Effect of aspect ratio on convection in a porous enclosure with partially active thermal walls

The aim of the present numerical investigation is to understand the effect of aspect ratio and partially thermally active zones on convective flow and heat transfer in a rectangular porous enclosure. Five different heating and cooling zones are considered along the vertical walls while the remaining portions of the sidewalls and top and bottom of the enclosure are adiabatic. The Brinkman-Forchheimer extended Darcy model is used in the study. The governing equations are solved by the finite volume method with the SIMPLE algorithm. The computations are carried out for a wide range of parameters and the results are presented graphically. The results reveal that the location of heating and cooling zones has a significant influence on the flow pattern and the corresponding heat transfer in the enclosure. The rate of heat transfer approaches to a constant value for very low values of the Darcy number. The heat transfer rate is decreased on increasing the aspect ratio.     

A level-set method for steady-state and transient natural convection problems

This paper introduces a topology optimization method for 2D and 3D, steady-state and transient heat transfer problems that are dominated by natural convection in the fluid phase and diffusion in the solid phase. The geometry of the fluid-solid interface is described by an explicit level set method which allows for both shape and topological changes in the optimization process. The heat transfer in the fluid is modeled by an advection-diffusion equation. The fluid velocity is described by the incompressible Navier-Stokes equations augmented by a Boussinesq approximation of the buoyancy forces. The temperature field in the solid is predicted by a linear diffusion model. The governing equations in both the fluid and solid phases are discretized in space by a generalized formulation of the extended finite element method which preserves the crisp geometry definition of the level set method. The interface conditions at the fluid-solid boundary are enforced by Nitsche’s method. The proposed method is studied for problems optimizing the geometry of cooling devices. The numerical results demonstrate the applicability of the proposed method for a wide spectrum of problems. As the flow may exhibit dynamic instabilities, transient phenomena need to be considered when optimizing the geometry. However, the computational burden increases significantly when the time evolution of the flow fields needs to be resolved.     

Effect of heat generation on mixed convection in porous cavity with sinusoidal heated moving lid and uniformly heated or cooled bottom walls

Effect of heat generation and absorption on mixed convection flows in a sinusoidal heated lid-driven square cavity filled with a porous medium is investigated numerically. Both the vertical walls of the enclosure are insulated while the bottom wall is uniformly heated or cooled. The top wall is moving at a constant speed and is heated sinusoidally. The governing equations and boundary conditions are non-dimensionalized and solved numerically by using finite volume method approach along with SIMPLE algorithm together with non-uniform grid system. The effect of Darcy and heat generation parameters are investigated in terms of the flow, heat transfer, and Nusselt number. The results for stream function and isotherm are plotted and it is found that there have significant influence with the presence of heat generation and porous medium.

     

Phase change process with free convection in a circular enclosure: numerical simulations

A numerical study of unsteady natural convection flow during freezing of water in a circular enclosure is presented. Mathematical model for phase change is based on apparent capacity method formulation and the governing equations are discretized on a fixed grid by means of finite element method. Water’s temperature is initially higher than its freezing temperature. Then, the temperature of the enclosure’s boundary is dropped to a temperature lower than freezing temperature. Ice forms at the enclosure boundary while natural convection flow is induced in the liquid region. Calculations have been made for the rate of change of solid fraction and temperature distributions, for conduction and conduction plus convection modes of heat transfer, and density inversion near freezing temperature phenomenon of water is considered. High resolution capturing of solid/liquid moving boundary as well as the details of flow structure is presented. The results indicate that the effect of natural convection is dominant over conduction if the Rayleigh number is higher than 5 × 10⁶ and relatively insignificant if the Rayleigh number is less than 1 × 10⁶.     

A time marching,finite area calculation of general quasi-one-dimensional flows

A numerical technique for calculating the quasi-one-dimensional steady flow of a viscous heat conducting compressible fluid, with allowance for wall friction and external heat transfer, is discussed. The time-dependent flow equations for a control volume are solved by an explicit, time marching, finite area method, basically of first order. The scheme's versatility is illustrated by calculating flows in convergent-divergent nozzles and, Fanno and Rayleigh line processes. The effects of the initial conditions and a relaxation factor used for increasing stability are assessed.     

Numerical investigation of heat and mass transfer flow under the influence of silicon carbide by means of plasma-enhanced chemical vapor deposition vertical reactor

The effect of characteristics flow (contour of velocity), mass transfer (Sherwood number) and heat transfer (Nu number) on the growth rate of silicon carbide by means of plasma-enhanced chemical vapor deposition vertical reactor is investigated. The species transport and thermal fluid transport with chemical reaction are taken into account. The steady-state laminar fluid flow and gas flow having ideal behavior are considered. A mixture of silane and propane (2% molar) as main reactant gases and hydrogen (96% molar) as propellant gas are injected into the reactor. Four different diameters of shower head, three different substrate rotation speeds and five different temperatures of the substrate are used. The finite volume method is employed to solve the problem. The governing equations are solved by upwind differencing scheme. The assumption of speed–pressure coupling leads to use of semi-implicit method for pressure-linked equations to solve the governing equation. It is found that the deposition rate reduces with the shower head diameter and value of substrate temperature and enhances with rotational speed of the substrate. Furthermore, the best shower head diameter to achieve maximum rate of deposition is 1 mm. At the end, a comparison as a limiting case of the considered problem with the existing studies is made. Comparing the results of this experiment with prior studies has shown acceptable consistency.

     

Natural convection in a porous trapezoidal enclosure with an inclined magnetic field

The effect of a magnetic field on steady convection in a trapezoidal enclosure filled with a fluid-saturated porous medium is studied numerically by the finite difference method. The inclined sloping boundaries is treated by adopting staircase-like zigzag lines. The sloping walls are maintained isothermally at different temperatures. The top and bottom horizontal straight walls are kept adiabatic. The results indicate that the heat transfer performance decreases by decreasing the angle of sloping wall. Optimum reducing of the heat transfer rate was obtained for an acute trapezoidal enclosure and large magnetic field in the horizontal direction.     

MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation $\epsilon$ is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.     

A two-dimensional study on natural convection and heat transfer in the enclosure with heat transfer and radiation coupled in natural convection

Numerical investigation using SIMPLE algorithm with QUICK scheme for natural convection and heat transfer in the enclosure bounded by a solid wall and with heat transfer and radiation coupled in natural convection has been conducted.The various parameters are:Rayleigh number(from 103 to 105),dimensionless conductivity of bounding wall(from 0 to 100),dimensionless wall thickness(from 0 to 0.6) and radiation emissivity of all surfaces(from 0 to 1).The results suggest that flow and heat transfer are influenced...     

MHD flow of radiative micropolar nanofluid in a porous channel: optimal and numerical solutions

The flow of a radiative and electrically conducting micropolar nanofluid inside a porous channel is investigated. After implementing the similarity transformations, the partial differential equations representing the radiative flow are reduced to a system of ordinary differential equations. The subsequent equations are solved by making use of a well-known analytical method called homotopy analysis method (HAM). The expressions concerning the velocity, microrotation, temperature, and nanoparticle concentration profiles are obtained. The radiation tends to drop the temperature profile for the fluid. The formulation for local Nusselt and Sherwood numbers is also presented. Tabular and graphical results highlighting the effects of different physical parameters are presented. Rate of heat transfer at the lower wall is seen to be increasing with higher values of the radiation parameter while a drop in heat transfer rate at the upper wall is observed. Same problem has been solved by implementing the numerical procedure called the Runge–Kutta method. A comparison between the HAM, numerical and already existing results has also been made.

     

A level set embedded interface method for conjugate heat transfer simulations of low speed 2D flows

This study examines the use of a level set based embedded interface method to simulate fluid-solid heat transfer processes using Cartesian grids. The flow field is described by the incompressible 2D Navier-Stokes equations using a vorticity-streamfunction approach. A fluid-solid coupling formulation for the thermal and momentum fields is developed that is robust, computationally efficient and second-order accurate. Solutions for several example problems are presented for flow over stationary and moving cylinders to bench mark the current approach. Heat transfer for an isolated cylinder and two cylinders in series are then examined to explore the Nusselt number dependence on cylinder spacing and unsteady conjugate heat transfer processes.     

Upstream-weighted differencing schemes and their application to elliptic problems involving fluid flow

A flexible finite-difference formulation is developed for the system of elliptic equations normally encountered in problems involving heat, mass, and momentum transfer. The flexible formulation allows selection of one of four different solution procedures for the steady-state equations (one of which is the solution of the time-dependent equations). A special diffrencing of the connective and diffusive terms is introduced (containing free parameters) which allows any several differencing schemes to be represented by the same formulae. Two example problems are solved which permit a comparison of central, upstream, and several upstream-weighted, differencing schemes. Emphasis is placed on obtaining efficient and accurate steady- state solutions of the example problems     

Finite difference solution to the flow between two rotating spheres

This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.     

The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet

The effects of variable viscosity and thermal conductivity on the flow and heat transfer in a laminar liquid film on a horizontal shrinking/stretching sheet are analyzed. The similarity transformation reduces the time independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The resulting five-parameter problem is solved by the homotopy perturbation method. The results are presented graphically to interpret various physical parameters appearing in the problem.     

Numerical determination of boundary heat fluxes in an enclosure dynamically with natural convection through Fletcher-Reeves gradient method

An iterative Fletcher-Reeves conjugate gradient method (CGM) is adopted to estimate the boundary heat fluxes in a fluid-saturated enclosure, where the fluid flow is dynamically coupled with the heat convection. The sets of direct, sensitivity and adjoint equations required for the solution of the inverse problem are formulated in terms of an arbitrary domain in two dimensions. The methodology of conjugate gradient method solves the inverse natural convection problem satisfactorily without any a priori information about the unknown heat fluxes. The pressure-correction method is utilized to solve the continuum direct, sensitivity and adjoint problems by enforcing global mass and energy conservations. Effects of boundary heat flux profile and thermal Rayleigh number on the convective heat transport are investigated. The effects of position and number of temperature sensors on the inverse problem solution are also addressed in this paper. Inverse solutions of noise data are regularized with the Discrepancy Principle of Alifanov; otherwise, the high frequency components of the random noise were reproduced.     

Simulation of Stokes flow over microelectrodes with least-squares meshfree method

Microelectrode structures in alternating current (AC) electrokinetics can generate high electric field strength to manipulate, characterize and separate particles in suspending medium. It has been widely used in biological, pharmaceutical and medical fields. In this paper, a least-squares meshfree method (LSMFM) based on the first-order velocity–pressure–vorticity formulation for the Stokes flow, electric potential–electric field strength expression for electric field and temperature–heat flux equations for heat transfer problem is presented to study two-dimensional electrothermally induced fluid flow over microelectrodes. Joule heat generated from electric field acts as heat source and gives rise to electric force and buoyancy force acting on the fluid. The discretization of all system of equations is completed by the least-squares method. The equal-order moving least-squares (MLS) approximation is used with Gaussian quadrature in the background cells constructed by the quadtree algorithm. A matrix-free element-by-element Jacobi preconditioned conjugate gradient method is applied to solve the resulting systems. Finally, an example of steady heat transfer problem with analytical solution is devised to analyze the error estimates of the LSMFM, and the examples of electric field of shielded microstrip line and Stokes flow over microelectrode are also solved to investigate the features of the LSMFM.     

Enhanced natural convection in an isosceles triangular enclosure filled with a nanofluid

Natural convection is studied in an isosceles triangular enclosure with a heat source located at its bottom wall and filled with an Ethylene Glycol–Copper nanofluid. This paper examines the effects of pertinent parameters such as the Rayleigh number, the solid volume fraction, the heat source location, and the enclosure apex angle on the thermal performance of the enclosure. The thermal performance of the enclosure is improved with an increase in the Rayleigh number and solid volume fraction. The results also show that the variation of heat transfer rate with respect to the enclosure apex angle and heat source position and dimensions is different at low and high Rayleigh numbers. A comparison is also presented between the results obtained from the modified and original Maxwell models. The results show that the heat transfer is generally higher based on the modified Maxwell model.     

二维稳态传热问题的计算机仿真

热量传输现象在工程技术领域中广泛存在,对二维稳态传热情形下温度场分布的研究有重要现实意义。对于复杂几何形状的物体和非线性的边界条件,分析解法显得无能为力;相比之下,建立在有限元基础上的数值计算是有效和准确的。在传热和流体流动问题的数值计算方面,SIMPLE算法被广泛采用。通过VC和Matlab的混合编程用SIMPLE算法实现了对二维稳态传热问题的计算仿真,描述了温度场的分布。     

Analysis of radiation-natural convection in a divided enclosure using the lattice Boltzmann method

A numerical hybrid lattice-Boltzmann equation finite-difference is used to study the radiation-natural convection phenomena in a square cavity, differentially heated, with partition of finite thickness and varying height located vertically at the center of the cavity. The results obtained show that the radiation exchange produces a rise in the heat transfer. The average Nusselt number and the throughflow strength increase when the gap width (w), maximal difference temperature (ΔT) and cavity width (L) get larger. Effects of different parameters on streamlines, temperature fields, average Nusselt number and throughflow strength are discussed.