The quantity of liquid available in its storage tank is often monitored by a cap

Question

The quantity of liquid available in its storage tank is often monitored by a capacitive level sensor. This sensor is a vertically aligned cylindrical capacitor with outer and inner conductor radii Ra and Rb, whose length l spans the height of the tank. When a nonconducting liquid fills the tank to a height h l from the tank’s bottom, the dielectric in the lower and upper region between the cylindrical conductors is the liquid (Kliq) and its vapor (KV), respectively (Fig. 24-33). (a) Determine a formula for the fraction f of the tank filled by liquid in terms of the level-sensor capacitance C. [Hint: consider the sensor as a combination of two capacitors; are they in series or parallel?] (b) By connecting a capacitance- measuring instrument to the level sensor, f can be monitored. Assume the sensor dimensions are l = 2.0 m, Ra = 5.0 mm, and Rb = 4.5 mm. For liquid nitrogen (Kliq = 1.4, KV = 1.0), what values of C (in pF) will correspond to the tank being completely full and completely empty?

Explanation / Answer

Capcitance for a cylinder: C = (2piL)/ln(Ra/Rb)

and = K0 and K = 1/4 pi epsilon

C = l/ln(Ra/Rb)

C = 9.5F

F = h/l

given l is less than or equal to h

for l = h

C = Cliq + Cv

= (2piKliq0h)/ln(Ra/Rb) + (2piKv0(L-h)/ln(Ra/Rb)

after some simplification:

= (2pi0(hKliq - hKv + LKv)/ln(Ra/Rb)

= 2 pi 0 / [ln (Ra/Rb)]       

= 1482pF

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