**O. Akpolat**

Department of Bioengneering, Faculty of Engineering, Ege University, 35100, Bornova,Izmir, Turkey

S. Eristurk

Graduate School of Natural and Applied Sciences, Bioengineering Branch,Ege University, 35100, Bornova,Izmir, Turkey

Department of Bioengneering, Faculty of Engineering, Ege University, 35100, Bornova,Izmir, Turkey

S. Eristurk

Graduate School of Natural and Applied Sciences, Bioengineering Branch,Ege University, 35100, Bornova,Izmir, Turkey

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O. Akpolat and S. Eristurk, 2007. An Algorithm for Modeling and Simulation of Microalgea Production. *Journal of Applied Sciences, 7: 2848-2855.*

**DOI:** 10.3923/jas.2007.2848.2855

**URL:** https://scialert.net/abstract/?doi=jas.2007.2848.2855

Microalgea has great importance by prducing of high value chemicals and their proven antiviral activities and by ecologically role in climate modeling as the biggest primary producer of oxygen worldwide (Csogor et al., 1999). Microalgea can synthesis themselves organic materials to need for cell activities by photosynthesis. Intermediate metabolites by photosynthesis are basically proteins, fatty acids, keratenoides, vitamins, antibiotics and lots of high value chemicals. *Heamatococcus pluvialis* is also a green microalgea grown authotrofically and it has a four cell type following each other along its life cycle.

During the cultivation of algae like *Heamatococcus pluvialis *growth of culture is determined by quantitative and qualitative methods. Qualitative methods are basically cell numbering, optical density, dry end wet weight, amount of chlorophyll and directly measuring of carbon context (EriŞtürk and Akpolat, 2005; Boussiba, 2000). Biomass composition, that means growth rate and product spectra depend strongly on the cultivation conditions. These are composition of medium, temperature, pH, carbon dioxide and oxygen supply and most important of all: illumination. Microalgeas as photosynthetic organisms need carbon dioxide and light within the Photosynthetically Active Radiation (PAR) to obtain energy. The wave length range of the PAR is between 400 to 700 nm, which is equal to visible light. Additionally it could be remembered that the use of excessive light and oxygen in cultivation has harm full effects. The radiation can be quantified adequately by the photon flux density (quantum of photons per area and time) (Csögör *et al*., 1999; Molina *et al*., 2001).

As to modeling of the cultivation kinetics, the cultivation process and the scale up of the process, because of the complex interactions between the production parameters of microalgea there is a big demand to develop a model which predicts the behavior of the whole system and a software algorithm by using numerical and optimization techniques for solving mathematical equations in the model (Csögör *et al*., 1999; Lübbert* et al*., 2000, 2001; Bailey and Ollis, 1986; Smith, 1970; Türker, 2005).

Here it is discussed the biomass production of algae experimentally and its mathematical modeling. Improved algorithm to solve the model equations has some main and sub algorithms to be explained details later on.

**Experimental:** In this study *Heamatococcus pluvialis* was cultivated for modeling purposes in a panel glass photobioreactor of 1.5 L volume shown in the experimental set-up on Fig. 1 under 75 μE m^{–2} s^{–1} light flux at the vessel surface at 25°C room temperature. The width of the reactor is 5 cm and the air sending to the reactor consists of CO_{2} of 1.5% V. The aeration rate was measured as 1.5 L min^{–1} by flow meter and *Heamatococcus pluvialis* strain U was used for inoculation in BG11 medium given in Table 1 (Sukatar, 2002).

Strains stored in agar at carrying glasses was firstly inoculated on solid agar with BG11 medium and incubated along 15 days. Later stock culture was prepared by transferring of inoculated strain into tube of 70 mL, flask of 250 mL and pyrex glasses of 1 and 5 L, respectively.

The sample of the volume to need for inoculation of the reactor medium was taken into sterilized flask firstly. After removing supernatant phase by precipitation of cells, fresh medium was added on the cells. This inoculated culture was grown along 4 days under 75 μE m^{–2} s^{–1} light flux and 1 L min^{–1} aeration rate conditions. Starting culture is 4x10^{5} cell numbers L^{–1} or 0.48 g L^{–1} for all experiments to do as three parallels. Reactors were sterilized by sodium hypo chloride solution of 1% w/v and cell accounting, biomass, chlorophyll-a, pH and viscosity measurements were repeated at the same time interval for all cultivation. Biomass measurements for the reaction conditions were given in Table 2 (EriŞtürk and Akpolat, 2005).

Biomass amounts measured experimentally were evaluated statistically using main programs coded 01 and related sub programs explained in modeling algorithm in part 2.4 and growth curve of the cultivation was lined. Statistically evaluation of the experimental results was also carried out by student t-test in a probability level of 95%. This test for a sample group with N elements consists of basically following steps (Ikiz *et al*., 1996).

• | Zero hypothesis (H_{0}): It is test hypothesis, here, x_{m}, x_{0}, d_{f}, P_{b}, σ, t-Calc and t-test show mean value,median value, degrees of freedom, probability level, standard deviation, calculated t value (or distribution function) and theoretical test t value from t-Table, respectively. |

Table 1: | Composition of the culture medium |

(1) |

(2) |

(3) |

(4) |

• | Test statistics: t-statistics value (t-test) is determined from t-test table related to degrees of freedom and probability level. |

• | If t-Calc<t-test, (H_{0}), is on an acceptable area and the sample group has a meaningful statistically. |

• | Regression analysis is done for this sample group at last. |

**Modeling of cultivation kinetics and cultivation process:** A photosynthetic reaction in a microalgea cell is basically as follows:

6 CO_{2} + 6 H_{2}O + light →C_{6}H_{12}O_{6} + 6 O_{2}

Glucose produced by this reaction is used for energy to need later either in anabolic or katabolic reactions or other cell activities by breaking down into CO_{2} and H_{2}O, or directly starting materials for smaller molecules like alcohols and acids produced enzymatically in catabolic path ways or for larger molecules like **amino acids** and pigments produced enzymatically in anabolic path ways (Bailey and Ollis, 1986; Türker, 2005).

Fig. 1: | Schematic view of the experimental set-up |

Table 2: | Biomass measurements for the cultivation of Heamatococcus pluvialis |

Biomass productivity or specific growth rate (μ) (h^{–1}) for microalgea was expressed as follow kinetic model basically depending on average irradiance (μE m^{–2} s^{–1}) (Molina, 2001)

(5) |

where, μ_{max}: maximum specific growth rate (h^{–1}), I_{AV}: an average irradiance constant (μE m^{–2} s^{–1}), I_{AK}: a constant dependent on algal species and cultivation conditions (μE m^{–2} s^{–1}) and n is an empirically established exponent.

Modeling of cultivation bases on batch process according to cell and cultivation rate in this model is shown by an ordinary differential equation as an initial value problem.

(6) |

where, μ: specific growth rate (h^{–1}), X: measured biomass concentration (g L^{–1}) and dX/dt: cultivation rate (g L^{–1} h^{–1}) (EriŞtürk and Akpolat, 2005).

Euler method for numerical solution of these types of equations is the simplest integration technique, which goes on following steps.

(7) |

(8) |

where, t_{i} = t_{0} and X_{i} = X_{0} are initial values of the problem and Δt is the time interval chosen for the numerical solution (Lübert, 2000). Calculated values of the variables by solving this differential equation depending on the initial values are used to plot simulation curves of the process.

Calculating the data of simulation needs to determine kinetic parameters in the kinetic model proposed for the experimental results in this work, which are μ_{max}, I_{AV}, I_{AK} and n, but I_{AV} was only measured directly here. For that reason, an optimization function based on biomass concentration was written depending on μ_{max}, I_{AK} and n values and minimized this function to give the sum of differences between the calculated (or simulated) by the solution of the differential equation and the experimental results.

(9) |

During the optimization of the calculated values for P function related with the case depending on chosen μ_{max}, I_{AK} and n values for minimization, minimal P to be found is optimal case and the chosen values for it are the kinetic parameters in the kinetic model proposed for the experimental results. Additionally, the optimization function results for all cases were evaluated statistically by student t-test as described earlier.

Finally, biomass-time simulation curves for different initial biomass concentration were plotted by using the kinetic parameters optimized.

Fig. 2: | Algorithm for modeling |

Here it is accepted that maximal productivities all simulations are the same as that of the experimental but reached in the shorter time than those of the experimental linearly. Calculation of the kinetic parameters, statistically evaluation of the optimization results and simulation of the cultivation process were carried out by using main programs coded 02 and 03 and related sub programs in modeling algorithm.

**Modeling of scale-up:** Because the objectives are different, operating conditions and/or reactor types are usually different for laboratory and large scale units and the global rate from the laboratory reactor can not be used directly for large scale apparatus. If all resistances to all transfers, internal or external, like heat and mass are significant, there will be a different global rate in the commercial-scale reactor for the conversion to be same as that of laboratory reactor (Smith, 1970).

Basically the conversions of chemical or biochemical reaction in the experimental conditions can be done also for larger volume reactors by scale up. In this part modeled of the scale up for microalgea cultivation because of the changes in reaction rates along with scale-up, as being the volumes larger by α_{i} times according to that of the experimentally the specific growth rates were accepted to be diminished by γ_{i}% according to that of laboratory conditions for the same conversions or the same growth rates of the microalgea cultivation. Model equations for this case are as follows, (i = 1 to m and m = 3);

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

Calculating of I_{AV} depending on changes of volume and of maximum specific growth rate during scale-up can be carried out by optimizing of I_{AV} similar to those of I_{AK} and n calculating. The optimization procedure was given earlier in details.

The numerical solution of the **mathematical model** equations explained for scale-up were conducted by using main programs coded 03 and related sub programs in modeling algorithm.

**Algorithm for modeling:** An algorithm for modeling which of steps given in earlier for algea cultivation kinetics, cultivation process and scale-up of the process was shown in Fig. 2.

This algorithm consists of six main programs and four sub programs. The programs in the algorithm are as follows;

Main_Program_01_0_Data_Save

Main_Program_01_1_Data_Load

Main_Program_01_2_Data_Statistics

Main_Program_02_0_Kinetic_Process_Model

Main_Program_03_0_Process_Simulation

Main_Program_04_0_Process_Scale_Up

Sub_Program_01_0_Statistics_Evaluation

Sub_Program_02_0_Simulation_Basic

Sub_Program_03_0_Euler_Solution

Sub_Program_04_0_Optimization_Procedure

A computation program based on the modeling algorithm given earlier can be written by any software like Visual Basic versions, MatLab versions, Matematica versions or etc. Calculated results by the algorithm were explained as follows.

**Statistically evaluation of microalgea growth:** The growth rate of *Heamatococcus pluvialis* plotted by using the data experimentally was given in Fig. 3. It is understood that biomass amounts in the 9th day reaches to a maximal of 4.2 g L^{–1} by an initial cell concentration of 0.48 g L^{–1} accepting of an average irradiance inside the reactor of 75 μE m^{–2} s^{–1} to be the same as that of on the surface of the reactor and it tends toward being lower following the 9th day. Measured all biomass values experimentally was evaluated by student-t-test for 0.95 probability and it was found them meaningful statistically.

**Simulation of cultivation kinetics and cultivation process:** As explained earlier details, the values of maximum specific growth rate (μ_{max}; h^{–1}), irradiance constant dependent on algal species and cultivation conditions (I_{AK}: μE m^{–2} s^{–1}) and empirically established exponent (n) for the kinetic model were chosen for the parameter groups in optimization procedure and each group were numbered optimization case.

Fig. 3: | The growth rate of microalgea for ten days |

Fig. 4: | Optimization function results |

Biomass amounts were simulated and optimization function values for all cases were calculated using by simulation results. The parameters in the case with minimal p-value are optimal for kinetic model proposed for the experimental results. The graphic of P function values dependent on kinetic parameters was plotted on Fig. 4.

Chosen parameters for optimization were in the following intervals.

(18) |

(19) |

Fig. 5: | Simulation curves of the microalgea productivity for different initial biomass concentrations |

Fig. 6: | Variation of average irradiation amounts depends on volumes changed in scale-up |

(20) |

As understood from the Fig. 4 determined kinetic parameters for optimal P (minimal P) value are μ_{max} = 0.04 g L^{–1}, I_{AK} = 35 and n = 0.8. Additionally the group of the calculated optimization function values was evaluated by student-t-test for 0.95 probability level and it was found that it has meaningful statistically.

Simulation curves of the microalgea productivity for different initial biomass concentrations based on determined kinetic parameters earlier were lined on Fig. 5. As shown in Fig. 5 initial biomass concentration was increased by 50% from 0.48 to 0.72 g L^{–1} depending on that of experimental.

**Scale-up:** Chosen reactor volumes for scale up were 1.5, 15 and 150 L increasing by 10 times and it was accepted that the specific growth rates depending on the changed reactor volumes decrease by 10% for each step of scale up. Calculated average irradiation amounts for the same conversions or the same growth rates for three different volumes of the reactors by scale up were given in Fig. 6 and these were 75, 110 and 210 μE m^{–2} s^{–1}, respectively, that means they increases of 30 and 50% approximately in average irradiations.

In this study biomass amounts were measured for the production *Heamatococcus pluvialis*, a green microalgea, in an experimental reactor of 0.5 L volume with an initial cell concentration 0.48 g L^{–1} and under an average irradiance of 75 μE m^{–2} s^{–1}. First a kinetic model as follows was proposed as given in literature

then the cultivation process was mathematically modeled as a function of growth rate and biomass production expressed by an ordinary differential equation depending on an initial microalgea concentration as follows

Simulation curves of biomass production were lined solving the kinetic and the process model equations numerically to optimize μ or μ_{max}, I_{AK} and n by minimization of P function consisting of their different values in a chosen interval. The optimization function was stated as the sum of differences between calculated and experimental biomass values as follows:

Additionally a scale-up modeling based on experimentally results was improved and finally, an algorithm was written for all numerical solution and statistically evaluation.

Calculated kinetic parameters by the algorithm were found as μ_{max} = 0.026 h^{–1}, I_{AK} = 35 μE m^{–2} s^{–1} and n = 0.8.

This study as a part of master thesis was financially supported by Ege University Research Fund, project number 2003 Muh 040. We also thankfully acknowledge for the financial support to Ege University.

**NOTATION**

d_{f} | : | Degrees of freedom (-) |

H_{0} | : | Zero hyphothesis (-) |

I_{AK} | : | Irradience constant (μE m^{–2} s^{–1}) |

I_{AV} | : | Average irradience constant (μE m^{–2} s^{–1}) |

N | : | No. of element in sample group (-) |

n | : | Emprically estabilished exponent (-) |

P_{b} | : | Probability (%) |

P | : | Optimization function (-) |

t-Calc | : | Calculated t-value (-) |

t-test | : | Test t-value (-) |

t | : | Time (Day) |

V | : | Volume (L) |

x | : | Conversion (%) |

X | : | Biomass concentration (g L^{–1}) |

x_{m} | : | Mean value (-) |

x_{0} | : | Median value |

α | : | Proportion invariable (-) |

β | : | Proportion invariable (-) |

Δ | : | Delta symbol (-) |

σ | : | Standard deviation (-) |

Σ | : | Sum operator |

μ | : | Specific growth rate (h^{-1}) |

μ_{max} | : | Maximum specific growth rate (h^{-1}) |

- Boussiba, S., 2000. Carotenogesis in the green algea
*Heamatococcus Pluvialis*: Cellular physiology and stress response. Physiologyia Pllantorum, 108: 111-117.

Direct Link - Csogor, Z., M. Herrenbauer, I. Perner, K. Schmidt and C. Posten, 1999. Design of a photo-biorector for modeling purposes. Chem. Eng. Proc., 38: 517-523.

Direct Link - Molina, E., J. Fernandez, F.G. Acien and Y. Chisti, 2001. Tubular photobioreactor design for algal cultures. J. Biotechnol., 92: 113-131.

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