Bubble drag force

There are three types of closure models in CFD simulation of gas—hquid flow in bubble columns, i.e., drag force, bubble-induced turbulence, and kernel functions of bubble breakup and coalescence. We will show how we utilize the EMMS approach to derive new models and integrate them into CFD simulation. 

The term mist generally refers to liquid droplets from submicron size to about 10 /xm. If the diameter exceeds 10 /xm, the aerosol is usually referred to as a spray or simply as droplets. Mists tend to be spherical because of their surface tension and are usually formed by nucleation and the condensation of vapors (6). Larger droplets are formed by bursting of bubbles, by entrainment from surfaces, by spray nozzles, or by splash-type liquid distributors. The large droplets tend to be elongated relative to their direchon of mohon because of the action of drag forces on the drops. 

Entrainment from fluidized beds is also affected by temperature and pressure. Increasing system pressure increases the amount of solids carried over with the exit gas because the drag force on the particles increases at higher gas densities. May and Russell (1953) and Chan and Knowlton (1984) both found that pressure increased the entrainment rate from bubbling fluidized beds significantly. The data of Chan and Knowlton are shown in Fig. 13. 

Note that the lubrication effect due to particle collisions in liquid is significant. The liquid layer dynamics pertaining to the lubrication effect was examined by Zenit and Hunt (1999). Zhang et al. (1999) used a Lattice-Boltzmann (LB) simulation to account for a close-range particle collision effect and developed a correction factor for the drag force for close-range collisions, or the lubrication effect. Such a term has been incorporated in a 2-D simulation based on the VOF method (Li et al., 1999). Equation (36) does not consider the lubrication effect. Clearly, this is a crude assumption. However, in the three-phase flow simulation, this study is intended to simulate only the dilute solids suspension condition (ep = 0.42-3.4%) with the bubble flow time of less than 1 s starting when bubbles are introduced to the solids suspension at a prescribed ep. 

In previous work, we have mainly used the DPM model to investigate the effects of the coefficient of normal restitution and the drag force on the formation of bubbles in fluidized beds (Hoomans et al., 1996 Li and Kuipers, 2003, 2005 Bokkers et al., 2004 Van der Floef et al., 2004), and not so much to obtain information on the constitutive relations that are used in the TFMs. In this section, however, we want to present some recent results from the DPM model on the excess compressibility of the solids phase, which is a key quantity in the constitutive equations as derived from the KTGF (see Section IV.D.). The excess compressibility y can be obtained from the simulation by use of the virial theorem (Allen and Tildesley, 1990). 

Thus, if the liquid is highly viscous, then the drag force is so predominant that the bubble volume is highly sensitive to viscosity. Again, if the viscosity and the flow rate are both small, then variations in the viscosity do not affect the bubble volume appreciably. 

These authors have assumed the bubble to be expanding at the orifice, and have used the force balance equation at the time of detachment. The various forces considered by these authors are buoyancy, force due to the addition of mass (P2), excess pressure force, surface tension force, drag force, and force due to the inertia of the liquid. 

Assuming the bubbles to be spherical, the frontal area Af of the bubble becomes nd2/4. Further, by making use of the expression v = Qjnd2, the drag force can be written in terms of Froude number, the drag coefficient, and the volume of the bubble. Thus... 

The reduction in bubble volume due to the continuous phase velocity is attributed to an extra upward drag force, which adds to the buoyancy. Thus,... 

If the continuous fluid has a net vertical velocity component, the additional drag causes earlier or later detachment and hence reduces or increases the volume of particle formed according to whether the drag force assists or impedes detachment. Significantly smaller bubbles or drops can be produced by causing the continuous fluid to flow cocurrently with the dispersed phase (Cl). 

Thakre SS, Joshi JB. CFD simulation of bubble-column reactors importance of drag-force formulation. Chem Eng Sci 1999 54 5055-5060. 

Unlike formation in a liquid the boundary of a fluidised bed bubble can only expand by gas flowing across it to produce the drag force that will cause the particles to move appropriately. During the time that a bubble grows to the size shown in Figure 9 the gas that produced it has advanced to fill the volume indicated by the outer broken line. The annular region above and around the bubble now contains an excess of gas and so the powder void-age must increase. This is unstable and as the bubble detaches and rises through the expanded dense phase the powder relaxes and and returns the excess gas to the bubble. This appears to be completed by the time it has risen about one diameter (of order 1/10 second) and thereafter is of constant volume until it coalesces. 

A height of three to four meters above the bed is required to allow solids entrained by the bubble wakes into the freeboard to return to the bed surface. The initial velocity of the solids that splash into the freeboard is 4 to 8 times the bubble rise velocity (100) and the freeboard height is determined by the kinetic energy of the larger particles for which drag forces are relatively unimportant. For a bubble rise velocity of 3.2 m/s (estimated from + 0.71 /gd (29) and a bubble diameter of... 

For the immobile case, if the pressure drop in the liquid phase across a bubble is largely due to the drag force in the film region, then... 

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