Microsegregation and Rayleigh number variation during the solidification of superalloy Inconel 718

The microstructure and composition of the residual liquid at different temperatures were investigated by scanning electron microscopy （SEM） and energy dispersive X-ray spectrometer （EDX） associated with the Thermo-calc software calculation of the equilibrium phase diagrams of Inconel 718 and segregated liquid. The liquid density difference and Rayleigh number variation during solidification were estimated as well. It is found that the heavy segregation of Nb in liquid prompts the precipitation of δ and Laves phase directly from liquid and the resultant quenched liquid microstructure consists of pro-eutectic γ＋eutectic,or complete eutectic according to the content of Nb from low to high. The liquid density increases with decreasing temperature during the solidification of Inconel 718 and the liquid density difference is positive. The largest relative Rayleigh number occurs at 1320°C when the liquid fraction is about 40vol%.

Effects of Rayleigh Number on MHD Heat Transfer and Fluid Flow in Two-Dimensional Open Square Cavity with Heat Generation

The present work is devoted to investigating heat transfer and fluid flow in a two dimensional square open cavity containing a heated circular cylinder at the centre. A constant heat flux is set at the left sidewall;high and low temperatures are fixed at the bottom and top walls of the cavity respectively. The right side of the cavity is open. Galerkin Weighted Residual Finite Element Analysis is used to visualize the heat transfer and fluid flow solving two-dimensional governing mass, momentum and energy equations for steady state, heat transfer and fluid flow in presence of magnetic field in side an open square cavity. A uniformly heated circular cylinder is placed at the centre of the cavity as a heat source. To find the effects of Rayleigh number (Ra) on the thermal fields and fluid flow in presence of magnetic field and a heated circular cylinder as heat source by visualization and line graphs is the objective of this study. Numerical results are presented in graphical and tabular form. The study is conducted for different values of Rayleigh number, some fixed Hartmann numbers (Ha) and heat flux (q). In the conclusion it has been observed that the temperature field and fluid flow pattern are functions of the parameters Rayleigh number and Hartmann number.

Influence of Anisotropic Permeability and Soret Effect on the Convective Heat and Mass Transfer through a Porous Cavity Saturated by a Non-Newtonian Fluid

In this work, an analytical study is carried out on double-diffusive natural convection through a horizontal anisotropic porous layer saturated with a non-Newtonian fluid by using the Darcy model with the Boussinesq approximations. The horizontal walls of the system are subject to vertical uniform fluxes of heat and mass, whereas the vertical walls are assumed to be adiabatic and impermeable. The Soret effect is taken into consideration. Based on parallel flow approximation theory, the problem is solved in the limit of a thin layer and documented the effects of the physical parameters describing this investigation.

Convective Instability and Appearance of Structured Flows for Diffusion in Multicomponent Gas Mixtures

The main objective of this article is to investigate the behavior of gaseous systems with two and more independent gradients or thermodynamic forces exhibiting complicated behavior,when the convective flows occur.The existence of structural formations in these systems is shown by the schlieren method and the fast-response transducers.The linear analysis of stability can explain reasons of the appearance of convective instability in multicomponent gas mixtures.

APPLICATIONS OF THE BERNSTEIN-DURRMEYER OPERATORS IN ESTIMATING THE NORM OF MERCER KERNEL MATRICES

The paper is related to the norm estimate of Mercer kernel matrices. The lower and upper bound estimates of Rayleigh entropy numbers for some Mercer kernel matrices on [0, 1] × [0, 1] based on the Bernstein-Durrmeyer operator kernel are obtained, with which and the approximation property of the Bernstein-Durrmeyer operator the lower and upper bounds of the Rayleigh entropy number and the l2 -norm for general Mercer kernel matrices on [0, 1] x [0, 1] are provided.

Natural convection heat transfer enhancement of different nanofluids by adding dimple fins on a vertical channel wall

In this numerical study,natural flow and heat transfer of nanofluids with Al2O3,TiO2,Cu and CNT nanoparticles in a vertical channel with dimpled fins at Rayleigh number(Ra)of Ra=3.25×107 to Ra=1×108 are investigated by using the finite volume method.The obtained results revealed that,using CNT in volume fractions of 2%and 4%leads to significant heat transfer and atφ=6%,using TiO2 nanoparticles has a great effect on Nu number enhancement.Also,using solid nanoparticles in base fluid causes more uniform heat transfer distribution,especially in areas close to heated surface and by adding more volume fraction in base fluid,temperature level reduces.In general,according to temperature contours,reduction of wall temperature depends on the increase of Ra and volume fraction and the type of solid nanoparticles.

CONTROL OF THE LORENZ CHAOS BY THE EXACT LINEARIZATION

Controlling chaos in the Lorenz system with a controllable Rayleigh number is investigated by the state space exact linearization method. Based on proving the exact linearizability, the nonlinear feedback is utilized to design the transformation changing the original chaotic system into a linear controllable one so that the control is realized. Numerical examples of control are presented.

Natural convection of SiO_2-water nanofluid in square cavity with thermal square column

A square with a thermal square column is a simple but nontrivial research prototype for nanofluid research. However, until now, the effects of the temperature of the square column on the heat and mass transfer of nanofluids have not been revealed comprehensively, especially on entropy generation. To deepen insight into this important field, the natural convection of the SiO2-water nanofluid in a square cavity with a square thermal column is studied numerically in this study. The effects of the thermal column temperature （T = 0.0, 0.5, 1.0, 1.5）, the Rayleigh number （ranging from 103 to 106）, and the volume fraction of the nanoparticle （varying from 0.01 to 0.04） on the fluid flow, heat transfer, and entropy generation are investigated, respectively. It is found that, no matter at a low or high Rayleigh number, the volume fraction of the nanoparticle shows no considerable effects on the flow field and temperature field for all the temperatures of the thermal column. With an increase in the volume fraction, the mean Nusselt number increases slightly. At the same time, it is found that, with an increase in the temperature of the thermal column, the average Nusselt number gradually decreases at all values of the Rayleigh number. Meanwhile, it is found that, at a high Rayleigh number, the heat transfer mechanism is the main parameter affecting the increase in the total entropy generation rather than the volume fraction. In addition, no matter at a high or low Rayleigh number, when T = 0.5, the total entropy generation is the minimum.

Convective stability analysis of a micropolar fluid layer by variational method

This paper studies Rayleigh-B’enard convection of micropolar fluid layer heated from below with realistic boundary conditions.A specific approach for stability analysis of a convective problem based on variational principle is applied to characterize the Rayleigh number for quite general nature of bounding surfaces.The analysis consists of replacing the set of field equations by a variational principle and the expressions for Rayleigh number are then obtained by using trial function satisfying the essential boundary conditions.Further,the values of the Rayleigh number for particular cases of large and small values of the microrotation coefficient have been obtained.The effects of wave number and micropolar parameter on the Rayleigh numbers for onset of stationary instability for each possible combination of the bounding surfaces are discussed and illustrated graphically.The present analysis establishes that the nature of bounding surfaces combination and microrotation have significant effect on the onset of convection.

Numerical Simulation of Natural Air Convection in Inclined Eccentric Hemispheres Enclosure

The subject of natural convection heat transfer is motivated by a wide range of applications in engineering technology. The hemispherical cavity is a part of basic geometries although it is not widely studied. The effect of inclinaison on natural convection fluid motions in the gap between two eccentric hemispheres is numerically studied. The inner hemisphere is subjected to a heat flux of a constant density and the outer one is maintened isothermal. The walls separating the two hemispheres are thermally adiabatic. Equations are formulated with vorticity and stream-functions variables. It is also assumed the fluid incompressible and obeys the approximation of Boussinesq. These equations are written by using bispherical coordinates system and solved by using a finite difference method. The results show the topology of flow is strongly dependent on the inclinaison because the flow can change from a unicellular regime to a multicellular regime by varying the inclination from 0 to π. By increasing the Rayleigh number (103<Ra<107), the flow intensifies. The results are shown in terms of streamlines and isotherms during their transient evolution.

Finite Element Analysis of Magnetohydrodynamic Natural Convection within Semi-Circular Top Enclosure with Triangular Obstacles

The phenomena of magneto-hydrodynamic natural convection in a two-dimensional semicircular top enclosure with triangular obstacle in the rectangular cavity were studied numerically. The governing differential equations are solved by using the most important method which is finite element method (weighted-residual method). The top wall is placed at cold Tc and bottom wall is heated Th. Here the sidewalls of the cavity assumed adiabatic. Also all the wall are occupied to be no-slip condition. A heated triangular obstacle is located at the center of the cavity. The study accomplished for Prandtl number Pr = 0.71;the Rayleigh number Ra = 103, 105, 5 × 105, 106 and for Hartmann number Ha = 0, 20, 50, 100. The results represent the streamlines, isotherms, velocity and temperature fields as well as local Nusselt number.

Simulation of Natural Convection in a Cavity:Cooling with Top versus Bottom Partition`

Natural convection in a square cavity is studied numerically with vertical hot walls and a cold partition located at the bottom or top walls. The effect of orientation of partition on isotherms, flow patterns, and heat transfer is analyzed for different fluids and Rayleigh numbers. The placement of partition is found to considerably influence the isotherms, flow patterns, and heat transfer. Heat transfer rate is observed always to increase when the partition is moved from bottom to top wall.

Heat Transfer in Tubing-Casing Annulus during Production Process of Geothermal Systems

In geothermal systems, the temperature distribution of heat flow in the wellbore is dependent on the well structure and the geological conditions of the surrounding formation. Understanding of heat transfer in the tubing-casing annulus can reduce the heat losses of wellbore fluid during the production process. The present study discusses the possible means of heat transfer in the annulus, and develops a piecewise equation for estimating the convective heat transfer coefficient with a wider valid condition of 0〈Ra〈7.17×10^8. By converting the radiation and natural convection into equivalent thermal conduction, their sum is defined as a total thermal conductivity to describe the heat transfer in the annulus. The results indicate that the annulus filled with gas can be utilized as a good thermal barrier for the fluid in the wellbore. Additionally, the contribution of radiation will increase to occupy a majority proportion in the total thermal conductivity when the annular size increases and the materials have high emissivity. Otherwise, thermal radiation is just the second factor.

Experimental research on nonlinear characteristics of natural convection in a 3-D shallow cavity

Experiments were conducted to study the temperature field,flow field,3-D characteristics and self-sustained oscillation of the natural air convection in a 3-D shallow cavity which was heated from underneath. The experiments were conducted by the methods of laser holographic interferometry photography and smoke visualization. To ensure the temperature of both plates to be constant and the accuracy of laser interferometer,the instrument was calibrated and error has was analyzed. The results showed that the flow field was stable at lower Rayleigh numbers. When the Rayleigh number increased,the flow field became instable and the isotherms distorted. The rolls merged at Ra=12500 and formed along both axes when Ra was over 18500. The air rose from the middle and descended in the circumference while the flow field and heat transfer converted to 3-D characteristic from 2-D characteristic. When the Rayleigh number increased,the flow field became more instable. The rolls became irregular and time dependent when Ra≥Rac(=30500) ,which is nonlinear. When lateral walls were heated or cooled,the rolls merged along the long axis and two rolls formed along the short axis. Three rolls formed occasionally in the process.

Galerkin finite element analysis of magneto-hydrodynamic natural convection of Cu-water nanoliquid in a baffled U-shaped enclosure

In this paper,single-phase homogeneous nanofluid model is proposed to investigate the natural convection of magneto-hydrodynamic(MIID)flow of Newtonian Cu—H20 nanoli­quid in a baffled U-shaped enclosure.The Brinkman model and Wasp model are considered to measure the effective dynamic viscosity and effective thermal conductivity of the nanoliquid coreespondingly.Nanoliquid's effective properties such as specific heat,density and thermal expansion coefficient are modeled using mixture theory.The complicated PDS(partial differ­ential system)is treated for numeric solutions via the Galerkin finite element method.The perti­nent parameters Hartmann number(1≤Ha≤60),Rayleigh number(10^(3)≤Ra≤10^(6))and nanoparticles volume fraction (0% ≤Ф≤4%) are taken for the parametric analysis, and it is conducted via streamlines and isotherms. Excellent agreement between numerical results and open literature. It is ascertained that heat transfer rate enhances with Rayleigh number Ra and volume fraction 0, however it is diminished for laiger Hartmann number Ha.

Natural Convection in a Horizontal Cylinder with Partial Heating: Energy Efficiency Analysis

The current study reports a numerical analysis of free convection of air in an isothermal horizontal cylinder,cooled and heated at different wall locations.Three heater sizes are discussed in this study.The first heated zone is spread across one-quarter of the sidewall;the second is uniformly distributed over the half of the wall and the third active wall covers three-quarters of the cylinder.Five various locations are considered and examined for each active zone of the sidewall.The computation is carried out for Rayleigh number ranging from 102 to 106.Numerical results characterizing heat transfer and flow features are achieved using an iterative model developed in COMSOL Multiphysics.The effect of Rayleigh number on heat transfer and fluid flow characteristics within the cavity are investigated.Particular attention is paid to the influence of heater location and heater size on energy efficiency.It is found that the mean Nusselt number and dimensionless velocity increase when increasing the Rayleigh number.Moreover,the optimal level of energy efficiency is achieved if the heating zone is centered at the upper part of the cylinder,regardless of the heater size.It is also shown that the optimal configuration providing higher energy efficiency is obtained when three-quarters of the sidewall are locally heated,and more precisely,if the active zone is centered at the top of the cylinder.

Mathematical Modeling of Onset of Convection in a Porous Layer with Viscosity Variation

Instability theory is applied to diffusion-convection phenomenon in porous media, where the area in direction of transfer is large and viscosity of the oil varies due to gas dissolution. An important application of this theory arises where diffusion-convection is employed as an EOR technique in oil reservoirs. As a bed of gas is formed below a colunm of oil, gas starts to diffuse into the oil. Therefore, the oil becomes lighter and an inverse gradient of density is developed as more gas diffuses in. Although this inverse density gradient is potentially unstable, convection will not initiate until the gradient extends to a certain value. The condition at which convection begins is known as "the onset of convectin" and is well specified by the dimensionless Rayleigh number. In this study, an instability analysis is made for convection-diffusion in large porous media. Unlike Other studies where viscosity is assumed constant in this work viscosity is postulated to be a function of gas concentration. It is shown that the mathematical model developed reduces to previous models if the viscosity vacations are ignored.

Heat Convection Between Two Confocal Elliptic Tubes Placed at Different Orientations

In this paper,transient and steady natural convection heat transfer in an elliptical annulus has been investigated.The annulus occupies the space between two horizontal concentric tubes of elliptic cross-section.The resulting velocity and thermal fields are predicted at different annulus orientations assuming isothermal surfaces.The full governing equations of mass,momentum and energy are solved numerically using the Fourier Spectral method.The heat convection process between the two tubes depends on Rayleigh number,Prandtl number,angle of inclination of tube axes and the geometry and dimensions of both tubes.The Prandtl number and inner tube axis ratio are fixed at 0.7 and 0.5,respectively.The problem is solved for the two Rayleigh numbers of 104 and 105 considering a ratio between the two major axes up to 3 while the angle of orientation of the minor axes varies from 0 to 90◦.The results for local and average Nusselt numbers are obtained and discussed together with the details of both flow and thermal fields.For isothermal heating conditions,the study has shown an optimum value for major axes ratio that minimizes the rate of heat transfer between the two tubes.Another important aspect of this paper is to prove the successful use of the Fourier Spectral Method in solving confined flow and heat convection problems.