Introduction to Monolith Reactor Modelling - KIT - ITCP · Introduction to Monolith Reactor...

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Introduction to Monolith Introduction to Monolith Reactor Modelling:Reactor Modelling:

Mass, Energy and Momentum TransportMass, Energy and Momentum Transport

R.E. HayesR.E. HayesDepartment of Chemical and Materials Department of Chemical and Materials

Engineering, University of Alberta, Engineering, University of Alberta, Edmonton, Alberta, CanadaEdmonton, Alberta, Canada

MODEGAT 14-15 September 2009

22

Objectives of this presentationObjectives of this presentation

•• To give an introductory overview to the To give an introductory overview to the area of monolith reactor modellingarea of monolith reactor modelling

•• To high light some of the factors that To high light some of the factors that must be taken into consideration when must be taken into consideration when choosing a model.choosing a model.

33

Why use computer modelling?Why use computer modelling?

•• Experiments are expensive and take time.Experiments are expensive and take time.

•• Analysis of reactors can be difficult, especially for Analysis of reactors can be difficult, especially for fast highly exothermic reactions. Nonfast highly exothermic reactions. Non--intrusive intrusive measurement methods may not be available.measurement methods may not be available.

•• Modelling can eliminate scenarios that do not Modelling can eliminate scenarios that do not offer promise.offer promise.

• Faster design and optimisation.

Models are not intended to eliminate experiments completely.

44

TypicalTypical catalyticcatalytic converterconverter

Heck and Farrauto, Catalytic Air Pollution Control, 2002

••The catalytic converter uses a metal or ceramic monolith The catalytic converter uses a metal or ceramic monolith support structure.support structure.

••Typical automotive converters use 400 CPSI cell densityTypical automotive converters use 400 CPSI cell density

Hayes and Kolaczkowski, Chem. Eng. Sci., 49, 3587, 1994.

55

ObservationsObservations•• The problem involves the transport of mass, energy The problem involves the transport of mass, energy

and momentum. It is a and momentum. It is a multimulti--physicsphysics problem.problem.

•• Transport occurs at different scales in space.Transport occurs at different scales in space.

•• The transient response of the gas phase and solid The transient response of the gas phase and solid phase occurs at different time scales.phase occurs at different time scales.

•• Modelling of monolith reactors is thus an exercise in Modelling of monolith reactors is thus an exercise in multimulti--scale modelling.scale modelling.

The monolith reactor presents a challenging modelling problem.The monolith reactor presents a challenging modelling problem.

66

Heterogeneous catalytic reactionsHeterogeneous catalytic reactionsHeterogeneous gas phase catalytic reactions typically comprise the same steps.

Fluid flow over the solid results in a boundary layer through which heat and mass diffuse to the gas/solid interface.

Reactants diffuse into the porous catalyst.

Reactants adsorb on the active sites.

The reaction occurs on the active sites, located on the walls of the tortuous pores.

Products desorb and diffuse back to the surface, and then through the boundary layer to the bulk fluid.

Bulk gasExternal catalyst surface

catalyst pore

77

Model ScalesModel ScalesMicro scale: At the atomic/molecular level, the basic chemistry involves a set of adsorption/desorption and surface reaction steps. The genuine mechanism may be very complicated.

Gas Phase

Catalyst

Gas Phase

Catalyst

Meso scale: Single channel of a monolith. Includes gas phase flow, transport to the surface of the washcoat by diffusion, diffusion and reaction in the washcoat. The basic process in a heterogeneous catalytic reaction are present at the meso scale.

Macro scale: Entire catalytic converter, which includes transport and interaction among channels.

88

Model complexity: Points to noteModel complexity: Points to note

•• How much complexity is wanted and or needed?How much complexity is wanted and or needed?– At what level do you wish to model?

– How much information do you want to include?

– Do you need to include all of the scales?

•• The model should be as simple as possible, but not The model should be as simple as possible, but not simpler (Albert Einstein).simpler (Albert Einstein).

•• Models do not have to be perfect to be useful.Models do not have to be perfect to be useful.

Gas Phase

Catalyst

Gas Phase

Catalyst

99

Mathematical formulationMathematical formulation•• The fundamental model is represented by a set of The fundamental model is represented by a set of

equations, which include key assumptionsequations, which include key assumptions–– Algebraic equations, linear or nonAlgebraic equations, linear or non--linearlinear–– Ordinary differential equations (IVP or BVP)Ordinary differential equations (IVP or BVP)–– Partial differential equationsPartial differential equations

•• Practical monolith modelling problems do not have Practical monolith modelling problems do not have analytical solutions; models are solved numerically.analytical solutions; models are solved numerically.

•• The model selected and the way it is represented will The model selected and the way it is represented will determine the numerical method used for its solution.determine the numerical method used for its solution.

1010

Mathematical formulationMathematical formulation

•• Reactors are fundamentally Reactors are fundamentally distributed parameter distributed parameter systems systems ; Concentration, temperature and velocity ; Concentration, temperature and velocity vary with spatial coordinate.vary with spatial coordinate.

•• The dimensionality (and hence complexity) can be The dimensionality (and hence complexity) can be reduced by using average parameter values over reduced by using average parameter values over specified model dimensions. The result is referred specified model dimensions. The result is referred to as a to as a lumped parameter modellumped parameter model..

•• Lumped parameter models are approximations.Lumped parameter models are approximations.

1111

Basic models in common useBasic models in common use

•• Single channel of monolith structureSingle channel of monolith structure–– 1D, 2D or 3D1D, 2D or 3D–– Diffusion in the washcoat may be consideredDiffusion in the washcoat may be considered

•• Whole reactor with variable channel conditions Whole reactor with variable channel conditions and channel interactionsand channel interactions–– 2D or 3D , several model options exist2D or 3D , several model options exist–– The upstream and downstream duct work might The upstream and downstream duct work might

be includedbe included

1212

Single channel model (SCM)Single channel model (SCM)

•• Single channel of monolith structureSingle channel of monolith structure

•• For a For a radiallyradially uniform monolith, the SCM is uniform monolith, the SCM is representative of the whole reactorrepresentative of the whole reactor

•• There are many approaches to writing a single There are many approaches to writing a single channel model:channel model:–– PDE models (all 2D and 3D models, some 1D)PDE models (all 2D and 3D models, some 1D)–– ODE models (some 1D models)ODE models (some 1D models)–– Algebraic equations only (some 1D models)Algebraic equations only (some 1D models)

1313

The 3D SCMThe 3D SCM• The real monolith does not have

axial symmetry.

• The washcoat can have a variety of possible shapes.

• 3D is required for exact channel representation

1414

The 3D SCMThe 3D SCM

• A minimum of a one-eighth section must be considered.

• For complex shapes such as the “sinusoidal” channel, the entire cross section must be considered.

• There is no analytical solution for the velocity.

• Must solve the transport equations in three dimensions.

1515

The 2D approximate SCMThe 2D approximate SCM•• The 2D model approximates the The 2D model approximates the

channel as a right circular cylinder channel as a right circular cylinder with a uniform annular washcoat with a uniform annular washcoat layer, thus giving layer, thus giving axiaxi--symmetry.symmetry.

•• The flow area, washcoat area and The flow area, washcoat area and substrate area should be the same substrate area should be the same for each system.for each system.

•• The 2D model is still the most The 2D model is still the most complex model that is used complex model that is used routinely.routinely.

•• The fully developed flow field can The fully developed flow field can be approximated, thus reducing the be approximated, thus reducing the number of equations to solve.number of equations to solve.

1616

2D and 3D SCM2D and 3D SCM•• 2D and 3D SCM are both distributed parameter 2D and 3D SCM are both distributed parameter

models.models.

•• Temperature, concentrations and velocities vary with Temperature, concentrations and velocities vary with both axial (flow direction) and radial (transverse to both axial (flow direction) and radial (transverse to flow) directions.flow) directions.

•• The most general model requires the solution of a set The most general model requires the solution of a set of coupled nonof coupled non--linear partial differential equations.linear partial differential equations.

•• The equations are the same in each case The equations are the same in each case –– only the only the number of space dimensions changes.number of space dimensions changes.

1717

The 2D SCMThe 2D SCM•• All of the physical/chemical processes present All of the physical/chemical processes present

in 2D and 3D can be illustrated using a two in 2D and 3D can be illustrated using a two dimensional model.dimensional model.

washcoat Diffusion and reaction Conduction

Conductionsubstrate

Centreline of wall

Mass Transfer Heat TransferConvection RadiationDirection of flow

1818

Basic solution procedureBasic solution procedure

•• Each physical space (fluid, washcoat and substrate) Each physical space (fluid, washcoat and substrate) is a computational domain governed by the is a computational domain governed by the conservation equations.conservation equations.

•• Write the governing differential equations for each Write the governing differential equations for each domain (fluid, washcoat and substrate)domain (fluid, washcoat and substrate)

•• Often only the solid state energy balance equations Often only the solid state energy balance equations retain the transient terms, owing to the relative retain the transient terms, owing to the relative magnitude of the time scales for fluid and solid.magnitude of the time scales for fluid and solid.

1919

Modelling the fluid domainModelling the fluid domainVelocity and pressure are obtained from the Navier-Stokes equation and the equation of continuity:

The mass balance (assuming Fick’s law) is:

The energy balance is:Species conservation should be expressed in terms of the mass fraction because mass is conserved.

The density depends on temperature.

( ) ( )( ) ( )23

Tv v p I v v v I⎡ ⎤η⎛ ⎞ρ ∇ = ∇ − + η ∇ + ∇ − − κ ∇⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦i i i

( ) 0v∇ ρ =i

( ) ( ) 0A A AD w v w∇⋅ ρ ∇ − ρ⋅∇ =

( ) ( ) 0f p fk T C v T∇ ∇ − ρ ∇ =i i

2020

ObservationsObservations•• The fluid equations are very classical and can be The fluid equations are very classical and can be

readily solved.readily solved.

•• The mass and energy balances are coupled to the The mass and energy balances are coupled to the washcoat at the gas solid interface.washcoat at the gas solid interface.

•• A zero slip condition is usually imposed at the gas A zero slip condition is usually imposed at the gas solid interface and a zero velocity boundary solid interface and a zero velocity boundary condition imposed.condition imposed.

•• One mass balance equation needed for each One mass balance equation needed for each species. species.

2121

Modelling the washcoat domainModelling the washcoat domain•• The washcoat is a porous matrix containing solid The washcoat is a porous matrix containing solid

and fluid fractions.and fluid fractions.•• The diffusion/reaction occurs in the fluid fraction, The diffusion/reaction occurs in the fluid fraction,

which is comprised of a network of tortuous pores.which is comprised of a network of tortuous pores.•• It is too complicated to model the actual diffusion It is too complicated to model the actual diffusion

in the porous network (and the structure is in the porous network (and the structure is unknown in any case)unknown in any case)

•• Diffusion is thus assumed to occur across the entire Diffusion is thus assumed to occur across the entire matrix and is governed by an effective diffusion matrix and is governed by an effective diffusion coefficient, with a coefficient, with a FickianFickian type of approach.type of approach.

2222

Washcoat balance equations:Washcoat balance equations:

The mass balance equations are solved for the washcoat:

The energy balance is:

Normally the momentum balance equation is not solved in the washcoat.

One mass balance for each species( )eff A A A( ) ( ) 0D w R∇⋅ ρ ∇ − − =

( ) ( ) ( )eff A( )R p S

Tk T H R Ct

∂∇ ∇ + −Δ − = ρ

∂i

2323

Diffusion in the washcoatDiffusion in the washcoat•• Diffusion in the pore is governed by a combination Diffusion in the pore is governed by a combination

of bulk diffusion (molecular collisions) and Knudsen of bulk diffusion (molecular collisions) and Knudsen diffusion (collisions of molecules with the wall).diffusion (collisions of molecules with the wall).

•• The relative importance of the two depends on the The relative importance of the two depends on the gas pressure and the pore size.gas pressure and the pore size.

•• The effective diffusion coefficient is then computed The effective diffusion coefficient is then computed by adjusting for catalyst porosity and by adjusting for catalyst porosity and tortuositytortuosity..

•• For a real catalyst with a given pore size For a real catalyst with a given pore size distribution, the calculation of effective diffusion distribution, the calculation of effective diffusion coefficients is not obvious.coefficients is not obvious.

2424

Diffusion in the washcoatDiffusion in the washcoat•• For binary For binary eqieqi--molar counter diffusion of molecules molar counter diffusion of molecules

A and B we write, for the diffusion of A:A and B we write, for the diffusion of A:

Bulk diffusion: Fuller correlation

Knudsen diffusion

Diffusion coefficient in the poreEffective diffusion coefficient in the washcoat

( ) ( )( )

0.5

1.75A B

AB 0.333 0.333A B

1 10.01013

i i

M M TD

P

⎛ ⎞+⎜ ⎟

⎝ ⎠=+∑ ∑ν ν

( )AA

48.5 pKTdD

M=

( ) ( )AB AA

1 1 1pore KD D D

= +( ) ( )eff A AporeDD ε

=τ

2525

Modelling the substrate domainModelling the substrate domain

The energy balance is:

The substrate is a porous medium and is modelled using effective properties.

( ) ( )eff p S

Tk T Ct

∂∇ ∇ = ρ

∂i

2626

Remarks: 2D and 3D modelsRemarks: 2D and 3D models•• The basic equations are the same for 2D and 3D The basic equations are the same for 2D and 3D

models.models.

•• Solution can be done using commercial or custom Solution can be done using commercial or custom software 3D models will take much longer to solve.software 3D models will take much longer to solve.

•• Normally, a finite element or finite volume method Normally, a finite element or finite volume method will be used.will be used.

•• For steady state models the substrate equations For steady state models the substrate equations can be eliminated.can be eliminated.

2727

Comparing 2D and 3D modelsComparing 2D and 3D models

•• The channel reconfiguration can cause model The channel reconfiguration can cause model differences. differences.

Reactant concentration at two temperatures in a washcoat fillet.

Increasing boundary temperature

2828


Reactant concentration in a complex shape

2929

Comparing 2D and 3D modelsComparing 2D and 3D models•• The following lightThe following light--off simulations show some off simulations show some

comparisons between 2D and 3D models.comparisons between 2D and 3D models.

•• 2D model uses a circular washcoat and a 2D model uses a circular washcoat and a circular substrate (circle/circle)circular substrate (circle/circle)

•• 3D model 1 uses a circular washcoat in a 3D model 1 uses a circular washcoat in a square substrate (circle/square)square substrate (circle/square)

•• 3D model 2 uses a square washcoat in a 3D model 2 uses a square washcoat in a square substrate (square/square).square substrate (square/square).

More et al., Topics in Catalysis, 37 155 (2006)

3030


Inlet Gas Temperature /K

475 500 525 550

Per

cent

age

Con

vers

ion

0

20

40

60

80

100

Circle/circleCircle/squareSquare/square

Inlet temperature ramp rate of 30 K/min at a GHSV of 20 000 h-1 and 80 000 h-1.

Inlet Gas Temperature /K

475 500 525 550 575

Perc

enta

ge C

onve

rsio

n

0

20

40

60

80

100

Circle/circleCircle/squareSquare/square


3131


Inlet temperature ramp rate of 300 K/min at a GHSV of 80 000 h-1.

Inlet temperature / K500 520 540 560 580 600 620 640

Per

cent

age

Con

vers

ion

0

20

40

60

80

100Circle/circleCircle/squareSquare/square


3232

The 1D SCM approximationThe 1D SCM approximation•• 2D and 3D models can be time consuming to run 2D and 3D models can be time consuming to run

and may not be necessary.and may not be necessary.

•• The radial dimension is eliminated using radial The radial dimension is eliminated using radial average properties. The model is transformed into a average properties. The model is transformed into a lumped parameter model in the radial direction.lumped parameter model in the radial direction.

•• Parameter lumping introduces a discontinuity at the Parameter lumping introduces a discontinuity at the gas solid interface which is handled using mass and gas solid interface which is handled using mass and heat transfer coefficients.heat transfer coefficients.

•• The washcoat and substrate are collapsed into a The washcoat and substrate are collapsed into a single one dimensional equation.single one dimensional equation.

3333

Review of boundary layer flowReview of boundary layer flow

δ

U∞U∞

δT

T∞

TS

T∞

δC

C∞C∞

CS

Flow over a flat plate with exothermic reaction at the surface:

Boundary layers develop for velocity, temperature and concentration.

The boundary layer thickness increases with length

The boundary layers have different thickness:

ScSc PrPr

nn n T

nC T C

δδ δ= = =

δ δ δ

3434

Thermal boundary layerThermal boundary layer

δT

T∞

TS

T∞

The heat transfer at the surface is :

The heat transfer can be defined as:

The heat transfer coefficient h assumes a value to make the equality true. If the temperature gradient at the surface is known, then h can be calculated.

0y

Tq k

y =

∂′′ = −∂

( )Sq h T T∞′′ = −

3535

Boundary layers in a tubeBoundary layers in a tube

With reaction at the wall, the concentration boundary layer development is similar.

•• At the entrance, boundary At the entrance, boundary layers grow from the surface.layers grow from the surface.

•• After the entry length, the After the entry length, the boundary layers meet.boundary layers meet.

•• The fully developed region can The fully developed region can be considered a boundary be considered a boundary layer flow.layer flow.

•• For laminar flow in a tube, the For laminar flow in a tube, the velocity, temperature and velocity, temperature and concentration fields can be concentration fields can be obtained from the solution of obtained from the solution of the governing PDE.the governing PDE.

3636

Heat transfer in a tubeHeat transfer in a tube•• In a tube, the flow becomes In a tube, the flow becomes fully developedfully developed after after

an entry length.an entry length.

•• There is no well defined reference temperature There is no well defined reference temperature external to the boundary layer, and therefore the external to the boundary layer, and therefore the heat transfer coefficient is defined in terms of the heat transfer coefficient is defined in terms of the mixing cup temperatures (average radial mixing cup temperatures (average radial temperature)temperature)

•• The mass transfer coefficient is defined in a The mass transfer coefficient is defined in a similar way.similar way.

3737

Heat and mass transferHeat and mass transfer•• Heat and mass transfer coefficients can be Heat and mass transfer coefficients can be

expressed in terms of dimensionless groups, the expressed in terms of dimensionless groups, the NusseltNusselt and Sherwood numbers.and Sherwood numbers.

•• For a circular tube:For a circular tube:

•• The heat and mass transfer coefficients can be The heat and mass transfer coefficients can be calculated by solving the 2D or 3D flow problems.calculated by solving the 2D or 3D flow problems.

Nuh Dk

=A

Shmk DD

=

3838

1D SCM: Model equations1D SCM: Model equations•• The form of the equation depends on the channel The form of the equation depends on the channel

geometry and the basis for the reaction rate:geometry and the basis for the reaction rate:

•• Example is shown for a circular tube with the rate Example is shown for a circular tube with the rate expressed in terms of washcoat volume:expressed in terms of washcoat volume:

3939

1D SCM mass balances1D SCM mass balances

The fluid phase mass balance expressed in terms of the mixing cup concentration and the mass transfer coefficient is:

The axial diffusion coefficient is now a dispersion coefficient.

The velocity is an average velocity.

( ) ( )A, A,A A, A,

4 0f fff m f f S

H

w wvD k w w

z z z D∂ ∂⎛ ⎞ ⎛ ⎞∂ ρρ − − ρ − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ⎝ ⎠⎝ ⎠

i

4040

1D SCM mass balances1D SCM mass balances

The solid phase mass balance is:

Washcoat diffusion is included via the effectiveness factor.

The rate here is evaluated at the surface temperature and concentration.

The rate is expressed in terms of washcoat volume.

( ) ( ) ( )A, A, A2 24

m f f S SWC

D k w w RD D

− = −−

ρ η

4141

1D SCM energy balances1D SCM energy balances

The fluid phase energy balance is:

The solid phase energy balance is:

The fluid thermal conductivity is an effective value that accounts for laminar flow.

( ) ( )4 0f fpI m S ff

dT d Td Ck v h T Td z d z d z D

⎛ ⎞− + − =⎜ ⎟

⎝ ⎠ρ

( ) ( ) ( ) ( ) ( )2 2

A2 22 24S WC S

S S f R P SSSS

T D D TDk h T T H R Cz z tD DD D

⎛ ⎞⎛ ⎞∂ − ∂∂− − − η Δ − = ρ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂−−⎝ ⎠ ⎝ ⎠

The solid thermal conductivity is an average value for washcoat and substrate.

4242

NusseltNusselt and Sherwood numbersand Sherwood numbers

•• The correct values of Nu and The correct values of Nu and ShSh are not obvious.are not obvious.

•• In the fully developed region, the heat and mass In the fully developed region, the heat and mass transfer coefficients are constant. The value transfer coefficients are constant. The value depends on the wall boundary condition.depends on the wall boundary condition.

•• The classical boundary conditions are constant wall The classical boundary conditions are constant wall temperature and constant wall flux.temperature and constant wall flux.

•• With a reaction on the surface, neither boundary With a reaction on the surface, neither boundary condition is true.condition is true.

4343

NusseltNusselt and Sherwood numbersand Sherwood numbers•• For flow in a circular tube with constant physical For flow in a circular tube with constant physical

properties, the classical result is:properties, the classical result is:

4444


•• Equations for the local Nu for Pr=0.7 with combined Equations for the local Nu for Pr=0.7 with combined entry length have been proposed. They can be entry length have been proposed. They can be expressed generically as:expressed generically as:

The value with reaction is found by interpolation between the values for constant wall temperature and constant wall flux.:

2

2 4H HH H H

T T

Nu NuNu Nu Da Nu Da Da NuNu Nu

⎛ ⎞= − + − +⎜ ⎟

⎝ ⎠

1 2501 expNu C C GzGz

⎡ ⎤⎛ ⎞= × + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦The constants depend on channel shape and wall boundary condition

4545


•• However, when the properties are not constant, the However, when the properties are not constant, the entry length can be more complicated:entry length can be more complicated:

1/Gz

0.001 0.01 0.1 1

Nu

3

4

5

6

7

T = 500 KT = 450 KT = 550 K

Local Nusselt number obtained from entry length solutions for laminar flow in a channel with different wall temperature. No reaction.

More, MSc Thesis, University of Alberta, 2007

4646

NusseltNusselt and Sherwood numbersand Sherwood numbers•• So, what should we choose for the correct value?So, what should we choose for the correct value?

•• On the bright side, for circular channels, it might not be On the bright side, for circular channels, it might not be that important:that important:

Axial conversion for a typical monolith channel for CO oxidation.

More, MSc Thesis, University of Alberta, 2007

Axial Distance, m

0.00 0.02 0.04 0.06 0.08 0.10

Per

cent

age

Con

vers

ion

0

20

40

60

80

100

Nu = 4Nu = 3Nu = 5Nu = 30

4747

NusseltNusselt and Sherwood numbersand Sherwood numbers•• However, for nonHowever, for non--circular channels, circular channels,

or nonor non--uniform washcoat, the Nu and uniform washcoat, the Nu and ShSh can vary around the washcoat can vary around the washcoat perimeter.perimeter.

Sherwood number along the washcoat/channel interface for the case of a circular channel in a square substrate. Interface length shown corresponds to 1/8th of the total interface. Fully developed flow.

Hayes et al. Chem. Eng. Sci., 59 3169 (2004).Distance along the interface, microns

0 50 100 150 200 250 300 350 400

She

rwoo

d nu

mbe

r

1

2

3

4

5

6

550K600K650K700K750K800K850K900K

(a)

4848


•• 1D models can give very close agreement to 1D models can give very close agreement to 2D models in many situations.2D models in many situations.

•• Generally the most difficult region to map Generally the most difficult region to map exactly will be the developing flow region, but exactly will be the developing flow region, but that length is often not significant.that length is often not significant.

Several studies in the literature comparing 1D and 2D models,

e.g. Groppi and Tronconi, Aiche Journal 1996.

4949

Simplified 1D modelsSimplified 1D models

•• Dispersion in the gas phase is sometimes ignored, Dispersion in the gas phase is sometimes ignored, and plug flow assumed.and plug flow assumed.

•• If axial conduction in the solid phase is ignored, the If axial conduction in the solid phase is ignored, the model can be reduced to an initial value problem.model can be reduced to an initial value problem.

•• With these two assumptions, the channel can be With these two assumptions, the channel can be modelled as a finite number of stirred tanks in modelled as a finite number of stirred tanks in series.series.

5050

The effectiveness factorThe effectiveness factor

η =Average reaction rate in washcoat

Rate evaluated at surface conditions

Because the washcoat thickness is not explicitly modelled, the washcoat diffusion is incorporated using an effectiveness factor, defined as:

The effectiveness factor can be estimated from a sub-model that usually involves solving a 1D diffusion-reaction problem. Washcoat geometry effects may be included.

Hayes, et al Chemical Engineering Science, 62 2209-2215, 2007.Hayes, et al Chemical Engineering Science, 59 3169-3181, 2004.

5151

Models for the converterModels for the converter

•• In reality the channels in the converter may not In reality the channels in the converter may not all have the same operating conditions.all have the same operating conditions.

•• The substrate constrains the velocity and The substrate constrains the velocity and concentrations to each channel, but heat transfer concentrations to each channel, but heat transfer can occur among the channels.can occur among the channels.

•• A model of the entire converter may therefore be A model of the entire converter may therefore be required. There are two general methods for required. There are two general methods for modelling a converter:modelling a converter:–– Discrete modelsDiscrete models–– Continuum modelsContinuum models

5252

Discrete converter modelsDiscrete converter models

•• In a discrete model both the fluid and solid In a discrete model both the fluid and solid domains are solved:domains are solved:

End view of a quarter section of a circular monolith with imposed temperature gradient.

Heat transfer occurs through a combination of gas and solid with very different properties.

Hayes et al., Catalysis Today, 2009

5353

Discrete converter modelsDiscrete converter models

•• A passenger car converter has A passenger car converter has about 7 about 7 –– 10 thousand channels10 thousand channels

•• Laboratory tests often use 1Laboratory tests often use 1--22””diameter cores with 300 diameter cores with 300 –– 1200 1200 channelschannels

Discretizing the entire physical space result in a problem too large to solve by conventional means. Until new methods are proven typically use ….

5454

Continuum converter modelsContinuum converter models

•• Classical porous media models treat the domain as Classical porous media models treat the domain as a continuum containing fluid and solid phases.a continuum containing fluid and solid phases.

•• The common term for developing the resulting The common term for developing the resulting equations is called volume averaging equations is called volume averaging –– the balances the balances are made on a representative elemental volume are made on a representative elemental volume that contains both fluid and solidthat contains both fluid and solid

•• Each node in the computational domain represents Each node in the computational domain represents both a fluid and a solid both a fluid and a solid –– the spatial identity is lost.the spatial identity is lost.

•• The conservation equations are then fairly classical:The conservation equations are then fairly classical:

Zygourakis, K. (1989). Chemical Engineering Science, 44, 2075-2086.

5555

Momentum balance equationMomentum balance equationVelocity and pressure are obtained from the Volume Averaged Navier-Stokes equation and the equation of continuity:

The velocity is the superficial velocity.

The source term S accounts for the porous inertia.

The Brinkman equation is a classical version of the VANS

( ) ( )( ) ( )23

Tiv v p I v v v I S⎡ ⎤η⎛ ⎞ρ ∇ = ∇ − + η ∇ + ∇ − − κ ∇ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

i i i

( ) 0v∇ ρ =i

212i i fS v C v

Kμ⎛ ⎞= − + ρ⎜ ⎟

⎝ ⎠iv

K is the permeability.

For laminar flow, the second term is zero.

5656

Energy and mass balancesEnergy and mass balances•• There are two main classes of porous media models.There are two main classes of porous media models.•• Heterogeneous model:Heterogeneous model:

–– The fluid and solid temperatures and concentrations The fluid and solid temperatures and concentrations are taken to be different. The two phases are coupled are taken to be different. The two phases are coupled using heat and mass transfer coefficients.using heat and mass transfer coefficients.

•• PseudoPseudo--homogeneous modelhomogeneous model–– The fluid and solid temperatures and concentrations The fluid and solid temperatures and concentrations

are taken to be the same. This situation corresponds are taken to be the same. This situation corresponds to infinitely large heat and mass transfer coefficients.to infinitely large heat and mass transfer coefficients.

For transient simulations the heterogeneous model is recommended.

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Energy balancesEnergy balances

The fluid phase energy balance is:

The solid phase energy balance is:

The effective thermal conductivity for each phase depends on theporosity and material properties.

see Hayes et al., Catalysis Today 2009

( ), , eff, ( )ff P f f P f f f f v f S

TC C v T k T ha T T

t∂

φρ + ρ ∇ = ∇ ∇ − −∂

i i

( ) ( ) ( ), eff, 1

1 ( ) ( )m

SS P S S S v f S R i i

i

TC k T ha T T H R

t =

∂− φ ρ = ∇ ∇ + − + Δ η

∂ ∑i

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Mass balancesMass balances

The fluid phase mass balance is:

The solid phase mass balance is:

Mass diffusion occurs only in the flow direction

Washcoat diffusion is included via the effectiveness factor

( ) ( ) ( ), , , , , ,f i f f i f i m i f m v f i f i Sw v w D w k a w wt∂

φ ρ + ρ ∇ = ∇ φρ ∇ + ρ −∂

i i

( ), ,0 i i m v f i f i SR k a w w= η + ρ −

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Continuum modelContinuum model•• In effect, the continuum model solves a set of 1D In effect, the continuum model solves a set of 1D

single channel models with channel coupling via single channel models with channel coupling via solid phase heat transfer. solid phase heat transfer.

•• However, every channel is not modelled.However, every channel is not modelled.

•• Heat and mass transfer coefficients are the same Heat and mass transfer coefficients are the same as used in the 1D SCM.as used in the 1D SCM.

•• The continuum model consists of a set of coupled The continuum model consists of a set of coupled PDE, which are solved using the FEM or the FVM.PDE, which are solved using the FEM or the FVM.

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Example: Example: 3D reverse flow 3D reverse flow CondorCondorTMTM converterconverter

3D heterogeneous model. 10 litre converter for methane IC engine with reverse flow. Fluent was used, dual zone model.

Liu et al. Computers and Chemical Engineering, 31 292 (2007)

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Model validationModel validation•• All models must be validated for:All models must be validated for:

–– (1) Coding (errors in model implementation) (1) Coding (errors in model implementation) –– (2) Numerical errors (2) Numerical errors –– (3) Simplifying assumptions (3) Simplifying assumptions

•• Models can be validated byModels can be validated by–– Comparison to experimentsComparison to experiments–– Analytical solutionsAnalytical solutions

•• No model can be completely validated.No model can be completely validated.

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Model validation Model validation -- ExampleExample

•• Oxidation of methane in airOxidation of methane in air

•• Reactor wall temperature measured along reactor Reactor wall temperature measured along reactor lengthlength

•• Step changes in methane concentrationStep changes in methane concentration

•• Kinetics determined in separate experimentsKinetics determined in separate experiments

•• 2D single channel model with washcoat diffusion2D single channel model with washcoat diffusion

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Model validation Model validation -- ExampleExample

Hayes et al, I&EC Research, 35 406 (1996)

Heating curve with step increase in methane concentration

Cooling curve when methane switched off

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Concluding remarksConcluding remarks

•• An appropriately chosen model can provide a An appropriately chosen model can provide a useful design aid.useful design aid.

•• Models can be used as a tool to investigate the Models can be used as a tool to investigate the phenomena occurring in the reactor.phenomena occurring in the reactor.

•• Recognize the limitations imposed by the Recognize the limitations imposed by the simplifying assumptions.simplifying assumptions.

•• Good values for the physical properties are Good values for the physical properties are essential.essential.

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R.E. Hayes and S.T. KolaczkowskiR.E. Hayes and S.T. Kolaczkowski

IIntroduction to Catalytic ntroduction to Catalytic CombustionCombustion

Taylor and Francis, 1997Taylor and Francis, 1997

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