Brinkman-Forchheimer Extended Darcy Model Research Articles

A parallel computational study of power-law non-Newtonian nanofluid in a C-shaped enclosure by multiple-relaxation-time lattice Boltzmann simulation

ABSTRACT This study aims to numerically examine the heat transfer and entropy generation of non-Newtonian power-law nanofluids in a C-shaped cavity filled with porous media considering the Brinkman-Forchheimer extended Darcy model (BFEDM). The graphics processing unit (GPU) based multiple-relaxation-time lattice Boltzmann method (MRT-LBM) was used through computing unified device architecture (CUDA) C/C++ platform. It is a known fact that the fluid viscosity varies with the shear rate and nanoparticle sizes and the thermal conductivity also varies with the fluid temperature. However, the extension of such phenomena needs to be further investigated to understand any specific threshold in heat transfer application. To leverage such characteristics, it is permissible to add certain input parameters. As a result, the power-law indices ( n ), the Darcy number ( Da ), the volume fraction of nanoparticles ( ϕ ), the Rayleigh number ( Ra ), and the fixed porosity ( ε = 0.4 ) were the five key parameters considered to analyze the numerical results. The GPU-based numerical code was validated with the available benchmark results, and good agreements were obtained. The results showed that the average rate of heat transfer in terms of the average Nusselt number ( Nu ‾ ) increased due to an increase in Ra due to the enhanced buoyancy force and a reduction in the .

Computational study of magneto-convective flow of aqueous-Fe3O4 nanoliquid in a tilted cylindrical chamber partially layered by porous medium: Entropy generation analysis

In various industrial applications, the main objective is to enhance thermal efficiency by minimizing the generation of entropy. Specifically, achieving optimal thermal efficiency in a tilted cylindrical chamber poses significant challenges due to the combined effects of tangential and normal gravity components. Our study focuses on the flow dynamics, thermal transport, and entropy generation of Fe3O4/H2O nanoliquid within a cylindrical annular enclosure by incorporating the synergistic effects of magnetic force, geometric inclination angle, and thickness of the porous region. The Brinkman–Forchheimer-extended Darcy model for ferrofluid motion and the one-equation model for heat transfer are applied in the porous region, while the conventional Navier–Stokes and energy equations are used in the fluid-only region. A series of computations is performed for various key parameters, such as Hartmann number (0≤Ha≤60), Darcy number (10−5≤Da≤10−1), porous layer thickness (0.1≤ε≤0.9), and angle of inclination (−60°≤γ≤60°). Our results reveal that the heat transport rate is enhanced by 48.6% with an increase in the Darcy number from 10−5 to 10−1. Moreover, the flow circulation and heat transport can be optimized by tilting the enclosure anticlockwise. It has been found that 91.8% of flow strength can be enhanced by rotating the enclosure from −60° to 60°. Finally, this study suggests that the inclination angle of 30° and a porous layer thickness of 0.3 emerge as the ideal configuration to obtain optimal performance, particularly for lower Hartmann and higher Darcy numbers. Our findings will provide insight into optimizing thermal processes in nanoliquid-filled enclosures subjected to magnetic force.

Free Convection Heat Transfer in Composite Enclosures with Porous and Nanofluid Layers

This work conducts a numerical investigation of convection heat transfer within two composite enclosures. These enclosures consist of porous and nanofluidic layers, where the porous layers are saturated with the same nanofluid. The first enclosure has two porous layers of different sizes and permeabilities, while the second is separated by a single porous layer. As the porous layer thickness approaches zero, both enclosures transition to clear nanofluid enclosures. The study uses the Navier–Stokes equations to govern fluid flow in the nanofluid domain and the Brinkman–Forchheimer extended Darcy model to describe flow within the saturated porous layer. Numerical solutions are obtained using an iterative finite difference method. Key parameters studied include the porous thickness ( 0.0 ≤ S ≤ 1.0 ), the nanoparticle volume fraction ( 0.0 ≤ ϕ ≤ 0.05 ), the thermal conductivity ratio ( 0.5 ≤ R k ≤ 10 ), and the Darcy number ( 10 − 5 ≤ D a ≤ 10 − 2 ). Key findings include the observation that the highest heat transfer is achieved at the highest concentration, regardless of the porous layer configuration, permeability value, or thermal conductivity ratio. Specifically, an augmentation in values of N u ― I up to 22% is obtained as concentration is adjusted from 1% to 5%. Similarly, an augmentation in values of N u ― II up to 25% is obtained as concentration is adjusted from 1% to 5%.

Convective nanofluid flow subjected to variable porosity, inclined magnetic field, and thermal radiations

Nanofluids pique the attention of researchers and scientists because of their excellent thermal transfer rates. They have numerous technological and industrial implementations in daily life. Nonsimilar analysis in this research is performed to study the consequences of nonuniform porosity of a porous medium on the radiative flow of nanofluid across a vertically positioned stretching surface. Moreover, the effects of viscous dissipation and inclined magnetic field are also taken into account. Single phase nanofluid model with latest thermophysical features of nanofluid are considered to devise the physical problem. Hamilton–Crosser model is used to investigate the various shapes of nanoparticles. Titanium oxide (TiO 2), Zirconium dioxide (ZrO2 ), water and Borosilicate ceramic are regarded as nanoparticles, base fluid, and porous media, respectively. The Brinkman–Forchheimer-extended Darcy model is assumed to represent the flow through porous medium. The governing partial differential equations (PDEs) with accompanying boundary conditions are transformed into the dimensionless system by deploying the appropriate transformations. The local nonsimilarity (LNS) approach with MATLAB bvp4c algorithm is executed to solve the dimensionless nonlinear coupled differential system. Results demonstrate that the augmentation in Hartmann number and porosity parameter diminished the velocity profiles of the considered nanofluids. Moreover, maximum temperature profile is noticed for the lamina shaped nanoparticles. Local Nusselt number for considered cases diminishes against the augmented estimations of Richardson number, radiation parameter and nanoparticles shape factor. A comparative study between the current study and previously published results for a specific case is conducted to validate the results, and good agreement is found between them.

Thermal-hydraulic characteristics analysis of steam–air condensation in vertical porous-wall corrugated tubes

In the passive containment cooling systems (PCCS) of Generation III nuclear power plants, the condensation performance of steam containing non-condensable gas (NCG) in condensate tubes is critical for removing heat from the nuclear power plant containment. By applying a novel porous-wall corrugated structure in the condenser tubes of this system, it was found that under certain conditions, a double vortex is formed inside and outside the porous layer, which is conductive to the transportation and condensation of steam to the condensing wall. The VOF multiphase relation and Brinkman–Forchheimerextended Darcy model in combination of local heat balance, condensation and species transport equations are employed to describe the heat and mass transfer in the porous-wall corrugated tube during condensation. The numerical analysis are conducted to investigate the effects of waveform channel, porous layer and its thicknesses, subcooling of cooling wall and inlet air mass fractions on flow and condensation under different inlet velocities, in which simulations agree with the experimental data. The results show that the condensation heat transfer is greatly influenced by the vortex distribution and air accumulation. At inflowing velocity of 1 m/s, the waveform with amplitude of 1 mm and wavelength of 18 mm can achieve the highest performance evaluation factor, φ (about 67.5 % enhancement than the smooth tubes). The ratio of waveform amplitude to wavelength needs to be chosen properly in consideration of convection between solid matrix and steam–air mixture in porous layer, and non-condensation gas distribution near cooling wall of condensation tube.

Analyzing the pressure drop in spacer stacks for electrodialytic processes considering the effect of mechanical compression: Experimental vs. free and porous flow model prediction

There is a growing interest in using electromembrane processes in sustainable applications. However, it involves designing efficient cells with low pressure drops, which is particularly important in processes requiring high energy efficiency. In order to analyze the fluid dynamics in spacer-filled channels, manifolds and stacks used primarily in acid base flow batteries a free and porous flow model was developed. The predictive model composed by Navier-Stokes equations and Brinkman Forchheimer extended Darcy model closely follows the experimental pressure drop in a single spacer-filled channel. According to the model results, the entrance and exit zones of the spacer produce the highest pressure drop per axial longitude, highlighting the strong effect of the design of spacer in these zones. In the case of stacks the mechanical compression among spacers as revealed by measurement with compression film suggests some spacer deformation on entrance and exit zones that intensified the inlet and outlet effects.

Three-dimensional transient CFD modeling of multiple finned aluminum foam heat sinks in a horizontal channel

Finned metal foam heat sinks are well-known because of their excellent performance in cooling of powered electronics. In this study, three-dimensional transient numerical simulations of finned aluminum foam heat sinks in a forced convection of air were carried out using commercial COMSOL. The geometry under consideration consists of an array of finned aluminum foam heat sinks mounted on heater blocks and placed on a plate in a horizontal channel. Heat sink aluminum foam regions were considered as porous media with a local non-equilibrium thermal model to evaluate thermal characteristics, while the Forchheimer-Brinkman extended Darcy model is considered for the flow analysis. Our main concern in the present study is to evaluate the transient thermal–hydraulic behavior and the cooling performance under constant flux heat sources while varying the Reynolds number and variable morphological parameters of the aluminum foam, i.e., porosity (ε) varied from 0.85 to 0.95. The thermal performance ratio and the average Nusselt number of the finned aluminum foam heat sinks are 23.14% and 30%, respectively, larger than the finned aluminum heat sinks. As the Reynolds number increases, the thermal characteristics are enhanced, and the pressure drop is increased. An increase in porosity causes a reduction in heat transfer rate and an elevation of pressure drop.

Mathematical modeling of unsteady convective flow analysis of water and nano-encapsulated phase change particles in composite enclosure subject to rotation

Free convection heat transfer in water and hybrid particles in an enclosure is studied. A layer of porous material with various thickness and permeability is attached to the enclosure. Both fluid and porous layers rotate with a specific speed. Hybrid particles are formed by a polyurethane (PU) and an N-nonadecane (ND). PU acts as shell and ND goes through a phase change and it capables of preserving and releasing a considerable amount of latent heat. The enclosure movement can be controlled through a rotation speed. The Brinkman-Forchheimer extended Darcy model is used for governing the flow within a fully saturated porous layer. The governing equations for the convective flow and phase change heat transfer, including specific heat capacity and permeability, were introduced and solved by FEM. The impact of the porous thickness, permeability, fusion temperature and rotation speed on the heat capacity ratio, streamline and isotherms structures were investigated. The results showed that a high Darcy number and thin porous thickness were a good candidate for the improvement of average heat transfer rate since reinforcing the melting process and heat transmitting.

Heat Transfer and Performance Enhancement of Porous Split Elliptical Fins

To augment the heat transfer phenomenon, the infusion of various fin geometry over the heated plate is being investigated by various researchers. Solid fins of the porous medium can enhance the convective heat transfer, providing a higher surface area-to-volume ratio for heat transfer. In this work, the fluid flow pattern and thermodynamic analysis of porous-based split elliptical fins mounted staggered over a heated base plate is numerically studied with the Reynolds numbers in the range of 783 to 1839, which is dependent on the fin dimension. The variable parameters were dimensionless transverse offset ( TO ∗ = transverse offset / diameter ), which varied from 0 to 0.5; dimensionless longitudinal offset ( L O ∗ = longitudinal offset / diameter ), which varied from 0 to 0.25; porosity (ɸ), which varies from 0.8 to 0.92; pores per inch (PPI), which was 10; permeability ( P n ); and inertial parameter ( F ). To count the viscous and inertial effect inside the porous zone, the Forchheimer–Brinkman extended Darcy model was adopted. The associated parameters, the Nusselt number (Nu), frictional coefficient ( C f ), and performance evaluation criteria (PEC), are thoroughly analyzed over the TO ∗ and LO ∗ combination. The results of the investigation revealed that the highest value of Nu and PEC was obtained by TO ∗ = 0.5 and L O ∗ = 0 at ɸ = 0.92 , which were approximately 54% and 79% higher than the solid circular fin at Re = 1839 . Additionally, a response function based on Nu was obtained using the response surface method, and cuckoo optimization was assessed to identify the optimal Nu. The optimal Nu is established at TO ∗ = 0.4141 and L O ∗ = 0 ( ɸ = 0.92 and Re = 1839 ) and was validated with the present investigation with an accuracy of 1.20%.

Exact analytical solution of forced convection through a porous‐saturated duct

AbstractIn the current paper, the effect of a magnetic field on the fully developed forced convective flow and heat transfer is studied. An exact solution is extracted when the flow in the porous medium is governed by the Brinkman–Forchheimer Extended Darcy model. First, the problem formulation is explained to obtain a new system of mathematical formulation. Then, by utilizing the properties which are imposed into the problem, the exact closed‐form analytical solution of the problem is explored. Finally, the main results are illustrated to show the impact of the porous media‐shaped parameter, magnetic parameter, Forchheimer number, and viscosity ratio. It should be mentioned that the asymptotic results achieved in this study were compared with the exact results and it is found that they are in good agreement.

Performance analysis of porous-based taper-shaped fin for enhanced heat transfer rate using cuckoo search algorithm

This paper deals with the thermal and fluid flow behavior of porous and non-porous fins. Initially, a solid heat sink with circular fins is analyzed and validated. Further work is done making circular tapered porous fin of the same volume and height as of uniform circular fin. The geometrical parameters are porosity (Ø = 0.1–0.92), pore diameter ( dp), fiber diameter ( df), radius ratio ( R*), and pores per inch (PPI = 10). Four different cases viz., case I ( R* = 1.2), case II ( R* = 1.5), case III ( R* = 2), and case IV ( R* = 4) of taper porous fins are considered in a computational domain with Reynolds number ( Re) ranging 400–1900 (based upon channel inlet parameter) and air as working fluid. The Forchheimer–Brinkman extended Darcy model is considered to characterize the flow in porous media. The value of Nusselt number ( Nu), friction factor ( f), and thermal performance factor (TPF) is evaluated. Results reveal that, for case I at Ø = 0.92 and PPI = 10, the value of Nu and TPF is approximately 63% and 159% higher than solid circular fin at Re = 1876.11. For case IV, the reduction in pressure drop is approximately 21.15% compared to solid circular fin at Re = 1876.11. For optimum Nu, a cuckoo search optimization is evaluated using the response surface method. Upon examination, the optimum Nu is obtained at the combination of R* = 1.2, Ø = 0.92, and Re = 1876.11, which got validated against the numerical work with 0.15% accuracy.

Determination of Three-Dimensional Brinkman—Forchheimer-Extended Darcy Flow

The aim of this study is to determine a 3D incompressible Brinkman–Forchheimer-extended Darcy fluid flow. Based on global well-posedness and regularity of solutions with a periodic boundary condition, the determining modes for weak and regular solutions is achieved via the generalized Grashof number for a 3D non-autonomous Brinkman–Forchheimer-Darcy fluid flow in porous medium. Furthermore, the asymptotic determination of the complete trajectories inside an attractor via Fourier functionals is shown for a 3D autonomous Brinkman–Forchheimer-extended Darcy model.

Numerical Investigation on Natural Convection in Inclined Channels Partially Filled With Asymmetrically Heated Metal Foam

Abstract A numerical study on natural convection in an inclined channel partially filled with open metal foam is investigated in two-dimensional laminar and incompressible and steady-state condition. The upper wall is heated at uniform heat flux, and the lower wall is adiabatic. The governing equations for the metal foam are written assuming the Brinkman–Forchheimer-extended Darcy model and the local thermal equilibrium (LTE) hypothesis. The inclination angle with respect to the horizontal direction ranges between 0 deg and 90 deg. The metal foam structure is homogenous and isotropic and the thermophysical properties are assumed constant with the temperature. The main aim of the present study is to analyze the effect of inclination angle, pore density, and porosity to improve the knowledge on the thermal behaviors of natural convection in partially filled channel with metal foam. Moreover, this study aims to evaluate the possible improvement with respect to the clean inclined channel. Results are presented in terms of velocity and temperature fields, and both temperature and velocity profiles at different significant sections are shown, to obtain a description of the natural convection inside the inclined channel. Finally, average Nusselt number values are evaluated. The presence of metal foam affects the fluid flow in the channel with different motion behaviors. The heat transfer rate increases with the increase in inclination angle, and the average Nusselt number shows different trends for lower and higher Rayleigh numbers, Ra. A significant enhancement is detected for the higher Ra and it depends on the porosity and pore density.

Effects of different viscous dissipation and Joule heating on electromagnetohydrodynamic flow of power-law fluid in the porous microchannel

Of concern in this paper is a theoretical investigation of Electromagnetohydrodynamic (EMHD) flow transport of a non-Newtonian power-law fluid through a porous microchannel based on the Brinkman-Forchheimer extended Darcy model. The impact of joule heating and different viscous dissipations (the Al-Hadharami term, Darcy term and Forchheimer term) on thermal transport was analysed by considering the EMHD flow through the porous microchannel. The governing nonlinear equations are solved numerically to analyse the characteristics of velocity and temperature distributions. Observing a Nusselt number variation for different parameter values is interesting. It also indicated from the effects that the heat transfer rate performed better for the pseudoplastic fluid than the dilatant fluid. It was also significantly influenced by different viscous dissipation terms. The apprehensible changes in the temperature profiles are found due to the additional viscous dissipation terms and joule heating shown in the channel’s middle layers for all the fluids.

Achievement of wall temperature uniformity by axially graded porous materials

An analysis based on local thermal non-equilibrium was made on thermally developing forced convective flows in a channel filled with axially graded porous materials so as to find a way to control its wall temperature distribution. Both analytical and numerical methods were exploited and compared to confirm the validity of the results. A collection of axially graded metal foams were considered under equal solid volume fraction with local porosity decreasing downstream following a power function of axial distance with an exponent n. The Brinkman-Forchheimer extended Darcy model was employed to investigate the axial development of the velocity field, which reveals a weak secondary flow towards the heated wall of the channel, working favorably in view of wall cooling. The wall temperature variation is found quite sensitive to the axial gradient of the local porosity. Depending on the value of the exponent n controlling the axial gradient of the porosity, the axial variation of the wall temperature changes drastically. By adjusting the exponent n, the wall temperature can stay nearly constant throughout the channel for the case of n = 1.0, or can increase from the inlet for the case of n = 1.5, attain its maximum, and then, decreases towards the exit. The results obtained in this study provides useful information for designing effective cooling systems, where control of not only the desired working temperature but also its uniformity are required such as in EV batteries and PEMFC.

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 Simulation of Solid and Porous Fins’ Impact on Heat Transfer Performance in a Differentially Heated Chamber

The development of different industrial fields, including mechanical and power engineering and electronics, demands the augmentation of heat transfer in engineering devices. Such enhancement can be achieved by adding extended heat transfer surfaces to the heated walls or heat-generating elements. This investigation is devoted to the numerical analysis of natural convective energy transport in a differentially heated chamber with isothermal vertical walls and a fin system mounted on the heated wall. The developed in-house computational code has been comprehensively validated. The Forchheimer–Brinkman extended Darcy model has been employed for the numerical simulation of transport phenomena in a porous material. The partial differential equations written, employing non-primitive variables, have been worked out by the finite difference technique. Analysis has been performed for solid and porous fins with various fin materials, amounts and heights. It has been revealed that porous fins provide a very good technique for the intensification of energy removal from heated surfaces.

Effect of Couple Stress Fluid in an Irregular Couette Flow Channel: An Analytical Approach

The work embodied in this paper presents the combined effects of Soret, chemical reaction, and Dufour on Couette flow in an irregular channel for dusty viscoelastic couple stress fluid. The behavior of the boundary layer is studied with the help of Brinkman-Forchheimer's extended Darcy model as a momentum equation for the unsteady, incompressible dusty viscoelastic fluid. The heat transfers are considered due to the radiation absorption parameter and Dufour effect, the mass transfer influenced by the chemical reaction, and the Soret effect. The boundary conditions of the problem and leading equations of the physical problem are solved by a similarity transformation, and the consequent ordinary linear differential equations are solved by the perturbation method. The obtained results are shown graphically. The computational results show a good agreement between our values and a particular case of the earlier work.

Effect of foam insertion in aneurysm sac on flow structures in parent lumen: relating vortex structures with disturbed shear.

Numerous studies suggest that disturbed shear, causing endothelium dysfunction, can be related to neighboring vortex structures. With this motivation, this study presents a methodology to characterize the vortex structures. Precisely, we use mapping and characterization of vortex structures' changes to relate it with the hemodynamic indicators of disturbed shear. Topological features of vortex core lines (VCLs) are used to quantify the changes in vortex structures. We use the Sujudi-Haimes algorithm to extract the VCLs from the flow simulation results. The idea of relating vortex structures with disturbed shear is demonstrated for cerebral arteries with aneurysms virtually treated by inserting foam in the sac. To get physiologically realistic flow fields, we simulate blood flow in two patient-specific geometries before and after foam insertion, with realistic velocity waveform imposed at the inlet, using the Carreau-Yasuda model to mimic the shear-thinning behavior. With homogenous porous medium assumption, flow through the foam is modeled using the Forchheimer-Brinkman extended Darcy model. Results show that foam insertion increases the number of VCLs in the parent lumen. The average length of VCL increases by 168.9% and 55.6% in both geometries. For both geometries under consideration, results demonstrate that the region with increased disturbed shear lies in the same arterial segment exhibiting an increase in the number of oblique VCLs. Based on the findings, we conjecture that an increase in oblique VCLs is related to increased disturbed shear at the neighboring portion of the arterial wall.

Numerical investigation of transient mixed convection of nanofluid in a cavity with non-Darcy porous inner block and rotating cylinders with harmonic motion

Mixed convection of nanofluid in a 2D square enclosure with a porous block in its center and four rotating cylinders, which are forced by a simple harmonic function, was studied numerically. The porous zone was studied by considering the Forchheimer–Brinkman-extended Darcy model. Effects of various parameters including Darcy number (10–5 ≤ Da ≤ 10–2), porosity (0.2 ≤ ɛ ≤ 0.7), Richardson number (0.1 ≤ Ri ≤ 10), and volume fraction of nanoparticles (0 ≤ ϕ ≤ 0.03), on heat transfer, entropy generation, PEC, velocity, streamline and isotherm contours were demonstrated. The results show that decreasing the Darcy number as well as reducing the Richardson number leads to an increase in the average Nusselt number. However, porosity changes had no decisive effect on heat transfer. Maximize the volume fraction of copper nanoparticles in the base fluid enhanced heat transfer. In the case of the high permeability of the porous medium, the impact of the harmonic rotation of the cylinders on the flow patterns became more pronounced.