LoudGuns explains it here:
Trade Secrets - Saving time & reducing risk when delivering to the TDD - YouTube. The whole vid is worth watching, but if you
skip to 04:00, he explains that specific point.
Edit: You can sell your quantanuim at New Babbage's TDD, but the demand there is kind of low. If other players have been selling their quant there, not just on your server but anywhere on any server, you might find you have to wait ages before they have any demand for it - and the error message in the terminal is confusing, as it will say 'No inventory' or something to that effect, on the SELL tab of the terminal. That means they don't want any, rather than they haven't got any. It's a bit confusing!
Anyway, you're better off taking more than a very small amount of Quant to Bajini Point or all the way down to Area 18 on ArcCorp. Area 18's TDD has MUCH more demand for quantanium (5,000,000 units + 50,000/min globally across all servers, compared with 400,000 units + 4,000/min at the other three places you can sell it). This means you're much more likely to be able to sell all your quantanium at once, and not have to hang around at the TDD for a long time, selling it off a bit at a time. Source for this is
SC Trade Tools (sc-trade.tools):
View attachment 22316
Edit 2: Oh, the 'units' you sell mineables in are confusing too: 1 'unit' is 1 cSCU. See
Standard Cargo Unit - Star Citizen Wiki. There are 100 cSCU (or 100 'units' on the trade screens) per SCU of cargo space, so if you have e.g. an MSR with 114 SCU cargo space, you can carry a max of 11,400 'units' of quant (or of anything else), assuming you didn't put any boxes or anything else in the cargo space. If you put a box in there, it's a bit less. So selling 1 completely full MSR of quant for 80.0 aUEC per 'unit' (i.e. per cSCU) will give you 80 x 11,400 = 912,000 aUEC gross. Why 80.0/unit when the table above says 88.0/unit? The TDD usually offers you less than the maximum price, based on the supply of quant. Back in 3.14 I usually got a little over 80/unit for mine. I have barely mined quant since then, so maybe it changed a bit. You might get a little bit more, but probably not the full 88.0/unit. (Net profit is that minus your refining costs, fuel and consumable costs for all the ships you used to mine and transport the quant, and it's wise to track the time you spent doing all of that to allow you to estimate your aUEC/hr, even roughly).