Falling film diffusion

Relative Kga valid for all systems controlled by mass transfer coefficient (Kg) and wetted area (a) per unit volume of column. Some variation should be expected when liquid reaction rate is controlling (not liquid diffusion rate). In these cases liquid hold-up becomes more important. In general a packing having high liquid hold-up which is clearly greater than that in the falling film has poor capacity. 

Figure 12-13 A falling film catalytic wall reactor in which reactant in the gas must diffuse through a liquid film to react L...

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). 

Particles are transported by turbulent diffusion from the core region to the surface of the falling film, where they are trapped and are carried downward. The implication of this assumption is that the net upward particle flow will decrease along the riser height in the core. 

Ammonia is absorbed in a falling film of water in an absorption apparatus and the film is disrupted and mixed at regular intervals as it flows down the column. The mass transfer rate is calculated from the penetration theory on the assumption that all the relevant conditions apply. It is found from measurements that the mass transfer rate immediately before mixing is only 16 per cent of that calculated from the theory and the difference has been attributed to the existence of a surface film which remains intact and unaffected by the mixing process. If the liquid mixing process takes place every second, what thickness of surface film would account for the discrepancy Diffusivity of ammonia in water = 1.76 x 10 9 m2/s. 

Example 6.3 Diffusion of species from a gas mixture to a falling liquid film In a wetted wall column, the falling liquid film consists of a mixture of acetone (1) and benzene (2). This falling film is in contact with a downward flowing gas mixture of acetone, benzene, and helium (3). The pressure, temperature, and thickness of the film are P = 1 atm, T = 28.1°C, and 8 = 0.044 cm. The gas entering at the top of the column contains 10 mol% acetone and no benzene, and the composition of the vapor flow at the interface is... 

Zelvenskij et al. [75 a] labelled low-boiling components with the radionuclides C-14 and Cl-36 in order to measure diffusion and material transfer rates in falling-film distillations. It should be possible to test columns by similar methods [75]. 

Alvarez (26) measured by physical absorption (without reaction) in a wetted wall falling film absorber, the diffusivity of non reacting gas N2O in different aqueous and organic liquids and solutions at 20 C. These systems are given in Table (I). 

For the practical design of a diffusion separation unit a tube bundle apparatus consisting of vertical double pipes may be used. The outer pipes are heated while inner pipes are cooled. In the annulus, two falling films, separated by an air gap, flow downward. Special attention should be paid to the generation of the falling film, which... 

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.

Penetration theory. The penetration theory derived by Higbie and modified by Danckwerts (D3) was derived for diffusion or penetration into a laminar falling film for short contact times in Eq. (7.3-23) and is as follows ... 

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... 

Let us now consider the behavior of the small amplitude disturbances when the diffusion part of the general problem ( onsidered. The base state corresponds to the diffusion boundary layer near the surface of the falling film. The linear or nonlinear behavior... 

This set of parameter values fits well a liquid metal with 7 = 29.2, Ca = 0.2. The calculations of eigenvalues are completed for various values Ma, T, a. In Figure 10 typical curves for = Cr oi) and aci = aci a) are plotted. As amplification factor aci a) of various instability modes could differ to several orders, a normalized amplification factor f = maCi where m is appropriate scale is used in figures. One of these growing modes is easily identified as the hydrodynamic mode of the falling film with small wavenumber and is indeed the same when Ma = 0. The phase velocity Cr of this wave mode diminishes from Cr = 3 as a grows form a = 0, takes a minimum value, and then increases. Amplification factor aCi is positive in the interval 0 < o < a and has maximum value aci)m inside of this interval. Other growing modes, which are referred to as diffusion modes appear only at Ma 7 0. The term diffusion could be applied for any mode which disappears at... 

Veriable surface excess concentration. In general case F const of a falling film of weak volatile surfactant solution in which the surfactant mass transfer is governed by the diffusion, evaporation and adsorption-desorption processes in the near-surface layer the development of instability depends on nine dimensionless external parameters. We can take 7, 6, Cq, Pe, G, Bi, T, Di, and Ma as these independent parameters. If the parameters are given, the problem reduces to the numerical solution of the dispersion relation for various of the wave number a and the spectral analysis of cj = u a). 

The wavenumber interval on which new diffusion instability modes exist is bounded from below and not bounded from above. Consequently, these perturbations are mainly shortwave. At the same time, the formation of regular nonlinear wave structures in falling film of a pure liquid is associated with the fact that only perturbations on the bounded interval 0 < a < are unstable. 

Our discussion about the penetration theory will, first of all, be based on the concept of diffusion into a falling film. The problem is well known and solved in many texts of transport phenomena and diffusion (e.g., Cussler, 2001). We will not go into the details of the problem and the solution we will rather give the final result and the equation for the flux of the species at the interface for the coordinates shown in Figure 6.2 when the bulk concentration of [A] is negligible ... 

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