A.I.Karpov (Khabarovsk Polyteohnioal Institute, Khabarovsk 680035

ON THE PREDICTION OF FLAME SPREAIl RATE OVER"
THE COIlBUBTIBLE IU.TERIALS
J ..
A.I.Karpov
(Khabarovsk Polyteohnioal Institute,
Khabarovsk 680035, Russia)
.ABSTRACT
The propagation of two-dimensional diffusive flame over combuatible material is studied numerioally by solving appropriate
oons~rvation equations for the reaoting
flow · ooupled with the
energy equation in the solid fuel. Two kinde of mathematioal
desoriptions are -oonsidered. Firstly, boundary layer approaoh is
taken into aooount but in some oases (speoifioly, when freestream velooi ty is not suffioient to damp the mo.Leou.Lar- heat and
mass transfer in the upwind direotion), the flame spread features
oan not be desoribed oorreotly by this approximation. Henoe, full
t wo-d ime ns i onal elliptio equations are solved. Among
other
aspeots of flame spread prooess, presented study has been fooused
mainly on the oentral issue of theoretioal analysis of oonsidered
phenomena that oonsists in the prediotion of steady flame spread
rate. Beoause the solving of unsteady elliptio equations is
f a irl y diffioult from the view-point of oomputational efforts,
ma thematioal model is formulated in the steady-state ooordinate
sys tem fixed on the flame front. Here, set of steady equations is
t ake n into aooount where steady flame spread rate appears as an
eigenValue. Generally aooepted method of eigenValue problem's
solution relates to t h e ue Ing of integral mass balanoe in the
s ol id fuel. As it was shown, two-dimensional eigenValue problem
has no t unique solution if regression of burn~ surface
is not
n e g l e o ted . It means that fuel's mass balanoe 1S kept for any
assigned value of flame spread rate. A new algorithm for the
prediotion of steady flame spread rate is proposed by using the
non-equilibrium thermodynamio approaoh.
The flame spread over PYKA. slabs and thin samples of paper is
studied. The influenoe of some parameters suoh a free-stream
velooity and ambient mass fraotion of oxygen on the flame spread
rate is investigated.
INTRODUCTION
The problem of theoretioal investigation of flame spread over
solid fuels was widely studied by different authors [1-5]. Among
lot of physioal aspeots surrounding this phenomena, present study
has been mainly oonoentrated on the theoretioal prediction of the
s teady flame spread rate. As it was notioed [3], quite diffioult
t wo-d ime nsi onal eigenValue problem has been arisen if steadys t ate ooordinate system fixed on flame front is oonsidered.
Al ternative approaoh is based on the unsteady equations in the
- 319Copyright © International Association for Fire Safety Science
ooordinate ayatem ~ixed on aolid ~uel. Uaing thia approaoh [4.5]
one oan avoid eigenvalue problem but aigni~ioant di~:fioultiea are
ariaen to solve two-dimensional unatea~ equations.
At presented paper only stea~ equations in the ooordinate
system :fixed on :flame :front are taken into aooount and eigenValue
problem ia atudied eapeoially. In the originating work [2] the
solution o~ eigenValue problem was reoeived by using the integral
maBa balanoe ot: aolid ~uel. In this paper the ratio o~ eunraoe
regreasion to the ~lame apread rate was assumed negligible that
allowed to reduoe ~ull two-dimensional ·eigenValue problem to onedimensional. In thia regard. preaent atudy has been intended to
develop an algori thm ~or the prediotion o:f :flame spread rate
basing on the analyaia o~ ~ull two-dimensional eigenValue problem
where ~lame spread rate and aolid :fuel's pYrolysis rate relate to
eaoh other as a two eigenvalues.
li'LAME SPREAD MODEL
The oo~iguration o~ oonsidered ~lame apread model ia shown in
li'ig.1. The horizontal ~lame apread ia ahown here but preaented
analyaia oan be appl i ed also 1'0 downward and even upward ~lame
apread ainoe the exiatenoe o~ ateady propagation ia t h e Bingle
oondition ~or auitability. The general ~orm o~ equations ia:
gas-phase
lJT
{J A {JT
{J A {JT
Q
+ .:JI pW
A+
{Jy C {Jy
a
8
{Jx C {Jx
{JT
+ pv--
ptL-
{Jx
{Jy
{J Le A (JY
{JYi.
{J Le A {JYi.
{JYi.
i
pu- + pv-- = A- - - - + - - - {Jy . C
{Jy
{JX C
{JX
{Jy
iJx
lJu.
{J
lJu.
pu- + PtJ-
{JX
{Jy
tJX
{Ju
{JU
Apu- + AptJ{JX
{Jy
tJptL
{Jpu
+ --
{Jx
p
{Jy
{J
lJu.
A-
~
{J
{JX
+
{JU
{Jy
{JX
{Jy
o
1)
i
pW
8
(2 )
(3 )
{JX
{Jy
{JU
A-~-+A-~-
{JX
-
{Jp
lJu.
~-
{J
(1)
{Jy
{Jp
(4 )
{Jy
(5 ) .
pRT
(6)
- 320-
solid
~el
2
BT
u -
I Bx
+ va
By
=;
[
2
A:;
+
B T ]
By2
+
Q
-!! pW
G
(7 )
a
oonditions
boundary
y=o:
BT
,
tL--U,
T=T a
PV=Pava
aT
- 'A. -g + (pvGT)
gay
e
-IE 'A. BY i
- -- + pvY ~
G ay
~
(8)
e
=-
'A.
aT
~
aay
+ (pvGT)a
(9 )
= pa v a Y a.~~
(10)
Here T is temperature, u,v are velooity oomponents, Y i is mass
fi>aotions where i=(O,P), 0 and P denote oxygen and gasi:!ied
:fuel respeotively, p is density,
A. is thermal oonduotivity,
~ is visoousity,
C is speoi:!io heat, p is pressure, R is -s pe oi:!io gas oonstant, Q is e:!:!eotive heat o:! reaotion,
18 is
Lewis number, U , and Va are steady :!lame spread rate and linear
pyrolysis rate respeotively, A=O :!or boundary l~er approaoh
and A=1 :!or:!ull two-dimensional elliptio equations.
The gas-phase oombustion and solid :!uel pyrolysis both are
desoribed by Arrhenius type :!ormula:
1.
1'1&
WJ = YPYerg
1l exp(-E /R T )
JOg
(11)
Ws = Rs exp(-Es /RoT)
s
(12 )
EIGENVALUE PROBLEM:
Assuming pyrolysis produot is gasified from the burning surfaoe
as soon as it i s generated, the linear rate of pyrolysis is
determL~ted from the fuel's mass conservation in the form [3]:
o
v =fW dy
S
-L
(13 )
S
The value of pyrolysis rate Vs in Eq. (13) relates to some orossseotion o:! :!uel sample in the x-ooordinate. The mass oonservation equation desoribing the whole :!uel sample is (see ~ig.1):
x
ii L
r
~
1'1&
= f v (x)dx +
0
a
U,(~
m
(14)
)
-3Z1-
whioh yields
:J:
TIl
aTIl = f0 v S (x)dx
,
(15 )
ti
Here XTIl is burnout point on t he ~uel sur~aoe beyond whioh PYrOlysis no ooours.
The thiokness o~ virgin material deoreases along the x-ooordinate
L(x) = L(O)-o(x)
(16 )
o~ gaBi~ied
and thiokness
material is
de~ined
as
t
O(x) = f v (x)dt
o S
(17 )
On the other hand, steady-state ooordinate system is
~lame ~ront
o~
steady
~lame
~ixed on the
spread rate yields
= Ujlt
dx
Replaoing
O(x) =
and
and determination
~or
(18)
dt
~rom
Eq. (18) to Eq. (17) we reoeive
:Ii
L
f
V
o
ti,
s
(x)dx
(19 )
the whole sample
:Ii
o
TIl
U,
= 1
fTll v (x)dx
0
(20)
S
:Finally, replacing 0 TIl from Eq. (2 0) .to Eq. (15) we ge t an ident ity
that means that ~el' a maaa oonservation is kept ~or any assigned
value o~ ~lame spread rate ti, and eigenValue problem has not
unique solution relatively to the spread rate. There~ore, some
additional oorrelat ions are needed.
BOUNDARY LAYER APPROACH
Considering the edgenvatue problem presented in the previous
section, the following algorithm is proposed for the prediotion
of flame spread rate. The investigation [6] of dependenoe Vs(U,)
has shown that there is an extremum at whioh the pYrOlysis rate
V s reaches its maxiInuJlJ
oorresponding to some value of flame
spread rate
The last one is oonsidered as a sought s teady
value. Beoause boundary l~er equations are taken into aooount
and terms describing heat t ransf e r in the upwind direction are
negleoted, Eqs. (1 )-(4) oan not be integrated ~m t h e point x =O
in ],ig. 1 [7 ]. So, to begin the oaloulat ions some pro~iles o~
predioted variables must be assigned at some oross~eotion
X
o
in li'ig.1. Aotually i t means t hat in the c ase o~ opposed ~low
~lame propagation the flame spread rate oan not be
predioted by
t heor etioal s tudy p ure l y because boundary l ayer equat i ons do not
U, .
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inolude any terms that desorib~ meohanism o~ heat transfer in the
spread direotion. There~ore, only an ~luenoe o:! Borne
parameters on the ~lame spread rate oan be studied by boundary
layer approaoh while one point o~ this dependenoe in any oase
must be guessed by ohoosing the initial pro~iles that satis~y
experimental result.
Some re3ult3 of the inve3tigation of flame 3pread over PMMA
slabs are presented in ~ig.2-J. The data o~ ~ig.2 are quite
satis~aotory wnile ~ig.J has not shown so good agreement.
Author
does not hesitate to show aohived results even i~ its have a
manner suoh as presented in ~ig.J beoause the limit o~ proposed
method ~or ~lame spread rate prediotion must be settled olearly.
~lame
:ffiLL TWO-DDlENSIONAL EQUATIONS
Here, a mathematioal model that Ino.Iudee full two-dfmenafonat
equation3 are investigated. The study o~ ~luenoe o~ ~ree-stream
velooity on the ~lame 3pread ra"Ge by using the method desoribed
in previous seotion (searohing the extremum o~ Vs(ti ) distribut
tion) was unsucoearuf, as it was shown in ~i~.J ~or boundary layer '
approaoh. Then, a new approaoh ~or the predJ.otion o~ ~lame spread
rate Was developed by usdng the non-equilibrium thermodYnamio
analysis [8]. By thi3 approaoh the steady ~lame propagation is
identi:!ied with the stationary state o~ irreversible thermodynamio system that is oharaoterized, aooording to theorem of
Prigo~ine, by minimal entropy produotion [9] .The result
aohived
by thJ.3 approach i3 shown :L.'"l li'ig. 4.
REli'ERENCES
1. De Ris J.N., 12th Symp. (Int.) on Combustion, Combustion In3t.
Pittsburgh, 1969, p.241.
2. Sirignano W.A., Aota ABtronautioa, .1 (1974), 1285.
3. ~ernandez-Pello A.C., Williams ~.A., Combust. ~lame, 28(1977),
251.
4. li'rey A.E., T'ien J.S., Combust. li'lame, 36(1979), 263.
5. Di BlaBi C. , Cresoi telli S., Ru330 G., Combust. li'lame,
72 (1988 ), 205.
6. Bulgakov V.K., Karpov A.I., in: Air and GaB DYnamios ot
Un3 teady Prooesses, Tomak Universi ty, 1988, p .121 (in Ru3sian).
7. Chen C.H., T'ien J .S. , Tra.na . . ABO: J. o~ Heat Tra.na~er,
106(1984), 4, 31.
8. Karpov, A.I., Bulga.kov V.K. Soviet Union - Japan Seminar on
Combustion, hplosion and li'ire Research, Khabarov3k Polytechnioal In3t., 1991, p.91.
9. Glan3dorff P. ,Prigogine, I., Thermodynamio Theory of Struoture, Stability and li'luotuatioI13, Wiley, New York, 1971.
10. McAlevy R.li'.III, ~e R.S., 13th Symp. (Int.) on Combustion,
Combustion In3t., PJ.ttsburgh, 1971, p.215. · .
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u
~
x
o
U-f
li'ig.1 li'lame spread model
0.2
li'ig.2 The intluence ot ambient
oxygen mass traction on
the tlame spread rate.
1 - present oaloulation,
2 - empirio tormula [10].
6--------....,
o
0.1
0.2
o.?t
•
o
U, m/S
li'ig.3 The In!luenoe ot !reestream velooity on the
tlame spread rate.
1 - present calculation
2 - measurements [6 ]
0.%
U,
O.It
tn/S
li'ig.4 The intluenoe ot !reestream ve1bcity on the
tlame spread rate.
1 - present caloulation
2 - oaloulation [4]
-32.4-
O.G