Speaker
Description
Computational Fluid Dynamics (CFD) analysis of reactive flows over heterogeneous catalysts is a challenging task even for “simple” laminar flows. This involves solving a system of Partial Differential Equations (PDEs) with highly non-linear boundary conditions imposed by the surface chemistry which leads to stiffness and bad conditioning of the overall equation system requiring extensive computational effort. To address this challenge, we recently developed reduced order models (ROMs) for heterogeneous catalytic systems (a,b). These ROMs have two advantages: (1) Their pre-processing or offline part is cheaper than other model order reduction methods or machine learning approaches as it does not require solving the high-fidelity simulations and (2) the online part requires only solving non-linear equations (NLEs) rather than a partial differential equation (PDE) system. Since the ROMs enable solving the governing non-linear PDE system with minimal computational effort, it becomes feasible to couple them directly with an optimizer as opposed to full scale CFD. Therefore, one can think of extending these ROMs to applications such as reactor design optimization. In the context of heterogeneous catalysis, this could revolutionize how catalytic reactors are designed especially with the onset of 3D printing techniques.
To this end, we couple the ROMs with the shape optimization toolbox Fireshape (c), which implements a moving-mesh approach to update the computational domain and minimize a target function, which in our case could be the product yield or selectivity. Fireshape has already been used effectively for optimizing geometries where the flow is described by incompressible Navier Stokes.
The complete workflow for optimal reactor design is as follows: The optimizer passes the initial shape to the finite element solver – Firedrake (d) for computation of the non-reactive part of flow comprising the incompressible Navier Stokes equations. The transport properties and velocity fields are computed in this step. Then for the same shape, snapshots (which are PDE systems with piecewise constant Neumann boundary conditions) are generated using Firedrake. The reactive part of the solution is assumed to be a linear combination of these snapshots. The coefficients for each snapshot are then computed via a non-linear algebraic equations’ solver. The non-reactive and reactive solutions are combined to estimate the ROM solution. This solution is then passed back to the optimizer which computes the next iterate (shape) based on the gradients computed via automatic differentiation. In this work, we apply this reactor design workflow on different kinetic systems for catalytic monolith geometries.
References:
a) Muhammad Uzair Qureshi et al. Reduced order CFD modeling approach based on the asymptotic expansion An application for heterogeneous catalytic systems : Chemical Engineering Journal 504 (Jan. 2025), p. 158684.
b) S. Matera, C. Merdon, and D. Runge. Reduced Basis Approach for Convection-Diffusion Equations with Non-linear Boundary Reaction Conditions””. In: Finite Volumes for Complex Applications X Volume 1, Elliptic and Parabolic Problems. Springer Nature Switzerland, 2023, pp. 335 343.
c) Alberto Paganini and Florian Wechsung. Fireshape: a shape optimization toolbox for Firedrake””. In: Structural and Multidisciplinary Optimization 63.5 (Feb. 2021), pp. 2553 2569.
d) Ham, David A., et al. Firedrake User Manual. First ed., Imperial College London; University of Oxford; Baylor University; University of Washington, May 2023.