How to calculate the minimum vessel sizes for all grain brewing

Draining from all the answers and comments above:

Boil Kettle = (((batch size + deadspace) + (boil off/h * boil length) ) + expansion) + extra Space for Boil over

With 9% boil off, 4% expansion, 90 min boil, a 20l batch and 5l dead space:

BK = (((20l + 5l) + (9% * 1.5)) + 4% )+ extra for boil over
BK = 29.5l extra for boil over

From the rule of thumb which says the brew kettle should be between 1.5 to 2 times the batch size we get:

Boil over = 20l * 1.5 = 30l

Or

Boil over = 20l * 2 = 40l

The 1.5 times figure looks dangerously close to the volume of the liquid, while the 2 times provide 10l of extra space, or 35% extra, which could be applied to the formula above.

Mash Tun= (Water to Grist Ratio) * (Largest Grist Bill) + (Grain Volume) * (Largest Grist Bill)

MT = 4 * 5 + 0.652 * 5 = 23.3 l

Hot Liquor tun = Strike water + sparge water

While that is correct, if we assume that the strike water is heated in the mash tun, the HLT needs only hold sparge water:

HLT = sparge water

Which is, with the numbers in the question :

Sparge water = Pre boil volume - first runnings
Sparge water = 28.5 - ( strike water - absorption - dead space)
Sparge water = 28.5 - ( 20 - 8.3 - 2)= 28.5 - 10.3 = 18l

Or if you want to heat all the water in the HLT:

HLT = 20 + 18 = 38l

This, a believe is the closest one can get to a set of formulas, which while having the limitations pointed out in the answers/comments, gives useful numbers, backed up by some logic.

While the formulae can give you some insight into the appropriate sizes, there are quite a few unknowns. I would instead take into account the common wisdom for kettle sizes.

The general rule of thumb is to have the boil kettle around twice the volume of your target batch. I brew 38 liter/10 gallon batches and use a 75 liter/20 gallon blichmann kettle. Pre-boil volume is typically just shy of 3/4 of the kettle capacity so boilovers are seldom, but it has happened once. The absolute minimum would be 1.5x your batch size, but the closer you get to 2x batch size the more convenient and pleasurable it will be to brew. Even with 2x batch size, you cannot totally eliminate boilovers, but they are seldom. As you reduce the kettle size, you have to be more attentive to boilovers, so bigger is definitely better here if you can accommodate it.

For the MLT, you may go smaller, down to 1.2x the batch size if you are sure you won't be making anything above 1.060. In my experience, the volume in my mash is never more the final batch size for regular strength beers.

For the HLT, you generally need about twice as much water as the final batch volume. My HLT is 2x my batch bolume - fill the kettle up at the start of brewday and use almost all of it - maybe a gallon left at most. Having more water also helps with the HERMS heat exchange. You could in principle go smaller if you're not using HERMS or if you're prepared to heat the water multiple times, e.g. heat the strike water, then any additional infusions, and then the water for the sparge. The HLT gives you the most flexibility and could be as small as 0.75x your batch size, but the smaller size will make the system less convenient.

Summarizing these figures for minimum sizes and useful range:

• BK: 1.5-2 x batch size = 30-40l. (Go for the largest you can accommodate.)
• MLT: 1.2+ x batch size = 24-40l.
• HLT: 0.75-2 x batch size = 15-40l.

These figures tie in well with your original estimates.

EDIT: I'm assuming you'll be mounting an outlet valve on the kettles. If so, I doubt you'll be able to store the kettles inside each other. It may be better instead to just get 3 kettles the same size 35-40l and stack them vertically when not in use.

EDIT 2: If you don't have the space for 3 kettles now, when why not just go with a single 40l kettle and do BIAB until you have space. I have a 3 kettle setup, but sometimes wish I could do the same with 1 - they take up a lot of room (see my profile pic.) The danger of reducing the size down to the absolute smallest so that the kettles can be stacked is that you spend quite a bit on the kettles, yet you may still end up with a hampered brewday due to inadequate kettle sizes. Also you'll need a pretty big stove top/hob to have room for all three kettles, if you're brewing in the kitchen. If you go with a single kettle solution now, you'll be able to brew all grain without needing space for 3 kettles, and also be sure you have stovetop space and power to be able to brew effectively. You can then always add 2 additional kettles later when your circumstances change.

Required Volume for each vessel:
Boil Kettle: (Batch Size + 4% expansion) + (Boiloff) + (Extra Space for Boilover)
Mash Tun: (Water to Grist Ratio) * (Largest Grist Bill) + (Grain Volume) * (Largest Grist Bill)
HLT: (Strike Water) + (Sparge Water)

Grain absorption is easy to estimate. I have always used 0.1 gallons of absorption per 1 pound of grain, or roughly .8 liters per kilo. This is the figure used by most of the software, and it has always been very accurate for my purposes.

Boilover varies significantly by the shape of the vessel and the nature of the heat source. I like to have at least 2-3 gallons (~8-12 liters) of spare space. You can get by with less (1-2 gallons; ~4-8 liters), especially if you are willing to use something like Fermcap to limit foam production that would lead to a boilover. There is no absolute answer here, but a common rule of thumb is to buy a boil kettle that is double the volume of your intended batch size

Your pre-boil volume is simply your batch size plus the amount of water you expect to lose due to boil off. The exact amount that you will lose to boil off is extremely complex, and it will depend by a number of different factors that cannot be sufficiently measured. Your best bet is to run a couple of tests to see what your average boil off will be. Until you figure out the values for your system, it is usually best to assume a low boil off, because it is always much easier to add water than to take it away.