Dimensionless frequency

Figure A2.2.3. Planck spectral density fimction as a fimction of the dimensionless frequency /)oi/(/rj 7). A2.2.4.7 APPLICATION TO IDEAL SYSTEMS ELASTIC WAVES IN A SOLID...

Fig. 2.8. The absorption coefficient of CHBr3 in CCL as a function of dimensionless frequency [113] (— —) theory 2. (—) experiment.

For small frequencies T, Reiss and Hanratty (R12, R13) deduced a simple relationship between fluctuating mass transfer and velocity gradient. This relationship makes use of the dimensionless frequency n ... 

A common simplification is to assume constant (equivalent to assuming that Re is always in the Newton s law range). It is then convenient to define a dimensionless frequency and amplitude ... 

We can make the above discussion more quantitative by introducing the parameter B = c /2aMQ and the dimensionless frequency... 

The presence of shock waves in atmospheres of pulsating stars is due to the existence of a critical frequency a for standing wave oscillations. For example, in the plane-parallel isothermal atmosphere oscillations exist in the form of standing wave if the dimensionless frequency... 

In general, the total emission from a particular burning zone is proportional to the y-ray optical depth through the zone, Aty = Kyi p(r) dr. The density structure is thus important for the relative line strengths. Fortunately, both this and the core velocity may be obtained directly from observations of the line profiles at late epochs. The velocity field has then relaxed to a V =r law, and for an optically thin line the emission per volume, j(r), is related to the intensity, 1(e), of the line at the dimensionless frequency e by... 

The apparent kinetic data as discussed in the previous section is given in terms of the dimensionless frequency factors Ai and the activation energy ), measured in kJ/kmol with... 

Fig. 1.1. Electrochemical impedance (x) obtained from Eq. (1-4) with the experimental measurements of ZEHD o and ZEHD P. Electrochemical impedance (o) directly measured at the half wave potential. Curve in full line represents the theoretical variation. The coordinates are normalized by the electrochemical impedance value at zero frequency Zac(0). The parameter is the dimensionless frequency pScwl. After [29].

By using Newman s method [33, 34], the solution of the set of six equations is derived for each dimensionless frequency. 

Fig. 3-1. Variation of W0 versus the dimensionless frequency p for different Sc values. In the high frequency range the amplitude is proportional to p 2.

In Fig. 5-7, the amplitude and phase shift corresponding to Eq. (5-17) for different angular velocities show that in contrast with the simple behavior of a bare electrode, the data are no longer reducible by the dimensionless frequency p. In fact 0C... 

Fig. 5-7. From Eq. (5-17), EHD impedance versus the dimensionless frequency p for different rotation speeds (------10.47rd-s 1,-----------65.4rd-s l,. 262rd-s ).

It was shown that the mass transfer problem is identical to that for a newtonian fluid when adequate dimensionless quantities are used. In particular, a generalized dimensionless frequency can be defined ... 

Fig. 6-14. Potentiostatic EHD impedance plots, in Bode representation (reduced amplitude A(pSc /3)/A(0) and phase shift, versus dimensionless frequency pSc / ) for the oxidation of hydroquinone on a 360 nm thick poly(TV-ethylcarbazole) film at E - 0.7 V (diffusion plateau).

Dimensionless well depth Velocity of a particle Steady-state distribution function Dimensionless frequency (for electrolyte solutions)... 

From Eq. (34a) we obtain the formula for the dimensionless frequency xD of the loss maximum ... 

Changing the order of integration and using jc = (o/fl and y = (fix) 1 as a real part and an imaginary part of the dimensionless frequency z, we have... 

The following molecular constants are used in further calculations density p of a liquid the static (es) and optical (n ) permittivity moment of inertia /, which determine the dimensionless frequency x in both HC and SD models the dipole moment p the molecular mass M and the static permittivity 1 referring to an ensemble of the restricted rotators. The results of calculations are summarized in Table XXIII. In Fig. 62 the dimensionless absorption around frequency 200 cm 1, obtained for the composite model, is depicted by dots in the same units as the absorption Astr described in Section B. Fig. 62a refers to H2O and Fig. 62b to D2O. It is clearly seen in Fig. 62b that the total absorption calculated in terms of the composite model decreases more slowly in the right wing of the R-band than that given by Eq. (460). Indeed, the absorption curve due to dipoles reorienting in the HC well overlaps with the curve generated by the SD model, which is determined by the restricted rotators. 

Figure 4.12. Real (a) and imaginary (b) components of the cubic susceptibility of a superparamagnetic assembly with coherently aligned easy axes the direction of the probing field is tilted with respect to the alignment axis at cos P = 0.5 the dimensionless frequency is (DTo = 10-6. Solid lines show the proposed asymptotic formulas taken with the accuracy a 3 circles present the result of numerically exact evaluation dashed lines correspond to the zero derivative approximation (4.167). The discrepancy of the curves is mentioned in the text following Eq. (4.220).

Figure 4.13. Real (a) and imaginary (b) components of the fifth-order susceptibility of a random superparamagnetic assembly the dimensionless frequency is coio = 1CT6. Solid lines show the proposed asymptotic formulas with the accuracy a 3 circles present the result of a numerical evaluation.

Figure 4.18. Position of the maximum of the function 4>(r) against the dimensionless frequency.

A dimensionless frequency ratio, such as f(lS-2S)/f(2S-nS), on the other hand, is independent of the Rydberg constant. Its measurement can serve as a sensitive test of quantum electrodynamic level shifts and as a means to determine the size of the proton or deuteron, provided QED is correct. 

The function e(i ) itself is dimensionless. Frequency is in radians per second, but for compactness is often written in units of electron volts (the energy of a photon with that same radial frequency). If a quantity is tabulated in units of electron volts, it can be converted to radians per second by multiplying by 1.519 x 1015 (e.g., Level 1, the table on the frequency spectrum). This holds for the quantities coj and gj, which are given in electron volts. The numerator f is given in electron volts squared [to keep e(/ ) dimensionless] and can be converted to (radians/second)2 if one chooses to work in those units through multiplying by (1.519 x 1015)2. In the Debye form d -/( 1 + r ), the numerator dj is dimensionless and the inverse relaxation time 1 /r - is in electron volts. 

On a RDE, in the absence of a surface layer, the EHD impedance is a function of a single dimensionless frequency, pSc1/3. This means that if the viscosity of the medium directly above the surface of the electrode and the diffusion coefficient of the species of interest are independent of position away from the electrode, then the EHD impedance measured at different rotation frequencies reduces to a common curve when plotted as a function of p. In other words, there is a characteristic dimensionless diffusional relaxation time for the system, pD, strictly (pSc1/3)D, which is independent of the disc rotation frequency. However, if v or D vary with position (for example, as a consequence of the formation of a viscous boundary layer or the presence of a surface film), then, except under particular circumstances described below, reduction of the measured parameters to a common curve is not possible. Under these conditions pD is dependent upon the disc rotation frequency. The variation of the EHD impedance with as a function of p is therefore the diagnostic for... 

If the disturbances are assumed to grow at the linear rate, the dimensionless frequency of lamellae generation is... 

The moments of normalized distributions are products of dimensionless frequencies and dimensionless molecular weights or of gram-moles with dimensions of mass. The former moments will be unitless, and the units of the latter will depend on the moment number and on the units of the distribution. Most equations in polymer science imply use of gram-moles, but this is not universal and the dimensions of the particular equation should be checked to determine which units, if any, are being used for molecular weight and concentration quantities. 

In both models the only model parameter used is the mean residence time tpp and tpsR F gtire 6 shows the reactor dynamics of the PFR and the PSR in the normalised time and frequency domain (dimensionless time 0 = t/T, dimensionless frequency fg = l/2jt9). In the time domain the step response F(9) as well as the impulse response E(0) (which is the RTD) can be discussed. This type of data presentation is normally used in chemical engineering application. But the same data can also be presented in the frequency domain, the so called Bode plot. This type of presentation allows to identify effects which are not visible in the classical used plot in the time domain. The Bode plot consists of the magnitude Gj 030)1 and the phase arg G(jO)0). ... 

---