Modeling and Dynamic Optimization of Microalgae Cultivation in

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Modeling and Dynamic Optimization of Microalgae Cultivation in Outdoor Open Ponds Abdulla Malek, Luca Zullo, and Prodromos Daoutidis Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b03209 • Publication Date (Web): 27 Oct 2015 Downloaded from http://pubs.acs.org on November 4, 2015

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Industrial & Engineering Chemistry Research

Modeling and Dynamic Optimization of Microalgae Cultivation in Outdoor Open Ponds Abdulla Malek a, Luca C. Zullo b, Prodromos Daoutidis a,* a

Department of Chemical Engineering & Materials Science, University of Minnesota, Minneapolis, MN. 55455, USA b

VerdeNero LLC, 955 Highway 169 North, Plymouth, MN. 55441, USA

Keywords: Microalgae, Open Pond, Dynamic Optimization, Dilution Rate, Modeling. ABSTRACT This paper presents a model-based dynamic optimization study of the operation of an outdoor open pond for microalgae cultivation. A nonlinear mathematical model based on first principles for predicting the growth of microalgae in open ponds is developed and validated against literature data. To account for the impact of weather vagaries on the cultivation of microalgae, data for local climatic conditions is incorporated into the model. The supply of dissolved CO2 to the algal culture from a CO2 rich gas is modeled as well. Optimal monthly operating profiles for the dilution rate, CO2 gas flowrate, and makeup water flowrate are determined based on minimization of the cultivation cost. The case study included in the analysis is for cultivating Nannochloropsis Salina over an annual production cycle in California, United States of America.

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The dynamic optimization identified a set of operation profiles that reduced the cultivation cost by at least 15% as opposed to relying on heuristic approaches for improving operation.

1. INTRODUCTION In 2013, global CO2 emissions reached 35.3 billion tonnes.1 Cutting greenhouse gas emissions towards achieving the targets pledged under the United Nations Framework Convention on Climate Change (UNFCCC) is a universal challenge.2 This is stimulating a global interest in finding renewable alternatives to fossil fuels, thus ensuring environmental and energy sustainability. A potential replacement for petroleum is biofuels, which can be integrated with the current fuel infrastructure. Satisfying 50% of the transportation fuel demand in the United States of America (USA) with biofuels derived from palm oil would require devoting approximately 24% of USA existing cropping land.3 Thus, relying on energy crops as feedstock for biofuels production might disrupt the food supply chain. On the other hand, utilizing algal biofuels would reduce that cultivation land requirement to 2.5%, making algal biomass a promising feedstock for biofuels production.3 Moreover, carbon constitutes up to 50% of dry weight algal biomass (DW), hence, microalgae cultivation would assist in mitigating CO2 as approximately 1.83 kg of CO2 is required to produce each kg of DW.4 A well-to-pump net CO2 emissions of -20.9 kg per GJ of energy generated has been reported in a life cycle analysis study for an algal biodiesel process.5 Microalgae are widely cultivated for a variety of products including food additives, pigments, antibiotics, and nutraceuticals.6 However, the production cost remains too high for high-volume, low-value markets; for example, a conservative estimate for the cost of producing algae-based green diesel is around $10 gal-1.7 In parallel with advances in algal biotechnology, process systems engineering can play an important role in optimizing the economics of algae-based


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commodity products such as biofuels. Several optimization studies have appeared in the literature focusing on the design and synthesis of downstream processes for biofuels production and CO2 mitigation using microalgae.8,9 Studies on bioreactor design and optimization based on computational fluid dynamics (CFD) modeling have been carried out as well.10 The economics of algae cultivation are highly sensitive to the productivity of microalgae and maintaining steady operation in outdoor cultivation at commercial scale is challenging, because varying weather conditions greatly affect the growth rate.7 The identification of optimal operating conditions mitigating the impact of such environmental factors can maximize the productivity of microalgae over a production cycle. Most of the studies on optimization of microalgae cultivation are based on trial-and-error and/or design heuristics.11-14 Some modelbased optimization studies have appeared focusing on laboratory algae growth systems without accounting for weather variations.15,16 The objective of this study is to develop a dynamic optimization formulation for determining optimal operating conditions for improving outdoor algae production over a production period. Specifically, a first principles model is developed for algae cultivation in outdoor open ponds considering the effect of daily varying local climatic conditions on algae growth. The model accounts for the effect of medium temperature, irradiance level, and nutrient availability on the growth of microalgae as well as the transfer of CO2 from a CO2 rich gas to the growth culture. Model validation against experimental results from the literature is then conducted. Finally, a dynamic optimization problem is formulated to determine the optimal dilution rate, makeup water flowrate and CO2 gas flowrate monthly profiles that minimize the cost of producing microalgae in a representative location (Imperial County in California, USA) over the course of a year.


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2. PROCESS DESCRIPTION AND MODELING The economics of mass cultivation of microalgae are more favorable for the open pond system than the photobioreactor configurations.7,17 Several open pond modeling studies in the literature18,19 have adopted the concept proposed and validated in references20,21 where the hydrodynamics of an open pond are modeled by a cascade of continuous stirred-tank reactors (CSTRs). In this approach, an individual open pond is approximated as a series of n compartments where the content leaving the last compartment is recirculated to the first compartment as demonstrated in Figure 1. Assuming each compartment to be well-mixed, it can be modeled as a CSTR and the corresponding mass and energy balances can be constructed for the species considered in this model: microalgae, water and CO2. Models for the growth of microalgae in open ponds can be found in references21-23, whereas references19,24 also incorporate modeling of the CO2 transfer. The microalgae growth kinetics formulation presented herein is based on the work in reference23 and the CO2 transfer modeling is adopted from reference24. Harvest

Compartment n

Compartment n-1

Compartment i

Internal recycle

Flow direction Compartment 1

Compartment 2

Compartment i-1

Feed

Figure 1. Approximation of the open pond model as a cascade of compartments.


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2.1. Water Material and Energy Balances. In an insulated open pond, water enters through the pond feed (Ffeed), or from precipitation, and leaves via evaporation (Fevap), harvest (Fharvest), or consumption by microalgae in photosynthesis. Typically algae are grown in geographic areas lacking rains, hence the amount of water added by precipitation is hereby neglected. Water consumed during photosynthesis is also negligible. For any CSTR in the pond, water flows in (Fin) from the preceding CSTR and flows out (Fout) to the succeeding CSTR in the direction of flow. Therefore, the depth of water (H) in an arbitrary CSTR in a segmented open pond can be found from LW

dH i = Fin, i − Fout, i − Fevap, i dt

(1)

where L is the length and W is the width of each CSTR, which are assumed constant and identical for all of the CSTRs. The following relations govern the flow within the pond between the i-th CSTR and the preceding CSTR

Fin, i = Fout, i -1 = W H i -1 v

(2)

where v is the water velocity, and i ∈ [1,n]. The pond feed and harvest are taken into account by adding Ffeed and subtracting Fharvest from eq 1 for the 1st and nth CSTRs, respectively. The dilution rate (D) dictates the rate of algal biomass harvest n

Fharvest = D LW ∑ H i

(3)

i =1

To compensate for evaporation losses, after harvesting the algal biomass the water is combined with makeup water (Fmakeup) and recycled back to the pond as Ffeed as shown in Figure 2. Regardless of the microalgae type, i.e. freshwater vs. marine, only freshwater is considered for Fmakeup, because even for a seawater pond to maintain the salinity at the desired level freshwater


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has to be added. Water blowdown for preserving culture quality is neglected. The amount of water evaporating from each CSTR is calculated accordingly Fevap, i =

LW Ev, i

(4)

ρ Le, i

where ρ is the water density. The latent heat flux (Ev), according to Dalton’s Law, and the latent heat of evaporation (Le) are given by Ev, i = (19.0 + 0.95 U 2 ) ( Psat, i − Pair )

(5)

Le, i = 597.3 − 0.57 Tw, i

(6)

where U is the wind speed above the pond water and Tw is the water temperature.25 The saturation vapor pressure at the water temperature (Psat), and the vapor pressure in the overlaying air (Pair) are computed using Antoine’s equation. The temperature of the water greatly affects the growth rate of algae. Hence, an energy balance for water is derived to track the water temperature in each compartment

LW ρ cp

d ( H iTw,i ) = ρ cp ( Fin,iTw,i −1 − Fout,iTw,i ) dt + LW Ei

(7)

where cp is the specific heat capacity of water. The term ‫ ܧ‬represents the heat exchanged through the water surface of each compartment. It accounts for the heat addition from the absorbed solar irradiance (Es) and atmospheric long-wave (Ea), and the heat loss due to evaporation (Ev), water long-wave (Ew), and conduction to atmosphere (Ec). These terms are calculated based on the model developed in reference25

(

Ea = σ (Tair + 273 ) α1 + 0.031 Pair 4

Ew, i = ε σ (Tw, i + 273)

) (1 − α ) 5

4

(8) (9)


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Ec, i = α 2 (19 + 0.95U 2 ) (Tw, i − Tair )

(10)

Es = α3 I a

(11)

where Ia is the daily average solar irradiance at the pond surface and Tair is the air temperature. The other parameters are defined in Table 1 including their assigned values as well as values for the mass balance parameters. Internal recycle Makeup water Harvest

Feed CSTR 1

CSTR i

CSTR n

External recycle Wet biomass

Figure 2. Modeling the open pond as a cascade of continuous stirred-tank reactors (CSTRs) with an internal and an external recycle stream. Table 1. Parameter definitions and values for the water mass and energy balances. Parameter

Description

Value

v

water velocity in the open pond (cm s-1)

25 26

ρ

density of water (g cm-3)

1.0

cp

specific heat of water (cal K-1 g-1)

1.0

σ

Stefan-Boltzmann constant (cal cm-2 d-1 K-4) 11.7×10-8

ε

water emissivity

0.97 25

α1

atmospheric attenuation coefficient

0.6 25


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α2

Bowen’s coefficient (mmHg oC-1)

0.47

α3

radiation absorption factor

0.9 27

α5

reflection coefficient

0.03 25

2.2. Microalgae Mass Balance and Growth Kinetics. In the absence of algae grazers, the concentration of algae in the pond (Calgae) would depend on the growth rate (μ), the harvest rate and the rate of algal biomass deterioration due to respiration and other basal metabolism processes (B) LW

d ( H i Calgae,i ) = Fin,i Calgae,i -1 − Fout,i Calgae,i dt + ( µ i − Bi ) LW H i Calgae, i

(12)

After harvesting the algal biomass, the water is recycled back to the pond and the concentration of algae in the feed is given by

Calgaefeed = (1 − H eff ) Calgae, n

Fharvest Ffeed

(13)

where Heff is the harvest efficiency which is around 95% for harvest of algal biomass.28 The areal productivity of the system (Pr) is often used as an evaluation criterion and could be defined as

Pr = H eff

D n ∑ H i Calgae,i n i =1

(14)

The basal metabolism rate increases exponentially with the water temperature

Bi = Bm e

(

kB Tw, i −TB

)

(15)

where Bm is the metabolic rate at a reference temperature (TB), and kB is a fitting constant.23 Factors affecting the growth rate of algae considered in the model are the temperature of the culture, the irradiance level, and the nutrient concentrations. The maximum algae growth rate


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(μmax) is achieved when those factors are at their optimal levels, otherwise the actual growth rate depends on the deviations from these levels as µ i = µ max f N, i f T, i f I, i

(16)

average

where fN, fT, and fI, are the attenuation factors for nutrient, temperature and light limitations, respectively.23 Most of these factors are species-specific, and so their detailed forms depend on the particular algae strain in study. The marine alga Nannochloropsis Salina (N. Salina) chosen for this study is a promising strain for algal biofuels production because of its high lipid content and robustness in outdoor cultivation.29 Since there are limited studies on outdoor cultivation of N. Salina, the mass and energy balances in the model are validated using data from the literature for the protein rich freshwater Spirulina which is widely cultivated outdoors at commercial scale for food-grade products.30 Microalgae proliferate within a certain range around an optimal temperature and growth stops completely away from such range of temperature. The term fT can be estimated using the following exponential relation

f T, i

 − kT,1 (Tw, i −Topt )2 , Tw, i > Topt e = 2 e− kT,2 (Topt −Tw, i ) , T ≤ T w, i opt 

(17)

where Topt is the optimal temperature for algae growth and kT,1, and kT,2 are fitting constants.23 The optimal temperature for Spirulina is within the range of 24-42 oC depending on the particular strain of Spirulina, and for N. Salina it is around 28 oC.28,31 Autotrophic microalgae drive photosynthesis by the energy from light photons and due to the shading effect the microalgae closer to the pond surface are exposed to higher light intensities, hence, the mean value of fI is considered


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f I, i

average

=

1 Hi

∫

Hi

0

1   tp

∫

β tp

0

 f I, i dt  dz  

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(18)

where β is the fractional length of a day having daylight (photoperiod), tp is the length of a day and z is the vertical distance from the water surface. Since autotrophic microalgae do not grow at night, this time integral is formulated to generate an average over the daylight hours; using a daily average instead would lead to overestimation of the growth rate. Various formulas have been developed in the literature32 for modeling fI and the model proposed in reference33 fits the experimental data presented in reference34 for N. Salina f I, i = 1 − e

−

Ii I max

(19)

where I is the photosynthetically active radiation (PAR) experienced by the microalgae and Imax is the maximum radiance above which algae growth does not increase any further. However, for some microalgae strains, including Spirulina, exceeding Imax actually damages the photosynthetic reaction, known as the photoinhibition effect. This phenomenon is captured in the relation proposed by Steele35

f I, i =

Ii I max

1−

e

Ii I max

(20)

The Beer-Lambert law is commonly used to model I as corrected for the attenuating effect of the growth culture

I i = α3 α 4

I a − ke,i z e β

(21)

where α4 is a factor for converting from total radiation to PAR. The extinction coefficient (ke) is related to the water/background turbidity (kw) and the concentration of algae in the pond


10

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ke, i = k w + .0088 Calgae, i RChl + .054 ( Calgae, i RChl )

2/3

(22)

where RChl is the chlorophyll content of the algal biomass.36 There are at least 30 chemical elements required for growing microalgae of which the most important ones, i.e. usually the growth limiting ones, are nitrogen, phosphorus, and carbon.37 However, for the purpose of determining fN in this analysis, only carbon is considered, because it constitutes a big fraction of the algal biomass.38 Assuming that all nutrients are abundant except for carbon, a Monod type equation modified for the inhibitory effect of oversupply of CO2 can be used to estimate fN

CCO2, i

f N, i =

K C + CCO2, i +

(23)

2 CCO 2, i

Ks

where KC and Ks are the half saturation and inhibition constants for CO2, respectively.24 Autotrophic microalgae acquire carbon by consuming dissolved CO2 supplied artificially and/or from diffusion of atmospheric CO2, hence the balance for the CO2 in the pond water (CCO2) is

(

Fin CCO2 , i -1 − Fout CCO2 , i + Gi d H i CCO2 ,i = dt LW

(

)

(

)

− ( µ i − Bi ) H i RCO2 Calgae, i − K atm CCO2 , i − CCO2 ,atm

(24)

)

where RCO2 is the CO2 requirement and G is the rate of CO2 supplementation by bubbling CO2 rich gas into the pond water. The mass transfer coefficient Katm regulates the diffusion of CO2 to/from the atmosphere as driven by the difference between CCO2 and the equilibrium concentration of atmospheric CO2 in water (CCO2,atm).


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2.3. CO2 Sump Stations. Sump stations located at the pond middle are used for bubbling CO2 gas into the growth culture in a concurrent or countercurrent arrangement. Modeling the CO2 transfer from the gas bubbles to the pond water in a sump station allows analyzing the effect of bubbling rate on algae growth which is useful for the optimization problem. For a given CO2 mole fraction in the inlet gas (yin), the term G can be calculated from determining the CO2 mole fraction in the outgassing bubbles (yout) Gi =

Pg Qg, i Rg Tg

(y

in

− yout, i )

(25)

where Rg is the universal gas constant, and the gas flowrate (Qg), temperature (Tg), and pressure (Pg) are assumed constant throughout the entire water column. Assuming plug flow for the gas phase, yout is estimated based on the model derived in reference24

yout, i =

  y P  H e  CCO2 , i +  in g − C CO2 ,i   Pg  H e Rg Tg   − K L ai (1− ε g, i ) W H i Ws  Qg, i H e  e  

Rg Tg

(26)

where He is the dimensionless Henry’s constant, KL is the mass transfer coefficient for the CO2 transfer from the gas phase to the liquid phase, Ws is the width of the sump station and εg is the gas hold up. For countercurrent flow, the gas bubbles’ total interfacial area (a) can be approximated from ai = π d b2

N o, i

(27)

Ws W (vb − v )

where db is the bubble diameter, and vb is the gas bubble terminal velocity.24 According to reference39, the number of bubbles of gas formed at the sump bottom (No) is


12

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N o, i =

Qg, i

(28)

d3 π b (1 − ε g, i ) 6

The gas hold up is determined from the volume ratio of gas in the sump ε g, i =

α 6 Qg, i

(29)

α 6 Qg, i + vWs W

where α6 is a correction factor for the compression of gas under water.40

2.4. Process Economics. The algal biomass production cost (PC), a criterion to evaluate different scenarios in this work, is defined as Tc

costCO2, t + costwater, t + costenergy, t

t =1

harvestt

PC = ∑

(30)

where t is an integer denoting the number of the day starting from the beginning of the production cycle, and Tc is the length of the time horizon. The daily amount of algal biomass harvested, cost of CO2, cost of water, and cost of energy are calculated by

harvestt = H eff Calgae,n, t Fharvest, t

(31)

n

costCO2, t = xCO2 ρCO2 ∑ Qg, i , t

(32)

costwater, t = xw Fmakeup, t

(33)

costenergy,t = xe Emixing, t

(34)

i =1

where xCO2, xw, and xe are the prices of CO2, freshwater and electricity, respectively. The CO2 gas density (ρCO2) is calculated using the ideal gas law. The daily energy requirement for mixing the open pond water using a paddle wheel (Emixing) is given by


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n

Emixing =

α 8 α 9 ρ ( hfriction + ∑ hbend ) v W ∑ H i i =1

n Meff

(35)

where Meff is the efficiency of the mixing system, α8 is a unit conversion factor, and α9 is the number of hours the paddle wheel is running daily.26 From the Gauckler-Manning formula for open channel flow one can calculate the head losses from friction of the pond bottom (hfriction) and flow around each bend (hbend) hfriction = v 2 no2

hbend =

nL r 4/3

(36)

3α 7 v 2 g

(37)

where no is a roughness factor also known as the Gauckler-Manning coefficient, r is the channel hydraulic radius, α7 is the kinetic loss coefficient and g is the acceleration of gravity.41 The mathematical model described above was coded in gPROMS ModelBuilder v4.0 installed in a 64-bit Windows 7 CPU equipped with an i7 processor at 3.4 GHz and 16 GB of RAM.42

3. MODEL VALIDATION 3.1. Experimental Setup and Model Assumptions. Spirulina was cultivated in a 450 m2 outdoor open pond for 10 months in Málaga, Spain.43,44 The experiment resembles a scenario for commercial scale algal biomass production, especially given the prolonged cultivation period and varying environmental conditions. Therefore, the water temperature, biomass areal concentration, and productivity profiles reported in references43,44 were used to validate the ones predicted by the developed model. In the experiment, the depth of water was maintained at 30 cm and the growth culture was prepared with modified Zarrouk’s Medium providing adequate nutrient levels.43,45 Moreover, the pond was inoculated with an areal concentration of 15 g DW m-2 and harvest was started after 13


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days and only interrupted during February due to heavy rains.44 The corresponding model assumptions and parameters are: (1) the open pond was discretized into n = 18 compartments each having a length and width of L = W = 5 m; (2) the makeup water flowrate was set to match the evaporation losses to fix the depth at H = 0.3 m; (3) the nutrient concentrations were at their optimal values, therefore eq 23 was set to fN,i = 1 and eqs 24-29 were excluded; (4) after the inoculation period (13 days), the dilution rate was set to an estimate D = 0.10 day-1 based on reference46 except for 17 days during February where there was no harvest. The initial conditions and values assigned to the kinetic parameters for Spirulina growth are shown in Table 2. Figure 3 shows the solar radiation and air temperature data for the period of the experiment at Málaga, which were obtained from the European Database of daylight and Solar Radiation47 and the Tutiempo Network48, respectively. These data, including data for humidity, wind speed, and photoperiod, were used in the simulation of the model consisting of eqs 1-18 and 20-23. The time integration of the ordinary differential equations was performed using the DASOLV solver which is based on variable time step Backward Differentiation Formulae (BDF).49 This solver embeds the MA48 sub-solver which uses a direct LUfactorization algorithm to solve a set of linear algebraic equations. To increase numerical stability the default value of the “PivotStabilityFactor” was changed to 0.9, which is one of the setting parameters of the MA48 sub-solver.


15


Solar Radiation (left axis)

Air Temperature (right axis) 45 40

20

35 30 25 2

oC

J m-2 (Millions)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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20 15 10

0

5 0

100 200 Time (Days from September 23, 1997)

300

Figure 3. Daily global solar radiation and average air temperature at Málaga from September 23, 1997 to July 31, 1998.47,48

Table 2. Parameter definitions and values for the microalgae growth kinetics model. Parameter

Description

Bm

metabolic rate at reference temperature (day-1) reference temperature for metabolic rate (oC) metabolic rate exponential fitting constant (oC-1) fractional length of a day having daylight length of a day (h) water (background) turbidity (m-1) coefficient for photosynthetically active radiation (PAR) Species-specific parameters maximum algae growth rate (day-1) optimal temperature for algae growth (oC) temperature limitation fitting constant (oC-2) temperature limitation fitting constant (oC-2) maximum irradiance for algae growth (µE m-2 s-1) algal biomass chlorophyll content (g Chl Kg-1 DW) Initial conditions depth of water in a CSTR in the open pond (m) temperature of the water in the CSTR (oC)

TB kB β tp kw α4

μmax

Topt kT,1 kT,2 Imax RChl H Tw

Value 0.04 23 20 23 0.069 23 0.5 24 0.3 25 2.05 50 Spirulina N. Salina 51 1.4 1.3 34 27.5 23 27 52 0.005 23 0.01 52 0.004 23 0.03 52 200 53 58 34 6.6 43 17 54 Spirulina N. Salina 0.3 0.3 * 22.1 12.4 *


16

Page 17 of 34

Calgae CCO2 *

concentration of microalgae in the pond (g DW m-3) molar concentration of CO2 in the pond (mol m-3)

15 -

15 0.02 *

assumed in equilibrium with the ambient environment.

3.2. Simulation Results. The predicted water temperature profile shown in Figure 4 is slightly lower than the one determined experimentally which is due to the fact that the air temperature data used herein are marginally lower than the measurements at the experiment.43,44 As a result, the growth was inhibited in the winter, but advanced in the summer and consequently the areal concentration profile was slightly altered. However, as shown in Figure 5 the predicted productivity is in good agreement with the experimentally determined productivity with a mean percent error of 16.3%, which demonstrates the adequacy of the proposed model. Algal Biomass (left axis) Water Temperature (right axis)

180 160

70 60

140

50

120 100

40

80

30

60

oC

g DW m-2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20

40

10

20 0

0 0

100 200 300 Time (days from September 23, 1997)

Figure 4. Simulation results for Spirulina cultivation in Málaga, Spain from September 1997 to July 1998: Algal biomass areal concentration and temperature of the pond water.


17


Percent Error (right axis) Simulation (left axis) Experiment (left axis)

Productivity (g DW m-2 day-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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16

100%

14 75%

12 10

50%

8 6

25%

4 2 0

0%

Figure 5. Comparison between monthly-averaged values of the predicted productivity and the productivity determined experimentally in reference44.

4. DYNAMIC OPTIMIZATION 4.1. Problem Definition. Imperial County in California is one of the suitable places for cultivating N. Salina in USA owing to the warmer weather and availability of resources including land, water and CO2.26 This site was selected for the optimization case study and the daily weather conditions in a typical meteorological year for this site were obtained from the National Solar Radiation Data Base55. The 4 ha open pond proposed in reference26 for the production of algal biofuels at commercial scale was adopted herein. Therefore, the pond was discretized into 44 compartments with L = W = 30 m and the CO2 gas was set to be introduced at the bottom of the 12th and 34th compartments. The parameter values selected for the growth kinetics of N. Salina are shown in Table 2. The values assigned to the parameters of modeling the CO2 transfer and process economics are shown in Tables 3 and 4, respectively. For a production cycle of one year the program, comprising eqs 1-19 and 21-37, contains 1043 variables, and 1876 parameters.


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Table 3. Values assigned to the parameters of the CO2 mass balance and transfer model. Parameter KC Ks CCO2,atm RCO2 vb Tg Pg yin Rg He Ws db α6 Katm KL

Description

Value

half saturation constant for CO2 (mol CO2 m-3) inhibition constant for CO2 (mol CO2 m-3) equilibrium concentration of atmospheric CO2 in water (mol m-3) CO2 requirement per unit of algae (mol CO2 g-1 DW)

9×10-4 24 180 24 0.02

CO2 gas bubble terminal velocity (cm s-1) temperature of CO2 gas (oC) pressure of CO2 gas (atm) mole fraction of CO2 in the bubbling gas at inlet universal gas constant (atm m3 mol-1 K-1) dimensionless Henry’s constant width of the CO2 gas sump station (m) diameter of CO2 gas bubble (mm) Coefficient for gas compression under water depth of 30 cm mass transfer coefficient for diffusion of CO2 to/from atmosphere (m day-1) mass transfer coefficient for CO2 transfer from gas bubbles to water (m day-1) Table 4. Parameter definitions and values for the process economics model.

30 assumed 316 26 1.2 26 1 8.2×10-5 0.8317 0.3 26 2 57,58 0.96 40 2.4 5 9.59 59

Parameter xw xe xCO2 Meff no r α7

Description price of agricultural water ($ m-3) price of electricity ($ kWh-1) price of pure CO2 gas ($ tonne-1)

0.042 56

Value 0.016 60 0.04 61 40 62

efficiency of the paddle wheel mixing system 40% assumed Gauckler-Manning coefficient 2.08×10-7 26 channel hydraulic radius 0.29 26 kinetic loss coefficient 2 26

4.2. Base Case. The dilution rate (D) is one of the main operating parameters affecting the productivity of an algal system. A low D creates a dense culture where light availability becomes limited due to the shading effect. This reduces the growth rate of microalgae and consequently leads to a lower productivity. On the other hand, a high D would reduce the concentration of microalgae in the pond also resulting in a lower productivity. Moreover, in an extremely sunny day this could be


19


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damaging to the growth rate for a microalga that experiences photoinhibition which would lead to an even lower productivity. Therefore, local climatic conditions have to be considered when searching for the optimal D. Another important operating parameter is the rate of bubbling of the CO2 gas (Qg) into the pond culture. Increasing Qg provides more dissolved CO2 to the growth culture, but it increases the loss of CO2 through degassing as well, and CO2 is a costly feedstock. The operating depth of water in the pond, controlled by Fmakeup, influences the system productivity and economics through affecting the light penetration, CO2 absorption and mixing energy requirement. For the base case scenario, the operation was set based on best practice as reported in the literature for commercial scale cultivation. The optimal D was determined by employing the empirical harvest scheme proposed in reference46. It starts by holding off harvest and allowing the algae to grow till the stationary phase of growth is reached followed by ramping up D in steps of 0.05 day-1 as demonstrated in Figure 6. Each time the concentration of algae in the pond is stabilized for several days the dilution rate is increased and the optimum is the one yielding the maximum algae productivity. The next step is to determine the algal biomass areal concentration, which when reached after inoculation, harvest at the identified D should commence. Typically the CO2 is added to the culture according to the demand for carbon as determined by the pH of the culture, hence Qg was set to the level just enough to prevent any limitation on growth due to CO2 deficiency.63 The depth of culture is usually maintained at a certain level, therefore, Fmakeup was set to the level that compensates for evaporation losses.4 The productivity profile shown in Figure 6 is consistent with the experimental results reported in reference46; note that the highest productivity was achieved when D is 0.10 day-1. From Figure 7, it was observed that the growth rate is mostly inhibited by the limited availability of sunlight


20

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and this limitation is higher for sunnier days. Considering that N. Salina does not experience photoinhibition, this could only mean that the shading effect of the high algal biomass concentration is the main factor responsible for limiting the productivity of the system. Comparing Figures 6 and 7, it was noticed that the growth rate peaks when the areal concentration is around 35 g DW m-2. Therefore, the base case was simulated starting with no harvest followed by D = 0.10 day-1 once the concentration exceeded 50 g DW m-2 and PC was found to be $240 tonne-1. Sensitivity analysis can provide a better insight into the effect of the dilution rate on PC as demonstrated in Figure 8, suggesting an optimal D of 0.31 day-1 which improves the base case PC to $196 tonne-1. An alternative to finding a single dilution rate is to aim for the optimal areal concentration and adjust the dilution rate accordingly.64 However, in outdoor cultivation the optimal areal concentration is a function of the uncontrolled environmental conditions, therefore, the dilution rate profile needs to be determined using dynamic optimization. Furthermore, in the previous scenarios around 22% of the CO2 fed to the pond was lost to the atmosphere due to degassing and Chart 1 shows that the cost of the CO2 gas constitutes 59% of the improved base case PC, hence, Qg needs to be optimized as well. Also, although the cost of makeup water is insignificant as Chart 1 shows, optimizing Fmakeup would affect the CO2 absorption.


21


0.25

0.2

105

0.15 70 0.1 35

0 January

Dilution Rate (day-1)

Algal Biomass (g DW m-2) Productivity (10-1 g DW m-2 day-1)

140

0.05

February

April

June

August

October

0 December

Figure 6. Demonstration of the heuristic approach for finding the optimal dilution rate. The dilution rate is increased in steps of 0.05 day-1 each time the microalgae reach a stationary phase.

Nutrient Limitation

Light Limitation

Temperature Limitation

Growth Rate

0.9

0.45

0.8

0.4

0.7

0.35

0.6

0.3

0.5

0.25

0.4

0.2

0.3

0.15

0.2

0.1

0.1

0.05

0 January

February

April

June

August

October

µ (day-1)

0.5

1

, fN, fT

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0 December

Figure 7. Effect of the attenuation factors on the growth rate during the search for the optimal dilution rate and biomass algal concentration in the pond using the heuristic approach.


22

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600 Production Cost ($ tonne-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


450

300

150 0

0.1

0.2

0.3

0.4

0.5

Dilution Rate (day-1)

Figure 8. Sensitivity analysis demonstrating the effect of the dilution rate (D) on the production cost (PC). The minimum PC is at D of 0.31 day-1.

Makeup Water 11%

Carbon Dioxide 59%

Energy 30%

Chart 1. Breakdown of the Production Cost (PC) of $196 tonne-1 for the improved base case scenario using a dilution rate of 0.31 day-1.

4.3. Optimization Problem. The optimization problem considered determines the profiles of the dilution rate, CO2 gas flowrate, and makeup water flowrate that minimize D , Qg , Fmakeup

PC “eq 30”

subject to the following constraints:


23


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(1) mass and energy balances “eqs 1-14” (2) growth kinetics “eqs 15-19 and 21-23” (3) CO2 transfer “eqs 24-29” (4) process economics “eqs 30-37” Note that this formulation contains nonlinear growth kinetics and transient balances making it a nonlinear dynamic optimization problem. Assuming that the optimal areal concentration varies only on a monthly basis, the time horizon was divided into 12 control intervals creating 36 optimization decision variables which are subject to the following bounds:

Qg ≥ 0, Fmakeup ≥ 0, 0 ≤ D ≤ 0.5, 0.2 ≤ H ≤ 0.4 assuming that the CO2 gas and makeup water availability are unlimited. The bounds on the dilution rate and depth were set based on typical operation in commercial algae facilities.4 The improved base case values for the decision variables were used as an initial guess and the dynamic optimization was performed using the CVP_SS solver which implements a control vector parametrization algorithm based on the single-shooting method.65 It took the optimizer 1443 seconds to find the optimal operating profiles shown in Figures 9b-d corresponding to a PC of $167 tonne-1 which is 15% less than in the improved base case. Interestingly, Figure 9a shows that the algal biomass areal concentration in the improved base case and the optimal case are almost identical after around three months of cultivation; hence, it seems that the optimal areal concentration is indeed close to 30 g DW m-2. As demonstrated in Figures 9a,b, the optimal dilution rate profile follows the algal biomass areal concentration during the year, but at the end of the production cycle D goes up so that the remaining algae in the pond is completely harvested. Note from Figure 9c that the optimal CO2 gas flowrate is on average less than in the improved base case scenario, which reduces the CO2 loss from the pond to the atmosphere from


24

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an annual average of 1.1 to 0.58 kg CO2 per kg DW produced. Although, this causes growth limitation due to CO2 deficiency as shown in figure 10, apparently the gain on algae growth does not justify the increased loss of CO2 to the atmosphere from an economic point of view. Optimal Case

Improved Base Case

Optimal Case

(a)

Improved Base Case

0.5 0.45

50

0.4

D (day-1)

Algal Biomass (g DW m-2)

60

40 30

0.35 0.3 0.25 0.2

20

0.15 0.1

10

0.05

0 January

April

Optimal Case

August

December

(b)

0 January

Improved Base Case

April

Optimal Case

August

December

Improved Base Case

300

Fmakeup (m3 day-1)

600

Qg (m3 day-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


200

100

(c)

0 January

April

August

December

(d)

400

200

0 January

April

August

December

Figure 9. Comparing results from the improved base case and the optimization study. (a) Algal biomass areal concentration. (b) Dilution rate. (c) CO2 gas flowrate. (d) Makeup water flowrate.


25


Nutrient Limitation

Light Limitation

Temperature Limitation

Growth Rate

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.45 0.4 0.35 0.3 0.25 0.2 0.15

µ (day-1)

, fN, fT

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.1 0.05 0 Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Figure 10. Effect of attenuation factors on the growth rate for the optimal case scenario. 5. CONCLUSIONS A mathematical model for estimating the growth of microalgae in an outdoor open pond based on local climatic conditions was developed in this work. The model was validated against literature data for the production of Spirulina in an outdoor open pond in Málaga, Spain. The simulated algal biomass productivity agreed with experimental data, with a mean percent error of approximately 16%. A dynamic optimization problem was formulated for determining the location-specific optimal monthly operating profiles for the dilution rate, CO2 gas flowrate, and makeup water flowrate. A case study was conducted for the cultivation of N. Salina in California, USA. The operating profiles generated by the optimization lowered the cultivation cost by at least 15% when compared with other case scenarios where best practice operation and sensitivity analysis were employed. Based on the analysis presented, algae facilities adopting outdoor open ponds and having access to local meteorological data can use the formulation presented in this work to lower the cultivation cost by identifying optimal operating conditions. Furthermore, the proposed formulation can be useful for determining facility locations for algal biomass production by comparing candidate sites based on optimized operations.


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ASSOCIATED CONTENT AUTHOR INFORMATION Corresponding Author: *E-mail: [email protected]. Notes: The authors declare no competing financial interest. ACKNOWLEDGMENTS The authors acknowledge the financial support from Abu Dhabi National Oil Company (ADNOC).

NOMENCLATURE a α1 α2 α3 α4 α5 α6 α7 α8 α9 B Bm β Calgae Calgaefeed CCO2 CCO2,atm costCO2 costwater costenergy cp D db E

total interfacial area of CO2 gas bubbles (m-1) atmospheric attenuation coefficient Bowen’s coefficient (mmHg oC-1) radiation absorption factor fraction accounting for the visible portion of solar irradiance reflection coefficient correction factor for the compression of gas under water kinetic loss coefficient conversion factor (9.8 Watt s kg-1 m-1) daily hours the paddle wheel is operating (24 h day-1) rate of basal metabolism processes (day-1) metabolic rate at reference temperature (day-1) fractional length of a day with daylight mass concentration of microalgae in the pond (g DW m-3) concentration of algae in the feed (g DW m-3) molar concentration of CO2 in the pond (mol m-3) equilibrium concentration of atmospheric CO2 in water at 20 oC (mol m-3) cost of CO2 supplied to the open pond ($ day-1) cost of freshwater supplied to the pond ($ day-1) cost of energy supplied to the pond ($ day-1) specific heat capacity of water (cal K-1 g-1) dilution rate (day-1) diameter of CO2 gas bubble (mm) heat exchanged through the water surface of the CSTR (cal cm-2 day-1)


27


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Ea Ec Emixing Es Ev Ew ε εg Fevap Ffeed Fharvest Fin Fmakeup Fout fI,fN,fT g G H harvest He Heff hfriction/bend i I Ia Imax Katm kB KC ke KL Ks kw kT,1,kT,2 L Le Meff μ μmax n

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heat added to the CSTR water from the atmospheric long-wave (cal cm-2 day-1) heat lost from the CSTR water by conduction to atmosphere (cal cm-2 day-1) energy requirement for mixing the open pond water (kWh day-1) heat added to the CSTR water from the absorbed solar irradiance (cal cm-2 day-1) latent heat flux (cal cm-2 day-1) heat lost from the CSTR water through water long-wave (cal cm-2 day-1) water emissivity gas hold up water evaporating from a CSTR in the open pond (m3 day-1) flowrate of the open pond feed stream (m3 day-1) flowrate of the open pond harvest stream (m3 day-1) water entering a CSTR from the previous CSTR within the open pond (m3 day-1) makeup water flowrate (m3 day-1) water leaving a CSTR to the following CSTR within the open pond (m3 day-1) attenuation factors for light, nutrient, and temperature limitations acceleration of gravity (m s-2) rate of CO2 bubbling into the pond water (mol day-1) depth of water in a CSTR in the open pond (m) algal biomass harvested daily (tonne day-1) dimensionless Henry’s constant efficiency of algal biomass harvest (%) head loss from friction and from flow around the bends and sumps (m) index for the location of the CSTR relative to the other CSTRs in the open pond visible irradiance absorbed at the pond surface over daylight hours (µE m-2 s-1) daily average solar irradiance at the pond surface (W m-2) maximum irradiance above which algae growth does not increase (µE m-2 s-1) mass transfer coefficient for diffusion of CO2 to/from the atmosphere (m day-1) exponential fitting constant for metabolic rate (oC-1) half saturation constant for CO2 (mol CO2 m-3) extinction coefficient related to water turbidity and algae concentration (m-1) mass transfer coefficient for CO2 transfer from gas phase to liquid phase (m day-1) inhibition constant for CO2 (mol CO2 m-3) water (background) turbidity (m-1) fitting constants of attenuation factor for temperature limitation (oC-2) length of the CSTR in the open pond (m) latent heat of evaporation (cal g-1) efficiency of the paddle wheel mixing system growth rate of microalgae (day-1) maximum algae growth rate (day-1) number of compartments the open pond is segmented to


28

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


no No Pair PC Pg Pr Psat Qg r Rg RCO2

Gauckler-Manning coefficient; a roughness factor (day m-1/3) number of bubbles of gas formed at the sump bottom (s-1) vapor pressure in the overlaying air (mmHg) algal biomass production cost ($ tonne-1) pressure of CO2 gas (atm) productivity of algal biomass in the open pond (g DW m-2 day-1) saturation vapor pressure for a given water temperature (mmHg) CO2 gas flowrate (m3 day-1) hydraulic radius of pond channel (m) universal gas constant (atm m3 mol-1 K-1) carbon dioxide requirement (mol CO2 g-1 DW)

RChl ρ ρCO2

algal biomass chlorophyll content (g Chl kg-1 DW) density of water (g cm-3) density of CO2 gas (g cm-3)

σ tp Tair TB Tc Tg Topt Tw U v vb W Ws xCO2

Stefan-Boltzmann constant (cal cm-2 d-1 K-4) length of a day in hours (24 h) daily average local air temperature (oC) reference temperature for metabolic rate (oC) time horizon of the production cycle temperature of CO2 gas (oC) optimal temperature for algae growth (oC) temperature of the water in the CSTR (oC) wind speed above the open pond water (m s-1) water velocity in the open pond (cm s-1) gas terminal velocity (cm s-1) width of the CSTR in the open pond (m) width of the CO2 gas sump station (m) price of CO2 ($ tonne-1)

xe xw yin/out z

price of electricity ($ kWh-1) price of agricultural water ($ m-3) CO2 mole in the inlet/outgassing bubbles vertical distance from the water surface in the pond (m)

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Modeling and Dynamic Optimization of Microalgae Cultivation in

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