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On power laws for describing the mass flux in the near field of fires
Fire Safety Journal 27 (19%) 159-178 copyright © 1997 Published by Elsevier Science Ltd. Printed in Northern Ireland 0379-7112/%/$15.00 PI!:S0379-7112(96)00039-2 Crown
tl' ELSEVIER
O n P o w e r Laws for D e s c r i b i n g the Mass Flux in the N e a r Field o f Fires Michael Poreh a & Howard P. Morgan b aTechnion--lsrael Institute of Technology,Department of Civil Engineering, Haifa 32000, Israel bFRS, Building Research Establishment, Garston, Watford WD2 7JR, UK (Received 25 January 1996; revised version received 24 April 1996; accepted 10 May 1996)
ABSTRACT A general, dimensionally homogeneous power law for describing the vertical variation of the mass flux M within limited regions in the near field of fire-generated turbulent flames and plumes is derived. The derivation assumes that M(z ) is determined by the geometric size of the fire--D, the buoyancy flux generated by the heat flux of the fire--B, and the to~tal mass flux of fresh air required for stoichiometric combustion of fuel--.Ms, which implies that, in general, the variation of M depends on a dimensionless number N, which is approximately proportional to Q2/DS, where Q is the heat flux of the fire. Simpler forms of this general law are then derived for the asymptotic cases of small and large values of N and for geometrically similar fires whose heat output Q is determined by D and Ms, which are shown to satisfy the condition N = constant. The required dimensionless specification of regions where such laws may be valid is also derived. Previously suggested power laws, some of which appear to be contradictory to each other, are then examined. Only two of them have been found to be incompatible with the derived power laws. The remainder are found to be of the simpler forms derived for small, large and constant values of N. Crown copyright © 1997 Published by Elsevier Science Ltd. NOTATION Ai
B Bi Cm
Cp
Numbers which vary with N Mean buoyancy flux (m 4 s -3) Numbers A coefficient defined by eqn (1) Specific heat of air (and smoky gases) at constant pressure (kW kg -1 K -1) 159
160
d D ei
F(x) g
tic hi ki L m
Mv Ms N P q
O Q* F
T W Z
Zo AT Ap P P~
M. Poreh, H. P. M o r g a n
Power of D Diameter of fire (m) Numbers Designates 'a function of x' Acceleration due to gravity (m s -1) Heat of combustion (kJ kg-1) Numbers Constant numbers Mean flame height (m) Power of z Mass flux in an axisymmetric plume (kg s-1) Mass flux of air needed for stoichiometric combustion (kg s-I) Dimensionless number defined by eqn (24) Perimeter of fire (m) Power of Q Heat flux (kW) Dimensionless number defined by eqn (26) Mass stoichiometric ratio of air to volatile fire gases Absolute local temperature of gases (K) Absolute ambient temperature (K) Local vertical velocity of gases (m s -l) Vertical distance (m) Height of virtual source (m) r - r. (K) 191 -- O (kg m - 3 ) Local density of gases (kg m -3) Density of ambient air (kg m -3)
1 INTRODUCTION The vertical variation of the mass flux M in the far field of an axi-symmetrical plume, where the density and temperature of the plume differ only slightly from the ambient density and temperature, is usually described by the power law 1 M = Cmp.[gQl(pT~Cp)l'/3Z 5/3 ~ Q'/3zS/3
(1)
where Q = fA CopATw d A is the mean heat flux across a horizontal plane A, z is the height of that plane above a 'virtual point source', w, p, and T are the instantaneous vertical velocity, density and temperature at dA, p~, and T~ are the ambient density and temperature, AT = T - E, C., is a
Mass flux in near field of fires
161
coefficient of the order of 0.21 z and ff denotes the time average of any variable w. The structure of strongly buoyant plumes from a point source is more complicated. However, it has been proposed by Cetegen et a l . , 2 on the basis of the work of Morton, 3 that the mass flow in such plumes is given by precisely the same equation. That is, the algebraic equations for the plume mass flux for both weakly and strongly buoyant plumes have the same form. Power laws are very convenient for engineering calculations. In addition, they suggest the existence of dynamic similarity in the region where the,y are valid. For these reasons, many attempts have been made to look for power laws that could describe the mass flux of flames and plumes in the near field, although it is recognised that power laws that correlate M with powers of z, Q, and D are inherently approximate, as they neglect the assumed secondary effect of other variables. However, while the uniqueness and validity of the above far field power law have been well established, many different and apparently contradictory power laws have been proposed for describing the variation of the mass flux within specific regions of the near field. It is also noted that the boundaries of these regions have not been specified in a uniform way. Thoma~s et al. 4 have proposed that in the flaming region of large fires M = 0.188Pz 3/~
(2)
100
M/P = 0.188z 1"5
J
o.
0.1
0,01 10
0.1
100
z (m)
Fig. 1.
The variation of M/P with z, the height from the floor, in Hinkley's5 data.
162
M. Poreh, 1t. P. Morgan
where P---4D is the perimeter of the fire and z is the height above the floor. Hinkley 5 has shown (see Fig. 1) that this correlation is in good agreement with a large set of measurements above (mostly) wooden, square crib fires in the range 0.2 - D (m) -< 4.5 and 0.01 -< Q (MW) -< 30. As a result, it has since been widely used for fire engineering design in the UK. Cox and Chitty 6 measured the properties of u n b o u n d e d fire plumes. Following McCaffrey, 7 the flaming plume was divided into three regions, whose boundaries were defined in terms of z ' = z / Q 2/5, and a different power law of the form
M / Q ~x (z/Q2/5) ''
(3)
was proposed for each region. It was then proposed that close to the base of the flame m = 1/2, so that M ~ O4/Sz'/2
(4)
A value of m = 5/3 was proposed for the region near the top of the flame, which gives
M = k'Ql/3z5/3
(5)
The value of the coefficient k' in the above equation is, however, smaller than the corresponding coefficient calculated from (1) for the far field. The authors did not suggest a definite value for m in the intermediate region, but it appears from Fig. 6(b) in their paper that m in that region was in the range 1 < m < 3/2. In a later paper, McCaffrey and Cox 8 proposed that in the intermediate region the data could be approximately described by:
M / Q ~ (z/QZ/5) L3
(6)
It is interesting to note that only a single, square, porous refractory burner was used in the experiments of Cox and Chitty 6 and that its diameter, D = 0.3 m, does not appear in any of the proposed laws. In a later study, aimed at identifying the 'secondary effects of the source dimension', Cox and Chitty 9 doubled the size of the burner and found that these effects were 'far greater than expected'. No explicit power laws for M were proposed in the second study, which focused on the variation of the axial velocity W and the temperature rise AT in the flame. It is noted, however, that all the measurements of W/(Q~/SD ~/2) could be described by 'universal' functions of either z / L or (z + Zo)/L, where L, the flame height, was given by: L = 0.2Q 2/5
(7)
163
Mass flux in near field of fires
Heskestad 1° proposed that in the flaming region M ~ Q z / L , where L is the flame height, which yields for large flame height/diameter ratios: (8)
M o¢ Q3,5 z
Cetegen et al. 2 have measured the mass flux in several plumes created by three gas burners. Their analysis showed that very close to the fire base, where 'a highly wrinkled, thin, laminar diffusion flame surrounds the burner perimeter and the diffusion flame separates air on one side and fuel on the other', (9)
M ~ O z 3/4
In the 'Turbulent Flame Region', which is defined in that paper, the mass flux was described by a power law similar to eqns (1) and (5), except that the vertical distance z" was measured to the time averaged top of the flame: (10)
M ~ Q1/3z"5/3
Delichatsios H analysed the dynamics of turbulent buoyant diffusion flames. His analysis of air entrainment into the burning region of fires suggests that, in general, M F r , [ [ ( r + 1)Mf] = ( z / O ) "
(11)
where Frj, is a Froude Number, based on the buoyancy flux, diameter of the fire and the adiabatic temperature of the flames and Mf is the mass flux of the fuel. He then suggested that n = 1/2 near the base of pool fires or mass fire's, n = 3/2 near the neck-in area of pool fires and n = 5/2 for buoyant flames or in the upper part of pool fires. He also showed that, since for a given fuel Frf is proportional to Mr, M is independent of Q and, thus, M ~ DZz ~lz near the base of pool fires or mass fires
(12)
M oc D z 3~z near the neck-in area of pool fires
(13)
and M ~ z 5/z for buoyant flames or the upper part of pool fires
(14)
Delichatsios 11 has also analysed the vertical variation of air entrainment into 'buoyant plumes that persist after the burning ends' and proposed that in such plumes M [ ( p , B " 3 D s'3) = F ( z [ D )
He then argued that within certain regions, F ( z / D )
(15) in eqn (15) is
164
M. Poreh, 1t. P. Morgan
proportional t o (z/D) m, where m = 1/3, 1 and 5/3, near the base, near the neck and in the far field of the plume, respectively. Finally, Zukoski 12 has shown that measurements in an intermediate zone, which is defined in his paper, could be described by the power law Mo~zD
(16)
Since all the above mentioned laws have been supported by measurements in some regions within the near field, at least two basic questions should be asked. The first one relates to the unexplained observation that the proposed power laws for the near field, which are summarized in Table 1, are of two distinct forms: M oc Ddz m and independent of Q, and M ~ Qqz m and independent of D. The second question concerns the observation that five powers of z, in the range 1/2-
2 ANALYSIS
2.1 Derivation of general dimensionless relationships Equation (1), which was initially proposed by Rouse et al., ~ may also be written as M / p l = CraB 1/3Z5/3
(17)
B = fA ( g / p , ) A p w d A = Ql[p, T~Cplg]
(18)
where
is the total mean buoyancy flux generated by the heat flux of the fire (it is assumed throughout the paper that the heat release rate lost by radiation is negligible).
D 2 z xl2
Q4/Szl/2
Base
(Dz3/4) * ( D z )*
Z 512
Q113z"5/3
__
a113D413zl/3
--
---
__ --
a t / 3 zS/3
Base
Q3/5 z D Z 312
D Z 3/2
Top
Q°52zl3
B u r n i n g region
* These laws are incompatible with the derived general p o w e r law, eqn (29).
Thomas and Hinkley 4'5 Cox and Chitty 6'9 McCaffrey and Cox 8 Heskestad 13 Cetegen et al. z Delichatsios ~ Zukoski 12
Author
TABLE 1 Functional F o r m of P r o p o s e d Mass Flux P o w e r Laws
__
Q1/3D213 z
__ --
--
---
Neck
Buoyant plume
Q1/3 z513
Q1/3z513
Q~13zS13
Q 1/3z5/3
al/3zS/3
Q1/3 zS/3 Q1/azS/3
Far f i e l d
M. Poreh, H. P. Morgan
166
It is noted that eqn (17) can be derived by dimensional analysis using a single assumption: that the volumetric flux M/p~ in the far field is determined by B and z, and is independent of the nature and size of the buoyancy source, the molecular properties of the fluids, and other parameters. Namely, M / p , = F(B, z)
(19)
This assumption is not valid, however, in the near field of a finite size fire, where the effects of other variables on the entrainment are expected. The geometric size of the fire, e.g. its diameter D or its perimeter P, is obviously one of these additional variables. Another variable, which will be assumed to affect the structure of the flaming plume, is the mass flux of fresh air required for stoichiometric combustion of Q. Its value is given by Ms = Q I ( H J r ) = B ( p , T~Cplg)l(H¢/r)
(20)
where Hc is the heat of combustion and r is the mass stoichiometric ratio of air to volatile. Now, the present analysis focuses on turbulent plumes and flames. It is, thus, assumed throughout the paper that the entrainment processes are independent of the molecular properties of the fuel and the air (the validity of this assumption is discussed by Delichatsios It) as well as of other initial conditions, such as the initial velocity of the fuel at the source. To sum up, it is assumed that in the near field M / p , = F(B, z, D, Ms)
(2!)
Dimensional analysis yields the following dimensionless relationship between the variables in eqn (21): M / (p, B'/3D 5/3) = F[z / D, N]
(22)
M / M s = F [ z / D , N]
(23)
N = [(MJpl)/(B'/3D513)] 3
(24)
= {CpTJ[gp~(Hc/r)3]}QZ/D 5
(24a)
= ksQZ/D 5
(24b)
or, equivalently,
where
It is also noted that N = [Cp T~/ (HJr)]3Q*2
(25)
Mass flux in nearfield of]ires
167
where
Q* = Q* = Q/[p~CpT~D2(gD) 1/2]= B/[gO2(gD) 1/2]
(26)
is a dimensionless parameter that is often used in studies of buoyant plumes. The dependence of the structure of fire flames on N is not surprising. It has been used extensively by Heskestad, 13-15 who showed that the mean flame height, L, is given by:
L / D = -1.02 + 15.6N "5
(27)
whose valiidity has been confirmed down to N = 10 -5. Now, if eqn (22) can be approximated within a certain region by a power law, the general form of such a power law must be:
M/pl = kB ~/305/3(Z/D)'n(N)"
(28)
where k is a dimensionless constant. It turns out that the ratio Hc/r in the expression for N does not vary appreciably among a n u m b e r of combustibles 15 and that its value is of the order of 3100 kJ kg -l, as for methanol. Thus, assuming that Pl, Cp and T~ do not vary appreciably either, ks in eqn (24b) is also an approximate constant, of the order of 7 x l 0 - ~ ° m S / ( k W ) 2. For such cases N is approximately proportional to Q2/D5 and one may write that
M ~ Q 1/305/3(Z/O)"(Q2/DS)"
(29)
The form of the above general power laws for specific cases will now be derived. 2.2 The far field In the far field, M should be independent of both N and D. These requirements are satisfied when n = 0 and m = 3/5, for which eqn (28) reduces to eqn (17). 2.3 Fires with small values of N The limit of eqn (22) for N--~ 0 is identical to the equation which has been suggested by Delichatsios ~1 for describing the vertical variation of air entrainment into 'buoyant plumes which persist after the burning'. Namely,
M/(p, B"3D"3) = F ( z / D )
(30)
M. Poreh, 11. P. Morgan
168
Clearly, as claimed by Delichatsios, ~ power laws for this asymptotic case should be of the form:
M/(O, B'/3D~/3) = k ( z / D ) m
(31)
2.4 Fires with large values of N As seen from eqn (27), the flame length of fires with large values of N is relatively large and independent of D. Similarly, it is expected that the mass flux in such fires would also be independent of D. Dividing the first term on the RHS of eqn (23) by [Ms/(plBl/3OS/3)] 3/5, o n e finds that M/M~ = F{z[plBl/3/M~] 3/s, Ms/(plB1/3DSl3)}, Since only the last term in this equation is a function of D, its limit for N---->~, or D --* 0, is:
M/Ms = F{z [0l B'/3/ M~]3/5}
(32)
M/Ms = F { ( z / D ) / N "5}
(32a)
or, using eqn (24),
As seen from eqn (27), the flame length L / D for large values of N is proportional to N 1~5.One may, thus, write that for this limit
M/Ms = F ( z / L )
(33)
M = QF(z/L)/(Hc/r)
(33a)
or
The above equations are consistent with the above mentioned observations that the structure of fires with large values of N is similar and scales proportionally to L. Using eqns (18) and (20), one may also write that
M(Hc/r)/Q = F{(z/Q2/5)[gp~(Hc/r)3/(CoT~)] ~t5}
(34)
The corresponding power law for large values of QZ/D5 is, thus,
M / Q = ( r / Hc){ ( z / Q2~5)[gp~( H J r )3/ ( Cp T~) ] "5}m
(34a)
M / Q ~ (z/Q2'5) " ~ (z/ L)"
(35)
or
Mass flux in near field of fires
169
as in eqn (3), or M ~ Qqz"
(35a)
q = 1 - 2m/5
(35b)
where
Note that eqn (35) can be obtained directly from eqn (29) by letting the power of D in that equation equal zero ( 5 / 3 - m - 5 n =0). Such a procedure does not show, however, the dependence of M on the other variables. 2.5 Fires with N = constant
It has been assumed earlier that both Q and D are independent parameters. This is correct, for example, for gas burners where their heat output is determined by the supply of the fuel. In many other types of fires, however, Q might depend in some way on D and on other factors. One particularly interesting example is the class of geometrically similar fuel beds whose burning rate, namely Q or equivalently B, is determined only by their geometric size and the mass flux required for stoichiometric combustion of Q. In other words, (36)
B = F(D, M.Jpl)
It follows from dimensional considerations that for this class of fires, M s / ( p ~ B1/3D 5/3) = N 1/3 = (k~Q2/O 5)
= constant
(37)
and Q ~ D 5/2
(38)
As N = constant for such fires, n in eqn (29) may be assigned any desired value. It is, thus, possible in such cases to express eqn (29) by different, equivalent power laws. Letting n = 1 / 3 - m / 5 , for example, eqn (29) becomes identical to eqn (35) which has been derived earlier for large values of N and in which D does not appear. Another interesting form of power law for this class of similar fires with N = constant is one in which Q does not appear. Multiplication of both sides of eqn (22) by N 1/6 = ( k s O 2 / D S ) 1/6 gives ( M / p l D5'2){T~Cp/[g(Hc/r)]} 1'2 = N " 6 F ( z / O , N)
(39)
Now, if F in this equation is a power law, one may write that ( M / p , D 5'2) = K{[g(Hc/r)]/T~ Cp}'aN "+ I/6(Z/O )'~
(40)
M. Poreh, H. P. Morgan
170
where K is a universal constant, which gives for n = - 1 / 6 : M / D 5/2 oc ( z / D ) m
(41)
M oc D,~z ,.
(42)
2d/5 = 1 - 2m/5
(43)
or
where d = 5/2 - m, or
Note that eqn (41) can be obtained directly from eqn (29) by letting the power of Q in that equation equal zero (1/3 + 2n = 0). Again, such a procedure does not show the dependence of M on the other variables. It appears that this class of fires, for which Q o~D 5~2, is not a hypothetical one. As shown in Fig. 2, the variation of Q with P (the fire perimeter) in the large data set of square, (mostly) wooden fires, which was used by Hinkley 5 to confirm eqn (2), satisfies this power law. It is, thus, expected that this relationship between Q and D characterises other geometrically similar, free burning fuel beds with ideal turbulent combustion and no significant radiation losses, for which the rate of combustion is controlled by the entrainment of ambient fresh air by the fire. Such a relationship does not exist, for example, in hydrocarbon pool fires, which are Reynolds number dependent up to a diameter of the order of i m and their burning rate for larger diameters is affected by radiative heat
°/
IOO
0 0.1 ¸
0.01'
/
/
0.001 0.1
10
100
P (ml
Fig. 2.
The dependence of the heat flux on the perimeter of the fire in Hinkley's5 data.
Mass flux in near field of fires
171
transfer fi:om the fire to the liquid pool and is proportional to their area) 6 Nevertheless, since in such fires Q ocD 2, the above results, which are based orL the assumption that Q ~ D 5/2, might also describe in an approximate manner some pool fire data.
2.6 Specification of boundaries Since power laws are expected to be valid in limited regions of the near field, it is important to correctly specify the boundaries of such regions. If a height zi, which describes a characteristic location in the near field of a dimensionless property, is a function of D, B and MJpl, one may write, on the basis of dimensional considerations, that zJD = E(N)
(44)
or, equivalently, using eqn (24b), z i / ( k s O 2 ) 1/5 = N-I/SE(N)
(45)
For large N, z~ is expected to be independent of D. It implies that in the limit of large N, F(N) in eqn (44) is proportional to N "5, so that zi/D = k N "5
(46)
zi/(ksQ2) l's = constant
(47)
or
As seen from eqn (27), the equation for the flame height, L, complies with these forms. Similarly, it follows from eqns (44) and(45) that for small values of N, zi/D = constant
(48)
It is concluded from the above analysis that, in general, the domain of each re~;ion within the near field should be specified by At(N) < z / D < Az(N)
(49)
B,(N) < z/(ksQ2) ''5 < B2(N)
(50)
or equivalently
172
M. Poreh, H. P. Morgan
where Ai and B~ are dimensional functions of N, which may vary, of
course, from one region to the other. For the asymptotic case of large values of N, the boundaries of each region should be specified by el < z/(ksQ2) '/5 < e2
(51)
k~ < z / L < k2
(52)
or
where ei and ki are constant numbers, and L is a characteristic height of the fire, such as the flame height, which satisfies the equation L / D = constant. The boundaries of each region for the case of small N, on the other hand, should be specified as h, < z / D < h z
(53)
where hi are constant numbers. Any one of the last equations (51)-(53) may be used for specifying regions in fires with N = constant, as in such fires L / D = constant. It should be noted that the values of the constants in the above equations might be affected by the dimensionless shape and other characteristics of the similar fuel beds. A summary of the results obtained in the analysis is presented in Table 2.
3 DISCUSSION A N D CONCLUSIONS A general form for dimensionally homogeneous power laws that may describe the vertical variation of the mass flux in limited regions of the near field of turbulent flames and plumes has been derived (see summary in Table 2). The derivation, which is based on the assumption that the dynamics of the fire is determined by the total buoyancy flux of the fire, the required supply of fresh air for stoichiometric combustion and the geometric size of the fire, suggests that the entrainment of ambient air into fire flames and plumes depends on the dimensionless number N, which is approximately proportional to Q2/DS. Simpler forms for such power laws were then derived for classes of fires with small, large and constant values of N.
IV
III
II
I
Specification of domains
General Specification of domains Asymptotic form for Q2/DS---> 0 (Buoyant plumes) Specification of domains Asymptotic form for Q 2 / D ~ oo Specification of domains Similar fires with QZ/D5 = constant
Case
29 49 31 53 35 52, 47
Q~/3DS/3(z/D )m(Q2/DS)~ A~(N) < z / D < A z ( N ) M ~ Q1/3DS/3(z/D)m
ht < z / D
41 35 51, 52
Equation no.
Dimensional power law
TABLE 2 Summary of Derived Power Laws
e~
5"
174
M. Poreh, H. P. Morgan
Of particular interest is the result that the vertical variation of M for large values of N may be described by M/Ms = F(z/L)
(54)
where L is the flame size, and that mass flux power laws for the flaming region will be of the form M / Q ~ (z/Q2/5) m ~ ( z / L ) m
(55)
which had been used by McCaffrey 7 and Cox and Chitty. 6 Equation (54), it is noted, is also compatible with a similar conclusion by Heskestad 1° that M / M L = F ( z / L ) , where Mc is the mass flux at z = L. It has also been shown that geometrically similar fires, whose heat output Q is determined by D and Ms/p~, satisfy the condition N = constant. As a result, different but equivalent forms of power law may be used to correlate the dependence of M on Q, D and z. A m o n g those are the forms M ~ D a z " and M ~ Q q z m. In addition, the correct specification of the dimensionless boundaries of regions, where such power laws may be valid, has been determined. Of particular interest is the finding that, in general, the boundaries of each region d e p e n d on N. However, for large values of N, the near field scales proportionally to the flame height L, which in turn is proportional to N 1/5. On the other hand, the different regions within the near field of similar fires with N = constant may be specified relative to either D or L. The previously reviewed power laws will now be compared with the results of the analysis. The power laws proposed by Delichatsios H for the special case of entrainment into buoyant plumes have been shown to be identical to the derived limit of the general law for the limit of small N, eqn (30), which is independent of Ms. On the other hand, it is easily seen that the power laws M ~ D z 3/4 [eqn (9)] and M ~ D Z [eqn (16)] are incompatible with the present results. This observation does not necessarily imply that these two power laws are incorrect, as it is quite possible that the entrainment in the experiments on which they were based had been affected by variables which had not been included in the above dimensional analysis. In fact, as noted by Cetegen et al., 2 eqn (9) describes experiments where a thin laminar diffusion flame was observed, whereas the present analysis is restricted to turbulent flames. It was noted that the form of each of the rest of the proposed power laws for the burning region of the near field is either M ~ Q q z " or M ~ D a z " , which have been derived for the asymptotic cases of large and constant values of N. To examine whether the correlation between the
175
Mass flux in near field of fires 1.0
O 2d/5
o.8
~,,NCC, DE
/
o.~
~"
N,,~,HE
0.4
0.0
+ q
" I ~ "I'N,DE \
.
,
.
1
2 m
Fig. 3. The values of the powers q, d, and m proposed by Cox and Chitty~'9 (CC), Delichatsios 1 (DE), Heskestad "~-'5 (HE), McCaffrey and Cox~ (MC) and Thomas/Hinkley 4"~(TH) (see Table 1). values of the powers in these laws is compatible with the results of the previous analysis, the values of the pairs (re,q) and ( m , 2 d / 5 ) , which appear in these laws, were plotted in Fig. 3. As seen from the figure, all the values of q and 2 d / 5 are equal to 1 - 2m/5, as suggested by eqns (35b) and (43). The agreement between the M ~ Q q z " type power laws with the result of the previous analysis is not surprising. Examination of the analysis and experimental data of Heskestad, 14''5 McCaffrey and Cox 8 and Cox and Chitty 6 clearly shows that they were proposed for fires with large values of N. The reason that the data measured by Cox and Chitty 9 in the experiments with the D = 0.6 m burner could not be matched with the earlier data is, perhaps, that the value of N in some of the experiments was not sufficiently large. Now, it has already been shown that the value of N in the data of Hinkley 5 was an approximate constant. Thus, the power law M = D z 3/2 proposed by Thomas et al. 4 is equivalent to the power law M oc Q2/Sz3/2" Both forms are compatible with our analysis. The d~Lta presented in Fig. 5 of the paper of Delichatsios I' for confirming the M = D 2 z ~12 and M ocDz3/2 laws are from experiments with relatively large values of N. Thus, they too should have assumed the form M = Qqz m. Since these measurements have been made in pool fires, one
M. Poreh, H. P. Morgan
176
explanation is that the values of N in this particular data set were approximately constant. If this hypothesis is correct, one could also express these laws in the M o c O q z m form: the M o c D 2 z in law could be replaced by M ~ oa/szl/2, which is similar to the law proposed by Cox and Chitty, 6 and M ~ D z 3/2, which is identical to the power law proposed by T h o m a s et al., 4 could be replaced by M ~ O2/Sz3/2. Table 1 shows that five values have been suggested for the power m, which increases from m = 1/2 at the base of the fire to m = 5/3 in the far field. The only exception is the M ~ z 5n law suggested by Delichatsios. 11 Does this observation suggest that the near field should be divided into four or five regions or, perhaps, that the different power laws are actually tangents (in log-log graphs) to a more general, continuous variation of M in which d M / d z is proportional to z p and p varies continuously from - 1 / 2 , near the base of the fire, to 2/3, in the far field? It is difficult to answer this question. It should be realised that experimental determination or verifications of such power laws are inherently very difficult. In addition to the scatter of the data and the limited span of each region, the origin of z is not uniquely defined. By changing this origin, the value of m can be modified. For example, in the power law proposed by T h o m a s et al. 4 and confirmed by Hinkley, 5 M = 0.188 PZ 3n, Z is the height from the floor on which the w o o d e n cribs were built. This choice is natural, but the use of another origin for z is possible. In Fig. 4, the variation of M p / Q 1/3 versus (z + 0.1P) was plotted. As seen from this figure, these same data 100 0.5(z+O.I p)5/3
#~
10
F
~
°
0.1 o
0.01 0.1
10
100
z+O.1P (m)
Fig. 4. T h e variation of M / Q tl3 with z + 0.1P in Hinkley's 5 data.
177
Mass flux in near field of fires
could also be described, with equal accuracy, by the form of the far field power law M = CnQ1/3(z + Z0) 5/3
(56)
where z0 :--0.1P and a slightly smaller proportionality coefficient, Cn = 0.5 (kg s-l)/(J~/lWl/3mS/3). The four data points shown in this figure by circles, which deviate from this correlation, are the same points that deviated from the 3/2 power law shown in Fig. 1. One may, similarly, assume that the data near the top of the flame that were described by the power laws M oc Ql13zS/3 might also be described by a 3/2 power law, similar to those proposed by Thomas et al. 4 Similarly, it is quite possible that the value of m = 1.3 in the power law of McCaffrey and Cox s might be modified to m = 3/2. If these assumptions are confirmed, the flame region for large values of N and for fires with N = constant, could be divided into three regions. The entrainment in each region will be given by eqn (35): M / Q oc (z/Q2/5) m, suggested by Cox and Chitty, 6 with m = 1/2, 1 and 3/2. As mentioned earlier, the dimensionless parameter Q* is often used in the analysis of buoyant plumes and flames. Cetegen et al. 2 suggest that the relative flame height L / D is proportional to Q,2/3, and to Q,2/5, for Q* < 1 and Q* ~-"1 respectively, and show that the entrainment in the near field depends ,an z / L . Similarly, eqn (27) is often written as L / D = - 1 / 0 2 + k ' O .2/5
(57)
where k' is a dimensionless number. Zukoski 12 writes that 'some consensus has been reached that a parameter such as Q*, or some variation which is proportional to Q*, may be used for modeling diffusion flames', which suggests that M / ( p l B'/3D 5'3) = F ( Z / O , Q*)
(58)
Although, for most cases, N is approximately proportional to Q,2 [see eqn (25)], it :should be noted that there is a fundamental difference between eqns (58) and (22) and between eqns (57) and (27). For example, eqn (27), contrary to eqn (57), suggests that the flame height is affected by the oxygen level in the ambient air, an effect which might be important for the analysis of fires burning in closed compartments.
ACKNOWLEDGEMENTS This work started during the stay of the first author at the FRS. Partial support for the continuation of the work was obtained from the Fund for the Promoting of Research at the Technion. Thanks are due to P.
178
M. Poreh, H. P. Morgan
L. Hinkley, for his help in retrieving the original data from his 1986 paper, and to G. Cox from the FRS and S. Trebukov from the Technion, who have reviewed the draft of this work.
REFERENCES 1. Rouse, H., Yih, C. S. & Humphreys, H. W., Gravitational convection from a boundary source. Tellus, 4 (1952) 201-210. 2. Cetegen, B. M., Zukoski, E. E. & Kubota, T., Entrainment in the near field and far field of fire plumes. Combustion Science and Technology, 39 (1984) 305-331. 3. Morton, B. R., Modelling fire plumes. Tenth Symposium (International) on Combustion, p. 973. Academic Press, New York, 1965. 4. Thomas, P. H. et al., Investigation into the flow of hot gases in roof venting. Fire Research Technical Paper No. 7, HMSO, London, 1963. 5. Hinkley, P. L., Rates of production of gases in roof venting experiments. Fire Safety Journal, 10 (1986) 57-65. 6. Cox, G. & Chitty, R., A study of the deterministic properties of unbounded fire plumes. Combustion and Flame, 39 (1980) 191-202. 7. McCaffrey, B. R., Purely buoyant diffusion flames--some experimental results in chemical and physical processes in combustion. Fall 1978 Meeting of the Eastern Section of the Combustion Institute, 1978. 8. McCaffrey, B. R. & Cox, G., NBSIR 82-2473, National Bureau of Standards (NIST), Washington, DC, USA, 1982. 9. Cox, G. & Chitty, R., Some source dependence effects of unbounded fires. Combustion and Flame, 60 (1985) 219-232. 10. Heskestad G., Fire plume air entrainment according to two competing assumptions. Twenty-first Symposium (International) on Combustion, pp. 111-120. Academic Press, New York, 1986. 11. Delichatsios, M. A., Air entrainment into buoyant jet flames and pool fires. Combustion and Flame, 70 (1987) 33-46. 12. Zukoski, E. E., Mass flux in fire plumes. Fire Safety Science--Proceedings of the Fourth International Symposium, pp. 137-147, 1984. 13. Heskestad, G., Luminous heights of turbulent diffusion flames. Fire Safety Journal, 5 (1983) 103-108. 14. Heskestad, G., Virtual origins of fire plumes. Fire Safety Journal, 5 (1983) 109-114. 15. Heskestad, G., Engineering relations for fire plume. Fire Safety Journal, 7 (1984) 25-32. 16. Mudan, K. S. & Crose, P. A., Hazard calculations for large open hydrocarbon fires. SFPE Handbook of Fire Protection Engineering, Sect. 2, Ch. 4. National Fire Protection Association, USA, 1988.
tl' ELSEVIER
O n P o w e r Laws for D e s c r i b i n g the Mass Flux in the N e a r Field o f Fires Michael Poreh a & Howard P. Morgan b aTechnion--lsrael Institute of Technology,Department of Civil Engineering, Haifa 32000, Israel bFRS, Building Research Establishment, Garston, Watford WD2 7JR, UK (Received 25 January 1996; revised version received 24 April 1996; accepted 10 May 1996)
ABSTRACT A general, dimensionally homogeneous power law for describing the vertical variation of the mass flux M within limited regions in the near field of fire-generated turbulent flames and plumes is derived. The derivation assumes that M(z ) is determined by the geometric size of the fire--D, the buoyancy flux generated by the heat flux of the fire--B, and the to~tal mass flux of fresh air required for stoichiometric combustion of fuel--.Ms, which implies that, in general, the variation of M depends on a dimensionless number N, which is approximately proportional to Q2/DS, where Q is the heat flux of the fire. Simpler forms of this general law are then derived for the asymptotic cases of small and large values of N and for geometrically similar fires whose heat output Q is determined by D and Ms, which are shown to satisfy the condition N = constant. The required dimensionless specification of regions where such laws may be valid is also derived. Previously suggested power laws, some of which appear to be contradictory to each other, are then examined. Only two of them have been found to be incompatible with the derived power laws. The remainder are found to be of the simpler forms derived for small, large and constant values of N. Crown copyright © 1997 Published by Elsevier Science Ltd. NOTATION Ai
B Bi Cm
Cp
Numbers which vary with N Mean buoyancy flux (m 4 s -3) Numbers A coefficient defined by eqn (1) Specific heat of air (and smoky gases) at constant pressure (kW kg -1 K -1) 159
160
d D ei
F(x) g
tic hi ki L m
Mv Ms N P q
O Q* F
T W Z
Zo AT Ap P P~
M. Poreh, H. P. M o r g a n
Power of D Diameter of fire (m) Numbers Designates 'a function of x' Acceleration due to gravity (m s -1) Heat of combustion (kJ kg-1) Numbers Constant numbers Mean flame height (m) Power of z Mass flux in an axisymmetric plume (kg s-1) Mass flux of air needed for stoichiometric combustion (kg s-I) Dimensionless number defined by eqn (24) Perimeter of fire (m) Power of Q Heat flux (kW) Dimensionless number defined by eqn (26) Mass stoichiometric ratio of air to volatile fire gases Absolute local temperature of gases (K) Absolute ambient temperature (K) Local vertical velocity of gases (m s -l) Vertical distance (m) Height of virtual source (m) r - r. (K) 191 -- O (kg m - 3 ) Local density of gases (kg m -3) Density of ambient air (kg m -3)
1 INTRODUCTION The vertical variation of the mass flux M in the far field of an axi-symmetrical plume, where the density and temperature of the plume differ only slightly from the ambient density and temperature, is usually described by the power law 1 M = Cmp.[gQl(pT~Cp)l'/3Z 5/3 ~ Q'/3zS/3
(1)
where Q = fA CopATw d A is the mean heat flux across a horizontal plane A, z is the height of that plane above a 'virtual point source', w, p, and T are the instantaneous vertical velocity, density and temperature at dA, p~, and T~ are the ambient density and temperature, AT = T - E, C., is a
Mass flux in near field of fires
161
coefficient of the order of 0.21 z and ff denotes the time average of any variable w. The structure of strongly buoyant plumes from a point source is more complicated. However, it has been proposed by Cetegen et a l . , 2 on the basis of the work of Morton, 3 that the mass flow in such plumes is given by precisely the same equation. That is, the algebraic equations for the plume mass flux for both weakly and strongly buoyant plumes have the same form. Power laws are very convenient for engineering calculations. In addition, they suggest the existence of dynamic similarity in the region where the,y are valid. For these reasons, many attempts have been made to look for power laws that could describe the mass flux of flames and plumes in the near field, although it is recognised that power laws that correlate M with powers of z, Q, and D are inherently approximate, as they neglect the assumed secondary effect of other variables. However, while the uniqueness and validity of the above far field power law have been well established, many different and apparently contradictory power laws have been proposed for describing the variation of the mass flux within specific regions of the near field. It is also noted that the boundaries of these regions have not been specified in a uniform way. Thoma~s et al. 4 have proposed that in the flaming region of large fires M = 0.188Pz 3/~
(2)
100
M/P = 0.188z 1"5
J
o.
0.1
0,01 10
0.1
100
z (m)
Fig. 1.
The variation of M/P with z, the height from the floor, in Hinkley's5 data.
162
M. Poreh, 1t. P. Morgan
where P---4D is the perimeter of the fire and z is the height above the floor. Hinkley 5 has shown (see Fig. 1) that this correlation is in good agreement with a large set of measurements above (mostly) wooden, square crib fires in the range 0.2 - D (m) -< 4.5 and 0.01 -< Q (MW) -< 30. As a result, it has since been widely used for fire engineering design in the UK. Cox and Chitty 6 measured the properties of u n b o u n d e d fire plumes. Following McCaffrey, 7 the flaming plume was divided into three regions, whose boundaries were defined in terms of z ' = z / Q 2/5, and a different power law of the form
M / Q ~x (z/Q2/5) ''
(3)
was proposed for each region. It was then proposed that close to the base of the flame m = 1/2, so that M ~ O4/Sz'/2
(4)
A value of m = 5/3 was proposed for the region near the top of the flame, which gives
M = k'Ql/3z5/3
(5)
The value of the coefficient k' in the above equation is, however, smaller than the corresponding coefficient calculated from (1) for the far field. The authors did not suggest a definite value for m in the intermediate region, but it appears from Fig. 6(b) in their paper that m in that region was in the range 1 < m < 3/2. In a later paper, McCaffrey and Cox 8 proposed that in the intermediate region the data could be approximately described by:
M / Q ~ (z/QZ/5) L3
(6)
It is interesting to note that only a single, square, porous refractory burner was used in the experiments of Cox and Chitty 6 and that its diameter, D = 0.3 m, does not appear in any of the proposed laws. In a later study, aimed at identifying the 'secondary effects of the source dimension', Cox and Chitty 9 doubled the size of the burner and found that these effects were 'far greater than expected'. No explicit power laws for M were proposed in the second study, which focused on the variation of the axial velocity W and the temperature rise AT in the flame. It is noted, however, that all the measurements of W/(Q~/SD ~/2) could be described by 'universal' functions of either z / L or (z + Zo)/L, where L, the flame height, was given by: L = 0.2Q 2/5
(7)
163
Mass flux in near field of fires
Heskestad 1° proposed that in the flaming region M ~ Q z / L , where L is the flame height, which yields for large flame height/diameter ratios: (8)
M o¢ Q3,5 z
Cetegen et al. 2 have measured the mass flux in several plumes created by three gas burners. Their analysis showed that very close to the fire base, where 'a highly wrinkled, thin, laminar diffusion flame surrounds the burner perimeter and the diffusion flame separates air on one side and fuel on the other', (9)
M ~ O z 3/4
In the 'Turbulent Flame Region', which is defined in that paper, the mass flux was described by a power law similar to eqns (1) and (5), except that the vertical distance z" was measured to the time averaged top of the flame: (10)
M ~ Q1/3z"5/3
Delichatsios H analysed the dynamics of turbulent buoyant diffusion flames. His analysis of air entrainment into the burning region of fires suggests that, in general, M F r , [ [ ( r + 1)Mf] = ( z / O ) "
(11)
where Frj, is a Froude Number, based on the buoyancy flux, diameter of the fire and the adiabatic temperature of the flames and Mf is the mass flux of the fuel. He then suggested that n = 1/2 near the base of pool fires or mass fire's, n = 3/2 near the neck-in area of pool fires and n = 5/2 for buoyant flames or in the upper part of pool fires. He also showed that, since for a given fuel Frf is proportional to Mr, M is independent of Q and, thus, M ~ DZz ~lz near the base of pool fires or mass fires
(12)
M oc D z 3~z near the neck-in area of pool fires
(13)
and M ~ z 5/z for buoyant flames or the upper part of pool fires
(14)
Delichatsios 11 has also analysed the vertical variation of air entrainment into 'buoyant plumes that persist after the burning ends' and proposed that in such plumes M [ ( p , B " 3 D s'3) = F ( z [ D )
He then argued that within certain regions, F ( z / D )
(15) in eqn (15) is
164
M. Poreh, 1t. P. Morgan
proportional t o (z/D) m, where m = 1/3, 1 and 5/3, near the base, near the neck and in the far field of the plume, respectively. Finally, Zukoski 12 has shown that measurements in an intermediate zone, which is defined in his paper, could be described by the power law Mo~zD
(16)
Since all the above mentioned laws have been supported by measurements in some regions within the near field, at least two basic questions should be asked. The first one relates to the unexplained observation that the proposed power laws for the near field, which are summarized in Table 1, are of two distinct forms: M oc Ddz m and independent of Q, and M ~ Qqz m and independent of D. The second question concerns the observation that five powers of z, in the range 1/2-
2 ANALYSIS
2.1 Derivation of general dimensionless relationships Equation (1), which was initially proposed by Rouse et al., ~ may also be written as M / p l = CraB 1/3Z5/3
(17)
B = fA ( g / p , ) A p w d A = Ql[p, T~Cplg]
(18)
where
is the total mean buoyancy flux generated by the heat flux of the fire (it is assumed throughout the paper that the heat release rate lost by radiation is negligible).
D 2 z xl2
Q4/Szl/2
Base
(Dz3/4) * ( D z )*
Z 512
Q113z"5/3
__
a113D413zl/3
--
---
__ --
a t / 3 zS/3
Base
Q3/5 z D Z 312
D Z 3/2
Top
Q°52zl3
B u r n i n g region
* These laws are incompatible with the derived general p o w e r law, eqn (29).
Thomas and Hinkley 4'5 Cox and Chitty 6'9 McCaffrey and Cox 8 Heskestad 13 Cetegen et al. z Delichatsios ~ Zukoski 12
Author
TABLE 1 Functional F o r m of P r o p o s e d Mass Flux P o w e r Laws
__
Q1/3D213 z
__ --
--
---
Neck
Buoyant plume
Q1/3 z513
Q1/3z513
Q~13zS13
Q 1/3z5/3
al/3zS/3
Q1/3 zS/3 Q1/azS/3
Far f i e l d
M. Poreh, H. P. Morgan
166
It is noted that eqn (17) can be derived by dimensional analysis using a single assumption: that the volumetric flux M/p~ in the far field is determined by B and z, and is independent of the nature and size of the buoyancy source, the molecular properties of the fluids, and other parameters. Namely, M / p , = F(B, z)
(19)
This assumption is not valid, however, in the near field of a finite size fire, where the effects of other variables on the entrainment are expected. The geometric size of the fire, e.g. its diameter D or its perimeter P, is obviously one of these additional variables. Another variable, which will be assumed to affect the structure of the flaming plume, is the mass flux of fresh air required for stoichiometric combustion of Q. Its value is given by Ms = Q I ( H J r ) = B ( p , T~Cplg)l(H¢/r)
(20)
where Hc is the heat of combustion and r is the mass stoichiometric ratio of air to volatile. Now, the present analysis focuses on turbulent plumes and flames. It is, thus, assumed throughout the paper that the entrainment processes are independent of the molecular properties of the fuel and the air (the validity of this assumption is discussed by Delichatsios It) as well as of other initial conditions, such as the initial velocity of the fuel at the source. To sum up, it is assumed that in the near field M / p , = F(B, z, D, Ms)
(2!)
Dimensional analysis yields the following dimensionless relationship between the variables in eqn (21): M / (p, B'/3D 5/3) = F[z / D, N]
(22)
M / M s = F [ z / D , N]
(23)
N = [(MJpl)/(B'/3D513)] 3
(24)
= {CpTJ[gp~(Hc/r)3]}QZ/D 5
(24a)
= ksQZ/D 5
(24b)
or, equivalently,
where
It is also noted that N = [Cp T~/ (HJr)]3Q*2
(25)
Mass flux in nearfield of]ires
167
where
Q* = Q* = Q/[p~CpT~D2(gD) 1/2]= B/[gO2(gD) 1/2]
(26)
is a dimensionless parameter that is often used in studies of buoyant plumes. The dependence of the structure of fire flames on N is not surprising. It has been used extensively by Heskestad, 13-15 who showed that the mean flame height, L, is given by:
L / D = -1.02 + 15.6N "5
(27)
whose valiidity has been confirmed down to N = 10 -5. Now, if eqn (22) can be approximated within a certain region by a power law, the general form of such a power law must be:
M/pl = kB ~/305/3(Z/D)'n(N)"
(28)
where k is a dimensionless constant. It turns out that the ratio Hc/r in the expression for N does not vary appreciably among a n u m b e r of combustibles 15 and that its value is of the order of 3100 kJ kg -l, as for methanol. Thus, assuming that Pl, Cp and T~ do not vary appreciably either, ks in eqn (24b) is also an approximate constant, of the order of 7 x l 0 - ~ ° m S / ( k W ) 2. For such cases N is approximately proportional to Q2/D5 and one may write that
M ~ Q 1/305/3(Z/O)"(Q2/DS)"
(29)
The form of the above general power laws for specific cases will now be derived. 2.2 The far field In the far field, M should be independent of both N and D. These requirements are satisfied when n = 0 and m = 3/5, for which eqn (28) reduces to eqn (17). 2.3 Fires with small values of N The limit of eqn (22) for N--~ 0 is identical to the equation which has been suggested by Delichatsios ~1 for describing the vertical variation of air entrainment into 'buoyant plumes which persist after the burning'. Namely,
M/(p, B"3D"3) = F ( z / D )
(30)
M. Poreh, 11. P. Morgan
168
Clearly, as claimed by Delichatsios, ~ power laws for this asymptotic case should be of the form:
M/(O, B'/3D~/3) = k ( z / D ) m
(31)
2.4 Fires with large values of N As seen from eqn (27), the flame length of fires with large values of N is relatively large and independent of D. Similarly, it is expected that the mass flux in such fires would also be independent of D. Dividing the first term on the RHS of eqn (23) by [Ms/(plBl/3OS/3)] 3/5, o n e finds that M/M~ = F{z[plBl/3/M~] 3/s, Ms/(plB1/3DSl3)}, Since only the last term in this equation is a function of D, its limit for N---->~, or D --* 0, is:
M/Ms = F{z [0l B'/3/ M~]3/5}
(32)
M/Ms = F { ( z / D ) / N "5}
(32a)
or, using eqn (24),
As seen from eqn (27), the flame length L / D for large values of N is proportional to N 1~5.One may, thus, write that for this limit
M/Ms = F ( z / L )
(33)
M = QF(z/L)/(Hc/r)
(33a)
or
The above equations are consistent with the above mentioned observations that the structure of fires with large values of N is similar and scales proportionally to L. Using eqns (18) and (20), one may also write that
M(Hc/r)/Q = F{(z/Q2/5)[gp~(Hc/r)3/(CoT~)] ~t5}
(34)
The corresponding power law for large values of QZ/D5 is, thus,
M / Q = ( r / Hc){ ( z / Q2~5)[gp~( H J r )3/ ( Cp T~) ] "5}m
(34a)
M / Q ~ (z/Q2'5) " ~ (z/ L)"
(35)
or
Mass flux in near field of fires
169
as in eqn (3), or M ~ Qqz"
(35a)
q = 1 - 2m/5
(35b)
where
Note that eqn (35) can be obtained directly from eqn (29) by letting the power of D in that equation equal zero ( 5 / 3 - m - 5 n =0). Such a procedure does not show, however, the dependence of M on the other variables. 2.5 Fires with N = constant
It has been assumed earlier that both Q and D are independent parameters. This is correct, for example, for gas burners where their heat output is determined by the supply of the fuel. In many other types of fires, however, Q might depend in some way on D and on other factors. One particularly interesting example is the class of geometrically similar fuel beds whose burning rate, namely Q or equivalently B, is determined only by their geometric size and the mass flux required for stoichiometric combustion of Q. In other words, (36)
B = F(D, M.Jpl)
It follows from dimensional considerations that for this class of fires, M s / ( p ~ B1/3D 5/3) = N 1/3 = (k~Q2/O 5)
= constant
(37)
and Q ~ D 5/2
(38)
As N = constant for such fires, n in eqn (29) may be assigned any desired value. It is, thus, possible in such cases to express eqn (29) by different, equivalent power laws. Letting n = 1 / 3 - m / 5 , for example, eqn (29) becomes identical to eqn (35) which has been derived earlier for large values of N and in which D does not appear. Another interesting form of power law for this class of similar fires with N = constant is one in which Q does not appear. Multiplication of both sides of eqn (22) by N 1/6 = ( k s O 2 / D S ) 1/6 gives ( M / p l D5'2){T~Cp/[g(Hc/r)]} 1'2 = N " 6 F ( z / O , N)
(39)
Now, if F in this equation is a power law, one may write that ( M / p , D 5'2) = K{[g(Hc/r)]/T~ Cp}'aN "+ I/6(Z/O )'~
(40)
M. Poreh, H. P. Morgan
170
where K is a universal constant, which gives for n = - 1 / 6 : M / D 5/2 oc ( z / D ) m
(41)
M oc D,~z ,.
(42)
2d/5 = 1 - 2m/5
(43)
or
where d = 5/2 - m, or
Note that eqn (41) can be obtained directly from eqn (29) by letting the power of Q in that equation equal zero (1/3 + 2n = 0). Again, such a procedure does not show the dependence of M on the other variables. It appears that this class of fires, for which Q o~D 5~2, is not a hypothetical one. As shown in Fig. 2, the variation of Q with P (the fire perimeter) in the large data set of square, (mostly) wooden fires, which was used by Hinkley 5 to confirm eqn (2), satisfies this power law. It is, thus, expected that this relationship between Q and D characterises other geometrically similar, free burning fuel beds with ideal turbulent combustion and no significant radiation losses, for which the rate of combustion is controlled by the entrainment of ambient fresh air by the fire. Such a relationship does not exist, for example, in hydrocarbon pool fires, which are Reynolds number dependent up to a diameter of the order of i m and their burning rate for larger diameters is affected by radiative heat
°/
IOO
0 0.1 ¸
0.01'
/
/
0.001 0.1
10
100
P (ml
Fig. 2.
The dependence of the heat flux on the perimeter of the fire in Hinkley's5 data.
Mass flux in near field of fires
171
transfer fi:om the fire to the liquid pool and is proportional to their area) 6 Nevertheless, since in such fires Q ocD 2, the above results, which are based orL the assumption that Q ~ D 5/2, might also describe in an approximate manner some pool fire data.
2.6 Specification of boundaries Since power laws are expected to be valid in limited regions of the near field, it is important to correctly specify the boundaries of such regions. If a height zi, which describes a characteristic location in the near field of a dimensionless property, is a function of D, B and MJpl, one may write, on the basis of dimensional considerations, that zJD = E(N)
(44)
or, equivalently, using eqn (24b), z i / ( k s O 2 ) 1/5 = N-I/SE(N)
(45)
For large N, z~ is expected to be independent of D. It implies that in the limit of large N, F(N) in eqn (44) is proportional to N "5, so that zi/D = k N "5
(46)
zi/(ksQ2) l's = constant
(47)
or
As seen from eqn (27), the equation for the flame height, L, complies with these forms. Similarly, it follows from eqns (44) and(45) that for small values of N, zi/D = constant
(48)
It is concluded from the above analysis that, in general, the domain of each re~;ion within the near field should be specified by At(N) < z / D < Az(N)
(49)
B,(N) < z/(ksQ2) ''5 < B2(N)
(50)
or equivalently
172
M. Poreh, H. P. Morgan
where Ai and B~ are dimensional functions of N, which may vary, of
course, from one region to the other. For the asymptotic case of large values of N, the boundaries of each region should be specified by el < z/(ksQ2) '/5 < e2
(51)
k~ < z / L < k2
(52)
or
where ei and ki are constant numbers, and L is a characteristic height of the fire, such as the flame height, which satisfies the equation L / D = constant. The boundaries of each region for the case of small N, on the other hand, should be specified as h, < z / D < h z
(53)
where hi are constant numbers. Any one of the last equations (51)-(53) may be used for specifying regions in fires with N = constant, as in such fires L / D = constant. It should be noted that the values of the constants in the above equations might be affected by the dimensionless shape and other characteristics of the similar fuel beds. A summary of the results obtained in the analysis is presented in Table 2.
3 DISCUSSION A N D CONCLUSIONS A general form for dimensionally homogeneous power laws that may describe the vertical variation of the mass flux in limited regions of the near field of turbulent flames and plumes has been derived (see summary in Table 2). The derivation, which is based on the assumption that the dynamics of the fire is determined by the total buoyancy flux of the fire, the required supply of fresh air for stoichiometric combustion and the geometric size of the fire, suggests that the entrainment of ambient air into fire flames and plumes depends on the dimensionless number N, which is approximately proportional to Q2/DS. Simpler forms for such power laws were then derived for classes of fires with small, large and constant values of N.
IV
III
II
I
Specification of domains
General Specification of domains Asymptotic form for Q2/DS---> 0 (Buoyant plumes) Specification of domains Asymptotic form for Q 2 / D ~ oo Specification of domains Similar fires with QZ/D5 = constant
Case
29 49 31 53 35 52, 47
Q~/3DS/3(z/D )m(Q2/DS)~ A~(N) < z / D < A z ( N ) M ~ Q1/3DS/3(z/D)m
ht < z / D
41 35 51, 52
Equation no.
Dimensional power law
TABLE 2 Summary of Derived Power Laws
e~
5"
174
M. Poreh, H. P. Morgan
Of particular interest is the result that the vertical variation of M for large values of N may be described by M/Ms = F(z/L)
(54)
where L is the flame size, and that mass flux power laws for the flaming region will be of the form M / Q ~ (z/Q2/5) m ~ ( z / L ) m
(55)
which had been used by McCaffrey 7 and Cox and Chitty. 6 Equation (54), it is noted, is also compatible with a similar conclusion by Heskestad 1° that M / M L = F ( z / L ) , where Mc is the mass flux at z = L. It has also been shown that geometrically similar fires, whose heat output Q is determined by D and Ms/p~, satisfy the condition N = constant. As a result, different but equivalent forms of power law may be used to correlate the dependence of M on Q, D and z. A m o n g those are the forms M ~ D a z " and M ~ Q q z m. In addition, the correct specification of the dimensionless boundaries of regions, where such power laws may be valid, has been determined. Of particular interest is the finding that, in general, the boundaries of each region d e p e n d on N. However, for large values of N, the near field scales proportionally to the flame height L, which in turn is proportional to N 1/5. On the other hand, the different regions within the near field of similar fires with N = constant may be specified relative to either D or L. The previously reviewed power laws will now be compared with the results of the analysis. The power laws proposed by Delichatsios H for the special case of entrainment into buoyant plumes have been shown to be identical to the derived limit of the general law for the limit of small N, eqn (30), which is independent of Ms. On the other hand, it is easily seen that the power laws M ~ D z 3/4 [eqn (9)] and M ~ D Z [eqn (16)] are incompatible with the present results. This observation does not necessarily imply that these two power laws are incorrect, as it is quite possible that the entrainment in the experiments on which they were based had been affected by variables which had not been included in the above dimensional analysis. In fact, as noted by Cetegen et al., 2 eqn (9) describes experiments where a thin laminar diffusion flame was observed, whereas the present analysis is restricted to turbulent flames. It was noted that the form of each of the rest of the proposed power laws for the burning region of the near field is either M ~ Q q z " or M ~ D a z " , which have been derived for the asymptotic cases of large and constant values of N. To examine whether the correlation between the
175
Mass flux in near field of fires 1.0
O 2d/5
o.8
~,,NCC, DE
/
o.~
~"
N,,~,HE
0.4
0.0
+ q
" I ~ "I'N,DE \
.
,
.
1
2 m
Fig. 3. The values of the powers q, d, and m proposed by Cox and Chitty~'9 (CC), Delichatsios 1 (DE), Heskestad "~-'5 (HE), McCaffrey and Cox~ (MC) and Thomas/Hinkley 4"~(TH) (see Table 1). values of the powers in these laws is compatible with the results of the previous analysis, the values of the pairs (re,q) and ( m , 2 d / 5 ) , which appear in these laws, were plotted in Fig. 3. As seen from the figure, all the values of q and 2 d / 5 are equal to 1 - 2m/5, as suggested by eqns (35b) and (43). The agreement between the M ~ Q q z " type power laws with the result of the previous analysis is not surprising. Examination of the analysis and experimental data of Heskestad, 14''5 McCaffrey and Cox 8 and Cox and Chitty 6 clearly shows that they were proposed for fires with large values of N. The reason that the data measured by Cox and Chitty 9 in the experiments with the D = 0.6 m burner could not be matched with the earlier data is, perhaps, that the value of N in some of the experiments was not sufficiently large. Now, it has already been shown that the value of N in the data of Hinkley 5 was an approximate constant. Thus, the power law M = D z 3/2 proposed by Thomas et al. 4 is equivalent to the power law M oc Q2/Sz3/2" Both forms are compatible with our analysis. The d~Lta presented in Fig. 5 of the paper of Delichatsios I' for confirming the M = D 2 z ~12 and M ocDz3/2 laws are from experiments with relatively large values of N. Thus, they too should have assumed the form M = Qqz m. Since these measurements have been made in pool fires, one
M. Poreh, H. P. Morgan
176
explanation is that the values of N in this particular data set were approximately constant. If this hypothesis is correct, one could also express these laws in the M o c O q z m form: the M o c D 2 z in law could be replaced by M ~ oa/szl/2, which is similar to the law proposed by Cox and Chitty, 6 and M ~ D z 3/2, which is identical to the power law proposed by T h o m a s et al., 4 could be replaced by M ~ O2/Sz3/2. Table 1 shows that five values have been suggested for the power m, which increases from m = 1/2 at the base of the fire to m = 5/3 in the far field. The only exception is the M ~ z 5n law suggested by Delichatsios. 11 Does this observation suggest that the near field should be divided into four or five regions or, perhaps, that the different power laws are actually tangents (in log-log graphs) to a more general, continuous variation of M in which d M / d z is proportional to z p and p varies continuously from - 1 / 2 , near the base of the fire, to 2/3, in the far field? It is difficult to answer this question. It should be realised that experimental determination or verifications of such power laws are inherently very difficult. In addition to the scatter of the data and the limited span of each region, the origin of z is not uniquely defined. By changing this origin, the value of m can be modified. For example, in the power law proposed by T h o m a s et al. 4 and confirmed by Hinkley, 5 M = 0.188 PZ 3n, Z is the height from the floor on which the w o o d e n cribs were built. This choice is natural, but the use of another origin for z is possible. In Fig. 4, the variation of M p / Q 1/3 versus (z + 0.1P) was plotted. As seen from this figure, these same data 100 0.5(z+O.I p)5/3
#~
10
F
~
°
0.1 o
0.01 0.1
10
100
z+O.1P (m)
Fig. 4. T h e variation of M / Q tl3 with z + 0.1P in Hinkley's 5 data.
177
Mass flux in near field of fires
could also be described, with equal accuracy, by the form of the far field power law M = CnQ1/3(z + Z0) 5/3
(56)
where z0 :--0.1P and a slightly smaller proportionality coefficient, Cn = 0.5 (kg s-l)/(J~/lWl/3mS/3). The four data points shown in this figure by circles, which deviate from this correlation, are the same points that deviated from the 3/2 power law shown in Fig. 1. One may, similarly, assume that the data near the top of the flame that were described by the power laws M oc Ql13zS/3 might also be described by a 3/2 power law, similar to those proposed by Thomas et al. 4 Similarly, it is quite possible that the value of m = 1.3 in the power law of McCaffrey and Cox s might be modified to m = 3/2. If these assumptions are confirmed, the flame region for large values of N and for fires with N = constant, could be divided into three regions. The entrainment in each region will be given by eqn (35): M / Q oc (z/Q2/5) m, suggested by Cox and Chitty, 6 with m = 1/2, 1 and 3/2. As mentioned earlier, the dimensionless parameter Q* is often used in the analysis of buoyant plumes and flames. Cetegen et al. 2 suggest that the relative flame height L / D is proportional to Q,2/3, and to Q,2/5, for Q* < 1 and Q* ~-"1 respectively, and show that the entrainment in the near field depends ,an z / L . Similarly, eqn (27) is often written as L / D = - 1 / 0 2 + k ' O .2/5
(57)
where k' is a dimensionless number. Zukoski 12 writes that 'some consensus has been reached that a parameter such as Q*, or some variation which is proportional to Q*, may be used for modeling diffusion flames', which suggests that M / ( p l B'/3D 5'3) = F ( Z / O , Q*)
(58)
Although, for most cases, N is approximately proportional to Q,2 [see eqn (25)], it :should be noted that there is a fundamental difference between eqns (58) and (22) and between eqns (57) and (27). For example, eqn (27), contrary to eqn (57), suggests that the flame height is affected by the oxygen level in the ambient air, an effect which might be important for the analysis of fires burning in closed compartments.
ACKNOWLEDGEMENTS This work started during the stay of the first author at the FRS. Partial support for the continuation of the work was obtained from the Fund for the Promoting of Research at the Technion. Thanks are due to P.
178
M. Poreh, H. P. Morgan
L. Hinkley, for his help in retrieving the original data from his 1986 paper, and to G. Cox from the FRS and S. Trebukov from the Technion, who have reviewed the draft of this work.
REFERENCES 1. Rouse, H., Yih, C. S. & Humphreys, H. W., Gravitational convection from a boundary source. Tellus, 4 (1952) 201-210. 2. Cetegen, B. M., Zukoski, E. E. & Kubota, T., Entrainment in the near field and far field of fire plumes. Combustion Science and Technology, 39 (1984) 305-331. 3. Morton, B. R., Modelling fire plumes. Tenth Symposium (International) on Combustion, p. 973. Academic Press, New York, 1965. 4. Thomas, P. H. et al., Investigation into the flow of hot gases in roof venting. Fire Research Technical Paper No. 7, HMSO, London, 1963. 5. Hinkley, P. L., Rates of production of gases in roof venting experiments. Fire Safety Journal, 10 (1986) 57-65. 6. Cox, G. & Chitty, R., A study of the deterministic properties of unbounded fire plumes. Combustion and Flame, 39 (1980) 191-202. 7. McCaffrey, B. R., Purely buoyant diffusion flames--some experimental results in chemical and physical processes in combustion. Fall 1978 Meeting of the Eastern Section of the Combustion Institute, 1978. 8. McCaffrey, B. R. & Cox, G., NBSIR 82-2473, National Bureau of Standards (NIST), Washington, DC, USA, 1982. 9. Cox, G. & Chitty, R., Some source dependence effects of unbounded fires. Combustion and Flame, 60 (1985) 219-232. 10. Heskestad G., Fire plume air entrainment according to two competing assumptions. Twenty-first Symposium (International) on Combustion, pp. 111-120. Academic Press, New York, 1986. 11. Delichatsios, M. A., Air entrainment into buoyant jet flames and pool fires. Combustion and Flame, 70 (1987) 33-46. 12. Zukoski, E. E., Mass flux in fire plumes. Fire Safety Science--Proceedings of the Fourth International Symposium, pp. 137-147, 1984. 13. Heskestad, G., Luminous heights of turbulent diffusion flames. Fire Safety Journal, 5 (1983) 103-108. 14. Heskestad, G., Virtual origins of fire plumes. Fire Safety Journal, 5 (1983) 109-114. 15. Heskestad, G., Engineering relations for fire plume. Fire Safety Journal, 7 (1984) 25-32. 16. Mudan, K. S. & Crose, P. A., Hazard calculations for large open hydrocarbon fires. SFPE Handbook of Fire Protection Engineering, Sect. 2, Ch. 4. National Fire Protection Association, USA, 1988.