The Red Moat/Just dump it in the river like everything else

Well, if it's a medieval society, they're probably in the "siege warfare" stage of civilization. And, so, you need a nice way to stop invaders swimming across into your castle.

Just fill the moat with this brand new, caustic, toxic substance we're producing in vast quantities. As a bonus, no nasty biting insects, and you can discharge as much human waste as you want into it, and it's not going to cause any disease issues. Probably. You might want to line the moat with clay or something impermeable, or you're going to mess up all the groundwater for miles around, but hey!

Plus, it looks badass, and even if the alkalinity of it isn't enough for serious burns (only a bit above 12), mud is harder to not drown in than water.

Seriously, medieval environmental practices would have no problem dealing with this, considering the default method of sewage treatment was "Dump it in the river" or "Dump it in the street". It wouldn't even register as a hazard compared to all the other hazards.

Like, sure, you'd blight the countryside for miles around. But when the default approach to a forest was "Fell the whole thing for farmland, and shoot anything that comes out of it for dinner", I don't think anyone is going to be remotely worried about a little soil contamination, or the odd "river whose waters now run blood red and have no fish in them anymore". Or they might, but they're probably peasants. And so don't really count.

You have options! Kinda...

Disclaimer: You tagged your post science-based. From that perspective, you're hosed. There is no safe way using medieval technology to be rid of Red Mud. It's toxic and is basically stored for the Earth to reduce at some distant future date. Even in the smaller quantities your people would be dealing with, over time there would be a catastrophic problem with it inevitably getting into ground water and contaminating soil. Initially your people would be hailed as wizards for their wondrous aluminum! Their descendants would be the pariahs of the whole country for making everybody sick. Go watch the movie Dark Waters (2019) to see just what that can look like (and it's remarkably on-point for this discussion).

On the other hand, if we shift to science-fiction, we have some options! Options that will make your people rich beyond the dreams of avarice! So, this answer is from the science-fiction perspective. So say we all.

I'm not going to address the alumina byproduct in Red Mud. Let's assume you've extracted enough that the byproduct can be ignored. What's left?

Iron Oxide

Up to 60% of Red Mud is iron oxide (Fe₂O₃). While today we use carbon and a high-temperature blast furnace to get the iron, the ancient peoples of the world (back as far as 2,000 BC) simply smelted the ore using Coke or another carbon source to separate the oxygen from the iron. In other words, at a medieval level technology, this process already exists. Mazel'tov! Iron is obviously useful in an iron-age society.

Calcium Oxide

Next is a somewhat nasty substance that has remarkable uses. Calcium oxide, also known as quicklime, which happens to be an important component of hydraulic cement, used by the Romans. And do you know what else is needed for cement?

Silicon Dioxide

Silicon dioxide (aka sand) is famous today for its use in semiconductor manufacturing — but in the Good Old Days it was good for creating cement, concrete and plaster. It's also great for making glass and can be used as the base for cosmetics. And do you know what else is good for making glass, at least if we squint a bit?

Sodium Oxide

Technically, glass is made using sodium carbonate and the resulting glass (due to the chemistry of its construction) has sodium oxide. But, science-fiction, right? I'll explain that momentarily.

And that leaves...

titanium(III) oxide

It's a very good thing that your people are adepts at advanced science! They're not just making alumina! In that locked shack in the back is a crude but effective electrolysis machine that's allowing them to convert Ti₂O₃ into pure titanium! But if that's too much to swallow, titanium(III) oxide just happens to be good for solar panels and desalination. But you might want to consider that electrolysis sleight-of-hand because that gets you to titanium dioxide (TiO₂), which just happens to be the fundamental component of a great paint pigment! While not used in Real Life until the mid-19th century, but your advanced people have the ability to revolutionize the medieval art world!

All we need is one more thing: a little Clarkean Magic

"Clarkean Magic" refers to the third of Arthur C. Clarke's three laws.

Any sufficiently advanced technology is indistinguishable from magic.

Your travelers arrived at a new planet by space ship! In other words, they can do things we cannot do nor can we imagine how to do. Proof? Look at the 1989 Rockwell Integrated Space Plan. I love hat poster! I have one of the originals! And what do we learn from it? After 37 years, it's proven to be completely inaccurate.

That doesn't mean it doesn't have use to worldbuilders! It's a fantastic plan! It just failed to predict the right things at the right times — proving that Humans are pretty bad at predicting the future. But that's where 99.9% of all science fiction writers make their money.

You can't store Red Mud with medieval technology that won't result in huge negative consequences in the reasonably near future (<100 years). And you really don't want detailed specifics about how to process Red Mud because anyone practiced in the art would know you can't do it economically. If anyone could (economically, that is), they'd be doing it right now.

What you want is suspension of disbelief. What that means is that your advanced science survivors know how to process Red Mud. This isn't as unreasonable as it sounds. Humanity is constantly learning how to do new and clever things with chemistry. There are things we do today that we didn't know how to do 50 years ago. As a worldbuilder you never, ever want to get hung up on the fact that there's something cool that science hasn't figured out yet. Never get caught in the trap of believing that at any moment "science" has all the answers. It never will, thank goodness!

If you want actual process names, invent them. Take a lesson from the movie The Age of Adeline.

(Adeline's car slides off a wintry road into a body of water.) The immersion in the fridgid water caused Adeline's body to go into an anoxic reflex, instantly stopping her breathing and slowing her heartbeat. Within two minutes, Adeline Bowman's core temperature had dropped to 87 degrees. Her heart stopped beating. At 8:55 a bolt of lightning struck the vehicle, discharging half a billion volts of electricity and producing 60,000 amperes of current.

It's effect was three-fold:

First, the charge defibrillated Adeline Bowmans' heart. Second, she was jolted out of her anoxic state, causing her to draw her first breath in two minutes. Third, based on Von Layman's Principle of Electron Compression in Deoxyribonucleic Acid, which will be discovered in the year 2035, Adeline Bowman will henceforth be immune to the ravages of time. She will never age another day.

Most of what you just read isn't scientifically plausible. Some of it is pure technobabble! But it reads with the basic believability needed to suspend disbelief to get on with the story the author actually wants to tell.

Conclusion

Your survivors know how to do Cool Chemical JuJu that permits them to extract iron, calcium oxide, silicon dioxide, sodium oxide and titanium(III) oxide from Red Mud. This allows them to...

• Sell high-grade iron into an iron-age society.
• Create Portland cement, Portland concrete and plaster.
• Create quality glass.
• Create better than average cosmetics.
• Create solar panels for some really useful purposes, including...
• White paint pigment, which is a bigger deal than it sounds like.

In short, don't store the Red Mud, process it!

The premise of this question is flawed in multiple ways, some already mentioned in comments.

Usefulness of alumina

In the modern world, aluminium oxide is required in vast quantities for production of metallic aluminium, but that process is energy-hungry and specifically requires electricity. With that out of the equation, they wouldn't find enough demand. Yes, aluminium oxide / emery is also good by itself as an abrasive, but that can also be done with other substances.

Feasibility of using the Bayer process

The Bayer process is designed for a heavy industrial setting, where you require automation but have access to bulk chemicals. In particular, you need sodium hydroxide which we produce in a process that requires - again - lots of electricity.

Even if they are be able to carry out the Bayer process, it will be very difficult to procure the reagents, so this would be a bad use of their time and resources. Much better to focus on improvements to the already existing steelmaking processes.

Danger of red mud

Don't get me wrong – it is nasty stuff. But it's a common fallacy to think of substances as either harmless or deadly. Some chemicals are toxic even in tiny doses, but most start out as just somewhat irritating and then only get gradually worse with increasing concentration and time of exposure. Red mud is one of those – the reason it is a big environmental problem is that humans produce it in vast heaps, with often millions of tons getting dumped at a single site. But due to the above reasons, there is no chance your people could produce anywhere near that much. Most of the nastiness is simply that it's highly alkaline - incidentally not really a property of the alumina process, but rather carried on from the sodium hydroxide you put in. And that aspect is readily mitigated by mixing with something acid or even just water. Most realistically they would just dump their kilogram per week or so in the next fast-flowing river. Even a small river discharges so much water that this would not noticeably raise the pH.

Need of disposal

As JBH discussed, the constituents of red mud are themselves useful chemicals. I find the relevance of this somewhat doubtful though, because separating those from the nasty lye is its own challenge - not only would-have-been in a medieval setting, it's also challenging in modern times which is the reason red mud accumulates in the first place.

If you have the Bayer process, you probably want electricity

To get the chemicals needed for the Bayer process, namely Sodiumhydroxide, in large amounts, you probably want the Chloralkali process. This is a rather simple process, for which you would need electricity and also get chlorine gas as a side product.

In the alternative, you might use Sodium carbonate from the Leblanc Process (using a series of acids on salt and calcium carbonate) or Solvay Process (using limestone and salt brine, but requires ammonia) and react that with calcium hydroxide, aka slaked lime, which is made by using quicklime. Those processes were used during the industrial revolution and needed a lot of heat and fuel, and in effect were inferior in production numbers.

Red mud is valuable for construction

Since we found the Bayer Process, red mud is known as a valuable precursor for other products. Even without a sophisticated chemical industry to separate out the raw products, it is a somewhat easy to work with base. It's used en-masse in cement manufacture, creating a pinkish product. The dried mud can be mixed into clay to make brick or other ceramic construction material. As it is alcalic, it can take up any production waste that is acidic, which is most notably the waste water from many mines.

If you are more into processing, the red mud could be a valuable iron ore, with up to 60% iron content. That is more than some iron sands, if the aluminium and sodium didn't mess up the metallurgy. Neutralizing the solution with mine wastewater allows to wash out a lot of the sodium over time, and make it processable at least in theory - though not into iron without some higher tech processing.

Pelletized and dried, the sodium-reduced mud could be decently processable into a porous artificial rock in a bloomery forge by deliberately tuning the conditions to make mostly iron-rich slag. This material should be decently crushable, making it usable as gravel or rocky infill in concrete, and should be mostly inert if treated right.

So you can hinge an increased output of the gravel, cement, brick and concrete production on using the (processed) waste of the Alumina making humans,

Lowest grade iron as byproduct of slag making

However with the technical knowledge of the humans, they could in turn build a crude blast furnace instead. The product from that could be, under the right conditions and under using a lot of slag forming flux, a (very) low grade cast iron with possibly considerable aluminium slag inclusions. Still, it would reduce the amount of red mud considerably, and unlock a source for semi cheap metal, and produce more slag aggregates for cement manufacture at the side. Depending on the purity of the red mud, the balance might be severely lopsided to making slag and only a trickle of iron. Yet, I remind that the task was getting rid of large amounts of red mud, so even with mostly slag aggregate, that'd be mission accomplished.

This cast iron might in turn be refined further, especially if you manage to reduce the amount of aluminium slag in it. Do note that many of those processes, such as arc furnaces, would require high voltage electricity. The least technologically challenging method one would require oxygen separation to process the metal with the Oxygen converter process, as the venerable Bessemer Process would not cope well with the Aluminum still in the slag.

The chemistry aspect is a little bit outside of my wheelhouse, but the bit that sticks out most to me is that red mud is its high Alkalinity.

Perhaps the aliens on that planet has trouble growing crops in certain areas, and part of the issue with that is very low (or niche, highly concentrated areas) of Alkaline. Too much can be bad for growing conditions, but generally speaking you do want a certain amount of acidity for growing things. Sometimes that is provided naturally through acidic plants like nightshade (e.g., tomatoes). However, in certain places it may be too: hard, inconvenient, or infeasible to make it work for various practical reasons (limited spacing, relative location to the sun, time of year, water constraints, concerns with attracting certain pests, etc.)

After a little searching, it does sound like it isn't completely impossible to work with red mud in the modern world at small scale, but it does take planning and effort over the course of several years. Given that that this is scaling issue that your characters are facing, I could see them being highly motivated in making it work. Ref: https://www.riotinto.com/en/news/stories/the-slow-journey-from-red-mud-to-green-plants

From the article:

[...] Lukash and Jinte [...] explained the lengthy process, which was filled with trial and error. Their focus was revegetating the site, but before that could happen, a separate civil engineering team had to remove and treat the water as well to transform the site so that rainwater could drain instead of collecting on top of the red mud. This was accomplished through largely mechanical means, moving and shaping the mud so that water could run off.

Once this was completed, Lukash, Jinte and their team proceeded to the chemical conditioning phase. They decided to add gypsum, a mineral with many uses, among which is as a fertiliser that improves soil structure and the amount of water it can absorb. However, the main usage was to lower soil pH. They also added chicken manure as an additional fertiliser. These two products helped to transform the high-sodium, high pH medium into something more amenable to growing vegetation.

However, it wasn’t simply a matter of adding some fertiliser and suddenly vegetation returned. It was a slow process, beginning with the planting of certain shrubs, grasses and other plants that were more tolerant of this still-challenging medium. Organic matter was very low in the red mud, and it took years of planting hardy vegetation to build up the soil. Throughout, Lukash and Jinte continued to experiment with introducing different plants to the site. They took special care to use only vegetation that were native to Jamaica.

If we're to assume that the plants on the alien planet are anything like the ones on earth, a few different searches seemed to suggest that the following plants will put up with higher Alkaline in the soil, but do not necessarily love it. Ref: https://eutrema.co.uk/growing-crops-in-alkaline-conditions

Barley is highly tolerant of alkaline soil

Sugar beet excels in alkaline and saline soils

Cotton thrives in sodic, calcareous, high-pH soils

alfalfa performs surprisingly well in alkaline soils

Buffalo grass and Bermuda grass tolerate alkaline

Grass may be useful for animal husbandry, if your aliens don't care about lawn care! :)

If you do go the red mud-to-plant growing route in your story, you'll want to be well aware of NPK, and how the impact the growing of plants. See: https://www.almanac.com/n-p-k-ratio-what-do-numbers-fertilizer-mean

The general way I remember it is:

N = growth + speed + "greener"

P = root development (If N is high, this component is very important, unless you are trying to feed just prior to its dormancy cycle.)

K = fight disease + cold resistance