Transient Conduction Problems Research Articles

Modeling of the heterogeneous hypergolic ignition with particle packing model

The ignition process of a hypergolic hybrid rocket fuel presents a significant challenge due to its complex and interconnected nature. However, understanding its underlying physics is crucial for accurately estimating the delay time and ignition performance. In this study, we employed an approximate analytical method to directly address the transient conduction problem with nonlinear surface heat generation. By using the Arrhenius equation and considering dual heat flux directions in both the liquid and solid domains, we propose a model for determining the delay time of heterogeneous hypergolic ignition. Our solution is straightforward and computationally efficient. Notably, when subjected to identical initial conditions and property settings, our results exhibit similarity with the solutions derived from other models. Moreover, we advance our approach by developing a particle packing model through the utilization of a well-established algorithm. By simulating the particle arrangement within the pellet, the surface layer properties and the thermal time constant of certain fuel designs can be modeled. This simulation serves the purpose of incorporating the particle size effect of the fuel pellet into our ignition model. The results generated by our heterogeneous hypergolic ignition model, in conjunction with the particle packing model, demonstrated impressive agreement with the experimental data derived from droplet tests. Novelty and significance statementThis study utilized a novel solving method to address a transient conduction problem with non-linear heat generation on the surface during a heterogeneous hypergolic ignition process. The study successfully derived an analytical solution, providing a simpler method for calculating ignition delay times. A scenario involving different initial temperatures was also examined. Additionally, the research integrated particle packing model data and thermal time constants to assess the effect of particle size. The model results were in agreement with the experimental data, confirming its accuracy and offering a reliable method for estimating the heterogeneous hypergolic ignition delay time.

The Effect of Biot Number on a Generalized Heat Conduction Solution

Abstract Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solution for a transient conduction problem in rectangular coordinates. The general solution is based on the principle that the three primary types of boundary conditions (prescribed temperature, prescribed heat flux, and convective) can all be handled using convective boundary conditions. A large convection coefficient closely approximates a prescribed temperature boundary condition and a very low convection coefficient closely approximates an insulated boundary condition. Since a dimensionless solution is used in this research, the effect of various values of dimensionless convection coefficients, or Biot number values, are explored. An understandable concern with a general analytical solution is the effect of the choice of convection coefficients on the precision of the solution, since the primary motivation for using analytical solutions is the precision offered. An investigation is made in this study to determine the effects of the choices of large and small convection coefficients on the precision of the analytical solutions generated by the general convective formulation. Results are provided, in tabular and graphical form, to illustrate the effects of the choices of convection coefficients on the precision of the general analytical solution.

Contact resistance analysis applied to simultaneous estimation of thermal properties of metals

This work aims to present a method for simultaneously estimating the thermal conductivity, k, and specific heat, cp, in samples of stainless steel 304 and cemented carbide by accounting the imperfect thermal contact at the heater–sample interface. A transient one-dimensional heat conduction model with constant thermal properties is used. The metallic sample is placed in the middle of a polyimide heater and a thermal insulator. A constant heat flux is applied on the specimen upside surface and a thermal insulation condition is maintained on the opposing surface, where a type T thermocouple measures the temperature response. Contact resistance is determined and accounted for as a reducing factor on heat flux. Instead of considering a lumped model, a microscopic approach of the contacting regions is used to describe both the surface roughness and the fluid gap. With the intention of guaranteeing simultaneous and reliable estimates for both thermal properties, sensitivity analysis is used to establish heat flux intensity, experiment duration, time step for data acquisition, and other characteristics of the experiments. To simultaneously achieve both thermal properties, the optimization method BFGS (Broyden–Fletcher–Goldfarb–Shanno) is employed to minimize a least-squares objective function comparing experientially measured and numerically calculated temperatures. The numerical solution for the transient conduction problem is determined with COMSOL, which solves the transient heat diffusion problem by discretizing it applying the finite element method (FEM). The linear system of equations is solved using the PARDISO solver and backward differentiation formula (BDF) method. Heat flux is estimated employing the sequential function specification method (SFSM) as a means of verifying the robustness of parameter estimation and experimental procedure. Moreover, uncertainty analysis is performed to confirm the quality of the results obtained. Finally, the uncertainty analysis outcomes and thermal properties estimated are compared with literature.

Numerical assessment of brick walls` use incorporating a phase change material towards thermal performance in buildings during a passive cooling strategy

The integration of new building materials incorporating phase change material (PCM) into the building envelope leads to an increase of the heat storage capacity, which may have an influence on minimizing the cooling demand and heating of the building. This work addresses a thermal performance enhancement of brick walls with incorporated PCM. The improvement has been assessed through a numerical approach in dealing with a 1-D transient conduction problem with phase change, while leaning on experimental results from a transient guarded hot plates method. The simulations have been fulfilled using a hybrid method combining both the finite volume method and an enthalpy porosity technique. The results of this combined approach are in good agreement. In the light of the findings obtained, it appears that PCM incorporation into a brick masonry can both reduce peak temperatures up to 3?C and smooth out daily fluctuations. Thereby, the evaluation achieved can turns out useful in developing brick walls with an incorporated PCM for passive cooling, thus improving buildings thermal performance.

Modelling of temperature distribution along PCB thickness in different substrates during reflow

Purpose In this paper, analytical modelling of heat distribution along the thickness of different printed circuit board (PCB) substrates is presented according to the 1 D heat transient conduction problem. This paper aims to reveal differences between the substrates and the geometry configurations and elaborate on further application of explicit modelling. Design/methodology/approach Different substrates were considered: classic FR4 and polyimide, ceramics (BeO, Al2O3) and novel biodegradables (polylactic-acid [PLA] and cellulose acetate [CA]). The board thicknesses were given in 0.25 mm steps. Results are calculated for heat transfer coefficients of convection and vapour phase (condensation) soldering. Even heat transfer is assumed on both PCB sides. Findings It was found that temperature distributions along PCB thicknesses are mostly negligible from solder joint formation aspects, and most of the materials can be used in explicit reflow profile modelling. However PLA shows significant temperature differences, pointing to possible modelling imprecisions. It was also shown, that while the difference between midplane and surface temperatures mainly depend on thermal diffusivity, the time to reach solder alloy melting point on the surface depends on volumetric heat capacity. Originality/value Results validate the applicability of explicit heat transfer modelling of PCBs during reflow for different heat transfer methods. The results can be incorporated into more complex simulations and profile predicting algorithms for industrial ovens controlled in the wake of Industry 4.0 directives for better temperature control and ultimately higher soldering quality.

Experimental and analytical study of transient convection at film flow-wall interface using infrared thermography

The convection heat transfer of transient flow usually involves transient temperature and transient heat flux at the flow-wall interface. For experimental studies on this type of convection heat transfer problems, it is difficult to take measurement at the flow-wall interface, particularly for liquid flow. A substrate with small thickness can be used. One surface is exposed to the flow, while, on the free surface, there is a thin-skin heat source providing a constant and uniform heat flux, and the temperature at the free surface is measured using infrared thermography. Then the heat transfer coefficient of the transient flow can be obtained based on the temperature measured on the free surface. Two experimental tests of water droplets impacting a water film flow are conducted using this method. We solve a 1-D transient conduction problem, and the derived solution is used to calculate the heat transfer coefficient for the two experiments. Additionally, the lumped capacitance method (LCM) and the present analytical method are compared for calculating convection heat transfer.

Analytical Solution Based on a New Series along with the Numeric Solution of the Non-Homogeneous Transient Heat Conduction Equation, in Hollow Cylinder under General Mixed Boundary Conditions

In this study, the problem of transient conduction heat transfer for an infinite hollow cylinder under non-homogeneous mixed boundary conditions at the both surfaces simultaneously in radial direction with general heat source depend on time and radius, also general initial condition by the method of superposition and separation variables, is solved and temperature distribution is obtained analytically. A new series based on the Bessel functions is obtained for the problem of transient heat conduction without heat source by using separation of variables. Any function that has expanded conditions by the Fourier series can be expanded by this new series. Then, by expanding the heat source function according to this new series, the problem of transient heat transfer involving the thermal source has been solved and the radial temperature distribution is obtained. Due to the limited case studies, numerical solution of heat conduction equation with implicit finite difference method also is presented. Finally, a numerical example is given to compare between analytical and numerical solutions.

Modeling the Heat Transfer Coefficient Between a Surface and Fixed and Fluidized Beds With Phase Change Material

The objective of this work is to model the heat transfer coefficient between an immersed surface and fixed and bubbling fluidized beds of granular phase change material (PCM). The model consists of a two-region model with two different voidages in which steady and transient conduction problems are solved for the fixed and fluidized bed cases, respectively. The model is validated with experimental data obtained under fixed and fluidized conditions for sand, a common material used in fixed and fluidized beds for sensible heat storage, and for a granular PCM with a phase change temperature of approximately 50 °C. The superficial gas velocity is varied to quantify its influence on the convective heat transfer coefficient for both the materials. The model proposed for the PCM properly predicts the experimental results, except for high flow rates, which cause the contact times between the surface and particles to be very small and lead the model to overpredict the results.

3D Radiative Transfer Equation Coupled with Heat Conduction Equation with Realistic Boundary Conditions Applied on Complex Geometries

This paper presents the solution of coupled radiative transfer equation with heat conduction equation in complex three-dimensional geometries. Due to very different time scales for both physics, the radiative problem is considered steady-state but solved at each time iteration of the transient conduction problem. The discrete ordinate method along with the decentered streamline-upwind Petrov-Galerkin method is developed. Since specular reflection is considered on borders, a very accurate algorithm has been developed for calculation of partition ratio coefficients of incident solid angles to the several reflected solid angles. The developed algorithms are tested on a paraboloid-shaped geometry used for example on concentrated solar power technologies.

A direct numerical integration (DNI) method to obtain wall thermal response factors

The solution of the problem of transient conduction in multi-layer walls through response factor method is well-known and has been studied widely and for a long time, and recent studies show that research in this field is not closed and keeps improving calculation methods of this fundamental issue in building energy performance.The aim of this work is to present an alternative to classical methods of inversion avoiding the process of root finding and approximating directly the integral Laplace's inversion formula. Previous attempts have been developed recently in this spirit, and a comparison of these methods with the one proposed will be shown in the results section.The numerical inversion considered in this work computes directly the inversion integral, using a quadrature rule, reason why this new method has been named direct numerical integration (DNI). It uses a specific integration path which makes the error diminish exponentially with the number of approximation points.The case studies will show the accuracy of the obtained results with very small computational effort and high programming simplicity.

A methodology for the estimation of ignition delay times in forest fire modelling

A methodology for the estimation of ignition times on solid materials is presented. It is based on the observation that the time to ignition is proportional to the squared time integral of the incident heat flux. This relationship can be readily demonstrated for the classical solutions for time to ignition which consider constant incident heat fluxes. Thus, ignition times can be calculated only with the knowledge of the incident heat flux on the sample (gas-phase calculations) and experimental ignition results. This method is particularly useful when modelling wildfire propagation, where grid resolution is larger than some solid objects. Analytical solutions of the transient conduction problem for constant and ramping incident heat fluxes are discussed. The proportionality between the time to ignition and the squared integral of the incident heat flux was verified experimentally for constant and time-varying incident heat fluxes, showing that the proposed methodology can be applied to both cases. The methodology is outlined and an example is presented.

Analytic solutions of initial–boundary-value problems of transient conduction using symmetries

Lie symmetry method is applied to find analytic solutions of initial–boundary-value problems of transient conduction in semi-infinite solid with constant surface temperature or constant heat flux condition. The solutions are obtained in a manner highlighting the systematic procedure of extending the symmetry method for a PDE to investigate BVPs of the PDE. A comparative analysis of numerical and closed form solutions is carried out for a physical problem of heat conduction in a semi-infinite solid bar made of AISI 304 stainless steel.

A parallel domain decomposition boundary element method approach for the solution of large-scale transient heat conduction problems

This paper develops a parallel domain decomposition Laplace transform BEM algorithm for the solution of large-scale transient heat conduction problems. In order to tackle large problems the original domain is decomposed into a number of sub-domains, and a Laplace transform method is utilized to avoid time marching. A procedure is described which provides a good initial guess for the domain interface temperatures, and an iteration procedure is carried out to satisfy continuity of temperature and heat flux at the domain interfaces. The decomposition procedure significantly reduces the size of any single problem for the BEM, which significantly reduces the overall storage and computational burden and renders the application of the BEM to large transient conduction problems on modest computational platforms. The procedure is readily implemented in parallel and applicable to 3D problems. Moreover, as the approach described herein readily allows adaptation and integration of traditional BEM codes, it is expected that the domain decomposition approach coupled to parallel implementation should prove very competitive to alternatives proposed in the literature such as fast multipole acceleration methods that require a complete re-write of traditional BEM codes.

Coupled electrical and thermal transient conduction problems with a quadratic interpolation cell method approach

The Cell Method is an approach to the numerical calculation of fields that has been recently applied to the coupled computation of electric and thermal conduction using a linear interpolation of both the electric and temperature fields. In this paper a quadratic interpolation approach is presented. It is aimed at the achievement of an accurate field distribution computation particularly in complex domains such as biological structures. Such condition are encountered in the prediction of the thermal distribution obtained in hyperthermic treatments. Some results for a real case are presented in the paper

Simulation de la conduction non linéaire en régime variable: décomposition sur les modes de branche

A Rayleigh-Ritz method is presented, for the achievement of a reduced model for nonlinear thermal problems. The modal basis that is used consists of ‘branch’ eigenfunctions, originally used in mechanical problems. After the definition of the branch eigenbasis, the eigenfunctions are determined analytically in the case of a one-dimensional cylindrical problem. An analysis of these modes is carried out, and the feasibility of the reduction of the modal basis is proved. This is done by separating the modes into two sets: the ‘slow’ modes and the ‘fast’ modes, each one of the latter being uncoupled from all the others and having a contribution that vanishes quickly in time. The method is tested in the case of a nonlinear transient conduction problem, which shows its great interest.

A simplified approach to the transient conduction in a two-dimensional fin

In this study, after applying an averaging technique in the transverse direction, the problem of transient conduction in a two-dimensional rectangular fin is reduced to that of a one-dimensional fin problem. The averaging method used in the improved one-dimensional formulation takes into account in an approximate manner the temperature variations across the fin. The classical formulation of the one-dimensional fin problem neglects the transversal temperature gradients. The resulting partial differential equation is analyzed using Laplace transforms when fin base is subjected to a step change in base temperature. Results of averaging technique are compared to one-dimensional transient solutions. These results indicate that accuracy on heat transfer at the fin base and average temperature profiles along the fin is significantly improved. The present analysis extends the range of applicability of one-dimensional formulation to larger values of Biot number based upon the lateral fin surface. Typical results are presented in graphical form for various values of pertinent parameters.

Some analytical solutions of time-dependent, continuously operating heat sources

The amount of heat released in a volume whose extent can be ignored as compared with the dimensions of the surroundings of interest, may be classified as (i) the point-heat source, (ii) the line-heat source, and (iii) the plane-heat source. The temperature solutions of these transient conduction problems in an extended, homogeneous, and isotropic medium, are discussed in closed form. These analytical solutions are presented for the case of time-dependent, continuously operating heat source of the form\(\dot Q\), wherev is a positive real number. The dimensionless temperature solutions for all the problems are presented in terms of the Whittaker function. It is found that the present analysis covers well known (classical) solutions as well as some new solutions.

Green’s Function Partitioning in Galerkin-Based Integral Solution of the Diffusion Equation

A procedure to obtain accurate solutions for many transient conduction problems in complex geometries using a Galerkin-based integral (GBI) method is presented. The nonhomogeneous boundary conditions are accommodated by the Green’s function solution technique. A Green’s function obtained by the GBI method exhibits excellent large-time accuracy. It is shown that the time partitioning of the Green’s function yields accurate small-time and large-time solutions. In one example, a hollow cylinder with convective inner surface and prescribed heat flux at the outer surface is considered. Only a few terms for both large-time and small-time solutions are sufficient to produce results with excellent accuracy. The methodology used for homogeneous solids is modified for application to complex heterogeneous solids.

A new approach to the transient conduction in a 2-D rectangular fin

The purpose of this work is to develop a method that enables one to reduce the problem of transient conduction in a 2-D rectangular fin to that of a 1-D problem. After applying the perturbation technique and an averaging method, the problem of transient conduction in a 2-D rectangular fin is reduced to that of a 1-D problem. A linear operator method can then be applied to solve the resulting partial differential equation if the associated Sturm-Liouville problem is manageable. Actual calculation of transient temperature profiles in a 2-D rectangular fin confirms the validity of the method developed in this work.

First‐order design sensitivities for transient conduction problems by an adjoint method

AbstractAn adjoint approach is presented for the design sensitivity analysis of transient conduction problems. Variations of a general design functional are expressed in explicit form with respect to all design quantities, i.e. material properties, shape, applied thermal loads including convection, and initial conditions. The methodology incorporates the mutual energy between the real and adjoint thermal problems and a geometric mapping to describe shape variations. Finite element implementation of the method is discussed and an example is provided. Some potential difficulties that might be encountered when using the adjoint method with the finite element method are addressed. These involve the application of impulse loadings and Dirac temperature fields in the adjoint loadings.