Stagnant film diffusion

Equimolar Counterdiffusion. Just as unidirectional diffusion through stagnant films represents the situation in an ideally simple gas absorption process, equimolar counterdiffusion prevails as another special case in ideal distillation columns. In this case, the total molar flows and are constant, and the mass balance is given by equation 35. As shown eadier, noj/g factors have to be included in the derivation and the height of the packing is... 

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. 

There have been many modifications of this idealized model to account for variables such as the freezing rate and the degree of mix-ingin the liquid phase. For example, Burton et al. [J. Chem. Phy.s., 21, 1987 (1953)] reasoned that the solid rejects solute faster than it can diffuse into the bulk liquid. They proposed that the effect of the freezing rate and stirring could be explained hy the diffusion of solute through a stagnant film next to the solid interface. Their theoiy resulted in an expression for an effective distribution coefficient k f which could be used in Eq. (22-2) instead of k. 

The partial pressures in the rate equations are those in the vicinity of the catalyst surface. In the presence of diffusional resistance, in the steady state the rate of diffusion through the stagnant film equals the rate of chemical reaction. For the reaction A -1- B C -1-. . . , with rate of diffusion of A limited. 

At a particular location in a distillation column, where the temperature is 350 K and the pressure 500 m Hg, the tnol fraction of the more volatile component in the vapour is 0.7 at the interface with the liquid and 0.5 in the bulk of the vapour. The molar latent heat of the more volatile component is 1.5 times that of the less volatile. Calculate the mass transferrates (kmol m s-11 of the two components. The resistance to mass transfer in the vapour may be considered to lie in a stagnant film of thickness 0.5 mm at the interface. The diffusivity in the vapour mixture is 2 x )() ° mV. ... 

This treatment may be compared with that given in Chapter 4. The top of the stagnant film is assumed to have a gas concentration in equilibrium with the overlying air (i.e., Cg = fCnTg). The unknown values are the flux and the thickness of the diffusive layer 2. The thickness 2 has been determined by analyses of isotopes and Rn) that can be used to obtain the flux (Broecker and Peng, 1974 Peng et al., 1979). The... 

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 steady-state continuity equations which describe mass balance over a fluid volume element for the species in the stagnant film which are subject to uniaxial diffusion and reaction in the z direction are... 

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 ... 

Mathematical approaches used to describe micelle-facilitated dissolution include film equilibrium and reaction plane models. The film equilibrium model assumes simultaneous diffusive transport of the drug and micelle in equilibrium within a common stagnant film at the surface of the solid as shown in Figure 7. The reaction plane approach has also been applied to micelle-facilitated dissolution and has the advantage of including a convective component in the transport analysis. While both models adequately predict micelle-facilitated dissolution, the scientific community perceives the film equilibrium model to be more mathematically tractable, so this model has found greater use. 

Two-film theory (Lewis and Whitman, 1924) the theory is based on molecular diffusion through two stagnant films, a liquid and a gas film, at the air-water interface. 

External diffusion of products. The last step is the diffusion of product(s) through the stagnant film into the bulk gas under conditions similar to those in step 1. 

One can roughly estimate the effects of gas-phase diffusion at steady state using a simple ID diffusion model, which has been employed (in some form) by numerous workers. 343 This approach yields the following expression for the linearized steady-state chemical resistance due to binary diffusion of O2 in a stagnant film of thickness... 

When a biocatalyst is immobilized on or within a solid matrix, mass transfer effects may exist because the substrate must diffuse from the bulk solution to the immobilized biocatalyst. If the biocatalyst is attached to non-porous supports there are only external mass transfer effects on the catalytically active outer surface in the reaction solution, the supports are surrounded by a stagnant film and substrate and product are transported across this Nemst layer by diffusion. The driving force for this diffusion is the concentration difference between the surface and the bulk concentration of substrate and product. 

Clearly, the elimination of the unknown concentration Cs between Equations (27), (28), and (29-31) is difficult. However, since the effective diffusion coefficient within the pores of carbon is considerably smaller than the free diffusion coefficient in the stagnant film (109) and since the thickness of the stagnant film is usually much smaller than R, it can be assumed that for large specimens the reaction in the solid will be mainly in Zone II lief ore (Cg — Cr) becomes appreciable. Therefore, at low rates of reaction... 

Relative viscosities are calculated from viscosities for the individual components at 0° (II7), weighting them on a mole fraction basis. The change in diffusivities and viscosities with temperature and pressure is assumed to be independent of gas mixture. If desired, more accurate calculations of diffusivities and viscosities of gas mixtures can be made using the approaches of Wilke (IIS) and Bromley and Wilke (II0), respectively. Table V presents relative values for Dfree, m, and p across the stagnant film for the gas-carbon reactions. Substituting these values in Equation (42), the relative reaction rates in Zone III for the gas-carbon reactions are calculated and also presented in Table V. Qualitatively, the rates of the carbon-oxygen and carbon-steam reactions are predicted to be about twice the rate... 

Consider the steady state ideal-gas diffusion of A through a stagnant film of B surrounding a sphere of radius ri (as in the preceding example). The temperature is assumed to vary according to... 

The example furthermore shows that diffusion from the bulk fluid phase toward the volume near the IRE, which is probed by the evanescent field, has to be accounted for because it may be the limiting step when fast processes are investigated. The importance of diffusion is more pronounced when a catalyst layer is present on the IRE, because of the diffusion in the porous film is much slower than that in the stagnant liquid film. Indeed, the ATR method, because of the measurement geometry, is ideally suited to characterization of diffusion within films (50,66-68). Figure 16 shows the time dependence of absorption signals associated with cyclohexene (top) and i-butyl hydroperoxide (TBHP, bottom). Solutions (with concentrations of 3mmol/L) of the two molecules in cyclohexane and neat cyclohexane were alternately admitted once to... 

Diffusion through a stagnant film, as in absorption or stripping processes involving the transfer of a single component between liquid and vapor phases. Since there is a concentration gradient... 

Figure 13.44. Factors in Eqs. (13.239) and (13.240) for HTUs of liquid and vapor films and slopes m and m" of the combining Eqs. (13.235) and (13.236) [Bolles and Fair, Inst. Chem. Eng. Symp. Ser. 56(2), 3.3/3.S, (1979)]. (a) Definitions of slopes m and m" in Eqs. (13.235) and (13.236) for combining liquid and gas film HTUs / = 1 for equimolal counter diffusion / = (jtB)mean for diffusion through a stagnant film, (b) Factor (j> of the liquid phase Eq. (13.239). (c) Factor C of the liquid phase, Eq. (13.239). (d) Factor ip of the gas phase, Eq. (13.240), for metal pall rings.

There is diffusion of salt away from both the solid-liquid interface and the vapor-liquid interface, in each case toward the brine. Water moves counterflow to the salt. Heat must transfer from solid to liquid to gas through stagnant films at the solid surface and through the turbulent liquid. An additional resistance to the formation of ice exists at the ice surface, where water molecules must orient themselves and find positions of low energy before being incorporated into the crystal lattice. When inadequate ice surface or foreign particles exist in the freezer, nucleation may control or affect the rate of ice production. 

An enzyme is immobilized by copolymerization technique. The diameter of the spherical particle is 2 mm and the number density of the particles in a substrate solution is 10,000/L. Initial concentration of substrate is 0.1 mole/L. A substrate catalyzed by the enzyme can be adequately represented by the first-order reaction with k0 = 0.002 mol/Ls. It has been found that both external and internal mass-transfer resistance are significant for this immobilized enzyme. The mass-transfer coefficient at the stagnant film around the particle is about 0.02 cm/s and the diffusivity of the substrate in the particle is 5 x 10-6 cm2/s. 

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 the absence of a diffusion-barrier films on the steel surface the corrosion current density, i (A/cm2) of steel in stagnant air-saturated water is given by ... 

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