## 1 Introduction  
Microalgae are small organisms that convert sunlight into energy in the presence of CO\({}_{2}\) and nutrients. Microalgae have been studied intensively in the past because of their application in food, medicines, and compounds for biofuels [1; 2]. Different mass culture systems have been used for the production of microalgae. Mainly, they are divided into open and closed photobioreactors. Closed photobioreactors are used for high-quality biomass production, but they require greater power consumption, thus making it economically expensive [3]. Open raceway ponds are being widely used for the large-scale culturing of microalgae in the world. The advantages of raceways are their relatively low cost and simple design. However, because of the high culture thickness in raceways (15-30 cm), the photo inhabitation can affect the photosynthesis phenomenon and consequently the algal productivity. They operate at low fluid velocity, and thereby, a paddle wheel is required to keep algal cells in suspension [4; 5]. An alternate approach for the mass production of microalgae is a thin-layer cascade reactor first introduced by Dr. [PERSON] in the 1960s [6]. A thin-layer cascade reactor is an open system for the commercial-scale cultivation of microalgae [7; 8]. A thin-layer cascade (TLC) reactor consists of shallow channels with a thin layer (thickness \(\leq 10\) mm) of algal culture medium. The thin layer of algal suspension improves light utilization and results in high biomass density (25-35 g/L) [8; 9]. Photinhibition is a major challenge in TLC reactors because it considerably affects the photosynthetic efficiency of algal cells and consequently reduces biomass production [9; 10; 11]. Light attenuation occurs as the liquid layer becomes thicker in a TLC reactor, which decreases the local light penetration. A steep gradient in light attenuation is expected along the culture depth. Thus, effective culture mixing is necessary to prevent photoinhibition in TLC reactors. Culture mixing also prevents cell sedimentation in the TLC reactor. The increase in layer thickness escalates the chances of cell sedimentation by increasing the residence time of algal cells in a TLC reactor. Precisely, an increase in layer thickness results in a twofold increase in the residence time of algae cells/decrease in algae cell mixing [12; 13]. Culture mixing significantly depends on various design parameters of TLC reactors, such as channel dimensions and slope angles and flow rates. At high cultural thickness, algae cell mixing can be improved by increasing flow velocity [7]. However, the power required for mixing the suspension increases with the increase in cultural thickness and inclination of the slope [14]. Therefore, TLC reactors with small culture thicknesses are favorable for high biomass densities of microalgae. The small suspension layer limits the night biomass loss by respiration and promotes faster warming of the culture at the start of cultivation, thereby preventing the strong photoinhibition of algal growth [15; 16; 17]. The algal productivity is strongly influenced by the inclination angle of the reactor and water depth. Under optimal climate conditions, the increase in channel length and slope angle causes a substantial increase in algal biomass productivity because of the increase in photosynthetic efficiency. However, the night biomass loss sufficiently increases with the increase in culture thickness [14]. The algal productivity in TLC is considerably affected by the surface-to-volume ratio. A TLC reactor unit with a large surface-to-volume ratio is a more efficient system for biomass density (about 30%) as compared to the unit with a small surface-to-volume ratio [8]. However, at low volumes of suspension, the utilization of light energy and algal growth can be increased by vigorous culture mixing [18]. The photon energy received by each cell is also considerably affected by the length of the optical path (thickness of culture layer), the design of the cultivation unit, and the rate of mixing.