The functions in Tables 1-4 were derived using simple mensuration formulas for circles in 2-D or spheres in 3-D. The mensuration formulas can be derived easily or found in reference books (for example, see Ref. [9]). As an example of the procedure, the results for the functions Aij(R) and R(V ) are derived for the 2-D base configuration.
Fig. 7. 2-D geometry for a base configuration used in deriving the areas, Aij of the various interfaces and the radius, R, as a function of volume.
In the base configuration, the area of the interface between the void and the wall surface, AWG, is zero by definition. The area of the interface between the void, having radius R, and the base surface, ABG, is twice the length of the lower leg of the triangle shown in Fig. 7:
The area of the interface between the void (radius R) and the liquid, ALG, is the difference between the perimeter of the circle and the arc length of the imaginary cap that lies below the base (see Fig. 7):
The volume of a void of radius R in the base configuration is the difference between the volume of a circle of radius R and the volume of the imaginary cap that lies below the base (see Fig. 7):
Solving for R produces
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