Shear laminar steady model

The quasi-steady laminar model is now employed to describe the heat transfer near the wall. Note that while the shear stress at the wall can be related easily to the pressure drop for the flow in a tube, it is more difficult to establish a relation between these two quantities for a packed or fluidized bed. However, while for the flow in a tube the dissipated energy is not uniform over the section... 

Steady Flow in Packed Beds of Monosized Spherical Particles. Steady incompressible fully developed flow in porous media confined in a circular pipe can be treated with a single differential equation as given by equation 111. The inertial effects are only reflected in the shear factor term. Two purposes are served in this section to verify the integrity of the models presented earlier, including the passage model on shear factor and wall effects on the flow, and to show the flow behavior itself. The flow problem is solved numerically with a central difference method. An abundance of experimental data are available in the literature. However, we confine ourselves to the laminar flow regime for a packed bed of spherical particles. We make use of the latest available data presented by Fand et al. (110) for a packed bed with weak wall effects and the experimental data of Liu et al. (32). 

The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. 

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