New approaches to numerical modelling of droplet transient heating and evaporation

New approaches to numerical modelling of droplet heating and evaporation by convection and radiation from the surrounding hot gas are suggested. The finite thermal conductivity of droplets and recirculation in them are taken into account. These approaches are based on the incorporation of new analytical solutions of the heat conduction equation inside the droplets (constant or almost constant h) or replacement of the numerical solution of this equation by the numerical solution of the integral equation (arbitrary h). It is shown that the solution based on the assumption of constant convective heat transfer coefficient is the most computer efficient for implementation into numerical codes. This solution is applied to the first time step, using the initial distribution of temperature inside the droplet. The results of the analytical solution over this time step are used as the initial condition for the second time step etc. This approach is applied to the numerical modelling of fuel droplet heating and evaporation in conditions relevant to diesel engines, but without taking into account the effects of droplet break-up. It is shown to be more effective than the approach based on the numerical solution of the discretised heat conduction equation inside the droplet, and more accurate than the solution based on the parabolic temperature profile model. The relatively small contribution of thermal radiation to droplet heating and evaporation allows us to take it into account using a simplified model, which does not consider the variation of radiation absorption inside droplets.     

Thermal decay of an initially hot isothermal water body with application to thermal energy storage

The thermal decay of an initially hot isothermal water body contained in a tank was studied. Analytical solutions (including a group-theoretic solution) were obtained for the one-dimensional heat conduction equation with heat loss from the sides of the tank. The convective heat transfer coefficient was assumed not to be constant over the surface of the tank but to vary with space and time. A very good agreement was obtained between the group-theoretic solution and a numerical solution using the Crank-Nicholson finite difference scheme. The analytical results are discussed in terms of the underlying physical mechanisms.     

Temperature in thermally nonlinear pad–disk brake system

The influence of the thermal sensitivity of pad and disk materials on temperature at braking is under investigation. A mathematical model of process of frictional heating in a pad–disk brake system, which takes into account the temperature-sensitive materials, is proposed. The basic element of this model is the thermal problem of friction—a one-dimensional boundary-value heat conduction problem with temperature-dependent thermal conductivity and specific heat. Contrary to the prior studies of authors, where a simple nonlinearity was considered, in this article the arbitrary nonlinearity of the thermophysical properties of materials is studied. The solution of a nonlinear boundary-value heat conduction problem is obtained by the method of successive approximations. The numerical analysis of temperature is executed for some materials of a pad and a disk with and without taking into account their thermal sensitivity.     

Periodic heat transfer through inhomogeneous media. Part 1. Slab

The paper presents exact analytical solutions of one-dimensional periodic heat conduction through an inhomogeneous slab for a certain class of thermal conductivity profiles (including linear and exponential). The exact analytical solutions for some of these profiles have been compared with those obtained by considering the slab to be made up of a number of homogeneous layers with different thermal conductivities varying from layer to layer and using the layered structure (or matrix multiplication) method. The numerical results arrived at by the layered-structure method converge rapidly (with increasing number of layers considered) to the values obtained from the exact analytical solutions. This gives confidence in the application of the layered-structure method to periodic heat conduction through inhomogeneous slabs. The numerical results have been presented in the form of elements of a 2 × 2 matrix, relating the sinusoidal steady-state temperature and heat flux on the two sides of the slab.     

Analytical heat diffusion models for heat sinks with circular micro-channels

This study explores the prediction of temperature distribution in a heat sink containing an array of circular micro-channels, which is found mostly in electronic cooling applications. The analytical heat diffusion models for most common micro-channel shapes are based on one-dimensional fin models with varying degrees of complexity. Because of a singularity in the governing one-dimensional heat diffusion equation for a fin with circular profile, no exact solution is possible for the circular heat sink geometry. In this paper, an alternative analytical power series solution technique is presented in which the differential equation is recast in polynomial form. Predictions of the power series solution are validated for different channel diameters and spacings and both one-sided and two-sided heating conditions using one-dimensional and two-dimensional numerical simulations. Overall, maximum percent differences in temperature and heat transfer rate between the analytical and two-dimensional numerical results of 0.23% and 1.33%, respectively, prove that the present analytical models are very accurate and effective tools for the design and thermal resistance prediction of micro-channel heat sinks found in electronic cooling applications.     

Lattice Boltzmann method applied to the solution of the energy equations of the transient conduction and radiation problems on non-uniform lattices

Application of the lattice Boltzmann method (LBM) to solve the energy equations of conduction–radiation problems is extended on non-uniform lattices. In the LBM on non-uniform lattices, the single relaxation time based on the minimum velocity is used. This minimum velocity corresponds to the smallest size lattice. Because information propagates with the same minimum velocity in the prescribed directions from all the lattice centers, in a given time step, they are not equidistant from the neighboring lattices. Collisions in the LBM take place at the same instant. Therefore, in the LBM on non-uniform lattices, in every time step, interpolation is required to carry the information to the neighboring lattice centers. To validate this very concept in heat transfer problems involving thermal radiation, transient conduction and radiation heat transfer problems in a 1-D planar and a 2-D rectangular geometries containing absorbing, emitting and scattering medium are considered. The finite volume method (FVM) is used to compute the radiative information. In both the geometries, results for the effects of various parameters are compared for LBM–FVM on uniform and non-uniform lattices. To establish the LBM–FVM on non-uniform lattices for the combined conduction and radiation heat transfer problems, numerical experiments were performed with different cluster values. The accurate results were found in all the cases.     

TRANSIENT THERMOVTSCOELASTIC RESPONSES: STABILITY AND ACCURACY OF FINITE-ELEMENT SOLUTIONS

This paper is concerned with the stability and accuracy of finite-element analyses of transient thermoviscoelasticity subjected to irreversible thermodynamic processes. The study includes the use of an incremental free energy that incorporates internal state variables in terms of a discretized generalized Maxwell model. Numerical integration is then performed, with past histories being updated in each discretized time step. The effects of thermomechanical coupling, internal dissipation, boundary conditions, and various temporal operators are studied with respect to spectral radii, phase errors, numerical damping, etc. A stability criterion predicted by spectral radii for combined equations of motion and heat conduction confirms the consequent numerical results for transient response according to the estimated degree of stability. Stable solutions generally lead to convergence to the exact solution, but with lower rates of convergence than those of linear-uncoupled equations. For simplicity, the numerical examples deal with one-dimensional structures.     

ANALYSIS/FINITE-ELEMENT COMBINED METHODOLOGY ON TRANSIENT HEAT CONDUCTION OF A FINITE DOMAIN WITH PLANAR-DISCRETE SOURCES

Abstract

A useful method, involving the combined use of the analysis and the finite-element methods, is successfully extended to the transient heat conduction problem with isolated heat sources. The results are compared in tables with exact solutions and other numerical data, and the agreement is found to be good. Previously reported analysis /finite-element combined method has been confined to the slow convergence in series solution of analytical method. By using the third Aitken's delta-squared process for accelerating the convergence of infinite series, this restriction is removed, and the new method provides a more powerful solution to transient problems with heat sources     

Freezing of Water in a Slab with Boundary Conditions of the Third Kind

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Heat flow rate at a bore-face and temperature in the multi-layer media surrounding a borehole

Assessment of the heat either delivered from high temperature rocks to the borehole or transmitted to the rock formation from circulating fluid is of crucial importance for a number of technological processes related to borehole drilling and exploitation. Normally the temperature fields in the well and surrounding rocks are calculated numerically by finite difference method or analytically by applying the Laplace-transform method. The former approach requires tedious and, in certain cases, non-trivial numerical computations. The latter method leads to rather bulky formulae that are inconvenient for further numerical evaluation. Moreover, in previous studies where the solution is obtained analytically, the heat interaction of the circulating fluid with the formation was treated on the condition of constant bore-face temperature. In the present study the temperature field in the rock formation disturbed by the heat flow from the borehole is modeled by a heat conduction equation, assuming the Newton model for the convective heat transfer on the bore-face, with boundary conditions that account for the thermal history of the borehole exploitation. The problem is solved analytically by the generalized heat balance integral method. Within this method the approximate solution of the heat conduction problem is sought in the form of a finite sum of functions that belong to a complete set of linearly independent functions defined at the finite interval bounded by the radius of thermal influence and that satisfy the homogeneous boundary conditions on the bore-face. In the present study first and second order approximations are obtained for the composite multi-layer domain. The numerical results illustrate that the second approximation is in a good agreement with the exact solution. The only disadvantage of this solution is that it depends on the radius of thermal influence, which is an implicit function of time and can only be found numerically by iterative algorithms. In order to eliminate this complication, in this study an approximate explicit formula for the radius of thermal influence and new close-form approximate solution are proposed on the basis of the approximate solution obtained by the integral-balance method. Employing the non-liner regression method the coefficients for this simplified solution are obtained. The accuracy of the approximate solution is validated by comparison with the exact analytical solution found by Carslaw and Jaeger for the homogeneous domain.     

ANALYSIS/FINITE-ELEMENT COMBINED METHODOLOGY ON TEMPERATURE DISTRIBUTION OF A FINITE DOMAIN WITH VARIOUS HEAT SOURCES

Abstract

A new method, involving the combined use of analysis and the finite-element method, is applicable to the heat conduction problem with isolated heat sources. Unlike the finite-element method, the analysis/finite-element combined method is able to discretize the distributed sources with discontinuities into course elements, and the solution is still calculated accurately. The results are compared in tables with exact solutions and other numerical data, and the agreement is found to be good.     

Application of CESE method to simulate non-Fourier heat conduction in finite medium with pulse surface heating

This study employs the space–time conservation element and solution element (CESE) method to simulate the temperature and heat flux distributions in a finite medium subject to various non-Fourier heat conduction models. The simulations consider three specific cases, namely a single phase lag (SPL) thermal wave model with a pulsed temperature condition, a SPL model with a surface heat flux input, and a dual phase lag (DPL) thermal wave model with an initial deposition of thermal energy. In every case, the thermal waves are simulated with respect to time as the thermal wave propagates through the medium with a constant velocity. In general, the simulation results are found to be in good agreement with the exact analytical solutions. Furthermore, it is shown that the CESE method yields low numerical dissipation and dispersion errors and accurately models the propagation of the wave form even in its discontinuous portions. Significantly, compared to traditional numerical schemes, the CESE method provides the ability to model the behavior of the SPL thermal wave following its reflection from the boundary surface. Further, a numerical analysis is performed to establish the CESE time step and mesh size parameters required to ensure stable solutions of the SPL and DPL thermal wave models, respectively.     

A DISCONTINUOUS FINITE-ELEMENT FORMULATION FOR INTERNAL RADIATION PROBLEMS

This article presents a discontinuous finite-element formulation for internal thermal radiation problems. In contrast to the conventional finite-element formulation, the discontinuous algorithm permits the discontinuity of field variables across the internal interelement boundaries and is useful for integral-differential equations describing thermal radiation in absorbing/scattering media. Mathematical formulation and numerical implementation are given. The convergence rate and local mesh adaptivity are discussed. Two approaches for coupling of the discontinuous and conventional methods for mixed heat transfer calculations are presented. Numerical results are given for internal radiation and combined conduction/radiation problems and are compared with analytical solutions whenever available.     

SOLUTION OF TWO-DIMENSIONAL HYPERBOLIC HEAT CONDUCTION BY HIGH-RESOLUTION NUMERICAL METHODS

Studies of hyperbolic heat conduction have so far been limited mostly to one-dimensional frameworks. For two-dimensional problems, the reflection and interaction of oblique thermal waves and complicated geometries present a challenge. This paper describes a numerical solution of two-dimensional hyperbolic heat conduction by high-resolution schemes. First, the governing equations are transformed from Cartesian coordinates into generalized curvilinear coordinates. Then the dependent variables are cast in a characteristic form that decouples the original system equation into scalar equations. Two-dimensional high-resolution numerical schemes, suck as total variational diminishing ( TVD) are built up by forming symmetrical products of one-dimensional difference operators on each individual wave. Three examples are used to demonstrate the unique feature of complicated interaction of two-dimensional thermal waves.     

Transient temperature profile inside thermoacoustic refrigerators

The linear theory used to calculate the thermal quantities inside the stack in the classical thermoacoustic refrigerators always overestimates those measured. The causes of these discrepancies have to be found in the complex processes of thermal exchanges. The analytical study of the transient response should provide an interpretation of these complex processes. This present paper provides such analytical modelling. This modelling remains within the framework of the classical linear theory. It includes the effects of the thermoacoustic heat flux carried along the stack, the conductive heat flux returning in the solid walls of the stack and through the fluid inside the stack, the transverse heat conduction in the stack and the heat leakages through the duct walls, the heat generated by viscous losses in the stack, the heat generated by vorticity at the ends of the stack, and the heat transfer through both ends of the stack. A modal analytical solution for the temperature profile is proposed, assuming the usual approximations in such thermal problems to avoid intricate calculations and expressions. The theoretical transient response of a thermoacoustic refrigerator is compared with experimental data. A good qualitative agreement is obtained between analytical and experimental results after fitting empirical coefficients.     

Analytical solution of the inverse unsteady wall heat conduction problem and experimental application

Based on the analytical solution of the unsteady heat conduction differential equation, a solution procedure is presented for the inverse unsteady wall heat conduction problem, i.e. for the calculation of the thermal properties of structural elements of existing buildings under real transient conditions, using on-site temperature measurements. Previous procedures, which were based on the finite-difference method, required a considerable number of temperature measurements in space and time within the wall. The advantage of the present analytical procedure is that it requires only two temperature measurements, apart from some information on the outdoor and indoor temperature variations. The two temperature measurements may be taken on the outdoor and indoor wall surfaces at the same time level, or on one of these surfaces at two different time levels. The proposed analytical procedure provides the values of the thermal conductivity and heat capacity of structural elements, and therefore it may be used in practice for ex post checking of the materials used by the constructor, or for load calculation when heating or cooling systems are to be installed in old buildings of unknown wall properties. Experimental examples are presented which show that the proposed analytical procedure may be applied in practice with very good accuracy.     

Calculation Error of Numerical Solution for a Boundary—Value Inverse Heat Conduction Problem

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Performance Analysis of Heat Sinks With Phase-Change Materials Subjected to Transient and Cyclic Heating

Phase-change cooling technique is a suitable method for thermal management of electronic equipment subjected to transient or cyclic heat loads. The thermal performance of a phase-change based heat sink under cyclic heat load depends on several design parameters, namely, applied heat flux, cooling heat transfer coefficient, thermophysical properties of phase-change materials (PCMs), and physical dimensions of phase-change storage system during melting and freezing processes. A one-dimensional conduction heat transfer model is formulated to evaluate the effectiveness of preliminary design of practical PCM-based energy storage units. In this model, the phase-change process of the PCM is divided into melting and solidification subprocesses, for which separate equations are written. The equations are solved sequentially and an explicit closed-form solution is obtained. The efficacy of analytical model is estimated by comparing with a finite-volume-based numerical solution for both transient and cyclic heat loads.     

SOLUTION OF STEADY PERIODIC HEAT CONDUCTION BY THE FINITE-ELEMENT METHOD

This paper describes the use of the finite-element technique to solve problems of steady periodic heal conduction. The concept of a complex variable can be used to reduce the governing unsteady heat conduction equation to two noncoupled Poisson equations. The validity of the present method is confirmed with a one-dimensional problem for which an analytic solution exists. Numerical solutions for a three-dimensional problem are presented to illustrate the capability of the method.     

Stability analysis of generalized mass formulation in dynamic heat transfer

ABSTRACT

In this article, the generalized mass formulation is developed in an explicit analysis of transient transport problems. It has been well known that the time step is typically smaller in explicit analysis than in implicit analysis when the same size mesh is used. Further, the over-stiffness of conventional finite-element model may result in poor accuracy with linear triangular or tetrahedral elements. In order to improve the computational efficiency and numerical accuracy, this article proposes a generalized mass formulation by matching the mass matrix to the smoothed stiffness matrix using linear triangular elements in 2-D problems. The proposed mass matrix can be obtained by simply shifting the integration points from the conventional locations. Without loss of generality, several 2-D examples, including conduction, convection, and radiation heat transfer problems, are presented to demonstrate that the generalized mass formulation allows a larger time step in explicit analysis compared with the lumped and consistent mass matrices. In addition, it is found that the maximum allowable time step is proportional to the softened effect of the discretized model in an explicit analysis.