Orthogonal Collocation Technique Research Articles

Simulation of biodegradation process of phenolic wastewater at higher concentrations in a fluidized-bed bioreactor

Experiments were carried out to study the biodegradation of phenolic wastewater using microorganisms in a fluidized-bed bioreactor (FBR). Synthetic wastewater has been used in the study. Experiments were conducted at various conditions of wastewater flow rate, dissolved oxygen (DO) concentration and inlet concentration. The wastewater with feed concentration of phenol as high as 1254 ppm has been successfully biodegraded to 50 ppm in the fluidized-bed bioreactor, whereas feed with concentration up to 1034 ppm could be completely biodegraded (to zero ppm). Biokinetic parameters for the growth of the cells were determined in batch experiments and they were found to be μ max = 1.436 × 10 −4 s −1, K s = 21.92 × 10 −3 kg/m 3, K i = 522 × 10 −3 kg/m 3. Model equations describing the simultaneous diffusion and reaction of phenol and oxygen in the biofilm of the solid particles, were solved using the orthogonal collocation technique to calculate the rate of biodegradation of phenol. The rate of biodegradation predicted by the model was compared with that observed experimentally and the difference between the two was found to be about 12%.

Modelling and Simulation of Enzymatic Membrane Reactor for Hydrolysis of Ibuprofen Ester: Separation of Products

AbstractThe enzymatic hydrolysis of racemic ibuprofen ester by lipase from Candida rugosa using a hollow‐fiber membrane reactor was studied. The modeling and simulation of an enzymatic membrane reactor was based on the diffusion of (S)‐ibuprofen acid product which leaves the reactor from the lumen side of the capillary fibers. The mathematical model containing a set of nonlinear partial differential equations was solved using orthogonal collocation technique. Orthogonal collocation technique is a weighted residual method based on Jacobi polynomials incorporating orthogonal trial functions. This paper presents a numerical technique of high efficiency, based on the orthogonal collocation of weighted residuals in solving the nonlinear partial differential equations of product diffusion in the capillary fibers. The modelling simulation study provided a better understanding of the process dynamics and product separation in the integrated enzymatic membrane reactor. The simulation study shows that noticeable product diffusion and transport occurred throughout the region of the fiber length, with 63.0% of product concentration (solute concentration) obtained at the exit of the fiber. The input parameters investigated were Peclet number and weighing function parameters, α and β. The best simulation results were achieved at the optimal 10 collocation points.

Analysis of modified surface force pore flow model with concentration polarization and comparison with Spiegler–Kedem model in reverse osmosis systems

The Modified Surface Force Pore Flow (MD-SF-PF) model is used for predicting the performance of the sodium chloride–water and sodium sulphate–water reverse osmosis systems. Unlike the previous analysis available in literature on this model, the present work takes concentration polarization into account explicitly. The required mass transfer coefficient is estimated by relating it to the feed flow rate through two additional parameters. The model equations being non-linear are solved using the orthogonal collocation technique. The model solution code is validated by comparing results with those available in literature. Simulation studies indicate observed rejection versus flux has a similar trend to that predicted by the Spiegler–Kedem model. From some of the experimental observations of Murthy [Studies on membrane transport models, Ph.D. thesis, I.I.T. Delhi, July 1996], model solution and mass transfer parameters are estimated through a combination of the Downhill Simplex method and Monte Carlo search. Simulated results using above parameters are found to be in very good agreement with the remaining experimental observations. Prediction of the membrane performance and mass transfer coefficient are also found to be very close to those estimated through the Spiegler–Kedem model. The membrane specific parameters such as the membrane thickness and pore radius are calculated from experimental data for the two different systems on a similar cellulose acetate membrane. The closeness of these parameter values again shows the validity of MD-SF-PF model.

Co-current and counter-current modes for water gas shift membrane reactor

In this study the performances of a membrane reactor (MR) are estimated when both shell side stream (sweep gas) and lumen side stream are continuously either in parallel flow configuration (co-current mode) or in counter-flow configuration (counter-current mode). Two mathematical models have been formulated and steady-state mass-balance gave two-dimensional differential equations, which were solved by using the orthogonal collocation technique. Simulation results for both co-current mode and counter-current mode have been compared in terms of hydrogen molar fraction (in the shell side) vs. axial co-ordinate at different hydrogen permeances, temperatures, and lumen pressures. At the operative conditions considered, a very similar CO conversion value has been obtained for both modes.

Modeling and simulation of enzymatic membrane reactor for kinetic resolution of ibuprofen ester

The enzymatic hydrolysis of racemic ibuprofen ester by lipase from Candida rugosa using hollow fiber membrane reactor was studied. In process modeling and simulation of the enzymatic reaction in a horizontal hollow fiber membrane reactor system, two distinctive regions were studied: (i) diffusion in the fiber lumen; and (ii) reaction in the membrane matrix support. The model equations for both regions were developed and solved numerically by different approaches. In the first part, the modeling and simulation work was based on the diffusion of ( S)-ibuprofen acid in which it leaves the reactor from the lumen side of the capillary fibers. The proposed mathematical model containing non-linear partial differential equations was solved using the orthogonal collocation technique. The second part involved an enzymatic reaction taking place with the immobilized enzyme in the porous matrix support of the membrane, at which the substrate flows in from the shell side. The driving force was predominantly induced by pressure difference across the membrane where radial convection was the only mass transport in the porous support. The model was solved using two numerical techniques, the first technique based on the orthogonal collocation of weighted residuals in solving the non-linear partial differential equations of product diffusion in the capillary fibers; and the second technique was a collocation technique for solving ordinary differential equations (ODEs) of the enzymatic reaction in the porous support structure. The simulation was studied over a wide range of process parameters to investigate their effects on separation efficiency, in terms of enantiomeric excess, ee S and enantiomeric ratio, E, respectively. The model parameters include Peclet number, weight function parameters α, β and collocation points ( n), Thiele Modulus, Φ 2, dimensionless Michaelis constant, Θ and Bodenstein numbers, B 0. The performance of hollow fiber membrane reactor was found to be sufficiently effective at Θ=0.55, B 0=0.174 and ee S=74% at Φ 2<1.

Kinetics of sorption in deactivated zeolite crystal adsorbents

Dehydration adsorbents deactivate with time due to either pore mouth closure, window blocking, coking and/or hydrothermal decrystallization. These mechanisms of deactivation may be significant under severe hydrothermal conditions and/or in presence of reactive hydrocarbons at elevated temperatures and/or pressures. A mathematical model for predicting the uptake rates for crystal particles partially blocked either due to coking or pore mouth closure has been developed and numerical results, using the orthogonal collocation technique, have been obtained. Concentrated blockage of the crystal surface greatly reduces the mass transfer rate in comparison to uniformly distributed surface blockage.

Numerical Simulation of Continuous Stirred Ultrafiltration Process: An Approach Based on Moving Boundary Layer Concept

A mass transfer model based on unsteady state mass balance over concentration boundary layer coupled with gradual development of so called "gel" or "cake" layer has been formulated in this study. The model considers the boundary layer problem simultaneously with the film theory of mass transfer and resistance in series model. The resulting equations are solved first by reducing the set of partial differential equations to nonlinear equations utilizing orthogonal collocation technique and then applying multidimensional Newton–Raphson method with suitable seeding scheme to solve the above-mentioned nonlinear equations together with some additional constraints. The problem of determination of membrane surface concentration is eliminated by this simultaneous solution of boundary layer equation together with gel layer. The model is found to be capable of predicting permeate flux and rejection under different experimental conditions. The main feature of this model is that it takes into account realistically the effect of gel layer formation within concentration boundary layer whereas in most of the previous studies this effect has not been considered. The dependency of concentration profile, gel thickness, and permeate concentration with different operating variables are also studied and the variations are found to be quite reasonable.

Mathematical modeling for chemical vapor deposition in a single-wafer reactor: Application to low-pressure deposition of Tungsten

A mathematical model for low pressure chemical vapor deposition in a single-wafer reactor in stagnation point flow has been developed to investigate the reactor performance. The transient transport equations for a simulated reactor include continuity, momentum, energy, and gaseous species balances. The model equations are simultaneously solved by using a numerical technique of orthogonal collocation on finite element method. Simulation studies have been performed to gain an understanding of tungsten low pressure chemical vapor deposition process. The model is then used to optimize the deposition rate and uniformity on a wafer, and the effects of operating conditions on deposition rate are studied to examine how system responses are affected by changes in process parameters. Deposition rate and uniformity calculated at the steady state are observed to be very sensitive to both temperature and total pressure. In addition, the model predictions for tungsten deposition from hydrogen reduction of tungsten hexafluoride have been compared with available experimental data in order to demonstrate the validity of the model.

A New Simplified and Accurate Approximation for a Batch Adsorber(Short communication)

A new approximate method derived from the technique of orthogonal collocation of finite elements with moving junction was much better than the other methods proposed in the past. With one point collocation on each subdomain, the relative error was less than 2% throughout the time domain for wide ranges of partition ratio and Biot number.

Voltammetry of Surface Redox Processes Perturbed by Dimerization and Adsorption of the Products : Application to the Oxidation of 3-Mercaptopropanol on Mercury

Electrochemical evidence shows that the oxidation of thiols on mercury is characterized by an extensive adsorption of the oxidized products on the electrode surface, and by the formation of dimeric R-S-Hg-S-R organometallic complexes as the final oxidation products. The orthogonal collocation technique has been used in this work to solve the simplest voltammetric boundary value problem accounting for these observations. The proposed oxidation mechanism includes a following surface dimerization step, and it considers simultaneously the adsorption of the oxidized products. Analytical expressions for the voltammetric wave have been derived when the adsorption strength of the products is either very strong or weak. When the adsorption strength is moderate, numerical results have been used to explore the influence of the dimerization step on the voltammetric shape. Theoretical predictions have been applied to interpret the voltammetric behavior of 3-mercaptopropanol on mercury as a function of the solution pH. Quantitative agreement with the proposed mechanism is only found in the most acid (pH 2) solution, whereas a progressive distortion of the voltammetric waves is observed as the solution pH increases, which is tentatively ascribed to the existence of two low- and high-density surface states of the adsorbed products. © 2002 The Electrochemical Society. All rights reserved.

The effect of latent heat of fusion on heat transfer in fluidized-bed coating of thin plates

This paper presents a numerical study of fluidized-bed coating on thin plates using an orthogonal collocation technique. Inclusion of the latent heat of fusion term in the boundary conditions of the mathematical model accounts for the fact that some polymer powders used in coating may be partially crystalline. Predictions of coating thickness on flat plates were made with actual polymers used in fluidized-bed coating. Reasonably good agreement between numerical predictions of the coating thickness and experimental coating data of Richart was obtained for steel panels preheated to 316 °C. A good agreement was also obtained between numerical predictions and our coating thickness data for nylon-11 and polyethylene powders. Predicted coating thickness for polyethylene powder on flat plates were obtained with values of heat transfer coefficient closer to those obtained from our experiments.

Simulation of mercury capture by activated carbon injection in incinerator flue gas. 1. In-duct removal.

A detailed model for the in-duct mercury capture in incinerator flue gas by powdered activated carbon injection is presented. Material balances on mercury in both gaseous and adsorbed phases are carried out along the duct length and inside the activated carbon particles, taking into account mass transfer resistances and adsorption kinetics. The set of the coupled partial differential equations is transformed by means of an orthogonal collocation technique and integrated using a Runge-Kutta method with adaptive stepsize control. The model has been applied to several sorbents of practical interest, whose parameters have been evaluated from available literature data. The values and range of the operating variables have been chosen in order to simulate typical incinerators operating conditions. Results of simulations indicate that large sorbent loadings in the duct are needed to obtain high mercury removal efficiencies, due to the short residence times. As a consequence very low utilization of the sorbents is achieved in any case. To minimize the sorbent feed rate it is particularly advisable to use a reactive sorbent and to lower the operating temperature as much as possible. Improvements in the mercury capture performance can be obtained also by increasing the in-duct particles residence time and by decreasing the sorbent particles size. Model results are compared with available relevant full scale data.

Application of the Unequal Distances of Closest Approach Theory to the Analysis of Double-Layer Effects on Zn(II) Reduction at a Mercury Electrode

The orthogonal collocation technique has been applied to the development of a numerical solution of the unequal distances of closest approach (UDCA) theory for any number of electrolytes and planes of closest approach to the electrode surface. On the basis of theoretical analysis of the reaction coordinate of Zn2+ reduction by Schmickler et al. (Chem. Phys. 2000, 252, 349), a UDCA analysis of the Frumkin effect on the observed charge-transfer rate constants has been performed. The supporting electrolytes LiClO4, Mg(ClO4)2, and Al(ClO4)3 were chosen to vary systematically the distance of closest approach of the electrolyte cation with respect to that of the electroactive species. UDCA distances of closest approach derived from electrocapillary data are shown to provide a reasonable account of the Frumkin effect. However, more consistent results are obtained when variations of the Zn2+ activity coefficient, computed from the MSA theory, are also considered.

Influence of Spatial Redox Distribution on the Electrochemical Behavior of Electroactive Self-Assembled Monolayers

Packing restrictions and hydrophobic interactions are likely to lead to a spatial distribution of redox centers in electroactive monolayers. A mean field analysis of the electrochemical implications of spatial redox dispersion in SAMs, including the possibility of surface ion pair formation, has been carried out. The boundary value problem associated with a layered distribution of potential-induced charges has been solved by using the orthogonal collocation technique under equilibrium conditions. Spreading of the redox centers into a 3D dielectric slab results in broader and asymmetric voltammograms, reflecting a layer-by-layer redox conversion. It is also shown that the voltammetric shape is sensitive to the specific features of the spatial redox distribution, and theoretical requirements for the appearance of asymmetric broadening are examined in terms of the electrostatic properties of the monolayer. It is suggested that this type of spatial inhomogeneity may cause some of the broad and asymmetric volt...

Modeling molecular weight distribution in emulsion polymerization reactions with transfer to polymer

AbstractA mathematical model was developed for the computation of the dynamic evolution of molecular weight distributions (MWDs) during nonlinear emulsion polymerization reactions. To allow the direct computation of the whole MWD, an adaptive orthogonal collocation technique was applied. The model was validated with experimental methyl methacrylate/butylacrylate (BuA) semicontinuous and vinyl acrylate (VA)/Veova10 continuous emulsion polymerization results. Both systems considered introduce significant chain‐transfer reactions to polymer chains as a result of the presence of BuA and VA, respectively. The model developed was able to represent quite properly the kinetics and MWD of polymer samples during emulsion polymerizations. © 2001 John Wiley &amp; Sons, Inc. J Polym Sci Part A: Polym Chem 39: 3513–3528, 2001

Comparison of a reduced order model for packed separation processes and a rigorous nonequilibrium stage model

The large dimensionality of the system of algebraic (and differential) equations of the models for packed distillation columns makes their solution difficult to be achieved. Aiming the solution of this problem, a reduced order model was developed for steady state packed separation processes. The model was developed transforming the differential equations of the rigorous model into algebraic equations through the use of the orthogonal collocation technique. The heat and mass transfer rates through the vapor-liquid interface were rigorously computed. The performance of this reduced order model was assessed comparing its results with those obtained with a rigorous rate based model. The results of the reduced order model were in good agreement with those from the rigorous model. Furthermore, the solution of the new model was the easiest to be obtained due to the robustness of the reduced order model.

Modelling of the catalytic combustion of lean fuel flowing over a fin with heat extraction from the trailing edge

The focus of this work was the study of catalytic combustion over a flat plate, or fin, from which heat was extracted from the trailing edge of the fin. This enabled those factors that effect the heat transfer to the sink to be elucidated. Experimental work was conducted that involved passing a heated lean fuel (either carbon monoxide or propane) in air mixture over a catalytic fin. The temperatures along the fin and across the fluid boundary layer were measured. Experimental conditions varied were bulk gas (or free stream) fuel concentration, coolant temperature, the temperature of the bulk gas and velocity of the bulk gas. A model of the fin was derived using the basic equations of momentum, energy and conservation of chemical species for boundary layer flow. The model equations were solved numerically using a combination of orthogonal collocation and finite difference techniques and theoretical predictions were compared with experimental results. Theoretical predictions of the fin and boundary layer temperature were, generally, in excellent agreement with experimental values.

Mathematical modelling of fixed bed adsorption of aromatics and sulphur compounds in kerosene deodorisation

A mathematical model that adequately predicts the effluent concentration and breakthrough profiles of aromatic and sulphur compounds in kerosene deodorisation has been developed. The contributions of radial transport (pore and surface diffusion) were incorporated in the mathematical formulations. Thus, the final model took into account the overall effect of both the solid and liquid phase mass transfer resistances. The resulting model expressions were coupled partial differential equations which were resolved into first order ordinary differential equations using the orthogonal collocation technique. The roots of the Jacobi orthogonal polynomials ( P N α, β ) with N=8 and α= β=0 were taken as the interior collocation points while the exterior points were ζ=1, z=0 and z=1. The fourth-order Runge Kutta method was then used to integrate the 4 N differential equations and the resulting functions were solved simultaneously to obtain the effluent and breakthrough profiles. Theoretical predictions from the model were compared with column adsorption data to ascertain the authenticity of the model. The agreement was good for both cases of aromatics and sulphur breakthroughs. The experimental breakthrough time of 8 h was predicted by the model. The breakthrough profiles also confirmed the formation of multiple adsorption layers.

Quantitative characterization of desorptive stripping voltammograms complicated by surface dimerization reactions. Application to the reductive desorption of thiols from mercury

Abstract The influence of analyte adsorption and of a follow-up surface dimerization reaction on desorptive stripping voltammograms has been investigated. A modified version of the spline orthogonal collocation technique has been developed to compute the voltammetric response under Langmuirian reversible conditions. Simple analytical expressions derived for surface-confined redox processes are shown to reproduce the stripping voltammograms only in the presence of significant ( θ * R >0.01) analyte adsorption. In the absence of analyte adsorption, asymmetrical waves are obtained as an intrinsic characteristic of the stripping protocol. Empirical equations relating the stripping peak potential and the voltammetric half-height width with scan rate and adsorbed product coverage are presented. Voltammetric scan rate dependence is shown to provide a quantitative estimate of the analyte adsorption extent, whereas its dependence on adsorbed product coverage allows one to identify the presence of surface dimerization processes. Maximum surface excesses and dimerization equilibrium constants can readily be obtained by plotting the reciprocal half-height width versus the surface concentration of stripped species. In the absence of analyte adsorption, a complementary convolutive analysis is also proposed. Application is made to the cathodic stripping of three mercaptocarboxylic acids differing in their hydrocarbon chain-length. The extent of oxidized product dimerization is shown to display a strong dependence on the solution pH, so that higher peak current sensitivities are found in acidic solutions, where oxidized products are present as dimers.

Dynamic simulation of bioreactor systems using orthogonal collocation on finite elements

The dynamics of continuous biological processes is addressed in this paper. Numerical simulation of a conventional activated sludge process shows that despite the large differences in the dynamics of the species investigated. the orthogonal collocation on finite element technique with three internal collocation and four elements (OCFE-34) gives excellent numerical results for bioreactor models up to a Peclet number of 50. It is shown that there is little improvement in numerical accuracy when a much larger internal collocation point is introduced. Over and above Peclet number of 50, considered to be large for this process. simulation with the global orthogonal collocation (GOC) technique is infeasible. Due to the banded nature of its structural matrix, the method of lines (MOL) technique requires the lowest computing time, typically four times less than that required by the OCFE-34. Validation of the hydraulics of an existing pilot-scale subsurface flow (SSF) constructed wetland process using the aforementioned numerical techniques suggested that the OCFE is superior to the MOL and GOC in terms of numerical stability.