© 2023 Elsevier Masson SASWhen a well does not fulfil its objectives, it is repurposed or permanently plugged, a shift in budget from revenue to Plug and Abandonment (P&A) expenditures occurs. New technologies being developed for P&A make use of a powerful heat emitter for melting the surrounding of the well forming a solid plug after the cool down. However, there is a gap in understanding the heat transport process in the well and much need for appropriate mathematical tools and solutions for estimating the effectiveness of Thermal P&A. The thermal analysis of the process requires eigenvalues in polar coordinates, which return only real quantities due to its implicitly dependence on the angular eigenvalues. Here, a hybrid analytical/numerical method is applied to an asymmetric transient heat conduction problem. The oil well is conceived as a 2-D multi-layer disc cast in polar coordinates, where the Separation of Variables Method (SVM) was applied to achieve a closed-form solution. Asymmetric boundary conditions of first, second and third kind can be implemented utilizing the proposed framework. A Finite Volume Method (FVM) numerical solution was produced for code verification. Lastly, research results show that temperature levels arising from the thermite reaction throughout the composite cylinder domain in radial and azimuthal ranges are enough for surpassing the melting point of the production tube steel. The results evidenced the thermal efficiency of the TP&A procedure and suggested that the production column may be destroyed by fusion, thus reducing tamponing expenses significantly.

© 2023 Elsevier LtdThe end of the production phase of an offshore oil well represents a remarkable shift in field operations from extraction to plug and abandonment. The international normative requirements for permanent well plugging demand a series of technical maneuvers to avoid structural failures and the formation of leakage paths during or after the sealing process. The current closure technique requires a complete or partial removal of the production column before the borehole may be plugged with Portland cement. However, the tube removal process frequently results in an increase in the involved time and costs. Hence, the current research was aimed at investigating the prototype approach of Thermal Plug and Abandonment of wells by applying a thermite heat emitter device to melt the production column's steel. Here, the Finite Integral Transform analytical method was computationally implemented through an inhouse code to calculate the resultant transient temperature fields at the multilayered medium. The results were compared to a Finite Volume Method numerical solution to enhance the study's reliability. The research outcomes provided insight that even in the event of a partial thermite reaction failure and a highly nonuniform heat pattern, the resultant molten azimuthal length of the production column may still allow enough room for cement flow. It was estimated through the temperatures achieved that the heat emitter is capable of melting at least nearly three-quarters of the production tube's azimuthal length, thus eliminating the need for its removal and significantly reducing the sealing process operational costs.

Copyright © 2024 by ASME.This article proposes a new formulation for a phase change model based on the enthalpy-porosity idea. A general one-energy equation model (1EEM) is extended to deal with the melting and solidification of pure substances and alloys. Before melting and after solidification, solid material is seen as a porous media with low porosity and very small permeability. During phase change, thermal equilibrium in the mushy zone is assumed. Viscous and form drag in the volume-Averaged momentum equation are reduced as the temperature rises above the melting point. In the energy equation, latent heat is treated implicitly in the accumulation term instead of explicitly as in most works in the literature. Liquid fraction for the entire field is updated after a new temperature field is calculated. Thermophysical properties are updated with the new liquid fraction field. Governing equations are discretized according to the control-volume method. Algebraic equation sets are relaxed with the Simple Method. Inner iterations make use of the Strong Implicit Procedure. Preliminary results indicate good agreement with the literature for pure substances.

© 2024 by Begell House, Inc.The thermite reaction is a self-sustained exothermic reaction commonly employed in welding processes of railway tracks, material synthesis, pyrotechnics, etc. More recently, this reaction has been assessed to plug depleted oil wells. The investigated geometry is modeled as a two-dimensional axisymmetric domain with a thermite mixture compressed between a polymethylmethacrylate (PMMA) lid and a stainless steel disk. First-order kinetic is assumed for the chemical kinetics model. The governing equations are discretized with the finite-volume approach. Experimental validation is performed by comparing numerical combustion velocities and peak temperatures with the experimental data in the literature. The results demonstrate a remarkable thermal gradient through the longitudinal direction, displaying higher thermal losses next to the thermite-steel interface. These heat losses also affect the melting of species, as a small portion of alumina remains entirely solid during the reaction.

© 2023 Elsevier LtdMotivated by a groundbreaking proposal to plug depleted oil wells using an exothermic reaction to melt the wellbore components, this work investigates the thermal behavior associated with the longitudinal propagation of a stoichiometric Fe2O3/Al thermite reaction. The primary objective of this study is to develop a reliable macroscopic numerical model capable of accurately estimating the heat generation and propagation during the reaction. A small-scale experiment is used to validate the numerical model, which approaches the experiment as a 2-D axisymmetric geometry within multiple regions. The reaction is modeled with a simplified zero-order kinetic model assuming a constant kinetic rate for all chemical species. A porous model assesses the impact of porosity on the overall heat diffusion, and a source-based phase change model is employed to evaluate the melting of the chemical species and the outer tube. Also, a disruptive model is included to consider the reaction between only condensed phases. The experimental validation demonstrated a good agreement between the numerical results with the disruptive model and transient temperature profiles measured experimentally. Varying the kinetic rate and porosity suggests that a slower reaction and denser mixture can enhance the heat transfer towards surrounding materials, potentially benefiting future applications in well sealing.

© 2023 by ASME.Land is a limited commodity that has always been fought over. Its use and allocation for various purposes have been the subject of much debate and for good reason. It is necessary for most industries. It is becoming more and more a topic of conversation as available land is used up. This review article explores land competition as it relates to the production of food and energy, as well as the ramifications of taking natural land and converting it to human use for these purposes. It also discusses the policies that some countries are enacting to deal with the ever-shrinking availability of free land and ways that society can decrease the necessity for more land.

© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.This research investigates the effects of thermodynamic and kinetic parameters on simulated Fe 2 O 3 –2Al thermite reaction propagation. For that, a full-factorial design was applied. Five parameters were investigated: mixture density (A), thermal conductivity (B), specific heat (C), activation energy (D), and pre-exponential factor (E). Among these factors investigated, the activation energy, the specific heat, and their two-factor interaction had by far the highest percentage contribution of effects in the five responses observed: burning velocity, thickness of the reaction zone, peak temperature, ignition temperature, and ignition delay. Higher activation energy and specific heat resulted in a slower and thicker reaction propagation wave that required a longer time to ignite and reached a lower peak temperature. However, while activation energy affected the ignition temperature positively, the specific heat presented a negative effect. The remaining parameters had less pronounced effects but were significant in all five responses. Moreover, regression models of burning velocity, thickness, and ignition delay responses were estimated, which allowed mapping effects on these responses through contour plots of the main two-factor interactions.

© 2023 Elsevier LtdThermite reactions are self-sustained exothermic reactions commonly employed in welding processes of railway tracks, material synthesis and pyrotechnics, to mention a few applications. More recently, this reaction has been assessed to plug depleted oil wells. Motivated by the foregoing, this work numerically investigates a Fe2O3/Al thermite reaction. A two-dimensional axisymmetric domain with a thermite layer compressed between a PMMA lid and a stainless-steel disk is considered. A first-order kinetic is assumed and the reaction is controlled by the hematite consumption. A computational solver is developed based on the open-source OpenFOAM® software. Numerical results showed good agreement with experimental data for temperature levels. Numerical results further indicated thermal losses next to the thermite-steel interface. These heat losses affected the melting of the species as a small portion of alumina remained entirely solid during the reaction.

© 2023 Elsevier Masson SASIn this work, a hybrid analytical and numerical solution for transient heat conduction across a composite cylindrical sector is presented. A two-dimensional domain consisting of a multi-layer circular sector of angle φ was investigated (φ<2π). The Separation of Variables Method (SVM) was applied to solve the partial differential equation with non-homogeneous boundary conditions of the first, second and third kinds prescribed in the radial direction. Homogeneous boundary conditions of first and second kinds were arbitrated in the angular direction. A spatial time-independent source term gi(r,θ) was considered. The radial eigenvalues problem for the (r,θ) domain returns only real quantities and depends implicitly on the angular eigenvalues. Results for time dependent temperatures using the Separation of Variables Method were compared with numerical results, showing good accuracy. A second set of results was developed to investigate boundary conditions, material properties and the thermal source power required to rise temperature levels (mainly around the mid-angle φ/2) high enough to promote melting of certain layers of materials. These results might be useful for investigating a novel technology for the decommissioning of oil wells using thermal sources, often referred to in the literature as Thermal Plug and Abandonment (TP&A).

© 2023 Elsevier LtdThis paper presents an improved one-dimensional nonstationary model to simulate the reaction propagation of aluminum and iron-oxide in thermite mixtures. This model is motivated by the application of thermite mixtures for thermal plug and abandonment of oil wells. The main improvements of this model include the chemical source term correction in the energy conservation equation, and the imposition of a temperature limit to account for aluminum vaporization. A simplified, first-order, one-step mechanism governed by the Arrhenius relation was assumed, and different pairs of activation energy and pre-exponential factor were analyzed, including some pairs that reproduce the experimental propagation speed reported in the literature. Numerical simulations were done to generate contour plots that map the effects of the kinetics parameters, alumina dilution, and aluminum addition to the initial mixture in the main characteristics of the reaction wave, such as velocity, thickness, ignition delay, and initiation temperature. These simulations indicate that, at alumina dilution of 20% or more, the simulated thermite reaction does not reach the aluminum vaporization temperature and may not present disruption of the system. The model shows that aluminum addition to the initial mixture accelerates the propagation and the numerical results reproduces experimental data from literature. Also, below a burning velocity of 26 mm/s and alumina-dilution higher than 40%, the reaction does not self-propagate.

© 2023, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.Global change from carbon-based to carbon-free economy has driven the development of a number of innovative technologies for decommissioning oil wells in mature fields. The common technology in use nowadays relies on cementing the borehole to permanently seal and abandon old wells. However, this procedure has a high cost and takes several weeks to be concluded, which constitutes a burden for offshore wells. This work investigates an innovative technology for plug and abandonment based on the release of large amounts of heat from thermite reactions at the sealing location in the borehole. Tubing and casing are melt forming a plug after the cool down period. Transitory transport of heat generated by a thermite mixture is numerically investigated by solving the energy equation using the control volume method. Time required to melt and subsequent solidification of the molten mass is estimated.

© 2023 Begell House Inc.. All rights reserved.This paper investigates the effects of Arrhenius parameters on the Fe2O3-2Al thermite system. Assuming a single-step kinetics mechanism, contour plots were generated to investigate the effects of the activation energy and pre-exponential factor on the velocity and thickness of the reaction wave. Higher activation energies and lower pre-exponential factors resulted in slower and thicker reaction waves. Also, the effect of activation energy on the burning velocity is enhanced at higher levels of the pre-exponential factor whereas the effect of pre-exponential factor is increased at lower levels of the activation energy. The opposite trend was observed on the thickness of the reaction wave. Finally, an exponential relationship between thickness and velocity of the reaction wave was identified regardless of the Arrhenius parameters.

© 2022 Elsevier B.V.Moving from a carbon-based to a carbon-free economy has driven the development of groundbreaking new technologies for permanent plugged and abandoned (P&A) of mature oil wells, including the use of thermites as the energetic material for the so-called “Thermal P&A” technology. Better knowledge is then much needed on such chemical reactions. Accordingly, this research presents an in-depth kinetic study of the Fe2O3-2Al thermite reaction by analyzing differential scanning calorimetry (DSC) data at three heating rates. After an endothermic peak corresponding to the aluminum melting process (∼660.3 °C), two exothermic peaks were identified corresponding to thermal stages of the overall thermite reaction: the first stage at 800–1000 °C and second stage at 1000–1300 °C. The apparent activation energy of each reaction stage was calculated using several isoconversional kinetics methods. All methods revealed significant variation of activation energy with the extent of conversion. However, the differential method of Friedmann and the flexible-integral methods of Popescu and Vyazovkin identified higher variations than the rigid-integral methods, with EA values between 188 and 356 kJ/mol for the first reaction stage and 280 and 509 kJ/mol for the second one. These high variations indicated a multi-step mechanism that requires multiple kinetic triplets. The pre-exponential factor at each extent of conversion and the reaction mode of each reaction stage were estimated by an approach based on Popescu's equation and the compensation effect. A contracting sphere and a random nucleation mechanism were identified as suitable models to describe the first and second reaction stage, respectively. Modeled data showed an excellent agreement with the experimental data of the first reaction stage, with average deviations up to 1.2 %. However, modeled data of the second stage presented more notable variations with average deviations up to 9.5 %.

© 2022 Elsevier LtdThe use of air as heat transfer fluid and a packed bed of rocks as storage medium for a thermal energy system (TES) can be a cost-effective alternative for thermal applications. Here, a porous media turbulent flow (standard k-ε) and heat transfer (local thermal non-equilibrium) model is used to simulate the discharge cycle of such system. Temperature fields of corresponding charging cycles are used as initial conditions. Effects of varying mass flow rates (Re number), porosity, permeability (Da number), thermal conductivity ratio and thermal capacity ratio on the effectiveness of the discharge are compared. The examination of these effects indicated that increasing the mass flow rate improved the effectiveness of the discharge, which was not seen for the charging cycle. Also, increasing porosity improved discharge efficiency more significantly than it did in the charging cycle. In both charge and discharge cycles the effect of permeability is significant and reducing Da number improved temperature stratification and efficiencies. The effect of the thermal conductivity ratio was mostly seen on the outlet temperatures, where lower ratios allowed for higher temperature values. Increasing the thermal capacity ratio improved charging effectiveness but, on the discharge cycle, cycle this effect was reduced. Moreover, for lower Re number flows, increasing this ratio reduced efficiency indicating that the mass flow rate should be matched carefully with the thermal capacity of the system. All these effects have important implications which should be taken into consideration when designing an effective thermal energy storage system.

© 2022 Elsevier LtdIn this work, the process of filtration of particles carried out by turbulent flows is investigated. The applied mathematical framework considers turbulent flow regime in the clear region and within the filter. The motivation for this work is to get insight on possible use of filters to separate thermite mixtures from a carrier phase and their use in thermal plug and abandonment of oil wells. Axisymmetric simulations are performed using the finite volume method. A macroscopic k-ε model is applied to handle turbulence. Pressure drop, stream function and turbulence field are computed. Using correlations in the literature, filter efficiencies are estimated based on calculated flow and pressure fields. Reynolds numbers varied from 2.3 × 103 to 4.6 × 104 for different permeabilities K ranging from 3.47 × 10−9 m2 to 8.89 × 10−9 m2, corresponding to filter particle diameters dp = 1.0 × 10−2 m and 1.0 × 10−3 m and different porosities (0.5 to 0.8). The results showed that pressure drop is strongly affected by porosity, permeability and Reynolds number; the stream function maps indicated wider stagnant zones for filters with larger particle diameters (higher permeabilities); turbulence fields showed slight generation of turbulence inside the filtration zone. The results for efficiency illustrated that, in case of filtrating thermite, higher inlet velocities, filters with particles of size 1.0 × 10−3 m and lower porosity filters favor the collection mechanism.

© 2022 by the author. Licensee MDPI, Basel, Switzerland.In this article, a concept named double decomposition, which is used to model turbulent flows in porous media, is examined. This concept is based on the idea that in a turbulent flow through a porous matrix, local instantaneous variables can be averaged in time and space, simultaneously. Depending on how these operators are applied, averaged equations take different forms. In this article, instantaneous local equations are averaged using both operators and a different set of equations resulting from such operations are commented upon. Additional terms proposed for the averaged equations are discussed.

© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.Thermites are powerful energetic materials able to release large amounts of energy in a self-propagating reaction. They have been widely applied in rail welding, pyrotechnics, and material synthesis, as they are highly exothermic. In recent years, there has been an increased interest on applying a thermite reaction in the plug and abandonment of wells due to the possibility of reducing the high cost of this process. However, some primary studies are required to understand these energetic materials and to select the most efficient thermite mixtures to be applied in a plug and abandonment scenario. Although they do not react as simple condensed-phase reactions because of all the complex physicochemical mechanisms involved, they can be characterized and understood by simple principles of thermodynamics. As so, this research presents the importance of the Gibbs free energy concept to determine the candidates of a thermite reaction, in addition to showing how important characteristics of these reactions such as adiabatic temperature and heat released can be calculated using thermodynamic principles. Lastly, the minimization of Gibbs free energy method for determining the final products of a reaction, considering chemical equilibrium, is presented and applied to predict the final products, as a function of temperature, for some of the most powerful thermite mixtures. The conclusion is that, although the 2Al–Fe2O3 thermite reaction has the lower mass and volumetric heat of combustion in comparison with the 2Al–3CuO and 3Be–Fe2O3 thermites, it can reach the highest adiabatic temperature observed due to the lower gaseous mass fraction in the products, which means fewer heat losses due to phase changes. So, the thermite mixture 2Al–Fe2O3 is a promising candidate for the plug and abandonment of mature oil wells.

© 2021 Elsevier Masson SASThe continuous exploration of oil wells has increased the demand for plug and abandonment procedures. Current techniques used for well plugging involve the cementing process, which is extremely expensive and difficult to perform. Aiming to overcome these challenges, a new approach in this area has been investigated. This novel technology, named here as Thermal Plug and Abandonment (TP&A), proposes a chemical mixture that would be introduced through the production tube or the steel casing. Then, this mixture is ignited and the exothermic reaction generates enough heat to melt the wellbore components. After the cooling stage, the solidified mass composed by these components and the products of the reaction will serve as a seal to plug the well. Under such circumstances, this work aims to investigate this new technology assuming a thermite mixture that is introduced through the steel casing. For that, a numerical analysis is employed to investigate the heat conduction and the phase change through the oil well structure, which is modelled as a two-dimensional axisymmetric domain. The heat generated by the reaction is approached as a spatial and time-dependent heat flux profile that was estimated based on experiments found in literature. The thermal behavior is assessed to discover if the heat generated by the reaction is enough to form a plug composed by all components found in an oil well environment. It was found that temperatures are high enough to melt most part of the steel and a significant amount of the cement layer. The investigation also found that temperatures heavily drop through the cement layer, which avoids any notable melting of the cap rock. Finally, the heat fluxes’ profiles were increased, and it was found that a TP&A procedure that accounts for a seal composed by only the thermite products and the melted steel layer might be a more practical approach.

© 2021 Elsevier Masson SASThis work presents a hybrid analytical/numerical approach for transient heat conduction through composite hollow cylinder structures applied for Plug and Abandonment (P&A). The Distribution Transfer Function Method (DTFM) is a mathematical framework able to solve both nonhomogeneous boundary conditions and a nonlinear partial differential equation with spatial-temporal source term, presenting good agreement with Finite Element Method (FEM) and other analytical approaches. A new technology of Thermal Plug and Abandonment (TP&A) devoted for P&A operation is studied in this work, where a heat generator is used to melt the first layer of the well structure. The heat source will be modeled using thermite within the oil well manifold structure, which must be descended to desirable depths along the borehole through a cylindrical container. After an exothermic reaction the temperature fields will be determined by means of DTFM method. A novel adaptative step-wise procedure applied for DTFM allows one to solve non-differentiable heat flux, thermal and volumetric heat source profiles which exact quadrature regression are not able to solve. This method provides simple replicability, which can be reproduced for many emerging fields where transient heat conduction is an important parameter to be accounted for. Using experimental data as input in DTFM provides insight that the temperature response overcame the design melting point of the production tube, being capable of melting down the production tubing wall, which is a valuable achievement for oil industries since not removing the production tubing can save a great amount of time, and hence save costs for oil industries.

© 2021 Elsevier Masson SASIn this study, a thermal energy storage system is modeled as an axisymmetric ventilated cavity partially filled with a porous medium that is subject to turbulent flow. Local thermal non-equilibrium is considered to analyze the heat transport and the turbulent k−ε model is used to account for the high inlet flow velocities. Finite volume method is employed for discretization of the equations that are relaxed with the SIMPLE method. The investigation is concerned with the turbulence field, heat fluxes and pressure drop in the system relative to variations in Reynolds number (from 8.3 × 103 to 5 × 104), porosity (ϕ from 0.6 to 0.8) and Da number (from 4x10−5 to 4x10−7). Results indicate that porosity effects slight changes in the turbulence field while the temperatures increase significantly faster for higher porosity cases. Turbulence in higher porosity cases was lower, accompanying the increase in thermal efficiencies. Variations in Da number with fixed porosity showed that, for lower Da number, recirculation in the clear region increased while for higher Da number porous region turbulence increased. Also, higher Da decreased heat exchange between fluid and solid phases. Finally, an increase in thermal efficiency for lower Da number flows was followed by increased average turbulent kinetic energy and relative pressure drop.

© 2021 Wiley-VCH GmbHA thermite reaction is a self-propagating exothermic reaction with many practical applications in welding processes, material synthesis, pyrotechnic and initiator technologies. Motivated by the above-mentioned, the present study involves modeling and simulation of common hematite-aluminum thermite reaction with the aim of predicting temperature levels and radial burning speeds in a thin disk ignited at the center. Balance equations of species and energy conservation were solved in one dimension space by applying a finite difference method, considering no species transport and a one-step mechanism. The Arrhenius equation was adopted to model the kinetics rate. Phase change and temperature dependence of the thermochemical properties were also considered. Analyses of spatial and temporal meshes revealed that numerical results were independent of the grid used. Predictions show that the ignition procedure affects the formation of the reaction-front, higher temperatures, and longer ignition zones can start the self-sustained reaction earlier. However, once the reaction wave is established, its velocity and peak temperature are the same, independent of the initial temperature profile. Simulations herein also show that an increase of the activation energy and decrease of the pre-exponential factor slows down the reaction speed considerably, impacting on accurate prediction of reaction-wave velocity. Further, the activation energy influences the burning velocity much more drastically than the pre-exponential factor. The maximum temperature observed in the model is around the melting temperature of alumina (2327 K), which is in agreement with the experimental results reported in the literature.

© 2021 by Begell House, Inc.A laminar natural convection local thermal nonequilibrium model of porous media is used to simulate the standby cycle of a thermal energy storage system that uses hot air as a heat transfer fluid. For this, an axisymmetric cavity, partially filled with a porous medium, is considered. The initial temperature field of the system is taken as being the final one for a charging cycle with the same properties. Results for the two-dimensional temperature fields, axis line temperatures, and energy loss after the standby cycle are compared for different properties, namely, porosity (0.6 < φ < 0.8), Da number (4 × 10-7 < Da < 4 < 10-5), thermal conductivity ratio (3.5 < ks / kf < 1062), and thermal capacity ratio (1483 < ρscps / ρf cpf < 7415). It was found that higher porosities result in higher rates of energy loss. Also, an increase in the Da number affects a significant thermal destratification of the system although the rate of energy loss does not vary. Increasing the thermal conductivity ratio meant increasing losses through the tank wall. Finally, the higher the thermal capacity ratio the slower the energy loss of the system.

© 2021 by Begell House, Inc.This work investigates the use of variable fluid properties when simulating the behavior of solar volumetric receivers (SVR) via the thermal nonequilibrium approach. Energy balances for both air and porous ceramic materials are numerically solved using the SIMPLE method. The system of algebraic equations is relaxed by the SIP procedure. Thermal efficiency increases for lower inlet velocities and higher solid-to-fluid thermal conductivity ratios. For variable properties cases, a further increase in thermal efficiency is higher for lower solid-to-fluid thermal conductivity ratios and less noticeable for higher solid-to-fluid thermal conductivity values and lower velocities. Thermal efficiency is reduced for more permeable structures and higher porosities and for variable properties cases higher values are calculated due to increase in equilibrium temperatures. No appreciable dependence on Darcy number was detected when efficiencies were calculated with either constant or variable properties.

© 2021 Begell House Inc.. All rights reserved.Ducts with walls having a sinusoidal shape can be used to enhance heat and mass transfer in a number of industrial flows. Modern solar energy systems can benefit by the use of such wavy surfaces resulting in improvements on their overall thermal efficiencies. If a layer of porous material is attached to the walls, further increase in transfer rates of mass and energy can be achieved due to the increase of surface contact area between the fluid and the walls. This article investigates the pressure drop in wavy ducts having different aspect ratios and distinct porous layer thicknesses, porosities, and permeabilities. A mathematical model for flow in a unique computational domain, encompassing both the porous region and the centered clear passage, was applied and numerically resolved. Results indicate that pressure losses were mostly affected by the thickness of the layers rather than the properties of the porous substrate.

© 2020 Elsevier Masson SASWhen the oil and gas extraction in a well is over or ended by wellbore issues, a Plugging and Abandoning (P&A) operation is required. The usual method is the well cementation. A recent process is the rock fusion method through a thermite reaction for wellbore buffering operation. This paper presents a numerical thermal study for this new process. The developed model is based on the solution of the three-dimensional heat diffusion equation by a seven points Finite Difference scheme. A multi-layer geometry, composed of dolomite, cement, carbon iron steel, and thermite, is used to simulate the oil well. The phase change problem is approached through the enthalpy function. In order to optimize the problem solution, an in-house parallel algorithm in CUDA-C language was developed to solve the problem in a Graphical Process Unity (GPU). A modify Successive Over-Relaxation (SOR-M) scheme was applied to minimizing the computational time. The thermal fields are analyzed to determine if the thermite generation heat is enough to create the plug. The study found that high-power thermite is required to make the P&A process. The study also found that a coupling between the cracks in the cement at high-temperature, liquid dolomite, and the mix of the liquid steel duct, and thermite are responsible to break the cement and create a plug to perform the P&A operation.

© 2020 Elsevier LtdThermocline energy storage systems can be adapted to store energy in a tank where the hot fluid is pumped in during a charging cycle and cold fluid during a discharging cycle, substituting the need for two storage tanks. Here, a numerical investigation of a hot air flow transferring thermal energy to a solid porous bed is performed to evaluate the influence of the thermal conductivity and capacity ratios on the efficiency of a charging cycle of a thermocline storage tank. The numerical model considers the mixed convection turbulent flow (k-ε model) through clear and porous media using the Local Thermal Non-Equilibrium approach to provide solutions for the solid and fluid phases separately. Macroscopic equations are obtained by the method of volume averaging and numerically processed by finite volume method. The system was modelled as a vented axisymmetric cavity, partially filled with a porous medium, with hot fluid inflow from the top and outflow at the bottom. The investigation included changing Reynolds number (Ret from 8.3 × 103 to 5 × 104), thermal conductivity ratios (ks/kf from 3.5 to 1062) and thermal capacity ratio (ρscps/ρfcpf from 1483 to 7415). It was found that lower ks/kf ratios decrease the heat loss throughout the charging cycle which allow for higher temperatures in the later stages of the cycle and thus improve charging efficiency. However, this effect decreases in importance as the system undergoes higher Ret number flows. On the other hand, increasing the ρscps/ρfcpf ratio affects the entire cycle, increasing the temperature difference between phases, lowering the velocity in which the solid raises its temperature but storing heat more efficiently. Finally, a design consideration is highlighted as the importance of the ks/kf ratio on the thermal efficiency is predominant at lower Ret flows, whereas as Ret increases, ρscps/ρfcpf becomes the dominant parameter providing efficiency gains.

© 2020This work investigates the thermal behavior of the charging cycle of a thermal energy storage system for a concentrated solar power plant filled with solid porous material. A transient model that describes turbulent flow in a hybrid medium (porous/clear) with both forced and natural convection was used. The mean flow macroscopic equations are developed based on the concept of double-decomposition. Governing equations were discretized using the SIMPLE method and the system of algebraic equations was relaxed by the SIP procedure. The k-ε turbulence model was used for modeling turbulence. The two-energy equation model was used to evaluate heat transfer between the solid and fluid phases. Simulations are based on an axisymmetric tank with external convection, hot fluid inlet at the top and distributor regions at the top and bottom of the tank. Effects of Reynolds number (Ret), porosity (ϕ) and permeability (K) were investigated. Additionally, storage (ηst) and charging (ηchg) efficiencies were compared to quantify the effectiveness of the charge. Consequently, a new thermal charge efficiency (ηth) was proposed to evaluate the charging cycle. Finally, it was found that the most efficient charging parameters were the ones with higher porosity and lower permeabilities. Moreover, increasing Ret values increased the thermal charge efficiency up to a maximum and thereafter presented a stabilized trend with further increasing of Ret values.

© 2019Volumetric heat absorbers made of porous materials allow for a greater contact area between the porous matrix and the working fluid. In such devices, heat is collected in a volume rather than a surface. This work deals with transient behavior of Volumetric Heat Receivers using the thermal non equilibrium approach. Balances for energy amounts for the porous ceramic materials and air are solved numerically using the backward Euler discretization method and application of the SIMPLE method. After obtaining the algebraic equation system, relaxation by the SIP method is applied. We investigate here the effects of permeability, thermal conductivity ratio, inlet velocity and porosity on the temperature reached after thermal equilibrium. Higher inlet velocities attain quicker stabilization times and decreases equilibrium temperature, Teq. In addition, increasing porosity lower Teq and shorten time for temperature stabilization. Less permeable solid matrices, i.e. porous structures with lower Da, result in slightly higher Teq due to better enhancement of energy exchange between phases. Increasing kskf/ increases stabilization time as well as equilibrium temperatures. Thermal efficiency increases for lower inlet velocities and higher thermal conductivity ratios, whereas η is reduced for more permeable structures and higher porosities.

© 2018, © 2018 Taylor & Francis Group, LLC.A numerical study of the combustion of lean methane/air mixtures in a porous media burner is performed using novelty geometry, cylindrical annular space. The combustion process takes place in the porous space located between two pipes, which are filled with alumina beads of 5.6 mm diameter forming a porosity of 0.4. The outer tube diameter of 3.82 cm is isolated; meanwhile the inner tube of 2 cm in diameter is covered by a continuous set of thermoelectric elements (TE) for transforming heat energy into electricity. To achieve and maintain the proper temperature gradient on TE, convective heat losses are considered from the TE. Computer simulations focus on the two-dimensional (2D) temperature analysis and displacement dynamics of the combustion front inside the reactor, depending on the values of the filtration velocity (0.1 to 1.0 m/s), the heat loss coefficient from the internal cylinder (400–1500 W/m2/K), and the fuel equivalence ratio (0.06– 0.5). The conditions that maximized the overall performance of the process of energy conversion are: 0.7 m/s of the filtration velocity, 0.363 of the fuel equivalence ratio and 1500 W/(m2·K) of the heat transfer coefficient from the internal cylinder, to obtain 2.05 V electrical potential, 21 W of electrical power, and 5.64% of the overall process efficiency. The study shows that the cylindrical annular geometry can be used for converting the energy of combustion from lean gas mixtures into electricity, with a performance similar to the specified by manufacturers of thermoelectric elements (TE).

© 2019 Elsevier Masson SAS Flow past a sudden expansion is found in a number of engineering equipment of practical relevance. This article presents numerical results for turbulence structure and heat transfer in flow past a two-dimensional backward-facing-step channel with a porous insert using linear and non-linear eddy viscosity macroscopic models. The expansion ratio is 1:3. The non-linear turbulence models are known to perform better than classical eddy-diffusivity models due to their ability to simulate important characteristics of the flow. Parameters such as porosity, permeability and thickness of the porous insert are varied in order to analyze their effects on the flow pattern, particularly on the damping of the recirculating bubble after the porous insertion. The numerical technique employed for discretizing the governing equations is the control-volume method. The SIMPLE algorithm is used to correct the pressure field. Wall functions for velocity and temperature are used in order to bypass fine computational close to the wall. Results showed that the recirculating bubbles simulated with the linear model were shorter than those calculated with non-linear theories. Thickness of the insert had a more pronounced effect in suppressing the recirculating bubble than permeability or porosity. Results for the statistical field indicate that using porous inserts dampens generation of turbulence along the channel and concentrate conversion of mean mechanical energy into turbulence inside the porous material. Inserting a porous substrate past the expansion seems to be a practical way to decrease the sudden variations on C f and St.

© 2019 Elsevier Ltd This work presents numerical results for the thermal performance of a Solar Volumetric Receiver (SVR). The Thermal Non-Equilibrium Model and Rosseland approximation were used. Radiation boundary condition was implemented at the absorber inlet. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Effects of inlet velocity (u in ), porosity (ϕ), medium permeability (K), and thermal conductivity ratio (k s /k f ) on the solid and fluid temperatures were investigated. Reduction of temperatures as porosity increases or thermal conductivity decreases was observed, in addition to an increase in entry length for lower porosities or higher thermal conductivity ratios. Increase in inlet solid temperature as permeability increases was accompanied by a longer entry length and reduced final equilibrium temperature.

© 2018 Elsevier LtdKnowledge on the effects of porosity and thermal conductivity ratio on double-diffusive transport are much needed for optimum design and analysis of a number of engineering equipment. So far, the open literature seems to lack specific investigations on the effects of those two parameters on overall heat and mass transfer in cavities. This work contributes to such much-needed study on double-diffusive convection in a porous square cavity. Turbulent flow regime and aiding drive cases were considered. Governing equations were time- and volume averaged. Turbulence was handled with a macroscopic two-equation model. The thermal non-equilibrium hypotheses was employed to analyze energy transport across the enclosure. Mass transport assumes a binary mixture with solute characterized by its mass fraction. Equations were discretized with the control volume method numerically relaxed using the SIMPLE method. Here, two situations are investigated regarding the effect of porosity. First, porosity is varied along with permeability. Second, permeability is fixed while porosity takes different values. Results indicated that reducing both porosity and permeability induced flow recirculation and increased overall heat and mass transfer, leading to higher levels of turbulent kinetic energy. Such effects are less pronounced when permeability was kept constant while varying porosity. Further, increasing the thermal conductivity ratio substantially affected flow recirculation in the cavity, enhancing, ultimately, turbulence and mass transfer.

© 2019 Begell House Inc. All rights reserved.Solar energy is an abundant source of clean and renewable energy for heat and power production. In this work we present numerical results for the thermal performance of a Solar Volumetric Receiver (SVR). The Thermal Non-Equilibrium Model was employed along with radiation boundary conditions. The numerical technique used for discretizing the governing equations was the control volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was applied to handle the pressure-velocity coupling. Effects of inlet velocity, medium permeability and solid-to-fluid thermal conductivity ratio on temperature distributions within the absorber were investigated. Reduction of temperatures as thermal conductivity ratio decreases was observed in addition to an increase in entry length for higher thermal conductivity ratios. Increase in inlet solid temperature as permeability increases was accompanied by a longer entry length and reduced final equilibrium temperature.

© 2019 by ASME.This work presents a study of double-diffusive free convection in a porous square cavity under turbulent flow regime and with aiding drive. The thermal nonequilibrium model was employed to analyze the energy and mass transport across the enclosure. Governing equations were time- and volume-averaged according to the double-decomposition concept. Analysis of a modified Lewis number, Lem, showed that for porous media, this parameter presents opposite behavior when varying the thermal conductivity ratio or the Schmidt number, while maintaining the same value for Lem. Differently form free flow, the existence of the porous matrix contributes to the overall thermal diffusivity of the medium, whereas mass diffusivity is only effective within the fluid phase for an inert medium. Results indicated that increasing Lem through an increase in Sc reduces flow circulation inside porous cavities, reducing Nuw and increasing Shw. Results further indicate that increasing the buoyancy ratio N promotes circulation within the porous cavity, leading to an increase in turbulence levels within the boundary layers. Partial contributionsof each phase of the porous cavity (solid and fluid) to the overall average Nusselt number become independent of n for higher values of the thermal conductivity ratio, ks/ kf. Further, for high values of ks/kf, the average Nusselt number drops as N increases.

© 2018, © 2018 Taylor & Francis Group, LLC. Walls made of layers of different materials and sizes, including empty spaces, can be used for modulating energy transfer across such composite structures. By changing layer thicknesses and by using distinct porous and solid materials, the overall cavity Nusselt number can be modified in regard to its traditional behavior found in cavities fully fitted with porous materials or with no obstructions. Numerical simulations of transport equations for such composite systems poses an additional difficulty if one considers the different layers, appropriate interface conditions between them and the need to use one single computational domain for simplicity. Motivated by such engineering application and numerical need, this work presents the analysis about natural convection in a two-dimensional horizontal composite square cavity, using laminar and k-ε turbulence models. Both the one energy equation (1EEM) and two energy equation (2EEM) closures are applied. The composite square cavity is equally divided and formed by three distinct regions. Non-dimensional temperatures are proposed such that the entire computational domain is handled in a single numerical scheme. It was found that the fluid begins to permeate the porous medium for values of Ra greater than 10 6 . Nusselt number values show that for the range of Ra analyzed there are no significant variation between the laminar/turbulent and 1EEM/2EEM model solution. When comparing the effects of Ra, thermal conductivity ratio k s /k f and Da on Nu, results indicate that (Formula presented.) has a greater influence in controlling heat transfer rates across the composite cavity.

© 2018 Taylor & Francis.The aim of this work is to estimate the permeability of porous enclosures for numerical solutions of turbulent natural convection in a square cavity. The motivation is that available permeability correlations were proposed based on force rather than natural convection through permeable media. Although commonly seen as a medium property, permeability is measured with a flow through the permeable structure and, as such, its value may carry a flow type dependency. Here, it is assumed that a fixed amount of a solid conducting material is distributed within the cavity and two mathematical models are used and compared when calculating the cavity Nusselt number. First, a porous-continuum model is considered based on the assumption that the solid and the fluid phases are observed as a single medium, over which volume- and time-averaged transport equations apply. Second, a continuum model is used to solve local momentum and energy equations, in both the solid and void spaces, through a conjugate heat transfer solution. The average Nusselt number at the hot wall obtained from the porous-continuum model for several Darcy numbers are compared with those obtained with the continuum model using up to N = 1,024 obstacles within the cavity. When comparing the two methodologies, this study shows that the average Nusselt number calculated by each approach differs by as much as 32% when the number of obstacles N is increased to 1,024. Based on that, an adjustment on the used correlation for calculating the porous medium permeability is proposed to match the Nusselt numbers calculated with the two models. Results indicate that the use of the new correlation gives results for Nu that differ less than about 4% for the range 4 < N < 1,024.

© 2018 by Begell House, Inc.This work presents a study on double-diffusive free convection in a porous square cavity saturated with a Newtonian fluid under laminar flow simulated with the thermal equilibrium model. Transport equations are discretized using the control-volume method and the system of algebraic equations is relaxed via the SIMPLE algorithm. The effect of Ram and porosity on average Nusselt and Sherwood values was investigated. Results show that as Ram increases, both Nusselt and Sherwood numbers increase, indicating enhancement of heat and mass transfer across the cavity. Further, when the Lewis number is increased while keeping the same thermal properties, reduction of mass diffusivity further enhances flow recirculation for aiding flows (N = 1) within the cavity, which leads to a further increase in Nuw and Shw. When varying the buoyancy ratio N from aiding (N > 0) to opposing flow (N < 0), simulations indicate that when both drives are of equal strength, minimum values for Nusselt and Sherwood occur for N = −1, regardless of Ram. For larger values of |N|, aiding drives will promote fluid rotation in the clockwise direction, for the gradients of T and C applied here, whereas for opposing flows, the fluid rotates in the counterclockwise direction for opposed conditions at the lateral walls. Porosity and thermal conductivity ratio also affect Nuw and Shw.

Copyright © 2017 by ASME.This work presents a study on double-diffusive free convection in a porous square cavity using the thermal equilibrium model. Transport equations are discretized using the control-volume method, and the system of algebraic equations is relaxed via the SIMPLE algorithm. The effect of ks/kf on average Nusselt and Sherwood values was investigated. Results show that increasing ks/kf affects Nuw and Shw boosting mass transfer at the expense of reducing overall heat transport across the enclosure.

© 2017 Taylor & Francis.Composite cavities formed by a clear space, a layer of porous material, and a solid plate can be engineered for controlling the overall heat transfer across the enclosure. Using different layer dimensions, as well as distinct porous and solid materials, the value of the cavity Nusselt number can be modified with regard to traditional Nu∝Ran behavior, which is encountered either in completely empty cavities or in cavities fully fitted with porous materials. Motivated by such novel application, this work presents a study about turbulent natural convection in a composite concentric annulus. The annulus is assumed to be two-dimensional and positioned horizontally, being isothermally heated at the inner cylinder and cooled from the outer surface. Laminar flow is considered in addition to the turbulent regime, which is handled via the standard k–ε model. The wall treatment applied is the High Reynolds approach. The Two-Energy Equation Model (2EEM) is utilized in the porous section. The transport equations are discretized using the control-volume method. The system of algebraic equations is relaxed via the Semi Implicit Pressure-Linked Equations (SIMPLE) algorithm. A new numerical methodology is applied to resolve all three layers in a single computational domain by establishing two temperature sets, defined according to the location inside the composite structure. Nusselt number behavior shows that for Rayleigh number up to 104 there is no significant variation between the laminar and turbulence models, although the differences increase when the flow gets more intense and/or the porous material becomes more permeable. When comparing the effects of Rayleigh number, Darcy number, porosity, and thermal conductivity ratio between the solid and the fluid on Nu, the results indicate that the solid-phase properties have a greater influence in enhancing the overall heat transferred through the cavity.

© 2017 Elsevier LtdNumerical simulations for laminar double-diffusive free convection in a porous square cavity using the Thermal Non-Equilibrium Model were presented. Vertical surfaces were maintained at constant temperature and concentration whereas horizontal walls were kept insulated. The cavity was filled with a rigid and isotropic porous matrix, which was saturated with an incompressible fluid. Transport equations were discretized by means of the control volume method leading to a coupled algebraic equation set that was solved via the SIMPLE method. Results pointed that both Nuw and Shw are dependent on porosity ϕ and on the thermal conductivity ratio ks/kf. Nuw decreases as ϕ decreases or ks/kf increases due to enhancement of conduction transport across the cavity. On the other hand, Shw and wall mass flux increases as porosity decreases or ks/kf increases. Such dependence of Shw arises from the intensification of recirculating motion in the cavity as ϕ is reduced or ks/kf is of a higher value, which affects heat exchange between phases and, consequently, wall mass fluxes. Finally, this study shows that both average Nusselt and Sherwood numbers diverge from published correlation when ks/kf > 1 for same Da value.

© 2017 Taylor & Francis.This work presents numerical investigations for turbulent flow and heat transfer in a backward-facing step with and without porous inserts. Two classes of the model were employed, namely linear and nonlinear turbulence closures. The entire set of transport equations was discretized by means of the control volume method and the system of algebraic equations obtained was relaxed using the SIMPLE (Semi Implicit Pressure-Linked Equations) method. Results were first validated against the experimental data and the simulations follow experimental values and trends. Computations further indicated that when using the porous insert, the size, shape, and length of the recirculating region were drastically reduced in addition to being pushed toward the channel exit, leading eventually to a complete bubble suppression for thicker inserts. A more permeable medium gave better results in quickly suppressing the circulatory motions. By including porous inserts in the channel, turbulence generated due to the shear inside the recirculating region was damped, whereas high levels of k were concentrated within the permeable structure. Large variations for the skin friction factor along the bottom wall were also smoothed out by placing inserts, spanning from a typical distribution for an unobstructed back-step flow to a standard parallel channel flow distribution as the inserts got ticker. On the other hand, at the upper wall, flow pushed toward the top surface gave rise to a sudden increase of the skin friction factor, which was later stabilized downstream the flow. Heat transfer analysis followed showing damping for Nu at the bottom wall as the thickness of the porous substrate was increased. Overall, the thickness of the insert played a dominant role in changing the final flow and heat transfer characteristics rather than the porosity or permeability of the porous material. Finally, this work indicated that the sudden increase of Nu around the reattachment point, known to be undesirable in many practical situations for causing additional thermomechanical loads on the surface, may by avoided by the use of a porous obstacle past the back-step.

© 2017 Begell House Inc.. All rights reserved.Channels with contractions and obstructions are configurations that can model a number of flows in living bodies and in pipes, in general, and their investigation is of importance in medicine and engineering. Blood obstruction in arteries and clogged flows in ducts due to fouling are examples were such study might be useful. This article deals with flow in a channel having a sinusoidal contraction, which is covered from inside with a layer of porous material. Heat Transfer and turbulence flow are investigated by four distinct models, namely linear high Reynolds, linear low Reynolds, non-linear high Reynolds and non-linear low Reynolds. The equations of motion and mass continuity are discretized by means of the control volume method. The system of algebraic equations is relaxed via the SIMPLE method and the SIP Strong implicit procedure. Results for the size of the recirculating bubble past the contraction indicated that its value seems to be a function of the model used as well as the thickness and properties of the porous material covering the internal walls. The structure of the turbulent thermal field along the channel is also shown to be a function of the turbulence model used as well as the porosity, and permeability of the porous layer.

© 2017 Begell House Inc.. All rights reserved.This paper deals with simulation of turbulent forced convection heat transfer in a channel containing solid and porous baffles. Governing equations are written in their conservative form. Turbulence is simulated using four distinct models, namely linear high Reynolds, linear low Reynolds, non-linear high Reynolds and non-linear low Reynolds. Transport equations of motion and mass continuity are discretized by means of the control volume method. The system of algebraic equations is relaxed via the strong implicit procedure. Results for the recirculating flows past the baffles are shown to be a strong function of the models used as well as the characteristics of the porous material, namely the porosity, and permeability of the porous plates.

© 2017, Begell House Inc. All Rights Reserved.This work presents a study on double-diffusive free convection in a porous square cavity saturated with a Newtonian fluid under laminar flow simulated with the thermal equilibrium model. Transport equations are discretized using the control-volume method and the system of algebraic equations is relaxed via the SIMPLE algorithm. The effect of Ram and porosity on average Nusselt and Sherwood values were investigate. Results show that as Ram increases, both Nusselt and Sherwood numbers increase, indicating enhancement of heat and mass transfer across the cavity. Further, when the Lewis number is increased keeping the same thermal properties, reduction of mass diffusivity further enhances flow recirculation for aiding flows (N=1) within the cavity, which leads to further increase in Nuw and Shw. When varying the buoyancy ratio N from aiding (N>0) to opposing flow (N<0), simulations indicates that when both drives are of equal strength, minimum values for Nusselt and Sherwood occur for N =-1, regardless of Ram. For larger values of |N|, aiding drives will promote fluid rotation in the clockwise direction, for the gradients of T and C here applied, whereas for opposing flows, the fluid rotates in the counter-clockwise direction for opposed conditions at the lateral walls. Results show that porosity also affects Nuw and Shw.

© 2017 Elsevier LtdThis work presents numerical results for two-dimensional combustion of an air/methane mixture in inert porous media using a macroscopic turbulence model. Conservation equations for mass, momentum, energy and chemical species are obtained based on volume-and-time double averaging concept. Distinct energy balances are considered for the porous burner and the gas mixture. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure–velocity coupling. Effects of inlet mass flow rate, excess air, porosity and thermal conductivity ratio, on both the preheating section and combustion region, were investigated. Increasing the mass flow increases peak gas temperatures and pushes the flame front from the preheating zone towards the burner exit. Stoichiometric mixture provokes undesirable combustion in the preheating zone while lean mixtures lower temperatures and pushes flame front forward. Low porosity in the preheating zone promotes more conduction of heat in that region whereas low ϕ values in the combustion zone raises gas and solid temperatures everywhere in that zone.

© 2015 Elsevier Masson SAS.This paper presents a mathematical model and corresponding numerical results for a power-law fluid flowing in a channel partially filled with a homogeneous and isotropic porous medium. At the interface between the clear fluid and the porous material, a model for the stress jump condition takes into consideration the behavior of a power-law fluid. This study shows that the use of a modified permeability, K∗, satisfactorily describes the friction factor of the flow for Reη∗ ≤ 1 (Darcy regime). The mathematical modeling presented, supported by comparisons with analytical and numerical results, also shows that the form drag must be taken into account in the momentum equation, even for a power-law fluid. The mathematical modeling presented has been used to simulate Newtonian as well as power-law fluids flowing in both porous and unobstructed media. For a channel partially filled with porous material and under a fixed mass flow rate, results indicated that the pressure drop is a function of porosity, Darcy number, shear jump coefficient, β, and flow behavior index, n.

© 2016, The Author(s).This book presents, in a self-contained fashion, a series of studies on flow and heat transfer in porous media, in which distinct energy balances are considered for the porous matrix and for the permeating fluid.

© 2016, The Author(s).There is an increasing interest in the use of moving bed technology for chemical compound separation, recuperation of petrochemical processes, drying of grains and seeds and removal of organic matter in affluents, to mention a few applications. The advantages of using a moving bed configuration are low investment, low energy consumption, low maintenance and improvement process performance.

© 2016, The Author(s).Modeling of flows in inert porous media has attracted the attention of scientists and engineers worldwide and in the last decade a number of outstanding books, handbooks and edited books have been written on the subject [Pop I, Ingham DB, Convective heat transfer: mathematical and computational modeling of viscous fluids and porous media (2001)–Nield DA, Bejan A, Convection in porous media, 4th edn (2013)].

© 2016, The Author(s).Convection heat transfer in porous media has been extensively investigated due to its many important engineering applications. The wide applications available have led to numerous investigations in this area. Such applications can be found in solar receiver devices, building thermal insulation, heat exchangers, energy storage units, etc. From the point of view of the energy equation there are two different models, local thermal equilibrium model and two energy approach.

© 2016, The Author(s).Analyses of double-diffusive phenomena in free convection in permeable media has many environmental and industrial applications, such as in oil and gas extraction, movement of gas concentration into the ground, contaminant dispersion in soils, grain storage and drying, petrochemical processes, electrochemical processes, to mention a few [Nithiarasu P, Sundararajan T, Seetharamu KN, Int Commun Heat Mass Transf 24(8):1121 (1997)–Khadiri A, Amahmid A, Hasnaoui M, Rtibi A, Numer Heat Transf Part A, 57(1 I):848–868 (2010)].

© 2016, The Author(s).This book presented, in a self-contained fashion, a series of studies on flow and heat transfer in porous media, in which distinct energy balances are considered for the porous matrix and for the permeating fluid.

Copyright © 2016 by ASME.This work investigates the influence of thermal conductivity ratio on energy and mass transport across a porous square cavity. Modeling of heat transfer from side to side of the enclosure assumed the hypothesis of thermal nonequilibrium between the solid matrix and the fluid phase. Transport equations were discretized using the control-volume method and the system of algebraic equations obtained was relaxed via the SIMPLE algorithm. Results showed that Shw, mass flux of chemical species and heat flux in the solid phase are strongly dependent of ks/kf, significantly increasing their values as such ratio increases.

Copyright © 2016 by ASME.Turbulent natural convection in a two-dimensional horizontal composite square cavity is numerically analyzed using the finite volume method and the thermal non-equilibrium approach. Distinct energy equations for the working fluid and for the porous matrix are proposed reflecting different energy balances for each phase. The composite square cavity is formed by three distinct regions, namely, clear, porous and solid region. It was found that the fluid begins to permeate the porous medium for values of Ra greater than 106. Nusselt number values show that for the range of Ra analyzed there are no significant variation between the laminar and turbulent model solution. When comparing the effects of Ra and Da on Nu, results indicate that the solid phase properties have a greater influence in enhancing the overall heat transferred trough the cavity.

Copyright © 2016 by ASME.We present numerical results for turbulent heat transfer past a backward-facing-step channel with a porous insert. A non-linear eddy viscosity model was applied to handle turbulence. For a constant Darcy number, the thickness of the porous insert was varied in order to analyze its effects on the flow pattern, particularly the damping of the recirculating bubble past the insert. Further, the reduction of the Nusselt number along the bottom heated surface, when using porous materials inside the channel, was investigated. The numerical technique employed for discretizing the governing equations was the control-volume method. The SIMPLE algorithm was used to correct the pressure field and the classical wall function approach was utilized in order to handle flow calculations near the wall. Comparisons of results simulated with different porous materials were presented.

© 2015 Elsevier Ltd. All rights reserved.This work investigates transition to turbulence using laminar kinetic energy modeling based in single-point RANS approach. Transport equations are discretized using the cell centered control-volume method and the system of algebraic equations is relaxed using SIMPLE algorithm. A modified version of the most known laminar kinetic energy model is proposed and compared with the original version and with an experimental correlation transition model. The numerical results show that the laminar kinetic energy approach has reasonable experimental correlation. The modified version presents improvements in prediction of skin friction coefficient in flows subjected to pressure gradients. However, the use of the laminar kinetic energy concept has shown weakness when dealing with detached induced transition. Probably the cause of that is the lack of physical modeling of this phenomenon. Laminar kinetic energy modeling is in continuous evolution and is a good alternative to deal with engineering simulation that needs a good prediction of wall shear stress in transitional flows.

© 2014 American Society of Civil Engineers.Double-averaging methodology is very convenient for investigating spatially heterogeneous flows such as boundary-layer flows over permeable walls. However, spatial averaging volumes suitable for boundary-layer flows over rough walls are very thin in the wall-normal direction whereas those for porous-media flows usually have similar length in all three directions. This scale mismatch can be addressed by allowing the averaging volume to vary in space so that its size can be adjusted to the physical characteristics of particular flow regions. This paper presents a new spatial averaging theorem derived for a spatially variable averaging volume that may contain a stationary solid phase and that may also extend beyond the boundary of the problem domain. The theorem provides the expression for the difference between the average of a spatial derivative and the derivative of the spatial average (of a general flow quantity), here named the commutation correction (CC) term. The CC term contains three parts accounting for (1) the presence of the solid phase in the averaging volume, (2) the averaging volume extending beyond the flow-domain boundary, and (3) the spatial variation of the averaging volume. The first two parts have been acknowledged in the literature; the third part is introduced in this paper and named the volume variation (VV) term. The averaging theorem is used to derive large-scale continuity and momentum equations, which contain the new VV term. The equations are applied to steady, uniform, microscopically two-dimensional flow over a permeable wall. The averaging volume for the boundary-layer flow above the wall is a thin wall-parallel layer; on crossing the wall surface, the volume grows until its height reaches the size required for the homogeneous porous layer. The averaging procedure and the magnitude of the VV term are illustrated by an example adopted from the literature that involves numerical simulation of two-dimensional open-channel flow over a bundle of circular cylinders.

© 2015 by Taylor & Francis Group, LLC.Modeling of flows in inert porous media has attracted the attention of scientists and engineers worldwide, and in the last decade, a number of outstanding books, handbooks, and edited books have been written on the subject (Pop and Ingham 2001, Ingham and Pop 2005, Vafai 2005, Vadasz 2008, Nield and Bejan 2013).

This work presents a study on laminar free convection within a square cavity filled with a fluid saturated porous medium. Macroscopic flow equations are obtained by volume-averaging local instantaneous continuity and momentum equations. The so-called "two-energy equation model" is used, in which distinct macroscopic equations are applied to the working fluid and the solid material. Transport equations are discretized using the control-volume method and the system of algebraic equations is relaxed via the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm. The effect of Ram on Nuw correctly predicted the enhancement of passive heat transfer across the cavity for increasing Ram. Increasing kskf enhances the conduction transport through the solid material and, consequently, dampens the overall Nusselt number, defined here as the ratio between conduction and convection mechanisms over conduction transport only. Further, results indicate that by increasing the void space within the porous material the overall Nusselt number is reduced rather than increased. Individual contributions to the average Nusselt number indicate that, although convection is enhanced with increasing porosity, the reduction of conduction heat transfer through the solid material is the controlling mechanics for Nuw as porosity increases. The results herein might contribute to design and optimization of passive heat transfer systems. © 2014 Taylor and Francis Group, LLC.

This work investigates the influence of physical properties on heat transfer in a turbulent counterflow in a moving bed using the High and Low Reynolds number turbulence models, in which the working fluid flows in opposite direction to that of the steady movement of the permeable rigid medium. Transport equations for flow and heat transfer in a moving bed equipment are applied and discretized using the control-volume method. The system of algebraic equations obtained is relaxed via the SIMPLE algorithm. The effects of Reynolds number, solid-to-fluid velocity ratio, permeability, porosity, ratio of solid-to-fluid thermal capacity and ratio of solid-to-fluid thermal conductivity on heat transport are investigated. Results indicate that motion of solid material, contrary to the direction of the fluid, enhances heat transfer between phases. The same effect was observed for smaller Darcy number and porosity, as well as for higher solid-to-fluid thermal capacity and thermal conductivity ratios. When the intrinsic fluid velocity increases there is a greater conversion of mechanical kinetic energy into turbulence, increasing the final levels of the turbulent kinetic energy for both High and Low Reynolds number models. © 2013 Elsevier Ltd. All rights reserved.

This paper presents an analysis of macroscopic heat and mass transport for turbulent flow in permeable structures, which is based on the thermal non-equilibrium assumption between the porous matrix and the working fluid. Two driving mechanisms are here considered to contribute to the overall momentum transport, namely fluid-temperature driven and concentration driven mass fluxes. The fluid temperature, however, is also affected by the solid temperature distribution as the two phases exchange heat through their interfacial area. Essentially, here the double-diffusive natural convection mechanism is investigated for the fluid phase in turbulent regime. Equations are presented based on the double-decomposition concept, which considers both time fluctuations and spatial deviations about mean values. This work intends to demonstrate that additional transport mechanisms are mathematically derived if velocity, fluid temperature and mass concentration simultaneously present time fluctuations and spatial deviations about average values. A modeled form for the entire set of transport equations is presented where turbulent transfer is based on a macroscopic version of the k-ε model. © 2013 Elsevier Ltd.

Copyright © 2014 by ASME.This article investigates the influence of porosity and thermal conductivity ratio on the Nusselt number in a heated vertical cavity. Heat transfer modeling across the enclosure assumed the hypothesis of thermal equilibrium between the solid matrix and the fluid phase. Transport equations were discretized using the control-volume method and the system of algebraic equations was relaxed via the SIMPLE algorithm. Results showed that, when using the one temperature model under the turbulent regime, the cavity Nusselt number is reduced for higher values of the ratio ks/kf as well as when the material porosity is increased. In both cases, conduction through the solid material overwhelms convection across the medium.

Copyright © 2014 by ASME.This article presents simulations for turbulent flows in a moving permeable bed making use of a macroscopic turbulence model. Intra-pore turbulence is considered by means of a two-equation closure. Governing equations for mean and turbulent flows are volume-averaged. The resulting set of transport equations is discretized using the control-volume method and the obtained algebraic equation set is relaxed via the SIMPLE algorithm. Results indicate that for larger values of Reynolds number, a greater amount of available mechanical energy is converted into turbulence. Simulations further indicate that for lower values of Darcy number and bed porosity, higher levels of turbulence kinetic energy are calculated.

This work presents a study on free convection in a porous square cavity saturated with a Newtonian fluid. Computations for laminar and turbulent flow are performed. Governing equations were time- and volume averaged according to the double-decomposition concept. Discretization of governing equations was obtained with the control-volume approach and the system of algebraic equation was relaxed via the SIMPLE method. Two energy models were employed, namely the one- and two-temperature models. Results indicated that when the ratio of thermal conductivities equals unity, both models give similar results. However, the overall Nusselt number across the cavity is reduced as porosity or the thermal conductivity ratio increases. A critical value for the Rayleigh number, understood as that when laminar and turbulent solution differ by a substantial amount, was found to be a function of the thermal conductivity ratio. © 2014 Elsevier Ltd. All rights reserved.

This paper presents a coupled surface-subsurface flow model which simulates flow over a permeable floodplain during a flooding event. A simple hypothetical example is used for illustrating the effects of the floodplain permeability (i.e. surface-subsurface exchange) and air entrapment in the subsurface on the surface flow characteristics. The results show that, for initially unsaturated conditions of a soil which forms a floodplain, neglecting both water and air movement in the subsurface may lead to significant errors in predicted characteristics of the surface flow during a flooding event which involves overbank flow. © 2014 Taylor & Francis Group, London.

This article presents a thermo-mechanical approach to investigate heat transfer between solid and fluid phases in a model gasifier. A two-temperature equation approach is applied in addition to a macroscopic model for laminar flow through a porous moving bed. Transport equations are discretized using the control-volume method and the system of algebraic equations is relaxed via the SIMPLE algorithm. The effects on inter-phase heat transfer due to variation of medium permeability, thermal conductivity and thermal capacity are analyzed. Results indicate that for smaller medium permeabilities, as well as for higher solid-to-fluid thermal capacity and thermal conductivity ratios, enhancement of heat transfer between phases is observed. © (2014) Trans Tech Publications, Switzerland.

This work investigates the influence of porosity and thermal conductivity ratio on the Nusselt number of a cavity filed with a fluid saturated porous substrate. The flow regime considered intra-pore turbulence and a macroscopic k-ε model was applied. Heat transfer across the cavity assumed the hypothesis of thermal equilibrium between the solid and the fluid phases. Transport equations were discretized using the control-volume method and the system of algebraic equations was relaxed via the SIMPLE algorithm. Results showed that when using the one energy equation model under the turbulent regime, simulated with a High Reynolds turbulence model, the cavity Nusselt number is reduced for higher values of the ratio ks/kf as well as when the material porosity is increased. In both cases, conduction thorough the solid material becomes of a greater importance when compared with the overall transport that includes both convection and conduction mechanisms across the medium. © 2013 Elsevier Ltd.

This paper deals with numerical simulation of turbulence in a parallel flow moving bed, in which turbulence is considered in the void spaces occupied by the fluid phase. Volume averaging techniques are applied to both time-mean and statistical flow fields. The set of resulting governing equations is discretized via the control-volume method and the resulting algebraic equation set is solved via the SIMPLE method. Results indicate that for lower values of slip ratio, Darcy number and bed porosity, higher levels of turbulence kinetic energy are computed. © 2013 Elsevier Ltd.

This study investigates the influence of physical properties on heat transfer between solid and fluid phases in a cross flow moving porous bed, in which the fluid moves with longitudinal and transversal components with respect to the permeable bed. For simulating flow and heat transfer, a two-energy equation model is applied in addition to a mechanical model. Transport equations are discretized using the control-volume method and the system of algebraic equations is relaxed via the SIMPLE algorithm. The effects of flow properties, such as Reynolds number, solid-to-fluid velocity ratio, permeability and porosity, as well as the effects of thermal properties, namely solid-to-fluid thermal capacity and solid-to-fluid thermal conductivity ratio, are analyzed. The numerical results show that the Reynolds number affects strongly the flow behavior and for high values of the solid-to-fluid velocity ratio, solid-to-fluid thermal capacity ratio and solid-to-fluid thermal conductivity ratio, there is a decrease in temperature gradients everywhere in the domain and the fluid temperature reaches higher values mainly in the symmetry region of the channel. © 2013 Elsevier Ltd. All rights reserved.

This work presents one-dimensional numerical results for combustion of an air/methane mixture in inert porous media using laminar and radiation models. Comparisons with experimental data are reported. The burner is composed by a preheating section followed by a combustion region. Macroscopic equations for mass, momentum and energy are obtained based on the volume average concept. Distinct energy equations are considered for the porous burner and the flowing gas. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to relax the entire equation set. Inlet velocity, excess air, porosity and solid-to-fluid thermal conductivity ratio were varied in order to investigate their effect on temperature profiles. Results indicate that higher inlet velocities result in higher gas temperatures, following a similar trend observed in the experimental data used for comparisons. Burning of mixtures close to the stoichiometric conditions also increased temperatures, as expected. Increasing the thermal conductivity of the preheating section reduced peak temperature in the combustion region. The use of porous material with very high thermal conductivity on the combustion region did not affect significantly temperature levels in the combustion section. © 2012 Elsevier Ltd.

This work investigates the influence of physical properties on heat transfer between the solid and fluid phases in a porous reactor, in which both the permeable bed and the working fluid move in the same direction with respect to fixed bounding walls. For simulating laminar flow and heat transfer, a two-energy equation model is applied in addition to a mechanical model. Transport equations are discretized using the control-volume method and the system of algebraic equations is relaxed via the SIMPLE algorithm. The effects of Reynolds number, solid-to-fluid velocity ratio, permeability, porosity, ratio of solid-to-fluid thermal capacity and ratio of solid-to-fluid thermal conductivity on flow and heat transport are analyzed. The laminar model is validated by means of an analytical solution. Results for concurrent laminar flow indicate that, when the speed of the solid approaches that of the fluid, the strong axial convection of the solid, as well as the reduction of the relative velocity, cause an increase in the axial length needed for thermal equilibrium between phases to occur. Longer thermal developing lengths are also found for higher permeabilities and higher porosities. For higher solid-to-fluid thermal capacities and higher solid-to-fluid thermal conductivity ratios, the temperature of the solid phase shows less axial variation regardless of its velocity in relation to the fluid phase. © 2011 Elsevier Ltd. All rights reserved.

© 2012 Elsevier Ltd. All rights reserved.Turbulence in Porous Media introduces the reader to the characterisation of turbulent flow, heat and mass transfer in permeable media, including analytical data and a review of available experimental data. Such transport processes occurring a relatively high velocity in permeable media are present in a number of engineering and natural flows. This new edition features a completely updated text including two new chapters exploring Turbulent Combustion and Moving Porous Media. De Lemos has expertly brought together a text that compiles, details, compares and evaluates available methodologies for modelling and simulating flow, providing an essential tour for engineering students working within the field as well as those working in chemistry, physics, applied mathematics, and geological and environmental sciences.

© 2012, The Author(s).This book investigated the influence of the presence of a porous layer covering a surface where a jet collides. This work reviewed and compiled a systematic study on impinging jets on bare and covered walls, which was carried out in the last few years at ITA, Brazil, and considered both laminar [Graminho and de Lemos (Numer Heat Transf Part A Appl 54(2):151–177, 2008), de Lemos and Fischer (Numer Heat Transf Part A Appl 54:1022–1041, 2008), Dórea and de Lemos (Inter J Heat Mass Transf 53:5089–5101, 2010)] and turbulent flow regimes [Graminho and de Lemos (Inter J Heat Mass Transf 52:680–693, 2009), Fischer and de Lemos (Numer Heat Transf Part A Appl 58:429–456, 2010), de Lemos and Dórea (Numer Heat Transf Part A Appl 59(10):769–798, 2011)]. By that, a self-contained text was put together in order to convey to the interested reader the major steps and results achieved on such research topic. Two energy modes were applied, namely 1EEM and 2EEM, based respectively on the Local Thermal Equilibrium (LTE) and Local Thermal Non-Equilibrium hypotheses (LNTE). It was observed that the Reynolds number and porosity strongly influences the stagnation Nusselt value while the porous layer thickness affects more intensely the distribution of Nu along the plate. Cases with low porosity and highly permeable layers of porous material tend to yield better heat absorption/release rates when compared with a bare wall case. Regardless of the model used, increasing the thermal conductivity ratio is always beneficial to heat transfer enhancement form the hot wall. Ultimately, results in this work might be useful to engineers designing systems that make use of impinging jets over thermally conducting porous materials.

© 2012, The Author(s).Impinging jets are often used in industrial applications for enhancing or damping localized heat transfer rates. When the flow is turbulent, thin boundary layers are located inside the stagnation zone, promoting even further cooling, heating or drying processes. Applications of such systems include metals cooling, glass tempering, electronics cooling, drying of textiles products and paper, to mention a few. In this book, two flow configurations are investigated, namely axisymmetric confined arrangements and two-dimensional planar jets. A fluid jet enters a cylindrical chamber through an aperture in an upper disk. An annular clearance between the cylinder lateral wall and the disc allows fluid to flow out of the enclosure.

© 2012, The Author(s).In this book, two flow configurations are investigated, namely axi-symmetric confined arrangements and two-dimensional planar jets. A fluid jet enters a cylindrical chamber through an aperture in an upper disk. An annular clearance between the cylinder lateral wall and the disc allows fluid to flow out of the enclosure.

© 2012, The Author(s).For an impinging jet, the flow is considered to be turbulent for (Formula presented.), where the Reynolds number is given by (Formula presented.), where (Formula presented.) is the incoming jet velocity and (Formula presented.) when calculating Re for adequate comparisons with similar simulations in the literature (see Fig. 1.2).

© 2012, The Author(s).First, for code validation, initial simulations were conducted in the clear chamber of height H, where free flow occurs. This first set of simulations considers that a solid wall is located at depth H and no porous layer is positioned at the bottom of the chamber. Therefore, streamlines, velocity profiles and turbulence kinetic energy contours for an empty enclosure are presented prior to showing computations considering the porous layer.

© 2012, The Author(s).For running the LTNE Model, the Reynolds number was also defined by Eq. (4.1). As mentioned before, for an impinging jet the flow is considered to be turbulent for Re > 1,000.

'Turbulence in Porous Media' introduces the reader to the characterisation of turbulent flow, heat and mass transfer in permeable media, including analytical data and a review of available experimental data. Such transport processes occurring a relatively high velocity in permeable media are present in a number of engineering and natural flows. This new edition features a completely updated text including two new chapters exploring Turbulent Combustion and Moving Porous Media. De Lemos has expertly brought together a text that compiles, details, compares and evaluates available methodologies for modelling and simulating flow, providing an essential tour for engineering students working within the field as well as those working in chemistry, physics, applied mathematics, and geological and environmental sciences. Brings together groundbreaking and complex research on turbulence in porous media Extends the original model to situations including reactive systems Now discusses movement of the porous matrix. © 2012 Elsevier Ltd. All rights reserved.

This paper presents two-dimensional numerical simulations of combustion of an air/methane mixture in a radial porous combustor using a model that explicitly considers the intra-pore levels of turbulent kinetic energy. Transport equations are written in their time-and-volume averaged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. A cylindrical porous combustor is analyzed, in which the mixture flows inside it in the axial direction, being the flue gases ejected through the lateral surface. Combustion is modeled via a simple closure. For high excess air, the flame front moves towards the lateral exit of the burner. Also, increasing the inlet flow rate for stoichiometric mixture pushes the flame out of the porous material. Copyright © 2012 by ASME.

This paper presents computations for natural convection within a porous cavity filled with a fluid saturated permeable medium. The finite volume method in a generalized coordinate system is applied. The walls are maintained at constant but different temperatures, while the horizontal walls are kept insulated. Governing equations are written in terms of primitive variables and are recast into a general form. Flow and heat transfer characteristics are investigated for two energy models and distinct solid-To-fluid thermal conductivity ratio. Copyright © 2012 by ASME.

This work investigates the influence of physical properties on heat transfer between solid and fluid phases in a moving porous bed, in which the working fluid flows in the opposite direction with respect to the permeable medium. A two-energy equation model is applied in addition to a macroscopic mechanical model for laminar flow. Transport equations are discretized using the control-volume method and the system of algebraic equations is relaxed via the SIMPLE algorithm. The effects on inter-phase heat transfer due to variation of Reynolds number, solid-to-fluid velocity ratio, solid-to-fluid thermal capacity ratio, permeability, porosity, and solid-to-fluid thermal conductivity ratio are analyzed. Results for a counterflow moving bed indicate that motion of solid material, contrary to the direction of the fluid, enhances heat transfer between phases. The same effect was observed for smaller Darcy numbers and porosity, as well as for higher solid-to-fluid thermal capacity and thermal conductivity ratios. © 2012 Copyright Taylor and Francis Group, LLC.

This paper presents a mathematical model for treating turbulent combusting flows in a moving porous bed, which might be useful to design and analysis of modern and advanced biomass gasification systems. Here, one explicitly considers the intra-pore levels of turbulent kinetic energy and the movement of the rigid solid matrix is considered to occur at a steady speed. Transport equations are written in their time-and-volume-averaged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. The rate of fuel consumption is described by an Arrhenius expression involving the product of the fuel and oxidant mass fractions. Results indicate that fixing the gas speed and increasing the speed of the solid matrix pushes the flame front towards the end of the reactor. Also, since the rate of production of turbulence is dependent on the relative velocity between phases, as the solid velocity approaches that of the gas stream, the level of turbulence in the flow is reduced. © 2011 Elsevier Ltd.

This paper presents two-dimensional numerical simulations of combustion of an air/methane mixture in a radial porous combustor using a model that explicitly considers the intra-pore levels of turbulent kinetic energy. Transport equations are written in their time-and-volume averaged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. A cylindrical porous combustor is analyzed, in which the mixture flows inside it in the axial direction, being the flue gases ejected through the lateral surface. Combustion is modeled via a unique simple closure. For high excess air, the flame front moves towards the lateral exit of the burner. Also, increasing the inlet flow rate for stoichiometric mixture pushes the flame out of the porous material. © 2011 by ASME.

This work shows simulations for a turbulent jet impinging against a flat plane covered with a layer of permeable and thermally conducting material. Distinct energy equations are considered for the porous layer attached to the wall and for the fluid that impinges on it. Parameters such as Reynolds number, porosity, permeability, thickness and thermal conductivity of the porous layer are varied in order to analyze their effects on the local distribution of Nu. The macroscopic equations for mass, momentum and energy are obtained based on volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Results indicate that inclusion of a porous layer eliminates the peak in Nu at the stagnation region. For highly porous and highly permeable material, simulations indicate that the integral heat flux from the wall is enhanced when a thermally conducting porous material is attached to the wall. Copyright © 2011 by ASME.

The objective of this chapter is to present one- and two-dimensional numerical simulations of combustion of an air/methane mixture in porous materials using a mathematical model that explicitly considers the intra-pore levels of turbulent kinetic energy. Transport equations are written in their time-and-volumeaveraged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. Four different thermomechanical models are compared, namely Laminar, Laminar with Radiation Transport, Turbulent, Turbulent with Radiation Transport. Combustion is modeled via a unique simple closure. Preliminary testing results indicate that a substantially different temperature distribution is obtained depending on the model used. In addition, for high excess air peak gas temperature are reduced and the flame front moves towards the exit of the burner. Also, increasing the inlet flow rate for stoichiometric mixture pushes the flame out of the porous material. © Springer-Verlag Berlin Heidelberg 2010.

Numerical solutions for turbulent flow in a composite channel are presented. Here, a channel with a centered porous material is considered. The interface between the porous medium and the clear flow was assumed to have different transversal positions and the porous matrix was simulated with distinct permeabilities. Governing equations were discretized and solved for both domains making use of one unique numerical methodology. Increasing the size of the porous material pushes the flow outwards, increasing the levels of turbulent kinetic energy at the macroscopic interface. For high permeability media, a large amount of mechanical energy is converted into turbulence inside the porous structure. © 2011 Elsevier Ltd.

Non-linear turbulence models can be seen as an improvement of the classical eddy-viscosity concept due to their better capacity to simulate characteristics of important flows. However, application of non-linear models demand robustness of the numerical method applied, requiring a stable discretization scheme for convergence of all variables involved. Usually, non-linear terms are handled in an explicit manner leading to possible numerical instabilities. Thus, the present work shows the steps taken to adapt a general non-linear constitutive equation using a new semi-implicit numerical treatment for the non-linear diffusion terms. The objective is to increase the degree of implicitness of the solution algorithm to enhance convergence characteristics. Flow over a backward-facing step was computed using the control volume method applied to a boundary-fitted coordinate system. The SIMPLE algorithm was used to relax the algebraic equations. Classical wall function and a low Reynolds number model were employed to describe the flow near the wall. The results showed that for certain combination of relaxation parameters, the semi-implicit treatment proposed here was the sole successful treatment in order to achieve solution convergence. Also, application of the implicit method described here shows that the stability of the solution either increases (high Reynolds with non-orthogonal mesh) or preserves the same (low Reynolds number applications). Additional advantages of the procedure proposed here lie in the possibility of testing different non-linear expressions if one considers the enhanced robustness and stability obtained for the entire numerical algorithm. © 2010 John Wiley & Sons, Ltd.

This work presents numerical results for two-dimensional combustion of an air/methane mixture in inert porous media using turbulence and radiation models. Distinct energy equations are considered for the porous burner and for the fuel in it. Inlet velocity and excess air-to-fuel ratio are varied in order to analyze their effects on temperature and flame front location. The macroscopic equations for mass, momentum and energy are obtained based on the volume average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Results indicate that for high excess air values, the gas temperature peaks are reduced. Also, for the same conditions the flame front moves towards the exit of the burner. Results also indicate that the same flame front behavior occurs as the inlet velocity increases. Copyright © 2011 by ASME.

The objective of this work is to study the influence of physical and geometrical properties on heat transfer between solid and fluid phases in a counter-flow porous bed, for cases where the fluid moves in opposite direction with respect to the permeable matrix. For simulating the flow and heat transfer, a two-energy equation model is applied in addition to a mechanical model. Transport equations are discretized using the control-volume method. The effects of solid-to-fluid velocity ratio, permeability, porosity, ratio of solid-to-fluid thermal capacity and ratio of solid-to-fluid thermal conductivity on flow and heat transport are analyzed. Results for a counterflow, that is similar to the heat exchangers in a countercurrent, indicate that there is more heat exchange for the smaller values of the parameters analyzed resulting in more uniform heat transfer between phases along the channel. Copyright © 2011 by ASME.

This article presents numerical results for a turbulent jet impinging against a flat plane covered with a layer of permeable and thermally conducting material. Distinct energy equations are considered for the solid porous material attached to the wall and for the fluid that impinges on it. Parameters such as Reynolds number, porosity, permeability, thickness, and thermal conductivity of the porous layer are varied in order to analyze their effects on the local distribution of Nu. The macroscopic equations for mass, momentum, and energy are obtained based on volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Results indicate that inclusion of a porous layer eliminates the peak in Nu at the stagnation region. For highly porous and highly permeable material, simulations indicate that the integral heat flux from the wall is enhanced when a thermally conducting porous material is attached to the surface. Copyright © Taylor &Francis Group, LLC.

Interface heat transfer in a moving porous bed is analyzed. This work proposes a set of transport equations for solving problems involving turbulent flow and heat transfer in a moving bed equipment. The device is modeled as a saturated porous matrix in which the solid phase moves with a steady imposed velocity. Additional drag terms appearing in the momentum equation, as well as interfacial heat transfer between phases, are assumed to be a function of the relative velocity between the fluid and solid phases. Results indicate that, as the phases attain velocities of equal order, heat transfer between solid and fluid occurs mainly by the conduction mechanism. © 2010 by ASME.

This paper presents simulations for a jet impinging against a flat plane covered with a layer of a porous material. Macroscopic equations for mass, momentum and energy, for the fluid and for the porous matrix, are obtained based on the volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. The effect of porosity and energy model on the local distribution of Nu was analyzed. Results indicate that for low porosity materials, a substantially different Nu number is calculated depending on the energy model applied. © 2010 by ASME.

This work shows numerical simulations of an impinging jet on a flat plate covered with a layer of a porous material. Macroscopic equations for mass and momentum are obtained based on the volume-average concept. Two macroscopic models are employed for analyzing energy transport, namely the one-energy equation model, based on the Local Thermal Equilibrium assumption (LTE), and the two-energy equation closure, where distinct transport equations for the fluid and the porous matrix follow the Local Non-Thermal Equilibrium hypothesis (LNTE). The numerical technique employed for discretizing the governing equations was the finite volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Parameters such as porosity, porous layer thickness, material permeability and thermal conductivity ratio were varied in order to analyze their effects on flow and heat transport. Results indicate that for low porosities, low permeabilities, thin porous layers and for high thermal conductivity ratios, a different distribution of local Nusselt number at the wall is calculated depending on the energy model applied. The use of the LNTE model indicates that it is advantageous to use a layer of highly conducting and highly porous material attached to the hot wall. © 2010 Elsevier Ltd. All rights reserved.

This work shows numerical results for a turbulent jet impinging against a flat plane covered with a layer of permeable material, which is kept at a higher temperature than that of the incoming fluid. Parameters such as porosity, permeability, thickness, and thermal conductivity of the porous layer are varied in order to analyze their effects on the local distribution of Nu. The macroscopic equations for mass, momentum, and energy are obtained based on volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Results indicate that inclusion of a porous layer decreases the peak in Nu avoiding excessive heating or cooling at the stagnation point. Also found, was that the integral heat flux from the wall is enhanced for certain ranges of values of porosity, layer thickness, and thermal conductivity ratio. Copyright © Taylor & Francis Group, LLC.

Engineering equipment design and environmental impact analyses can benefit from proper and more accurate modeling of turbulent transport in porous media. Several natural and engineering systems can be seen as porous structures through which a working fluid permeates. Turbulence models proposed for such flows depend on the order of application of time and volume average operators. Two methodologies, following the two orders of integration, lead to different governing equations for the statistical quantities. The concept of double-decomposition is discussed and models are classified in terms of the order of application of time and volume averaging operators, among other peculiarities. For hybrid media, involving both a finite porous structure and a clear flow region, difficulties arise due to the proper mathematical treatment given at the interface. This paper presents and discusses numerical solutions for such hybrid medium. © 2010 American Institute of Physics.

In this paper, a model for turbulent flow and heat transfer in a highly porous medium is proposed and applied to a porous channel bounded by parallel plates. Macroscopic continuity, momentum and energy equations are presented. Local non-thermal equilibrium is considered by means of independent equations for the solid matrix and the working fluid. The numerical methodology used is based on the control-volume approach. The effects of thermal dispersion, Reynolds number, dimensionless particle diameter, thermal conductivity ratio and Darcy number, on the Nusselt number, are presented. For laminar and turbulent flows the thermal dispersion mechanism leads to larger local temperature differences. Increase in Re number causes values for Nu, of both phases, to increase. Porosity increase causes the solid phase Nusselt number to decrease whereas the fluid Nusselt number in augmented. In general, an increase in the particle diameter increases Nusselt number. Also, the thermal conductivity ratio causes the most pronounced effect on Nusselt numbers. © 2010 Elsevier Ltd. All rights reserved.

The objective of this paper is to present an extension of a simplified reaction kinetics model that, combined with a thermo-mechanical closure, entails a full-generalized turbulent combustion model for flow in porous media. In this model, one explicitly considers the intra-pore levels of turbulent kinetic energy. Transport equations are written in their time-and-volume-averaged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. The rate of fuel consumption is described by an Arrhenius expression involving the product of the fuel and oxidant mass fractions. These mass fractions are double decomposed in time and space and, after applying simultaneous time-and-volume integration operations to them, distinct terms arise, which are here associated with the mechanisms of dispersion and turbulence. Modeling of these extra terms remains an open question and the derivations herein might motivate further development of models for turbulent combustion in porous media. © 2010 Elsevier Ltd. All rights reserved.

The objective of this paper is to show numerical simulations of combustion of an air/methane mixture in porous materials. Here, a model that considers the intrapore levels of turbulent kinetic energy is used. Transport equations are written in their time-and-volume-averaged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. Different thermo-mechanical models are compared, namely Laminar, Laminar with Radiation Transport, Turbulent, Turbulent with Radiation Transport. Combustion is modeled via a unique simple closure. Results indicate that a substantially different temperature distribution is obtained depending on the model used. In addition, for high excess air, peak gas temperatures are reduced. Copyright © 2010 by ASME.

The objective of this paper is to present numerical simulations of combustion of an air/methane mixture in porous materials using a model that considers the intra-pore levels of turbulent kinetic energy. Transport equations are written in their time-and-volume-averaged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. Four different thermo-mechanical models are compared, namely Laminar, Laminar with Radiation Transport, Turbulent, Turbulent with Radiation Transport. Combustion is modeled via a unique simple closure. Preliminary testing results indicate that a substantially different temperature distribution is obtained depending on the model used. In addition, for high excess air peak gas temperature are reduced. © (2010) Trans Tech Publications.

In this paper, numerical simulation of a jet impinging against a flat plane covered with a layer of a porous material is presented. The plate is kept at a temperature higher than that of the incoming fluid. Macroscopic transport equations are obtained based on a volume average concept. Discretization of such governing equations is accomplished by means of the control volume method applied with a boundary-fitted non-orthogonal coordinate system. Pressure-velocity coupling is treated with the use of the SIMPLE algorithm. Parameters such as permeability, thickness of the porous layer and thermal conductivity ratio are varied in order to analyze their effects on the local distribution of Nu. Results indicate that inclusion of a porous layer decreases the peak in Nu avoiding excessive heating or cooling at the stagnation point. Also found was that the integral heat flux from the wall is enhanced for certain range of values of layer thickness, porosity, and thermal conductivity ratio. Copyright © 2009 by ASME.

This paper presents one-dimensional simulations of combustion of an air/methane mixture in porous materials using a model that explicitly considers the intra-pore levels of turbulent kinetic energy. Transport equations are written in their time-and-volume-averaged form and a volume-based statistical turbulence model is applied to simulate turbulence generation due to the porous matrix. Four different thermo-mechanical models are compared, namely Laminar, Laminar with Radiation Transport, Turbulent, Turbulent with Radiation Transport. Combustion is modeled via a unique simple closure. Preliminary testing results indicate that a substantially different temperature distribution is obtained depending on the model used. In addition, for high excess air peak gas temperature is reduced and the flame front moves towards the exit of the burner. Also, increasing the inlet flow rate for stoichiometric mixture pushes the flame out of the porous material. © 2009 Elsevier Ltd. All rights reserved.

Laminar heat transfer in a porous channel is numerically simulated with a two-energy equation model for conduction and convection. Macroscopic equations for continuity, momentum and energy transport for the fluid and solid phases are presented. The numerical methodology employed is based on the control volume approach with a boundary-fitted non-orthogonal coordinate system. Fully developed forced convection in a porous channel bounded by parallel plates is considered. Solutions for Nusselt numbers along the channel are presented for laminar flows. Results simulate the effects Reynolds number Re, porosity, particle size and solid-to-fluid thermal conductivity ratio on Nusselt sumber, Nu, which is defined for both the solid and fluid phases. High Re, low porosities, low particle diameters and low thermal conductivity ratios promote thermal equilibrium between phases leading to higher values of Nu. © 2009 Elsevier Ltd. All rights reserved.

This paper presents results for coupled heat and mass transport under laminar and turbulent flow regimes in porous cavities. Two driving mechanisms are considered to contribute to the overall momentum transport, namely temperature driven and concentration driven mass fluxes. Aiding and opposing flows are considered, where temperature and concentration gradients are either in the same direction or of different sign, respectively. Modeled equations are presented based on the double-decomposition concept, which considers both time fluctuations and spatial deviations about mean values. Turbulent transport is accounted for via a macroscopic version of the k-ε model. Variation of the cavity Nusselt and Sherwood numbers due to changes on N, where N is the ratio of solute to thermal Grashof numbers, is presented. Results indicate that for adding cases, mass and heat transfer across the cavity are enhanced faster than for cases with opposing temperature and concentration gradients. For the conditions here investigated, the use a turbulence model gave results for Nu and Sh that were nearly double when compared with laminar results for the same conditions. © 2009 Elsevier Ltd. All rights reserved.

This work shows numerical results for a jet impinging onto a flat plane covered with a layer of a porous material. Porosity of the porous layer is varied in order to analyze its effect on the local distribution of Nu. Macroscopic equations for mass and momentum ae obtained based on the volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Results indicate that inclusion of a porous layer decreases the peak in Nu avoiding excessive heating or cooling near the stagnation region. Copyright © 2008 by ASME.

This paper shows a proposition of a set of transport equations and their boundary conditions for solving problems involving flow and heat transfer in a moving bed equipment. The reactor is seen as a porous matrix in which the solid phase is moving. Additional drag terms appearing the momentum equation are a function of the relative velocity between the fluid and solid phases. Turbulence equations are also influenced by the speed of the solid phase. Results show the decrease for turbulent kinetic energy as the solid speed approaches the fluid speed. Heat transfer rate between phases is also damped as the solid speed increases. Copyright © 2008 by ASME.

This paper presents numerical solutions for turbulent flow in a channel containing a layer of porous material with wavy form. One unique set of transport equations, for mass and momentum, is applied to both regions, namely the clear and porous domains. Effects of interface wave number on mean and turbulence fields are investigated. Results indicate that around the peaks of the sinusoidal layer values of the turbulent kinetic energy are higher for lower values of n, where n is the wave number associated with the wavy interface shape. Also, as the surface gets rough (high n), the use of a jump condition for the diffusion flux across the interface does not affect the pressure drop along the channel. Copyright © 2008 by ASME.

Flow over vegetation and bottom of rivers can be characterized by some sort of porous structure of irregular surface through which a fluid permeates. Also, in engineering systems, one can have components that make use of a working fluid flowing over irregular layers of porous material. This article presents numerical solutions for such hybrid medium, considering here a channel partially filled with a flat porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both the regions. A diffusion-jump model for both the turbulent kinetic energy and its dissipation rate, across the interface, is presented and discussed upon. The discretization steps taken for numerically accommodating such model in the system of algebraic equations are presented. Numerical results show the effects of Reynolds number, porosity, and permeability on mean and turbulence fields. Results indicate that when negative values for the stress jump coefficient are applied, the peak of the turbulent kinetic energy distribution occurs at the macroscopic interface. © Springer Science+Business Media B.V. 2009.

Turbulent natural convection in a two-dimensional horizontal composite square cavity, isothermally heated at the left side and cooled from the opposing surface, is numerically analyzed using the finite volume method. The composite square cavity is formed by three distinct regions, namely, clear, porous and solid region. The development of a numerical tool able to treat all these regions as one computational domain is of advantage for engineering design and analysis of passive thermal control systems. Governing equations are written in terms of primitive variables and are recast into a general form. It was found that the fluid begins to permeate the porous medium for values of Ra greater than 106. Nusselt number values show that for the range of Ra analyzed there is no significant variation between the laminar and turbulent model solution. When comparing the effects of Ra, ks/kf and Da on Nu, results indicate that the solid phase properties have a greater influence in enhancing the overall heat transferred trough the cavity. © 2008 Elsevier Ltd. All rights reserved.

Turbulent impinging jets on heated surfaces are widely used in industry to modify local heat transfer coefficients. The addition of a porous substrate covering the surface contributes to a better flow distribution, which favors many engineering applications. Motivated by this, this work shows numerical results for a turbulent impinging jet into a cylindrical enclosure with and without a porous layer at the bottom. The macroscopic time-averaged equations for mass and momentum are obtained based on a concept called double decomposition, which considers spatial deviations and temporal fluctuations of flow properties. Turbulence is handled with a macroscopic k-ε model, which uses the same set of equations for both the fluid layer and the porous matrix. The numerical technique employed is the control volume method in conjunction with a boundary-fitted coordinate system. One unique computational grid is used to compute the entire heterogeneous medium. The SIMPLE algorithm is applied to relax the system of algebraic equations. Results indicate that the permeability of the porous layer and the height of the fluid layer significantly affect the flow pattern. The effect of the porous layer thickness was less pronounced in affecting the flow behavior in the fluid layer. © 2008 Elsevier Ltd. All rights reserved.

Convective heat-transfer coefficients in foam-like materials, modelled as an array of elliptic rods, are numerically determined. An incompressible fluid is considered, flowing through an infinite foam-like material with an arbitrary solid temperature. A repetitive cell is identified and periodic boundary conditions are applied. Turbulence is handled with both low and high Reynolds number formulations. The interfacial heat-transfer coefficient is obtained by volume integrating the distributed variables obtained within the cell. The results indicate that, for the same mass-flow rate, materials formed by elliptic rods have a lower interfacial heat-transfer coefficient compared to other media modelled as staggered arrays of square rods. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

This work examines the performance of linear and nonlinear eddy-viscosity models when used to predict the turbulent flow in periodically sinusoidal-wave channels. Two geometries are investigated, namely a converging-diverging channel and a channel with concave-convex walls. The numerical method employed for the discretization of the equations is the control-volume method in a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm is used for correcting the pressure field. The classical wall function and a low Reynolds model are used to describe the flow near the wall. Comparisons between those two approaches using linear and nonlinear turbulence models are done. Here, a new implicit numerical treatment is proposed for the nonlinear diffusion terms of the momentum equations in order to increase the robustness. Results show that by decomposing and treating terms as presented, solutions using nonlinear models and the high Reynolds wall treatment, which combine accuracy and economy, are more stable and easier to be obtained.

This paper reports a numerical study of the Soret effect on steady-state flows, which are induced by double-diffusion in an enclosure. Convection takes place in a square cavity filled with a porous medium. Horizontal walls are impermeable and subjected to a vertical gradient of temperature. Vertical surfaces are adiabatic and subjected to a horizontal gradient of concentration. The physical model for momentum conservation equation makes use of the Forchheimer extension of the classical Darcy model. Governing parameters of the problem under study are thermal and solutal Rayleigh (Ra), Buoyancy ratio (N), Lewis numbers (Le) and Soret parameters (M). Computations using the finite-volume method cover the range 100<Ra<1000, -40<M<40 for N=0.1. Copyright © 2008 by ASME.

The objective of this paper is to simulate turbulent flow and heat transfer in industrial porous burners. Transport equations are written in their time-and-volume-averaged form and a volume-based statistical turbulence model is applied to simulate the intra-porous turbulence generation. Combustion is modeled via a simple closure. Preliminary testing results indicate that a substantially different flow pattern is obtained depending on the model used. In addition, for high inlet flow rates or high excess air, the flame front moves towards the exit of the chamber. Copyright © 2008 by ASME.

Moving beds are present in a number of engineering equipment. Their analyses contribute to improvement of many energy and material production processes. Here, transport equations for flow in a moving bed reactor are presented. Such device is modeled as a saturated porous matrix with a steady speed. Transport equations are time-and-volume averaged and additional form and viscous drags, due to the porous structure, are assumed to be a function of the relative velocity between phases. Turbulence equations reflect their dependence on the speed of the solid substrate. Results show the decay of turbulent kinetic energy levels as the solid speed approaches the speed of the moving fluid. Copyright © 2008 by ASME.

Heat transfer in a porous reactor under cross flow is investigated. The reactor is modeled as a porous bed in which the solid phase is moving horizontally and the flow is forced into the bed in a vertical direction. Equations are time-and-volume averaged and the solid phase is considered to have a constant imposed velocity. Additional drag terms appearing the momentum equation are a function of the relative velocity between the fluid and solid phases. Turbulence equations are also affected by the speed of the solid matrix. Results show temperature distributions for several ratios of the solid to fluid speed. Copyright © 2008 by ASME.

Heat transfer between phases in a moving porous bed is analyzed. This work proposes a set of transport equations for solving problems involving turbulent flow and heat transfer in a moving bed equipment. The device is modeled as a saturated porous matrix in which the solid phase moves with a steady imposed velocity. Additional drag terms appearing the momentum equation, as well as interfacial heat transfer between phases, are assumed to be a function of the relative velocity between the fluid and solid phases. Turbulence transport equations are here also dependent on the speed of the solid material. Results indicate that, as the phases attain velocities of equal order, turbulence in damped and heat transfer between solid and fluid occurs mainly by conduction mechanism. © 2008 Elsevier Ltd. All rights reserved.

This paper presents a set of transport equations for solving problems involving turbulent flow in a moving bed reactor. The reactor is seen as a porous matrix with a moving solid phase. Equations are time-and-volume averaged and the solid phase is considered to have an imposed constant velocity. Additional drag terms appearing in the momentum equation are assumed to be a function of the relative velocity between the fluid and solid phase. Turbulence equations are influenced by the speed of the solid phase in relation to that of the flowing fluid. Results show the decrease of turbulent kinetic energy levels as the solid speed approaches the speed of the moving bed. © 2008 Elsevier Ltd. All rights reserved.

Thermal dispersion in porous media is an import phenomenon in combustion and in steam injection systems for Enhanced Oil Recovery methods, among several others engineering applications. In this work, thermal dispersion tensors were calculated within an infinite porous medium formed by a spatially periodic array of longitudinally-displaced elliptic rods. Two different thermal conductivity ratios between the solid and fluid phases were used for analyzing their effect on the thermal dispersion tensor, following a systematic analysis of several porous media modeled by different unit-cell geometry. As such, just one unit-cell, together with periodic boundary conditions for mass, momentum and energy equations, was used to represent the medium. The numerical methodology herein employed is based on the control-volume approach. Turbulence was assumed to exist within the fluid phase and a low Reynolds k-ε closure was used to model it. The flow equations at the pore-scale were numerically solved using the SIMPLE method on a non-orthogonal boundary-fitted coordinate system. Cell-integrated results for the longitudinal dispersion coefficient showed little sensitiveness on porosity, boundary condition type, medium morphology and solid-fluid conductivity ratio, whereas for the transversal direction, all of these parameters modified the numerical value obtained for the dispersion coefficient. © 2008 Elsevier Ltd. All rights reserved.

Comparisons of computations for turbulent natural convection within clockwise and counter-clockwise inclined cavities, filled with a fluid saturated porous medium, are presented. The finite volume method in a generalized coordinate system is applied. Oblique walls are maintained at constant but different temperatures, whereas horizontal surfaces are kept insulated. Flow and heat transfer characteristics are investigated for Rayleigh number up to 104 and inclination angles up to 45°, in both directions of rotation. Turbulent is handled using a macroscopic two-equation model with a wall function. In this work, the turbulence model is first switched off and the laminar branch of the solution is obtained. Subsequently, the turbulence model is included and the solution merges to the laminar branch for a reducing value of Ram. Present computations are compared with published results and the influence of the inclination angle on Racr is analyzed, for both the left and right rotating directions. For Ram greater than around 104, both laminar and turbulent flow solutions deviate, possibly indicating that a critical value for Ram was reached. Both left and right rotation of the hot wall reduce Nu, but rotating the hot wall on the counter-clockwise direction decreases Nu at a faster rate than when bending the cavity to the right. © 2008 Elsevier Ltd. All rights reserved.

The ability to realistically model flows through heterogeneous domains, which contain both solid and fluid phases, can benefit the analysis and simulation of complex real-world systems. Environmental impact studies, as well as engineering equipment design, can both take advantage of reliable modelling of turbulent flow in permeable media. Turbulence models proposed for such flows depend on the order of application of volume-and time-average operators. Two methodologies, following the two orders of integration, lead to distinct governing equations for the statistical quantities. This paper reviews recently published methodologies to mathematically characterize turbulent transport in permeable media. A new concept, called double-decomposition, is here discussed and instantaneous local transport equations are reviewed for clear flow before the time and volume averaging procedures are applied to them. Equations for turbulent transport follow, including their detailed derivation and a proposed model for suitable numerical simulations. The case of a moving porous bed is also discussed and transport equations for the mean and turbulent flow fields are presented. © Versita Warsaw and Springer-Verlag Berlin Heidelberg 2008.

Providing the reader with a solid understanding of the fundamentals as well as an awareness of recent advances in properties and applications of cellular and porous materials, this handbook and ready reference covers all important analytical and numerical methods for characterizing and predicting thermal properties. In so doing it directly addresses the special characteristics of foam-like and hole-riddled materials, combining theoretical and experimental aspects for characterization purposes. © 2008 Wiley-VCH Verlag GmbH & Co. KGaA. All rights reserved..

This work shows numerical results for a jet impinging against a flat plane covered with a layer of a porous material, which is maintained at a higher temperature than the incoming fluid. Parameters such as permeability and thickness of the porous layer and thermal conductivity ration are varied in order to analyze their effects on the local distribution of Nu. The macroscopic equations for mass, momentum, and energy are obtained based on a volume-average concept. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted nonorthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Results indicate that inclusion of a porous layer decreases the peak in Nu avoiding excessive heating or cooling at the stagnation point. Also found was that the integral heat flux from the wall is enhanced for certain range of values of layer thickness, porosity, and thermal conductivity ratio.

This work aims at studying laminar impinging jets on surfaces covered with a layer of a porous material. This contribution may provide insight into the design and optimization of heat and mass transfer processes over surfaces. Numerical simulations are presented and the porous substrate is treated as a rigid, homogeneous, and isotropic medium. Macroscopic transport equations are written for a representative elementary volume (REV), yielding a set of equations that is valid for the entire computational domain, including both the porous layer attached to the surface and the fluid layer over the porous substrate. These equations are discretized using the control-volume method and the resulting system of algebraic equations is relaxed using the Strongly Implicit Procedure (SIP) methods. The SIMPLE algorithm is used to handle the pressure-velocity coupling. Results for flow, in both clear and porous flow domains, are given in terms of streamlines patterns, velocity profiles, pressure contours, and friction coefficient along the impinging wall. The influence of porosity on the flow pattern is shown to be very low in comparison to the effects caused by varying permeability, fluid-layer height, and porous-layer thickness. These finding could be used to advantage when designing engineering equipment, since the use of selected porous materials could reduce undesirable recirculation zones, promote quick flow redistribution, and adjust pressure to required levels.

Combustion in inert porous media has been extensively investigated due to the many engineering applications and demand for developing high efficiency power production devices. The growing use of efficient radiant burners can be encountered in the power and process industries and, as such, proper mathematical models of flow, heat and mass transfer in porous media under combustion can benefit the development of such engineering equipment. This paper proposes a new mathematical model for computing temperature and flow variables inside a porous burner. A new concept called "double-decomposition" is used to represent all transported variables. A set of governing equations is presented and the numerical solution method proposed is discussed. Computations are carried out for a test case considering a simple one-energy equation model and one-step reaction rates. Simulations are presented comparing the inclusion of turbulence and radiation transfer in the model. It is shown that for high Re flows, inclusion of turbulence is as important as modeling radiation for obtaining reliable temperature distribution within the porous material. Copyright © 2007 by ASME.

Copyright © 2007 by ASME.This paper presents computations for turbulent natural convection within an inclined cavity filled with a fluid saturated permeable medium. The finite volume method in a generalized coordinate system is applied. The inclined walls are maintained at constant but different temperatures, while the horizontal walls are kept insulated. Governing equations are written in terms of primitive variables and are recast into a general form. Flow and heat transfer characteristics are investigated for a wide range of values of Rayleigh number and inclined angles. The turbulent model used is the macroscopic k-eps model with a wall function. In this work, the turbulence model is first switched off and the laminar branch of the solution is found. Subsequently, the turbulence model is included so that the solution merges to the laminar branch for a reducing Ram. This convergence of results as Ram decreases can be seen as an estimate of the well known laminarization phenomenon. Present solutions are compared with published results and the influence of the inclination angle on Racr is analyzed. For Ram greater than around 104, both laminar and turbulent flow solutions deviate, indicating that such critical value for Ram was reached.

Impinging jets are widely used in industry to modify local heat transfer coefficients. The addition of a porous substrate covering the surface contributes to better flow distribution, which favors many engineering applications. Motivated by that, this work shows numerical results for a turbulent jet impinging against a cylindrical enclosure with a porous substrate at the bottom. Macroscopic time-averaged equations for mass and momentum are obtained based on a concept called double decomposition, which considers spatial deviations and temporal fluctuations of flow properties. The numerical technique employed for discretizing the governing equations is the control volume method in conjunction with a boundary-fitted coordinate system. The SIMPLE algorithm is used to handle the pressure-velocity coupling. The influence of the cylinder height on the mean and statistical flow fields within the entire cavity is presented. Copyright © 2006 by ASME.

Turbulent natural convection in a vertical two-dimensional square cavity, isothermally heated from below and cooled at the upper surface, is numerically analyzed using the finite volume method. The enclosure has a thin horizontal porous obstruction, made of a highly porous material and extremely permeable, located at the cavity midheight. Governing equations are written in terms of primitive variables and are recast into a general form. For empty cavities, no discrepancies result for the Nusselt number when laminar and turbulent model solutions are compared for Rayleigh numbers up to 107. Also, in general the porous obstruction decreases the heat transfer across the heated walls showing overall lower Nusselt numbers when compared with those without the porous obstruction. However the presence of a porous plate in the cavity seems to force an earlier separation from laminar to turbulence model solutions due to higher generation rates of turbulent kinetic energy into the porous matrix. Copyright © 2006 by ASME.

This work presents numerical computations for laminar and turbulent natural convection within a horizontal cylindrical annulus filled with a fluid saturated porous medium. Computations covered the range 25 < Ram < 500 and 3.2 × 10-4 > Da > 3.2 × 10-6 and made use of the finite volume method. The inner and outer walls are maintained at constant but different temperatures. The macroscopic k-ε turbulence model with wall function is used to handle turbulent flows in porous media. First, the turbulence model is switched off and the laminar branch of the solution is found when increasing the Rayleigh number, Ram. Subsequently, the turbulence model is included and calculations start at high Ram, merging to the laminar branch for a reducing Ram. This convergence of results as Ram decreases can be seen as an estimate of the so-called laminarization phenomenon. Here, a critical Rayleigh number was not identified and results indicated that when the porosity, Prandtl number, conductivity ratio between the fluid and the solid matrix and Ram are kept fixed, the lower the Darcy number, the higher is the difference of the average Nusselt number given by the laminar and turbulent models. © 2006 Elsevier Ltd. All rights reserved.

© 2006 Elsevier Ltd. All rights reserved.'Turbulence in Porous Media' introduces the reader to the characterisation of turbulent flow, heat and mass transfer in permeable media, including analytical data and a review of available experimental data. Such transport processes occurring a relatively high velocity in permeable media, are present in a number of engineering and natural flows. De Lemos has managed to compile, detail, compare and evaluate available methodologies for modelling simulating purposes, providing an essential tour for engineering students working within the field. The hotly debated topic of heterogeneity and flow turbulence has never before been addressed in book format. Offers an experimental approach to turbulence in porous media as it discusses disciplines that have been traditionally developed apart from each other.

Interfacial heat transfer coefficients in a porous medium modeled as a staggered array of square rods are numerically determined. High and low Reynolds k-ε turbulence models are used in conjunction of a two-energy equation model, which includes distinct transport equations for the fluid and the solid phases. The literature has documented proposals for macroscopic energy equation modeling for porous media considering the local thermal equilibrium hypothesis and laminar flow. In addition, two-energy equation models have been proposed for conduction and laminar convection in packed beds. With the aim of contributing to new developments, this work treats turbulent heat transport modeling in porous media under the local thermal nonequilibrium assumption. Macroscopic time-average equations for continuity, momentum, and energy are presented based on the recently established double decomposition concept (spatial deviations and temporal fluctuations of flow properties). The numerical technique employed for discretizing the governing equations is the control volume method Turbulent flow results for the macroscopic heat transef coefficient, between the fluid and solid phase in a, periodic cell, are presented. Copyright © 2006 by ASME.

Simulations are presented for laminar flow in a channel containing baffles made with solid (impermeable) and porous materials. The equations of mass continuity, momentum and energy are written for an elementary representative volume, yielding a set of equations valid for the entire computational domain. These equations are discretized using the control-volume method and the resulting system of algebraic equations is relaxed with the SIMPLE method. The numerical results for the friction factor f and for the Nusselt number Nu are compared with available data, indicating that results herein differ by less than 5% in relation to published results. Further simulations comparing the effectiveness of the porous material used show that no advantages are obtained when using low-porosity baffles in the laminar flow regime investigated here. © 2006, Taylor & Francis Group, LLC. All rights reserved.

Flow over a finite porous medium is investigated using different interfacial conditions. In such configuration, a macroscopic interface is identified between the two media. In the first model, no diffusion-flux is considered when treating the statistical energy balance at the interface. The second approach assumes that diffusion fluxes of turbulent kinetic energy on both sides of the interface are unequal. Comparing these two models, this paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a porous layer through which fluid flows in turbulent regime. One unique set of transport equations is applied to both regions. Effects of Reynolds number, porosity, permeability and jump coefficient on mean and turbulence fields are investigated. Results indicate that depending on the value of the stress jump parameter, substantially dissimilar fields for the turbulence energy are obtained. Negative values for the stress jump parameter give results closer to experimental data for the turbulent kinetic energy at the interface. © 2005 Elsevier Ltd. All rights reserved.

'Turbulence in Porous Media' introduces the reader to the characterisation of turbulent flow, heat and mass transfer in permeable media, including analytical data and a review of available experimental data. Such transport processes occurring a relatively high velocity in permeable media, are present in a number of engineering and natural flows. De Lemos has managed to compile, detail, compare and evaluate available methodologies for modelling simulating purposes, providing an essential tour for engineering students working within the field. - The hotly debated topic of heterogeneity and flow turbulence has never before been addressed in book format. - Offers an experimental approach to turbulence in porous media as it discusses disciplines that have been traditionally developed apart from each other. The hotly debated topic of heterogeneity and flow turbulence has never before been addressed in book format. Offers an experimental approach to turbulence in porous media as it discusses disciplines that have been traditionally developed apart from each other. © 2006 Elsevier Ltd All rights reserved.

Impinging jets are widely used in industry to modify local heat transfer coefficients. The addition of a porous substrate covering the surface contributes to better flow distribution, which favors many engineering applications. Motivated by that, this work shows numerical results for a turbulent jet impinging against a cylindrical enclosure with a porous substrate at the bottom. Macroscopic time-averaged equations for mass and momentum are obtained based on a concept called double decomposition, which considers spatial deviations and temporal fluctuations of flow properties. The numerical technique employed for discretizing the governing equations is the control volume method in conjunction with a boundary-fitted coordinate system. The SIMPLE algorithm is used to handle the pressure-velocity coupling. The influence of the cylinder height on the mean and statistical flow fields within the entire cavity is presented. Copyright © 2006 by ASME.

Environmental impact analyses as well as engineering equipment design can both benefit from the reliable modeling of turbulent flow in porous media. A number of natural and engineering systems can be characterized by a permeable structure through which a working fluid permeates. Turbulence models proposed for such flows depend on the order of application of time- and volume-average operators. Two methodologies, following the two orders of integration, lead to different governing equations for statistical quantities. The chapter reviews recently published methodologies to mathematically characterize turbulent transport in porous media. It also introduces a new concept called double-decomposition and classifies models for turbulent transport in porous media in terms of the order of application of the time- and volume-averaging operators, among other peculiarities. The chapter also reviews instantaneous local transport equations for clear flow before time- and volume-averaging procedures are applied to them. The double-decomposition concept is presented and thoroughly discussed prior to the derivation of macroscopic governing equations. Equations for turbulent transport follow, showing a detailed derivation for mean and turbulent field quantities. The statistical k-e model for clear domains, used to model macroscopic turbulence effects, also serves as the basis for heat transfer modeling. Mass transfer in porous matrices is further reviewed in the light of the double-decomposition concept. © 2005 Elsevier Ltd All rights reserved.

Turbulent impinging jets on heated surfaces are widely used in industry to modify local heat transfer coefficients. The addition of a porous substrate covering the surface contributes to a better flow distribution, which favors many engineering applications. Motivated by this, the present work shows numerical results for a turbulent impinging jet against a cylindrical enclosure with and without a porous layer at the bottom. The macroscopic time-averaged equations for mass, momentum and energy are obtained based on a concept called double decomposition, which considers spatial deviations and temporal fluctuations of flow properties. The numerical technique employed for discretizing the governing equations is the control volume method in conjunction with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm is used to handle the pressure-velocity coupling. The influence of characteristics of the porous layer on the mean and statistical flow fields within the cylinder is presented. Copyright © 2005 by ASME.

Simulations are presented for turbulent flow in a channel containing baffles made with solid and porous materials. The equations of mass continuity, momentum and energy are written for an elementary representative volume yielding a set of equations valid for the entire computational domain. These equations are discretized using the control volume method and the resulting system of algebraic equations is relaxed with the SIMPLE method. The presented numerical results for the friction factor f and for the Nusselt number Nu were compared with available data. Further simulations comparing the effectiveness of the porous material used showed that no advantages are obtained when using low porosity baffles in the turbulent flow regime. Copyright © 2005 by ASME.

This work compares two different approaches for obtaining numerical solutions for laminar natural convection within a square cavity, which is filled by a fixed amount of a solid conducting material. The first model considered, namely, porous-continuum model, is based on the assumption that the solid and the fluid phases are seen as the same medium, over which volume-averaged transport equations apply. Secondly, a continuum model is considered to solve the momentum equations for the fluid phase that would resemble a conjugate heat transfer problem in both the solid and the void space. In the continuum model, the solid phase is composed of square obstacles, equally spaced within the cavity. In both models, governing equations are numerically solved using the finite volume method. The average Nusselt number at the hot wall, obtained from the porous-continuum model, for several Darcy numbers, are compared with those obtained with the second approach, namely the continuum model, with different number of obstacles. When comparing the two methodologies, this study shows that the average Nusselt number calculated for each approach for the same Ra m differs between each other and that this discrepancy increases as the Darcy number decreases, in the porous-continuum model, or the number of blocks increases and their size decreases, in the continuum model. A correlation is suggested to modify the macroscopic thermal expansion coefficient in order to match the average Nusselt numbers calculated by the two models for Ra m=const=104 and Da ranging from 1.2060×10 -4 to 1. Copyright © 2005 by ASME.

This work compares heat transfer characteristics across a square cavity partially filled with a fixed amount of conducting solid material. The solid phase is shaped into two different geometries, namely square and cylindrical rods, which are horizontally displaced inside the cavity. Comparisons are obtained by numerically solving a conjugate heat transfer problem that considers both the solid and the fluid space. Governing equations are solved using the finite volume method and the algebraic equation set is relaxed with the SIP procedure. The average Nusselt number at the hot wall, obtained from the cavity with square obstacles and for several Darcy numbers, are compared with those calculated with circular obstacles. When comparing the two geometries considering the same modified Rayleigh number Ram, this study shows that the average Nusselt number for cylindrical rods are slightly lower than those for square rods. © 2005 Elsevier Ltd. All rights reserved.

This work compares two different approaches for obtaining numerical solutions for laminar and turbulent natural convection within a cavity filled by a fixed amount of a solid conducting material. In the first model, a porous-continuum, homogeneous or macroscopic approach is considered based on the assumption that the solid and the fluid phases are observed as a single medium, over which volume-averaged transport equations apply. Secondly, a continuum, heterogeneous or microscopic model is considered to solve the momentum equations for the fluid phase resulting in a conjugate heat transfer problem in both the solid and the void space. In the continuum model, the solid phase is composed of square obstacles, equally spaced within the cavity. In both models, governing equations are numerically solved using the finite volume method. The average Nusselt number at the hot wall, obtained from the porous-continuum, homogeneous or macroscopic model, for several Darcy numbers, are compared with those obtained with the second approach, namely the continuum model, with different number of obstacles. When comparing the two methodologies, this study shows that the average Nusselt number calculated for each approach for the same Ra m differs from each other and that this discrepancy increases as the Darcy number decreases, in the porous-continuum model, or the number of blocks increases, in the continuum model. Inclusion of turbulent transfer raises Nusselt for both the continuum and the porous-continuum models. A correlation is suggested to modify the macroscopic Rayleigh number in order to match the average Nusselt numbers calculated by the two models for Ram = const = 104 and Da ranging from 1.2060 × 10-4 to 1. © 2005 Elsevier Ltd. All rights reserved.

Environmental impact analyses as well as engineering equipment design can both benefit from reliable modeling of turbulent flow in porous media. A number of natural and engineering systems can be characterized by a permeable structure through which a working fluid permeates. Turbulence models proposed for such flows depend on the order of application of time and volume average operators. Two methodologies, following the two orders of integration, lead to different governing equations for the statistical quantities. This paper reviews recently published methodologies to mathematically characterize turbulent transport in porous media. A new concept, called double-decomposition, is here discussed and models for turbulent transport in porous media are classified in terms of the order of application of the time and volume averaging operators, among other peculiarities. Within this paper Instantaneous Local Transport Equations are reviewed for clear flow before Time and Volume Averaging Procedures are applied to them. The Double-Decomposition Concept is presented and thoroughly discussed prior the derivation of macroscopic governing equations. Equations for Turbulent Transport follow, showing detailed derivation for the mean and turbulent field quantities. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The literature has documented proposals for macroscopic energy equation modeling for porous media considering the local thermal equilibrium hypothesis and laminar flow. In addition, two-energy equation models have been proposed for conduction and laminar convection in packed beds. With the aim of contributing to new developments, this work treats turbulent heat transport modeling in porous media under the local thermal non-equilibrium assumption. Macroscopic time-average equations for continuity, momentum and energy are presented based on the recently established double decomposition concept (spatial deviations and temporal fluctuations of flow properties). Interfacial heat transfer coefficients are numerically determined for an infinite medium over which the fully developed flow condition prevails. The numerical technique employed for discretizing the governing equations is the control volume method. Preliminary laminar flow results for the macroscopic heat transfer coefficient, between the fluid and solid phase in a periodic cell, are presented. © 2005 Elsevier Ltd. All rights reserved.

This work presents a numerical investigation of turbulent flow past a backward-facing-step channel with a porous insert using linear and nonlinear eddy viscosity macroscopic models. The nonlinear turbulence models are known to perform better than classical eddy-diffusivity models due to their ability to simulate important characteristics of the flow. Turbulence-driven secondary motion and the effects of streamline curvature on turbulence cannot be fully accounted for with simpler Isotropic models. Parameters such as porosity, permeability, and thickness of the porous insert are varied in order to analyze their effects on the flow pattern, particularly on the damping of the recirculating bubble after the porous insertion. The numerical technique employed for discretizing the governing equations is the control-volume method. The SIMPLE algorithm is used to correct the pressure field. The classical wall function is utilized in order to handle flow calculation near the wall. Comparisons of results simulated with both linear and nonlinear turbulence models are shown. Copyright © 2005 Begell House, Inc.

The purpose of this work is to investigate the influence of a porous insert in an incompressible turbulent flow in a pipe that suffers a sudden contraction. The Reynolds number considered is 158,000 based on the pipe outlet diameter. The flow equations are discretized by using the control volume method and the SIMPLE algorithm is applied for the velocity-pressure coupling. In all cases, the macroscopic k - ε Low-Reynolds turbulence model is employed. For an initial numerical validation a simulation is carried out without the porous insert in order to be compared with an experimental result. Subsequently, a porous insert is considered in the numerical simulations. The flow losses obtained with the porous insert are calculated and compared with those obtained from the calculations without the porous insert.

Interfacial convective heat transfer coefficient is calculated for turbulent flow in a porous medium formed by square rods. Such information is needed for turbulent heat transport modeling in porous media when local thermal non-equilibrium is considered. The model considers fluid flowing through a packed bed with arbitrary bed temperature. This adjustment is obtained by solving the microscopic flow governing equations, using high Reynolds formulation and periodic boundary conditions. The numerical methodology employed is based on the control-volume approach with a boundary-fitted non-orthogonal coordinate system. This work intends to obtain functional relationships for the interfacial convective heat transfer coefficient for turbulent low in packed beds.

© 2005 by Taylor & Francis Group, LLC.Engineering equipment design and environmental impact analyses can benefit from appropriate modeling of turbulent flow in porous media. Accordingly, a number of natural and engineering systems can be characterized by some sort of porous structure through which a working fluid permeates. Turbulence models proposed for such flows depend on the order of application of time and volume-average operators. Two developed methodologies, following the two orders of integration, lead to different governing equations for the statistical quantities. This chapter reviews recently published methodologies to mathematically characterize turbulent transport in porous media. For hybrid media, involving both a porous structure and a clear flow region, difficulties arise due to the proper mathematical treatment given at the interface. This chapter also presents and discusses numerical solutions for such hybrid media, here considering a channel partially filled with a wavy porous layer through which fluid flows in turbulent regime. In addition, macroscopic forms of buoyancy terms are also considered in both the mean and the turbulent fields. Cases reviewed include heat transfer in cavities partially filled with porous material. In summary, within this chapter local instantaneous governing equations are reviewed for clear flow before volume and time-average operators are applied to them. The double-decomposition concept is presented and thoroughly discussed prior to the derivation of macroscopic governing equations. Equations for turbulent momentum transport in porous media follow showing detailed derivation for the mean and turbulent field quantities. The statistical k–e model for clear domains, used to model macroscopic turbulence effects, also serves as the basis for turbulent heat transport modeling. Turbulent mass transport in porous matrices is further reviewed in the light of the double-decomposition concept. A section on applications in hybrid media covers flow over porous layers in channels and in cavities partially filled with porous material.

For hybrid media, involving both a porous substrate and an unobstructed flow region, difficulties arise due to the proper mathematical treatment given at the macroscopic interface. The literature proposes a jump condition in which shear stresses on both sides of the interface are not of the same value. This paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a porous layer through which an incompressible fluid flows in turbulent regime. Here, diffusion fluxes of both momentum and turbulent kinetic energy across the interface present a discontinuity in their values, which is based on a certain jump coefficient. Effects of such parameter on mean and turbulence fields around the interface region are numerically investigated. Results indicate that depending on the value of the stress jump parameter, a substantially different structure for the turbulent field is obtained. © 2004 Elsevier Ltd. All rights reserved.

Detailed numerical computations for laminar and turbulent natural convection within a square cavity filled with a fluid saturated porous medium are presented. Heated vertical walls are maintained at constant but different temperatures, while horizontal surfaces are kept insulated. The macroscopic κ-ε turbulence model with wall function is used to handle turbulent flows in porous media. In this work, the turbulence model is first switched off and the laminar branch of the solution is found when increasing the Rayleigh number, Ram. Computations covered the range 10 < Ram < 106 and 10-7 < Da < 10-10 and made use of the finite volume method. Subsequently, the turbulence model is included and calculations start at high Ram, merging to the laminar branch for a reducing Ram and for Ram less than a certain critical Rayleigh number, Racr. This convergence of results as Ram decreases can be seen as a characterization of the laminarization phenomenon. For Ram values less than around 104, both laminar and turbulent flow solutions merge, indicating that such critical value for Ra m was reached, Results further indicate that when the parameters porosity, Pr, conductivity ratio between the fluid and the solid matrix and the Ram are kept fixed, the lower the Darcy number, the higher the average Nusselt number at the hot wall. © 2004 Elsevier Ltd. All rights reserved.

This paper presents an analysis of the macroscopic heat and mass transport equations for turbulent flow in permeable structures. Two driving mechanisms are considered to contribute to the overall momentum transport, namely temperature driven and concentration driven mass fluxes. Double-diffusive natural convection mechanism is investigated for the fluid phase in turbulent regime. Equations are presented based on the double-decomposition concept, which considers both time fluctuations and spatial deviations about mean values. This work intends to demonstrate that additional transport mechanisms are mathematically derived if temperature, concentration and velocity present simultaneously time fluctuations and spatial deviations within the domain of analysis. A modeled form for the final mass transport equation is presented where turbulent transfer is based on a macroscopic version of the k-ε model. Stability analysis of mixtures, composed of lighter or heavier components under gradients of temperature and concentration, is discussed. © 2004 Elsevier Ltd. All rights reserved.

This work presents numerical prediction for the turbulent flow field confined in a circular duct past a segment of gradually varying cross section. Both expanding and contracting sections are investigated. Equations of boundary-layer type are used and the linear k-ε model, in its high Reynolds form, is applied. A new correlation for treating the grid point closest to the wall is proposed. A marching-forward method is employed for sweeping the computational domain. Computations are first performed for developing and fully developed constant-area ducts in order to assess the reliability of the code. Results are then presented for contractions and diffusers, where comparisons with experimental data for air and water are carried out. Turbulence damping in contractions and its enhancement in diffusers are calculated correctly. Further, for contractions with angles of up to 21°, the use of a parabolic solver shows good agreement with experimental values for the mean and statistical quantities. For diffusers, adverse pressure gradient along the flow limits the quality of the predictions as the angle and length of diffuser increase past 5° and 10 duct radii, respectively.

The present work investigates the efficiency of the multigrid numerical method when used to solve two-dimensional laminar velocity and temperature fields inside a rectangular domain. Numerical analysis is based on the finite volume discretization scheme applied to structured orthogonal regular meshes. Performance of the correction storage (CS) multigrid algorithm is compared for different inlet Reynolds number (Rein) and number of grids. Up to four grids were used for both V- and W-cycles. Simultaneous and uncoupled temperature-velocity solution schemes were investigated. Advantages in using more than one grid are discussed. For simultaneous solution, results further indicate an increase in the computational effort for higher inlet Reynolds number Rein. Optimal number of intermediate relaxation sweeps for within both V- and W-cycles is discussed upon. © 2003 Elsevier Inc. All rights reserved.

In this work, mass dispersion tensors were calculated within an infinite porous medium formed by a spatially periodic array of longitudinally-displaced cylindrical rods. For the sake of simplicity, just one unit-cell, together with periodic boundary conditions for mass and momentum equations, and Neumann conditions for the mass concentration, was used to represent such medium. The numerical methodology herein employed is based on the control volume approach. Turbulence is assumed to exist within the fluid phase. High and low Reynolds k-e models were used to model such non-linear effects. The flow equations at the pore-scale were numerically solved using the SIMPLE method applied to a non-orthogonal boundary-fitted coordinate system. Integrated mass fraction results were compared with existing data in the literature. Copyright © 2004 by ASME.

The study of important environmental and engineering flows can benefit from more realistic modeling. Accordingly, grain storage and drying as well as flows over layers of vegetation can be characterized by some sort of porous structure through which a fluid permeates. For such hybrid media, involving both a porous structure and a clear flow region, difficulties arise due to the proper mathematical treatment given at the macroscopic interface. The literature proposes a jump condition in which shear stresses on both sides of the interface are not of the same value. This paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a porous layer through which fluid flows in turbulent regime. Here, diffusion fluxes of both momentum and turbulent kinetic energy across the interface present a discontinuity in their values, which is based on a certain jump coefficient. Effects of such jump parameter on mean and turbulence fields around the interface regions are numerically investigated. Results indicate that depending on the value of the stress jump parameter, a substantially different structure for the turbulent field is obtained. Copyright © 2004 by ASME.

Detailed numerical computations for steady-state laminar natural convection within in oblique cavities totally filled with a fluid saturated porous medium is numerically analyzed using the finite volume method in a generalized coordinate system. The inclined walls are maintained at constant but different temperatures, while the horizontal walls are kept insulated. Governing equations are written in terms of primitive variables and are recast into a general form. Flow and heat transfer characteristics, (streamlines, isotherms and average Nusselt number), are investigated for Rayleigh number ranging from 10 3 to 104 and inclined angles ranging from 0° to 45°. In general, present results show good agreement with previous works. Analyses of important environmental and engineering flows can benefit from the derivations herein and, ultimately, it is expected that additional research on this new subject be stimulated by the work here presented. Copyright © 2004 by ASME.

This work presents derivations of macroscopic heat and mass transport equations for turbulent flow in permeable structures. Two driving mechanisms are considered to contribute to the overall momentum transport, namely temperature driven and concentration driven mass fluxes. Double-diffusive natural convection mechanism is investigated for the fluid phase in turbulent regime. Equations are presented based on two distinct procedures. The first method considers time averaging of the local instantaneous mass transport equation before the volume average operator is applied. The second methodology employs both averaging operators but in a reverse order. This work is intended to demonstrate that additional transport mechanisms are mathematically derived if temperature, concentration and velocity present simultaneously time fluctuations and spatial deviations within the domain of analysis. A modeled form for the final mass transport equation is presented where turbulent transfer is based on a macroscopic version of the k-ε model. Copyright © 2004 by ASME.

This work presents a numerical investigation for the turbulent flow and heat transfer in an abrupt contraction channel with a porous material placed in a flow passage. The channel has a contraction rate of 3:2. Results for the hybrid medium were obtained using linear and non-linear k-ε macroscopic models. It was used an inlet Reynolds number of Re = 132000 based on the height of the step. Parameters such as porosity, permeability and thickness of the porous insert were varied in order to analyze their effects on the flow pattern. The results of local heat transfer, friction coefficient and stream lines obtained by the two turbulence models were compared for the cases without and with porous insertion of thickness a/H=0.083, 0.166 and 0.250, where H is the step height. Insert porosity of varied between 0.85 and 0.95 with permeability in the range 10-6-10-2 m2. Copyright © 2004 by ASME.

In this work, results for a macroscopic mass transport model are presented for a parallel plate channel filled with a fluid saturated heterogeneous porous medium. The numerical methodology herein employed is based on the control volume approach. Turbulence is assumed to exist within the fluid phase. High and low Reynolds k-e models were used to model such non-linear effects. The flow equations at the pore-scale were numerically solved using the SIMPLE method applied to a non-orthogonal boundary-fitted coordinate system. Integrated mass fraction results were compiled leading to correlations for the mass dispersion coefficients in the x and y directions. Application of the macroscopic model using the proposed correlations showed the role of dispersion mechanism in the overall transport in porous media. Copyright 2004 by ASME.

Computations for turbulent natural convection within an inclined cavity totally filled with a fluid saturated porous medium are presented. The finite volume method in a generalized coordinate system is applied. The inclined walls are maintained at constant but different temperatures, while the horizontal walls are kept insulated. Governing equations are written in terms of primitive variables and are recast into a general form,. Flow and heat transfer characteristics, (streamlines, isotherms and average Nusselt number), are investigated for a wide range of values of Rayleigh number and inclined angle. The turbulent model used is the standard k-ε model with a wall function. In this work, the turbulence model is first switched off and the laminar branch of the solution is found. Subsequently, the turbulence model is included so that the solution merges to the laminar branch for a reducing Ram. This convergence of results as Ram decreases can be seen as an estimate of the so-called relaminarization phenomenon. Present solutions are compared with published results and the influence of the inclination angle on Ram is analyzed. For Ram greater than around 104, both laminar and turbulent flow solutions deviate, indicating that such critical value for Ram was reached. Copyright © 2004 by ASME.

Simulations are presented for laminar flow in a channel containing fins made with solid (impermeable) and porous materials. The equations of mass continuity, momentum and energy are written for an elementary representative volume yielding a set of equations valid for the entire computational domain. These equations are discretized using the control volume method and the resulting system of algebraic equations is relaxed with the SIMPLE method. The presented numerical results for the friction factor/and for the Nusselt number Nu were compared with available data indicating that results herein differ by less than 5% in relation to published results. Further simulations comparing the effectiveness of the porous material used showed that no advantages are obtained for using low porosity baffles in the laminar flow regime. Copyright © 2004 by ASME.

This paper deals with the problem of heat transfer in square cavities partially filled with porous material. Local flow and energy equations are integrated in a representative elementary volume in order to obtain a set of equations valid in both the clear flow region and in the porous matrix. A unique set of equations is discretized with the control volume method and solved with the SIMPLE algorithm. Enhancement of convective currents within the porous substrate is detected as the Rayleigh number increases. Thin boundary layers along the cavity vertical walls and stratification of the thermal field are observed for Ra>109.

This work reports numerical results for the case of incompressible laminar heated flow with a swirl in a vertical cylindrical chamber. Computations are obtained with a point-wise block-implicit scheme. Flow governing equations are written in terms of the so-called primitive variables and are recast into a general form. The discretized momentum equations are applied to each cell face and then, together with the mass-continuity, tangential velocity and energy equations, are solved directly in each computational node. The effects of Rayleigh, Reynolds and Swirl numbers on the temperature field are discussed. Flow pattern and scalar residual history are reported. Further, it is expected that more advanced parallel computer architectures can benefit from the error smoothing operator here described. © 2003 John Wiley and Sons, Ltd.

This paper applies the volume-average mathematical operator on the buoyancy term in the flow equations governing turbulent flow. Volume averaging is taken on both mean and turbulent fields. Derivations are carried out under the recently established double-decomposition concept. Results show that additional buoyancy generation term appears if both time and volume averaging procedures are simultaneously applied. Final modeled equations are based on a macroscopic k-ε model for porous media. © 2003 Elsevier Science Ltd.

A number of natural and engineering systems can be characterized by some sort of porous structure through which a working fluid permeates. Boundary layers over tropical forests and spreading of chemical contaminants through underground water reservoirs are examples of important environmental flows that can benefit form appropriate mathematical treatment. For hybrid media, involving both a porous structure and a clear flow region, difficulties arise due to the proper mathematical treatment given at the interface. The literature proposes a jump condition in which stresses at both sides of the interface are not of the same value. The objective of this article is to present a numerical implementation for solving such a hybrid medium, considering here a channel partially filled with a porous layer through which fluid flows in laminar regime. One unique set of transport equations is applied to both regions. Numerical results are compared with available analytical solutions in the literature for two cases, namely, with and without the nonlinear Forchheimer term. Results are presented for the mean velocity across both the porous structure and the clear region. The influence of medium properties, such as porosity and permeability, is discussed.

Through the volumetric averaging of the microscopic transport equations for the turbulent kinetic energy, k, and its dissipation rate, ε, a macroscopic model is proposed for flow in porous media. As an outcome of the volume-averaging process, additional terms appeared in the equations for k and ε. These terms are adjusted assuming the porous structure to be modeled as an infinity array of transversally displaced elliptic rods. This adjustment is obtained by solving the microscopic flow governing equations numerically, using a law-Reynolds formulation, in the periodic cell composing the infinite medium. Different porosity and aspect ratios are investigated. The adjusted model is compared with similar results found in the literature. A general view of the effect of the medium morphology on model assumptions is obtained by comparing results for elliptic, cylindrical, and square rods.

Numerical results for swirling flowe obtained by a point-wise locally-implicit scheme are here reported. Computations are presented for incompressible laminar flow inside a model combustor. Governing equations are written in terms of the so-called primitive variables and are recast into a general form. Finite-differencing is obtained by means of the widely-used control-volume approach. Discretized cross-flow equations are applied to each cell face and then, together with the mass-continuity and the tangential momentum equations, are simultaneously solved by means of a direct method in each computational node. Residue histories for governing equations are presented. Advantages in using a coupled procedure when compared with standard segregated schemes are discussed. © 2003 Elsevier Science Ltd.

This work presents a numerical investigation of turbulent flow past a porous structure in a channel using linear and nonlinear eddy viscosity macroscopic models. Parameters such as porosity and permeability of the porous material are varied in order to analyze their effects on the flow pattern, particularly on the damping of the recirculating bubble after the entrance and exit regions. The numerical technique employed for discretizing the governing equations is the control-volume method. The SIMPLE algorithm is used to correct the pressure field. The classical wall function is utilized in order to handle flow calculation near the wall. A discussion on the use of this technique for simulating the flow in question is presented. Comparisons of results simulated with both linear and nonlinear turbulence models are shown.

The present work investigates the efficiency of the multigrid method when applied to solve laminar flow in a two-dimensional tank filled with a porous material. The numerical method includes finite volume discretization with the flux blended deferred correction scheme on structure orthogonal regular meshes. Performance of the correction storage (CS) multigrid algorithm is compared for different numbers of sweeps in each grid level. Up to four grids, for both multigrid V- and W- cycles, are considered. Effects of medium permeability on converged rates are presented. Results indicate that W-cycles perform better in reducing the required computational effort and that the lower the permeability, faster solutions are obtained.

This paper presents derivations of mass transport equations for turbulent flow in permeable structures. Equations are developed following two distinct procedures. The first method considers time averaging of the local instantaneous mass transport equation before the volume average operator is applied. The second methodology employs both averaging operators but in a reverse order. This work is intended to demonstrate that both approaches lead to equivalent equations when one takes into account both time fluctuations and spatial deviations of velocity and mass fraction. A modeled form for the final transport equation is presented where turbulent transfer is based on a macroscopic version of the k-ε model.

In this work, numerical solutions are presented for turbulent flow in a channel containing fins made with porous material. The condition of spatially periodic cell is applied longitudinally along the channel. A macroscopic two-equation turbulence model is employed in both the porous region and the clear fluid. The equations of mass continuity, momentum and turbulence transport equations are written for an elementary representative volume yielding a set of equations valid for the entire computational domain. Results are presented for the velocity field as a function of Reynolds, porosity and permeability of the fins. Pressure drop along the channel is compared with the case of solid material.

In this work, numerical solutions are presented for turbulent flow in a channel containing fins made with porous material. The condition of spatially periodic cell is applied longitudinally along the channel. A macroscopic two-equation turbulence model is employed in both the porous region and the clear fluid. The equations of momentum, mass continuity and turbulence transport equations are written for an elementary representative volume yielding a set of equations valid for the entire computational domain. These equations are discretized using the control volume method and the resulting system of algebraic equations is relaxed with the SIMPLE method. Results are presented for the velocity field as a function of Reynolds number, porosity and permeability of the fins.

Turbulent natural convection in a two-dimensional horizontal composite square cavity, isothermally heated at the left side and cooled from the opposing surface, is numerically analyzed using the finite volume method. The composite square cavity is formed by three distinct regions, namely, clear, porous and solid region. Accordingly, the development of a numerical tool able to treat all these regions as one computational domain is of advantage for engineering design of thermal systems. Governing equations are written in terms of primitive variables and are recast into a general form. It was found that the fluid begins to permeate the porous medium for values of Ra greater than 10 6. Nusselt number values show that for the range of Ra analyzed there are no significant variation between the laminar and turbulent model solution.

A number of natural and engineering systems can be characterized by some sort of porous structure through which a working fluid permeates. Atmospheric boundary layers over tropical forests and vegetation can be modeled as flow over a porous layer of irregular surface. In addition, in engineering systems one can have components that make use of a working fluid flowing over irregular layers of porous material. This paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a sinusoidal porous layer saturated by a fluid flowing in laminar regime. One unique set of transport equations is applied to both regions. Effects of Reynolds number, porosity and permeability on mean and turbulence fields are investigated. For a fixed inlet mass flow rate, increase of either porosity or permeability reduced the strength of the recirculating motion over the porous layer.

Turbulent natural convection in a horizontal two-dimensional square cavity, isothermally heated from below and cooled at the upper surface, is numerically analyzed using the finite volume method and a generalized coordinate system. The enclosure has a thin horizontal porous obstruction located at the cavity mid height. Governing equations are written in terms of primitive variables and are recast into a general form. In general, the porous obstruction decreases the heat transfer across the heated walls showing an overall lower Nusselt numbers when compared with those without the same porous obstruction. However, the presence of a porous obstruction in a square cavity seems to force an earlier transition from laminar to turbulent regime due to higher generation rates of turbulent kinetic energy into the porous matrix.

Horizontally-layered porous media in enclosures represents an important configuration with many technological applications in mechanical and aerospace engineering. This work presents numerical solutions for flow and heat transfer in square cavities partially obstructed with porous material. The microscopic flow and energy equations are integrated in a representative elementary volume in order to obtain a set of equations valid in both the clear flow region and in the porous matrix. A unique set of equations is discretized with the control volume method and solved with the SIMPLE algorithm. Heat transfer enhancement across the porous cavity is calculated as the permeability or the porosity of the porous substrate increase.

Flow over forests and vegetation can be characterized by some sort of porous structure of irregular surface through whicha fluid permeates. Also, in engineering systems one can have components that make use of a working fluid flowing over irregular layers of porous material. This paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a sinusoidal porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both regions. Effects of Reynolds number, porosity and permeability on mean and turbulence fields are investigated. Results indicate that around the peaks of the sinusoidal layer values of the turbulent kinetic energy are higher.

For hybrid media, involving both a porous structure and a clear flow region, difficulties arise due to the proper mathematical treatment given at the interface. The literature proposes a jump condition in which shear stresses on both sides of the interface are not of the same value. This paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a porous layer through which fluid flows in turbulent regime. One unique set of transport equations is applied to both regions. Effects of Reynolds number, porosity, permeability and jump coefficient on mean and turbulence fields are investigated. Results indicate that depending on the value of the stress jump parameters, a substantially different structure for the turbulent field is obtained. © 2003 Elsevier Ltd. All rights reserved.

Multigrid methods are known to reduce computational time of iterative solutions. In this paper, a multigrid technique is implemented following a correction storage (CS) formulation and a V-cycle strategy to numerically solve steady-state two-dimensional incompressible laminar recirculating flows. Structured, orthogonal and irregular meshes are employed to perform a finite volume discretization. Pressure-velocity is accomplished through the SIMPLE method and the TDMA and Gauss-Seidel algorithms are used to relax the resulting algebraic equations. The solution method is tested against the laminar flow between parallel plates and recirculating flow patterns are qualitatively presented. The advantages of using more than one grid level and the CS approach are discussed upon. © 2003 Elsevier Inc. All rights reserved.

This paper presents derivations of mass transport equations for turbulent flow in permeable structures. Equations are developed following two distinct procedures. The first method considers time averaging of the local instantaneous mass transport equation beford the volume average operator is applied. The second methodology employs both averaging operators butin a reverse order. This work is intended to demostrate that both approaches lead to equivalent equations when one takes into account both time fluctuations and spatial deviations of velocity and mass fraction. A modeled form for the final transport equation is presented where turbulent transfer is based on a macroscopic version of the k-ε model. © 2003 Elsevier Science Ltd.

Environmental flows of extreme importance, such as turbulent atmospheric boundary layer over thick rain forests, may benefit from more realistic mathematical models. Accordingly, flow over layers of dense vegetation can be characterized by some sort of porous structure through which a fluid permeates. For hybrid media, involving both a porous structure and a clear flow region, difficulties arise due to the proper mathematical treatment given at the interface. The literature proposes a jump condition in which shear stresses on both sides of the interface are not of the same value. This paper presents numerical solutions for such hybrid medium, considering here a channel partially filled with a porous layer through which fluid flows in turbulent regime. One unique set of transport equations is applied to both regions. Effects of Reynolds number, porosity, permeability and jump coefficient on mean and turbulence fields are investigated. Results indicate that depending on the value of the stress jump parameters, a substantially different structure for the turbulent field is obtained.

A number of natural and engineering systems can be characterized by some sort of porous structure through which a working fluid permeates. Boundary layers over tropical forests the spreading of chemical contaminants through underground water reservoirs are examples of important environmental flows that can benefit form appropriate mathematical treatment. For hybrid media, involving both a porous structure and a clear flow region, difficulties arise due to the proper mathematical treatment given at the interface. The literature proposes a jump condition in which stresses at both sides of the interface are not of the same value. The objective of this paper is to present a numerical implementation for solving such hybrid medium, considering here a channel partially filled with a porous layer through which fluid flows in laminar regime. One unique set of transport equations is applied to both regions. Numerical results are compared with available analytical solutions in the literature for two cases, namely, with and without the non-linear Forchheimer term. Results are presented for the mean velocity across both the porous structure and the clear region. The influence of medium properties, such as porosity and permeability, is discussed.

Many engineering and environmental system analyses can benefit from appropriate modeling of turbulent flow in porous media. Through the volumetric averaging of the microscopic transport equations for the turbulent kinetic energy, k, and its dissipation rate, epsilon, a macroscopic model was proposed for such media (IJHMT, 44(6), 1081-1093, 2001). In that initial work, the medium was simulated as an infinite array of cylindrical rods. As an outcome of the volume averaging process, additional terms appeared in the equations for k and epsilon. These terms were here adjusted assuming now the porous structure to be modeled as an array of elliptic rods instead. Such an adjustment was obtained by numerically solving the microscopic flow governing equations, using a low Reynolds formulation, in the periodic cell composing the medium. Different porosity and Reynolds numbers were investigated. The fine turbulence structure of the flow was computed and integral parameters were presented. The adjusted model constant was compared to similar results for square and cylindrical rods. It is expected that the contribution herein provide some insight to modelers devoted to the analysis of engineering and environmental systems characterized by a porous structure saturated by a fluid flowing in turbulent regime. © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

This work examines the performance of linear and non-linear eddy-viscosity models when used to predict the turbulent flow in periodically sinusoidal-wave channels. Two geometries are investigated, namely a converging-diverging channel and a channel with concave-convex walls. The numerical method employed for the discretization of the equations is the control-volume method in a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm is used for correcting the pressure field. The classical wall function and a low Reynolds model are used to describe the flow near the wall. Comparisons between those two approaches using linear and non-linear turbulence models are done. Here, an implicit numerical treatment was proposed for the non-linear diffusion terms of the momentum equations in order to increase the robustness of the solution method.

Steady laminar and turbulent natural convection in a two-dimensional square cavity, isothermally heated from the left side and cooled from the opposing side, is numerically analyzed using the finite volume method. Benchmark results for laminar and turbulent flows are compared with similar numerical solutions in the literature. The cases of clear and porous media are considered. Governing equations are written in terms of primitive variables and are recast into a general form. The effects of Rayleigh number on flow pattern and energy transport are investigated for Ra ranging from 103 to 1010 for clear media and 101 to 106 for porous media. The turbulence model used was the standard k-ε along with the wall function approach. All results presented herein showed reasonable agreement with calculations presented in the literature. Critical values for the Rayleigh number for the onset of turbulence are suggested. The main objective of this work is to validate a numerical tool for simulating turbulent natural convection in both clear and porous media.

This work presents numerical results for heat transfer in turbulent flow past a backward-facing step. It is shown that non-linear k-ε models perform better than their linear counterparts when simulations are compared with experimental values. Wall functions are used for simplicity of the simulations. The finite-volume technique is employed for discretizing the transport equation set on a non-orthogonal grid system. The SIMPLE method is used for correcting the pressure field. Results for the reattachment length using the non-linear model are closer to the experimental values when compared with similar calculations using the standard linear closure.

This work presents numerical results for heat transfer in turbulent flow past a backward-facing-step channel with a porous insert using linear and non-linear eddy viscosity macroscopic models. The non-linear turbulence models are known to perform better than classical eddy-diffusivity models due to their ability to simulate important characteristics of the flow. Parameters such as porosity, permeability and thickness of the porous insert are varied in order to analyze their effects on the flow pattern, particularly on the damping of the recirculating bubble after the porous insertion. The numerical technique employed for discretizing the governing equations is the control-volume method. The SIMPLE algorithm is used to correct the pressure field. Wall functions for velocity and temperature are used in order to bypass fine computational close to the wall. Comparisons of results simulated with both linear and non-linear turbulence models are presented.

This work presents a numerical investigation of fully developed turbulent flow in a triangular sub-channel of a bare rod bundle using a Non-Linear Eddy Viscosity Model (NLEVM). The numerical technique employed for discretizing the governing equations is the control-volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to correct the pressure field. The classical wall function and a low Reynolds model were used in order to handle flow calculations near the wall. In this work, the influence of constants of calibration existing in the non-linear terms of the model is analyzed.

This paper presents numerical results for turbulent heat transfer in channels presenting a sudden expansion and obstructed by a porous layer. The turbulence model of Pedras and de Lemos (2001) is further developed to consider heat transfer analysis. The concept of double decomposition introduced by Pedras and de Lemos (2000) is extended to the energy equation and a set of macroscopic transport equations for flow and heat transfer analysis is proposed. Within the porous inserts, additional terms are considered for the turbulence kinetic energy equation and its dissipation rate. Governing equations for the mean and turbulent fields are written in their high Reynolds form and are discretized by means of the control volume method. Wall proximity is treated with the wall function approach and the algebraic equation set is relaxed with the SIMPLE method. Solution is sought for both the clear fluid region and the porous material in a unique computational domain. The effects of thickness and permeability of the inserts on flow pattern and heat transfer features are assessed. It is found that for some combinations of thickness and permeability, the recirculating bubble right after the step is completely suppressed, improving the heat transfer characteristics for the lower wall. © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Steady laminar and turbulent natural convection in twodimensional concentric and eccentric annular cavities, isothermally heated from the inner cylinder and cooled from the outer wall, is numerically analyzed using the finite volume method. Benchmark results for laminar and turbulent flows are compared with similar calculations by Cho et al. (1982)(JHT, 104), Kenjeres & Hanjalic (1995)(IJHFF, 16) and the experimental data of Kuehn & Goldstein (1978)(JHT, 100) and McLeod & Bishop (1989)(IJHMT, 32). Governing equations are written in terms of primitive variables and are recast into a general form. For laminar flows, isotherms and streamlines for annuli with the same eccentricity, but located at different angular positions, are presented for RaL=104 and Ri/Ro=0.3846, where Ro and Ri are the cylinder radii and RaL is the Rayleigh number based on a characteristic length Ro-Ri. Turbulence flow calculated for RaL=2.5×106 and 1.22×107, for both concentric and eccentric annuli, show reasonable agreement with calculations presented in the literature. © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

This work examines the performance of linear and nonlinear eddy-viscosity models when used to predict the turbulent flow in periodically sinusoidal-wave channels. Two geometries are investigated, namely a converging-diverging channel and a channel with concave-convex walls. The numerical method employed for the discretization of the equations is the control-volume method in a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm is used for correcting the pressure field. The classical wall function and a low Reynolds model are used to describe the flow near the wall. Comparisons between those two approaches using linear and non-linear turbulence models are done. Here, an implicit numerical treatment was proposed for the non-linear diffusion terms of the momentum equations in order to increase the robustness of the solution method. Copyright © 2002 by ASME.

This work presents numerical results for heat transfer in turbulent flow past a backward-facing-step channel with a porous insert using linear and non-linear eddy viscosity macroscopic models. The non-linear turbulence models are known to perform better than classical eddy-diffusivity models due to their ability to simulate important characteristics of the flow. Parameters such as porosity, permeability and thickness of the porous insert are varied in order to analyze their effects on the flow pattern, particularly on the damping of the recirculating bubble after the porous insertion. The numerical technique employed for discretizing the governing equations is the control-volume method. The SIMPLE algorithm is used to correct the pressure field. Wall functions for velocity and temperature are used in order to bypass fine computational close to the wall. Comparisons of results simulated with both linear and non-linear turbulence models are presented. Copyright © 2002 by ASME.

Steady laminar and turbulent natural convection in a two-dimensional square cavity, isothermally heated from the left side and cooled from the opposing side, is numerically analyzed using the finite volume method. Benchmark results for laminar and turbulent flows are compared with similar numerical solutions in the literature. The cases of clear and porous media are considered. Governing equations are written in terms of primitive variables and are recast into a general form. The effects of Rayleigh number on flow pattern and energy transport are investigated for Ra ranging from 103 to 1010 for clear media and 101 to 106 for porous media. The turbulence model used was the standard k-ε along with the wall function approach. All results presented herein showed reasonable agreement with calculations presented in the literature. Critical values for the Rayleigh number for the onset of turbulence are suggested. The main objective of this work is to validate a numerical tool for simulating turbulent natural convection in both clear and porous media. Copyright © 2002 by ASME.

This work presents numerical results for heat transfer in turbulent flow past a backward-facing step. It is shown that non-linear k-ε models perform better than their linear counterparts when simulations are compared with experimental values. Wall functions are used for simplicity of the simulations. The finite-volume technique is employed for discretizing the transport equation set on a non-orthogonal grid system. The SIMPLE method is used for correcting the pressure field. Results for the reattachment length using the non-linear model are closer to the experimental values when compared with similar calculations using the standard linear closure. Copyright © 2002 by ASME.

Turbulence models proposed for porous media follow different approaches depending on the order of application of time and volume average operators. The two different methodologies lead to different governing equations for the statistical quantities. The turbulence kinetic energy resulting from application of the two averaging operators, following both orders of integration, is different. The connection between these two quantities is here discussed in light of the double-decomposition (time and volume) concept. Transport equations following both orders of integration are developed and compared.

The literature presents two different methodologies for developing turbulent models for flow in a porous medium. The first one starts with the macroscopic equations using the extended Darcy-Forchheimer model. The second method makes use, first, of the Reynolds-averaged equations. These two methodologies lead to distinct set of equations for the κ-ε model. The present work details a mathematical model for turbulent flow in porous media following the second path, or say, space-integrating the equations for turbulent flow in clear fluid. In order to account for the porous structure, an additional term is included in the sources for κ and ε. A methodology is followed for determining the additional constant proposed. The equations for the microscopic flow were numerically solved inside a periodic elementary cell. The porous structure was approximated by an infinite array of circular rods. The method SIMPLE and a non-orthogonal boundary-fitted coordinate system were employed. Integrated parameters where compared to the existing data for fully developed homogeneous flow through porous media. Preliminary results are in agreement with numerical experiments presented in the literature. © 2001 Elsevier Science Ltd. All rights reserved.

A spatially periodic array is used to simulate the turbulent flow field inside an elementary control volume representing a porous medium. The low Reynolds (Re) version of the k - ε model is employed. Mean flow and turbulence equations are discretized by means of the control-volume approach. Boundary treatment includes symmetry lines and spatially periodic conditions. A generalized coordinate system is used to generate the computational grid. Solution of the flow equations is accomplished through the SIMPLE method. Detailed computations are used to close the proposed macroscopic turbulence model. Overall pressure drop and volume-averaged turbulence kinetic energy (TKE) are presented. For the condition analyzed here the integral turbulence energy increases with reduction of the medium porosity.

Turbulence models proposed for flow through permeable structures depend on the order of application of time and volume average operators. Two developed methodologies, following the two orders of integration, lead to different governing equations for the statistical quantities. The flow turbulence kinetic energy resulting in each case is different. This paper reviews recently published mathematical models developed for such flows. The concept of double decomposition is discussed and models are classified in terms of the order of application of time and volume averaging operators, among other peculiarities. A total of four major classes of models are identified and a general discussion on their main characteristics is carried out. Proposed equations for turbulence kinetic energy following time-space and space-time integration sequences are derived and similar terms are compared. Treatment of the drag coefficient and closure of the interfacial surface integrals are discussed. © 2001 by ASME.

Turbulent heat transport in porous media has been an extensively discussed issue in the pertinent literature. Recently, Rocamora and de Lemos (2000) have developed a macroscopic energy equation using the double decomposition concept presented by Pedras and de Lemos (2000). This development includes an extra term for the convective heat transport therein called turbulent thermal dispersion. This extra term arises from the spatial deviations and temporal fluctuations of both velocity and temperature. This work aims at proposing a gradient type diffusion model for the convective terms involving dispersion and turbulent heat flux. Also considered is the tortuosity derived from the diffusive term of the energy equation.

The present work investigates the existence of optimal algorithm parameters for multigrid numerical solutions of a two-dimensional steady-state conductive-convective problem. The velocity field inside the rectangular domain and the temperature distribution at its four boundaries are known and kept constant. The numerical method includes finite volume discretization and Weighted Upstream Differencing Scheme (WUDS) interpolation on structured, orthogonal and regular meshes. Multigrid is implemented according to the correction storage (CS) formulation. Minimum computational effort is sought as a function of control-volume Peclet number, different numbers of grids, number of smoothing sweeps in each level and distinct combinations of iterative solution algorithm. © 2001 Elsevier Science Inc. All rights reserved.

Many engineering and environmental system analyses can benefit from appropriate modeling of turbulent flow in porous media. Through the volumetric averaging of the microscopic transport equations for the turbulent kinetic energy, k, and its dissipation rate, ε, a macroscopic model was proposed for such media (IJHMT, 44(6), 1081-1093, 2001). In that initial work, the medium was simulated as an infinite array of cylindrical rods. As an outcome of the volume averaging process, additional terms appeared in the equations for k and ε. These terms were here adjusted assuming now the porous structure to be modeled as an array of elliptic rods instead. Such an adjustment was obtained by numerically solving the microscopic flow governing equations, using a low Reynolds formulation, in the periodic cell composing the medium. Different porosity and Reynolds numbers were investigated. The fine turbulence structure of the flow was computed and integral parameters were presented. The adjusted model constant was compared to similar results for square and cylindrical rods. It is expected that the contribution herein provide some insight to modelers devoted to the analysis of engineering and a environmental systems characterized by a porous structure saturated by a fluid flowing in turbulent regime. © 2001 by ASME.

This work presents numerical predictions for turbulent flow field confined in a circular duct past a gradually varying cross section segment. Both expanding and contracting sections were investigated. Equations of boundary-layer type were used and the standard linear k-ε model was applied. A forward marching method was employed for sweeping the computational domain. Results are presented for contractions and diffusers in addition to comparisons with experimental data for air. Turbulence damping in contractions and its enhancement in diffusers were correctly calculated. Further, for contractions with angles of up to 21° degrees, the use of a parabolic solver showed good agreement with experimental values for the mean and statistical quantities. For diffusers, adverse pressure gradient along the flow limits the quality of the predictions as the angle and length of diffuser increase past 5° and 10 duct radius, respectively.

Turbulent flow in a channel, totally and partially filled with a porous medium, is simulated with a proposed turbulence model. Two cases are analyzed, namely clear flow past a porous obstacle and flow through a porous medium having a cavity with a higher porosity. Mean and turbulence quantities were solved within both computational domains using a single numerical technique. The control volume approach was used to discretize the governing equations. In the first case analyzed, the flow penetration into the porous substrate is accompanied by generation of turbulence kinetic energy within the obstacle. In the second geometry, the flow is pushed towards the cavity as porosity increases.

This paper presents numerical results for laminar heat transfer and turbulent flow past a backward-facing step channel with and without a porous insert. The effects of thickness and permeability of the inserts on flow pattern and heat transfer features are assessed. It is found that for some combinations of thickness and permeability, the recirculating bubble right after the step is completely suppressed, improving the heat transfer characteristics for the lower wall.

The literature documents two procedures for modeling turbulent heat transport in incompressible flows through homogeneous rigid porous media. The first method considers time averaging of the energy equation before the volume average operator is applied. The second methodology also employs both averaging operators, but in the reverse order. Resulting equations in both cases are different, leading to controversies and interesting discussions in the literature. This work is intended to demonstrate that both approaches lead to equivalent equations when one takes into account both time fluctuations and spatial deviations of velocity and temperature. (C) 2000 Elsevier Science Ltd.The literature documents two procedures for modeling turbulent heat transport in incompressible flows through homogeneous rigid porous media. The first method considers time averaging of the energy equation before the volume average operator is applied. The second methodology also employs both averaging operators, but in the reverse order. Resulting equations in both cases are different, leading to controversies and interesting discussions in the literature. This work is intended to demonstrate that both approaches lead to equivalent equations when one takes into account both time fluctuations and spatial deviations of velocity and temperature.

In the literature, there are two distinct approaches for developing turbulent models for flow in a porous medium. The first one starts with the macroscopic equations using the extended Darcy-Forchheimer model. The second method considers first the microscopic balance equations. In both cases, time and volume averaging operators are applied in a different order. The turbulence kinetic energy equation resulting from application of the two averaging operators, following both orders of integration, are different. In this work, a new double-decomposition (time and volume) methodology is suggested and the differences between those two mathematical treatments are highlighted. (c) 2000 Elsevier Science Ltd.In the literature, there are two distinct approaches for developing turbulent models for flow in a porous medium. The first one starts with the macroscopic equations using the extended Darcy-Forchheimer model. The second method considers first the microscopic balance equations. In both cases, time and volume averaging operators are applied in a different order. The turbulence kinetic energy equation resulting from application of the two averaging operators, following both orders of integration, are different. In this work, a new double-decomposition (time and volume) methodology is suggested and the differences between those two mathematical treatments are highlighted.

Copyright © 2000 by ASMEThis paper presents numerical results for laminar heat transfer and turbulent flow past a backward-facing step channel with and without a porous insert The effects of thickness and permeability of the inserts on flow pattern and heat transfer features are assessed. It is found that for some combinations of thickness and permeability, the recirculating bubble right after the step is completely suppressed, improving the heat transfer characteristics for the lower wall.

This work reports a numerical investigation on buoyancy-induced flows occurring in enclosures of small aspect ratio and inclined with respect to the horizontal direction. The numerical method used consists of the control-volume approach and a new block-implicit error-smoothing operator. Governing equations are written in terms of primitive variables and are recast into a general form. In the proposed method, all governing equation are relaxed locally, in contrast with commonly used segregated schemes. The effects of Rayleigh number, aspect ratio, and cavity inclination on temperature and velocity patterns are discussed. It is expected that more advanced parallel computer architectures can benefit from the error-smoothing operator described here. © 2000, Taylor & Francis Group, LLC. All rights reserved.

Copyright © 2000 by ASMETurbulent flow in a channel, totally and partially filled with a porous medium, is simulated with a proposed turbulence model. Two cases are analyzed, namely clear flow past a porous obstacle and flow through a porous medium having a cavity with a higher porosity. Mean and turbulence quantities were solved within both computational domains using a single numerical technique. The control volume approach was used to discretize the governing equations. In the first case analyzed, the flow penetration into the porous substrate is accompanied by generation of turbulence kinetic energy within the obstacle. In the second geometry, the flow is pushed towards the cavity as porosity increases.

In the literature, turbulence models proposed for porous media follow two contradictory approaches. In the first one, governing equations for the mean and turbulent fields are obtained by time-averaging the volume-averaged equations. In the second method, volume averaging is applied to the time-averaged equations. The two different approaches lead to different governing equations and, ultimately, to contradicting overall conclusions. In this work, a new double-decomposition (time and volume) methodology is suggested and the differences between those two mathematical treatments are highlighted.

The present work investigates the efficiency of the multigrid numerical method applied to solve two-dimensional laminar velocity and temperature fields inside a rectangular domain. Numerical analysis is based on the finite volume discretization scheme applied to structured orthogonal regular meshes. Performance of the correction storage (CS) multigrid algorithm is compared for different inlet Reynolds number (Rein) and number of grids. Up to four grids were used for both V- and W-cycles. Simultaneous and uncoupled temperature-velocity solution schemes were also applied. Advantages in using more than one grid is discussed. Results further indicate an increase in the computational effort for higher Rein and an optimal number of relaxation sweeps for both V- and W-cycles.

© 1999 American Society of Mechanical Engineers (ASME). All rights reserved.The present work investigates the efficiency of the multigrid numerical method applied to solve two-dimensional laminar velocity and temperature fields inside a rectangular domain. Numerical analysis is based on the finite volume discretization scheme applied to structured orthogonal regular meshes. Performance of the correction storage (CS) multigrid algorithm is compared for different inlet Reynolds number (Rein) and number of grids. Up to four grids were used for both V- and iV-cycles. Simultaneous and uncoupled temperaturevelocity solution schemes were also applied. Advantages in using more than one grid is discussed. Results further indicate an increase in the computational effort for higher Re,„ and an optimal number of relaxation sweeps for both V- and W-cycles.

This work reports numerical results for the case of incompressible laminar heated flow with a swirl in a vertical cylindrical chamber. Computations are obtained with a point-wise block-implicit scheme. Flow governing equations are written in terms of the so-called primitive variables and are recast into a general form. The discretized momentum equations are applied to each cell face and then, together with the mass-continuity, tangential velocity and energy equations, are solved directly in each computational node. The effects of Rayleigh, Reynolds and Swirl numbers on the temperature field are discussed upon. Flow pattern and scalar residual history are reported. Further, it is expected that more advanced parallel computer architectures can benefit from the error smoothing operator here described.

© 1998 American Society of Mechanical Engineers (ASME). All rights reserved.This work reports numerical results for the case of incompressible laminar heated flow with a swirl in a vertical cylindrical chamber. Computations are obtained with a point-wise block-implicit scheme. Flow governing equations are written in terms of the so-called primitive variables and are recast into a general form. The discretized momentum equations are applied to each cell face and then, together with the mass-continuity, tangential velocity and energy equations, are solved directly in each computational node. The effects of Rayleigh, Reynolds and Swirl numbers on the temperature field are discussed upon. Flow pattern and scalar residual history are reported. Further, it is expected that more advanced parallel computer architectures can benefit from the error smoothing operator here described.

© 1998 The American Institute of Aeronautics and Astronautics Inc. All rights reserved.The present work investigates the efficiency of the multigrid numerical method when applied to solve two-dimensional steady-state conductive-convective problems. The velocity field inside a rectangular domain and the temperature distribution at its four boundaries are known and kept constant. The numerical method includes finite volume discretization and the Weighted Upstream Differencing Scheme interpolation on structured orthogonal regular meshes. The correction storage (CS) multigrid algorithm performance is compared for different Peclet numbers and the number of sweeps in each grid level. Up to six grids for both multigrid V-and W-cycles are considered. Results indicate a better performance of the W-cycle and reduction in computational effort for larger Peclet numbers.

Turbulent flow field calculations for confined coaxial streams are presented. The cases of gradual duct enlargement and contraction are analyzed. Turbulence is handled with the standard k-ε model. A marching-forward numerical integration technique is used to sweep the computational domain. Within contractions, turbulence is damped, whereas in expansions the valued of k is increased. Also, general turbulence kinetic energy levels are greater when the internal jet is faster than the annular stream.

This paper reports numerical results obtained with a point-wise block-implicit scheme. Computations are presented for the case of incompressible laminar heated flow with swirl in a vertical cylindrical chamber. Governing equations are written in terms of the so-called primitive variables and are recast into a general form. The discretized momentum equations are applied to each cell face and then, together with the mass-continuity, tangential velocity and energy equations, are solved directly in each computational node. Results are obtained with a Personal Computer under reasonable computing times. Flow pattern and mass residual behavior are reported. Further, it is also expected that more advanced parallel computer architectures can benefit from the error smoothing operator here described.

Numerical results for natural convection flows obtained by a point-wise locally-implicit scheme are reported. Computations are presented for incompressible laminar thermally-driven flow inside a square cavity. Governing equations are written in terms of the so-called primitive variables and are recast into a general form. Finite-differencing is obtained by means of the widely-used control-volume approach. The discretized momentum equations are applied to each cell face and then, together with the mass-continuity and energy equations, are simultaneously solved by means of a direct method in each computational node. An Alternating Symmetrical Coupled Gauss-Siedel procedure is employed in which iterations are performed alternating the most varying index in every sweep over the computational domain. Flow pattern and mass residual behavior are reported.

This work consists of a numerical investigation on the effect of buoyancy on the thermal developing mean and turbulence fields for buoyancy-aided and buoyancy-opposing channel flows. The geometry considered was vertical pipe flow and an Algebraic Stress Model for turbulence was used in conjunction with a marching-forward finite-difference numerical scheme. Calculations were performed for mercury (Pr=0.025) and for Ra/Re2=1.2×105, where Ra is the Rayleigh number and Re the Reynolds number. An adiabatic entry length of 67 diameters was used before a heated section of another 74 diameters was computed. Results for the thermal developing region are presented for the friction factor ff the Nusselt number Nu, the axial velocity U and for the turbulent kinetic energy k. Predictions are also reported for the axial and radial turbulent fluxes.

Numerical results for swirling flows obtained by a point-wise locally-implicit scheme are here reported. Computations are presented for incompressible laminar flow inside a model combustor. Governing equations are written in terms of the so-called primitive variables and are recast into a general form. Finite-differencing is obtained by means of the widely-used control-volume approach. The discretized momentum equations are applied to each cell face and then, together with the mass-continuity equation, are simultaneously solved by means of a direct method in each computational node. Extension to complete heat transfer analysis is commented upon. Flow pattern and mass residual behavior are reported.

This work presents numerical predictions for the incompressible laminar axial flow over bare rod-bundles. Discretization of the governing equations is accomplished through the control volume method applied to a boundary fitted curvilinear coordinate system. © 1992 Society of Automotive Engineers, Inc.