Modeling of mass transfers in a porous green compact with two-component binder during thermal debinding

Mass transfers and phase changes of two-component binder in a porous green compact during thermal debinding process are modeled. The evaporation of low molecular weight (LMW) component and volatile fragments, the thermal degradation of high molecular weight (HMW) component, the capillary driven and pressure driven liquid phase transports, the binary diffusion in solutions, the convection and diffusion of gas phases, and the heat transfer in a porous medium are captured in the model. The model is validated with experimental data. The simulated results show that mass transfers during the early stage of thermal debinding are mainly due to capillary driven and pressure driven liquid transports. During the final stage of thermal debinding, both convective liquid and gas transports are important in binder removal. The developed model provides physical understanding of binder removal mechanisms that are essential for process optimization.     

Numerical and experimental investigations on thermal debinding of polymeric binder of powder injection molding compact

A mathematical model is established to describe the thermal debinding process of polymeric binder from a powder injection molding compact. The model takes into account of the thermal degradation of liquid polymer into liquid volatile fragment, the evaporation of liquid volatile fragment, the capillary driven liquid phase transport, the binary diffusion in solution, the convection and diffusion of gas phases, and the heat transfer in a porous medium. The proposed model is solved numerically based on a finite volume method and validated with experimental data. Based on the numerical results, the binder removal, the pressure buildup, the binder distribution, the mass transfers, and the removal mechanisms during thermal debinding are studied.     

A new approach for the characterization of the pore structure of dual porosity rocks

For the realistic representation of the pore space of dual porosity rocks, a new method of pore structure characterization is developed by combining experimental Hg intrusion/retraction curves with back-scattered scanning electron microscope (BSEM) images and inverse modeling algorithms. The pore space autocorrelation function measured by processing the digitized BSEM images is combined with the surface fractal dimension estimated from the high pressure Hg intrusion (MIP) data to derive a synthetic small-angle neutron scattering (SANS) intensity function, the inversion of which provides a volume-based pore body radius distribution (PBRD). The volume-based PBRD is fitted with a multimodal number-based PBRD consisting of two component distributions: one representing the macroporosity and another one representing the microporosity. Based on arguments of percolation theory, analytical mathematical models are developed to describe the Hg intrusion in and retraction from dual pore networks in terms of the complete PBRD, pore throat radius distribution (PTRD) of macroporosity, drainage accessibility functions (DAFs) of both porosities, and imbibition accessibility functions (IAFs) of both porosities. Inverse modeling of the Hg intrusion data set enables us to estimate the PTRD and DAFs. Inverse modeling of the Hg retraction datasets enables us to estimate a set of primary and secondary IAFs. The method is demonstrated by the pore structure characterization of four outcrop samples of carbonate and sandstone rocks. Analytic approximate equations developed from the critical path analysis (CPA) of percolation theory enable us to calculate explicitly the absolute permeability and the formation factor of the porous rocks using the estimated parameters (PBRD, PTRD, DAF) of the macroporosity. The measured permeability of cores is predicted satisfactorily and observed discrepancies may be attributed to large length-scale macro-heterogeneities which are not evident in BSEM images and Hg porosimetry data.     

From pore scale to wellbore scale: Impact of geometry on wormhole growth in carbonate acidization

Acid injection in carbonate reservoir is commonly used in the oil industry to improve, or at least recover, its productivity. The aim of this stimulation technique is to create empty channels called wormholes which, if successful, would bypass the damaged area near the wellbore. During production, wormholes become pathways for the reservoir oil to reach the well. This technique increases near-wellbore permeability, and therefore improves oil production. The interaction between the transport of acid, chemical reaction, and heterogeneities encountered at different scales, controls the unstable behaviour of wormholing and, thus, the success of the treatment. Most of the experimental and numerical studies done on this subject in the past have been limited in their observations because they only considered the dissolution process at a small scale (from pore scale to core scale). The purpose of this work is to study how the geometry of the domain can constrain wormhole competition, and influence wormholing dynamics in a core submitted to acidizing.After a short review of the literature on wormholing to see how the geometry effect could have influenced previous experiments, we study specifically the question of wormhole density. We emphasize that two mechanisms are involved in wormhole competition, with one of them being effective only at small scale. Thus we conclude that wormholing is not a full-scale independent process. We describe differences in the wormhole growth dynamics between “confined” and “unconfined” domains for different dissolution regimes. We focus on optimum conditions and their transition from “confined” to “unconfined” domain to realize that the flow rate in the dominant wormhole does not depend on geometric effects. We conclude by a comparison between 2D and 3D simulations, in both linear and radial flow, and observe changes in the wormholing process. All our results serve as a discussion about definitions of optimum conditions in the literature.     

Transport analysis and model for the performance of an ultrasonically enhanced filtration process

This paper presents an analysis of a filtration technique that uses ultrasound to aid the collection of small particles (tens of microns in diameter) from suspension. In this method, particles are retained within a porous mesh that is subjected to a resonant ultrasonic field, even though the pore size of the mesh is two orders of magnitude greater than the particle diameter. The role of acoustic forces in driving the retention phenomena has previously been studied on a micro-scale, which included modeling and experimental verification of particle motion and trapping near a single element of the mesh. Here, we build on this work to develop an overall transport model to predict macroscopic performance criteria such as breakthrough times and the dynamics of the filtration performance. Results from this model compare favorably to experimental studies of the filtration phenomena; simulation results scale appropriately with experimental results in which inlet feed concentration and flow rate are varied.     

Supercritical fluid flow in porous media: modeling and simulation

The present work aims the modeling and simulation of supercritical fluid flow through porous media. This type of flow appears in several situations of interest in applied science and engineering, as the supercritical flow in porous materials employed in chromatography, supercritical extraction and petroleum reservoirs. The fluid is constituted of one pure substance, the flow is monophasic, highly compressible and isothermal. The porous media is isotropic, possibly heterogeneous, with rectangular format and the flow is two-dimensional. The heterogeneities of porous media are modeled by a simple power law, which describes the relationship between permeability and porosity. The modeling of the hydrodynamic phenomena incorporates the Darcy's law and the equation of mass conservation. Appropriated correlations are used to model, in a realistic form, the density and the viscosity of the fluid. A conservative finite-difference scheme is used in the discretization of the differential equations. The nonlinearity is treated by Newton method, together with the conjugate gradient method. The results of the simulation for pressure and mobility of supercritical and liquid propane flowing through porous media are presented, analyzed and graphically depicted.     

Modeling of a reaction network and its optimization by genetic algorithm

Continuous endeavors are going on in many research works to find out the strategy to mathematically model and optimize complex reaction networks in order to maximize the main product and at the same time keeping the reactor dimensions within some acceptable limits. The aim of this work is to provide with a strategy for efficient modeling and optimization of reaction networks for reaction controlled processes. Genetic algorithm (GA) has been used for optimizing complex search spaces with multiple optima. Formation of styrene monomer from the ethylbenzene dehydrogenation, with several by-products in a fixed bed reactor, is taken as an example for this study. Two activation energies are found to be the best in term of maximizing styrene productivity.     

Thin-layer drying of porous materials: Selection of the appropriate mathematical model and relationships between thin-layer models parameters

Experimental investigation of the drying kinetics of various types of materials was carried out in laboratory-scale dryers under different conditions of temperature, microwave heating power and pressure. Leather samples (mechanically and vacuum-dewatered bull napa and wet blue cutting), paperboards (grafopack, testliner), wood (alder, birch, willow) and two pharmaceutical powders (chlorpropamide and hydrochlorotiazide) were dried in a microwave dryer. Thin clay slabs, Al–Ni catalyst and chlorpropamide were dried in a convection dryer, while chlorpropamide and ketoprofen were dried in a vacuum dryer. In order to compare drying kinetics, experimentally obtained data, X = f(t), were correlated with the Lewis “thin-layer” equation, the modified Page equation and Fick's second law. The drying constant, effective diffusion coefficient, mass transfer coefficient and modified Page model parameters were estimated by fitting the selected mathematical models to experimental data. High levels of correlation between measured and calculated data were obtained for all materials and dryers using modified Page model. The application of the Lewis and Fick's equation is justified only for drying of clay, catalyst and leather. Mass transfer coefficient depends linearly on the drying constant. The relation between the modified Page model parameter and the drying constant can be represented by a unique power function.     

Numerical analysis of silo behavior using non-coaxial models

Granular solids in silos experience considerable principal stress rotations, which result in the non-coaxiality between principal stresses and plastic strain rates. This paper discusses the influences of the use of elastoplastic non-coaxial models for granular solids on predictions of wall pressure distributions in silos by using the finite element method. A well established non-coaxial model in geomechanics, the yield vertex model, is employed. Simulations are performed on a steep hopper characterized with a mass flow and a flat-bottomed silo with a semi-mass flow. The simulations indicate that the non-coaxiality does not influence predictions of wall pressures after filling. On the other hand, the predicted discharge wall pressures with non-coaxial considerations are larger than those without it. Its mechanism is discussed in this paper. The suppressed shear-dilatancy of granular solids in silos leads to a larger increase of normal stress with non-coaxial models.     

Sherwood number in porous microtube due to combined pressure and electroosmotically driven flow

Mass transfer of a neutral solute in a porous microtube is quantified in this study. An analytical expression of the Sherwood number is developed from first principles for combined flow of pressure driven and electroosmotic flow. Similarity solution method is adopted for solution of convective-diffusive species balance equation with coupled velocity profile, within the mass transfer boundary layer. It is observed that the Sherwood number increases with decrease in the Debye length (as the electric double layer becomes more compact) and it becomes constant beyond scaled Debye length of 60. Effects of the Reynolds number, dimensionless suction velocity, ratio of driving force and scaled Debye length have been investigated in detail. The analysis is useful for efficient design of microfluidic devices and flow through porous media.     

Modeling and numerical analysis of counter-current dialyzer at steady state

A mathematical model of a continuous counter-current dialyzer in the case of the transport of one component, which is based on the mass balance and balance of transported component, has been presented. It enables users to calculate the component recovery yield if parameters of the dialyzer (height, dimensions of compartments), parameters of the membrane (thickness, partition coefficients, diffusivity of the component in the membrane, flux of solution through the membrane), parameters of liquids in both the compartments (density, viscosity, diffusivity of the component) and parameters of streams entering the dialyzer (component concentration, volumetric liquid flow rate) are specified.     

Transport in deformable food materials: A poromechanics approach

A comprehensive poromechanics-based modeling framework that can be used to model transport and deformation in food materials under a variety of processing conditions and states (rubbery or glassy) has been developed. Simplifications to the model equations have been developed, based on driving forces for deformation (moisture change and gas pressure development) and on the state of food material for transport. The framework is applied to two completely different food processes (contact heating of hamburger patties and drying of potatoes). The modeling framework is implemented using total Lagrangian mesh for solid momentum balance and Eulerian mesh for transport equations, and validated using experimental data. Transport in liquid phase dominates for both the processes, with hamburger patty shrinking with moisture loss for all moisture contents, while shrinkage in potato stops below a critical moisture content.     

Analysis of pressure fluctuations in fluidized beds. II. Reconstruction of the data by the Fokker–Planck and Langevin equations

In this part of the sequel we develop a continuum representation of the pressure fluctuation time series p(t) for a fluidized bed (FB), analyzed in part I, by using stochastic methods based on the Markov processes. It is shown that the data may be represented by Markov series with a Markov time scale t_M that is estimated based on the data. Using the relation between the Markov processes and the Kramers–Moyal (KM) expansion that is a continuum equation that involves, in principle, an infinite number of coefficients, we represent the pressure fluctuation time series by a KM expansion. However, since the third and higher-order coefficients of the expansion are very small, the KM expansion reduces to a Fokker–Planck (FP) equation that represents p(t) in terms of a drift and a diffusion coefficients that are computed based on the data. The FP equation is, in turn, equivalent to a Langevin equation, which is used to reconstruct the data, i.e. generate the time series that mimic, in a statistical sense, the original data. Thus, the Langevin equation may also be used to make probabilistic predictions for the future values of the pressure over time scales that are of the order of the Markov time scale t_M. The accuracy of the reconstructed series and, hence, their continuum representation, is demonstrated. We also compute the frequency of level-crossing at a given level α, i.e. the relative frequency (probability) of occurrence of the event defined, for two times t_i−1 and t_i, by, , where P(x) is the probability of the event. Thus, yields the frequency that a given pressure fluctuation level may be expected. The average time that one should wait in order to observe the pressure at a given level again is also computed. The two quantities may be particularly important for modeling of the FBs. A relation is presented between the drift and diffusion coefficients of the FP equation and the Hurst exponent that has previously been used to describe the pressure fluctuation time series in terms of self-affine stochastic distributions.     

Porometry, porosimetry, image analysis and void network modelling in the study of the pore-level properties of filters

We present fundamental and quantitative comparisons between the techniques of porometry (or flow permporometry), porosimetry, image analysis and void network modelling for seven types of filter, chosen to encompass the range of simple to complex void structure. They were metal, cellulose and glass fibre macro- and meso-porous filters of various types. The comparisons allow a general re-appraisal of the limitations of each technique for measuring void structures. Porometry is shown to give unrealistically narrow void size distributions, but the correct filtration characteristic when calibrated. Shielded mercury porosimetry can give the quaternary (sample-level anisotropic) characteristics of the void structure. The first derivative of a mercury porosimetry intrusion curve is shown to underestimate the large number of voids, but this error can be largely corrected by the use of a void network model. The model was also used to simulate the full filtration characteristic of each sample, which agreed with the manufacturer's filtration ratings. The model was validated through its correct a priori simulation of absolute gas permeabilities for track etch, cellulose nitrate and sintered powder filters.     

Application of Danckwerts-type boundary conditions to the modeling of the thermal behavior of metal hydride reactors

The paper presents a model-based investigation of a metal hydride reactor applied as a solid state hydrogen storage device. The elements of a metal hydride reactor are hydrogen supply duct, internal hydrogen distribution, hydride bed, reactor shell and the flow domain of the heat transfer fluid. Internal hydrogen distribution and hydride bed are porous media. Therefore, hydrogen flows through non-porous and porous regions during its reversible exothermic absorption and endothermic desorption, respectively. The interface between porous and non-porous regions is a discontinuity with respect to energy transport mechanisms. Hence, Danckwerts-type boundary conditions for the energy balance equation are introduced. Application of the first and second law of thermodynamics to the interface reveals that temperature jumps may occur at the hydrogen inlet but are not allowed at the hydrogen outlet. Exemplarily the loading behavior of a metal hydride storage tank based on sodium alanate is analyzed. It is demonstrated and experimentally validated that only Danckwerts-type boundary conditions predict the important cooling effect of the inlet hydrogen on the exothermic absorption process correctly.     

Analysis of pressure fluctuations in fluidized beds. I. Similarities with turbulent flow

Experimental data for pressure fluctuation time series in a fluidized bed are analyzed by several distinct, but complementary methods. First, we compute a multiscale probability density function (PDF) for the successive increments of the data. The results demonstrate the evolution of the PDF from the short to long time scales, and indicate striking similarity to the Castaing equation that has been proposed for modeling velocity fluctuations in turbulent flows. Next, to further check the results we compute the structure function of the successive increments of the data. We find that the fluctuations exhibit multifractal behavior, which is also prevalent in turbulence. The multifractality implies that the pressure fluctuates differently over distinct time scales. To understand the origin of the multifractality, we use a powerful method of analysis, namely, the multifractal detrended fluctuation analysis (MF-DFA) in order to analyze the data. The results confirm the multifractal property of the data. To better understand the similarities between the pressure fluctuations in fluidized beds and velocity fluctuations in turbulent flow, and whether the multifractality is due to extended correlations in the data or because the PDF of the successive increments is broad, we also construct the shuffled and surrogate series for the data and analyze them by the MF-DFA method. Comparison of results for the original data with the shuffled and surrogate time series indicates that the correlations in the pressure are responsible for the multifractality of the data, rather than the broadness of the PDF.     

Parametric investigations of premixed methane-air combustion in two-section porous media by numerical simulation

Motivated by detailed designs of industrial porous burners published in patents, the combustion of methane-air mixtures in a two-section porous burner has been studied numerically. The software FLUENT is used to solve a two-dimensional transient mathematical model of the combustion. In order to reveal the reality of the combustion in porous media, the user defined function (UDF) is used to extend the ability of FLUENT and enable two-dimensional distributions of temperature and velocity to be obtained. Some operating or property parameters, which mainly affect the functions and quality of the industrial burner design, such as the inlet velocity of the reactants, the equivalence ratio, the extinction coefficient and the thermal conductivity of porous media, have been investigated. The results show that the contours of temperature and velocity change considerably at the interface of the porous media and near the wall, the gas temperature at the low inlet velocity limit is higher than that for the high velocity limit, the thermal conductivity in the upstream section has more influence on the temperature than that in the downstream section and finally, the temperature profiles of both the gas and the porous skeleton vary considerably with changes of the radiative extinction coefficient of the large-pore porous media.     

Modeling the continuous melt transesterification of bisphenol-A and diphenyl carbonate in a continuous stirred tank reactor

A model of continuous melt transesterification of bisphenol-A and diphenyl carbonate in a continuous stirred tank reactor is developed using phase equilibria assumption and the method of molecular weight moments. The model equations can be simplified into a polynomial system that has 17 equations and 17 unknowns. Solution of the polynomial system gives out almost every aspects of the continuous transesterification process. Molecular weight and polydispersity index, end group ratio of hydroxyl to phenyl carbonate, contents of molecular species, and lost diphenyl carbonate fractions are studied in different operation parameters.     

Fully resolved numerical simulation of reactive mixing in a T-shaped micromixer using parabolized species equations

This paper introduces and exploits a hybrid numerical approach for fully resolved numerical simulations of reactive mixing in T-shaped microreactors and thereby enables a computational analysis of how chemical reactions interact with convective and diffusive transport. The approach exploits the fast redirection of the flow inside the mixing channel, resulting in a flow field with positive axial flow component everywhere after a short entry zone. This allows handling the axial flow direction as a pseudo-time variable, so that the evolution of the concentration profile can be computed consecutively on successive cross sections, following the main axial flow direction. With this approach the finest length scales, given by the Batchelor length scale, can be resolved for such a reactive mixing process inside a T-microreactor at stationary flow conditions. This allows for a detailed analysis of the mixing state as well as important characteristics of the reactive mixing process like yield and selectivity. The concrete numerical simulations yield local diffusion times inside the reactor, reveal the influence of the strength of the secondary flow on the progress of the chemical reaction and show how local selectivities result from the species transport.