[Paper Review] Topography optimization for enhancing microalgal growth in raceway ponds

Olivier Bernard, Liu‐Di Lu+2 more|arXiv (Cornell University)|Nov 9, 2020

Algal biology and biofuel production17 references4 citations

TL;DR

This study proposes a topography optimization framework for microalgal raceway ponds using a coupled hydrodynamic-biological model based on the Han photosynthesis model and shallow water equations. The optimization, solved via adjoint-based methods, reveals that while a flat topography is optimal in periodic laminar flow, introducing a paddle wheel mixer yields a non-flat optimal topography with only a 0.217% increase in average biomass productivity.

ABSTRACT

Modelling the evolution process for the growth of microalgae in an artificial pond is a huge challenge, given the complex interaction between hydrodynamics and biological processes occurring across various timescales. In this paper, we consider a raceway, i.e., an oval pond where the water is set in motion by a paddle wheel. Our aim is to investigate theoretically and numerically the impact of bottom topography in such raceway ponds on microalgae growth. To achieve this goal, we consider a biological model based on the Han model, coupled with the Saint--Venant systems that model the fluid. We then formulate an optimization problem, for which we apply the weak maximum principle to characterize optimal topographies that maximize biomass production over one lap of the raceway pond or multiple laps with a paddle wheel. In contrast to a widespread belief in the field of microalgae, we show that a flat topography in a periodic regime satisfies the necessary optimality condition, and observe in the numerical experiments that the flat topography is actually optimal in this case. However, non-trivial topographies may be more advantageous in alternative scenarios, such as when considering the effects of mixing devices within the model. This study sheds light on the intricate relationship between bottom topography, fluid dynamics, and microalgae growth in raceway ponds, offering valuable insights into optimizing biomass production.

Motivation & Objective

• To investigate whether non-flat pond topographies can enhance microalgal biomass productivity in raceway systems.
• To develop a coupled physical-biological model integrating shallow water dynamics with the Han photosynthesis model for accurate light and growth prediction.
• To design an adjoint-based optimization framework that determines the optimal topography to maximize average biomass growth rate.
• To evaluate the impact of mixing devices, such as paddle wheels, on the optimality of topography and productivity gains.
• To assess whether the marginal gains from non-flat topographies justify the engineering complexity of implementation.

Proposed method

• Coupling the Han model for photosystem dynamics with shallow water equations to simulate fluid flow and light exposure in raceway ponds.
• Modeling the net specific growth rate as a function of light intensity and photo-inhibition state C, derived via singular perturbation theory.
• Formulating an optimization problem to maximize the time-averaged growth rate over one or multiple laps under periodic boundary conditions.
• Implementing an adjoint-based optimization scheme to compute gradients of the objective functional with respect to topography parameters.
• Parameterizing the bottom topography using a truncated Fourier series and solving the optimization with multiple initial guesses to ensure convergence.
• Simulating multiple laps with a paddle wheel modeled as a permutation matrix to enforce periodic reordering of algal trajectories.

Experimental results

Research questions

• RQ1 Is a non-flat topography more effective than a flat one for enhancing microalgal productivity in raceway ponds?
• RQ2 How does the inclusion of a paddle wheel mixer affect the optimality of pond topography and biomass growth rate?
• RQ3 What is the magnitude of productivity gain achievable through topography optimization in realistic mixing scenarios?
• RQ4 Does the optimal topography depend on initial conditions such as the initial photo-inhibition state C?
• RQ5 How do numerical discretization and Fourier truncation order affect the stability and convergence of the optimization results?

Key findings

• In a periodic laminar regime without mixing, a flat topography is proven to be optimal, as it eliminates the gradient of the objective functional.
• When a paddle wheel is introduced to enforce periodic mixing, the optimization yields a non-flat optimal topography.
• The non-flat optimal topography results in a 0.217% increase in the average growth rate compared to a flat topography.
• The improvement over a non-permuting (non-mixing) case is approximately 0.265%, indicating minimal gains from topography optimization under mixing.
• The convergence of the objective functional is observed as spatial resolution increases, with N_z = 50 selected for final simulations.
• The optimal topography is sensitive to initial photo-inhibition state C₀, with distinct shapes emerging for C₀ = 0.1 and C₀ = 0.9, though differences persist even at high resolution.