4 Behaviour of natural fires
4.5 FIRE SEVERITY AND TIME EQUIVALENCE .1 Fire severity
4.5.2 Time equivalence
4.5.2.2 Normalized heat load base
(4.35)
where αvis the ratio between the area of vertical openings and the floor area and should lie between 0,025 and 0,25, αh by the ratio between the area of the horizontal openings and the floor area, H is the compartment height and bvis given by
bv= 12,5
1+ 10αv− α2v
≥ 10,0 (4.36)
For small compartments with no roof openings and a floor area less than 100 m2and no horizontal openings, the ventilation factor wf may be taken as
wf =Af
At
O−0,5 (4.37)
where O is the opening factor.
4.5.2.2 Normalized heat load base
The theory behind this approach was developed by Harmathy and Mehaffey (1985) and is presented below.
The normalized heat load h in a furnace test to the standard furnace curve can be related to the duration of the test te,dby
te,d = 0,11 + 0,16 × 10−4h+ 0,13 × 10−9 h
2
(4.38)
The heat flow from a compartment normalized with respect to the thermal boundaries of the compartment h is given by
h=
where Af is the floor area, At is the total area of compartment bound-aries, ÷(λρc) is the surface averaged thermal inertia of the compartment boundary, is a ventilation factor related to the rate of mass inflow of air into the compartment, Lf,dis the fire load (kg of wood equivalent) per unit area of floor and δ is a parameter defining the amount of fuel energy released through the openings and is given by
δ= 0,79
H3
(4.40)
where H is the compartment height.
The value of may be taken as its minimum value min given by Eq. (4.41) as this produces a conservative answer
min= ρairAv
gheq (4.41)
where heqand Avare the height and area of the window opening respec-tively, ρair is the density of air and g is acceleration due to gravity (9,81 m/s2).
The normalized heat load to the furnace h can be related to the normalized heat load in a compartment fire hby the following equation
h= hexp
where β is the statistical acceptance/rejection limit on the variables and is 1,64 for a 5% limit, and the ratio of the variance of the normalized heat load to the normalized heat load in the compartment is given by
σh
h = σL Lf ,d
At√
λρc+ 467,5
minLf ,dAf
At√
λρc+ 935
minLf ,dAf
(4.43)
where σL/ Lf ,dis the coefficient of variation in the combustible fire load in the compartment.
The coefficient of variation for the normalized heat input to the furnace is given by
σh
h = 0,9σt
e,d
te,d (4.44)
where σte,d/te,d is the coefficient of variation in the test results from a furnace and can typically be taken in the order of 0,1.
Thus when values of the fire load and compartment geometry are determined, the equivalent furnace test value can be found.
The alternative methods of determining the behaviour characteristics of a fire compartment are compared in the following example.
Example 4.1: Determination of the behaviour characteristics of a fire compartment
Consider a compartment 14 m by 7 m by 3 m high with 6 windows each 1,8 m wide by 1,5 m high with a fire load related to floor area of 60 kg/m2 of wood equivalent.
Compartment construction:
dense concrete(÷(λρc)= 32 Wh0,5/m2K)
Total area of windows:
Av= 6 × 1,5 × 1,8 = 16,2 m2 Af = 14 × 7 = 98 m2
At= 2 × 98 + 2 × 14 × 3 + 2 × 7 × 3 = 322 m2
Calorific value of fire load= 18 MJ/kg Fire load per unit surface area qt,d:
qt,d= 60 × 18 × Af/At= 60 × 18 × 98/322 = 329 MJ/m2
(This is equivalent to 1080 MJ/m2based on floor area) 1. Equivalent fire durations
(a) Equation (4.29):
so Av is greater than 10%Af, therefore the approximate method may be used.
(i) Exact method:
Determine the value of w from Eq. (4.32):
w=
Determine the boundary conditions:
÷ρcλ= 32 Wh0,5/m2K, hence from Table 4.5, c= 0,07
From Eq. (4.31),
(d) EN 1991-1-2 Annex F Take kc= 1,0.
From the data in the text with
÷(ρcλ)= 32 × 60 = 1920 J/m2s0,5K, kb = 0,055.
As there are no horizontal openings, the value of b is not required.
αv= Aw/Af = 16,2/98 = 0,165
The value of wf satisfies the limiting condition of 0,5.
Note: using the approximate Eq. (4.37) for wf gives
wf =Af
From Eq. (4.33)
te,d = qf ,dkbwf = 0,055 × 1,101 × 1080 = 65 min
The conservative value of wf gives te,d as 73 min.
Using the kbfactor value of 0,07 recommended in PD 7974-3 gives values for te,d of 83 and 93 minutes, respectively.
(e) Harmathy and Mehaffey Boundary conditions:
÷(ρcλ)= 32 × ÷3600 = 1920 Js0,5/m2degK Lf ,d= 60 kg/m2; Lf ,dAf = 5880 kg.
From Eq. (4.41) calculate min:
min= ρairAv
g heq= 1,21 × 16,2
9,81× 1,5 = 75,2 kg/s
From Eq. (4.40) calculate δ:
δ= 0,79
H3
min
0,5
= 0,79
33 75,2
0,5
= 0,473
To ease subsequent calculations:
÷Lf ,dAf = ÷75,2 × 60 × 98 = 665 At÷(ρck)= 322 × 1920 = 6 18 240
From Eq (4.40) calculate h:
h= (11,0δ+ 1,6)(Lf ,dAf)× 106 At√
λρc+ 935
Lf ,dAf
= (11,0× 0,473 + 1,6)5880 × 106 6 18 240+ 935 × 665
= 32 260 s0,5K
The normalized standard deviation for the fire load will be taken as 0,3. This is reasonable if the data for office loading in Table 4.3 are examined.
As recommended by Harmathy and Mehaffey, σte,d/te,d is taken as 0,1, thus from Eq. (4.44) From Eq. (4.38) calculate te,d:
te,d= 0,11 + 0,16 × 10−4h+ 0,13 × 10−9(h)2
= 0,11 + 0,16 × 10−4× 48003 + 0,13 × 10−9× 480032
= 1,18 h = 71 min
A comparison between the values of te,dis presented in Table 4.8, where it is noted that with the exception of the conservative approach adopted by the second of the two CIB approaches the answers are reasonably consistent, but with the method in EN 1991-1-2 giving a lower (therefore unconservative) value than earlier methods.
Table 4.8 Comparison between calculated equivalent fire durations (Example 4.1)
Method Equivalent fire duration (min)
Equation (4.29) 81
Equation (4.30) 89
CIB Workshop: Exact 93
Approximate 162
EN 1991-1-2 65 (73)
PD 7974-3 83 (93)
Harmathy and Mehaffey 71
2. Maximum temperatures (a) Equations (4.23) and (4.24)
Calculate η from Eq. (4.23):
η= At− AV
Av
heq = 322− 16,2 16,2√
1,5 = 15,4
Calculate θf,maxfor Eq. (4.24)
θf,max= 60001− e−0,1η
√η = 60001− e√ −1,54
15,4 = 1201◦C
Correct θf,max for the type of fire:
From Eq. (4.26) calculate ψ:
ψ= Lfi,k
√AV(At− Av)= 60× 98
16,2
322− 16,2 = 83,5
Calculate θmaxfrom Eq. (4.25):
θmax= θf,max
1− e−0,05ψ
= 1201
1− e−0,05×83,5
= 1183◦C
(b) Theory due to Lie using Eqs (4.9)–(4.12) Opening factor, O:
O= Av÷h/At= 16,2 × ÷1,5/310 = 0,0616
Fire load (in kg/m2) of compartment= 60×98/322 = 18,26 kg/m2 Fire duration tdusing Eq. (4.11):
td= Lfi,k/(330O)= 18,26/(330 × 0,0616) = 0,898 h = 53,9 min
Maximum allowable value of t using Eq. (4.10):
tmax= 0,08/O + 1 = 0,08/0,0616 + 1 = 2,30 h
Thus Eq. (4.9) will hold up to the total fire duration, and is evaluated in Table 4.9 and plotted in Fig. 4.9.
Table 4.9 Comparison between parametric curves due to Lie and EN 1991-1-2
Time (min) LIE (◦C) EN 1991-1-2 (◦C)
0 20 20
5 568 683
10 768 716
15 844 747
20 877 774
25 895 799
30 910 822
35 925 842
40 939 861
45 954 878
50 970 894
55 951 909
60 895 922
65 840 934
70 784 946
75 729 944
continued
Table 4.9—Cont’d
Time (min) LIE (◦C) EN 1991-1-2 (◦C)
80 673 909
85 618 874
90 562 839
95 506 804
100 451 769
105 395 734
110 340 699
115 284 664
120 229 630
125 173 595
130 118 560
135 62 525
140 20 490
145 20 455
150 20 420
155 20 385
160 20 350
165 20 315
170 20 280
175 20 245
180 20 210
185 20 176
190 20 141
195 20 106
200 20 71
205 20 36
210 20 20
215 20 20
220 20 20
225 20 20
230 20 20
235 20 20
240 20 20
The maximum temperature θmaxattained in the fire is 970◦C.
On the decay phase the fire reaches ambient at 2,31 h (139 min), and the temperature profile is linear between the maximum and the point at which ambient is reached.
Time (min)
Temperature (°C)
LIE EN1991-1-2 0
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 200
400 600 800 1000 1200
Figure 4.9 Comparison between parametric curves due to Lie and EN 1991-1-2.
(c) EN 1991-1-2 using Eq. (4.13)–(4.19):
From Eq. (4.15) calculate :
O= 0,0616 m0,5and ÷(ρcλ)= 1920 Ws0,5/m2◦C
= (O/÷(ρcλ))2/(0,04/1160)2= 0,866
Calculate tmaxfrom Eq. (4.17):
tmax= 0,20 × 10−3qtd/(O)= 1,223 h(=74 min)
The values of the increase in gas temperature θgover ambient may now be calculated up to the design time td. These values are given in Table 4.9 and plotted in Fig. 4.9. The maximum gas temperature θmax(above 0◦C) at a real time of 74 min is 953◦C.
As the fire load density is typical of an office, then from Table 4.4, tlim is 20 min (0,333 h). This is less than tmax, therefore x = 1,0 in the decay phase.
Since the parametric fire duration is between 0,5 and 2,0 h, Eq. (4.16) is used to calculate the delay phase. The fire decays to a temperature of 20◦C at 2,96 h parametric time or 3,42 h real time.
It is noticed in Fig. 4.9 that whilst the maximum temperatures are similar, Lie predicts a shorter time period to maximum tempera-ture, a faster cooling rate and a shorter total overall duration of the complete fire of around two-thirds that of EN 1991-1-2.
3. Fire duration:
From Eqs (4.27) and (4.28) calculate the rate of burning:
(i) Approximate Eq. (4.27)
From Eq. (4.22) R= 0,1×16,2×÷1,5 = 1,98 kg/s
td= Lfi,k/R= 60 × 98/1,98 = 2970 s or 49,5 min.
(ii) More exact equation, Eq. (4.26):
R= 0,18 × 16,2 × ÷1,5 × ÷(14/7) × (1 − e−0,036×15,4)= 2,15 kg/s
thus the fire duration is 46 min.
Table 4.10 gives the maximum temperatures reached in both paramet-ric curves and the value predicted by Eq. (4.21) where it will observed that both Lie and EN 1991-1-2 predict similar maximum temperatures and that Eq. (4.23) gives a value some 20% higher.
Table 4.11 gives the fire duration predicted by both parametric curves and the value predicted by Eqs (4.25) and (4.26) where it will be observed Lie, Eqs (4.25) and (4.26) predict similar values of around 50 min and EN 1991-1-2 a value around 50% longer.
Table 4.10 Comparison between maximum fire temperatures
Method Maximum temperature (◦C)
Lie 970
EN 1991-1-2 953
Equation (4.23) 1183
Table 4.11 Comparison between fire durations
Method Fire duration (min)
Lie 54
EN 1991-1-2 74
Equation (4.27) 50
Equation (4.28) 46