Water Tank System Response

8/13/2019 Water Tank System Response

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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


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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


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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


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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

Water Tank System Response

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