Exact Closure Procedures of Hierarchical VAT Capillary Thermo-Convective Problem for Turbulent and Laminar Regimes

Volume Averaging Theory (VAT), an effective and rigorous approach for study of transport (laminar and turbulent) phenomena, is used to model flow and heat transfer in porous media. The modeling is based on a simple pore level network. The primary difficulties in applying VAT to straight capillary networks, the many unknown integral and differential terms that are needed for closure, are overcome. VAT based modeling of pore level transport in straight capillaries results in two sets of scale governing equations. One scale is the upper scale VAT equations which describe ensemble properties for flow and heat transfer in porous media. The other scale is the lower scale laminar and turbulent transport equations that represent flow and heat transport in each straight pore capillary. It is how the unknown VAT terms in the upper scale equations can be estimated using the relationships between upper scale properties and lower scale properties. Exact closures and mathematical procedures are developed for the turbulent regime, extending the previous laminar regime work. Numerical results for turbulent and laminar transport in straight capillary porous media are shown in this paper.