1

1. THEORY

1.1 Process Control

Control in process industries refers to the regulation of all aspects of the process. Precise

control of level, temperature, pressure and flow is important in many process applications.

This module introduces you to control in process industries, explains why control in

important, and identifies different ways in which precise control is ensured. Refining,

combining, handling, and otherwise manipulating fluids to profitably produce end products

can be a precise, demanding, and potentially hazardous process. Small changes in a process

can have a large impact on the end result. Variations in proportions, temperature, flow,

turbulence, and many other factors must be carefully and consistently controlled to produce

the desired end product with a minimum of raw materials and energy. Process control

technology is the tool that enables manufacturers to keep their operations running within

specified limits and to set more precise limits to maximize profitability, ensure quality and

safety.[1]

Figure 1.1 An example of process control[2]

1.2 Transfer Functions

The transfer function of a linear dynamical system is the ratio of the Laplace transform of its

output to the Laplace transform of its input. In systems theory, the Laplace transform is called

the “frequency domain” representation of the system.[3]

2

Transfer function G(s) is ratio of output x to input f, in s-domain (via Laplace trans.):

G(s)=X(s)/F(s)[4]

(1)

Transfer functions are;

Describes dynamics in operational sense

Dynamics encoded in G(s)

Ignore initial conditions (I.C. terms are “transient” & decay quickly)

Transfer function, for input-output operation, deals with steady state terms

Figure 2. Block diagram of transfer functions[4]

1.2.1 Step Functions

Before proceeding into solving differential equations we should take a look at one more

function. Without Laplace transforms it would be much more difficult to solve differential

equations that involve this function in g(t).

The function is the Heaviside function and is defined as,

(2)

Heaviside functions are often called step functions. Here is some alternate notation for

Heaviside functions.[5]

3

1.2.2 Impulse Functıons

In some applications, it is necessary to deal with phenomena of an impulsive nature.

For example, an electrical circuit or mechanical system subject to a sudden voltage or force

g(t) of large magnitude that acts over a short time interval about t0. The differential equation

will then have the form;

(3)

- Measuring Impulse

In a mechanical system, where g(t) is a force, the total impulse of this force is measured by

the integral;

(4)

Note that if g(t) has the form

(5)

Then

(6)

Suppose the forcing function d(t) has the form

(7)

4

Then as we have seen, I() = 1. We are interested d(t) acting over shorter and shorter time

intervals (i.e., 0). See graph on right.Note that d(t) gets taller and narrower as 0.

Thus for t 0, we have

(8)

1.3 Dynamic Behavior of First-order and Second-order Systems

1.3.1. Analysis of first-order systems

Consider the system shown in Figure 1 which consists of a tank of uniform cross

sectional area A to which is attached a flow resistance R. Assume that qo is related to the

head by linear relationship

qo=h/R

In a general form, Equation (4.8) can be written in a general form:

(9)

5

Response of a first-order system to a step change in the input.The term τ (time constant) and

K (steady state gain) characterize the first-order system.

Note that the both parameters depend on operating conditions of the process and that the

transfer function does not contain the initial conditions explicitly.

1.3.2. Analysis of second-order systems

A second-order system is one whose output, y(t), is described by a second-order

differential equation. For example, the following equation describes a second-order linear

system:

(10)

If ao≠ 0, then Equation (4.24) yields

(11)

Equation (4.25) is in the standard form of a second-order system, where;

The very large majority of the second- or higher-order systems encountered in a

chemical plant come from multicapacity processes, i.e. processes that consist of two or

more first-order systems in series, or the effect of process control systems.

Laplace transformation of Equation (4.25) yields;

(12)

Dynamic response

For a step change of magnitude M, U(s) = M/s, Equation (4.26) yields

6

The two poles of the second-order transfer function are given by the roots of the

characteristic polynomial,

(12,13,14)

The form of the response of y(t) will depend on the location of the two poles in the

complex plane. Thus, we can distinguish three cases:

Case A: (over-damped response), when ζ > 1, we have two distinct and real poles.

In this case the inversion of Equation (4.30) by partial fraction expansion yields

(15)

Where cosh(.) and sinh(.) are the hyperbolic trigonometric functions defined by

Case B: (critically damped response), when ζ = 1, we have two equal poles (multiple

pole).

In this case, the inversion of Equation (4.30) gives the result

(16)

Case C: (Under-damped response), when ζ < 1, we have two complex conjugate poles.

The inversion of Equation (4.30) in this case yields;

7

(17)

Figure 3 : Characteristics of Underdamped Systems

- Overshoot: Is the ratio of a/b, where b is the ultimate value of the response and a is the

maximum amount by which the response exceeds its steady state value. It can be shown

that it is given by the following expression:

(18)

Rise time: tr

is the the process output takes to first reach the new steady state value.

- Time to first peak: tp is the time required for the output to reach its first maximum

value.

- Settling time: ts is defined as the time required for the process output to reach and

8

remain inside a band whose width is equal to ± 5 % of the total change in the output.

- Period: Equation (4.34) defines the radian frequency, to find the period of oscillation

P (i.e. the time elapsed between two successive peaks), use the well-known

relationship ω = 2π/P; thus:

(19)

Comparison between first-order and second-order responses.

9

2. EXPERIMANTAL METHOD

The experimental set-up consists of the various U-manometers in different diameters and

length that contains everalk inds of liquids with the different physicochemicapl ropertiess uch

as water, glycerol and their mixtures.The pressure difference in the U-manometer is created

by a vacuum pump.

2.1.DESCRIPTION OF APPARATUS

The experimental set-up consists of the various U-manometers in different diameters that

contain several kinds of liquids with different physical properties such as water, glycerol and

their miztures. The pressure difference in the U-manometer is created by a vacuum pump.

Figure 4. U tube manometer[8]

2.2. EXPERIMENTAL PROCEDURE

Apply a pressure difference on the U-manometer by vacuum .pump and determine the

variation of the liquid level with time until the steady state is reached. Stop the vacuum pump

when the constant liquid level is observedi n U-manometer, and determine again the variation

10

of the liquid level with time.

After the constant liquid level is reached in the U-manometer, compress the silicone tube

between the connection points of U-manometer and give up the silicone tube suddenly and

then record the variation of the liquid level with time. Determine the variation of oscillation

amplitude according to initial liquid level with time, if oscillation is observed on the U-

manometer liquid.Repeat these steps at least twice for each U-manometer.

11

3. RESULTS & DISCUSSION

3.1 U-TUBE MANOMETERS

Table 3.1.1 Properties of U-Manometers

Properties Manometer

1

Manometer

2

Manometer

3

Manometer

4

Manometer

5

Manometer

6

ρ(g/cm3) 0,885 0,997 1,261 0,885 1,058 1,261

µ(cP) 137,6 0,894 902,9 137,6 1,362 902,9

D(cm) 0,600 1,100 0.600 1,100 1,100 1,100

L(cm) 88,00 95,00 102,0 98,00 85,00 116,0

τ(s) 0.212 0.220 0.228 0.224 0.208 0.243

ξ 14,64 0,026 72,52 4,600 0,035 23,03

As it can seen from Table 3.1.1. the damping factors ( ξ ) of 1st, 3

rd, 4

th and 6

th

manometers are bigger than 1; and 2nd

and 5th

manometers are smaller than 1. Because of this

we can say these first four manometers gives overdamped responses and the last two

manometers gives underdamped responses, as we expected.In addition to these;when damping

factor is less then 1 (as it seen 2. and 5.) ,then root is complex and response have

oscillation.When damping factor is greater than 1,then root is real (as it seen 1,3,4,6) and

response having no oscillation. If we analyze the time constants of four manometers (M-1,

M-3, M-4, M-6) we can say that M-1 named manometer gives the fastest response and M-6

named manometer gives the slowest response between this group, because response rate of

any process depends on the time constant (τ). Also when we analyze the time constants of

second group manometers between each other (M-2, M-5) we can say that the response of M-

5 is faster than M-2. Finally if we analyze the time constants of M-1, M-4 and M-3, M-6

which contains same fluids, we saw that τ1< τ4 and τ3< τ6.That means time constants, so

responses of U-tube manometers depens on its dimensions.When M-3 and M-6 compared,it

is obviously seen that even if they are same fluid (also same density,viscosity) time constants

are different.As a result, small diameter has fastest response time.When M-2 and M-4 are

compared(different fluid same diameter),it is seen that viscose one(M-4:engine oil) is slower

than M-2.

12

3.2. RESULTS for OVERDAMPED U-MANOMETERS

Table 3.2.1 Experimental Responses of Overdamped U-manometers to step change

t

(s)

Manometer 1 Manometer 3 Manometer 4 Manometer 6

t/τ hr/Kp hf/Kp t/τ hr/Kp hf/Kp t/τ hr/Kp hf/Kp t/τ hr/Kp hf/Kp

0 0,0000 0,0000 1,0000 0,0000 0,0000 1,0000 0,0000 0,0000 1,0000 0,0000 0,0000 1,0000

3 14,1509 0,3959 0,6143 13,1579 0,0300 0,6818 13,3929 0,2714 0,5929 12,3457 0,5263 0,5263

6 28,3019 0,6348 0,3413 26,3158 0,5455 0,3818 26,7857 0,5571 0,3714 24,6914 0,7368 0,2368

9 42,4528 0,7918 0,2184 39,4737 0,6818 0,2273 40,1786 0,6929 0,2143 37,0370 0,8947 0,1316

12 56,6038 0,8771 0,0853 52,6316 0,7455 0,1455 53,5714 0,8143 0,1214 49,3827 0,9474 0,0658

15 70,7547 0,9283 0,0614 65,7895 0,8636 0,0773 66,9643 0,8857 0,0571 61,7284 1,0000 0,0000

18 84,9057 0,9556 0,0205 78,9474 0,9091 0,0182 80,3571 0,9286 0,0143 - - -

21 99,0566 0,9795 0,0068 92,1053 0,9455 - 93,7500 0,9571 - - - -

24 113,2075 1,0000 0,0000 105,2632 0,9636 - 107,1429 0,9714 - - - -

27 - - - 118,4211 1,0000 - 120,5357 0,9857 - - - -

30 - - - - - - 133,9286 1,0000 - - - -

When M-1 and M-4 are compared it is seen that hr/Kp values are different eventhough

they are both same fluid(engine oil).This is because diameter differences.M-1 which has

smaller diamer than M-4 falls more much than M-4.Also the height values of the liquids in the

manometers can be seen. It is seen that when the diameter and the length values of tubes are

changed, the height values will change, for the same liquid. The M3 and M6 manometers

were filled with the same liquid, glycerol. However, it is seen that the diameter of the pipe

value changes the Kp value, and so the ratio of the hr/Kp value. Also it is seen that t/ τ values

of the different U-manometers are nearly equal to each other.

13

Figure 3.2.1. Experimental hr/kp versus t/τ values

In this graph a comparison of the hr/Kp and t/τ values of different manometers can be

seen. “hr” values are the height of the manometers , and the Kp values are the height value

that the maximum one the liquids reached. It is seen that the experimental results are nearly

equal to each other except M-6. M6 manometer has shorter response time than other

manometers. In addition M-6 rising faster than others.Reason of these differences can be

intrinsic properties of fluid(glycerol) in M-6.In addition it is seen from graph height values in

the manometers are firstly increasing rapidly, and then this slope is decreasing and finally

they reach Kp value. In the graph, it is seen that, this stationary point for both manometers are

not so different than each other.

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

0 20 40 60 80 100 120 140 160

hr/

Kp

t/ּז

M1

M3

M4

M6

14

Figure 3.2.2. Experimental hf/Kp versus t/τ values

There is a falling when we compare figure 3.2.2 and 3.2.1; same approach is valid in this

graph the hf/Kp and the t/τ values of different manometers are seen. “hf” value is the value

that the raising height reached at least when the pump is not working. It is seen from the graph

that the curves belonged to other manometers are very close to each other except M-6 because

it’s intrinsic properties cause this differences.

Table 3.2.2 Theoretical responses of Overdampded U-manometers to step change

t

(s)

Manometer 1 Manometer 3 Manometer 4 Manometer 6

t/τ hr/Kp hf/Kp t/τ hr/Kp hf/Kp t/τ hr/K

p hf/Kp t/τ hr/Kp hf/Kp

0 0,0000 0,0000 1,0000 0,0000 0,0000 1,0000 0,0000 0,0000 1,0000 0,00 0,0000 1,000

3 14,150 0,3840 0,6160 13,157 0,0867 0,9133 13,3929 0,7710 0,2290 12,4 0,2380 0,762

6 28,301 0,6200 0,3800 26,315 0,1659 0,8341 26,7857 0,9476 0,0524 24,7 0,4193 0,581

9 42,452 0,8110 0,1890 39,473 0,2383 0,7617 40,1786 0,9880 0,0120 37,0 0,5575 0,443

0

0,2

0,4

0,6

0,8

1

1,2

0 20 40 60 80 100 120

hf/

kp

t/Ʈ

M1

M3

M4

M6

15

12 56,603 0,8912 0,1088 52,631 0,3043 0,6957 53,5714 0,9973 0,0027 49,4 0,6628 0,337

15 70,754 0,9380 0,0620 65,789 0,3646 0,6354 66,9643 0,9994 0,0006 61,7 0,7431 0,257

18 84,905 0,9640 0,0360 78,947 0,4197 0,5803 80,3571 0,9999 0,0001 - - -

21 99,056 0,9790 0,0210 92,105 0,4701 0,5299 93,7500 1,0000 3,30E-05 - - -

24 113,207 0,9880 0,0120 105,263 0,5160 0,4840 107,143 1,0000 7,56E-06 - - -

27 - - - 118,421 0,5580 0,4420 120,536 1,0000 1,73E-06 - - -

30 - - - - - - 133,929 1,0000 3,97E-07 - - -

In this table the theoretical results can be seen. It is seen that t/τ values are equal to the

experimental values. However the hr/Kp values and the hf/Kp values are different than the

experimental ones. Although there is a difference in the values, they are not so different than

each other. These hr/Kp values and hf/Kp values were calculated by using some formulas.

Figure 3.2.3. Theoretical hr/Kp versus t/τ values

In Figure 3.2.3, it is seen that the curves are not closer like the experimental ones.

However, again all the values are behaving similar with the experimental ones.

0

0,2

0,4

0,6

0,8

1

1,2

0 50 100 150

hr/

kp

t/Ʈ

M1

M3

M4

M6

16

Figure 3.2.4. Theoretical hf/Kp versus t/τ values

This figure, like Figure 3.2.3, this graph which gives a comparison of hf/Kp versus t/τ

values of all manometers results are different from experimental one. It is seen that the curves

are far from each other than the experimental ones.

3.3 RESULTS FOR UNDERDAMPED U-MANOMETERS (TO STEP CHANGE)

Table 3.3.1 Period of Oscillation and Radian Frequency of Underdamped U-Manometers

Manometer-2 Manometer-5

Period of Oscillation

T(s)

1,383 1,33

Radian Frequency

W(s-1

)

4,544 4,802

In this table the period of oscillation and the radian frequency values can be seen. These

values were calculated for only M2 and M5 U-manometers. Because their damping factor

0

0,2

0,4

0,6

0,8

1

1,2

0 50 100 150

hf/

Kp

t/to

M1

M3

M4

M6

17

values were smaller than 1 so the system behaves as underdamped and oscillated. The period

values are nearly equal to each other. However, M2 manometer has higher period value than

M5. Reason of these difference may be that the density and viscosity values aren’t so close to

each other although the diameter of the pipes are equal. Alsothe heights are not equal. The

difference may be caused from there. The frequency value is conversely proportional to

period value. Therefore, it is seen that the frequency value of M2 manometer is smaller than

M5 manometer.

Table 3.3.2 Experimental Responses of Underdamped U-manometers to Step Change

Manometer 2 Manometer 5

Oscillation Texp (s) t/τ h/Kp Texp (s) t/τ h/Kp

0 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000

1 0,9900 4,5000 1,0000 0,9000 4,3269 1,0000

2 1,6300 7,4091 0,6099 1,5700 7,5481 0,5778

3 2,3500 10,6818 0,8936 2,2500 10,8173 0,8778

4 3,3000 15,0000 0,6667 3,1500 15,1442 0,6333

5 3,8400 17,4545 0,8298 3,9100 18,7981 0,8111

6 4,6000 20,9091 0,7092 4,6300 22,2596 0,7000

7 5,2300 23,7727 0,7943 5,4000 25,9615 0,7667

8 5,9500 27,0455 0,7376 6,2500 30,0481 0,7222

9 6,7600 30,7273 0,7730 6,9000 33,1731 0,7444

10 7,3500 33,4091 0,7518 - - -

As it is seen from Table 3.3.2. with increasing time “t/ ” values increase both for M-2

and M-5. Since 2 and 5 manometers are underdamped the oscillating responses are expected

and “h/Kp” values show both decreasing and increasing that it is the proof of oscillation.

Experimental responses are close to each other for M-2 and M-5.In addition to these

,experimental set up is based on step change means that pump is worked along the

experiment.

18

Table 3.3.3 Theoretical Responses of Underdamped U-manometers to Step Change

Manometer 2 Manometer 5

Oscillation Texp (s) t/τ h/Kp Texp (s) t/τ h/Kp

0 0,3270 1,4864 0,1379 0,4170 2,0048 0,1540

1 1,0180 4,6273 1,7952 1,0810 5,1971 1,7403

2 1,7080 7,7636 0,2648 1,7450 8,3894 0,3540

3 2,3980 10,9000 1,6796 2,4090 11,5817 1,5621

4 3,0880 14,0364 0,3717 3,0730 14,7740 0,5125

5 3,7780 17,1727 1,5808 3,7370 17,9663 1,4214

6 4,4680 20,3091 0,4632 4,4010 21,1587 0,6370

7 5,1580 23,4455 1,4962 5,0650 24,3510 1,3114

8 5,8480 26,5818 0,5414 5,7290 27,5433 0,7340

9 6,5380 29,7182 1,4238 6,3930 30,7356 1,2262

10 7,2280 32,8545 0,6083 - - -

In Table 3.3.3. as in Table 3.3.2. “t/ ” values increase with increasing time. Again

decreasing and increasing “h/Kp” values show there is an oscillation for M-2 and M-5 since

they are underdamped. The close values of ttheo. , t/ and h/Kp are obtained for M-2 and M-5.

19

Figure 3.3.1. Comparison of Experimental and Theoretical Responses for M-2

In Figure 3.3.1. it is shown that experimental values differ from the theoretical values.

Since M-2 is an underdamped manometer it is common to see oscillation for both

experimental and theoretical values. The main reason of this big difference between the

experimental and theoretical values may be the wrong readings of the data.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

0 5 10 15 20 25 30 35

h/k

p

t/to

M2-exp

M2-theo

20

Figure 3.3.2. Comparison of Experimental and Theoretical Responses for M-5

Since M-5 is an underdamped manometer again the oscillation is observed in Figure

3.3.2. The big difference between the experimental and theoretical values are seen. The main

reason of big difference between the experimental and theoretical values may be the wrong

readings of the data.

Table 3.3.4 Comparison of Theoretical and Experiment Overshoot, Decay Ratio and

Response Time to Step Change

Manometer-2 Manometer-5

Experimental Theoretical Experimental Theoretical

Overshoot 0,186 0,922 0,179 0,897

Decay Ratio 1,77 0,850 2 0,805

Response time 7,350 7,228 6,900 6,393

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

0 5 10 15 20 25 30 35

h/k

p

t/to

M5-exp

M5-theo

21

In Table 3.3.4. there is a comparison between M-2 and M-5 and according to this

comparison it can be said that the overshoot values of M-2 both experimentally and

theoretically is higher than that of M-5 while response time of M-2 is higher than that of M-5

by experimentally. When experimental and theoretical values are compared it is seen that

overshoot and decay ratio values are close to each other(exception of experimentally decay

ratio and reason of this may be about personal mistakes)while response times are differ from

each other. Reason of differences between M-2 and M-5 can be intrinsic properties of water

(M-2)and glycerol solution(M-5).

3.4. RESULTS FOR UNDERDAMPED U-MANOMETERS (TO IMPULSE CHANGE)

Table 3.4.1 Experimental Responses of Underdamped U-manometers to Impulse Change

Manometer 2 Manometer 5

Oscillation texp (s) t/τ h/Kp texp (s) t/τ h/Kp

0 0,0000 0,0000 0,0000 0,0000 0,0000

1 0,1400 0,6364 1,0000 0,5400 2,5962 1,0000

2 1,5800 7,1818 -0,7091 1,4800 7,1154 -0,7209

3 2,1600 9,8182 0,4909 2,3400 11,2500 0,5349

4 2,7900 12,6818 -0,4364 3,1500 15,1442 -0,4651

5 3,5100 15,9545 0,2909 4,0000 19,2308 0,3256

6 4,2800 19,4545 -0,2636 4,7700 22,9327 -0,1395

7 4,9500 22,5000 0,2364 5,4700 26,2981 0,1628

8 5,7600 26,1818 -0,1727 6,4100 30,8173 -0,0698

9 6,4800 29,4545 0,1455 7,0400 33,8462 0,0930

10 7,2500 32,9545 -0,1091 7,9000 37,9808 -0,0465

11 7,8800 35,8182 0,0818 8,6500 41,5865 0,0465

12 8,9400 40,6364 -0,0727 - - -

13 9,6600 43,9091 0,0545 - - -

14 10,3800 47,1818 -0,0364 - - -

22

15 11,1900 50,8636 0,0364 - - -

16 11,9900 54,5000 -0,0273 - - -

17 12,8000 58,1818 0,0091 - - -

In this table, experimental responses are seen for M-2 and M-5. Time (t) and height (h)

values have been normalized by obtaining t/ and h/Kp. Here, Kp is the highest value of

oscillations. h/Kp values exhibit successive positive-negative results because experimental set

up based on impulse change means that pump is worked then shut down. Two manometers

give nearly the same responses. This may be related to the fluids that is water for M-2 and

15% glycerol solution for M-5. Since the densities of these fluids are not so different from

each other, the results are close.

Table 3.4.2 Theoretical Responses of Underdamped U-manometers to Impulse Change

Manometer 2 Manometer 5

Oscillation Texp (s) t/τ h/Kp Texp (s) t/τ h/Kp

1 -0,3460 -1,5727 -1,0421 -0,3325 -1,5986 -1,0570

2 0,3460 1,5727 0,9602 0,3325 1,5986 0,9466

3 1,0380 4,7182 -0,8848 0,9975 4,7957 -0,8453

4 1,7300 7,8636 0,8152 1,6625 7,9928 0,7526

5 2,4220 11,0091 -0,7512 2,3275 11,1899 -0,6682

6 3,1140 14,1545 0,6921 2,9925 14,3870 0,5915

7 3,8060 17,3000 -0,6377 3,6575 17,5841 -0,5220

8 4,4980 20,4455 0,5876 4,3225 20,7813 0,4592

9 5,1900 23,5909 -0,5414 4,9875 23,9784 -0,4026

10 5,8820 26,7364 0,4988 5,6525 27,1755 0,3518

11 6,5740 29,8818 -0,4596 6,3175 30,3726 -0,3064

12 7,2660 33,0273 0,4234 6,9825 33,5697 0,2658

13 7,9580 36,1727 -0,3901 - - -

23

14 8,6500 39,3182 0,3594 - - -

15 9,3420 42,4636 -0,3311 - - -

16 10,0340 45,6091 0,3051 - - -

17 10,7260 48,7545 -0,2811 - - -

In Table 3.4.2., theoretical responses are seen for M-2 and M-5. Again, time (t) and

height (h) values have been normalized by obtaining t/τ and h/Kp. These values have been

gained by formulas that consist of ξ, τ and T (period). Since these manometers give

underdamped responses, oscillation is observed on h/Kp values. Again, two manometers give

nearly the same responses as the densities of these fluids are not so different from each other.

Figure 3.4.1. Theoretical and experimental values for M-2

In this figure, experimental and theoretical responses of M-2 are seen for comparison.

Since M-2 gives underdamped responses, oscilllation is observed. Theoretical curve exhibit

oscillation with high peaks that are taller than the ones for experimental. This result may be

related to the wrong reading of data and also usage of formula.

-1,5

-1

-0,5

0

0,5

1

1,5

-10 0 10 20 30 40 50 60 70

h/k

p

t/to

M2 exp

M2-theo

24

Figure 3.4.2. Theoretical and experimental values for M-5

In this figure, experimental and theoretical responses of M-5 are seen for comparison.

Because M-5 gives underdamped responses, oscillation is observed in the curves. Again, it is

seen that theoretical values are so different from experimental ones. This result may be related

to the wrong reading of data and also usage of formula.

Table 3.4.3 Comparison of Theoretical and Experimental Overshoot, Decay Ratio and

Response time to Impulse change

Manometer-2 Manometer-5

Experimental Theoretical Experimental Theoretical

Overshoot 0,461 0,922 0,535 0,897

Decay Ratio 0,6 0,850 0,607 0,805

Response time 12,80 11,42 8,650 6,980

-1,5

-1

-0,5

0

0,5

1

1,5

-10 0 10 20 30 40 50

h/k

p

t/to

M5-exp

M5-theo

25

In this table, comparison of theoretical and experimental overshoot, decay ratio and

response time values are seen for both M-2 and M-5. For M-2 and M-5 theoretical and

experimental overshoot value is far from each other.Reason of this may be mistake of

measurement. In both manometers, experimental response times differ from the theoretical

ones. Observing the experimental response times different from the theoretical ones can be

related to the wrong reading or recording.These different values(decay

ratio,overshoot,response time)are related to fluids properties(viscosity vs.).

26

4. CONCLUSIONS

The dynamic behaviour of U-Manometer systems, which are Manometer-1 with engine oil,

Manometer-2 with water, Manometer-3 with glycerol, Manometer-4 with engine oil,

Manometer-5 with 15% glycerol solution and Manometer-6 with glycerol,are investigated for

step and and impulse input a overdamped or underdamped.

Firstly;Overdamped systems(M-1:Engine oil,M-3:Glycerol,M-4:Engine oil,M-6:Glycerol)

have damping factor higher than 1 and diverge from the steady state value at higher damping

factors.

Secondly;Underdamped systems(M-2:Water,M-5:%15 glycerol solution)have damping factor

smaller than 1 and the response is observed as oscillating around the steady state value.

Thirdly;For overdamped systems, systems with high damping factors require more to reach

the ultimate value when it is compared with underdamped systems.

Fourtly;For overdamped systems, Manometer-6 is the first system reaching its steady state

value for both experimentally and theoretically . Also, Manometer-3 is the system reaching

last to its uştimate value for experimental result and theoretical observations shows that M-

1,M-3 and M-4 is reaching almost same time to their steady state value .

Fiftly;For underdamped systems, the response of manometers are more stable to step change

in comparison with an impulse change. Because, the response of the systems to impulse

change is diverging and converging around the ultimate value with both positive and negative

values.

Finally;For underdamped systems, the variation between the experimental overshoot, decay

ratio and response time is greater for impulse change than step change due to the variations in

responses.In theoretical step change greater than impuse change.

27

5. NOMENCULATURE

A, B Constants in the transfer function

At Surface area of bulb for heat transfer (m2)

g Acceleration of gravity (m/s2)

Kp Static gain or gain (m)

L Total length of the liquid in U-manometer (m)

m Mass of liquid in the monometer (kg)

r Liquid lever difference at any time in U-manometer (m)

t Time (s)

tr Rise time (s)

T period of oscillation (s/cycle)

Q Volumetric flow rate of the liquid (m3/s)

p Time constant (s) ح

µ Viscosity of the liquid (Pa.s)

ρ Density of the liquid (kg/m3)

ω Radian frequency (radian/s)

ξ Damping factor

28

5.REFERENCES

1. http://www.pacontrol.com/download/Process%20Control%20Fundamentals.pdf

2. http://bin95.com/training_software/fluid_process_systems.htm

3. http://planetmath.org/transferfunction

4. http://www.me.utexas.edu/~bryant/courses/me344/DownloadFiles/LectureNotes/L

aplace+TransferFunctions.pdf

5. http://tutorial.math.lamar.edu/Classes/DE/StepFunctions.aspx

6. http://www.math.ust.hk/~mamu/courses/151/Lectures/Mu/ch06_5.ppt

7. http://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20Fir

st_Second%20Order%20Systems.pdf

8. http://www.3bscientific.com/U-Tube-Manometer-S

U8410450,p_83_110_856_14309.html

9. "Viscosity of Glycerol and its Aqueous Solutions". Retrieved 2011-04-19.

10. http://edge.rit.edu/edge/P13051/public/Research%20Notes/Viscosity%20of%20Aq

ueous%20Glycerol%20Solutions.pdf