Frequency Response for Flow System

Size: px

Start display at page:

Download "Frequency Response for Flow System"

Annis Jackson
6 years ago
Views:

1 Frequency Response for Flow System Report By: Ben Gordon Red Squad: Ben Klinger, Dianah Dugan UTC, Engineering 329 October 7, 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. Benjamin Gordon 2

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

3 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 Benjamin Gordon 3

4 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 c(t) Average Output (lb/min m(t) Power Input (% ) Figure 3: Steady-State Operating Curve for Flow System Benjamin Gordon 4

5 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 c(t) Average Output (lb/min y = 0.25x m(t) Power Input (% ) 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. Benjamin Gordon 5

6 (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, t 0 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. G s ( s) = t0s Ke τ s + 1 Figure 6: Equation for Transfer Function of a System Benjamin Gordon 6

7 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 Seconds Lower Middle Upper SSOC Region 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. Benjamin Gordon 7

8 (τ) Positive Step Test 1 Test 2 Test 3 Average Std. Dev. Lower Section (sec) Middle Section (sec) Upper Section (sec) 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) Middle Section (sec) Upper Section (sec) 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. Benjamin Gordon 8

9 Dead Time Values Seconds Lower Middle Upper SSOC Region 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) Middle Section (sec) Upper Section (sec) 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) Middle Section (sec) Upper Section (sec) Figure 13: Calculated Data on the Dead Time for Negative Step Response Benjamin Gordon 9

10 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 lb/ min % Lower Middle Upper SSOC Region 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%) Middle Section (lb/min%) Upper Section (lb/min%) Figure 15: Calculated Data on the Gain for Positive Step Responses Benjamin Gordon 10

11 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%) Middle Section (lb/min%) Upper Section (lb/min%) Figure 16: Calculated Data on the Gain for Positive Step Responses Benjamin Gordon 11

12 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 Input (%) Output (lb/min) Time (sec) Figure17. Example of 0.03 Frequency in the lower region. Benjamin Gordon 12

13 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 Output (lb/min) Input (%) 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 Benjamin Gordon 13

14 Results Bode Plot - Amplitude Ratio Frequency (Hz) Amplitude Ratio, Dc/Dm (units) 0.01 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 Benjamin Gordon 14

15 Bode Plot - Phase Angle Frequency (Hz) Phase Angle (degrees) 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 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 Input (%) Output (lb/min) Time (sec) Figure 21. Example of what it looks like close to ultimate frequency. Benjamin Gordon 15

16 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 Figure 22 Lower Region amplitude ratio and phase angle. Frequency Average AR AR Uncert. Average Phase Angle PA Uncert Figure 23. Middle Region amplitude ratio and phase angle. Frequency Average AR AR Uncert. Average Phase Angle PA Uncert Figure 24. Upper Region amplitude ratio and phase angle. Benjamin Gordon 16

18 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/%) order Fu (cycles/sec) 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/%) To (sec) Tau (sec) 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 Frequency (Hz) Amplitude Ratio (lb/min*%) Benjamin Gordon 17

19 Figure 28. Lower Region of Amplitude Ratio Modeling (Pink) and Experimental (Blue). Bode Phase Angle Frequency (Hz) Phase Angle (degrees) Figure 29. Lower Region of Phase Angle Modeling (Red) and Experimental (Green). Bode Amplitude Ratio Frequency (Hz) Amplitude Ratio (lb/min*%) Figure 30. Middle Region of Amplitude Ratio Modeling (Pink) and Experimental (Blue). Benjamin Gordon 18

20 Bode Phase Angle Frequency (Hz) Phase Angle (degrees) Figure 31. Middle Region of Phase Angle Modeling (Red) and Experimental (Green). Bode Amplitude Ratio Frequency (Hz) Amplitude Ratio (lb/min*%) Figure 32. Upper Region of Amplitude Ratio Modeling (Pink) and Experimental (Blue). Benjamin Gordon 19

21 Bode Phase Angle Frequency (Hz) Phase Angle (degrees) Figure 33. Upper Region of Phase Angle Modeling (Red) and Experimental (Green). Benjamin Gordon 20

22 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 Benjamin Gordon 21

23 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. Benjamin Gordon 22

24 Appendices Explanation for how to get experimental values Benjamin Gordon 23

25 Bode Plot - Amplitude Ratio Frequency (Hz) Amplitude Ration, Dc/Dm (units) 0.01 Bode Amplitude Ratio for middle region Bode Plot - Amplitude Ratio Frequency (Hz) Amplitude Ration, Dc/Dm (units) 0.01 Bode Phase Angle for middle region Benjamin Gordon 24

26 Bode Plot - Amplitude Ratio Frequency (Hz) Amplitude Ration, Dc/Dm (units) 0.01 Bode Amplitude Ratio for upper region. Bode Plot - Phase Angle Frequency (Hz) Phase Angle (degrees) Bode Phase Angle for upper region. Benjamin Gordon 25

27 References Figure 1 and Figure 2 were obtained by the following website, as well as information regarding the background and theory: Figure 5 and Figure 6 were obtained by the following website: The internet site where the tests were performed: Benjamin Gordon 26

Steady-State and Step Response for the Flow System

Steady-State and Step Response for the Flow System Report By: Dianah Dugan Red Squad: Ben Klinger, Ben Gordon UTC, Engineering 329 September 19, 2007 Introduction: The objectives of this experiment are