Frequency Response for Flow System - web2.utc.eduweb2.utc.edu/...Frequency-Response-for-Flow-System... · maintained by a variable speed centrifugal pump. Figure 1, below, shows the

Frequency Response for Flow System

Report By: Ben Gordon

Red Squad: Ben Klinger, Dianah Dugan

UTC, Engineering 329

October 7, 2007

Benjamin Gordon October 8, 2007 2

Introduction

The objective of this experiment is to observe the output function of the flow

system to a sine function input at several different frequencies. From this output function

we can observe the system’s amplitude ratio and the system’s phase lag. To do this ten

frequencies were chosen and where tested at each operating range; lower, 40-60%,

middle 60-80%, and upper 80-95%. From the results of the experiment the amplitude

ratio, Tau, and the Dead time can be calculated. Tau and the Dead Time can be used to

calculate the phase angle. The uncertainty was also calculated for the amplitude ratio and

the phase angle.

This report explains the background and theory of the filter wash flow system, as

well as the steady-state operating curve. The experiment is theorized on how the behavior

of the tests should respond. A detailed explanation of the processes is explained, and then

results from the procedure are graphed to show the relationship between each of the

sections in the operating region. A discussion summarizes the results observed, and then

conclusions were made about the experiment as a whole, in terms of how the filter wash

flow system performs under conditions of frequency inputs.


Background and Theory

The filter wash flow system is used at Publicly Owned Treatment Works to filter

out the sewerage sludge solids, in order to send the sewerage to the landfill and the

filtrate water back into the Tennessee River. The filter presses, which operate in batch

mode, must be washed between each batch. The flow rate of the nozzles in each of the

filter presses is required to operate between 20 and 23 pounds per minute, which is

maintained by a variable speed centrifugal pump. Figure 1, below, shows the diagram for

this control system, nozzles, and pump.

Figure 1. Schematic diagram of the POTW Filters

For this flow system, the input, expressed in terms of a percentage of power over

a course of time, is a function called the manipulated variable, represented by m(t). The

output of the flow, expressed in pounds per minute, is a function called the control


variable, represented by c(t). The operational diagram is represented in Figure 2, shown

below. Note that the filter wash pump system is also recognized as the transfer function,

G(s) for the flow system.

Figure 2. Block diagram of Filter Wash System

A previous experiment required manipulation of various power inputs, which

produced a correlated output. This enabled a steady-state operating curve of the flow

system to be determined. By producing this curve, the normal operating region was

obtained, which allowed each group member to focus on a specific region of the curve.

The operating region for this curve ranged from 40 to 100 percent power input. Figure 3,

below, shows the steady-state operating curve for the given flow system.

Steady-State Operating Curve

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90 100

m(t) Power Input (% )

c(t)

Ave

rage

Out

put (

lb/m

in

Figure 3: Steady-State Operating Curve for Flow System


Figure 4, below, shows the steady-state operating curve for the given operating region.

The slope of the curve is 0.25 pounds per minute per percent, which should be equivalent

to the gain when determining the first order plus dead time parameters. The slope remains

steady throughout the operating region.

Steady-State Operating Curve for Operating Region

y = 0.25x - 1.4

7

9

11

13

15

17

19

21

23

25

40 50 60 70 80 90 100

m(t) Power Input (% )

c(t)

Ave

rage

Out

put (

lb/m

in

Figure 4: Steady-State Operating Curve for the Operating Region

In theory, when a step function input is given a specific value, the output will

have a response. Note that the step input can be given a negative or positive value based

on whether the step needs to step up or down with the power input. The step occurs at the

time and power input specified in the test. At the specific time the step occurs, the

response occurs at Δm, which is the percent power. The output response is expressed in

terms of Δc, which are pounds per minute. Below, Figure 5a shows a positive step input,

and Figure 5b shows the systems response.


(a) Step Input

(b) Step Response (Output)

Figure 5. Step response input and output functions

The transfer function, a Laplace domain expression, enables one to

determine the first order plus dead time parameters for a system. The equation

for the transfer function is shown below in Figure 6. K represents the gain, 0t

represents the dead time, and τ is the time constant. K is determined by dividing

Δc by Δm. The dead time is found by using a tangential line at the steepest slope

on the response curve, and cross-referencing it with a line of minimum output.

Subtract this cross-referenced line from the start of the step to achieve dead

time. To determine the time constant, a maximum output is drawn, while using

the same tangential line. Then, that cross-referenced point is subtracted by the

minimum cross-referenced point.

1)(

0

+=

−

sKesG

st

s τ

Figure 6: Equation for Transfer Function of a System


Figure 8 shows the time constant with a 95 percent confidence level, as described by each

section in the operating region. The areas in red represent a positive step, and the areas in

blue represent a negative step.

Tau Values

0

0.5

1

1.5

2

2.5

3

Lower Middle Upper

SSOC Region

Seco

nds

Figure 8: Time Constant Values for Step Response with 95% Confidence

As shown in Figure 9, the upper region of the positive step response produced the

greatest average time constant, at a value of 1.7 seconds, with the smallest standard

deviation, a value of zero; whereas the lower region produced the smallest average time

constant, at a value of 1.2 seconds, with the largest standard deviation, a value of 0.17

seconds. The entire operating region for the positive step response averages a time

constant value of 1.4 seconds.


(τ) Positive Step Test 1 Test 2 Test 3 Average Std. Dev. Lower Section (sec) 1 1.3 1.3 1.2 0.17 Middle Section (sec) 1.5 1.5 1.2 1.4 0.17 Upper Section (sec) 1.7 1.7 1.7 1.7 0

Figure 9: Calculated Data on the Time Constant for a Positive Step Response

As shown in Figure 10, the lower region produced the greatest time constant for

the negative steps, at a value of 2.0 seconds with a standard deviation of 0.65 seconds.

The entire operating region for the negative step response averages a value of 1.5

seconds.

(τ) Negative Step Test 1 Test 2 Test 3 Average Std. Dev. Lower Section (sec) 2 2.6 1.3 2 0.65 Middle Section (sec) 1.5 1.7 1.6 1.6 0.1 Upper Section (sec) 0.7 1 0.9 0.87 0.15 Figure 10: Calculated Data on the Time Constant for a Negative Step Response

Figure 11, below, shows the results of dead time values with a 95 percent

confident range. The bars in red represent a positive step, and the bars in blue represent a

negative step.


Dead Time Values

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Lower Middle Upper

SSOC Region

Seco

nds

Figure 11: Dead Time Values for Step Response with 95% Confidence

The dead time values, as shown in Figure 12, were consistent throughout the

entire operating range, averaging approximately 0.5 seconds for the positive step.

(to) Positive Step Test 1 Test 2 Test 3 Average Std. Dev. Lower Section (sec) 0.39 0.54 0.5 0.49 0.078 Middle Section (sec) 0.75 0.75 0.4 0.63 0.2 Upper Section (sec) 0.2 0.5 0.5 0.4 0.17

Figure 12: Calculated Data on the Dead Time for Positive Step Response

The negative step response was also consistent for dead time values, as shown in

Figure 13. The range for the entire operating region was between 0.47 and 0.80 seconds.

The average was 0.61 seconds, which is consistent with the 60 to 100 percent power

input regions for the positive step response.

(to) Negative Step Test 1 Test 2 Test 3 Average Std. Dev. Lower Section (sec) 0.5 0.15 0.75 0.47 0.3

Middle Section (sec) 0.75 0.85 0.8 0.8 0.05

Upper Section (sec) 0.6 0.5 0.6 0.57 0.058 Figure 13: Calculated Data on the Dead Time for Negative Step Response


Figure 14, below, shows the average gain values for both a positive and negative

step in the operating region. Bars in red represent a positive step response, whereas bars

in blue show a negative step response.

K Values

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lower Middle Upper

SSOC Region

lb/ m

in %

Figure 14: Gain Values for Step Response with 95% Confidence

Throughout the operating range of the positive step response, the gain averaged a

value of 0.25 pounds per minute percent, as shown in Figure 15. This is expected because

it should be close to the value of the slope of the SSOC which is also 0.25 pounds per

minute percent.

K Positive Step Test 1 Test 2 Test 3 Average Std. Dev.

Lower Section (lb/min%) 0.25 0.22 0.21 0.23 0.021

Middle Section (lb/min%) 0.22 0.25 0.26 0.24 0.021

Upper Section (lb/min%) 0.27 0.25 0.24 0.25 0.015 Figure 15: Calculated Data on the Gain for Positive Step Responses


The negative step response held consistent throughout the operating range of 40 to

100 percent power input, as shown in Figure 16. The average of all sections was 0.25

pounds per minute percent, which is consistent with the gain in the positive step response.

It also follows with the SSOC slope of 0.25 pounds per minute per percent.

K Negative Step (lb/min%) Test 1 Test 2 Test 3 Average Std. Dev.

Lower Section (lb/min%) 0.25 0.25 0.24 0.25 0.0058

Middle Section (lb/min%) 0.25 0.24 0.25 0.25 0.0058

Upper Section (lb/min%) 0.23 0.25 0.26 0.25 0.015 Figure 16: Calculated Data on the Gain for Positive Step Responses


Procedure

This experiment could have been performed at the first floor lab of the

engineering department. But the lab was done by accessing the computer that runs the

experiment through the internet. So no one was actually present at the time the

experiment took place. This website where the experiment is done is mentioned in the

appendix. The website allows you to adjust the baseline input frequency, frequency, the

amplitude of the sine wave and the time. Filters Two and Three were left open for the

entire experiment. The experiment was run for ten different frequencies at the lower,

middle, and upper regions of the steady state operating curve. Once all of the tests were

performed excel was used to show the relationship between the Input at that frequency

and the output in pounds per minute for a length of time, as shown in figure 17.

Frequencey f=0.03 Sine Response

35

40

45

50

55

60

65

20 40 60 80 100 120

Time (sec)

Inpu

t (%

)

5

7

9

11

13

15

17

19

Out

put (

lb/m

in)

Figure17. Example of 0.03 Frequency in the lower region.


A Lissajous Graph was also created (Figure18) to show the relationship between only the

input and output. The quicker the output responded to the input the straighter the line.

The more the output lags the input the graph starts to take on a circular shape. In this

graph it looks like a straight line because the output doesn’t lag far behind the input.

Lissajous f=0.03

7

89

101112

1314

1516

40 45 50 55 60

Input (%)

Out

put (

lb/m

in)

Figure18. Example of 0.03 Frequency Lissajous Graph.

By using Figure 17, The amplitude ratio was calculated by the difference between

two upper peak output heights in pounds per minute divided by twice the amplitude. Tau

was calculated by the time difference between upper peaks in the input. The dead time

was calculated by the time difference between upper peak of the output minus the input

upper peak behind it. After calculating three of each the Amplitude Ratio, Tau, and Dead

Time. The phase angle was calculated by Dead time divided by Tau times three hundred

and sixty. The average and uncertainties were plotted on two graphs, Bode Amplitude

Ratio and Phase Angle Graphs were made in excel, Figure 19 and Figure 20


Results

Bode Plot - Amplitude Ratio

0.01

0.10

1.000.01 0.10 1.00

Frequency (Hz)

Am

plitu

de R

atio

, Dc/

Dm

(uni

ts)

Figure 19. Example of Bode Plot for Lower Region.

This graph shows that at ultimate frequency of 0.5 Hz the amplitude ratio is

0.1.This 0.1 is called the ultimate controller gain which will be helpful to use as a starting

place when trying to find the ultimate controller gain in the proportional only controller

experiment. It also tells you that the gain for the system is about 0.25.


Bode Plot - Phase Angle

0.00

50.00

100.00

150.00

200.00

250.00

0.01 0.10 1.00

Frequency (Hz)

Phas

e A

ngle

(deg

rees

)

Figure20. Example of Phase Angle for Lower Region.

This bode phase angle graph is useful because you need to know what the

frequency is when the phase angle is -180. The frequency at that phase angle is the

ultimate frequency that was mentioned earlier. At this frequency when the output was at

the lowest point the input would be at the its highest and vice versa. on the

Frequency 0.6 Sine Response

35

40

45

50

55

60

65

23 24 25 26 27 28 29 30

Time (sec)

Inpu

t (%

)

9

9.5

10

10.5

11

11.5

12

12.5

13O

utpu

t (lb

/min

)

Figure 21. Example of what it looks like close to ultimate frequency.


The average amplitude Ratio and Phase angle as well as their uncertainties are

listed below for the experimental values. The amplitude ratio is amplitude ratio vs

frequency and the phase angle is phase angle vs frequency. There are graphs

corresponding to this later in the result section.

Frequency Average AR AR Uncert. Average Phase Angle PA Uncert. 0.02 0.25 0.01 7 3 0.03 0.26 0.03 15 6 0.04 0.23 0.02 17 2 0.08 0.26 0.01 39 9 0.10 0.24 0.02 48 10 0.20 0.19 0.01 72 25 0.30 0.15 0.01 109 52 0.40 0.12 0.01 132 21 0.50 0.10 0.01 199 18 0.60 0.07 0.01 185 27

Figure 22 Lower Region amplitude ratio and phase angle.

Frequency Average AR AR Uncert. Average Phase Angle PA Uncert. 0.02 0.34 0.00 8 2 0.03 0.34 0.01 16 6 0.04 0.31 0.01 6 0 0.08 0.34 0.02 45 15 0.1 0.31 0.01 66 11 0.2 0.28 0.02 35 2 0.3 0.23 0.01 102 19 0.4 0.19 0.01 118 10 0.5 0.15 0.01 178 23 0.6 0.14 0.01 190 22

Figure 23. Middle Region amplitude ratio and phase angle.

Frequency Average AR AR Uncert. Average Phase Angle PA Uncert. 0.02 0.29 0.01 9 0 0.04 0.28 0.00 13 2 0.06 0.28 0.01 20 1 0.08 0.28 0.01 21 2 0.1 0.27 0.00 45 9 0.2 0.25 0.01 72 8 0.3 0.20 0.02 97 9 0.6 0.12 0.02 169 30 0.8 0.11 0.01 209 38

1 0.07 0.01 228 42 Figure 24. Upper Region amplitude ratio and phase angle.


In the above Figure 23 has higher over all values for there amplitude ratio. But the

phase angle seems to be more consistent with the rest of the regions. For the experimental

part the ultimate frequency, gain, order and Kcu can be found. This explanation for how

to obtain these calculations is in the appendix.

lower middle upper k (lb/min/%) 0.25 0.34 0.29 order 0.6 1.1 .7 Fu (cycles/sec) 0.5 0.5 0.6 Kcu (lb/min/%) 1/0.11 1/0.15 1/0.11

Figure 25. the results for each region are given.

Table 26. Shows the results of the modeling for each of the three operating inputs. These

K, To (Dead time), and Tau can be manipulated to create the modeled line.

Lower Middle Upper K (lb/min/%) 0.25 0.35 0.29 To (sec) 0.6 0.5 0.5 Tau (sec) 0.7 0.6 0.6

Figure 27. Variables to make the modeling fit the experimental results

Using the experimental values of the frequency, amplitude ratio, and phase angle

a model result can be made by altering Gain (k), Tau and Dead time, example from the

different regions below.

Bode Amplitude Ratio

0.01

0.1

10.001 0.01 0.1 1

Frequency (Hz)

Am

plitu

de R

atio

(lb

/min

*%)


Figure 28. Lower Region of Amplitude Ratio Modeling (Pink) and Experimental (Blue).

Bode Phase Angle

050100150200250300350400

0.001 0.01 0.1 1

Frequency (Hz)

Phas

e A

ngle

(d

egre

es)

Figure 29. Lower Region of Phase Angle Modeling (Red) and Experimental (Green).


0.01

0.1

10.001 0.01 0.1 1

Frequency (Hz)A

mpl

itude

Rat

io

(lb/m

in*%

)

Figure 30. Middle Region of Amplitude Ratio Modeling (Pink) and Experimental (Blue).


Bode Phase Angle

050100150200250300350400

0.001 0.01 0.1 1

Frequency (Hz)

Phas

e A

ngle

(d

egre

es)

Figure 31. Middle Region of Phase Angle Modeling (Red) and Experimental (Green).


0.01

0.1

10.001 0.01 0.1 1

Frequency (Hz)A

mpl

itude

Rat

io

(lb/m

in*%

)

Figure 32. Upper Region of Amplitude Ratio Modeling (Pink) and Experimental (Blue).


Bode Phase Angle

050100150200250300350400

0.001 0.01 0.1 1

Frequency (Hz)

Phas

e A

ngle

(d

egre

es)

Figure 33. Upper Region of Phase Angle Modeling (Red) and Experimental (Green).


Discussion

The graphs of the Bode amplitude ratio tell us what the output per input is at

different frequencies of a sine response input. Where this line flattens out is the Gain (K)

which is where every frequency below this has the same value, K. For all regions the

output per input goes down around the frequency of 0.1. The K values for each region

were close between model results and the experimental K values. All of the phase angles

also drop sharply around the 0.1 frequency. If you were to do a Frequency Sine Response

Output, input vs. time at the ultimate frequency then the top peaks of the output would

match up with the bottom peaks of the input. The ultimate controller gain (Kcu) can be

found at the ultimate frequency on the amplitude ratio graph. The ultimate controller gain

is the 1/(output per input) at the ultimate frequency. All of these things plus the order of

the system will be helpful in building a proper controller for the system. The lower region

k value for step up and step down matches the k value in this experiment at 0.25

lb/min%. The time constant in the step up and step down was about twice of what the

value is for the frequency response. The dead time however is similar at around 0.6

seconds


Conclusion and Recommendations

Since all these experiments are designed to find the system gain, the dead time

and the time constant, we can compare them to the step up and step down and see if the

results are similar. Since these two experiments gave similar values we would expect that

the ultimate frequency, the order and the ultimate controller gain to be accurate for the

system as well. Having these values will assist in the development of a proportional only

controller experiment to be done for the flow system. The ultimate controller gain is very

important, it is the highest value that the system can run at, around this value the

experiment will never settle down and look like these frequency response graphs.


Appendices

Explanation for how to get experimental values



0.01

0.10

1.000.01 0.10 1.00

Frequency (Hz)

Am

plitu

de R

atio

n, D

c/D

m (u

nits

)

Bode Amplitude Ratio for middle region


0.01

0.10

1.000.01 0.10 1.00

Frequency (Hz)

Am

plitu

de R

atio

n, D

c/D

m (u

nits

)

Bode Phase Angle for middle region



0.01

0.10

1.000.01 0.10 1.00

Frequency (Hz)

Am

plitu

de R

atio

n, D

c/D

m (u

nits

)

Bode Amplitude Ratio for upper region.

Bode Plot - Phase Angle

0.00

50.00

100.00

150.00

200.00

250.00

300.00

0.01 0.10 1.00

Frequency (Hz)

Phas

e A

ngle

(deg

rees

)

Bode Phase Angle for upper region.


References

Figure 1 and Figure 2 were obtained by the following website, as well as information

regarding the background and theory:

http://chem.engr.utc.edu/green-engineering/Filter-Wash/Filter-Wash-System-Detail.htm

Figure 5 and Figure 6 were obtained by the following website:

http://chem.engr.utc.edu/green-engineering/Filter-Wash/Filter-Wash-System-Step.htm

The internet site where the tests were performed:

http://chem.engr.utc.edu/green-engineering/Filter-Wash/.htm

Documents

Frequency Response for Flow System - web2.utc.eduweb2.utc.edu/...Frequency-Response-for-Flow-System... · maintained by a variable speed centrifugal pump. Figure 1, below, shows the