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Abstract This study investigates an innovative hybrid system that combines hydroponics and microalgae in a compact portable shipping container. This container is divided into two parts; one contains the microalgae system and the other contains the hydroponic system. This combined system works by dividing the 24 hours into 12 hours of light and 12 hours of dark for each part. Both parts are connected using light-impermeable pipes that pass carbon dioxide from the dark side to the lit side and oxygen from the lit side to the dark side. In this paper, the authors developed a validated mathematical model for hydroponic and microalgae to evaluate the system’s performance. Results found by the model show the optimum parameters for the split photobioreactor and hydroponic system. The first investigated parameter is the sparger diameter for split the photobioreactor and the second is the number of plants that give the better and optimum result. The optimum modelling design for the combined hydroponic and microalgae system was using 100 plants for lettuce and three photobioreactors with a 0.009-m diameter for sparger to a photobioreactor system with an area of 15.6 m2. Open in new tabDownload slide hydroponic system, microalgae system, split photobioreactor, nutrient film technique (nft), microalgae lipids, biofuels Introduction The increase in greenhouse gas emissions released into the atmosphere by industrial and transportation activity contributes to climate change [1]. According to the UN’s Food and Agriculture Organization (FAO), climate change causes drought, food shortage, energy shortage and poverty. Changes in temperature and rainfall will affect countries already struggling with water and energy scarcity. The effects of climate change force 800 million humans to go hungry daily [2]. Traditional agriculture practices can cause soil corruption and poor water utilization [3]. Recent research has targeted new and more eco-friendly agriculture methods as finding new agricultural methods becomes necessary. Two possible options to alleviate adverse issues caused by traditional agricultural practices are hydroponics to produce foods and microalgae to produce biofuels. Hydroponics is an eco-friendly system designed to grow plants in a solution of nutrients with or without soil. Hydroponic agriculture produces food faster than conventional agriculture and could provide a sustainable solution for agricultural needs in the future [4]. Microalgae, which are photosynthetic living micro-organisms found in freshwater, brackish water and oceans, can generate biofuel. There are two methods for cultivating and harvesting microalgae: open pond and closed photobioreactor [5]. The closed photobioreactor is often preferred to the open pond method since the open pond method is outdoor cultivation where there is a high probability of contamination [6]. Many published articles are dedicated to discussing the biofuel of hydroponics and microalgae of agricultural plants. Literature shows that due to the depletion of fossil fuel reserves, there is a need for renewable energy resources [7]. Researchers offer two options: biofuel production from plants and biofuel production from microalgae. It was found that growing microalgae is the most efficient way to produce biofuel [8]. Growing microalgae was studied in Arabian Gulf countries using wastewater, which presents an alternative method to achieving self-sufficiency for freshwater farming in countries lacking freshwater supply [9]. Growing 19 microalgae species in Morocco was investigated and it was found that the system could be an alternative source of energy for the country [10]. Researchers found that growing microalgae in India could satisfy its need for diesel and 30% of the energy needs worldwide [11]. Moreover, the chances of large-scale production for microalgae in Europe are high [12]. As for Australia, growing microalgae could be a fuel source that can sufficiently meet 11% of Australia’s diesel needs [13]. The co-cultivation of plants and microalgae in hydroponic units has drawn the attention of researchers in the sustainability field over the past few years. The goals for the combined research are to attempt to reduce fertilizer load and limit the consequences of eutrophication. When growing plants using a hydroponic system, most nutrients in the media are wasted and drained, polluting the environment. Aquaponic and Flocponics are common techniques to growth lettuce; it is theoretically viable if specific fertilization management is followed [14]. Several potential solutions were identified that save water and energy in studies producing Nordic microalgae using a hydroponic system [15]. Studies on the use of the hydroponic system in plants demonstrate the benefits by saving 12 times more water and a high yield of produce [16]. A study in Australia used wastewater as a nutrient and shows how this method conserves water and nutrients [16]. Likewise, similar results were found in studies of an organic hydroponic system in New Zealand [17]. Researchers successfully grew beets using a hydroponic system that utilizes genetically modified vegetable varieties [18]. The literature review showed that hydroponics and microalgae systems were thoroughly investigated; some studies examined the combined hydroponic and microalgae system. Experimentally growing tomatoes using algae yielded an increase in the efficiency for nitrogen and potassium, reaching 85% [19]. Hydroponics could be upgraded by using the production of microalgae, resulting in a more environmentally and economically sustainable production system [20]. Photoautotrophic Bio floc Technology (P-BFT) and the nutrient film technique (NFT) showed that they can cover the cost and it could make some profit, especially in arid or semi-arid areas [21]. This study presents a new innovative portable system that combines hydroponic and microalgae systems. The hydroponic system produces food and the microalgae system produces energy, biofuel or biogas. This system serves as a sustainable method for growing food and bioenergy in the same unit by combining the two systems. The unique feature of the proposed system, if compared with others, is that it integrates hydroponic and microalgae systems in a portable, compact and mobile way; this system is hybrid and autotrophic because it transfers the excess carbon from the microalgae to hydroponic and the excess oxygen from hydroponic to microalgae. 1 Methodology The proposed design is a prefabricated shipping container with photovoltaic (PV) panels installed on top, as illustrated in Fig. 1a, to sustainably produce the required energy to meet the system’s demands. The container is divided into two compartments to accommodate the hydroponics and microalgae systems as depicted in Fig. 1b. The microalgae produce biofuel and the hydroponics grow the plants. During daylight, the photosynthesis process consumes CO2 and produces O2. At night, the photosynthesis process consumes O2 and releases CO2. The lighting in the shelter compartments alternates between light and dark to achieve 12 hours of even split. The dark compartment provides CO2 to the other one while the lit side provides O2. The total area of the container is 18 m2, the microalgae side’s area is 1.9 m2 and the hydroponics side’s area is 15.6 m2. Fig. 1: Open in new tabDownload slide (a) A design of a prefabricated shipping container fitted with PV panels to house the hydroponics and microalgae system and (b) the interior design of the container showing the two sides of the system. The light and dark sides of the proposed system are separated and alternate every 12 hours. The microalgae system uses three photobioreactors while the hydroponic system is utilized to grow lettuce plants using the NFT. Both sides are connected using pipes that pass air but not light. Those pipes are designed to be installed at the top since carbon dioxide is lighter than oxygen. The CO2 pipe is Z-shaped to prevent light from reflecting in the pipe, thus it would not pass as shown in the design in Fig. 2. The pipe contains a fan to withdraw the air and an automated valve to control the CO2 flow. The O2 Z-shaped pipe is located at the bottom since O2 is denser than CO2. Fig. 2: Open in new tabDownload slide Piping configuration inside the container. 1.1 Mathematical equations for the hydroponic system For a hydroponic system, plants need water, light, oxygen and organic nutrients to grow. A hydroponic system can grow plants across all seasons, similar to glasshouses, as both systems have a controlled environment, light and temperature conditions for plants to grow [22]. Mathematical equations to calculate the different factors needed to operate the hydroponic system are presented in Table 1. Table 1: Mathematical equations to calculate the different factors needed to operate the hydroponic system Equation . Reference . Equation number . Ep=Eo−2,303(RuTF)pH [23] (1) Ep: the potential measurement in the unit (mV) Eo: reference potential (mV) Ru: Universal gas constant T: temperature (K) F: Faraday constant pH: pH value 1 mS = 10 cF= 700 ppm [24] (2) mS: milliSiemens cF: conductivity factor ppm: parts per million C= 98.19E−1.462 [22] (3) C: salt concentration (mg/L) E: electric conductivity (mS/cm) ETc =KcETo . [24] (4) ETc: actual evapotranspiration (mm/day) Kc: crop coefficient ETo: maximum evapotranspiration rate (mm/day) CWR=KsKc [24] (5) CWR: crop water requirement Ks: effect of water stress in transpiration Kc: crop coefficient IW=CWR−Pr [24] (6) IW: irrigation of water needed (mm) Pr: precipitation (mm) NIR=ETc−Er−Ge [24] (7) NIR: the net irrigation water requirement (mm) Er: the rainfall water with a unit (mm) Ge: the groundwater with a unit (mm) LRmeas= RmeasVU [25] (8) LRmeas: leaching requirement measured (L/m2) Rmeas: water run-off measured (L/m2) VU: total crops measured the unit (L/m2) LRpred=VfVUEC, max [25] (9) LRpred: single flushing event (L/m2) Vf: volume of water discharged (m3) VUEC,max: cumulative for crop water (L/m2) H= Ul×(Tin−Tout)−bτs−Eh . (10) H: heat energy per square metre of floor area (kW/m2) Ul: heat loss factor for every metre square (kW/m2K) Tin: temperature in the greenhouse (°C) Tout: the temperature out of the greenhouse (°C) b: percentage of solar insolation τ: average transmissivity for the glazing in a greenhouse s: solar radiation received per square metre (kW/m2) Eh: heat energy input to the greenhouse per square metre of floor area by equipment (e.g. lights) (kW/m2) Ta=Tout+( bτs +Eh)/ (Ul+pVCp) . (11) Ta: air temperature for the greenhouse (°C) p: cost of fuel ($/kWh) V: volume of the greenhouse (m3) Cp: specific heat for air temperature (kJ/kg) U=N(At−At−1) [26] (12) U: total uptake for N plants added peat block during interval time for a week (mm mol/week) N: number of plants At: content of nutrients in plant plus peat block at the end of the week (mmol/plant) At-1: content of nutrients in plant plus block at the start week (mmol/plant) Ln=(Ct−1− Ct +Dn)V [26] (13) Ln: loss of nutrient in a solution for a wk (mmol/ week) Ct–1: nutrient concentration at the start week (mmol/L) Ct: nutrient concentration at the end of a week (mmol/L) Dn: number of nutrients added to the nutrient solution during the week (mmol/L) V: volume of a system (L) Vo=Vmax (IlIopt)exp(1−IlIopt)−r . [27] (14) Vo: specific oxygen release (mg-O2kg-wet–1h–1) Vmax: maximum gross rate of oxygen release rate (mg-O2kg-wet–1h–1) Il: light intensity (lux) Iopt: optimum light intensity (lux) r: respiration rate (mg-O2kg-wet–1h–1) Vmax= Vmax (Ts)exp(θv (T−Ts)) [27] (15) Ts: standard temperature(°C) θv: respective temperature coefficient (°C–1) Iopt= Iopt(Ts)exp((T−Ts)) 27] (16) r=r(Ts)exp(θr (T−Ts)) 27] (17) θr: respective temperature coefficient (°C–1) PSW= b0+b1PAD [28] (18) PSW: plant-specific weight [g/(mm2)] b0: constant value equal 0.003 (mm–1) b1: constant values equal 0.02 (g/day) PAD: plant age (days) SDW= C0+A.C1+AC2 HD [28] (19) SDW: mass of plants (g) C0: constant value (0.54 g) A: area (mm2) C1: –0.001(g/mm2) C2: 9.882 × 10–5 (g/daym2) HD: plant age after planting (days) Equation . Reference . Equation number . Ep=Eo−2,303(RuTF)pH [23] (1) Ep: the potential measurement in the unit (mV) Eo: reference potential (mV) Ru: Universal gas constant T: temperature (K) F: Faraday constant pH: pH value 1 mS = 10 cF= 700 ppm [24] (2) mS: milliSiemens cF: conductivity factor ppm: parts per million C= 98.19E−1.462 [22] (3) C: salt concentration (mg/L) E: electric conductivity (mS/cm) ETc =KcETo . [24] (4) ETc: actual evapotranspiration (mm/day) Kc: crop coefficient ETo: maximum evapotranspiration rate (mm/day) CWR=KsKc [24] (5) CWR: crop water requirement Ks: effect of water stress in transpiration Kc: crop coefficient IW=CWR−Pr [24] (6) IW: irrigation of water needed (mm) Pr: precipitation (mm) NIR=ETc−Er−Ge [24] (7) NIR: the net irrigation water requirement (mm) Er: the rainfall water with a unit (mm) Ge: the groundwater with a unit (mm) LRmeas= RmeasVU [25] (8) LRmeas: leaching requirement measured (L/m2) Rmeas: water run-off measured (L/m2) VU: total crops measured the unit (L/m2) LRpred=VfVUEC, max [25] (9) LRpred: single flushing event (L/m2) Vf: volume of water discharged (m3) VUEC,max: cumulative for crop water (L/m2) H= Ul×(Tin−Tout)−bτs−Eh . (10) H: heat energy per square metre of floor area (kW/m2) Ul: heat loss factor for every metre square (kW/m2K) Tin: temperature in the greenhouse (°C) Tout: the temperature out of the greenhouse (°C) b: percentage of solar insolation τ: average transmissivity for the glazing in a greenhouse s: solar radiation received per square metre (kW/m2) Eh: heat energy input to the greenhouse per square metre of floor area by equipment (e.g. lights) (kW/m2) Ta=Tout+( bτs +Eh)/ (Ul+pVCp) . (11) Ta: air temperature for the greenhouse (°C) p: cost of fuel ($/kWh) V: volume of the greenhouse (m3) Cp: specific heat for air temperature (kJ/kg) U=N(At−At−1) [26] (12) U: total uptake for N plants added peat block during interval time for a week (mm mol/week) N: number of plants At: content of nutrients in plant plus peat block at the end of the week (mmol/plant) At-1: content of nutrients in plant plus block at the start week (mmol/plant) Ln=(Ct−1− Ct +Dn)V [26] (13) Ln: loss of nutrient in a solution for a wk (mmol/ week) Ct–1: nutrient concentration at the start week (mmol/L) Ct: nutrient concentration at the end of a week (mmol/L) Dn: number of nutrients added to the nutrient solution during the week (mmol/L) V: volume of a system (L) Vo=Vmax (IlIopt)exp(1−IlIopt)−r . [27] (14) Vo: specific oxygen release (mg-O2kg-wet–1h–1) Vmax: maximum gross rate of oxygen release rate (mg-O2kg-wet–1h–1) Il: light intensity (lux) Iopt: optimum light intensity (lux) r: respiration rate (mg-O2kg-wet–1h–1) Vmax= Vmax (Ts)exp(θv (T−Ts)) [27] (15) Ts: standard temperature(°C) θv: respective temperature coefficient (°C–1) Iopt= Iopt(Ts)exp((T−Ts)) 27] (16) r=r(Ts)exp(θr (T−Ts)) 27] (17) θr: respective temperature coefficient (°C–1) PSW= b0+b1PAD [28] (18) PSW: plant-specific weight [g/(mm2)] b0: constant value equal 0.003 (mm–1) b1: constant values equal 0.02 (g/day) PAD: plant age (days) SDW= C0+A.C1+AC2 HD [28] (19) SDW: mass of plants (g) C0: constant value (0.54 g) A: area (mm2) C1: –0.001(g/mm2) C2: 9.882 × 10–5 (g/daym2) HD: plant age after planting (days) Open in new tab Table 1: Mathematical equations to calculate the different factors needed to operate the hydroponic system Equation . Reference . Equation number . Ep=Eo−2,303(RuTF)pH [23] (1) Ep: the potential measurement in the unit (mV) Eo: reference potential (mV) Ru: Universal gas constant T: temperature (K) F: Faraday constant pH: pH value 1 mS = 10 cF= 700 ppm [24] (2) mS: milliSiemens cF: conductivity factor ppm: parts per million C= 98.19E−1.462 [22] (3) C: salt concentration (mg/L) E: electric conductivity (mS/cm) ETc =KcETo . [24] (4) ETc: actual evapotranspiration (mm/day) Kc: crop coefficient ETo: maximum evapotranspiration rate (mm/day) CWR=KsKc [24] (5) CWR: crop water requirement Ks: effect of water stress in transpiration Kc: crop coefficient IW=CWR−Pr [24] (6) IW: irrigation of water needed (mm) Pr: precipitation (mm) NIR=ETc−Er−Ge [24] (7) NIR: the net irrigation water requirement (mm) Er: the rainfall water with a unit (mm) Ge: the groundwater with a unit (mm) LRmeas= RmeasVU [25] (8) LRmeas: leaching requirement measured (L/m2) Rmeas: water run-off measured (L/m2) VU: total crops measured the unit (L/m2) LRpred=VfVUEC, max [25] (9) LRpred: single flushing event (L/m2) Vf: volume of water discharged (m3) VUEC,max: cumulative for crop water (L/m2) H= Ul×(Tin−Tout)−bτs−Eh . (10) H: heat energy per square metre of floor area (kW/m2) Ul: heat loss factor for every metre square (kW/m2K) Tin: temperature in the greenhouse (°C) Tout: the temperature out of the greenhouse (°C) b: percentage of solar insolation τ: average transmissivity for the glazing in a greenhouse s: solar radiation received per square metre (kW/m2) Eh: heat energy input to the greenhouse per square metre of floor area by equipment (e.g. lights) (kW/m2) Ta=Tout+( bτs +Eh)/ (Ul+pVCp) . (11) Ta: air temperature for the greenhouse (°C) p: cost of fuel ($/kWh) V: volume of the greenhouse (m3) Cp: specific heat for air temperature (kJ/kg) U=N(At−At−1) [26] (12) U: total uptake for N plants added peat block during interval time for a week (mm mol/week) N: number of plants At: content of nutrients in plant plus peat block at the end of the week (mmol/plant) At-1: content of nutrients in plant plus block at the start week (mmol/plant) Ln=(Ct−1− Ct +Dn)V [26] (13) Ln: loss of nutrient in a solution for a wk (mmol/ week) Ct–1: nutrient concentration at the start week (mmol/L) Ct: nutrient concentration at the end of a week (mmol/L) Dn: number of nutrients added to the nutrient solution during the week (mmol/L) V: volume of a system (L) Vo=Vmax (IlIopt)exp(1−IlIopt)−r . [27] (14) Vo: specific oxygen release (mg-O2kg-wet–1h–1) Vmax: maximum gross rate of oxygen release rate (mg-O2kg-wet–1h–1) Il: light intensity (lux) Iopt: optimum light intensity (lux) r: respiration rate (mg-O2kg-wet–1h–1) Vmax= Vmax (Ts)exp(θv (T−Ts)) [27] (15) Ts: standard temperature(°C) θv: respective temperature coefficient (°C–1) Iopt= Iopt(Ts)exp((T−Ts)) 27] (16) r=r(Ts)exp(θr (T−Ts)) 27] (17) θr: respective temperature coefficient (°C–1) PSW= b0+b1PAD [28] (18) PSW: plant-specific weight [g/(mm2)] b0: constant value equal 0.003 (mm–1) b1: constant values equal 0.02 (g/day) PAD: plant age (days) SDW= C0+A.C1+AC2 HD [28] (19) SDW: mass of plants (g) C0: constant value (0.54 g) A: area (mm2) C1: –0.001(g/mm2) C2: 9.882 × 10–5 (g/daym2) HD: plant age after planting (days) Equation . Reference . Equation number . Ep=Eo−2,303(RuTF)pH [23] (1) Ep: the potential measurement in the unit (mV) Eo: reference potential (mV) Ru: Universal gas constant T: temperature (K) F: Faraday constant pH: pH value 1 mS = 10 cF= 700 ppm [24] (2) mS: milliSiemens cF: conductivity factor ppm: parts per million C= 98.19E−1.462 [22] (3) C: salt concentration (mg/L) E: electric conductivity (mS/cm) ETc =KcETo . [24] (4) ETc: actual evapotranspiration (mm/day) Kc: crop coefficient ETo: maximum evapotranspiration rate (mm/day) CWR=KsKc [24] (5) CWR: crop water requirement Ks: effect of water stress in transpiration Kc: crop coefficient IW=CWR−Pr [24] (6) IW: irrigation of water needed (mm) Pr: precipitation (mm) NIR=ETc−Er−Ge [24] (7) NIR: the net irrigation water requirement (mm) Er: the rainfall water with a unit (mm) Ge: the groundwater with a unit (mm) LRmeas= RmeasVU [25] (8) LRmeas: leaching requirement measured (L/m2) Rmeas: water run-off measured (L/m2) VU: total crops measured the unit (L/m2) LRpred=VfVUEC, max [25] (9) LRpred: single flushing event (L/m2) Vf: volume of water discharged (m3) VUEC,max: cumulative for crop water (L/m2) H= Ul×(Tin−Tout)−bτs−Eh . (10) H: heat energy per square metre of floor area (kW/m2) Ul: heat loss factor for every metre square (kW/m2K) Tin: temperature in the greenhouse (°C) Tout: the temperature out of the greenhouse (°C) b: percentage of solar insolation τ: average transmissivity for the glazing in a greenhouse s: solar radiation received per square metre (kW/m2) Eh: heat energy input to the greenhouse per square metre of floor area by equipment (e.g. lights) (kW/m2) Ta=Tout+( bτs +Eh)/ (Ul+pVCp) . (11) Ta: air temperature for the greenhouse (°C) p: cost of fuel ($/kWh) V: volume of the greenhouse (m3) Cp: specific heat for air temperature (kJ/kg) U=N(At−At−1) [26] (12) U: total uptake for N plants added peat block during interval time for a week (mm mol/week) N: number of plants At: content of nutrients in plant plus peat block at the end of the week (mmol/plant) At-1: content of nutrients in plant plus block at the start week (mmol/plant) Ln=(Ct−1− Ct +Dn)V [26] (13) Ln: loss of nutrient in a solution for a wk (mmol/ week) Ct–1: nutrient concentration at the start week (mmol/L) Ct: nutrient concentration at the end of a week (mmol/L) Dn: number of nutrients added to the nutrient solution during the week (mmol/L) V: volume of a system (L) Vo=Vmax (IlIopt)exp(1−IlIopt)−r . [27] (14) Vo: specific oxygen release (mg-O2kg-wet–1h–1) Vmax: maximum gross rate of oxygen release rate (mg-O2kg-wet–1h–1) Il: light intensity (lux) Iopt: optimum light intensity (lux) r: respiration rate (mg-O2kg-wet–1h–1) Vmax= Vmax (Ts)exp(θv (T−Ts)) [27] (15) Ts: standard temperature(°C) θv: respective temperature coefficient (°C–1) Iopt= Iopt(Ts)exp((T−Ts)) 27] (16) r=r(Ts)exp(θr (T−Ts)) 27] (17) θr: respective temperature coefficient (°C–1) PSW= b0+b1PAD [28] (18) PSW: plant-specific weight [g/(mm2)] b0: constant value equal 0.003 (mm–1) b1: constant values equal 0.02 (g/day) PAD: plant age (days) SDW= C0+A.C1+AC2 HD [28] (19) SDW: mass of plants (g) C0: constant value (0.54 g) A: area (mm2) C1: –0.001(g/mm2) C2: 9.882 × 10–5 (g/daym2) HD: plant age after planting (days) Open in new tab The electric potential of the hydroponic system is calculated using Equation (1). Electrical conductivity measures how many free electrolytes (dissolved salts) are in the fluid and is calculated in different units as shown in Equation (2). Electric conductivity, in Equation (3), depends on the salt concentration. The amount of water required for irrigation in the hydroponic system and the total amount of water for irrigation can be determined using Equations (6) and (7), while the amount of leaching water and the single flushing of water can be determined by Equations (8) and (9). The temperature, a critical component in a hydroponic system, parameters were calculated using Equations (10) and (11). The nutrient increase in plants and decrease in the solution were considered and Equation (12) was used to determine the balance between the two while the loss of nutrients in the solution was calculated using Equation (13). The amount of oxygen released from the hydroponic system is vital in this study as excess oxygen will be used in the microalgae system. The Equations (14–17) govern the oxygen released by the effect of temperature and light intensity. The plant-specific weight for the hydroponic system depends on the age of the plant as used in Equation (18). The shoot dry weight (SDW), in Equation (19), is used to calculate the weight of productive biomass. 1.2 Mathematical equations for the split photobioreactor The mathematical equations to calculate the different factors needed to operate the hydroponics–microalgae system are listed in Table 2. Table 2: Mathematical equations to calculate the different factors needed to operate the microalgae system Equation . Reference . Equation number . V= Vlr+Vld2=ULr2((11−εr)+(11−εd) ) [29] (20) V: velocity in split photobioreactor (m/s) Vlr: velocity in the photobioreactor’s riser (m/s) Vld: the velocity in the downcomer of the photobioreactor (m/s) ULr: average superficial liquid velocity for the riser and downcomer (m/s) ε r: efficiency in the riser region ε d: efficiency in the downcomer region εr=0.4(Fr )0.51+0.4(Fr)0.5(1+(ULrUGr)) [29] (21) Fr: Froude number UGr: gas velocity for the photobioreactor (m/s) Fr= UGr2gdo [29] (22) g: gravitational acceleration (m/s2) do: diameter for sparger diameter (m) UGrεr= 1.607( UGr+ULr)+0.298 [29] (23) εd=0.841 εr [29] (24) ULr= Vlr(1− εr)= Vld(1−εd) [29] (25) UGr=FgA [30] (26) Fg: volumetric flow rate (m3/s) A: area of the photobioreactor (m2) A=2πrh+2π r2 [31] (27) v= Fgn14Πd2 [30] (28) v: entrance velocity, sparger, for photobioreactor (m/s) n: number of nozzles (assumed to be one in this study) d: nozzle diameter (mm) Re= 4 ρFg(ndη) [30] (29) Re: Reynold number ρ: aqueous density (kg/m3) (assumed to be 1000 kg/m3) η: aqueous dynamic viscosity (kg/ms) (assumed to be 0.001 kg/ms) ε=Arεr +AdεdAr+Ad [32] (30) Ar: cross-section area for downcomer region (m2) Ad: cross-section area for riser region (m2) kLa(co2)=0.837UGr−0.971−1 [29] (31) kLa(CO2): mass transfer coefficient of carbon dioxide (h–1) kLa(co2)=0.83 kLa (o2) [33] (32) kLa(O2): mass transfer coefficient of oxygen (h–1) Rco2= Cc×μl×(Mco2Mc) [34] (33) RCO2: CO2 fixation rate (mg/Ld) Cc: average carbon content (0.63 g carbon/g dry cell weight) μl: volumetric growth rate in a unit (mg/Ld) MCO2: molecular weight of carbon dioxide (mg/mole) Mc: molecular weight for carbon (mg/mole) μ=(lnx2−lnx1)t2−t1 [35] (34) μ: growth rate (g/Ld) x2: biomass concentration at day t2 x1: biomass concentration at day t1 θm=TmTc [36] (35) θm: mixing time Tm: mixing time (s) Tc: cycling time (s) Bo=VlLEZ [36] (36) Bo: Bodenstein number Vl: velocity of the liquid (cm/s) L: the separation between the tracer infusion and discovery focuses (cm) EZ: pivotal scattering coefficient (m2/s) Vl=(LcTc) (37) Lc: length of the photobioreactor (cm) Bo=k(TmTc) [36] (38) k: constant value of split photobioreactor is (9.6 ± 0.2) Bolg= UGdEz [36] (39) Bolg: Bodenstein number depends on the superficial gas velocity (cm/s) UG: gas velocity (cm/s) d: diameter of the sparger photobioreactor (m) Bolg=3.77(Fr13)1.2 [36] (40) dXdt=kcX(1−(XXmax)) [34] (41) dX/dt: biomass concentration rate (g/Lh) Kc: specific growth rate(h–1) X: biomass concentration of microalgae (g/L) Xmax: maximum concentration of microalgae (g/L) X=Xmax1+(XmaxXo−1) e(−Kct) [34] (42) Xo: initial concentration of microalgae/L dpdt= α(dXdt)+β X [34] (43) dp/dt: lipid concentration rate (mg/Lh) α: growth coefficient β: non-growth coefficient α= K24+KXffao [37] (44) K: kinetic constant Xffao: initial free fatty acid (kcal/g) β=αK [37] (45) K= k1k2 [37] (46) k1: kinetic rate constant for the forward reaction (s–1) k2: active rate consistent for the reverse response (s–1) p=po−αXo+(αXoXmineμmaxtXmin−Xo+Xoeμmaxt)+β(Xminμmax)lnXmin−Xo+XoeμmaxtXmin [34] (47) p: lipid concentration for the microalgae in (g/L) po: initial lipids concentration (g/L) Xmin: maximum concentration of microalgae (g/L) μmax: the maximum growth rate (g/Ld) S˙=Dsin−ρ(s)x−Ds [38] (48) Ṡ: nitrogen cell concentration (g/m3) Dsin: influent concentration of nitrogen (gmole/L) ρ(s): retention rate (g/m3) x: nitrogen biomass (g) Ds: dilution or weakening rate (d–1) q˙=ρ(s)−μ(q)q [38] (49) q̇: constrained nutrients rate μ(q): development rate q: constrained quotient x˙=μ (q)x−Dx [38] (50) ẋ: nitrogen biomass rate (g/m3) dIdz=−ka Ir X [34] (51) dI/dz: light intensity rate (μmol/m2s) ka: specific light retention coefficient Ir: irradiance light needed to grow microalgae (μmol/m2s) X: biomass concentration (g/L) Z: distance between light and the photobioreactor (m) 1Lln(IoIr)=kaX [34] (52) L: length for the light path for the photobioreactor (m) Io: irradiance light on the surface (μmol/m2s) X= aka [34] (53) a: absorbance factor a=1Lln(IoIr) [34] (54) −dsdt= 1Ydxdt+mX [34] (55) ds/dt: soduinm nitrate consumption rate to (g/Ls) Y: extreme microalgae coefficient dx/dt: biomass concentration rate (g/Ls) m: maximum maintenance coefficient S=So−1yx(XoXmaxeμmaxt Xmax−Xo+ Xoe μmaxt−xo)m Xmax μln(Xmax−Xo+XoeμmaxtXmax) [34] (56) S: nitrate sodium concentration (g/L) So: initial consumption for nitrate sodium at zero time (g/L) yx: maximum lipid coefficient (g/g) t: time (days) h(pH)=h [H+]=k0k1+1K1[H+]+k2k1K1K2[H+]21+1K1[H+]+1K1K2[H+]2 [39] (57) h(pH): pH concentration h[H+]: proton concentration (gmole/L) k0: parameters of the pH-dependent kinetic (h–1) k1: parameters of the pH-dependent kinetic (h–1) k2: parameters of pH-dependent kinetic (h–1) K1: parameters of the pH-dependent kinetic (molH+/L) K2: parameters of the pH-dependent kinetic (molH+/L) Harvesting Efficency (%)= A1−B1A1×100 [40] (58) A1: the first estimation value of biomass (g/L) B1: the final estimation value of biomass (g/L) Equation . Reference . Equation number . V= Vlr+Vld2=ULr2((11−εr)+(11−εd) ) [29] (20) V: velocity in split photobioreactor (m/s) Vlr: velocity in the photobioreactor’s riser (m/s) Vld: the velocity in the downcomer of the photobioreactor (m/s) ULr: average superficial liquid velocity for the riser and downcomer (m/s) ε r: efficiency in the riser region ε d: efficiency in the downcomer region εr=0.4(Fr )0.51+0.4(Fr)0.5(1+(ULrUGr)) [29] (21) Fr: Froude number UGr: gas velocity for the photobioreactor (m/s) Fr= UGr2gdo [29] (22) g: gravitational acceleration (m/s2) do: diameter for sparger diameter (m) UGrεr= 1.607( UGr+ULr)+0.298 [29] (23) εd=0.841 εr [29] (24) ULr= Vlr(1− εr)= Vld(1−εd) [29] (25) UGr=FgA [30] (26) Fg: volumetric flow rate (m3/s) A: area of the photobioreactor (m2) A=2πrh+2π r2 [31] (27) v= Fgn14Πd2 [30] (28) v: entrance velocity, sparger, for photobioreactor (m/s) n: number of nozzles (assumed to be one in this study) d: nozzle diameter (mm) Re= 4 ρFg(ndη) [30] (29) Re: Reynold number ρ: aqueous density (kg/m3) (assumed to be 1000 kg/m3) η: aqueous dynamic viscosity (kg/ms) (assumed to be 0.001 kg/ms) ε=Arεr +AdεdAr+Ad [32] (30) Ar: cross-section area for downcomer region (m2) Ad: cross-section area for riser region (m2) kLa(co2)=0.837UGr−0.971−1 [29] (31) kLa(CO2): mass transfer coefficient of carbon dioxide (h–1) kLa(co2)=0.83 kLa (o2) [33] (32) kLa(O2): mass transfer coefficient of oxygen (h–1) Rco2= Cc×μl×(Mco2Mc) [34] (33) RCO2: CO2 fixation rate (mg/Ld) Cc: average carbon content (0.63 g carbon/g dry cell weight) μl: volumetric growth rate in a unit (mg/Ld) MCO2: molecular weight of carbon dioxide (mg/mole) Mc: molecular weight for carbon (mg/mole) μ=(lnx2−lnx1)t2−t1 [35] (34) μ: growth rate (g/Ld) x2: biomass concentration at day t2 x1: biomass concentration at day t1 θm=TmTc [36] (35) θm: mixing time Tm: mixing time (s) Tc: cycling time (s) Bo=VlLEZ [36] (36) Bo: Bodenstein number Vl: velocity of the liquid (cm/s) L: the separation between the tracer infusion and discovery focuses (cm) EZ: pivotal scattering coefficient (m2/s) Vl=(LcTc) (37) Lc: length of the photobioreactor (cm) Bo=k(TmTc) [36] (38) k: constant value of split photobioreactor is (9.6 ± 0.2) Bolg= UGdEz [36] (39) Bolg: Bodenstein number depends on the superficial gas velocity (cm/s) UG: gas velocity (cm/s) d: diameter of the sparger photobioreactor (m) Bolg=3.77(Fr13)1.2 [36] (40) dXdt=kcX(1−(XXmax)) [34] (41) dX/dt: biomass concentration rate (g/Lh) Kc: specific growth rate(h–1) X: biomass concentration of microalgae (g/L) Xmax: maximum concentration of microalgae (g/L) X=Xmax1+(XmaxXo−1) e(−Kct) [34] (42) Xo: initial concentration of microalgae/L dpdt= α(dXdt)+β X [34] (43) dp/dt: lipid concentration rate (mg/Lh) α: growth coefficient β: non-growth coefficient α= K24+KXffao [37] (44) K: kinetic constant Xffao: initial free fatty acid (kcal/g) β=αK [37] (45) K= k1k2 [37] (46) k1: kinetic rate constant for the forward reaction (s–1) k2: active rate consistent for the reverse response (s–1) p=po−αXo+(αXoXmineμmaxtXmin−Xo+Xoeμmaxt)+β(Xminμmax)lnXmin−Xo+XoeμmaxtXmin [34] (47) p: lipid concentration for the microalgae in (g/L) po: initial lipids concentration (g/L) Xmin: maximum concentration of microalgae (g/L) μmax: the maximum growth rate (g/Ld) S˙=Dsin−ρ(s)x−Ds [38] (48) Ṡ: nitrogen cell concentration (g/m3) Dsin: influent concentration of nitrogen (gmole/L) ρ(s): retention rate (g/m3) x: nitrogen biomass (g) Ds: dilution or weakening rate (d–1) q˙=ρ(s)−μ(q)q [38] (49) q̇: constrained nutrients rate μ(q): development rate q: constrained quotient x˙=μ (q)x−Dx [38] (50) ẋ: nitrogen biomass rate (g/m3) dIdz=−ka Ir X [34] (51) dI/dz: light intensity rate (μmol/m2s) ka: specific light retention coefficient Ir: irradiance light needed to grow microalgae (μmol/m2s) X: biomass concentration (g/L) Z: distance between light and the photobioreactor (m) 1Lln(IoIr)=kaX [34] (52) L: length for the light path for the photobioreactor (m) Io: irradiance light on the surface (μmol/m2s) X= aka [34] (53) a: absorbance factor a=1Lln(IoIr) [34] (54) −dsdt= 1Ydxdt+mX [34] (55) ds/dt: soduinm nitrate consumption rate to (g/Ls) Y: extreme microalgae coefficient dx/dt: biomass concentration rate (g/Ls) m: maximum maintenance coefficient S=So−1yx(XoXmaxeμmaxt Xmax−Xo+ Xoe μmaxt−xo)m Xmax μln(Xmax−Xo+XoeμmaxtXmax) [34] (56) S: nitrate sodium concentration (g/L) So: initial consumption for nitrate sodium at zero time (g/L) yx: maximum lipid coefficient (g/g) t: time (days) h(pH)=h [H+]=k0k1+1K1[H+]+k2k1K1K2[H+]21+1K1[H+]+1K1K2[H+]2 [39] (57) h(pH): pH concentration h[H+]: proton concentration (gmole/L) k0: parameters of the pH-dependent kinetic (h–1) k1: parameters of the pH-dependent kinetic (h–1) k2: parameters of pH-dependent kinetic (h–1) K1: parameters of the pH-dependent kinetic (molH+/L) K2: parameters of the pH-dependent kinetic (molH+/L) Harvesting Efficency (%)= A1−B1A1×100 [40] (58) A1: the first estimation value of biomass (g/L) B1: the final estimation value of biomass (g/L) Open in new tab Table 2: Mathematical equations to calculate the different factors needed to operate the microalgae system Equation . Reference . Equation number . V= Vlr+Vld2=ULr2((11−εr)+(11−εd) ) [29] (20) V: velocity in split photobioreactor (m/s) Vlr: velocity in the photobioreactor’s riser (m/s) Vld: the velocity in the downcomer of the photobioreactor (m/s) ULr: average superficial liquid velocity for the riser and downcomer (m/s) ε r: efficiency in the riser region ε d: efficiency in the downcomer region εr=0.4(Fr )0.51+0.4(Fr)0.5(1+(ULrUGr)) [29] (21) Fr: Froude number UGr: gas velocity for the photobioreactor (m/s) Fr= UGr2gdo [29] (22) g: gravitational acceleration (m/s2) do: diameter for sparger diameter (m) UGrεr= 1.607( UGr+ULr)+0.298 [29] (23) εd=0.841 εr [29] (24) ULr= Vlr(1− εr)= Vld(1−εd) [29] (25) UGr=FgA [30] (26) Fg: volumetric flow rate (m3/s) A: area of the photobioreactor (m2) A=2πrh+2π r2 [31] (27) v= Fgn14Πd2 [30] (28) v: entrance velocity, sparger, for photobioreactor (m/s) n: number of nozzles (assumed to be one in this study) d: nozzle diameter (mm) Re= 4 ρFg(ndη) [30] (29) Re: Reynold number ρ: aqueous density (kg/m3) (assumed to be 1000 kg/m3) η: aqueous dynamic viscosity (kg/ms) (assumed to be 0.001 kg/ms) ε=Arεr +AdεdAr+Ad [32] (30) Ar: cross-section area for downcomer region (m2) Ad: cross-section area for riser region (m2) kLa(co2)=0.837UGr−0.971−1 [29] (31) kLa(CO2): mass transfer coefficient of carbon dioxide (h–1) kLa(co2)=0.83 kLa (o2) [33] (32) kLa(O2): mass transfer coefficient of oxygen (h–1) Rco2= Cc×μl×(Mco2Mc) [34] (33) RCO2: CO2 fixation rate (mg/Ld) Cc: average carbon content (0.63 g carbon/g dry cell weight) μl: volumetric growth rate in a unit (mg/Ld) MCO2: molecular weight of carbon dioxide (mg/mole) Mc: molecular weight for carbon (mg/mole) μ=(lnx2−lnx1)t2−t1 [35] (34) μ: growth rate (g/Ld) x2: biomass concentration at day t2 x1: biomass concentration at day t1 θm=TmTc [36] (35) θm: mixing time Tm: mixing time (s) Tc: cycling time (s) Bo=VlLEZ [36] (36) Bo: Bodenstein number Vl: velocity of the liquid (cm/s) L: the separation between the tracer infusion and discovery focuses (cm) EZ: pivotal scattering coefficient (m2/s) Vl=(LcTc) (37) Lc: length of the photobioreactor (cm) Bo=k(TmTc) [36] (38) k: constant value of split photobioreactor is (9.6 ± 0.2) Bolg= UGdEz [36] (39) Bolg: Bodenstein number depends on the superficial gas velocity (cm/s) UG: gas velocity (cm/s) d: diameter of the sparger photobioreactor (m) Bolg=3.77(Fr13)1.2 [36] (40) dXdt=kcX(1−(XXmax)) [34] (41) dX/dt: biomass concentration rate (g/Lh) Kc: specific growth rate(h–1) X: biomass concentration of microalgae (g/L) Xmax: maximum concentration of microalgae (g/L) X=Xmax1+(XmaxXo−1) e(−Kct) [34] (42) Xo: initial concentration of microalgae/L dpdt= α(dXdt)+β X [34] (43) dp/dt: lipid concentration rate (mg/Lh) α: growth coefficient β: non-growth coefficient α= K24+KXffao [37] (44) K: kinetic constant Xffao: initial free fatty acid (kcal/g) β=αK [37] (45) K= k1k2 [37] (46) k1: kinetic rate constant for the forward reaction (s–1) k2: active rate consistent for the reverse response (s–1) p=po−αXo+(αXoXmineμmaxtXmin−Xo+Xoeμmaxt)+β(Xminμmax)lnXmin−Xo+XoeμmaxtXmin [34] (47) p: lipid concentration for the microalgae in (g/L) po: initial lipids concentration (g/L) Xmin: maximum concentration of microalgae (g/L) μmax: the maximum growth rate (g/Ld) S˙=Dsin−ρ(s)x−Ds [38] (48) Ṡ: nitrogen cell concentration (g/m3) Dsin: influent concentration of nitrogen (gmole/L) ρ(s): retention rate (g/m3) x: nitrogen biomass (g) Ds: dilution or weakening rate (d–1) q˙=ρ(s)−μ(q)q [38] (49) q̇: constrained nutrients rate μ(q): development rate q: constrained quotient x˙=μ (q)x−Dx [38] (50) ẋ: nitrogen biomass rate (g/m3) dIdz=−ka Ir X [34] (51) dI/dz: light intensity rate (μmol/m2s) ka: specific light retention coefficient Ir: irradiance light needed to grow microalgae (μmol/m2s) X: biomass concentration (g/L) Z: distance between light and the photobioreactor (m) 1Lln(IoIr)=kaX [34] (52) L: length for the light path for the photobioreactor (m) Io: irradiance light on the surface (μmol/m2s) X= aka [34] (53) a: absorbance factor a=1Lln(IoIr) [34] (54) −dsdt= 1Ydxdt+mX [34] (55) ds/dt: soduinm nitrate consumption rate to (g/Ls) Y: extreme microalgae coefficient dx/dt: biomass concentration rate (g/Ls) m: maximum maintenance coefficient S=So−1yx(XoXmaxeμmaxt Xmax−Xo+ Xoe μmaxt−xo)m Xmax μln(Xmax−Xo+XoeμmaxtXmax) [34] (56) S: nitrate sodium concentration (g/L) So: initial consumption for nitrate sodium at zero time (g/L) yx: maximum lipid coefficient (g/g) t: time (days) h(pH)=h [H+]=k0k1+1K1[H+]+k2k1K1K2[H+]21+1K1[H+]+1K1K2[H+]2 [39] (57) h(pH): pH concentration h[H+]: proton concentration (gmole/L) k0: parameters of the pH-dependent kinetic (h–1) k1: parameters of the pH-dependent kinetic (h–1) k2: parameters of pH-dependent kinetic (h–1) K1: parameters of the pH-dependent kinetic (molH+/L) K2: parameters of the pH-dependent kinetic (molH+/L) Harvesting Efficency (%)= A1−B1A1×100 [40] (58) A1: the first estimation value of biomass (g/L) B1: the final estimation value of biomass (g/L) Equation . Reference . Equation number . V= Vlr+Vld2=ULr2((11−εr)+(11−εd) ) [29] (20) V: velocity in split photobioreactor (m/s) Vlr: velocity in the photobioreactor’s riser (m/s) Vld: the velocity in the downcomer of the photobioreactor (m/s) ULr: average superficial liquid velocity for the riser and downcomer (m/s) ε r: efficiency in the riser region ε d: efficiency in the downcomer region εr=0.4(Fr )0.51+0.4(Fr)0.5(1+(ULrUGr)) [29] (21) Fr: Froude number UGr: gas velocity for the photobioreactor (m/s) Fr= UGr2gdo [29] (22) g: gravitational acceleration (m/s2) do: diameter for sparger diameter (m) UGrεr= 1.607( UGr+ULr)+0.298 [29] (23) εd=0.841 εr [29] (24) ULr= Vlr(1− εr)= Vld(1−εd) [29] (25) UGr=FgA [30] (26) Fg: volumetric flow rate (m3/s) A: area of the photobioreactor (m2) A=2πrh+2π r2 [31] (27) v= Fgn14Πd2 [30] (28) v: entrance velocity, sparger, for photobioreactor (m/s) n: number of nozzles (assumed to be one in this study) d: nozzle diameter (mm) Re= 4 ρFg(ndη) [30] (29) Re: Reynold number ρ: aqueous density (kg/m3) (assumed to be 1000 kg/m3) η: aqueous dynamic viscosity (kg/ms) (assumed to be 0.001 kg/ms) ε=Arεr +AdεdAr+Ad [32] (30) Ar: cross-section area for downcomer region (m2) Ad: cross-section area for riser region (m2) kLa(co2)=0.837UGr−0.971−1 [29] (31) kLa(CO2): mass transfer coefficient of carbon dioxide (h–1) kLa(co2)=0.83 kLa (o2) [33] (32) kLa(O2): mass transfer coefficient of oxygen (h–1) Rco2= Cc×μl×(Mco2Mc) [34] (33) RCO2: CO2 fixation rate (mg/Ld) Cc: average carbon content (0.63 g carbon/g dry cell weight) μl: volumetric growth rate in a unit (mg/Ld) MCO2: molecular weight of carbon dioxide (mg/mole) Mc: molecular weight for carbon (mg/mole) μ=(lnx2−lnx1)t2−t1 [35] (34) μ: growth rate (g/Ld) x2: biomass concentration at day t2 x1: biomass concentration at day t1 θm=TmTc [36] (35) θm: mixing time Tm: mixing time (s) Tc: cycling time (s) Bo=VlLEZ [36] (36) Bo: Bodenstein number Vl: velocity of the liquid (cm/s) L: the separation between the tracer infusion and discovery focuses (cm) EZ: pivotal scattering coefficient (m2/s) Vl=(LcTc) (37) Lc: length of the photobioreactor (cm) Bo=k(TmTc) [36] (38) k: constant value of split photobioreactor is (9.6 ± 0.2) Bolg= UGdEz [36] (39) Bolg: Bodenstein number depends on the superficial gas velocity (cm/s) UG: gas velocity (cm/s) d: diameter of the sparger photobioreactor (m) Bolg=3.77(Fr13)1.2 [36] (40) dXdt=kcX(1−(XXmax)) [34] (41) dX/dt: biomass concentration rate (g/Lh) Kc: specific growth rate(h–1) X: biomass concentration of microalgae (g/L) Xmax: maximum concentration of microalgae (g/L) X=Xmax1+(XmaxXo−1) e(−Kct) [34] (42) Xo: initial concentration of microalgae/L dpdt= α(dXdt)+β X [34] (43) dp/dt: lipid concentration rate (mg/Lh) α: growth coefficient β: non-growth coefficient α= K24+KXffao [37] (44) K: kinetic constant Xffao: initial free fatty acid (kcal/g) β=αK [37] (45) K= k1k2 [37] (46) k1: kinetic rate constant for the forward reaction (s–1) k2: active rate consistent for the reverse response (s–1) p=po−αXo+(αXoXmineμmaxtXmin−Xo+Xoeμmaxt)+β(Xminμmax)lnXmin−Xo+XoeμmaxtXmin [34] (47) p: lipid concentration for the microalgae in (g/L) po: initial lipids concentration (g/L) Xmin: maximum concentration of microalgae (g/L) μmax: the maximum growth rate (g/Ld) S˙=Dsin−ρ(s)x−Ds [38] (48) Ṡ: nitrogen cell concentration (g/m3) Dsin: influent concentration of nitrogen (gmole/L) ρ(s): retention rate (g/m3) x: nitrogen biomass (g) Ds: dilution or weakening rate (d–1) q˙=ρ(s)−μ(q)q [38] (49) q̇: constrained nutrients rate μ(q): development rate q: constrained quotient x˙=μ (q)x−Dx [38] (50) ẋ: nitrogen biomass rate (g/m3) dIdz=−ka Ir X [34] (51) dI/dz: light intensity rate (μmol/m2s) ka: specific light retention coefficient Ir: irradiance light needed to grow microalgae (μmol/m2s) X: biomass concentration (g/L) Z: distance between light and the photobioreactor (m) 1Lln(IoIr)=kaX [34] (52) L: length for the light path for the photobioreactor (m) Io: irradiance light on the surface (μmol/m2s) X= aka [34] (53) a: absorbance factor a=1Lln(IoIr) [34] (54) −dsdt= 1Ydxdt+mX [34] (55) ds/dt: soduinm nitrate consumption rate to (g/Ls) Y: extreme microalgae coefficient dx/dt: biomass concentration rate (g/Ls) m: maximum maintenance coefficient S=So−1yx(XoXmaxeμmaxt Xmax−Xo+ Xoe μmaxt−xo)m Xmax μln(Xmax−Xo+XoeμmaxtXmax) [34] (56) S: nitrate sodium concentration (g/L) So: initial consumption for nitrate sodium at zero time (g/L) yx: maximum lipid coefficient (g/g) t: time (days) h(pH)=h [H+]=k0k1+1K1[H+]+k2k1K1K2[H+]21+1K1[H+]+1K1K2[H+]2 [39] (57) h(pH): pH concentration h[H+]: proton concentration (gmole/L) k0: parameters of the pH-dependent kinetic (h–1) k1: parameters of the pH-dependent kinetic (h–1) k2: parameters of pH-dependent kinetic (h–1) K1: parameters of the pH-dependent kinetic (molH+/L) K2: parameters of the pH-dependent kinetic (molH+/L) Harvesting Efficency (%)= A1−B1A1×100 [40] (58) A1: the first estimation value of biomass (g/L) B1: the final estimation value of biomass (g/L) Open in new tab The mixing time is an essential factor in growing microalgae in a split photobioreactor. The mixing time is the time needed to arrive at 95% of complete blending. The cycling time is required to blend one section through the split photobioreactor. Two models investigated the light requirement for the microalgae system, the Droop model and the Beer–Lambert model. The Droop model represents the development of biomass and this dynamic model can limit nitrogen and light by using Equations (48–50). Equation (51) is used for the Beer–Lambert model calculations. The nitrate required for the closed photobioreactor to grow microalgae, especially sodium nitrate, are calculated using Equation (56). Once the amount of nitrate needed to grow microalgae is established, the proportional amount of phosphorous required to assimilate the nitrogen must then be calculated; the optimal N/P for the microalgae strain Botryococcus braunii is in the range of 11–25 [40]. The harvesting efficiency for microalgae is defined as the percentage of the microalgae’s optimum amount of biomass or lipids; the efficiency of the harvesting is calculate using Equation (58). 1.3 Solving the mathematical equations MATLAB® was utilized to solve the mathematical equations that were presented in the last two sections to determine the optimal results for the hydroponic–microalgae combined system as illustrated in Fig. 3. Fig. 3: Open in new tabDownload slide Steps for solving the mathematical equations for the hydroponic–microalgae system. The equations for the hydroponic system, which are linear, were solved using MATLAB’s for loop function; it is a repetition control structure method that allows the user to design a loop that has to run a certain number of times to solve a linear equation. The microalgae system equations, which are non-linear, were solved using MATLAB’s fsolve and fval functions that were used to determine the roots of non-linear equations. 2 Results Two parameters were changed to study their effect on the hydroponic–microalgae system. The first parameter was the number of plants in the hydroponic system, which varied between 40 and 105. The second parameter was the sparger diameter for the microalgae system, which varied between 0.005 and 0.01 m. The 10 cases for which both parameters were changed are listed in Table 3. Table 3: Number of plants and the sparger diameter (m) for the 10 investigated cases Case number . Number of plants . Sparger diameter (m) . Case 1 40 0.005 Case 2 50 0.006 Case 3 70 0.004 Case 4 75 0.003 Case 5 80 0.002 Case 6 85 0.001 Case 7 90 0.007 Case 8 95 0.008 Case 9 100 0.009 Case 10 105 0.01 Case number . Number of plants . Sparger diameter (m) . Case 1 40 0.005 Case 2 50 0.006 Case 3 70 0.004 Case 4 75 0.003 Case 5 80 0.002 Case 6 85 0.001 Case 7 90 0.007 Case 8 95 0.008 Case 9 100 0.009 Case 10 105 0.01 Open in new tab Table 3: Number of plants and the sparger diameter (m) for the 10 investigated cases Case number . Number of plants . Sparger diameter (m) . Case 1 40 0.005 Case 2 50 0.006 Case 3 70 0.004 Case 4 75 0.003 Case 5 80 0.002 Case 6 85 0.001 Case 7 90 0.007 Case 8 95 0.008 Case 9 100 0.009 Case 10 105 0.01 Case number . Number of plants . Sparger diameter (m) . Case 1 40 0.005 Case 2 50 0.006 Case 3 70 0.004 Case 4 75 0.003 Case 5 80 0.002 Case 6 85 0.001 Case 7 90 0.007 Case 8 95 0.008 Case 9 100 0.009 Case 10 105 0.01 Open in new tab For all 10 cases, some assumptions were made. For the hydroponic system: the crop coefficient (Kc) was 0.3, temperature was 22°C, pH range was between 5 and 7, and the crop water requirement (CWR) was <400 L/m2. For the microalgae system: the velocity for the photobioreactor (V) was assumed to be <2.9 m/s, the average superficial liquid velocity for the riser region (ULr) was assumed to be <2 m/s and the pH values were assumed to be ~7 or less. 2.1 Results for the microalgae system Ten different cases, listed in Table 4, were simulated. Case 10, shown in Fig. 4, has 105 lettuce plants. The change with time for two parameters was monitored for 30 days of operation; those parameters were the plant-specific weight (PSW) and the SDW. Results of the PSW and SDW for Case 10 showed that the PSW increased linearly by 95% for the period of 0.5–29.5 days with a rate of 0.01 (g/mm2day) and a linear increase in the SDW of almost 24% for a period of 0.5–29.5 days with a rate of 0.44 (g/day). A linear behaviour in those two parameters was found in all 10 cases. Table 4: The modelling results for plant-specific weight (PSA) and shoot dry weight (SDW) for the hydroponic system for 10 cases Case number . Number of plants . Plant-specific weight (g/mm2) . Shoot dry weight (g) . Case 1 40 0.14–1.3 21.9–27.4 Case 2 50 0.18–1.6 23.9–28.5 Case 3 70 0.25–2.3 35.1–43.1 Case 4 75 2.3–4.5 38.6–47.1 Case 5 80 2.4–4.8 40.2–49.3 Case 6 85 2.6–5.1 46–55.5 Case 7 90 2.6–5.1 46.8–57.5 Case 8 95 2.9–5.7 49.2–60.8 Case 9 100 3.1–6 52.2–65 Case 10 105 3.2–6.2 54.2–68.2 Case number . Number of plants . Plant-specific weight (g/mm2) . Shoot dry weight (g) . Case 1 40 0.14–1.3 21.9–27.4 Case 2 50 0.18–1.6 23.9–28.5 Case 3 70 0.25–2.3 35.1–43.1 Case 4 75 2.3–4.5 38.6–47.1 Case 5 80 2.4–4.8 40.2–49.3 Case 6 85 2.6–5.1 46–55.5 Case 7 90 2.6–5.1 46.8–57.5 Case 8 95 2.9–5.7 49.2–60.8 Case 9 100 3.1–6 52.2–65 Case 10 105 3.2–6.2 54.2–68.2 Open in new tab Table 4: The modelling results for plant-specific weight (PSA) and shoot dry weight (SDW) for the hydroponic system for 10 cases Case number . Number of plants . Plant-specific weight (g/mm2) . Shoot dry weight (g) . Case 1 40 0.14–1.3 21.9–27.4 Case 2 50 0.18–1.6 23.9–28.5 Case 3 70 0.25–2.3 35.1–43.1 Case 4 75 2.3–4.5 38.6–47.1 Case 5 80 2.4–4.8 40.2–49.3 Case 6 85 2.6–5.1 46–55.5 Case 7 90 2.6–5.1 46.8–57.5 Case 8 95 2.9–5.7 49.2–60.8 Case 9 100 3.1–6 52.2–65 Case 10 105 3.2–6.2 54.2–68.2 Case number . Number of plants . Plant-specific weight (g/mm2) . Shoot dry weight (g) . Case 1 40 0.14–1.3 21.9–27.4 Case 2 50 0.18–1.6 23.9–28.5 Case 3 70 0.25–2.3 35.1–43.1 Case 4 75 2.3–4.5 38.6–47.1 Case 5 80 2.4–4.8 40.2–49.3 Case 6 85 2.6–5.1 46–55.5 Case 7 90 2.6–5.1 46.8–57.5 Case 8 95 2.9–5.7 49.2–60.8 Case 9 100 3.1–6 52.2–65 Case 10 105 3.2–6.2 54.2–68.2 Open in new tab Fig. 4: Open in new tabDownload slide The shoot dry weight and plant-specific weight modelling results for the hydroponic system for Case 10. 2.2 Results for the microalgae system The simulation of the microalgae systems was performed using three photobioreactors, as listed in Table 5; the maximum value for the lipid and CO2 fixation rate was observed in Case 9. The harvesting efficiency for microalgae is defined as the percentage of microalgae’s optimum amount of biomass or lipids. The lowest harvesting efficiency was observed in Case 10 at 70% in which an inverse behaviour between lipids and the CO2 fixation rate was observed. Table 5: Modelling results for the lipid and the CO2 fixation rate for 10 cases Case number . Sparger diameter (m) . Lipid (g/L) . CO2 fixation rate (mg/Lh) . Harvesting efficiency (%) . Case 1 0.005 0.013–4 0.033–0.90 78 Case 2 0.006 0.24–5.7 0.07–2.04 80 Case 3 0.004 0.05–2.4 0.01–0.2 77 Case 4 0.003 0.4–1.3 0.005–0.08 76 Case 5 0.002 0.3–1.1 0.005–0.07 74 Case 6 0.001 0.004–0.8 0.0005–0.06 71 Case 7 0.007 0.2–6.1 2–7.3 82 Case 8 0.008 0.3–7.5 2–7.7 90 Case 9 0.009 0.3–8.3 2–8.1 95 Case 10 0.01 0.04–0.3 0.08–0.8 70 Case number . Sparger diameter (m) . Lipid (g/L) . CO2 fixation rate (mg/Lh) . Harvesting efficiency (%) . Case 1 0.005 0.013–4 0.033–0.90 78 Case 2 0.006 0.24–5.7 0.07–2.04 80 Case 3 0.004 0.05–2.4 0.01–0.2 77 Case 4 0.003 0.4–1.3 0.005–0.08 76 Case 5 0.002 0.3–1.1 0.005–0.07 74 Case 6 0.001 0.004–0.8 0.0005–0.06 71 Case 7 0.007 0.2–6.1 2–7.3 82 Case 8 0.008 0.3–7.5 2–7.7 90 Case 9 0.009 0.3–8.3 2–8.1 95 Case 10 0.01 0.04–0.3 0.08–0.8 70 Open in new tab Table 5: Modelling results for the lipid and the CO2 fixation rate for 10 cases Case number . Sparger diameter (m) . Lipid (g/L) . CO2 fixation rate (mg/Lh) . Harvesting efficiency (%) . Case 1 0.005 0.013–4 0.033–0.90 78 Case 2 0.006 0.24–5.7 0.07–2.04 80 Case 3 0.004 0.05–2.4 0.01–0.2 77 Case 4 0.003 0.4–1.3 0.005–0.08 76 Case 5 0.002 0.3–1.1 0.005–0.07 74 Case 6 0.001 0.004–0.8 0.0005–0.06 71 Case 7 0.007 0.2–6.1 2–7.3 82 Case 8 0.008 0.3–7.5 2–7.7 90 Case 9 0.009 0.3–8.3 2–8.1 95 Case 10 0.01 0.04–0.3 0.08–0.8 70 Case number . Sparger diameter (m) . Lipid (g/L) . CO2 fixation rate (mg/Lh) . Harvesting efficiency (%) . Case 1 0.005 0.013–4 0.033–0.90 78 Case 2 0.006 0.24–5.7 0.07–2.04 80 Case 3 0.004 0.05–2.4 0.01–0.2 77 Case 4 0.003 0.4–1.3 0.005–0.08 76 Case 5 0.002 0.3–1.1 0.005–0.07 74 Case 6 0.001 0.004–0.8 0.0005–0.06 71 Case 7 0.007 0.2–6.1 2–7.3 82 Case 8 0.008 0.3–7.5 2–7.7 90 Case 9 0.009 0.3–8.3 2–8.1 95 Case 10 0.01 0.04–0.3 0.08–0.8 70 Open in new tab MATLAB results after 30 days of operation show that two parameters have been observed to change with time, which were the lipids concentration and the CO2 fixation rate, as shown in Fig. 5. Fig. 5: Open in new tabDownload slide The model results of the lipid concentration and the CO2 fixation rate for (a) Case 6 using a sparger diameter of 0.01 m, (b) Case 9 using a sparger diameter of 0.009 m and (c) Case 10 using a sparger diameter of 0.01 m. The lipid concentration and the CO2 fixation rate for Case 6, which used a sparger diameter of 0.01 m, is shown in Fig. 5a. The lipid concentration increased almost 2-fold for a period of 1–25 days. A slight reduction in the lipids concentration was observed for Days 26–30. The CO2 fixation rate slightly changed in the first few days and it remained level thereafter. Fig. 5b shows the results of the lipid concentration and CO2 fixation rate for Case 9, which uses a sparger diameter of 0.009 m. The lipid concentration increased by >24% for the duration of the 30 days of operation. The CO2 fixation rate significantly changed in the first 5 days and it remained level thereafter. Fig. 5c shows the results of the lipid concentration and CO2 fixation rate for Case 10, which uses a sparger diameter of 0.01 m. The lipid concentration increased by >7% during the 30 days. The CO2 fixation rate significantly changed in the first 5 days and it remained level thereafter. Case 9 result was the optimum result since the lipids production was 8084 g/month. Accordingly, the optimum ratio for the hydroponics and microalgae was achieved when using three splits of photobioreactors to 100 lettuce plants for the hydroponic system. 2.3 Combined hydroponic and microalgae system model A model was proposed to predict CO2 lipids and SDW for the hydroponic–microalgae system. The model was based on two variables: the sparger diameter and the number of lettuce plants. To estimate the lipids and the CO2 fixation rate for the microalgae system and SDW for the hydroponic system, the three predicted models Equations (59–61) from the Sigma Plot Program were used: Fixation carbon=126.92exp(−0.5(X−0.00790.0013)2+(y−891.655345.938)2)(59) Lipids= 66544.009exp(−0.5(X−6.00720.0021)2+(y−2254.3451035.49)2)(60) SDW=98.683exp(−0.5(X−28.5389893310.5878)2+(y−178.205984.9572)2)(61) 2.4 Combined hydroponic and microalgae system model’s sensitivity analysis The sensitivity analysis measures the effect of changing any variables and assumptions on the model results [41]. The sensitivity was measured using the regression-based method on the mathematical model for outputs from the hydroponic and microalgae systems. These outputs are: lipids, CO2 fixation, PSW and SDW. When using the regression-based method, a sensitivity of <40 has a marginal effect on outcomes. For this study, the lipids’ highest sensitivity in all parameters did not exceed 18 and the lowest sensitivity was found in the PSW where the numeric value was 3. The SDW and the CO2 fixation rate were both ~10. Therefore, the combined hydroponic and microalgae outputs were not very sensitive to changes in those parameters making the results reliable. 3 Conclusion This study investigates an innovative portable hydroponic–microalgae system. The results for 10 different cases found using a mathematical model developed in this work showed that changing the sparger split photobioreactor diameter to between 0.001 and 0.01 m altered the number of lettuce plants. 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For commercial re-use, please contact journals.permissions@oup.com © The Author(s) 2022. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy
Clean Energy – Oxford University Press
Published: Jun 1, 2022
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