, 0.3 and 0.4 g vanilla/mL solvent (60% w/w ethanol-water) and 30, 40 and 50 °C; and extraction kinetics were developed at 0.1 g/mL and 30, 40 and 50 °C. Apparent equilibrium constant and solute effective diffusivity estimation were performed by linear regression on experimental results. Apparent equilibrium constant was 0.86, and diffusivities were 1.22, 1.98, and 2.43 × 10−11 m2/s at 30, 40 and 50 °C respectively. Diffusivities and analytical solutions predicted the experimental results at 30 and 50 °C with 0.3, and 0.4 g/mL respectively.

Conditions for reducing spherical thermal aberrations in transversally pumped YAG rods

Temperature dependences of the thermal conductivity and thermo-optic coefficient are often neglected when thermal lensing in laser crystals is investigated, though their influence is very significant. In this paper, the general solution of heat transfer equation with temperature dependent thermal conductivity is found in an integral form, which is very convenient for analytical and numerical analysis. Using this solution, the possibility to eliminate spherical aberration by a proper choice of the pump parabolicity parameter is investigated in detail. The inaccuracies in the definition of optical path differences used in a few previous works for the case of temperature dependent thermo-optic coefficient are explained. It is shown that the use of a correct definition increases the value of the parabolic coefficient at which the elimination of spherical aberrations may be realized as compared with the original work of Hodgson and Weber. It is also found that nearly the same shapes of pumping are required for elimination of spherical aberrations for both radial and tangential polarization.

Thermofluid Analysis in Magnetic Hyperthermia Using Low Curie Temperature Particles

Magnetic hyperthermia uses the targeted therapeutic heat from magnetic particles (MPs) in an alternating magnetic field to kill cancer cells. MPs with low Curie temperature (Tc) (in the range 42 °C–45 °C), high magnetization, and magnetic permeability/susceptibility are good candidates for their use in hyperthermia therapy. This paper analyzes the hyperthermic effects determined by the MPs with low Tc within a tumoral configuration from healthy tissue when an alternating magnetic field is applied. The temperature field was determined as a solution of the Pennes bioheat transfer equation. The spatial distribution of the particles after their injection within the tissues was computed by solving the convection–diffusion equation in porous tissues. Results show that the MP injection rate significantly influences the spatial distribution of the particles and the temperature field of the tumor. Higher values of the ferrofluid flow rate push more particles to the tumor center, thus also elevate its central temperature. The temperature field becomes uniform in a percentage of 90% of tumor volume if a blood vessel is localized at the tumor center.