03 FIRE BOOST Aftertreatment UsersGuide

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Users Guide

FIRE BOOST Aftertreatmentv2014


2/3182 FIRE BOOST Aftertreatment

Contents

1. Introduction........................................................................................................ 41.1. Scope................................................................................................................................4

1.2. Symbols............................................................................................................................ 4

1.3. Configurations...................................................................................................................4

2. Overview............................................................................................................. 5

3. Theory................................................................................................................. 63.1. Catalytic Converter Model................................................................................................63.1.1. Principle of Heterogeneous Catalytic Reactions................................................ 6

3.1.2. General Approaches and Assumptions..............................................................7

3.1.3. FIRE Balance Equations.................................................................................. 10

3.1.4. BOOST Balance Equations, Single Channel Model.........................................13

3.1.5. Washcoat Layer Pore Diffusion........................................................................20

3.1.6. General Chemical Reaction Rate Calculation.................................................. 26

3.1.7. Transfer Coefficients.........................................................................................28

3.1.8. Spray - Reactive Porosity Interaction...............................................................31

3.1.9. Nomenclature....................................................................................................33

3.2. Particulate Filter Model.................................................................................................. 39

3.2.1. Introduction....................................................................................................... 39

3.2.2. Overall Modeling Concept................................................................................ 403.2.3.Filter Flow Model.............................................................................................. 48

3.2.4. Deposition and Regeneration of Soot and Ash................................................ 52

3.2.5. Soot Migration...................................................................................................56

3.2.6. Modeling a Partial Wall Flow Filter...................................................................57

3.2.7. Modeling Glueing Zones in SIC PFs................................................................57

3.2.8.Particulate Filter Model Integration in FIRE and BOOST................................. 58

3.2.9. Nomenclature....................................................................................................60

3.3. Pipe Model..................................................................................................................... 64

3.3.1. Gas Phase Balance Equation.......................................................................... 64

3.3.2.Multi-Layered Wall Model.................................................................................65

3.3.3. Nomenclature....................................................................................................68

3.4. Injector Model.................................................................................................................70

3.4.1. Injector Model................................................................................................... 70

3.4.2. Injection Process.............................................................................................. 70

3.4.3. Wallfilm Modeling..............................................................................................71

3.4.4. Nomenclature....................................................................................................72

3.5. Temperature Sensor Model............................................................................................73

3.5.1. Nomenclature....................................................................................................74

3.6. Liquid Species Transport................................................................................................75

3.7. ThermalCoupling of Exhaust Aftertreatment Components............................................75

3.8. Kinetic Models................................................................................................................ 77

3.8.1. DOC Catalyst Reactions...................................................................................77

3.8.2. TWC Catalyst Reactions.................................................................................. 78

3.8.3. HSO-SCR Catalyst Reactions, Steady-State Approach................................... 81

3.8.4. HSO-SCR Catalyst Reactions, Transient Approach.........................................833.8.5. Lean NOx Trap.................................................................................................84

3.8.6. NOx Trap Catalyst Reactions...........................................................................89

3.8.7. Filter Regeneration with Oxygen .....................................................................90

3.8.8. Filter Regeneration with Oxygen and Nitric Dioxide.........................................91

3.8.9.Filter Regeneration with Oxygen, Nitric Dioxide and NO-Oxidation..................91

3.8.10. Filter CSF Catalytic Reactions....................................................................... 92

3.8.11. Nomenclature..................................................................................................93

3.9. Literature.........................................................................................................................95

3.10. Appendix.......................................................................................................................98

3.10.1. Analysis Formulae.......................................................................................... 98

3.10.2. Conversion of Mole and Volume Fractions and ppm's to Mass Fractions

and Vice Versa...................................................................................................... 99


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4. FIRE Aftertreatment.......................................................................................1004.1. Input Data.....................................................................................................................100

4.1.1. Run Mode.......................................................................................................100

4.1.2. Module Activation........................................................................................... 100

4.1.3. Aftertreatment................................................................................................. 100

4.1.4. Catalyst Specification..................................................................................... 100

4.1.5. Particulate Filter Specification........................................................................ 144

4.1.6. Reactive Porosity Specification...................................................................... 165

4.1.7. 3D Output Specification..................................................................................174

4.1.8. Mesh Requirements and MPI Decomposition................................................ 1754.1.9. Aftertreatment-Device Import from BOOST....................................................178

4.1.10. FIRE Aftertreatment User Functions ........................................................... 178

4.1.11. Homogenous Gas Phase Reactions - Input data......................................... 179

5. BOOST Aftertreatment ................................................................................. 1805.1. Input Data.....................................................................................................................181

5.1.1. Aftertreatment Solver .....................................................................................181

5.1.2. Boundary Conditions...................................................................................... 185

5.1.3. Catalyst .......................................................................................................... 187

5.1.4. Particulate Filter .............................................................................................224

5.1.5.Aftertreatment Pipe ........................................................................................240

5.1.6. Aftertreatment Injector.................................................................................... 243

5.1.7. Control Elements ........................................................................................... 246

5.1.8. Solid Materials ............................................................................................... 2495.1.9. Liquid Materials...............................................................................................249

5.1.10. Homogenous Gas Phase Reactions - Input data......................................... 250

5.1.11. InputData Checklist: Catalytic Converter and Particulate Filter................... 250

5.1.12. Best Practice ................................................................................................252

5.2. Databus Channels........................................................................................................258

5.2.1. Aftertreatment Boundary Databus Channels.................................................. 258

5.2.2. Catalyst Databus Channels............................................................................ 259

5.2.3. Particulate Filter Databus Channels............................................................... 263

5.2.4. Aftertreatment Pipe Databus Channels..........................................................265

5.2.5. Aftertreatment Injector Databus Channels......................................................266

5.2.6. SolverDatabus Channels...............................................................................267

5.3. Simulation Results........................................................................................................2675.3.1. Catalyst Results..............................................................................................267

5.3.2. Particulate Filter Results.................................................................................274

5.3.3. Aftertreatment Pipe Results............................................................................284

5.3.4. Aftertreatment Injector Results....................................................................... 290

5.3.5. Aftertreatment Boundary Results....................................................................296

5.3.6. Temperature Sensor Results..........................................................................297

5.3.7. Solver Results.................................................................................................298

5.4. Simulation Messages................................................................................................... 301

5.4.1. Message Analysis...........................................................................................301

5.4.2. Preprocessing ................................................................................................ 302

5.4.3. Calculation ..................................................................................................... 310

5.4.4. Postprocessing ...............................................................................................314

5.4.5.Reaction Library .............................................................................................316


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Introduction

4 FIRE BOOST Aftertreatment

1. Introduction

This manual describes the usage, files and the theoretical background of aftertreatment modelingand simulation using the AVL simulation codes BOOST and FIRE.

1.1. ScopeThis document is for users of the FIRE/BOOST Aftertreatment Module and anyone interested in

catalyst theory and modeling.

1.2. Symbols

The following symbols are used throughout this manual. Safety warnings must be strictlyobserved during operation and service of the system or its components.

Caution:Cautions describe conditions, practices or procedures which could result in damage to,or destruction of data if not strictly observed or remedied.

Note:Notes provide important supplementary information.

Convention Meaning

Italics For emphasis, to introduce a new term.

monospaceTo indicate a command, a program or a file name, messages, input/output on a screen, file contents or object names.

MenuOptA MenuOpt font is used for the names of menu options, submenus and

screen buttons.

1.3. Configurations

Software configurations described in this manual were in effect on the publication date of this

manual. It is the user's responsibility to verify the configuration of the equipment before applyingprocedures in this manual.


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2. Overv

FIRE BOOST Aftertreatment 5

2. Overview

The FIRE/BOOST Aftertreatment Module enables the simulation of the chemical and physicalprocesses occurring in various types of

honeycomb type catalytic converter wall-flow type particle filter

pipes (for BOOST).The models account for the simulation of the fluid flow within these elements, for heterogeneous

chemical reaction, for adsorption and desorption of species on the catalysts' surface and alsofor heterogeneous soot regeneration reactions. The solution of continuity, momentum, speciesand energy balances in the gas phase coupled with the solid phase energy conservation andchemical reactions models delivers detailed results resolved in time and space. Typical results

are for example:

flow velocities inside the channels and overall pressure drop species mass fractions and pollutant conversion gas/solid temperatures and thermal behavior reaction rates and chemical behavior heat and mass transfer

soot decomposition and regenerationWith the FIRE/BOOST aftertreatment models and their results, a broad range of aftertreatmentapplications can be investigated, developed and optimized:

Catalytic Converter Particle Filter

Three-way catalyst Particle filter loading

Diesel oxidation catalyst Bare trap regeneration

NOx storage catalyst Fuel additive regeneration

Selective Catalytic Reduction (SCR) catalyst Low temperature NO2 regeneration

Reformer catalyst Catalytic supported regeneration

In order to model all the different chemical reactions given by these various types of applications,FIRE offers a general chemical reaction input language which has similar functionality to the

CHEMKIN software package. Thus the user can set up his own chemical reaction modelscontaining gas phase species and species stored on the surface. The kinetic rate equationsare defined via a standard Arrhenius type rate law or via user models. The chemical equilibriumand sticking coefficient formulation is also considered. The FIRE Aftertreatment Module allows

definition of different kinetic parameter sets that can be assigned to any number of differentcatalysts in one geometric model. Additionally, FIRE and BOOST offer pre-defined reaction sets.For the simulation of catalytic reactions Langmuir-Hinshelwood approaches were setup. The userhas access to all kinetic parameters and therefore can adapt all pre-defined models to different

types of catalysts. In the same way pre-defined soot regeneration models are implemented for allthe regeneration types listed above. Free access to any reaction model, with an arbitrary numberof reactions and species, is offered by user-routines that can be linked to BOOST and FIRE.


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Theory


3. Theory

3.1. Catalytic Converter Model

Availability

BOOST AT: Catalystpage [187]

FIRE:Catalyst Specificationpage [100]

3.1.1. Principle of Heterogeneous Catalytic ReactionsIn this section effects are discussed that should be considered when a mathematical formulationfor the description of surface kinetics is developed.

Catalytic combustion reactors are heterogeneous reactors because they contain a gas phase(reactants and products) and solid catalyst. Since the catalytic reactions occur on the catalyst,the reactants have to be transported to the external gas-solid interface. Modeling the overallcombustion process therefore requires the consideration of both the physical transport and

chemical kinetic steps.

Generally there is a boundary layer between the bulk fluid stream and the solid surface.Within this boundary layer there are variations in velocity, concentration and temperature.Species transport from the bulk fluid stream to the solid surface can have limiting effect onthe rate of the catalytic reaction.

Most catalysts are porous materials. Much of the chemical reactions occur inside the porouscatalyst, which in some cases can have significant effect on the complexity of the problem.

Figure 1. Steps of a Catalytic Reaction

The above figure (adapted from Hayes et al. [21page [95]

]) shows the individual steps taking place

during a heterogeneous catalytic reaction. As discussed by Froment and Bischoff [14page [95]

] thefollowing steps can bedistinguished:

1. Transport of the reactants from the bulk gas phase to the external solid surface across theboundary layer.

2. Diffusion of the reactants into the porous catalyst. Since the main part of the catalyst islocated inside the porous material (washcoat) the reactants must diffuse into it.

3. Adsorption of the reactants onto the surface.4. Catalytic reaction at the surface.5. Desorption of the products of the reaction.6. Diffusion of the products to the surface of the catalyst.


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3. The


7. Transport of the products into the bulk gas phase.

Steps 1, 2, 6 and 7 are mass transport steps while steps 3, 4 and 5 are chemical kinetic steps.To account for these effects properly, the FIRE/BOOST Aftertreatment Module distinguishes thefollowing types of species:

Gas phase species:

: Concentration in the bulk gas flow (Species transport equation) : Concentration directly above the surface of the catalyst

Stored (adsorbed) species:

A stored species occupies one 'site' of the catalytic surface. The number of sites isconserved.

This allows to model steps 4, 5 and 6 either separately (i.e. Oxygen storage on the surface) or inone step (i.e. Langmuir-Hinshelwood-Hougen-Watson reaction model for 3 way catalysts).FIRE Example: Three-way catalyst:

CO + 0.5*O2 = CO2

C3H6 + 4.5*O2 = 3*CO2 + 3*H2O

H2 + 0.5*O2 = H2O

This mechanism accounts for the catalytic oxidation of CO, C3H6 and H2 as proposed bynumerous authors in literature (i.e. Voltz et al. [65

page [97]], Chen et al. [11

page [95]] and Wanker

et al.[67page [97]

]). The reactions are global reactions and do not contain any stored species.Therefore the influence of adsorption and desorption of species on the surface has to beconsidered in the formulation of the reaction rates (kinetics). Most commonly the Langmuir-

Hinshelwood-Hougen-Watson type rate equations are used in literature for these reactions.FIRE Example: Oxygen storage:

O2 + 2*PT_s = 2*O_s

The above reaction accounts for the effect of Oxygen storage on the catalyst. The Oxygenmolecule dissociates to two Oxygen atoms that are stored on the surface, which is indicated by

the identifier "_s"added to "O". Since two surface sites are occupied by the two Oxygen atoms,the expression "2*PT_s"must appear on the left hand side of the reaction definition line. PT isa dummy identifier for one surface site.

3.1.2. General Approaches and AssumptionsIn the following section general approaches considering catalytic converter modeling are brieflysummarized. For more detailed information please refer to the literature cited.

3.1.2.1. Cell Specification of Honeycomb-Type Catalytic ConverterThe Honeycomb-type catalytic converter consists of hundreds (thousands) of individual channels.The exhaust gas flows through these channels and reacts catalytically. The catalytic reactions

take place at active sites that are spread within the so-called washcoat of the monolith. Thiswashcoat is a porous solid layer that covers the solid substrate as shown in the following figure.


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Theory


Figure 2. Structure of a Squared Cell Monolith

As shown, the total thickness of the monolith's wall results to

(1)

where wallis the thickness of the substrate wall and wcl,totis the thickness of the washcoatlayers. The repeat distance sof the monolith can be derived from the cell density CPSMaccording to:

(2)

where CPSM is defined as the number of channels per square meter cross sectional area.Catalysts are often specified with the CPSI number determining the number of channels per

square inch. With given CPSI number one obtains CPSM with equation

(3)

Based on this information (CPSI, wall and washcoat thickness) FIRE/BOOST calculates thehydraulic channel diameter dhyd, open frontal area OFA and the geometric surface area GSA asshown below.Hydraulic channel diameter:

(4)

Monolith's open frontal area (= fluid volume fraction g) results from:

(5)

Geometric surface area (= channel wetted perimeter) GSA given in surface per monolith volume

is calculated as:

(6)

In the same way as the dhyd, OFA and GSA are derived from the cell density CPSM and the totalwall thickness , the latter can be calculated from the first three data. Therefore the above givenequations have to be inverted. The cell density is given by

(7)

and the total wall thickness of the monolith is

(8)


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3. The


The washcoat layer thickness ( wcl,tot) of the monolith is assumed to be zero and therefore the

total thickness is equal to the substrate thickness wall. Eq.7page [8]

and Eq.8page [8]

show thatthree different equations can be used for the evaluation of the cell density and the wall thickness.The difference between them is that only a pair of two values out of the three data (dhyd, OFA

and GSA) is required. FIRE/BOOST uses the first term on the right hand side of Eq.7page [8]

and

Eq.8page [8]

where the hydraulic diameter dhydand the open frontal area OFA are needed.The above calculated values of CPSM and wallare exact for squared cells. If other geometries(round, sinusoidal channel) are given, the derived values of CPSM and wallhave to be

understood as approximate values. Deviations do not matter since the calculation kernel of FIRE/BOOST use the values of dhyd, OFA and GSA in any case.

3.1.2.2. Conservation Equations of Mass or MolesIn general the balance of mass or moles is equivalent and therefore leads to the exact sameresults. Due to chemical reactions the number of moles in the system changes, but their overall

mass remains constant. Therefore mass balances are often preferred. In a mole balanceequation, the change of the total number of moles has to be taken into account by an additionalcorrection term.A second reason to use mass balances is the fact that many physical properties such as

enthalpies or caloric values of combustibles are given as a function of their mass. The molarmass which is necessary to transform mass specific values to mole specific data is not alwayscompletely accessible.

3.1.2.3. Volume Fraction, Density and Mass FractionCatalytic converter models have to describe a system consisting of two different phases (gas and

solid substrate) with two different volumes. The volume of the gas phase in this system is givenby means of an overall volume fraction. This volume fraction of gas phase in the entire system is

defined as follows:(9)

where gis the volume fraction of the phase g(as) in the entire volume V. The volume of the solidphase Vsis evaluated by Vs=(1- g)V = sV. Note, the fluid volume fraction gis identical to theopen frontal area OFA.

If one phase comprises several different species, a cumulative density consisting of the densitiesof all species can be defined. For this purpose the next relation is used:

(10)

The densityof the entire phase gis the sum of the densities of all different species kin it. In anadditional step the mass fraction wk,gof one species in a system can be defined as the fraction of

the density of the species k,gand the total density g(11)

The relation given by these two equations defines that the sum of all mass fractions always hasto be equal to one.


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Theory


3.1.2.4. Equation of State and Ideal Gas LawIf conservation equations for a gaseous phase are given, a general relation between theintensive variables of the gas is necessary. Pressures and temperatures observed during typical

catalytic converter applications lie within moderate ranges (p


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3. The


3.1.3.1.1.2. Effective Thermal Conductivity including RadiationThis approach intends to model the cross-stream thermal conductivity based on the heat transfer

modes that in reality take place in the monolith: conduction and radiation. The following figure

shows the heat transfer modes within a catalyst squared unit cell. The walls have a width and

dis the hydraulic diameter. The length sis the unit cell width of the catalyst derived from the

density number (cpsi) Nas: .

The heatflux exchanged between faces at temperature T1and T2can be written

(16)

where Lis the catalyst length and is the effective radial thermal conductivity.

Figure 3. Heat Transfer within a Catalyst Squared Unit Cell

The heatflux is composed of the heat conduction within the wall along length s and width(flux QL, orange zone in the above figure), the conduction along length and width d(flux Qs1,green zone) and the radiation through the channel (flux Qs2, blue zone). Following composition ofthermal resistance rules, the last two are treated in serial and are in parallel with the first one, i.e:

(17)

In the above equation, the radiation term has been linearized and the term between parentheses

is the effective thermal conductivity . This relation describes the heat exchange within a unitcell. If one assumes thermal equilibrium of all unit cells contained into a mesh cell, the relationextends to mesh cells as:

(18)

where his the distance between two mesh cell centers. Based on relation (4) one can build theanisotropic heat conduction matrix as follows:

(19)

The model can be applied as it is on catalyst or particulate filters. No specific modeling is

associated to the channel shapes.


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Theory


3.1.3.1.1.3. Solid SurfacesThe diffusion fluxes must be computed along the solid surface of the cells As . For monoliths witha preferential flow direction (e.g. monoliths with channel shaped geometry: DPFs, catalysts) thesolid surface vectors are calculated different than for catalytic blocks without any preferential flowdirection (e.g. undirected porosities: packed beds). For the first case one can create the followingassumption:

If one considers a cell face A normal to the main catalyst direction, then is for the fluid and

the complement for the solid. If one considers a face tangent to the main direction, all the surface

is solid. The solid surface vectors are then computed by the general relation: (20)

where is the surface reduction matrix in the genuine catalyst reference frame. For the secondcase of catalytic blocks with undirected porosities, the surface reduction is uniform in alldirections. Thus, the solid surface vectors are computed by the relation:

(21)

3.1.3.1.2. Diffusion Terms CalculationThe following formula is used to compute the diffusion fluxes on the cell face j. It is derived fromthe isotropic relation generally used in FIRE.

(22)

The first term on the right-hand side determines the diffusion coefficient, while the second term

is the cross-diffusion part and is added in the source terms. is the interpolated cell-face

temperature gradient. djis the distance between cell centers Pjand Pi.

3.1.3.1.3. Boundary ConditionsAs the walls are in contact with the solid part of the catalyst, the thermal wall boundary conditions

are removed from the gas enthalpy equation and added to the solid temperature equation. The

boundary fluxes are computed according to the boundary version of the relation (Eq.22page [12]

).The local wall heat transfer coefficient is then proportional to the solid thermal conductivity andinversely proportional to the wall distance. Post-processing the solid heat transfer coefficient can

be confusing as it can be very high due to the dependence on the wall distance. But it is physical.When reducing the wall distance the solid heat transfer coefficient increases but the wall to cell

solid temperature difference decreases, giving a wall heat flux of same order.The interfaces between the catalyst and the gas are presumed adiabatic for the solid

temperature.

3.1.3.2. Source Terms in the Gas Phase Balance Equations

3.1.3.2.1. Sources in the Enthalpy Conservation EquationThe term Sr(W) accounts for heat sources due to catalytic chemical reactions. It is calculatedusing the species' reaction rates and the corresponding enthalpies of formation using thefollowing formula:

(23)


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3. The


where is the reaction rate of species k(kmol/(m3s)) and is the formation enthalpy of

species kat 298 K.

3.1.3.2.2. Sources in the Species Conservation EquationsThe following sources are added for each species kto the right-hand side of the corresponding

species transport equation:(24)

where is the reaction rate of species k(kmol/(m3s)).

3.1.4. BOOST Balance Equations, Single Channel ModelUnder the assumption that radial transport effects of a honeycomb-type catalytic converter aresmall compared to the heat transport in axial direction, the entire converter can be represented

by one single channel. The physical situation of such a channel is sketched in the followingfigure. The effects taking place are convective, diffusive and conductive transport in the gasphase, mass and energy transfer through the boundary layer, diffusion and catalytic conversion

in the wash-coat, and conduction in the solid phase. Neglecting radial gradients in the channel,transient and 1D (in axial direction) conservation equations suffice to describe the thermo- andfluid dynamics.

Figure 4. Scheme of One Single Channel in a Honeycomb-type Catalytic Converter

The differential conservation equations for mass momentum and energy of a single channel canbe written as shown in the following section.The continuity equation of the gas phase is

(25)

where gis the density of the gas phase, t is the time, vgis the interstitial gas velocity and z is thespatial coordinate in axial direction.The momentum conservation equation is given by the steady-state Darcy equation (see Kaviani

[26page [96]

])

(26)

where pgis the pressure of the system. The Darcy constant ADcan be described by:

(27)

dhydrepresents the hydraulic channel diameter and is a friction coefficient. The factor is calledFanning friction factor and takes into account deviations from round channel cross sections. It

has values as summarized in the following table.

The friction factor is typically described as a function of the Reynolds Number Re and changesdepending on the flow regime (laminar, transition or turbulent):

(28)


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Theory


The bounds for the transition region from laminar to turbulent are set by Reynolds numbers of

Relam= 2300 and Returb= 5000. In the turbulent region, turbis considered as a constant input

value. In the laminar region lamis given by

(29)

where a and b are input values. These two parameters are supplied with default values (a=64,

b=-1) according to the Hagen-Poisseuille-Law for laminar tube flow.Table 3-1: Fanning Friction Factor (see VDI ,Lb7 [64page [97]

]

Channel Cross Section

Round 1.00

Square 0.89

Equilateral Triangle 0.83

Sinusoidal (duct open height to open width ratio 0.425) 0.69

The species conservation equation is given by(30)

wk,g is the mass fraction of species kand Deffis an effective diffusion coefficient. Diffusion is

usually small compared to convection but becomes important for small Peclet numbers of masstransfer.

represents the molar reaction rate of the catalytic surface reactions with their

stoichiometric coefficients vi,k.Homogeneous gas phase reactions are not considered, since their rates are negligible in the

temperature range which is typical for automotive applications.Assuming that viscous dissipation can be neglected, the energy balance of the gas phase iswritten as

(31)

where Tgis the gas temperature and hk the total enthalpy of the component k. Conductive heattransport in the gas phase is modeled by Fourier's law using the thermal conductivity g. Thiseffect is usually small compared to convection but becomes important for small Peclet numbersof heat transfer. The third term on the right side takes into account the enthalpy transport due to

species diffusion. khis the heat transfer coefficient between the gas phase and the solid walls,and GSA represents the total channel surface area per unit of substrate volume. The heat of

reaction of the catalytic surface reactions is represented by hi. This heat is released in the

solid phase and convected into the gas phase. Thus, the heat of reaction that is implicitly takeninto account by the combined solution of the gas species and energy conservation equations has

to be deducted from the gas phase (minus sign before the last term) and subsequently added tothe solid phase energy balance equation.


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3. The


The solid phase energy balance equation is given by

(32)

where Tsis the temperature of the catalyst wall, sis its thermal conductivity, and considersa general radial heat transport between radially distributed channels as they are defined by the

Discrete Channel Method (see Section Total and Diffusive Velocitypage [15]

).

The heat loss to the surrounding is captured with . There are two different models available:

1. a simplified heat loss model as described in section Boundary Conditionspage [19]

, where theheat loss of the overall canning and insulation is lumped into a 0D model.

2. In the second modeling approach a 1D model for the multi-layered wall is set-up according

to section Multi-Layered Wall Modelpage [65]

.

Thermal radiation is not taken into account in the energy conservation equation due to the lowtemperature range, as it is given by 'standard' operation conditions. Thus, radiation does not

significantly affect the exit conversion and ignition/extinction bounds.Due to the chemical reactions occurring on the surface of a catalyst, the concentrations of the

species directly above the catalytic surface are not equal to the concentration of species inthe bulk. This effect is accounted for by solving additional balance equations for the individual

species concentrations at the solid surface. Therefore it is possible to take into account for thetwo cases of chemical and mass transfer limitation.Under the assumption of quasi steady-state conditions, the rates of the catalytic surface

reactions balance the diffusive transport from the bulk gas to the surface. Thus, the molar surfaceconcentration (ck

Lof the component j can be evaluated using

(33)

where ck,gB

is the molar concentration of species kin the bulk gas, and kk,mis the mass transfercoefficient of the individual species.The amount of a certain species stored on the surface is represented by a surface fraction . The

conservation of this species on the surface is accounted for by the following equation,

(34)

where the product ( GSA) of the site density and the geometrical reaction surface GSA is

a measure for the entire storage capacity. The right hand side of the equations represents ageneral reaction term depending on the applied storage model.

3.1.4.1. Total and Diffusive Velocity

In systems where the fluid flow is modeled, the velocity of the system and of different species init is an important property. In the current model, the following definition is chosen. One species

moves with its proper velocity vkin one direction of the space domain. The mean velocity (see

Bird et al. [6page [95]

]) of all the species or the entire continuum is given by the following equation:

(35)

The mean velocity vgis the sum of all species velocities vk,gmultiplied by their mass fraction

wk,g. In that way, a mass specific mean velocity or 'mass-averaged velocity' is determined. The


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Theory


difference between the velocity of the mass continuum and that of one single species is calleddiffusive velocity vk

D. The mathematical relation is given by:

(36)

vD

k,grepresents a general diffusive velocity that has to be quantified by additional diffusion

models. In the presented model Fick's first law of diffusion is used (see Taylor and Krishna [62page [97]

]). This decision was made due to the fact that in typical catalytic converter applications,convective fluxes have more influence than diffusive. Thus, errors in the modeling of diffusionhave only minor importance and simplified models are sufficient. Fick's law states that thediffusive velocity vk

Dof a component k of concentration wk,g, across a surface of unit area, is

proportional to the concentration differential multiplied by a system constant Dk, and is expressed

by:

(37)

The system constant Dk,gis called diffusion coefficient of the species k.

3.1.4.2. Enthalpy and Heat CapacityIf a considered phase consists of different species, the mass-specific enthalpy hg of the entirephase can be described as the weighted sum of all theenthalpies hk,g of the different species k:

(38)

The heat capacity of the entire gas is defined as partial derivative of the total enthalpy withrespect to temperature assuming constant composition and pressure

(39)

and the species heat capacity is given by (40)

Assuming ideal gas mixtures (see Barin [4page [95]

]) the species enthalpy hk,galso can be defined

as the partial derivative of the total enthalpy with respect to the species mass fraction:

(41)

3.1.4.3. Heat Conduction and Fourier's Law

Fourier's law states that the area specific heat flow q through a homogeneous phase is directlyproportional to the temperature difference along the path of heat flow multiplied by a system

constant g. In order to comply with the second law of thermodynamics, the negative sign in thefollowing equation is used. Heat only flows from higher to lower temperature:

(42)

Where

Specific heat flow

Thermal conductivity


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3. The


Temperature

3.1.4.4. BOOST Multi-Channel Model and Discrete Channel Method (DCM)The Discrete-Channel-Method (DCM), developed in BOOST (see Wurzenberger and Peters

[77page [97]

, 76page [97]

]) describes the spatial distribution of the converter by locating severalchannels along each radial direction as sketched in the following figure.

Figure 5. Setup of Four Radially Distributed Single Channels for 2D Catalyst Simulation

The thermal and fluid dynamic behavior of each channel in the above figure is represented byconservation equations for mass, momentum and energy as summarized in Section BOOST

Balance Equations, Single Channel Modelpage [13]

. Hence, the solution of these differential

balance equations describes the catalytic converter locally very accurately. This can beunderstood as solution at fine scaleof the individual channel.The distribution of the temperature (Ts) in the radial directions of an entire catalytic converter as

the coarse scaleis assumed to depend on the heat flux through the web walls as shown in Fig. 6page [17].

Figure 6. Radial Heat Transfer in a Catalytic Converter

The comparison of the heat conductivity of the wall material ( s) and the gaseous phase ( g),respectively, shows that the transport of heat in radial direction through gas and ring walls is

negligible.

On this coarse scale, therefore, the converter can be treated as a homogeneous reactor withlocally dependent heat sources. These heat sources are determined by the catalytic conversionreactions as described by the single channel model and the fine scale. An analytical investigation

of such radial heat conduction reaction problems, as given by

(43)

delivers a shape function for radial temperature profiles:

(44)

The radial distribution of the solid temperature Ts(r) is determined by a polynomial function of theorder M that corresponds to the number of single channels models considered. The polynomial

coefficients amare determined by solving Eq.44page [17]

with the known temperatures given at

each single channel. Once the radial temperature profile is known, the heat fluxes at arbitrarypositions can be estimated by applying the gradient of this spatial temperature distribution. Thefluxes entering and leaving each channel in radial directions complete the energy balance andthus, couple the channels. This makes it possible to take into account the spatial behavior of a


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converter through a spatial distribution of temperatures. The benefit of using the above sketched

shape function (Eq.44page [17]

) is computational efficiency. By using an analytically derived shape

function within the numerical solution procedure, the solution of the radial temperature profile isa priori 'pushed' intothe right direction and therefore only very few radial grid points (i.e. singlechannel simulations) are required to get converged results.

3.1.4.5. Thermodynamic and Transport PropertiesThermodynamic and transport properties are required for the simulation of catalytic converters

and the solution of all model equations summarized in Section FIRE Balance Equations

page [10]

.In the present model all physical properties of the fluid change with the temperature, pressureand composition of the gas. The following table briefly summarizes how properties are calculatedand on which reference they are based. For more detailed information see the cited references

and basic literature of fluid mechanics.

Table 3-2: Physical Properties and Calculation Approach

Species Unit Reference

Molecular weight (kmol/kg) tabulated from literature

Specific heat/ Enthalpy/Entropy

(kJ/(kgK)) Polynomial fits from Barin [4page [95]

]

Thermal conductivity (W/(mK)) Polynomial fits from VDI [64page [97]

], and

Reid et al. [60page [97]

]

Viscosity (Pas) Polynomial fits from VDI [64page [97]

], and

Reid et al. [60page [97]

]

Diffusion coefficients (m2/s) Binary acc. to Fuller et al. [15

page [95]],

mixture acc. to Perry et al. [56page [97]

] (WilkeMethod)

The properties given above are internally stored by BOOST for a list of 34 species, as given in

the following table. A detailed description of how the fluid properties are treated by FIRE is givenin the Species Transport Manual.

Table 3-3: Gas Species of Internal Database

Species

C2H2 C5H12 H2 NO3

C2H4 C6H10 H2O O

C2H6 C6H14 HCl O2

C3H4 C6H6 N OH

C3H6 CH3OH N2 SO

C3H8 CH4 N2O SO2

C4H10 CO NH3 SO3

C4H6 CO2 NO

C4H8 H NO2

3.1.4.6. Initial and Boundary Conditions

The equations given in Section FIRE Balance Equationspage [10]

and Section Total and Diffusive

Velocitypage [15]

represent a set of coupled partial differential equations with independent


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3. The


variables time (t), axial position (z) and radial position (r). In order to solve the entire system,initial and boundary conditions have to be defined.

3.1.4.6.1. Boundary ConditionsThe boundary conditions at the catalyst inlet/outlet in axial directions have to be defined by the

user. For the solution of the continuity and momentum balance equations, the model is set upin a way that at one side (inlet) a mass flux has to be defined and at the other side (outlet) apressure has to be given. If the direction of the flow should change, negative mass fluxes can

be applied. The restriction here is that the simulated pressure drop over the entire catalyst is notbigger than the pressure at the outlet. Inlet-temperatures and species mass fractions have to begiven for the solution of the energy and species balance equations.At the outlet either an adiabatic back flow option can be chosen or also outlet temperatures

and species mass fraction can be set. For the solution of the solid energy balance adiabaticconditions were chosen at the inlet and outlet of the converter.In radial direction adiabatic boundaries can be chosen or 'heat loss conditions' have to bedefined.

Figure 7. Radial Heat Loss to the Ambient

The overall heat transfer in radial direction, as sketched in the above figure, is evaluatedconsidering transfer through an insulation material, a shell and a boundary layer. Therefore thefollowing correlation is applied for overall heat flux given in Watt:

(45)

Where

Overall heat flux

Solid temperature at the border (in 1D simulation this radial dependency is

not required)

Overall heat loss coefficient

Diameter of the monolith

Environment temperature

The overall heat loss coefficient , is defined by:

(46)

Where


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Thermal conductivity of the material

Thermal conductivity of the shel

Material position

Shell position

Heat transfer coefficient between the outer surface of the shell and environment

3.1.4.6.2. Initial ConditionsIn the present catalytic converter model all initial conditions are derived from the inlet boundaryconditions and set automatically. Thus, if constant boundary conditions of temperatures orspecies mass fractions are given, these values are used in order to initialize the entire spatial

domain of the converter. If the boundary conditions change as a function of time, the valuecorresponding to the start of integration time is used for the initialization. The initial temperatureof the solid is assumed to be identical to the one of the gas phase. The initial pressure and

velocity field is evaluated using the inlet mass flux, the outlet pressure and the pressure drop ofthe entire converter.

3.1.5. Washcoat Layer Pore Diffusion

3.1.5.1. Pore Diffusion Model

Fig. 1page [6]

in section Principle of Heterogeneous Catalytic Reactionspage [6]

describes theprinciple of the heterogeneous catalytic reactions. Most catalysts are porous materials where thechemical reactions take place in a certain catalytically active layer, the washcoat. Other catalystsconsist of extruded ceramics where the whole porous material is catalytically active. The noble

metals responsible for the catalytic reactions are distributed in the porous reactive material, andthe reactants must diffuse into it. As an example, the following figure shows a catalyst coated

with three different washcoat layers. According to Hayes et al. [21page [95]

], mass transfer of thereactants takes place from the bulk gas onto the solid surface across the boundary layer. Via

pore diffusion the reactants are further transported through and into the washcoat layers wherethe adsorption of the reactants, the chemical reactions and the desorption of the products take

place. Further diffusion causes the transport of the products back to the solid surface, and themass transfer through the boundary layer transports the products back to the bulk gas phase.

Figure 8. Square Cell Catalyst with Washcoat Layers

BOOST/FIRE offers two different approaches to model heterogeneous reactions. In the standardmodel approach, the pore diffusion through the washcoat layer(s) is neglected. This assumptionis valid for unlimited diffusion, where pore diffusion is so fast that every reactant and every

product is uniformly distributed over the whole washcoat layer. This is the reason why the

chemical reaction rate of any reaction ican be related to the catalytic surface area in [kmol/

(sm2_cat)]. By multiplication with the geometrical surface area GSA, the reaction rate


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3. The


is related to the catalyst volume in [kmol/(sm3_cat)], as solved in Eq.32

page [15].

In the advanced model approach, pore diffusion is taken into account. Therefore, every washcoat

layer is discretized in the direction perpendicular to the catalyst solid surface (y-direction). Thefollowing assumptions are made:

Uniform temperature Tsover the whole washcoat layer in y-direction. Diffusion of the species through the washcoat layer in y-direction is the only transport

mechanism, convective transport is neglected. Diffusive transport in axial direction (z-direction) is not accounted for. No species diffusion in the monolith, since the ceramic substrate is assumed to be

catalytically inert. Transport of the species from the bulk gas to the solid surface (y=0) across the boundary

layer is modeled via a Sherwood number based on mass transfer correlation.

The balance equation for species k, solved for every computational cell and obtained from thewashcoat layer discretization over all layers, is described by

(47)

where wclis the porosity (gas void fraction) of the considered washcoat layer.Lis the density

of the gas mixture in the washcoat layer cell, and wkLis the mass fraction of species k. The left

hand side describes the transient change of mass of species kin the washcoat layer. The secondterm on the right hand side is the species source/sink through chemical reactions, where Mk is

the molecular weight of species k, i,kis the stoichiometric coefficient of species kin reaction i,

and is the reaction rate per unit volume washcoat [kmol/(sm3_wcl)]. The first term on the right

hand side is the diffusive transport of the species. The transport model, as described in section

Transport Modelspage [21]

, is used to determine the effective diffusion coefficient Dk,eff.

The boundary condition at the solid surface (y=0) is determined by the balance of diffusive fluxand mass transfer through the boundary layer from the bulk gas to the solid surface and vice

versa, as described by

(48)

kk,mis the mass transfer coefficient of the individual species k,Bis the bulk gas density, and w

kBis the mass fraction of the species in the bulk gas. The second boundary condition at the total

washcoat layer thickness (y= wcl,tot) is simply described by

(49)

leading to no diffusive flux out of the last washcoat layer. The total or entire washcoat layerthickness wcl,totis the sum of the individual layer thicknesses wcl,ilay, as described by

(50)

3.1.5.2. Transport ModelsIn a simplifying way the porous structure of a catalyst washcoat can be seen as complex networkof individual channels of different diameters, lengths and shapes. Diffusive transport in suchsystems can be described by the general Fick's law where the applied diffusion coefficients

need to be appropriately chosen. The pure gas phase diffusion coefficients do no longer applyin small pores with diameters in range of the mean free path of the diffusing molecules. Here


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diffusion is not dominated by fluid-fluid collisions but it changes to fluid-solid collision drivendiffusion where according to kinetic gas theory the 'Knudsen' diffusion takes place, Froment

and Bischoff [14page [95]

]. As discussed by the same authors, models for effective pore diffusioncoefficients in porous systems are widely spread in the literature. They reach from simple modelsincorporating solely the porosity and tortuosity of solid the structure to complex descriptions ofthe pore-network including multi-component diffusion considerations as used in the dusty gas

model applied by Khinast [29page [96]

].

3.1.5.2.1. Effective Pore Diffusion Model

A simple approach to take into account the hindered molecular movement in the porous medium

is described by the effective pore diffusion model. The interaction of the gas molecules with thesolid walls result in a higher diffusion resistance and longer diffusion paths. The tortuosity wcldescribes the locally averaged ratio of actual diffusion length to direct diffusion length. Thus, theeffective diffusion coefficient Dk,eff of the species is smaller than the free gas diffusion coefficient

Dk,g of species k, as described by the equation

(51)

In the numerical implementation for two-component mixtures Dk,g

is the binary diffusion

coefficient, and for multi-component mixtures Dk,gis calculated according to Wilke's approach

(see Froment and Bischoff [14page [95]

]) assuming diffusion of species kthrough the stagnantother species

3.1.5.2.2. Parallel Pore ModelAn often cited model for the effective diffusion coefficient in porous structures is the parallel pore

model (PPM) described by Wheeler [73page [97]

].

The model composes the transport effects of the pure gas phase and Knudsen diffusion

assuming both transport effects taking place in parallel. With this, the effective diffusioncoefficient is defined as

(52)

where DKnis the Knudsen diffusion coefficient depending on pore diameter dpor, molar mass Mofthe considered species and temperature Ts, as described by the equation

(53)

3.1.5.2.3. Random Pore ModelA more complex approach to describe an effective diffusion coefficient is given by the random

pore model (RPM) developed by Wakao and Smith [66page [97]

].

Assuming that the washcoat features two distinct characteristic pore size diameters, called

macro- and micro-pores, the approach of the PPM is first applied to both pores sizes individually.In a second step, the two macro and micro pore diffusion coefficients, DMand , are combined

applying probabilistic and geometrical considerations. This leads to an effective diffusioncoefficient according to the equations

(54)


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3.1.5.3. Reference for Chemistry DataThis topic describes why and how reaction mechanisms formulated with respect to converter

surface are to be converted to washcoat volume.

Conversion of reaction rates from converter surface based to washcoat volume based

The reaction rates in the available Kinetic Modelspage [77]

for catalytic conversions are

formulated with respect to the inner surface area of a converter in units of . On

the other hand, the reaction rate in Eq.47page [21]

is related to the washcoat layer volume and

has units of . Consequently, the converter surface based reaction rates need

to be converted from converter surface based to washcoat volume based units.This conversion is done by multiplying the converter surface based reaction rates with the

specific reactive surface area per unit volume of washcoat, in units of

:

(55)

This conversion is valid, but for any calibrated reaction mechanism related to converter surface

its application in a different catalyst model (variation of converter type and/or washcoat thickness)using the WCL model does not correctly predict the conversion behavior.In order to resolve this issue either all kinetic parameters would have to be transformed

to washcoat layer volume, which would be a huge effort, or a complete new set of kineticparameters would be necessary for the WCL model, which would make the comparison with thesurface reaction model very difficult.

Hence a characteristic number - the reference washcoat layer volume - may beintroduced with which the conversion of converter surface based reaction rates shall be

simplified.

The reference washcoat layer volumeThe reference washcoat layer volume is used to convert reaction rates and a particular setof kinetic parameters considering the reference converter whose conversion behavior is

characterized by this parameter set. It is denoted by and shall be defined as the ratio of

washcoat volume to total monolith volume of the reference converter. can be interpreted

as a reciprocal measure of the noble metal density in the washcoat layer volume.By taking into account some geometrical transformations, the reference washcoat layer volume

for layer ilaycan be calculated with (56)

where CPSM is the cell density per square meter calculated with CPSM = CPSI / (0.0254)2.

The reference washcoat layer volume is used to scale the geometric surface area of a converter

to account for the reference converter's washcoat volume:

(57)


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Tip: The reference washcoat layer volume determines the reference washcoat layervolume (thickness) for which the kinetic parameters are valid. If one uses the samekinetic parameter set, but varies the washcoat layer thicknesses and consequently

the washcoat layer volumes, one has to have the same value of to obtain

reasonable conversion rates.

Example

An example shall demonstrate the effect of the reference washcoat layer volume in differentlayer configurations. Fig. 9

page [24]shows three catalysts A, B, and C with different washcoat

layer coatings. Only the species conversion in the coating called Diesel Oxidation Catalyst (DOC)is considered. Unlimited diffusion is assumed and the same set of kinetic reaction parameters

of the DOC is applied for all three catalysts. Eq.55page [23]

is solved for all species and the

conversion of species k, e.g. C3H6is compared. In coating INERT present in catalyst C onlydiffusion takes place (no chemical reactions).

Figure 9. Example for catalytic conversion in three different washcoat layers

The three samples have the following geometrical parameters:

Parameter Catalyst A(ReferenceConverter)

Catalyst B Catalyst C

Specific ConverterSurface (m

2conv)

DOC Washcoat LayerVolume (m

3wcl)

2

INERT WashcoatLayer Volume (m

3wcl)

- -

To characterize the conversion of the three samples the following two cases shall be highlighted:

1. The absolute amount of noble metals is the same inall samples:When assuming that the absolute amount of noble metals is the same in catalyst A, B and Cit can be expected that the conversion of C3H6is the same for all three samples.The reference washcoat layer volumes for the three samples are:


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3. The


Comparing the overall reaction rate in the DOC layer one finds that all samples lead to the

same C3H6conversion:

The same result would have been achieved if the reaction rate wouldn't have been

converted using the reference washcoat layer volume of the related reference converter:

2. The noble metal density is the same in all samples:In the case of having the same noble metal density in the DOC coating of catalyst A, B andC, the C3H6conversion of catalyst A and C will be the same, whereas it will be twice as large

for catalyst B.As the noble metal density is the same in all catalysts, the reference washcoat layer volumefor the three samples is the same, namely the one from catalyst A:

Comparing the overall reaction rate in the DOC layer one finds that the expectation is metwhen using the proper reference washcoat layer volume:

The reason for this behavior can be found in the way how the kinetic parameters related to thecatalyst surface are transferred into the rates of reactions which take place in a certain washcoatlayer volume. The kinetic parameters contain the information of how many noble metals are

located within the washcoat layer. Using the same parameter set and calculating the specific


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reference volume for both catalysts, A and B, with Eq.56page [23]

, means that the same amount of

noble metals is distributed for catalyst A in volume and for catalyst B in volume 2 . Althoughthe washcoat layer volume of catalyst B is twice as big as that of catalyst B, the noble metal

density is only half. Thus for comparison of washcoat layer coatings of varying thickness with thesame set of kinetic parameters, it is indispensable to use the same value of the specific referencewashcoat volume.

Related Information

Where can I find the reference washcoat layer volume of a catalyst?

3.1.6. General Chemical Reaction Rate Calculation

According to Coltrin et al. [12page [95]

] a chemical reaction can be written in the general form

(58)

where are stoichiometric coefficients and is the chemical symbol for the kth

species. K is thetotal number of species (gas phase and stored) in the system, I is the total number of chemicalreactions considered.

The stoichiometric coefficient of species kin reaction iis defined as:(59)

The rate of production of species kis:

(60)

The reaction rate of reaction iis defined by the difference of forward and backward reactionrates:

(61)

, and are the exponents of concentration of the gas phase species in reaction i. For

elementary reactions these exponents are equal to the stoichiometric coefficients:

(62)

The definition of ck,gdepends on the phase the species is part of. For gas phase species ideal

gas is assumed.

(63)

For stored species the following definition is used:

(64)

The forward reaction rate constant is defined by the following Arrhenius temperature

dependence:

(65)
http://localhost/var/www/apps/conversion/tmp/BOOST_AT_Examples/topics/BOOST_AT_Examples_where_to_find_refWCLVolume.html


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3. The


For irreversible reactions the backward rate constant is zero by definition. For reversible

reactions, the backward reaction rate is evaluated with the forward reaction rate and the

equilibrium constants as:

(66)

is the equilibrium constant in concentration units for reaction i. Coltrin et al. [12page [95]

] notes

that in some cases there are experimental data that indicate the Arrhenius expression for the

reaction rate constant is modified by the coverage (concentration) of some surface species, as

described by:

(67)

, , and are the three coverage parameters for the surface site species kand the reactioni. The -term enhances the Arrhenius expression so that the pre-exponential factor A and the

activation energy E can be written as: (68)

In general, the equilibrium constant is obtained from the standard state Gibbs free energy of

formation:

(69)

where

(70)

(71)

Finally is obtained from via:

(72)

For the cases where no stored species are considered the second term in this equation becomes

'1'.


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3.1.6.1. Sticking CoefficientsFor some simple surface reactions, the rate of the reaction can be calculated via the 'stickingcoefficient' formulation. The sticking coefficient expresses the probability that adsorption of the

molecule on the surface takes place (sticking) when a collision occurs.The sticking coefficient form of the rate equation is allowed for the simple case of a surfacereaction in which there is exactly one gas phase reactant species.The sticking coefficient is calculated via the following expression:

(73)

The three parameters Ai, bi, Ei are the Arrhenius parameters, but in this case Ai and bi are

dimensionless and Ei is in (kJ/(kmolK)).

In order to convert the rate constants given in sticking coefficient formulation to the kinetic rate

constants the following equation is used:

(74)

where Mk,gis the molecular weight of the reaction gas phase species, totis the total surface site

concentration and m is the sum of all the stoichiometric coefficients of reactants that are surface

species. The rate of progress is calculated using Eq.61page [26]

.

3.1.7. Transfer Coefficients

The FIRE/BOOST Aftertreatment Module calculates the transfer coefficients for mass ( j) and

heat ( ) inside the catalytic monoliths based on empirical relations for Nusselt and Sherwoodnumbers.Generally the following functional relations apply

(75)

where Re is the Reynolds number, Pr is the Prandtl number, and Sc is the Schmidt number. For

channel shaped monoliths dhydrepresents the hydraulic channel diameter and l is the channellength.For granulated materials (undirected porosities) dhydrepresents the characteristic pore length,e.g. the solid particle diameter while l is meaningless in that case. The transport coefficients for

heat khand species mass kk,mfinally result from

(76)

where, gis thermal conductivity of the gas mixture and Dk,gis the diffusion coefficient of species

kin the gas mixture.

3.1.7.1. Transfer Coefficients for Directed PorositiesFor laminar flow in circular catalyst channels, literature offers a plethora of functionalrelationships to calculate the actual Nusselt and Sherwood numbers as a function of catalystlength and operating conditions. Most of them are based on the definition of the dimensionlessGraetz numbers for heat and mass transfer:

(77)


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3. The


3.1.7.1.1. Sieder/Tate

FIRE/BOOST suggests the Sieder/Tate relationship (see Perry[56page [97]

]) as a default:

(78)

In addition to the Sieder/Tate approach, FIRE offers additions to the Nusselt/Sherwood relations.

3.1.7.1.2. Hausen

The more general Hausen equation (Perry [56page [97]

]) is described by

(79)

3.1.7.1.3. Hawthorn

The Hawthorn's equation which is suggested by more recent papers (i.e. Ahn et al. [1page [95]

]) isdescribed by:

(80)

3.1.7.1.4. Martin modelVDI [64]

page [97](Chapter Gb) suggests for the heat transfer of a hydraulic and thermal

developing flow in a pipe a correlation from Martin. Kirchner and Eigenberger [30]page [96]

extendthis correlation also to describe the mass transfer between the gas phase and the solid wallsurface. The approaches are given by:

(81)

3.1.7.1.5. Constant and User Defined Transfer CoefficientsIn addition to these, FIRE/BOOST offer the possibility to set constant values for khandkk,m. The FIRE user can also define his own correlation by using the user subroutine

use_cattra.f.The BOOST user can define his own correlation by using the user subroutine

cat_dimless_numbers.f90.


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3.1.7.2. Transfer Coefficients for Undirected PorositiesFIRE offers the possibility to simulate the reactive flow through undirected porosities like packedbeds or granulated materials. These materials are represented by undirected porosities with

arbitrary flow direction through the pores. The following models for the heat and mass transfercoefficients are available:

3.1.7.2.1. VDI Packed BedThe transfer coefficient in a packed bed increases within the first particle layers and approachesa final value rapidly. The heat transfer coefficient in packed beds consisting of spheres of uniformsize is much higher than that of a single sphere. The reason for this is the production of swirl

when the fluid flows through the interstices between the spheres.

As described in the book for Baehr and Stephan [2]page [95]

, the averaged Nusselt number in thepacked bed is proportional to the Nusselt number of a single sphere Nusph:

(82)

The shape factor depends on the fluid volume fraction gaccording to equation

(83)

Eq.82page [30]

can be also applied for packed beds consisting of non-spherical particles. In

VDI [64]page [97]

(Chapter Gh) one can found a list for the shape factors of different particlegeometries:

Table 3-4: Shape Factor of Packed Beds

Particle valid for

Cylinder length L, diameter d 1.6 0.24 < L/d < 1.2

Cube 1.6 0.6 Pr, Sc 1300

Raschig ring 2.1 Pr = 0.7, Sc = 0.6

According to Baehr and Stephan [2]page [95]

the Nusselt number for a single sphere Nusphrequired for Eq.82

page [30]is calculated by

(84)

The Reynolds number Re is calculated with the equivalent particle diameter dPand the interstitial

velocity vg, as described by(85)

For non-spherical particles, dPis defined as the diameter of a sphere with the same surface areaas the particle. If the specific surface area GSA and the number of particles per unit volume nPare known, dPis simple determined by

(86)


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3. The


The Sherwood number for the mass transfer coefficient is calculated by applying the analogyof heat and mass transfer by replacing Nusselt with Sherwood number as well as Prandtl with

Schmidt number. The Sherwood number of the packed bed is proportional to the Sherwoodnumber of the single sphere, as described by

(87)

The Sherwood number of the single sphere can be calculated by

(88)

3.1.7.2.2. Constant and User Defined Transfer CoefficientsFIRE offers the possibility to set constant values for khand kk,m. Furthermore, the FIRE user can

also define his own correlation by using the user subroutine use_cattra.f.

3.1.8. Spray - Reactive Porosity InteractionAs previously mentioned in section Transfer Coefficients for Undirected Porosities

page [30]FIRE

offers the possibility to simulate the reactive flow through undirected porosities. This model alsocalled Reactive Porosity can be used to simulate devices such as coated wiremesh mixers orcatalysts where gas can flow, to some extent, in a radial direction. In such devices the interaction

with urea-water liquid sprays can be complex and requires models more detailed than a simplestop of Lagrangian particles at porosity inlet.The spray - reactive porosity interaction model is composed of three submodels:

The collision submodel checks the probability of collision between the Lagrangian particlesand the solid part of the porous medium.

The interaction submodel, when a collision occurs, considers the type of interactionperformed (deviation, splashing, deposition, )

The enhancement of evaporation and thermolysis in the porous medium and theredistribution of evaporation energy sources to the solid part of the medium.

A user function cyuse_rpor.fhas been added allowing self-modeling of spray-porositysubmodels.

3.1.8.1. The Collision SubmodelThe modeling of the collision of a Lagrangian particle with the solid part of porous medium

follows the lines of the O'Rourke particle collision model [55page [97]

]. More details about thismodel can be found in the Spray Manual.

The porous surface is assimilated to a sphere of diameter where

GSA is the geometrical surface area of the porosity and Vthe volume of the cell where the

droplet is located. The collision frequency between the wall and a droplet of diameter is thenevaluated as:

(89)

where is the droplet velocity. During a time step , the spray particle motion is calculatedwithin a subcycling loop, each iteration of this loop being associated with a specific parcel time

step .

Following O'Rourke, one assumes that the probability that n collisions occur between thedroplet and the porous medium follows a Poisson distribution

(90)


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Thus the probability of no collision is where cis a calculation parameter. A random

number R is then computed to decide whether a collision takes place or not.

If 0 R P0then no collision is calculated. If R > P0then the spray-porosity interaction submodel is applied.

The higher the parameter cis, the closer to zero is P0and therefore the more probable is thecollision.

3.1.8.2. The Interaction SubmodelOnce a collision between a particle and the porous medium is assessed, the interaction is treated

following the lines of the Kuhnke 46page [96]

wall-interaction model (especially developed for theinteraction of urea-water mixture, see the Spray Manual).The Kuhnke model considers four alternative treatments of the interaction depending on thevalues of two parameters (as shown in the following figure):

(91)

Where We is the Weber number and La the Laplace number.

Figure 10. Regime Map for Spray-Wall Interaction According to Kuhnke

In the porous medium, the role of Twallis played by the solid temperatureTs.One cannot use the wallfilm model within the reactive porosity because the liquid film must

be generated on the wall boundary faces of the mesh while the porous domain is essentiallycomposed of internal faces and cells. Therefore the modeling of deposition is not easy. In afirst approximation, the regimes lying below the adimensioned temperature T* - which includedeposition - are neglected and we assume that a particle undergoes only rebound or thermalbreakup.

In the case of thermal breakup, one assumes that the whole mass of the incident particlesgoes into the secondary droplets. One then uses the Kuhnke correlations to estimate the mass,diameter and velocity of these droplets. Mass conservation is ensured by adapting the number of

droplets in parcel in the secondary droplets. The number nsof secondary droplets per collision isa calculation parameter.The angle of the collision is calculated randomly by a Gaussian law assuming the normal-

to-the-wall collision is the most probable (top of the Gaussian curve). This angle is used forthe determination of several characteristics of the rebound/secondary droplets. However, thedirection of the droplet extracted from this estimation is not directly used because the probabilityto obtain droplets moving back to the inlet would be too high and this is not realistic. Instead atransformation is performed, imposing that the direction of the droplet(s) after collision is oriented

with a maximum angle maxfrom the gas direction. The max angle is reached when normalcollision is selected from the Gaussian law (a in the following figure). On the other hand the angleis zero when a tangential collision occurs (b in the following figure). This generates a cone of

possible directions with the gas direction as the centerline and the angle selected randomly fromthe Gaussian law. Furthermore the direction of the rebound/secondary droplets orthogonal to thegas direction is calculated randomly.


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3. The


Figure 11. Droplet Direction after Collision in Porous Medium

left: angles calculated from Gaussian law, right: angles used in the interaction submodel

3.1.8.3. Enhancement of Evaporation & Energy RedistributionIn some devices that can be modeled via the reactive porosity approach - such as wiremeshmixers - the spray evaporation is enhanced compared to the evaporation level in free gas flow. InFIRE the spray model includes evaporation and thermolysis enhancement parameters which acton reactive porosity regions and can be modified by the user.

Using these evaporation enhancement parameters can lead to a rapid decrease of gastemperature and consequently to a quite limited thermolysis of urea. In order to reduce this side

effect, one takes into consideration the idea that the energy used to evaporate the liquid mightcome partly from the solid and partly from the gas.

The spray energy source added to the gas enthalpy ( ) equation is calculated as follows:

(92)

The first term on the right hand side is the enthalpy source associated to the gain of vapor

mass. Its numerical discretization is implicit. The second term Senergis the enthalpy sourceexchanged between the liquid and gas phases, including the heat exchange due to the differenceof temperature between liquid and gas, and to the latent heat of evaporation. The numerical

treatment is explicit.The redistribution of the source term to gas and solid enthalpy equations reads:

(93)

where is the porosity (gas volume fraction) and eis a tuning parameter. The function f is

continuous with regards to both and e. For example, for =0 (full solid) the function is zero andall energy required for the evaporation is extracted from the solid and vice versa for =1 (full gas).For e=0, the function equals 1 and all energy is extracted from the gas. For high values of e,the function tends to zero.

Note that this treatment influences only the sources terms of the gas and solid enthalpyequations. The spray evaporation routine, which makes use of the gas enthalpy and gastemperature and modifies them locally, is not changed with respect to the use of solid enthalpy.

3.1.9. Nomenclature

Units

a Constant in the laminar friction approach (-)

ai Arrhenius parameter (variable)

am Polynomial coefficient of radial temperature shape function (variable)


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A Temperature Dependency Factor (-)

Ai Pre-exponential factor of the rate constant of reaction i (variable)

AD Darcy Constant (kg/(m3s))

AS Surface of the solid part in a computation cell (m2)

Awcl,spec Specific reactive surface area per unit volume washcoat (m2_cat/m

3_wcl)

b Constant in the laminar friction approach (-)bi Temperature exponent in the rate constant of reaction i (-)

ci Arrhenius parameter (kJ/kmol)

ckL

Concentrations of species kin the reactive surface layer (kmol/m3)

ck,g Concentration of species kin the gas phase (kmol/m3)

cp,k Specific heat at constant pressure of species kin the gasphase

(kJ/(kmolK))

cp Specific heat at constant pressure of the entire gas phase (kJ/(kmolK))

cp,s Specific heat of the solid phase (kJ/(kgK))

CPSI Number of channels per square inch (1/in2)

CPSM Number of channels per square meter (1/m2)

d Diameter (m)

dmat Diameter of the insulation mat (m)

dmon Diameter of the monolith (m)

dP Solid particle diameter (m)

dshell

Diameter of the shell (m)

Dk,g Diffusion coefficient of species kin gas mixture (m2/s)

Deff Effective mean diffusion coefficient (m2/s)

dd Droplet diameter (m)

dp Equivalent diameter of solid part in the porous medium (m)

Ei Activation energy of the rate constant of reaction i (kJ/kmol)

Shape factor for packed beds (-)

Fo Forward reaction order (-)

G Anisotropic heat conduction factor (-)

GSA Geometrical surface area (=reactive surface area) of the

catalyst

(m2/m

3)

Gzmass Graetz number for mass transfer (-)

GSA Geometry surface area of the catalyst (=atrans) (m2/m

3)

hg Enthalpy of the entire gas phase (kJ/kmol)

hk,g Enthalpy of species kin the gas phase (kJ/kmol)


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Hf,k Heat of formation of the species k (kJ/kmol)

Hk Enthalpy of species kin the gas phase (kJ/kmol)

hi Heat of reaction i (kJ/kmol)

Hi Enthalpy of reaction i (kJ/kmol)

kh Heat transfer coefficient (W/(m2K))

kk,m Mass transfer coefficient of species k (m/s)

kfi Forward rate constant of reaction i (variable)

kri Backward rate constant of reaction i (variable)

kout Heat transfer coefficient to the environment (W/(m2K))

K K-number (adimensioned) (-)

K Anisotropic heat conduction matrix (W/(mK))

Kci Equilibrium constant in concentration units for reaction i (variable)

Kpi Equilibrium constant in pressure units for reaction i (variable)l Channel, monolith length (m)

La Laplace number (-)

m Sum of all stoichiometric coefficients (-)

m Evaporated liquid mass (kg)

Mg Molar mass of the entire gas phase (kg/kmol)

Mk,g Molar mass of the species kin the gas phase (kg/kmol)

n Normal Vector to the surface cell (-)

ns Number of secondary droplets after collision (-)

nP Particle number density (1/m3)

N Cell density (1/in2), (1/cm

2)

Nu Nusselt Number (-)

OFA Open frontal area of the catalyst (=fluid volume fraction g) (m3/m

3)

pg Pressure (Pa)

P0 Probability of no collision (-)

Pn Probability that n collisions occur (-)Pr Prandtl number (-)

Rate of progress of reaction i (kmol/(m2s))

Heat flux (W/(m2K))

Q Rotation matrix (-)

Radial heat loss flux (W/m)


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r Radial coordinate (m)

General reaction source term (kmol/(m3s))

Reaction rate of reaction iper catalyst surface (kmol/(m2_cats))

Reaction rate of reaction iper washcoat unit volume (kmol/(m3_wcls))

Reaction rate of reaction iper catalyst unit volume (kmol/(m3_cats))

Reaction rate of species k (kmol/(m3s))

R Radius of the monolith (m)

R Universal gas constant (kJ/(kmolK))

Re Reynolds number (-)

Ro Backward reaction order (-)

s Monolith repeat thickness (m)

S Surface of the solid part of a computational cell (m2)

Sg Gas enthalpy source term (from spray evaporation) (W)

Ss Solid enthalpy source term (from spray evaporation) (W)

Senerg Spray enthalpy loss due to evaporation (W)

Sk Entropy of species k (kJ/(kmolK))

Sr Energy source term (W)

Sw,k Mass source term of the species k (kg)

Sc Schmidt number (-)

Sh Sherwood number (-)

Si Entropy of reaction i (kJ/kmol)

t Time (s)

td Droplet calculation time step (s)

Tenv Environment temperature (K)Tg Gas temperature (K)

Ts Solid temperature (K)

Tsat Saturation temperature (K)

Twall Wall temperature (K)

T* Adimensioned solid temperature (-)

ud Droplet velocity (m/s)

vg Mean mass weighed gas velocity (m/s)


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vk,g Velocity of the species kin the gas phase (m/s)

vk,gD

Diffusive velocity of the species kin the gas phase (m/s)

V Volume of a computational cell (m3)

Vg Volume of a computational cell (gaseous part) (m3)

Vs Volume of a computational cell (solid part) (m3)

Specific reference volume of the washcoat layer (m3

/m3

)

Specific volume of the washcoat layer (per catalyst unitvolume)

(m3/m

3)

We Weber number (-)

yk,g Molar fraction of species kin the gas phase (mol/mol)

z Spatial coordinate in Cartesian coordinates (m)

z Length of a calculation cell (m)

Greek Letters

c User-defined collision factor (-)

e User-defined energy redistribution factor (-)

out Heat transfer coefficient between shell and environment (W/m2)

Sticking coefficient of the reaction i (-)

tot Total surface site concentration (mol/m2)

Monolith total wall thickness (m)

wall Monolith wall thickness (m)

wcl Washcoat layer thickness (m)

g Fluid volume fraction in catalyst (=open frontal area OFA) (m3/m

3)

k Surface coverage parameter of surface site species k (-)

s Solid volume fraction in catalyst (m3/m

3)

wcl Porosity of the washcoat

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03 FIRE BOOST Aftertreatment UsersGuide