u Manometers
-
Upload
goezde-salkic -
Category
Documents
-
view
142 -
download
3
Transcript of u Manometers
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