∂z

∂Tw

∂z

- λs w

s

∂2 T

(17)

∂z2

in th e es timat ion of th ese values ar e not critical becau se th e operat ing conditions stu died here ar e below th e temperatur e region in which reaction with O2 becomes important 。

In regar ds to th e reaction with NO2, th e expe rimenta l data obta ined in th e engine tes ts will be used to fit th e two rat e param eters (activat ion energy an d preexpo- nent ial factor)。 The activat ion energy should be with in th e  ran ge of values report ed in th e  relat ed li teratur e。

Pressu re Drop Mode l。 The press ur e drop thr ough

th e soot layer is describe d by Darcy’s law with th e inclusion of th e nonlinear Forchh eimer term , which becomes important at high flow rat es。 In th e genera l case, th e different ial press ur e drop over a porous medi a of length  dx is

dp ) µv + þFv2 (18) dx kp

Although th e ma ss flow rat e is steady thr oughout th e soot layer, th e gas velocity var ies along th e soot layer becau se of chan ges in th e gas density an d filtrat ion ar ea, as shown by th e cont inu ity equat ion

FwvwD

v(x) ) (19)

F(x) b(x)

The gas density var iat ion along th e soot layer is a function of th e local press ur e。 With th e ass um ption of ideal gas, th e  density can  be  exp ressed as

p(x) M

lines of th e gas flow  in  th e  filter  in  th e  axial  an d tran sverse directions。 Therefore, th e rat e of soot ac- cumu lat ion in each axial node along th e filter will be directly proport iona l to th e gas flow rat e at th e respe c- tive node in th e tran sverse  direction。  Becau se th e solut ion of th e above balance equat ions provides th e tran sverse gas flow rat e distr ibut ion along th e filter chann el, it is stra ight forwar d to comput e th e rat e of soot accumu lat ion  at  each axial node。

Mode l Vali dation

The math emat ical model prese nt ed  above still lacks

F(x) )

RT

(20)

th e informat ion on th e  rat e  param eters that describe th e   NO2   reaction。  Neverth eless ,  by  simu lat ing  th e

Becau se of th e tra pezoidal sha pe of th e soot layer deposit, th e soot layer  wid th  increases  as  th e  gas app roaches th e wall surface, according to th e followi ng relat ionship:

b(x) ) D - 2(w - x) (21)

The perm eability of soot is depe ndent on th e gas mean free path becau se of slip phenomena 。 This depe ndence can  be  exp ressed as26

engine tes ts, it is possible to obta in th e rat e constant s that produce good agreement betwee n expe riment an d simu lat ion。 A typical exam ple is prese nt ed in Figur e 5, which refers to an engine tes t consisting of six 10-min steady-stat e operat ing point s in series。

In a first step, it is necess ar y to ensur e  that  th e part iculat e filter temperatur e is  predicted  corr ectly thr oughout th e tes t。 This is obvious from th e uppe r gra ph, which compar es th e measur ed an d comput ed temperatur es at th e filter exit。 It ha s to be ment ioned that  th e  heat of  th e  NO2  reaction at  th ese  operat ing

p0

kp ) k0 1 + C µ

T  (22)

conditions produces negligible temperatur e chan ges in th e  filter。 In fact, almost isoth erma l  conditions  occur