Erratum: “Homogenization of the linearized ionic transport equations in rigid periodic porous media” [J. Math. Phys. 51, 123103 (2010)]

2011 ◽  
Vol 52 (6) ◽  
pp. 063701 ◽  
Author(s):  
Grégoire Allaire ◽  
Andro Mikelić ◽  
Andrey Piatnitski
2006 ◽  
Vol 65 (1) ◽  
pp. 107-131 ◽  
Author(s):  
Jason R. Looker ◽  
Steven L. Carnie

2010 ◽  
Vol 51 (12) ◽  
pp. 123103 ◽  
Author(s):  
Grégoire Allaire ◽  
Andro Mikelić ◽  
Andrey Piatnitski

2011 ◽  
Vol 133 (10) ◽  
Author(s):  
Ivan Catton

Optimization of heat exchangers (HE), compact heat exchangers (CHE) and microheat exchangers, by design of their basic structures is the focus of this work. Consistant models are developed to describe transport phenomena in a porous medium that take into account the scales and other characteristics of the medium morphology. Equation sets allowing for turbulence and two temperature or two concentration diffusion are obtained for nonisotropic porous media with interface exchange. The equations differ from known equations and were developed using a rigorous averaging technique, hierarchical modeling methodology, and fully turbulent models with Reynolds stresses and fluxes in the space of every pore. The transport equations are shown to have additional integral and differential terms. The description of the structural morphology determines the importance of these terms and the range of application of the closure schemes. A natural way to transfer from transport equations in a porous media with integral terms to differential equations with coefficients that could be experimentally or numerically evaluated and determined is described. The relationship between computational fluid dynamics, experiment and closure needed for the volume averaged equations is discussed. Mathematical models for modeling momentum and heat transport based on well established averaging theorems are developed. Use of a “porous media” length scale is shown to be very beneficial in collapsing complex data onto a single curve yielding simple heat transfer and friction factor correlations. The general transport equations developed for a single phase fluid in a heat exchange medium have many more integral and differential terms than the homogenized or classical continuum mechanics equations. Once these terms are dealt with by closure, the resulting equation set is relatively simple and their solution is obtained using simple numerical methods quickly enough for multiple parameter optimization using design of experiment or genetic algorithms. Current efforts to significantly improve the performance of an HE for electronic cooling, a two temperature problem, and of a finned tube heat exchanger, a three temperature problem, are described.


2011 ◽  
Vol 688 ◽  
pp. 219-257 ◽  
Author(s):  
Manav Tyagi ◽  
Patrick Jenny

AbstractA probabilistic approach to model macroscopic behaviour of non-wetting-phase ganglia or blobs in multi-phase flow through porous media is proposed. The key idea is to consider a set of stochastic Markov processes that can mimic the microscopic multi-phase dynamics. These processes are characterized by equilibrium probability density functions (PDFs) and correlation times, which can be obtained from micro-scale simulation studies or experiments. A Lagrangian viewpoint is adopted, where stochastic particles represent infinitesimal fluid elements and evolve in the physical and probability space. Ganglion mobilization and trapping are modelled by a two-state jump process with transition probabilities given as functions of ganglion size. Coalescence and breakup of ganglia influence the ganglion size distribution, which is modelled by a Langevin type equation. The joint probability density function (JPDF) of the chosen stochastic variables is governed by a high-dimensional Chapman–Kolmogorov equation. This equation can be used to derive moment (e.g. saturation, mean mobility etc.) transport equations, which in general do not form a closed system. However, in some special cases, which arise in the limit of one time scale being smaller or larger than the others, a closed set of moment transport equations can be obtained. For slowly varying and quasi-uniform flows, the saturation transport equation appears in closed form with the mean mobility fully determined, if the equilibrium PDFs are known. Furthermore, it is shown how statistical parameters such as mobilization and trapping rates and equilibrium PDFs can be obtained from the birth–death type approach, in which ganglia breakup and coalescence are explicitly considered. A two-equation transport model (one equation for the total saturation and one for the trapped saturation) is obtained in the limit of very fast coalescence and breakup processes. This model is employed to mimic hysteresis in relative permeability–saturation curves; a well known phenomenon observed in the successive processes of imbibition and drainage. For the general case, the JPDF-equation is solved using the stochastic particle method, which was proposed in our previous paper (Tyagi et al. J. Comput. Phys. 227, 2008, 6696–6714). Several one- and two-dimensional numerical simulation results are presented to show the influence of correlation times on the averaged macroscopic flow behaviour.


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