transient phenomena in boundary layer ignition with finite plate thermal resistance

Eighteenth Symposium (International) on Combustion The Combustion Institute, 1981

T R A N S I E N T P H E N O M E N A IN B O U N D A R Y LAYER I G N I T I O N W I T H F I N I T E P L A T E T H E R M A L R E S I S T A N C E

CI~SAR TREVINO MIHIR SEN

Divisidn de Ingenieria Mecdnica y El~ctrica, Divisidn de Estudios de Posgrado, Facultad de Ingenieria, Universidad Nacional Autdnoma de Mdxico, M~xico 20, D.F., Mdxico

In this paper the transient ignition of a gaseous combustible mixture in a flat plate boundary layer is studied. The thermal capacitance of the plate is neglected but its thermal resistance is taken into account. The lower surface of the plate is given a sudden increase in temperature and the processes leading to a permanent flame structure are analyzed through numerical solutions of the parabolic governing equations. The events leading to ignition are characterized by the formation and propagation downstream of a high reaction rate spot. The generation of this spot is related to the appearance of a zero heat transfer point on the wall. Three characteristic times can then be defined for the ignition process: (a) the time necessary for the formation of the adiabatic point on the wall, (b) the time necessary for the high reaction rate spot to reach the steady state flame position, and (c) the time after which a removal of the temperature perturbation at the plate lower surface would still lead to flame formation within a reasonable distance. These characteristic times are found to be strongly affected by the plate thermal resistance.

Introduction

The process of ignition of a gaseous combustible mixture in boundary layer flows has been the object of numerous studies. This is of great interest in practical problems, such as flame stabilization and fire prevention. A great number of experimental studies have been carried out for different flow situations, the flame stabilization mechanism being one of the important aspects of the problem' 6. Theoretical studies include those of Dooley 7 and Toong ~ who solved the reacting boundary layer equations to determine the ignition distance. Sharma and Sirignano ~ analyzed the ignition of a combustible mixture by a hot projectile, dividing the problem into three parts: (a) a stagnation point flow, (b) flow over a flat plate, and (c) wake flow. The parabolic equations for the fiat plate problem were solved numerically using a quasilinearization technique. The ignition distance was calculated for different flow parameters. Berman and Ryazantsev'~ and Law and Law'~ used matched asymptotic expansions for the high activation energy limit to deduce an expres- sion for the ignition distance. Law and Law '2 used

a combined numerical-perturbation method to ob- tain the ignition distance considering the effects of reactant consumption. The flammability limits of the system were identified. Lifi~n and Williams 1:~ included higher order terms in the asymptotic expansion series for a more precise calculation of the ignition process. They found a small but finite difference between the results obtained from the adiabaticity ignition criterion and the runaway ignition criterion.

Through numerical calculations, Trevifio and Sen ~4 found the existence of a high reaction rate spot in the transient boundary layer ignition process generated by a sudden temperature increase at the upper surface of a flat plate. The time of flame formation was defined to be the time necessary for this spot to reach the steady flame position. They also investigated the influence of plate thermal resistance under steady flow conditions ''~ and found that the effect of this was to move the adiabatic point on the wall downstream while the lower edge of the flame could shift upstream.

The present paper analyzes the effect of a sudden increase in temperature of the plate lower surface

1781

1782 IGNITION

and considers the influence of plate thermal resistance on the ignition process, assuming the thermal capacitance of the plate to be negligible.

A n a l y s i s

A gaseous stoichiometric mixture flows along an impermeable, non-catalytic fiat plate of thickness t, length L and thermal conductivity h w (Fig. I). The free stream conditions are constant and correspond to a velocity u| temperatre T~, density p~, pressure p=, viscosity Ix= and fuel concentration Y~. The temperature of the lower surface of the plate is raised in a stepwise fashion from T= to Twt at time t = o. The initial condition considered is that for time t -< o the rate of chemical reaction is negligible and so the species concentration everywhere is equal to that in the free stream. An one-step, irreversible chemical reaction is considered, where the rate of generation of any species is given by the Arrhenius law.

For the gas phase we assume the Prandtl boundary layer approximation for the momentum equation while retaining the longitudinal diffusion terms in the energy and species balance equations. The following simplifying assumptions are also made: negligible buoyancy forces and radiation effects; mixture composed of ideal gases with constant specific heats equal for all chemical species within the mixture; mass diffusion given by Fick's law; constant mean molecular weight of the mixture.

Under these conditions the non-dimensional governing equations can be shown to be of the form )4

a~ \ on2/ on 2

{ a~f = 2~ a~0~q

{" Of a2f Of 02f /

+ J a~l 040"q 04 a~l 2

02f 02f ]

a~04 a~O~ (I)

u ya

k

j l

'l 'f

L" ,

Fl(;. I. Schematic of flow field.

- - = 8 2 (kM~) a~q

�9 +24

o ( oo/+ oo &q \Pr a~l / 0~1

a2f 02f } - ~ - - + ~ , , (2)

a'r04 0"r0~

( P ) { 2 4 - ~ ( F ( 0 ) a 0 ~ - , - - [ - --P~--~ ~-r 0 (F(0) a0~ +32

_ ~__(F(0). a O ) + 3 ~ F(0)a0

04 \ Pr a~l 2 4 Pr &q

l 'q~ 0 (F(O) 00 ) _ _ } + + $3

2 4 0n \ P r 0n

{ a O O f a O d f O O = 2 4 +

on a4 o4 an _ C 4 0 a ' r - a F - a o + l p ~ F + ' % -1

aF+a �9 m F "exp - - fECk0,q 2 ]

~2EcAz (Of+ 24 a2f a2f ) - - - - ~ + 6 , (3) \a~ 0T04 0@~1

0 (AOmFS+famF 0~1 \ S c 0~1 / 0-q

g2 + ~ \ Se 04

- .q 0 (F(0)OmI...I -'rl __a (17(0)OmF)

0n \ SC 04 / 0~ \ Sc ao

+ 2 4 Sc an 2 4 a.q \ S c &q

I amF Of amF 0f 0m F + G 24 + - - I o~ o~ a4 o4 o~

q- C OU'r- '*v-% + l mr +%,

�9 (4)

where the non-dimensional variables are defined in the Nomenclature. Here, 4 corresponds to the first DamkShler number.

The initial and boundary conditions for the combustible mixture transform to:

(i) ~ l ~ , d f / & q = 0 = m F= 1 (ii) "q = o, Of/O"q = f = o

For "r -< o, 0 = 1 and m v = 1 everywhere.

TRANSIENT PHENOMENA IN BOUNDARY LAYER IGNITION 1783

For v > o , OmF/& q = O at "q = o.

Neglecting the thermal capacitance of the plate as well as the axial heat flux, the energy balance equation for the plate becomes:

~20 w - o ( 5 )

where the nondimensional transverse coordinate in the plate is given by:

= h~ P-----7 \ 2L V (6)

On applying suitable boundary conditions, the solution to Eq. (5) is:

1 0w(r T) = 0 , + - - {0w,(r -0~,}({ +{m) (7)

w ~ m

where ~m represents the thermal resistance of the plate given by

1 / 2 C~ ( . C u ~ p ~ ~,,. ~ (8)

~= h~ Pr \ 2L

For low Mach number flows, Eq. (2) shows that the small ~ approximation means an uniform pressure field. That is

P = 1 (11)

For non-negligible values of 8, strong pressure gradients may exist which may effect the (hydro- dynamic) stability of the flow, 6 leading to an oscilla- tion of the flame.

If we also assume that the viscosity coefficient is proportional to the absolute temperature, we have f = 1. Under these approximations the longitudinal momentum equation, Eq. (1), decouples itself from the other governing equations. Also, given that the boundary conditions on fare time independent, this equation reduces to the Blasius form

where

f " + f f" = o (12)

f = f(~q)

Assuming the Prandtl and Schmidt numbers to be constant, and neglecting the Eckert number, the energy balance and fuel conservation equations, Eqs. (3) and (4) become

At the gas-solid interface, we have the heat flux matching condition represented by:

--=OOw _ I (00__~_~ (ow. O~,)/~m = ~ (9)

All the governing equations, Eqs. (1) to (4), involve the nondimensional parameter 8 which plays an important role in the stabilization mechanism of the flame. 8 is a local parameter proportional to the flame velocity relative to the fluid velocity. The largest value of 8 occurs at the lower edge of the flame. This point is then critical to the small approximation. If 8 is supposed to be small everywhere, the governing equations become parabolic in the ~ direction and the Prandtl boundary layer approximation is valid even close to the flame. In other words, in the 8 ~ o limit, longitudinal temperature and concentration gradients become negligible compared to the transverse ones due to the inclina- tion of the flame towards the wall. In this case, the continuous ignition mechanism is the one which

stabdlzatmn. governs the flame . . . 5 The longitudinal momentum balance equation,

Eq. (i), is coupled to the other balance equations through $~ and A The parameter f i s given by

1020 00 {00 f , O0 - - - - + f - - = 2 ~ + - - Pr 0"q 2 O'rl 04

1 O2mF Om~ ~Om F Om F + f - - = 2~ "~ + f ' - -

Sc 0rl z O'q ( 0-r 0~

+ C 0 o my oexp - (14)

Results and Discussion

Equations (13) and (14) were solved numerically using a finite difference scheme based on the quasilinearization technique. T M The calculations were carried out for the flow of a stoichoimetric propane- air mixture. Parameters for this mixture are the following:

0, = 28.55

= 8.84

Pr = 0.75

Sc = 1.32

Ixp F(0) P a v = a, = 1 . . . . (10)

Ix~p~ 0 a T = o

1784 IGNITION

Figures 2, 3 and 4 show constant reaction rate (W) curves at different instants of time for a plate lower temperature 0wt --- 3.5 and with ~m = O. These curves indicate the processes leading up to flame formation. A high reaction rate spot is generated and is observed to propagate in the downstream direction (increasing first Damk6hler number), gradually growing in intensity. The ignition process, therefore, includes the events in the time interval between the formation of the high reaction rate spot, and the instant it reaches the steady state flame position. Between these limits the thermal runaway ignition point should lie, beyond which a removal of the temperature perturbation would still lead to a flame formation in a reasonably short time. Similar processes can be observed for other values of the plate thermal resistance.

Figure 5 shows the heat transfer rate at the wall at different instants of time as a function of the Damk6hler number ~ for 0w~ = 3.5 and ~,,, = o. The non-dimensional heat transfer rate (q') is given by

1 ~0 I q ' - (15) 0n +,=o

The generation of the high reaction rate spot at the wall is seen to be associated with the appearance of a zero wall heat transfer point. Before this instant, the flow is dominated by non-reactive diffusion processes. After the establishment of the adiabatic point on the wall, reaction becomes important and the spot moves away from the wall. The same information is plotted in Fig. 6, which shows the location of the zero wall heat transfer point at different times. Just after the first appearance of this adiabatic point at time "r = %, there are two such points at any instant of time. The zone in between these two can be considered to be reaction affected.

The effect of plate thermal resistance L, and plate lower temperature 0~ on % is shown in Fig. 7.

97=.5

8wt = 3.5 ~m: 0 'r : 7

110 115 2 /

F1c. 2. Constant reaction rate curves at "r = 7 for O wt = 3.5 and ~m = o.

Inclusion of the plate thermal capacitance in the numerical calculations would have meant corre- spondingly larger values of %. In this paper the thermal capacitance of the plate has been neglected, meaning that the characteristic time for the plate

2 7#

0wt=3 .5

r ,18

2 1

I I I I 5 10 15 20

t :

FIG. 3. Constant reaction rate curves at �9 = 18 for 0wl = 3.5 and ~,, = o.

2

1 w~.5

5 10 15 20

Fi(;. 4. Steady state constant reaction rate curves for 0wl = 3.5 and ~,,, = o.

\ OwL= 3.5 +m= 0

I I I I I 2 4 6 8 10

FIG. 5. Nondimensional heat transfer rate at the wall at different instant of time for 0w~ = 3.5 and ~m = O.


T

7

ra.

6

5 I I 2

Ow~ 3.5

~m~ 0

I I I 4 6 8

r

Fic. 6. Position of zero wall heat transfer points ('r, being the time of first appearance) at different instants of time for Ow~ = 3.5 and ~,,, = o.

is considered to be much smaller than that in the mixture.

There is no general agreement in the literature regarding an exact definition of the flame position. In this paper we consider the flame to coincide with a portion of the maximum reaction rate line. Figure 8 shows the steady state maximum reaction rate line for 0~,~ = 4.5 and for different values of the plate thermal resistance. The flame does not include the whole line since quenching effects become important near the wall, leading to a rapid

20

ra 3.5

10

~r 4.5

I I I I I I 0 10 20 30 40 50

F]c. 7. Effect of plate thermal resistance ~,,, and plate lower temperature 0,~ t on "r.

2

7/

0

0

Bwt = 4.5

/ ; . . . -

I ! I I 2 4 6 8

r

FIG. 8. Maximum steady state reaction rate line for 0,, t = 4.5 and for different values of the plate thermal resistance.

decrease in the maximum reaction rate there. This is shown in Fig. 9 where the value of the maximum reaction rate is plotted at each { position. In these curves the maximum wall heat transfer point is indicated by a dot and at this point the gradient of the reaction rate is seen to be near its maximum value. For these reasons, the maximum wall heat transfer point can be taken to be a good indicator of the lower edge of the flame. The rest of the maximum reaction rate line which does not corre-

J

20

O~ 4 .5 T : at)

3 4 5 6

F[(;. 9. Maximum steady state reaction rate for 0,~ t = 4.5 and for different values of the plate thermal resistance. The dot indicates the position of the maximum heat transfer rate at the wall.

1786 IGNITION

spond to a flame is indicated by a broken line in Fig. 8. These data show that for higher plate thermal resistances, the lower edge of the flame moves .9 upstream even though the steady state adiabatic point on the wall can be shown to move downstream.~5

The flame formation time (rss) can be taken to be the time necessary for the high reaction rate spot .8 to reach the maximum steady state wall heat transfer

position. This is indicated as a dot in Fig. 10 which shows the movement of this spot for % <-

<-- tee, calculated for different values of ~ and 0w~. The two broken lines would correspond to a .7 nondimensional spot velocity (~/u| of 0.5 and 1.0.

Finally, Fig. 11 shows the maximum value of the reaction rate within the high reaction rate spot as a function of time for 0w~ = 4.5 and ~,,, = o. This .6 curve is similar to the one shown in Fig. 9. To study thermal runaway, the computer program was modified to include, instead of a stepwise temperature increase, a temperature pulse at the plate lower surface of the form:

"r--<o,0~ = 1

o < "r _<-rp, 0w~ = 4.5

" r > ' r , 0 ~ = l

ra

I, 2

8wt: 4,5 /

i ",

/,.75 I t , , ; f f I | I I

~r

Fie. 11. Maximum reaction rate of the high reaction rate spot with the application of a temperature pulse of duration "rp at the plate lower surface for ~m ~ O.

20

'T

15

10

/ (3.5.0)

' / /

////c4.5,o, I I I 5 10 15

FIG. 10. Propagation of the high reaction rate spot for different values of the plate lower temperature 0~, and the plate thermal resistance ~m" Broken lines indicate Vs/u= = 0.5 and 1.0.

Figure 11 also shows the variation with time of the maximum value of the reaction rate within the spot for different values of % (broken lines). Thermal runaway is seen to approximate to a "rp of 5 + 0.25. On inclusion of the plate thermal resistance, the thermal runaway time tends to %.

Law and Law 1~ have shown in a steady state analysis, that for small values of the parameter 0~/0, , the thermal runaway point coincides with the adiabatic point on the wall. This is due to the fact that, in this case, the Damk6hler number at the adiabatic point is very high and so diffusion effects can be neglected as compared to reaction effects. For this reason, the steady state flame position would also coincide with the adiabatic point. For larger values of 02w/0, (in the present study values are near unity) the difference in the three positions is considerable.

C o n c l u s i o n

The governing equations are taken to be parabolic in the combustible mixture as well as in the plate for vanishingly small values of B which can be readily obtained in practice. This parabolic approximation may be difficult to justify for the plate because of the existence of an adiabatic point on

TRANSIENT PHENOMENA 1N BOUNDARY LAYER IGNITION 1787

the wall. However, before the first appearance of f the adiabatic point, the plate equation is parabolic so that the % values are not essentially altered. For F(0)

> % a longitudinal heat transfer in the plate would h ~" affect the propagation of the high reaction rate spot, except in the case where the plate thermal resistance k is zero. f

The generation of the high reaction rate spot is L basic to the ignition process, even when 8 is large, mr In this case, as the spot intensity increases so does the flame velocity corresponding to it. If this flame M~ velocity is larger than the fluid velocity, the spot N moves upstream. It crosses the steady flame position since the flame velocity corresponding to the spot p is still larger than the fluid velocity at that point. P The oscillatory nature of the ignition process may thus be characteristic of flows with high 8 values, q'

The complete boundary layer ignition mechanism for the small 8 approximation can be summarized R in the following sequence: Pr

a) An adiabatic point appears at the wall. Reaction Sc t effects become important even though diffusion

is significant as indicated by the local Damk6hler T number {, which decreases as 0~/0~ increases. T~ A high reaction rate spot is simultaneously gen-

erated. b) Diffusive effects and then convective effects u,v

transport this spot downstream towards higher Damk6hler numbers with increasing reactive el- V, fects.

W ~ c) Reactive effects are dominant from the point at which the reaction rate curve is at a minimum

W ~ in the Fig. 11. This point would be a logical ,ty r choice for the definition of an ignition point.

d) On increasing the Damk6hler number a moment x, y is reached after which a removal of the plate y~ temperature perturbation would still lead to flame formation within a reasonable time. Calculations of this point require large computation times and on this basis, an ignition point as defined in (c) above may be preferred.

e) The high reaction rate spot increases in intensity in a downstream direction. It crosses the steady state flame position and continues downstream forming the rest of the flame.

Nomenclature

a reaction order with respect to the species a 0

a T temperature exponent in preexponent reaction rate 0,

B frequency factor h C numerical constant taken to be I000 C,, specific heat at constant pressure of the h~

mixture D binary mass diffusion coefficient A 1,Az Ec Eekert number tx

nondimensional stream function defined b y f = O/{2xp=tt| 1/2

defined by F(0) = ix/ix= formation enthalpy of species c~ at refer-

ence temperature T ~ ratio of specific heats defined by f = Ixp/p~p| plate length nondimensional fuel concentration de-

fined by m v = y V / y ~ free stream Mach number number of chemical species taking part

in the reaction pressure nondimensional pressure defined by P

= p/p| nondimensional heat flux at the wall

defined in Eq. (15) universal gas constant Prandtl number defined by Pr = ttCp/h Schmidt number defined by Sc = Ix/pD time plate thickness temperature activation temperature of the chemical

reaction Cartesian components of the fluid veloc-

ity longitudinal velocity of the high reaction

rate spot rate of generation of the species a by

the chemical reaction molecular weight of the species a nondimensional reaction rate defined by

~V = w F / w ~ (T = T ,, yF = y~) Cartesian coordinates mass concentration of species a

Greek letters

8 defined by 8 = ( ix~ , ,p~/p2u~L) 1/2 ~ , ' t l nondimensional coordinates defined by

= ~ m x / L ; ~1 = ( n ~ / 2 P ~ i x ~ x ) l / 2 I ~ o p(x ,y ' , t )dy '

~ , , , maximum first Damk6hler number defined by

Lv ~ B T~T(Y~) a~-'(Y2)~176174 aP+% '

C U~ (W F) aF l ( w ~ a~

second Damk6hler number defined by -~ = x~_, Ih~176176176

nondimensional temperature defined by 0 = T/T~

nondimensional activation temperature coefficient of thermal conductivity of the

mixture eoefficient of thermal conductivity of the

plate operators defined in reference (14) viscosity coefficient

1788

~m

+~,+~,+~

P "r

Ta

Tp

IGNITION

stoichiometric coefficient for the species a

nondimensional transverse plate coordinate defined in Eq. (6)

nondimensional plate thermal resistance defined in Eq. (8).

compressibility terms defined in reference (14).

density nondimensional time defined by r =

~,,,u| nondimensional time corresponding to

first appearance of an adiabatic point on the wall

nondimensional time for the formation of the flame

nondimensional temperature pulse duration

stream function defined by pu = b+/by; Y X ov = - ( o , / O x + o /o t I, ,o( , y', t) d y ' }

lndices

refers to the free stream F refers to fuel o refers to oxidizer w refers to the plate wl refers to the plate lower surface wu refers to the plate upper surface

Acknowledgment

One of the authors (C. T.) acknowledges partial support from the Consejo Naeional de Ciencia y Tecnologla of Mexico for this research.

REFERENCES

1. GRoss, R. A.: Boundary layer flame stabilization, Jet Propulsion, Vol. 25, p. 288. 1955.

2. HOTTEL, H. C., TOONC, T. Y., ANn MaRTIN, J. J.: Flame stabilization in a boundary layer, Jet Propulsion, Vol. 27, p. 28, 1957.

3. ZIEMER, R. W. AND CaM~EL, A. B.: Flame stabilization in the boundary layer of heated plates, Jet Propulsion, Vol. 28, No. 9, p. 592, 1958.

4. TooNc, T. Y.: Flame stabilization in boundary layers, Combustion and Propulsion, 3rd AGARD Colloquium, Pergamon Press, p. 581, 1958.

5. Turcotte, D. L.: Stable combustion of a high velocity gas in a heated boundary layer, Journal of Aerospace Science, Vol. 27, p. 509, 1960.

6. Wu, W. S., TooNc, T. Y.: Further study on flame stabilization in a boundary layer--A mechanism of flame oscillations, IX (International) Sympo- sium on Combustion, p. 49, 1963.

7. Dooley, D. A.: Ignition in the laminar boundary layer of a heated plate, 1957 Heat Transfer and Fluid Mechanics Institute, Stanford University Press, p. 321, 1957.

8. TOONG, T. Y.: Ignition and Combustion in a laminar boundary layer over a hot surface, VI (International) Symposium on Combustion, Reinhold, p. 532, 1957.

9. SHARMA, O. P., S1R1GNANO, W. A.: On the ignition of premixed fuel by a hot projectile, Combustion Science and Technology, Vol. 1, p. 481, 1970.

10. BERMaN, V. S., RYAZaNTSEV, Yu. S.: Ignition of a gas in a boundary layer at a heated plate, Fluid Dynamics, Vol. 12, p. 758, 1978.

11. LAw, C. K., Law, H. K.: Thermal ignition analysis in boundary layer flows, Journal of Fluid Me- chanics, Vol. 92, part 1, p. 97, 1979.

12. Lnw, C. K., Law, H. K.: Flat plate ignition with reactant consumption, To be published.

13. LI~AN, A., WILLiaMS, F. A.: Ignition of a reactive solid exposed to a step in surface temperature, SIAM Journal of Applied Mathematics, Vol. 36, p. 587, 1979.

14. TREVl~O, C., SEN, M.: Transient ignition in a flat plate boundary layer, 27th Heat Transfer and Fluid Mechanics Institute, U.S,C., p. 92, 1980.

15. TRwvi~o, C., SEN, M.: Effect of Plate thermal resistance on boundary layer ignition, Paper WSS 80-26, 1980 Spring Meeting of the Western States Sect ion/CI U.C.I., 1980.

COMMENTS

A. Nemeth, Computer Center for Universities, Hungary. Have you any evidence how your calculations are effected by the assumption that thermal capacitance has been neglected?

Author's Reply. We have not made any numerical calculations taking the plate thermal capacitance into

account. However, we feel that the qualitative features of the ignition process would not be affected even though the characteristic times for the ignition might be significantly larger. One difficulty in including the plate thermal capacitance is the existence of two different response times, one for the reacting mixture and another for the plate. Since


this work was aimed at unders tanding the ignition mechanism, no effort was made to include the thermal capacitance effect.

F. Robben, Lawrence Berkeley Lab, USA. Have you considered catalytic surfaces and the effect this would have on ignition? Can you extend the calculations to consider a case that could be realized in experimental measurements?

Author's Reply. No, Catalytic surface boundary

conditions were not considered, though here again the qualitative features of the ignition process remain the same. We see no basic problem in analyzing the catalytic surface case.

Direct application of these numerical studies to experimental situations is at the moment difficult. The large number of assumptions made during the analysis, frustrate all attempts to quantitatively relate numerical results to experimental data. Moreover, experimental difficulties associated with raising the plate lower surface temperature suddenly are not trivial.

transient phenomena in boundary layer ignition with finite plate thermal resistance

Documents