Post on 04-Jun-2018
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Abstract
Two tanks connected in series were studied to determine their responses to step,
ramp, and on-off changes to inlet flow rate under linear and non-linear discharge control.
Time constants for the small tank were calculated and compared for a step input yielding
an average of 15.4 s for filling tank and 21.8 s for emptying tank. The theoretical step
response for a second order system was compared with the experimental response giving
percent errors ranging from 10.3% or less at times above 18 s and below 100 s, while
deviations of up to 8.1% occur above 100 s. Additionally, the experimental on-off
responses of the single-capacity and two-capacity systems were in agreement with theory,
except in the case where the tank is empties. Also, it was determined that switching from
linear to square root discharge profiles caused the time constant of the small tank to
increase, while the time constant for the large tank decreased.
It was recommended to use digital control to precisely adjust the on-off valve and
the flow rate valve in replacement of manual operation. This change could potentially
improve the precision in ramp input and on-off control responses, thus allowing a more
accurate calculation of the time constants.
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Table of Contents
Abstract ................................................................................................................................ i
Introduction ..........................................................................................................................1
Theory ..................................................................................................................................1
Results and Discussion ........................................................................................................5
Evaluation of the Time Constants ................................................................................................ 5
Ramp Input ................................................................................................................................... 7
On-off Control ............................................................................................................................ 11
Conclusions and Recommendations ..................................................................................13
Appendix ............................................................................................................................15
Original Data .............................................................................................................................. 15
Sample Calculations ................................................................................................................... 25
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Introduction
The system under consideration consisted of two tanks in series. Water flowed
first into tank one, and then out of tank one into a larger tank two. The height of each
tank was monitored with pressure sensors and transmitted to a computer for data
collection. A discharge rate proportional to the height in tank one was used to observe
linear control, and a rate proportional to the square root of height was used for non-linear
control. Step deviations of gpm in the flow rate as well as ramp inputs with slopes
of +0.01 gpm/s and 0.02 gpm/s were implemented using a flow valve and rotameter. The
one and two-tank liquid level responses to these deviations were analyzed. On-off control
was also applied for the one and two-tank systems and responses were evaluated.
Theory
The mass balance for a single tank yields (1), where m is the mass accumulated in
the tank, is the mass flow rate in and out of the tank.
(1)
Dividing both sides of (1) by the density of the fluid and using the definition of
the volume accumulated in the tank (2) is obtained. The volume of the fluid inside the
tank is C, the cross-sectional area, multiplied by h, the height of the fluid.
(2)
The volumetric flow rate in and out of the tank is represented by q in and q
respectively. The height of the fluid can be set equal to the product of the outlet
volumetric flow rate and the outlet resistance, R, yielding (3), where , the time constant
parameter, equals RC.
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(3)
For time t = 0, (4) is obtained from (3).
(4)
Taking the difference of (4) and (3), deviation variables can be defined as
, giving (5). For a unit step change in the inletflow rate, is equal to the Heaviside function, H(t).
(5)
(6) is obtained by taking the Laplace transform of (5).
(6)
For the deviation variable x at t = 0, x(0) = 0. Plugging this into (6) and solving
for gives (7). (7)
Laplace inversion of (7) yields (8).
(8)
Defining , it can be found that . As time goes toinfinity, y becomes where h ss is the steady state height at infinite timeand h(0) is the height at time zero. With these relationships the height as a function of
time in response to a step input can be shown as (9).
(9)
(9.1) shows the fractional change in height, f, as a function of time.
(9.1)
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(9.2) is another useful form of (9).
(9.2)
For a ramp input where A is the ramp slope, (5) becomes (10).
(10)(11) is obtained through Laplace transform of (10).
(11)
From (11) it follows that
(12)
Using y = Rx and performing Laplace inversion produces (13).
(13)
As time goes to infinity, a linear form of (13) is obtained and .
For the two-tank system response, the form of (3) can be used to find equations for each
tank with time constants and given in (14) and (15). (14)
(15)
In (14) and (15) q 1 and q 2 represent the flow rates out of the first and second tanks
respectively. Defining deviation variables and , (16) and (17) can be obtained.
(16)
(17)
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Laplace transform of (16) and (17) yields (18) and (19).
(18)
(19)
With the relation , the height of fluid in the second tank can be found by (20).
(20)
For a step input (20) becomes (21) after partial fraction expansion.
Laplace inversion of (21) gives (22) which is the time domain response of the two-tank
system.
[ ] (21)* + (22)
For a ramp input with slope A, and the two-tank response is seen in
(23). Laplace inversion gives (24).
[ ] (23)
* + (24)For and (22) takes the form of (9) and (24) becomes
(13). As time goes to infinity, the value of for a ramp response can be
detrmined. For (24) gives which subsequentlyyields (25). (25)
can be determined from (25).
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Results and Discussion
Evaluation of the Time Constants
The response of a small tank to the sudden change in flow rate from 0.3 gpm to
0.7 gpm was measured and plotted over time. According to theory, the time constant can
be found by noting the time where the response is 63.2% of its final value, or 0.632 when
plotting the fraction of the total change in height. Using this approach and linear
interpolation, the time constant calculated for filling the tank was 16.1 seconds. Similarly,
emptying the tank yielded a time constant of 21.7 seconds. An alternative is to use the
initial value theorem (IVT) for the slope of the response, which says that the initial slope
is inversely proportional to the time constant of a first-order system. The IVT method
yielded time constants of 96.2 s for filling and 357.1 s for emptying the small tank.
Figure 1 shows the response of filling and emptying the small tank after a step input, as
well as the theoretical solution for filling the tank using an estimated time constant of
14.6 s.
Figure 1. Response for filling and emptying the small tank using a step input.
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Another method calls for plotting the log of the fraction of the total change in
height, and performing a linear regression the slope of the linear fit then corresponds to
a value of the time constant. This method yielded 14.6 s and 21.8 s as time constants for
filling and emptying, respectively. Figure 2 shows a semi-log plot of the fraction of the
total change in height over time, which follows a linear trend for the first 40 seconds. It is
worth nothing that the time constants yielded by this linear regression method and by
noting the time at 63.2% completion are similar a relative difference of 9.3% for filling
and 0.46% for emptying.
Figure 2. Semi-log plot of the fraction of the total change in height over time.
The step response of a second-order system was analyzed using the data collected
from the larger tank. Unlike the step response of the small tank, the initial response of the
second-order system is much slower. At larger times, the large tank reaches its final
height faster than predicted by a theoretical step response ( = 30 s, = 60 s).
Nevertheless, the theoretical model for the second-order system is fairly accurate (within
10.3% or less from the experimental data) at times above 18 s and below 100 s. Above
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100 seconds, the total change in height for the large tank starts to deviate (up to 8.1%)
from the theoretical model. Figure 3 shows the step response of the large tank, as well as
the theoretical step response with = 30 s and = 60 s.
Figure 3. Step response of the large tank.
Ramp Input
For analysis of the system response to a ramp input, two different magnitudes of
ramp inputs were used. The system was allowed to reach steady state for a flow rate of
0.3 gpm. The ramp input was manually approximated by continuously adjusting the flow
rate until a flow of 0.8 gpm was reached. For the slow ramp, the flow rate was increased
at a rate of 0.1 gpm every 10 seconds, while the flow rate for the fast ramp was increased
at a rate of 0.2 gpm every 10 seconds. The analysis was performed for both linear
discharge profile and square root discharge profile. The deviation variable, y is defined as
(26)
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The time constants for both tanks were determined using the methods outlined in
the Theory section. As t approaches infinity, the response of the small tank reaches a
long-time asymptote of the form
(27)
This asymptote is a linear function with slope Rb and displaced by time units
from the origin. In the same manner, the response of the second-order large tank reached
a long-time asymptote of the form
( ) (28)By plotting the linear region of the response and setting y = 0, the time constants
of the two tanks can be obtained from experimental data. Figure 4 shows the ramp
response to the slow ramp input with a linear discharge profile. The equations for the fit
to the linear region of the response are given in the figure.
Figure 4. The height, h h(0) , as a function of time for the slow ramp with a linear discharge profile.
y = 0.0953x - 0.9752
y = 0.0619x - 1.9702
0
0.5
1
1.5
2
2.5
3
3.5
0 10 20 30 40 50 60
y ( i n
)
Time (s)
Small Tank -Slow RampLarge Tank -Slow RampFit 1
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From equations (27) and (28), it was determined that the time constants for the
small and large tanks are 10.23 s and 21.60 s, respectively. Figure 5 shows the ramp
response to the fast ramp input with a linear discharge profile. The equations for the fit to
the linear region of the response are given in the figure. From the linear regressions, it
was determined that the time constants for the small and large tanks are 6.79 s and 5.07 s,
correspondingly.
Figure 5. The height, h h(0) , as a function of time for the fast ramp with a linear discharge profile.
Figure 6 shows the ramp response to the slow ramp input with a square
root discharge profile. Once again, the equations for the fit to the linear region of the
response are given in the figure. From the linear regressions and equations (27) and (28),
it was determined that the time constants for the small and large tanks are 12.33 s and
12.48 s, in that order.
y = 0.1401x - 0.9514
y = 0.0127x - 0.1506
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 5 10 15 20
y ( i n
)
Time (s)
Small Tank - Fast RampLarge Tank - Fast
RampFit Small Tank
Fit Large Tank
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Figure 6. The height, h h(0) , as a function of time for the slow ramp with a square root dischargeprofile.
Figure 7 shows the ramp response to the fast ramp input with a square root
discharge profile. The equations for the fit to the linear region of the response are given
in the figure. From the linear regressions, it was determined that the time constants for
the small and large tanks are 5.92 s and 9.02 s, respectively.
Figure 7. The height, h h(0) , as a function of time for the fast ramp with a square root dischargeprofile.
y = 0.1116x - 1.3762
y = 0.0285x - 0.707
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30 35 40
y ( i n
)
Time (s)
Small Tank -Slow RampLarge Tank -Slow RampFit 1
Fit 2
y = 0.14x - 0.8285
y = 0.0138x - 0.2062
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 5 10 15 20
y ( i n
)
Time (s)
Small Tank -Fast RampLarge Tank -Fast RampFit 1
Fit 2
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A faster ramp input, corresponding to a larger rate of increase of flow rate, has
smaller estimated time constants for both tanks. In other words, the system responded
faster to a ramp input of larger magnitude. Its worth noting that , in general, ramp input
time constants are smaller than the estimates derived from step inputs.
On-off Control
In on-off control analysis, the tanks were first allowed to reach steady state at a
higher flow rate of 0.5 gpm. The flow rate was then rapidly reduced to 0.3 gpm. The level
in the tanks were then regulated using a bypass valve on the inlet, which has a very high
flow rate. A deadband of 2 inches was used when performing the analysis, where the
bypass valve was opened when the liquid level drops 1 inch below the steady state level
and closed when the liquid level reaches 1 inch above the steady state level. The system
response was observed and recorded for single and two-capacity systems. In single
capacity system analysis, the water level was regulated for the first, smaller tank. In two-
capacity system analysis, the water level was instead regulated for the second, larger
tank. Analysis was performed only for a linear discharge profile.
From Figure 8, it can be seen that the height in the first tank always exceeds the
upper height limit when filling. This corresponds to the theoretical result obtained in the
Theory section, which states that the derivative of the height is positive when the height
hits the upper limit. However, the response when emptying does not match up withtheory. Because there is no input term when emptying, the derivative of the height only
has a negative discharge term, which means that the lower limit should be exceeded. The
experimental data shows otherwise.
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Figure 8. On-off control response for a single capacity system with a deadband of 2 inches.
The upper and lower height limits of the second tank is always exceeded, which is
in agreement with theory. Figure 9 shows this on a plot of on-off control response over
time. A large height in the first tank was required to regulate the level in the second tank.
The values in the vicinity of t = 350 s correspond to a mistake in performing the analysis,
and should not be used as valid values in the system response.
Figure 9. On-off control response for a two-capacity system with a deadband of 2 inches.
-1
-0.5
0
0.5
1
1.5
0 20 40 60 80 100 120 140 160
y ( i n
)
Time (s)
Tank 1 - SingleCapacity
-4
-2
0
2
4
6
8
0 100 200 300 400 500 600 700 800
y ( i n
)
Time (s)
Tank 1 - TwoCapacity
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The single and two-capacity system are quite different in their responses. The
single capacity system behaves as a first-order system with multiple step inputs, and the
second tank displays gradually increasing values of oscillatory behavior. For the two-
capacity system, there is less control over the height of water in the second tank, and the
system responds very slowly to any step inputs. This corresponds to a larger time scale,
where each filling and emptying cycle takes about 150 s, while the single capacity system
only takes about 25 s.
Conclusions and Recommendations
The time constants obtained from the step and ramp inputs agree in the case
where the ramp input is slow and the discharge profile is set by the square root of the
height (i.e. flow rate that was increased at a rate of 0.1 gpm every 10 seconds). When
going from a slow to a fast ramp input, the time constant for the small tank decreases by
33.6% for the linear discharge profile and by 52.0% for the square root profile. In terms
of the larger tank, the time constant decreases by 76.5% for the linear discharge profile
and by 27.7% for the square root profile. Tables 1 and 2 show a summary of all the time
constants calculated using step and ramp inputs, respectively.
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Table 1. Summary of time constants calculated from the step response of the small tank.
Method Filling Emptying
Time at 63.2% completion 16.1 21.7
IVT for slope 96.2 357.1Linear regression on semi-log plot 14.6 21.8
Table 2. Summary of time constants for ramp input.
Linear DischargeInput Tank 1 Tank 2
Slow Ramp 10.2 21.6Fast Ramp 6.8 5.1
Square Root DischargeInput Tank 1 Tank 2Slow Ramp 12.3 12.5Fast Ramp 5.9 9.0
It follows from Table 2 that the time constants for both tanks are a function of the
discharge profile, as noted in the Theory section. At the same point in time, the level in
the first tank for a square root discharge profile is higher than the level for a linear
discharge profile. However, due to the smaller magnitude of the discharge, the level in
the second tank for a square root discharge profile is smaller than the level for a linear
discharge profile.
On-off control for the single capacity system has similar behavior as a first-order
system with multiple step inputs at a set interval, meanwhile the second tank displays
oscillatory behavior that increases as a whole over time . In the two-capacity system,
control over the height of water in the second tank is minimal so the system obtains a
larger time constant.
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A significant source of error is the presence of bubbles in the inlet lines. This
means that the actual value of the flow rate is not the value indicated by the rotameter.
Reading and adjusting the rotameter could also contribute to the error. Another important
source of error is the manual approximation of a ramp input, which might result in a ramp
that is not smooth. There was also a slight delay between the time when the level in the
tanks hit the height limits and the time when the bypass valve was opened or closed in the
on-off analysis.
For future studies, its recommended to use automatic valve control in lieu of
manual operation. This improvement would allow precise control of the ramp input,
hence the effect a ramp inputs magnitude on time constants would be clearer.
Appendix
Original Data
ElapsedTime
(s)
Linear Step Response Linear Ramp Response On-off Response
Filling Emptying Slow Fast Single-capacity Two-capacityTank 1Level
Tank 2Level
Tank 1Level
Tank 2Level
Tank 1Level
Tank 2Level
Tank 1Level
Tank 2Level
Tank 1Level
Tank 2Level
Tank 1Level
Tank 2Level
0 3.226 0.554 5.276 5.433 2.553 0.66 0.097 2.671 5.723 1.334 3.106 2.686
0.5 3.228 0.557 5.276 5.429 2.552 0.644 0.098 2.651 5.719 1.357 3.08 2.677
1 3.217 0.559 5.276 5.431 2.552 0.654 0.101 2.624 5.677 1.384 3.066 2.661
1.5 3.201 0.573 5.277 5.437 2.553 0.654 0.113 2.586 5.632 1.407 3.052 2.64
2 3.185 0.583 5.284 5.427 2.552 0.657 0.141 2.541 5.594 1.437 3.046 2.62
2.5 3.168 0.593 5.283 5.421 2.552 0.66 0.186 2.5 5.553 1.46 3.029 2.611
3 3.153 0.594 5.283 5.421 2.552 0.66 0.226 2.459 5.513 1.494 3.014 2.6
3.5 3.139 0.59 5.281 5.414 2.549 0.663 0.269 2.416 5.476 1.521 3.004 2.584
4 3.126 0.591 5.277 5.417 2.552 0.656 0.316 2.376 5.434 1.566 2.993 2.571
4.5 3.113 0.601 5.283 5.416 2.552 0.661 0.361 2.333 5.397 1.587 2.983 2.554
5 3.098 0.599 5.283 5.413 2.552 0.649 0.406 2.294 5.359 1.617 2.972 2.541
5.5 3.086 0.607 5.283 5.423 2.552 0.644 0.446 2.256 5.331 1.639 2.964 2.524
6 3.08 0.596 5.279 5.411 2.546 0.644 0.487 2.21 5.297 1.676 2.952 2.51
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6.5 3.063 0.59 5.27 5.407 2.546 0.641 0.531 2.179 5.263 1.704 2.937 2.493
7 3.051 0.589 5.212 5.41 2.545 0.64 0.571 2.136 5.233 1.717 2.924 2.477
7.5 3.039 0.586 5.15 5.401 2.548 0.636 0.608 2.096 5.204 1.741 2.916 2.459
8 3.022 0.59 5.08 5.406 2.55 0.637 0.65 2.056 5.175 1.751 2.906 2.443
8.5 3.017 0.586 5.017 5.409 2.545 0.651 0.686 2.024 5.143 1.78 2.895 2.427
9 3.003 0.59 4.952 5.396 2.545 0.649 0.72 1.987 5.117 1.799 2.884 2.414
9.5 2.999 0.591 4.894 5.389 2.545 0.649 0.759 1.949 5.088 1.82 2.879 2.393
10 2.986 0.599 4.834 5.387 2.545 0.65 0.792 1.909 5.059 1.829 2.868 2.381
10.5 2.974 0.587 4.774 5.374 2.545 0.654 0.826 1.874 5.033 1.853 2.855 2.366
11 2.968 0.591 4.724 5.369 2.545 0.653 0.858 1.84 5.011 1.869 2.852 2.353
11.5 2.953 0.591 4.67 5.356 2.545 0.653 0.89 1.804 4.986 1.874 2.846 2.329
12 2.942 0.587 4.611 5.34 2.545 0.646 0.926 1.77 4.963 1.894 2.834 2.314
12.5 2.931 0.589 4.563 5.336 2.546 0.646 0.954 1.731 4.942 1.921 2.828 2.297
13 2.924 0.59 4.514 5.337 2.549 0.64 0.989 1.704 4.921 1.909 2.819 2.28
13.5 2.92 0.584 4.461 5.323 2.545 0.639 1.019 1.667 4.898 1.931 2.815 2.263
14 2.906 0.586 4.414 5.304 2.545 0.647 1.048 1.639 4.88 1.949 2.814 2.25414.5 2.919 0.584 4.366 5.296 2.545 0.653 1.08 1.599 4.861 1.964 2.821 2.234
15 2.977 0.581 4.321 5.283 2.552 0.647 1.11 1.569 4.84 1.979 2.834 2.214
15.5 3.052 0.593 4.277 5.26 2.552 0.636 1.142 1.544 4.825 1.991 2.862 2.196
16 3.124 0.581 4.233 5.246 2.552 0.646 1.166 1.513 4.808 2.004 2.89 2.181
16.5 3.186 0.599 4.194 5.23 2.552 0.651 1.197 1.476 4.789 2.019 2.92 2.169
17 3.261 0.604 4.15 5.214 2.552 0.639 1.228 1.45 4.774 2.024 2.952 2.15
17.5 3.327 0.589 4.106 5.197 2.552 0.65 1.25 1.416 4.761 2.05 2.986 2.137
18 3.386 0.597 4.069 5.169 2.552 0.641 1.279 1.38 4.741 2.056 3.022 2.124
18.5 3.457 0.607 4.029 5.157 2.552 0.641 1.308 1.343 4.721 2.059 3.054 2.107
19 3.51 0.601 3.993 5.144 2.556 0.631 1.332 1.324 4.71 2.07 3.084 2.0919.5 3.568 0.607 3.959 5.124 2.552 0.631 1.357 1.296 4.694 2.084 3.117 2.074
20 3.625 0.616 3.92 5.104 2.552 0.643 1.386 1.271 4.679 2.079 3.143 2.064
20.5 3.68 0.619 3.89 5.084 2.552 0.639 1.41 1.24 4.659 2.096 3.178 2.053
21 3.735 0.624 3.861 5.067 2.556 0.64 1.436 1.211 4.65 2.109 3.199 2.039
21.5 3.783 0.616 3.829 5.049 2.552 0.64 1.458 1.184 4.634 2.111 3.234 2.029
22 3.836 0.619 3.799 5.033 2.552 0.631 1.48 1.15 4.623 2.129 3.259 2.017
22.5 3.879 0.633 3.767 5.004 2.552 0.63 1.502 1.124 4.611 2.141 3.287 1.997
23 3.927 0.624 3.738 4.99 2.552 0.634 1.526 1.1 4.596 2.153 3.317 1.987
23.5 3.974 0.634 3.706 4.967 2.552 0.651 1.548 1.076 4.589 2.159 3.342 1.981
24 4.022 0.639 3.681 4.943 2.557 0.631 1.57 1.047 4.572 2.154 3.374 1.969
24.5 4.066 0.646 3.655 4.924 2.559 0.634 1.592 1.024 4.564 2.164 3.393 1.959
25 4.109 0.667 3.626 4.897 2.559 0.627 1.612 0.997 4.554 2.177 3.417 1.943
25.5 4.145 0.661 3.6 4.88 2.559 0.627 1.637 0.971 4.542 2.179 3.436 1.943
26 4.189 0.674 3.574 4.856 2.559 0.629 1.657 0.947 4.535 2.181 3.461 1.934
26.5 4.23 0.686 3.548 4.83 2.563 0.62 1.677 0.931 4.523 2.187 3.48 1.923
27 4.266 0.699 3.524 4.813 2.566 0.626 1.694 0.92 4.51 2.199 3.503 1.916
27.5 4.302 0.704 3.503 4.783 2.566 0.636 1.712 0.88 4.502 2.211 3.526 1.907
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28 4.345 0.711 3.479 4.75 2.566 0.633 1.731 0.859 4.495 2.207 3.546 1.897
28.5 4.382 0.724 3.454 4.741 2.566 0.641 1.749 0.843 4.484 2.211 3.564 1.889
29 4.412 0.734 3.432 4.724 2.567 0.626 1.767 0.826 4.474 2.224 3.585 1.881
29.5 4.446 0.744 3.41 4.693 2.567 0.627 1.781 0.8 4.465 2.22 3.607 1.874
30 4.476 0.757 3.39 4.669 2.566 0.629 1.797 0.78 4.459 2.23 3.621 1.87
30.5 4.51 0.769 3.368 4.623 2.566 0.629 1.814 0.76 4.452 2.241 3.637 1.864
31 4.548 0.78 3.348 4.611 2.566 0.614 1.832 0.739 4.448 2.251 3.657 1.859
31.5 4.579 0.799 3.332 4.586 2.564 0.63 1.846 0.72 4.436 2.244 3.677 1.851
32 4.604 0.816 3.312 4.56 2.564 0.629 1.866 0.703 4.434 2.253 3.691 1.851
32.5 4.632 0.824 3.297 4.539 2.566 0.627 1.883 0.683 4.428 2.267 3.709 1.844
33 4.659 0.834 3.279 4.507 2.56 0.637 1.892 0.66 4.426 2.264 3.724 1.843
33.5 4.687 0.85 3.259 4.48 2.566 0.634 1.91 0.64 4.419 2.276 3.741 1.833
34 4.712 0.871 3.243 4.46 2.566 0.647 1.926 0.621 4.41 2.286 3.749 1.829
34.5 4.731 0.887 3.228 4.427 2.566 0.629 1.937 0.611 4.407 2.283 3.766 1.83
35 4.753 0.911 3.21 4.407 2.566 0.629 1.953 0.597 4.399 2.28 3.782 1.821
35.5 4.777 0.924 3.192 4.377 2.577 0.636 1.964 0.563 4.39 2.297 3.793 1.82336 4.803 0.947 3.178 4.354 2.59 0.634 1.978 0.561 4.383 2.293 3.811 1.821
36.5 4.826 0.961 3.161 4.319 2.596 0.64 1.993 0.546 4.378 2.293 3.821 1.813
37 4.851 0.974 3.149 4.304 2.6 0.637 2.006 0.533 4.377 2.297 3.836 1.807
37.5 4.868 0.996 3.141 4.281 2.611 0.636 2.018 0.524 4.37 2.311 3.846 1.804
38 4.891 1.017 3.13 4.253 2.625 0.643 2.033 0.507 4.37 2.307 3.858 1.807
38.5 4.913 1.036 3.117 4.226 2.641 0.64 2.054 0.51 4.361 2.31 3.874 1.797
39 4.934 1.057 3.102 4.191 2.662 0.636 2.065 0.504 4.359 2.309 3.886 1.797
39.5 4.945 1.073 3.086 4.186 2.677 0.634 2.074 0.507 4.359 2.321 3.899 1.799
40 4.964 1.099 3.074 4.154 2.691 0.63 2.083 0.504 4.352 2.333 3.91 1.793
40.5 4.983 1.116 3.062 4.127 2.702 0.636 2.099 0.501 4.341 2.329 3.916 1.79341 5 1.136 3.048 4.103 2.723 0.626 2.112 0.513 4.345 2.337 3.928 1.794
41.5 5.01 1.151 3.036 4.071 2.745 0.629 2.121 0.507 4.337 2.334 3.945 1.796
42 5.032 1.173 3.023 4.049 2.774 0.627 2.137 0.504 4.331 2.336 3.952 1.789
42.5 5.05 1.194 3.008 4.029 2.79 0.63 2.145 0.506 4.328 2.344 3.956 1.789
43 5.057 1.221 3 3.996 2.811 0.617 2.153 0.51 4.321 2.347 3.968 1.783
43.5 5.07 1.239 2.992 3.98 2.837 0.631 2.163 0.5 4.324 2.353 3.975 1.79
44 5.09 1.263 2.983 3.959 2.862 0.631 2.177 0.506 4.324 2.359 3.989 1.794
44.5 5.101 1.279 2.97 3.937 2.883 0.651 2.188 0.5 4.319 2.351 3.997 1.787
45 5.108 1.313 2.957 3.914 2.912 0.643 2.196 0.513 4.312 2.361 4.001 1.783
45.5 5.128 1.331 2.945 3.887 2.937 0.649 2.204 0.513 4.309 2.366 4.007 1.786
46 5.139 1.353 2.932 3.85 2.959 0.63 2.214 0.511 4.305 2.364 4.015 1.786
46.5 5.15 1.377 2.926 3.834 2.99 0.633 2.222 0.514 4.303 2.374 4.022 1.787
47 5.163 1.403 2.919 3.817 3.023 0.644 2.233 0.51 4.298 2.379 4.032 1.787
47.5 5.171 1.414 2.91 3.793 3.052 0.633 2.248 0.514 4.299 2.374 4.039 1.779
48 5.182 1.444 2.902 3.763 3.087 0.64 2.248 0.506 4.297 2.374 4.05 1.781
48.5 5.194 1.469 2.89 3.734 3.113 0.653 2.254 0.504 4.29 2.376 4.057 1.781
49 5.207 1.491 2.88 3.717 3.139 0.644 2.263 0.513 4.29 2.379 4.063 1.781
8/13/2019 Water Tank System Response
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18
49.5 5.221 1.523 2.872 3.691 3.175 0.65 2.27 0.504 4.283 2.38 4.069 1.781
50 5.23 1.547 2.865 3.667 3.208 0.656 2.281 0.51 4.28 2.383 4.076 1.793
50.5 5.236 1.571 2.858 3.644 3.243 0.65 2.294 0.513 4.276 2.386 4.083 1.787
51 5.239 1.596 2.851 3.624 3.277 0.671 2.297 0.504 4.276 2.384 4.083 1.786
51.5 5.248 1.627 2.841 3.606 3.316 0.66 2.308 0.514 4.273 2.38 4.088 1.794
52 5.258 1.649 2.834 3.586 3.345 0.666 2.314 0.509 4.272 2.384 4.097 1.789
52.5 5.268 1.681 2.823 3.559 3.386 0.673 2.32 0.514 4.269 2.389 4.103 1.789
53 5.277 1.699 2.815 3.524 3.432 0.679 2.33 0.511 4.262 2.384 4.106 1.787
53.5 5.284 1.724 2.808 3.503 3.468 0.673 2.334 0.507 4.262 2.39 4.114 1.793
54 5.287 1.759 2.804 3.49 3.514 0.667 2.341 0.516 4.259 2.389 4.12 1.794
54.5 5.297 1.783 2.8 3.467 3.56 0.671 2.345 0.511 4.251 2.387 4.124 1.8
55 5.303 1.809 2.793 3.437 3.608 0.673 2.352 0.516 4.251 2.39 4.126 1.803
55.5 5.31 1.834 2.786 3.423 3.648 0.671 2.359 0.516 4.247 2.397 4.131 1.806
56 5.312 1.863 2.778 3.403 3.695 0.68 2.363 0.504 4.241 2.4 4.138 1.806
56.5 5.314 1.879 2.771 3.381 3.743 0.677 2.368 0.514 4.241 2.391 4.143 1.807
57 5.319 1.904 2.766 3.363 3.778 0.683 2.372 0.514 4.239 2.4 4.145 1.81357.5 5.328 1.941 2.76 3.331 3.822 0.684 2.379 0.523 4.234 2.399 4.146 1.817
58 5.332 1.969 2.75 3.31 3.872 0.694 2.386 0.513 4.233 2.396 4.152 1.82
58.5 5.337 1.993 2.743 3.294 3.924 0.697 2.393 0.516 4.228 2.39 4.159 1.82
59 5.338 2.001 2.738 3.273 3.959 0.711 2.399 0.511 4.226 2.396 4.159 1.821
59.5 5.301 2.026 2.734 3.246 4.003 0.719 2.4 0.516 4.222 2.409 4.166 1.819
60 5.32 2.03 2.723 3.226 4.057 0.717 2.406 0.504 4.222 2.404 4.166 1.82
60.5 5.302 2.044 2.723 3.211 4.098 0.737 2.41 0.514 4.221 2.404 4.171 1.819
61 5.326 2.07 2.717 3.19 4.148 0.734 2.407 0.506 4.221 2.406 4.174 1.821
61.5 5.314 2.07 2.712 3.169 4.208 0.744 2.414 0.52 4.218 2.414 4.177 1.819
62 5.337 2.093 2.699 3.146 4.259 0.75 2.419 0.523 4.212 2.411 4.186 1.82462.5 5.328 2.12 2.695 3.12 4.305 0.769 2.421 0.521 4.214 2.399 4.186 1.831
63 5.354 2.156 2.69 3.091 4.349 0.783 2.425 0.517 4.208 2.404 4.192 1.83
63.5 5.337 2.16 2.69 3.069 4.397 0.799 2.428 0.51 4.21 2.41 4.192 1.829
64 5.356 2.173 2.687 3.053 4.441 0.817 2.432 0.509 4.208 2.411 4.203 1.84
64.5 5.354 2.19 2.676 3.027 4.498 0.821 2.436 0.513 4.207 2.411 4.204 1.846
65 5.364 2.213 2.665 3.009 4.542 0.821 2.441 0.517 4.206 2.41 4.207 1.854
65.5 5.354 2.23 2.662 2.999 4.592 0.841 2.448 0.516 4.204 2.41 4.207 1.846
66 5.367 2.26 2.662 2.97 4.64 0.85 2.451 0.504 4.201 2.413 4.207 1.851
66.5 5.366 2.256 2.661 2.956 4.699 0.873 2.451 0.511 4.2 2.409 4.207 1.856
67 5.371 2.276 2.652 2.937 4.75 0.884 2.455 0.51 4.2 2.407 4.211 1.854
67.5 5.372 2.293 2.643 2.91 4.79 0.904 2.461 0.523 4.199 2.409 4.214 1.856
68 5.383 2.323 2.641 2.89 4.847 0.926 2.462 0.509 4.197 2.404 4.214 1.863
68.5 5.372 2.34 2.641 2.871 4.886 0.943 2.472 0.516 4.167 2.407 4.214 1.86
69 5.386 2.337 2.641 2.857 4.927 0.969 2.473 0.51 4.123 2.41 4.218 1.874
69.5 5.378 2.369 2.637 2.834 4.977 0.989 2.476 0.523 4.081 2.407 4.225 1.863
70 5.39 2.379 2.634 2.809 5.019 0.994 2.476 0.523 4.048 2.406 4.222 1.876
70.5 5.381 2.406 2.63 2.79 5.061 1.014 2.481 0.523 4.012 2.399 4.228 1.874
8/13/2019 Water Tank System Response
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19
71 5.39 2.414 2.626 2.769 5.099 1.031 2.483 0.523 3.974 2.4 4.228 1.881
71.5 5.381 2.431 2.625 2.743 5.145 1.056 2.487 0.514 3.95 2.399 4.23 1.883
72 5.406 2.441 2.628 2.726 5.188 1.079 2.49 0.524 3.91 2.393 4.234 1.893
72.5 5.388 2.473 2.619 2.703 5.229 1.099 2.494 0.524 3.877 2.389 4.234 1.89
73 5.397 2.47 2.618 2.689 5.263 1.121 2.497 0.514 3.844 2.39 4.241 1.887
73.5 5.382 2.49 2.61 2.673 5.291 1.144 2.497 0.523 3.817 2.377 4.24 1.889
74 5.399 2.52 2.614 2.651 5.324 1.173 2.503 0.516 3.786 2.38 4.24 1.887
74.5 5.408 2.543 2.611 2.623 5.357 1.196 2.503 0.52 3.756 2.37 4.241 1.893
75 5.397 2.55 2.603 2.61 5.39 1.227 2.509 0.509 3.726 2.364 4.241 1.899
75.5 5.414 2.566 2.599 2.586 5.422 1.256 2.51 0.504 3.697 2.354 4.246 1.906
76 5.399 2.576 2.599 2.564 5.45 1.276 2.51 0.517 3.665 2.351 4.243 1.909
76.5 5.426 2.591 2.597 2.546 5.477 1.304 2.514 0.507 3.637 2.346 4.248 1.913
77 5.41 2.609 2.593 2.521 5.508 1.337 2.517 0.506 3.614 2.344 4.248 1.921
77.5 5.418 2.626 2.586 2.496 5.531 1.369 2.517 0.511 3.588 2.339 4.248 1.923
78 5.415 2.646 2.582 2.49 5.561 1.394 2.517 0.513 3.563 2.326 4.243 1.916
78.5 5.423 2.657 2.586 2.467 5.575 1.417 2.521 0.494 3.546 2.319 4.24 1.9279 5.428 2.676 2.586 2.45 5.606 1.441 2.524 0.507 3.524 2.309 4.248 1.92
79.5 5.425 2.677 2.579 2.431 5.632 1.477 2.527 0.516 3.499 2.304 4.255 1.92
80 5.426 2.699 2.579 2.414 5.651 1.51 2.531 0.507 3.473 2.294 4.255 1.926
80.5 5.415 2.713 2.578 2.393 5.676 1.543 2.531 0.5 3.535 2.286 4.244 1.931
81 5.432 2.72 2.572 2.373 5.694 1.576 2.531 0.509 4.211 2.277 4.212 1.939
81.5 5.422 2.737 2.572 2.354 5.719 1.607 2.535 0.503 5.043 2.28 4.179 1.937
82 5.43 2.739 2.572 2.324 5.735 1.636 2.538 0.504 5.583 2.286 4.145 1.934
82.5 5.425 2.767 2.566 2.309 5.753 1.674 2.538 0.501 5.55 2.3 4.103 1.939
83 5.433 2.779 2.566 2.29 5.766 1.7 2.545 0.507 5.498 2.31 4.066 1.937
83.5 5.43 2.797 2.568 2.274 5.79 1.739 2.542 0.509 5.433 2.34 4.034 1.93984 5.447 2.806 2.566 2.256 5.804 1.773 2.543 0.501 5.36 2.354 3.994 1.939
84.5 5.44 2.823 2.56 2.233 5.821 1.811 2.546 0.497 5.292 2.376 3.957 1.936
85 5.44 2.834 2.559 2.219 5.833 1.846 2.552 0.506 5.215 2.393 3.921 1.937
85.5 5.437 2.853 2.559 2.196 5.848 1.883 2.552 0.503 5.146 2.414 3.887 1.939
86 5.437 2.864 2.559 2.173 5.869 1.913 2.553 0.513 5.088 2.42 3.852 1.94
86.5 5.447 2.889 2.554 2.154 2.559 0.513 5.017 2.429 3.822 1.933
87 5.436 2.906 2.556 2.134 2.559 0.516 4.949 2.446 3.792 1.94
87.5 5.439 2.921 2.556 2.109 2.559 0.516 4.894 2.46 3.763 1.933
88 5.421 2.939 2.554 2.1 2.561 0.507 4.833 2.473 3.731 1.926
88.5 5.433 2.947 2.549 2.087 2.561 0.497 4.772 2.476 3.702 1.923
89 5.43 2.959 2.545 2.066 2.564 0.509 4.717 2.484 3.673 1.914
89.5 5.432 2.969 2.542 2.04 2.566 0.5 4.665 2.494 3.648 1.907
90 5.415 2.987 2.545 2.024 2.568 0.521 4.608 2.497 3.625 1.906
90.5 5.433 3 2.545 2.009 2.566 0.506 4.553 2.504 3.594 1.9
91 5.434 3.017 2.542 1.991 2.572 0.501 4.508 2.509 3.571 1.89
91.5 5.417 3.027 2.532 1.973 2.572 0.511 4.459 2.513 3.549 1.88
92 5.443 3.044 2.531 1.941 2.578 0.506 4.411 2.519 3.523 1.879
8/13/2019 Water Tank System Response
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92.5 5.417 3.074 2.534 1.927 2.579 0.504 4.366 2.516 3.498 1.877
93 5.433 3.091 2.532 1.914 2.586 0.509 4.319 2.52 3.476 1.867
93.5 5.421 3.096 2.532 1.897 2.592 0.499 4.277 2.529 3.455 1.857
94 5.426 3.109 2.535 1.877 2.612 0.499 4.228 2.529 3.432 1.841
94.5 5.421 3.139 2.531 1.854 2.632 0.499 4.189 2.536 3.412 1.831
95 5.426 3.156 2.526 1.843 2.658 0.491 4.145 2.527 3.39 1.824
95.5 5.43 3.156 2.528 1.826 2.681 0.497 4.103 2.533 3.367 1.821
96 5.426 3.187 2.527 1.807 2.708 0.496 4.066 2.534 3.35 1.809
96.5 5.423 3.2 2.524 1.791 2.734 0.494 4.028 2.529 3.332 1.8
97 5.417 3.217 2.524 1.773 2.761 0.499 3.99 2.523 3.312 1.796
97.5 5.425 3.224 2.524 1.757 2.797 0.503 3.952 2.526 3.292 1.787
98 5.419 3.24 2.523 1.734 2.826 0.509 3.921 2.519 3.274 1.781
98.5 5.422 3.249 2.524 1.716 2.862 0.497 3.89 2.507 3.261 1.767
99 5.417 3.273 2.517 1.699 2.903 0.506 3.858 2.513 3.24 1.761
99.5 5.43 3.276 2.52 1.679 2.935 0.509 3.823 2.507 3.222 1.747
100 5.429 3.294 2.517 1.664 2.967 0.503 3.793 2.501 3.208 1.733100.5 5.408 3.309 2.517 1.649 3.006 0.521 3.764 2.499 3.189 1.727
101 5.426 3.331 2.517 1.639 3.054 0.52 3.732 2.491 3.17 1.714
101.5 5.399 3.351 2.517 1.616 3.108 0.529 3.708 2.48 3.159 1.703
102 5.418 3.357 2.517 1.596 3.154 0.524 3.68 2.481 3.143 1.694
102.5 5.397 3.381 2.51 1.581 3.203 0.526 3.657 2.474 3.132 1.683
103 5.415 3.394 2.51 1.563 3.252 0.529 3.629 2.464 3.117 1.674
103.5 5.41 3.414 2.51 1.549 3.31 0.521 3.604 2.463 3.101 1.656
104 5.417 3.427 2.509 1.536 3.361 0.524 3.582 2.459 3.087 1.646
104.5 5.417 3.444 2.51 1.519 3.422 0.527 3.553 2.443 3.076 1.637
105 5.41 3.456 2.513 1.5 3.483 0.539 3.53 2.441 3.057 1.626105.5 5.411 3.47 2.51 1.484 3.546 0.521 3.508 2.43 3.046 1.611
106 5.418 3.48 2.51 1.466 3.606 0.537 3.484 2.424 3.029 1.597
106.5 5.406 3.493 2.51 1.451 3.67 0.536 3.622 2.417 3.015 1.591
107 5.418 3.516 2.51 1.434 3.732 0.549 4.219 2.411 3.003 1.577
107.5 5.423 3.527 2.51 1.417 3.797 0.547 5.105 2.4 2.99 1.569
108 5.432 3.543 2.512 1.406 3.865 0.561 5.575 2.413 2.977 1.556
108.5 5.407 3.551 2.513 1.384 3.937 0.56 5.545 2.431 2.961 1.54
109 5.422 3.576 2.51 1.366 4.015 0.567 5.492 2.447 2.945 1.533
109.5 5.399 3.581 2.51 1.351 4.079 0.577 5.426 2.473 2.93 1.516
110 5.419 3.596 2.509 1.34 4.148 0.589 5.349 2.477 2.921 1.504
110.5 5.4 3.604 2.508 1.323 5.283 2.501 2.914 1.49
111 5.412 3.629 2.505 1.311 5.211 2.517 2.903 1.473
111.5 5.406 3.636 2.503 1.299 5.141 2.53 2.89 1.466
112 5.421 3.657 2.503 1.284 5.083 2.547 2.874 1.459
112.5 5.412 3.664 2.503 1.267 5.01 2.55 2.863 1.437
113 5.421 3.677 2.499 1.251 4.939 2.571 2.854 1.42