Characteristic time scales for diffusion processes through layers and across interfaces

Abstract

This paper presents a simple tool for characterizing the time scale for continuum diffusion processes through layered heterogeneous media. This mathematical problem is motivated by several practical applications such as heat transport in composite materials, flow in layered aquifers, and drug diffusion through the layers of the skin. In such processes, the physical properties of the medium vary across layers and internal boundary conditions apply at the interfaces between adjacent layers. To characterize the time scale, we use the concept of mean action time, which provides the mean time scale at each position in the medium by utilizing the fact that the transition of the transient solution of the underlying partial differential equation model, from initial state to steady state, can be represented as a cumulative distribution function of time. Using this concept, we define the characteristic time scale for a multilayer diffusion process as the maximum value of the mean action time across the layered medium. For given initial conditions and internal and external boundary conditions, this approach leads to simple algebraic expressions for characterizing the time scale that depend on the physical and geometrical properties of the medium, such as the diffusivities and lengths of the layers. Numerical examples demonstrate that these expressions provide useful insight into explaining how the parameters in the model affect the time it takes for a multilayer diffusion process to reach steady state.

Rear-surface integral method for calculating thermal diffusivity from laser flash experiments

“…The boundary conditions, Eq (11), are derived by making use of the boundary conditions satisfied by T (x, t), Eq (3), and noting that u (x) = ∞ 0 − ∂T ∂x (x, t) dt (Carr, 2017(Carr, , 2018. The continuity conditions, Eq (12), follow from the definition of u(x), Eq (8), and continuity of T (x, t) and ∂T ∂x (x, t).…”

Drug delivery from microcapsules: How can we estimate the release time?

“…involving the (k − 1)th moment, u i,k−1 (r). Similar boundary and interlayer conditions to those in Eqs (3.6)-(3.12) apply [14] giving the following boundary value problem for the kth moment: The release time is computed iteratively by solving the sequence of boundary value problems (4.12)-(4.18) for increasing values of k [18,20]. First, an initial estimate of t * r is calculated using k = 1 in Eq (4.9) by solving Eqs (4.12)-(4.18) with k = 1 and by using the zeroth order moment u i,0 (r) computed previously (Section 3) in the right-hand side of Eqs (4.12)-(4.14).…”

“…Therefore, it is reasonable to conclude that t A major attraction of working with Eq (3.1) is that a simple closed-form expression can be derived for t (1) c involving the model parameters without requiring the full expression for c 0 (0, t). This is achieved by extending and modifying similar ideas presented elsewhere (see, e.g., [14,15]). First, we define:…”

“…Supplementary boundary and interlayer conditions corresponding to Eqs (2.4)-(2.7) are derived using the definition of u i (r) in Eq (3.2) (see, e.g. [14,18]). In summary, recalling the initial conditions (2.8)-(2.10) and the steady-state solution (2.11), we have the following boundary value problem satisfied by u i (r) (i = 0, 1, .…”

“…Here, the transition from initial to equilibrium state is represented as a cumulative distribution function, with the mean of the corresponding probability density function defining the characteristic timescale. The attraction of the mean action time is that it completely avoids any calculation of the transient solution and produces simple explicit algebraic formulas for the timescale in terms of the parameters in the model [14,15]. Finally, we note that the notion of release time is different but related to the concept of penetration time, which that has been studied in a number of configurations for thermal disturbances in heat conduction problems [17].…”