{Eqn. 9.88}
Nanomedicine, Volume I: Basic Capabilities
© 1999 Robert A. Freitas Jr. All Rights Reserved.
Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999
9.5.3.4 Nanoflyer Force and Power Requirements
What forces must be applied, and what power must be expended, in order for a flying nanorobot to maintain continuous and controlled progression through the air? An exact calculation requires a detailed knowledge of the mode of locomotion, the shape and dimensions of wing or body, airfoil surface characteristics, and many other factors. However, the force needed to drive a spherical flyer of mass mnano through the air at a velocity vnano may be approximated by summing the three drag components of skin friction (Fviscous), pressure drag (Finertial), and induced drag due to lift (Finduced), as:
while conservatively taking the dimensionless coefficient of drag CD ~ 2. Induced drag, associated with the shedding of transient vortexes from each wing tip, is an important consideration in macroscale winged vehicles but may be ignored here (e.g., Finduced ~ 0) for microscale flyers operating in the viscous or transitional flight regimes.
The nanorobot aeromotive energy plant, of efficiency e%, must develop a continuous power of:
giving a whole-device power density of:
These formulas give somewhat conservative results (e.g., overestimates of required force and power) because they do not take into account special body shapes, dynamic wing motions (e.g., delayed stall, rotational circulation, and wake capture3268), virtually complete pressure recovery in creeping flow, and so forth that may be used to improve flight efficiency in an optimized design. On the other hand, flight efficiency may be reduced by the accretion of water molecules and other environmental substances on working surfaces, and by the power expended in shearing air (dominant at the micron scale) or in imparting net kinetic energy to the air (at higher Reynolds numbers).
Table 9.5 shows force, power, and power density for spherical flying nanorobots of various sizes and airspeeds, computed using the above relations. Note that power scales as ~vnano2 in the viscous regime, ~vnano3 in the transitional regime. (The maximum sustainable velocity for transitional-regime insects is ~11 m/sec.)*
* The first well-documented report of maximum insect flight speed was by Tillyard,3157 who used a stopwatch to time the flight of the dragonfly Australophlebia costalis along a downhill slope at 27 m/sec; Hocking3158 subsequently calculated that the maximum speed of A. costalis would only be 16 m/sec on a level surface. Unpublished slow-motion cinematography research by Butler3159 reported an unconfirmed 40 m/sec burst of speed by the male Hybomitra hinei wrighti during an Immelmann Turn maneuver3160 at the beginning of female pursuit.
For an Rnano = 1 micron aerial nanorobot, taking hair = 0.0183 x 10-3 kg/m-sec and rair = 1.205 kg/m3 in dry air at 20°C and at 1 atm, with rnano ~ 1000 kg/m and e% = 0.10 (10%), a vnano = 1 cm/sec airspeed requires Fnano ~ 4 pN and a Pnano ~ 0.4 pW powerplant (Dnano ~ 82,000 watts/m3). At vnano = 1 m/sec, then Fnano ~ 350 pN and Pnano ~ 3500 pW powerplant (Dnano ~ 8 x 108 watts/m3). A much larger Rnano = 1 mm nanorobot flying at vnano = 1 m/sec requires Fnano ~ 4 microN and Pnano ~ 41 microW (Dnano ~ 10,000 watts/m3). Again, an optimized design might reduce some of the power figures by a factor of 10 or more.
Aerobot power density Dnano scales as ~vnano2 and ~Rnano-2 in the viscous regime. In the transitional regime, power density scales as ~vnano3, and as ~Rnano-2 at low velocities and ~Rnano-1 at high velocities. Thus, to minimize power density and therefore conserve energy, circumcorporeal clouds of aerial nanorobots may coalesce into progressively larger but fewer tightly-packed clumps as the collective velocity of the cloud moves to higher airspeeds, assuming that the aeromotive mechanism design is largely scale-invariant over the full size range of the progressive aggregations. This strategy is most effective in the viscous regime.
For example, consider a cloudlet consisting of 1 million nanorobots, each of size Rnano = 1 micron. With individual nanorobots traveling at vnano ~ 30 cm/sec, the cloudlet consumes ~0.4 milliwatts and operates at a power density of ~108 watts/m3. Now assume that the cloudlet must speed up to 10 m/sec to track a fast-moving object around which it is stationkeeping, or to compensate for a heavy wind. If the individual nanorobots comprising the cloudlet simply increase their airspeed to 10 m/sec, then power density in each nanorobot increases to ~1011 watts/m3 and cloudlet power consumption rises to ~400 milliwatts (a 1000-fold increase). However, if the nanorobots temporarily aggregate into a single collective approximating a single device of Rnano = 100 microns, then the power consumption of the collective can be held to the original 0.4 milliwatts and power density remains constant at ~108 watts/m3. Facultative aggregation may permit stationkeeping over a wide range of velocities without significantly increasing power. (Other power-conserving behaviors, such as preferential migration into the downwind slipstream of a rapidly-moving tracked object, are not considered further here but may be useful.)
How fast can nanoflyers accelerate? The answer is highly design-dependent but a crude generalization may be made. Consider a spherical nanorobot of radius Rnano that uses circumferential wings of similar size to impart momentum to a nearby mass of air by speeding up that mass of air to a higher velocity. The wings sweep air from a circular cross-section of radius 2Rnano. The nanoflyer accelerates with constant acceleration anano = vnano2 / 2 Xaccel from a standing start to a final velocity vnano in a time taccel = vnano/anano and a running distance Xaccel. The mass of the swept-out air is Mair = 4 p rair Xaccel Rnano2 and the mass of the nanorobot is Mnano = (4/3) p rnano Rnano3. To conserve momentum, Mair vair = Mnano vnano, hence vair = rnano vnano Rnano / 3 rair Xaccel; vair < vsound = 343 m/sec in dry air at 20°C and 1 atm to remain subsonic (Section 9.5.3.5). The kinetic energy imparted to the sweptout air is KEair = (1/2) Mair vair2 and to the nanorobot is KEnano = (1/2) Mnano vnano2, with KEtotal = KEair + KEnano. Neglecting drag losses (e.g., ~3500 pW for Rnano = 1 micron and vnano = 1 m/sec; Table 9.5) and assuming negligible losses within the air-accelerating mechanism itself and any drag on the air used as reaction mass, then propulsive efficiency is e% ~ KEnano / KEtotal and total power consumption is Pnano ~ Pdrag + (KEtotal / e% taccel). In the examples below we take Rnano = 1 micron, vnano = 1 m/sec, Pdrag ~ 3500 pW (Table 9.5), rair = 1.205 kg/m3 for dry air at 20°C and 1 atm, and rnano = 1000 kg/m3; here the Reynolds number is near unity.
A 1 micron3 high-density (~1012 W/m3; Chapter 6) powerplant develops Pnano ~ 1 million pW. At this high power level, a nanorobot can accelerate at anano ~ 12,000 g's for taccel ~ 9 microsec, reaching vnano = 1 m/sec after crossing a running distance of Xaccel ~ 4.4 microns with an efficiency of e% ~ (0.016) 1.6% and vair ~ 64 m/sec. As an alternative approach, note that a 5-microsec gas discharge from a simple pressure-release pump (Section 9.2.7.1) of volume vreservoir ~ 0.1 micron3 having a pressure differential of 1000 atm (storing ~108 J/m3; Section 6.2.2.3) produces a mean discharge power of Pnano ~ 2 million pW, which allows anano ~ 15,000 g's of acceleration with e% ~ (0.01) 1% for a maximum gas exit velocity of vmax ~ 70 m/sec using a nozzle of length ltube = 1 micron and radius rtube ~ 10 nm (Eqn. 9.36).
At a more modest Pnano ~ 4500 pW, the nanoflyer accelerates at anano ~ 1100 g's for taccel ~ 96 microsec, reaching vnano = 1 m/sec after crossing a running distance of Xaccel ~ 48 microns with an efficiency of e% ~ 0.15 (15%) and vair ~ 6 m/sec.
Attitude control of flying machines is a problem often mentioned in MEMS aerobotics work;1576 airborne nanorobots may employ gravity-based dynamic stabilization (Section 8.3.3), gyrostabilization (Section 9.4.2.2), or other means. In 1998, the energetics and control of steering maneuvers in highly-maneuverable insects was being investigated.1581,1582
Last updated on 22 February 2003
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