Design and optimization of artificial cultivation units for algae production

lable at ScienceDirect

Energy xxx (2014) 1e17

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Energy

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Design and optimization of artificial cultivation units for algaeproduction

Soumya Yadala, Selen Cremaschi*

Department of Chemical Engineering, The University of Tulsa, 800 South Tucker Drive, Tulsa, OK 74104, USA

a r t i c l e i n f o

Article history:Received 1 November 2013Received in revised form1 March 2014Accepted 1 June 2014Available online xxx

Keywords:Algae cultivationPhotobioreactor designBiodiesel productionGrowth optimization

Abbreviations: PBR, photobioreactor; MARR, mreturn; DA, dry algae biomass; WIA, water present iwater; WRD, water remain in dryer; trans, transesterimethanol; gly, glycerol; LBTD, lower bound tube dibiomass concentration; LBCD, lower bound column diflat plate width; UBFPL, upper bound flat plate length* Corresponding author. Tel.: þ1 918 631 3422; fax

E-mail addresses: [email protected](S. Cremaschi).

http://dx.doi.org/10.1016/j.energy.2014.06.0010360-5442/© 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Yadala S,(2014), http://dx.doi.org/10.1016/j.energy.20

a b s t r a c t

This paper focuses on finding the optimum design of artificial cultivation units for biomass productiondepending on geographical location and kind of algal species selected for growth. Here, the optimum isdefined as the design that yields the lowest net present sink for the lifetime of the cultivation unit.Models are developed for tubular, column, and flat plate photobioreactors by considering diurnal patternof sunlight and temperature fluctuations. As part of the case study, algae growth is modeled for 10 yearsin each cultivation unit using two species and four locations, resulting in twenty-four optimizationproblems. Each optimization model is implemented in GAMS 23.6.5 and the solution is obtained usingCONOPT (version 3.14W) solver. The results indicate that algae species with higher oil content requiressmaller reactor volume to produce the desired amount of biomass. The results also reveal that thegeographical location with higher incident solar irradiance may not necessarily be the optimal locationfor algae culturing because higher irradiance may lead to cell damage, and hence, lower growth rates.Among the options considered in the case study, the design of tubular photobioreactor for culturingPhaeodactylum tricornutum at Hyderabad, India yields the minimum net present sink.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Excessive usage of fossil fuels has not only led to depletion ofworld reserves but also emission of greenhouse gases [1]. Theseconcerns have enhanced the interest in developing first generationbiofuels extracted from food crop feedstocks including soy beans,palm, canola, and rape seeds using conventional technologies [2].First generation biofuels, however, are limited in their ability tomeet the existing demand for transport fuels besides causing atremendous strain on global food markets and endangering hunger[3]. To accommodate some of these short falls, second generationbiofuels emerged from non-food crop feedstocks including wheatstraw, corn stover, and wood using advanced technologies [4].These biofuels might still not be abundant enough to replace morethat 20e25% of our total transportation fuels because of concerns

inimum acceptable rate ofn algae biomass; WW, wasteficator; BD, biodiesel; MeOH,ameter; UBBC, upper boundameter; LBFPW, lower bound.: þ1 918 631 3268., [email protected]

Cremaschi S, Design and op14.06.001

over competing land use. The major draw backs associated withfirst and second generation biofuels are addressed by third gener-ation biofuels derived from algae [5].

The main advantage of algae is that they create their own foodthrough photosynthesis by combining light, carbon dioxide, andwater. This food is then stored as carbohydrates and lipids. Majorityof algal species exhibit much higher growth rates and pro-ductivities than conventional forestry, agricultural crops, andaquatic plants, which makes it possible to use algae to fulfill theoverall fuel demand while using limited land resources [6] and [7].Furthermore, algae can be cultivated in saline water on non-arableland [8]. One of the common uses of algal biomass is to producebiodiesel because lipid or oil content present in algae may be quitehigh, with individual species containing anywhere between 2% and40% on a dry weight basis [9] and [10]. The most common pro-duction route for biodiesel includes the following steps: the culti-vated cells are separated from the growth medium and dried, thelipid content of the cells is extracted, and subsequently biodiesel isproduced via transesterification reaction. Algae-based biodiesel ishighly biodegradable and contains no sulfur; hence, it is seen as aclean and more environmentally-friendly fuel source [8]. Consid-ering these benefits, algae appears to be a viable alternative feed-stock for producing biodiesel that is capable of meeting the demandfor transportation fuels.

timization of artificial cultivation units for algae production, Energy

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Nomenclature

Greek symbolsr density of water (g m�3)m viscosity of water (g m�1 s�1)s surface tension (g s�2)w kinematic viscosity of water (m2 s�1)

Parametersg acceleration due to gravity (m s�2)

Sc solar constant (mE m�2 s�1)Cp specific heat (J g�1�C�1)EmpA empirical constant for interior and coastal regionshdryer efficiency of dryerhextractor efficiency of extractorhtrans efficiency of transesterificatorDemand biodiesel demand (g y�1)nT number of tube diameters of separation[O2]out outlet concentration of dissolved oxygen (g m�3)[O2]in inlet concentration of dissolved oxygen (g m�3)Tmax maximum temperature attained at a location in a day

(�C)Tmin minimum temperature attained at a location in a day

(�C)Latitude latitude (degree)Longitude longitude (degree)Time zone time zoneEqCost equipment cost of a node at a location ($ m�3)ElCost electric cost of pump ($ kWh�1)%s percentage of dry algae present in algae species (%)OC lipid content present in an algae species (dry weight

basis)Ka extinction coefficient of algae biomass (m2 g�1)Ik species dependent constant (mE m�2 s�1)n empirical constantUlmax maximum permissible liquid velocity inside the PBR

(m h�1)sunrise time time of sunrise (h)q daily zenith angle at a location (radians)It extraterrestrial solar radiation on horizon surface

(mE m�2 s�1)Io incident solar radiation on horizontal surface

(mE m�2 s�1)Tsurr surrounding temperature (�C)

VariablesZ objective function ($)O mass flow rates of reactant or product entering or

leaving transesterificator or extractor at a location for aspecies in a year (g y�1)

N mass flow rates of reactant or products entering orleaving PBR at a location for a species in a year (g y�1)

X mass flow rates of reactant or product entering andleaving PBR at a location for a species and at certaintimes (g h�1)

ṁ mass flow rate of products produced in PBR at alocation for a species and at certain times (g h�1)

Iavg average irradiance inside the PBR (mE m�2 s�1)Øeq length of the light path from the surface to any point in

the PBR (m)BC biomass concentration (g m�3)Treactor reactor temperature (�C)AS Surface area of the cultivation unit (m2)mmax maximum specific growth rate (day�1)m specific growth rate (h�1)PrV volumetric productivity of the PBRs (g m�3 h�1)PrVO2 volumetric rate of oxygen generation by

photosynthesis (rate of photosynthesis) (g m�3 h�1)V volume occupied (m3)Ul velocity of fluid flow in PBR (m h�1)AC Cross sectional area of the cultivation unit (m2)Re Reynolds numberHØ Hydraulic diameter of PBR (m)PP Power of pump (kW)f tube diameter (m)TL length of solar loop of a tubular PBR (m)A Land area occupied by tubular PBR (m2)

Variables related to column PBRdC column diameter (m)ε gas hold upUg superficial gas velocity (m h�1)dB bubble diameter (m)CH height of column PBR (m)

Variables related to flat plate PBRW width of flat plate (m)FPH height of flat plate (m)FPL length of flat plate (m)ε gas hold upUg superficial gas velocity (m h�1)Ub bubble rise velocity (m h�1)Ra Rayleigh number

Subscriptss speciesl locationPi process i (P1, P2, P3, P4, and P5 represent the series of

processes involved in biodiesel production)c componenty yeard day of the yeart time of the dayin stream flowing in to unit operationout stream flowing out from unit operationmax maximummin minimum

S. Yadala, S. Cremaschi / Energy xxx (2014) 1e172

Despite their inherent potential as a source of biofuel, algae-based applications have scarcely reached industrial scale. Themain reason underlying such a low practical implementation is thehigh costs associated with algae cultivation [11]. Practical methods

Please cite this article in press as: Yadala S, Cremaschi S, Design and op(2014), http://dx.doi.org/10.1016/j.energy.2014.06.001

for cultivating algae in large scale are open ponds and closedphotobioreactors (PBRs) [12]. Open ponds are less efficient whencompared to PBRs [13] because of the difficulty to controlcontamination, temperature fluctuations, and evaporative losses.


S. Yadala, S. Cremaschi / Energy xxx (2014) 1e17 3

They are also prone to inefficient mixing and light limitations [12]and [14]. On the other hand, PBRs offer a contamination freeenvironment that is appropriate for the growth of sensitive strains.They allow better control of cultivation conditions whichcontribute to the production of strains rich in high value products[15]. However, PBRs are substantially more expensive compared toopen ponds. They are difficult to scale up, and it is challenging tocontrol light flux [12]. Design of artificial cultivation units is mainlyinfluenced by three features: (1) algae strain that will be grown, (2)geographical location for cultivating the chosen algae strain, and (3)type of cultivation unit for producing the biomass.

More than 50,000 algal species are estimated to exist [16] andnot all of these species are appropriate for producing biodieselbecause each individual algae species has unique physiological andgrowth characteristics. In an earlier study, Aquatic Species Programof US Department of Energy screened more than 300 micro algalspecies on the basis of growth rate, oil content, and nutrient defi-ciency under different environmental conditions [17]. Theyconcluded that during nutrient deficiency, rates of oil productionare lower. The possibility of large scale production of algae in openponds was investigated by conducting experiments at various lo-cations, and these studies confirmed the viability of long term,reliable production of algae. Studies were conducted in laboratoryto select the algae strain for oil production [18]. Thirty micro algalstrains were screened for biomass productivity and lipid content.One strainwith relatively high lipid content was selected to grow inan outdoor PBR to study the influence of irradiance, nutrient supplyon fatty acid accumulation and to determine its lipid productionrate. The experimental results revealed that micro algal straineustigmatophyte has the potential for an annual production of 20tons of lipid per hectare in Mediterranean climates and of morethan 30 tons of lipid per hectare in sunny tropical areas. Algaespecies with high oil content, growth rate, productivity, ability tosurvive the shear stress in PBRs, robust, tolerable to a wide range oftemperatures, are desirable to produce biodiesel efficiently [14].Therefore, it is crucial to select the right algae species, which ulti-mately may lead to a cost effective production of biodiesel.

The second feature that should be considered is the location ofthe PBR. Certain regions of the world are better suited for algaegrowth than others [19]. According to the observations of Wogan[20], southwestern and southern states of United States are goodlocations to cultivate algae. Borowitzka [21] argued that selection of

Table 1Main pros and cons of tubular, column, and flat plate PBRs.

Closed systems Advantages

Tubular PBR 1. Large illumination surface area2. Suitable for outdoor cultures3. Good biomass productivities

Column PBR 1. High mass transfer, photosyntheticpotential for scalability

2. Reduced photoinhibition and photo3. Low cost, compact, and easy to oper4. Greater gas hold ups5. Best exposure to light/dark cycles6. Least land use7. Promising for large scale cultivation

Flat plate PBR 1. Large illumination surface area2. High area to volume ratios3. Suitable for outdoor cultures4. High biomass productivities5. Uniform distribution of light6. Inexpensive7. Easy to construct, maintain, and cle8. High photosynthetic efficiency9. Mass production of microalgae


reactor type is also influenced by geographical factors such as landcosts and climate at the reactor site, which is correlated with theavailable sunlight and temperature at that location. Availability ofsunlight at a location influences cell growth because algae absorblight to grow [22]. The amount absorbed greatly depends onbiomass concentration. The higher the algae concentration, the lesslight is absorbed [22]. Effect of average light intensity on cell con-centrationwas examined formicro algal cultivation in PBRs [23]. Anempirical hyperbolic equation, analogous to LamberteBeer typeequation, was proposed where light intensity and algal cell con-centration are calculated by numerical integration. It was foundthat identification of the appropriate shape and size of cultivationunit for the selected location is critical for attaining optimum ab-sorption of light by algal biomass. Temperature is another impor-tant environmental factor that strongly regulates cell growth [24]and [25]. Influence of temperature on growth of various species ofalgae was experimentally examined by several investigators[24,26e29]. Correlations were proposed to describe the relation-ship between growth rate and temperature using Arrheniusequation [30], Van't Hoff equation [25], and Berthelot's equation[31]. Studies show that algal growth rates increase as temperatureincreases up to an optimal temperature, beyond which theydecrease [32] and [33]. It was also found that different algal speciesrespond differently to temperature changes because of differencesin cell sizes [34] and [35], concentration of photosynthetic pig-ments within the cells [35], and tolerance levels. Hence, thegeographic location of the cultivation unit plays a vital role in itsdesign.

The third important aspect that influences the design of culti-vation units is the type of PBR chosen for cultivation. Variety of PBRdesigns have been proposed and tested with the three main cate-gories for large scale cultivation of algae being tubular/horizontal,column/vertical, and flat plate/rectangular PBRs [36]. The pros andcons of different reactor types are summarized in Table 1 [37]. Thealgae cultivation unit must be designed to utilize maximumamount of light (sunlight if placed outside e which is the consid-ered approach in the current work) for growth, minimizing pho-toinhibition and photolimitation affects. In a PBR, light intensitydecreases exponentially with distance from the illuminated surface[22]. Due to this effect, the algae cells near the illuminated surfaceof the reactor are exposed to high light intensities. Prolongedexposure to such high light intensities may cause cell death leading

Disadvantages

1. Requires large land space2. Photoinhibition is common3. Poor mass transfer

efficiency,

-oxidationate

of algae

1. Small illumination area2. Low surface to volume ratio3. Expensive compared to open ponds

an up

1. Difficult to scale-up2. Algae adheres to walls3. Low photosynthetic efficiency



to photoinhibition [38]. The algae cells away from the illuminatedsurface receive less light due to mutual shading which leads tolower growth rates. This phenomena is called photolimitation [39]and [40]. Hence, the design of PBRs should consider all these effectsand enable effective utilization of sunlight by algae. In general, ithas been suggested that effective mixing in the reactor andincreasing the illumination surface area to volume ratio of thereactor leads to better sunlight utilization [13] and [41].

Many PBR models and designs have been proposed and/ortested in literature, e.g. [42e47]. For example, Chisti, Fernandezand Grima [36] designed an airlift tubular PBR with continuousrun solar loops to grow Phaeodactylum tricornutum at Almeria,Spain. They studied the effects of culture productivity on theobserved solar irradiance, mixing, O2 accumulation, O2 removal,and flow velocity through solar loop, and recommended a reactorwith a capacity of 0.2 m3 which yielded 1.2 g l�1 d�1 of dry algae.They recommended the tubes be less than 0.1 m in diameter and80 m long. Fernandez and Grima [48] developed a macro modelto estimate the year-long Phaeodactylum tricornutum pro-ductivities in two outdoor tubular PBRs, located in Almeria,Spain. The inputs to the model were the day of the year, thereactor geometry (diameter), the kinetic parameters of the algaestrain, and the biomass concentration, which was measuredthroughout the experiments. The temperature was maintainedaround 20 �C using heat exchangers. Solar irradiance inside thewater, where tubes were immersed, was measured using quan-tum light meter. Daily solar irradiance on the earth's surface wasestimated using equations from literature [49]. They reportedthat the biomass productivities ranged from 1.08 to 2.76 g l�1 d�1

for tube diameters of 0.06 m and 0.03 m, respectively. Also,Fernandez and Grima [50] completed a comparative analysis ofthe concentration and productivity of Haematococcus pluvialis intubular and bubble column PBRs to determine the best reactorfor outdoor mass cultivation. According to their results, tubularPBRs were preferable compared to bubble columns. The meandaily irradiance on tubular PBR was 2.5 times higher than inbubble column. The biomass concentration and overall produc-tivity in tubular PBR were 7 g l�1 (dry weight) and 0.55 g l�1 d�1,as opposed to 0.41 g l�1 and 0.12 g l�1 d�1 in bubble column.Chisti and Grima [47] used engineering analysis and experi-mental observations to evaluate vertical bubble column andhorizontal tubular PBRs for large scale outdoor cultivation ofPhaeodactylum tricornutum. They compared performance of thetwo PBRs in terms of the observed gaseliquid hydrodynamics,mass transfer, internal irradiance, and overall productivity. Theirresults revealed that vertical bubble columns perform better thanhorizontal loops, and that bubble columns and airlift vesselsappear to be the only ones that can be used effectively in large-scale cultivation of microalgae. They concluded that the optimaldimensions of vertical bubble column PBRs are about 0.2 m indiameter and 4 m in column height with a spacing of about3.5 m. Merchuk [51] simulated the growth of Porphyridium sp. inbubble-column PBRs. The proposed model integrated the kineticsof photosynthesis and photoinhibition with the fluid dynamics ofbubble column. The model's predictions were in agreement withthe experimental results from laboratory. The results indicatedthat high biomass concentrations are achieved at low columndiameters, and that areal productivity increases with increasingdiameter up to a point and then decreases. A mathematicalmodel was developed by Bosma, Zessen, Reith, Tramper andWijffels [52] to predict the volumetric productivity of MonodusSubterraneus cultivated in an outdoor pilot-plant bubble columnlocated in Netherlands. Two models, one with no light integra-tion and one with full light integration, were evaluated atdifferent natural light conditions and different temperatures. The


results showed that the model with no light integration under-estimated the productivity, while the model with full lightintegration over-estimated it. Characterization of flat plate pho-tobioreactor such as orientation, hydrodynamics, heat and masstransfer, and mixing for the production of microalgae was pre-sented by Sierra [53]. It was concluded that productivity isdependent on location and orientation of the PBR, and comparedto tubular and column PBRs, flat plate PBRs require lower powersupply to achieve similar mass transfer, mixing, and heat trans-fer. Meiser [54] investigated the biomass productivity ofPhaeodactylum tricornutum in flat panel airlift loop reactor. Theinfluence of carbon dioxide aeration rates and irradiance onbiomass productivity was studied. Productivity increased byhigher aeration rates and higher light intensities. Slegers [55]developed a model to predict biomass production in single andparallel placed flat plate PBRs. The effect of sunlight on produc-tivity was studied, and the results indicate that productivityvaries between locations, reactor layout, algae species, andvarying light intensities over the day and the year. It wasconcluded that vertical and east-west oriented single panelsproduce the most amount of biomass.

As per available literature, physical parameters considered indesigning the outdoor cultivation units are temperature, solar irra-diance, area to volume ratio, oxygen build up, carbon dioxide uptakeefficiency, mass transfer, mixing, pH, and nutrient supply [7]. Theimpact of reactor geometry (e.g. reactor diameter), operation (e.g.average velocity, mixing), and temperature fluctuations on produc-tivity are not fully understood. Most of the literature focuses ondesigning pilot scale PBRs by considering the impact of one or a fewphysical parameters on algae growth at one time. In order to replicatethe actual behavior of outdoor PBR, it is necessary to understand therelationship between all thephysical parameters. Therefore, there is acritical need for a model that considers all physical parameters, andthe resulting analysis that investigates the impact of each parameterover the rest. Such an examination would not only generate anoptimal PBR design but also help in understanding which physicalparameter(s) have major influence(s) on the biomass productivity,growth rate, and hence the economics of the PBR. Furthermore, se-lection of an optimal PBR design to suit the particular characteristicsof micro algal strain and geographical location is important for costeffective production of biomass because a cultivation unit suitable forone type of algae species at a location,may not necessarily be optimalfor another species at the same or different location.

The present work was undertaken with an objective toanalyze the influence of algae species, geographical location, andreactor type on biomass productivity, growth rate, and reactorgeometry. It presents a unique approach that combines theexperimentally validated models (growth, temperature, andirradiance models) available in the open literature for optimallydesigning outdoor PBRs. Three main PRB designs, tubular, col-umn, and flat plate, were considered. The impact of physicalparameters such as solar irradiance, temperature, oxygen buildup, mass transfer, nutrient supply, and mixing on growth of algaebiomass were modeled for each reactor type. Then, the modelswere used to determine the optimum reactor geometries forgrowing two different algae species (Phaeodactylum tricornutumand Isochrysis galbana) on four different locations (Tulsa, USA;Hyderabad, India; Cape Town, South Africa; and Rio-de-Janeiro,Brazil). The optimum reactor is defined as the one that yieldsthe minimum total production cost (net present sink) assumingthe algae grown was used to produce a certain amount of bio-diesel. In Section 2, the design models of the PBRs are explained.The solution approach is detailed in Section 3. Section 4 illus-trates the case study. Results are summarized in Section 5, fol-lowed by conclusions and future work given in Section 6.



2. Biodiesel production and PBR design models

Fig. 1 presents the steps involved in the production of algae-based biodiesel. The first step is the cultivation of suitable algaespecies in a PBR. The cultivated algae at the selected location is thenharvested and sent for dryingwhere excess water is removed. Then,the lipids are extracted from the dried algae yielding algae oil. Thisalgae oil is sent to transesterification reactor for biodiesel produc-tion. Transesterification is a chemical reaction between tri-glycerides and alcohol to produce biodiesel, and glycerol isproduced as byproduct. In the current work, given a biodieselproduction rate, we performed a material balance separately foreach step of Fig. 1 to determine the necessary rate of algae growth.

Algae cultivation: In the cultivation unit, the algal biomassproduction via photosynthesis is approximated with

aCO2þbH2Oþ cPþgN/CaHeOfNgPczfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{dry algae

þhH2Ozfflffl}|fflffl{water in algae

þiO2þ jH2Ozfflffl}|fflffl{wastewater

a moles of carbon dioxide, b moles of water, c moles of phospho-rous, and g moles of nitrogen enter the cultivation unit to produce1 mol of DA (dry algae), h moles of WIA (water in algae), i moles ofoxygen, and jmoles of WW (waste water). In the current work, it isassumed that nitrogen and phosphorus are supplied in the appro-priate form in excess for algal growth in the growth medium.

As noted in Section 1, the algae growth is a function of availablelight and temperature among other things. If the algae cultivationunits are built outdoors as envisioned by many (e.g.[23,36,50,52,56e58]), both of these factors change based on thetime of the day, and the day of the year. Therefore, the algaecultivation units operate in a dynamic mode throughout the day,never reaching a steady-state behavior. This dynamic behavior canbe captured by differential equations; however repeated solution ofthese differential equations may be computationally expensive.Therefore, the following assumptions were made in order to cap-ture this dynamic behavior and reduce the computational burdenin this model: (1) Time is discretized using time buckets of equalsize, mainly hours. (2) All properties are assumed to be constantwithin a time bucket, e.g., the algae growth rate within a timebucket is fixed. However, it may change from one time bucket to thenext.

Algae Cultivation

Harvesting & Concentration

Drying

Oil Extraction

Transesterification

Fig. 1. Process flow chart.


Based on the above assumptions, the elemental balances ofcarbon (Eq. (1)), hydrogen (Eq. (2)), and oxygen (Eq. (3)) writtenaccording to the algal-biomass-growth reaction stoichiometry,yield three relationships between the inlet and outlet flow ratesof the algae cultivation unit for each day, d and the time of theday, t.

XCO2;in;P1;d;t

MWCO2

¼ aXDA;out;P1;d;t

MWalgaecd;ct (1)

2XH20;in;P1;d;t

MWH2O¼ e

XDA;out;P1;d;t

MWalgaeþ 2

XWIA;out;P1;d;t

MWH2O

þ 2XWW;out;P1;d;t

MWH2Ocd;ct (2)

2XCO2;in;P1;d;t

MWCO2

þ XH2O;in;P1;d;t

MWH2O¼ f

XDA;out;P1;d;t

MWalgaeþ XWIA;out;P1;d;t

MWH2O

þ 2XO2;out;P1;d;t

MWO2

þ XWW;out;P1;d;t

MWH2Ocd;ct

(3)

In this formulation, X is the mass flow rate (g h�1). The firstsubscript of X represents the chemical species: carbon dioxide(CO2), water (H2O), DA (dry algae),WIA (water in algae),WW (wastewater), and oxygen (O2), respectively. The second subscript of Xdenotes whether a stream is an input (in) or an output (out orremoved) for a process. The third subscript of X specifies whichprocess (P) the stream is attached to. Subscripts P1, P2, P3, P4, andP5 refer to the five stages of biodiesel production which are culti-vation, harvesting and concentration, drying, extraction, andtransesterification, respectively (Fig. 1). The fourth subscript of Xspecifies the day (d) of the operation, and the last subscript cor-responds to the hour (t) of the operation. In Eq. (1) through (3),MWdenotes the molecular weight, and its subscript corresponds to thechemical species. For example, MWCO2

is the molecular weight ofcarbon dioxide.

Given the percentage of dry algae in wet algae biomass forspecies s, %s, Eq. (4) calculates themass flow rate of water present inalgae. The total algae mass is the sum of dry algae and water pre-sent in algae.

XDA;out;P1;d;t

XWIA;out;P1;d;t þ XDA;out;P1;d;t� 100 ¼ %s cd;ct (4)

Mass flow rate of products (dry algae, water in algae and wastewater), ṁout,P1,d,t, in the PBR on day d at time t, is computed via Eq.(5).

_mout;P1;d;t ¼ XDA;out;P1;d;t þ XWIA;out;P1;d;t

þ XWW;out;P1;d;t cd;ct (5)

Harvesting & Concentration: During this step, algae biomass isharvested and is separated from excess waste water. In this work, itis assumed that the process(es) used for concentration step suc-cessfully removes all growth medium water, i.e., waste water,collected during the algal biomass harvesting step. Hence, onlywet algae, i.e., dry algae and water present in algae, continue to thenext step. Eq. (6) shows the overall mass balance of harvesting andconcentration step.XDA;in;P2;d;t þ XWIA;in;P2;d;t þ XWW;in;P2;d;t

¼ XDA;out;P2;d;t þ XWIA;out;P2;d;t þ XWW;removed;P2;d;t cd;ct

(6)



Drying: In this step, water present in algae is removed fromwet algae with the dryer efficiency, hdryer. Eq. (7) calculates theoutlet mass flow rate of remaining water, XWRD,out,P3,d,t, that leaves,out, the dryer on day d at time t. Eq. (8) gives the overall materialbalance for drying step.

XWRD;out;P3;d;t ¼ XWIA;in;P3;d;t � hdryer cd;ct (7)

XDA;in;P3;d;t þ XWIA;in;P3;d;t ¼ XDA;out;P3;d;t þ XWRD;out;P3;d;t

þXWIA;removed;P3;d;t cd;ct (8)

Extraction: In the extractor, lipid is extracted from dry algaebiomass leaving cake as residue. Eq. (9) calculates the flow rate oflipid using the efficiency of extractor, hextractor, and oil content ofalgae species, OCs. Eq. (10) gives the overall mass balance forextraction step where mass flow rates of dry algae and waterremaining in algae after the dryer step, is equal tomass flow rates oflipid, Xlipid,out,P4,d,t, and cake, Xcake,out,P4,d,t, leaving, out, the extractoron day d at time t.

Xlipid;out;P4;d;t ¼ XDA;in;P4;d;t � hextractor � OCs cd;ct (9)

XDA;in;P4;d;t þ XWRD;in;P4;d;t ¼Xlipid;out;P4;d;t

þ Xcake;out;P4;d;t cd;ct(10)

Transesterification reactor: The biodiesel transesterification re-action is defined as [59]:

Lipidþ 3ðMethanolÞ5catalyst Glycerolþ 3ðBiodieselÞAll lipid (assumed to be all triglyceride in this article) leaving the

extractor, enters the transesterification reaction where it reactswith methanol in the presence of catalyst to produce glycerol andbiodiesel.

The rate of biodiesel production on each day, d, and time, t,XBD,out,P5,d,t, can be calculated using the transesterification reactionstoichiometry, and the efficiency of transesterification reactor,htrans, and is given in Eq. (11).

XBD;out;P5;d;t

MWBD¼ Xlipid;in;P5;d;t

MWlipid� htrans � 3 cd;ct (11)

where,

MWBD: Molecular weight of biodiesel [60]MWlipid: Molecular weight of lipid [61]

The necessary mass flow rate of methanol, XMeOH,in,P5,d,t, and theproduction rate of glycerol, Xgly,out,P5,d,t, on day d at time t can becalculated similar to rate of biodiesel production, and these re-lationships are given in Eq. (12) for methanol and Eq. (13) forglycerol.

X MeOH;in;P5;d;t

MWmeth¼ Xlipid;in;P5;d;t

MWlipid� 3 cd;ct (12)

Xgly;out;P5;d;t

MWgly¼ Xlipid;in;P5;d;t

MWlipid� htrans cd;ct (13)

where,MWMeOH: Molecular weight of methanol.MWgly: Molecular weight of glycerol.


2.1. PBR design models

The cultivation units considered in this work are tubular, col-umn, and flat plate PBRs. The design models relate the rate of algaegrowth to the necessary volume for that specific reactor. There aretwo sets of design equations: one set is common for the three PBRsand the other set is specific to each reactor type.

2.1.1. Common design equations

2.1.1.1. Algal production. Eq. (14) calculates volume, V, of the culti-vation unit when productivity, PrVd,t, and production, XDA,out,P1,d,t, ofalgae biomass are known.

XDA;out;P1;d;t ¼ PrVd;t � V cd;ct (14)

Productivity, PrVd,t, of algae biomass in a PBR on day d at time t isdefined as the reactor's product generation rate (g m�3 h�1), andcan be calculated as a function of specific growth rate (h�1), md,t, andbiomass concentration (g m�3), BCd,t, on day d at time t, as shown inEq. (15).

PrVd;t ¼ md;t � BCd;t cd;ct (15)

In Eq. (16), mass flow rate of waste water, XWW,out,P1,d,t, and massflow rate of dry algae biomass, XDA,out,P1,d,t, produced in PBR are usedto calculate the concentration of dry algae biomass in water, BCd,t.

BCd;t ¼"XDA;out;P1;d;t

XWW;out;P1;d;t

#� r cd;ct (16)

In Eq. (16), r is the density of water.

2.1.1.2. Temperature and light dependence of algae growth rate.The maximum specific growth rates, mmax, of algae species dependon the temperature and the species itself [62]. One approach tomodel this relationship has a form inspired by the Arrheniusequation: mmax ¼ a:expðb:TÞ where a and b are species dependentconstants, and T is temperature. Response of growth rate to changesin temperaturewasmodeled in Ref. [63], for eight species of marinephytoplankton, where maximum growth rate tends to increasewith temperature and also varies with species [62]. Eq. (17) showsthe empirical relationship between the maximum specific growthrate, mmaxd;t

, and temperature inside the PBR, Treactord;t , on day d attime t.

mmaxd;t¼ a� eb�Treactord;t cd;ct (17)

The temperature inside the PBR (K), Treactord;t , can be estimatedby a simple energy balance around the reactor at each time bucket.Assuming that the energy stored in thewalls of the PBR is negligiblecompared to the energy stored by algae growth medium, and thetemperature in the PBR was initially equal to the surroundingtemperature, Eq. (18) describes the relationship between theaverage solar irradiance inside the culture (mE m�2 h�1), Iavgd;t , andthe reactor temperature on day d at time t.

Iavgd;t �Ephoton�As ¼ _mout;P1;d;t �Cp��Treactord;t �Tsurrd;t�cd;ct

(18)

where,Ephoton: Photon energyCp: Specific heat of water (J g�1�C�1)Tsurrd;t : Surrounding air temperatureAs: Surface area of the cultivation unit (it is the area of the

reactor that is exposed to sunlight)



Algal growth is also a function of the available light. Eq. (19) iscalled light-limited growth equation which was developed andtested by Grima, Camacho, P�erez, Sevilla, Fern�andez, and G�omez[64]. It models the effect of light attenuation on the observed algaegrowth rate in the cultivation unit. It defines a hyperbolic rela-tionship between the specific algae growth rate and average solarirradiance inside the culture when algae species and its light ab-sorption coefficient (mE m�2 s�1), Ik, are known. Based on the study,material balance performed in a chemostat where specific growthrate is expressed as a function of irradiance, it was observed thatthe algae growth rate predictions of this model exhibit a goodagreement with the measured ones.

md;t ¼ mmaxd;t�"

Inavgd;tInk þ Inavgd;t

#cd;ct (19)

where,n: Exponent that describes the abruptness of the transition from

weekly-illuminated to strongly-illuminated regions.In a PBR, light intensity is attenuated because of biomass con-

centration [23], light absorption, andmutual-shading [36]. The algaecells closer to light source receive higher irradiance than cells furtheraway. These effects are modeled in Eq. (20) which was developed byFern�andez, Camacho, P�erez, Sevilla, and Grima [23], who proposedan empirical relationship between light attenuation and biomassconcentration. They verified the model's accuracy using severaltypes of freshwater microalga, Phaeodactylum tricornutum. Thismodel was later validated for different algae species and PBR ge-ometries in Ref. [57]. Their results indicate that horizontal tubularPBR provide better light distribution than other conventional PBRs.Eq. (20) estimates the average solar irradiance, Iavgd;t , experienced bya single cell moving inside the culture based on the position of thesun relative to the PBR and Beer-Lambert relationship [65]. The lightabsorption coefficient, Ka, varies with algae species.

Iavgd;t ¼Iod;t

4eqd;t� Ka� BCd;t

1� e� 4eqd;t

�Ka�BCd;t

� �" #cd;ct

(20)

In Eq. (20), 4eqd;tis the length of light path from reactor surface

to any point inside the culture on day d at time t, and it is calculatedby Eqs. (21) and (22) depending on the orientation of PBR relative tothe position of the sun [23]. In these equations, the position of sunis defined by the solar zenith angle, q, the angle of declination ofsun from the vertical. For horizontal orientation of PBR, the higherthe sun's position from earth's surface, the smaller the length oflight path. For vertical orientation of PBR, the higher the sun's po-sition from earth's surface, the greater the length of light path. Thisphenomenon is displayed in Fig. 2.

4eqd;t¼"Designcos qd;t

#for Horizontal PBR cd;ct (21)

4eqd;t¼"Designsin qd;t

#for Vertical PBR cd;ct (22)

Here, Design is the relevant design parameter of the PBR such asdiameter in case of circular geometry and thickness in case of non-circular geometry.

Concentration of algae biomass should be kept below a pre-determined value, firstly, to inhibit light attenuation because whenlight passes through dense cultures, it would decay along the depth,and secondly, to maintain the Newtonian behavior of the growth


mediumwhile keeping the fluid properties similar to that of water[66]. This upper bound on biomass concentration (UBBC) is addedto the model in Eq. (23).

BCd;t � UBBC cd;ct (23)

2.1.1.3. Maximum dissolved oxygen concentration in PBR. It is shownthat high levels of dissolved oxygen inhibit photosynthesis [67] andbecome toxic to most microalgae [16]. Eq. (24) ensures that, thelevel of dissolved oxygen in the PBR stays below the acceptablelimit throughout the operation of the PBR. Given the maximumpermissible liquid velocity, Ulmax, rate of photosynthesis(g m�3 h�1), PrVO2d;t

, on day d at time t, and the acceptable limit ofthe concentration of dissolved oxygen in the reactor, Eq. (24) placesan upper bound on the relevant design parameter of the PBR(Design), which is the diameter in case of circular geometry andthickness in case of non-circular geometry.

Design � Ulmax ��½O2�out � ½O2�in

�PrVO2d;t

cd;ct (24)

Here, Ulmax is 0.5 ms-1 and is taken from literature [36]. [O2]in isthe dissolved oxygen concentration at the entrance of the reactortaken to be 8.2 mg l�1, which is the saturation value when water isin equilibriumwith atmosphere. [O2]out is the exit dissolved oxygenconcentration that does not inhibit photosynthesis. In literature, ithas been stated that this value should not generally exceed about400% of air saturation value [13], and in this study, it is taken as32.8 mg l�1. Rate of photosynthesis, PrVO2d;t

, is estimated from theamount of oxygen, XO2,out,P1,d,t, produced in cultivation unit andreactor volume, V as shown in Eq. (25).

PrVO2d;t¼ XO2;out;P1;d;t

Vcd;ct (25)

Reynolds number of the fluid flow inside the PBR is calculated byEq. (26).

Red;t ¼�H∅� Uld;t � r

m

�cd;ct (26)

where,HØ: Hydraulic diameter of PBR and is calculated depending on

the type of the cultivation unit used m: Viscosity of the fluid.Eq. (27) calculates the liquid velocity, Uld,t, in the PBR on day d at

time t when cross sectional area, Ac, of the cultivation unit isknown.

Uld;t ¼_mout;P1;d;t

r� Accd;ct (27)

2.1.1.4. Pumping costs. Pumps not only help in creating mixing ef-fect but also help in transporting the fluid from the inlet to theoutlet of the PBR. Good mixing circulates algal cells between lightand dark regions of PBRs, avoids cell attachment to reactor walls,reduces the degree of mutual shading, and lowers the probability ofphotoinhibition [68]. Pumping costs may play a significant role inPBR design especially as the diameter of the reactor becomessmaller. Power, PPd,t, required for pumping the fluid throughout thePBR is calculated with Eqs. (28) and (29), for horizontal-orientationPBRs and vertical-orientation PBRs, respectively. Eq. (28) takes intoaccount the pressure losses due to friction. Here, friction factor iscomputed using Haaland equation for circular and noncircularsmooth pipes [69], whereDesign is the relevant design parameter ofthe PBR (diameter in case of circular geometry and thickness in caseof non-circular geometry) in whose direction frictional loss occurs.


Fig. 2. Zenith angle and length of light path for horizontal and vertical oriented PBRs.


Eq. (29) takes into account both the pressure losses due to frictionand gravitational affects. The first component of the summation inEq. (29) calculates the pressure drop due to gravitational affectswhere g is acceleration due to gravity, and Design is the relevantdesign parameter of the PBR in whose direction gravitational lossoccurs. The second component calculates the pressure drop due tofriction similar to Eq. (28).

PPd;t ¼ _mout;P1;d;t�"�1:8 log

6:9Red;t

#��DesignH∅

��"Ul2d;t2

#cd;ct

(28)

PPd;t ¼_mout;P1;d;t � g � Design

þ(

_mout;P1;d;t �"� 1:8 log

6:9Red;t

#

��DesignH∅

��"Ul2d;t2

# )cd;ct

(29)

2.1.2. Reactor-type specific model equations

2.1.2.1. Tubular PBR. Volume of tubular PBR, V, is computedassuming that the tubular PBR is a perfect cylinder, viz. Eq. (30).

V ¼ p

4�∅2 � TL (30)

where,Ø: Diameter of tubes in a tubular PBR.TL: Length of tube.Eq. (31) calculates land area occupied by the tubular reactor, A,

when volume is known. Separation between the tubes of thereactor is expressed as a function of tube diameter,separation ¼ nT*∅, where nT is obtained from literature [70].

A ¼ separation� VAc

(31)

For tubular PBRs, high surface to volume ratio ensures highgrowth rates and productivities [22]. In order to obtain high surfaceto volume ratio, tube diameter, Ø, should be kept as low as possible.However, pumping the culture medium at such low diametersbecome expensive. Hence, a realistic lower bound on diameter isadded to the model via Eq. (32).


∅ � LBTD (32)

where,LBTD: Lower bound on tube diameter.

2.1.2.2. Column PBR. Eq. (33) calculates the volume occupied by thebubble column PBR, V.

V ¼ p

4� dC2 � CH (33)

where,dC: Diameter of column PBRCH: Height of column PBR.Gas hold up, which is defined as the percentage of gas in a

multiphase mixture, is considered to be one of the most importantparameters of column PBR design because it characterizes the hy-drodynamics within the bubble column. Gas hold up, εd,t, of columnPBR on day d at time t mainly depends on superficial gas velocity,Ugd,t, of the fluid flow on day d at time t, column diameter, gravity,and fluid properties such as density, surface tension, and viscosity.The effect of superficial gas velocity on gas hold up was experi-mentally examined in Ref. [71], and the best correlation for pre-dicting the gas hold up within liquidegas columns was found to beEq. (34) and Eq. (35). Their experimental data suggested that thetotal gas hold up increase with increasing superficial gas velocities.

εd;t ¼ 1:335� Ug0:449d;t cd;ct (34)

εd;t ¼ 0:20�g � dC2 � r

s

��0:13�g � dC3 � r2

m2

�0:11

Ugd;tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig � dC

p!0:54

cd;ct (35)

where,g: Acceleration due to gravity s: Surface tension of water inside

the PBRm: Viscosity of water inside the PBRAnother important parameter that greatly influences the hy-

drodynamics with the column PBR is bubble diameter. Bubble sizeis controlled by the balance between coalescence and breakupforces. The magnitudes of these forces depend on superficial gasvelocity and liquid physiological properties [72]. Hence, bubblediameter, dBd,t, on day d at time t is affected by column diameter,acceleration due to gravity, superficial gas velocity, and liquidproperties such as density, surface tension, kinematic viscosity.Dimensionless correlation for the estimation of bubble size as afunction of gas velocity and liquid physiological properties wasproposed in Ref. [72] based on experimental data using photo-graphic technique. Their correlation agreed within 20% of thepreviously published data of Towell [73].

dBd;t ¼ 26� dC�dC2 � g � r

s

��0:5�g � dC3

w2

��0:12

Ugd;tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig � dC

p!�0:12

cd;ct (36)

Lastly, to obtain uniform distribution of light and to ensureheterogeneous bubble flow inside the column, diameter, dC, shouldbe kept above a realistic lower bound [47] and [74] as shown in Eq.(37).



dC � LBCD (37)

where,LBCD: Lower bound on column diameter.

2.1.2.3. Flat plate PBR. Eq. (38) allows calculating the volume, V,occupied by the flat plate PBR.

V ¼ FPH�W � FPL (38)

where,W: Width of flat plate PBRFPL: Diameter of flat plate PBR FPH: Height of flat plate PBR.Flat plate bioreactor is similar to a rectangular bubble column

PBR. Hence, the design of flat plate PBR utilizes similar correlationsused for gas hold up and bubble diameter. Gas hold up is calculatedusing Eqs. (39) and (40) where Ubd,t is bubble rise velocity of thefluid flow on day d at time t [26]. In heterogeneous flow regime, thelarger the bubble size, the higher the bubble rise velocity and thelower the gas hold up.

εd;t ¼ 1:335� Ug0:449d;t cd;ct (39)

εd;t ¼Ugd;tUbd;t

cd;ct (40)

Bubble diameter, dBd,t, on day d at time t is estimated by meansof Sauter mean bubble diameter as shown in Eq. (41). Correlation ofbubble diameter to the liquid physiological properties (such asdensity, viscosity, surface tension), and superficial gas velocity wasdeveloped in Ref. [75] based on twenty numerical experimentsconducted on three virtual liquids. They concluded that the pre-dictions of their correlation were in good agreement with experi-mental bubble diameters observed in the laboratory settings.

dBd;t ¼ 0:289� r�0:552 � m�0:048 � s0:442 � Ug�0:124d;t cd;ct

(41)

Unlike column and tubular PBRs, better mixing inside a verticalflat plate PBR, is achieved at very highRayleigh numbers (the ratio ofbuoyancy and viscous forces [76]). To ensure that the flat plate PBRoperates at right Rayleigh number, Rad,t, ranges, Eq. (42) is used.

105 � Rad;t � 109 (42)

Eq. (43) calculates Rayleigh number in rectangular bubble col-umn PBR as a function of bubble properties such as bubble diam-eter and bubble rise velocity.

Rad;t ¼g � r� Ugd;t � dB3d;t

m�W � Ub2d;tcd;ct (43)

Width of flat plate PBR, W, should be maintained at a lowervalue to acquire good surface to volume ratio without considerablyincreasing the pumping costs as shown in Eq. (44). Length of flatplate PBR, FPL, is also constrained to avoid high levels of dissolvedoxygen concentrations as shown in Eq. (45).

W � LBFPW (44)

FPL � UBFPL (45)

where,LBFPW: Lower bound on width of flat plate.UBFPL: Upper bound on length of flat plate.


2.2. Air temperature and irradiance models

As explained in Section 2.1.1.2, the outdoor cultivation of algae iseffected by the available sunlight (Eq. (19)) and the temperature(Eq. (17)), which depend on the location of the PBR. The governingequations to estimate the surrounding temperature, Tsurrd;t , andincident light availability, Iod;t , on day d at time t are explained in thefollowing sections.

2.2.1. Surrounding temperature equationsTemperature simulation model presented in Ref. [77], and given

in Eqs. (46)e(48), is used to calculate the surrounding temperature,Tsurrd;t , on day d at time t on earth surface. This model is validated byBerninger [78] in his study by comparing the simulated data andthe instantaneous measured data on a sample day for a period of 45days. The results indicated that the model was a good fit to theinstantaneous temperature measurements. The model assumesthat minimum temperature, Tmind

, occurs at the time of sunrise andmaximum temperature, Tmaxd , at 14:00 h Eq. (46) calculates thesurrounding temperature before the time of sunrise, sunriseTimed.Eq. (47) calculates the surrounding temperature between sunrisetime and time whenmaximum temperature occurs, i.e., 14:00 h Eq.(48) calculates the surrounding temperature after the time whenmaximum temperature occurs.

Tsurrd;t ; ¼Tmaxd þTmind

2þTmaxd �Tmind

2

�cos�

180�ðtþ10Þ10þ sunriseTimed

�c t if t<sunriseTimed;cd

(46)

Tsurrd;t ; ¼Tmaxd þ Tmind

2� Tmaxd � Tmind

2

� cos�180� ðt � sunriseTimedÞ

14� sunriseTimed

�ct if sunriseTimed � t � 14 h; cd

(47)

Tsurrd;t ; ¼ Tmaxd þ Tmind

2þ Tmaxd � Tmind

2

� cos�

180� ðt � 14Þ10þ sunriseTimed

�ct if t >14 h;cd

(48)

2.2.2. Incident light equationsLight intensity is a function of the time of the day, the day of the

year, and the location on the earth surface. In order to find theavailability of sunlight at a location, extraterrestrial solar radiation,Itd;t , on day d at time t on the earth's surface is calculated based onthe position of sun, q, using Eq. (49) [79]. This model has beendeveloped and validated by Maxwell [80] to estimate the hourlyprofile of irradiance. It is widely accepted and considered to berelatively “universal”. The model was validated using data fromthree locations at widely varying latitudes with significantlydifferent climates and comparing these measurements with exist-ing models. Their results show that their model performs betterwithout compensating for seasonal, climatic, or geographicdifferences.

Itd;t ¼ Sc� cos qd;t cd;ct (49)

Here, Sc is solar constant and is equal to 1370 W m�2. Zenithangle, qd,t, can be calculated when latitude, longitude, and timezone of the location are known. It should be noted that, in thismodel, solar radiations at a given location are calculated between



sunrise time and sunset time, i.e., when cosine of zenith angle ispositive. Hence, algae biomass production is modeled only duringthe day.

The global solar radiation, Iod;t , on day d at time t can be esti-mated as a function of the daily maximum temperature, Tmaxd ,minimum temperature, Tmind

, and the extraterrestrial solar radia-tion viz. Eq. (50), based on the empirical correlation presented byHargreaves [81]. The empirical model was developed using avail-able meteorological observations.

Iod;t ¼ EmpA�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTmaxd � Tmind

q� Itd;t cd;ct (50)

where,EmpA: empirical coefficient which is 0.19 for coastal regions and

0.16 for interior regions [82].It should be noted that, all parameters in Eq. (46)e(50), i.e.,

Tmind, Tmaxd , sunriseTimed, Itd;t , qd,t, and Iod;t , are location dependent.

Sunrise time and cosine of solar zenith angle both depend on thelatitude, longitude, and time zone. They are calculated using themodel developed by NOAA called NOAA Solar Calculator [83]. Theseequations are provided as supplementary material for this article.The temperature and irradiance models employed in this articleassume no cloud cover. Therefore, the algae growth rates andproductivities obtained using the model presented in this article isa theoretical upper bound for the actual growth rates and pro-ductivities that can be expected with the suggested PBR designs.

3. Solution approach

Given an algae species and a location, the tubular PBR can bedesigned using common design equations: Eq. (1)e(29) and Eq.(46)e(50), and reactor specific design equations: Eq. (30)e(32),making a total of 91,032 equations and 94,361 variables. It yields3329 degrees of freedom. The column PBR can be designed usingcommon design equations: Eq. (1)e(29) and Eq. (46)e(50), andreactor specific design equations: Eq. (33)e(37), making a total of96,661 equations and 97,690 variables, which yields 1029 degreesof freedom. Similarly, flat plate PBR can be designed using commondesign equations: Eq. (1)e(29) and Eq. (46)e(50), and reactorspecific design equations: Eq. (38)e(45). There are 114,053 equa-tions and 126,169 variables which lead to 12,116 degrees offreedom. In order to obtain a unique solution to these models, thevalues of the degrees of freedom should be specified. One approachto specify these variables is optimization (the approach used in thispaper), in which the extra degrees of freedom can be used to ach-ieve a specific objective. In this study, we set the objective as theminimization of the net present sink of biomass production for aplant life of 10 years. This cost is calculated in Eq. (51). First part ofthe summation in Eq. (51) calculates the equipment costs of culti-vation unit (PBR) where EqCostl represents the cost of PBR ($ m�3)with unit capacity. This calculation assumes that the cost of the PBRchanges linearly with its volume. Second part calculates the oper-ating costs of the cultivation unit for continuous mixing andpumping large volumes of algae biomass and growth medium. Costof operating the pumps is calculated assuming a depreciation of10% using MARR (Minimum Acceptable Rate of Return).

ZZ ¼ EqCostl � V þX10

t¼0

11þMARRð Þt

!� Elcostl � EP

(51)

Here, ElCostl represents the cost of electricity ($ kWh�1) used forpumping algae biomass and EP represents the annual energy re-quirements of the pump and is calculated by Eq. (52).


EP ¼d t

PPd;t � 30 (52)
XX
Other utilities such as water and macronutrient (N and P) areassumed to be supplied in abundant, and the macronutrient cost isnegligible compared to pumping costs.

The time discretization, which is necessary to capture thedynamic nature of algae growth, results in 8640 time buckets(24 h � 360 days e assuming a 360 day operation period) in ayear. Hence, equation sets (1) through (50) result in a total of100,582 equations and 106,073 variables on an average for thethree reactor types. Given the exponential increase of number ofequations and variables with the number of time buckets, thesolution of the resulting nonlinear optimization problem (i.e.,nonlinear program) requires considerable computational re-sources. In order to reduce the computational burden and tomake the resulting optimization problem tractable, the followingsimplification approach is adapted in this paper: It is assumedthat the climate and available light (which are the main sources ofdynamics of algal growth in the model) within each month of ayear can be approximated by one specific day of each month. Thisreduces the number of days from 360 to 12. During that selectedday, all dynamic variables are calculated using the hourly time-bucket approach. The variables obtained for that one specificday of each month are replicated for all other days in that month.This reduces the number of equations and variables to approxi-mately 3350 and 3535, respectively. Eq. (53)e(55) give the totalmonthly and yearly mass flow rates obtained using the aboveapproximation.

In Eq. (53), daily mass flow rate of component, c, produced, out,in PBR, on day d at time t. Nc,out,P1,d is computed from Xc,out,P1,d,t.

Nc;out;P1;d ¼Xt

Xc;out;P1;d;t

cd2f1;2;3;4;5;6;7;8;9;10;11;12g; cc2fDA; WIAg(53)

The yearly mass flow rate, Oc,out,P1,y, is calculated using Eq. (54).

Oc;out;P1;y ¼X12d¼1

30� Nc;out;P1;d cy;cc2fDA;WIAg (54)

where,Nc,out,P1,d: Mass flow rate of component, c, produced, out, in PBR

on day d.Eq. (55) ensures that the total biodiesel produced by the

transesterification reaction in a year, y, satisfies the demand.

OBD;out;P5;y ¼ Demand cy (55)

Eq. (1)e(32) and Eq. (46)e(55) form a Nonlinear Programming(NLP) formulations representing the tubular PBRs. Similarly, Eq.(1)e(29), Eq. (33)e(37), and Eq. (46)e(55) represent modelformulation for column PBR and Eq. (1)e(29), Eq. (38)e(45), andEq. (46)e(55) represent model formulation for flat plate PBR. Thesemodels are implemented and solved in the General AlgebraicModeling System (GAMS 23.6.5) using CONOPT (version 3.14W) assolver. Here, it should be noted that the CONOPT is a local solver,and, hence, yields a local minimum solution of a non-convexoptimization problems like the models we present in this paper.One approach to overcome the shortcoming of using a local opti-mizer is to solve the problem starting from different initializationpoints. In this paper, the NLPs were solved 1000 times withdifferent initializations. The initial guesses were generated by Latin



hypercube sampling technique. The best of the 1000 solutions arepresented in Section 5.

4. Case study

In the case study, the influence of two marine-water species(Phaeodactylum tricornutum and Isochrysis galbana) and fourgeographical locations (Tulsa, USA; Hyderabad, India; CapeTown, South Africa; and Rio-de-Janeiro, Brazil) on the design ofPBRs was investigated. The necessary parameters for the modeldescribed in Section 2 are lipid content (OCs), light absorptioncoefficient (Ka), percentage of dry algae present in algae biomass(%s), species dependent constant (Ik), latitude, longitude, andtime zone. The relevant data for the two marine-water speciesconsidered in the case study are listed in Table A1. The speciesspecific constants a and b shown in Eq. (17) are estimated bynonlinear regression analysis on experimental data given in Ref.[84] where eight species of marine phytoplankton were grown atdifferent temperatures ranging from 10 to 25 �C. Fig. A1 showsthe graphical representation of our regression analysis forPhaeodactylum tricornutum and Isochrysis galbana species.Figure shows that maximum specific growth rate is exponen-tially dependent on reactor temperature. The four locations andtheir corresponding data are listed in Table A2. Maximum andminimum temperatures of the selected 12 days (obtained from aweather website called WeatherSpark [85]) at these locations areshown in Table A3. The model parameters and cost coefficientssuch as equipment costs and electricity costs that are used forthe case studies are summarized in Table A4 and Table A5,respectively. The biodiesel demand in Eq. (55) is 34,386 g y�1

based on [36].Given the above data, a total of twenty-four case problems

(nonlinear programs e NLPs) were generated for tubular (twospecies and four locations), column (two species and four loca-tions), and flat plate (two species and four locations) PBRs usingthe models presented in Section 2. These twenty-four NLPs weresolved separately to determine the best combination of species,location, and reactor type that produces the desired amount ofbiodiesel at the minimum net present sink. This combination isconsidered to be the best case. Sensitivity analysis was per-formed on the best case results to understand the influence ofmaximum permissible liquid velocity in the PBR (Ulmax), andmaximum acceptable dissolved oxygen concentration along thedirection of flow ([O2]out), on the production costs. All theseoptimization problems were solved for a production period of 10years.

5. Results and discussion

5.1. Comparison of our design and Fern�andez FG, Fern�andez JM,P�erez, Grima, Chisti [36] tubular PBR design

Here, we compare the tubular PBR design obtained by themodeling approach presented in our paper to the one presentedin Ref. [36], when the same demand is used. The discussion inthis subsection serves as a verification of our modeling approach

Table 2Comparison of tubular PBR results.

Design F (m) TL (m) V (m3) BC (g m

Base Case 0.06 80 0.20 23801 0.043 141.75 0.207 41302 0.054 29.03 0.066 3624


because our overall model uses the algal growth model devel-oped in Ref. [36], and extends it by considering the diurnalpatterns of temperature and irradiance for the whole year.

The growth model was used to design a 0.2 m3 tubular PBR togrow algae and to predict the volumetric productivity, liquidvelocity, and biomass concentration of this reactor [36]. Theproposed tubular PBR design was tested for growingPhaeodactylum tricornutum species at Almeria, Spain. The avail-able light on culture surface and maximum specific growth rateused in this design were 820 mE m�2 s�1 and 0.063 h�1, respec-tively. The authors recommended the tubes be less than 0.1 m indiameter and 80 m long. The Base Case row in Table 2 gives thetubular PBR design proposed in Ref. [36] for producing algae at arate of 238 g d�1 (dry weight basis). The information provided inthe table is the reactor configuration, daily averages of biomassconcentration, growth rate, irradiance inside the reactor, andvolumetric productivity.

Rows Design 1 and Design 2 in Table 2 were obtained bysolving the optimization problem presented in our paper withthe minimum net sink objective for the same species(Phaeodactylum tricornutum), location (Almeria, Spain), type of PBR(tubular), and algal production rate (238 g d�1 (dry weight basis)).Design 1 is the solution of the optimization problem if the effect oftemperature on the maximum specific algae growth rate andavailable light are ignored, i.e., Eqs. (17) and (18) and (46)e(50) areremoved from our model. The model solved in this case uses thesame light intensity and maximum specific growth rate of the BaseCase. Design 2 is obtained by solving the model for a whole yearrather than a single day for designing a tubular PBR to grow algae ata rate of 238 g d�1 (dry weight basis) in Almeria, Spain. It takes intoaccount the diurnal pattern of sunlight and temperature during theyear, and the effect of temperature on the maximum specific algaegrowth rate and available light (i.e. temperature and light modelsare considered).

We can see from Table 2 that the overall reactor volumes ofthe Base Case and Design 1 are comparable. The reactor config-uration suggested in Design 1 yields higher biomass concentra-tion, and productivity compared to the reactor configuration ofthe Base Case. Design 1 requires a smaller diameter to meet thesame biomass production rate, given an objective such as mini-mum net present sink. Lower diameter leads to higher biomassconcentration and productivity because of more light penetra-tion. However, growth rate obtained from Design 1 is lowercompared to that of Base Case because of dense cultures and celldeath due to increased light penetration. If the temperaturevariations and the available light changes throughout the yearare taken into consideration, i.e. Design 2, a lower reactor volume(compared to the Base Case) is obtained to meet the samebiomass production rate. The model used to obtain the reactorconfiguration of Design 2 calculates the hourly irradiance basedon zenith angle, and maximum and minimum temperatures in aday. This hourly irradiance is used to estimate the average irra-diance inside the reactor yielding more accurate average irradi-ance than the ones used for designing the reactors of the BaseCase and Design 1. Design 2 estimates the change in the tem-perature of the growth medium throughout the day due to

�3) m (h�1) Iavg (mE m�2 s�1) PrV (g m�3 h�1)

0.04 e 49.580.022 73.40 88.260.093 140.35 302.24



hourly irradiance changes and the surrounding temperaturethroughout the year. The effect of this temperature change on themaximum specific growth rate is considered via the Arrhenius-equation type empirical model, thus yielding more accurateoverall growth compared to Base Case and Design 1. It can beseen from Table 2 that the specific growth rate and averageirradiance are higher compared to the ones observed in the BaseCase and Design 1 reactors, hence requiring a smaller overallreactor volume for producing algae at the same rate.

5.2. Results of the case problems

Twenty-four case problems were generated using the originalmodel presented in Section 2 for two species and four locations foreach PBR type with the minimum net present sink objective. Theresulting optimization problems (NLPs) include the temperatureand light variations for the whole year. The NLPs for tubular, col-umn and flat plate cases have 3948, 3820, and 4667 equations, and4091, 4234, and 5101 variables, respectively. As explained in Section3, each model was initialized 1000 times using Latin-HypercubeSampling technique for each case separately. Out of the 1000 ini-tializations, some problems yielded infeasible solutions, somefeasible, and a few locally-optimal solutions. The best of theselocally-optimal solutions are reported in Table 3 for each kind ofPBR.

Table 3 shows the results of three PBRs used in the casestudy: tubular, column, and flat plate PBRs. The variables suchas cost (Z), tube diameter (F), TL (tube length), column diameter(dC), CH (column height), flat plate width (W), FPH (flat plateheight), FPL (flat plate length), volume of the reactor (V), areaoccupied by the reactor (A), bubble diameter (dB), averageirradiance inside the reactor (Iavg), superficial gas velocity (Ug),average liquid velocity (Ul), BC (biomass concentration), specificgrowth rate (m), volumetric productivity (PrV) are summarized.In Table 3, the Cases one (1) through four (4) in the first columnfor each type of PBR represent the results for growing

Table 3PBR case results.

Tubular cases Z ($) F (m) TL (m) V (m3) A (m2) Iavg (

1 283 0.056 22.27 0.055 3.55 1332 147 0.049 15.12 0.029 2.11 1613 262 0.053 23.51 0.051 3.51 1954 175 0.052 16.05 0.034 2.37 1325 474 0.071 23.78 0.093 4.75 2066 216 0.060 14.71 0.042 2.52 2577 452 0.065 26.97 0.088 4.94 3078 270 0.064 16.26 0.053 2.96 200

Column cases Z ($) dC (m) CH (m) V (m3) dB (m) Iavg (mE m�2 s

1 167 0.1 7.08 0.056 0.108 812 156 0.1 6.61 0.052 0.107 763 184 0.1 7.82 0.061 0.107 814 167 0.1 7.10 0.056 0.107 765 214 0.1 9.05 0.071 0.105 1076 179 0.1 7.56 0.059 0.107 1007 247 0.1 10.46 0.082 0.102 1068 200 0.1 8.47 0.067 0.104 101

Flat plate cases Z ($) W (m) FPH (m) FPL (m) V (m3) dB (m) Iavg (mE

1 239 0.05 0.097 10 0.049 0.001 1312 196 0.05 0.079 10 0.039 0.001 1163 283 0.05 0.12 10 0.058 0.001 1294 221 0.05 0.09 10 0.045 0.001 1155 398 0.05 0.162 10 0.081 0.001 1426 251 0.05 0.102 10 0.051 0.001 1307 558 0.05 0.227 10 0.114 0.001 1418 314 0.05 0.128 10 0.064 0.001 129


Phaeodactylum tricornutum species at Tulsa, USA; Hyderabad,India; Cape Town, South Africa; and Rio-de-Janeiro, Brazil;respectively. Similarly, the results for growing Isochrysis galbanaspecies at each location (same order) are given in Cases five (5)through eight (8) for each type of PBR.

The optimal PBR designs presented in Table 3 suggest thatgrowing Phaeodactylum tricornutum species at Hyderabad, India(Case 2 of each PBR type) meet the demand of 34,386 g y�1

biodiesel at the minimum production costs of $147, $156, and$196, respectively for a plant life of 10 years. The lipid content ofPhaeodactylum tricornutum species (31 DW%) is higher than thelipid content of the Isochrysis galbana species (21 DW%). Becausethe amount of lipid production directly impacts the amount ofbiodiesel production, the solutions favor the growth ofPhaeodactylum tricornutum species over the Isochrysis galbanaspecies for biodiesel production. Higher lipid content requireslower reactor volumes to produce the given demand, and hence,yields high biomass concentrations. This behavior can also beobserved in Table 3. For a given location and reactor type, Cases1 through 4 where Phaeodactylum tricornutum species iscultured requires a smaller PBR when compared to Cases 5through 8 where Isochrysis galbana species is cultured. Forexample, for tubular PBR at Tulsa, OK, when Cases 1 and 5 arecompared, the volume of the PBR in whichPhaeodactylum tricornutum is cultured (55 L), is lower than thevolume (93 L) of the PBR where Isochrysis galbana is cultured.

From Table 3, it can be observed that Hyderabad, India haslower average incident irradiance when compared to CapeTown, South Africa. However, it is considered to be the optimallocation for algae culturing because higher irradiance damagethe algae cells and lower the growth rates and concentration.Lower concentrations may lead to larger reactor volumes. For agiven algae species and reactor type, Cases 2 and 6, cultured atHyderabad, India require lower reactor volume when comparedto Cases 1, 3, and 4, and Cases 5, 7, and 8, respectively. Forexample, for tubular PBR and Phaeodactylum tricornutum

mE m�2 s�1) Ul (m h�1) BC (g m�3) m (h�1) PrV (g m�3 h�1)

2.03 4215 0.090 3582.11 5548 0.136 6882.63 4478 0.110 3871.74 5412 0.105 5793.79 2415 0.159 3164.13 3565 0.278 6945.40 2564 0.199 3313.17 3401 0.193 554

�1) Ug (m h�1) Ul (m h�1) BC (g m�3) m (h�1) PrV (g m�3 h�1)

25002 0.348 7452 0.047 35725002 0.396 6526 0.058 38225002 0.359 7103 0.044 32325002 0.385 6446 0.053 35625002 0.662 5694 0.072 41225002 0.759 4953 0.099 49325002 0.680 5478 0.064 35725002 0.728 4889 0.085 440

m�2 s�1) Ug (m h�1) Ul (m h�1) BC (g m�3) m (h�1) PrV (g m�3 h�1)

616 0.004 9101 0.043 408616 0.004 8313 0.057 497

2218 0.005 8900 0.037 3442538 0.004 8361 0.049 4412556 0.007 8406 0.042 3622218 0.023 7504 0.076 5732218 0.028 8066 0.031 2582538 0.008 7458 0.057 458



species, Case 2 requires a lower reactor volume of 29 L whencompared to Cases 1, 3, and 4 which require higher reactorvolumes of 55 L, 51 L, and 34 L, respectively. When it comes todeciding which reactor type to select for growing algae at alower production cost, the results in Table 3 reveal that hori-zontally oriented tubular PBR is the best choice because it hashigher surface to volume ratio when compared to other PBRreactor types. Higher surface to volume ratio in tubular PBRleads to lower tube diameter as shown in Table 3. Hence, ourresults suggest that species with higher oil content cultured inlocations with moderate light availability yields lower biodieselproduction costs. Among the 24 cases considered, a combinationof Phaeodactylum tricornutum species grown at Hyderabad, Indiain a tubular PBR with a reactor configuration of 0.05 m tubediameter and 15.12 m tube length yields the lowest biodieselproduction cost with algae biomass concentration, specificgrowth rate, and volumetric productivity of 5.55 g l�1, 0.14 h�1,and 0.69 g l�1 h�1, respectively.

Fig. 3. Variation of decision variables with time


5.2.1. Effect of orientation of PBR on the culture propertiesFig. 3 shows how the culture properties (average irradiance

(Iavg), volumetric productivity (PrV), BC (biomass concentration)and specific growth rate (m)) change with solar hour for a wholeyear depending on the orientation of PBR. Only the results of thebest case, Case 2, for tubular, column and flat plate PBRs arepresented in Fig. 3. The graphs in Fig. 3 reveal that the cultureproperties greatly depend on the orientation of PBR. In tubularPBR cases, the peak observed in irradiance, productivity, biomassconcentration, and growth rates around 12 p.m. or solar noon,occurs when the sun is at its apex and most nearly aligned withthe vertical axis of the horizontal tubular PBR. In this position,the direct sunlight flux on the tubular PBR is at its highestresulting in the highest growth rates, and hence, the highestbiomass concentrations and productivities. On the contrary, forthe vertical column [47] and the rectangular flat plate PBRs, thepeaks observed in the internal irradiance, productivity, concen-tration, and growth rates around sunrise and sunset, occurs when

for best case (Case 2) of each reactor type.


Table 4Sensitivity analysis performed on best case, tubular PBR e Case 2.

Tubular case Z ($) F (m) TL (m) V (m3) Ul (m h�1) BC (g m�3) m (h�1) PrV (g m�3 h�1)

SA1 189 0.06 13.47 0.037 1.61 5267 0.12 536SA2 111 0.04 17.37 0.022 2.99 5938 0.17 913SA3 94 0.04 18.76 0.018 3.67 6316 0.02 1078SA4 196 0.06 13.26 0.038 1.55 5140 0.11 518SA5 318 0.09 10.07 0.062 0.85 4175 0.08 318SA6 555 0.15 6.34 0.109 0.33 3318 0.05 183


sun is low on the horizon and most nearly aligned with thehorizontal axis of the vertical PBRs. While, the depressionobserved around solar noon occurs when sun is at its apexbecause, in this position, the direct sunlight flux on the verticalPBRs is low. Hence, culture properties are at their peak duringsolar noon for horizontal PBRs while for vertical PBRs, they are atpeak during sunrise and sunset times.

5.3. Sensitivity analysis

Sensitivity analysis is performed to determine the variables withmajor influences on the production cost and to understand theirimpact on the variation of other variables. The analysis is carriedout on two model parameters that may be considered important:maximum permissible liquid velocity (Ulmax), and maximumacceptable dissolved oxygen concentration ([O2]out) addressed inEq. (24). From the equation it can be noticed that both the chosenmodel parameters have direct influence on reactor geometry(Design) and hence, the production cost. Table 4 summarizes sixsensitivity analysis cases and their corresponding results obtainedby changing the chosen model parameters.

In order to investigate the impact of maximum permissibleliquid velocity, three different liquid velocities are tested on thebest case of tubular PBR, Case 2, using Latin hypercube samplinginitializations. Cases SA1, SA2, and SA3 shown in Table 4 are theresults obtained using three randomly chosen velocities namely,0.35 m s�1, 0.75 m s�1, and 0.95 m s�1, respectively. The resultsreveal that the higher the liquid velocity, the higher the growthrate and volumetric productivity of algae biomass becomes,yielding the highest growth rate and productivity of Case SA3 witha velocity of 0.95 m s�1. This increase in growth rates and pro-ductivities are due to better mixing of the culture at higher liquidvelocities, which facilitates better light distribution inside thePBR. This enhances biomass concentration which leads to asmaller reactor volume for satisfying the algal biomass demand ina cost effective way.

Sensitivity analysis was also performed onmaximum acceptabledissolved oxygen concentration to assess its influence on reactorgeometry. Cases SA4, SA5, and SA6 shown in Table 4 are the caseswith three randomly chosen dissolved oxygen concentrationsnamely, 24.6 g m�3, 16.4 g m�3, and 12.3 g m�3, respectively. It canbe observed that when dissolved oxygen concentration in PBR ishigh, (Case SA4), the allowable length of tubes is longer. Since thedemand (biomass production rate) is constant, the change in lengthdirectly affects the tube diameter. Hence, the tube diameter de-creases to meet the same demand. Lower diameter (lower surfaceto volume ratio) leads to higher biomass concentration and volu-metric productivity.

6. Conclusion and future work

There has been a growing interest in the production of bio-diesel from algae because of global warming and scarcity in fossil


fuels. However, substantial challenges are encountered inderiving fuels from algae biomass. Primary amongst them is thehigher costs of cultivating algae and the lack of efficient culti-vation units. Design of the cultivation units is strongly influencedby type of algal species selected for cultivation and preference oflocation.

In this paper, a novel mathematical model is developed toestimate the best combination of algae species, geographicallocation, and type of outdoor cultivation unit by combiningexperimentally validated temperature, irradiance, and algaegrowth models with optimization. This work differs from existingapproaches in that it considers all the physical parameters such astemperature, solar irradiance, oxygen build up, mass transfer, andmixing to design the optimal cultivation unit and to analyze theimpact of one physical parameter over the other. In order torepresent the actual behavior of the outdoor cultivation unit, thecurrent model takes into account the diurnal pattern of sunlight,temperature fluctuations, and the dynamic behavior of solarzenith angle. Impact of this pattern on culture properties such asbiomass concentration, productivity, and growth rate has beenaddressed in this study. The estimated biomass concentration,productivity, and growth rate during outdoor cultivation facili-tates in assessing the reactor geometry. The benefits of thismethod include but not limited to a realistic design of photo-bioreactors that produce the desired amount of biodiesel costefficiently. It has been found that horizontal-oriented PBRconfiguration provides better yield than vertical-oriented PBRconfiguration, given a desired demand, because of higher surfaceto volume ratio. In the current work, the influence of two marine-water species (Phaeodactylum tricornutum and Isochrysis galbana)and four geographical locations (Tulsa, USA; Hyderabad, India;Cape Town, South Africa; and Rio-de-Janeiro, Brazil) on the designof horizontal/vertical PBRs was investigated. A combination ofPhaeodactylum tricornutum species, Hyderabad location and hor-izontal tubular PBR configuration provided cost effective pro-duction of biodiesel.

It should be noted, however, that our current optimizationmodel ignores the effects of cloud cover on the growth rate, andhence, gives an absolute upper bound for the biomass production.The incorporation of cloud cover effects, requires the introductionof stochastic modeling of the weather data, and is left as a futurestudy. This work is a first step towards understanding the combinedimpact of all physical parameters on the reactor geometry and thusthe economics of algae cultivation.

Acknowledgments

The authors gratefully acknowledge the support from The Uni-versity of Tulsa.

Appendix A

Case study data tables


Table A1Species dependent parameters used in the case study [86]

Species Percentage of dry algae (%) Lipid content (DW) Light absorption coefficientof biomass (m2 g�1) [36]

Constant (mE m�2 s�1)

P. tricornutum 20 0.31 0.0369 114.67 [36]I. galbana 30 0.21 0.0369 170.68

Table A2Latitude, longitude and time zone for the selected locations [83]

Location Latitude Longitude Time zone

Tulsa, USA 37�N 96�W UTC-6Hyderabad, India 17�N 78�E UTCþ5Cape Town, South Africa 34�S 18�E UTCþ2Rio de Janeiro, Brazil 23�S 43�W UTC-3

Table A3Temperature parameters for the locations considered in the case study [85].

Date Tulsa Hyderabad Cape Town Rio de Janeiro

Tmax (�C) Tmin (�C) Tmax (�C) Tmin (�C) Tmax (�C) Tmin (�C) Tmax (�C) Tmin (�C)

January 15, 2012 17.78 0.00 27.78 7.22 33.89 20.00 35.00 25February 15, 2012 11.11 5.56 35.00 20.00 23.33 16.67 32.22 22.78March 15, 2012 26.67 18.89 36.67 20.56 27.22 20.00 35.00 23.89April 15, 2012 24.44 16.11 37.22 24.44 18.89 12.22 36.11 25May 15, 2012 28.89 12.22 40.56 28.89 21.11 6.11 21.11 17.78June 15, 2012 30.00 18.89 36.11 27.78 20.00 6.11 27.22 20.00July 15, 2012 34.44 18.89 28.89 22.78 11.11 8.89 21.11 18.89August 15, 2012 36.11 18.89 31.11 22.78 15.00 8.89 27.22 17.22September 15, 2012 22.78 16.67 30.00 22.22 27.22 7.78 27.78 18.89October 15, 2012 29.44 11.67 32.22 18.33 21.67 16.11 25.00 17.22November 15, 2012 16.67 1.11 28.33 15.56 22.78 16.67 23.89 17.78December 15, 2012 17.22 8.33 30.00 16.11 22.78 17.78 33.89 22.78

Fig. A1. Nonlinear regression analysis on experimental data for Phaeodactylum tricornutum and Isochrysis galbana [84].


Please cite this article in press as: Yadala S, Cremaschi S, Design and optimization of artificial cultivation units for algae production, Energy(2014), http://dx.doi.org/10.1016/j.energy.2014.06.001

Table A4Values of the model parameters used in the case study

Scalar Value Unit

LBTD 0.01 mUBBC 10000 G m�3

LBCD 1 mLBFPW 1 mUBFPL 10 mhdryer 0.9hextractor 0.948htrans 0.962 [87]MWalgae 259 g mol�1

MWlipid 634 [61] g mol�1

MWBD 300 [60] g mol�1

MWgly 92.1 g mol�1

EmpA 0.16, 0.19 [82]Ephoton 225.3 kJ mol�1

a 14.28 molesb 49066.67 molesc 0.032 molese 25.51 molesf 3.34 molesg 0.55 molesh 57.56 molesi 18.99 molesj 48996.36 moles


In Table A4, molecular weight of algae (MWalgae) is obtained byconverting the protein, carbohydrate, lipid, and nucleic acid con-tents of fifteen most commonly available algae species into ele-ments (carbon, hydrogen, oxygen, and nitrogen) using averageformula for each compound. The estimated composition of algaecontains 14.28 mol of elemental carbon (a), 25.51 mol of elementalhydrogen (e), 3.34 mol of elemental oxygen (f), 0.55 mol ofelemental nitrogen (g), and 0.03 mol of elemental phosphorous (c).Using the estimated molecular formula of microalgae, a mass bal-ance was performed on photosynthetic reaction where carbon di-oxide, water, elemental phosphorous, and elemental nitrogen reactto produce the dry microalgae, water in algae, oxygen, and largeamount of waste water.

aCO2þbH2Oþ cPþgN/CaHeOfNgPczfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{dry algae

þhH2Ozfflffl}|fflffl{water in algae

þiO2þ jH2Ozfflffl}|fflffl{wastewater

According to stoichiometry,14.28mol (a) of carbon dioxide reactwith 49066.67 mol (b) of water, 0.032 mol (c) of elemental phos-phorous, and 0.55mol (g) of elemental nitrogen to produce 1mol ofdry algae, 57.56 mol (h) of water in algae, 18.99 mol (i) of oxygen,and 48996.36 mol (j) of excess waste water.

Table A5Design parameters comparison of two primer sets for P. intermedia.

Tulsa Hyderabad Cape Town Rio de Janeiro

Tubular PBR [88]EqCostl 5110 5110 5110 5110ElCostl 0.07 0.07 0.07 0.07Column PBR [89]EqCostl 3000 3000 3000 3000ElCostl 0.07 0.07 0.07 0.07Flat Plate PBR [90]EqCostl 4919 4919 4919 4919ElCostl 0.07 0.07 0.07 0.07


Appendix B. Supplementary data

Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.energy.2014.06.001.

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