The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically for the case of weak heterogeneity. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. For the case of conducting impermeable top and bottom boundaries and a square box, the effects of permeability heterogeneity and conductivity heterogeneity each cause a reduction in the critical value of Ra, while for the case of a tall box there can be either a reduction or an increase.     

The effect of combined vertical and horizontal heterogeneity on the onset of convection in a bidisperse porous medium

The effects of both horizontal and vertical hydrodynamic and thermal heterogeneity, on the onset of convection in a horizontal layer of a saturated bidisperse porous medium uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. The thermal heterogeneity of the p-phase can be quite significant when the thermal diffusivity of that phase is large relative to that of the f-phase.     

The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium: Moderate heterogeneity

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are now studied analytically for the case of moderate heterogeneity (rather than the weak heterogeneity previously studied), for the case of a square box where the properties vary in a piecewise constant or linear fashion, with conducting impermeable top and bottom boundaries and insulating impermeable sidewalls. In order to allow for the moderate heterogeneity the order of the Galerkin expansion employed has been increased, and the expansion of a determinant of high order has been avoided by the use of a least squares methodology to find the critical value of the Rayleigh number Ra. It is found that the effects of permeability heterogeneity and conductivity heterogeneity each cause a reduction in the critical value of Ra in all cases, and the effects of horizontal and vertical heterogeneity are still approximately additive.     

The onset of convection in a bidisperse porous medium

The classical Rayleigh–Bénard theory, for the onset of convection in a horizontal layer uniformly heated from below, has been applied to a bidisperse porous medium. The linear stability analysis leads to an expression for the critical Rayleigh number as a function of a Darcy number, two volume fractions, a permeability ratio, a thermal capacity ratio, a thermal conductivity ratio, an inter-phase heat transfer parameter and an inter-phase momentum transfer parameter.     

A numerical study of laminar natural convective heat transfer around a horizontal cylinder inside a concentric air-filled triangular enclosure

A numerical investigation of steady-state laminar natural convective heat transfer around a horizontal cylinder to its concentric triangular enclosure was carried out. The enclosure was filled with air and both the inner and outer cylinders were maintained at uniform temperatures. The buoyancy effect was modeled by applying the Boussinesq approximation of density to the momentum equation and the governing equations were iteratively solved using the control volume approach. The effects of the Rayleigh number and the aspect ratio were examined. Flow and thermal fields were exhibited by means of streamlines and isotherms, respectively. Variations of the maximum value of the dimensionless stream function and the local and average Nusselt numbers were also presented. The average Nusselt number was correlated to the Rayleigh number based on curve-fitting for each aspect ratio. At the highest Rayleigh number studied, the effects of different inclination angles of the enclosure and various cross-section geometries of the inner cylinder were investigated. The computed results indicated that at constant aspect ratio, both the inclination angle and cross-section geometry have insignificant effects on the overall heat transfer rates though the flow patterns are significantly modified.     

Effect of nanofluid variable properties on natural convection in enclosures

In this work, the heat transfer enhancement in a differentially heated enclosure using variable thermal conductivity and variable viscosity of Al₂O₃–water and CuO–water nanofluids is investigated. The results are presented over a wide range of Rayleigh numbers (Ra = 10³–10⁵), volume fractions of nanoparticles (0 ≤ φ ≤ 9%), and aspect ratios (½ ≤ A ≤ 2). For an enclosure with unity aspect ratio, the average Nusselt number of a Al₂O₃–water nanofluid at high Rayleigh numbers was reduced by increasing the volume fraction of nanoparticles above 5%. However, at low Rayleigh numbers, the average Nusselt number was slightly enhanced by increasing the volume fraction of nanoparticles. At high Rayleigh numbers, CuO–water nanofluids manifest a continuous decrease in Nusselt number as the volume fraction of nanoparticles is increased. However, the Nusselt number was not sensitive to the volume fraction at low Rayleigh numbers. The Nusselt number demonstrates to be sensitive to the aspect ratio. It was observed that enclosures, having high aspect ratios, experience more deterioration in the average Nusselt number when compared to enclosures having low aspect ratios. The variable thermal conductivity and variable viscosity models were compared to both the Maxwell-Garnett model and the Brinkman model. It was found that at high Rayleigh numbers the average Nusselt number was more sensitive to the viscosity models than to the thermal conductivity models.     

Conjugate natural convection in a fluid-saturated porous enclosure with two solid vertical partitions

Conjugate natural convection in a fluid-saturated square porous enclosure with two solid vertical partitions of finite and equal thickness equispaced from center of enclosure is investigated in this paper. The primary objective is to attenuate the Nusselt number (Nu) and hence the heat transfer rate across a differentially heated enclosure. Darcy's model is considered. Numerical computation is performed using successive accelerated replacement and explicit scheme. Partition ratio, partition length, thermal conductivity ratio, and modified Rayleigh number are the parameters under study. Fluid flow is analyzed by observing transient changes of streamlines and isotherms for partition length 0.3-1, thermal conductivity ratio 0.5-2, partition ratio 0.1-0.3 and modified Rayleigh number 100 and 1000 where partition ratio is the ratio of distance between center of enclosure and either of the partition center to the total length of the enclosure; while Nusselt number is calculated to estimate the heat transfer rate for each configuration. It is found that, employing a solid partition within the enclosure most definitely reduces the Nusselt number. The drop in Nusselt number is more for partition length 0-0.6 after which it does show a drop in Nu but only very subtle. Further, Nu is the least for partition ratio 0.2. Also, Nusselt number is proportional to thermal conductivity ratio which is the ratio of thermal conductivity of solid to porous medium.     

Thermal and Moisture Transport Inhibitions in a Moist Air Saturated Enclosure Attached with Protruding Partitions for Built Energy Conservations

ABSTRACT

Combined heat and moisture transportation in an enclosure has been numerically investigated, which could benefit the sustainable building energy conservations and electronic cooling designs. An adiabatic and impermeable partition of finite thickness is considered, placed in the enclosure following an ordered arrangement. Effects of length and location of the partition, buoyancy ratio and thermal Rayleigh number on convective heat and moisture transfer rates in the enclosure are discussed. Firstly, this situation of the partition placed in the horizontal wall is studied, where inhibition effect of partition is observed. It is seen that the location of partition put relatively weaker influences on the heat and mass transfer in the regime of thermal-driven flow, when its length exceeds the critical value. Additionally, inhibition effect is more pronounced as the partition is fixed in center of vertical wall. Furthermore, local heat and mass transfer rates could be suppressed when the buoyancy ratio becomes negative. Finally, thermal Rayleigh number greatly affects the transport structures of fluid, heat and moisture, whatever aiding flow or opposing flow situations. Heat and mass transfer potentials could be promoted with increasing thermal Rayleigh numbers. Present work could be adopted to optimize the enclosure flows simultaneously with heat and moisture transport.     

Magnetohydrodynamic Natural Convection of Nanofluids in U-shaped Enclosures

This article presents the results of a numerical study of laminar natural convection in a U-shaped enclosure that is filled with a water-Cu nanofluid and is under the influence of a horizontal magnetic field. A computational domain was defined and a numerical scheme based on the control volume formulation using the SIMPLE algorithm was developed. The convection-diffusion terms were discretized using a power-law scheme. The effects of the Rayleigh number, the solid volume fraction, the Hartmann number, and the enclosure aspect ratio on the heat transfer performance of the enclosure were examined. The thermal performance of the enclosure was found to be a function of the enclosure aspect ratio. The results also showed that the heat transfer rate increased with an increase of the Rayleigh number and the solid volume fraction, but it decreased with an increase of the Hartmann number.     

Effect of anisotropic permeability on double‐diffusive bidisperse porous medium

The model of thermosolutal convection in a fluid‐saturated bidisperse porous medium of Darcy type is studied in this paper. The permeability is allowed to be horizontally isotropic for both the macro‐ and microphases. The linear instability and nonlinear stability are analyzed by taking the Soret effect into account. Furthermore, the effect of anisotropy parameter, Soret coefficient, and other physical parameters on the stability of the system are investigated. It is shown that the linear instability boundaries and the energy stability boundaries do not coincide when the layer is heated and salted from below, where a region of potential subcritical instability occurs. The results reveal that the horizontal to vertical permeability ratio plays a crucial role in the stability of the system. It is also observed that for large values of the salt Rayleigh number, the onset of thermal convection is more likely to be via oscillatory convection rather than stationary convection. Furthermore, the onset of stationary convection is significantly influenced by the presence of the Soret coefficient.     

Effects of Wall-Located Heat Barrier on Conjugate Conduction/Natural-Convection Heat Transfer and Fluid Flow in Enclosures

The effects of a heat barrier, located in the ceiling wall of an enclosure, on conjugate conduction/natural convection are investigated numerically. The vertical walls of the enclosure are differentially heated and the horizontal walls are adiabatic. Heatline technique is used to visualize heat transport. The variations of average Nusselt number, dimensionless heat transfer rate through the ceiling wall, and dimensionless overall heat transfer rate are studied. Calculations are performed for different Rayleigh numbers (10³ ≤ Ra ≤ 10⁶), thermal conductivity ratios (1 ≤ K ≤ 100), dimensionless locations of the heat barrier (0 < X _h  < 1),and two dimensionless ceiling wall thicknesses (D = 0.05 and D = 0.20). For high thermal conductivity ratio (K = 100), the heat barrier considerably reduces the dimensionless overall heat transfer rate. The effect of the heat barrier on dimensionless heat transfer rate through the enclosure increases as the Rayleigh number decreases. For low Rayleigh number (i.e., Ra = 10³), a location exists in the ceiling wall for which the dimensionless overall heat transfer rate is minimum.     

Natural convection in an enclosure with localized heating and salting from below

Two-dimensional, double diffusion, natural convection in a rectangular enclosure filled with binary fluid saturating porous media is investigated numerically. Multiple motions are driven by the external temperature and concentration differences imposed across horizontal walls with the simultaneous presence of discrete heat and contaminant sources. The general Brinkman-extended Darcy model is adopted to formulate the fluid flow in the cavity. The fluid, heat and moisture transport through the isotropic porous layer are analyzed using the streamlines, heatlines and masslines, and the heat and mass transfer potentials are also explained by the variations of overall Nusselt and Sherwood numbers. The numerical simulations presented here span a wide range of the main parameters (thermal Rayleigh numbers, strip pitches and Darcy number) in the domain of destabilizing solutal buoyancy forces. It is shown that the heat and mass transfer potential can be promoted or inhibited, depending strongly on the permeability of porous medium, the strip pitch, the thermal and solutal Rayleigh numbers.     

Three-Dimensional Numerical Study of Natural Convective Heat Transfer of Liquid in a Cubic Enclosure

ABSTRACT

Steady-state laminar natural convection in a cubic enclosure with a cold vertical wall and two hot square heaters with constant temperature on the opposite wall is studied numerically. The enclosure is filled with various liquids. Three-dimensional Navier–Stokes Equations are solved by employing the SIMPLE algorithm. Computations are performed for a range of Rayleigh number from 10³ to 10⁷ while enclosure aspect ratio varies from 0.05 to 1.6. The effects of Rayleigh number, enclosure aspect ratio, and Prandtl number on heat transfer characteristics are studied in detail. The results show that the flow field is very complex and heat transfer from the two heaters is not the same. The effects of Prandtl number are negligible in the range from 5 to 140 with other parameters kept constant. This allows the use of liquids such as water for studying other dielectric liquids, provided the flow geometry and other nondimensional parameters are similar. The overall Nusselt number increases markedly with Rayleigh number. It is also affected by enclosure aspect ratio. It attains the maximum value when aspect ratio is in the range of 0.1–0.2 and decreases as enclosure aspect ratio varies from 0.2 to 1.6. Also, various settings of cooling face and arrangement of heaters are investigated, and the results show that they have considerable effects on heat transfer of both heaters.     

Critical Rayleigh number of natural convection in high porosity anisotropic horizontal porous layers

The critical Rayleigh number Ra_C at the onset of natural convection was studied by linear stability analysis for high porosity anisotropic horizontal porous layers of uniformly arraying vertical thin circular wires stretched across a hot and a cold surface. Navier–Stokes equations including flow resistance by the wires were solved since Darcy’s law cannot be applied due to high porosity. Ratio of permeability in the horizontal direction to the vertical direction is constant to be 0.5 but ratio of effective thermal diffusivity ξ is changed dependent on wire materials. The critical Rayleigh number Ra_C for the case of ξ = 1 obtained by the analysis agreed well with the experiment.     

Linear and nonlinear stability of Soret‐Dufour Lapwood convection near double codimension‐2 points

In this study, the Dufour and Soret effects on natural double‐diffusion convection in a horizontal porous layer was studied numerically using FORTRAN 90 programming and analytically near various convection onset thresholds. The porous layer was subject to a uniform heat and mass fluxes on the horizontal walls while the vertical walls were impermeable and adiabatic. The Darcy model along with the Boussinesq approximation was assumed in the problem formulation. The governing parameters of the problem are the thermal Rayleigh number, R_T, the buoyancy ratio, N, the Lewis number, Le, the aspect ratio of the cavity, A, and the Dufour, D_u, and Soret, S_r, numbers. For a shallow enclosure, the analytical solution was derived assuming zero convection wave number, which is valid near and above criticality. The onset of subcritical, supercritical and oscillatory convection was investigated. Two linear and nonlinear codimension‐2 points were found to exist. Whether the system was subject to constant fluxes and heat and solute, regardless of the aspect ratio of the layer, the subcritical convection behavior remained the same with similarity in the thresholds expressions for subcritical bifurcation.     

The onset of convection in a tridisperse porous medium

This paper develops a theory of mass, momentum, and heat transfer in a tridisperse porous medium. Coupling between three different scales present in this medium is accounted for by introducing momentum and interphase heat transfer coupling coefficients. The developed theory is then applied to solve the classical Rayleigh–Bénard problem, for the onset of convection in a horizontal layer uniformly heated from below, for this new type of a porous medium. The formulation uses the Darcy law, which now results in three different filtration velocities in three porosity scales present in this medium. The linear stability analysis leads to an expression for the critical Rayleigh number as a function of three volume fractions, two permeability ratios, two thermal capacity ratios, two thermal conductivity ratios, two inter-phase heat transfer parameters and two inter-phase momentum transfer parameters. The dependence of the critical Rayleigh number on these parameters is investigated.     

Entropy generation due to conjugate natural convection in enclosures bounded by vertical solid walls with different thicknesses

Entropy generation due to conjugate natural convection heat transfer and fluid flow has been studied inside an enclosure with bounded by two solid massive walls from vertical sides at different thicknesses. Enclosure is differentially heated from vertical walls and horizontal walls are adiabatic. Governing equations which are written in streamfunction-vorticity form solved by finite difference technique for the governing parameters as Rayleigh number, 10³ ≤ Ra ≤ 10⁶, length ratio of solid walls as ₁ (for left vertical wall) and ₂ (for right vertical wall) and thermal conductivity ratio of solid to fluid (k), 1 ≤ k ≤ 10. Entropy generation contours due to fluid friction and heat transfer irreversibility, isotherms, streamlines, Nusselt numbers and velocity profiles were obtained. It is found that entropy generation increases with increasing of thermal conductivity ratio and thicknesses of the walls. Entropy generation due to heat transfer is more significant than that of fluid flow irreversibility for all values of thickness of the solid vertical walls.     

Transient Natural Convection in Vertical Annuli: Numerical Modeling and Heat Transfer Correlation

A finite volume model is developed for analyzing transient natural convection in vertical annuli, which are assumed to be isothermally heated (or cooled) from the inner surface and insulated from horizontal and outer surfaces. Based on the numerical results, the dependence of transient heat transfer on governing dimensionless parameters is investigated. It is found that the dimensionless accumulated heat transfer as a function of dimensionless time can be satisfactorily approximated by a simple correlation, which explicitly includes the effects of the Rayleigh and Prandtl numbers as well as the enclosure aspect ratio and height-inner radius ratio.     

Effect of anisotropy on the development of convective boundary layer flow in porous media

The effects of anisotropy on the development of thermal boundary layer flow in a rectangular porous cavity is studied. The side walls of the cavity are respectively heated and cooled isothermally. Top and bottom walls are insulated. The porous medium is anisotropic both in permeability and thermal conductivity with its principal axes oriented in a direction that is oblique to the gravity vector. Scale analysis is applied to predict the orders of magnitude involved in the boundary layer regime. In the large Rayleigh number limit, the governing boundary layer equations are solved in closed form, using an intergral approach. A finite difference method is used to obtain numerical solutions of the full governing equations. The effects of the anisotropy in permeability and thermal conductivity on the development of free convective boundary layer flow are found to be significant.     

Brownian motion of nanoparticles in a triangular enclosure with natural convection

This paper presents the results of a numerical study on the natural convection in a right triangular enclosure, with a heat source on its vertical wall and filled with a water–CuO nanofluid. The effects of parameters such as Rayleigh number, solid volume fraction, heat source location, enclosure aspect ratio and Brownian motion on the flow and temperature fields as well as the heat transfer rate, are examined. The results show that when Brownian motion is considered in the analysis, the solid volume fraction, the heat source location and the enclosure aspect ratio affect the heat transfer performance differently at low and high Rayleigh numbers. At high Rayleigh numbers, an optimum value for the solid volume fraction is found which results in the maximum heat transfer rate. This is in contradiction to the results of the analysis in which Brownian motion is neglected.