Evaluation of weighted residual methods for the solution of the pellet equations: The orthogonal collocation,Galerkin, tau and least-squares methods

In recent years a number of publications have adopted the least-squares method for chemical reactor engineering problems such as the population balance equation, fixed bed reactors and pellet equations. Evaluation of the performance of the least-squares method compared to other weighted residual methods is therefore of interest. Thus, in the present study, numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau, and least-squares methods, have been adopted to solve a non-linear comprehensive and highly coupled pellet problem. The methanol synthesis and the steam methane reforming process have been adopted for this study.Based on the residual of the governing equations, the orthogonal collocation method obtains the same accuracy as the Galerkin and tau methods. Moreover, the orthogonal collocation method is associated with the simplest algebra theory and thus holds the simplest implementation. Another benefit of the orthogonal collocation method is that the technique is more computational efficient than the other methods evaluated. The least-squares method does not obtain the same accuracy as the other weighted residual methods. In particular, the least-squares method is not suitable for strongly diffusion limited systems that give rise to steep gradients in the pellet. On the other side, considering the spectral–element framework, the least-squares method is less sensitive to the placement of the element boundaries than observed for the orthogonal collocation, Galerkin and tau methods.The present paper outlines the algebra of the weighted residual methods and illustrate the numerical solution techniques through a simplified pellet problem.     

The Effect of the Estimation on Goodness‐of‐Fit Tests in Time Series Models

Abstract. We analyze, by simulation, the finite‐sample properties of goodness‐of‐fit tests based on residual autocorrelation coefficients (simple and partial) obtained using different estimators frequently used in the analysis of autoregressive moving‐average time‐series models. The estimators considered are unconditional least squares, maximum likelihood and conditional least squares. The results suggest that although the tests based on these estimators are asymptotically equivalent for particular models and parameter values, their sampling properties for samples of the size commonly found in economic applications can differ substantially, because of differences in both finite‐sample estimation efficiencies and residual regeneration methods.     

Simulation and Analysis of a Tubular Fixed‐Bed Fischer‐Tropsch Synthesis Reactor with Co‐Based Catalyst

A two‐dimensional pseudohomogeneous reactor model is proposed to simulate the performance of fixed‐bed Fischer‐Tropsch synthesis (FTS) reactors by lumped thought. A CO consumption kinetics equation and a carbon chain growth probability model were incorporated into the reactor model. The model equations discretized by a two‐dimensional orthogonal collocation method were solved by the Broyden method. Concentration and temperature profiles were obtained. The validity of the reactor model against the pilot plant test data was investigated. Satisfactory agreements between model prediction values and experiment results were obtained. Further simulations were carried out to investigate the effect of operating conditions on the reaction behavior of the fixed‐bed FTS reactor.     

Model for the outer cavity of a dual‐cavity die with parameters determined by two‐dimensional finite‐element analysis

A coating die forms liquid layers of uniform thickness for application to a substrate. In a dual‐cavity coating die an outer cavity and slot improves flow distribution from an inner cavity and slot. A model for axial flow in the outer cavity must consider the ever‐present cross flow. A 1‐D equation for the pressure gradient for a power‐law liquid is obtained as a small departure from a uniform flow distribution and no axial flow. The equation contains a shape factor dependent on cavity shape, Reynolds number, and power‐law index. The shape factor for five triangular cavity shapes is obtained by finite‐element analysis and correlated for application to die design up to the onset of flow recirculation which arises at the junction of the cavity and outer slot. The performance of the combined cavity and slot is considered and the most effective design determined. © 2017 American Institute of Chemical Engineers AIChE J, 64: 708–716, 2018     

Fuzzy multi‐model based adaptive predictive control and its application to thermoplastic injection molding

Many chemical processes are inherently nonlinear. A single linear model is ineffective for these processes. Several local linear models may be developed for different operating conditions. A combination of these local models, through a fuzzy logic representation, results in an overall model for a wider operation range. In this paper, on‐line improvements and a fuzzy multi‐model have been proposed for predictive control implementation. Firstly, assuming that the premises of the fuzzy rules keep their original structures, the linear parameters in the rule consequents are on‐line updated by a weighted recursive least squares algorithm at each sample interval. Secondly, a batch learning algorithm is proposed to tune the fuzzy rule premises using a competitive learning algorithm. The effectiveness of the proposed improvements is demonstrated with experimental applications to the filling velocity control of thermoplastic injection molding     

Raman spectra of high‐density,low‐density,and linear low‐density polyethylene pellets and prediction of their physical properties by multivariate data analysis

Raman spectra have been measured for pellets of five samples of high‐density polyethylene (HDPE), seven samples of low‐density polyethylene (LDPE), and six samples of linear low‐density polyethylene (LLDPE). The obtained Raman spectra have been compared to find out characteristic Raman bands of HDPE, LDPE, and LLDPE. Principal component analysis (PCA) was applied to the Raman spectra in the 1600–650 cm^?1 region after multiplicative scatter correction (MSC) to discriminate the Raman spectra of the three different PE species. They are classified into three groups by a score plot of PCA factor 1 vs. 2. HDPE with high density and high crystallinity gives high scores on the factor 1 axis, while LDPE with low density and low crystallinity yields negative scores on the same axis. It seems that factor 1 reflects the density or crystallinity. A PC weight loadings plot for factor 1 shows six upward peaks corresponding to the bands arising from the crystalline parts or all‐trans ? (CH₂)_n? groups and seven downward peaks ascribed to the bands of the amorphous or anisotropic regions and those arising from the short branches. Partial least‐squares (PLS‐1) regression was applied to the Raman spectra after MSC to propose calibration models that predict the density, crystallinity, and melting points of the polyethylenes. The correlation coefficient was calculated to be 0.9941, 0.9800, and 0.9709 for the density, crystallinity, and melting point, respectively, and their root‐mean‐square error of cross validation (RMSECV) was found to be 0.0015, 3.3707, and 2.3745, respectively. The loadings plot of factor 2 for the prediction of melting point is largely different from those for the prediction of density and crystallinity. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 86: 443–448, 2002     

Explosive Random‐Coefficient AR(1) Processes and Related Asymptotics for Least‐Squares Estimation

Abstract. Large sample properties of the least‐squares and weighted least‐squares estimates of the autoregressive parameter of the explosive random‐coefficient AR(1) process are discussed. It is shown that, contrary to the standard AR(1) case, the least‐squares estimator is inconsistent whereas the weighted least‐squares estimator is consistent and asymptotically normal even when the error process is not necessarily Gaussian. Conditional asymptotics on the event that a certain limiting random variable is non‐zero is also discussed.     

Minimum α‐divergence estimation for arch models

Abstract. This paper considers a minimum α‐divergence estimation for a class of ARCH(p) models. For these models with unknown volatility parameters, the exact form of the innovation density is supposed to be unknown in detail but is thought to be close to members of some parametric family. To approximate such a density, we first construct an estimator for the unknown volatility parameters using the conditional least squares estimator given by Tjøstheim [Stochastic processes and their applications (1986) Vol. 21, pp. 251–273]. Then, a nonparametric kernel density estimator is constructed for the innovation density based on the estimated residuals. Using techniques of the minimum Hellinger distance estimation for stochastic models and residual empirical process from an ARCH(p) model given by Beran [Annals of Statistics (1977) Vol. 5, pp. 445–463] and Lee and Taniguchi [Statistica Sinica (2005) Vol. 15, pp. 215–234] respectively, it is shown that the proposed estimator is consistent and asymptotically normal. Moreover, a robustness measure for the score of the estimator is introduced. The asymptotic efficiency and robustness of the estimator are illustrated by simulations. The proposed estimator is also applied to daily stock returns of Dell Corporation.     

A Bayesian framework for real‐time identification of locally weighted partial least squares

Just‐in‐time (JIT) learning methods are widely used in dealing with nonlinear and multimode behavior of industrial processes. The locally weighted partial least squares (LW‐PLS) method is among the most commonly used JIT methods. The performance of LW‐PLS model depends on parameters of the similarity function as well as the structure and parameters of the local PLS model. However, the regular LW‐PLS algorithm assumes that the parameters of the similarity function and structure of the local PLS model are known and do not fully utilize available knowledge to estimate the model parameters. A Bayesian framework is proposed to provide a systematic way for real‐time parameterization of the similarity function, selection of the local PLS model structure, and estimation of the corresponding model parameters. By applying the Bayes' theorem, the proposed framework incorporates the prior knowledge into the identification process and takes into account the different contribution of measurement noises. Furthermore, Bayesian model structure selection can automatically deal with the model complexity problem to avoid the overfitting issue. The advantages of this new approach are highlighted through two case studies based on the real‐world near infrared data. © 2014 American Institute of Chemical Engineers AIChE J, 61: 518–529, 2015     

Akaike’s information criterion correction for the least‐squares autoregressive spectral estimator

In this article we propose a new correction for the penalty term of the Akaike’s information criterion (AIC), when it is used in the context of order selection for an autoregressive fit of the spectral density of a stationary time series. The classical AIC penalty term may be viewed as an approximation of an appropriate target quantity. Simulations show that the quality of this approximation strongly depends on the type of autoregressive estimator used, as well as on the discrepancy used. Therefore here we consider the least squares autoregressive estimator and the Whittle discrepancy only. In this context we propose a closed formula correction of the AIC penalty term. We also develop asymptotic theory which justifies this proposal: an asymptotically valid second‐order expansion of a stochastic approximation of the target quantity. This expansion assumes a non‐parametric framework: it does not assume gaussianity of the process and only requires its spectral density to be smooth enough. Simulations show that, as compared to previously introduced corrections, this new correction performs similarly to finite sample information criterion, while they both outperform AIC corrected and AIC.     

Effect of the Mg/Al Atomic Ratio of Ni‐Mg‐Al Catalysts for the Hydrodealkylation of 1,2,4‐Trimethylbenzene

The hydrodealkylation of 1,2,4‐trimethylbenzene (1,2,4‐TMB) to benzene, toluene and xylenes (BTX) was investigated on Ni‐Mg‐Al catalysts prepared by the coprecipitation method. The catalytic performances of these catalysts were considerably influenced by the Mg content of the catalyst. The catalysts were characterized via X‐ray diffraction, H₂‐temperature‐programmed reduction, NH₃‐temperature‐programmed desorption (TPD), CO₂‐TPD, and Fourier transform infrared spectroscopy. The results demonstrated that the appropriate amount of Mg species significantly affected the structural properties and caused the Ni nanoparticles to become highly dispersed. The higher activity of the catalysts might be ascribed to the homogenous distribution of the Ni nanoparticles, and the synergetic effects between Ni⁰, NiAl₂O₄ and MgAl₂O₄ were the key factor for obtaining the BTX.     

Synthesis and spectral characterization of surface‐enriched polymer‐supported phase transfer catalysts and their effects on alkylation of phenylacetone

Polymer‐supported phase transfer catalysts with active sites mostly on the surface were prepared by suspension copolymerization of styrene (St), divinylbenzene (DVB), and vinylbenzyl chloride (VBC) with AIBN, followed by the quaternization of the resulting copolymer beads with triethylamine. Active sites on the surface were achieved by the delayed addition of functional monomer (VBC) to the partially copolymerized St/DVB. Polymer beads enriched with active sites were characterized by SEM, EDAX, FTIR, and ESCA. The electron micrographs showed that the exterior surface of delayed‐addition functional monomer catalysts (type 1) has a large number of nodules attached to the surface compared to the smooth surface exhibited by the conventional type 2 catalyst upon the simultaneous addition of all three monomers. In the EDAX analysis up to a depth of 100 Å, the surface chloride of type 1 peak intensity is greater (compared with type 2), indicating the  CH₂Cl enrichment on the surface. In FTIR, the peak intensities of the C N stretching (quaternary onium group) in type 1 are greater than those of type 2, confirming the evidence of more quaternization on the surface than in the bulk. From ESCA analysis to a depth of about 30 Å, it was found that type 1 (beads) contains 26% and type 2 contains 14% of covalent chloride on the surface, which strongly supports the grafting of VBC on St/DVB. In the estimation found by the Volhard method, type 1 has 4.73 m eq g⁻¹ and type 2 has 2.29 m eq g⁻¹ of ionic chloride, thus supporting the surface grafting of VBC. The catalytic activity of these two different catalysts was tested by studying the reaction, that is, the C‐alkylation of phenylacetone. The rate constants of this reaction for type 1 are almost twofold greater than those of type 2, a finding that could uphold the preceding experimental observations. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 76: 408–418, 2000     

Short‐ and long‐term loss of carvacrol from polymer/clay nanocomposite film – a chemometric approach

A combination of partial least squares (PLS) regression and design of experiments was used to analyse and define the short‐ and long‐term loss mechanisms of carvacrol from polymer/clay nanocomposite films used as an antimicrobial food packaging material. Three formulation x variables were evaluated: type of nanoclay, concentration of nanoclay and concentration of compatibilizer agent. Five experimental y responses were measured: loss of carvacrol from the films stored at room conditions on the third and fifth days (short‐term loss); on the forty‐fifth and sixtieth days (long‐term loss); and the polymer/clay level of intercalation. All variables and responses were simultaneously calibrated by an ANOVA‐validated (P < 0.05) comprehensive PLS interaction model with a satisfactory R² = 0.956 and Q² = 0.505. A differentiation and discrimination of the mechanisms that control the loss of carvacrol over different periods of time was observed. Short‐term loss seems to be controlled mainly by direct molecular interactions in the interlayer space between carvacrol and the nanoclay organomodifier. Long‐term loss seems to be mainly controlled by diffusion within the nanocomposite matrix once carvacrol has left the nanoclay interlayer space, and it is closely related to the levels of intercalation reached in the nanocomposite. © 2016 Society of Chemical Industry