Thermal diffusivity

Using the data given in sections 5.1.1.2 and 5.1.1.3 and the standard den-sity values, the thermal diffusivity of steel (m²/h) shows a sensibly linear relationship with temperature up to 750^◦C according to the following equation (Malhotra, 1982a):

aa= 0,87 − 0,84 × 10⁻³θ_a (5.10)

5.1.2 Concrete

With concrete, the situation is much more complex, in that values of the thermal parameters required are dependant on the mix proportions, the type of aggregate, the original moisture content of the concrete and

the age of the concrete. The data presented in this section can thus only be taken as representative of typical concretes.

5.1.2.1 Density

Even though when concrete is heated there will be a loss in weight caused by the evaporation of both free and combined water, this loss is not generally enough to cause substantial changes in density and thus it may be considered accurate enough to take ambient values. However, EN 1992-1-2 suggests that change in density with temperature to be used in thermal calculations may be taken as

For 20^◦C≤ θc≤ 115^◦C

ρ (θ_c)= ρ(20^◦) (5.11)

For 115^◦C≤ θc≤ 200^◦C

ρ (θ_c)= ρ(20^◦C)

1− 0,02θ_c− 115 85



(5.12)

For 200^◦C≤ θc≤ 400^◦C

ρ (θ_c)= ρ(20^◦C)

0,98− 0,03θ_c− 200 200



(5.13)

For 400^◦C≤ θc≤ 1200^◦C

ρ (θ_c)= ρ(20^◦C)

0,95− 0,07θ_c− 400 800



(5.14)

where ρ(20^◦C) is the ambient density.

For structural calculations the density of concrete must be taken as its ambient value over the whole temperature range.

5.1.2.2 Specific heat

Figure 5.3 presents values of specific heat for a variety of concretes (Schneider, 1986a), where it will be noted that the type of aggregate has a substantial effect on the values.

00 0,4 0,8 1,2 1,6 2,0

200 400 600 800 1000

c_p (kJ/kgK)

Θ (°C) Wet concretes

Calcerous concrete

Gravel concrete

Granite concrete Carbonate

Lightweight

Eqs (5.15)–

(5.18)

Figure 5.3 Variation of specific heat of concrete with temperature (Schneider, 1986a, by permission).

EN 1992-1-2 gives the following equations for the specific heat of dry normal-weight concrete (siliceous or calcareous aggregates) (J/kgK):

For 20^◦C≤ θc≤ 100^◦C

cp(θ_c)= 900 (5.15)

For 100^◦C≤ θc≤ 200^◦C

cp(θ_c)= 900 + (θc− 100) (5.16)

For 200^◦C≤ θc≤ 400^◦C

cp(θ_c)= 1000 +θ_c− 200

2 (5.17)

For 400^◦C≤ θc≤ 1200^◦C

cp(θ_c)= 1100 (5.18)

Table 5.1 Values of c_p,peak

Moisture content (%) c_p,peak(J/kgK)

0 900

1,5 1470

3,0 2020

10,0 5600

where the moisture content is not explicitly evaluated in the thermal anal-ysis, then a peak c_p,peakis added to Eq. (5.16) at 100–115^◦C before decaying linearly to 200^◦C. The values of c_p,peakare given in Table 5.1.

Equations (5.15)–(5.18) are also plotted in Fig. 5.3.

For lightweight concrete a constant value of 840 J/kg^◦C may be taken (EN 1994-1-2).

5.1.2.3 Thermal conductivity

Figure 5.4 presents values of thermal conductivity for various concretes (Schneider, 1986a). It will be observed that the normal-weight aggregate concretes fall into a band with the values for lightweight concrete being substantially lower.

EN 1992-1-2 gives the following equations as limits between which the values of thermal conductivity (W/mK) of siliceous aggregate normal-weight concretes lie:

A country’s National Annexe is likely to specify which curve is to be used. However, the discrepancy between the analytical curves and the results from Schneider plotted in Fig. 5.4 is not explained.

EN 1994-1-2 indicates that it is permissible to take a constant value of 1,6 W/m^◦C.

0 0 1,0 2,0

200

Concrete thermal conductivity (λc) (W/m°C) ^Siliceous

Lightweight

400 600 800 1000

Temperature (°C) Lower bound

(Eq. 5.20)

Upper bound (Eq. 5.19)

Figure 5.4 Variation of thermal conductivity of concrete with temperature (Schneider, 1986a, by permission).

For lightweight concrete EN 1994-1-2 gives the following relationship:

For 20^◦C≤ θc≤ 800^◦C

λ_c= 1,0 − θ_c

1600



(5.21)

For θc>800^◦C

λ_c= 0,5 (5.22)

0 0 0,001 0,002 0,003 0,004

200 Lightweight

Concrete thermal diffusitivity (ac) (m2/h)

Normal weight

400 600 800

Temperature (°C)

Figure 5.5 Variation of the thermal diffusivity of concrete with temperature (Schneider, 1986a, by permission).

5.1.2.4 Thermal diffusivity

For the results plotted in Fig. 5.5 it will be observed, as expected, that two distinct bands of results for normal-weight and lightweight concrete exist (Schneider, 1986a).

Using the values of λc = 1,60 W/mK and cc = 1000 J/kgK recom-mended in EN 1993-1-2 for simple calculation methods together with a density ρc = 2400 kg/m³, it is suggested that an approximate value of the thermal diffusivity ac can be determined using Eq. (5.2) to give ac= λc/ρcc= 1,6/(1000 × 2400) = 0,67 m²/s. This value may be high as Hertz (1988) suggests acmay be taken as 0,35× 10⁻⁶m²/s for granite (or sea gravel) and 0,52× 10⁻⁶m²/s for quartzite concretes, and Wickström (1985a) suggests a value of ac = 0,417 m²/s (see section 5.7 for a further discussion).

5.1.3 Masonry

The most significant variable characterizing the high temperature perfor-mance of masonry is the density rather than the type of brick (clay or calcium silicate) as the density is a measure of the porosity of the brick.

5.1.3.1 Density

Again the density should be taken as the ambient value.

5.1.3.2 Specific heat

As shown in Fig. 5.6, the specific heat is sensibly independent of the den-sity of the brick (Malhotra, 1982a). Harmathy (1993) gives the following expression for the specific heat of masonry (kJ/kg^◦C):

cpm= 0,851 = 0,512 × 10⁻³θ_m−8,676× 10³

(θ_m+ 273)² (5.23)

where θmis the temperature of the masonry.

5.1.3.3 Thermal conductivity

As shown in Fig. 5.7, the thermal conductivity of masonry is dependant on the density of the brick with high density bricks having higher values of thermal conductivity (Malhotra, 1982b). Welch (2000) indicates the effect

0 0 0,4 0,8 1,2 1,6

200 400 600 800 1000 1200

Temperature (°C)

Masonry specific heat (kJ/Kg°C)

Figure 5.6 Variation of the specific heat of masonry with temperature (Malhotra, 1982a, by kind permission of the author).

0 0 0,4 0,8

0,6

0,2 1,0

200 400 600

Density 2100 kg/m³

1600 kg/m³

1100 kg/m³ 700 kg/m³

800 1000 1200

Temperature (°C)

Clay brick thermal conductivity (W/m°C)

Figure 5.7 Variation of the thermal conductivity of masonry with temperature (Malhotra, 1982b, by permission Messrs Dunod).

of moisture on the effective thermal conductivity λ^is given by

λ^= λ0(1+ M)^0,25 (5.24)

where λ₀ is the dry thermal conductivity (W/mK) and M the moisture content (%).

5.1.4 Timber

As mentioned in the introduction to this section, generally, the only ther-mal property needed to determine temperatures within the uncharred core is either the thermal diffusivity (mm²/s) which is given in Schaffer (1965) as

aw= 0,2421 − 0,1884S (5.25)

where S is the specific gravity of the timber or, the thermal conductivity (W/m^◦C)

λ_w = (2,41 + 0,048M)S + 0,983 (5.26) where M is the moisture content in per cent by weight.

It appears that both the thermal diffusivity and the thermal conductiv-ity are independent of temperature whereas the specific heat of oven dry wood given in White and Schaffer (1978) is temperature dependant and is given in kJ/kg^◦C as

cpw= 1,114 + 0,00486θw (5.27)

where θw is the temperature of the wood.

5.1.5 Aluminium

Owing to the lower softening and melting points of aluminium compared with steel materials data are required over a more limited temperature range, i.e. up to 300^◦C. This is true for aluminium in both its pure state and when alloyed.

5.1.5.1 Density

The density used in calculations may be taken as that pertaining at ambient conditions (i.e. 2700 kg/m³).

5.1.5.2 Specific heat

Touloukian and Ho (1973) and Conserva, Donizelli and Trippodo (1992) suggest that over the temperature range 0–300^◦C the specific heat may be taken as constant with a value between 90 and 100 J/kg^◦C with the slight scatter in the values being due to the effect of the various amounts, and identity, of the trace elements used in the various alloys.

ENV 1999-1-2 gives the following equation for specific heat c_al (J/kg^◦C) for an aluminium temperature θalfor 0^◦C < θal<500^◦C

cal= 0,41θal+ 903 (5.28)

5.1.5.3 Thermal conductivity

Touloukian and Ho (1973) and Conserva, Donizelli and Trippodo (1992) suggest that over the temperature range 0–300^◦C the thermal conductivity may be taken as constant with a value of 180–240 W/m^◦C. The scatter in the values quoted is again due to both the effect of the various amounts

and identity of the trace elements used in the various alloys and there is also a very slight temperature dependence. ENV 1999-1-2 gives two equa-tions for the thermal conductivity of aluminium λ_al (W/m^◦C) dependant upon the alloy:

For alloys in the 1000, 3000 and 6000 series,

λ_al= 0,07θal+ 190 (5.29)

For alloys in the 2000, 4000, 5000 and 7000 series,

λ_al= 0,1θal+ 140 (5.30)

In document Fire Safety Engineering - Design of Structures Malestrom (Page 116-125)