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A numerical model for the prediction of radiation-controlled turbulent wall fires
A NUMERICAL MODEL RADIATION-CONTROLLED
FOR THE PREDICTION TURBULENT WALL
OF FIRES
FRANCESCO TAMANINI Factory Mutual Research Corporation, Factory Mutual System, Norwood, Massachusetts
A modified version of the k-~-g model of turbulence is used to calculate the rate of burning of large-scale vertical walls. The standard k-~-g procedure is generalized by the introduction of algebraic formulas for the stresses and the mass/energy turbulent fluxes, and by the use of wall correction factors. Two models for predicting flame radiation are verified against experiment: 1) the "'constant-• approach, according to which the radiated power is a constant fraction of the energy liberated per unit time by chemical reaction; and 2) the "soot-band" concept, which postulates that radiation is emitted by a thin, constant-temperature layer of particles at the flame fronts. The predictions of pyrolysis rate and flame radiance, when compared with available data on PMMA wall fires, show the latter model to be more realistic. Calculated values for amount of unburned fuel, entrainment coefficient, contribution by radiation to total wall flux and fuel mass fraction at the wall are also reported as a function of height along the wall.
Introduction The rate of upward flame spread over a vertical combustible surface is an important parameter in the ranking of the fire hazard offered by different materials. Typically, when the flame height reaches about .2 m, the flow becomes turbulent and radiative heat transfer starts to play an important role in the overall energy budget. Since the time scales associated with transients in the solid are much greater than those of the gas phase, the process of fire spread can be treated as a succession of quasi-steady burning configurations. This consideration motivates our interest in the steady-state burning problem discussed in the paper. Models dealing with the laminar burning of a vertical surface in natural convection have already been presented [1,2]. Little attention has been given so far to the turbulent case and, even then, only in the absence of radiation [3]. The turbulence model used in this paper consists essentially of the modified version of the k-~-g technique which we adopted for computing buoyancy-controlled fuel-jet flames [4]. The addition of the wall correction terms, proposed by Gibson and Launder [5], is the only change to the turbulence model of Ref. [4]. With regard to the modeling of flame radiation, the paper considers two scenarios: 1) the radiated power is proportional to the local rate of energy release due
to chemical reaction; and 2) radiation is emitted by an infinitesimal layer of soot particles "near" the wrinkled laminar flame sheets in the turbulent flame brush.
Definition of the Problem The physical situation addressed by this work is schematically shown in Fig. 1. A vertical combustible wall pyrolyzes due to the convective (4~) and radiative (4',') energy flux received from the high temperature gases in the flame. The wall surface temperature (T w) and the mass fraction of combustibles in the pyrolysis gases (Ysw) are both prescribed and assumed to be constant. ~ It should be noted that the mass fraction of combustibles at the surface (Yz,) is not equal to (actually less than) Ys~ and, in general, varies along the wall. Oxidizer is entrained by the flame from a quiescent environment
*A physically more realistic model of pyrolysis could allow for a variable surface temperature, related, for example, to the pyrolysis rate through an Arrhenius expression. In the case considered in the paper, the neglect of surface temperature variations has a negligible effect on the results and is dictated purely by considerations of simplicity.
1075
1076
COLLOQUIUM ON FIRE AND EXPLOSION T: mean absolute temperature, and Qr: energy of combustion per unit mass of fuel.
xI
, / T u r b u l e n t Flame Brush ,
UI
YOX=YOX,r =0 Yf =Yfw T=T w
L..J~,,Q~
"' t/
x=
Yf =Yf,=o = 0 T =T~176 = 298'
Fro. 1. Schematic representation of wall burning problem.
of ambient air (Y..... = .232; Yf.~ = 0). At steady state, the rate of pyrolysis (m") is given by the following energy balance: rh"Qa + q:, = q" + q'~
(1)
where Q ~is the constant effective heat of gasification per unit mass of pyrolysis products (heat of vaporization plus heat required to bring solid from room to vaporization temperature) and q','r = ~r(T ~ - T~) is the radiative loss from the surface. This formula for q~r implicitly assumes an opaque solid with unit surface emissivity. The mean motion of the flame gases as well as the turbulent fluctuations are driven by the action of gravity.
Theoretical Model
Mean Flow Equations
Conservation of vertical momentum, species and energy is satisfied by writing the boundary layer form of the differential transport equations for the following time-averaged quantities: 1) longitudinal velocity, U , ; 2) composition parameter, F = Y,,x sYs; and 3) total enthalpy, O = c T + QeYs. The quantities appearing in the last two variables are: Yox and Ys: mean oxygen and fuel mass fractions, s: oxygen / fuel stoichiometric mass ratio, cp: specific heat,
Even though we neglect it in this model, incomple. teness of combustion could be taken into account by modifying appropriately the values of s and Q,, Conservation of total mass is achieved by introduc. ing a stream function; the system of parabolic equations is solved using the Patankar-Spalding procedure [6]. The details of the equations are given in Ref. [4]; here, we will only discuss an issue which has so far received little attention. The choice of the Schvab-Zeldovich variables F and O is advantageous when: 1) all molecular species can be assumed to have the same diffusion coefficient; 2) combination of fuel and oxidizer to yield products of combustion takes place in constant stoichiometric proportions; 3) mass and thermal diffusivity are equal (Lewis number equal to 1); and 4) a constant amount of energy is released by chemical reaction of unit mass of fuel. Assumptions 1) and 2) are sufficient to make F a conserved quantity; O is conserved if 3) and 4) are verified and if there is no emission/absorption of radiation by the gas. For situations where F and O are both conserved quantities, these two variables obey the same differential equation. If the boundary conditions are also the same, it follows that F and O are linearly related through a relationship which involves the boundary values for the two variables. In that case, only one differential equation (for F or O) needs to be solved. This simple result can be used, for example, when dealing with a fuel jet either unconfined [4] or along an adiabatic wall [3], and in the case of the pure convective burning of a wall t 1, 21. In the general wall fire problem considered here, where radiation cannot be neglected, the proportionality between F and O breaks down, unless one can assume that: 1) the net surface radiative flux is proportional to the convective heat flux; and 2) the flame radiative heat loss is a constant fraction of the energy liberated by chemical reaction. We have made the second assumption in one of the two radiation models used in this work. The first assumption is almost never true in practice: as a result, the solution of separate transport equations for F and O is necessary when modeling radiationcontrolled fires. Turbulence
In our study of the fuel-jet problem [4], we made use of a modified version of the k-~-g model, originally developed at Imperial College [7]. The modification was brought about by the adoption of a suggestion for algebraic stress modeling (ASM) put forth by Rodi [8] and later extended to the compo-
RADIATION-CONTROLLED TURBULENT WALL FIRES nents of the turbulent mass/heat flux by Gibson and Launder [9]. In order to generalize the procedure to the case of flows along solid boundaries, wall correction terms (also proposed by Gibson and Launder [5]) have been added to the model. The formulas actually used are reported in the Appendix. In those expressions, the Reynolds stresses (u,%) and the components of the turbulent mass flux (~ u~) are given as functions of the turbulence kinetic energy (k), its dissipation (e), the mean square fluctuations (g = ~/2) of the composition parameter F, and the spatial gradients of mean quantities. Values for the three variables k, e and g are calculated by solving their respective transport equations. The values of the constants appearing in the model equations for k, e and g are the same as those used in Ref. [4] and are listed in the Appendix. For simplicity, we have made our calculations of wall fires by neglecting the secondary contributions to the production of k and g, which were included in the fuel-jet model. However, we are currently investigating the importance of these terms in the case of buoyant flows along solid boundaries.
1077
due to emission of radiation is easily taken into account by modifying the definition of the total enthalpy to read: 0 = cpT + (l - •
(a)
Yj
Since we neglect absorption of radiation within the gas, O, as defined in eq (3), is a conserved quantity. By further assuming that the flame is optically thin, the heat flux to the surface is given by:
((;=7
q*mx2 2l x Q f
Iim'Tdx2
2. Soot-Band Emission Model The only emission of radiation is assumed to originate from a thin layer of soot located on the fuel side of the flame sheet, as indicated in Fig. 2. The heat loss due to radiation from unit volume of soot can be written as: #~; = 4'~klR
Radiation In keeping with the spirit of a boundary layer treatment, the radiative heat flux to the surface is calculated assuming that at a given height the wall "sees" a slab of gas with uniform properties parallel to the wall. The variation in the direction perpendicular to the wall is given by the profile of temperature, species, etc., calculated at the height in question. Two radiation models have been tested in the study: their applicability is restricted to situations involving luminous turbulent diffusion flames, where radiation is mostly emitted by soot concentrated in narrow regions near the flame front. In both cases we have neglected absorption of radiation by the gas which, in wall fires, becomes important only at very large scales (i.e., "~3 m).
1. Constant-x Model
(t Z = - x Q e m 7
(2)
The subtraction of energy from the convective field
(5)
where r I~ =
(6)
is the radiance emitted by a blackbody at the soot temperature, T,,, and k is the soot absorption coeffi-
Yfs 8---
/
r
u
The radiative power generated by unit volume of the flame is assumed to be a constant fraction, • of the chemical energy liberated per unit time by combustion. This assumption is inspired by the observation, which Markstein [10] has verified in the case of propane, that for buoyancy-controlled, luminous diffusion flames from a given fuel the total radiated power is a constant fraction of the total chemical energy liberated in the reaction. The radiated power is calculated as:
(4)
I I
Xw
X,~'Yo~, | "sYfw
~-sY~._~
7
"
it 0
Yox,=o
Fro. 2. Instantaneous compositions and temperature as a function of the composition parameter, % for two values of the fuel mass fraction at the surface. Also shown is the location of the soot band.
1078
COLLOQUIUM ON FIRE AND EXPLOSION
cient ((r is the Stefan-Boltzmann constant). In order to obtain the average volumetric heat loss, 4"~' must be multiplied by the probability that soot be actually present; this can be written as:
4':'~ = 4,";n~ ( r , , ) a L ,
(v)
where the probability density function, I-1, is defined in such way that I I (F)dF is the fraction of total time during which F lies between F and F + dF. In eq (7), F,, is the location of the soot band and AFt, is its width in terms of the composition parameter F. From eqs (5), (6) and (7) one can write:
AF~t
4Y,. = 4XLzT~.II ( F ~ , ) -
(8)
L
where L is the physical thickness of the soot layer. An estimate of the ratio AF~t/L can be obtained from the dissipation, %, of the mean square fluctuations of the composition parameter [11] :
= \ 2~, "/
(9)
(n is the direction perpendicular to the flame front and 5 is the Prandtl-Schmidt number). Since II ~ (Fs,) can be calculated from an assumed shape for the probability de__nsityfunction and from the values of F and g = ~t2 [4], and % is obtained from a model expression (see Appendix), eqs (8) and (9) allow one to evaluate 4~'. Again, postulating optically thin flames, the radiant heat flux to the surface is given by (see first part of eq (4)): 4" = 2 k L ~ T 2 , N,,
(10)
where N,,=
II ( r , )
Jo
er
\2~
dx 2 (11)
/
is the average number of flame sheets at the height in question. It is convenient to introduce the emissivity of the flame sheet, defined as: ~s, = 2 h L
(12)
With this radiation model, the total enthalpy is again defined as O = cpT + QsYsand emission of radiation is taken into account by introducing - q',, (see eqs (8) and (9)) as a source term in the O - equation. In the calculations carried out for this work we
assumed Ts, = 1400~ as indicated by Markstein's results for PMMA [12] : with regard to ~,,, we selected values that gave reasonable agreement between numerical predictions and experiment." While we are aware that the soot is located on the rich side of the flame front, we have set F,, = 0 for simplicity. This further assumption is not expected to significantly affect the results.
Combustion
The combustion model is based on the assumption of infinite chemical kinetics. The mean fuel mass fraction Ye is calculated from the mean value of the composition parameter F, th__emagnitude of its mean square fluctuations, g = -/~, and the probability density function 11~(~/), which we assume to have a cusp shape given by two parabolae (see Fig. 2). The volumetric rate of burning is then computed by solving the fuel conservation equation for its source term. The details of the procedure have already been given in Ref. [4]. In the simulation of wall fires with radiation, a problem arises because the fuel concentration at the wall surface (Yf~)varies with distance along the wall: more specifically, Yf, increases as the contribution by flame radiation to the total heat transfer to the wall increases, approaching Ys~ as the ratio of radiation to convection becomes large. The effect of a variation in Ys, on the relationship between the values of instantaneous temperature and composition parameter is shown in Fig. 2, for Yf, = .5 and
Ys~ = .8. The variation of Yss along the surface leads to the result that fuel eddies no longer have a unique composition, since different eddies originate from different locations on the wall. A rigorous solution to the problem would require a procedure which, by keeping track of the decay of individual fuel eddies, provides the distribution of compositions among fuel eddies at any point in the flow from the time of birth to final dissipation. In the calculations which used the constant-• model of radiation (Model 1), we have postulated the existence of an equivalent fuel eddy, characteristic of a given point in the flow. The composition of such an eddy, Ys~,, is obtained by assuming that the instantaneous values of ~ and 0 are linearly related so that:
"Note that the neglect of absorption of radiation by the flame gases implies that the soot-band model contains only one parameter: the radiant flux emitted by one soot layer, %t(~T4t. T h e effect of a change in the value of T~t can be exactly offset by an appropriate change in %t.
RADIATION-CONTROLLED TURBULENT WALL FIRES eo,-e.
~-e|
r.,-r|
~-r.
e-e~ = - -
(13)
r-r.
where F and 0 are the mean values of the ShvabZeldovich variables obtained from the solution of their respective differential equations, F| and O| are values in the ambient, F and O,, pertain to the "equivalent" fuel eddy. Since Yf.,, appears only in F,, and 0~,, eq (13) can be used to calculate the value of Ys.~, at every point in the flow. Once the assumption is made of a linear relationship between the instantaneous quantities -~ and 0, the actual value of Yf.-, (and, therefore, F,,) is of relevance only in that part of the flow where there is intermittency of fuel, typically very near the wall. In practice, the "equivalent" fuel eddy cannot be defined in the manner indicated above if radiation is modeled by using the soot-band concept (Model 2). In this case we simply assume that the fuel eddies at a given longitudinal position in the boundary layer have the wall composition. The instantaneous relation between temperature and composition parameter is still assumed to be linear, with the flame temperature now chosen so that it is compatible with the mean temperature calculated from O as T = (O - Q i r j ) / c , , . Wall Functions
The transfer of momentum, mass and energy at the wall is computed using the wall function approach. In this scheme, the first grid point away from the wall is chosen in the fully turbulent region and the convective transfer to the wall is calculated through a Couette flow analysis of the region between that point and the wall. To compute the shear stress, %, and the diffusive flux, J~, of the composition parameter (or total enthalpy) at the wall, we have adopted the following recommendations by Patankar and Spalding [6]: %/(pU~)c = [K/In (E(p U, x~/p.)c 9
~-
2
~/(pu,)o}]
2
(14)
- j , / [ ( r o - r,)(ou,)ol = [ % / ( o U , ) c ] / [8.(1 + P~ X/%/(pU~)c)]
(15)
where the subscripts c and s refer to the first grid point away from the surface and the surface itself respectively. The constants K and E take the values .4 and 9 while the function P ~ is given by: P *= 9
- 1
(16)
1079
the mean shear "~instead of "re, where § is the average shear stress in the layer of fluid between the wall and the first grid point away from the wall. With regard to the three variables k, ~, and g, we follow the usual practice of neglecting diffusion to the wall. Furthermore, k and g near the wall are computed through the regular balance among convection, diffusion, production and dissipation, while for the dissipation of turbulence kinetic energy the value near the wall is set equal to:
a(ka/~ I_
(17)
where a is a constant equal to .12. These prescriptions have undergone so far very limited testing for the case of buoyant flows with mass addition. It is likely that future research will improve this aspect of the model. Solution Procedure
The model requires the solution of six partial differential equations for the variables U~, F, O, k, ~ and g. Arbitrary linear profiles of U~, F and O are assigned at the bottom of the wall to initiate the calculation. The laminar boundary layer is allowed to develop up to the location (x,) where the turbulent calculation is started by introducing a distribution of small values for k and g, and initializing ~ so that the correction factor F(x~) (see eq (A-12) in the Appendix) equals 1 at every grid node. The level of turbulence is then allowed to grow as dictated by the equations. The marching nature of the computational procedure makes it possible to satisfy the surface energy balance expressed by eq (1) only approximatelY. Before the execution of the forward step a guess is made of the magnitude of the pyrolysis rate at the wall, th". The guessed value of rh" is calculated from the radiative (q',') and convective (ti")heat flux components at the previous step, corrected for the energy excess or defect accumulated up to that point in the overall surface energy budget updated after execution of each forward step. This scheme leads to a very stable calculation and conserves energy. The calculations in the following discussion have been carried out using 22 grid points across the boundary layer. Typically 550 steps and 4.5 minutes of CPU time on 1BM 370/158 are necessary to compute the flow over a 3.6-m high wall, where the turbulent calculation is started at .04 m from the bottom of the wall. Wall Fire Prediction for PMMA
(8, is the turbulent Prandtl-Schmidt number). Since the shear stress is not constant near the wall, we have introduced in the right hand side of eq (14)
The technique was tested for the experimental situation of the PMMA wall fire described by Orloff
1080
C O L L O Q U I U M ON F I R E AND EXPLOSION
et al [13]. In particular, we were interested in the performance of the two radiation models described earlier. The calculations were carried out with the parameters listed in Table I. Some of the numerical results are discussed in the following sections.
.030
Model I
.025
~
2
......
,020
Pyrolysis Rate
/. -E
-
Calculated values for the wall pyrolysis rate as a function of height are shown in Fig. 3. Each of the two radiation models was run for two values of the appropriate free parameter: Model 1 for X = .25 and .28 and Model 2 for %, = .027 and .030. It is apparent that the curves obtained with the soot-band model of radiation (Model 2) more closely follow the trend shown by the data (note that comparison should be made with the solid circles). At this point we cannot rule out the possibility that the poor performance of the constant-x approach (Model 1) is due to inaccuracies in the fluid dynamic part of the model, which affect the evaluation of the burning rate distribution in the gas phase. Until this issue is resolved, all we can say is that Model 2 overpredicts the pyrolysis rate in the lower parts of the wall and underpredicts it near the top. The curve corresponding to % = .030 is very close to the points calculated by Orloff et al [13] in the extrapolation of their data to the case of an infinitely wide wall. Since the model neglects self-absorption of radiation by the gas, caution should be exercised before using this result to determine values of the soot absorption coefficient, k, and the width of the soot band, L. Furthermore, while it is likely that the order of magnitude of %t can be guessed correctly from Fig. 3, the accuracy of the inferred value may be affected by the possible existence of an additional term in eq (9), where we used the approximate equal sign. The early part of the turbulent calculation suffers the same problems already encountered in our fueljet study [4]: as was done there, we have again used the turbulence model in the region of transition from laminar to turbulent flow, even though this stretches the procedure beyond its limits of applica-
TABLE I Thermodynamic properties of PMMA Heat of Reaction Heat of Pyrolysis Oxygen / Fuel Stoichiometric Ratio Specific Heat Fuel Mass Fraction in Pyrolysis Gases Pyrolysis Temperature Fuel Molecular Weight
Qf
= 2.53 10 v J / k g Q,, = 1.61 106 J / k g
s = 1.92 c v = 1300 J / k g / ~ Ys~=I T~ = 636 ~ Mj = I00
,lEst =DSO i~.,~-
j' Colculated
~/ _
_
-
of
-Q
-
" 9~ ~ . . . .
.015
o {It: .,,,
,-I"I'""
o
~..olo
E..e.~.,o, do,o [,~
"
o 9
.005
00o.0
o'.s
L
,.o
h
,.5
~io
-q-m wide woll (o-wide woll (estimoted
i
2~
io
i
35
,.o
Heighl, x~(m)
FIG. 3. Pyrolysis rate as a function of height calculated using two radiation models. Also shown are the measurements by Orloff et al [13] for a .9-m wide wall (9 and the expected rate of pyrolysis of an infinitely wide wall (0).
bility. Additional uncertainties in the model predictions at low heights can be attributed to the use of the viscosity of propane to represent that (unknown) of the pyrolysis gases. These questions, along with the already mentioned need for further testing of the prescription for wall functions (eqs (14)-(17)), do not justify extended attempts to improve agreement between experiment and the predictions obtained from the present version of the model. However, the results in Fig. 3 do indicate that the soot-band model is not only realistic, but also accurate in reproducing the trend of the data.
Radiance The positive impression of the performance of Model 2, based on the results in Fig. 3, is confirmed by the comparison between calculated radiant intensity and experimental values. The radiance received by a ray radiometer at 60 ~ from the perpendicular to the surface is given by: I = 4';/~
(18)
for both the optically thin and thick limits. Values calculated from eq (18) using Model 1 with • = .28 and Model 2 with ~, = .030 are shown in Fig. 4 with the measurements of Orloff et al [13], The superiority of Model 2 is apparent in the success with which it reproduces the trend of the data. The initial horizontal stretch of the calculated line is due to the presence of a cutoff in the model, which prevents N ~, (the average number of flame sheets) from being less than 1 when the flow is
RADIATION-CONTROLLED T U R B U L E N T WALL FIRES
9[
.-/'"
, i
o
,I
..--
9 od,ot,on
9
Model
6~[ :.,," 9 / "/
r i
co~ulotedl"-t.oleuloleol__
l i ~ 3
The curves for Yts and q~/(tl'c + (7~)simply reiterate the statement made earlier, namely that Yss tends to approach YI-, (in our case 1) as radiation becomes the dominant mode in the transfer of energy from the flame to the surface. The fraction of unburned fuel reaches a minimum of about .42 at a height of about 1.5 m, then increases slightly. For a large part of the flow, roughly half of the fuel pyrolyzed up to a given height is still unreacted. The entrainment coefficient E , is defined as:
.~'~
15t
Experimentol 9
x:.2.e^ 2~ ~st--O30 Ref [15]
.
m' Eo -
F
%
o'.~
,b
,.~ ;o ~ Height, Xi (m)
~'.o
3'.~
1081
,.o
Fio. 4. Radiance at 60 ~ from the normal to the surface obtained from the two radiation models and measured by Orloff et al [13].
laminar or the level of turbulent fluctuations is not sufficiently developed.
dm'
- -
p=W', dxl
(19)
where m ' is the mass flow and W'1 is the vertical momentum per unit width of the fire plume. Entrainment increases as turbulence develops, first rapidly then more slowly. The average value of .1 taken by E o for heights between 2. and 4. m is very close to the value of .095 measured by Grella and Faeth [14] for a thermal plume along a vertical adiabatic wall. Given the difference between the two flows in question, the full implications of the agreement are hard to assess and will require further study.
Other Quantities Figure 5 presents the results of the predictions obtained with Model 2 (%, = .030) for four quantities: 1) fuel mass fraction at the surface, Yf~; 2) contribution by radiation to the total heat flux, tl~/(tT~ + 0'~); 3) fraction of the fuel gasified up to a given height unreacted at that height, f , ; and 4) entrainment coefficient, E , . ion
.Tp
Yfs / /
Model with ~ , = .030
fu
15
.05" OL
O
'
I
.5
I0
I
I
I
1.5 2D 2.5 Height, x I (m)
J
~
3.0
3.5
I0 4.0
FIG. 5. Surface fuel mass fraction (Y~), radiative contribution to total flame heat flux to the surface ((~"/(tT~ + 0',')), unburnt fuel fraction (f,) and entrainment coefficient (Eo) as a function of height. Values predicted using Model 2 with %t = .030.
Discussion and Conclusions Calculations of a radiation-controlled wall fire have been used in the paper to rate the performance of two models for the radiation emitted by luminous flames. Even though some question still remains as to the effect of numerical anomalies, it appears that better agreement with experiment can be obtained with the soot-band model rather than with the constant-• model of radiation. The fact that absorption of radiation by the gas was neglected should not alter that conclusion. Work is in progress to take gas absorption into account. Furthermore we are studying a modification where the temperature of the soot is allowed to vary following the flame temperature. Other parts of the modeling procedure still need to be improved: one important aspect, particularly for applications with significant convective heat transfer, is that concerning the choice of appropriate wall functions. A test of these components of the numerical procedure will require comparison with well documented cases of natural convection flows over vertical and inclined walls. The inescapable conclusion, at least for models which attempt to predict large-scale fires along walls, is that an accurate treatment of the radiative processes remains the condition for achieving success. The non-linear nature of radiation, combined with the intrinsic time-dependence of turbulence, makes it difficult to envision models which deal with gas
1082
C O L L O Q U I U M ON F I R E AND EXPLOSION
absorption/emission other than through integration of some mean species and temperature profiles. A more serious limitation, however, lies in the empiricism with which we have to conceal our ignorance of soot formation processes. This latter question probably represents the critical obstacle now facing fire researchers. Nomenclature
a a cv E E~ f. g
constant in eq (17), acceleration of gravity, m / s 2 specific heat, J / k g / ~ constant in eq (14), entrainment coefficient, unburnt fuel fraction, = ~ 2, mean square fluctuations of composition parameter, I radiant intensity, W/mU/sr J~ diffusive flux of F, k g / m 2 / s k turbulence kinetic energy, m2/s 2 K yon Karman constant (eq (17)), L thickness of soot band, m rh' flame mass flow per unit width, k g / m / s m" wall pyrolysis rate, k g / m ~ / s rh~' volumetric fuel consumption rate, k g / m a / s MI fuel molecular weight, k g / k g mole Na average number of flame sheets, P* function defined in eq (16), q'~', q'; flame convective and radiative heat flux, W/m ~ q~ surface reradiation, W / m ~ q" volumetric emission of radiated power, W/m ~ heat of combustion, J / k g Qy effective heat of gasification, J / k g Q~ $ oxygen/fuel stoichiometric ratio, mean and fluctuating temperature, ~ T,t US~US mean and fluctuating velocity components, m/s xs coordinates, m W'~ flame vertical momentum, per unit width kg/s ~ Ys, ~/, mean and fluctuating specie mass fraction, F, ~/
AFt,
~a O, 0 h Ix 1-1 p
mean and fluctuating composition parameter, Schmidt-Prandtl number, width of soot band, dissipation of turbulence kinetic energy, m2/s a emissivity of soot band, mean and fluctuating total enthalpy, J / k g soot absorption coefficient, mviscosity, k g / m / s probability density function, density, k g / m ~
Stefan-Boltzmann constant (= 5.67'10-8), W/m2/OK" shear stress, k g / m / s 2 radiant fraction, -
~r "r •
Subscripts B c eq f ox
blackbody first grid point away from the wall equivalent fuel oxygen
$
surface
st w o0 1,2,3 9r
soot wall ambient conditions coordinate directions (see Fig 1) turbulent
Superscripts instantaneous value average value
Appendix Algebraic Stress Modeling (ASM) Expressions for U~U~ and "yu s
The underlying assumption of the ASM approach as introduced in Refs [8, 9] is expressed by the following two statements on the balance between convection (A) and diffusion (A) of the Reynolds stress u su~ and turbulent flux component ~/us :
(A - A) k (A - A)~-7~~= u!uj k (A - A)v'a7= - ~tus (A - a ) , + ~tus (A - A)k
2g
--if-
(A-l)
(A-2)
Introduction of the above ass___umptions in the model equations for usu~ and "tu, proposed by Gibson and Launder [5] and algebraic manipulation leads to: U~Uj
= [ ( I - ca)P . + (1 - c3)G,, + 8 . Z ~ k
+ W,~]/(~XD
(A-3)
where: 2 Z ~ = - - [r 3
- 1) + c2P + c3G]
(A-4)
RADIATION-CONTROLLED TURBULENT WALL FIRES
3 ( u~u~ W~=2c
f e
1 (P+G_ X~ = ~ - - - - 1 )
czp~ z
k
1083
2c~2\
% + c~
- caG,~)F(x~)
21
W,, = ---
(
W ~ = W ~ = c~ e - k
2 P ) - ca(G~2
3 1
P =2E OU1 uz--; c3xz
P,2=-
G
G)
2
P~z=P3~=O; P+G +c,-1;
-
(A-21)
The three variables k, ~ and g are computed from a model equation of the type: C,,
,
(A-7)
- -
p
o+
o+
o
o3Xl
0X~
aX~
(A-8) 9
Ix +~3c,
+1~ (P, + G,
l,r
a~,p' u~
(A-9)
Gz~=Ga3=O
(A-IO)
~p' p'u,=--~---~tu,
(A-11)
- %)
(A-12)
Gk=G;
~k=~
% = c~e~/k
E
+ (1 - c3~)G. + W . ] / X , ~
(A-13)
where:
c , ~ = c , a = 1.44;
c~2=cg2=1.82
c~= 1.8; c z = . 6 ;
W~ = - c f [
(A-24)
and P,, G~, and % have already been given. The values assigned to the different constants are as follows:
p ', ~ + ( 1 - c 2 ~ ) P .
- OF u , u 2 -OqX2 - . ~ P'2~ =
OU, Pl~ = - ~u~ ~ . dxz '
(A-23)
E
and
P'l'v = -
(A-22)
where the dummy variable ~b indicates either k, or g. The production/dissipation terms are given by: Pk=P:
a k a/2 F (x2) = - - ~ K ex~
k "tu, = - [
(A-20)
F(x~) (A-6)
2 =
G, = o
c~ k g
1 G t 2 = - _-7%p'uz p GI~
m OF Pg = - 2 "/u2 - - ' bx 2 '
(A-19)
- c~(P=
=L Z P,,;
- OU~ P , = - 2 u~uz - - " Ox~ '
X~=
;-
(A-5)
--
- - OF ~ (A-14) U2 - Ox~
c1~=3.5;
%=.5
c~ =c3~=.5
c~=.5;
a=.12;
ck=.21;
c~=.15;
K=.4 cg=.30
P~ = 0
(A-15)
1-G,~ = - _---~/pt a g; G~ = 0 P
(A-16)
Acknowledgments
Wl~ = 0
(A-17)
The author is indebted to John de Ris for his support and encouragement, to Helen G. Hurley and Louis A. Post for contributing their editorial skills to the preparation of the manuscript. The work described in the paper was internally sponsored: the support of Factory Mutual Research Corporation is gratefully acknowledged.
k.tu~-c~p~ 3 - c ~ Gz~ / F(x2) /
(A-IS)
8~,k = 8 ~ , ~ = l . ;
St.~ = . 7
1084
C O L L O Q U I U M ON F I R E AND EXPLOSION REFERENCES
1. KosooN, F. J., WlLLi^ras, F. A. AND BUM^N, C., Twelfth Symposium (International) on Combustion, p. 253, The Combustion Institute, 1969. 2. KIM, J. S., DE Rls, J. AND KROESSER, F. W., Thirteenth Symposium (International) on Combustion, p. 949, The Combustion Institute, 1971. 3. KENNEt)Y, L. A., p. 683 in "Heat Transfer and Turbulent Buoyant Convection," Vol II, Spalding, D. B. and Afgan, N., editors, Hemisphere, 1977. 4. TaraXNIN1,F., Combustion and Flame, 30, 85-101 (1977). 5. GIBSON, M. M. AND LAUNDER, B. E., J. Fluid Mech., 86, 3, 491-511 (1978). 6. P^TArqKAR,S. V. ^ND SPALDING,D. B., "Heat and Mass Transfer in Boundary Layers," Intertext Books, 1970. 7. L^tJrqDEr~,B. E. ANDSPALDIrq~;,D. B., "'Mathemat-
8. 9. 10.
11.
12. 13.
14.
ical Models of Turbulence," Academic Press, 1972. ROD1,W., Z. Angew. Math. Mech., 56, 219-221 (1976). Gmsorq, M. M. AND LAUNDER, B. E., J. Heat Transfer, 98C, 81 (1976). MASKS'rEIN, G. H., Sixteenth Symposium (International) on Combustion, p. 1407, The Combustion Institute, 1977. Bll~CEr~, R. W., "Turbulent Jet Diffusion Flames," in Progress in Energy and Combustion Science, Vol 1, p. 87, Pergamon, 1976. MARKSTEIN,G. H., This Symposium. ORLOFF, L., MODA~:, A. T. ANt) ALPERT, R. L., Sixteenth Symposium (International) on Combustion, p. 1345, The Combustion Institute, 1977. GRELLA,J. J. aND FAETH, G. M., J. Fluid Mech., 71, 4, 701-710 (1975).
COMMENTS 1. H. Kent, University of Sydney, Australia. The difference between your two radiation models appears to be in the way you formulate the l:eaction rate for the flame. Since the time average reaction rate at any point can only have one value, doesn't this point to an inconsistency between the methods? Author's Reply. The reaction rate is computed by differentiating the mean fuel fraction profiles in both cases: in this respect the two models are identical. Where they differ is in the procedure used to calculate the power radiated from the combustion region. Model 1 considers it to be proportional to the computed reaction rate. Model 2 assumes that each flame sheet in the turbulent brush produces a constant, pre-assigned radiant flux. If the burning rate per unit surface of the flame sheet were constant throughout the flame, the two models would be identical. In general this is not true: the calculations for the wall fire discussed in the paper show that the burning rate per unit flame surface decreases with height. The implication of this finding is that the characteristic length for diffusion of fuel toward oxygen increases with height, which is qualitatively in accord with the increase in size of the turbulence eddies.
1. G. Quintiere, National Bureau of Standards, USA. Both of your radiation models appear to d e p e n d on the emissive power and a radiation length
scale which is less than the fluid dynamic boundary layer thickness. Your turbulent model should predict the boundary layer thickness very well, and your emissive power is based on an empirically suitable method in Model 1 and an experimentally derived soot temperature in Model 2. Since your wall fire experiment was dominated by radiative effects, and agreement with theory is good, do the above comments suggest a less sophisticated model for the convective aspects would suffice? However, a field model for temperature would still be required to relax the assumption of a specified soot temperature.
Author's Reply. I concur with your assessment that, for the case considered in the paper, a simpler treatment of the convective part of the problem could have been used satisfactorily. In fact our philosophical approach is that the kind of detailed numerical procedure used in the paper should be pursued with the main objective of using the results to guide the development of less sophisticated rnodels, which would be more convenient for engineering calculations. So far, our attempts in this direction have been frustrated by inability to devise a simple, and yet realistic, formulation of the turbulent combustion aspect of the problem. With regard to your comment on the choice of the value of the soot temperature, I do agree that the current formulation of the model leaves much space for improvement. There is indication L~ that sootier fuels have lower equivalent radiation temperatures. It is possible that in the not too distant
RADIATION-CONTROLLED future we will be able to u n d e r s t a n d t h e r e l a t i o n s h i p b e t w e e n t h e t e n d e n c y of different f u e l s to p r o d u c e soot in b u o y a n t d i f f u s i o n f l a m e s a n d the soot temperature. At that stage, it m a y b e c o m e p o s s i b l e to relate actual soot p r o d u c t i o n a n d soot t e m p e r a t u r e to a " s o o t i n e s s " parameter, d e t e r m i n e d for the particular fuel u n d e r c o n s i d e r a t i o n , t h r o u g h a s i m p l e m o d e l for soot p r o d u c t i o n w h i c h a c c o u n t s for flow
TURBULENT
WALL FIRES
1085
c o n d i t i o n s . U n t i l then, i m p r o v e m e n t s to f o r m u l a tions, s u c h as that d i s c u s s e d in the paper, are likely to be m o r e of a cosmetic t h a n c o n c e p t u a l nature. REFERENCES 1. DE RIS, J., T h i s S y m p o s i u m . 2. MARI(STEIN, G. H., T h i s S y m p o s i u m .
FOR THE PREDICTION TURBULENT WALL
OF FIRES
FRANCESCO TAMANINI Factory Mutual Research Corporation, Factory Mutual System, Norwood, Massachusetts
A modified version of the k-~-g model of turbulence is used to calculate the rate of burning of large-scale vertical walls. The standard k-~-g procedure is generalized by the introduction of algebraic formulas for the stresses and the mass/energy turbulent fluxes, and by the use of wall correction factors. Two models for predicting flame radiation are verified against experiment: 1) the "'constant-• approach, according to which the radiated power is a constant fraction of the energy liberated per unit time by chemical reaction; and 2) the "soot-band" concept, which postulates that radiation is emitted by a thin, constant-temperature layer of particles at the flame fronts. The predictions of pyrolysis rate and flame radiance, when compared with available data on PMMA wall fires, show the latter model to be more realistic. Calculated values for amount of unburned fuel, entrainment coefficient, contribution by radiation to total wall flux and fuel mass fraction at the wall are also reported as a function of height along the wall.
Introduction The rate of upward flame spread over a vertical combustible surface is an important parameter in the ranking of the fire hazard offered by different materials. Typically, when the flame height reaches about .2 m, the flow becomes turbulent and radiative heat transfer starts to play an important role in the overall energy budget. Since the time scales associated with transients in the solid are much greater than those of the gas phase, the process of fire spread can be treated as a succession of quasi-steady burning configurations. This consideration motivates our interest in the steady-state burning problem discussed in the paper. Models dealing with the laminar burning of a vertical surface in natural convection have already been presented [1,2]. Little attention has been given so far to the turbulent case and, even then, only in the absence of radiation [3]. The turbulence model used in this paper consists essentially of the modified version of the k-~-g technique which we adopted for computing buoyancy-controlled fuel-jet flames [4]. The addition of the wall correction terms, proposed by Gibson and Launder [5], is the only change to the turbulence model of Ref. [4]. With regard to the modeling of flame radiation, the paper considers two scenarios: 1) the radiated power is proportional to the local rate of energy release due
to chemical reaction; and 2) radiation is emitted by an infinitesimal layer of soot particles "near" the wrinkled laminar flame sheets in the turbulent flame brush.
Definition of the Problem The physical situation addressed by this work is schematically shown in Fig. 1. A vertical combustible wall pyrolyzes due to the convective (4~) and radiative (4',') energy flux received from the high temperature gases in the flame. The wall surface temperature (T w) and the mass fraction of combustibles in the pyrolysis gases (Ysw) are both prescribed and assumed to be constant. ~ It should be noted that the mass fraction of combustibles at the surface (Yz,) is not equal to (actually less than) Ys~ and, in general, varies along the wall. Oxidizer is entrained by the flame from a quiescent environment
*A physically more realistic model of pyrolysis could allow for a variable surface temperature, related, for example, to the pyrolysis rate through an Arrhenius expression. In the case considered in the paper, the neglect of surface temperature variations has a negligible effect on the results and is dictated purely by considerations of simplicity.
1075
1076
COLLOQUIUM ON FIRE AND EXPLOSION T: mean absolute temperature, and Qr: energy of combustion per unit mass of fuel.
xI
, / T u r b u l e n t Flame Brush ,
UI
YOX=YOX,r =0 Yf =Yfw T=T w
L..J~,,Q~
"' t/
x=
Yf =Yf,=o = 0 T =T~176 = 298'
Fro. 1. Schematic representation of wall burning problem.
of ambient air (Y..... = .232; Yf.~ = 0). At steady state, the rate of pyrolysis (m") is given by the following energy balance: rh"Qa + q:, = q" + q'~
(1)
where Q ~is the constant effective heat of gasification per unit mass of pyrolysis products (heat of vaporization plus heat required to bring solid from room to vaporization temperature) and q','r = ~r(T ~ - T~) is the radiative loss from the surface. This formula for q~r implicitly assumes an opaque solid with unit surface emissivity. The mean motion of the flame gases as well as the turbulent fluctuations are driven by the action of gravity.
Theoretical Model
Mean Flow Equations
Conservation of vertical momentum, species and energy is satisfied by writing the boundary layer form of the differential transport equations for the following time-averaged quantities: 1) longitudinal velocity, U , ; 2) composition parameter, F = Y,,x sYs; and 3) total enthalpy, O = c T + QeYs. The quantities appearing in the last two variables are: Yox and Ys: mean oxygen and fuel mass fractions, s: oxygen / fuel stoichiometric mass ratio, cp: specific heat,
Even though we neglect it in this model, incomple. teness of combustion could be taken into account by modifying appropriately the values of s and Q,, Conservation of total mass is achieved by introduc. ing a stream function; the system of parabolic equations is solved using the Patankar-Spalding procedure [6]. The details of the equations are given in Ref. [4]; here, we will only discuss an issue which has so far received little attention. The choice of the Schvab-Zeldovich variables F and O is advantageous when: 1) all molecular species can be assumed to have the same diffusion coefficient; 2) combination of fuel and oxidizer to yield products of combustion takes place in constant stoichiometric proportions; 3) mass and thermal diffusivity are equal (Lewis number equal to 1); and 4) a constant amount of energy is released by chemical reaction of unit mass of fuel. Assumptions 1) and 2) are sufficient to make F a conserved quantity; O is conserved if 3) and 4) are verified and if there is no emission/absorption of radiation by the gas. For situations where F and O are both conserved quantities, these two variables obey the same differential equation. If the boundary conditions are also the same, it follows that F and O are linearly related through a relationship which involves the boundary values for the two variables. In that case, only one differential equation (for F or O) needs to be solved. This simple result can be used, for example, when dealing with a fuel jet either unconfined [4] or along an adiabatic wall [3], and in the case of the pure convective burning of a wall t 1, 21. In the general wall fire problem considered here, where radiation cannot be neglected, the proportionality between F and O breaks down, unless one can assume that: 1) the net surface radiative flux is proportional to the convective heat flux; and 2) the flame radiative heat loss is a constant fraction of the energy liberated by chemical reaction. We have made the second assumption in one of the two radiation models used in this work. The first assumption is almost never true in practice: as a result, the solution of separate transport equations for F and O is necessary when modeling radiationcontrolled fires. Turbulence
In our study of the fuel-jet problem [4], we made use of a modified version of the k-~-g model, originally developed at Imperial College [7]. The modification was brought about by the adoption of a suggestion for algebraic stress modeling (ASM) put forth by Rodi [8] and later extended to the compo-
RADIATION-CONTROLLED TURBULENT WALL FIRES nents of the turbulent mass/heat flux by Gibson and Launder [9]. In order to generalize the procedure to the case of flows along solid boundaries, wall correction terms (also proposed by Gibson and Launder [5]) have been added to the model. The formulas actually used are reported in the Appendix. In those expressions, the Reynolds stresses (u,%) and the components of the turbulent mass flux (~ u~) are given as functions of the turbulence kinetic energy (k), its dissipation (e), the mean square fluctuations (g = ~/2) of the composition parameter F, and the spatial gradients of mean quantities. Values for the three variables k, e and g are calculated by solving their respective transport equations. The values of the constants appearing in the model equations for k, e and g are the same as those used in Ref. [4] and are listed in the Appendix. For simplicity, we have made our calculations of wall fires by neglecting the secondary contributions to the production of k and g, which were included in the fuel-jet model. However, we are currently investigating the importance of these terms in the case of buoyant flows along solid boundaries.
1077
due to emission of radiation is easily taken into account by modifying the definition of the total enthalpy to read: 0 = cpT + (l - •
(a)
Yj
Since we neglect absorption of radiation within the gas, O, as defined in eq (3), is a conserved quantity. By further assuming that the flame is optically thin, the heat flux to the surface is given by:
((;=7
q*mx2 2l x Q f
Iim'Tdx2
2. Soot-Band Emission Model The only emission of radiation is assumed to originate from a thin layer of soot located on the fuel side of the flame sheet, as indicated in Fig. 2. The heat loss due to radiation from unit volume of soot can be written as: #~; = 4'~klR
Radiation In keeping with the spirit of a boundary layer treatment, the radiative heat flux to the surface is calculated assuming that at a given height the wall "sees" a slab of gas with uniform properties parallel to the wall. The variation in the direction perpendicular to the wall is given by the profile of temperature, species, etc., calculated at the height in question. Two radiation models have been tested in the study: their applicability is restricted to situations involving luminous turbulent diffusion flames, where radiation is mostly emitted by soot concentrated in narrow regions near the flame front. In both cases we have neglected absorption of radiation by the gas which, in wall fires, becomes important only at very large scales (i.e., "~3 m).
1. Constant-x Model
(t Z = - x Q e m 7
(2)
The subtraction of energy from the convective field
(5)
where r I~ =
(6)
is the radiance emitted by a blackbody at the soot temperature, T,,, and k is the soot absorption coeffi-
Yfs 8---
/
r
u
The radiative power generated by unit volume of the flame is assumed to be a constant fraction, • of the chemical energy liberated per unit time by combustion. This assumption is inspired by the observation, which Markstein [10] has verified in the case of propane, that for buoyancy-controlled, luminous diffusion flames from a given fuel the total radiated power is a constant fraction of the total chemical energy liberated in the reaction. The radiated power is calculated as:
(4)
I I
Xw
X,~'Yo~, | "sYfw
~-sY~._~
7
"
it 0
Yox,=o
Fro. 2. Instantaneous compositions and temperature as a function of the composition parameter, % for two values of the fuel mass fraction at the surface. Also shown is the location of the soot band.
1078
COLLOQUIUM ON FIRE AND EXPLOSION
cient ((r is the Stefan-Boltzmann constant). In order to obtain the average volumetric heat loss, 4"~' must be multiplied by the probability that soot be actually present; this can be written as:
4':'~ = 4,";n~ ( r , , ) a L ,
(v)
where the probability density function, I-1, is defined in such way that I I (F)dF is the fraction of total time during which F lies between F and F + dF. In eq (7), F,, is the location of the soot band and AFt, is its width in terms of the composition parameter F. From eqs (5), (6) and (7) one can write:
AF~t
4Y,. = 4XLzT~.II ( F ~ , ) -
(8)
L
where L is the physical thickness of the soot layer. An estimate of the ratio AF~t/L can be obtained from the dissipation, %, of the mean square fluctuations of the composition parameter [11] :
= \ 2~, "/
(9)
(n is the direction perpendicular to the flame front and 5 is the Prandtl-Schmidt number). Since II ~ (Fs,) can be calculated from an assumed shape for the probability de__nsityfunction and from the values of F and g = ~t2 [4], and % is obtained from a model expression (see Appendix), eqs (8) and (9) allow one to evaluate 4~'. Again, postulating optically thin flames, the radiant heat flux to the surface is given by (see first part of eq (4)): 4" = 2 k L ~ T 2 , N,,
(10)
where N,,=
II ( r , )
Jo
er
\2~
dx 2 (11)
/
is the average number of flame sheets at the height in question. It is convenient to introduce the emissivity of the flame sheet, defined as: ~s, = 2 h L
(12)
With this radiation model, the total enthalpy is again defined as O = cpT + QsYsand emission of radiation is taken into account by introducing - q',, (see eqs (8) and (9)) as a source term in the O - equation. In the calculations carried out for this work we
assumed Ts, = 1400~ as indicated by Markstein's results for PMMA [12] : with regard to ~,,, we selected values that gave reasonable agreement between numerical predictions and experiment." While we are aware that the soot is located on the rich side of the flame front, we have set F,, = 0 for simplicity. This further assumption is not expected to significantly affect the results.
Combustion
The combustion model is based on the assumption of infinite chemical kinetics. The mean fuel mass fraction Ye is calculated from the mean value of the composition parameter F, th__emagnitude of its mean square fluctuations, g = -/~, and the probability density function 11~(~/), which we assume to have a cusp shape given by two parabolae (see Fig. 2). The volumetric rate of burning is then computed by solving the fuel conservation equation for its source term. The details of the procedure have already been given in Ref. [4]. In the simulation of wall fires with radiation, a problem arises because the fuel concentration at the wall surface (Yf~)varies with distance along the wall: more specifically, Yf, increases as the contribution by flame radiation to the total heat transfer to the wall increases, approaching Ys~ as the ratio of radiation to convection becomes large. The effect of a variation in Ys, on the relationship between the values of instantaneous temperature and composition parameter is shown in Fig. 2, for Yf, = .5 and
Ys~ = .8. The variation of Yss along the surface leads to the result that fuel eddies no longer have a unique composition, since different eddies originate from different locations on the wall. A rigorous solution to the problem would require a procedure which, by keeping track of the decay of individual fuel eddies, provides the distribution of compositions among fuel eddies at any point in the flow from the time of birth to final dissipation. In the calculations which used the constant-• model of radiation (Model 1), we have postulated the existence of an equivalent fuel eddy, characteristic of a given point in the flow. The composition of such an eddy, Ys~,, is obtained by assuming that the instantaneous values of ~ and 0 are linearly related so that:
"Note that the neglect of absorption of radiation by the flame gases implies that the soot-band model contains only one parameter: the radiant flux emitted by one soot layer, %t(~T4t. T h e effect of a change in the value of T~t can be exactly offset by an appropriate change in %t.
RADIATION-CONTROLLED TURBULENT WALL FIRES eo,-e.
~-e|
r.,-r|
~-r.
e-e~ = - -
(13)
r-r.
where F and 0 are the mean values of the ShvabZeldovich variables obtained from the solution of their respective differential equations, F| and O| are values in the ambient, F and O,, pertain to the "equivalent" fuel eddy. Since Yf.,, appears only in F,, and 0~,, eq (13) can be used to calculate the value of Ys.~, at every point in the flow. Once the assumption is made of a linear relationship between the instantaneous quantities -~ and 0, the actual value of Yf.-, (and, therefore, F,,) is of relevance only in that part of the flow where there is intermittency of fuel, typically very near the wall. In practice, the "equivalent" fuel eddy cannot be defined in the manner indicated above if radiation is modeled by using the soot-band concept (Model 2). In this case we simply assume that the fuel eddies at a given longitudinal position in the boundary layer have the wall composition. The instantaneous relation between temperature and composition parameter is still assumed to be linear, with the flame temperature now chosen so that it is compatible with the mean temperature calculated from O as T = (O - Q i r j ) / c , , . Wall Functions
The transfer of momentum, mass and energy at the wall is computed using the wall function approach. In this scheme, the first grid point away from the wall is chosen in the fully turbulent region and the convective transfer to the wall is calculated through a Couette flow analysis of the region between that point and the wall. To compute the shear stress, %, and the diffusive flux, J~, of the composition parameter (or total enthalpy) at the wall, we have adopted the following recommendations by Patankar and Spalding [6]: %/(pU~)c = [K/In (E(p U, x~/p.)c 9
~-
2
~/(pu,)o}]
2
(14)
- j , / [ ( r o - r,)(ou,)ol = [ % / ( o U , ) c ] / [8.(1 + P~ X/%/(pU~)c)]
(15)
where the subscripts c and s refer to the first grid point away from the surface and the surface itself respectively. The constants K and E take the values .4 and 9 while the function P ~ is given by: P *= 9
- 1
(16)
1079
the mean shear "~instead of "re, where § is the average shear stress in the layer of fluid between the wall and the first grid point away from the wall. With regard to the three variables k, ~, and g, we follow the usual practice of neglecting diffusion to the wall. Furthermore, k and g near the wall are computed through the regular balance among convection, diffusion, production and dissipation, while for the dissipation of turbulence kinetic energy the value near the wall is set equal to:
a(ka/~ I_
(17)
where a is a constant equal to .12. These prescriptions have undergone so far very limited testing for the case of buoyant flows with mass addition. It is likely that future research will improve this aspect of the model. Solution Procedure
The model requires the solution of six partial differential equations for the variables U~, F, O, k, ~ and g. Arbitrary linear profiles of U~, F and O are assigned at the bottom of the wall to initiate the calculation. The laminar boundary layer is allowed to develop up to the location (x,) where the turbulent calculation is started by introducing a distribution of small values for k and g, and initializing ~ so that the correction factor F(x~) (see eq (A-12) in the Appendix) equals 1 at every grid node. The level of turbulence is then allowed to grow as dictated by the equations. The marching nature of the computational procedure makes it possible to satisfy the surface energy balance expressed by eq (1) only approximatelY. Before the execution of the forward step a guess is made of the magnitude of the pyrolysis rate at the wall, th". The guessed value of rh" is calculated from the radiative (q',') and convective (ti")heat flux components at the previous step, corrected for the energy excess or defect accumulated up to that point in the overall surface energy budget updated after execution of each forward step. This scheme leads to a very stable calculation and conserves energy. The calculations in the following discussion have been carried out using 22 grid points across the boundary layer. Typically 550 steps and 4.5 minutes of CPU time on 1BM 370/158 are necessary to compute the flow over a 3.6-m high wall, where the turbulent calculation is started at .04 m from the bottom of the wall. Wall Fire Prediction for PMMA
(8, is the turbulent Prandtl-Schmidt number). Since the shear stress is not constant near the wall, we have introduced in the right hand side of eq (14)
The technique was tested for the experimental situation of the PMMA wall fire described by Orloff
1080
C O L L O Q U I U M ON F I R E AND EXPLOSION
et al [13]. In particular, we were interested in the performance of the two radiation models described earlier. The calculations were carried out with the parameters listed in Table I. Some of the numerical results are discussed in the following sections.
.030
Model I
.025
~
2
......
,020
Pyrolysis Rate
/. -E
-
Calculated values for the wall pyrolysis rate as a function of height are shown in Fig. 3. Each of the two radiation models was run for two values of the appropriate free parameter: Model 1 for X = .25 and .28 and Model 2 for %, = .027 and .030. It is apparent that the curves obtained with the soot-band model of radiation (Model 2) more closely follow the trend shown by the data (note that comparison should be made with the solid circles). At this point we cannot rule out the possibility that the poor performance of the constant-x approach (Model 1) is due to inaccuracies in the fluid dynamic part of the model, which affect the evaluation of the burning rate distribution in the gas phase. Until this issue is resolved, all we can say is that Model 2 overpredicts the pyrolysis rate in the lower parts of the wall and underpredicts it near the top. The curve corresponding to % = .030 is very close to the points calculated by Orloff et al [13] in the extrapolation of their data to the case of an infinitely wide wall. Since the model neglects self-absorption of radiation by the gas, caution should be exercised before using this result to determine values of the soot absorption coefficient, k, and the width of the soot band, L. Furthermore, while it is likely that the order of magnitude of %t can be guessed correctly from Fig. 3, the accuracy of the inferred value may be affected by the possible existence of an additional term in eq (9), where we used the approximate equal sign. The early part of the turbulent calculation suffers the same problems already encountered in our fueljet study [4]: as was done there, we have again used the turbulence model in the region of transition from laminar to turbulent flow, even though this stretches the procedure beyond its limits of applica-
TABLE I Thermodynamic properties of PMMA Heat of Reaction Heat of Pyrolysis Oxygen / Fuel Stoichiometric Ratio Specific Heat Fuel Mass Fraction in Pyrolysis Gases Pyrolysis Temperature Fuel Molecular Weight
Qf
= 2.53 10 v J / k g Q,, = 1.61 106 J / k g
s = 1.92 c v = 1300 J / k g / ~ Ys~=I T~ = 636 ~ Mj = I00
,lEst =DSO i~.,~-
j' Colculated
~/ _
_
-
of
-Q
-
" 9~ ~ . . . .
.015
o {It: .,,,
,-I"I'""
o
~..olo
E..e.~.,o, do,o [,~
"
o 9
.005
00o.0
o'.s
L
,.o
h
,.5
~io
-q-m wide woll (o-wide woll (estimoted
i
2~
io
i
35
,.o
Heighl, x~(m)
FIG. 3. Pyrolysis rate as a function of height calculated using two radiation models. Also shown are the measurements by Orloff et al [13] for a .9-m wide wall (9 and the expected rate of pyrolysis of an infinitely wide wall (0).
bility. Additional uncertainties in the model predictions at low heights can be attributed to the use of the viscosity of propane to represent that (unknown) of the pyrolysis gases. These questions, along with the already mentioned need for further testing of the prescription for wall functions (eqs (14)-(17)), do not justify extended attempts to improve agreement between experiment and the predictions obtained from the present version of the model. However, the results in Fig. 3 do indicate that the soot-band model is not only realistic, but also accurate in reproducing the trend of the data.
Radiance The positive impression of the performance of Model 2, based on the results in Fig. 3, is confirmed by the comparison between calculated radiant intensity and experimental values. The radiance received by a ray radiometer at 60 ~ from the perpendicular to the surface is given by: I = 4';/~
(18)
for both the optically thin and thick limits. Values calculated from eq (18) using Model 1 with • = .28 and Model 2 with ~, = .030 are shown in Fig. 4 with the measurements of Orloff et al [13], The superiority of Model 2 is apparent in the success with which it reproduces the trend of the data. The initial horizontal stretch of the calculated line is due to the presence of a cutoff in the model, which prevents N ~, (the average number of flame sheets) from being less than 1 when the flow is
RADIATION-CONTROLLED T U R B U L E N T WALL FIRES
9[
.-/'"
, i
o
,I
..--
9 od,ot,on
9
Model
6~[ :.,," 9 / "/
r i
co~ulotedl"-t.oleuloleol__
l i ~ 3
The curves for Yts and q~/(tl'c + (7~)simply reiterate the statement made earlier, namely that Yss tends to approach YI-, (in our case 1) as radiation becomes the dominant mode in the transfer of energy from the flame to the surface. The fraction of unburned fuel reaches a minimum of about .42 at a height of about 1.5 m, then increases slightly. For a large part of the flow, roughly half of the fuel pyrolyzed up to a given height is still unreacted. The entrainment coefficient E , is defined as:
.~'~
15t
Experimentol 9
x:.2.e^ 2~ ~st--O30 Ref [15]
.
m' Eo -
F
%
o'.~
,b
,.~ ;o ~ Height, Xi (m)
~'.o
3'.~
1081
,.o
Fio. 4. Radiance at 60 ~ from the normal to the surface obtained from the two radiation models and measured by Orloff et al [13].
laminar or the level of turbulent fluctuations is not sufficiently developed.
dm'
- -
p=W', dxl
(19)
where m ' is the mass flow and W'1 is the vertical momentum per unit width of the fire plume. Entrainment increases as turbulence develops, first rapidly then more slowly. The average value of .1 taken by E o for heights between 2. and 4. m is very close to the value of .095 measured by Grella and Faeth [14] for a thermal plume along a vertical adiabatic wall. Given the difference between the two flows in question, the full implications of the agreement are hard to assess and will require further study.
Other Quantities Figure 5 presents the results of the predictions obtained with Model 2 (%, = .030) for four quantities: 1) fuel mass fraction at the surface, Yf~; 2) contribution by radiation to the total heat flux, tl~/(tT~ + 0'~); 3) fraction of the fuel gasified up to a given height unreacted at that height, f , ; and 4) entrainment coefficient, E , . ion
.Tp
Yfs / /
Model with ~ , = .030
fu
15
.05" OL
O
'
I
.5
I0
I
I
I
1.5 2D 2.5 Height, x I (m)
J
~
3.0
3.5
I0 4.0
FIG. 5. Surface fuel mass fraction (Y~), radiative contribution to total flame heat flux to the surface ((~"/(tT~ + 0',')), unburnt fuel fraction (f,) and entrainment coefficient (Eo) as a function of height. Values predicted using Model 2 with %t = .030.
Discussion and Conclusions Calculations of a radiation-controlled wall fire have been used in the paper to rate the performance of two models for the radiation emitted by luminous flames. Even though some question still remains as to the effect of numerical anomalies, it appears that better agreement with experiment can be obtained with the soot-band model rather than with the constant-• model of radiation. The fact that absorption of radiation by the gas was neglected should not alter that conclusion. Work is in progress to take gas absorption into account. Furthermore we are studying a modification where the temperature of the soot is allowed to vary following the flame temperature. Other parts of the modeling procedure still need to be improved: one important aspect, particularly for applications with significant convective heat transfer, is that concerning the choice of appropriate wall functions. A test of these components of the numerical procedure will require comparison with well documented cases of natural convection flows over vertical and inclined walls. The inescapable conclusion, at least for models which attempt to predict large-scale fires along walls, is that an accurate treatment of the radiative processes remains the condition for achieving success. The non-linear nature of radiation, combined with the intrinsic time-dependence of turbulence, makes it difficult to envision models which deal with gas
1082
C O L L O Q U I U M ON F I R E AND EXPLOSION
absorption/emission other than through integration of some mean species and temperature profiles. A more serious limitation, however, lies in the empiricism with which we have to conceal our ignorance of soot formation processes. This latter question probably represents the critical obstacle now facing fire researchers. Nomenclature
a a cv E E~ f. g
constant in eq (17), acceleration of gravity, m / s 2 specific heat, J / k g / ~ constant in eq (14), entrainment coefficient, unburnt fuel fraction, = ~ 2, mean square fluctuations of composition parameter, I radiant intensity, W/mU/sr J~ diffusive flux of F, k g / m 2 / s k turbulence kinetic energy, m2/s 2 K yon Karman constant (eq (17)), L thickness of soot band, m rh' flame mass flow per unit width, k g / m / s m" wall pyrolysis rate, k g / m ~ / s rh~' volumetric fuel consumption rate, k g / m a / s MI fuel molecular weight, k g / k g mole Na average number of flame sheets, P* function defined in eq (16), q'~', q'; flame convective and radiative heat flux, W/m ~ q~ surface reradiation, W / m ~ q" volumetric emission of radiated power, W/m ~ heat of combustion, J / k g Qy effective heat of gasification, J / k g Q~ $ oxygen/fuel stoichiometric ratio, mean and fluctuating temperature, ~ T,t US~US mean and fluctuating velocity components, m/s xs coordinates, m W'~ flame vertical momentum, per unit width kg/s ~ Ys, ~/, mean and fluctuating specie mass fraction, F, ~/
AFt,
~a O, 0 h Ix 1-1 p
mean and fluctuating composition parameter, Schmidt-Prandtl number, width of soot band, dissipation of turbulence kinetic energy, m2/s a emissivity of soot band, mean and fluctuating total enthalpy, J / k g soot absorption coefficient, mviscosity, k g / m / s probability density function, density, k g / m ~
Stefan-Boltzmann constant (= 5.67'10-8), W/m2/OK" shear stress, k g / m / s 2 radiant fraction, -
~r "r •
Subscripts B c eq f ox
blackbody first grid point away from the wall equivalent fuel oxygen
$
surface
st w o0 1,2,3 9r
soot wall ambient conditions coordinate directions (see Fig 1) turbulent
Superscripts instantaneous value average value
Appendix Algebraic Stress Modeling (ASM) Expressions for U~U~ and "yu s
The underlying assumption of the ASM approach as introduced in Refs [8, 9] is expressed by the following two statements on the balance between convection (A) and diffusion (A) of the Reynolds stress u su~ and turbulent flux component ~/us :
(A - A) k (A - A)~-7~~= u!uj k (A - A)v'a7= - ~tus (A - a ) , + ~tus (A - A)k
2g
--if-
(A-l)
(A-2)
Introduction of the above ass___umptions in the model equations for usu~ and "tu, proposed by Gibson and Launder [5] and algebraic manipulation leads to: U~Uj
= [ ( I - ca)P . + (1 - c3)G,, + 8 . Z ~ k
+ W,~]/(~XD
(A-3)
where: 2 Z ~ = - - [r 3
- 1) + c2P + c3G]
(A-4)
RADIATION-CONTROLLED TURBULENT WALL FIRES
3 ( u~u~ W~=2c
f e
1 (P+G_ X~ = ~ - - - - 1 )
czp~ z
k
1083
2c~2\
% + c~
- caG,~)F(x~)
21
W,, = ---
(
W ~ = W ~ = c~ e - k
2 P ) - ca(G~2
3 1
P =2E OU1 uz--; c3xz
P,2=-
G
G)
2
P~z=P3~=O; P+G +c,-1;
-
(A-21)
The three variables k, ~ and g are computed from a model equation of the type: C,,
,
(A-7)
- -
p
o+
o+
o
o3Xl
0X~
aX~
(A-8) 9
Ix +~3c,
+1~ (P, + G,
l,r
a~,p' u~
(A-9)
Gz~=Ga3=O
(A-IO)
~p' p'u,=--~---~tu,
(A-11)
- %)
(A-12)
Gk=G;
~k=~
% = c~e~/k
E
+ (1 - c3~)G. + W . ] / X , ~
(A-13)
where:
c , ~ = c , a = 1.44;
c~2=cg2=1.82
c~= 1.8; c z = . 6 ;
W~ = - c f [
(A-24)
and P,, G~, and % have already been given. The values assigned to the different constants are as follows:
p ', ~ + ( 1 - c 2 ~ ) P .
- OF u , u 2 -OqX2 - . ~ P'2~ =
OU, Pl~ = - ~u~ ~ . dxz '
(A-23)
E
and
P'l'v = -
(A-22)
where the dummy variable ~b indicates either k, or g. The production/dissipation terms are given by: Pk=P:
a k a/2 F (x2) = - - ~ K ex~
k "tu, = - [
(A-20)
F(x~) (A-6)
2 =
G, = o
c~ k g
1 G t 2 = - _-7%p'uz p GI~
m OF Pg = - 2 "/u2 - - ' bx 2 '
(A-19)
- c~(P=
=L Z P,,;
- OU~ P , = - 2 u~uz - - " Ox~ '
X~=
;-
(A-5)
--
- - OF ~ (A-14) U2 - Ox~
c1~=3.5;
%=.5
c~ =c3~=.5
c~=.5;
a=.12;
ck=.21;
c~=.15;
K=.4 cg=.30
P~ = 0
(A-15)
1-G,~ = - _---~/pt a g; G~ = 0 P
(A-16)
Acknowledgments
Wl~ = 0
(A-17)
The author is indebted to John de Ris for his support and encouragement, to Helen G. Hurley and Louis A. Post for contributing their editorial skills to the preparation of the manuscript. The work described in the paper was internally sponsored: the support of Factory Mutual Research Corporation is gratefully acknowledged.
k.tu~-c~p~ 3 - c ~ Gz~ / F(x2) /
(A-IS)
8~,k = 8 ~ , ~ = l . ;
St.~ = . 7
1084
C O L L O Q U I U M ON F I R E AND EXPLOSION REFERENCES
1. KosooN, F. J., WlLLi^ras, F. A. AND BUM^N, C., Twelfth Symposium (International) on Combustion, p. 253, The Combustion Institute, 1969. 2. KIM, J. S., DE Rls, J. AND KROESSER, F. W., Thirteenth Symposium (International) on Combustion, p. 949, The Combustion Institute, 1971. 3. KENNEt)Y, L. A., p. 683 in "Heat Transfer and Turbulent Buoyant Convection," Vol II, Spalding, D. B. and Afgan, N., editors, Hemisphere, 1977. 4. TaraXNIN1,F., Combustion and Flame, 30, 85-101 (1977). 5. GIBSON, M. M. AND LAUNDER, B. E., J. Fluid Mech., 86, 3, 491-511 (1978). 6. P^TArqKAR,S. V. ^ND SPALDING,D. B., "Heat and Mass Transfer in Boundary Layers," Intertext Books, 1970. 7. L^tJrqDEr~,B. E. ANDSPALDIrq~;,D. B., "'Mathemat-
8. 9. 10.
11.
12. 13.
14.
ical Models of Turbulence," Academic Press, 1972. ROD1,W., Z. Angew. Math. Mech., 56, 219-221 (1976). Gmsorq, M. M. AND LAUNDER, B. E., J. Heat Transfer, 98C, 81 (1976). MASKS'rEIN, G. H., Sixteenth Symposium (International) on Combustion, p. 1407, The Combustion Institute, 1977. Bll~CEr~, R. W., "Turbulent Jet Diffusion Flames," in Progress in Energy and Combustion Science, Vol 1, p. 87, Pergamon, 1976. MARKSTEIN,G. H., This Symposium. ORLOFF, L., MODA~:, A. T. ANt) ALPERT, R. L., Sixteenth Symposium (International) on Combustion, p. 1345, The Combustion Institute, 1977. GRELLA,J. J. aND FAETH, G. M., J. Fluid Mech., 71, 4, 701-710 (1975).
COMMENTS 1. H. Kent, University of Sydney, Australia. The difference between your two radiation models appears to be in the way you formulate the l:eaction rate for the flame. Since the time average reaction rate at any point can only have one value, doesn't this point to an inconsistency between the methods? Author's Reply. The reaction rate is computed by differentiating the mean fuel fraction profiles in both cases: in this respect the two models are identical. Where they differ is in the procedure used to calculate the power radiated from the combustion region. Model 1 considers it to be proportional to the computed reaction rate. Model 2 assumes that each flame sheet in the turbulent brush produces a constant, pre-assigned radiant flux. If the burning rate per unit surface of the flame sheet were constant throughout the flame, the two models would be identical. In general this is not true: the calculations for the wall fire discussed in the paper show that the burning rate per unit flame surface decreases with height. The implication of this finding is that the characteristic length for diffusion of fuel toward oxygen increases with height, which is qualitatively in accord with the increase in size of the turbulence eddies.
1. G. Quintiere, National Bureau of Standards, USA. Both of your radiation models appear to d e p e n d on the emissive power and a radiation length
scale which is less than the fluid dynamic boundary layer thickness. Your turbulent model should predict the boundary layer thickness very well, and your emissive power is based on an empirically suitable method in Model 1 and an experimentally derived soot temperature in Model 2. Since your wall fire experiment was dominated by radiative effects, and agreement with theory is good, do the above comments suggest a less sophisticated model for the convective aspects would suffice? However, a field model for temperature would still be required to relax the assumption of a specified soot temperature.
Author's Reply. I concur with your assessment that, for the case considered in the paper, a simpler treatment of the convective part of the problem could have been used satisfactorily. In fact our philosophical approach is that the kind of detailed numerical procedure used in the paper should be pursued with the main objective of using the results to guide the development of less sophisticated rnodels, which would be more convenient for engineering calculations. So far, our attempts in this direction have been frustrated by inability to devise a simple, and yet realistic, formulation of the turbulent combustion aspect of the problem. With regard to your comment on the choice of the value of the soot temperature, I do agree that the current formulation of the model leaves much space for improvement. There is indication L~ that sootier fuels have lower equivalent radiation temperatures. It is possible that in the not too distant
RADIATION-CONTROLLED future we will be able to u n d e r s t a n d t h e r e l a t i o n s h i p b e t w e e n t h e t e n d e n c y of different f u e l s to p r o d u c e soot in b u o y a n t d i f f u s i o n f l a m e s a n d the soot temperature. At that stage, it m a y b e c o m e p o s s i b l e to relate actual soot p r o d u c t i o n a n d soot t e m p e r a t u r e to a " s o o t i n e s s " parameter, d e t e r m i n e d for the particular fuel u n d e r c o n s i d e r a t i o n , t h r o u g h a s i m p l e m o d e l for soot p r o d u c t i o n w h i c h a c c o u n t s for flow
TURBULENT
WALL FIRES
1085
c o n d i t i o n s . U n t i l then, i m p r o v e m e n t s to f o r m u l a tions, s u c h as that d i s c u s s e d in the paper, are likely to be m o r e of a cosmetic t h a n c o n c e p t u a l nature. REFERENCES 1. DE RIS, J., T h i s S y m p o s i u m . 2. MARI(STEIN, G. H., T h i s S y m p o s i u m .