Crossover temperature

In the limit of reasonably high temperatures (above the so-called crossover temperature), i.e. < 2k, the above fomnila in A3.8.21 is best simplified fiirther and approximately written as... 

Makarov D E and Topaler M 1995 Quantum transition-state theory below the crossover temperature Phys. Rev. E 52 178... 

Efficiency. Since only 35 to 50% of fired duty is absorbed in the radiant section, the flue gas leaving the radiant chamber contains considerable energy that can be extracted efficiently in the convection section of the furnace. In the convection section, the feed is preheated along with dilution steam to the desired crossover temperature. Residual heat is recovered by generating steam. The overall thermal efficiency of modem furnaces exceeds 93%, and a value of 95% is not uncommon. 

Kramers formula for classical escape out of a metastable well in the case of moderate and strong damping [Kramers 1940]. In accord with the multidimensional theory predictions, the crossover temperature should be equal to... 

On the other hand, it is clear that in the classical regime, T> (T i is the crossover temperature for stepwise transfer), the transition should be step-wise and occur through one of the saddle points. Therefore, there should exist another characteristic temperature. r 2> above which there exist two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points atlT = T i. The existence of the second crossover temperature, 7, 2, for two-proton transfer has been noted by Dakhnovskii and Semenov [1989]. 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

The temperature dependences of k, calculated by Hancock et al. [1989], are given in fig. 48. The crossover temperature equals 25-30 K. The weak increase of k T) with decreasing temperature below is an artefact caused by extending the gas-phase theory prefactor to low temperatures without taking into account the zero-point vibrations of the H atom in the crystal. For the same reason the values of the constants differ by 1-2 orders of magnitude from the experimental ones. 

Management decisions might not be very definitive. For example, how does one maximize air-cooling Normally, a study will reveal the optimum crossover temperature between air- and water-cooling (often, air-cooling will be shown to be economical above 140°F). What management really wants in this case is air-cooling wherever it can be used economically. 

These results would be satisfactory, but for the arbitrary nature of the adjustment to the coil 2 cracking temperature. The gross error could also be eliminated by adjusting either the flow rates through the coil 2 and 3 or the inlet temperature to the radiant zone (crossover temperature) for coil 2. As it happens, the best reduction in... 

An indication of errors in the measurement of the coil flow rate can be found in the relative value of the crossover temperature for that coil. The absolute value of the crossover temperature is of less value because it depends on the firing rate for the furnace, as well as the coil flow rate. With this in mind, an analysis of the daily averages of the furnace data was undertaken. 

FIGURE 5 Crossover temperature data clusters—Side A (from Islam et al., 1994). 

The answer to our question at the beginning of this summary therefore has to be as follows. When you want to locate the glass transition of a polymer melt, find the temperature at which a change in dynamics occurs. You will be able to observe a developing time-scale separation between short-time, vibrational dynamics and structural relaxation in the vicinity of this temperature. Below this crossover temperature, one will find that the temperature dependence of relaxation times assumes an Arrhenius law. Whether MCT is the final answer to describe this process in complex liquids like polymers may be a point of debate, but this crossover temperature is the temperature at which the glass transition occurs. 

Horita and Wesolowski (1994) have summarized experimental results for the hydrogen isotope fractionation between liquid water and water vapor in the temperature range 0-350°C (see Eig. 2.3). Hydrogen isotope fractionations decrease rapidly with increasing temperatures and become zero at 220-230°C. Above the crossover temperature, water vapor is more enriched in deuterium than liquid water. Fractionations again approach zero at the critical temperature of water (Fig. 2.3). 

We also know that the one imaginary frequency associated with the transition state has the form rvfm. These quantities can be combined to form a crossover temperature, Tc, defined by... 

C. Size of Cooperatively Rearranging Regions and the Average Degree of Polymerization Evidence for Universality at the Crossover Temperature Ti... 

The two different temperature regimes of occurring below 7a are separated from each other by a precisely determined crossover temperature T) (denoted by crosses in Fig. 6), which is defined by an inflection point in the product J c(T ) T as a function of T. As discussed later, this crossover or inflection temperature T) appears to conform to the phenomenology of the experimentally estimated mode coupling temperature T p (denoted also as T] ). 

In summary, Fig. 6 exhibits the four characteristic temperatures of glass formation the Arrhenius temperature 7a, the crossover temperature Tj, the ideal glass transition temperature Tb, and a kinetic glass-formation temperature Tg (dehned in Section VI), to illustrate their relative locations with respect to the temperature variation of s. ... 

Our estimates of typical values for Ap for both F-F and F-S high molar mass polymers (Ap/ B 2000K and 2600 K, respectively) are comparable in magnitude with Ap obtained for high molar mass alkanes by Tabor [84] (Ap/ B 2700 K). The interrelation between Ap and has implications regarding the magnitude of the structural relaxation time r at the crossover temperature Recent investigations [102, 137] indicate that r at the... 

Figure 11. Same as in Fig. 10, but the configurational entropy ScT is normalized by the product of the critical entropy si and the activation energy Ap (estimated from Eq. (42) and the computed crossover temperature 7i). According to Eq. (41), the slope of the resulting curves defines the fragility parameter K. (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas,... 

Figure 22. The configurational entropy Sc per lattice site as calculated from the LCT for a constant pressure, high molar mass (M = 40001) F-S polymer melt as a function of the reduced temperature ST = (T — To)/Tq, defined relative to the ideal glass transition temperature To at which Sc extrapolates to zero. The specific entropy is normalized by its maximum value i = Sc T = Ta), as in Fig. 6. Solid and dashed curves refer to pressures of F = 1 atm (0.101325 MPa) and P = 240 atm (24.3 MPa), respectively. The characteristic temperatures of glass formation, the ideal glass transition temperature To, the glass transition temperature Tg, the crossover temperature Tj, and the Arrhenius temperature Ta are indicated in the figure. The inset presents the LCT estimates for the size z = 1/of the CRR in the same system as a function of the reduced temperature 5Ta = T — TaI/Ta. Solid and dashed curves in the inset correspond to pressures of P = 1 atm (0.101325 MPa) and F = 240 atm (24.3 MPa), respectively. (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas, Journal of Physical Chemistry B 109, 21350 (2005). Copyright 2005, American Chemical Society.)...

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