(98a) Dimensional Analysis and Geometric Scaling Relations for the Viscous Flow Resistance of Exhaust Particulate Filters

Diesel and gasoline particulate filters, hereinafter exhaust particulate filters, are used to remove soot and ash from the exhaust gases of Diesel and gasoline direct-injection engines, thereby lowering particulate matter emissions and improving air quality.  A large pressure drop across the filter during operation adversely affects fuel efficiency as it corresponds to increased engine power required to discharge the exhaust gas through the filter.  Hence an objective in designing exhaust particulate filters is to minimize the pressure drop, subject to mechanical, thermal, filtration efficiency, particulates storage capacity, and other constraints.  Experimental data, when plotted as the Euler number versus the Reynolds number, show that under on-engine conditions approximately 70% to 90% of the pressure drop arises from viscous effects and 10% to 30% is due to inertial effects.  Our present work accordingly focuses on the analysis of the viscous flow resistance of exhaust particulate filters; for the inertial loss coefficient one may refer to prior literature [1–4].

Exhaust particulate filters are typically made of gas-permeable porous ceramic substrates, extruded as honeycombs consisting of hundreds to thousands of parallel square channels.  Adjacent channels are closed at alternate ends with plugs to form pairs of inlet and outlet channels.  Soot and ash are filtered onto the side of the partitioning ceramic wall facing the inlet channel as exhaust gas passes from the inlet channel through the wall to the outlet channel.  The substrate permeability κ [dimensions L²] and the four principal variables that describe the filter geometry—the diameter D [L], the channel length L [L], the channel density (the reciprocal honeycomb unit cell cross-sectional area) σ [L⁻²], and the wall thickness w [L]—determine the viscous flow resistance of a clean filter.  Adding to the above five variables are the mass density ρ [ML⁻³] and the dynamic viscosity μ [ML⁻¹t⁻¹] of the exhaust gas, and the volumetric flow rate Q [L³t⁻¹] and the pressure drop Δp [ML⁻¹t⁻²].  Analysis of the pressure drop of a clean exhaust particulate filter thus involves nine variables [5,6].

Previous investigations on optimizing the design of exhaust particulate filters have relied on empirical rules such as fixing the aspect ratio and the volume of the filter to make the problem more tractable [7].  Such approaches, however, do not easily lend to a general understanding of the scaling relations between the filter geometry and the viscous flow resistance.  We instead prefer to apply dimensional analysis in our study, both to reduce the number of degrees of freedom and to find the dimensionless groups that best describe the underlying physics.

By drawing inspiration from the analysis of the classical Z-type parallel-flow manifold [8–10], we construct a ladder-shaped resistor network to model gas flow through a pair of adjacent inlet and outlet channels in an exhaust particulate filter.  This changes the flow basis of the problem from Q to the average volumetric flow rate per filter channel q = 4Q/πD²σ [L³t⁻¹], effectively reducing the number of degrees of freedom by two.  It follows from Buckingham’s Π theorem [2,11] that seven variables spanning three basis dimensions can be reorganized into four dimensionless groups.  We make the customary choice of selecting two of these to be the Reynolds number Re = ρκ^1/2v/μ and the Euler number Eu = 2Δp/ρv², where v = 4Q/πD² [Lt⁻¹] is the superficial velocity of the exhaust gas flow through the entire filter.  The remaining two dimensionless groups will then define the dimensionless viscous flow resistance R_vκ^1/2.  Several choices are possible for formulating the two remaining dimensionless groups:

• (1) Since the solution to the resistor network model, R_v = R_ch{1 + (2R_w/R_ch)^1/2coth[(R_ch/2R_w)^1/2]}, gives the overall viscous flow resistance R_v [L⁻¹] as a function of the channel flow resistance R_ch = s_chL/d⁴σ [L⁻¹] and the wall flow resistance R_w = s_ww/dσLκ [L⁻¹], where d = σ^−1/2 − w [L] is the filter channel width, and s_ch and s_w are constants, a logical choice for the two groups is the respective pair of dimensionless resistances R_chκ^1/2 = s_chLκ^1/2/d⁴σ and R_wκ^1/2 = s_ww/dσLκ^1/2.  Two exhaust particulate filters are similar for the purpose of comparing or scaling their viscous flow resistances if they have the same value for R_chκ^1/2 and the same value for R_wκ^1/2.

• (2) An alternative is to use the wall flow-to-channel flow resistance ratio α = (s_ch/s_w)(R_w/R_ch) = wd³/L²κ along with the honeycomb cross-sectional open area fraction β = d²σ = d²/(d + w)², where β is related to the product of the dimensionless channel flow and wall flow resistances by [(1/β^1/2) − 1]/β² = R_chR_wκ/s_chs_w = w/d⁵σ².  Hence two exhaust particulate filters that are said to be similar have the same resistance ratio α and the same cross-sectional open area fraction β.

• (3) A third possibility is to use the dimensionless channel density 1/ζ² = σLκ^1/2 and the dimensionless wall thickness η = w/L^1/2κ^1/4.  Two exhaust particulate filters have scale-similar viscous flow resistances if they have the same dimensionless channel density 1/ζ² and the same dimensionless wall thickness η.  From this choice of dimensionless groups we also see that Lκ^1/2 scales as 1/σ, w², or d² for scale-similar exhaust particulate filters.

Having identified the above dimensionless groups that can be used to determine the overall dimensionless viscous flow resistance of an exhaust particulate filter, we can construct a universal viscous flow resistance diagram for exhaust particulate filters, where curves of constant R_vκ^1/2 are plotted in a plane with ln(R_chκ^1/2) as the abscissa and ln(R_wκ^1/2) as the ordinate.  On this diagram, lines of constant α have a slope of +1 and lines of constant β have a slope of −1, and curves of constant ζ and constant η form a curvilinear grid system.  The aforementioned geometric scaling relations for the viscous flow resistance of the exhaust particulate filters can be demonstrated on this diagram, and comparisons between different filter designs are easily made.

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