View source for Limits of power density imposed by limits of cooling


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

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

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

== Summary of results (tl;dr) ==

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

'''Very much possible and certainly far from the limit:''' <br>
Still for a concrete example of a wheel hub motor in a car sized wheel <br>
total power can easily exceed limits of current day tech (electromagnetic motors) by more than an order for magnitude. <br>
* todays powerful car scale wheel-hub-motors are about 100kW typiclal 200kW (near limit?) 
* atomically precise gemstone based nano-electrostaic [[shear drives]] <br>4000kW with intentionally wide margins upwards. <br> If this powerful of a wheel hub motor even still makes sense is an other questions. <br> 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.''' <br>

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

With all these assumptions: '''The question we want to ask is:''' <br>
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. <br>
Undispersed bulk flow. That is: We ask for the speed through one fat coolant supply channel of the cubes cross sectional area. <br>
Dispersed flow will need to be a bit faster but only by a small constant factor so we'll ignore that here. <br>

== Symbolic analysis ==

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

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

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

== Numbers ==

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

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

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

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

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

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

== Cooling of the coolant outside the nanomachinery volume ==

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

== Example: Macroscopic wheel with electromechanical metamaterial drive ==

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

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

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

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

== 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 <br>
resistance against flow grows with the fourth power of channel diameter (inversely). <br>

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

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

== Relation to exploratory engineering ==

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

== Related ==

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