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Ok, this is not a new topic, but the most recent such topic I could find on Dakka was from 2012, so I think enough new information has appeared to dig back into this again.

Primarily, what is the population range for a Hive world? I do not have the 3rd edition rulebook, but apparently that gives the range as 25 billion to 500 billion. The population of Minea in the 5th edition rulebook is 154 billion, which fits this range. However, I don't think that 500 billion is a hard limit, but more a typical upper range.

Consider Necromunda. The population of Necromunda appears to be much greater than 500 million- just the upper levels of a single Hive (Hive Trazior) is given as a billion, and there are over a thousand hive clusters on the surface of Necromunda. Now, Hive Trazior is one of the largest Hives on Necromunda, but only a fraction of the Hive seems to have been measured, and these is something like 10 miles of vertical habitable space in the largest hives. If most of the hive clusters (nevermind individual hives) only had a population of 500 million, that would still give a population over 500 billion when including the largest hives. Necromunda is a particularly old and important Hive world, which was a Hive world before the Imperium even discovered it. I don't think it is at all unreasonable for it to break the 500 billion "barrier".

I believe (although cannot confirm personally) that some of the recent Black Library works set upon Terra (Carrion Throne) describe Terra as having a population within the quadrillions. Now, as Terra is not strictly speaking just a Hive world, but the Throneworld, probably it technically doesn't break the 3rd edition fluff. It is also not implausible, as the cities on Terra are described as being planet-wide and there are also known to be floating Hives as well. If you take a population density equivalent to Kowloon's Walled City, and extrapolate it to a planet-wide, miles thick city, that population is obtainable. The constant food, water, and oxygen supply needed to maintain it must be collossal though! Now, Terra is the capital of a Galaxy-wide empire, and capitals are typically much larger than other cities within the empire (look at Paris and London in comparison to other cities in their countries). However, this is typically only a single order of magnitude or two greater than the next largest settlements, which would suggest at least a handful of other planets had populations in the trillions or even low quadrillions.

Note that for the following paragraph, I am ignoring the usual GW numbers for recruitment, as they are ludicrously low- therefore I am assuming that Armageddon was recruiting more than "at least a hundred million men at arms". If we recruited this from the population of Terra, it wouldn't even feature within a reasonable margin of error of the total population... Necromunda and Terra are in the Segmentum Solar (the most densely populated segmentum). Armageddon is also a Hive world within Segmentum Solar, and at the height of the Second War for Armageddon, more regiments were being raised from Armageddon than any other two worlds within the segmentum combined. Both Necromunda and Terra are known to raise Imperial Guard regiments, and whilst neither planet is at war, if we take the population of Terra to be in the quadrillions (and adjacent to the largest Forge world in the galaxy for equipment supplies), their standard peacetime recruitment of regiments must be enormous! For Armageddon to at least beat Terra plus Necromunda/another very populous Hive world, it must have a collossal population of a scale approaching those worlds- likely over 500 billion (at least before the Third War for Armageddon devastated the planet).


Now, my question is whether these few notable worlds having or likely having a population above the 500 billion "limit" constitutes a general retcon of the old fluff, or whether they are simply part of a handful of exceptionally important and populous worlds?

Is a world with a population above 25 billion still considered a Hive world, or has the whole range been shifted higher?

Does this increase previous estimates for the population of the wider Imperium from quadrillions to quintillions perhaps?

Using the density of the old Hong Kong Kowloon Walled City is not an accurate assumption to estimate population density. The reason is any depiction of hives also shows large public spaces such as roads, plazas, cathedrals, or factories. All of these lower the overall density. The Imperium is an institution where public spaces are vast in order to awe the population and make them feel tiny and insignificant. The underhive also has a lower population density. It is shown as a post-industrial wasteland with scattered settlements eking a living out of the scraps, so the density is much lower than the teeming inhabited sections above them.

Iracundus wrote:
Using the density of the old Hong Kong Kowloon Walled City is not an accurate assumption to estimate population density. The reason is any depiction of hives also shows large public spaces such as roads, plazas, cathedrals, or factories. All of these lower the overall density. The Imperium is an institution where public spaces are vast in order to awe the population and make them feel tiny and insignificant. The underhive also has a lower population density. It is shown as a post-industrial wasteland with scattered settlements eking a living out of the scraps, so the density is much lower than the teeming inhabited sections above them.

Yeah, but it really doesn't need to. Hive Primus on Necromunda has 12 miles of habitable city. If the entirety of Terra was covered in 12 miles of city, even discounting the Imperial Palace and Inquisitorial Fortress (which will likely have millions or billions of workers living within them each anyway), then it wouldn't need to be at the density of Kowloon to approach quadrillions, but that gives us a real-world idea of how closely people can live together. Many of the Imperial citizens on Terra are likely to be living in equivalently packed quarters between the grand plazas and cathedrals.

If you had a layer of city covering Terra of ~200m depth (basically twice the height of Kowloon Walled City), at twice the population density of Kowloon Walled City (because twice the height: ~2,510,000 people /km2), you reach a population of almost 1.3 quadrillion. Terra likely has 19km (12 miles) depth of inhabited cities covering the surface, or more. There is plenty of space to have a population over 1 quadrillion in size, and still have large areas of public spaces or industry, with the population crammed into the spaces.

Anyway, that is beside the point- Terra has been stated to have a population in the quadrillions, so that is what we have to work with.

I honestly find it hard to believe there's a quadrillion humans on Terra. A trillion? Possible but still a bit far-fetched in my eyes, but a quadrillion? A thousand, trillion? To me it sounds like one of those bits of fluff they've written without fully understanding how big a number that is.

 Valkyrie wrote:
I honestly find it hard to believe there's a quadrillion humans on Terra. A trillion? Possible but still a bit far-fetched in my eyes, but a quadrillion? A thousand, trillion? To me it sounds like one of those bits of fluff they've written without fully understanding how big a number that is.

Well, if you go on raw space, it is easily possible. The logistics required to support such a population are absolutely mind boggling though. Terra would need round-the-clock resupply shipments and a near-total recycling of organic material on the planet itself. The sheer number of people could probably be used for some kind of power generation through ambient heat.

I think it would be most feasible if the orbital stations of the Sol system contained a large number of hydroponic farms. They would need a constant supply of organic material to account for the material being lost from Terra through unrecycled deaths and recruited troops (Terra is essentially a factory producing humans in this situation), but having hydroponics in-system would provide a buffer to the relative unreliability of warp travel.

Anyone fancy calculating the daily food requirements of a quadrillion people?

This is a bit of a 'how long is a piece of string' question, no?

It would depend entirely on how much land is available to build on and how far you could build up on said ground.

Someone may want to math this, but from what I understand we can fit the entire 7B world population into Texas with a population density of ~270,000/mile (think Manhattan). Won't be pretty but the Emperor's Imperium ain't perty...

Calculating population density is easy enough. The earth has a land surface area around 150 million km². For one quadrillion people, you get 10^15/1.5*10^8 = 6.6*10^6 people/km².
Using more than strict land surface (are there still oceans in the grim dark future?) lowers that a bit, but the order of magnitude stays around the million/km².

That's roughly a hundred times the density of Manila and it's planet wide, so that sure is a lot of people crammed together. It's probably possible to give that many people a lttle living space with the usual sci-fi architecture where everything is a humongous skyscraper. I don't know about the logistics, though. The population probably couldn't move around that much. It's hard enough having millions of people commuting at the same time, I can't imagine billions upon billions doing it. Regarding food, there's obviously not enough room for both people and the agriculture to support them. I'd like to see some estimated numbers for that, but shipping food in and waste out to support that kind of population sounds insane...

I'd say a quadrillion people on one planet is a bit much to be believable (without some magic tech we've never heard of making it possible). A trillion seems more likely.

Waste management and commute are the bane of every large cities. Traffic jams in Hive Cities must be absolutly insane. Considering the difficulties of Warp Travel, waste management and providing, food, air and water to those Hives must be a nightmare.

So...

Food consumption. The daily average for men is 10,500 kJ, and 8,400kJ for women (according to NHS.uk). Assuming the population of Terra is 50:50 split (it may weight towards, say women if men are more likely to be recruited into the Imperial Guard), that averages at 9,450kJ per person. I'm going to round this to 9,500 as a reasonable average energy intake required to sustain an active citizen per day.

So a quadrillion citizens... needs a lot of energy: 9.5*10^18 kJ of energy per day. Yikes! How much food is required to sustain that? Well, assuming the Imperium is feeding most of its population on algae-nutrient paste with any waste organics reprocessed back into the paste (somewhat supported by the fluff), we can assume Imperial gruel is likely to be as nutrient and energy dense as possible to reduce the energy and space expenditure of carrying the stuff around. This stuff is going to be something like calogen- if you have ever smelt or tasted this stuff, you know it is rank.

Lets use something incredibly energy dense as a starting point. Butter has an energy density of 30,000kJ/kg (almost twice the energy density of the gross calogen stuff, I dunno why they don't just use butter). If we fed our population of Terra on just butter for energy needs, it would require 317,000,000,000 metric tons of butter every day to feed Terra (heavily rounded). 317,000,000,000 billion tons of butter is a lot. Butter is ~81% fat, so it could be substantially more energy dense, but I think our Imperial gruel needs the other 19% to be essential nutrients other than fat

Now, necessary water intakes for humans are all over the place- it varies between humans and no one is really sure. However, most health authorities seem to recommend somewhere between 2 and 4 litres a day. I shall go for 3.5, on the assumption that Terra will be hot and many citizens will be working manual labour. That is 3.5*10^15L per day, which would weigh 3.5*10^12 metric tons.

So every day we need a minimum of around 3.82*10^12 metric tons of nutrient paste to sustain a 1 quadrillion population on Terra, assuming the nutrient gruel has both the required liquid and energy for sustenance mixed together.

Better load up those mass conveyors!


Regarding the transportation and waste handling- I'd think that most of the Terran Hives are built around standardised modules, each one being a self-contained population hub and city in it's own right, with housing directly attached to places of worship, work (industrial output/admin/etc) and necessary conveniences. These modules would each likely have a centralised food and waste processing system, which would be connected to a larger system between modules and ultimately to the orbital docks.

Much of the lower levels probably forms an underhive of disgusting slums, like on Necromunda, and the upper levels will be opulent palaces and the like, but having each area be a self-contained unit of people living their productive lives removes much of the logistical hassle, and you are then just left with the issue of getting all unprocessable waste out, and all fresh food/water/oxygen/power in to each module. Transport would be contained within each. The slums and pilgrim ways are going to be enormously congested though.

 Valkyrie wrote:
I honestly find it hard to believe there's a quadrillion humans on Terra. A trillion? Possible but still a bit far-fetched in my eyes, but a quadrillion? A thousand, trillion? To me it sounds like one of those bits of fluff they've written without fully understanding how big a number that is.

It's easy actually to pack that many people onto a planet, the issue is that you're not grasping just how big a planet is in surface area and how tall 40k buildings are. The oceans are completely gone and the entire planet is covered in a global city with buildings reaching into the stratosphere. The only reason why the typical Hive World doesn't encroach upon Terra is merely because most Hive Worlds are not in fact covered by a planet-spanning city. Look at Hong Kong if you want to see what maximum population density looks like with extreme poverty.

 Wyzilla wrote:

 Valkyrie wrote:
I honestly find it hard to believe there's a quadrillion humans on Terra. A trillion? Possible but still a bit far-fetched in my eyes, but a quadrillion? A thousand, trillion? To me it sounds like one of those bits of fluff they've written without fully understanding how big a number that is.

It's easy actually to pack that many people onto a planet, the issue is that you're not grasping just how big a planet is in surface area and how tall 40k buildings are. The oceans are completely gone and the entire planet is covered in a global city with buildings reaching into the stratosphere. The only reason why the typical Hive World doesn't encroach upon Terra is merely because most Hive Worlds are not in fact covered by a planet-spanning city. Look at Hong Kong if you want to see what maximum population density looks like with extreme poverty.

Why that's positively spacious compared to some of the so called 'coffin' apartments in Singapore!

And as you go up, each layer reduces the population density with enough layers, the number of people you can cram into one sqKm of ground really starts to get large. And with 40K buildings being multiple miles upwards, and entire planet of several hundred story building its not completely unfeasible to get that many people in. The logistics of food/water in and waste out though - now that becomes a challenge, where does all the poop go?

The waste (if Necromunda is any indicator) gets recycled too. Hive Worlds produce a lot of food locally in the form of fungal farms and the like. That is for the commoners. The nobility get food imports from the stars, and maybe hydroponic greenhouses on the upper spires. However if the imports stop, starvation hits, and the nobility probably start confiscating even the fungal food for themselves, leaving the commoners to starve.

The one big never addressed issue is what happens to all the heat. The population itself would produce a lot of waste heat, along with any industrial processes. The planet can only radiate so much heat into space, and I imagine Hive Worlds to have strong greenhouse effects from all the unconstrained emissions from industry over centuries. Even if one started with a frigid icy planet, centuries of industrial emissions and the waste heat from so many bodies should have heated things up to the point where life is unsustainable. Before someone says "air conditioning", that would only pump heat out from a hive into the surrounding atmosphere. It still would not increase the heat the planet sheds into space as a whole.

Iracundus wrote:
The one big never addressed issue is what happens to all the heat. The population itself would produce a lot of waste heat, along with any industrial processes. The planet can only radiate so much heat into space, and I imagine Hive Worlds to have strong greenhouse effects from all the unconstrained emissions from industry over centuries. Even if one started with a frigid icy planet, centuries of industrial emissions and the waste heat from so many bodies should have heated things up to the point where life is unsustainable. Before someone says "air conditioning", that would only pump heat out from a hive into the surrounding atmosphere. It still would not increase the heat the planet sheds into space as a whole.

Yeah, that is the biggest problem, but I'd imagine it can be managed. Ship all water and food in completely frozen, maybe even to zero kelvin, you then lose heat warming them up. Ship in blocks of steel or other dense metal, pump heat into them until they melt, ship to Martian forges or something. Alternatively, do reverse geothermal, namely tap into mantle, heat up the lava then pump it back in. I'd imagine heat management would be one of the most important sectors of planetary management...

 Haighus wrote:
So a quadrillion citizens... needs a lot of energy: 9.5*10^18 kJ of energy per day. Yikes! How much food is required to sustain that? Well, assuming the Imperium is feeding most of its population on algae-nutrient paste with any waste organics reprocessed back into the paste (somewhat supported by the fluff), we can assume Imperial gruel is likely to be as nutrient and energy dense as possible to reduce the energy and space expenditure of carrying the stuff around. This stuff is going to be something like calogen- if you have ever smelt or tasted this stuff, you know it is rank.

Lets use something incredibly energy dense as a starting point. Butter has an energy density of 30,000kJ/kg (almost twice the energy density of the gross calogen stuff, I dunno why they don't just use butter). If we fed our population of Terra on just butter for energy needs, it would require 317,000,000,000 metric tons of butter every day to feed Terra (heavily rounded). 317,000,000,000 billion tons of butter is a lot. Butter is ~81% fat, so it could be substantially more energy dense, but I think our Imperial gruel needs the other 19% to be essential nutrients other than fat

Now, necessary water intakes for humans are all over the place- it varies between humans and no one is really sure. However, most health authorities seem to recommend somewhere between 2 and 4 litres a day. I shall go for 3.5, on the assumption that Terra will be hot and many citizens will be working manual labour. That is 3.5*10^15L per day, which would weigh 3.5*10^12 metric tons.

So every day we need a minimum of around 3.82*10^12 metric tons of nutrient paste to sustain a 1 quadrillion population on Terra, assuming the nutrient gruel has both the required liquid and energy for sustenance mixed together.

Better load up those mass conveyors!

Water probably can be recycled though, you'd only need to top up the losses in the system, not ship fresh water in every day. Even so, there is plenty of it in-system. Neptune alone has what, ten times the volume of entire Earth of water on its own? So, very little need to ship over interstellar distances.

317,000,000,000 metric tons is not a lot, though. Two years ago, container shipping alone was 1,900,000,000 metric tons*, around two orders of magnitude less. That's just with container ships after 15 years of economic slowdown. I'd imagine Holy Terra generates easily economy like dozen orders of magnitude larger than our own, simply multiplying carrying capacity of what Earth could do 38.000 years earlier by mere 100 times is going to be child's play to them, even if space travel is more expensive, and it can actually be less expensive considering grav tech.

* https://www.statista.com/topics/1367/container-shipping/

The calculation for these 317 billion tons was per day, though. That shipping number is per year, so it's not a 100 times more, but 36,500 times.
One huge problem would be storage, too. It would be ridiculous to bring every morning enough food to last until dinner... Ships would more likely bring many times more and then all of that would need to be stored somewhere. Mountain ranges hollowed out and turned into warehouses sound very 40kish, but that means even less room for people.
The more I think about it, the more it seems ridiculous to have more than a tiny fraction of Terra's food and water being imported. I know it sounds cool to imagine a thousand agri-worlds shipping their produce to feed the untold masses, but it's more plausible that everyone is eating and drinking recycled waste. A portion of which must be corpses, too.

Tiennos wrote:
A portion of which must be corpses, too.

Eh, I never liked that meme. Sure, they probably recycle dead instead of burying them, but even if typical inhabitant of Terra lives something laughably short like 40 years, the amount of meat on their body would sustain another human for like 2 weeks tops, mere 1/1000 of his/her lifespan assuming simple 1 to 1 population replacement. Double the lifespan to get more reasonable 80 years and the corpse meat would only last you 0.0004% of your life, so would be virtually insignificant part of the diet. Modern human probably swallows more bugs in his/her lifetime than that, yet no one says western died is partly insects.

Iracundus wrote:


The one big never addressed issue is what happens to all the heat. The population itself would produce a lot of waste heat, along with any industrial processes. The planet can only radiate so much heat into space.

There are strategies to help, including:
- heat reclamation to convert back to useful energy (presumably fairly efficient)
- tall spires reaching high into the atmosphere (and beyond) to radiate heat more quickly
- blocking light from the sun with a big sheet of foil (Terra is probably always night-time)
- shipping in frozen (probably liquid) hydrogen and dumping the heat into that, which promptly shoots into space (hydrogen geysers). Hydrogen is stupid abundant and a byproduct of many kinds of mining. If Terra has orbital elevators, this gets extra easy, you pour liquid hydrogen down from the top and let the hotter gas escape from the top of the orbital elevator with escape velocity.

 Irbis wrote:

Tiennos wrote:
A portion of which must be corpses, too.

Eh, I never liked that meme. Sure, they probably recycle dead instead of burying them, but even if typical inhabitant of Terra lives something laughably short like 40 years, the amount of meat on their body would sustain another human for like 2 weeks tops, mere 1/1000 of his/her lifespan assuming simple 1 to 1 population replacement. Double the lifespan to get more reasonable 80 years and the corpse meat would only last you 0.0004% of your life, so would be virtually insignificant part of the diet. Modern human probably swallows more bugs in his/her lifetime than that, yet no one says western died is partly insects.

Well, I guess if the population remains stable that means everyone will eat the equivalent of another person during their life. It's the circle of life and everything, but it must be weird for the guys whose job it is to collect corpses and dump them into the food processors

 Irbis wrote:

Tiennos wrote:
A portion of which must be corpses, too.

Eh, I never liked that meme. Sure, they probably recycle dead instead of burying them, but even if typical inhabitant of Terra lives something laughably short like 40 years, the amount of meat on their body would sustain another human for like 2 weeks tops, mere 1/1000 of his/her lifespan assuming simple 1 to 1 population replacement. Double the lifespan to get more reasonable 80 years and the corpse meat would only last you 0.0004% of your life, so would be virtually insignificant part of the diet. Modern human probably swallows more bugs in his/her lifetime than that, yet no one says western died is partly insects.

They don't use meat, they grind your entire body down into a protein paste.

The real problem with planetary populations is that GW loves to add zeroes for shock value, but with no idea of what they actually mean.

 AegisGrimm wrote:
The real problem with planetary populations is that GW loves to add zeroes for shock value, but with no idea of what they actually mean.

The funny part is that GW usually doesn't use enough zeroes. Space Marine chapters are so pathetically small as to be irrelevant. Humanity should be inhabiting billions of star systems, not "one million worlds".

 John Prins wrote:

Iracundus wrote:


The one big never addressed issue is what happens to all the heat. The population itself would produce a lot of waste heat, along with any industrial processes. The planet can only radiate so much heat into space.

There are strategies to help, including:
- heat reclamation to convert back to useful energy (presumably fairly efficient)
- tall spires reaching high into the atmosphere (and beyond) to radiate heat more quickly
- blocking light from the sun with a big sheet of foil (Terra is probably always night-time)
- shipping in frozen (probably liquid) hydrogen and dumping the heat into that, which promptly shoots into space (hydrogen geysers). Hydrogen is stupid abundant and a byproduct of many kinds of mining. If Terra has orbital elevators, this gets extra easy, you pour liquid hydrogen down from the top and let the hotter gas escape from the top of the orbital elevator with escape velocity.

Terra has two polar heat sinks/heat management units. Think there is an inquisitorial fortress amidst them as well.

I've read that there could be up to trillions in a hive world (might be wrong on that as I'm not 100%). Makes sense because hive worlds are not just flat like our cities, they are stacked like mountains, nor is there so much empty land. You are talking about hundreds of billions at the lowest level.

Tiennos wrote:

Spoiler:

 Irbis wrote:

Iracundus wrote:
The one big never addressed issue is what happens to all the heat. The population itself would produce a lot of waste heat, along with any industrial processes. The planet can only radiate so much heat into space, and I imagine Hive Worlds to have strong greenhouse effects from all the unconstrained emissions from industry over centuries. Even if one started with a frigid icy planet, centuries of industrial emissions and the waste heat from so many bodies should have heated things up to the point where life is unsustainable. Before someone says "air conditioning", that would only pump heat out from a hive into the surrounding atmosphere. It still would not increase the heat the planet sheds into space as a whole.

Yeah, that is the biggest problem, but I'd imagine it can be managed. Ship all water and food in completely frozen, maybe even to zero kelvin, you then lose heat warming them up. Ship in blocks of steel or other dense metal, pump heat into them until they melt, ship to Martian forges or something. Alternatively, do reverse geothermal, namely tap into mantle, heat up the lava then pump it back in. I'd imagine heat management would be one of the most important sectors of planetary management...

 Haighus wrote:
So a quadrillion citizens... needs a lot of energy: 9.5*10^18 kJ of energy per day. Yikes! How much food is required to sustain that? Well, assuming the Imperium is feeding most of its population on algae-nutrient paste with any waste organics reprocessed back into the paste (somewhat supported by the fluff), we can assume Imperial gruel is likely to be as nutrient and energy dense as possible to reduce the energy and space expenditure of carrying the stuff around. This stuff is going to be something like calogen- if you have ever smelt or tasted this stuff, you know it is rank.

Lets use something incredibly energy dense as a starting point. Butter has an energy density of 30,000kJ/kg (almost twice the energy density of the gross calogen stuff, I dunno why they don't just use butter). If we fed our population of Terra on just butter for energy needs, it would require 317,000,000,000 metric tons of butter every day to feed Terra (heavily rounded). 317,000,000,000 billion tons of butter is a lot. Butter is ~81% fat, so it could be substantially more energy dense, but I think our Imperial gruel needs the other 19% to be essential nutrients other than fat

Now, necessary water intakes for humans are all over the place- it varies between humans and no one is really sure. However, most health authorities seem to recommend somewhere between 2 and 4 litres a day. I shall go for 3.5, on the assumption that Terra will be hot and many citizens will be working manual labour. That is 3.5*10^15L per day, which would weigh 3.5*10^12 metric tons.

So every day we need a minimum of around 3.82*10^12 metric tons of nutrient paste to sustain a 1 quadrillion population on Terra, assuming the nutrient gruel has both the required liquid and energy for sustenance mixed together.

Better load up those mass conveyors!

Water probably can be recycled though, you'd only need to top up the losses in the system, not ship fresh water in every day. Even so, there is plenty of it in-system. Neptune alone has what, ten times the volume of entire Earth of water on its own? So, very little need to ship over interstellar distances.

317,000,000,000 metric tons is not a lot, though. Two years ago, container shipping alone was 1,900,000,000 metric tons*, around two orders of magnitude less. That's just with container ships after 15 years of economic slowdown. I'd imagine Holy Terra generates easily economy like dozen orders of magnitude larger than our own, simply multiplying carrying capacity of what Earth could do 38.000 years earlier by mere 100 times is going to be child's play to them, even if space travel is more expensive, and it can actually be less expensive considering grav tech.

* https://www.statista.com/topics/1367/container-shipping/

The calculation for these 317 billion tons was per day, though. That shipping number is per year, so it's not a 100 times more, but 36,500 times.
One huge problem would be storage, too. It would be ridiculous to bring every morning enough food to last until dinner... Ships would more likely bring many times more and then all of that would need to be stored somewhere. Mountain ranges hollowed out and turned into warehouses sound very 40kish, but that means even less room for people.
The more I think about it, the more it seems ridiculous to have more than a tiny fraction of Terra's food and water being imported. I know it sounds cool to imagine a thousand agri-worlds shipping their produce to feed the untold masses, but it's more plausible that everyone is eating and drinking recycled waste. A portion of which must be corpses, too.

See my calculations above- a population of 1 quadrillion could be squeezed into a planet-covering layer of city less than 500m thick (closer to 200m). Terra likely has 12 miles (19km) thick of inhabited city. Therefore there is plenty of room for vast stores, huge manufactorums, structural supports and great plazas and cathedrals, especially if that population is distributed throughout the strata, not just in a single packed layer.

I think most food would be produced from recycled products on Terra and within the wider Sol system. The Agri world replacements will be needed to replenish water and organic losses of recruited humans. Every regiment raised or Navy ship crewed means lost biomass from Terra that cannot be recycled. Depending on how much recruitment is occurring, this could require significant offworld resupply to maintain.

Regarding heat- the simple, and most efficient option is to use that heat as an energy source. For example, the planet will have a huge oxygen use with a quadrillion humans on it, plus whatever the industry requires. Use some of that heat energy to drive a carbon-fixing process to produce oxygen and food. Excess energy could also be used to power foundries and utilities, or even to charge batteries and move the energy offworld if needed. Homothermal power generation

 Haighus wrote:

Regarding heat- the simple, and most efficient option is to use that heat as an energy source.

Energy reclamation from heat generally requires a temperature gradient to work across - a hot side, and a cold side. Since your 'hot side' can't exceed livable temperatures in most areas, you still have to move and concentrate the heat, and that takes up more and more space the thicker your inhabited layers become. Heat management is a big issue regardless of whether you're reclaiming or just venting it into space. Given how power generation is fairly trivial in the Imperium of Man (fusion reactors are prevalent), you can afford to use a lot of power to get rid of waste heat - space efficiency is probably a bigger priority than power efficiency. Reclaiming it fast enough to avoid deadly heat build up might be a real issue.