**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

**6.4.3.3 Hydraulic and
Acoustic Tethers**

Power is easily conveyed to medical nanodevices through tethers
by simple hydraulic means. For example, a virtually leakproof thick-walled fullerene
nanotube measuring 2r_{tube} = 1 micron in diameter and l_{tube}
= 1 meter long could safely transport pressurized fluid of absolute viscosity
h = 6.9 x 10^{-4} kg/m-sec (310 K water)
at a fluid pressure p_{tube} = 5 atm to drive a mechanical turbine or
valved reciprocating piston system, establishing an energy density of 0.5 pJ/micron^{3}
and delivering P_{n} = p r_{tube}^{4}
p_{tube}^{2} / 8 h l_{tube}
~ 10 pW to a V_{n} = 1 micron^{3} nanorobot power plant at a
fluid flow velocity of v_{fluid} = r_{tube}^{2} p_{tube}
/ 8 h l_{tube} ~ 20 microns/sec and producing
a power density of ~10^{7} watts/m^{3}, assuming Poiseuille
flow.

Acoustic waves could be delivered by tether, but slightly
larger tubes are required. Consider a water-filled pipe with dimensions as defined
in the previous paragraph which is excited at one end by a vibrating piston
operating at a frequency n_{p} << v_{sound}/r_{tube}
(~1 GHz for 2r_{tube} = 1 micron). For such small tubes, the inertia
and kinetic reaction of the fluid may be neglected in favor of the frictional
force (e.g., Poiseuille flow; Eqn. 9.25), since
the compressions and rarefactions of the fluid are practically isothermal on
account of the almost perfect heat conduction, and the power transmission is
given by:

with the attenuation coefficient given by Rayleigh's classical
formulation:^{649,650}

^{}
{Eqn. 6.45}

where g is the ratio of specific
heats (1.004 for water, 1.009 for seawater), v_{sound} = 1500 m/sec
and r = 993.4 kg/m^{3} for water at 310 K.
Maximum energy transfer (like a rigid bar) occurs at integral multiples of half-wavelengths
of the incident sonic waves, in other words, at n_{p}
= n v_{sound} / 2 l_{tube}, where n = 1 is the fundamental frequency
or first harmonic, n = 2 is the first overtone or second harmonic, and so forth.
Minimum attenuation occurs at n = 1, the first harmonic.

A periodic source pulse of an intensity low enough to avoid
cavitation in pure water (~10^{4} watts/m^{2}; Section
6.4.1) applied as input power P_{0} across a tube area of pr_{tube}^{2}
produces a 10 pW output power for l_{tube} __<__ 300 microns at
n_{p} = 2.5 MHz and r_{tube} = 0.5
microns with P_{0} = 8000 pW and a_{tube}
~11,000 m^{-1}. To get l_{tube} = 1 meter length requires r_{tube}
= 15 microns and n_{p} = 750 Hz to deliver
10 pW at the output. (A three-phase "alternating current" acoustical power transmission
system using 100 Hz waves traversing three water-filled pipes was patented by
M. Constantinesco in 1920.)^{650}

Diamond rods can also make nearly lossless acoustic power transmission lines (Section 7.2.5.3).

Last updated on 18 February 2003