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The purpose of this paper is to first present the key features of hybrid numerical methods that enable cost-effective simulations of complex thermofluid systems, and then demonstrate the formulation and application of such a method.

A hybrid numerical method is formulated for simulations of a closed-loop thermosyphon operating with slurries of a micro-encapsulated phase-change material suspended in distilled water. The slurries are modeled as homogeneous mixtures, with inputs of effective properties and overall heat-loss coefficients. Combinations of an axisymmetric two-dimensional (2D) control-volume finite-element method and a segmented-quasi-one-dimensional (1D) model are used to achieve cost-effective simulations. Proper matching of the solutions at the interfaces between adjacent axisymmetric 2D and quasi-1D zones is ensured by incorporating and heuristically determining suitable lengths of pre- and post-heating (and also pre- and post-cooling) sections.

In the demonstration problem, which would strictly require full three-dimensional simulations of the fluid flow and heat transfer phenomena, the proposed hybrid 1D/2D numerical method produces results that are in very good agreement with those obtained in a complementary experimental investigation.

The hybrid numerical methods discussed in this paper allow cost-effective computer simulations of complex thermofluid systems. These methods can therefore serve as very useful tools for the design, parametric studies, and optimization of such systems.

The purpose of this paper is to present outcomes of efforts made over the last 20 years to extend the applicability of the Richardson extrapolation procedure to numerical predictions of multidimensional, steady and unsteady, fluid flow and heat transfer phenomena in regular and irregular calculation domains.

Pattern-preserving grid-refinement strategies are proposed for mathematically rigorous generalizations of the Richardson extrapolation procedure for numerical predictions of steady fluid flow and heat transfer, using finite volume methods and structured multidimensional Cartesian grids; and control-volume finite element methods and unstructured two-dimensional planar grids, consisting of three-node triangular elements. Mathematically sound extrapolation procedures are also proposed for numerical solutions of unsteady and boundary-layer-type problems. The applicability of such procedures to numerical solutions of problems with curved boundaries and internal interfaces, and also those based on unstructured grids of general quadrilateral, tetrahedral, or hexahedral elements, is discussed.

Applications to three demonstration problems, with discretizations in the asymptotic regime, showed the following: the apparent orders of accuracy were the same as those of the numerical methods used; and the extrapolated results, measures of error, and a grid convergence index, could be obtained in a smooth and non-oscillatory manner.

Strict or approximate pattern-preserving grid-refinement strategies are used to propose generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions. Such extrapolation procedures play an indispensable role in the verification and validation techniques that are employed to assess the accuracy of numerical predictions which are used for designing, optimizing, virtual prototyping, and certification of thermofluid systems.