Dimensionless mechanical energy balanc

The mechanical energy balances for the center-tube and outer annulus are linked, through the pressure terms, by the equation for pressure drop across the bed. This may be integrated over the bed width and the equations combined to give a single equation for the dimensionless axial velocity in the center-tube, Vz, which may be written in the general form [5]... 

The exponents were found to be independent of the impeller type, vessel size, impeller clearance and impeller to tank diameter ratio. The dimensionless constant Ci accounted for variations in the system geometry (e.g. on dc/di). This would indicate that the basic mechanism leading to minimum suspension may be the same for rather different stirrer geometries. Table 2, which is an update of the one given by Nienow [19], indicates different exponents found in a few other investigations. Baldi et al. [26] made a relatively successful theoretical approach by assuming that the suspension of particles is mainly due to eddies of a certain critical scale comparable to the particle size. From an energy balance it follows that... 

The design equations and the species concentration relations contain another dependent variable, 6, the dimensionless temperature, whose variation during the reactor operation is expressed by the energy balance equation. For ideal batch reactors with negligible mechanical shaft work, the energy balance equation, derived in Section 5.2, is... 

Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid. When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value A 7. The flow has a stationary cellular character with a spatial periodicity of about 2d. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T in temperature creates warmer and cooler regions, and due lO buoyancy effects the former tends to move upwards and the latter downwards. When AT < AT, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T and the velocity (normal to the layer) vary as exp (i j,y) with x ji/rf, the threshold is given by the dimensionless Rayleigh number... 

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