Fick's First Law is still valid even when steady-state conditions do not exist. However, it is not very convenient to use it in this form for solving transient diffusion problems, for which the concentration at any position depends on both time and position. It is therefore more convenient to derive a second version of this equation, Fick's Second Law, which contains an explicit dependence on time. In this chapter we will first derive the equation; a second-order partial differential equation. We will then learn how solutions to this equation can be obtained for a large variety of situations depending on the problem geometry, the initial and boundary conditions and the time-frame over which we need a valid solution.

Fick's Second Law

We consider the situation illustrated in Fig. 3.1. At two positions along the y-axis at y1 and y2, a distance Δy apart, the concentration of the solute is C1 and C2 respectively. We can use Fick's First Law to determine the amount of solute per unit time which enters this element at y1 and which leaves it at y2. Because matter must be conserved, the difference between these must be equal to the rate at which solute accumulates within the element.

In this chapter we introduce several advanced topics. The first deals with different approaches that can be taken to develop overall mass balances within a vessel. This can be used to simplify the analysis of a complex process, as we will see. We then turn to the problem of multi-phase resistances. Up to now we have been able to make assumptions about which of two phases in contact dominates the mass transfer kinetics across the interface between them. However, there are situations in which both phases may contribute to the overall mass transfer, at least over a limited range of conditions. We will develop an approach for dealing with this situation. Finally, we will consider a specific case involving topochemical reactions in porous solids.

Overall mass balance

We have used mass balances extensively throughout this book. Most often we have used them to develop an expression for the motion of an interface separating two reacting phases. In fluids, however, this process can become quite complex. Consider, for example, what happens when a gas stream is passed through or over a bed containing some reactive species. The reaction occurs slowly as the gas stream passes.

This chapter introduces a wide range of examples related to the heat treatment of binary alloys. The process involved is very simple–an alloy of fixed overall composition is subjected to a given temperature–time cycle. However, the analysis can be quite complex. Our guide to the various possibilities is the appropriate binary phase diagram, which summarizes the equilibrium conditions for the system. We will almost always assume that local equilibrium is established at interfaces, the boundary conditions thus being given by the phase diagram. The problems of interest include the dissolution and the growth of precipitates and the growth of lamellar structures such as pearlite. We will briefly consider how the analysis method can be modified to treat systems involving a third component.

Introduction

We now wish to consider what happens when a multi-component material, initially at equilibrium, is subjected to a change of temperature. This is clearly related to the process of heat treatment of materials. Depending on the temperature change involved particles may dissolve, they may be precipitated or they may change in size and volume fraction.

In this chapter we address a range of issues related to mass transport when counter diffusion is possible. This can occur in solid alloys containing a relatively large concentration of a substitutional solute, such that solute diffusion requires a significant compensating diffusion of the solvent. It also occurs in fluids in which both counter diffusion and convective flow can occur. In this chapter we will consider only quiescent liquids (i.e. those in which convection due to external forces is absent). We will see that counter diffusion and convection are in fact similar in their impact on mass transport. This will enable us to develop a general framework in which it is possible to treat a wide range of problems. One process we will encounter in this chapter involves the use of diffusion couples, in which two materials are placed in contact such that inter diffusion occurs. Diffusion couples represent the second most common process for micro structural manipulation in the solid state (the first being heat treatment, as discussed in Chapter 6). We will also consider a range of problems in which a species diffuses into a binary mixture and reacts with one of the elements in this mixture at a well-defined front. Examples include evaporation and internal oxidation.

In this chapter we continue our study of mass transfer in fluids by relaxing the assumption of quiescent flow. This raises considerably the level of mathematical complexity involved in obtaining solutions, to the point where analytical solutions are not available in most cases. Instead we introduce the concept of a mass transfer coefficient, analogous to the diffusion coefficient for simple diffusion. We will see that a wide range of correlations can be made which predict the mass transfer coefficient as a function of the geometry of the interface, the nature of fluid flow and other material parameters. These correlations are generally obtained by a combination of experimentation and numerical simulation. However, in some simple cases rudimentary analytical models provide useful approximations that we can use. In the latter part of the chapter we will return to the problem of reactions occurring within fluids. We will see that analytical solutions are not possible unless the reaction occurs at a well-defined front.

Transient diffusion in fluids

In the last chapter we worked mostly with flux equations analogous to Fick's First Law. We have noted previously, however, that this form of the diffusion law is most useful when working on steady-state problems. For transient problems, it is more convenient to use an equation which contains an explicit time dependence of concentration.

The transport of matter within materials can occur either by diffusion or by convective flow. Diffusion can occur in both solids and fluids while convective flow is found only in fluids. This chapter provides a brief overview of these processes. It also offers a summary of the mechanisms involved, for diffusion in solids, liquids and gases. While these are presented in a highly simplified fashion they do offer sufficient insight to enable many mass transport problems of practical interest to be solved.

Mass transport processes

When a drop of dye is added to a beaker of still water the highly concentrated dye spreads throughout the liquid until a uniform pale colour results. There are two processes which can contribute to this. The first is called diffusion. This process is driven by differences in the concentration of a substance (in this case the molecules that make up the dye) from one region to another. Diffusion occurs until the concentration becomes uniform, i.e. the concentration gradient goes to zero everywhere. The same process happens in the solid state when two soluble substances are mixed together.