International Journal of Heat and Mass Transfer
Critical times in multilayer diffusion. Part 1: Exact solutions
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,is applicable in each layer where is distance, is the temperature in layer i at time t, and is the diffusivity of layer i, as shown in Fig. 1. For clarity, in subsequent sections the notation 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 such thatwhere is the temperature, is a chosen constant, and 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 expressionwhere is the commonly averaged diffusivity for layered materials, given byHere L is the total medium length, and are the lengths of each layer with material diffusivity . 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, , is calculated at and integrated over time. The asymptotic line is found, as , and is rearranged for t when . See Crank [26, pp. 47–48], for a more detailed explanation. Eq. (3) corresponds to the critical time definition given in Eq. (2) for , a result calculated in Section 2. Here 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, , also referred to as a ‘biperiodic region’. Their complicated derivation is equivalent to choosing 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, wherefor some initial condition, , and mixed boundary conditions,
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, , and transient, , components.
In many situations the contact between layers is imperfect giving rise to a jump condition in U:at for where is the contact transfer coefficient and is the conductivity, is the density and is the
Numerical results
To explore the behaviour of the multilayer critical time a biperiodic region is considered, with repeating layers with ‘A’ and ‘B’ properties such as layer width, and , and diffusivity, and . That is, there are n layers in total with repeated layers , denoted in shorthand by or equivalently . The region is defined with and , hence . For simplicity here, equal widths for both layers are used, so , 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
References (34)
Transient temperature distribution in a multilayer medium subject to radiative surface cooling
Appl. Math. Model.
(1994)- et al.
One layer, two layers, etc. An introduction to the EIS study of multilayer electrodes. Part 1: Theory
J. Electroanal. Chem.
(2005) Diffusion impedance and equivalent circuit of a multilayer film
Electrochem. Commun.
(2005)- et al.
Mass diffusion through two-layer porous media: an application to the drug-eluting stent
Int. J. Heat Mass Transfer
(2007) - et al.
Analysis of diffusion delay in a layered medium
Biophys. J.
(1988) Transient heat conduction in one-dimension composite slab. A natural analytic approach
Int. J. Heat Mass Transfer
(2000)- et al.
Temperature profiles in composite plates subject to time-dependent complex boundary conditions
Compos. Struct.
(2003) An analytic approach to the unsteady heat conduction processes in one-dimensional composite media
Int. J. Heat Mass Transfer
(2002)- et al.
On transient heat conduction in a one-dimensional composite slab
Int. J. Heat Mass Transfer
(2004) - et al.
Transient analytical solution to heat conduction in a composite circular cylinder
Int. J. Heat Mass Transfer
(2006)