IMHO i think your numbers are off on the sodium
The cost I cited came from here:
ELEMENT: SODIUM
Metallic sodium is priced at about 15 to 20 cents/lb in quantity. Reagent grade (ACS) sodium in January 1990 cost about $35/lb. On a volume basis, it is the cheapest of all metals.
They say 4th most common element, other places say 6th. I erred on the side of caution and cited the lower number.
From the same link:
Sodium is present in fair abundance in the sun and stars. The D lines of sodium are among the most prominent in the solar spectrum. Sodium is the fourth most abundant element on earth, comprising about 2.6% of the earth's crust; it is the most abundant of the alkali group of metals.
...and i don't think you need to look at cost so much as energy in in the form of refined sodium V energy out. If you are not capturing more CO2 than is generated making the electric for the constituents - consumables of this cell your not benefiting the world!
I think there is a misunderstanding of the efficiency and the purpose. This is just my take- I may be wrong, but the 50% efficiency I read as the conversion of CO2 to Sodium Bicarb. Meaning half of the CO2 input will precipitate out, and half will stay in solution or gas off.
The only electrical requirement of the system would be the pumps. The energy comes from the chemical reaction, so assuming the cost to produce the sodium is correct, the sodium component cost would be low. The CO2 generated in producing the sodium should not that much, based on the cost of the sodium itself.
Admit I am engaging in some speculation there.
The beauty of this idea (to me) is that it is not about generating power. It is about sequestering CO2 in an efficient manner, and it has the happy by-product of producing useful stuff in the process. I don't really care if it is a net winner or loser wrt the power generated as long as the losses are lower than the cost of current CO2 sequestration technology.
What I like best is that it generates no waste products. The Na ions used in the conversion are essentially free, the other input (C02) is a waste product that we want to get rid of anyway.
So the calculus becomes this: Does the cost of the process, minus the revenue generated by the production of electricity, hydrogen, and sodium bicarb, exceed the cost of current CO2 sequestration technologies?
I don't know that answer- but the elegance of the design and simplicity of the process leads me to believe that is would not only be less expensive than what we do now, it could potentially turn a tidy profit.
I was a Chemistry major, so I may be biased, but I really like this idea.