The photosynthetic efficiency can be calculated as the energy content of the glucose produced during photosynthesis divided by the incident ra­diation. This value is different than the overall energy efficiency of growth, which includes the cost of living (i. e., respiration to enable cell functions), conversion of glucose to biomass, and required energy inputs (e. g., mix­ing, nutrient supply, etc.) [29]. The energy content of the glucose produced (per liter processed) can be calculated as:

ЁСН20 = (/ ■ PAR ■ PTE ■ PUE ■ a)

where tc is the volumetric irradiance (in joules per liter of processed volume), PAR is the photosynthetically active radiation fraction (0.46), PTE is the photon transmission efficiency, and PUE is the photon utiliza­tion efficiency [30]. The volumetric irradiance can be converted to an areal irradiance, I, according to:

where tc is the cultivation time (123 days for the Experimental Case and 12.5 days for the Highly Productive Case) and d is the pond depth (0.2 m for both cases). In Equation 3, the variable a characterizes the effi­ciency by which photons used for photosynthesis are converted to glucose through Z-scheme photosynthesis. With a quantum requirement of 8 mol photons per mol of glucose (the energy content of glucose is 467.5 kJ/mol) and an average photon energy content of 225 kJ/mol [30], a = (467.5/8 •

225) = 0.26 [29]. The photosynthetic efficiency, PE, is the ratio of EDCH2) to І, which becomes:

(5)

The amount of energy contained in the growth volume (as algal bio­mass), EDgv, can be calculated from:

where HHVGM is the higher heating value of the grown algal biomass. The amount of energy contained in the growth volume can also be calcu­lated from:

tDGV = / ■ PAR ■ PTE ■ PUE ■ a ■ (1 — CoL) ■ т

where CoL is the cost of living, which is defined as the fraction of glu­cose consumed for cellular operations [30]. The energy conversion of glu­cose to biomass energy can be grossly simplified as a single-step process, represented by t. Assuming algae have the Redfield stoichiometry defined by Clarens et al. [15] ( C106H181O45N15P ), the conversion of glucose to algal biomass can be approximated as:

106CH2O + 15NaNO3 + 0.5P2O5 + C106H181O45N15P + 8H2O + 42.75O2 + 15NaOH

2 3 2 5 106 181 45 15 2 2

(8)

The higher heating value (HHVGM) for algae can be estimated as stated by Clarens et al.:

HHVnh, = 35160 • x + 116225 — 11090 • x„ + 6280 • x [kJ/kg]

GM c H 0 n

where x is the mass fraction of each element (carbon, hydrogen, oxygen, and nitrogen) [15]. The HHVGM for the algae considered here is 24.49 MJ/kg (59.12 MJ/mol). The energy conversion of glucose (106 mol with a HHV of 467.5 kJ/mol) to biomass energy is represented by т = 1.19. Com­bining several of these relations, the PE can be calculated as:

5.2.5 ENERGY RETURN ON INVESTMENT FORMULAE

The second-order energy return on investment, 2nd O EROI is calcu­lated as:

„nd ЁОоШ ЁОво + ЁОвмр (11)

Eg + Ёр + Ёк Ёс + Ёр + Ёк

As a second-order analysis, direct and indirect energy inputs are in­cluded, while a first-order analysis would include only direct energy inputs [5]. An apostrophe accent is used to denote units that are reported with respect to the growth volume processed, such as the energy inputs in units of J per L processed (J/Lp).

To account for differences in energy quality among the inputs and out­puts, the quality-adjusted second-order energy return on investment (QA 2nd EROI) was calculated by multiplying each input and output by priced — based quality factors. The quality factors (QF) were calculated for energy flows based on the energy price (EP), which is the price of each energy source per joule, and correlates the relative value of each fuel [31]. U s­ing coal as the arbitrary standard with a quality factor equal to 1 ($1.5/ MMBtu, $1.4/GJ), the quality factors used in this study were: electricity

19.5 ($27.8/GJ, 010/kWh), petroleum 14.5 ($20.6/GJ, $0.66/L), and natu­ral gas 2.7 ($3.8/GJ, $4/MMBtu) [32]. The bio-oil was assigned the QF of petroleum and methane was assigned the QF of natural gas. For materials, the quality-factor was determined as:

where MP is the price (in $/kg), EE is the energy equivalent (in MJ/kg), and EPcoal is the energy price for coal ($1.4/GJ).