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3.3 Process Model of the TWC
3.3.1 Thermodynamic Model of the TWC
The thermodynamic model has been derived from [6]. Figure 3.3 shows a sketch of the heat and mass transfer terms and of the cross sectional geometry of the TWC. Two gas phases and one solid phase have been considered. The first (outer) gas phase includes the gas in the core of the channel. Hence, convective mass flow and heat transfer in the axial direction are considered. Additionally, mass is exchanged with the second gas phase by means of radial diffusion. The second (inner) gas phase contains the washcoat and the boundary layers. Here, no axial convection occurs. Mass is exchanged in the radial direction with the outer gas phase on the one hand by means of diffusion, and with the solid surface on the other hand by means of adsorption or desorption, respectively.
One mass balance per gaseous species can be formulated for the inner and the outer gas phases. Additionally, mass balances for the adsorbed species establish the kinetic model. The axial temperature distribution is assumed to be identical for the two gas phases. Hence, two energy balances, one for the gas and one for the solid phase, can be derived.
The mass balance for the gas phase species iin the channel consists of a convective term, an axial diffusion term and a radial diffusion term, which de- scribes the mass exchange between the two gas phases. Mathematically, this can be expressed as follows:
̺gε∂wi
∂t = εDef f
∂2wi
∂z2 −m˙exh
Acs
∂wi
∂z −DiAgeo(̺gwi−̺wcvi) +wi
X
j
DjAgeo(̺gwj−̺wcvj) (3.1) wi denotes the mass fraction of the species i in the channel along the axial coordinate z, vi the mass fraction in the washcoat. On the right-hand side of the equation, the first and the second terms stand for the diffusive and the convective mass transport. The former is small as compared to the latter and has mainly been accounted for to increase the numerical stability of the solver.
The gas dispersion coefficientDef f and the exhaust gas mass flowm˙exh are assumed to be constant throughout the TWC. The last two terms on the right denote the radial mass exchange between the channel and the washcoat. The
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Outer Gas Phase (Channel) - Gas void fraction
- Channel gas density c - Specific heat capacity e
rg p,g
Inner Gas Phase (Washcoat) - Washcoat porosity - Washcoat gas density e
r
wc wc
Solid Phase (Substrate) - Solid density c - Specific heat capacity rs
s
Mass Balance Terms 1 Exhaust Mass Flow
(Convection/Diffusion) 2 Mass Transfer to/from Pores
(Radial Diffusion) 3 Adsorption/Desorption
1
6 5
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Dwc Dchan
Energy Balance Terms 1 Exhaust Mass Flow 4 Heat Transfer solid/gas phase 5 Reaction Heat
6 Heat Conduction in solid phase
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x y
x z
y
x z
A - Specific geometric catalyst surface
geo
A - Specific catalytic active surface
cat
A - Cross-sectional areacs
Figure 3.3: Sketch of the TWC structure and the terms of the balance equations. View from the front (on the left) and from the side (on the right) of the TWC.
last term is necessary to keep the sum of all mass fractions at 1, see also Ap- pendixB.1. The geometry parameters are illustrated in Figure3.3. Di is the radial mass transfer coefficient. With mass flows ranging from5to50 g/sand a viscosity of 3.4·10−5Ns/m2(at 800 K), the Reynolds number hardly exceeds 500. Hence, the flow is laminar and a constant Sherwood numberShDof2.47 can be assumed for a triangular shaped tube, see [50]. Di can be calculated from the Sherwood number:
ShD= 2.47 = DiDchan
DiN2
(3.2) DiN2 is the binary diffusion coefficient of the speciesiin N2. It is calculated as proposed by Füller et. al., see [84]:
DiN2 = 143·Tg1.75 pexhp
Mi,N2
hΣ1/3v,i + Σ1/3v,N2i2 (3.3)
whereΣvis the diffusion volume and pMi,N2 =
√2 q 1
Mi +M1
N2
. (3.4)
Notice that the units ofMi,N2andDiN2are g/mol and cm2/s, respectively.
The mass balance for the speciesiin the inner gas phase does not contain any convective or axial diffusion terms, only radial mass transport occurs. On the channel side, mass is exchanged with the channel phase by means of radial diffusion. On the solid surface side, mass is exchanged by means of adsorption and desorption. Since the shape of one channel is roughly an isosceles triangle, the reference volumes for the washcoat and the gas phase have been assumed as follows:
dVwc = 3DwcDchandz dVg =
√3
4 D2chandz
Hence, the mass balance equation for speciesiin the washcoat/boundary layer reads as follows:
̺wcεεwc4√ 3Dwc
Dchan
∂vi
∂t = DiAgeo(̺gwi−̺wcvi)
−vi
X
j
DjAgeo(̺gwj−̺wcvj) + ∆ri−vi
X
j
∆rj (3.5)
Notice that no gas phase reactions have been assumed to occur. The first two terms on the right-hand side denote the mass exchange with the channel phase. The subsequent terms describe the mass exchange with the solid phase by means of sorption. Thereby, ∆ri stands for the mass transfer of species i. Again, a balancing term is required, which ensures that the sum of all mass fractions remains 1. The mass transfer term is calculated from the reaction rates involving sorption. For speciesiwith a total sorption ratersorptione. g. from the noble metal, the mass transfer rate is obtained as follows:
∆ri =rsorption,iLsAcatMi (3.6) The calculation ofrsorptionwill be discussed in Section3.3.2.Lsis the surface storage capacity of the noble metal (LN M) or the ceria (LCer).
The energy balance for the gas phase incorporates a heat conduction term and a convective heat transport term in the axial direction (1stand 2ndterm in (3.7)). In the radial direction, heat is exchanged with the solid phase by means of “convective” heat transfer (3rdterm in (3.7)). Thus, the energy balance for the gas phase can be formulated accordingly:
̺gεcp,g∂Tg
∂t = ελg∂2Tg
∂z2 −m˙exh
Acs
cp,g∂Tg
∂z +αAgeo(Ts−Tg) (3.7) Notice again that no gas phase reactions have been assumed to occur. λg de- notes the heat conductivity in the gas phase,αthe heat-transfer coefficient be- tween the gas and the solid phase. Again, a constant Nusselt number of2.47 can be assumed in the laminar flow, see also [50]. Thus,αcan be calculated using the following expression:
N uD= 2.47 = αDchan
λg
(3.8) The energy balance for the solid phase only includes axial heat conduction (1st term in (3.9) with the heat conductivity λs) and the heat transfer (2nd term) in the radial direction. Additionally, heat production from the chemical reactions on the solid surface occurs (3rdterm):
̺s(1−ε)cs
∂Ts
∂t = (1−ε)λs
∂2Ts
∂z2 −αAgeo(Ts−Tg) +Acat
X
j
−∆Hj·Rj (3.9)
∆Hjdenotes the enthalpy of the speciesj,Rjthe net production rate.
Since the heat production of each reaction is somewhat difficult to estimate, only the net production (if negative: consumption) rates of CO, CO2, H2O, C3H6, and NO have been accounted for. The enthalpies of O2, H2, and N2are zero. Additionally, the reaction enthalpy of the ceria oxidation has been taken into account:
Ce2O3+1
2O2(g) → 2CeO2 (3.10)
The values for the enthalpies used in the model are listed in AppendixB.2.
This simple approach of course influences the transient behaviour of the TWC temperature. Since temperature changes on the solid phase occur relatively slowly, as long as phenomena such as light-off are omitted, this method for the calculation of the heat production has been assumed to be sufficiently accurate.