Energy spectrum model

Use SHMO to obtain the energy spectrum for the models methylenepentadiene. bicyclohexatriene, and styrene. IDraw all three energy level diagrams.. Are there degeneracies for these molecules ... 

In conclusion note that for a sufficiently dense energy spectrum the caustic segments have been shown [Benderskii et al. 1992b] to disappear after statistical averaging, which brings one back to the instanton and, for the present model, leads to eqs. (2.80a, b). 

In the Keilson-Storer model of J-diffusion, non-adiabatic relaxation is assumed to extend to the whole energy spectrum of a rotator. Actually, for large J the relaxation becomes adiabatic. The considerable difference between the times appears in the adiabatic limit since xe = 00, while xj is defined by m-diffusion according to Eq. (1.12). As is seen from Eq. (1.5) and Eq. (1.6), both J- and m-diffusion are just approximations which hold for low- and high-excited rotational levels, respectively. In general 0 < xj/xE < 1 + y. 

Fig. 10 Electrochemical energy level model for orbital mediated tunneling. Ap and Ac are the gas-and crystalline-phase electron affinities, 1/2(SCE) is the electrochemical potential referenced to the saturated calomel electrode, and provides the solution-phase electron affinity. Ev, is the Fermi level of the substrate (Au here). The corresponding positions in the OMT spectrum are shown by Ar and A0 and correspond to the electron affinity and ionization potential of the adsorbate film modified by interaction with the supporting metal, At. The spectrum is that of nickel(II) tetraphenyl-porphyrin on Au (111). (Reprinted with permission from [26])...

To estimate the amount of turbulent kinetic energy lost when filtering at a given grid size, it is useful to introduce a normalized model energy spectrum (Pope, 2000) as follows ... 

Fig. 1. The normalized model turbulent energy spectrum for a range of Reynolds numbers.

In a RANS simulation of scalar mixing, a model for i ,/, must be supplied to compute (4>a). In fully developed turbulence, t,p can be related to tu by considering the scalar energy spectrum, as first done by Corrsin (1964). 

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 ... 

In Fig. 2, the normalized model scalar energy spectrum is plotted for a fixed Reynolds number (ReL = 104) and a range of Schmidt numbers. In Fig. 3, it is shown for Sc = 1000 and a range of Reynolds numbers. The reader interested in the meaning of the different slopes observed in the scalar spectrum can consult Fox (2003). By definition, the ratio of the time scales is equal to the area under the normalized scalar energy spectrum as follows ... 

Fig. 5.3. Energy spectrum of solar neutrinos predicted from a standard solar model (e.g. Bahcall et al. 1982), omitting the undetectably small flux due to the CNO cycle. Fluxes are in units of cm-2 s-1 MeV-1 for continuum sources and cm-2 s-1 for line sources. Detectors appropriate in various energy ranges are shown above the graph. Courtesy J.N. Bahcall.

Figure 2.4. Sketch of model turbulent energy spectrum at Rk = 500.

Pope (2000) developed the following model turbulent energy spectrum to describe fully developed homogeneous turbulence 12... 

The model turbulent energy spectrum for Rk = 500 is shown in Fig. 2.4. Note that the turbulent energy spectrum can be divided into roughly three parts ... 

The model turbulent energy spectrum given in (2.53) was introduced to describe fully developed turbulence, i.e., the case where / , (/<. t) does not depend explicitly on t. The case where the turbulent energy spectrum depends explicitly on time can be handled by deriving a transport equation for the velocity spectrum tensor 4> (k, t) starting from the Navier-Stokes equation for homogeneous velocity fields with zero or constant mean velocity (McComb 1990 Lesieur 1997). The resultant expression can be simplified for isotropic turbulence to a transport equation for / ,(/<. t) of the form14... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum E k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998). 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

As in Section 2.1 for the turbulent energy spectrum, a model scalar energy spectrum can be developed to describe lop(n). However, one must account for the effect of the Schmidt number. For Sc < 1, the scalar-dissipation wavenumbers, defined by19... 

We have defined two diffusion cut-off wavenumbers in terms of /cdi and in order to be consistent with the model turbulent energy spectrum introduced in Chapter 2. 

A model scalar energy spectrum can be developed by combining the various theoretical spectra introduced above with appropriately defined cut-off functions and exponents ... 

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.

Having defined the model scalar energy spectrum, it can now be used to compute the scalar mixing time as a function of Sc and Rk. In the turbulent mixing literature, the scalar mixing time is usually reported in a dimensionless form referred to as the mechanical-to-scalar time-scale ratio R defined by... 

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. 

The model scalar energy spectrum was derived for the limiting case of a fully developed scalar spectrum. As mentioned at the end of Section 3.1, in many applications the scalar energy spectrum cannot be assumed to be in spectral equilibrium. This implies that the mechanical-to-scalar time-scale ratio will depend on how the scalar spectrum was initialized, i.e., on E (k. 0). In order to compute R for non-equilibrium scalar mixing, we can make use of models based on the scalar spectral transport equation described below. 

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. 

Figure 4.9. Sketch of CSTR representation of the SR model for 1 < Sc. Each wavenumber band is assumed to be well mixed in the sense that it can be represented by a single variable

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