Compressible Fluids in Packed Beds

A problem arises with the models of Section 9.1.1 when the density of the fluid changes appreciably with temperature or pressure. To understand the difficulty, refer to Section 3.1, where variable-density PFRs are treated. We consider only the case where the tube cross section is constant. A more general version of Equation 9.1 is 

The fact that T varies with radial position causes a dilemma. If T varies in the radial direction, so must Us, and the assumption of a uniform velocity profile is violated. However, the radial variation in density depends on the ratio of wall and centerline temperatures, T si z)/T 0, z), and will typically be reasonably small. A suggested approximation is to use the mixing-cup average temperature in Equation 

Corresponding to Equation 9.11 is a modified equation for femperafure fhaf is based on the enthalpy flux  

An important embellishment to the foregoing treatment of packed-bed reactors is to allow for temperature and concentration gradients within the catalyst pellets. Intrapellet diffusion of heat and mass is governed by differential equations that are about as complex as those governing the bulk properties of the bed. See Section 

A set of simultaneous PDEs (ODEs if the pellets are spherical) must be solved to estimate the extent of reaction and conversion occurring within a single pellet. These local values are then substituted into Equations 9.1 and 9.3 so that we need to solve a set of PDEs that are embedded within a set of PDEs. The resulting system truly reflects the complexity of heterogeneous reactors, but practical solutions rarely go to this complexity. Most industrial reactors are designed on the basis of pseudohomogeneous models as in Equations 9.1 and 9.3, and the local catalyst behavior is described by the effectiveness factor defined in Chapter 10. 

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