process-model-based control of wastewater ph a …

PROCESS-MODEL-BASED CONTROL OF WASTEWATER pH

by

GAYLON LYNN WILLIAMS, B.S.

A THESIS

IN

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CHEMICAL ENGINEERING

Approved

Accepted

August, 1989

• f • •«

T5

ACKNOWLEDGEMENTS

I am deeply indebted to Dr. R. Russell Rhinehart, chairman of

my advisory committee, for his valuable advice, direction, and

financial support. I also express a sincere appreciation to Dr. J.

B. Riggs, my other committee member, for his suggestions and criti

cisms throughout this work.

I want to thank Dr. Rhinehart and Dr. Riggs for giving me

permission to quote portions of their works. Appreciation is

extended to Mrs. Sue Willis for her clerical assistance throughout

my graduate program and, finally, I wish to take the opportunity to

thank ray parents for their constant encouragement and dedicated

guidance.

11

/

TABLE OF CONTENTS

PAGE

ACKNOWLEDGEMENTS ii

LIST OF TABLES vi

LIST OF FIGURES vii

LIST OF SYMBOLS ix

CHAPTER 1 INTRODUCTION 1

CHAPTER 2 FUNDAMENTALS 5

2 .1 The Interpretation of pH 5

2.2 The Titration Curve and Its

Influence on the Process 6

2.3 Rangeability and Sensitivity 10

2.4 Characteristics of Strong and Weak Acids (or Bases) and Buffering Effect 14

CHAPTER 3 LITERATURE REVIEW 18

3.1 Conventional Approaches to pH Control 18

3.2 Process Model-Based Control Strategy 21

3.3 Overview of Process Model-Based

Control 30

3.3.1. PMBC Advantages 31

3.3.2 PMBC Disadvantages 34

3.4 Previous PMBC Wastewater pH

Control Work 36

CHAPTER 4 PROCESS SIMULATOR 44

4.1 Introduction to the Simulator

Development 44 iii

PAGE

CHAPTER 5

CHAPTER 6

4.2 Simulating the Reaction Mechanics...

4.3 Simulating the Mixer Fluid Dynamics

4.4. Simulating Instnomentation Noise....

4.5 Simulating Instrumentation Response Time Lag

4.6 Simulating Nonstationary Wastewater Disturbances

CONTROLLER MODEL AND STRATEGY

5.1 Model Parameter Identification

5.2 Control Strategy

5.3 Options and Variations

RESULTS AND DISCUSSION

6.1 Regulatory Control

6.2 Step Upset

6.3 Ramp Upset

6.4 Large Finishing Blender

6.5 Model Mismatch and Calibration Errors

6.6 Varied Set Ratios for Base Split...

6.7 Variable Base Split Ratio

6.8 Weighting Factors in the Least Squares Minimization

6.9 pH Setpoints Other than Seven

44

45

47

49

50

51

51

53

56

60

60

69

76

79

82

84

86

90

95

IV

PAGE

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 97

7.1 Conclusions 97

7 .2 Reconmiendations 98

LIST OF REFERENCES 101

APPENDIX A IONIZATION CONSTANT OF WATER 9CRC, 1983) 104

APPENDIX B MULTI-VARIABLE NEWTON' S METHOD 106

APPENDIX C RANDOM NOISE 108

APPENDIX D AUTOREGRESSIVE DRIFT 109

APPENDIX E FILTERING PROCESS VARIABLES AND

SIMULATING INSTRUMENTATION TIME LAG 110

APPENDIX F LEAST SQUARES MINIMIZATION 112

APPENDIX G SPLIT CALCULATOR 116

APPENDIX H FILTERED STANDARD DEVIATION 117

APPENDIX I COMPUTER CODE 118

LIST OF TABLES

PAGE

Table 2.1: Titration Curve Slopes (process gain)

Table 6.1: Initial Wastewater Composition for Figure 6.1

Table 6.2

Table 6.3

Table 6.4

Wastewater Compositions for Figure 6.7.

Wastewater Compositions for Figure 6.10

lAE Results for Modeling Errors Introduced Into the Step Change Conditions of Section 6.2

Table 6.5: lAE Results for Various Base Split Ratios...

Table 6.6: lAE Results for Various Tuning Parameters for the Step Change Conditions Presented in Section 6.2

12

63

71

78

85

87

Table A.l Water Ionization Constant.

94

105

VI

LIST OF FIGURES

PAGE

Figure 2.1: The Process Gain Illustrated for a Simple Weak Acid Plus Salt in Water Titrated with NaOH at 2980K 8

Figure 2.2: The Characteristic Curves of Several Wastewater Compositions 9

Figure 2.3: The Process Gain Change for the Weak Acid System of Figure 2.1 11

Figure 3.1: The Chemistry of a Hypothetical Acidic Wastewater 38

Figure 3.2: The Chemistry of a Single Weak Acid and Strong Base Reaction System 41

Figure 4.1: The Actual Wastewater Neutralization System 48

Figure 5.1: Functional Diagram for PMBC of an In-line pH Neutralization System 56

Figure 6.1: Simulator Results for Controlled Effluent pH for a System with Normal Nonstationary Process Behavior 61

Figure 6.2: Comparison of Titration Curves for the

Process versus the Model for Figure 6.1 64

Figure 6.3: Weak Acid Concentration for Figure 6.1 65

Figure 6.4: Strong Acid Concentration for Figure 6.1 66

Figure 6.5: Coimnon Ion Salt Concentration for Figure 6.1 67

Vll

PAGE

Figure 6,6: Weak Acid AG of Dissociation for Figure 6.1 68

Figure 6.7: Controlled Effluent pH Response to Step Changes 70

Figure 6.8: Comparison of Titration Curves for the Process versus the Model for Times of 30 and 600 Seconds on Figure 6.7 74


Figure 6.10: Controlled Effluent pH Response to a Ramp Change 77


Figure 6.12: Comparison of pH of Finishing Blender to pH of Effluent from the Neutralization Process 81

Figure 6.13: Base Flow Split Ratio Calculated by Controller versus Time 89

Figure 6.14: Comparison of Effluent pH's for Two Different Weighting Factor Sets 92

Vlll

LIST OF SYMBOLS

Alphabetical

ajj - activity of H+ ion in solution

X - amount of weak acid dissociated, mol/lit

Y - vector of measurable process outputs

Af - fictitious acid concentration, mol/lit

Ka - chemical reaction equilibrium constant for the acid-base

reaction

K^ - water ionization constant

pH - wastewater pH

R - universal gas constant

T - wastewater or reagent base temperatures

Greek

a - initial concentration of strong acid, mol/lit

^ - initial concentration of weak acid, mol/lit

8 - mount of hydrogen ionized from H2O, mol/lit

e - base concentration, mol/lit

7 - initial concentration of salt, mol/lit

AG - Gibbs free energy of change, cal/gmol

AGf - fictitious Gibbs free energy of change, cal/gmol

p - density

IX

Acronvms

AF - acid flow rate

BOH - base

CSTR - continuous stirred tank reactor

HA - acid

HW - weak acid

MW - salt

PI - proportional-integral

PMBC - process model based control

SISO - single input single output

^

CHAPTER 1

INTRODUCTION

Chemical process wastewater typically contains several weak

and strong acids and their salts and must be neutralized before

discharge into the environment. The level of acidity in the ef

fluent is conventionally measured with a pH meter and the acid is

easily neutralized by the addition of a relatively small amount of

a strong but inexpensive base.

The composition of a wastewater stream can vary considerably.

The freestanding acids and bases (sulfuric acid, sodium hydroxide,

etc.) are typically encountered, yet some industries also produce

"combined" acidic or basic salts, which form weak acids or bases by

hydrolysis upon dilution of the waste by the receiving water

(Patterson, 1975). Therefore, for control of the effluent stream

pH, the acidic and basic salts must be treated along with the

freestanding acids and bases in the wastewater stream. The waste

water treatment processes employed for the neutralization of the

freestanding acids and bases are simultaneously effective in con

trolling the acidic and basic salts. However, control of waste

water neutralization is subject to several difficulties, including:

1) The pH response to base addition is a highly nonlinear 'S'

shaped curve and process gain can change by 10 orders of magnitude.

2) The multicomponent system is a buffer solution making the

process gain a complicated function, a) dependent on the non-

stationary wastewater composition, b) and dependent on temperature.

Finally, 3) the rate of base addition depends on the plant effluent

rate. For these reasons, control by a conventional PID feedback

mechanism is ineffective and, industrially, neutralization of such

waste requires either sequential neutralization steps and interme

diate hold ponds or batch operations (Shinskey, 1983; Moore, 1978).

The capital and expense associated' with such techniques make a

continuous flow-through process attractive. Additionally, a better

controller would reduce the magnitude and frequency of pH discharge

events.

Some of the many pH control applications include: boiler

water treatment, chemical and biological reactions, municipal waste

digestion, acid pickling and etching, cooling tower water treat

ment, electrohydrolysis, and coagulation/precipitation. However,

wastewater treatment is the most difficult due to the pH - 7 target

and the unknown and non-stationary fluid composition. Consequent

ly, demonstration of a strategy for wastewater neutralization

suggest effectiveness for pH control of these other processes.

The subject of this study is the simulation of a new in-line

pH control strategy for a multicomponent process wastewater.

Neutralization is accomplished by splitting the base flow into two

streams and sequentially injecting it into the wastewater line.

The supervisory controller then generates a titration curve from

the on-line measurable data and uses this titration curve to pre

dict the required base flow rate setpoint. A conventional Single

Input Single Output (SISO) Proportional-Integral (PI) controller

is used to control the base flow rate. The supervisory controller

generates the titration curve by using a process model that is

based on the phenomena of the process, and is coined nonlinear

Process Model Based Control (nonlinear PMBC).

New opportunities are developing in process control because of

the increasing potential and availability of computers in the work

place. One of these opportunities is the nonlinear PMBC control

ler. Nonlinear PMBC controllers use a phenomenological model of

the process with adaptive parameters; overcome many of the disad

vantages of the 3-term PID controllers; and have advantages over

other modern linear controllers in speed, accurate decoupling,

setpoint tracking, and both steady-state and dynamic optimization.

Specifically, this study is directed toward the application of

a PMBC control strategy for pH control of wastewater. Current pH

control strategies are difficult to implement and are either labor

or capital intensive and prone to pH discharge "events." There

fore, an "intelligent" controller for a continuous process could

reduce the cost of maintaining pH control and reduce the frequency

and magnitude of environmental upsets. This intelligent controller

would require adaptive parameters, be computationally fast, and

reflect the true phencnicna of the actual process.

In general, process control applications are made difficult by

the process characteristics of nonlinearity, dead-time, and multi-

variable coupling. The successful technique development and demon

stration of PMBC on wastewater pH control would intimate its effi

cacy in other process control applications. Some of the chemical

process applications would include reactors, distillation columns,

extractors, heat exchangers, and pH neutralization; some of the

non-chemical process control applications would include robotics

and missile guidance.

CHAPTER 2

FUNDAMENTALS

Modeling of acid/base neutralization requires xise of kinetic,

thermodynamic, and process dynamic behaviors; and, before

attempting to solve a pH neutralization problem, these fundamental

aspects must be understood. This section will also review control

valve rangeability and characteristics of reagents in solution,

both of which are pertinent to the research.

2.1 The Interpretation of pH

The operational definition of pH is:

pH - -Logio(aH * [H+]) (2.1.1)

where ajj is the hydrogen ion activity, and [H^] is the hydrogen ion

concentration (gmole/liter). The hydrogen ion activity is

approximately equal to 1.0 liters/gmole, and is usually not shown

in pH calculations; leaving us with the familiar definition of pH

as

pH - -Logio[H+]. (2.1.2)

Therefore, pH is, a unitless measure because of the hidden units

left from the hydrogen ion activity, which we chose not to show in

the definition of pH.

The measurement of pH is a meaningful measure of hydrogen ion

concentration only for dilute concentrations of an acid or alkali

dissolved in water. This limitation on concentration exists

because only at high dilution is Henry's law valid, and only then

does concentration become equal to hydrogen ion activity (which is

the characteristic actually being measured in a pH probe).

Industrial wastewater, however, is usually sufficiently dilute;

therefore, the application of Henry's law can be justified.

Some acids dissociate readily in water. These are known as

strong acids (hydrochloric acid, sulfuric acid, etc.). There are

some acids, however, that resist hydrolysis and do not dissociate

readily in water. These are known as weak acids. Water also

dissociates and adds hydrogen ions such that [H''"][OH'] = lO' ' .

Thus, as an equilibrium constraint, the dissociation of water

molecules participates in the equilibrium balance of all reactions

in aqueous solution. Pure water has a pH of 7 at 24°C, 0.IN HCl

has a pH of 1.1 at 25^0, and 0. IN NaOH has a pH of 13.0 at 25^0,

for example (CRC Handbook, 1983).

2.2 The Titration Curve and Its Influence on the Process

There is usually a single waste treatment facility for an

entire plant, and, therefore, the treatment facility must handle a

mixture of many streams that contain a variety of chemicals. For

single strong acids and bases the reagent demand can be

theoretically calculated from a single pH measurement. Most

treatment facilities, however, must handle a mixture of reagents

and various metal ions. For these cases, the reagent demand can

best be determined by evaluating an actual wastewater titration

curve. Therefore, the basic information required for pH control

system design is the titration curve, sometimes referred to as the

characteristic curve of the equalization curve.

If the identification, concentration, and equilibrium

constraints of each specie in the wastewater stream were known, the

titration curve could be calculated. Since acid/base/salt

reactions are extremely rapid compared to the time scales of other

dynamic process effects, such as mixing, one can assume that the

reactions are in thermodynamic equilibrium. Using thermodynamic

equilibrium, the initial component concentrations can be used to

calculate hydrogen ion concentrations, which can be converted to pH

by the logarithmic relationship. The titration curve of Figure 2.1

shows pH for a simple acid solution as a function of base addition.

Effluent pH is the measured variable and base flow rate is the

manipulated quantity in a neutralization control system; and the

slope of the titration curve at any pH setpoint is, therefore, the

apparent steady state process gain. The titration curves of Fig

ures 2.1 and 2.2 illustrate the extreme nonlinearity (change in

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gain) which may be encountered in pH control. Figure 2.3 shows

chac this gain changes from 81.2 to 1588.7 when pH changes from 6.0

to 8.3. Around pH 7 small pH measurement error can lead Co big

mistakes in Judging process gain. As wastewater composition

varies, both the shape and location of the titration curve (and

consequently gain) change must be updated. A good example of

process gain change for an actxial plant wastewater is presented in

Table 2.1 (Piovoso and Williams, 1985).

2.3 Rangeability and Sensitivity

The control of pH, as countless articles, papers, and

textbooks point out, is characterized by extreme rangeability and

sensitivity. Rangeability is the ratio of the maximxim to minimum

flow rate which must be controllably delivered. Flow control

devices xised in manufacturing a product ordinarily do not require a

wide range of flow rate. They are operated at some process optimum

condition, profitable production rate, and use feedstocks of

relatively consistent quality (Shinskey, 1983). Treatment of

waste, however, is another matter. If the plant manufactures

piecework, or if some of its areas operate discontinuously, wide-

ranging composition and flow rate of the plant effluent are likely

to be encountered. Often the base flow rate required for

11

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12

Table 2.1: Titration Curve Slopes (process gain)

Sample pH - 6.5 pH - 8.5

1 2.5 2.15

2 1.0 0.53

3 0.47 0.46

4 0.91 0.62

5 1.42 0.75

6 1.49 0.56

7 1.49 0.63

8 0.82 0.87

9 0.27 0.26

Note: where gain is in pH (meq/1) and samples were taken at nine consecutive hourly intervals.

13

neutralization varies over a 50:1 range, which is a problem because

control valves normally show effective control only over a 10:1

flow range.

The pH scale corresponds to hydrogen ion concentration from

10° to 10'^^ moles per liter. The measuring electrodes can respond

to changes as small as 0.001 pH, so that some instruments can track

hydrogen ion concentration changes as small as 5*10"^^ moles per

liter at 7 pH (McMillan, 1984). No other common measurement has

such tremendous sensitivity. Sensitivity is the ratio of the

maximum value of a measurement to the minimxim detectable change,

5*10"^0/100 - 5*10-12, for the above example. The implication of

such great rangeability and sensitivity can be illustrated by

considering a continuous feedback neutralization system for a

strong acid and a strong base. The reagent flow should essentially

be proportional to the difference between the hydrogen ion

concentration of the process fluid and that of the set point. A

reagent control valve must therefore have a rangeability greater

than 10:1 for a set point of pH 7 when the influent stream

fluctuates between 0 and 7 pH. Moreover, uncertainties in the

control valve stroke translate directly into pH errors, such that a

valve hysteresis of only 0.00005Z can cause an offset of 1 pH unit

for a 7 pH set point (McMillan, 1984).

14

2.4 Characteristics of Strong and Weak Acids (or Bases) and Buffering Effect

The words "strong" and "weak" have dual meaning when applied

to acidic and basic materials. As previously discussed, "strong"

when applied to pH measurement refers to the degree of completion

of the hydrolysis (ionic dissociation) of the material when added

to water. Another definition of "strong" refers to the

concentration of the material, e.g., a 95Z acid solution is

"stronger" than a 4Z acid solution. Increasing acid strength, in

the sense of the latter definition, is seen to move the titration

curve horizontally to the right if the acid becomes stronger in

concentration, or to the left if the acid becomes weaker (Moore,

1978). Increasing strength in the former definition, however, adds

a severe complexity to the regulation of pH by moving the titration

curve vertically upward if the degree of ionization (hydrolysis)

decreases, or downward if the degree of ionization (hydrolysis)

increases. From now on, to prevent the confusion about the degree

of acidity, the following terms will be used, "strong" as "easily

hydrolyzed," "weak" as "not easily hydrolyzed," "concentrated" as

"high concentration," and "dilute" as "low concentration."

It has been pointed out that the measurement of pH is not only

a measure of the total acid components or total base components of

the solution, nor of only the degree of ionization (Moore, 1978);

15

rather, pH is a measure of hydrogen ion concentration which itself

depends on many factors, including degree of ionization and

component concentrations. Strong acids and strong bases ionize

almost completely when dissolved in water. Therefore, if

equivalent amounts of the strong acid and strong base were added to

the water, the resulting solution would be neutral. The hydroxyl

ions from the base would combine with the hydrogen ions from the

acid, and would form salts, which would have a pH of 7, or are

neutral.

The situation is quite different when a weak acid is titrated

with a strong base. A weak acid dissociates in an aqueous solution

to yield a small amount of hydrogen ions.

(i) HA -^—^ H+ + A".

When hydroxyl ions are added, they are neutralized by the

hydrogen ions to form water.

(ii) OH- + H+ -*-^ H2O.

The removal of the hydrogen ions disturbs the equilibrium

between the weak acid and its ions. Consequently, more HA ionizes

to reestablish the equilibrium. The newly produced hydrogen ions

can then be neutralized by more hydroxyl ions and so on until all

16

of the acid hydrogen originally present is neutralized. The

overall result, the sum of reactions (i) and (ii), is the

titration of HA with hydroxyl ions.

(iii) HA + OH" -•—^ H2O + A".

The number of equivalents of hydroxyl ions stoichiometrically

required, equals the total number of equivalents of hydrogen

present (as hydrogen ions plus HA) and not simply the number of

hydrogen ion equivalents initially present. However, the pH at the

exact indolent (equivalence) of the titration is not 7. The pH is

higher because of the hydrolysis of the A" ion, because reaction

(iii) itself is an equilibrium reaction. In the absence of any

remaining HA, the A" ion reacts with the water to produce hydroxyl

ions and the undissociated weak acid, HA. (Because thermodynamic

equilibrium conditions must always be satisfied in solutions of

acids and bases, the hydrogen ion concentration and pH during the

titration can be calculated from K^ expressions.)

Similarly, if equivalent amounts of a strong base and a weak

acid (or a strong acid and a weak base) are mixed together, the

resulting solution will not be neutral because the undissociated

weak acid (or base) will not neutralize the "excess" strong base

(or acid) . As a result of this phenomenon, it will always require

more of the strong reagent to neutralize the weak reagent than the

17

original pH of the weak reagent would indicate. For example, a

sample of wastewater with a pH of 2 will require more of a strong

basic reagent to neutralize if the acidity is caused by acetic acid

(a weak acid) than if by hydrochloric acid (a strong acid).

Buffering is the capacity of a solution to resist changes in

pH. A weak reagent buffers by holding a reserve of undissociated

acid or base. Further, the amount of reserve ions is dependent on

the availability of companion ions. For example, consider a weak

acid being neutralized by a strong base. At low pH values, the

acid has a great capacity to absorb hydroxyl ions from the base

because of the reserve of the undissociated acid. As the base is

titrated into the solution, hydroxyl ions are neutralized, the

equilibrium of the reaction shifts, releasing more hydrogen ions,

and resulting in little change in pH. Thus buffered solutions

resist change in pH. However, as more caustic is added, pH will

finally approach a point where the reserve capacity of ions is used

up. At this point, pH will begin to change very rapidly with the

addition of caustic. Buffering, then, provides a built-in pH

regulation that is both helpful and detrimental to control (Moore,

1978). It is helpful wherein resistance to pH change makes the pH

titration curve much less sensitive. However, it is detrimental

because the solution is seldomly neutralized with the base flow set

point which is evaluated from the apparent process pH.

CHAPTER 3

LITERATURE REVIEW

3.1 Conventional Approaches to pH Control

Linearization attempts, adaptive control, and gain scheduling

are three methods that have been discarded as likely control strat

egies for pH neutralization since those strategies do not compen

sate well for the extreme nonlinearities of the process. If the

wastewater titration curve was known, a supervisory controller

could then feedforward a base flow rate setpoint from the measured

input pH (Gray, 1984). Such a controller is now commercially

available (Leeds and Northrup, 1984). However, it is inefficient

because the input pH is relatively insensitive to acid

concentration, and the titration curve must be determined off-line.

Their controller cannot automatically respond to these temporal

composition changes.

Another commercially available controller divides the pH

region into three sections with a different controller gain chosen

for each pH section (LFE, 1987; OMEGA, 1987). As the measured

effluent pH moves among the three sections, the controller switches

to the appropriate controller gain for optimum control. Again,

however, as the influent composition changes, the gain for each of

the three sections must be manually reset. The fact that special

18

19

controllers are commercially available shows the widespread impor

tance and difficulty of pH control.

Balhoff and Corripio (1983) and Jeffreson (1983), in separate

works, addressed some of the control related aspects through simu

lations , of different adaptive proportional-integral-derivative

(PID) controllers. However, both works retained the PID controller

structure (and its limitations) and neither addressed typical

industrial complications of measurement and modeling errors.

Shinskey (1983) discusses several PID controller applications. He

thoroughly documented the process gain characteristics and dis

cussed the control difficulties. He uses conventional linear

control theory (transfer function notation) and indicated some

additional measures (sequential controllers, holding tanks) re

quired for adequate control.

Such conventional PID control schemes show several problems:

(i) The controller responds only after an upset occurs,

(ii) Detection of a pH upset causes the response of

increasing the base addition rate until the pH is at

target (pH—7). The final base rate is unknown, so

for process/controller stability the response is

slow. Until the final addition rate is reached, the

pH is out of specification.

20

(iii) Several controller parameters have to be adjusted,

most of them manually, to accommodate the changes in

the process.

Gustafsson and Waller (1983, 1986) reported on the problems

and dangers of feedforward pH control and attempted to find a

reaction invariant model suitable for a linear control strategy.

Since a prior knowledge of both the reaction invariant model and

some of the chemical content is needed, an off-line identification

step is necessary. Because the reaction invariant model is

constant their controller is not adaptive to residence time

changes. The changes in the time constant are caused by such

factors as the varying reactor throughput or aging electrodes.

Albert and Kurz (1985) experimented with a model reference

adaptive control system which adjusted PI controller gain. A

superimposed adaptive control loop alters the gain of the main

control algorithm so that in the case of a control deviation the

pH-value of the effluent stream approaches its set point by a pre

scribed trajectory. The prescribed trajectory has nothing to do

with the prediction of the titration curve, but is based on the

speed of the control deviation. For stability reasons the integral

decreasing function of the trajectory curve must decrease slowly;

therefore, it cannot handle a fast changing process.

21

Self-tuning PI control with a linear first order process model

was investigated by Proudfoot and McGreavy (1983) who developed a

conventional direct-digital control algorithm using Z-transform

notation. Since their linearized model does not represent the

highly nonlinear process, it does not take into account the fast

pH-dependent variations of the process gain.

3.2 Process Model-Based Control Strategy

Although the majority of the advanced control literature

pertains to strategies that use linear process models, such as

Internal Model Control (IMC) and Djmamic Matrix Control (DMC),

there is a growing body of Process Model-Based Control (PMBC)

publications in the open literature. Additionally, there are

considerable applications working within industry for which the

open literature presents only a glimpse. This subsection is to

first highlight multivariable control based upon linear models,

then to highlight the PMBC literature and our knowledge of indus

trial activities. By doing so, we illustrate the need for a re

search program focused on the unique PMBC aspects. Throughout the

reference citings, an asterisk (*) indicates an author's affilia

tion to the industrial sector and is an indication of the practi

tioner's interest.

Garcia and Morari (1981) presented a unifying review of sev

eral IMC-type control schemes. Among them are: Dynamic Matrix

22

Control (Cutler* and Ramaker*, 1980), Inferential Control (Joseph

and Brosilow, 1978), and Model Predictive Control (Richalet, et

al., 1978). Note that each scheme is based on a linear process

model.

An advantage of linear controllers is the computational speed

and definable stability; however, accurate identification of para

meters of a generic linear model requires statistically significant

process upsets. As an alternative, phenomenological process models

have been used to update constants for the linear process model of

a linear IMC type controller. In addition, the process model can

also be adaptively updated from measurable process data. Klumper*

and Tobias (1986) have applied this approach to simulate the con

trol of an in situ leaching process using IMC. Lee (1987) demon

strated that for control of wastewater pH a reduced phenomenologic

al model could be used for the prediction of the process gain, and

for adjusting the controller parameters.

Along a similar vein, McDonald (1987) investigated the use of

a steady-state model of a high purity distillation column to pre

dict process gain for a DMC controller on a detailed dynamic column

simulator. She found that such a method of "gain scheduling" of

the dynamic controller allowed it to adapt to the process dynamics

rapidly and effectively, and set point tracking control was much

improved over the set point tracking obtained through conventional

recursive least squares DMC adaptation.

23

Adaptive model control usually uses a linear process model or

possibly a pre-set nonlinear form in which the parameters are

periodically updated. A thorough description of adaptive model

control was presented by Seaborg (1983).

Juba* and Hamer* (1986) applied PMBC to the control of a batch

reactor for a highly exothermic polymerization reaction. Using an

energy balance on the reactor and an estimate of the activation

energy of the reaction, they formulated a model of the heat genera

tion from the reaction mixture. However, because of the difficulty

and complexity of deriving an accurate heat transfer model, the

heat removal system was modeled empirically. Their PMBC equations

comprise both phenomenological and empirical model approaches as

they sought to use the simplest model that was sufficiently accu

rate to represent the phenomena. They found that a steady-state

(asymptotic) model of the process was sufficient. With their

controller, they found they were able to effectively contain tem

perature runaway. The authors state that the use of time-varying

nonlinear process models is a key element in the future advance of

batch reactor control.

The growing importance of process-based models in the indus

trial practice of process control is stated by Richalet* and

Congalidis* (1983) as they describe their efforts to synthesize a

combination feedforward and feedback controller of a continuous

recycle polymerization reactor. The feedforward portion was defined

24

by component mass balances and the PI feedback parameters and

input-output pairings were defined from modeled dynamic system

behavior. Simulated tests demonstrated control ability and econom

ic benefits. Such use of models eliminates the on-line, often

expensive, generation of process transfer functions and controller

sjmthesis and tuning.

Parrish and Brosilow (1986) simulate the results of a nonlin

ear inferential controller (NLIC) to control the output concentra

tion of a CSTR. In their strategy, a phenomenological model was

used to infer the non-measurable output concentration from other

measurable data, and control action was then taken on the inferred

value. The model contained an adjustable coefficient which was

adjusted to match measurable data. One interesting aspect of their

investigation was that in spite of intentional model/mismatch, the

simulation indicated that NLIC control was much better than PI

control.

Gardner* (1984) reported the results of an application of PMBC

for a commercial-scale reactor system which consisted of three

fixed-bed reactors with intercoolers. One control difficulty of

this system was the 1.8:1 ratio of dead-time to reactor response

time. Any disturbance that entered the reactor caused its effect

before the results were detected at the exit of the reactor.

Another difficulty was the process gain changed by a factor of ten

over the operating range due to the strongly exothermic nature of

25

the reaction. The third difficulty was the unpredictability of the

catalyst activity decay. Gardner used a process model in a Smith

predictor configuration. In addition, the model was used in a

supervisory steady-state optimizer to choose set points to produce

optimum yield from the system. From on-line data, the process

model was adaptively updated every 15 minutes to reflect the

catalyst activity of each reactor bed. The model was then used for

economic optimization and determination of the process input

setpoints. The control system is privately reported to have been

in operation for over 3 1/2 years with greatly improved control and

efficiency for the unit. In fact, Gardner cites a severe control

action example in which the unit feed was changed from 70% to 60%

to 20% to 40% in a 30-minute period and the process was still under

stable control. Additionally, when output instrumentation fails or

is being calibrated the controller continues to function normally

(except that the parameterization step is bypassed).

Cott*, et al. (1986) applied PMBC to a binary distillation

column simulator with a first order response and a 40-minute time

constant. They compared the control of two PMBC models to PI

control, and internal material balance (1MB) strategy, and DMC.

Their choice for the adjustable model parameters were the number of

stages in the stripping and in the rectification sections of the

column. Since the model parameterization and control calculation

steps were on a 10-minute schedule, they added two normal PI SISO

26

loops to trim higher frequency disturbances and to guarantee no

off-set. PMBC out-performed the other three methods in their

simulations. In fact, since a significant feed rate change was

used to compare methods, DMC actually yielded very poor oscillatory

behavior. Subsequently, Cott*, et al. (1986) applied the PMBC

algorithm to a commercial depropanizer unit and found it "to pre

dict new steady-state operating conditions, reduce response time

and effectively eliminate overshoot." An interesting aspect of

their work was the use of steady state as opposed to dynamic models

for PMBC. Their idea being steady state models are computationally

faster and, therefore preferred for on-line control calculations.

However, the steady-state models do not reflect process dynamics

and, during a transient, the column input-output state does not

match the steady state model pairing. To account for the error,

their algorithm used the measurable column output data to calculate

a fictitious input composition which was decayed in a first-order

response to the actual value. In this way the steady-state model

dynamically compensated itself to match the process. The choice of

a first order decay and the decay rate were made from observed

column behavior.

Lu (1987) reported on the application of a PMBC strategy to

control a continuous casting step in metal processing. He over

views the development of a phenomenological dynamic model of the

distributed parameter, moving boundary layer, metal solidification

27

step of an ingot handling process. Then he describes the results

when the model is used for a supervisory setpoint calculation for

DDC controllers of a production pusher type reheating furnace of a

steel mill: Heating uniformity improved significantly, fuel con

sumption was reduced 9%, and steel losses were reduced 0.18%. He

does not comment on the ability of the model to use process data

feedback to adapt the model, so one presumes that the model is

stationary.

Proceedings of the 1987 American Control Conference report on

the development of phenomenological dynamic models of automobile

engines toward the objective of engine efficiency optimization at

local driving conditions. One explicitly stated that the model

parameters change with operating state and that the model must be

locally parameterized to be valid. An objective explicitly stated

in another, is that the model is "compact enough to run in real

time and can be used as an imbedded model with a control algorithm

or an observer." In both works the nonlinear form of the process

is preserved in the model, but the methods of on-line adaption,

choice of adjustable parameters, method of model inverse

calculation or process optimization are not addressed.

Economou and Morari (1986) present an extension of IMC to

nonlinear systems. Their approach uses a nonlinear process model,

an inverse of the nonlinear model, and a nonlinear filter in the

28

IMC configuration. The calculation of the inverse from the process

model can become quite tedious. Specifically, the model inversion

procedure that they used involved a linearization of the nonlinear

process model.

Rhinehart (1985) and Rhinehart, et al. (1986) have applied

PMBC for setpoint tracking of a pressurized pilot scale fluidized

bed coal gasification reactor. There were several specific problem

sin the four by four coupled nonlinear MIMO reaction system: One

of the controlled variables could neither be measured during the

continuous fed batch run nor inferred directly from measurable

data; and secondly, the process characteristics changed over a

several hour period as leak rates, heat loss rates, feed coal

moisture, and instrument calibration changed. In their trials,

three parameters in a dynamic process model were determined to make

the modeled output match the measurable process output, then with

the locally valid (timewise) inverse of the process model the

reactor was moved to the desired state in one control action.

Lee and Sullivan (1988) have presented a method for implement

ing PMBC that has a proportional type and an integral type action.

Their approach is called Generic Model Control (GMC) and is a

single-step control law that usually requires the solution of a

system of algebraic equations for its implementation. Lee and

Sullivan show that GMC is similar in many ways to single-loop PI

control, feedforward and decoupling control, time horizon matrix

29

controllers, and IMC; but GMC uses the dynamic process model di

rectly without any linearization steps.

Riggs and Rhinehart (1988) compared GMC and the NLIMC strategy

of Economou and Morari (1986) for a wide range of exothermic CSTR

control problems and found that GMC and NLIMC yielded essentially

the same control performance. They pointed out that the GMC

control law has an explicit numerical formulation while NLIMC has

an implicit one; therefore, GMC is considerably easier to implement

and requires less computational effort. Another comparison by

Riggs showed insignificant differences between the performances of

GMC and the (nonlinear predictive model control) NLPMC strategy of

Parrish and Brosilow (1986) applied to the start-up of an open

loop unstable, exothermic CSTR. Therefore, results to date

indicate that there is insignificant difference between performance

of these PMBC methods. In fact, the major difference between the

various PMBC methods is the way in which offset is removed: GMC-

integral term; NLIMC-setpoint bias; and NLPMC adjustment to

disturbances. These results suggest that using the approximate

model is more important than the way it is applied. Therefore,

since GMC is the most advanced form of PMBC, and since other PMBC

methods are actually subsets of GMC, GMC will be used for the

research proposed here.

30

In a review of industrial reactor control Seaborg, (1983)

conclude with the statement,

Problems of recycle, interaction between units, pathological disturbances, as well as start-ups and shut-downs are important industrial problems. Feedforward control is a tool heavily used to solve these disturbance problems... Nonlinear model-based control strategies are also becoming important in industry.

Published work from academia and industry have shown the

potential for PMBC. The work proposed here is aimed at studying

some of the fundamental questions associated with PMBC so that it

can be effectively applied in the future.

3.3 Overview of Process Model-Based Control

PMBC has only a few adjustable parameters so that it is rapid

ly parameterized with statistical confidence and continuously true

to the process. Compared with other model-based controllers, the

model of a PMBC controller expresses the functional form of the

process response over a broad operating range and is excellent for

setpoint tracking and instrument fault detection.

The majority of process control problems can be solved effec

tively using conventional PID controllers, but there is a signifi

cant class of process control applications that are better handled

by more advanced control strategies. These processes are usually

highly nonlinear with a significant degree of coupling between

31

process variables. Currently, in these cases, PID controllers are

usually used. Even though control performance may be marginal, PID

controllers are a familiar technology to the practicing engineer

and operator, are standard practice, are inexpensive, and to the

industrial manager represent a balancer of cost to performance and

reliability. The technology of many of the modern internal model

controllers is not familiar to industry and creates a wary view in

a "can't afford to fail" atmosphere. By contrast, process models

are the college's tool of unit operations, are used for design and

optimization by the practicing engineer, and are familiar to him.

Since the performance to cost ratio of computing power has

increased dramatically in the past five years, it is now economi

cally possible to use a model-based control algorithm. One class

of such advanced control algorithms are based upon a linearized

model of the process, have been applied to some nonlinear process,

and are currently the basis for much academic investigation. But,

the low cost computing power has also made it feasible to use

control strategies based upon phenomenological process models

(i.e., PMBC). Advantages and disadvantages of PMBC compared with

control algorithms based upon linearized models are listed below.

3.3.1 PMBC Advantages

(i) PMBC should perform better on highly nonlinear, coupled

nrocesses. For example, if a controller "knows" that the rate of

32

chemical reaction varies with the exponential of inverse tempera

ture, it should be able to more efficiently track setpoint changes

and handle larger load changes, whereas an algorithm based upon

linear process models can seriously underestimate the effect of

temperature on reaction rate because of the lack of knowledge of

the process nonlinearity. Similar statements can be made with

regard to processes with coupling between control variables as well

as those with large process dead-times that might change with

operating conditions. The following two models visualize the

difference between a linear model and a nonlinear

(phenomenological) model:

Y - ax (i)

Y - exp(bx) (ii)

where a and b are model parameters.

Suppose Equation (ii) is a phenomenological model of a real pro

cess, while Equation (i) is a linearized process model. Due to the

closeness of the phenomenological model to the real process,

modeled output Y is "globally valid" for any input x once model

parameter b is evaluated. Equation (i), however, is only valid for

a particular input x or small x deviation. It is "locally valid."

For a highly noisy process or where state changes occur, output y

33

of Equation (i) is always deviated from the real process. For this

reason, PMBC can be superior to IMC.

(ii) PMBC interfaces well with process optimization schemes

an plant-wide optimization algorithms. PMBC uses a currently valid

process model for control; therefore, the process model can also be

used for optimization purposes. Since the model is not simply

locally valid, it is useful for global optimization of an inte

grated process.

(iii) PMBC should generate acceptance by the practicing

engineer. Because he is familiar with the model and because the

control strategy is simple, it should gain industry acceptance, and

more easily become a part of standard practice.

(iv) The model is quickly and easily parameterized. General

ly there are only two or three adjustable model parameters which

must be evaluated to keep the process-based model true to the

process, and parameterization can be achieved on-line without

process upsets. Most parameters, even rate parameters, can be

obtained by fitting the model to steady state process data. Time

constants and process dead-time can be estimated from flow rates

and process geometry.

By contrast, values of the many parameters of the linear model

must be obtained by process upsets. Commonly, at installation and

at intervals appropriate to account for changes in process charac

teristics, the process must be repeatedly and significantly upset

34

to parameterize the linear controller model with a necessary degree

of statistical confidence. Even so, the model is only locally

valid, and when a nonlinear process makes a significant state

change the model can become invalid. To prevent the need for

process changes, one scheme is to use a recursive least squares

method to update model parameters from process input "noise" and

output trends. Such practice, however, often results in a model of

low statistical confidence, and again, locally valid at best.

(v) Process and instrument fault detection are facilitated.

Because the model is based on the process phenomena the model

parameters, even the adjustable parameters, have phenomenologically

expected values. If model parameters must be too far from their

expected values to make the model reproduce the measured process

data, then a fault is indicated. The model conservation equations

can locate the fault: "there is an inconsistency in the mass

balance" the computer might tell the operator, "please check the

calibration of the flow and concentration meters."

3.3.2 PMBC Disadvantages

(i) Each application of PMBC must be "tailor-fit" to the

particular purpose. The advanced control strategies which are

based upon linear process models are generic in their approach and,

as a result, can be applied in nearly the same manner to each

35

process. By contrast, a PMBC controller requires the engineer to

obtain and maintain a model specific to the process. Although

apparently a major disadvantage, such a model is probably available

from the process design stage.

(ii) Little experience with PMBC is available to guide the

control engineer. Although the idea of PMBC is not new, develop

ments in computer hardware have only recently made it feasible. As

a result, there is little experience to guide the practitioner,

e.g. ,

How much mismatch between the model and the process can one

tolerate?

How does one choose an approximate model and what parameters

should one choose to be adjustable?

How often should one re-parameterize the model?

How does one tune (if that be the appropriate term) the con

troller?

How is the controller configured to handle constraints?

How nonlinear does the process have to be to warrant use of

PMBC?

How to account for dead-time (transport delay)?

How to incorporate anti-windup features?

With regard to the first disadvantage, it becomes a question

as to whether PMBC provides improvements that outweigh the effort

of implementation. (This is also true of any advanced control

\

36

strategy.) Considering that the PMBC model is a better description

of the nonlinear, coupled, and dead-time characteristics of the

process, it would seem that PMBC would offer significant potential

for improved control performance over locally valid models. Since

a large number of chemical processes are highly nonlinear, have

large throughputs, are subject to significant drifts, and are

subject to daily state changes; PMBC should have significant poten

tial for improving process efficiency and controllability. Addi

tionally, the phenomenological model of a PMBC controller can be

used in supervisory on-line calculations to improve the accuracy of

both process optimization and data reconciliation efforts.

Part of the overall research program at Texas Tech University

is the investigation of such implementation concerns as were men

tioned above. Once these implementation concerns have been ad

dressed, and satisfactory results have been obtained, such control

lers may gain user acceptance.

3.4 Previous PMBC Wastewater pH Control Work

Rhinehart and Choi (1988) investigated an equilibrium reaction

model that adequately described the actual wastewater neutraliza

tion reactions, and then simplified this model for use as the

phenomenological model of a PMBC control strategy. This reduced

model retained the pertinent phenomena of the process, yet was

37

simple enough to facilitate the mathematical manipulations required

for development of a control algorithm.

Figure 3.1 illustrates the chemistry of a hypothetical acidic

wastewater. The letters x and 8 represent the amount of weak

acid and water dissociated, respectively. The weak acid concentra

tion, therefore becomes fi-x. The strong acid, the salt and

strong base each fully dissociate so that their initial concentra

tions stoichiometrically describe ion concentrations. From the

definition of chemical reaction equilibrium:

K - Exnr AG/RT^ - ["" H 'l . (a+X+g)(x+7) /o 4 .) K^ - Exp(-AG/RT) - ^ j - ^^_^^ . (3.4.1)

Regardless of the source of each species, molecules and ions of

that species are indistinguishable, and the individual total amount

of [H"*"] , [ W ] , and [HW] must be used in the K^ expression. The

ionization constant of water, K , is available from tabulated data

as a function of temperature (refer to Appendix A):

K - 10'^^*^^ - [H'^][0H'] - (a+x+5)(5+c) (3.4.2) w

where f(T) at T - 25°C is the familiar 14 giving neutral water a pH

of 7. From the definition of pH:

pH - -Log^QlH"^] - - Log^Q(a+x+5) . (3.4.3)

38

Initial Concentration Reaction

HS • aK^ + aS

0

7

S

(fi-x) HW < > xH+ + xW

MW < > 7M+ + 7W

H2O < > 5H+ + 50H-

BOH •7 B + OH

where HS is a strong acid HW is a weak acid MW is a common ion salt BOH is a strong base (titrant)

Figure 3.1: The Chemistry of a Hypothetical Acidic Wastewater

39

A generalized function is then developed which relates the equilib

rium pH phenomena to the species concentrations and solution tem

perature. From Equation (3.4.3):

a+x-l-5 - 10"P" (3.4.4)

Inserting Equation (3.4.4) into Equation (3.4.2), gives

6 - K * loP" - e . (3.4.5)

w

and from Equation (3.4.4),

X - 10"P" - a - 6 - 10"P" - a - K * 10^" + e . (3.4.6)

w

Substituting Equations (3.4.4), (3.4.5), and (3.4.6) into Equation

(3.4.1), one gets (loP" )(10"P" - a - K /10"P" + € + 5)

Exp(-AG/RT) - — — (3.4.7)

a + ;9 V^° • ' - °

To use Equation (3.4.7) for a control model one would simply find

the value of e, the base concentration, that satisfies Equation

(3.4.7) when pH-7. However, the controller cannot know either a,

fi, -y or AG (measures of wastewater composition) and therefore

cannot use Equation (3.4.7) for control.

In the work done by Rhinehart and Choi (1988), however, the

process was represented by a single weak acid and base reaction

40

system, and such a reduced phenomenological model is used to simu

late the real system for control purposes. The reduced model

suggested is in Figure 3.2. Paralleling the development of Equa

tion (3.4.7), for the single acid system

•" IjW I _ 1X4 4U1 " r\

K^ - EKp(..C,/RX - i a ^ - ^ ^ ^

where AGf is the Gibbs free energy of dissociation of the ficti

tious weak acid. Additionally,

k - 10'^^^^ - [H'*"][0H'] - (x+5)(5+e) . (3.4.9) w

pH - - Log^Q[H"^] - - Log^Q(x+5) . (3.4.10)

Combining Equations (3.4.8), (3.4.9) and (3.4.10), gives the

equilibrium behavior of the simplified neutralization process

(10"P")(10"P" -K^*10P" + 0 f(A., AG )=Exp(-AG /RT) - — — - 0 .

^ ^ ^ (A^ - lO'P - K *10"P-£) (3.4.11) f w

Equation (3.4.11) has two unknowns (model parameters): Af, the

concentration of the single fictitious weak acid and AGf, the

Gibbs free energy of dissociation of the fictitious acid. Each

were determined in the Rhinehart/Choi work from the measurable

input pH (at no base addition) and output pH (at the base concen

tration in the outlet) values and the base and acid flow rates.

41

Initial Concentration

Af

8

Reaction

(Af-x) HW ^ > xH+ -H xW-

H2O < > 5H+ + 50H"

BOH -> €B+ + €0H-

where Af is the fictitious weak acid concentration.

Figure 3.2: The Chemistry of a Single Weak Acid and Strong Base Reaction System

42

While their controller worked, sometimes either dramatic

changes in the process composition or measurement noise would lead

to invalid parameter identification. Consequently, base flow rate

setpoints were occasionally considerably different from those

actually required.

The rigorous model developed to describe the actual wastewater

neutralization reactions, however, as stated earlier, was shown to

be an adequate simulator of the experimental system. This same

type reaction model was also investigated by McAvoy, et al. (1972)

for acetic acid titrated with a base in a CSTR. They, too, showed

that a model developed with such equilibrium thermodynamic basics

adequately simulated the experimental system results.

This controller research investigation will therefore utilize

this same rigorous reaction model. This work will also follow the

same development of the reduced model as was done by Rhinehart and

Choi, except that the base flow stream will be split into two

portions and sequentially inject them into the acidic wastewater

line to achieve neutralization. This technique will yield three

data pairs (inlet-pH, flow rate, and temperature; after first base

injection-pH and flow rate (from mass balances); and outlet-pH,

flow rate (from mass balances), and temperature) vs the two data

pairs used by the controller developed by Rhinehart and Choi. With

three data pairs and only two unknowns, there is enough data to

43

allow the use of a least squares optimization procedure for calcu

lation of the unknown parameters.

^

CHAPTER 4

PROCESS SIMULATOR

4.1 Introduction to the Simulator Development

This section develops a rigorous process simulator for the

investigation of a nonlinear PMBC pH neutralization controller

strategy. This simulator must yield results as close to the actual

process as possible so that relatively accurate process responses

to many different system compositions and upsets may be obtained.

With a reliable process simulator, many different aspects of the

proposed nonlinear PMBC pH neutralization controller can be studied

without the expense and time associated with running a wide variety

of tests on a pilot scale operation.

The process simulator will use the reaction mechanism and

equilibrium equations developed by Rhinehart and Choi (1988), as

shown in Chapter III. Many of the other phenomena of a wastewater

pH neutralization system will be modeled; such as flow dynamics,

instrumentation noise and lags, nonstationary composition effects,

and others.

4.2 Simulating the Reaction Mechanics

Usually the reaction rates for acid-base reactions between

dissolved components are extremely high (Gustafsson and Waller,

1983) and such rates are much faster than the mixing rate. For

44

45

such fast acid-base reactions, equilibrium conditions are ap

proached sufficiently to justify these conditions for use as addi

tional algebraic state equations. Figure 3.1 illustrates the

chemistry of the hypothetical acidic wastewater system, and since,

as stated, this system is assumed to be near equilibrium, we can

use Equations (3.4.1) and (3.4.2) to describe the relevant reaction

phenomena of the system. If the influent compositions are fixed

then a and 7 become known values. Equations (3.4.1) and

(3.4.2), thus, have only two unknowns, x and 8. We can then

solve for the unknowns x and 8 by iteratively applying the multi-

variable Newton's Method to the system functions. Equations (4.2.1)

and (4.2.2), which are developed from Equations(3.4.1) and (3.4.2)

refer to Appendix B).

fl - Ka(;3-x) - (a+x-H5)(x+7) (4.2.1)

and f2 - Kv - (a+x+S)(S+€) (4.2.2)

With the variables x and 8 now known, we can determine the pH of

the system by applying Equation (3.4.3).

4.3 Simulating Instrumentation Noise

We must also model the fluid dynamics of mixing the reaction

invariant species due to adding titrant into the wastewater stream.

46

This phenomena will be modeled as a small ideal CSTR with a volume

of 0.02 liters and the wastewater and titrant streams as inputs.

In general the mass balance is:

V ^ir^ - F. * [x]. -F ^ * [x] ^ (4.3.3) dt in "• •• m out *• ' out

where x represents any reaction invariant species. Equation

(4.3.3) is used to simulate the composition transients of weak

acid, strong acid, coimnon ion salt, and the reagent base added for

neutralization.

Additionally, a similar energy balance is used to simulate

process temperature. If the assumptions of constant density,

constant heat capacity, and negligible heat of reaction are made

(all due to the dilute nature of the wastewater system), the gen

eral energy balance reduces to:

(AF*T + BF*T )

V^^ - . T * fAF + BF) C4 3 4") dt (AF + BF) 3 ^ ^ k"*-^-"*)

where AF represents the wastewater flow rate in, BF represents the

strong base (titrant) flow rate in, and T^, T2, and T3 represent

the temperatures of the wastewater stream in, the titrant stream

in, and the resulting outlet stream, respectively. Equation

(4.3.4) is used to simulate the temperature behavior of the neu

tralization system.

47

The rigorous process simulator will model the actual waste

water neutralization system shown in Figure 4.1. This simulator

will first calculate the pH of the influent by the method stated

above using influent compositions and temperature supplied by the

programmer. The simulator will then model the injection of a

portion of the strong base (titrant) and use the fluid dynamic

modeling equations to determine the resulting compositions and

temperature. Next, the reaction equilibrium equations are reap

plied to determine the pH of the waste stream after the first base

injection. The remainder of the strong base is then injected, and

again the mixing effects are calculated for the resulting stream

after the final base injection. The reaction equilibrium equations

are reapplied to determine the effluent pH of the wastewater

stream.

The basics of the neutralization process have, therefore, been

modeled and now other factors that come into play in an actual

system can be added to the simulator. These factors can be very

important in the investigation of the controller and must be mod

eled if we are to obtain realistic results for comparison of con

troller performance.

4.4 Simulating Instrumentation Noise

Measurement noise is common to all process instruments. This

simulator incorporates the measurement noise as a Gaussian

48

3

<u

<u

3

a (U 4J CO >

CO

C

o

cd

(0 V4 4J 3 <U

z lU 4-)

<u 4J (0 CO 3

cd 3 4J o <

X H

0)

u 3 00

49

distributed random variable with a mean of zero which is added to

all measured process variables (see Appendix C). A standard

deviation of 0.12 pH units is used and results in the simulated pH

meters having a 95% confidence interval of 0.48 pH units. The

simulated flow meter standard deviation is 0.0005 liters/minute

which results in a 95% confidence interval of 0.002 liters/minute.

Measurement noise was not added to the simulated temperature probes

because the noise for these instruments is small and the

temperature effect on the process pH is small, therefore simulated

temperature noise would be inconsequential.

4.5 Simulating Instrumentation Response Time Lag

Another phenomenon common to all process instruments is re

sponse time lag (the time the instrument reading takes to travel

from its beginning value to the new value) . The turbine flow

meters simulated for this system have a first order response with a

time constant of approximately 2.5 seconds (see Appendix E). The

pH meters simulated, on the other hand, have a first order time

constant of approximately 7 seconds. Another factor that must be

modeled is the response of the final control element to set point

changes. Due to valve dynamics, the base flow rate follows the

base flow rate set point in a first order lag with a 2.5-second

time constant.

50

4.6 Simulating Nonstationary Wastewater Disturbances

Earlier it was stated that the programmer supplies the initial

influent composition, flow rate, and temperature. However, the

nonstationary nature of these process variables must also be simu

lated. The weak acid, strong acid, common ion salt concentrations

drift with time, as does the Gibbs energy of dissociation of the

weak acid. The influent wastewater flow rate and temperature also

drift with time. This nonstationary condition is simulated by

applying a second order auto-regressive moving average disturbance

to each variable (refer to Appendix D). This results in a nonsta

tionary process nature in which each input independently drifts

about an average value, and therefore more realistically simulates

the real process.

CHAPTER 5

CONTROLLER MODEL AND STRATEGY

If the rigorous process simulator were used as the control

model, one would use Equation (3.4.7) and simply find the value

of c, the base concentration, that satisfies the equation when

pH-7. However, the controller cannot know either a, fi, l or AG

(measures of the wastewater composition) and cannot exactly know T

(a measured variable) and therefore cannot use Equation (3.4.7) for

control. Rhinehart and Choi (1988), however found that, for con

trol purposes, the process can be represented by the single weak

acid and base reaction system illustrated in Figure 3.2. Equation

(3.4.11) will, therefore, be used as the controller model equation,

and as stated, a dual base injection scheme will be used in the

control strategy.

5.1 Model Parameter Identification

Equation (3.4.11) has two unknowns (model parameters): Af,

the concentration of the single fictitious weak acid and AGf, the

Gibbs free energy of dissociation of the fictitious weak acid. The

measurable information from the dual base injection scheme has

three distinct sets of data: (1) pH, wastewater flow rate, and

temperature at a point before the first base injection point, (2)

pH and base flow rate at a point after the first.base injection

51

52

point, and (3) pH, and base flow rate at a point after the second

base injection point. From simple energy and mass balances, as

developed in Section 4.3 for use in the process simulator, one can

obtain three T/pH/base concentration data sets.

At this point, however, the possibility of process to model

mismatch must be acknowledged, for the actual volumes of the mixing

vessels and the exact concentration of the strong base (titrant)

may be different from those used in the controller model. The

effects of these process to model mismatches must therefore be one

of the areas investigated in this work. Another potential problem

could exist if the noise on the wastewater flow rate and pH values

caused unnecessary and unwanted fluctuations in the base flow rate

setpoint. Therefore these values are filtered to give a more

stable base flow rate setpoint response. We used a first order

filter with a two second time constant (see Appendix E for filter

equations). Another potential problem that must be acknowledged at

this point is the possibility of pH probes or flow rate meter

calibration error.

However, there is, with respect to the above limitations,

enough information to obtain the two parameters, Af, AGf, through

a least squares minimization of the filtered pH values and those

calculated using Equation (3.4.11).

The next question to be answered is when to calculate the

model paramp-t'=»rs. There are many different schemes to determine

53

when is the appropriate time for model parameterization. One

scheme is to parameterize the model when the effluent pH deviates

more than one pH unit from the 7.0 setpoint. Another is to para

meterize the model when a steady state condition is acknowledged,

and still another is to parameterize the model only when a steady

state condition has been lost and then a "new" steady-state condi

tion is reached.

Model parameterization in this work will be performed each

time that the controller acknowledges that the process is at steady

state. The controller identifies a steady state condition when the

filtered standard deviation of the output pH is below a threshold

value of 1.5 times the true steady state value (0.18 pH units in

this case). The filtered standard deviation is the square root of

the filtered variance which is calculated from the filtered efflu

ent pH at each sampling interval. We used a first order filter

with a one second time constant (refer to Appendix H for the equa

tions used to calculate the filtered standard deviation).

5.2 Control Strategy

The GMC (Lee and Sullivan, 1988) control law based on a steady

state model is used in this study. Applied to the pH controller,

this law first calculates a temporary steady state pH target for

the next control interval:

54

pH - pH ^ •»- Kl * e + K2 * r e dt (5.2.1) S3 * out ''

where pHgg is the temporary steady state target, pHout ^^ ^ ® P^ °^

the effluent from the neutralization process, e is the error term

(7.0 - PHQ^C)» Kl is the tuning constant that relates to the rate

of return from pHout ^° pH-7, K2 is the tuning constant that

weights the process/model mismatch correction between parameteriza-

tions, and J e dt is the integral since the last parameteriza

tion. This pHgs value is then substituted into Equation (3.4.11)

as the target pH value along with the current values of the model

parameters and filtered flow rates. Upon rearrangement, the ratio

of the base flow rate set point to the acid flow rate can be deter

mined. This ratio is then used to calculate the base flow rate

setpoint values until the next time the ratio is changed by the

controller (controller period is 1 second). This ratio action

allows the controller to respond quickly (with a filtered lag and

at most a 1 second control period delay) to acid wastewater flow

rate changes.

A difficulty to overcome at this point is the sensitivity of

the base flow rate setpoint to small changes to pHgg when pHgg

target is less than 4 or greater than 11 (the extremes of the

titration curve). Such a value of pHgg can be calculated after a

severe upset where the effluent pH becomes either relatively low or

high. We chose to keep controller responsiveness high by keeping

55

Kl and K2 near the controller stability limits. Then we used very

mild "gain scheduling" to offset this sensitivity. This is accom

plished in the following manner:

Kl - 1.25 - (0.02 * I e I) (5.2.2)

K2 - 0 014 - (0.002 * I e I) (5.2.3)

where lei is the absolute value of the error term, 1.25 is the

value of Kl when pHout"^» 0.014 is the value of K2 when pHout"^»

and 0.02 and 0.002 are the slopes of the linear decrease in Kl and

K2, respectively, as the e term increases.

The controller will continue calculating ratios and setpoints

using old parameter values and Equations (3.4.11), (5.2.1), and

(5.2.3) until a steady state condition is determined. When this

condition is determined, the integral term in Equation (5.2.1) is

reset to zero, and using the current pH, flow rate, and temperature

data, new values for the model parameters are determined by the

least squares minimization procedure mentioned above (refer to

Appendix F for the above least squares minimization procedure).

Figure 5.1 illustrates the process and nonlinear PMBC control

ler. At the bottom of the figure, the wastewater flows from left

to right, is monitored for pH, temperature, and flow rate, enters a

small mixer where a fraction of the desired base is injected (see

56

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57

Appendix G for the equations used in determining the fraction of

base to be injected), exits, is monitored for pH, enters another

small mixer where the remaining fraction of the base is injected,

exits, is monitored for pH and temperature, and then is discharged.

From the reservoir at the top of the figure, the concentrated base

flows through a flow monitor, a flow valve, a flow splitter, and

into the mixers. The mixer residence times are approximately 1

second.

The supervisory nonlinear PMBC controller and the primary base

flow controller are each grouped within dashed lines within the

digital controller of Figure 5.1. The primary base flow controller

is conventional "velocity mode" digital SISO PI controller, tuned

for an over-damped, closed-loop response of about 3 seconds and

operating 5 times per second.

5.3 Options and Variations

There are advantages and disadvantages to the options and

variations presented in the preceding sections. For instance., the

choice of parameterization "trigger" (when to reparameterize the

model parameters) can have a strong influence on controller per

formance. If the controller parameterizes the model when the

effluent pH deviates more than one pH unit from the 7.0 setpoint,

there is a possibility of invalid parameter identification upon a

58

severe step upset due to the time lag on measured variables

(because of mixing effects and instrumentation response time lag).

Since the instruments do not all have the same response time lag

and because of transport delays, the measured variables of pH and

flow rate could be out of "sync" and therefore the parameterization

would be based on inappropriate information. Other parameteriza

tion trigger scheme advantages and disadvantages will be discussed

in Chapter 6.

The form of the objective function used in the multivariable

Newton's search for parameter identification is also a very impor

tant aspect of this investigation. Many different forms of the

objective function can be obtained by simple algebraic manipula

tion, such as multiplication (or division) by a variable (or group

of variables) from the original objective function (Equation

(3.4.11)) and by simply inverting the entire objective function.

The form of the function used in this work is unique and stable,

characteristics that many other forms of the function do not pos

sess. Another advantage can be gained by assuming a constant

temperature through the neutralization process. This assumption

does not significantly affect the model's process characteristics

and will simply be absorbed into the model as another process-to-

model mismatch. This assumption does, however, greatly affect the

59

ease with which this form of the objective function can be imple

mented by allowing the simplification of solving for one of the

model parameters, AGf, in terms of the other model parameter, Af.

This reduces the model parameter search from a two equation/two

unknown system to a one equation/one unknown system, thereby great

ly reducing the computational effort required to parameterize the

model.

The choice of the fraction (portion of the base to be injected

into the first mixing vessel divided by the total amount of base to

be injected for neutralization) also has some importance. This

fraction must be large enough to allow for valve rangeability

(especially in the second fraction). It must be large enough so

that the pH value of the waste stream after the first base injec

tion point needs to be near the neutral portion of the titration

curve. Yet, this pH value must also be far enough away to be able

to obtain some indication of the location of the titration neutral

ization region in the event that too much base is added.

Another possible variation would be to add weighting factors

to the least squares minimization procedure for parameter identifi

cation. This variation could lead to greater accuracy in the

effluent pH value, due to the possibility of better modeling of the

titration curve in the setpoint region (pH-7.0 for neutralization,

or pH of less than or greater than 7.0 for other control applica

tions) .

CHAPTER 6

RESULTS AND DISCUSSION

This chapter will discuss the simulated responses of the

acidic wastewater neutralization process (1) in regulatory mode (no

process upsets other than normal nonstationary wastewater distur

bances) , (2) for a ramp upset in wastewater composition, and (3)

for a step upset in wastewater composition. Additionally, (4) the

effects of adding a large blending tank at the effluent of the

neutralization process will also be examined. Other aspects stud

ied in this controller investigation are the effects of: (5)

process to model mismatch and calibration errors; and in less

detail (6) varying the fraction of the base to be injected at the

first mixing vessel, (7) incorporating weighting factors into the

least squares minimization procedure for model parameter identifi

cation, and (8) effluent pH setpoints other than 7.

6.1 Regulatory Control

Figure 6.1 illustrates the simulated controlled system re

sponse in regulatory mode. In this case, the controller must

respond to normal noise, lags, and auto-regressive drifts that are

inherent in the process. The initial wastewater composition is

60

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detailed in Table 6.1, and the process and model titration curves

are illustrated in Figure 6.2. The nonstationary nature of the

process compositions are illustrated in Figures 6.3, 6.4, 6.5, and

6.6. The controller, as would be expected of any effective con

troller in regulatory mode, shows adequate control response over

the simulated 1800 seconds.

The controller (and its tuning parameters and parameterization

trigger strategy) used in the case will hereafter be called the

base case controller. The gain factors, Kl and K2 of Equation

(5.2.1), will be the same as shown in Equations (5.2.2) and

(5.2.3). The amount of the base to be injected into the first

mixing vessel will be a set value of 60% of the total amount of

base required for neutralization. The parameterization trigger

strategy (scheme for determination of the appropriate time to

reparameterize the model) will be, as discussed in section 5.1, to

parameterize the model each time a steady state condition is iden

tified and to zero the integral term of the control law at that

time. The threshold value of the filtered standard deviation of

the outlet pH required for identification of this steady state will

be 0.18, as stated in section 5.1. Also, the pH points will be

equally weighted for the least squares minimization procedure used

in obtaining the model parameters.

63

Table 6.1: Initial Wastewater Composition for Figure 6.1

Weak Acid Concentration 0.5 gmoles/liter

Strong Acid Concentration 0.0055 gmoles/liter

Salt Concentration 0.005 gmoles/liter

Weak Acid G of Dissociation 5000 cal/gmole

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6.2 Step Upset

Figure 6.7 illxistrates the controller response to step changes

in composition as well as normal noise, lags, and auto-regressive

drifts. At time equals 400 seconds the weak acid, strong acid, and

common ion salt, and the Gibbs energy of dissociation of the weak

acid make an instantaneous change (see Table 6.2). This change

results in the process changing from a system consisting of mainly

a highly buffered weak acid to one consisting of a weak acid, a

strong acid and very little buffering. Initially the pH rises and,

in response, the -controller drops the base flow rate, but the model

parameters are not valid and the pH begins to level at 12.3 pH

units with a steady state offset of 5.3 pH units. At that time

steady state is recognized. After three parameterizations the

model is corrected and control is regained. Events at time equals

1200 seconds are similar; however, this change results in the

system changing from one consisting of a weak acid, a strong acid,

and very little buffering to one consisting of mainly a buffered

weak acid (refer to Table 6.2). Initially the pH drops and, in

response, the controller raises the base flow rate, but the model

parameters are not valid and again the pH begins to level at 4.5 pH

units with a steady state offset of 2.5 pH units. At that time

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71

Table 6.2: Wastewater Compositions for Figure 6.7

Simulated Wastewater Upset Changes at 400 and 1200 Seconds

Component or Characteristic 400 1200

Weak Acid Concentration 0.050 03 0.06 gmoles/liter

Strong Acid Concentration 0.05 0.03 0.004 gmoles/licar

Salt Concentration 0.005 0.00005 0.002 gmoles/liter

Weak Acid G Dissociation 5000 7000 6000 cal/gmole

72

steady state is recognized. After four parameterizations the model

is corrected and control is regained.

It should be noted that the controller seems to respond slower

to the second upset. This is caused by the lack of information

available to the controller regarding the neutrality region of the

titration curve immediately after the upset, when all three pH

points (influent, midpoint, and effluent) lie on the lower region

of the titration curve. This causes slightly invalid model para

meter identification. The process then moves to a new steady state

of 4.9 pH units, where the model parameters again are parameterized

on somewhat limited information. This is repeated again at the

steady state condition identified at 5.7 pH units, yet now the

controller has better information (closer to the neutrality region)

and can yield a better estimate for model parameters.

This parameterization sequence should be compared to that of

the first upset in which the effluent pH immediately after the

upset lies on the upper region of the titration curve and the

midpoint pH lies near the neutral region, thereby yielding excel

lent information to the controller for model parameter identifica

tion. The parameters are therefore very close to those actually

required for control. The resulting base flow rate setpoint calcu

lated by the controller is also very close to that required for

control, and the response therefore is very rapid.

Note that before, between, and after the step changes the

controller must continuously work to account for drifts. Although

pH is held at 7 during the 600 to 1200 second interval, the base

flow rate had to increase due to composition changes.

Titration curves of both the simulated process and the

controller's fictitious acid at times of 600 to 1400 seconds are

shown on Figure 6.7. Figxires 6.8 and 6.9 show the titration curves

of both the simulated process and the controller's fictitioxis acid

at times corresponding to 100 and 600 seconds, respectively . The

fictitious titration curves at 100 and 1400 seconds, Figures 6.8

and 6.9, show a close approximation to the actual titration curve

of the system over the entire concentration range^ especially the

desired neutrality region aroxind pH equal to 7 pH units. This is

to be expected because the simulator is essentially a weak acid

system and can be closely approximated by the single acid model of

the controller. The fictitious titration curve at time equals 600

seconds shows a close approximation of the process titration curve

in the desired neutrality region yet shows considerable mismatch in

the lower pH regions. This, too, is to be expected because the

single acid model of the controller is trying to mimic a system

that has both strong and weak acid characteristics. This model

mismatch, however, does not effect the base setpoint calculations

of the controller because of the close fit of the model titration

curve to the actual one in the desired neutrality region.

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While not shown, conventional and advanced PID controllers are

completely ineffective. While these PID controllers can be tuned

for one particular system of compositions and acid strengths, when

these factors change, even slightly (which is a common occurrence

for an actual wastewater system), the PID controllers typically

oscillate about the pH setpoint or have a steady state offset.

These conditions result because the changing conditions of the

wastewater system cause the tuning parameters of the PID control

lers to become invalid.

6.3 Ramp Upset

Ramp disturbances are as likely as step changes, and Figure

6.10 illustrates such an event superimposed on the auto-regressive

drifts. From time 30 to 1500 seconds the weak acid concentration

decreases 22.5%, the strong acid concentration increases 82X, the

common ion salt decreases 96.5%, and the Gibbs energy of dissocia

tion of the weak acid increases 40% (see Table 6.3). All these

changes occur gradually over a 25 minute period, and are intention

ally independent. During the upset the simulated influent pH

decreases by approximately 0.5 pH units. While the wastewater flow

and temperature remain essentially constant. One might expect that

this requires more base; however, the controller must reduce the

base flow rate 17% to hold the effluent pH within the control

77

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Table 6.3: Wastewater Compositions for Figure 6.10

Simulated Wastewater Upset Change Period

30 to 1500 Seconds

Component or Characteristic 300 1500

Weak Acid Concentration 0.053 0.01 gmoles/liter

Strong Acid Concentration 0.005 0.01 gmoles/liter

Salt Concentration 0.0075 0.0005 gmoles/liter

Weak Acid G Dissociation 5000 7000 cal/gmole

79

specifications. This counter intuitive feedforward nature is the

result of the process shift from a weak acid to a strong acid which

the controller parameterization fully adequately characterized.

Figure 6.10 shows the titration curves of both the simulated

process and the controller's fictitious acid at times corresponding

to 30 and 1500 seconds on Figure 6.11. And again, similar conclu

sions to those for the step upset in section 6.2 can be made about

the approximations of the fictitious titration curve to the actual

process.

6.4 Large Finishing Blender

A CSTR blender added to the end of the effluent line of the

base case controller system will act as a first order lag on compo

sition changes, consequently damping out temporal changes and

reducing both the rate and amplitude of upsets. The effects of

adding a large blender at the end of the pH neutralization effluent

line can be seen in Figure 6.12. The finishing blender used in

this investigation has a residence time of approximately 100 sec

onds. The wastewater compositions and upsets used in this compari

son were the same as those used for the step upset controller

response examined in section 6.2. Figure 6.12 shows a damping of

the pH deviations from the 7.0 pH setpoint; especially for the

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crucial regions after the process upsets, where control specifica

tions have been violated. It can, therefore, be noted that the

effluent blending tank reduced both the total amount of time that

the effluent pH was out of specification and the magnitude of the

excursion.

A similar type response would be expected for all upset types,

since the CSTR acts only as a mechanism for blending a large amount

of "in-spec" effluent with a small amount of "out-of-spec" efflu

ent. It should be noted that a blender with a smaller residence

time would dampen the upset less, while one with a larger residence

time would dampen the upset more. Therefore, this one case is

enough to demonstrate the effect of a finishing blender.

6.5 Model Mismatch and Calibration Errors

The control model contains the significant simplification of

considering the wastewater as containing a single weak acid when it

is actually a multicomponent acid system. In this section, the

base case controller used in the previous sections will have the

added responsibility of correcting for additional intentional

modeling and calibration errors. The errors to be added in this

section are those which are the maximum expected in a normal model

ing effort because no process measurement can be exactly known.

83

The additional modeling errors were the values of the mixer

volumes and the concentration of the reagent base used as the

neutralizing agent. The controller model uses a mixer volume of

0.02 liters, while for this case the actxial mixer volume was set at

0.0221 liters: a 10.5Z error. The controller model also uses a

reagent base concentration of 0.1 Molar, while for this case the

actual concentration was set at 0.1093 Molar. And again, this 9.3X

deviation is a typically expected maximum variation.

The controller, in this case, also had to account for calibra

tion errors in effluent pH and influent wastewater flow rate. The

effluent pH value read by the controller had a continuous bias of

0.67 pH units added (or subtracted, depending on the direction of

the intended bias) to the measurable value. The wastewater flow

rate also had a continuous bias of 0.095 liters/minute added (or

subtracted) to the measurable value. These calibration errors are

relatively slightly higher than those expected, yet are used to

illustrate the controller efficacy in handling such errors.

For the base case controller acting on the dual step change

the average lAE (Integral of the Absolute Error, a controller

performance criteria) was 586.1 pH*seconds (for the time interval

of 0 through 1700 seconds) for the controller response with no

calibration error and no modeling errors in mixer vol\ime or reagent

84

base concentration. The average lAE (for the base case controller

acting over the same time interval) was 609.6 pH*seconds for the

controller response with the above mentioned modeling mismatches

and calibration errors (acting in both the positive and negative

direction). Table 6.4 illustrates the actual lAE's obtained-for

progressively adding the modeling mismatches and calibration er

rors . This yields only a 4Z decrease in controller performance

(increase in overall average lAE), and there are no visual differ

ences between the controller effluent pH responses with or without

the errors, therefore curves will not be shown.

It should also be noted that the errors are sometimes slightly

beneficial to control and then sometimes detrimental; yet overall,

reasonable calibration and modeling errors are inconsequential.

6.6 Varied Set Ratios for Base Split

Another effect that had to be examined before further investi

gation, was what effect the different ratio settings (percent of

total base required to be injected in the first mixer divided by

the percent of total base required to be injected into the second

mixer) would have on controller performance. The desired ratio had

to be small enough to yield a midpoint pH far enough away from the

pH neutrality region to yield some information concerning the

location of the middle section of the titration curve, yet also has

85

Table 6.4: IA£ Results for Modeling Errors Introduced Into the Step Change Conditions of Section 6.2

Base Casa * "+•• Calibration "-" Calibration With Errors Errors

No Additional x x errors 666.3,625.3,563.2 587.8.594.8,479.4

Base Concentration Mismatch

Mixer Volume Mismatch

655.1,620.9,572.9

675.1,634.6,573.3

597.7,590.4,475.7

606.2,587.3,480.3

Calibration Error in pH of the Ef f Ixient 6 3 1 . 0 , 577 . 9 , 50^ . 1 721. 9 , 660 . 5 , 589 . 2

Calibration Error in Base Flow meter 585.3,570.7,520.1 708.5,669.1,603.8

*Each error is progressively included into the simulation.

xSimulation runs made with 3 different random number seeds

86

to be large enough to allow for valve rangeability (especially in

the second injection).

Many trials (refer to Table 6.5) for the base case controller

acting on the ramp upset presented in section 6.3 were made at

varied values of base flow ratio and the operating region for this

ratio was set between 60/40 and 80/20. This region was obtained by

a subjective balance of lowest response lAE's under control condi

tions where the nonstationary process compositions varied strongly

(Seed 137) and where they varied weakly (Seeds 747, 274, & 3117).

Another factor considered when bounding this region at 60/40

and 80/20 was consideration of actual plant implementation. The

valve required for the secondary base injection should be smaller

than the first for more precision in the flow, yet must also be

large enough to accommodate higher possible flow rates required for

some wastewater systems.

6.7 Variable Base Split Ratio

In the base case controller, the base flow rate was split into

two streams and sequentially injected into the pH neutralization

line. The ratio used in the base case was 60/40, or in other words

60% of the desired base flow rate was injected into the first mixer

and the remaining 402 was injected into the second mixer. This

ratio was chosen in order to try and obtain some meaningful measure

87

Table 6.5: IA£ Results for Various Base Split Ratios

Random Ntomber Seed

137

747

274

3117

60/40

827.9

556.7

578.2

560.2

Base 70/30

822.0

565.9

591.8

564.8

Split Ratio 75/25 80/20

728.6

563.2

598.4

573.5

762.6

584.8

617.8

587.4

85/15

758.0

604.8

637.9

620.0

88

of where the middle section (between influent pH and effluent pH)

of the titration ctirve was located over a wide range of effluent

pH's.

For this case, however, the controller will be allowed to

calculate the base flow ratio to be used for control of the ramp

upset process of section 6.3. This will be accomplished (refer to

Appendix G) by having the controller calculate the amount of base

required to bring the intermediate pH of the fictitious acid system

to a target pH of 5.0. This calculated base requirement is then

divided by the base requirement calculated for pH neutralization to

yield the desired base flow ratio.

However, because of instrumentation noise and other factors,

this calculated base flow ratio will fluctuate rapidly around the

desired ratio. This calculated ratio will therefore, be filtered

using a first order filter in order to make the transition from one

ratio setting to another. The factor or the first order filter

used in this case was 0.99.

Therefore, based on the results of section 6.6, the ratio

calculated by the controller was bounded between 60/40 and 80/20.

Figure 6.13 illustrates the ratio calculated by the controller for

the same compositions and upset used in the ramp upset case of

section 6.3. It should be noted, however, that the effluent pH

89

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90

responses, for the controller acting under a set ratio of 60/40 and

for the controller calculating the ratio to be used, had no visual

differences (and only slight variations in LAW; 594.8 for the base

case controller versus 612.3 for this variable ratio controller)

and will therefore not be shown. It should also be noted than even

though this effect was not studied extensively, based on the incon

sequential effect foiind in this one investigation and from the

results of section 6.6, it would be recommended to use a set ratio

of 75/25. This would reduce the number of computations and com

plexity of the controller as well as be simpler to implement in a

real system.

6.8 Weighting Factors in the Least Sqxiares Minimization

Another possible method to reduce lAE might be obtained by

varying the weight put on each pH point for obtaining the titration

curve (see Appendix F for minimization procedure). This may seem

to be only slightly beneficial for the cases where the pH setpoint

is 7, because only slight variations in the base flow rate setpoint

cause large variations in the effluent pH; therefore naturally

weighting the effluent pH point significantly more than the first

two pH points. Manual weighting may prove to be entirely necessary

for pH setpoints less than 7, however, because the effluent pH

point is no longer located on the neutrality region (near vertical

91

portion) of the titration curve and therefore has lost the natural

weighting found for pH setpoints of 7.

In the previous sections, the three pH points had equal

weighting in the least squares minimization procedure. Now varied

weighting factors will be used and the resulting effluent pH re

sponses examined.

For this case, the wastewater compositions and upsets from

section 6.2 (the dxial step upset) will be used. Figure 6.14 illus

trates two controlled effluent pH responses. The top curve is for

a case with a weighting of 0.1, 0.4, and 0.5 (for pH points 1, 2,

and 3, respectively), and the bottom curve is for a case with a

weighting of 0.25, 0.25, and 0.5. The controller response for the

weighting of 0.1, 0.4, and 0.5 shows adequate control over the

entire time region. The major difference in the two controller

strategies presented in the investigation is in decreasing the

weight on the influent pH point and increasing the weight of the

midpoint pH for the second control strategy. The results in less

ening the effect of the deviation of the influent pH of the process

from the influent pH of the model. This may be necessary for there

is the possibility of large deviation at the influent even for

adequate parameters. The controlled response for the weighting of

0.25, 0.25, and 0.5 shows adeqxiate control only over the time

92

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O O in

o o

o o

o o 0>

o o 1^

o o in

o o

o o

30

c

00 •«« 4) 3

C V u 41

O 3 H u O

irt

(A •3 C 0 u 4) Ul

4} s

.^ H

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" • V^rt

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o c o •Ji

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4> cn

fl o a. — S y o fl

4) u 3 30

(O CN ^ O 0> 00 CO in rO CN

93

region of 100 to 1200 seconds. At 1200 seconds an extreme step

upset upward in acid composition occurs that causes the pH measured

values at the influent, the midpoint, and the effluent to all drop

dramatically. When this occurs, the controller has information

concerning the lower portion of the titration curve but no informa

tion concerning the neutrality region or above. This causes the

controller model to parameterize the model coefficients on insuffi

cient data and thereby resulting, in this case, in an invalid

parameter identification. It can also be seen by the 2.5 pH unit

offset that this identification resulted in an almost unrecoverable

condition (it can be seen in the 1500 to 1800 second region that

the controller is slowly recovering and bring the effluent pH back

toward the setpoint of pH-7. It should also be noted that the

choice of "tioning parameters" (Kl, K2, weighting factors, standard

deviation value used to identify steady state, "trigger" that

initiates the model parameterization procedure, etc.) all have an

effect on the controller's performance in this type situation. The

lAE's presented in Table 6.6 illustrate the effect of varying the

K2 value and the parameterization trigger mechanism on the above

controller performance.

94

Table 6.6: lAE Results for Various Tuning Parameters for tha Step Change Conditions Presented in Saction 6.2

Effluent pH Set Point

7.0

7.0

7.0

7.0

5.0

5.0

5.0

5.0

5.0

Midpoint pH Target

5.0

4.0

5.0

5.0

3.5

3.5

3.5

3.5

3.5

K2

0.014

0.014

0.04

0.07

0.014

0.014

0.014

0.04

0.07

Para. * Trigger

1

1

2

2

1

1

2

2

2

Weighting 1st,2nd,3rd

.33,.33,.33

.25,.25,.5

.1, .4, .5

.1, .4, .5

.33,.33,.33

.1, .4, .5

.1, .4, .5

.1, .4, .5

.1, .4, .5

lAE

603.1

1744.1

830.1

846.9

439.5

1688.2

671.4

342.3

267.9

•Parameterization Trigger

1--Parameterize model each time steady state is identified.

2--Parameterize the model only once per steady state identification and identify the loss of steady state as reaching a threshold value of the effluent pH standard deviation of 0.22.

95

6.9 pH Setpoints Other than Seven

Another potential controller application investigated was the

use of setpoints other than 7, such as in chemical reactors, cool

ing tower water treatment, waste digestion, etc., where special pH

conditions need to be maintained.

For the system composition and upset of section 6.3, the ramp

upset, a pH setpoint of 9.0 was implemented. The controller re

sponses showed adequate control with practically no deviation in

the lAE's (882.6 for weighting factors of 0.25, 0.25, 0.5 versus

889.3 for equal weighting factors of 0.33). This is to be expected

since this type situation lends itself to this controller strategy

because of the inherent positions of the pH points. These posi

tions yield information about all the regions of the titration

curve: (1) pH at the influent--information about the location of

the lower titration curve region, (2) pH after the first base

injection-information about the location of the neutrality region

of the titration curve, and (3) pH at the effluent--information

about the upper titration curve region.

The runs made at a pH setpoint of 5 (refer to Table 6.6),

however, did not show such favorable results. This too is to be

expected though because of the inherent lack of needed information

concerning the titration curve. This lack of information results

from all three pH points being in the lower titration curve region

96

and therefore there is no indication of the location of the neu

trality region of tha curve much less the upper region. It can be

noted however, for the case with a pH setpoint of 5 that different

(Para. Trigger 2 of Table 6.6) parameterization "trigger" mech

anisms (indication of when to parameterize the model) and different

values of K2 (increasing the K2 value from the 0.014 used in the

base case controller) made significant improvements in the control

ler performance.

CHAPTER 7

CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions

A new strategy for the control of wastewater pH neutralization

has been simulated under many varying control conditions and in the

presence of noise and nonstationary process behavior. In both ramp

and step upsets, its performance suggests rapid and effective

control. The process model based controller only requires adapta

tion of two model parameters, is rapidly parameterized, remains a

true process simulator during rapid process changes, does not

require process upsets for parameterization, and controls stably

for a wide variety of upsets.

Control is not degraded with maximum expected model and cali

bration errors.

The effect of the finishing blender was, as expected, to damp

out the temporal composition changes and to reduce both the rate

and the amplitude of the pH upsets.

The least squares minimization procedure for calculation of

the model parameters has shown improved robustness to noise over

the parameterization procedure used in the Rhinehart and Choi work

(1988). The investigation of variable base flow split ratio, on

the other hand, seems to add little to control performance.

97

98

The study of the effects of weighting the pH points in the

least squares minimization for parameter identification, and the

effect of pH setpoints other than 7 illustrates the potential

ability of the controller to more effectively handle the extreme

step upsets in the neutralization process and to also adequately

control the process for pH setpoints less than 7.

7.2 Recommendations

Even though the method applied in the control strategy works,

there are several factors that need to be investigated for their

possible improvement to the control performance.

The most important factor for future study would be the choice

of the parameterization trigger strategy. This factor is important

becatose the controller's response to extreme step upsets is highly

dependent on the trigger strategy.

The trigger strategy expected to yield the best results would

be to parameterize the model only once per steady state and identi

fy the loss of steady state by reaching a pH of greater or less

than 0.5 pH units from the pH steady state value. Another possi

bility would be to again parameterize the model only once per

steady state, yet identify the loss of steady state by exceeding a

threshold value for the filtered standard deviation of the effluent

pH. This threshold value would be expected to be about 0.22. And

still another possibility would be to parameterize the model when

9v

the effluent pH violates plus or minus 1 pH unit from the pH set

point.

The next area for future study would be the effect of the

weighting factors of the least squares minimization procedure on

model parameterization. These factors have been shown, in Chapter

VI, to play a major role in the controller's response to extreme

step upsets and also in the controller's response for pH setpoints

of less than 7.

It is expected that the weighting factor for the influent pH

point should be reduced, to around 0.1 or 0.05. And the weighting

factors for the midpoint and effluent pH points should be in

creased, to around 0.4 and 0.5 or 0.45 and 0.5, respectively.

These weighting factors should result in improved control,

especially for cases with pH setpoints of less than 7.

Another possibility for improvement would be to study the

effect of the intercept value for the K2 value of the control law.

An increase in K2 would be expected to increase the integral recov

ery of offset due to model mismatch. This offset can be somewhat

devastating as seen in Figure 6.14.

And still another possibility worth further study would be to

not zero the integral term of the control law at parameterization

as was done in this work. This would be expected to help the

controller recovery from extreme step upsets as in Figure 6.14, yet

this could also be detrimental to control in regulatory mode if the

100

integral term is high enough (large enough K2) to cause small

oscillations about the pH setpoint.

Additionally, a constant base flow split ratio of 75/25 would

be recommended.

And finally, the controller should be experimentally vali

dated.

LIST OF REFERENCES

Albert, W. and H. Kurz: "Adaptive Control of a Wastewater Neutralization Process-Control Concepts, Implementation and Practical Experiences," IFAC Adaptive Control Chemical Processes, Frankfurt am Main, FRG, 1985.

Balhoff, R. A. and A. B. Corripio: "An Adaptive Feedforward Control Algorithm for Computer Control of Wastewater Neutralization," IFAC Real Time Digital Control Applications, Guadalajara, Mexico, 1983.

Gardner, D. V.: "Model Inferential Optimizing Computer Control of Three Series Reactors," ISA/84 Conference Proceedings. ISA/84 paper C.I.84-4722, Houston, TX, October, 1984.

Cott, B. J., P. M. Reilly and G. R. Sullivan: "Selection Techniques for Process Model-Based Controllers," AIChE Annual Meeting, Miami Beach, FL, November, 1986.

CRC Handbook of Chemistry and Physics. 62nd Edition, Robert C. Weast and Melvin J. Astle, editors, 1983.

Cutler, C. R. and B. L. Ramaker: "Dynamic-Matrix Control--A Computer Control Algorithm," 1980 Am. Control Conference Proceedings. San Francisco, CA, 1980.

Economou, C. and C. Morari: "Internal Model Control. 5. Extension to Nonlinear Systems," Ind. Eng. Chem. Proc. Des. Dev.. Vol. 25, 1986.

Garcia, C. E. and M. Morari: "Internal Model Control. l.A Unifying Review and Some New Results," Ind. Eng. Chem. Proc. Des. Dev.. Vol. 21, 1981.

Gray, D. M.: "New Solution to pH Control Problems," Pollution Enp;ineering. April, 1984.

Gustafsson, T. K. and K. V. Waller: "Dynamic Modeling and Reaction Invariant Control of pH," Chemical Engineering Science. Vol. 38, No. 3, 1983.

Gustafsson, T. K. and K. V. Waller: "Myths About pH and pH Control," AIChE J.. Vol. 32, No. 2, February, 1986.

101

102

Jeffreson, C. P.: "Computer Control of Simple Variable Flow Processes," IFAC Real Time Digital Control Applications, Guadalajara, Mexico, 1983.

Joseph, B. and C. B. Brosilow: "Inferential Control of Processes," AIChE J.. Vol. 24, No. 3, March 1978.

Juba, M. R. and W. Hamer: "Progress and Challenges in Batch Process Control," Chemical Process Control - CPCII, Morari and McAvoy editors, CACHE-Elsevier, Proceedings of 3rd Intl. Conf. on Chem. Process Control. Asilomar, CA, June, 1986.

Klumper, I.V. and D. Z. Tobias: "Simulation and Control of Inaccessible Complex Non-Steady State Processes," AIChE Summer National Meeting, Boston, MA, August, 1986.

Lee, P. L.: "Real Time Multivariable Control," IFAC Conference, 1987.

Lee, P. L. and G. R. Sullivan: "Generic Model Control - GMC," Computers and Chemical Engineering. Vol. 12, No. 6, p. 573, 1988.

Leeds and Northrup Instruments: 7084 Microprocessor pH Analyzer/Controller, Product Bulletin C2.1212-DS, North Wales, PA, 1984.

LFE Instruments: Model-3031 pH Controller, Product Bulletin 2409-SBl, Clinton, MA, 1987.

Lu, Y. Z.: "Application of Control Strategies to Thermal Processes in the Metal Industry," 1986 Am. Control Conference Proceedings. ACC Meeting, Minneapolis, MN, June, 1987.

McAvoy, T. J., H. Hsu and S. Lowenthal: "Dynamics of pH in Controlled Stirred Tank Reactor," Ind. Eng. Chem. Proc. Des. Dev. . Vol. 11, No. 1, 1972.

McDonald, K. : "Performance Comparison of Methods for On-Line Updating of Process Models for High Purity Distillation Control," AIChE Spring Meeting, Houston, TX, Session 59, March, 1987.

McMillan, J. M.: "Multivariable Adaptive Predictive Control of a Binary Distillation Column," Automatica. 20, no. 5, 1984.

Moore, R. L. : "Neutralization of Waste Water by pH Control," ISA. Monograph Series 1. 1978.

OMEGA Engineering, Inc.: New Horizons in Process Measurenienu and Control, Model PHCN-2031, Vol. V, Stamford, CT, 1987.

103

Parrish, J. and C. Brosilow: "Nonlinear Inferential Control of Reactor Effluent Concentration from Temperature and Flow Measurements," 1986 Am. Control Conference Proceedings. Seattle, WA, June, 1986.

Patterson, M. J.: "A Survey of Model Reference Adaptive Techniques--Theory and Applications," Automatica, 10, 1975.

Piovoso, G. and A. W. Williams: "Modeling a Low Pressure Steam-Oxygen Fluidized Bed Coal Gasifying Reactor," Chem. Eng. Sci.. 36, 1985.

Proudfoot, R. D., and C. McGreavy: "Computational Techniques in Optimal State Estimation," J. Dynamic Systems. Measurement & Control, 1983.

Rhinehart, R. R.: Dynamic Modeling and Control of a Pressurized Fluidized Bed Coal Gasification Reactor. PhD Dissertation, NC State Univ., Raleigh, NC, 1985.

Rhinehart, R. R. and J. Y. Choi: "Process Model-Based Control of Wastewater pH Neutralization," Advances in Instrumentation. Vol. 43, pp. 351-358, 1988.

Rhinehart, R. R., R. M. Felder and J. K. Ferrell: "Internal Adaptive-Model Control of a Coal Gasification Reactor," AIChE National Meeting, Boston, MA, paper 273, August, 1986.

Richalet, J. A. and I. H. Congalidis: "New Challenges for Process Control," AIChE Annual Meeting, Washington, DC, November, 1983.

Richalet, J., A. Rault, J. L. Testud and J. Papon: "Model Predictive Heuristic Control: Application to Industrial Processes," 4th IFAC Symposium in Identification and System Parameter Estimation, 1976; also Automatica, Vol. 14, September, 1978.

Riggs, J. B. and R. R. Rhinehart: "Comparison Between Process Model-Based Controllers," Proceedings of the 1988 Am. Control Conference. Atlanta, GA, 1988.

Seaborg, D. A.: "Stability and Response of the Analytical Reactor," Tnd. Eng. Chem. Proc. Des.. Dev.. Vol. 22, 1983.

Seaborg, D. A., R. Kelly and R. W. Ferrell: "Internal Adaptive Model Control," Proc. of Amer. Control Conf.. Minneapolis, June 10-12, 1982.

Shinskey, G.: pH and oION Control in Process and Waste Streams. Wiley, 1983.

APPENDIX A

IONIZATION CONSTANT OF WATER (CRC, 1983)

Table A.l shows the variation of water ionization constant (K)

for temperatures ranging from 273 K to 333 K.

The functional form of K (T) is obtained by fitting data to a

second degree polynomial:

K - 10**(-C1-C2*T-C3*T**2) (i)

where CI is equal to 38.42828, C2 is equal to -0.1305219, and C3 is

equal to 1.628-4.

Equation (i) is used to calculate K values at given tempera

tures. At a temperature of 297 K (24 C), the K value calculated

from Equation (i) deviates only 0.2% from tabulated data.

104

105

Table A.l: Water Ionization Constant

-Log (K) Temperature

14.9435

14.7338

14.5346

14.3463

14.1669

14.0000

13.9965

273

278

283

288

293

297

298

-Log (K)

13.8330

13.6081

13.5348

13.3960

13.2617

13.1369

13.0171

Temperature

303

308

313

318

323

328

333

APPENDIX B

MULTI-VARIABLE NEWTON'S METHOD

Newton's Method is used to evaluate the two unknowns of Equa

tions 4.2.1 and 4.2.2. Newton's Method is applied by setting

the solution vector V - and then applying the following

iterative steps:

Step 1:

Step 2:

Calculate f(V)

J(V) * AV - - f(V)

where J - Zf.,/Zx

Zf^/Zx

Zf^/Zd

Zf2/Zd

Solving this linear system, yields

AX - (f2*dfi/88 - fi*af2/a5)/DET,

A8 - (fi*af2/ax - f2*afi/3x)/DET

where DET - 8fi8-x*df2/<p8 - d^i/d8*dd2/dx

Step 3: Check the stopping criteria

Step 4: Repeat the loop until the criteria is satisfied.

106

107

*Stopping criteria is: ABS (Ax/x) > 1 * 10"5

and

ABS (AJ > 1 X 10-5

Execution will be repeated until AV satisfies the stopping cri

teria.

i

APPENDIX C

RANDOM NOISE

A Gaussian Distribution Random Number is generated by use of the

following equation developed by Dr. R. R. Rhinehart of Texas Tech

University.

X - RND

Noise - SD * 1.9607 * (x-.5)/((x + .002432) * (1.002432-x))•203

where x - A uniform distributed random number generated by the Computer

Internally.

SD - the standard deviation of the Gaussian distributed numbers

to be generated.

Noise - A Gaussian distributed random nximber with a mean of zero

and a standard deviation of SD.

The random noise on the process instruments is simulated by simply

adding the noise variable from above to the instrument's simulated

reading value calculated by the computer program.

108

APPENDIX D

AUTOREGRESSIVE DRIFT

The non-stationary process characteristics are simulated by

adding a disturbance to the value calculated by the program. This

disturbance is simply a second order autoregressive sequence gener

ated in the following manner.

Step 1: Time - 0 PV - PV o

Step 2: drt - b3*drt + (l-b3)(.5-RND)

dist - dist + b2*drt

where: b2 and b3 are constants

dist - the disturbance to be added to PV

Step 3: Time - t^ PV^ - PVQ + dist^

Step 4: Repeat Steps 2 and 3.

109

APPENDIX E

FILTERING PROCESS VARIABLES AND SIMULATING

INSTRUMENTATION TIME LAG

To reduce the effect of the random noise associated with the

process variable we will simply apply a first order filter in the

following form:

PVp - aPVc + (1-a) PVp

where: PV^ is the current process variable reading

PVp is the filtered process variable

a is the filtering constant associated with each

process variable.

This filtering will dampen the noise effects, yet will also cause a

time lag in recognition of an actual change in the process vari

able. Therefore, the value chosen for a must be small enough to

dampen out the noise, yet be large enough to allow for recognition

of an actual change within a reasonable amount of time. This first

order filter equation will also be used to simulate the time lag

associated with the process instrumentation readings. In this

case, the a value will be determined as follows:

a - At/r

no

HI

where At is the sampling time interval

r is the process time constant

^

APPENDIX F

LEAST SQUARES MINIMIZATION

The original function developed in Equation (3.4.11) is as

follows:

-pH f(AfAGf) - Exp(-AGf/RTi) " dO ) (Ci +ei)/(A - C - e .)-0

1

where: Af - fictitious weak acid concentration

AGf - fictitious DG of the fictitious weak acid

R = gas constant

T - temperature

pH - pH

€ — base concentration

i - denote t he point in the system

1 = inlet

2 = point after 1st base injection

3 - effluent and

C - 10-pH - Kv,*10P i .

The term (A^ - C. - c.) was seen to be zero at certain f. 1 1

1

conditions and, therefore, for control ler robustness the function

wi l l be inverted, yielding:

112

113

f(A^G^) Exp(AG^/RT^) - ^^-jj-

(A^^ - C^ - e.) I.

(10 '•)(C + e^) - 0

The least squares minimization function will therefore be: 2

3 (Af. • i • î) S - S Exp(AG^/RT^)

i-1 - - -P«i (10 '•)(C + 6 ) (i)

Expanding Equation (i) yields:

^ 2 Exi^^f^/^î^^î ' ^ ' '^^ S - S Exp(2\5\g /RT ) - ^""^ l^

i-1 "Pî (10 "•) (C. + € )

2 (A. - C. - €.)"•

f L 1 1

+ -pH (10 '•)^(C^ + €^)^

Taking the derivatives with respect to Af and AGf and setting

them equal to zero gives:

3 2(A^^ - C. - . . )

ff- - R u 0 (ii) 3Af i _ i -pH.

^ "• "• ( 1 0 ^ ) ( C ^ + e^)

3 Exp(AG /RT )(A - C - e )

I f^ - 2 i T Exp(2AG /RT.) - ^ ' ' , ^ — ' f i - 1 i i -pH.

(10 "• ) (C^ + c^) ( i i i )

114

Now assume T. ? constant - T.

Divide Equation (ii) by 2.

Divide Equation (iii) by 2/RT.

Equation (ii) yields 3 (A - C - 6 ) 2 i ^ ^ i-1

-PH.)2 (10 "- (C. + e.r

Exp(AG^/RT) - - ^ ^

i-1 ,, P»i (10 ^)(C^ + e^) . (iv)

Equation (iii) yields

3 Exp(AG^/RT)(A^ - C^ - €/)

^ - f k - .2 Exp(2AG^/RT) - — ^ - 0

(10 '•)(C^ + e/)

From Equation (iv), AGf is expressed in terms of Af, therefore,

leaving us with one independent equation with one unknown (Equation

(v) and unknown Af. Af can therefore be determined by simply

applying Newton's Method to Equation (v) . Then AGf can be

determined by (inserting into Equation (iv)) the Af value

determined by Newton's Method. (Note: The Newton's Method search

uses the analytical derivative of Equation (v) yielding

115

3 -Exp(AG-/RT) N - 2 ^

i - 1 -P» i (10 ^ ) ( C . + € )

APPENDIX G

SPLIT CALCULATOR

To determine the split (base injected into first port/total

base injected) required at each control action, the following steps

are applied:

Step 1: Call subprogram SSINV to determine the total base

required for the pH steady state target (phss).

Step 2: Call subprogram SSINV to determine the base required

for some desired intermediate pH (say pH-5.0).

Step 3: Divide the amount of base calculated in Step 2 by

that calculated in Step 1. (Note: Since this is a

highly nonlinear process, this split can vary some

what severely from control action to control action.

Therefore, apply Step 4.

Step 4: Apply a first order filter to the calculated split

(from Step 3), with a recommended a of 0.6.

116

APPENDIX H

FILTERED STANDARD DEVIATION

To determine the filtered standard deviation used to identify

steady state the following equations are applied.

xf - a * pHout + (l-a)xf (1)

vf - a * (xf - pHout)^ + (l-a)vf (2)

sv - (vf)0.5 (3)

where xf is the filtered effluent pH

a is 0.3

vf is the filtered variance in the effluent pH

sv is the filtered standard deviation in the effluent pH

A threshold value of 0.175 for sv was used in this work for identi

fication of steady state.

117

APPENDIX I

COMPUTER CODE

•PMBC.BAS •July 25, 1989 'Gaylon L. Williams

'THE SIMULATION OF A LABORATORY-SCALE WASTEWATER 'NEUTRALIZATION PROCESS

'The QuickBasic program PMBC.BAS consist of two parts, 'process simulator (simulation of the acidic wastewater 'stream) and controller part (PMBC). The process 'simulator generates simulated data of the wastewater, 'temperature, flowrate, and pH for the influent, the 'midpoint, and effluent. The process simulator also 'simulates the instrumentation noise present on the pH 'probes and the flow meters; and also the measurement 'lag present with these instruments. The controller •model is periodically adapted to match the real process ' in the controller part so that the controller 'calculates the proper base flow rate set point. This 'simulation program does not consider any transport 'delay between the mixing vessels. All used variables 'are commented in each subroutine.

'The code for PMBC.BAS follows:

118

119

•k-k-k-k-k MAIN PROGRAM *****

The main program supervises the entire simulation program and this part provides the required constants and initial values needed in the other sections. The process simulator subprograms are:

DFT - this subprogram generates the numbers used in simulating the nonstationary nature of the component compositions

DataGen - this sxibprogram generates the pH values of the 3 different positions (influent, midpoint, and effluent)

Noise - this subprogram generates the noise which is added to the pH and flow rate measurements

PrcsLag - this subprogram simulates the time lag present in the measurement devices.

Pictl - this subprogram simulates the dynamics of the PI control valve on the base (titrant)

The controller part subprograms are: CalBase - this subprogram calculates the base

concentration of the influent, midpoint, and effluent based upon the simulated measured data.

Suprvsr - this subprogram calculates control actions and decides when to reparameterize the model

AdaptPar - this subprogram parameterizes the model when called upon by Suprvsr subprogram

DEFDBL a-h,k-z DEFINT i,j

COMMON SHARED time, dt, r, cl, c2, c3, vol, be, tb

RANDOMIZE (747)

OPEN "C:\lotus\PMBCl.PRN" FOR OUTPUT AS #1 OPEN "C:\lotus\PMBC2.PRN" FOR OUTPUT AS #2 OPEN "C:\lotus\PMBC3.PRN" FOR OUTPUT AS #3 OPEN "C:\lotus\SPLIT.PRN" FOR OUTPUT AS #4

DIM E(3),C(3),PH(3),AFICT(3),N(3),D(3),A(3)

120

************ Set up All Constants And Initial Values Cl= 38.42818D0 C2=-0.1305219D0 C3= 0.00016287D0 r=1.987D0 bc=0.IDG tinc=298D0 t2c=298D0 toutc=298D0 Vol=0.02D0 dt=0.2D0 bfsp=l.llD0 bfc-bfsp afc=2.0D0 ratio=bfsp/afc i=l ebasel-O.ODO ebase2=0.02498124531132783D0 ebase3=0.03569131832797428D0 wa2=0.03750937734433608D0 sa2=0.004126031507876969D0 salt2=0.0005626406601650413D0 base2=0.02498124531132783D0 wa3=0.03215434083601286D0 sa3=0.003536977491961415D0 salt3=0.0004823151125401929D0 base3=0.03569131832797428D0 tb=298D0 Xl=.lD-4 dell=0D0 x2=.lD-4 del2=0D0 x3=.lD-4 del3=0D0 time=0.ODO distwa=0.ODO distsa=0.ODO distsal=O.ODO distg=O.ODO disttl=O.ODO distaf=O.ODO ctltime=O.ODO sd=0.12D0 proIAE=0.0D0 ebfsp=l.llD0 ebfcf=l.llD0 eafcf=2.0D0 ephoutcf=7.ODO eph2cf=3.5D0 ephincf=2.ODO

121

irp=0 afic=5.55D-2 gfic=4.75D3 Xf=7.0D0 vf=0.15D0 af1=2.ODO bfl=:l.llDO phinl=2.ODO ph21=3.5D0 phoutl=7.ODO icount=6 split=0.75D0

' Split the base flow into two parts

100 bfcl=bfc*split bfc2=bfc*(1.ODO-split)

' Calculate the amount of drift to be added to each ' process variable

CALL DFT (dftwa,.997D0,.00004D0,distwa) CALL DFT (dftsa,.997D0,.000009D0,distsa) CALL DFT (dftsal,.997D0,.00002D0,distsal) CALL DFT (dftg,.997D0,10.0D0,distg) CALL DFT (dftaf,.997D0,.00025D0,distaf) CALL DFT (dfttl,.997D0,0.5D0,disttl)

' Add the drifts to each process variable ' and make step or ramp changes

IF (time < 400) THEN af c=2. ODO-i-distaf wal=0.05D0-»-distwa sal=0.0055D0-i-distsa saltl=0.005D0-Hdistsal g=5.0D3-Hdistg tinc=298D0-i-disttl

ELSEIF (time < 1200) THEN af c=2. ODO-i-distaf wal=0.03D0-i-distwa sal=0. OlDO-Hdistsa saltl=0.00005D0-t-distsal g=7.0D3+distg tinc=298D0-hdisttl

122

ELSE af c = 2 . ODO-J-distaf wal=0 .06D0+dis twa sa l=0 .004D0-t -d is t sa s a l t l = 0 . 0 0 2 D 0 - i - d i s t s a l g = 6 . 0D3-f-distg t inc=298D0- t -d i s t t l

END IF

• Concentrations cannot drift below zero

IF (wal < O.ODO) THEN wal=O.ODO

END IF

IF (sal < O.ODO) THEN sal=O.ODO

END IF

IF (saltl < O.ODO) THEN saltl=O.ODO

END IF

CALL DataGen (i,l,bfcl,tine,afc,phinc,basel,wal,sal, saltl,xl,dell,tine,t2c,toutc,bfc2,wa2, sa2,salt2,base2,g)

CALL DataGen (i,2,bfcl,t2c,afc,ph2c,base2,wa2,sa2, salt2,x2,del2,tine,t2e,toutc,bfc2,wa2, sa2,salt2,base2,g)

CALL DataGen (i,3,bfcl,toute,afe,phoutc,base3,wa3,sa3, salt3,x3,del3,tine,t2c,toutc,bfc2,wa2, sa2,salt2,base2,g)

CALL PrcsLag (afc,bfc,phine,ph2e,phoutc,afl,bfl,phinl, ph21,phoutl)

CALL Noise (tine,t2c,toutc,bfl,afl,phoutl,ph21,phinl, etinc,et2e,etoutc,ebfc,eafc,ephoutc,eph2c, ephine,sd)

• Filter the process variable readings

ebfcf=.6D0*ebfc-i-.4D0*ebfef eaf cf =. 6D0*eaf c-f-. 4D0*eaf cf ephoutcf =. 6D0*ephoute-i-. 4D0*£phoutcf eph2cf =. 6D0*eph2c-i-. 4D0*eph2ef ephinef=.6D0*ephine+.4D0*ephincf

123

CALL CalBase (ebasel,ebase2,ebase3,ebfef,eafcf, split)

CALL Sprvsr (ebasel,ebase2,ebase3,ratio,ephoutc, ephoutcf,eph2ef,ephincf,etinc,et2e,etoutc, eafcf,ebfef,proIAE,ctltime,afie,gfic, ebfsp,eint,xf,vf,sv,irp,split)

bfsp=ebfsp

CALL pictl (bfsp,bfc)

'prepare numeric format for lotus

pafic=afic*100D0 peint=eint*100D0 psv=sv*1000D0 pwal=wal*100D0 psal=sal*1000D0 psaltl=saltl*10000D0 pg=g/1000D0

'print every fifth time to lotus

IF (icount>4) THEN

PRINT #1,time,ephinc,ephoutc,bfsp PRINT #2,pwal,psal,pafic,gfic ieount=0

PRINT "pH in = ";ephinc;" @ time = ";time

PRINT "pH 2nd = ";eph2c;" pH out = "; ephoutc

END IF

icount=icount+l

time=i*dt i=i-Hl

IF (i < 9000) THEN GO TO 100

END IF

CLOSE #1 CLOSE #2 CLOSE #3 CLOSE #4

150 END

124

Subprogram DataGen

This Subprogram simulates the actual acid wastewater stream characteristics. It is used to simulate all three points in the stream, influent - midpoint -effluent, and is therefore called three times at each process time step, which is 0.2 seconds.

Variables

unless otherwise noted flow rate are in liters/minute, concentrations are in gmoles/liter and temperatures are in degrees K

ithpoint = the stream point 1 is the influent 2 is the midpoint 3 is the effluent

bfel = the base flow amount injected in the first mixing vessel

bfc2 = the amount injected in the second mixing vessel t = the temperature at each respective ithpoint af = the acid wastewater flow rate ph = the pH at each respective ithpoint basek = the base concentration at each respective ithpoint wa = the weak acid concentration at each respective

ithpoint sa = the strong acid concentration at each respective

ithpoint salt = the salt concentration at each respective ithpoint X = amount of weak acid dissociated del = the amount of water that has dissociated tine = current influent temperature t2c = current midpoint temperature toutc = current effluent temperature g = the Gibbs free energy of the wastewater ka = The equilibrium constant of weak acid dissociation kw = the water ionization constant

125

SUB DataGen (i,ithpoint,bfcl,t,af,ph,basek,wa,sa,salt,x, del,tine,t2c,toutc,bfc2,wa2,sa2,salt2,base2, g) STATIC

IF (ithpoint = 1) THEN

basek=0.ODO wal=wa sal'ssa saltl=salt t=tinc

ELSEIF (ithpoint = 2) THEN

itn=0 tau=60.0D0*vol/(af+bfcl) base2=basek-i- (bc*bf el/ (af-Kbf cl) -basek) *dt/tau basek==base2 wa2=wa-i- (af *wal/ (af+bfel) -wa) *dt/tau wa=wa2 sa2=sa-t- (af *sal/ (af+bfel) -sa) *dt/tau sa=sa2 salt2=salt-»- (af *saltl/ (af-i-bfel) -salt) *dt/tau salt=salt2 t2c=t+((bfcl*tb+af*tine)/(af+bfel)-t)*dt/tau t=t2c

ELSE wf2=af+bfel itn=0 tau=60.0D0*vol/(wf2+bfe2) wa3=wa+(wf2*wa2/(wf2+bfc2)-wa)*dt/tau wa=wa3 sa3=sa+(wf2*sa2/(wf2+bfe2)-sa)*dt/tau sa=sa3 salt3=salt+(wf2*salt2/(wf2+bfe2)-salt)*dt/tau salt=salt3 base3=basek+((be*bfe2+base2 *wf2)/(wf2+bfc2)-basek)*dt/tau basek=base3 toutc=t+((wf2*t2e+bfe2*tb)/(wf2+bfe2)-t)*dt/tau t=toutc

END IF

itn=0

Ka=exp(-g/(r*t))

kw=(10.0D0)^(-el-c2*t-c3*t^2)

126

200 fl=ka*(wa-x)-(sa+x+del)*(x+salt) f2=kw-(sa+x+del)*(del+basek) dfldx=-ka-2.0D0*x-sa-del-salt dflddel=-x-salt df2dx=-del-basek df2ddel=-sa-x-2.0D0*del-basek det=dfldx*df2ddel-dflddel*df2dx

IF (det = O.ODO) THEN x=x*0.03D0 del=del*0.03D0 GO TO 200

ELSE

dx=(f2*dflddel-fl*df2ddel)/det ddel=(fl*df2dx-f2*dfIdx)/det x=x+dx del=del+ddel h=sa+x+del

END IF

IF (h < 0) THEN x=x*1.5D0 del=del*0.03D0 GO TO 200

ELSE

itn=itn+l

END IF

IF (ABS(dx) > lD-14) AND (ABS(ddel) > lD-14) THEN

GO TO 200

ELSE

z=log(10.0D0) ph= -l*LOG(h)/z

END IF

END SUB

127

Subprogram Noise

This subprogram adds noise to the process variable from the DataGen subprogram in order to simulate the noise present on actual process instrumentation.

Variables

An e added to the front of a process variable signifies that the variable has noise added

An 1 added to the rear of a process variable signifies that the variable is a lag variable which simulates the time lag associated with process instrumentation

sd = the standard deviation of the Gaussian random number added to the process variable

SUB Noise (tine,t2c,toutc,bfl,afl,phoutl,ph21,phinl,etinc, et2c,etoutc,ebfc,eafc,ephoutc,eph2e,ephinc, sd) STATIC

etinc=tine et2e=t2c etoutc=toutc

CALL GAUSS (noise,0.00025D0) ebfc=bfl+noise

CALL GAUSS (noise,0.0005D0) eafc=afl+noise

CALL GAUSS (noise,sd) ephoute=phoutl+noise

CALL GAUSS (noise,sd) eph2c=ph21+noise

CALL GAUSS (noise,sd) ephinc=phinl+noise

END SUB

128

Subprogram CalBase

Since the controller cannot actually know the base concentrations at each point, this subprogram calculates these concentrations from noisy process data which it can know.

Variables

ebase = the base concentration calculated for each respective point in the stream

ebfcf = the filtered base flow rate measurement eafcf = the filtered acid flow rate measurement split = the base injected in the first mixing vessel

devided by the total base injected

SUB CalBase (ebasel,ebase2,ebase3,ebfcf,eafcf,split) STATIC

ebfcl=ebfcf*split ebfc2«ebfcf*(1.ODO-split)

ebasel^^O.ODO ebase2=ebase2+(bc*ebfcl-ebase2*(eafef+ebfcl)) *dt

/(60.0D0*vol)

ewf2=eafef+ebfel

ebase3=ebase3+((bc*ebfe2+ebase2*ewf2)-ebase3*(ewf2+ebfc2)) *dt/(60.0D0*vol)

END SUB

Subprogram Suprvsr 129

This supervisory controller looks at the process every 1.0 seconds and makes a control action based on the current conditions of the filtered lagging noisy process data measurements. If a steady state condition is recognized then the supervisory controller also calls the AdaptPara S\ibprogram and adapts the model parameters to fit the corrent conditions.

Varaibles

ebase = the base concentration calculated by the controller for each point in the wastewater stream

ratio = the base flow rate setpoint devided by the acid rate measurement (This allows for response to temporal acid flow rate changes between control actions)

ephoutc = the current effluent pH measurement ephoutcf = the filtered effluent pH measurement proIAE = the control proformanee measurement of lAE

(Integral of the Absolute Error) ctltime = variable used to determine when the controller

should take another action e = the error in effluent pH from the pH setpoint eint = the integral of the error xf = the filtered effluent pH used to determine steady state vf = the variance in the effluent pH used to determine

steady state sv = the standard deviation in the effluent pH used to

determine steady state irp = a varaible used when it is desired to determine if a

new pH steady state has been reached a2 = the alpha constant in the filter equations phss = the pH steady state target

SUB Sprvsr (ebasel, ebase2, ebase3 , ratio, ephoutc, ephoutcf, eph2cf,ephincf,etinc,et2c,etoutc,eafcf,ebfcf, proIAE,ctltime,afic,gfic,ebfsp,eint,xf,vf,sv, irp,split) STATIC

IF (time > ctltime) THEN

ctldt=1.0D0 ctltime=ctltime+l.ODO a2=0.3D0 e=7.ODO-ephoute

130

eint=eint + e*ctldt kl=l.250D0-0.02D0*ABS(e) k2-0.014D0-0.002D0*ABS(e)

• SS identification

xf=0.3D0*ephoute + 0.7D0*xf vf=a2*((xf-ephoute)^2.0D0) + (l-a2)*vf sv=vf^0.5D0

PRINT "sv =: ";sv

phss=ephoutc + kl*e + k2*eint

IF (phss > 14.ODO) THEN phss=14.0D0

ELSEIF (phss < O.ODO) THEN phss^O.ODO

END IF

PRINT "phss = ";phss

CALL SSINV (phss,afic,gfic,eafcf,ebfsp,etoutc,delbfsp)

IF (ebfsp < O.ODO) THEN ebfsp=0.ODO

ELSEIF (ebfsp > 1.5D0) THEN ebfsp=1.5D0

END IF

ratio=ebfsp/eafef

IF (sv < 0.175) AND (delbfsp < 0.01) THEN

CALL AdaptPar (ebase2,ebase3,afie,gfic,ephoutcf,eph2cf, ephincf,etinc,et2c,etoutc,ebfcf,eafcf, split)

irp=0 eint=0.ODO

END IF

CALL SSINV ( 5 . O D O , a f i c g f i e , e a f c f , e b f p h 5 , e t o u t c , d e l b f p h 5 )

I F (ebfsp=O.ODO) THEN s p l i t n = 1 . 0 D 0 131

ELSE splitn=ebfph5/ebfsp

END IF

IF (splitn<0.6D0) THEN splitn=0.6D0

ELSEIF (splitn>0.8D0) THEN splitn=0.8D0

END IF

split=split*0.6D0+splitn*0.4D0

PRINT "split = ";split

IF (time>200) THEN

proIAE=proIAE+ABS(e)

END IF

ebfsp=ratio*eafef

PRINT "lAE = ";proIAE;"bfsp = ";ebfsp PRINT "afic = ";afic;"gfic = ";gfic PRINT #3,proIAE,phss PRINT #4,time,split END IF

END SUB

Subprogram AdaptPar 132

A two parameter phenomenological model is used for the model. This subprogram parameterizes the controller model so that the model matches the real process data. This allows the model "inverse" to be used to calculate the desired base flow rate setpoint.

Variables

wf = weighting factor exgrt = the variable used to convert the two equation

two unknown system to a single equation with one unknown

T = the average of the influent and effluent temperatures

SUB AdaptPar (ebase2,ebase3,afic,gfic,ephoutcf,eph2cf, ephincf,etinc,et2c,etoutc,ebfcf,eafcf, split) STATIC

wf(1)=0.333333D0 wf(2)=0.333333D0 wf(3)=0.333333D0

T = (etinc+etoute)/2.0D0 E(l) = O.ODO E(2) = ebase2 E(3) = ebase3 PH(1) = ephincf PH(2) = eph2cf PH(3) = ephoutcf

FOR i = 1 TO 75 STEP 1

kw = (10.0D0^(°-el-c2*T-c3*T^2.0D0) )

FOR j = 1 TO 3 STEP 1

C(j) = (10.0D0^(-PH(j)))-kw*(10.0D0^PH(j))

NEXT j

AFICT(l) AFICT(2) AFICT(3)

afic (afic*eafcf)/(eafef+split*ebfef) (afic*eafef)/(eafcf+ebfcf)

133

FOR j =1 TO 3 STEP 1

N(j) = ((AFICT(j)-C(j)-E(j))/((((10.0D0-(-PH(j)))-2.0D0) *((C(j)+E(j))-2.0D0))))

D(j) - (l/((10.0D0-(-PH(j)))*(C(j)+E(j))))

NEXT j

N = 0.0 D = 0.0

FOR j =1 TO 3 STEP 1

N = N+N(j) D = D+D(j)

NEXT j

exgrt = n/d


A(j) = (AFICT(j)-C(j)-E(j))

NEXT j

F = 0.0 DF = 0.0


F = (exgrt^2.0D0)*wf(j)-exgrt*A(j)*D(j)*wf(j) + F DF = DF - exgrt*D(j)*wf(j)

NEXT j

dafic = F/DF

afic = afic+dafie

134

IF (ABS(dafic) < lD-12) THEN

GO TO 2000

END IF

NEXT i

2000 IF (exgrt < 1 ) THEN exgrt=1.0

END IF

gfic = r*T*LOG(exgrt)

END SUB

135

Subprogram SSINV

This subprogram is the model inverse and is used to calcualte the base flow rate setpoints.

SUB SSINV (phss,afic,gfic,eafcf,ebfsp,etoutc,delbfsp) STATIC

obfsp=ebfsp

kw3=10.0D0^(-el-c2*etoutc-c3*etoute^2) ka=EXP(-gfic/(r*etoute)) ebfsp=(ka*(10.ODO^phss)*afic*eafcf/(ka*(10.ODO^phss)+

1.0D0)+kw3*(10.0D0^phss)*eafcf-(10.ODO^-phss)* eafcf)/(-kw*(10.ODO^phss)+(10.ODO^-phss)+bc)

delbfsp=ABS(ebfsp-obfsp)

END SUB

'Subprogram PICTL t

'This subprogram simualtes the lag associated with the 'SISO PI valve

SUB pictl (bfsp,bfc) STATIC

dt=0.2D0 taui=.4D0 kc=2O.ODO cv=.7D0

bfc=bfc+dt*kc*cv/(taui*(12.ODO+kc*cv))*(bfsp-bfe)

END SUB

136

Subprogram DFT

This subprogram cauculates the drifting disturbances to be added to each process variable.

SUB DFT (drift,b3,b2,dist) STATIC

drift=b3 *drift+(l-b3)*(.5D0-RND) dist=dist+b2 *drift

END SUB

'Subprogram GAUSS

'This subprogram genertate the Gaussian random numbers 'to be added to each process variable to simulate the 'process noise. f

SUB GAUSS (noise,sd) STATIC

n=RND noise=sd*1.9607D0*(n-.5D0)/((n+.002432D0)*(1.002432-n)

)^.203D6

END SUB

137

Subprogram PrcsLag

This subprogram simualtes the lag associated with process instrumentation by simply applying a first order filter to the variable.

SUB PrcsLag (afe,bfe,phine,ph2e,phoute,afl,bfl,phinl,

ph21,phoutl) STATIC

alpha = 0.08D0

beta = 0.0285D0

afl = afc*beta + afl*(1.ODO-beta)

bfl = bfc*beta + bfl*(1.ODO-beta)

phinl = phinc*alpha + phinl*(l.ODO-alpha)

ph21 = ph2e*alpha + ph21*(l.ODO-alpha)

phoutl = phoute*alpha + phoutl*(l.ODO-alpha)

END SUB

PERMISSION TO COPY

In presenting this thesis in partial fulfillment of the

requirements for a master's degree at Texas Tech University, I agree

that the Library and my major department shall make it freely avail

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process-model-based control of wastewater ph a …

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