Thermal length

Consider a transversally lumped and axially integral system (Step 1) as shown in Fig. 2.30. For this system, the first law of thermodynamics (Step 2) gives 

Note that Eq. (2.123) includes the effect of boundaries (Step 5). That is, a governing equation resulting from an integral formulation always combines Steps 4 and 5. In the 

Clearly, the origin for x selected in Fig. 2.30 leads to the simplest possible form of this parabola. Inserting Eq. (2.124) into Eq. (2.123) results in 

This distance denotes approximately the longitudinal penetration depth (boundary layer) of heat by conduction through the fin. Also, on dimensional grounds, replacing l with 8 in Eq. (2.113), we have the same distance within a numerical constant, 

For a thickness l and width L, A — It and P —2L +21 = 2L, hence m = s/hP/kA = J2k/kl and 8 = Jikl/h. Assume the engine fins of a motorbike have l 0,2 cm, k = 180 W/m-K (an aluminum alloy), and h = 300 W/m2-K (forced convection to air). These give 5 S 6 cm. However, the fin is usually cut somewhat shorter than 5 consequently some material and weight are saved without appreciably affecting the heat transfer from the fin. (In Fig. 2.30 note the slope of the parabola, which is a measure of the heat loss near the origin.) 

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In hydrocarbon liquids other than n-hexane, the procedure for obtaining the thermalization distance distribution could conceivably be the same. However, in practice, a detailed theoretical analysis is rarely done. Instead, the free-ion yield extrapolated to zero external field (see Chapter 9) is fitted to a one-parameter distribution function weighted with the Onsager escape probability, and the mean thermalization length (r ) is extracted therefrom (see Mozumder, 1974 ... 

While the variation of the mean thermalization length among different liquid hydrocarbons under high-eneigy irradiation has been well documented (see, e.g., Schmidt and Allen, 1968,1970), the question of the dependence of thermalization... 

Effective thermalization length, the b value for origin-centered gaussian distribution (see text) only when G[( < 0.2 is a truncated power law distribution used by Freeman and his associates. For the free-ion yield in kcal/mole. 

Braun and Scott (1987) used two-photon ionization of benzene and azulene in n-hexane and followed the e-ion recombination process by monitoring the transient absorption of the electron. The results are not very different from those obtained by the IR stimulation technique. A mean thermalization length of 5.0 nm was inferred at 223 K using a two-photon excitation at 266 nm. Hong and Noolandi s theory was used for the analysis. The absorption technique was... 

In photoionization experiments, there is evidence that the thermalization length increases with the photon energy. 

The mean thermalization length in high-energy irradiation tends to be somewhat longer than in the photoionization case, possibly due to higher initial energy. 

Jay-Gerin et al. (1993) also found an empirical correlation between G. and the most probable thermalization length b as follows ... 

FIGURE 9.3 A low-LET track in a liquefied rare gas (LRG). Even at a minimum LET, the electron thermalization length ( 103 nm) greatly exceeds the inter-positive-ion separation of R2+ ( 102 nm). Thus, the geometry approximates cylindrical symmetry rather than a collection of isolated ionizations. Reproduced from Mozumder (1995a), with the permission of Elsevier . 

Values of rh in LAr, LKr, and LXe at their respective temperatures are 1568, 3600, and 4600 nm. Wide variation of thermalization length around the rms value is of course expected. However, the escape probability has been found to be insensitive to rg in these cases around their rms values. Therefore, averaging Pesc over the distribution of thermalization length has been deemed unnecessary when r() is taken equal to rth. 

The last exponential factor takes into account a possibility of the free-positron annihilation occuring during the Ps formation time, ips (on the order of some picoseconds) with the annihilation rate 1/t2 (< 2 ns-1). Obviously, the contribution from this factor is negligible. In nonpolar molecular media at room temperature rc is 300 A. Typical thermalization lengths b of electrons are < 100 A. Thus, the Onsager factor, 1 — err, is also very close to unity. Therefore, to explain observable values of Ps yields, which never exceed 0.7, we must conclude according to Eq. (11) that the terminal positron spur contains on average 2 to 3 ion-electron pairs. 

Geminate recombination is suppressed when the density of excited electron-hole pairs is large. For example, a pair density of 10 cm results in an average separation of the carriers of 50 A. The geminate pairs overlap when this distance is less than the thermalization length, Z.T, and non-geminate recombination between carriers from different pairs occurs. Geminate recombination is therefore most likely at low temperatures and weak excitation intensities and in a-Si H it is only observed under these conditions. 

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