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The purpose of this study is to investigate the applicability of volume average which is extensively used for analyzing the heat and fluid flow (both for single-phase and solid/liquid-phase change) in a closed cell porous medium numerically.

Heat conduction equations for the solid frame and fluid (or phase change material) are solved for pore scale and volume average approaches. The study mainly focuses on the effect of porosity and the number of porous media unit cell on the agreement between the results of the pore scale and volume average approaches.

It is observed for the lowest porosity values such as 0.3 and the number of porous media unit cell as 4 in heat transfer direction, the results between two approaches may be questionable for the single-phase fluid. By increasing the number of porous media unit cell in heat transfer direction, the agreement between two approaches becomes better. In general, for high porosity values (such as 0.9) the agreement between the results of two approaches is in the acceptable range both for single-phase and solid/liquid-phase change. Two charts on the applicability of volume average method for single-phase and solid/liquid-phase change are presented.

The authors’ literature survey shows that it is the first time the applicability of volume average which is extensively used for analyzing the heat and fluid flow in a closed cell porous medium is investigated numerically.

Presents a physical model for determining the effective thermal conductivity of a two‐phase composite medium with fixed or moving interfaces. A rigorous numerical method for removing oscillations in the thermal field is proposed. The methodology is based on the volume averaging technique with the assumption that the phases may coexist at a temperature different from that of fusion. The analysis reveals that the effective conductivity of a two‐phase medium is dependent on the phase volume fractions, on their thermal conductivities and on a constitutive constant which determines the geometric structure of the medium and the nature of the interface (fixed or moving). The results for the one and two dimensional conduction‐dominated phase change problem show that the oscillations produced by previous fixed‐grid methods are eliminated.

This paper aims to tackle the problem of some ambiguity of the momentum equation formulation in the commonly used macroscopic models of two‐phase solid/liquid region, developing during alloy solidification. These different appearances of the momentum equation are compared and the issue is addressed of how the choice of the particular form affects velocity and temperature fields.

Attention is focused on the ensemble averaging method, which, owing to its stochastic nature, is a new promising tool for setting up the macroscopic transport equations in highly inhomogeneous multiphase micro‐ and macro‐structures, with morphology continuously changing in time when the solidification proceeds. The basic assumptions of the two other continuum models, i.e. based on the classical mixture theory and on the volume‐averaging technique, are also unveiled. These three different forms of the momentum equation are then compared analytically and their impact on calculated velocity and temperature distribution in the mushy zone is studied for the selected test problem of binary alloy solidification driven by diffusion and thermal natural convection in a square mould.

It is found that a chosen appearance of the momentum equation mildly affects temporal velocity/temperature, and shapes of the phase interface at longer times of the solidification.

This mainly results from small variations of the liquid fraction across the mushy zone and from a low solidification rate, and it may change drastically when anisotropic properties of the mushy zone, solutal convection, different phase densities and cooling conditions are considered. Therefore, further comprehensive study is needed.

The paper addresses how the different focus of the momentum equation for liquid flow is compared.

The purpose of this paper is to focus on modeling buoyancy driven viscous flow and heat transfer through saturated packed pebble‐beds via a set of homogeneous volume‐averaged conservation equations in which local thermal disequilibrium is accounted for.

The local thermal disequilibrium accounted for refers to the solid and liquid phases differing in temperature in a volume‐averaged sense, which is modeled by describing each phase with its own governing equation. The partial differential equations are discretized and solved via a vertex‐centered edge‐based dual‐mesh finite volume algorithm. A compact stencil is used for viscous terms, as this offers improved accuracy compared to the standard finite volume formulation. A locally preconditioned artificial compressibility solution strategy is employed to deal with pressure incompressibility, whilst stabilisation is achieved via a scalar‐valued artificial dissipation scheme.

The developed technology is demonstrated via the solution of natural convective flow inside a heated porous axisymmetric cavity. Predicted results were in general within 10 per cent of experimental measurements.

This is the first instance in which both artificial compressibility and artificial dissipation is employed to model flow through saturated porous materials.

The purpose of this paper is to present a new numerical approach for modeling the multi‐phase flow during an alloy solidification process. In many solidification processes, advection of solid may have a dramatic effect on bulk convection field as well as on the solid front growth and hence on the macro‐segregation pattern. In the present work, a numerical model is developed to simulate directional solidification in presence of melt convection as well as solid advection in the form of sedimentation. A 2D cavity filled with hyper‐eutectic aqueous ammonium chloride solution (25 wt.% of ammonium chloride) being chilled from one of the side walls has been chosen as the model problem for the numerical simulation.

A fixed grid volume averaging technique has been used for solving mass, momentum, energy, and species equation while taking into account the solid phase advection and local re‐melting. Two different criteria have been identified for the solid particles in the mushy zone to be mobile. These two criteria are represented by a critical solid fraction, and a critical velocity. Based on these two criteria, the mushy zone has been subdivided into two different regions namely, an immobile coherent zone consisting of packed equiaxed crystals and a mobile non‐coherent zone where the solid crystals are able to move.

The numerical results are compared with corresponding experimental observations.

The solid advection velocity and source terms dealing with solid velocity have been calculated using an explicit scheme, whereas the main conservation equations are solved using an implicit scheme.

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.

In the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.

Numerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.

Accurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.

The present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.

The paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

Fluid flow and heat transfer in a dual scale porous media is investigated to determine the interfacial convective heat transfer coefficient, numerically. The studied porous media is a periodic dual scale porous media. It consists of the square rods which are permeable in an aligned arrangement. It is aimed to observe the enhancement of heat transfer through the porous media, which is important for thermal designers, by inserting intra-pores into the square rods. A special attention is given to the roles of size and number of intra-pores on the heat transfer enhancement through the dual scale porous media. The role of intra-pores on the pressure drop of air flow through porous media is also investigated by calculation and comparison of the friction coefficient.

To calculate the interfacial convective heat transfer coefficient, the governing equations which are continuity, momentum and energy equations are solved to determine velocity, pressure and temperature fields. As the dual scale porous structure is periodic, a representative elementary volume is generated, and the governing equations are numerically solved for the selected representative volume. By using the obtained velocity, pressure and temperature fields and using volume average definition, the volume average of aforementioned parameters is calculated and upscaled. Then, the interfacial convective heat transfer coefficient and the friction coefficient is numerically determined. The interparticle porosity is changed between 0.4 and 0.75, while the intraparticle varies between 0.2 and 0.75 to explore the effect of intra-pore on heat transfer enhancement.

The obtained Nusselt number values are compared with corresponding mono-scale porous media, and it is found that heat transfer through a porous medium can be enhanced threefold (without the increase of pressure drop) by inserting intraparticle pores in flow direction. For the porous media with low values of interparticle porosity (i.e. = 0.4), an optimum intraparticle porosity exists for which the highest heat transfer enhancement can be achieved. This value was found around 0.3 when the interparticle porosity was 0.4.

The results of the study are interesting, especially from heat transfer enhancement point of view. However, further studies are required. For instance, studies should be performed to analyze the rate of the heat transfer enhancement for different shapes and arrangements of particles and a wider range of porosity. The other important parameter influencing heat transfer enhancement is the direction of pores. In the present study, the intraparticle pores are in flow direction; hence, the enhancement rate of heat transfer for different directions of pores must also be investigated.

The application of dual scale porous media is widely faced in daily life, nature and industry. The flowing of a fluid through a fiber mat, woven fiber bundles, multifilament textile fibers, oil filters and fractured porous media are some examples for the application of the heat and fluid flow through a dual scale porous media. Heat transfer enhancement.

The enhancement of heat transfer is a significant topic that gained the attention of researchers in recent years. The importance of topic increases day-by-day because of further demands for downsizing of thermal equipment and heat recovery devices. The aim of thermal designers is to enhance heat transfer rate in thermal devices and to reduce their volume (and/or weight in some applications) by using lower mechanical power for cooling.

The present study might be the first study on determination of thermal transport properties of dual scale porous media yielded interesting results such as considerable enhancement of heat transfer by using proper intraparticle channels in a porous medium.

The purpose of this paper is to simulate two macrosegregation benchmarks with a newly developed stabilized finite element algorithm based on a semi-implicit pressure correction scheme.

A streamline-upwind/Petrov–Galerkin (SUPG) stabilized finite element algorithm is developed for the coupled conservation equations of mass, momentum, energy and species. A semi-implicit pressure correction method combined with SUPG stabilization technique is proposed to solve the convection flow during solidification. An analytically derived enthalpy method is adopted to solve the energy conservation equation. The nonlinearities of the energy and species equations are tackled by Newton–Raphson method. Two macrosegregation benchmarks considering the solidification of an Al-4.5 per cent Cu alloy and a Sn-10 per cent Pb alloy are simulated.

A very good agreement is achieved by comparison with the classical finite volume method and a novel meshless method for the Al-4.5 per cent Cu alloy solidification benchmark. Moreover, a unique reference numerical solution has been obtained. Besides, it is demonstrated that the stabilized finite element algorithm can capture the flow instability and channel segregation during solidification of the Sn-10 per cent Pb alloy.

A semi-implicit pressure correction method combined with SUPG stabilization technique is adopted to develop robust stabilized finite element algorithm for the macrosegregation model. A new enthalpy formulation for heat transfer problems with phase change is derived analytically.

The purpose of this paper is to improve the volume-averaged models for free convection flow in porous media.

A pore scale simulation is conducted against which an independent volume-averaged solver is fine-tuned.

Micro and macro scale results can merge if proper choice of local thermal non-equilibrium and thermal dispersion models are selected. This depends on the range of Ra values investigated.

This is the first time a work like this is published in the literature.

The objective of the present work was to perform a detailed numerical study of laminar forced convection in a three‐dimensional square duct packed with an isotropic granular material and saturated with a Newtonian fluid. Hydrodynamic and heat transfer results are reported for three different thermal boundary conditions. The flow in the porous medium was modeled using the semi‐empirical Brinkman‐Forchheimer‐extended Darcy model which also included the effects of variable porosity and thermal dispersion. Empirical models for variable porosity and thermal dispersion were determined based on existing three‐dimensional experimental measurements. Parametric studies were then conducted to investigate the effects of particle diameter, Reynolds number, Prandtl number and thermal conductivity ratio. The results showed that channeling phenomena and thermal dispersion effects are reduced considerably in a three‐dimensional duct compared with previously reported results for a two‐dimensional channel. It was found that the Reynolds number affects mainly the velocity gradient in the flow channeling region, while the particle diameter affects the width of the flow channeling region. As the Reynolds number increases or as the particle diameter decreases (i.e., when the inertia and thermal dispersion effects are enhanced), the Nusselt number increases. The effects of varing the Prandtl number on the magnitude of the Nusselt number were found to be more significant than those of the thermal conductivity ratio. Finally, the effects of varing the duct aspect ratio on the friction factor can be neglected for small particle diameter (D_p ≤ 0.01) or for high particle Reynolds number (Re_d ≥ 1000) due to the dominant bulk damping resistance from the porous matrix (Darcy term) or strong inertia effects (Forchheimer term), respectively.