DYNAMICS of U-Tube Manometer

IIT MADRAS

CH4510:Process Control Lab DYNAMICS OF U-TUBE MANOMETER

Batch - I 10/30/2013

Submitted by: Shubham Jhanwar

CH10B092

Simple Kumar

CH10B093

J Vijay Prasad

CH10B094

V Avinash

CH10B095

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 ( where

)

(1)

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)

τ2

Define

, 2ζτ =

and Kp =

(4)

pP

(5)

Now equ.(3) becomes

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=

√

=

√

Period of oscillation T=

(11)

√ √

=

= t2-t1

(12) (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

Raised

First

First

Second

Second

Time

level

level

peak

valley

peak

valley

between

(cm)

(cm)

(cm)

(cm)

(cm)

(cm)

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

Raised

First

First

Second

Second

Time

level

level

peak

valley

peak

valley

between

(cm)

(cm)

(cm)

(cm)

(cm)

(cm)

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.

A

B

C

Overshoot

Decay

Period of

No.

(cm)

(cm)

(cm)

A/B

ratio

oscillation

C/A

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

τ (sec)

ζ

(b) For uncoiled tube

Sl.

A

B

C

Overshoot

Decay

Period of

No.

(cm)

(cm)

(cm)

A/B

ratio

oscillation

C/A

T (sec)

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

height (cm) vs time (sec) for coiled 700 0, 660 600

500

4, 500

400 height (cm)

6, 330

300 2, 240 200

100

0 0

1

2

3

4

5

6

7

(b) For uncoiled tube

height (cm) vs time (sec) for uncoiled 700 0, 660 600

500

4, 500

400 height (cm) 300

6, 290

200

2, 180

100

0 0

1

2

Result: 1. From experiment (a) For coiled

τ =0.635 ζ = 0.148

(b) For uncoiled τ =0.641 ζ = 0.1088

3

4

5

6

7

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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