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.
Making synthetic fuel via the Fischer–Tropsch 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. World consumption of oil is around 100 million bbl/day so it would take around 3000 of these plants to replace 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
It's not obvious this is the best way to go. It's possible to electrolyze CO2 into oxygen and carbon monoxide. If this is less costly (energy and capital) than making the extra hydrogen, that's the way to go.
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 half a ton, a billion 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 presumable 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.
Hydrogne 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. 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/3MWh or about 56%. But you are turning inexpensive electrical energy that can't be stored into expensive fuel which can be stored and shipped, so it might make a profit.
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. The CO2 separation at a GJ/ton would take about 145 MW, about 3.5% 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, 10 hours per day, will have to run at 24/10 times this rate or 82.8 * 2.4 * 50 MW or 10 GW. 10 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 30 Oryx-sized plants.
The waste heat from electrolysis would be 2 GW. The total waste heat during the day would be 2.9 GW.
How much will 2.9 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 2.9 GW will heat a 1.2 cubic km of air per hour by about 7.3 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 draw 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. At $13.50 per MW and 50 MWh/ton, the hydrogen energy cost for 14 tons of fuel would be $4050. This is a bit less than $40/bbl.
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 degraded 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. 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.
At 50 bar hydrogen is about 5 kg/m^3 or 200 cubic meters per ton. 1500 tons of hydrogen at this pressure would take 300,000 cubic meters. As a sphere that's 83 m in diameter. The bisected area would be 5410 m^2 and the force would be area * 100,000 Ps * 50 or ~27 billion newtons. That's spread along the circumference of ~260 m. Per meter, it would be 104 million newtons. For steel stressed at 200 million Pa, the shell would be half a meter thick and with an area of 21642 square meters would mass around 86,000 tons, close to the mass of an aircraft carrier.
This is not impossible. Some steels can be stressed 5 times as much, but another solution such as an empty gas field would be more economical. If there is excess CO2, it could be mixed with the hydrogen.
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 as 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