Film models

In a well-fluidized gas-solid system, the bulk of the bed can be approximated to be isothermal and hence to have negligible thermal resistance. This approximation indicates that the thermal resistance limiting the rate of heat transfer between the bed and the heating surface lies within a narrow gas layer at the heating surface. The film model for the fluidized bed heat transfer assumes that the heat is transferred only by conduction through the thin gas film or gas boundary layer adjacent to the heating surface. The effect of particles is to erode the film and reduce its resistive effect, as shown by Fig. 12.3. The heat transfer coefficient in the film model can be expressed as 

The model based on the concept of pure limiting film resistance involves the steady-state concept of the heat transfer process and omits the essential unsteady nature of the heat transfer phenomena observed in many gas-solid suspension systems. To take into account the unsteady heat transfer behavior and particle convection in fluidized beds, a surface renewal model can be used. The model accounts for the film resistance adjacent to the heat transfer 

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The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Jn.st. Chem. Eng., 31,365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theoiy. 

The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surface-renewal models. Thus, in view of its relative simphcity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for maldug a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-28. must be employed. 

Effects of High Solute Concentrations on Ug and As discussed previously, the stagnant-film model indicates that fcc should be independent of ysM and/cc should be inversely proportional to The data of Vivian and Behrman [Am. Tn.st. Chem. Eng. J., 11, 656 (1965)] for the absorption of ammonia from an inert gas strongly suggest that the film model s predicted trend is correct. This is another indication that the most appropriate rate coefficient to use is fcc. nd the proper driving-force term is of the form (y — yd ysM-... 

On tbe basis of tbe two-film model for mass transfer, and relating all efficiencies to gas-pbase concentrations (for convenience only a similar development can be made on tbe basis of bquid concentrations), point efficiency can be expressed in terms of transfer units ... 

Although our simple oxide film model explains most of the experimental observations we have mentioned, it does not explain the linear laws. How, for example, can a material lose weight linearly when it oxidises as is sometimes observed (see Fig. 21.2) Well, some oxides (e.g. M0O3, WO3) are very volatile. During oxidation of Mo and W at high temperature, the oxides evaporate as soon as they are formed, and offer no barrier at all to oxidation. Oxidation, therefore, proceeds at a rate that is independent of time, and the material loses weight because the oxide is lost. This behaviour explains the catastrophically rapid section loss of Mo and W shown in Table 21.2. 

A more rigorous seheme of gas-liquid ehemieal reaetion and absorption followed by preeipitation is deseribed based on the film model (Englezos etai, 1987a,b Waehi and Jones, 1990 1991a Skovborg and Rasmussen, 1994). The eoneept is illustrated in Figure 8.13. 

The data plotted in the figure clearly support the predicted positive dependence of crystal size on agitation rate. Precipitation in the crystal film both enhances mass transfer and depletes bulk solute concentration. Thus, in the clear film model plotted by broken lines, bulk crystal sizes are initially slightly smaller than those predicted by the crystal film model but quickly become much larger due to increased yield. Taken together, these data imply that while the initial mean crystal growth rate and mixing rate dependence of size are... 

Jss Flux obtained experimentally or from the film-model... 

A. Absorption in Agitated Liquid with Simultaneous Chemical Reaction—Film Model... 

Fig. 6. Film model for diffusion with simultaneous irreversible first-order chemical reaction [after Lightfoot (L5)].

In a process where mass transfer takes place across a phase boundary, the same theoretical approach can be applied to each of the phases, though it does not follow that the same theory is best applied to both phases. For example, the film model might be applicable to one phase and the penetration model to the other. This problem is discussed in the previous section. 

Using a steady-state film model, obtain an expression for the mass transfer rate across a laminar film of thickness /. in the vapour phase for the more volatile component in a binary distillation process ... 

The gas phase mass transfer coefficient for the absorption of ammonia into water from a mixture of composition NHj 20%, N2 73%, Hj 7% is found experimentally to be 0.030 m/s. What would you expect the transfer coefficient to be for a mixture of composition NH3 5%, N2 60%, Hj 35% All compositions are given on a molar basis. The total pressure and temperature are die same in both cases. The transfer coefficients are based on a steady-state film model and the effective film thickness may be assumed constant. Neglect the solubility of Ny and Hi in water. 

Surface Renewal Theory. The film model for interphase mass transfer envisions a stagnant film of liquid adjacent to the interface. A similar film may also exist on the gas side. These h5q>othetical films act like membranes and cause diffu-sional resistances to mass transfer. The concentration on the gas side of the liquid film is a that on the bulk liquid side is af, and concentrations within the film are governed by one-dimensional, steady-state diffusion ... 

We have undertaken a series of experiments Involving thin film models of such powdered transition metal catalysts (13,14). In this paper we present a brief review of the results we have obtained to date Involving platinum and rhodium deposited on thin films of tltanla, the latter prepared by oxidation of a tltanliua single crystal. These systems are prepared and characterized under well-controlled conditions. We have used thermal desorption spectroscopy (TDS), Auger electron spectroscopy (AES) and static secondary Ion mass spectrometry (SSIMS). Our results Illustrate the power of SSIMS In understanding the processes that take place during thermal treatment of these thin films. Thermal desorption spectroscopy Is used to characterize the adsorption and desorption of small molecules, In particular, carbon monoxide. AES confirms the SSIMS results and was used to verify the surface cleanliness of the films as they were prepared. 

GL 16] ]R 12] ]P 15] Using a simple thin-film model for mass transfer, values for the overall mass transfer coefficient were determined for both micro-channel processing and laboratory trickle-bed reactors [11]. The value for micro-reactor processing (fCL = 5-15 s ) exceeds the performance of the laboratory tool Ki a = 0.01-0.08 s ) [11, 12], However, more energy has to be spent for that purpose (see the next section). 

Usually, mass transfer in gas-liquid models is based on a film model (see Fig. 5.4-13). The rate of transfer of A from the gas bubble to the liquid, i.e. absorption of A in the liquid phase, equals ... 

It has been observed that under reaction conditions mass transfer is often significantly faster than would be expected based on the film model. This is modelled by introducing an enhancement factor, E. In case the concentration in the bulk liquid, ca, is zero, the rate of mass transfer of A now becomes ... 

For external mass transfer again a film model is usually applied ... 

This can be further integrated from the wall to the boundary layer thickness y = 8, where the component is at the bulk concentration Cj,. Substituting / = - o and k = D/o, the mass-transfer coefficient yields the stagnant film model [Brian, Desalination by Reverse Osmosis, Merten (ed.), M.I.T. Press, Cambridge, Mass., 1966, pp. 161-292] ... 

Lacy, C. E., A. E. Dukler, 1994, Flooding in Vertical Tubes—I Experimental Studies of the Entrance Region, Int. J. Multiphase Flow 20 219-233 Flooding in Vertical Tubes—II A Film Model for Entrance Region Flooding, Int. J. Multiphase Flow 20 235-247. (3)... 

Figure 1 Stagnant film model for dissolution of acidic drug (HA) in solution with base (B). (From Ref. 2.)...

The boundary conditions for this early dissolution model included saturated solubility for HA at the solid surface (Cha ) with sink conditions for both HA and A at the outer boundary of a stagnant film (Cha = Ca = 0). Since diffusion is the sole mechanism for mass transfer considered and the process occurs within a hypothesized stagnant film, these types of models are colloquially referred to as film models. Applying the simplifying assumption that the base concentration at the solid surface is negligible relative to the base concentration in the bulk solution (CB CB(o)), it is possible to derive a simplified scaled expression for the relative flux (N/N0) from HPWH s original expressions ... 

The derivation and experimental verification of the MMHS model represented a significant accomplishment and a natural plateau for film models. To be sure, there are general criticisms of film models and more specific criticisms of the MMHS model [6], However, overall the MMHS model should be recognized as a robust but simply applicable model which serves to demonstrate how factors such as intrinsic solubility of the acid drug, ionization and pA of the drug, and concentration of the reactive base all contribute to increasing the dissolution rate and mass transfer. 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

Other researchers used flow between two parallel plates as the experimental and theoretical system to incorporate diffusion plus convection into their dissolution modeling and avoid film model approximations [10]. Though they did not consider adding reactions to their model, these workers did show that convection was an important phenomenon to consider in the mass transfer process associated with solid dissolution. In fact, the dissolution rate was found to correlate with flow as... 

A paper by Ozturk, Palsson, and Dressman (OPD), reporting a refinement of the MMSH model, did create some controversy. OPD developed a film model with reaction in spherical coordinates and applied quasi-steady-state assumptions to the boundary conditions at the solid surface [11], They theorized that the flux of all species at the solid surface must be zero, except for HA, or the other species (A-, H+, OH ) would penetrate the solid surface. A debate by correspondence in the Letters to the Editor columns of Pharmaceutical Research ensued [12,13], The reader is invited to evaluate which author s arguments are more convincing. What is difficult to evaluate is whether the OPD model produces dissolution results which are different from those which would be predicted using the MMSH model cast in comparable spherical geometry. Simply, these authors never graphically demonstrate how their model predictions compare to the MMSH model. Algebraically, the solutions to both models appear comparable. 

Figure 7 Stagnant film model (top) and reaction plane model (bottom) for micelle-facilitated dissolution.

Dissolution film models, though conceptually simple and instructive, are inherently unsatisfying in their application to complex dissolution phenomena or systems. 

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