Cheap synthetic fuel
Chemistry; and Math-heavy Draft
Hydrocarbon fuels are just too useful as portable energy sources not to mention being the basis for plastics and lubricants. All serious energy proposals (such as StratoSolar or power satellites) include making some synthetic hydrocarbons. Even without considering climate problems, natural oil will eventually run out. The problem is making synthetic fuel for a cost close to the current price of oil. (Design-to-cost target of $100/bbl.)
Making synthetic fuel via the Fischer–Tropsch (F/T) reaction is well developed. Sasol has a plant in Qatar (Oryx GTL) that makes 34,000 bbl/day from reformed natural gas. It has been running since 2007 and cost a billion dollars to build. World consumption of oil is around 100 million bbl/day so it would take around 3000 of these plants to replace the current use of natural oil. That's a large number, but not impossible.
The Sasol plant uses reformed natural gas as a feedstock, made into carbon monoxide and hydrogen. Synthetic fuel is roughly CH2. If you incorporate the step of reducing CO2 to CO, the overall reaction is:
CO2 + 3H2 yields CH2 + 2H2O
44 + 6 -> 14 + 36 (mols, kg, tons, etc.)
It's possible to electrolyze CO2 into oxygen and carbon monoxide. Unfortunately, this is more costly in energy than making the extra hydrogen. (161 kWh to 100 kWh) Still, if the CO2 electrolyzers are less expensive than the hydrogen electrolyzers, they may be worth investigating.
34,000 bbl/day is 1417 bbl/hr. At 7.33 bbl/ton, 193 tons of hydrocarbons per hour of which 12/14 or 166 tons per hour is carbon The CO2 input for an Oryx-sized plant would be 44/12 times this figure or 607 tons per hour.
Air is about 1.2 kg/m^3 at 20 deg C and about 418 ppm CO2. A cubic meter contains close to half a gram of CO2, 1000 cubic meters about half a kg, a million cubic meters half a ton, a billion (a cubic km) 500 tons. Depending on the separation efficiency, upwards of 1,2 cubic km of air will have to be processed per hour to feed an Oryx-sized plant.
This is incidentally about 5.3 million tons of CO2 per year. It would presumably displace oil that put 5.3 million tons per year of CO2 in the atmosphere. If the X-prise people could be talked into crediting this, it would qualify because oil left in the ground would certainly stay there for more than 100 years.
Hydrogen and Energy
The hydrogen would be 6/14 of 193 tons or 82.8 tons per hour. 2/3rd of the hydrogen combines with oxygen from the CO2 making water. At 50 MWh/ton, this requires about 4.14 GW to make the hydrogen.
The energy in hydrogen is about 40 MWh/ton (at 80% electrolysis efficiency). At 1.7 MWh/bbl (12.46 MWh/ton) for the fuel, the F/T efficiency is 193t*12.46 MWh/ton/(82.8t*40 MWh/ton) = 72.6%. The waste heat from the reaction is about 907 MW) The electricity to fuel efficiency would be 1.7MWh/bbl of fuel /2.92MWh per bbl input or about 58%. But you are turning inexpensive electrical energy that can't be stored into valuable fuel which can be stored and shipped, so it might make a profit, especially in places trying to reach carbon neutrality (like California).
Waste Heat and a Possible Use
Combining CO and H2 is exothermic. As above, 27 percent of the energy in the feedstock has to be removed from the reaction vessel as waste heat. Some of this can be recovered with steam turbines, though the temperature is not high enough for good efficiency. However, the waste heat can be used to drive a chimney which sucks this huge volume of air through the CO2 separators and exhaust it far above the separator intake.
Verdox, the company developing https://news.mit.edu/2019/mit-engineers-develop-new-way-remove-carbon-dioxide-air-1025 says to assume 5 m/s gas velocity and 200 Pa pressure drop through the CO2 separators. A square meter of intake at 5 m/s will admit 5*3600 m^3/hr. The input area for one cubic km/hr at 5 m/s would be about 60,000 square meters. At 30 m high, this would be a wall 2000 meters long. Bent in a circle around the chimney, this would be about 640 meters in diameter tapering into a 180 m diameter chimney. The CO2 separation at a GJ/ton would take about 169 MW, about 4% of the power needed for hydrogen production.
During the day, the waste heat from the electrolysis can be added to the F/T waste heat. The hydrogen production of 82.8 tons per hour has a waste heat of 10 MW per ton. To provide enough hydrogen for the F/T to run full time, the hydrogen production, while the sun is up (an average of 9 hours a day for Qatar), will have to run at 24/9 times this rate or 82.8 * 24/9 * 50 MW or 11 GW. 11 GW is a large block of power, but the Saudis have signed a deal for 300 GW of solar by 2030. That's enough for 27 Oryx-sized plants. That may be what they have in mind.
The waste heat from electrolysis would be 2.2 GW. The total waste heat during the day would be 3.1 GW.
How much will 3.1 GW heat the air going up the stack? At 300 K (27 C), the specific heat of air is 1.005 kJ/kg⋅K. I.e., a kW-s (kJ) will heat a kg of air close to a deg K. A GWh is 3.6 billion kW-s, a cubic km of air has a mass of 1.2 billion kg. Thus, a GW will heat 1.2 cubic km of air per hour by 2.5 deg K and 3.1 GW will heat a 1.2 cubic km of air per hour by about 7.75 deg K. Nighttime heating will be much lower at around 2.3 deg or the flow could be reduced and most CO2 extracted during the day.
From https://en.wikipedia.org/wiki/Flue-gas_stack the stack will need to be about 180 m in diameter and a km tall to get a flow of a cubic km per hour. The velocity up the stack will be 11.5 m/s and the delta pressure 171 Pa. Wall friction for this size turns out to be about 0.2 Pa for a 1000-meter stack. The air going up the stack has been stripped of CO2, making it slightly lighter. The stripped CO2 at half a gr/cubic meter or half a kg in 1000 m would lower the pressure by 5 newtons per square meter or 5 Pa.
The flow up the stack would be restricted by the pressure drop of 200 Pa in the CO2 separators. An extra 1000 meters of the stack would about compensate. The lower stack flow at night might require fans to provide the pressure.
PV power in the Mideast has fallen to as low as 1.35 cents per kWh at the 900 MW scale. It may fall to less than a cent. Making 14 tons of fuel takes 6 tons of hydrogen (2/3rd of it combining with the oxygen from the CO2_. At $13.50 per MWh and 50 MWh/ton, the hydrogen energy cost for 14 tons of fuel would be $4050. This is a bit less than $40/bbl.
If you write the billion-dollar Oryx plant down in ten years, the capital cost is about $8/bbl.
The current electrolyzer cost can be found here https://www.hydrogen.energy.gov/pdfs/19009_h2_production_cost_pem_electrolysis_2019.pdf. At the cited, $1000/kW, that is a billion dollars per GW. It would be 4 billion if the power was on all the time, but (since it is intermittent PV) it is more like 11 billion dollars when you are making hydrogen only in the daytime.
need to fix here. cost per bbl is ok, need better number on capital
At 82.8 tons per hour, the production of hydrogen would be about 725,000 tons per year, 7.25 million tons over 10 years. If the capital cost is $11 B over ten years the cost is about 88 dollars per bbl. This is over twice the cost of energy consumption. The cost of electrolyzers will have to drop by a factor of ten for the per bbl capital cost of the hydrogen plant to reach the same as the per bbl capital for the F/T plant. At this scale, such a cost reduction seems possible.
The capital cost for the chimney is unknown at this time. 6/22/21
Verdox expects the cost of the CO2 separators to be no more than $800 per ton per hour. That's less than half a million dollars, which in this context is negligible.
The CO2 separator can run all the time, but there must be storage for hydrogen while the sun is not shining. 3/4 of a day at 83 tons per hour is almost 1500 tons of hydrogen.
An above-ground tank for the hydrogen would mass as much as an aircraft carrier.
Another solution such as an empty gas field would be much more economical. If there is excess CO2, it could be mixed with the hydrogen and the feed to the F/T reactor modulated to get the correct ratio.
A PV farm producing hydrogen for an Oryx-sized plant would be big. At a GW/km^2, 4 GW would take 4 square km. This has to be increased by about 5 for the PV efficiency and another factor of 4-5 for the peak to average. This means the PV farm would cover around 90 square km of land.
It may be worth considering having the PV farm deliver ~1000 V DC. This would take some huge conductors, at 1000 V, a GW is a million amps. Aluminum smelters use 250,000 A. The conductors (from memory) were about 3 inches by 10 inches.
Other useful links
The take-home is that in some places PV has gotten so inexpensive that it would be possible to make carbon-neutral synthetic hydrocarbons to replace natural oil for about the same price.
The U.S. Internal Revenue Service defines a BOE is equal to 5.8 million BTU. (5.8×106 BTU59°F equals 6.1178632×109 J, about 6.1 GJ [HHV], or about 1.7 MWh