Fluidized beds force balance equations

Batchelor (1988) developed a force balance equation for the solid phase in fluidized beds using the ensemble average method. With the help of the resulting equation and the theory of linear stability the following criterion was obtained by Ham et al. (1990) ... 

For the discrete bubble model described in Section V.C, future work will be focused on implementation of closure equations in the force balance, like empirical relations for bubble-rise velocities and the interaction between bubbles. Clearly, a more refined model for the bubble-bubble interaction, including coalescence and breakup, is required along with a more realistic description of the rheology of fluidized suspensions. Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermophysical properties, to study heat transport in large-scale fluidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors. 

From the analysis presented in the last two paragraphs, it is evident that the gravitational force acting upon the particle is used for the derivation of the equations for the terminal velocity and the pressure drop in the fluidized bed. Then, it is clear that the hydraulic density should be used in these equations as well as in any other equations that are derived from a similar force-balance analysis. For instance, this is the case of the Foscolo-Gibilaro criterion for determining the fluidization pattern (Section 3.8.2). 

Equation (32) is the generalized force balance differential equation for non-fluidized gas-particle flow. In this equation, for an inclined cylindrical moving bed, a = 0 for a vertical conical moving bed, 0 = 0 for vertical cylindrical moving bed, a = 0 and 0 = 0 and for a vertical cylindrical moving bed without interstitial gas flow, a = 0,0 = 0, and dp/dz = 0 simultaneously. 

While the particle diameter of fixed bed reactors is on the order of 1-5 mm, the particle size used in another type of reactor—fluidized bed reactors—is on the order of 50-200 p,m. Because of the small diameters, the effectiveness factor is close to 1 in most cases. The Ergun equation characterizes the pressure drop across a bed of solids—at low gas velocities (relative to the particle terminal velocity). When the drag force of the upward moving fluid exceeds the weight of the particles, the particles become fluidized—they begin to move up and down and the solids bed itself behaves like a fluid objects that are denser than the bed will fall through the bed while objects that are less dense will be remain at the top. Based on a force balance, the pressure drop across the bed, AF/L, will be equal to the head of solids (neglecting frictional forces) ... 

Write down the equation for the force balance across a fluidized bed and use it to come up with an expression for the pressure drop across a fluidized bed. 

Newton s law is applied in order to derive the linear momentum balance equation. Newton s law states that the sum of all forces equals the rate of change of linear momentum. The derivation/application of this law ultimately provides information critical to any meaningful analysis of most chemical reactors. Included in this information are such widely divergent topics as flow mechanisms, the Reynolds number, velocity profiles, two-phase flow, prime movers such as fans, pumps and compressors, pressure drop, flow measurement, valves and fittings, particle dynamics, flow through porous media and packed beds, fluidization— particularly as it applies to fluid-bed and fixed bed reactors, etc. Although much of this subject matter is beyond the scope of this text, all of these topics are treated in extensive detail by Abulencia and Theodore. ... 

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