First-pass effect definition

Darton and Harrison (1975) derived a criterion for the point of transition to predict whether a solid-liquid fluidized bed will expand or contract when the gas is first introduced. The definition of Pa used by Darton and Harrison was the ratio of upper clear (particle-free) wake volume to the bubble volume. But since they did not consider the circulation of sohds associated with the lower nonclear portion of the wake, their Pa was effectively the same as that of Bhatia and Epstein (1974). The use of the Wallis drift flux approach by Darton and Harrison (1975) also represents no real difference from the relative velocity approach taken by Bhatia and Epstein (1974), since the two methods are rigorously interrelated. It is therefore not surprising that the final criteria of Bhatia and Epstein (1974) and Darton and Harrison (1975) are identical. 

In the operator L, the first term represents convection and the second diffusion. Equation (44) therefore describes a balance of convective, diffusive, and reactive effects. Such balances are very common in combustion and often are employed as points of departure in theories that do not begin with derivations of conservation equations. If the steady-flow approximation is relaxed, then an additional term, c5(pa)/3t, appears in L this term represents accumulation of thermal energy or chemical species. For species conservation, equations (48) and (49) may be derived with this generalized definition of L, in the absence of the assumptions of low-speed flow and of a Lewis... 

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