Sequential statistical design strategy in a kinetic study of propylene oxidation

A sequential design strategy was used to direct an experimental kinetic study of propylene oxidation over bismuth molybdate. Two statistical criteria were used to design experiments. The first is a joint criterion for the dual problem of model discrimination and parameter estimation which was proposed by Hill et al. (1968), but which to the best of our knowledge has not been tested in an experimental program. Using this joint criterion design, it was demonstrated that both the objectives of model discrimination and precise parameter estimation were effectively accomplished. After model discrimination had been achieved, a design strategy for further improvement in parameter precision was found, as expected, to be effective for that purpose.     

Using a statistical multiresponse method of experimental design in a reaction network study

A statistical design method for multiresponse models was used in an experimental study of a reaction network. The study involved the vanadia-catalysed oxidation of o-xylene using a recirculation reactor. Concentrations of reactants and temperature were used as operating variables, and the concentrations of products were the measured responses. The published design procedure was modified by conducting several runs after each design evaluation to reduce computing costs while retaining the benefits of statistical design and analysis. This enabled network analysis to be conducted efficiently to the limits of the equipment, and pointed the way to improved estimates of the parameters.     

Sequential experimental design based on multiobjective optimization procedures

Model-based sequential experimental designs are frequently applied for discrimination of rival models and/or estimation of precise model parameters. Although the development and use of a single design criterion to perform the simultaneous model discrimination and precise parameter estimation seem appealing, published material indicates that previous attempts to develop such a single design criterion have not been successful. Despite that, this problem has rarely been analyzed with the help of multiobjective optimization procedures. In this work, a multiobjective optimization method based on the particle swarm optimization procedure is used to build the Pareto fronts in experimental design problems where distinct design criteria used for discrimination of rival models and/or estimation of precise model parameters are considered simultaneously. It is shown through the rigorous analysis of the Pareto sets that both design objectives are frequently conflicting, which means that optimum discrimination of rival models and estimation of precise model parameters cannot be performed simultaneously in many cases. However, it is also shown that the use of the posterior covariance matrix of estimated model parameters for model discrimination makes the design of experiments for the simultaneous optimum model discrimination and estimation of model parameters possible in many experimental design problems.     

Development of rapid-set high-strength cement using statistical experimental design

Many applications like aircraft runway, busy roads, highway or motorways, water tank repair, etc. demand a cement that sets fast and gains the required strength in a few hours. Though there are few cements available to meet the requirements given above, most of them are very costly, like magnesium phosphate cement, jet cement, geopolymeric cement, etc. So, an attempt has been made to make cost-effective rapid-set high-strength cement having initial setting time of ∼15 min, final setting time of ∼30 min, 4 h cold compressive strength (CCS) of ∼12 MPa (minimum), 8 h CCS of ∼24 MPa and 1 day CCS of ∼40 MPa for the neat cement. The experiments were designed using orthogonal array technique in L₉ array with three factors, namely OPC/high-alumina cement/anhydrous calcium sulphate, fineness of the cement, and type of additives, at three levels each. The responses studied are initial setting time, final setting time, and CCS after 4, 8, and 24 h curing. The response data were analysed using analysis of variance (ANOVA) technique with a software package, ANOVA by Taguchi Method (ATM). In the case of setting time, fineness of the cement and OPC/HAC/anhydrous calcium sulphate ratio plays a significant role. Additive type and the OPC/HAC/anhydrous calcium sulphate are significant factors affecting the CCS at different ages. The confirmatory trial results clearly indicate that the setting time and CCS at different ages targeted were achieved using design of experiments.     

Sequential design of initial experiments for chemical kinetic modeling-isomerization of n-pentane

A procedure for the sequential design of experiments for model discrimination and precise parameter estimation is presented and applied to the published example of isomerization of n-pentane. The present paper outlines a method for designing the preliminary experiments which were chosen arbitrarily by the earlier workers. The procedure leads to selection of the correct model for all the cases considered here. Well-designed initial experiments improve the degree of precision in parameter estimation also.     

Sequential experimental design procedures for precise parameter estimation in ordinary differential equations

In this paper, it is shown how an experimental program for precise parameter estimation can be designed sequentially for the case that the mathematical model is given in the form of a set of ordinary differential equations. Two strategies are proposed. The first aims at minimizing the volume of the joint confidence region associated with the parameter estimates. The second attempts to alter as much as possible the shape towards a spherical region, by shortening the length of the longest principal axis of the confidence region to the maximum extent. The application of both criteria is illustrated by means of examples, representative for real problems in chemical reaction engineering. The techniques are easily applicable with our present day computing facilities. Qualitative indications are derived concerning the question when the use of an experimental design will result in an appreciable gain in significance for the parameter estimates.     

Sequential experimental design for model discrimination in the case of multiple responses

The sequential experimental design procedure for discriminating among single response rival models outlined in Chem. Engng Sci 1983 38, 225 is here extended to the case of multiresponse models. The method is illustrated with an example.     

Application of statistical experimental design in Ziegler–Natta polymerization of butadiene

The results of butadiene polymerization initiated with CoCl₂ · 4Py/Et₂AlCl/H₂O are presented. The effectiveness of statistical experimental design techniques is demonstrated in identifying the individual and joint effects of polymerization variables on rate constants, molecular weights, and polydispersity. The proposed conversion model, representing polymerization with instantaneous initiation and bimolecular termination, was found to provide an adequate representation of experimental conversion data. Evidence indicates that water is not only directly involved in the initiation reaction but also plays an important role in the chain transfer reaction.     

Application of an experimental design method to study the performance of electrochlorination cells

A study of the effects of some parameters on the active chlorine production from aqueous sodium chloride solutions in the hypochlorite electrochemical production cells was undertaken. These varying parameters included the anodic surface area (S_a), the ratio of anodic and cathodic surface areas () the inter-electrode gap, and the type of the cathode used. In addition, a study of the performance of some electrochemical cells that differ in the type of anodes (platinum-coated titanium, ruthenium-coated titanium, and graphite) was made. By means of the experimental design method employing a full factorial of2² the effects ofthe most influencing parameters, the set-up of optimum conditions, and the formation of optimal concentration of active chlorine were assessed. Under the following conditions, a concentration of as high as 65.67 g/L of active chlorine was gained: ruthenium-coated titanium anode (S_a = 24 cm²); titanium cathode, ; inter-electrode gap, 0.5 cm; current density, 35 A/dm² ; temperature, 20°C; concentration of NaCl aqueous solution, 3 M; time, 2 h.     

An example of rigid PVC foam formulating using a statistical experimental design approach

A study was undertaken to understand the effect of PVC molecular weight, modifier molecular weight, and modifier loading on die swell and foam density in a rigid PVC formulation. We used a statistical experimental design to separate the effects from the noise of normal experimentation. Die swell and foam density of extruded pieces were measured and analyzed in a commercially available software package. Analysis of the experimental data shows that the die swell response is linear over the experimental range, while the foam density response is curved. The mathematical model developed in the data analysis is then used to predict the performance responses when the experimental factors are changed.     

An experimental design strategy for distinguishing among rival mechanistic models an application to the catalytic hydrogenation of propylene

Box and Hill's method was applied to discriminate between two mechanistic models for the catalytic hydrogenation of propylene. A sequential experimental design procedure, this method makes use of a discrimination criterion obtained from information theory.     

Optimizing the synthesis and properties of Al-modified anatase catalyst supports by statistical experimental design

The important variables in the synthesis of stable, high surface area, aluminum-modified anatase TiO₂ catalyst supports were identified and optimized using statistically designed experiments (DOEs). The first DOE examined ten variables at two levels and a second DOE studied eight variables at three levels. Equations were developed to predict the conditions to obtain the highest surface area and pore volume at the desired pore diameter and predict the pore diameter range that may be obtained. Confirmation trials closely matched predicted surface areas, pore volumes, and pore diameters in all but one trial. Rinsing order (before or after calcination) was the most significant factor. Other important factors were calcination temperature, mol% aluminum, and water addition speed. The results of this study demonstrate (a) the power of DOEs in identifying and controlling synthesis variables in relatively few experiments and (b) how analysis of factor effects can provide insight into the formation mechanism.     

A joint model-based experimental design approach for the identification of kinetic models in continuous flow laboratory reactors

Continuous flow laboratory reactors are typically used for the development of kinetic models for catalytic reactions. Sequential model-based design of experiments (MBDoE) procedures have been proposed in literature where experiments are optimally designed for discriminating amongst candidate models or for improving the estimation of kinetic parameters. However, the effectiveness of these procedures is strongly affected by the initial model uncertainty, leading to suboptimal design solutions and higher number of experiments to be executed. A joint model-based design of experiments (j-MBDoE) technique, based on multi-objective optimization, is proposed in this paper for the simultaneous solution of the dual problem of discriminating among competitive kinetic models and improving the estimation of the model parameters. The effectiveness of the proposed design methodology is tested and discussed through a simulated case study for the identification of kinetic models of methanol oxidation over silver catalyst.     

Protein ultrafiltration: an experimental study

An experimental study was carried out on the ultrafiltration (UF) of protein-containing solutions under different conditions, as compared with a solution of a linear synthetic polymer. Three different fluidynamic regimes were investigated, namely, unstirred batch system, stirred batch system, and recirculating system. The results obtained substantially agree with the predictions of the gel polarization model. A significant effect of the electrolytes on the UF flux has been observed, which can be attributed to solute–solute interaction. The influence of different UF membranes has also been investigated.     

Quantifying segregation in heaps: an experimental study

The phenomena of segregation of nonhomogeneous particulate materials is a troublesome and common occurrence in the process industries. The work presented is a report of an experimental study of segregation in heap formation together with a statistical analysis of the results which compares several methods of quantifying the segregation that occurs. Of the nine quantifiers investigated, one method based on the work of Rollins et al. gave an indication of “structure” in the sample variance.     

Rheometric study of chitosan/activated carbon composite hydrogels for medical applications using an experimental design

Composite chitosan/activated carbon hydrogels were prepared with the vapor‐induced phase separation process. A rheometric study was performed with a factorial fractional design to determine the formulation and process parameters significantly influencing the mechanical properties of the gels. The results revealed that three factors played a key role in the storage modulus of the gels. According to the model, these factors could be classified with respect to their relative influence on the storage modulus in the following descending order: chitosan concentration > gel time of exposure to ammonia vapors > temperature of the reactor. Increasing these parameters led to an increase in the physical crosslinking density within the matrices and resulted in a reinforcement of the mechanical properties of the hydrogels. Two interactions were also shown to be significant and promoted the formation of supplementary junction zones within the matrix: the first one corresponded to the interaction between the chitosan concentration and the exposure time to ammonia vapors, and the second one concerned the interaction between the chitosan concentration and the temperature of the reactor. A second‐order model was obtained from statistical analysis. Because of the determination coefficient (89.4%) and the P value related to the lack of adjustment of the model (0.043), which was associated with a 95% confidence level, this model could be considered to be of good quality. Three gels were used to validate the model, and good accuracy was obtained. The maximum elastic modulus was obtained with the highest chitosan concentration [4% (w/v)], the highest temperature in the gelation chamber (50°C), and the longest time of exposure to ammonia vapors (24 h). © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

Second-order statistical regression and conditioning of replicate transient kinetic data

A new method has been developed to estimate physico-chemical parameters from transient kinetic data: second-order statistical regression (SOSR). It allows to account for heteroskedasticity and nonwhiteness of the noise in the time series measured. SOSR makes use of replicates to estimate the second-order statistics, i.e. the autocovariance pattern of the noise. A sample principal noise component analysis of the experimental time series allows nonlinear least-squares (NLSQ) regression of the latter. The method has been validated by regression of artificially generated experimental data and the results have been compared with those obtained with direct NLSQ regression. The SOSR has also been applied to the irreversible adsorption of oxygen on a reduced vanadia/silica catalyst and the interaction of propane with a copper/ceria catalyst, as studied with a temporal analysis of products (TAP) setup. In general, compared with those obtained with direct NLSQ regression, the parameter estimates and their confidence intervals are more accurate.