Volume-averaged Model Research Articles

Estimating pressure drop and Ergun/Forchheimer parameters of flow through packed bed of spheres with large particle diameters

Characterization of flow through packed bed of spheres with large particle diameters (O ~ m) is important in industrial applications, such as mining, geothermal, oil and gas, and construction industries. In this study, a three-dimensional pore-scale mathematical model of packed bed of spheres with large particle diameters considering uniform and non-uniform particle size distributions is developed and validated against experimental data. It was found that the available correlations fail to predict flow behavior in packed bed with larger particle sizes. Hence, Ergun and Forchheimer theories along with pore-scale mathematical model are utilized to simulate and develop permeability (Carman-Kozeny) and inertial resistance (Forchheimer) correlations which are valid for flow through packed bed of spheres with large particle sizes, i.e. sphere diameter from 0.04 to 1.2 m and porosity between 0.2 and 0.7. The developed correlation can be used in either analytical or volume-averaged computational models which is useful for industrial/practical.

Effect of the Coriolis force on the macrosegregation of aluminum in the centrifugal casting of Ti-Al alloys

Within the framework of the ESA GRADECET project, experiments of directional solidification of cylindrical Ti-Al samples were conducted in hypergravity. The experiments were performed in a centrifuge with the apparent gravity (sum of centrifugal and terrestrial gravity) aligned along the cylinder centerline. 3D numerical simulations of aluminum macrosegregation in these samples are presented. A volume-averaging solidification model is used that accounts for centrifugal and Coriolis accelerations in a non-inertial rotating reference system. We compare the melt flow pattern and the macrosegregation formation under terrestrial gravity and under centrifugation. The results show that the Coriolis acceleration, although very weak, breaks the symmetry of the thermosolutal convection, having an important impact on the final macrosegregation pattern. The macrosegregation is entirely modified in comparison with a sample solidified under terrestrial gravity conditions. Besides the aluminum segregation intensity increases with the centrifugation level.

On the Modelling of Macrosegregation during Twin-Roll Casting

Solidification during twin-roll casting happens by cooling of the melt between two counter-rotating rolls, where the molten alloy is constantly fed in. For an inoculated Al-melt, grain growth leads to a gradual increase of solid fraction, so that a coherent solid network forms. Depending on the process condition, this solid network might be subjected to compression within the gap between the two rolls. By using a two-phase volume average model that accounts for (i) transport and growth of spherical grains within a flowing melt, (ii) formation of a coherent solid network above a specific solid fraction and (iii) viscoplastic flow of the solid network saturated with interstitial melt during casting and compression, the process is numerically analysed. It is found that an optimum process with minimum macrosegregation can be achieved for conditions where the kissing point of the two viscoplastic semi-solid shells nearly coincides with the roll nip. It is demonstrated how casting speed, cooling intensity and strand thickness must be related to hit the optimum process window.

A Comprehensive Analysis of Macrosegregation Formation During Twin-Roll Casting

A two-phase Eulerian–Eulerian volume-averaged model is used to predict the formation of macrosegregation during the twin-roll casting of inoculated Al-4 wt pct Cu alloys. For low solid fractions, the equiaxed crystals are modeled according to the submerged object approach. However, above a given transition limit, they are assumed to behave like a viscoplastic material. This means that the solid phase behaves as a coherent structure that can influence the motion of the liquid. Such approach allows one to take into account the flow dynamics arising from the occurrence of both solidification and hot rolling simultaneously, which usually occurs in twin-roll casting. Therefore, it is possible to explain the origin of the macrosegregation patterns obtained in the simulations based on the relative motion between the phases. Compression-induced expulsion of segregated melt is observed as a result of the deformation of the solidifying shells. Such occurrence leads to a negative macrosegregation region in the outer part of the as-cast strip. Then, because the solute-enriched melt is squeezed out toward adjacent regions, two positively segregated bands can be found near the center of the domain. Furthermore, it is shown how solidification-induced feeding weakens the absolute value of the negative and positive segregation bands.

Volume-Averaged Modeling of Multiphase Flow Phenomena during Alloy Solidification

The most recent developments and applications in volume-averaged modeling of solidification processes have been reviewed. Since the last reviews of this topic by Beckermann and co-workers [Applied Mech. Rev. 1993, p. 1; Annual Rev. Heat Transfer 1995, p. 115], major progress in this area has included (i) the development of a mixed columnar-equiaxed solidification model; (ii) further consideration of moving crystals and crystal dendritic morphology; and (iii) the model applications to analyze the formation mechanisms of macrosegregation, as-cast structure, shrinkage cavity and porosity in different casting processes. The capacity of computer hardware is still a limiting factor. However, many calculation examples, as verified by the laboratory casting experiments, or even by the casting processes at a small industrial scale, show great application potential. Following the trend of developments in computer hardware (projection according to Moore’s law), a full 3D calculation of casting at the industry scale with the multiphase volume-averaged solidification models will become practically feasible in the foreseeable future.

Modeling of argon–acetylene dusty plasma

The properties of an Ar/C2H2 dusty plasma (ion, electron and neutral particle densities, effective electron temperature and dust charge) in glow and afterglow regimes are studied using a volume-averaged model and the results for the glow plasma are compared with mass spectrometry measurements. It is shown that dust particles affect essentially the properties of glow and afterglow plasmas. Due to collection of electrons and ions by dust particles, the effective electron temperature, the densities of argon ions and metastable atoms are larger in the dusty glow plasma comparing with the dust-free case, while the densities of most hydrocarbon ions and acetylene molecules are smaller. Because of a larger density of metastable argon atoms and, as a result, of the enhancement of electron generation in their collisions with acetylene molecules, the electron density in the afterglow dusty plasma can have a peak in its time-dependence. The results of numerical calculations are in a good qualitative agreement with experimental results.

Gas rarefaction in high power impulse magnetron sputtering: comparison of a particle simulation and volume-averaged models

We calculate the rarefaction of Ar background gas in front of a magnetron caused by sputtered target material atoms (so called ‘sputtering wind effect’) during pulsed sputtering. For a detailed analysis, we use a three-dimensional particle simulation using the direct simulation Monte Carlo (DSMC) method. We compare the results with those of two volume-averaged (V-A) models describing the same problem but with significantly lower computational demands. The comparison is made for three values of flux of sputtered atoms (expressed as ‘sputtering current’ in the units of amperes) and for three target materials (Zr, Al and C) sputtered from a circular target of diameter placed in a chamber with realistic geometry. Ar rarefaction is more pronounced for target atoms with higher mass, but the difference between the three target materials is surprisingly small. The region where Ar is significantly rarefied extends much further from the target than a typical extent of the high-density plasma confined by the magnetic field. The V-A models provide good approximation of the time evolution of the target material density in front of the target compared to the DSMC simulation. However, the presented V-A models underestimate (in all but one case) the magnitude of Ar rarefaction during the pulse-on time and also predict faster return to equilibrium during the pulse-off time comared to the DSMC simulation.

Modeling of the Twin-Roll Casting Process: Transition from Casting to Rolling

During twin-roll casting, an alloy melt is passing the gap between two counter-rotating rolls, where cooling and solidification leads to the continuous formation of a solid strand. In order to describe this process, a two-phase Eulerian–Eulerian volume-averaging model is presented that accounts for (1) transport and growth of spherical grains within a flowing melt, (2) the formation of a coherent solid network above a specific solid fraction and (3) the viscoplastic flow of the solid network with the interstitial melt during casting and compression. For the considered case of an inoculated Al–4wt%Cu alloy, the process conditions are chosen such that two relatively thick viscoplastic semi-solid shells meet between the rolls, and thus, the material is pressed together and squeezed against the casting direction. The squeezed out material consists of segregated melt and some solid that quickly disappears after melting. It is observed during this study that macrosegregation distributions are inherently connected to the mush deformation that is enforced during the hot rolling process.

Implications of inherent inhomogeneities in thin carbon fiber-based gas diffusion layers: A comparative modeling study

Thin porous media are present in multiple electrochemical energy devices, where they provide key transport and structural functions. The prototypical example is gas diffusion layers (GDLs) in polymer-electrolyte fuel cells (PEFCs). While modeling has traditionally been used to explore PEFC operation, this is often accomplished using volume-averaged (VA) formulations, where the intrinsic inhomogeneities of the GDL are smoothed out and the lack of defining a representative elementary volume is an ever-present issue. In this work, the predictions of a single-phase VA PEFC model are compared to those of a pore-scale PEFC model using GDL tomograms as a part of the meshed domain to delineate important aspects that VA models cannot address. The results demonstrate that while VA models equipped with suitable effective properties can provide a good average estimate for overall performance, the lack of accounting for real structures limits their predictive power, especially for durability and degradation behavior where large deviations are found in the spatial distributions. Furthermore, interfacial effects between the GDL and the microporous layer are explored with the pore-scale model to understand the implications of the layered geometry. It is shown that the actual microstructure of the GDL/MPL transition region can significantly affect the fluxes across the sandwich, something that VA models cannot easily consider. Interfacial design is recognized as a key quality control parameter for large-scale MEA manufacturing and assembly.

Numerical Simulation of V-shaped Segregation in Continuous Casting Blooms Based on a Microsegregation Model

This study establishes a volume-averaged model for transport phenomena to study V-shaped segregation in continuous casting blooms with different superheating and cooling rates by coupling fluid flow, macro-scale heat transfer, and solute transport. The results demonstrate that the modified Won–Thomas model considering back diffusion in the solid phase corresponds well with the measured carbon segregation. V-shaped segregation decreases with decreasing superheating and increases slightly with decreases in the cooling rate. The results of original position statistical distribution analysis correspond well with the simulation results. This phenomenon can be exploited to improve the quality of bloom.

A volume-averaged model of nitrogen–hydrogen plasma chemistry to investigate ammonia production in a plasma-surface-interaction device

Nitrogen impurity seeding is a promising technique for increasing the radiative power dissipation rate in the edge plasma of a fusion device. It will be required in future fusion devices such as ITER to reduce the directed heat flux on the divertor strike-points to within erosion limits. However, chemical reactions between nitrogen and fuel isotopes may complicate tritium control measures by increasing in-vessel retention and impacting the gas-handling plant. To gain insight into the nitrogen–hydrogen plasma chemistry a volume-averaged (global) model is developed and compared with experimental measurements in the MAGnetised Plasma Interaction Experiment plasma device. A set of 702 reactions is compiled and used to model the population dynamics of 51 relevant neutral, ionic, electron, surface and metastable excited state species. Stable equilibrium values are compared to results from an experimental investigation in which a combination of mass spectrometry, Langmuir probe analysis and optical emission spectroscopy is used to determine neutral and positive-ionic trends under the same conditions. The dominant ammonia production mechanism is found to be the Langmuir–Hinshelwood reaction between adsorbed atomic hydrogen and above 25% hydrogen concentration. For lower hydrogen proportions the Eley–Rideal reaction between free atomic hydrogen and is found to dominate. The dominant loss mechanism (for all compositions) is found to be electron impact dissociation into neutral fragments.

Development and Validation of Simplified 1D Models for Hollow Cathode Analysis and Design

Several space electric propulsion devices, such as ion engines and Hall effect thrusters, use hollow cathodes as the electron sources for providing the necessary electrons for the ionization of the propellant and to neutralize the ion beam leaving the thruster. Since cathode performance is strongly tied with its geometry and size and plasma diagnostics is very difficult to be performed, simple numerical tools are required in order to determine optimal geometries and operative conditions for a given mission profile. The paper presents a preliminary design tool for orificed hollow cathodes. Two physical models compose the tool: a plasma model and a thermal model. A time-independent, volume-averaged model has been developed to determine plasma properties in the emitter and orifice regions. A Lumped Element Thermal Model has been also developed to compute temperature distributions and the respective gradients within the main cathode’s element. The study, conducted to validate the results of the models, shows that there is a good agreement with values and trends found in the available literature. The tool is able to estimate performances of new devices by the calculation of cathode working conditions for different geometries given device operating condition and insert material.

Premature melt solidification during mold filling and its influence on the as-cast structure

Premature melt solidification is the solidification of a melt during mold filling. In this study, a numerical model is used to analyze the influence of the pouring process on the premature solidification. The numerical model considers three phases, namely, air, melt, and equiaxed crystals. The crystals are assumed to have originated from the heterogeneous nucleation in the undercooled melt resulting from the first contact of the melt with the cold mold during pouring. The transport of the crystals by the melt flow, in accordance with the socalled “big bang” theory, is considered. The crystals are assumed globular in morphology and capable of growing according to the local constitutional undercooling. These crystals can also be remelted by mixing with the superheated melt. As the modeling results, the evolutionary trends of the number density of the crystals and the volume fraction of the solid crystals in the melt during pouring are presented. The calculated number density of the crystals and the volume fraction of the solid crystals in the melt at the end of pouring are used as the initial conditions for the subsequent solidification simulation of the evolution of the as-cast structure. A five-phase volume-average model for mixed columnar-equiaxed solidification is used for the solidification simulation. An improved agreement between the simulation and experimental results is achieved by considering the effect of premature melt solidification during mold filling. Finally, the influences of pouring parameters, namely, pouring temperature, initial mold temperature, and pouring rate, on the premature melt solidification are discussed.

Charging and Transport Dynamics of a Flow-Through Electrode Capacitive Deionization System.

We present a study of the interplay among electric charging rate, capacitance, salt removal, and mass transport in "flow-through electrode" capacitive deionization (CDI) systems. We develop two models describing coupled transport and electro-adsorption/desorption which capture salt removal dynamics. The first model is a simplified, unsteady zero-dimensional volume-averaged model which identifies dimensionless parameters and figures of merits associated with cell performance. The second model is a higher fidelity area-averaged model which captures both spatial and temporal responses of charging. We further conducted an experimental study of these dynamics and considered two salt transport regimes: (1) advection-limited regime and (2) dispersion-limited regime. We use these data to validate models. The study shows that, in the advection-limited regime, differential charge efficiency determines the salt adsorption at the early stage of the deionization process. Subsequently, charging transitions to a quasi-steady state where salt removal rate is proportional to applied current scaled by the inlet flow rate. In the dispersion-dominated regime, differential charge efficiency, cell volume, and diffusion rates govern adsorption dynamics and flow rate has little effect. In both regimes, the interplay among mass transport rate, differential charge efficiency, cell capacitance, and (electric) charging current governs salt removal in flow-through electrode CDI.

Evolution of a mushy zone in a static temperature gradient using a volume average approach

A volume average model to study the transition of a semi-solid mushy zone to a planar solid/liquid interface in a static temperature gradient is presented. This model simulates the principal phenomena governing mushy zone dynamics including solute diffusion in the interdendritic and bulk liquids, migration of both the solid-liquid interface and the mushy-liquid boundary at the bottom and top of the mushy zone, and solidification.The motion of the solid-liquid interface is determined analytically by performing a microscopic solute balance between the solid and mushy zones. The motion of the mushy-liquid boundary is more complex as it consists of a transition between the mushy and bulk liquid zones with rapidly changing macroscopic properties. In order to simulate this motion, a control volume characterized by continuity in the solute concentration and a jump in both the liquid fraction and the solute concentration gradient was developed.The volume average model has been validated by comparison against prior in-situ X-ray radiography measurements [1], and phase-field simulations [2] of the mushy-to-planar transition in an Al-Cu alloy. A very good similarity was achieved between the observed experimental and phase-field dynamics with this new model even though the described system was only one-dimensional. However, an augmentation of the solute diffusion coefficient in the bulk liquid was required in order to mimic the convective solute transport occurring in the in situ X-ray study. This new model will be useful for simulating a wide range of natural and engineering processes.

On the Limitations of Volume-Averaged Descriptions of Gas Diffusion Layers in the Modeling of Polymer Electrolyte Fuel Cells

As a complement to experimentation, numerical modeling is a key tool to optimize the design of Polymer Electrolyte Fuel Cells (PEFCs). Moreover, it plays an important role for the study of multiphysics, multiphase and multiscale phenomena, which are difficult to explore experimentally due to the small characteristic dimensions of the Membrane Electrode Assembly (MEA). In particular, thorough understanding of the coupled mass, charge and heat transport processes that occur in thin, heterogeneous, anisotropic, mix-wettability Gas Diffusion Layers (GDLs) is essential to develop new MEA designs with improved performance and enlarged durability [1,2]. In the PEFC literature, the most extended approach to model GDLs is volume averaging, where effective properties are used to describe the multiple transport processes that take place in this component. However, the applicability of volume-averaged formulations when modeling thin GDL materials has been questioned many times, and the error introduced in the numerical predictions is unclear. In this work, we explore the limitations introduced by volume-averaged formulations of GDLs under single-phase conditions. To this end, Lattice Boltzmann (LB) simulations were performed on X-ray computed tomography images of various commercial GDLs to assess the existence of a Representative Elementary Volume (REV). The tomographic images of the dry GDLs were acquired at Lawrence Berkeley National Lab’s Advanced Light Source (ALS) synchrotron (Beamline 8.3.2) with a resolution of 1.3 µm/voxel, and then segmented into void and solid phases using the algorithms described in [1,3]; the model was implemented in the open-source LB solver Palabos. Relevant effective transport properties were determined in both the in- and through-plane directions by direct numerical simulation, including effective diffusivity, permeability and effective electrical/thermal conductivity. Figure 1(a) shows the results computed for the through-plane effective diffusivity and thermal conductivity on an untreated Toray® TGP-H-120 sample using boxes of material with different size in the x, y, z directions (L x, L y, L z); the center of the boxes was always taken in the middle position of the original GDL sample. The in-plane size to obtain representative and consistent effective properties was found to be around 1 mm, in agreement with the microstructural analysis reported by Zenyuk et al. [3]. This length scale is comparable to the rib/channel width used in PEFCs, which is typically around 1 mm. Regarding the through-plane direction, there is not a representative size much smaller than the thickness of the material. Numerical results are only similar to previous experimental data when the full thickness (or almost the full thickness) of the material is considered. Therefore, it can be concluded that a REV cannot be rigorously defined in a GDL. In other words, there is not a piece of material with representative properties, whose dimensions are much smaller than the characteristic size of the cell in any direction. Given the impossibility to define a REV in GDLs, the error introduced by volume-averaged formulations was then evaluated by using a non-isothermal, single-phase, steady-state model of a 2x1 mm2 differential PEFC implemented in ANSYS® Fluent. In particular, we compared the predictions of a fully volume-averaged model with those of a pore-scale model accounting for the GDL microstructure. Only the core region of the GDL was included, and a macroscopic description was considered for the remaining components of the MEA (catalyst layers, microporous layers and membrane). The operating conditions were established at 80 ○C, 50% relative humidity, 1.5 bar and 0.5 V. As shown in Figure 1(b), the average results predicted by both models are similar, even though the spatial distributions differ significantly. The impact of the GDL microstructure on the current density distribution arises from the lack of separation of scales discussed above. The preliminary results show that GDL volume-averaged models are a good approximation to determine overall cell performance, as it has been traditionally assumed in the literature. This holds, at least, when the effect of interfacial regions is not considered. However, the local distributions predicted by volume-averaged models can be quite different from the actual scenario at the pore-scale level, in which heterogeneities can lead, e.g., to the formation of hotspots that may affect MEA durability. Further implications of the use of GDL volume-averaged models under multiphase conditions should also be investigated in future work.

On the Limitations of Volume-Averaged Descriptions of Gas Diffusion Layers in the Modeling of Polymer Electrolyte Fuel Cells

Understanding of the coupled transport processes that occur in thin gas diffusion layers (GDLs) is necessary to develop improved designs. The traditional technique used to model GDLs is the volume-averaged approximation. However, the applicability of this approach has been long questioned, and the error in the results is unclear. In this work, the limitations of GDL volume-averaged models are examined under single-phase conditions. The lattice Boltzmann method is combined with tomography images of carbon-paper GDLs to assess the existence of a representative elementary volume (REV) in terms of various effective properties. Then, the predictions of GDL volume-averaged and pore-scale formulations are compared by using a CFD model of a differential cell. The results show that a REV cannot be clearly defined. This leads to inhomogeneities in the pore-scale model that the volume-averaged model is not able to capture despite the overall flux through the GDL is similar in both cases.

NUMERICAL STUDY ON THE INFLUENCE OF INFILTRATION ON SWASH HYDRODYNAMICS AND SEDIMENT TRANSPORT IN THE SWASH ZONE

Infiltration and exfiltration processes have a significant influence on the hydrodynamics of the swash zone. Such processes need to be taken into account in the modelling of cross-shore sediment transport and the prediction of beach profile evolution. This paper presents a numerical study of the swash hydrodynamics using a 2D Volume-Averaged Reynolds-Averaged Navier-Stokes model, which was calibrated and validated against new experimental data. The model was used to simulate wave run-up from regular waves over permeable and impermeable fixed slopes. Swash flow velocities and water depth data were obtained from the simulations and used to estimate bed shear stresses at three different locations on the beach slope. The results show that infiltration can have opposing effects on the bed shear stress when compared to equivalent swash on an impermeable slope. During the uprush phase, stresses are directly increased due to boundary layer thinning, whereas, during the backwash phase, there is a significant reduction of flow leading to a decrease in the bed shear stresses.

A global model of cylindrical and coaxial surface-wave discharges

A volume-averaged global model is developed to investigate surface-wave discharges inside either cylindrical or coaxial structures. The neutral and ion wall flux is self-consistently estimated based on a simplified analytical description both for electropositive and electronegative plasmas. The simulation results are compared with experimental data from various discharge setups of either argon or oxygen, measured or obtained from the literature over a wide range of pressure and power, for a continuous and a pulse-modulated power input. A good agreement is observed between the simulations and the measurements. The contribution of the wall flux on the net loss rates is quantified for a variety of species in different discharge setups. A coaxial plasmaline is further investigated to reveal the detailed behaviour of plasma properties with respect to input power and pressure.

Investigation of the Solidification Behavior of NH4Cl Aqueous Solution Based on a Volume-Averaged Method

Based on solidification theory and a volume-averaged multiphase solidification model, the solidification process of NH4Cl-70 pct H2O was numerically simulated and experimentally verified. Although researchers have investigated the solidification process of NH4Cl-70 pct H2O, most existing studies have been focused on analysis of a single phenomenon, such as the formation of channel segregation, convection types, and the formation of grains. Based on prior studies, by combining numerical simulation and experimental investigation, all phenomena of the entire computational domain of the solidification process of an NH4Cl aqueous solution were comprehensively investigated for the first time in this study. In particular, the sedimentation of equiaxed grains in the ingot and the induced convection were reproduced. In addition, the formation mechanism of segregation was studied in depth. The calculation demonstrated that the equiaxed grains settled from the wall of the mold and gradually aggregated at the bottom of the mold; when the volume fraction reached a critical value, the columnar grains stopped growing, thus completing the columnar-to-equiaxed transition (CET). Because of solute partitioning, negative segregation occurred at the bottom region of the ingot concentrated with grains, whereas a wide range of positive segregation occurred in the unsolidified, upper part of the ingot. Experimental investigation indicated that the predicted results of the sedimentation of the equiaxed grains in the ingot and the convection types agreed well with the experimental results, thus revealing that the sedimentation of solid phase and convection in the solidification process are the key factors responsible for macrosegregation.