Thermal thickness

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

For a Prandtl number, Pr. less than unity, the ratio of the temperature to the velocity boundary layer thickness is equal to Pr 1Work out the thermal thickness in terms of the thickness of the velocity boundary layer... 

Ignition. For thermally thick materials, time to ignition is found to follow the following relationship as external heat flux is varied ( 4,6) ... 

The thermally thin case holds for d of about 1 mm. Let us examine when we might approximate the ignition of a solid by a semi-infinite medium. In other words, the backface boundary condition has a negligible effect on the solution. This case is termed thermally thick. To obtain an estimate of values of d that hold for this case we would want the ignition to occur before the thermal penetration depth, <5T reaches x d. Let us estimate this by... 

Hence, we might expect solids to behave as thermally thick during ignition and to be about 8 cm for t-lg = 300 s and about 2.5 cm for 30 s. Therefore, a semi-infinite solution might have practical utility, and reduce the need for more tedious finite-thickness solutions. However, where thickness and other geometric effects are important, such solutions must be addressed for more accuracy. 

Let us consider the semi-infinite (thermally thick) conduction problem for a constant temperature at the surface. The governing partial differential equation comes from the conservation of energy, and is described in standard heat transfer texts (e.g. Reference [13]) ... 

We will develop an analytical formulation of the statement in Equation (8.2). This will be done for surface flame spread on solids, but it can be used more generally [1], As with the ignition of solids, it will be useful to consider the limiting cases of thermally thin and thermally thick solids. In practice, these solutions will be adequate for first-order approximations. However, the model will not consider any effects due to... 

It should be pointed out that Equation (8.6), and its counterpart for thermally thick materials, will hold only for Ts > 7 smm, a minimum surface temperature for spread. Even if we include the heat loss term in Equation (8.4) by a mean-value approximation for the integrand,... 

Surface Flame Spread for a Thermally Thick Solid... 

Let us now turn to the case of a thermally thick solid. Of course thickness effects can be important, but only after the thermal penetration depth due to the flame heating reaches the back face, i.e. t = ff, 6j(t ) = d. As in the ignition case, if tf is relatively small, say 10 to even 100 s, the thermally thick approximation could even apply to solids of d < 1 cm. Again, we represent all of the processes by a thermal approximation involving the effective properties of Tlg, k, p and c. Materials are considered homogeneous and any measurements of their properties should be done under consistent conditions of their use. Other assumptions for this derivation are listed below ... 

Consider the control volume in Figure 8.9 where the thermally thick case is drawn for a wind-aided mode, but results will apply in general for the opposed case. The control... 

Figure 8.9 Control volume energy conservation for a thermally thick solid with flame spread steady velocity, Up...

SURFACE FLAME SPREAD FOR A THERMALLY THICK SOLID 201... 

We might just as well use the exact thermally thick high heat flux (short time) result, where 0.81 is replaced by 7t/4 = 0.785. We can use... 

Conservation of energy in the virgin solid for steady, thermally thick conditions... 

This shows that this modified heat of gasification includes all effects that augment or reduce the mass loss rate. Recall that the term in the [ ] becomes zero if the solid is thermally thick and the virgin solid equilibrates to the steady state. Equating Equations (9.107) and (9.108) gives an equation for the flame temperature ... 

Since the thermally thick case will predominate under most fire and construction conditions, the conductance can be estimated from Equation (11.10). Values for the materials characteristic of Table 11.1 are given in Table 11.2. As time progresses, the conduction heat loss decreases. 

The solution to this boundary value problem was approximated by Rosencwaig and Gersho for six different cases (4) one of which, a thermally thick but optically thin sample, often applies to layers adsorbed on heterogeneous catalysts. The photoacoustic signal arises from the chemisorbed species and the support. 

The typical polymer or rubber sample would be classified as optically transparent or opaque and thermally thick except possibly for the strongest bands. In this case the signal intensity would be proportional to the product of the optical absorption coefficient (P) and the thermal diffusion length and show a - 3/2 dependence on the modulation frequency (to). The angular modulation frequency is a product of the interferometer mirror velocity and the wavenumber ... 

As observed earlier, the material thickness plays a significant role in the rate of flame spread, either through the mode of heat transfer or through the losses that can occur through the solid. Materials are generally defined as thermally thin, where Sp = L or thermally thick, where 5p [Pg.60]

For thermally thick materials, 8y can be extracted from Equation 3.24 ... 

Thus, the following expressions for the propagation velocity can be obtained for a thermally thick solid ... 

Because it is very tedious to measure Tig and kpc directly, it is much more common to determine ignition properties on the basis of an analysis of time-to-ignition data obtained over a range of heat fluxes. The analysis is usually based on a simple heat conduction model, which assumes that the solid is inert (negligible pyrolysis prior to ignition) and thermally thick (heat wave does not reach the back surface prior to ignition). An example of this type of analysis is discussed in Section 14.3.2.3.2. 

Thermal thick charring with additional peak at the end of burning... 

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