Critical Times in One- and Two-Layered Diffusion

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An important consideration in applications modelled by diffusion is: how long does the diffusive process take? For example, in steel annealing (Barry & Sweatman, 2009; McGuinness et al, 2009; Yuen, 1994), a steel coil is heated by imposing an external temperature; an important consideration is when the coldest point in the coil reaches a given temperature. Similar applications include determining the effectiveness of drug carriers inserted into living tissue (Pontrelli & de Monte, 2007) and the performance of electrodes (Freger, 2005). Numerous other applications are referenced in Hickson et al (2009b). The essential elements of finding this heating time, or critical time, are well illustrated by considering heat diffusion through either one or two layers. The single-layer analysis involves traditional methods taught in second-year engineering mathematics classes with some interesting extensions. The two-layer solution in section 4 of this paper, although more algebraically complicated, involves few new concepts and makes a suitable project extension for more advanced undergraduate students.

The solution to the heat equation through a single medium with imposed temperature on the boundary (see figure 1(a)) is a classical application of separation of variables. Explicitly separating the solution into "steady-state" and "transient" components is not necessarily seen at an undergraduate level, but is conceptually the same as splitting a second-order inhomogeneous equation into the homogeneous and particular solutions. Traditional engineering mathematics textbooks usually stop when the solution is found, despite this solution being an infinite series with little meaning to the students. By considering the critical time, this solution can be used to provide both motivation and a better understanding of what the solution means.

The "critical time" for diffusion can be defined in many ways. In the context of the steel annealing application, the critical time is defined as the amount of time taken for the coldest point in the coil to reach the desired temperature. A common definition is the time when the average temperature reaches a proportion of the average steady-state. That is, the value of t = [t.sub.c] such that:

[[integral].sup.L.sub.0] U(x,[t.sub.c])dx = [alpha][[integral].sup.L.sub.0] w(x)dx (1)

where U(x, t) is the temperature, 0 < [alpha] < 1 is a chosen constant, and w(x) is the steady-state. Landman & Australasian journal of Engineering Education, Vol 15 No 2 McGuinness (2000) summarised previous work and applications using this critical time definition, also called the mean action time (McNabb, 1993). An additional more complex definition (Crank, 1975) is found by calculating the asymptote of the net heat flux. For a simple scenario of imposing a temperature at the boundary of a single material layer, this gives a critical time [t.sub.av] of:

[t.xub.av] = [L.sup.2]/6[D.sub.av] (2)

which is shown in section 3 to be when [alpha] [approximately equal to] 0.8435 in equation (1). Here L is the length of the medium and [D.sub.av] is the diffusivity. When two layers are considered this equation is often used (see for example Ash et al (1965) and Graff et al (2004)) with [D.sub.av] defined by:

L/[D.sub.av] = [l.sub.1]/[D.sub.1] + [l.sub.2]/[D.sub.2] (3)

where [l.sub.1] and [l.sub.2] are the lengths of layers one and two, respectively, and [D.sub.1] and [D.sub.2] are the diffusivities of layers one and two. The limitations of this approximation are more fully investigated and discussed in Absi et al (2005) and Hickson et al (2009a; 2009b; 2009c).


In this paper we first outline, in section 2, the model and solution method for a single layer with two different boundary conditions. These solutions are then used to find the critical time in section 3, and the two different boundary condition cases are compared. …