Title: Design and implementation of minimum time computer control schemes for start-up of a double effect evaporator
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Title: Design and implementation of minimum time computer control schemes for start-up of a double effect evaporator
Physical Description: xv, 208 leaves. : illus. ; 28 cm.
Language: English
Creator: Nayak, Santosh, 1946-
Publication Date: 1974
Copyright Date: 1974
 Subjects
Subject: Control theory   ( lcsh )
Electronic data processing   ( lcsh )
Automatic control   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 206-207.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00097556
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000580682
oclc - 14072058
notis - ADA8787

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DESIGN AND IMPLEMENTATION OF MINIMUM TIME COMPUTER CONTROL SCHEMES
FOR START-UP OF A DOUBLE EFFECT EVAPORATOR











By

SANTOSH NAYAK


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY







UNIVERSITY OF FLORIDA

1974













ACKNOWLEDGMENTS


The author wishes to express his sincere appreciation to

Dr. A. W. Westerberg, chairman of his supervisory committee and

principal research advisor, for his invaluable guidance and en-

couragement.

The author also wishes to thank the following people who

have helped him at various times throughout the duration of his

research.

Dr. T. E. Bullock, a member of his supervisory committee,

for his valuable suggestions and for introducing the author to

optimal control theory.

Dr. U. H. Kurzweg who served as a member of his supervisory

committee.

The North-East Regional Data Center of the State University

system of Florida and the IBM Customer Engineers who helped out

with the working of the IBM 1070 interface and its communication

with the IBM 370/165.

The Department of Chemical Engineering for financial support,

and the faculty and technicians of the department for their suggestions

and help.

Mr. S. S. Sriram for his help in preparing the figures.

Miss Sara McElroy, the author's fiancee, for her encouragement

during the major part of the work and for her typing of the manuscript.











TABLE OF CONTENTS



ACKNOWLEDGMENTS ............................. .................

LIST OF TABLES .............................................

LIST OF FIGURES ................................................

NOMENCLATURE ...................................................

ABSTRACT .......................................................

CHAPTERS:

I. INTRODUCTION ..........................................

II. DESCRIPTION OF EVAPORATOR AND COMPUTER INTERFACING
EQUIPMENT .............................................

11.1 Evaporator Layout and Description ...............

11.2 Operating Notes .................................

11.3 Evaporator Instrumentation .....................

11.4 Transducing and Controlling Equipment ...........

11.5 IBM 1070 Interface ..............................

11.6 Software

III. DYNAMIC MODEL AND PARAMETER ESTIMATION ................

III.1 Dynamic Model ..................................

III.1.1 State Equations .........................

111.1.2 Connection Equations ....................

111.1.3 Heat Transfer Equations .................

111.1.4 Decision Variables ......................

111.1.5 Assumptions .............................

111.2 Parameter Estimation ...........................

111.2.1 Stochastic versus Deterministic
Estimation ..............................


Page


i
vi

viii

x

xiii



1


5

5

8

9

13

16

17

22

23

23

26

26

29

30

31


31








111.2.2 Experimental Work for Determining 1ea ... 34

111.2.3 Calculations and Results for 1a ........ 40

111.2.4 Experimental Work for Determining elb
and 02 ........................ .... .. 51

111.2.5 Calculations and Results for elb and
2 ............................... ...52

IV. MINIMUM TIME CONTROL POLICY ........................... 63

IV.1 Statement of the Problem for the Evaporator ..... 63

IV.1.1 State and Control variables .............. 63

IV.1.2 State and Control variable Constraints ... 66

IV.1.3 Control Scenarios ........................ 67

IV.1.4 Summary of the Problem Statement ......... 72

IV.2 A Minimum Time Algorithm ........................ 73

IV.2.1 General Problem .......................... 73

IV.2.2 Lagrange Formulation and Necessary
Conditions ............................... 74

IV.2.3 Comments on the Necessary Conditions ..... 79

IV.2.4 Minimum Time Algorithm ................... 81

IV.3 Solution to the Evaporator Problem .............. 85

IV.3.1 Problem 1.' Constraint on the Second
Effect Hold-up ........................... 85

IV.3.2 Problem 2. Fixed Feed Rate .............. 102

IV.3.3 Problem 3. No Bound on the Second
Effect Hold-up ........................... 110

IV.4 Experimental Runs ............................... 123

FOLDOUT NOMENCLATURE LIST ...................................... 148

V. COMMENTS AND RECOMMENDATIONS .......................... 149

V.1 Model ............................................ 149

V.2 Experimental Setup ............................... 150








V.3 Theory ......................................... 152

V.4 Conclusions .................................... 152

APPENDICES:

A. HEAT TRANSFER EQUATIONS AND OTHER RELATIONSHIPS ..... 154

A.1 Relation between Temperatures and Enthalpies ... 154

A.2 Heat Transfer Equations--First Effect .......... 154

A.2.1 Sensible Heating Zone ................... 155

A.2.2 Vaporizing Zone ......................... 157

A.3 Heat Transfer Equations--Second Effect ......... 160

B. LISTING OF COMPUTER PROGRAMS ........................ 162

LITERATURE CITED ........................................... 206

BIOGRAPHICAL SKETCH ......................................... 208













LIST OF TABLES


Table

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13


1 Data for Run A1..........................

1 Data for Run A2..........................

1 Data for Run A3..........................

1 Data for Run A4..........................

1 Data for Run A5..........................

versus Observed Values of T1 for Run Al....

versus Observed Values of T1 for Run A2....

versus Observed Values of T1 for Run A3....

versus Observed Values of T1 for Run A4....

versus Observed Values of T1 for Run A5....

1 Data for Run B1..........................

1 Data for Run B2..........................

1 Data for Run B3..........................


3.14 Calculated versus Observed Values of T, and T2 for
Run B1 ..............................................

3.15 Calculated versus Observed Values of T1 and T2 for
Run B2 ................................ ..............

3.16 Calculated versus Observed Values of T1 and T2 for
Run B3 ................................ ..............

4.1 State Variables for Problem 1, Iteration 1...........

4.2 Adjoint Variables for Problem 1, Iteration 1..........

4.3 State Variables for Problem 1, Iteration 2...........

4.4 Adjoint Variables for Problem 1, Iteration 2..........


Experimenta

Experimenta

Experimenta

Experimenta

Experiment

Calculated

Calculated

Calculated

Calculated

Calculated

Experimenta

Experimenta

Experimenta


Page

35

36

37

38

39

46

47

48

49

50

53

54

55


59


60


61

92

93

94

95








Table Page

4.5 State Variables for Problem 1, Iteration 3............ 96

4.6 Adjoint Variables for Problem 1, Iteration 3.......... 97

4.7 State Variables including Concentration Dynamics...... 100

4.8 State Variables for Problem 2, Iteration 1............ 107

4.9 State Variables for Problem 2, Iteration 4............ 108

4.10 State Variables for Problem 2, Iteration 5........... 109

4.11 Adjoint Variables for Problem 3, Iteration 1.......... 117

4.12 State Variables for Problem 3, Iteration 3............ 118

4.13 Adjoint Variables for Problem 3, Iteration 3.......... 119

4.14 State Variables for Problem 3, Iteration 5............ 120

4.15 Adjoint Variables for Problem 3, Iteration 5.......... 121

4.16 Experimental Data for Run C1 .......................... 130

4.17 Theoretical Minimum Time Simulation for Run Cl........ 131

4.18 Actual Minimum Time Simulation for Run C1............. 132

4.19 Experimental Data for Run C2........................... 134

4.20 Actual Minimum Time Simulation for Run C2............. 135

4.21 Experimental Data for Run C3.......................... 137

4.22 Actual Minimum Time Simulation for Run C3............. 138

4.23 Experimental Data for Run C4.......................... 140

4.24 Actual Minimum Time Simulation for Run C4............. 141

4.25 Experimental Data for Run C5.......................... 143

4.26 Actual Minimum Time Simulation for Run C5............. 144

4.27 Experimental Data for Run C6.......................... 146

4.28 Actual Minimum Time Simulation for Run C6............. 147












LIST OF FIGURES


Figure Page

2.1 Schematic Diagram of the Double Effect Evaporator..... 7

2.2 Evaporator Instrumentation.. ......................... 11

2.3 Evaporator Instrumentation.......................... 12

2.4 Layout of Transducer and Controller Cabinet and IBM
1070 Cabinets......................................... 15

2.5 Process Interface Computer Information Flow....... 18

2.6 Software Setup........................................ 20

3.1 Variables for Material and Energy Balances............ 24

3.2 Calculated versus Observed Values of T1 for Run Al.... 41

3.3 Calculated versus Observed Values of T1 for Run A2.... 42

3.4 Calculated versus Observed Values of T1 for Run A3.... 43

3.5 Calculated versus Observed Values of T1 for Run A4.... 44

3.6 Calculated versus Observed Values of T1 for Run A5.... 45

3.7 Calculated versus Observed Values of T1 and T2 for
Run B1 .............................................. 56

3.8 Calculated versus Observed Values of T1 and T2 for
Run B2 ............................................... 57

3.9 Calculated versus Observed Values of T1 and T2 for
Run B3 ............................................... 58

4.1 Control, State and Adjoint Variables for Problem 1,
Iteration 1 ......................................... 89

4.2 Control, State and Adjoint Variables for Problem 1,
Iteration 2 .......................................... 90

4.3 Control, State and Adjoint Variables for Problem 1,
Iteration 3 .......................................... 91

4.4 Optimal Simulation including Concentration Dynamics... 99







Figure Page

4.5 State Variables for Problem 2, Iterations 1, 4 and 5.. 106

4.6 Control, State and Adjoint Variables for Problem 3,
Iteration 1........ .................................. 114

4.7 Control, State and Adjoint Variables for Problem 3,
Iteration 3 .......................................... 115

4.8 Control, State and Adjoint Variables for Problem 3,
Iteration 5 ......................................... 116

4.9 Filtered versus Actual Flow Rate...................... 127

4.10 Experimental versus Actual Minimum Time for Run C1.... 128

4.11 Experimental versus Optimal Minimum Time for Run Cl... 129

4.12 Experimental versus Actual Minimum Time for Run C2.... 133

4.13 Experimental versus Actual Minimum Time for Run C3.... 136

4.14 Experimental versus Actual Minimum Time for Run C4.... 139

4.15 Experimental versus Actual Minimum Time for Run C5.... 142

4.16 Experimental versus Actual Minimum Time for Run C6.... 145









NOMENCLATURE


A = Heat transfer area, ft2

C = Solute concentration, Ibs/(lb solution)

C = Specific heat, Btu/lb

D = Diameter, ft

G = Mass velocity, lbs/(ft2)(hr)

Gr = Grashof number, dimensionless

H = Hold-up, Ibs

I = Index set

K = Index set

L = Length, ft

N = Number of tubes

P = Pressure, lbs/(in2)

P(t) = Covariance of estimate, vector

Pr = Prandtl number, dimensionless

Q = Heat transfer rate, Btu/(hr)

Q(t) = Covariance of process noise, vector

R(t) = Covariance of measurement noise, vector

Re = Reynold's number, dimensionless

T = Temperature, F

AT = Temperature difference, F

U = Overall heat transfer coefficient, Btu/(hr)(ft2)(OF)

U(t) = Control vector

V(t) = Process noise, vector

V' = Vapor volume, ft3

V = Vapor flow rate, Ibs/min









W = Liquid flow rate, Ibs/min

X = State variables, vector

X = Estimate of state variables, vector

Xt = Lockhart-Martinelli factor

Y = Calculated observations, vector

Y = Actual observations, vector

f = Function of

g = Acceleration due to gravity, ft/(hr2)
h = Liquid enthalpy, Btu/lb

hv = Vapor enthalpy, Btu/lb

h = Film coefficient, (Btu)(ft)/(hr)(ft2)(OF)

k = Thermal conductivity, (Btu)(ft)/(hr)(ft2)(F)

p = Variance of estimate

t = Time, minutes

u = Control variable

v = Process noise

w = Measurement noise

x = State variable

x = Estimate of state variable


Subscripts

o = Outside world

1 = First effect

2 = Second effect

i j = From unit j to unit i

a = Before

c = Condensate









s = Sensible heating zone

t = Tube

B = Boiling zone

w = Wall conditions

F = Feed

i/j = At "i" given conditions at "j"

f = Film conditions or final condition


Superscripts

i = Inside

o = Outside

v = Vapor

* = Optimality


Greek Letters

a = Point constraint multiplier

B = Inequality constraint multiplier

o = Estimated parameters, vector

6 = Estimated parameter

p = Density, lbs/(ft3)

A = Lagrange multiplier or latent heat of vaporization,

Btu/lb

= Viscosity, Ibs/(ft)(hr)







Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy in Chemical Engineering



DESIGN AND IMPLEMENTATION OF MINIMUM TIME COMPUTER CONTROL SCHEMES
FOR START-UP OF A DOUBLE EFFECT EVAPORATOR

By

Santosh Nayak

March, 1974


Chairman: Dr. Arthur W. Westerberg
Major Department: Chemical Engineering


The application of optimal control theory to a chemical engi-

neering problem is investigated by the development and implementation

of a control policy for the minimum time start-up of a double effect

evaporator.

The particular evaporator on which experimental runs were made

was a laboratory scale double effect evaporator with reverse feed. It

was completely instrumented for control by the installation of orifices

for measuring flow rates, thermocouples for measuring temperatures,

pressure taps for measuring pressures and hold-ups, and pneumatic con-

trol valves for manipulating flow rates. Transducing and controlling

instruments were installed. In order to do on-line computerized data

logging and control, interfacing of the process with the IBM 370/165

computer on campus was provided by an existing IBM 1070 interface. A

part of the existing user circuitry associated with the interface had

to be rewired and modified to function appropriately.

The dynamic model of the evaporator consisted of six differential

or state equations and about sixty algebraic equations. This latter







group consisted of connection equations between the effects, property

correlations and heat transfer equations. To overcome the uncertainty

in the empirical relationships for the inside film coefficient two

unknown parameters were introduced, one for each effect. These para-

meters were estimated by correlating model predictions with data

collected on experimental runs. A nonlinear least-squares technique

was utilized to get the best fit.

The algorithm used for the theoretical development of a minimum

time policy is one in which Hamiltonian minimizations result in control

policy updates on successive iterations. Control variable constraints

and point constraints are accounted for along the trajectory. The

utility of the algorithm is enhanced by assuming a control scenario and

determining whether it is optimal when compared to other likely scenarios

This approach keeps the number of active state and adjoint variables to

a minimum at any particular time resulting in simple Hamiltonians and

less computational expense for the integration of the state and adjoint

equations and Hamiltonian minimizations.

The minimum time algorithm was used on the evaporator model to

determine the optimal policy under three different sets of conditions.

In the first case it was assumed that there was a constraint on the

maximum value of the second effect hold-up. The second case dealt with

a different set of control variables in that the feed rate to the second

effect was assumed to be fixed. In the third case the assumption of a

constrained maximum second effect hold-up was done away with. The

simulation results indicated that the third case resulted in the smallest

start-up time with the optimal policy calling for an overfilling of the

second effect followed by a gradual decrease in the second effect hold-up







to the desired value which took place when boiling just started in

the second effect.

For all three cases it was found that the control policy is

bang-bang in nature and that the control switches occur at times at

which the point constraints are met on the assumed scenario. Because

of this the switching times can be related to the state variables

and a feedback control policy is obtained.

Experiments were run to try out the optimal control policy and

to test the model. On an average, the simulations resulted in final

times which were between ten and fifteen percent within those obtained

experimentally. This accuracy was reasonable considering the experi-

mental problems associated with hold-up measurements and analog control

of the hold-ups and the theoretical problems associated with the as-

sumption of the heat transfer mechanisms.












CHAPTER I

INTRODUCTION


Optimal control theory has been developed to a fairly sophis-

ticated level in the field of electrical engineering. However, the

uses of the theory and possible applications in chemical engineering

have been virtually unexplored. The reasons for this rather limited

progress on both the theoretical and applied fronts are many (Foss,

1973).

The starting point in the applications of control theory is

a good dynamic model of the process. Most chemical processes have

been modeled poorly due to an incomplete understanding of the complex

interactions among numerous variables. High dimensionality and non-

linearities in behavior require the use of sophisticated numerical

techniques for the simulation and design of control schemes. Many

chemical processes have inherently large time constants which make

them unsuitable for control. It is generally not possible to make

all the measurements that are required for a feedback control scheme,

and, in addition, measurements are subject to noise which is not

readily filtered in nonlinear systems. Since most of the states are

not measurable, there is a need for building reduced order observers

which again is a formidable problem for nonlinear systems.

In spite of the above drawbacks, quite a few articles on the

subject of optimal control have appeared in the chemical engineering







literature during the last five years. The early investigations dealt

with the control of simplified lumped parameter linear processes. The

linear quadratic loss problem resulted in feedback control which was

particularly useful in regulatory control; i.e. control in the face of

disturbances (Nieman and Fisher, 1970; Newell and Fisher, 1971; Newell

et al., 1972). The arbitrary nature of the correlation of the weighting

matrices to the actual physical problem often makes the linear quadratic

loss problem unrealistic. Later investigators extended these techniques

to the control of nonlinear lumped parameter systems by using various

forms of linearization on the system equations (Weber and Lapidus, 1971a,

1971b; Siebenthal and Aris, 1964; Tsang and Luus, 1973). Others worked

on nonlinear systems with one or two control and state variables (Joffe

and Sargent, 1971; Jackson, 1966). The control of distributed parameter

systems is still in its infancy. Some investigators have reported sub-

optimal control of distributed systems in which some other criterion,

such as minimization of a Lyapunov functional, is used (Vermeychuk and

Lapidus, 1973a, 1973b; Chant and Luus, 1968). Simulated start-up studies

have been made on plate distillation columns (Pollard and Sargent, 1966)

and autothermic reaction systems (Jackson, 1966).

This work was directed towards applying existing control theory

to a useful chemical process. The aim was to study the dynamics and

control of a simple, yet reasonably complex, piece of equipment commonly

found in the chemical industry. A double effect evaporator was chosen

as the subject of study for the above reason and also because a laboratory

scale double effect evaporator was available for experimental work. The

study involved,

a) Developing a nonlinear model for the evaporator.







b) Estimating model parameters to fit experimental data.

c) Developing a minimum start-up time control policy taking into

account constraints on the state and control variables and putting

the optimal policy in feedback form in terms of switching times.

d) Experimentally determining the effect of the policy.

This approach differs from previous ones in a few respects.

The model is highly nonlinear and is treated as such. No linearization

is resorted to as start-up involves large changes in the state variables

and linearized equations would be inaccurate. The mathematics involved

in obtaining the minimum time policy is simplified as the approach

adopted presupposes a start-up scenario and then verifies that it is

optimal. The algorithm leading to the optimal policy handles con-

straints on control and state variables in a logical fashion by directly

holding the state or control variable on the constraint and changing

the equation set and its solution procedure as a result. This avoids

the use of penalty function methods and the like. Finally, the control

policy is experimentally verified.

The minimum time objective was chosen primarily from the point

of view of economics. Control costs during start-up are minimal com-

pared to the start-up time in batch processes. Reducing the start-up

time results in reduced down time thus improving cycle efficiency and

increasing profits. The food industry is an example of an industry

which must shut down frequently to have the processing equipment cleaned.

Orange juice is concentrated in multiple effect evaporator systems, and

these systems are cleaned about three times a day. A second reason for

minimum time start-up was more specific to the ultimate use of the parti-

cular double effect evaporator investigated. It is to be used in an







undergraduate laboratory experiment in computer control, and past

experiences indicated that it took a very long time to bring it to

steady state under manual control. Thus, in order to reduce the

start-up time and consequently to reduce the amount of on-line com-

puter time for steady state observations, it was imperative to have

start-up in a minimum time.

Chapter II contains a description of the experimental evapo-

rator, the instrumentation and the interfacing equipment with the

IBM 370/165 (which is the main computer on campus). Chapter III deals

with the building of a dynamic model for the evaporator and also the

estimation of parameters from experimental data. A derivation of the

optimal control algorithm is given in Chapter IV. It also contains the

simulated and experimental results of the application of the control

algorithm. Some comments and proposals for further work are given in

Chapter V. Appendix A contains all the heat transfer equations which

supplement the main model equations in Chapter III. A listing and

description of the computer program implementing the optimal control

algorithm is the subject of Appendix B.












CHAPTER II

DESCRIPTION OF EVAPORATOR AND COMPUTER INTERFACING EQUIPMENT



II.1 Evaporator Layout and Description


The double effect evaporator is located in the unit operations

laboratory of the chemical engineering department. Figure 2.1 is a

schematic of the double effect, showing the arrangement of the two

effects, EV1 and EV2, and the basic process and vapor lines. Note

that backward feed is used; that is, the vapor flow and process fluid

flow are in opposite directions.

The first effect is a long tube vertical (LTV) evaporator.

It contains 3 tubes, each 9 feet, 6 inches (2.90 m) long and 1 inch

(0.0254 m) O.D. with heating steam at about 20 psig (2.39 bar) on the

outside of the tubes. The process fluid flows upward through the

tubes either by natural or forced circulation. The latter method is

almost always used because of the increased heat transfer coefficients

obtainable. The pressure on the process side is at or slightly above

atmospheric.

The mixture of process fluid and vapor formed in the first

effect enters a vapor-liquid separator, SE1, which is at the same

pressure as the first effect. The liquid is drawn off the bottom of

the separator and is recirculated back into the first effect by pump

PU3 after some liquid product is withdrawn. Fresh feed to the first









































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effect is pumped by pump PU2 from the second effect. The vapor from

separator SE1 is used as the steam input in the second effect. This

leads to steam economy as one pound of heating steam used in the first

effect should evaporate more than one pound of water from the first

and second effects combined.

The second effect, EV2, is a calandria type effect in which

there are 15 tubes, each 2 feet, 4 inches (0.711 m) long and 1 inch

(0.0254 m) O.D. The effect also has a 2 inch (0.0508 m) O.D. central

downtake. Heat transfer is by natural convection only, resulting in

much lower heat transfer coefficients compared to the first effect.

Fresh preheated feed is pumped into the bottom of the second effect

from the feed tank by pump PU1. The heating medium is the vapor from

the first effect on the outside of the tubes. Above the calandria is

a vapor body which separates the vapor from the liquid. The vapor is

drawn into a condenser, CDI, by means of a vacuum produced by a steam

jet ejector. The ejector maintains the pressure on the process side

in the effect at around 10 inches mercury vacuum (0.675 bar).

The vapor condensate from the first effect is collected in tank

T1 and that from the second effect is collected in tank T2, both of

which are maintained at a vacuum by the same steam jet ejector.


11.2 Operating Notes


There were a few precautions which had to be observed during

operation.

1) The feed rate was kept at around 2-3 gpm (0.12 to 0.18 kg/s).

2) The recirculation rate in the first effect was kept at a maximum

of 15 gpm (0.9 kg/s). A higher rate caused entrainment of liquid with







the vapor in the vapor-liquid separator SE1. This separator has no

baffling of any kind and is very inefficient at high flow rates.

3) To avoid cavitation in the recirculation pump PU3, care was taken

to see that the vertical suction leg from the separator to the re-

circulation pump was always filled with liquid. This was particularly

critical when the pump was first started. Incomplete filling of the

vertical leg led to pulsating flows resulting in large upsets in the

evaporator operation. A recirculation rate higher than 15 gpm (0.9

kg/s) also caused a high discharge head on pump PU3, much higher than

the maximum discharge head on pump PU2 (which is of a smaller capacity),

eliminating all flow of fresh feed to the first effect.

4) The liquid level in the second effect was maintained around the

top of the tubes for best utilization of the heat transfer area.



11.3 Evaporator Instrumentation


As part of this work the evaporator piping had to be modified

to accommodate the instrumentation required for control. The work con-

sisted mainly of installing pneumatic control valves, orifices, pressure

taps, thermocouples and extra manual valves. Figures 2.2 and 2.3 show

the detailed instrumentation of the evaporator. The legend of Figure

2.2 also applies to Figure 2.3.

Three pneumatic control valves, CV1, CV2 and CV3, were installed

in the feed, inter-effect and recirculation lines respectively. CV2

and CV3 were normally closed (air-to-open) valves and were installed in

bypasses on the lines, whereas CV1 was a normally open valve and was in-

stalled in the feed line as such. The purpose of the by-passes was to












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allow for complete or partial manual control of experimental runs

when desired.

Flow rates were measured with square-edged orifices, OR1,

OR2, OR3 and OR4, installed in the feed, inter-effect, recirculation

and product lines respectively. The pressure drop across an orifice

is indicative of the flow through it.

Liquid levels (proportional to hold-ups) were measured by

taking the difference in total pressure between the bottom and top

of each of the two effects. Pressure taps were installed at both

ends of the sight glasses for this purpose. The upper taps were

also used to measure absolute pressure in the effects.

Temperatures were measured by jacketed copper-constantan

thermocouples, TC1, TC2, TC3 and TC4. These were installed in the

feed line, at the exit of the second effect, at the entrance to the

first effect, and in the steam chest of the first effect, severally.

All liquid lines from the pressure taps and air lines to the

valves were brought to a central panel in the front of the evaporator

with 1/4-inch poly-flo tubing. Quick-connect fittings were used at

the panel so that leads to the interfacing equipment could be con-

nected quickly when required. The thermocouple wires also terminated

with special thermocouple outlets at the panel.



11.4 Transducing and Controlling Equipment


The transducing and controlling equipment was installed in a

19-inch relay rack on casters. All air and liquid lines were of poly-

flo tubing with quick-connect fittings. This made the rack very ver-

satile as it can be moved to a number of different pieces of equipment







if desired. A layout of the cabinet is shown in Figure 2.4.

The pneumatic controllers have adjustable proportional and

reset action, motorized set point control, and indication facilities.

One Fischer and Porter model 51 and three Taylor model 662R control-

lers were installed. The set point motor of the Fischer and Porter

controller operates on a pulse train input and Taylor controllers on

a 24-volt DC signal. The controllers are equipped with feedback

potentiometers which indicate their set point positions. The pneu-

matic input signal range to each controller (from the DP cells) is

3-15 psig and that of the output pressure to the associated valve

is also a 3-15 psig signal.

The EMF to pneumatic converters (not used in the current ex-

periments) are Foxboro model 33A converters. They can transduce

either a millivoltage or voltage signal into a 3-15 psig pneumatic

signal.

The differential pressure (DP) cells used are Foxboro Model

13A DP cells. The adjustable range of the differential input signal

is 0-500 inches water and the proportional pneumatic output is in the

3-15 psig range. These DP cells were used to transduce the pressure

drops across the orifices and the pressure differences corresponding

to the liquid levels. The outputs were thus proportional to the flow

rates or to the liquid levels (hold-ups). The DP cells were also

used for measuring absolute pressures by venting the high pressure

side when measuring pressures above atmospheric. The output pressure

in this case was proportional to the vacuum or above-atmospheric

pressure.













-- z w )I
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11. C)I Z ) I Q C Q lC'


SCD o UJ l < 0 oQ




.a:: C, V) F- U) Im


= F-- 0
01 Z I I 0 QT I U





F o 0 C' >< rJ O



:K LU CD : L3 C

I- __ a _- F -




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0
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lu 1 2 I- C F


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J LU
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a )

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of 0 F- 0 -J (-J Q)
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11.5 IBM 1070 Interface


An IBM 1070 interface was used which consisted of an IBM 1071

central terminal unit, two IBM 1072 multiplexing units, an IBM 1073

Model 3 digital to pulse converter and an IBM 1075 decimal display.

The interface with all the auxiliary equipment resides in 3 relay racks

as shown in Figure 2.4. These cabinets were originally assembled by

R. C. Eschenbacher and are described in his Ph.D. thesis (Eschenbacher,

1970). However, as part of this work, some of the equipment had to be

rewired to accommodate pulse duration outputs and most of the relays

had to be rewired and changed to double coil relays in order to isolate

the IBM equipment from the user circuitry. The entire set up is de-

scribed in detail in the GIPSI (General Interface for Process Systems

Instrumentation) hardware manual (GIPSI, 1973) and is summarized very

briefly here.

1) Input

The pressure-voltage transducers converted the 3-15 psig air

signals from the DP cells to 0-5 volt DC signals which were fed as

analog inputs to the 1070. The millivoltage thermocouple signals

were also fed as analog inputs through the special thermocouple in-

put feature in the 1070. The unit also has a facility for digital

input which was not used.

Another convenient input facility frequently used was a form

of digital input through pre-designated demand functions which were

dialled into the system through rotary switches.

2) Output

Output from the 1070 was in digital and in pulse form. The

digital output was used mainly to ring a bell to alert the operator







to possible alarm conditions in the hardware and software. The

pulsed output was obtained from the 1073 and was used to move the set

points on the controllers. The digital to pulse converter (1073)

has outputs in the form of a pulse train as well as a duration pulse.

The 1075 decimal display is a feature which utilizes digital

outputs and was used to display particular variable values or error

codes.

3) Process Alerts

The 1070 interface is linked with the IBM 370/165 computer on

campus. The seven process alerts attached to a process alert (PA)

bus in the 1070 provide a hardware interrupt capability of the com-

puter by the process. The software issues a conditional read of the

1070 terminal to the computer. It is then in a hardware wait stage.

When the PA bus is activated, by one of the process alerts on the

1070, the IBM 370/165 computer senses the closure and reactivates

the software which then determines which PA was set. The software

then resets the PA and executes the program associated with the PA.

Process alert 1 has the added facility of being set automatically by

a hardware poller on which the timing is adjustable. This enables

one to have PA1 periodically and automatically set after a predeter-

mined time interval has elapsed.

Figure 2.5 is a schematic illustrating the flow of information

among the various hardware components of the experiment.



11.6 Software


The GIPSI software package was originally written by R. C.

Eschenbacher. Version 2, the version used, was written by L. A. Delgado












































- LU

0:

PI
F-










0













LO




c-.
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CD



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(GIPSI, 1973) and was marginally modified as part of this work to

extend its capabilities. It is written entirely in Fortran (with

the exception of certain input/output routines in BTAM provided by

IBM), has an extensive debugging facility, and an extensive error

handling facility to flag user software and hardware errors. A

simplified flow chart is reproduced from Westerberg and Eschenbacher

(1971) and is shown in Figure 2.6. It is described in greater detail

in Westerberg and Eschenbacher (1971) and GIPSI (1973).

The heart of the software is the concept of the execute and

delay stacks. When a PA is set, it is identified and the program

(or programs) associated with it are stacked by the program stacker

in the order of priority on the execute stack. Control then passes

to the Execute subprogram which then examines the execute stack and

passes control one at a time to the programs that are due for execu-

tion. If the sequence began with PAl, its response program CLOCK

removes programs from the delay stack if their delay time has expired

and puts them on the execute stack. The Execute subprogram then finds

additional programs on the execute stack which it continues to remove

and cause to be executed. This is done until the execute stack is

empty whence control returns to the PA handler which issues a condi-

tional read and the IBM 370/165 again waits for a PA to be set to

start the cycle again. All delay times are compared to the computer

clock. Data on program priorities, delay time, etc. are specified

in the Program Descriptive Data.

All the user has to do is to provide his specific programs

associated with the various process alerts, the program descriptive

data for all his programs, and a subroutine GOTO which the Execute














































II
II















L~II
>< -- )




bj I








auJ
I- , \

w ~

(I)I




21


subprogram uses to pass control to CLOCK and the user and system

subprograms.

The computer costs are extremely low when based mainly on

central processing unit (CPU) time. The software utilizes very

little CPU time. Typical costs are in the range of $3-5 an hour

provided no elaborate computations are called for in the user pro-

grams. However, costs for core residency charges dominate as the

basic software package requires around 20,000 words or 80,000

bytes of core.












CHAPTER III

DYNAMIC MODEL AND PARAMETER ESTIMATION



The dynamic modeling of multiple-effect evaporators has been

extensively investigated in recent years at the University of Alberta

(Andre and Ritter, 1968), (Newell, 1970). In simulation and experi-

mental work high order, linear models have been found to be satisfactory.

However, linear models are not realistic when the operating conditions

change drastically as in start-up. In the first part of this chapter

a nonlinear, first order, lumped parameter model is proposed. The

first order and lumped parameter nature of the model was resorted to

for two main reasons:

1) The model was simple and adequately described the data.

2) The model was used to devise an optimal control policy for minimum

time start-up. Optimal control theory has been rigorously developed

for lumped parameter systems and its extension to distributed systems

has not yet been extensively investigated.

In addition, the model presented here takes into account heat

transfer dynamics from the viewpoint of film coefficients. Although

this leads to complicated algebraic equations, it has the advantage of

leading to a better understanding of the heat transfer dynamics. It

also gives rise to two constant correction parameters. The necessity

for these parameters is due to the uncertain coefficients that are used

in the film coefficient equations. The second part of this chapter

deals with the estimation of these parameters to fit the experimental data.







III.1 Dynamic Model

The dynamic model is a collection of the material and energy

balances for each effect. For a double effect evaporator concen-

trating a solution with one major solute, there are two material

balances (one for the solution and one for the solute) and one energy

balance for each effect, giving rise to a total of six dynamic or

state equations for the two effects. In addition, there are dynamic

equations for the vapor phases and metal but the time constants of

these are negligible compared to the six mentioned earlier (Andre,

1968) so that these dynamic equations could be reduced to be alge-

braic equations. This procedure of setting the derivatives of the

equations with small time constants to zero reduces the order of

the system. The full model will be presented here. In later chapters,

appropriate simplifications will be applied as some of the model

states are held fixed (for example, as boiling does or does not take

place). A summary of all the assumptions made is presented at the

end of the model. Refer to Figure 3.1 for the symbols used for the

flows, hold-ups and temperatures. There is also a foldout nomenclature

list on page 148.

III.1.1 State Equations

1) First and second effect hold-ups, H1 and H2.

dH1
dt- 12 V21 1 l 301

dHl
dt -=IF V2 2 (3.2)
2) First and second effect enthalpies.







































L)

u








ra




cr






a-







U.

ct
L






















-0
r-













r-
C

VI







Since the evaporator was to be used ultimately for the con-
centration of dilute solutions, it was assumed that there would be
no boiling point elevations in the effects. Also, perfect mixing
is assumed which would be close to the case for small hold-ups and
dilute solutions.

d(Hlh)
dt- = L'12h2 V21h (1 11+ l01)h1 + 1

where Q1 is the heat transferred from the steam. Simplifying this
using equation (3.1), we get

dh1 1
dt I 12(h2 h1) + (h] (3.3)

Similarly, an energy balance on the second effect under the
same assumptions gives rise to
dh2
[W2 F (hF h 2(h h 2 (3.4)
at h2) + V02h2 2+Q2

3) First and second effect solute material balances.
Again, assuming perfect mixing we have for the first effect
solute,
d(H1C1)
dt 12C12 (11 + "01)C1


Simplifying this with equation (3.1) we have
dC I
dC1 1
dt 1 1-[2(C12 C1) + V21C1] (3.5)

Similarly, a balance on the second effect solute yields,

dC2 1
-t = T [ F(CF C2) + V02C2] (3.6)







111.1.2 Connection Equations

In addition to the state equations listed above, there are

algebraic equations which arise due to the mixing of the two streams

between the second and first effects. One energy and two material

balances describe the mixing as follows:

W12 = '2 + W11 (3.7)

W12C12 = 12C2 + "UIIC1 (3.8)


1 12h2 = 2h2 + lllhl (3.9)
111.1.3 Heat Transfer Equations

These equations arise in computing the terms Q1 and Q2 that

arise in the enthalpy equations (3.3) and (3.4).

The heat transfer rate Q1 is a function of the steam temper-

ature, the first effect temperature, the inside and outside film

coefficients, the wall resistance including fouling and the heat

transfer area in the first effect. The inside film coefficient is

a function of the flow rate through the tubes, the vaporization,

the inner wall and bulk temperatures and the entrance temperature.

The heat transfer mechanism is initially simple--a combination of

the Dittus-Boelter equation for the inside and the Nusselt equation

for the condensing steam. However, when boiling takes place two-

phase heat transfer occurs because of the vapor formed. The complete

boiling mechanism is a topic for further investigation. Approximate

correlations were obtained from (Fair, 1960, 1963a, 1963b), (Hughmark,

1969) and more recently from (Tong, 1965). The complete list of

equations leading to the determination of 01 from the state variables

and flow rates is given in Appendix A. Due to the uncertainty in







the empirical equations which predict the inside film coefficients,

a parameter 01 was introduced in the overall heat transfer equations

(A.17) and (A.40). It is assumed that the outside film coefficient

is predicted by the Nusselt equation, (A.14) and (A.28) to a rea-

sonably high degree of accuracy as is borne out later by experiment.

It is also assumed that the parameter 91 has two different values

depending upon the heat transfer mechanism in the first effect. This

depends upon the stage of start-up as follows:

1) e1 = Ola, when the first effect liquid is being heated. The

dominant equation for the inside film coefficient is solely the Dittus-

Boelter equation (A.16).

2) 61 = alb, when the liquid in the first effect is boiling. The

inside film coefficient is a combination of many factors including a

coefficient due to nucleate boiling (A.38) and a two-phase convective

coefficient (A.33).

The equation for the first effect heat transfer rate is given

in functional form as:

Q1= Q (Ts'T1T 12,'H W12',V21' 1) (3.10)

where the temperatures, T, are functions of the enthalpies, h, in the

form
T = f(h)

Equation (3.10) has implicitly used the fact that the overall

coefficient, film coefficients and heat transfer area are functions of

temperatures (enthalpies) and hold-ups.

Strictly speaking, the equations describing the steam (vapor)

temperatures or enthalpies, hs, hI and h2 are differential equations,







dp
1 21 c2
v v
d(p hl) v
Vi E- = V21h1 Wc2hc2 2

where Vi is the volume of the vapor space in the first effect, the

vapor-liquid separator and the tubes of the second effect. Note that

two assumptions have been made here--the steam (vapor) is saturated

and there is no subcooling of the condensate.

However, it has been shown by Andre and Ritter (1968) that

the response rate of the steam enthalpy is negligible compared to

that of the hold-up, concentration and enthalpy equations (3.1) to

(3.6) in the two effects. The differential equations describing

the steam density and temperature are so replaced with the steady

state equations

V21 =c2
and = V21(h hc2

or Q2 = V21 (3.11)

where = f(T)


The heat transfer rate in the second effect, Q2, is also a

function of the film coefficients, wall resistance including fouling,

area and temperatures in the second effect. The heat transfer

mechanism is purely natural convection. Here again, it is assumed

that the Nusselt equation (A.47) is reasonably accurate in predicting

the outside film coefficient. The inside film coefficient is pre-

dicted by the natural convection equation (A.51) and the overall

coefficient (A.52) has an undetermined parameter 82 which again can




29

have two values. One value (e2a) is for the heating of the liquid

and the other (e2b) is for the boiling of the liquid in the second

effect. The functional form for Q2 is:

Q2 = Q2(TIT2TFH2'82) (3.12)
where the temperatures have been determined from the corresponding

enthalpies.

Note that there are no liquid flow and vapor flow terms here

as the natural convection overall coefficient is not a function of

these variables.

A subroutine called HEAT has been written to calculate the

heat transfer rates Q1 and Q2 from the temperatures, hold-ups and

flow rates. It is included in Appendix B. The rates 01 and 02 are

found by using all the heat transfer equations in Appendix A. The

inner and outer wall temperatures that figure in the film coefficient

calculations are unknown. These temperatures are initially guessed

and the film coefficients are calculated. The wall temperatures are

adjusted until the equations predict the same heat transfer rate

per unit area across the inside and outside films and the wall. With

the final wall temperatures, the film coefficients and heat transfer

rates are estimated using the appropriate equations depending upon

the nature of boiling and the mechanism.

111.1.4 Decision Variables

Equations (3.1) to (3.6) are the differential equations and

(3.7) to (3.12) are the algebraic equations describing the dynamic

response of the evaporator. After enumerating the number of variables

and the number of equations it is found that there are eight more

variables than there are equations. These eight decision variables

are chosen in a natural way making them manipulative or control







variables. These are:

1) feed flow rate, WF

2) feed temperature, TF

3) feed concentration, CF

4) inter-effect flow rate, Wi2

5) recirculation in first effect, W11

6) product flow rate, W01

7) steam temperature in first effect, T

8) total pressure in second effect, P2

111.1.5 Assumptions

A summary of the assumptions made in writing the model is pre-

sented here.

1) Time responses of the vapor phase and the tube walls are negligible

compared to that of the liquid phase. This results in simpler algebraic

equations for the vapor phase and tube walls and also decreases the

dimensionality of the model.

2) The vapor is saturated and is in equilibrium with the liquid at the

same temperature.

3) Condensate on the vapor side of the first and second effects is not

subcooled and condensate hold-up is negligible.

4) Boiling point elevations due to the presence of solute in the two

effects is negligible. This is justified in the case of dilute solutions.

5) There is perfect mixing in the two effects resulting in lumped

parameter concentration and heat transfer equations.

6) The heat transfer mechanism in the first effect is single phase con-

vection followed by two-phase convection and nucleate boiling and that

in the second effect is natural convection.







7) Slug flow is the predominant flow pattern in the first effect when

boiling takes place.

8) Heat losses are negligible.

9) The inaccuracy of the heat transfer rates is due mainly to the

uncertainty of the inside film coefficient leading to undetermined

parameters to correct for the inside coefficients alone.


111.2 Parameter Estimation

111.2.1 Stochastic versus Deterministic Estimation

An extensive review of parameter estimation techniques in

differential equations is available in Nieman et al., (1971). In

the deterministic case the simplest and most effective method is a

least squares fit. The problem is stated as follows:

Given the state equations
X = f[X(t), U(t), B(t)]

where U(t) are the control or manipulative variables and o(t) are the

parameters to be estimated from experimental data Y(t). The obser-

vations are related to the states and controls by
Y(t) = h[X(t), U(t)]

The problem is to determine the parameters o(t) such that the model

"best fits" the given experimental data Y(t).

Assuming that the parameters are constant, as in the case of

the evaporator, e(t) = 0, the problem can be reduced to a least-squares

estimation
S2
Min i (Yi Yi)2
0 1=1


where Yi is a calculated observation at time t. (i=1 ,...,N) and Y. is







the actual observation.
The alternative to the above deterministic estimation is the

problem of stochastic estimation. It seemed that the model would fit
the data far better if the parameters 0 were updated as each measure-

ment was made. Further, if the states and measurements were subject
to process and measurement noise it would be necessary to estimate the
states using a nonlinear or a linearized Kalman filter. It was apparent

all along that this would require an appreciable amount of computer
time; however, the estimation technique was investigated off-line from

an academic viewpoint.
The approach follows that of Padmanabhan (1970). The model

is assumed to have process and measurement noise v(t) and w(t)
X(t) = f(X(t), t) + V(t)
Y(t) = h(X(t), t) + W(t)

where V(t) and W(t) are white Gaussian noise sequences with zero mean

and covariances Q(t) and R(t) respectively. Note that the control

vector U(t) is expressed in terms of the state vector X(t) in the state

and observation equations above. To the state equations could be
augmented the parameter equations
e(t) = 0 + V'(t)

thus treating the parameters 0 as states.
The problem reduces to estimating the states X(t) and the

parameters 0(t) from the experimental data Y(t). A recursive estimate
X(tk/tk) representing an estimate of X at time tk based on all the

data collected until time tk is given by


d t) f(X(tk/tk), t) P(tk)(tk/tk)







with X(O/O) = p = initial estimate of X(t )

where Z(tk/tk) = hR(tk)-1 ((tk) h(X(tk/tk), tk))

and the covariance of the estimate P(tk) satisfies a matrix Ricatti

equation

P (tk) = P(tk) + P(tk)fx + P(tk)ZxP(tk) + Q(tk)


with P(0) = n = initial covariance of estimate X(t0). In the linear

case the covariance equation can be integrated off-line and stored for

use by the state estimation equations.

To study the effectiveness of the scheme an example was run

off-line in which it was assumed that the first and second effect hold-

ups and the first effect temperature were constant. It was desired to

estimate the second effect temperature and the parameter 6' in the

equation
= hdh2) Q2
S- = H- DF(hF h2 +e2


while measurements were obtained on the second effect temperature

T2 = f(h2)

The state equation and the Ricatti equations were integral

for eight minutes of real time and this took 100 seconds of IBM 3i

CPU time. The results for the first four minutes were as follows


Time
30.7
31.7
32.7
33.7
34.7


X4,measured
118.36
122.63
124.32
125.82
126.62


ted

70/165


x4 x4,predicted e'
1.0
121.4 118.73 0.08
124.1 122.37 0.18
126.76 125.05 0.19
127.34 126.35 0.17







This example showed that it was not practical to try on-line

stochastic estimation for a problem of this nature especially when

there are more state variables. Because of this the deterministic

least-squares estimate was resorted to and the results obtained were

acceptably good.

To account for noisy flow variables a linear filter was used

on the flow measurements in the form

ui(j) = ci(j 1) + (1 a)ui(j)

where O
ui(j) was the filtered value of the flow ui at t.. ui(j) was the

measurement of the flow u. at t.. a=0.3 was a value commonly used.

It was assumed that the temperatures were measured with a high degree

of accuracy.

111.2.2 Experimental Work for Determining 8la

The experimental runs conducted for determining 81 and 02 were

made in two sets, A and B. The runs in set A were made when the first

effect was being heated and hla was estimated.

Effects 1 and 2 were initially filled to their steady state

hold-ups. The product flow, inter-effect flow and feed to the second

effect were cut off. The recirculation flow, W ,ll was held between

10 and 15 gpm (0.6 to 0.9 kg/s). Heating steam was then started to

the first effect. Recordings of the inter-effect temperature T12,

steam temperature, Ts, and recirculation flow, W,11, were made along

with the other variables but it was only these variables that figured

in the ensuing calculations. Tables 3.1 to 3.5 contain the data from

Runs Al to A5.
























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111.2.3 Calculations and Results for 01a

A computer program was written to estimate ,la from the data

collected in Runs Al to A5. The least-squares program used was sub-

routine RMINSQ (Westerberg, 1969). This program was based on a program

coded by M. J. D. Powell and described in Powell (1964). This routine

has the capability of performing a least-squares search over several

functions in several variables. The search routine does not require

evaluation of derivatives.

The equation describing the enthalpy rise in the first effect

is equation (3.3).

dh
dh1 [12(h2 hl) + h +) + 1(3.3)
dt [12(h12 + V21 (h1 Q1]

Runs Al to A5 were conducted when the first effect was not

boiling and with constant hold-ups in the first and second effects.

Thus, the liquid entering the first effect was only the recirculated

liquid, W12 = 11 and the vaporization was zero, V21 = 0. The obser-

vation T12 = T1 since all the liquid entering the first effect was

recirculated. For every value of 61a which subroutine RMINSQ searched

over, equation (3.3) was integrated from the initial to the final

time for each run. Ten functions of the form (T,calc T,observed)2

over the time span were minimized by RMINSQ. The functions toward the

end of the time interval were weighted one hundred times more than those

at the start. This was because the final time at which the first effect

liquid started to boil was more important from the point of view of

possible switching times in the control variables at this point in time.

The results of the minimization are tabulated in Tables 3.6 to 3.10 and

these values are plotted in Figures 3.2 to 3.6. It can be seen that the

















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indicated) which converge on the observed values toward the end times.

On the basis of these results a value for ela was taken as

0.1395.

111.2.4 Experimental Work for Determining 61b and e2

The runs in Set B were made with the first effect boiling and

the second effect initially in the heating and later in the boiling stage.

Three parameters were actually estimated from the data for each run.

81b was a parameter for the boiling of the liquid in the first effect,

62a was a similar parameter for the heating of the liquid in the second
effect and 02b was a parameter for the boiling of the liquid in the

second effect. Note that 62b is, in effect, the revised value of 02a

when there is boiling in the second effect.

The experimental procedure consisted of bringing the hold-ups

in the two effects to their steady state values. The controllers in

the Transducing and Controlling cabinet (Chapter II) were used to main-

tain the hold-ups constant. One controller was used on the first effect

and another on the second. The pneumatic input signal to the first

controller was the output from the DP cell which measured the height

(hold-up) in the first effect. The pneumatic output from this controller

was directed to valve CV2 (Figure 2.2). Thus analog control of the first

effect hold-up was achieved by manipulating flow W12 which is the liquid

input stream to the first effect from the second effect. Likewise, analog

control of the second effect hold-up was achieved by manipulating flow

WF which is the input stream to the second effect.

When the hold-ups were about constant, steam was let in to the

first effect for heating and a vacuum of around ten inches'mercury







(0.675 bar) was maintained in the second effect. Data were recorded when

the first effect temperature was near boiling. The results for three

runs B1, B2 and B3 are shown in Tables 3.11, 3.12 and 3.13 respectively.

111.2.5 Calculations and Results for 81b and 02

A similar least-squares search using subroutine RMINSQ was used

to estimate Olb' 82a, and 82b from the data obtained in Runs B1 to B3.

The function evaluations for RMINSQ entailed integration of all four of

the differential equations (3.1) to (3.4) as neither the hold-ups nor

the temperatures were held constant. Whenever the second effect tem-

perature corresponded to the temperature of boiling in the second effect

(which was found from the pressure observed) parameter 82b was used

instead of 02a. The calculated temperature of the first effect solution

T1 and that of the second effect solution T2 were obtained from the

integration of the state equations (3.1) and (3.3) respectively. The

criteria for minimization were the functions

10
f(l) = (Tlcalc Tlobserved) (3.3)


10
f(2) = (T T )2 (3.4)
i= 2,calc 2, observed


Ten functions of each type were evaluated in the time span of each run

resulting in a total of 20 functions for the evaluation of 1lb' 02a and

82b.
The correspondence between the observed and calculated values is

shown in Tables 3.14 to 3.16, while Figures 3.7 to 3.9 are plots of these

values. The minimization took an average of three minutes CPU time on

the IBM 370 for each run. This was mainly due to the three-dimensional






53















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62



search and the integration of the four differential equations for

every function evaluation. Because of this only three runs were

analyzed. The results for runs B1 and B2 were much better than those

for run B3. On this basis the mean values for elb, 62a and e2b were

taken to be 0.0928, 0.1359 and 0.1763 respectively. Physically,

this meant that the model predicted an inside film coefficient which

was from 80 to 90 percent higher than that obtained experimentally.

This could be either due to the assumptions made in the model or to

the empirical nature of the film coefficient correlations (A.16), (A.39)

and (A.51).









CHAPTER IV

MINIMUM TIME CONTROL POLICY


This chapter deals with the development of a minimum start-

up time control policy for the double effect evaporator using the model

equations of Chapter III. The problem is stated in Section IV.1 and

this involves identifying state and control variables, equality

constraints in the form of algebraic equations, state and control

variable inequality constraints, and possible start-up scenarios.

Section IV.2 contains the derivation of a general algorithm useful

for solving minimum time problems similar to that for the evaporator.

The actual use of the algorithm for solving the present problem is

described in Section IV.3. It contains the results of model simulations

in arriving at the optimal policy for three problems, all of which are

minimum time start-up problems under various conditions. Experimental

verification of one of the minimum time policies and the effectiveness

of the model is presented in Section IV.4.

Refer to the foldout nomenclature list of the more important

symbols at the end of this chapter (page 148) to aid in interpreting

statements made using these symbols in the succeeding sections.

IV.1 Statement of the Problem for the Evaporator


IV.1.1 State and Control Variables

The state variables, which are necessary to describe completely

the state of the process at any particular time, are the differential

variables in the model differential equations (3.1) to (3.6) of








Section III.1.1. For uniformity, x. will be used for a state variable

and X will be the state vector with components xi. These are assigned

as follows:

x1 First effect hold-up (H1)

x2 First effect liquid enthalpy (hl)

x3 Second effect hold-up (H2)

x4 Second effect liquid enthalpy (h2)

X5 First effect solute concentration (C1)

x6 Second effect solute concentration (C2).

The decision variables listed in Section 111.1.4 have to be

defined so that the model is complete. The control or manipulative

variables are chosen from this set depending upon the controllability

of the process and the physical realizability of the control. For

example, the second effect vacuum pressure is not capable of being

manipulated physically on this system and so it is not chosen to be a

control variable. The feed temperature and concentration are not used

as control variables in this problem either. The remaining decision

variables, comprising four flow rates and the steam pressure to the

first effect, can be easily manipulated physically and can be used to

force the process in any desired direction. For example, the feed to

the second effect, WF, and inter-effect flow rate, Wi2, can control

the inventories in the first and second effects. The steam temperature,

Ts, and recirculation rate, W11, have an effect on the first effect

temperature and the rate of increase of the second effect temperature.

The product flow rate, W01, is used to control the product concentration.

These control variables are capable of keeping the process at steady








state and the steady state values for these variables are governed by
the steady state solution of the differential equations (3.1) to
(3.6).
For uniformity, let ui denote a control variable and let U be
the control vector with components ui. The assignment is as follows:
uI Feed to the second effect (1F)
u2 Intereffect flow rate (WI2)
u3 Recirculation flow rate (U11)
u4 Temperature of steam to first effect (TS)
U5 Product flow rate out of first effect (H01).
Rewriting the state equations (3.1) to (3.6) and the algebraic
equations (3.7) to (3.12) in terms of the state and control variable
nomenclature defined above, we have

1 = W12 V21 u3 u5 (4.1)

1
x = W2 (h2 x2) + V21 (x2 h) + Q1] (4.2)

3 = Ul u2 V02 (4.3)

1 2 (4.4)
4 = [l(hF x4) + V2(x4 h) + Q] (4.4)

x5 12(C12 x5) + V21x5]
X1


x6= [u(CF x6) + V02x6] (4.6)
3
Connection equations,

W12 = u2 + u3 (4.7)
+ux(48


W12h12 = u3x2 + u2x4


(4.8)








V12C12 = u3x5 + u2x6 (4.9)

Heat transfer equations,


Q1= Ql(x1' x2 u3' u4, h12' '12' V21) (4.10)

Q2 2(x2, x3' x4, hF) (4.11)

V21 = Q2/A (4.12)


IV.1.2 State and Control Variable Constraints

It is evident that in a real system the control variables
cannot take on all values as there are physical limitations on the

maximum and minimum flow rates and temperatures. The lower limit for
all the flow variables is zero. The lower limit for the steam

temperature is 2120F as steam cannot be supplied at a lower pressure

than atmospheric in the first effect. The upper limit depends upon
the pipe size and the valve size for the flow rates and on the steam

supply pressure for the steam temperature. Thus, all the control
variables are subject to lower and upper bounds of the form

u. < u. < u. (4.13)
Ui,min- -i i,max (4.13)

In a like manner some of the state variables are constrained.
At steady state all the state variables should be greater than or equal
to their steady state (desired) values (xi)--the steady state hold-ups
are the desired operating hold-ups, the steady state temperature in

the first effect should be at least the boiling temperature of water at
1 atmosphere, the steady state temperature in the second effect should









be the boiling temperature of water at the pressure in the second

effect, and the steady state concentration in the first effect should

be equal to the desired concentration. The upper bounds are less

clearly defined; for example, the liquid level for the second effect

(hold-up) should not exceed the overflow limit. The upper bounds

on the temperatures are dictated by the design specifications and by

characteristics of the solution being concentrated. In general, the

state constraints are given by


x X i,max (4.14)

IV.1.3 Control Scenarios

In the start-up of the evaporator, it is useful to visualize

the change with time of the state variables for certain values of the

control variables. It is evident that the control variables determine

the order in which the state variables reach their steady state or

desired values. Intuitively, the optimal policy will endeavor to

force each state variable directly to its final steady state value

and then maintain it while bringing the others to steady state. The

order in which the state variables reach their steady state values

completes the scenario. Theoretically, there should be a total of n

factorial scenarios for a system with n state variables. But most of

these are not possible as certain state variables can reach steady

state only if certain others have. For example, in the case of the

evaporator, the first effect liquid has to boil before the second

effect liquid does. The equations describing a stage in a scenario

are different from those describing another stage. For each stage they








are simplifications of the general equations. A typical scenario
for the start-up of the evaporator (and the resulting simplifications
in the general equations (4.1) to (4.12)) is shown below.
Stage A: tO s time t < t1. Filling and heating first effect.
Control variables:

Feed to second effect at maximum flow rate, ul = ul,max

All feed delivered directly to first effect, u2 = u1
Temperature of steam for first effect at maximum value, u4 = u4,max

No recirculation or product flow possible, u3 = u5 = 0.
Resulting state equations:
First effect hold-up increasing; x1 = u2, x1(t0) = 0
1
Heating of first effect; x2 = ) + h

No increase in second effect hold-up; x3 = 0, x3 = x3(to) = 0
No heating of second effect; x4 = 0, x4 = x4(t0) = hF

No concentration occurring in either effect; x5 = x6 = 0;

x5 = x6 = 5(t0) = x6(t0) = CF'
Time tl, signifying the end of Stage A, is determined when the first
effect is filled, i.e., when xl(tl) = xl"
Stage B: t,1 time t < t2. First effect solution is being heated

and second effect is being filled.
Control variables:

Feed flow to second effect at maximum, ul = ul,max
Flow to first effect stopped, u2 = 0

Recirculation flow set at maximum, u3 = u3,max
Temperature of steam to first effect set at maximum value,


u4 = "4,max








No product withdrawn, u5 = 0.
Resulting state equations:
No change in first effect hold-up; xl = 0, x1 = x1(tl) = xl
Heating of first effect; x2 = Q0/xl
Second effect hold-up increasing; x3 = ul, x3(tl) = 0
No heating of second effect; x4 = 0, x4 = x4(t1) = hF
No concentration occurring in either effect; x5 = x6 = 0,

x5 = x6 = x5(tl) = x6(tl) = CF'
Time t2, signifying the end of Stage B, is determined when the second
effect is filled, i.e., when x3(t2) = x3.
Stage C: t2 < time t < t3. First effect solution is heated to boiling.
Control variables:
Feed flow stopped, u1 = 0
Flow to first effect stopped, u2 = 0
Recirculation flow set at maximum, u3 = u3,max
Temperature of steam to first effect set at maximum value,

u4 = U4,max
No product withdrawn, u5 = 0.
Resulting state equations:
No change in first effect hold-up; x1 = 0, x1 = xl(t2) = x
First effect is being heated; x2 = Ql/xl
No change in second effect hold-up; x3 = 0, x3 = x3(t2) = x3
No heating of second effect; x4 = 0, x4 = x4(t2) = hF
No concentration occurring in either effect; x5 = x6 = 0;

x5 = x6 = 5(t2) = x6t2) = CF'
Time t3, signifying the end of Stage C, is determined when the first
effect starts to boil, i.e., when x2(t3) = x2'








Stage D: t3 < time t < t4. First effect solution is boiling (and
becoming concentrated). Second effect solution is being heated.
Control variables:
Feed to first effect is equal to vaporization from first
effect to maintain constant hold-up, u2 = V21

Feed to second effect set to maintain constant hold-up in
second effect, ul = u2

Recirculation at maximum, u3 = U3,max

Temperature of steam to first effect at maximum, u4 = u4,max

No product withdrawn, u5 = 0.

Resulting state equations:
No change in first and second effect hold-ups; x = x3 = 0;
x = x (t3) = 1 X; x2 = x2(t3) = 2
First effect is being heated; 2 = [u2x4 + V21h ,
Xl
x2(t3) = x2
Second effect is being heated; x4 = [ul(hF x) + ];
x4t3) = hF 3
First effect solution is being concentrated; x (u2CF);

x5(t3) = CF
No concentration of second effect solution; x6 = 0; x6 = x6(l

Time t4 signifying the end of Stage D is determined when the second

effect starts to boil, i.e., when x4(t4) = x4'

Stage E: t4 < time t < tf. Solution in both effects is boiling and
being concentrated.

Control variables:

Feed to first effect set equal to vaporization in first
effect to maintain constant hold-up, u2 = V21


3) = CF









Feed to second effect set to maintain constant hold-up in
second effect, ul = u2 + V02
Recirculation at maximum, u3 = u3,max
Temperature of steam to first effect at maximum, u4 = u4,max

No product withdrawn, u5 = 0.
Resulting state equations:

No change in first or second effect hold-ups; xl = x3 = 0;
Xl = xl(t4) = X', x3 = x3(t4) = x3
First effect is being heated; = (u2x + Vp1h)
xl.
No change in second effect enthalpy; x4 = 0, x4 = x4(t4) = x4
First effect solution is being concentrated; x5 = (lx6)
x1
Second effect solution is being concentrated;

6 = (ulCF u2x6), x(t4) = CF.
x3
Time tf, signifying the end of Stage E, and consequently the final

time, is determined when the first effect concentration reaches a
desired value, i.e., when x5(tf) = x5'
After the final time, tf found above, the control variables

Ul, U2 and u5 are obtained from the steady state solutions of the
differential equations (4.1), (4.3) and (4.5). The feed to the first
effect is such that a constant hold-up is maintained in the first
effect, i.e., u2 = u5 + V21. The feed to the second effect is such that

a constant hold-up is maintained in the second effect, i.e., ul = u2 + V02.

The product rate is such that a constant product concentration is
x6
obtained, i.e., u5 = u2 x5 Simplifying the above three relationships
we can obtain a sequence for determining the control variables ul,

u2 and u5 at steady state as follows:








x6 / x6
u2 V21 1 = u2 + 02; u5 = u2 5


Note that certain point constraints of the form xi(tj) = xi

separate the various stages of the scenario. The control variables,
state and algebraic equations change when these points in time are
encountered.
It is assumed that certain state variables such as the first

and second effect hold-ups will be maintained at their steady state
values once these are attained. That is, the point constraint
xi(t.) = xi is followed by the equality constraint xi = 0. This

latter equality is maintained by calculating the required value of an
appropriate control variable. This is similar to the concept of a
first order state variable inequality constraint (Bryson et al., 1963),
(Bryson and Ho, 1969).
It is clear that another scenario will give rise to a different
ordering of the point constraints, xi(t.) = xi. depending upon the
order in which the state variables reach their steady states. The
optimal scenario is the one which will result in a minimum final time.
This scenario approach simplifies the mathematical problem

to a great extent as a minimum number of state equations have to be
integrated. The equations simplify considerably leading to simpler
adjoint equations and Hamiltonian minimizations.

IV.1.4 Summary of the Problem Statement

The problem is to minimize the final time, that is


Min J = tf
U









subject to the state equations (4.1) to (4.6), the connection equations

(4.7) to (4.9), and the heat transfer equations (A.1) to (A.52) where

the control variables are constrained,

U < U < U
min - max

and point constraints pertaining to the particular scenario are to be

satisfied

xi(tj) = xi ; i = 1,n
j = l,n

IV.2 A Minimum Time Algorithm

IV.2.1 General Problem

The objective is to minimize the final time by selection of

the controls U

Min J = tf (4.15)
U
subject to:

the state equations

X = f(X, U, Z) ; X(O) given (4.16)

the algebraic equations

g(X, U, Z) = 0 (4.17)

the control constraints

Umin < U < Umax (4.18)

or
hli(u.) = ui i,max
1 i i = 1 ,mXi i
(u Uiin h(U) < 0 (4.19)
h(u,) = Ui.min Ui < 0








and the point constraints

Xo(to) = X

Xkl(tkl) = Xkl

xk2(tk2) = k2 (4.20)



Xf(tf) = xf

t = fixed

IV.2.2 Lagrange Formulation and Necessary Conditions

Assume that the point constraints are met at times tO, tkl ...'

tf. Also assume the times ti correspond to all those times when one
(or more) of the controls or states reaches or leaves a constraint.
Next, form the following index sets,

10 = {0, 1, 2, ...., kl, k2, ......... f)
I1 = {1, 2, ..... .. kl, k2, ......... f}

12 = {0, ......., kl, k2, ........, f-l}

Within each set, order the indices so the times ti, i e I0 or I1 or 12
are in increasing order. Also form the index set

K = {kl, k2, ........ }

Introducing multipliers a, and A we can write the Lagrangian
for the problem
L(X, U, ti, i E I, a, B, X) tf + af(xf Xf(tf)) + 'k(xk xk(tk))
t- keK
+ (AT(f x) + BTh)dt (4.21)
isl1 i-l








The algebraic equations g(X, U, Z) = 0 are not included.
These will always be satisfied by solving for the dependent variables
Z and substituting the resulting values into the state equations.
Also the equations at tO, X(t0) = XO, will always be satisfied and are
not included. Note that the fourth term accounts for the changes in
the state equations, algebraic equations, and control variable
constraints along the trajectory. This term has been written to allow
for possible discontinuities in the Lagrange multipliers, A, at the

entry and exit corners of constraints.
We define the Hamiltonian H = ATf and rewrite equation (4.21)


L(X, U, ti, i 10c a, a, x) = tf + af(xf Xf(tf))


+ ak(xk Xk(tk))+ i I (H- TX + gTh)dt (4.22)
kcK iel
Si-l

On an extremal solution, the Lagrangian L must be stationary
with respect to small, arbitrary perturbations of the times ti(i e Il);
the multipliers a, , and X; and the states X and controls U. To
derive the necessary conditions for a stationary point of the Lagrangian,
we take first order variations of L with respect to ti(i E Ii), a, B,
A, X and U,

6L = (1 + H H T + Th fxf)t tf af xf(tf) + (x^ xf(tf))6af



+ k [- ck6xk(tk) + (xk xk(tk))6k (akXk)tk6tk

+ [(H- AT + JTh) t6ti (H ATX + OTh)+ 6t -l
ieI2 1 i-l 1








ieI + DX I X T
i-I


2T ah Ta 6X
aX


+ BT Uh- JU + hT6b dt
3U1 1


Integrating the term


ATsxdt = (AT6x) +
i-i


+ HuT



(4.23)


- AT6xdt by parts,


- (T6x)
ti


+
t+
i-I


AT6xdt


(4.24)


Substituting (4.24) into (4.23) and collecting coefficients of
6c, 6B, 6x, 6X, 6U, 6xk and 6ti separately, we have


6L = (1 + H T + Th a fxf)tftf

af xf(tf) AT(tf)6X(tf)

+ AT(t0)X(t0)

+ (xf Xf(tf))6af


i l2
ilK


(AT) 6X(ti) (AT) 6X(t.)
it ti


+ I [(H ATx + Th) 6ti (H -
id82 ti
i/K

+ I [(T ) +6X(tk) (AT) 6X(tk) +
kcK tk tk


ATX + Th) +6ti
t.
(f)

ck6xk(tk)] (g)








+ I (xk xk(tk))6k
kEK


+ I [(H XT + BTh) _6tk (H XT + B Th) +tk
kK tk tk


+ (kk)t tk]

+ i H 6

ti-


+ -

ti


ti-
+

i-1


[iT +2 -T + T ah ]6xdt
T T 2XT
ax ax


[ H + BT 3h ]Udt
aU aU


t
+ 1 hTSadt (s)
1+
i-1 (4.25)

The Kuhn-Tucker conditions arise from the Kuhn-Tucker multipliers B and

the inequality constraints h,

B.h. = 0 (t)

> 0 (u)

For stationarity of the Lagrangian all the coefficients of 6X,

6U, 6xk, 6a, 62, 6X and 6ti occurring in equation (4.25) must be zero.
By equating these coefficients to zero we arrive at the following

necessary conditions (NC).









From term (q);


From term (r);


From term (p);


From term (t);


From term (u);


From term (h);

From term (d);

From term (b);



From term (c);

From term (e);



From term (g);


From

From

From


T aH T ah +
A T t < t < t
1- T 1 i- 1

aH T h +
-- -- = 0; ti-1 < t
SaU

X = f(X, U); t < t

.jh = 0; t < t( t
Sj i- 1

> 0; t < t i-1l t


^k xk(tk) = 0; k K

xf Xf(tf) = 0

f(tf) = af(tf)
Xj(tf) = 0; j=1,...,n; j/f

xj(tO) = unknown since 6x(to) = 0

Xj(tt) = xj(ti); j=1,.. ,n
icI2, ijK

x.(t+) = Xj(tk); j=1,...,n; j/k; kcK

xk(tk) = k(tk) ak; ksK

(NC4), (NC5) and (NC8); H(tf) = -1

(NC4) and (NC11); H(t) = H(tI); iCI2, i/K

(NC4) and (NC12); H(tk) = H(tk); keK


(NC1)


(NC2)


(NC3)


(NC4)


(NC5)


(NC6)

(NC7)



(NC8)

(NC9)



(NC10)




(NC11)

(NC12)

(NC13)

(NC14)









IV.2.3 Comments on the Necessary Conditions

The state equations and point constraints are the necessary
conditions NC3, NC6 and NC7. The Kuhn-Tucker conditions NC4 and
NC5 indicate that, when a control constraint h is encountered, that
constraint is held by choosing U such that h(X, U) = 0 provided that

the Kuhn-Tucker multiplier B is non-negative.
The problem is set up such that the point constraints arise
when there is a change in the form of the state equations and,
consequently, a change in the adjoint equations and Hamiltonian.
For example, before time t1 (where x,(t) = X) let f(1) represent

the state equations, or

X = f(1)(X, U) t < t1

After time tI let f(2) represent the state equations, that is

= f(2)(X, U) t > t1

The Hamiltonians are

H() = Tf(1) and H(2) = XTf(2)

The functional forms f(1) and f(2) reflect the change in the state
equations. Condition NC14 shows that the Hamiltonian is continuous
across the times tk where point constraints are encountered. For
example, at t = tl, xl(tl) = xl and I() = H(2)
Condition NC11 states that the Lagrange multiplier (or adjoint
variable), corresponding to the particular state on which there is a
point constraint at ti, has a discontinuity at ti, whereas the









multipliers corresponding to the other states are continuous at ti.

At time tI, for example,

x (t+) = (t) -

Xj(t+) = Xj(t-) ; j=2,...,n

The value of the "jump," a1, in the multiplier Xl,is readily determined
from the condition H() = 1(2)
At times ti, i E I, when control constraints have to be

satisfied, all the multipliers are continuous as the trajectories

enter and leave the constraints--IC10. The Hamiltonian is also

continuous at such points in time as shown by condition NC13.
Necessary conditions NC8 and NC12 provide values for the adjoint

variables and Hamiltonian at the final time. The adjoint variables

corresponding to the states that are unconstrained at the final time

have a value zero. If only one state is constrained at the final

time, the adjoint variable corresponding to this state can be

determined from the final condition on the Hamiltonian

H(tf) = -1

or ff

However, if more than one state variable is constrained at the final
time, the values for all but one adjoint variable have to be guessed

and the remaining one adjoint variable can then be determined from the

final condition on the Hamiltonian. Let us suppose that the adjoint

variable corresponding to the stopping condition Xf(tf) = xf is
determined from the final value of the Hamiltonian and that another









variable A. has been guessed. On subsequent iterations this value of

A. is updated by noting that the gradient of L with respect to Ai(tf)

is (xi xi(tf)). The Saddle Point Theorem requires us to maximize

L with respect to Xi(tf) which is equivalent to driving the gradient to

zero. So the value of Ai(tf) on following iterations should be in such

a direction that at the final time the deviation of xi(tf) from the

desired value xi, i.e., (xi xi(tf)) is driven to zero.

This same iterative technique is used if more than one point

constraint is encountered at an intermediate point in time tk. One

of the adjoint variables at time tk can be determined by using the

continuity of the Hamiltonian at time tk. The other variables will

have to be guessed initially and then updated using the gradients

available on subsequent iterations.

Necessary condition NC2, along with the Kuhn-Tucker conditions

NC4 and NC5 enables us to replace condition NC2 by another condition

made possible by invoking the strong minimum principle. Denn (1969)

has lucidly shown how the Hamiltonian takes on its minimum value for

the optimal decision function U(t), both on and off the inequality

constraints. Utilizing this result, we can replace the stationary

condition NC2 by

Min H(X, U, X)
U

subject to Umn U Umax
min -- max


An algorithm for arriving at the minimum time policy was


IV.2.4 Minimum Time Algorithm








developed based on the necessary conditions arrived at in Section

IV.2.2. It can be classified as a "Minimum H" algorithm as it deals

with direct Hamiltonian minimizations along the trajectory. Proposing

a control scenario to start with is essential to simplify the problem

and to use the algorithm effectively.

Let the various elements of the state vector be divided into

3 groups as follows:

Group A: States which remain unconstrained through start-up.

Group B: States which meet their steady state values during

start-up and which define the point constraints.

Group C: States which meet their steady state values at the

final time.

The state equations are modified along the trajectory when

the point constraint conditions are met by the variables in Group B.

Also, all algebraic equations are satisfied throughout the trajectory,

and this is implied in the state equations. When a variable belonging

to Group B arrives at its steady state value, it is assumed that for

subsequent time the state equation is replaced by the corresponding

algebraic equation x = 0. Thus the Hamiltonian H = T f is different

for points on the trajectory depending upon which of the state

equations are active; so also are the adjoint equations

3H T ah +
^1 B ; t t < t.
aXT 3XT i-l t I 1

The algorithm proceeds as follows:

1) Guess a nominal control policy U which will cause a stopping

condition, xf = xf, to be satisfied at the final time.








2) Integrate the state equations forward until the stopping
condition is satisfied. This implicitly determines the final time.
The proper set of state equations should be integrated depending upon
the point constraints, the control constraints, and the algebraic
equations which have to be satisfied along the trajectory.

3) At the final time determined in step (2), let all the
multipliers of the variables in Group A be zero. Guess the multipliers
of all the variables in Group C with the exception of that corresponding
to the stopping condition. Determine this latter multiplier from the
final condition on the Hamiltonian,

fxf = 1 ixi
icC
iff
4) With the values of the adjoint variables at the final time
determined in step (3), integrate the adjoint equations in the reverse
direction. At times tk, when point constraints were met on the forward
integration, determine the values X(tk) by utilizing the continuity
and jump conditions on the adjoint variables and Hamiltonian. For
example, if only one point constraint of the form xk(tk) = k is met
at time tk, then Ak(tk) can be determined from the continuity of the
Hamiltonian

Xk(tk)fk(tk) + Xj (tk)fj(tk) = kxj(t )fj k)
jEn jen
jik jUk
and Xj(tk) = Aj(tk); jen, jUk. If more than one point constraint is
met at time tk, then the values of all but one of these multipliers
should be guessed at time tk and the last one determined from the
continuity of the Hamiltonian.








5) Simultaneously on the reverse iteration, minimize the
T *
Hamiltonian H = Tf at each point to determine the optimal U

Min H(X, U, X)
U
6) On reaching time tO, update the control policy used on
the forward integration with that found on the reverse integration in
step (5)
Ui+ = Ui + C(t)(U* Ui)

c(t) is chosen to limit the change in U if too large a change is
indicated.
7) Integrate the state equations forward as in step (2).
At the final time determine the difference in the states for the
variables in Group C from their desired values and update the guess on
the multipliers .i such that the gradient of L with respect to
Xi(tf) is driven to zero, i.e., to drive (xi xi(tf)) to zero. As
before, determine the X corresponding to the stopping condition from
the final value of the Hamiltonian.
8) Integrate the adjoint equations in the reverse direction
as in step (4). Update the guess on Xj(tk); jen, j/k in a similar
manner as in step (7). Determine Xk as before from the continuity
of the Hamiltonian.

9) Repeat steps (5)-(8) until
(1) 6J < eI no significant improvement in the final time
(2) IUi+l Ui 1< 2 no significant change in the control policy
and (3) 116Al I < n at tf
and (4) 116XXII < n at tk.
.k'








The optimal policy is chosen as the one which satisfies the
above conditions.

IV.3 Solution to the Evaporator Problem

IV.3.1 Problem 1. Constraint on the Second Effect Hold-up

The problem was solved for the scenario described in
Section IV.1.3. The concentration equations (4.5) and (4.6) were not
used in this simulation. The Hamiltonians and adjoint equations for the
various stages of the scenario are given below.
The general Hamiltonian for the problem is:

H = (W12 V21 u3) + [W12(h12 x2) + V21(x2 h ) + 1


+ 13(u1 u2 V02) + 4 [u(hF x4) + V2(x4 h) + 2

This simplifies for the various stages as follows:
Stage A: t < t < t

H = Au +2
Ha = 2 + [ 2(4 x2) + Q1] + 3(ul "2)

Stage B: t1 < t < t2. x (tl) = x
A2
Hb = X + 3(u1 u2)
x1

W12 = u3 ; u2 = 0 or fI = 0

Stage C: t2 < t < t3 x3(t2) = x3


x1
ul = u2 or f3 = 0 and f = 0




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