Multilayer one-dimensional Convection-Diffusion-Reaction (CDR) problem: Analytical solution and imaginary eigenvalue analysis
Introduction
Heat and mass transport in a multilayer body [1] has been investigated in several past papers for a variety of engineering and biomedical applications. For example, mass transfer in a multilayer structure has been used for modeling drug delivery in drug eluting stents [2]. Thermal transport in biological tissue has been modeled using multilayer Pennes equation that contains a temperature-dependent perfusion term [3]. Heat transfer in multilayered bodies is also relevant for several traditional engineering fields, such as manufacturing [4], atmospheric re-entry [5], extended surfaces [6], microelectronics cooling [7] and nuclear power generation [8].
Key physical processes that occur in multilayer heat and mass transfer problems include diffusion, advection due to fluid flow and generation/consumption. The rate of generation or consumption of heat/mass is often proportional to the local temperature/concentration. For example, a chemical reaction with first-order kinetics consumes/generates species at a rate proportional to the local concentration of the reactant. Heat generated by such a chemical reaction is also often approximated as linearly temperature-dependent [9,10], even though, strictly speaking, the dependence is exponential in nature, as modeled by Arrhenius kinetics [11]. In addition, the perfusion term in the Pennes bioheat transfer equation [12] may also be interpreted as an energy consumption term proportional to the local temperature. Finally, the fin equation used for analysis of multilayer segmented fin may also be interpreted to contain a negative heat generation term that is proportional to the local temperature [6].
A transient energy conservation equation representing a balance between diffusion, advection and generation in each layer of an M-layer one-dimensional body may be written as [2]
where Tm is the temperature field relative to ambient in the mth layer.
A similar equation can be written for the concentration field in a mass transfer problem. This equation is often referred to as the Convection-Diffusion-Reaction (CDR) equation [13,14], and has been heavily researched for both heat and mass transfer problems. This problem is distinct from the pure-diffusion multilayer problem analyzed in textbooks [1] due to the appearance of convection and reaction terms. The eigenvalues and orthogonality relationships for CDR problems are likely to be very different from the ones for pure-diffusion multilayer problems discussed in textbooks. Analytical approaches for solving CDR problems include an eigenfunction-based series solution [1] that uses quasi-orthogonality of the eigenfunctions [15,16] and an appropriate transformation to account for the effect of advection and generation [17]. The nature of orthogonality of eigenfunctions depends strongly on the presence of advection and the specific nature of boundary conditions. A number of unique interface conditions between layers in mass transfer problems involving porous media have been presented [17]. Analytical solutions for a variety of convection-diffusion problems – a subset of the CDR problem – have also been presented [18,19].
Several previous studies have developed analytical, semi-analytical, and numerical solutions of problems governed by the CDR equation. For example, the separation of variables technique was used to derive an exact solution for mass diffusion through a two-layer porous media with pure diffusion in one layer and all three phenomena in the other one [17,20]. A similar technique was also used to solve a more general CDR problem in multilayer porous media [21]. Laplace transformation technique has been used to derive an analytical solution to a two-layer CDR problem for drug-eluting stent problems [22]. A semi-analytical model for solute transport in multilayer porous media using Laplace transformation has been presented [23]. Several studies have used numerical techniques such as Additive Runge–Kutta [14,24], Positivity-Preserving Variational (PPV) [25], and Boundary Element Method [26] to solve more complicated CDR problems involving higher dimensions, non-linear terms, and variable coefficients. Various types of convection-diffusion problems have been solved using the method of eigenfunction expansions [27], integral transform [28,29] and separation of variables [30].
A limited amount of past work suggests that imaginary eigenvalues may be encountered in multilayer heat/mass transport problems. The first class of such problems pertains to 2D [31,32,33] and 3D [34] multilayer diffusion problems. In such problems, imaginary eigenvalues appear in the thickness direction due to real eigenvalues in the orthogonal direction(s). The second class of problems with imaginary eigenvalues involves multiple transport and generation processes in each layer, even for a one-dimensional body. For example, it has been suggested that eigenvalues may become imaginary in a one-dimensional two-layer mass transfer problem with diffusion in one layer and a combination of diffusion, advection and generation in the second layer [17]. However, a detailed analysis of conditions in which such eigenvalues may appear and their physical interpretation is missing. In recent work, an analysis of imaginary eigenvalues appearing in multilayer diffusion-reaction problems [35] has been presented. While this work derived the conditions in which imaginary eigenvalues appear and discussed their physical interpretation, the impact of advection was not considered. A complete analysis including the effect of advection is needed for modeling systems where advection plays a key role.
Imaginary eigenvalues are not merely a theoretical curiosity because certain imaginary eigenvalues in one-dimensional multilayer problems are directly related to the phenomenon of thermal runaway [35], in which, the temperature field diverges at large times. Prediction of imaginary eigenvalues and thermal runaway is of much practical importance for the safety of engineering systems such as Li-ion cells and battery packs. Specifically, in a flow battery [36] where the electrolyte is circulated, diffusion, advection and heat generation due to electrochemical reactions all occur simultaneously. In order to ensure the safety and reliability of such systems, it is important to develop a robust theoretical understanding of the regimes in which an imbalance between diffusion, advection and generation may lead to imaginary eigenvalues, and therefore, thermal runaway. In addition, understanding imaginary eigenvalues is important because a series solution must include all eigenvalues, whether real or imaginary, and standard methods for computing eigenvalues may miss an imaginary eigenvalue.
This paper presents the solution of a multilayer one-dimensional CDR heat transfer problem and specifically investigates the conditions that result in imaginary eigenvalues in such a problem. A solution is first presented for a general multilayer problem, followed by discussion of a special case of a two-layer body. A physical interpretation of imaginary eigenvalues in terms of a balance between conduction, advection and generation is presented. Within the context of a two-layer body, conditions for appearance of two types of imaginary eigenvalues are discussed. It is shown that the first type of imaginary eigenvalues results in thermal runaway in the body. The analysis of conditions that lead to imaginary eigenvalues may help in better design of multilayer systems in practical problems such as a flow battery.
This paper is organized as follows: the next section presents the general, M-layer problem and its solution, including derivation of the eigenequation and the orthogonality of eigenfunctions. The specific case of a two-layer body is discussed in Section 3. Section 4 then derives a mathematical requirement for the occurrence of imaginary eigenvalues. Several aspects of the conditions in which imaginary eigenvalues may appear – and their relationships with parameters associated with diffusion, advection and generation – are discussed in Section 5.
Section snippets
General M-layer body
Consider a one-dimensional M-layer body such as shown in Fig. 1(a). Heat is generated within each layer at a rate proportional to the local temperature. Heat transfer occurs within this body due to diffusion and due to advection driven by an imposed one-dimensional fluid flow in each layer from left to right. Each layer has distinct thermal properties, flow velocity and rate of heat generation. General convective boundary conditions are assumed on the left and right boundaries, respectively.
Special case: two-layer body
This section considers the special case of the CDR equations for a two-layer body, which is particularly relevant for several engineering applications. Analysis of the two-layer body also reduces the number of parameters, and makes it easier to understand the interplay between diffusion, advection, heat generation and heat removal from the boundaries.
In this case, a solution for temperature fields in the two layers, θ1 and θ2 can be written as follows:
Eigenvalue analysis
This section presents an analysis of the eigenvalues for the two-layer case described in Section 3. Determining the conditions that result in imaginary eigenvalues is of particular interest from both theoretical and practical perspectives, since an imaginary value of λn will cause exponential rise in temperature at large times, which corresponds to thermal runaway. Therefore, in order to ensure safety and reliability of engineering systems modeled by these equations, it is critical to
Discussion
Before discussing the impact of various problem parameters on the occurrence of imaginary eigenvalues in the problem, the accuracy of the analysis presented in this work is established. Specifically, the predicted temperature distribution for a two-layer body based on Eq. (28) and (29) is compared against numerical simulations based on the finite-difference method. For this purpose, the governing equation and boundary conditions in each layer are discretized using a second-order central
Conclusions
Theoretical modeling of multilayer CDR problems is important from both theoretical and practical perspectives. The occurrence of imaginary eigenvalues in such problems, even for a one-dimensional geometry, determines whether thermal runaway is encountered or not. In addition, the prediction of conditions in which imaginary eigenvalues appear is important because standard algorithms for computing eigenvalues only focus on real eigenvalues and may miss an imaginary eigenvalue. The analytical
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This material is based upon work supported by CAREER Award No. CBET-1554183 from the National Science Foundation.
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2023, International Journal of Heat and Mass TransferCitation Excerpt :Multilayer problems with more complicated features, such as spatially-dependent [13] or time-dependent [14] boundary conditions, convective transport [15], reaction [16], multispecies advection-dispersion [17] and a large number of layers [18] have also been analyzed in the past. Imaginary eigenvalues are known to appear in a subset of such problems, which have been related to divergence of the temperature field at large times [15,16,19]. While one-dimensional analysis is sufficient for a large number of engineering problems, in some cases, a two- or three-dimensional analysis is necessitated.
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