Schulzes superposition model

Schulze (1993) notices that the overall collision probability of single particles should be the algebraic sum of individual collision efficiencies . These are given as the interception effect E j, the gravitational effect Eg and inertial effect E. The quality of this approach can be 

The representation of a four-dimensional function by a superposition of functions of smaller dimension is not generally justified. However, in extreme cases, at very small values of one of the parameters, the dependence on it can be negligible. For instance, under condition (10.76) 

The process of particle rebound is not incorporated in the superposition model by Schulze so that it can only be used for particles which attach at the first collision. 

The hydrodynamic field used does not comprise the interesting case of large rising bubbles under the potential flow. Ej and Ej are sensitive to the hydrodynamic field and the bubble surface state, which was not yet been taken into account. 

To evaluate Schulze s equation as a semi-empirical relationship, let us analyse the experimental method and data. Schulze used flow tube experiments with bubbles of 2 mm in diameter and Re = 30. There is a significant difference between experiments by Schulze (1993) and Hewitt et al. (1994) who used single rising bubble with 2 mm diameter. Re = 600 and St = 0.25. It is unclear if collisions occur in Schulze s experiment at St 0.07. 

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The data in Fig. 11.9 correspond to results given in Fig. 3 by Hewitt et al. (1994) where the curves are calculated by means Schulze superposition model. The solid line is calculated by Hewitt et al. using Schulze s superposition and for a retarded bubble surface of 2 mm diameter. If the bubble surface is free of surfactant Sutherland s Eq. (10.11) yields E , in Schulze s approximation and Langmuir s equation can be used to estimate the effect by inertia (Section... 

The number of captured particles per single bubble can be expressed by the collision efficiency. This quantity, calculated by Hewitt et al. (1994), using Schulze s superposition model (solid line in Fig. 11.9), is also shown in Fig. 11.10. We followed the same procedure using the dashed line of Fig 11.9 and taking into account the unwanted influence of particle rebound (dashed line in Fig. 11.10). The ordinates of dashed line of Fig. 11.10 is calculated from the ordinate of the solid line in Fig. 11.10 as the product with the ratio of ordinates of... 

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