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A simple algebraic model for turbulent wall fires
Twenty-firstSymposium(International)on Combustion/TheCombustionInstitute, 1986/pp. 53-64
A SIMPLE A L G E B R A I C M O D E L FOR T U R B U L E N T W A L L FIRES MICHAEL A. DELICHATSIOS
Factory Mutual Research Corporation* Propagation of flames over vertical walls is a key phenomenon for 1) predicting fire growth in rooms, and 2) characterizing the flammability properties of practical materials. Because of the complexity of the fire problem and the lack of values for key material properties, simple, yet comprehensive, models are sought for modeling turbulent wall fires. In this paper, simple algebraic expressions for turbulent wall fires are presented including expressions for the wall pyrolysis rate, the mass flow rate, the maximum mean velocity and the maximum mean gas temperature. These expressions evolved from a more complicated integral wall fire model presented previously. In the proposed relationships, mass blowing at the wall owing to pyrolysis and flame radiation feedback is accounted for. An extensive discussion is also presented for the fundamental fluid mechanical and combustion principles of the model including comparison with experimental property profiles and wall pyrolysis rates. Limitations of the model regarding the modeling of flame radiation are clearly noted.
Introduction Much effort has been expended d u r i n g the last ten years by Orloff et al 1, Ahmad and Faeth 2, and Delichatsios3for predicting flame spread and b u r n i n g intensities in fires on vertical walls. Such wall fire models are needed as components of zone models for evaluating the fire hazard in buildings. I n addition, turbulent wall fire models are essential for interpreting and classifying material flammability based on laboratory property measurements or standardized performance tests such as the Room Corner Test or the Factory Mutual Corner Test. In a recently published paper 4 we presented an extension of a novel integral turbulent model 3 to predict 1) wall fire intensities with significant flame radiation, and 2) the extent of burning, i.e., the flame height, in wall fires. Such predictions are key inputs for calculating upward fire spread 1 rates on a vertical wall wherein flame spread depends 1 on 1) the flame extent beyond the pyrolizing region and 2) the heat feedback from the flames to the yet unignited fuel between the pyrolysis front and the flame tip. In the present paper, we derive a simplified algebraic model for t u r b u l e n t wall fires, which has evolved from a more complicated integral
*Funded in part by the Federal Aviation Administration 53
model 4. For completeness, we present a review of previous models as well as an evaluation of the physical principles and experimental verification of the present turbulent wall fire model including a description of its limitations. Subsequently, we develop simple expressions for mass pyrolysis rates, air e n t r a i n m e n t and gas velocities along the pyrolizing wall zone of a turbulent wall fire; flame radiation is not calculated by the present model but it is incorporated in the model as a given input property of the b u r n i n g gases.
Evaluation of the New Integral Model for Wall Fires The principles of the integral model are presented in order to demonstrate the soundness of its foundation; however, the ultimate justification of a new model must be based on a comparison of its predictions with experimental data. Such a comparison is also included in the following discussion. There are two components to the integral model: Fluid Dynamics and Combustion.
Fluid Dynamics Of The Model A flow on a vertical b u r n i n g wall is driven by buoyancy forces which are associated with the lower densities of combustion products compared to the s u r r o u n d i n g ambient air. Such flows are usually turbulent, unless their vertical size is less than say ten centimeters. T u r b u l e n t
54
FIRE
wall fires belong to the general class of turbulent natural convective flows next to vertical walls. T h e principle features of the proposed integral model for turbulent convective flows next to a vertical wall are listed in Table I. Perhaps the major innovation of the new model for turbulent convective flows is the introduction and use of wall laws (i.e., profiles close to the wall) for flow properties different from the logarithmic wall laws proposed by von K a r m a n for forced b o u n d a r y layer flows. These laws (see References 3 and 4 for details) have been developed from the recognition that buoyancy in turbulent convective flows generates the turbulence close to the wall,* in contrast to forced b o u n d a r y layer flows where turbulence is generated by principally local wall shear stress. Recognition o f this key difference allows one to predict the correct heat transfer in natural convective situations. T h e natural convective wall laws agree well with detailed measurements 5 of velocity a n d temperature profiles in turbulent convective flows next to walls having a constant t e m p e r a t u r e or a constant heat flux. In addition, these new natural convective wall laws are consistent with the experimental fact that the heat transfer coefficient for turbulent free convection from a wall at constant t e m p e r a t u r e is i n d e p e n d e n t of location along the wall s . It is i m p o r t a n t to discuss at this point previously developed integral models for turbulent natural convective flows. All previous models including turbulent wall fire integral models 2 have been based on the classical Eckert and Jackson (see Rohsenow and Choi 6) integral model approach. This model solves the m o m e n t u m and heat conservation equations by assuming 1) similar profiles for velocity and temperature across the entire wall layer; and 2) wall friction (and heat transfer) relations identical to the respective relations for forced b o u n d a r y layer flows. It is difficult to justify the second assumption given the fundamental differences between forced b o u n d a r y layer flows and free convective flows on vertical walls. Moreover, the Eckert and Jackson model (and its variations) predicts a convective heat tranfer coefficient that varies as x vS, where x is the distance along the wall from the origin o f the flow; this prediction contradicts the experimental evidence s which convincingly shows no d e p e n d e n c e o f convective heat transfer coefficients on x. *In similarity to von Karman's development, we have assumed that the inner flow close to the wall is not affected by the outer flow properties.
In the present integral model, (see Table I) we divided the natural convective boundary layer into two layers ~. For each layer different flow p r o p e r t y profiles were employed. The inner layer close to the wall is nonsimilar, while the o u t e r layer b o r d e r i n g the inner layer and extending to the ambient fluid is self-similar, i.e., the profiles have similar shapes at different heights. T h e p r o p e r t y values for the inner and outer layers are matched at an intermediate position which we take to be the position of m a x i m u m velocity across the b o u n d a r y layer. T h e two-layer a p p r o a c h allows us to carefully consider the specific thermal b o u n d a r y conditions at the wall which have a significant impact on the development o f the natural convective flow. T h e detailed consideration of wall thermal b o u n d a r y conditions includes mass transfer effects which are particularly i m p o r t a n t for the description o f turbulent wall fires wherein pyrolysis at the wall occurs. It should be noted that the width of the inner layer is small relative to the total flow width so that flow p r o p e r t y profiles may a p p e a r selfsimilar across most of the width of the boundary layer. (For comparison, note that the forced b o u n d a r y layers are treated in integral models as if they were self-similar over their entire thickness.) It is well known that wall and nearby buffer regions ("wall laws") are in general nonsimilar for turbulent b o u n d a r y layers (Cebeci and Bradshaw7). Integral conservation equations 3 were employed over the outer layer as well as over the total width of the b o u n d a r y layer. These conservation equations together with the matching conditions between the inner and outer layers provide a closed system of integral equations. Regarding our integral equations (see Table I), we employ a m o m e n t o f m o m e n t u m conservation equation in place o f the usual m o m e n t u m equation to take advantage o f its simpler bound a r y condition. First we will present the new conservation equation for forced flows, and
TABLE I Distinct Characteristics of the Integral Model 1. New wall laws for natural convective flows. 2. Two layer modeling; inner layer (nonsimilar) close to wall overlaps with outer layer (self-similar). 3. Conservation of moment of momentum for thin wall flows as a replacement of momentum conservation (see equ (2)). 4. Average stream function in place of distance normal to wall, for rigorous incorporation of variable density effects.
SIMPLE MODEL FOR TURBULENT WALL FIRES
55
b o u n d a r y laver. It is expected, hut must be shown by using e x p e r i m e n t a l data, that the RHS of equ (1) is sufficiently small so that P (see equ (2a)) can be a s s u m e d to be i n d e p e n d e n t ofx even for t u r b u l e n t (forced) wall flows. T h e conservation equ (1) can be extended to n a t u r a l convective flows, which include buovancy forces, with a small effort (Delichatsios, 1982);
Edge o f W ~ I I J e t
u =
d
-"77"7.,
~t~
oa
o
FIG. 1. la: A Thin Turbulent Wall Jet-Without Buoyancy 1B: Velocity and Shear Stress Profiles t h e n we will e x t e n d it for b u o y a n t flow a n d also p u t forward some advantages of its application in the present model. F i g u r e 1 shows a thin n o n b u o y a n t wall j e t t o g e t h e r with profiles of velocity a n d shear stress. It is a simple exercise (see Batchelor ~, p 377, 1967, exercise 2) which shows that: 0
oo u
0 f,-f,
oa
r
Equation (1) was derived by 1) using the b o u n d a r y layer a p p r o x i m a t i o n for wall jets a n d 2) neglecting l o n g i t u d i n a l velocity fluctuations for t u r b u l e n t flow conditions followed by muhiplying the m o m e n t u m e q u a t i o n by d a n d i n t e g r a t i n g over y n o r m a l to the surface. Equation (1) is valid for both l a m i n a r or t u r b u l e n t wall jets. A n expression for the shear stress in a t u r b u l e n t flow is given in F i g u r e 1. It is obvious (by integration) that the RHS o f e q u (1) is zero for l a m i n a r flows -r = Ix ~ T h e r e f o r e , the quantity (if2 dy) = P
(2a)
J~-i -g~y-
,
o
For the same reasons as in torced flows, the second term at the RHS of equ (2b) is expected to be small a n d can be conveniently set to zero for natural convective flows adjacent to vertical walls. Equation (2b) was used over the outer (self-similar) layer of o u r t u r b u l e n t wall model (see Reference 3 for m o r e details). T h e new m o m e n t of m n m e n t u m e q u a t i o n replaced the m o m e n t u m equation, which would have req u i r e d an expression of the wall Diction in o r d e r to be solved. Such an expression for the friction is not available from m e a s u r e m e n t s or first principles. Finally, in the d e v e l o p m e n t of the integral equations, we used the average stream ftmction, 0 as an i n d e p e n d e n t variable in place of the n o r m a l distance from the wall (see Table I); this formttlation allows a straightforward incorporation of variable density effects, wlnich are particularly i m p o r t a n t in wall fres.
o
t.0 .9 .8 .7
Veloetty (Ref. 8) I ~ n s l t y Defect
.6 .5
~
;::u2;,7:~;2.
.4 n E
0 o
.2
is i n d e p e n d e n t of x for a thin l a m i n a r forced wall jet. T h e conserved quantity, P, may be characterized as m o m e n t (w.r.t. ~) of m o m e n t u m of the wall flow. For a t u r b u l e n t forced wall jet the RHS of equ (1) is not zero. T h e r e are no reliable data for estimating its precise magnitude; however, observe (see, also, Figure 1) that the i n t e g r a n d at the RHS o f e q u (1) 1)becomes zero at three positions across the b o u n d a r y layer a n d 2) changes sign at a point inside the
.1,
1
,00
.02
t
.04
a
L
I
,06
,08
.10
I
.12
y/x
FIG. 2. Comparison of Velocity and Temperature Profiles in an Adiabatic Wall Plume with Profiles Used in the Integral Model (Ref. 8 for x = 1.22 m, u,,, = .935 m/s, A~ = .06)
56
FIRE
We conclude by presenting in Figure 2 a comparison between profiles at the outer layer used in the integral model with experimental profiles. For the implementation of any integral model for b o u n d a r y layer flows, it is required to introduce profiles o f the flow properties in o r d e r to evaluate the integrals 7. In the present approach, we e m p l o y e d different profiles for the inner and outer layers. For the inner layer we used the new wall laws; it is important to re-emphasize strongly that these profiles agree well with experimental data 5. For the outer wall layer, we assumed that the property profiles vary linearly with the value of the stream function. That is: ..., -
r162
a.,, -
r162
(3)
where u,,, is the m a x i m u m velocity; A is the mean t e m p e r a t u r e rise,
T-T•. T= ' tb is the stream function
~bt is the total flow =--
~dy; p~
0
and ~ is the stream function at the m a x i m u m value of the velocity. In Figure 2, an approximate agreement is shown between the profiles represented by equ (3) with the experimental profiles in weak turbulent wall plumes (Grella and Faeth, 1975). To evaluate the reasonableness of our assumed outer profiles (given by equ (3)) we use the definition o f ~ o r ~ = ~ / p ~ which upon integration yields
um
A A,,, = exp[--(Y--Ym)/6~]
in the Boussinesq limit (i.e. A,, < where 8r = (tbt-tb,~)/U~. Figure 2 compares the resulting calculated profiles with experimental data for an adiabatic vertical wall plume u n d e r Boussinesq conditions (~,, = 0.06). The linear fit appears quite satisfactory. We presented in this section a detailed discussion and evaluation of the fundamental characteristics (see Table I above) of the new integral model that we developed for natural turbulent convective flows next to vertical walls. In addition, we d e m o n s t r a t e d that the p r o p e r t y profiles used for implementing the integral
model are very close to experimental profiles, both for the inner a n d outer layers. We have used the present integral model s to calculate the flow in a t u r b u l e n t wall plume 9 where the predictions agree very well with the experimental data 3. In o r d e r to demonstrate more convincingly the superiority and soundness o f the new integral model, we also intend to compare, in a subsequent report, its predictions with existing detailed a n d extensive experimental data in natural convective flows next to heated walls.
Modeling Of Burning In The Integral Wall Fire Model The integral wall model for natural convective flows was e x t e n d e d s to turbulent wall fires, which involve pyrolysis at the wall and subsequent b u r n i n g o f the pyrolyzing gases with ambient air. T h e details of the analysis, as well as its further extension to wall fires with flame radiation, may be f o u n d in the literature 3'4, respectively. O u r intention in this section is to discuss: 1) some difficulties for developing integral models for turbulent diffusion flames (fires) and 2) the drastic simplifications introduced in the development of the present wall fire model 3'4. The cause o f the flow in the wall fires is the high t e m p e r a t u r e s (low densities) o f the combustion products relative to ambient air. These temperatures, however, are related, in a rather complicated way, to the average characteristics of the flow such as, for example, the heat release rate. A significant simplification o f the analysis has been obtained by introducing a conserved scalar quantity based on the classical Schvab-Zeldovich formulation. T h e instantaneous concentrations and t e m p e r a t u r e s are then related in a rather simple way with the instantaneous value of the conserved scalar 1~ However, the level of turbulent fluctuations is also required in o r d e r to find the average values of concentrations or temperature in terms o f the average values o f the conserved scalar in turbulent diffusion flames. An integral model for turbulent diffusion flames must provide for a relation between the average t e m p e r a t u r e (required in the flow equations) and the average conserved scalar (required for describing the burning). Such a relation may be obtained either by j u d g m e n t and use of e x p e r i m e n t a l facts or preferably, by introducing an additional integral equation for the fluctuations o f the conserved scalar 11. It must be emphasized that considerable complication is the penalty for using a conserved scalar fluctuation equation. We also considered using an integral fluctua-
SIMPLE MODEL FOR TURBULENT WALL FIRES tion equation in the present turbulent wall fire model. During the model development, it became apparent that not only an integral equation but also an accurate profile of the fluctuations is required, in o r d e r to predict a correct value of the local average t e m p e r a t u r e (especially close to the wall) in terms of the local average value of the conserved scalar. Within the constraints of an integral model, such an "accurate" profile for the conserved scalar correlations would have to be lifted from experimental data, which unfortunately are nonexistent. We decided to abandon this route. T h e principal characteristics of the new integral wall fire model may be described with reference to Figure 3. In this figure, the p r o p e r t y profiles (velocity, t e m p e r a t u r e rise, conserved scalar) used in the model are shown in terms of the stream function,
~/=
1
dy
p~ where y is the normal distance to the wall and 9~ is the ambient air density. T h e main features of the model are: 1) T h e outer layer profiles were assumed to be self-similar and to have a linear relationship with the stream function. 2) T h e inner layer profiles were determined by extending the methodology used for developing the inner layer profiles in turbulent (nonburning) buoyant flows next to vertical walls. For determining the velocity profile, we assumed that the buoyant force in the inner layer is driven by a constant temperature rise equal to the average of m a x i m u m (AT,,) rise. For determining the inner profile of the conserved scalar, we accounted for both the effects of blowing and heat transfer at the wall 3'4. It should be noted that we also assmned a linear relationship of the t e m p e r a t u r e rise in terms of + within the inner layer (see Figure 3) for evaluating convective heat flow parallel to the wall. However, since the inner layer is quite small, this latter assumption is not important. 3) Finally, we decided to uncouple the temperature rise from the conserved scalar by assuming that, along the wall fire, the maxim u m average t e m p e r a t u r e is i n d e p e n d e n t of x (but depends on the wall material). T h e maxim u m average t e m p e r a t u r e in a turbulent wall fire is considerably smaller than the maximum t e m p e r a t u r e in a laminar wall fire having the same wall material; the main reasons for the lower average t e m p e r a t u r e in a turbulent fire are: 1) turbulent fluctuations which smooth peak values, and 2) increased radiative losses. (Incompleteness of combustion probably has a
57
= T e m p e r s t u r e Rise "~ = Consecved Scalar
-fi,
/ ,.~\ /
/
~
Inner Layer
~Tw
*= FIG. 3. Profiles for the Inner and Outer Layers of the Turbulent Wall Fire Model smaller effect on the m a x i m u m average temperature rise.) Experimental results agree with the assumption that the maximum teinperatures remain approximately constant along a wall fire; there is no quantitative explanation for this experimental fact. We emphasized previously in this section the difficulties in modeling the turbulent fluctuations within the context of an integral model. However, even detailed k-e-g models for wall fires 12 overpredict the temperature rise for a vertical wall fire. (The author lz claimed that incompleteness of combustion, which is not included in his model, may be the reason for this discrepancy.) In Figure 4 we have assembled some recent data l~ for turbulent wall fires; these data include 1) velocity (Figure 4a) and temperature (Figure 4b) profiles along a wall fire for a fixed mass transfer n u m b e r (B = .7), and 2) temperature profiles (Figure 5) at a fixed location for varying mass transfer numbers (B = .5, .7, 1.0). In all these figures, the abscissa is the stream function that was calculated from the temperature and velocity data at several positions normal to the wall: 1 ~=T= fy0 7~dy. T h e experimental profiles (see Figure 4) can now be c o m p a r e d with the model profiles in Figure 3 and with the main features of the present integral wall-fire model. We can make the following remarks: 1) T h e outer profiles show an almost linear relationship of velocity and temperature with
FIRE
58
4a
r
(~
OOO ~/kA & 0 0
0
&
O
XX•
~***
'
I= 0 . 6 0
0
O
&0
~
l=
O A
XX
O &
..
& X
§
&
§ X
O O
.4-
& X
ae
X= 1.2~
O
& §
1=02
O
X
§ 9
0.80
6&
X
0
Xx
"k
X
P==~ ( kg/m 3
. m 2Is)
4b
~
I.S
"*-, ~•176176176
I=
0=80
I:
1=02
X=
1=25
8 ,,o o XX
+
~~
I
I
/,.
0
§
& X
O
o
&
§
&
O
X
O
§ §
X
&
A
O
A
O
A ~5
X
O A&
+
A X
+
X
A
O
XX X
@,@
P==~ ( kg/m 3 . m 2 / s ) FI(;. 4. Velocity and Temperature Profiles at Four Locations along an Ethane Wall Fire with Mass Transfer Number B = .7 the stream function; these plots do not show ~ h e t h e r the o u t e r profiles are self-similar. However, similarity o f the o u t e r profiles has been clearly d e m o n s t r a t e d previously ~4: 2) T h e e x p e r i m e n t a l velocity and t e m p e r a ture profiles are flat near the m a x i m u m values of the variables in contrast to the m o d e l profiles; this d i f f e r e n c e i n t r o d u c e s an uncertainty in d e f i n i n g the extent of the inner-layer
(i.e., value o f 4,~, see Figure 3). Notice that the velocity profiles are flatter than the t e m p e r a ture profiles. An i m p r o v e m e n t o f the profiles could be o b t a i n e d by p r e s e r v i n g a linear profile for the t e m p e r a t u r e while m o d i f y i n g the instead o f ~ = ~ (4, - 4) (see Figure 3),(4a) use t~ ~ +J.'3 (4, - 4).
(4b)
SIMPLE MODEL FOR TURBULENT WALL FIRES
B:
0.35
+
B=
0.70
x
B:
1.00
A
59
Profile Produced by Integral Model for B = . 7
' 0.|
,.,~
,d,
,.,~
,.,~
.-4,
,.~
.d,
,.,~
,d,
,.-~
Owl~ ( kg/m3 . m2 1 ~ )
TEnPERATURE FIr 5. Temperature Profiles at a Fixed Location (0.6 m from the origin) for Three Different Mass Transfer Numbers, B = .35, .7, 1.00. Note that adding more fuel has little effect on temperature or the total convective upward heat flow. (Note: for the i n n e r layer 3 ~ = ~ $J'3.) T h e implications of this modification will be investigated in a s u b s e q u e n t paper. 3) Finally, Figures 4b a n d 5 clearly d e m o n strate that the m a x i m u m average t e m p e r a t u r e is constant not only along a wall fire but also for a small variation of the mass transfer n u m b e r (B = .35 to 1.0). It is expected that this t e m p e r a t u r e will vary somewhat for different wall materials. No complete quantitative explan a t i o n for this e x p e r i m e n t a l fact is known. While f o r m u l a t i n g the integral model, we employed the following p r o c e d u r e for d e t e r m i n ing the " t u r b u l e n t " gas temperature3'4: 1) we calculated the laminar flow t e m p e r a t u r e with a correction for the radiative heat loss fraction; 2) we assumed that the m a x i m u m t u r b u l e n t flame t e m p e r a t u r e is a c o n s t a n t fraction of this peak l a m i n a r flame t e m p e r a t u r e ; a n d 3) we suitably modified the wall b o u n d a r y conditions. T h e e x p e r i m e n t a l velocity profiles are compared with the calculated integral model profiles in Figure 6 at five d i f f e r e n t locations.
T h e r e is good a g r e e m e n t for the outer-layer profiles, i n c l u d i n g the m a x i m u m velocity and the total flow rate values, b u t the a g r e e m e n t is only fair* for the profiles at the i n n e r layer; it is expected that the a g r e e m e n t between experim e n t a l profiles a n d the profiles used in the model will i m p r o v e by m o d i f y i n g the outerlayer velocity profile as shown in equ (4). Flame radiation in t u r b u l e n t wall fires affects both the rate of pyrolysis a n d the rate of fire spread. In a t u r b u l e n t wall fire model, flame radiation can be i n t r o d u c e d by two distinct methods4: I) by using a constant radiative fraction, XR, along the wall, or 2) by using a soot emission model a s s u m i n g representative flame radiation t e m p e r a t u r e a n d absorption coefficient. Either m e t h o d requires appropriate meas u r e m e n t s of radiative properties for materials 4. *It should be noted that, in our LDV measurements 13, we observed an uncertainty (due to laser beam deflection) of 1.5 nun in the location of the measuring volume; maximum temperature and velocities are observed at distances 7 to 10 mm from the wall ~:~.
6o
FIRE
Profiles Produced by I n t e g r a l
Model
.
!
it
x=
n~
o
O = ~ ( k g l m 3 . m2 1 s ) Fro. 6. Comparison of .Measured Velocit~ P,-ofiles in a Wall Fire with Protiles Used in the lmegral Model
( B - .7)
Mass Pyrolysis Rate
We conclu(te this section by pointing out the m@)r shortcomings of the p,esem integral model: I) constant maximum "turbulent" temperature depe,~ding on wall material; and 2) incomplete ttame radiation modeling. Extensive wo,-k is cttrrentlv underway concerning the modeling of wall flame radiation 1:'. Additional work is required to shed light on tile insensitivity of Inaxillltll~l "turbulelU" temperature to hication ;doug a wall fire or to small variations of the mass transfer n u m b e r (see Figures 4b and 5).
where p:~ is tile ambient air density and v~ is the ambient air kinematic viscosity. Moreover,
S i m p l e E x p r e s s i o n s in T u r b u l e n t Wall Fires and Discussion
where AT,,, the m a x i m u m gas temperature rise, will be determined later. The "convective" mass transfer number, Be, is given by
The integral model produces a system of ordinary nonlinear diftmential equati(ins which can be s o h e d numerically by using standard solutio,l algorithms :~'4. The complete svste,n of equations for m , b u l e n t wall fires is inciuded in I)elichatsios~ and in Delichatsios 4. We present here some new simplified expressions tbr turbulent wall fires. T h e p,oposed expressions were obtained from an analysis of the solutions for the complete system of equations.
,it"/p= =
.088( v~ogS,,,) v:~en(1 + Be).
a,,, =
T~
(5)
(6)
B'
] ] -
9 ~t
1+
,tt
q'--qq rh"AH,.
(AH,(xA--xR)Y~,~-h,)/AH, _
v~,M~,
q';
""
1 + - - -q,i
vh"Att,
(7)
SIMPLE MODEL FOR TURBULENT WALL FIRES
61
where B' is the mass transfer number of the wall material corrected for radiation losses Xn, 4 and incompleteness o f combustion, XA; q',', is the wall reradiation loss; ~} is the flame radiation flux to the wall; AH~ is the effective heat of vaporization and h~ = @ (Tp - T~) where Tp is the so called pyrolysis temperature. The effective heat of vaporization AH~, is equal to the latent heat of vaporization plus the heat conducted into the wall p e r unit mass of vaporized fuel. In general, a one-dimensional unsteady heat conduction equation for the solid fuel is required together with the gas phase equations for a calculation of the heat conducted into the solid. For homogeneous noncharring materials and quasi-steady pyrolysis, the heat conducted into the solid is equal to the sensible heat required to bring the solid from its initial t e m p e r t a u r e to the pyrolysis temperature: rh" C~ (Tp - T~), where G is the specific heat for the wall material. We note that the wall pyrolysis rate expression (see equ (5)) is similar to pyrolysis rate relations that can be derived from a stagnant film theory 16.
In accordance with the experimental data, the maximum turbulent t e m p e r a t u r e rise AT,, was assumed to be i n d e p e n d e n t of downstream distance along the wall. It can be determined by using the recipe p r o p o s e d in the previous section. First, find the m a x i m u m gas temperature in a laminar wall fire having the same radiation loss as a turbulent fire, assuring a classical thin flame model for combustion4: "
Mass Flow Rate T h e original integral equations included two assumed turbulent correlation constants which lead to a solution which closely approximates the result,
M,, ~o YFC '
d~bt rh" = E,,,,, + - Ox p~
q;-r L~ :- AH,, + ~ rh
where h~ = C~(Tp- T.)
r
YET L~
•
(llb)
: specific enthalpy of the gas at the pyrolysis ten> p e r a t u r e (see definition after equ (7)); : stoichiometric ratio
" fuel mass concentration in transferred gas; : "convective" heat of vaporization (see equ (11 b)) : heat o f combustion per mass of fuel; and 9 (local) radiative fraction.
T h e n , the m a x i m u m turbulent temperature is calculated from:
fi~dy;
E = .045
(9)
T h e entrainment coefficient in turbulent wall fire plumes is less than half the entrainment coefficient in two-dimensional turbulent free buoyant plumes. The maximum velocity was also deduced from the integral model solution: 2
AmgX
l+r
[A H,.(XA--XR)YFT+ Ct,( 7 ; - To~)-L,I/Cp T+ (lla)
AH~
E is the entrainment coeficient; u~ is the maxim u m velocity and the second term on the RHS is the "blowing" velocity o f the pyrolyzing material at the surface, which is usually negligible. T h e entrainment coefficient, E, was determined from the solution of the integral equations,
u,,,
To~
(8)
Here, Ot is the total mass flow rate in the b o u n d a r y layer, i
A~
_ .64
(lo)
al;,, AT,[ = c~7.
(12)
where ~T is assumed constant. From the ethane wall fire experiments ~'~4, c~7-was found to vary from .4 to .6. Two other parameters remain to be defined: 1) reradiation loss, ~)','~, and 2) flame radiation flux to the surface, @~ T h e reradiation loss is:
~; =~oT?
(13)
where e is the surface emissivity and o~ the g o l t z m a n n constant. T h e flame radiation to the surface @, may be obtained from the conservation of energy across the layer:
62
FIRE
rh",5 H~.+ ~;', = q~)+ 0;' surface energy balance (14a)
flow energy balance (14b) @ = XkO;'/2
definition of XR (14c)
2 gm
or
2
--
p~CpT~
7%gx
~x ~~0l'r Ad~q-rh AHvq-q~"r
--1 --
.,,
XR
(14d)
q,f
One can either specify XR or @. T h e integral in the n u m e r a t o r o f e q u (14) is proportional to the convective heat flux across the layer; based on the profiles shown in Figure 3, we obtain: Z~w
~m
(15)
The first m e m b e r at the RHS o f e q u (15) may be neglected, since it is about three percent of the second number. So, equ (14) may be written ~
d~b,
P~GT~
2 --
dx F
rh. . . . . AH,.+q,
--1 --
(16)
x,~
r
Equations (5), (6), (7), (8), (10), (1 1), (12), (13) and (16) provide a complete set of equations if the local radiative fraction XR in equ (16) is known or the flame radiation heat flux to the surface, @, can be d e t e r m i n e d from a sootemission model 4. A comparison o f predictions for mass pyrolysis rates with experimental data for ethane and 9 1114
,
*
,
i
,
i
,
I
,
t,/,-r
qo
/4
, O2'
|0
SO
m
~" -1
PMMA (plexiglass) turbulent wall fires has been made in Figure 7 taken from Delichatsios 4. Moreover, Figure 5 in this p a p e r shows the predicted values o f m a x i m u m velocity and total flow rates s u p e r i m p o s e d on the experimental data: there is good agreement with the experimental data. T h e F r o u d e n u m b e r
,01'
0 ~,~
Y
pT~olTJ~,
/ / O. O0
1'
~ rtlSTAHr'E
i
,i
o
FROM ~ I R I N XQ4)
FIG. 7. Predicted and Measured Values of Wall Pyrolysis Rates and Radiative Heat Fluxes to the Wall in a Large PMMA Fire
was measured to be .591~, which compares favorably with the predicted value .64 in equ (10). It is obvious by inspecting equs (5) to (16) that the local or total radiative fraction, • plays a very significant rote in controlling the pyrolysis rate (see equ (5)). Its main effect is derived from a modification of the mass transfer number (see equ (7)); a secondary effect is due to the modification o f m a x i m u m t e m p e r a t u r e (see equ (lla)). Unfortunately, there are very few experimental measurements of flame radiation in wall fires; a research p r o g r a m is in progress at FMRC to completely characterize and model wall flame radiation in terms of the fuel's smoke point ]5. Such modeling together with a p p r o p r i ate experimental data will allow the closure of the wall fire p r o b l e m (see equ (16)). Conclusions
In this paper, we proposed an essentially algebraic system o f equations for d e t e r m i n i n g pyrolysis rates in wall fires, see equs. 5 - 1 6 . These equations evolved from a detailed integral wall model ~'4 and agree with existing experimental data. T h e main uncertainties of the model concern the determination of flame t e m p e r a t u r e (see equ (12)) and flame radiation (see equ (16)). T h e fundamental questions r e m a i n - - w h y is the peak mean gas temperature insensitive to height and B-number, and how sensitive is it to the type o f fuel. While the model results may not be sensitive to ;rm our u n d e r s t a n d i n g o f turbulent b u r n i n g surely requires some f u n d a m e n t a l explanation. Indeed, it is the recognition of this insensitivity which allowed a significant b r e a k t h r o u g h in providing this simple algebraic model of turbulent burning. Although the uncertainty in flame t e m p e r a t u r e has little effect on the predicted results (see equ (15) or equ (10)), wall flame radiation is very i m p o r t a n t and deserves intensive investigation in o r d e r to complete the solution o f the wall fire problem for a given wall material.
SIMPLE MODEL FOR T U R B U L E N T WALL FIRES REFERENCES
16. SPALDINg,
63
D.B., Proc. the Inst. of Mech. Engs.,
168, 545, No 19 (1954).
1. ORLOVF, L, DE RIS, J., AND MARKSTEIN, G.H., "Upward Flame Spread and Burning of Fuel Surface," Fifteenth Symposium on Combustion, The Combustion Institute, pp 183-192, (1975). 2. AnMAD, T. AND FAETH, G.M., "Turbulent Wall Fires," Seventeenth Symposium (International) on Combustion, The Combustion Institute, p 1149, 1979. 3. DELICHATSIOS, M.A., "Turbulent Convective F!ows and Burning on Vertical Walls," Nineteenth Symposium (International) on Combustion, The Combustion Institute, p 855, 1982. 4. DELICHAXSmS,M.A., "Flame Heights in Turbulent Wall Fires With Significant Flame Radiation," Combustion Science and Technology in press. 5. GEORGE,W.K., JR. AND CAPP, I.P., "A Theory for Natural Convective Turbulent Boundary Layers Next to Heated Vertical Walls," Int. J. of Heat and Mass Transfer, 22, p 813, 1979. 6. ROHSENOW,W.M. AND Cnoi, H.Y., "Heat, Mass and Momentum Transfer," p 202, Prentice Hall, Inc., Englewood Cliffs, NJ, 1961. 7. CrBECI, T. AND BRADSHAW, P., "Momentum Transfer in Boundary Layers," Hemisphere Publishing Corp., London, 1977. 8. BATCHELOR, G.K. "An Introduction to Fluid Mechanics", Cambridge University Press, 1967. 9. GRELLA, T.T. AND FAETH, G.M., "Measurements in a Two-Dimensional Thermal Plume Along a Vertical Adiabatic Wall," J. Fluid Mech., 71, p 701, 1975. 10. BILGER, R.W., "Turbulent Jet Diffusion Flows," Prog. Energy Combust. Sci., p 87, 1976. 11. TAMANINI, F., Eighteenth Symposium (International) on Combustion, The Combustion Institute, 1981. 12. TAMANINI, F., "A Numerical Model for the Prediction of Radiation Controlled Turbulent Wall Fires," Seventeenth Symposium (International) on Combustion, The Combustion Institute, 1979. 13. MOST, J.M., SZTAL, B. AND DELICHATSIOS,M.A., "Turbulent Wall Fires--LDV and Temperature Measurements and Implications," accepted for presentation at the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 2-4, 1984. 14. AHMAD, T., "An Experimental Study of the Pyrolysis Zone of Turbulent Wall Fires," Factory Mutual Research Corporation, Report 0AOE7.BU-1, 1978. 15. MARKSTEIN,G.H., "Radiant Emission from Wall Fires," Eastern Section Meeting of the Combustion Institute, Nov. 2 - 3 - 4 (1985).
Nomenclature
B'
Be
cp C, E
mass t r a n s f e r n u m b e r c o r r e c t e d for radiation (see n u m e r a t o r o f equ (7)) "convective" mass transfer n u m b e r (see equ. (7)) specific h e a t o f gases specific h e a t o f wall material e n t r a i n m e n t rate coefficient (see equ.
(8)) Lc ~ll r162
Mo MF r
~' T
rp /./ o r /~ x
Y
Yo
Frr
"convective" heat o f vaporization (see equ. (1 lb)) mass pyrolysis rate p e r unit area o x y g e n m o l e c u l a r weight (gaseous) fuel m o l e c u l a r weight mass stoichiometric ratio (fuel to air) convective heat flux to the wall flame radiation flux to wall r e r a d i a t i o n wall losses m e a n absolute gas t e m p e r a t u r e wall pyrolysis t e m p e r a t u r e m e a n gas velocity vertical c o - o r d i n a t e c o o r d i n a t e n o r m a l to wall o x y g e n mass c o n c e n t r a t i o n transfer fuel c o n c e n t r a t i o n
Greek Letters k J-/( ~lHv
A or ~
v "r • • tb
heat o f c o m b u s t i o n effective h e a t o f vaporization (latent heat plus heat c o n d u c t e d into the wall) 7-r~ m e a n t e m p e r a t u r e rise normalT~ ized by the absolute ambient temperature surface emissivity dynamic viscosity kinematic viscosity B o l t z m a n n ' s r a d i a t i o n constant shear stress c o m b u s t i o n efficiency coefficient local radiative fraction stream f u n c t i o n c o o r d i n a t e (see definition after equ. (3))
Subscript m t 0r overbars
m a x i m u m value total value a m b i e n t conditions r e p r e s e n t t u r b u l e n t m e a n values
64
FIRE
COMMENTS T. Kashiwagi, National Bureau of Standards (USA). Could you give Some description as to how you measured the stream function in your turbulent flame? These are very difficult measurements.
Author'~s Reply. We made simultaneous measurements of velocity and temperature profiles in a simulated (ethane) turbulent wall fire by using a Laser Doppler Velocimeter and a fine (25 p~m) thermocouple located 1 mm upstream of the focus of the laser beam (see paper by J.M. Most, B. Sztal and M.A. Delichatsios, "Turbulent Wall Fires--LDV and Temperature Measurements and Implications", from the book "Laser Anemometry in Fluid Mechanics" published by Ladoan--Instituto Superior Technico, 1095 Lisboa Codex--Portugal, 1984). Note that the gas temperatures were corrected for radiation errors. (See cited paper). We calculated the average density by assunfing equal molecular weights everywhere and using the perfect gas law. Then the stream function for the mean velocity can be calculated from: where ~' is the distance from the wall.
*=fl The assumption of a constant molecular weight gas mixture is violated only close to the wall (a few millimeters), especially for fuels heavier or lighter than ethane.
@
M. Delichatsios, Factory Mutual Research (USA). You had an excellent correlation between your theory and your data. Could you comment on how many adjustable parameters you employ in your model? Specifically, how do you specify the mean peak temperature ATe? Author's Reply. One may find out from References 3 and 4 of the present paper that we used five constant parameters to implement the integral model, including the new wall laws which are applicable for natural convective turbulent flows. Three of these parameters (pertinent to the flow) were determined by using detailed data in natural convective flows without combustion, while the other two (pertinent to combustion) were determined from results in wall fires having negligible flame radiation. One may observe that these parameters are reflected in the proportionality constants of the simple wall fire equations presented in this paper, namely eqs. 5, 8, and 10. Regarding the mean peak temperature, we point out that equations (lla), (llb) and (12) provide a recipe for evaluating its value. Justification for the proposed recipe is presented in the main text and corroborated by the results in Figures 4b and 5, which show that the peak mean temperature is insensitive to height and B-number.
A SIMPLE A L G E B R A I C M O D E L FOR T U R B U L E N T W A L L FIRES MICHAEL A. DELICHATSIOS
Factory Mutual Research Corporation* Propagation of flames over vertical walls is a key phenomenon for 1) predicting fire growth in rooms, and 2) characterizing the flammability properties of practical materials. Because of the complexity of the fire problem and the lack of values for key material properties, simple, yet comprehensive, models are sought for modeling turbulent wall fires. In this paper, simple algebraic expressions for turbulent wall fires are presented including expressions for the wall pyrolysis rate, the mass flow rate, the maximum mean velocity and the maximum mean gas temperature. These expressions evolved from a more complicated integral wall fire model presented previously. In the proposed relationships, mass blowing at the wall owing to pyrolysis and flame radiation feedback is accounted for. An extensive discussion is also presented for the fundamental fluid mechanical and combustion principles of the model including comparison with experimental property profiles and wall pyrolysis rates. Limitations of the model regarding the modeling of flame radiation are clearly noted.
Introduction Much effort has been expended d u r i n g the last ten years by Orloff et al 1, Ahmad and Faeth 2, and Delichatsios3for predicting flame spread and b u r n i n g intensities in fires on vertical walls. Such wall fire models are needed as components of zone models for evaluating the fire hazard in buildings. I n addition, turbulent wall fire models are essential for interpreting and classifying material flammability based on laboratory property measurements or standardized performance tests such as the Room Corner Test or the Factory Mutual Corner Test. In a recently published paper 4 we presented an extension of a novel integral turbulent model 3 to predict 1) wall fire intensities with significant flame radiation, and 2) the extent of burning, i.e., the flame height, in wall fires. Such predictions are key inputs for calculating upward fire spread 1 rates on a vertical wall wherein flame spread depends 1 on 1) the flame extent beyond the pyrolizing region and 2) the heat feedback from the flames to the yet unignited fuel between the pyrolysis front and the flame tip. In the present paper, we derive a simplified algebraic model for t u r b u l e n t wall fires, which has evolved from a more complicated integral
*Funded in part by the Federal Aviation Administration 53
model 4. For completeness, we present a review of previous models as well as an evaluation of the physical principles and experimental verification of the present turbulent wall fire model including a description of its limitations. Subsequently, we develop simple expressions for mass pyrolysis rates, air e n t r a i n m e n t and gas velocities along the pyrolizing wall zone of a turbulent wall fire; flame radiation is not calculated by the present model but it is incorporated in the model as a given input property of the b u r n i n g gases.
Evaluation of the New Integral Model for Wall Fires The principles of the integral model are presented in order to demonstrate the soundness of its foundation; however, the ultimate justification of a new model must be based on a comparison of its predictions with experimental data. Such a comparison is also included in the following discussion. There are two components to the integral model: Fluid Dynamics and Combustion.
Fluid Dynamics Of The Model A flow on a vertical b u r n i n g wall is driven by buoyancy forces which are associated with the lower densities of combustion products compared to the s u r r o u n d i n g ambient air. Such flows are usually turbulent, unless their vertical size is less than say ten centimeters. T u r b u l e n t
54
FIRE
wall fires belong to the general class of turbulent natural convective flows next to vertical walls. T h e principle features of the proposed integral model for turbulent convective flows next to a vertical wall are listed in Table I. Perhaps the major innovation of the new model for turbulent convective flows is the introduction and use of wall laws (i.e., profiles close to the wall) for flow properties different from the logarithmic wall laws proposed by von K a r m a n for forced b o u n d a r y layer flows. These laws (see References 3 and 4 for details) have been developed from the recognition that buoyancy in turbulent convective flows generates the turbulence close to the wall,* in contrast to forced b o u n d a r y layer flows where turbulence is generated by principally local wall shear stress. Recognition o f this key difference allows one to predict the correct heat transfer in natural convective situations. T h e natural convective wall laws agree well with detailed measurements 5 of velocity a n d temperature profiles in turbulent convective flows next to walls having a constant t e m p e r a t u r e or a constant heat flux. In addition, these new natural convective wall laws are consistent with the experimental fact that the heat transfer coefficient for turbulent free convection from a wall at constant t e m p e r a t u r e is i n d e p e n d e n t of location along the wall s . It is i m p o r t a n t to discuss at this point previously developed integral models for turbulent natural convective flows. All previous models including turbulent wall fire integral models 2 have been based on the classical Eckert and Jackson (see Rohsenow and Choi 6) integral model approach. This model solves the m o m e n t u m and heat conservation equations by assuming 1) similar profiles for velocity and temperature across the entire wall layer; and 2) wall friction (and heat transfer) relations identical to the respective relations for forced b o u n d a r y layer flows. It is difficult to justify the second assumption given the fundamental differences between forced b o u n d a r y layer flows and free convective flows on vertical walls. Moreover, the Eckert and Jackson model (and its variations) predicts a convective heat tranfer coefficient that varies as x vS, where x is the distance along the wall from the origin o f the flow; this prediction contradicts the experimental evidence s which convincingly shows no d e p e n d e n c e o f convective heat transfer coefficients on x. *In similarity to von Karman's development, we have assumed that the inner flow close to the wall is not affected by the outer flow properties.
In the present integral model, (see Table I) we divided the natural convective boundary layer into two layers ~. For each layer different flow p r o p e r t y profiles were employed. The inner layer close to the wall is nonsimilar, while the o u t e r layer b o r d e r i n g the inner layer and extending to the ambient fluid is self-similar, i.e., the profiles have similar shapes at different heights. T h e p r o p e r t y values for the inner and outer layers are matched at an intermediate position which we take to be the position of m a x i m u m velocity across the b o u n d a r y layer. T h e two-layer a p p r o a c h allows us to carefully consider the specific thermal b o u n d a r y conditions at the wall which have a significant impact on the development o f the natural convective flow. T h e detailed consideration of wall thermal b o u n d a r y conditions includes mass transfer effects which are particularly i m p o r t a n t for the description o f turbulent wall fires wherein pyrolysis at the wall occurs. It should be noted that the width of the inner layer is small relative to the total flow width so that flow p r o p e r t y profiles may a p p e a r selfsimilar across most of the width of the boundary layer. (For comparison, note that the forced b o u n d a r y layers are treated in integral models as if they were self-similar over their entire thickness.) It is well known that wall and nearby buffer regions ("wall laws") are in general nonsimilar for turbulent b o u n d a r y layers (Cebeci and Bradshaw7). Integral conservation equations 3 were employed over the outer layer as well as over the total width of the b o u n d a r y layer. These conservation equations together with the matching conditions between the inner and outer layers provide a closed system of integral equations. Regarding our integral equations (see Table I), we employ a m o m e n t o f m o m e n t u m conservation equation in place o f the usual m o m e n t u m equation to take advantage o f its simpler bound a r y condition. First we will present the new conservation equation for forced flows, and
TABLE I Distinct Characteristics of the Integral Model 1. New wall laws for natural convective flows. 2. Two layer modeling; inner layer (nonsimilar) close to wall overlaps with outer layer (self-similar). 3. Conservation of moment of momentum for thin wall flows as a replacement of momentum conservation (see equ (2)). 4. Average stream function in place of distance normal to wall, for rigorous incorporation of variable density effects.
SIMPLE MODEL FOR TURBULENT WALL FIRES
55
b o u n d a r y laver. It is expected, hut must be shown by using e x p e r i m e n t a l data, that the RHS of equ (1) is sufficiently small so that P (see equ (2a)) can be a s s u m e d to be i n d e p e n d e n t ofx even for t u r b u l e n t (forced) wall flows. T h e conservation equ (1) can be extended to n a t u r a l convective flows, which include buovancy forces, with a small effort (Delichatsios, 1982);
Edge o f W ~ I I J e t
u =
d
-"77"7.,
~t~
oa
o
FIG. 1. la: A Thin Turbulent Wall Jet-Without Buoyancy 1B: Velocity and Shear Stress Profiles t h e n we will e x t e n d it for b u o y a n t flow a n d also p u t forward some advantages of its application in the present model. F i g u r e 1 shows a thin n o n b u o y a n t wall j e t t o g e t h e r with profiles of velocity a n d shear stress. It is a simple exercise (see Batchelor ~, p 377, 1967, exercise 2) which shows that: 0
oo u
0 f,-f,
oa
r
Equation (1) was derived by 1) using the b o u n d a r y layer a p p r o x i m a t i o n for wall jets a n d 2) neglecting l o n g i t u d i n a l velocity fluctuations for t u r b u l e n t flow conditions followed by muhiplying the m o m e n t u m e q u a t i o n by d a n d i n t e g r a t i n g over y n o r m a l to the surface. Equation (1) is valid for both l a m i n a r or t u r b u l e n t wall jets. A n expression for the shear stress in a t u r b u l e n t flow is given in F i g u r e 1. It is obvious (by integration) that the RHS o f e q u (1) is zero for l a m i n a r flows -r = Ix ~ T h e r e f o r e , the quantity (if2 dy) = P
(2a)
J~-i -g~y-
,
o
For the same reasons as in torced flows, the second term at the RHS of equ (2b) is expected to be small a n d can be conveniently set to zero for natural convective flows adjacent to vertical walls. Equation (2b) was used over the outer (self-similar) layer of o u r t u r b u l e n t wall model (see Reference 3 for m o r e details). T h e new m o m e n t of m n m e n t u m e q u a t i o n replaced the m o m e n t u m equation, which would have req u i r e d an expression of the wall Diction in o r d e r to be solved. Such an expression for the friction is not available from m e a s u r e m e n t s or first principles. Finally, in the d e v e l o p m e n t of the integral equations, we used the average stream ftmction, 0 as an i n d e p e n d e n t variable in place of the n o r m a l distance from the wall (see Table I); this formttlation allows a straightforward incorporation of variable density effects, wlnich are particularly i m p o r t a n t in wall fres.
o
t.0 .9 .8 .7
Veloetty (Ref. 8) I ~ n s l t y Defect
.6 .5
~
;::u2;,7:~;2.
.4 n E
0 o
.2
is i n d e p e n d e n t of x for a thin l a m i n a r forced wall jet. T h e conserved quantity, P, may be characterized as m o m e n t (w.r.t. ~) of m o m e n t u m of the wall flow. For a t u r b u l e n t forced wall jet the RHS of equ (1) is not zero. T h e r e are no reliable data for estimating its precise magnitude; however, observe (see, also, Figure 1) that the i n t e g r a n d at the RHS o f e q u (1) 1)becomes zero at three positions across the b o u n d a r y layer a n d 2) changes sign at a point inside the
.1,
1
,00
.02
t
.04
a
L
I
,06
,08
.10
I
.12
y/x
FIG. 2. Comparison of Velocity and Temperature Profiles in an Adiabatic Wall Plume with Profiles Used in the Integral Model (Ref. 8 for x = 1.22 m, u,,, = .935 m/s, A~ = .06)
56
FIRE
We conclude by presenting in Figure 2 a comparison between profiles at the outer layer used in the integral model with experimental profiles. For the implementation of any integral model for b o u n d a r y layer flows, it is required to introduce profiles o f the flow properties in o r d e r to evaluate the integrals 7. In the present approach, we e m p l o y e d different profiles for the inner and outer layers. For the inner layer we used the new wall laws; it is important to re-emphasize strongly that these profiles agree well with experimental data 5. For the outer wall layer, we assumed that the property profiles vary linearly with the value of the stream function. That is: ..., -
r162
a.,, -
r162
(3)
where u,,, is the m a x i m u m velocity; A is the mean t e m p e r a t u r e rise,
T-T•. T= ' tb is the stream function
~bt is the total flow =--
~dy; p~
0
and ~ is the stream function at the m a x i m u m value of the velocity. In Figure 2, an approximate agreement is shown between the profiles represented by equ (3) with the experimental profiles in weak turbulent wall plumes (Grella and Faeth, 1975). To evaluate the reasonableness of our assumed outer profiles (given by equ (3)) we use the definition o f ~ o r ~ = ~ / p ~ which upon integration yields
um
A A,,, = exp[--(Y--Ym)/6~]
in the Boussinesq limit (i.e. A,, < where 8r = (tbt-tb,~)/U~. Figure 2 compares the resulting calculated profiles with experimental data for an adiabatic vertical wall plume u n d e r Boussinesq conditions (~,, = 0.06). The linear fit appears quite satisfactory. We presented in this section a detailed discussion and evaluation of the fundamental characteristics (see Table I above) of the new integral model that we developed for natural turbulent convective flows next to vertical walls. In addition, we d e m o n s t r a t e d that the p r o p e r t y profiles used for implementing the integral
model are very close to experimental profiles, both for the inner a n d outer layers. We have used the present integral model s to calculate the flow in a t u r b u l e n t wall plume 9 where the predictions agree very well with the experimental data 3. In o r d e r to demonstrate more convincingly the superiority and soundness o f the new integral model, we also intend to compare, in a subsequent report, its predictions with existing detailed a n d extensive experimental data in natural convective flows next to heated walls.
Modeling Of Burning In The Integral Wall Fire Model The integral wall model for natural convective flows was e x t e n d e d s to turbulent wall fires, which involve pyrolysis at the wall and subsequent b u r n i n g o f the pyrolyzing gases with ambient air. T h e details of the analysis, as well as its further extension to wall fires with flame radiation, may be f o u n d in the literature 3'4, respectively. O u r intention in this section is to discuss: 1) some difficulties for developing integral models for turbulent diffusion flames (fires) and 2) the drastic simplifications introduced in the development of the present wall fire model 3'4. The cause o f the flow in the wall fires is the high t e m p e r a t u r e s (low densities) o f the combustion products relative to ambient air. These temperatures, however, are related, in a rather complicated way, to the average characteristics of the flow such as, for example, the heat release rate. A significant simplification o f the analysis has been obtained by introducing a conserved scalar quantity based on the classical Schvab-Zeldovich formulation. T h e instantaneous concentrations and t e m p e r a t u r e s are then related in a rather simple way with the instantaneous value of the conserved scalar 1~ However, the level of turbulent fluctuations is also required in o r d e r to find the average values of concentrations or temperature in terms o f the average values o f the conserved scalar in turbulent diffusion flames. An integral model for turbulent diffusion flames must provide for a relation between the average t e m p e r a t u r e (required in the flow equations) and the average conserved scalar (required for describing the burning). Such a relation may be obtained either by j u d g m e n t and use of e x p e r i m e n t a l facts or preferably, by introducing an additional integral equation for the fluctuations o f the conserved scalar 11. It must be emphasized that considerable complication is the penalty for using a conserved scalar fluctuation equation. We also considered using an integral fluctua-
SIMPLE MODEL FOR TURBULENT WALL FIRES tion equation in the present turbulent wall fire model. During the model development, it became apparent that not only an integral equation but also an accurate profile of the fluctuations is required, in o r d e r to predict a correct value of the local average t e m p e r a t u r e (especially close to the wall) in terms of the local average value of the conserved scalar. Within the constraints of an integral model, such an "accurate" profile for the conserved scalar correlations would have to be lifted from experimental data, which unfortunately are nonexistent. We decided to abandon this route. T h e principal characteristics of the new integral wall fire model may be described with reference to Figure 3. In this figure, the p r o p e r t y profiles (velocity, t e m p e r a t u r e rise, conserved scalar) used in the model are shown in terms of the stream function,
~/=
1
dy
p~ where y is the normal distance to the wall and 9~ is the ambient air density. T h e main features of the model are: 1) T h e outer layer profiles were assumed to be self-similar and to have a linear relationship with the stream function. 2) T h e inner layer profiles were determined by extending the methodology used for developing the inner layer profiles in turbulent (nonburning) buoyant flows next to vertical walls. For determining the velocity profile, we assumed that the buoyant force in the inner layer is driven by a constant temperature rise equal to the average of m a x i m u m (AT,,) rise. For determining the inner profile of the conserved scalar, we accounted for both the effects of blowing and heat transfer at the wall 3'4. It should be noted that we also assmned a linear relationship of the t e m p e r a t u r e rise in terms of + within the inner layer (see Figure 3) for evaluating convective heat flow parallel to the wall. However, since the inner layer is quite small, this latter assumption is not important. 3) Finally, we decided to uncouple the temperature rise from the conserved scalar by assuming that, along the wall fire, the maxim u m average t e m p e r a t u r e is i n d e p e n d e n t of x (but depends on the wall material). T h e maxim u m average t e m p e r a t u r e in a turbulent wall fire is considerably smaller than the maximum t e m p e r a t u r e in a laminar wall fire having the same wall material; the main reasons for the lower average t e m p e r a t u r e in a turbulent fire are: 1) turbulent fluctuations which smooth peak values, and 2) increased radiative losses. (Incompleteness of combustion probably has a
57
= T e m p e r s t u r e Rise "~ = Consecved Scalar
-fi,
/ ,.~\ /
/
~
Inner Layer
~Tw
*= FIG. 3. Profiles for the Inner and Outer Layers of the Turbulent Wall Fire Model smaller effect on the m a x i m u m average temperature rise.) Experimental results agree with the assumption that the maximum teinperatures remain approximately constant along a wall fire; there is no quantitative explanation for this experimental fact. We emphasized previously in this section the difficulties in modeling the turbulent fluctuations within the context of an integral model. However, even detailed k-e-g models for wall fires 12 overpredict the temperature rise for a vertical wall fire. (The author lz claimed that incompleteness of combustion, which is not included in his model, may be the reason for this discrepancy.) In Figure 4 we have assembled some recent data l~ for turbulent wall fires; these data include 1) velocity (Figure 4a) and temperature (Figure 4b) profiles along a wall fire for a fixed mass transfer n u m b e r (B = .7), and 2) temperature profiles (Figure 5) at a fixed location for varying mass transfer numbers (B = .5, .7, 1.0). In all these figures, the abscissa is the stream function that was calculated from the temperature and velocity data at several positions normal to the wall: 1 ~=T= fy0 7~dy. T h e experimental profiles (see Figure 4) can now be c o m p a r e d with the model profiles in Figure 3 and with the main features of the present integral wall-fire model. We can make the following remarks: 1) T h e outer profiles show an almost linear relationship of velocity and temperature with
FIRE
58
4a
r
(~
OOO ~/kA & 0 0
0
&
O
XX•
~***
'
I= 0 . 6 0
0
O
&0
~
l=
O A
XX
O &
..
& X
§
&
§ X
O O
.4-
& X
ae
X= 1.2~
O
& §
1=02
O
X
§ 9
0.80
6&
X
0
Xx
"k
X
P==~ ( kg/m 3
. m 2Is)
4b
~
I.S
"*-, ~•176176176
I=
0=80
I:
1=02
X=
1=25
8 ,,o o XX
+
~~
I
I
/,.
0
§
& X
O
o
&
§
&
O
X
O
§ §
X
&
A
O
A
O
A ~5
X
O A&
+
A X
+
X
A
O
XX X
@,@
P==~ ( kg/m 3 . m 2 / s ) FI(;. 4. Velocity and Temperature Profiles at Four Locations along an Ethane Wall Fire with Mass Transfer Number B = .7 the stream function; these plots do not show ~ h e t h e r the o u t e r profiles are self-similar. However, similarity o f the o u t e r profiles has been clearly d e m o n s t r a t e d previously ~4: 2) T h e e x p e r i m e n t a l velocity and t e m p e r a ture profiles are flat near the m a x i m u m values of the variables in contrast to the m o d e l profiles; this d i f f e r e n c e i n t r o d u c e s an uncertainty in d e f i n i n g the extent of the inner-layer
(i.e., value o f 4,~, see Figure 3). Notice that the velocity profiles are flatter than the t e m p e r a ture profiles. An i m p r o v e m e n t o f the profiles could be o b t a i n e d by p r e s e r v i n g a linear profile for the t e m p e r a t u r e while m o d i f y i n g the instead o f ~ = ~ (4, - 4) (see Figure 3),(4a) use t~ ~ +J.'3 (4, - 4).
(4b)
SIMPLE MODEL FOR TURBULENT WALL FIRES
B:
0.35
+
B=
0.70
x
B:
1.00
A
59
Profile Produced by Integral Model for B = . 7
' 0.|
,.,~
,d,
,.,~
,.,~
.-4,
,.~
.d,
,.,~
,d,
,.-~
Owl~ ( kg/m3 . m2 1 ~ )
TEnPERATURE FIr 5. Temperature Profiles at a Fixed Location (0.6 m from the origin) for Three Different Mass Transfer Numbers, B = .35, .7, 1.00. Note that adding more fuel has little effect on temperature or the total convective upward heat flow. (Note: for the i n n e r layer 3 ~ = ~ $J'3.) T h e implications of this modification will be investigated in a s u b s e q u e n t paper. 3) Finally, Figures 4b a n d 5 clearly d e m o n strate that the m a x i m u m average t e m p e r a t u r e is constant not only along a wall fire but also for a small variation of the mass transfer n u m b e r (B = .35 to 1.0). It is expected that this t e m p e r a t u r e will vary somewhat for different wall materials. No complete quantitative explan a t i o n for this e x p e r i m e n t a l fact is known. While f o r m u l a t i n g the integral model, we employed the following p r o c e d u r e for d e t e r m i n ing the " t u r b u l e n t " gas temperature3'4: 1) we calculated the laminar flow t e m p e r a t u r e with a correction for the radiative heat loss fraction; 2) we assumed that the m a x i m u m t u r b u l e n t flame t e m p e r a t u r e is a c o n s t a n t fraction of this peak l a m i n a r flame t e m p e r a t u r e ; a n d 3) we suitably modified the wall b o u n d a r y conditions. T h e e x p e r i m e n t a l velocity profiles are compared with the calculated integral model profiles in Figure 6 at five d i f f e r e n t locations.
T h e r e is good a g r e e m e n t for the outer-layer profiles, i n c l u d i n g the m a x i m u m velocity and the total flow rate values, b u t the a g r e e m e n t is only fair* for the profiles at the i n n e r layer; it is expected that the a g r e e m e n t between experim e n t a l profiles a n d the profiles used in the model will i m p r o v e by m o d i f y i n g the outerlayer velocity profile as shown in equ (4). Flame radiation in t u r b u l e n t wall fires affects both the rate of pyrolysis a n d the rate of fire spread. In a t u r b u l e n t wall fire model, flame radiation can be i n t r o d u c e d by two distinct methods4: I) by using a constant radiative fraction, XR, along the wall, or 2) by using a soot emission model a s s u m i n g representative flame radiation t e m p e r a t u r e a n d absorption coefficient. Either m e t h o d requires appropriate meas u r e m e n t s of radiative properties for materials 4. *It should be noted that, in our LDV measurements 13, we observed an uncertainty (due to laser beam deflection) of 1.5 nun in the location of the measuring volume; maximum temperature and velocities are observed at distances 7 to 10 mm from the wall ~:~.
6o
FIRE
Profiles Produced by I n t e g r a l
Model
.
!
it
x=
n~
o
O = ~ ( k g l m 3 . m2 1 s ) Fro. 6. Comparison of .Measured Velocit~ P,-ofiles in a Wall Fire with Protiles Used in the lmegral Model
( B - .7)
Mass Pyrolysis Rate
We conclu(te this section by pointing out the m@)r shortcomings of the p,esem integral model: I) constant maximum "turbulent" temperature depe,~ding on wall material; and 2) incomplete ttame radiation modeling. Extensive wo,-k is cttrrentlv underway concerning the modeling of wall flame radiation 1:'. Additional work is required to shed light on tile insensitivity of Inaxillltll~l "turbulelU" temperature to hication ;doug a wall fire or to small variations of the mass transfer n u m b e r (see Figures 4b and 5).
where p:~ is tile ambient air density and v~ is the ambient air kinematic viscosity. Moreover,
S i m p l e E x p r e s s i o n s in T u r b u l e n t Wall Fires and Discussion
where AT,,, the m a x i m u m gas temperature rise, will be determined later. The "convective" mass transfer number, Be, is given by
The integral model produces a system of ordinary nonlinear diftmential equati(ins which can be s o h e d numerically by using standard solutio,l algorithms :~'4. The complete svste,n of equations for m , b u l e n t wall fires is inciuded in I)elichatsios~ and in Delichatsios 4. We present here some new simplified expressions tbr turbulent wall fires. T h e p,oposed expressions were obtained from an analysis of the solutions for the complete system of equations.
,it"/p= =
.088( v~ogS,,,) v:~en(1 + Be).
a,,, =
T~
(5)
(6)
B'
] ] -
9 ~t
1+
,tt
q'--qq rh"AH,.
(AH,(xA--xR)Y~,~-h,)/AH, _
v~,M~,
q';
""
1 + - - -q,i
vh"Att,
(7)
SIMPLE MODEL FOR TURBULENT WALL FIRES
61
where B' is the mass transfer number of the wall material corrected for radiation losses Xn, 4 and incompleteness o f combustion, XA; q',', is the wall reradiation loss; ~} is the flame radiation flux to the wall; AH~ is the effective heat of vaporization and h~ = @ (Tp - T~) where Tp is the so called pyrolysis temperature. The effective heat of vaporization AH~, is equal to the latent heat of vaporization plus the heat conducted into the wall p e r unit mass of vaporized fuel. In general, a one-dimensional unsteady heat conduction equation for the solid fuel is required together with the gas phase equations for a calculation of the heat conducted into the solid. For homogeneous noncharring materials and quasi-steady pyrolysis, the heat conducted into the solid is equal to the sensible heat required to bring the solid from its initial t e m p e r t a u r e to the pyrolysis temperature: rh" C~ (Tp - T~), where G is the specific heat for the wall material. We note that the wall pyrolysis rate expression (see equ (5)) is similar to pyrolysis rate relations that can be derived from a stagnant film theory 16.
In accordance with the experimental data, the maximum turbulent t e m p e r a t u r e rise AT,, was assumed to be i n d e p e n d e n t of downstream distance along the wall. It can be determined by using the recipe p r o p o s e d in the previous section. First, find the m a x i m u m gas temperature in a laminar wall fire having the same radiation loss as a turbulent fire, assuring a classical thin flame model for combustion4: "
Mass Flow Rate T h e original integral equations included two assumed turbulent correlation constants which lead to a solution which closely approximates the result,
M,, ~o YFC '
d~bt rh" = E,,,,, + - Ox p~
q;-r L~ :- AH,, + ~ rh
where h~ = C~(Tp- T.)
r
YET L~
•
(llb)
: specific enthalpy of the gas at the pyrolysis ten> p e r a t u r e (see definition after equ (7)); : stoichiometric ratio
" fuel mass concentration in transferred gas; : "convective" heat of vaporization (see equ (11 b)) : heat o f combustion per mass of fuel; and 9 (local) radiative fraction.
T h e n , the m a x i m u m turbulent temperature is calculated from:
fi~dy;
E = .045
(9)
T h e entrainment coefficient in turbulent wall fire plumes is less than half the entrainment coefficient in two-dimensional turbulent free buoyant plumes. The maximum velocity was also deduced from the integral model solution: 2
AmgX
l+r
[A H,.(XA--XR)YFT+ Ct,( 7 ; - To~)-L,I/Cp T+ (lla)
AH~
E is the entrainment coeficient; u~ is the maxim u m velocity and the second term on the RHS is the "blowing" velocity o f the pyrolyzing material at the surface, which is usually negligible. T h e entrainment coefficient, E, was determined from the solution of the integral equations,
u,,,
To~
(8)
Here, Ot is the total mass flow rate in the b o u n d a r y layer, i
A~
_ .64
(lo)
al;,, AT,[ = c~7.
(12)
where ~T is assumed constant. From the ethane wall fire experiments ~'~4, c~7-was found to vary from .4 to .6. Two other parameters remain to be defined: 1) reradiation loss, ~)','~, and 2) flame radiation flux to the surface, @~ T h e reradiation loss is:
~; =~oT?
(13)
where e is the surface emissivity and o~ the g o l t z m a n n constant. T h e flame radiation to the surface @, may be obtained from the conservation of energy across the layer:
62
FIRE
rh",5 H~.+ ~;', = q~)+ 0;' surface energy balance (14a)
flow energy balance (14b) @ = XkO;'/2
definition of XR (14c)
2 gm
or
2
--
p~CpT~
7%gx
~x ~~0l'r Ad~q-rh AHvq-q~"r
--1 --
.,,
XR
(14d)
q,f
One can either specify XR or @. T h e integral in the n u m e r a t o r o f e q u (14) is proportional to the convective heat flux across the layer; based on the profiles shown in Figure 3, we obtain: Z~w
~m
(15)
The first m e m b e r at the RHS o f e q u (15) may be neglected, since it is about three percent of the second number. So, equ (14) may be written ~
d~b,
P~GT~
2 --
dx F
rh. . . . . AH,.+q,
--1 --
(16)
x,~
r
Equations (5), (6), (7), (8), (10), (1 1), (12), (13) and (16) provide a complete set of equations if the local radiative fraction XR in equ (16) is known or the flame radiation heat flux to the surface, @, can be d e t e r m i n e d from a sootemission model 4. A comparison o f predictions for mass pyrolysis rates with experimental data for ethane and 9 1114
,
*
,
i
,
i
,
I
,
t,/,-r
qo
/4
, O2'
|0
SO
m
~" -1
PMMA (plexiglass) turbulent wall fires has been made in Figure 7 taken from Delichatsios 4. Moreover, Figure 5 in this p a p e r shows the predicted values o f m a x i m u m velocity and total flow rates s u p e r i m p o s e d on the experimental data: there is good agreement with the experimental data. T h e F r o u d e n u m b e r
,01'
0 ~,~
Y
pT~olTJ~,
/ / O. O0
1'
~ rtlSTAHr'E
i
,i
o
FROM ~ I R I N XQ4)
FIG. 7. Predicted and Measured Values of Wall Pyrolysis Rates and Radiative Heat Fluxes to the Wall in a Large PMMA Fire
was measured to be .591~, which compares favorably with the predicted value .64 in equ (10). It is obvious by inspecting equs (5) to (16) that the local or total radiative fraction, • plays a very significant rote in controlling the pyrolysis rate (see equ (5)). Its main effect is derived from a modification of the mass transfer number (see equ (7)); a secondary effect is due to the modification o f m a x i m u m t e m p e r a t u r e (see equ (lla)). Unfortunately, there are very few experimental measurements of flame radiation in wall fires; a research p r o g r a m is in progress at FMRC to completely characterize and model wall flame radiation in terms of the fuel's smoke point ]5. Such modeling together with a p p r o p r i ate experimental data will allow the closure of the wall fire p r o b l e m (see equ (16)). Conclusions
In this paper, we proposed an essentially algebraic system o f equations for d e t e r m i n i n g pyrolysis rates in wall fires, see equs. 5 - 1 6 . These equations evolved from a detailed integral wall model ~'4 and agree with existing experimental data. T h e main uncertainties of the model concern the determination of flame t e m p e r a t u r e (see equ (12)) and flame radiation (see equ (16)). T h e fundamental questions r e m a i n - - w h y is the peak mean gas temperature insensitive to height and B-number, and how sensitive is it to the type o f fuel. While the model results may not be sensitive to ;rm our u n d e r s t a n d i n g o f turbulent b u r n i n g surely requires some f u n d a m e n t a l explanation. Indeed, it is the recognition of this insensitivity which allowed a significant b r e a k t h r o u g h in providing this simple algebraic model of turbulent burning. Although the uncertainty in flame t e m p e r a t u r e has little effect on the predicted results (see equ (15) or equ (10)), wall flame radiation is very i m p o r t a n t and deserves intensive investigation in o r d e r to complete the solution o f the wall fire problem for a given wall material.
SIMPLE MODEL FOR T U R B U L E N T WALL FIRES REFERENCES
16. SPALDINg,
63
D.B., Proc. the Inst. of Mech. Engs.,
168, 545, No 19 (1954).
1. ORLOVF, L, DE RIS, J., AND MARKSTEIN, G.H., "Upward Flame Spread and Burning of Fuel Surface," Fifteenth Symposium on Combustion, The Combustion Institute, pp 183-192, (1975). 2. AnMAD, T. AND FAETH, G.M., "Turbulent Wall Fires," Seventeenth Symposium (International) on Combustion, The Combustion Institute, p 1149, 1979. 3. DELICHATSIOS, M.A., "Turbulent Convective F!ows and Burning on Vertical Walls," Nineteenth Symposium (International) on Combustion, The Combustion Institute, p 855, 1982. 4. DELICHAXSmS,M.A., "Flame Heights in Turbulent Wall Fires With Significant Flame Radiation," Combustion Science and Technology in press. 5. GEORGE,W.K., JR. AND CAPP, I.P., "A Theory for Natural Convective Turbulent Boundary Layers Next to Heated Vertical Walls," Int. J. of Heat and Mass Transfer, 22, p 813, 1979. 6. ROHSENOW,W.M. AND Cnoi, H.Y., "Heat, Mass and Momentum Transfer," p 202, Prentice Hall, Inc., Englewood Cliffs, NJ, 1961. 7. CrBECI, T. AND BRADSHAW, P., "Momentum Transfer in Boundary Layers," Hemisphere Publishing Corp., London, 1977. 8. BATCHELOR, G.K. "An Introduction to Fluid Mechanics", Cambridge University Press, 1967. 9. GRELLA, T.T. AND FAETH, G.M., "Measurements in a Two-Dimensional Thermal Plume Along a Vertical Adiabatic Wall," J. Fluid Mech., 71, p 701, 1975. 10. BILGER, R.W., "Turbulent Jet Diffusion Flows," Prog. Energy Combust. Sci., p 87, 1976. 11. TAMANINI, F., Eighteenth Symposium (International) on Combustion, The Combustion Institute, 1981. 12. TAMANINI, F., "A Numerical Model for the Prediction of Radiation Controlled Turbulent Wall Fires," Seventeenth Symposium (International) on Combustion, The Combustion Institute, 1979. 13. MOST, J.M., SZTAL, B. AND DELICHATSIOS,M.A., "Turbulent Wall Fires--LDV and Temperature Measurements and Implications," accepted for presentation at the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 2-4, 1984. 14. AHMAD, T., "An Experimental Study of the Pyrolysis Zone of Turbulent Wall Fires," Factory Mutual Research Corporation, Report 0AOE7.BU-1, 1978. 15. MARKSTEIN,G.H., "Radiant Emission from Wall Fires," Eastern Section Meeting of the Combustion Institute, Nov. 2 - 3 - 4 (1985).
Nomenclature
B'
Be
cp C, E
mass t r a n s f e r n u m b e r c o r r e c t e d for radiation (see n u m e r a t o r o f equ (7)) "convective" mass transfer n u m b e r (see equ. (7)) specific h e a t o f gases specific h e a t o f wall material e n t r a i n m e n t rate coefficient (see equ.
(8)) Lc ~ll r162
Mo MF r
~' T
rp /./ o r /~ x
Y
Yo
Frr
"convective" heat o f vaporization (see equ. (1 lb)) mass pyrolysis rate p e r unit area o x y g e n m o l e c u l a r weight (gaseous) fuel m o l e c u l a r weight mass stoichiometric ratio (fuel to air) convective heat flux to the wall flame radiation flux to wall r e r a d i a t i o n wall losses m e a n absolute gas t e m p e r a t u r e wall pyrolysis t e m p e r a t u r e m e a n gas velocity vertical c o - o r d i n a t e c o o r d i n a t e n o r m a l to wall o x y g e n mass c o n c e n t r a t i o n transfer fuel c o n c e n t r a t i o n
Greek Letters k J-/( ~lHv
A or ~
v "r • • tb
heat o f c o m b u s t i o n effective h e a t o f vaporization (latent heat plus heat c o n d u c t e d into the wall) 7-r~ m e a n t e m p e r a t u r e rise normalT~ ized by the absolute ambient temperature surface emissivity dynamic viscosity kinematic viscosity B o l t z m a n n ' s r a d i a t i o n constant shear stress c o m b u s t i o n efficiency coefficient local radiative fraction stream f u n c t i o n c o o r d i n a t e (see definition after equ. (3))
Subscript m t 0r overbars
m a x i m u m value total value a m b i e n t conditions r e p r e s e n t t u r b u l e n t m e a n values
64
FIRE
COMMENTS T. Kashiwagi, National Bureau of Standards (USA). Could you give Some description as to how you measured the stream function in your turbulent flame? These are very difficult measurements.
Author'~s Reply. We made simultaneous measurements of velocity and temperature profiles in a simulated (ethane) turbulent wall fire by using a Laser Doppler Velocimeter and a fine (25 p~m) thermocouple located 1 mm upstream of the focus of the laser beam (see paper by J.M. Most, B. Sztal and M.A. Delichatsios, "Turbulent Wall Fires--LDV and Temperature Measurements and Implications", from the book "Laser Anemometry in Fluid Mechanics" published by Ladoan--Instituto Superior Technico, 1095 Lisboa Codex--Portugal, 1984). Note that the gas temperatures were corrected for radiation errors. (See cited paper). We calculated the average density by assunfing equal molecular weights everywhere and using the perfect gas law. Then the stream function for the mean velocity can be calculated from: where ~' is the distance from the wall.
*=fl The assumption of a constant molecular weight gas mixture is violated only close to the wall (a few millimeters), especially for fuels heavier or lighter than ethane.
@
M. Delichatsios, Factory Mutual Research (USA). You had an excellent correlation between your theory and your data. Could you comment on how many adjustable parameters you employ in your model? Specifically, how do you specify the mean peak temperature ATe? Author's Reply. One may find out from References 3 and 4 of the present paper that we used five constant parameters to implement the integral model, including the new wall laws which are applicable for natural convective turbulent flows. Three of these parameters (pertinent to the flow) were determined by using detailed data in natural convective flows without combustion, while the other two (pertinent to combustion) were determined from results in wall fires having negligible flame radiation. One may observe that these parameters are reflected in the proportionality constants of the simple wall fire equations presented in this paper, namely eqs. 5, 8, and 10. Regarding the mean peak temperature, we point out that equations (lla), (llb) and (12) provide a recipe for evaluating its value. Justification for the proposed recipe is presented in the main text and corroborated by the results in Figures 4b and 5, which show that the peak mean temperature is insensitive to height and B-number.