Diesel Particulate Filter Model With Detailed Permeability Analysis

Abstract

Recent legislation of engine exhaust Particulate Matter (PM) emission levels cannot be met with in-cylinder PM reduction techniques, thus resulting in the need for a Diesel Particulate Filter (DPF). Modern DPFs use a honeycomb of long channels with porous walls in order to filter PM with near 100% efficiency. They must be designed to balance trapping efficiency and pressure drop, as flow restriction decreases engine efficiency. This paper describes the construction of two Matlab models in order to predict properties within the filter. Two methods for simultaneously solving the differential conservation equations along with the algebraic ideal gas law in the inlet and outlet channels have been developed. The first method solves the channel equations by transforming the differential algebraic equations (DAEs) into an ordinary differential equation (ODE) system. In addition, a second method is developed that directly solves DAE systems of index-one. In order to link the inlet and outlet channel profiles, modeling of the wall flow is necessary. Four permeability models from different disciplines are used in Darcy’s law to determine their applicability in calculating DPF wall velocity profiles. The resulting inlet, wall, and outlet parameters are compared with published results to demonstrate each model’s accuracy.

Combining the Classical and Lumped Diesel Particulate Filter Models

“…The volume of PM per pore involves the entire PM that is stored in the wall zone that is currently being analyzed, divided by the total number of pores (Np) in the wall at the current location: (15) (16) where the denominator is an equivalent pore volume and Veo is the empty volume of the wall at the location under analysis: (17) It is anticipated that Np is grossly overestimated as it assumes that all pores will be available for PM capture, which is untrue. This is remedied through ρ sw , the packing density of the PM particles in the filter structure.…”

“…(15), the average wall PM mass per pore is found: (25) where the total number of pores in the zone is calculated as a fraction of the total number of pores in the filter: (26) and Npa is the total number of pores in the filter: (27) with Vft equal to the total solid filter volume. Now, from Eqn.…”

“…Similar to previous efforts [15], the density and velocity in each channel (I and II) is combined into an effective flow term: (36) Drawing from experience with the two-channel model, a slight adaptation is made here with respect to this variable for the inlet channel. In specific, the velocity in the inlet channel (u I ) is equal to zero at the end of the channel.…”

“…Moreover, in order to reduce the numerical difficulty in resolving a differential-algebraic set of equations, the derivative of the ideal gas law is employed [15]. As a result, the governing ordinary differential equations for the inlet channel are: with the PM, wall density and velocity updated within the ODE solver routine using the computed values of pressure for the inlet and outlet channels.…”

Catalyzed diesel particulate filter modeling

“…They employ the implicit Nystr ö m method to solve for the flow equations and incorporate a NetwonRaphson method for convergence. Recent work by Sprouse et al (2011) includes two different numerical solvers incorporated in MATLAB: one for the differential algebraic equation (DAE) system of equations and one through the differentiation of the algebraic constraint equation (ideal gas law) to complete an ordinary differential equation (ODE) system:…”