In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by 
$$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot (\nabla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$

 The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.

The Forchheimer equation : A theoretical development

Developing a new form of permeability and Kozeny-Carman constant for homogeneous porous media by means of fractal geometry

An investigation of the permeability for viscous flow through random fiber structures used as preforms in composite fabrication processes is presented. A method derived from the electrical conduction principles is employed to predict the viscous permeability of the fiber structures directly from their formation factor, specific surface area, and porosity. The recent Brownian diffusion random-walk simulation results on the molecular survival time are used to derive an upper bound and an approximation to the viscous permeability of these structures. The results are compared to experimental data and theoretical models of the literature. It is found that the conduction-based method provides very good permeability estimates in most cases, resulting in an overall ratio of the theoretical prediction to experimental measurement in the proximity of 1 for the over 500 experimental points utilized. This is one of the most comprehensive experimental validations of the conduction-based method to date and its first val...

Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates with Experimental and Analytical Results

We report the results of physical experiments that demonstrate the strong influence of the thermal conductivity of the substrate on the evaporation of a pinned droplet. We show that this behaviour can be captured by a mathematical model including the variation of the saturation concentration with temperature, and hence coupling the problems for the vapour concentration in the atmosphere and the temperature in the liquid and the substrate. Furthermore, we show that including two ad hoc improvements to the model, namely a Newton's law of cooling on the unwetted surface of the substrate and the buoyancy of water vapour in the atmosphere, give excellent quantitative agreement for all of the combinations of liquid and substrate considered.

/pdf/the-strong-influence-of-substrate-conductivity-on-droplet-3az2v2hqhh.pdf

The strong influence of substrate conductivity on droplet evaporation

The growing interest in energy applications of molten salts is justified by several of their properties. Their possibilities of usage as a coolant, heat transfer fluid or heat storage substrate, require thermo-hydrodynamic refined calculations. Many researchers are using simulation techniques, such as Computational Fluid Dynamics (CFD) for their projects or conceptual designs. The aim of this work is providing a review of basic properties (density, viscosity, thermal conductivity and heat capacity) of the most common and referred salt mixtures. After checking data, tabulated and graphical outputs are given in order to offer the most suitable available values to be used as input parameters for other calculations or simulations. The reviewed values show a general scattering in characterization, mainly in thermal properties. This disagreement suggests that, in several cases, new studies must be started (and even new measurement techniques should be developed) to obtain accurate values.

Molten salts database for energy applications

Experimental studies have been carried out on fluid flow through porous media made up of randomly packed monodisperse fibers. The permeability and the Kozeny-Carman parameter K k are deduced from experimental results. The variations of the permeability increase exponentially with the porosity. The parameter K k is a decreasing function of the porosity ∈ and tends asymptotically to a value close to that deduced from a modified Ergun relation. The important decrease, observed for small aspect ratios, is certainly an effect of the cut sections of fibers. The results in terms of parameter K k are systematically compared to those deduced from various theoretical models. The variation laws of the parameter K k , deduced from different models, present important discrepancies with our experimental results.

Fluid Flow Through Randomly Packed Monodisperse Fibers: The Kozeny-Carman Parameter Analysis

Etude expérimentale de l'évaporation d'une goutte posée sur une plaque chauffante. Influence de la mouillabilité

Measurement of thermal conductivity of molten salts in the range 100–500°C

Experimental study of the particle flow in a circulating fluidized bed using a phase Doppler particle analyser : A new post-processing data algorithm

The 3D periodic state, following the steady thermocapillary convection state, around an air bubble immersed in a low-Prandtl-number silicone oil layer under a heated wall was experimentally investigated on the ground and during parabolic flights. This oscillation was observed under reduced gravity conditions for the first time. Consequently, the initiation of this oscillation seems to be independent of gravity and so of buoyancy convection. The reduced and increased gravity conditions showed that the gravity level modifies the oscillation. Its frequency increases with the gravity level. The comparison with the results obtained on the ground shows the bubble aspect ratio is not a relevant parameter when the gravity varies.

R. Santini

Papers

Fluid Flow Through Randomly Packed Monodisperse Fibers: The Kozeny-Carman Parameter Analysis

Etude expérimentale de l'évaporation d'une goutte posée sur une plaque chauffante. Influence de la mouillabilité

Measurement of thermal conductivity of molten salts in the range 100–500°C

Experimental study of the particle flow in a circulating fluidized bed using a phase Doppler particle analyser : A new post-processing data algorithm

Experimental study of the gravity influence on the periodic thermocapillary convection around a bubble