UTC

Engineering 3280L

Matthew Addison

Green Team

(Michael Hansen)

9/4/12

Aerator Mixer Speed Control System

Step Response Modeling

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 200-1700 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.

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.

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

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.

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.

Average output= 300 motor RPMs

Standard Deviation=4.45 RPMs

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

10 1.65 4.09 0.08

15 2.52 3.68 0.07

20 3.4 4.45 0.09

25 4.28 3.75 0.08

30 5.15 3.53 0.07

60 10.4 2.5 0.05

65 11.2 2.4 0.05

70 12.1 2.5 0.05

75 13.0 1.9 0.04

80 13.8 3.9 0.08

85 14.6 4.7 0.09

90 15.5 4.6 0.09

95 16.3 4.3 0.09

100 17.1 4.2 0.08

AVG= 0.07

Input (%)OutPut (RPM of

the mixer)STDev STDev x 2

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: �������� � ���� � �����

where � 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�� �

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 ��� � �������� � 1�� �

Here t0 is the time delay or dead time. (Smith & Armando , 1997).

For this experiment the input function is a step function of the form below, where A is the

magnitude of the step, and td is the time at which the step occurs ���� � ���� � ���

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� ����������� ��

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 td is the time when the step response

occurs, and t0 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 � �� . The first-order time constant,�, is the constant that forces the

model response to coincide with the actual response: ��� � �� � ����1� ��� � .632��

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.

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 100-600 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 1100-1600 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%.

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.

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.

0

5

10

15

20

25

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12

Inp

ut

(%)

Ou

tpu

t (R

PM

)

Time (sec)

Output(RPM)

Output

Input Value(%)

Input

Time(sec)

Input

Value(%) Output(RPM) Input Output

td= 5

0 20 0 20 340

A= -10

0.102 20 33.027 20 340

K= 18

0.206 20 160.527 20 340

to= -0.02

0.309 20 250.251 20 340

tau= 0.15

0.413 20 287.45 20 340

inbl= 20

0.516 20 308.812 20 340

outbl= 340

0.62 20 322.438 20 340

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

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

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.

Appendix:

Step Up: 10% to 20%

70% to 80%

0

5

10

15

20

25

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12

Inp

ut

(%)

Ou

tpu

t (R

PM

)

Time (sec)

Step Up 10% to 20%

Output(RPM)

Output

Input Value(%)

Input

68

70

72

74

76

78

80

82

1100

1150

1200

1250

1300

1350

1400

1450

0 2 4 6 8 10 12

Inp

ut

(%)

Ou

tpu

t R

esp

osn

se (

RP

M)

Time (seconds)

Step Response: 70 to 80

Output(RPM)

Model Output

Input Value(%)

Model Input

Step Down: 20% to 10%

80% to 70%

0

5

10

15

20

25

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12

Inp

ut

(%)

Ou

tpu

t (R

PM

)

Time (sec)

Step Down 20% to 10%

Output(RPM)

Output

Input Value(%)

Input

68

70

72

74

76

78

80

82

1100

1200

1300

1400

1500

1600

1700

1800

2 4 6 8 10 12

Inp

ut

(%)

Ou

tpu

t R

esp

osn

se (

RP

M)

Time (seconds)

Step Response 80% To 70%

Output(RPM)

Model Output

Input Value(%)

Model Input