NUMERICAL ANALYSIS OF NATURAL CONVECTION IN CAVITIES WITH VARIABLE POROSITY

This work analyzes the effects of variable porosity on the heat transfer by natural convection, in a cavity with isothermal vertical walls and adiabatic horizontal ones and a porous medium inside. The hydrodynamic field in the porous medium is modelled according to the general model obtained by Brinkman's and Forchheimer's terms. An exponential variation of the porosity near the walls was considered. The equations in terms of the real variables were numerically solved by the finite-volume method, with a staggered variables arrangement. The pressure-velocity coupling was treated with the PRIME algorithm. The simulations involved both Darcian (Da = 10 -7 ) and non-Darcian flows (Da = 10 -6 ). In each case, the modified Rayleigh range (Ra m = Ra 2 Da) was from 10 to 1,000, and from 100 to 5,000, respectively. For the numerical simulation we considered Pr = 1, Rk = 1, A = 1, and k X = 0.36. The results are shown through streamlines and isotherms and velocity and temperatures profiles. This shows that the generalized method with variable porosity means an increase in the average Nusselt number. For physical validation, some experimental sets were simulated in which glass was the porous medium, and water, alcohol, and transformer oil were used as the fluids. The simulated results indicate that the adopted model reduces the discrepancy in the experimental results obtained with water and alcohol. We proposed a correlation to evaluate the average Nusselt number as a function of six parameters: Rayleigh number, Prandtl number, the dimensionless particle diameter, thermal conductivity ratio between solid and fluid phase, porosity, and the aspect ratio of the cavity.     

NATURAL CONVECTION FROM A BURIED PIPE WITH EXTERNAL BAFFLES

Numerical solutions are presented for the natural convection heat transfer from a pipe with two baffles attached along its surface buried beneath a semi-infinite, saturated, porous medium. The surface of the medium is assumed to be permeable. The governing equations for Darcy flow are solved using finite differences. The complicated geometry is handled through the use of a body-fitted curvilinear coordinate system. Results are presented for three baffle lengths and a range of burial depths and Rayleigh numbers. The numerical simulations indicate that substantial energy savings can be realized if baffles are used. The results obtained in terms of the Nusselt number for the case of no-baffle are in excellent agreement with analytical and experimental results available in the literature. A simple correlation for Nu has been developed as a function of Ra, pipe burial depth h/R, and baffle length l/R.     

NATURAL CONVECTION IN OPEN CAVITIES AND SLOTS

The present work is concerned with natural convection from open cavities or heated plates attached with parallel vertical strips. The bottom of the cavity is heated, and the vertical walls are assumed adiabatic. Numerical results are presented for steady, laminar natural convection for the geometry described. Effects of Rayleigh numbers from 1 × 10 ³ to 1 × 10 ⁷ , inclination angles from 10^° to 90^°, and aspect ratios of 0.5, 1.0, and 2.0 are investigated for a fixed Prandtl number (0.7). It is found that the average Nusselt number is not very sensitive to the inclination angle. Flow becomes unstable at high Rayleigh numbers and at low inclination angles. Flow pattern and heat transfer results are presented and discussed.     

具有完整隔板的双向倾斜矩形封闭空腔内自然对流数值解

对具有完整隔板的双向倾斜矩形封闭容器中的流动及传给出数值解。讨论了相关比、倾角、隔板位置及Ｒａ数对传热的影响，并给出了相应曲线。所得结果与已有的试验结果符合甚佳，其有关结论对太阳能集热器的设计和改进有一定的指导意义。     

THREE-DIMENSIONAL NATURAL CONVECTION OF AIR IN AN INCLINED CUBIC CAVITY

Three-dimensional steady and transient natural connective flow in a differentially heated inclined cubic cavity of air was numerically simulated by the projection method combined with the power law scheme, Various three-dimensional flow structures were induced, depending on the Rayleigh number and inclined angle, although the main flow is nearly two-dimensional. It is of interest to note that the flow is stronger when a vertical cavity is inclined, implying significant flow acceleration by the buoyancy component normal to the hot and cold walls. Change in the flow structure during the transient causes wavy variation of the space-averaged Nusselt number with time.     

NATURAL CONVECTION AIR COOLING OF HEATED COMPONENTS MOUNTED ON A VERTICAL WALL

The present paper discusses a numerical study of natural-convection air cooling of single and multiple uniformly heated devices. A two-dimensional, conjugate, laminar-flow model is used. The solid and fluid physical properties are assumed constant (not varying with temperature). For the multicomponent cooling, the effects of component thickness, the spacing between components, nonpowered components, and highly powered components are studied to arrive at qualitative suggestions that may improve the overall cooling of a multicomponent system.     

HYDROMAGNETIC NATURAL CONVECTION FROM AN INCLINED POROUS SQUARE ENCLOSURE WITH HEAT GENERATION

The problem of unsteady, laminar, two-dimensional hydromagnetic natural convection heat transfer in an inclined square enclosure filled with a fluid-saturated porous medium in the presence of a transverse magnetic field and fluid heat generation effects is studied numerically. The walls of the enclosure are maintained at constant temperatures. The flow in the porous region is modeled using the Brinkman-extended Darcy's law to account for the no-slip conditions at the walls. The control volume method is used to solve the governing balance equations for different values of the Darcy number, Hartmann number, and the inclination angle. Favorable comparisons with previously published work are performed. These comparisons confirmed the correctness of the numerical results. The obtained numerical results are presented graphically in terms of streamlines and isotherms as well as velocity and temperature profiles at midsections of the cavity to illustrate interesting features of the solution.     

MIXED CONVECTION IN AN INCLINED CHANNEL WITH LOCALIZED HEAT SOURCES

Fully developed opposing mixed convection is numerically studied in an inclined channel that has discrete heating on the bottom and is insulated on the top. The numerical approach is based on the hypothesis that the solution is periodic according to the imposed wavelength of the heating elements. Considering that Ike heat produced by the heating elements is totally carried downstream, the temperature increment from one heating element to the other is defined on the basis of an energy balance. To verify the accuracy of the computational code, an analytical study of the extreme case with an entirely heated wall is investigated. Also, to validate that the solution of the problem is periodic with a wavelength corresponding to the imposed perturbation, a channel with entrance and exit sections containing four to six heating elements is simulated numerically. In the present study, the relative strength of the forced flow and buoyancy effects is examined for a broad range of Rayleigh numbers, Reynolds numbers, and inclination angles. Both overall and local recirculating flows are observed that are caused by buoyancy effects on the forced flow.     

NATURAL CONVECTION IN CUBICAL ENCLOSURES WITH THERMAL SOURCES ON ADJACENT VERTICAL WALLS

Natural convection in a cubical enclosure with a hot source centered on a vertical wall and with an adjacent, fully cooled vertical wall was numerically investigated for Rayleigh numbers ( Ras ) of 10 3 - 4 2 10 6 . At Ra < 10 5 the heat transfer strongly depends on the hot sector side and its dependence on Ra is weak. Opposite characteristics occur at very high Ra , and there is a long transition between these two regimes. This behavior is caused by a mixing pattern, which, at low Ra , is similar to the one found in side heated cavities, whereas at high Rayleigh numbers lateral mixing is dominant.     

NATURAL CONVECTION ABOVE AN ARRAY OF OPEN CAVITIES HEATED FROM BELOW

Abstract

The effect of buoyancy on the flow and heat transfer that develop between a horizontal cold surface and an infinite two-dimensional array of open cavities heated from below is studied numerically. In earlier investigations the steady-state features of this problem were studied for the case of unbounded flow above the cavities. The resulting flow pattern was found to be symmetrical with respect to the centerlines of the cavities. In the present work it is shown that the symmetry of the flow can be destroyed due to the presence of an upper wall. The evolutionary path to steady-state flow is examined, and sustained oscillatory behavior has been observed in several cases. The solution structure is governed by five parameters, i.e., the geometric parameters A = l'/H', B = h'/H', and C = L'/H', the Rayleigh number Ra = gβ ΔT' H' ³/av, and the Prandtl number Pr = v/α. For a geometry with A = ½z, B = ¼, and C = 1, a complicated solution structure is observed upon increasing the Rayleigh number. For Ra ≤ 4 × 10³, a steady symmetric two-cell pattern is observed. This pattern becomes asymmetric for 4 × 10³ < Ra ≤ 9 × 10⁴, periodic for Ra ≤ 3 × 10⁵ and chaotic above that. The transition to periodic convection occurs at lower Rayleigh numbers with decreasing B.     

NUMERICAL ANALYSIS OF HEAT TRANSFER BY NATURAL CONVECTION AND RADIATION IN PARTICIPATING FLUIDS ENCLOSED IN SQUARE CAVITIES

The influence of thermal radiation on natural convection in a participating fluid contained in a square cavity is studied numerically. The radiative transfer process is solved from the PI approximation. The Navier-Stokes equations are solved by a finite difference scheme integrated over control volumes. A numerical study of the so-called window problem (thermally driven cavity) shows the influence of thermal radiation on this reference problem for Rayleigh numbers in the range of 10³-10⁷ and Planck numbers varying from 1 to 0.05. The isotherms, streamlines, and heat lines show an increase of the dynamical effects in the central part of the cavity and a significant modification of the boundary layers. Results obtained from the simulation of an isotropically scattering medium are given.     

MIXED CONVECTION IN CAVITIES WITH A LOCALLY HEATED LOWER WALL AND MOVING SIDEWALLS

An optimal control algorithm for cryopreservation of cells using ultrarapid freezing technique is applied successfully in the present study to determine the strength of optimal laser heating based on the desired limited temperature distribution of the cell. The validity of this optimal control analysis utilizing the conjugate gradient method of minimization is examined using numerical experiments. Three different heating times are given, and the corresponding optimal control heat fluxes are determined. Results show that the optimal boundary heat fluxes can be obtained with any arbitrary initial guesses within a very short CPU time on a Pentium III 600-MHz PC.     

MAXIMAL HEAT TRANSFER DENSITY IN VERTICAL MORPHING CHANNELS WITH NATURAL CONVECTION

In this article we show numerically that the entire flow geometry of a vertical diverging or converging channel with laminar natural convection can be optimized for maximal heat transfer rate density (total heat transfer rate per unit of flow system volume). The geometry is free to change in three ways: (1) the spacing between the walls, (2) the distribution of heating along the walls, and (3) the angle between the two walls. Numerical simulations cover the Rayleigh number range 10⁵ ≤ Ra_H ≤ 10⁷, where H is the channel height. Nonuniform wall heating is modeled as an isothermal patch of varying height H ₀ (≤H) on each wall, which is placed either at the bottom (entrance) end of the channel, or at the top (exit) end. The results confirm that the use of upper unheated sections enhances the chimney effect and the heat transfer. The new aspect is that the heat transfer rate density decreases because the unheated sections increase the total volume. It is shown that for maximal heat transfer rate density it is better to place the H ₀ sections at the channel entrance. It is also shown that the optimal angle between the two walls is approximately zero when Ra _H is large, i.e., for maximal heat transfer rate density the walls should be parallel or nearly parallel. Finally, the optimized spacing (1) developed in the presence of (2) and (3) as additional degrees of freedom is of the same order of magnitude as the optimal spacing reported earlier for parallel isothermal walls, i.e., in the absence of features (2) and (3). The robustness of the optimized flow architecture is discussed. Additional degrees of freedom and global objectives that may be incorporated in this constructal approach are the curvature of the facing walls and the mechanical strength and stiffness of the confining walls.     

TRANSIENT NATURAL CONVECTION IN A CAVITY WITH WALLS OF FINITE THICKNESS

Numerical studies are made of transient natural convection in a square cavity. The top and bottom end walls are thermally insulated. The vertical solid side walls are of finite thickness and of finite thermal conductivity. Flow is driven, from the motionless isothermal initial state, by impulsively increasing (decreasing) the temperature at the outer surface of one (the other) vertical side wall. Numerical solutions art sought to the time-dependent Navier-Stokes equations for the fluid and the solid regions. The ratios of thermophysical properties between solid and fluid are the significant parameters. As the thermal capacity ratio increases, the development of flow in the fluid region is retarded. The conductive and convective timescales are estimated. The effects of the thermal conductivity ratio and of the thickness of the side wall are delineated. The effect of the system Rayleigh number on transient heat transport is analyzed. The applicability of the approximate one-dimensional thermal conductance model to the transient features is scrutinized.     

MULTIGRID COMPUTATION OF NATURAL CONVECTION IN ENCLOSURES WITH A CONDUCTIVE BAFFLE

Conjugate heat transfer has been investigated in a two-dimensional square enclosure with a conducting vertical baffle of finite thickness and varying height. The horizontal end walls are assumed to be adiabatic, and the vertical watts are at constant but different temperatures. Calculations are made by a staggered, finite volume multigrid procedure. The performance of the multigrid method in accelerating the convergence rate is remarkable by comparison with the usual iterative method. The influence of inserting a baffle into the buoyancy-driven square cavity on the heat transfer, as well as on the temperature distribution and velocity field, has been obtained for various Rayleigh numbers Ra, solid /fluid conductivity ratio k? dimensionless baffle height H, and baffle location L. Predicted flow patterns and isotherms indicate that the effect of inserting baffles of varying height upon the overall heat transfer and local temperature profiles in the cavity is limited, except when the height of baffle is large (H > 0.5). The effect of conductivity is also found to be marginal. However, both the height and the conductivity become very important when the baffle is located very near the hot wall or cold walls.     

MAGNETIC AND GRAVITATIONAL CONVECTION OF AIR WITH A COIL INCLINED AROUND THE X AXIS

Air is filled in a cubic enclosure whose one vertical wall is isothermally heated and the opposite one is cooled while the other four walls are thermally insulated. A large coil is placed outside of this enclosure with the coil center coinciding with the cube center. An electric current in the coil generates a magnetic field to affect the convection of air, because the air contains oxygen whose magnetic susceptibility is exceptionally large among gases. The coil is further inclined around the X axis, which is horizontal and perpendicular to the hot and cold walls through the wall center. The heat transfer rate changes depending on the inclination angle. This system is studied numerically for convection for the following combination of parameters: Ra = 1.51 × 10⁴, 9.06 × 10⁴; Pr = 0.71; γ = 0 ? 100; x_Euler = 0–π/2, where γ represents the strength of magnetic field and x_Euler is the angle of inclination of the coil. For example, at Ra = 1.51 × 10⁴ and γ = 30, the average Nusselt number 2.535 at x_Euler = 0 increased to 2.823 at x_Euler = π/2. This study suggests that the coil inclination affects the heat transfer rate extensively.     

WALL HEAT CONDUCTION EFFECT ON NATURAL CONVECTION IN AN ENCLOSURE FILLED WITH A NON-DARCIAN POROUS MEDIUM

ABSTRACT

A numerical analysis has been made of the conjugate natural convection in a rectangular enclosure filled with a fluid-saturated porous medium and surrounded with four solid walls. The conductance of the walls is assumed to be much greater than that of the cavity filled with a porous medium. The main objective was to investigate the influences of the ratio of thermal conductivity of the wall to that of the fluid-porous matrix composite, the Darcy-modified Rayleigh number, the Prandtl number, and the aspect ratio. The streamlines and isotherms are presented; also, the local and average Nusselt numbers are presented along the interface between walls and cavity. A non-Darcian model was employed and the numerical method was SIMPLE-C. The numerical results indicate that the wall heat conduction effects decrease the heat transfer rate. When the wall heat conduction is considered, the greater the conductance of the solid walls surrounding the cavity, the greater is the rate of heat transfer.     

TRANSIENT NATURAL CONVECTION INSIDE POROUS CAVITIES: HYBRID NUMERICAL-ANALYTICAL SOLUTION AND MIXED SYMBOLIC-NUMERICAL COMPUTATION

A hybrid numerical-analytical solution for two-dimensional transient-free convection is presented and applied to a vertical porous cavity, based on application of the ideas in the generalized integral transform technique (GITT). The integral transformation process reduces the original coupled partial differential equations (PDEs), for stream function and temperature, into an infinite system of nonlinear ordinary differential equations (ODEs) for the transformed potentials, which is adaptively truncated and numerically solved through subroutine NDSolve from the Mathematica software system. All the analytical steps in the solution procedure are symbolically evaluated through the Mathematica package, mixing with the numerical computations and graphic representation.     

NUMERICAL SIMULATION OF NATURAL CONVECTION WITH DENSITY INVERSION IN A SQUARE CAVITY

Water has its density maximum at 4^oC, and natural convection around the temperature is complicated. In this study, natural convection with density inversion in a two-dimensional vertical square cavity is investigated numerically. To express the average Nusselt number nondimensionally, some parameters are chosen by observing the calculated flow-temperature fields. Then a simple expression of heat transfer correlation as a function of Galilei number, wall temperature ratio, and density function is proposed. The present correlation has a deviation of about 10%.