Fick molecular 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 ... 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

As shown in Fig. 1.12, diffusional flow contributions in engineering situations are usually expressed by Fick s Law for molecular diffusion... 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

Diffusion in solution is the process whereby ionic or molecular constituents move under the influence of their kinetic activity in the direction of their concentration gradient. The process of diffusion is often known as self-diffusion, molecular diffusion, or ionic diffusion. The mass of diffusing substance passing through a given cross section per unit time is proportional to the concentration gradient (Fick s first law). 

Hydrodynamic dispersion is in many cases taken to be a Fickian process, one whose transport law takes the form of Fick s law of molecular diffusion. If flow is along x only, so that vx = v and vy = 0, the dispersive fluxes (mol cm-2 s-1) along x and y for a component i are given by,... 

The steady-state transport of A through the stagnant gas film is by molecular diffusion, characterized by the molecular diffusivity DAg. The rate of transport, normalized to refer to unit area of interface, is given by Fick s law, equation 8.5-4, in the integrated form... 

Mass transfer by laminar (molecular) diffusion is directly analogous to conduction with the analog of Fourier s law as Fick s law describing the mass flux (mass flow rate per unit area) of species i due to diffusion ... 

Equation (2.19), which concerns a situation without processes in the biofilm, can be extended to include transformation of a substrate, an electron donor (organic matter) or an electron acceptor, e.g., dissolved oxygen. If the reaction rate is limited by j ust one substrate and under steady state conditions, i.e., a fixed concentration profile, the differential equation for the combined transport and substrate utilization following Monod kinetics is shown in Equation (2.20) and is illustrated in Figure 2.8. Equation (2.20) expresses that under steady state conditions, the molecular diffusion determined by Fick s second law is equal to the bacterial uptake of the substrate. 

Each film is in stagnant or laminar flow, such that mass transfer across the films is by a process of molecular diffusion and can therefore be described by Ficks Law. 

Ficks Law states that the flux j (mol/s m2) for molecular diffusion, for any given component is given by... 

We have used Fick s law of diffusion with separate molecular diffusivities for each species. However, most PDF models for molecular mixing do not include differential-diffusion effects. 

We will now describe the application of the two principal methods for considering mass transport, namely mass-transfer models and diffusion models, to PET polycondensation. Mass-transfer models group the mass-transfer resistances into one mass-transfer coefficient ktj, with a linear concentration term being added to the material balance of the volatile species. Diffusion models employ Fick s concept for molecular diffusion, i.e. J = — D,v ()c,/rdx, with J being the molar flux and D, j being the mutual diffusion coefficient. In this case, the second derivative of the concentration to x, DiFETd2Ci/dx2, is added to the material balance of the volatile species. 

It is of interest to consider first what is happening in pipe flow. Random molecular movement gives rise to a mixing process which can be described by Fick s law (given in Volume 1, Chapter 10). If concentration differences exist, the rate of transfer of a component is proportional to the product of the molecular diffusivity and the concentration gradient. If the fluid is in laminar flow, a parabolic velocity profile is set up over the cross-section and the fluid at the centre moves with twice the mean velocity in the pipe. This... 

The flux caused by molecular diffusion follows Fick s first law, which states that the diffusive flux (jf ) is directly proportional to the concentration gradient of the chemical undergoing net diffusive transport id C]Jdz). Thus, the larger the concentration difference, or gradient, the larger the flux. For a diffusive flux occurring in the vertical... 

The effect of turbulent mixing has been shown to follow the same behavior as molecular diffusion as previously shown in Eq. 3.6 (Fick s first law), where the diffusive flux, T (jiff(mol m s ), of a solute, C (mol/m ), is given by ... 

Since the mixing process involves a shuffling or redistribution of material either by slippage or eddies, and since this is repeated many, many times during the flow of fluid through the vessel we can consider these disturbances to be statistical in nature, somewhat as in molecular diffusion. For molecular diffusion in the x-direction the governing differential equation is given by Fick s law ... 

Fick s or molecular diffusion in large pores (DF), also called bulk diffusion ... 

One well-known example of the gradient-flux law is Fick s first law, which relates the diffusive flux of a chemical to its concentration gradient and to the molecular diffusion coefficient ... 

If diffusivity D is independent of x, which for molecular diffusion is usually the case, then dD/dx is zero and the second term on the far right-hand side drops. We are left with Fick s second law. 

Turbulent flow means that, superimposed on the large-scale flow field (e.g., the Gulf Stream), we find random velocity components along the flow (longitudinal turbulence) as well as perpendicular to the flow (transversal turbulence). The effect of the turbulent velocity component on the transport of a dissolved substance can be described by an expression which has the same form as Fick s first law (Eq. 18-6), where the molecular diffusion coefficient is replaced by the so-called turbulent or eddy diffusion coefficient, E. For instance, for transport along the x-axis ... 

When comparing Eqs. 19-1 and 19-3, the reader may remember the discussion in Chapter 18 on the two models of random motion. In fact, these equations have their counterparts in Eqs. 18-6 and 18-4. If the exact nature of the physical processes acting at the bottleneck boundary is not known, the transfer model (Eqs. 18-4 or 19-3) which is characterized by a single parameter, that is, the transfer velocity vb, is the more appropriate (or more honest ) one. In contrast, the model which started from Fick s first law (Eq. 19-1) contains more information since Eq. 19-4 lets us conclude that the ratio of the exchange velocities of two different substances at the same boundary is equal to the ratio of the diffusivities in the bottleneck since both substances encounter the same thickness 5. Obviously, the bottleneck model will serve as one candidate for describing the air-water interface (see Chapter 20). However, it will turn out that observed transfer velocities are usually not proportional to molecular diffusivity. This demonstrates that sometimes the simpler and less ambitious model is more appropriate. 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

The rate of mass transfer depends on the inlerfacial contact area and on the rate uf mass transfer per unit inlerfacial area, i.e.. the mass flux. Lie mass flux very close to ihe liquid-liquid interface is determined by molecular diffusion in accordance with Ficks First law ... 

Dispersion. The movement of aggregates of molecules under the influence of a gradient such as concentration, temperature, density, etc. The effect is represented by Fick s diffusion equation with a dispersion coefficient substituted for molecular diffusivity. Thus, Rate of Transfer = -DeOC/3z). 

Fick s law, which describes the steady-state bi-molecular diffusion, is the simplest and most used form. It is described in Section 3.2, but it is also reported below for the reader s convenience ... 

Fick s law is derived only for a binary mixture and then accounts for the interaction only between two species (the solvent and the solute). When the concentration of one species is much higher than the others (dilute mixture), Fick s law can still describe the molecular diffusion if the binary diffusion coefficient is replaced with an appropriate diffusion coefficient describing the diffusion of species i in the gas mixture (ordinary and, eventually, Knudsen, see below). However, the concentration of the different species may be such that all the species in the solution interact each other. When the Maxwell-Stefan expression is used, the diffusion of... 

Knudsen diffusion can be taken into account also when Fick s law or the Maxwell-Stefan rate of mass transport are employed [34, 59] by combining the molecular diffusion, DLm, and the Knudsen diffusion as follows ... 

Fick s diffusion law is used to describe dispersion. In a tubular reactor, either empty or packed, the depletion of the reactant and non-uniform flow velocity profiles result in concentration gradients, and thus dispersion in both axial and radial directions. Fick s law for molecular diffusion in the x-direction is defined by... 

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