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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 64, 2018 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Enrico Bardone, Antonio Marzocchella, Tajalli Keshavarz
Copyright © 2018, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-56-3; ISSN 2283-9216 

Development of a Semi-Theoretical Light Radiation and 
Photosynthetic Growth Model for the Optimal Exploitation of 

Wastewaters by Microalgae 

Alessio Mazzellia,b*, Agnese Cicci b, Giorgia Seda,Marco Bravia. 
a
Dept. Chemical Engineering Materials Environment, University of Rome La Sapienza; via Eudossiana 18, 00184 Rome 

(IT). 
b
Bio-P s.r.l. via di Vannina 88, 00156 Rome (IT) 

 
alessio.mazzelli@uniroma1.it  

In the last decade, interest toward the potential application of microalgae has grown considering their potential 
use in industrial sectors as human nutrition and health, animal feed and biopolymers. Their ability to use light 
or/and organic carbon as energy source, makes them able to grow in a wide range of conditions. Because of 
that, the possibility to use alternative nutrients and water sources for their cultivation has been investigated. 
The microalgal cultivation using wastewaters mixed with synthetic medium might be a good combination that 
could reduce costs of water, nutrients and wastewater treatment. Anyway, wastewaters are frequently dark 
colored and contain toxic compounds that could have a negative impact on microalgal light uptake and 
metabolism. In this study, an experimental-first principles hybrid method for the estimation of microalgal 
growth in non-transparent media was developed as a guide in the choice of the best formulation of 
wastewater-based culture media for microalgae. To carry out several experimental runs in parallel with 
different conditions (dilution of the wastewater, different light sources, etc.) a cylindrical bubble column 
PhotoBioReactor (PBR) was adopted. Its simple geometry allows the analysis of inside light fluxes. A non-
metabolizable and non-toxic dye, in condition of purely light-radiative growth limitation, was added to the 
medium mimicking the reduced transparency of wastewaters. As final step to test the model, culture mediums 
with wastewater addiction were used for microalgal cultivation, showing their nutritive effects on growth.  

1. Introduction 

Microalgae are a source of high added value products with potential applications in the food, cosmetic and 
pharmaceutical industries (Sed et al., 2017). Anyway, microalgal cultivation in phototrophic ways presents still 
problems of high costs. The use of wastewaters as nutrients source for microalgal cultivation may reduce the 
culture management and water treatment costs but promotion and demotion effects complicate data analysis 
and exploitation (Gouveia et al., 2016). An hybrid approach to data analysis, coming out from the post-
treatment of experimental data with a semi-theoretical light-centered microalgal growth model, enabled the 
split of metabolic effects due to dissolved substances (substrates, nutrients, toxins) from that of light 
(considered as a substrate). Furthermore, an improvement to the previous work (Cicci et al. 2014b) for a 
better data interpretation has been added, considering light-scattering phenomena and introducing an 
absorption efficiency calculated in all wavelengths (λ) of visible spectrum. The model would allow the 
microalgal process developer to find the ideal composition of a wastewater-incorporating medium at the 
optimal scale for laboratory array testing. In the present work, some models of irradiance distribution (Kumar 
et al. 2013, Cornet et al. 1995) and phototrophic growth (Monod et al. 1949, Pruvost et al. 2002), with different 
level of accuracy and complexity, were considered. Combination models were benchmarked for their ability to 
describe the effect of light supply reduction on observed specific growth rate adopting the concept of semi-
theoretical normalized growth rate. The best performing models came out from the combination of the mono-
dimensional approach of the Radiative Transfer Equations described in the Two-Flux model with the growth 
model described by Pruvost et al. (2002). However, a satisfactory compliance with the obtained experimental 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1864115

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Mazzelli A., Cicci A., Sed G., Bravi M., 2018, Development of semi-theoretical light radiation and photosynthetic 
growth model for the optimal exploitation of wastewaters by microalgae, Chemical Engineering Transactions, 64, 685-690  
DOI: 10.3303/CET1864115 

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results also required the analysis of the irradiance effects at a spectral level, i.e., taking into account the 
absorption efficiency of microalgae and the photosynthetic action at each wavelength in the 
(photosynthetically active) spectrum. The experimental growth rate obtained in light limitation condition inside 
the photobioreactor, using first a dye simulating non transparent water, and then wastewaters mixed with a 
synthetic medium (Olive Oil Mill WasteWater and digestate) were used to test and validate the developed 
model (Cicci et al. 2014a). 

2. Materials and methods 

2.1 Microalgal cultivation and dye addition. 
Scenedesmus dimorphus (UTEX 1237) was obtained from the Culture Collection of Algae at the University of 
Texas at Austin, USA. It was inoculated into a Bold 3N modified (3NB) culture medium; a liter of medium 
contains: 0.025 g NaCl, 0.75 g NaNO3 , 0.075 g MgSO4*7H2O, 0.075 g K2HPO4, 0.175 g KH2PO4 , 0.019 g 
CaCl2 and 6 mL of PVI metal solution composed by: 0.7 g/L of Na2EDTA*2H2O, 0.097 g/L of FeCl3*6H2O, 
0.041g/L of MnCl2*4H2O, 0.005 g/L of ZnCl2, 0.002 g/L  of CoCl2*6H2O, 0.04g/L of Na2MoO4*2H2O. It was 
sterilized by autoclaving at 121 °C for 20 minutes. A black water-based color (Pigment PBk6, PBk7, vehicles 
Arabic Gum) was first diluted in distilled water and after sterilized; S. Dimorphus 1237 is unable to metabolize 
Arabic gum. The series of colored media was prepared by diluting the mother solution in synthetic medium to 
a final absorbance of: 0.1, 0.2, 0.4, 0.6, and 0.8 measured at 690 nm. The cultures were grown in a 
Polymethyl methacrylate (PMMA) cylindrical photobioreactors with a volume of 500 mL and a diameter of 6 
cm, insufflated with filtered and pre-humidified air to prevent evaporation. Air flowrate was set at 5.4*10-3 vvm 
(130 NL/ h) to gain also a good mixing. These photobioreactors were kept in a climatic cabin where lighting 
was provided by eight cold white fluorescent lamps (400-700 nm, 865 K, 32 W, 80 μmol photons*m-2s-1) with a 
light/dark period of 16:8 hours. Temperature inside the cultures was maintained constant at 23 ± 2 °C. The 
wastewaters used in substitution of dyed medium were: a cattle digestate from a plant located in Cicerale 
(Southern of Italy); OOMWW (Olive Oil Mill WasteWater) from a olive oil plant located in Sabina area (Center 
of Italy). The digestate was sieved at a 710 μm aperture to get rid of suspended solids and micro-filtered in a 
polyamide column (type JX) (pore size 0.3 μm), then it was thermally sterilized (20 minutes at 120 °C) (Cicci et 
al. 2014a). The raw OMWW was sieved at a cut-size of 300 micron, then it was flocculated by acidification 
with nitric acid to pH values near 3.0, centrifuged and photo-catalyzed (Cicci et al. 2013). 

2.2 Microalgal harvesting and growth monitoring 
The cultures were carried on for a semi-continuous growth time of 17 days. At the end, the biomass was 
collected, centrifuged and washed twice with distilled water and dispatched for chemical assays. Afterwards 
the collected algae were transferred to a dehumidified flask and placed for one night to dry in an oven at a 
temperature of 110 ° C. The dried biomass was used to calculate the dry weight net of the tare. The 
relationship found between cell density C (g/l) and absorbance measured at the wavelength of 690 nm 
corresponds to the linear correlation: C = 0.92 ∗ A 											with				R2 = 	0.99				                                                                  (1) 
The absorbance measures were collected with a spectrophotometer UV-1800PC (Shanghai Mapada 
Instruments Co., Ltd).  

3. Light and growth modelling 

3.1 Introduction to the general method of analysis   
Before introducing models for growth and light distribution, the general method of data analysis already used 
in the work of Cicci et al.(2014b) is described. In order to separate the nutritional/toxic effects (e.g. 
wastewaters medium) from the energetic ones (e.g. illumination source) on microalgal growth, the irradiance 
inside the cylindrical photobioreactor was expressed as a function of the radial position I(r). Afterward, the 
instantaneous specific growth rate (μ ) was calculated  by integration of μ(r) in all volume of reactor, making it 
only dependent on the radial position after substitution of the irradiance function within it: μ = μ(r)2πrdr  (2) 
The average specific growth rate (semi-theoretical) at a certain dilution ratio (D) is obtained by dividing the Eq 
(2) for the reactor’s volume: 

μ = μ (D) = ( ) 	 (3)	

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This estimated growth rate it also normalized by dividing it for the estimated value of the growth rate at infinite 
dilution in the same medium without the dye (μ (Ctr)): μ = ( )( ) (4) 
With the same procedure, the value of experimental growth at the average absorbance of the culture (medium 
+ microalgae) was determined, assuming the cultural medium absorbance (A ) to be constant along the 
culture lifetime. The average absorbance, with these assumptions was calculated as: A = A + ( ) ( ) (5) 
Obtaining a normalized experimental growth rate: μ = ( )( ) (6) 
Finally, the relative growth rate (μ ) was calculated, by dividing Eq. (6) by Eq. (4): μ = 	 (7) 
Our aim was to demonstrate, with the help of a mathematical model, that in the normal experimental 
conditions with a dyed culture medium without surplus of nutrients (μ >μ ) or toxic elements (μ <μ ) 
in the medium,	μ =1 will be obtained. This method provides the microalgal process prospector with a suitable 
data analysis tool to choose the “ideal” composition (Dilution Ratio) of a wastewater in microalgal cultivation 
with a limited set of small-scale experiments.    

3.2 Growth model   
The model used for representing the growth in function of irradiance is the ‘Light-inhibition model’ described by 
Pruvost (Pruvost et al. 2002): μ(I) = ( )( )( ) ( )  (8) 
With	ρ = , ρ = , and μ  that are in order: the normalized light intensity, the normalized light intensity 
compensation and the specific growth rate at saturation light (I ); the values of which were assume to be: 	I = 10 ,	I = 150  (Liu et al. 2013). The Pruvost model expresses the main aspects of the 
photosynthetic response of the microalgae to light, namely, the photoinhibition of the cells to high values of I, 
and the weight loss that occurs below a certain irradiance value, called compensation irradiance	I .  
3.3 Light distribution model  
For light distribution inside the reactor two models were analyzed, both considering scattering phenomena 
inside the reactor. The first one is an empiric model obtained with the image analysis used as a tool in 
studying the light distribution, (Kumar et al. 2013). The irradiance function written in Eq(9) is actually a 
modified Lambert-Beer equation:  

I(r, t) = ∙ e ∙ ∙( )( ) (9) 
where ε = 23.86, K = 4.03	g/l, and K = 3.78	cm are respectively: coefficients of maximal light absorption, 
light scattering by cells, and light scattering by light path-length taken from the quoted paper (Kumar et al. 
2013); besides C is the cell density (g/l) inside the reactor. The second one is a more rigorous model, based 
on the solution of the Radiative Transfer Equations (RTEs), which can be derived by considering a small 
element of culture suspension and performing a radiation balance on this element, considering the incident, 
absorbed, transmitted, and scattered sources of radiation. The solution of RTEs in its tridimensional form is an 
integro-differential problem in a six-dimensional Euclidean space needing sophisticated numerical methods, 
so Cornet (Cornet et al. 1995) introduced a monodimensional approximation, called ‘two flux approximation’, 
splitting the field of radiation into two opposite and different fluxes I  and I . These considerations were 
applied to the cylindrical geometry used in the experiments and the flux balance obtained is schematized in 
Figure 1.    
 
 
 
 

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Figure 1: Flux balance schematization in cylindrical geometry applying the two flux approximation; is the 

scattering parameter ( ) and 	is the incident irradiance ( ). 
In Figure 1 a back scattering coefficient b =  is considered because, after simulations at different b (data not 
shown), the Isotropic Scattering has been chosen as the best solution to represent the experimental light 
conditions; indicating that half of the irradiance follows the initial path and the other half is scattered in the 
opposite direction. The flux balance is summarized in Eq(10), where	C ,E ,E 	are in order: the cell density 
inside the reactor (	 ), the mean absorption and the scattering mass coefficients on the visible spectrum ( ). I 2πrΔz| I 2πrΔz| + E C I 2πrΔrΔz| E C I 2πrΔrΔz| = E C I 2πrΔrΔz|I 2πrΔz| I 2πrΔz| + E C I 2πrΔrΔz| E C I 2πrΔrΔz| = E C I 2πrΔrΔz| 													(10)                             	
The Eq (10) after some simplifications gave the system that was actually implemented and solved in MATLAB 
(Eq(11)) using the function bvp5c, that is a finite difference code that implements the four-stage Lobatto IIIa 
formula to solve ‘Boundary Value Problems’. = I E C I + E C 	(I I )= I + E C I + E C (I I )	 (11)	
With the relevant Boundary Conditions: r = R 		I = Ir = R 		I = I  (12) 
3.4 Absorption Efficiency introduction 
For a better representation of the mechanisms that occurs in the phototrophic microorganisms, the 
considerations done by Senger (1970) were studied, and two new parameters were introduced: the Quantum 
Yield Spectrum (Φ=moles of O2 produced/moles of absorbed photons), and the Action Spectrum (Ψ=moles of 
O2 produced/moles of incident photons). With these preconditions a new parameter named ‘Absorption 
Efficiency’ was introduced (Eq (13)), E = = 	 	 	 ∗ 	 		 	 	 ∗ 	 	 = 	 	 		 	 	  (13) 

                           

Figure 2: Absorption Efficiency calculated in all wavelengths (λ) of visible spectrum for the microalga 
Scenedesmus obliquus. 

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In addition to that, the Absorption Efficiency was calculated in all wavelengths of visible spectrum by data 
obtained from Senger’s work (Senger 1970) (Figure 2), adding its effect to the absorbance of microalgae 
during the growth without the value of the dye, because this parameter affects only the algae themselves. At 
last, the value of the dye’s absorbance was added to the value of absorbance modified with the Absorption 
Efficiency, considering also the coloring effects on microalgal growth. 

4. Results 

4.1 Application to artificial colored medium 
Taking the above considerations into account, the results achieved by applying the previously described 
models to the growth data obtained with a balanced medium conditioned by a non-toxic and non-metabolized 
dye were plotted (Figure 3). It can be seen that the Cornet light model, associated with the Pruvost growth 
model predicts the experimental growth rate significantly better than the Modified Lambert-Beer model, 
although at the price of a higher complexity. Therefore, the Cornet and Pruvost models will be considered as 
the reference models to use for the future application of the model in predictive mode.  

 

Figure 3: Relative specific growth rate  in function of dye dilution ratio for both light distribution models: Two 
flux model(Cornet et al. 1995) and Modified Lambert Beer (Kumar et al. 2013). Both light models were coupled 
with the same growth model described by Pruvost (Pruvost et al. 2002). 

 
                              

Figure 4: Relative specific growth rate  in function of wastewater dilution ratio for both wastewaters: 
digestate(dig) and olive oil mill wastewater(oomww). 

4.2 Application to wastewaters 
As further implementations of this work, two types of wastewaters in substitution of the inert dye were used, 
showing their positive (μ 1) or negative (μ 1) effects on growth. The aim of this part of the work was to 
find the ideal composition (or dilution) of a wastewater-incorporating medium, coincident to the highest value 
of μ , using it as a base for a future tests without any further experimental analysis. The results are shown in 
Figure 4 both for the digestate and for OOMWW, and the positive effect of compounds (nitrogen and 
phosphorous) contributed by the wastewaters (μ 1 at all dilutions)	can be seen particularly in the case of 
the digestate (Cicci et al., 2014a); for OOMWW the positive effects are less stronger revealing also a value 
less than one. It should be noted that this conclusion cannot be drawn upon the raw measured specific growth 

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rates only, which exhibit a continuous decrease of specific growth rate when the wastewater volume fraction in 
the formulated media is increased. 

5. Conclusions 

This work showed a method of analysis of the phototrophic microalgal growth inside a non-transparent media 
using a mathematical model that could be easily implemented and reused by any final users. This model 
incorporates all the principal phenomena that occur in a phototrophic cultivation, but also considers physical 
light phenomena like scattering that has been shown being indispensable for a proper representation of the 
problem. The results showed above confirmed the hypothesis made for a synthetic medium with an inert dye 
( ≅ 1), giving also positive response for using wastewaters in addition to growth medium. However the 
method needs to be studied more accurately with further specific tests in a wider range of dilution ratio, 
analyzing also the substitution, as diluent, of the synthetic media with tap water, reducing costs further.  
In that case growth of the photobacteria would rely only on light and nutriments present in wastewater. 
All of these considerations were done because wastewaters have many advantages in terms of diminishing 
treatment costs and raw material costs, but could also have to face with disadvantages like the presence of 
toxic elements for growth, and the model could help to preliminarily investigate that without too many 
experiments. To improve μ  and biomass yield, a new type reactor with an optimized geometry that diminish 
the microalgal thickness exposed to light could be designed (Moroni et al. 2014). This exposure has to be 
controlled, avoiding the photo-inhibition and fully exploiting the “metabolically compliant” medium formulated 
with the aid of the tool presented in this study. 

Reference  

Cicci A., Bravi M., 2014a, Production of the freshwater microalgae Scenedesmus Dimorphus and Arthrospira 
Platensis by using cattle digestate, Chemical Engineering Transactions, 38, 85-90. 

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cost alternative cultural media, Chemical Engineering Transactions, 38, 79-84. 

Cicci A., Stoller M., Bravi M., 2013, Microalgal biomass production by using ultra- and nanofiltration membrane 
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Monod J., The growth of bacterial cultures, 1949, Annu. Rev. Microbiol., 3, 371–394. 
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