Semi-infinite transient diffusion bodies

Transient Interdiffusion in Two Semi-Infinite Bodies The transient diffusion problem illustrated in Figure 4.8, which involves the interdiffusion of two semi-infinite bodies in contact with one another, is closely related to the previous semi-infinite transient diffusion problem. In fact, if you consider just one-half of the problem domain (e.g., consider the evolution of the diffusion profiles for species A for X > 0), diffusion proceeds exactly like the previous semi-infinite diffusion problem. The only difference is that in this case the interfacial concentration of species A is assumed to be pinned at half of its bulk (i.e., pure material A) value. 

In section 2.5.3 it was shown that the differential equation for transient mass diffusion is of the same type as the heat conduction equation, a result of which is that many mass diffusion problems can be traced back to the corresponding heat conduction problem. We wish to discuss this in detail for transient diffusion in a semi-infinite solid and in the simple bodies like plates, cylinders and spheres. 

We will consider transient diffusion of a substance in a semi-infinite body B. At time t = 0, substance A is stored in the body at a concentration cAa. The desired concentration profile cA = cA(x,t) satisfies the following differential equation, under the assumption cD = const... 

FIGURE 4.8 Transient interdiffusion of two semi-infinite bodies. Material A on the left (t < 0), assumed to be initially composed purely of species A, is in contact with material B on the right (x > 0), which is assumed to be initially composed purely of species B. At time f = 0, species A and B begin to interdiffuse. If the initial bulk concentrations and diffusivities of A and B are equal, their concentrations at the interface will be pinned at 1/2 c. The materials will slowly diffuse into one another, but the position of the interface between them will not move. 

Solution To answer this question, it is helpful to first establish a sketch of the problem, as shown in Figure 4.9. Considering first the diffusion of A into B from left to right, we wish to determine the right-side boundary of the interdiffusion region. As detailed in the problem statement, this corresponds to the location (x value) where the concentration of species A falls to 1/10 of its initial bulk value. As shown on the schematic illustration, we will call this location x = 8. Applying this criteria to the solution for the transient interdiffusion of two semi-infinite bodies gives... 

FIGURE 4.11 Transient diffusion of a thin layer between two semi-infinite bodies. 

Kirkendal Effect When we previously discussed the transient interdiffiision of two semi-infinite bodies (material A and material B, respectively), we explicitly specified that the diffusion of A in B and that of B in A were identical and could therefore be described by a single diffusion coefficient. In many solids, however, this is not true. For example, the diffusivity of zinc in copper is much larger than the diffiisivity of copper in zinc. If a block of brass (a copper-zinc alloy) and a block of pure copper are bonded together at high temperatures, the zinc atoms will diffuse out of the brass and into the copper at a much faster rate than the copper atoms diffuse into the brass block. The net result is that the effective interface between the brass and copper blocks moves toward the brass, as illustrated schematically in Figure 4.17. This phenomenon is known as the Kirkendal effect and it occurs in many solid-state systems. 

Problem 4.2. Equation 4.40 in the text provides the solution for the transient diffusion of a thin layer of material between two semi-infinite bodies ... 

The computer display then shows the steady-state values for characteristics such as the thermal conductivity k [W/(mK)], thermal resistance R [m K/W] and thickness of the sample s [mm], but also the transient (non-stationary) parameters like thermal diffusivity and so called thermal absorptivity b [Ws1/2/(jti2K)], Thus it characterizes the warm-cool feeling of textile fabrics during the first short contact of human skin with a fabric. It is defined by the equation b = (Xpc)l, however, this parameter is depicted under some simplifying conditions of the level of heat flow q [ W/m2] which passes between the human skin of infinite thermal capacity and temperature T The textile fabric contact is idealized to a semi-infinite body of the finite thermal capacity and initial temperature, T, using the equation, = b (Tj - To)/(n, ... 

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