Step Response of a Manometer

height versus time

manometer

τ

0.1756

γ

0.04

f

0.25

A manometer is an instrument for measuring pressure. It consists of a U-shaped tube filled with liquid (usually mercury because of its high density). The difference, 2

h

, between the heights of the liquid is a function of the pressure difference between the two sides.

The step response of a manometer obeys the following equation:

2

τ

2

d

h

d

2

t

+2τγ

dh

dt

+h=fU(t)

,

where

U(t)

is the unit-step function,

h

is the height above the rest position,

2

τ

=

L

2g

(

τ

is a time constant),

L

is the total height of the manometer, and

g=9.81

2

m

/s

is the acceleration due to gravity. Also,

γ=

2Lμ

ρ

2

R

g

2g

L

is a damping coefficient, where

μ

is the viscosity,

ρ

is the density,

R

is the radius of the manometer, and

f=

P

g

c

2ρg

is proportional to the applied gas pressure, where

g

c

=32.1740

is a unit conversion factor.

This Demonstration shows the height above the rest position versus time. The observed behavior is typical of a second-order process. The overshoot observed in the step response of the manometer, for small values of the damping coefficient, makes it easy for manometers to overflow.