A. Influence of the Wall on Porosity and Flow

The structure and randomness of packed beds are disturbed in the vicinity of a rigid wall. While this effect has been neglected in the discussion of stagnant beds (Section 200) it has to be taken into account in connection with fluid flow. Radial porosity profiles in circular tubes filled with almost perfect spheres are depicted in Figure 1. As expected, porosity is very high at the wall and reaches its minimum at a distance of half a particle diameter from it. Thereafter, damped oscillations are observed. Computer simulations of packed beds (Cheng et al., 2000) lead to results similar to those of Figure 1. However, increasing deviations from sphericity, size dispersity, and surface roughness of the particles surpress the oscillations. For such imperfect spheres, which are typical for many practical applications, the behavior of Figure 2 is obtained. Now, a monotonic function is sufficient for the approximation of the porosity profile, namely the function

\[\label{eq1} \psi\,(r)=\psi_{\infty} \Bigg [ 1+ 1.36 \,\exp \left( -5.0\frac{R-r}{d} \right) \Bigg] \tag{1}\]

after Giese (1997). According to Equation 1 the void fraction of the bed is increased considerably in a relatively narrow zone in the vicinity of the wall. Outside this zone, the porosity of the undisturbed (infinitely extended) bed ψ_∞ prevails. For technical (imperfect) cylinders with a length that does not differ very much from their geometrical diameter (l ≈ d_a), the analogous relationship

\[\label{eq2} \psi\,(r)=\psi_{\infty} \Bigg [ 1+\left( {\frac{0.65}{\psi_{\infty } }-1} \right) \,\exp \left( -6.0\frac{R-r}{d} \right) \Bigg] \tag{2}\]