Difference between revisions of "Power density"

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(added * Limits of power density imposed by limits of cooling added a note on lower limits)
 
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{{Template:Stub}}
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Power densities are covered in [[Nanosystems]] in the early chapter about [[scaling law]]s. <br>
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{{wikitodo|Add reference to relevant Nanosystems section}}
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Lower bounds for maximally achievable power densities are [[exploratory engineering|explored]] in <br>
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later chapters covering [[electromechanical converters]]. They are ''very'' high.
  
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There are two kinds of power densities  
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Power-densities for [[mechanical energy transmission]] are explored here on this wiki a bit (follow the link).
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* aerial power density
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* volumetric power density
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{{todo|In how far can they be made comparable? And is there a way to get an intuitive grasp on them?}}
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The power densities mentioned in [[Nanosystems]] seem to exceed the maximum what seems possibly of force times speed
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== SI Units ==
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in [[mechanical energy transmission cables]]. <br> {{todo| Resolve this mystery}}
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There are two kinds of power densities. <br>
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Per area (areal) and per volume (volumetric).
 +
* areal power densities are relevant for power transmission
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* volumetric power densities are relevant for power conversion
 +
----
 +
'''Volumetric power density:'''
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* electromechanical conversion: >10^15 W/m³ ([[Nanosystems]] 11.7.3. Motor power and Power density)
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* mechanical transmission example: 1.7*10^19 W/m³ (1.0 m/s) ([[Nanosystems]] equation (2.13) 10nN 1nm³)
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* mechanical transmission example: 1.7*10^22 W/m³ (1.7 km/s) ([[Nanosystems]] equation (2.13) 10nN 1nm³) - huge dynamic friction!
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Example: The power transmitted by a gear divided by its volume. <br>
 +
 
 +
Please note that there are lower '''[[limits of power density imposed by limits of cooling]]''' <br>
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for bulk volumes of densely packed nanomachinery running steady state.
 +
----
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'''Areal power density:'''
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* – W/m² … See: [[Mechanical energy transmission]]
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{{todo|How & in how far can the two types be made comparable? And is there a way to get an intuitive grasp on them?}} <br>
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{{todo|If possible get power densities from [[electromechanical converters]] and [[mechanical energy transmission]] in a form that is comparable. Despite different units ...}}
 +
 
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== Ultimate limits & cooling ==
 +
 
 +
Details in [[Nanosystems]] 11.7.4. Energy dissipation and efficiency
 +
 
 +
While for
 +
* small volumes and steady state
 +
* big volumes and brief times (burst mode operation)
 +
We already have lower bound [[conservative estimation]] numbers, for
 +
* bigger volumes and longer operation times
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waste heart removal becomes the limiting bottleneck. <br>
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Even with very high efficiencies given extreme power densities there will still be produce way too much waste heat.
 +
 
 +
To get a better handle on the ultimate limits of [[gem-gum system]]s <br>
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lower bounds for ultimate limits for power-densities for waste heat removal need closer investigation. <br>
 +
That is ultra high performance cooling systems. Likely based on solid state capsules in machine phase. <br>
 +
 
 +
== Math for onset of cooling limiting power density by system geometry ==
 +
 
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[[Nanosystems]] only covers:
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* cooling systems sufficient for proposes the [[nanofactories]] – 11.5. Convective cooling systems
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* cooling for nanomechanical computation – 12.8. Cooling and  computational capacity
 +
 
 +
'''Questions:''' At which size scale does heat removal become the limiting factor for
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* a volume (cube or sphere shape)
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* a thin layer of specified thickness
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'''Volumetric (cube) case:'''
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* P_in = Q_V * V = Q_V * s³
 +
* P_out = Q_A * 6A = Q_A * s²
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* P_in = P_out
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* s = 6Q_A / Q_V
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This gives a maximal cube side-length.<br>
 +
Spherical case is left as exercise for the so inclined reader.
 +
 
 +
'''Thin layer case (s >> t):'''
 +
* P_in = Q_V * (A*t) = Q_V * (s²*t)
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* P_out = Q_A * (2*A + 4*s*t) ~ Q_A * (2*s²+0)
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* P_in = P_out
 +
* t = 2Q_A / Q_V
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This gives a maximal layer thickness. <br>
 +
The layer side-length s can be arbitrarily large.
 +
 
 +
----
 +
 
 +
'''TODO:''' That leaves Q_A for to be determined. <br>
 +
A lower bound for the maximally achievable heat removal power density.
 +
 
 +
'''Complications:'''
 +
* Thin layer case: Heat conduction alone does not work. When assuming s as infinitely large then the carry-heat-away power density never drops with distance to the layer  and the longer the (not widening) carry-heat-away-path gets the higher the cumulative thermal resistance gets. Thus active heat removal (material with heat capacity moving cold in & hot out) is a necessity. E.g. Boosted heat removal by fast active motion of thermal mass capsules.
 +
* volumetric case: The heat gradient is nonlinear since spherical hear flow. Thermal resistance integral to infinite radius is finite though.
 +
----
 +
* Going too far to the limits the heat removal system will itself start to produce waste heat complicating the analysis more.
 +
----
 +
* Analyzing '''[[burst mode operation]]''' it gets even more complicated needing additional assumptions on heat capacity of the power conversion machinery itself.
  
 
== Related ==
 
== Related ==
  
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* [[Energy conversion]] – especially high max power in [[electromechanical converters]] (to check)
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* [[Energy conversion]] – There's an especially high lower bound for the maximal power density in [[electromechanical converters]]
 
* [[Energy density]]
 
* [[Energy density]]
  
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* [[Higher throughput of smaller machinery]]
 
* [[Higher throughput of smaller machinery]]
 
* [[High performance of gem-gum technology]]
 
* [[High performance of gem-gum technology]]
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----
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* [[Limits of power density imposed by limits of cooling]]
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== External Links ==
 +
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Wikipedia
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* [https://en.wikipedia.org/wiki/Surface_power_density Surface power density] – W/m²
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* [https://en.wikipedia.org/wiki/Power_density Power density] – W/m³
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[[Category:Pages with math]]

Latest revision as of 19:36, 15 September 2024

Power densities are covered in Nanosystems in the early chapter about scaling laws.
Lower bounds for maximally achievable power densities are explored in
later chapters covering electromechanical converters. They are very high.

Power-densities for mechanical energy transmission are explored here on this wiki a bit (follow the link).

SI Units

There are two kinds of power densities.
Per area (areal) and per volume (volumetric).

  • areal power densities are relevant for power transmission
  • volumetric power densities are relevant for power conversion

Volumetric power density:

  • electromechanical conversion: >10^15 W/m³ (Nanosystems 11.7.3. Motor power and Power density)
  • mechanical transmission example: 1.7*10^19 W/m³ (1.0 m/s) (Nanosystems equation (2.13) 10nN 1nm³)
  • mechanical transmission example: 1.7*10^22 W/m³ (1.7 km/s) (Nanosystems equation (2.13) 10nN 1nm³) - huge dynamic friction!

Example: The power transmitted by a gear divided by its volume.

Please note that there are lower limits of power density imposed by limits of cooling
for bulk volumes of densely packed nanomachinery running steady state.

Areal power density:

(TODO: How & in how far can the two types be made comparable? And is there a way to get an intuitive grasp on them?)
(TODO: If possible get power densities from electromechanical converters and mechanical energy transmission in a form that is comparable. Despite different units ...)

Ultimate limits & cooling

Details in Nanosystems 11.7.4. Energy dissipation and efficiency

While for

  • small volumes and steady state
  • big volumes and brief times (burst mode operation)

We already have lower bound conservative estimation numbers, for

  • bigger volumes and longer operation times

waste heart removal becomes the limiting bottleneck.
Even with very high efficiencies given extreme power densities there will still be produce way too much waste heat.

To get a better handle on the ultimate limits of gem-gum systems
lower bounds for ultimate limits for power-densities for waste heat removal need closer investigation.
That is ultra high performance cooling systems. Likely based on solid state capsules in machine phase.

Math for onset of cooling limiting power density by system geometry

Nanosystems only covers:

  • cooling systems sufficient for proposes the nanofactories – 11.5. Convective cooling systems
  • cooling for nanomechanical computation – 12.8. Cooling and computational capacity

Questions: At which size scale does heat removal become the limiting factor for

  • a volume (cube or sphere shape)
  • a thin layer of specified thickness

Volumetric (cube) case:

  • P_in = Q_V * V = Q_V * s³
  • P_out = Q_A * 6A = Q_A * s²
  • P_in = P_out
  • s = 6Q_A / Q_V

This gives a maximal cube side-length.
Spherical case is left as exercise for the so inclined reader.

Thin layer case (s >> t):

  • P_in = Q_V * (A*t) = Q_V * (s²*t)
  • P_out = Q_A * (2*A + 4*s*t) ~ Q_A * (2*s²+0)
  • P_in = P_out
  • t = 2Q_A / Q_V

This gives a maximal layer thickness.
The layer side-length s can be arbitrarily large.

TODO: That leaves Q_A for to be determined.
A lower bound for the maximally achievable heat removal power density.

Complications:

  • Thin layer case: Heat conduction alone does not work. When assuming s as infinitely large then the carry-heat-away power density never drops with distance to the layer and the longer the (not widening) carry-heat-away-path gets the higher the cumulative thermal resistance gets. Thus active heat removal (material with heat capacity moving cold in & hot out) is a necessity. E.g. Boosted heat removal by fast active motion of thermal mass capsules.
  • volumetric case: The heat gradient is nonlinear since spherical hear flow. Thermal resistance integral to infinite radius is finite though.
  • Going too far to the limits the heat removal system will itself start to produce waste heat complicating the analysis more.
  • Analyzing burst mode operation it gets even more complicated needing additional assumptions on heat capacity of the power conversion machinery itself.

Related

High power energy transmission as a combination of:

External Links

Wikipedia

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