Heat transfer—a review of 1981 literature

The present work considers the application of the generalized integral transform technique (GITT) in the solution of a class of linear or nonlinear convection–diffusion problems, by fully or partially incorporating the convective effects into the chosen eigenvalue problem that forms the basis of the proposed eigenfunction expansion. The aim is to improve convergence behavior of the eigenfunction expansions, especially in the case of formulations with significant convective effects, by simultaneously accounting for the relative importance of convective and diffusive effects within the eigenfunctions themselves, in comparison against the more traditional GITT solution path, which adopts a purely diffusive eigenvalue problem, and the convective effects are fully incorporated into the problem source term. After identifying a characteristic convective operator, and through a straightforward algebraic transformation of the original convection–diffusion problem, basically by redefining the coefficients associated with the transient and diffusive terms, the characteristic convective term is merged into a generalized diffusion operator with a space-variable diffusion coefficient. The generalized diffusion problem then naturally leads to the eigenvalue problem to be chosen in proposing the eigenfunction expansion for the linear situation, as well as for the appropriate linearized version in the case of a nonlinear application. The resulting eigenvalue problem with space variable coefficients is then solved through the GITT itself, yielding the corresponding algebraic eigenvalue problem, upon selection of a simple auxiliary eigenvalue problem of known analytical solution. The GITT is also employed in the solution of the generalized diffusion problem, and the resulting transformed ordinary differential equations (ODE) system is solved either analytically, for the linear case, or numerically, for the general nonlinear formulation. The developed methodology is illustrated for linear and nonlinear applications, both in one-dimensional (1D) and multidimensional formulations, as represented by test cases based on Burgers’ equation. [DOI: 10.1115/1.4037576]

Convective Eigenvalue Problems for Convergence Enhancement of Eigenfunction Expansions in Convection–Diffusion Problems

The formulation and testing of a control-volume finite element method (CVFEM) for the prediction of three-dimensional, linear and nonlinear, diffusion-type phenomena in anisotropic media in irregular calculation domains are presented and discussed in this paper. In this CVFEM, the calculation domain is discretized into four-node tetrahedral elements. Contiguous, non-overlapping, polyhedral control volumes are then associated with each node, and the governing differential equation is integrated over these control volumes. The dependent variable is interpolated linearly in each four-node tetrahedral element. Centroidal values of the diffusion coefficients are stored and assumed to prevail over the corresponding tetrahedral element. The source term is linearized, and nodal values of its coefficients are stored and assumed to prevail over the polyhedral sub-control volumes. Using these interpolation functions, the discretized equations, which are algebraic approximations to the integral conservation equations, are derived. The discretized equations, which in general, are nonlinear and coupled, are solved using an iterative procedure. The proposed CVFEM for the solution of anisotropic diffusion- type problems appears to be the first such method that is based on tetrahedral elements and vertex-centered polyhedral control volumes. These features make it particularly attractive for amalgamation with adaptive-grid schemes and applications to problems with complex irregular geometries, such those encountered in the general areas of drying, ground-water flows, conduction in composite materials, injection molding in heterogeneous porous media, and solidification. The proposed three-dimensional CVFEM and its computer implementation were tested using several steady conduction-type problems, for which analytical solutions were constructed using a special technique. In all cases, the agreement between the numerical and analytical solutions was excellent.

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A control-volume finite element method for the prediction of three-dimensional diffusion-type phenomena in anisotropic media

An integral transform approach to the solution of the problem on conjugate heat transfer, combining the singledomain formulation with the convective eigenfunction expansion basis within the total integral transformation framework, which leads to a nonclassical eigenvalue problem, is presented. The problem on the conjugate heat transfer in the transient two-dimensional incompressible laminar flow of a Newtonian fluid in a parallel-plate channel is considered to illustrate the hybrid numerical-analytical approach. To demonstrate the improvement of the convergence rate achieved with the methodology proposed, a critical comparison against the traditional total integral transformation solution of the diffusive eigenvalue problem is provided, and results are presented and discussed for three representative situations realized with different Peclet numbers: Pe = 1, 10 and 100. A remarkable improvement of the convergence rate, obtained especially with the large Peclet numbers, offers evidence of the validity of the expansion constructed upon the nonclassical eigenvalue problem proposed.

Conjugate Heat Transfer: Analysis Via Integral Transforms and Eigenvalue Problems

A special matrix is introduced, the elements of which are zero or first-order operators This matrix is used to define the initial boundary value problem which covers a wide class of engineering applications To bring the solution of the problem within the realm of integral transform theory, a new finite integral transform and the corresponding inversion formula are developed The necessary orthogonality relations is derived using an extension of the Gauss-Green theorem The formal solution obtained here can be applied to study heat conduction problems, elasticity problems, etc

M.D. Mikhailov

Papers

On general solution of heat conduction in an anisotropic medium