# Modeling of the Interactions Between Catalytic Surfaces and Gas-Phase

### Abstract

The catalytic surface interacts with the gas-phase by a variety of chemical and physical processes. Hence, optimization of design and operation conditions of catalytic reactors do not only require the understanding of the catalytic reaction sequence but also its coupling with mass and heat transport and potential homogeneous reactions. The chemical, thermal, and mass-transport interactions between the catalytic surface and the gas-phase are discussed in terms of the individual and combined interactions. The state-of-the-art modelling of reactive flows and its coupling with the catalytic surface is summarized. The interactions are illustrated by a number of examples such as reforming of hydrocarbons, catalytic combustion, exhaust-gas after-treatment, each focusing on a special aspect of catalyst–gas interactions. The potentials and limitations of the numerical simulations will be discussed including experimental techniques for model validation.

### Graphical Abstract

### Keywords

Heterogeneous catalysis Gas-phase kinetics Surface reaction kinetics Auto-thermal reforming Combustion Partial oxidation Kinetic modeling## 1 Introduction

_{2}in NO

_{x}storage catalysts [12, 13]. Desorbing intermediates and radicals may trigger chemical reactions in the gas-phase and by this open a competing reaction pathway. For instance, desorbing OH from Pt can enhance homogeneous ignition in the gas phase in catalytically supported combustion [14]. On the other hand, adsorbing radicals may quench chemical reactions in the gas-phase such as in the suppression of explosions in catalytic micro reactors [15]. Depending on the species’ partial pressures in the gas phase a strong adsorbate may cover the entire catalytic surface and inhibit catalytic reactions (CO on Pt in low-temperature fuel cells) [16]. Oxidizing and reducing species may lead to oxidation (reduction) of the few top monolayers of the catalyst and even oxidize (reduce) the entire catalytic particle. Examples are CO oxidation [17] as well as partial and selective oxidation of hydrocarbons over noble metals and metal oxides. Here catalyst size and morphology as well as support and additives matter as well. The catalyst oxidation state may change in time, e.g. due to higher temperature caused by the ignition of an exothermic reaction such as in reduction of Rh in partial oxidation of methane [18] or due to the availability of a catalytically formed product such as in the oxidation of Pt with NO

_{2}in diesel oxidation catalysts [19]. Depending on the surrounding gas-phase the catalyst may need an initiation time to reach the state of catalytic activity such as in Fischer–Tropsch catalysts [20]. The catalyst may also store certain gaseous components and release them upon modified conditions in the gas-phase such as NO

_{x}storage catalysts [21, 22]. Adsorbed species can undergo structural changes leading to undesired deposits such as carbonaceous overlayers, e.g. catalyst coking by the Boudouard reaction, and sulfates, e.g. in exhaust-gas cleaning, which eventually stops the catalytic cycle. However, blocking of catalytic sites may be selective and by this open alternate beneficial reaction pathways. Chemical reactions in the gas-phase may produce species that potentially threat the catalyst cycle when deposited on the surface such as in coking of catalysts due to gas-phase molecular growth processes [23]. Gas-phase species may lead to catalyst loss due to evaporation at lower temperatures upon oxidation of the catalyst. Homogeneous gas-phase reactions may also produce gas-phase species that then interact with the catalytic cycle: For instance, olefins and aldehydes are formed in the gas-phase in partial oxidation of propane [24] and ethanol [25], respectively, over Rh; these species are then converted catalytically again. Catalytic conversion may trigger desired homogeneous gas-phase reactions such as in the formation of olefins in oxy-dehydrogenation of alkanes over Pt [24, 26, 27].

Aside from the chemical interactions between gas-phase and catalytic surface, physical interactions occur by mass and heat transfer. The optimization of mass and heat transfer are decisive steps to transfer a novel catalytic process from the laboratory to the technical scale. On the larger length scale, the convective flow determines the transport of reactants and products in the catalytic reactor. When pressure drop becomes an issue, laminar flows are preferred, e.g. in automotive catalytic converters. When mixing is of more importance, then turbulent flow is beneficial, e.g. in fluidized beds. On a much smaller length scale, convective transport can be introduced by catalytic reactions with large volumetric changes (e.g. in cracking of hydrocarbons) and by net mass fluxes at the surface (e.g. in chemical vapor deposition, combustion of solids, etching, ignition of catalytic reactions).

The overall reaction rate is often limited by diffusion processes and not by the intrinsic turn-over-frequency. At intermediate temperatures, internal diffusion in the pores of the catalytic structure matters. In automotive catalytic converters, the reduced overall conversion due to internal mass transfer limitations at intermediate temperatures [28, 29] is a threat for licensing the devices, because any legislative test cycle covers this temperature range in the first few minutes after cold start-up of the engine. At higher temperatures, external diffusion in the catalyst-near gas-phase becomes the rate-limiting step. Diffusive transport may also be introduced by temperature gradients, significant for instance in catalytic reformers for hydrogen production or catalytic sensors for detection of hydrogen gases.

Thermal interactions between catalytic surfaces and the surrounding gas-phase do not only occur due to the reaction enthalpy and the heat needed to be transported from (exothermic reaction) and to (endothermic reaction) the catalyst but also due to varying external conditions (e.g. in automotive catalytic converters). Conductive and convective heat transport in the gas phase can significantly influence the catalytic conversion indirectly aside from conductive heat transport in the solid structure and radiative heat transfer.

Aside from experimental studies, modeling and numerical simulations can significantly support our understanding of the interactions between the gas phase and the catalytic surface. Actually, numerical simulations are more and more used as the only tractable tool to study the complex coupling between flow, mass and heat transfer and multi-phase chemical reactions in the process of designing and optimizing of technical catalytic reactors. Modeling and simulation over the years became a decisive toolbox in saving the engineers’ costs and time in scale-up of the catalytic process. In the next two sections, the current approaches of modeling of gas-surface interactions will summarized also giving the fundamental governing equations for reactive flows. In the remainder, several studies from literature will be discussed to highlight certain aspects of the interactions between catalytic surface and gas phase. This article will neither consider molecular simulations such as Density Functional Theory (DFT), Molecular Dynamics (MD), and Monte-Carlo (MC) simulations nor liquid and three-phase flows, for which it is referred to the literature, e.g. for multi-scale modeling of heterogeneously catalyzed gas-phase reactions [30] and for three-phase flow modeling [31, 32, 33].

## 2 Modeling of Gaseous Reactive Flows

### 2.1 Governing Equations for Multi-component Reactive Flows

As long as a fluid can be treated as continuum, the most accurate description of the flow field of multi-component mixtures is given by the transient three-dimensional (3D) Navier–Stokes equations coupled with the energy and species governing equations [34, 35, 36, 37, 38]. Alternative concepts such as Lattice-Boltzmann models (LBM) [39, 40] are still difficult to handle in multi-component mixtures but are promising for the simulation of complex flow structures for instance in fixed beds [41, 42, 43]; they will not be covered here.

*control volume*. The principle of mass conservation then leads to the mass continuity equation

*ρ*, the time

*t*, Cartesian coordinates

*x*

_{ i }(

*i*= 1,2,3), velocity components

*v*

_{ i }. The source term

*S*

_{m}vanishes unless mass is either deposited on or ablated from the solid surfaces (e.g. chemical vapor deposition, filters, etching, combustion of solids). The Einstein convention is used here, i.e., whenever the same index appears twice in any term, summation over that index is implied, except if the index refers to a chemical species. The principle of momentum conservation for Newtonian fluids leads to three scalar equations for the momentum components

*ρ v*

_{ i }

*p*, stress tensor

*τ*

_{ ij }, gravity

*g*

_{ i }. The stress tensor can be written as

*μ*, respectively. The coupled mass continuity and momentum governing equations have to be solved for the description of the flow field.

*m*

_{ i }of each of the

*N*

_{g}gas-phase species obeys a conservation law

*Y*

_{ i }is the mass fraction of species

*i*in the mixture (

*Y*

_{ i }=

*m*

_{ i }/

*m*) and

*m*the total mass.

*R*

_{ i }

^{hom}is the net rate of production due to homogeneous chemical reactions in the gas phase. The components

*j*

_{ i,j }of the diffusion mass flux caused by concentration and temperature gradients are often modeled by the mixture-average formulation [44]:

*D*

_{ i }

^{M}is the effective diffusion coefficient of species

*i*in the mixture,

*D*

_{ i }

^{T}is the thermal diffusion coefficient, which is significant only for light species, and

*T*is the temperature. The molar fraction

*X*

_{ i }is related to the mass fraction

*Y*

_{ i }using the species molar masses

*M*

_{ i }by

*h*:

*S*

_{h}, e.g. due to thermal radiation. In multi-component mixtures, diffusive heat transport is significant due to heat conduction and mass diffusion, i.e.

*h*

_{ i }. Heat transport in solids such as reactor walls and catalyst materials can also be modeled by an enthalpy equation, for instance in the form of

*h*is the specific enthalpy and

*λ*the thermal conductivity of the solid material.

*S*

_{h}accounts for heat sources.

*ρ*,

*p*, and

*T,*the simplest model being the ideal gas equation

*R*= 8.314 J mol

^{−1}K

^{−1}.

*μ*,

*D*

_{ i }

^{M},

*D*

_{ i }

^{T}, and

*λ*appearing in Eqs. (3, 5, 8) depend on temperature and mixture composition. They can be derived from the transport coefficients of the individual species and the mixture composition by applying empirical approximations [34, 35, 37]. The temperature is commonly implicitly derived from Eq. (9) by

*c*

_{ p,i }is the specific heat capacity at constant pressure.

The Eqs. (1–11) can principally be used in the numerical simulation of laminar as well as turbulent flow fields; the so-called Direct Numerical Simulations, DNS. In practice, however, the solution of the Navier–Stokes equations for turbulent flows demands a prohibitive amount of computational time due to the huge number of grid points needed to resolve the small scales of turbulence. Therefore, several concepts were developed to model turbulent flows by the solution of averaged governing equations. However, the equation system is not closed, that means a model has to be set up to describe the so-called Reynold stresses that are the correlations between the velocity fluctuations and the fluctuations of all the quantities of the flow (velocity, enthalpy, mass fractions). The *k*–*ε*—model [45] is one of the most widely used concepts for modeling the Reynold stresses at high Reynolds numbers, which adds two additional partial differential equations for the description of the turbulent kinetic energy, *k,* and the dissipation rate, *ε,* to the governing equations. Although the model has well-known deficiencies, it is today implemented in most commercial CFD (Computatial Fluid Dynamcis) codes and also widely used for the simulation of catalytic reactors. Modeling of turbulent flows is still a very active field of research. A promising approach is represented by Large-Eddy-Simulations (LES), which combine DNS for the larger scales with a turbulence model, e.g. *k*–*ε*—model, for the unresolved smaller scales. Aside from this closure problem, one still has to specify the averaged chemical reaction rates [37, 46, 47]. Because of the strong non-linearity of the rate coefficients due to the exponential dependence on temperature and the power-law dependence on partial pressures, the source terms of chemical reactions in turbulent flows should never (!) be computed using average concentrations and temperature. Here, probability density functions (PDFs) [37], either derived from transport equations [48] or empirically constructed [49], are used to take the turbulent fluctuations into account when calculating the chemical source terms. For the simulation of reactions on catalysts it is also important to use appropriate models for the laminarization of the turbulent flow near the solid surface [47, 50]. Multi-phase dispersed flows such as in fluidized beds are another class of flow systems in catalytic converters, which is still a very active field of research. Here, it is referred to two excellent books by Marchisio and Fox [51, 52].

### 2.2 Modeling Gaseous Transport in Porous Catalysts

*α*is the permeability (Darcy’s law) and

*C*is the inertial resistance, which can be viewed as a loss per unit length along the flow direction. Concerning the temperature profile in porous media, the enthalpy Eqs. (7) and (10) have to be adapted. The approach assuming constant unidirectional flow breaks down for fixed bed reactors with reactor diameters less than ten times the particle size. Thus the model cannot predict the velocity maximum in the vicinity of the wall observed experimentally for those reactors [56]. An averaged velocity with a radially varying axial component can be provided by a further modification of the momentum balance [56, 57, 58] as an improvement of the classical model. The reader is referred to the textbook by Keil [54] and references therein.

### 2.3 Coupling of the Gas-Phase with the Catalytic Surface

The coupling of the species mass fluxes at the gas-surface interface depends on the spatial size and structure of the catalyst, e.g. flat surface, gauzes, pellets, washcoats.

*Infinitely fast internal diffusion*usually occurs at low temperatures (kinetically controlled regime) and in rather thin catalyst layers. Then no concentration gradients occur in the porous structure and, hence, its spatial resolution is not needed. If the catalytic structure is spatially-resolved, obtained e.g. by magnetic resonance imaging (MRI) [59], no homogenizing model is needed either. In both cases, the catalytic surface is coupled to the gas-phase by the boundary conditions for the species-continuity Eqs. (4) [34, 60]:

Here \(\vec{n}\) is the outward-pointing unit vector normal to the surface, \(\vec{j}_{i}\) is the diffusion mass flux of species *i* as discussed in Eqs. (4), and \(R_{i}^{\text{het}}\) is the heterogeneous surface reaction rate (desorption minus adsorption), which is given per unit geometrical surface area, *A* _{geo}, in kg m^{−2} s^{−1}. The molar net production rate of gas phase species *i*, \(\dot{s}_{i}\), is given in mol m^{−2} s^{−1}; the area now refers to the actual catalytically active surface area, *A* _{cat}, to be determined experimentally, e.g. by chemisorption measurements. The ratio \(F_{\text{cat/geo}} = \frac{{A_{\text{cat}} }}{{A_{\text{geo}} }}\) is also used to describe the dependence of the overall reaction rate on catalyst loading and effects of hydro-thermal aging for structure-insensitive catalysts [61]. Practically, it was recently applied to model the performance of on-road aged three-way catalysts [62].

At steady-state conditions, this mass flux vanishes unless mass is deposited on the surface, e.g. chemical vapor deposition, or ablated, e.g. material etching. Equation (14) basically means that for \(\vec{v}_{\text{Stef}}\) = 0 the amount of gas-phase molecules of species *i*, which are consumed/produced at the catalyst by adsorption/desorption, have to diffuse to/from the catalytic wall (Eq. 5). Only for fast transient (<10^{−4} s) adsorption/desorption processes, e.g. during ignition of catalytic oxidation, Eq. (14) breaks down and special treatment of the coupling is needed [34, 63, 64]. Furthermore, those fast transient processes may lead to heat accumulation terms [63] and also to additional convective transport and associated pressure gradients in the fluid phase above the catalyst [64].

*effectiveness factor approach*is the simplest model to account for finite diffusion by scaling the intrinsic reaction rate at the fluid–solid interphase:

*Φ*

_{ i }[38, 65]:

Here, *L* is the thickness of the porous medium (washcoat), *V* _{wc} the washcoat volume, and *c* _{ i,0} are the species concentrations at the fluid/porous media interface. Since mass conservation has to be obeyed (Eq. 16), the same effectiveness factor has to be applied for all chemical species. Therefore, this simple model can only be applied at conditions, at which the reaction rate of one species *i* determines overall reactivity, i.e. in Eq. (16), \(\eta = \eta_{i}\). Furthermore, this model then implies that mass diffusion inside the porous media can be described by the same diffusion coefficient for all species.

^{−3}s

^{−1}; the volume here refers to the volume of a computational cell in the in the geometrical domain of fluid flow. Then \(R_{i}^{\text{het}}\) simply represents an additional source term on the right side of the species-continuity equation, Eq. (4), and is computed by

^{−1}.

*reaction–diffusion equations*:

Fluxes within porous media which are driven by gradients in concentration and pressure, i.e. diffusion and convection, can be described by the *Dusty Gas Model* (DGM) [54, 66]. This model, which is also applicable for three-dimensional and larger porous media, is superior to the ones discussed above but also leads to large computational efforts.

## 3 Modeling the Rates of Chemical Reactions

### 3.1 Surface Reaction Rate

Since the rate of catalytic reactions is very specific to the catalyst formulation, global rate expressions have been used for many years. These reaction rates are based on catalyst mass, catalyst volume, reactor volume, or catalyst external surface area. The implementation of this *macrokinetics* approach into reactor simulations is straightforward; the reaction rate can in general be expressed by any arbitrary function of gas-phase concentrations and temperature at the catalysts surface calculated at every computational cell containing either catalytically active particles or walls. It is evident that this approach cannot account for the complex variety of phenomena of catalysis and that the rate parameters must be evaluated experimentally for each new catalyst and various external conditions.

The direct computation of surface reaction rates from the molecular situation leads more and more to a comprehensive description, at least for idealized systems. However, DFT, MD, and MC simulations cannot be implemented in complex flow field simulations of technically relevant systems due to missing algorithms and, more importantly, due to the immense amount of computational time needed. Treating the catalytic system as a *black box* is not the alternative, the knowledge gained from experimental and theoretical surface science studies can be implemented in the chemical models used in reaction engineering simulation. A tractable approach is the treatment of surface chemistry by rate equations [34, 67, 68]. This so-called mean-field approximation (MF) was then first computationally realized in the SURFACE CHEMKIN [60] module of the CHEMKIN [69, 70] software package; later other packages such as CANTERA [71] and DETCHEM [72] as well as many of the commercially available multi-purpose CFD codes adopted this methodology or directly implemented those modules.

*T*and a set of surface coverages,

*θ*

_{ i }, depending on time and the meso/macroscopic position in the reactor, but averaged over microscopic local fluctuations. The surface structure of the catalyst is associated with a surface site density

*Γ*that describes the maximum number of species adsorbing on a unit surface area, given, e.g., in mol m

^{−2}. Each surface species is associated with a coordination number

*σ*

_{ i }describing the number of surface sites which are covered by this species. Under the assumptions made, multi-step (quasi-elementary) reaction mechanisms can be set up. The local chemical source term,

*R*

_{ i }

^{het}, is then defined by

*K*

_{s}is the number of surface reactions,

*c*

_{i}are the species concentrations, which are given, e.g., in mol m

^{−2}for the

*N*

_{s}adsorbed species and in, e.g., mol m

^{−3}for the

*N*

_{g}and

*N*

_{b}gaseous and bulk species. According to Eq. (22) and the relation \(\varTheta_{i} = c_{i} \sigma_{i} \varGamma^{ - 1}\), the variations of surface coverages follow

*A*

_{ k }is the pre-exponential factor,

*β*

_{ k }is the temperature exponent, \(E_{{{\text{a}}_{k} }}\) is the activation energy. Because thermodynamic data, in particular entropies, for adsorbes species are difficult measure and compute, thermodynamic consistency of surface kinetics for reaction networks is a crucial issue; two methods have been proposed enforcing thermodynamic consistency without the necessity to know the thermodynamics of all adsorbates explicitly [74, 75].

### 3.2 Gas-Phase Reaction Rate

*R*

_{ i }

^{hom}in Eq. (4), is commonly expressed by

Various sets of elementary reactions are available for modeling homogeneous gas phase reactions that may matter in catalytic reactors [37, 76, 77].

Even though the implementation of Eqs. (22) and (25) into CFD codes is straightforward, an additional highly nonlinear coupling is introduced into the governing equations leading to considerable computational efforts. The nonlinearity, the large number of chemical species, and the fact that chemical reactions exhibit a large range of time scales render the solution of those equation systems challenging. In particular for turbulent flows, but sometimes even for laminar flows, the solution of the system is too CPU time-consuming with current numerical algorithms and computer capacities.

## 4 Examples for Interactions Between Catalytic Surface and Gas-Phase

Recent and challenging numerical simulations of catalytic reactors will be discussed now with an emphasis on the interactions of processes in the gas-phase and on the catalytic surface. The order of the examples will follow the sketch shown in Fig. 2 with a number of different applications. In most examples, however there are multiple interactions between catalyst and gas phase.

### 4.1 Chemical Interactions

#### 4.1.1 Adsorption and Desorption: Partial Oxidation of CH_{4} over Rh

#### 4.1.2 Modification of the State of the Catalyst: Partial Oxidation of CH_{4} over Noble Metals

In-operando modifications of the catalyst due to temporarily changing concentrations is a well-known phenomenon as revealed by many experiments. Modeling of this aspect on a reactor scale is rather challenging as soon as the modification goes beyond the variation of surface coverages. At first, the intrinsic kinetics of the chemical modification (e.g., oxidation, reduction, hydrogenation, and carbonization) of the bulk of the catalytic particle is not only difficult to measure and simulate from first-principles but also to implement in CFD models, because it depends on catalyst size, morphology, porosity. Currently, models use simple parameters such as an overall oxidation state or a capacity to model bulk modifications in reactor simulations. Attempts to include a spatial resolution of the state of the catalytic particle are rare and only used for idealized particles.

#### 4.1.3 Deposits from the Gas-Phase: Gas-Phase Initiated Coking in Dry Reforming of Methane

For instance, Kahle et al. [89] recently investigated why strong coking was observed in dry reforming of methane at the entrance of a laboratory pilot plant flow reactor. The results revealed that coking in the upper part of the catalyst bed and at the entrance of the reactor depends on the temperature. Olefins are formed in front of the catalytic bed due to the high temperature and pressure leading slowly but with increasing rate to coke formation inside the bed in the reforming of methane in the presence of carbon dioxide over 900 °C at 20 bar. Hydrogen addition can inhibit coke formation more efficiently than water addition. In this study, the model used an elementary-step reaction mechanism of 4,238 gas-phase reactions among 1,034 species and 58 heterogeneous reactions among eight gas-phase and 14 surface-adsorbed species. The study eventually led to a better positioning of the catalyst in the reformer tube.

#### 4.1.4 Undesired Side Reactions in the Gas-Phase: Catalytic Reforming of Gasoline

_{2}, the oxygen deficiency at the catalytic surface leads to the formation of hydrogen by steam reforming. Due to the high temperature of approximately 1,000 °C, some remaining fuel is pyrolyzed by gas-phase reactions to form the coke precursors ethylene and propylene (Fig. 8), a relatively slow process that is kinetically controlled but presenting a threat for any downstream system such as fuel cell devices [94, 95, 96].

### 4.2 Mass Transport Interactions

#### 4.2.1 Complex Flows Structures: Catalytic Fixed Bed Reactors

Several attempts have been made to resolve the flow field in the actual configuration of fixed bed reactors, i.e. by a direct numerical simulation (DNS). Even though the governing equations are relatively simple for laminar flows, this usually can only be applied for small and periodic regions of the reactor due to the huge number of computational cells needed to resolve all existing boundary layers [97, 98, 99, 100, 101, 102, 103].

*k*–

*ε*model [109], and two different wall functions (standard [110] and non-equilibrium) were applied to model the flow field near solid surfaces. Attempts to correlate the local wall heat flux with local properties of the flow field, such as velocity components, velocity gradients, and components of vorticity led to the conclusion that local heat transfer rates do not correlate statistically with the local flow field. Instead, a conceptual analysis was used to suggest that local patterns of wall heat flux are related to larger-scale flow structures in the bed. Recently, the same group studied the interplay of 3D transport and reaction occurring inside cylindrical pellets and in the gas flow around the pellets used for propane dehydrogenation to better understand catalyst deactivation by carbon deposition (Fig. 9) [111].

Lattice–Boltzmann-methods (LBM) have also been applied for a better understanding of fluid flow in complex reactor configurations [42, 43, 112]. The packing of spheres in cylindrical columns can be created either from experimental observations, such as magnetic resonance imaging (MRI), or by computer simulations. The created topology is then divided into a Cartesian grid, where individual elements are labeled as solid or fluid regions. A high resolution of the grid leads to accurate flow profiles. Zeiser et al. [43] generated the geometrical structures of the fixed bed with a Monte-Carlo method. This allowed to efficiently simulate the placement of randomly packed spheres in a cylinder and to obtain detailed information of statistical properties, such as the distribution of the void fraction. This geometrical information was the basis for subsequent numerical flow simulation using LBM. This approach allowed the prediction of the local fluid velocity distribution in the bed as well as the transport and rate of simple chemical reactions. Yuen et al. [112] studied correlations between local conversion and hydrodynamics in a 3D fixed-bed esterification process by applying a LBM and comparing its results with data from in situ magnetic resonance visualization techniques.

#### 4.2.2 Laminar Flow Fields in Channels: Capillary Sampling Technique

There are wide varieties of catalytic reactors, in which the reactive mixtures flow through channel-like devices such as tubular chemical reactors, honeycomb structured monoliths, and foams. Depending on the channel size the flow field can be laminar or turbulent, or an initially turbulent flow field is laminarized in the entrance region of the channel. The latter situation can be found in most automotive catalytic converters.

#### 4.2.3 Turbulent Flow: Catalytic Combustion Monoliths

The flow in pipes with diameters ranging from a few centimeters up to meters is in most cases turbulent, guaranteeing a good mixing of the reactants. A fine resolution of flow field details is rarely of interest, and, aside from that, such a task exceeds today’s computer capacities. Therefore, averaged equations and turbulence models are applied. Mantzaras et al. [116] applied the *k*–*ε* model, a presumed (Gaussian) probability density function for gaseous reactions, and a laminar-like closure for surface reactions to study turbulent catalytically stabilized combustion of lean hydrogen-air mixtures in plane platinum-coated channels; a system very relevant for catalytic combustion stages in gas turbines, which reduces NO_{x} emissions and increases efficiencies. They also examined different low-Reynolds number near-wall turbulence models and compared the numerically predicted results with data derived from planar laser-induced fluorescence measurements of OH radicals, Raman measurements of major species and laser doppler velocimetry measurements of local velocities and turbulence [50]. They found that discrepancies between predictions and measurements can be ascribed to the capacity of the various turbulence models to capture the strong flow laminarization induced by the heat transfer from the hot catalytic surfaces.

#### 4.2.4 Internal Diffusion: Three-Way Catalyst

### 4.3 Thermal Interactions

The exponential dependence of the reaction rate on temperature calls for accurate predictions of the temperature profiles in catalytic reactors, which is a formidable task due to the variety in time scales of heat transport processes. In addition, the heat transfer at the outer boundaries of catalytic reactors, spatially varying inlet conditions, and catalyst coatings as well as aging effects demand heat transfer models that consider the entire catalytic reactor.

#### 4.3.1 Chemical Heat Release and Heat Loss: Faster Hydrogen Production at Shorter Residence Times

^{MONOLITH}[72] combines a transient three-dimensional simulation of the solid structure with a 2D model of the single-channel flow field and detailed models for homogeneous gas-phase chemistry, heterogeneous surface chemistry as well as internal diffusion processes. In Fig. 13, the impact of flow rate on the temperature distribution in the monolithic sections of the short contact time reactor reveals that higher flow rates lead to an increase in temperature, conversion and consequently higher hydrogen yields [6]. With increasing flow rate the ratio of chemical heat release (exponential dependence on temperature) to heat loss (linear dependence for conductive but fourth order dependence for radiative losses) increases for the monolithic structures. Consequently, the temperature increases (Fig. 13), the rate of reaction increases, fuel conversion increases. In addition, higher temperature favor hydrogen yields from a thermodynamic point of view. Unfortunately, higher temperatures may also lead to undesired side-reactions in the gas-phase towards coke-precursors [96] as discussed above.

#### 4.3.2 Convective Heat Transport: Light-Off of Catalytic Converters

#### 4.3.3 Chemical Release of Heat and Heat Transport: Ammonia Oxidation over Wire Gauzes

## 5 Summary and Outlook

The ultimate objectives of modeling and simulation of catalytic reactors are (1) a better understanding of the interactions of chemistry (heterogeneous and potential homogeneous reactions) and physics (mass and heat transport), (2) the support of reactor design and engineering, and (3) the optimization of operating conditions for the maximization of the yield of the desired product and minimization of undesired side-products. The numerous interactions between the catalytic surface reactions and the processes in the gas-phase play a central role. Depending on the chemical system, reactor configuration, and operating conditions, chemical, thermal, and mass-transfer induced interactions or often a combination of them will determine the observable reaction rate and reactor performance.

From a reaction engineering perspective, computational fluid dynamics simulations have matured into a powerful tool for understanding mass and heat transport in catalytic reactors. Initially, CFD calculations focused on a better understanding of mixing, mass transfer to enhance reaction rates, diffusion in porous media and heat transfer. Over the last decade, the flow field and heat transport models have also been coupled with models for heterogeneous chemical reactions. So far, most of these models are based on the mean field approximation, in which the local state of the surface is described by its coverage with adsorbed species averaged on a microscopic scale. The currently increasing research activities on surface reactions at practical conditions will certainly boost the application of CFD codes that combine fluid flow and chemistry. New insights into the complexity of heterogeneous catalysis, however, will also reveal the demand for more sophisticated chemistry models and their implementation into CFD simulations, which then requires sophisticated numerical algorithms and computer hardware. Hence, modeling the interactions of catalytic surface and surrounding gas-phase will remain a very active field and the implementation of more adequate and complex models will continue.

Despite this development, it should be noted that the results of the computations will always remain a reflection of the chemical and physical models and inlet and boundary conditions applied. The careful choice of the sub models for reaction kinetics, spatial dimension and resolution, turbulence, external and internal diffusion, thermal radiation etc. and the physical parameters and transport coefficients is a precondition for reliable simulation results. Furthermore, numerical algorithms never provide the accurate solution of the model equations but only an approximated one. Hence, error estimation is needed. Having these crucial issues in mind, computations can really serve as a powerful tool in understanding the behavior in catalytic reactors and in supporting the design and optimization of reactors and processes.

In this sense, understanding of the interaction of the catalyst with the surrounding gas-phase presents one of the top challenges and potentials in catalysis research today, aside from the development of new catalysts of course. The most powerful tool on this way is the combination of experimental techniques for the analyses of the state of the catalyst and the surrounding gas-phase in operando combined with numerical simulations using models based on the molecular behavior. Impressive progress has been made in the last 25 years but more is needed.

## Notes

### Acknowledgments

The author would like to thank his former and current co-workers at the Karlsruhe Institute of Technology for their dedicated work that led to many of the results shown here. Financial support by the German Research Foundation (DFG), the Helmholtz Association, the European Union, and many partners in industry is gratefully acknowledged.

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