Self-Conveoting Flows 



assumptions we shall depend finally upon a comparison between 

 theoretical predictions and the results of systematic experiments. 



The energy balance is expressed as follows. 



9E _ 8 



8t at 



^[1 (u^ + v^ + w^) + (p - Pe) g(\ de dii dC 



= - (Rate of Dissipation of Energy) (27) 



See Fig. 14 for nomenclature. 



For a self-similar, self-convecting flow, the dissipation of 

 kinetic energy per unit volume which occurs due to the action of 

 turbulent shear stresses must for dimensional reasons be of the 

 form. 



Dissipation q. p W {Z&\ 



Unit Volume R ^ ' 



As a result, the energy balance, Eq. (27), for a self-con- 

 vecting mass in a homogeneous medium of the same density takes 

 the form, 



where 



j = (two-dimensional) 



j = 1 (axisymmetrical) 



and K is a constant of virtual energy, defined by the identity, 



I 



^'+7'^^^ de dTi d; = ^ wV*^ (30) 



and where Cq is a dissipation coefficient. 



Making use of (29), together with the relationship R = Pz , 

 it may be shown that the height of rise follows the law, 



^(2*j/2*Cd/^K)^^ (31) 



343 



