Times characteristic

As is evident from the fomi of the square gradient temi in the free energy fiinctional, equation (A3.3.52). k is like the square of the effective range of interaction. Thus, the dimensionless crossover time depends only weakly on the range of interaction as In (k). For polymer chains of length A, k A. Thus for practical purposes, the dimensionless crossover time is not very different for polymeric systems as compared to the small molecule case. On the other hand, the scaling of to is tln-ough a characteristic time which itself increases linearly with k, and one has... 

The characteristic time of the tliree-pulse echo decay as a fimction of the waiting time T is much longer than the phase memory time T- (which governs the decay of a two-pulse echo as a function of x), since tlie phase infomiation is stored along the z-axis where it can only decay via spin-lattice relaxation processes or via spin diffusion. 

Since the distance from the source to the detector is fixed, the time taken for an ion to traverse the analyzer in a straight line is proportional to its velocity and hence its mass (strictly, proportional to the square root of its m/z value). Thus each m/z value has its characteristic time of flight from the source to the detector. 

The superpositioning of experimental and theoretical curves to evaluate a characteristic time is reminiscent of the time-tefnperature superpositioning described in Sec. 4.10. This parallel is even more apparent if the theoretical curve is drawn on a logarithmic scale, in which case the distance by which the curve has to be shifted measures log r. Note that the limiting values of the ordinate in Fig. 6.6 correspond to the limits described in Eqs. (6.46) and (6.47). Because this method effectively averages over both the buildup and the decay phases of radical concentration, it affords an experimentally less demanding method for the determination of r than alternative methods which utilize either the buildup or the decay portions of the non-stationary-state free-radical concentration. 

If the dominant mode of heat transfer to the soHds is convection between the wall and the soHds, then the characteristic time for a dry system is... 

In a radiation dominated kiln environment, with hot combustion gases and reradiating walls, the characteristic time is... 

If the rate of moisture vaporization is controlled bv the rate of heat transfer to the wet soHd, then for convection dominated heat transfer, h, at the boiling poiat of water, the characteristic time is... 

The characteristic times for waste destmction to an efficiency of 99.99%, for water in a nitrogen atmosphere, where the residue is a typical hydrocarbon breaking into two large fragments, eg, / -butane decomposing into two ethyl radicals would be droplet heatup, 0.073 s droplet evaporation,... 

Whether a viscoelastic material behaves as a viscous Hquid or an elastic soHd depends on the relation between the time scale of the experiment and the time required for the system to respond to stress or deformation. Although the concept of a single relaxation time is generally inappHcable to real materials, a mean characteristic time can be defined as the time required for a stress to decay to 1/ of its elastic response to a step change in strain. The... 

The main stages of coal combustion have different characteristic times in fluidized beds than in pulverized coal combustion. Approximate times are a few seconds for coal devolatilization, a few minutes for char burnout, several minutes for the calcination of limestone, and a few hours for the reaction of the calcined limestone with SO2. Hence, the carbon content of the bed is very low (up to 1% by weight) and the bed is 90% CaO in various stages of reaction to CaSO. About 10% of the bed s weight is made up of coal ash (91). This distribution of 90/10 limestone/coal ash is not a fixed ratio and is dependent on the ash content of the coal and its sulfur content. 

A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For veiy large Deborah numbers, the behavior closely resembles that of an elastic solid. 

The characteristic time for adjustment of p to the current applied stress t is... 

Solving Newton s equation of motion requires a numerical procedure for integrating the differential equation. A standard method for solving ordinary differential equations, such as Newton s equation of motion, is the finite-difference approach. In this approach, the molecular coordinates and velocities at a time it + Ait are obtained (to a sufficient degree of accuracy) from the molecular coordinates and velocities at an earlier time t. The equations are solved on a step-by-step basis. The choice of time interval Ait depends on the properties of the molecular system simulated, and Ait must be significantly smaller than the characteristic time of the motion studied (Section V.B). 

When the characteristic time of vibrational relaxation is much shorter than tr, the rate constant is independent of Zy. For molecules consisting of not too many atoms, the inequality (2.58) is not fulfilled. Moreover, Zy may even become larger than tr. This situation is beyond our present consideration. The total set of resonant sublevels partaking in RLT consists of a small number of active acceptor modes with nonzero matrix elements (2.56) and many inactive modes with Vif = 0. The latter play the role of reservoir and insure the resonance = f. 

The classical bath sees the quantum particle potential as averaged over the characteristic time, which - if we recall that in conventional units it equals hjk T- vanishes in the classical limit h- Q. The quasienergy partition function for the classical bath now simply turns into an ordinary integral in configuration space. 

Suppose a small characteristic time interval exists such that y i changes without strongly affecting - i(y 2b 0 so that the latter may be expanded in a Taylor series as... 

The main predictions of the scaling theory [40], concerning the dynamics behavior of polymer chains in tubes, deal with a number of characteristic times the smallest time rtube measures the interval of essentially Rouse relaxation before the monomers feel the tube constraints significantly, 1 < Wt < Wrtube = and diffusion of an inner monomer is... 

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