Towards Integration of CFD and Photosynthetic Reaction Kinetics in Modeling of Microalgae Culture Systems

Abstract

Despite the growing interest in photosynthetic microalgae, e.g., as a source of biofuels, the mass cultivation of microalgae still exhibits many drawbacks and successful industrial applications are scarce. Reliable model integrating all relevant phenomena and enabling to assess the system performance, its in silico scale-up and eventually optimization, is still lacking. Here, we present a unified modeling framework for microalgae culture system. We propose a multidisciplinary modeling framework to bridge biology (cell growth) and physics (hydrodynamics and optics) together. This framework consists of (i) the state system (mass balance equations in form of advection-diffusion-reaction PDEs), (ii) the fluid flow equations (Navier-Stokes equations), and (iii) the irradiance distribution. To validate the method, the Couette-Taylor reactor, which hydrodynamically induces the fluctuating light conditions, was chosen. The results of numerical simulation of microalgae growth in this device show good agreement with experimental data.

Appendix

Model of Photosynthetic Factory – PSF Model

Here we describe the multi-timescale three-state model of photosynthetic factory (PSF model) proposed by Eilers and Peeters [8] and further developed by Wu and Merchuk [26]. As follows, PSF model is used for the reaction term $R(c_i)$ derivation, cf. the transport equation (1). PSF model (9) describes the dynamics of three basic phenomena occurring simultaneously in three largely separated time-scales: (i) cell growth (including the shear stress effect), (ii) photoinhibition, and (iii) photosynthetic light and dark reactions. The state vector of the PSF model is three dimensional, $y=(y_R,y_A,y_B)^{\top }$, where $y_R$ represents the probability that PSF is in the resting state R, ${y_A}$ the probability that PSF is in the activated state A, and ${y_B}$ the probability that PSF is in the inhibited state B.

$$\begin{aligned} \left[ \begin{array}{c} \dot{y_R} \\ \dot{y_A} \\ \dot{y_B} \end{array} \right] = \left[ \begin{array}{rrr} 0 &{} \gamma &{} \delta \\ 0 &{} - \gamma &{} 0 \\ 0 &{} 0 &{} - \delta \end{array} \right] \left[ \begin{array}{c} y_R \\ {y_A} \\ {y_B} \end{array} \right] + u(t) \left[ \begin{array}{rrr} - \alpha &{} 0 &{} 0 \\ \alpha &{} - \beta &{} 0 \\ 0 &{} \beta &{} 0 \end{array} \right] \left[ \begin{array}{c} y_R \\ {y_A} \\ {y_B} \end{array} \right] \end{aligned}$$

(9)

The values of model parameters, i.e., $\alpha ,~\beta , ~\gamma ,~\delta$, and $\kappa$, cf. 6, are published in Wu and Merchuk (2001) [26]. Concerning the experimental design for parameter estimation and the identifiability study, see [18].

For given input variable, i.e., the irradiance u(t), the ODE system (9) can be solved either by numerical methods or by asymptotic methods. For the special case of the periodic piecewise constant input, the state trajectories were calculated explicitly in [15].