Energy spectrum integral

The time-of-flight spectrum of the H-atom product from the H20 photodissociation at 157 nm was measured using the HRTOF technique described above. The experimental TOF spectrum is then converted into the total product translational distribution of the photodissociation products. Figure 5 shows the total product translational energy spectrum of H20 photodissociation at 157.6 nm in the molecular beam condition (with rotational temperature 10 K or less). Five vibrational features have been observed in each of this spectrum, which can be easily assigned to the vibrationally excited OH (v = 0 to 4) products from the photodissociation of H20 at 157.6 nm. In the experiment under the molecular beam condition, rotational structures with larger N quantum numbers are partially resolved. By integrating the whole area of each vibrational manifold, the OH vibrational state distribution from the H2O sample at 10 K can be obtained. In... 

In order to see the effect of the rotational excitation of the parent H2O molecules on the OH vibrational state distribution, the experimental TOF spectrum of the H atom from photodissociation of a room temperature vapor H2O sample has also been measured with longer flight distance y 78 cm). By integrating each individual peak in the translational energy spectrum, the OH product vibrational distribution from H2O photodissociation at room temperature can be obtained. 

In the normalized energy spectrum, k— 1 corresponds to the inverse of the local integral length scale and k — Re 4 to the inverse of the local Kolmogorov length scale. The range of wavenumbers in Fig. 1 over which the slope is —5/3 is... 

To determine how the scalar time scale defined in Eq. (15) is related to the turbulence integral time scale given in Table I, we can introduce a normalized model scalar energy spectrum (Fox, 2003) as follows ... 

For a continuous energy spectrum the sums may be replaced by the integrals... 

As before, the summation over n is replaced by integration over the energy spectrum and introduction of the density factor p(E). Thus, finally... 

However, because the energy spectrum is discrete, both before and after chemisorption, AEa does not take the form of the energy-integral (4.99), but is instead evaluated via finite sums (4.86) of the energies, i.e.,... 

The turbulent energy spectrum is defined in terms of the velocity spectrum tensor by integrating out all directional information ... 

By definition, the turbulent kinetic energy k can be found directly from the turbulent energy spectrum by integrating over wavenumber space ... 

For isotropic turbulence, the longitudinal integral length scale Ln is related to the turbulent energy spectrum by... 

Because the integral scale is defined in terms of the energy spectrum, an appropriate starting point would be the scalar spectral transport equation given in Section 3.2. 

Thus, E k, t) Ak represents the amount of scalar variance located at wavenumber k. For isotropic turbulence, the scalar integral length scale is related to the scalar energy spectrum by... 

Figure 3.12. Model scalar energy spectra at Rk = 500 normalized by the integral scales. The velocity energy spectrum is shown as a dotted line for comparison. The Schmidt numbers range from Sc = 10 4 to Sc = 104 in powers of 102.

In Fig. 3.14, the mechanical-to-scalar time-scale ratio computed from the model scalar energy spectrum is plotted as a function of the Schmidt number at various Reynolds numbers. Consistent with (3.15), p. 61, for 1 Sc the mechanical-to-scalar time-scale ratio decreases with increasing Schmidt number as ln(Sc). Likewise, the scalar integral scale can be computed from the model spectrum. The ratio L Lu is plotted in Fig. 3.15, where it can be seen that it approaches unity at high Reynolds numbers. 

In many reacting flows, the reactants are introduced into the reactor with an integral scale L that is significantly different from the turbulence integral scale Lu. For example, in a CSTR, Lu is determined primarily by the actions of the impeller. However, is fixed by the feed tube diameter and feed flow rate. Thus, near the feed point the scalar energy spectrum will not be in equilibrium with the velocity spectrum. A relaxation period of duration on the order of xu is required before equilibrium is attained. In a reacting flow, because the relaxation period is relatively long, most of the fast chemical reactions can occur before the equilibrium model, (4.93), is applicable. 

This equation shows that only the closed classical trajectories (x(f) = x(0) and x(t) = x(0)) should be taken into account, and the energy spectrum is determined by these periodic orbits [Balian and Bloch, 1974 Gutzwiller, 1967 Miller, 1975b Rajaraman, 1975]. Finally, the stationary-phase integration over time yields the identity... 

Figure 3 Linearized electronic energy spectrum around the Fermi surface (k = k,r). The degrees of freedom in the energy shells of width dEJ2 at the extrema of the spectrum are integrated over in each step of the renormalization group.

The reaction rate constant is determined with statistic averaging of transfer rate w( ) across the initial-state energy spectrum. When this spectrum is continuous, the averaging comes down to integration [6, 7] ... 

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