Integral Expressions for Component Pressure Losses in Wall Flow Filters
James Pakko, Ford Motor Company
A global trend in automotive emissions control is an ongoing reduction in particulate emission standards. Since diesel engines usually create more particulate emissions than gasoline engines, diesels have been affected first by these rules, resulting in widespread use of diesel particulate filters (DPF). To meet future regulations, many gasoline engines will also be unable to comply without additional aftertreatment. This is expected to lead to use of gasoline particulate filters (GPF) in many of the world’s largest automotive markets. The most common type of exhaust filter is formed from a porous ceramic monolith with square parallel flow channels. Alternate channels are blocked, forcing exhaust gas to flow through porous walls. While there are many operational differences between DPFs and GPFs, in the limited scope of the present work, they are treated the same.
Since both DPFs and GPFs require exhaust gas to pass through a porous medium, exhaust backpressure is markedly increased over traditional flow-through catalyst monoliths. Understanding and managing pressure drop is one of the more important aspects of filter design. The total pressure drop through a filter can be separated into component contributions accounting for inlet and outlet channel frictional losses, and flow through the porous wall which is characterized by Darcy’s law. While expressions for these component losses have been published previously, they assume that wall flow velocity is constant along the axial dimension. However, under certain flow conditions, wall flow can deviate significantly from uniform flow. Starting from one-dimensional statements of mass and momentum conservation, an analytical model has been developed to express inlet channel, exit channel and wall flow contributions to pressure loss as integral equations that can be applied to arbitrary flow profiles. The model accommodates both symmetric and asymmetric flow channel sizes, and takes into account gas compressibility. Finally, by applying common simplifying assumptions, the integral equations are reduced to familiar forms, elucidating the physical origins of these equations.