Topography optimization for enhancing microalgal growth in raceway ponds

TL;DR

This work develops a coupled hydrodynamic-biological framework to optimize bottom topography in raceway ponds for maximizing microalgal biomass. By integrating a 1D Saint-Venant flow model with the Han photosynthesis dynamics and using adjoint-based optimization under a weak maximum principle, the authors show that a flat topography is optimal in a periodic laminar regime, with only modest gains possible from non-flat geometries when mixing devices are present. Numerical experiments, parameterized by Fourier series for the depth, reveal convergence with vertical discretization and highlight the conditions under which non-flat shapes can marginally improve productivity, especially when paddle-wheel mixing is considered. The study provides practical guidance for bathymetry design in laminar raceways and lays groundwork for exploring more complex, turbulent regimes and richer mixing strategies.

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.

Figures (5)

• Figure 1: Representation of the one dimensional hydrodynamic model.
• Figure 2: Han's model, describing the state transition probability, as a function of the photon flux.
• Figure 3: Values of the functional $\bar{\mu}_{N_z}$ for $N_z = [1, 80]$.
• Figure 4: Optimal topography for $C_0=0.1$ (left) and $C_0=0.9$ (right). The red thick line represents the topography $z_b$, the blue thick line represents the free surface $\eta$, and all the other curves between represent the different trajectories. $\bar{\mu}_{N_z}(0)$: flat topography, $\bar{\mu}_{N_z}(\boldsymbol{a}^*)$: optimal topography.
• Figure 5: Optimal topography (left) and evolution of the photoinhibition state $C$ (right) over two laps.

Theorems & Definitions (17)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.1
• Remark 3.1
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Remark 3.2