Stretch times

Ultrafast proton transfer. The diffusion-controlled limit for second-order rate constants (Section A3) is 1010 M 1 s 1. In 1956, Eigen, who had developed new methods for studying very fast reactions, discovered that protons and hydroxide ions react much more rapidly when present in a lattice of ice than when in solution.138 He observed second-order rate constants of 1013 to 1014 M 1 s These represent rates almost as great as those of molecular vibration. For example, the frequency of vibration of the OH bond in water is about 1014 s . The latter can be deduced directly from the frequency of infrared light absorbed in exciting this vibration Frequency v equals wave number (3710 cm-1 for -OH stretching) times c, the velocity of light (3 x 1010 cm s ). 

The rate at which x2 approaches x2 can be very large, since dx2/dt = (l/e)g, and e —> 0. Singular perturbation theory relies on defining a stretched time variable r = t/e, with r = 0 at t = 0, to analyze such fast transient phenomena. The term stretched refers to the behavior of the new time variable r, which tends to 00 even for t only slightly larger than 0. Note that, while x2 and r vary very rapidly, xi stays near its initial value x°. 

In order to capture the fast component of the dynamics, we define the stretched time scale r = t/e and consider the limit s — 0 (i.e., an infinitely high heat-transfer coefficient between Bi and B2). We thus obtain a description of the fast dynamics as... 

This chapter has reviewed existing results in addressing the analysis and control of multiple-time-scale systems, modeled by singularly perturbed systems of ODEs. Several important concepts were introduced, amongst which the classification of perturbations to ODE systems into regular and singular, with the latter subdivided into standard and nonstandard forms. In each case, we discussed the derivation of reduced-order representations for the fast dynamics (in a newly defined stretched time scale, or boundary layer) and the corresponding equilibrium manifold, and for the slow dynamics. Illustrative examples were provided in each case. 

We define the fast, stretched time scale r = t/e. On rewriting Equation (3.10) in this time scale and considering the limit case e —> 0 (which physically corresponds to an infinitely large recycle number or, equivalently, an infinitely high recycle flow rate), we obtain a description of the fast dynamics of the process ... 

Let us define the new time variable r = 1/e i, which is of the order of magnitude of the residence time in an individual process unit. In this fast ( stretched ) time scale, the model of Equation (5.10) becomes... 

We proceed with the derivation of approximate models for the process dynamics in each time scale, beginning with the fastest. To this end, we define the fast, stretched time scale r1=t/e1, in which the process model takes the form of Equation (5.11), and, in the limit i -> 0, corresponding to an infinitely large... 

As we anticipated at the beginning of this section, owing to the presence of the small parameter , the model in Equation (7.18) is still stiff. We will follow the developments in Section 3.5 to investigate its dynamics. To this end, let us define the intermediate stretched time scale iq, and consider the limit of an infinite recycle flow rate, or, equivalently, 1 —> 0 ... 

The descriptions of the fast subsystems are obtained hierarchically, starting from the fastest fast time scale. On introducing a stretched time variable tm = t/eM, the system in Equation (B.l) takes the form... 

Continuing this line of reasoning, the introduction of the fcth stretched time scale (Vfc S [1, M]), t/, = i/e, results in a description of system (B.l) of the form... 

In another development, PVC bottles of thin-wall constmction were produced by biaxial stretching of preform by rapid blow molding to the mold s dimensions. The stretch rates were between 100 and 1000%. The best result was obtained when stretching was 225% and stretch time 3 seconds (Hafiier, W. Huf-nagel, W, US Patent 3,980,192, Sept. 14, 1976.)... 

Figure 7.1.15. The mass uptake of TCE by PEEK vs. Figure 7.1.16. The frequency of carbonyl stretching time. [Adapted, by permission, from B H Stuart, mode of PEEK vs. temperature. [Adapted, by...

When the load is applied spring 2 extends by 0/M2 and remains stretched. Time dependency is due entirely to the Kelvin unit, in which the total strain = viscous strain, v = elastic strain e-... 

Using Eq. (32a), we get Brown 0=1.00 and iV=100, in good agreement with Fig.5. Another time of interest is the "stretching time", which gives the rate of stretching of the initial tube. By definition... 

Equations 11.23 through 11.26 are the counterparts to Eqs. 11.14 through 11.17 of the MED theory. In Eq. 11.23, the CCR term is just Ir S, similar to the CCR term K S - XlA) in the MED theory, but without the transient chain retraction rate A/ /I. (In Eq. 11.23, an absolute value must be taken of the CCR term at S to keep its value positive, while in the MED theory, this term is kept positive through the stretch equation 11.16.) The expression Eq. 11.23 for the orientational relaxation time contains not only the reptation time and the rate of convective constraint release k S, but also the stretch time t. This guarantees that even for velocity gradients greater than 1 /Tj, the rate of orientational relaxation remains bounded by 1 /Tj. This effectively switches off the CCR effect for fast flows, and so functions in much the same way as the switch function/(A) in the MED theory. Hence, no explicit switch function is present in Eq. 11.23. 

Recently, McLeish and Larson [95] developed a nonlinear viscoelastic theory for an idealized branched polymer with multiple branches but only two branch points. This molecular structure, called the pom-pom (described in Section 10.9.2), is a generalization of the H polymer in that each of the two branch points of the pom-pom is permitted to have an arbitrary number of branches, q see Fig. 9.4. The pom-pom model contains three basic time constants the backbone reptation time T, the backbone stretch time T, and the arm relaxation time x,. These time constants are given in terms of the molecular parameters of the pom-pom molecule as ... 

---