Film Theories

Film theory goes back to work by Lewis and Whitman [1.27] from 1924. In order to explain the principles we will assume that a substance A is transferred from a quiescent solid or liquid surface, shown as a hat plate in Fig. 1.48, to flowing fluid B. The concentration of A drops from cA0 at the plate surface to cAi in the fluid. Film theory comes from the assumption that mass transfer takes place in a thin him of thickness S near the wall, hence the name. Concentration and velocity should only change in the y direction, but not, as is further assumed, with time or in any other coordinate direction. In steady how this results in a constant molar flux hA = cAwA of A being transferred in the y direction. If this were not the case, more A would how into a volume element of the fluid than out of it, and therefore the concentration of substance A would change with the time, or material A could also be howing in the x direction which would therefore cause a concentration difference in another coordinate direction. However neither of these scenarios are admissible in terms of the prerequisites for the application of him theory. Therefore according to him theory 

As the film thickness 8 is not normally known, the mass transfer coefficient / , cannot be calculated from this equation. However the values for the cases used most often in practice can be found from the relevant literature (i.e. [1.23] to [1.26]) which then allows the film thickness to be approximated using (1.189). In film theory the mass transfer coefficient / for vanishing convection flux h( — 0 is proportional to the diffusion coefficient D. 

A different result is obtained when a finite convective flux h is permitted. As before 

The concentration profile for vanishing convection h — 0 as given in (1.187) is obtained from (1.190). A Taylor series of the exponential function at h = 0 can be developed to indicate this. The material flux transferred to the surface y = 0 (index 0 = wall) in the y-direction is 

According to film theory this is constant and equal to the value at the wall, where hA = hAo and n = h0. Differentiating (1.190) and introducing the result into (1.191) yields 

The stagnant film theory was developed by Nemst in 1904. Because no flow can exist at the solid interface, a stagnant or laminar film of fluid (i.e., hydraulic film) of thickness, 5, was assumed to exist, and beyond this film the fluid was assumed to be turbulent. 

For mass transfer the effective film (i.e., concentration film) thickness is 8c, in meters, and the chemical flux is regulated by molecular diffusion. Being highly turbulent, the fluid beyond is rapidly mixed and providing no resistance to mass transfer. 

Based on these assumptions the final conceptual model equation is 

The advantage of nsing Equation 6.13 for the mass transfer coefficient lies in the assumption that under similar hydrodynamic conditions, the film thickness 5 is constant, so that the flux across the film, on the one hand, is equal to the convective flux, with a mass transfer coefficient of k l, and, on the other hand, is equal to a diffusive flux around a linear concentration gradient given by 

Astarita (1967) has combined the Higbie and Danckwerts approaches to give the following general equation for the mass transfer coefficient  

The total flux at the gas-liquid interface, indicated by then is equal to= 0. The total flux at the interface given by Equation 6.13 is a point value, when averaged over x  

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Films, anodic oxide Films, passivating Films, plastic Film theory Film wrappers Filter Filter aid Filter aids Filter fabrics Filtering centrifuges Filter media Filters... 

Twin-fluid atomizer Twisted pair cable Twitchell splitting Twitchell s reagents Two-film theory... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

Neither the penetration nor the surface renewal theory can be used to predict mass transfer coefficients directiy because T and s are not normally known. Each suggests, however, that mass transfer coefficients should vary as the square root of the molecular diffusivity, as opposed to the first power suggested by the film theory. 

Equation 39 can often be simplified by adopting the concept of a mass transfer unit. As explained in the film theory discussion eadier, the purpose of selecting equation 27 as a rate equation is that is independent of concentration. This is also tme for the Gj /k aP term in equation 39. In many practical instances, this expression is fairly independent of both pressure and Gj as increases through the tower, increases also, nearly compensating for the variations in Gj. Thus this term is often effectively constant and can be removed from the integral ... 

Fig. 2. Schematic representation for the two-film theory of gas transfer = partial pressure of gas Pj = partial pressure of the gas at the interface Cj = concentration of gas at time t, Cj = initial concentration of gas at the interface Cg = initial concentration of gas at t = 0 and S = gas saturation.

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

Several theories have appeared in the Hterature regarding the mechanism of protection by -PDA antiozonants. The scavenger theory states that the antiozonant diffuses to the surface and preferentially reacts with ozone, with the result that the mbber is not attacked until the antiozonant is exhausted (25,28,29). The protective film theory is similar, except that the ozone—antiozonant reaction products form a film on the surface that prevents attack (28). The relinking theory states that the antiozonant prevents scission of the ozonized mbber or recombines severed double bonds (14). A fourth theory states that the antiozonant reacts with the ozonized mbber or carbonyl oxide (3) in Pig. 1) to give a low molecular weight, inert self-healing film on the surface (3). 

Penetration theoiy often is used in analyzing absorption with chemical reaction because it makes no assumption about the depths of penetration of the various reacting species, and it gives a more accurate result when the diffusion coefficients of the reacting species are not equal. When the reaction process is veiy complex, however, penetration theoiy is more difficult to use than film theory, and the latter method normally is preferred. 

Effects of Total Pressure on Uq and The influence of total system pressure on the rate of mass transfer from a gas to a licniid or to a solid has been shown to be the same as would be predicted from stagnant-film theory as defined in Eq. (5-285), where... 

The parameter values for the curves of Fig. 14-14 originally were defineci from film theory as (Dg/D )(B /vCi) but later were refined by the results of penetration theory to the definition (( ) — 1), where... 

F = Function of the molecular volume of the solute. Correlations for this parameter are given in Figure 7 as a function of the parameter (j), which is an empirical constant that depends on the solvent characteristics. As points of reference for water, (j) = 1.0 for methanol, (j) = 0.82 and for benzene, (j) = 0.70. The two-film theory is convenient for describing gas-liquid mass transfer where the pollutant solute is considered to be continuously diffusing through the gas and liquid films. 

Consider a vessel containing an agitated liquid. Heat transfer occurs mainly through forced convection in the liquid, conduction through the vessel wall, and forced convection in the jacket media. The heat flow may be based on the basic film theory equation and can be expressed by... 

There are various theories on how passive films are formed however, there are two commonly accepted theories. One theory is called the oxide film theory and states that the passive film is a diffusion-barrier layer of reaction products (i.e., metal oxides or other compounds). The barriers separate the metal from the hostile environment and thereby slow the rate of reaction. Another theory is the adsorption theory of passivity. This states that the film is simply adsorbed gas that forms a barrier to diffusion of metal ions from the substrata. 

Because of the difficulties in determining x, the thickness of the film between the two vapor pressures, an overall transfer coefficient is introduced. Based on the two film theory, the overall transfer coefficient is used. In the case of water evaporation, the gas film is the controlling mechanism and the resulting equation is... 

In view of the fact that there are two opposing views on the mechanism of passivity it is not surprising that a similar situation prevails concerning the mechanism of breakdown of passivity. The solid film theory of passivity and breakdown of passivity is dealt with in some detail in Section 1.5, so that it is appropriate here to discuss briefly the views based on the adsorption theory. 

Based on film theory, the oxygen flux in the gas film is equal to flux in the liquid film ... 

The simplest theory involved in mass transfer across an interface is film theory, as shown in Figure 3.10. In this model, the gas (CO) is transferred from the gas phase into the liquid phase and it must reach the surface of the growing cells. The rate equation for this case is similar to the slurry reactor as mentioned in Levenspiel.20... 

Increase in mass-transfer rate per unit area. As stated above, agitated gas-liquid contactors are used, in general, when it is necessary to deal with sparingly soluble gases. According to the terminology of the film theory, absorption is then controlled by the liquid resistance, and agitation of the liquid phase could increase the mass-transfer rate per unit area. As will be... 

In evaluating their results they assumed the film theory, and, because the oxygen is sparingly soluble and the chemical reaction rate high, they also assumed that the liquid film is the controlling resistance. The results were calculated as a volumetric mass-transfer coefficient based, however, on the gas film. They found that the volumetric mass-transfer coefficient increased with power input and superficial gas velocity. Their results can be expressed as follows ... 

The penetration theory holds for the region where t is much less than L2jD, the film theory for the region where t is much greater than L2/D. This comparison is shown in Fig. 8, which clearly shows that the film and penetration theories are asymptotes of the film-penetration model. 

The two-film theory of WHITMAN119 was the first serious attempt to represent conditions occurring when material is transferred from one fluid stream to another. Although it does not closely reproduce the conditions in most practical equipment, the theory gives expressions which can be applied to the experimental data which are generally available, and for that reason it is still extensively used. 

From equation 10.22 the rate of transfer per unit area in terms of the two-film theory for equimolecular counterdiffusion is given for the first phase as ... 

If it is assumed that each element resides for the same time interval te in the surface, equation 10.115 gives the overall mean rate of transfer. It may be noted that the rate is a linear- function of the driving force expressed as a concentration difference, as in the two-film theory, but that it is proportional to the diffusivity raised to the power of 0.5 instead of unity. 

Thus either the penetration theory or the film theory (equation 10.144 or 10.145) respectively can be used to describe the mass transfer process. The error will not exceed some 9 per cent provided that the appropriate equation is used, equation 10.144 for L2 jDt > n and equation 10.145 for L2/Dt < n. Equation 10.145 will frequently apply quite closely in a wetted-wall column or in a packed tower with large packings. Equation 10.144 will apply when one of the phases is dispersed in the form of droplets, as in a spray tower, or in a packed tower with small packing elements. 

When the film theory is applicable to each phase (the two-film theory), the process is steady state throughout and the interface composition does not then vary with time. For this case the two film coefficients can readily be combined. Because material does not accumulate at the interface, the mass transfer rate on each side of the phase boundary will be the same and for two phases it follows that ... 

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