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When I heard the words Goldman and Arbitrage, I was expecting the worst.
This is a pretty blatant case of greenwashing one of the most destructive financial firms on the face of the planet. Even if this system is utterly ridiculous in terms of cost/benefit, Goldman Sachs is a company that can afford to experiment with it in an effort to appear less evil.
They are also glazing over the fact that the system uses millions of gallons of water per year.
Why would they need to flush the water? If they did, and it's not coming into contact with any major contaminants, why couldn't it go more or less directly back into the water supply after maybe a trip through a treatment centre?

For that matter, wouldn't it be even more efficient to use something like seawater? They are on an island after all.

There's no reason it needs to. If you put some copper sulfate in the water it will keep the microbes at bay and allow you to re-use the water either indefinitely or very good approximation. You could also make a strong brine which would prevent microbial activity and also lower the freezing and melting point allowing you to move more heat with less water on the air conditioning side. It might make your chillers for making ice less efficient, though.
How is this economically efficient? It takes a massive amount of space in their manhattan building but only saves $50k per month?
The article did say that the ice is kept below ground. Perhaps, if it is economically efficient, it is because below-grade real estate is less valuable? I'd imagine the ice could be in places that are difficult to use for other purposes.
I doubt they'r renting out office space in the sub-basements.
There were probably other deciding factors beyond economics, such as carbon footprint.