To understand the flame flash-back phenomena more extensively, an analytical model was formed and a numerical program was written and tested to solve the set of differential equations describing the model. Results show that under a given set of conditions flame propagates in the boundary layer on a flat plate when the free stream is at or below 1.8 m/s

Feng, C. C.

English

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ANALYTICAL MODELING OF FLASH-BACK PHh~C~ENA 
(H1Sl-CR-162389) ANALYTICAL ftODBLIIG or H80~103l0 
PL1SH-B1Ck PREIOR!ll (Trenton state ColI., 
H. J.) 17 P He 102/R' 101 CSCL 21B 
BY 
C. C. Ft:mg 
Trenton State College 
Hillwood Lakes 
P.O. Box 940 
Trenton, NJ 08625 
Nobcmber 1, 1979 
Onclas 
G3/25 45959 
Research suppoIted by the National Aeronautics and Space 
Administration under Grant No. NSG-3l80, Supplement No. 1 
From November 5, 1978 approximately 12 months 
• 
I. Introduction 
To reduce j~t-en9ine~produced NOx andeCO simultaneously, 
one of the advanced combustor concepts i~vloVes lean premixed/ 
prp.yaporized combustion system. In. the systel:', since liquid' 
fuel droplets evaporati~n in present-day combustion chambers 
iS,absent, locally fuel-rich and high temperature re~ions are 
removed. As a result, substantial reduction of MOx is possible 
without.u~dcceptable CO emissions. Furthermore, because of 
lower flame temperature and more uniform combustion, smoke and 
radiation may be reduced, turbine inlet temperature may be 
iEProved and turbine erosion from carbon particles may be 
avoided. 1 
Acompanying lean prernixed/prevaporized combustion system, 
however, are problems such as much narrower flame stabilization 
limit, which result in blowoff, i~nition and altitude reliqht 
difficulties, and flash-back and auto-ignition, which result in 
the presence of flame in the premixinq/prevaporizinq section 
wjth potential subsequent hardware damage. 2,3,4,S,6 
To understand the flame flash-back phenomena more extensively, 
an analytical model has been formed and a numerical program has 
been written and tested to solve the set of differential 
equations describing the model. Initial results show that under 
a given set of conditions flame propagates in the boundary layer 
on a flat plate when the free stream is at or below 1.8 m/s. 
More effort is needed in workin~ out with a turbulant reaction 
model and conducting various parametric studies. 
: 2: 
The formulation of the problem, a general discussion of 
the finite difference solution of the qoverning equations and 
the results so far obtained and their discussion are contained 
.! in the following sections. 
II. Formulation of the Problem 
As shown in Figure 1, it is assumed that a uniform flow 
proceeds from left to right passin~ on top of a flat plate. 
Downstream a flame is established by a hot segment of the plate. 
Cartesian coordinates are used for the flow field. 
A. Equations of Continuity and Momentum 
Continuity 
Momentum 
In the above, u and l1- are x- and y-direction component 
velocities respectively, ~4is the effective viscosity which is 
discussed in II. E. below, r the density, t the time, and p 
the pressure. 
To obtain the variation of r, the equation of the state 
is used. 
. ...... 
._. __ ~"_"_~. - :----=c,-....------_.-F_iiiiiIIliii1iii_ ... --.... ; mllliajil'liTQiIlll;i'C~_--~~i..:l" .o;.::--e~:liii""~*-... -:o;j;"'...;::;;;:.;;..;.;;;.~;";";O;'~~ ....... ;...;.;.;~_"~_.--... ".~.,·.:;., ;;':'-0 ;;';;"-;.;,," - ...;....;.;..:.~ "' 
.3. 
B. Species !quati~. 
~ ... ~ a ~ ~ ~ IIr T~'j ... ..J.. [ tx ( Ifl) ).J1H 1 ... )~[ (rD{,1t * lj 
... 't ~ J. . ~t 'l -lr I r I 
Where l-F (fuel) or 0 (oXYgen), Yi is ~'e speci.s mas. function, 
1 the density, (fl>.l, is the effective trana~rt coefficientJ D 
1\ 
1s the diffusion coefficient ,and Wi is \:he time-mean volumetric 
rate of fuel consumption depending on the combustion model 
described in section II. F. below. 
c. Enthalpy Equation 
~(fh) ~(!Ilh) ~(f~·L.) =~[&J. ~1 +.& [(-t)fO.Il*J 
.;at. -to ~x. ~ ~ r aX c:r ~g.~"J. .;,) TTl 
Where Hfu is the heat of combustion; cp specific heat of the 
gas mixture; T the temperature and 7\ the conductivity; and 
h = 'p T + Hfw Yf 
D. Boundary Conditions 
Boundary conditions are applied at all form boundaries as 
described below. 
Far upstream, u is specified,~ is zero, the fuel fraction is 
0.059, and the enthalpy is calculated from the specified heat of 
combustion and ambient temperature. The incoming air is 
stoichiometrically mixed with methane. 
At tr.e upper boundary, u, p and h have Neumann conditions, 
~ is zero and YF.has Dirichlet condition there. 
At the lower boundary u has Neunann and Dirichlet conditions 
before and after the leading edge of the flat plate respectively. 
~ is zero and p, hand YF have Neumann conditione except that 
along the hot segment of the plate h has Dirichlet condition 
there. 
- ~ 
:4: 
At the downstream end, ~ is zero, p, hand YF have Neumann 
conditions, and u is modified such that the mass flow rate is 
consistant with its incoming rate far upstream. 
E. Transport Coefficients 
It is assumed that the transport of momentum, mass and 
heat by molecular and turbulent mixing obeys laws of the 
"coefficient-times-gradient" type associated with the names 
of Newton, Ficks and Fourier. The subscripts "eff" implies 
that the turbulent f.luid is being treated as a laminar orte 
with a set of transport coefficients having augmented values. 
Strictly speaking, the "effective" transport coefficients may 
be expected to posses directional properties: for turbulence 
is rarely isotropic in practically interesting situations. 
However physical knowledge of these matters is still at an 
early stage. tole shall assume isotropici ty ,U~ frequently treated. 
a) The quantity effective viscosity is defined as 
where the eddy viscosity for momentum, fH' assumes the 
well-known Prandtl mixing-len~th model, i.e. 
where 1 is the mixino length and, in the viscous reClion, it 
is expressed, according to Van Driest7 , by 
: i 
, 1 
;j 
1IIioIL-____ .--........ _______________________ -_~"_,~,~, ' ,. <, _.-,_.,,_, ,_. ________ JJ 
'~-
~ .. ""'/T~/~# 
r. ::s~. Sf/" 
1St 
¥here the sUbacript 0 .. ana 
that the quantity ia evaluated at the wall, K • 0.41, and 
~+. 25 (Reference No.8). 
b) To express the quantity, (f,-+, onedefines a turbulent 
heat conduction in parallel with Fourier's law of heat 
conduction 
1t = ,~AlCt~; , 
, . ~- ~.~--
A,= "t./~ 
-', 
Then one may forinaneffective heat conduction as 
By defining a turbulent Prandtlnumber as 
it can be shown that 
(~)<tll ::: A ... t fn 
r 11' Pr ~ 
The above equati~~ governs the heat transport coefficients 
in ~he energy~alance equation. Pr = 0.7 and Prt = 1.0 
are frequently assumed8,9. 
c) To express the quantitY,(rD~H/it is assumed that, for 
both laminar and turbulent flows, Lewis number is equal 
to 1 as frequently th~ practicelO , i.e. 
" Le"t ==( c;lrO~,t = I 
Then, 
(fl> J.a- = (r P)y .... (f P)I: = (~)/ of' (~)t =: (~)'if 
i 
! 
I 
,I 
Jtj 
16: 
F. Reaction Models 
A one step forward overall gas phase reaction of •• cond 
order is taken to represent the chemistry: 
C3HS + 502 + 5ttH2 -t- 4H20 + 3C02 + 5aH2 
molar ratio. 
Ca' Laminar reaction rate 
To complete the species equation, 
" W = 7=P~ rF r· ;).1' (- ~/RT ) 
where a is the 
where F is the pre-exponential constant: p, pressure1 
YF& Yo' the species concentrations of fuel and oxidizer~ 
E, activation energy~ and R, the gas constant. 
In the initial results discussed in Section III only 
laminar reaction rate is used. 
(b) Turbulent reaction rate 
The eddy-break-up model ll for the turbulent flame is 
used as follows. 
First, the reactedness of the gas, ~ , is defined as 
T- TIo( 
= Tt,-T..« 
Then the time-mean volumetric rate of fuel consumption 
may be expressed as j-t 
kiF = [ (T. i,. ..... ) -... ( c (.-!:) r ':ti i' 
where 
, :J 
5_ ..... 1. ~--~ ,---- ..... '" - .... i .-; ~ 
! ~ F" 
"I 
I~ 
III. 
:7 I 
-F- • air fuel mass ratio 
t f • flame temperature 
p -= pressure 
c • 0.35 
In this model, the local reaction rate is taken to 
depend not only on a bimolecular Arrenious expression but 
also on the rate of break-up of the eddies by the action 
of turbulence. 
G. Viscosity 
A simple power law9 is used. 
1A. T (). oz.> /.:..:- = (-::;:) )40 I. 
-> 
where .Me:::}' {, 7 5 B i 10 k rim. ~"eL 
,~ =- ).1 3 . 3 ~k 
Finite Difference Solution of the Governing Equations 
A. Coordinates and Grid Distribution 
A rectangular field of 4.374cm x l.4l3cm is imposed on the 
main region of the flow field shown in Fia. 1. In both x- and 
y-directions, uniform fine grid lines prevail over the interested 
region centered on the established flame front. Far away from 
the flame front, i.e. outside of the interest4e region, the 
grid lines follow a geometric series withthe exception of the 
first and second grid lines near the boundary. 
The magnitude of grid space (space between grid lines) is 
r.rucial to the proper propagation of the flame. It should be 
a fraction of the flame thickness which can be estimated by 
the thermal diffusivity divided by the reference flow velocity. 
E71 ~'.C • 
= a.ne m _ 
1 
.j 
.~..:.....:.~~...;.;.;.~ __ , _________ ...... ______ ~ __ ,_~~ __ •__ ....... ___ ,_, ___ "'-_--.;.._JT~.;~._.__~.t, 
:8: 
In the program, b.X is chosen to be 0.1 _ in the interested 
reoion resulting in a grid field of 93 x 31 points. 
B. Control Volume 
Figure 2 shows a grid point and its control volume. The 
arrangement is based on the work of Patankar and spalding. 12 
The control volume, indicated by the dashed lines, is the 
region traversed by a rectangle of sides Al and il't. The area 
of the control surface normal to the x-direction is simply At 
since a unit thickness is assumed everywhere in the two dimensional 
region. The surface normal to the x-direction and located at e 
is midway between p and E. As a result p is not always the mid-
point of e and w. 
With the above arrangment a staggered grid which is similar 
to the one used by Harlow et. al. 13 is then used. It has the 
convenient feature that the velocity components, W. and lJ. , are 
stored at just the points at which they are needed for the cal-
culation of the convective contribution to the balance of the 
quantity of question. 
A finite difference equation for a node P is calculated by 
integrating ~he differential equation over the control volume 
indicated in the above. For evaluating the convection and 
diffusion fluxes through a control volume surface the arithmetic 
mean of the property values on either side of that face is taken. 
For other purposes a step-wise variation with discontinuties at 
the control volume surfaces is assumed. 
~---~~~----------------------------,------ ----------,-._._---
19: 
c. Modifications to Finite Difference Equations 
To ensure that the difference equations behave realistically 
when the particular method of computation leads to physically 
unreasonable results, the high-lateral-flux modificationl4 is 
used. Under-rela~ation is also used to ensure proper convergence 
of the s~lution. 
D. Solution of the Finite-Difference Equations by TOMA 
The finite-difference equations are solved by the successive 
use of the tri-diagonal matrix algorithm in the x and y directions. 12 
The TDMA used is similar to the one developed by Roache. lS The 
interior-point equation is written as 
-Ajv!.· ... , -t Do' Wj - Bj .. J,'-, - C . I , 
The solution is of the following form 
w; - A: ",1 .. .,., ...,. 8; 
where 
Ai - A' D; - B; i-I 
and 
For the left-hand boundary condition, one use: 
(a) Dirichlet condition 
A! = 0 1 
B' = W i I 
(b) Neumann condition 
A' = 1 i 
B1 = 0 
For the right-hand boundary condition, 
I = I, 11 
........... _-_....--...._-------------------------,----'''-- --,--
! I, 
t. 
I 
-- - -- --
---=>-.... -~._--_.~_ .. _.=~= __ =-~-.... .. --=o.'--_.;;..:.;~~ , 
.10 I 
Ca) Dirichlet condition 
W • a M M 
Cb) Neumann condition 
"M • B~_l I (l-~l' 
IV. Results and Discussion 
The values of various parameters in the numerical computation 
are listed in Table 1. The programs were run on IBM 3033 at 
Princeton University Computer Center. 
It is assumed that the steady stream on the flat plate in 
Figure 1 is a stoichiometric methane - air mixture under 
atmospheric pressure and temperature. Using Prandtl mixing length 
for the eddy diffusivity for momentum ~nd using the bi-molecular 
Arrhenius expression for the chemical reaction the program was 
run for various upstream velocities to determine approximately 
the maximum flow velocity beyond which a flame, which is initiated 
by the hot segment (at 20~O·~ on the plate, would not propapa~e 
upstream. It was found that at a critical upstream speed of 
1.8 mIs, the flame propagated upstream and that at 1.9 mls the 
flame remains stationary. This compares with the experimental 
result of GrossS who reported that flash-backs occu~red around a 
flow speed of 3 m/s using propane-air gas mixture over an airfoil 
plate. The critical upstream speed in the present program is 
expected to increase if eddy-break-up reaction model is u~eu. 
Figure 3 shows the flame envelope and typical velocity, 
temperatur£ and concentration profiles at x • 2.2 em for an 
upstream speed of 1.9 m{s. At the entrance (far upstream) it is 
rll: 
at one atmospheric pressure of 101. 325 N,Im~ while at the exit. 
2 (Far down-stream) 101.189 KN/m. Near the upper boundary the 
fuel (.~rqcentration approaches t~ the up-str.am value and the 
velocity exceeds the up-stream value by about 10'. The temp-
erature profile shows a maximum in the middle but returns to tne 
fixed wall temperature of 20000 K on the plate on the one side 
and approaches the up-stream tempera.ture on the other. 
To reach a steady state from a given set of initial profiles 
such as the one in the above 40 time steps ( t-O.l sec) and 1 
minute and 35 seconds of computer time lire needed to run the 
program. 
The eddy-breAk-up reaction model has yet to be incorporated 
into the program and a series of paramatric studies need to be 
conducted. 
• 
~ 
.12 • 
1. . Verkamp, F. J ., Verdoui v,· A. J. and 'l'oa1inaon, J. G., It Ir.( ... act of 
legulatim~s on Future Gas Turbine Engine Combustors" p. 340, 
J. Aircraft, June 1974. 
2. Marek, C.J., Papathakos, L.C. and Verbulecl, P.W., "Preliminary 
Studies of Autoignition and Flash-back in a Premixing/Prevaporilin9 f 
Flane TUbe Using Jet-A Fuel at Lean Equivalence Ratios" NASA ~ 
TN X-3526, May 1977. ~ 
3. Blalowski, W.S. and Bresowar, G.E., AFAPL-TR-74-32, May 1974. 
4. Anderson, D.N., NASA TN X-3301, 1975. 
5. Gross, R.A., "Boundary Layer Flame Stabilization" p. 288, Jet 
Propulsion, 1955. 
6. Plee, S.L. and Mellor, A.M., "Preliminary Investigation of Flash-
back in Prevaporizing/Premixinq Combustors" WSS/CI 77-22. 
7. Van Driest, E.R., "On Turbulent Flow Near a Wall" p. 1007, J. 
Aeronaut~cal Sciences, November 1956. 
8. Crawford, M.E. and Kays, W.M., NASA Report CR-2742, December 1976. 
9. Schlichting, H. "Boundar~ -layer Theory" p. 260, 6th ed. McGraw 
Hill, 1968. 
10. Bir~, Steward and Li9htfact, "Transport Phenomena" p. 629, Wiley, 
1960 ed. 
11. Spalding, D.B., "Mixing and Chemical Reaction in Steady Confined 
Turbulent Flames" p. 649, Thirteenth Syrn. on Combustion, 1970. 
12. Pantakar, S.V. and Spaldinq, O.B., "A Calculation Procedure for 
Heat, Mass and Momentum Transfer in Three-dimensional Parabolic 
rl~~s" Int. J. Heat Mass Transfer, 15, p. 1787-1806, 1972. 
13. Harlow, F.H. and Welch, J.E., "Numerical Calculation of Time-
dependent Viscous Inc~mpressible Flow of Fluid With Free Surface" 
Physics Fluids, 8, p. 2182, 1965. 
14. Patankar, S.V. and Spalding, O.B., "Heat and Mass Transfer ;~ 
Boundary Layers" 2nd ed. Intertext Books, London, 1970. 
15. Roache, P.J., "Computional Fluid Dynamics· Hermosa Publishers, 1976. 
• 
L.r.r.?~'~ .... -.tt~-~.W?~ •• ··~=~~.~n.·~'~;.:*;.;.;.t.·~st~·~;.··~ •• --~$ ••• ··~ ••• '.2 •• I~M~·.r.·.t~.~·_.~.~_~~ __ 
1 
J 
:13: 
Table 1 
Numerical Values Used in the Program 
Parameter 
A+ (Defined in Section II.E. (a» 
Cp Specific Heat 
E Activation energy 
J( (Defined in Section II .E. (a» 
Le Lewis no. 
Pr Prandtl no. 
Prt Prandtl no. for turbulent flow 
Q Heat of combustion 
R Gas constant for air 
TB Temperature of burned gas 
TU Temperature of unburned gas 
Yf,uFuel concentration in the upstream 
z Pre-exponential constant 
Numerical Value 
25 
265 cal/Kg-oJ( 
40000 cal/mole 
0.41 
1 
0.7 
1.0 
1.1062xl07 cal/~g 
286.8 N-M/Kg-oJ( 
20000 K 
3000 K 
0.0594 
0.5 
• 
~ I , 
, 
.. 
t-~-j 
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.... 
~ 
.... 
o 
L4 
.... 
C 
o 
u 
::& 0-------
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:1'-,~ 
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J 
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u. 
~ 
yLFlat 
x 
Plate Hot segment, of Plate 
Figure 1 Schematic Diagram of a Flame on a Flat Plate 
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, 
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t' 
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, 
i 
f 
k, 
1.0 
~ 
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ai°. S 
u 
c ,., 
~ 
Ul 
" o 
0.0 
u = 1.9 mls 
0.5 1.0 
Figure 3 
and T/T, 0.0 0.5 1.0 
T/'I«» TW 
Flame Front ... 
YF/YF.," 
1.5 2.0 2.5 3.0 3.5 
Distance, x cm 
Steady Flame Attached on a Flat Plate 
.. 
4.0 
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1 
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Analytical modeling of flash-back phenomena

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