### Inoculum preparation and cultivation system

The microalgae strains were cultivated at a temperature of 20 ± 1 °C, aeration flow rate of 1 *V*/*V* and irradiance of 20.3 μmol·m^{− 2}·s^{− 1} of white cool fluorescent light for 10 days in flasks using BG11 medium. The inoculum concentration was defined for each strain depending on the chlorophyll content, which was determined via methanol extraction (Fogg and Thake 1987). Chlorophyll a is used as an algal biomass indicator, and its constituents are on the average, 1.5% of the dry weight of organic matter of algae (Association APHA 1915). Microalgae were harvested via settling without additives for (30–60 min), and the aeration and irradiance cultivation conditions accelerate the growth rate and increase the pH to 10.5 by the end of the cultivation period, thereby facilitating settling without additives. The sediments were collected and washed twice with water and then centrifuged at 3000 rpm for 10 min. The algal cells were dried at 60 °C overnight. Oil was extracted from dried weighed samples using hexane and isopropanol in a ratio of 3:2 according to the method applied by Halim et al. (2012); 300 mL of the mixture was added to 4 g of microalgae powder; and the mixture of algal powder and solvents was homogenized at 800 rpm for 10 min to rupture the cell wall. The extraction process was applied at 40 °C for 2 h, and the mixture was filtered. The filtrate was washed with water to allow the miscible solution to separate according to polarity.

### Strain selection

The four isolated strains of *S. platensis*, *S. obliquus*, *Nannochloropsis sp.*, and *M. aeruginosa* were supplied by the Water Pollution Research Department, Environmental Research Division, National Research Centre in Cairo, where the toxicity test of using microcystin was applied to *M. aeruginosa* because certain species of Microcystis are toxic. The four strains were cultivated using BG11 medium in an airlift bubble column photobioreactor (PBR, as shown in Fig. 1) at a temperature of 20 ± 1 °C, aeration flow rate of 1 *V*/*V* and irradiance of 20.3 μmol·m^{− 2}·s^{− 1} for 10 days. After harvesting, the oil concentration and biomass yield of the four strains were compared to select the most suitable strain for oil production.

### Optimization of PBR dimensions and cultivation conditions

The PBR design is based on optimizing the parameters that affect microalgae growth. The main parameter that maximizes algal growth is capturing optimal light, which implies a high illuminated surface area/culture volume ratio (S/V) (Richmond 2004). Thus, four duplicated experiments at various S/V ratios (1.8, 0.9, 0.63, and 0.48) using the previous cultivation conditions were performed to optimize the biomass yield.

### Selected models describing the algal growth kinetics

The biomass yield can be expressed as the photosynthetic yield, which can be predicted by growth models. Each species has its own growth characteristics, which differ from that of the others. Thus, the optimal growth model for describing the selected strain *M. aeruginosa* must be identified. One such model is the hyperbolic tangent model (Kurano and Miyachi 2005):

$$ P={P}_{\mathrm{max}}.\tanh \left(\alpha I\right) $$

(1)

where

*P* represents the growth rate (mg·L^{− 1}·d^{− 1});

*I* represents the irradiance (μmol·m^{− 2}·s^{− 1}); and

*α* is the model constant, which was obtained experimentally (mg·L^{− 1}·d^{− 1}/μmol·m^{− 2}·s^{− 1}).

The second model is the Monod model (Jalalizadeh 2012; Kurano and Miyachi 2005; Perez et al. 2008; Sundstrom and Klei 1979; Tamiya 1951), which expresses the growth rate in the following form:

$$ \mu ={\mu}_{\mathrm{max}}I/\left(I+{K}_I\right) $$

(2)

where

*μ* represents the specific growth rate (d^{− 1});

*I* represents the irradiance (μmol·m^{− 2}·s^{− 1}); and

*K*_{I} is the model constant, which was obtained experimentally (μmol·m^{− 2}·s^{− 1}).

The third model is a modified Monod model that considers light irradiances as the substrate (Bechet et al. 2013; Jeon et al. 2005):

$$ P={P}_{\mathrm{max}}I/\left(I+{I}_k\right) $$

(3)

where

*I*_{k} is the saturated light intensity (μmol·m^{− 2}·s^{− 1}).

The experimental data and constants obtained from eight duplicated runs at irradiance values ranging from 20.3 to 176 μmol·m^{− 2}·s^{− 1} were used to verify the most applicable model for the selected algal strain. The three models were evaluated by calculating the absolute average deviation (AAD) in the following form (Mejia et al. 2013):

$$ \mathrm{AAD}=\left(1/N\right)\sum \mid \left(\mathrm{Act}.-\mathrm{Pred}.\right)/\mathrm{Act}.\mid $$

(4)

where

*AAD* means absolute average deviation;

Act. means actual value;

Pred. means predicted value; and

*N* means number of set values.

### Effect of irradiance and initial inoculum concentration on the biomass yield of *M. aeruginosa* and oil concentration

The irradiance and initial inoculum concentration are considered the most significant parameters affecting algal growth that obey phototrophic cultivation (Posten 2009). The range of the examined irradiance was 27–81 μmol·m^{− 2}·s^{− 1}, and the maximum value of irradiance was determined from the photosynthetic yield–irradiance relation (P–I curve) via the intersection between the initial tangent of the curve and the maximum photosynthetic rate. A dense culture retards light penetration via the PBR wall (Bezerra et al. 2011; Coles and Jones 2000; Jeon et al. 2005). Thus, a range of inoculum concentrations (15.5–67 mg biomass/L) was selected for study. The RSM was chosen as a suitable route for optimizing the interactive effect of irradiance (*I*) and the initial inoculum concentration (*C*) on the oil concentration and biomass yield (Montgomery 2003; Silva et al. 2013). The experimental results were statistically analyzed and modeled using the RSM according to Eq. 5 (Montgomery 2003), which was applied using Design Expert-6.0.8 software during a trial period. The extent of the fit of the model was evaluated using the coefficient of determination and analysis of variance (ANOVA).

$$ Y={a}_0+{a}_1{X}_1+{a}_2{X}_2+{a}_{11}{X_1}^2+{a}_{22}{X_2}^2+{a}_{12}{X}_1{X}_2 $$

(5)

where

*a*_{0} = the regression constant;

*a*_{0}, *a*_{1}, *a*_{2}, *a*_{11}, *a*_{22}, *a*_{12} = regression coefficients; and

*X*_{1} and *X*_{2} = independent variables investigated here.