Thermal Diffusivity in Fire Protection

Thermal Diffusivity has been used as an indicator for the utility of materials as an insulator or heat conductor. It is most useful in unsteady state conditions, typically when heat is being added (fire protection) or removed (refrigeration). Thermal Diffusivity is defined as the ratio of Thermal Conductivity to Volumetric Heat Capacity.

To explain why Thermal Diffusivity is used as a Heat Transfer Characterization parameter we need to look at the Thermal Conductivity and the Volumetric Heat Capacity individually first.

Definition and Components of Thermal Diffusivity

Thermal Diffusivity can be expressed in the following manner:


Where

Thermal Conductivity

Thermal Conductivity is a material property that describes how fast that material can conduct energy. A high Thermal Conductivity indicates that the material transfers energy though itself, from hot-side to cold-side quickly. Cookware, pressing irons, and soldering equipment all take advantage of materials with this characteristic. Low Thermal Conductivity materials resist the flow of energy though itself. Potholders, ironing boards, space shuttle tiles all have materials in them with low Thermal Conductivity.

Density

Density is important to Thermal Diffusivity, since it is indicative of how much (mass) material is available to absorb energy.

Specific Heat Capacity

Specific Heat Capacity is the amount of energy (in SI it’s usually in Joules) needed to raise the temperature of specific amount of material 1 degree. This all depends on the unit system used. In SI, it’s 1 Kilogram or one Gram of material 1 degree Kelvin (Celsius). The product of Density and Specific Heat Capacity is the Volumetric Heat Capacity.

Using Thermal Diffusivity in Unsteady-State Applications

Thermal Conductivity can be misleading when used solely to judge a material's fitness as an insulator. Thermal Conductivity is useful in steady-state applications, but many thermal applications are unsteady-state. In fire protection we are never concerned with steady state. For some applications, the requirement is to minimize the temperature increase inside of an enclosed area completely surrounded by fire or heat for a specific time. There is no method for removing energy from the system.

We must look at it as a dynamic process, since the energy inside of the protected area will be accumulating. To minimize the resulting temperature increase we need a material that functions in two ways. First it must reduce the amount of energy transferred into the system (a classical "good insulator" with a low Thermal Conductivity).

Second it must absorb the accumulated energy with as little increase in temperature as possible. This is a material with a high Specific Heat Capacity and high Density (Volumetric Heat Capacity). An ideal insulator in this non-steady-state application is one that combines both a low Thermal Conductivity and a high Volumetric Heat Capacity. By definition this means we are looking for a material with a low Thermal Diffusivity.

Unsteady state calculations are tricky at best, and usually require solving differential equations with the known boundary conditions. For an example of a fire protection calculation, we will use the special condition of heat through a solid slab of known thickness, and infinite surface area. This is already solved for us in an equation that will tell us how long it will take the slab to come to an average temperature with known exterior temperatures.

Where:

The important thing to note in this equation is the term. Since we are seeing how long it takes to reach a certain temperature, with all other conditions being equal, the lower the thermal diffusivity, the larger the time will be to obtain that temperature.

For comparison, contrast a two-inch air gap with two inches of refractory ceramic fiber insulation. For these calculations, we need to determine some values first for Thermal Diffusivity of air and for RCF board at 18 pounds per cubic foot.

The value for air was found in a reference3, while the value for the RCF board was calculated using equation 1, and the appropriate thermal values. The values for Thermal Conductivity and Heat Capacity had to be calculated using a weight fraction distribution of the thermal properties of the individual components of the fiber, Silica and Alumina, and combined with the thermal diffusivity of the air, based on the board’s density. Interesting to note is the Thermal Diffusivity of the base material the alumina-silicate fibers are made of is lower than the thermal diffusivity of air. These calculations are beyond the scope of this paper, but are available for further study.

S2 the slab thickness = 2 inches = 0.051 meters

We will assume the temperature on the outside of the slab to be a constant value of 1500 F (1088 Kelvin), and we will find how long it takes to get the average temperature of the slab to 813F (707 Kelvin), the middle temperature between the cold face at 125F and the hot face of 1500F. Our initial temperature will be 60F (288K). In the equations notation:

Solving first for a "panel" composed of a two-inch air gap:


Equation 3

Yields a heating time of 25.3 seconds.

Next, we will calculate for 2 inches of ceramic fiberboard insulation, with all of the values remaining constant except the Thermal Diffusivity. By observation, the Thermal Diffusivity calculated for the ceramic fiberboard is less than the Thermal Diffusivity for air, which should result in a longer time to achieve the average temperature of 707 degrees Kelvin, even though the fiber has a much greater Heat Transfer Coefficient. This is indeed the case:


Equation 4

Yields a heating time of 27.0 seconds, 6% longer.

Thermal Diffusivity and Density

As we increase the density of the insulation, the Thermal Diffusivity drops, due to a higher concentration of fiber strands than air as density increases. As you recall, the Thermal Diffusivity of the base material is lower than the Thermal Diffusivity of air, so more weight fraction of alumina-silicate fiber will yield a comparable drop in Thermal Diffusivity. In this case, RCF board at 24 PCF and 32 PCF will have Thermal Diffusivities of :

These in turn yield heating times, using the same values as the previous calculations, of:

Although the differential heat equation for specific physical parameters needs to be solved to make predictive results, the general solution we have used is enough to demonstrate the peculiarities of using a "Thermal Conductivity Only" approach to unsteady-state heating problems, and the value of increased density in thermal insulation. >

References

1 Bird, R., Stewart, W., and Lightfoot, E. (1960). Transport Phenomena (p 246). New York: John Wiley & Sons.

2 McCabe, W., Smith, J., and Harriot, P. (1993). Unit Operations of Chemical Engineering, Fifth Edition (p 301). New York: McGraw-Hill, Inc.

3 Holman, P.(2002) Heat Transfer, 9th Ed., McGraw-Hill (which in turn cites Brown and Marco, Introduction to Heat Transfer, 3rd Ed, McGraw-Hill, 1958 and Eckert & Drake, Heat and Mass Transfer, McGraw-Hill, 1959). Retrieved on September 24th, 2008 from: http://en.wikipedia.org/wiki/Thermal_diffusivity