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The spill plume in smoke control design
Fire Safety Journal 30 (1998) 21—46 ( 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Northern Ireland 0379—7112/98/$19.00 PII: S0379–7112(97)00037–4
The Spill Plume in Smoke Control Design P. H. Thomas, H. P. Morgan & N. Marshall Building Research Establishment, Fire Research Station, Garston, Herts WD2 7JR, UK (Received 4 December 1996; revised version received 28 April 1997; accepted 21 May 1997)
ABS¹RAC¹ ºsing recent data, obtained by Morgan, Poreh and colleagues, we produce correlations for the mass flow of a two-dimensional plume emerging normal to the straight edge of a flat horizontal surface—the balcony—and rising up into a uniform atmosphere (the spill plume). A comparison is made with an earlier correlation of the same data by Poreh et al. which required values of the layer depth, D , in addition to those of the layer flow per unit length of B line plume, M@ . ¹he treatment of Poreh et al. followed others assuming the B linear relationship typical of far-field line plumes between the mass flow M@ and the height z with a correction *—the virtual source. ¹his linearity is a theoretical consequence of self-similarity (and a constant entrainment coefficient) in the velocity and temperature profiles across the plume, but recent, as yet unpublished, studies including some by computational fluid dynamics (CFD) cast doubt on the existence of self-similarity for these plumes at the low heights relevant in practice. However, a dimensional analysis of the flow does not require the assumption of self-similarity and we have demonstrated the linearity as a conclusion and not an assumption. ¹he effective entrainment coefficient is, as found by Poreh et al., less than the value 0)16 found by ¸ee and Emmons and used in early work by Morgan and Marshall. ¹he lower figure of 0)11 is consistent with other recent work on line plumes. ¹he experimental values of D , the layer depth reported by B Poreh et al., are in reasonable agreement with theoretical values for small increases in temperatures only. Experiments in model atria by Hansell, Morgan and Marshall which are not fully two-dimensional are discussed. Our correlation of them can be reconciled with that obtained by ¸aw and subsequently used by the Chartered Institution of Building Service Engineers (CIBSE). ( 1998 Published by Elsevier Science ¸td. All rights reserved. Keywords: smoke control, spill plume, dimensional analysis, data correlation.
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NOTATION a A b 0 B C D C P D B E g H k K M M@ Q Q@ ¹ u w ¼ x z Greek a b c D h i Q i . j k l o t
Constant [see eqn (6)] Constant (kg/m3) Source half-width (m) Constant [see eqn (3)] Discharge coefficient Specific heat (kJ/kg K) Layer depth (K2b ) (m) 0 Entrainment coefficient Acceleration due to gravity (m/s2) Height of opening (m) Thermal diffusivity (m2/s) Constant [see eqns (20)] Mass flux (kg/s) ("M/¼) (kg/m s) Rate of convective heat release (kW) ("Q/¼) (kW/m) Temperature (K) Characteristic velocity (m/s) Velocity (m/s) Width (m) Horizontal distance (m) Vertical distance (m) letters Regression coefficient Regression coefficient or i /i . Q Regression coefficient or mean value of (d¹ /dz) in ambient surroundings 0 Depth of virtual source (below z"0) Temperature difference Layer profile factor for heat flux12 Layer profile factor for mass flux Factor depending on k and v Dynamic viscosity (N s/m) Kinematic viscosity Density Angle of inclination
Subscripts 0 Initial, ambient B Balcony
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1 INTRODUCTION There are several published procedures for designing smoke control systems1—4 and many published comments on them. This paper is concerned with but one feature of these, viz., the flow from under a wide horizontal balcony over a vertical opening into an atmosphere without any restraint, e.g. no adherence to a vertical wall above the balcony and no allowance for effects at the edges of the plume: it is a two-dimensional system, the core of all UK design methodologies. Because the Building Research Establishment method1 of design has been the subject of some controversy within the UK and because there are many differences between it and NFPA4 procedures, we felt it useful to clarify and, one hopes, to resolve some of these diverse views. There are several recent studies5 which we shall exploit and a recent correlation by Poreh et al.6 is discussed and the similarities to our proposal and differences from it examined.
2 THE PROBLEM Today, it is possible to solve problems of the kind described above, and represented in Fig. 1, by means of computational fluid dynamics (CFD). However the solutions referred above are all based on classical zone models and it is with these we are concerned here. CFD can assist in resolving some of the ambiguities, but is not discussed here. When confronted in the mid-1970s with the problem of estimating the mass flow of contaminated but diluted air rising from a fire in a compartment, the
Fig. 1. The spill plume.
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P. H. Thomas et al.
roof of which was extended as a balcony, Morgan and Marshall7 approached the initial analysis as follows. (i) The plume and the fire were considered separately, i.e. decoupled, so that the source of the spill plume could be considered as a wide layer emerging from under the balcony under its own buoyancy with given properties independently calculated. This implicitly assumes that the layer is not so deep as to suffer interference from inflowing air. This is only fully justified a priori for the early, important stages of fire growth. (ii) Till very recently, there was little information about spill plumes or even about wholly vertical line or strip plumes. This is contrary to the situation for axisymmetric plumes. The approximation of coupling a layer emerging from a narrow opening to an axi-symmetric plume cannot, or in view of a common tendency to do so, should not be applied to wide openings. Of course, any real finite length two-dimensional plume is eventually eroded by entrainment at the edges and becomes a three-dimensional plume. We do not deal with this problem here in detail but will briefly refer to it below. (iii) The main analysis of two-dimensional vertical plumes, on which Morgan and Marshall7 drew was the work of Lee and Emmons.8 This theory of vertical plumes disregarded density differences in all but the buoyancy. It assumed the entrainment coefficient to be a constant and the distribution of velocity and density differences (temperature rise) across each horizontal section of the plume to be Gaussian. Lee and Emmons also allowed for a difference between the velocity and density difference profiles—reflecting differences in molecular diffusivity—and allowed for arbitrary initial conditions of mass and mean momentum (or mass and mean kinetic energy) but they did not allow initial cross-sectional distributions to be non-Gaussian. In their use of this work, Morgan and Marshall7 coupled the conditions at what Lee and Emmons had defined as the equivalent Gaussian source to properties of the layer emerging from below the balcony and made allowance for density differences in matching the conditions of the layer to this equivalent Gaussian source. All theories of vertical plumes have at least one empirical constant built into the analysis and this is determined from experiments. In the oldest plume theories, this was related to the angle, observed to be constant, at which the plume spread upward in a uniform atmosphere (Fig. 2). In modern theories it is the entrainment coefficient invented by Morton et al.9 (Fig. 3). A long way from an arbitrary source the angle of spread in a uniform atmosphere is constant in both treatments, the assumption of overall similarity being replaced by that of local similarity. Profiles which are Gaussian remain Gaussian. Profiles treated as ‘top hat’ remain so. These are the main examples of self-similar profiles.
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Fig. 2. Self-similar straight sided plumes in a uniform atmosphere.
Plumes in which density differences are neglected except in the buoyancy force are termed ‘weak’ plumes. Without that assumption, i.e. near a hot source, the plume is described as ‘strong’. The Lee and Emmons theory of the weak strip or line plume gives, far from the source, the basic result:
A
B
1@3 M gQ@ M@" "A (z#D) ¼ o C¹ 0 p 0 1@3 gQ "A (z#D) o ¼C ¹ 0 p 0
A
B
Fig. 3. Local self-similarity. ¼ /¼ constant. % #
(1) (2)
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P. H. Thomas et al.
where A and D are constants to be calculated, estimated or fitted from experiments. M"M@¼, the total mass flow, Q"Q@¼, the total convected heat release, ¼ is the width of the plume with no allowance for edge effects, Q@ the convected heat release rate per unit width, g the acceleration due to gravity, o is the density of the surrounding air, ¹ its temperature, and C the 0 0 p specific heat assumed to be constant for all the gas phase. ‘A’ depends on the assumed shape of the cross-sectional profiles and also on the numerical value of the entrainment coefficient (which was, in effect, determined experimentally). Near to the source, the relationship between M@ and z is not linear and the theoretical solution involves a tabulated function. Far from the source a firstorder correction D can be evaluated from the theory or from experiments. Equation (1) can be written as M"BQ1@3¼2@3(z#D) where
A
(3)
B
1@3 g o C¹ 0 p 0 Law10 and Thomas11 have used this to correlate data (see also Morgan18 and Law19). Law plotted ‘M’ against Q1@3¼2@3z to find the best straight line without using Lee and Emmons’ theory in detail only the form of its asymptotic result. Law appears to have been the first to decouple the spill plume from the Lee and Emmons value of the entrainment coefficient. We shall discuss such correlations below. Poreh et al. have extended these treatments. B"A
3 DIMENSIONAL ANALYSIS Equation (1) is derived from an analysis making conventional assumptions concerning local similarity of cross-sectional distributions of velocity and temperatures (or density differences) in turbulent flow. Lee and Emmons made the conventional weak plume approximation neglecting departures of density from ambient except in the buoyancy term. If a dimensional analysis is to be made, one must identify the basic physical quantities involved and the first set of these is Q@, g, o , C , ¹ , and c the 0 p 0 gradient of temperature rise of the air surrounding the plume, and the coordinates x, y and z. Velocity and temperature (and their distributions) depend these independent variables. The temperature affects the density in the mass, momentum and the buoyancy. But these effects of temperature can, in principle, be accommodated in the dependent variable even if the dependence is not explicit.
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The spill plume in smoke control design
A second set of terms are those describing the initial source conditions and the terms t , where t is the inclination of the source,; w , b , o , respectively, 0 0 0 0 B velocity, source size, and initial density, are sufficient*. A third factor is implicit in the Lee and Emmons analysis which distinguishes between the velocity and temperature profiles (or density difference profiles). This is due to the difference between their molecular diffusivities. This introduces the viscosity k and the thermal diffusivity k (and in principle the mass diffusion coefficient) into a full dimensional analysis. Given that the flow is such that a time mean flux of mass can be defined, the analysis of the dimensions involved leads to denoting the upward plume flow ‘M’ by the equation
A
M@C ¹ gz gb w g¼ w b l o cu2 p 0"function , 0, 0, , 0 0, t , B, 0k o g¹ Q@ u2 u2 u u2 l 0 0
B
(4)
where
A
B
1@3 gQ o C¹ ¼ 0 p 0 is a characteristic velocity, k is the thermal diffusivity and l is the kinematic viscosity, k/o. There is a velocity vector at all positions z, x (‘y’ having no special role in two-dimensional flow) but the maximum values of velocity define a trajectory z(x) (e.g. the locus of the maximum velocity), so eqn (4) need contain only z. For a purely two-dimensional system, Q and ¼ have no significance except as Q/¼("Q@), similarly with M, M/Q is M@/Q@. We note that gb /u2 and w /u can be described by two other terms, one of 0 0 which (their product) reduces to o M C ¹ /Qo , where M is the mass flow 0 B p 0 B B from the source. One must, of course retain one or other of the above pair (or some combination other than their product). M C ¹ /Q is in effect ¹ /h and can be used as an alternative to o /o B p 0 0 B B 0 amongst the list of independent variables. Where helium is employed, as its specific heat is different from that of air, other property ratios may need to be considered. j in Lee and Emmons’ work can be regarded as a surrogate for l/k, which will be taken here as a constant and omitted, being absorbed in the function. Again, the helium data may be different from the others if molecular diffusivities have a significant role. We note the group w b /l in eqn (4) is the 0 0 u"
* Temperature rises can be substituted for density differences. In weak plumes, the Boussinesq approximation applies and the plume temperature and density are, respectively, equal to the surrounding values in the mass and momentum terms.
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P. H. Thomas et al.
source Reynolds number. We shall neglect the limitations imposed by this (see Section 4). Hence for a given fluid
A
o cu2 M@C ¹ gz gb w p 0"function , 0, 0, t , B, 0 o g¹ Q@ u2 u2 u 0 0 and its trajectory
A
gz gx w gb o cu2 "function , 0, 0, t , B, u2 u2 u u2 0 o0 g¹0
B
B
(5a)
(5b)
We have not assumed a characteristic length for the source flow except that defined by b . The functional relationship will, of course, be different for ‘plug 0 flow’, triangular distributions, etc. Other profiles may be explored by the following methodology, but it may be necessary to recognise that some profiles may introduce a characteristic dimension. One could rearrange the variables by combining any two, or more, of the above groups. Thus, the left-hand side could be M@ o Jgz)z 0 which is a combination of M@C ¹ /Q@ and p 0 gz gQ@ 2@3 o C ¹ 0 p 0 In the spill plume problem, b is not necessarily known, and since it is not 0 an independent variable and depends on Q@, M@ , and o for stable layers, B B b can often be omitted. 0 Note that no assumption has been made regarding similarity and entrainment except that entrainment—expressed as a coefficient—is dependent only on the variables included in the functional relationship, eqn (4). Constancy of the entrainment coefficient E is a simplifying approximation based on observations of straight side plumes or similarity of profiles—be they Gaussian or ‘top hat’. This important removal of the assumptions of constancy in entrainment allows one to treat linearity (yet to be demonstrated) as a conclusion, and not an assumption based on a constant E. This is one difference between this development of the treatment and that of Poreh et al. It is perhaps in retrospect not so important for this problem since linearity is obtained, in
A
B
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practice over the practical range of z, although early ideas on this problem doubted it. Linearity is not obtained a priori for the effects of entrainment near a finite source. Another matter is the question of how to assess nearness to the source. The dimensional analysis shows that one should assess nearness by z or b 0
gz gQ@ 2@3 o C¹ 0 p 0
A
B
not by Q@1@3z, an opposite dependence on Q@. This is an important consideration when dealing with data ‘near the source’ about which there has been some controversy because this is the region where the equivalent Gaussian source lies. Data have been plotted in Fig. 4 as required by eqn (4), omitting all terms in l, k, c and including only data where there are no edge effects. We omit a discussion on stratification. Before analysing Fig. 4a—c, we must discuss the layer flow because we shall regard the layer depth D as dependent on M@ and Q@ and not treat it as an B B independent variable as Poreh et al. did. 4 THE LAYER FLOW12 Figure 5 shows observed values of D ("2b ) for all the experiments and it is B 0 seen that apart from the data from the experiments with helium, there is little or no obvious discernable correlation between D and M@ /Q@1@3. B B An analysis by Morgan12 calculates a theoretical value for the layer depth, i.e.
A
B
D Q@1@3 Q@ 2@3 B "a 1#b M@ C ¹ M@ B p 0 B
(6)
where
A
a"
B
9C i ¹ 1@3 1 p Q 0 i C 8o2g . $ 0
and i b" . i Q The values of i and i are dependent on the cross-sectional profile of the Q . velocity and temperature distribution across the layer but the layers in these
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P. H. Thomas et al.
Fig. 4. (a) Experimental data for Series I in Marshall and Harrison,5 see also Poreh et al.6 (b) Experimental data for Series II and III in Marshall and Harrison,5 see also Poreh et al.6 (c) Experimental data from helium and later studies, see Marshall and Harrison.5
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The spill plume in smoke control design
Fig. 5. Values of layer depth (D ). B
experiments were similar to those studied earlier12 for which conventional values were i "1)30 and i "0)95. . Q With values for o , g and ¹ we obtain 0 0 D Q@1@3 D Q@ 1) 4Q 2@3 B (7) " B "2)16 1# M@ C¹M M@ Q@2@3 B p 0 B B
A
B
The empirical correlation from Fig. 6 is
A
B
D Q@1@3 Q B "2)50 1# (8) M@ C¹M B p 0 B and, whilst agreement with eqn (7) is good at low values of Q@/C ¹ M@ , p 0 B D differs from that theoretically expected in eqn (7) progressively as the B temperature rises. The layer in reduced scale experiments has a Reynolds number lower than the prototype has and the horizontal flow and its transition into the rising spill plume may be locally influenced by this but we have not pursued this feature of the spill plume, since we have been able to correlate the available data (admittedly small-scale experiments) without involving effects attributable to it.
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P. H. Thomas et al.
Fig. 6. The variation of D with temperature. B
5 PRELIMINARY ANALYSIS OF FIG. 4a—c We seek correlations of M/Q in terms of M /Q and z/Q@2@3. We note the B existence of linearity with a small but perhaps significant intercept. Poreh et al. included calculated points corresponding to z"!D ("2b ) B 0 in the virtual downward extrapolation of the far-field plume. We discuss this later. (M!M ) is the total mass of entrained air but some is entrained below B z"0 (see Fig. 1) so that we expect M/Q to increase as aM /Q where a51. It B will be seen that there are few data in the range 0(z/Q@2@3(0)03, but these are important in considering the intercept at z"0. Several sets of data5 have been analysed separately, and Series I—IV as described by Poreh et al.6 are similar and can be treated together, so we can seek a correlation of the form M@ M@ z "a B#b #c Q@ Q@ Q@2@3
(9)
where, here and henceforth, a, b and c are regression coefficients. One set of data with a small collecting hood has to be treated separately. This distinction was noted by Poreh et al.6 and has been demonstrated by Miles et al.13 using CFD. Another set of data from a model atrium14 which are not purely two-dimensional are discussed separately below.
6 LINEAR CORRELATIONS AFTER LEE AND EMMONS We have mentioned above that the treatment of Lee and Emmons leads to a linear relationship between M@ and z. The coefficient of proportionality led
The spill plume in smoke control design
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Lee and Emmons to declare a value of 0)16 for the entrainment coefficient and, in the absence of other data, this was used by Morgan and Marshall to develop design methodology. The earliest of the experiments by Morgan and his collaborators also used the assumption of a Gaussian distribution to infer mass from measurements of temperature rise. Lee and Emmons assumed a Gaussian distribution e~x2@j2b2 for temperature rise and e~x2@b2 for velocity and found j"0)9. The ratio of the mean temperature rise h ("Q@/C M@) to h is given by .%!/ p .!9 j h .%!/" O0 ) 67 h J1#j2 .!9
(10)
whereas it is unity for a top hat profile. In the absence of direct measurements of mass flow or of temperature and velocity distribution, one must anticipate some uncertainty in the precision of estimates of mass flow from temperature measurements. Also, in the early experiments entrainment was possible into the ends of the plume, so that a methodology was devised7 to modify the Lee and Emmons plume theory. This problem is of considerable design importance, but we shall leave the discussion of end effects until the simple two-dimensional spill plume is dealt with. Work15—17 later than Morgan and Marshall’s early work has been interpreted by employing entrainment coefficients E differing from 0)16. Although one published value is larger than 0)16 and two other are lower.15,17 Law10,18,19 correlated data from Morgan et al. by using the same linear form of correlation as that produced by Lee and Emmons but did not use an a priori value for E. Thomas11 attempted to use the value of 0)16 with adjustments to the end-effect correction, viz; combining a line plume and an axi-symmetric plume so that they were asymptotic forms to a line plume with end entrainment which must eventually approximate to an axi-symmetric plume. The position of the virtual source D was also treated as an experimentally determined quantity, and D was expressed as a fraction of the opening height H. For small values of D /H, D is expected, from dimensional grounds, to be B related primarily to the depth D ("2b ) of the layer of the balcony flow with B 0 secondary temperature effects. However, when D /H becomes larger the layer B interacts with the inflow and D needs to be expressed as a fraction of H, temperature and D /H, viz., B
A
D Q@ D B, "f H H C ¹ M@ p 0 B
B
(11)
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P. H. Thomas et al.
In the absence of sufficient design data to determine D or M@ , one can B B relate D to H as a design parameter only on statistical grounds.
7 STATISTICAL ANALYSIS The experiments analysed in this paper mostly only use recent measurements of M under a ‘large’ hood. The data in Fig. 4a—c have been analysed " statistically, each separate set being shown to be part of a homogeneous set described by M@ M M@ z B#0)159 " "1)20 #0)0027 Q@ Q Q@ Q@2@3
(12)
The coefficient of z/Q@2@3 is 0)16, whereas, in the same units, the coefficient of Lee and Emmons is 0)22. M is proportional to E2@3 so the Fig. 4 correlation corresponds to E"(0)16/0)21)3@2]0)16, viz., 0)11 which is closer to two values obtained15,16 since Lee and Emmons obtained 0)16 and to some previous values when recalculated.17 The correlation obtained by Poreh et al. using a database almost identical to that used here was in terms of M@ and Q@ rather than M and Q: M@!M@ B"0)16(z#2b ) or 0)16(z#D ) 0 B Q@1@3
(13)
The difference between the two forms, eqns (12) and (13), arises from our use of M@/Q@ and z/Q2@3 and Poreh et al.’s use of M/Q1@3 and z (or M and zQ1@3). Direct comparison of the regression equations is somewhat complicated by the different nature of the residual constant in a linear relation. Poreh et al., in effect, accommodated the terms 0)2M and the 0)0027 in the B assumption regarding D and that it could be simply added to ‘z’. They did not B separate so as to compare the regression coefficients of z and D , but they B demonstrated a close agreement between measured and calculated M (standard error $6%). The agreement is good but there is a redundancy in the formula requiring values of both M@ and D s. B B We can introduce eqn (8) into eqn (13) and so obtain M@ M@ 0)16z "1)40 B# #0)0014 Q@ Q@ Q@2@3
(14)
s Of course, it is possible to arrange D to be independent of M@ and Q@— but it would require B B different kinds of experiments.
The spill plume in smoke control design
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The standard deviation (S.D) of the regression coefficients 1)20 and 0)159 are respectively 0)092 and 0)034. The S.D. of the constant term 0)027 is 0)006. There is error on D too. The 1)40 is thus on the border of acceptability. M@ /Q@ B B for these experiments varies from 0)0025 to 0)012t so the (1)40!1)20) M@ /Q@#0)0014 is close enough to 0)0027 and hence the equations can, in B practice, be regarded as the same. Poreh et al. did not include in their analysis the experiment with helium at z"0 and it is seen in Fig. 7 that this experiment is slightly anomalous, perhaps because our description of the behaviour is incomplete: its density difference makes it simulate a flame more than it does the hot smoke. Figure 7 shows that the four results for z"0 are described equally well by M@"1)4M@ B
(15a)
and M@"1)2M@ #0)0027Q@ (15b) B and there is a convenience in using the simpler equation, an alternative which will have little consequence away from the balcony.
Fig. 7. Correlation between M and M at Z"0. B t If the one experiment with helium is included the upper limit is 0)019.
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P. H. Thomas et al.
Poreh et al.6 extrapolated a far-field plume downwards. However, there are a few data for z"0. The values of M for the three experiments for which z is zero are reported by Poreh et al. in terms of the value of M calculated from their eqn (15). The #!-# three ratios M /M are 1)09, 1)09 and 1)15, and they have an average of 1)11 #!-# which also suggests that M is slightly too large. We note that if z"0 Poreh #!-# et al.’s equation gives
A
B
¹ M@"M@ #0)16D Q@1@3"M@ 1#0)4 B (16) B B B ¹ 0 There is some ambiguity about the effect of the temperature of the layer yet to be resolved. One expects upward entrainment into the emerging layer to be less and downward entrainment to be greater when the layer temperature is higher and density is lower. One would need to involve D independently of M@ and Q@ if there were B B other factors, e.g. surface friction or an imposed initial momentum. However, one would require the study of a range of layer conditions, wider than in these experiments, the layer being presumed to be or nearly in equilibrium with its own buoyancy. To sum up, our analysis of almost the same set of data as Poreh et al. shows that their correlation can be simplified by substituting for D , the experiB mental relationship between D and M@ and Q@. It is then equivalent to what B we have obtained for the whole range of z within experimental error. However, our presentation of data, derived from dimensional analysis, leads to our conclusion that, despite any lack of self-similarity found by the CFD studies, the M(z) relationship is in practice linear. 8 EXPERIMENTS WITH PLUME EDGE EFFECTS: MODELLING ATRIA Some data on spill plumes in a reduced-scale model atrium have been published by Hansell et al.14 They were subsequently discussed by Law20 and Thomas21 and were incorporated into a report by CIBSE, the Chartered Institution of Building Service Engineers.22 The data shown in Fig. 8 are not general: the value of ¼ was close to being constant (c"0)5 m). They refer to a scaled-down version of a real atrium. This is not strictly two-dimensional but the data exhibit linearity. Morgan and Marshall7 had earlier on considered the near-field erosion of the edges of a finite source. One expects their procedure to provide a first approximation. Thomas11 combined two- and three-dimensional axi-symmetric plume formulae to provide an expression which was appropriate at the limits zP0
The spill plume in smoke control design
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Fig. 8. Experimental data from atrium study14.
and zPR. The equation is not linear but the connection between the two limits is 1#0)22(z/¼) where ¼ is the length of the line source and the coefficient 0)22 comes from the ratio of Lee and Emmons’ line plume formula and an axisymmetric formula. The two components are equally weighted when z/¼+5: above this height this model suggests that the centreline temperature is affected significantly by edge entrainment. The highest experimental value of z/¼ was about 3 so the above expression implies only a 50% enhancement. When the data shown in Fig. 8 was subjected to a more detailed statistical analysis, the resulting regression gives a very different picture, viz., M@ M@ z "2)64 B#0)342 !0)0083 Q@ Q@ Q@2@3
(17)
To compare eqns (17) and (12), we note that so long as ¼/D is not too B small the two sets of data should give the same value of M at z"0 before the atrium edge effects become significant. The value 2)64 in eqn (17) has a large standard deviation p of 0)75 and it is plausible to reduce its value and raise the intercept (!0)0083) which is an implausibly negative intercept (p is 0)004 but there are covariances with the 2)64). The mean experimental value of M@ /Q@ is 0)0067, so we can obtain a first B approximation to the effect of removing the intercept by reducing 2)64M@ /Q@ B
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P. H. Thomas et al.
by (0)0083/0)0067)M@ /Q@, i.e. to 1)40M@ /Q@. This is close to the coefficient in B B eqn. (15) and is within 2p of it and this is satisfied at a value of M@/Q@"0)0076, about 12% above the mean experimental value 0)0067. At M@ /Q@ equal to B 0)0076 both sides of the above equation give M@/Q@K0)012, a little larger than the 1)4]0)0067 found in the tests. We conclude that, apart from the regression coefficients of ‘z’, the difference between eqns (12) and (17) is more apparent than real. Without more data, one cannot evaluate the role of the dimensionless width as expressed by ¼/Q@2@3, ¼/D or z/¼. Equation (17) must be regarded as an B empirical correlation. Law’s20 equation is M@"0)31Q@1@3(z#0)102)
(18)
The difference between the regression coefficients for z/Q@2@3 is small. We discuss the significance of the constant 0)102 in Section 9. Our statistical analysis of the data for the small-scale model atrium is confined to the condition where the layer of hot gas is channelled. Poreh et al. report these results as they report the others, as M versus z and Q and subsequent accommodation of the plume width ¼. This is as appropriate for an expanding plume (i.e. one with edge effects) as the two-dimensional formulation of M@. They note a tendency in M to increase faster than z but not as fast as z5@3. They state that instead of M/Q increasing as 0)15z/Q2@3 (which becomes 0)16z/Q@2@3) it increases as 0)20z/Q2@3 for the unchannelled flow. Our correlation produces a regression of 0)34z/Q@2@3 and is only a little larger than Law’s. The values of ¼ for the atrium are about 0)47—0)53 m not 0)91 m, so the use of Q instead of Q@ lessens the difference between the correlation for channelled two-dimensional flow and the plume expanding into the atrium. Whichever formulation is used, the problem of dealing with edge effects remains. Morgan and Marshall’s correction for entrainment into line edges as ends of a plume was obtained by using the mean vertical velocity of the plume (in its initial stages) to which the entrainment velocity is proportional and an estimate of the area over which entrainment takes place. For a very thick layer this is plausibly proportional to D and z and for a very thin layer to z2. B Whether or not the values evaluated for the coefficients of M@ in the atrium B data eqn (17) and the wholly channelled flow data eqn (12) are statistically identical, the value of 1)4 is plausible for both and, even if this is not the optimum value, it is not plausible to assign significantly different values at z"0. The difference between the correlations away from z"0 is thus roughly M !M 0)18z !53*6. 2$*.%/4*0/!-" Q Q@2@3
(19)
The spill plume in smoke control design
39
where the 0)18 is the difference between the 0)34 in eqn (17) and the 0)16 in eqn (14). The difference between the two mass flows is 0)18zQ@1@3. W and this is twice the end entrainment. However this end entrainment is not dependent (not to a first approximation) on ¼, and to generalise this result we need to speculate!! We generalise the result by taking the limits of ¼ in the experiments, i.e. 0)47 m(¼(0)53 m. Our best estimate for each end lies between 0)18 zQ@1@3]0)47 2 and 0)18 zQ@1@3]0)53 2 with a mean of 0)045zQ@1@3. The finding that the entrainment in these experiments is proportional to z implies the other component of area is a constant, i.e. independent of ‘z’. It is, e.g. possibly proportional to D or D itself, which in these experiments varied B B from 0)10 m to 0)17 m. Let us write M "KD Q@1@3z %$'% B
(20a)
0)045 0)045 KO P 0)10 0 ) 17
(20b)
where
i.e. M "0)33D Q@1@3z$25% %$'% B From eqn (8) this becomes ¹ (21) M "0)83M@ z B %$'% B ¹ 0 a quantity dimensionally consistent. Although this expression is plausible it is highly speculative and is clearly not applicable for z large compared to D or ¼. B The range—indeed the entire formulation—might be tested by a more detailed statistical analysis but a satisfactory confirmation of this procedure would require data over a wider range of D and ¼. B
40
P. H. Thomas et al.
9 THE VIRTUAL SOURCE In response to an invitation from Law20 to comment on the position of the virtual source, Thomas21 has shown that eqn (18) implies a virtual source at a level near the base of the layer. In order to formulate a working hypothesis, Poreh et al. assumed that at z"!D , M@ equalled M@ . At low zP0, edge B B effects or none, the two situations would be expected to be very similar for small D /¼, which for these experiments was about 1/5. B We identify 0)102 as D, and at z"0 we have M@"0)31Q@1@3D Substituting for Q@1@3 from eqn (8) gives ¹ M@ D "1)27 0 (22a) ¹ M@ D B B B which, since both D and D scale the same way for a given temperature and B equal values of any other dimensionless variables, would seem to be a general result, becoming D ¹ "1)7 0 (22b) D ¹ B B for M@/M@ equal to 1)4. In these experiments, the data have a mean value of B 0)0067 for M@ /Q@ which corresponds to a mean h of about 150°C, whilst the B experimental values of D vary from 0)10 m to 0)17 m. Hence, our estimate of B D from eqn (22b) lies between 0)11 m and 0)19 m. The lower end of the range of these estimates is consistent with the 0)102 in eqn (18). Had we used the regression coefficient 0)16 instead of 0)31, the value 0)102 would be increased to 0)102]0)31 &0)20, 0)16 and this is near the upper end of these estimates. The comparisons are limited by the different forms of eqns (12) and (18).
10 THE UK—US DIFFERENCE If the gases from a fire emerge from a door, the depth of the layer will be deeper in relation to the width of the opening than for gases emerging from a wide opening. The gases are therefore likely to become a rising axisymmetric plume relatively sooner than those from a wide opening. As
41
The spill plume in smoke control design
Fig. 9. The emerging plume.
discussed above, one expects that the height at which three-dimensional effects become significant must increase with the width ‘¼’ of the source of the plume, and this means that the major part of a plume rising from a doorway to a high ceiling is more nearly three-dimensional than that from the wider one typical of a shop front; see Fig. 9. An equation given in NFPA 92B-14,
C
M@"0)41Q@1@3(z#0)3H) 1#0)063
D
(z#0)6H) 2@3 ¼
(23)
is described as ‘based on Law’s interpretation of data given by Morgan and Marshall, a subsequent re-analysis by Morgan and modifications of a kind suggested by Thomas to make the calculated rate of entrainment approach that for an axi-symmetric plume at large heights’. The virtual source D (the correction to z) can only be associated as in eqn (22b) with H (for D ;H) as an average value for a particular design fire and B opening geometry for design procedures. As discussed above, the value of D is physically related to D and M@ . B B The value of 0)41 is larger than the 0)34 in eqn (17) and the 0)31 in eqn (18). Substituting either of these would necessarily raise the value to 0)063. This figure implies z/¼&15, where the two-dimensional and three-dimensional
42
P. H. Thomas et al.
expressions have equal weight instead of the lower value in the earlier discussion, and seems implausible. The discussion of this question in a later edition (1995) ignores the effect of entrainment into the ends of the spill plume for all z below 13¼, which also seems implausible.
11 THE ‘SMALL HOOD’ DATA One set of data5 has been recognised as very different from the others, and are shown in Fig. 10. The small hood appears to lead to higher entrainment and this has been thought to be due to some recirculation under the hood. Apart from the difference between the mean correlations there is a markedly greater scatter of the data about the average. The variation with ‘z’ is 50% greater than when the large hood was used, the regression coefficient between M@ and Q@1@3z was 0)25 instead of 0)16 but the scatter in the data is also quite different from that of the other data (cf. Figs. 4 and 10). The regression produces a regression coefficient on M@ /Q@ of 0)22 with a rather large error, as B implied by the scatter in Fig. 10. Poreh et al. recognised the apparently anomalous behaviour: we await a CFD study and more data to resolve the problems.
12 DISCUSSION AND CONCLUSIONS We have analysed two-dimensional data for the plume of hot gases rising from under a horizontal flat surface when they pass around the balcony edge.
Fig. 10. Plot of experimental ‘small hood’ data.1
The spill plume in smoke control design
43
Most of these data have been studied by Poreh et al. but our analysis—derived from a dimensional analysis—has a form different from theirs but it gives an equally good prediction of the mass flow M. It is simpler as it does not use layer depth D as well as M and Q. B B For h ;¹ , the layer depth, as reported by Poreh et al., is only slightly B 0 higher than a simplified theoretical prediction of D from values of M and B B Q—the heat flow, but the difference increases significantly for h &¹ . B 0 In the absence of self-similarity, one cannot expect a constant entrainment coefficient independent of height, only an average value over all the height. However, the data do produce an effectively constant value and—as Poreh et al. observed—it is less than the Lee and Emmons value originally taken by Morgan and his colleagues. The value is near to 0)11. This agrees with other estimates in the recent literature. We have separately examined data from experiments on a model atrium and reconciled our regression with the correlation in the CIBSE report (or vice versa), i.e. Poreh et al.’s assumption regarding the virtual source position and our interpretation of the 0)102 in eqn (18) are very similar and the coefficients 0)31 and 0)34 for the term in ‘Z’ are also very close. However, these data merit further comment. In our discussion of the problem of edge, i.e. end effects on a finite line plume, we have speculated as to how to scale these data. However, further data are required. We found the results are: (a) The large hood data are consistent broadly with the analysis of Poreh et al. (b) The large hood data are consistent with an entrainment coefficient less than 0)16 used in earlier work. This use of 0)16 had necessitated the introduction of various procedures and coefficients. The procedure became complex and the complexity can be removed by decoupling early BRE work from the experiments of Lee and Emmons. (c) Our correlation [eqns (12) and (14)] seems to be no less accurate than that of Poreh et al. and does not require all three terms M@ , Q@ and B D between which there is a relationship [eqn (8)]. B (d) The value of dimensional analysis has been demonstrated because the structure of data correlations can be deduced without reference to certain assumptions regarding similarity and hence the constancy of the entrainment coefficient. (e) The formula for the data on two-dimensional spill plumes rising into large reservoirs (hoods) is, for z/Q@2@3 less than 0)7: M@"1)20M@ #0)16zQ@1@3#0)0027Q@ B
(24)
44
P. H. Thomas et al.
An acceptable alternative is M@"1)40M@ #0)16zQ@1@3#0)0014Q@ (25) B Formulae of the same form but with different coefficients have been obtained for experiments with a small hood and a model atrium. The difference between the atrium formula and eqn (11) suggests that, provided z/¼ is not too large, the mass entrained into one edge in these experiments is 0)045zQ@1@3. We have discussed and reconciled our correlation of Hansell et al.’s data with that of Law. Care has to be taken in extrapolating the formulae for the two latter situations if only because the behaviour exhibits three-dimensional characteristics which are not recognised in the correlations which must be seen to be confined to a range of values of ¼/D or ¼/Q@2@3. B However, a somewhat speculative correction for edge effects has been proposed. It differs from that proposed by Morgan and Marshall and that proposed by Thomas4 and it can only be applied for z/¼ less than some, as yet undetermined, value. Further data are required. (f ) Poreh et al. have shown ‘that an acceptable agreement can be obtained between the BRE Spill Plume Model using the lower value° of the entrainment constant (and no adjustment to the height of rise) and the experimentally measured entrainment’. However, the range of the calculations seems to be relatively narrow, viz., 0)16—0)18 kg/s. Our model, being based on small-scale data effectively the same as Poreh et al.’s, which have been claimed6 to support the BRE Spill Plume Model, can presumably also be considered to be consistent with the BRE calculations in that range. What these models demonstrate is that the plume mass flow tends to be less than hitherto proposed for shallow smoke layers (the early stages of a fire). There has been a tendency to distinguish between small-scale data for z(0)3 m and for z'0)3 m but we have not found this. Dimensional analysis would anyway make any distinction occur at a particular value of z/Q@2@3. However, despite these remarks, the number of data available for analysis at low z/Q@2@3 is rather small to support—or contradict—conjectures concerning special entrainment processes into a deep smoke layer (zP0) when the dilution is low and the smoke control system is nearer the limit of its capability. Likewise, this work is confined to analysing data for flow from
°
That is 0)11 as represented by the coefficient 0)16 for z/Q@2@3 instead of the Lee and Emmons value of 0)16 which gives circa 0)22 z/Q@2@3.
The spill plume in smoke control design
45
beneath a horizontal balcony with no walls above which the plume might, if wide enough, adhere.
REFERENCES 1. Hansell, G. O. & Morgan, H. P., Design approaches for smoke control in atrium buildings. Buildings Research Establishment Report BR 258, Building Research Establishment, UK, 1994. 2. Morgan, H. P. & Gardiner, J. P., Design principles for smoke ventilation in enclosed shopping centres. BR 186, Building Research Establishment, 1990. 3. Butcher, G. & Parnell, A., Smoke Control in Fire Safety Design. E. & F. N. Spon, 1989. 4. Guide for smoke management systems in malls, atria and large areas. NFPA 92B, National Fire Protection Association, Quincy, MA, USA, 1991 (see also the 1995 edition). 5. Marshall, N. R. & Harrison, R., Experimental studies of thermal spill plumes. Building Research Establishment Occasional Paper OP1, April 1996. 6. Poreh, M., Morgan, H. P., Marshall, N. R. & Harrison, R., Entrainment by two-dimensional spill plumes. Fire Safety J., 30 (1998) 1—19. 7. Morgan, H. P. & Marshall, N. R., Smoke hazards in covered multi-level shopping malls: on experimentally based theory for smoke production. Building Research Establishment Current Paper CP 48/75, Fire Research Station, Building Research Establishment, UK, 1975. 8. Lee, S.-L. & Emmons, H. W., A study of natural convection above line fires. J. Fluid Mechanics, 11 (1961) 353—68. 9. Morton, B. R., Taylor, G. I. & Turner, J. S., Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. (¸ondon), 234A (1956) 1—23. 10. Law, M., A note on smoke plumes from fires in multi-level shopping malls. Fire Safety J., 10 (1986) 197—202. 11. Thomas, P. H., On the upward movement of smoke and related shopping mall problems. Fire Safety J., 12 (1987) 191—203. 12. Morgan, H. P., The horizontal flow of buoyant gases toward an opening. Fire Safety J., 11 (1986) 193—200. 13. Miles, S. D., Kumar, S. & Cox, G., The balcony spill plume—some CFD simulations. Proc. 5th Int. Symp. on Fire Safety Science, IAFSS, 1997 (to be published). 14. Hansell, G. O., Morgan, H. P. & Marshall, N. R., Smoke flow experiments in a model atrium. Building Research Establishment (BRE) Occasional Paper OP 55, Fire Research Station, BRE, 1993. 15. Ramaprian, B. R. & Chandresekra, M. S., Measurements in a vertical plane turbulent plume. J. Fluid Engng, 111 (1989). 16. Kotsovinos, N. E., A study of entrainment and turbulence in a plane buoyant jet. W. M. Keck Laboratory of Hydraulics and Water Resources Report KH-R-32, California Institute of Technology, Pasadena, 1975. 17. Yuan, L. & Cox, G., An experimental study of some line fires. Fire Safety J., 27 (1997) 123—39.
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18. Morgan, H. P., Letter to the Editor. Fire Safety J., 12 (1987) 83. 19. Law, M., Letter to the Editor. Fire Safety J., 12 (1987) 85. 20. Law, M., Measurements of balcony smoke flow (short communication). Fire Safety J., 24 (1995) 189—95. 21. Thomas, P. H., Discussion of measurements of balcony smoke flow (comment). Fire Safety J., 24 (1995). 22. Relationships for smoke control calculations. Chartered Institute for Building Service Engineers TM 19, 1995.
The Spill Plume in Smoke Control Design P. H. Thomas, H. P. Morgan & N. Marshall Building Research Establishment, Fire Research Station, Garston, Herts WD2 7JR, UK (Received 4 December 1996; revised version received 28 April 1997; accepted 21 May 1997)
ABS¹RAC¹ ºsing recent data, obtained by Morgan, Poreh and colleagues, we produce correlations for the mass flow of a two-dimensional plume emerging normal to the straight edge of a flat horizontal surface—the balcony—and rising up into a uniform atmosphere (the spill plume). A comparison is made with an earlier correlation of the same data by Poreh et al. which required values of the layer depth, D , in addition to those of the layer flow per unit length of B line plume, M@ . ¹he treatment of Poreh et al. followed others assuming the B linear relationship typical of far-field line plumes between the mass flow M@ and the height z with a correction *—the virtual source. ¹his linearity is a theoretical consequence of self-similarity (and a constant entrainment coefficient) in the velocity and temperature profiles across the plume, but recent, as yet unpublished, studies including some by computational fluid dynamics (CFD) cast doubt on the existence of self-similarity for these plumes at the low heights relevant in practice. However, a dimensional analysis of the flow does not require the assumption of self-similarity and we have demonstrated the linearity as a conclusion and not an assumption. ¹he effective entrainment coefficient is, as found by Poreh et al., less than the value 0)16 found by ¸ee and Emmons and used in early work by Morgan and Marshall. ¹he lower figure of 0)11 is consistent with other recent work on line plumes. ¹he experimental values of D , the layer depth reported by B Poreh et al., are in reasonable agreement with theoretical values for small increases in temperatures only. Experiments in model atria by Hansell, Morgan and Marshall which are not fully two-dimensional are discussed. Our correlation of them can be reconciled with that obtained by ¸aw and subsequently used by the Chartered Institution of Building Service Engineers (CIBSE). ( 1998 Published by Elsevier Science ¸td. All rights reserved. Keywords: smoke control, spill plume, dimensional analysis, data correlation.
21
22
P. H. Thomas et al.
NOTATION a A b 0 B C D C P D B E g H k K M M@ Q Q@ ¹ u w ¼ x z Greek a b c D h i Q i . j k l o t
Constant [see eqn (6)] Constant (kg/m3) Source half-width (m) Constant [see eqn (3)] Discharge coefficient Specific heat (kJ/kg K) Layer depth (K2b ) (m) 0 Entrainment coefficient Acceleration due to gravity (m/s2) Height of opening (m) Thermal diffusivity (m2/s) Constant [see eqns (20)] Mass flux (kg/s) ("M/¼) (kg/m s) Rate of convective heat release (kW) ("Q/¼) (kW/m) Temperature (K) Characteristic velocity (m/s) Velocity (m/s) Width (m) Horizontal distance (m) Vertical distance (m) letters Regression coefficient Regression coefficient or i /i . Q Regression coefficient or mean value of (d¹ /dz) in ambient surroundings 0 Depth of virtual source (below z"0) Temperature difference Layer profile factor for heat flux12 Layer profile factor for mass flux Factor depending on k and v Dynamic viscosity (N s/m) Kinematic viscosity Density Angle of inclination
Subscripts 0 Initial, ambient B Balcony
The spill plume in smoke control design
23
1 INTRODUCTION There are several published procedures for designing smoke control systems1—4 and many published comments on them. This paper is concerned with but one feature of these, viz., the flow from under a wide horizontal balcony over a vertical opening into an atmosphere without any restraint, e.g. no adherence to a vertical wall above the balcony and no allowance for effects at the edges of the plume: it is a two-dimensional system, the core of all UK design methodologies. Because the Building Research Establishment method1 of design has been the subject of some controversy within the UK and because there are many differences between it and NFPA4 procedures, we felt it useful to clarify and, one hopes, to resolve some of these diverse views. There are several recent studies5 which we shall exploit and a recent correlation by Poreh et al.6 is discussed and the similarities to our proposal and differences from it examined.
2 THE PROBLEM Today, it is possible to solve problems of the kind described above, and represented in Fig. 1, by means of computational fluid dynamics (CFD). However the solutions referred above are all based on classical zone models and it is with these we are concerned here. CFD can assist in resolving some of the ambiguities, but is not discussed here. When confronted in the mid-1970s with the problem of estimating the mass flow of contaminated but diluted air rising from a fire in a compartment, the
Fig. 1. The spill plume.
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P. H. Thomas et al.
roof of which was extended as a balcony, Morgan and Marshall7 approached the initial analysis as follows. (i) The plume and the fire were considered separately, i.e. decoupled, so that the source of the spill plume could be considered as a wide layer emerging from under the balcony under its own buoyancy with given properties independently calculated. This implicitly assumes that the layer is not so deep as to suffer interference from inflowing air. This is only fully justified a priori for the early, important stages of fire growth. (ii) Till very recently, there was little information about spill plumes or even about wholly vertical line or strip plumes. This is contrary to the situation for axisymmetric plumes. The approximation of coupling a layer emerging from a narrow opening to an axi-symmetric plume cannot, or in view of a common tendency to do so, should not be applied to wide openings. Of course, any real finite length two-dimensional plume is eventually eroded by entrainment at the edges and becomes a three-dimensional plume. We do not deal with this problem here in detail but will briefly refer to it below. (iii) The main analysis of two-dimensional vertical plumes, on which Morgan and Marshall7 drew was the work of Lee and Emmons.8 This theory of vertical plumes disregarded density differences in all but the buoyancy. It assumed the entrainment coefficient to be a constant and the distribution of velocity and density differences (temperature rise) across each horizontal section of the plume to be Gaussian. Lee and Emmons also allowed for a difference between the velocity and density difference profiles—reflecting differences in molecular diffusivity—and allowed for arbitrary initial conditions of mass and mean momentum (or mass and mean kinetic energy) but they did not allow initial cross-sectional distributions to be non-Gaussian. In their use of this work, Morgan and Marshall7 coupled the conditions at what Lee and Emmons had defined as the equivalent Gaussian source to properties of the layer emerging from below the balcony and made allowance for density differences in matching the conditions of the layer to this equivalent Gaussian source. All theories of vertical plumes have at least one empirical constant built into the analysis and this is determined from experiments. In the oldest plume theories, this was related to the angle, observed to be constant, at which the plume spread upward in a uniform atmosphere (Fig. 2). In modern theories it is the entrainment coefficient invented by Morton et al.9 (Fig. 3). A long way from an arbitrary source the angle of spread in a uniform atmosphere is constant in both treatments, the assumption of overall similarity being replaced by that of local similarity. Profiles which are Gaussian remain Gaussian. Profiles treated as ‘top hat’ remain so. These are the main examples of self-similar profiles.
The spill plume in smoke control design
25
Fig. 2. Self-similar straight sided plumes in a uniform atmosphere.
Plumes in which density differences are neglected except in the buoyancy force are termed ‘weak’ plumes. Without that assumption, i.e. near a hot source, the plume is described as ‘strong’. The Lee and Emmons theory of the weak strip or line plume gives, far from the source, the basic result:
A
B
1@3 M gQ@ M@" "A (z#D) ¼ o C¹ 0 p 0 1@3 gQ "A (z#D) o ¼C ¹ 0 p 0
A
B
Fig. 3. Local self-similarity. ¼ /¼ constant. % #
(1) (2)
26
P. H. Thomas et al.
where A and D are constants to be calculated, estimated or fitted from experiments. M"M@¼, the total mass flow, Q"Q@¼, the total convected heat release, ¼ is the width of the plume with no allowance for edge effects, Q@ the convected heat release rate per unit width, g the acceleration due to gravity, o is the density of the surrounding air, ¹ its temperature, and C the 0 0 p specific heat assumed to be constant for all the gas phase. ‘A’ depends on the assumed shape of the cross-sectional profiles and also on the numerical value of the entrainment coefficient (which was, in effect, determined experimentally). Near to the source, the relationship between M@ and z is not linear and the theoretical solution involves a tabulated function. Far from the source a firstorder correction D can be evaluated from the theory or from experiments. Equation (1) can be written as M"BQ1@3¼2@3(z#D) where
A
(3)
B
1@3 g o C¹ 0 p 0 Law10 and Thomas11 have used this to correlate data (see also Morgan18 and Law19). Law plotted ‘M’ against Q1@3¼2@3z to find the best straight line without using Lee and Emmons’ theory in detail only the form of its asymptotic result. Law appears to have been the first to decouple the spill plume from the Lee and Emmons value of the entrainment coefficient. We shall discuss such correlations below. Poreh et al. have extended these treatments. B"A
3 DIMENSIONAL ANALYSIS Equation (1) is derived from an analysis making conventional assumptions concerning local similarity of cross-sectional distributions of velocity and temperatures (or density differences) in turbulent flow. Lee and Emmons made the conventional weak plume approximation neglecting departures of density from ambient except in the buoyancy term. If a dimensional analysis is to be made, one must identify the basic physical quantities involved and the first set of these is Q@, g, o , C , ¹ , and c the 0 p 0 gradient of temperature rise of the air surrounding the plume, and the coordinates x, y and z. Velocity and temperature (and their distributions) depend these independent variables. The temperature affects the density in the mass, momentum and the buoyancy. But these effects of temperature can, in principle, be accommodated in the dependent variable even if the dependence is not explicit.
27
The spill plume in smoke control design
A second set of terms are those describing the initial source conditions and the terms t , where t is the inclination of the source,; w , b , o , respectively, 0 0 0 0 B velocity, source size, and initial density, are sufficient*. A third factor is implicit in the Lee and Emmons analysis which distinguishes between the velocity and temperature profiles (or density difference profiles). This is due to the difference between their molecular diffusivities. This introduces the viscosity k and the thermal diffusivity k (and in principle the mass diffusion coefficient) into a full dimensional analysis. Given that the flow is such that a time mean flux of mass can be defined, the analysis of the dimensions involved leads to denoting the upward plume flow ‘M’ by the equation
A
M@C ¹ gz gb w g¼ w b l o cu2 p 0"function , 0, 0, , 0 0, t , B, 0k o g¹ Q@ u2 u2 u u2 l 0 0
B
(4)
where
A
B
1@3 gQ o C¹ ¼ 0 p 0 is a characteristic velocity, k is the thermal diffusivity and l is the kinematic viscosity, k/o. There is a velocity vector at all positions z, x (‘y’ having no special role in two-dimensional flow) but the maximum values of velocity define a trajectory z(x) (e.g. the locus of the maximum velocity), so eqn (4) need contain only z. For a purely two-dimensional system, Q and ¼ have no significance except as Q/¼("Q@), similarly with M, M/Q is M@/Q@. We note that gb /u2 and w /u can be described by two other terms, one of 0 0 which (their product) reduces to o M C ¹ /Qo , where M is the mass flow 0 B p 0 B B from the source. One must, of course retain one or other of the above pair (or some combination other than their product). M C ¹ /Q is in effect ¹ /h and can be used as an alternative to o /o B p 0 0 B B 0 amongst the list of independent variables. Where helium is employed, as its specific heat is different from that of air, other property ratios may need to be considered. j in Lee and Emmons’ work can be regarded as a surrogate for l/k, which will be taken here as a constant and omitted, being absorbed in the function. Again, the helium data may be different from the others if molecular diffusivities have a significant role. We note the group w b /l in eqn (4) is the 0 0 u"
* Temperature rises can be substituted for density differences. In weak plumes, the Boussinesq approximation applies and the plume temperature and density are, respectively, equal to the surrounding values in the mass and momentum terms.
28
P. H. Thomas et al.
source Reynolds number. We shall neglect the limitations imposed by this (see Section 4). Hence for a given fluid
A
o cu2 M@C ¹ gz gb w p 0"function , 0, 0, t , B, 0 o g¹ Q@ u2 u2 u 0 0 and its trajectory
A
gz gx w gb o cu2 "function , 0, 0, t , B, u2 u2 u u2 0 o0 g¹0
B
B
(5a)
(5b)
We have not assumed a characteristic length for the source flow except that defined by b . The functional relationship will, of course, be different for ‘plug 0 flow’, triangular distributions, etc. Other profiles may be explored by the following methodology, but it may be necessary to recognise that some profiles may introduce a characteristic dimension. One could rearrange the variables by combining any two, or more, of the above groups. Thus, the left-hand side could be M@ o Jgz)z 0 which is a combination of M@C ¹ /Q@ and p 0 gz gQ@ 2@3 o C ¹ 0 p 0 In the spill plume problem, b is not necessarily known, and since it is not 0 an independent variable and depends on Q@, M@ , and o for stable layers, B B b can often be omitted. 0 Note that no assumption has been made regarding similarity and entrainment except that entrainment—expressed as a coefficient—is dependent only on the variables included in the functional relationship, eqn (4). Constancy of the entrainment coefficient E is a simplifying approximation based on observations of straight side plumes or similarity of profiles—be they Gaussian or ‘top hat’. This important removal of the assumptions of constancy in entrainment allows one to treat linearity (yet to be demonstrated) as a conclusion, and not an assumption based on a constant E. This is one difference between this development of the treatment and that of Poreh et al. It is perhaps in retrospect not so important for this problem since linearity is obtained, in
A
B
The spill plume in smoke control design
29
practice over the practical range of z, although early ideas on this problem doubted it. Linearity is not obtained a priori for the effects of entrainment near a finite source. Another matter is the question of how to assess nearness to the source. The dimensional analysis shows that one should assess nearness by z or b 0
gz gQ@ 2@3 o C¹ 0 p 0
A
B
not by Q@1@3z, an opposite dependence on Q@. This is an important consideration when dealing with data ‘near the source’ about which there has been some controversy because this is the region where the equivalent Gaussian source lies. Data have been plotted in Fig. 4 as required by eqn (4), omitting all terms in l, k, c and including only data where there are no edge effects. We omit a discussion on stratification. Before analysing Fig. 4a—c, we must discuss the layer flow because we shall regard the layer depth D as dependent on M@ and Q@ and not treat it as an B B independent variable as Poreh et al. did. 4 THE LAYER FLOW12 Figure 5 shows observed values of D ("2b ) for all the experiments and it is B 0 seen that apart from the data from the experiments with helium, there is little or no obvious discernable correlation between D and M@ /Q@1@3. B B An analysis by Morgan12 calculates a theoretical value for the layer depth, i.e.
A
B
D Q@1@3 Q@ 2@3 B "a 1#b M@ C ¹ M@ B p 0 B
(6)
where
A
a"
B
9C i ¹ 1@3 1 p Q 0 i C 8o2g . $ 0
and i b" . i Q The values of i and i are dependent on the cross-sectional profile of the Q . velocity and temperature distribution across the layer but the layers in these
30
P. H. Thomas et al.
Fig. 4. (a) Experimental data for Series I in Marshall and Harrison,5 see also Poreh et al.6 (b) Experimental data for Series II and III in Marshall and Harrison,5 see also Poreh et al.6 (c) Experimental data from helium and later studies, see Marshall and Harrison.5
31
The spill plume in smoke control design
Fig. 5. Values of layer depth (D ). B
experiments were similar to those studied earlier12 for which conventional values were i "1)30 and i "0)95. . Q With values for o , g and ¹ we obtain 0 0 D Q@1@3 D Q@ 1) 4Q 2@3 B (7) " B "2)16 1# M@ C¹M M@ Q@2@3 B p 0 B B
A
B
The empirical correlation from Fig. 6 is
A
B
D Q@1@3 Q B "2)50 1# (8) M@ C¹M B p 0 B and, whilst agreement with eqn (7) is good at low values of Q@/C ¹ M@ , p 0 B D differs from that theoretically expected in eqn (7) progressively as the B temperature rises. The layer in reduced scale experiments has a Reynolds number lower than the prototype has and the horizontal flow and its transition into the rising spill plume may be locally influenced by this but we have not pursued this feature of the spill plume, since we have been able to correlate the available data (admittedly small-scale experiments) without involving effects attributable to it.
32
P. H. Thomas et al.
Fig. 6. The variation of D with temperature. B
5 PRELIMINARY ANALYSIS OF FIG. 4a—c We seek correlations of M/Q in terms of M /Q and z/Q@2@3. We note the B existence of linearity with a small but perhaps significant intercept. Poreh et al. included calculated points corresponding to z"!D ("2b ) B 0 in the virtual downward extrapolation of the far-field plume. We discuss this later. (M!M ) is the total mass of entrained air but some is entrained below B z"0 (see Fig. 1) so that we expect M/Q to increase as aM /Q where a51. It B will be seen that there are few data in the range 0(z/Q@2@3(0)03, but these are important in considering the intercept at z"0. Several sets of data5 have been analysed separately, and Series I—IV as described by Poreh et al.6 are similar and can be treated together, so we can seek a correlation of the form M@ M@ z "a B#b #c Q@ Q@ Q@2@3
(9)
where, here and henceforth, a, b and c are regression coefficients. One set of data with a small collecting hood has to be treated separately. This distinction was noted by Poreh et al.6 and has been demonstrated by Miles et al.13 using CFD. Another set of data from a model atrium14 which are not purely two-dimensional are discussed separately below.
6 LINEAR CORRELATIONS AFTER LEE AND EMMONS We have mentioned above that the treatment of Lee and Emmons leads to a linear relationship between M@ and z. The coefficient of proportionality led
The spill plume in smoke control design
33
Lee and Emmons to declare a value of 0)16 for the entrainment coefficient and, in the absence of other data, this was used by Morgan and Marshall to develop design methodology. The earliest of the experiments by Morgan and his collaborators also used the assumption of a Gaussian distribution to infer mass from measurements of temperature rise. Lee and Emmons assumed a Gaussian distribution e~x2@j2b2 for temperature rise and e~x2@b2 for velocity and found j"0)9. The ratio of the mean temperature rise h ("Q@/C M@) to h is given by .%!/ p .!9 j h .%!/" O0 ) 67 h J1#j2 .!9
(10)
whereas it is unity for a top hat profile. In the absence of direct measurements of mass flow or of temperature and velocity distribution, one must anticipate some uncertainty in the precision of estimates of mass flow from temperature measurements. Also, in the early experiments entrainment was possible into the ends of the plume, so that a methodology was devised7 to modify the Lee and Emmons plume theory. This problem is of considerable design importance, but we shall leave the discussion of end effects until the simple two-dimensional spill plume is dealt with. Work15—17 later than Morgan and Marshall’s early work has been interpreted by employing entrainment coefficients E differing from 0)16. Although one published value is larger than 0)16 and two other are lower.15,17 Law10,18,19 correlated data from Morgan et al. by using the same linear form of correlation as that produced by Lee and Emmons but did not use an a priori value for E. Thomas11 attempted to use the value of 0)16 with adjustments to the end-effect correction, viz; combining a line plume and an axi-symmetric plume so that they were asymptotic forms to a line plume with end entrainment which must eventually approximate to an axi-symmetric plume. The position of the virtual source D was also treated as an experimentally determined quantity, and D was expressed as a fraction of the opening height H. For small values of D /H, D is expected, from dimensional grounds, to be B related primarily to the depth D ("2b ) of the layer of the balcony flow with B 0 secondary temperature effects. However, when D /H becomes larger the layer B interacts with the inflow and D needs to be expressed as a fraction of H, temperature and D /H, viz., B
A
D Q@ D B, "f H H C ¹ M@ p 0 B
B
(11)
34
P. H. Thomas et al.
In the absence of sufficient design data to determine D or M@ , one can B B relate D to H as a design parameter only on statistical grounds.
7 STATISTICAL ANALYSIS The experiments analysed in this paper mostly only use recent measurements of M under a ‘large’ hood. The data in Fig. 4a—c have been analysed " statistically, each separate set being shown to be part of a homogeneous set described by M@ M M@ z B#0)159 " "1)20 #0)0027 Q@ Q Q@ Q@2@3
(12)
The coefficient of z/Q@2@3 is 0)16, whereas, in the same units, the coefficient of Lee and Emmons is 0)22. M is proportional to E2@3 so the Fig. 4 correlation corresponds to E"(0)16/0)21)3@2]0)16, viz., 0)11 which is closer to two values obtained15,16 since Lee and Emmons obtained 0)16 and to some previous values when recalculated.17 The correlation obtained by Poreh et al. using a database almost identical to that used here was in terms of M@ and Q@ rather than M and Q: M@!M@ B"0)16(z#2b ) or 0)16(z#D ) 0 B Q@1@3
(13)
The difference between the two forms, eqns (12) and (13), arises from our use of M@/Q@ and z/Q2@3 and Poreh et al.’s use of M/Q1@3 and z (or M and zQ1@3). Direct comparison of the regression equations is somewhat complicated by the different nature of the residual constant in a linear relation. Poreh et al., in effect, accommodated the terms 0)2M and the 0)0027 in the B assumption regarding D and that it could be simply added to ‘z’. They did not B separate so as to compare the regression coefficients of z and D , but they B demonstrated a close agreement between measured and calculated M (standard error $6%). The agreement is good but there is a redundancy in the formula requiring values of both M@ and D s. B B We can introduce eqn (8) into eqn (13) and so obtain M@ M@ 0)16z "1)40 B# #0)0014 Q@ Q@ Q@2@3
(14)
s Of course, it is possible to arrange D to be independent of M@ and Q@— but it would require B B different kinds of experiments.
The spill plume in smoke control design
35
The standard deviation (S.D) of the regression coefficients 1)20 and 0)159 are respectively 0)092 and 0)034. The S.D. of the constant term 0)027 is 0)006. There is error on D too. The 1)40 is thus on the border of acceptability. M@ /Q@ B B for these experiments varies from 0)0025 to 0)012t so the (1)40!1)20) M@ /Q@#0)0014 is close enough to 0)0027 and hence the equations can, in B practice, be regarded as the same. Poreh et al. did not include in their analysis the experiment with helium at z"0 and it is seen in Fig. 7 that this experiment is slightly anomalous, perhaps because our description of the behaviour is incomplete: its density difference makes it simulate a flame more than it does the hot smoke. Figure 7 shows that the four results for z"0 are described equally well by M@"1)4M@ B
(15a)
and M@"1)2M@ #0)0027Q@ (15b) B and there is a convenience in using the simpler equation, an alternative which will have little consequence away from the balcony.
Fig. 7. Correlation between M and M at Z"0. B t If the one experiment with helium is included the upper limit is 0)019.
36
P. H. Thomas et al.
Poreh et al.6 extrapolated a far-field plume downwards. However, there are a few data for z"0. The values of M for the three experiments for which z is zero are reported by Poreh et al. in terms of the value of M calculated from their eqn (15). The #!-# three ratios M /M are 1)09, 1)09 and 1)15, and they have an average of 1)11 #!-# which also suggests that M is slightly too large. We note that if z"0 Poreh #!-# et al.’s equation gives
A
B
¹ M@"M@ #0)16D Q@1@3"M@ 1#0)4 B (16) B B B ¹ 0 There is some ambiguity about the effect of the temperature of the layer yet to be resolved. One expects upward entrainment into the emerging layer to be less and downward entrainment to be greater when the layer temperature is higher and density is lower. One would need to involve D independently of M@ and Q@ if there were B B other factors, e.g. surface friction or an imposed initial momentum. However, one would require the study of a range of layer conditions, wider than in these experiments, the layer being presumed to be or nearly in equilibrium with its own buoyancy. To sum up, our analysis of almost the same set of data as Poreh et al. shows that their correlation can be simplified by substituting for D , the experiB mental relationship between D and M@ and Q@. It is then equivalent to what B we have obtained for the whole range of z within experimental error. However, our presentation of data, derived from dimensional analysis, leads to our conclusion that, despite any lack of self-similarity found by the CFD studies, the M(z) relationship is in practice linear. 8 EXPERIMENTS WITH PLUME EDGE EFFECTS: MODELLING ATRIA Some data on spill plumes in a reduced-scale model atrium have been published by Hansell et al.14 They were subsequently discussed by Law20 and Thomas21 and were incorporated into a report by CIBSE, the Chartered Institution of Building Service Engineers.22 The data shown in Fig. 8 are not general: the value of ¼ was close to being constant (c"0)5 m). They refer to a scaled-down version of a real atrium. This is not strictly two-dimensional but the data exhibit linearity. Morgan and Marshall7 had earlier on considered the near-field erosion of the edges of a finite source. One expects their procedure to provide a first approximation. Thomas11 combined two- and three-dimensional axi-symmetric plume formulae to provide an expression which was appropriate at the limits zP0
The spill plume in smoke control design
37
Fig. 8. Experimental data from atrium study14.
and zPR. The equation is not linear but the connection between the two limits is 1#0)22(z/¼) where ¼ is the length of the line source and the coefficient 0)22 comes from the ratio of Lee and Emmons’ line plume formula and an axisymmetric formula. The two components are equally weighted when z/¼+5: above this height this model suggests that the centreline temperature is affected significantly by edge entrainment. The highest experimental value of z/¼ was about 3 so the above expression implies only a 50% enhancement. When the data shown in Fig. 8 was subjected to a more detailed statistical analysis, the resulting regression gives a very different picture, viz., M@ M@ z "2)64 B#0)342 !0)0083 Q@ Q@ Q@2@3
(17)
To compare eqns (17) and (12), we note that so long as ¼/D is not too B small the two sets of data should give the same value of M at z"0 before the atrium edge effects become significant. The value 2)64 in eqn (17) has a large standard deviation p of 0)75 and it is plausible to reduce its value and raise the intercept (!0)0083) which is an implausibly negative intercept (p is 0)004 but there are covariances with the 2)64). The mean experimental value of M@ /Q@ is 0)0067, so we can obtain a first B approximation to the effect of removing the intercept by reducing 2)64M@ /Q@ B
38
P. H. Thomas et al.
by (0)0083/0)0067)M@ /Q@, i.e. to 1)40M@ /Q@. This is close to the coefficient in B B eqn. (15) and is within 2p of it and this is satisfied at a value of M@/Q@"0)0076, about 12% above the mean experimental value 0)0067. At M@ /Q@ equal to B 0)0076 both sides of the above equation give M@/Q@K0)012, a little larger than the 1)4]0)0067 found in the tests. We conclude that, apart from the regression coefficients of ‘z’, the difference between eqns (12) and (17) is more apparent than real. Without more data, one cannot evaluate the role of the dimensionless width as expressed by ¼/Q@2@3, ¼/D or z/¼. Equation (17) must be regarded as an B empirical correlation. Law’s20 equation is M@"0)31Q@1@3(z#0)102)
(18)
The difference between the regression coefficients for z/Q@2@3 is small. We discuss the significance of the constant 0)102 in Section 9. Our statistical analysis of the data for the small-scale model atrium is confined to the condition where the layer of hot gas is channelled. Poreh et al. report these results as they report the others, as M versus z and Q and subsequent accommodation of the plume width ¼. This is as appropriate for an expanding plume (i.e. one with edge effects) as the two-dimensional formulation of M@. They note a tendency in M to increase faster than z but not as fast as z5@3. They state that instead of M/Q increasing as 0)15z/Q2@3 (which becomes 0)16z/Q@2@3) it increases as 0)20z/Q2@3 for the unchannelled flow. Our correlation produces a regression of 0)34z/Q@2@3 and is only a little larger than Law’s. The values of ¼ for the atrium are about 0)47—0)53 m not 0)91 m, so the use of Q instead of Q@ lessens the difference between the correlation for channelled two-dimensional flow and the plume expanding into the atrium. Whichever formulation is used, the problem of dealing with edge effects remains. Morgan and Marshall’s correction for entrainment into line edges as ends of a plume was obtained by using the mean vertical velocity of the plume (in its initial stages) to which the entrainment velocity is proportional and an estimate of the area over which entrainment takes place. For a very thick layer this is plausibly proportional to D and z and for a very thin layer to z2. B Whether or not the values evaluated for the coefficients of M@ in the atrium B data eqn (17) and the wholly channelled flow data eqn (12) are statistically identical, the value of 1)4 is plausible for both and, even if this is not the optimum value, it is not plausible to assign significantly different values at z"0. The difference between the correlations away from z"0 is thus roughly M !M 0)18z !53*6. 2$*.%/4*0/!-" Q Q@2@3
(19)
The spill plume in smoke control design
39
where the 0)18 is the difference between the 0)34 in eqn (17) and the 0)16 in eqn (14). The difference between the two mass flows is 0)18zQ@1@3. W and this is twice the end entrainment. However this end entrainment is not dependent (not to a first approximation) on ¼, and to generalise this result we need to speculate!! We generalise the result by taking the limits of ¼ in the experiments, i.e. 0)47 m(¼(0)53 m. Our best estimate for each end lies between 0)18 zQ@1@3]0)47 2 and 0)18 zQ@1@3]0)53 2 with a mean of 0)045zQ@1@3. The finding that the entrainment in these experiments is proportional to z implies the other component of area is a constant, i.e. independent of ‘z’. It is, e.g. possibly proportional to D or D itself, which in these experiments varied B B from 0)10 m to 0)17 m. Let us write M "KD Q@1@3z %$'% B
(20a)
0)045 0)045 KO P 0)10 0 ) 17
(20b)
where
i.e. M "0)33D Q@1@3z$25% %$'% B From eqn (8) this becomes ¹ (21) M "0)83M@ z B %$'% B ¹ 0 a quantity dimensionally consistent. Although this expression is plausible it is highly speculative and is clearly not applicable for z large compared to D or ¼. B The range—indeed the entire formulation—might be tested by a more detailed statistical analysis but a satisfactory confirmation of this procedure would require data over a wider range of D and ¼. B
40
P. H. Thomas et al.
9 THE VIRTUAL SOURCE In response to an invitation from Law20 to comment on the position of the virtual source, Thomas21 has shown that eqn (18) implies a virtual source at a level near the base of the layer. In order to formulate a working hypothesis, Poreh et al. assumed that at z"!D , M@ equalled M@ . At low zP0, edge B B effects or none, the two situations would be expected to be very similar for small D /¼, which for these experiments was about 1/5. B We identify 0)102 as D, and at z"0 we have M@"0)31Q@1@3D Substituting for Q@1@3 from eqn (8) gives ¹ M@ D "1)27 0 (22a) ¹ M@ D B B B which, since both D and D scale the same way for a given temperature and B equal values of any other dimensionless variables, would seem to be a general result, becoming D ¹ "1)7 0 (22b) D ¹ B B for M@/M@ equal to 1)4. In these experiments, the data have a mean value of B 0)0067 for M@ /Q@ which corresponds to a mean h of about 150°C, whilst the B experimental values of D vary from 0)10 m to 0)17 m. Hence, our estimate of B D from eqn (22b) lies between 0)11 m and 0)19 m. The lower end of the range of these estimates is consistent with the 0)102 in eqn (18). Had we used the regression coefficient 0)16 instead of 0)31, the value 0)102 would be increased to 0)102]0)31 &0)20, 0)16 and this is near the upper end of these estimates. The comparisons are limited by the different forms of eqns (12) and (18).
10 THE UK—US DIFFERENCE If the gases from a fire emerge from a door, the depth of the layer will be deeper in relation to the width of the opening than for gases emerging from a wide opening. The gases are therefore likely to become a rising axisymmetric plume relatively sooner than those from a wide opening. As
41
The spill plume in smoke control design
Fig. 9. The emerging plume.
discussed above, one expects that the height at which three-dimensional effects become significant must increase with the width ‘¼’ of the source of the plume, and this means that the major part of a plume rising from a doorway to a high ceiling is more nearly three-dimensional than that from the wider one typical of a shop front; see Fig. 9. An equation given in NFPA 92B-14,
C
M@"0)41Q@1@3(z#0)3H) 1#0)063
D
(z#0)6H) 2@3 ¼
(23)
is described as ‘based on Law’s interpretation of data given by Morgan and Marshall, a subsequent re-analysis by Morgan and modifications of a kind suggested by Thomas to make the calculated rate of entrainment approach that for an axi-symmetric plume at large heights’. The virtual source D (the correction to z) can only be associated as in eqn (22b) with H (for D ;H) as an average value for a particular design fire and B opening geometry for design procedures. As discussed above, the value of D is physically related to D and M@ . B B The value of 0)41 is larger than the 0)34 in eqn (17) and the 0)31 in eqn (18). Substituting either of these would necessarily raise the value to 0)063. This figure implies z/¼&15, where the two-dimensional and three-dimensional
42
P. H. Thomas et al.
expressions have equal weight instead of the lower value in the earlier discussion, and seems implausible. The discussion of this question in a later edition (1995) ignores the effect of entrainment into the ends of the spill plume for all z below 13¼, which also seems implausible.
11 THE ‘SMALL HOOD’ DATA One set of data5 has been recognised as very different from the others, and are shown in Fig. 10. The small hood appears to lead to higher entrainment and this has been thought to be due to some recirculation under the hood. Apart from the difference between the mean correlations there is a markedly greater scatter of the data about the average. The variation with ‘z’ is 50% greater than when the large hood was used, the regression coefficient between M@ and Q@1@3z was 0)25 instead of 0)16 but the scatter in the data is also quite different from that of the other data (cf. Figs. 4 and 10). The regression produces a regression coefficient on M@ /Q@ of 0)22 with a rather large error, as B implied by the scatter in Fig. 10. Poreh et al. recognised the apparently anomalous behaviour: we await a CFD study and more data to resolve the problems.
12 DISCUSSION AND CONCLUSIONS We have analysed two-dimensional data for the plume of hot gases rising from under a horizontal flat surface when they pass around the balcony edge.
Fig. 10. Plot of experimental ‘small hood’ data.1
The spill plume in smoke control design
43
Most of these data have been studied by Poreh et al. but our analysis—derived from a dimensional analysis—has a form different from theirs but it gives an equally good prediction of the mass flow M. It is simpler as it does not use layer depth D as well as M and Q. B B For h ;¹ , the layer depth, as reported by Poreh et al., is only slightly B 0 higher than a simplified theoretical prediction of D from values of M and B B Q—the heat flow, but the difference increases significantly for h &¹ . B 0 In the absence of self-similarity, one cannot expect a constant entrainment coefficient independent of height, only an average value over all the height. However, the data do produce an effectively constant value and—as Poreh et al. observed—it is less than the Lee and Emmons value originally taken by Morgan and his colleagues. The value is near to 0)11. This agrees with other estimates in the recent literature. We have separately examined data from experiments on a model atrium and reconciled our regression with the correlation in the CIBSE report (or vice versa), i.e. Poreh et al.’s assumption regarding the virtual source position and our interpretation of the 0)102 in eqn (18) are very similar and the coefficients 0)31 and 0)34 for the term in ‘Z’ are also very close. However, these data merit further comment. In our discussion of the problem of edge, i.e. end effects on a finite line plume, we have speculated as to how to scale these data. However, further data are required. We found the results are: (a) The large hood data are consistent broadly with the analysis of Poreh et al. (b) The large hood data are consistent with an entrainment coefficient less than 0)16 used in earlier work. This use of 0)16 had necessitated the introduction of various procedures and coefficients. The procedure became complex and the complexity can be removed by decoupling early BRE work from the experiments of Lee and Emmons. (c) Our correlation [eqns (12) and (14)] seems to be no less accurate than that of Poreh et al. and does not require all three terms M@ , Q@ and B D between which there is a relationship [eqn (8)]. B (d) The value of dimensional analysis has been demonstrated because the structure of data correlations can be deduced without reference to certain assumptions regarding similarity and hence the constancy of the entrainment coefficient. (e) The formula for the data on two-dimensional spill plumes rising into large reservoirs (hoods) is, for z/Q@2@3 less than 0)7: M@"1)20M@ #0)16zQ@1@3#0)0027Q@ B
(24)
44
P. H. Thomas et al.
An acceptable alternative is M@"1)40M@ #0)16zQ@1@3#0)0014Q@ (25) B Formulae of the same form but with different coefficients have been obtained for experiments with a small hood and a model atrium. The difference between the atrium formula and eqn (11) suggests that, provided z/¼ is not too large, the mass entrained into one edge in these experiments is 0)045zQ@1@3. We have discussed and reconciled our correlation of Hansell et al.’s data with that of Law. Care has to be taken in extrapolating the formulae for the two latter situations if only because the behaviour exhibits three-dimensional characteristics which are not recognised in the correlations which must be seen to be confined to a range of values of ¼/D or ¼/Q@2@3. B However, a somewhat speculative correction for edge effects has been proposed. It differs from that proposed by Morgan and Marshall and that proposed by Thomas4 and it can only be applied for z/¼ less than some, as yet undetermined, value. Further data are required. (f ) Poreh et al. have shown ‘that an acceptable agreement can be obtained between the BRE Spill Plume Model using the lower value° of the entrainment constant (and no adjustment to the height of rise) and the experimentally measured entrainment’. However, the range of the calculations seems to be relatively narrow, viz., 0)16—0)18 kg/s. Our model, being based on small-scale data effectively the same as Poreh et al.’s, which have been claimed6 to support the BRE Spill Plume Model, can presumably also be considered to be consistent with the BRE calculations in that range. What these models demonstrate is that the plume mass flow tends to be less than hitherto proposed for shallow smoke layers (the early stages of a fire). There has been a tendency to distinguish between small-scale data for z(0)3 m and for z'0)3 m but we have not found this. Dimensional analysis would anyway make any distinction occur at a particular value of z/Q@2@3. However, despite these remarks, the number of data available for analysis at low z/Q@2@3 is rather small to support—or contradict—conjectures concerning special entrainment processes into a deep smoke layer (zP0) when the dilution is low and the smoke control system is nearer the limit of its capability. Likewise, this work is confined to analysing data for flow from
°
That is 0)11 as represented by the coefficient 0)16 for z/Q@2@3 instead of the Lee and Emmons value of 0)16 which gives circa 0)22 z/Q@2@3.
The spill plume in smoke control design
45
beneath a horizontal balcony with no walls above which the plume might, if wide enough, adhere.
REFERENCES 1. Hansell, G. O. & Morgan, H. P., Design approaches for smoke control in atrium buildings. Buildings Research Establishment Report BR 258, Building Research Establishment, UK, 1994. 2. Morgan, H. P. & Gardiner, J. P., Design principles for smoke ventilation in enclosed shopping centres. BR 186, Building Research Establishment, 1990. 3. Butcher, G. & Parnell, A., Smoke Control in Fire Safety Design. E. & F. N. Spon, 1989. 4. Guide for smoke management systems in malls, atria and large areas. NFPA 92B, National Fire Protection Association, Quincy, MA, USA, 1991 (see also the 1995 edition). 5. Marshall, N. R. & Harrison, R., Experimental studies of thermal spill plumes. Building Research Establishment Occasional Paper OP1, April 1996. 6. Poreh, M., Morgan, H. P., Marshall, N. R. & Harrison, R., Entrainment by two-dimensional spill plumes. Fire Safety J., 30 (1998) 1—19. 7. Morgan, H. P. & Marshall, N. R., Smoke hazards in covered multi-level shopping malls: on experimentally based theory for smoke production. Building Research Establishment Current Paper CP 48/75, Fire Research Station, Building Research Establishment, UK, 1975. 8. Lee, S.-L. & Emmons, H. W., A study of natural convection above line fires. J. Fluid Mechanics, 11 (1961) 353—68. 9. Morton, B. R., Taylor, G. I. & Turner, J. S., Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. (¸ondon), 234A (1956) 1—23. 10. Law, M., A note on smoke plumes from fires in multi-level shopping malls. Fire Safety J., 10 (1986) 197—202. 11. Thomas, P. H., On the upward movement of smoke and related shopping mall problems. Fire Safety J., 12 (1987) 191—203. 12. Morgan, H. P., The horizontal flow of buoyant gases toward an opening. Fire Safety J., 11 (1986) 193—200. 13. Miles, S. D., Kumar, S. & Cox, G., The balcony spill plume—some CFD simulations. Proc. 5th Int. Symp. on Fire Safety Science, IAFSS, 1997 (to be published). 14. Hansell, G. O., Morgan, H. P. & Marshall, N. R., Smoke flow experiments in a model atrium. Building Research Establishment (BRE) Occasional Paper OP 55, Fire Research Station, BRE, 1993. 15. Ramaprian, B. R. & Chandresekra, M. S., Measurements in a vertical plane turbulent plume. J. Fluid Engng, 111 (1989). 16. Kotsovinos, N. E., A study of entrainment and turbulence in a plane buoyant jet. W. M. Keck Laboratory of Hydraulics and Water Resources Report KH-R-32, California Institute of Technology, Pasadena, 1975. 17. Yuan, L. & Cox, G., An experimental study of some line fires. Fire Safety J., 27 (1997) 123—39.
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18. Morgan, H. P., Letter to the Editor. Fire Safety J., 12 (1987) 83. 19. Law, M., Letter to the Editor. Fire Safety J., 12 (1987) 85. 20. Law, M., Measurements of balcony smoke flow (short communication). Fire Safety J., 24 (1995) 189—95. 21. Thomas, P. H., Discussion of measurements of balcony smoke flow (comment). Fire Safety J., 24 (1995). 22. Relationships for smoke control calculations. Chartered Institute for Building Service Engineers TM 19, 1995.