Thiele Parameter Research Articles

Mathematical Modeling of the Stefan Flow Effect on Process in a Catalyst Grain Using Mathcad

Weekman and Gorring [2] considered the problem of the influence of the Stefan flow in a catalyst grain for mth order gas-phase chemical reaction A → nB. Special numerical methods were used to solve the problem. This paper proposes a simple method for solving the problem of a first-order gas-phase chemical reaction A → 2B using the Mathcad program. As an illustration of the method, the profiles of dimensionless reagent concentration and gas velocity of the Stefan flow at the Thiele parameter ψ = 1 and the molar fraction of reagent A outside the grain θ = 1, the dependence of the catalyst effectiveness factor on the parameters ψ and θ in the ranges ψ = 0.1–10 and θ = 0.1–1, and the dependence of a ratio between the catalyst effectiveness factor at θ = 1 and an analogous parameter in the absence of the Stefan flow on the Thiele parameter were calculated.

Mathematical Modeling of the Catalyst Deactivation Process inside a Grain Using Mathcad

The problem of the mathematical modeling of the catalyst deactivation process inside a spherical grain with a parallel first-order deactivation mechanism has been solved in the work [9] by the finite difference method. This paper presents a simpler method for the solution of this problem. It is shown that the set of nonlinear partial differential equations for planar, cylindrical, and spherical grains can be reduced to a boundary problem for two ordinary differential equations with respect to the spatial variable, where time is a parameter. The obtained equations are solved by the shooting method using Mathcad functions. For illustration, the profiles of relative catalyst activity and dimensionless reagent concentration are calculated for a spherical grain at a Thiele parameter of 5 and different time moments, together with the dependence of the degree of internal grain surface utilization on dimensionless time. Some asymptotic dependences are proposed for these parameters over a long time period.

Formation of a “dead zone” in porous structures during processes that proceeding under steady-state and unsteady-state conditions

The problem of the origination of a “dead zone” in porous materials during the proceeding of the arbitrary chemical reactions within them under steady- and unsteady-state conditions is investigated. As the “dead zone”, we understand the region in a porous medium that cannot be penetrated by the reagents that participate in the chemical transformations. In this study, the necessary conditions of the origination of the dead zone in the central region of the spherical, cylindrical, or lamellar porous catalyst granule in the course of the arbitrary catalytic reaction in it are found. The resistances to the mass-and-heat exchange in the outer surface of the porous material and the external reaction mixture are taken into account. It should be noted that the proposed algorithm for deriving the necessary conditions of the dead zone is also valid for porous materials of any other geometric shape. It is also shown that, under definite process conditions in the unsteady-state conditions (involving the artificially formed conditions), the phenomenon of pore locking is observed in certain time intervals. This phenomenon, which involves the duration of pore locking, can be controlled. The locking phenomenon cannot be used in technologies developed based on applying porous media and the repeated origination of the dead zone in the porous material. The analytical expression for determining the critical value of the Thiele parameter at which the dead zone originates is only derived for the simple cylindrical or lamellar shape in the steady- and quasi-steady-state conditions. For the catalytic reaction of the mentioned type, which proceeds under steady- and unsteady-state conditions, the nonisothermality of the porous granule on the critical value of the Thiele parameter is also investigated. Similar investigations are also performed for the catalytic processes occurring by the parallel, consecutive, and consecutive-parallel schemes. For example, it should be noted that catalysts with porous granules are widely used under the conditions of chemical technologies. Additionally, in connection with the rapid development of the microelectronics and nanoelectronics, the problem of the development and use of effective miniature devices for energy generation and storage also becomes very important. Currently, porous materials are widely used in various technologies of the development of accumulators as the electrodes. In recent years, the porous materials are of considerable interest in the projects associated with the development of the alternative power engineering, in particular hydrogen power engineering. In particular, the technologies of storing hydrogen that can successfully be used in the future bond hydrogen in metal hydrides, metals, alloys, intermetallic compounds of composites, and its storage in carbon nanostructures.

Solid‐solid reactions in series: A modeling and experimental study

AbstractReactions among particulate solid phases are important and abundant in many materials, chemical, and metallurgical process industries. Many of these are reaction networks, and not single‐step reactions as normally assumed. There is no theoretical framework available for the analysis of such systems, and single‐reaction models derived from the gas–solid literature continue to be used. Formation of cement clinker in the rotary cement kiln is a prime example of the genre, in which mechanistic aspects play an important role in determining energy efficiency and the composition and nature of the phases that form. In the present study, we formulate a model within the ambit of the “shrinking core” class of models, for reactions in series among solid phases. The model shows the presence of one or two moving fronts in the reacting particle, depending on the relative rates of the processes involved. A single Thiele‐type parameter controls the model behavior, at once describing the relative rates of the intermediate formation and consumption processes, and the diffusion‐reaction competition for the product formation step. The model has been shown to reduce to the well known single reaction models at the limits of low and high values of the Thiele parameter. Experimental data have been obtained on the calcia‐alumina system, an important one in cement manufacture, in the temperature range 1150–1250°C. The model has been fitted to these data and the kinetic parameters determined. The comparison bears out the salient features of the theory, and shows that a degree of diffusion limitation exists for the intermediate conversion step under these conditions. The diffusivity values estimated are in the range of 10−19 to 10−18 m2/s and agree with values found in the literature for similar systems. The rate constant for the intermediate conversion step is of the order of 10−6 s−1. This being among the first such determinations, this value awaits confirmation from other studies. © 2009 American Institute of Chemical Engineers AIChE J, 2009

Effect of oxygen mobility in solid catalyst on transient regimes of catalytic reaction of methane partial oxidation at short contact times

Mathematical modeling of the effect of the oxygen mobility in a solid oxide catalyst on the dynamics of transients of fast catalytic reactions has been carried out. The analysis was based upon the redox mechanistic scheme with a due regard for diffusion of oxygen from the bulk of catalyst to its surface. Parameters of kinetic and mathematical models were selected via fitting of the experimental data for methane selective oxidation into syngas on 1.4%Pt/Gd0.2Ce0.4Zr0.4O x catalyst. The range of the Thiele parameter (φ) where the oxygen bulk diffusion affects the most strongly reaction transients corresponds to φe [0.3 ÷ 7]. For high-surface-area oxide catalysts, the bulk oxygen diffusion coefficients corresponding to this range of the Thiele parameter are in the range of 10–18 ÷ 10–13cm2/s.

Sulfur dioxide capture under PFBC conditions: the influence of sorbent particle structure

An unreacted shrinking core model with variable effective diffusivity was applied to sulfation test data from a pressurised thermogravimetric apparatus (P-TGA) for various limestone and dolomite samples. The particle size was 250–300 μm for all sorbents. The tests were carried out under conditions typical for a pressurised fluidised bed combustor, i.e. 850 or 950°C, 1.5 MPa. Under these conditions the limestone remains uncalcined, whilst the dolomite is half-calcined. The sorbents were characterised by chemical composition analysis, particle density measurement, mercury porosimetry and BET internal surface measurement. A key parameter in this study is the conversion-dependent Thiele parameter, which reveals a shift in rate-determining mechanism during the sorbents' sulfation. The extended model allows the effect of particle structure on the reaction rate constant and the product layer diffusivity to be evaluated. Estimates for the apparent activation energy of the kinetics and the product layer diffusion can also be given, based on data from two temperatures.

Effectiveness factor of a plate catalyst grain for reactions with degree kinetics

The exact analytical interrelations connecting the effectiveness factor with the Thiele parameter on catalyst grains of plate forms for reactions with degree kinetics are generally determined.

EFFECT OF PORE DIFFUSION IN METHANOL SYNTHESIS

Abstract Pore diffusion limitations dampen the sensitivity of temperature runaway to variations in the coolant temperature in methanol synthesis reactors. For an isothermal pellet and low product concentration, the relationship between catalyst effectiveness and the Thiele parameter can be represented with a single curve. However, when equilibrium is approached or at high product concentrations, the relationship between catalyst effectiveness and the Thiele parameter becomes a function of catalyst surface temperature. For large temperature gradients in the catalyst, the effectiveness factor can significantly increase and parametric sensitivity in the relationship between catalyst effectiveness and the Thiele parameter may result.

Effectiveness of reaction proceeding with degree kinetics in a plate catalyst grain

The dependences of the effectiveness factor of a plate catalyst grain on the Thiele parameter for reactions with degree kinetics in isothermal and non-isothermal conditions were investigated. Analysis of a possible stationary distribution of concentration in the thickness of a catalyst grain was performed. The condition under which “dead zones” arise and their sizes in the central part of a catalyst grain depending on the reaction order were established.

TG study on the reaction of γ-Al 2O 3 by CCl 4. Part II. Influence of the mass transport processes

The role of the mass transport processes in the chlorination of γ-alumina with gaseous CCl 4 was studied by thermogravimetry in the temperature range 650–820 K at 0.1–10 kPa CCl 4 partial pressure. The effect of the sample mass, the temperature and the CCl 4 concentration, as well as the nature of the inert carrier gas on the reaction rate was measured and discussed with respect to the mass transport processes. The observed role of the pore diffusion is characterized by the evaluation of the dimensionless Thiele parameter. Results show that the upper temperature limit of the chemical control varies between 700 and 740 K as a function of the concentration and the effective diffusivity of CCl 4 at a fixed pore structure of the γ-alumina.

Calculation of effectiveness factor for multiple claus reactions with single‐step rate‐controlling

AbstractMost of the bountiful literature on the calculation of effectiveness factors for catalyst particles deals with relatively simple reactions and kinetics. The performance of catalyst particles in the industrially important, modified Claus reaction introduces multiple reactions, sulphur vapour equilibria, non‐linear kinetics and limiting thermodynamic conversions into the calculation of local effectiveness factors. This analysis demonstrates a means for generating local effectiveness factors, during reactor design calculations by the use of a single calculated curve of versus a modified Thiele parameter for spherical particles.

Transient response of three-phase slurry reactors

Analytical time domain solutions are provided for the transient response of three-phase slurry reactors, with diffusion and first-order irreversible ch input of the reactant in the inlet gas stream; and (b) when the catalyst particles are suddenly introduced into the reactor which has otherwise reached obtained for the general case of continuous slurry reactors but can be easily reduced for application to semi-batch slurry reactors in which there is n and are expected to be useful for evaluating the rate and equilibrium constants from dynamic response experiments. Asymptotic solutions have also been obtained for the case of large Thiele parameter (φ ≥ 3) and some other simpler expressions have been derived as only for gaseous reactants of moderate or high solubility in the liquid phase and are not applicable to those with low solubility.

Transient response of continuous-flow stirred reactors containing heterogeneous systems for catalysis or sorption

Time domain solutions are presented for the transient response of an isothermal CFSR with one-dimensional diffusion and first-order irreversible chemical reaction in catalyst particles or liquid pools of regular geometry and with mass transfer resistance at the fluid-catalyst interface. Three different cases of feed perturbations have been considered, namely, step input, impulse input and the sudden introduction of pellets into the reactor. The solutions have been derived in a general manner, without specifying the particle shape, and are in terms of characteristic functions defined for each of the three particle geometries considered, i.e. slab with sealed edges, cylinder with sealed ends, and sphere. Asymptotic solutions suitable for large Thiele parameters (gf≥3) have also been derived and are particularly convenient when the series solutions converge slowly. Experimental verification of the exact and asymptotic solutions for a specific case of absorption and diffusion in a liquid pool is presented.

Catalyst effectiveness in sulfur dioxide oxidation

The sensitivity of the relationship between catalyst effectiveness in SO 2 oxidation and the Thiele parameter to: (1) the kinetic expression with respect to the presence or absence of strong SO 3 inhibition, (2) the activation energy in the range of 16–32 kcal/g mol, (3) the conversion of SO 2 up to 90%, (4) the temperature at the exterior surface of the pellet in the range of 450–600°C, and (5) the feed composition ranging from 6–10% SO 2 in air, was investigated. A nonisothermal model was used and the calculations were carried close to equilibrium. Far from equilibrium, the relationship is essentially unaffected except by conversion at the pellet surface for the case of strong product inhibition. Close to equilibrium, the relationship is significantly influenced by conversion, temperature, and the inlet composition but the form of the kinetics is not important. The activation energy is not a significant variable. Catalyst size, surface area, and density were not specified while it was necessary to fix the ratios of the transport properties for the calculations. The properties of the American Cyanamid V 2O 5 catalyst were used for this purpose.

A method for the study of performance of a single spherical particle with nonuniform catalytic activity

A method for the study of the performance of a single spherical particle with nonuniform catalytic activity consists of the deuterium exchange reaction with neopentane on supported platinum; the exchange can be followed in the mass spectrometer and it resembles industrially important reactions, such as the isomerization of certain olefins and the hydrogenolysis of polynuclear aromatics. The selectivity parameter, M, ranges from 1 to 3. The selectivity is greatly influenced by activity distribution at Thiele parameters of 1-100. This analysis refers to catalyst activity which increases toward the particle surface. However, the analysis can be applied similarly to reactions in which catalyst poisoning starts at the pore mouth and the activity increases toward the particle center.

Analytical solutions of limiting poisoning mechanisms for a zero-order main reaction

The single-pellet diffusion reactor has proven to be a particularly useful tool to diagnose the mechanism of catalyst poisoning. Previous work was done with main reactions of first, second and 1/2 order and behavior was found to be similar for reactions of different orders. However, a zero order main reaction in this reactor presents a singular behavior not observed previously. These singularities are illustrated using the limiting poisoning models referred to as pore-mouth, uniform and core poisoning: cases which can be solved analytically. From these analyses, it is clear that the single-pellet diffusion reactor can still be used to discriminate among the various poisoning mechanisms causing deactivation. However, the range of Thiele parameter over which the measured centerplane concentration gives useful additional information is reduced to values of the effective Thiele parameter less than √2.

Catalyst effectiveness factor for redox model kinetic equation

A generalized isothermal effectiveness factor correlation has been proposed for catalytic reactions whose intrinsic kinetics are based on the redox model. In this correlation which is exact for asymptotic values of the Thiele parameter the effect of the parameters appearing in the model, the order of the reaction and particle geometry are incorporated in a modified form of Thiele parameter. The relationship takes the usual form: ▪ and predicts effectiveness factor with an error of less than 2% in a range of Thiele parameter that accommodates both the kinetic and diffusion control regimes.

Experimental study of the poisoning of a single catalyst pellet in a diffusion reactor

Catalyst poisoning experiments were carried out in an isothermal single-pellet reactor under conditions where both kinetic and diffusion processes contribute to the overall reaction rate (Thiele parameter between about 0·5 and 5). The concentration of the reacting species was monitored at the pellet's external surface and at its center plane. Impurity poisoning and parallel, series and triangular self-poisoning processes could be qualitatively discriminate by a sequence of three experiments, once the order of the main reaction was established. It was found that the hydrogenolysis of cyclopropane to propane between 50 and 75°C over an ν-Al 2O 3 supported Pt catalyst is simultaneously poisoned by the reactant cyclopropane and the product propane (triangular self-poisoning). Quantitative values of the poisoning kinetic exponents were evaluated by decoupling the time variable from the poisoning process, using relative overall rate versus normalized center-plane concentration data. Superimposing time on the results (center-plane concentration versus time and relative overall rate versus time data) detected an initial time-delay in the poisoning process. This time-delay was shown to depend upon the hydrogen-pretreatment of the catalyst. Conventional bulk rate-time studies would have been seriously hampered by the complicated time-behavior of the system, while the approach presented in this paper allowed the resolution of the mechanism and kinetics of the poisoning process.

An optimal catalyst activation policy for poisoning problems

Pontryagins minimum principle is used to calculate the optimum distribution of active material throughout a single pellet that is uniformly experiencing activity decay. The definition of optimality is based upon balancing catalyst costs against the net return from reactant conversion. The optimal policy is full activation to a fractional depth with an inert core, the depth depending upon the Thiele parameter, poisoning time constant, operating time, and an economic parameter. An isothermal, first order, irreversible reaction is considered. Auxiliary calculations of the effectiveness factor are given for the reaction rate in the nonisothermal pellet with the catalyst distribution found to be optimal in the isothermal case. The results of the pellet calculations are utilized in numerically calculating a uniform activation policy that is optimal for a homogeneously poisoned bed. The definition of optimality is on the same basis as that for the single pellet and the results depend upon the same physical parameters in addition to the time constant for convection in the tube relative to that for diffusion in the pellets. Four regimes of behavior arise depending upon catalyst costs: (i) the reactor cannot be operated economically, (ii) only partial activation policies are economical, a single one being optimal, (iii) both partial and full activation schemes are economical, with one of the former being optimal, and (iv) full activation is optimal but the reactor can be operated economically with partially activated pellets. The results for both the single pellet and the packed tube are viewed as the homogeneous poisoning limit of many practical problems and are believed to reflect the major characteristics of more complicated poisoning mechanisms.

Control regimes in experimentation of heterogeneous kinetics

Based on the generalized treatment of isothermal mean effectiveness factor presented by Aris, an experimental factor Φ exp has been obtained, the numerical value of which could be ascertained from experimental measurements of reaction rate and other pertinent properties of the fluid-particle system. It is shown that if Φ exp ⩽ 0·03, the experimental data reflect solely the influence of intrinsic kinetics of the system. Based on available kinetic information pertaining to a phthalic anhydride process, the variation of Φ exp with temperature, catalyst activity and external diffusion has been presented. Finally a plot of Φ exp against generalized Thiele parameter for low orders of reaction, has been used to check the consistency and absence or otherwise of pore diffusion limitation in experimental data reported in existing literature.