Aerator Mixer Speed Control System Step Response Modeling

Transcription

1 UTC Engineering 3280L Matthew Addison Green Team (Michael Hansen) 9/4/12 Aerator Mixer Speed Control System Step Response Modeling

2 Introduction In this experiment a program that models the aerator mixing system for treatment waste in Chattanooga was used to produce data used to show the relationship for the input function and the output function of the system. The ultimate goal of this project was to create a model for the Aerator system s output for the ranges of 100 to 600 and 1100 to 1600 RPMs, given a certain percentage of the motors total power. This report will describe all relevant background information, the procedure taken and the results of this experiment. The results will be followed by a discussion of the results and recommendations. Background Apparatus The waste-water treatment plant uses aerobic microbes to digest waste in water. It uses an Aerator Mixer to provide the circulation needed for aerobic life. The basic layout consists of a controls system that receives feedback from the motor and provides output to it. This allows the motor to maintain a certain operating speed (Henry). The mixer speed operates on a range of 2-17 RPM. The gearbox between the motor and the aerator is a 1 to 100 ratio, so the motor operates on a range of RPM (Henry). The output signal from the control system to the motor varies from 0 to 100% of the rated motor power of the motor. The Schematic of the system can be seen below. The SRC/247, SCZ/247, and ST/247 bubbles are the control system and the cylinder labeled M-247 is the motor.

3 Figure 1: Aerator System (Henry). This diagram describes a feedback control loop. In this loop the transducer ST 247 reads the RPM produced by the motor and transfers this into an electrical signal that is processed by the analog controller, SRC 247. Once the controller has become aware that a deviation from set point has occurred, the controller sends a signal, corresponding with the necessary changes in input to meet the target output of the system, to the final control element SCZ-247. This element then manipulates the power input to the motor. The aerator mixer is generalized to a much simpler system as seen below. Where M(t) is the manipulated variable and the input to our system. It is the percentage of power given to the motor. The controlled variable, C(t), is the output of our system, or the RPM of the mixer.

4 Figure 2: Input-output relation (Henry) Data Acquisition This experiment used LabView for data acquisition, and excel for analysis and presentation. The information path is shown in the schematic below. Figure 3: Information paths for experiments As the schematic describes the user gives inputs the LabView program which then tells the equipment what actions to carry out. In this particular experiment the operator inputs are the percentage of max motor power, duration of the experiment, step size, and when the step takes place. The equipment (the aerator mixer system) provides feedback or data measurements to LabView. LabView then outputs this data in the form of spreadsheets. The

5 data generated by the equipment consists of sampling signals received by the ST-247 transmitter and the signals sent out by the final control element. Previous Experimentation Previous experiments on this system have been done to find the steady state operating curve. This was done by running the system at a constant input, then graphing the output generated as a function of time. An example of this can be seen below. Average output= 300 motor RPMs Standard Deviation=4.45 RPMs Figure 4 : SSOC for 20% input power This graph shows the constant input as the green bar and the motor speed as the gold points. After about 2 seconds the output becomes relatively constant. All the output data after this point is averaged. Once enough steady state averages were obtained they were graphed against their respective input power. A table of these averages and the graph are featured below.

6 Input (%) OutPut (RPM of the mixer) STDev STDev x AVG= 0.07 Figure 5: Average mixer speed at steady state for a constant input The table shows the input on the left as a percentage of total motor power, and the output in following column. A confidence interval of 95% is shown in the last column. Figure 6: steady state operating curve of aerator mixer system

7 The operating curve shown in the graph above indicates a linear relation; almost all the data points lie on the trend line. Theory In this experiment it is assumed that the relation between the input function, m (t), and the output function c (t) is a linear first-order differential equation of the form shown below: Ê ¹¹» ¹» ¹» º» where Ê is the time constant and is the steady state gain. The transfer function of this equation is obtained by taking the Laplace transform, which results in: ¹» º Ê» 1 º» Because the control system for the aerator mixer is a feedback loop there could be a time delay between the input and output. Once there is a change in the input, this change will travel through the entire system before it reaches the controller. This time delay will modify the transfer function to the one shown below º¹ ¹» Ê» 1 º» Here t 0 is the time delay or dead time. (Smith & Armando, 1997).

8 For this experiment the input function is a step function of the form below, where A is the magnitude of the step, and t d is the time at which the step occurs º»»»» The model for the first order plus dead time is derived by taking the inverse Laplace of our transfer function substituting the previous equation in. ¹»»»»» º 1 ¹ The gain, dead time, and the first-order time constant were found by analyzing step response of the system. As seen on the graph below t d is the time when the step response occurs, and t 0 is the time from when the step occurs to when there is an actual response of the system. A is the size of the input step. Dc is the change in of the steady state output after the step. The gain is given by model response to coincide with the actual response:. The first-order time constant,ê, is the constant that forces the ¹» Ê º 1 ¹.632 ¹ This equation is taken from the second edition of Principles and Practice of Automatic Process Control. Its derivation can be found on pages 310 to 313.

9 Figure 7: Step response showing model variables Procedure In order to produce an accurate model, multiple parameters for different steps covering each section of the operating range. These parameters were found for positive step input functions and negative step input functions. For the motor RPM operating range, six total trials were done. Three of these models were of a step up situation. Three more were of a step down situation. Each step was of 10% and was of a power percentage input of 10% to 40%. For the motor RPM operating range six total trials were done. Three were step increases and three were step decreases. The step size was 10% each trial, ranging from 70% to 100%.

10 Using the labview software, we inputted different power percentages to run the motor in order to produce outputs for the different runs within our individual operating ranges. The labview then collected all of our output RPM s from the motor and organized it into data that could be exported to an Excel spreadsheet. From this point it was then organized and plotted. The output Rpm s and input power percentages were plotted against the time to observe the comparison between a FOPDT model to actual recovered data. Results When observing the constructed plots comparing the FOPDT models to actual collected data for our ranges. It was obvious that the collected data of the motor driven aerator was very close to the predicted behavior of the FOPDT model. Below is an example of one of the plots overlayed with the FOPDT model.

11 Output(RPM) Output Input Value(%) 20 Output (RPM) Input Input (%) Time (sec) 0 Figure 8. Represents the plot of actual data with the FOPDT model of this system overlaid. The FOPDT input and output lines a colored green and purple respectively. The output plot of the calculate FOPDT model is very close in relation to the actual output of the motor system (red line). Since the input observed is constantly regulated almost no difference is noticed between it and the model and can be considered negligible. Below is a table of some of the sample data that was used to construct the plot in figure 8. Table 1: The left three columns are experimentally collected data, while the right two columns are calculated data for the FOPDT model overlaid in figure 8. The box to the far right is values found from the fit 2 analysis of the experimental data.

12 Input Time(sec) Value(%) Output(RPM) Input Output td= A= K= to= tau= inbl= outbl= The FOPDT model calculated data from table 1 uses variable from the box on the far right. These values are the time step, power step percentage, gain, dead time, and time constant respectively. The inbl and outbl are the base layer values used to correct position the model to the actual collected data more accurately for better comparison. Discussion In all output ranges management wanted to be analyzed, the accuracy of the FOPDT model to the actual recorded data was very accurate (refer to the example for one range in Figure 8). Table 1 quickly illustrates the values used to construct a FOPDT model for the system s actual data. The formulas for the curves that were used to plot the FOPDT model overlaying the actual data

13 plot was used to represent a perfect operating condition of the motor. In figure 8 it is obvious that the actual data plot is not exactly matching the FOPDT model, there are several possibilities that would cause this behavior. The most probable explanation of this non-matching behavior is that the motor in the aerator assembly is spinning waste water using a fan at a 100:1 ration through a gearbox. The waste water is most likely not consistent throughout, and would cause an non-constant impedance in the motor through fan drag. Additionally, the motor RPMs are reduced through a gear reduction, and depending on the condition of the gearbox and backlash in gears, there is a chance that there is unseen impedance through the gearbox. Also, the momentum of turning such a large mixer at this speed is so large that it would be difficult to slow down as quickly as the FOPDT model requires it too. Conclusion If all impendences can be removed from the aerator systems and it allows completely fluid motion startups and slow downs, only then will it be possible to match the FOPDT model for a unit step system. However, for the measured system output RPM range that management required us to run; the aerator system was very close to the FOPDT model estimate for a unit step operating condition. Considering our analyzed data plot was so accurately close to the FOPDT model, in most running conditions or operating RPM s the model could be used to estimate future conditions without having to run the motor in order to determine data. Such a situation as ours is very helpful when trying to diagnose a problem with the system; such as if the motor is running at a RPM lower than the FOPDT model calculated it to be at for a set power percentage. A case like this could be due to a running

14 problem or a worn out part of the system causing the motor to run lower than normal. Only way to tell if the system is not performing up to full capacity would be to calculate the projected RPM it should be running at using the FOPDT model.

15 Appendix: Step Up: 10% to 20% Output (RPM) Step Up 10% to 20% Time (sec) Output(RPM) Output Input Value(%) Input Input (%) 70% to 80% 1450 Step Response: 70 to Output Resposnse (RPM) Output(RPM) Model Output Input Value(%) Model Input Input (%) Time (seconds) 68

16 Step Down: 20% to 10% Output (RPM) Step Down 20% to 10% Output(RPM) Output Input Value(%) Input Input (%) Time (sec) 80% to 70% 1800 Step Response 80% To 70% 82 Output Resposnse (RPM) Output(RPM) Model Output Input Value(%) Model Input Input (%) Time (seconds) 68

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