Steady Diffusion Across a Thin Film

His third effort was more successful. He used a glass cylinder containing crystalline sodium chloride in the bottom and a large volume of water in the top, shown as the lower apparatus in Fig. 2.1-3. By periodically changing the water in the top volume, he was able to establish a steady-state concentration gradient in the cylindrical cell. He found that this gradient was linear, as shown in Fig. 2.1-3. Because this result can be predicted either from Eq. 2.1-1 or from Eq. 2.1-2, this was a triumph. 

But this success was by no means complete. After all, Graham s data for liquids anticipated Eq. 2.1-1. To try to strengthen the analogy with thermal conduction, Fick used the upper apparatus shown in Fig. 2.1-3. In this apparatus, he established the steady-state concentration profile in the same manner as before. He measured this profile and then tried to predict these results using Eq. 2.1-2, in which the funnel area A available for diffusion varied with the distance z. When Fick compared his calculations with his experimental results, he found good agreement. These results were the initial verification of Tick s law. 

Useful forms of Tick s law in dilute solutions are shown in Table 2.1-2. Each equation closely parallels that suggested by Fick, that is, Eq. 2.1-1. Each involves the same phenomenological diffusion coefficient. Each will be combined with mass balances to analyze the problems central to the rest of this chapter. 

One must remember that these flux equations imply no convection in the same direction as the one-dimensional diffusion. They are thus special cases of the general equations given in Table 3.2-1. This lack of convection often indicates a dilute solution. In fact, the assumption of a dilute solution is more restrictive than necessary, for there are many concentrated solutions for which these simple equations can be used without inaccuracy. Nonetheless, for the novice, I suggest thinking of diffusion in a dilute solution. 

In the previous section we detailed the development of Tick s law, the basic relation for diffusion. Armed with this law, we can now attack the simplest example steady 

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Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

Figure 2 illustrates steady diffusion across a thin film of thickness h. The solutions on both sides of the film are dilute, so the diffusion coefficient can be considered constant. The solute molecules diffuse from the well-mixed higher concentration Ci to the well-mixed lower concentration c2. The concentrations on both sides of the film are kept constant. After sufficient time, a steady state is reached in which... 

Steady diffusion across a thin film is mathematically straightforward but physically subtle. Dissolution film theory, suggested initially by Nernst and Brunner, is essentially based on steady diffusion across a thin film. 

It may be appropriate here to introduce film theory. As mentioned in reference to the steady diffusion across a thin film, we often hypothesize a film called an unstirred layer to account for the aqueous diffusion resistance to mass transfer. Film theory is valuable not only because of its simplicity but also because of its practical utility. However, the thickness of the film is often difficult to determine. In the following, we try to answer the question, What does the thickness of the film represent ... 

Steady diffusion across a thin film with a fixed boundary... 

Steady diffusion across a thin film is illustrated schematically in Fig. 2.2-1. On each side of the film is a well-mixed solution of one solute, species 1. Both these solutions are dilute. The solute diffuses from the fixed higher concentration, located at z 0 on the left-hand side of the film, into the fixed, less concentrated solution, located at z / on the right-hand side. 

This example is a mainstay of the analysis of diffusion. It is a good mathematical introduction of spherical coordinates, and it gives a result which is much like that for steady diffusion across a thin film. After all, Eq. 2.4-25 is the complete parallel of Eq. 2.2-10, but with the sphere radius i o replacing the film thickness /. Thus most teachers repeat this example as gospel. 

The classic steady-state diffusion problem is diffusion across a thin film at constant pressure and temperature and with no convection in the direction of diffusion (z direction), as shown in Figure 15-1. At steady state, there is no accumulation in the film and the concentration profile does not change with time. Over a segment of the film of thickness Az, the mass balance is input = output, which can be written in this case as = Ja,z+Az> leads to... 

This emphasis on dilute solutions is found in the historical development of the basic laws involved, as described in Section 2.1. Sections 2.2 and 2.3 of this chapter focus on two simple cases of diffusion steady-state diffusion across a thin film and unsteady-state diffusion into an infinite slab. This focus is a logical choice because these two cases are so common. For example, diffusion across thin films is basic to membrane transport, and diffusion in slabs is important in the strength of welds and in the decay of teeth. These two cases are the two extremes in nature, and they bracket the behavior observed experimentally. In Section 2.4 and Section 2.5, these ideas are extended to other examples that demonstrate mathematical ideas useful for other situations. 

However, we must also see a different and broader blueprint based on physics, not mathematics. This blueprint includes the two limiting cases of diffusion across a thin film and diffusion in a semi-infinite slab. Most diffusion problems fall between these two limits. The first, the thin film, is a steady-state problem, mathematically easy and sometimes physically subtle. The second, the unsteady-state problem of the thick slab, is harder to calculate mathematically and is the limit at short times. 

Example 7.3-2 Steady-state multicomponent diffusion across a thin film In steady-state binary diffusion, we found that the solute s eoncentration varied linearly across a thin film. Will solute concentrations vary linearly in the multieomponent ease What will the flux be Solution By comparison with Eq. 2.2-9, we see that... 

The diffusion across this thin film is considered to be a steady-state problem. There are no concentration changes with time, as indicated in Fig 5.13. 

The purpose behind this design is to establish gradients in the bridge channels by allowing diffusion of molecules between the two main streams. Diffusion across the bridge channels is theoretically equivalent to diffusion in a thin film, in which a linear gradient is established at steady state. The chamber can thus be seen as... 

Figure 15-1. Steady-state diffusion across a thin layer of film...

Consider a very thin film between two well-mixed fluids. Each of the fluids are dilute binary mixtures, consisting of the same solvent and solute having different concentrations. The solute diffuses from the higher concentrated solution into the less concentrated one. The diffusion across this thin film is considered to be a steady-state problem. There are no concentration changes with time, as indicated in Fig. 5.13. 

The first of the problems of concern here, sketched in Fig. 2.5-1, involves steady-state diffusion across a thin, moving hquid film. The concentrations on both sides of this film are fixed by electrochemical reactions, but the film itself is moving steadily. I have chosen this example not because it occurs often but because it is simple. I ask that readers oriented toward the practical will wait with later examples for results of greater applicability. 

Fig. 20.1-1. Steady heat conduction across a thin film. Heat conduction across a thin film is like diffusion across a membrane (see Section 2.2). The resulting temperature profile is linear, and the flux is constant and inversely proportional to the film thickness /.

We now imagine that a solute present at high dilution is slowly diffusing across this film. The restriction to high dilution allows us to neglect the diffusion-induced convection perpendicular to the interface, so we can use the simple results in Chapter 2, rather than the more complex ones in Chapter 3. The steady-state flux across this thin film can be written in terms of the mass transfer coefficient ... 

The film theory was originally proposed by Whitman,195 who obtained his idea from the Nernst117 concept of the diffusion layer. It was first applied to the analysis of gas absorption accompanied by a chemical reaction by Hatta.85,86 It is a steady-state theory and assumes that mass-transfer resistances across the interface are restricted to thin films in each phase near the interface. If more than one species is involved in a multiphase reaction process, this theory assumes that the thickness of the film near any interface (gas-liquid or liquid-solid) is the same for all reactants and products. Although the theory gives a rather simplified description of the multiphase reaction process, it gives a good answer for the global reaction rates, in many instances, particularly when the diffusivities of all reactants and products are identical. It is simple to use, particularly when the... 

The CP model, which is based on the film theory, was developed to desaibe the back diffusion phenomenon during filtration of macromolecules. In this model, the rejection of particles gives rise to a thin fouling layer on the membrane surface, overlaid by a CP layer in which particles diffuse away from the membrane surface, where solute concentration is high, to the bulk phase, where the solute concentration is low [162], At steady state, convection of particles toward the membrane surface is balanced by diffusion away from the membrane. Thus, integrating the ID convection-diffusion equation across the CP layer gives... 

Por steady-state diffusion occurring across flat and thin diffusion films, only one dimension can be considered and Eq. (5.1) is greatly simplified. Moreover, by replacing differentials with finite increments and assuming a linear concentration profile within the film of thickness 5, Eq. (5.1) becomes... 

When the oxygen sensor is immersed in a flowing or stirred solution of the analyte, oxygen diffuses through the membrane into the thin layer of electrolyte immediately adjacent to the disk cathode, where it diffuses to the electrode and is immediately reduced lo water. In contrast with a normal hydrodynamic electrode, two diffusion processes are involved — one through the membrane and the other through the solution between the membrane and the electrode surface. For a steady-state condition to be reached in a reasonable period (10 to 20 s), the thickness of the membrane and the electrolyte film must be 20 pm or less. Under these conditions, it is the rale of equilibration of the transfer of oxygen across the membrane that determines the steadv-stalc current that is reached. 

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