Jan T. Duda

A comparison of theoretical and empirical expansion of coals in the high pressure sorption of methane

Abstract Empirical data on coal expansion due to high pressure sorption of methane were compared with the linear expansion ratio predicted on the basis of the theoretical dual sorption model. The model is based on a thermodynamical treatment of the sorption process, viewed as consisting of absorption and adsorption subprocesses, with a submicroporous and heterogeneous copolymeric nature of coal being assumed. While regarding the expansion of coal as a result of the absorption subprocess only, this model may be used to evaluate the theoretical linear expansion ratio. Three coal-methane systems have been examined. Coals with different degrees of metamorphism (wt% C (daf): 80.9, 86.5, 92.4) were used. Sorption and expansion measurements were carried out at 298 K, over the pressure range 0.5–4.5 MPa. Very good agreement of theoretical and empirical values was found in the lower pressure range. Differences observed in the higher pressure range suggest compression of pores was significant.

Optimizing control of uncertain plants with constrained feedback controlled outputs

On-line set-point optimization of technological processes under uncertainty is the subject of the paper for the case with important active constraints on certain process outputs. A miltilayer structure with additional upper layer follow-up constraint controller responsible for keeping the output constraints satisfied both in steady states and during transient processes is considered. The integrated system optimization and parameter estimation (ISOPE) method is investigated, i.e. iterative algorithms using an on-line steady-state feedback, designed to determine optimum operation of the plant in the presence of inaccurate models and disturbance uncertainties. A new formulation of the ISOPE method suitable for the considered structure is derived and discussed, followed by a development of an efficient dual-type algorithm. The algorithms are applied to a case study examplean industrial styrene production unit consisting of a series of distillation columns. Results of simulation runs for cases with and without errors in the feedback information from the process to the optimizer are reported.

New BET-like models for heterogeneous adsorption in microporous adsorbents

Abstract The paper presents a package of isotherm equations for heterogeneous adsorption aimed at the analysis of pore structure of sub- and microporous materials. One considers adsorption of small nearly spherical molecules in irregular pores of molecular size. The generalized BET theory is exploited respecting restrictions for multilayer adsorption (LBET approach). The model is based on thermodynamic relationships expressing changes of internal energy and configurational entropy due to the process. The adsorption energy is evaluated by using the Berthelot rule, and corrected with a factor Za representing a fraction of effective contacts enabling full adsorbent–adsorbate interaction. Side adsorbate–adsorbate interactions are neglected and constrained multilayer adsorption is considered. One assumes the values for Za to be uniformly distributed over the first layer adsorption sites within a range depending on the pore size. New models make it possible to obtain information on structure of pores and adsorption mechanisms on the basis of adsorption isotherms of small molecule adsorbates. Exemplary results of new models application for adsorption of CO2 and CH4 in an activated carbon are discussed.

A theoretical model for evaluation of configurational entropy of mixing with respect to shape and size of particles

A general method of evaluation of configurational entropy of a liquid mixture is presented. It is based on a generalized lattice model with no restrictions due to particle shape being introduced. A general formula for the entropy is derived. Achieved results open a way for a rigorous analysis of particle shape effect on mixing process. As an example, a new formula for the entropy of mixing of hard spheres in continuous space is derived which may respect a physical bound for packing ratio. A systematic approach to improve the model accuracy is proposed. The resultant alternative models are discussed in details. A comparison with literature data and the Mansoori-Carnahan-Starling formula is presented. Very good agreement is shown.

A non-BET adsorption and its LgBET model☆

The paper proposes new mathematical tools for interpretation of adsorption isotherms at relative pressures ranging to 0.95. Adsorption of small (nearly spherical) molecules in microporous adsorbents containing hole-like micropores is considered. In such materials formation of adsorbate clusters more compact than BET stacks is possible that leads to non-BET configurational effects. Additionally, geometrical and energetic constraints for the cluster size may be of significance. The paper shows that such effects can be handled with a rigorous mathematical model based on the BET approach. The derived general formulas may be useful in theoretical studies of effects of pores geometry on the adsorption isotherm shape. The neglecting of the geometrical constraints leads to a BET-like analytical formula (gBET equation). Moreover, an approximating formula is proposed (LgBET equation), taking into account the cluster size constraints represented by an exponential distribution function. The gBET and LgBET models may be used to draw semi-quantitative information on the shape of pores by the examination of adsorption isotherms at high relative pressures (ranging to 0.9). They provide also more reliable information on the adsorbent surface area and adsorption energy than the BET equation. Exemplary results of the new model application to analysis of N2 adsorption on activated carbons are discussed.

High-Accuracy PVT Relationships for Compressed Fluids and Their Application to BET-like Modelling of CO2 and CH4 Adsorption

This paper presents a set of high-accuracy formulae enabling the evaluation of the compression factor, activity and cohesion energy of fluids at near-critical and super-critical temperatures, with reduced molar volumes ranging from 0.4. Such data are necessary in BET-like theoretical modelling of adsorption process. The proposed fluid state equation (PVT relationship) combines a theoretical description of the fluid entropy (based on a hard-sphere model) with cohesion energy relationships obtained via high-accuracy approximations of universal compression factor data and vapour-liquid equilibria. The resultant relationships are incorporated into a BET-like description of adsorption in materials of irregular microporous structure (LBET model). The set of formulae gathered together in this paper allows the possible effective calculation of adsorption isotherms at near- and super-critical temperatures relative to the pore structures. A multi-variant fitting of the LBET model to the empirical data is proposed to detect active constraints for multilayer adsorption and, hence, to obtain information on the structure of the pores. Application of the formulae to the analysis of methane and carbon dioxide adsorption onto an active carbon is discussed.

A geometrical model of random porous structures to adsorption calculations

This paper describes a new approach to identification of random porous structures (e.g. present in cheap natural adsorbents, active carbons). It comes from a clustering based description of adsorption process assuming an exponential distribution of adsorbate stack size (the LBET model), combined with the new consistent mathematical relationships between the pore geometry, adsorption isotherm parameters and physical properties of adsorptive. The newly derived formulae are discussed, and results of their application to analysis of an active carbon structure are shown.

Enhancements of Moving Trend Based Filters Aimed at Time Series Prediction

One of the techniques recommended for calculating the nonparametric trend (nonstationary, low-frequency time series component) is the moving trend based smoothing [10], [3], [4] typically based on linear Least Square (LS) approximates of the series in a moving window. Such algorithm splits input time series into three sections, starting, central, final ones. The procedures used in each section may be viewed as Moving Trend based Filters (MTF). In the paper MTFs properties in frequency domain are considered, from seasonal time series decomposition, smoothing and prediction efficiency perspectives. A number of MTFs enhancements is proposed, involving different approximating polynomials and final section signal corrections. In particular, a compression of the final section MTF output signal is proposed to reduce the filter delay, and then reconstruction of the missing signal by a special multiple LS approximation. It improves significantly the final section signal shape evaluation, which are essential for further prediction purposes.

A qualitative approach to adsorption mechanism identification on microporous carbonaceous surfaces

The paper presents results of research on identification of localized and other adsorption mechanisms, on geometrically heterogeneous graphite-like carbonaceous surfaces. It attempts to get an insight into properties of individual adsorptive molecule movement near attractive adsorption sites, arising from adsorbent surface geometrical heterogeneities. In particular, a shape and volume of space occupied by the continuously moving molecule mass center are investigated. To this aim, kinematic equilibrium of the particle moving near a hypothetical microporous carbonaceous adsorbent wall is considered, and then compared with thermodynamic equilibrium. The proposed approach enables to examine effects of certain surface geometry on the shape and volume of space occupied by adsorbed particles, and so to outline temperature conditions for the localized adsorption mechanism predomination. Thus, it provides a cognitive basis to answer the question, what particular mechanism (localized or other—e.g. mobile) should be assumed for a class of adsorption systems in order to select the most appropriate mathematical adsorption model. Hence, it makes it possible for more reliable examination of real porous structures, based on adsorption measurements.

Effect of adsorbate properties on adsorption mechanisms: computational study

Abstract The paper presents results of research on identification of localized and mobile adsorption mechanisms of several selected adsorbates, on geometrically heterogeneous graphite-like carbonaceous surface. The proposed approach is intended to examine effects of surface geometrical heterogeneity on shape and volume of space occupied by the selected adsorbate molecules. In particular, the continuously moving individual molecule mass center properties, near adsorbent surface are investigated. When compared to the corresponding liquid phase properties it enables to outline the conditions for localized and mobile adsorption mechanisms. To this aim, the kinematic and thermodynamic equilibrium conditions are taken under the study, providing information on particular mechanism predomination. Therefore, the approach gives a cognitive basis for selection of the most appropriate mathematical adsorption model to reliable examination of material porous structure (comprising similar geometrical heterogeneity). Numerous simulation results for selected adsorbates H2, CO2, CH3OH and C6H6 are presented, and adsorption mechanism identification is discussed.