Power density
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).
Contents
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:
- – W/m² … See: Mechanical energy transmission
(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
- Energy conversion – There's an especially high lower bound for the maximal power density in electromechanical converters
- Energy density
High power energy transmission as a combination of:
- Chemical energy transmission
- Mechanical energy transmission
- Entropic energy transmission
- Thermal energy transmission – other quite different constraints here
External Links
Wikipedia
- Surface power density – W/m²
- Power density – W/m³
