Limits of power density imposed by limits of cooling

In the book "Nanosystems: Molecular Machinery, Manufacturing, and Computation"
There is a table of power densities that can be quite misleading.
It is stating electromechanical power conversion at power densities > 10^15 W/m³
This is meant to be possible in very small volumes (or thin layers) only.
Or, if in big volumes, then only for very brief bursts in time.
A bit like excimer lasers can feature exorbitantly high power densities but only for brief burst in time.

Beside cooling limits mechanical material strength
can also pose limits to macroscopically applyable power density.
Not the focus of this page though.

The absolute limits of steady state power are largely unexplored.
And also not yet accessible for exploration.
They lie well beyond the limits of exploratory engineering.

Summary of results (tl;dr)

Not (predictably) possible:
These power densities of ~1MW/mm³ when within the example geometry of a cube of 1mm side-length (or bigger)
are not predictable to be possible to cool (hard statement) and seem quite impossible to cool (soft statement).
One can make some fun speculations about ultra advanced cooling techniques but that's about it. Soft SciFi at this point.

Very much possible and certainly far from the limit:
Still for a concrete example of a wheel hub motor in a car sized wheel
total power can easily exceed limits of current day tech (electromagnetic motors) by more than an order for magnitude.

• todays powerful car scale wheel-hub-motors are about 100kW typiclal 200kW (near limit?)
• atomically precise gemstone based nano-electrostaic shear drives
  4000kW with intentionally wide margins upwards.
  If this powerful of a wheel hub motor even still makes sense is an other questions.
  Also limits of mechanical shear strength are not discussed here.

Assumptions for the following Analysis

We take the approach of a steady state coolant flow analysis.

We run a cube of densely packed electromechanic nanomachinery metamaterial at the
(in Nanosystems identified) maximal possible power density for such nanomachinery.
We consider not only the volume of a cube but also the specific geometry of a cube.
This giving us a cross-sectional area.
(An infinite layer would work too but a cube seems more intuitive.)
– We feed in cold water (or encapsulated ice) on one side of the cube
– We take out hot water on an other side of the cube
(the opposing side seem most natural but we make no such detailed assumptions).
Cooling of the coolant water is mostly outside the scope of this analysis.
We take no assumptions on …
– cooling channel geometry
– whether there are cooling flow forks outside the analyzed volume or not
– whether coolant is a liquid in channels or solid state pellets ins a special transport system

With all these assumptions: The question we want to ask is:
At what speed of the coolant do we need to move the coolant to get all the heat out and achieve a steady state operation.
Undispersed bulk flow. That is: We ask for the speed through one fat coolant supply channel of the cubes cross sectional area.
Dispersed flow will need to be a bit faster but only by a small constant factor so we'll ignore that here.

Symbolic analysis

Volumetric flow [math]Q_V[/math] of cooling substance (e.g. water/ice/…)
is speed times area to flow through like so:
[math]Q_V = v \times s^2[/math]

Speed dependent cooling power is that volumetric flow times
volumetric energy across heating and phase transitions like so:
[math]P_{cooling}(v) = v s^2 (c_V \Delta T + \epsilon_{melt})-P_{coolantfriction}(v)[/math]

The total power loss is the volumetric power density
times the inefficiency times the volume.
[math]P_{loss} = \pi' (1-\eta) s^3[/math]
Condition for sufficient cooling
(assuming a cube as the chosen geometry):
[math]P_{cooling}(v_{min}) \gt P_{loss} + P_{coolantfriction}(v)[/math]
Substituting in and
dropping the significant term [math]P_{coolantfriction}(v)[/math], which we have to remember:
[math]v_{min} s^2 (c_V \Delta T + \epsilon_{melt}) \gt \pi' (1-\eta) s^3[/math]
Expressing the minimal coolant speed:
[math]v_{min} \gt \pi' (1-\eta) s / (c_V \Delta T + \epsilon_{melt})[/math]
[math]v_{min}/s \gt \pi' (1-\eta) / (c_V \Delta T + \epsilon_{melt})[/math]
Now lets run some numbers …

Numbers

Nanosystems 11.7.3. Motor power &and power density
(claimed power densities of nano-electrostatic motors)
[math]\pi' = 10^{15} W/m^2[/math] Nanosystems 11.7.4. Energy dissipation and efficiency –> d. Contact drag. & e. Summary.
(bearing drag dominant, contact drag to investigate, >99% efficiency @ max power-densities)
[math]\eta \gt 99\%[/math]
Melting heat of ice:
[math]\epsilon_{melt} = 333.55 kJ/kg[/math]
[math]\epsilon_{melt} \approx 300 MJ/m³[/math] (ignoring ice expansion)
Heat capacity of water:
[math]c_{V,water} = 4.1845MJ/(1000kg K) ~~ @+20C°[/math]
[math]c_{V,water} \approx 4MJ/(m^3 K) ~~ @+20C°[/math]
Heat capacity of ice (for the curious, we don't use it):
[math]c_{V,ice} = 2.05MJ/(m^3K) ~~ @-10C°[/math]
Ignoring evaporation heat as it comes with a huge change in volume.
But we go all the way till there: [math]\Delta T = 100K[/math]

Putting in the numbers in:
[math]v_{min}/s \gt \pi' (1-\eta) / (c_V \Delta T + \epsilon_{melt})[/math]
Let's assume for now [math]\epsilon_{melt}=0[/math]
That is: We feed water not ice.
Then evaluating gives:
[math]v_{min}/s \gt 10^{15} W/m^3 (0.01) / (4 \times 10^{+6} J/(m^3 K) 100K)[/math]
[math]v_{min}/s \gt 10^{13} W/m^3 / (4 \times 10^{+8} J/m^3)[/math]
[math]v_{min}/s \gt (1/4) \times 10^5 (m/s)/m[/math]
[math]v_{min}/s \gt 0.25 \times 10^5 (m/s)/m[/math]
[math]v_{min}/s \gt 25000 (m/s)/m[/math]
Thus:
Cooling a 1m side-length cube of densely packed nanoelectrostaic motors …
running full throttle ([math]\pi = 10^{15} W/m^2[/math] … but steady state!!) would require to shoot
water through faster than Earths escape velocity. Not happening.
Or let's say very likely not feasibly by any means.
Exotic means for ultra high speed flow cooling may be discussed in a dedicated section.
This is nicely linear. Lets explore less extreme cases:

Cooling a cube with side-lengh of 1mm?
[math]v_{min}/s \gt 15 (m/s)/mm[/math]
This might work with some super advanced cooling technology perhaps?
Keep in mind that fractal branching will be necessary increasing speeds and adding losses from coolant friction.

Cooling a cube with side-lengh of 1µm?
[math]v_{min}/s \gt 15 (mm/s)/µm[/math]
This seems very much feasible as this lies around proposed speeds for nanomachinery of ~5mm/s.
This can be extended to a sheet of this thickness to some degree.

Keep in mind that this is still overly optimistic as we not at all
take into account the additional heating from the friction of the coolant
moving through the volume of electrostatic nano-motors.
remember that we dropped the term [math]P_{coolantfriction}(v)[/math]

Using ice (which will require some unspecified form of solid state coolant transport)
[math]v_{min}/s \gt 10^{15} W/m^3 (0.01) / (4 \times 10^{+6} J/(m^3 K) 100K + 3 \times 10^{+8} J/m^3)[/math]
[math]v_{min}/s \gt 10^{13} W/m^3 / (7 \times 10^{+8} J/m^3)[/math]
[math]v_{min}/s \gt (1/7) \times 10^5 (m/s)/m[/math]
[math]v_{min}/s \gt 0.15 \times 10^5 (m/s)/m[/math]
[math]v_{min}/s \gt 15000 (m/s)/m[/math]

Cooling of the coolant outside the nanomachinery volume

When assuming a closed coolant cycle to keep contaminants out easily
then at some point convective or radiative radiators are needed. Larger in area than the flow cross-section.
The flow of coolant is "easy" to spread out in the case of a cube as the geometry of choice.
In case of a thin layer this becomes increasingly challenging the bigger the area of the layer is.
Gauss' law. Just like the electric field of an infinite plate capacitor does not drop with distance
the coolant flow of an infinite area nano-machinery layer does not drop either.

Example: Macroscopic wheel with electromechanical metamaterial drive

Assuming a very thin shearing drive layer in a wheel about the size of a car wheel.
Let's go for a layer of thickness of one micron as that seemed viably coolable in the preceding analysis.
Getting to some exploratory engineering numbers rather asking for the ultimate limits.

Wheel width w, layer diameter d, layer thickness t
[math]A = d \cdot \pi \cdot w = 0.6m \cdot \pi \cdot 0.2 m = 0.377m^2[/math]
[math]V = A \cdot t = 0.377m^2 \cdot 1µm = 3.77 \times 10^{-7}m^3[/math]
[math]P_{work} = V \cdot \pi' = 3.77 \times 10^{-7}m^3 * 10^{15} W/m^3 =[/math] 377MW – Yikes!
[math]P_{loss} = P_{work} (1-\eta) = 3.77MW = 3.77MJ/s[/math]
[math]Q_{cool} = P_{loss} / (c_V \Delta T + \epsilon_{melt})[/math]
[math]Q_{cool} = (3.77MJ/s) /(4 \times 10^{+6} J/(m^3 K) \cdot 100K) = 9.425 \times 10^{-3}m^3/s = 9.425 ~ liter/s[/math]
[math]v_{cool} = Q_{cool}/A = 25mm/s[/math]
Odd that we did not get 15mm/s. Gotta go investigate, but the OOM is right.
Practically this thin layer would be spread out over a bigger volume of more dispersed nano-machinery.

Now imagine this would have been a 10 cm thick layer instead.
– 10^5 times thicker thus …
=> 10^5 times more total power (37.7 TW)
=> 10^5 times more losses (377GW)
This would most likely not be feasibly steady state coolable even by the most advanced means.

The message to remember here is:
The in Nanosystems identified maximal power density value of [math]\pi' = 10^{15} W/m^3[/math]
is not meant to be taken naively at face value.
It is not meant to be feasible at steady state in bulk macroscopic volumes.
At best at mesoscale near the limit of human vision.

Exotic means for ultra high speed flow cooling

See main page: The limits of cooling

Dynamic friction in liquids is really bad for nanoscale channels as
resistance against flow grows with the fourth power of channel diameter (inversely).

Solid state transfer (like pellets filled with ice or water) can completely avoid that
channel diameter problem but high speeds are still a huge problem.
Superlubrication does not help for high speeds. On the contrary.
Dynamic superlubric friction grows with the square of speed. 10x speed 100x the heat.
One can try to cheat using infinitesimal bearings and larger diameter roller gearberings
but for cooling purposes this comes at the cost of
hampering the heat transport across the thicker more porous interface.

One can try going sort of physically contactless by using electrostatic levitation.
This limits heat transport modes to radiative transmission only. No phonons.
If the nanomachinery can tolerate high temperatures
then this may be effective as radiative heat transfer
scales with the fourth power of temperature
Difficult nontrivial questions here.

Relation to exploratory engineering

The exploratory engineering question to answer would be:
What cooling power densities could be highly predictably reached using reliable basic models and large safety margins
that are still significantly bigger than what we can do today?
Not: What are the maximal cooling capabilities that we bight expect.
This would not be exploratory engineering.

Related

• Power density
• The limits of cooling (Cooling)
• Mechanical energy transmission, Thermal energy transport (Energy transmission)
• Common misconceptions about atomically precise manufacturing