[adrian_b]

In living beings things can be 3-dimensional, like a brain, because they are penetrated by a fine network of tubes through which fluid is circulated, taking the heat away.

This is likely to also be the only solution for truly 3D integrated circuits (i.e. of unlimited thickness), but it is very difficult to ensure that such a solution has high reliability, because at least for now it cannot be self-repairable, like living tissues, so the cooling network can become clogged or it can start leaking.

[sandworm101]

Aka "porous circuits". - Red Dwarf 1988

[don_esteban]

This does not scale, due to square/cube law (the energy is generated in a volume, but has to leave via surface -> the energy evacuated per surface area has to grow)

[adrian_b]

It scales, because the surface of the network of tubes that fills the volume also grows with the volume, unlike the external surface that encloses that volume. When the volume is scaled, the tubes (i.e. capillaries) are not scaled in size, but they grow in number per a given fraction of the volume. So the scaled devices are not geometrically similar, thus the law of variation of the area per volume with scaling does not apply.

This is why a brain or a muscle can scale from a mite with a volume of one thousandth of cubic millimeter to a whale.

The scaling is not ideal because the smallest capillaries must be aggregated into vessels of increasing size, so some part of the volume becomes wasted, but still the ratio of area per volume does not decrease linearly, as in the case when the external surface is used for cooling.

The same technique is applied in many engineering domains. For instance, the maximum current for a power transistor depends on the perimeter of the source or emitter, not on the die area. Therefore, if one would scale a power transistor maintaining geometric similarity, the switched power per die area would drop quickly, leading to very high costs for the devices.

Instead of this, a bigger power transistor does not have bigger sources or emitters, but it has more of them, with the same size as in the smallest transistors, which keeps constant the power switched per die area.

[esterna]

Capillaries cannot defeat the divergence theorem: If you have a volume where heat is produced but the temperature does not increase, the heat has to leave via the surface. It is true that the surface area available to diffusion can scale at any rate (eg. Menger sponge), but the heat still has to leave the volume via its boundary. In the case of capillaries, this is by convection which means that the product of coolant temperature differential and flow speed has to scale.

[don_esteban]

Thanks, this is what I wanted to say, but perhaps I was a bit too terse.

Perhaps a good coolant with a good flow speed can gain you a decent constant factor versus a classical heat sink, but once the third dimension of your chip gets bigger, you will hit a barrier. Just do the math.

[dgacmu]

But it does, at least to some degree: For the cpu (or brain) you want high density to minimize the latency between components. For the heat sink, you want high surface area, which you can actually do to some degree three-dimensionally, particularly when you have active cooling. Look at a typical heat sink with a fan attached to it -- it has some depth, because that depth allows more heat to be transferred to the air to it by increasing the surface area exposed to the air. Lungs do the same thing, and they do function as part of our cooling system. So if you have a way that the flow of the exchange medium is not limited by the external surface area of your heat exchanger (a fan, a pump, a diaphragm), you can go pretty far.

[zamadatix]

It doesn't scale forever in the extra dimension but it does at least scale a hell of a lot better than just using the envelope of the volume or using a single flat layer.