Steady-State and Step Response for the Flow System

Transcription

1 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

2 Introduction: The objectives of this experiment are to observe the time response of the output function with a given input step function and to observe steady-state gain, dead time, and response time for the flow system. These objectives will allow for better determination of the first-order plus dead time parameters. To achieve the objectives described, eleven tests were performed based on various inputs found on the flow s steady-state operating curve. The operating region, found by the steady-state operating curve, was separated into a lower, middle, and upper section. Each test was then analyzed to better determine the steady-state gain, dead time, and time constant of the flow system. Each test performed was graphed through excel and then analyzed by hand. 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 steady-state and step response.

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 the filter presses is required to operate between seven and ten 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

4 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 25 c(t) Average Output (lb/min) m(t) Power Input (%) Figure 3: Steady-State Operating Curve for Flow System

5 Figure 4, below, shows the steady-state operating curve for the given operating region. The slope of the curve is 0.25, 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 = x 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.

6 (a) Step Input (b) Step Response (Output) Figure 5. Step response input and output functions The transfer function 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 crossreferencing 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 crossreferenced 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

7 Procedure: Eleven tests were performed on the filter wash flow system based on the assigned regions of power. Ben Gordon performed three tests on region one, from 40 to 60 percent power; Ben Klinger performed five tests on region two, from 60 to 80 percent power, and Dianah Dugan performed three tests on region three, from 80 to 100 percent power. The time at which the step occurs was based on when the function reached steadystate at the baseline. The experiment length was determined appropriately by the time at which the steady-state occurred after the step input function was induced. The valves all remained open for the duration of each test. Once all of the tests were complete, Excel was used to graph the data in terms of the input of the step function, the output of the flow and the duration of the test, as shown below in Figure 7. Step Response for Region 3 Input (%) Output (lb/min) Time (sec) Figure 7: Example of Step Response Performed in Region 3 Once each graph was created, the region after the step input was focused to manually determine the gain, dead time, and time constant. These first order plus dead time parameters were then averaged together for each section with a 95 percent

8 confidence level. To gain a better understanding of step response, a step down test was then performed. This was done by using a negative as apposed to a positive number for the input step.

9 Results: Figure 8 shows the time constant with a 95 percent confidence level, as described by each section in the operating region. The upper region produced the greatest time constant, at a value of 1.71, with the smallest standard deviation, a value of zero; whereas the lower region produced the smallest time constant, at a value of 1.20, with the largest standard deviation, a value of The entire operating region averages a time constant value of 1.4. Tau Values Seconds Lower Middle Upper SSOC Region Figure 8: Time Constant Values for Step Response with 95% Confidence The dead time values, as shown in Figure 9, were consistent between 60 and 100 percent power input averaging approximately 0.5; however, the lower region, between 40 and 60 percent power produced an average dead time of 2.3 and a greater standard deviation. The entire operating region averages a value of 1.1.

10 Dead Time Values Seconds Lower Middle Upper SSOC Region Figure 9: Dead Time Values for Step Response with 95% Confidence The average gain for each section of the operating region is shown below in Figure 10. From 60 to 100 percent power input, the gain maintained an average of 0.24, which is consistent with the slope of the steady-state operating curve. In the lower section from 40 to 60 percent power input, the gain averaged a value of 0.48 and had a standard deviation of 0.15.

11 K Values lb/ min % Lower Middle Upper SSOC Region Figure 10: Gain Values for Step Response with 95% Confidence **Talk about step down**

12 Discussion: The results show that for the upper section of the operating region, which is between 80 and 100 percent power input, there is a 95 percent confidence that the time constant will be between zero and 3.4 seconds, the dead time will be between zero and 0.86 seconds, and the gain will be between 0.03 and The middle section of the operating region, which is between 60 and 80 percent power input are 95 percent confident of falling between 0.01 and 2.65 seconds for the time constant, 0.14 and 1.3 seconds for the dead time, and and 0.52 for the gain. The results for the lower section of the operating region, which is between 40 and 60 percent power input, there is a 95 percent confidence that the time constant will be between 0.9 and 1.2 seconds, the dead time will be between 0.4 and 4.9 seconds, and the gain will be between 0.33 and For the range of 60 to 100 percent power, the gain of the first order plus dead time parameter agrees with the slope of the steady-state operating curve of approximately However, for the lower section, the gain was about twice as much. * relate k for down step

13 Conclusions: With a given input step, there was a time response of the output function. Steadystate gain, dead time, and response time were able to be determined for all regions in the designated operating range, as shown in Figure 11. By determining all the values for the first order plus dead time parameters, a transfer function, as described in the background, of the filter wash flow system can successively be solved for. SSOC Region K Uncertainty Dead Time Uncertainty Tau Uncertatinty Lower Middle Upper Figure 11: Tabulated Values for First Order plus Dead Time Parameters

14 Appendices: The following three graphs were obtained and analyzed in the upper region by Dianah Dugan. (1) Step Response Graph Input (%) Output (lb/min) Time (sec) K = 0.27 To = 0.20 Tau = 1.71

15 (2) Step Response for Experiment # Input (%) Output (lb/min) Time (sec) K = 0.25 To = 0.5 Tau = 1.71 (3) Step Response for Experiment # Input (%) Output (lb/min) Time (sec) K = 0.24

16 To = 0.5 Tau = 1.71 The following five graphs were obtained and analyzed in the middle region by Ben Klinger. (4) Step Response Graph Power Input (%) Flow Output (lb/min) Time (s) K =.215 To =.75 Tau = 1.5

17 (5) Step Response Graph Input (%) Output (lb/min) Time (sec) K =.25 To =.75 Tau = 1.5

18 (6) Step Response Graph Input (%) Output (lb/min) Time (sec) K =.25 To =.5 Tau = 1.2

19 (7) Step Response Graph Input (%) Output (lb/min) Time (sec) K =.26 To =.4 Tau = 1.2

20 (8) Step Response Graph Input (%) Output (lb/min) Time (sec) K =.26 To = 0.5 Tau = 1.2

21 The following graphs were obtained and analyzed for the lower region by Ben Gordon. (9) Step Response Graph Input (%) Time (sec) Output (lb/min) K = 2.5 To = 0.39 Tau = 1 (10) Step Response Graph Input (%) Time (sec) Output (lb/min) K= 2.20 To = 0.54 Tau = 1.3

22 (11) Step Response Graph Input (%) Time (sec) Output (lb/min) K = 2.1 To = 0.5 Tau = 1.3

23 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: