Darcy Model Research Articles

Combined effects of viscous and Darcy dissipation on mixed convection flow in a composite vertical channel partially filled with a porous material: Analytical approach

In this paper, the combined influence of thermal dissipation and Darcy dissipation on the steady motion of a fully developed mixed convection flow of viscous, incompressible fluid through a vertical composite channel partially filled with porous material is investigated in depth. The motion of the fluid within the channel is triggered by the sudden application of a pressure gradient in the direction of flow as well as natural convection due to the temperature gradient caused by applied heat through the channel boundaries. The setup of the system is done in such a way that the clear fluid and porous regions are separated by an interface between the two regions. The Brinkman-extended Darcy model is used to simulate momentum transfer. By matching their velocities and accounting for shear stress jump situations at the interface, the clear fluid and porous areas are linked. In addition, the thermal energy equation in the clear fluid region considers the viscous dissipation effect, whereas, in the porous region, the Darcy dissipation effect is considered in addition to the viscous dissipation effect. The homotopy perturbation method (HPM) was applied to solve the momentum and energy equations. The effects of various flow parameters introduced into the problem are discussed using line graphs. The study discovered that increasing the Brinkmann number increased both the temperature and velocity profiles. Furthermore, as the thickness of the porous medium increases, so does the pressure gradient, whereas increasing the porosity reduces the pressure gradient within the composite channels. Also, when the viscosity ratio goes up, the pressure gradient needed to move the flow through the channel goes up as well.

Numerical study on the hybrid nanofluid (Co3O4-Go/H2O) flow over a circular elastic surface with non-Darcy medium: Application in solar energy

In recent years, the conversion of solar radiation to thermal energy has received a significant attention as the demand for renewable heat and power increases. Nanofluids can play an important role in improving the performance of solar-thermal systems due to their capabilities for heat transfer enhancement. This paper investigates numerically the flow of a hybrid nanofluid through a porous medium. A water-based hybrid nanofluid is propagating across a circular elastic surface. Water flow is incompressible, irrotational and electrically conducting. The deployment of an external magnetic field ignores the induced magnetic field due to the small magnetic Reynolds number. Cobalt oxide (Co3O4) and Graphene (Go) nanoparticles (NPs) are suspended in the base fluid. For the porous media, the Darcy model is employed, while the viscous dissipation effects are also incorporated in the energy equation. Similarity variables are used to develop the mathematical modeling of momentum and energy equations. The numerical solution of the finalized forms of nonlinear differential equations is accomplished by the use of the Successive Linearization Method (SLM). The simulation results are then validated against the previously published data.

Influence of cavity inclination on mixed convection in a double-sided lid-driven cavity with a centrally inserted hot porous block

A numerical investigation is made to understand the influence of seven distinct cavity inclination angles (α = 0°, 15°, 30°, 45°, 60°, 75° and 90°) on the fluid flow behaviour during mixed convection in a double-sided lid-driven cavity for varied Richardson numbers (0.01≤Ri≤100). A hot porous square block with a blockage ratio of 0.25 is inserted at the cavity centre. We employ the multiple-relaxation-time lattice Boltzmann technique, which incorporates the Brinkman-Forchheimer-extended Darcy model for simulating flow through the permeable structure at different Darcy numbers (10−2≤Da≤ 10−6). The validation of the present developed code provides reasonable agreement with the previously reported findings acquired using conventional CFD approaches. It is noticed from the results that the cavity inclination angles provide a substantial impact on the generation of entropy and the rate of heat transfer. Furthermore, the findings reveal that the values of Ri and Da influence the optimal energy-efficient condition for a flow configuration.

Simulating Nanofluid Forced Convection Flow by Thermal Lattice Boltzmann Approach

Improving heat transfer using nanofluids has proven to be a promising option with many practical applications. However, the behavior of particles conveying energy for thermal transport depends closely on the dimensions of systems and channels where the flow evolves. Thereby, any fine thermal analysis should lean on a mesoscale approach applied at a microscale level. To this end, the multi-distribution functions–thermal lattice Boltzmann method has been taken to deal with convective heat flow and entropy generation in a channel with isothermal top–bottom walls and filled with a nanofluid (Cu/water). It was extended to simulate the flow governed by the Brinkman–Forchheimer Darcy model using the local thermal equilibrium assumption. The effects of nanoparticles’ volume fraction, Darcy number, porosity, heat capacity ratio and thermal conductivity ratio on heat transfer, entropy generation, average Nusselt number, and Bejan number are investigated. Among the salient results, it can be stated that the nanoparticles’ volume fraction increases heat transfer and entropy generation, but such a propensity can be affected by the porous medium permeability used. To sum up, the findings confirm the potential of the multi-distribution functions–lattice Boltzmann formalism to tackle forced nanofluid flows with heat transfer in porous media.

Optimization of heat transfer rate in a moving porous fin under radiation and natural convection by response surface methodology: Sensitivity analysis

The remarkable advancement has been made in area of investigation and extensive demands for permeable fin in recent past. Permeable fins have numerous prospective employments including industrial applications specifically in engineering implementations such as in heat exchangers and super heaters. Additionally, the permeable fin performs a key role in aeronautic engineering. In this study the sensitivity analysis of natural convection and radiation transmission of heat in permeable fin is established. The main objective of the study is to optimize the rate of transmission of heat in a moving permeable fin in natural convection and radiation environment by utilizing Response Surface Methodology (RSM). Also, sensitivity of heat transfer rate is evaluated using RSM. The Darcy model is applied to imitate the impacts of permeable medium and convective heat. Midpoint (midrich) method is utilized to solve the energy equation numerically. The outcomes represents that the temperature of the fin enhance to the neighboring temperature as the value of permeable parameter rises. It is investigated that the heat along the length of fin decline for finite fin as the value of the dimensionless ratio of the surrounding temperature T∞to the difference of the base temperature Tb and surrounding temperature and radiation parameter rises. Also, rate of transmission of heat at the fin tip bring down as the value of Peclet number increased. The outcomes of sensitivity analysis represents that expansion in porosity parameter increases the heat transmission rate and its highest value is observed in level of (-1). Also, heat transfer rate has a non-negative sensitivity towards permeable parameter. However, the decline in dimensionless ratio of the surrounding temperature T∞ to the difference of the base temperature Tb and surrounding temperature and radiation parameter increases the heat transmission rate and its highest value is observed in level of (-1) and (+1). Furthermore, heat transmission rate has a non-positive sensitivity towards dimensionless ratio of surrounding temperature T∞ to the difference of base temperature Tb and surrounding temperature and radiation parameter.

Magneto-bioconvection flow of hybrid nanofluid in the presence of oxytactic bacteria in a lid-driven cavity with a streamlined obstacle

This work explores the magneto-bioconvection flow of silver(Ag)‑magnesium oxide(MgO)-water hybrid nanoliquid in a porous cavity considering the effect of gyrotactic microorganisms. The novelty of this work is to examine the magnetic force on bioconvection flow in a porous cavity containing obstacle, oxytactic bacteria and occupied by a hybrid nanofluid. A streamlined square obstacle is inserted in the flow field. For the modeling of porous medium, Brinkman-Forchheimer extended Darcy's model is considered. In order to simulate the governing system of dimensionless equations, a higher Galerkin finite element method (GFEM) is implemented. The resulting discrete algebraic systems are treated using the adaptive Newton's method. The validity and reliability of designed solver is ensured by recomputing the numerical and experimental data available in the literature. The flow structure, heat and mass transport are analyzed for the controlling parameters such as Hartmann number (Ha = 0 − 100), bioconvection Rayleigh number (Rb = 10 − 100), Peclet number (Pe = 0.1 − 1), Richardson number (Ri = 0.1 − 5), Lewis number (Le = 1 − 10) and Darcy number (Da = 10−5 − 10−2). The obtained numerical approximations demonstrated that the variation of heat transfer becomes almost constant for varying the Peclet number. Whilst the Sherwood number is a decreasing function of Peclet number. The dimensions of streamlined obstacle can serve as a control parameter and it makes opposite impact on heat and mass transport.

Permeability estimation of a porous structure in cancer treatment based on sampled velocity measurement* *Submitted to the editors 26 July 2021.

The problem of parameter identification appears in many physical applications. A parameter of particular interest in cancer treatment is permeability, which modulates the fluidic streamlines in the tumor microenvironment. Most of the existing permeability identification techniques are invasive and not feasible to identify the permeability with minimal interference with the porous structure in their working conditions. In this paper, a theoretical framework utilizing partial differential equation (PDE)-constrained optimization strategies is established to identify a spatially distributed permeability of a porous structure from its modulated external velocity field measured around the structure. In particular, the flow around and through the porous media are governed by the steady-state Navier–Stokes–Darcy model. The performance of our approach is validated via numerical and experimental tests for the permeability of a 3D printed porous surrogate in a micro-fluidic chip based on the sampled optical velocity measurement. Both numerical and experimental results show a high precision of the permeability estimation.

Secondary instabilities on double‐diffusive convection in an anisotropic porous layer with Soret effect

AbstractIn an anisotropic porous matrix with a Soret coefficient, the onset of double‐diffusive convection is investigated analytically using weakly nonlinear analysis. The momentum equation is expressed using a generalized Darcy model with a time derivative term. The Newell–Whitehead–Segel equation is acquired, thereby examining the Eckhaus and zigzag secondary instabilities. Nusselt and Sherwood numbers are used to examine convection onset by quantifying heat and mass movement. Heat and mass transmission dynamics are graphically depicted as a consequence of several parameters. An increase of the positive value of the Soret parameter enhances heat transport, whereas, an increase of the negative value of the Soret parameter reduces it.

Mixed bioconvection flow of Ag-MgO/water in the presence of oxytactic bacteria and inclined periodic magnetic field

In this paper, a numerical investigation on mixed bioconvection flow of silver(Ag)-magnesium oxide(MgO)-water hybrid nanofluid in a trapezoidal porous cavity under the effect of inclined periodic magnetic field is carried out. Brinkman–Forchheimer-extended Darcy model for the porous medium is adopted. Galerkin weighted residual finite element method is implemented to simulate the governing steady dimensionless equations. The resulting discrete nonlinear algebraic systems are treated using the adaptive Newton's method. The fluid flow, heat and mass transfer behavior are examined in variation of parameters as Hartmann number (Ha=0−100), angle of periodic magnetic field (θ=0∘−90∘), period of periodic magnetic field (Λ=0.1−1), bioconvection Rayleigh number (Rb=10−100), Richardson number (Ri=0.1−5), Lewis number (Le=1−10) and Peclet number (Pe=0.1−1). The rise in Hartmann number has a weakening effect on both transfer of heat and mass. When θ=0∘, convective heat and mass transfer is the smallest at Λ=1 where the most reduction in the average Nusselt and Sherwood numbers is 9.14% and 6.98% as Λ is raised from 0.1 to 1, respectively, and occurs at Ha=100. The significant decrement in the average Nusselt number is 42.57% at Ha=75 and in the Sherwood number is 10.4% at Ha=50 when θ is changed from 0∘ to 90∘.

Numerical Investigation On Thermal Behaviours Of Latent Thermal Energy Storages With Pcm Partially Filled With Aluminum Foam

In this paper a numerical investigation on the effects of the metal foam into the Latent Heat Thermal Energy Storage System, based on a phase change material, is accomplished. The geometry of the system is a vertical shell and tube LHTESS made with two concentric aluminum tubes. The internal surface of the hollow cylinder is assumed at a constant temperature above the melting temperature of the PCM to simulate the heat transfer from a hot fluid. The other external surfaces are assumed adiabatic. The PCM is completely embedded in the volume between the two coaxial cylinders, it is pure paraffin wax with a low value of thermal conductivity and it melts in a range of temperatures. The metal foam is made of aluminum and it partially fills the volume starting from the internal cylinder with a thickness lower the differences between the external cylinder radius and internal one. The enthalpy-porosity theory is employed to simulate the phase change of the PCM and the metal foam is modelled as a porous media that follow the Brinkman-extended Darcy model. The governing equations are solved employing the Ansys-Fluent code. The results are shown in term of melting time, temperature and energy at varying of time and at varying the filling ratio.

Numerical study of natural convection in an annulus between two concentric cylinders provided with metal foam fins

Natural convection in an annular space provided with metal foam fins attached to the inner cylinder is studied numerically. The metal foam fins made of copper were inserted in different axial sections with three fins in each section. The temperature of the inner cylinder is kept constant while the annular outer surface is adiabatic. The thickness effect of the inner pipe wall was considered. Naiver Stokes equation with Boussinesq approximation is used for the fluid regime while Brinkman-Forchheimer Darcy model is used for metal foam. In addition, the local thermal non-equilibrium condition in the energy equation of the porous media is presumed. The effect of Rayleigh number and number of foam fins in the axial direction, on fluid flow and heat transfer characteristics, were examined. The current model was valid with the available published results and good agreement is noticed. Results showed that as the Rayleigh number increases the dominated of convection mode increases and average Nusselt increases. It was found that at Rayleigh of 106 Nusselt reached its higher value which is 4.6 for the case of adding seven axial metal foams. A comparison between adding foam fins and copper fins was established for a range of Rayleigh numbers between 104 and 106. It showed a good enhancement in Nusselt number and the greatest enhancement percentage was 45.9% at Rayleigh equal 106 for the case of using seven sections of foam fins.

Numerical investigation on effects of induced magnetic field and viscous dissipation on MHD mixed convection in a vertical micro porous channel using Brinkmann – Forchheimer extended Darcy model

Impact of viscous dissipation and induced magnetic field in Brinkmann- Forchheimer extended Darcy flow of MHD mixed convectionsin a vertical microporous channel is investigated. Three different viscous dissipation models along with three different wall ambient temperature difference ratios are analysed. Projected results evidently indicate that the kind of viscous dissipation model has a notable impact on velocity distribution, induced magnetic field, current density, skin friction, and Nusselt number. It is found that an increase in Brinkman number increases velocity in the channel for distinct values of wall ambient temperatures difference ratios. Due to the domination of convection, an increase in Darcy number increases skin friction. The effect of induced current density on Darcy number, Brinkmann number, and inertial parameters are also discussed. The CFD data is used to train the built ANN model for predicting the heat transfer characteristics in the considered channel with 98.96% accuracy.

Discrimination between Pore and Throat Resistances against Single-Phase Flow in Porous Media

This study investigates the critical agents that cause non-Darrian flow in porous media. Four porous media different in morphology but similar in topology were studied numerically. By varying the throat diameters, the distinct roles of pores and throats in total dissipation were investigated using direct numerical simulation. Forchheimer model was selected to analyze the non-Darcian flow. In our simplified geometry, the ratio KappKD can best be correlated by non-Darcy effect (E). Total dissipation is directly related to the porous medium resistance against fluid flow. The energy dissipated in pores and throats was calculated by summing the dissipation in each computational segment. Pores are more prone to disobey the Darcy model than throats due to irregularity in fluid flow, and they are introduced as the cause of Darcy-model cessation. By increasing the pore-to-throat ratio, the non-Darcian flow in the pores begins sooner. The results show that the energy dissipation due to eddies is negligible. The dissipation in pores and throats was simulated through separate power-law equations, and their exponents were also extracted. The exponent for the pore body is equal to two when the viscous forces are dominant, and it increases by increasing the inertia force. The dissipation due to pore bodies is more apparent when the size of pore and throats are of the same order of magnitude. The relative losses of pore body increase as the velocity increases, in contrast to throats.

Computational Framework of Magnetized MgO-Ni/Water-Based Stagnation Nanoflow Past an Elastic Stretching Surface: Application in Solar Energy Coatings.

In this article, motivated by novel nanofluid solar energy coating systems, a mathematical model of hybrid magnesium oxide (MgO) and nickel (Ni) nanofluid magnetohydrodynamic (MHD) stagnation point flow impinging on a porous elastic stretching surface in a porous medium is developed. The hybrid nanofluid is electrically conducted, and a magnetic Reynolds number is sufficiently large enough to invoke an induced magnetic field. A Darcy model is adopted for the isotropic, homogenous porous medium. The boundary conditions account for the impacts of the velocity slip and thermal slip. Heat generation (source)/absorption (sink) and also viscous dissipation effects are included. The mathematical formulation has been performed with the help of similarity variables, and the resulting coupled nonlinear dimensionless ordinary differential equations have been solved numerically with the help of the shooting method. In order to test the validity of the current results and the convergence of the solutions, a numerical comparison with previously published results is included. Numerical results are plotted for the effect of emerging parameters on velocity, temperature, magnetic induction, skin friction, and Nusselt number. With an increment in nanoparticle volume fraction of both MgO and Ni nanoparticles, the temperature and thermal boundary layer thickness of the nanofluid are elevated. An increase in the porous medium parameter (Darcy number), velocity slip, and thermal Grashof number all enhance the induced magnetic field. Initial increments in the nanoparticle volume fraction for both MgO and Ni suppress the magnetic induction near the wall, although, subsequently, when further from the wall, this effect is reversed. Temperature is enhanced with heat generation, whereas it is depleted with heat absorption and thermal slip effects. Overall, excellent thermal enhancement is achieved by the hybrid nanofluid.

Convective instabilities in binary mixture 3He-4He in porous media

The present study attempts to simulate and analyze the Rayleigh-Bénard convection of superfluid mixture (3He-4He) kept in the sparsely packed porous medium with stress-free boundary conditions. Theoretically the linear and nonlinear analysis are carried out near the onset of stationary convection. The nonlinear governing equations describing the motion with the Darcy model, temperature and concentration fields are expanded as the sequence of non-homogeneous linear equations. These equations are solved by employing the Fourier analysis of perturbations in terms of the non-dimensional expansion parameter ε until O(ε8) as proposed by Kuo (1961). The flow field and heat transfer characteristics are analyzed for different control parameters arising in the system such as the Rayleigh number (R), the separation ratio (ψ), which is the coupling between the temperature and concentration fields, and the Lewis number (Le). Apart from streamlines and isotherms, the novel flow visualization technique for the heat flow patterns in terms of heatlines is derived and plotted. Also, the kinetic energy, potential energy, and minimum entropy generation are analyzed. The Nusselt number is found to be get enhanced as the values of R and ψ are increased whereas it gets inhibited as Le increases.

Numerical study on influence of titling position and density extremum on convection in a porous cavity with sinusoidal heating

The aspiration of the present problem is to investigate numerically the 2D unsteady buoyancy convection of cold water near its highest density value in an inclined square porous enclosed space. Left sidewall temperature varies in sinusoidal manner whereas the opposing right sidewall is sustained with constant lower temperature. The walls along x axis and parallel to x axis are kept in adiabatic condition. The range of angle of inclination is considered from 0 degree to 90 degree. The finite volume method (FVM) was applied to portray all mathematical equations. The special outcome of inclination angle, Darcy number, maximum density parameter and porosity are obtained in the present study with the use of Brinkmann–Forchheimer extended Darcy model. The change in the angle of inclination is noteworthy in the structure and direction of the flow of the fluid. The bicellular structure of the fluid is identified because of the higher density effect.

A Comparison between Klinkenberg and Maxwell-Stefan Formulations to Model Tight Condensate Formations

Summary Cyclic solvent (gas) injection is an efficient recovery method for condensate reservoirs. However, in tight, unconventional formations, the added complexity of low permeability results in more physics at play, beyond the widely used Darcy model for conventional reservoirs. In this work, a rigorous mass transfer model is implemented considering the real gas version of the Maxwell-Stefan formulation to evaluate cyclic injection schemes in tight condensate reservoirs. This model is then compared to the more widespread used Klinkenberg formulation, which does not include molecular diffusion. An evaluation is performed to check if a simplified formulation can be used to provide reasonable results in modeling production and enhanced recovery in tight condensate formations. Verification of the implemented equations is performed using experiments (Maxwell-Stefan model) and a commercial reservoir simulator (Klinkenberg model). Furthermore, the cell length used for the numerical studies is selected based on a sensitivity study to evaluate how numerical dispersion impacts recovery factor and liquid saturation for different cell sizes. By comparing the Klinkenberg model with different tangential momentum accommodation coefficient (TMAC) values to the Maxwell-Stefan model during primary production, it is possible to select a value of TMAC that can match closely the recovery values of lighter components when using the Maxwell-Stefan equations. However, for heavier hydrocarbon fractions, difference in recovery is more accentuated owing to increased molecule size (more molecular friction). This results in differences in condensate yield during primary production that may be relevant in a field scale. In the cyclic injection scheme, the importance of accounting for frictional effects between molecules is demonstrated using the Maxwell-Stefan formulation. In this case, molecular diffusion fluxes are influenced by high composition gradients. This results in differences between the Maxwell-Stefan and Klinkenberg models in terms of gas stored and hydrocarbon produced during cyclic injection simulations. Furthermore, a sensitivity study on operational parameters in the cyclic injection stage demonstrated that increasing the length of production cycles may be more beneficial than increasing the length of injection or soaking cycles. For the simulations in this study, the gas is injected above the dewpoint and pressure diffusivity is at least one order of magnitude higher than the other physics present in the process. Therefore, increasing the length of production cycles allows for recovery of heavier hydrocarbon fractions that remain in the gas phase. In this work, it is demonstrated that using a rigorous mass transfer formulation, such as the Maxwell-Stefan equations, can provide more information on a per component basis when evaluating cyclic injection schemes in tight condensate reservoirs.

Gas permeability and compressive strength evaluation of porous Al2O3 supported membrane

The aim was to investigate the feasibility of using a device to characterize permeability in porous substrates with low gas flux. Al2O3 supports were prepared by dry pressing using 1 to 3 wt% of polyethylene glycol (PEG) as a pore-forming agent. The green body was partially sintered at 1300 ºC. CO2 gas was used to measure both the pressure loss and the gas flux passing through the porous substrate. The permeability behavior was analyzed by the Darcy and Forchheimer models. The correlation between Darcy’s and Forchheimer’s permeabilities showed that the porous substrates may be used as membranes for micro, ultra, and nanofiltration. The mechanical resistance and the gas permeability exhibited antagonistic dependencies in relation to porosity, being identified as an optimized condition at 2 wt% PEG, in which the two properties presented high values.

Wetting Dynamics of Spontaneous Imbibition in Porous Media: From Pore Scale to Darcy Scale

AbstractSpontaneous imbibition plays an important role in many subsurface and industrial applications. Unveiling pore‐scale wetting dynamics, and particularly its upscaling to the Darcy model, are still unresolved. We conduct image‐based pore‐network modeling of cocurrent spontaneous imbibition and the corresponding quasi‐static imbibition in homogeneous sintered glass beads and heterogeneous Estaillades carbonate. We find that pore‐scale heterogeneity significantly influences entrapment of the nonwetting fluid, which in Estaillades is mainly because of the poor connectivity of pores. We show that wetting dynamics significantly deviates capillary pressure and relative permeability away from their quasi‐static counterparts. Moreover, we propose a nonequilibrium model for wetting permeability that well incorporates flow dynamics. We implement the nonequilibrium model into two‐phase Darcy modeling of a 10 cm long medium. Sharp wetting fronts are numerically predicted, which are in good agreement with experimental observations. Our studies provide insights into developing a two‐phase imbibition model with measurable material properties.

A mechanically consistent model for fluid–structure interactions with contact including seepage

We present a new approach for the mechanically consistent modelling and simulation of fluid–structure interactions (FSI) with contact. The fundamental idea consists of combining a relaxation of the contact conditions with the modelling of seepage through a porous layer of co-dimension one during contact. For the latter, a Darcy model is considered in a thin porous layer attached to a solid boundary in the limit of infinitesimal thickness. The resulting computational model is both mechanically consistent and simple to implement. The FSI coupling and the contact conditions are imposed weakly by means of a unified Nitsche approach. We analyse the approach in detailed numerical studies with both thick- and thin-walled solids, within a fully Eulerian and an immersed approach and using fitted and unfitted finite element discretisations.