Unsteady Diffusion in a Semi-infinite Slab

We now turn to a discussion of diffusion in a semi-infinite slab, which is basic to perhaps 10 percent of the problems in diffusion. We consider a volume of solution that starts at an interface and extends a long way. Such a solution can be a gas, liquid, or solid. We want to find how the concentration varies in this solution as a result of a concentration change at its interface. In mathematical terms, we want to find the concentration and fiux as a function of position and time. 

This type of mass transfer is sometimes called free diffusion simply because this is briefer than unsteady diffusion in a semi-infinite slab. At first glance, this situation may seem rare because no solution can extend an infinite distance. The previous thin-film example made more sense because we can think of many more thin films than semi-infinite slabs. Thus we might conclude that this semi-infinite case is not common. That conclusion would be a serious error. 

This example points to an important corollary, which states that cases involving an infinite slab and a thin membrane will bracket the observed behavior. At short times. 

The diffusion in a semi-infinite slab is schematically sketched in Fig. 2.3-2. The slab initially contains a uniform concentration of solute ci oo. At some time, chosen as time zero, the concentration at the interface is suddenly and abruptly increased, although the solute is always present at high dilution. The increase produces the time-dependent concentration profile that develops as solute penetrates into the slab. 

We want to find the concentration profile and the flux in this situation, and so again we need a mass balance written on the thin layer of volume AAz  

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Unsteady diffusion in a semi-infinite slab with a fixed boundary... 

S.2.3.2 Unsteady Diffusion in a Semi-Infinite Slab with a Fixed Boundary... 

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. 

The penetration theory can be viewed as the original surface-renewal model. This model was formulated by Higbie [51]. This model is based on the assumption that the liquid surface contains small fluid elements that contact the gas phase for a time that is equal for all elements. After this contact time they penetrate into the bulk liquid and each element is then replaced by another element from the bulk liquid phase. The basic mechanism captured in this concept is that at short contact times, the diffusion process will be unsteady. Considering that the fluid elements may diffuse to an infinite extend into the liquid phase, the model formulation developed earlier for diffusion into a semi-infinite slab can be applied describing this system. After some time the diffusion process will reach a steady state, thus the penetration theory predictions will then correspond to the limiting case described by the basic film theory. However, when the transient flux development is determining a notable amount of the total flux accumulated, the two models will give rise to different mass transfer coefficients. 

This variable can be used to estimate which limiting case is more relevant. If it is much larger than unity, we can assume a semi-infinite slab. If it is much less than unity, we should expect a steady state or an equilibrium. If it is approximately unity, we may be forced to make a fancier analysis. For example, imagine that we are testing a membrane for an industrial separation. The membrane is 0.01 centimeters thick, and the diffusion coefficient in it is 10 cm /sec. If our experiments take only 10 seconds, we have an unsteady-state problem like the semi-infinite slab if they take three hours, we approach a steady-state situation. 

Fig. 20.1-2. Unsteady heat conduction into a semi-infinite slab. The temperature profile in this case is an error function, just like the concentration profile in Section 2.3. This profile depends on the variable z/v/4at, where a (= kr/pCp) is the thermal diffusivity.

This variable is the argument of the error function of the semi-infinite slab, it determines the standard deviation of the decaying pulse, and it is central to the time dependence of diffusion into the cylinder. In other words, it is a key to all the foregoing unsteady-state problems. Indeed, it can be easily isolated by dimensional analysis. 

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