DYNAMICS OF U-TUBE MANOMETER

Objective:

(a) To study the dynamic response of an U-Tube manometer following a step change

(b) To study the characteristics of an under-damped second order response like overshoot,

rise time, decay ratio, response time etc.

Theory:

Fig 1. A U-tube manometer

Systems with inherent second order dynamics can exhibit oscillatory behavior

(under-damped).Examples of these physical systems are simple manometers, externally

mounted level indicators, pneumatic control valve, variable capacitance differential

pressure transducer.

U-tube manometer is a classic example of a second order system. The basic

equation is the force balance

(

)

(1)

where A = cross-sectional area

ρ = liquid density (density of gas above fluid is negligible)

P = applied pressure

R = fractional resistance

With laminar flow, the resistance is given by Hagen-Poiseuille equation.

or R

(2)

Substituting in Equ (1) and rearranging gives

(3)

Define τ2

, 2ζτ =

and Kp =

(4)

Now equ.(3) becomes

pP (5)

Thus the transfer function between h and P is

(6)

Equation (4) and (5) represents the inherent second order dynamics of the manometer.

Equation (3) may be written in a standard form

(7)

Where ωn = natural frequency, rad/sec

ζ = damping coefficient

For a step change in input pressure, when damping coefficient less than 1, the

output overshoots the final value and oscillates before coming to equilibrium. The

system is said to be under damped.

For < 1.0,

√ ( √ )

Where √

With a damping coefficient of zero, the response is an under damped sine wave

of frequency and amplitude 2hi.

For = 1.0, (critically damped)

For > 1.0, (over damped)

(

( ⁄ )

(

⁄ )

)

Experimental values of and can be easily obtained from the under damped

response curve. The damping coefficient can be found either from the decay ratio which

is the ratio of successive peak heights or from maximum overshoot.

Decay ratio=

√ =

(11)

√

√ =

(12)

Period of oscillation T=

√ = t2-t1 (13)

Fig1: An under damped response

Procedure:

1. Before starting the experiment note down the level of liquid column in the U-tube

manometer. This is the base level.

2. Give a pressure input by blowing air into one of the limbs of the manometer and

close the corresponding limb air tight with your thumb.

3. Note the level in the other limb.

4. Release the pressure by loosening your thumb.

5. When the level reaches the first lower position start the stop watch and note the

time at which it reaches the second lower position. Also note the first peak, first

valley (first lower position), second peak height and second valley.

6. Repeat the experiment for two different waves.

7. By using equ. (11), (12) and (13) the value of τ and ζ can be calculated

experimentally.

8. Using equ. (4), for the given value of L and D the value of τ and ζ can be obtained

theoretically.

9. The values of τ and ζ obtained experimentally and theoretically are to be compared.

Observations:

(a) Tube in coiled position

Sl.No. Base

level

(cm)

Raised

level

(cm)

First

peak

(cm)

First

valley

(cm)

Second

peak

(cm)

Second

valley

(cm)

Time

between

two valleys

(cm)

1 400 260 660 240 500 330 4 seconds

2 400 240 640 250 490 335 4.1 seconds

3 380 235 615 230 480 320 4 seconds

(b) Tube in uncoiled position

Sl.No. Base

level

(cm)

Raised

level

(cm)

First

peak

(cm)

First

valley

(cm)

Second

peak

(cm)

Second

valley

(cm)

Time

between

two valleys

(cm)

1 380 280 660 180 500 290 4 seconds

2 380 270 650 190 500 290 4.1 seconds

Calculations:

1. Experiment:

(a) For coiled tube

Sl.

No.

A

(cm)

B

(cm)

C

(cm)

Overshoot

A/B

Decay

ratio

C/A

Period of

oscillation

T (sec)

τ (sec) ζ

1 160 260 70 0.615 0.44 4 seconds 0.629 0.153

2 150 240 65 0.625 0.43 4.1 seconds 0.645 0.148

3 150 235 60 0.638 0.4 4 seconds 0.630 0.143

(b) For uncoiled tube

Sl.

No.

A

(cm)

B

(cm)

C

(cm)

Overshoot

A/B

Decay

ratio

C/A

Period of

oscillation

T (sec)

τ (sec) ζ

1 200 280 90 0.714 0.45 4 seconds 0.633 0.1065

2 190 270 90 0.704 0.474 4.1 seconds 0.649 0.1112

2. Theoretically:

L = 980 cm

D = 1.2 cm

ρ = 1000 kg/m3

g = 9.8 m/(sec) 2 ;

µ = 10-3 kg/m.(sec)

τ2

= 0.5

τ = 0.707

2ζτ =

=0.111

ζ = 0.0786

Sample Calculation :

For 1st reading (without coil) :

1. Calculation of ξ

Now,

√

√

( (

)

)

( (

)

)

√(

)

(

)

2. Calculation of τ :

√

Substituting T = 4 sec,

We get τ = 0.633 sec-1

Graphs

(a) For coiled tube

0, 660

2, 240

4, 500

6, 330

0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7

height (cm) vs time (sec) for coiled

height (cm)

(b) For uncoiled tube

Result:

1. From experiment

(a) For coiled

τ =0.635

ζ = 0.148

(b) For uncoiled

τ =0.641

ζ = 0.1088

0, 660

2, 180

4, 500

6, 290

0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7

height (cm) vs time (sec) for uncoiled

height (cm)

2. From theoretical calculation

τ = 0.707

ζ = 0.0786

Are the values of τ and ζ from experimental and theoretical calculation matching? If not

explain why?

No, the values of τ and ζ from experimental and theoretical calculation are not matching. This is

because theoretical calculations consider pipe to be straight and does not account for the extra

pressure developed due to coiling of pipe. This is quite evident from the fact that uncoiled pipe

has lesser ζ value in comparison to the coiled one.

References:

1. Process Control - Peter Harriot.

2. Chemical Process Control - George Stephanopoulos

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dynamics of u-tube manometer

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