A Bethe Tree Model

The Bethe tree shown in Fig. 3.5.3.1-1 is a branching network of pores with a coordination number of 3 and no closed loops. Three pores merge in each node. From this node 2 pores emerge. Higher coordination numbers can also be considered and the pores can have a variable diameter. The main advantage of the Bethe tree shown here is that it can yield analytical solutions for the fluxes, but the absence of closed loops is not entirely realistic. Its use will be illustrated in Chapter 5 on catalyst deactivation. What will be illustrated here is how to derive a tortuosity factor from this type of geometry. 

Beeckman and Froment [1982] represent the catalyst texture by a network of pores structured as a tree with a degree of branching equal to 2 and connectivity Z = 3. The bonds of the tree correspond to the pores of the catalyst the nodes correspond to their intersection. Each pore is characterized by two stochastic variables the pore length L and the pore diameter. The probability density function of the former, X I), is derived from the assumption that the 

The flux of a component A toward the center and through a sphere at a distance r, measured from the center onwards, may be written 

The product E i) N can be expressed in terms of the basic parameters involved in the characterization of the network structure. This is not required for 

What remains to be done is to find an expression for p. From Fig. 3.5.3.1-2, the relation between dr, the distance along the radial, and dx, the distance along a pore making an angle 6 with the radial at a distance r from the center of the spherical particle, is easily derived  

---