Thrust needed for Ion Thruster Lift off from Earth surface

Question

I've been calculating the needed mass flow rate for a hypothetical ion rocket, lifting off from Earth's surface and propelled with a Dual-Stage 4-Grid ion thruster. Data given: mass of the spaceship /inkl. propellant/ is 1000 t, Isp=19300s. So, the thrust I need is F=mg=9,81m/s2 x 1000000kg=9810000 N. For the exhaust velocity I come to Ve=Isp*g=19300s x 9,81m/s2=189333 m/s. Finally, for the mass flow rate I come to F/Ve=(9810000kgm/s2)/(189333m/s)=51.813 kg/s. This mass flow rate of approx. 52 kg/s seems pretty low to me, even considering the high Isp. Is there any error in the logic and calculations? Thanks guys!

Answer 1

Thrust needed for Ion Thruster Lift off from Earth surface... Is there any error in the logic and calculations?

calculation:

$$m = 1 \times 10^6 \ \text{kg}$$ $$F = 9.8 \times 10^6 \ \text{kg m/s}^2 \approx v_E \frac{dm}{dt}$$

If "Isp=19300s" then:

$$v_E = 9.8 \times 1.93 \times 10^4 \approx 1.89 \times 10^5 \text{m/s}$$

That definitely gives $dm/dt$ of 51.8 kg/s, so your math checks out.

logic:

You're going to need a whole lot of electrical power to make this work. Usually the biggest power hog is the production of the plasma from neutral gas, not the acceleration. It's a highly inefficient process to heat electrons enough and contain them efficiently enough to ionize a flowing gas with a high ionization efficiency.

The traditional xenon is way way to expensive and rare to use in a launch vehicle at this rate (52 kg/s times say 154 seconds is about 8 tons of propellant) The price of xenon varies dramatically with purity, so you'll spend some time checking how well your particular engine design can deal with impurities that try to reabsorb your free electrons.

Instead you'll walk up the table of elements like SpaceX to krypton and then to argon which is about 1% of our atmosphere so much cheaper to produce and purify. It's probably a sweet spot though you may consider other alternatives.

The atomic mass of argon is about 40, so you'll be releasing about 1.3 moles per second or $1.3 \times 6 \times 10^{23} = 7.8 \times 10^{23} \ \text{atoms/sec}$. Each atom will be charged +1e so divide that by 1 Coulomb; that's $7.8 \times 10^{23} / 6.24 \times 10^{18} = 125,000 \ \text{amps}$ of charged argon!

I'm not going to calculate the coulomb repulsion in that ion beam, but I have a hunch it's not going to be held together by a normal nozzle. I think we're talking super-duper superconducting solenoids.

Let's calculate the voltage needed to accelerate our ions to $1.9 \times 10^5 \text{m/s}$. For one atom, the mass is $40 \times 938 \times 10^6 \text{eV/c}^2$. With $v/c$ of $6.3\times 10^{-4}$ and an atomic mass of $40 \times 938 \times 10^6 \text{eV/c}^2$ I get a kinetic energy of about 7500 eV per atom, or an accelerating voltage of 7.5 kV which is the right ballpark for an ion engine.

Note that 125,000 amps times 7500 volts is about a gigawatt of electrical power just to do the acceleration. I have a hunch you'll need a few orders of magnitude more than that to produce and control an absurdly large plasma and flow rate.