Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.     

Thermal protection characteristics of a vertical rectangular cell filled with PCM/air layer

The present paper presents a numerical analysis concerning thermal protection characteristics of a vertical rectangular composite cell filled with a solid-liquid phase change material (PCM) and air layer. Inside the composite cell the PCM layer is separated from air layer by a solid partition of negligible thickness. The buoyancy-induced flows developed in both the air-filled layer and the molten PCM zone inside the PCM layer were modeled as two-dimensional laminar Newtonian fluid flow adhering to the Boussinesq approximation. Meanwhile, two-dimensional conduction heat transfer was accounted for the unmelted solid PCM region. Delineation is made via a parametric simulation of the effects of the pertinent parameters:Ste (Stefan number),Sc (subcooling factor),Ra (Rayleigh number), aspect ratio of composite cell,A, and relative thickness ratioA _p /A _a , on the transient thermal protection performance of the composite cell. Results demonstrate that by means of the latent-heat absorption inside the PCM layer, heat penetration across the composite cell can be greatly retarded over an effective duration until a critical instant until the melting front of PCM reaches the partition wall. Such an effective thermal protection duration is found to be a strong function ofRa, Ste, A _p /A _a , andA. In addition, the results of the transient heat transfer rate penetrating through the composite cell are examined as a function of the pertinent parameters of the problem.     

Thermophoresis particle deposition in a non-Darcy porous medium under the influence of Soret, Dufour effects

Thermophoresis particle deposition in free convection on a vertical plate embedded in a fluid saturated non-Darcy porous medium is studied using similarity solution technique. The effect of Soret and Dufour parameters on concentration distribution, wall thermophoretic deposition velocity, heat transfer and mass transfer is discussed in detail for different values of dispersion parameters (Ra _γ, Ra _ξ) inertial parameter F and Lewis number Le. The result indicates that the Soret effect is more influential in increasing the concentration distribution in both aiding as well as opposing buoyancies. Also, the non-dimensional heat transfer coefficient and non-dimensional mass transfer coefficient changes according to different values of thermophoretic coefficient k.     

Heat transfer enhancement of PCM melting in 2D horizontal elliptical tube using metallic porous matrix

In this study, the melting process of ice as a phase-change material (PCM) saturated with a nickel–steel porous matrix inside a horizontal elliptical tube is investigated. Due to the low thermal conductivity of the PCM, it is motivated to augment the heat transfer performance of the system simultaneously by finding an optimum value of the aspect ratio and impregnating a metallic porous matrix into the base PCM. The lattice Boltzmann method with a double distribution function formulated based on the enthalpy method, is applied at the representative elementary volume scale under the local thermal equilibrium assumption between the PCM and porous matrix in the composite. While reducing or increasing the aspect ratio of the circular tubes leads to the expedited melting, the 90\(^{\circ }\) inclination of each elliptical tube in the case of the pure PCM melting does not affect the melting rate. With the reduction in the porosity, the effective thermal conductivity and melting rate in all tubes promoted. Although the natural convection is fully suppressed due to the significant flow blockage in the porous structure, the melting rates are generally increased in all cases.     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

On cooling behavior of a vertical plate in a phase change material/water composite enclosure under pulsating heat load

This paper presents a numerical and experimental study concerning cooling characteristics of a pulsating heated vertical plate sandwiched between a substrate of phase change material (PCM) and an enclosure of water, forming a composite vertical rectangular enclosure. The vertical plate is assumed to have a uniform pulsating (on/off) volumetric heat source. The PCM considered in the present study is n-Octadecane. In the finite-difference simulation, the two-dimensional buoyancy-driven fluid flows developed in both the water-filled subenclosure and the molten PCM region of the PCM-filled subenclosure were modeled as laminar Newtonian fluid flow adhering to the Boussinesq approximation. Meanwhile, two-dimensional conduction is accounted for the plate heater as well as the solid PCM zone. Numerical results are presented to unveil the cooling behavior of the pulsating heat-generating plate through the PCM substrate and water-filled enclosure. Results of the parametric simulations reveal that the water layer has the better capability of heat dissipation than the PCM substrate. Heat dissipation from the plate through the PCM substrate is mainly via the latent heat absorption as associated with melting phenomenon. Moreover, numerical results obtained are compared with the corresponding experiments.     

Conjugate natural convection in a square porous cavity

 Steady-state conjugate natural convection in a square cavity filled with a porous medium is studied numerically in this paper. The enclosure consists of two horizontal conductive walls of finite thickness and two vertical walls at different uniform temperatures. The focus is on the role of solid-fluid conductivity ratio, k, on the flow and heat transfer characteristics and the average Nusselt number, , over the vertical hot and cold walls of the cavity for a limited set of particular parameters. It was shown that the interface temperature, θ_w, along the top of the solid wall decreases with the increase in the wall conductivity k. Also, the values of decreases with the increase of the values of the parameter k. Comparison with known results from the open literature when the wall thickness of the horizontal solid walls is neglected (non-conjugate problem) is excellent. Received on 4 April 2000     

Lattice Boltzmann investigation for enhancing the thermal conductivity of ice using Al2O3 porous matrix

An enthalpy-based Lattice Boltzmann method (LBM) with double-distribution function (DDF) model is used to investigate numerically the effects of inserting a porous matrix on the heat transfer performance of the phase change material (PCM). Simulations are carried out for melting of ice in saturated Al₂O₃ porous matrix encapsulated in a concentric annulus. The process is considered as a conduction/convection controlled phase change problem at a representative elementary volume (REV) scale. The present results are validated by previous published numerical simulations of melting with and without porous media. In this research paper, the effects of decreasing the porosity on the temperature contours, flow patterns within the melt zone, complete melting time of the PCM and average Nusselt number are discussed qualitatively and quantitatively.     

Magnetic and Gravitational Convection of Air in a Porous Cubic Enclosure with a Coil Inclined Around the Y Axis

The natural convection heat transfer of air in a porous media can be controlled by gradient magnetic field. Thermomagnetic convection of air in a porous cubic enclosure with an electric coil inclined around the $Y$ axis was numerically investigated. The Biot–Savart law was used to calculate the magnetic field. The governing equations in primitive variables were discretized by the finite-volume method and solved by the SIMPLE algorithm. The flow and temperature fields for the air natural convection were presented and the mean Nusselt number on the hot wall was calculated and compared. The results show that both the magnetic force and coil inclination have significant effect on the flow field and heat transfer in a porous cubic enclosure, the natural convection heat transfer of air can be enhanced or controlled by applying gradient magnetic field.     

Thermal Non-equilibrium Natural Convection in a Square Enclosure with Heat-Generating Porous Layer on Inner Walls

Differentially heated enclosure with heat-generating porous layer on inner walls is studied computationally for non-Darcy flow and thermal non-equilibrium models. In this study, this problem is investigated for different internal and external Rayleigh numbers, Darcy numbers, porosity-scaled thermal conductivity ratio, solid-/fluid-scaled heat transfer coefficient and dimensionless thickness of the porous layer. The results indicate that the dimensionless thickness of the porous layer has an important effect on the heat transfer in the enclosure. It was found that the thermal non-equilibrium model is needed for small values of the porosity-scaled thermal conductivity ratio and the solid-/fluid-scaled heat transfer coefficient. It is shown that the convection of heat due to internal heat generation is increased in the enclosure when the ratio of internal Rayleigh number to external Rayleigh number is larger.     

Mixed convection flow in a lid-driven inclined square enclosure filled with a nanofluid

This work is focused on the numerical modeling of steady laminar mixed convection flow in a lid-driven inclined square enclosure filled with water–Al₂O₃ nanofluid. The left and right walls of the enclosure are kept insulated while the bottom and top walls are maintained at constant temperatures with the top surface being the hot wall and moving at a constant speed. The developed equations are given in terms of the stream function–vorticity formulation and are non-dimensionalized and then solved numerically subject to appropriate boundary conditions by a second-order accurate finite-volume method. Comparisons with previously published work are performed and found to be in good agreement. A parametric study is conducted and a set of graphical results is presented and discussed to illustrate the effects of the presence of nanoparticles and enclosure inclination angle on the flow and heat transfer characteristics. It is found that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by inclination of the enclosure at moderate and large Richardson numbers.     

Numerical simulation of natural convection of latent heat phase-change-material microcapsulate slurry packed in a horizontal rectangular enclosure heated from below and cooled from above

A two-dimensional numerical simulation of natural convection in a rectangular enclosure heated from below and cooled from above has been conducted with non-Newtonian phase-change-material (PCM) microcapsulate slurry with latent heat capacities. The formulation of the mathematical model in dimensionless co-ordinates and discretization of the governing equations have been done using the finite volume method. Both natural convection and heat transfer characteristics are discussed about natural convection with PCM microcapsulate slurry, which exhibits the pseudoplastic non-Newtonian fluid behavior and a peak value in the specific heat capacity with latent heat. The viscosity of the present PCM microcapsulate slurry is assumed to follow the Ostwald-de Waele power law fluid model with the power-law index n and the consistency coefficient K. The effects of phase-change material, the mass concentration, and the aspect ratio Ar on the natural convection heat transfer are described, respectively. By comparing with the results of microcapsule slurry without phase change, the enhancement in heat transfer is found in microcapsule slurry with phase change during the phase change temperature range. Numerical simulations are performed in the following parametric ranges: the width–height aspect ratio of the enclosure Ar from 2 to 20, the mass concentrations C _m of the slurry from 10 to 40%, power law index n of the slurry from 0.89 to 1.0 and Rayleigh numbers Ra ranges from 10³ to 10⁷.     

Natural convection due to heat and mass transfer in a composite system

A numerical study is performed to analyse heat and mass transfer phenomena due to natural convection in a composite cavity containing a fluid layer overlying a porous layer saturated with the same fluid. The flow in the porous region is modelled using Brinkman–Forchheimer-extended Darcy model that includes both the effect of macroscopic shear (Brinkman effect) and flow inertia (Forchheimer effect). The vertical walls of the two-dimensional enclosure are isothermal whilst the horizontal walls are adiabatic. The two regions are coupled by equating the velocity and stress components at the interface. The resulting coupled equations in non-dimensional form are solved by an alternating direction implicit method by transforming them into parabolic form by the addition of false transient terms. The numerical results show that the amount of fluid penetration into the porous layer depends strongly upon the Darcy, thermal and solutal Rayleigh numbers. Average Nusselt number decreases while average Sherwood number increases with an increase of the Lewis number. The transfer of heat and mass on the heated wall near the interface depends strongly on the Darcy number. Received on 11 May 1998     

Conjugate conduction-natural convection in an enclosure with time-periodic sidewall temperature and inclination

The unsteady conjugate conduction-natural convection in enclosure is of great theoretical significance and is widely encountered in engineering applications in the areas of fluid dynamics and heat transfer. However, there are relatively few efforts to investigate the unsteady flow physics and heat transfer characteristics in the inclined enclosure of finite thickness walls. In the present work, this problem is numerically investigated by a high accuracy multidomain temporal-spatial pseudospectral method. The enclosure is filled with Boussinesq fluid and is bounded by four finite thickness and conductive walls; one of the vertical sidewall is exposed to time-periodic temperature environment while the opposite sidewall holds constant temperature; the top and bottom walls are assumed to be adiabatic. Particular efforts are focused on the effects of three types of influential factors: the wall thermophysical properties, the time-periodic temperature patterns and the inclination, and the time-periodic flow patterns and heat transfer characteristics are presented. Numerical results reveal that within the present parameter range, the heat transfer rate increases almost linearly with the thermal conductivity ratio and thermal diffusivity ratio but decreases with the inclination angle. Moreover, the heat transfer could be enhanced or weakened by selecting different temperature pulsating period in the case of finite thickness wall, while it is always enhanced if the walls are zero thickness. The back heat transfer and heat transfer resonance phenomena are observed, and their relationships with the time-periodic flow patterns and temperature distributions are analyzed. The findings are helpful to the understandings of the fluid flow and heat transfer mechanisms in the related enclosure configurations, and may be of engineering use in thermal design improvement.     

Natural convection in an inclined rectangular porous cavity with uniform heat flux from the side

The problem of natural convection in an inclined rectangular porous layer enclosure is studied numerically. The enclosure is heated from one side and cooled from the other by a constant heat flux while the two other walls are insulated. The effect of aspect ratio, inclination angle and Rayleigh number on heat transfer is studied. It is found that the enclosure orientation has a considerable effect on the heat transfer. The negative orientation sharply inhibits the convection and consequently the heat transfer and a positive orientation maximizes the energy transfer. The maximum temperature within the porous medium can be considerably higher than that induced by pure conduction when the cavity is negatively oriented. The peak of the average Nusselt number depends on the Rayleigh number and the aspect ratio. The heat transfer between the two thermally active boundaries is sensitive to the effect of aspect ratio. For an enclosure at high or low aspect ratio, the convection is considerably decreased and the heat transfer depends mainly on conduction.     

Conduction and convection heat transfer in composite solar collector systems with porous absorber

Steady natural convection and conduction heat transfer has been studied in composite solar collector systems. The system consists of a glazing, a porous layer and a massive wall installed in a room. The heat transfer in this system is studied by assuming the glazing and the vertical bounding wall isothermal at different temperatures, two horizontal bounding walls adiabatic and porous layer without vents. The aspect ratioA was from 0.1 to 1.0 but the detailed study was carried out withA=1. The thickness of the porous wallF _p varied from 1/3 to 1, while the solid wall thickness was kept constant. The conductivity ratio of porous layer was from 10^–2 to 10²,Ra from 10³ to 10⁷. The results are presented in terms of thermal parameters as function ofRa and non-dimensional geometrical parameters. The isotherms and stream lines within the system are produced.

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Wärmetransport infolge Leitung und Konvektion durch ein zusammengesetztes Solarkollektor-System mit porösem Absorber

Zusammenfassung Die Untersuchung bezieht sich auf den stationären Wärmetransport infolge natürlicher Konvektion und Leitung in zusammengesetzten Solarkollektor-Systemen. Ein solches System besteht aus einer Verglasung, einer porösen Schicht und der massiven Außenwand eines Raumes. Der Wärmetransport in diesem System wird unter der Annahme ermittelt, daß sich die Verglasung und die Hinterwand auf verschiedenen konstant Temperaturen befinden, Boden und Decke des Raumes adiabat sind und daß die poröse Schicht luftundurchlässig ist. Das VerhältnisA (Höhe zu Tiefe des Raumes) variierte von 0,1 bis 1,0, wobei die eingehenderen Untersuchungen fürA=1,0 erfolgten. Die bezogene DichteF _p der porösen Schicht bewegte sich von 1/3 bis 1, die Dicke der Außenwand blieb konstant. Das Wärmeleitfähigkeitsverhältnis bezüglich der porösen Schicht lag im Bereich 10^–2 bis 10² und die Rayleigh-ZahlRa reichte von 10³ bis 10⁷. In der Ergebnisdokumentation sind die thermischen Parameter als Funktionen vonRa und den Geometrieverhältnissen wiedergegeben. Stromlinien und Isothermen im Inneren des Systems wurden generiert und dargestellt.

     

Fully Developed Forced Convection in a Parallel Plate Channel with a Centered Porous Layer

In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu _cl = 4.12 and fRe _cl = 24) are used to define heat transfer increment ratio (e_th = Nu_p/Nu_cl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (e_p = fRe_p/fRe_cl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer and pressure drop increment ratios are used to define an overall performance (e = e_th/e_p)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k _r) when k _r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls.     

Experimental and numerical study on laminar natural convection in a cavity heated from bottom due to an inclined fin

Natural convection heat transfer in an inclined fin attached square enclosure is studied both experimentally and numerically. Bottom wall of enclosure has higher temperature than that of top wall while vertical walls are adiabatic. Inclined fin has also adiabatic boundary conditions. Numerical solutions have been done by writing a computer code in Fortran platform and results are compared with Fluent commercial code and experimental method. Governing parameters are Rayleigh numbers (8.10⁵ ≤ Ra ≤ 4 × 10⁶) and inclination angle (30° ≤ and ≤ 120°). The temperature measurements are done by using thermocouples distributed uniformly at the wall of the enclosure. Remarkably good agreement is obtained between the predicted results and experimental data. A correlation is also developed including all effective parameters on heat transfer and fluid flow. It was observed that heat transfer can be controlled by attaching an inclined fin onto wall.     

Natural convection heat transfer in enclosures with microemulsion phase change material slurry

This paper has dealt with the natural convection heat transfer characteristics of microemulsion slurry composed of water, fine particles of phase change material (PCM) in rectangular enclosures. The microemulsion slurry exhibited non-Newtonian pseudoplastic fluid behavior, and the phase changing process can show dramatically variations in both thermophysical and rheological properties with temperature. The experiments have been carried out separately in three subdivided regions in which the state of PCM in microemulsion is in only solid phase, two phases (coexistence of solid and liquid phases) or only liquid phase. The complicated heat transfer characteristics of natural convection have appeared in the phase changing region. The phase change phenomenon of the PCM enhanced the heat transfer in natural convection, and the Nusselt number was generalized by introducing a modified Stefan number. However, the Nusselt number did not show a linear output with the height of the enclosure, since a top conduction lid or stagnant layer was induced over a certain height of the enclosure. The Nusselt number increased with a decrease in aspect ratio (width/height of the rectangular enclosure) even including the side-wall effect. However, the microemulsion was more viscous while the PCM was in the solid phase, the side-wall effect on heat transfer was greater for the PCM in the solid region than that for the PCM in the liquid region. The correlation generalized for the PCM in a single phase is $ Nu = 1/3(1 - C_1 )Ra^{{1 \over {3.5n + 1}}} , $ where C ₁ = e ^–0.09AR for the PCM in solid phase and C ₁ = e ^–0.33AR for the PCM in liquid phase. For the PCM in the phase changing region, the correlation can be expressed as $ Nu = CRa^{{1 \over {7n + 2}}} Ste^{ - (1.9 - 1.65n)} , $ where C = 1.22 – 0.035AR for AR > 10 and C = 0.55 – 16.4e ^–1.1AR for AR < 10. The enclosure height used in the present experiments was varied from H = 5.5 [mm] to 30.4 [mm] at the fixed width W = 120 [mm] and depth D = 120 [mm]. The experiments were done in the range of modified Rayleigh number 7.0 × 10² ≤ Ra ≤ 3.0 × 10⁶, while the enclosure aspect ratio AR varied from 3.9 to 21.8.     

Unsteady entrance heat transfer in laminar pipe flows with convection from the ambient

The characteristics of unsteady entrance heat transfer in the combined entrance heat transfer region of laminar pipe flows resulting from time-varying inlet temperature are numerically investigated. Three non-dimensional parameters,Nu ₀, a^*, andf are identified in the study. Also, their effects on the non-dimensional duct wall temperature, fluid bulk temperature, and duct wall heat flux are discussed in great detail. Comparisons are made with the zero thermal capacity wall solution.