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Radiation effects on large fire plumes
RADIATION EFFECTS ON LARGE FIRE P L U M E S R. K. SMITH
Department of Mathematics, The University of Manchester, Manchester, England Recent attempts to model the flow in very hot fire plumes where radiative transport of heat may significantly modify both the dynamics of the flow and the processes of combustion have met with only partial success. This paper gives an account of a model for the flow in a turbulent fire plume at the levels above which combustion is not appreciable but where strong buoyancy and radiation effects are still important. The model includes an analysis of the radiation field in which some strong approximations are made, but the main features are retained. In the treatment, the radiative flux is divided into lateral and vertical components which are considered separately. An opacity is defined for the plume as the ratio of mean radius to mean free path of radiation, and the intermediate case in which this ratio is of order unity is discussed in detail. The role of large eddies which carry very hot gas to the edge of the smoke plume is emphasized, and an estimate is made of their substantial contributions to the radiative heat loss to the surroundings. taking account of dynamic effects due to the radiation field and representing the combustion zone as a source of mass, momentum, and heat upon which depends the flow above.
Introduction The air flow above a large fire is driven by tile strong buoyancy forces which result from the high temperatures produced by combustion. In the lower regions, temperatures are high and heat transfer by radiation plays an essential role. High-temperature effects retain their importance for some distance above the level at which heat generation by combustion can be neglected, although cooling both by radiation and by entrainment of cold ambient air is vigorous, and the plume quickly attains a weakly buoyant state in which density variations are small and are manifest only through the buoyancy forces they produce. Thus, the plume may be divided approximately into three regions: a combustion zone, a residual high-temperature zone, and a weakly buoyant zone. A model taking account of combustion and variations in chemical composition in the lowest region has been given by Nielsen and Ta@ but their treatment of the radiation field goes only so far as to consider opaque radiative heat loss from the plume edge. Murgai ~ examines radiation but not combustion effects in eases when the plume is opaque and transparent, but there appear to be some limitations in the equations he uses. An outline of some of the principles necessary for a proper modeling of the flow is given by Morton, a who discusses separately the effects of large density differences across the plume and of radiation, but he does not attempt to link these in the equations. The present paper is an attempt to make a combined study of all but the lowest region,
The Dynamic The flow coordinates momentum, equations of ponents are
Equations
is represented in cylindrical polar (X, R, 0) by equations of mass, and heat conservations, and an state. Corresponding velocity com-
(u, v, w) = (~ + ~;', ? + v',-~ + w'), where U, V, W denote mean velocities with respect to time, and U', V', W' are turbulent fluctuations about these. The flow is taken to be axi-symmetric (i.e., I7" = 0 on R = 0 and 0/a8 ------0) with zero mean swirl component of velocity (i.e., ~" = 0). Variations with height of ambient pressure are normally small over heights for which high temperatures persist and will be neglected. Inside the plume, the only part of the pressure field contributing to the net driving force is that included in the buoyancy term. The lateral pressure variation due to turbulent diffusion of momentum radially plays no dynamic role in the flow, although it has a small effect on the heat content via the specific heat cp. Thus, the pressure is assumed constant and equal to its ambient value. Following Morton, ~ we use the perfect gas law
507
508
FIRE RESEARCH
as the equation of state (which is reasonable since temperatures are not so high as to produce appreciable dissociation of the plume gas) and we suppose the specific beat to be constant. The equations then take the forms continuity:
(op/oO + (o/ox) (pu) + R-~(O/OR) (RpV) = 0;
(1)
X momentum:
state:
pT = poT0 = piT1;
(4)
where e = %T is the enthalpy per unit mass; Q is the net absorption of heat due to the radiation field; p, T are the density and temperature inside the plume, p0 (X), To (X) the corresponding ambient quantities at height X, and pl, T1 at the source level (X -- 0), respectively. The enthalpy per unit volume, pe = p~cpT~is constant and Eqs. (3) and (4) together give a volume conservation equations
(ou/ot) + u ( o u / o x ) + V(oU/OR)
(0 U/OX) q.- R-1 (O/OR) (RV) = pQ/c~,mT1. = a[(p0 -
p)/p];
(2)
(5)
q-- R-I(O/OR) (RpeV) ~ pQ; (3)
If we now express the variables p, U, V, Q, in terms of their mean and fluctuating parts, and take a time average over each equation, the steady-state equations for mean quantities become
energy:
(0100 (pe) + (OIOX) (peG)
(O/OX) (~U q.- p'U') --{-R-I(O/OR)ER(~? -'k p'U')-] --- O, (O/OX) (~7' -k ~U'~ Jr- 2~7p~ q- ~ )
• [R(~g7? +
(6)
-{- R-'(O/OR) ~ i ~ ~, + COb~' + 0 7 W + r
(o~/ox) + >I(O/OR) (Rg) = (~0. + p~Q,)/c,,~rl.
= g(po - ~),
(7) (s)
I t is assumed that U% p'U', p'U'2 are small in comparison with the mean quantities U=, (p0 - ~) U, (p0 - ~)/J~, and are thus neglected. This imposes limitations on the turbulent intensity x, where
x = ~(x,
o)/gT=(x, o),
but should be adequate providing the latter does not exceed about 20%. The axial velocity component and all turbulent fluctuation terms are small at large distances from the axis, and are taken to be zero at R --- ~. Thus, integrating across a plume section, the equations for mean quantities become
(9)
~-~
~U2R dR = [-R~UFT]R=~ -q-
g(po - ~) R dR,
(10)
(11)
Observation shows that plumes exhibit slow rates of spread in relation to their vertical dimensions in neutral or stratified environments (except for the rather rapid spread in a stably stratified environment near the highest level of
ascent). For the most part, therefore, the turbulent structure of the flow will be expected to be approximately self-preserving and profiles of mean quantities may be assumed to be approximately similar at all heights. A similarity
RADIATION EFFECTS ON LARGE FIRE PLUMES solution of Eqs. (9), (10), and (11) may be obtained by taking ~3 = V . u ( x ) f ( , ) , = 500 -
(~ -
where E is called the entrainment constant. This form for -- I~R finds support in the experiments of Ricon and Spalding. ~ The mass inflow per unit height is then
p.)m(~)g(,)]/~,
X = zR,
(--2wR~?)R-.~r
. = R/Ror(x), where r(x), m ( z ) , u ( x ) are local scale ratios of length, density difference, and velocity across the flow at height x, nondimensionalized with respect to the source radius, velocity, and density difference, R,, Us, pl -- p~, and f ( , ) , g(7/) satisfy f(0) = g(0) = 1 a n d / ( , ) , g(~/) ~ 0 as ~/---~ oo. The rate of entrainment into the plume can be represented by assuming an entrainment velocity - V n across some suitably defined "mean plume edge", taken here as R,r(x). Morton 3 and Thomas 4 used simple dimensional arguments to obtain the expression -~
509
= E (~(R,z, O)/po(R.~) )'~V.~(x),
(d/dx)lur~E1- (z/O -"Y(a2/al)-]}
= 2~p0ES~(R.x,
O)/po(R.~) J~ U,R,u (x) r (x).
(12)
Take
\
C~plT1 ]
'
(13)
and assume a stable linear density stratification, p0 = p1(1 -- x / l ) , in the environment, with I >> 1. In terms of nondimensional variables r, x, u, m, Eqs. (9), (10), and (11), take the form
= (E/al)ur{[l-
(x/l)~['l-
(x/l)-
"ym]} 112, (14)
( d/dx) {u2r~E1 -- (x/l) -- 7(b2/bl)m-] I = (ba/b,) ArZrn,
(15)
(d/dx)Eur 2-] = (E/el) ur{[-1 -- (x/l) -- 3'm-Ill-1 -- (x/l)-]} 1/2 q- h(x),
(16)
where -y = (pl ps)/Pl is the relative density difference at the source level with A = g'yR,/U, ~ a nondimensional parameter comparing buoyancy forces with inertial forces at the source h ( x ) = H (X)/alU~R,, and -
-
al =
fil &h
a~ =
--land,,
bl = fo~f~. &l, b~ = ba =
/? /:
--J~gy dy, g'q d . .
For Gaussian profiles f(~/) = g(7/) = exp (_~/z), these profile integrals have the values al----- 89 as = - - 88 bl = } , b ~ - - - - - ~,ba---- 3. The Radiation
Field
A simplified model for radiation can be obtained by dividing the radiation flux into radial
and vertical components with respective contribution hL(X) and b y ( x ) to h ( x ) . In this context, plumes may be classified broadly into three types: opaque, intermediate, and transparent, according to their opacity to radiation z, defined as the local ratio of mean radius to mean free path of radiation. These three types correspond to opacities z >> 1, z -- 0(1), z <( 1. Most interest is centered about the intermediate case, which is considered here in detail. Fuels which burn incompletely, with the production of large amounts of carbon particles, give rise to a particularly opaque plume. Figure 1 shows a fire on a pool of burning petroleum waste in calm atmospheric conditions. The plume consists of a dense column of black smoke amid which visibly hot eddies of gas appear intermittently, producing bursts of radiation as they are exposed to the environment. These eddies arise from bulk convection of very hot gas from the central regions of the column and their length scales are typically those of the energycontaining eddies. They transport hot gas into the outer regions of the plume, and because of their high temperature compared with other regions at the plume edge, they make a major contribution to the total radiative heat loss and must be taken account of in any radiation model. This effect is less marked in the intermediate regime, where deeper optical penetration
510
F I R E RESEARCH The net rate of gain of heat per unit height of the plume across a whole section at height X , due to lateral radiation, is o~ (~(2 + p-r~) R dR = -- 2~-o-R~(aTR 4 -- ~To4).
(17)
We proceed b y stages to construct an expression for a, treating separately each of the effects mentioned. (i) A preliminary study, which should at least provide an order-of-magnitude estimate of the effects due to radiation bursts, can be made by considering the effect of large positive sinusoidal temperature fluctuations at the mean plume edge, on the outgoing mean radiation flux. A t any one instant, a proportion of the plume surface only is at temperature significantly above average, and so we take an instantaneous edge temperature 0 = TR[1 -~ q sin 2 (lrt/r)], FIG. 1. A fire plume formed in a still atmosphere by a pool of petroleum waste with a base diameter of about 20 meters. The photograph clearly shows eddies of hot gas which appear intermittently at the plume edge and contribute significantly to the outgoing radiation flux.
where q models the relative magnitude of the maximum fluctuation, r is an effective period for the appearance of "hot eddies" of plume gases at the edge, and t is the time variable. This leads to a mean radiation rate per unit area ( 0~4 ) . . . .
of radiation through the outer and cooler layers of the column gives rise to an increase in the direct radiative loss from the fire plume to the environment.
(18)
~" T -1
{TTR4
X [1 -}- q sin~(~t/v)] t dt = a a l ( q ) T R 4, where al(q) =-- 1 .-~ 2q + 9qV4 ~- 5qZ/4 -~- 35q4/128.
Lateral Radiation Lateral radiation from the plume to the environment is represented by defining a mean radiating edge together with an associated mean radiating temperature TR(X), and an effective emissivity a, which is introduced to allow for the emission properties of the hot combustion products and for the effects due to radiation bursts. The outward radiation flux is then taken as aaTR 4 per unit area of a cylindrical element of the mean plume surface, where ~ is Stefans constant. In addition to heat loss, a small amount of radiation will be received by the plume from the environment, which is assumed to be in radiative equilibrium a t temperature To. To allow for this, we define an absorption coefficient for the section.
(19) The maximum relative temperature fluctuation q should be proportional to (Taxi~ -- TR)/TI~; it seems likely that the constant of proportionality may be of order 1, and we can do no better at this stage than assume unit constant. The expression O~l(q) may be interpreted as an emission coefficient which takes account of radiation bursts. I n regions of the fire plume where radiation is important, q varies between about 2 and 3 and al(q) between about 30 and 80; thus, the effects on heat transfer of radiation bursts may be very large indeed, and it must be stressed that the use of a mean outer temperature for the plume m a y lead to a serious underestimate of lateral radiative flux. (ii) An a t t e m p t to estimate a (as a~) in the
RADIATION EFFECTS ON LARGE FIRE PLUMES case of intermediate opacity may be made by comparing the calculated radial intensity at the mean plume edge, based on an integration of the one-dimensional radiation equation for the grey case, with the radial intensity due to blackbody emission at the mean edge temperature. The intensity of radiation I~(y) of frequency v at distancy y from the axis along a radius of the cylindrical plume cross section under consideration, is given by
function, defined as the ratio of emission coefficient to absorption coefficient (see, e.g., Chandrasekhar6). I t may be shown that if the system is in local thermodynamic equilibrium, the source function
J, = B,( T), where
B , ( T ) = (2h~Vc2)/[exp (h,/kT) -- 17
I~(y) = I~(0) exp [--7"(y, 0)']
+
J,(s) exp r--~'(y, s)'R,p(s) d~,
with
T(y, s) =
f
y
K,p(t) dr,
where ~, is the absorbtivity of the medium per unit mass at frequency ~, and J~ is the source
I(R,r) =
511
is Planck's function (see, e.g., KondratievT). We have already introduced a number of approximations for the radiation field at intermediate opacity, and it will be consistent with this approach to assume that conditions of local thermodynamic equilibrium exist and to make the grey radiation approximations that the absorhtivity K~ is independent of frequency and takes the value K. The integrated intensity at the plume edge is then
I, R~r) dv
f0~ (
= (aR,/Tr)
fRsrT4(R,s)
exp [--v(R,r, R,s)~Kp(Rss) ds,
"0
where the symmetry condition I ( 0 ) = 0 has been used. The local mean free path of radiation h = (Kp0)-1 is related to that at the source level },8 = (Kpl)-1 b y h ---- h, (1 -- x/l)-1. Hence, the opacities z --- R,r/h at height x, and z, at source level, are related b y z ---- z,r(1 -- x/l). For a Gaussian density profile P ---- Po -- (Pl -- Ps) exp
(--sVr2),
it may then be shown that ae = I ( R ~ r ) / [ ( a / ~ ) TRS
= (ToVT.')z~rO -- ~/l)~ f~ exp -- { (1 -- t) (I -- ~/Z) -- ~-~-~rerf (1) -- erf (t)-l}z.rdr, "o
~1
x / l -- 3~m exp (--t2)] ~
where eft(x) = r-57~
(iii) In the intermediate case, the emissivity a2 will be somewhat larger than unity, accounting for direct optical penetration more deeply into the plume. Thus, we may take Taa21/4 as an effective mean radiating temperature. Radiation bursts are allowed for in the model by using
exp ( - - t 2) dt.
TRO~21/4instead of TR in the expression for q in Eq. (19), and taking the emissivity al(q) in Eq. (17). This approximation is worse for small values of a~, and in these conditions may result in growing values of q, which are accompanied b y a rapid increase in al. This happens when the
512
F I R E RESEARCH
opacity is small, as in a hydrogen flame, or when the temperature differences across the plume are small, as will always be the case far above the source. In both cases, radiation bursts are of no consequence, and lateral radiation terms are excluded in the computations for a~ < 1. When the relative density deficiency becomes small, the integral for a: tends to that for a cylindrical section at uniform temperature (i.e., m = 0), which is simply 1 -- e-z. To complete the model, we suppose that the rate of absorbtion by the plume section, of radiation from the environment, is modified by an absorbtion coe~cient fl = 1 -- e-~, equal to the emission coefficient in the case when m = 0. This means that the lateral radiation will be in equilibrium at heights where temperature variations over the plume are negligible. Collecting these results together, we find that h~(~) =
-
(B/aO~(1
-
x/O-%(r,
~ , z., ~ ) ,
where B = 27raTl~/crplU~ is a nondimensional parameter, and B F i ( 1 , 1, z,, 0) provides a measure of the ratio of relative cooling rate to mean conective heat flux at the source, and F l ( r , m, z~, x) is the lateral radiation function which takes into account opaqueness and the effects due to radiation bursts on the cooling rate. Thus,
height, taken as the axial temperature (as lateral radiation has been treated separately). This is equivalent to the use of a "top-hat" profile. The integrated axial intensity I ( X ) satisfies the equation dI/dX
=
-- Kp(X)[I(X)
di//dx
=
--
z,[l
--
(x/l) -- ~/m] 4
•
i-
]_xfl_.rm
"
The heating of plume gas per unit volume at height X is -- ( d / d X ) I ( X ) . Thus, the net absorbtion of heat per unit height of the plume due to vertical radiation is, at height X, 0"(~O -~- ~()') R dR = -- (dI/dX)TrRfl and we may take
hv(x) = ( C/al) F2(i, ,n, x), say, where C = z, zT.~t/czplTiUs
q- exp [--z~r(1 -- x / l ) ] -- 1 with I -- m/e
(a/r) T4(X)3.
In terms of the nondimensional intensity, i ( x ) = IrI(X)/(~T~ 4, where T, is the source temperature; then
r l ( r , m, z., x) = o~l(q)o~e I --
q-
-
is a nondimensional parameter representing the ratio of heat flux due to vertical radiative transport and vertical convective transport, respectively, and
i.
(1 -- m)a~ 1/4
F2(i, m, x) =
(1 - - z/Z -
"ym) 4
Vertical Radiation I t remains to estimate the contribution of b y ( x ) due to the vertical flux of radiation. The model described here is greatly oversimplified, but is of interest because it allows some estimate of the effects of local heating due to the vertical flux. A first very rough approximation is achieved b y considering the one-dimensional equation of transfer applied to a cylinder of plume gas, extending from the source upwards and having a radius R, Efor heights over which radiation is important, the radius varies only slowly so that R~r(x) differs little from R,-]. One m a y calculate the heat absorbed b y a horizontal slice of this cylinder at height x, and with thickness dx, in terms of a characteristic temperature at each
The plume is assumed to be in thermal equilibrium at temperature T~ before it enters the region under consideration so that di/dx = 0 for x ~ 0 a n d i ( 0 ) = 1. Combining these results we have, for the radiation term in Eq. (16),
h(x) = hL(x) + by(x) =
- -
( A F i r ( 1 -- x/l) ~ -
CF2)/ai.
(22)
Solution of the Equations The equations of motion, Eqs. (14), (15),
RADIATION EFFECTS ON LARGE FIRE PLUMES and (16), are rearranged in the form dr/dx
-
m;,
du/dx
= m~,
dm/dx
513
3 w
= m3,
w
E
where m~ are known, but complicated, functions of the independent and dependent variables x, r, u, m, i, and h ( x ) . These are solved numerically together with Eq. (21) for i ( x ) , using a Runge-Kutta integration program on the Manchester Atlas Computor, taking starting values x -~ O, r = u = i = 1, m -~ ps/pl. The nondimensional parameters A, B, C, which determine the system are specified in terms of source values (Re, Us, T~, i~), environmental conditions (pl, T1, R~l), physical constants (cp, z), and flow properties (z,, E ) . Solutions have been computed for the following system: Re Ue = T, = pl = T1 = Rs1 = =
10 meters, 10 meters/see, 1500~ 0.0012 g/cc, 300~ 2 X 104 meter -1
(equivalent to a typical lapse rate of 1.5~ 100 meters), and E = 0.1.
per
:~2.E c2
u
z 1.5
FIG. 3. The variation of nondimensional mean axial temperature 0 = ( T -- To)/T1, with height for various opacities above the standard source. Numbers by each curve denote the opacity.
Discussion
E
c
";3
u
I0
Fro. 2. The variation of nondimensional mean axial velocity and mean plume radius with height for various opacities (indicated by numbers alongside curves) above the standard source. The mean radius curve varies little with opacity, and only one curve is drawn.
The essential feature of all these flows is the existence of a region of strong buoyancy close to the source, in which the gases are suffering rapid acceleration (Figs. 2 and 3); hence, the radius decreases initially, though this rate of decrease is governed strongly by entrainment, which quickly reverses the trend, and the plume begins to spread with steady decrease in upward velocity. Vigorous cooling by entrainment continues, and the plume has attained a weakly buoyant state by a height of about 10 source diameters. Lateral radiation enhances the cooling rate, but transfer of heat along the column by radiation prolongs the strongly buoyant region to some extent. Radiation is generally important only over heights of a few source diameters, as indicated in Fig. 4, where the cooling due to lateral radiation and heating due to vertical radiation are compared with the cooling by entrainment. The relative importance of vertical radiation is maintained over larger height ranges only when the plume is moderately transparent (ze "-~ 0.1). Lateral radiation will have a significant effect on the cooling rate, and may dominate the contribution by entrainment if large axial temperatures are maintained, for instance by combustion, but otherwise its effect
514
F I R E RESEARCH TABLE I The maximum height xm~x, attained by a plume for various source temperatures and opacities, T, and zo.
w E ci
Xmaz
T,(~
.E
1000 1200 1400 1600 1800 1500 1500 1500
w 3c
\
0:5
l[O
1~5
FIG. 4. The comparative importances of lateral and vertical radiation terms with respect to entrainment on the relative cooling rate T -I dT/dx are shown as function of height for varying opacities above the standard source; mean radius, 10 meters; mean axial velocity, 10 meters/sec; mean axial temperature, 1500~ Numbers indicate the opacity; unbracketed numbers, vertical radiation/entrainment; bracketed numbers (dashed lines) = lateral radiation/entrainment.
is soon outweighed by mixing with cold ambient fluid. The maximum height attained should be governed, to some extent, by the opacity. If the strongly buoyant region is maintained by the net effect of radiation, the range of initial acceleration is prolonged and, consequently, a greater maximum velocity is reached. If the plume is fairly opaque--say, zs ~ 10--the vertical radiative heating is vigorous in the neighborhood of the source, b u t rapidly dies away with height (Fig. 4). This causes a higher initial acceleration, compared with the case z8 = 0.1, for example, but the higher velocities reached give rise to a larger rate of entrainment from the environment. Thus, beyond about 2 source diameters in height, cooling is faster in the opaque case (Fig. 3), and this has the over-all effect of reducing the height attained b y the top of the plume in a stably stratified environment. This demonstrates further the importance of plume penetration discussed previously b y M o r t o n ) I t is interesting to note t h a t there will be an optimum value of the opacity for which a
z,
(source diameters)
1.0 1.0 1.0 1.0 1.0 0.1 1.0 10
29.8 30.9 32.4 34.1 36.1 31.8 33.2 32.0
For a fixed opacity, the maximum height increases steadily with the source temperature. The optimum opacity (see text) for which a plume will rise to a maximum height is of order unity. Source conditions are: mean radius 10 meters; mean axial velocity, 10 meters/sec. The nondimensional stratification parameter 1 = 2 X 104, corresponding to an atmospheric lapse of about 1.5~ per 100 meters.
plume with given source radius, velocity, and temperature will reach a maximum height, and this suggests the possibility that a buoyant plume of hot noxious products might be removed to greater heights by suitable seeding with carbon dust. The temperature at the source is the largest single factor in determining the maximum height to which a plume will rise, although variations in opacity can still produce effects of about 5%-10%. This is indicated in Table I, where heights of zero vertical velocity are tabulated for different source temperatures and opacities.
ACKNOWLEDGMENT
The author is indebted to Dr. B. R. Morton for his encouragment and stimulating discussions throughout the preparation of this paper.
REFERENCES 1. NIELSEN,H. J. AND TAO, L. N.: Tenth Symposium (International) on Combustion, p. 965, The Combustion Institute, 1965. 2. MURGAI, M. P.: J. Fluid Mech. 12, 441 (1962). 3. MORTON, B. R.: Tenth Symposium (International) on Combustion, p. 973, The Combustion Institute, 1965.
R A D I A T I O N E F F E C T S ON L A R G E F I R E P L U M E S 4. THOMAS, P. H.: Ninth Symposium (International) on Combustion, p. 844, Academic Press, 1963. 5. Ricou, F. P. AND SPALDING, D. B.: J. Fluid Mech. 11, 21 (1961).
515
6. CHANDRASEKHAR, S..* Radiative Transfer, p. 9, Dover, 1950.
7. KONDRATIEV, K. YA.: Radiative Heat Exchange in the Atmosphere, p. 3, Pergamon, 1965. 8. MORTON, B. R.: J. Fluid Mech. 5, 151 (1959).
COMMENTS Dr. W. Unterberg (Rocketdyne): How were the results of this investigation influenced by the nature of the radiating gas, i.e., the presence of luminous solid particles such as soot? Mr. R. K. Smith: The presence or absence of luminous solid particles, such as soot, is accounted for through the opacity z,, defined at the source level. Thus, one must ask what effect these have on the opacity.
Plumes which contain large amounts of solid particles will be fairly opaque (i.e., z, >> 1) and will generally have a dark edge except for patches of hot gas which appear intermittently at the plume edge. These are radiation bursts, which are described in the paper, and their effect is taken into account in the model depending on zs. Plumes having a lesser concentration of solid particles will be less opaque and will fit into the intermediate (z8 ~ 1) or transparent (z8 << 1) classifications also described.
Department of Mathematics, The University of Manchester, Manchester, England Recent attempts to model the flow in very hot fire plumes where radiative transport of heat may significantly modify both the dynamics of the flow and the processes of combustion have met with only partial success. This paper gives an account of a model for the flow in a turbulent fire plume at the levels above which combustion is not appreciable but where strong buoyancy and radiation effects are still important. The model includes an analysis of the radiation field in which some strong approximations are made, but the main features are retained. In the treatment, the radiative flux is divided into lateral and vertical components which are considered separately. An opacity is defined for the plume as the ratio of mean radius to mean free path of radiation, and the intermediate case in which this ratio is of order unity is discussed in detail. The role of large eddies which carry very hot gas to the edge of the smoke plume is emphasized, and an estimate is made of their substantial contributions to the radiative heat loss to the surroundings. taking account of dynamic effects due to the radiation field and representing the combustion zone as a source of mass, momentum, and heat upon which depends the flow above.
Introduction The air flow above a large fire is driven by tile strong buoyancy forces which result from the high temperatures produced by combustion. In the lower regions, temperatures are high and heat transfer by radiation plays an essential role. High-temperature effects retain their importance for some distance above the level at which heat generation by combustion can be neglected, although cooling both by radiation and by entrainment of cold ambient air is vigorous, and the plume quickly attains a weakly buoyant state in which density variations are small and are manifest only through the buoyancy forces they produce. Thus, the plume may be divided approximately into three regions: a combustion zone, a residual high-temperature zone, and a weakly buoyant zone. A model taking account of combustion and variations in chemical composition in the lowest region has been given by Nielsen and Ta@ but their treatment of the radiation field goes only so far as to consider opaque radiative heat loss from the plume edge. Murgai ~ examines radiation but not combustion effects in eases when the plume is opaque and transparent, but there appear to be some limitations in the equations he uses. An outline of some of the principles necessary for a proper modeling of the flow is given by Morton, a who discusses separately the effects of large density differences across the plume and of radiation, but he does not attempt to link these in the equations. The present paper is an attempt to make a combined study of all but the lowest region,
The Dynamic The flow coordinates momentum, equations of ponents are
Equations
is represented in cylindrical polar (X, R, 0) by equations of mass, and heat conservations, and an state. Corresponding velocity com-
(u, v, w) = (~ + ~;', ? + v',-~ + w'), where U, V, W denote mean velocities with respect to time, and U', V', W' are turbulent fluctuations about these. The flow is taken to be axi-symmetric (i.e., I7" = 0 on R = 0 and 0/a8 ------0) with zero mean swirl component of velocity (i.e., ~" = 0). Variations with height of ambient pressure are normally small over heights for which high temperatures persist and will be neglected. Inside the plume, the only part of the pressure field contributing to the net driving force is that included in the buoyancy term. The lateral pressure variation due to turbulent diffusion of momentum radially plays no dynamic role in the flow, although it has a small effect on the heat content via the specific heat cp. Thus, the pressure is assumed constant and equal to its ambient value. Following Morton, ~ we use the perfect gas law
507
508
FIRE RESEARCH
as the equation of state (which is reasonable since temperatures are not so high as to produce appreciable dissociation of the plume gas) and we suppose the specific beat to be constant. The equations then take the forms continuity:
(op/oO + (o/ox) (pu) + R-~(O/OR) (RpV) = 0;
(1)
X momentum:
state:
pT = poT0 = piT1;
(4)
where e = %T is the enthalpy per unit mass; Q is the net absorption of heat due to the radiation field; p, T are the density and temperature inside the plume, p0 (X), To (X) the corresponding ambient quantities at height X, and pl, T1 at the source level (X -- 0), respectively. The enthalpy per unit volume, pe = p~cpT~is constant and Eqs. (3) and (4) together give a volume conservation equations
(ou/ot) + u ( o u / o x ) + V(oU/OR)
(0 U/OX) q.- R-1 (O/OR) (RV) = pQ/c~,mT1. = a[(p0 -
p)/p];
(2)
(5)
q-- R-I(O/OR) (RpeV) ~ pQ; (3)
If we now express the variables p, U, V, Q, in terms of their mean and fluctuating parts, and take a time average over each equation, the steady-state equations for mean quantities become
energy:
(0100 (pe) + (OIOX) (peG)
(O/OX) (~U q.- p'U') --{-R-I(O/OR)ER(~? -'k p'U')-] --- O, (O/OX) (~7' -k ~U'~ Jr- 2~7p~ q- ~ )
• [R(~g7? +
(6)
-{- R-'(O/OR) ~ i ~ ~, + COb~' + 0 7 W + r
(o~/ox) + >I(O/OR) (Rg) = (~0. + p~Q,)/c,,~rl.
= g(po - ~),
(7) (s)
I t is assumed that U% p'U', p'U'2 are small in comparison with the mean quantities U=, (p0 - ~) U, (p0 - ~)/J~, and are thus neglected. This imposes limitations on the turbulent intensity x, where
x = ~(x,
o)/gT=(x, o),
but should be adequate providing the latter does not exceed about 20%. The axial velocity component and all turbulent fluctuation terms are small at large distances from the axis, and are taken to be zero at R --- ~. Thus, integrating across a plume section, the equations for mean quantities become
(9)
~-~
~U2R dR = [-R~UFT]R=~ -q-
g(po - ~) R dR,
(10)
(11)
Observation shows that plumes exhibit slow rates of spread in relation to their vertical dimensions in neutral or stratified environments (except for the rather rapid spread in a stably stratified environment near the highest level of
ascent). For the most part, therefore, the turbulent structure of the flow will be expected to be approximately self-preserving and profiles of mean quantities may be assumed to be approximately similar at all heights. A similarity
RADIATION EFFECTS ON LARGE FIRE PLUMES solution of Eqs. (9), (10), and (11) may be obtained by taking ~3 = V . u ( x ) f ( , ) , = 500 -
(~ -
where E is called the entrainment constant. This form for -- I~R finds support in the experiments of Ricon and Spalding. ~ The mass inflow per unit height is then
p.)m(~)g(,)]/~,
X = zR,
(--2wR~?)R-.~r
. = R/Ror(x), where r(x), m ( z ) , u ( x ) are local scale ratios of length, density difference, and velocity across the flow at height x, nondimensionalized with respect to the source radius, velocity, and density difference, R,, Us, pl -- p~, and f ( , ) , g(7/) satisfy f(0) = g(0) = 1 a n d / ( , ) , g(~/) ~ 0 as ~/---~ oo. The rate of entrainment into the plume can be represented by assuming an entrainment velocity - V n across some suitably defined "mean plume edge", taken here as R,r(x). Morton 3 and Thomas 4 used simple dimensional arguments to obtain the expression -~
509
= E (~(R,z, O)/po(R.~) )'~V.~(x),
(d/dx)lur~E1- (z/O -"Y(a2/al)-]}
= 2~p0ES~(R.x,
O)/po(R.~) J~ U,R,u (x) r (x).
(12)
Take
\
C~plT1 ]
'
(13)
and assume a stable linear density stratification, p0 = p1(1 -- x / l ) , in the environment, with I >> 1. In terms of nondimensional variables r, x, u, m, Eqs. (9), (10), and (11), take the form
= (E/al)ur{[l-
(x/l)~['l-
(x/l)-
"ym]} 112, (14)
( d/dx) {u2r~E1 -- (x/l) -- 7(b2/bl)m-] I = (ba/b,) ArZrn,
(15)
(d/dx)Eur 2-] = (E/el) ur{[-1 -- (x/l) -- 3'm-Ill-1 -- (x/l)-]} 1/2 q- h(x),
(16)
where -y = (pl ps)/Pl is the relative density difference at the source level with A = g'yR,/U, ~ a nondimensional parameter comparing buoyancy forces with inertial forces at the source h ( x ) = H (X)/alU~R,, and -
-
al =
fil &h
a~ =
--land,,
bl = fo~f~. &l, b~ = ba =
/? /:
--J~gy dy, g'q d . .
For Gaussian profiles f(~/) = g(7/) = exp (_~/z), these profile integrals have the values al----- 89 as = - - 88 bl = } , b ~ - - - - - ~,ba---- 3. The Radiation
Field
A simplified model for radiation can be obtained by dividing the radiation flux into radial
and vertical components with respective contribution hL(X) and b y ( x ) to h ( x ) . In this context, plumes may be classified broadly into three types: opaque, intermediate, and transparent, according to their opacity to radiation z, defined as the local ratio of mean radius to mean free path of radiation. These three types correspond to opacities z >> 1, z -- 0(1), z <( 1. Most interest is centered about the intermediate case, which is considered here in detail. Fuels which burn incompletely, with the production of large amounts of carbon particles, give rise to a particularly opaque plume. Figure 1 shows a fire on a pool of burning petroleum waste in calm atmospheric conditions. The plume consists of a dense column of black smoke amid which visibly hot eddies of gas appear intermittently, producing bursts of radiation as they are exposed to the environment. These eddies arise from bulk convection of very hot gas from the central regions of the column and their length scales are typically those of the energycontaining eddies. They transport hot gas into the outer regions of the plume, and because of their high temperature compared with other regions at the plume edge, they make a major contribution to the total radiative heat loss and must be taken account of in any radiation model. This effect is less marked in the intermediate regime, where deeper optical penetration
510
F I R E RESEARCH The net rate of gain of heat per unit height of the plume across a whole section at height X , due to lateral radiation, is o~ (~(2 + p-r~) R dR = -- 2~-o-R~(aTR 4 -- ~To4).
(17)
We proceed b y stages to construct an expression for a, treating separately each of the effects mentioned. (i) A preliminary study, which should at least provide an order-of-magnitude estimate of the effects due to radiation bursts, can be made by considering the effect of large positive sinusoidal temperature fluctuations at the mean plume edge, on the outgoing mean radiation flux. A t any one instant, a proportion of the plume surface only is at temperature significantly above average, and so we take an instantaneous edge temperature 0 = TR[1 -~ q sin 2 (lrt/r)], FIG. 1. A fire plume formed in a still atmosphere by a pool of petroleum waste with a base diameter of about 20 meters. The photograph clearly shows eddies of hot gas which appear intermittently at the plume edge and contribute significantly to the outgoing radiation flux.
where q models the relative magnitude of the maximum fluctuation, r is an effective period for the appearance of "hot eddies" of plume gases at the edge, and t is the time variable. This leads to a mean radiation rate per unit area ( 0~4 ) . . . .
of radiation through the outer and cooler layers of the column gives rise to an increase in the direct radiative loss from the fire plume to the environment.
(18)
~" T -1
{TTR4
X [1 -}- q sin~(~t/v)] t dt = a a l ( q ) T R 4, where al(q) =-- 1 .-~ 2q + 9qV4 ~- 5qZ/4 -~- 35q4/128.
Lateral Radiation Lateral radiation from the plume to the environment is represented by defining a mean radiating edge together with an associated mean radiating temperature TR(X), and an effective emissivity a, which is introduced to allow for the emission properties of the hot combustion products and for the effects due to radiation bursts. The outward radiation flux is then taken as aaTR 4 per unit area of a cylindrical element of the mean plume surface, where ~ is Stefans constant. In addition to heat loss, a small amount of radiation will be received by the plume from the environment, which is assumed to be in radiative equilibrium a t temperature To. To allow for this, we define an absorption coefficient for the section.
(19) The maximum relative temperature fluctuation q should be proportional to (Taxi~ -- TR)/TI~; it seems likely that the constant of proportionality may be of order 1, and we can do no better at this stage than assume unit constant. The expression O~l(q) may be interpreted as an emission coefficient which takes account of radiation bursts. I n regions of the fire plume where radiation is important, q varies between about 2 and 3 and al(q) between about 30 and 80; thus, the effects on heat transfer of radiation bursts may be very large indeed, and it must be stressed that the use of a mean outer temperature for the plume m a y lead to a serious underestimate of lateral radiative flux. (ii) An a t t e m p t to estimate a (as a~) in the
RADIATION EFFECTS ON LARGE FIRE PLUMES case of intermediate opacity may be made by comparing the calculated radial intensity at the mean plume edge, based on an integration of the one-dimensional radiation equation for the grey case, with the radial intensity due to blackbody emission at the mean edge temperature. The intensity of radiation I~(y) of frequency v at distancy y from the axis along a radius of the cylindrical plume cross section under consideration, is given by
function, defined as the ratio of emission coefficient to absorption coefficient (see, e.g., Chandrasekhar6). I t may be shown that if the system is in local thermodynamic equilibrium, the source function
J, = B,( T), where
B , ( T ) = (2h~Vc2)/[exp (h,/kT) -- 17
I~(y) = I~(0) exp [--7"(y, 0)']
+
J,(s) exp r--~'(y, s)'R,p(s) d~,
with
T(y, s) =
f
y
K,p(t) dr,
where ~, is the absorbtivity of the medium per unit mass at frequency ~, and J~ is the source
I(R,r) =
511
is Planck's function (see, e.g., KondratievT). We have already introduced a number of approximations for the radiation field at intermediate opacity, and it will be consistent with this approach to assume that conditions of local thermodynamic equilibrium exist and to make the grey radiation approximations that the absorhtivity K~ is independent of frequency and takes the value K. The integrated intensity at the plume edge is then
I, R~r) dv
f0~ (
= (aR,/Tr)
fRsrT4(R,s)
exp [--v(R,r, R,s)~Kp(Rss) ds,
"0
where the symmetry condition I ( 0 ) = 0 has been used. The local mean free path of radiation h = (Kp0)-1 is related to that at the source level },8 = (Kpl)-1 b y h ---- h, (1 -- x/l)-1. Hence, the opacities z --- R,r/h at height x, and z, at source level, are related b y z ---- z,r(1 -- x/l). For a Gaussian density profile P ---- Po -- (Pl -- Ps) exp
(--sVr2),
it may then be shown that ae = I ( R ~ r ) / [ ( a / ~ ) TRS
= (ToVT.')z~rO -- ~/l)~ f~ exp -- { (1 -- t) (I -- ~/Z) -- ~-~-~rerf (1) -- erf (t)-l}z.rdr, "o
~1
x / l -- 3~m exp (--t2)] ~
where eft(x) = r-57~
(iii) In the intermediate case, the emissivity a2 will be somewhat larger than unity, accounting for direct optical penetration more deeply into the plume. Thus, we may take Taa21/4 as an effective mean radiating temperature. Radiation bursts are allowed for in the model by using
exp ( - - t 2) dt.
TRO~21/4instead of TR in the expression for q in Eq. (19), and taking the emissivity al(q) in Eq. (17). This approximation is worse for small values of a~, and in these conditions may result in growing values of q, which are accompanied b y a rapid increase in al. This happens when the
512
F I R E RESEARCH
opacity is small, as in a hydrogen flame, or when the temperature differences across the plume are small, as will always be the case far above the source. In both cases, radiation bursts are of no consequence, and lateral radiation terms are excluded in the computations for a~ < 1. When the relative density deficiency becomes small, the integral for a: tends to that for a cylindrical section at uniform temperature (i.e., m = 0), which is simply 1 -- e-z. To complete the model, we suppose that the rate of absorbtion by the plume section, of radiation from the environment, is modified by an absorbtion coe~cient fl = 1 -- e-~, equal to the emission coefficient in the case when m = 0. This means that the lateral radiation will be in equilibrium at heights where temperature variations over the plume are negligible. Collecting these results together, we find that h~(~) =
-
(B/aO~(1
-
x/O-%(r,
~ , z., ~ ) ,
where B = 27raTl~/crplU~ is a nondimensional parameter, and B F i ( 1 , 1, z,, 0) provides a measure of the ratio of relative cooling rate to mean conective heat flux at the source, and F l ( r , m, z~, x) is the lateral radiation function which takes into account opaqueness and the effects due to radiation bursts on the cooling rate. Thus,
height, taken as the axial temperature (as lateral radiation has been treated separately). This is equivalent to the use of a "top-hat" profile. The integrated axial intensity I ( X ) satisfies the equation dI/dX
=
-- Kp(X)[I(X)
di//dx
=
--
z,[l
--
(x/l) -- ~/m] 4
•
i-
]_xfl_.rm
"
The heating of plume gas per unit volume at height X is -- ( d / d X ) I ( X ) . Thus, the net absorbtion of heat per unit height of the plume due to vertical radiation is, at height X, 0"(~O -~- ~()') R dR = -- (dI/dX)TrRfl and we may take
hv(x) = ( C/al) F2(i, ,n, x), say, where C = z, zT.~t/czplTiUs
q- exp [--z~r(1 -- x / l ) ] -- 1 with I -- m/e
(a/r) T4(X)3.
In terms of the nondimensional intensity, i ( x ) = IrI(X)/(~T~ 4, where T, is the source temperature; then
r l ( r , m, z., x) = o~l(q)o~e I --
q-
-
is a nondimensional parameter representing the ratio of heat flux due to vertical radiative transport and vertical convective transport, respectively, and
i.
(1 -- m)a~ 1/4
F2(i, m, x) =
(1 - - z/Z -
"ym) 4
Vertical Radiation I t remains to estimate the contribution of b y ( x ) due to the vertical flux of radiation. The model described here is greatly oversimplified, but is of interest because it allows some estimate of the effects of local heating due to the vertical flux. A first very rough approximation is achieved b y considering the one-dimensional equation of transfer applied to a cylinder of plume gas, extending from the source upwards and having a radius R, Efor heights over which radiation is important, the radius varies only slowly so that R~r(x) differs little from R,-]. One m a y calculate the heat absorbed b y a horizontal slice of this cylinder at height x, and with thickness dx, in terms of a characteristic temperature at each
The plume is assumed to be in thermal equilibrium at temperature T~ before it enters the region under consideration so that di/dx = 0 for x ~ 0 a n d i ( 0 ) = 1. Combining these results we have, for the radiation term in Eq. (16),
h(x) = hL(x) + by(x) =
- -
( A F i r ( 1 -- x/l) ~ -
CF2)/ai.
(22)
Solution of the Equations The equations of motion, Eqs. (14), (15),
RADIATION EFFECTS ON LARGE FIRE PLUMES and (16), are rearranged in the form dr/dx
-
m;,
du/dx
= m~,
dm/dx
513
3 w
= m3,
w
E
where m~ are known, but complicated, functions of the independent and dependent variables x, r, u, m, i, and h ( x ) . These are solved numerically together with Eq. (21) for i ( x ) , using a Runge-Kutta integration program on the Manchester Atlas Computor, taking starting values x -~ O, r = u = i = 1, m -~ ps/pl. The nondimensional parameters A, B, C, which determine the system are specified in terms of source values (Re, Us, T~, i~), environmental conditions (pl, T1, R~l), physical constants (cp, z), and flow properties (z,, E ) . Solutions have been computed for the following system: Re Ue = T, = pl = T1 = Rs1 = =
10 meters, 10 meters/see, 1500~ 0.0012 g/cc, 300~ 2 X 104 meter -1
(equivalent to a typical lapse rate of 1.5~ 100 meters), and E = 0.1.
per
:~2.E c2
u
z 1.5
FIG. 3. The variation of nondimensional mean axial temperature 0 = ( T -- To)/T1, with height for various opacities above the standard source. Numbers by each curve denote the opacity.
Discussion
E
c
";3
u
I0
Fro. 2. The variation of nondimensional mean axial velocity and mean plume radius with height for various opacities (indicated by numbers alongside curves) above the standard source. The mean radius curve varies little with opacity, and only one curve is drawn.
The essential feature of all these flows is the existence of a region of strong buoyancy close to the source, in which the gases are suffering rapid acceleration (Figs. 2 and 3); hence, the radius decreases initially, though this rate of decrease is governed strongly by entrainment, which quickly reverses the trend, and the plume begins to spread with steady decrease in upward velocity. Vigorous cooling by entrainment continues, and the plume has attained a weakly buoyant state by a height of about 10 source diameters. Lateral radiation enhances the cooling rate, but transfer of heat along the column by radiation prolongs the strongly buoyant region to some extent. Radiation is generally important only over heights of a few source diameters, as indicated in Fig. 4, where the cooling due to lateral radiation and heating due to vertical radiation are compared with the cooling by entrainment. The relative importance of vertical radiation is maintained over larger height ranges only when the plume is moderately transparent (ze "-~ 0.1). Lateral radiation will have a significant effect on the cooling rate, and may dominate the contribution by entrainment if large axial temperatures are maintained, for instance by combustion, but otherwise its effect
514
F I R E RESEARCH TABLE I The maximum height xm~x, attained by a plume for various source temperatures and opacities, T, and zo.
w E ci
Xmaz
T,(~
.E
1000 1200 1400 1600 1800 1500 1500 1500
w 3c
\
0:5
l[O
1~5
FIG. 4. The comparative importances of lateral and vertical radiation terms with respect to entrainment on the relative cooling rate T -I dT/dx are shown as function of height for varying opacities above the standard source; mean radius, 10 meters; mean axial velocity, 10 meters/sec; mean axial temperature, 1500~ Numbers indicate the opacity; unbracketed numbers, vertical radiation/entrainment; bracketed numbers (dashed lines) = lateral radiation/entrainment.
is soon outweighed by mixing with cold ambient fluid. The maximum height attained should be governed, to some extent, by the opacity. If the strongly buoyant region is maintained by the net effect of radiation, the range of initial acceleration is prolonged and, consequently, a greater maximum velocity is reached. If the plume is fairly opaque--say, zs ~ 10--the vertical radiative heating is vigorous in the neighborhood of the source, b u t rapidly dies away with height (Fig. 4). This causes a higher initial acceleration, compared with the case z8 = 0.1, for example, but the higher velocities reached give rise to a larger rate of entrainment from the environment. Thus, beyond about 2 source diameters in height, cooling is faster in the opaque case (Fig. 3), and this has the over-all effect of reducing the height attained b y the top of the plume in a stably stratified environment. This demonstrates further the importance of plume penetration discussed previously b y M o r t o n ) I t is interesting to note t h a t there will be an optimum value of the opacity for which a
z,
(source diameters)
1.0 1.0 1.0 1.0 1.0 0.1 1.0 10
29.8 30.9 32.4 34.1 36.1 31.8 33.2 32.0
For a fixed opacity, the maximum height increases steadily with the source temperature. The optimum opacity (see text) for which a plume will rise to a maximum height is of order unity. Source conditions are: mean radius 10 meters; mean axial velocity, 10 meters/sec. The nondimensional stratification parameter 1 = 2 X 104, corresponding to an atmospheric lapse of about 1.5~ per 100 meters.
plume with given source radius, velocity, and temperature will reach a maximum height, and this suggests the possibility that a buoyant plume of hot noxious products might be removed to greater heights by suitable seeding with carbon dust. The temperature at the source is the largest single factor in determining the maximum height to which a plume will rise, although variations in opacity can still produce effects of about 5%-10%. This is indicated in Table I, where heights of zero vertical velocity are tabulated for different source temperatures and opacities.
ACKNOWLEDGMENT
The author is indebted to Dr. B. R. Morton for his encouragment and stimulating discussions throughout the preparation of this paper.
REFERENCES 1. NIELSEN,H. J. AND TAO, L. N.: Tenth Symposium (International) on Combustion, p. 965, The Combustion Institute, 1965. 2. MURGAI, M. P.: J. Fluid Mech. 12, 441 (1962). 3. MORTON, B. R.: Tenth Symposium (International) on Combustion, p. 973, The Combustion Institute, 1965.
R A D I A T I O N E F F E C T S ON L A R G E F I R E P L U M E S 4. THOMAS, P. H.: Ninth Symposium (International) on Combustion, p. 844, Academic Press, 1963. 5. Ricou, F. P. AND SPALDING, D. B.: J. Fluid Mech. 11, 21 (1961).
515
6. CHANDRASEKHAR, S..* Radiative Transfer, p. 9, Dover, 1950.
7. KONDRATIEV, K. YA.: Radiative Heat Exchange in the Atmosphere, p. 3, Pergamon, 1965. 8. MORTON, B. R.: J. Fluid Mech. 5, 151 (1959).
COMMENTS Dr. W. Unterberg (Rocketdyne): How were the results of this investigation influenced by the nature of the radiating gas, i.e., the presence of luminous solid particles such as soot? Mr. R. K. Smith: The presence or absence of luminous solid particles, such as soot, is accounted for through the opacity z,, defined at the source level. Thus, one must ask what effect these have on the opacity.
Plumes which contain large amounts of solid particles will be fairly opaque (i.e., z, >> 1) and will generally have a dark edge except for patches of hot gas which appear intermittently at the plume edge. These are radiation bursts, which are described in the paper, and their effect is taken into account in the model depending on zs. Plumes having a lesser concentration of solid particles will be less opaque and will fit into the intermediate (z8 ~ 1) or transparent (z8 << 1) classifications also described.