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.     

Mathematical simulation of mass transfer processes under heterogeneous reactions in polydisperse porous media

Submitted is a theoretical study of mass transfer processes in polydisperse porous media in the presence of chemical reactions. Kinetic regime of methane pyrolysis in a porous carbon skeleton considering external and internal diffusion resistances for different initial distributions of particles forming the porous medium is investigated. Derived is a general analytical expression describing the influence of the inner reaction surface variation on the degree of the pore filling for an arbitrary initial particle size distribution. Expressions defining the time of pores filling by pyrocarbon based on approximate and exact solutions of the equation for the probability density function (PDF) of particle size distribution are received. Dependence of pore filling time on effective diffusion coefficient and initial particle size distribution using both solutions for PDF-equation is compared. It is shown, that the dominant factor influencing the pore filling time is the dispersion of particle size distribution.     

Simulations of vibrated fine powders

During the past few decades, several studies have been conducted to understand the behaviour of powders in vibrated beds. This paper introduces a technique of incorporating the agglomeration and deagglomeration phenomena in the simulation of vibrated fine powders. Two-dimensional direct simulations are performed using 300 spheres 2.99 mm in diameter in a trapezoidal container vibrated vertically at an amplitude of 2.5 mm and 20 Hz frequency as preliminary conditions. Under non-cohesive conditions, the results are in agreement with those found in the literature. As a preliminary effort to predict the behaviour of cohesive fine powders under vibrated conditions, agglomeration and deagglomeration processes are modelled as the formation and destruction, respectively, of interparticle bonds during particle collisions. Two parameters used to model agglomeration and deagglomeration are the ease of cohesion and cohesivity of the powder. Dependencies of these parameters on certain physical properties of cohesive powders have been suggested. Simulation results reveal two aggregate populations, one with uniform size aggregates and another population with multi-sized aggregates. The former aggregates were more prevalent in weakly cohesive powders while the latter in highly cohesive powders. Interesting macroscopic bed behaviour such as alternating cycles of agglomeration and deagglomeration were also observed. Further work is needed in which the aerodynamic forces are taken into account and cohesion mechanisms at the particle surface are modelled.     

Convective heat and mass transfer coefficient measurements and correlations for particle beds using temperature and humidity step changes with air flow through a test bed

In this paper, transient responses of a porous urea particle bed subject to a step change in the inlet temperature or humidity for a forced convective air flow through the particle bed are investigated to determine the convective heat and mass transfer coefficients inversely by comparing the measured time constant with the predicted characteristic time constant, which is a function of the convection coefficients and Reynolds number. The experimental results show, that although both the time constants for temperature and humidity step changes are dependent on Reynolds number, the temperature response time constant (35-1300 s) is much larger than the humidity response time constant (4-25 s) for the Reynolds number range of 300-5. The surface adsorption of water vapor is very rapid but the absorption inside the porous urea particle is slowed by a very low internal effective diffusion coefficient within the particles whereas the very low Biot number for heat transfer in the particles implies a complete thermal interaction with the air flow throughout each particle and a much larger time constant. Empirical correlations of the Chilton-Colburn j-factor and Nusselt number versus Reynolds number are compared with the correlations of other researchers. These new correlations, which include an uncertainty analysis, imply much lower convective coefficients than those reported previously in the literature.     

Mathematical modelling of Sec pathway mechanism in Escherichia coli: a case study for periplasmic translocation of maltose binding protein-glucose isomerase fusion protein

Within the framework of reported information on the Sec pathway mechanism, a mathematical model for the periplasmic translocation of fusion proteins in bacteria was developed. The mathematical model includes all stages of the targeting stage and assume that the ATP-driven translocation stage is completed in a single step. The equations for the targeting stage involved cytoplasmic folding rate and SecB binding kinetics. Rate equations for the translocation stage were derived using King-Altman and network reduction techniques. Experimental data for maltose binding protein-glucose isomerase fusion protein (MBP-GI) translocation and reported data for MBP translocation were used to estimate the parameters. The simulation results show that the model fits well to the experimental data of cytoplasmic and periplasmic MBP-GI distributions. When the values of the targeting stage parameters, k₁ and k₃ or the concentration of SecB are changed, the model correctly predicts the expected changes in the MBP-GI distribution to the cytoplasmic and periplasmic spaces. The SecB complex and the preprotein concentrations predicted attain steady state immediately, within seconds, and their amounts are very low when compared to MBP-GI in either compartment. The model can be made applicable to any protein that uses the Sec pathway, and with ATP and Sec pathway protein limitations.     

Transmission probabilities and particle-wall contact for Knudsen diffusion in pores of variable diameter

Knudsen dynamics simulations were performed in cylindrical pores with multiple sections of different diameters. The number and location of hits between particles and the pore walls can be related to the known distributions in the uniform diameter cases. Both sweep-gas and pass-through modes of operation were examined. The contact of the particles traveling inside the pores with the pore walls can be regulated by adjusting the aspect ratio (section length/section diameter) of the pore sections and the ratio of their cross-sectional areas. The results were found to be independent of the transition angle between sections except for very smooth transitions. Analytical expressions were developed to relate the transmission probability in pores of multiple sections to the transmission probabilities of the constituent sections.     

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.     

Comparison of Wicke-Kallenbach and Graham's diffusion cells for obtaining transport characteristics of porous solids

Counter-current gas diffusion measurements on a series of porous solids covering a broad range of pore sizes (mean pore radii between 78 nm and ) were performed in the Wicke-Kallenbach and Graham's diffusion cells. Mutual agreement of diffusion fluxes from both cells was found in the whole range of tested pore radii and inert gas systems. For pore materials with mean pore radii exceeding the experimentally unavoidable tiny total pressure gradient induces additional permeation flow which precludes the use of Graham's law for evaluation of transport parameters of the porous solids. Transport parameters together with 95% confidence regions were determined for porous materials with pore radii up to and the prevailing diffusion mechanism, intimately connected with the shape of confidence regions, was estimated.     

Dynamic modelling of an industrial R2R FCC unit

The aim of this study is to obtain a model that can simulate the performance of an industrial fluid catalytic cracking (FCC) unit in steady and dynamic state, and which will subsequently be used in studies of control and real time optimisation. In this paper, a dynamic model for a R2R type FCC unit is presented. The model includes the riser, the stripper/disengager, the regeneration system and the catalyst transport lines. Mass, energy and pressure balances are performed for each of these sections.Simulation results for steady state are presented and compared qualitatively to those obtained from previous FCC models. The dynamic behaviour of the system is explored through two perturbations in open loop, one on the fresh feed flow rate and one on the air flow rate to the first regenerator. The results illustrate the consistency of the model and are in agreement with what has been observed in studies available in the open literature.     

Measurement of lyophilisation primary drying rates by freeze-drying microscopy

The mass transfer processes that influence the rate of primary drying during lyophilisation are characterised by freeze-drying microscopy. A simple diffusion model is shown to describe the observed progress of drying fronts through isothermal, one-dimensional systems of slab and cylindrical geometry. Measured drying rates are modelled by an effective diffusion coefficient, D_eff, which describes the diffusion of water vapour from the inner frozen core through the outer dried cake of the lyophilising mass. In this way, the relationship between the instantaneous thickness of the dried cake and the drying time is determined for different drying temperatures. Values of D_eff determined by analysis of freeze-drying microscopy data for two simple model systems are shown to be in satisfactory agreement with those predicted from theory. Furthermore, the lyophilisation behaviour of a citrate and a tris-HCl buffer are well described by the model and values of D_eff are constant at temperatures below the collapse temperatures of the dried cakes. However, values of D_eff increase approximately four- and six-fold, respectively as the collapse temperatures are approached. Microscopic examination of the citrate buffer dried cake shows its structure to be homogeneous at temperatures well below the collapse temperature, as required by the model, but indicates significant cracking of the cake as the collapse temperature for this buffer is approached. As a result, channeling of the water vapour through the dried cake causes enhanced drying rates. Finally, we show that the total time required to sublime water from aqueous slurries of glass beads in a conventional laboratory lyophiliser are in reasonable agreement with those times estimated using the values of D_eff determined by freeze-drying miscroscopy.     

Application of power addition as modelling technique for flow processes: Two case studies

In many of the continuum processes typically found in chemical engineering, the functional dependency of the dependent variable is only known for large and small values of the independent variable. Exact solutions in the transitional regime are often obscure for various reasons (e.g. narrow band within which the transition from one regime to the other occurs, inadequate knowledge of the physics in this area, etc.). An established method for the matching of limiting solutions is reviewed and subsequently applied. The method regards the known solutions as asymptotes and proposes addition to a power of such asymptotes. It yields a single, adjustable correlating equation that is applicable over the entire domain. This procedure circumvents the introduction of ad hoc curve fitting measures for the different regions and subsequent, unwanted discontinuities in piece-wise fitted correlative equations for the dependent variables. Experimental data of two diverse processes, namely flow in a straight-through diaphragm valve and the fluidisation of a packed bed, are analysed as case studies. Empirical results are investigated for possible asymptotic bounds whereafter power addition is applied to the functional dependencies. The outcome is compared to those of the empirical models and the results discussed. The procedure is revealed to be highly useful in the summarising and interpretation of experimental data in an elegant and simplistic manner. It may also, in general, aid the setup of experimental apparatus for investigation of continuum processes.     

A fast algorithm for mass transfer on an unstructured grid formed by DEM particles

The Discrete Element Method (DEM) was used to generate a particle packing and transient mass transfer in the particle layer was simulated using a novel algorithm for determining the mass fluxes between DEM particles. This method is intended for simulating diffusion phenomena occurring during the dissolution of swelling polymers as the DEM particles can change size and distort the grid, whilst maintaining sharp boundaries between mesh elements. In this work, the robustness and accuracy of the algorithm was tested by solving a diffusion problem over the DEM mesh and comparing it to an accurate solution obtained by a finite difference (FD) approximation, whose discretisation was 10 times finer than that of the unstructured grid. Parametric studies were conducted where the random packing and the size distribution of the DEM particles were altered and the differences in concentration profiles were compared with the FD reference solution through the use of the mean squared error. Both steady state and transient cases were compared. Two methods to improve the accuracy of the DEM unstructured grid were devised and tested using porosity and tortuosity data. In one case the porosity and tortuosity were obtained from the steady-state simulations and in the other case, local convex hulls were used to calculate porosity. Although both these methods decreased the mean squared error up to a factor of two, they also resulted into increased complexity of the simulation. It was concluded that the use of an effective packing algorithm and a narrow particle size distribution are key to maintaining the accuracy of the DEM-generated unstructured grid method.     

Theoretical probing of the phenomenon of the formation of the outermost surface layer of a multi-component particle, and the surface chemical composition after the rapid removal of water in spray drying

Spray drying is a primary process for the manufacture of powders, which satisfy a vast array of societal demands in the areas of nutrition, health and medicine. The functionality of a spray-dried product begins with its incorporation into water (wetting followed by dispersion) Therefore, as its surface chemical composition and structure determine its first contact with water (that is, its hydrophilic nature), these are of prime concern. Laboratory studies on this first layer, which is in the order of several nm in depth from the surface, have been extensive but there is still a lack of a fundamental quantitative explanation of it. This has hampered the development of any sort of approach, which would enable industries to predict what the product may be like before conducting costly trials. This current study is an attempt to describe the on-set of solid formation around the outermost layer of a single droplet during the drying process using an innovative ‘conventional’ continuum approach, i.e. diffusion–convection equations, with a few innovative derivations. Though some complexities such as multi-component equations deduced from irreversible thermodynamics, are avoided for simplicity, some comparisons with experimental/industrial results are made. The main feature of this work is that the multi-component effect is combined with the viscosity (at the surface) effect upon the diffusivities of individual components in the solution droplets or suspension droplets. The derivations allow some extended analytical procedures to proceed in order to help make sense of the experimental observations. Comparisons are made against the data published on dairy fluids. This work provides a good basis for a fruitful area of study that will have a positive impact for spray drying industries. In particular, to help this kind of industry to forge ahead into high performing functional particle production, which are becoming increasingly popular.     

Taylor dispersion in finite-length capillaries

The solute transport in a core-annular geometry is studied. A Newtonian or non-Newtonian (i.e., power-law) liquid flows through the core, while the solute can exchange between the liquid and the surrounding tissue. The permeability of the phase interface depends on the nature of the solute, i.e., relatively low for lipids and macromolecules but high for ions and gases. We analyse the moment’s equations of the residence time distribution (RTD). The solution of the equation for the second moment provides the exact formula for the Taylor dispersion coefficient. Unlike previous studies using a perturbation procedure where coefficient of axial dispersion cannot be defined at low permeability, the current study gives Taylor coefficient of dispersion for any value of the permeability. It is found that the coefficient in shear-thinning fluid is lower than in the Newtonian one, although the relative importance of non-Newtonian effects decreases when other factors, e.g. inter-phase transport and solubility, become dominant. The equations for the higher moments are analysed and the general structure of the solution is obtained in the form of integrals, which can be easily evaluated numerically. When the analysis is applied to solute transport between capillaries and surrounding tissues, it is shown that the classic Taylor expression does not describe dispersion of solute, e.g. glucose and albumin, in the capillary, except in situations where the Péclet number is very low. For the range of parameters typical for microvascular circulation in tissues, the higher moments play an important role and need to be considered.     

Optimization of square microneedle arrays for increasing drug permeability in skin

Microneedles array is a new transdermal drug delivery technique designed to create holes in the epidermis and penetrate the stratum corneum, thus avoiding the high resistance of this barrier. Microneedles have been shown to increase the skin permeability of drugs with no or little pain. However, the skin permeability of epidermis while using microneedle arrays has yet to be fully studied. In some cases, microneedle and microneedle array designs which were developed based on certain criteria (e.g., material of the microneedles) have to be related to other criteria (e.g., drug permeability in skin, skin thickness, etc.). Therefore, in order to determine the optimum design of the microneedle arrays, the effect of different factors (e.g., number of the microneedle, surface area of the patch, etc.) along with skin permeability by using microneedles should be determined accurately. In this work, an optimization framework for transdermal delivery of high molecular weight drug from microneedle is presented. The outputs of this framework have allowed us to identify the optimum design of various microneedles. Data from this optimization algorithm is then used to predict skin permeability of high molecular weight injected into the skin from a microneedle system. The effect of the optimized microneedles on blood drug concentration has been determined. The outcome of this study is useful to propose an optimum design based on different measurement (e.g., variation of skin thickness) for transdermal delivery of drugs.     

Analysis and extension of the theory of multicomponent fluid diffusion

Recently we have presented an extended theory for molecular mass transport (Kerkhof and Geboers, 2005). We have indicated the possibility, and the need, to define a new framework for studying and teaching the theory of multicomponent molecular transport in fluids (Kerkhof and Geboers, 2004). Here a new, structured, overview of the subject is presented. An analysis is given of complicated issues in textbooks and review papers.     

Hierarchical modelling of polymeric materials

Within the last 20 years, computer simulations of materials have evolved from an academic curiosity to a predictive tool for addressing structure-property-processing-performance relations that are critical to the design of new products and processes. Chemical engineers, with their problem-oriented thinking and their systems approach, have played a significant role in this development.The computational prediction of physical properties is particularly challenging for polymeric materials, because of the extremely broad spectra of length and time scales governing structure and molecular motion in these materials. This challenge can only be met through the development of hierarchical analysis and simulation strategies encompassing many interconnected levels, each level addressing phenomena over a specific window of time and length scales.In this paper we will briefly discuss the fundamental underpinnings and example applications of new methods and algorithms for the hierarchical modelling of polymers. Questions to be addressed include: How can one equilibrate atomistic models of long-chain polymer melts at all length scales and thereby predict thermodynamic and conformational properties reliably? How can one quantify the structure of entanglement networks present in these melts through topological analysis and relate it to rheological properties? Are there ways to predict the microphase-separated morphology and stress-strain behaviour of multicomponent block copolymer-based materials, such as pressure sensitive adhesives? Is it possible to anticipate changes in the barrier properties of glassy amorphous polymers used in packaging applications as a consequence of modifications in the chemical constitution of chains?     

Mathematical modelling of a reverse flow reactor with catalytic surface dynamics

This paper studies a system of partial differential equations modelling the behaviour of a reverse flow reactor. For the parameters appropriate for the oxidation of ammonia on a Pt/Al₂O₃ catalyst in a typical laboratory set-up, the reactor may be split into regions where approximate formulas that determine its behaviour are deduced. Numerical calculations are presented and can be used to compare with the analytical formulas. The physical insight gained from the asymptotic analysis suggests a new switching strategy which is the subject of numerical experiments. The switching strategy is found to be efficient at minimising the ammonia exiting the reactor after reversal.     

Finite element modelling of concentration profiles in flow domains with curved porous boundaries

Domains containing porous boundaries are a feature of important processes such as crossflow filtration commonly used in a variety of industries ranging from medical to oil and gas to water treatment. Prediction of concentration profiles of solids being carried by the fluid stream in these types of flow domains is a necessary step in the design of efficient systems used in such operations. The novel approach of the present paper is shown to provide a very convenient technique for simulating concentration profiles within geometrically complex domains. The main feature of this approach is the construction of a very simple technique for the handling of concentration boundary conditions along the porous walls. Therefore this model provides a necessary and useful step towards the development of a fast engineering technique for the quantitative analysis of crossflow filters. The developed algorithm is based on the Streamline Upwind Petrov Galerkin finite element scheme (SUPG) for the solution of flow and convective dispersion equations in two dimensional domains with solid and porous walls. Using iterative algorithms the model can take into account process dependent changes of physical parameters that inevitably occur as a portion of the fluid seeps through the porous wall in these domains. The numerical tests prove the stability of the developed scheme and show that the model can produce results that are consistent with theoretical considerations.