Optimal design of a model energy conversion device

Lincoln Collins; Kaushik Bhattacharya

doi:10.1007/s00158-018-2072-6

Structural and Multidisciplinary Optimization

February 2019, Volume 59, Issue 2, pp 389–401 | Cite as

Optimal design of a model energy conversion device

Authors
Authors and affiliations

Lincoln Collins
Kaushik Bhattacharya

Research Paper

First Online: 06 September 2018

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Abstract

Fuel cells, batteries, and thermochemical and other energy conversion devices involve the transport of a number of (electro-) chemical species through distinct materials so that they can meet and react at specified multi-material interfaces. Therefore, morphology or arrangement of these different materials can be critical in the performance of an energy conversion device. In this paper, we study a model problem motivated by a solar-driven thermochemical conversion device that splits water into hydrogen and oxygen. We formulate the problem as a system of coupled multi-material reaction-diffusion equations where each species diffuses selectively through a given material and where the reaction occurs at multi-material interfaces. We introduce a phase-field formulation of the optimal design problem and numerically study selected examples.

Keywords

Energy convergence device Phase field approach Reaction-diffusion equations Thermal hydrolysis Interfacial reactions

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Notes

Acknowledgments

This work draws from the doctoral thesis of LC at the California Institute of Technology. It is a pleasure to acknowledge many interesting discussions with Sossina M. Haile, Robert V. Kohn and Patrick Dondl. We gratefully acknowledge the financial support of the National Science Foundation through the PIRE grant: OISE-0967140.

Funding information

We gratefully acknowledge the financial support of the National Science Foundation through the PIRE grant: OISE-0967140.

Appendix

We have seen above that the optimal (phase field) design often involves oscillatory interfaces or a diffuse interface (which we argue is also an indication of oscillatory interfaces). To understand when such rough interfaces can occur, we study the problem without a phase field penalization:

$$\begin{array}{@{}rcl@{}} O&=& \sup_{\chi\in \mathcal{X}}\inf_{u\in\mathcal{V}}\left\{L(u,\chi) = {\int}_{\Omega} \frac{1}{2} \sum\limits_{i = 1,2} k_{i} |\nabla u_{i}|^{2}\right.\\ &&\left.\quad\quad\quad\quad\quad\vphantom{{\int}_{\Omega} \frac{1}{2} {\sum}_{i = 1,2}}+ \frac{1}{2} \chi(1-\chi)u \cdot A u -\lambda\chi d x\right\}. \end{array} $$

(10)

where $\lambda \in {\mathbb R}$ is given and

$$\begin{array}{@{}rcl@{}} \mathcal{X} &=& L^{2}({\Omega};[0,1]) \quad \text{and} \quad \mathcal{V}=\{v\in H^{1}({\Omega};\mathbb{R}^{2}):\\ v_{i}&=&u_{i}^{*}\text{ on } \partial_{i}{\Omega},i = 1,2\}. \end{array} $$

We have the following explicit characterization of the optimal design problem (10).

Theorem

We have

$$O = \min_{v\in\mathcal{V}} {\int}_{\Omega} \overline{W}(v,\nabla v)dx. $$

where

$$\overline{W}(v,\xi) = \left\{\begin{array}{lll} \frac{1}{2} \left( |\xi_{1}|^{2} k_{21}+|\xi_{2}|^{2} k_{22} \right) & \quad(v,\xi) \in {\mathcal R}_{0}, \\ \\ \displaystyle{ \frac{ \left( {\sum}_{i} {\Delta} k_{i} |\xi_{i}|^{2}\right)^{2} + 2 {\sum}_{i} |\xi_{i}|^{2} (k_{v}(k_{i1}+k_{i2}) - 2\lambda {\Delta} k_{i}) + (k_{v} - 2 \lambda)^{2}}{8k_{v}}} &\quad(v,\xi) \in {\mathcal R}, \\ \\ \frac{1}{2} \left( |\xi_{1}|^{2} k_{11}+|\xi_{2}|^{2} k_{12}-2 \lambda \right) & \quad(v,\xi) \in {\mathcal R}_{1} \end{array}\right. $$

with

$$\begin{array}{@{}rcl@{}} {\mathcal R}_{0} &=& \{ (v,\xi): \sum\limits_{i} {\Delta} k_{i} |\xi_{i}|^{2} - 2\lambda \leq -k_{v} \}, \\ {\mathcal R} &=& \{ (v,\xi): -k_{v} < \sum\limits_{i} {\Delta} k_{i} |\xi_{i}|^{2} - 2\lambda < k_{v} \}, \\ {\mathcal R}_{1} &=& \{ (v,\xi): \sum\limits_{i} {\Delta} k_{i} |\xi_{i}|^{2} - 2\lambda \geq k_{v} \}, \end{array} $$

and

$${\Delta} k_{i} = k_{i1} - k_{i2} , \quad k_{v} = k_{s}(v_{1}-v_{2})^{2}. $$

This explicit characterization provides insights into when we see individual phases and when we see diffuse intermediate phases. The region ${\mathcal R}_{0}$ is where χ = 0 and we have pure phase 2 while the region ${\mathcal R}_{1}$ is where χ = 1 and we have pure phase 1. The region ${\mathcal R}$ is where we have χ ∈ (0, 1) or the mixed phase region (which we interpret as fine scale structure). This region is shown as the shaded region in Fig. 10 along with the contours of the function $\overline {W}$ for various parameters. Note that when λ = 0 and the two diffusivities and gradients are equal k₁₁ = k₂₂, |ξ₁| = |ξ₂|, then we are in the mixed phase (or rough interface) regime. This is consistent with the numerical examples studied earlier.

Fig. 10
Contour plot of $\overline {W}$ for fixed v, λ. The shaded regions indicate where mixed phase (χ ∈ (0,1)) occurs. ak₁₁ = k₂₂ = k_s = 1, k₁₂ = k₂₁ = 0.1,(v₁ − v₂)² = 1, λ = 0. b Parameters as in (a) except k₁₁ = 5, c parameters as in (a) except λ = 1, d parameters as in (a) except (v₁ − v₂)² = 10

Proof

For $v \in {\mathbb R}^{2}, \xi \in {\mathbb R}^{2\times 2}, \chi \in {\mathbb R}$, set

$$\begin{array}{@{}rcl@{}} W(v,\xi,\chi) &=& \frac{1}{2} \sum\limits_{i = 1,2} (\chi k_{i1} + (1-\chi)k_{i2}) |\xi_{i}|^{2}\\ &&+ \frac{k_{s}}{2} \chi(1-\chi) v \cdot A v -\lambda\chi \end{array} $$

and

$$ \overline{W}(v,\xi) = \max_{\chi \in [0,1]} W(v,\xi,\chi) . $$

(11)

Note that

$$O = \sup_{\chi\in \mathcal{X}} \inf_{v\in \mathcal{V}} {\int}_{\Omega} W(v, \nabla v, \chi) dx $$

We can show (see Collins (2017) for details) using Proposition 2.4 of Ekeland and Témam (1999) that the saddle point problem is attained and the order can be reversed. Specifically, there exists $\bar {v} \in {\mathcal V}$, $\bar {\chi } \in {\mathcal {X}}$ such that

$$\begin{array}{@{}rcl@{}} O &=& {\int}_{\Omega} W(\bar v, \nabla \bar v, \bar \chi) dx = \sup_{\chi\in \mathcal{X}} \inf_{v\in \mathcal{V}} {\int}_{\Omega} W(v, \nabla v, \chi)\\ dx &=& \inf_{v\in \mathcal{V}}\sup_{\chi\in \mathcal{X}} {\int}_{\Omega} W(v, \nabla v, \chi) dx \\ &=& \inf_{v\in \mathcal{V}} {\int}_{\Omega} \overline{W} (v,\nabla v) dx. \end{array} $$

It remains to compute $\overline {W}$. To that end, note that for a fixed v, ξ, W(v, ξ, χ) is quadratic in χ. Therefore, the equation

$$\frac{\partial W}{\partial \chi}(v,\xi,\chi) = 0 $$

has a unique solution χ = χ^∗. A simple calculation reveals

$$\chi^{*}(v,\xi)=\frac{ {\sum}_{i} {\Delta} k_{i} |\xi_{i}|^{2} + k_{s}(v_{1}-v_{2})^{2} -2 \lambda }{2 k_{s} (v_{1}-v_{2})^{2}}. $$

Notice that

$$\chi^{*}(v,\xi) \left\{\begin{array}{lll} \le 0 &\quad (v,\xi) \in {\mathcal R}_{0}, \\ \in (0,1) & \quad(v,\xi) \in {\mathcal R},\\ \ge 1 &\quad (v,\xi) \in {\mathcal R}_{1}. \end{array}\right. $$

A longer, but straightforward, calculation reveals that

$$W(v,\xi,\chi^{*}) = \frac{ \left( {\sum}_{i} {\Delta} k_{i} |\xi_{i}|^{2}\right)^{2} + 2 {\sum}_{i} (|\xi_{i}|^{2} k_{v}(k_{i1}+k_{i2}) - 2\lambda {\Delta} k_{i}) + (k_{v} - 2 \lambda)^{2}}{8k_{v}} . $$

Similarly,

$$\begin{array}{@{}rcl@{}} W(v,\xi,0) &=& \frac{1}{2} \left( |\xi_{1}|^{2} k_{21}+|\xi_{2}|^{2} k_{22} \right), \\ W(v,\xi,1) &=& \frac{1}{2} \left( |\xi_{1}|^{2} k_{11}+|\xi_{2}|^{2} k_{12}-2 \lambda \right) . \end{array} $$

Now, we can verify by explicit calculation that

$$\begin{array}{@{}rcl@{}} W(v,\xi, \chi^{*}) - W(v,\xi, 0) &=& \frac{k_{v}}{2} (\chi^{*}(v,\xi))^{2} \\ W(v,\xi, \chi^{*}) - W(v,\xi, 1) &=& \frac{1}{4} (\chi^{*}(v,\xi)-1 )^{2} \\ W(v,\xi, 1) - W(v,\xi, 0) &=& \frac{1}{2} \left( {\sum}_{i} {\Delta} k_{i} |\xi_{i}|^{2} - 2 \lambda \right) \end{array} $$

(12)

We obtain the desired result by recalling (11), rewriting

$$\overline{W} (v,\xi) = \max \{ {\Psi}(v,\xi) W(v,\xi, \chi^{*}), W(v,\xi, 0), W(v,\xi,1) \} $$

where

$${\Psi}(v,\xi) = \left\{\begin{array}{ll} 1 & \quad(v,\xi) \in {\mathcal R}\\ - \infty &\quad else \end{array}\right. $$

and using (12). □

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Authors and Affiliations

Lincoln Collins
- 1
Kaushik Bhattacharya
- 2
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1.Sandia National LaboratoriesAlbuquerqueUSA
2.California Institute of TechnologyPasadenaUSA

Optimal design of a model energy conversion device

Abstract

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Acknowledgments

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