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.
Double Diffusive Convection and the Improvement of Flow in Square Porous AnnulusIJERA Editor
There has been increased interest shown in recent years to investigate the behavior of heat and mass transfer in a square annulus with a porous medium fixed between the inner and outer walls. This paper aims to evaluate the Soret effect arising in the case of heat and mass transfer in a porous medium bounded by a square annulus and subjected to isothermal heating of the inner surfaces as well as the outer horizontal surfaces. The phenomenon is governed by 3 partial differential equations, the momentum, energy and concentration equations, that are coupled together and result in a situation where change in one variable affects the other equations and vice versa. The partial differential equations are converted into finite element equations with the help of the Galerkin method and then solved to predict solution variables such as temperature, stream function and concentration in the porous medium. It is found that the heat transfer rate at the hot wall decreases with increasing viscous dissipation effect in the porous medium.
Unsteady MHD Flow Past A Semi-Infinite Vertical Plate With Heat Source/ Sink:...IJERA Editor
In the present paper a numerical attempt is made to study the combined effects of heat source and sink on unsteady laminar boundary layer flow of a viscous, incompressible, electrically conducting fluid along a semiinfinite vertical plate. A magnetic field of uniform strength is applied normal to the flow. The governing boundary layer equations are solved numerically, using Crank-Nicolson method. Graphical results of velocity and temperature fields, tabular values of Skin-friction and Nusselt are presented and discussed at various parametric conditions. From this study, it is found that the velocity and temperature of the fluid increase in the presence of heat source but they decrease in the presence of heat absorption parameter.
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.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed
MHD Free Convection from an Isothermal Truncated Cone with Variable Viscosity...IJERA Editor
This paper presents a study of MHD free convection flow of an electrically conducting incompressible fluid with
variable viscosity about an isothermal truncated cone in the presence of heat generation or absorption. The fluid
viscosity is assumed to vary as a inverse linear function of temperature. The non-linear coupled partial
differential equations governing the flow and heat transfer have been solved numerically by using an implicit
finite - difference scheme along with quasilinearization technique. The non-similar solutions have been obtained
for the problem, overcoming numerical difficulties near the leading edge and in the downstream regime. Results
indicate that skin friction and heat transfer are strongly affected by, both, viscosity-variation parameter and
magnetic field. In fact, the transverse magnetic field influences the momentum and thermal fields, considerably.
Further, skin friction is found to decrease and heat transfer increases near the leading edge. Also, it is found that
the direction of heat transfer gets reversed during heat generation.
Double Diffusive Convection and the Improvement of Flow in Square Porous AnnulusIJERA Editor
There has been increased interest shown in recent years to investigate the behavior of heat and mass transfer in a square annulus with a porous medium fixed between the inner and outer walls. This paper aims to evaluate the Soret effect arising in the case of heat and mass transfer in a porous medium bounded by a square annulus and subjected to isothermal heating of the inner surfaces as well as the outer horizontal surfaces. The phenomenon is governed by 3 partial differential equations, the momentum, energy and concentration equations, that are coupled together and result in a situation where change in one variable affects the other equations and vice versa. The partial differential equations are converted into finite element equations with the help of the Galerkin method and then solved to predict solution variables such as temperature, stream function and concentration in the porous medium. It is found that the heat transfer rate at the hot wall decreases with increasing viscous dissipation effect in the porous medium.
Unsteady MHD Flow Past A Semi-Infinite Vertical Plate With Heat Source/ Sink:...IJERA Editor
In the present paper a numerical attempt is made to study the combined effects of heat source and sink on unsteady laminar boundary layer flow of a viscous, incompressible, electrically conducting fluid along a semiinfinite vertical plate. A magnetic field of uniform strength is applied normal to the flow. The governing boundary layer equations are solved numerically, using Crank-Nicolson method. Graphical results of velocity and temperature fields, tabular values of Skin-friction and Nusselt are presented and discussed at various parametric conditions. From this study, it is found that the velocity and temperature of the fluid increase in the presence of heat source but they decrease in the presence of heat absorption parameter.
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.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed
MHD Free Convection from an Isothermal Truncated Cone with Variable Viscosity...IJERA Editor
This paper presents a study of MHD free convection flow of an electrically conducting incompressible fluid with
variable viscosity about an isothermal truncated cone in the presence of heat generation or absorption. The fluid
viscosity is assumed to vary as a inverse linear function of temperature. The non-linear coupled partial
differential equations governing the flow and heat transfer have been solved numerically by using an implicit
finite - difference scheme along with quasilinearization technique. The non-similar solutions have been obtained
for the problem, overcoming numerical difficulties near the leading edge and in the downstream regime. Results
indicate that skin friction and heat transfer are strongly affected by, both, viscosity-variation parameter and
magnetic field. In fact, the transverse magnetic field influences the momentum and thermal fields, considerably.
Further, skin friction is found to decrease and heat transfer increases near the leading edge. Also, it is found that
the direction of heat transfer gets reversed during heat generation.
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.
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
Numerical Simulation on Mixed Convection Flow within Triangular Enclosures Ha...IOSR Journals
This paper is conducted to investigate mixed convection flow within triangular enclosures having heatline concept. Here the left vertical wall which is moving from the bottom corner of the enclosure is also kept at a uniform constant cold temperature and the bottom wall is heat generating, while the other inclined wall is adiabatic. A finite element analysis is performed for solving the governing equations which are discretized by Galerkin weighted residual method of finite element formulation. The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters Prandtl number, Pr (Pr = 0.71 - 6) varying Rayleigh number (Ra) from 103 to 104 while Reynolds number, Re (Re = 50) is fixed. The Result also indicates streamlines, isotherms, average Nusselt number along the hot wall, average temperature of the fluid in the enclosure. The computational results also indicate that the average Nusselt number at the hot wall of the Enclosure is depending on the dimensionless parameters.
Non-NewtonianFluid Flow and Heat Transfer over a Non- Linearly Stretching Sur...IJERA Editor
This paper investigates the MHD flow and heat transfer of an electrically conducting non-newtonian power-law
fluid over a non-linearly stretching surface along with porous plate in porous medium. The governing equations
are reduced to non-linear ordinary differential equations by means of similarity transformations. These
equations are then solved numerically with the help ofRunge – Kutta shooting method. The effect of various
flow parameters in the form of dimensionless quantities on the flow field are discussed and presented
graphically.
NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFE...IAEME Publication
In the present paper an analysis of mixed convection of an unsteady magneto hydrodynamic (MHD) flow of an incompressible viscous fluid through porous media due to a vertical porous stretching sheet in the presence of viscous dissipation, thermal radiation and heat source /sink has been carried out. The fluid considered is viscous and incompressible. The governing partial differential equations of the flow, mass and heat transfer are highly non linear hence are converted into a system of ordinary differential equations using suitable similarity transformations. These ordinary differential equations are further converted into 7 first order ordinary differential equations and are solved numerically by Matlab ode-45 solver via shooting method.
Visualization of Natural Convection in a Vertical Annular Cylinder with a Par...IJERA Editor
In this work, we visualize the effect of varying wall temperature on the heat transfer by supplying the heat at
three different positions to the vertical annular cylinder embedded with porous medium. Finite element method
has been used to solve the governing equations. Influence of Aspect ratio 𝐴𝑟 , Radius ratio 𝑅𝑟 on Nusselt
number 𝑁 𝑢 is presented. The effect of power law exponent effect for different values of Rayleigh number is
discussed. The fluid flow and heat transfer is presented in terms of streamlines and isotherms.
The results show that, with proper selection of physical parameters, significant heat transfer
enhancements and pressure drop reductions can be achieved simultaneously with porous pin fins and
the overall heat transfer performances in porous pin fin channels are much better than those in
traditional solid pin fin channels. The effects of pore density are significant. As PPI increases, the
pressure drops and heat fluxes in porous pin fin channels increase while the overall heat transfer
efficiencies decrease and the maximal overall heat transfer efficiencies are obtained at PPI 20.
Furthermore, the effects of pin fin form are also remarkable. With the same physical parameters, the
overall heat transfer efficiencies in the long elliptic porous pin fin channels are the highest while they
are the lowest in the short elliptic porous pin fin channels
THE EFFECT OF TRANSVERSE VIBRATION ON THE NATURAL CONVECTION HEAT TRANSFER IN...IAEME Publication
The effect of transverse vibration on the natural convection heat transfer in a
rectangular enclosure with an aspect ratio of 0.5 filled with air as a working fluid
aligned horizontally on a mechanical shaker generating a sinusoidal transverse
vibrational displacement was studied experimentally. The study was carried for a
Raghiely number between (3.77 - 10.8)*107 with applied heat flux between (20 - 45)
Watt. The vibrational experimental measurements were carried out for different
frequency ratio (0.87-1.6) and vibrational Rayleigh number ranged between (0.12 -
2.7)*107. The results of the heat transfer inside the enclosure without vibration show
a very close agreement with the published one. The vibrational heat transfer results
show that the behavior of different heat transfer convection parameters can be
affected by applying a forced vibration condition. It is shown that the high heat
transfer can be achieved at frequencies near to the system natural frequency at
constant heat flux. Also, it is concluded that a careful attention should be given to the
proper selection of heat flux and frequency ratio results in obtaining maximum values
of heat transfer parameters with low cost of power consumption.
Magnetic field effect on mixed convection flow in a nanofluid under convectiv...IAEME Publication
An analysis is carried out to investigate the influence of the prominent magnetic effect on mixed convection heat and mass transfer in the boundary layer region of a semi-infinite vertical flat plate in a nanofluid under the convective boundary conditions. The transformed boundary layer,ordinary differential equations are solved numerically using Runge-Kutta Fourth order method.
In theoretical computer science and mathematics , the theory of computation is the branch that
deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is
divided into three major branches: automata theory, computability theory, and computational complexity
theory.
In order to perform a rigorous study of computation, computer scientists work with a mathematical
abstraction of computers called a model of computation. There are several models in use such as Lambda
calculus, Combinatory logic, mu-recursive functions , but the most commonly examined is the Turing machine.
Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to
prove results, and because it represents what many consider the most powerful possible "reasonable" model of
computation . It might seem that the potentially infinite memory capacity is an unrealizable attribute, but
any decidable problem solved by a Turing machine will always require only a finite amount of memory. So in
principle, any problem that can be solved (decided) by a Turing machine can be solved by a computer that has a
bounded amount of memory.
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.
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
Numerical Simulation on Mixed Convection Flow within Triangular Enclosures Ha...IOSR Journals
This paper is conducted to investigate mixed convection flow within triangular enclosures having heatline concept. Here the left vertical wall which is moving from the bottom corner of the enclosure is also kept at a uniform constant cold temperature and the bottom wall is heat generating, while the other inclined wall is adiabatic. A finite element analysis is performed for solving the governing equations which are discretized by Galerkin weighted residual method of finite element formulation. The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters Prandtl number, Pr (Pr = 0.71 - 6) varying Rayleigh number (Ra) from 103 to 104 while Reynolds number, Re (Re = 50) is fixed. The Result also indicates streamlines, isotherms, average Nusselt number along the hot wall, average temperature of the fluid in the enclosure. The computational results also indicate that the average Nusselt number at the hot wall of the Enclosure is depending on the dimensionless parameters.
Non-NewtonianFluid Flow and Heat Transfer over a Non- Linearly Stretching Sur...IJERA Editor
This paper investigates the MHD flow and heat transfer of an electrically conducting non-newtonian power-law
fluid over a non-linearly stretching surface along with porous plate in porous medium. The governing equations
are reduced to non-linear ordinary differential equations by means of similarity transformations. These
equations are then solved numerically with the help ofRunge – Kutta shooting method. The effect of various
flow parameters in the form of dimensionless quantities on the flow field are discussed and presented
graphically.
NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED CONVECTIVE MASS AND HEAT TRANSFE...IAEME Publication
In the present paper an analysis of mixed convection of an unsteady magneto hydrodynamic (MHD) flow of an incompressible viscous fluid through porous media due to a vertical porous stretching sheet in the presence of viscous dissipation, thermal radiation and heat source /sink has been carried out. The fluid considered is viscous and incompressible. The governing partial differential equations of the flow, mass and heat transfer are highly non linear hence are converted into a system of ordinary differential equations using suitable similarity transformations. These ordinary differential equations are further converted into 7 first order ordinary differential equations and are solved numerically by Matlab ode-45 solver via shooting method.
Visualization of Natural Convection in a Vertical Annular Cylinder with a Par...IJERA Editor
In this work, we visualize the effect of varying wall temperature on the heat transfer by supplying the heat at
three different positions to the vertical annular cylinder embedded with porous medium. Finite element method
has been used to solve the governing equations. Influence of Aspect ratio 𝐴𝑟 , Radius ratio 𝑅𝑟 on Nusselt
number 𝑁 𝑢 is presented. The effect of power law exponent effect for different values of Rayleigh number is
discussed. The fluid flow and heat transfer is presented in terms of streamlines and isotherms.
The results show that, with proper selection of physical parameters, significant heat transfer
enhancements and pressure drop reductions can be achieved simultaneously with porous pin fins and
the overall heat transfer performances in porous pin fin channels are much better than those in
traditional solid pin fin channels. The effects of pore density are significant. As PPI increases, the
pressure drops and heat fluxes in porous pin fin channels increase while the overall heat transfer
efficiencies decrease and the maximal overall heat transfer efficiencies are obtained at PPI 20.
Furthermore, the effects of pin fin form are also remarkable. With the same physical parameters, the
overall heat transfer efficiencies in the long elliptic porous pin fin channels are the highest while they
are the lowest in the short elliptic porous pin fin channels
THE EFFECT OF TRANSVERSE VIBRATION ON THE NATURAL CONVECTION HEAT TRANSFER IN...IAEME Publication
The effect of transverse vibration on the natural convection heat transfer in a
rectangular enclosure with an aspect ratio of 0.5 filled with air as a working fluid
aligned horizontally on a mechanical shaker generating a sinusoidal transverse
vibrational displacement was studied experimentally. The study was carried for a
Raghiely number between (3.77 - 10.8)*107 with applied heat flux between (20 - 45)
Watt. The vibrational experimental measurements were carried out for different
frequency ratio (0.87-1.6) and vibrational Rayleigh number ranged between (0.12 -
2.7)*107. The results of the heat transfer inside the enclosure without vibration show
a very close agreement with the published one. The vibrational heat transfer results
show that the behavior of different heat transfer convection parameters can be
affected by applying a forced vibration condition. It is shown that the high heat
transfer can be achieved at frequencies near to the system natural frequency at
constant heat flux. Also, it is concluded that a careful attention should be given to the
proper selection of heat flux and frequency ratio results in obtaining maximum values
of heat transfer parameters with low cost of power consumption.
Magnetic field effect on mixed convection flow in a nanofluid under convectiv...IAEME Publication
An analysis is carried out to investigate the influence of the prominent magnetic effect on mixed convection heat and mass transfer in the boundary layer region of a semi-infinite vertical flat plate in a nanofluid under the convective boundary conditions. The transformed boundary layer,ordinary differential equations are solved numerically using Runge-Kutta Fourth order method.
In theoretical computer science and mathematics , the theory of computation is the branch that
deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is
divided into three major branches: automata theory, computability theory, and computational complexity
theory.
In order to perform a rigorous study of computation, computer scientists work with a mathematical
abstraction of computers called a model of computation. There are several models in use such as Lambda
calculus, Combinatory logic, mu-recursive functions , but the most commonly examined is the Turing machine.
Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to
prove results, and because it represents what many consider the most powerful possible "reasonable" model of
computation . It might seem that the potentially infinite memory capacity is an unrealizable attribute, but
any decidable problem solved by a Turing machine will always require only a finite amount of memory. So in
principle, any problem that can be solved (decided) by a Turing machine can be solved by a computer that has a
bounded amount of memory.
Fluorescence quenching of 5-methyl-2-phenylindole (MPI) by carbon tetrachlori...IOSR Journals
The fluorescence quenching of 5-methyl-2-phenylindole (MPI) by carbon tetrachloride by steady state in different solvents, and by transient method in benzene has been carried out at room temperature. The Stern–Volmer (SV) plot has been found to be non-linear with a positive deviation for all the solvents studied. In order to interpret these results we have invoked the ground state complex and sphere of action static quenching models. Using these models various rate parameters have been determined. The magnitudes of these parameters imply that sphere of action static quenching model agrees well with the experimental results. Hence the positive deviation in the SV plots is attributed to the static and dynamic quenching. Further, from the studies of temperature dependence of rate parameters and lifetime measurements, it could be explained that the positive deviation is due to the presence of a small static quenching component in the overall dynamic quenching. With the use of finite sink approximation model, it was possible to check whether these bimolecular reactions as diffusion limited and to estimate independently distance parameter R′ and mutual diffusion coefficient D. Finally an effort has been made to correlate the values of R′ and D with the values of the encounter distance R and the mutual diffusion coefficient D determined using the Edward's empirical relation and Stokes–Einstein relation.
Unification Scheme in Double Radio Sources: Bend Angle versus Arm Length Rati...IOSR Journals
From radio source unification scheme, asymmetries are more pronounced in quasars than in radio galaxies. Using a sample of 625 double radio sources (316 radio galaxies and 309 quasars), we investigated the variations between bend angle, arm-length ratio and redshift. We find no significant correlation (0.045) between bend angle and redshift for the entire sample and at low z for radio galaxies while a weak correlation (0.121) between bend angle and redshift for quasars for the entire sampleis observed. A weak correlation (radio galaxies, r = 0.173 and quasars, r = 0.102) between the bend angle and arm-length ratio for the double radio sources used in this study is observed. Kharbet al. (2008) in their study of powerful classical double radio galaxies noted that this correlation could suggest that the environmental asymmetries that give rise to the arm-length ratio could be contributory to the misalignment angles in these sources. Quasars appear much more bent and misaligned. This is consistent with quasars being radio galaxies viewed at small angles.
Nearest Adjacent Node Discovery Scheme for Routing Protocol in Wireless Senso...IOSR Journals
The broad significance of Wireless Sensor Networks is in most emergency and disaster rescue
domain. The routing process is the main challenges in the wireless sensor network due to lack of physical links.
The objective of routing is to find optimum path which is used to transferring packets from source node to
destination node. Routing should generate feasible routes between nodes and send traffic along the selected path
and also achieve high performance. This paper presents a nearest adjacent node scheme based on shortest path
routing algorithm. It is plays an important role in energy conservation. It finds the best location of nearest
adjacent nodes by involving the least number of nodes in transmission of data and set large number of nodes to
sleep in idle mode. Based on simulation result we shows the significant improvement in energy saving and
enhance the life of the network
Interfacing of Java 3D objects for Virtual Physics Lab (VPLab) Setup for enco...IOSR Journals
During this paper we target the combination of web accessible physics experiments (VPLabs) combined with the Sun’s toolkit for making cooperative 3D virtual worlds. Among such a cooperative setting these tools give the chance for academics and students to figure along as avatars as they control actual instrumentation, visualize natural phenomenon generated by the experiment, and discuss the results. Especially we'll define the steps of integration, future goals, yet because the price of a collaboration area in Wonderland's virtual world.
Effect of Different Physico-Chemical Parameters on Production ofAmylase by Ba...IOSR Journals
The present study is concerned with the production of amylase by Bacillus species strain. In this
study 12 bacterial strains were isolated and screened for their α-amylase activity. These strains were
maintained on nutrient agar medium. Fermentation for the production of amylase was carried out in the enzyme
production medium (EPM). All the 12 strains were tested for amylase production. On the basis of maximum
amylase activity strain no.1 was selected for further studies. Different starch concentrations, 0.75,1.00,1.25%,
pH labels 6.5,7.0,7.5,8.0, aeration (RPM), 100,120,140, temperatures 250C,280C,370C, and 400C and inoculums
level 0.5%,1.0%, 1.5% and 2.0% were studied
Effects of Aqueous and Methanolic Leaf Extracts of Vitex doniana on Lipid Pro...IOSR Journals
The effect of aqueous and methanolic extracts of Vitex doniana leaves in serum lipid profile and liver enzymes in normal and alloxan-induced diabetic rats were investigated using standard analytical protocols. A total of 35 albino rats divided into seven groups of five rats each comprising one normal untreated group as animal control, one diabetic untreated group as diabetic control, one normal treated with 750mg/kg body weight as reference group, three diabetic groups treated with 250, 500 and 750mg/kg body weight respectively and one diabetic group treated with 5mg/kg Glibenclamide as standard. The result of acute toxicity test obtained indicated lethal dose (LD50) of greater than 5000mg/kg extract. The results showed that induction of diabetes caused significant (P<0.05)><0.05)><0.05)><0.05) increase in high density lipoprotein in the reference and diabetic groups when compared to normal and diabetic control groups respectively after oral administration of Vitex doniana leaf extracts. It could therefore be concluded that Vitex doniana leaf extract is safe, medicinal and have anti-lipidemia properties and hepato-protective effects.
A study of serum Cadmium and lead in Iraqi postmenopausal women with osteopor...IOSR Journals
Postmenopausal status is an independent risk factor for osteoporosis. Several studies have reported that heavy metals, including lead, mercury, cadmium, and arsenic, have harmful effects on bone. The aim of this study was to evaluate the effect of heavy metals, including Cadmium and Lead on osteoporosis in postmenopausal Iraqi women. This prospective study included a total of 70 postmenopausal women divided as 40patients with osteoporosis compared to 30 apparently healthy women as controls during 2011. Serum levels of Cadmium and Lead were measured using atomic absorption while serum Calcium, Phosphorus and Alkaline phosphatase were measured by spectrophotometry.The results showed that there was no significant difference between patients and controls regarding age, Body Mass Index, Calcium, Phosphorous, and Alkaline phosphatase. Serum levels of Cadmium and Lead were higher in patients compared to controls, p < 0.001 and p< 0.01 respectively. It is concluded that increased serum levels of cadmium and lead maybe associated with higher risk of osteoporosis in postmenopausal women.
The Study of Heat Generation and Viscous Dissipation on Mhd Heat And Mass Dif...IOSR Journals
The present work is devoted to the numerical study of magneto hydrodynamic (MHD) natural convection flow of heat and mass transfer past a plate taking into account viscous dissipation and internal heat generation. The governing equations and the associated boundary conditions for this analysis are made non dimensional forms using a set of dimensionless variables. Thus, the non dimensional governing equations are solved numerically using finite difference method Crank-Nicolson’s scheme. Numerical outcomes are found for different values of the magnetic parameter, Modified Grashof number, Prandtl number, Eckert number, heat generation parameter and Schmidt number for the velocity and the temperature within the boundary layer as well as the skin friction coefficients and the rate of heat and mass transfer along the surface. Results are presented graphically with detailed discussion.
Moving Lids Direction Effects on MHD Mixed Convection in a Two-Sided Lid-Driv...A Behzadmehr
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1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 3 (May. - Jun. 2013), PP 64-73
www.iosrjournals.org
www.iosrjournals.org 64 | Page
Finite Element Analysis of MHD Free Convection within
Trapezoidal Enclosures with Uniformly Heated Side Walls
M. S. Hossain1*
, Mohammed Nasir Uddin2
, M. A. Alim3
, M. Shobahani4
1
Department of Arts and Sciences, Ahsanullah University of Science & Technology (AUST),
Dhaka-1208, Bangladesh
2
Department of Mathematics, Bangladesh University of Business & Technology (BUBT),
Dhaka, Bangladesh
3
Department of Mathematics, Bangladesh University of Engineering & Technology (BUET),
Dhaka-1000, Bangladesh
4
Department of Mathematics and Natural Sciences, Brac University, Dhaka-1212, Bangladesh
Abstract: 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.
Keywords: Free convection, Finite element method, Trapezoidal Enclosures, Uniform heating.
Nomenclature
B0 Magnetic Induction
Cp Specific Heat At Constant Pressure (J/Kg K)
G Gravitational Acceleration (M/S2
)
Gr Grashof Number
H Convective Heat Transfer Coefficient
(W/M2
K)
Ha Hartmann Number
K Thermal Conductivity Of Fluid(W/M K)
L Height Or Base Of Trapezoidal Cavity (M)
K Thermal Conductivity Ratio Fluid
N Total Number Of Nodes
Nuav Average Nusselt Number
Nulocal Local Nusselt Number
P Non-Dimensional Pressure
P Pressure
Pr Prandtl Number
Ra Rayleigh Number
T Non-Dimensional Temperature
Th Temperature Of Hot Bottom Wall (K)
Tc Temperature Of Cold Bottom Wall (K)
U X Component Of Dimensionless Velocity
U X Component Of Velocity (M/S)
V Y Component Of Dimensionless Velocity
V Y Component Of Velocity (M/S)
V0 Lid Velocity
X, Y Cartesian Coordinates
X, Y Dimensionless Cartesian Coordinates
Greek Symbols
Thermal Diffusivity (M2
/S)
Coefficient Of Thermal Expansion (K-1
)
Density Of The Fluid (Kg/M3
)
∆Θ Temperature Difference
Θ Fluid Temperature
Μ Dynamic Viscosity Of The Fluid (Pa S)
Π Heatfunction
Ν Kinematic Viscosity Of The Fluid (M2
/S)
Σ Fluid Electrical Conductivity(Ω-1
m-1
)
Subscripts
B Bottom Wall
L Left Wall
R Right Wall
S Side Wall
2. Finite Element Analysis of MHD Free Convection within Trapezoidal Enclosures with Uniformly Heated Side Walls
www.iosrjournals.org 65 | Page
I. Introduction
The well-known buoyancy driven Phenomena of convection motion of fluid has attracted many
researchers over the past few years. The phenomenon of heat and mass transfer frequently exist in chemically
processed industries such as food processing and polymer production. The phenomenon of free convection flow
involving coupled heat and mass transfer occurs frequently in nature. In these studies the magnetohydrodynamic
phenomenon is applied. Magnetohydrodynamics (MHD) has attracted the attention of a large number of
scholars due to its diversified applications. The study of the effects of magnetic field on free convection flow is
important in liquid-metals, electrolytes and ionized gases. Magnetohydrodynamic flows have applications in
meteorology, solar physics, cosmic fluid dynamics, astrophysics, geophysics and in the motion of earthes core.
Shanker and Kishan [1] presented the effect of mass transfer on the MHD flow past an impulsively started
infinite vertical plate. Elabashbeshy [2] studied heat and mass transfer along a vertical plate in the presence of
magnetic field. However, Free convection in enclosed cavities has received significant attention due to many
engineering applications [3–7]. The extensive studies for rectangular and square enclosures using various
numerical simulations reported by Patterson and Imberger [8], Nicolette et al.[9], Hall et al. [10], Hyun and Lee
[11], Fusegi et al. [12], Lage and Bejan [13,14], Xia and Murthy [15] and Al-Amiri et al.[16]ensure that several
attempts have been made to acquire a basic understanding of natural convection flows and heat
transfercharacteristics in an enclosure. The majority of works dealing with convection in enclosures are
restricted to the cases of simple geometry e.g., rectangular, square, cylindrical and spherical cavities. But the
configurations of actual containers occurring in practice are often far from being simple. A few studies on
natural convection on triangular enclosures filled with a viscous fluid or a porous medium have been carried out
by earlier researchers [17–19]. In recent years, most of the enclosures commonly used in industries are
cylindrical, rectangular, trapezoidal, triangular etc. Trapezoidal enclosures have also received a considerable
attention for their application in various fields.
A comprehensive understanding of energy flow and entropy generation is needed for an optimal
process design via reducing irreversibilities in terms of „entropy generation‟. In this study, analysis on entropy
generation during natural convection in a trapezoidal cavity with various inclination angles (φ = 45°, 60° and
90°) have been carried out for an efficient thermal processing of various fluids of industrial importance
(Pr = 0.015, 0.7 and 1000) in the range of Rayleigh number (103
− 105
) by Basak et. al [20]. Basak et al. [21]
studied a comprehensive heatline based approach for natural convection flows in trapezoidal enclosures with the
effect of various walls heating. The present numerical study deals with natural convection flow in closed
trapezoidal enclosures. Anandalakshmi and Basak [22] studied for the energy distribution and thermal mixing in
steady laminar natural convective flow through the rhombic enclosures with various inclination angles, φ for
various industrial applications. Here simulations are carried out for various regimes of Prandtl (Pr) and Rayleigh
(Ra) numbers. Dimensionless streamfunctions and heatfunctions are used to visualize the flow and energy
distribution, respectively. Basak et al. [23] also investigated the numerical investigation of natural convection in
a porous trapezoidal enclosures for uniformly or non-uniformly heated bottom wall. Penalty finite element
analysis with bi-quadratic elements is used for solving the Navier–Stokes and energy balance equations. The
numerical solutions are studied in terms of streamlines, isotherms, heatlines, local and average Nusselt numbers
for a wide range of parameters Da(10−5
–10−3
), Pr(0.015–1000) and Ra(Ra = 103
–106
). At low Darcy number
(Da = 10−5
), heat transfer is primarily due to conduction for all φ‟s as seen from the heatlines which are normal
to the isotherms. Basak et al. [24] also performed heat flow patterns in the presence of natural convection within
trapezoidal enclosures with heatlines concept. In this study, natural convection within a trapezoidal enclosure
for uniformly and non-uniformly heated bottom wall, insulated top wall and isothermal side walls with
inclination angle have been investigated. Momentum and energy transfer are characterized by streamfunctions
and heatfunctions, respectively, such that streamfunctions and heatfunctions satisfy the dimensionless forms of
momentum and energy balance equations, respectively. Finite element method has been used to solve the
velocity and thermal fields and the method has also been found robust to obtain the streamfunction and
heatfunction accurately. The unique solution of heatfunctions for situations in differential heating is a strong
function of Dirichlet boundary condition which has been obtained from average Nusselt numbers for hot or cold
regimes. Natarajan et. al [25] presented a numerical study of combined natural convection and surface radiation
heat transfer in a solar trapezoidal cavity absorber for Compact Linear Fresnel Reflector (CLFR) . The
numerical simulation results are presented in terms of Nusselt number correlation to show the effect of these
parameters on combined natural convection and surface radiation heat loss.
In the present investigation, visualization of heat flows via heatlines for magneto-hydrodynamics
(MHD) free convection with uniformly heated side walls were reported for trapezoidal enclosures. Results are
obtained to display the circulations and for different physical parameters in terms of streamlines, stream
functions, total heat flux, isotherms and heat transfer rates for the walls in terms of average and local nusselt
numbers.
3. Finite Element Analysis of MHD Free Convection within Trapezoidal Enclosures with Uniformly Heated Side Walls
www.iosrjournals.org 66 | Page
II. Model Specification
The geometry for the configuration with the system of co-ordinates is schametically shown in Fig. 1.
The model considered here is a two-dimensional trapezoidal enclosures of height L with the left wall inclined at
an angle ф = 450
, 300
, 00
with Y axis. Here left wall and right (side) walls are subjected to hot Th temperature;
bottom wall is subjected to cold Th temperature while the top wall is kept insulated. The boundary conditions for
velocity are considered as no-slip on solid boundaries.
Figure 1: Schematic diagram of the physical system for (a) ф = 450
(b) ф = 300
and (c) ф = 00
2.1 Mathematical Formulation
The flow inside the cavity is assumed to be two-dimensional, steady, laminar and incompressible and the fluid
properties are said to be constant. For the treatment of buoyancy term in the momentum equation, Boussinesq
approximation is used.The dimensionless governing equations describing the flow are as follows:
Continuity Equation
0
U V
X Y
(1)
Momentum Equations
2 2
2 2
Pr
U U P U U
U V
X Y X X Y
(2)
2 2
2
2 2
Pr( ) Pr Pr
U U P V V
U V Ra Ha V
X Y Y X Y
(3)
Energy Equation
2 2
2 2
U V
X Y X Y
(4)
The Following definitions and dimensionless parameters are used in writiing equations (1-4) as,
2
2
3 3 2 2
2 0
2 2
, , , , , , Pr ,
Pr
, , ,
c
h c
h c h c
p
T Tx y uL vL pL
X Y U V P
L L T T
g L T T g L T T B L k
Gr Ra Ha
C
Th
Thermal insulated
Th
Tc
X
Y
BA
CD
(a) ф = 450
ф
B0 B0
Thermal insulated
Th
Y
X
BA
CD
(b) ф = 300
Th
Tc
ф
B0
Tc
Thermal insulated
D
Th
X
Y
C
A B
(c) ф = 00
Th
4. Finite Element Analysis of MHD Free Convection within Trapezoidal Enclosures with Uniformly Heated Side Walls
www.iosrjournals.org 67 | Page
Here Ha, Ra, Pr, Gr are Hartmann number , Rayleigh number, Prandtl number and Grashof number
respectively. Thermal diffusivity, volumetric thermal expansion coefficient, dynamic viscosity, kinematic
viscosity, electrical conductivity, density and dimensional temperature difference of the fluid are represented by
the symbols α, β, μ, ν, σ, ρ, ΔT, respectively.
2.2 Boundary Conditions
The boundary conditions (also shown in Fig. 1), for the present problem are specified as follows:
At the bottom wall:
0, 0, 1, 0, 0 1U V Y X
At the left wall:
0, 0, 0, cos sin 0, 0 1U V X Y Y
At the right wall:
0, 0, 0, cos sin cos , 0 1U V X Y Y
At the top wall:
0, 0, 0, 1, tan 1 tanU V Y X
Y
where X and Y are dimensionless coordinates varying along horizontal and vertical directions, respectively; U
and V are dimensionless velocity components in X and Y directions, respectively; is the dimensionless
temperature.
The local Nusselt number at the heated surface of the cavity which is defined by the following expression:
l r b sNu Nu Nu Nu
n
,
where n denotes the normal direction on a plane.
The average Nusselt number at the cold bottom wall, uniformly heated left and right (side) walls and insulated
top walls of the enclosures based on the non-dimensional variables may be expressed as,
1 1 1 1
0 0 0 0
l r s bNu Nu dX Nu dX Nu dX Nu dX
.
The non-dimensional stream function is defined as, ,U V
Y X
III. Research Methodology
Finite element analysis is a method to solve differential equations numericallywhich can be applied to
many problems in engineering and scientific fields. Finite element simulation of free convection in a two-
dimensional trapezoidal enclosures has been studied.This research starts from two-dimensional Navier-Stoke‟s
equations together with the energy equation to obtain the corresponding finite element equations.Galerkin‟s
weighted resudal method is applied to discretize the non-dimentional governing equations. Triangular mesh is
used to obtain the solution. Because this type of mesh can be used in any shape of domain .Details of method are
available in Taylor and Hood [26] and Dechaumphai [27].
IV. Grid Independence Test
In order to determine the proper grid size for this study, a grid independence test are conducted with Pr
= 0.7, Phi =450
, Ha=50 and Ra = 105
. The following five types of mesh are considered for the grid
independence test.These grid densites are 1527, 2541, 3573, 4563, 5858 nodes and 216, 365, 527, 668, 864
elements. Average Nusselt numbers at the heated surface study of trapezoidal enclosures are used as a measure
of accuracy of solution. From the table 1, a grid size of 4563 nodes and 668 elements is chosen for better
accuracy.
Table1: Grid Sensitivity Check at Pr = 0.7, Phi = 450
, Ha= 50 and Ra = 105
.
Nodes
(Elements)
1527
(216)
2541
(365)
3573
(527)
4563
(668)
5858
(864)
Nu 1.7517261.747969 1.834902 1.848767 2.050656
Time (s) 4.609 5.672 7.204 8.016 9.625
5. Finite Element Analysis of MHD Free Convection within Trapezoidal Enclosures with Uniformly Heated Side Walls
www.iosrjournals.org 68 | Page
V. Code Validation
The present numerical solution is validated by comparing the current code results against the numerical
result of Basak et al. [24] for free convection in a trapezoidal cavity for streamlines, isotherms and heatflux. For
three different Rayleigh numbers (Ra = 103
, 104
and 105
), while the prandtl number and angle are fixed i.e. Pr =
0.7 , ф = 450
for uniform heating of side wall, average Nusselt number is calculated. The numerical solutions
(present work and Basak et al. [24]) are in good agreement.
Ra
Average Nusselt Number, ( Nuav )
Present
work
Basak et al. (March
2009)
ф = 450
ф = 450
103
1.672972 1.27778
104
1.842988 1.83453
105
2.797346
2.71105
Table 2: Code Validation For Uniform Heating of Side Wall With Pr = 0.7.
VI. Results And Discussion
In this section we present numerical results for streamlines, isotherms and heat function or heatflux for
different Rayleigh number Ra = 103
- 105
and Prandtl number, Pr = 0.026, 0.7 for the fluid with various angles, ф
= 450
, 300
, 00
. These are shown in figure (2-4). Addition, heat transfer rate for local and average nusselt numbers
have been shown for various values of Rayleigh and Prandtl numbers and angles ф.
6.1 Uniform Heating
Figure 2 shows the effects of streamlines, isotherms and heat function for Rayleigh numbers. Here the
magnitudes of streamfunction and heat transfers are primarily due to conduction at low Rayleigh number.
Isotherms with θ = 0.10 - 0.20 take place symmetrically along side (left or right) walls and with θ ≥ 0.30 are
smooth curves symmetric with respect to vertical symmetrical line for Ra = 103
, Pr = 0.026 and ф = 00
(square
cavity) (Fig. 2a). Again, for Ra = 103
, Pr = 0.026 and ф = 300
the temperature contours with θ = 0.10 – 0.40
come about symmetrically near the side walls of the enclosure and with θ ≥ 0.50 are smooth curves symmetric
with respect to central symmetrical line (Fig. 2b). Also for Ra = 103
, Pr = 0.026 and ф = 450
isotherms
(temperature) with θ = 0.10 – 0.50 arise symmetrically near the side walls of the enclosure and with θ ≥ 0.60 are
smooth curves symmetric with respect to vertical symmetrical line (Fig. 2c).The heatlines or total heat flux or
heat function are shown in panels of fig 2a-c. The heatlines illustrate similar attribute that were observed for
uniform heating cases. Besides, we see that vortices are obtained for streamlines in fig 2a-c.
The interesting message is that at the bottom corner point ф = 00
(square cavity) is superior to ф = 450
and 300
. It is evident that heatlines near the bottom portion of side walls are more dense for ф = 450
and less
dense for ф = 00
(square cavity). The dense heatlines are also indicating enhanced rate of heat transfer from the
bottom to the side walls. Therefore for ф = 450
isotherms with θ = 0.05 – 0.35 are shifted toward the side walls.
It is also observed that at the top portion of the cavity for ф = 450
and 300
heat transfer is higher compressed to
ф = 00
(square cavity) based on value of heatfunction (Π). At the corners of bottom wall as the heat transfer is
quite large, the thermal boundary layer is found to develop near the bottom edges and thickness of boundary
layer is bigger at the top portion of the cold wall signifying less heat transfer to the top portion.
0.48
4.52
1.90
7.03
0.30
4.06
1.65
1.75
0.48
3.62
2.46
1.52
1.27
П
П
6. Finite Element Analysis of MHD Free Convection within Trapezoidal Enclosures with Uniformly Heated Side Walls
www.iosrjournals.org 69 | Page
a b c
Figure 2: Stream function (Ψ), temperature (θ), heat function or total heat flux (П) for
uniform bottom heating θ(X,0) = 1 with Pr = 0.026, Ha = 50 and Ra = 103
(a) Φ = 0o
(b) Φ = 30o
(c) Φ = 45o
Figure 3 shows that the magnitudes of streamfunction are smaller for Ra = 105
, Pr = 0.026 and
isotherms (temperature) with θ = 0.05 – 0.15, θ = 0.05 – 0.30, θ = 0.05 – 0.35 occur symmetrically near the
side walls of the enclosure and with θ ≥ 0.25, θ ≥ 0.30, θ ≥ 0.35 are smooth curves symmetric with respect to
central symmetrical line for Ra = 105
, Pr = 0.026 and ф = 450
, 300
, 00
(square cavity) ( Fig 5.9a-c, 5.10a-c) and
heatlines enhanced the rate of heat transfer from the bottom to side walls for Ra = 104
, Pr = 0.026. It is observed
that at critical Ra the middle portion of isotherms starts getting deformed and the maximum value of ψ is at the
eye of vortices. As Ra increases, the buoyancy driven circulation inside the cavity is also increased as seen from
greater magnitudes of stream function (fig. 3). It is also observed that the greater circulation in bottom regime
follows a progressive wrapping and isotherms are more compressed towards the side wall as can be seen figure
3.
Figure 4 demonstrates that the magnitudes of streamfunction are circular or elliptical near the core but
the streamlines near the wall is almost parallel to wall exhibiting large intensity of flow for Pr = 0.7 and Ra =
105
. Vortices are also enhanced for every case of enclosures. Also for Pr = 0.7 isotherms with θ = 0.05 – 0.15, θ
= 0.05 – 0.45, θ = 0.05 – 0.50 for 0.7 and Ra = 105
transpire symmetrically near the side walls of the enclosure
and θ ≥ 0.20, θ ≥ 0.50, θ ≥ 0.55 are flat curves symmetric with respect to central symmetrical line for Ra = 105
,
Pr = 0.7 and ф = 450
, 300
, 00
(square cavity). The detection is that for
irrespective of angles ф the intensity of flow has been increased as seen in fig 3. Although for intensity
of flow streamlines near the wall is almost parallel to wall but streamlines look like circular or elliptical near the
core (see figure 3). It is fascinating that multiple correlations are absent for Pr = 0.7 and Ra = 105
. Due to
enhanced flow circulations the isotherms are highly compressed near the side walls except near the bottom wall
especially for ф = 450
and 300
. The large temperature gradient near the side walls are due to noteworthy number
of heatlines with a large variation of heatfunction as seen in figure 4a-c .
0.10
0.50
1.601.10
0.60
0.20
0.20
0.20 0.100.50
0.80
1.60
0.10
0.20
0.50
0.60
0.80
1.30
1.80
7.82
0.002
0.004
3.53
7.57
9.4
0.011
0.001
7.4010.002
0.01
9.11
5.58
3.63
0.0
0.025
0.0
0.601 0.6
0.21
0.15
8.74
1.62
3.71
0.2
0.78 0.68
5.5
3.6
0.2
0.7
0.6
Ψ
θ
θ
П
П
7. Finite Element Analysis of MHD Free Convection within Trapezoidal Enclosures with Uniformly Heated Side Walls
www.iosrjournals.org 70 | Page
a b c
Figure 3: Stream function (Ψ), temperature (θ), heat function or total heat flux(П) for
uniform heating of side wall θ(X,0) = 1 with Pr = 0.026, Ha = 50 and Ra = 105
(a) Φ = 0o
(b) Φ = 30o
(c) Φ = 45o
a b c
Figure 4: Stream function (Ψ), temperature (θ), heat function or total heat flux(П)
contours for uniform bottom heating θ(X,0) = 1 with Pr = 0.7, Ha = 50 and Ra = 105
(a)
Φ = 0o
(b) Φ = 30o
(c) Φ = 45o
0.05
0.10
0.15
0.20 0.250.35
0.55
0.55
0.10
0.20 0.30
0.25
0.15
0.100.35
0.25
0.75
0.55
0.80
0.95
0.15
0.10
0.40
0.15
0.40
0.90
0.60
0.25
0.10
0.20
0.0601
0.0
0.0
0.04
0.0
0.5
0.758
0.3
0.22
0.40
0.11
0.03
0.01
0.19
4.74
0.870.94
5.15
3.32
2.65
6.92
4.294.03
0.10
0.10
0.30
0.20
0.75
0.35
0.25
0.95
0.10
0.100.45
0.40
0.25
0.55
0.25
0.65
0.75
0.95
0.10
0.35
0.50
0.55
0.70
0.90
0.20
0.15
3.19
0.88
1.32
6.91
7.467.04
6.96
9.869.70
θ
θ
Ψ
Ψ
θ
θ
П
П
8. Finite Element Analysis of MHD Free Convection within Trapezoidal Enclosures with Uniformly Heated Side Walls
www.iosrjournals.org 71 | Page
6.2 Heat Transfer Rates: Local And Average Nusselt Numbers
Figure 5(a-c) shows the effect of local heat transfer rates (Nus) vs distance for various inclination tilt
angles i.e. for ф = 00
, 300
, 450
when Ra = 103
and Pr = 0.7for uniform heating of side wall. It is observed that
local heat transfer rate is maximum at the bottom edge of side wall and thereafter decreases sharply upto a point
which is very near to the bottom edge. It is seen that Nus increase upto a point near to the top wall and also
decreases with distance near to the bottom wall. The boundary layer starts to form at the bottom edge of the side
wall and the boundary layer thickness is quite large near the bottom wall for all фs. Due to large intensity of
convection at Ra = 103
the thickness of the boundary layers are small at the middle portions of side walls and is
found to be larger near the top portion. But also figure 5(b-c) shows the similar effects of local heat transfer
rates (Nus) with distance for same Pr and different Ra respectively. As Ra increases, magnitudes of local heat
transfer rates become smaller and maximum heat transfer occurs near the top portion. It may be mentioned that
the larger degree of compression of isotherms for uniform heating case results in larger and Nus(local) is quite
large near to the bottom wall.
(a) (b)
(c)
Figure 5: Variations of local Nusselt numbers (Nub) with distance for different Rayleigh
numbers, (a) Ra = 103
, (b) 104
, (c)105
and angles Φ = 0o
, 30o
, 45o
when Pr = 0.7
The heat transfer rates are presented in figure 6(a)-(c), where distributions of average Nusselt number
are plotted vs the logarithmic Rayleigh number respectively. Here figure 6(a)-(c) illustrates uniform heating of
side walls respectively. It may be noted that average Nusselt number is obtained by considering temperature
gradient. It also be noted that as Ra increases then the average Nusselt number increases. It is seen in figure 6(a)
that as Ra increases from 103
-105
then average Nusselt number is straightly moving for Φ = 0o
, 300
but for Φ =
45o
Ra increases more when Pr = 0.026. As Pr increases (figure 6(b)) then conduction dominant heat transfer is
narrowed down. It is also seen from figure 6(c) that, as Pr increases more than from uniform heating case it is
analyzed that average Nusselt number for bottom wall is also slightly increasing during the entire Rayleigh
number regime.. As Pr increases then for conduction dominant heat transfer, the average Nusselt number is
generally constant irrespective of Ra. It is observed that Nus at the middle portion of bottom wall for Φ = 0o
is
larger for uniform heating case whereas for Φ = 30o
and 45o
heat transfer rates are identical. It is also observed
that when Pr = 0.026, heat transfer rates for Φ = 45o
is swiftly increasing except Φ = 30o
and 0o
.
Distance, X
LocalNusseltNumber,NuL
0 0.2 0.4 0.6 0.8
0
10
20
30
40 Phi = 0, Ra = 1e3
Phi = 30, Ra = 1e3
Phi = 45, Ra = 1e3
Distance, X
LocalNusseltNumber,NuL
0 0.2 0.4 0.6 0.8
0
10
20
30
40 Phi = 0, Ra = 1e4
Phi = 30, Ra = 1e4
Phi = 45, Ra = 1e4
Distance, X
LocalNusseltNumber,NuL
0 0.2 0.4 0.6 0.8
0
10
20
30
40 Phi = 0, Ra = 1e5
Phi = 30, Ra = 1e5
Phi = 45, Ra = 1e5
9. Finite Element Analysis of MHD Free Convection within Trapezoidal Enclosures with Uniformly Heated Side Walls
www.iosrjournals.org 72 | Page
(a) (b)
Figure 6: Variations of Average Nusselt Number vs Rayleigh number for (a) Pr =
0.026, (b) Pr = 0.7 and of angles Φ = 0o
, 30o
, 45o
for uniform heating of side wall.
VII. Conclusions
The problem of MHD free convection within trapezoidal enclosures for uniformly heated side wall
with heatlines concept has been studied numerically. Flow and temperature field in terms of streamlines and
isotherms and heat function or total heat flux have been displayed. The results of the numerical analysis lead to
the following conclusions:
Local Nusselt number of uniform heating of side wall is largest at the bottom edge of the side wall and
thereafter that decreases sharply upto a point which is very near to the bottom edge.
The heat transfer rate average Nusselt Number, Nuav increases with the increase of Rayleigh number, Ra,
for uniform heating of side wall.
The maximum rate of heat transfer is obtained for the highest Pr.
Various vortices entering into the flow field and secondary vortex at the vicinity boundary wall and bottom
wall of the cavity is seen in the streamlines.
Acknowledgements
Authors would like to express their gratitude to the Department of Mathematics, Bangladesh University
of Engineering and Technology (BUET) and Department of Arts and Sciences, Ahsanullah University of
Science and Technology (AUST), Dhaka, Bangladesh, for providing computing facility during this work.
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