Mathematic model for design of reactor

the mathematic models for reactor design are also classified into continuous heat exchange bed and adiabatic one. Usually, the design of reactor adopts one-dimension quasi-homogeneous model which considers that when reactive gas passes the catalyst bed like a plug-flow, there exist no radial and axial return mixture, and microkinetics can be treated in intrinsic kinetics multiplied by an effective factor that involves the effects of transfer processes, and by an active coefficient that involves the effects of reduction, poisoning and declining etc. Macrokinetics can be 

Consider a micro-volume segment in catalyst bed, the area for fluid flow (height dL) in axial bed is, 

For radial bed, there are two directions for fluid flow From the center to the wall of reactor (from inside to outside) and from the wall to the center of reactor (from outside to inside). Usually, gas flows from the wall to the center of reactor, and other parameters can be obtained as axial bed except area  

With the mass balance in this micro-volume segment and considering active coefficient and effective factor for internal diffusion, we get  

Prom heat balance in this micro-volume segment, we get  

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