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Ignition of a combustible gas near heated vertical surfaces
COMBUSTION AND FLAME 42: 77-92 (1981)
77
Ignition of a Combustible Gas near Heated Vertical Surfaces L-D. CHEN and G. M. FAETH Department of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania ]6802
A theoretical investigation of the ignition of a combustible gas by a heated vertical surface is described. Laminar, natural convection conditions were treated by numerical solution of the nonsimilar boundary layer equations, assuming an ideal gas and a one-step reaction and neglecting radiation. The plume above the surface was also analyzed, so that the ignition criterion could be specified as the least severe surface condition where a deflagration wave develops in the plume. Heated surfaces included both isothermal and constant heat flux boundary conditions. Plume conditions involved free-standing surfaces and surfaces along a wall. In the latter case, wall boundary conditions included adiabatic and isothermal (equal ambient and wall temperatures) wall plumes. Isothermal surfaces required less severe conditions for ignition than constant heat flux surfaces having the same maximum temperature. The required severity of surface conditions for ignition increased as plume conditions varied in this order: adiabatic wall plume, free-standing plume, isothermal wall plume. The frequently used ignition criterion for heated surfaces, which assumes that ignition occurs where the temperature gradient normal to the surface becomes zero, is shown to overestimate the severity of surface conditions required for ignition in some circumstances. Other limitations of this earlier criterion are that it cannot be applied to either constant heat flux or high-temperature surfaces.
INTRODUCTION Exposure of a combustible gas to a heated surface poses an obvious ignition hazard. Such faults can occur due to fires external to storage or processing equipment, or as a result of a process upset. Evaluation of the ignitability of combustibles, particularly when ignition energies are large, can also involve exposing the material to a heated surface. The objective of the present study was to complete a theoretical investigation of this process, limited to heated vertical surfaces, under laminar natural convection conditions. The earliest theoretical studies of ignition near heated surfaces considered forced convection conditions. Toong [1] examined this problem for laminar flow along a heated constant temperature flat plate, assuming a one-step reaction and neglecting radiation. The nonsimilar boundary layer equations were solved using a series expansion technique. Toong adopted the criterion that ignition occurs when the temperature increase of Copyright © 1981 by The Combustion Institute Published by Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, NY 10017
the flow due to reaction is sufficient to cause the temperature gradient normal to the surface to become zero at some point. Although this criterion is somewhat arbitrary, agreement between predictions and measurements was satisfactory; therefore, most subsequent investigators have employed the same definition of ignition. Sharma and Sirignano [2] have extended the analysis of forced convection ignition to other geometries, solving the nonsimilar boundary layer equations by direct numerical integration. Ono et al. [3] examined ignition of gases along a vertical constant temperature surface, under laminar natural convection conditions. Their general approach was similar to the forced convection studies [1, 2]. Extensive calculations were completed to determine the relation between ignition and surface height, surface and ambient temperature, pressure, and the nature of the combustible. The trends of these predictions were generally supported by measurements of ignition near free-standing heated foils.
0010-2180/81/07077+ 16502.50
78 Application of the findings of Ono et al. [3] to ignition of gases under natural convection conditions reveals several limitations, as follows: 1. When the flame temperature of the combustible is less than the surface temperature, the temperature gradient at the surface generally cannot become zero. Therefore, the zero-gradient criterion cannot be used to estimate ignition conditions in this case. 2. Heated surfaces are often best approximated by constant heat flux surfaces--a case that also cannot be treated with the zero-gradient ignition criterion. 3. Ignition measurements for some combustibles indicate that the distance of quenching surfaces above the heated surface can influence the results [4, 5]. This suggests that processes in the plume above the heated surfaces can affect ignition and should be considered in the criterion for ignition. The present investigation examines these limitations, for a vertical fiat plate under natural convection conditions. The new feature of the present study involves continuing the computation of flow properties along the heated surface into the plume region. In this manner a more rational definition of ignition can be prescribed. The computations also provide more insight concerning the development of a propagating flame above an ignition source and the potential effect of quenching surfaces in the plume region. The effect of a time variation of plate conditions is not examined. Only steady-state solutions are sought, with ignition specified as the least severe heated surface condition for which a deflagration wave is formed. This analysis bears some resemblance to the classical MarbleAdamson problem [6-8], where ignition resulting from the contact of a semiinfinite combustible gas and a heated gas stream, in a forced flow shear layer, is examined. The present problem differs, however, due to consideration of a natural convection flow and the fact that the heated material only involves partially reacted combustible gas in the plume above the heated surface.
L-D. CHEN and G. M. FAETH Similar to past work [2, 3], the nonsimilar laminar boundary layer equations are numerically integrated assuming a one-step chemical reaction and an ideal gas and neglecting radiation. Constant temperature and constant heat flux surfaces are treated, allowing for surface temperatures both above and below the adiabatic flame temperature of the combustible. The bulk of the results are for free-standing surfaces, due to their importance for ignition tests [3-5]. Some computations were also completed for heated surfaces along a wall. In this case, both adiabatic and isothermal surfaces adjacent to the plume were examined. THEORY
The analysis is applied to ignition near a smooth vertical flat surface which is heated. Both flow along the surface and the plume above the surface are considered. The following assumptions apply in both regions: the flow is laminar, twodimensional, and steady; the boundary layer approximations apply; ambient conditions are constant, except for pressure, which varies hydrostatically (however, the hydrostatic pressure variation is neglected in the reaction rate expression); the ambient velocity is negligible; the combustible is premixed and undergoes a one-step chemical reaction which can be represented by a global Arrhenius expression of order n; the flow is an ideal gas mixture with constant and equal molecular weights and specific heats, equal binary diffusion coefficients, and constant values of p/a and p;~ (these assumptions imply constant Prandtl, Schmidt, and Lewis numbers, although the Lewis number is not assumed to be unity); only concentration diffusion is considered; and radiation is neglected. These assumptions have generally been adopted during past investigations of ignition along surfaces [1-3, 6-8]. The advantage of this approach is that it yields a relatively compact formulation which includes most important phenomena of the ignition process. The origin of the coordinate system is taken at the base of the vertical surface, with x and y denoting distance along and normal to the surface, respectively. Adopting the previous as-
IGNITION NEAR HEATED SURFACES
79
sumptions, the governing equations for both the heated surface and plume regions are as follows [9]:
plume region: x > H ~c
y=O; apu
+--
p
(uU
ax
ay
apv
--
ay
= 0,
(1)
--+v ax
(
=
--
P\
+
+g~p(T--Too),
7yy/g g =
v/=gy
+ ~Q,
Pgy -"'
(4)
(5)
= r ( p c ) n exp (--E/RT).
Application of Eqs. (1)-(4) to the plume region over a free-standing surface requires the assumption of an infinitely thin surface since the formulation is not suitable for a recirculating flow behind a finite-width body. The ambient boundary conditions are the same in all instances: u=O,
c=c**,
T=Too.
ay ay
isothermal wall
u = 0,
adiabatic wall
aT u=--=0.
(12)
T = Too
(13)
ay
A wall temperature equal to the ambient temperature has been chosen for the isothermal wall plume in Eq. (12) since this represents the maximum realistic wall quenching effect. The adiabatic wall plume, on the other hand, represents a realistic upper bound for wall temperatures. A stream function is defined which satisfies Eq. (1) as follows:
acJ - - =
ay
af pu/ooo,
ax
-
(14)
pv/O~.
The transformation of variables is completed by combining a Howarth-Dorodnitzyn transformation for variable properties with a similarity transformation appropriate for natural convection along a vertical surface in the absence of reaction. The new independent variables are
~=x/~,
heated surface region: 0 < x < H ~c
(11)
- 0
(6)
The remaining boundary conditions vary for the two types of heated surfaces (isothermal and constant heat flux) and three types of plumes (free, isothermal wall, adiabatic wall), as follows:
y=0;
aT
-
(3)
where
y~oo;
au
free
(2) 0%
(10)
v=--=O
(7)
T = Tw
(8)
fOy
(p/p=) dy.
(15) The dependent variables are chosen as follows: C =c/coo,
u=v=--=O
rl = (Grx/4x4) 114
f = ~/(4v~o(Grx/4)ll4),
0 = ( T - - Too)/(Tref -- Too),
(16)
ay
where isothermal
constant heat flux
aT -- X - - = qw ; •
ay
It
(9)
Grx = g[3oo(Tre~ _ To~)xafl, 2.
(17)
For isothermal heated surfaces, Tref = T,. For constant heat flux surfaces, T~efwas a constant
80
L-D. CHEN and G. M. FAETH
temperature chosen arbitrarily (but not equal to Too). This approach was employed in order to unify the notation for the various cases and regimes considered in the analysis since the wall temperature is not constant for constant heat flux surfaces. Substituting Eqs. (14)-(17) into Eq s. (1)-( 5) and applying the approximations p~t = poo/a~, p), = poo~ yields
where .
,,
@w
Zlw"H
=
- -
X.o(T,e, -- T**) \ a m /
plume region: ~ > 1
free
f = C' = 0 f ' = 0' = 0
isothermal wall Pr---lO '' + 3.1'0'+ ~I/2Ad(TredT~
i)O =4,~( f O~
-
-
I)
0 , i)_~) '
Sc-xC '' + 3I'C' - ~II2Ar/B = 4~
adiabatic wall (19)
a~i (20)
where C" exp [E/(RT..(O(T,,dT.. -- 1) + 1))] 7"=
(Tret/T** -- 1)llZ(O(T,e~/T.. - 1) + 1~ - 1 (21)
A = 2KB(co.p**)n-X(H/g) xlu,
(22)
B = c**Q/CpT**.
(23)
In Eqs. (18)-(20), partial derivatives with respect to r/ are denoted by primes. The boundary conditions, Eqs. (6)--(13), are transformed as follows: 7/-+ ~';
.f' = 0 = O, C = i:
(24)
heated surface region: 0 < ~ < 1 n = 0;
isothermal
f=f
= c' = 0
0=1
constant heat flux 0' = -- ~1/4~ w'',
(28)
and Grn is evaluated by setting x = H in Eq. (17);
r/= 0;
f " + af.f"-- 2(f )= + O = 4~ ( f ' O-YT--.t (f I s -"a]) ) -~d E,
( 4 ~ 1/4
(29) (30)
f' = 0 = 0
(31)
f = 0' = 0.
(32)
Equations (18)--(32) were solved by numerical integration, employing the Keller "box" algorithm described by Cebeci and Bradshaw [10]. With this approach, the nonlinear implicit difference equations are solved using Newton's method with block-tridiagonal factorization. The method of solution was checked using existing results for forced and natural convection along surfaces [li-13]. During these comparisons ambient conditions were fixed at ~/oo = 6.0 and 61-71 cross stream grid nodes were employed, with a grid spacing ratios (AT/i+JA~/i) in the range 1-1.05. Maximum differences between the present calculations and the results of Refs. [11-13], for wall friction factors, Nusselt numbers, maximum velocities, and stream function values were less than 0.22~. Streamwise stability was also verified by checking that known similarity solutions did not vary with ~ when Eqs. (18)--(20) were integrated in the ~ direction. Good agreement was also obtained with the results of Ono et al. [3], for ignition along surfaces, as discussed in the next section.
RESULTS AND DISCUSSION (25)
Isothermal Free-Standing Sad'ace
(26)
Figure 1 is an illustration of profiles of dimensionless temperature, concentration, and velocity at several positions near a free-standing, heated,
(27)
81
IGNITION NEAR HEATED SURFACES
1.0
I
I
I
I
I
i
i
I
I
i
i
i
i
i
I
\N~....-----~ 0133
0.6
4.5
i
PLUME REGION
WALL REGION
0.8
0
I
!
3.8 3.1
z•4 T/T,,
0.4 0.2
1.7
0.0
1.0
0.8
0.0
C 0"6 f 0.4
1.0
0.2 0.0
n
~'1.11 " 1.01
=2
E/RT** = 34.3 A = 2.0 x 105 B =9.4
0.4
0;0
0.3
f,
Pr =
O.C
0.0
~
1.0
0.7, Sc = 0.65
~ 7
T./T=, = 4.5
1"= = 293K
I •79 I•11 .01
1.0 0.33
2.0
3.0
4.0
0.0
1.0
2.0
3.0
4.0
5.0
Fig• 1. Flow properties near an isothermal free-standing surface whose temperature is below the flame temperature (no ignition).
isothermal surface. Both the wall or heated surface region, and the plume region, are illustrated. Parameters for these results have been selected to agree with a computation reported by Ono et al. [3]. In fact, this condition corresponds to an ignition condition by the criterion that ignition occurs when the temperature gradient normal to the wall is zero at the top of the surface. Typical of the other results reported in Ref. I-3], the wall temperature is lower than the adiabatic flame temperature at this condition•
Near the base of the wall, chemical residence times are small and reaction has a negligible influence on the process. Thus profiles of 0 and f , in Fig. 1, correspond to conventional natural convection flows, and C = 1 for all 1/, at 1~ = O. With increasing ~, reaction becomes more important; the concentration of the combustible decreases near the wall and temperature increases in the region a short distance from the wall. The increased buoyancy, due to the temperature rise, causes the flow to accelerate
82 slightly as ~ increases along the wall. As noted earlier, this condition just corresponds to a case where the temperature gradient normal to the wall is zero at the top of the wall. Considering the results in the plume region, however, shows that this condition will not sustain a deflagration wave. As distance increases in the plume, the reaction quenches and the temperature increase and concentration defect decay away. After a period of adjustment, the velocity profile approaches that of a plume, with the maximum velocity at 77 = 0. In order to obtain a deflagration wave, more severe surface conditions than those employed in Fig. 1 must be present. The results for a condition leading to ignition are illustrated in Fig. 2. Most parameters are the same as Fig. 1, except that the reaction rate parameter A has been increased. This change can be accomplished in practice by increasing the height of the surface or the pressure. In this case, the extent of reaction is much more pronounced along the wall than in Fig. 1. Temperature levels well in excess of the wall temperature are developed along the surface. Temperatures continue to rise in the plume, eventually approaching the adiabatic flame temperature. A flame front forms and begins to propagate into the unburned mixture; however, even at ~ = 2.72 the wave is not yet fully developed. Thus, ignition in this circumstance can be influenced by quenching surfaces a significant distance above the ignition source. The true incipient ignition condition occurs for a value of A somewhere between the values used in Figs. 1 and 2. This limit was determined by selecting trial values of A to find the conditions which separated quenching and development of a deflagration wave in the plume. These results will be considered later. Many combustibles have relatively low flame temperatures and possible fault conditions could result in the combustible contacting a surface having a temperature greater than the flame temperature. Figures 3 and 4 consider such conditions, for no ignition and ignition, respectively. In neither instance does the temperature gradient at the wall reach zero. When the value of A is too small, the temperature perturbation
L-D. CHEN and G. M. FAETH caused by the wall decays, passing through the adiabatic flame temperature, and the process is quenched. However, for sufficiently high values of A, the temperature decay is arrested at levels slightly below the adiabatic flame temperature. Temperatures then increase once again as the flame front develops. As before, an appreciable distance above the heated surface is required to complete flame front development. Figure 5 is an illustration of the variation in the value of A required for ignition near a freestanding isothermal surface as Tw/T~ is varied. The plot is for a fixed ambient temperature; therefore, the results primarily illustrate the effect of varying surface temperature. Both the zero gradient (denoted Ono et al.) and plume ignition criteria (denoted present) are illustrated, with activation energy as a parameter. The parameters used in the computations correspond to values employed by Ono et al. [3]. Present calculations, using the zero gradient criterion, agree with those of Ono et al. [3], to the accuracy that the figures of Ref. [3] can be read (the range considered in Ref. [3] is limited to 3.9 < T,,/,To~< 5.7). Both criteria vary in a similar fashion for values of Tw/T~ < 5.5, which corresponds to the range of existing experimental data for ignition near vertical surfaces [3]. This helps explain why the zero gradient criterion has been successful for purposes of data correlation. Kinetic parameters are generally unknown; therefore, a minor adjustment of their values from the true values could bring the two predictions into agreement in the low Tw'/T~regime. It is evident, however, that this would not be the case as the wall temperature approaches the adiabatic flame temperature of the combustible. In this region, the zero-gradient criterion predicts an increasing resistance to ignition with increasing wall temperature, before breaking down entirely at the adiabatic flame temperature. In contrast, the plume ignition criterion shows that ignition resistance continues to decrease with increasing surface temperature. Figure 6 is an illustration of the variation of A for ignition when the surface temperature is kept constant, but the temperature of the combustible is varied. The parameter values for the com-
IGNITION NEAR HEATED SURFACES
2.8 2.4
2.0
.
.
.
.
.
.
.
83
.
.
.
.
.
.
.
.
.
.
.
••.•
ADIABATIC J FLAME J TEMP. n =2 E/RT= = 34.3 A = 8.81 x 105
10.8 9.4
2.72 =
~ z o 4
\ - ~,
B =9.4
L3--~
~.55
8.0
V - ~ I . 1 5
Pr = 0.7, Sc = 0.65 Tw/T= = 4.5
1.6
8
6.6
T/T=
T** = 2 9 3 K
1.2
5.2
i.oo = ¢ 0.61 0.37 0.072
0.8
3.8
0.4
2.4
0.0
1.0
1.0
C
0.8 0.6
0.0
0.4 ~ Z 0.2
-'--- o.61 1.00
i .oo
0.0 0.8
WALL REGION .
0.6
PLUMEREGION 2.72 2 .04
~
f ' 0.4
~
1.00 0.61 0 37 0.072 0.00
1.55
i~
,)~.~\\
1.15
0.2 0.0 O.0
,
1.0
2D
3.0
4.0
0.0
1.0
2.0
3.0
I
4.0
I
5.0
Fig. 2. Flow properties near an isothermal free-standing surface whose temperature is below the flame temperature (ignition).
84
L-D. CHEN and G. M. FAETH
1.0
,
,
;
,
,
k
I , ; , WALL REGION
| ~
,
,
,
,
0.8 _~AD_IABATIFTE C TEMP. ~ ~ O04 •
/~IL_/o3
I-
°.°'~k
I
o.o-
o 2
~
0"0 ~
.
~
~ O.Ol
,o
F 0,30)
0.3[" ~
12.2
\
\
~
66T/T=
~.Ol=~,
\ X~~2-o3 \ k ~ e.73
•
3.8
~~<.-'//29.5
1.0
~ ~ 2 9 . 5 ~ -
-
-
--
2.03 ,.0,
I
P,:o.7,Sc:O.65~
=15.0
,:o,
i- /~\~J~29.5
TWIT.
f' 0.21 o~oo
-
~~8.73 I ~
n=2 E / R T . = 34.3 A=2XlO 4
°'F
150
~
0.6 I- ~ ' ~ 0"2 1 O0 L
I ; ; I , PLUME REGION
=
0.1 0.0
0.0
1.0
20
3.0
4.0
0.0
1.0
20
30
40
5.0
Fig. 3. Flow properties near an isothermal free-standing surface whose temperature is above the flame temperature (no ignition).
IGNITION NEAR HEATED SURFACES
I.om, ,
.
.
.
.
.
.
.
85 .
t
0.6
0
\\\~--------1.00 = (
~-
o.L a41
.
.
.
.
.
.
.
.
i 15.°
.
~ \ ~ ~ 3 . 0 = ¢
19.4
(_
-I
\ ' k - ~ O.OI
~
~..---~44.6
I
T/T. 66
O.O
IO
1.0 0.8
C
0.6 0.4 O.2
"1.00
11=2 E/RT** = 54.5 I O0 A= I x 105 ~ /2.83 / / ~ . ; 50 B=9.4 ~ 1 5 0 ~ , " Pr = 0.7, S c = 0.65 44.6
QO
0.5 Q4
f, o3 0.2
t
OJ 0.0 O0
S.S I
I
I.O
I
I
20
I
I
3.O
'
I
I
4.0
OO
IO
20
30
4.0
50
~q
Fig. 4. Flow properties near an isothermal free-standing surface whose temperature is above the flame temperature (ignition).
bustible are the same as in Fig. 5. A wall temperature of 1500K is employed for the calculations since this is in the range of maximum temperatures for metal foils and wires used in ignition experiments [3-5]. As before, results for both the zero-gradient and plume ignition criteria are illustrated. For the range of conditions shown, either criterion could be used to correlate data, although the effective value of A needed for ignition would be underestimated by the zerogradient criterion. The results show that the severity of plate conditions required for ignition is reduced as the ambient temperature of the combustible is increased.
Isothermal Surfaces along Walls Since plume conditions are clearly important for predicting ignition near heated surfaces, several boundary conditions which might be encountered in practice were considered. The results are summarized in Table 1, which provides the value of A required for ignition as a function of E, Tw/T~, and plume configuration. The plume configurations include the plume over a free-standing surface and wall plumes adjacent to adiabatic and isothermal (T, = T~) surfaces. The latter conditions would be encountered for ignition near a heated patch on a vertical surface, with the
86
L-D. CHEN and G. M. FAETH i
!
i I
i
t
E, 4o KCAL/C,~_~
I
P~ESENT
~/ !~{/
I --'-I [ ....
ONO, ET AL EXTRAPOLA-
I rio. I 8=9.4
~.\ ~\ \ \
I Pr "0.7; So= 0.65 I T=" L~93K
\\
i I I I"'ADIAIBATIC FLAME I TEMPERATURE
IOs ~ : , 0 KCAL/GMOLE /!
105' '°'o-,
~ I '
80 '
'
12.0
16.0
'
Tw/T® Fig. 5. R e l a t i o n b e t w e e n A and TWIT=for i g n i t i o n near i s o t h e r m a l f r e e - s t a n d i n g surfaces w i t h c o n s t a n t a m b i e n t temperature.
I
I
=
,I
I
I
I
=1
two cases essentially bounding the realistic range of surface temperatures in the plume region. Results for the zero-gradient ignition criterion are also shown for comparison. The parameters of the combustible are similar to those used in Figs. 5 and 6. Within the range of parameters considered in Table 1, ignition resistance increases in this order: adiabatic wall plume, free-standing plume, isothermal wall plume. The isothermal wall plume requires the most severe heated surface conditions for ignition since the flow in the plume is quenched by both entrained fluid and heat loss to the cold wall. The free-standing plume and the adiabatic wall plume are similar to the extent that quenching only occurs due to entrainment. The greater ignition resistance of the free-standing surface is due to the fact that velocities are higher in the free plume than for a wall plume, where wall friction retards the flow. In a Lagrangian sense, the higher velocity reduces the residence time of the combustible in high temperature portions of the flow, which increases ignition resistance. In the Eulerian sense, the increased velocity increases convection effects, causing temperature and concentration profiles to steepen in the direction normal to the surface. Since an ignition process is largely chemical rate controlled, the increased heat loss due to more abrupt gradients increases ignition resistance. Similar arguments concerning the effect of ve-
I
I
1
=1
I
I
I
I
I
I
I
I
I
0.4
/GMOLE
-20 KGAL/GMOLEn • 2
~
10.3
0.2 103
\
\ \~
BT** = 2754 Pr " 0.7; Sc- 0.65 ,..,,ooK
I
I
io 4
I
I
I
-• ..
PRESENT (~ll~, E r 1~1..
I
I
I
1o5
I
I
I
I
i0 e_
107
108
Ax(T=/T.) z Fig. 6. R e l a t i o n b e t w e e n A and Tw/T. for i g n i t i o n near i s o t h e r m a l f r e e - s t a n d i n g surfaces with constant surface t e m p e r a t u r e .
IGNITION NEAR HEATED SURFACES
87
TABLE1 Comparison of the Values of A Required for Ignition by Constant Temperature Surfaces with Different Plume Conditionsa E (kcal/gmol)
Tw/To.
=
40
3.6 X 105 5.6 X 105 3.6 X 105
3.1 X 109 4.2 X 109 3.1 X 109
1.0 1.7 8.6 6.5
8.3 2.0 8.3 3.4
4.0
zero gradient free standing adiabatic wallb
T,,,/r,o =
20
5.5
zero gradient free standing adiabatic wallb isothermal wallb
X × × X
105 10ft 104 105
X x × X
107 108 107 108
a T** = 293 K, n = 2,B = 9.4,Pr = 0.7, Sc = 0.65. b Wall condition in plume region: wall and ambient temperatures equal for the isothermal wall plume.
locity have been given for forced flow ignition along surfaces [1, 2]. The main difference for natural convection is that the flow configuration itself influences the magnitude of the velocities. The results summarized in Table 1 provide a further indication that the zero-gradient criterion does not always underestimate the severity of surface conditions required for ignition. Generally, the adiabatic wall plume requires A values for ignition that are less than or equal to the values provided by the zero-gradient criterion even for the relatively low values of surface temperature considered in Table 1.
Constant Heat Flux Free-Standing Surfaces Figures 7 and 8 are illustrations of flow properties near free-standing constant heat flux surfaces. Figure 7 is typical of quenched reaction in the plume, while Fig. 8 is a case where a deflagration wave develops. In both circumstances, f = 0= 0 and C = 1, for all values of r/ at I~= 0. With increasing I~, the wall temperature and the maximum velocity increase with increasing distance along the surface. Similar to the findings for isothermal heated surfaces, a large
distance above the source is required for a flame to develop, as the limiting conditions for ignition are approached. The minimum value of A required for ignition along free-standing constant heat flux surfaces is illustrated in Fig. 9 as a function of dimensionless heat flux. Only results for the plume ignition criterion are shown, since the zero-gradient criterion can never be satisfied for a constant~heat flux surface. At low values of ¢ w", the value of A required for ignition decreased with increasing w". This behavior is expected, due to the higher wall temperatures resulting from higher heat fluxes. An interesting feature of the results, however, is that the minimum value of A for ignition becomes relatively independent of ¢,," at higher values of this parameter. At least two factors contribute to this behavior. First of all, the temperature sensitivity of the chemical reaction rate begins to decrease at higher temperatures, with the effect occurring sooner for lower values of activation energy. This is shown by the fact that the slope of the curve of minimum A decreases sooner, and more rapidly, for the lower value of E (the same trend is observed for the isothermal wall results; cf. Fig. 5). Thus, at high temperature levels the rate of increase of reactivity, in comparison to heat losses, is reduced. The second factor involves the larger flow velocities encountered for high heat fluxes. This tends to reduce chemical residence times while increasing quenching effects due to steeper thermal gradients and increased entrainment, similar to effects discussed when wall and free plume ignition were compared; cf. Figs. 7 and 8. Results for free-standing isothermal and constant heat flux surfaces are compared in Table 2. The table provides the value of A, required by the plume ignition criterion, as a function of the ratio of the maximum surface temperature and the ambient temperature. At a given temperature ratio, the required values of A are larger for constant heat flux surfaces. This is expected since high temperature levels are only reached near the top of constant heat flux surfaces, which reduces residence times at high states of reactivity, in comparison to isothermal surfaces. The effect is less at large values of Twm~x/To~,due to reduced
88
L-D. CHEN and G. M. FAETH
~---~0
/
i
i
!
i
6 Ill I
!
i
i
i
!
I
I
I
I
I
I
5.0
I
PLUME REGION
WALL REGION
42
\_
O 0.8
~ .
5.4
1.00 =~ 1.13 1.89
0.24
~
T/T**
2.6
0.4
1.8
0.0
1.0
1.0 0.8 0.6 C 0.4
~
1, 0.24
~
10.6 \
1.89
\1.13
i.oo
1.00 n=2
0.2 0.0 0.41/ =.00
E/RT= = 34.3 A = 1.84 x 105 Pr = 0.7, Se = 0.65 Tref/T==3.0 •
0.3
~ ~ ~
~
1.89 ~1.13
=
1o.6
f, 0.2 0.1 0.0 0.0
1.0
2.0
3.0
4.0
0.0
1.0
2.0
3.0
4.0
5.0
I/ Fig. 7. Flow properties near a free-standing constant heat flux surface (no ignition, ~w" = 0.82).
temperature sensitivity of the reaction rate expression at such conditions.
LimitationsoftheResults A major limitation of the present and earlier studies concerning ignition along surfaces I-1-3] is neglect of radiation. This limits the results to
small surfaces, relatively low pressures, and weakly absorbing gases. Even in this case, it should be noted that the present constant heat flux computations are limited to constant c o n v e c t i v e heat flux. Surface radiation is normally important for high-temperature surfaces in a natural convection environment. Therefore, constant heat flux surfaces would generally require
IGNITION NEAR HEATED SURFACES ,0
i
i
i
i
f
i
89
|
i
i
I
I
I
I
I
I
I
I
I1.0
I
T
ADIABATICFLAMETEMP.
4.0
9.0 ~ L O 0 J._4--4---u
={ 7.0
I
~ ~2.2a I ~ L..4~6.4a
0 3.0 ~ - - - 1.00=~' 2.0 x,L1-o.95
5"° T/T=
,~O.al
l.O
:5.0
0.0
1.0
C '-,.oy
~-o.gs
n=2
J/
~ 1 . 0 0
~
/"/--../~l.lW
REGION
".1.0(0 E/RT== 34.3 [" / ~ 2 . 2 3 A = e.os, Io 5 l/" / /...:'-6.42 B=9.4 ~ -"" 2 44 Pr = 0.7, Sc = 0.65
0.41 o.z ~ 0.0 L
OAr ~
i OoTter/T®=3.0
r ~~'5
aa ~//fk-
L/// I///
R
0.0095 0.81
^ ~ ~"/~-
O.Sl
1.6
r
T.=2,3.
1-'
K.
~/0.0095
"
k~ ~
1.2 244
I \V~...~6.42 ~'x ~ . . ~ 2.25
A
f' 0"2I / / / / ~ " ~
o.8 f' 0.4
.'lo
, ,,
o.o
,.o
, io
~.o
,.o
o.o
,o
2o
3.0
io.o io
5o
Fig. 8. Flow properties near a free-standing constant heat flux surface (ignition, ~w". = 0.82).
more energy input for ignition than the values prescribed by this analysis. Significant radiation effects would also influence the surface temperature distribution limiting the applicability of the present analysis in such cases. A second limitation of the analysis, already pointed out by Sharma and Sirignano [2] for ignition along a surface under forced convection conditions, involves the use of boundary layer equations in the analysis. It is clear from the
results illustrated in Figs. 1-4, 7, and 8 that there are significant streamwise gradients in flow properties. Thus the effect of streamwise diffusion of heat and mass can be important in some circumstances, particularly when G r x is low in the region of flame development. A quantitative indication of this effect must await more complete analysis. It is suggested that streamwise transport would tend to increase quenching; therefore, somewhat higher values of A would be required for ignition
90
L-D. CHEN and G. M. FAETH
I0 9
I
I
I
n=2 B= 9.4 Pr = 0.7; So = 0.65 T= • 293 K
i0 e
10 7
E = 40 K C A L / G M O L E % .
I0 6
10 5
104 0.0
I 1.0
t 20
I 30
(~'WII Fig. 9. Relation between A and ~w" for ignition near constant heat flux, free-standing surfaces with ambient temperature maintained constant.
when G r x is small. Streamwise diffusion would also influence the rate at which the flame spreads from the surface. Transition to turbulence also has interesting implications for ignition along surfaces. The question has not been considered, to the authors' knowledge, and the conditions for transition in natural convection reacting flows have not been generally established. Thus the Grashoff number
limit of the present analysis, and the impact of transition on ignition properties, cannot be stated without further study.
S U M M A R Y AND C O N C L U S I O N S A theoretical study of ignition near heated vertical surfaces under laminar natural convection
IGNITION NEAR HEATED SURFACES
91
TABLE 2
properties vary in this order: adiabatic wall plume, free plume, isothermal wall plume (ambient and wall temperatures the same). Wall quenching, in the case of isothermal wall plumes, and higher velocities, in the case of free plumes, tend to reduce the hazard when compared to adiabatic wall plumes. . Near limiting conditions for ignition, substantial distances above the surface are required for a deflagration wave to develop. Therefore, the effect of quenching surfaces above the ignition source must be carefully evaluated during ignition experiments; otherwise, the ignition hazard of the combustible will be underestimated. . Constant temperature surfaces require less severe surface conditions for ignition (lower values of A) than constant heat flux surfaces having the same maximum temperature.
Comparison of Ignition Requirements for Constant Temperature and Constant Heat Flux Surfacesa
Twmax/Too 4.5 6.5 9.5 11.5 14.0 15.0
A × 10--4 Constant heat flux Isothermal 95 20 8,5 4.4 3,4 3.2
35 10 5.0 3.7 2.9 2.7
a Free-standing surfaces: E = 20 kcal/gmol, T** = 293 K, n = 2, B = 9.4, Pr = 0.7, Sc = 0.65.
conditions was completed. Flow properties along the surface and in the plume were determined for steady conditions. Ignition was prescribed by the heated surface condition which separated quenching and flame development in the plume. Constant temperature and constant heat flux heated surfaces were considered. Plume conditions included free plumes and adiabatic and isothermal wall plumes. The major conclusions of the study are as follows: 1. Plume properties should be considered when estimating the ignition properties of heated surfaces. The earlier zero-gradient criterion (which prescribes ignition when the temperature gradient normal to the surface is zero at some point) provides only a qualitative indication of ignition properties when surface temperatures are well below the adiabatic flame temperature of the combustible. In some circumstances, the zero-gradient criterion overestimates the severity of surface conditions required for ignition. The zero-gradient criterion also breaks down as the surface temperature nears the flame temperature and cannot be applied at all for constant heat flux surfaces or for a surface whose temperature is greater than the flame temperature. . The severity of surface conditions required for ignition generally increases as plume
These conclusions are based on solution of laminar boundary layer equations for an ideal gas, assuming a one-step reaction and neglecting radiation. The effect of streamwise diffusion, real gas properties, radiation, and turbulence are matters that should receive further attention in order to better understand the ignition properties of heated surfaces.
This work was supported by Union Carbide Corporation, Chemicals and Plastics, South Charleston, West Virginia, with P. Wadia of the Research and Development Department serving as technical monitor for the project. NOMENCLATURE A B c C Cp D E f g
reaction parameter, Eq. (22) parameter, Eq. (23) mass fraction of combustible dimensionless mass fraction, Eq. (16) specific heat at constant pressure binary diffusivity activation energy dimensionless stream function, Eq. (16) acceleration of gravity Grx Grashoff number, Eq. (17)
92
L-D. CHEN and G. M. FAETH
H K n
surface height preexponential factor reaction order Pr Prandtl n u m b e r w" surface heat flux Q heat of c o m b u s t i o n R Universal gas constant Sc Schmidt n u m b e r T temperature u velocity parallel to surface v velocity n o r m a l to surface x distance along surface y distance normal to surface t~ reaction rate, Eq. (5) /3 coefficient of volumetric expansion r/ dimensionless distance, Eq. (15) 0 dimensionless temperature, Eq. (16) X thermal conductivity absolute viscosity v kinematic viscosity dimensionless distance, Eq. (15) p density r dimensionless reaction rate, Eq. (21) w" dimensionless heat flux, Eq. (28) ~0 stream function
SUBSCRIPTS ref w
reference value wall value ambient value
REFERENCES 1. Toong, T.-Y., Sixth Symposium (lnternationaO on Combustion, Reinhold, New York, 1955, pp. 532540. 2. Sharma, O. P., and SMgnano, W. A., Combustion Science and Technology 1:95-104 (1969); ibid. 1:481-494 (1970). 3. Ono, S., Kawano, H., Niho, H., and Fukuyama, G., Bull. JSME 19:676-683 (1976). 4. Chen, L.-D., and Faeth, G. M., Combustion and Flame, 40:13-28 (1981). 5. Cawse, J. N., Pesetsky, B., and Vyn, W. T., '~Fhe Liquid Phase Decomposition of Ethylene Oxide," AIChE LossPrevention Symposium, Houston (1979). 6. Marble, F. E., and Adamson, T. C., Jr., Jet Propulsion 24:85-94 (1954). 7. Cheng, S. I. and Kovitz, A. A., Sixth SymPosium (InternationalJ on Combustion, Reinhold, New York, 1955, pp. 418-427. 8. Cheng, S. I., and Chiu, H. H., Int. J. Heat Mass Transfer 1:280-293 (1961). 9. Williams, F. A., Combustion Theory, Addison-Wesley, Reading, MA, 1965, Chap. 1. 10. Cebeci, T. and Bradshaw, P., Momentum Transfer in Boundary Layers, McGraw-Hill, New York, 1977, Chap. 7. 11. Rosenhead, L. (ed.), Laminar Boundary Layers, Oxford University Press, Oxford, 1963, pp. 224-234. 12. Cohen, C. B. and Roshotko, E., Similar Solutions for the Compressible Laminar Boundary Layer with Heat Transfer and Pressure Gradient, NACA TN-3325, 1955, p. 54. 13. Ostrach, S., Analysis of Laminar Free-Convection Flow and Heat Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force, NACA TN-2635, 1952, pp. 36-37. Received 28 May 1980; revised 28 August 1980
77
Ignition of a Combustible Gas near Heated Vertical Surfaces L-D. CHEN and G. M. FAETH Department of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania ]6802
A theoretical investigation of the ignition of a combustible gas by a heated vertical surface is described. Laminar, natural convection conditions were treated by numerical solution of the nonsimilar boundary layer equations, assuming an ideal gas and a one-step reaction and neglecting radiation. The plume above the surface was also analyzed, so that the ignition criterion could be specified as the least severe surface condition where a deflagration wave develops in the plume. Heated surfaces included both isothermal and constant heat flux boundary conditions. Plume conditions involved free-standing surfaces and surfaces along a wall. In the latter case, wall boundary conditions included adiabatic and isothermal (equal ambient and wall temperatures) wall plumes. Isothermal surfaces required less severe conditions for ignition than constant heat flux surfaces having the same maximum temperature. The required severity of surface conditions for ignition increased as plume conditions varied in this order: adiabatic wall plume, free-standing plume, isothermal wall plume. The frequently used ignition criterion for heated surfaces, which assumes that ignition occurs where the temperature gradient normal to the surface becomes zero, is shown to overestimate the severity of surface conditions required for ignition in some circumstances. Other limitations of this earlier criterion are that it cannot be applied to either constant heat flux or high-temperature surfaces.
INTRODUCTION Exposure of a combustible gas to a heated surface poses an obvious ignition hazard. Such faults can occur due to fires external to storage or processing equipment, or as a result of a process upset. Evaluation of the ignitability of combustibles, particularly when ignition energies are large, can also involve exposing the material to a heated surface. The objective of the present study was to complete a theoretical investigation of this process, limited to heated vertical surfaces, under laminar natural convection conditions. The earliest theoretical studies of ignition near heated surfaces considered forced convection conditions. Toong [1] examined this problem for laminar flow along a heated constant temperature flat plate, assuming a one-step reaction and neglecting radiation. The nonsimilar boundary layer equations were solved using a series expansion technique. Toong adopted the criterion that ignition occurs when the temperature increase of Copyright © 1981 by The Combustion Institute Published by Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, NY 10017
the flow due to reaction is sufficient to cause the temperature gradient normal to the surface to become zero at some point. Although this criterion is somewhat arbitrary, agreement between predictions and measurements was satisfactory; therefore, most subsequent investigators have employed the same definition of ignition. Sharma and Sirignano [2] have extended the analysis of forced convection ignition to other geometries, solving the nonsimilar boundary layer equations by direct numerical integration. Ono et al. [3] examined ignition of gases along a vertical constant temperature surface, under laminar natural convection conditions. Their general approach was similar to the forced convection studies [1, 2]. Extensive calculations were completed to determine the relation between ignition and surface height, surface and ambient temperature, pressure, and the nature of the combustible. The trends of these predictions were generally supported by measurements of ignition near free-standing heated foils.
0010-2180/81/07077+ 16502.50
78 Application of the findings of Ono et al. [3] to ignition of gases under natural convection conditions reveals several limitations, as follows: 1. When the flame temperature of the combustible is less than the surface temperature, the temperature gradient at the surface generally cannot become zero. Therefore, the zero-gradient criterion cannot be used to estimate ignition conditions in this case. 2. Heated surfaces are often best approximated by constant heat flux surfaces--a case that also cannot be treated with the zero-gradient ignition criterion. 3. Ignition measurements for some combustibles indicate that the distance of quenching surfaces above the heated surface can influence the results [4, 5]. This suggests that processes in the plume above the heated surfaces can affect ignition and should be considered in the criterion for ignition. The present investigation examines these limitations, for a vertical fiat plate under natural convection conditions. The new feature of the present study involves continuing the computation of flow properties along the heated surface into the plume region. In this manner a more rational definition of ignition can be prescribed. The computations also provide more insight concerning the development of a propagating flame above an ignition source and the potential effect of quenching surfaces in the plume region. The effect of a time variation of plate conditions is not examined. Only steady-state solutions are sought, with ignition specified as the least severe heated surface condition for which a deflagration wave is formed. This analysis bears some resemblance to the classical MarbleAdamson problem [6-8], where ignition resulting from the contact of a semiinfinite combustible gas and a heated gas stream, in a forced flow shear layer, is examined. The present problem differs, however, due to consideration of a natural convection flow and the fact that the heated material only involves partially reacted combustible gas in the plume above the heated surface.
L-D. CHEN and G. M. FAETH Similar to past work [2, 3], the nonsimilar laminar boundary layer equations are numerically integrated assuming a one-step chemical reaction and an ideal gas and neglecting radiation. Constant temperature and constant heat flux surfaces are treated, allowing for surface temperatures both above and below the adiabatic flame temperature of the combustible. The bulk of the results are for free-standing surfaces, due to their importance for ignition tests [3-5]. Some computations were also completed for heated surfaces along a wall. In this case, both adiabatic and isothermal surfaces adjacent to the plume were examined. THEORY
The analysis is applied to ignition near a smooth vertical flat surface which is heated. Both flow along the surface and the plume above the surface are considered. The following assumptions apply in both regions: the flow is laminar, twodimensional, and steady; the boundary layer approximations apply; ambient conditions are constant, except for pressure, which varies hydrostatically (however, the hydrostatic pressure variation is neglected in the reaction rate expression); the ambient velocity is negligible; the combustible is premixed and undergoes a one-step chemical reaction which can be represented by a global Arrhenius expression of order n; the flow is an ideal gas mixture with constant and equal molecular weights and specific heats, equal binary diffusion coefficients, and constant values of p/a and p;~ (these assumptions imply constant Prandtl, Schmidt, and Lewis numbers, although the Lewis number is not assumed to be unity); only concentration diffusion is considered; and radiation is neglected. These assumptions have generally been adopted during past investigations of ignition along surfaces [1-3, 6-8]. The advantage of this approach is that it yields a relatively compact formulation which includes most important phenomena of the ignition process. The origin of the coordinate system is taken at the base of the vertical surface, with x and y denoting distance along and normal to the surface, respectively. Adopting the previous as-
IGNITION NEAR HEATED SURFACES
79
sumptions, the governing equations for both the heated surface and plume regions are as follows [9]:
plume region: x > H ~c
y=O; apu
+--
p
(uU
ax
ay
apv
--
ay
= 0,
(1)
--+v ax
(
=
--
P\
+
+g~p(T--Too),
7yy/g g =
v/=gy
+ ~Q,
Pgy -"'
(4)
(5)
= r ( p c ) n exp (--E/RT).
Application of Eqs. (1)-(4) to the plume region over a free-standing surface requires the assumption of an infinitely thin surface since the formulation is not suitable for a recirculating flow behind a finite-width body. The ambient boundary conditions are the same in all instances: u=O,
c=c**,
T=Too.
ay ay
isothermal wall
u = 0,
adiabatic wall
aT u=--=0.
(12)
T = Too
(13)
ay
A wall temperature equal to the ambient temperature has been chosen for the isothermal wall plume in Eq. (12) since this represents the maximum realistic wall quenching effect. The adiabatic wall plume, on the other hand, represents a realistic upper bound for wall temperatures. A stream function is defined which satisfies Eq. (1) as follows:
acJ - - =
ay
af pu/ooo,
ax
-
(14)
pv/O~.
The transformation of variables is completed by combining a Howarth-Dorodnitzyn transformation for variable properties with a similarity transformation appropriate for natural convection along a vertical surface in the absence of reaction. The new independent variables are
~=x/~,
heated surface region: 0 < x < H ~c
(11)
- 0
(6)
The remaining boundary conditions vary for the two types of heated surfaces (isothermal and constant heat flux) and three types of plumes (free, isothermal wall, adiabatic wall), as follows:
y=0;
aT
-
(3)
where
y~oo;
au
free
(2) 0%
(10)
v=--=O
(7)
T = Tw
(8)
fOy
(p/p=) dy.
(15) The dependent variables are chosen as follows: C =c/coo,
u=v=--=O
rl = (Grx/4x4) 114
f = ~/(4v~o(Grx/4)ll4),
0 = ( T - - Too)/(Tref -- Too),
(16)
ay
where isothermal
constant heat flux
aT -- X - - = qw ; •
ay
It
(9)
Grx = g[3oo(Tre~ _ To~)xafl, 2.
(17)
For isothermal heated surfaces, Tref = T,. For constant heat flux surfaces, T~efwas a constant
80
L-D. CHEN and G. M. FAETH
temperature chosen arbitrarily (but not equal to Too). This approach was employed in order to unify the notation for the various cases and regimes considered in the analysis since the wall temperature is not constant for constant heat flux surfaces. Substituting Eqs. (14)-(17) into Eq s. (1)-( 5) and applying the approximations p~t = poo/a~, p), = poo~ yields
where .
,,
@w
Zlw"H
=
- -
X.o(T,e, -- T**) \ a m /
plume region: ~ > 1
free
f = C' = 0 f ' = 0' = 0
isothermal wall Pr---lO '' + 3.1'0'+ ~I/2Ad(TredT~
i)O =4,~( f O~
-
-
I)
0 , i)_~) '
Sc-xC '' + 3I'C' - ~II2Ar/B = 4~
adiabatic wall (19)
a~i (20)
where C" exp [E/(RT..(O(T,,dT.. -- 1) + 1))] 7"=
(Tret/T** -- 1)llZ(O(T,e~/T.. - 1) + 1~ - 1 (21)
A = 2KB(co.p**)n-X(H/g) xlu,
(22)
B = c**Q/CpT**.
(23)
In Eqs. (18)-(20), partial derivatives with respect to r/ are denoted by primes. The boundary conditions, Eqs. (6)--(13), are transformed as follows: 7/-+ ~';
.f' = 0 = O, C = i:
(24)
heated surface region: 0 < ~ < 1 n = 0;
isothermal
f=f
= c' = 0
0=1
constant heat flux 0' = -- ~1/4~ w'',
(28)
and Grn is evaluated by setting x = H in Eq. (17);
r/= 0;
f " + af.f"-- 2(f )= + O = 4~ ( f ' O-YT--.t (f I s -"a]) ) -~d E,
( 4 ~ 1/4
(29) (30)
f' = 0 = 0
(31)
f = 0' = 0.
(32)
Equations (18)--(32) were solved by numerical integration, employing the Keller "box" algorithm described by Cebeci and Bradshaw [10]. With this approach, the nonlinear implicit difference equations are solved using Newton's method with block-tridiagonal factorization. The method of solution was checked using existing results for forced and natural convection along surfaces [li-13]. During these comparisons ambient conditions were fixed at ~/oo = 6.0 and 61-71 cross stream grid nodes were employed, with a grid spacing ratios (AT/i+JA~/i) in the range 1-1.05. Maximum differences between the present calculations and the results of Refs. [11-13], for wall friction factors, Nusselt numbers, maximum velocities, and stream function values were less than 0.22~. Streamwise stability was also verified by checking that known similarity solutions did not vary with ~ when Eqs. (18)--(20) were integrated in the ~ direction. Good agreement was also obtained with the results of Ono et al. [3], for ignition along surfaces, as discussed in the next section.
RESULTS AND DISCUSSION (25)
Isothermal Free-Standing Sad'ace
(26)
Figure 1 is an illustration of profiles of dimensionless temperature, concentration, and velocity at several positions near a free-standing, heated,
(27)
81
IGNITION NEAR HEATED SURFACES
1.0
I
I
I
I
I
i
i
I
I
i
i
i
i
i
I
\N~....-----~ 0133
0.6
4.5
i
PLUME REGION
WALL REGION
0.8
0
I
!
3.8 3.1
z•4 T/T,,
0.4 0.2
1.7
0.0
1.0
0.8
0.0
C 0"6 f 0.4
1.0
0.2 0.0
n
~'1.11 " 1.01
=2
E/RT** = 34.3 A = 2.0 x 105 B =9.4
0.4
0;0
0.3
f,
Pr =
O.C
0.0
~
1.0
0.7, Sc = 0.65
~ 7
T./T=, = 4.5
1"= = 293K
I •79 I•11 .01
1.0 0.33
2.0
3.0
4.0
0.0
1.0
2.0
3.0
4.0
5.0
Fig• 1. Flow properties near an isothermal free-standing surface whose temperature is below the flame temperature (no ignition).
isothermal surface. Both the wall or heated surface region, and the plume region, are illustrated. Parameters for these results have been selected to agree with a computation reported by Ono et al. [3]. In fact, this condition corresponds to an ignition condition by the criterion that ignition occurs when the temperature gradient normal to the wall is zero at the top of the surface. Typical of the other results reported in Ref. I-3], the wall temperature is lower than the adiabatic flame temperature at this condition•
Near the base of the wall, chemical residence times are small and reaction has a negligible influence on the process. Thus profiles of 0 and f , in Fig. 1, correspond to conventional natural convection flows, and C = 1 for all 1/, at 1~ = O. With increasing ~, reaction becomes more important; the concentration of the combustible decreases near the wall and temperature increases in the region a short distance from the wall. The increased buoyancy, due to the temperature rise, causes the flow to accelerate
82 slightly as ~ increases along the wall. As noted earlier, this condition just corresponds to a case where the temperature gradient normal to the wall is zero at the top of the wall. Considering the results in the plume region, however, shows that this condition will not sustain a deflagration wave. As distance increases in the plume, the reaction quenches and the temperature increase and concentration defect decay away. After a period of adjustment, the velocity profile approaches that of a plume, with the maximum velocity at 77 = 0. In order to obtain a deflagration wave, more severe surface conditions than those employed in Fig. 1 must be present. The results for a condition leading to ignition are illustrated in Fig. 2. Most parameters are the same as Fig. 1, except that the reaction rate parameter A has been increased. This change can be accomplished in practice by increasing the height of the surface or the pressure. In this case, the extent of reaction is much more pronounced along the wall than in Fig. 1. Temperature levels well in excess of the wall temperature are developed along the surface. Temperatures continue to rise in the plume, eventually approaching the adiabatic flame temperature. A flame front forms and begins to propagate into the unburned mixture; however, even at ~ = 2.72 the wave is not yet fully developed. Thus, ignition in this circumstance can be influenced by quenching surfaces a significant distance above the ignition source. The true incipient ignition condition occurs for a value of A somewhere between the values used in Figs. 1 and 2. This limit was determined by selecting trial values of A to find the conditions which separated quenching and development of a deflagration wave in the plume. These results will be considered later. Many combustibles have relatively low flame temperatures and possible fault conditions could result in the combustible contacting a surface having a temperature greater than the flame temperature. Figures 3 and 4 consider such conditions, for no ignition and ignition, respectively. In neither instance does the temperature gradient at the wall reach zero. When the value of A is too small, the temperature perturbation
L-D. CHEN and G. M. FAETH caused by the wall decays, passing through the adiabatic flame temperature, and the process is quenched. However, for sufficiently high values of A, the temperature decay is arrested at levels slightly below the adiabatic flame temperature. Temperatures then increase once again as the flame front develops. As before, an appreciable distance above the heated surface is required to complete flame front development. Figure 5 is an illustration of the variation in the value of A required for ignition near a freestanding isothermal surface as Tw/T~ is varied. The plot is for a fixed ambient temperature; therefore, the results primarily illustrate the effect of varying surface temperature. Both the zero gradient (denoted Ono et al.) and plume ignition criteria (denoted present) are illustrated, with activation energy as a parameter. The parameters used in the computations correspond to values employed by Ono et al. [3]. Present calculations, using the zero gradient criterion, agree with those of Ono et al. [3], to the accuracy that the figures of Ref. [3] can be read (the range considered in Ref. [3] is limited to 3.9 < T,,/,To~< 5.7). Both criteria vary in a similar fashion for values of Tw/T~ < 5.5, which corresponds to the range of existing experimental data for ignition near vertical surfaces [3]. This helps explain why the zero gradient criterion has been successful for purposes of data correlation. Kinetic parameters are generally unknown; therefore, a minor adjustment of their values from the true values could bring the two predictions into agreement in the low Tw'/T~regime. It is evident, however, that this would not be the case as the wall temperature approaches the adiabatic flame temperature of the combustible. In this region, the zero-gradient criterion predicts an increasing resistance to ignition with increasing wall temperature, before breaking down entirely at the adiabatic flame temperature. In contrast, the plume ignition criterion shows that ignition resistance continues to decrease with increasing surface temperature. Figure 6 is an illustration of the variation of A for ignition when the surface temperature is kept constant, but the temperature of the combustible is varied. The parameter values for the com-
IGNITION NEAR HEATED SURFACES
2.8 2.4
2.0
.
.
.
.
.
.
.
83
.
.
.
.
.
.
.
.
.
.
.
••.•
ADIABATIC J FLAME J TEMP. n =2 E/RT= = 34.3 A = 8.81 x 105
10.8 9.4
2.72 =
~ z o 4
\ - ~,
B =9.4
L3--~
~.55
8.0
V - ~ I . 1 5
Pr = 0.7, Sc = 0.65 Tw/T= = 4.5
1.6
8
6.6
T/T=
T** = 2 9 3 K
1.2
5.2
i.oo = ¢ 0.61 0.37 0.072
0.8
3.8
0.4
2.4
0.0
1.0
1.0
C
0.8 0.6
0.0
0.4 ~ Z 0.2
-'--- o.61 1.00
i .oo
0.0 0.8
WALL REGION .
0.6
PLUMEREGION 2.72 2 .04
~
f ' 0.4
~
1.00 0.61 0 37 0.072 0.00
1.55
i~
,)~.~\\
1.15
0.2 0.0 O.0
,
1.0
2D
3.0
4.0
0.0
1.0
2.0
3.0
I
4.0
I
5.0
Fig. 2. Flow properties near an isothermal free-standing surface whose temperature is below the flame temperature (ignition).
84
L-D. CHEN and G. M. FAETH
1.0
,
,
;
,
,
k
I , ; , WALL REGION
| ~
,
,
,
,
0.8 _~AD_IABATIFTE C TEMP. ~ ~ O04 •
/~IL_/o3
I-
°.°'~k
I
o.o-
o 2
~
0"0 ~
.
~
~ O.Ol
,o
F 0,30)
0.3[" ~
12.2
\
\
~
66T/T=
~.Ol=~,
\ X~~2-o3 \ k ~ e.73
•
3.8
~~<.-'//29.5
1.0
~ ~ 2 9 . 5 ~ -
-
-
--
2.03 ,.0,
I
P,:o.7,Sc:O.65~
=15.0
,:o,
i- /~\~J~29.5
TWIT.
f' 0.21 o~oo
-
~~8.73 I ~
n=2 E / R T . = 34.3 A=2XlO 4
°'F
150
~
0.6 I- ~ ' ~ 0"2 1 O0 L
I ; ; I , PLUME REGION
=
0.1 0.0
0.0
1.0
20
3.0
4.0
0.0
1.0
20
30
40
5.0
Fig. 3. Flow properties near an isothermal free-standing surface whose temperature is above the flame temperature (no ignition).
IGNITION NEAR HEATED SURFACES
I.om, ,
.
.
.
.
.
.
.
85 .
t
0.6
0
\\\~--------1.00 = (
~-
o.L a41
.
.
.
.
.
.
.
.
i 15.°
.
~ \ ~ ~ 3 . 0 = ¢
19.4
(_
-I
\ ' k - ~ O.OI
~
~..---~44.6
I
T/T. 66
O.O
IO
1.0 0.8
C
0.6 0.4 O.2
"1.00
11=2 E/RT** = 54.5 I O0 A= I x 105 ~ /2.83 / / ~ . ; 50 B=9.4 ~ 1 5 0 ~ , " Pr = 0.7, S c = 0.65 44.6
QO
0.5 Q4
f, o3 0.2
t
OJ 0.0 O0
S.S I
I
I.O
I
I
20
I
I
3.O
'
I
I
4.0
OO
IO
20
30
4.0
50
~q
Fig. 4. Flow properties near an isothermal free-standing surface whose temperature is above the flame temperature (ignition).
bustible are the same as in Fig. 5. A wall temperature of 1500K is employed for the calculations since this is in the range of maximum temperatures for metal foils and wires used in ignition experiments [3-5]. As before, results for both the zero-gradient and plume ignition criteria are illustrated. For the range of conditions shown, either criterion could be used to correlate data, although the effective value of A needed for ignition would be underestimated by the zerogradient criterion. The results show that the severity of plate conditions required for ignition is reduced as the ambient temperature of the combustible is increased.
Isothermal Surfaces along Walls Since plume conditions are clearly important for predicting ignition near heated surfaces, several boundary conditions which might be encountered in practice were considered. The results are summarized in Table 1, which provides the value of A required for ignition as a function of E, Tw/T~, and plume configuration. The plume configurations include the plume over a free-standing surface and wall plumes adjacent to adiabatic and isothermal (T, = T~) surfaces. The latter conditions would be encountered for ignition near a heated patch on a vertical surface, with the
86
L-D. CHEN and G. M. FAETH i
!
i I
i
t
E, 4o KCAL/C,~_~
I
P~ESENT
~/ !~{/
I --'-I [ ....
ONO, ET AL EXTRAPOLA-
I rio. I 8=9.4
~.\ ~\ \ \
I Pr "0.7; So= 0.65 I T=" L~93K
\\
i I I I"'ADIAIBATIC FLAME I TEMPERATURE
IOs ~ : , 0 KCAL/GMOLE /!
105' '°'o-,
~ I '
80 '
'
12.0
16.0
'
Tw/T® Fig. 5. R e l a t i o n b e t w e e n A and TWIT=for i g n i t i o n near i s o t h e r m a l f r e e - s t a n d i n g surfaces w i t h c o n s t a n t a m b i e n t temperature.
I
I
=
,I
I
I
I
=1
two cases essentially bounding the realistic range of surface temperatures in the plume region. Results for the zero-gradient ignition criterion are also shown for comparison. The parameters of the combustible are similar to those used in Figs. 5 and 6. Within the range of parameters considered in Table 1, ignition resistance increases in this order: adiabatic wall plume, free-standing plume, isothermal wall plume. The isothermal wall plume requires the most severe heated surface conditions for ignition since the flow in the plume is quenched by both entrained fluid and heat loss to the cold wall. The free-standing plume and the adiabatic wall plume are similar to the extent that quenching only occurs due to entrainment. The greater ignition resistance of the free-standing surface is due to the fact that velocities are higher in the free plume than for a wall plume, where wall friction retards the flow. In a Lagrangian sense, the higher velocity reduces the residence time of the combustible in high temperature portions of the flow, which increases ignition resistance. In the Eulerian sense, the increased velocity increases convection effects, causing temperature and concentration profiles to steepen in the direction normal to the surface. Since an ignition process is largely chemical rate controlled, the increased heat loss due to more abrupt gradients increases ignition resistance. Similar arguments concerning the effect of ve-
I
I
1
=1
I
I
I
I
I
I
I
I
I
0.4
/GMOLE
-20 KGAL/GMOLEn • 2
~
10.3
0.2 103
\
\ \~
BT** = 2754 Pr " 0.7; Sc- 0.65 ,..,,ooK
I
I
io 4
I
I
I
-• ..
PRESENT (~ll~, E r 1~1..
I
I
I
1o5
I
I
I
I
i0 e_
107
108
Ax(T=/T.) z Fig. 6. R e l a t i o n b e t w e e n A and Tw/T. for i g n i t i o n near i s o t h e r m a l f r e e - s t a n d i n g surfaces with constant surface t e m p e r a t u r e .
IGNITION NEAR HEATED SURFACES
87
TABLE1 Comparison of the Values of A Required for Ignition by Constant Temperature Surfaces with Different Plume Conditionsa E (kcal/gmol)
Tw/To.
=
40
3.6 X 105 5.6 X 105 3.6 X 105
3.1 X 109 4.2 X 109 3.1 X 109
1.0 1.7 8.6 6.5
8.3 2.0 8.3 3.4
4.0
zero gradient free standing adiabatic wallb
T,,,/r,o =
20
5.5
zero gradient free standing adiabatic wallb isothermal wallb
X × × X
105 10ft 104 105
X x × X
107 108 107 108
a T** = 293 K, n = 2,B = 9.4,Pr = 0.7, Sc = 0.65. b Wall condition in plume region: wall and ambient temperatures equal for the isothermal wall plume.
locity have been given for forced flow ignition along surfaces [1, 2]. The main difference for natural convection is that the flow configuration itself influences the magnitude of the velocities. The results summarized in Table 1 provide a further indication that the zero-gradient criterion does not always underestimate the severity of surface conditions required for ignition. Generally, the adiabatic wall plume requires A values for ignition that are less than or equal to the values provided by the zero-gradient criterion even for the relatively low values of surface temperature considered in Table 1.
Constant Heat Flux Free-Standing Surfaces Figures 7 and 8 are illustrations of flow properties near free-standing constant heat flux surfaces. Figure 7 is typical of quenched reaction in the plume, while Fig. 8 is a case where a deflagration wave develops. In both circumstances, f = 0= 0 and C = 1, for all values of r/ at I~= 0. With increasing I~, the wall temperature and the maximum velocity increase with increasing distance along the surface. Similar to the findings for isothermal heated surfaces, a large
distance above the source is required for a flame to develop, as the limiting conditions for ignition are approached. The minimum value of A required for ignition along free-standing constant heat flux surfaces is illustrated in Fig. 9 as a function of dimensionless heat flux. Only results for the plume ignition criterion are shown, since the zero-gradient criterion can never be satisfied for a constant~heat flux surface. At low values of ¢ w", the value of A required for ignition decreased with increasing w". This behavior is expected, due to the higher wall temperatures resulting from higher heat fluxes. An interesting feature of the results, however, is that the minimum value of A for ignition becomes relatively independent of ¢,," at higher values of this parameter. At least two factors contribute to this behavior. First of all, the temperature sensitivity of the chemical reaction rate begins to decrease at higher temperatures, with the effect occurring sooner for lower values of activation energy. This is shown by the fact that the slope of the curve of minimum A decreases sooner, and more rapidly, for the lower value of E (the same trend is observed for the isothermal wall results; cf. Fig. 5). Thus, at high temperature levels the rate of increase of reactivity, in comparison to heat losses, is reduced. The second factor involves the larger flow velocities encountered for high heat fluxes. This tends to reduce chemical residence times while increasing quenching effects due to steeper thermal gradients and increased entrainment, similar to effects discussed when wall and free plume ignition were compared; cf. Figs. 7 and 8. Results for free-standing isothermal and constant heat flux surfaces are compared in Table 2. The table provides the value of A, required by the plume ignition criterion, as a function of the ratio of the maximum surface temperature and the ambient temperature. At a given temperature ratio, the required values of A are larger for constant heat flux surfaces. This is expected since high temperature levels are only reached near the top of constant heat flux surfaces, which reduces residence times at high states of reactivity, in comparison to isothermal surfaces. The effect is less at large values of Twm~x/To~,due to reduced
88
L-D. CHEN and G. M. FAETH
~---~0
/
i
i
!
i
6 Ill I
!
i
i
i
!
I
I
I
I
I
I
5.0
I
PLUME REGION
WALL REGION
42
\_
O 0.8
~ .
5.4
1.00 =~ 1.13 1.89
0.24
~
T/T**
2.6
0.4
1.8
0.0
1.0
1.0 0.8 0.6 C 0.4
~
1, 0.24
~
10.6 \
1.89
\1.13
i.oo
1.00 n=2
0.2 0.0 0.41/ =.00
E/RT= = 34.3 A = 1.84 x 105 Pr = 0.7, Se = 0.65 Tref/T==3.0 •
0.3
~ ~ ~
~
1.89 ~1.13
=
1o.6
f, 0.2 0.1 0.0 0.0
1.0
2.0
3.0
4.0
0.0
1.0
2.0
3.0
4.0
5.0
I/ Fig. 7. Flow properties near a free-standing constant heat flux surface (no ignition, ~w" = 0.82).
temperature sensitivity of the reaction rate expression at such conditions.
LimitationsoftheResults A major limitation of the present and earlier studies concerning ignition along surfaces I-1-3] is neglect of radiation. This limits the results to
small surfaces, relatively low pressures, and weakly absorbing gases. Even in this case, it should be noted that the present constant heat flux computations are limited to constant c o n v e c t i v e heat flux. Surface radiation is normally important for high-temperature surfaces in a natural convection environment. Therefore, constant heat flux surfaces would generally require
IGNITION NEAR HEATED SURFACES ,0
i
i
i
i
f
i
89
|
i
i
I
I
I
I
I
I
I
I
I1.0
I
T
ADIABATICFLAMETEMP.
4.0
9.0 ~ L O 0 J._4--4---u
={ 7.0
I
~ ~2.2a I ~ L..4~6.4a
0 3.0 ~ - - - 1.00=~' 2.0 x,L1-o.95
5"° T/T=
,~O.al
l.O
:5.0
0.0
1.0
C '-,.oy
~-o.gs
n=2
J/
~ 1 . 0 0
~
/"/--../~l.lW
REGION
".1.0(0 E/RT== 34.3 [" / ~ 2 . 2 3 A = e.os, Io 5 l/" / /...:'-6.42 B=9.4 ~ -"" 2 44 Pr = 0.7, Sc = 0.65
0.41 o.z ~ 0.0 L
OAr ~
i OoTter/T®=3.0
r ~~'5
aa ~//fk-
L/// I///
R
0.0095 0.81
^ ~ ~"/~-
O.Sl
1.6
r
T.=2,3.
1-'
K.
~/0.0095
"
k~ ~
1.2 244
I \V~...~6.42 ~'x ~ . . ~ 2.25
A
f' 0"2I / / / / ~ " ~
o.8 f' 0.4
.'lo
, ,,
o.o
,.o
, io
~.o
,.o
o.o
,o
2o
3.0
io.o io
5o
Fig. 8. Flow properties near a free-standing constant heat flux surface (ignition, ~w". = 0.82).
more energy input for ignition than the values prescribed by this analysis. Significant radiation effects would also influence the surface temperature distribution limiting the applicability of the present analysis in such cases. A second limitation of the analysis, already pointed out by Sharma and Sirignano [2] for ignition along a surface under forced convection conditions, involves the use of boundary layer equations in the analysis. It is clear from the
results illustrated in Figs. 1-4, 7, and 8 that there are significant streamwise gradients in flow properties. Thus the effect of streamwise diffusion of heat and mass can be important in some circumstances, particularly when G r x is low in the region of flame development. A quantitative indication of this effect must await more complete analysis. It is suggested that streamwise transport would tend to increase quenching; therefore, somewhat higher values of A would be required for ignition
90
L-D. CHEN and G. M. FAETH
I0 9
I
I
I
n=2 B= 9.4 Pr = 0.7; So = 0.65 T= • 293 K
i0 e
10 7
E = 40 K C A L / G M O L E % .
I0 6
10 5
104 0.0
I 1.0
t 20
I 30
(~'WII Fig. 9. Relation between A and ~w" for ignition near constant heat flux, free-standing surfaces with ambient temperature maintained constant.
when G r x is small. Streamwise diffusion would also influence the rate at which the flame spreads from the surface. Transition to turbulence also has interesting implications for ignition along surfaces. The question has not been considered, to the authors' knowledge, and the conditions for transition in natural convection reacting flows have not been generally established. Thus the Grashoff number
limit of the present analysis, and the impact of transition on ignition properties, cannot be stated without further study.
S U M M A R Y AND C O N C L U S I O N S A theoretical study of ignition near heated vertical surfaces under laminar natural convection
IGNITION NEAR HEATED SURFACES
91
TABLE 2
properties vary in this order: adiabatic wall plume, free plume, isothermal wall plume (ambient and wall temperatures the same). Wall quenching, in the case of isothermal wall plumes, and higher velocities, in the case of free plumes, tend to reduce the hazard when compared to adiabatic wall plumes. . Near limiting conditions for ignition, substantial distances above the surface are required for a deflagration wave to develop. Therefore, the effect of quenching surfaces above the ignition source must be carefully evaluated during ignition experiments; otherwise, the ignition hazard of the combustible will be underestimated. . Constant temperature surfaces require less severe surface conditions for ignition (lower values of A) than constant heat flux surfaces having the same maximum temperature.
Comparison of Ignition Requirements for Constant Temperature and Constant Heat Flux Surfacesa
Twmax/Too 4.5 6.5 9.5 11.5 14.0 15.0
A × 10--4 Constant heat flux Isothermal 95 20 8,5 4.4 3,4 3.2
35 10 5.0 3.7 2.9 2.7
a Free-standing surfaces: E = 20 kcal/gmol, T** = 293 K, n = 2, B = 9.4, Pr = 0.7, Sc = 0.65.
conditions was completed. Flow properties along the surface and in the plume were determined for steady conditions. Ignition was prescribed by the heated surface condition which separated quenching and flame development in the plume. Constant temperature and constant heat flux heated surfaces were considered. Plume conditions included free plumes and adiabatic and isothermal wall plumes. The major conclusions of the study are as follows: 1. Plume properties should be considered when estimating the ignition properties of heated surfaces. The earlier zero-gradient criterion (which prescribes ignition when the temperature gradient normal to the surface is zero at some point) provides only a qualitative indication of ignition properties when surface temperatures are well below the adiabatic flame temperature of the combustible. In some circumstances, the zero-gradient criterion overestimates the severity of surface conditions required for ignition. The zero-gradient criterion also breaks down as the surface temperature nears the flame temperature and cannot be applied at all for constant heat flux surfaces or for a surface whose temperature is greater than the flame temperature. . The severity of surface conditions required for ignition generally increases as plume
These conclusions are based on solution of laminar boundary layer equations for an ideal gas, assuming a one-step reaction and neglecting radiation. The effect of streamwise diffusion, real gas properties, radiation, and turbulence are matters that should receive further attention in order to better understand the ignition properties of heated surfaces.
This work was supported by Union Carbide Corporation, Chemicals and Plastics, South Charleston, West Virginia, with P. Wadia of the Research and Development Department serving as technical monitor for the project. NOMENCLATURE A B c C Cp D E f g
reaction parameter, Eq. (22) parameter, Eq. (23) mass fraction of combustible dimensionless mass fraction, Eq. (16) specific heat at constant pressure binary diffusivity activation energy dimensionless stream function, Eq. (16) acceleration of gravity Grx Grashoff number, Eq. (17)
92
L-D. CHEN and G. M. FAETH
H K n
surface height preexponential factor reaction order Pr Prandtl n u m b e r w" surface heat flux Q heat of c o m b u s t i o n R Universal gas constant Sc Schmidt n u m b e r T temperature u velocity parallel to surface v velocity n o r m a l to surface x distance along surface y distance normal to surface t~ reaction rate, Eq. (5) /3 coefficient of volumetric expansion r/ dimensionless distance, Eq. (15) 0 dimensionless temperature, Eq. (16) X thermal conductivity absolute viscosity v kinematic viscosity dimensionless distance, Eq. (15) p density r dimensionless reaction rate, Eq. (21) w" dimensionless heat flux, Eq. (28) ~0 stream function
SUBSCRIPTS ref w
reference value wall value ambient value
REFERENCES 1. Toong, T.-Y., Sixth Symposium (lnternationaO on Combustion, Reinhold, New York, 1955, pp. 532540. 2. Sharma, O. P., and SMgnano, W. A., Combustion Science and Technology 1:95-104 (1969); ibid. 1:481-494 (1970). 3. Ono, S., Kawano, H., Niho, H., and Fukuyama, G., Bull. JSME 19:676-683 (1976). 4. Chen, L.-D., and Faeth, G. M., Combustion and Flame, 40:13-28 (1981). 5. Cawse, J. N., Pesetsky, B., and Vyn, W. T., '~Fhe Liquid Phase Decomposition of Ethylene Oxide," AIChE LossPrevention Symposium, Houston (1979). 6. Marble, F. E., and Adamson, T. C., Jr., Jet Propulsion 24:85-94 (1954). 7. Cheng, S. I. and Kovitz, A. A., Sixth SymPosium (InternationalJ on Combustion, Reinhold, New York, 1955, pp. 418-427. 8. Cheng, S. I., and Chiu, H. H., Int. J. Heat Mass Transfer 1:280-293 (1961). 9. Williams, F. A., Combustion Theory, Addison-Wesley, Reading, MA, 1965, Chap. 1. 10. Cebeci, T. and Bradshaw, P., Momentum Transfer in Boundary Layers, McGraw-Hill, New York, 1977, Chap. 7. 11. Rosenhead, L. (ed.), Laminar Boundary Layers, Oxford University Press, Oxford, 1963, pp. 224-234. 12. Cohen, C. B. and Roshotko, E., Similar Solutions for the Compressible Laminar Boundary Layer with Heat Transfer and Pressure Gradient, NACA TN-3325, 1955, p. 54. 13. Ostrach, S., Analysis of Laminar Free-Convection Flow and Heat Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force, NACA TN-2635, 1952, pp. 36-37. Received 28 May 1980; revised 28 August 1980