FEEP, MEMS thrusters — micro-propulsion

WHAT are we even trying to do?

WHY high exhaust speed matters — recall the rocket equation logic. To change velocity by $\Delta v$ using exhaust velocity $v_e$, the propellant fraction is set by $\frac{m_p}{m_0} = 1 - e^{-\Delta v/v_e}.$ A small satellite carries almost no propellant, so we make $v_e$ huge (tens of km/s) — then even a teaspoon of propellant lasts the whole mission. Electric thrusters win here.

FEEP — Field Emission Electric Propulsion

HOW: deriving the physics from scratch

Step 1 — Why does a liquid form a sharp cone? A conducting liquid at the tip of an emitter feels two competing effects:

• Surface tension $\gamma$ tries to keep the surface smooth (pulls inward).
• Electric field creates an outward electrostatic pressure $\frac{1}{2}\varepsilon_0 E^2$.

Why this step? When the electric pull balances surface tension, the liquid self-organizes into a sharp cone — the Taylor cone — with half-angle $\approx 49.3^\circ$. From the tip, ions are emitted.

Step 2 — What speed do the ions reach? An ion of charge $q$ and mass $m$ dropped through a potential difference $V$ gains kinetic energy $qV = \tfrac{1}{2} m v_e^2.$ Why this step? Energy conservation: electric potential energy → kinetic energy. Solve for exhaust speed: $\boxed{v_e = \sqrt{\dfrac{2qV}{m}}}$

Step 3 — What thrust do we get? Thrust is momentum per second. If the ion mass flow rate is $\dot m$, $F = \dot m\, v_e.$ The ion beam current is $I = \dfrac{q}{m}\dot m$ (charge per second), so $\dot m = \dfrac{m}{q} I$. Substitute: $\boxed{F = \dot m v_e = \frac{m}{q} I \sqrt{\frac{2qV}{m}} = I\sqrt{\frac{2mV}{q}}}$

Why FEEP is superb for precision

• Thrust set by beam current, which we can tune electronically down to nanoamps → µN thrust resolution.
• Very high $I_{sp}$ (thousands of seconds) → almost no propellant.
• Downside: needs kilovolt-level high voltage, thrust is tiny (µN), and cesium can contaminate surfaces.

MEMS Thrusters

HOW a cold-gas MEMS thruster works (derivation): Gas at chamber pressure $p_0$, temperature $T_0$ expands through a micro-nozzle. For an ideal expansion, energy conservation of a gas parcel gives exhaust speed (from enthalpy → kinetic energy): $v_e = \sqrt{\frac{2\gamma}{\gamma-1}\frac{R T_0}{M}\left[1-\left(\frac{p_e}{p_0}\right)^{\frac{\gamma-1}{\gamma}}\right]}.$ Why this step? A hot high-pressure gas has stored thermal (enthalpy) energy; the nozzle converts it into directed kinetic energy. The bracket is the fraction of enthalpy released across the pressure drop. Thrust: $F = \dot m v_e + (p_e - p_a)A_e.$

The MEMS catch — the Reynolds number. At micro-scale, channels are tiny, so $Re = \frac{\rho v L}{\mu}$ is small → viscous (friction) losses dominate and the boundary layer eats much of the flow. This is why real MEMS efficiency is lower than the ideal formula predicts.

Worked Examples

Common Mistakes

Flashcards

Connections

• Rocket Equation — why high $v_e$ saves propellant.
• Specific Impulse — $I_{sp}=v_e/g_0$, the efficiency metric.
• Ion Thrusters — same charge-accelerate principle, larger scale.
• Electrospray & Colloid Thrusters — cousin of FEEP using charged droplets.
• Taylor Cone — electrohydrodynamic surface shape.
• Reynolds Number — why micro-nozzles lose efficiency.
• CubeSats & Attitude Control — the customers of micro-propulsion.

Concept Map

Hinglish (regional understanding)