A single-phase, ID mathematical formulation is developed in radial/cylindrical coordinates to examine unsteady-state micropore sorption in a composite micropore/fracture, coalbed-methane transport problem. In the formulation, the micropore transport equation accounts for unsteady-state sorption and diffusion in the primary porosity. Gas entering the fracture network is considered a source term in the fracture-transport equation. The micropore and fracture systems are coupled by equating the gas pressure at the surface of the micropore elements to the pressure in the fracture network.
A single-phase, ID mathematical model, formulated in Part 1 of this study, is used to study unsteady-state micropore sorption in the composite micropore/fracture coalbed-methane-transport problem. The mathematical model is solved numerically by writing the transport equations in finite-difference form and linearizing the residual form of the difference equations with the generalized Newton-Raphson procedure. The numerical model is used to compare methane production rates predicted by unsteady-and quasisteadystate sorption formulations. Results indicate that the two models give different rates during early degasification periods. The high rates predicted by the unsteady-state model, however, generally approached lower quasisteady-state rates within the first few months of simulation.The numerical model is also used to examine the time-dependent response of concentration gradients in the micropores to changes in fracture pressure, to compare diffusion rates predicted by spherical and cylindrical micropore elements, and to construct dimensionless type-curve solutions to the coalbed-methane flow problem.
Purpose
– The purpose of this paper is to numerically investigate steady, laminar natural and mixed convection heat transfer in a two-dimensional cavity by using a finite volume method with a fourth-order approximation of convective terms, with and without the presence of nanoparticles. Highly accurate benchmark results are also provided.
Design/methodology/approach
– A finite volume method on a non-uniform staggered grid is used for the solution of two-dimensional momentum and energy conservation equations. Diffusion terms, in the momentum and energy equations, are approximated using second-order central differences, whereas a non-uniform four-point fourth-order interpolation (FPFOI) scheme is developed for the convective terms. Coupled mass and momentum conservation equations are solved iteratively using a semi-implicit method for pressure-linked equation method.
Findings
– For the case of natural convection problem at high-Rayleigh numbers, grid density must be sufficiently high in order to obtain grid-independent results and capture reality of the physics. Heat transfer enhancement for natural convection is observed up to a certain value of the nanoparticle volume fraction. After that value, heat transfer deterioration is found with increasing nanoparticle volume fraction.
Originality/value
– Developed a non-uniform FPFOI scheme. Highly accurate benchmark results for the heat transfer of Al2O3-water nanofluid in a cavity are provided.
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