Biofuels take flight
Sometimes we get so wrapped up in the fight to bring alternative fuels to our cars and trucks that we forget about another mainstream vehicle that is trapped by dependence on oil — planes.
Sometimes we get so wrapped up in the fight to bring alternative fuels to our cars and trucks that we forget about another mainstream vehicle that is trapped by dependence on oil — planes.
John Hofmeister went on CNBC on Thursday to talk about the current and future price of oil, and he gave viewers more to think about when he elevated the discussion to include the need for alternatives to meet the world’s growing appetite for transportation fuel.
United Airlines took a giant step toward cutting its reliance on foreign fuels last week when it made $30 million investment in Fulcrum BioEnergy, one of the leading manufacturers of aviation biofuels made from municipal waste.
The move is being touted as a step toward reducing carbon emissions, although there are some doubts about its impact in that respect. But reducing consumption of jet fuel certainly will have a significant effect in reducing our dependence on foreign oil.
Last year, United’s fleet of aircraft consumed 3.9 billion gallons of jet fuel, at a cost of $11.6 billion. Fuel costs represent 40 percent of any airline’s total expenses, and any move that cuts into that expense would be huge. Jet fuel currently sells for $2.11 a gallon, whereas Fulcrum says it can provide biofuel for less than $1 per gallon. More than 12 percent of our oil goes to making jet fuel.
Fulcrum has developed and certified a technology that can turn municipal waste, like household trash, into a sustainable aviation fuel that can be blended with existing jet fuel. The company is currently building a refinery called the Sierra BioFuels Plant near Reno that is scheduled to begin operation during the third quarter of 2017. The company also has plans for five more refineries around the country.
Biofuels are having some difficulty penetrating the automobile market, for a variety of reasons. But they’re perfectly suited for airlines. For one, they are a “drop-in” fuel that can be substituted for jet fuel without any changes. It will not require a whole new national infrastructure.
Second, airlines do most of their fueling at centralized locations. This eliminates a lot of difficulty in transporting and distributing the fuel. United, for instance, can fuel a very high percentage of its flights from its hub in Los Angeles.
Third, with jet biofuel there’s no risk of hitting the “blend wall” that supposedly limits ethanol to 10 percent of the gasoline mix. United says it will begin using Fulcrum’s fuel in 30 percent of its fuel mix for the first two weeks of flights between Los Angeles and San Francisco this summer. After that, the biofuels will be mixed in with its entire fuel stock.
United’s deal with Fulcrum is just one of several recent efforts by airlines to get into the biofuels business. Alaska Airlines aims to use biofuels at one of its airports by 2020. Southwest Airlines announced last year it would purchase 3 million gallons of jet fuel made from wood residues and produced by Red Rock Biofuels. And last year British Airways joined with Solena Fuels to build a biofuel refinery near London’s Heathrow Airport for completion by 2017.
United is on its third venture into the field. In 2009 the company made an unsuccessful attempt to introduce jet fuel manufactured from algae. Then in 2013 it agreed to buy 15 million gallons over three years from California-based AltAir Fuels, which makes biofuels out of inedible natural oils and agricultural waste. United is expecting the first 5 million gallons of Fulcrum fuel to be delivered to its LAX hub this summer.
The decision comes at a good time for the airlines, because the Environmental Protection Agency is starting to make noise about regulating the emissions of jet planes. Jet planes account for only 3 percent of our carbon emissions, but the number is growing rapidly. The Obama administration is proposing to set limits for airliner emissions. The International Civil Aviation Organization, a United Nations agency, is also expected to complete its own deliberations on setting standards to limit airline emissions by next February.
Fulcrum claims its technology will reduce the airlines’ carbon emissions by 80 percent, but this is based on dubious math that says carbon emissions count for zero if they do not come from fossil fuels. This premise has been challenged by a growing number of scientists who say that the whole logic of biofuels is flawed. Professor Timothy Searchinger of Princeton University has become a gadfly to the industry, arguing that if a forest is cut for biofuels consumption, it will be 90 years before this carbon can be replaced by new growth. A group of 78 scientists recently sent a letter to EPA Administrator Gina McCarthy warning against the new EPA policy of encouraging the substitution of wood for coal. They said there would be no savings in carbon emissions.
The same logic applies, to some degree, to the use of municipal waste for biofuels. If the waste remained in landfills, it would be stored and not feeding its carbon content to the atmosphere. Therefore, it doesn’t make much difference if they are substituted for fossil fuels – the carbon output is the same. There is some benefit to using it, however, since some carbon from municipal waste ends up escaping from landfills as methane, and many facilities are required to capture it.
As far as gaining an advantage in cutting the level of foreign fuel imports, however, there is no question that biofuels can substitute for jet fuel on a 1-to-1 basis. Airlines are at a disadvantage in that they cannot be powered by electrification or natural gas, as is starting to occur in the automotive sector. Therefore, the amount of municipal waste-based fuel that can be substituted for oil-based jet fuel will be significant. And after all, the nation is certainly not going to run out of household trash.
(Photo from Hub.United.com)
Tesla is trying to convert the world to the electric car. The Japanese are pushing hydrogen. But Audi, the German carmaker, has a different idea. It’s trying to synthesize fuel from the simplest of elements – water, carbon dioxide and solar energy.
Audi’s research facility in Dresden has produced what the company calls an e-diesel – a net-zero-carbon-footprint fuel made from carbon dioxide and water. The company announced the project to great fanfare on April 21. In May, it unveiled another advance – e-benzine, a fuel that acts just like gasoline.
The two are the latest of a suite of six fuels developed by Audi that behave just like traditional gasoline or diesel, but burn without releasing any sulfur or aromatic hydrocarbons, the stuff that produce air pollution. The fuels also can be labeled as carbon-neutral, since the carbon dioxide they’re removing from the atmosphere perfectly matches the CO2 they put back in when they burn. E-benzine currently derives its carbon from organic material – biofuels made from rapeseed, sunflower oil or corn. But Audi officials say they soon hope to switch to atmospheric carbon dioxide.
“To me, this is a historic moment,” said Marc Delcourt, CEO of Global Bioenergies, the French company that is partnering with Audi on the e-benzine project. “It is the first time that we have produced real gasoline from plants.”
The e-diesel process works like this: Audi begins by splitting water by electrolysis into hydrogen and oxygen. The electricity is provided by wind or solar energy, which makes it completely fossil-fuel free. The oxygen is released into the atmosphere. Meanwhile, Audi filters carbon dioxide out of the atmosphere. The C02 is stripped down to carbon monoxide, and the CO and hydrogen are then mixed together under high pressure to produce a long-chained hydrocarbon that Audi calls “blue crude.” It has all the properties of crude oil and can be refined down to commercial fuels like e-diesel. “We’re thinking we’re bringing green-ness to a field that desperately needs green-ness,” said Rick Bockrath, vice president for chemical engineering at Global Bioenergies. “It’s basically how we’re moving away from an oil-based economy towards something that has a renewable, sustainable future to it.”
Johanna Wanka, Germany’s Minister of Education and Research, attended the ceremony at which the first batch of Audi e-diesel, five liters’ worth, was put into her official car, an Audi A8 3.0 TDI clean diesel Quattro (that’s her in the photo above). “This synthetic diesel, made using CO2, is a huge success for our sustainability research,” she said. “If we can make widespread use of CO2 as a raw material, we will make a crucial contribution to climate protection and the efficient use of resources, and put the fundamentals of the ‘green economy’ in place.”
The product has a 100 octane rating and can be used either as an additive or as a stand-alone fuel. Audi says cars run much smoother on the product because of the lack of aromatic compounds, sulfur and other impurities. It also converts to energy at 70 percent efficiency, which is much better than regular diesels.
Audi’s pilot project in Dresden is currently producing 160 liters of e-diesel per day. Obviously, that isn’t enough to shake the world. But the long-term plan is to scale up to a level that will make the product available to the public. The estimated price will be 1 to 1.5 euros per liter, which comes to about $3.75 per gallon. This would not offer any price advantage in the United States, where diesel is selling at $2.88 per gallon, but it would be competitive in Europe, where diesel currently sells for about 1.4 euros per liter.
The problem with all such inventions, of course, is whether they can scale up at a price that remains competitive. Robert Rapier, the highly respected energy analyst, is skeptical. In a lengthy piece in GreentechMedia, Rapier did a step-by-step analysis, including all the chemical reactions. He concluded that the price is going to be $3.76 per gallon, which would put it above the current price of diesel in the United States, but perhaps not in Europe. But that doesn’t include any price increases that may come with scaling up the process. In addition, several critics have wondered whether solar and wind electricity will be available on a scale capable of supporting such a commercial operation.
“To sum up, can Audi produce fuel from thin air? Sure. There is no question about technical viability,” Rapier wrote. But “The question boils down to economic viability, which appears to be challenging given what has been released about the process.”
All this doesn’t mean Audi shouldn’t continue experimenting. There’s always room for improvement, and there may be other breakthroughs down the road. A carbon tax would also benefit the process, particularly if Audi could be given credit for the carbon it takes out of the atmosphere. There is also the possibility of combining the procedure with a carbon-capture and storage operation at a fossil-fuel plant, where carbon dioxide is currently regarded as a noxious waste material.
A system that would manufacture automotive fuel out of carbon dioxide in the atmosphere would be like the philosopher’s stone of the transport sector. Audi should keep trying.
(Photo credit: Audi)
Biofuels have been taking their lumps lately. After almost seven years of controversy, the European Parliament has acted to limit the amount of biofuels that can be garnered from land that could be used to grow food.
The EU has set itself a goal of getting 10 percent of its transport fuel from biofuels by 2020. Last week the Parliament voted to reduce this to 7 percent. The concern is that biofuels are taking food out of people’s mouths. Biofuels are also accused of leading to deforestation, both in Europe and in countries such as Brazil and Argentina, where Amazon rainforest and Argentinian pampas are being put under cultivation for growing biofuels for export.
“Let no one be in doubt, the biofuels bubble has burst,” Robbie Blake of Friends of the Earth Europe said in a statement. “These fuels do more harm than good for people, the environment and the climate. The EU’s long-awaited move to put the brakes on biofuels is a clear signal to the rest of the world that this is a false solution to the climate crisis. This must spark the end of burning food for fuel.”
Ironically, it was soft-energy guru Amory Lovins, who at the time was British representative of Friends of the Earth, who originally suggested the biofuels idea in his 1976 book, Soft Energy Paths. Lovins used an elaborate comparison with the beer and wine industry to show that it would be possible to produce a good one-third of the United States’ gasoline requirements through biofuels. Unfortunately, Lovins did not take account of the amount of land that would be required to grow these crops. This oversight has dogged the biofuels effort ever since.
In the U.S., criticism is mounting as well. A study published last month by researchers at the University of Wisconsin-Madison shows that corn and soy crops for biofuels are expanding into previously un-farmed prairie land in the Midwest. Using high-resolution satellite photographs, the authors identified the expansion of cropland from 2008 to 2012, the four years following the passage of the Renewable Fuels Act that mandated the use of biofuels. The authors estimate that 40 percent of the corn crop grown in the U.S. is now used to make ethanol for use in vehicles. Ironically, environmentalists who originally celebrated ethanol are among its biggest detractors.
So does this mean that American biofuels will soon be facing the same limitations they’ve encountered in Europe? Probably not. The reason, once again, is technology.
From the beginning, the dream of biofuels enthusiasts has been that ways could be found for breaking down the refractory cellulose molecule and turning it into basic sugars that can be synthesized into ethanol. This is a very difficult task. It can only be accomplished in two ways: 1) heating corn stover and other cellulosic materials to a very high temperature, which consumes more energy than is produced; and 2) taking advantage of bacteria in the guts of cows and termites that can break down cellulose. These bacteria are highly temperamental, however, and have proved to be extremely difficult to cultivate on a commercial scale.
Nevertheless, progress has been made, and there are several commercial operations now approaching successful operations. Among them are:
• Abengoa Bioenergy (Hugoton, Kansas). This Spanish company’s cellulosic-ethanol facility came online in 2014 and is expected to produce 25 million gallons per year from corn stover, wheat straw, milo stubble and switchgrass.
• DuPont (Nevada, Iowa). Its 30 million-gallon-per-year cellulosic plant is scheduled to begin production this year. The plant will get corn stover from 500 farmers who are participating in the company’s Feedstock Harvest Program.
• Poet-DSM Advanced Biofuels (Emmetsburg, Iowa). Co-funded by a Dutch company, Project Liberty opened in September 2014 and is producing ethanol from corn cobs, leaves, husk and stalk. It is shooting for 25 MMGY.
• Quad County Corn Processors (Galva, Iowa) started production last year. Its Quad County facility can produce 2MMGY. The company says its patented technology has the ability to generate 1 billion gallons per year, without consuming any more corn, by adding bolt-on technology to existing corn-ethanol refineries.
So ethanol is not standing still. The EPA is expected to issue its renewable fuel standard sometime next month, after dodging the issue for two years. The threshold likely will be below the 14 billion gallons that was originally scheduled for 2014. But the law’s requirement for Gen-2 biofuels has barely been scratched, since these cellulose efforts have not borne fruit to date. With cellulosic operations now gearing up, it appears that ethanol may be ready to take on a second life.
(Photo: Corn-stover harvest. Posted to Flickr by Idaho National Laboratory)
Jim Lane, editor and publisher of Biofuels Digest, is one person who thinks alternative fuels aren’t necessarily going to be hurt by the huge drop in the price of crude oil.
In a post on the Digest Jan. 6, Lane lays out the rather complicated case of why it doesn’t pay right now to be dumping your alternate-energy stocks. That’s been the reaction so far to anything related to the price of oil. But Lane says there are special aspects of alternatives like ethanol that will be affected in a different way.
In the first place, Lane notes that while crude oil prices have been falling, ethanol prices have been falling, too. Since last June, crude oil has fallen from $115 a barrel to under $50, a remarkable 60 percent drop. Yet ethanol has fallen as well, from $2.13 a gallon to $1.55 a gallon, a formidable 27 percent drop. This is due mainly to the falling price of corn, which has been at its lowest level in recent years. A bushel of corn fell over the same period from $4.19 a bushel to $3.78, a 10 percent drop. In this way, ethanol is only marginally dependent on the price of oil and can show its own price pattern.
One thing worth noting is that there is a certain amount of elasticity in American driving. People tend to increase their driving range when the price of gasoline goes down. This is particularly true when it comes to taking vacations, which tend to be a long-term planning effort. If the price of gasoline stays down through next summer, people are more likely to increase gas consumption. The fact is that gasoline demand has actually reached its highest point in the last few months since the price of oil began to fall, as the following graph indicates:
Now drivers are required to include 10 percent ethanol in each gallon of gas. Therefore, ethanol has a fixed market. Driving has been declining in recent years, which is one reason that the Renewable Fuel Standard has been under fire – because the absolute amount of ethanol required has exceeded the 10 percent requirement in relation to the amount of gasoline consumed. Refiners and oil companies must buy this amount of ethanol. This is the reason the Environmental Protection Agency has been holding back on setting an RFS for 2014 — because the original amount prescribed was going to exceed the 10 percent figure. If people start taking advantage of lower gas prices and start consuming more gasoline, the amount of ethanol required will grow. “(W)e should be seeing a 2+% increase in gasoline demand, and that will take some pressure off the ethanol blend wall,” Lane writes. It might make EPA’s decision easier, if it ever gets around to setting a number.
Just to emphasize this point, an RIN — Renewable Index Number — is required by the EPA to prove that a refinery has been adding ethanol up to the 10 percent mark. The price of RINs has actually been rising as gas prices have fallen. As Lane writes: “Part of the reason that the ethanol market is holding up relatively well in tough times is the impact of the Renewable Fuel Standard, and its traded RIN system. RIN prices have jumped as oil prices have slumped — and a $0.76 increase in the RIN value of a gallon of fuel is a striking increase in value.”
So all is not dark for the future of alternatives. Ethanol’s place is secure, despite the fall in gasoline prices. Remember, it’s not that demand for gas is falling, but people are spending less for what they get. If methanol is given a chance, it might turn out to be more invulnerable, since it’s not tied to corn prices but to natural gas, which we seem to have in even greater abundance than oil. Electric cars also don’t lose their appeal, since much of their appeal is getting off gas entirely and unbuckling from the oil companies. It may not be time to abandon your stock in alternative energies quite yet.
The federal government’s new threshold for the amount of ethanol blended into America’s gasoline supply was already 10 months overdue. So officials have gone ahead and delayed the decision further, into 2015.
The Environmental Protection Agency announced Friday that it would defer an announcement on the renewable fuel standard (RFS), which stipulates that ethanol should make up 10 percent of gasoline.
(The Des Moines Register has some of the day’s best reporting on this issue. Agriculture.com also has a good explanation of the granular details.)
The standard, first established under a 2005 law, calls for the amount of renewable fuels in gasoline to progressively increase each year. But the law was written at a time when demand for gasoline was expected to keep going up. Slackened demand around the world, combined with stepped-up U.S. production, has dropped domestic prices below $3 a gallon.
Based on that reality, the EPA recommended, in November 2013, that the amount of corn ethanol in the should be reduced, from 14.4 billion gallons a year to 13.01 billion gallons.
This upset the corn growers and ethanol producers, most of them clustered in the Midwest and Great Plains. They said the delays deterred investment in biofuels, and even the oil companies complained that the regulatory vacuum created too much uncertainty in the fuels market.
The EPA’s recommendations had not been finalized. They had been sent to the White House Office of Budget and Management for review, but that office “ran out the 90-day clock to review the agency’s proposed standards, which for the first time signaled a retreat by the EPA on the percentage of biofuels that must be blended,” The Hill reported.
Since the EPA was already so late in setting the 2014 guidelines, the agency “intends to get back on track next year, though details on how it would do that weren’t available Friday,” The Wall Street Journal wrote. The EPA statement said: “Looking forward, one of EPA’s objectives is to get back on the annual statutory timeline by addressing 2014, 2015, and 2016 standards in the next calendar year.”
The reaction among the affected parties was mixed Friday. The WSJ tries to untangle the various interests:
The debate over the biofuels mandate triggers strange bedfellows, with trade groups representing the oil and refining companies, car manufacturers, livestock and even some environmental interests all opposed to the policy for different reasons. Proponents of the standard include the corn industry, which is the most common way ethanol is produced, and producers of ethanol.
The EPA’s announcement gave cautious hope to ethanol-industry leaders that the agency will fundamentally rethink how it proposes the annual biofuels levels. The draft 2014 biofuels levels, which the agency proposed almost a year ago, were much lower than the ethanol industry lobbied for.
“I am truly pleased that they’re pulling away from a rule that was so bad,” said Bob Dinneen, president and CEO of the Renewable Fuels Association, a trade group representing biofuels companies. “But I recognize as well we have to work with the agency to try to figure out a path forward that everybody can live with.”
Executives in the oil-refining industry criticized the delay, and said it was evidence the renewable-fuel standard was itself inherently flawed and should be repealed.
“Each year is dependent upon the previous year, and to some extent dependent upon the following year,” said Charlie Drevna, president of the American Fuel & Petrochemical Manufacturers, a trade association representing the nation’s refining industry. “The problem is, every year EPA is late in getting this out, it exacerbates it. They’re never going to be able to catch up.”
The lure of the oceans has always had a special appeal for advocates of biofuel. The vast reaches of the deep speak of a promise that unlimited amounts of space will be able to bring forth completely sustainable forms of energy.
“Two-thirds of the globe is covered with water,” says Khanh-Quang Tran, a Norwegian researcher who has published papers on the possibility of growing algae as a biofuel on an industrial basis. “If we used only a tiny portion of that space, we’d have enough to supply ourselves with all the fuel we needed.”
Of particular interest to researchers is one species, laminaria sacceyarina (“sugar kelp”), which grows along the coast of many countries, including Scandinavia. It is the “seaweed” that seems to be a flower but is actually all one undifferentiated cellular structure that takes on various forms in competing for sunlight. As the name implies, it contains lots of sugar – three times as much as the sugar beet. Scandinavian scientists have been especially intent on harvesting this plant for food and fuel use.
“It’s actually regarded as a nuisance, since it grows everywhere and clogs the beaches,” says Fredrik Grondahl, a researcher at the KTH Royal Institute of Technology in Sweden who heads the Seafarm project. “But it absorbs nitrogen out of the water, effectively as a wastewater treatment plant. It’s regarded as an environmental problem, but it’s actually a valuable resource.”
The big question will be this: Can a weed that grows so prolifically in the sea be domesticated so that it can grow in large quantities under controlled conditions?
Sweden and Norway seem to have taken the lead on this project, mainly because of their long coastlines, where the algae grows intensely in a cold climate. The Seafarm project involves growing underwater algae farms on ropes. The team collects excess algae from the Baltic Sea and cultivates it as food and fuel. One technique is called the “sporophyte factory farm.” The algae spores are sown onto ropes. They sink and grow in the sea. In about six months, they have grown onto the ropes and are harvested and processed on land covering two hectares. From there it can be converted to eco-friendly food, medicine, plastics and energy fuels such as methanol. The city of Trelleborg, where the farm is located, estimates that 2.8 million liters of fuel can be extracted from its algae resources.
Kahnh-Quang Tran of Norway has been following another line of research. He mixes a slurry of kelp biomass and water and heats it rapidly to 350 degrees Centigrade. Tran says the fast hydrothermal liquefaction gives him a product that is 79 percent bio-oil. A similar experiment on the U.K. was only able to produce 19 percent oil, but Tran claims that the rapid heating improves the process tremendously. “What we are trying to do it mimic the natural process that produces oil,” he said. “Whereas it takes geological time in nature to produce oil, we can do it in a matter of minutes.”
Tran is now looking for partners who can help him move up to an industrial scale.
Another plan developed in France and the Netherlands is to line highways with algae pools in the hope that they will immediately absorb the carbon exhaust that comes from automobiles. This will remove CO2 from the atmosphere and recycle the fuel as well. An experimental installation was demonstrated at the summer garden festival at Genève Villes et Champs this year.
Another country that is experimenting with algae is Australia. This October, the Muradel Corporation opened a $101. 7 million demonstration plant in Whyalla designed to produce 30,000 liters of green crude every year. The company is employing its Greeen2Black technology, designed to produce a continuous stream of environmentally sustainable crude equivalent.
Muradel CEO and University of Adelaide Associate Professor David Lewis said if the demonstration plant were successfully scaled to a commercial plant, it would produce 500,000 barrels of refinable green crude a year by 2019 – enough petrol and diesel to fuel 30,000 vehicles for a year. The planned 1,000-hectare commercial plant would create at least 100 new skilled jobs in the Whyalla region.
“This is world-leading technology which can be scaled up exponentially to help steer our fossil fuel-dependent economy toward a more sustainable future,” Lewis said.
Not everyone is enthusiastic about algae. “It will take anywhere from 5 to 15 years to produce on a scale that would be meaningful to the nation’s every needs,” says Jim Rekoske, general manager of Honeywell’s UOP division. He likened it to trying to maintain the water balance in a fish tank.
“You have to have just the right temperature and the right amount of carbon dioxide to get these growth spurts,” he said. Algae farms are also very susceptible to invasive species and have to be monitored constantly. Still, an acre of algae can ideally produce 15,000 gallons of biofuel per year, as opposed to only 420 gallons per acre from corn ethanol. “We could replace all the diesel we consume now on half of 1 percent of our current farmland,” says Douglas Henston, CEO of Solix Biosystems of Fort Collins, Colo. Solix is supplying the military with biofuels at a whopping $33 per gallon.
So, will algae make the same progress in the United State that it seems to be making in Sweden and Norway? American researchers may take up the challenge as well. The long coastal lines are not there to tempt us, but research breakthroughs may finally make algae biofuels more practical and economically viable everywhere.
Scientists are using biotechnology to chip away at barriers to producing biofuels from woody plants and grasses instead of the corn and sugarcane used to make ethanol. NC State’s Forest Biotechnology Group, which has been responsible for several research milestones published this year, summed up biofuel research progress and challenges for a special issue of the Plant Biotechnology Journal.
Read more at: Phys
Image rights: Phys.org
While the debate rages about what the threshold for biofuels should be in the government’s next (and long-delayed) Renewable Fuel Standard, Breaking Energy’s Jared Anderson has a timely post about the makeup of the current RFS, as it was proposed by the EPA last November.
There are thresholds within the larger thresholds, and it looks like the cellulosic ethanol target will go down. But as Anderson notes:
“While the battle over the RFS continues, the cellulosic ethanol industry took a major step forward today with the inauguration of a commercial-scale plant in Hugoton, Kansas. The biorefinery has the capacity to produce 25 million gallons of cellulosic ethanol per year, which alone exceeds EPA’s proposed 17 mm gallon blending target under RFS. The plant also generates 25 MW of electricity, which supplies its own needs and provides excess power to the local community.”
Anderson signs off with:
“The RFS will remain controversial, but this new plant is a big win for the cellulosic ethanol portion of the equation.”
(Photo credit: Shutterstock)