Concentration profiles stagnant film diffusion

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 ... 

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. 

Figure 5.1. Diffusion mechanisms and corresponding concentration profiles, where C is the overall solid concentration, r0 is the mean radius of the particle, r is the linear distance along the particle radius from the radius surface, and 5 is the thickness of the stagnant fluid film. [From Liberti and Passino (1983), with permission.]...

Figure 9.7. Concentration profiles of reactants and products for diffusion in a stagnant film with reversible homogeneous chemical reaction Das = 1.0, DaP = 0.5, y = 0 bold dashed line is for 0S gray dashed line is for 6P Das = 50.0, DaP = 40.0, 7 = 0 ...

A heterogeneous reaction A -> 2B with nth order kinetics. /rA = k( A (n > 0) takes place on a catalyst surface. The component A with initial concentration CA0 diffusses through a stagnant film on the catalyst surface at isothermal and isobaric conditions. Assume one-dimensional diffusion, and determine the concentration profile of component A within the film of thickness 8 if the k is constant. 

The mass transfer flux across the stagnant film can thus be described as a steady diffusion flux. It can be shown that within this steady-state process the mass flux will be constant as the concentration profile is linear and independent of the diffusion coefficient. 

In the previous sections, stagnant films were assumed to exist on each side of the interface, and the normal mass transfer coefficients were assumed proportional to the first power of the molecular diffusivity. In many mass transfer operations, the rate of transfer varies with only a fractional power of the diffusivity because of flow in the boundary layer or because of the short lifetime of surface elements. The penetration theory is a model for short contact times that has often been applied to mass transfer from bubbles, drops, or moving liquid films. The equations for unsteady-state diffusion show that the concentration profile near a newly created interface becomes less steep with time, and the average coefficient varies with the square root of (D/t) [4] ... 

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. 

In the film theory description of the mass-transfer process occurring between two fluid phases or between a solid and a fluid phase, the complex mass-transfer phenomenon is substituted by the notion of simple molecular diffusion of the species through a stagnant fluid fUm of thickness <5. The actual concentration profiles of species A being transferred from phase 2 to phase 1 are shown in Figures 3.1.6 (a) and (b) in one phase only for a solid-liquid and a gas-iiquid system, respectively. The concentration of A in the liquid phase at the solid-liquid or the gas-Uquid interface is C. Far away from the interface it is reduced to a low value in the liquid phase. In turbulent flow, the curved profile of species A shown would correspond to the time-averaged value (Bird et al, 1960, 2002). According to the... 

The fllm theory is the simplest model for interfacial mass transfer. In this case it is assumed that a stagnant fllm exists near the interface and that all resistance to the mass transfer resides in this fllm. The concentration differences occur in this film region only, whereas the rest of the bulk phase is perfectly mixed. The concentration at the depth I from the interface is equal to the bulk concentration. The mass transfer flux is thus assumed to be caused by molecular diffusion through a stagnant fllm essentially in the direction normal to the interface. It is further assumed that the interface has reached a state of thermodynamic equilibrium. The mass transfer flux across the stagnant film can thus be described as a steady diffusion flux. It can be shown that within this steady-state process the mass flux will be constant as the concentration profile is linear and independent of the diffusion coefficient. Consider a gas-liquid interface, as sketched in Fig. 5.16. The mathematical problem is to formulate and solve the diffusion flux equations determining the fluxes on both sides of the interface within the two films. The resulting concentration profiles and flux equations can be expressed as ... 

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