Critical times in multilayer diffusion. Part 1: Exact solutions

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Abstract

While diffusion has been well studied, diffusion across multiple layers, each with different properties, has had less attention. This type of diffusion may arise in heat transport across composite materials or layered biological material. Usually of most interest is a critical time, such as the time for a material to heat up. Here an exact solution is found which is used to numerically demonstrate the critical time behaviour for transport across multiple layers with imperfect contact. This solution illustrates the limitations of traditional averaging methods, which are only good for a large number of layers.

Introduction

Diffusion through multiple layers has applications to a wide range of areas in heat and mass transport. Industrial applications include annealing steel coils [1], [2], [3], the performance of semiconductors [4] and electrodes [5], [6], and geological profiles [7]. Biological applications include determining the effectiveness of drug carriers inserted into living tissue [8], the probing of biological tissue with infrared light [9], and analysing the heat production of muscle [10].

For multilayer diffusion across n layers the standard diffusion equation,Uit=Di2Uix2,i[1,n],is applicable in each layer where xi-1xxi is distance, Ui is the temperature in layer i at time t, and Di is the diffusivity of layer i, as shown in Fig. 1. For clarity, in subsequent sections the notation diDi is also used.

Exact solutions for diffusion in layered media have been found for diverse applications and geometries. These include solutions in Cartesian coordinates for two layers [8], [11], [12] and n layers [13], [14], [15], [16], [17], and cylindrical n layer solutions [18], [19]. Many of the publications use mathematical separation methods similar to that outlined in Sections 2 Single layer solution, 3 Multilayer solutions of this article. However, previous publications that use separation methods, and also consider Cartesian coordinates, assume perfect contact between the layers [13], [14], [15], [16], [17] and some [13], [14], [17] have less general boundary conditions. Laplace transform approaches are also used [5], [20], but are less common due to the difficulty of the inverse transforms, which are often only numerically found.

An important aspect of multilayer diffusion is the ‘critical time’, which is a measure of how long the diffusive process takes. There are multiple definitions of critical time since mathematically, an infinite amount of time is required to reach steady state [21]. One definition, appropriate in the annealing of steel coils [1], [2], [3], is when the coldest point in the coil reaches a given temperature. A common and more general definition we consider here is the time when the average temperature reaches a given or specified proportion of the average steady state. That is, the value of t=tc such thatx=0LU(x,tc)dx=αx=0Lw(x)dx,where U(x,t) is the temperature, 0<α<1 is a chosen constant, and w(x) is the steady state. Landman and McGuinness [21] summarise previous work and applications using this critical time definition, also called the mean action time [22], [23].

The most common approximation of critical time (see for example [24], [25], [26], [27], [28]), is the simple expressiontav=L26Dav,where Dav is the commonly averaged diffusivity for layered materials, given byLDav=i=1nliDi.Here L is the total medium length, and li are the lengths of each layer with material diffusivity Di. This series-averaged diffusivity is a valid measure when calculating heat fluxes at steady state, or for a large number of layers. The critical time, Eq. (3), is found using the exact solution for the one layer problem, with different surface concentrations [26]. The flux, Qt, is calculated at x=x0 and integrated over time. The asymptotic line is found, as t, and is rearranged for t when Qt=0. See Crank [26, pp. 47–48], for a more detailed explanation. Eq. (3) corresponds to the critical time definition given in Eq. (2) for α0.8435, a result calculated in Section 2. Here α corresponds to the asymptotic analysis conducted by Crank [26] to quantify a ‘breakthrough’ or ‘lag’ time. However, Eq. (3) is only strictly valid for a single layer of material. Absi et al. [29] describe a brief numerical and experimental comparison using Eq. (3) versus the full coupled numerical system for two layers. Their results indicate the limitations of this formula, a result we corroborate in our numerical simulation shown in Fig. 3.

Several publications have attempted to calculate a diffusive critical time through composites, in Cartesian, cylindrical and spherical geometries with Ash et al. [24], [30] giving detailed solutions. These are summarised in Barrer [25] for some of the usual layer configurations, such as two repeated layers, ABAB, also referred to as a ‘biperiodic region’. Their complicated derivation is equivalent to choosing α0.8435 in Eq. (2). However, as discussed in detail in our companion paper [31], their result is only as accurate as the approximate result given by Eq. (3), that is of the order of 10–50%. Aguirre et al. [4] determined a solution for sinusoidally imposed temperature, calculating an effective diffusivity for a composite material. Their result is an improved version of the series-averaged diffusivity given in Eq. (4). The effective diffusivity was found in terms of the imposed frequency where Eq. (4) is reflected in the low frequency limit when the material is in quasi-steady state.

Previous work [14] explored a different definition of critical time, where the temperature at the end of the region reached a critical threshold. This definition is only applicable for an insulated boundary condition, whereas a more general definition which is not dependent on the boundary conditions is now considered. Also, we consider here the more general case of imperfect contact between the layers.

We will show standard Eqs. (3), (4) give inaccurate results. The exact solution is found for diffusion in a one-dimensional Cartesian material with only one layer in Section 2. This is extended to the more complicated multilayer case in Section 3 where the solutions are also verified. The critical time is calculated numerically in Section 4 and discussed in Section 5.

Section snippets

Single layer solution

In this section we find an exact solution for the single layer case. Whilst not original (see for example [32]), this will demonstrate the solution method used for the more difficult multilayer diffusion problem in Section 3. Additionally, these results will assist in understanding the definition and behaviour of critical time.

The single layer case is depicted in Fig. 2, whereUt=D2Ux2,for some initial condition, U(x,0)=f(x), and mixed boundary conditions,a1U+b1Ux=θ1atx=x0,a2U+b2Ux=θ2atx=

Multilayer solutions

In this section the multilayer solution for general boundary conditions are found using the same method as the single layer, by splitting the solution into the steady state, wi(x), and transient, vi(x,t), components.

In many situations the contact between layers is imperfect giving rise to a jump condition in U:κiUix=Hi(Ui+1-Ui),κi+1Ui+1x=Hi(Ui+1-Ui),at x=xi for i=1,2,,(n-1) where Hi is the contact transfer coefficient and κi=Diρici is the conductivity, ρi is the density and ci is the

Numerical results

To explore the behaviour of the multilayer critical time a biperiodic region is considered, with n/2 repeating layers with ‘A’ and ‘B’ properties such as layer width, lA and lB, and diffusivity, DA and DB. That is, there are n layers in total with repeated layers ABABAB, denoted in shorthand by [A,B] or equivalently [DA,DB]. The region is defined with x0=0 and xn=1, hence L=1. For simplicity here, equal widths for both layers are used, so li=1/n, but in general this is not necessary. Different

Discussion and conclusion

The most interesting point illustrated with this work is that layered materials do not exhibit symmetric properties. That is, the time taken to diffuse depends greatly on which order the materials are layered. Hence this work can be used to consider the inverse problem, where it is possible to apply the critical time to determine the individual properties of the layered materials. That is, two measurements of critical time, with the material direction switched, are sufficient to determine the

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