INTRODUCTION
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.
These are phase of microzoides, phase of macrozoides with big flagelies, phase of immobile palmella form and phase of hematokists having bigger red cells with thick cell membrane. As the culture medium having lower level concentration of nutrients during the cultivation, the number of cells in palmella phase change their structure and they go into hematokist phase and then astaxhantin accumulation in the cells follows this phase. Astaxhantin is a ketakeratoneid with red color and high value pigments to give pink color to bodies of lots of sea animals like salmon, shrimp or crab (EriŞtürk and Akpolat, 2005).
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).
MATERIALS AND METHODS
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 setup 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, chlorophylla, 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 ttest 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, tCalc
and ttest show mean value,median value, degrees of freedom, probability
level, standard deviation, calculated t value (or distribution function)
and theoretical test t value from tTable, respectively. 
Table 1: 
Composition of the culture medium 

• 
Test statistics: tstatistics value (ttest) is determined
from ttest table related to degrees of freedom and probability level. 
• 
If tCalc<ttest, (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 setup 
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)
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.
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.
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.
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 ttest as described earlier.
Finally, biomasstime 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 scaleup: 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 commercialscale 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 scaleup, 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);
Calculating of I_{AV} depending on changes of volume and of maximum specific growth rate during scaleup 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 scaleup 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 scaleup
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
RESULTS
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 studentttest 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 pvalue 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.

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 scaleup 
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 studentttest 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.
Scaleup: 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.
CONCLUSION
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 scaleup 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.
ACKNOWLEDGMENT
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 () 
tCalc 
: 
Calculated tvalue () 
ttest 
: 
Test tvalue () 
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}) 