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Effective thermal inertia in relation to normalized heat load
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EFFECTIVE THERMAL INERTIA I N RELATION T O NORMALIZED HEAT LOAD
by G. Williams-Leir
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La c h a r g e t h e r m i q u e n o r m a l i s ' e e , c a l c u l ' e e en d i v i s a n t l ' a b s o r p t i o n t h e r m i q u e p a r l ' i n e r t i e t h e r m i q u e , permet d e l i e r l a v i o l e n c e d ' u n i n c e n d i e l o r s d e f e u x d e compartiment e n v r a i e g r a n d e u r
B
l a v i o l e n c e d'un i n c e n d i e mesur'ee l o r s d ' e s s a i s c o u r a n t s d e r g s i s t a n c e a u f e u . Harmathy a montr'e comment l a c h a r g e t h e r m i q u e n o r m a l i s ' e e p o u v a i t C t r e u t i l i s ' e e p o u r d l t e r m i n e r l a r l s i s t a n c e a u f e u r e q u i s e d a n s l e s b 3 t i m e n t s p r o j e t ' e s . Cet a r t i c l e e x p l i q u e comment on p e u t o b t e n i r d e s v a l e u r s e f f i c a c e s d e l ' i n e r t i e t h e r m i q u e p o u r d e s mat'eriaux t e l s q u e l a b r i q u e e t l e b l t o n , p o u r l e s q u e l s l e s propri'et'es thermiques dCpendent f o r t e m e n t de l a t e m g r a t u r e .Effective Thermal Inertia in Relation to
Normalized Heat Load
G . Williams-Leir
Fire Research Section, Division of Building Research, National Research Council of Canada, Division of Building Research, Ottawa, Canada KIA OR6
Normalized heat load, obtained by dividing heat absorption by thermal inertia, is a quantity useful in bnilding design for relating fire severity in M y developed compartment fires to fire severity in standard fue resistance tests. Harmathy has shown how normalized heat load may be used for determining required fire resistance in projected buildings. The present work d e s c 4 i how effective values of thermal inertia can be calculated for such important materials as brick and concrete, both normal and lightweight, for which thermal properties depend strongly upon temperatore.
INTRODUCIlON and other variables, but within the range relevant to
fire protection it is found to be essentially constant for To ensure that a fire wall will withstand a fire, two
steps are necessary; the severity of the fire must be assessed and so must the fire resistance of the wall. Until the present century, both were estimated, solely from experience of accidental fires, in terms of hours of duration. The introduction of fire-resistance testing of walls and other elements of construction did not alter the logical basis, but it defined a standard meas- ure of severity of fire.'
A change began when efforts were made to estimate what fire resistance future construction would need. Ingberg's work2 related fire-resistance requirement to fire load, the mass of combustibles per unit of floor area. The next advance was the recognition that the ventilation of an enclosure was relevant. The fire- resistance requirement estimated by Law3 and Lie4 was not necessarily the duration of fire that could be expected in given circumstances. Instead, it took ac- count of fire load and of the geometry of the enclosure in estimating the resistance necessary in terms of the standard fire, and allowed for fires more or less severe than the standard.
The concept of fire severity as an additional variable was thus implicit before Harmathys showed that it could be characterized in terms of a parameter that he called the 'normalized heat load,' H. The gain from this innovation was that it became possible to take into account the thermal properties of the construction. Where some part of the compartment boundaries was able to absorb heat because of high thermal conductiv- ity and capacity, the compartment as a whole suffered a correspondingly less severe fire.
The mode of calculation of H assumed that the thermal properties were independent of temperature, which is rarely, if ever, true in p r a ~ t i c e . ~ The purpose of the present paper is to show how H should be calculated from temperature-dependent properties. This will be facilitated by defining an 'effective thermal
each material.
To demonstrate this, 8 has been calculated for each of a variety of materials, for a range of fire duration times, and for distinct time-temperature schedules. One of these schedules is the standard curve used in fire resistance testing.' Another curve is needed to represent a developed compartment fire. It is, how- ever, a matter of controvery which
of
many published compartment fire models7 best fits the many published experirnents."~, it is the implication of the nor- malized heat load doctrine that, fora
given value ofH,
the actual form of the temperature curve is unimpor- tant, a view justified by the observation that the calculated effective thermal inertia is little affected by the choice of curve. To prove this, the property has been calculated using three models, those published by Lie,g Wickstrom1° and Babrauskas.ll For each, the governing parameters were varied through a range of values intended to represent the range of practical conditions.
Normalized heat load H has been defined,12 for materials having constant thermal properties, as
H = (?7IJ(kw) (1)
If an infinitesimal time interval dt is considered, it follows that
d H = q dtlJ(kpc) (2)
and for such an interval the equation remains true even if k, p and c are temperature-dependent and q varies with time. It follows that for such materials the normalized heat load is
Ir H = [ q(t) dt
JCk(T)w(T)I (3)
Since this is a less convenient form to apply than Eqn (1) above, there is reason to consider the analog- ous form
-
inertia', 8. H = i j ~ / e
This quantity is a property of each material. Strictly, (4)
it is also dependent on temperature range, timescale where 8, entitled 'effective thermal inertia', is defined
by Eqns (3) and (4) taken together. Once values of 8
have been established (by a procedure to be de- scribed), Eqn (4) may be used as readily as Eqn (1). From Eqns (3) and (4) it follows that
NORMALIZED HEAT LOAD IN
STANDARD
FIRE
TE!3TSTable 1 presents the results of computations in which (ASTM E-119) standard fire tests on a series of slabs of graduated thicknesses were simulated. Two kinds of quartz-anorthosite concrete of widely differing ther- mal properties, normal and lightweight, clay bricks, and two kinds of insulating firebricks13 were consi- dered. As the efficiency of heat transfer from furnace to specimen might be expected to influence the results, two extreme conditions were considered, namely (a) sf, emissivity of furnace gas = 0.9; E, emissivity of
specimen = 0.6; (b) both emissivities unity.
Only the concretes are included in Table 1. Both radiative and convective heat transfer were taken into account. The heat flow and temperature distribution in the specimen were calculated using thermal properties dependent on temperature.14 At each time step the
Table 1. N o m d h d heat loads and thermal fire endurances of two concretes
Standard fire Normalized Effective thermal Emissivity Thickness enduranca. heat load, inertia
product (m) T (h) H (s''~K) (J m-2s-'n K - ~ ) (1) Normal density quartz-anorthosite concrete
(furnace gas emissivity 0.9, specimen 0.6)
0.54 0.0254 0.203 6 058 2061 0.0508 0.397 16010 2137 0.0762 0.668 29 596 2023 0.1016 1.027 45 251 1952 o.ino 1.46s 62 079 1910 0.1524 1.995 79 225 1 888
(furnace gas and specimen emissivities 1.0)
1 0.0254 0.175 6 671 2076 0.0508 0.350 18 531 2049 0.0762 0.605 32 456 1 964 0.1016 0.950 48 079 1910 0.1270 1.383 64 865 1 883 0.1524 1.905 81 934 1 869
(2) Lightweight quartz-anorthosite concrete
(furnace gas emissivity 0.9, specimen 0.6)
0.54 0.0254 0.238 12787 864
0.0508 0.583 32 106 852
0.0762 1.123 54 246 846
0.1016 1.870 78 290 849
(furnace gas and specimen emissivities 1 .O)
1 0.0254 0.217 13 702 856 0.0508 0.552 33 106 849 0.0762 1.088 55 265 845 0.1016 1.837 79 458 849 0.1270 2.800 104232 855 0 0 0.5 1.0 1. 5 2 . 0 2 . 5
STANDARD FlRE T E S T DURATION, h
Figure 1. Normalized heat load requirements, H, of standard fire tests.
quantity q(t) dt/ J[k(T)pc(T)] was summed, so that at the end of each computation an estimate of the in- tegral, i.e. the normalized heat load, H, in Eqn (4),
was available; H is tabulated in column 4. The heat flow, q(t)dt, was also summed. From these sums, 8,
the effective thermal inertia, was calculated using Eqn
( 5 ) and is given in column 5. This quantity depends on the material of the slab. It is only slightly influenced by test duration, or by emissivity of furnace and speci- men. The values are not far from those proposed by H a r m a t h ~ . ~
Figure 1 is a graph of H, normalized heat load calculated as described, against duration of fire test. The product of furnace and specimen emissivities ap- pears as a parameter. The graph may be compared with curves 4 and 6 in Fig. 1 of Reference 15 in order to visualise the implications of Eqn (3).
In addition, a curve is given for the theoretical limiting case of thermal inertia tending to zero, as calculated by the method of Harmathy and Mehaffey.15 The consistency of the limiting curve with the others, derived by a very different procedure, is supporting evidence of their validity.
NORMALIZED HEAT LOAD IN REAL
COMPARTMENT FIRES
It has been shown above how to derive 8, the effective thermal inertia, from heat flow calculations using the temperature curve of the ASTM E-119 standard fire test. In real fires the temperature curve may be very different. To examine the influence of the temperature curve on the effective value of the thermal inertia, additional calculations were performed by the same method as before, except that the ASTM E-119 stan- dard temperature curve was replaced by curves in- tended to represent real fires. Any published tempera- ture curves for compartment fires could have been used for this purpose, but those of Lie,g Wickstrom1° and Babrauskasl' were chosen for the examples that follow.
EFFECTIVE THERMAL INERTIA IN RELATION TO NORMALIZED HEAT LOAD
Table 2. Effective thermal inertia in compartment &res F, opening factor (m'?
Concrete Thickness Duration, 0.01 0.02 0.04 0.08 0.12 0.15
n/pe (mm) ~ ( h ) 9, sftsemn, thermal inertia {J rn-'sj IC-1
Normal
Lightweight
TIME TO FUEL EXHAUSTION. h
Figurg 2. Normalized heat load, H, on ceilings of 152mm thickness for temperature curves characteristic of compartment fires.
For convenience, Lie's temperature curves are de- scribed in terms of two parameters, namely fire dura-
tion, T ( s ) , and ventilation factor, ~(rn'3:
When these are specified, fuel load is determined. The curves were used, as described previously for standard
fire tests, as input to a computation to sum the heat
flow through the surfaces and the normalized heat
load. The quotient of these is 8, the effective thermal
inertia. The summations were continued to the instant of heat flow reversal, i.e. the time at which the surface begins to lose heat to the compartment gases as they
cool. This was done for six values of F from 0.01 to
0.15 m1I2; for four fire durations from 15 min to 2 h; and for three slab thicknesses, 51, 101 and 152 mm, of
Table 3. Effective thermal inertia for five materials
Effective thermal inertia (J m-2 s-"' K-'1
Insulating fire brick: G23 Insulating fire brick: G26 Lightweight concrete Clay bricks Normal concrete
Computed using compartment temperature
curve due to Sug(lested
Standard working
Ref. fire test (Ref. 1) Lie (Ref. 9) Wickstmm (Ref. 10) Babrauskas (Ref. 11) value
G. WILLIAMS-LEIR the same five materials
as
before. Thus 360 cornbina-CONCLUSION
lions of Darameter values were tried. The results for the two concretes at temperatures corresponding to the Lie curves are given in Table
2;
Fig. 2 illustrates them for 152-mm slabs.As before, 8, the effective thermal inertia, is found to depend principally on the material of the compart- ment walls, ceiling and ffoors, but it is also slightly influenced by the values of T, F and slab thickness.
The ranges of 8 values are shown in Table 3 for the three models conqidered as well as for the standard f i e test.
The B values for compartment fires are, to the accuracy expected in fire studies, the same as those for standard tests. From these ranges, rounded working values have been arbitrarily chosen and shown in coIumn 7. Given these working values for effective thermal inertia, required fire endurance may be calcu- lated by Harmathy's method%ith greater precision.
Normalized heat load in either real fires or standard fire tests may be obtained by dividing the tola1 heat absorbed through a bounding surface of a compart- ment by 8, the effective thermal inertia of the material of the surface. Calculated values of 8 are given in Table 3.
Acknowledgments
The author takes pleasure in acknowledging a valuable suggestion
from J. R. Mehaffey, and comments and encouragement from other colleagues.
'This paper is a contribution from the Division of Building Re-
search. National Research Council of Canada, and is published with thc approval of the Ditector of the Division.
REFERENCES
1. American Society for Testing and Materials. Standard methods of fire tests of building construction and materi- als, E 119.
2. S. H. Ingberg, Tests of severity of building fires. NFPA Quarterly 22, 43 (1928).
3. M. Law, A relationship between fire grading and building
design and contents. Fire Research Station, Borehamwood, England, IN 385 (January 1971).
4. T. T. Lie, Fire resistance of structural steel. Eng. J, Amer.
Inst. Steel Constr, 15, 116-25 (1978).
5. T. 2. Harmathy, Fire severity: basis of fire safety design.
Presented at International Symposium on Fire Safety of
Concrete Structures, American Concrete Institute, San Juan, Puerto Rico (1980).
6. T. 2. Harmathy, Thermal properties of concrete at elevated
temperatures. ASTM, J. Materials 5.47-74 (1970).
7. T. 2. Harmathy and J. R. Mehaffey, Review paper: post-flash-
over compartment fires. Fires and Materials, 7,
49-61 1983).
8. P. H. Thomas and A. J. M. Heselden, Fully-developed fires in single compartments. CIB Report No. 20, Fire Research Note 923, Building Research Establishment, Boreham- wood, Harts, August 1972.
9. T. T. Lie, Characteristic temperature curves for various fire severities. Fire Technol. 10, 315-26 (1974).
10. U. Wickstrom, Temperature calculation of insulated steel
columns exposed to natural fire. Fire Safety J. 4, 219 (1981 121.
11. V. Babrauskas, A closed-form approximation for post-
flashover compartment fire temperatures. Fire Safety J. 4, 63, (1981).
12. J. R. Mehaffey and T. 2. Harmathy, Assessment of fire resistance requirements. Fire Technol. 17, 221 (1981). 13. T. 2. Harmathy, Properties of building materials at elevated
temperatures. Prepared for members of CIB W14, February 1982.
14. T. T. Lie, Calculations of the fire resistance of composite concrete floor and roof slabs. Fire Technol. 14, 2845, 50
(1978).
15. T. 2. Harmathy and J. R. Mehaffey, Normalized heat load, a key parameter in fire safety design. Fire and Materials 6, No. 1, 27-31 (1982).
Received 19 July 1982; accepted (revised) 24 May 1983
NOMENCLATURE c (J kg-' K-') F (m'I2) hv (m) H (s1I2 K)
k
(W m-' K-') q (W m-3Area of openings (doors, windows, etc.)
Area of bounding surfaces of com- partment (ceiling, floor and walls, excluding openings)
Specific heat
Opening factor A, J(h,)IA, Opening height
Normalized heat load Thermal conductivity
Instantaneous value of heat flux through the surface
4
(W m-2) Mean value of q over the interval from 0 to TT PC) Temperature
t (s) Time
E Emissivity of specimen
E f Emissivity of furnace and its gases
8 (J m-2 s-'I2 KP')Effective thermal inertia of the com- partment surface materials
P (kg m-3) Density
(s) Duration of test or compartment
fire
T h i s p a p e r , w h i l e b e i n g d i s t r i b u t e d i n r e p r i n t form by t h e D i v i s i o n of B u i l d i n g Research, remains t h e c o p y r i g h t of t h e o r i g i n a l p u b l i s h e r . It s h o u l d n o t be r e p r o d u c e d i n whole o r i n p a r t w i t h o u t t h e p e r m i s s i o n of t h e p u b l i s h e r . A l i s t of a l l p u b l i c a t i o n s a v a i l a b l e from t h e D i v i s i o n may be o b t a i n e d by w r i t i n g t o t h e P u b l i c a t i o n s S e c t i o n , D i v i s i o n o f B u i l d i n g R e s e a r c h , N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a , O t t a w a , O n t a r i o , K I A OR6.
