where C D is the drag coefficient, and A is the projected area. For conven- 

 ience, A can be redefined to be the projected area of a sphere of equal vol- 

 ume. In addition, the drag force for a bubble at its terminal velocity is 

 equal to the buoyant force V/>g where V is the volume of the bubble. Hence, 

 in terras of the equivalent radius, the expression for the drag coefficient 

 reduces to 



S=f^ [3] 



The above expression for C^ is, except for a numerical factor, the 

 first dimensionless parameter of Equation ["!]. More generally, since the es- 

 sential effect of gravity is to produce a uniform pressure gradient Pg, the 

 drag coefficient for a bubble at terminal velocity may be written 8/3 r e^P/pU £ 

 where vp is any pressure gradient in which the bubble may be placed. The 

 second and third ratios are the Reynolds and Weber numbers respectively. 

 Therefore the relation sought is of the form 



f 2 (c D , Re, W) = [4] 



No attempt was made to determine theoretically the form of f_ except 

 in the Stokes region of flow, that is, for very small spherical bubbles, where 

 inertial terms may be neglected. 5 ' 6 These have not proven satisfactory beyond 

 a Reynolds number of 1 . Actually, up to a Reynolds number of 70, bubbles in 

 water behave exactly like rigid spheres. 



Since a theoretical solution to the hydrodynamic equation of motion 

 for a bubble at its terminal velocity in a liquid does not appear feasible, 

 methods for the empirical solution of Equation [4] are indicated. However, 

 the form of these parameters is inconvenient for presenting the data as ordi- 

 narily obtained. The experimental procedure is to vary the size of the bubble 

 and measure the resulting velocity of rise. If, instead of the Weber number, 

 we chose a dimensionless ratio not involving the bubble size or speed, presen- 

 tation and interpretation of the results are facilitated. A suitable param- 

 eter is gn /py 3 . This new parameter will be denoted by the symbol M. As ex- 

 plained before, the function of the gravitational field is to produce a pres- 

 sure gradient. Therefore M can be rewritten as ft\p/p 2 y 3 . 



The experimental data may then be presented as 



f 3 (c D , Re, M) = [ 5 ; 



