Cooling rate estimation

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. 

Taylor L.A., Onorato P.I., and Uhlmann D.R. (1977) Cooling rate estimations based on kinetic modeling of Fe-Mg diffusion in olivine. Proc. Lunar Sd. Conf. 8th, 1581-1592. 

Thermal metamorphism in chondrites and melting in differentiated asteroids are driven by heat produced by the decay of short-lived radionuclides (especially 26A1). Thermal models can reproduce the peak temperatures and cooling rates estimated for meteorites, as well as... 

The essence of cryomicroscopy lies in the ability to vitrify the sample using the thermal fixation technique that alters the sample microstructure to the least extent. Cooling rate estimates of 10 K/s [14], 10 K/s [15], and 10 K/s [16] are quoted as necessary to vitrify water or dilute aqueous suspensions. The ability to achieve the desired cooling rate depends on the sample, the cryogen, and the fixation technique. 10 K/s is probably the most accurate estimate. This evaluation is based on estimates of the cooling rate in freezing techniques that are known to produce vitreous specimens. 

Eig. 5. The Widmanstatten pattern ia this poHshed and etched section of the Gibbeon iron meteorite is composed of iatergrown crystals of kamacite and taenite, NiFe phases that differ ia crystal stmcture and Ni content. Ni concentration gradients at crystal boundaries ia this 3-cm-wide sample can be used to estimate the initial cooling rates and corresponding size of the asteroid from which the meteorite was derived. 

Using the fluxing technique, Lau and Kui [33] determined that the critical cooling rate for forming a 7-mm diameter bulk amorphous Pd4QNi4()P2o cylinder was 0.75 K/sec. From this value, they estimated that the steady-state nucleation frequency was on the order of lO" m s. On the other hand, Drehman and Greer [34] estimated that the steady state nucleation frequency at 590 K is 10 m" s, which is also the maximum... 

Were all of these newly discovered substances also new elements This question would not be answered for some years but there was a flurry of other major discoveries to keep the protagonists occupied. Pierre Curie discovered that radioactivity released large quantities of heat (Curie and Laborde 1903) which appeared mysterious—as if the heat was coming from nowhere. This discovery provided an extra heat source for the Earth and reconciled the estimates of a very old Earth, based on geological estimates, with the young age calculated by Lord Kelvin from cooling rates. The year 1903 also witnessed the first demonstration that a-decay released He (Ramsay and Soddy 1903). The build up of He was soon put to use to date geological materials, initially by Rutherford in 1905 who calculated the first ever radiometric age of 500 Myr for a pitchblende sample, and then by Strutt who examined a wide variety of minerals (Strutt... 

The Laser-spin-atomized droplets are usually spherical, clean, and homogeneous in composition. A mass median diameter of 100 pm has been obtained for a Ni-Al-Mo alloy. Cooling rates are estimated to be in the order of magnitude of 105 °C/s. Similarly to other centrifugal atomization techniques, droplet properties (shape, size, cooling rate, etc.) are dependent on the rotation speed, ingot diameter, superheat, and material properties. 

The temperature of a liquid metal stream discharged from the delivery tube prior to primary breakup can be calculated by integrating the energy equation in time. The cooling rate can be estimated from a cylinder cooling relation for the liquid jet-ligament breakup mechanism (with free-fall atomizers), or from a laminar flat plate boundary layer relation for the liquid film-sheet breakup mechanism (with close-coupled atomizers). 

Table 11.5 Some estimates of closure temperature, related to dilfusion of daughter isotope. Large discrepancies and wide ranges can be ascribed to differences in cooling rates of system and grain size dimensions.

The reaction rate constant and the diffusivity may depend weakly on pressure (see previous section). Because the temperature dependence is much more pronounced and temperature and pressure often co-vary, the temperature effect usually overwhelms the pressure effect. Therefore, there are various cooling rate indicators, but few direct decompression rate indicators have been developed based on geochemical kinetics. Rutherford and Hill (1993) developed a method to estimate the decompression (ascent) rate based on the width of the break-dovm rim of amphibole phenocryst due to dehydration. Indirectly, decompres-... 

Figure 2-13 Schematic drawing of (a) density as a function of temperature, and (b) entropy as a function of temperature for glasses with different cooling rates and hence different glass transition temperature (Martens et al., 1987). The entropy of the undercooled liquid is estimated assuming constant heat capacity.

The inferred cooling rate (7.7 K/d) is within a factor of two of the experimental cooling rate (13.7 K/d). The difference of a factor 1.8 is due to (i) the inaccuracy of Equation 5-125, which is likely minor, (ii) uncertainty in the calculation of Tae from species concentrations (Equation 5-129a), and (iii) errors in the dependence of the kinetic coefficient on temperature (Equation 5-127). This difference of a factor of 1.8 is considered small, taking into consideration of the various uncertainties. (Usually, when cooling rate can be estimated to within a factor of 2, it is considered excellent agreement.)... 

Next we turn to the inference of cooling history. The length of the concentration profile in each phase is a rough indication of (jDdf) = (Dot), where Do is calculated using Tq estimated from the thermometry calculation. If can be estimated, then x, Xc and cooling rate q may be estimated. However, because the interface concentration varies with time (due to the dependence of the equilibrium constants between the two phases, and a, on temperature), the concentration profile in each phase is not a simple error function, and often may not have an analytical solution. Suppose the surface concentration is a linear function of time, the diffusion profile would be an integrated error function i erfc[x/(4/Ddf) ] (Appendix A3.2.3b). Then the mid-concentration distance would occur at... 

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