Liquid-phase velocity

As mentioned above, this approach treats each phase as a constituent to a mixture. Thus, all parameters are mixture parameters and must be averaged, usually by the saturation. Unlike the models mentioned at the end of the previous section, the models here use capillary phenomena. Furthermore, although the mixture moves at a mass-average velocity, interfacial drag between the phases and other conditions allow each separate phase velocity to be determined. The liquid-phase velocity is found by 9... 

Since the liquid-phase velocity is obtained, for example, from the model as follows... 

The settling velocity, p, is relative to the continuous liquid phase where the particle or drop is suspended. If the liquid medium exhibits a motion other than the rotational velocity, CO, the vector representing the liquid-phase velocity should be combined with the settling velocity (eq. 2) to obtain a complete description of the motion of the particle (or drop). 

Once you calculate Vc in Eq. (7.20), you can easily find the dispersed-phase velocity using Eq. (7.23). Note that you now have the 100% flood values determined. This means you can now calculate the actual liquid-phase velocities from Eqs. (7.21) and (7.22) and ratio them to these 100% flood velocities. Thus this ratio times 100 is the flood condition of your extractor column. Please see Eqs. (7.24) and (7.25), where this is done. 

Figure 14 Schematic diagram of a hybrid PIV/LIF/SIT system used for measuring liquid-phase velocity distribution and bubble s shape and motion (Tokuhiro et al, 1998).

U Dimensionless liquid phase velocity averaged in y-direction... 

Ux Dimensionless liquid phase velocity in x-direction Dimensionless liquid phase velocity in z-direction V Gas (dispersed phase) velocity, m/s... 

Fig. 11. Typical computational results obtained by Lapin and Liibbert (1994) with a mixed Eulerian-Lagrangian approach. Liquid phase velocity pattern (left) and the bubble positions (right) in a wafer column (diameter, 1.0 m height, 1.5 m) where the bubbles are generated uniformly over its entire bottom. (Reprinted from Chemical Engineering Science, Volume 49, Lapin, A. and Liibbert, A., Numerical simulations of the dynamics of two-phase gas-liquid flows in bubble columns, p. 3661, copyright 1994 with permission from Elsevier Science.)...

Fig. 28. Positions of two coaxial gas bubbles generated subsequently at the same orifice together with the distribution of the relative (i.e., with respect to the velocity of the leading gas bubble) liquid phase velocity.

Liquid phase velocities are related to the volume flows in each section while the adsorbent movement in the case of SMB is equal to the column volume moved per shifting time ... 

In these equations, CfJ and Cfj are the concentrations of component i at the outlet and the inlet of column /, respectively. Qj is the actual volumetric flow rate of mobile phase through column j. It is related to the liquid phase velocity, Uj, by Qj = eAuj, where A is the column cross-section area, which is assumed, without loss of generality, to be the same for all the columns including the feed and drawoff columns. 

To avoid negative coefficients, the relation for the coefficient ap can be modified using the continuity equation, as shown for the liquid phase velocity equations. The alternative a p and h coefficients are defined by ... 

The transport equation for the pressure corrections is obtained substituting all the velocity components in (C.430) by the sum of the guessed and corrected velocities, and then substituting the velocity corrections by the corresponding pressure corrections employing the liquid phase velocity correction formulas. The resulting algebraic equation to be solved is written ... 

FIGURE 9.18 Film condensation at a vertical impermeable surface with 8,/d > 1 and 5tg/d > 1. Distributions of saturation, temperature, and liquid phase velocity are also depicted. 

Solid/liquid flows are commonly found in industrial processes to avoid flow obstruction, nonintrusive flowmeters are generally preferred. Flowmeters based on ultrasonic techniques are ideal nonintrusive instruments because, in most applications, the ultrasonic transducers are simply clamped on the outside pipe wall. In this section, we describe two ultrasonic flowmeters based on the Doppler and cross-correlation methods. Both require an inherent flow tag thus both are directly applicable to solid/liquid flows because of the presence of solid particles. Both flowmeters measure mainly particle velocity liquid-phase velocity, if different from the particle velocity, is not determined. 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

One should realize that these calculations are based on an expression for Vr which corresponds to potential flow past a stationary nonde-formable bubble, as seen by an observer in a stationary reference frame. However, this analysis rigorously requires the radial velocity profile for potential flow in the Uquid phase as a nondeformable bubble rises through an incompressible liquid that is stationary far from the bubble. When submerged objects are in motion, it is important to use liquid-phase velocity components that are referenced to the motion of the interface for boundary layer mass transfer analysis. This is accomplished best by solving the flow problem in a body-fixed reference frame which translates and, if necessary, rotates with the bubble such that the center of the bubble and the origin of the coordinate system are coincident. Now the problem is equivalent to one where an ideal fluid impinges on a stationary nondeformable gas bubble of radius R. As illustrated above, results for the latter problem have been employed to estimate the maximum error associated with the neglect of curvature in the radial term of the equation of continuity. 

Liquid-phase velocity ySMB yXMB ySMB ... 

In most cases, the models start out from coarse assumptions (e.g. plug flow or straight-line flow) and do not consider the spatial and time-line changes of the gas and liquid phase velocities and the different holdups. 

The liquid phase velocities in a laboratory scale bubble column of 140 mm in diameter have been measured with a constant temperature anemometer. Data acquisition and analysis have been done by means of a modern process computer. 

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