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

a) They are in parallel,

So equivalent Capacitance is sum of these two capacitors.

Ceq = Cliq +Cvap

C = Kliq. 20h/ln(ra/rb) + Kv 20(l-h)/ln(ra/rb)

Divide both side by l

C/l=  Kliq. 20(h/l)/[ln(ra/rb)] + Kv 20(1-h/l)/[ln(ra/rb)]

now,h/l =f which is fraction

C/l= Kliq. 20f/[ln(ra/rb)] + Kv 20(1-f)/[ln(ra/rb)]

[ln(ra/rb)] /20 = Kliq.f +  Kv.(1-f)= [Kliq- Kv]f +Kv

f = [((ln(ra/rb)) /20) -Kv] /[Kliq- Kv]   

b) l=2m, Ra=5mm, Rb=4.5mm, Kliq=1.4, Kv=1

C=Kliq. 20h/ln(ra/rb) + Kv 20(l-h)/ln(ra/rb)

For full, h=L

C=1.4*20*2/ln(5/4.5) +0

= 17.59 * 8.85 x 10-12  /0.105

= 1.48 * 10-9 F = 1.48 uF

For empty, h=0,

C=1*20*2/ln(5/4.5) = 1.06  * 10-9 F = 1.06 uF

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