process control experiment

7/30/2019 process control experiment

1/24

1

1-THEORY

1.1-PROCESS CONTROL

Process control refers to the methods that are used to control process variables when

manufacturing a product. For example, factors such as the proportion of one ingredient to

another, the temperature of the materials, how well the ingredients are mixed, and the

pressure under which the materials are held can significantly impact the quality of an end

product. Manufacturers control the production process for three reasons(1):

Reduce variability

Increase efficiency

Ensure safety1

In controlling a process there exist two type of classes of variables(2).

1. Input VariableThis variable shows the effect of the surroundings on the process. It

normally refers to those factors that influence the process. An example of this would be the

flow rate of the steam through a heat exchanger that would change the amount of energy put

into the process. There are effects of the surrounding that are controllable and some that are

not. These are broken down into two types of inputs.

a.Manipulated inputs: variable in the surroundings can be control by an operator or the

control system in place.

b.Disturbances: inputs that can not be controlled by an operator or control system. There

exist both measurable and immeasurable disturbances.

2. Output variable- Also known as the control variable These are the variables that are

process outputs that effect the surroundings. An example of this would be the amount of CO2

gas that comes out of a combustion reaction. These variables may or may not be measured.

As we consider a controls problem. We are able to look at two major control structures.

1. Single input-Single Output (SISO)- for one control(output) varible there exist one

manipulate (input) variable that is used to affect the process


2/24

2

2. Multiple input-multiple output(MIMO)- There are several control (output) variable that are

affected by several manipulated (input) variables used in a given process (2).

1.1.1- Transfer Functions

A Transfer Function is the ratio of the output of a system to the input of a system, in the

Laplace domain considering its initial conditions and equilibrium point to be zero. If we have

an input function ofX(s), and an output function Y(s), we define the transfer functionH(s) to

be(3):

(1)

Figure 1.1 :Block diagram of Transfer functions

For comparison, we will consider the time-domain equivalent to the above input/output

relationship. In the time domain, we generally denote the input to a system asx(t), and the

output of the system asy(t). The relationship between the input and the output is denoted as

the impulse response, h(t).

We define the impulse response as being the relationship between the system output to its

input. We can use the following equation to define the impulse response:

(2)

Impulse Function

It would be handy at this point to define precisely what an "impulse" is. The Impulse

Function, denoted with (t) is a special function defined piece-wise as follows:


3/24

3

(3)

The impulse function is also known as the delta function because it's denoted with the Greek

lower-case letter . The delta function is typically graphed as an arrow towards infinity, as

shown below:

Figure 1.2 : mpulse (delta) function

1.1.2- Step Response

Similarly to the impulse response, the step response of a system is the output of the system

when a unit step function is used as the input. The step response is a common analysis tool

used to determine certain metrics about a system. Typically, when a new system is designed,

the step response of the system is the first characteristic of the system to be

analyzed.However, the impulse response cannot be used to find the system output from the

system input in the same manner as the transfer function(3).
http://en.wikibooks.org/wiki/File:Delta_Function.svg


4/24

4

1-2 DYNAMIC BEHAVIOURS OF FIRST ORDER AND SECOND ORDER

SYSTEMS

1.2.1 First-Order Systems

A one-degree-of-freedom first-order system is governed by the first-order ordinary

differential equation(4,5,6)

(4)

where y(t) is the response of the system (the output) to some forcing function F(t) (the input).

Eq. (4) may be rewritten as

(5)

where =a1/a0 has the dimension of time and is the time constant for the system and k =1/a0

is the gain.

Response of a First-order System to a Step Input

Consider a first-order system subjected to a constant force applied instantaneously at the

initial time t = 0 (4,5,6)

(6)

The initial condition is y(0) = 0. The solution to Eq. (5) with the step input Eq. (6) is then

(7)

The response approaches the final value y= kA exponentially. By using the boundary

conditions equation (7) then may be rewritten as


5/24

5

(8)

The rate at which the response approaches the final value is determined by the time constant.

When t = , y has reached 63.2% of its final value as illustrated in Figure 3. When t =5, y has

reached 99.3% of its final value.

Figure 1.3 : First Order systems

The time constant of a system can be determined from the measured response using a linear

regression. Taking the natural log Eq. (8) yields

(9)

The slope s of the natural log term plotted against t gives the time constant through the

relation s = -1/(4,5,6).


6/24

6

Transient Response of a Thermocouple

The dynamic response of a sensor is often an important consideration in designing a

measurement system. The response of a temperature sensor known as a thermocouple (TC)

may be modeled as a first-order system. When the TC is subjected to a rapid temperature

change, it will take some time to respond. If the response time is slow in comparison with the

rate of change of the temperature that you are measuring, then the TC will not be able to

faithfully represent the dynamic response to the temperature fluctuations(6).

A model of the response of a TC is based on a simple heat transfer analysis. The rate at which

the sensor exchanges heat with its environment must equal the rate of change of the internal

energy of the sensor. If the dominant mechanism of heat exchange is convection (neglecting

conduction and radiation), as it is for a TC in a fluid, then this energy balance is

(10)

h is the convection coefficient, A is the surface area of the sensor, T is the temperature, m is

the TC mass, and c is the heat capacity. Writing Eq. (11) in the form of Eq. (5)

(11)

where the time constant is

(12)

1.2.2 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(7):

(13)

If ao 0, then Equation (13) yields


7/24

7

(14)

Equation (14) is in the standard form of a second-order system, where

= natural period of oscillation of the system

= damping factor

K = steady state gain

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 (14) yields

(15)

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

case the inversion of Equation (15) by partial fraction expansion yields

(16)

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

(17)

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

In this case, the inversion of Equation (15) gives the result

(18)

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

inversion of Equation (15) in this case yields


8/24

8

(19)

Figure 4 : 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:

(20)

- Decay ratio: Is the ratio of the amount above the stead state value of two successive peaks,

c/a. it can be shown that it can be calculated by the following equation:

(21)


9/24

9

- 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 remain

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

- Period: Equation (21) 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 = 2/P;

(22)


10/24

10

2. EXPERIMANTAL METHOD

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

that contains diffrent type of liquids via their properties such as water, glycerol and their

mixtures.

The pressure difference in the U-manometer was created by a vacuum generator.

2.1.DESCRIPTION OF APPARATUS

Figure 2.1. U tube manometer[8]

2.2. EXPERIMENTAL PROCEDURE

Pressure difference was applied on the U-manometer by vacuum generator and determine the

variation of the liquid level with time until the manometer balanced. The vacuum pump was

stoped when the constant liquid level was observed. This process was repeated for all

overdamp U-manometer, and determine again the variation of the liquid level with time.

For underdamped U manometer the vacuum generator was opened and then oscilation wasobserved . The liquid level and their times was determined for step and impulse function.


11/24

11

3.0 RESULTS AND DISCUSSION

3.1 U TUBE MANOMETERS

Table 3.1 U tube manometers properties

properties Manometer

1

Manometer

2

Manometer

3

Manometer 4 Manometer5 Manometer 6

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

(Cp) 137,6 0,894 902,85 137,6 1,362 902,85

D (cm) 0.6 1,1 0.6 1,10 1,10 1,10

L( cm) 88 95 102 98 85 116

(s) 0.212 0.220 0.228 0.224 0.208 0.243

14,64 0,026 72,52 4,6 0,0354 23,03

According to Table 3.1 the viscosity of liquid in manometer 2 and 5 were realy smaller than

other and their diameter were same or bigger. This conditions effected the damping factor to

be smaller than 1. As a result manometer 2 and 5 could not absorb the effect of disturbition

like others so that their response will to be underdamped conditions. To determine the

response time we must look their time constant. The time constant was proportional with

square root of their lenght.As a result manometer 4 and 6 had a fast response time.

3.2. RESULTS for OVERDAMPED U-MANOMETERS

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

Manometer 1 Manometer 3 Manometer 4 Manometer 6

t(S) t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp

0 0,000 0,000 1,000 0,000 0,000 1,000 0,000 0,000 1,000 0,000 0,000 1,000

3 14,151 0,415 0,510 13,158 0,430 0,589 13,393 0,420 0,594 12,346 0,571 0,457

6 28,302 0,701 0,238 26,316 0,645 0,336 26,786 0,623 0,319 24,691 0,771 0,229

9 42,453 0,844 0,143 39,474 0,766 0,206 40,179 0,754 0,145 37,037 0,886 0,114

12 56,604 0,918 0,068 52,632 0,850 0,131 53,571 0,841 0,072 49,383 0,943 0,057

15 70,755 0,952 0,041 65,789 0,916 0,075 66,964 0,884 0,029 61,728 0,971 0,029

18 84,906 0,980 0,007 78,947 0,944 0,047 80,357 0,928 0,014 74,074 1,000 0,000

21 99,057 0,993 0,000 92,105 0,972 0,019 93,750 1,000 0,000

24 113,208 1,000 105,263 0,991 0,000

27 118,421 1,000


12/24

12

According to table 3.2.1 as predicted at table 3.1.1 fast response occured in manometer 4 and

6. Because of the tube lenght and diameter of the tube was bigger than other tubes so that

manometer 6 can easily absorp the effect of distirubition and give us fast response. But

manometer 4 must had a fast response time because its viscosity was smaller than manometer

6s liquid maybe some personal mistake in the experiment.

Figure 3.2.1. Experimental hr/kp versus t/ values

According to Figure 3.2.1 we can determine the response time . Kp values were the ultimate

values. Manometer 6 was reach their ultimate values faster than others when fluid was rising.

0,000

0,200

0,400

0,600

0,800

1,000

1,200

0,000 20,000 40,000 60,000 80,000 100,000 120,000 140,000

hr/Kp

t/to

M1

M3

M4

M6


13/24

13

Figure 3.2.2. Experimental hf/Kp versus t/ values

When the fluid was falling again the manometer 6 had a fast response time others .

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

Manometer 1 Manometer 3 Manometer 4 Manometer 6

t(s) t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp

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

3 14,151 0,384 0,616 13,158 0,087 0,913 13,393 0,771 0,2290 12,346 0,238 0,762

6 28,302 0,620 0,380 26,316 0,166 0,834 26,786 0,948 0,0524 24,691 0,419 0,581

9 42,453 0,811 0,189 39,474 0,238 0,762 40,179 0,988 0,0120 37,037 0,558 0,442

12 56,604 0,891 0,109 52,632 0,304 0,696 53,571 0,997 0,0027 49,383 0,663 0,337

15 70,755 0,938 0,062 65,789 0,365 0,635 66,964 0,999 0,0006 61,728 0,743 0,257

18 84,906 0,964 0,036 78,947 0,420 0,580 80,357 1,000 0,0001 74,074 0,804 0,196

21 99,057 0,979 0,021 92,105 0,470 0,530 93,750 1,000 0,0000

24 113,208 0,988 0,012 105,263 0,516 0,484

27 118,421 0,558 0,442

This table show the theoretical responses of overdamped u manometers t/ values must be

same with the experiment . hr/Kp values were different with experimental because of the

persanol mistakes.

0,000

0,200

0,400

0,600

0,800

1,000

1,200

0,000 20,000 40,000 60,000 80,000 100,000 120,000

hf/kp

t/to

M1

M3

M4

M6


14/24

14

Figure 3.2.3. Theoretical hr/Kp versus t/ values

M1 and M4 included same fluid and their viscoty values were smaller so that their response

times must be faster than others and also M6s lenght was bigger than M3 so that M6 must

gives us fast response time.

Figure 3.2.4. Theoretical hf/Kp versus t/ values

Same approach with the Figure 3.2.3 when the fluid was falling

0

0,2

0,4

0,6

0,8

1

1,2

0 50 100 150

hr/kp

t/to

M1

M3

M4

M6

0

0,2

0,4

0,6

0,8

1

1,2

0 50 100 150

hf/Kp

t/to

M1

M3

M4

M6


15/24

15

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) 4,544 4,802

Period of oscilation of manometer 2 and 5 were nearlly close together but manometer 2 little

bit long. The reason maybe the viscoty of liquid in manometer 2 was small so it rised more

than manometer 5 and that effected the raidan frequency .

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

Manometer 2 Manometer 5

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

0 0,000 0,000 0,000 0,000 0,000 0,000

1 1,510 6,864 1,000 1,430 6,875 1,000

2 1,780 8,091 0,522 1,890 9,087 0,500

3 2,590 11,773 0,882 2,470 11,875 0,890

4 3,230 14,682 0,676 3,460 16,635 0,646

5 3,960 18,000 0,809 4,220 20,288 0,768

6 4,900 22,273 0,728 5,190 24,952 0,720

7 5,810 26,409 0,699 6,110 29,375 0,720

8 6,890 31,318 0,743 7,160 34,423 0,744

As we expected the h /Kp values shows us the oscillation was occured because of the their

damping factor . And also the input was step function so that the osicalliton was reach one

point


16/24

16

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

Manometer 2 Manometer 5

ttheo(s) t/ h/Kp ttheo(s) t/ h/Kp

0 0,327 1,486 0,138 0,417 2,005 0,154

1 1,018 4,627 1,795 1,081 5,197 1,740

2 1,708 7,764 0,265 1,745 8,389 0,354

3 2,398 10,900 1,680 2,409 11,582 1,562

4 3,088 14,036 0,372 3,073 14,774 0,513

5 3,778 17,173 1,581 3,737 17,966 1,421

6 4,468 20,309 0,463 4,401 21,159 0,637

7 5,158 23,445 1,496 5,065 24,351 1,311

8 5,848 26,582 0,541 5,729 27,543 0,734

9 6,538 29,718 1,424 6,393 30,736 1,226

10 7,228 32,855 0,608

The cause of reading mistakes the experimental values was not close with the experimental

values.

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

Experimental values were not correctly readed .

0,000

0,200

0,400

0,600

0,800

1,000

1,200

1,400

1,600

1,800

2,000

0,000 5,000 10,000 15,000 20,000 25,000 30,000 35,000

h/kp

t/to

M2-exp

M2-theo


17/24


18/24

18

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

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

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

0 0,000 0,000 0 0,000 0,000

1 0,650 2,955 1,000 1,1 5,288 1,000

2 1,280 5,818 -0,693 1,79 8,606 -0,791

3 1,890 8,591 0,511 2,5 12,019 0,674

4 2,550 11,591 -0,341 3,12 15,000 -0,372

5 3,270 14,864 0,295 4 19,231 0,186

6 4,170 18,955 -0,239 4,89 23,510 -0,140

7 5,010 22,773 0,295 5,75 27,644 0,070

8 5,770 26,227 -0,239 6,59 31,683 -0,023

9 6,670 30,318 0,193 7,71 37,067 0,012

10 7,870 35,773 -0,114

8,690 39,500 0,091

The input was the impulse function so that the h/Kp values changes positive to negative. The

lenight of oscicallation should reach 0.

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

ttheo(s) t/ h/Kp theo(s) t/ h/Kp

0 -0,346 -1,573 -1,042 -0,333 -1,599 -1,057

1 0,346 1,573 0,960 0,333 1,599 0,947

2 1,038 4,718 -0,885 0,998 4,796 -0,845

3 1,730 7,864 0,815 1,663 7,993 0,753

4 2,422 11,009 -0,751 2,328 11,190 -0,6685 3,114 14,155 0,692 2,993 14,387 0,591

6 3,806 17,300 -0,638 3,658 17,584 -0,522

7 4,498 20,445 0,588 4,323 20,781 0,459

8 5,190 23,591 -0,541 4,988 23,978 -0,403

9 5,882 26,736 0,499 5,653 27,175 0,352

10 6,574 29,882 -0,460 6,318 30,373 -0,306

11 7,266 33,027 0,423 6,983 33,570 0,266


19/24

19

12 7,958 36,173 -0,390

13 8,650 39,318 0,359

14 9,342 42,464 -0,331

15 10,034 45,609 0,305

16 10,726 48,755 -0,281

17 11,418 51,900 0,259

Figure 3.4.1. Theoretical and experimental values for M-2

According to Figure 3.4.1 the experimental and theoretical curve was close early but than

some of the mistakes maybe reading mistakes was effectted the phase of the oscillation. But

both of them was aproach to zero because of the impulse function.

-1,5

-1

-0,5

0

0,5

1

1,5

0,000 10,000 20,000 30,000 40,000 50,000 60,000h/kp

t/to

M2 exp

M2-theo


20/24

20

Figure 3.4.2. Theoretical and experimental values for M-5

According to Figure 3.4.2 the experimental and theoretical curve was close early but than

some of the mistakes maybe reading mistakes was effectted the phase of the oscillation. But

both of them was aproach to zero because of the impulse function.

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

Response time to Impulse change

Monometer 2 Monometer 5

Experimental Theoretical Experimental Theoretical

Overshoot 0,511 0,922 0,674 0,897

Decay ratio 0,577 0,85 0,276 0,805

Response

Time

8,6 11,42 7,71 6,98

-1,500

-1,000

-0,500

0,000

0,500

1,000

1,500

0,000 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000h/kp

t/to

M5-exp

M5-theo


21/24

21

4. CONCLUSIONS

In this experiment ,to determine the effects of liquid properties and shape of U-tube

manometers on response time by using step and impulse input, 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 were used.

In the overdamped systems (m-1,m-3,m-4,m-6), the damping factor was calculated and it was

observed that their damping factors were greater than 1. These systems can easily absorb the

energy of disturbiton and the reason of this viscosity of liquids that contained in these

manometers were high enough according to their diameter and length.Furtheremore, to

compare their response time, it was observed that higher length and higher diameter cause the

response time to get low for same liquid.

In the underdamped systems (m-2 ,m-5), the damping factor was calculated again and it was

observed that their damping factor were smaller than 1. As we expected they relased their

energy with doing oscillation step by step. Our experimental values was different from the

theoretical values.The reason of this the oscillation was realy fast so that the reading mistakes

was done. Howewer, according to theoretical and experimental response time, we could

observed that the impulse system had a higher response time than step system. The reason of

this, while they relasing their energy which comes from disturbition from vacuum generator,

the potential energy differences at step function was small than impulse function.


22/24

22

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)

Viscosity of the liquid (Pa.s)

Density of the liquid (kg/m3)

Radian frequency (radian/s)

Damping factor


23/24

23

6.REFERENCES

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

2-

https://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminolo

gy

3-. http://en.wikibooks.org/wiki/Control_Systems/Transfer_Functions

4- J.P. Holman, Experimental Methods for Engineers, 7th Ed., McGraw-hill, New York,

2001: First-order systems, p. 19-23; Thermocouples p. 368-377; Linear regression p. 91-

94; Signal conditioning (RC Circuits) p. 183-190.

5-R.S. Figliola and D.E. Beasley, Theory and Design for Mechanical Measurements, Wiley,

New York, 1991, p. 63, 73.

6- Omega Technologies Handbook, Thermocouple Reference Tables, Omega Engineering

Inc., 1993, p. B172.

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

%20Order%20Systems.pdf

8.http://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmecha

nicsgeneral
http://www.pacontrol.com/download/Process%20Control%20Fundamentals.pdfhttps://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminologyhttps://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminologyhttp://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second%20Order%20Systems.pdfhttp://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second%20Order%20Systems.pdfhttp://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmechanicsgeneralhttp://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmechanicsgeneralhttp://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmechanicsgeneralhttp://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmechanicsgeneralhttp://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second%20Order%20Systems.pdfhttp://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second%20Order%20Systems.pdfhttps://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminologyhttps://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminologyhttp://www.pacontrol.com/download/Process%20Control%20Fundamentals.pdf


24/24

7. APPENDIX

process control experiment

Documents