The Japanese have been refining the processes for extracting Uranium from seawater for decades. The japanese have extracted 1 kg of uranium from seawater.
If 2g-U/kg-adsorbent is submerged for 60 days at a time and used 6 times, the uranium cost is calculated to be 88,000 yen/kg-U, including the cost of adsorbent production, uranium collection, and uranium purification. When 6g-U/kg-adsorbent and 20 repetitions or more becomes possible, the uranium cost reduces to 15,000 yen. This price level is equivalent to that of the highest cost of the minable uranium. The lowest cost attainable now (2006) is 25,000 yen with 4g-U/kg-adsorbent used in the sea area of Okinawa, with 18 repetitionuses.
This is about $220 per kg (114 yen to 1 US Dollar in 2007) The price of Uranium is currently in the $150-190/kg range.
The japanese process is to use irradiated polymers and stick a braided net of it into the ocean and basically “fish” for 30-90 days for Uranium.
There is 4 to 4.6 billion tons of Uranium in seawater. At the $160/kg price, this would be equal to $720 trillion for 4.5 billion tons.
Here is some more detail on the preparation of the polymer
I would propose two ways that should be investigated to radically reduce the cost of removing metals from the ocean and for increasing to very large volumes.
1. Functionalize an algae bloom to concentrate Uranium
See the work of Matt Francis at Berkeley for functionalizing virus shells and microbes for anti-cancer or for solar power. Many others are trying to engineer microbes using synthetic biology.
The goal would be to increase the concentration of Uranium from 3 parts per billion to 300 parts per million. The higher concentration allows regular methods of Uranium mining to take over. It is an increase of 100,000 times.
Scoop up the algae that concentrated the uranium and process it.
There may be a need to cross the algae with extremophiles to toughen them up or there may be a need to pick another starting microbe.
2. Nanomembrane Filtering
Nanomembrane filtering is starting to be used for desalinization of water at 100,000 gallons per day using a 6 inch diameter membrane.
If one could filter 1 billion gallons per day then there would be $1.92 million/day worth of Uranium. (3 mg per ton of water. 1 billion gallons is 4 million tons. 12,000 kg of Uranium in 1 billion gallons) Ten thousand of the 6 inch diameter nanomembrane enabled filtration pipes would be needed.
There are other rare metals in seawater. Lithium seems like a valuable target. Its price is in the $80/kg range and there is concern about having enough Lithium for lithium ion batteries. There is 56 times as much Lithium as Uranium in seawater. At $80/kg it would be a $14,000 trillion resource.