91Ó°ÊÓ

A formulation for the exit velocity in an MPD that allows for a simple estimate of the accelerator length is shown below; these equations relate the accelerator distance to the velocity implicitly through the acceleration time \(t\). Considering a flow at a constant plasma of density \(\rho_{m}\) (which does not choke), solve Newton's second law first for the speed \(v(t)\) and then for the distance \(x(t)\) and show that $$ \begin{aligned} &v(t)=\left(E_{y} / B_{z}\right)\left[1-e^{-t / \tau}\right]+v(0) e^{-t / \tau} \\ &x(t)=\left(E_{y} / B_{z}\right)\left[t+r e^{-t / r}-r\right]+x(0) \end{aligned} $$ where \(\tau=\rho_{m} / \sigma B_{z}^{2}\) and has units of seconds. For this simplified plasma model of an MPD accelerator, calculate the distance needed to accelerate the plasma from rest up to \(v=0.01(E / B)\) and the time involved. Take the plasma conductivity as \(\sigma=100 \mathrm{mho} / \mathrm{m}, B_{z}=10^{-3}\) tesla web \(/ \mathrm{m}^{2}\) ), \(\rho_{m}=10^{-3} \mathrm{~kg} / \mathrm{m}^{3}\), and \(E_{y}=1000 \mathrm{~V} / \mathrm{m} .\) Answer: \(503 \mathrm{~m}, 0.1005 \mathrm{sec}\).

An arcjet delivers \(0.26 \mathrm{~N}\) of thrust. Calculate the vehicle velocity increase under gravitationless, dragless flight for a 28-day thrust duration with a payload mass of \(100 \mathrm{~kg}\). Take thruster efficiency as \(50 \%\), specific impulse as \(2600 \mathrm{sec}\), and specific power as \(200 \mathrm{~W} / \mathrm{kg}\). This is not an optimum payload fraction; estimate an \(I_{s}\) which would maximize the payload fraction with all other factors remaining the same.