**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 m_{nano} through the air at a velocity v_{nano} may
be approximated by summing the three drag components of skin friction (F_{viscous}),
pressure drag (F_{inertial}), and induced drag due to lift (F_{induced}),
as:

_{}
{Eqn. 9.88}

while conservatively taking the dimensionless coefficient
of drag C_{D} ~ 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., F_{induced} ~ 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 capture^{3268}), 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 ~v_{nano}^{2} in the viscous regime, ~v_{nano}^{3}
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; Hocking^{3158}
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 Butler^{3159} reported
an unconfirmed 40 m/sec burst of speed by the male *Hybomitra hinei wrighti*
during an Immelmann Turn maneuver^{3160}
at the beginning of female pursuit.

For an R_{nano} = 1 micron aerial nanorobot, taking
h_{air} = 0.0183 x 10^{-3} kg/m-sec
and r_{air} = 1.205 kg/m^{3} in dry
air at 20°C and at 1 atm, with r_{nano} ~
1000 kg/m and e% = 0.10 (10%), a v_{nano} = 1 cm/sec airspeed requires
F_{nano} ~ 4 pN and a P_{nano} ~ 0.4 pW powerplant (D_{nano}
~ 82,000 watts/m^{3}). At v_{nano} = 1 m/sec, then F_{nano}
~ 350 pN and P_{nano} ~ 3500 pW powerplant (D_{nano} ~ 8 x 10^{8}
watts/m^{3}). A much larger R_{nano} = 1 mm nanorobot flying
at v_{nano} = 1 m/sec requires F_{nano} ~ 4 microN and P_{nano}
~ 41 microW (D_{nano} ~ 10,000 watts/m^{3}). Again, an optimized
design might reduce some of the power figures by a factor of 10 or more.

Aerobot power density D_{nano} scales as ~v_{nano}^{2}
and ~R_{nano}^{-2} in the viscous regime. In the transitional
regime, power density scales as ~v_{nano}^{3}, and as ~R_{nano}^{-2}
at low velocities and ~R_{nano}^{-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 R_{nano} = 1 micron. With individual nanorobots
traveling at v_{nano} ~ 30 cm/sec, the cloudlet consumes ~0.4 milliwatts
and operates at a power density of ~10^{8} watts/m^{3}. 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 ~10^{11}
watts/m^{3} 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 R_{nano} = 100
microns, then the power consumption of the collective can be held to the original
0.4 milliwatts and power density remains constant at ~10^{8} watts/m^{3}.
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 R_{nano} 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 2R_{nano}. The nanoflyer accelerates with constant acceleration
a_{nano} = v_{nano}^{2} / 2 X_{accel} from a
standing start to a final velocity v_{nano} in a time t_{accel}
= v_{nano}/a_{nano} and a running distance X_{accel}.
The mass of the swept-out air is M_{air} = 4 p**
**r_{air} X_{accel} R_{nano}^{2}
and the mass of the nanorobot is M_{nano} = (4/3) p
r_{nano} R_{nano}^{3}. To conserve momentum,
M_{air} v_{air} = M_{nano} v_{nano}, hence v_{air}
= r_{nano} v_{nano} R_{nano}
/ 3 r_{air} X_{accel}; v_{air}
< v_{sound} = 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 KE_{air} = (1/2) M_{air} v_{air}^{2}
and to the nanorobot is KE_{nano} = (1/2) M_{nano} v_{nano}^{2},
with KE_{total} = KE_{air} + KE_{nano}. Neglecting drag
losses (e.g., ~3500 pW for R_{nano} = 1 micron and v_{nano}
= 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% ~ KE_{nano} / KE_{total} and total power consumption is
P_{nano} ~ P_{drag} + (KE_{total} / e% t_{accel}).
In the examples below we take R_{nano} = 1 micron, v_{nano}
= 1 m/sec, P_{drag} ~ 3500 pW (Table
9.5), r_{air} = 1.205 kg/m^{3}
for dry air at 20°C and 1 atm, and r_{nano}
= 1000 kg/m^{3}; here the Reynolds number is near unity.

A 1 micron^{3} high-density (~10^{12} W/m^{3};
Chapter 6) powerplant develops P_{nano} ~ 1
million pW. At this high power level, a nanorobot can accelerate at a_{nano}
~ 12,000 g's for t_{accel} ~ 9 microsec, reaching v_{nano} =
1 m/sec after crossing a running distance of X_{accel} ~ 4.4 microns
with an efficiency of e% ~ (0.016) 1.6% and v_{air} ~ 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 v_{reservoir}
~ 0.1 micron^{3} having a pressure differential of 1000 atm (storing
~10^{8} J/m^{3}; Section 6.2.2.3)
produces a mean discharge power of P_{nano} ~ 2 million pW, which allows
a_{nano} ~ 15,000 g's of acceleration with e% ~ (0.01) 1% for a maximum
gas exit velocity of v_{max} ~ 70 m/sec using a nozzle of length l_{tube}
= 1 micron and radius r_{tube} ~ 10 nm (Eqn.
9.36).

At a more modest P_{nano} ~ 4500 pW, the nanoflyer
accelerates at a_{nano} ~ 1100 g's for t_{accel} ~ 96 microsec,
reaching v_{nano} = 1 m/sec after crossing a running distance of X_{accel}
~ 48 microns with an efficiency of e% ~ 0.15 (15%) and v_{air} ~ 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