Design scenario analysis for porous substrate photobioreactor assemblies

Abstract

For commercial-scale production of microalgae, in the recent years, biofilm bioreactors have been proven to solve some of major technical problems in this field. Among different approaches, porous substrate photobioreactor (PSBR) is one of the most promising bioreactor types. However, before the actual construction of such a system and application, various design parameters need to be determined. For this purpose, mathematical modeling is considered to be an efficient tool. In the present study, a model for estimating design parameters of PSBRs is proposed. The proposed model estimates the effects of different design parameters on the productivity of PSBR systems based on (1) global solar irradiance data, (2) geographical location of the simulated system, and (3) empirical functions describing the relationship between light intensity on the module surface and module productivity, derived from experimental data. In the present study, to demonstrate the capacity of the proposed model, production of Halochlorella rubescens using a PSBR (Twin-Layer technology) was modeled for various scenarios. Also, simulated production of astaxanthin using Haematococcus pluvialis was performed. The results demonstrated the ability of the proposed model to estimate various design parameters for PSBR systems under various conditions. Based on the prediction made by the proposed model, these parameters can be individually optimized for different geographical locations and/or applications.

Appendices

Appendix 1. Estimation of ground solar radiation on a horizontal surface

The ground solar radiation on a horizontal surface was estimated using methods as summarized by Duffie and Beckman (2013), and Slegers et al. (2011) applied this method for calculating the productivity of photobioreactors. First, solar constant (G_sc) was acquired from ASTM Standard Extraterrestrial Spectrum Reference E-490-00 for the photosynthetically active radiation (PAR), with a value of 1366 W m⁻². Then, based on the reactor location, solar radiation outside of the atmosphere on a horizontal plane (G_o) was calculated for different time points in a year as

$$ {G}_{\mathrm{o}}={G}_{\mathrm{sc}}\bullet \left[1.00011+0.034221\bullet \cos (B)+0.00128\bullet \sin (B)+0.000719\bullet \cos (2B)+0.000077\bullet \sin (2B)\right]\bullet \cos \left({\theta}_{\mathrm{z}}\right) $$

(2)

where

$$ B=\frac{\left(i-1\right)\bullet 360}{365} $$

(3)

on the ith day of the year. θ_z is the zenith angle of the solar radiation (Fig. 7, Table 1) and was calculated as

$$ \cos \left({\theta}_{\mathrm{z}}\right)=\sin \left(\phi \right)\bullet \sin \left(\delta \right)+\cos \left(\phi \right)\bullet \cos \left(\delta \right)\bullet \cos \left(\omega \right) $$

(4)

Fig. 7

A section from a parallel arranged system with infinitely wide modules as simulated in the present study, and variables (definitions given in Table 1) required for the calculation of irradiance distribution on the module surface. Here, a vertical system (i.e., β = 90°) is depicted; for better depiction, the first module was made transparent. Thick solid line with arrow indicates the direction of incident direct radiation. Dotted-dashed lines with arrow represent cardinal directions; dashed lines with arrow show the normal of the module surface; dotted lines represent the top surface of the shadowed volume between the first and the second module. Symbol ∟ represents right angles

Table 1 Definitions of symbols presented in Fig. 7

In Eq. 4, ϕ is the latitude of the preselected reactor location and δ is the daily solar delination angle and is calculated as

$$ \delta =23.45\bullet \sin \left(\frac{360\bullet \left(284+i\right)}{365}\right) $$

(5)

on the ith day of the year, and ω is the solar hourly angle (i.e., displacement angle of the sun from local meridian)

$$ \omega =15\bullet \left({t}_{\mathrm{solar}}-12\right) $$

(6)

In Eq. 6, t_solar is the local solar time (in hours) and is calculated based on the location and the factor e

$$ {t}_{\mathrm{solar}}=t+\frac{\left[4\bullet \left(\lambda -\kappa \right)+e\right]}{60} $$

(7)

where λ and κ are the longitude and the local meridian of the preselected reactor location, respectively; t is the local standard time (in hours); and

$$ e=229.2\bullet \left[0.000075+0.001868\bullet \cos (B)-0.032077\bullet \sin (B)-0.014615\bullet \cos (2B)-0.04089\bullet \sin (2B)\right] $$

(8)

Total solar irradiance on a horizontal plane on the ground at the reactor location was calculated from G_o and the attenuation caused by the earth’s atmosphere

$$ {G}_{\mathrm{cb}}={G}_{\mathrm{o}}\bullet {\tau}_{\mathrm{b}} $$

(9)

where τ_b is the attenuation coefficient

$$ {\tau}_{\mathrm{b}}={a}_0+{a}_1\bullet \exp \left(-\frac{k}{\cos \left({\theta}_{\mathrm{z}}\right)}\right) $$

(10)

and for atmosphere with 23 km visibility (Duffie and Beckman 2013)

$$ \left\{\ \begin{array}{c}{a}_0=0.4337-0.00821\bullet {\left(6-\sigma \right)}^2\\ {}{a}_1=0.5055-0.00595\bullet {\left(6.5-\sigma \right)}^2\\ {}k=0.2711-0.01858\bullet {\left(2.5-\sigma \right)}^2\end{array}\right. $$

(11)

and σ is the altitude (in m) of the preselected reactor location.

To calculate the direct and diffuse components of the total ground solar irradiance, average daily clearness index (0 ≤ k_T ≤ 1) is defined as the ratio between hourly ground direct radiation to extraterrestrial radiation: k_T = 0 indicates a completely cloud-covered sky, and k_T = 1 represents a cloud-free sky. The ground irradiance under clouded condition (G_{cb, cloud}) is estimated using

$$ {G}_{\mathrm{cb},\mathrm{cloud}}={G}_{\mathrm{cb}}\bullet \left({a}_1+{b}_1\bullet {k}_{\mathrm{T}}\right) $$

(12)

where a₁ and b₁ are equal to 0.22 and 0.78, respectively (Duffie and Beckman 2013). Then, direct and diffuse irradiances on the ground (G_{cb,  direct} and G_{cb, diffuse}, respectively) are calculated as (Erbs et al. 1982)

$$ \frac{G_{\mathrm{cb},\mathrm{diffuse}}}{G_{\mathrm{cb},\mathrm{cloud}}}=\left\{\begin{array}{c}1-0.09\bullet {k}_{\mathrm{T}},\mathrm{for}\ {k}_{\mathrm{T}}\le 0.22\\ {}0.9511-0.1604\bullet {k}_{\mathrm{T}}+4.388\bullet {k_{\mathrm{T}}}^2-16.638\bullet {k_{\mathrm{T}}}^3+12.336\bullet {k_{\mathrm{T}}}^4,\mathrm{for}\ 0.22<{k}_{\mathrm{T}}\le 0.8\\ {}0.165,\mathrm{for}\ 0.8<{k}_{\mathrm{T}}\end{array}\right. $$

(13)

and

$$ {G}_{\mathrm{cb},\mathrm{direct}}={G}_{\mathrm{cb},\mathrm{cloud}}-{G}_{\mathrm{cb},\mathrm{diffuse}} $$

(14)

Appendix 2. Calculation of irradiance distribution on reactor module

Irradiance distribution on the module surface was calculated using the estimated solar radiation data as described above. Based on the method described by Duffie and Beckman (2013), first, direct radiation on the surface of a module (G_{surface, direct}) was calculated as

$$ {G}_{\mathrm{surface},\mathrm{direct}}={G}_{\mathrm{cb},\mathrm{direct}}\bullet \frac{\cos \left(\theta \right)}{\cos \left({\theta}_{\mathrm{z}}\right)} $$

(15)

θ is the angle of incidence (Fig. 7) and is calculated using

$$ \cos \left(\theta \right)=\sin \left(\delta \right)\bullet \sin \left(\phi \right)\bullet \cos \left(\beta \right)-\sin \left(\delta \right)\bullet \cos \left(\phi \right)\bullet \sin \left(\beta \right)\bullet \cos \left(\gamma \right)+\cos \left(\delta \right)\bullet \cos \left(\phi \right)\bullet \cos \left(\beta \right)\bullet \cos \left(\omega \right)+\cos \left(\delta \right)\bullet \sin \left(\phi \right)\bullet \sin \left(\beta \right)\bullet \cos \left(\gamma \right)\bullet \cos \left(\omega \right)+\cos \left(\delta \right)\bullet \sin \left(\beta \right)\bullet \sin \left(\gamma \right)\bullet \sin \left(\omega \right) $$

(16)

In Eq. 16, β and γ are the inclination and orientation of the modules, respectively (Fig. 7). Then, the height of the shadowed region (i.e., distance from module bottom to the upper edge of the shadow) (H_shadow) on the module surface was calculated (Fig. 7) as follows:

$$ {H}_{\mathrm{shadow}}=H-\tau \bullet \frac{\sin \left({\alpha}_{\mathrm{p}}\right)}{\sin \left(180-\beta -{\alpha}_{\mathrm{p}}\right)} $$

(17)

where 퐻 is the height of the module, τ is the distance between modules, and α_p is the profile angle of the direct solar radiation and can be calculated using

$$ {\alpha}_{\mathrm{p}}={\tan}^{-1}\frac{\tan \left({\alpha}_{\mathrm{s}}\right)}{\cos \left({\gamma}_{\mathrm{s}}-\gamma \right)} $$

(18)

where α_s is solar altitude angle and is equal to 90° − θ_z and γ_s is the solar azimuth angle which is calculated as

$$ {\gamma}_{\mathrm{s}}=\sin \left(\omega \right)\bullet \left|{\cos}^{-1}\left(\frac{\cos \left({\theta}_{\mathrm{z}}\right)\bullet \mathit{\sin}\left(\phi \right)-\sin \left(\delta \right)}{\sin \left({\theta}_{\mathrm{z}}\right)\bullet \cos \left(\phi \right)}\right)\right| $$

(19)

Then, diffuse light on the module surface (G_{surface, diffuse}) is calculated as

$$ {G}_{\mathrm{surface},\mathrm{diffuse}}={G}_{\mathrm{cb},\mathrm{diffuse}}\bullet \frac{1+\cos \left(\beta +u\right)}{2} $$

(20)

where

$$ u={\tan}^{-1}\left(\frac{h\bullet \cos \left(\beta \right)}{\tau}\right) $$

(21)

h is the distance between a position on the module surface and the bottom of the module. The total module surface irradiance (G_surface) is calculated as

$$ \left\{\begin{array}{c}{G}_{\mathrm{surface}}={G}_{\mathrm{surface},\mathrm{diffuse}}+{G}_{\mathrm{surface},\mathrm{direct}},\left(h>{H}_{\mathrm{shadow}}\right)\\ {}{G}_{\mathrm{surface}}={G}_{\mathrm{surface},\mathrm{diffuse}},\left(h\le {H}_{\mathrm{shadow}}\right)\end{array}\right. $$

(22)

Irradiance was converted then from W m⁻² to I_surface in μmol photons m⁻² s⁻¹

$$ {I}_{\mathrm{surface}}=0.00836\bullet {G}_{\mathrm{surface}}\bullet \lambda $$

(23)

where λ is the wavelength in nm and 0.00836 is the conversion coefficient based on the solar spectrum (considered not to change with time of the day). Daytime average irradiance for 1 day is calculated as

$$ {I}_{\mathrm{surface},\mathrm{DA}}=\frac{\int_{T_{\mathrm{sunrise}}}^{T_{\mathrm{sunset}}}{I}_{\mathrm{surface}} dt}{T_{\mathrm{sunset}}-{T}_{\mathrm{sunrise}}} $$

(24)

T_sunrise and T_sunset are the sunrise and sunset time (in hours) on a particular day of 1 year, respectively, and are calculated as

$$ \left\{\begin{array}{c}{T}_{\mathrm{sunrise}}=12-\frac{\cos^{-1}\left[-\tan \left(\phi \right)\bullet \tan \left(\delta \right)\right]}{15}\\ {}{T}_{\mathrm{sunset}}=12+\frac{\cos^{-1}\left[-\tan \left(\phi \right)\bullet \tan \left(\delta \right)\right]}{15}\end{array}\right. $$

(25)

and the day length for this particular day (T_L) is

$$ {T}_{\mathrm{L}}={T}_{\mathrm{sunset}}-{T}_{\mathrm{sunrise}} $$

(26)

For all calculations performed above, angles are measured in degrees (°). The productivity at position h on the ith day of the year on the PBR growth surface was calculated based on the day length of the ith day of the year and the light received by the position h at the different times of the day. The biomass concentration at the end of the ith day was calculated cumulatively using the biomass concentration of the previous day (i.e., (i − 1)th day as the starting value (Eq. 1).

Appendix 3. Estimation of biomass productivity of Halochlorella rubescens based on dry biomass concentration and light intensity

Light intensities, dry biomass concentrations, and associated growth rates as presented by Schultze et al. (2015) were used as the x-, y-, and z-axis data for a surface fit, respectively. The dry biomass concentrations data used in the present study was the mean value of triplicates; for the detailed calculation of the growth rates, refer to Schultze et al. (2015). A thin-plate segmented polynomial line (spline) interpolant fit was used (MATLAB fit function with a thin-plate spline interpolant method). This method was selected because it produces a smooth surface with an appropriate rigidness and performs well in extrapolation (MATLAB 2015b, MathWorks, USA). However, a direct mathematical description of the fitting function cannot be extracted from MATLAB; as a result, the raw data used in the present study can be found in the online supplementary material. For the surface fitting, only the data mentioned above is required, and the fitting function can be found by using the curve-fitting tool, in the interpolant tab with the method thin-plate spline. The fitted surface is presented in Fig. 8. The fitting function covers light intensity from 0 to 1000 μmol m⁻² s⁻¹, biomass concentration on the PBR surface from 5 to 200 g m⁻², and biomass productivity on the PBR surface from 0.5 to 11.9 g m⁻² day⁻¹.

Fig. 8

Data points (Schultze et al. 2015) used in this study to create the function (i.e., surface) describing the relationship between light intensity, biomass concentration, and biomass growth rate, and the graphical representation of the acquired function (fitted surface)

Appendix 4. Estimation of astaxanthin productivity using Haematococcus pluvialis based on light intensity

Light intensity and astaxanthin productivity during the linear accumulation phase (i.e., the first 10 days of cultivation) as presented by Kiperstok et al. (2017) were used as the x- and y-axis data for a curve fit, respectively. For the detailed calculation of the astaxanthin productivities, refer to Kiperstok et al. (2017). A shape-preserving interpolant fit was used (MATLAB fit function with a shaper-preserving interpolant method), i.e., a group of third-order polynomial equations, each equation for a range of light intensity values. The coefficients of these equations and their light intensity ranges are given in Table 2, in the form of productivity = a ∙ (light intensity)³ + b ∙ (light intensity)² + c ∙ (light intensity) + d. This method was selected because it produces a smooth curve with an appropriate rigidness and performs well in extrapolation (MATLAB 2015b, MathWorks, USA). The data used to create the fit surface and the fitted surface are presented in Fig. 9. The fitting function covers light intensity from 0 to 1000 μmol photons m⁻² s⁻¹ and astaxanthin productivity on the PBR surface from 0 to 0.41 g m⁻² day⁻¹. Values outside of these ranges were extrapolated based on the fitting function.

Table 2 Fitting coefficients for Haematococcus pluvialis astaxanthin productivity data

Fig. 9

Data points (Kiperstok et al. 2017) used in this study to create the function (i.e., curve) describing the relationship between light intensity and astaxanthin productivity when 5% CO₂ was applied, and the graphical representation of the acquired function (fitted curve)