Asymptotic cooling

To integrate the above equation, the functional form of the dependence of T on t, T(t), must be assumed. Usually, two functions are used, one is referred to as asymptotic cooling, and the other as exponential cooling. The two functions are... 

In particular, consider a thermal history of monotonic cooling (such as an igneous rock, especially a volcanic rock) represented by the asymptotic cooling model (Equation 2-41) ... 

Although Xc characterizes the cooling history, sometimes one would like to know the cooling rate q rather than Xc. For asymptotic cooling, the cooling rate q is... 

Assuming asymptotic cooling T= To/(l + f/xc), then q = -dT/dt t=o = To/xc = To/[E/(RTo) 2.5 Myr] = 19 K/Myr. Because part of the profile is likely due to growth, the cooling rate estimated above is the lower limit. Actual cooling rate must be greater than this. 

To estimate the closure temperature, it is necessary to estimate the diffusion distance. From earlier results (Section 3.2.8.1), for asymptotic cooling from the closure temperature (Equation 3-55),... 

Figure 5-14 Diffusive loss of Ar that was initially in hornblende during cooling after complete cooling down t=rx>) for asymptotic cooling history with (a) a fixed cooling timescale but varying the initial temperature and (b) a fixed initial temperature but varying the cooling rate.

The growth rate for °Ar is X Po where Pq is the initial concentration of The solution to this problem follows Dodson (1973). Assume asymptotic cooling is... 

For the general case of one-dimensional diffusion of a radiogenic component in a slab of half-thickness a under asymptotic cooling, the diffusion equation is, hence. 

Box 5.1 Derivation of Equation 5-1 25 for the special case of first-order reversible reactions with f = 2 b, and an asymptotic cooling history with Too = 0 K. 

The above results may be applied to infer the critical cooling rate for the concentration of the core to be affected by diffusion. It is necessary to define precisely what is meant when we say "the center is affected by diffusion." If we use center concentration of Ci + 0.01 (Cmid - Ci) as the criterion for center concentration to be affected by diffusion, then it would occur at z = 2(fDdt) /a = 0.3947. For an asymptotic cooling history, this means that 2 Dox) la = 0.3947, or Doxla = 0.0389. Combining with Equation 5-137 that J = RTq/ xE), we obtain the critical q ... 

Sometimes, the profile is so short that it cannot be resolved by the measurement technique. Such information may also be applied to constrain cooling rate. For example, if the spatial resolution of the measurement is I, the absence of a profile (i.e., a step-function profile) means that jDdt[Pg.536]

Figure 10-41. For many cooling waters, the fouling resistance increases rapidly, then decreases, and finally approaches an asymptotic value. (Used by permission Knudsen, J. G., Chemical Engineering Progress. V. 87, No. 4, 1991. American Institute of Chemical Engineers. All rights reserved.)...

Fig. 4.8. Colour-magnitude (HR) diagram of the globular cluster Messier 68 with [Fe/H] —2. In order of successive evolutionary stages, MS (sd) indicates the main sequence occupied by cool subdwarfs, with the position of the Sun shown for comparison, SGB indicates the subgiant branch, RGB the red giant branch, HB the horizontal branch including a gap in the region occupied by RR Lyrae pulsating variables and AGB the asymptotic giant branch. Adapted from McClure etal. (1987).

Example 5.19. This example shows how diffusivity may be inferred from natural samples. Suppose an Mg-Fe interdiffusion profile is measured in a mineral, and it can be modeled as a diffusion couple with I=60 pm. The temperature history is asymptotic with Tq = HOOK, and the cooling time-scale Tc= 10 Myr. Estimate the diffusivity at HOOK. 

For a bubble to grow, vapor must pass from the superheated liquid into the bubble. Thus latent heat of vaporization is removed from the surrounding liquid, and the liquid cools. The drop in liquid temperature near the bubble means a decrease in the driving force between liquid and bubble. This temperature drop strongly affects the bubble rate of growth. The rate can be shown to approach asymptotically a condition whereby the radius increases according to the square root of time. 

At time t = 0, the ribbon is cooled. The pressure rapidly drops from Po to a much lower value and remains steady for perhaps 100 sec. (see Figure 14 regions B to C and C to D). Then the pressure rises, slowly at first, then more rapidly, and then more slowly (see regions D to E and E to F). After about 10 or 20 min. the pressure asymptotically rises to p . At any time t during this experiment the ribbon can be reheated suddenly or flashed to its high temperature. The pressure suddenly rises to a maximum value pmai, which is noted or recorded. The difference (pmai — p) is called Ap. The amount of adsorbed gas can be calculated from Ap. 

An exothermal reaction is to be performed in the semi-batch mode at 80 °C in a 16 m3 water cooled stainless steel reactor with heat transfer coefficient U = 300 Wm"2 K . The reaction is known to be a bimolecular reaction of second order and follows the scheme A + B —> P. The industrial process intends to initially charge 15 000 kg of A into the reactor, which is heated to 80 °C. Then 3000 kg of B are fed at constant rate during 2 hours. This represents a stoichiometric excess of 10%.The reaction was performed under these conditions in a reaction calorimeter. The maximum heat release rate of 30Wkg 1 was reached after 45 minutes, then the measured power depleted to reach asymptotically zero after 8 hours. The reaction is exothermal with an energy of 250 kj kg-1 of final reaction mass. The specific heat capacity is 1.7kJ kg 1 K 1. After 1.8 hours the conversion is 62% and 65% at end of the feed time. The thermal stability of the final reaction mass imposes a maximum allowed temperature of 125 °C The boiling point of the reaction mass (MTT) is 180 °C, its freezing point is 50 °C. 

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