---

- Natural convection heat transfer in a square cavity induced by heated plate is investigated using the lattice Boltzmann method. A suitable forcing term is represented in the Boltzmann equation. With the representation, the Navier-Stokes equation can be derived from the lattice Boltzmann equation through the Chapman-Enskog expansion. Top and bottom of the cavity are adiabatic; the two vertical walls of the cavity have constant temperatures lower than the plate’s temperature. The flow is assumed to be two-dimensional. Air is chosen as a working fluid (Pr=0.71). The study is performed for different values of Grashof number ranging from 103 to 105 for different aspect ratios and position of heated plate. The effect of the position and aspect ratio of heated plate on heat transfer are discussed. With increase of the Grashof number, heat transfer rate is increased in both vertical and horizontal position of the plate. The obtained results of the lattice Boltzmann method are validated with those presented in the literature.

---

In this paper the problem of coupled radiation and natural convection is investigated in a square porous cavity using local thermal non-equilibrium model. The radiative transfer equation (RTE) is solved by the discrete ordinates method (DOM) and the energy and momentum equations are solved using finite element method. The results of the present study are compared with that of the other investigations which have used another method to solve radiative transfer equation. Effective parameters on heat transfer and fluid flow characteristics such as Planck number, inter-phase heat transfer coefficient and scattering albedo are studied and the results are presented in the group of dimensionless parameters. The results indicate that the solving method of the radiative transfer equation would have significant effect on the fluid flow and heat transfer characteristic. In the case of low optically thick media, discrete ordinates method (DOM) is more precise than the other methods which used in other literature.

---

Water-Alumina natural convection heat transfer in an inclined L shape cavity ABSTRACT A numerical investigation of water-alumina natural convection laminar flow in an inclined L shape cavity is performed. Two wall of the cavity are hot with the temperature of Th, two walls are cold with the temperature of Tc , and two other walls are thermally adiabatic. The two dimensional continuity, momentums and energy equations are solved numerically with a finite volume approach using the well known SIMPLE algorithm. The influence of pertinent parameters such as Rayleigh number, Ra, solid volume fraction, , inclination angle, , and cavity aspect ratio, A, on the fluid flow, fluid temperature and cavity heat transfer characteristics is studied. The results indicate that nanofluid with higher nanoparticles has better performance. The results also show that the inclination angle has a significant effect on the heat transfer performance at high Rayleigh numbers. Keywords: Natural convection, nanofluid, L shape cavity, Rayleigh number.

---

Abstract- This paper presents the results of a numerical study on the natural convection in a right triangular enclosure filled with a water- Cu nanofluid in presence of a constant magnetic field. A heat source embedded on the bottom wall of enclosure, the inclined wall is cold and the other walls are adiabatic. Discretization of the governing equations are achieved through a finite volume method and solved with SIMPLE algorithm. The effects of parameters such as the Reyleigh number, the solid volume fraction, the Hartman number, length and location of heat source on flow and temperature fields and the heat transfer rate have been examined. The results show that increasing of Hartman number caused decreasing velocity of flow and heat transfer. Also, increase in solid volume fraction causes increase in heat transfer but its change in different Reyleigh number and Hartman number is not same. Therefore, the location of heat source in bottom of enclosure affects on the rate of heat transfer from enclosure.

---

In this paper, natural convection heat transfer inside an enclosure which is partially filled with porous layer is reported using lattice Boltzmann method. Generalized equations in modeling flow in porous media have been employed which are coupled with the lattice Boltzmann formulation of the momentum and energy equations. The present study investigates the effect of position of porous layer on heat transfer rate for different dimensionless parameters, such as Rayleigh number, Darcy number and porosity of the porous layer. In addition, a modified Rayleigh number is presented as an effective parameter which affects the degree of penetration of the fluid into the porous layers. The obtained results showed that the heat transfer rate in the case of vertical layer is more than that of horizontal porous layers.

---

In the present paper,by performing a two-dimensional simulation,the heat transfer from a hot cylinder to a cold square enclosure has been studied parametrically and the consequent effect of changing in cylinder diameter has been investigated. The 2-D governing equations have been solved using the finite volume method and TDMA in an ADI procedure for different diameters of cylinder inside a square enclosure with a constant characteristic length for two different Rayleigh numbers of 104 and 105.Results showed that the patterns of streamlines, isotherms and the Nusselt number values depend strongly on the Rayleigh number and also ratio of cylinder diameter to characteristic length of enclosure (2R/H). In this case, the centers of vortices created around the cylinder appear in bottom half of enclosure in 2R/H=0.4 for Ra=104 and in 2R/H= 0.5 for Ra=105. Moreover, it is observed that increasing the Rayleigh number and 2R/H ratio, the heat transfer rate from the enclosure is also increased.For example,in 2R/H=0.5, by increasing the Rayleigh number from 104 to 105, the average Nusselt enhances about 30 percent of its initial value and in Ra=105, by changing the 2R/H ratio from 0.2 to 0.5, the average Nusselt climbs almost 35 percent of its initial value.

---

Abstract- In this paper, two-dimensional natural convection heat transfer in semi ellipse cavities is investigated using lattice Boltzmann method. The Prandtl number is taken as 0.71 that corresponds to that of air. Heat transfer and flow pattern are predicted at various Rayleigh numbers ranging from 104 to 106 for different aspect ratios. By increasing of the aspect ratio, the heat transfer rate in the cavity is increased for low Rayleigh numbers, but it is decreased for high Rayleigh numbers. The obtained results of the lattice Boltzmann method are validated with those presented in the literature and show that the lattice Boltzmann method can simulate heat transfer and flow pattern in complex cavities. Analysis of heat transfer in a semi-ellipse cavity using second order boundary condition on curved surfaces is among the novelties of the present work.

---

Abstract- The present study aims to investigate numerically the natural convection of various nanofluids inside a square enclosure with a central heat source at different aspect ratio. Also, some correlations are presented in order to calculate the Nusselt number in terms of Rayleigh number and volume fraction of nanoparticles. The heat source and cavity walls are kept at constant temperatures of Th and Tc, respectively. The nanofluids are considered to be water as the base fluid and different nanoparticles such as Cu, CuO, Ag, Al2O3, or Tio2. To discretize the governing equations, the control volume method and SIMPELER algorithm have been employed. The study has been carried out for aspect ratios from 0.2 to 0.8, Rayleigh numbers from 1e3 to 1e6 and the volume fractions of nanoparticles ranging in 0-0.05. The results indicated that the Nusselt number increases with increasing the volume fraction of nanoparticles as well as the aspect ratio. Furthermore, by increasing the Rayleigh number, some eddies, of kind of Rayleigh-Benard, are developed in the space between the heat source and the upper wall of the enclosure. Based on the obtained results, several correlations with high accuracy have been present in order to evaluate the Nusselt number.

---

Electrohydrodynamic (EHD) is one of the techniques for heat transfer enhancement. In current study, the enhancement of natural convection heat transfer inside a vertical tube is experimentally investigated under applying a strong electrical field (EHD). For this purpose, a wire electrode with positive polarity is used along the pipe axis while the inner surface of tube is connected to the ground. EHD disturbs the thermal boundary layer by generating ionic wind which flows from wire electrode to inner side of tube and causes the heat transfer enhancement. In this study, the effects of wire electrode diameter and also electrical field on heat transfer enhancement are investigated. Obtained data are reported as local Nusselt number along the pipe axis and mean Nusselt number. The results show that decreasing the wire electrode diameter increases the heat transfer of tube. In addition, increasing of electrical current due to strong electrical field, increases the Nusselt number. At the lowest wire electrode diameter, the highest Nusselt number was observed which was 2.03 times more than the case that no electrical field was applied.

---

In this paper, the result of a numerical study on the natural convection in an inclined T shap cavity filled with Water-Cu nanofluid with the presence of a constant magnetic field was investigated. A heat source embedded on the bottom wall of enclosure, the upper wall is cold and the other walls are adiabatic. Discretization of the governing equations are achieved through a finite volume method and solved with SIMPLE algorithm. The Hartmann number has been varied from 0 to 80 and the cavity has been twisted under the angles between 0 to 90 degrees. The findings of study show that the effect magnetic field on the average Nusselt number is higher in high Reyleigh number. In Ra=105, the increase in nanofluid, to the Hartman number 20, contributes to decrease of the average number and in the Hartman number 40 and more, causes the average Nusselt number to increase. In Ra=106 , the increase in nanofluid, to the Hartman number 20, contributes to increase of the average number and in the Hartman number 40 and more, causes the average Nusselt number to decrease. The results also indicate that, the maximum heat transfer, in Ra=105 and Ra=106 accurse at 67.5° angle. the minimum heat transfer, in Ra=105 and Ra=106 accurse at 0° and 22.5° angle respectively.

---

The significance of research on the specifications of the supercritical fluids becomes more evident with respect to the increase of their application in different food, chemical, polymer, oil, and gas industries. One of the major specification, is the coefficient of thermal expansion (β) where the ideal gas model was used in most of the processes in which this component is applied; the weakness of this model is that it is unable to make an accurate prediction of this parameter within the range of critical point. For this reason, in this study to determine the coefficient of thermal expansion, Redlich–Kwong equation of state is used and a new relation as a function of temperature, pressure, and compressibility is obtained. Comparing behavior of the curves obtained from this relation with experimental data, exhibits a favorable consistency. Moreover, natural convection heat transfer of the supercritical fluid in a vertical channel at constant temperature walls conditions were considered numerically. The governing equations were solved using the finite-volume method (FVM) and based on the SIMPLE Algorithm. After validation with the earlier studies. Then, the flow and heat transfer characteristics based on the obtained coefficient of thermal expansion were compared with the ideal gas assumptions. Finally, the trend of change in heat transfer coefficient away from the critical point was studied.

---

In this paper, natural convection of Cu-Water nanofluid inside an enclosure which is partially filled with porous media, with internal heat generation has been studied numerically. Cu-water nanofluid was used where Maxwell and Brinkmen models determine its properties. Due to the low velocity of nanofluid, Darcy-Brinkman equation used for the modeling of porous media. In order to gain the maximum energy from the temperature dependent heat source, different parameters such as Rayleigh number, volume fraction of nanoparticles, porosity of porous matrix and heat conduction ratio has been investigated. The results show that increasing the volume fraction of nanofluid increases Nusselt number at all porosities and Nusselt will further increases at lower porosities. Changes of thermal conductivity ratio were effective only at low porosities and causes to fast conduction of generated heat and two-fold increase in Nusselt number. Moreover the porosity changes at different thermal conductivity ratio Cause to minimum Nusselt at the porosity of 0.4 to 0.6. Increasing Rayleigh number will lead to nanofluid penetration increase into the porous matrix and with further matrix cooling more increase in Nusselt number in all porosity ranges will be achieved.

---

In this paper, natural convective heat transfer of nanofluids in a uniform magnetic field between the square cavity and inner cylinder, was simulated via Lattice Boltzmann Method. The inner cylinder in square shape, diamond, and circular has been examined. Square cavity walls and inner cylinder surfaces are at a constant cold and warm temperature, respectively. The flow, temperature, and magnetic field is calculated with solving flow, temperature, and magnetic distribution functions simultaneously. D2Q9 lattice arrangement for each distribution function is used. The results clearly show the behavior of fluid flow and heat transfer between the cavity and the cylinder. The results have been validated with available valid results showing relatively good agreement. The effects of Rayleigh number, Hartmann number, void fraction and type of nanoparticles on natural convective heat transfer are investigated. This study shows that for all three geometries used with the same void fraction, type of nanofluid, and Rayleigh number, natural convective heat transfer decreases with Hartmann number. Also, when Hartmann number was had fixed, natural convective heat transferwas increased with Rayleigh number. Thus, to select the right geometry for optimum natural convective heat transfer, our needs to pay special attention to Hartmann and Rayleigh numbers. In addition, viod fraction and type of nanofulid can affect heat transfer directly.

---

In this paper the mixed convection and entropy generation in a square cavity filled with Al2O3-water nanofluid with the presence of a constant axial magnetic field, is analyzed. The upper and bottom walls are adiabatic. Discretization of the governing equations were achieved through a finite volume method and solved with SIMPLE algorithm. In this research the effects of the Rayleigh number (103- 106), Hartmann number (0 - 100) and also inclination angle (0 - 90°) are investigated. When the cavity is rotated, it is observed that the mean Nusselt number and total entropy generation increase when the Rayleigh number increases in cavity. In square cavity, regardless of the Ha number, by increasing of the inclination angel, the mean Nusselt number and entropy generation rate, increase until inclination angel 30°, then decreases. Also when the magnetic field is rotated, it is observed that the mean Nusselt number decrease when the Hartmann number increases. The mean Nusselt number when the cavity rotates with specific inclination angel is less than state that the cavity rotates with specific magnetic field. For finding optimum condition of heat transfer, Artificial Neural networks (ANN) were used. The results from optimization show that as the Rayleigh number increases, the optimum angel decreases. Whatever the Rayleigh number more increases, the decrement in optimum angel more intenses. Also in low the Rayleigh number, as the Hartmann number increases, the optimum angel decreases firstly then increases. In high Rayleigh number, as the Hartmann number increases, the optimum angel increases too.

---

In this study, the numerical investigation of transient natural convection with respect to the effects of two-way fluid-structure interaction, is presented in a square enclosure containing a flexible baffle. The enclosure is filled with air of Prandtl number 0.71. Temperature is constant in both hot and cold vertical walls, while baffle and horizontal walls are adiabatic. Arbitrary Lagrangian-Eulerian (ALE) formulation is used to describe the fluid motion in the given model. Non-Dimensional fluid domain equations with relevant boundary conditions are discretized by the finite volume method (FVM), and PISO algorithm is used to solve the pressure-velocity coupling. Non-Dimensional equations of the baffle motion are solved by the finite element method (FEM) and Newton-Raphson iteration technique. Rayleigh number changes over the range of 10^3 to 10^6. Among the assessed cases in this study, 25 and 35 percentages of them indicate respectively, increase and decrease in the rate of heat transfer in compare with the enclosure containing a rigid baffle. Maximum and minimum values of Num,ss variation are respectively, 4.5 and -15.4 percent. In compare with the rigid baffle, about 90 percent of assessed cases indicate an increase in the time to reach the steady state situations, that it is not considered favorable.

---

In this study, the natural convection heat transfer is numerically examined in a square enclosure filled with a non-Newtonia power-law fluid. Two fixed temperature baffles are mounted on the left wall of the enclosure. The left wall of the enclosure and the baffles installed on it, are at a constant temperature of T_h and the right wall of the enclosure is at a constant temperature of T_c, while its horizontal walls are thermally insulated. The governing equations for the power-law fluid flow are solved with the numerical finite difference method based on the control volume formulation and SIMPLE algorithm. The study investigates the effects of relevant parameters such as the Rayleigh number (〖10〗^3≤Ra≤〖10〗^6), the power-law index (0.8≤n≤1.4), the baffles length (0≤B≤0.5) and the baffles distance from each other (0.1≤D≤0.8) on flow and temperature fields and the rate of heat transfer. The results show that an increase in Rayleigh number, particularly when n

---

The use of phase-change material to enhance the capacity of energy storage/release is the subject of many new researches on management of the energy supply. Study of these systems is directly related to the solid-liquid phase-change problem, in which the evaluation of temperature distribution, position of phase-change front and liquid or solid fraction becomes a basic problem. Study of freezing and melting process with regard to natural convection in the liquid phase is the main purpose of the present paper. For this purpose, a rectangular finned container of phase-change material is intended. Fins are used to enhance the heat transfer rate. This fact necessitates the use of immersed boundary condition on the solid phase. Hence, the melting process considering the both effects of natural convection and movement of solid phase is studied. The freezing process is also studied taking into accounts the natural convection with no need to impose the immersed boundary condition. Lattice Boltzmann method is used as a numerical method and results are reported based on the dimensionless parameters. Based on the results, the effects of natural convection is negligible during freezing process, while imposing the effects of natural convection provides a significant change in the required time for complete melting of the phase change material.

---

Supercritical fluids have substituted non-super critical fluids in some areas of industry because of their unique characteristics and have been the subject of numerous experimental, numerical and analytic studies since their discovery. In this study laminar natural convection between a hot vertical tube with constant temperature and supercritical carbon dioxide with uniform temperature at inlet is simulated by utilizing a numerical model. The simulation is a two-dimensional, pseudo-transient numerical model based on finite volume method. The main objective of this study is to investigate and analyze the effect of severe property variations of supercritical carbon dioxide on the flow and temperature field of natural convection that ultimately affect heat transfer rates with respect to non-critical natural convection. Numerical simulations have been carried out for temperature and pressure ranges of 305K to 312K and 7.5MPa to 9MPa respectively. Span and Wanger’s multi-parameter equation of state have been used directly to determine carbon dioxide properties around pseudo critical temperature for the first time. Results indicate an increased rate of total heat transfer up to 160% near pseudo-critical temperature and 118% in other temperatures for supercritical natural convection with respect to ideal gas assumption.

---

In this study, a Lattice Boltzmann Method (LBM) has been developed to calculate the distribution of a scalar quantity, like temperature, in a natural convection flow field under the condition of varying fluid thermal conductivity. The standard form of an LBM usually considers the fluid properties to be constant without any source term in conservation equations. The model developed is to account for variation of thermal conductivity with temperature in the presence of an external heat source. The proposed model has been examined against various case studies. It is shown that it is capable of modeling the extremely nonlinear problems. To magnify the nonlinear term in the natural convection case of under study, the radiation and other thermal sources have been used. The multiple relaxation time scheme has been applied to assure the solution stability. Using Chapman-Enskog analysis, the error associated with the proposed model has been estimated. The part of error which was not due to variations in the fluid properties, may be eliminated by introducing a correction term in higher order terms in Chapman-Enskog analysis. In addition, it has been shown that the correction term associated with the fluid conductivity variations, create an error of second order in terms of Knudsen number and is negligible. The present LBM model has an error of the second order of magnitude with respect to time.

---

In the present paper, the effect of fins presence on natural convection between coaxial annuli was investigated, numerically. The external duct was circular and the internal ducts included three circular, square, and triangular cross sections for discussed annuli. As a geometrical constrain, both cross section area and diameter of external duct of annuli were considered equivalent together for all investigated cases. The area of fins installed on the internal ducts was constant, and their effects on thermal behavior of annuli were compared with considering the constant wall temperature boundary condition for surfaces in the range of 105≤Ra≤108. The results showed that with increase of Rayleigh number and consequently velocity, the heat transfer coefficient was increased for both surfaces. However, presence of fins reduced the values of heat transfer coefficient of internal ducts about 50%, while they increased those values for external ducts. Also, in the case of circular annulus, with increase of Rayleigh number, the Nusselt number wincreased about 71% and 64% for non-finned and finned ones, respectively. As a result, fins increased the overall heat transfer rate of both surfaces of annuli about 13% in comparison of non-finned surfaces.