03 FIRE BOOST Aftertreatment UsersGuide

Users Guide

FIRE BOOST Aftertreatment

2

Contents

1. Introduction... 4

2. Overview... 5

3. Theory... 6

3.1. Catalytic Converter Model... 6

3.1.1. 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... 40

3.2.3. 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. Thermal Coupling 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... 83

3.8.5. 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

4.1. 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... 175

4.1.9. 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 ... 180

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

5.1.11. Input Data 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. Solver Databus Channels... 267

5.3. Simulation Results...267

5.3.1. 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

4

1. Introduction

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

1.1. Scope

This 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 strictly observed 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.

monospace To indicate a command, a program or a file name, messages, input/

output on a screen, file contents or object names.

MenuOpt A 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 applying procedures in this manual.

2. Overview

The FIRE/BOOST Aftertreatment Module enables the simulation of the chemical and physical processes 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 also for heterogeneous soot regeneration reactions. The solution of continuity, momentum, species and energy balances in the gas phase coupled with the solid phase energy conservation and chemical 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 regeneration

With the FIRE/BOOST aftertreatment models and their results, a broad range of aftertreatment applications 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 models containing gas phase species and species stored on the surface. The kinetic rate equations are defined via a standard Arrhenius type rate law or via user models. The chemical equilibrium and 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 different catalysts 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 user has 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 all the regeneration types listed above. Free access to any reaction model, with an arbitrary number of reactions and species, is offered by user-routines that can be linked to BOOST and FIRE.

6

3. Theory

3.1. Catalytic Converter Model

Availability

BOOST AT: Catalyst page [187]

FIRE: Catalyst Specification page [100]

3.1.1. Principle of Heterogeneous Catalytic Reactions

In this section effects are discussed that should be considered when a mathematical formulation for 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 overall combustion 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 on the rate of the catalytic reaction.

• Most catalysts are porous materials. Much of the chemical reactions occur inside the porous catalyst, 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. [21 page [95]]) shows the individual steps taking place during a heterogeneous catalytic reaction. As discussed by Froment and Bischoff [14 page [95]] the following steps can be distinguished:

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

boundary layer.

2. Diffusion of the reactants into the porous catalyst. Since the main part of the catalyst is

located 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.

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 the following 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 is conserved.

This allows to model steps 4, 5 and 6 either separately (i.e. Oxygen storage on the surface) or in one 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 by numerous authors in literature (i.e. Voltz et al. [65 page [97]], Chen et al. [11 page [95]] and Wanker et al.[67 page [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 be considered 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 Oxygen molecule 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 is a dummy identifier for one surface site.

3.1.2. General Approaches and Assumptions

In the following section general approaches considering catalytic converter modeling are briefly summarized. For more detailed information please refer to the literature cited.

3.1.2.1. Cell Specification of Honeycomb-Type Catalytic Converter

The 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. This washcoat is a porous solid layer that covers the solid substrate as shown in the following figure.

8

Figure 2. Structure of a Squared Cell Monolith

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

(1) where wall is the thickness of the substrate wall and wcl,tot is the thickness of the washcoat

layers. The repeat distance s of the monolith can be derived from the cell density CPSM according 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 the hydraulic channel diameter dhyd, open frontal area OFA and the geometric surface area GSA as

shown 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 total

wall thickness , the latter can be calculated from the first three data. Therefore the above given equations have to be inverted. The cell density is given by

(7)

and the total wall thickness of the monolith is

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.7

page [8]

and Eq.8 page [8] show that

three 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.7 page [8] and

Eq.8 page [8] where the hydraulic diameter d_hyd and the open frontal area OFA are needed.

The above calculated values of CPSM and wall are exact for squared cells. If other geometries

(round, sinusoidal channel) are given, the derived values of CPSM and wall have 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 Moles

In general the balance of mass or moles is equivalent and therefore leads to the exact same results. 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 balance equation, the change of the total number of moles has to be taken into account by an additional correction 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 molar mass which is necessary to transform mass specific values to mole specific data is not always completely accessible.

3.1.2.3. Volume Fraction, Density and Mass Fraction

Catalytic 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 given by means of an overall volume fraction. This volume fraction of gas phase in the entire system is defined as follows:

(9)

where g is the volume fraction of the phase g(as) in the entire volume V. The volume of the solid

phase Vs is evaluated by Vs=(1- g)V = sV. Note, the fluid volume fraction g is identical to the

open frontal area OFA.

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

(10)

The density of the entire phase g is the sum of the densities of all different species k in it. In an

additional step the mass fraction wk,g of one species in a system can be defined as the fraction of

the density of the species k,g and the total density g

(11)

10

3.1.2.4. Equation of State and Ideal Gas Law

If conservation equations for a gaseous phase are given, a general relation between the

intensive variables of the gas is necessary. Pressures and temperatures observed during typical catalytic converter applications lie within moderate ranges (p<10bar, T<3000K). Thus, the ideal gas law is sufficient as equation of state in the present models.

(12)

The mass density g is directly proportional to the pressure pg, the total molar mass Mg, and it is

indirectly proportional to the temperature Tg and the ideal gas constant R. The molar mass is a

function of the composition of the different species k in the considered phase:

(13)

M_k,g represents the molar mass of the species k in the gas phase.

3.1.3. FIRE Balance Equations

The modeling of the balance equation for the solid energy is presented in this section. These equations are solved in addition to all other transport equations (momentum, gas phase enthalpy, turbulence quantities, species transport ...) if the aftertreatment module is activated.

3.1.3.1. Solid Energy Balance Equation

The following basic equation rules the anisotropic heat conduction within the solid part of the porous medium:

(14)

where K is the anisotropic heat conduction matrix, kh is the gas-solid heat transfer coefficient,

GSA the geometric surface area per unit volume, Tg and Ts the gas and solid temperature, Sr

the chemical reaction source, V the cell volume, Vs the solid volume part of the cell and As the

surface of the solid part of the cell.

3.1.3.1.1. Anisotropic Heat Conduction Matrix

The presence of channels implies that the conduction does not have the same magnitude in cross-stream (radial) as in streamwise direction; in other words in the porous medium, the solid heat conduction is anisotropic.

3.1.3.1.1.1. Anisotropic Conduction Factor

The default approach assumes that crosstream and streamwise solid thermal conductivity are linearly linked; i.e. they differ only from a so-called anisotropy factor. The matrix solid heat conduction K reads

(15)

where is the conduction matrix in the genuine catalyst reference frame and Q and its inverse are transfer matrices from the genuine catalyst reference frame to the Cartesian reference frame.

3.1.3.1.1.2. Effective Thermal Conductivity including Radiation

This 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 d is the hydraulic diameter. The length s is the unit cell width of the catalyst derived from the density number (cpsi) N as: .

The heatflux exchanged between faces at temperature T1 and T2 can be written

(16)

where L is 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 of

thermal 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 unit cell. If one assumes thermal equilibrium of all unit cells contained into a mesh cell, the relation extends to mesh cells as:

(18)

where h is the distance between two mesh cell centers. Based on relation (4) one can build the anisotropic 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.

12

3.1.3.1.1.3. Solid Surfaces

The diffusion fluxes must be computed along the solid surface of the cells As . For monoliths with

a preferential flow direction (e.g. monoliths with channel shaped geometry: DPFs, catalysts) the solid surface vectors are calculated different than for catalytic blocks without any preferential flow direction (e.g. undirected porosities: packed beds). For the first case one can create the following assumption:

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 second case of catalytic blocks with undirected porosities, the surface reduction is uniform in all directions. Thus, the solid surface vectors are computed by the relation:

(21)

3.1.3.1.2. Diffusion Terms Calculation

The following formula is used to compute the diffusion fluxes on the cell face j. It is derived from the 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. dj is the distance between cell centers Pj and Pi.

3.1.3.1.3. Boundary Conditions

As 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.22 page [12]). The local wall heat transfer coefficient is then proportional to the solid thermal conductivity and inversely 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 Equation

The term Sr (W) accounts for heat sources due to catalytic chemical reactions. It is calculated

using the species' reaction rates and the corresponding enthalpies of formation using the following formula:

where is the reaction rate of species k (kmol/(m3·s)) and is the formation enthalpy of species k at 298 K.

3.1.3.2.2. Sources in the Species Conservation Equations

The following sources are added for each species k to the right-hand side of the corresponding species transport equation:

(24) where is the reaction rate of species k (kmol/(m3·s)).

3.1.4. BOOST Balance Equations, Single Channel Model

Under the assumption that radial transport effects of a honeycomb-type catalytic converter are small 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 following figure. The effects taking place are convective, diffusive and conductive transport in the gas phase, 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- and fluid 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 can be written as shown in the following section.

The continuity equation of the gas phase is

(25)

where g is the density of the gas phase, t is the time, vg is the interstitial gas velocity and z is the

spatial coordinate in axial direction.

The momentum conservation equation is given by the steady-state Darcy equation (see Kaviani [26 page [96]])

(26)

where pg is the pressure of the system. The Darcy constant AD can be described by:

(27)

dhyd represents the hydraulic channel diameter and is a friction coefficient. The factor is called

Fanning 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 changes depending on the flow regime (laminar, transition or turbulent):

14

The bounds for the transition region from laminar to turbulent are set by Reynolds numbers of Re_lam = 2300 and Re_turb = 5000. In the turbulent region, _turb is considered as a constant input value. In the laminar region _lam is 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 [64 page [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 k and Deff is an effective diffusion coefficient. Diffusion is

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

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 is written as

(31)

where Tg is the gas temperature and h k the total enthalpy of the component k. Conductive heat

transport in the gas phase is modeled by Fourier's law using the thermal conductivity g. This

effect is usually small compared to convection but becomes important for small Peclet numbers of heat transfer. The third term on the right side takes into account the enthalpy transport due to species diffusion. k_h is 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 h i . This heat is released in the

solid phase and convected into the gas phase. Thus, the heat of reaction that is implicitly taken into 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 to the solid phase energy balance equation.

The solid phase energy balance equation is given by

(32)

where T_s is the temperature of the catalyst wall, _s is its thermal conductivity, and considers a general radial heat transport between radially distributed channels as they are defined by the Discrete Channel Method (see Section Total and Diffusive Velocity page [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 Conditions page [19], where the heat 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 Model page [65].

Thermal radiation is not taken into account in the energy conservation equation due to the low temperature 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 in the 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 the two 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 surface concentration (c k

L

of the component j can be evaluated using

(33)

where ck,g

B

is the molar concentration of species k in the bulk gas, and kk,m is the mass transfer

coefficient 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 a general 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 in it is an important property. In the current model, the following definition is chosen. One species moves with its proper velocity vk in one direction of the space domain. The mean velocity (see

Bird et al. [6 page [95]]) of all the species or the entire continuum is given by the following equation: (35)

16

difference between the velocity of the mass continuum and that of one single species is called diffusive velocity v_kD. The mathematical relation is given by:

(36) vDk,g represents 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 [62

page [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 diffusion have only minor importance and simplified models are sufficient. Fick's law states that the diffusive velocity vk

D

of 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,g is called diffusion coefficient of the species k.

3.1.4.2. Enthalpy and Heat Capacity

If a considered phase consists of different species, the mass-specific enthalpy hg of the entire

phase can be described as the weighted sum of all the enthalpies hk,g of the different species k:

(38)

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

(39)

and the species heat capacity is given by

(40)

Assuming ideal gas mixtures (see Barin [4 page [95]]) the species enthalpy hk,g also 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 directly proportional 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 the following equation is used. Heat only flows from higher to lower temperature:

(42)

Where

Specific heat flow

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 [77 page [97], 76 page [97]]) describes the spatial distribution of the converter by locating several channels 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 by conservation equations for mass, momentum and energy as summarized in Section BOOST Balance Equations, Single Channel Model page [13]. Hence, the solution of these differential balance equations describes the catalytic converter locally very accurately. This can be understood as solution at fine scale of the individual channel.

The distribution of the temperature (T_s) in the radial directions of an entire catalytic converter as the coarse scale is assumed to depend on the heat flux through the web walls as shown in Fig. 6

page [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 with locally dependent heat sources. These heat sources are determined by the catalytic conversion reactions 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 the

order M that corresponds to the number of single channels models considered. The polynomial coefficients am are determined by solving Eq.44

page [17]

with the known temperatures given at each single channel. Once the radial temperature profile is known, the heat fluxes at arbitrary positions can be estimated by applying the gradient of this spatial temperature distribution. The

18

converter through a spatial distribution of temperatures. The benefit of using the above sketched shape function (Eq.44 page [17]) is computational efficiency. By using an analytically derived shape function within the numerical solution procedure, the solution of the radial temperature profile is a priori 'pushed' into the right direction and therefore only very few radial grid points (i.e. single channel simulations) are required to get converged results.

3.1.4.5. Thermodynamic and Transport Properties

Thermodynamic 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, pressure and composition of the gas. The following table briefly summarizes how properties are calculated and 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/(kg·K)) _{Polynomial fits from Barin [4} page [95]

]

Thermal conductivity (W/(m·K)) _{Polynomial fits from VDI [64} page [97]

], and Reid et al. [60 page [97]]

Viscosity (Pa·s) _{Polynomial fits from VDI [64} page [97]

], and Reid et al. [60 page [97]]

Diffusion coefficients (m2/s) _{Binary acc. to Fuller et al. [15} page [95]

],

mixture acc. to Perry et al. [56 page [97]] (Wilke Method)

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 given in 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 Equations page [10] and Section Total and Diffusive Velocity page [15] represent a set of coupled partial differential equations with independent

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 Conditions

The 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 up in a way that at one side (inlet) a mass flux has to be defined and at the other side (outlet) a pressure 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 not bigger than the pressure at the outlet. Inlet-temperatures and species mass fractions have to be given 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 adiabatic conditions were chosen at the inlet and outlet of the converter.

In radial direction adiabatic boundaries can be chosen or 'heat loss conditions' have to be defined.

Figure 7. Radial Heat Loss to the Ambient

The overall heat transfer in radial direction, as sketched in the above figure, is evaluated considering transfer through an insulation material, a shell and a boundary layer. Therefore the following 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:

20

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 Conditions

In the present catalytic converter model all initial conditions are derived from the inlet boundary conditions and set automatically. Thus, if constant boundary conditions of temperatures or species 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 value corresponding to the start of integration time is used for the initialization. The initial temperature of 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 of the entire converter.

3.1.5. Washcoat Layer Pore Diffusion

3.1.5.1. Pore Diffusion Model

Fig. 1 page [6]in section Principle of Heterogeneous Catalytic Reactions page [6] describes the

principle of the heterogeneous catalytic reactions. Most catalysts are porous materials where the chemical reactions take place in a certain catalytically active layer, the washcoat. Other catalysts consist 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, and the 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. [21 page [95]], mass transfer of the reactants 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 where the 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 the mass 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 standard model approach, the pore diffusion through the washcoat layer(s) is neglected. This assumption is 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 i can be related to the catalytic surface area in [kmol/ (s·m2_cat)]. By multiplication with the geometrical surface area GSA, the reaction rate

is related to the catalyst volume in [kmol/(s·m3_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). The following assumptions are made:

• Uniform temperature Ts over 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 the washcoat layer discretization over all layers, is described by

(47)

where wcl is the porosity (gas void fraction) of the considered washcoat layer. L

is the density of the gas mixture in the washcoat layer cell, and w k

L

is the mass fraction of species k. The left hand side describes the transient change of mass of species k in the washcoat layer. The second term on the right hand side is the species source/sink through chemical reactions, where M k is

the molecular weight of species k, i,k is the stoichiometric coefficient of species k in reaction i,

and is the reaction rate per unit volume washcoat [kmol/(s·m3_wcl)]. The first term on the right hand side is the diffusive transport of the species. The transport model, as described in section

Transport Models page [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 flux and mass transfer through the boundary layer from the bulk gas to the solid surface and vice versa, as described by

(48)

kk,m is the mass transfer coefficient of the individual species k,

B

is the bulk gas density, and w

k B is 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 layer thickness wcl,tot is the sum of the individual layer thicknesses wcl,ilay, as described by

(50)

3.1.5.2. Transport Models

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

22

diffusion is not dominated by fluid-fluid collisions but it changes to fluid-solid collision driven diffusion where – according to kinetic gas theory – the 'Knudsen' diffusion takes place, Froment and Bischoff [14 page [95]]. As discussed by the same authors, models for effective pore diffusion coefficients in porous systems are widely spread in the literature. They reach from simple models incorporating solely the porosity and tortuosity of solid the structure to complex descriptions of the pore-network including multi-component diffusion considerations as used in the dusty gas model applied by Khinast [29 page [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 the solid walls result in a higher diffusion resistance and longer diffusion paths. The tortuosity wcl

describes the locally averaged ratio of actual diffusion length to direct diffusion length. Thus, the effective diffusion coefficient D_k,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 D_k,g is the binary diffusion coefficient, and for multi-component mixtures Dk,g is calculated according to Wilke's approach

(see Froment and Bischoff [14 page [95]]) assuming diffusion of species k through the stagnant other species

3.1.5.2.2. Parallel Pore Model

An often cited model for the effective diffusion coefficient in porous structures is the parallel pore model (PPM) described by Wheeler [73 page [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 diffusion coefficient is defined as

(52)

where DKn is the Knudsen diffusion coefficient depending on pore diameter dpor, molar mass M of

the considered species and temperature T_s, as described by the equation

(53)

3.1.5.2.3. Random Pore Model

A more complex approach to describe an effective diffusion coefficient is given by the random pore model (RPM) developed by Wakao and Smith [66 page [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, DM and , are combined

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

3.1.5.3. Reference for Chemistry Data

This 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 Models page [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.47 page [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 kinetic

parameters would be necessary for the WCL model, which would make the comparison with the surface reaction model very difficult.

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

The reference washcoat layer volume

The reference washcoat layer volume is used to convert reaction rates and a particular set of 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 ilay can 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:

24

Tip: The reference washcoat layer volume determines the reference washcoat layer

volume (thickness) for which the kinetic parameters are valid. If one uses the same kinetic 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 different layer 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.55 page [23] is solved for all species and the conversion of species k, e.g. C3H6 is compared. In coating INERT present in catalyst C only

diffusion 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 (Reference Converter) Catalyst B Catalyst C Specific Converter Surface (m2conv)

DOC Washcoat Layer Volume (m3wcl)

2·

INERT Washcoat Layer Volume (m3wcl)

-

-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 in all samples:

When assuming that the absolute amount of noble metals is the same in catalyst A, B and C it can be expected that the conversion of C3H6 is the same for all three samples.

Comparing the overall reaction rate in the DOC layer one finds that all samples lead to the same C₃H₆ conversion:

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 and C, the C3H6 conversion 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 volume for 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 met when using the proper reference washcoat layer volume:

26

reference volume for both catalysts, A and B, with Eq.56 page [23], means that the same amount of noble metals is distributed for catalyst A in volume and for catalyst B in volume 2· . Although the 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 the same set of kinetic parameters, it is indispensable to use the same value of the specific reference washcoat 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. [12 page [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 the total number of species (gas phase and stored) in the system, I is the total number of chemical reactions considered.

The stoichiometric coefficient of species k in reaction i is defined as:

(59) The rate of production of species k is:

(60)

The reaction rate of reaction i is defined by the difference of forward and backward reaction rates:

(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,g depends 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:

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. [12 page [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 k and the reaction i. 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'.

28

3.1.6.1. Sticking Coefficients

For some simple surface reactions, the rate of the reaction can be calculated via the 'sticking coefficient' 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 surface reaction in which there is exactly one gas phase reactant species.

The sticking coefficient is calculated via the following expression:

(73)

The three parameters A i , b i , E i are the Arrhenius parameters, but in this case A i and b i are

dimensionless and E _i is in (kJ/(kmol·K)).

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,g is the molecular weight of the reaction gas phase species, tot is 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.61 page [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 Sherwood numbers.

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 dhyd represents the hydraulic channel diameter and l is the channel

length.

For granulated materials (undirected porosities) d_hyd represents the characteristic pore length, e.g. the solid particle diameter while l is meaningless in that case. The transport coefficients for heat kh and species mass kk,m finally result from

(76)

where, _g is thermal conductivity of the gas mixture and D_k,g is the diffusion coefficient of species k in the gas mixture.

3.1.7.1. Transfer Coefficients for Directed Porosities

For laminar flow in circular catalyst channels, literature offers a plethora of functional

relationships to calculate the actual Nusselt and Sherwood numbers as a function of catalyst length and operating conditions. Most of them are based on the definition of the dimensionless Graetz numbers for heat and mass transfer:

3.1.7.1.1. Sieder/Tate

FIRE/BOOST suggests the Sieder/Tate relationship (see Perry[56 page [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 [56 page [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. [1 page [95]]) is described by:

(80)

3.1.7.1.4. Martin model

VDI [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] extend this correlation also to describe the mass transfer between the gas phase and the solid wall surface. The approaches are given by:

(81)

3.1.7.1.5. Constant and User Defined Transfer Coefficients

In addition to these, FIRE/BOOST offer the possibility to set constant values for kh and

30

3.1.7.2. Transfer Coefficients for Undirected Porosities

FIRE offers the possibility to simulate the reactive flow through undirected porosities like packed beds 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 transfer coefficients are available:

3.1.7.2.1. VDI Packed Bed

The transfer coefficient in a packed bed increases within the first particle layers and approaches a final value rapidly. The heat transfer coefficient in packed beds consisting of spheres of uniform size 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 the packed bed is proportional to the Nusselt number of a single sphere Nusph:

(82) The shape factor depends on the fluid volume fraction g according to equation

(83)

Eq.82 page [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 particle geometries:

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 Nu_sph required for Eq.82 page [30] is calculated by

(84)

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

velocity vg, as described by

(85)

For non-spherical particles, dP is defined as the diameter of a sphere with the same surface area

as the particle. If the specific surface area GSA and the number of particles per unit volume nP

are known, dP is simple determined by

The Sherwood number for the mass transfer coefficient is calculated by applying the analogy of 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 Sherwood number 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 Coefficients

FIRE offers the possibility to set constant values for k_h and k_k,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 Interaction

As 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 – also called Reactive Porosity – can be used to simulate devices such as coated wiremesh mixers or catalysts 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 simple stop 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 particles and the solid part of the porous medium.

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

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

A user function cyuse_rpor.f has been added allowing self-modeling of spray-porosity submodels.

3.1.8.1. The Collision Submodel

The 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 [55 page [97]]. More details about this model 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 V the volume of the cell where the droplet is located. The collision frequency between the wall and a droplet of diameter is then evaluated as:

(89)

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

32

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

number R is then computed to decide whether a collision takes place or not. • If 0 R P0 then no collision is calculated.

• If R > P₀ then the spray-porosity interaction submodel is applied.

The higher the parameter c is, the closer to zero is P0 and therefore the more probable is the

collision.

3.1.8.2. The Interaction Submodel

Once a collision between a particle and the porous medium is assessed, the interaction is treated following the lines of the Kuhnke 46 page [96] wall-interaction model (especially developed for the interaction of urea-water mixture, see the Spray Manual).

The Kuhnke model considers four alternative treatments of the interaction depending on the values 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 Twall is played by the solid temperature Ts.

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 essentially composed of internal faces and cells. Therefore the modeling of deposition is not easy. In a first approximation, the regimes lying below the adimensioned temperature T* - which include deposition - are neglected and we assume that a particle undergoes only rebound or thermal breakup.

In the case of thermal breakup, one assumes that the whole mass of the incident particles goes 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 ns of secondary droplets per collision is

a 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 for the determination of several characteristics of the rebound/secondary droplets. However, the direction of the droplet extracted from this estimation is not directly used because the probability to obtain droplets moving back to the inlet would be too high and this is not realistic. Instead a transformation is performed, imposing that the direction of the droplet(s) after collision is oriented with a maximum angle max from the gas direction. The max angle is reached when normal

collision is selected from the Gaussian law (a in the following figure). On the other hand the angle is 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 from the Gaussian law. Furthermore the direction of the rebound/secondary droplets orthogonal to the gas direction is calculated randomly.

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 Redistribution

In some devices that can be modeled via the reactive porosity approach - such as wiremesh mixers - the spray evaporation is enhanced compared to the evaporation level in free gas flow. In FIRE the spray model includes evaporation and thermolysis enhancement parameters which act on reactive porosity regions and can be modified by the user.

Using these evaporation enhancement parameters can lead to a rapid decrease of gas

temperature 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 might come 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 Senerg is the enthalpy source

exchanged between the liquid and gas phases, including the heat exchange due to the difference of 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 e is a tuning parameter. The function f is

continuous with regards to both and e. For example, for =0 (full solid) the function is zero and

all 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 enthalpy equations. The spray evaporation routine, which makes use of the gas enthalpy and gas temperature and modifies them locally, is not changed with respect to the use of solid enthalpy.

3.1.9. Nomenclature

Units