© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

9.2.7.2 Positive Displacement Pumps

Mechanical energy may be converted to fluid flow using a side-mounted piston as a single-action reciprocating positive displacement pump. Consider a pipe of radius r_tube and length l_tube, with valves at either end and a piston mounted in a stub cylinder of radius r_stub mounted perpendicular to the pipe axis at the midpoint of the pipe length (Fig. 9.1). The valves may be purely passive, or may be actively controlled in synchronization with piston motions.

During a single pump cycle, the valve at one end of the pipe is opened and the piston in the stub cylinder is drawn out, sucking a fluid volume V_cycle from the external environment into the pipe through the open end. The first valve is then closed, the second valve at the other end is opened, and the piston is pushed back into the stub cylinder, forcing an equal volume V_cycle of fluid out through the opposite end of the pipe which is now open. Cycling this pump at a frequency n_pump establishes a semicontinuous flow of V_pump = n_pump V_cycle, with pumped-fluid velocity v_flow and pump power P_flow given by:

{Eqn. 9.37}

{Eqn. 9.38}

where P_diss is the power dissipation due to piston wall friction, inertia, and fluid drag (Eqn. 6.13); for submicron pistons and n_pump <~ 1 MHz, P_diss / n_pump << kT in good designs. From Eqn. 9.25, pumping pressure is given by Dp = 8 h l_tube v_flow / r_tube².

Operating frequency is restricted by at least four factors:

1. The mechanical response time of the fluid defines a maximum operating frequency of n_max <~ v_sound / (l_tube/2); for l_tube = 100-1000 nm and v_sound ~ 1500 m/sec in water at 310 K, n_max = 30-3 GHz. (Note also the requirements that v_max << v_sound to avoid acoustic radiation losses, and that pressure and frequency must be sufficiently limited to avoid cavitation (Section 6.4.1).)

2. The maximum mechanical operating speed of nanoscale pistons is, conservatively, n_max ~ GHz.¹⁰

3. The maximum acceptable flow velocity v_max defines a maximum operating frequency n_max = p r_tube² v_max / V_cycle = v_max / 4 r_tube taking V_cycle ~ (2 r_tube)³ (e.g., one full piston throw opens a volume equal to one cubic pipe diameter); taking v_max = 4 m/sec, then n_max = 1-100 MHz for r_tube ~ r_stub = 1000-10 nm.

4. The maximum valving speed of v_valve = 0.01-1 m/sec (Sections 3.2.4 and 3.3.2) when applied to valves of radius r_tube defines a maximum valving frequency of n_max = v_valve / r_tube; for r_tube = 1000-10 nm, n_max = 0.01-100 MHz.

These considerations suggest that as a conservative limit, n_max ~ 1 MHz may be appropriate in nanomedical piston pump designs that are intended to interact with an aqueous or protoplasmic external biological environment. Nanorobot piston pumps transferring durable fluids internally may safely operate at frequencies up to ~GHz and pressures >1000 atm.

From Eqns. 9.37 and 9.38, and taking r_tube = 10 nm, l_tube = 100 nm, n_pump = 1 MHz, h = 0.6915 x 10^-3 kg/m-sec for 310 K water, and V_cycle ~ (2 r_tube)³ = 8000 nm³, then v_flow ~ 3 cm/sec, Dp = 1.6 atm, and P_flow ~ 1 pW assuming P_diss << P_flow, which will normally be the case. The mean pressure amplitude developed during each piston cycle is Dp_piston ~ 8 h l_tube v_flow / r_tube² ~ 1 atm. Reynolds number N_R ~ 0.0004 (Eqn. 9.29), giving laminar flow throughout the cycle; power losses due to differential compression of the streamlines as fluids curve through an elbow to enter and exit the piston stub should be negligible. If precise control of flow volume is not necessary, the active valves described above may be replaced by passive one-way check valves or flap valves analogous to those found in human veins (Fig. 8.3).

Many other mechanical positive displacement pumps are readily imagined -- including diaphragm pumps, rotary pumps (valveless), hydraulic vane pumps (valveless), lobe pumps (valveless), screw pumps (valveless), reciprocating flap-valve or plunger pumps, bellows pumps, multiple piston pumps, continuous ciliary²⁶⁹⁶ or peristaltic pumps (valveless),^1215,1216 and paddlewheel or waterwheel "gear" pumps (valveless) -- all with similar scaling and performance parameters. The dominance of viscous forces over inertial forces in fluid flow at small scales favors the use of these positive displacement or static-type pumps over the use of rotodynamic or kinetic-type pumps such as centrifugal pumps (with impeller blades) and venturi or jet pumps (e.g., using a motive fluid, nozzle and diffuser).

Last updated on 20 February 2003