**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

**6.5.3 Nanorobot Power
Scaling**

As an initial, crude order-of-magnitude estimate of nanorobot
power consumption, allometric scaling of metabolism in biology for whole organisms
follows a 3/4 power law.^{698,3242}
Normalizing to P = 100 watts for an m = 70 kg human body mass and assuming ~water
density for nanorobots, then P = (4.13) m^{3/4} = 23 pW for a 1 micron^{3}
nanorobot, a power density of D_{n} ~ 2 x 10^{7} watts/m^{3}.

Surface power intensity considerations drive maximum nanorobot
onboard power density. For instance, the maximum safe intensity for ultrasound
is 100-1000 watts/m^{2} (pain threshold for human hearing ~100 watts/m^{2})
(Section 6.4.1) and the conservative maximum safe electromagnetic
intensity is also ~100 watts/m^{2} (Section 6.4.2).
Additionally, a surface at 373 K (boiling water) relative to a 310 K environment
radiates 100-1000 watts/m^{2} at emissivities e_{r} = 0.1-1
(Eqn. 6.19). Conservatively taking 100 watts/m^{2}
as the maximum safe energy flux across the surface of a spherical nanorobot
of radius r_{n}, then the maximum nanorobot power density is

for a device ~1 micron in diameter -- the largest safe whole-device
power density that should be developed in vivo. A 10^{9} watt/m^{3}
maximum implies a ~1000 pW limit for 1 micron^{3} nanorobots. Interestingly,
the energy dissipation rate required to disrupt the plasma membrane of ~95%
of all animal cells transported in forced turbulent capillary flows is on the
order of ~10^{8}-10^{9} watts/m^{3}.^{1185}

For chemically powered foraging nanorobots, another fundamental
constraint on power density is imposed by diffusion limits on fuel molecules.
From Eqn. 3.4, the maximum diffusion current
of glucose molecules of energy content E_{fuel} = E_{glu} (4765
zJ; Eqn. 6.16), burned at efficiency e% ~
0.50 (50%) in oxygen, to the surface of a spherical nanorobot of radius r_{n}
will support a maximum onboard power density of:

For D = 7.1 x 10^{-10} m^{2}/sec for glucose
in water at 310 K (Table
3.3), C = (0.67) 3.5 x 10^{24} molecules/m^{3} in (newborn
and) adult human blood plasma (Appendix B), and
r_{n} ~ 0.5 micron, D_{diff} = 17 x 10^{10} watts/m^{3}.
However, because oxygen dissolves only slightly in blood plasma and interstitial
fluid, the oxyglucose engine is more severely diffusion-limited by its oxygen
requirements than by its glucose requirements.* Applying Eqn.
6.58 and using (for oxygen) D = 2.0 x 10^{-9} m^{2}/sec
(Table
3.3), C = 7.3 x 10^{22} molecules/m^{3} in arterial blood
plasma (Appendix B), E_{fuel} ~ E_{glu}/6,
and r_{n} ~ 0.5 micron, D_{diff} = 7 x 10^{8} watts/m^{3}.
Once again, ~10^{9} watts/m^{3} appears to be a correct upper
limit for whole nanorobots.**

* Since oxyglucose foraging nanorobots are
not seriously glucose-limited, there is little to be gained by enabling energy
organs or cooperative nanorobot populations to secrete insulin/glucagon hormones
(mimicking the pancreas), cortisol hormones, etc. to artificially manipulate
serum glucose levels. Available oxygen also may be artificially manipulated
using nanorobotic compressed gas dispensers^{1400}
or by other means (Chapter 22).

** Note that as nanodevices get smaller, their surface/volume ratio expands. Thus the Square/Cube law predicts that smaller nanodevices can admit more energy (e.g., chemical, acoustic, electromagnetic) through their surfaces per unit enclosed volume of working nanomachinery, hence can have higher power densities, as illustrated by Eqn. 6.58.

The disposition of combustion byproducts, particularly CO_{2},
may provide another weak constraint on chemical systems. For example, a 10pW
glucose-burning nanorobot generates ~10^{7} molecules/sec of CO_{2}.
Pressurized to 1000 atm, this production rate fills ~0.001 micron^{3}/sec
of onboard storage space, assuming no venting. If CO_{2} is vented from
a population of 10^{12} 10-pW nanorobots uniformly distributed throughout
a ~0.1 m^{3} human body volume, then local CO_{2} concentration
rises by ~2 x 10^{-7} M/sec, reaching 0.0003 M after 1 hour of continuous
operation assuming no physiological removal -- still well below the normal ~0.001
M blood plasma CO_{2} concentration.

By implication, these limits also drive the maximum number
density of nanodevices deployable in human tissue. For example, at 10^{9}
watts/m^{3} the hottest 1-micron nanorobot develops 1000 pW; assuming
a ~0.1 watt power budget when restricted to the thyroid gland (Section
6.5.2), 100 million nanodevices may be deployed in the gland giving a maximum
number density of ~10^{13} nanorobots/m^{3}. Nanorobot power
consumption may range from ~0.1 pW for simple respirocytes^{1400}
(Chapter 22) up to ~10,000 pW or more for the largest
and most sophisticated repair and defensive in vivo devices (Chapter
21), but the typical simple micron-scale nanorobot may develop ~10 pW (roughly
in line with biologically-derived allometric scaling laws giving D_{n}
~ 10^{7} watts/m^{3}) and thus could safely achieve a number
density of ~10^{15} nanorobots/m^{3} (~10 micron mean interdevice
separation). Recall that the maximum diffusion-limited total power draw for
a population of oxygen-unrestricted glucose-energized foraging nanorobots in
cyto is ~70,000-300,000 pW (Section 6.3.4.1).

Minimum powerplant size varies with requirements. Assuming
>~10^{12} watts/m^{3} energy conversion for chemical (Section
6.3.4) or electrical (Section 6.4.1) power transducers,
then a 100 pW power supply subsystem inside a working nanorobot may be as small
as (~50 nm)^{3} in size.

Macroscale masses of working nanodevices may grow extremely
hot, placing major scaling limits on artificial nano-organs and other large-scale
nanomachinery aggregates (Chapter 14). As a somewhat
fanciful example, consider a macroscopic ball of radius R consisting of N tightly-packed
nanodevices each of mass density r and whole-nanorobot
power density D_{p} ~ 10^{9} watts/m^{3}, of which nanodevices
some fraction f_{n} are active, all suspended in mid-air. The ball grows
hotter as R (~N^{1/3}) rises, until at some "critical combustible mass"
M_{crit} = (4/3) p r
R_{crit}^{3} the surface temperature exceeds the maximum combustion
point for diamond in air (T_{burn} = 1070 K)^{691}
and the solid ball of nanorobots bursts into flame. (Sapphire devices cannot
burn, but have a T_{melt} ~ 2310 K;^{1602}
as a practical matter, nanomachinery may fail at temperatures significantly
below T_{burn}.)

From simple geometry and neglecting ~2% air conduction losses,
the maximum noncombustible aggregate radius R_{crit} is:

For e_{r} = 0.97 (e.g., carbon black) to maximize
heat emission at the lowest possible temperature and T_{environ} = 300
K, then R_{crit} = 0.22 mm for f_{n} = 100%. Assuming a full
cold start, critical time to incineration is t_{crit} = C_{V}
(T_{burn} - T_{environ}) / f_{n} D_{p} = 1.4
sec for f_{n} = 100% if nanorobot heat capacity C_{V} = 1.8
x 10^{6} joules/m^{3}-K (~diamondoid). Decreasing D_{p}
to a more reasonable 10^{7} watts/m^{3} or simply switching
off 99% of the nanorobots (f_{n} = 1%) increases R_{crit} to
~22 cm.

Last updated on 18 February 2003