Darcy Forchheimer Law Research Articles

Characterization of laminar flow in periodic open-cell porous structures

The present work deals with the numerical simulations of 3D incompressible flow through periodic porous structures, consisting of cubic cells with different strut geometry and structure tilting with respect to the flow direction. Simulation results are obtained for different porosities, while covering a range of Reynolds numbers from laminar-steady flow (Darcy regime) up to the transition to laminar-unsteady flow (moderate Forchheimer regime). These results are post-processed in order to characterize the influence of the periodic porous structures on the flow statistics related to the Darcy-Forchheimer law. Based on this analysis, along with 3D simulation results previously obtained for different random foams, correlations for the Darcy and Forchheimmer permeability coefficients are developed from a theoretical background, being expressed only as functions of the cell geometrical parameters and tortuosity estimation. However, the influences of flow tortuosity and cell orientation on the Darcy coefficient are found to cancel each other. The proposed correlation for the Darcy coefficient performs very well for both periodic and random porous structures, when compared with recent correlations available in the literature. The performance of the correlation proposed for the Forchheimmer coefficient is found to be satisfactory, however, different correlation constants were found for periodic and random structures. The proposed correlation for the Forchheimer permeability coefficient depends on the flow tortuosity inside the porous structure, which must be estimated either through 3D fluid flow simulations or using tortuosity correlations. In this respect, a correlation is proposed for predicting the flow tortuosity inside porous structures, as an alternative for avoiding 3D simulations.

CFD-modelling of activated carbon fibers for indoor air purification

Activated carbon fibers for indoor air purification were investigated by means of pressure drop and adsorption capacity. The Darcy-Forchheimer law combined with Computational Fluid Dynamics (CFD) modelling was deployed to simulate the pressure drop over an activated carbon fiber (ACF) filter with varying filter thickness. The CFD model was later combined with adsorption modelling to simulate breakthrough profiles of acetaldehyde adsorption on the ACF-filter. The adsorption model incorporates mass transfer resistance and adsorption equilibrium. It assumes local equilibrium between gas phase and solid phase. The latter was investigated for three different adsorption isotherms: linear, Langmuir and Freundlich adsorption. Successful agreement between model simulations and experimental data was obtained, using the Freundlich adsorption model. The numerical model could provide valuable insights and allows to continuously improve the design of filtration devices.

Thermal computational analysis of microclimates for optimal crop production in controlled atmosphere

This study aims to perform a thermal computational analysis of the indoor environment with a view to identifying a promising method to improve the level of harvest in a regulated indoor space or modified version of controlled environment agriculture (CEA) space. A modified closed cultivated greenhouse model is considered based on a customary model. The operational environment of the refined CEA space model is simulated in computational fluid dynamics environment. In the study, CEA space is assumed to be heated by a constant heat source from the same path for a limited time. The CEA space has two air inlets on one side and two outlets on the other side. A standard k-epsilon model was implemented in the simulation model to define the turbulent transport phenomena. On the virtual environment, the crop is defined using a porous medium approach that was proposed by Darcy-Forchheimer Law. The results are recorded in the form of temperature, velocity and pressure distribution.

Effect of Nonlinear Thermal Radiation, Heat Source on MHD 3D Darcy-Forchheimer Flow of Nanofluid Over Aa Porous Medium with Chemical Reaction

The present work nonlinear thermal radiation and chemical reaction effect on three-dimensional MHD flow of permeable medium analysed. We are considering introduce the Darcy-Forchheimer law along with axial and transverse velocity. Using suitable transportations the nonlinear partial differential equations are converted into ordinary differential equations. These equations are solved numerically by 4th Runge-Kutta-Fehlberg scheme with shooting procedure. We are getting unique numerical solution for distinct physical variables temperature and concentration fields are depicted. Also the heat transfer and skin friction coefficients drawn through numerical data. We are finding great results of the velocity profiles behaviors opposite trend of porosity and Forchheimer parameters, the profiles of and behavior reverse trend follows other than chemical reaction parameter, both directions of skin friction coefficient and heat transfer rates reduction.

Porous media representation of louvers in building simulations for natural ventilation

Many buildings have louvers on their facades or windows, as they reduce penetration of wind-driven rain, indoor glare and irradiance from the sun and external noise. However, these louvers typically have millimetre-scale blade thicknesses and slat pitch, while spanning metres across the building in a full simulation domain of several kilometres. Consequently, computational fluid dynamic models require vastly increased cell counts to explicitly resolve these fine features for accuracy, which is computationally expensive. An alternative is to utilize a porous media model based on the Darcy–Forchheimer law to reduce the cell count. We present a method to determine parameters for a porous media representation of louvers, and quantify the associated deviation in velocity and pressure in a simplified building simulation. We show errors in the air velocities after passage through the porous media zone ranges from <10% in the simplest models to 38% in more complex cases.

Effects of manufacturing errors on the static characteristics of aerostatic journal bearings with porous restrictor

Numerical and experimental research on the effects of manufacturing errors on the static performance of aerostatic porous journal bearings is presented. The effects of circumferential waviness, taper, concavity, and convexity on the bearing film thickness were modeled based on the experimental results. The effects of amplitude and spatial wave length on the bearing film pressure, load capacity and stiffness were investigated by solving the Darcy-Forchheimer law and Navier-Stokes equations. Both the numerical results and experimental data show that the bearing static characteristics were significantly influenced by the manufacturing errors. Circumferential waviness errors caused the obvious inhomogeneity of the flow field and the transformation of morphology of the high-pressure region, whereas axial errors had a significant impact on the area and location of the high-pressure region. The bearing load capacity and stiffness can be improved by increasing the amplitude of the manufacturing errors, except for the concavity errors. Both the bearing load capacity difference and stiffness difference increased with an increase in the axial errors' amplitude. For the manufacturing errors, the calculation results are close to the measurement data. Therefore, it is necessary to consider the influence of manufacturing errors on the bearing static characteristics in numerical calculations.

Fluid flow through replicated microcellular materials in the Darcy-Forchheimer regime

We extend here a “bottleneck” flow model derived earlier for incompressible fluids flowing under creeping flow conditions [Despois, J. and Mortensen, A: Acta Materialia 53 (2005) 1381] to flow regimes where inertial losses are no longer negligible, causing the governing flow law to deviate from Darcy's law and become the Darcy-Forchheimer law. The proposed law is compared with measurements of the Darcian permeability KD and of the Forchheimer coefficient C in forced-flow of air through microcellular aluminium made by the replication process. The geometrical features of the cellular medium are varied in terms of volume fraction of porosity (in the range of 0.66–0.86) and the average cell diameter from (108–425 μm). As found previously in measurements with water, the Darcy permeability of the foams for airflow is also reasonably well captured by the model. In the Forchheimer-regime the model gives good quantitative agreement with data if one assumes that the amount of air kinetic energy that is dissipated when passing across each bottleneck linking one pore to its neighbour along the fluid flow path corresponds to the difference, in a stream of constant cross-sectional area, between a uniform fluid velocity profile and the non-uniform profile that is created by the no-slip condition along the window boundary.

A Numerical Analysis on a Compact Heat Exchanger in Aluminum Foam

A numerical investigation on a compact heat exchanger in aluminum foam is carried out. The governing equations in two-dimensional steady state regime are written in local thermal non-equilibrium (LTNE). The geometrical domain under investigation is made up of a plate in aluminum foam with inside a single array of five circular tubes. The presence of the open-celled metal foam is modeled as a porous media by means of the Darcy-Forchheimer law. The foam has a porosity of 0.93 with 20 pores per inch and the LTNE assumption is used to simulate the heat transfer between metal foam and air. The compact heat exchanger at different air flow rates is studied with an assigned surface tube temperature. The results in terms of local heat transfer coefficient and Nusselt number on the external surface of the tubes are given. Moreover, local air temperature and velocity profiles in the smaller cross section, between two consecutive tubes, as a function of Reynolds number are showed. The performance evaluation criteria (PEC) is assessed in order to evaluate the effectiveness of the metal foam.

Numerical simulation and analysis of fluid flow hydrodynamics through a structured array of circular cylinders forming porous medium

Numerical analysis of the mechanisms that govern the flow in a porous region is crucial for modeling porous media flows. This study describes an adapted and efficient turbulence modeling technique for this category of porous media flows. The main objective of the present study is to provide a detailed pore scale description of fluid flow and to analyze the formation of coherent structures in the wake region close to the solid wall subject to the effects of fine-scale turbulence. The computations were performed in a three-dimensional representative elementary volume (REV) of a porous matrix, which comprised a periodic array of circular cylinders. Two flow-modeling strategies were employed: the steady Reynolds averaged Navier–Stokes (RANS) and transient large eddy simulation (LES) approaches. In the RANS modeling framework, both standard k–ε and low Re k–ε turbulence models were used. The porosity (ϕ) of the porous REV and Reynolds number (ReD) varied from 0.3 to 0.8 and 10 to 40,000, respectively. We investigated the effects of porosity and ReD on the pore scale velocity distribution, overall macroscopic pressure gradient, and turbulent dynamics, and performed comparisons using RANS and LES calculations within the porous REV. The Darcy parameter (i.e., medium permeability, K) was computed and a correlation was determined between the medium porosity and permeability. The macroscopic pressure gradients in the three-dimensional REV were also compared with the Kozeny–Carman and Darcy–Forchheimer law.

Experimental study of single-phase pressure drops in coarse particle beds

Motivated by uncertainty reduction in nuclear debris beds coolability, experiments have been conducted on the CALIDE facility in order to investigate single-phase pressure losses in representative debris beds, i.e., high sphericity (>80%) particle beds with small size dispersion (from 1mm to 10mm), for which no validated model exists. In this paper, experimental results are presented and analyzed in order to identify a simple correlation for single-phase flow pressure losses generated in this kind of porous media in reflooding flowing conditions, which cover Darcy to weakly turbulent regimes. In the literature, it has been observed that their behavior can be accurately described by a Darcy–Forchheimer law, involving the sum of a linear term and a quadratic non-linear deviation, with respect to the filtration velocity. Expressions for the coefficients of the linear and quadratic terms are determined by assessing the possibility to evaluate equivalent diameters, i.e., characteristic lengths allowing correct predictions of the linear and quadratic terms by the Ergun equation. It has been observed that the Sauter diameter of particles allows a very precise prediction of the linear term, while the quadratic term can be predicted using the product of the Sauter diameter and a sphericity coefficient as an equivalent diameter.

Experimental investigation on single-phase pressure losses in nuclear debris beds: Identification of flow regimes and effective diameter

During a severe nuclear power plant accident, the degradation of the reactor core can lead to the formation of debris beds. The main accident management procedure consists in injecting water inside the reactor vessel. Nevertheless, large uncertainties remain regarding the coolability of such debris beds. Motivated by the reduction of these uncertainties, experiments have been conducted on the CALIDE facility in order to investigate single-phase pressure losses in representative debris beds. In this paper, these results are presented and analyzed in order to identify a simple single-phase flow pressure loss correlation for debris-bed-like particle beds in reflooding conditions, which cover Darcean to Weakly Turbulent flow regimes.The first part of this work is dedicated to study macro-scale pressure losses generated by debris-bed-like particle beds, i.e., high sphericity (>80%) particle beds with relatively small size dispersion (from 1mm to 10mm). A Darcy–Forchheimer law, involving the sum of a linear term and a quadratic deviation, with respect to filtration velocity, has been found to be relevant to describe this behavior in Darcy, Strong Inertial and Weak Turbulent regimes. It has also been observed that, in a restricted domain (Re=15 to Re=30) between Darcy and Weak Inertial regimes, deviation is better described by a cubic term, which corresponds to the so-called Weak Inertial regime.The second part of this work aims at identifying expressions for coefficients of linear and quadratic terms in Darcy–Forchheimer law, in order to obtain a predictive correlation. In the case of monodisperse beds, and according to the Ergun equation, they depend on the porosity of the medium, empirical constants and the diameter of the particles. Applicability of the Ergun equation for debris-bed-like particle beds has been investigated by assessing the possibility to evaluate equivalent diameters, i.e., characteristic length allowing correct predictions of linear and quadratic terms by the Ergun equation. It has been observed that the Sauter diameter of particles allows a very precise prediction of the linear term, by less than 10% in most cases, while the quadratic term can be predicted using the product of the Sauter diameter and a sphericity coefficient as an equivalent diameter, by about 15%.

Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture

We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy-Forchheimer law while that in the surrounding matrix is governed by Darcy's law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy-Forchheimer law is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy's law in the matrix is the weak limit of solutions of the model with the Darcy-Forchheimer law in the entire domain when the Forchheimer coefficient in the matrix tends toward zero.

Computation of Flow Through a Three-Dimensional Periodic Array of Porous Structures by a Parallel Immersed-Boundary Method

A parallel implementation of an immersed-boundary (IB) method is presented for low Reynolds number flow simulations in a representative elementary volume (REV) of porous media that are composed of a periodic array of regularly arranged structures. The material of the structure in the REV can be solid (impermeable) or microporous (permeable). Flows both outside and inside the microporous media are computed simultaneously by using an IB method to solve a combination of the Navier–Stokes equation (outside the microporous medium) and the Zwikker–Kosten equation (inside the microporous medium). The numerical simulation is firstly validated using flow through the REVs of impermeable structures, including square rods, circular rods, cubes, and spheres. The resultant pressure gradient over the REVs is compared with analytical solutions of the Ergun equation or Darcy–Forchheimer law. The good agreements demonstrate the validity of the numerical method to simulate the macroscopic flow behavior in porous media. In addition, with the assistance of a scientific parallel computational library, PETSc, good parallel performances are achieved. Finally, the IB method is extended to simulate species transport by coupling with the REV flow simulation. The species sorption behaviors in an REV with impermeable/solid and permeable/microporous materials are then studied.

The Darcy-Forchheimer law for modelling fluid flow in biological tissues

The motion of the interstitial uid of a biological tissue is studied by employing the Darcy-Forchheimer law, a correction to standard Darcy's law. The tissue is modelled as a saturated biphasic medium comprising the fluid and a deformable matrix. The reason for undertaking this study is that a description of the tissue's dynamics based on the Darcy-Forchheimer law might be more complete than the one based on Darcy's law, since the former provides a better macroscopic representation of the microscopic fluid-solid interactions. Through numerical simulations, we analyse the influence of the Forchheimer's correction.

Lattice Boltzmann simulations of flow past a circular cylinder and in simple porous media

The single relaxation time variant of the Lattice-Boltzmann Method (LBM) is applied to the two–dimensional simulations of viscous fluids. Two flow geometries are considered: a single obstacle (circular cylinder) and a system of obstacles making up a simple, computer-created porous medium. A comparative study of two boundary schemes at the fluid–solid interface is performed. Although reasonable results can be achieved with the usage of (non-equilibrium) half-way bounce–back conditions, the interpolation-free scheme is recommended because of its better accuracy and stability. For flow past a circular cylinder, the lift and drag coefficients are computed together with the Strouhal numbers for periodically shedding vortices and validated against empirical relationships for a wide range of Reynolds numbers. The single cylinder case was also used for a novel comparison of two outlet schemes. Next, the system of obstacles that makes up a porous medium is created in two ways: by a regular arrangement of identical obstacles and by a simple randomization of cylinder centres and radii. The pressure loss in function of volumetric flow rate is compared with empirical relationships: the Darcy–Forchheimer law and the Ergun correlation. The impact of the LBM boundary schemes and the Reynolds number on the permeability and Forchheimer coefficients is studied for the two arrangements.

Extension of the Darcy–Forchheimer Law for Shear-Thinning Fluids and Validation via Pore-Scale Flow Simulations

Flow of non-Newtonian fluids through porous media at high Reynolds numbers is often encountered in chemical, pharmaceutical and food, as well as petroleum and groundwater engineering, and in many other industrial applications. Under the majority of operating conditions typically explored, the dependence of pressure drops on flow rate is non-linear and the development of models capable of describing accurately this dependence, in conjunction with non-trivial rheological behaviors, is of paramount importance. In this work, pore-scale single-phase flow simulations conducted on synthetic two-dimensional porous media are performed via computational fluid dynamics for both Newtonian and non-Newtonian fluids and the results are used for the extension and validation of the Darcy–Forchheimer law, herein proposed for shear-thinning fluid models of Cross, Ellis and Carreau. The inertial parameter β is demonstrated to be independent of the viscous properties of the fluids. The results of flow simulations show the superposition of two contributions to pressure drops: one, strictly related to the non-Newtonian properties of the fluid, dominates at low Reynolds numbers, while a quadratic one, arising at higher Reynolds numbers, is dependent on the porous medium properties. The use of pore-scale flow simulations on limited portions of the porous medium is here proposed for the determination of the macroscale-averaged parameters (permeability K, inertial coefficient β and shift factor α), which are required for the estimation of pressure drops via the extended Darcy–Forchheimer law. The method can be applied for those fluids which would lead to critical conditions (high pressures for low permeability media and/or high flow rates) in laboratory tests.

Analysis of permeability and effective viscosity by CFD on isotropic and anisotropic metallic foams

Hydrodynamic properties of open-cell metallic foams are analyzed at the pore scale on a microtomographied sample from creeping flow to unsteady inertial flow. The influence of the anisotropy, obtained by shearing the initial sample, is also analyzed. The simulations show that the Darcy–Forchheimer law is valid at R e dl > 1 and that the inertial coefficient tensor can be asymmetric as proved by Whitaker [1996. The Forchheimer equation: a theoretical development. Transp. Porous Media 25, 27–61] and checked by the lattice Boltzmann method. Despite this property, the eigen vectors are nearly orthogonal to each other and their orientation follows the shear angle. To evaluate the accuracy of the results, the analysis of the REV leads to a REV size lower than 2.5 d p for the velocity and is lower than 4.5 d p for the inertial flow permeability tensor. It is shown also that the CFD approach, validated by MRI investigation [Graf von den Schulenburg et al., 2007. Flow through an evolving porous media-compressed foam. J. Mater. Sci. 42 6541–6548], matches the 3D-Brinkman–Forchheimer model if the effective viscosity is close to two.

Geometric framework for modeling nonlinear flows in porous media, and its applications in engineering

Abstract Our work is focused on certain theoretical aspects of non-linear non-Darcy flows in porous media, and their application in reservoir and hydraulic engineering. The goal of this paper is to develop a mathematically rigorous framework to study the dynamical processes associated to nonlinear Forchheimer flows for slightly compressible fluids. Using fundamental geometric methods, we have proved the existence of a nonlinear scaling operator which relates constant mean curvature surfaces and time invariant pressure distribution graphs constrained by the Darcy–Forchheimer law. The hereby obtained properties of fast flows and their geometric interpretation can be used as analytical tools to evaluate important technological parameters in reservoir engineering.

Simulation of air permeation through a moving plug in a vertical pipe

Air permeation through a plug, moving in a vertical pipe with a constant velocity, was investigated theoretically. The preliminary experiments confirmed the feasibility of Darcy-Forchheimer's law for air flow through a plug. Although the experiments were conducted with granular materials, the theoretical investigation was expanded to include cohesive materials (powders). On this basis, a theoretical model that predicts the air velocity, pressure drop and plug velocity for conveying a cohesive plug in a vertical pipe was developed. Further experiments were conducted to characterize the conveying parameters of different particulate solids and to confirm the theoretical model. Pressure drop, air and plug velocities were examined as functions of different parameters. Parametric investigation showed that the plug velocity is lower than the gas velocity, due to the gas flow through the plug.