Von Karman analogy

W. Tubes, turbulent, smooth tubes. Constant surface concentration. Von Karman analogy... 

With the universal velocity profile Eqs. 10.2.14-10.2.16, we obtain the multicomponent form of the von Karman analogy... 

In the past, there have been two major approaches to analyze the problem of heat and mass transfer across the liquid-solid interface. The first approach can be broadly classified as "Analogies." This method essentially consists of (von Karman ( ) and Wasan and Wilke ( )) (i) development of velocity profile near the interface, (ii) suitable assumption for the variation of eddy diffusivity with respect to the distance from the interface, and... 

Figure 10.4. Stanton number, (St) as a function of Schmidt number (Sc) from the von Karman velocity profile and the Chilton-Colburn analogy. The friction factor is maintained at 0.073 for this illustration.

In Figure 10.8 we have plotted the variation of the ratios of mass transfer coefficients 12/ 11 k i/k22 for an acetone-benzene-helium system considered in Example 11.5.3. The Chilton-Colburn analogy predicts that these ratios would be independent of Re, as shown by the horizontal lines in Figure 10.8. The von Karman turbulent model, on the other hand, predicts that the influence of coupling should decrease with increase in Re. The latter trend is in accord with our physical intuition. Depending on the driving forces for mass transfer, the Chilton-Colburn and the von Karman turbulent models could predict different directions of transfer of acetone (see, e.g., Krishna, 1982). 

Figure 10.8. Ratio /C12A115 which are elements of the zero-flux matrix of mass transfer coefficients [/c], as a function of the gas-phase Reynolds number. Mass transfer between a gaseous mixture of acetone (l)-benzene (2)-helium (3) and a liquid film containing acetone and benzene. Calculations by Krishna (1982) based on the von Karman turbulent film model and the Chilton-Colburn analogy.

Experiments show [56, 212, 289, 427] that the turbulent boundary layer on a flat plate includes two qualitatively different regions, namely, the wall region (adjacent to the plate surface) and the outer region (bordering the unperturbed stream). By analogy with the flow through a circular tube, it is common to subdivide the thin wall region into three subdomains (von Karman s scheme) ... 

Sherwood modified Von Karman s analysis to retain the ratio EglEy rather than assuming it to be unity. By using experimental results that indicated EglEy to be about 1.6, he was able to correlate data for the evaporation of nine different liquids into air in a wetted-wall column. The Von Katman analogy is not as successftil at high Schmidt numbers where most of the resistance is concentrated near the wall. 

Von Karman further modified the Prandtl analogy by considering the buffer region in addition to the viscous sublayer and the turbulent core. These three regions are shown in the universal velocity profile in Fig. 3.10-4. Again, an equation is written for molecular diffusion in the viscous sublayer using only the molecular diffusivity and a Reynolds analogy equation for the turbulent core. Both the molecular and eddy diffusivity are used in an equation for the buffer layer, where the velocity in this layer is used to obtain an... 

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