This paper focuses on mixing strategies to enhance the growth rate in an algal raceway system. A mixing device, such as a paddle wheel, is considered to control the rearrangement of the depth of the algae cultures hence the light perceived at each lap. The dynamics of the photosystems after a rearrangement is accounted for by the Han model. Our approach consists in considering permanent regimes where the strategy is parametrized by a permutation matrix which modifies the order of the layers at the beginning of each lap. It is proven that the dynamics of the photosystems is then periodic, with a period corresponding to one lap of the raceway whatever the order of the considered permutation matrix is. An objective function related to the average growth rate over one lap is then introduced. Since N ! permutations (N being the number of considered layers) need to be tested in the general case, it can be numerically solved only for a limited number of layers. Consequently, we propose a second optimization problem associated with a suboptimal solution of the initial problem, which can be determined explicitly. A sufficient condition to characterize cases where the two problems have the same solution is given. Some numerical experiments are performed to assess the benefit of optimal strategies in various settings.
翻译:本文侧重于混合战略,以提高藻类赛道系统中的增长率。 混合装置, 如桨轮等, 被视为控制藻类培养深度的重新排列, 从而控制每圈的光线。 汉型模型计算了重新安排后光系的动态。 我们的方法是考虑永久制度, 使战略在每圈开始时改变层的顺序。 事实证明, 光系系统的动态是周期性的, 与赛道的一圈相对应的时段, 不论考虑的变换矩阵的顺序如何。 与每圈的平均增长率有关的客观功能随后被引入。 由于一般情况下需要测试 N! 变换( 是所考虑的层数 ), 因此, 只能在数量有限的层次上用数字方式解决 。 因此, 我们提出第二个最优化的问题与最初问题的次优化解决方案有关, 这个问题可以明确确定。 一个充分的条件, 用来确定两个问题在两种情况下具有相同解决方案的案例的特征 。 一些最优化的实验将进行。