106-2018192 Practice for Midterm 1

Question 1

The disc with a hole

Your team is working on Saturn V's F1 engines. You are tasked with finding a way to measure the amount of liquid Nitrogen left in the tank which is responsible for filling the space the oxidizer is spent in a precise and quick manner without igniting a spark in a very very volatile situation. A capacitive level sensor is the way to go. The tank can be modeled as a vertically aligned cylindrical capacitor with outer and inner conductor radii $R_{a}$ and $R_{b}$ , whose length $l$ spans the height of the tank. When a nonconducting liquid fills the tank to a height $h(\leq l)$ from the tank’s bottom, the dielectric in the lower and upper region between the cylindrical conductors is the liquid ($K_{\textrm {liq}}$) and its vapor (Failed to parse (syntax error): {\displaystyle K_V}} ), respectively

(a) Determine a formula for the fraction F of the tank filled by liquid in terms of the level-sensor capacitance C. (b) By connecting a capacitance-measuring instrument to the level sensor, F can be monitored. Assume the sensor dimensions are l = 2.0 m, R a = 5.0 mm, and R b = 4.5 mm. For liquid nitrogen AK liq = 1.4, K V = 1.0B, what values of C (in pF) will correspond to the tank being completely full and completely empty?