Volume Averaged Conservation Equations Research Articles

Simulation of macrosegregation in direct-chill casting—A model based on meshless diffuse approximate method

A multiphysics numerical model has been developed for prediction of macrosegregation in direct-chill casting. The model uses volume averaged mass, momentum, energy, and species conservation equations to simulate the solidification of axisymmetric aluminium-alloy billets. The boundary conditions for the heat transfer include the effects of hot top, mould chill, and direct chill. The volume averaged conservation equations are solved with the explicit local meshless diffuse-approximate method. The approximation coefficients are evaluated on subdomains containing 14 nodes with the second order polynomial basis and the Gaussian weight function. Problems with instabilities due to strong convection are successfully eliminated with an adaptive shift of the weight function and evaluation position in the upstream direction. The automatic adaptive generation of computational node arrangement decreases the calculation time and in a straightforward way enables investigation of the complex flow for geometrically different inflow structures, including sharp and curved edges. The model is used for casting simulation of an Al-5.25wt%Cu alloy billet with a 120 mm radius for various inlet designs. The effect of inlet geometry on the temperature, solid fraction, melt flow, and macrosegregation is investigated. The described developments represent a first meshless numerical approach for simulation of macrosegregation of an important industrial process.

Thermal Performance of a PCM-Based Thermal Energy Storage with Metal Foam Enhancement

The energy transport inside a phase change material (PCM) based thermal energy storage system using metal foam as an enhancement technique is investigated numerically. The paraffin is used as the PCM and water as the heat transfer fluid (HTF). The transient heat transfer during the charging and discharging processes is solved, based on the volume averaged conservation equations. The flow in PCM/foam and HTF/foam composites is modelled by the Forchheimer-extended Darcy equation, while the two-temperature model is employed to account for the local thermal non-equilibrium effect between the foam matrix and fluid phase. The results show that the overall performance is greatly improved by inserting metal foam in both HTF and PCM sides. A nearly 84.9% decrease in the time needed for the total process is found compared with the case of pure PCM, and 40% compared with the case of metal foam insert only in the PCM side. Foam porosity and HTF inlet temperature greatly affect the dynamic heat storage/release process.

A level-set method for droplet impact and penetration into a porous medium

Abstract A level-set method is presented for computation of droplet impact and penetration into a porous medium. The volume averaged conservation equations of mass and momentum are employed for the porous region including the effects of porosity and drag force caused by the porous solid matrix. They are coupled to the conservation equations in the external fluid region through the matching conditions of velocity and stress on the porous surface. Computations are conducted to investigate the effects of initial droplet radius, impact velocity, contact angles, particle size and porosity on the droplet spreading and penetration. A simple analytical formulation is developed for the initial droplet penetration depth and compared with the numerical results. The present level-set method is also applied to the computation of droplet-droplet interaction in the porous medium.

Numerical simulation of droplet impact and evaporation on a porous surface

Numerical simulation is performed for the evaporation of a droplet impacted on a porous surface. A level-set formulation for tracking the droplet deformation is extended to include the effects of evaporation coupled to heat and mass transfer, porosity and porous drag and capillary forces. The local volume averaged conservation equations of mass, momentum, energy and vapor fraction for the porous region are simultaneously solved with the conservation equations for the external fluid region. The computations demonstrate not only the evolution of the liquid-gas interface during the whole period of droplet penetration and evaporation in a porous medium, but also the associated flow, temperature and vapor fraction fields. The effects of impact velocity, porosity and particle size on the droplet deformation and evaporation are quantified.

Reticulated porous ceria undergoing thermochemical reduction with high-flux irradiation

A numerical and experimental analysis is performed on the solar-driven thermochemical reduction of ceria as part of a H2O/CO2-splitting redox cycle. A transient heat and mass transfer model is developed to simulate reticulated porous ceramic (RPC) foam-type structures, made of ceria, exposed to concentrated solar radiation. The RPC features dual-scale porosity in the mm-range and μm-range within its struts for enhanced transport. The numerical model solves the volume-averaged conservation equations for the porous fluid and solid domains using the effective transport properties for conductive, convective and radiative heat transfer. These in turn are determined by direct pore-level simulations and Monte-Carlo ray tracing on the exact 3D digital geometry of the RPC obtained from tomography scans. Experimental validation is accomplished in terms of temporal temperature and oxygen concentration measurements for RPC samples directly irradiated in a high-flux solar simulator with a peak flux of 1200 suns and heated to up to 1940K. Effective volumetric absorption of solar radiation was obtained for moderate optically thick structures, leading to a more uniform temperature distribution and a higher specific oxygen yield. The effect of changing structural parameters such as mean pore diameter and porosity is investigated.

Liquid drainage from high-expansion (HiEx) aqueous foams during and after filling of a container

An aqueous foam drains interstitial liquid as it fills a container after an induction period. We obtain solutions of quasi-steady and volume-averaged conservation equations containing the moving foam front to describe the induction period and the subsequent drainage, respectively. The quasi-steady theory predicts that the induction period increases with decreasing foam injection velocity. After the induction period, the theory shows a constant liquid drainage rate when the foam front propagates at a constant velocity. The space-averaged liquid fraction of the foam decreases with time and reaches a constant (pseudo-steady state) value because of the drainage. The theory shows that the liquid drainage is significant when the foam injection velocity is less than or equal to the terminal velocity of the liquid in a channel (Plateau border), which is shown to depend on foam parameters including initial liquid volume fraction and bubble diameter. At large injection velocity, the theory predicts induction time and drainage rate corresponding to free drainage from stationary foams. We also present experiments measuring liquid drained from dry foams (high-expansion foams), which are generated by forcing air through a liquid-covered metal screen at bench-scale. The experimental data show significant liquid drainage during filling in good agreement with the theoretical predictions for foams with liquid volume fraction less than 0.01. After the container is filled with the foam and the foam addition is stopped, both the theory and experiments show that the drainage rate decreases exponentially with time until equilibrium between capillarity and gravitational forces is reached and drainage ceases to occur. Thus, the theory shows striking differences in drainage behavior during and after filling in agreement with the experimental data.

Prediction of Solidification Paths for Fe–C–Cr Alloys by a Multiphase Segregation Model Coupled to Thermodynamic Equilibrium Calculations

A microsegregation model for the solidification of multicomponent alloys is developed. It couples the volume-averaged conservation equations for total mass, solute mass and energy assuming a uniform temperature. The diffusion in the liquid and solid phases, the growth kinetics of the solidifying microstructures and the velocity of the solid/liquid and solid/solid phase interfaces are considered in the model. Equilibrium between phases is taken into account and computed using dedicated thermodynamic software. The thermodynamic properties and their evolutions during solidification are directly retrieved from a database. Illustration is provided by the solidification of a Fe–C–Cr alloy. The occurrence of the recalescence due to the growth of the microstructure and the progress of solidification are predicted. The solidification behavior near to recalescence is evaluated. By adjusting the cooling intensity and the solute diffusivities, extreme approximations are retrieved. The model shows potentials to be coupled with a macrosegregation model for application to the solidification of multicomponent alloys.

Novel porous media formulation for multiphase flow conservation equations

Multiphase flows consist of interacting phases that are dispersed randomly in space and in time. An additional complication arises from the fact that the flow region of interest often contains irregularly shaped structures. While, in principle, the intraphase conservation equations for mass, momentum, and energy, and their initial and boundary conditions can be written, the cost of detailed fluid flow and heat transfer analysis with explicit treatment of these internal structures with complex geometry and irregular shape often is prohibitive, if not impossible. In most engineering applications, all that is required is to capture the essential features of the system and to express the flow and temperature field in terms of local volume-averaged quantities while sacrificing some of the details. The present study is an attempt to achieve this goal by applying time averaging after local volume averaging. Local volume averaging of conservation equations of mass, momentum, and energy for a multiphase system yields equations in terms of local volume-averaged products of density, velocity, energy, stresses, and field forces, together with interface transfer integrals. These averaging relations are subject to the following length scale restrictions: d ≪ ℓ ≪ L , where d is a characteristic length of the pores or dispersed phases, ℓ a characteristic length of the averaging volume, and L is a characteristic length of the physical system. Solutions of local volume-averaged conservation equations call for expressing these local volume-averaged products in terms of products of averages. In nonturbulent flows, this can be achieved by expressing the “point” variable as the sum of its intrinsic volume average and a spatial deviation. In turbulent flows, the same can be achieved via subsequent time averaging over a duration T such that τ HF ≪ T ≪ τ LF , where τ HF is a characteristic time of high-frequency fluctuation and τ LF is a characteristic time of low-frequency fluctuation. In this case, and instantaneous “point” variable ψ k of phase k is decomposed into a low-frequency component ψ kLF and a high-frequency component ψ ′ k , similar to Reynolds analysis of turbulent flow. The low-frequency component consists of the sum of the local intrinsic volume average 3 i 〈 ψ k〉 LF and its local spatial deviation ψ ˜ kLF . Time averaging then reduces the volume-averaged products to products of averages plus terms representing eddy and dispersive diffusivities of mass, Reynolds and dispersive stresses, and eddy and dispersive conductivities of heat, etc. These terms arise from both high-frequency fluctuations and local spatial deviations. This procedure of time averaging after local volume averaging leads to a set of differential–integral equations of conservation for multiphase flow. This set of multiphase flow conservation equations is particularly suitable for numerical analysis with staggered grid computational systems. Attention is focused on multiphase flow in a region containing fixed and dispersed heat-generating and absorbing solid structures. The novel porous media formulation employs the concept of volume porosity, directional surface porosities, distributed resistance and distributed heat source and sink which is derived through local volume averaging of conservation of mass, momentum and energy equations. The directional surface porosities are defined as a fraction of free flow surface area to control surface area in three principal directions which are readily calculable quantities. The conventional porous media formulation employs the concept of volume porosity, distributed resistance and distributed heat source and sink. Most of engineering problems involve many complex shapes and sizes of structures which are impossible to quantify their distributed resistance accurately. The concept of directional surface porosities reduced the reliance of empirical estimate of distributed resistance and improved the resolution and modeling accuracy. The novel porous media formulation represents a significant advance for solving real engineering problems.

Numerical study for enhancing the thermal conductivity of phase change material (PCM) storage using high thermal conductivity porous matrix

In this paper, the melting process inside an irregular geometry filled with high thermal conductivity porous matrix saturated with phase change material PCM is investigated numerically. The numerical model is resting on solving the volume averaged conservation equations for mass, momentum and energy with phase change (melting) in the porous medium. The convection motion of the liquid phase inside the porous matrix is solved considering the Darcy, Brinkman and Forchiemer effects. A local thermal non-equilibrium assumption is considered due to the large difference in thermal properties between the solid matrix and PCM by applying a two energy equation model. The numerical code shows good agreement for pure PCM melting with another published numerical work. Through this study it is found that the presence of the porous matrix has a great effect on the heat transfer and melting rate of the PCM energy storage. Decreasing the porosity of the matrix increases the melting rate, but it also damps the convection motion. It is also found that the best technique to enhance the response of the PCM storage is to use a solid matrix with high porosity and high thermal conductivity.

Modelling of macrosegregation during solidification processes using an adaptive domain decomposition method

A numerical method for the simulation of buoyancy-induced macrosegregation during solidification processes is presented.The physical model is based on volume averaged conservation equations for energy, mass, momentum, and solute. The resulting partial differential equations are solved by a finite element method, which considers two different length-scales: on the one hand, the scale of the overall process, on the other, a small critical zone near the solidification front where solutal inhomogeneities are initiated and the fluid velocity is non-zero. A domain decomposition method using two adapted grids has been developed. The overall computational domain is discretized using a `coarse' finite element mesh adapted to the process scale. At each time step, the energy conservation equation is solved using this discretization and new values of temperature and solid fraction are calculated at each finite element node. Based on these values, the computational domain is subdivided into three subdomains: the so-called solid, mushy, and liquid regions. The mushy subdomain corresponds to the critical zone near the solidification front and is adaptively discretized with a finer finite element mesh, whereas the liquid domain uses the initial coarse grid and the solid is no longer considered. The fluid flow and solute transport equations are then solved on the different meshes using a Dirichlet–Neumann substructuring iterative method, together with the mortar technique to deal with the non-conforming discretizations at the subdomain interfaces.The method is applied to several test problems such as the Hebditch–Hunt macrosegregation experiment or the prediction of freckles, and the performances and limits of the approach are pointed out and discussed.

Two-phase modeling of mushy zone parameters associated with hot tearing

A two-phase continuum model for an isotropic mushy zone is presented. The model is based upon the general volume-averaged conservation equations, and quantities associated with hot tearing are included, i.e., after-feeding of the liquid melt due to solidification shrinkage is taken into account as well as thermally induced deformation of the solid phase. The model is implemented numerically for a one-dimensional model problem with some similarities to the aluminium direct chill (DC) casting process. The variation of some key parameters that are known to influence the hot-tearing tendency is then studied. The results indicate that both liquid pressure drop due to feeding difficulties and tensile stress caused by thermal contraction of the solid phase are necessary for the formation of hot tears. Based upon results from the one-dimensional model, it is furthermore concluded that none of the hot-tearing criteria suggested in the literature are able to predict the variation in hot-tearing susceptibility resulting from a variation in all of the following parameters: solidification interval, cooling contraction of the solid phase, casting speed, and liquid fraction at coherency.

Role of inter- and intraparticle diffusion in nonuniform particle size gasless compacted-powder combustion synthesis—I: formulation

The propagation-mode, compacted-powder, gasless combustion synthesis of intermetallics is modeled through a combined three-scale, particle-level⧹specimen-level treatment. The specimen-level treatment is based on the local volume-averaged conservation equations for species and energy. The particle-level treatment considers a uniform temperature, the formation of products through a diffusion-controlled heterogeneous reaction, and allows for melting of the product assuming a distinct interface between the phases. The extent of conversion to the final product depends on the local availability of the reactants and through a statistical model and assuming that the interparticle mass transfer resistance is very large, it is determined a priori from the powder particle-size distribution. The influence of the particle-size distribution on the flame structure and the propagation speed is determined using the reaction rate of an average-size particle and also the ensemble average of the reaction rates experienced by particles of different sizes. In a follow-up article (Part II) , the results of these models are compared with each other and with experimental results.

II. Recirculation flow induced by a bubble stream rising in a viscous liquid

The recirculation flow induced by the rising motion of a bubble stream in a viscous fluid within an open-top rectangular enclosure is studied. The three-dimensional volume averaged conservation equations are solved by a control-volume method using a hybrid finite differencing scheme to describe the liquid phase hydrodynamics. The momentum exhange between the bubbles and the liquid phase is modeled with a source term equals to the volumetric buoyancy force acting on the gas in the bubble stream. The volumetric buoyancy force accounts for in line interactions between bubbles through the average gas volume fraction in the gas liquid column which depends on the size and the rising velocity of bubbles. The fluid flow within an open-top rectangular enclosure is further investigated by particle image velocimetry for a bubble stream rising in a water-glycerol solution. The measured fluid velocities in a vertical plane are compared with the predictions of the numerical model over a wide range of fluid viscosity (43 mPa s-800 mPa s) and gas flow rates. Finally, the recirculation flows resulting from the interaction of two neighbouring vertical bubble streams are studied.

Numerical studies of forced convection heat transfer from a cylinder embedded in a packed bed

Forced convection heat transfer from a cylinder embedded in a packed bed was studied numerically. The local volume-averaged conservation equations were used to examine the effects of the governing dimensionless parameters: the Reynolds, Darcy, Forchheimer and effective Prandtl numbers. An increase in either the Reynolds number or effective Prandtl number resulted in heat transfer enhancement. The effect of decreasing Darcy number ( Da) was an increase in the Nusselt number. The effect of Forchheimer number on heat transfer was found to depend on the product of Darcy and Reynolds numbers ( Da Re D ). Through a comparison of numerical predictions with experimental data, effective conductivities were extracted and average thermal dispersion effects were quantified.

Direct simulation vs volume-averaged treatment of adiabatic, premixed flame in a porous medium

For the case of flame thickness being of the order of the pore linear dimension, the flame structure and speed in adiabatic, premixed methane-air combustion in porous media are examined. The local, volume-averaged conservation equations that assume a local thermal equilibrium between the solid and the gas phases (i.e. the single-medium treatment) or allow for a thermal nonequilibrium (i.e. the two-medium treatment) are used along with the direct application of the pointwise conservation equation to a two-dimensional porous medium model (ordered arrangement of discrete or connected square cylinders). The effective properties of the porous medium in the volume-averaged treatments, including the interfacial Nusselt number, are found by applying the local volume-averaging principles. The results show that, although significant variations of the temperature and species concentrations occur over a pore, the flame structure, thickness, speed, and excess temperature (i.e. local gas temperature in excess of the adiabatic temperature) are fairly well predicted by the two-medium model (the single-medium treatment is unable to predict the local excess temperature). However, the volume-averaged treatments are unable to predict the pore-level, local high temperature region in the gas phase (which can be up to 40% above the adiabatic temperature), and the pore-level variation in the flame speed with respect to the flame location in the pore (which can vary by up to 20%). Other shortcomings of the volume-averaged treatments are also revealed through a parametric examination involving the pore-geometry variables, solid to gas conductivity ratio, equivalence ratio, porosity, and flame location within the pore.

Natural convection flow and heat transfer within a rectangular enclosure containing a vertical porous layer

An experimental and theoretical study of natural convection flow and heat transfer within a rectangular enclosure partially filled with an anisotropic porous medium is conducted as a step to understand the effects of the mushy region flow characteristics on the interacting flows of the melt pool and the mushy zone of solidifying alloys. The test cavity is filled with a porous medium for half the lateral distance to the cold wall and is heated and cooled at the vertical end walls by imposing uniform but different temperatures. The solid matrices of the porous media are constructed with perforated plates, and measured flow characteristics are used in the predictions. Flow visualization in the porous region is made possible by the latticed structure of the porous matrices and electrolysis dye generation. The effect of anisotropic flow characteristics of the porous medium on the flow and heat transfer is modeled mathematically using volume-averaged conservation equations. The mathematical model was validated by comparing the predicted velocity and temperature fields with the visualized streamlines and measured temperatures. To investigate the effect of anisotropic permeability for wider ranges of dimensionless parameters, numerical experiments were conducted. The importance of incorporating the mathematical models which account for the anisotropic characteristics of the mushy zone is emphasized.

Funicular state in drying of a porous slab

Drying of a non-hygroscopic porous slab initially saturated with liquid, up to the time of the first appearance of dried patches (critical time), is studied experimentally using glass beads and convective heating, and analytically using the volume averaged conservation equations for capillary driven liquid flow and empirical constitutive relations. Good agreement has been found between the predicted and measured results and the results show that: 1. (a) The surface mass transfer rate depends on the surface saturation. This surface saturation coefficient, which is also a function of both surface geometry and the free stream velocity, is determined experimentally. 2. (b) The critical time is a ratio of the internal liquid transport conductance (Péclet number) to the external vapor transport conductance (Biot number).

Non-isothermal liquid flow and heat transfer in sintered metallic porous media

Abstract This paper presents an experimental and analytical study of non-isothermal liquid flow in sintered fiber metallic porous media. The purpose of the paper is to present a self-consistent procedure for determining heat-transfer characteristics of porous metals. The analysis uses the volume averaged conservation equations. Darcy's law is assumed to hold and local thermal equilibrium is assumed to exist between the solid and liquid phases. The coupled conservation equations are solved for one and two dimensional temperature distributions using finite difference schemes. Different flow and thermal boundary conditions are used. The experiments involve the measurement of temperature distributions and effective thermal conductivities of water saturated copper and nickel sintered fiber metal wicks. The effective thermal conductivities are measured using the steady-state method of comparison and the resulting values are used in the volume averaged energy equation. The measured temperature distributions and those predicted by the analysis show good agreement. Thus the procedure presented in this paper for predicting heat transfer seems to be valid.