Mass transfer stagnant-film model

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] ... 

A stagnant film model is used for two-phase boundaries (1-2), which in effect, isolates the mass transfer process to a thin region at the interface stagnant film. Once the expressions for entropy production in terms of pressure, temperature, and composition are available a transformation is made to process variables such as reflux ratio, column height, packing or tray geometry, column diameter and column efficiency. Results of this design optimization model are compared with the results obtained via traditional methods. 

In this stagnant film model, we consider all the resistance to mass transfer to be lumped into the thickness 5. The reciprocal of the mass transfer coefficient can be thought of as this resistance... 

Four of the simplest and best known of the theories of mass transfer from flowing streams are (1) the stagnant-film model, (2) the penetration model, (3) the surface-renewal model, and (4) the turbulent boundary-layer model... 

Mass transfer, an important phenomenon in science and engineering, refers to the motion of molecules driven by some form of potential. In a majority of industrial applications, an activity or concentration gradient serves to drive the mass transfer between two phases across an interface. This is of particular importance in most separation processes and phase transfer catalyzed reactions. The flux equations are analogous to Ohm s law and the ratio of the chemical potential to the flux represents a resistance. Based on the stagnant-film model. Whitman and Lewis [25,26] first proposed the two-film theory, which stated that the overall resistance was the sum of the two individual resistances on the two sides. It was assumed in this theory that there was no resistance to transport at the actual interface, i.e., within the distance corresponding to molecular mean free paths in the two phases on either side of the interface. This argument was equivalent to assuming that two phases were in equilibrium at the actual points of contact at the interface. Two individual mass transfer coefficients (Ld and L(-n) and an overall mass transfer coefficient (k. ) could be defined by the steady-state flux equations ... 

Let us consider the gas-liquid mass transfer. In the stagnant film model, it is postulated that mass transfer proceeds via steady-state molecular diffusion in a hypothetical stagnant film at the interface with thickness while the bulk of the... 

The application of suitable models to various systems must be determined on a case-by-case basis. This could be judged from the behavior of experimental mass transfer coefficient with respect to the contact time of two phases. For dynamic systems, the penetration model is physically more realistic than the stagnant film model. Flowever, the mixing in different phases is important to describe the overall mass transfer performance, and, therefore, the above models are usually combined with fluid flow models, which includes detailed flow description. 

The gas-phase mass transfer is described by the stagnant film model whilst for the liquid phase Higbie s penetration model was used. The process of diffusion and simultaneous reaction in the liquid-phase penetration zone is given by the following balances [4] ... 

Figure4.4.1 Two-film concept (stagnant film model) for gas-liquid mass transfer of component A.

The solution of such an equation for an actual membrane device for ultrafiltration is difficult to obtain (see Zeman and Zydney (1996) for background information). One therefore usually falls back on the stagnant film model for determining the relation between the solvent flux and the concentration profile (see result (6.3.142b)). To use this result, we need to estimate the mass-transfer coefficient kit = Dit/dt), for the protein/macromolecule. One can focus on the entrance region of the concentration boundary layer, assume to be constant for a dilute solution, V = V, Vj, = 0 in the thin boundary layer, v = y ,y (where is the wall shear rate of magnitude AVz/Ay ) and obtain the result known as the Leveque solution at any location z in terms of the Sherwood number ... 

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 ... 

This approach to the problem is purely theoretical, since this model is based on a characteristic stagnant film thickness which is difficult to estimate or to measure. In addition, this model does not give any information as to the value of A, which must be determined separately by some other method. As a result, it is impossible to estimate the total mass-transfer rate in the disperser with the aid of this model only. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

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 ... 

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 ... 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness <5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

First, we must consider a gas-liquid system separated by an interface. When the thermodynamic equilibrium concentration is not reached for a transferable solute A in the gas phase, a concentration gradient is established between the two phases, and this will create a mass transfer flow of A from the gas phase to the liquid phase. This is described by the two-film model proposed by W. G. Whitman, where interphase mass transfer is ensured by diffusion of the solute through two stagnant layers of thickness <5G and <5L on both sides of the interface (Fig. 45.1) [1—4]. 

The mass and heat transport model should be able to predict mass and energy fluxes through a gas/vapour-liquid interface in case a chemical reaction occurs in the liquid phase. In this study the film model will be adopted which postulates the existence of a well-mixed bulk and a stagnant transfer zone near the interface (see Fig. 1). The equations describing the mass and heat fluxes play an important role in our model and will be presented subsequently. 

A conservative estimate of the disc s mass transfer performance may be obtained from the Nusselt model, assuming that there is no film mixing as it proceeds to the edge of the disc. For unsteady diffusion into a finite stagnant slab, the plot shown in Figure 8 from (8) gives the relative concentration distribution within the slab at various times, with a zero initial concentration and a surface concentration C0 imposed at time t = 0. The parameter on the curves is the Fourier number, Fo, where... 

In the two-film model (Figure 13), it is assumed that all of the resistance to mass transfer is concentrated in thin stagnant films adjacent to the phase interface and that transfer occurs within these films by steady-state molecular diffusion alone. Outside the films, in the bulk fluid phases, the level of mixing is so high that there is no composition gradient at all. This means that in the film region, only one-dimensional diffusion transport normal to the interface takes place. 

In most common separation processes, the main mass transfer is across an interface between a gas and a liquid or between two liquid phases. At fluid-fluid interfaces, turbulence may persist to the interface. A simple theoretical model for turbulent mass transfer to or from a fluid-phase boundary was suggested in 1904 by Nernst, who postulated that the entire resistance to mass transfer in a given turbulent phase lies in a thin, stagnant region of that phase at the interface, called a him, hence the name film theory.2 4,5 Other, more detailed, theories for describing the mass transfer through a fluid-fluid interface exist, such as the penetration theory.1,4... 

The transfer of mass from the bulk of a fluid phase to the external surroundings can be described by the so-called film model, see Fig. 7.6. The film model assumes the existence of a stagnant layer of thickness 8 along the external surface of the... 

Gas-liquid interfacial areas, a, and volumetric liquid-side mass transfer coefficients, kLa, are measured in a high pressure trickle-bed reactor. Increase of a and kLa with pressure is explained by the formation of tiny bubbles in the trickling liquid film. By applying Taylor s theory, a model relating the increase in a with the increase in gas hold-up, is developed. The model accounts satisfactorily for the available experimental data. To estimate kLa, contribution due to bubbles in the liquid film has to be added to the corresponding value measured at atmospheric pressure. The mass transfer coefficient from the bubbles to the liquid is calculated as if the bubbles were in a stagnant medium. 

A two-parameter model is obtained if wetting is incomplete (tice < 1) > but the inactively wetted surface is assumed to have negligible mass transfer resistance (Bi - °°). This latter condition was used by Mata and Smith (13) and physically corresponds to the inactively wetted area being dry, or to the presence of stagnant liquid film which is at equilibrium with the gaseous reactant. The expression for the conversion given by Equation 10 reduces to ... 

The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k . For two-film theory, n = 1 as discussed above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460-1467 (1951)] allow for surface renewal or penetration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0 the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with internal circulation, or as oscillating drops. These theories are discussed by Skelland [ Tnterphase Mass Transfer, Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)]. 

---