Silicon is produced in submerged arc furnaces which are heated by electric currents passing through the furnace. It is important to understand the distribution of heating within the furnace in order to accurately model the silicon production process, yet many existing studies neglect aspects of this current flow. In the present paper, we formulate a model that couples the electrical current to thermal, material flow and chemical processes in the furnace. We then exploit disparate timescales to homogenise the model over the timescale of the alternating current, deriving averaged equations for the slow evolution of the system. Our numerical simulations predict a minimum applied current that is required in order to obtain steady-state solutions of the homogenised model and show that for high enough applied currents, two spatially heterogeneous steady-state solutions exist, with distinct crater sizes. We show that the system evolves to the steady state with a larger crater radius and explain this behaviour in terms of the overall power balance typically found within a furnace. We find that the industrial practice of stoking furnaces increases the overall rate of material consumption in the furnace, thereby improving the efficiency of silicon production.

We study the extended Stefan problem which includes constitutional supercooling for the solidification of a binary alloy in a finite spherical domain. We perform an asymptotic analysis in the limits of large Lewis number and small Stefan number which allows us to identify a number of spatio-temporal regimes signifying distinct behaviours in the solidification process, resulting in an intricate boundary layer structure. Our results generalise those present in the literature by considering all time regimes for the Stefan problem while also accounting for impurities and constitutional supercooling. These results also generalise recent work on the extended Stefan problem for finite planar domains to spherical domains, and we shall highlight key differences in the asymptotic solutions and the underlying boundary layer structure which result from this change in geometry. We compare our asymptotic solutions with both numerical simulations and real experimental data arising from the casting of molten metallurgical grade silicon through the water granulation process, with our analysis highlighting the role played by supercooling in the solidification of binary alloys appearing in such applications.

In this paper, numerical simulations of solid particle erosion phenomena on H-type finned circular/elliptic tube surface, which is of great significance to the antiwear design of heat exchanger, are presented. The Eulerian-Lagrangian approach is applied to simulate the dilute gas-solid flow through H-type finned circular/elliptic tubes. A semi-empirical model is adopted to predict the erosion rate. The dynamics behavior of the entrained solid particles in the flow is presented. The geometry of eroded tube surface is changed with the predicted erosion which is taken into account by a UDF and the flow field is re-solved for the eroded tube surface at every time step. The influences of ten parameters (the tube bundle arrangement, particle size, particle concentration, fluid Reynolds number, fin thickness, fin pitch, fin length, fin width, slit width and the transverse tube pitch) on the maximum erosion depth of the H-type circular/elliptic finned tube surface are investigated. Using H-type finned elliptic tube surface can effectively reduce the erosion rate of tube surface comparedwith that using H-type finned circular tube surface. The erosion in in-line arrangement is less severe than that in staggered arrangement. With the increase of particle size, particle concentration and the fluid Reynolds number, the erosion rate of the tube surface rises. The numerically predicted effect of Reynolds number is in good agreement with previous test data. Among the six geometry parameters, the most influential parameter is the transverse tube pitch.

An investigation of natural convective flow and heat transfer inside a three dimensional rectangular cavity containing an array of discrete heat sources is carried out. The array consists of a row and columnwise regular arrangement of identical square shaped isoflux discrete heaters and is flush mounted on a vertical wall of the cavity. A symmetrical isothermal sink condition is maintained by cooling the cavity uniformly from either the opposite wall or the side walls or the top and bottom walls. The other walls of the cavity are maintained adiabatic. A finite volume method based on the SIMPLE algorithm and the power law scheme is used to solve the conservation equations. The parametric study covers the influence of pertinent parameters such as the Rayleigh number, the Prandtl number, side aspect ratio of the cavity and cavity heater ratio. A detailed fluid flow and heat transfer characteristics for the three cases are reported in terms of isothermal and velocity vector plots and Nusselt numbers. In general it is found that the overall heat transfer rate within the cavity for Ra=107 is maximum when the side aspect ratio of the cavity lies between 1.5 and 2. A more complex and peculiar flow pattern is observed in the presence of top and bottom cold walls which in turn introduces hot spots on the adiabatic walls. Their location and size are highly sensitive to the side aspect ratio of the cavity and hence offers more effective ways for passive heat removal.

Effects of viscous dissipation and radiation on MHD natural convection in oblique porous cavity with constant heat flux is studied numerically in the present article. The right inclined wall is maintained at a constant cold temperature Tc and the left inclined wall has a constant heat flux q with length S, while the remainder of the left wall is adiabatic. The horizontal walls are assumed to be adiabatic. The governing equations are obtained by applying the Darcy model and Boussinesq approximations. COMSOL's finite element method is used to solve the non-dimensional governing equations together with specified boundary conditions. The governing parameters of this study are Rayleigh number (Ra=10,100,200,250,500 and 1000), Hartmann number (0≤ Ha ≤20), inclination angle of the magnetic field (0° ≤ ω ≤ π/2), Radiation (0≤R≤15), the heater flux length (0.1≤H≤1) and inclination angle of the sloping wall (–π/3≤ϕ≤π/3). The results are considered for various values of the governing parameters in terms of streamlines, isotherms and averageNusselt number. It is found that the intensity of the streamlines and the isotherm patterns decrease with an increment in Hartmann number. The overall heat transfer is significantly increased with the increment of the viscous dissipation and the radiation parameters.

A mathematical multi-zone ice accretion model used in the numerical simulation of icing on airfoil surface based on three water states, namely, continuous film, rivulets and beads is studied in this paper. An improved multi-zone roughness model is proposed. According to the flow state of liquid water and film flow, rivulets flow governing equations are established to calculate film mass distribution, film velocity, rivulet wetness factor and rivulet mass distribution. Force equilibrium equations of droplet are used to establish the critical conditions of water film broken into rivulets and rivulets broken into beads. The temperature conduction inside the water layer and ice layer is considered. Using the proposed model ice accretion on a NACA0012 airfoil profile with a 4° angle of attack under different icing conditions is simulated. Different ice shapes like glaze ice, mixed ice and rime ice are obtained, and the results agree well with icing wind tunnel experiment data. It can be seen that, water films are formed on the surface, and heights of the films vary with icing time and locations. This results in spatially-temporally varying surface roughness and heat transfer process, ultimately affects the ice prediction. Model simulations indicate that the process of water film formation and evolution cannot be ignored, especially under glaze ice condition.

High intensity focused ultrasound is a rapidly developing technology for the ablation of tumors. Liver cancer is one of the most common malignancies worldwide. Since liver has a large number of blood vessels, blood flow cooling can reduce the necrosed volume and may cause regeneration of the tumor to occur. All cancer cells should be ablated without damaging of the critical tissues. Today, treatment planning tools consider liver as a homogeneous organ. This paper is a step towards the development of surgical planning platform for a non-invasive HIFU tumor ablative therapy in a real liver geometry based on CT/MRI image. This task requires coupling of different physical fields: acoustic, thermal and hydrodynamic. These physical fields can influence each other. In this paper we illustrate how a computational model can be used to improve the treatment efficiency. In large blood vessel both convective cooling and acoustic streaming can change the temperature considerably near blood vessel. The whole tumor ablation took only 30 seconds in the considered simulation case, which is very small comparing with the current treatment time of several hours. Through this study we are convinced that high ultrasound power and nonlinear propagation effects with appropriate treatment planning can sufficiently reduce the treatment time.

A heated triangular cylinder is suddenly cooled in a constant temperature bath. The transient heat conduction problem is transformed to the Helmholtz equation related to the vibration of membranes. Using the membrane analogy, exact analytic solutions for the transient heat conduction problem for three triangular cross sections are found.

We show that a single particle distribution for the “energy-conserving” D2Q13 lattice Boltzmann scheme can simulate coupled effects involving advection and diffusion of velocity and temperature. We consider various test cases: non-linear waves with periodic boundary conditions, a test case with buoyancy, propagation of transverse waves, Couette and Poiseuille flows. We test various boundary conditions and propose to mix bounce-back and anti-bounce-back numerical boundary conditions to take into account velocity and temperature Dirichlet conditions. We present also first results for the de Vahl Davis heated cavity. Our results are compared with the coupled D2Q9-D2Q5 lattice Boltzmann approach for the Boussinesq system and with an elementary finite differences solver for the compressible Navier-Stokes equations. Our main experimental result is the loss of symmetry in the de Vahl Davis cavity computed with the single D2Q13 lattice Boltzmann model without the Boussinesq hypothesis. This result is confirmed by a direct Navier Stokes simulation with finite differences.

This paper investigates the laminar boundary layer flow of nanofluid induced by a radially stretching sheet. Nanofluid model exhibiting Brownian motion and thermophoresis is used. Series solutions for a reduced system of nonlinear ordinary differential equations are obtained by homotopy analysis method (HAM). Comparative study between the HAM solutions and previously published numerical results shows an excellent agreement. Velocity, temperature and mass fraction are displayed for various values of parameters. The local skin friction coefficient, the local Nusselt number and the local Sherwood number are computed. It is observed that the presence of nanoparticles enhances the thermal conductivity of base fluid. It is found that the convective heat transfer coefficient (Nusselt number) is decreased with an increase in concentration of nanoparticles whereas Sherwood number increases when concentration of nanoparticles in the base fluid is increased.

Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

A diesel particulate filter (DPF) is a key technology to meet future emission standards of particulate matters (PM), mainly soot. It is generally consists of a wall-flow type filter positioned in the exhaust stream of a diesel vehicle. It is difficult to simulate the thermal flow in DPF, because we need to consider the soot deposition and combustion in the complex geometry of filter wall. In our previous study, we proposed an approach for the conjugate simulation of gas-solid flow. That is, the gas phase was simulated by the lattice Boltzmann method (LBM), coupled with the equation of heat conduction inside the solid filter substrate. However, its numerical procedure was slightly complex. In this study, to reduce numerical costs, we have tested a new boundary condition with chemical equilibrium in soot combustion at the surface of filter substrate. Based on the soot oxidation rate with catalysts evaluated in experiments, the lattice Boltzmann simulation of soot combustion in the catalyzed DPF is firstly presented to consider the process in the after-treatment of diesel exhaust gas. The heat and mass transfer is shown to discuss the effect of catalysts.

The governing equations for heat transfer and fluid flow are often formulated in a general form for the simplification of discretization and programming, which has achieved great success in thermal science and engineering. Based on the analysis of the popular general form of governing equations, we found that energy conservation cannot be guaranteed when specific heat capacity is not constant, which may lead to unreliable results. A new concept of generalized density is put forward, based on which a new general form of governing equations is proposed to guarantee energy conservation. A number of calculation examples have been employed to verify validation and feasibility.

Two-phase flow and heat transfer, such as boiling and condensing flows, are complicated physical phenomena that generally prohibit an exact solution and even pose severe challenges for numerical approaches. If numerical solution time is also an issue the challenge increases even further. We present here a numerical implementation and novel study of a fully distributed dynamic one-dimensional model of two-phase flow in a tube, including pressure drop, heat transfer, and variations in tube cross-section. The model is based on a homogeneous formulation of the governing equations, discretized by a high resolution finite difference scheme due to Kurganov and Tadmore.

The homogeneous formulation requires a set of thermodynamic relations to cover the entire range from liquid to gas state. This leads a number of numerical challenges since these relations introduce discontinuities in the derivative of the variables and are usually very slow to evaluate. To overcome these challenges, we use an interpolation scheme with local refinement.

The simulations show that the method handles crossing of the saturation lines for both liquid to two-phase and two-phase to gas regions. Furthermore, a novel result obtained in this work, the method is stable towards dynamic transitions of the inlet/outlet boundaries across the saturation lines. Results for these cases are presented along with a numerical demonstration of conservation of mass under dynamically varying boundary conditions. Finally we present results for the stability of the code in a case of a tube with a narrow section.

This paper summarizes suitable material models for creep and damage of concrete which are coupled with heat and moisture transfer. The fully coupled approach or the staggered coupling is assumed. Governing equations are spatially dis-cretized by the finite element method and the temporal discretization is done by the generalized trapezoidal method. Systems of non-linear algebraic equations are solved by the Newton method. Development of an efficient and extensible computer code based on the C++ programming language is described. Finally, successful analyses of two real engineering problems are described.

A review is presented on our recent Vlasov-Fokker-Planck (VFP) simulation code development and applications for high-power laser-plasma interactions. Numerical schemes are described for solving the kinetic VFP equation with both electron-electron and electron-ion collisions in one-spatial and two-velocity (1D2V) coordinates. They are based on the positive and flux conservation method and the finite volume method, and these two methods can insure the particle number conservation. Our simulation code can deal with problems in high-power laser/beam-plasma interactions, where highly non-Maxwellian electron distribution functions usually develop and the widely-used perturbation theories with the weak anisotropy assumption of the electron distribution function are no longer in point. We present some new results on three typical problems: firstly the plasma current generation in strong direct current electric fields beyond Spitzer-Härm’s transport theory, secondly the inverse bremsstrahlung absorption at high laser intensity beyond Langdon’s theory, and thirdly the heat transport with steep temperature and/or density gradients in laser-produced plasma. Finally, numerical parameters, performance, the particle number conservation, and the energy conservation in these simulations are provided.

A mathematical model for the steady, mixed convection heat and mass transfer along a semi-infinite vertical plate embedded in a micropolar fluid in the presence of Soret and Dufour effects is presented. The non-linear governing equations and their associated boundary conditions are initially cast into dimensionless forms using local similarity transformations. The resulting system of equations is then solved numerically using the Keller-box method. The numerical results are compared and found to be in good agreement with previously published results as special cases of the present investigation. The non-dimensional velocity, microrotation, temperature and concentration profiles are displayed graphically for different values of coupling number, Soret and Dufour numbers. In addition, the skin-friction coefficient, the Nusselt number and Sherwood number are shown in a tabular form.

Fluids engineering is extremely important in a wide variety of materials processing systems, such as soldering, welding, extrusion of plastics and other polymeric materials, Chemical Vapor Deposition (CVD), composite materials manufacturing. In particular, mixed convection due to moving surfaces is very important in these applications. Mixed convection in a channel, as a result of buoyancy and motion of one of its walls has received little research attention and few guidelines are available for choosing the best performing channel configuration, particularly when radiative effects are significant. In this study a numerical investigation of the effect of radiation on mixed convection in air due to the interaction between a buoyancy flow and an unheated moving plate induced flow in a uniformly heated convergent vertical channel is carried out. The moving plate has a constant velocity and moves in the buoyancy force direction. The principal walls of the channel are heated at uniform heat flux. The numerical analysis is accomplished by means of the commercial code Fluent. The effects of the wall emissivity, the minimum channel spacing, the converging angle and the moving plate velocity are investigated and results in terms of air velocity and temperature fields inside the channel and wall temperature profiles, both of the moving and the heated plates, are given. Nusselt numbers, both accounting and not for the radiative contribution to heat removal, are also presented.

In the present paper a combined procedure for the quasi-dimensional modelling of heat transfer, combustion and knock phenomena in a “downsized” Spark Ignition two-cylinder turbocharged engine is presented. The procedure is extended to also include the effects consequent the Cyclic Variability. Heat transfer is modelled by means of a Finite Elements model. Combustion simulation is based on a fractal description of the flame front area. Cyclic Variability (CV) is characterized through the introduction of a random variation on a number of parameters controlling the rate of heat release (air/fuel ratio, initial flame kernel duration and radius, laminar flame speed, turbulence intensity). The intensity of the random variation is specified in order to realize a Coefficient Of Variation (COV) of the Indicated Mean Effective Pressure (IMEP) similar to the one measured during an experimental campaign. Moreover, the relative importance of the various concurring effects is established on the overall COV. A kinetic scheme is then solved within the unburned gas zone, characterized by different thermodynamic conditions occurring cycle-by-cycle. In this way, an optimal choice of the “knock-limited” spark advance is effected and compared with experimental data. Finally, the CV effects on the occurrence of individual knocking cycles are assessed and discussed.

Cooling methods are needed for turbine blade tips to ensure a long durability and safe operation. A common way to cool a tip is to use serpentine passages with 180-deg turn under the blade tip-cap taking advantage of the three-dimensional turning effect and impingement like flow. Improved internal convective cooling is therefore required to increase the blade tip lifetime. In the present study, augmented heat transfer of an internal blade tip with pin-fin arrays has been investigated numerically using a conjugate heat transfer method. The computational domain includes the fluid region and the solid pins as well as the tip regions. Turbulent convective heat transfer between the fluid and pins, and heat conduction within pins and tip are simultaneously computed. The main objective of the present study is to observe the effect of the pin material on heat transfer enhancement of the pin-finned tips. It is found that due to the combination of turning, impingement and pin-fin crossflow, the heat transfer coefficient of a pin-finned tip is a factor of 2.9 higher than that of a smooth tip at the cost of an increased pressure drop by less than 10%. The usage of metal pins can reduce the tip temperature effectively and thereby remove the heat load from the tip. Also, it is found that the tip heat transfer is enhanced even by using insulating pins having low thermal conductivity at low Reynolds numbers. The comparisons of overall performances are also included.