Investigation of gas transport through porous membranes based on nonlinear frequency response analysis

Theoretical development of a new experimental method for investigation of mass transport in porous membranes, based on the principle of the modified Wicke-Kallenbach diffusion cell and the nonlinear frequency response analysis is presented. The method is developed to analyze the transport of a binary gas mixture in a porous membrane. The mixture is assumed to consist of one adsorbable and one inert component. Complex mass transfer mechanism in the membrane, where bulk or transition diffusion in the pore volume and surface diffusion take place in parallel, is assumed. Starting from the basic mathematical model equations and following a rather standardized procedure, the frequency response functions (FRFs) up to the second order are derived. Based on the derived FRFs, correlations between some characteristic features of these functions on one side, and the whole set of equilibrium and transport parameters of the system, on the other, are established. As the FRFs can be estimated directly from different harmonics of the measured outputs, these correlations give a complete theoretical basis for the proposed experimental method. The method is illustrated by quantifying the transport of helium (inert gas) and C₃H₈ and CO₂ (adsorbable gases) through a porous Vycor glass membrane.     

Estimation of single solute adsorption isotherms applying the nonlinear frequency response method using non-optimal frequencies

In order to estimate single solute adsorption isotherms, the nonlinear frequency response (FR) of a chromatographic column is analyzed experimentally and evaluated using the concept of higher order frequency response functions (FRFs) based on the Volterra series and generalized Fourier transform. In this case study, it has been investigated the adsorption of ethyl benzoate on octadecyl silica from a mixture of methanol and water (60:40) as a solvent. Experiments are performed using a standard gradient HPLC unit. For estimation of adsorption isotherms by the nonlinear FR method the column inlet concentration is changed in a nearly sine waveform around several steady-state concentrations. Using this method the first three local derivatives of a single solute adsorption isotherm are estimated from the low frequency asymptotes of the corresponding functions, i.e. the phase and first order derivative of the FRFs. For an accurate estimation of isotherm coefficients periodical experiments should be preformed for frequencies below a certain critical frequency. This is the frequency needed for approaching the low frequency asymptotic behaviour of the corresponding functions close enough, so that errors due to the non-feasibility of experiments with zero frequency can be neglected. Unfortunately, depending on the properties of the system, it can happen (as for the system investigated here) that experiments for the critical frequency would be too long and cannot be realized. In order to study the loss of accuracy of the nonlinear FR method, when it is applied for non-optimal frequencies, experiments are performed for frequencies approximately one order of magnitude higher than the critical frequency required to evaluate the FRF phases. The obtained isotherm model coefficients are compared with the ones estimated using conventional frontal analysis as a reference method. The isotherms determined by two methods are similar, however a closer look reveals that peaks predicted under overloading conditions differ.     

Nonlinear FR-ZLC method for investigation of adsorption equilibrium and kinetics

A new method for investigation of adsorption equilibrium and kinetics, named Nonlinear Frequency Response-Zero Length Column (NFR-ZLC) method, is introduced. It combines the advantages of the Nonlinear FR method (the potential to identify a model corresponding to the most probable kinetic mechanism and to estimate the equilibrium and kinetic parameters of the identified model) and of the ZLC method (the potential to derive direct information about the processes on the particle level, by eliminating the influence of the adsorber). The frequency response functions of a ZLC system, up to the third order, and for three simple kinetic mechanisms (film resistance control, micropore diffusion control and pore-surface diffusion control) are derived and simulated. The procedure for estimation of the equilibrium and kinetic parameters is defined and illustrated based on numerical simulations.     

Determination of competitive adsorption isotherms applying the nonlinear frequency response method: Part II. Experimental demonstration

This paper demonstrates an experimental application of the nonlinear frequency response (FR) method extension to determine adsorption isotherms of binary mixtures. This method, based on the analysis of the response of a chromatographic column subjected to the sinusoidal inlet concentration changes, is shown to be an alternative for isotherm determination. The critical issue related to the successful application of the method is to reach experimentally the low frequency asymptotic behaviour of the corresponding frequency response functions (FRFs). Although, there are different possibilities to perform periodical inlet concentration changes, in this paper only simultaneous changes for both components were considered. The adsorption of phenol and 2-phenylethanol on octadecyl silica was analyzed experimentally using a mixture of methanol and water as a solvent. Parameters of competitive isotherms were also estimated for comparison using the classical perturbation method. Despite certain differences between competitive isotherms estimated with the two methods that were found, the obtained results show the potential of the nonlinear FR method for measuring competitive isotherms.     

Kinetic models for solution imidization of polyamic acid containing naphthalene‐pendant group

The kinetics and mechanisms of the solution imidization of polyamic acid resulting from a diamine, bis(4‐aminophenoxy‐3,5‐dimethylphenyl)naphthylmethane, and a dianhydride, 3,3′4,4′‐diphenylsulfonetetracarboxylic dianhydride, were studied at three various temperatures (145, 165, and 180 °C). The results were confirmed by means of ¹H NMR and gel permeation chromatography (GPC). Kinetic parameters were obtained by an isothermal study, and the results were quite close to second‐order kinetics for the homogeneous solution imidization. In addition, Carother's equation, Mark–Houwink theory, and GPC were used to explain the molecular weight of the imidization processes. The apparent activation energy (E_a) was 104 KJ/mol, and the pre‐exponential factor (k₀) was 3.48 × 10¹⁴. The proposed kinetic mechanism is in good agreement with the kinetic models. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 4139–4151, 2001     

Determination of competitive adsorption isotherms applying the nonlinear frequency response method: Part I. Theoretical analysis

In this work adsorption equilibria of binary mixtures are quantified analyzing the nonlinear frequency response of a chromatographic column. Local partial derivatives of an isotherm model can be estimated for certain steady-states from the low frequency asymptotes of the corresponding frequency response functions (FRFs). The required FRFs correspond to two different compounds and the type of the imposed inlet concentration changes, e.g. periodical inlet concentration changes of only one compound or of both of them. For an accurate determination of isotherm parameters, it is necessary to approach as close as possible the low frequency asymptotic behaviour of these functions. Based on principles valid for the FRFs corresponding to the adsorption of a single solute, frequencies needed to reach the low frequency asymptotes of the functions of interest for estimating competitive isotherms are defined in this paper. The relation between the accuracy of the isotherm parameters determined and numbers and types of periodical inlet concentration changes and steady-states analyzed is also evaluated.     

Kinetics of non-isothermal decomposition of three IRGANOX-type antioxidants

In the present work a comparative kinetic study was performed on the thermal behavior of three antioxidants of IRGANOX-type (L101, L109 and L115) in dynamic air atmosphere under non-isothermal conditions. The TG-DTG data were obtained at heating rates of 5, 7, 10 and 15 K min⁻¹. The kinetic parameters were obtained by processing these data with strategies corresponding to Flynn-Wall-Ozava (FWO), Friedman (FR), Budrugeac-Segal (BS) and non-parametric kinetic (NPK) methods. The thermal degradation by all the three compounds take place in melted state, so that any kinetic models regarding the decomposition of solids are inapplicable. Only with the NPK method it was possible a separation between the two functions of the reaction rate. For the temperature dependence, f(T), an Arrhenius-type model was searched; for the conversion dependence, the Ŝestak-Berggren equation was suggested in order to discriminate between physical (m) and chemical (n or p) steps of a complex thermodegradation process.     

Nonisothermal decomposition kinetics of [CoC<Subscript>2</Subscript>O<Subscript>4</Subscript>·2.5H<Subscript>2</Subscript>O]<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The authors present their results concerning the decomposition in air of the homopolynuclear coordination compound [CoC₂O₄·2.5H₂O] _n . In the temperature range 20–300 °C, the heating curves TG, DTG and DTA allowed to evidence three decomposition steps. The kinetic analysis was performed on the second step which proved to be the only workable one. The application of nonlinear regression procedure shows that this is a complex process consisting in three successive steps. The checking of the mechanism and corresponding kinetic parameters for quasi-isothermal data (T = 150 °C) shows that the obtained results could be used for prediction of the thermal behaviour of the investigated compound in both isothermal and non-isothermal conditions.     

Comparison of Dinitrogen,Methane, Carbon Monoxide,and Carbon Dioxide Mass‐Transport Dynamics in Carbon and Zeolite Molecular Sieves

Equilibrium (based on Henry constants) and kinetic (based on relaxation‐time constants or rather macropore transport diffusivities) selectivities for commercial zeolite and carbon‐molecular‐sieve (CMS) adsorbents were compared. Adsorption isotherms were recorded at ?20°. The frequency‐response (FR) sorption‐rate spectra were determined in the range of ?78 and 70° at 133 Pa. In particles of a larger size than 1.0 mm, macropore diffusion governed the rate of sorption mass transport in both types of microporous materials. The differences in the intercrystalline diffusivities established the kinetic separation of the gases notwithstanding the essential importance of interactions in the micropores. Zeolites seem to be more advantageous for a dynamic separation of CO₂ and CH₄ than CMS 4A. With the CO₂ and CO pair, the CMS is characterized by short characteristic times which, together with a good separation factor, is a double advantage in a short‐cycle adsorption technology. Upon comminution of the carbon pellets, intercrystalline‐diffusion resistance can be completely removed by using CMS 4A adsorbent particles with a diameter smaller than 1 mm. The carbonization of spruce‐wood cubes resulted in an excellent carbon honeycomb structure, which seems to be ideal from a dynamic point of view for applications in short‐cycle adsorption‐separation technologies. In the development of adsorbents, the use of the FR method can be beneficial.     

Equivalency of Kinetic Schemes: Causes and an Analysis of Some Model Fitting Algorithms

The same experimental data can often be equally well described by multiple mathematically equivalent kinetic schemes. In the present work, we investigate several model‐fitting algorithms and their ability to distinguish between mechanisms and derive the correct kinetic parameters for several different reaction classes involving consecutive reactions. We have conducted numerical experiments using synthetic experimental data for six classes of consecutive reactions involving different combinations of first‐ and second‐order processes. The synthetic data mimic time‐dependent absorption data as would be obtained from spectroscopic investigations of chemical kinetic processes. The connections between mathematically equivalent solutions are investigated, and analytical expressions describing these connections are derived. Ten optimization algorithms based on nonlinear least squares methods are compared in terms of their computational cost and frequency of convergence to global solutions. Performance is discussed, and a preferred method is recommended. A response surface visualization technique of projecting five‐dimensional data onto the three‐dimensional search space of the minimal function values is developed.     

Transport of Carbamazepine,Ciprofloxacin and Sulfamethoxazole in Activated Carbon: Solubility and Relationships between Structure and Diffusional Parameters

The transport of carbamazepine, ciprofloxacin and sulfamethoxazole in the different pores of activated carbon in an aqueous solution is a dynamic process that is entirely dependent on the intrinsic parameters of these molecules and of the adsorbent. The macroscopic processes that take place are analyzed by interfacial diffusion and reaction models. Modeling of the experimental kinetic curves obtained following batch treatment of each solute at 2 µg/L in tap water showed (i) that the transport and sorption rates were controlled by external diffusion and intraparticle diffusion and (ii) that the effective diffusion coefficient for each solute, with the surface and pore diffusion coefficients, were linked by a linear relationship. A statistical analysis of the experimental data established correlations between the diffusional parameters and some geometrical parameters of these three molecules. Given the major discontinuities observed in the adsorption kinetics, the modeling of the experimental data required the use of traditional kinetic models, as well as a new kinetic model composed of the pseudo first or second order model and a sigmoidal expression. The predictions of this model were excellent. The solubility of each molecule below 60 °C was formulated by an empirical expression.     

Approximate Analytical Expressions for the Steady‐State Concentration of Substrate and Cosubstrate over Amperometric Biosensors for Different Enzyme Kinetics

Mathematical models of amperometric biosensors at three basic types of enzyme kinetics in nonstationary diffusion conditions are discussed. The models are based on nonstationary diffusion equations containing a linear term related to the first‐order and nonlinear term related to the Michaelis–Menten and ping–pong of the enzymatic reaction mechanism. In this paper, we obtain approximate closed‐form analytical solutions for the nonlinear equations under steady‐state condition by using the homotopy analysis method. Analytical expressions for concentrations of substrate and cosubstrate and corresponding current response have been derived for all possible values of parameters. Furthermore, in this work, the numerical simulation of the problem is also reported using Scilab/MATLAB program. An agreement between analytical and numerical results is noted.     

Polynomial multivariate least‐squares regression for modeling nonlinear data applied to in‐depth characterization of chromatographic resolution

Well‐established, linear multivariate calibration methods such as multivariate least‐squares regression (MLR), principal component regression (PCR), or partial least squares (PLS) have two limitations: (i) measured data must be linearly related to the response variables and (ii) predictor variables x_{n = 1, …, N} cannot be coupled to each other. For evaluation of nonlinear data, however, these restrictions need to be overcome and thus polynomial multivariate least‐squares regression (PMLR or “response surfaces”) has been introduced here. PMLR is based on multivariate least squares but incorporates all combinations of predictor variables up to a user‐selected polynomial order (e.g., including u or v = 0). Because of the inclusion of such coupled terms and their powers, PMLR models are better adapted to model nonlinear data and can help to enhance the prediction step's accuracy and precision. PMLR has been based on MLR because it facilitates—unlike PCR or PLS—a physical and chemical interpretation of the predictors. Hence, the origins and the relevance of nonlinear and/or coupled predictors can be investigated. The details of the PMLR algorithm and its implementation are presented along with a method for model optimization utilizing gradients of response surfaces. Newly developed PMLR models up to quintic order have been applied to predict a chromatograph's peak resolution as a function of six‐instrument parameters. It has been demonstrated that PMLR is better capable than MLR and PCR to describe these nonlinear and coupled instrument parameters. In addition, the novel software tool has been utilized for model optimization to determine instrument parameters, which result in the best chromatographic resolution. Copyright © 2011 John Wiley & Sons, Ltd.     

Kinetic analysis of thermal decomposition in liquid and solid state of 3-nitro and 4-nitro-benzaldehyde-2,4-dinitrophenylhydrazones

A kinetic study on the thermal decomposition of 3-nitro and 4-nitro-benzaldehyde-2,4-dinitrophenylhydrazones was carried out. The isothermal and dynamic differential scanning calorimetric curves were recorded for solids and melts, respectively. The standard isoconversional analysis of the obtained curves from both isothermal and nonisothermal analysis suggests an autocatalytic decomposition mechanism. This mechanism is also supported by the temperature dependence of the observed induction periods. Based on the results of the model-free method from nonisothermal regime, the kinetic model was derived and the kinetic parameters were obtained by means of a multivariate nonlinear regression.     

Mechanisms of Molecular Mobility of Oxygen and Nitrogen in Carbon Molecular Sieves

Molecular mobility of oxygen, O₂, and nitrogen, N₂, in Carbon Molecular Sieves, CMS, was investigated using the Frequency Response, FR, technique to identify mass-transfer mechanisms and related kinetic time constants. The FR data showed that O₂ mobility in four types of CMS was dominantly controlled by surmounting surface-barrier resistances, whereas the mobility of both O₂ and N₂ in pellets of a fifth CMS type obeyed the Fickian diffusion model. Temperature and pressure dependences of surface-barrier penetration time constants were obtained for O₂ and N₂ in several of those CMS materials. The kinetic time constants of surface-barrier penetration were related to Langmuir-type rate constants, which indicates that kinetic behavior of O₂ therein could also be interpreted in terms of a Langmuir-kinetics equation.     

Removal of Thiram from Aqueous Solutions

In this study activated carbon was used for the removal of thiram from aqueous solutions. Adsorption experiments were carried out as a function of time, initial thiram concentration and temperature. Equilibrium data fitted well to the Freundlich and Langmuir equilibrium models in the studied concentration range. Adsorption kinetics followed a pseudo second‐order kinetic model rather than pseudo first‐order model. The results from kinetic experiments were used to describe the adsorption mechanism. Both boundary layer and intraparticle diffusion played important role in the adsorption mechanism of thiram. Thermodynamic parameters (ΔG₀, ΔH₀, and ΔS₀) were determined and the adsorption process was found to be an endothermic one. The negative values of ΔG⁰ at different temperatures were indicative of the spontaneity of the adsorption process.     

Significance of Non-isothermal Kinetic Data. A statistical study

Arrhenius parameters values, in non-isothermal kinetic vaporisation processes for a series of compounds with related structures, have been calculated. This was made using a method of calculation that allows to find the most probable vaporisation mechanisms. According to this method DTG curves were compared with some theoretical ones reported in literature, whose shape results to be only a function of the mechanisms. In this way the choice of the mathematical functions which can be inserted in the kinetic equations, was influenced by the shape of the DTG plots and other thermal analysis signals thus allowing to choose the most probable mechanisms. The kinetic parameters derived from these mechanisms were compared, using statistical analysis, with those obtained from another method of calculation based on ‘a priori’ vaporisation mechanism chosen for the investigated liquid–gas transition. The standard deviations of the slope and of the intercept, together with the standard deviation and the square correlation coefficient (r ²) of the linear regression equations related to the mechanisms of the two methods were calculated. Student t-test, Fisher F-test, confidence intervals (c.i.) and residuals valueswere also given. Statistical analysis shows that the mechanisms obtained with the former method (diffusive and geometrical models) and the related Arrhenius parameters result to be more significant (in terms of probability) than the corresponding quantities of the latter for which a first-order model was chosen. This revised version was published online in August 2006 with corrections to the Cover Date.     

Curing reaction of o-cresol novolac epoxy resin according to hardener change

The investigations of cure kinetics and glass transition temperature (T_g) versus reaction conversion (α) of o-cresol novolac epoxy resin with the change of hardener were performed. All kinetic parameters of the curing reaction such as the reaction rate order, activation energy, and frequency factor were calculated. The curing mechanisms were classified into two types. One was an autocatalytic mechanism and the other was a nth order kinetic mechanism. The constants related to the chain mobility of polymer segments were obtained by using the DiBenedetto equation. We have tried to correlate the relationships between curing mechanism and molecular structures of hardeners from these results. © 1993 John Wiley & Sons, Inc.