Steady-state assumption, uniform velocity

A useful simphfication of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m through a single planar entrance and a single planar exit (Fig. 6-4), to whi(m the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tu is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + pgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. 

The ideal plug flow reactor PFR is a simplified picture of the motion of a fluid in a tubular reactor as it is assumed that all fluid elements move with a uniform velocity along parallel streamlines and thus have a fixed residence time r. Strictly speaking, this assumption breaks the hydrodynamic rule that the velocity is zero at the wall (no slip condition. Figure 3.2.22). The steady-state mass balance of a PFR for a constant volume reaction can be deduced from the one-dimensional mass balance for a differential small element with thickness Az in direction of flow ... 

Steady wave A propagating transition region that connects two uniform states of a material. The wave velocities of all parts of the disturbance are the same, so the profile does not change with time, and the assumptions that go into the jump conditions are valid. 

Typically, there are two ways to inject tracers, steady tracer injection and unsteady tracer injection. It has been verified that both methods lead to the same results (Deckwer et al., 1974). For the steady injection method, a tracer is injected at the exit or some other convenient point, and the axial concentration profile is measured upward of the liquid bulk flow. The dispersion coefficients are then evaluated from this profile. With the unsteady injection method, a variable flow of tracer is injected, usually at the contactor inlet, and samples are normally taken at the exit. Electrolyte, dye, and heat are normally applied as the tracer for both methods, and each of them yields identical dispersion coefficients. Based on the assumptions that the velocities and holdups of individual phases are uniform in the radial and axial directions, and the axial and radial dispersion coefficients, E and E, are constant throughout the fluidized bed, the two-dimensional unsteady-state dispersion model is expressed by... 

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