Thermal penetration depth

The thermal penetration depth of the cooled liquid can be approximated as... 

Finally, we estimate the order of magnitude of the time of the heated solid to achieve Tpy at the surface. This is primarily a problem in heat conduction provided the decomposition and gasification of the solid (or condensed phase) is negligible. We know that typically low fuel concentrations are required for piloted ignition (XL 0.01-0.10) and by low mass flux (mv 1-5 g/m2 s) accordingly. Thus, a pure conduction approximation is satisfactory. A thermal penetration depth for heat conduction can be estimated as... 

First, we will consider thin objects - more specifically, those that can be approximated as having no spatial, internal temperature gradients. This class of problem is called thermally thin. Its domain can be estimated from Equations (7.11) to (7.12), in which we say the physical thickness, d, must be less than the thermal penetration depth. This is illustrated in Figure 7.7. For the temperature gradient to be small over region d, we require... 

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... 

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 ... 

Next we integrate Eq. E5.3-3 over from the outer surface (x = 0) to a certain, yet unknown depth 8(t), which is defined as the thermal penetration depth... 

Boundary conditions T(x, 0) = T(oo,t)= Tq are both taken care of by assuming a time-dependent thermal penetration depth of finite thickness. 

The second microscale heat transfer issue considered in this paper deals with short time scales and their influence on the dimensions required for good heat transfer. Many cryocoolers use oscillating flows and pressures with frequencies as high as about 70 Hz. Heat flow at such high frequencies can penetrate a medium only short distances, known as the thermal penetration depth temperature amplitude of a thermal wave decays as it travels within a medium. The distance at which the amplitude is 1/e of that at the surface is the thermal penetration depth, which is given by... 

Figure I. Schematic showing the decay of temperature amplimde inside a solid and the deftnition of thermal penetration depth.

Figure 2. Thermal penetration depths at 10 Hz in helium and several pure metals.

The absorbed energy of the pulse leads to a heating of the material. Based on the thermal conductivity k of the material and the duration x of the pulse, the thermal penetration depth can be... 

Based on the definitions of energy and thermal penetration depth, in material processing, pulses can be classified as short and ultrashort. Pulses for which > 8 are considered to be short. If d < S they are classified as ultrashort (Dirscherl 2008). This definition depends strongly on the material processed, as both thermal conductivity and the energy deposition characteristics... 

The ignition resistance of polymers has been investigated in detail both experimentally and theoretically [14, 15, 21, 22, 34, 35], It is well recognized that the ignition behavior of a polymer is governed by its actual thickness (d) relative to the thermal penetration depth (5) [14, 15, 34, 35], The thermal penetration depth... 

The actual experimental system uses abridge circuit and a lock-in amplifier, and the AT values are measured as a function of frequency. The theoretical value of AT is given by the solution of the diffusion equation for a radial heat flow from the surface electrode. At low frequencies in which the thermal penetration depth is much larger than the PS layer thickness, AT is the summation of two components one from the PS layer, ATps, and the other from the c-Si substrate, ATs- The ATps value depends on the thermal conductivity of the PS layer and the experimental parameters. Since the ATs value is obtained from the known thermal parameters of c-Si, a value of the PS layer can be determined from the analysis of the measured AT At high frequencies, on the other hand, the thermal penetration depth is smaller than the PS layer thickness, and then the contribution of the substrate to AT is negligible. Under this situation, the experimental AT data simply relate to the D value that is given as a/C. So the C value of the PS layer can be deduced from a measured at low frequencies. Owing to the insensitivity to errors from black-body radiation, this method makes it possible to determine the thermal constants more precisely rather than the method based on simple thermal flow measurements. 

The above quantity is called as thermal penetration depth or the wavelength of the diffusive thermal wave. Either the real or the imaginary part of the temperature oscillations AT can be used to determine the thermal conductivity. The imaginary part (out-of-phase oscillations) gives the thermal conductivity directly. The slope of the real part (in-phase oscillations) versus In also gives the thermal conductivity. 

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