**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

**10.3.1 Fluid Storage
Tank Scaling**

The ability of a vessel to store fluids at high pressure is
determined by the rupture strength of the valve and pipe systems and by the
tensile strength of the tank walls. A symmetric structural shell transmitting
only normal stresses in orthogonal directions may employ either a stressed skin
or a ribbed design. The stressed skin is more efficient because the same material
carries stress in both directions and there is integral resistance to secondary
torsional and bending loads.^{2023}
For a stressed skin spherical pressure vessel of radius R with wall material
of density r_{wall} and working stress s_{w},
containing fluid at pressure differential p_{fluid}, the required wall
thickness t_{wall} is given by:^{2023}

_{}
{Eqn. 10.12}

where g = 9.81 m/sec^{2} (acceleration of gravity).
For all but the most enormous macroscale tanks, the second term in the denominator
is negligible and so the maximum pressure differential that can be restrained
by a spherical microscale tank without bursting is:

_{}
{Eqn. 10.13}

Thus a tank of radius R = 1 micron made of diamondoid walls
which are t_{wall} = 5 nm (~30 carbon atoms) thick with s_{w}
= 10^{10} N/m^{2} (~0.2 times the failure strength of diamond;
Table
9.3) can store fluids up to a maximum pressure differential of p_{maxsph}
= 1000 atm. Empty tank mass is M_{tank} ~ 4 p
r_{wall} t_{wall} R^{2} =
(4 p r_{wall} / 2 s_{w})
p_{maxsph} R^{3}, for t_{wall} << R. A spherical
fullerene nanotank with defect-free single-carbon walls (t_{wall} ~
0.34 nm thick), pressurized to p_{maxsph} ~ 1000 atm differential, must
have R <~ 67 nm or it may burst.

Similarly, for a cylinder or pipe of radius R:

_{}
{Eqn. 10.14}

Note that a hemispherical-capped cylindrical tank, pressurized
until it explodes, will usually split its sides before blowing off its ends
because p_{maxsph} = 2 p_{max cyl}. Wall stress for a cylindrical
tube with Young's modulus E, inside radius R and wall thickness t_{wall}
upon which is imposed a pressure differential of p_{cyl} is ~ p_{cyl}
R / t_{wall,}^{362} giving
a wall strain of S ~ p_{cyl} R / t_{wall} E; hence a single-walled
carbon nanotube with R = 20 nm, t_{wall} ~ 0.2 nm, and E ~ 10^{12}
N/m^{2} pressurized to p_{cyl} = 1000 atm stretches by S ~ 1%.
(For comparison, aluminum soda pop cans are pressurized to ~2 atm differential,
beer cans ~1 atm.)^{3703}

For a toroid tank of small meridional radius r and large circumferential
(hoop direction) radius R, the burst pressure in the meridional (p_{max merid})
and hoop (p_{max hoop}) directions are:^{2023}

_{}
{Eqn. 10.15}

_{}
{Eqn. 10.16}

Hence as pressure is raised, the toroidal tank fails first in the meridional direction, much like the capped cylinder.

Spherical pressure vessels are the most efficient. The maximum
mass of gas that can be contained within a spherical tank at ideal gas pressures
(Section 10.3.2) is M_{gas} ~ (4 p
N_{A} m_{gas} / 3 R_{gas} T_{gas}) p_{max}
R^{3}, where N_{A} = 6.023 x 10^{23} molecules/mole
(Avogadro's number), m_{gas} is the mass per gas molecule, R_{gas}
= 8.31 J/mole-K (universal gas constant), and T_{gas} is gas temperature.
Thus we observe that the ratio of gas mass to structural mass (M_{gas}/M_{tank})
at a given pressure is constant and independent of the size of the tank. This
ratio reaches a maximum value at maximum pressurization; the maximum value is
independent of both tank size and the maximum pressure selected (within the
ideal gas range). In particular, M_{gas}/M_{tank} ~ 2 s_{w}
N_{A} m_{gas} / 3 R_{gas} T_{gas} r_{wall}
~ 21 for nitrogen gas molecules (m_{gas} = 4.65 x 10^{-26} kg/molecule)
stored at temperature T_{gas} = 310 K. That is, the tank can store up
to ~21 times its weight of gas.

Other important microtank design constraints include diffusion
leakage and vessel flammability (Section 10.3.4), vapor
pressure of tank materials at high temperatures (Section
10.3.5), thermal cycling during filling and discharging operations (Section
5.3.3), resistance to mechanical crushing, and susceptibility to acoustic
resonances (e.g., the natural frequency of an r_{sph} ~ 1 micron hollow
diamondoid sphere is of order ~v_{sound} / r_{sph} ~ 20 GHz,
fortunately much higher than most nanomedically useful frequencies).

Last updated on 24 February 2003