Unsteady Heat Conduction Equation Research Articles

Novel Two-Dimensional Transient Heat Conduction Calculation in a Cooled Rotor: Ventilation Preheating—Blow-Down Flux

This contribution presents an alternative to classical data reduction techniques to measure the heat transfer using thin-film gauges. A finite-element model of the two-dimensional unsteady heat conduction equation is solved in the cross-sectional area of a metallic airfoil bounded with a polyamide sheet on which thermal sensors are deposited. This novel methodology allows capturing all 2D heat conduction effects that are irremediably neglected with the 1D data reduction technique. The application of this technique in a compression tube facility allows an exact evaluation of the initial wall heat flux into cooled rotor blades. During the spinning-up period, the rotor is spun up to nearly its nominal speed (from 0 rpm to 6200 rpm) resulting in preheating due to drag losses. The long duration of this experiment (∼450 s) and the magnitude of the wall temperature increase result in significant 2D conduction effects that are not accounted for using the 1D approach. In addition, short-duration experiments confirm the existence of 2D effects at smaller time scales (∼0.5 s), as well as the influence of the initial nonuniform temperature distribution in the rotor blade. The resulting flux with such an initial condition appears to be the superposition of the wall heat flux at the end of the spinning up before the test and the flux due to the blow-down itself.

An Accelerated Two-Dimensional Unsteady Heat Conduction Calculation Procedure for Thermal-Conductivity Measurement by the Transient Short-Hot-Wire Method

A fast and accurate procedure is proposed for solution of the two-dimensional unsteady heat conduction equation used in the transient short-hot-wire method for measuring thermal conductivity. Finite Fourier transforms are applied analytically in the wire-axis direction to produce a set of one-dimensional ordinary differential equations. After discretization by the finite-volume method in the radial direction, each one-dimensional algebraic equation is solved directly using the tri-diagonal matrix algorithm prior to application of the inverse Fourier transform. The numerical procedure is shown to be very accurate through comparison with an analytical solution, and it is found to be an order of magnitude faster than the usual numerical solution.

Numerical-analytical method of solution of a nonlinear unsteady heat-conduction equation

A method of solution of a one-dimensional nonlinear unsteady heat-conduction equation has been proposed. The use of the method of Green’s functions made it possible to transform the resulting equation to a nonlinear Volterra integral equation of the second kind for temperature, which is solved by the quadratic-form method. A system of recurrence relations, which is solved numerically, has been obtained. The influence of the nonlinearity on the temperature profiles has been analyzed. A comparison to the numerical finite-element method has shown that the numerical-analytical technique allows a reduction of more than 103 times in the calculation time.

Modelling of ‘thermal shocks’ in composite materials using a meshfree FEM

Abstract This work discusses an application of a meshfree, semidiscrete finite element method to solving the ‘thermal shock’ problem for a thin, cylindrical plate made of functionally graded ceramics. The unsteady heat conduction equation is discretized in space with the partition of unity FEM. The temporal discretization is realized by using an explicit finite difference method. The computations are performed on two model samples: the homogenous one, made of the pure Al 2 O 3 material, and the composite plate, made of the alumina/zirconium layers with variable weight content of ZrO 2 . The heat transfer coefficient on the surface subjected to the ‘thermal shock’, as well as thermal conductivity and thermal diffusivity of the material, are modelled with the theoretical distribution function based on the experimental findings. The discrete solution is first checked for accuracy against the analytical solution to an exemplary test problem, yielding very advantageous results. Finally, it is verified in comparison with the jet-impingement-cooling experimental data, showing good accordance.

A Two-Dimensional Analytical Solution for the Transient Short-Hot-Wire Method

Unlike the conventional transient hot-wire method for measuring thermal conductivity, the transient short-hot-wire method uses only one short thermal-conductivity cell. Until now, this method has depended on numerical solutions of the two-dimensional unsteady heat conduction equation to account for end effects. In order to provide an alternative and to confirm the validity of the numerical solutions, a two-dimensional analytical solution for unsteady-state heat conduction is derived using Laplace and finite Fourier transforms. An isothermal boundary condition is assumed for the end of the cell, where the hot wire connects to the supporting leads. The radial temperature gradient in the wire is neglected. A high-resolution finite-volume numerical solution is found to be in excellent agreement with the present analytical solution.

Motion and heat and mass transfer in a disperse admixture in turbulent nonisothermal jets of a gas and a low-temperature plasma

The motion and heat and mass transfer of particles of a disperse admixture in nonisothermal jets of a gas and a low-temperature plasma are simulated with allowance for the migration mechanism of particle motion actuated by the turbophoresis force and the influence of turbulent fluctuations of the jet flow velocity on heat and mass transfer of the particle. The temperature distribution inside the particle at each time step is found by solving the equation of unsteady heat conduction. The laws of scattering of the admixture and the laws of melting and evaporation of an individual particle are studied, depending on the injection velocity and on the method of particle insertion into the jet flow. The calculated results are compared with data obtained with ignored influence of turbulent fluctuations on the motion and heat and mass transfer of the disperse phase.

Heat Transfer Characteristics for Practical Hydrogen Pressure Vessels Being Filled at High Pressure

Experiments have been conducted to measure the rise in temperature of hydrogen and vessel wall during filling of commercially available, practical tanks to 35 and 70 MPa. Three test vessels with volumes 205, 130 and 39 liters are investigated. The filling time ranges from 5 to 20 minutes. The heat transfer process is modeled using a one-dimensional unsteady heat conduction equation for the wall coupled with a flow and heat balance for the compressed gas. The model requires heat transfer coefficients between the hydrogen and the wall and the wall and surrounding air. Values of 500 W/(m2K) during filling, 250 W/(m2K) after filling for the inside wall and 4.5 W/(m2K) for the outside tank wall are tentatively assumed based on results from a previous study on a smaller vessel. The measured temperatures for the hydrogen gas and the wall are in good agreement with the calculations.

Thermal shock response of FGM cylindrical plates with various grading patterns

The work focuses on the numerical study of FGM grading-pattern impact on the temperature distribution in the cylindrical plates made of Al 2O 3/ZrO 2, under the thermal shock. Three patterns of material grading are studied, each composed of 9 layers with different volume content of ZrO 2. It is assumed that in each layer of the cylindrical specimens the heat conduction along the length of layer and in direction of its thickness is different. Thus in the paper the effect of thermal orthotropy on the temperature distribution is studied. Simulations of the temperature shock propagation in the composite material are performed by using the unsteady heat conduction equation solver based on the alternating direction implicit (ADI) method. The results are illustrated in figures.

A 3D model for simulating the spatial and temporal distribution of temperature within ellipsoidal fruit

A physical model simulating the spatio-temporal distribution of temperature in an ellipsoidal fruit has been developed. It is based on the numerical resolution of the three-dimensional (3D) unsteady heat conduction equation with unsteady and non-homogeneous heat fluxes as boundary conditions. The numerical scheme and the physical models have been tested by comparing with an analytical solution for simple configurations. The model quality has been assessed by comparing model outputs to temperature measured with thermocouples at several locations in isolated peach and apple fruits. The root mean square error (RMSE) for temperature simulated at a 1 min time step was about 0.8 °C. A sensitivity analysis showed that accurate estimations of the surface conductance to water vapor diffusion and of the thermal conductivity are necessary. The model has also been used for identifying the main microclimate variables governing temperature dynamics within fruits.

Exact variational principle for 3-D unsteady heat conduction with second sound

The exact variational formulation of the extended unsteady heat conduction equation with finite propagation speed (the 2nd sound speed) of hyperbolic type is derived herein via a systematic and natural way. Moreover, the boundary-and the physically acceptable initial-value conditions are accommodated in the variational principle by a novel method suggested just recently. In this way a perfect justification of the variational theory of transient heat conduction and a rigorous theoretical basis for the finite element analysis of heat conduction are provided.

A close-form solution to predict the total melting time of an ablating slab in contact with a plasma

An exact melt-through time is derived for a one-dimensional heated slab in contact with a plasma when the melted material is immediately removed. The plasma is composed of a collisionless presheath and sheath on a slab, which partially reflects and secondarily emits ions and electrons. The energy transport from plasma to the surface accounting for the presheath and sheath is determined from the kinetic analysis. This work proposes a semi-analytical model to calculate the total melting time of a slab based on a direct integration of the unsteady heat conduction equation, and provides quantitative results applicable to control the total melting time of the slab. The total melting time as a function of plasma parameters and thermophysical properties of the slab are obtained. The predicted energy transmission factor as a function of dimensionless wall potential agrees well with the experimental data. The effects of reflectivities of the ions and electrons on the wall, electron-to-ion source temperature ratio at the presheath edge, charge number, ion-to-electron mass ratio, ionization energy, plasma flow work-to-heat conduction ratios, Stefan number, melting temperature, Biot number and bias voltage on the total melting time of the slab are quantitatively provided in this work.

Three-dimensional temperature field in a line-heater embedded by a spiral electric resistor

In this study, three-dimensional temperature fields induced by AC and DC through a spiral electric resistor in a line-heater are numerically investigated. Electric heaters have been widely found, for example, in houses as dryers, stoves, and water heaters, and in industrial and research institutions as elements of equipments. The line-heater in the present model is realistically considered to have multiple regions composed of a spiral electric resistor, electrically insulated region, covering outer layer, and two terminal pins with distinct thermal and electrical properties. Solving unsteady three-dimensional heat conduction equations in distinct regions, the surface temperatures predicted as a function of time in this model are confirmed by the measured data. The calculated results quantitatively show that high surface temperatures of the heater can be reached by increasing dimensionless joule heat parameter, radius of the spiral electric resistor or pins, thermal diffusivity of the insulation region, and decreasing Biot number and radius of insulation region. The effects of the pins on surface temperature are also studied. Aside from showing that DC produces higher temperature than AC, the results indicate that the effects of current frequency on temperature fields are insignificant. The findings can be generalized to a curved heater, because any local location can be considered as a small line-segment. The present work provides general and quantitative data valuable for designing an efficient heater/furnace.

Computer Simulation of Heat Transfer in Regenerative Chambers of Self-preheating Hot Blast Stoves

Heat transfer in chambers of self-preheating hot blast stoves in Anshan Iron and Steel Co., China, has been numerically simulated with a three-dimensional unsteady heat conduction equation and an energy conservation equation for fluid flowing through channels in the chambers. The blast temperature and air preheating temperature in the numerical simulation study are in good agreement with those in measurement. The numerical results show that hot blast temperature can reach 1 200°C by using a self-preheating process and low calorific blast furnace gas under the studied operational conditions. Amount of heat exchange between the checkerwork and the fluids and usable heat increase in the middle and low parts of the chambers of the hot blast stoves due to the application of the self-preheating process.

Simulation of heat transfer during rotational molding

AbstractIn rotational molding, polymer powders are subjected to heating, melting, cooling, and subsequent solidification in biaxially rotating molds. Heat transfer phenomena during rotational molding are significantly affected by the presence of endothermic and exothermic transitions. In this paper instead of using the traditional moving interface method, a new approach is presented which is applicable to semicrystalline materials like linear low‐density polyethylene. Melting is described by a statistical model and crystallization by a kinetic model. The model parameters are determined from differential scanning calorimetry measurements. The one‐dimensional unsteady heat conduction equation is solved by a finite difference method. The numerical predictions are in good agreement with experimental data. The overall heat transfer model can be used for process optimization purposes. © 2003 Wiley Periodicals, Inc. Adv Polym Techn 22: 271–279, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/adv.10055

Solution of the Unsteady Heat-Conduction Equation for a System of Two Bounded Heterogeneous Cylinders with the Use of The Integral Transformations of Hankel and Laplace

Consideration has been given to the problem of uniform unsteady lateral heating of two bounded cylinders having dissimilar thermophysical characteristics and being in ideal thermal contact. The exact analytical solution for determination of a three‐dimensional temperature field in real space has been found with the use of the integral‐transformation method. The distinctive features of the solution obtained have been investigated and examples of specific calculations have been given.

Creating compact models of complex electronic systems: an overview and suggested use of existing model reduction and experimental system identification tools

The electronic system packaging community has a great need to reduce the size of its heat transfer simulations so that it can: simulate and analyze more complex systems, include additional physical phenomena, and improve its ability to search the electronic systems packaging design space. Aside from further improvements in machine speed and numerical algorithm efficiency, this is basically a question of model reduction and experimental identification: one would like to know how to dramatically reduce the size of heat transfer simulations when they are available, and one would further like to identify models directly from experiment when accurate, computationally feasible, numerical simulations are not available (as in the case of turbulent flows through complex geometries). Fortunately, the topic of low-order modeling for design has been widely studied and successfully applied in other fields. This paper gives a brief overview of these techniques, it suggests how these tools might be effectively used for electronic systems including cases that involve unsteady fluid dynamics, and it summarizes some of the reduced-order modeling lessons learned in other fields. The paper includes some initial work in model reducing the unsteady heat conduction equation, a result on component model inter-connections, and an outline of a systems level model for an air cooled personal computer.

Propagation of boundary of inhomogeneous heat conduction equation

We consider the problem of determining analytically the exact solutions of the heat conduction equation in an inhomogeneous medium, described by the diffusion equation ∂ t T( x, t)= r 1− s ∂ r ( k( r) r s−1 ∂ r T( r, t)) with a position-dependent thermal diffusivity K( r). The unsteady one-dimensional heat conduction equation is transformed into an ordinary differential equation called Kummer’s equation unifiedly in the linear, cylindrical and spherical coordinate systems. Kummer’s equation is solved in terms of the confluent hypergeometric functions. These solutions exist on the conditions that boundaries move with their positions proportional to some functions of time. Progress has been made in this direction by introducing similarity variables and transformations.

Green's function approach to unsteady thermal stresses in an infinite hollow cylinder of functionally graded material

A Green's function approach based on the laminate theory is adopted for solving the two-dimensional unsteady temperature field (r, z) and the associated thermal stresses in an infinite hollow circular cylinder made of a functionally graded material (FGM) with radial-directionally dependent properties. The unsteady heat conduction equation is formulated as an eigenvalue problem by making use of the eigenfunction expansion theory and the laminate theory. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the unsteady temperature. The associated thermoelastic field is analyzed by making use of the thermoclastic displacement potential function and Michell's function. Numerical results are carried out and shown in figures.

Some Algebraically Explicit Analytical Solutions of Unsteady Nonlinear Heat Conduction

The analytical solutions of nonlinear unsteady heat conduction equation are meaningful in theory. In addition, they are very useful to the computational heat conduction to check the numerical solutions and to develop numerical schemes, grid generation methods and so forth. However, very few explicit analytical solutions have been known for the unsteady nonlinear heat conduction. In order to develop the heat conduction theory, some algebraically explicit analytical solutions of nonlinear heat conduction equation have been derived in this paper, which include one-dimensional and two-dimensional unsteady heat conduction solutions with thermal conductivity, density and specific heat being functions of temperature.

Entrance of a body with a lifting force and a heat‐conducting surface into the earth's atmosphere

The problem of gliding descent of a smooth blunted body possessing a lifting force and a heat‐conducting surface in the Earth's atmosphere is solved. The descent trajectory is represented not only by the altitude and velocity as functions of the flight time but also by angles of attack and sideslip varying with time. Three‐dimensional equations of a parabolized viscous shock layer for a multispecies mixture of gases are solved jointly with a three‐dimensional equation of unsteady heat conduction in the solid phase.