Computationally efficient solution techniques for adsorption problems involving steep gradients in bidisperse particles

A piecewise uniform fitted mesh method turns out to be sufficient for the solution of a surprisingly wide variety of singularly perturbed problems involving steep gradients. The technique is applied to a model of adsorption in bidisperse solids for which two fitted mesh techniques, a fitted-mesh finite difference method (FMFDM) and fitted mesh collocation method (FMCM) are presented. A combination (FMCMD) of FMCM and the DASSL integration package is found to be most effective in solving the problems. Numerical solutions (FMFDM and FMCMD) were found to match the analytical solution when the adsorption isotherm is linear, even under conditions involving steep gradients for which global collocation fails. In particular, FMCMD is highly efficient for macropore diffusion control or micropore diffusion control. These techniques are simple and there is no limit on the range of the parameters. The techniques can be applied to a variety of adsorption and desorption problems in bidisperse solids with non-linear isotherm and for arbitrary particle geometry.

Adsorption in mesopores: A molecular-continuum model with application to MCM-41

The classical model of surface layering followed by capillary condensation during adsorption in mesopores, is modified here by consideration of the adsorbate solid interaction potential. The new theory accurately predicts the capillary coexistence curve as well as pore criticality, matching that predicted by density functional theory. The model also satisfactorily predicts the isotherm for nitrogen adsorption at 77.4 K on MCM-41 material of various pore sizes, synthesized and characterized in our laboratory, including the multilayer region, using only data on the variation of condensation pressures with pore diameter. The results indicate a minimum mesopore diameter for the surface layering model to hold as 14.1 Å, below which size micropore filling must occur, and a minimum pore diameter for mechanical stability of the hemispherical meniscus during desorption as 34.2 Å. For pores in-between these two sizes reversible condensation is predicted to occur, in accord with the experimental data for nitrogen adsorption on MCM-41 at 77.4 K.

Enzyme deactivation in fixed bed reactors with michaelis-menten kinetics

Rate expression for enzyme poisoning which are consistent with a Michaelis-Menten main reaction are used to analyze the performance of a fixed bed reactor containing immobilized enzyme. When enzyme deactivation results from the irreversible bonding of a product molecule to an existing substrate-enzyme complex, it is shown that minimum enzyme activity can occur in the interior of the bed, well away from the ends. This suggests that bed sectioning techniques may enable direct evaluation of fundamental poisoning mechanisms.

Fixed bed reactors with catalyst poisoning - first order kinetics.

The long performance of an isothermal fixed bed reactor undergoing catalyst poisoning is theoretically analyzed using the dispersion model. First order reaction with dth order deactivation is assumed and the model equations are solved by matched asymptotic expansions for large Peclet number. Simple closed-form solutions, uniformly valid in time, are obtained.

A finite integral transform technique for solving the diffusion-reaction equation with Michaelis-Menten kinetics

An approximate analytical technique employing a finite integral transform is developed to solve the reaction diffusion problem with Michaelis-Menten kinetics in a solid of general shape. A simple infinite series solution for the substrate concentration is obtained as a function of the Thiele modulus, modified Sherwood number, and Michaelis constant. An iteration scheme is developed to bring the approximate solution closer to the exact solution. Comparison with the known exact solutions for slab geometry (quadrature) and numerically exact solutions for spherical geometry (orthogonal collocation) shows excellent agreement for all values of the Thiele modulus and Michaelis constant.

Substrate-inhibited kinetics with catalyst deactivation in an isothermal CSTR

Analytical expressions are developed for the time-dependent reactant concentration and catalyst activity in an isothermal CSTR with Langmuir-Hinshelwood kinetics of deactivation and reaction. Several parallel and series posioning mechanisms are considered for a reactor which, without poisoning, would operate at a unique steady state. The use of matched asymptotic expansions and abandonment of the usual initial-steady-state assumption give results, valid from startup to final loss of activity, whose accuracy can be improved systematically.

Substrate-inhibited kinetics with catalyst deactivation in an isothermal CSTR. II. Multiple pseudosteady states and reactor failure

Analytical expressions are derived for the time and magnitude of failure of an isothermal CSTR with substrate-inhibited kinetics, caused by slow catalyst deactivation under three types of parallel and series mechanisms. Reactors operating at high space velocity are found to be most susceptible to early failure and poisoning by product is more dangerous than by reactant. The magnitude of the jump across steady states depends solely on the Langmuir-Hinshelwood kinetic parameters and a detailed analysis of reactor behavior during the jump itself is given.

Gypsum solubility in seawater, and its application to bauxite residue amelioration

The solubilities and dissolution rates of three gypsum sources (analytical grade (AG), phosphogypsum (PG) and mined gypsum (MG)) with six MG size fractions ((mm) > 2.0, 1.0-2.0, 0.5-1.0, 0.25-0.5, 0.125-0.25, and < 0.125) were investigated in triple deionised water (TDI) and seawater to examine their suitability for bauxite residue amelioration. Gypsum solubility was greater in seawater (3.8 g L 1) than TDI (2.9 g L 1) due to the ionic strength effect, with dissolution in both TDI and seawater following first order kinetics. Dissolution rate constants varied with gypsum source (AR > PG > MG) due to reactivity and surface area differences, with 1:20 gypsum:solution suspensions reaching saturation within 15 s (AR) to 30 min (MG > 2.0). The ability of bauxite residue to adsorb Ca from solution was also examined. The quantity of the total solution Ca adsorbed was found to be small (5 %). These low rates of solution Ca adsorption combined with the comparatively rapid dissolution rates preclude the application of gypsum to the residue sand/seawater slurry as a method for residue amelioration. Instead, direct field application to the residue would ensure more efficient gypsum use. In addition, the formation of a sparingly soluble CaCO3 coating around the gypsum particles after mixing in a highly alkaline seawater/supernatant liquor (SNL) solution greatly reduced the rate of gypsum dissolution.

The kinetics of NO and N2O reduction over coal chars in fluidised-bed combustion

This paper presents a comprehensive and critical review of the mechanisms and kinetics of NO and N2O reduction reaction with coal chars under fluidised-bed combustion conditions (FBC). The heterogeneous reactions of NO and N2O with char/carbon surface have been well recognised as the most important processes in reducing both NOx and N2O in situ FBC. Compared to NO-carbon reactions in FBC, the reactions of N2O with chars have been relatively less understood and studied. Beginning with the overall reaction schemes for both NO and N2O reduction, the paper extensively discusses the reaction mechanisms including the effects of active surface sites. Generally, NO- and N2O-carbon reactions follow a series of step reactions. However, questions remain concerning the role of adsorbed phases of NO and N2O, and the behaviour of different surface sites. Important kinetics factors such as the rate expressions, kinetics parameters as well as the effects of surface area and pore structure are discussed in detail. The main factors influencing the reduction of NO and N2O in FBC conditions are the chemical and physical properties of chars, and the operating parameters of FBC such as temperature, presence of CO, O-2 and pressure. It is shown that under similar conditions, N2O is more readily reduced on the char surface than NO. Temperature was found to be a very important parameter in both NO and N2O reduction. It is generally agreed that both NO- and N2O-carbon reactions follow first-order reaction kinetics with respect to the NO and N2O concentrations. The kinetic parameters for NO and N2O reduction largely depend on the pore structure of chars. The correlation between the char surface area and the reactivities of NO/N2O-char reactions is considered to be of great importance to the determination of the reaction kinetics. The rate of NO reduction by chars is strongly enhanced by the presence of CO and O-2, but these species may not have significant effects on the rate of N2O reduction. However, the presence of these gases in FBC presents difficulties in the study of kinetics since CO cannot be easily eliminated from the carbon surface. In N2O reduction reactions, ash in chars is found to have significant catalytic effects, which must be accounted for in the kinetic models and data evaluation. (C) 1997 Elsevier Science Ltd.

Adsorption characteristics of carbonized bark for phenol and pentachlorophenol

The potential of using carbonized slash pine bark as a substitute for activated carbon was examined in this study. The bark was carbonized by slow heating in nitrogen for 6.5 h to 672 degrees C. The BET-N-2 surface area, average micropore and mesopore diameter, and micropore volume were 332 m(2) g(-1) 21.7 Angstrom, and 0.125 cm(3) g(-1), respectively. The adsorption capacities for phenol and pentachlorophenol (PCP) at pH 2 and pH 8 were evaluated. The Langmuir equation provided a slightly better fit than the Freundlich equation to two sets of phenol data. The calculated Freundlich constants, K = 0.41 - 0.58 mmol/g/(mmol dm(-3))(1/n) and 1/n = 0.30 - 0.41, were lower and higher, respectively, than literature values for activated carbons. The adsorption capacity of the carbonized bark was much lower for PCP than for phenol. The protonated and anionic PCP isotherms were Type II or III, respectively, in the Brunauer classification. The BET equation provided the best fit to protonated PCP isotherm data. The anionic PCP data were fitted to both the BET model and an equation used in the literature to represent phosphate adsorption on activated carbons. Nonlinear regression of the data for both phenol and PCP adsorption with the Freundlich, Langmuir and BET equations generally gave more accurate parameters, compared with the use of linearized equations to obtain the parameters. (C) 1998 SCI.

Modification of MCM-41 by surface silylation with trimethylchlorosilane and adsorption study

Siliceous MCM-41 samples were modified by silylation using trimethylchlorosilane (TMCS). The surface coverage of functional groups was studied systematically in this work. The role of surface silanol groups during modification was evaluated using techniques of FTIR and Si-29 CP/MAS NMR. Adsorption of water and benzene on samples of various hydrophobicities was measured and compared. It was found that the maximum degree of surface attachments of trimethylsilyl (TMS) groups was about 85%, corresponding to the density of TMS groups of 1.9 per nm(2). The degree of silylation is found to linearly increase with increasing pre-outgassing temperature prior to silylation. A few types of silanol groups exist on MCM-41 surfaces, among which both free and geminal ones are responsible for active silylation. Results of water adsorption show that aluminosilicate MCM-41 materials are more or less hydrophilic, giving a type IV isotherm, similar to that of nitrogen adsorption, whereas siliceous MCM-41 are hydrophobic, exhibiting a type V adsorption isotherm. The fully silylated Si-MCM-41 samples are more hydrophobic giving a type III adsorption isotherm. Benzene adsorption on all MCM-41 samples shows type IV isotherms regardless of the surface chemistry. Capillary condensation occurs at a higher relative pressure for the silylated MCM-41 than that for the unsilylated sample, though the pore diameter was found reduced markedly by silylation. This is thought attributed to the diffusion constriction posed by the attached TMS groups. The results show that the surface chemistry plays an important role in water adsorption, whereas benzene adsorption is predominantly determined by the pore geometry of MCM-41.

Experimental and theoretical investigations of adsorption hysteresis and criticality in MCM-41: Studies with O-2, Ar, and CO2

MCM-41 materials of six different pore diameters were prepared and characterized using X-ray diffraction, transmission electron microscopy, helium pycnometry, small-angle neutron scattering, and gas adsorption (argon at 77.4 and 87.4 K, nitrogen and oxygen at 77.4 K, and carbon dioxide at 194.6 K). A recent molecular continuum model of the authors, previously used for adsorption of nitrogen at 77.4 K, was applied here for adsorption of argon, oxygen, and carbon dioxide. While model predictions of single-pore adsorption isotherms for argon and oxygen are in satisfactory agreement with experimental data, significant deviation was found for carbon dioxide, most likely due to its high quadrupole moment. Predictions of critical pore diameter, below which reversible condensation occurs: were possible by the model and found to be consistent with experimental estimates, for the adsorption of the various gases. On the other hand, existing models such as the Barrett-Joyner-Halenda (BJH), Saito-Foley, and Dubinin-Astakhov models were found to be inadequate, either predicting an incorrect pore diameter or not correlating the isotherms adequately. The wall structure of MCM-41 appears to be close to that of amorphous silica, as inferred from our skeletal density measurements.

Solution of adsorption problems involving steep moving profiles

The moving finite element collocation method proposed by Kill et al. (1995) Chem. Engng Sci. 51 (4), 2793-2799 for solution of problems with steep gradients is further developed to solve transient problems arising in the field of adsorption. The technique is applied to a model of adsorption in solids with bidisperse pore structures. Numerical solutions were found to match the analytical solution when it exists (i.e. when the adsorption isotherm is linear). The method is simple yet sufficiently accurate for use in adsorption problems, where global collocation methods fail. (C) 1998 Elsevier Science Ltd. All rights reserved.