Growth-dependent radiative properties of <Emphasis Type="Italic">Chlorella vulgaris</Emphasis> and its influence on prediction of light fluence rate in photobioreactor

C. Y. Ma; J. M. Zhao; L. H. Liu; L. Zhang

doi:10.1007/s10811-018-1499-x

Journal of Applied Phycology

February 2019, Volume 31, Issue 1, pp 235–247 | Cite as

Growth-dependent radiative properties of Chlorella vulgaris and its influence on prediction of light fluence rate in photobioreactor

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Authors and affiliations

C. Y. Ma
J. M. Zhao
L. H. Liu
L. Zhang

Article

First Online: 09 May 2018

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Abstract

The radiative properties of microalgae (the absorption cross-section, scattering cross-section, and scattering phase function) are essential for analyzing the light transfer in photobioreactors (PBRs). Due to the growth of microalgal cells, their radiative properties will vary with time. However, the growth-dependent radiative properties of microalgal cells have been rarely considered in the literature in analyzing light fluence rate in the PBRs and its influence on the accuracy of light fluence rate prediction is not well evaluated. In this work, the growth-dependent radiative properties of Chlorella vulgaris are studied via a semi-analytical approach. The growth-dependent cell size distribution of C. vulgaris is taken from experimental data in the literature. The temporal evolution of the scattering and absorption cross-sections and scattering phase function of C. vulgaris is determined based on the Lorenz-Mie theory combined with the growth-dependent cell size distributions and pigment content. The relative deviation of the radiative properties of microalgae at different growth phase with that at stationary phase microalgae is investigated. The deviation of predicted light fluence rate in a flat-plate PBR using the growth-dependent radiative properties and the radiative properties at stationary phase is evaluated. The results show that the relative deviation of absorption, scattering, and extinction cross-sections is significant, especially at exponential growth phase (over 18%). The relative deviation of the predicted light fluence rate is over 40% at exponential growth phase. As a result, the growth-dependent radiative properties have to be applied in order to accurately predict the light fluence rate in PBRs during different growth phases. A simple model is presented and demonstrated good performance to predict the growth-dependent radiative properties of C. vulgaris over the whole growth phase. The presented analysis will facilitate the design and optimization of the optical configuration of PBRs for microalgal cultivation.

Keywords

Microalgae Chlorella vulgaris Growth-dependent radiative properties Photobioreactors Growth model

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Notes

Funding information

The supports by the National Nature Science Foundation of China (Nos. 51276051, 51336002) are greatly acknowledged.

Appendix

The microalgal suspension is a strongly forward scattering medium in the PAR region (Pilon et al. 2011), which causes additional difficulty in the solution of light fluence rate in PBR. The discretization of angular domain using a finite set of discrete directions is known to break the conservation of the asymmetry factor of the scattering phase function for strongly forward scattering media (Boulet et al. 2007; Hunter and Guo 2012). The problem can be solved by using the Delta-Eddington approximation (DE), in which the highly forward scattering phase function is represented by the sum of a Dirac delta function and a smooth function Φ^∗, as follows (Joseph et al. 1976):

$$ \Phi \left(\cos \Theta \right)=2 f\delta \left(1-\cos \Theta \right)+\left(1-f\right){\Phi}^{\ast}\left(\cos \Theta \right)={\Phi}_1\left(\cos \Theta \right)+{\Phi}_2\left(\cos \Theta \right) $$

(A.1)

where f denotes the fraction of forward scattering. Here, the approach by Coelho (2016) is applied to solve the light field in PBR with collimated light irradiation. The total light intensity I is decomposed into a collimated component I_c and a diffuse component I_d

$$ I\left(r,\boldsymbol{\Omega} \right)={I}_c\left(r,\boldsymbol{\Omega} \right)+{I}_d\left(r,\boldsymbol{\Omega} \right) $$

(A.2)

The two components of the light intensity satisfy the following two equations (Coelho 2016):

$$ \Omega \cdot \nabla {I}_c+\beta {I}_c=\frac{\kappa_s}{4\pi}\underset{4\pi }{\int }{I}_c\left(r,{\boldsymbol{\Omega}}^{\hbox{'}}\right)\kern0.5em {\Phi}_1\left({\boldsymbol{\Omega}}^{\hbox{'}}\to \boldsymbol{\Omega} \right)d{\Omega}^{\hbox{'}} $$

(A.3a)

$$ {\displaystyle \begin{array}{l}\Omega \cdot \nabla Id+\beta Id=\frac{\kappa_s}{4\pi}\int Id\left(r,{\boldsymbol{\Omega}}^{\hbox{'}}\right)\Phi \left({\boldsymbol{\Omega}}^{\hbox{'}}\to \boldsymbol{\Omega} \right)d{\Omega}^{\hbox{'}}\\ {}\frac{\kappa_s}{4\pi}\int Id\left(r,{\boldsymbol{\Omega}}^{\hbox{'}}\right){\Phi}_2\left({\boldsymbol{\Omega}}^{\hbox{'}}\to \boldsymbol{\Omega} \right)d{\Omega}^{\hbox{'}}\end{array}} $$

(A.3b)

Substituting the definition of Φ₁ and Φ₂ into Eq. (A.3) yields the following equations with scaled parameters:

$$ \boldsymbol{\Omega} \cdot \nabla {I}_c+{\beta}^{\hbox{'}}{I}_c=0 $$

(A.4a)

$$ {\displaystyle \begin{array}{l}\boldsymbol{\Omega} \cdot \nabla {\mathbf{I}}_d+{\beta}^{\hbox{'}}{I}_d=\frac{\kappa_s^{\hbox{'}}}{4\pi}\int {I}_d\left(r,{\boldsymbol{\Omega}}^{\hbox{'}}\right){\Phi}^{\ast}\left({\boldsymbol{\Omega}}^{\hbox{'}}\to \boldsymbol{\Omega} \right)d{\Omega}^{\hbox{'}}\\ {}+\frac{\kappa_s^{\hbox{'}}}{4\pi }{I}_c\left(r,{\boldsymbol{\Omega}}_c\right){\Phi}^{\ast}\left({\boldsymbol{\Omega}}_c\to \boldsymbol{\Omega} \right)\end{array}} $$

(A.4b)

where Ω_c denotes the direction of the collimated beam, and β^' and $ {\kappa}_s^{\hbox{'}} $ are the scaled extinction coefficient and scattering coefficient defined as

$$ {\beta}^{\hbox{'}}=\beta \left(1-\omega f\right) $$

(A.5a)

$$ {\kappa}_s^{\hbox{'}}={\kappa}_s\left(1-f\right) $$

(A.5b)

Equation (A.4) is finally solved to obtain the light fluence rate in PBRs. In the DE approximation, the phase function Φ^∗ is defined as (Joseph et al. 1976)

$$ {\Phi}^{\ast}\left(\cos \Theta \right)=1+3{g}^{\hbox{'}}\cos \Theta $$

(A.6)

where the parameters f and g^' are given by (Joseph et al. 1976)

$$ f={g}^2 $$

(A.7)

$$ {g}^{\hbox{'}}=\frac{g}{1+g}. $$

(A.8)

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Authors and Affiliations

C. Y. Ma
- 1
J. M. Zhao
- 1
Email authorView author's OrcID profile
L. H. Liu
- 1
- 2
- 3
Email author
L. Zhang
- 4

1.School of Energy Science and EngineeringHarbin Institute of TechnologyHarbinChina
2.Department of PhysicsHarbin Institute of TechnologyHarbinChina
3.School of Energy and Power EngineeringShandong UniversityQingdaoChina
4.College of Mechanical and Electrical EngineeringNortheast Forestry UniversityHarbinChina