Investigation of the effect of magnetic field on melting of solid gallium in a bottom-heated rectangular cavity using the lattice Boltzmann method

ABSTRACT

A numerical study is presented for two-dimensional convection melting of solid gallium in a rectangular cavity. The bottom wall of the cavity is uniformly heated and a uniform magnetic field is applied separately in both horizontal and vertical directions. The lattice Boltzmann (LB) method considering the magnetic field force is employed to solve the governing equations. The effects of magnetic field on flow and heat transfer during melting are presented and discussed at Rayleigh number Ra = 1 × 10⁵ and Hartmann number Ha = 0, 15, and 30. The results show that the magnetic field with an inclination angle has a significant impact on the flow and heat transfer in the melting process. For a small Hartmann number, similar melting characteristics are observed for both horizontally applied and vertically applied magnetic fields. For a high value of Hartmann number, it is found that in the earlier stage of melting process, the flow retardation effect caused by the horizontally applied magnetic field is less obvious than that caused by the vertically applied magnetic field. However, the opposite is true in the later stage.     

Numerical Simulation of Double Diffusive Mixed Convection in a Lid-Driven Square Cavity Using Velocity-Vorticity Formulation

In this article, convection driven by combined thermal and solutal concentration buoyancy effects in a lid-driven square cavity is examined using velocity-vorticity form of Navier-Stokes equations. The governing equations consist of vorticity transport equation, velocity Poisson equations, energy equation, and concentration equation. Validation results are discussed for convection due to heat and mass transfer in a lid-driven square cavity at Re = 500, Le = 2, and G_RT  = G_RS  = 100. These results indicate that the present velocity-vorticity formulation could predict the characteristic parameters of flow, temperature, and solutal concentration fields using a much coarser mesh compared to the mesh used in a stream function-vorticity formulation. The capability of the proposed algorithm to handle complex geometry is demonstrated by application to mixed convection in a lid-driven square cavity with a square blockage. The effect of buoyancy ratio on the convection phenomenon is discussed for buoyancy ratio varying from ? 100 to 100 at Re = 100. Under opposing temperature and concentration gradients along the vertical direction, the negative buoyancy ratios give rise to aiding flows.     

Two phase simulation of nanofluid flow and heat transfer using heatline analysis

In this study Control Volume based Finite Element Method is applied to solve the problem of natural convection heat transfer in an enclosure filled with nanofluid. The important effect of Brownian motion and thermophoresis has been included in the model of nanofluid. The inner sinusoidal and outer circular walls are maintained at constant temperatures while the two other walls are thermally insulated. The heat transfer between cold and hot regions of the enclosure cannot be well understood by using isotherm patterns so heatline visualization technique is used to find the direction and intensity of heat transfer in a domain. Effects of thermal Rayleigh number (Ra), buoyancy ratio number (Nr) and Lewis number (Le) on streamline, isotherm, isoconcentration and heatline are examined. The results indicate that the average Nusselt number decreases as buoyancy ratio number increases until it reaches a minimum value and then starts increasing. As Lewis number increases, this minimum value occurs at higher buoyancy ratio number.     

Effects of Wall-Located Heat Barrier on Conjugate Conduction/Natural-Convection Heat Transfer and Fluid Flow in Enclosures

The effects of a heat barrier, located in the ceiling wall of an enclosure, on conjugate conduction/natural convection are investigated numerically. The vertical walls of the enclosure are differentially heated and the horizontal walls are adiabatic. Heatline technique is used to visualize heat transport. The variations of average Nusselt number, dimensionless heat transfer rate through the ceiling wall, and dimensionless overall heat transfer rate are studied. Calculations are performed for different Rayleigh numbers (10³ ≤ Ra ≤ 10⁶), thermal conductivity ratios (1 ≤ K ≤ 100), dimensionless locations of the heat barrier (0 < X _h  < 1),and two dimensionless ceiling wall thicknesses (D = 0.05 and D = 0.20). For high thermal conductivity ratio (K = 100), the heat barrier considerably reduces the dimensionless overall heat transfer rate. The effect of the heat barrier on dimensionless heat transfer rate through the enclosure increases as the Rayleigh number decreases. For low Rayleigh number (i.e., Ra = 10³), a location exists in the ceiling wall for which the dimensionless overall heat transfer rate is minimum.     

Effect of Radiative Heat Transfer on Three-Dimensional Double Diffusive Natural Convection

This work presents a numerical study of the effect of the radiative heat transfer on the three-dimensional double diffusive convection in a differentially heated cubic cavity for different optical parameters of the medium. This numerical study is conducted for fixed Prandtl, Rayleigh, and Lewis numbers, Pr = 13.6, Ra = 10⁵, Le = 2, and buoyancy ratio N in the range [–2, 0]. The natural convection equations, using the Boussinesq approximation for the treatment of buoyancy term in the momentum equation, are expressed using the vorticity–stream function formulation. These equations and the radiative transfer equation are discretized, respectively, with the control volume finite difference method and the FTn finite volume method. The influences of the optical thickness and the conduction–radiation parameter of the semitransparent fluid on heat and mass transfer are depicted. Results show different transitions of the structure of the main flow when varying the conduction–radiation parameter and the optical thickness.     

Mesoscopic simulation of CuOH2O nanofluid in a porous enclosure with elliptic heat source

In this article, mesoscopic approach has been utilized to investigate magnetic field impact on CuOH₂O nanofluid free convection inside a porous cavity with elliptic heat source. Simulations have been done via LBM. KKL model is employed to consider Brownian motion impact on nanofluid properties. Influences of Rayleigh number (Ra), nanofluid volume fraction (?), Hartmann number (Ha), Darcy number (Da) on heat transfer treatment are demonstrated. Outputs demonstrate that temperature gradient reduces with increase of Ha while it increases with augment of Da,Ra.     

Natural Convection Heat Transfer in an Enclosure Filled with an Ethylene Glycol—Copper Nanofluid Under Magnetic Fields

This article presents the results of a numerical study on natural convection in a square enclosure filled with ethylene glycol–copper nanofluid in the presence of magnetic fields. Two opposite horizontal walls of the enclosure are insulated and the two vertical walls are kept constant at different temperatures. A uniform horizontal magnetic field is externally imposed. The governing equations (mass, momentum, and energy) are formulated and solved numerically with a finite element using COMSOL Multiphysics. The effects of pertinent parameters such as Rayleigh number (10³ ≤ Ra ≤ 10⁷), Hartmann number (0 ≤ Ha ≤ 120), and solid volume fraction (0 ≤ φ ≤ 0.06) on the flow and the heat transfer performance of the enclosure are examined when the Prandtl number is assumed to be Pr = 151.     

Analytical and Numerical Study of Combined Effects of a Magnetic Field and an External Shear Stress on Soret Convection in a Horizontal Porous Enclosure

The fluid flow induced by combined actions of Soret effect and shear stress applied on the top horizontal free surface (the lower one being rigid) in a horizontal porous layer, under an external magnetic field, is studied analytically and numerically. The horizontal walls of the porous layer are subject to uniform heat fluxes. The porous layer is sparsely packed then the flow is governed by the Brinkman model assuming the Boussinesq approximation. The governing parameters are the thermal Rayleigh number, R_T, the Lewis number, Le, the separation parameter, ?, the effective Darcy number, Da, the Hartmann number Ha, the dimensionless shear stress, τ, and the aspect ratio of the enclosure, A_r. An analytical solution is derived on the basis of the parallel flow approximation, assuming enlarge aspect ratio layer, and validated numerically using a finite-difference method. The critical Rayleigh numbers for the onset of stationary, subcritical, and oscillatory convection are determined explicitly as functions of the governing parameters for infinite layers with a zero shear stress, τ = 0. The codimension-2 point is identified and different flow behaviors are observed and discussed. The effects of the governing parameters on the fluid flow intensity and heat and mass transfer characteristics are also discussed.     

Sensitivity analysis for MHD effects and inclination angles on natural convection heat transfer and entropy generation of Al2O3-water nanofluid in square cavity by Response Surface Methodology

The natural convection heat transfer and entropy generation of Al₂O₃-water nanofluid, in a square cavity with inclination angle θ and the presence of a constant axial magnetic field B₀ are examined in this paper. The governing equations are solved numerically by finite volume method. Also an effective parameters analysis was performed by using of the Response Surface Methodology (RSM). The effects of the Rayleigh number (10³, 10⁴, 10⁵ and 10⁶), Hartmann number (0, 10, 30 and 50) and also inclination angles (0°, 30°, 60° and 90°) are investigated. It is observed that the mean Nusselt number and the total entropy generation increase when the Rayleigh number increases. It is also found that, regardless of the Ha parameter, by increasing of the inclination angles, the mean Nusselt number and entropy generation rate increase until inclination angle 30° and then they decrease. Also, for low Ra numbers, by increasing the Ha parameter, the mean Nusselt number increases until Ha = 10 and then decreases. The analysis showed that the sensitivity of the Nusselt number and the entropy generation to Ha parameter was too small, and as a result it was negligible. Also, the sensitivity of the mean Nusselt number and the entropy generation to inclination angle, θ, increases by increasing of this angle. It is also observed that the mean Nusselt number and the entropy generation were more sensitive to the inclination angle θ than the Ha parameter.     

Role of Entropy Generation On Thermal Management During Natural Convection in a Tilted Square Cavity with Isothermal and Non-Isothermal Hot Walls

Entropy generation during natural convection within tilted square cavity inclined with different angles (? = 30°and 75°) for various thermal boundary conditions (case 1: isothermal heating and case 2: non-isothermal heating) has been studied. Simulations are carried out over a range of parameters: Rayleigh number (10³ ≤ Ra ≤ 10⁵) and Prandtl numbers (Pr = 0.025 and 998.24). The numerical results are presented in terms of isotherms (θ), streamlines (ψ), entropy generation due to heat transfer (S _θ ) and fluid friction (S _ψ ). Heating strategy is energy efficient for case 2 (non-isothermal heating) due to its less total entropy generation with reasonable heat transfer rate, irrespective of Pr.     

MHD Mixed Convection in a Channel with a Triangular Cavity

A numerical study has been carried out in an open channel, which have a heated triangular cavity at the bottom wall. The remaining walls of the channel are adiabatic. Flow inlets to the channel with uniform velocity and fully developed flow are accepted at the exit of the channel. Steady state mixed convection by laminar flow has been studied by numerically solving governing equations to obtain flow field and temperature distribution under the magnetic field and Joule effect. Equations are solved via the Galerkin weighted residual finite element technique. Calculations are performed for different governing parameters such as Hartmann number (10 ≤ Ha ≤ 100), Reynolds number (100 ≤ Re ≤ 2,000), Rayleigh number (10³ ≤Ra ≤ 10⁵), Joule parameter (0 ≤ J ≤ 5), and Prandtl number (1 ≤ Pr ≤ 10). It is found that heat transfer decreases with an increasing of the Hartmann number especially at higher values of Rayleigh number. Fluid temperature at the exit of the channel also decreases with increasing of Hartmann number. Fluid temperature at the outlet of the channel becomes higher at low Reynolds number and higher Rayleigh number. However, it decreases with the decreasing of the Reynolds number.     

Hydromagnetic double-diffusive convection in a rectangular enclosure with opposing temperature and concentration gradients

The finite-difference method is used to predict numerically the characteristics of hydromagnetic double-diffusive convective flow of a binary gas mixture in a rectangular enclosure with the upper and lower walls being insulated. Constant temperatures and concentrations are imposed along the left and right walls of the enclosure and a uniform magnetic field is applied in the x-direction. Consistent with what is reported by previous investigators, an oscillation in the flow is observed in the absence of the magnetic field for a specific range of buoyancy ratio values where the Prandtl number Pr=1, the Lewis number Le=2, the thermal Rayleigh number Ra_T=10⁵, and the aspect ratio A=2 for the enclosure. In the presence of the magnetic field, however, no oscillatory behavior is observed. Numerical results are reported for the effect of the heat generation or absorption coefficient and the Hartmann number on the contours of streamline, temperature, concentration and density. In addition, results for the average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. In this study, the thermal and compositional buoyancy forces are assumed to be opposite.     

Magnetic field influence on CuO–H2O nanofluid convective flow in a permeable cavity considering various shapes for nanoparticles

In this paper, CuO–H₂O nanofluid forced convection in a lid driven porous cavity is investigated under the impact of magnetic field. Shape effect of nanoparticles and Brownian motion impact on nanofluid properties are taken into consideration. Vorticity stream function formulation is utilized. The solutions of final equations are obtained by CVFEM. Graphs are shown for different values of Darcy number (Da), CuO–H₂O volume fraction (?), Reynolds (Ra) and Hartmann (Ha) numbers. Outputs indicate that selecting Platelet shaped nanoparticles results the highest heat transfer rate. Nusselt number augments with rise of Darcy and Reynolds number while it decreases with augment of Lorentz forces.     

Entropy generation at the onset of natural convection

The entropy generation due to heat transfer and friction has been determined in transient state for laminar natural convection by solving numerically the mass, momentum and energy balance equations, using a control volume finite-element method. The variations of the total entropy generation as function of time for Rayleigh number and irreversibility distribution ratio set at 10³?Ra?10⁵ and 10⁻⁴???10⁻¹ were investigated. The evolution of the maximum of entropy generation with the Rayleigh number is studied. The effect of the irreversibility distribution ratio on the maximum entropy generation and the entropy generation in steady state are analyzed. The irreversibility maps for Rayleigh number set at 10³?Ra?10⁵ and irreversibility distribution ratio ?=10⁻⁴ are plotted.     

Visualization of Heat Transport during Natural Convection Within Porous Triangular Cavities via Heatline Approach

In this article, natural convection in a porous triangular cavity has been analyzed. Bejan's heatlines concept has been used for visualization of heat transfer. Penalty finite-element method with biquadratic elements is used to solve the nondimensional governing equations for the triangular cavity involving hot inclined walls and cold top wall. The numerical solutions are studied in terms of isotherms, streamlines, heatlines, and local and average Nusselt numbers for a wide range of parameters Da (10^?5–10^?3), Pr (0.015–1000), and Ra (Ra = 10³–5 × 10⁵). For low Darcy number (Da = 10^?5), the heat transfer occurs due to conduction as the heatlines are smooth and orthogonal to the isotherms. As the Rayleigh number increases, conduction dominant mode changes into convection dominant mode for Da = 10^?3, and the critical Rayleigh number corresponding to the on-set of convection is obtained. Distribution of heatlines illustrate that most of the heat transport for a low Darcy number (Da = 10^?5) occurs from the top region of hot inclined walls to the cold top wall, whereas heat transfer is more from the bottom region of hot inclined walls to the cold top wall for a high Darcy number (Da = 10^?3). Interesting features of streamlines and heatlines are discussed for lower and higher Prandtl numbers. Heat transfer analysis is obtained in terms of local and average Nusselt numbers (Nu _l , Nu _t ) and the local and average Nusselt numbers are found to be correlated with heatline patterns within the cavity.     

Numerical investigation on heat transfer of supercritical water in a roughened tube

ABSTRACT

The turbulent mixed convection heat transfer of supercritical water flowing in a vertical tube roughened by V-shaped grooves has been numerically investigated in this paper. The turbulent supercritical water flow characteristics within different grooves are obtained using a validated low-Reynolds number κ-ε turbulence model. The effects of groove angle, groove depth, groove pitch-to-depth ratio, and thermophysical properties on turbulent flow and heat transfer of supercritical water are discussed. The results show that a groove angle γ = 120° presents the best heat transfer performance among the three groove angles. The lower groove depth and higher groove pitch-to-depth ratio suppress the enhancement of heat transfer. Heat transfer performance is significantly decreased due to the strong buoyancy force at T_b = 650.6 K, and heat transfer deterioration occurs in the roughened tube with γ = 120°, e = 0.5 mm, and p/e = 8 in the present simulation. The results also show that the rapid variation in the supercritical water property in the region near the pseudo-critical temperature results in a significant enhancement of heat transfer performance.     

Entropy Generation During Natural Convection in a Porous Cavity: Effect of Thermal Boundary Conditions

Entropy generation plays a significant role in the overall efficiency of a given system, and a judicious choice of optimal boundary conditions can be made based on a knowledge of entropy generation. Five different boundary conditions are considered and their effect of the permeability of the porous medium, heat transfer regime (conduction and convection) on entropy generation due to heat transfer, and fluid friction irreversibilities are investigated in detail for molten metals (Pr = 0.026) and aqueous solutions (Pr = 10), with Darcy numbers (Da) between 10^?5–10^?3 and at a representative high Rayleigh number, Ra = 5 × 10⁵. It is observed that the entropy generation rates are reduced in sinusoidal heating (case 2) when compared to that for uniform heating (case 1), with a penalty on thermal mixing. Finally, the analysis of total entropy generation due to variation in Da and thermal mixing and temperature uniformity indicates that, there exists an intermediate Da for optimal values of entropy generation, thermal mixing, and temperature uniformity.     

Effect of the Buoyancy Ratio on Oscillatory Double-Diffusive Convection in Binary Mixture

In this work, a numerical study of double-diffusive convection in binary mixture has been presented. A square cavity filled with a binary mixture and exposed to opposition solute and thermal gradients has been considered. The following flow parameters were considered: Prandtl number Pr = 10, Lewis number Le = 10, and buoyancy ratio varies 0 ≤ N ≤ 2. The finite volume method with SIMPLER algorithm was used to solving numerically the mathematical model. Our computer code is validated and shows a good agreement with literature available results. The obtained results show a strongest dependence of the thermal structure and solute effect with the buoyancy ratio. The oscillatory double-diffusive flow appeared from periodic time-evolution where the phenomena retuned in each period time. A critical thermal Rayleigh number Ra_Tcr and corresponding dominated frequency for the onset of oscillatory double-diffusive convection were determined for each buoyancy ratio N, and the results show a strongest dependence between the buoyancy ratio and critical Rayleigh number. Also, the dominance of solute force increases the intensity of the flow better than the case of the dominance of thermal force.     

Double-Diffusive Natural Convection in a Closed Annulus

A numerical study for steady laminar double-diffusive natural convection within a vertical closed annulus is examined with constant temperature and mass species (concentration) differences imposed across the vertical walls. The annulus has an aspect ratio of 1 and a curvature ratio of 2, while the fluid Prandtl number is 7. In this paper the problem is defined and the numerical solution procedure is validated. Moreover, the effect of buoyancy ratio on the flow structure and rite resulting heal and mass transfer rates is presented. It is determined that buoyancy ratio is the primary factor that defines flow structure, including concentration—dominated (buoyancy force) opposing flow, transitional flow, thermal-dominated flow, or concentration-dominated aiding flow. The relationship for buoyancy ratios, in the range -10 ≤ n ≤ 10, and the average NusseU and Sherwood numbers have been obtained for a thermal Rayleigh number of 50,000 and a Lewis number of 5. Future papers wilt include the effect of thermal Rayleigh number, Lewis number, and various geometric parameters on the flow structure and heat and nusi transfer.     

MHD Mixed Convection in a Lid-Driven Cavity Including Heat Conducting Solid Object and Corner Heaters with Joule Heating

The hydromagnetic mixed convection flow and heat transfer in a top sided lid-driven square enclosure is numerically simulated in this paper following a finite volume approach based on the SIMPLEC algorithm. The enclosure is heated by corner heaters which are under isothermal boundary conditions with different lengths in bottom and right vertical walls. The lid is having lower temperature than heaters. The other boundaries of the enclosure are insulated. A uniform magnetic field is applied along the horizontal direction. A heat conducting horizontal solid object (a square cylinder) is placed centrally within the outer enclosure. Shear forces through lid motion, buoyancy forces due to differential heating and magnetic forces within the electrically conducting fluid inside the enclosure act simultaneously. Heat transfer due to forced flow, thermal buoyancy, Joule dissipation and conduction within the solid object are taken into account. Simulations are conducted for various controlling parameters such as the Richardson number (0.1 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 50) and Joule heating parameter (0 ≤ J ≤ 5) keeping the Reynolds number based on lid velocity fixed as Re = 100. The flow and thermal fields are analyzed through streamline and isotherm plots for various Ha, J and Ri. Furthermore, the pertinent transport quantities such as the drag coefficient, Nusselt number and bulk fluid temperature are also plotted to show the effects of Ha, J and Ri on them.