Correlations Based on Dynamic Similarity

1 Empirical Correlations Using Conventional Dimensionless Groups The 

Equations 6.3 and 6.4 assume that at equal Re and Sc the value of Shp should be the same irrespective of the type and size of the contacting device/reactor. However, the definitions of Re and Shp pose a formidable problem. The linear dimension term in Re and Shp and the velocity term in Re need to be defined with relevance to the type of system/reactor. Most of the investigators used simple definitions for the velocity term. For instance, in bubble columns, the superficial gas velocity was used, whereas for stirred vessels the tip speed (NxD) was used. The discussion presented in Chapter 5 clearly indicates that these simplified definitions cannot form the basis of convective mass transfer in a highly turbulent multiphase reactor. The hnear dimension to be used in Shp and Re is similarly elusive, particularly in the case of 

2 Approach Based on Kolmogorov s Theory The first attempt at putting forth a credible theory of turbulence is attributed to Kohnogorov (1941). A simple 

Using dimensional reasoning, Kolmogorov s length scale, rj, and velocity, u, can be defined, respectively, as follows  

Further, according to Kolmogorov, if distances and velocities are referred to these scales, a universal function p exists such that 

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The representative characteristic velocity to be used in Re can be obtained from Equation 10.7. Unfortunately, unlike the linear dimension which is well defined in the case of it is not possible to define a representative linear dimension for the present case. For instance, the bubble diameter, d, which should be the preferred choice for representative linear dimension, exhibits a wide variation, particularly in the heterogeneous regime. Thus, in the absence of a representative value for d, an approach similar to Equation 10.9 cannot be adopted. In view of this difficulty, most investigators resorted to correlations based on dynamic similarity involving dimensionless groups (Section 6.3.1) as is evident from Table 10.2. 

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