Multi-scale analysis of exotic dynamics in surface catalyzed reactions—II: Quantitative parameter space analysis of an extended Langmuir-Hinshelwood reaction scheme

The multi-time-scale approach to modelling heterogeneously catalysed reaction systems developed in a companion paper [1] is applied to a simple Langmuir-Hinshelwood mechanistic scheme for an overall reaction 2A(g) + B(g)→products supplemented by a slow buffering step involving a chemisorbed inert species. Based on leading-order approximations to the full governing equations and the use of non-linear algebraic equations in the definition of system manifolds and prediction lociin appropriate phase planes, regions of different dynamic behaviors (stationary states, single and multipeak oscillations) are qualitatively and quantitatively delineated as a function of residence time. The bifurcation predictions from the leading-order approximations achieve remarkable agreement both with numerical integration results as well as at local bifurcation points obtained via Hopf analysis of the full equations. Numerical simulations reveal some very complex (“chaotic”?) behavior in the multipeak region, with an indication that the onset of chaos may be via intermittency. It is expected that the application of multi-time-scale analysis to specific well-characterized reaction systems will provide fuller understanding of past and adequate theoretical guidance for future work.     

The cyclic voltammetry of carbon steel in concentrated sodium hydroxide solution

The potentiodynamic behaviour of carbon steel in 1.0 M, 2.5 M and 5.0 M NaOH at 20 ± 2°C is reported. The peak potentials and currents were found to vary linearly with the square root of the potential sweep-rate. It is shown that the “pore resistance” model cannot account quantitatively for the cyclic voltammetric behaviour of iron in sodium hydroxide solution. However, the sweep-rate dependencies of i_p and E_p suggest that the growth of the film on the surface occurs via an essentially resistive mechanism, which possibly involves the low field migration of ions through an oxide/hydroxide lattice.     

Experimental investigation of controller-induced bifurcation in a fixed-bed autothermal reactor

“Snap-through” Hopf bifurcations are induced in a laboratory autothermal reactor by the destabilizing effects of a single proportional controller. These unexpected and dramatic phenomena are predicted a priori by a bifurcation analysis of the model equations with the controller gain as the bifurcation parameter. The experimental bifurcation data are used in a parameter estimation study which yields an extremely accurate dynamic model for the nonlinear system.     

Predicting Thermal Efficiency in Timber Radio Frequency Vacuum Drying

Abstract

In this work, the efficiency of transforming dielectric energy into evaporated water is analyzed for the case of timber radio frequency vacuum drying. Based on well-known heat and mass transfer equations, a simplified mathematical model is proposed that estimates the drying efficacy in regards to the thermo-physical properties of wood. Although not exact, the theoretical results are close to the experimental observations and elucidate some phenomena like the tendency of the timber to dry from inside to outside, and the drying rate increase with the rise of the timber gas permeability. The theoretical efficiency model also predicts a range of wood permeability values for which the drying efficiency changes from 100 to 0%, thus providing a quantitative scale for classifying the spectrum of “difficult-to-dry” all the way to “easy-to-dry” wood species when using radio frequency vacuum technology.     

Logic hybrid simulation-optimization algorithm for distillation design

In this paper, we propose a novel algorithm for the rigorous design of distillation columns that integrates a process simulator in a generalized disjunctive programming formulation. The optimal distillation column, or column sequence, is obtained by selecting, for each column section, among a set of column sections with different number of theoretical trays. The selection of thermodynamic models, properties estimation, etc. is all in the simulation environment. All the numerical issues related to the convergence of distillation columns (or column sections) are also maintained in the simulation environment. The model is formulated as a Generalized Disjunctive Programming (GDP) problem and solved using the logic based outer approximation algorithm without MINLP reformulation. Some examples involving from a single column to thermally coupled sequence or extractive distillation shows the performance of the new algorithm.     

Free radical polymerization engineering—I: A new method for modeling free radical homogeneous polymerization reactions

This paper deals with a new appraisal of free radical polymerization modeling. Classical properties of the distribution of polymerization degrees and methods for studying them thanks to the z-transform are recalled. When simple mechanisms are involved, the kinetics may be represented by a “detailed analytic” model relying on macromolecule balances. This is no longer possible when complex processes are occurring, e.g. transfer to polymer, β-scission, terminal double bond propagation… In order to deal with these more complex cases, a “tendency model” is proposed, relying on balances of quantities such as free radicals, macromolecules and the moments of the distribution of polymerization degrees. The quality of the polymer is described by chemical characters such as double bonds, long and short branching points, terminal double bonds… for which kinetic equations are established. Equations are given for calculating the moments of free radicals and those of instantaneously produced macromolecules via various processes. The simplicity and the usefulness of the method are illustrated by several examples and a comparison is made, when possible, with the detailed approach. Finally, equations are given, making it possible to calculate the quality of polymer produced in any kind of reactor and for different states of segregation.     

Mixing and chemical reactions: A contrast between fast and slow reactions

The effect of fluid mixing on chemical reactions is analyzed by means of a model which treats simultaneously the complex interactions of mechanical mix diffusion and reaction. The model assumes that reactants diffuse from adjacent sheets of fluid undergoing a stretching motion, contact one another and This stretching motion causes deformation of the intermaterial surface and is designated mechanical mixing. A history of striation thickness s(t) contains all information about fluid mechanical mixing. Numerical methods were employed to solve the simultaneous partial differential equations of the model for mixing and single-step irreversible reactions. A criterion is established for limits of reaction control of bimolecular reactions by mixing, including diffusion, and by chemical kinetics in terms of a single dimensionless group φ  kC_B0S²/D_A. The present model can be applied to reactions in both laminar and turbulent flows.     

Mass transfer and biochemical reaction in enzyme membrane reactor systems—II: Expanded analysis for single enzyme systems: effects of enzyme intermediates,denaturation and elution

A detailed study of the dynamics of a packed-bed reactor containing immobilized enzyme particles is presented. The analysis consists of (i) transient state behavior; (ii) models for interphase and interfacial mass transfer between fluid and solid phases and intraphase mass transfer for the solid phase; (iii) detailed reaction rate model for the Bodenstein intermediates; (iv) mass balances for substrates, Bodenstein intermediates, unoccupied enzyme active sites, and products; and (v) models for enzyme denaturation and elution. The general reactor model consists of a set of nonlinear, coupled, partial differential equations. Numerical solutions of the system equations were obtained, using the discrete-space, continuous-time method of lines and realistic parameter values. A generalized map of the range of validity of the Steady-State Hypothesis was established under conditions where multiple mass transfer gradients were present within the reactor.A detailed analysis of the computational errors was performed. It was conclusively shown that the computer simulation solutions obtained in the analyses were not disguised to any significant degree as a result of employing finite difference approximations to the spatial derivatives.It was shown that the level of “error” involved in invoking the Steady-State Hypothesis depends on the relative magnitude of the kinetic parameters and also on the level of “disturbance” at the reactor inlet (i.e. per cent change in substrate inlet concentration). The “error”, however, did appear to be strikingly insensitive to the magnitude of the resistances to mass transfer, as characterized by the Modified Sherwood Number. It was concluded that, given any complete set of kinetic parameters, a transient, heterogeneous, isothermal reactor model based on the Steady-State Hypothesis may be used for predicting time-varying concentration profiles for minor (i.e., less than 5 per cent change in substrate inlet concentration) “disturbances” at the reactor inlet. The corresponding “errors” would be at an acceptable level (i.e., less than 2 per cent in the concentration and less than 10 per cent in the time lag) under these conditions.Further, various mechanisms for enzyme denaturation and elution were incorporated in the general reactor model. Numerical solutions of the resulting system of partial differential equations were obtained, using hypothetical parameter values. Through extensive simulation research, it was shown that the loss in activity of immobilized enzyme reactors cannot be uniquely ascribed to any one particular set of mechanistic deactivation modes.     

Theory of electroluminescence of Al/anodic alumina/electrolyte system

A relationship between the intensity of luminescence, L, and the film thickness, d, was obtained theoretically for the electroluminescence of the Al/anodic alumina/electrolyte system. According to the model whose principal assumption is the increasing probability of the excitation of luminescent centres by electron multiplication within the oxide with thickening of the oxide, and the relationship is represented by

where a is a constant associated with the injecting current density of electrons at the oxide-electrolyte interface and the mean range for ionization of the lattice by collision of electrons, α is the cationic transport number and is assumed to be constant throughout anodizing and b is a constant which is equal to the reciprocal of the mean range for ionization by collision.The experimental results on “flaw free” films fit the theoretical curve well, and the mean range for ionization by collision obtained from the luminescence shows good agreement with that estimated from the experimental results on electric breakdown of anodic alumina according to O'dwyer's “space charge modified theory of avalanche breakdown of solid”.Although the theoretical equation is exactly the same as the empirical one already presented by van Geel in 1956, the latter is a fortuitous relation coming from an increasing probability of scintillation at “flaws” with thickening of the film and, in his case, the luminescence is not the “electroluminescence” in its true sense, as previously pointed out by us.     

The simulation of binary adsorption in activated carbon columns using estimates of diffusional resistance within the carbon particles derived from batc

A computational procedure is presented for solving the set of coupled parabolic partial differential equations describing the simultaneous adsorption oOrthogonal collocation is used to solve for the first time the general form of these equations for simultaneous adsorption which include the effects ofSimulated results obtained from this model are compared with experimental data for the adsorption of mixtures of 2-butanol and t-amylalcohol. When     

The potential dependence of electrochemical transients in electrocrystallisation experiments

Experimental results are presented for the electrocrystallisation of 2D and 3D anodic films on liquid metal substrates. The data clearly show the inadequacy of “empirical” equations which have been commonly used to characterise the potential dependences of 2D and 3D nucleation—growth rates; trends in the experimental dependences are here examined and presented in detail. Analysis of the data raises some doubts about the validity of the model of 3D growth as it currently stands. Successful constant-potential geometric models of 2D electrocrystallisation are extended using ad hoc empirical equations that closely describe the experimental potentiostatic data. It is shown that useful “semi-empirical” models are generated. These models are internally self- consitent and, for the first time, quantitatively describe experimental responses over a wide range of potential perturbation.     

Characterization and non-isothermal crystallization behavior of biodegradable poly(ethylene sebacate)/SiO2 nanocomposites

Poly(ethylene sebacate) (PESeb) and PESeb/silica nanocomposites (PESeb/SiO₂) were prepared by in situ polymerization from the direct esterification of ethylene glycol with sebacic acid in the presence of proper amounts of silica nanoparticles. The non-isothermal crystallization behavior of PESeb/SiO₂ nanocomposites has been studied using different theoretical equations such as Avrami, Ozawa and combined Avrami and Ozawa equations. It is found that the addition of nanoparticles of SiO₂ influenced the mechanism of nucleation and the growth of PESeb crystallites. Also, the nanocomposites show a higher Avrami value than the neat PESeb, implying a more complex crystallization configuration. Moreover, the combined Avrami and Ozawa equation can successfully describe the crystallization model under the non-isothermal crystallization. The crystallization activation energies, E _a, calculated from “Kissinger’s equation” have shown that the synthesized PESeb/SiO₂ nanocomposites have lower energy than the neat PESeb, reflecting the much lower energy barrier for the rapid heterogeneous nucleation.     

Optimal design of engineered gas adsorbents: Pore-scale level

This study presents an optimization methodology and results for the structure of gas adsorbents at a pore level by evaluating the effect of pore geometry, size and overall adsorbent porosity on ultimate working capacity of adsorbents used in pressure swing adsorption (PSA) processes. Three model pore network topologies are studied: parallel, grid-like and branched structures. These are “near” optimal structures for an adsorbent particle and their relative performance is compared in a two-step PSA cycle. The macropore network of such structured adsorbents is optimized through maximization of an objective function i.e. working capacity WC, defined as the number of moles adsorbed per unit volume of slab. Molecular and Knudsen diffusion are considered as the sole transport mechanisms in the macropore channels. An unexpected finding of this optimization technique is that the branched structure with a porosity of less than 50% represents an optimum structure with highest working capacity for the system considered. Furthermore, for faster cycles the advantage of branched structures is more obvious, reflecting the advantages of the pore network in facilitating diffusion more efficiently than other structures. A macropore channel density (defined as the density of macropores per metre of external surface) of below 10 is suggested for optimum performance for both “fast” and “slow” PSA cycles. The theoretical results of this study will be used as a priori results for the design of adsorbents at the macro-scale (bed) level.     

Calculation of multicomponent multiphase equilibria

An algorithm which gives very good first trial values for the computation of chemical equilibrium composition is proposed. It is based on writing the chemical reactions in “canonical form” and changing the independent component set until it is formed by the species present in greatest quantity at equilibrium. The advancement degrees ξ_r of the reactions are calculated considering the reactions to occur independently of any other. By applying a similar procedure to the series reactor method, a new powerful solution algorithm of the Rosenbrook's type is obtained. In the first method we linearize the equilibrium equations written making null the gradient of the free energy of the system with respect to the corresponding ξ_r whilst in the second one we make successive unidimensional searches solving the equations for ξ_r     

Absorption of oxygen in countercurrent multistage bubble columns—III Viscoelastic liquids. Comparison of systems with high viscosity

The volumetric mass transfer coefficients, k_La, were measured by a steady state method in the first stage of a multistage countercurrent column 20 cm in diameter. Perforated plate trays with different bore hole diameters and free cross sectional area were employed. The height of the stage was varied.Viscoelastic fluids (PAA) solutions) were used as media. In general k_La increases with increasing superficial gas velocity, with decreasing bore hole diameter and free cross sectional area of the perforated plate trays as well as with decreasing concentration and viscosity.With increasing height of the bubbling layer k_La diminishes. This effect is lower with an increase of PAA concentration. A qualitative comparison of the rheologically corresponding media indicates that under similar conditions PAA solutions yield the highest, and the glycerol solution the lowest k_La values. With increasing specific power input k_La becomes larger. At constant power input the highest k_La is attained with the smallest bore hole diameter of the trays for CMC and PAA solutions. For glycerol no such effect was found.     

Amine plant simulation by mass transfer and kinetic approach

A computer program has been developed for the simulation of gas-liquid absorption columns. The calculations are based on known assumptions and methods – the two-film theory. Mass and energy balances together with chemical reactions are carried out for each tray of the column. The results obtained from the simulation procedure are compared with those measured at industrial plants. For the items “rich solution temperature” and “calculated number of trays” the agreement of calculated and measured results are satisfactory. However, for the CO₂ and H₂S content of the streams only few comparisons could be made due to the fact that no measured values are available.     

Fluid Mechanics of Distillation Trays (I): Depth-Averaged Theory and One-Dimensional Flows

Abstract

Rigorous design of a distillation column requires a better fundamental understanding of the fluid mechanics of bubble formation and global flows on trays than that currently available. To progress beyond the empirical-or correlation-based state of understanding that currently exists, a theoretical and computational framework is described here that is based on reducing the governing set of three-dimensional conservation equations to a two-dimensional set by averaging them across the depth of the fluid film flowing across the tray. In contrast to related previous works, realistic boundary conditions to the flow problem are provided in this paper by solving simultaneously for the flow on the tray and its inlet and outlet downcomers. In this first of a series of papers, attention is focused on situations in which the flow is invariant in the direction perpendicular to the main flow direction. By means of such a set of one-dimensional, depth-averaged equations, predictions are made in several interesting and practically important situations in which the flow is either steady or time dependent.     

Gas absorption with or without chemical reaction in laminar non-newtonian falling liquid films

An analytical solution is presented for gas absorption with or without a first-order or zero-order chemical reaction in a laminar non-Newtonian power-l model falling liquid film. For physical absorption, the first ten eigenvalues, series coefficients and related quantities are computed accurately by a quasinumerical method which shows considerable improvement over previous investigations. The range of applicability of the penetration theory solution is also established to indicate in what regions will the finite film thickness and complete velocity profile be important in determining the absorption rate. It is found that the range of dimensionless axial contact length X* in which the penetration theory is valid diminishes rapidly with increasi values of the power-law index n. For chemical absorption, the solution can be obtained by a linear superposition principle in terms of a “transie part” in which the effect of hydrodynamics within the liquid film is of importance and a “steady part” in which the reaction rate is controlling. In the “transient part” solution, the first ten eigenvalues and related quantities are reported for a variety of values of n and the dimensionl reaction rate parameter k_l* or k₀*. Certain asymptotic solutions from the penetration theory are also given and their range of applicab estimated. For any given n, it is estimated that only when k₁* or k₀* is less than approximately 10 will the finite film thickness and velocity profile have any effect on the absorption rate as compared to that calculated from the penetration theory with chemical reaction. The non-Newt character of the liquid film also has a significant influence on the absorption rate. At a fixed X*, the absorption enhancement due to reaction is when n = ∞ and is smallest when n = 0. The solutions obtained in this work are useful either for predicting absorption rates or for deter molecular diffusivity (and reaction rate constant) of gases in non-Newtonian falling liquid films.     

Solute–solvent interactions governing preferential solvation phenomena of acetaminophen in CO2-expanded organic solutions: A spectroscopic and theoretical study

The present work aims to characterize the nature and intensity of the specific and non-specific solute–solvent interactions responsible for the different solvating behaviour of “CO₂-expanded ethanol” and “CO₂-expanded acetone” towards acetaminophen, an analgesic drug commercially known as paracetamol. The intermolecular interactions between acetaminophen and solvent molecules involved in these expanded media and its sensitivity to solvent composition changes, have been analyzed by high-pressure IR spectroscopy, LSER analysis, and theoretical ab initio calculations performed with acetaminophen–(ethanol)_n and acetaminophen–(acetone)_n complexes. It comes out that the distinct perturbation experienced by the cybotactic region of acetaminophen in “CO₂-expanded ethanol” and “CO₂-expanded acetone” when CO₂ content increases is basically a consequence of the higher sensitivity to solvent composition changes of the dipole–dipole interactions, established between the acetaminophen carbonyl group and acetone solvent molecules, in relation to that of the hydrogen bond interactions existing between this group and ethanol solvent molecules.     

Toughness measurement on ball specimens. Part II: Experimental procedure and measurement uncertainties

The “Surface Crack in Flexure” method is widely used for fracture toughness (K_Ic) determination of ceramics. In part I of the paper we developed the theoretical fundamentals to apply this procedure to ceramic balls by using the stress application as developed for the so-called “Notched ball test”. The new test (SCF-NB) can be used to test spherical components without the need to cut out special specimens such as bending bars. In this work the practical part is presented including suggestions for crack introduction and specimen preparation and possible measurement errors are discussed. It is concluded that a measurement error less than ±5% is possible.Experiments on balls and bars made from the same silicon nitride ceramic indicate that SCF-NB delivers the same K_Ic-values as standardised measurements on bars. Additionally, K_Ic-values obtained for silicon carbide, alumina and zirconia ceramics are presented. They coincide with K_Ic-data from the literature.