Flow regime Differential models

Gas jets in fluidized beds were reviewed by Massimilla (1985). A more recent review is by Roach (1993) who also developed models to differentiate three jet flow regimes jetting, bubbling and the transition. However, most of the data were from jets smaller than 25 mm. The discussion here will emphasize primarily large jets, up to 0.4 m in diameter, and operation at high temperatures and high pressures. The gas jets can also carry solids and are referred to as gas-solid two-phase jets in this discussion. 

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). 

CPBR behavior under pseudo-steady-state and plug-flow regime is described by the resolution of the system of differential Eqs. 5.76 and 5.79. This model was experimentally validated in a laboratory packed-bed reactor with chitin-immobilized... 

They subsequently (2) developed a one-dimensional mathematical model in the form of coupled differential and integro-differential equations, based on a gross mechanism for the chemical kinetics and on thermal feedback by wall-to-wall radiation, conduction in the tube wall, and convection between the gas stream and the wall. This model yielded results by numerical integration which were in good agreement with the experimental measurements for the 9.53-mm tube. For this tube diameter, the flows of unbumed gas for stable flames were in the turbulent regime. 

At the lower temperature (783 K open symbols in Fig. 70) a substantially different behavior is observed. The imide band (A in Fig. 69 bottom) decreases quasi-linearly with the elapsed time (see Eq. 24). The aromatic band (V in Fig. 70 top) is complex, revealing two distinct decomposition patterns. At the beginning (first half) of the normalized time a slow linear decrease is observed, followed by a fast decrease. The decrease of the imide band and the change of the aromatic band in the second part of the curve are typical for a film diffusion-controlled reaction of shrinking particles in a gas flow in the Stokes regime. To confirm this observation a new mathematical model is used to fit the curves [321]. Starting from Eq. 20, the reaction velocity ks is substituted with kg=D Rf1 [321]. D is the diffusion velocity and kg the mass transfer coefficient between fluid and particle. The differential equation is solved and the time necessary to reduce a particle from a starting radius R0 to Rt is obtained [see Eq. (22)] [321],... 

In the case of transient operation, an accumulation term, that is, a differential term with respect to time, has to be added to equations 79 and 80 for being able to describe the observations. A batch reactor with uniform concentrations throughout the entire reactor but without continuous feed addition and effluent removal, is inherently operating in a transient regime. The corresponding reactor model equation is analogous to that of a plug flow reactor with the derivative taken with respect to time rather than with respect to position ... 

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