Mathematical modelling of landfill degradation

This paper presents a mathematical model for the primary and secondary stages of biodegradation within domestic landfills. Quantitative relations are proposed to describe the rates of mass conversion between the major landfill chemicals and the effects of gas evolution. Particular attention is given to the methanogenic stage with the associated bacterial population being modelled explicitly. The processes are phrased in terms of a system of coupled differential equations, and examples of their numerical solution are given for cases in which spatial dependency is neglected. The main inadequacies of this model and the need for more appropriate data are indicated.     

Mathematical modelling of lignin pyrolysis

Seven lignins from different sources were pyrolysed (i) isothermally in vacuum over the temperature range 300–1300 °C and (ii) at a constant heating rate of 30 °C min^?1 and a pressure of 0.1 MPa over the temperature range 150–900 °C. The mass fraction of each product—char, tar and gas species—and the elemental composition of the char and the tar were determined for the flash pyrolysis experiments. The evolution rates of the gas species and the tar versus the dynamic temperature of pyrolysis were determined for the constant heating rate pyrolysis experiments. Although the amount of each product species varied from lignin to lignin, the evolution rates were insensitive to the lignin source and the extraction process. To model the data, modifications were made to a recently developed model of coal pyrolysis. The model proved to be successful in simulating both the data from vacuum flash pyrolysis and constant heating rate pyrolysis of Iotech lignin.     

Mathematical modelling of avascular tumour growth based on diffusion of nutrients and its validation

In this paper, a mathematical model based on the diffusion of nutrients is developed by considering the physiological changes accompanying the growth of avascular tumour. Avascular tumour growth involves the formation of three different zones namely proliferation, quiescent and necrotic zones. The main processes on which avascular tumour growth depends are: (i) diffusion of nutrients through the tumour from the contiguous tissues, (ii) consumption rate of the nutrients by the cells in the tumour, and (iii) cell death by apoptosis and necrosis. In the model, we consider the tumour to be spherical and the principal nutrients responsible for its growth are oxygen and glucose. By solving for the concentration profiles using the model developed, we are able to compute the radii of the quiescent and necrotic zones as well as that of the tumour. The proposed model is also validated using in vitro tumour growth data and Gompertzian empirical relationship parameters available in the literature. Our model is also successful in capturing the saturated volume of the avascular tumour for different nutrient concentrations at the tumour surface.     

Mathematical modelling of cross-flow heat exchangers

The models of cross-flow heat exchangers are n-channel systems through which two heat-exchanging media are flowing, the cross-sections of the channels being rectangular. Based on the pattern of channel interconnections and on the character of cross-mixing of the media, nine basic variants can be distinguished. Mathematical models of these variants are given in the form of differential equations, and the methods of solving the equations are proposed, leading to the temperature profiles and the values of the outlet temperatures in the exchangers. Consequently, a possibility is provided to calculate the mean heat transfer driving force for each of the cases analysed and to give (in analytical, graphical or tabular form) the value of the ratio of this force to the logarithmic mean temperature difference (the so-called 'counter-current correction').     

Mathematical modelling of the MCFC cathode

Different models for the NiO cathode have been compared with respect to their abilities to predict polarization curves and the influence of the amount of electrolyte on the electrode performance. It has been shown that the agglomerate model for the MCFC cathode gives more reasonable results when the exterior agglomerate surface area is specifically taken into account. In the cathode only the outermost layer of nickel oxide particles in the agglomerate is utilized for the electrochemical reaction. The pseudohomogeneous approach is questionable for these agglomerates since the individual particles constituting the agglomerate are of the same size as the reaction zone thickness. A thin-film model with a roughness factor for the electrode surface appears to be as good a model as the agglomerate model. A model based on a chain of spherical agglomerates and the partially drowned agglomerate model are physically more realistic models than the homogeneous agglomerate model for the prediction of the influence of electrolyte fill on the electrochemical performance.     

Mathematical modelling of formation heat treatment process

A novel matrix stimulation concept, formation heat treatment (FHT), which involves the application of intense heat around the near-wellbore region for the treatment of water blockage and clay related formation damage in water sensitive formations previously was developed and presented in the literature. The FHT process involves the application of intense heat around the wellbore using a downhole heater. The heat is conveyed to the near-wellbore region using an inert gas flowing through a downhole heater. To understand the heat transfer and fluid-flow characteristics of the FHT process, a transient two-dimensional mathematical model has been developed and is presented in this paper. The model is based on coupling the momentum and energy-balance equations for the wellbore gas with the surrounding porous formation. The presence of the heater across the net pay (sandface) is taken into account in the energy equation as a localized volumetric heat source. A control-volume based finite-difference scheme is used to solve the model equations on a staggered grid. Parametric studies indicate that by injecting a suitable quantity of gas through the tube and annulus, and by adjusting the power input to the downhole heater, the temperature near the wellbore can be favourably controlled.     

Mathematical modelling of heterogeneous—homogeneous reactions

Mathematical models have been developed for a reaction network involving concurrent and consecutive heterogenous and homogeneous reactions. Comparisons have been drawn between models of decreasing complexity involving two dimensions—two phases, two dimensions—one phase and one dimension—one phase. The catalytic oxidation of methanol to formaldehyde coupled to the gas phase oxidation of formaldehyde to carbon dioxide is used as an example of a consecutive reaction: the decomposition of nitrous oxide in the gas phase or over gold catalysts is used as an example of a concurrent reaction. Differences between the prediction of the three models are discussed, as is the effect on the catalytic reaction of the gas phase reaction.     

Mathematical modelling of gas-solid reactions with solid product

Random pore structural models, that can be used with any type of pore size distribution, are developed for the study of gas-solid reactions with solid product under reaction conditions controlled by kinetics and diffusion in the product layer. An intraparticle diffusion and reaction model is also developed and used, along with the structural models, to study the transient behaviour of solid particles reacting under significant intraparticle diffusional limitations. Simulation results for the limestone-SO₂ system show that the transient behaviour of the reacting particles is strongly influenced by the type of the pore size distribution. Various pore structures characterized by the same values of internal surface area and porosity can lead to completely different reaction trajectories. The predictions of the mathematical models compare well with experimental data from the literature.     

Mathematical modelling of industrial pilot-plant penicillin-G fed-batch fermentations

Penicillin-G fermentation with industrial media in 1 m³ stirred tank bioreactors was studied. A model based on the Bajpai–Reuss model structure was developed. Under typical production conditions catabolite repression is nonidentifiable and extensive mycelium differentiation occurs. Thus, the original model was reformulated, neglecting glucose repression of penicillin production and including biomass autolysis. The multi-substrate nature of industrial media was critically analysed. By combining the two most important carbon substrates present, a simple and applicable model was obtained. Model predictions agreed well with experimental data and reproduced the general characteristics observed in the fermentations. The predictive power of the model was tested for fermentations with different sugar feed rate profiles and raw materials (corn-steep liquor and sugar syrup). Several aspects of parameter estimation and model development are discussed on the basis of direct experimental data inspection and a sensitivity analysis of model parameters.     

Mathematical modelling of fire development in cable installations

In 1996 DG XII of the European Commission (Research and Development) approved a 3 year project on the fire performance of electrical cables. Within this FIPEC project, a major part of the work involved correlation and mathematical modelling of flame spread and heat release rate in cable installations. The FIPEC project has developed different levels of testing ranging from a small‐scale, cone calorimeter test procedures developed for cables and materials, a full‐scale‐test procedure based on the IEC 60332‐3, but utilizing HRR and SPR measurements, and a real scale test conducted on model cable installations. Links through statistical correlations and mathematical fire modelling between these levels were investigated and the findings are presented in this paper. These links could form the scientific foundations for standards upon which fire performance measurements can be based and for new fire engineering techniques within fire performance based codes. Between each testing level correlation, numerical and mathematical models were performed. All of the models were based on the cone calorimeter test method. The complexity of the models varied from correlation models to advanced physical pyrolysis models which can be used in CFD codes. The results will allow advanced prediction of cable fires in the future. Also a bench mark was established for the prediction of cable performance by means of data obtained from the constituent materials. Copyright © 2002 John Wiley & Sons, Ltd.     

Mathematical modelling of gas purification with getter-based purifiers

The getter-based purification of a helium stream doped with nitrogen has been studied in a bench scale purifier. The absorption of nitrogen by a Zr-V-Fe alloy is the fundamental mechanism of this purification process. The data collected from performing several experiments under different conditions were simulated by a mathematical model.     

An equivalent circuit method for sulphur electrode modelling

A method is presented for calculating the reaction-rate distributions and resistances of sulphur electrodes by means of an equivalent circuit treatment. This method can be used to model both flat-plate and tubular cells (of either central sodium or central sulphur configuration). The effects of pole activity and pole—matrix contact resistance can be included in the treatment, which can also be used to model electrodes containing more than one variety of carbon matrix.     

Dust explosions in connected vessels: Mathematical modelling

This paper is devoted to the topic of dust explosions in connected vessels, a research topic that has gained a lot of interest in recent years, not only because of its safety applications in processing units, but also because of the growth of mathematical models and computer simulations of such phenomena. The Eulerian-Eulerian approach has been used for modelling the propagation of explosions from one vessel to another via a duct. Different cases have been considered, where the size of the particles and the height of the duct have been varied. It is shown that the moment at which an explosion can occur is a function of the duct height: for bigger channels an explosion is more likely to occur in the second vessel. Similar results have been reported in literature on basis of experiments.     

Fractal growth modelling of nanoparticles

The kinetics of iodine oxide aerosol production and growth was studied in an aerosol flow reactor by the photolysis of I₂ in an excess of O₃, at a temperature of 295 K and a total pressure of 1 atm. The time-resolved evolution of the particle size distribution was fitted using a model which assumes that the initial period of particle growth (in the free molecular flow regime) is dominated by collision-coalescence, maintaining spherical shape and compact structure. This leads to the formation of primary particles of about 3 nm radius, which trigger fractal (agglomerative) growth in the transition regime resulting in particle aggregates characterised by lower mass fractal dimensions (D_f) in the range 2.2–2.5. Enhancement of the particle pair collision kernels due to competing van der Waals and hydrodynamic forces is treated within the model. The densities of the fractal aggregates are lower than that of the bulk material, recently identified as I₂O₅ [Saunders, R. W., & Plane, J. M. C. (2005). Formation pathways and composition of iodine oxide ultrafine particles. Environmental Chemistry, 2, 299], as a result of internal void space within the aggregate structures.     

Mathematical modelling of coalescence of viscous particles: An overview

Viscous particles of polymer melts and glasses coalesce under the action of surface tension. Resistance is due to viscosity, while inertia is not a contributing factor, with the Ohnesorge number being very high. Russian physicist Yakov Frenkel developed a model for neck growth during the initial stage of the merging process of two spherical particles, assuming uniform biaxial extensional flow. Frenkel's model was extended for prediction of neck size as a function of time to the completion of coalescence, expressed by an ordinary differential equation. The time t is expressed in dimensionless form as (tγ/ηR), where η, γ, and R denote the viscosity, surface tension, and particle radius, respectively. Models were also developed for viscoelastic effects, non-isothermal conditions, and unequal diameter particles. For the coalescence of infinitely long cylinders, planar extensional flow is assumed. Other investigators presented numerical solutions of the Navier–Stokes equations, which include shear flow components, but the predictions of neck growth are not much different from those of the Frenkel-based models. Comparisons to experiments are also discussed, involving polymers, glasses, animal tissue cells, and biomacromolecules. The models are also used in additive manufacturing applications for the determination of bonding and pore shrinkage.     

Convolutive modelling of the disk electrode geometry under reversible conditions

The disk electrode is one of the most readily constructed electrode geometries but the perimetric discontinuity complicates the theoretical analysis. Under reversible conditions, this system simplifies to a situation that can be described by a convolution expression and can therefore be modelled by the method known as convolutive modelling. Convolutive modelling is here employed to simulate cyclic voltammograms at a disk electrode; those results are compared to fast quasi-explicit finite difference digital simulations. The inaccuracy of the latter method is thereby exposed. In addition, current reversal chronopotentiometry at a disk is modelled.     

Mathematical modelling of the combustion of a single wood particle

A mathematical model describing the thermal degradation of densified biomass particles is presented here. The model uses a novel discretisation scheme and combines intra-particle combustion processes with extra-particle transport processes, thereby including thermal and diffusional control mechanisms. The influence of structural changes on the physical–thermal properties of wood in its different stages is studied together with shrinkage of the particle during its degradation. The model is used to compare the predicted data with data on the mass loss dynamics and internal temperature of several particles from previous works and relevant literature, with good agreement.     

Multiple time constant modelling of a printed conducting polymer electrode

Constant phase elements (CPE) are routinely used to describe the frequency response of electrochemical systems. However, this approach is often scientifically unsatisfactory because the physical origin of the phase is unclear. Here we observe CPE-like behaviour in a conducting polymer poly(3,4-ethylenedioxythiophene)/poly(styrene-4-sulfonate) (PEDOT:PSS) film that was inkjet printed onto paper to form a flexible electrochemical double layer capacitor electrode. We show that the response of the electrochemically active film can also be described using a physical model with multiple parallel finite RC (resistor–capacitor) transmission lines whose lengths and time constants are determined by the distribution of the measured film thickness. The modeled volumetric capacitance and ionic conductivity match those determined experimentally, suggesting that the physical origin of the constant phase response is a distribution of mass transport limited time constants.