A Report on My Recent Investigations of Solar Energy Harvested by Photosynthesis in a Controlled Environment
Table of Contents
Solar Energy to Chemical Energy in Biomass by Photosynthesis
Solar Energy to the Combustion Chamber
Purpose of this Exercise
The purpose of this exercise is to place a reasonable upper bound on the rate at which useful energy can be harvested from the sun by photosynthesis in a controlled environment like a greenhouse. I shall not be concerned here with the investment of embodied energy in the form of equipment, physical plant, chemicals, etc.; but, rather, concern myself solely with the yield per unit area regardless of the infusion of money and materials from outside the project. This is not a study of sustainability; but, rather, a study to determine if such a controlled environment can generate all of the energy it consumes on the premises, which might include some liquid or gaseous bio-fuels and some electricity. Therefore, the issue of Energy Returned over Energy Invested (EROI) will not arise. The principal effort will be to look for the maximum and average values of the appropriate conversion efficiencies on the Internet.
Solar Energy
Sunlight consists of ultraviolet light (UV), visible light, and infrared (IR). The energy distribution within the solar spectrum is approximately 2 percent ultraviolet, 47 percent visible light, and 51 percent infrared according to http://home.wanadoo.nl/paulschils/03.04.html. The global-average incident radiation is about 70% of one-quarter of the solar constant; i.e., 240 watts/m2. However, the average insolation that one gets by dividing the solar constant, which is about 1370 watts/m2, by the ratio of the surface of the earth to the projected area seen by the sun represents an average over the uninhabited parts of the globe as well as the highly populated regions. According to http://www.solarpowerfor.us/solar-power.html “ … in North America the average insolation at ground level over an entire year (including nights and periods of cloudy weather) lies between 125 and 375 W/m2 … .” I will use the upper limit of that range first, then the average value. Also, it might be useful to note that, according to http://www.lifesciences.napier.ac.uk/teaching/Eu/muI.htm, 1 J of PAR energy = 4.15 µE of PAR photons, where E stands for Einstein and 1.0 E is Avogadro’s number or 6.023 · 1023 photons.
I found the following remarks at http://www.ihtec.org/fileadmin/archives/IHTEC/documents/Physical_Limits_060919.doc.
The central part of the solar spectrum is photosynthetically active radiation. Only 45% of solar radiation energy is carried by this part of the spectrum. A further reduction of biological solar energy conversion efficiency is due to the fact that some of the qualified photons absorbed by the plant fail to perform photosynthesis; the quantum efficiency is given as 25%, which reduces the conversion efficiency to 11%. In addition, some of the solar radiation is reflected, and photosynthesis requires respiration which requires energy. Thus, a realistic expectation for the efficiency by which solar radiation energy can be converted into biomass energy is 3% to 6% [9]. This theoretical efficiency is 10 times lower than the technical conversion efficiency. Hence some 300 m2/person of biologically productive land is required to supply the total present energy needs of humankind. In addition, transpiration of water is required for this photosynthesis to take place. Water needs for transpiration depend on conditions; the University of Prince Edward Island website states that between 250 g to 700 g water are needed for the photosynthesis of 1 g of dry biomass [10].
In practice, the efficiency of biomass conversion is much less than the theory predicts. An energy crop data base developed by the Oak Ridge National Laboratory [11] offers realistic yields of un-irrigated switch grass and hybrid poplar plantations. The data for Barbor, Alabama may serve as an example. The median annual yield for switch grass, planted on former cropland, is 8.6 dry tons/acre; for hybrid poplar it is 4.1dry tons/acre. In SI units this represents an average dry matter production rate of 61µg/s, and 29 µg/s respectively. Using a heating value of 15 kJ/g, the biomass power generation rate is 0.92 W/m2 for switch grass, and 0.44 W/m2 for hybrid poplar. These values represent the energy harvested. The net overall efficiency is further reduced by the energy requirements to plant, harvest, dry, transport, process the crop into a suitable transportation fuel, and by the thermodynamic efficiency in electricity generation. In the end, the realistic overall power of biological conversion of solar energy to satisfy present human needs is less than 0.5 W/m2. Therefore, replacing the 2080 W/person presently derived from fossil fuels and nuclear energy with biomass energy requires more than 4000 m2/person of biologically productive land.
The author is emeritus, so he can tell the truth without incurring corporate wrath. This is depressing data indeed. For example, if these numbers applied to controlled growing environments, which they don’t, we could not hope to harvest enough energy from a greenhouse with a roof area of 100 square meters to feed even one person or to run a hundred-watt light bulb, let alone a computer or a refrigerator. However, I have decided to use the most optimistic values for the most important numbers; therefore, I shall look elsewhere.
Solar Energy to Biomass
I began by looking at Monteith's approach as elucidated at http://www.gardenwithinsight.com/help100/00000283.htm: According to http://www-eosdis.ornl.gov/VEGETATION/lai_des.html, one thousand values of the leaf area index (LAI) were tabulated that “varied from 0.1-0.18 (minimum; desert and tundra) to 47.0 (maximum; a peculiarity of one allometric method for estimating all-sided LAI in coniferous tree stands). Units are m2/m2 or dimensionless. However, only 14% of the records have LAI greater than 8.0 (a more typical maximum value for one-sided or projected LAI, unlikely to be exceeded except with peculiar conditions or methodology.)” The highest energy harvest corresponds to an LAI of 47. Let us use the highest value of the solar radiation (RA) to compute photosynthetically active radiation (PAR) using Equation 255 of http://www.gardenwithinsight.com/help100/00000283.htm:
Thus, we might simply set the PAR to one-half of RA since we are looking for upper limits to what can be expected. The following Equation 258 is employed to compute the average growth of biomass under that PAR:
ΔB(p) = 0.001 * BE * PAR,
where
BE = BE* - bc(3) * (VPD - 1.0), VPD > 0.5,
BE* = 100 * CO2 / (CO2 + exp(bc(1) - bc(2) * CO2)).
The variables and parameters are as follows:
ΔB(p) = potential increase in biomass
BE = biomass conversion factor (same units as BE*)
CO2 = carbon dioxide in atmosphere (ppm)
bc(1) = unknown constant
bc(2) = unknown constant
bc(3) = unknown constant
VPD = vapor pressure deficit (kPa)
Unfortunately, http://www.gardenwithinsight.com/help100/00000283.htm does not explain all of the units and the three constants bc(1), bc(2), and bc(3) nor does it provide optimal or average values for them. The simulator can be downloaded free of charge, but it does not have a version for Windows XP. Nevertheless, I installed the Windows NP version; but, I couldn’t get it to work. It doesn’t really matter because this computation does not make sense except for a specific growing climate that is completely known to the investigator; therefore, I shall look for overall efficiencies for the conversion of solar energy to biomass, to the chemical energy in biomass, to biodiesel, to hydrogen, directly to the combustion chamber, and directly to electricity. In the case of the production of fuel or electricity (resp.), I may assume that the biomass is algal or bacterial (resp.).
Below Figure 1, Photosynthetic conversion of CO2 to biomass and oxygen, at http://www.ent.ohiou.edu/~ohiocoal/projects/algae.pdf, we read:
Photosynthesis reduces carbon in the gas stream by converting it to biomass. As shown in Figure1, if the composition of "typical" cyanobacteria (normalized with respect to carbon) is CH1.8N0.17O0.56, then one mole of CO2 is required for the growth of one mole of cyanobacteria based on the relative molar weights, the carbon from 1 kg of CO2 could produce an increased cyanobacteria mass of 25/44 kg, with 32/44 kg of O2 released in the process, assuming O2 is released in a one-to-one molar ratio with CO2. Therefore, a photosynthetic system provides critical oxygen renewal along with the recycling of carbon into potentially beneficial biomass. Photosynthesis reduces carbon in the gas stream by converting it to biomass. As shown in Figure 1, if the composition of "typical" cyanobacteria (normalized with respect to carbon) is CH1.8N0.17O0.56, then one mole of CO2 is required for the growth of one mole of cyanobacteria. Based on the relative molar weights, the carbon from 1 kg of CO2 could produce an increased cyanobacteria mass of 25/44 kg, with 32/44 kg of O2 released in the process, assuming O2 is released in a one-to-one molar ratio with CO2. Therefore, a photosynthetic system provides critical oxygen renewal along with the recycling of carbon into potentially beneficial biomass.
“A Look Back at the U.S. Department of Energy’s Aquatic Species Program: Biodiesel from Algae”, published by the National Renewable Energy Lab, at http://www1.eere.energy.gov/biomass/pdfs/biodiesel_from_algae.pdf, tells us that a growth rate for algae in an outdoor pond was found to be about 2.1 grams/hour/m2. But, the growth rate in a controlled environment should be much higher. We do not have to decide which growth rate to use if we use overall conversion rates of solar energy to chemical energy in biomass. The proper term for this chemical energy is exergy as discussed at http://www.dematerialism.net/Chapter%202.html#_Definitions. Nevertheless, it would have been useful to compute this conversion rate by two different routes, which I might do in the future. (I read some useful data for growth rate per unit of incident solar power, but I can’t find the URL no matter what I google, which is a bit frustrating.)
Solar Energy to Chemical Energy in Biomass by Photosynthesis
This conversion rate can be computed as the chemical energy per unit mass of biota times the rate of conversion of solar energy to biomass computed in the last section. However, instead of calculating a value for the rate of increase of the mass of the plants, I can use the highest value in the literature I can find for the efficiency with which photosynthesis converts the incident radiant energy to chemical energy within the plant. The following paragraph comes from http://www.life.uiuc.edu/govindjee/paper/gov.html#58:
The theoretical minimum quantum requirement for photosynthesis is 8 quanta for each molecule of oxygen evolved (four quanta required by photo-system II and four by photo-system I). Measurements in algal cells and leaves under optimal conditions (e.g., low light) give quantum requirements of 8-10 photons per oxygen molecule released (see Emerson, 1958). These quantum yield measurements show that the quantum yields of photo-system II and photo-system I reaction centers under optimal conditions are near 100%. These values can be used to calculate the theoretical energy conversion efficiency of photosynthesis (free energy stored as carbohydrate/light energy absorbed). If 8 red quanta are absorbed (8 mol of red photons are equivalent to 1,400 kJ) for each CO2 molecule reduced (480 kJ/mol), the theoretical maximum energy efficiency for carbon reduction is 34%. Under optimal conditions, plants can achieve energy conversion efficiencies within 90% of the theoretical maximum. However, under normal growing conditions the actual performance of the plant is far below these theoretical values. The factors that conspire to lower the quantum yield of photosynthesis include limitations imposed by biochemical reactions in the plant and environmental conditions that limit photosynthetic performance. One of the most efficient crop plants is sugar cane, which has been shown to store up to 1% of the incident visible radiation over a period of one year. However, most crops are less productive. The annual conversion efficiency of corn, wheat, rice, potatoes, and soybeans typically ranges from 0.1% to 0.4% (Odum, 1971).
From http://www1.eere.energy.gov/biomass/pdfs/biodiesel_from_algae.pdf:
The maximum efficiency can be estimated at about 10% of total solar (Bolton 1996)*. Such efficiencies have been used in the projections for micro-algae biodiesel production (see Section III.D.). However, high sunlight conversions are observed only at low light intensities. Under full sunlight, typically one-third or less of this maximal efficiency, biomass productivity is obtained, because of the light saturation effect. Light saturation is simply the fact that algae, like many plants, can use efficiently rather low levels of light, typically only 10% of full sunlight (and often even less). Above this level, light is wasted. In fact, full sunlight intensities can damage the photosynthetic apparatus, a phenomenon known as photo-inhibition. Light saturation and photo-inhibition result from several hundred chlorophyll molecules collaborating in light trapping, an arrangement ideally suited for dense algal cultures, where on average a cell receives little light. However, exposed to full sunlight, the photosynthetic apparatus cannot keep up with the high photon flux and most of the photons are wasted, as heat and fluorescence, and can damage the photosynthetic apparatus in the process. One possibility, suggested by Neidhardt et al. (1998), is that photosynthetic productivity and light utilization could be maximized in micro-algae by reducing the size of the light-harvesting antenna through mutation or genetic engineering. This is an interesting idea that will be discussed further in the next section.
*Bolton, J.R. (1996) “Solar photo production of hydrogen.” Report to the Int. Energy Agency under Agreement on the Production and Utilization of Hydrogen, IEA/H2/TR-96.
From http://www.fao.org/docrep/w7241e/w7241e01.htm#TopOfPage
Plant photosynthesis takes place only in the presence of visible light (400-700 nm). However, solar light contains both visible and infrared components. Since visible light accounts for about 45% of all solar energy, the maximum achievable energy efficiency for CO2 fixation using solar radiation is approximately 13%.
I am not sure how the rate of CO2 fixation is related to the rate of conversion of solar energy to chemical energy in biomass. However, I will use 13% for a maximum value, which is higher than the 10% given above; and I will use 4.5% for the average value, which is also high probably.
Solar Energy to Biodiesel
A maximum yield of 20,000 gallons/acre/year of biodiesel from photosynthesis in algae, the energy content of which was 35.7 MJ/gallon, was found in the Wikipedia http://en.wikipedia.org/wiki/Algaculture. This amounts to 0.02 kilowatts per square meter:
Solar Energy to Hydrogen
From http://www.fao.org/docrep/w7241e/w7241e01.htm#TopOfPage:
Hydrogen gas is seen as a future energy carrier by virtue of the fact that it is renewable, does not evolve the "greenhouse gas" CO2 in combustion, liberates large amounts of energy per unit weight in combustion, and is easily converted to electricity by fuel cells. Biological hydrogen production has several advantages over hydrogen production by photo-electrochemical or thermo-chemical processes. Biological hydrogen production by photosynthetic microorganisms for example, requires the use of a simple solar reactor such as a transparent closed box, with low energy requirements. Electrochemical hydrogen production via solar battery-based water splitting on the [other] hand, requires the use of solar batteries with high energy requirements.
Low conversion efficiencies of biological systems can be compensated for by low energy requirements and reduced initial investment costs. Moreover, in laboratory experiments, a light energy conversion efficiency as high as 7% has been obtained using a photo-heterotrophic process (Fig. 5-1). The basic characteristics of biological hydrogen production and experiments designed to improve the feasibility of biological hydrogen production, particularly through the use of photosynthetic microorganisms, are described in this chapter. Though not described in the text of this chapter, progress has also been made in research on anaerobic fermenters (see, for example, ref. 1).
Ibid (“Ibid” refers to http://www.fao.org/docrep/w7241e/w7241e01.htm#TopOfPage always.)
Since an 850-nm photon has an energy content of 141 kJ energy (sic), the photon energy required for the production of hydrogen is 1269 kJ/mol H2 if monochromatic light of 850 nm is used. Energy is also required for the decomposition of organic substances (Equation 2-13). This energy is however, much lower (8.5 kJ/mol H2 in the case of lactic acid) than the energy required for nitrogenase-mediated reactions.
The energy of one 850-nm Einstein, that is, Avogadro’s number of 850-nm photons, is 141 kJ – not one photon.
Ibid
Hydrogen gas is seen as a future energy carrier by virtue of the fact that it is renewable, does not evolve the "greenhouse gas" CO2 in combustion, liberates large amounts of energy per unit weight in combustion, and is easily converted to electricity by fuel cells**. Biological hydrogen production has several advantages over hydrogen production by photo-electrochemical or thermo-chemical processes. Biological hydrogen production by photosynthetic microorganisms for example, requires the use of a simple solar reactor such as a transparent closed box, with low energy requirements. Electrochemical hydrogen production via solar battery-based water splitting on the hand, requires the use of solar batteries with high energy requirements.
**According to http://www.evworld.com/article.cfm?storyid=730, fuel cells have an efficiency of 40% under normal operating conditions.
Low conversion efficiencies of biological systems can be compensated for, by low energy requirements and reduced initial investment costs. Moreover, in laboratory experiments, a light energy conversion efficiency as high as 7% has been obtained using a photo-heterotrophic process (Fig. 5-1). The basic characteristics of biological hydrogen production and experiments designed to improve the feasibility of biological hydrogen production, particularly through the use of photosynthetic microorganisms, are described in this Chapter. Though not described in the text of this Chapter, progress has also been made in research on anaerobic fermenters (see, for example, ref. 1).
Solar Energy to the Combustion Chamber
Although there are few reports of precisely determined hydrogen production efficiencies of cyano-bacteria, Miyamoto et al. determined that outdoor solar incubation for a period exceeding one month in California, resulted in an average energy conversion efficiency (energy yielded by combustion of produced hydrogen/incidence solar energy) of 0.2% (6).
A key factor in determining the commercial applicability of hydrogen production processes, is the rate at which hydrogen is produced. Bacteria have been widely investigated for their rates of hydrogen production. To date, R. sphaeroides has been identified as the bacterium having the highest hydrogen-producing rate (260 ml/mg/h) (7), with a photo-energy conversion efficiency (energy yielded by combustion of produced hydrogen/incident solar energy) of 7%, determined using a solar simulator (7, 8). Further strain development will potentially elevate the energy conversion efficiency of photosynthetic bacteria to levels comparable to those of solar batteries.
Please note that this result was obtained by computer simulation, which is much less reliable than experimental evidence. (Take it from a (former) professional developer of computer simulators. See http://www.dematerialism.net/Resume97.html.)
Solar Energy to Electricity
Ibid
As stated above, [the photosynthetic reaction center] (RC) is capable of converting photo-energy into electrical energy. It can therefore be utilized in either photoelectric converters or photo-semiconductors. Solar batteries containing chromatophore membranes or an RC are now available. The RC is a relatively stable protein which retains its photoelectric conversion function even in the dry state (13). Unlike ordinary enzymes, the RC neither binds to the substrate nor undergoes chemical reactions, except for its binding to, or dissociation from quinones. Upon exposure to photons within membranes, the RC only releases electrons and has no mobile portions. Unlike enzymes, the RC does not need to vibrate within water molecules. Using the Langmuir-Blodgett method, it is possible to prepare dried RC photoelectric devices (14).
The energy efficiency of 0.07 for hydrogen production must be multiplied by 0.4, the efficiency of the fuel cell that generates electricity, according to http://www.evworld.com/article.cfm?storyid=730 to get an overall efficiency of 0.028 in the maximum case. For the average case, the overall efficiency is 0.04 · 0.4 = 0.016.
Applications
I shall assume that a controlled environment with a roof area of one hundred square meters with the following special facilities:
1) a thick canopy of leafy plants hung from the ceiling and
2) a number of bio-chemical reactors containing an appropriate mix of algae and bacteria arrayed upon the floor.
These facilities are to provide food, motor fuel, and electricity for a family of four.
Maximum Case
Food
Let us assume that the canopy of leafy plants can supply us with food. The rest of the sunlight is allowed to pass through to be absorbed in the bio-chemical reactors. We may need more than 0.4 kilowatts (kW) of chemical energy in plants to supply the 0.4 kW nutritional budget of a family of four, but we shall assume that only 0.4 kW is needed. I am not sure what happens to the light that strikes the leaves and is not absorbed. Is it degraded and, therefore, no longer available for photosynthesis? Or, is it reflected as light that has a lower fraction of PAR? I shall assume that, if the maximum efficiency at which solar energy is converted to chemical energy in biomass is 13%, the leafy canopy will absorb or dissipate [or reflect] 0.4 kW/0.13 = 3.1 kW of solar energy, which leaves 37.5 – 0.77 · 3.1 = 34.6 kW of sunlight for conversion to fuel and electricity. It is good that the incident radiation is filtered through a leafy canopy because high-intensity sunlight is not good for algae.
Note. Upon reflection (no pun intended) it seems that some of the sunlight is reflected, but I don’t know how much. The rest is tied up in the latent heat of water and various plant functions that degrade it to junk heat eventually. I have more work to do because the reflected solar energy might be significant.; but, in any case, it cannot exceed 2.7 kW, i. e., 3.1 kW minus 0.4 kW. In the average case (below), it might amount to a large fraction of 8.5 kW, though. According to http://www.water.tkk.fi/wr/kurssit/Yhd-12.135/kirja/evapo.htm, the energy balance has three terms: (1) chemical energy captured by photosynthesis, (2) the latent heat of evaporated water, and (3) reflected light, which for green leaves amounts to about 23% of the incident light and for water amounts to only 5% of the incident light. Of course, its spectrum will be quite different; however, I will assume for it the same efficiencies with respect to the energy conversions of interest. Therefore, we must add 23% of 3.1 kW to the light available to algae and bacteria. I have changed the solar power remaining after food has been harvested from 34.4 to 34.6 kW or 0.346 kW/m2.
Certainly, the leafy canopy should capture more than 0.4 kW so as to ensure a surplus and for other reasons; however, I have not reduced the solar energy available for the production of fuel and electricity because I am trying to generate a best-case scenario.
Biodiesel
According to http://www.answers.com/topic/energy-content-of-biofuel, biodiesel has a maximum energy content of 35.7 megajoules/liter, which is equivalent to 35.7 · 0.2778 / 0.264 = 37.6 kW-hours/gallon. Let us suppose that, even after the incident insolation has passed through the leafy canopy, it is sufficient to produce 0.02 kW/m2 of biodiesel. This is more than we need fortunately, because we need to devote most of the bio-chemical reactors to the production of electricity. If our family of four drives a car that gets 50 miles/gallon on biodiesel and drives only 28 miles/week, the family needs
Therefore, if we use the biodiesel power density of 0.02 kW/m2 given above, we shall require only 6.25 m2 for bio-chemical reactors to supply biodiesel. The rest of the available area can be devoted to the production of electricity. However, the 6.25 square meters must have absorbed, reflected, or degraded 6.25 m2 · 0.346 kW/m2 = 2.16 kW. According to http://www.water.tkk.fi/wr/kurssit/Yhd-12.135/kirja/evapo.htm, 5% of the incident radiation is reflected, which leaves 34.6 – 0.95 · 2.16 = 32.55 kW of sunlight. The area of incidence would be 100 – 6.25 = 93.75 m2 corresponding to 0.347 kW/m2.
If we agree that we can double the power density to 0.04 kW/m2 in a controlled environment, in which, for example, the CO2 level can be raised artificially, we require only 3.125 m2. The intercepted solar power is 1.1 kW, which leaves 33.6 kW for conversion to electricity. A different scenario that is worth pursuing, however, is using the entire floor area for bio-reactors in which biodiesel is produced. Some of the biodiesel, then, is used to generate electric power.
In the Wikipedia at http://en.wikipedia.org/wiki/Future_energy_development#Solar_power, we read, “Researchers have estimated that algae farms could convert 10 percent of the energy of incident light into biodiesel energy;” but, since there is no supporting documentation, we cannot use this figure. If we did, though, we would have a power density of 0.0344 kW/m2, which is less than our assumed value of 0.04 kW/m2; therefore, I shall continue the computation with the larger figure.
Since we shall need some floor space for ancillary equipment and access aisles, we are computing an upper bound of 34.6 kW times 0.04 or 1.384 kW. If it is converted to electricity, I must assume that it cannot do better than the best power plants, which, nowadays, achieve an efficiency of 35%. This results in 0.48 kW, which is worse than the direct conversion to electricity by bacteria. The Covertech reference has the efficiency at only 20% for conventional methods, which results in only 0.28 kW electricity, which is subject to additional electrical losses.
If we use all of the biodiesel for motor fuel, however, we might travel a maximum of (1.384/0.125) · 4 = 44 miles per day at 50 miles per gallon. If there are four drivers in the family, that is not an extraordinary amount of driving. If we are using just one vehicle for business, it is quite ordinary. Certainly it is not enough to support interstate travel. The average case (below) is much more constrained.
Electricity
According to http://www.eia.doe.gov/emeu/recs/recs2001/enduse2001/enduse2001.html, the average American household electrical usage is 1.22 kW. However, if we consume only 50% of the typical American household budget, we will consume only 0.61 kW, which will require 0.61/0.028 = 21.79 kW of incident sunlight, which will require 21.79/0.347 = 62.8 m2 of bio-chemical reactors.
Result for Maximum Case
Thus, if we enjoyed the maximum values for every quantity that varies over a large range, we would be able to supply all of the household’s direct energy requirements. Of course, the household would be importing large amounts of embodied energy in the form of depreciation on equipment, consumable supplies such as soil nutrients, etc. The average American per capita energy budget is more than 10 kW; therefore, a family of four might expect to consume 0.5 · 40 kW = 20 kW under the severe constraint of 50% conserved. As we have seen, the most we could expect to harvest by photosynthesis from 37.5 kW is 4.875 kW as chemical energy in biomass that would have to be processed further. We have done very well to have harvested 1.135 kW at the end-use stage for food, fuel, and electricity. It is assumed that energy for cooking, space and water heating, and space cooling can be harvested from the infra-red portion of the sunlight and from the anaerobic digestion of plant debris, garbage, and sewage. Electricity can be made available to assist space cooling if necessary since the average American electric bill includes air-conditioning normally.
Average Case
Food
In the average case, the total insolation is 25 kW and the average efficiency at which solar energy is converted to chemical energy in biomass is 4.5%. Therefore, photosynthesis is able to capture 0.045 · 25 kW = 1.125 kW of which 0.4 kW must be reserved for food. This removes 77% of 0.4/0.045 = 8.9 kW, which leaves 25.0 – 0.77 · 8.9 = 18.1 kW.
Biodiesel
The biodiesel requirements are the same as in the maximum case, and we require 6.25 m2 of bio-chemical reactor for algae, which must degrade 0.95 · 6.25 m2 · 0.181 kW/m2 = 1.07 kW of sunlight, which leaves approximately 17.0 kW of sunlight over an area of 93.75 m2.
Electricity
So, now, in the average case, we might get an efficiency of 0.016 or less for the conversion of solar energy directly to electricity, in which case we would have only 0.016 · 17.0 kW = 0.27 kW electricity, which is less than 21% of the current American average. This much electricity will require 17.0 kW/0.181 kW per m2 = 94 m2 of bio-chemical reactors. Thus, we shall require the a little more than 100 square meters for reactor space, which leaves nothing for anything else – even a place to walk.
Result for Average Case
Thus, for average values of the important quantities that vary over a large range, we would be on a very tight budget indeed, and 100 square meters would not be adequate for all of the ancillary activities necessary for the production of energy even, let alone living space of any kind. Moreover, the household would be importing large amounts of embodied energy in the form of depreciation on equipment, consumable supplies such as soil nutrients, etc. The average American per capita energy budget is more than 10 kW; therefore, a family of four might expect to consume 0.5 · 40 kW = 20 kW under the severe restraint of 50% conserved. Or, we could accommodate the electrical generating capacity of the average case by reducing our entire energy budget, which is five times the world average, by a factor of five across the board. Regardless of the intelligence applied to such a formidable undertaking, we would be a very poor family. As we have seen, the most we could expect to harvest by photosynthesis from 25.0 kW is 1.125 kW as chemical energy in biomass, which would have to be processed further to meet our needs. The energy for cooking, space and water heating, and space cooling could be harvested from the infra-red portion of the sunlight, if it were still needed to service an auxiliary building. It would not be possible to do anything with the greenhouse space other than growing leafy plants for food, algae for fuel, and bacteria for electricity. A separate space would be needed for household activities and the ancillary equipment necessary for the operation of a greenhouse.
Suppose, for a moment, that the entire surface of the earth (509.6 · 1012 m2) were covered by biomass that converted solar energy into useful fuel at an overall thermodynamic efficiency of 3%. The entire harvest would provide (1370/4) · (1 – 0.3) · 0.03 · 509.6 · 1012 = 3665.3 terawatts of continuous power. If the energy efficiency were only 0.2 %, the harvest would be 244.4 TW, where 1370 W/m2 is the solar constant, 4 is the ratio of the surface area to the projected area “seen” by the sun, 0.3 is the earth’s albedo, and 509.6 · 1012 m2 is the surface area of Earth.
An Objection
Of course, it might be objected that the thermodynamic efficiency of photosynthesis in a greenhouse is very much higher than the values I have found in the literature. If so, I would like to see experimental evidence that it is. In any case, it cannot exceed the theoretical quantum efficiency.
Houston, Texas
March 11, 2007
Addendum
Suppose that there are seven levels of foliage that absorb sunlight of diminishing intensity such that the energy efficiency of photosynthesis is 0.05 at the top layer but increases linearly to 0.20 at the bottom layer. At each level 23% of the radiation is reflected and finds its way to the layer below where it is the incident radiation. The efficiency rises as the intensity diminishes. This is represented by a finite series involving the fractions converted to chemical energy at each level and the fraction reflected. Please see the table below.
|
Computation of overall efficiency in a greenhouse with seven layers of foliage |
||||
|
Level |
Incident power in kW |
Efficiency |
Captured power in kW |
Reflected power in kW |
|
Top |
375.000 |
0.050 |
18.750 |
86.250 |
|
2 |
86.250 |
0.075 |
6.470 |
19.840 |
|
3 |
19.840 |
0.100 |
1.980 |
4.560 |
|
4 |
4.560 |
0.125 |
0.570 |
1.050 |
|
5 |
1.050 |
0.150 |
0.160 |
0.240 |
|
6 |
0.240 |
0.175 |
0.040 |
0.060 |
|
Bottom |
0.060 |
0.200 |
0.010 |
NA |
|
|
|
0.075 |
27.980 |
|
The sum of all of the power captured on each of the seven layers (yellow) divided by 375 (in this case) is the overall thermodynamic efficiency of 7.5% (turquoise).
March 15, 2007