Application to a Model of the Bubbling Fluidized Bed

Reaction only takes place in the dense phase since that is where the catalyst particles are. Since the exchange is with a uniform environment where the concentration is cp, we can see that by the time the bubble has reached the top of the bed, the concentration of reactant in it is cp + (co - cp).exp -Tr, where c0 is the entering concentration, H the height of the bed and Tr = QH/UaV is a dimensionless transfer number. By doing a mass balance on the dense phase as a whole14 we obtain a linear equation for cp in terms of the 

Equation (18) with a square in the numerator of the second term looks a little odd at first, but, in fact it unfolds itself with unusual clarity. We first observe that even when the reaction is infinitely fast, Da — , a fraction /3 e-Tr of the inlet concentration will remain in the bubbles when they reach the top of the bed. Thus y can never be less that /3 e Tr so if we subtract this, to allow y — /3eTr to go to zero, and rescale by dividing by 1 - /3e Tr, the effective amount that is available to the dense phase, we have  

Mathematicians have their names attached to theorems, lemmas and even conjectures naturalists have species and genera called after them physicists have their principles and chemists their reactions and reagents. Engineers can claim the dimensionless parameters, for they, who, in their function as designers, have often to work with units, appreciate more than anyone the conceptual beauty of the dimensionless number, invested as it is with full contextual meaning and magnitude. I submit that it would be more than appropriate to call this transfer parameter the Davidson number, for who has done more to elucidate the mechanism of this transfer process than John Davidson Damkohler has preempted the initial letters Da, but that is no matter, for transfer gives us Tr and we immediately think of Trinity and its present Vice-Master. 

We wish to see what the overall conversion of a continuous mixture will be, but, first, we have to ask which parameters will depend on jc, the index variable of the continuous mixture. Clearly k the rate constant in the Damkohler number will be a function of jc, and, if monotonic, can be put equal to Da.x. The parameter /3 is clearly hydrodynamic and so, for the most part, are the terms in the Davidson number. The only term in the equation 6.21 of Davidson and Harrison that might depend on x is the gas phase diffusivity, 

The inequalities for the exponential integral stated above, equation (12), then give 

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APPLICATION OF THE GAMMA DISTRIBUTION, 107 A GENERAL THEOREM FOR SIMPLE LINEAR REACTOR MODELS, 108 APPLICATION TO A MODEL OF THE BUBBLING FLUIDIZED BED, 109 THE DAMKOHLER NUMBER, 111... 

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