The following figure shows a zoom on the different soot, ash and wall layers as they are used and understood in the present soot deposition and regeneration approach. The model considers three reactive layers where soot regeneration and catalytic reactions can take place. These are the soot cake, the soot depth layer and the wall itself. The ash layer is assumed to be inert with respect to chemical reactions and mass changes. It is reasonable to keep the ash mass constant (i.e. solve no dedicated ash balance equations) since no noticeable changes of the ash mass are expected within the time ranges given by soot loading and regeneration events.
Figure 21. 1D Slice of Soot Cake, Ash Cake, Soot Depth Layer and Filter Wall
3.2.4.1. Soot Cake and Depth Layer Balance Equation
Two distinct balance equations are applied to capture the transient changing soot mass in both the cake and depth layer. These are
(147)
(148)
where msc(z) and msd(z) are the volume specific soot mass (soot loading with respect to the filter volume) in the cake and depth layer at each axial position of the filter, respectively. Rsd and Rsc describe general soot reaction source terms in the different layers. vw,dl(z) represents a dimensionless wall velocity (see Eq.149 page [53]) at each axial position, and msoot,inl is the specific soot mass flow entering the filter. Ssd is a binary switch to steer soot deposition in the depth filtration layer. Soot deposition in the depth layer is switched off as soon as an ash or soot cake is present or the depth filtration capacity is reached. The binary switch Ssc controls the soot deposition in the cake layer. Soot deposition in the cake is switched on as soon as the depth filtration layer has reached its full capacity. The application of switches to steer the soot loading either in depth or cake layers needs to be understood as a coarse phenomenological approach.
Under the assumption that all soot particulates follow the streamlines in the inlet channel, the wall velocity can be used as 'weighting function' in order to distribute the entire incoming soot mass over the effective filter length. Therefore a dimensionless wall velocity defined by
(149)
is used in the present approach. The crucial characteristic of the new velocity is that its shape is similar to the original velocity and that its integral sum over the entire effective filter length is equal to one. With this approach it is possible to describe the effect that higher soot deposition rates are given at spatial locations with lower soot heights (i.e. lower pressure drops and higher wall velocities).
3.2.4.2. LLD Concept and Regeneration Reactions
The soot regeneration and catalytic wall reaction schemes summarized in Section Filter Regeneration with Oxygen page [90] - Filter CSF Catalytic Reactions page [92] comprises soot combustion and catalytic conversion reactions to describe:
• Bare trap regeneration with O2
• fuel additive regeneration
• low temperature regeneration with NO2
• catalytically supported NO2 regeneration
• CSF conversion of CO, C3H6, C3H8 and NO in the catalytic filter wall and in the outlet channel
• CSF Selective Catalytic Reduction in the catalytic filter wall and in the outlet channel 3.2.4.2.1. Soot Regeneration and CSF Reactions in the filter wall
The reaction schemes take place along the streamlines of gas flow passing through the soot cake layer, the soot depth layer and the wall. Provided that transport effects are negligible in directions other than those given by the wall velocity, a 1D isothermal steady-state fixed-bed model (direction x in Fig. 21 page [52]) can be applied for small soot slices (Local Layers) given by an axial discretization. The advantage of this Local Layer Discretization is its computational performance and its flexible application to various types of reaction mechanisms taking place with different reaction layers.
The balance equation for the gas phase continuity is given by
(150)
54
where vw is the wall velocity at each axial position, g is the gas density and x represents the spatial coordinate over the height of the soot cake, soot depth layer and filter wall. Mj is the molar mass of the species j and S denotes the total number of species. The stoichiometric coefficient of the species j in there action i is given by i,j that multiplies the molar reaction rate r i (yg,Ts) of the ith reaction.
The total number of reactions is represented by R. Catalytic conversion without surface storage reactions have no impact on the global continuity of the gas phase. Reactions where solid soot or stored solid species are involved contribute to the continuity source. Thus, the right-hand side of Eq.150 page [53] directly equals to changes of the solid soot/stored solid species mass.
The species conservation equation over the height of the soot cake, soot depth layer and wall is (151)
where wg,k represents the gas mass fraction of species k. The first term on the right side
considers changes in the species composition due to all reactions involved. The second term on the right side (chain rule) is the spatial derivative of the gas density at constant species fractions.
Eq.150 page [53] and Eq.151 page [54] represent an initial value problem that can be integrated from the top of the soot cake down to the bottom of the filter wall. The initial conditions at the topmost layer are given by
(152) (153) assuming that the species gas composition does not change significantly over the length of the inlet channel.
In low mass flow bare trap or fuel additive regeneration cases, where the assumption of a constant O2 concentration along the inlet channel is not valid due to O2 diffusion into the soot layer, a 1D isothermal steady-state model is applied for the inlet channel gas flow. Sinks for this model are the convective mass flow through the wall and a diffusive oxygen mass flow into the soot layer. The oxygen diffusion term, there, is proportional to an artificial diffusion coefficient and a concentration gradient which comes from the local oxygen concentration in the inlet channel and the corresponding concentration at the bottom of the soot layer, a result from the Local Layer model described above. This model approach represents an initial value problem that can be integrated over the particulate filter length giving an oxygen profile along the inlet channel as well as diffusive oxygen fluxes into local soot slices given by the axial discretization. These results are used as an input for the Local Layer regeneration model described above.
The solution of the 1D fixed-bed model (Eq.150 page [53] to Eq.154 page [54]) delivers overall soot reaction source terms for both the cake and the depth layer. Therefore the model is integrated
(154)
(155)
over the individual heights of the two layers. The sources Rsc and Rsd are applied in the transient balance equations of two soot layers, Eq.147 page [53] and Eq.149 page [53].
3.2.4.2.2. CSF Reactions in the PF Inlet and Outlet Channels
While passing through the inlet and outlet channels, the exhaust gas comes into contact with the catalytic surface of the filter wall. As an example in particulate filters (PF), which are (zone) coated in the rear part, the unconverted gas which passes the filter wall in the front part comes into contact with the catalytic surface in the outlet channel rear part and reacts there.
To describe the species conversion in the inlet and outlet channel, 1D isothermal steady-state models are applied. The discretization is given by the PF monolith discretization in axial direction.
Below the conservation equation for the outlet channel is derived as representative example.
The same applies to the inlet channel with one exception - the sign of the wall flow term, which is leaving the inlet channel.
Figure 22. 1D Slice of the Outlet channel
The steady-state species conservation equation of the PF outlet channel is given by
(156)
where j,g,2 is the density of the gas phase species j in the outlet channel and vg,2 is the gas velocity. z is the spatial coordinate in axial direction. AF,2 represents the free channel cross section that is available for the gas flow and PS,n is the wet perimeter of the free channel cross section of the outlet channel. j,g,w is the density of the gas phase species j at the bottom of the filter wall entering the outlet channel and vw,2 represents the corresponding wall velocity lateral to the axial direction. Mj is the molar mass of the species j and S notes the total number of species.
The stoichiometric coefficient of the species j in the reaction i is given by i,j that multiplies the molar reaction rate r i (cj,g,L,Ts) of the ith reaction.
Beside the temperature, the reaction rates depend on the species concentrations cj,g,L on the surface of the outlet channel filter wall. The total number of reactions is represented by R.
Due to the chemical reactions occurring on the surface of the catalytic filter wall and due to the gas flow coming from the inlet channel passing through the filter wall, the concentrations cj,g,L of the species directly above the catalytic outlet channel surface are not equal to the concentration of species in the outlet channel bulk.
This effect is accounted for by solving additional balance equations for the individual species concentrations at the outlet channel surface, taking into account the mass transfer limitation.
Under the assumption of quasi steady-state conditions and neglecting the gas flow through the filter wall in a first step, the rates of the catalytic surface reactions balance the diffusive transport from the bulk gas to the surface. Thus, the molar surface concentration (cj,g,L of the component j can be evaluated using
(157)
where cj,g,2 is the molar concentration of species j in the bulk gas, and kj,m,corr is a mass transfer
56
The impact of the gas flow passing through the filter wall on the species surface concentrations cj,g,L is considered in the calculation of the mass transfer coefficients kj,m,corr. These coefficients are evaluated with a Sherwood correlation and then corrected with a wall flow term
(158) The equations above represent an initial value problem that can be integrated from the beginning of the outlet channel to the end. The initial conditions at the first layer are given by
(159)
3.2.4.3. LLD Sources for Lumped Filter Model
By integrating the LLD model of Section LLD Concept and Regeneration Reactions page [53] over the entire height of all layers (from x=0 to x=xwall in Fig. 21 page [52] and over the outlet channel), source terms for the continuity, species and energy balance equations can be derived. The overall source term of species j is given by
(160)
This source term comprises the impact of reactions taking place within all different reaction layers (cake, depth, wall, outlet channel). The source term of the gas phase continuity equation is the sum of all species sources. It is given by
(161)
where this equation comprises the sum of the two soot sources from the depth (Eq.154 page [54]) and cake (Eq.155 page [54]) filtration layer as well as the sources from stored surface species. The enthalpy source term results to
(162)
where hg,j represents the enthalpy of species j.
The source terms summarized in this section are applied in the framework of transient non-isothermal, two-phase models solved by BOOST/FIRE in order to capture the transient behavior of loading and regeneration. More details on this integration are given in Section Modeling Glueing Zones in SIC PFs page [57].