gonzales_wesley_engr3406_final_project

ENGR3406

Process Control Systems Project: Level Control of a Twin Tank System

Wesley Gonzales

4/21/2014

ENGR3406 Final Project: Two Tank Level Control Wesley Gonzales

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Executive Summary The purpose of this project is to develop a model of level control for a two tank system in which an electric pump controls flow into the first tank, which subsequently affects by its own unique process dynamics the level in the second tank. The manipulated variable will be the voltage supplied to the pump, the controlled variable is the inlet flow to the first tank and the process variable is the level in the secondary tank. Additionally, the outflow from the second tank will flow into an overflow reservoir in which the pump shall draw and subsequently pump water back into the first tank. The process is capable of being controlled in two modes: manual, in which the voltage supplied to the pump is manipulated by an operator in open-loop mode, and automatic, in which the process shall be controlled in a closed feedback configuration based on PID parameters. Initially, a model shall be developed in order to numerically estimate the process dynamics and response of tank 2 level based on voltage supplied to the pump. An empirical model shall be devised based on actual data from the tank, utilizing PAS TuneWizard software, which shall provide the process gain and time constant as well as provide recommendations for PID tuning parameters. Different tuning methods shall be employed and compared for optimal performance until finally an operational test shall be employed using the PID tuning parameters with the best response by conducting closed loop performance tests. This project shall illustrate the key takeaways from the process control curriculum: numerical modelling, controller tuning, operational testing. Main Contributions/Results This project demonstrated the six-step procedure for developing a model for a two tank level control system. It involved analytical modeling, empirical modeling, experimental tuning and validation of results. The application of level control has many industrial applications, and this project demonstrated achieving level control with rather unsophisticated equipment and feedback control using a PID algorithm programmed into National Instruments LabVIEW software. This project revealed the functionality of the PAS TuneWizard software that enables process control technicians (or students) to accurately develop numerical models and provide tuning parameters for optimal performance. By the end of this project, analytical and empirical models were developed using different methods, all of which successfully contributed to the achievement of controlling fluid level in a second order system.


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Table of Contents

I. Introduction A. Literature Review 3 B. Project Objective and Description 4 C. Project Significance and Impact 4

II. Methods and Materials A. Alternative Approaches Considered

to Solve the Problem 5

B. Selected Approach to Solve the Problem

6

1. Materials 6 2. Data 6 3. Assumptions 7 4. Problem Formulation 7 5. Calculations 8 6. Experiments 14

III. Results A. Presentation of Results 15 B. Discussion/Interpretation 15

IV. Conclusions A. Contributions of Project 17 B. Significance of Work 17

V. Future Work 17 VI. Appendices VII. References 18


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Literature Review ‘Process Control, Designing and Processes and Control Systems for Dynamic Performance’ by Thomas E. Marlin provided the fundamental theories and practices that made this project possible. It established the modeling procedure that was adopted for this project and even included some handy tables that were often utilized for reference. One of the examples of level control in the text was indispensable and particularly valuable for the analytical modeling developed in this experiment. ‘Applied Fluid Mechanics’ by Robert L. Mott provided insight into the conservation of energy of incompressible fluids through Bernoulli’s equation. One of the essential aspects of this project entailed understanding the dynamics of the change in flow rate with respect to elevation changes.


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Project Objective and Description

The purpose of this endeavor is to develop analytical and empirical models for a two tank level control system. The numerical models shall attempt to capture the process dynamics in which the voltage applied shall control the inlet flow into the first tank, and the contents of the first tank shall freely flow into the second tank based on the hydrostatic pressure based on the level differential between the two tanks. Additionally, the second tank will have an outflow that will empty into a system return reservoir that continuously provides the pump flow into the first tank. The analytical model will reflect the theoretical process dynamics and furthermore the empirical model shall be developed using proprietary software that shall utilize actual process data from the ultrasonic level transducers in both tanks. The analytical model and empirical models shall be conducted simultaneously, independently in order to provide a comparison of the reliability of the two different modeling methods. Based on the process dynamics of the models, numerous tuning methods shall be employed, including IMC and Ziegler-Nichols in order to find the optimal set point response. This project shall demonstrate the general six-step modeling procedure and fulfill some of the essential skills in the study of process control systems. The physical layout of the project entails the two tanks and system reservoir, the ultrasonic sensors and the PC which functions as the controller based on a National Instruments LabVIEW software algorithm. The LabVIEW program is capable of functioning in manual (open loop) mode or automatic mode (closed feedback loop). In the manual mode, the operator can freely manipulate the voltage to the pump and the subsequent inlet flow into the first tank. In the automatic mode, the LabVIEW software shall increase and decrease the voltage and pump flow into the first tank based on PID tuning parameters entered by the operator. The manual mode shall be necessary for the development of the empirical model in which the operator shall conduct open loop test(s) in order to capture the output response due to a change in the input at steady state. The automatic mode shall be used as model and tuning parameter validation at the conclusion of the project analysis.

Project Significance and Impact

Level control has applications in many processes such as distillation systems, coolant return reservoirs for reactors and ballast control. The variable control demonstrated in this project is a mere component in what could otherwise be a complex system that involves multi-variable control such as pressure, flow, concentration and temperature. This system process involves a master and slave configuration in which the tank 2 controller determines the set point of tank 1. This cascaded control loop within a control loop can even be utilized between different variables as long as the slave directly influences the process variable of the master. The interface between the controls of different variables enables a process to reach higher levels of optimization and efficiency, and after further analysis, provides an opportunity for identification of diminished levels of performance and degraded system components. The capability to analyze and control at different levels empowers managers and supervisors of process systems to make better decisions and provide flexibility under budgetary and resource constraints.


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Alternate Approaches Considered to Solve Problem For this modelling problem, we used the six step modeling procedure as a baseline to guide our analysis. The six steps and their associated considerations are as follows:

1. Define goals – what variable are we measuring? 2. Prepare information – collect static data, define assumptions, draw block diagram 3. Formulate the model – conduct material / energy balance, conduct degree of freedom

analysis, develop differential equation(s), non-linear, linear models 4. Determine the solution – determine input / output response process dynamics and

parameters: process gain, time constant, time delay 5. Analyze the results – assess accuracy of process response with predicted, compare process

response with empirical (if possible) 6. Validate the model – conduct data collection that supports modelling results and present

post-analysis Generally, this six step procedure can apply exclusively to theoretical models, however, this exercise was intended to represent the process dynamics of an actual physical process, so we were presented with the luxury of developing a theoretical analytical model, and an empirical model. We would be able to carry out the theoretical modeling procedure and simultaneously collect empirical data using open loop testing and utilize PAS TuneWizard software to evaluate the process dynamics and recommend tuning parameters. One of the different approaches we considered was to conduct the empirical analysis first, and let the process dynamics and tuning parameters guide our analytical modelling process analysis. However, we wanted the analytical modelling to proceed independent of the empirical modelling in order to get a more theoretical ‘second opinion’ and allow the empirical results reinforce and validate our theoretical model. Due to the cascaded configuration of this process, we were also faced with the decision to pursue tuning parameters for the process dynamics of tank one first and then implement tuning for tank 2 or develop second order tuning for the integrated process, with the voltage input to the pump as the manipulated variable and tank 2 level as the controlled variable. A theoretical tuning spreadsheet had been developed as a class exercise that could be implemented to simulate manual tuning.


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Selected Approach to Solve Problem Materials Other than the physical components associated with this experiment, this project extensively utilized various software platforms, notably MS Excel to develop the analytical modeling, as well as PAS TuneWizard software that facilitated the empirical modeling of this process. PAS TuneWizard is used in order to conduct an analysis of the process dynamics and transient response of a particular process. It has been utilized in industrial applications for process tuning and performance assessment.

Figure 1. Physical layout of twin tank system

Data The physical dimensions of the plant determined the model parameters, and additionally, steady state data was available. The height of tank 1 and tank 2 are 37cm, however, the ultrasonic flow transmitters are calibrated such that the level is 100% at 30cm.

Table 1. Physical Dimensions and Equipment Specifications Length (cm) 14.2 Width (cm) 12.8 Height (cm) 30

Pipe diameter (in.) 0.5 Pump Rating (gpm) 1200

Ultrasonic level sensor range (mA) 4 - 20 Pump voltage operating range (V) 0-12 Pulse Width Modulation range (V) 0-5


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Table 2. Steady State Values Variable Value Units

Vps 2.75 V DC F1s 48.45 cm3/s F2s 6.5 cm L2s 3.5 cm

Assumptions Most of the assumptions we shall make will be based on the equipment we shall be using for this experiment. The LabVIEW program that contains the PID algorithm has been pre-conceived by another individual, so we shall assume that the algorithm operates with a reasonable level of satisfactory performance. We do not have a flowmeter available to measure the flow rate of this experiment, so we shall have to rely on empirical methods to estimate the flow rates that increase with voltage input and extrapolate flow rates from minimum values to maximum. The level sensors that we shall be using have been calibrated such that the level sensors will read 100% at 30cm (when the top of the tank is actually 37cm). One last caveat is that for every possible session in which open loop testing, closed loop tuning and performance verification occurred, every possible effort was made to ensure that operating conditions were the same as the previous sessions and that no alterations or equipment configuration changes were made. Problem Formulation To approach this problem, we decided to apply the six step modeling procedure as closely as possible, which entailed us developing an analytical model, an empirical model and developing tuning parameters based on the process dynamics of each model. The analytical and empirical models would be done in parallel, as independent as possible, in order to get independent verification and confirmation of the results. For the analytical model, the process gain and time constant will be found by numerical manipulation as well as importing the simulated transient response created in Excel into the PAS TuneWizard software. For the empirical model, the process gain and time constant will be found by conducting open loop testing from steady state and importing the csv file data into TuneWizard in order to get the process gain and time constant and recommended tuning parameters. In TuneWizard, one can select tuning parameters based on different applications, for disturbance rejection or setpoint response. For the purposes of this project, the tuning parameters shall be selected based on setpoint response using IMC tuning parameters.

Figure 2. Block diagram representation of two tank level control system


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Figure 3. Flowchart for Procedural Methodology

Calculations The development of the non-linear model started by developing a mass balance regarding the level as a function of inlet and outlet flow. The cross sectional area of the tank multiplied by the derivative of tank level with respect to time would give the volumetric displacement as a function of time. At any given time, the volumetric displacement would be function of the inlet flow minus the outlet flow.

𝑑𝐿1𝑑𝑡

=𝐹0𝐴−𝑘𝑓1√𝐿1 − 𝐿2

𝐴

The outlet flow is determined by Bernoulli’s equation, in this case, the cross sectional diameter of the pipe does not change from tank 1 to tank 2, so there is no change in kinetic energy. There is no differential pressure from tank 1 to tank 2 since both are at atmospheric pressure, so the outlet flow from tank 1 to tank 2 is determined by the height differential, with the level of tank 1 utilized as the datum. The value of the drain coefficient kf1 was found by determining the outlet flow rate of tank 1 to tank 2 at steady state which was measured using empirical data and solving for kf1.


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𝐹0 = 𝑘𝑓1√𝐿1 − 𝐿2

The differential equation for the non-linear model of tank 2 was completed similarly by measuring the flow rate by volumetric displacement for a given unit of time at a steady state tank level L2. In the case for the differential equation for tank 2, the inlet flow to tank 2 is the same as the outlet flow from tank 1.

𝑑𝐿2𝑑𝑡

=𝑘𝑓1√𝐿1 − 𝐿2

𝐴−𝑘𝑓2√𝐿2

𝐴

Figure 4. Non-linear model in MS Excel, voltage input step change and flow response

Figure 5. Non-linear model transient response from steady state 2.75V to 3.5V SS

0

20

40

60

80

100

120

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30

Flo

w (

cm^3

/s)

Vo

ltag

e(V

)

Time(s)

Manipulated Variable Step Δ

V

F

0

5

10

15

20

25

30

35

0 200 400 600 800

Leve

l (cm

)

Time (s)

Controlled Variable (PV)

Tank 1

Tank 2


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The linear model was developed by defining the non-linear term to linearize, which was the (L1 – L2)-0.5.

[(𝐿1 − 𝐿2)]′ =0.5𝑘𝑓1(𝐿1 − 𝐿2)

−0.5[(𝐿1 − 𝐿2) − (𝐿1𝑆 − 𝐿2𝑆)]

𝐴

𝑑𝐿1𝑑𝑡

=𝐹0𝐴−𝑘𝑓1√𝐿1 − 𝐿2

𝐴−0.5𝑘𝑓1(𝐿1 − 𝐿2)

−0.5[(𝐿1 − 𝐿2) − (𝐿1𝑆 − 𝐿2𝑆)]

𝐴

0 =𝐹0𝐴−𝑘𝑓1√𝐿1𝑆 − 𝐿2𝑆

𝐴−0.5𝑘𝑓1(𝐿1𝑆 − 𝐿2𝑆)

−0.5[(𝐿1𝑆 − 𝐿2𝑆) − (𝐿1𝑆 − 𝐿2𝑆)]

𝐴

𝑑𝐿1′

𝑑𝑡=𝐹0𝐴−0.5𝑘𝑓1(𝐿1 − 𝐿2)

−0.5

𝐴𝐿1′

𝑑𝐿′

𝑑𝑡=1

𝐴𝐹0 − (

0.5𝑘𝑓(𝐿1 − 𝐿2)−0.5

𝐴)𝐿′

𝜏𝑝𝑑𝐿′

𝑑𝑡= 𝑘𝑝𝐹0 − 𝐿′

𝜏 =𝐴

0.5𝑘𝑓(𝐿1 − 𝐿2)−0.5

𝑘𝑝 =1

0.5𝑘𝑓(𝐿1 − 𝐿2)−0.5

Figure 6. Linearized Model transient response from steady state 2.75V to new steady state value 3.5V

0

5

10

15

20

25

0 50 100 150 200

Leve

l (cm

)

Time (s)

Controlled Variable (PV)

Tank 1

Tank 2


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IMC Tuning Parameters (Analytical model)

𝑘𝑝𝐾𝑐 =𝜏 +

𝛩2

𝜏 +𝛩2

𝐾𝑐1 =1

𝑘𝑝1=

1

0.012= 83.3

𝐾𝑐2 =1

𝑘𝑝2=

1

0.016= 62.5

𝜏𝐼 = 𝜏 +𝛩

2

𝜏𝐼1 = 𝜏 +𝛩

2= 22.19 +

0

2= 22.19𝑠

𝜏𝐼2 = 𝜏 +𝛩

2= 29.74 +

0

2= 29.74𝑠

𝜏𝐷 =𝜏 ⋅ 𝛩

2𝜏 + 𝛩

𝜏𝐷1 =𝜏 ⋅ 𝛩

2𝜏 + 𝛩=

(22.19) ⋅ 0

2(22.19) + 0= 0

𝜏𝐷2 =𝜏 ⋅ 𝛩

2𝜏 + 𝛩=

(29.74) ⋅ 0

2(29.74) + 0= 0

Figure 7. Analytical Model parameters and recommended tuning in PAS TuneWizard software for Tank 1


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Figure 8. Analytical Model parameters and recommended tuning in PAS TuneWizard software for Tank 2

Figure 9. Empirical model parameters and tuning in PAS TuneWizard for Tank 1

The tuning parameters from the analytical and empirical models were imported into a MS Excel spreadsheet that simulated the closed loop transient impulse response for tank 1 and tank 2, which can be seen in Fig. 11-12. The independent confirmation of the tuning parameters in PAS TuneWizard and the MS Excel spreadsheet with the PID algorithm inspired our confidence in the tuning parameters generated by TuneWizard, and gave us at least some initial tuning parameters to implement into the two tank level controller. No conversion would be necessary because TuneWizard time constant and dead time was already in minutes as was the LabVIEW program for the two tank controller. There was an issue to consider when assessing performance: the LabVIEW program gave values of level in percentages, instead of actual data in centimeters, which necessitated some intermediary calculations for steady state values and step changes. However, this should not have affected the tuning parameters or process dynamics.


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Figure 10. Empirical model parameters and tuning in PAS TuneWizard for Tank 1

𝒌𝒑 𝝉 𝜣 P I D

Analytical TuneWizard Recommended Tuning

Tank 1 52.8 2.45 0 0.01 2.5 0

Tank 2 33.2 2.97 0.0875 0.029 3.0 0

Empirical TuneWizard Recommended Tuning

Tank 1 16.5 68.2 5.47 0.056 68 0

Tank 2 9.92 63.5 15.6 0.081 63 0

Figure 11. PID Simulation of closed loop empirical model in MS Excel using PAS tuning parameters

0

0.2

0.4

0.6

0.8

1

1.2

-50 50 150 250 350 450 550

Leve

l (cm

)

Time (s)

Tank 1

PV

Setpoint


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Figure 12. PID Simulation of closed loop empirical model in MS Excel using PAS tuning parameters

Experiments For the final tuning, the LabVIEW interface has two possible configurations: cascaded and non-cascaded. In cascaded, the operator must tune each tank individually and enter PID parameters that ultimately affect the set point response and offset. In the non-cascaded, the PID parameters that the operator enters only affects the tank one controller, bypassing the tank 2 controller but using the error inputs from the tank 2 ultrasonic level transmitter. For the cascaded configuration, tank 1 was

tuned first, and followed by optimal tuning for tank 2 set point response. Under close observation, the final results reveal that the non-cascaded tuning had better performance. Additionally, the non-cascaded configuration is obviously easier to tune considering that the operator only had to tune one set of tuning parameters and did not have to worry that optimal tuning parameters for tank 1 would create a conflict with the parameters for tank 2. One might

wonder why even bother have separate tuning for tank 1 at all.

With the non-cascaded configuration, for large set point increases, the transient response for tank 1 could potentially have an unstable response. For example, if a large set point change from 10% was increased to 80%, it would excessively cycle the pump and increase the flow rate to tank 1 in such a manner that the flow into tank 2 could not handle the increased volumetric flow into tank 1 and tank 1 could overflow. The cascaded configuration could be safer and more stable and offers more control in more sensitive processes. In the non-cascaded configuration, the operator is not too concerned what is going on in tank 1. Notice that the process response is slower due to the tank 1 controller basing its algorithm on the tank 2 level, which has a higher time constant. The cascaded configuration will be more responsive, due to the tight restrictions of PID tuning parameters being placed on both tanks.

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600

Leve

l (cm

)

Time (s)

Tank 2

PV

Setpoint

Figure 13. Experimental evaluation of closed loop response


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Results The results from this experiment revealed tuning parameters that were close to the ones recommended by TuneWizard PAS. They were close enough to reveal baseline values in which to fine tune by conventional methods. The method that is employed is to first set kp such that a set point change will provoke a slight overshoot (ensure no oscillatory behavior) and increase the integral term to get rid of the offset. The PAS TuneWizard did not recommend a derivative term, and one can see the satisfactory results one gets with PI tuning. One can see a slight offset in the cascaded configuration, which was acceptable in the LabVIEW front panel interface initially, however, importing the data into MS Excel and plotting the trend in a graph revealed disappointing level of offset in retrospect, which could have been eliminated in further tuning by increasing the integral term. However, the process response and settling time were impressive, reaching 2% steady state value in 3.38 minutes.

Figure 14. Closed Loop performance of cascaded PID tuning parameters

The initial analytical modeling revealed some deficiencies, particularly the steady state values following a step change. The low process gain was attributed to the fact that the voltage-inlet flow relationship had not been taken into account in the differential equation that was developed, and thus a process gain <1 was realized. When the data simulating a steady state value step change from 2.75V to 3.5V using the analytical method was imported into PAS, the voltage-inlet flow was taken into account, and the model parameters reflected numbers closer to the empirical model. The empirical model proved to be a reliable source of modeling parameters and yielded solid tuning, which guided the eventual final tuning parameters and revealed the deficiencies of the initial analytical model.

0

5

10

15

20

25

560 660 760 860 960 1060 1160

Leve

l (%

)

Time (s)

Cascaded

PV

Setpoint


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Figure 15. Closed loop performance of non-cascaded PID tuning parameters

Final Closed Loop Tuning Parameters

P I D

Cascaded

PID Controller Tank 1 0.15 5.13 0


Non-cascaded


0

5

10

15

20

25

30

35

7121 7321 7521 7721

Leve

l (%

)

Time (s)

Non-Cascaded

PV

Setpoint


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Conclusions

Contributions of Project The contributions of this project reinforce the ability of a feedback control loop utilizing a PI algorithm to control fluid level using rather unsophisticated equipment. The ability to control fluid level has much significance in industrial applications and control processes. It can be utilized in conjunction which other controlled variables, such as temperature, if the fluid is utilized as cooling water. For this application, the two would be utilized in a cascaded configuration, such that the temperature determines the set point of the cooling water fluid level, in a master-slave configuration. This project proved the functionality of the PAS TuneWizard software, which accurately developed modeling parameters and recommended fairly reliable tuning, which could potentially save much engineering man-hours and increase productivity if it replaced more conventional methods.

Future Work This project by no means explores the full range and capabilities of level control, and a more exhaustive study is possible. This project could have explored the possibility of multivariable control by controlling temperature and fluid level. This project could have further evaluated the disturbance rejection of the tuning parameters from the PAS TuneWizard software. It is highly probable that many students shall utilize level control as their capstone senior project, possibly by using more sophisticated control interfaces and measurement devices.


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References

Marlin, Thomas, E. “Process Control, Designing Processes and Control Systems for Dynamic

Performance”, 2nd Edition, McGraw-Hill, 2000.

Mott, Robert L. “Applied fluid mechanics”. 6th Edition,. Upper Saddle River, N.J.: Prentice Hall, 2005.

Print.

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