This work presents a numerical analysis of entropy generation in Γ-Shaped enclosure that was submitted to the natural convection process using a simple thermal lattice Boltzmann method (TLBM) with the Boussinesq approximation. A 2D thermal lattice Boltzmann method with 9 velocities, D2Q9, is used to solve the thermal flow problem. The simulations are performed at a constant Prandtl number (Pr = 0.71) and Rayleigh numbers ranging from 103 to 106 at the macroscopic scale (Kn = 10-4). In every case, an appropriate value of the characteristic velocity is chosen using a simple model based on the kinetic theory. By considering the obtained dimensionless velocity and temperature values, the distributions of entropy generation due to heat transfer and fluid friction are determined. It is found that for an enclosure with high value of Rayleigh number (i.e., Ra=105), the total entropy generation due to fluid friction and total Nu number increases with decreasing the aspect ratio.
One dim, steady-state, heat conduction_with_heat_generationtmuliya
This file contains slides on One-dimensional, steady-state heat conduction with heat generation.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
It is hoped that these Slides will be useful to teachers, students, researchers and professionals working in this field.
ANALYSIS OF POSSIBILITY OF GROWTH OF SEVERAL EPITAXIAL LAYERS SIMULTANEOUSLY ...ijoejournal
We analyzed nonlinear model with varying in space and time coefficients of growth of epitaxial layers
from gas phase in a vertical reactor with account native convection. We formulate several conditions to
increase homogeneity of epitaxial layers with varying of technological process parameters.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Heat transfer from extended surfaces (or fins)tmuliya
This file contains slides on Heat Transfer from Extended Surfaces (FINS). The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Governing differential eqn – different boundary conditions – temp. distribution and heat transfer rate for: infinitely long fin, fin with insulated end, fin losing heat from its end, and fin with specified temperatures at its ends – performance of fins - ‘fin efficiency’ and ‘fin effectiveness’ – fins of non-uniform cross-section- thermal resistance and total surface efficiency of fins – estimation of error in temperature measurement - Problems
The term phase transition (or phase change) is most commonly used to describe transitions between solid, liquid and gaseous states of matter, and, in rare cases, plasma (physics). A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of the change of some external condition, such as temperature, pressure, or others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition. Phase transitions are common in nature and used today in many technologies.
The microscopic effect of the exchange interaction parameter for the 2-Dimensional ising model with nearest neighbor interaction has been studied. By supposing simple temperature dependent relationship for the exchange parameter, graphs were straightforwardly obtained that show the reentrant closed looped phase diagrams symptomatic of some colloids and complex fluids and some binary liquids mixtures in particular. By parameter modifications, other phase diagrams were also obtained. Amongst which are the u-shapes and other exotic shapes of phase diagrams. Our results show that the exchange interaction parameter greatly influence the size of the ordered phase. Hence the larger the value of the constant, the larger the size of the ordered phase. This means that the higher values of the exchange parameter brings about phase transitions that straddle a wider range of polarizations and temperatures.
This file contains slides on Transient Heat conduction: Part-I
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010. Contents: Lumped system analysis – criteria for lumped system analysis – Biot and Fourier Numbers – Response time of a thermocouple - One-dimensional transient conduction in large plane walls, long cylinders and spheres when Bi > 0.1 – one-term approximation - Heisler and Grober charts- Problems
Stellar Measurements with the New Intensity FormulaIOSR Journals
In this paper a linear relationship in stellar optical spectra has been found by using a
spectroscopical method used on optical light sources where it is possible to organize atomic and ionic data.
This method is based on a new intensity formula in optical emission spectroscopy (OES). Like the HR-diagram ,
it seems to be possible to organize the luminosity of stars from different spectral classes. From that organization
it is possible to determine the temperature , density and mass of stars by using the new intensity formula. These
temperature, density and mass values agree well with literature values. It is also possible to determine the mean
electron temperature of the optical layers (photospheres) of the stars as it is for atoms in the for laboratory
plasmas. The mean value of the ionization energies of the different elements of the stars has shown to be very
significant for each star. This paper also shows that the hydrogen Balmer absorption lines in the stars follow
the new intensity formula.
By using the anharmonic correlated einstein model to define the expressions o...Premier Publishers
By using potential effective interaction in the anharmonic correlated Einstein model on the basis of quantum statistical theory with phonon interaction procedure, the expressions describing asymmetric component (cumulants) and thermodynamic parameters including the anharmonic effects contributions and by new structural parameters of cubic crystals have been formulated. These new parameters describe the distribution of atoms. The expansion of cumulants and thermodynamic parameters through new structural parameters has been performed. The results of this study show that, developing further the anharmonic correlated Einstein model it obtained a general theory for calculation cumulants and thermodynamic parameters in XAFS theory including anharmonic contributions. The expressions are described through new structural parameters that agree with structural contributions of cubic crystals like face center cubic (fcc), body center cubic (bcc).
Heat Capacity of BN and GaN binary semiconductor under high Pressure-Temperat...IOSR Journals
In this paper, we have calculated the molar heat capacity for cubic zinc blende (cZB) BN and GaN binary semiconductors at high pressure-temperature (PT). For the calculation of heat capacity, we firstly obtained the Debye temperature (ϴD) variation with temperature and at higher temperature it becomes constant with temperature in quasi-harmonic approximation limits. We have also calculated the static Debye temperature (ϴD) from elastic constant for the both BN and GaN binary semiconductors. The elastic constants are calculated from the energy-strain relation using plane wave method in DFT approach. All the calculated results are well consistence with experimental and reported data
Fouling, in technical language, it is the general term of unwanted material which is accumulating on surfaces, such as inside pipes, machines or heat exchanger.
One dim, steady-state, heat conduction_with_heat_generationtmuliya
This file contains slides on One-dimensional, steady-state heat conduction with heat generation.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
It is hoped that these Slides will be useful to teachers, students, researchers and professionals working in this field.
ANALYSIS OF POSSIBILITY OF GROWTH OF SEVERAL EPITAXIAL LAYERS SIMULTANEOUSLY ...ijoejournal
We analyzed nonlinear model with varying in space and time coefficients of growth of epitaxial layers
from gas phase in a vertical reactor with account native convection. We formulate several conditions to
increase homogeneity of epitaxial layers with varying of technological process parameters.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Heat transfer from extended surfaces (or fins)tmuliya
This file contains slides on Heat Transfer from Extended Surfaces (FINS). The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Governing differential eqn – different boundary conditions – temp. distribution and heat transfer rate for: infinitely long fin, fin with insulated end, fin losing heat from its end, and fin with specified temperatures at its ends – performance of fins - ‘fin efficiency’ and ‘fin effectiveness’ – fins of non-uniform cross-section- thermal resistance and total surface efficiency of fins – estimation of error in temperature measurement - Problems
The term phase transition (or phase change) is most commonly used to describe transitions between solid, liquid and gaseous states of matter, and, in rare cases, plasma (physics). A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of the change of some external condition, such as temperature, pressure, or others. For example, a liquid may become gas upon heating to the boiling point, resulting in an abrupt change in volume. The measurement of the external conditions at which the transformation occurs is termed the phase transition. Phase transitions are common in nature and used today in many technologies.
The microscopic effect of the exchange interaction parameter for the 2-Dimensional ising model with nearest neighbor interaction has been studied. By supposing simple temperature dependent relationship for the exchange parameter, graphs were straightforwardly obtained that show the reentrant closed looped phase diagrams symptomatic of some colloids and complex fluids and some binary liquids mixtures in particular. By parameter modifications, other phase diagrams were also obtained. Amongst which are the u-shapes and other exotic shapes of phase diagrams. Our results show that the exchange interaction parameter greatly influence the size of the ordered phase. Hence the larger the value of the constant, the larger the size of the ordered phase. This means that the higher values of the exchange parameter brings about phase transitions that straddle a wider range of polarizations and temperatures.
This file contains slides on Transient Heat conduction: Part-I
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010. Contents: Lumped system analysis – criteria for lumped system analysis – Biot and Fourier Numbers – Response time of a thermocouple - One-dimensional transient conduction in large plane walls, long cylinders and spheres when Bi > 0.1 – one-term approximation - Heisler and Grober charts- Problems
Stellar Measurements with the New Intensity FormulaIOSR Journals
In this paper a linear relationship in stellar optical spectra has been found by using a
spectroscopical method used on optical light sources where it is possible to organize atomic and ionic data.
This method is based on a new intensity formula in optical emission spectroscopy (OES). Like the HR-diagram ,
it seems to be possible to organize the luminosity of stars from different spectral classes. From that organization
it is possible to determine the temperature , density and mass of stars by using the new intensity formula. These
temperature, density and mass values agree well with literature values. It is also possible to determine the mean
electron temperature of the optical layers (photospheres) of the stars as it is for atoms in the for laboratory
plasmas. The mean value of the ionization energies of the different elements of the stars has shown to be very
significant for each star. This paper also shows that the hydrogen Balmer absorption lines in the stars follow
the new intensity formula.
By using the anharmonic correlated einstein model to define the expressions o...Premier Publishers
By using potential effective interaction in the anharmonic correlated Einstein model on the basis of quantum statistical theory with phonon interaction procedure, the expressions describing asymmetric component (cumulants) and thermodynamic parameters including the anharmonic effects contributions and by new structural parameters of cubic crystals have been formulated. These new parameters describe the distribution of atoms. The expansion of cumulants and thermodynamic parameters through new structural parameters has been performed. The results of this study show that, developing further the anharmonic correlated Einstein model it obtained a general theory for calculation cumulants and thermodynamic parameters in XAFS theory including anharmonic contributions. The expressions are described through new structural parameters that agree with structural contributions of cubic crystals like face center cubic (fcc), body center cubic (bcc).
Heat Capacity of BN and GaN binary semiconductor under high Pressure-Temperat...IOSR Journals
In this paper, we have calculated the molar heat capacity for cubic zinc blende (cZB) BN and GaN binary semiconductors at high pressure-temperature (PT). For the calculation of heat capacity, we firstly obtained the Debye temperature (ϴD) variation with temperature and at higher temperature it becomes constant with temperature in quasi-harmonic approximation limits. We have also calculated the static Debye temperature (ϴD) from elastic constant for the both BN and GaN binary semiconductors. The elastic constants are calculated from the energy-strain relation using plane wave method in DFT approach. All the calculated results are well consistence with experimental and reported data
Fouling, in technical language, it is the general term of unwanted material which is accumulating on surfaces, such as inside pipes, machines or heat exchanger.
Computational Analysis of Natural Convection in Spherical Annulus Using FEVIJMER
HEAT transfer by natural convection from a body to its finite enclosure is of importance
in nuclear reactor technology, electronic instrumentation packaging, aircraft cabin design, the
analysis of fluid suspension gyrocompasses, and numerous other practical situations. The steady
natural convection heat transfer of fluids between two concentric isothermal spheres is investigated
computationally with the help of FEV in ANSYS 14.5. The inner wall is subjected to a higher
temperature and outer is at room temperature. The steady behavior of the flow field and its
subsequent effect on the temperature distribution for different Rayleigh numbers and radius ratios
are analyzed.
Bossious boundary condition is taken for natural convection and which is solved in fluent
module. Steady solutions of the entire flow field is obtained for Rayleigh number (5x101<ra><105),><rr><3). The result shows that the Rayleigh number and
radius ratio have a profound influence on the temperature and flow fields and Prandlt number has
very negligible effect. The results of average Nusselt numbers are also compared with those of
previous numerical investigations. Excellent agreement is obtained.
FREE CONVECTION HEAT TRANSFER OF NANOFLUIDS FROM A HORIZONTAL PLATE EMBEDDED ...AEIJjournal2
In this paper the natural convection heat transfer from a horizontal plate embedded in a porous medium
saturated with a nanofluid is numerically analyzed. By a similarity approach the partial differential
equations are reduced to a set of two ordinary differential equations. In order to evaluate the influence of
nanoparticles on the heat transfer, Ag and Cuo as the nanoparticles were selected. Results show that heat
transfer rate (Nur) is a decreasing function of volume fraction of nanoparticles.
Fuzzy numbers, Nth - order Fuzzy Initial Value Problems, Runge-Kutta method, ...IOSR Journals
A numerical study is presented of two-dimensional laminar steady-state on megneto-hydrodynamics
(MHD) free convection for heat flow patterns within trapezoidal enclosures. A finite element analysis is
performed to investigate the effects of unifor heating and is also used for solving the Navier-Stokes and
Energybalance equations.In this study, cold bottom walls, uniformly heated left and right (side) walls and
insulated top walls with inclination angles (ф) are considered in a trapezoidal enclosure. The present numerical
procedure adopted in this investigation yields consistent performance over a wide range of parameters, Prandtl
numbers, (Pr = 0.026 - 0.7), and Rayleigh numbers (Ra = 103 – 105), Hartmann number (Ha = 50) with various
tilt angles Ф = 450, 300 and 00(square).Numerical results are presented in terms of streamlines, isotherms, heat
function (total heat flux) and nusselt numbers.for different Ra and Pr. As Ra increases conduction dominant
region changes for different Pr. Complete heat transfer analysis is performed in terms of local and average
nusselt numbers.
The Effects of Nanofluids on Forced Convection Heat Transfer Inside Parallel ...AI Publications
A numerical solution on forced convection of Al2O3-water nanofluid for different volume fractions is investigated for laminar flow through a parallel plate with flush mounted discrete heat sources. The model used for nanofluid mixture is a single-phase approach and fluid properties are considered constant with temperature. The finite difference method is used for solutions and four different volume fractions are considered varying from 0% to 4%. A fully developed laminar velocity profile is considered and the parallel plate is assumed as heated with three discrete heat sources flush mounted to the top and bottom plate with the same lengths. Uniform wall temperature boundary condition is taken for discrete heaters. Peclet numbers are in the range of 20-100. For comparison and validity of the solution the results for a classical problem, laminar flow through a parallel plate which is heated at the downstream region with constant temperature, are obtained. Results are presented in terms of bulk temperature, heat flux, and local Nusselt number. Heat transfer is enhanced with the particle volume concentration. For comparison, pure water results are also shown in the figures. At the locations where heat is applied the heat flux values decrease as the volume fraction increase and the bulk temperature values are higher for the higher volume fractions at the heated locations. As the volume fraction increases the local Nusselt number can increase up to 30% than to pure water.
Similarity Solution of an Unsteady Heat and Mass Transfer Boundary Layer Flow...iosrjce
The unsteady hydromagnetic boundary layer flow of an incompressible and electrically conducting
fluid through a porous medium bounded by a moving surface has been considered. It is assumed that the moving
surface has a velocity profile with respect to time and fluid flow is taken under the influence of a transverse
magnetic field. The similarity solution is used to transform the system of partial differential equations,
describing the problem under consideration, into a boundary value problem of coupled ordinary differential
equations and an efficient numerical technique is implemented to solve the reduced system. The effects of the
parameters such as Magnetic parameter, Prandtl number and Eckert number are discussed graphically on
velocity and temperature distributions
Using Lattice Boltzmann Method to Investigate the Effects of Porous Media on ...A Behzadmehr
A numerical investigation of forced convection in a channel with hot solid block inside a square porous block mounted on a bottom wall was carried out. The lattice Boltzmann method was applied for numerical simulations. The fluid flow in the porous media was simulated by Brinkman-Forchheimer model. The effects of parameters such as porosity and thermal conductivity ratio over flow pattern and thermal field were investigated. In this paper the effects of mentioned parameters were discussed in detail. The result show with increasing the thermal conductivity ratio and porosity the fluid temperature will reduce.
Thermodynamics analysis of diffusion in spark plasma sintering welding Cr3C2 ...AliFeiz3
In the thermodynamics analysis of diffusion in spark plasma sintering (SPS) welding of Cr3C2 (chromium carbide) and Ni (nickel), various thermodynamic principles and concepts are applied to understand the heat and mass transfer processes involved. SPS is a specialized technique used to consolidate powders into dense materials using pulsed direct current and pressure.
The focus of the analysis is on diffusion, which refers to the movement of atoms or molecules from regions of high concentration to regions of low concentration. Diffusion plays a crucial role in the welding process of Cr3C2 and Ni, as it influences the formation of interfacial bonds between the particles.
Thermodynamic analysis involves examining the energy changes and driving forces associated with diffusion during the SPS welding process. This analysis aims to determine the factors that govern the diffusion process, such as temperature, pressure, concentration gradients, and material properties.
By studying the thermodynamics of diffusion, researchers can gain insights into the kinetics and mechanisms of atomic or molecular movement, as well as the resulting microstructural changes and bonding at the interfaces between Cr3C2 and Ni particles. This knowledge helps optimize the SPS welding process parameters and improve the quality and properties of the welded material.
Key aspects explored in the thermodynamics analysis may include heat transfer mechanisms, such as Joule heating during SPS, and mass transfer phenomena, such as atomic diffusion of Cr, C, and Ni species. The analysis may also consider thermodynamic properties of the materials involved, such as melting points, phase diagrams, and chemical potential gradients, to understand the driving forces for diffusion.
Overall, the thermodynamics analysis of diffusion in spark plasma sintering welding of Cr3C2 and Ni provides a deeper understanding of the fundamental principles governing the welding process, aiding in the development of advanced materials with enhanced properties and performance.
NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFERFROM SQUARE CYLIND...ijmech
The enhancement of natural convection heat transfer using nanofluids from horizontal square cylinder
placed in a square enclosure is investigated numerically. Water-based Cu is used as the working nanofluid.
The investigation covered a range of Rayleigh numbers of 104
- 106
, nanoparticles volume fraction of
(0<ϕ≤0.2), enclosure width to cylinder height ratio, W/H of 2.5. The investigation includes the solution of
the governing equations in the Vorticity-Stream function space with the aid of a body fitted coordinate
system. Algebraic grid generation is used in the initial transformations, followed by an elliptic
transformation to complete the grid generation to computational domain. The resulting discretized system
of equations is solved using an ADI method. The built code is validated and the results showed an increase
in average Nusselt number with increasing the volume fraction of the nanoparticles for the whole range of
Rayleigh number. The isotherms are nearly similar when the volume fraction of nanoparticles is increased
from 0 to 0.2 for each Rayleigh number but a change in the streamlines is observed.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Melting Heat Transfer in MHD Boundary Layer Stagnation-Point Flow towards a S...iosrjce
An analysis is carried out to study the MHD steady two-dimensional stagnation-point flow and heat transfer from
a warm, laminar liquid flow to a melting stretching sheet. The governing partial differential equations are transformed into
ordinary differential equations by similarity transformation, before being solved numerically using Runge-Kutta-Fehlberg
method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are
presented for different values of the governing parameters. Effects of the Magnetic parameter, melting parameter, stretching
parameter and prandtl number on the flow and heat transfer characteristics are thoroughly examined.
Basic Study on Solid-Liquid Phase Change Problem of Ice around Heat Transfer ...IJERDJOURNAL
Abstract:- Phase change heat transfer around heat transfer tubes is one of the basic problem of an ice heat storage exchanger. It can lead to decrease of thermal storage efficiency and damage of heat transfer tubes if continued freezing further after the ice has bridged because of the generated ice thermal resistance and volume expansion. In this study, we focused on freezing phenomena of phase change material (PCM) between two heat transfer tubes, which can simulate an inside structure of ice heat storage exchangers. Bridging time between two heat transfer tubes was studied numerically. We used water as the PCM, which is filled in the water container. Two horizontal elliptical tubes were used as heat transfer tubes in order to observe the influence of natural convection. Single-domain calculation model was used to calculate arbitrary shape of the two tubes during the ice freezing process. We changed arranged angle and relative position of the tubes to investigate impact of the tube arrangement on freezing phenomenon. In order to confirm the accuracy of our analysis, analytical results were compared with experimental results at the same conditions. Results show that the bridging time was not simply in proportional to the initial temperature of water when considered the natural convection influenced by such as density inversion of water. Moreover, we found that when we set the temperature of tube wall and initial temperature of water as the parameters, bridging time has a similar trend with distance between the axes. Therefore, it is possible to predict the bridging time for elliptical heat transfer tube.
Experiment on single-mode feedback control of oscillatory thermocapillary con...IJERA Editor
Feedback control was carried out on nonlinear thermocapillary convections in a half-zone liquid bridge of a high
Prandtl number fluid under normal gravity. In the liquid bridge, the convection changed from a two-dimensional
steady flow to a three-dimensional oscillatory flow at a critical temperature difference. Feedback control was
realized by locally modifying the free surface temperature using local temperature measured at different
positions. The present study aims to confirm whether the control method can effectively suppress oscillatory
flows with every modal structure. Consequently, the control was theoretically verified to be effective for
oscillatory flows with every modal structure in a high Marangoni number range.
A three-dimensional numerical analysis of laminar natural convection with entropy generation in an open trapezoidal cavity filled with water has been carried out. In this investigation, the inclined wall is maintained at isothermal hot temperature while cold water enters into the cavity from its right open boundary and all other walls are assumed to be perfect thermal insulators. Attention is paid on the effects of buoyancy forces on the flow structure and temperature distribution inside the open enclosure. Rayleigh number is the main parameter which changes from 103 to 105 and Prandtl number is fixed at Pr =6.2. Obtained results have been presented in the form of particles trajectories, iso-surfaces of temperature and those of entropy generated as well as the average Nusselt number. It has been found that the flow structure is sensitive to the value of Rayleigh number and that heat transfer increases with increasing this parameter.
Mixed Convection of Variable Properties Al2O3-EG-Water Nanofluid in a Two-Dim...A Behzadmehr
In this paper, mixed convection of Al2O3-EG-Water nanofluid in a square lid-driven enclosure is investigated numerically. The focus of this study is on the effects of variable thermophysical properties of the nanofluid on the heat transfer characteristics. The top moving and the bottom stationary horizontal walls are insulated, while the vertical walls are kept at different constant temperatures. The study is carried out for Richardson numbers of 0.01–1000, the solid volume fractions of 0–0.05 and the Grashof number of 104. The transport equations are solved numerically with a finite volume approach using the SIMPLER algorithm. The results show that the Nusselt number is mainly affected by the viscosity, density and conductivity variations. For low Richardson numbers, although viscosity increases by increasing the nanoparticles volume fraction, due to high intensity convection of enhanced conductivity nanofluid, the average Nusselt number increases for both constant and variable cases. However, for high Richardson numbers, as the volume fraction of nanoparticles increases heat transfer enhancement occurs for the constant properties cases but deterioration in heat transfer occurs for the variable properties cases. The distinction is due to underestimation of viscosity of the nanofluid by the constant viscosity model in the constant properties cases and states important effects of temperature dependency of thermophysical properties, in particular the viscosity distribution in the domain.
Moving Lids Direction Effects on MHD Mixed Convection in a Two-Sided Lid-Driv...A Behzadmehr
Magnetohydrodynamic (MHD) mixed convection flow of Cu–water nanofluid inside a two-sided lid-driven square enclosure with adiabatic horizontal walls and differentially heated sidewalls has been investigated numerically. The effects of moving lids direction, variations of Richardson number, Hartmann number, and volume fraction of nanoparticles on flow and temperature fields have been studied. The obtained results show that for a constant Grashof number (), the rate of heat transfer increases with a decrease in the Richardson and Hartmann numbers. Furthermore, an increase of the volume fraction of nanoparticles may result in enhancement or deterioration of the heat transfer performance depending on the value of the Hartmann and Richardson numbers and the configuration of the moving lids. Also, it is found that in the presence of magnetic field, the nanoparticles have their maximum positive effect when the top lid moves rightward and the bottom one moves leftward.
Influence of Interface Thermal Resistance on Relaxation Dynamics of Metal-Die...A Behzadmehr
Nanocomposite materials, including noble metal nanoparticles embedded in a dielectric host medium, are interesting because of their optical properties linked to surface plasmon resonance phenomena. For studding of nonlinear optical properties and/or energy transfer process, these materials may be excited by ultrashort pulse laser with a temporal width varying from some femtoseconds to some hundreds of picoseconds. Following of absorption of light energy by metal-dielectric nanocomposite material, metal nanoparticles are heated. Then, the thermal energy is transferred to the host medium through particle-dielectric interface. On the one hand, nonlinear optical properties of such materials depend on their thermal responses to laser pulse, and on the other hand different parameters, such as pulse laser and medium thermodynamic characterizes, govern on the thermal responses of medium to laser pulse. Here, influence of thermal resistance at particle-surrounding medium interface on thermal response of such material under ultrashort pulse laser excitation is investigated. For this, we used three temperature model based on energy exchange between different bodies of medium. The results show that the interface thermal resistance plays a crucial role on nanoparticle cooling dynamics, so that the relaxation characterized time increases by increasing of interface thermal resistance.
Numerical Analysis of Inlet Gas-Mixture Flow Rate Effects on Carbon Nanotube ...A Behzadmehr
The growth rate and uniformity of Carbon Nano Tubes (CNTs) based on Chemical Vapor Deposition (CVD)
technique is investigated by using a numerical model. In this reactor, inlet gas mixture, including xylene as
carbon source and mixture of argon and hydrogen as carrier gas enters into a horizontal CVD reactor at
atmospheric pressure. Based on the gas phase and surface reactions, released carbon atoms are grown as CNTs on the iron catalysts at the reactor hot walls. The effect of inlet gas-mixture flow rate, on CNTs growth rate and its uniformity is discussed. In addition the velocity and temperature profile and also species concentrations throughout the reactor are presented.
Effect of Nanoporous Anodic Aluminum Oxide (AAO) Characteristics On Solar Abs...A Behzadmehr
Nanoporous anodic aluminum oxide (AAO) has been used in many different fields of science and technology, due to its great structural characteristics. Solar selective surface is an important application of this type porous material. This paper investigates the effect of nanoporous AAO properties, including; film thickness, pore area percentage and pore diameter, on absorption spectra in the range of solar radiation. The parameters were verified individually depending on anodization condition, and the absorption spectra were characterized using spectrophotometer analysis. The results showed that the absorptivity was increased with growth of the film thickness. Furthermore, increasing the pore diameter shifted the absorption spectra to the right range, and vice versa. The investigation revealed the presence of an optimum pore area percentage around 14% in which the absorptivity was at its maximum value.
MHD Nanofluid Flow Analysis in a Semi-Porous Channel by a Combined Series Sol...A Behzadmehr
In this paper, Least Square Method (LSM) and Differential Transformation Method (DTM) are used to solve the problem of laminar nanofluid flow in a semi-porous channel in the presence of transverse magnetic field. Due to existence some shortcomings in each method, a novel and efficient method named LS-DTM is introduced which omitted those defects and has an excellent agreement with numerical solution. In the present study, the effective thermal conductivity and viscosity of nanofluid are calculated by Maxwell–Garnetts (MG) and Brinkman models, respectively. The influence of the three dimensionless numbers: the nanofluid volume friction, Hartmann number and Reynolds number on non-dimensional velocity profile are considered. The results show that velocity boundary layer thickness decrease with increase of Reynolds number and nanoparticle volume friction and it increases as Hartmann number increases.
Numerical Study of Mixed Convection of Nanofluid in a Concentric Annulus with...A Behzadmehr
In this work, the steady and laminar mixed convection of nanofluid in horizontal concentric annulus with
rotating inner cylinder is investigated numerically. The inner and outer cylinders are kept at constant
temperature Ti and To respectively, where Ti>To. The annular space is filled with Alumina-water nanofluid.
The governing equations with the corresponded boundary conditions in the polar coordinate are discretized
using the finite volume method where pressure-velocity coupling is done by the SIMPLER algorithm.
Numerical results have been obtained for Rayleigh number ranging from 102 to 105, Reynolds number from 1 to 300 and nanoparticles volume fraction from 0.01 to 0.06. The effects of the Reynolds and Rayleigh numbers, average diameter of nanoparticles and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the annuli are investigated. According to the results, the average Nusselt number decreases with increasing the Reynolds number. However, the average Nusselt number increases by increasing the Rayleigh number. Moreover, the maximum average Nusselt number occurs for an optimal nanoparticle volume fraction except situations that heat conduction predominates over the heat convection. In these conditions the average Nusselt number is close to unity.
Study on Thermal and Hydrodynamic Indexes of a Nanofluid Flow in a Micro Heat...A Behzadmehr
The paper numerically presents laminar forced convection of a nanofluid flowing in a duct at microscale.
Results were compared with both analytical and experimental data and observed good concordance with
previous studies available in the literature. Influences of Brinkman and Reynolds number on thermal and
hydrodynamic indexes have been investigated. For a given nanofluid, no change in efficiency (heat dissipation
to pumping power) was observed with an increasing in Reynolds number. It was shown that the pressure was
decrease with an increase in Brinkman number. Dependency of Nu increment changes with substrate material.
Numerical Analysis of Inlet Gas-Mixture Flow Rate Effects on Carbon Nanotube ...A Behzadmehr
The growth rate and uniformity of Carbon Nano Tubes (CNTs) based on Chemical Vapor Deposition (CVD)
technique is investigated by using a numerical model. In this reactor, inlet gas mixture, including xylene as
carbon source and mixture of argon and hydrogen as carrier gas enters into a horizontal CVD reactor at
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Natural Convection and Entropy Generation in Γ-Shaped Enclosure Using Lattice Boltzmann Method
1. 1
Trans. Phenom. Nano Micro Scales, 1(1): 1-18, Winter - Spring 2013
DOI: 10.7508/tpnms.2013.01.001
ORIGINAL RESEARCH PAPER .
Natural Convection and Entropy Generation in Γ-Shaped Enclosure
Using Lattice Boltzmann Method
E. Fattahi1
, M. Farhadi1,*
, K. Sedighi1
Faculty of Mechanical Engineering, Babol University of Technology Babol, Iran
Abstract
This work presents a numerical analysis of entropy generation in Γ-Shaped enclosure that was submitted to the
natural convection process using a simple thermal lattice Boltzmann method (TLBM) with the Boussinesq
approximation. A 2D thermal lattice Boltzmann method with 9 velocities, D2Q9, is used to solve the thermal
flow problem. The simulations are performed at a constant Prandtl number (Pr = 0.71) and Rayleigh numbers
ranging from 103
to 106
at the macroscopic scale (Kn = 10-4
). In every case, an appropriate value of the
characteristic velocity i.e. y
V g THbD ؛ is chosen using a simple model based on the kinetic theory. By
considering the obtained dimensionless velocity and temperature values, the distributions of entropy generation
due to heat transfer and fluid friction are determined. It is found that for an enclosure with high value of
Rayleigh number (i.e., Ra=105
), the total entropy generation due to fluid friction and total Nu number increases
with decreasing the aspect ratio.
Keywords: Entropy Generation; Lattice Boltzmann Method; Natural Convection; Γ-Shaped enclosure
1. Introduction
The lattice Boltzmann (LB) method is a powerful
approach to hydrodynamics, with applications
ranging for vast Reynolds numbers and modeling the
physics in fluids [1–4]. Various numerical
simulations have been performed using different
thermal LB models or Boltzmann-based schemes to
investigate the natural convection problems [5–11].
The lattice Boltzmann equation (LBE) is a minimal
form of the Boltzmann kinetic equation, and the result
is a very elegant and simple evolution equation for a
number of distribution functions, which represent the
number of fluid particles moving in these discrete
__________
*
Corresponding author
Email Address: mfarhadi@nit.ac.ir
with speed ci .In LBM the domain is discretized in
uniform Cartesian cells which each one holds a fixed
directions. With respect to the more conventional
numerical methods commonly used for the study of
fluid
flow situations, the kinetic nature of LBM
(Lattice Boltzmann Method) introduces several
advantages, including easy implementation of
boundary conditions and fully parallel algorithms. In
addition, the convection operator ( .c fر
rr
) is linear,
no Poisson equation for the pressure must be solved
and the translation of the microscopic distribution
function into the macroscopic quantities consists of
simple arithmetic calculations. The phenomenon of
natural convection in enclosures has attracted
increasing attention in recent years.
2. E. Fattahi et al./ TPNMS 1 (2013) 1-18
2
Nomenclature
hT Hot temperature ( )K
ic Discrete lattice velocity in direction (i) cT
Cold temperature ( )K
sc Speed of sound in Lattice scale 0T Bulk temperature (K), (T0= (Th+Tc)/2)
iF External force in direction of lattice
velocity
v,u Horizontal and vertical components of
velocity
eq
if Equilibrium distribution 1
( . )m s−
yg Acceleration due to gravity, 2
( . )m s −
kw Weighting factor
H Height of enclosure ( )m w
non-dimensional length of step, (w′/ H)
h Non-dimensional height of step, (h′/H)
k
Thermal conductivity 1 1
( . . )W m K− −
Greek symbols
uN Mean Nusselt number β Thermal expansion coefficient ( )1
K −
yx NuNu , Local Nusselt number along surfaces µ Molecular viscosity 1 1
( . . )kg m s− −
Pr Prandtl number ( / )ν α ϕ Irreversibility distribution ratio
Ra
Rayleigh number 3
( / )g THβ αν∆ ρ Density 3
( . )kg m−
Kn Knudsen number τ Lattice relaxation time
genS ′′′ Total volumetric entropy generation rate t∆ Lattice time step
( )3 1
. .W m K− −
PS ′′′ Volumetric entropy generation rate due to
Subscript
friction ( )3 1
. .W m K
− −
C cold
TS ′′′ Volumetric entropy generation rate due to h hot
heat transfer ( )3 1
. .W m K
− −
i discrete lattice directions
Applications extending from the double paned
windows in buildings to the cooling of electronic
systems are examples of natural convection systems.
In natural convection processes, the thermal and the
hydrodynamic are coupled and both are, according to
Bejan [12], strongly influenced by the fluid thermo-
physical characteristics, the temperature differences
and the system geometry. The comprehensive
reviews of articles on natural convection were made
by Catton [13], Ostrach [14] and Kakac and Yener
[15]. In addition to the studies [16-20], Lage and
Bejan [21] investigated numerically the natural
convection in a square enclosure heated and cooled in
the horizontal direction in the Prandtl number range
0.01– 10 and the Rayleigh number range 102
–1011
.
Notable researches have been done to investigate
importance of entropy generation in thermal systems.
Entropy generation and its minimization were
investigated widely with Bejan [22-24]. Natural
convection in enclosure was summarized in
rectangular coordinates by Davis [25]. In his study,
he made a review about numerical studies and
investigated the effect of various non-dimensional
numbers and boundary conditions on natural
convection heat transfer. Additionally, an analysis of
the entropy generation in rectangular cavities was
3. E. Fattahi et al./ TPNMS 1 (2013) 1-18
3
performed by Oliveski et al. [26]. They found that for
the same aspect ratio, the entropy generation due to
the viscous effects increases with the Rayleigh
number and, for a certain Rayleigh number, the
entropy generation due to the viscous effects also
increases with the aspect ratio.
Ha and Jung [27] used LBM to investigate the
steady, three-dimensional, conjugate heat transfer of
natural convection and conduction in a vertical cubic
enclosure within which a centered, cubic, heat-
conducting body generates heat. They found that the
fluid flow and temperature distribution show very
complex three-dimensional pattern. Mezrhab et al.
[28] studied the radiation-natural convection
interactions of a square heat-conduction body within
a differentially heated square cavity. Dagtekin et al.
[29] dealt with the prediction of entropy generation
of natural convection in a Γ-shaped enclosure using
FDM (Finite Difference Method). They found that
the main entropy generation is formed due to heat
transfer for Ra<105
, while the contribution due to
fluid friction becomes stronger for Ra>105
.
In the present study natural convection and entropy
generation was simulated numerically in the Γ-shaped
enclosure using LBM. As the horizontal walls are
insulated perfectly, vertical walls heats. An in house
lattice BGK (Bhatnagar–Gross–Krook) scheme
FORTRAN code was used to simulate the present
problem. The contribution of this work is the analyses
of the variation of entropy generation in relation to
Rayleigh number, aspect ratio at fixed irreversibility
coefficient(φ=10-6
) in Γ-shaped enclosure. The results
are displayed graphically in term of the streamlines,
isotherms and local entropy generation contours to
show the effect of aspect ratio (AR) and Rayleigh
number. To calculate the entropy generation, a new
model [30] was used to determine the dimensionless
velocity. The results of the present study show that
this model is a suitable for calculating the entropy
generation in the natural convection problems.
2. Numerical Procedure
2.1 The Lattice Boltzmann Method
In investigating the natural convection problems,
the effect of viscous heat dissipation can be neglected
for applications in incompressible flow [10]. This
assumption can be used to simulate the natural
convection by LBM. The LB model used here is the
same as that employed in [9-11]. The thermal LB
model utilizes two distribution functions, f and g, for
the flow and the temperature field, respectively. It
uses modeling of movement of fluid particles to
capture macroscopic fluid quantities such as velocity,
pressure and temperature. In this approach the fluid
domain is discretized in uniform Cartesian cells.
Each cell holds a fixed number of distribution
functions, which represent the number of fluid
particles moving in specified discrete directions. For
this work, the most popular model for the 2D case,
the D2Q9 model, which consists of 9 distribution
functions, has been used (Fig. 1). The values of
0
4 9w = for 0
0c = (for the static particle),
1 4
1 9w −
= for 1 4
1c −
= and 5 9
1 36w −
= for
5 9
2c −
= are assigned for this model.
The density and distribution functions i.e. the f
and g (temperature distribution function), are
calculated by solving the Lattice Boltzmann equation
(LBE), which is a special discretization of the kinetic
Boltzmann equation. After introducing BGK
approximation, the general form of lattice Boltzmann
equation with external force can be written as:
(1)
( , )
( , ) ( , ) ( , )
.
i i
eq
ii i
i i
f x c t t t
t
f x t x t f x tf
tc F
υτ
+ ∆ +∆ =
∆ + −
+∆
For the flow field and
(2)
( , ) ( , )
( , ) ( , )
ii i
eq
ii
D
x c t t t x tg g
t
x t g x tg
τ
+ ∆ +∆ =
∆ + −
For the temperature field
Where t∆ denotes lattice time step, i
c is the discrete
lattice velocity in direction k , i
F is the external
force in direction of lattice velocity, υ
τ and D
τ
denotes the lattice relaxation time for the flow and
temperature field. The kinetic viscosity υ and the
thermal diffusivity α, are defined in terms of their
respective relaxation times, i.e. 2
( 1 2)s
c υυ τ= − and
2
( 1 2)s Dcα τ= − , respectively. The speed of sound
( s
c ) is a lattice-dependent quantity, which has the
value of 1 3 for the D2Q9 model. Note that the
4. E. Fattahi et al./ TPNMS 1 (2013) 1-18
4
limitation 0.5 < τ should be satisfied for both
relaxation times to ensure that viscosity and thermal
diffusivity are positive. Furthermore, the local
equilibrium distribution functions are calculated with
equations (3, 4) for the flow and temperature fields,
respectively.
(3)
( )2
2 4 2
.. 1 1 .
. . 1
2 2
ρ
= + + −
eq ii
ii
s s s
c uc u u u
wf
c c c
(4)2
.
. ( , ). 1
= +
eq i
ii
s
c u
T x tg w
c
Where i
w a weighting factor, u is velocity vector
andρ is the lattice fluid density.
In order to incorporate buoyancy force in the
model, the force term in the equation (1) need to be
calculated as below in vertical direction (y) [31]:
(5)3i i yF w g βθ=
The Boussinesq approximation is applied and
radiation heat transfer is negligible. θ is non-
dimensional temperature.
To ensure that the code works in near
incompressible regime, the characteristic velocity of
the flow y
V g THβ≡ ∆ must be small compared
with the fluid speed of sound. In the present study,
the characteristic velocity was selected as 0.1 of
speed of sound.
Finally, the following macroscopic variables can
be calculated in terms of these variables, with the
following formula.
(6)Flow density: i
i
fρ = ∑
(7)Momentum: i j ji
j
u f cρ = ∑
(8)
Temperature: i
i
T g= ∑
2.2 Entropy Generation Calculation
The following dimensionless variables (primed
quantities are dimensional) are used:
(9)
( )
( , ) ( , ) , , ( , ) / ,
, , /c h c
x y x y H u v u v H
T T T T T T T Pr
α
υ α
′ ′ ′ ′= =
′= − ∆ ∆ = − =
Fig. 1. Discrete velocity vectors for the D2Q9 model of
LBM
Volumetric entropy generation due to heat
transfer, T
S ′′′ , and friction, P
S ′′′ , are calculated as
below:
(10)
2'''
2
( )= ∇T
k
TS
T
(11)'''
PS
T
µ
ψ=
where ψ is defined by:
(12)
u ju ui i
x x xj i j
ψ
∂∂ ∂ = + ∂ ∂ ∂
And the total volumetric entropy generation can
be obtained by:
(13)SSS PTgen
'''''''''
+=
The dimensionless form of Eq. (13) is called Ns
(local EG number) which is defined as follow:
(14)
22
2 22
2
T
P
T T
Ns
x y
S
u v v u
x y x y
S
φ
∂ ∂ = + ∂ ∂
∂ ∂ ∂ ∂ + + + + ∂ ∂ ∂ ∂
5. E. Fattahi et al./ TPNMS 1 (2013) 1-18
5
The irreversibility distribution ratio, φ (the ratio
of entropy generated due to fluid friction to heat
transfer), is written as follows:
(15)
2
0T
k H T
µ α
φ
= ∆
0
T is bulk temperature ( ( )0
/ 2h c
T T T= + (K). The
dimensionless total entropy generation is the integral
volume of the computational domain:
(16)s
v
S N dV= ∫
2.3 Curved Boundary Treatment
Consider Fig. 2 is a part of an arbitrary curved
wall geometry, where the black small circles on the
boundary w
x , the open circles represent the boundary
nodes in the fluid region f
x and the grey circles
indicate those in the solid region b
x .In the boundary
condition ( , ), ( , )b b
f t g tx x are needed to perform the
streaming steps on fluid nodes f
x .
The fraction of an intersected link in the fluid
region ∆ is defined by:
f
f b
w−
∆ =
−
x x
x x
(17)
The standard (half-way) bounce back no-slip
boundary condition always assumes a delta value of
0.5 to the boundary wall (Fig.3a). Due to the curved
boundaries, delta values in the interval of (0, 1] are
now possible. Fig.3b shows the bounce back
behavior of a surface with a delta value smaller than
0.5 and Fig.3c shows the bounce back behavior of a
wall with delta bigger than 0.5. In all three cases, the
reflected distribution function ( , )f t tα + ∆x at f
x is
unknown. Since the fluid particles in the LBM are
always considered to move one cell length per time
step, the fluid particles would come to rest at an
intermediate node i
x . In order to calculate the
reflected distribution function in node f
x , an
interpolation scheme has to be applied. For treating
velocity field in curved boundaries, the method is
based on the method reported in [32]. For
temperature field, the method is based on an
extrapolation method of second-order accuracy
applied in [33].
Fig. 2. Schematic view of Γ-shaped enclosure
Fig. 3. Layout of the regularly spaced lattices and curved
wall boundary
2.3.1 Velocity in curved boundary condition
To calculate the distribution function in the solid
region ( , )b
f tα x based upon the boundary nodes in
the fluid region, the bounce-back boundary
conditions combined with interpolations including a
one-half grid spacing correction at the boundaries [3,
34].
6. E. Fattahi et al./ TPNMS 1 (2013) 1-18
6
Then the Chapman–Enskog expansion for the
post-collision distribution function on the streaming
step is conducted as:
2
( , ) (1 ) ( , )
( , )
3
2 ( , ) .
b f
b
f w
f t t f t t
f t t
w t t
c
α α
ο
α
α α
λ
λ
ρ
+∆ = − +∆
+ + ∆
− + ∆
x x
x
x e u
(18)
Where
( )2
( , ) ( , )
3
( , ) .
eq
b f
f bf f
f t t f t t
w t t
c
ο
α α
α αρ
+ ∆ = + ∆
+ + ∆ −
x x
x e u u (19)
,
2 1 1
, 0
2 2
bf ff
m
ifλ
τ
=
∆ −
= < ∆ ≤
−
u u
(20a)
3 3
(1 ) ,
2 2
2 1 1
, 1
1 2
2
bf f w
m
ifλ
τ
= − +
∆ ∆
∆ −
= < ∆ ≤
+
u u u
(20b)
w
u denotes the velocity of solid wall, bf
u is the
imaginary velocity for interpolations and α α≡ −e e .
Temperature in curved boundary condition
For temperature field in curved boundary this
study use the method is based on the method reported
in [34]. Distribution function for temperature divided
two parts, equilibrium and non equilibrium
( , ) ( , ) ( , )eq neq
b b b
g t g t g tα α α= +x x x (21)
By substituting Eq.21 into temperature streaming
step, we have
( , ) ( , )
1
(1 ) ( , )
eq
b b
neq
b
T
g t t g t
g t
α α
α
τ
+ ∆ =
+ −
x x
x
(22)
Obviously to calculate ( , )b
g t tα + ∆x , both
( , )eq
b
g tα x and ( , )neq
b
g tα x are required.
Equilibrium and non equilibrium parts of Eq.21
are define as:
*
2
3
( , ) 1 .eq
b b bg t w T
c
α α α
= +
x e u (23)
*
b
T is determined by linear extrapolation using either:
*
1, if 0.75b bT T= ∆ ≥ (24a)
*
1 2(1 ) , if 0.75b b bT T T= + − ∆ ∆ ≤ (24b)
Where ∆ is the fraction of the intersected link in the
fluid region (Eq. 17), which is illustrated in Fig. 3
and:
1 [ ( 1) ]/b w fT T T= + ∆ − ∆ (25a)
2 [2 ( 1) ]/(1 )b w ffT T T= + ∆ − + ∆ (25b)
Where Tf and Tff denote the fluid temperatures in
node xf and xff, respectively. The extrapolation
scheme is the same as Ref. [35].
The next task is to determine the ( , )neq
b
g tα x .
Second-order approximation is also used.
( , )neq
b
g tα x is evaluated as:
( , ) ( , )
(1 ) ( , )
neq neq
b f
neq
ff
g t g x t
g x t
α α
α
= ∆
+ − ∆
x
(26)
From the Chapman-Enskog analysis,
( , )neq
g tα x
can be expressed as:
1
( , ) ( , ) .neq
g t g t xα α δ=x x (27)
Where
0
( , )g tα x is the same order as ( , )eq
g tα x .
Since
1 1
( , ) ( , ) ( )g t g t O xα α δ− =w fx x ,
2
( , ) ( , ) ( )neq neq
g t g t O xα α δ− =w fx x .
By the same token, it can be proved that
2
( , ) ( , ) ( )neq neq
g t g t O xα α δ− =w ffx x (28)
That means the approximation ( , )neq
b
g tα x is of
second order in space which is in consistent with
thermal lattice Boltzmann equation.
7. E. Fattahi et al./ TPNMS 1 (2013) 1-18
7
3.2. Computational Domain and Numerical
Details
Figure 4 shows the Γ-Shaped enclosure analyzed
in the present study. h, w, AR are the height of the
step, the width of the step and the aspect ratio (h/w),
respectively. The step heater kept at a constant
temperature of Th and the vertical walls of the
enclosure are fixed at low isothermal of Tc. The top
and bottom walls are insulated.
No-slip boundary condition was imposed on all
the walls of the cavity. Dirichlet type boundary
conditions have been used that the insulated
boundary conditions were simulated by converting
them to Dirichlet type by using a second order
accurate finite-difference approximation. The
boundary conditions have been implemented by
using the counter-slip approach such as Dixit and
Babu [8]. Although the suitability of the counter slip
approach has only been established for the
hydrodynamic boundary conditions [36], but Dixit
and Babu [8] have shown that this technique is useful
for modeling the thermal boundary conditions to
simulate the flow and heat transfer in the cavity.
They found that “the traditional implementation of
the Dirichlet boundary condition gave rise to a
spurious gradient in the temperature between the wall
and the first point in the fluid. This gradient was a
steep drop in the temperature near the hot wall and
the cold wall and it persisted even after decreasing
the mesh spacing by a factor of 8. This clearly
implies that the source of this phenomenon is the
implementation of the boundary condition and not
the grid spacing especially for a lower value of
Rayleigh number. This, in turn, would yield incorrect
values of wall Nusselt number. To overcome this
difficulty, the counter-slip approach has been used
for simulating the Dirichlet boundary condition also”.
This method was investigated carefully in detail by
Dixit [37]. As mentioned above, this boundary
condition was used on the walls of the cavity.
The Rayleigh number (Ra) and Nusselt number
(Nu) for the current problem are defined as follow:
(29)3
Ra g THβ αυ= ∆
Local Nusselt numbers are defined on front and
top face of the step as Nux, Nuy
(30)
,x y
h w
T T
Nu Nu
x y
∂ ∂ =− = − ∂ ∂
The average Nusselt number is calculated by
integrating the local Nusselt number along surface of
step as:
(31)
0 0
1
h w
y xNu Nu dy Nu dx
h w
= + +
∫ ∫
Determination of characteristic velocity is
necessary to simulate the natural convection in LBM.
This velocity is defined as y
V g THβ≡ ∆ . The
kinetic viscosity and thermal diffusivity are
calculated from the characteristic velocity through
the following relationships, respectively:
(31)
2 2
2 Pr
Pr
V H
Ra
and
υ
υ
α
=
=
The relaxation times, υ
τ and D
τ , for flow and
temperature LB equations given in equation (1) and
(2) can then be determined. It implies that kinetic
viscosity (υ) and thermal diffusivity (α) can not be
considered as constants in LBM simulations if the
characteristic velocity (V) is kept constant. Kao et al.
[38] developed a new model for determining an
appropriate characteristic velocity value based on the
principle of kinetic theory. From the kinetic theory
[39, 40], the Knudsen number is defined as [41]:
(32)
Re
.
2
Ma
Kn
πγ
=
Where Ma is the Mach number, Re is the
Reynolds number, and the heat capacity ratio to be γ
= 5/3 for a monatomic ideal gas and γ = 7/5 for a
diatomic gas according to the definition of mean free
path,
2 s
c
πγ υ
λ = , the Knudsen number for this
geometry can be written as [38]:
(33)Hc
v
H
Kn
s .
.
2
πγλ
=≡
From the definition of Rayleigh number and Eq
(17):
(34)
3
where,
2Pr
.. 2
2
22
2 c
c
cKnRa
V s
s
==
πγ
By this definition, the characteristic velocity is a
function of the Rayleigh number, Knudsen number,
8. E. Fattahi et al./ TPNMS 1 (2013) 1-18
8
Prandtl number and the value of c, that all are
specified as a given values in LBM simulations
which includes the macroscopic and mesoscopic
scales at natural convection problems.
For grid independency, the average Nusselt
number over the step was calculated at high Ra
numbers for different grid points. As seen in table 1
for grid points passing from 80×80 and 100×100 for
Ra = 105
and 106
, respectively, no considerably
change in the average Nusselt number was observed
(maximum variation is less then 0.16%). According
to the table 1, the 80×80 grid points was used for Ra
≤ 105
and 100×100 grid points was used only for Ra
= 106
.
(a)
(b)
(c)
Fig. 4. Illustration of the bounce-back boundary
conditions. (a) ∆ = 1/2, the ‘‘perfect” bounce-back without
interpolation. (b) ∆ < 1/2, the bounce-back with
interpolations before the collision with the wall located at
xw. (c) ∆ > 1/2, the bounce-back with interpolations after
the collision with the wall
5. Results and discussion
To validate the numerical simulation, the results
of natural convection in square and Г-shaped
enclosure were compared with previous works ([25]
and [29]).
In the square cavity, flow was heated from the left
wall, while the right wall was maintained at a
constant low temperature. Meanwhile, the upper and
bottom walls were assigned adiabatic boundary
conditions. A vertical gravitational effect was applied
in the y-direction. Regarding the flow field, the
square cavity was assumed to be closed and the no-
slip boundary conditions were imposed at each of the
four solid walls. In this simulations, appropriate
values of V were obtained using the model presented
in Eq. (23) with a fixed Knudsen number of Kn = 10-
4
at the macroscopic scale and Rayleigh numbers of
Ra = 105
and 106
, respectively.
The results of the present simulation in
comparison with the previous study ([25]) are shown
in the table 2. The results of the present study show a
good agreement (maximum variation is less than 2%)
with the previous study.
Figure 5 shows the temperature contours and
streamlines in the Г-shaped enclosure in comparison
with the results of the FDM [29]. The results show a
good accuracy of present simulation. As mentioned
above, this model is a suitable technique for
simulation of fluid flow and heat transfer in natural
convection problems using LBM in the cavity. On
the other hand, to validate the calculated entropy
generation by this model, the results of the entropy
generation in the cavity was compared with the work
of the Oliveski et al. [26] in figure 6. This figure
shows the total entropy generation in the cavity
verses Rayleigh number at different irreversibility
distribution ratio, φ. Results show a good agreement
with the previous results that shows the ability of this
model to calculate of the entropy generation. Effect
of the aspect ratio and Rayleigh number over the
temperature distribution, flow field and entropy
generation in Γ-shaped enclosure has been analyzed
numerically using Lattice Boltzmann method at
constant irreversibility coefficient(ψ=10-6
). In this
simulation the Rayleigh number and aspect ratio are
changed from 103
to 106
and 0.25 to 3 respectively.
Results are presented as a form of streamlines,
temperature contours, entropy generation contours,
mean Nusselt number and total entropy generation.
Figure 7 shows the streamlines and temperature
contours for AR=1(h=0.5, w=0.5) at different
Rayleigh number. There exist two rotating cells in
the enclosure. The main and the larger rotating cell
on the left occupies region between left vertical wall
and front face of the hot step which circulates
counter clockwise (CCW). The small and secondary
9. E. Fattahi
Fig. 5. Streamlines and temperature contours for different Rayleigh numbers in comparison with the FDM [29]
E. Fattahi et al./ TPNMS 1 (2013) 1-18
9
Ref [29],Ra=10
Present study, Ra=10
Ref [29], Ra=10
Present study, Ra=10
Streamlines and temperature contours for different Rayleigh numbers in comparison with the FDM [29]
Ref [29],Ra=104
Present study, Ra=104
Ref [29], Ra=105
Present study, Ra=105
Streamlines and temperature contours for different Rayleigh numbers in comparison with the FDM [29]
10. E. Fattahi et al./ TPNMS 1 (2013) 1-18
10
Fig. 6. Validation of total entropy generation in the square cavity in comparison with Ref [26]
Table 1.
The averaged Nusselt Number on the step's wall of Γ-shaped enclosure for different grid points
Mesh size
Ra 40×40 80×80 100×100 110×110
105
18.52 18.66 18.69 -
106
26.93 27.15 27.23 27.26
Table 2.
The averaged Nusselt Number on the vertical Boundary at x=0 in comparison with the previous study
Nu Ref[25] Present study Ref[25] Present study Present study
Ra 40×40 40×40 80×80 80×80 100×100
105
4.487 4.449 4.523 4.508 4.512
106
8.798 8.623 8.928 8.883 8.891
cell forms over the step and circulates clockwise.
At Ra=103
, the heat transfer is mainly controlled by
conduction. The isotherms are nearly parallel to the
vertical wall which indicates domination of
conduction heat transfer in this case. At Ra=105
, the
intensity of the circulation becomes stronger, which
Ra
S
103
104
105
10610-1
10
0
101
102
103
104
ϕ=10−3
ϕ=10−4
ϕ=10−5
ϕ=10
−3
ϕ=10
−4
ϕ=10−5
Ref. [36]
Present
study
11. E. Fattahi et al./ TPNMS 1 (2013) 1-18
11
implies that the convection heat transfer begins
dominating the thermal flow field in the cavity. As
the Rayleigh number becomes large (Ra=106
), the
crowded streamlines and isothermal lines indicate
that the hydrodynamic and thermal boundary layers
have been developed along the hot and cold walls,
respectively, reflecting rigorous heat transfer rate
occurred .
Variation of the flow field and temperature
distribution has a different effect over the entropy
generation. Figure 8 shows the contours of local
entropy generation due to heat transfer and fluid
friction at different Rayleigh number for
AR=1(h=0.5, w=0.5) at ϕ =10-6
. A comparison of the
entropy generation maps for heat transfer and fluid
friction (Fig. 8a.) at Ra=103
confirms the less effect
of the fluid flow on entropy generation since the
strength of recirculation is relatively low. The
entropy generation due to fluid friction and heat
transfer is considerably high close to the step’s hot
walls. At higher Rayleigh number, buoyancy force
and consequently the fluid flow effects increases,
hence, the values of entropy generation due to fluid
friction increase. This phenomenon can be observed
near the walls of cavity with brighter color of the
figure 8(c, d).One of the main parameters that have a
major effect over the flow field and heat transfer
distribution in the enclosure is the aspect ratio (AR).
The effect of AR and length of horizontal and
vertical walls of the enclosure over the flow field and
temperature contours are shown in figures 9 and 10.
It is observed that when the isotherm lines are
parallel together, the main mechanism of the heat
transfer is conduction (Fig 9a.). This phenomenon
can be observed for the high length of the horizontal
hot wall (h). If the distance between the hot step and
cold wall increases, the convective effect on heat
transfer rate will increase. By decreasing the length
of vertical wall, the second cell (over the step) gets
bigger, so the heat transfer rate increases (Fig. 10c.).
It is observed from the temperature contours which
show the more diffusion of the heat to the main flow.
It should be mentioned that the constant aspect ratio
can be created by different length of the horizontal
and vertical walls of the enclosure. This variation on
the height of the walls has a special effect on heat
transfer rate which will be explain as follows.
The total entropy generation at various Rayleigh
numbers for different aspect ratio is plotted in figure
11. For all cases, the total entropy generation
increases with increasing Rayleigh number, though
the rate of increase is different as seen in the figure
11. It is observed that, for Rayleigh number lower
than 105
the effect of the aspect ratio on total entropy
generation is insignificant but for greater Rayleigh
number the total entropy generation is increased
considerably. It is due to the formation of the
recirculation areas and their speed which have a
direct effect over the entropy generation due to fluid
friction. The effect of the secondary recirculation
area over the entropy generation due to the fluid
friction is more important than the first bubble at
high Rayleigh number. This is occurred at lower
aspect ratio and length of the vertical hot wall (h).
One of the main parameters in natural convection in
the cavity is the rate of the heat transfer. In this study,
the variation of the mean Nusselt number over the step
walls of the cavity verses aspect ratio is plotted at
different Rayleigh number. With increasing the AR,
the size of the secondary recirculation over the
horizontal wall of the step reduces and fills the gap
between the horizontal surface of the step and
insulated wall of the cavity. This phenomenon
reduces the effect of the convection so the Nusselt
number decreases which is observed for all size of the
h (Fig. 12). On the other hand, at lower AR, the first
recirculation propagates in the cavity and fills
commonly with second bubble the gap between
thehorizontal surface of the step and insulated wall.
This phenomenon increases the effect of the
convective heat transfer in comparison with the
conduction and subsequently causes to increase the
Nusselt number. With increasing the Rayleigh
number, the speed of the flow in the cavity increases,
therefore the convective heat transfer increases. This
increase in the rate of the heat transfer is shown at the
mean Nusselt number curve. It should be mentioned
that the best heat transfer was observed for AR less
than unity at h=0.25. It is due to formation of the
bubbles in the cavity which was explained above.
With increasing the h form 0.25 to 0.75, the Nusselt
number decreases approximately 72% at all Rayleigh
numbers for AR=1. This result shows that the effect of
the h on heat transfer is more important than the
aspect ratio.
12. Fig. 7.Streamlines (top) and temperature contours (bottom) for h = w = 0.5 at different Rayleigh numbers
E. Fattahi et al./ TPNMS 1 (2013) 1-18
12
Streamlines (top) and temperature contours (bottom) for h = w = 0.5 at different Rayleigh numbers
(a)Ra= 103
(b)Ra=104
(c)Ra=105
(d)Ra= 106
Streamlines (top) and temperature contours (bottom) for h = w = 0.5 at different Rayleigh numbers
13. E. Fattahi
Fig. 8. Entropy generation due to heat transfer (on the top) and fluid friction (on the bottom) for h = w = 0.5 at
different Rayleigh number
E. Fattahi et al./ TPNMS 1 (2013) 1-18
13
Entropy generation due to heat transfer (on the top) and fluid friction (on the bottom) for h = w = 0.5 at
(a)Ra= 103
(b)Ra=104
(c)Ra=105
(d)Ra= 106
Entropy generation due to heat transfer (on the top) and fluid friction (on the bottom) for h = w = 0.5 at
14. E. Fattahi et al./ TPNMS 1 (2013) 1-18
14
6. CONCLUSION
In this study the effect of aspect ratio and
Rayleigh number on the heat transfer and entropy
generation in the Γ-Shaped enclosure was
investigated numerically using Lattice Boltzmann
Method. The results of this study show that new
model of the Kao et al. [32] for determining an
appropriate characteristic velocity value based on the
principle of kinetic theory is not only a suitable
model for calculating the heat transfer characteristic
but also a very useful technique to calculate the
entropy generation in the natural convection
problems. The accuracy of the present study for
calculating the entropy generation in comparison
with previous result [26] in the square cavity was
very well. In Γ-Shaped enclosure, the numerical
results show that increasing the Rayleigh number
causes to increase the Nusselt number for all cases.
At high aspect ratio, the formation of the bubbles
causes to increase the diffusion heat transfer in
comparison with convective heat transfer so Nusselt
number decreases for all cases. The best heat transfer
performance was observed at low AR. Also it was
observed that the lowest size of the vertical wall of
the step has a best heat transfer rate for low AR. The
entropy generation is independent of the AR for low
Rayleigh number which is due to the low speed of the
fluid and subsequently low entropy generation due to
the fluid friction. In natural convection in the cavity,
the main part of the entropy generation was created
by fluid friction so it was high at high Rayleigh
number. The size of the vertical wall of the step does
not have any observable effect on total entropy
generation except for high Rayleigh number.
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