Free convection in porous media filled right-angle triangular enclosures

Steady-state free convection heat transfer in a right-angle triangular enclosure, whose vertical wall insulated and inclined and bottom walls are differentially heated, is performed in this study. The governing equations are obtained using Darcy model. In this study, the governing equations were solved by finite difference method and solution of algebraic equations was made via Successive Under Relaxation method. The effect of aspect ratios ranging from 0.25 to 1.0 and Rayleigh numbers 50 ≤ Ra ≤ 1000 is investigated as governing parameters on heat transfer and flow field. It is observed that heat transfer is increased with the decreasing of aspect ratio and multiple cells are formed at high Rayleigh numbers.     

Natural convection in porous triangular enclosure with a centered conducting body

A numerical work was performed to examine the heat transfer and fluid flow due to natural convection in a porous triangular enclosure with a centered conducting body. The center of the body was located onto the gravity center of the right-angle triangular cavity. The Darcy law model was used to write the governing equations and they were solved using a finite difference method. Results are presented by streamlines, isotherms, mean and local Nusselt numbers for the different parameters such as the Rayleigh number, thermal conductivity ratio, and height and width of the body. It was observed that both height and width of the body and thermal conductivity ratio play an important role on heat and fluid flow inside the cavity.     

Natural convection flow in porous enclosures with heating and cooling on adjacent walls and divided by a triangular massive partition

The problem of steady, laminar, natural convection flow in a porous enclosure divided by a triangular massive partition has been formulated. The massive triangular partition is a solid adiabatic body which is located to the right and top wall. Bottom and left vertical wall of porous enclosure are isothermally heated and cooled, respectively. Remaining wall is adiabatic. Governing equations using Darcy model are solved numerically by the finite-difference method and the Successive Under Relaxation (SUR) technique is used to solve linear algebraic equations. Thanks to massive partition, two different enclosure are formed, depends on dimensions of the triangular body, as triangle and trapezoidal. Flow patterns and temperature distributions were presented at different aspect ratios (0 ≤ AR ≤ 1) and Rayleigh numbers (100 ≤ Ra ≤ 1000). Results are given for different aspect ratios (AR) for AR = 0, 0.25, 0.50, 0.75 and 1. A parametric study is conducted and a set of representative results for flow and temperature characteristics are presented and discussed.     

Two-dimensional natural convection in a porous triangular enclosure with a square body

A two-dimensional numerical solution for steady-state buoyancy induced convection in a right-triangular enclosure with a square body is obtained using finite difference technique. The solid body is located far from the origin with the distance of 0.3 in both directions. It is considered that the temperature of the bottom wall of triangular enclosure is higher than that of inclined wall while the vertical wall is insulated. To obtain the effects of the presence of a square body on heat transfer and fluid flow inside the enclosure, four different temperature boundary conditions were applied for the body as heated, cooled, neutral and adiabatic at different Ra numbers. It is observed that fluid flow and temperature fields strongly depend on thermal boundary conditions of the body.     

The effects of Prandtl number on natural convection in triangular enclosures with localized heating from below

The effect of Prandtl number on natural convection heat transfer and fluid flow in triangular enclosures with localized heating has been analyzed by solving governing equations of natural convection in streamfunction–vorticity form with finite-difference technique. Solution of linear algebraic equations was made by Successive Under Relaxation (SUR) method. Bottom wall of triangle is heated partially while inclined wall is maintained at a lower uniform temperature than heated wall while remaining walls are insulated. Computations were carried out for dimensionless heater locations (0.15 ≤ s ≤ 0.95), dimensionless heater length (0.1 ≤ w ≤ 0.9), Prandtl number (0.01 ≤ Pr ≤ 15) and Rayleigh number (10³ ≤ Ra ≤ 10⁶). Aspect ratio of triangle was chosen as unity. It is observed that both flow and temperature fields are affected with the changing of Prandtl number, location of heater and length of heater as well as Rayleigh number.     

Natural convection in right-angle porous trapezoidal enclosure partially cooled from inclined wall

A numerical study is conducted to investigate the steady free convection flow in a two-dimensional right-angle trapezoidal enclosure filled with a fluid-saturated porous medium. The left vertical wall of the cavity is heated; the inclined wall is partially cooled; and the remaining walls are insulated (adiabatic). Three different cases are considered. While in Case I the cooler wall is located adjacent to the top wall, in Case II it is located in the middle inclined wall. In Case III, it is located adjacent to the bottom wall. Flow and heat transfer characteristics are studied for a range of parameters: the Rayleigh number, Ra, 100 ≤ Ra ≤ 1000; and the aspect ration, AR = 0.25, 0.50 and 0.75. Numerical results indicate that there exist significant changes in the flow and temperature fields as compared with those of a differentially heated square porous cavity. These results lead, in particular, to the prediction of a position of minimum heat transfer across the cavity, which is of interest in the thermal insulation of buildings and other areas of technology.     

Maximum density effects on buoyancy-driven convection in a porous trapezoidal cavity

A numerical study of the steady buoyancy-induced flow and heat transfer in a trapezoidal cavity filled with a porous medium saturated with cold water at a temperature around 4 °C has been performed. The analysis has been done for a cavity with different aspect ratios ranging from 0.25 to 0.75 and Rayleigh numbers ranging from 100 to 1000 using a finite-difference method. It is found that four cells are formed inside the cavity independent of the Rayleigh number and aspect ratio. The existence of buoyancy force reversal resulting from the maximum density effect, leads to a reduction in the strength of the convective flow and also the average Nusselt number.     

Entropy production due to free convection in partially heated isosceles triangular enclosures

A numerical analysis of the entropy production has been performed due to natural convection heat transfer and fluid flow in isosceles triangular enclosures with partially heated from below and symmetrically cooled from sloping walls. Governing equations are solved by finite difference method. Governing parameters on flow and temperature fields are Rayleigh number (10³  Ra  8.8 × 10⁵), dimensionless length of heater (0.25  (ℓ′ = ℓ/L)  1.0), dimensionless location of heater (0.25  (c′ = c/L)  0.75) and inclination angle of slopping walls (30°  β  60°). Heat transfer results are presented in terms of local and mean Nusselt numbers (Nu) while entropy production results are shown with entropy production number (N_s) and Bejan number (Be). Isotherms, streamlines, contours of entropy production due to heat transfer and fluid friction irreversibility are plotted. It is observed that entropy production number increases but Bejan number decreases with increasing of Rayleigh number. However, both entropy production due to heat transfer and fluid friction irreversibility are affected by higher inclination angle of triangle and length of heater.     

Experimental and numerical analysis of buoyancy-induced flow in inclined triangular enclosures

The buoyancy‐induced heat transfer and fluid flow in a triangular enclosure are investigated both numerically and experimentally. The enclosure is heated from one wall and the adjacent wall is insulated. Hypotenuse of the triangle is cooled isothermally. The numerical tests and experiments covered a range of Rayleigh number, Ra, from 1.5 × 10⁴ to 1.5 × 10⁵. The local and average Nusselt numbers are given for different orientation angles. A code was written based on finite difference method in Fortran platform to solve governing equations of natural convection. Experimental and numerical results show good agreement. It is observed that inclination angle can be used as a control parameter for heat transfer.     

Heatline based thermal management for natural convection within right-angled porous triangular enclosures with various thermal conditions of walls

Analysis of natural convection in porous triangles have many important energy related applications in geophysical and solar energy fields. A numerical study on heat distribution and thermal mixing during steady laminar natural convective flow inside a right-angled triangular enclosure filled with porous media subjected to various wall boundary conditions is investigated in this study using Bejan’s heatlines approach. Influence of various thermal boundary conditions and inclination angles (φ) on evaluation of complex heat flow patterns are studied as a function of Darcy numbers (Da) for various regimes of Prandtl (Pr) and Rayleigh (Ra) numbers. Studies illustrate that maximum heat transfer occurs at the top vertex for lower top angle (φ=^°15) at higher Da(Da=10⁻³). As φ increases to ^°45, the maximum heat flux at the top vertex decreases and thermal mixing increases irrespective of Da and Pr. The enhanced convection at higher Da significantly affects the heat flow distribution, which is clearly depicted by high local Nusselt numbers at Da=10⁻³. It is also found that isothermal heating of walls enhances the heat distribution and thermal mixing. Overall, it is shown that heatlines provide suitable guideline on thermal management in porous right-angled triangular enclosures with various heating strategies.     

Mixed convection in a lid-driven triangular enclosure filled with nanofluids

This paper presents the results of a numerical study on the mixed convection in a lid-driven triangular enclosure filled with a water–Al₂O₃ nanofluid. A comparison study between two different scenarios of upward and downward left sliding walls is presented. The effects of parameters such as Richardson number, solid volume fraction and the direction of the sliding wall motion on the flow and temperature fields as well as the heat transfer rate are examined. The results show that the addition of Al₂O₃ nanoparticles enhances the heat transfer rate for all values of Richardson number and for each direction of the sliding wall motion. However, the downward sliding wall motion results in a stronger flow circulation within the enclosure and hence, a higher heat transfer rate.     

Numerical analysis of natural convection heat transfer from rectangular shrouded fin arrays on a horizontal surface

The main aim of this investigation is to discover the effects of clearance parameters on the steady-state heat transfer. In order to solve the three-dimensional elliptic governing equations, a finite volume based CFD code was used. The clearance gap between fin tips and shroud, the base and fin temperatures and the size and configuration of the finned surfaces were varied during the parametric study. The numerical results have been compared to existing experimental values from the literature and the comparison shows a good agreement. It is found that the heat transfer coefficient increases with the increase in the clearance parameter and it approaches to the value of heat transfer coefficient obtained for unshrouded fin arrays.     

Numerical analysis of natural convection in an inclined trapezoidal enclosure filled with a porous medium

A theoretical study of buoyancy-driven flow and heat transfer in an inclined trapezoidal enclosure filled with a fluid-saturated porous medium heated and cooled from inclined walls has been performed in this paper. The governing non-dimensional equations were solved numerically using a finite-difference method. The effective governing parameters are: the orientation or inclination angle of the trapezoidal enclosure , which varies between 0° and 180°, the Rayleigh number Ra, which varies between 100 and 1000, the side wall inclination angle θ_s and the aspect ratio A. The side wall inclination parameter θ_s is chosen as 67°, 72° and 81° and the calculations are tested for two different values of A=0.5 and 1.0. Streamlines, isotherms, Nusselt number and flow strength are presented for these values of the governing parameters. The obtained results show that inclination angle is more influential on heat transfer and flow strength than that of the side wall inclination angle θ_s. It is also found that a Bénard regime occurs around =90°, which depends on the inclination of the side wall, Rayleigh number and aspect ratio.     

Natural convection in three-dimensional porous cavities: integral transform method

A transient three-dimensional Darcy model of natural convection in porous medium filled cavities is studied, using a vorticity-vector potential formulation and the generalized integral transform technique (GITT). A general formulation and solution methodology for vertical cavities (insulated vertical walls with differential horizontal wall temperatures) is developed. Results for cubic cavities are presented while evaluating the Rayleigh number effects for stable situations, observing the transient evolution of the heat transfer process. The convergence behavior of the proposed eigenfunction expansion solution is investigated and comparisons with previously reported steady-state solutions are critically performed.     

Control of mixed convection in lid-driven enclosures using conductive triangular fins

Control of mixed convection (combined forced and natural convection) in a lid-driven square cavity is performed using a short triangular conductive fin. A numerical technique is used to simulate the flow and temperature fields. The vertical walls of the cavity are differentially heated. Both the top lid and the bottom wall are adiabatic. The fin is located on one of the motionless walls of the cavity. Three different cases have been studied based on the location of the fin. In this context, Cases I, II and III refer to the fin on the left, bottom and right walls, respectively. Results are presented for +x and −x directions of the top lid in horizontal axis and different Richardson numbers as Ri = 0.1, 1.0 and 10.0. It is observed that the triangular fin is a good control parameter for heat transfer, temperature distribution and flow field.     

Double-diffusive turbulent natural convection in a porous square cavity with opposing temperature and concentration gradients

This paper presents results for coupled heat and mass transport under laminar and turbulent flow regimes in porous cavities. Two driving mechanisms are considered to contribute to the overall momentum transport, namely temperature driven and concentration driven mass fluxes. Aiding and opposing flows are considered, where temperature and concentration gradients are either in the same direction or of different sign, respectively. Modeled equations are presented based on the double-decomposition concept, which considers both time fluctuations and spatial deviations about mean values. Turbulent transport is accounted for via a macroscopic version of the k–ε model. Variation of the cavity Nusselt and Sherwood numbers due to changes on N, where N is the ratio of solute to thermal Grashof numbers, is presented. Results indicate that for adding cases, mass and heat transfer across the cavity are enhanced faster than for cases with opposing temperature and concentration gradients. For the conditions here investigated, the use a turbulence model gave results for Nu and Sh that were nearly double when compared with laminar results for the same conditions.     

Prediction of flow fields and temperature distributions due to natural convection in a triangular enclosure using Adaptive-Network-Based Fuzzy Inference System (ANFIS) and Artificial Neural Network (ANN)

Artificial Neural Network (ANN) and Adaptive-Network-Based Fuzzy Inference System (ANFIS) were used to predict the natural convection thermal and flow variables in a triangular enclosure which is heated from below and cooled from sloping wall while vertical wall is maintained adiabatic. Governing equations of natural convection were solved using finite difference technique by writing a FORTRAN code to generate database for ANN and ANFIS in the range of Rayleigh number from Ra = 10⁴ to Ra = 10⁶ and aspect ratio of triangle AR = 0.5 and AR = 1. Thus, the results obtained from numerical solutions were used for training and testing the ANN and ANFIS. A comparison was performed among the soft programming and Computational Fluid Dynamic (CFD) codes. It is observed that although both ANN and ANFIS soft programming codes can be used to predict natural convection flow field in a triangular enclosure, ANFIS method gives more significant value to actual value than ANN.     

Sudden or smooth transitions in porous media natural convection

In porous media isothermal flow a transition from the Darcy regime, via an inertia dominated regime, towards turbulence is anticipated. In porous medium natural convection the transition to turbulence follows a different route. The first transition from a motionless-conduction regime to steady natural convection is followed by a direct second transition to a non-steady (time dependent) and non-periodic regime (referred to as weak turbulent), prior to the amplitude of the convection reaching such large values as to involve inertial, non-Darcy effects. The latter is due to an additional non-linear interaction that appears in natural convection as a result of the coupling between the equations governing the fluid flow and the energy equation. The present paper deals with identifying whether the transitions are sudden or possibly smooth. The latter is accomplished by using a truncated Galerkin representation of the natural convection problem in a porous layer heated from below (an extended Darcy model) leading to the familiar Lorenz equations for the evolution of the convection amplitudes with time. Two different formulations (named the “original” and the “modified” systems) are being used in an anticipation to obtaining a smooth transition in the form of an imperfect bifurcation from the “modified” system formulation. The results show that the transition remains sudden and the accuracy of the “modified” system results is being tested in comparison with the “original” system showing a sufficiently high degree of accuracy.     

Natural convection of non-Newtonian fluids through permeable axisymmetric and two-dimensional bodies in a porous medium

The present work concerns the natural convection of non-Newtonian power-law fluids with or without yield stress over the permeable two-dimensional or axisymmetric bodies of arbitrary shape in a fluid-saturated porous medium. Using the fourth-order Runge-Kutta scheme method and shooting method we obtain the local non-similarity solutions. The parameters that include the dimensionless yield stress Ω, permeable constant c and power index n are studied, and the heat flux and the wall temperature are taken into consideration as variables. The local non-similarity solutions are found to be in excellent agreement with the exact solution. It is found that the results depend strongly on the values of the yield stress parameter, the wall temperature distributions, the lateral mass flux rate, and the heat flux at the boundary.