Bounds on natural convective heat transfer in a porous layer with fixed heat flux

Using the background field variational method, bounds on natural convective heat transfer in a porous layer heated from below with fixed heat flux are derived from the primitive equations. The enhancement of heat transfer beyond the minimal conduction value (the Nusselt number Nu) is bounded in terms of the non-dimensional forcing scale set by the ‘effective’ Rayleigh number (Rˆ) according to Nu ≤ 0.354Rˆ^1/2 and in terms of the conventional Rayleigh number (Ra) defined by the temperature drop across the layer according to Nu ≤ 0.125Ra. It is presented that fixing the heat flux at the boundaries does not change the linear dependence between Nusselt number and Rayleigh number at high Rayleigh number region.     

Effect of Darcy,fluid rayleigh and heat generation parameters on natural convection in a porous square enclosure: A Brinkman-extended Darcy model

A Pressure-velocity solution for natural convection for fluid saturated heat generating porous medium in a square enclosure is analysed by finite element method. The numerical solutions obtained for wide range of fluid Rayleigh number, Ra_f, Darcy number, Da, and heat generating number, Q_d. The justification for taking these non-dimensional parameters independently is to establish the effect of individual parameters on flow patterns. It has been observed that peak temperature occurs at the top central part and weaker velocity prevails near the vertical walls of the enclosure due to the heat generation parameter alone. On comparison, the modified Rayleigh number used by the earlier investigators[4,6], can not explain explicitly the effect of heat generation parameter on natural convection within an enclosure having differentially heated vertical walls. At higher Darcy number, the peak temperature and peak velocity are comparatively more, resulting in better enhancement of heat transfer rate.     

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.71 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ⩽ 10⁻⁵ irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ⩾ 10⁻⁴. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Magnetic field effect on the unsteady free convection flow in a square cavity filled with a porous medium with a constant heat generation

The effects of an inclined magnetic field and heat generation on unsteady free convection within a square cavity filled with a fluid-saturated porous medium have been investigated numerically. The top and bottom horizontal walls of the enclosure are adiabatic whereas the vertical walls are kept at constant but different temperatures. The physical problems are represented mathematically by a set of partial differential equations along with the corresponding boundary conditions. By using an implicit finite-difference scheme, namely the ADI method (Alternative Direction Implicit), the non-dimensional governing equations are numerically solved. The influential parameters are the Rayleigh number Ra, the inclination angle γ of the magnetic field relative to the gravity vector g, the Hartmann number Ha and the heat generation parameter Q. In the present study, the obtained results are presented in terms of streamlines, isotherms and average Nusselt number along the hot wall. The result shows that with increasing Ha, the diffusive heat transfer become prominent even though the Rayleigh number increases.     

Cooling of a sphere by natural convection – The applicability of the lumped capacitance method

Two approaches to predicting the sphere cooling process by laminar natural convection were compared in terms of the accuracy of the volume averaged sphere temperature and the heat transfer rate between the sphere and the surrounding fluid. The first approach is based on the formulation of conjugate heat transfer (heat conduction in the sphere and laminar natural convection in the fluid). The second approach includes the lumped capacity method based on the assumption that the temperature in the sphere is spatially uniform and on the Churchill correlation function. The solution to the problem depends on the Rayleigh number (Ra), the Biot number (Bi), the Prandtl number (Pr), and the sphere-to-fluid thermal diffusivity ratio (A). The lumped capacitance method gives fairly accurate results with respect to the conjugate heat transfer method (discrepancy in the volume averaged sphere temperature less than 5%) when A · Bi/Ra^0.452 < 0.05, for Bi < 0.15 and Pr > 1.     

Effects of thermal boundary conditions on natural convection flows within a square cavity

A numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁵ and Prandtl number Pr, 0.7 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rates at the center of the bottom wall than the uniform heating case for all Rayleigh numbers; however, average Nusselt numbers show overall lower heat transfer rates for the non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Rayleigh–Bénard convection in a supercritical fluid along its critical isochore in a shallow cavity

We perform a numerical investigation of the Rayleigh–Bénard convection in supercritical nitrogen in a shallow enclosure with an aspect ratio of 4. The transient and steady-state fluid behaviors over a wide range of initial distances to the critical point along the critical isochore are obtained and analyzed in response to modest homogeneous bottom heating. On account of the fluid layer being extremely thin, density stratification is notably excluded from consideration herein, which leads to the dominating role of the Rayleigh criterion in the onset of convection. Following the Boussinesq approximation, we find the power law scaling relationships over five decades of the Rayleigh number (Ra) for various transient quantities including the exponential growth rate of the mean enstrophy in the cavity and the characteristic times of the development of convective motion. The correlation of the Nusselt number versus the Rayleigh number shows asymptotic features at the two ends of the Ra spectrum, which incidentally correspond to different convection patterns. Under the regime of high Ra, the heat transfer through the fluid cavity is enhanced by the turbulent bursts of thermal plumes from the boundary layers. On the other hand, under the regime of low Ra, it is the orderly multicellular flow that moves heat from the bottom of the layer to the top, which includes a transition from a four-cell structure to a six-cell structure with decreasing Ra.     

Natural convective flow in an inclined lid-driven enclosure with a heated thin plate in the middle

Natural convection heat transfer from a heated thin plate located in the middle of a lid-driven inclined square enclosure has been analyzed numerically. Left and right of the cavity are adiabatic, the two horizontal walls have constant temperature lower than the plate’s temperature. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using a fully higher-order compact (FHOC) finite difference scheme on the 9-point 2D stencil. Air was chosen as a working fluid (Pr = 0.71). Two cases are considered depending on the position of heated thin plate (Case I, horizontal position; Case II, vertical position). Governing parameters, which are effective on flow field and temperature distribution, are Rayleigh number values (Ra) ranging from 10³ to 10⁵ and inclination angles γ (0° ? γ < 360°). The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and Nusselt number (Nu). It is found that fluid flow and temperature fields strongly depend on Rayleigh numbers and inclination angles. Further, for the vertical located position of thin plate heat transfer becomes more enhanced with lower γ at various Rayleigh numbers.     

Unsteady conjugate natural convection in a porous cavity boarded by two vertical finite thickness walls

The objective of this study is to investigate unsteady conjugate natural convection in a porous cavity sandwiched by finite conductive walls considering time-periodic boundary conditions and local thermal non-equilibrium. The top and bottom boundaries are assumed to be isolated and the continuity of temperature and heat transfer are considered in interface boundaries. The effect of varying a plethora of parameters such as Rayleigh number, Thermal conductivity ratio, wall thickness, and non-dimensional frequency on the streamlines, isotherms, and Nusselt number has been studied. It is shown that, apart from non-dimensional frequency and wall thickness, the amplitude of periodic fluid Nusselt number is an increasing function of all aforementioned parameters. Furthermore, aside from Rayleigh number and heat transfer coefficient, the behavior of the solid Nusselt number is the same as fluid Nusselt number. Eventually, the time-averaged Nusselt number and heat transfer through the vertical walls for different values of non-dimensional frequencies are calculated.     

Conjugate natural convection in the roof cavity of heavy construction building during summer

This paper presents the results of a study of conjugate natural convection inside a building attic in the shape of a rectangular enclosure bounded by realistic walls made from composite construction materials under summer day boundary conditions. The effects of cavity aspect ratio, Rayleigh number (Ra), and orientation of the external surfaces on the flow and heat transfer characteristics were the main focus of the investigation. The problem was formulated in terms of the vorticity-stream function procedure, and the governing equations for steady, laminar, two-dimensional conjugate natural convection heat transfer were solved by employing the Alternating Direction Implicit (ADI) control volume method along with under-relaxation factors for temperature, vorticity, and stream functions. For Ra ranging from 10³ to 10¹⁰, steady state results of the streamline and temperature contours in addition to local and mean Nusselt number at all surfaces of the cavity were obtained. The results show that the values of Ra and the aspect ratio have significant effect on the temperature and stream function contours within the enclosure. Another important finding of the study is that heat flux into the room increases with the increase of both the aspect ratio and Rayleigh number.     

Heat transfer correlations for PCM-based heat sinks with plate fins

Characterization of melting process in a Phase Change Material (PCM)-based heat sink with plate fin type thermal conductivity enhancers (TCEs) is numerically studied in this paper. Detailed parametric investigations are performed to find the effect of aspect ratio of enclosure and the applied heat flux on the thermal performance of the heat sinks. Various non-dimensional numbers, such as Nusselt number (Nu), Rayleigh number (Ra), Stefan number (Ste) and Fourier number (Fo) based on a characteristic length scale, are identified as important parameters. The half fin thickness and the fin height are varied to obtain a wide range of aspect ratios of an enclosure. It is found that a single correlation of Nu with Ra is not applicable for all aspect ratios of enclosure with melt convection taken into account. To find appropriate length scales, enclosures with different aspect ratios are divided into three categories, viz. (a) shallow enclosure, (b) rectangular enclosure and (c) tall enclosure. Accordingly, an appropriate characteristic length scale is identified for each type of enclosure and correlation of Nu with Ra based on that characteristic length scale is developed.     

MHD natural convection in a laterally and volumetrically heated square cavity

A numerical study is presented of unsteady two-dimensional natural convection of an electrically conducting fluid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. The flow is characterized by the external Rayleigh number, Ra_E, determined from the temperature difference of the side walls, the internal Rayleigh number, Ra_I, determined from the volumetric heat rate, and the Hartmann number, Ha, determined from the strength of the imposed magnetic field. Starting from given values of Ra_E and Ha, for which the flow has a steady unicellular pattern, and gradually increasing the ratio S = Ra_I/Ra_E, oscillatory convective flow may occur. The initial steady unicellular flow for S = 0 may undergo transition to steady or unsteady multicellular flow up to a threshold value, Ra_I,cr, of the internal Rayleigh number depending on Ha. Oscillatory multicellular flow fields were observed for S values up to 100 for the range 10⁵-10⁶ of Ra_E studied. The increase of the ratio S results usually in a transition from steady to unsteady flow but there have also been cases where the increase of S results in an inverse transition from unsteady to steady flow. Moreover, the usual damping effect of increasing Hartmann number is not found to be straightforward connected with the resulting flow patterns in the present flow configuration.     

Hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface

The present numerical simulation is conducted to analyze the mixed convection flow and heat transfer in a lid-driven cavity with sinusoidal wavy bottom surface in presence of transverse magnetic field. The enclosure is saturated with electrically conducting fluid. The cavity vertical walls are insulated while the wavy bottom surface is maintained at a uniform temperature higher than the top lid. In addition, the transport equations are solved by using the finite element formulation based on the Galerkin method of weighted residuals. The implications of Reynolds number (Re), Hartmann number (Ha) and number of undulations (λ) on the flow structure and heat transfer characteristics are investigated in detail while, Prandtl number (Pr) and Rayleigh number (Ra) are considered fixed. The trend of the local heat transfer is found to follow a wavy pattern. The results of this investigation illustrate that the average Nusselt number (Nu) at the heated surface increases with an increase of the number of waves as well as the Reynolds number, while decreases with increasing Hartmann number.     

Heat generation effects in natural convection inside a porous annulus

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr = 5, through a porous matrix of volumetric porosity, ? = 0.4. The generalized momentum equation with Brinkman–Darcy–Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, Ra_I,cr^?, the convection transits from unicellular to bicellular mode, as the annulus T_max becomes higher than the fixed hot-wall temperature. The Ra_I,cr^? increases proportionately when the permeability based external Rayleigh number Ra_E^? and the solid–fluid thermal conductivity ratio γ are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of Ra_E^?, Ra_I^?, Da and γ. It predicts the results within ± 20% accuracy.     

Visualization of heat flow due to natural convection within triangular cavities using Bejan’s heatline concept

Natural convection and flow circulation within a cavity has received significant attention in recent times. The wide range of applicability of flow inside a cavity (food processing industries, molten metal industries, etc.) requires thorough understanding for cost efficient processes. This paper is based on comprehensive analysis of heat flow pattern using Bejan’s heatline concept. The key parameters for our study are the Prandtl number, Rayleigh number and Nusselt number. The values of Prandtl number (0.015, 0.026, 0.7 and 1000) have been chosen based on wide range of applicability. The Rayleigh number has been varied from 10² to 10⁵. Interesting results were obtained. For low Rayleigh number, it is found that the heatlines are smooth and perfectly normal to the isotherms indicating the dominance of conduction. But as Ra increases, flow slowly becomes convection dominant. It is also observed that multiple secondary circulations are formed for fluids with low Pr whereas these features are absent in higher Pr fluids. Multiple circulation cells for smaller Pr also correspond multiple cells of heatlines which illustrate less thermal transport from hot wall. On the other hand, the dense heatlines at bottom wall display enhanced heat transport for larger Pr. Further, local heat transfer (Nu_l, Nu_t) are explained based on heatlines. The comprehensive analysis is concluded with the average Nusselt number plots. A correlation for average heat transfer rate and Ra has been developed and the range of Rayleigh number is also found, to depict the conduction dominant heat transfer.     

Natural convection from narrow horizontal plates at moderate Rayleigh numbers

The present work deals with the natural convection flow and heat transfer from a horizontal plate cooled from above. Experiments are carried out for rectangular plates having aspect ratios between φ=0.036 and 0.43 and Rayleigh numbers in the range 290?Ra_w?3.3×10⁵. These values of Ra_w and φ have been selected below those commonly considered in previous research in view of a future application to the design of printed circuit boards. The plates are made of two different metals, copper and steel. The choice of a metal is relevant to the present problem because the plates are heated by means of an electric current. Important variations of the surface temperature are observed along the transverse direction for the steel plates. The surface of the copper plates is almost isothermal because of the high thermal conductivity of the metal.Calculations for a semi-infinite plate are carried out to predict the transverse profiles of the surface temperature and heat flux and to visualize the structure of the flow. Three-dimensional calculations are also used at a qualitative level to observe the changes in the flow structure due to the finite length of the plate. Present results are compared with both previous experimental work and analyses that are based on boundary layer theory. It is shown that analyses for an infinite boundary layer are not completely applicable to the present problem because of its different physics. The most relevant feature of the natural convection flow, which is not predicted by boundary layer analyses, is a thermal plume rising near the center of the plate.Present heat transfer results differ from previous experimental work because of the lower Rayleigh numbers and aspect ratios investigated here. The Nusselt number is found to depend on Ra_wⁿ, with the exponent n=0.17 being lower than most of the values reported in the literature. This comparatively low value is related to the transverse conduction of heat through the air, which becomes increasingly significant as Ra_w approaches zero. It is shown that such a low-Ra_w effect can be accounted for in a physically consistent manner by adding a constant term to the heat transfer correlation. On the other hand, it is found that the Nusselt number does not significantly depend on the aspect ratio in the range of φ investigated contrary to what has been previously reported for wider plates.     

Numerical analysis of conjugate heat transfer in a partially open square cavity with a vertical heat source

Conjugate heat transfer in partially open square cavity with a vertical heat source has been numerically studied. The cavity has an opening on the top with several lengths and three different positions. The other walls of cavity were assumed adiabatic. The heat source was located on the bottom wall of cavity and it has got a width such as Printed Circuit Boards (PCB). Steady state heat transfer by laminar natural convection and conduction is studied numerically by solving two dimensional forms of governing equations with finite difference method. The results were reported for various governing parameters such as Rayleigh number (10³ ≤ Ra ≤ 10⁶), conductivity ratio, opening position, opening length, PCB distance and PCB height. The numerical results were discussed with streamlines, isotherms, Nusselt number and velocity profiles on x- and y-directions. It is found that ventilation position has a significant effect on heat transfer.     

Steady natural convection flow in a square cavity filled with a porous medium for linearly heated side wall(s)

In this paper natural convection flows in a square cavity filled with a porous matrix has been investigated numerically when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Darcy–Forchheimer model without the inertia term is used to simulate the momentum transfer in the porous medium. Penalty finite element method with bi-quadratic rectangular elements is used to solve the non-dimensional governing equations. Numerical results are presented for a range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.2 ⩽ Pr ⩽ 100) in terms of stream functions and isotherm contours, and local and average Nusselt numbers.     

Thermal description of pseudosteady-state natural convection inside a vertical cylinder

The heat transfer correlation, minimum temperature, and mean temperature are presented for pseudosteady-state natural convection heat transfer to a fluid inside a vertical cylinder. The SIMPLER numerical method was used for calculation in the range 0.25 < H/D <2, Ra < 10⁷, and Pr = 7. This range includes conduction to weak turbulence. The overall heat transfer for the convection-dominated range was found to be correlated by Nu = 0.519Ra^0.255 where the temperature difference for both the Nusselt and Rayleigh numbers was the center temperature minus the inside wall temperature. Correlations using other temperature differences are also presented and provide a method for prediction of the mean temperature, minimum temperature, or center temperature by knowing any one of them.     

Optimal location of a pair heat source-sink in an enclosed square cavity with natural convection through PSO algorithm

In this paper, the optimum positions of a pair heat source-sink in an enclosure have been studied. The objective of this investigation is to minimize the maximum temperature (T_max) on the heat source with constant heat flux. For this, a Particle Swarm Optimization algorithm (PSOA) has been used. Continuity, momentum and energy equations with the Boussinesq approximation for a laminar and incompressible flow of a Newtonian fluid have been solved by finite volume method. The present study has been carried out for governing parameters like Rayleigh numbers (Ra) from 10³ to 10⁶, the cavity aspect ratio, A = H/L = 1 and the source and sink sizes D₀ from 0.2 to 0.5. Numerical results revealed that the optimum configurations are a function of Rayleigh numbers and the source and sink sizes.