Very cool!
It sounds like your sorbent requires calcium to precipitate calcium carbonate from sea water. What would be the source of this calcium? Hopefully it won't come from reducing lime!
How much seawater do you need to get one kg of CaCO3 precipitate?
Assuming that the new process stochiometrically generates CaO according to the amount of CO2 taken up from the ocean you'd produce 1.27 tonnes of CaO per tonne of CO2. (molar masses of CO2 and CaO: 44 g/Mol and 56 g/Mol)
Cementa website states [1] an annual output of 2.7 million tonnes of cement.
So in order to produce what is only a portion of Sweden's (a comparatively small country) annual demand for cement, Heimdal's process would require processing of ~ 23 billion tonnes of seawater.
That's 23 billion cubic metres or 23 cubic kilometres per year or ~ 730 litres of seawater per second.
But let's not forget that it's not about merely pumping the water around. All this water must be processed. And I figure the process is a little more involved than simple reverse osmosis for seawater desalination.
In the desalination business 1 m diameter pipes with water traveling at 1 m/s and the resulting volumetric flow rate isn't a big thing .. but I'd guess for more complex processes it indicates huge/expensive apparatuses.