Ratio scale

A scale ratio R is used for scale-up from the standard configuration as shown in Table 7-2. The procedure is ... 

Determine geometric similarity to develop a single scale ratio R, for the relative magnitudes for all linear dimensions [32]. 

To select a turbine, there must also be geometric similarities for the type of turbine, blade width, number of blades, impeller diameter, etc. From the geometric similarity determination of the turbine diameter, the mixer speed can be established to duplicate. The Scale Ratio R, ... 

For geometric similarity of liquid motion (n = 1.0) the linear scale ratio of volume is... 

Tables 4 and 5 show the values of the mean particle diameter for models of an atmospheric and pressurized commercial bed, respectively, for different selected linear scale ratios between the model and commercial bed.

Thus, from Figs. 2 and 3 we can conclude that the time-scale ratio will depend on ReL and Sc. 

In the literature on turbulent mixing, the mechanical-to-scalar time-scale ratio is defined by... 

Fig. 4. Mechanical-to-scalar time-scale ratio found from the normalized model scalar energy spectra.

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... 

Figure 3.14. The mechanical-to-scalar time-scale ratio for a fully developed scalar energy spectrum as afunction of the Schmidt number at various Reynolds numbers R> = 50,100, 200,400, and 800. The arrow indicates the direction of increasing Reynolds number.

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. 

Chasnov (1994) has carried out detailed studies of inert-scalar mixing at moderate Reynolds numbers using direct numerical simulations. He found that for decaying scalar fields the scalar spectrum at low wavenumbers is dependent on the initial scalar spectrum, and that this sensitivity is reflected in the mechanical-to-scalar time-scale ratio. Likewise, R is found to depend on both the Reynolds number and the Schmidt number in a non-trivial manner for decaying velocity and/or scalar fields (Chasnov 1991 Chasnov 1998). 

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. 

Despite the explicit dependence on Reynolds number, in its present form the model does not describe low-Reynolds-number effects on the steady-state mechanical-to-scalar time-scale ratio (R defined by (3.72), p. 76). In order to include such effects, they would need to be incorporated in the scalar spectral energy transfer rates. In the original model, the spectral energy transfer rates were chosen such that R(t) —> AV, = 2 for Sc = 1 and V

model parameter. DNS data for 90 < R-,. suggest that Re, is nearly constant. However, for lower... 

In the definition of b, Ro is the equilibrium mechanical-to-scalar time-scale ratio found with Sc = 1 and = 0.34 The parameters Cd, Cb, and Cd appear in the SR model for the scalar dissipation rate discussed below. Note that, by definition, xi + Yi + K3 + Kd = 1. ... 

The similarity transformation transforms a set of points S at position x = (xj,...,xE) in Euclidean E-dimensional space into a new set of points r(S) at position x = (rXj,...,rxE) with the same value of the scaling ratio 0self-similar with respect to a scaling ratio r if S is the union of N nonoverlapping subsets SU...,SN, each of which is congruent to the set r(S). Here congruent means that the set of points. S is identical to the set of points r(S) after possible translations and/or rotations. For the deterministic self-similar fractal, the selfsimilar fractal dimension dFss is clearly defined by the similarity... 

Figure 2. (a) A deterministic self-similar fractal, i.e., the triadic Koch curve, generated by the similarity transformation with the scaling ratio r = 1/3 and (b) a deterministic self-affine fractal generated by the affine transformation with the scaling ratio vector r = (1/4, 1/2). 

Scaling ratio vector with the different scaling ratios 0 < rj,...,rE < 1... 

Perimeter Polyacrylonitrile Poly-vinylidene fluoride Scaling ratio... 

Scale is a special ratio used for models of real life items, such as model railroads and model airplanes, or scale drawings such as blueprints and maps. On model airplanes, you will often find the scale ratio printed on the model as model real. For example, a toy car may have the ratio 1 62 printed on the bottom. This is the ratio of all of the dimensions of the actual toy to the corresponding dimensions of the real car. This scale ratio says that the real car is 62 times larger than the toy, since the ratio is 1 62. 

There is conflicting evidence regarding the extent to which imposed vibrations increase particle to fluid heat and mass transfer rates (G2), with some authors even claiming that transfer rates are decreased. For sinusoidal velocity variations superimposed on steady relative motion, enhancement of transfer depends on a scale ratio A/d and a velocity ratio Af /Uj (G3). These quantities are rather like the scale and intensity of turbulence (see Chapter 10). For Af /Uj < l/2n, the vibrations do not cause reversal in the relative motion and the enhancement of mass transfer has been correlated (G3) by... 

Avd Avn Avp c A AW Doppler half-width in hertz natural half-width l/47cyN pressure-broadened half-width 1 /4nyP variable of integration wavelength of radiant flux triangle function of unit height and half-width % scaled ratio (AvN -1- Avc) /ln 2/AvD... 

If the scale ratio is not the same along each axis, the relationship among the two bodies is of a distorted geometric similarity and the axial relationships are given by... 

Thus, equipment specifications can be described in terms of the scale ratio L or, in the case of a distorted body, two or more scale ratios X, Y, Z). Scale ratios facilitate the comparison and evaluation of different sizes of functionally comparable equipment in process scale-up. 

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