Falling film velocity profile

Pig. 4. Diffusion into a falling film (a) with flat velocity profile, and (b) with parabolic velocity profile. 

Another solution to Eq. (159) for the diffusion into a falling film with a flat velocity profile is obtained by taking into account the finite thickness of the liquid film 8 and its effect on the concentration profiles that is, the boundary conditions are taken to be... 

Next we consider a fluid flowing through a circular tube with material at the wall diffusing into the moving fluid. This situation is met with in the analysis of the mass transfer to the upward-moving gas stream in wetted-wall-tower experiments. Just as in the discussion of absorption in falling films, we consider mass transfer to a fluid moving with a constant velocity profile and also flow with a parabolic (Poiseuille) profile (see Fig. 5). 

A knowledge of the velocity profiles within falling films under various flow conditions would be of very great value, making it possible to calculate the rates of convective heat and mass transfer processes in flowing films without the need for the simplified models which must be used at present. For instance, the analyses of Hatta (H3, H4) and Vyazovov (V8, V9) indicate clearly the differences in the theoretical mass-transfer rates due to the assumption of linear or semiparabolic velocity profiles in smooth... 

Grimley (G10, Gil) used an ultramicroscope technique to determine the velocities of colloidal particles suspended in a falling film of tap water. It was assumed that the particles moved with the local liquid velocity, so that, by observing the velocities of particles at different distances from the wall, a complete velocity profile could be obtained. These results indicated that the velocity did not follow the semiparabolic pattern predicted by Eq. (11) instead, the maximum velocity occurred a short distance below the free surface, while nearer the wall the experimental results were lower than those given by Eq. (11). It was found, however, that the velocity profile approached the theoretical shape when surface-active material was added and the waves were damped out, and, in the light of later results, it seems probable that the discrepancies in the presence of wavy flow are due to the inclusion of the fluctuating wavy velocities near the free surface. 

For specificity let us first consider the soluble wall problem sketched in Fig. 4.2.3. Now in a channel (or pipe) flow u = 0, and near the surface the streamwise velocity component is given by u (lylh). This behavior of the velocity profile is the same as for a fully developed thin liquid film on a vertical wall, falling under gravity with a free surface at atmospheric pressure. The velocity profile is parabolic with the fall velocity and has a maximum at the free surface equal to... 

Figure 3.43. G L model reactors for systematic process kinetic analyses, (a) Falling liquid film with thickness Si. (b) Falling liquid jet. (c) Thin-layer reactor with S = penetration depth of gas, being always higher than (5l (see A. Moser, 1973c). In all three cases the velocity profile of liquid flow is simple enough to allow the application of simple mass transfer theories.

Flow of a falling film [6]. The velocity (length/time) profile of a fluid in an inclined flat surface can be expressed as... 

Velocity Profile in Falling Film and Differential Momentum Balance. A Newto-... 

Diffusion in a laminar falling film. In Section 2.9C we derived the equation for the velocity profile in a falling film shown in Fig. 7.3-la. We will consider mass transfer of solute A into a laminar falling film, which is important in wetted-wall columns, in developing theories to explain mass transfer in stagnant pockets of fluids, and in turbulent mass transfer. The solute A in the gas is absorbed at the interface and then diffuses a distance into the liquid so that it has not penetrated the whole distance x = <5 at the wall. At steady state the inlet concentration = 0. At a point z distance from the inlet the concentration profile of is shown in Fig. 7.3- la. 

Figure 7.3-1. Diffusion of solute A in a laminar falling film (a) velocity profile and concentration profile, (h) small element for mass balance.

Diffusion into a falling film (see Figure 7.6)[8]. Consider the diffusion of a solute A into a moving liquid film B. The liquid is in laminar flow. Assuming that (1) the film moves with a flat velocity profile vo, (2) the film may be... 

The basic model system (9) for vertically falling film are applied. It is easy to generalize this equations for slopped falling films with tangential force r acting along the free surface. For the velocity profile u x,y,t) and self-induced pressure p x,yj) we have... 

Here, is the maximum velocity of the laminar velocity profile of the falling liquid film. This equation still prevails, for the situation depicted in Figure 6.2 when we have appreciable bulk concentration of [A], the solution becomes... 

A wetted wall tower is a piece of process equipment that uses a liquid film of S meters thickness in laminar flow in the axial (Z direction). Find the velocity profile in the falling film. 

Solution. In this example we will illustrate the use of the microscopic balances in solving mass transfer problems. Figure 4.6 shows the falling polymer film, its velocity profile, the concentration of the dye, and a microscopic element dx dz. It is evident from this figure that dye is transferred in both the X and z directions, so that a molar balance on the differential element yields... 

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