M.N. Özişik

Propagation and reflection of thermal waves in a finite medium

Abstract For situations involving extremely short times following the start of transients or for very low temperatures near absolute zero, the classical diffusion theory of heat conduction breaks down since the wave nature of thermal energy transport becomes dominant. In this work, analytical solutions are developed for the hyperbolic heat conduction equation describing the wave nature of thermal energy transport in a finite slab with insulated boundaries subjected to a volumetric energy source in the medium. The exact analytical solutions developed for the temperature field and heat flux show that the release of a concentrated pulse of energy gives rise to a severe thermal wave front which travels through the medium at a finite propagation speed, dissipating energy in its wake and reflecting from the insulated surfaces.

ON THE NUMERICAL SOLUTION OF HYPERBOLIC HEAT CONDUCTION

Non-Fourier heat conduction is governed by the hyperbolic heat conduction equation, which involves the wave nature of thermal energy transport. In such cases, energy propagates in the medium as a wave with sharp discontinuities at the wave front. Difficulties encountered in the numerical solution of such problems include, among others, numerical oscillations and the representation of sharp discontinuities with good resolution at the wave front. In this work it is shown that a numerical technique based on MacCormacks predictor-corrector scheme can be used to handle discontinuities at the wave front with high resolution and little oscillation. Numerical predictions are compared with the exact analytic solutions for a wide variety of strict test conditions. For all the cases considered, the present numerical scheme remains stable and produces high resolution at the sharp wave front.

An inverse analysis to estimate linearly temperature dependent thermal conductivity components and heat capacity of an orthotropic medium

Abstract An inverse analysis is used to estimate linearly temperature dependent thermal conductivity components k x ( T ), k y ( T ) and specific heat capacity C ( T ) per unit volume for an orthotropic solid. Simulated measured transient temperature data are generated by adding random errors to the exact temperatures computed from the solution of the two-dimensional, direct transient heat conduction problem. An iterative procedure, based on minimizing a sum of squares function with the Levenberg-Marquardt iterative procedure is used to solve the inverse problem.

Hyperbolic heat conduction with surface radiation

Abstract A numerical approach is used to solve hyperbolic heat conduction in a semi-infinite medium where the boundary at x = 0 is subjected to a surface heat flux and dissipates heat by radiation into an ambient at temperature T ∞ . The results are compared with those obtained from the standard parabolic heat conduction equation.

Flux formulation of hyperbolic heat conduction

The development of the general flux formulation for heat conduction based on the modified Fourier’s law is presented. This new formulation produces a hyperbolic vector equation in heat flux which is more convenient to use for analysis in situations involving specified flux conditions than the standard temperature formulation. The recovery of the temperature distribution is obtained through integration of the energy conservation law with respect to time. The Green’s function approach is utilized to develop a general solution for hyperbolic heat conduction in a finite medium. The utility of the flux formulation and the unusual nature of heat conduction based on the hyperbolic formulation are demonstrated by developing analytical expressions for the heat flux and temperature distributions in a finite slab exposed to a pulsed surface heat flux.

DIRECT INTEGRATION APPROACH FOR SIMULTANEOUSLY ESTIMATING TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY AND HEAT CAPACITY

Abstract One of the difficulties in the solution of inverse heat conduction problems is that of making sufficiently accurate initial guesses for the unknowns in order to start the iterations. In this work a direct integration method is developed for determining good initial guesses for the unknown property coefficients within about 10% error. The Levenberg-Marquardt method is then applied to refine the results to within a specified convergence criterion. The problem studied here is concerned with simultaneous estimation of temperature dependent thermal conductivity and heat capacity from the multiple spatial and temporal measurements taken during transient heat conduction. Interior temperature sensors are found to be necessary when the properties vary with respect to temperature. A statistical analysis is performed to determine approximate confidence bounds for estimating the thermal conductivity and heal capacity per unit volume

Laminar forced convection inside ducts with periodic variation of inlet temperature

Abstract Laminar forced convection with periodic variations of inlet temperature is studied in both parallel-plate channels and circular ducts. The generalized integral transform technique is employed to reduce the original problem to a system of linear first-order differential equations, which is then solved utilizing the related complex matrix eigenvalue problem. Amplitudes and phase lags with respect to the inlet condition are determined for fluid bulk temperature and wall heat flux, and the results are presented in graphical form as a function of the dimensionless axial distance along the channels for different values of the dimensionless frequency of inlet oscillations.

Diffusion in composite layers with automatic solution of the eigenvalue problem

Abstract The analytical treatment of transient heat conduction problems for one-dimensional multilayered composites by the orthogonal expansion technique requires the solution of a corresponding eigenvalue problem if this analytical solution is to be implemented for practical purposes. Such an eigenvalue problem is not of the conventional Sturm-Liouville type because of the discontinuities of the coefficient functions. Its solution with conventional techniques is not guaranteed from missing eigenvalues in the course of the computation. An analytical solution of one transient heat conduction problem in one-dimensional multilayered slabs, cylinders and spheres is presented, which implements a safe algorithm for the automatic computation of the eigenvalues and the eigenfunctions of the resulting Sturm-Liouville type system.

Two‐dimensional inverse heat conduction problem of estimating the time‐varying strength of a line heat source

Inverse analysis utilizing the conjugate gradient method is used to estimate the timewise varying strength of a line heat source placed at a specified location in a rectangular region with insulated boundaries. No prior information was used on the functional form of the source strength with time. Transient temperature recordings taken at the boundaries of the region served as the simulated experimental data needed as input for the inverse analysis. In order to test the accuracy of the inverse analysis under most strict conditions, timewise variation of the source strength is chosen in the form of rectangular, triangular, and sinusoidal functions. Both the regular iterated and modified conjugate gradient method are used for solving the inverse problem. The modified conjugate gradient method often provided the answer with smaller number of iterations when the initial condition for the unknown function was available.

Conjugate gradient method for determining unknown contact conductance during metal casting

Abstract The air-gap formation between the casting and the metal mold during the casting of metals creates a thermal resistance that reduces the heat transfer and solidification rates. In this work, the inverse solution methodology based on the conjugate gradient method is developed for estimating the variation of air-gap resistance with time from the transient temperature measurements taken with thermocouples inside the casting region and at the outer mold surface. The advantage of the conjugate gradient method is that there is no need to assume a specific functional form for the unknown quantity beforehand, since the solution automatically determines the functional form over the domain specified. Furthermore the method is stable and converges over an order of magnitude faster than the least square method.