Dynamicmodelling of catalytic three-phase reactors for hydrogenation and oxidation processes

DYNAMIC MODELLING OF CATALYTIC THREE-PHASE REACTORS FOR HYDROGENATION AND OXIDATION PROCESSES

T.Salmi, J.Wärnå, S.Toppinen, M.Rönnholm and J.P.Mikkola

Abo Akademi, Process Chemistry Group, Laboratory of Industrial Chemistry,

FIN-20500 Turku-Abo, Finland, Fax [phone],

E-mail Tapio.Salmi @abo.fi

(Received: October 10, 1999 ; Accepted: May 18, 2000)

Abstract - The dynamic modelling principles for typical catalytic three-phase reactors, batch autoclaves and fixed (trickle) beds were described. The models consist of balance equations for the catalyst particles as well as for the bulk phases of gas and liquid. Rate equations, transport models and mass balances were coupled to generalized heterogeneous models which were solved with respect to time and space with algorithms suitable for stiff differential equations. The aspects of numerical solution strategies were discussed and the procedure was illustrated with three case studies: hydrogenation of aromatics, hydrogenation of aldehydes and oxidation of ferrosulphate. The case studies revealed the importance of mass transfer resistance inside the catalyst pallets as well as the dynamics of the different phases being present in the reactor. Reliable three-phase reactor simulation and scale-up should be based on dynamic heterogeneous models.

Keywords: catalytic three-phase reactor, kinetics, diffusion, dynamics, numerical strategies.

INTRODUCTION

Catalytic three-phase reactors are used in numerous industrial processes, such as hydrogenation and oxidation processes. In the past, three-phase reactors were usually described with pseudo-homogeneous models, where the existence of different phases were discarded: the catalytic reaction rates were described with the concentrations in the liquid bulk, and overall balance equations were used for the liquid bulk. The gas phase was discarded.

However, a realistic description of catalytic three-phase reactors should be based on the real mass and heat transfer characteristics of the system, i.e. on kinetic models, mass and heat transfer models for both the bulk phases of gas and liquid as well as for the catalyst particles. It is well-known that most of catalytic three-phase processes carried out in fixed beds are heavily influenced by diffusional resistance inside the catalyst particles. Typical examples are catalytic hydrogenation of aromatics, aldehydes, ketones and nitro groups as well as hydrodesulphurization and hydrodemetallation of aromatics and cyclic molecules.

Furthermore, dynamic models should be preferred to steady-state models, since dynamic models provide a realistic description of the transient states of three-phase reactors and the numerical solution strategy of dynamic models is more robust than the solution of steady-state models. The present paper discusses the dynamic modelling principles of batch and fixed bed three-phase reactors and the numerical solution strategies of the models. The approach will be illustrated with three-phase hydrogenation and oxidation processes.

MODELLING PRINCIPLES

Rate Models

The rates of catalytic processes are complex, since adsorption, surface reactions as well as product desorption take place simultaneously on the surface of the solid catalyst. Basically the surface processes are described with surface concentrations, coverages, but in the engineering approach, the surface concentrations are eliminated by using simplifying assumptions, such as quasi-equilibria and quasi-steady state hypotheses (Laidler 1987). Thus, each overall reaction of the process is described with a rate expression

(1)

where c denotes the concentration vector. Consequently, the generation rates of the components are obtained from the stoichiometry

(2)

In three-phase reactors, the catalysts are usually completely wetted, thus the kinetics is described with liquid-phase concentrations.

Models for Catalyst Particles

A schematic picture of the mass and heat transfer resistances in a catalytic three-phase reactor is given in Fig. 1. Typically the resistances are of different orders of magnitude, e.g. for mass transfer, the liquid-side mass transfer limitation in the gas-liquid film and inside the catalyst particle are of greatest importance. For heat transfer, the film resistances are very important, whereas the heat transfer inside the solid catalyst is rapid. However, in a generalized modelling approach, no one of the resistances is neglected a priori, but all of them are included in the balance equations.

A heterogeneous model for three-phase reactors consists of balance equations for the catalyst particles as well as for the bulk phases of gas and liquid. In the present treatment, the chemical reactions are assumed to take place on the surface of the solid catalyst, i.e. inside the pores of the catalyst and non-catalytic reactions are assumed to proceed in the liquid bulk and in the liquid phase existing inside the pores. No chemical reactions are assumed to take place in the gas phase.

For an infinitesimal volume element in a porous catalyst particle, the following general mass balance equation can be written

(3)

Where e_p denotes the porosity of the particle, N_i is the molar flux, r is the radial coordinate and s is the shape factor; s=O for a slab, 1 for an infinitely long cylinder, and 2 for a sphere; non-ideal geometries can be treated with non-integer values . The further development of eq. (3) depends on which diffusion model is used. Strictly speaking, multicomponent diffusion should be described with Stefan-Maxwell equations, according to which all of the fluxes are related to all concentration gradients,

(4)

where F is a coefficient matrix consisting of binary diffusion coefficients (Fott and Schneider 1984).

Comparisons between different diffusion models have, however, shown (Salmi and Wärnå1991) that a simpler approach based on the law of Fick is sufficient, provided that the diffusion coefficients are described in an approximative way, e.g. by using the approximation of Wilke and/or the correlations for liquid-phase diffusion coefficients (Reid et al. 1988). The law of Fick implies that individual fluxes are given by

(5)

Consequently, the mass balance equation (3) can be rewritten to

(6)

Heat transfer in porous particles is described with Fourier´s law leading to the following differential equation with respect to temperature:

(7)

The boundary conditions of the mass and energy balances relate the surface properties to the bulk properties. In reactor modelling, the balance equations for the catalyst particle are solved together with the balance equations for the bulk phases. In the sequel, we consider two principal reactor configurations, a (semi-)batchwise operating agitated vessel (autoclave) and a continuous tubular reactor (fixed bed/trickle bed). The former reactor configuration is used in laboratory scale, in kinetic experiments and in catalyst screening as well as in industrial scale in the production of fine and specialty chemicals, while the latter one is the work horse in the production of bulk chemicals, e.g. in petrochemical industry.

Models for Batch Reactors

For batchwise operating autoclaves, the gas-phase mass balances can be discarded in most cases, since the gas phase pressure is maintained constant by control. In the cases, when the volatility of the liquid phase is high, the gas-phase mass balances are included. We consider here just the liquid-phase balances. The liquid-phase mass balance of a component in an autoclave can be written as follows,

(8)

where n_iis the amount of substance in the liquid bulk; N_i and N_GLi denote the fluxes from the catalyst particle and to the gas phase, respectively. For non-volatile components the gas-liquid flux term is dropped. In the case that the liquid density is virtually constant, the derivative can be expressed with concentrations, and the balance equation for non-volatile components becomes

(9)

where is the ratio between the catalyst surface area and liquid volume. The initial condition of eq. (9) is c₁= c,(0) at t=0.

The fluxes Ni are principally obtained from the mass balance equations for the particle, but in practice, it is, however, more favorable to calculate the flux by integrating the mass balance of the particle, eq. (3):

(10)

Since eq. (10) utilizes the integral of the numerically computed concentration profile and not the derivatives, the numerical inaccuracies are effectively eliminated.

Models for Fixed Beds

For a fixed bed reactor, the mass balance equations for the gas and liquid phase can be written as follows:

(11)

(12)

The symbols are explained in Notation. Usually the balance equations presented above can be simplified. For instance, the gas phase is close to plug flow, which implies that the axial dispersion effects are discarded (D_G=0). For non-volatile components, the gas phase mass balance is ignored. The balance equations have the boundary conditions, usually the Danckwerts boundary conditions are used, but in case of numerical difficulties in the proximity of the reactor exit, a modified semi-empirical boundary condition is useful (Salmi and Romanainen 1995).

The interfacial flux can be expressed in different ways. The simple two-film theory gives the expression

(13)

where k_Gi and k_Lidenote the mass transfer coefficients for gas (liquid phases, respectively, and A is the gas-liquid equilibrium ratio. The signs -/+ in eq. (12) refer to concurrent and countercurrent flow, respectively. In an analogous manner, the energy balance equations for gas and liquid phases can be derived. Usually it is sufficient to consider the liquid and solid phases only, since the heat capacity of the gas phase is much lower compared to that of the solid and liquid phase. A description of the energy balances is given in ref. (Wärnå 1994) and is not considered here in detail. However, the energy balance has principally the same mathematical form as the mass balance. The numerical solution strategies of the models are considered in next section.

Numerical Strategies

The model for the catalyst particles, eqs (6) and (7) are solved together with the model for the bulk phases of the reactor, e.g. eq. (9) or eqs (1l)-(12). The balances are connected through the flux terms, e.g. eq. (10). The model equations form a system of coupled partial parabolic differential equations (PDEs) and ordinary differential equations (ODEs).

Some approximations can be introduced for the numerical solution: one could assume that the dynamics of catalyst particles is more rapid than the dynamics of the bulk phases and one could assume that the dynamics of the gas phase is more rapid than the dynamics of the liquid bulk phase. In steady state modelling of fixed beds, the time derivatives disappear in eqs (1l)-(12). Furthermore, the axial dispersion is sometimes negligible. These kind of simplifications change the mathematical character of the problem. The mathematical structures of models with various sophistication levels are summarized in Table I. We have investigated the numerical solution of the model variants from a practical viewpoint in order to find the most convenient and robust solution strategies (Wärnå1994, Wärnå and Salmi 1996). Our experience tells in an unequivocal way that simplifications are useful in few cases only.

The use of a pseudo-state model for the catalyst particles changes the PDE to an ODE, which unfortunately is a boundary value problem (BVP). The BVP has to be solved as a system of nonlinear equations (NLEs) after discretizing the derivatives either by finite differences or by collocation. The numerical solution of the NLEs, however, often suffers from convergence problems (Wärnå, 1994). The pseudo-steady state model works, when good initial guesses are provide to the NLE-solver. For the concurrent steady-state trickle bed reactor model, the concentration and temperature profiles in the particle obtained for the previous reactor length coordinate can be used as initial guesses at the next be coordinate. Analogously for the batch reactor, the profiles obtained at the previous time can be used as initial guesses for the subsequent time. These strategies, are, however, very sensitive for the actual case, i.e. how sensitive the reactions are with respect to the concentrations and temperature. Therefore, a general approach is to use the complete dynamic model.

The dynamic model equations give an initial value problem with respect to the time. This implies that the differential equations can simply be integrated forwards with an appropriate differential equation solver. Prior to the solution, the PDEs are converted to ODEs by discretization. Two basic methods exist for discretization: finite differences and approximation functions. Orthogonal collocation has been successfully used as an approximation method. Reliable computer codes exist for the calculation of collocation points, i.e. roots of orthogonal polynomials as well as for the calculation of first and second derivatives (Villadsen and Michelsen 1978). A global collocation method, where the entire concentration and temperature profiles are approximated by a single polynomial can be used in limited cases, provided that the profiles are not too steep. In three-phase systems this is not usually the case: the concentrations of the gas-phase components are limited by their solubilities, which results in prominent diffusional limitations inside the catalyst particles, and the concentration profiles of the gas-phase components become very steep. Therefore, global collocation is replaced by spline collocation in most cases: the profiles are approximated piecewise by orthogonal polynomials requiring a continuous function with a continuous first derivative. The method has turned out to be very accurate and numerically very efficient in the simulation of steep profiles in reactive films (Romanainen 1991).

A mathematically simpler, but numerically less efficient method for discretization is based on the use of finite differences to express the first and second derivatives. Expressions for finite differences can be found e.g. in (Abramowitz and Stegun 1970) and computer codes are provided in (Schiesser 1991). Multipoint finite differences have recently been derived by Romanainen and Salmi (1995). The crucial point in the use of finite differences is the selection of a correct difference formula: the derivatives which originate from plug flow should be described with backward differences, whereas central differences are used in the description of first and second derivatives originating from diffusion and dispersion. A rule of thumb can be given for the choice of the order of the difference formula: two- and three-point difference formulae are very rough for the description of diffusion and dispersion effects, whereas a five-point formula is in many cases sufficient.

After the discretization of the spatial coordinates in the pellet and in the reactor, a system of ODEs is obtained. A characteristic feature for the system is its stiffness. The stiffness has its origin in several features: the dynamics of the catalyst particles is more rapid than that of the bulk phases, the spatial discretization causes stiffness because of local variations in the velocities of kinetic and transfer processes, and, finally, the velocities of the chemical reactions being present in the system can be very different. This implies that all explicit methods for the solution of ODEs, such as explicit Euler method and explicit Runge-Kutta methods can be directly forgotten in the search of a suitable algorithm. There exist three kinds of methods, which are suitable for the solution of stiff systems arising in three-phase reactor simulations: implicit Runge-Kutta, semi-implicit Runge-Kutta and backward difference methods. Implicit and semi-implicit RungeKutta methods are absolutely stable (A-stable), and also the backward difference method has very good stability properties. Implicit Runge-Kutta methods are usually abandoned, since they result in the solution of cumbersome systems of NLEs, whereas semi-implicit Runge-Kutta methods have been developed to work very well for stiff systems (Hairer and Wanner 1996); particularly the Rosenbrock-Wanner modification of semi-implicit Runge-Kutta methods has turned out to be very robust. Computer codes are listed in ref. (Hairer and Wanner 1996).

Backward difference methods were published by Henrici (1962), but the first working computer code was developed by Gear (1971). Later on, a very efficient family of codes was developed by Hindmarsh (1983). Nowadays, his code LSODE has become the standard routine for the solution of stiff ODEs. The code exists as several modifications, for instance, for implicit differential equations and for sparse systems. The system of differential equations which is created the spatial discretization is very sparse by its nature. Our experience from the simulation of a three-phase trickle bed reactor model has shown that dramatic savings in the computing time can be achieved, when the sparse matrix version of LSODE is used.

CASE STUDIES

Hydrogenation of Aromatics

The first example concerns catalytic hydrogenation of aromatic compounds, such as benzene, toluene, xylenes, isopropylbenzene and mesitylene over a supported nickel catalyst. The aromatic ring is hydrogenated in an exothermal reaction

where R₁, R₂ and R₃ denote hydrogen or alkyl chains. The reactions are industrially carried out in fixed beds, but the kinetics is conveniently measured in laboratory-scale autoclaves. In order to directly obtain kinetic data which are relevant for industrial scale, catalyst particles of industrial size were used in the laboratory experiments. The experiments were carried out in an autoclave operating at 90-140ºC and at pressures of 20-40 bar hydrogen. The reactants and products were analysed with gas chromatoraphy. Typical results from the kinetic experiments are displayed in Fig. 2. As can be seen from the figure, the kinetic curve of the reactant decreases monotonicly. For xylenes, cis- and trans-isomers appear as parallel products. The product ratio was essentially independent of the hydrogen pressure and the kinetics was described with rate equations of the type

where k and K denote the rate and adsorption parameters. The rate model was coupled to the models of the catalyst particles and the batch reactor. The textural and transport parameters were estimated a priori, and the complete reaction-diffusion model was used in the estimation of kinetic parameters by non-linear regression. The results from parameter fitting are displayed in Fig. 2, where the continuous curves represent the model predictions. As revealed by the figures, the model gave an excellent fit to the experimental data and the parameter estimation statistics was good, the standard deviations and mutual correlation coefficients between the parameters were low (Toppinen et al. 1996).

The dynamic reaction diffusion model gives valuable information about the diffusional resistance inside the particles as well as the dynamics of the particle and bulk phases. Fig. 3 shows that the process is heavily influenced by the diffusional limitation of hydrogen in the beginning, whereas the diffusional limitation of the aromatic compound is negligible. The situation is however, changed during the course of the reaction: some diffusional limitation of hydrogen always remains, but also the diffusional limitation of the aromatic compound gets more importance, since its concentration is low at the end of the reaction.

The concentration profiles inside the pellet were simulated for different reaction times. The results are displayed in Fig. 4. The figure shows that the dynamics of the catalyst particles is very rapid compared to the dynamics of the bulk phases: a pseudo-state is established inside the particle within about 1 mm or less, which means that it is justified to approximate the catalyst particle with a pseudo-steady state model. From the numerical viewpoint this is, however, not very wise as discussed in the previous section.

Hydrogenation of Xylose to Xylitol

A typical example of the carbonyl group hydrogenation is the catalytic hydrogenation of xylose to xylitol over Raney nickel. Xylitol is an artificial sweetening agent in alimentary products such as chewing gum and chocolate. Besides the main reaction, side reactions proceed in the system producing xylulose, arabinitol and furfural as undesired by-products. The reaction scheme is displayed below:

The reaction carried out in batchwise operating autoclaves in industrial scale. Kinetic experiments in laboratory scale were carried out in an autoclave at 80-140o C and 40-70 bar hydrogen. Water was used as solvent. Typical kinetic results are displayed in Fig. 5.

The experiments were carried out with finely dispersed catalyst particles, and numerical simulation of the concentration profiles inside the particles revealed that the diffusional resistance in the particle was negligible, since the effectiveness factor always exceeded 0.9 (Mikkola et al. 1999). On the other hand, the external mass transfer resistance on the liquid side of the gas-liquid side can become of importance in industrial scale, if the agitation of the reactor is not efficient enough. The formation of xylulose is an isomerization reaction, which is favoured if the access of hydrogen is limited. Fig. 6 illustrates the effect of mass transfer limitation on the product distribution: the value of the external mass transfer coefficient of hydrogen has to be high in order to suppress the formation of xylulose. At ideal mixing conditions (Fig. 6) the formation of xylulose is minimized and the production of xylitol is maximized.

Oxidation of Ferrosulphate to Ferrisulphate

Oxidation of ferrosulphate (FeSO₄) to trivalent iron sulphate, ferrisulphate (Fe₂(50₄)₃) is an important process in the production of water purification chemicals, since ferrisulphate is a very efficient coagulation agent. The reaction is carried out in the presence of molecular oxygen and sulphuric acid. The reaction milieu is very acidic, the pH being clearly below 0. The overall reaction is

The reaction proceeds as a gas-liquid reaction even in the absence of a catalyst (Rönnholm et al. 1999), but it can be considerably enhanced by introducing a catalyst, for example, active carbon (Rönnholm et al. 1999) or active carbon impregnated with a noble metal, such as Pt (Valtakari 1999). The kinetics of the non-catalytic reaction is expressed with the rate equation

and the kinetics of catalytic oxidation in the presence of active carbon is given by the expression

The symbols are defined in Notation. The kinetic parameters of the non-catalytic oxidation process were determined from experiments carried out in a pressurized autoclave operating at 60-130ºC. The oxygen pressure was varied between 4 and 10 bar. The catalytic oxidation kinetics was determined under similar conditions, but in the presence of active carbon. Typical kinetic curves are displayed in Fig. 7. As revealed by the figure, the presence of the catalyst implies a tremendous improvement in the efficiency of oxidation. The kinetic parameters of catalytic oxidation were determined by using the a priori calculated non-catalytic oxidation rate parameters and applying non-linear regression analysis to the catalytic oxidation data. The parameters became well-determined, because the rate equations comprised just a limited number of adjustable parameters.

Again there remains the question, how important role the intraparticular diffusion plays in the kinetics. Textural and diffusional parameters for the catalyst and the diffusion coefficient of oxygen were estimated from available correlations (Reid et al. 1988) and the concentration profiles in the catalyst particles were simulated numerically. Some numerical simulation results are displayed in Fig.7. As revealed by the figures, the process is limited both by oxygen mass transfer and the transfer of Fe(II)-ions. Thus a complete diffusion-reaction model is necessary for the realistic description of the process.

A possibility to carry out ferrosulphate oxidation in larger scale is to place the active carbon catalyst inside structural packing elements. The standard fixed bed model was modified by introducing a new liquid phase, namely the liquid which is flowing inside the structural elements. The gas bubbles are not able to penetrate inside the network of structural packings. Thus the reactor model consisted of the following elements: the reaction-diffusion model for the liquid inside the catalyst particles, the axial dispersion model for the liquid flowing inside the structural packings and the axial dispersion and plug flow models for the liquid and the gas flowing outside the packings. The flow rates inside and outside the packings were evaluated a priori with CFD-calculations. It turned out that the liquid velocity is less than 10% inside the packings when it is compared with the velocity outside the packings. The values of the axial dispersion coefficients were estimated with separate pulse experiments with inert tracers. The transport through the network was described with an overall transport coefficient, analogously with the interfacial transport concept of the Kunii-Levenspiel model (Levenspiel 1999). All of the ingredients were inserted into the dynamic three-phase reactor model. Some simulation examples are displayed in Fig. 8a. The results were compared with independent experimental data obtained from a column reactor equipped with packing elements. The simulation results are shown in Fig.8b. The prediction provided by the model followed very closely the experimental points. Furthermore, it tumed out that the gas-liquid transport resistance played an important role in the reactor.

CONCLUSIONS

The general principles for the modelling and simulation of dynamic three-phase reactor models were considered. The most robust way is to use complete dynamic models for catalyst particles and the bulk phases of gases and liquid. The PDEs appearing in the models were discretized with respect to the spatial coordinate and thus converted to ODEs, to an initial value problem. The best discretization methods are spline collocation or the use of finite differences. The ODEs were solved with backward difference method suitable for stiff initial value problems. The case studies revealed the important role of internal mass transfer resistance in the catalyst particles as well as the dynamics of the different phases being present in three-phase reactors. Reliable three-phase reactor modelling, simulation and scale up should be based on true dynamic heterogeneous models.

NOMECLATURE

A interfacial area a interfacial arealvolume a₁,ai₂ rate parameters for ferrosuiphate oxidation c concentration c concentration vector c_p heat capacity D dispersion coefficient D_I diffusion coefficient of component I E activation energy F function F coefficient matrix for Stefan-Maxwell model Dr reaction enthalpy K equilibrium parameter k rate constant k_G gas-phase mass transfer coefficient k_L liquid-phase mass transfer coefficient N component flux n amount of substance n flow of amount of substance R reaction rate; gas constant in the rate equation for ferrosuiphate oxidation r radial coordinate in catalyst particles r_I component generation rate S shape factor T temperature e porosity, void fraction, hold-up l heat conductivity n stoichiometric coefficient r density

Subscripts and Superscripts

cat catalytic G gas i component index j reaction index L liquid P particle " lumped parameter

Abbreviations

A romatic compound Fe ferrous or ferric compound O oxygen ref reference

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Publication Dates

• Publication in this collection
  16 Mar 2001
• Date of issue
  Dec 2000

History

• Accepted
  18 May 2000
• Received
  10 Oct 1999