On the discrete ordinates method for radiative heat transfer in anisotropically scattering media

In the discrete ordinates method (DOM), the normalized condition for the numerical quadrature of some complex scattering phase functions may not be satisfied. In this paper, a revised discrete ordinates method (RDOM) is developed to overcome this problem, in which a renormalizing factor is added into the numerical quadrature of in-scattering term. The RDOM is used to solve the radiative transfer problem in one-dimensional anisotropically scattering media with complex phase function. The radiative heat fluxes obtained by the RDOM are compared with those obtained by the conventional discrete ordinates method (CDOM) and Monte Carlo method. The results show the RDOM can overcome the false scattering resulted from the numerical quadrature of in-scattering term and improve largely the accuracy of solution of the radiative transfer equation by comparison with the CDOM.     

Experimental and numerical study of heat and moisture transfers by natural convection in a cavity filled with solid obstacles

Heat, moisture transfers and airflow by natural convection in a rectangular cavity containing on line cylinders were studied. The work zone was arranged in such a way that 2D transfer and flow were established. At steady state, temperature, velocity and humidity fields on the symmetry plane were measured in un-humidified and humidified cavity. These results were then used to compare with CFD simulation. The thermal stratification and circular air flow in the cavity was observed. Humidification at the bottom face of cavity contributes to increase air velocity. The influence of radiation near the cold and warm walls is significant.     

Numerical simulation of mixed convection flows in a square lid-driven cavity partially heated from below using nanofluid

The present numerical study deals with mixed convection in a square lid-driven cavity partially heated from below and filled with water-base nanofluid containing various volume fractions of Cu, Ag, Al₂O₃ and TiO₂. Finite difference method was employed to solve the dimensionless governing equations of the problem. The effects of governing parameters, namely, Reynolds number, solid volume fraction, different values of the heat source length and different locations of the heat source on the streamlines and isotherms contours as well as Nusselt number and average Nusselt number along the heat source were considered. The present results are validated by favorable comparisons with previously published results. The results of the problem are presented in graphical and tabular forms and discussed.     

Three-dimensional numerical simulation of free convection and entropy generation in a cubic cavity containing a heat source

The present article provides a three-dimensional numerical investigation of thermal convection and entropy generation. The lattice Boltzmann method, coupled with the finite difference approach, is applied to perform numerical simulations. The validation of these numerical approaches for thermal convection simulation and entropy calculation is performed by comparing our numerical results with those in the published literature for the case of benchmark problems. The physical geometry studied in this paper concerns a hot obstacle having the shape of a plus sign (+) placed in the center of a cubic enclosure. This cube is filled with air of a Prandtl number of 0.71 and characterized by two cold vertical walls. The heat exchange between the fluid and the hot body is studied as a function of the Rayleigh number (${10}^3\le \mathit{Ra}\le {10}^7$). The performed simulations show that the heat transfer rate can be increased by about 429% by switching from $\mathit{Ra}={10}^3$ to ${10}^7$. The entropy generation due to fluid friction, heat transfer, and total entropy are also calculated and discussed. For an irreversibility coefficient $\mathit{\phi }={10}^{-4}$, the analysis of the results showed that for low values of the Rayleigh number ($\mathit{Ra}={10}^3$), the entropy production due to temperature gradients predominates over that produced by viscous effects. In the cases of $\mathit{Ra}={10}^4$ and ${10}^5$, entropy generation is due to both fluid friction and heat transfer. However, when the Rayleigh number becomes large ($\mathit{Ra}{\ge 10}^6$), entropy generation due to viscosity predominates over entropy production related to heat exchange. These results have important implications for the optimization and design of heat transfer systems in various industrial applications.     

Estimation of thermal and flow fields due to natural convection using support vector machines (SVM) in a porous cavity with discrete heat sources

Support vector machines (SVM), a soft programming technique, has been used to estimate the temperature distribution and flow fields in a square porous enclosure heated discretely by three isothermal heaters from the left vertical wall. Right vertical wall of the cavity was isothermal but it has colder temperature than the heaters while remaining walls were adiabatic. A database was prepared by solving the governing equations which were written using Darcy flow model. Using finite difference method to discretize the equation, a computational fluid dynamics (CFD) code was written. A correlation was developed between Nusselt and Rayleigh numbers. Using obtained database, further values of temperature and velocities were estimated by SVM technique at different Rayleigh numbers and locations of heater. It was observed that SVM was a useful technique on estimation of streamlines and isotherms. Thus, SVM reduces the computational time and helps to solve some cases when CFD fails to solve due to numerical instability.     

Lattice Boltzmann simulation of natural convection in an open ended cavity

Natural convection in an open ended cavity is simulated using Lattice Boltzmann Method (LBM). The paper is intended to address the physics of flow and heat transfer in open end cavities and close end slots. The flow is induced into the cavity by buoyancy force due to a heated vertical wall. Also, the paper demonstrated that open boundary conditions used at the opening of the cavity is reliable, where the predicted results are similar to conventional CFD method (finite volume method, FVM) predictions. Prandtl number (Pr) is fixed to 0.71 (air) while Rayleigh number (Ra) and aspect ratio (A) of the cavity are changed in the range of 10⁴–10⁶ and of 0.5–10, respectively. It is found that the rate of heat transfer deceases asymptotically as the aspect ratio increases and may reach conduction limit for large aspect ratio. The flow evaluation in the cavity starts with recirculation inside the cavity, as the time proceeds the flow inside the cavity communicates with the ambient.     

Experimental and numerical study of the natural convection from a heated horizontal cylinder

Natural convection heat transfer from a horizontal cylinder is studied experimentally and numerically. Experimental study had taken place in different environmental temperature in a conditioned room which can be maintained at a stable required value and inside a sufficiently designed test cabin. The environmental and cylinder surface temperatures varied between 10 °C–40 °C and 20 °C–60 °C respectively. In the experimental study, two cylinders having different diameters of 4.8 mm–9.45 mm are used and constant heat flux was applied. On the basis of the experimental data, a correlation for the average Nusselt number over the cylinder is proposed in the range of 7.4 10¹ < Ra < 3.4 10³. The proposed correlation is compared with the well known correlations on natural convection heat transfer from a horizontal cylinder in the specified range of Rayleigh number, and it is shown that the results are in satisfactory agreement. The problem is also investigated numerically. The experimental data and the numerical results fall in ± 20% band. The numerical results obtained in this study are also compared with the results of Merkin. The characteristics of trend lines are similar.     

Three-dimensional numerical computation for magnetic convection of air inside a cylinder heated and cooled isothermally from a side wall

Three-dimensional convection of air in a vertical cylinder isothermally heated and cooled from a side wall was numerically computed both in magnetic and gravity fields. A single electric coil was placed around a cylinder to generate a magnetic field. Convection was calculated for various coil levels and magnetic strengths. The gravity field, magnetic strength and Rayleigh number are shown to have substantial effect. Convection modes and heat transfer rates are also presented.     

A numerical investigation of transient natural convection heat transfer of aqueous nanofluids in a differentially heated square cavity

Transient natural convection heat transfer of aqueous nanofluids in a differentially heated square cavity is investigated numerically. The effective thermal conductivity and dynamic viscosity of nanofluids are predicted by using the proposed models that take the contribution of Brownian motion of nanoparticles into account. Three different Rayleigh numbers and five different volume fractions of nanoparticles are considered. The development of natural convection is presented through the evolutions of the average Nusselt number along the cold side wall. The predicted flow development times and time-averaged Nusselt numbers are scaled with Rayleigh number. In addition, the time-averaged Nusselt numbers are presented in terms of volume fraction of nanoparticles. It is shown that at constant Rayleigh numbers, the time-averaged Nusselt number is lowered with increasing volume fraction of nanoparticles.     

A numerical study on the effect of a heated hollow cylinder on mixed convection in a ventilated cavity

The present work is aimed to study mixed convection heat transfer characteristics within a ventilated square cavity having a heated hollow cylinder. The heated hollow cylinder is placed at the center of the cavity. In addition, the wall of the cavity is assumed to be adiabatic. Flows are imposed through the inlet at the bottom of the left wall and exited at the top of the right wall of the cavity. The present study simulates a practical system such as air-cooled electronic equipment with a heat component or an oven with heater. Emphasis is sited on the influences of the cylinder diameter and the thermal conductivity of the cylinder in the cavity. The consequent mathematical model is governed by the coupled equations of mass, momentum and energy and solved by employing Galerkin weighted residual method of finite element formulation. A wide range of pertinent parameters such as Reynolds number, Richardson number, cylinder diameter and the solid-fluid thermal conductivity ratio are considered in the present study. Various results such as the streamlines, isotherms, heat transfer rates in terms of the average Nusselt number and average fluid temperature in the cavity are presented for different aforesaid parameters. It is observed that the cylinder diameter has significant effect on both the flow and thermal fields but the solid-fluid thermal conductivity ratio has significant effect only on the thermal field.     

Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 10 - Part I: Numerical methods and time-averaged flow

A set of direct numerical simulations of a differentially heated cavity of aspect ratio 4 with adiabatic horizontal walls is presented. The five configurations selected here (Rayleigh numbers based on the cavity height and ) cover a relatively wide range of Ra from weak to fully developed turbulence. A short overview of the numerical methods and the methodology used to verify the code and the simulations is presented. The time-averaged flow results are presented and discussed in this first part. Significant changes are observed for the two highest Ra for which the transition point at the boundary layers clearly moves upstream. Such displacement increases the top and bottom regions of disorganisation shrinking the area in the cavity core where the flow is stratified. Consequently, thermal stratification values are significantly greater than unity (1.25 and 1.41, respectively).     

Numerical exploration of 1 + 2 type laminar natural convection in a differentially heated square cavity using Chebyshev spectral collocation method

In this paper, the Chebyshev spectral collocation method is applied to explore the unsteady two dimensional (1 + 2 type) laminar natural convection in a differentially heated square cavity at a Rayleigh number (Ra) of 10⁷. The method has embedded the traditional Chorin's algorithm so as to avoid the trouble of seeking the pressure field in the buoyancy driven wall-jet flow. The sensitivity of the δ− parameter has been numerically investigated. It is found that when the δ value is over 11.6173, numerical instability occurs. Comparing the maximum horizontal velocity component with the existing numerical data obtained by solving the Poisson's equation of pressure field reveals that the Chorin's algorithm should be inapplicable for the solution of the benchmark problem of natural convection at Ra = 10⁷ in thermal science.     

Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls

In the present study, entropy generation in rectangular cavities with the same area but different aspect ratios is numerically investigated. The vertical walls of the cavities are at different constant temperatures while the horizontal walls are adiabatic. Heat transfer between vertical walls occurs by laminar natural convection. Based on the obtained dimensionless velocity and temperature values, the distributions of local entropy generation due to heat transfer and fluid friction, the local Bejan number and local entropy generation number are determined and related maps are plotted. The variation of the total entropy generation and average Bejan number for the whole cavity volume at different aspect ratios for different values of the Rayleigh number and irreversibility distribution ratio are also evaluated. It is found that for a cavity with high value of Rayleigh number (i.e., Ra = 10⁵), the total entropy generation due to fluid friction and total entropy generation number increase with increasing aspect ratio, attain a maximum and then decrease. The present results are compared with reported solutions and excellent agreement is observed. The study is performed for 10² < Ra < 10⁵, 10^− 4 < ? < 10^− 2, and Pr = 0.7.     

Numerical simulation of natural convection in an open-ended square cavity filled with porous medium by lattice Boltzmann method

In the present work, natural convection in an open-ended square cavity packed with porous medium is simulated. The double-population approach is used to simulate hydrodynamic and thermal fields, and the Taylor series expansion and the least-squares-based lattice Boltzmann method has been implemented to extend the thermal model. The effect of a porous medium is taken into account by introducing the porosity into the equilibrium distribution function and adding a force term to the evolution equation. The Brinkman–Forchheimer equation, which includes the viscous and inertial terms, is applied to predict the heat transfer and fluid dynamics in the non-Darcy regime. The present model is validated with the previous literature. A comprehensive parametric study of natural convective flows is performed for various values of Rayleigh number and porosity. It is found that these two parameters have considerable influence on heat transfer.     

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.     

A numerical method for natural convection and heat conduction around and in a horizontal circular pipe

Natural convection around a horizontal circular pipe coupled with heat conduction in the solid structure is numerically investigated using a preconditioning method for solving incompressible and compressible Navier–Stokes equations. In this method, fundamental equations are completely reduced to an equation of heat conduction when the flow field is static (zero velocity). Therefore, not only compressible flows but also very slow flows such as natural convection in a flow field and heat conduction in a static field can be simultaneously calculated using the same computational algorithm. In this study, we first calculated the compressible flow around a NACA0012 airfoil with conduction in the airfoil and then simulated natural convections around a horizontal circular pipe with a different heat conductivity. Finally, we numerically investigated the effect of heat conductivity of the pipe on natural convection.     

Effect of radiation on natural convection flow on a sphere in presence of heat generation

Effect of radiation on natural convection flow around a sphere in presence of heat generation has been investigated. The governing equations are transformed into dimensionless non-similar equations by using a set of suitable transformations and solved numerically by the finite difference method along with Newton's linearization approximation. We have focused our attention on the evaluation of velocity profiles, temperature profiles, shear stress in terms of local skin friction and the rate of heat transfer in terms of local Nusselt number for different values of heat generation parameter, radiation parameter and the Prandlt number and the numerical results have been shown graphically.     

Experimental and numerical study of the natural convection from a heated horizontal cylinder wrapped with a layer of textile material

Natural convection heat transfer enhancement from a horizontal cylinder with a textile coating is studied experimentally and numerically. The coating layer may be used for two purposes. According to the thickness of the coating it may be used as an insulating material or for surface augmentation. In the experimental study, two cylinders having different diameters of 4.8 mm and 9.45 mm are used. The bare cylinders having a radius smaller than a certain critical size were wrapped with a textile material. Wrapped cylinder diameters were increased to 9 and 12.8 mm respectively after coating and constant heat flux was applied to all bare and wrapped cylinders. Experimental study was carried out at different ambient temperatures in a conditioned room which can be maintained at a stable required value and inside a sufficiently designed test cabin. The ambient and cylinder surface temperatures (T_∞ and T_w) varied between 10 °C – 40 °C and 20 °C – 60 °C respectively. Heat transfer rates from bare and wrapped horizontal cylinders were compared and heat transfer enhancement was observed. On the basis of the experimental data average Nusselt numbers were calculated and compared with the well known correlations on natural convection heat transfer from a horizontal cylinder in the specified range of Rayleigh number, and it is seen that the results are in good agreement. The problem is also investigated numerically. Experimental and the numerical results fall in ± 30% band.     

Numerical simulation of free convection heat transfer in a vertical annular cavity with discrete heating

The main objective of this article is to investigate the effect of discrete heating on convection heat transfer in a vertical cylindrical annulus. In this analysis, the inner wall of the cavity has two discrete flush-mounted heat sources and the outer wall is isothermally cooled at a lower temperature and top and bottom walls are thermally insulated. The governing equations are solved using an implicit finite difference technique to investigate the influence of each parameter and in particular the radii ratio. The numerical results reveal that the heat transfer rate is always higher at the bottom heater. Also, the rate of heat transfer increases with radii ratio but decreases with aspect ratio. Further, the present results are in excellent agreement with the existing benchmark solutions.