(308e) Geometrical Abstraction Using Mazes to Guide Rational Design of Engineered Systems

Engineers can control the internal geometry of materials down to the micro- and nano- scale; however, it is near impossible to know what geometrical configuration is ideal for a given, multi-phase transport process. An abstraction of the real geometry is thus necessary to develop a fundamental understanding of how geometry affects transport.

Methods

We use mazes to abstractly represent the geometries of two-phase transport systems. A plethora of maze generation algorithms allows for one to generate a maze suitable for a given transport system. The example transport system we consider is single-phase fluid flow in a commercial membrane (see Figure 1a). The maze representation of this membrane (see Figure 1b) is designed to capture the skin layers (regions of smaller pore size) present at the top and bottom of the membrane.

A further abstraction is performed by using a graph to represent the maze (see Figure 1c), where the nodes represent the center of the maze cells and edges represent flow connections (negligible edges/connections are removed). Three graph parameters can be extracted from the graph and used to gauge the transport performance of the maze. The first graph parameter is the effective tortuous resistance , and it represents the normalized, length resistance of the graph; can be used to directly calculate the permeability of the maze. The second graph parameter is the average tortuosity , and it is a normalized average path length of the graph; can provide an estimate of the average residence time within the maze. The third graph parameter is the minimum-cut-size κ, and it is a measure of the robustness of the graph to edge removal; κ is correlated to the resilience of the maze geometry to channel blockage (e.g., by a particle). The values of these graph parameters are shown in Figure 1c.

Results

Numerical simulations of the membrane and maze were compared. There is qualitative agreement between the flow field of the two systems. The nonlinear pressure drop caused by the skin layers in the membrane is also qualitatively present for the maze. The permeability of the maze calculated from the simulation was 14.9% less than that of the membrane. It is important to note that the goal here is not to achieve matchup of permeability – it is to show that mazes are able to capture the key features of a transport system.

Figure 1: (a) Commercial polyethersulfone membrane, with the fluid phase in grey. (b) Maze representation of membrane, with the fluid phase in white. (c) Graph representation of maze, with the fluid phase represented by the grey edges, and the extracted graph parameters.

The permeability of the maze calculated using the first graph parameter, , was 15.2% less than the permeability of the maze calculated using the full numerical simulation. While not as accurate, the graph theory approach is faster and will enable screening of many more geometries to find common or hidden patterns. For membrane separations, the selectivity and fouling of the maze are equally, if not more important than the permeability of the maze. Developing quantitative correlations between the second and third graph parameters to the selectivity and fouling of a maze, respectively, is the subject of future work.

Implications/Conclusions

A maze can serve as a high-fidelity, geometrical abstraction of a two-phase transport system. Mazes are the most suitable type of abstraction since maze generation algorithms are highly tunable and thus allow for a wide variety of maze configurations. Furthermore, a maze can be readily converted into a graph. The advantage of a graph representation is that relevant graph parameters can be extracted and related to key aspects of the maze’s transport performance without performing expensive simulations. For example, the effective tortuous resistance, , is a graph parameter which can be used to directly calculate the maze’s effective transport coefficient (e.g., permeability, effective thermal conductivity, or effective diffusion coefficient). For a given transport process, other metrics relevant to the maze’s transport performance should be identified and related to additional graph parameter. By leveraging the tunability of maze generation algorithms and the computational feasibility of the graph representation, a form of guided natural selection can be applied to find the optimal or "fittest" maze/graph. With the geometry of the optimal maze in hand, one can either use it to guide the rational design of engineered systems or turn to additive manufacturing to fabricate the maze for direct usage.