Darcy Regime Research Articles

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%.

Forced-Convection Measurements in the Fully Developed and Exit Regions of Open-Cell Metal Foam

Experimental heat transfer data for water flow in commercial, open-cell-aluminum-foam cylinder heated at the wall by a constant heat flux, are presented. The measurements include wall temperature along flow direction as well as average inlet and outlet temperatures of the water. Flow speeds were in the Darcy and non-Darcy (transitional and Forchheimer) regimes. Heat fluxes were 14,998 and \(26{,}347\,\hbox {W/m}^{2}\) for the Darcy and non-Darcy regimes, respectively. Measurements were focused on the thermally fully developed and an anticipated exit regions, with the latter region being often ignored in the literature. The experimental Nusselt number for the Darcy flow cases is compared to its analytical counterpart. A comparison shows good agreement, considering the approximations involved in the analytical solution and experimental errors. Previously unpublished phenomenon is presented in the behavior of Nusselt number for non-Darcy regimes. The experimental results and measuring technique can be used for validation of other analytical and numerical solutions, as well as in testing heat-exchange engineering designs based on metal foam.

Various Flow Regimes and Permeabilities for Packed-Spheres Porous Media

Flow in porous media occurs in many naturally-occurring and engineered systems. One of the key properties for understanding the fluid flow and pressure drop in porous media is permeability, which is varies widely among researchers. The current work presents systematic experimental data for packed spheres of uniform size (3 mm) having a porosity of 36.6% subjected to water flow. The experiments covered a sufficiently broad range of flow Reynolds number such that all flow regimes are encountered: pre-Darcy, Darcy, Forchheimer and Turbulent. The pre-Darcy regime is very scarce or non-present in the literature. As a necessary initial step, flow regimes were identified and different permeabilities exhibited by the porous medium in each flow regime were calculated. The length scales in defining the Reynolds number included the diameter of the sphere and the square root of the various permeabilities in order to study the transitional Reynolds numbers among the flow regimes. It is shown that the permeability in the Darcy regime is most appropriate and produces results consistent with accepted understanding in the literature of porous media.

Performance improvement of high power liquid-cooled heat sink via non-uniform metal foam arrangement

The present study proposes a novel concept by employing non-uniform arrangements of metal foams alongside the heat sink applicable for liquid cooling. A total of 7 arrangements by varying PPI (pores per inch) of metal foam are made and additional 4 multiple channel designs are fabricated for comparison. Test results indicate that non-linearity term in Forchheimer equation rises significantly with increase of PPI (or with the decrease of permeability). The results suggest that the transition from the linear Darcy regime to the quadratic Forchheimer regime depends on the metal foam structure. It is also found that metal foam appreciably outperforms conventional multiple channel heat sink. The thermal resistance can be reduced by more than 62% as compared to that of empty plate design. For a given pumping power, the non-uniform arrangement of metal foam shows the best performance by employing a larger PPI metal foam at the inlet, followed by a smaller PPI metal foam, and the least PPI metal foam should be placed at the rear part of the heat sink.

Thermal development in open-cell metal foam: An experiment with constant wall heat flux

Experimental heat transfer results for a commercial open-cell aluminum foam cylinder heated at the wall by a constant heat flux and cooled by water flow, are presented. The results cover thermal-entry and fully-developed regions. Measurements include wall temperature along flow direction as well as average inlet and outlet temperatures of the water. Flow rates are in the Darcy and non-Darcy (transitional and Forchheimer) regimes. The wall temperature along the foam clearly shows two distinct behaviors related to thermally-developing and fully-developed conditions, which is confirmed by the behavior of local Nusselt number. The thermal entry length is determined and discussed in detail; it is also compared to its analytical counterpart for Darcy flow. The thermal entry region in metal foam is found to be significant and much longer than its analytically-predicted value. A method for estimating the bulk fluid temperature is envisioned for calculating the local Nusselt number. Previously undiscussed phenomenon is captured in the behavior of Nusselt number for non-Darcy regimes, which suggests periodic thermal development along the foam. The fully-developed data for the Darcy flow cases is compared to its analytical counterpart, and a correlation for the Nusselt number as a function of Reynolds number is proposed for non-Darcy flows.

Magnetohydrodynamic flows in model porous structures

An interesting option for plasma-facing components such as divertors seems to be the use of liquid–metal saturated capillary porous structures, on which a stable liquid surface could be obtained by capillary forces. The aim of the present study is providing a first description of liquid metal flows in a capillary porous system (CPS) when an external uniform magnetic field is imposed, in order to judge the feasibility, from the magnetohydrodynamic (MHD) point of view, of the CPS technology for steady-state divertor applications. Numerical results show that, independently of the orientation of the magnetic field, the permeability strongly reduces in porous MHD flows compared to those in the corresponding hydrodynamic Darcy regime. Main parameters that influence MHD flows in CPS are identified.

Characteristics of oscillating liquid flow in foam-like highly-porous media: An experimental study

Characteristics of oscillating water flow in open-cell metal foam were obtained experimentally. The foam had 20pores per inch and a porosity of 87.6%. Three flow displacements 74.35, 97.2 and 111.53mm were applied in the flow frequency range 0.116–0.696Hz. Steady-state experiment was also conducted and the permeability and form/inertial drag coefficient were obtained. The range of flow velocity covered Darcy and Forchheimer flow regimes. The effect of flow displacement and frequency on important variables is presented and discussed. The appropriately defined friction factor correlated well with the Reynolds number for steady-state and oscillating flows, with the friction factor for oscillating flow being higher. The current results were compared to other studies from the literature employing oscillating air and water in various kinds of porous media. Some agreements and disagreements are stated and discussed.

Experimental flow in various porous media and reconciliation of Forchheimer and Ergun relations

Flow characteristics and pressure drop in traditional porous media, e.g., packed beds of spheres, and in modern man-made fibrous media, e.g., metal foam, are critical in many naturally-occurring and engineered applications. Pressure drop parameters such as permeability and form/inertial drag coefficients reported in the literature are very divergent for both classes of porous media. The choice of an appropriate characteristic length; and the selection of a way for correlating pressure drop data have also varied among researchers. In the current study a large set of experimental data for pressure drop of water flow in three different porous media was collected. The porous media were packed spheres of 1mm, packed spheres of 3mm and aluminum foam having 20 pores per inch. The porosity of both sets of packed spheres was practically the same at about 35%, while the porosity of the foam was 87.6%. The internal structure of the two classes (packed spheres and foam) of porous media investigated here are markedly different. The range of flow velocity covered Darcy, Forchheimer and turbulent flow regimes. It is shown that the same porous medium exhibited different values of permeability in different flow regimes. The widely-used equations of Ergun and Forchheimer for the post-Darcy regimes were revisited. An apparent difference between the two famous equations was presented and explained. The two equations were reconciled using the hydraulic radius theory, and the fact that the same porous medium exhibits different values of its permeability in different flow regimes. The multipliers of the viscous term and the inertial/form drag term in the post-Darcy regimes were shown to be connected. The square root of the permeability determined in the Darcy regime is shown to be appropriate length scale for defining and correlating the friction factor and the Reynolds number.

Flow Regimes in Packed Beds of Spheres from Pre-Darcy to Turbulent

Characteristics of flow regimes in porous media, along the processes of energy dissipation in each regime, are critical for applications of such media. The current work presents new experimental data for water flow in packed steel spheres of 1- and 3-mm diameters. The porosity of the porous media was about 35 % for both cases. The extensive dataset covered a broad range of flow Reynolds number such that several important flow regimes were encountered, including the elusive pre-Darcy regime, which is rarely or never seen in porous-media literature and turbulent regime. When compared to previous information, the results of this study are seen to add to the divergence of available data on pressure drop in packed beds of spheres. The divergence was also present in the coefficients of Ergun equation and in the Kozeny–Carman constant. The porous media of the current work were seen to exhibit different values of permeability and Forchheimer coefficient in each flow regime. The current data correlated well using the friction factor based on the permeability (measured in the Darcy regime) and the Reynolds number based on the same length scale. An attempt was made to apply recent theoretical results regarding the applicability of the quadratic and cubic Forchheimer corrections in the strong and weak inertia regime.

Investigation of fluid flow properties in open cell foams: Darcy and weak inertia regimes

Knowledge of flow properties induced by open cell foams is essential to the successful design and operation of high performance industrial systems. Flow properties are strongly dependent on the geometry of foams, but these relationships are not still fully characterized. This paper investigates the impact on flow properties of the geometrical characteristics of metal foams in the porosity range of 80–95%. An analytical model is proposed to characterize the foam matrix fully. 3-D direct numerical simulations at pore scale have been performed, and the Navier–Stokes equation has been solved for various Reynolds numbers (10−5<Re<500) in the fluid space. We determine permeability in the Darcy regime and distinguish it from permeability obtained in the inertia regime (Forchheimer Equation). The inertia coefficient is determined using both the Darcy and inertia regimes. The two approaches are fully discussed and the critical importance of the permeability of the Darcy regime to the friction factor is shown. We further derive an analytical solution determining fluid flow characteristics in order to incorporate the resulting macroscopic pressure and velocity gradients in the Ergun-like equation. The analytical results are thoroughly compared with the numerical and experimental data and an excellent agreement is observed.

Metal foam hydrodynamics: Flow regimes from pre-Darcy to turbulent

Metal foam is a relatively new class of porous media. The internal morphology of the foam is composed of connected cells each having many ligaments that form a web. In addition, metal foam has very high porosity (often greater than 90%) and a large surface area density. These properties are exploited in many applications, e.g., heat exchanger, reactors and filters. Flow regimes, and transition from one to another, are critical for understanding energy dissipation mechanisms for flow through the foam, and for heat transfer or reaction rates. While this topic is well studied in traditional porous media, e.g., packed beds, it is not well understood for foam-like porous media such as metal, graphite and polymeric foams. Pressure drop parameters such as permeability and form/inertial drag coefficients are very divergent for metal foam. The choice of an appropriate characteristic length for metal foam has also varied among researchers. In the current study a large set (88 points) of carefully-obtained experimental data for pressure drop of water flow in aluminium foam having 20 pores per inch and a porosity of 87.6% was collected. The range of flow Reynolds number covered all known flow regimes in porous media from pre-Darcy to turbulent. Flow regimes and transition between them were identified and compared to their counterparts in traditional porous media and to what is available for metal foam. The current data correlated very well using the friction factor based on the square root of permeability (measured in the Darcy regime) and the Reynolds number based on the same characteristic length. It is shown that the same foam exhibits different values of its permeability and Forchheimer coefficient in different flow regimes. The finding of this study can help in numerical and analytical work concerning flow and heat transfer in foam-like porous media.

Measuring static viscous permeability of porous absorbing materials.

Conventional acoustical methods for measuring the permeability or flow resistivity of a porous material require a priori estimation of the porosity. In this work, an acoustical method is presented in which a simplified expression (independent of both the frequency and porosity) for the transmitted waves at the Darcy's regime (low frequency range) is derived, and used for the inverse determination of both the viscous static permeability (or flow resistivity) and the thickness of air-saturated porous materials. The inverse problem is solved based on the least-square numerical method using experimental transmitted waves in time domain. Tests are performed using industrial plastic foams. Experimental and numerical validation results of this method are presented, which show the advantage of measuring the viscous permeability and thickness of a porous slab, without the required prior knowledge of the porosity, but by simply using the transmitted waves.

Bulk stress distributions in the pore space of sphere-packed beds under Darcy flow conditions.

In this paper, bulk stress distributions in the pore space of columns packed with spheres are numerically computed with lattice Boltzmann simulations. Three different ideally packed and one randomly packed configuration of the columns are considered under Darcy flow conditions. The stress distributions change when the packing type changes. In the Darcy regime, the normalized stress distribution for a particular packing type is independent of the pressure difference that drives the flow and presents a common pattern. The three parameter (3P) log-normal distribution is found to describe the stress distributions in the randomly packed beds within statistical accuracy. In addition, the 3P log-normal distribution is still valid when highly porous scaffold geometries rather than sphere beds are examined. It is also shown that the 3P log-normal distribution can describe the bulk stress distribution in consolidated reservoir rocks like Berea sandstone.

X-ray micro-tomography and pore network modeling of single-phase fixed-bed reactors

A three-dimensional (3D) irregular and unstructured pore network was built using local topological and geometrical properties of an isometric bead pack imaged by means of a high-resolution X-ray computed micro-tomography technique. A pore network model was developed to analyze the 3D laminar/inertial (non-Darcy) flows at the mesoscopic (pore level) and macroscopic (after ensemble-averaging) levels. The non-linear laminar flow signatures were captured at the mesoscale on the basis of analogies with contraction and expansion friction losses. The model provided remarkably good predictions of macroscopic frictional loss gradient in Darcy and non-Darcy regimes with clear-cut demarcation using channel-based Reynolds number statistics. It was also able to differentiate contributions due to pore and channel linear losses, and contraction/expansion quadratic losses. Macroscopic mechanical dispersion was analyzed in terms of retroflow channels, and transverse and longitudinal Péclet numbers. The model qualitatively retrieved the Péclet-Reynolds scaling law expected for heterogeneous networks with predominance of mechanical dispersion. Advocated in watermark is the potential of pore network modeling to build a posteriori constitutive relations for the closures of the more conventional macroscopic Euler approaches to capture more realistically single-phase flow phenomena in fixed-bed reactor applications in chemical engineering.

Slip-Flow and Heat Transfer in a Porous Microchannel Saturated with Power-Law Fluid

This study aims to numerically examine the fluid flow and heat transfer in a porous microchannel saturated with power-law fluid. The governing momentum and energy equations are solved by using the finite difference technique. The present study focuses on the slip flow regime, and the flow in porous media is modeled using the modified Darcy-Brinkman-Forchheimer model for power-law fluids. Parametric studies are conducted to examine the effects of Knudsen number, Darcy number, power law index, and inertia parameter. Results are given in terms of skin friction and Nusselt number. It is found that when the Knudsen number and the power law index decrease, the skin friction on the walls decreases. This effect is reduced slowly while the Darcy number decreases until it reaches the Darcy regime. Consequently, with a very low permeability the effect of power law index vanishes. The numerical results indicated also that when the power law index decreases the fully-developed Nusselt number increases considerably especially, in the limit of high permeability, that is, nonDarcy regime. As far as Darcy regime is concerned the effects of the Knudsen number and the power law index of the fully-developed Nusselt number is very little.

Numerical simulation of sorption/desorption processes in metal-hydride systems for hydrogen storage and purification. Part I: Development of a mathematical model

The pressure drops and the interfacial convective heat and mass transfer coefficients in packed beds saturated with a gaseous mixture containing hydrogen are studied by direct numerical simulation in the Darcy regime flow (Red⩽1) for both random and structured packings of the beds. For conditions with the Knudsen number Kn⩽0.01, the effective thermal conductivity of the random packings was calculated for various filling gases (H2, He, N2, and Ar). Additional correlations of the mathematical model for variable heat and mass transfer in porous hydrogen-absorbing media and based on the approximation of interpenetrating continua were specified from the received results. The developed mathematical model was used in numerical simulations of the heat and mass transfer processes in practical hydrogen storage devices. These results will be presented in the second part of the paper.

Measurement and Correlation of Pressure Drop Characteristics for Air Flow Through Sintered Metal Porous Media

Sintered metal porous media are currently used to replace conventional orifices as restrictors in air-bearing systems. The flow properties in porous media are generally approximated by Darcy and Forchheimer regimes in different flow regions. In this study, an ISO expanded expression is proven defective when it is used to represent flow properties through porous media under slight pressure drops (\({ 0.1\), rather than determining these two simultaneously. The theoretical mass flow rate in terms of the modified Forchheimer equation provides close approximations to the experimental data.

Fluid temperature measurements inside metal foam and comparison to Brinkman–Darcy flow convection analysis

Actual air temperatures were locally measured inside commercial aluminum foam cylinder heated at the wall by a constant heat flux, and cooled by forced air flow. The specially-developed experimental technique for such measurements is described in detail, and is shown to produce reasonably good data. Air speeds were in the Darcy regime. The permeability of the foam was directly determined from experimental pressure drop points that were obtained using the same experimental set-up. The experimental air temperatures are compared to their analytical counterparts. The volume-averaged analytical formulation employed the Darcy-extended Brinkman model for momentum, and the non-thermal-equilibrium two-energy-equation model for the temperatures of the solid and the fluid phases inside the foam. The solution steps, which are not new, are summarized. A comparison shows good agreement between the experimental and the analytical air temperatures, given the complexity of the foam’s morphology and the rounding nature of the volume-averaging technique. However, the analysis seems to under-predict the fluid temperature over most of the cross section. The experimental technique can be used for validation of other analytical solutions, numerical models and heat-exchange engineering designs based on metal foam and similar porous media.

Hydrophobically Modified Sulfonated Polyacrylamides for IOR: Correlations between Associative Behavior and Injectivity in the Diluted Regime

We report new experimental correlations between the injectivity through polycarbonate membranes and associative properties of random Hydrophobically Modified Water Soluble Polymers (HMWSP) with sulfonated polyacrylamides (SPAM) backbones and variable compositions in hydrophobic units. The investigations are focused on both their associative behavior in the diluted and semi-diluted regime and their injectivity under frontal filtration conditions in the diluted regime. Results from viscosimetric and dynamic light scattering measurements indicate the existence of thresholds in terms of amount (≥ 0.5 mol%) and mass (≥ C12) of alkyl hydrophobic units above which interchain interactions arise. These interactions are evidenced by the presence of multichain aggregates in diluted solutions and by enhanced thickening abilities in semi- diluted solutions. The filtration study was performed with capillary pore membranes (track-etched) in the Darcy regime under constant -flow rate and high Jamming Ratio conditions. Results show that: injection of diluted solutions of HMWSP without interchaininteractions (i.e. with composition in hydrophobic units belowthe above mentioned thresholds) does not lead to significantmobility and permeability reductions as compared to theinjection of a reference Water Soluble Polymer (WSP); injection of diluted solutions of HMWSP with interchain interactions leads to significant mobility and permeability reductions; HMWSP-induced mobility and permeability reductions are essentially due to irreversible polymer adsorption on the pore walls and not to the formation of filter-cakes; HMWSP adsorbed layers thicknesses are limited by the effective stress applied by the solution’s flow in the pores.

Prediction of Non-Darcy Coefficients for Inertial Flows Through the Castlegate Sandstone Using Image-Based Modeling

Near wellbore flow in high rate gas wells shows the deviation from Darcy’s law that is typical for high Reynolds number flows, and prediction requires an accurate estimate of the non-Darcy coefficient (β factor). This numerical investigation addresses the issues of predicting non-Darcy coefficients for a realistic porous media. A CT-image of real porous medium (Castlegate Sandstone) was obtained at a resolution of 7.57 μm. The segmented image provides a voxel map of pore-grain space that is used as the computational domain for the lattice Boltzmann method (LBM) based flow simulations. Results are obtained for pressure-driven flow in the above-mentioned porous media in all directions at increasing Reynolds number to capture the transition from the Darcy regime as well as quantitatively predict the macroscopic parameters such as absolute permeability and β factor (Forchheimer coefficient). Comparison of numerical results against experimental data and other existing correlations is also presented. It is inferred that for a well-resolved realistic porous media images, LBM can be a useful computational tool for predicting macroscopic porous media properties such as permeability and β factor.