Milan Kubíček

Modelling of chemical reactors — X Multiple solutions of enthalpy and mass balances for a catalytic reaction within a porous catalyst particle

As was already shown, equations describing mass and enthalpy balances for an exothermic reaction taking place inside a porous catalyst particle can have more than one stationary solution. In this paper, two conditions for the existence of multiple solutions were stated: (1) the value of parameter γβ has to be greater than a certain critical value, (γβ)*, (2) Thieles modulus has to be simultaneously in the definite range φ1 ⩽ φ ⩽ φ2 on the basis of the model, where the state of the particle is described by means of mean values of concentration and temperature. Values of (γβ)* and the critical limits of Thieles modulus φ1 and φ2 for reactions of power and of adsorption type of kinetics were then calculated. Values thus obtained, are in good agreement with correct values, obtained from numerical integration of conservation equations. In the last part of the paper a general procedure was described, which enables one to determine, whether for a given set of values of physical parameters more than one stationary profile of concentration in the catalyst particle can exist.

DERPER—an algorithm for the continuation of periodic solutions in ordinary differential equations

Abstract An algorithm for the continuation of periodic solutions of ODE was derived and tested. The algorithm is based on the shooting method and a standard continuation algorithm. Variational variables are used to compute the Jacobi matrix. The starting point on the periodic orbit is adaptively changed in the algorithm by fixing one component of the solution. Stability of periodic solutions along the continuous branch of solutions is determined by computing characteristic multipliers. Without difficulty the algorithm crosses limit points and at bifurcation points, double-period bifurcation points, and points of tori bifurcation it proceeds along the original branch of solutions. The usefulness of the algorithm is demonstrated in two examples. One of them describes two stirred tank reactors with mutual mass exchange, the other one is the Lorenz model of turbulence.

Multiplicity and stability in a sequence of two nonadiabatic nonisothermal CSTR

Abstract An analysis of mass and enthalpy balances describing dynamic behavior of two homogeneous CSTR with recirculation is performed. Methods of investigation of regions of existence of multiple steady states and stability are described. Effect of recirculation ratio, Damkohler number, cooling parameter and cooling temperature on the exit temperature is followed. The dependence of the exit temperature on the residence time is analyzed and it is shown that bizarre situations may occur (deformated “mushrooms”, two isolas, etc.). Plots are given to show the influence of system parameters on the reactor behavior.

Modelling of catalytic monolith converters with low- and high-temperature NOx storage compounds and differentiated washcoat

Two types of catalytic washcoats with different properties can be present simultaneously in a monolith channel, e.g. the flat foils and the corrugated ones in metallic monolith can be individually coated with specific type of the washcoat. The model of such multiphase, differentiated NOx storage and reduction (NSR) catalytic monolith converter has been developed. The important reactions as are the oxidation of carbon monoxide, hydrocarbons and hydrogen, the reduction of nitrogen oxides (NOx), the water gas shift and the steam reforming reactions, the NO/NO2 transformation, and the oxygen and NOx storage are considered. Unknown kinetic parameters of the NSR are evaluated from transient experiments with the samples of two different types of NSR catalysts. The first catalyst is of the Pt–γ-Al2O3–CeO2 type with alkali earth metals (e.g. barium) as NOx storage components (active at lower temperature). The second one is of the PtRh–γ-Al2O3–CeO2 type with both alkali earth metals and alkali metals (e.g. potassium) as NOx storage components (active at higher temperature). Simulations of periodic lean/rich operation of the low-temperature, the high-temperature and the differentiated (combined) NSR converters are performed. The results of the computations agree well with the experimental data. The dependences of integral NOx conversion on the lengths of the lean and rich phases and on the temperature of the inlet gas are discussed. The efficiencies of the low-temperature, the high-temperature and the differentiated NSR converters are compared at different operating conditions.

Mathematical modelling of catalytic monolithic reactors with storage of reaction components on the catalyst surface

Abstract Effects of periodic switching between lean and rich combustion conditions on CO, HC and NO x conversion on a monolithic catalyst with NO x storage were simulated by mathematical model. The model includes description of oxygen and NO x storage on the washcoat. Parametric study showed possibility to reach much improved time-averaged NO x conversion on a single monolith in comparison with steady-state operation, but lower CO and HCs conversions under reducing (rich) conditions. Sequence of monoliths, the first one with NO x storage catalyst and the second one with oxidizing catalyst, then enables to obtain satisfactory conversions of all pollutants.

Modeling of chemical reactors—XXV Cylindrical and spherical reaction with radial flow

Transport equations describing heat and mass transfer in radial flow reactors are developed. An analysis of these equations for both limit cases—piston flow and CSTR—is presented. A detailed examination indicates that the outlet conversion is not dependent on the flow direction in reactors of commercial size. The geometry of the reactor does not exhibit any marked effect on the exit conversion if the same amount of catalyst is used. The piston flow description may be employed for reactors when Pe(1−ξ0) > 50.

A computational tool for nonlinear dynamical and bifurcation analysis of chemical engineering problems

Abstract We present a program package cont for modelling and analysis of nonlinear problems and use it to solve chemical engineering problems including chemical reactors and adsorption columns. Model should be in the form of ordinary differential equations (ODEs), spatially distributed systems described by partial differential equations may also be treated upon transformation to ODEs by a built-in discretization. Continuation of a steady state, periodic solution or solution of a general boundary value problem with respect to a parameter provides a solution diagram; simultaneous stability analysis identifies bifurcation points and these may be continued with respect to a second parameter to obtain a bifurcation diagram. Additionally, the program performs a direct integration of ODEs, calculates Poincare orbits and Lyapunov exponents, and possesses other functionalities useful for practical analysis of complex dynamics. Three example problems are analyzed and discussed in detail: (1) dynamical and bifurcation patterns in a stirred reactor and a tubular reactor with a metabolic biochemical reaction; (2) shock waves in an adsorption column; (3) pulse waves in a model of intracellular calcium dynamics.

Solution of nonlinear boundary value problems—XI

Abstract The solution of a nonlinear boundary value problem may be found by solving a pertinent transient equation until the solution ceases to change significantly (the false transient method). The method is used to get a solution of a strongly nonlinear diffusion problem as well as towards solution of equations arising in the boundary layer theory. If multiple solutions occur the false transient method is not capable of calculating all profiles. For the boundary layer problems it is difficult to construct a reliable false transient equation if it is not based on a physically “sound” transient model.

On the modeling of PSA cycles with hysteresis-dependent isotherms

Abstract A four-step pressure swing adsorption (PSA) process for air drying on a silica gel dessicant is considered. The effects of hysteresis in adsorption/desorption equilibrium isotherms and of loading dependence of the effective diffusion coefficient on the process performance are investigated on a mathematical model. A comparative parametric study reveals that for identical operating conditions, hysteresis can be detrimental to product purity. Multiplicity of cyclic steady states is shown to exist as a result of hysteresis. A transient regime invoked by a gradual change in the inlet air humidity is discussed.

Pore-scale modeling of non-isothermal reaction phenomena in digitally reconstructed porous catalyst

Meso-scale mathematical model of local reaction and transport processes in a porous, supported heterogeneous catalyst with bimodal pore-size distribution is presented. The model takes into account individual reaction steps on active sites (microkinetics), diffusion of reactants in macro-pores and meso-/micro-pores (molecular and Knudsen-diffusion), and heat generation and transport. The processes are modelled within a three-dimensional domain ( ≈ 10 × 10 × 10 μm3) of computer-reconstructed porous catalyst. The methodology is demonstrated on CO oxidation on Pt/γ-Al2O3. Several 3D porous structures are digitally reconstructed by the methods of particle packing and Gaussian-correlated random fields from typical electron-microscopy images of the catalyst. Typical dependences of overall reaction rate and effectiveness factor on the temperature and properties of the porous catalyst structure are evaluated. Relative importance of diffusion in macro- and meso-pores under varying temperature is demonstrated. Local optimum of effectiveness factor is found for the mixing ratio of catalyst support particles with two different sizes. The resulting temperature gradients over the studied section of catalyst are very small (approximately 0.2 K for the CO concentrations in the order of 1% mol). The results represent the local situation on the meso-scale, which can be interpreted as one discretisation point in the full-scale model of the reactor.