Moving Bed Flows in a Feed Hopper

For the momentum equation of gas in a moving bed flow, the inertial and gravitational terms can be neglected. The pressure drop is therefore balanced by the drag forces, which give 

To close the problem, constitutive relations of powders must be introduced for the internal connections of components of the stress tensor of solids and the linkage between the stresses and velocities of solids. It is assumed that the bulk solid material behaves as a Coulomb powder so that the isotropy condition and the Mohr-Coulomb yield condition may be used. In addition, og has to be formulated with respect to the other stress components. 

The isotropy condition is that in the (R, p ) plane, the principal directions of stresses and the rate of deformation coincide [Jenike, 1961]. Thus, the isotropy condition is written as 

the Haar-von Karman hypothesis, which requires oe to be equal to one of the principal stresses in the (R, cp) plane, is adopted. Thus, it is reasonable to assume og to be equal to the 

So far, we have nine equations (Eqs. (8.31)-(8.39)) for nine variables (uR, uv, uRp, uw, p, or, o, ov, and rR(C). Therefore, the problem is closed and, in principle, can be solved numerically. However, analytical solutions are available with further simplifications of the model equations. 

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