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
22.214.171.124 Hovering Flight
In large helicopters, thrust is produced by imparting a downward velocity to the mass of air flowing through the rotor, with lift proportional to the change in momentum. In 1997, engineers at the Institute for Microtechnology in Mainz, Germany, constructed a 3-cm long, 1-cm tall mini-helicopter weighing 0.3 gm with two blades turning at ~1700 Hz, and flew it to a hovering altitude of 13 cm, then landed it safely. I. Kroo at Stanford University has built and flown a "mesicopter" whose motor is 3 mm in diameter and 5 mm long, weighing ~0.3 gm, that turns the oddly-shaped rotors at ~50,000 rpm.3246
As helicopter size shrinks through the transitional regime, drag grows at the expense of lift and the rotorblade microhelicopter becomes increasing inefficient. However, the viscous-lift helicopter (Fig. 9.22) is estimated to be able to hover by rotating its four wheels at a frequency of nwheel ~ 3 rnano g Rwheel / 32 hair ~ 1 KHz, taking rnano = 2000 kg/m3, g = 9.81 m/sec2, and Rwheel = 10 micron.1982
In the general case, the terminal velocity vterminal of a compact non-aerodynamic body of any size falling through fluid may be approximated by setting:
and solving the resulting quadratic for vterminal (= vnano), using Fviscous and Finertial for spherical bodies from Eqns. 9.89 and 9.90. This velocity may then be used in Eqns. 9.88 through 9.92 to conservatively estimate Fnano, Pnano, and Dnano for hovering. Representative values are given in Table 9.5, assuming flight in dry sea-level air and powerplant efficiency e% = 0.10 (10%). Hovering power Phover scales as ~Rnano5 for Rnano <~ 10 microns and as ~Rnano3.5 in the Rnano ~ 0.1-1 mm range; for Rnano >~ 1 mm, aerodynamic lift forces are available, the computation of which is beyond the scope of this book. For Rnano = 1 micron in air, vterminal = 120 microns/sec (Section 126.96.36.199) and Phover ~ 0.00005 pW (e% = 0.10 (10%)); for Rnano = 10 microns, vterminal = 1.2 cm/sec and Phover ~ 5 pW (e% = 0.10 (10%)).
In the special case of flapping winged aerobots in the transitional flight regime (e.g., 100 microns <~ Rnano <~ 10 cm) during steady-state hovering, T. Weis-Fogh1578,1583 calculates that the average aerodynamic power needed for small winged objects to remain airborne is:
where rair = 1.205 kg/m3 at 1 atm and 20°C, t is a shape factor that equals 0.5 for a rectangular wing, 0.1 for a triangular wing attached at the base, and 0.4 for a triangular wing attached at the apex; CD = 0.07-0.36 for bats, birds, and insects with wing length of Lwing = 0.25-13 cm and wing width (chord) of wwing = 0.7-55 mm (flight mass M = 0.001-20 gm); wingstroke frequency nwing ranges from 15 Hz at Lwing = 13 cm (large hummingbird) to 600 Hz at Lwing = 0.25 cm; and stroke angle jwing is the angle subtended by each flapping wing in the flapping plane during a complete stroke cycle, typically 2-3 radians (120°-180°).
Thus for example, the common honeybee Apis mellifera (NR ~ 1900) has M = 100 milligrams, Lwing = 1.0 cm, wwing = 0.43 cm, t = 0.27 (half-ellipse), CD = 0.09, nwing = 240 Hz, and jwing = 2.09 rad, giving Phover = 1 milliwatt with a dynamic efficiency of e% = 30%*; during level flight at peak speeds near ~6 m/sec,739 experimentally-measured honeybee metabolic demand is 20-60 milliwatts.1580 Honeybees can also carry a ~40 milligram payload of honey. At the lower extreme of the transitional flight regime, the parasitic chalcid wasp Encarsia formosa (NR ~ 15) has M = 25 microgram, Lwing = 620 microns, wwing = 230 microns, t = 0.50 (rectangle), CD = 3.20, nwing = 400 Hz, and jwing = 2.36 rad, giving Phover = 0.4 microwatt. Wingspeed vwing > 1.5 m/sec during the downstroke.1578 From the conservative assumption that biological wing muscles are limited to ~200 watts/kg (~300,000 watts/m3), Weis-Fogh1577 estimates than no flying animal with a mass larger than ~100 grams can hover continuously by means of wing flapping and wing twisting alone.
* A few other dynamic efficiency figures for hovering flight include the hornet wasp (31%), hummingbird (51%), mosquito (70%), fruit fly (95%), and butterfly (97%).1578
For artificial nanorobotic flyers near the edge of the design envelope, maximum wingbeat frequency nmax <~ vwing / Lwing. To avoid supersonic turbulence, vwing < vsound ~ 343 m/sec in air at 20°C and 1 atm, giving nmax <~ 10-100 MHz for Lwing = 3-30 microns and <~100 KHz for Lwing = 3 mm. For each small-wing cycle at 100 MHz, the boundary layer after an impulsive start is at most (modified from Prandtl1584) dlayer ~ (hair / rair nmax)1/2 ~ 0.4 micron ~ lgas ~ 0.2 microns, the mean free path of air molecules at 1 atm (Eqn. 9.23); faster cycling would allow insufficient time for mechanical coupling to the medium. Note also that vsound is significantly slower than the torsional deformation propagation velocity vtorsion ~ (G / rwing)1/2 ~ 12,000 m/sec along the wing structure, taking G ~ 5 x 1011 N/m2 and rwing ~ 3510 kg/m3 for diamondoid materials. For comparison, the slowest insectile wingbeat frequency is ~5 Hz for the swallowtail butterfly (Papilo machaon); the fastest is ~1046 Hz for the tiny midge Forcipomyia in natural conditions, up to ~2200 Hz at 310 K in laboratory experiments with truncated wings.739,2033
Last updated on 22 February 2003