Solving the Gaseous Velocity Field

Equations of Conservation of Mass of Exhaust Gas Mixture

Mass conservation equations used in the inlet channel, outlet channel and wall of the model are derived from conservation of mass of the exhaust gas mixture as it passes through different locations in the CPF. A detailed derivation of mass conservation equations in the inlet channel, cake+cat+wall and outlet channel control volumes are shown in AppendixA.

The steady form of conservation of mass of the exhaust gas mixture in a control volume in the inlet channel of the particulate filter can be expressed mathematically as:

d

dx(ρ₁v₁) = −4 a

a a^∗

2

ρ_wv_w (3.13)

where v₁ is the velocity of the exhaust gas mixture in the inlet channel and a is the width of the clean inlet channel before application of catalyst washcoat. The effective width of the inlet channel a^∗is calculated as a^∗ = a − 2w_cat−2w_pat a given axial location. Figure3.6is a diagram showing how the effective width of the inlet channel can be calculated knowing

the clean channel width (a), catalyst washcoat thickness (wcat) and instantaneous PM cake layer thickness (w_p). Note that a^∗ is a function of axial location since w_p is a function of the axial location.

Figure 3.6: Diagram showing the calculation of effective inlet channel width

The quasi-steady form of the conservation of mass of the exhaust gas mixture as it passes through the PM cake layer, catalyst washcoat and substrate wall can be expressed as:

ρ_wv^∗_wa^∗∆x = ρ_wv_wa∆x (3.14) where v_w^∗ is the ‘entry velocity’ of the exhaust gas mixture as it enters the cake+cat+wall control volume, vw is the ‘exit velocity’ of the exhaust gas, ρw is the density of the exhaust gas mixture (assumed constant) in the cake+cat+wall control volume and ∆x is the discretized dimension of each cake+cat+wall control volume in the axial (x) direction, which can be expressed mathematically as ∆x = ^L_j where j is the number of discretizations in the axial direction and L is the total length of the CPF.

The steady form of conservation of mass of the exhaust gas mixture in a control volume in the outlet channel of the particulate filter can be expressed mathematically as:

d

dx(ρ₂v₂) = 4

aρ_wv_w (3.15)

where v₂is the velocity of the exhaust gas mixture in the outlet channel.

Equations of Conservation of Axial Momentum of Exhaust Gas Mixture

Steady form of the conservation of axial momentum of the exhaust gas mixture as it passes through the inlet channel is derived in AppendixBand is:

d where P1 is the absolute pressure of the exhaust gas mixture in the inlet channel, F is the fanning friction factor (and assumed constant = 28.454 as discussed in reference [11]) and µ₁ is the dynamic viscosity of the exhaust gas mixture in the inlet channel. This equation results from balancing the convection of axial momentum (LHS term in Equation (3.16)) to the pressure forces (first term on RHS of Equation (3.16)) and frictional forces (second term in Equation (3.16)) that are counteracting this convection.

Steady form of the conservation of axial momentum of the exhaust gas mixture as it passes through the outlet channel is derived in AppendixB) and is:

d

dx ρ₂v²₂
= −dP₂

dx − Fµ₂v₂

a² (3.17)

where P₂ is the absolute pressure of the exhaust gas mixture in the outlet channel and µ₂is the dynamic viscosity of the exhaust gas mixture in the outlet channel.

Numerical Solution Procedure

Solution of the gaseous velocity field to obtain the velocity of the exhaust gas mixture at different axial locations in the inlet channel, substrate wall and outlet channel is carried out by solving a boundary value problem involving a system of two first order ordinary differential equations. AppendixBshows the derivation of the equations used for solving for the gaseous velocity field in the particulate filter. The system of Equations (B.34) is reproduced here for convenience.

where:

where µ_wis the dynamic viscosity of the exhaust gas mixture in the cake+cat+wall control volume, ρ_wis the density of the exhaust gas mixture in the cake+cat+wall control volume, w_p is the thickness of the PM cake layer, kp is the instantaneous permeability of the PM cake layer, w_s is the thickness of the substrate wall, k_s is the instantaneous permeability of the substrate wall, ˙m_in is the CPF inlet mass flow rate and n_cells is the number of inlet channels in the CPF

. In order to solve for the dependent variables N and G as given in the system of Equations (3.18), information that is available about the boundary

values for outlet channel is as follows:

N |_x=0 = 0 } no-slip condition due to end-plug (3.21a)

N |_x=L = M } from mass conservation (3.21b)

The shooting method in reference [67] is employed to convert the problem from a boundary value problem to an initial value problem and a ‘marching’ method is used to solve the system of equations. A 4^th-order Runge-Kutta method [67] is used in this case to march the solution of Equations (3.18a) and Equation (3.18b) simultaneously through the domain (x = 0) to (x = L). The value of G at (x = 0) is initially assumed to be equal to ^M_L. At (x = L), the value of a discrepancy function defined as f = N |_x=L− M is evaluated and a secant method is employed to find the ideal value of G|_x=0 such that f → 0.

Knowing the solution of N , we can recover outlet channel velocity v2as:

v₂ = N

ρ₂ (3.22)

Then, from equation (B.12) in AppendixB, we can find:

v₁ = M − N

ρ₁ (3.23)

Also, from equation (B.1b) in AppendixB, it follows that:

v_w = a Hence, gaseous velocities in the inlet channel (v₁), outlet channel (v₂) and wall (v_w) can be determined knowing the densities of the exhaust gas from the equation of state, Equation (3.1).