Multiphysics Modeling for Exhaust Gas Treatment
Henrik von Schenck, COMSOL AB, Sweden
© COPYRIGHT 2008, COMSOL, Inc
Contents
• What is Multiphysics?• Capabilities and opportunities of COMSOL Multiphysics• Multiphysics modeling for exhaust gas treatment
– Case 1: Selective catalytic reduction of NO– Case 2: Abatement of VOC in a packed bed– Case 2: Abatement of VOC in a packed bed– Case 3: Diesel particulate filter (DPF)
• Concluding Remarks
COMSOL
• Started in 1986 with agency products, markets only own products now.• Released COMSOL Multiphysics in1998.• 180 employees worldwide.• 16 offices, 12 in Europe, 3 in the US and 1 in India.• Distributors worldwide.
COMSOL Products
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What is Multiphysics?
• Reliable simulation requires accurate mathematical modelsElectromagnetics:
Maxwell’s equations
Structural Mechanics:Newton’s laws of motion
Thermal Analysis:Heat transfer equation
Fluid Flow:Navier-Stokes’ equations
• “Single physics” approach is limited since no phenomenon is isolated
• Today’s engineering challenges demand that multiphysics be addressed
Newton’s laws of motion Navier-Stokes’ equations
The Multiphysics Approach
Mass, Energy and Momentum Transport
Mass, Energy and Momentum Transport
Select from
Mass, Energy and Momentum Transport
Select from predefined modeling interfaces
Select from
Create your multiphysics model
Mass, Energy and Momentum Transport
Select from predefined modeling interfaces
model
Predefined Modeling Interfaces – Fluid Flow
• Example - Navier-Stokes equations for fluid flow
( ) ( )( )[ ] FuuIuuu +∇+∇+−⋅∇=∇⋅+
∂∂ Tpt
ηρρ
( )∂ρ ( ) 0=⋅∇+∂∂
uρρt
Predefined Modeling Interfaces – Fluid Flow
• Couplings– Transport properties (ρ, η) dependent
upon• Temperature• Fluid composition
– Flow field affects– Flow field affects• Convective mass and energy
transport• Turbulent mixing
Predefined Modeling Interfaces – Mass Transport
• Example - Convection, diffusion and reaction
( ) iiiii cRcDt
c ∇⋅−=∇−⋅∇+∂∂
ut∂
Predefined Modeling Interfaces – Mass Transport
• Couplings– Affected by
• Convective transport (u)• Temperature (reaction rates, Di)
– Affects• Local mixture composition• Chemical reactions generate or
consume energy
Predefined Modeling Interfaces – Heat Transfer
• Example – Energy transport by convection and conduction
( ) TCQTkt
TC pp ∇⋅−=∇−⋅∇+
∂∂
uρρt∂
Predefined Modeling Interfaces – Heat Transfer
• Couplings– Affected by
• Convective transport (u)• Chemical composition• Exothermic/endothermic reactions
– Temperature affects– Temperature affects• Reaction rate• Transport properties
Coupled Transport Processes
Flow
HeatMass
Convectivetransport
Chemical reactions
Exothermic reactions
Reaction rates
Gas expansion
Equation Based Modeling
• Enter any PDE in general or coefficient form
FΓ =⋅∇+∂∂+
∂∂
tt
φφ2
2
NO Reduction in a Catalytic Converter
• Competing reactions– NO reduction by NH3
– NH3 oxidation
• Eley-Rideal kinetics3
3
111NH
NHNO ac
acckr
+=
322 NHckr = )//(22
2 TRgEeAk −=
)//(11
1 TRgEeAk −=
• Honeycomb monolith with V2O5/TiO2 catalyst
• A single monolith channel• Circular cross-section
approximation
NO Reduction in a Catalytic Converter
approximation
catalytic wash-coat
channel inlet
0.36 m
Model Equations
• Fluid flow– Coupled free and porous media
flow– Navier-Stokes equations– Brinkman equations
Free flow
Porous media flow
Model Equations
• Mass transport– N2, NO, NH3, O2, and H2O
transport through convection and diffusion in the open channel
– Diffusion and chemical reaction in the catalytic wash-coat
Non-reactive transport
the catalytic wash-coat
• Energy transport– Convection and conduction in the
open channel– Conduction and heat source due to
reaction in the porous structure
Chemical reaction
Modeling in COMSOL Multiphysics
VOC Abatement in a Packed Bed Reactor
• Parallel reactions– Hydrocarbon conversion– CO oxidation
OHCOOHC 22263 6692 +→+
22 22 COOCO →+
• Kinetic expressions2
11 )1(
6363
2
HCHCcoco
COCO
cKcK
cckr
++=
2
22 )1(
6363
263
HCHCcoco
COHC
cKcK
cckr
++=
• Reactor Equations– Mass balance on the macro-scale;
convection, diffusion and reaction
Model Equations
( ) iiii cRcD ∇⋅−=∇−⋅∇ u
reactor pore scale ~mm
– Ri depends on the transport in the pellets, i.e. the flux into at the pellet surface times surface area per unit volume
– A pellet mass balance is required to calculate the flux
Model Equations
• Pellet Equations– Mass balance on the micro-scale;
diffusion and reaction
( ) iii RcD ′=′∇′−⋅∇
rp
pellet pore scale ~µm
– Boundary conditions
– The concentration distribution in the pellet gives the flux at all r => the reaction term for the catalyst bed is given by the solution of the micro-scale mass balance
0=⋅′∇′− nii cD 0=r
ii cc ε=′prr =
Model Equations
• 2 geometries– Reactor– Pellet
• Coupling variables connect the mass transport equations on mass transport equations on each geometry
– Reactor bulk concentrations are coupled to pellet surface concentrations.
– Pellet species surface flux is coupled to reactor mass source term
ii cc ε=′ nN ⋅== )( pipi rrAR
• Reactor mass transport– C3H6, CO, CO2, H2O, and O2
– Convection, diffusion and reaction
Modeling in COMSOL Multiphysics
( ) cRcD ∇⋅−=∇−⋅∇ u( ) iiii cRcD ∇⋅−=∇−⋅∇ u
Modeling in COMSOL Multiphysics
• Coupling variables– Couple dependent
variables on different geometries
– Pellet =>reactor
nN ⋅== )( pipi rrAR
Modeling in COMSOL Multiphysics
• Pellet mass transport– C3H6, CO, CO2, H2O, and O2
– Diffusion and reaction
( ) iii RcD ′=′∇′−⋅∇ ( ) iii RcD ′=′∇′−⋅∇
Modeling in COMSOL Multiphysics
• Coupling variables– Couple dependent
variables on different geometries
– Pellet =>reactor
– Reactor=>pellet
nN ⋅== )( pipi rrAR
ii cc ε=′
Results – Reactor Species Distribution
Results – Pellet Species Distribution
Results
xr
Capture and Combustion of Soot in a DPF
5.86x4.66x8 inches
Model Equations
• Fluid flow– 1000’s of channels– Assume fully developed laminar
flow in the channels– Average flow field if proportional
u1
u2w
Hp1
p2
H/2 ∆xvm
to the pressure difference– Overall mass balance gives the
velocity in the channels– The channels are connected by
mass transfer across the porous membrane
( ) mvH
pkt 1111 4 ρρρ −=∇−⋅∇+
∂∂
( ) mvH
pkt 1222 4 ρρρ =∇−⋅∇+
∂∂
( )21 ppvm
m −=ηδ
κ
Model Equations
• Soot balance– Soot enters channel 1– Deposition at the membrane
results in a sink term
u1
u2w
Hp1
p2
H/2 ∆xvm
( ) smsss cv
HccD
t
c 41 −=+∇−⋅∇+
∂∂
u
Model Equations
• Species mass balances– O2, CO, and CO2
– O2 sink terms– Soot oxidation– Transfer across membrane
u1
u2w
Hp1
p2
H/2 ∆xvm
– Transfer across membrane
( ) sOmOOoO R
Hcv
HccD
t
c 441,211,21,22
1,2 −−=+∇−⋅∇+∂
∂u
( ) 2,222,22,222,2 4
OmOOOO cv
HccD
t
c=+∇−⋅∇+
∂∂
u
Model Equations
• Soot layer thickness, δs
– Decreases through oxidation– Increases by deposition of soot
particles in the exhaust gas– Affects vm
u1
u2w
Hp1
p2
H/2 ∆xvm
ms
ss
s
ss vc
RM
t ρρδ +−=∂
∂
Model Equations
• Energy balances– Channels– Filter walls; this temperature field
is connected for the entire system such that heat flow between channels
u1
u2w
Hp1
p2
H/2 ∆xvm
channels
( ) ( )11111111111
11
444TTh
HQ
HTvC
HTCTk
t
TC msmppp −++−=+∇−⋅∇+
∂∂ ρρρ u
( ) ( )22222222222
22
44TTh
HTvC
HTCTk
t
TC mmppp −+=+∇−⋅∇+
∂∂ ρρρ u
( ) ( ) ( )2111122 2 TTTh
TCTCv
Tkt
TC m
mpmp
m
mmm
mpmm −−+−−=∇−⋅∇+
∂∂
δρρ
δρ
Predefined Modeling Interfaces
• 9 coupled partial differential equations
• Use predefined modeling interfaces
– Pressure driven flow; Darcy’s – Pressure driven flow; Darcy’s Law interface
– Mass transport; Convection and Diffusion interface
– Energy transport; Convection and Conduction interface
The General Form PDE
• The general form PDE
• The equation for the soot
FΓ =⋅∇+∂∂+
∂∂
tt
φφ2
2
• The equation for the soot layer thickness
ms
ss
s
ss vc
RM
t ρρδ +−=∂
∂
Equation System View
• All predefined equations are viewable and editable
– Modify anisotropic transport properties; permeability and thermal conductivities
• PDEs displayed on a general form
FΓ =⋅∇+∂∂+
∂∂
tt
φφ2
2
Results – Flow Field
Results – Temperature Distribution
Results – Oxygen Concentration
Results – Soot layer in a central channel
Results – Soot layer in a peripheral channel
Concluding Remarks
• COMSOL offers a simulation environment for unlimited Multiphysics couplings
• The Chemical Engineering Module provides many of the Module provides many of the equations describing fluid flow, mass, and energy transport in predefined modeling interfaces
• You can also type in your own equations directly into the graphical user intefarce
• Model library– NO reduction, VOC abatement,
DPF– Model set up and solved + Model
doc
Resources and Contact
– COMSOL Multiphysics + Chemical Engineering Module ~100 models
• Introduction to Chemical Engineering Simulations CD
• COMSOL Conference CD 2008• Contact, software trial, training and
support
Resources and Contact
support– www.comsol.com– [email protected]
Thank you for your attention!