Citation
Design and implementation of minimum time computer control schemes for start-up of a double effect evaporator

Material Information

Title:
Design and implementation of minimum time computer control schemes for start-up of a double effect evaporator
Creator:
Nayak, Santosh, 1946-
Publication Date:
Copyright Date:
1974
Language:
English
Physical Description:
xv, 208 leaves. : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Adjoints ( jstor )
Control variables ( jstor )
Enthalpy ( jstor )
Evaporators ( jstor )
Heat transfer coefficients ( jstor )
Liquids ( jstor )
Mathematical variables ( jstor )
Steam ( jstor )
Subroutines ( jstor )
Vapors ( jstor )
Automatic control ( lcsh )
Chemical Engineering thesis Ph. D
Control theory ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF
Electronic data processing ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

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.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
022774034 ( AlephBibNum )
14072058 ( OCLC )
ADA8787 ( NOTIS )

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Full Text









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.













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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












































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0:

PI
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0













LO




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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















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bj I








auJ
I- , \

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(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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calculated values are indeed least-squares values (with the weighting

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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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 IIVERSITY OF FLORIDA 1974

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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 encouragement. 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. Sri ram 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. n

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TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES . LIST OF FIGURES NOMENCLATURE ... ABSTRACT CHAPTERS: I II III INTRODUCTION 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 1 1. 6 Software DYNAMIC MODEL AND PARAMETER ESTIMATION III.l Dynamic Model 111. 1.1 State Equations 111. 1.2 Connection Equations 111. 1.3 Heat Transfer Equations 111. 1.4 Decision Variables 111. 1.5 Assumptions 1 1 1. 2 Parameter Estimation Pa^e i vi viii x x i i i III. 2.1 Stochastic versus Deterministic Estimation 5 5 8 9 13 16 17 22 23 23 26 26 29 30 31 31 m

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1 1 1. 2. 2 Experimental Work for Determining 8, ... 34 111. 2. 3 Calculations and Results for 0, 40 I a 111. 2. 4 Experimental Work for Determining 0,, and 2 51 III. 2. 5 Calculations and Results for 6,. and a lb 6 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 F0LD0UT NOMENCLATURE LIST 148 V . COMMENTS AND RECOMMENDATIONS 149 V.l Model 149 V.2 Experimental Setup 150 TV

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V.3 Theory 152 V.4 Conclusions 152 APPENDICES: A. HEAT TRANSFER EQUATIONS AND OTHER RELATIONSHIPS 154 A.l 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

PAGE 6

LIST OF TABLES Table Page 3.1 Experimental Data for Run Al 35 3.2 Experimental Data for Run A2 36 3.3 Experimental Data for Run A3 37 3.4 Experimental Data for Run A4 38 3.5 Experimental Data for Run A5 39 3.6 Calculated versus Observed Values of T, for Run Al 46 3.7 Calculated versus Observed Values of T-, for Run A2 47 3.8 Calculated versus Observed Values of T, for Run A3 48 3.9 Calculated versus Observed Values of T-, for Run A4 49 3.10 Calculated versus Observed Values of T, for Run A5 50 3.11 Experimental Data for Run Bl 53 3.12 Experimental Data for Run B2 54 3.13 Experimental Data for Run B3 55 3.14 Calculated versus Observed Values of T-, and T 9 for Run Bl ! 59 3.15 Calculated versus Observed Values of T-, and T ? for Run B2 I 7 60 3.16 Calculated versus Observed Values of T-, and T ? for Run B3 ! f 61 4.1 State Variables for Problem 1, Iteration 1 92 4.2 Adjoint Variables for Problem 1, Iteration 1 93 4.3 State Variables for Problem 1, Iteration 2 94 4.4 Adjoint Variables for Problem 1, Iteration 2 95

PAGE 7

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 CI 130 4.17 Theoretical Minimum Time Simulation for Run CI 131 4.18 Actual Minimum Time Simulation for Run CI 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 vn

PAGE 8

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 1 070 Cabi nets 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 T-, for Run Al 41 3.3 Calculated versus Observed Values of T-, for Run A2 42 3.4 Calculated versus Observed Values of T-, for Run A3 43 3.5 Calculated versus Observed Values of T-, for Run A4.... 44 3.6 Calculated versus Observed Values of T, for Run A5.... 45 3.7 Calculated versus Observed Values of T-, and T ? for Run Bl ! f 56 3.8 Calculated versus Observed Values of T-, and T ? for Run B2 ! : 57 3.9 Calculated versus Observed Values of T-, and T 9 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 Vlll

PAGE 9

Figure ?M± 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 1^5 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 CI 128 4.11 Experimental versus Optimal Minimum Time for Run CI... 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 IX

PAGE 10

NOMENCLATURE 2 A = Heat transfer area, ft C = Solute concentration, 1 bs/( 1 b solution) C = Specific heat, Btu/lb D = Diameter, ft G = Mass velocity, lbs/(ft )(hr) Gr = Grashof number, dimensionless H = Hold-up, lbs I = Index set K = Index set L = Length, ft N = Number of tubes 2 P = Pressure, lbs/(in ) 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)(ft )(°F) U(t) = Control vector V(t) = Process noise, vector 3 V = Vapor volume, ft V = Vapor flow rate, lbs/mi n

PAGE 11

W Liquid flow rate, lbs/mi n X = State variables, vector X = Estimate of state variables, vector X = Lockhart-Martinelli factor Y = Calculated observations, vector Y = Actual observations, vector f = Function of g = Acceleration due to gravity, ft/(hr ) h = Liquid enthalpy, Btu/lb h v = Vapor enthalpy, Btu/lb h = Film coefficient, (Btu)(ft)/(hr) (ft )(°F) k = Thermal conductivity, (Btu)(ft)/(hr)(ft )(°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 = Outside world 1 = First effect 2 = Second effect i j = From unit j to unit i a = Before c = Condensate xi

PAGE 12

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 * = Optimal ity Greek Letters a = Point constraint multiplier 3 = Inequality constraint multiplier = Estimated parameters, vector 6 = Estimated parameter p = Density, lbs/ (ft 3 ) A = Lagrange multiplier or latent heat of vaporization, Btu/lb u = Viscosity, lbs/(ft)(hr) xn

PAGE 13

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 engineering 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 control 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 x i i i

PAGE 14

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 parameters 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 xiv

PAGE 15

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 experimental problems associated with hold-up measurements and analog control of the hold-ups and the theoretical problems associated with the assumption of the heat transfer mechanisms. xv

PAGE 16

CHAPTER I INTRODUCTION Optimal control theory has been developed to a fairly sophisticated 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 nonlinearities 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

PAGE 17

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 suboptimal 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.

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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 constraints 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 compared 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 particular double effect evaporator investigated. It is to be used in an

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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 computer time for steady state observations, it was imperative to have start-up in a minimum time. Chapter II contains a description of the experimental evaporator, 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.

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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-liguid 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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K A ,\ A A A A V V m / v V V UJ Q Q o 1—

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8 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, CD1 , 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 Tl 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. 1 1. 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

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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 recirculation pump was always filled with liquid. This was particularly critical when the pump was first started. Incomplete filling of the vertical leg led to pulsatinq 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. II. 3 Evaporator Instrumentation As part of this work the evaporator piping had to be modified to accomodate the instrumentation required for control. The work consisted 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 Fiqure 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 installed in the feed line as such. The purpose of the by-passes was to

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I i -a

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11

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12

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13 allow for complete or partial manual control of experimental runs when desired. Flow rates were measured with square-edged orifices, 0R1 , 0R2, 0R3 and 0R4, 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, TCI, 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 connected quickly when required. The thermocouple wires also terminated with special thermocouple outlets at the panel. II. 4 Transducing and Controlling Equipment The transducing and controlling equipment was installed in a 19-inch relay rack on casters. All air and liguid lines were of polyflo tubing with quick-connect fittings. This made the rack very versatile as it can be moved to a number of different pieces of equipment

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14 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 controllers 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 pneumatic 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 experiments) 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.

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15

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16 II. 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 accomodate pulse duration outputs and most of the relays had to be rewired and chanqed to double coil relays in order to isolate the IBM equipment from the user circuitry. The entire set up is described in detail in the GIPSI (General .Interface for Process Systems Instrumentation) hardware manual (GIPSI, 197 3) 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 input 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 throuqh 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

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17 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 computer 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 predetermined time interval has elapsed. Figure 2.5 is a schematic illustrating the flow of information among the various hardware components of the experiment. 1 1. 6 Software The GIPSI software package was originally written by R. C. Eschenbacher. Version 2, the version used, was written by L. A. Delgado

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n_ C (DO ui to •!— cnr~3 CD +-> (D -M i — r» i — 3 •— r— C Z3 Q. C O C •rQ. C -r> T~ c c: SS_ 4-> oi oj a; o ti— o <-) Q. 4(T3 3T3XI'rO So to to 3 U C C CD CD "O O ro ro cn O E iSiO SSS_ +->+-> 4-> I— a. a. > 1,1, Q OO ^ or Q O M IIL 4 I/) CD CD Q> i — iTJi — a+J ro 3 > O CD O Sr— O 3 o E to S_ S_ 00 +-> CD CD C -c s_ o I— a. c_> , — s— r— O CD CD SO (_> +-> O0O Q Q C_> ha >
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19 ( G IPS 1 , 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 execution. If the sequence began with PA1 , 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 conditional 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

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20 —I
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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 programs. However, costs for core residency charges dominate as the basic software package requires around 20,000 words or 80,000 bytes of core.

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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 experimental 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 22

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23 HI .1 Dynamic Model The dynamic model is a collection of the material and energy balances for each effect. For a double effect evaporator concentrating 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 algebraic 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, H-, and bL. dT=W 12V 21 -«11 W 01 (3J) dl! 3T ' W F " V 02 " W 12 (3 ' 2) 2) First and second effect enthalpies.

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24 > CM Q. > CM N .. A CVi\ I CO 1 o o \ / \ / \ / r v v v so 4-

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25 Since the evaporator was to be used ultimately for the concentration 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(H 1 h ) -at— = w i2 h i2 v 2i h I< w n +1 W h i +n i where Q, is the heat transferred from the steam. Simplifying this using equation (3.1), we get dh 1 , dT" = M7 [H 12 (h 12 " V + V 21< h l " h l> + V (3-3 Similarly, an energy balance on the second effect under the same assumptions gives rise to dh 3) First and second effect solute material balances. Again, assuming perfect mixing we have for the first effect solute, ar = ftEw F (h F h 2 ) + v 02 (h 2 h p + o 2 ] (3.4) d(H 1 Cj -dT~ =W 12 C 12< w n + l 'W C l Simplifying this with equation (3.1) we have dC l 1 dT~TTj" CW 12^ C 12 " C l } + V 21 C 1 ] (3.5) Similarly, a balance on the second effect solute yields, dC ar-TfC^F c 2^ + v o 2 y (3.6)

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26 III. 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: W 12 = 1^ 2 + W n < 3 ' 7 ) w 12 c 12 = w-i2 c 2 + lJ n c i (3,8) W]2 h 12 = W] 2 h 2 + l! 11 h 1 (3.9) III. 1.3 Heat Transfer Equations These equations arise in computing the terms ] and 0. 2 that arise in the enthalpy equations (3.3) and (3.4). The heat transfer rate Q, is a function of the steam temperature, 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 twophase 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 C^ from the state variables and flow rates is given in Appendix A. Due to the uncertainty in

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27 the empirical equations which predict the inside film coefficients, a parameter 6-. 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 reasonably high degree of accuracy as is borne out later by experiment. It is also assumed that the parameter 9-, 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) 9-, = 9-, , when the first effect liquid is being heated. The i a dominant equation for the inside film coefficient is solely the DittusBoelter equation (A. 16). 2) 9, = 9-,,, 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: Q 1 = Q 1 (T s ,T 1 ,T 12 ,H T W 12 ,V 21 ,e 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, h , h, and h,> are differential equations,

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28 dp, V i HT = V 21 " W c2 Vn -^L -= V 21 h T W c2 h c2 ^2 where V] 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 V 21 = W c2 ^ Q 2 = V 2lK h c2> °r Q 2 =V 21 X 1 (3>11) where ^ = f{J ^ The heat transfer rate in the second effect, Q 2 » 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 Musselt equation (A. 47) is reasonably accurate in predicting the outside film coefficient. The inside film coefficient is predicted by the natural convection equation (A. 51) and the overall coefficient (A. 52) has an undetermined parameter 6 2 which again can

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29 have two values. One value (0„ ) is for the heating of the liquid and the other (8ok) is f° r the boiling of the liquid in the second effect. The functional form for Q ? is: Q 2 = Q 2 (T r T 2 ,T F ,H 2 ,e 2 ) (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 Q-, and Q ? from the temperatures, hold-ups and flow rates. It is included in Appendix B. The rates 0-, and Q„ 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. III. 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

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30 variables. These are: 1 ) feed flow rate, Wp 2) feed temperature, Tp 3) feed concentration, Cp 4) inter-effect flow rate, W^ 2 5) recirculation in first effect, W-j -| 6) product flow rate, Wq-| 7) steam temperature in first effect, T s 8) total pressure in second effect, P 2 III. 1.5 Assumptions A summary of the assumptions made in writing the model is presented 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 convection followed by two-phase convection and nucleate boiling and that in the second effect is natural convection.

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31 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. III. 2 Parameter Estimation II 1 .2.1 Stochastic versus Deterministic Estimation An extensive review of parameter estimation technigues in differential eguations is available in Nieman et al . , (1971). In the deterministic case the simplest and most effective method is a least sguares fit. The problem is stated as follows: Given the state eguations X = f[X(t), U(t), e(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 observations 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, O(t) = 0, the problem can be reduced to a least-sguares estimation N . 2 Min I (Y. Y.) G i=l where Yis a calculated observation at time t(i=l,...,N) and Y. is

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32 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 were updated as each measurement 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. Mote 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 O(t) + V (t) thus treating the parameters as states. The problem reduces to estimating the states X(t) and the parameters O(t) from the experimental data Y(t). A recursive estimate X(t,/t, ) representing an estimate of X at time t^ based on all the data collected until time t k is given by $(t k /t k ) =f(x(t k /t k ), t k ) + P(t k )z(t k /t k )

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33 with X(0/0) = \j = initial estimate of X(t ) where Z(yt k ) = h^R(t k ) _1 (Y(t k ) h(X(yt k ), t R )) and the covariance of the estimate P(t. ) satisfies a matrix Ricatti equation HP ^ ^T ^ & (t k ) f/(t k ) + P(t k )fJ + P(t k )Z x P(t k ) Q(t k ) with P(0) = tt = initial covariance of estimate X(t Q ). 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 holdups and the first effect temperature were constant. It was desired to estimate the second effect temperature and the parameter e 1 in the equation dh ? 1 ^. = ^[w F (h F -h 2 ) + e'Q 2 ] while measurements were obtained on the second effect temperature T 2 = f(h 2 ) The state equation and the Ricatti equations were integrated for eight minutes of real time and this took 100 seconds of IBM 370/165 CPU time. The results for the first four minutes were as follows: Time

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34 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 ^(j) = au.(j -!) + (! a) Ui (j) where 0
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35 U.t~-CO0OCO0OCOO0CO0O00CX3QOCO00CDQO000OCXDeO o lu 0(MN(\icococofMr\ooininO(fico!ii>}-0<0 Q •• • 2:1— r-f— r— t — i — r— r — r^-r— r— r— ^-p^-r— p~r~-i^-r-r — r— uj (mo^o^— «ir*mi > -i-«ir\r^v0^cor\jccLr\r\ior'~o* cc • » D(\)r-l>}-0[)-(0 , NjfM(Mrs)rg
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36 I— r\J (\J CO f\J f\J <\i — I •-< — lr-^•-lr*-r-^r^t^(\)<*•^-'>*-•-^ u.co v cocro > (r^o , o v o v t^tDO i eoo)flCiocO v a;o> U. ^_,^4^ r _ (r _l^,-4^-(,-4^^-Hr-lr-Jr-lr-Hr-l^-<---r— v0f-v0>Ov0 -o m un ir\ -J" ir\ m >! >!>!I/) lu fsjoor^^-ooroco^-icooroino^cvj^-to^-omo^ cc ,....••.•••••••••••• t— ^^r-cooocooocc^c^croooooocoO"-*!-* <|_^|^^|^^HrHrHr-l(\JM(\J(\IM(NNMN(\J(\J UJ lu mirivOsO>or-f^t*-oocro v coa*a N CT'CT , crCT > oa* (— oo^>l•<^>t^^^l>^" < *"^<•' ; '" > * >1 ,J ^ >1 ' >1 " Ln, *' h-(M(M(NCM(Nj(NJf\ic\jrgrgrsirsjcM(Nj(\jf\jCNjCsjr>jcM OOOOOOOOOOOOOOOCOOOO _l • • . » •••••••••••••••• c c c c c o o o o co o e> c c c c o c o c «/) l<0 2 co co z: _j _j •-' oo in(\iNir'Oi-iMHv0^o>oinouMfiff>o • •co^-i. ..»••••• ..»•••••••• r\JCM 3^0 0^0-^00^00000000000 II II •"• c a. r> z> i lu I O »Q. U j2:o •••••• .....•• wl.'3)'tMf f l'M>t^'lrteC«0ff^Hfni-l^o0 , OO r (HHrlr^^(MIMI\lM(MClJN[Mf r lWf r l' r l' f ll f l

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37 r — r^r— I s t^^i^o-0'0»c>o^vOi\iNNN(\iix)(fl U. _ tf _,,-»,-,,HfHfHf-l.««.-l.-.rIf-H—l—l,-,^.-,,-!,-. Q r\it^r-r^f-r-r— r-r-r— t*rrrrro & o o o -2»— r — r-i — r — r— r^-i — r — r— r^r— ) — i — r— r— r-r— i^r-rf-f — It— Ir-lt— It— If— If— If— If— I i— I f— lf-4.— It— If— If— Irtr-lfli— I uj -j , oo(MO>oirnrii-iin(MifiHO'fH>o -o>0"H>tr-ooOf-i-4-irif*-r>-oocT'0^-i »— — O^OOOOOOOOOCyf-HrH J a: a, a. o*ooooocoMni/\a>rf>r'i(DoocoM(\iNM[\i(M s: h<\J(\J(\JCMr\|(\iCMj(\J<\JCM(\J(\JC\l<\J I— 2 oooooooooooooooooooo o c c o o o o o o o e o co c o e o e o o CO z ~ _l _J — ' -j s: =3; 00^ (\i4 , o-t-cooci^in t~~ 0000 oomg; M-OJH;.iHm(00^' r l*OlDfl>ffflHOONNtOH(n I— I rOC\J 5!\J(M(Mf-l(M^HHH^i-IHfHf^HH^rJf-lpJ Z ,_<,H_l_,_ 4 .-* r ^.^_.,-*,W r H_t.^,-l.-«rH.-*,-*.-4 II •Q. 00 1 uj Q r— Q. j -^(\ivOO^vO(T'^>tf^-jroO'4-CO>4-cO<\JOO(NjO^C\l 002:0 • -HUJOfMCOvOOO^-il^cvjomrvjrO^-ioO^^O^r-C^f-i 1— ooi^-cocooOf-<'-'fN](tNO w^ r ^r-ifHN(M(\ifV(\c\irvj(\iiN(MNr'irOrfir'lro

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38 < roprirrioooornmooaooor*-cgr\i(MrsjsOvOvOCvi!Nj u. m m lo -J >t ir\ it, «i->tj->tir\ir\ir\Lnif>tr>inmm — oncnororcrenorcocrorcrorootroocC'Orororcri U. _,_,_ 4r -< r < ^,..,_<_i_ ( _i_.,-«,< ,H^H.-i,--i,-i— i O lu ^^j-inif\ir\HHHi-ioi^'-ip>ti-tvOa.i.-i-irvi*-aoo"c , ,--im,j-ir\vGf— aoc^O h-^iooaDaoc^oocrcrc^oooooooooOr-* <»— ^H^l^l^i-t^4i-t^-l^r-t<\JfMfM(Mf\J<\lf\J(\Jf\Jf\J a: LU a. ^^taooor--r>-— ii/\o , o>NMO'000'-o>or'-r-r"-r~cocooo:DO v CT > a s CT > C7 v C N I— <\JOJ0slOC^OO'i)LnrMOf r i>0 o oo mco>i '4'coor-r-f-«ooooo oo^i-«cDvrc>Lr\rrir-i(\jir\rOi--j(\jrgr\jr\ir\j(\)c\j^.-it\j.-*c\ir\j-j II •"-• CL 13 OO I LU CK Q. -J < (M O O O C O a£ — « • X 3 OOOOOOOOOOOOOOOOOOOO 3 ^ a O _J LU LL U. oooooooooooooooooooo LL. U... LU 3E o C o o o o o o o o o o o o o o o a o — r^fMinrOvOvOo-J"O s r^--'r-Lro<-i'OOf^---'co LULuoo>OLnroro^-o>OLOcooco>4"vO-4"'-»ir> W2U • • • • "iL'oONM^-t<(NJO'00 I— oo^-ir-trvjrri^-m ot^-cocoO'-'fN'r r i>*'LOvOr^-00

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39 OvOMNOvO'OO-O'O'O^OOCllAlfiirini^h-

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40 III. 2. 3 Calculations and Results for 8, i a A computer program was written to estimate 9, from the data collected in Runs Al to A5. The least-squares program used was subroutine 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) . 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, W,p = W-, , and the vaporization was zero, V ? , = 0. The observation T, 2 = T-, since all the liquid entering the first effect was recirculated. For every value of 9-, which subroutine RMINSQ searched over, equation (3.3) was integrated from the initial to the final ar"Jr CM i2< h i2h i> + v 2i< h i h P +
PAGE 56

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46 LUOOfViooirvr<"iir\ < orornsOO< ,r ><7*fM>l"^o , J" u.>cj > a*r-<«i-ir\f^-t-i>a-msOoco<\jf^irif\ic7*foa a. O OLU^oo^orM-^-or^-cjDcro^rvjromr^-ooocsj ujirtr»f>(ocoij'0>o>o , o , o , oooooooHr( Q a3 _,, J ,_i,^_»,-*,-i,-«,-i,-«c\jr\jf\J<\jc\if\if\jc\jr\J O sr a: >o ID • < LU aKrfilTiOOOO^^OOO^COi-iNOON^O^ II CL _l • • • crtrc7 , cru > cjO(_»o<-''-« I— ? i/ILUoOOl^l^^oOC^r^ a» rn >} ^ON( no «o •-< t— 2_ir\irvir\vO>o>ooNOh-^r*-r^r»-cocoaDaooo

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47 N > ONHcoooo jen o D uj >< ro O u_ > oo r*r»cc «•••••••••••• LUl/)0000C000CT>0>0'0 >> OOOOOOOOO'-«'-i o (M LU for o t• < o at LU II Q. Ka , c^-ro < fioo-ooi > ->05)'-*Ni-iin'Or s -co ooor^r-io>>or , ~-4-'-cj<' r >Lnr--c>orvJ<" r >Lr!>oo > om _i^,-*,-*,-i^^,-i^.-i^(Mc\j<\j(\jc\j(\j<\ir\j(\j — — vj---icoLnroLorMCT , rnor«l'<-'cr'OOfMO x coaor\j>rrn(MOvr »— s:r\i(Njf\immrrirocri>j-o

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48 o iumf\jcMoo^co^ir\vOcrff>cr>oir\oo-£Ocr(NJ a i5c^^mvr«ooocr0^mmcrtNJsrinr-corv)rn l_i i_l—J-J.-4^.-«.-t<-t»-«'^'-< r ^>Orr,^CO>OOCr<>C^COl^coi/> ^m^^<^i^r^rvj^^rr 1 rr 1 ^r\jcsi>roocor-.iorsjrg

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49 cL • Otuoao^cMCM^irir^cocrOr-tco^rifiyOh-coo^o* UJ0000000 , '<7 s , N (^CrOOOOOOOOOO QcQi-«>-li-li-«r-l»-»*"«i-«r-«i-tCVJ«\lfNJCNJC\J(\l<\|C\|f\jC\J O z -H IX) <_i OC K ^4LnOOOCT«TtN(NP->TOOOr-NT>t)sOr~CDtP ijj oinm<\)fOO , H(7>o»oo'Tu"\^r-cro j^»ov>Oirif r >rricomrsio^-OO s >i-'-'Ococs»"f^O ooi-uooooiriLOLn-j-^vroor^r-tO'Ooaairi^^evjo-o

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50 IS) u. >ooaooo-*».-ir r >c^sOir\»-mt , -.-«o\j (jujr-^>raoO'4 >*"^) 00 0'^ rn,4 ' ir,,K " aococoONC: uji^r-aoooaocrc^crcrooooooooooo^ Q C Q,_ < ^ t -*,-4r-«^r-<.-Hr-«
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51 calculated values are indeed least-squares values (with the weighting indicated) which converge on the observed values toward the end times. On the basis of these results a value for 6, was taken as i a 0.1395. 1 1 1. 2. 4 Experimental Work for Determining 6-,, and 6~ 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. 8,. was a parameter for the boiling of the liquid in the first effect, 9 ? was a similar parameter for the heating of the liquid in the second effect and 6 ? . was a parameter for the boiling of the liquid in the second effect. Note that 0-, is, in effect, the revised value of 9 2a 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 maintain 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 W-j 2 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 W F 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

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52 (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 Bl, B2 and B3 are shown in Tables 3.11, 3.12 and 3.13 respectively. III. 2. 5 Calculations and Results for e lb and 6 2 A similar least-squares search using subroutine RMINSQ was used to estimate e lb> Q z& , and e 2b from the data obtained in Runs Bl 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 temperature corresponded to the temperature of boiling in the second effect (which was found from the pressure observed) parameter 9 2b was used instead of B . The calculated temperature of the first effect solution T, and that of the second effect solution T 2 were obtained from the integration of the state equations (3.1) and (3.3) respectively. The criteria for minimization were the functions f(U-!l < T l,calcT l .observed^ < 3 3 ) n=l 10 f ^= .1 ( T 2,calc " T 2. observed^ (3 ' 4) i=l 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 9 lb> 6 2a and 9 2bThe 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

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53 •-I a 5: x O QII O < o 3 »O < ir»ooof*>r-^^vO<-tir>0' , oooo>orO'^«mojrsjrg^pHr>-r>-r-r»ll.*..«... .•••...«*... ...... (\j*. ......•••.••••. ••••••••• LUfsir-iricooor-^ryjoor^-oaooio-j-^^j-ooroovjcorvifNjrvjrvjh-f^-rO -* •• »-h rvj(\j<\j(Ni(Nr\)c\4rsjc\jrgoj(N(\l<\irsi(\j4-cooomir\mir\ir\oooo N or , -r-f -r^ {/)•••• •••••••••»•••••••••••• H-crcrc^ooaor->ot*-r~oooocooooDcoooQOQooocoa3a3C30cocooo rg(NfSi(NfNJf\l^(NI^OO^< N JfOOl'NO>} , 00 , oo(\jcrcrmir\inr~croocoaooof*-of^^-' v -tr s c^otra)ajaof N -C7' .2 a. • • • • r-l r-l >-ifor-f\)»rO', '-*irvir\-a irvoO'^)^ocr , fNjC7 N orn>5-r r >^-*f^h< -ioorvjr--oro>o<--oiri»-i<)oo<\i(Nj»^o , ''>4"'-rir\r-«o r-i... ................ ....... 3sroor-iofMTii-'r r io N oor~r-r-r— aor-oor-r^ooaop-ajcoascoajr-ooaotDoooooor" a. a» o >r srr^oooo— <•*•— *ooooocooocooooooooo >-H^|. ......... ................ X SOOOO^lTi^OOOOCOOOOOOCOOOOOOO —I OOOOOlOI'-rgOOOCOOOOOOOCOOOOOO u_... ............. .......... rSOOOOOCOOlTlfPOOOOOOOOOC'OOOOOnO -~a3rn«j-t\jr>-cD<-f0^oornc7 v roooinrn^-»o(\Jco^-tfn-T'4 i-t'--'-r UJoO... ............ .......•.*. iOrOf\jro(^-inrOi-i^-i-OvOr r > I -< < --irOvOO>J"(NJ fN Jr~, -, -4"<^'r*> , 00

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54 moooooo-oovOvO>OvOvOo^ONOO^rH>-trH.-«oir* coooeooocx5cocoooooaoooooooeooooooocooooooocooococo oo r>-mo»*fNiOir\'^P~''-^r«-i*-ir\0<-HOv004-oo-4--j| >-iMinrf\(MN_ l r>iriooo v aDaoi v -r--r-r^f , ~cOr-((NirONj-^irvNj-'J-miriLrisO t— ooooo»a , oooocDoocooocDcoooc^c^ocroo (? > a > o s -((j>vO^OifiOOcoMr\oci 1 OHHifiO>ir>m(\joo < oo > ZO.-I • ...•• •••• ••• mmir\tr\>ou*iirivOvOf--coo v O N ooc7 > o , oc<7 N C7 N > '0 > of v J'M i/> • • •••••••••••••••• •••• ••• c£ | jl i i i • • • « • • • i • • • i • • • • * * * ' < 2 Q r HOO > oo^cocoo > ^^(\i<\jrvjr\j(\jr\jc\jr\J(Nr\ir\i(\j<\jr<"tr<'> Of^ocfMoovO»-< Q 1/1 I • • • • huj I z •a crrgr-taoLr\0'Or^cMvOoovOir>vOO'3" ?^ O (M'4'»0lI1O%0O , in^Hi-l>}->0'-ltD^>T-J'IA0O v (NI ^••••••••••••••••••••t*«« •• 300r-ti-l(T'C3CO^OjF-lsO>TLnLOir\LOOf N JLn lfNfv J s OOf*> 3 fM lu^ •4-ir\inu'> | J^Lninir>minu>inir\oir\invoir\uAin < oOO uJZfMooOC^toorg^Nj-oooOOOOOOOOOOn 3: I— •— • •— « • • •••••••••••• ••••••••••• U«3.2.30000r^-0 , '(7>r-i>j-OOOOCOOOOOOOOOOCU. l/> 320 CI r((f J HCO -O r-u_i ooo»-icT'>ror r >r^mooooooooooooooo o — i ^••••••••••••••••••••••••« 3 LL 3 O O O O O O O O O O C a CL UJ00••••••••••••••••••••••• , • «UJfri(0(D'^4'inNC0O'-tNNtirih-C0a''-tin>0<»!7''-'(N i— ooLnmiriO-o^OvOor-r-r-r-r-r-r^r-coooooooaooocoo^a*

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55 CO

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61 O. Q LU D -J • ' • • 1«h Jhhh rocoro>toc\ic\jLr\mtLni-iro>*->i>j-rOrOf\j(\jf\i<\j<\ic\ic\jc\irMc\irorom ^ (M O 11 CC r> CO •Q c\j < lu ir\ ro o < a: > lp <\i o hLU Q£ • • • LU Olu >3" ro >fr I r woo coco hLU no i-> — * r-* hO omr-^ir-rnr-Loc^ooor-i^-h-^ CHIfiOMflOO , ClNH^>)-'tini-< OOCOCOOOCOCOCT > OC7 s C7*Cr , C*O < CX % s C v o It LU Q < < i-ic^Lr\sfr~rf>CT>>*-ir\r-ti-i^ >oi > -"-'0 N co( v -roinoo > <}-v0^fvj _!••••••••••••••»•••• 3(MN(Mrti-IHH(MHHHi-ti-ICMHHHH(M _j(Mc\jrsjrsjr>jc\jc\ioj(Mrvj(\jt\jcM UJ GO

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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 Bl and B2 v/ere much better than those for run B3. On this basis the mean values for 6-,, , O and OU were I b 2a 2b 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).

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CHAPTER IV MINIMUM TIME CONTROL POLICY This chapter deals with the development of a minimum startup 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 63

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64 Section III. 1.1. For uniformity, x. will be used for a state variable and X will be the state vector with components x. . These are assigned as follows: x, First effect hold-up (I-!-,) x ? First effect liquid enthalpy (h-j) x 3 Second effect hold-up (M 2 ) x 4 Second effect liquid enthalpy (h 2 ) x c First effect solute concentration (C, ) 5 x r Second effect solute concentration (C ? ). The decision variables listed in Section III. 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, W p , and inter-effect flow rate, W-j 2 , can control the inventories in the first and second effects. The steam temperature, T , and recirculation rate, W,-|, have an effect on the first effect temperature and the rate of increase of the second effect temperature. The product flow rate, kL, , is used to control the product concentration These control variables are capable of keeping the process at steady

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65 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 udenote a control variable and let U be the control vector with components u-. The assignment is as follows: u-, Feed to the second effect (W,-) u 2 Intereffect flow rate (W-jp) u 3 Recirculation flow rate (W-,-,) u, Temperature of steam to first effect (T ) iir Product flow rate out of first effect O'q-,). 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 = W 12 " V 21 " u 3 " U 5 (4J) X 2 = x~ [W 12 (h 12 " x 2 } + V 21 (x 2 " h l> + Q l ] (4 2) x 3 = u l " u 2 ' V 02 ^ 4 " 3 ^ X 4 = T [u l (h F " X 4 } + V 02 (x 4 " h 2 } + Q 2 ] (4>4) V ^12^12 " x 5> +V 21 X 5^ (4 5) V^l^F"^ W (4 6) Connection equations, lJ 12 = u 2 + u 3 ^"^ W 12 h 12 = U 3 X 2 + U 2 X 4 (4.8)

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66 W 12 C 12 = u 3 x 5 + u 2 x 6 (4.9) Heat transfer equations, Q 1 = Q-j (x-j , x 2 , u 3 , u 4 , h 12 , W 12 , V 21 ) (4.10) Q 2 Q 2 (x 2 , x 3 , x 4 , h F ) (4.11) V 21 = Q 2 /X 1 (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 212°F 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) i ,mm i — l ,max v ' 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 (x.)--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

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67 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 1 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

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68 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 : t n <_ time t < t-, . Filling and heating first effect. Control variables: Feed to second effect at maximum flow rate, u-, = u-. All feed delivered directly to first effect, u 9 = u-. Temperature of steam for first effect at maximum value, u» = u* No recirculation or product flow possible, u^ = u 5 = 0. Resulting state equations: First effect hold-up increasing; x-, = u 2 , x-iUq) = Heating of first effect; x 2 = — [u 2 (x 4 x 2 ) + Qi]» x 2 (t Q ) = lv No increase in second effect hold-up; x~ = 0, x= Xo(0 = No heating of second effect; x» = 0, x. = x»(t«) = h p No concentration occurring in either effect; kr = kr = 0; X 5 = X 6 = X 5 (t } = X 6 (t } = C FTime t-, , signifying the end of Stage A, is determined when the first effect is filled, i.e., when x, (t-,) = x, . Stage B : t-, <_ time t < t ? . First effect solution is being heated and second effect is being filled. Control variables: Feed flow to second effect at maximum, u^ = u n 1 1 ,max Flow to first effect stopped, iu = Recirculation flow set at maximum, u = u „„ 3 3, max Temperature of steam to first effect set at maximum value, u 4 = u 4,max

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69 No product withdrawn, Ur = 0. Resulting state equations: No change in first effect hold-up; x-, = 0, x-, = x-. (t-. ) = x-. Heating of first effect; x ? = Q-i/x-, Second effect hold-up increasing; x 3 = u, , x 3 (t, ) = No heating of second effect; x. = 0, x. = x,(t-,) = lv No concentration occurring in either effect; x 5 = x g = 0, X 5 = X 6 = X 5 ( V = X 6 ( V = C FTime t ? , signifying the end of Stage B, is determined when the second effect is filled, i.e., when x 3 (t 2 ) = x 3< Stage C : t 2 £ time t < t 3< First effect solution is heated to boiling, Control variables: Feed flow stopped, u-, = Flow to first effect stopped, u 2 = Recirculation flow set at maximum, u = u „ „ 3 3, max Temperature of steam to first effect set at maximum value, u 4 = u 4,max No product withdrawn, Ur = 0. Resulting state equations: No change in first effect hold-up; x, = 0, x, = x-,(t 2 ) = x. First effect is being heated; x ? = Q-./X-. No change in second effect hold-up; x 3 = 0, x 3 = x 3 (t 2 ) = x 3 No heating of second effect; x» = 0, x« = x*(t«) = hp No concentration occurring in either effect; x 5 = Xg = 0; x 5 = x 6 = X 5 (t 2 } = x 6 (t 2> = C FTime t~ s signifying the end of Stage C, is determined when the first effect starts to boil, i.e., when x 2 (t 3 ) = Xp«

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70 Stage D : t< time t < t 4 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, u 2 = V 21 Feed to second effect set to maintain constant hold-up in second effect, u-, = u^ Recirculation at maximum, u^ = u^ max Temperature of steam to first effect at maximum, u 4 = Uq tmx No product withdrawn, Ur = 0. Resulting state equations: No change in first and second effect hold-ups; x-j = X 3 = 0; x l = x 1^3^ = x l ' x 2 = x 2^3^ = X 2 First effect is being heated; x« = — [u 2 x 4 + Q ] V 21 h^], x l X 2 3 ~~ x 2 Second effect is being heated; x 4 = 7[u-,(h F x^) + Q 2 ] ; x 3 x 4 (t 3 ) h p First effect solution is being concentrated; x 5 = — (u 2 Cp); x l x 5 (t 3 ) = C F No concentration of second effect solution; x g = 0; x g = Xg(t 3 ) = Cp, Time t* signifying the end of Stage D is determined when the second effect starts to boil, i.e., when x 4 (t 4 ) = x 4> Stage E : t 4 <_ time t < t f . 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, u 2 = V 21

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71 Feed to second effect set to maintain constant hold-up in second effect, u, = u ? + V fi? Recirculation at maximum, u = u 3 3, max Temperature of steam to first effect at maximum, u» = u. No product withdrawn, Ur = 0. Resulting state equations: No change in first or second effect hold-ups; x-, = x 3 = 0; X-, X-j^t»J X-j , Xo Xo\t^j x. First effect is being heated; x ? = — (u ? x. + 0, V 9 ,h,) x l. No change in second effect enthalpy; x« = 0, x» = x«(t.) = x. First effect solution is being concentrated; x 5 = — (u-iXg) x l Second effect solution is being concentrated; x 6 = — (u^p u 2 x 6 ), x 6 (t 4 ) = Cp. X 3 Time t f , 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 Xr(t f ) = Xr. After the final time, t f found above, the control variables u-, , u 2 and Ur 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., \x ? = Ur + V ? , . The feed to the second effect is such that a constant hold-up is maintained in the second effect, i.e., u-, = u^ + V^' The product rate is such that a constant product concentration is x 6 obtained, i.e., Ur = u — . Simplifying the above three relationships 5 d x 5 we can obtain a sequence for determining the control variables u, , u ? and Ur at steady state as follows:

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72 u 2 = V 21 /(l + ^] ; U] =u 2 + V Q2 ; u 5 = u 2 ^ Note that certain point constraints of the form x^t-j) = x i 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 x.(t.) = x. is followed by the equality constraint x^ = 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, x^t.) = x. s 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 = t f U

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73 subject to the state equations (4.1) to (4.6), the connection equations (4.7) to (4.9), and the heat transfer equations (A.l) to (A. 52) where the control variables are constrained, U . < U < U mm — — max and point constraints pertaining to the particular scenario are to be satisfied x.j(t,0 = *j ; i = l,n J = 1 ,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 = t, U 1 subject to: the state equations (4.15) X = f(X, U, Z) ; x(0) given the algebraic equations g(X, U, Z) = the control constraints u . < u < u mm — max or h li (u i } u i " u i,max^ h 2i (u i } = u i,min u i ^° h(U) < [4.16) [4.17) (4.18) (4.19)

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74 and the point constraints x (t ) x Q x kl^kl^ x kl x k2 (t k2 ) = x k2 (4.20) t„ = fixed IV. 2. 2 Lagrange Formulation and Necessary Conditions Assume that the point constraints are met at times t«, t, ,,..., tj;. Also assume the times t. 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, I {0, 1, 2, kl , k2 , f} l } = {1, 2 kl, k2 , f} I 2 = {0, 1, , kl, k2, f-1} Within each set, order the indices so the times t., i £ L or I, or I« are in increasing order. Also form the index set K = {kl, k2 } Introducing multipliers a, 6 and A we can write the Lagrangian for the problem L(X, U, t., i £ I Q , a, 6, X) = t f + a f (x f x f (t f )) + I a k (x k x k (t R )) t" ^ e ^ + I (> T (f x) + B T h)dt (4.21) ieli + 1 'i-i

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75 The algebraic equations g(X, U, Z) = 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 t~, X(t fi ) = X~, 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 = A f and rewrite equation (4.21) L(X, U, t., i e I Q , a, 3, A) = t f + a f (x f x f (t f )) f *1 T T + I a.(x. x.(t.)) + I (H A'X + 3'h)dt (4.22) keK K K K K iel 1 j + T i-1 On an extremal solution, the Lagrangian L must be stationary with respect to small, arbitrary perturbations of the times t.(i e I,); the multipliers a, 6, and A; 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 t. (i e I-i), a, 3> A, X and U, 6L = (1 + H A T X + 3 T h a f x f ) t 6t f a f 6x f (t f ) + (x f x f (t f ))6a f + kU~ a k 5X k (t k } + ( "k " MVK " (Vk } t k 6t k^ + I [(H A T X + 3 T h).-6t. (H A T X + 3 T h).+ 6t. ,] iel 2 r i ^ r i-l 1_1

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76 s1 t + l-l + l . f|H _x]' 6A+ [3H +g T3h _ A T 6 x] 5X+ [ 3H ill, J I i 3X J i 3X T 8X T J I 3U T + 3 T ^V IfiU + h T 66 3U 1 (4.23) Integrating the term A SXdt by parts, '1-1 i 1 A T 6xdt = (A T 5x) + (A T 6x) _ + + t i-1 *1 t + r i-l r i-l A T 6xdt [4.24' Substituting (4.24) into (4.23) and collecting coefficients of 6a, 66, 6A, 6X, 6U, 6x, and 6t, separately, we have TV, . J, 6L = (1 + H AX + B h a f x f ) t 6t f a f 6x f (t f ) A T (t f )6X(t f ) + A T (t )6X(t Q ) + (x f x f (t f ))6a f + I (A T ) + 6X(t.) (A T ) 6X(t.) ieL i^K lei 2 t: t i (a) (b) (c) (d) (e) Tc . „T, + I [(H A T X + 6 T h) 6t. (H X'X + e'h) ,6t. id 2 t: t| f] + I [(\ T ) + 6X(t k ) (A T ) _6X(t k ) + a k 6x k (t k )J (g] keK

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77 + I (x k " x k (t k ))6a k (h) keK + I [(H A T X + 6 T h) 6t, (H A T X + B T h) ,6t, keK t" t k + (a k X R ) t 6t k ] (o) MtT (ix-*) Tsxdt (p) lei-, . r i-l t+ I [A T + ^V+ 3 T ^V]6xdt (q) 1eI,J + 3X 1 9X 1 1 t i-l t+ I [ 3" + g T 3h ]fiUdt (r) iel-, J + 3U 1 3U 1 t i-l ^ T + I h'63dt (s) ieI i ; + i ~ 1 (4.25) The Kuhn-Tucker conditions arise from the Kuhn-Tucker multipliers 3 and the inequality constraints h, e o h j ° (t) B>0 (u) For stationarity of the Lagrangian all the coefficients of 6X, 6U, 6x, , 6a, 63, 6X and 6toccurring in equation (4.25) must be zero. By equating these coefficients to zero we arrive at the following necessary conditions (NC).

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78 From term From term From term From term From term From term From term From term From term From term From term From term From term From term q) ; X j 3 — f ; tj , < t < t. dr 9X 1 1_l ] r); i"+ 6 T ^V= 0; tt < t < tT 3U 1 9U 1 1_l 1 p); X = |£= f(X, U); t|_ 1 < t < tT t); 3-h= 0; t j _ 1 < t < tT u); 6 > 0; t. , < t <_ t". h); x k x k (t k ) = 0; ke K d); x f x f (t f ) = b); A f (t f ) = a f (t f ) Xj(t f ) = 0; j=l,...,n; j7f c ); ^i(^n) = unknown since 6x(t ) = e); Aj (tt) = A^tT); j=l n i e 1 2 » H& g); X.(t k ) = A^t"); j=l,...,nj j/k; k £ K A k (t k ) = A k (t k ) a k ; k e K a), (NC4), (NC5) and (NC8); H(t f ) = -1 f), (NC4) and (NC11); H(tT) = H(t|); i e I 2> i£K o), (NC4) and (NC12); H(t~) = H(t£); keK (NCI) (NC2) (NC3) (NC4) (MC5) (NC6) (NC7) (NC8) (NC9) (NC10) (ricn) (NC12) (NC13) (NC14)

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79 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) = 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 tim the state equations, or For example, before time t, (where x-^tj) = x-| ) let f^ represent x = f( ]) (x, u) t < t 1 After time t, let f^ ' represent the state equations, that is X = f (2) (X, U) t > t 1 The Hamiltonians are hM-xV 1 ' and H< 2 '=A T f< 2 > The functional forms f^ and V 2 ' reflect the change in the state equations. Condition NCI 4 shows that the Hamiltonian is continuous across the times t, where point constraints are encountered. For example, at t = t-j , x-^t-j) = x-j and \v ' = \v . Condition NC11 states that the Lagrange multiplier (or adjoint variable), corresponding to the particular state on which there is a point constraint at t., has a discontinuity at t^ , whereas the

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80 multipliers corresponding to the other states are continuous at t y At time t., for example, X-,(t|) = X-^tf) a-, X.(tj) = X..(t~) ; j=2 n The value of the "jump," ^ , in the multiplier A-,, is readily determined from the condition H^ ' = \r K At times t., i e I Q , when control constraints have to be satisfied, all the multipliers are continuous as the trajectories enter and leave the constraints— NC10. The Hamiltonian is also continuous at such points in time as shown by condition NCI 3. flecessary 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(t f ) = -1 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 x^(t f ) = x f is determined from the final value of the Hamiltonian and that another

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81 variable A. has been guessed. On subsequent iterations this value of X. is updated by noting that the gradient of L with respect to X.(t f ) is (x. x.(tp)). The Saddle Point Theorem requires us to maximize L with respect to X-(t f ) which is equivalent to driving the gradient to zero. So the value of X.(t-) on following iterations should be in such a direction that at the final time the deviation of x. (t-) from the desired value x. , i.e., (x. x. (t-)) is driven to zero. This same iterative technique is used if more than one point constraint is encountered at an intermediate point in time t. . One of the adjoint variables at time tj" can be determined by using the continuity of the Hamiltonian at time t, . 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 U . < U < U J mm max IV. 2. 4 Minimum Time Algorithm An algorithm for arriving at the minimum time policy was

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82 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 = X f is different for points on the trajectory depending upon which of the state equations are active; so also are the adjoint equations A — y g — j , t. , <_ t <_ t. 3X 1 3X' 1_l 1 The algorithm proceeds as follows: 1) Guess a nominal control policy U which will cause a stopping condition, x. = x f , to be satisfied at the final time.

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83 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, Vf = ] .L Vi w 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 t, , when point constraints were met on the forward integration, determine the values A(tj") by utilizing the continuity and jump conditions on the adjoint variables and Hamiltonian. For example, if only one point constraint of the form x, (t. ) = x. is met at time t. , then ^(tj") can be determined from the continuity of the Hamiltonian J7k j/k and A.(tj") = A.(t. ); jen, j^k. If more than one point constraint is met at time t. , then the values of all but one of these multipliers should be guessed at time tj~ and the last one determined from the continuity of the Hamiltonian.

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84 5) Simultaneously on the reverse iteration, minimize the T * Hamiltonian H = A f at each point to determine the optimal U Min H(X, U, A) * U 6) On reaching time t Q , update the control policy used on the forward integration with that found on the reverse integration in step (5) U i+1 = U 1 ' + c(t)(U* U 1 ') 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 A. such that the gradient of L with respect to A.(t f ) is driven to zero, i.e., to drive (x. x.(t.)) to zero. As before, determine the A 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 A.(tJ~); jen, j7k in a similar manner as in step (7). Determine A, as before from the continuity of the Hamiltonian. 9) Repeat steps (5)-(8) until (1) 6J < e-j no significant improvement in the final time (2) IIU 1 U 1 W< £p no significant change in the control policy and (3) II6A.II < n at t f and (4) II6A.H < n at t" J K

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85 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 = Xl (W 12 V 21 u 3 ) + ^ [W 12 (h 12 x 2 ) + V 21 (x 2 h{) + Q] ] + X 3 ( Ul u 2 V Q2 ) + ji [ Ul (h p x 4 ) + V Q2 (x 4 hp + Q 2 ] This simplifies for the various stages as follows: Stage A : t n < t < t, A 2 H a = A 1 U 2 + ~ ^2^4 " x 2^ + *V + M U 1 " u 2^ Stage B : t, < t < t ? . x, (t,) = x, H b = ^l + A 3 (u l U 2 } x l H|2 = u« ; u~ = or f, = Stage C : t ? < t < t, x-,(t ? ) = x,, X l u-, = u 2 or f~ = and f -, =

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86 Stage D : t~ < t < t f x ? (t^) > x ? X ? H d = ~[W 12 (h 12 x 2 ) + V 21 (x 2 hp + Q^ x l + ^ [ Ul (h F x 4 ) + q 2 ] x 3 W-J2 = Vp-i + u 2 or f, = 0; u-, = u 2 or f_ = Time t. is determined when the second effect starts to boil, i.e., when x»(t f ) = x». The variables Q-, , Q 2 , V^-, , V Q2 , W-,,,, h,,,, h^, lv are determined from the algebraic equations in Appendix A. The adjoint equations are and H is either H , H, , H or H,. a b c d 9H The partial derivatives -rr are evaluated numerically for each point in time using the appropriate Hamiltonian valid at that time. Note that in Stage A, X^ = 0; this is because the liquid enthalpy in the second stage, x., is constrained to stay at x. = !v by x A = 0. In Stage B, X-j = since the first effect hold-up X-. = X-, = constant. That is, the objective function is made insensitive to changes in x-, by holding x 1 = x-j . Likewise, in Stage C, X 3 = and X 1 = because both holdups are fixed. The final time is determined when the second effect boils, i.e., when x 4 (t f ) = x^. At t f , X 2 and X 4 are active. X 2 = since the final condition on the liquid enthalpy in Stage 1, x , is unspecified. X. is

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87 determined from the final condition on the Hamiltonian. On the reverse integration of the adjoint variables at time t« the second effect holdup, x 3 , becomes constrained. The mulitplier Xo(tZ) is found from continuity of the Hamiltonian H. (t«) = H (t 2 ). At time t-, , the first effect holdup, x-, , becomes constrained and A-,(t7) is determined from the continuity of the Hamiltonian at time t-, , H (t7) = H, (t-,). Also, on the reverse integration, the appropriate Hamiltonian for each stage is minimized with respect to the control variables U. Note that in Stage A minimization of H is with respect to second effect feed u-, , intereffect flow u ? , and first effect steam temperature u*. The recirculation rate u, and the product flow u 5 cannot be started when the first effect is still filling. In Stage 3 the feed flow to the first effect, Up, has a set value as it maintains the first effect holdup x-, = x-, . So minimization of H, is with respect to Up > u 3 and u,. Likewise, in Stage C, the second effect feed flow, u-, , is fixed to maintain the second effect holdup at x, = x~ and minimization of H is with respect to the recirculation flow u, and the steam temperature u». The same holds for Stage D. Minimization of the Hamiltonian was achieved by a slightly modified version of the computer program VA04A originally coded by M. J. D. Powell. It is based on a conjugate gradient search technique which does not involve partial derivatives, and it is explained in Fletcher and Powell (1963). This routine was adapted for bounded variable minimizations by writing a package which accounted for the control variable constraints. When a control constraint was encountered, a perturbation of the control variable into the feasible region decides

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whether the constraint should be held or be released. This is basically a numerical evaluation of the associated Kuhn-Tucker multiplier. The forward and reverse integrations were done using Hamming's predictor-corrector method HPCG available in the IBM Scientific Subroutine Package. The adjoint equations required the numerical values of partial derivatives, and a subroutine PDERIV was written to do this. Tables 4.1 to 4.6 and Figures 4.1 to 4.3 are the results of the three iterations to determine the optimal policy. The control policy, shown at the top of each figure, is the one used during the forward iteration resulting in the states shown. The adjoint variables on the reverse iteration are also shown at the bottom of each figure. The final time decreased from 13.2 minutes to 10.27 minutes in three iterations. The maximum allowed change in the flow rates were 1 lb/mi n for u-j and u~ and 10 lbs/min for the recirculation rate u-,. The maximum allowed change in the steam temperature was 5°F. The minimum time for this scenario could not be reduced any further as all the variables were bounded. It can be noticed from the plots that the final control policy is bang-bang and that the switches occur at the point constraints assumed in the scenario. This is indeed a fortunate result as the control policy can be put in feedback form dependent upon whether the states are below or at their steady state values. This is not a typical result but is due to the assumed control scenario which is specific to this type of problem. The final control policy which resulted in a minimum time can thus be put into the following feedback form.

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r f u ; u 3 = 65.0 u 4 240.0 u 1 = u 2 = 10.0 u ] = 10.0 U l = U 2 3.0 LU _l -2.0 > gl.O o a ^0.0 -1.0 0.0 3.0 6.0 9.0 TIME(MINUTES) 40.0 30.0 20.0 -10.0 0.0 12.0 Figure 4.1 Control, State and Adjoint Variables for Problem 1, Iteration 1

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90 u 3 = 75.0 lu. u 1 = u ? = 11 .0 II .0 'u 3 = u 2 = u 4 245.0 U l = U 2 260r-.40.0 Figure 4.2 Control, State and Adjoint Variables for Problem 1, Iteration 2

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97 LU o CO — < Z ooooooooooooocooooo OOOOOOCi^O oooooooooo ooooooooooooooooooo f\J<\J(M(NI(NjrsJ(\|(\jrvJ(\jrj(\J(NJ(NJ © C C C o o co co oo oo co oo
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98 Intereffect liquid flow, u 2 = u 2 max for x-j < x-, (first effect filling) Intereffect liquid flow, u 2 = V 21 for Xi > x-| (first effect filled) Feed flow, u-| = u-j max for x 3 < x 3 (second effect filling) Feed flow, u, = u 2 + V Q2 for x 3 >_ x 3 (second effect filled' Recirculation rate, u 3 = u 3 max for x-| > x-| (recirculation after first effect filled) Steam temperature, u 4 = u 4 max for all time (input steam temperature at maximum) With the optimal policy as shown above a simulation of the state equations including the concentration equations was done and is shown in Figure 4.4 while Table 4.7 gives the simulated values. The product flow rate, u 5 , was kept shut off until the desired concentration was reached in the first effect. While concentration was taking place, the two hold-ups were kept steady by maintaining u 2 = V 2] and U] = u 2 + V Q2 After the desired concentration C, was achieved, the product flow rate was fixed at u,= u ? C ? /C, to keep the product concentration constant. In the simulation shown in Figure 4.4, it was assumed that the initial and final concentrations of the solute were very small so as to cause a negligible boiling point elevation and also so that the properties of water were not altered by the presence of the solute. For example, a 3 percent weight solution of sodium hydroxide would cause a boiling point elevation of approximately 3°F at 212°F which was considered to be an insignificant elevation for this problem.

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99 ^0.0219 I/O CO ^0.0214 ^0.0209 g 0.0204 40.0 m 30.0 _i D. o 20.0 _i o zn 10.0 0.0 235 F 215 195 175 0.0 3.0 6.0 9.0 TIME(MINUTES) 12.0 15.0 Figure 4.4 Optimal Simulation including Concentration dynamics

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102 IV. 3. 2 Problem 2. Fixed Feed Rate The minimum start-up time was again determined using the same algorithm for a problem similar to Problem 1 but with the added restriction that the feed rate was fixed at a nominal value. Such a start-up procedure is used by the students in the undergraduate unit operations laboratory. It was desired to determine the effect of this policy. (No verification was made that industrial practice uses this procedure, but it would be surprising if many start-up policies did not.) The nominal value chosen was 2.9 Ibs/min (0.025 kg/s) which is the steady state flow rate used for the laboratory experiment. It is clear that the minimum time must be at least 23 minutes as this is the time needed just to fill the two effects. The product flow rate from the first effect then became an active control variable and had to be used to maintain a constant hold-up in the first effect. The control scenario was modified for this problem as the final time was determined by the hold-up in the second effect reaching its steady state value. The second effect solution started boiling at an earlier time and so was not used as the stopping condition as in Problem 1. The scenario was as follows: Stage A : t Q < time t < t, . First and second effects are being filled. First effect is being heated. Control variables: Recirculation flow, u 3 , and product flow, u 5 , are not possible since the first effect hold-up has not reached its desired value. Feed to first effect, u 2 , and steam temperature, u 4 , are found from minimization of the Hamiltonian.

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103 Hamiltonian: H a = A 1 U 2 + x [u 2 (x 4 " X 2 J + Q l ] + A 3 (W F " u 2> The second effect enthalpy is constant; x, = or x, = lv resulting in X« = 0. Time t-, , signifying the end of Stage A, is determined when the first effect is filled, i.e., when x-,(t-,) = x, . Stage B : t, <_ time t < t~. First effect is filled and is being heated. Second effect is being filled. Control variables: Feed to the first effect, u ? = to maintain a constant hold-up. Product from first effect, Ur = 0. Minimization of the Hamiltonian determines the recirculation rate, u~> and the steam temperature, u«. Hamiltonian: H b = -rQi + *3 w f x i The first effect hold-up and the second effect enthalpy are unchanged; x-, = x» = 0; and A, = A, = 0. Time tp» signifying the end of Stage B, is determined when the first effect liquid starts to boil, i .e. , when x 2 (t 2 ) = x 2 . Stage C : t« 1 time t < t,. Second effect is being filled and being heated. Control variables: The feed to the first effect, u 2 , is such that it maintains a constant first effect hold-up, u 2 = Vpi . The product rate Ur = 0. The recirculation rate, u 3> and the steam temperature, u„, are determined from minimization of the Hamiltonian.

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104 Hamiltonian: X H = £ CW 12 (h 12 x 2 ) + V 21 (x 2 h\) + Q^ + X 3 (W F u 2 ) x l X A + ^ [W F (h F x 4 ) + Q 2 ] Time t,, signifying the end of Stage C, is determined when the second effect starts to boil, i.e., when x 4 (t 3 ) = x 4 Stage D : t 3 < time t < t f . Second effect is being filled. Control variables: Same as in Stage C. Hamiltonian: H. £ [W ]2 (h 12 x 2 ) + V 21 (x 2 h]) + Q^ + X 3 (W F u 2 V Q2 ) x l First effect hold-up and second effect enthalpy are maintained constant. Time t f , is determined when the second effect hold-up reaches its desired value, i.e., when x 3 (t f ) = x^. The final condition on the Hamiltonian yields a value for A 3 (t f ) X 2 (t f ) = 0, x 3 (t f )x 3 (t f ) = -1 determines x 3 (t f ) At intermediate points continuity of the Hamiltonian is used to determine the unknown multipliers. At time t 3 , H d (t 3 ) = H (t 3 ) determines A 4 (t~) At time t, , H b (t|) = H fl (t[) determines X-j(t^) It took 5 iterations for the final time to reach its minimum

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105 value of 24.53 minutes. The feed rate was fixed at 2.9 lbs/minute (0.025 kg/s). The control variable which contributed significantly to this reduction in the final time was the intereffect flow rate u 2 The state variables for three of the iterations are shown in Figure 4.5 and the values are tabulated in Tables 4.8 to 4.10. The results of the iterations were. Iteration # u^

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106 ITERATION 5 ITERATION 4 ITERATION 1 40. On 235 0.0 5.0 10.0 15.0 20.0 25.0 30.0 TIME(MINUTES) Figure 4.5 State Variables for Problem 2, Iterations 1 , 4 and 5 35.0

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no Steam temperature, u* = u* . for x ? £ x 2 (first effect boiling) Product rate, u^ = for x, < x., (second effect filling) Product rate, Ur = u~ V^-i for x~ >_ x, (second effect filled) IV. 3. 3 Problem 3. No Bound on the Second Effect Hold-up The minimum time problem was run again for the conditions as in Problem 1 but with the upper bound on the second effect hold-up removed. The second effect hold-up H ? (state variable x~) was permitted to vary with the restriction that it should be within one percent of its desired steady state value at the final time. Since the hold-up is allowed to vary the feed to the second effect, u, , need not be equal to the flow of liquid and vapor out of the second effect, Up + Vgp, and minimization has to be done with respect to u-, and u ? as opposed to Up alone as in problem 1. The control scenario is different from the one used in problem 1 as we now assume that x 4 (t f ) = x. and x-Jt.) = x\ ± 0.01 x\ at the final time. In other words two of the states reach steady state at the final time resulting in an iterative solution to update the guess on one of the adjoint variables at the final time. The scenario is as follows: Stage A : t Q <_ time t < t-, . The two effects are being filled while the first effect is being heated. Control variables: The first effect recirculation flow, u~, and product flow, Ur, are not possible until the first effect hold-up has reached its steady state value. Thus u 3 = u 5 = 0. Minimization of the Hamiltonian is with respect to the

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m feed to the second effect, u, , the intereffect flow rate, u ? , and the steam temperature, u.. Hamiltonian: h H a = A l u 2 + x~ ^ u 2^ x 4 " x 2^ + °^ + A 3^ u l " u 2^ The second effect enthalpy is unchanged, x. = 0, x. = lv, X. = 0. Time t, , signifying the end of Stage A, is determined when the first effect hold-up reaches its steady state value, i.e., x-,(t-,) = x, . Stage B : t-, <_ time t < t ? . First effect is being heated and the second effect is being filled. Control variables: The feed to the first effect is stopped, i.e., u 2 = 0, to maintain the first effect hold-up at x-, . The product rate Ur is kept off, i.e., u,= 0. Minimization of the Hamiltonian is with respect to the second effect feed rate, u-, , the first effect recirculation rate, u.,, and the steam temperature, u„, Hamiltonian: A 2 H b = — [u 2 (x 4 x 2 ) + Q-,] + X 3 (u-j u 2 ) x l The first effect hold-up, x-, , is unchanged; x, = 0, x, = x, , X-, = 0. The second effect enthalpy is also unchanged; x 4 = 0, x 4 = hp, A 4 = 0. Time t ? , signifying the end of Stage B is found from the condition x 2 (t 2 ) = x 2 , that is when the enthalpy of the solution in the first effect reaches its steady state value.

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112 Stage C : t ? <_ time t < t f . The second effect is being filled and heated. Control variables: The feed to the first effect, u ? , is set to maintain the first effect hold-up constant, u ? = Vp-i • The product rate, Ur, is kept at zero. Minimization of the Hamiltonian is with respect to the feed to the second effect, u, , the first effect recirculation rate, u,, and the steam temperature, u». Hamiltonian: X 2 v H c = ~ [W 12 (h 12 x 2 ) + V 2] (x 2 \\\) + Q^ + x l X 3 (u 1 u 2 ) + — [u^hp x 4 ) + Q 2 ] The first effect hold-up is unchanged; x-, = 0, x-, = x, , X-, = 0. Time t f , the final time is determined when the second effect solution starts to boil, i.e., x,(t f ) = x». At this time we also require that the second effect hold-up reach a certain value x^(t f ) = x^ i 0.01*x 3 . This change in the scenario requires that one of the adjoint variables at the final time, A 3 (t f ), be guessed initially and then updated on subsequent iterations. The multiplier A,(t f ), corresponding to the stopping condition x,(t f ) = x» (solution starts to boil), is found from the final value of the Hamiltonian. A 3 (t f ) is updated on successive iterations by choosing it to zero the gradient of L with respect to it, which is equivalent to driving >L x^(t f ) to zero.

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113 Tables 4.11 to 4.15 and Figures 4.6 to 4.8 show the results of three of the iterations. Of interest is that t f is decreased from 10.27 minutes to 9.80 minutes. The initial policy was taken to be the final policy from Problem 1. On the reverse iteration A 3 (t f ) (corresponding to the second effect hold-up) was guessed to be zero. This resulted in a control policy that was bang-bang on all the variables but which required the switching time (T ) on the second effect feed rate, u-, , to be increased. The results for iterations 2 through 7 are summarized below.

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115 u 3 = 80.0 I U. = 250.0 u 1 = u 2 = 12.0 ! Ul = 12.0 U l = ° U 2 = V 21 u 3 = u 2 = 215

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116 u 3 = 80.0 u 1 = u 2 = 12.0 u 3 = u 1 = 12.0 u 2 = u 4 = 250.0 U] = u 2 = V 21 0.0 2.0 4.0 6.0 TIME(MINUTES) co 2.0 LU CO S 1.0 < t 0.0 8.0 U\i

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122 in a shorter final time but at the expense of a larger deviation of the second effect hold-up, x 3 , from the steady state value, x 3 . The optimal policy for this problem can be put in a feedback form by getting rid of the dependence of the feed to the second effect, u, , on the switching time. It should be noted that the switching time for u, in Problem 1 was approximately 5.66 minutes which is close to that for Problem 2. This is not unusual as the switch from u, m ,„ 1 ,max to u-, in Problem 2 occurred when the excess hold-up in the second 1 ,rmn r effect was just sufficient to drain out by the time the second effect liquid started to boil. As the vaporization in the first effect is of the order of 0.1 lbs per minute (0.0008 kg/s), the second effect should be overfilled by about 0.5 lbs (0.23 kg), above the final hold-up of 32 lbs (14.56 kg) to compensate for the vaporization in the first effect. The major difference between the optimal policy of Problem 3 and that of Problem 1 is that the second effect feed flow rate is stopped after the second effect fills up whereas u-, = u 2 , the intereffect feed flow rate, in Problem 1 during the same stage of startup. Summary of the control policy: Feed flow, u = u „ for x-, < x-, (first effect filling) c Z, max 11 3 Feed flow, u 2 = V 21 for x, >_ x, (first effect filled) Intereffect liquid flow, u, = u n m , for x~ < x~ (second effect filling' i I »max o -5 Intereffect liquid flow, u, = u-, . for x 3 >_ x 3 (second effect filled) Recirculation rate, u, = for x-, < x-, (first effect filling) Recirculation rate, u, = u Q m=v for x, > x\ (first effect filled) Steam temperature, u, = u, for all time.

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123 IV. 4 Experimental Runs Start-up runs were made on the evaporator as a final test of the minimum time policy with a bound on the second effect hold-up (as in Problem 1, Section IV.3.1). These runs consequently served as a test for the evaporator model as well. The procedure for each run was as follows: 1 ) The feed to the second effect was preheated to around 180°F. 2) The temperature in the steam chest of the first effect, u», was kept at the maximum possible value throughout the run. 3) The feed rates to the two effects were at their maximum values initially. The feed to the first effect was cut off as soon as the first effect hold-up reached its steady state value. This hold-up, X-, , was then put on analog control by manipulation of the feed rate to the first effect, u ? , by one of the automatic controllers. 4) The recirculation pump was started and the recirculation flow rate, u~, rapidly built up to its maximum value. 5) The feed rate to the second effect, u-, , was kept at its maximum value until the second effect hold-up, x~, reached its steady state value. This hold-up was then put on analog control by manipulation of the feed flow rate to the second effect, u-, . 6) The flow rates, temperatures, hold-ups and vacuum pressure were sampled automatically by the IBM 1070 unit every 15 to 20 seconds. The 1070 unit also maintained the recirculation rate u 3 constant by moving the setpoint of the controller which operated on the recirculation valve CV3 (see Figure 2.2) depending upon the deviation

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124 of the measured flow iu from the desired flow. A linear Kalman filter was built for this particular flow as it was subject to a lot of pulsations and noise. The discrete filter was formulated as follows: F = F , + v , (4.26) n n-1 n-1 Y = F + w n n n where F is the flow rate u~ which we wish to maintain constant. Y is the measured value of the flow, v and w are random noise sequences of mean zero and covariances q and r, respectively. The filter equations were obtained from Jazwinski (1970). Given F , and P , , where F , is the estimate of F n/n n/n n/n and P , is the covariance of the estimate, we can predict the flow rate and covariance to the next sampling time by and F n+l/n F n/n P n+l/n " P n/n + q n On sampling the flow rate y , at the next sampling time, we can update the covariance and estimate P 2 D d n+l/n n+l/n+1 n+l/n P ... + r ., n+l/n n+1 F F + Pn+1 / n+1 (V F \ n+l/n+1 n+l/n r., u n+l " n+l/n ; P "n+1 This was the recursive scheme used to obtain an estimate of the intereffect flow rate, u 3 . It was found that regardless of what the initial value for the covariance P Q was the estimate converged rapidly—generally

PAGE 140

125 within three or four sampling points. To allow for this convergence to the real flow rate, control was not done at the first few sampling points. Figure 4.9 shows the behavior of the filtered flow rate u~ as opposed to the measured flow rate. In all three cases shown it can be seen that the filtered estimate is representative of the actual flow after about 6 sampling intervals regardless of the initial estimate of the variance of the estimate p. In most of the runs an initial estimate of unity was taken for p. The variances of the process and measurement noise were also taken as unity. The results of complete runs through start-up are shown in Figures 4.10 to 4.16. In these figures the experimental minimum time, t , is compared with either the theoretical minimum time predicted exp r by the model, t ., using idealized controls or the minimum time predicted by the model, t ,, using the actual controls measured in the experiment. The initial and final conditions for the model simulations were the same as those of the experimental runs. Figure 4.10 is a comparison of the experimental and actual minimum times for Run CI. The data and simulated values are listed in Tables 4.16 and 4.18. The control variable values used for the simulation are the smoothed values obtained in the experiment. Note that in this run and in the remaining runs the hold-ups measured are not quite near those predicted. Some possible explanations for this discrepancy are given in Chapter V. Figure 4.11 compares the experimental time with the theoretical minimum time, t . , using idealized controls--in other words, the control variables used on the

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126 simulation were shut off and on precisely. Tabulated values for this simulation can be found in Table 4.17. Figure 4.12 shows the results of experimental run C2 and the simulation using the actual controls. As these controls were more realistic than the idealized controls, all subsequent simulations used the actual controls. Data and simulated values for this run are listed in Tables 4.19 and 4.20. Run C3 was a start-up run in which the recirculation rate was held at around 100 lbs/mi n as opposed to 80 lbs/mi n for runs CI and C2. The experimental and simulated values are plotted in Figure 4.13 and listed in Tables 4.21 and 4.22. A similar recirculation rate was used in run C4 and these results can be found in Ficiure 4.14 and in Tables 4.23 and 4.24. Runs C5 and C6 were different from runs CI to C4 in that the recirculation rate was maintained at 65 Ibs/min. Figures 4.15 and 4.16 along with Tables 4.25 to 4.28 contain the experimental and simulated values for runs C5 and C6. It can be seen from Figures 4.10 to 4.16 that the model is fairly representative of the process. There is about a 15 percent discrepancy in the prediction of the final time by the model to the final time obtained experimentally. This could be either due to deficiencies in the model or in the experimental setup. The next chapter lists some of these deficiencies and a few suggestions are made to overcome them.

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127 90 80 o£ 7n — 70 325 "onRUN C2 P 5 o o o o o o OC -ir\ — 425 525 625 TIME(SECONDS) 400 500 TIME(SECONDS) 725 -3.0 TT o 600 825 4.0 3, 2. 1. 700 S 70 CQ 1

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139 tact 10 37 t = 9.95 t e*P ex£ „ og6 r act S" »— i S 20 CO _) > D? loo CO _J -T 90 **1 260ch 215 205 195 185 175 °J^y n q-9-a-oeP-o-i ,, O C | OO O O O OOP o — oo o o o o p o o o ^QQ o Q° ° ° ° T 2 (D) -Q_ 0.0 2.0 8.0 10.0 4.0 6.0 TIME(MINUTES) Figure 4.14 Experimental versus Actual minimum Time for Run C4

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142 t . = 10.48 act t n = 9.6 exp ? 15 | io > _ 70 Si g 60 i ^ 50 260 o ^ 250 40 920' s o 225 act =©=Q"o ^r. q MO _2=a3i^=<* O-i O O o — O O O O OQ O 6~ ^-Q Maximum observable H /TldA Imi .^(O) z AA-A-T^--g°-° 2 -0 4.0 6.0 8.0 10.0 TIME(MINUTES) Figure 4.15 Experimental versus Actual Minimum Time for Run C5

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145 tact = 10 ' 4 Sap = 10 ' 9 ^£ = 1.048 'act £ 20 r)OO O O => 2 20 £ 70 =• 60 '?~3-o225 215 £ 205 195 185 175 k^= ^^^^ Q— & i>-eQ -^~b-BJ> — e-A o, t^Wr^n^v^ op o Q -gi ooPOQ QqOo , , Maximum observable Hy /• A) __SO ^-^M^.§ T^O) 4.0 6.0 TIME(MINUTES) 10.0 11.0 Figure 4.16 Experimental versus Actual Minimum Time for Run C6

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148 LIST OF IMPORTANT VARIABLES USED IN CHAPTER IV State variables: X-, First effect hold-up (H-,) x ? First effect solution enthalpy (h,) x~ Second effect hold-up (H 2 ) x» Second effect solution enthalpy (h 2 ) Xr First effect solute concentration (C-.) Xr Second effect solute concentration (C^) Control variables: u-, Feed flow rate to second effect (top) u ? Feed flow (intereffect) rate to first effect (Wj 2 ) u, Recirculation rate in first effect (W-j-j) u, Steam temperature to first effect (T $ ) u r Product rate from first effect (Wq-j) Other variables: W, ? Flow rate of solution entering first effect V 21 Vapor rate out of first effect V n? Vapor rate out of second effect h 1? Enthalpy of solution entering first effect hV Enthalpy of vapor in first effect hi Enthalpy of vapor in second effect C, 9 Solute concentration in solution entering first u effect Q, Heat transfer rate in first effect Q ? Heat transfer rate in second effect

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CHAPTER V COMMENTS AND RECOMMENDATIONS V.l Model The lumped parameter nature of the model causes it to be somewhat inaccurate in predicting the temperatures. There is a temperature gradient in both effects and particularly in the first effect as is evidenced by the temperature gauges at the top and bottom of the tubes. Using a temperature gradient based on these two temperatures is probably better than using a bulk temperature. The heat transfer mechanisms assumed, particularly the twophase coefficients and the natural convection coefficient in the second effect, are doubtful. Developed correlations in the literature are of doubtful accuracy. The method for calculating an average heat transfer coefficient in two-phase flow is particularly suspect. All of these suffer some loss of credibility during highly transient conditions. It was hoped that the parameter estimates G-i and 9 ? would account for these inaccuracies, but experimental runs C described in Chapter IV indicate that the first effect temperature in particular is not predicted well once boiling has begun. A better model fit would have been possible by taking 9, and 6p to be time varying parameters, and these could have been optimally determined in the same fashion as the optimal control policy was determined in Chapter IV. The time varying nature of the parameters 149

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150 could possibly have accounted for varying heat loss rates throughout the experiment as well as for varying temperature gradients in the effects. All this, of course, would be at the expense of added computer time. V.2 Experimental Setup The measurements of the levels in the two effects were subject to a few inaccuracies. Measurement was based on the height of water column in the sight glasses. The actual height varied depending upon the intensity of boiling and the pulsating flow, particularly in the first effect. This drawback could have been alleviated, but certainly not eliminated, had the DP cells which were used on the hold-ups had an adjustable damping device to "filter" the readings. These momentary variations in the indicated height caused improper analog control of the hold-ups. The proportional gain on the automatic controllers had to be kept high enough to enable the control valves to respond reasonably fast to changes in the hold-ups, but, with a high gain, the momentary variations caused by the boiling in the two effects would cause the controller to respond to the noise. A compromise value of the proportional gain was used, but this did not completely overcome the problem of sensitivity to the momentary variations. A filter on the hold-ups and direct control of the setpoints by the IBM 1070 is a possible solution to the problem. This can be done if the vapor rates from the two effects could be directly determined or accurately estimated without a great deal of calculation as this increases the on-line computer costs.

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151 The recirculation flow rate in the first effect was one which did cause a number of problems. Starting the recirculation pump was critical as a certain amount of liquid head on the suction side was necessary to avoid causing the pump to suck in air. Thus the hold-up had to be maintained above a minimum value. In addition, the hold-up had to be kept below a certain value not much above the minimum required to avoid entrainment of the liquid with the vapor going to the second effect. This hold-up was critical and had to be maintained throughout the run. This problem of a critical hold-up could be avoided if the vapor-liquid separator were baffled or if an additional well-designed separator were installed in addition to the existing one. This would also ensure no entrainment at higher recirculation rates. Temperature measurement at the exit line of the second effect is not a proper means for obtaining a measure of the temperature in the second effect. This measurement is accurate only if there is a flow out of the second effect. The thermocouple should project directly into the bulk of the second effect liquid in order to get a good estimate of the temperature. The use of additional thermocouples at the top of the two effects may help to give a better temperature average in the two effects and could account for temperature gradients. The minimum time runs C shown in Chapter IV required the bleeding of the DP cells while the run was in progress. This led to inaccurate flow measurements, but only at the initial time.

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152 V.3 Theory The theoretical development of the minimum time solution presented in Chapter IV is not a generalized approach as it draws heavily on the concept of a scenario leading to particular orderings of the point constraints. Proposing an optimal scenario is intuitively obvious in some instances. Most start-up problems in chemical engineering processes can be solved by examining relatively few scenarios to determine the optimal one. This approach of ruling out most of the less likely scenarios makes the mathematical problem much less formidable. In the case of the double effect evaporator, the minimum time solution is bang-bang on the control variables. Although a bang-bang minimum time solution is the rule in linear systems, it is not necessarily true in the case of nonlinear systems such as the evaporator. The switching times found here were directly related to the states thus obtaining the control policy in feedback form. This form made the optimal policy easy to implement in practice. V.4 Conclusions In spite of all the drawbacks mentioned in the preceding sections, it can be concluded that the dynamic and algebraic equations of Chapter III together with the parameter estimates for 6-j and Q^ provided a reasonably accurate working model of the double effect evaporator. The experimental runs of Section IV. 4 show that the model is accurate to within 15 percent in the prediction of the final time. Moreover, with the control policies for minimum time start-up in

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153 feedback form as obtained in Chapter IV, the accuracy of the model does not really matter. The control policy was solely a function of the states and not explicitly a function of time, so that the precise state trajectories predicted by the model were not important from the viewpoint of implementation of the control policy. The scenario approach adopted in Chapter IV for obtaining the optimal control policy and the resulting use of point constraints not only simplified the mathematics but was a factor in obtaining the control policy in feedback form. This approach could possibly be used with success in other start-up problems in the chemical industry.

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APPENDIX A HEAT TRANSFER EQUATIONS AND OTHER RELATIONSHIPS In addition to the 6 differential equations (3.1) to (3.6) derived in Chapter III and the 4 algebraic equations (3.7) to (3.9) and (3.11) we have the following algebraic equations to complete the set of equations. A.I Relation Between Temperatures and Enthalpies T, = f(h-,) from solution data (A.l) To = f(ho) from solution data (A. 2) hi = f(T-,) from steam tables (A. 3) ho = f(To) from steam tables (A. 4) T = f(h ) from steam tables (A. 5) j = f(h-,p) from solution data (A. 6) T F = f(h F ) from solution data (A. 7) A. 2 Heat Transfer Equations—First Effect The mechanism for heat transfer is as presented in (Fair, 1960, 1963a, 1963b), (Hughmark, 1969) and (Tong, 1965). It is assumed that the first effect can be divided into two zones — the sensible heating zone (subscripted s) and the vaporizing zone (subscripted B). Single-phase convective heat transfer is the mechanism in the sensible heating zone whereas the mechanism in the vaporizing zone is a combination of twophase convective and nucleate boiling heat transfer. Slug flow is 154

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155 the predominant flow pattern in which the vapor bubbles coalesce into slugs of vapor rapidly once boiling begins. The slugs of gas accelerate more rapidly than the remaining liquid and the vapor has a higher velocity than the liquid at exit. The fractional lengths of the sensible heating zone and the vaporizing zone are obtained through a consideration of the pressure drops relative to the total pressure drop in each tube. The total heat transfer rate is QT = Q ls + Q 1B (A.8) A. 2.1 Sensible Heating Zone Q ls = U ls A ir ^AT ls ( A .9) where the mean temperature difference AT, is AT ls = T s < T 1 + T 12>' 2 < A 10) A, is the area of the sensible heating zone. (L ls h «1. " fi ?s A l rf< T , " W < A 11) and . where T , and T are the inside and outside wall temperatures, wis w2s respectively. Also "is fi isw A i rf< T „is W (AJ3) where h, , is the film coefficient due to the tube walls and fouling. I sw The outside film coefficient is obtained from the Mussel t equation

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156 h° s = 0.943 hsf p lsf g X 1s L ls u lsf (T s W _ hs f( V 0.25 (A. 14: (A. 15) The inside film coefficient is obtained from the Dittus-Boelter equation calculated at boiling zone conditions h n = n i d77 D ii G i ] hb f c ibhb 0.4 (A. 16) For Re = y lb > 5000; n, 0.023, n 2 = 0. 2000 < Re < 5000; n 1 = 0.0775, n 2 = 0.667 Re < 2000; n-, = 0.183, n 2 = 0.545 The overall heat transfer coefficient in the sensible heating zone is given by J ls (A. 17: -o 7TT D 10 J. "is e l h ls U h lsw This introduces an unknown parameter 6-, to account for the uncertainty of the constant used in equation (A. 16). 6 ] = ]a when no boiling is taking place and 0-, = e lb when there is boiling. Also, when there is no boiling L-j = L,. The length of the sensible heating zone is given by AT H AP Js 'Is (£M£M£ (A. 18)

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157 where Tp is the slope of the vapor pressure curve estimated from s steam tables § | f(V (A. 19) The term I tt I is obtained from a heat balance All * D li N lt h 1s( T wlB-V AL W 12 C 1B AP The term I -rrI is estimated from (A. 20) ^]=-P 1B g (A.21) The relationship between the properties of liquid occurring in equations (A. 14), (A. 16), (A. 20) and (A.21) at the corresponding temperatures , are T lsf = T s " °75(T s W * k lsf p lsf y lsf T lb (T^ + T^/2-.k^, y lb , C lb T l * P 1B' C 1B A. 2. 2 Vaporizing Zone The length is found from the remainder of the total length Hb " H " hs < A 22 > The heat transfer rate is QlB = U lB A l if AT 1B (A 23)

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158 The mean temperature difference is Also and AT 1B T s " T l QlB = R ?B A 1 Ef < T s"W L QlB fi lVl Ef < T wlB"V Q 1B h lBw A l U, ^ T w2B " T wlB^ (A. 24) (A. 25) (A. 26) (A. 27) The outside film coefficient is again obtained from the Nusselt equation h!J B = 0.943 " 3 2 k lBf p lBf g X ls p L lB y lBf( T s " T w2b' 0.25 (A. 28) The fraction of vapor to liquid is 12 W (A. 29; The variable x-jp "is introduced for simplifying the solution procedure ^12 0.4 x 12 (A. 30) The Lockhart-Martinell i parameter X.., which represents a ratio of kinetic energies of liquid and vapor, is

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159 For the steam-air system the two-phase convective heat transfer coefficient is h n P = 3 5h isl4J (A>33) The mass velocity is G l = N-T(A 34) 1 ,N irit The coefficients a used in the calculation of the average inside film coefficient are given by a 12 =f(G,,X tt ) (A.35) aj 2 =f(G r X it ) (A.36) A mean a is a 12 = (a 12 + a\ 2 )/2 (A. 37) The film coefficient due to nucleate boiling is given by fi lnb= 52 < T w 1B T l> (A 38) The average inside film coefficient is a combination of the nucleate boiling coefficient and the two-phase convective coefficient fi lB "l^lnb + fi ltp (A 39; The overall heat transfer coefficient is U 1B = 1— k (A.40) J_+ 1 10 + J_ fi lB e lb fi lB Dl1 ^IBw

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160 where is a combined resistance for the tube walls and fouling. h lBw The temperature-property relationships are T l "* M 1B' P 1B' y lB for 1ic ' uid T ]Bf = T s 0.75(T s T w2B ) k 1Bfi P 1Bf , y 1Bf for vapor A. 3 Heat Transfer Equations—Second Effect A-, = f(T 1 ) Q 2 = UgAgATg where the mean temperature difference is given by also AT 2 = T 1 (T 2 + T F )/2 ^2 = ^V T 1 " T w2> Q 2 = h 2 A 2 (T wl T 2 ) The outside film coefficient is found from the Nusselt equation (A. 41) (A. 42) (A. 43) (A. 44) (A. 45) (A. 46) h 2 = 0.943 2f fD 2f ^ 1 L 2 y 2f (T l " V 0.25 'A. 47' The inside film coefficient is found from the natural convection equation Gr 2 = LJJP29• T W1 " V (A. 48)

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161 Pr C 2 y 2 2 k, Y 2 = Gr 2 Pr 2 0.55 k ] f 0.13 k n ) 25 4 7 Y 2 for 10^ i Y 2 < 3.5 x 10 7 Y 2 * 33 for Y > 3.5 x 10 7 The overall heat transfer coefficient is (A. 49) (A. 50) (A. 51) U 2 = 1 h° e 2 h 2 D 2i h 2w (A. 52) The correction factor ? is again introduced to account for uncertainties in the inside film coefficient equation (A. 51) 6 ? = 9 ? when the liquid is not boiling and 2 = e 2 h w ^ en *' ie l 1a > u i c ' is boiling. The properties of vapor required in (A. 47) are found at the film temperature T 2f = T l " °' 75(T 1 " T W 2 ) ~* k 2f p 2f y 2f The liquid properties in (A. 48), (A. 49) and (A. 51) are found at the mean temperature T 2 = ^ T 2 + T w2^ /2 * k 2' p 2' M 2 5 C 2' 3 2

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APPENDIX B LISTING OF COMPUTER PROGRAMS This appendix contains a listing of the computer programs written for determining the minimum start-up time control policy for the double effect evaporator described in Chapter II. These programs utilize the algorithm of Section IV. 2. 4. Basically, each iteration consisted of two integrations and for simplicity, two programs were run separately with the output from one being the input to the other. The first program handled the forward integration of the state equations using either an initial guess on the control policy or an updated version of the previous control policy as input. Integration was done by subroutine HPCG (IBM Scientific Subroutine Package) which is based on a predictor corrector method. Subroutine XDOT supplied the derivatives of the state variables which are the right hand sides of the differential equations (3.1) to (3.6). The results of the forward integration comprising the state and control variables against time were output on cards and on the line printer. The input to the next main program consisted of these cards and the times at which point constraints were encountered on the forward pass together with derivatives of the states before and after these points in time. The second program integrated the adjoint equations in the reverse direction in time. The Hamiltonian was also minimized at selected points to obtain the optimal control policy. The final conditions and jump conditions on the adjoint variables were evaluated 162

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163 using the derivatives of the states from the forward integration. Subroutine HPC6 was again used for the integration of the adjoint equations and subroutine LAMDOT supplied the right hand sides of the adjoint equations. Subroutine VA04A (Powell, 1964) was the search program used for minimizing the hamiltonian. Evaluation of the time dependent variables, other than the state and control variables, was done by subroutine FUNCS. The film coefficients were estimated in the following subroutines with the associated references. 1) The overall coefficient was evaluated in subroutine HEAT based on the method presented in Fair (1960) and Tong (1965). 2) The forced convection coefficient was evaluated in subroutine FCIF using the Dittus-Boelter equation (Hughmark, 1969). 3) The natural convection coefficient was evaluated in subroutine FCIN using the natural convection equation (McCabe and Smith, 1967). 4) The natural convection coefficient for two phase flow was evaluated in subroutine FCINB using the Rohsenow equation (Tong, 1965). 5) The steam side coefficient was evaluated in subroutine FCO using the Nusselt equation (McCabe and Smith, 1967).

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164 C MAIN PROGRAM FOR INTEGRATION OF THE STATE EQUATIONS £*•««*•••#«*«*•««»«••••«•*««*•*****•**••*•*»•*»•••*•****•*•*••**• C PURPCSE C THIS IS THE MAIN PROGRAM WHICH FIRST CALLS SUBROUTINE C INPUT TO INPUT DATA. IT THEN INITIALIZES THE STATE C VECTCR AND CALLS SUBROUTINE HPCG FROM THE IBM SCIENTIFIC C SUBROUTINE PACKAGE TO INTEGRATE THE STATE EQUATIONS. C FINALLY, IT OUTPUTS THE RESULTS. C SUBPROGRAMS REQUIRED C SUBROUTINE INPUT C SUBROUTINE HPCG C DIMENSION X(4),PRMT( 5),.DERY( 4),AUX( 16, A) EXTERNAL XCOT,OUT COMMCN/TIMEl/TTT( 120 ) CONNCN/STATES/XK 120 ) , X 2 { 120 ) , X3 ( 120),XM 120) C0MM0N/TIMER/TIME1120)f.IT4MEi IMAX CC^yCN/STEADY/XHAT(4),XINITU),TF,HF,N0RDER,UlMAX,UlMIN, lU2MAX,U2MN,U3MAX,U3MlN,U4MAX r U4MlN COMMCN/CCNTRL/UH 120 ),U2( 120),U3( 120),U4(120) C C THE INPUT SUBROUTINE IS CALLED HERE. IT INITIALIZES THE C CCNMON BLOCKS CATAl, CATA2 AND STEADY. C CALL INPUT C C INITIALIZE ARRAY COUNTER ITIME AND VARIABLES TO BE INPUT C TO SUBROUTINE HPCG C ITIME=1 N=4 PRMT(1)=0. PRMT(2)=2C. PRMT( 3)=1.0 PRMT(4)=?.2 CO 13 J=1,N 10 X(J) = XIMTU) DERY(l)=3./8. DERY(2)=l./8. DERY(3)=3./8. DERY(h)=1./8. CALL HPCG(PRMT,X,CERY,N, I FLF , XDOT ,OUT, AUX ) C C OUTPUT RESULTS ON LINE PRINTER AND ON CARDS C wRITt(6,200C) I M=ITIME-1 WRITE(6,2010)(K,TTT{K),X1(K),X2(K),X3(K),X4(K),K=1,IM) WRITE(6,2020) WRITl(6,2u10)(K,TTT(K),U1{K),U2(K),U3(K),U4(K),K=1,IM)

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165 WRITE (7, 2040) (TTT(K),X1(K),X2(K),X3(K),X4{K),UI(K),.U2(K), 1U3(K) ,U4(K) ,K=1, IM) STOP 2000 F0RMAT(lHl t 20X, •STATE VARIABLES ON FIRST I TERATION •,//, 5X , 1 'STAGE NC ,5X, 'TIME' ,5X, 'XI' , 10X, 'X2',10X, 'X3' ,10X, 'X4' ) 2010 FORMAT (7X, I3,7X,F6.3,2X,F9.5,4X,F9.5,4X,F9.5,4X,F9.5) 2020 F0RMAT(1H1,20X, 'CONTROL VARIABLES ON FIRST I TERATION ',// , 15X t 'STAGE NO' ,5X, 'TIME' , 5X,«U1', 10X, 'U2',10X, • U3 • , 10X , • U4« ) 2040 F0RMAT(F6.2,2X,8F9.4 ) END

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166 C*« C c* » c c c c c c c c c c c c c c XCGT PURPOSE THIS SUBROUTINE SUPPLIES VALUES OF THE DERIVATIVES TO SUBROUTINE HPCG. USAGE CALL XCCT(T,X,DERY) NOTEO WILL BE CALLED ONLY BY SUBROUTINE HPCG PARAMETERS T = TIME, INPUT X =. STATE VECTOR, INPUT CERY = VECTOR OF DERIVATIVES OF X AT TIME T, OUTPUT SUBPROGRAMS REQUIREC SUBROUTINE MAPI SUBROUTINE FUNCS SUBROUTINE XDCT ( T , X , C ERY ) DIMENSION X(l ),DERY( 1),Y(8), IPOSTU) ,F(8),-XX(8),JPOSTU) , 1D(8) COMMON/INDEX/ I (A) COMMCN/TIMER/TIME(120),ITIME, IMAX COMMCN/TEMPU/UTEMP(A) CGMMCN/STEADY/XHATU),XINITU),TF,HF,N0RDER,U1MAX,U1MIN, 1U2MAX,U2MIN,U3MAX,U3 W IN,U4MAX,U4MIN COMMCN/CCNTRL/UK 120 ),U2( 120 ),.U3( 120),U4( 120) THE ELEMENTS OF VECTOR I INDICATE WHETHER THE STATES HAVE ARRIVED AT THEIR STEADY STATE VALUES XHAT. EXAMPLE 1(1) = 1 IF Xd).GE.XHAT(l) = OTHERWISE DO 2 N=l,4 I (N)=0 8 IF(X(N) .LT.XHAT(N) )GC TO 2 I (N)=l 2 CCNTINUt THE VECTOR UTEMP CONTAINS VALUES OF THE CONTROL VARIABLES TO BE USED IN THE PRESENT TIME STEP. THE OPTIMAL LAW IS UTILIZED HERE IN FEEDBACK FORM. IF A SIMPLE CONTROL LAW IS NOT AVAILABLE, THEN UTEMP SHCULC BE THE INTERPOLATED VALUES OF THE CONTROL VARIABLES INPUT IN FEEDFORWARD FORM UTEMP(1)=U1MAX IF(I(3).EC.1)UTEMP(1)=U1MIN UTEMP(2)=U2MAX UTEMP(3)=U3MAX IF( I ( 1) .EG.C)UTEMP( 3)=U3M,IN UTEMP(4)=U4MAX

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167 SUBROUTINE MAPI IS USED TO MAP THE STATE AND CONTRGL VARIABLES INTO A VECTOR D WITH THE STATE VARIABLES IN THE FIRST FOUR POSITIONS IN D AND THE CONTROL VARIABLES IN THE NEXT FOUR 40 DO 10 J=l,4 10 IPOST(J)=J CALL MAPl(XX,8,X, IP0ST,.4,Y ) DC 20 J=5,8 JJ=J-4 20 JPOST(JJ)=J CALL NAP1(Y,8,UTEMP, JPOST, 4, C) THE VALUES OF THE TIME-DEPENDENT VARIABLES (WHICH ARE FUNCTIONS OF THE STATE AND CONTROL VARIABLES) ARE FOUND BY CALLING SUBROUTINE FUNCS CALL FUNCS(D,8,F,8) RE-MAP THE CONTROL VARIABLES WHICH MAY HAVE BEEN ALTERED BY SUBROUTINE FUNCS DO 25 J=l,4 25 UTEMPl J)=C( J + 4) W12=F(1) V21=F(2) V02=F(3) Hl2=F(4) G1=F(5) G2=F(6) H1V=F17) H2V=F(8) THE RIGHT HAND SICES OF THE STATE EQUATIONS COMPRISE THE DERIVATIVES DERYU)=W12-V21-UTEMP(3) IF(Xll) .LT..1E-70JG0 TO 50 DERY(2)=(W12»(H12-X( 2 ) ) +V21* ( X( 2 )-HlV ) +Q 1 ) / X< 1) GC TC 60 50 DERY(2)=0. GO TG 70 60 DERY(3)=UTEMP(1)-UTEMP(2)-V02 70 IF(X(3) .LT..DGO TO 80 DERY (4)= (UTEMPl 1)*
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C OUT £••««»«»*«*«*••**««***»«*•««••••**«*••«••*••*»*•*•••*••»»*•***••• C PURPCSE C THIS SUBROUTINE OUTPUTS VALUES OF THE STATE VARIABLES C ANC DERIVATIVES AT TIME T C USAGE C CALL OUT(TT,X,DERY, I HL F,ND IM,PRMT ) C NCTEO THIS SUBROUTINE IS CALLED ONLY BY HPCG C PARAMETERS C ALL ARE INPUTS. REFER TO HPCG FOR DESCRIPTION C SUBROUTINE OUT { TT , X, CERY , IHL F r ND IM , PRMT ) DIMENSION X(l ) ,DERY( 1),PRMT( 1) C0MMCK/STATES/X1(12C),X2(120),-X3(120),X4{120) COMMCN/CCNTRL/UK 120),U2( 120),.U3( 120),U4(120) CQMMCN/TIMER/TIME(120), I TIME, IMAX COMMOK/TIMEl/TTTt 120) C0MMCN/STEADY/XHAT(4),XINIT( 4 ) , T F, HF , NORDER , U1MAX, U1M IN , 1U2MAX,U2MIN,U3MAX,U3MIN,U4MAX,.U4MIN COMMON/TiZMPU/UTEMPJA ) COMMON/INDEX/ 1(4) I 1 = I T IME THE ERROR PARAMETER PRMT<4) IS RE-ESTIMATED DEPENDING ON THE MAGNITUDE CF X AND THE WEIGHTING PRMTU) = (0.375»ABS(X( 1) ) +0. 125»ABS ( X { 2 ) ) +0. 3 75»AB S ( X ( 3 ) ) 1+0.125*ABS(XU) ) )*0.01 A POINT T(I) IS SAVED IN THE OUTPUT ARRAYS (COMMON BLOCK CCISTRL, STATES AND TIMED ONLY IF T(I)-T(I-2) IS GT.0.25 MIKUTES IF( I I .LT.3)G0 TO 10 J=II-1 JJ=I 1-2 IF(TT-TTTUJ) -0.25)5, 10, 10 5 K=J GC TC 20 10 K= I I 11=11+1 IF( II.LT.12DG0 TO 20 ^RITE(6,2010) I I STOP 20 X1(K)=X( 1) X2(K)=X(2) X3(K)=X(3) XMK)=.XU) Ul (K)=UTtMP(l ) U2(K)=UTEMP(2 )

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169 2010 2020 U3(K)=UTEMP(3 ) U4
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170 C»*» c c»»* c c c c c c c c c c c c MAIN PROGRAM FOR INTEGRATION OF THE ADJOINT EQUATIONS PURPOSE THIS PROGRAM CALLS THE INPUT ROUTINE TO INITIALIZE COMMON BLOCKS CATA1.DATA2 AND STEADY. IT THEN CALLS CN SUBROUTINE HPCG TO INTEGRATE THE ADJOINT ATIONS. WHEN A POINT CONSTRAINT IS ENCOUNTERED ON THE REVERSE INTEGRATION CONTROL RETURNS TO THIS PROGRAM WHICH DETERMINES THE VALUES OF THE ADJOINT VARIABLES SUBPRCGRAMS REQUIRED SUBROUTINE SEARCH SUBROUTINE CONVAR REAL LAMDA(A) CI MENS I CN PRMT(5 ) , CL AMDA ( 4 ) , AUX ( 16 , 4 ) „DER Y K ) EXTERNAL L AMDCT, L AMCLT CCMMCN/PCINTS/TFtT2tTl,F3TFM,F4TFM t F2TlPtF2TlM,F3TlP,F3Tl*S 1F1T1M COMMCN/STATES/XK 12C ) , X 2 ( 120 ) ,.X3 ( 120 ) , X4 { 120 ) COMMCN/STEADY/XHATU),XINIT( 4 ) , TF , HF , NORDER , U1MAX , UlMIN , lU2MAX,U2MIN,U3MAX t U3MIN,U4MAX,U4MIN COMMCN/CCNTRL/UH 120 )tU2< 120),.U3(120) ,UM120) C0MMCN/TlMER/TlME(12 o ),ITIME,IMAX„IC0UNT,TIM,IG0 COMMON/ C\EW/U1NEW< 120), U2NEW( 120 ) *U3NEW ( 120 ) ,U4NEW(120) CCCMCN/DITIME/NITIME READ IN VALUES OF THE STATES AND CONTROLS ON THE FORWARD INTEGRATION INTO COMMON BLOCKS STATES ,CONTRL AND TIMER. ALSO THE TIMES AT WHICH POINT CONSTRAINTS ARE ENCOUNTERE D ANC THE DERIVATIVES BEFORE AND AFTER THE POINT CONSTRAIN TS CALL INPUT READ(5,1000) IMAX REAC(5,1010)(TlME(K),Xl(K) t X2(K),X3(K) f X4(K) f Ul(K),U2(K), 1U3(K) ,U4 (K) ,K = 1, IMAX ) READ(5,10 20)F3TFM,F4TFM,T2,T1,F2T1P,F3T1P,F1T1M,F2T1P, lF2TlM t F3riM,TF WRITE(6,2030) WRITE (6, 2040) (K,TIME(K),X1(KUX2(K),X3
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171 LAMDM1 ) = • LAMDA(2)=0. LAMCA(3)=C008 LAMDA(4)=(-1.+F3TFM*LAMDA( 3) )/F4TFM ISTARTMMAX IEND=1 PRMT(1)=TIME( IMAX ) PRMT(2)=0. NITIME=3 PRMT(3)=-0.25 PRMT(4)=0.0025 N = 4 ITIME=ISTART 10 CONTINUE DO 20 J=l,4 20 OLANCM J)=0.25 IGO=IGOTC CALL HPCG(PRMT,LAMDA,DLAMCA,Nt IHLF ,L AMDOT ,LAMOLT , AUX ) IF(IHLF.GT.10)STOP C C CUTPUT RESULTS WRITE(6,200C) WRITE(6,2010)(J*TIME(J),X1(J),.X2{J),X3U),X4(J),U1(J), 1U2( J) ,U3{ J) ,U4( J ), J= I ST ART, I END) IGCTC=IGCTC+1 C C START NEXT INTEGRATION AT TIME WHEN PREVIOUS STAGE ENDED C PRMT( 1)=TIM GO TC (70,40*50,70), IGOTO C C CONDITION CN LAMDM4) AT TIME T2 C 40 CONTINUE LAMDA(4)=G. GO TC 10 C C CONDITION ON LAMCA(l) AT TIME Tl C 50 CONTINUE LAMDA(1)=( (F2T1P-F2T1M)*LAMDA(2)+(F3T1P-F3T1M)* 1LAMDA(3) 1/F1T1M GO TC 13 70 CONTINUE STOP 1000 F0RMATU3) 1010 F0RMAT(F6.0,2X,8F9.0 ) 1020 F0RMATI11F7.0) 10 FORMAT! 1H1.20X, 'STATE AND CONTROL VARIABLES', 1 //,2X,'STAGE',2X,'TIME',5X, 'XISIOX, • X2 • , 10X, • X3 % 210X, 'X4' ,10X, «U1' , 10X, 'U2', 10X, «U3', 10X, 'U4' )

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172 2010 FORMAT (3X,I3t2X, F6.3,2X,F6.3,5X,F7.3,5X,F7.3,5X,F7.3,5X f lF7.3,5X,F7.3,5X,F7.3,5X t F7.3) 2030 FORMAT! 1H1,30X,« STATE AND CONTROL VAR I A8LES • , // 2 X, • STAG • , 15X, »TIME« t 10X,»Xl«,lCX, •X2% 10X, 'X3« t 10X, 'X4',10X, 'Ul 1 ,10X, 2'02' .lOX.'US' ,10X,«U4' ) 2C40 F0RMAT(4X,I2,5X,F6.2,5X,F7.2,5X,F7.2,5X,F7.2,5X,F7.2,5X, 1F7.2,5X,F7.2,5X,F7.2»5X»F7.2) END

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173 C LAMOUT C PURPOSE OUTPUTS VALUES OF LAMCA AND THE DERIVATIVES AT DESIRED TINES USAGE CALL LAMOUT (TT , LAMCA, DERY , IHLF , ND IM , PRMT ) TC BE CALLEC ONLY BY SUBROUTINE HPCG PARAMETERS REFER TO SUBROUTINE HPCG FOR PARAMETER DESCRIPTION SUBRCUTINt LAMOUT (TT, LAMCA, DERY,. IHLF, NDIM, PRMT) REAL LAMDA(4),L1(4),L2(4),L3(4),L4(4),L(4) DIMENSION CERY(l) ,PRMT( 1),T( 4) C0MMCN/P0INTS/TF,T2,T1,F3TFM,F4TFM,F2T1P,F2T1M,F3T1P,F3T1M, 1F1T1M CGMMCN/DITIME/NITIME COMMCN/TIMER/TIMEC 12C), I TIME, IMAX, ICOUNT , TIM , I GO COMMCK/INCEX/ 1(4) COMMON/ CNEW/U1NEW( 120 ) , U2NEW ( 120 ) , U3NEW{ 120 ) ,U4NfcW(120) CHECK IF POINT CONSTRAINT IS ENCOUNTERED. IF IT IS ENCOUNTERED TERMINATE INT GRATION AND RETURN TO MAIN PROGRAM FOR NEW VALUES OF ADJOINT VARIABLES IGCTC=IGO GO TC (60,70,80) , IGOTO 60 CONTINUE IF(TT.LT.T2)PRMT(5)=1. GO TO 80 70 CONTIISUc IF(TT.LT.T1)PRMT( 5> = 1. 80 CONTINUE PRMT(4)=0.0025*{ ABS(LAMDA( 1) )+ABS(LAMDA( 2) ) +AB S t LAMDA ( 3 ) )+ 1ABS(LAMDA(4) ) ) IF(PRMT(4).LT..0009)PRMT( 4)=0.C0C9 TIM=TT T( ICCUNT)=TT L1(ICCUNT)=LAMCA( 1) L2( ICCUNT) = LAMDA( 2) L3(ICCUNT)=LAMCA( 3) L4(ICCUNT)=LAMDA(4) IF( ICCUNT.GT.3JG0 TO 10 1CCUNT=ICCUNT+1 RETURN 10 TOUT=TIME( ITIME) IFITCUT. GE.T(4). AND. TOUT. LE. T( 1) )G0 TO 30 DO 20 J=l,3 T( J)=T( J+l) Ll( J)=L1( J+l)

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174 L2( J)=L2( J+l) L3( J)=L3( J+l) 20 L4( J)=L4( J+l) RETURN 30 CALL SEARCMT, LI, TOUT, A, 3,L( 1) ) CALL SEARCH{T,L2,T0UT,4, 3,H 2) ) CALL SEARCMT,L3,T0UT,4, 3,L( 3) ) CALL SEA3Ch(T r L4,T0UT,4, 3,L( 4) ) CALL CCNVAR ONLY AT TIMES AT : ON THE FORWARD INTEGRATION CALL CCNVAR(TCUT,L) WRITE(6,2Q40)T0UT,(L(J), J=l,4) WRIT £(7, 2040) TOUT, ( L ( J ) , J = I, 4 ) 2040 F0RMAT(F7.3,3X,4F10.4) 2C30 FCRf"AT( I 2 , F8 . 3 , 9 F8 . 3 ) ITIME=ITI^E-NITIME GO TC 10 END WHICH VARIABLES WERE STORED

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175 £»«««•«**»*»#«»****«#***«* £»«««««***«•«*««•#»«*•**»* LAMDOT PURPOSE PRCVI SUGRC USAGE CALL PARAMETE TT LAMDA DERY SUBPRCGR SUBRG SUERG SUBRG SUBRO SUBROUTI REAL LAM CIMENSIC DES DERIVATIVES OF THE ADJOINT VARIABLES TO UTINE HPCG LAMCOT(TT,LA NGTEO CALLED *S TIME, MI INPUT VE OUTPUT V AMS REQUIRED UTINE SEARCh UTINE FUNCS UTINE MAPI UTINE PDERIV NE LAMDOT(TT DA14) N CERY(l) ,XX VCA,CERY ) ONLY BY SUBROUTINE HPCG NLTES, INPUT CTOR OF ADJOINT VARIABLES ECTOR OF DERIVATIVES lIP0SITI2) f IMEM(4) COMMCN/STATES/XK 120 CCMM0N/CELT/DELX1,0E C0MMCI\/TIMER/TIME( 12 COMMON/INCEX/ IU) CGNMCN/TEMP/X(8) CCCMCN/CCNTRL/UK 120 COMMON/STEADY/XHATU 1U2MAX,U2MIN,U3MAX,U3 WRITE(6,2060)TT,LAMC 2G60 FCRMATdOX, 'LAMDOT. 114) INTERPOLATE FOR V VARIABLES AND SET BE EXPRESSEC IN F CALL

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176 IIN)=0 IF(X(M .LT.XHAT(N) )GC TO 2 I IN)=1 CCNTINUE X{8)=U4MAX X(5)=U1MAX I F < I (3).EC.1)X(5)=U1MIN X(7)=U3MAX I F ( I (1) .Et.0)X(7)=U3NIN FEED=X(5) CALL FUNCS(X,8,F,8) wl2=F(l) V21=F(2) VC2=F(3) H12=F(4) G1=F(5) G2=F(6) H1V=F(7) H2V=F(8) IGCTC=I( 1)+2*I(2)+4*I(3 )+8»I (4)+l GC TG (10, 170,50, 170,50, 170, 5C, 170, 50,50, 50, 150, 50, 1IGCTC KC STATE VARIABLES AT STEADY STATE DETERMINE RIGHT HAND SICE OF CtLAM 50,50,12 5), MDAD/DT 10 DERY(4)=0. 13 XX(1)=X(1) KCUNT=0 CALL PDERlV(XX,FF,2,2,KOUNT,DELXl,DFDX,id) IF(KCUNT)5000, 16, 12 Y( 1)^XX(KCUNT ) 11 12 I POSIT (1 ) = 1 CALL l"APl(X,8»Y t IPOSIT, 1, D) CALL FUKCSIC, 8,F,9) FF(KCUNT,1)=F(5) 16 FF(KCUNT,1)=F(5) FF(KCLNT,2)=F(6) GC TC 11 CC1DX1=CFCX( 1 ) DG2DX1=CFCX(2 ) 1*(X(2)10X1) /X( 1) + /X<3)

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177 109 XX(1)=X(2) KCUNT=C 106 CALL PDERIV(XX,FF,2, 2 , K0UNT r D ELX2 ,DFDX , W ) IF(K0UNT)5000,108, 107 107 Y(1)=XX(KCUNT) IPCS1T(D=2 CALL fAPKX, 8,Y, IPOSIT, 1, D) CALL FUNCS(C,8,F,8) FF(K0UNT,1)=F(5) FF(K0UNT,2)=F(6> GC TC 106 108 DG1DX2=CFCX(1) 3000 FORMAT(15X,'LAMDA(2) = 'iF10.5,5X,. •0Q1DX2= , ,E15.7) DG2DX2=CFCX(2 ) IF(V21.LE.0.)CQ2DX2=0. DH1CX2=0. IF(W12.GT.C.)CH1DX2=X(7)/W12 DENR=1078.-0.6*D( 2) DV2DX2=(CENR*CG2DX2+C.6*Q2)/(DENR*DENR) DHVDX2=0.4 DV0DX2=CQ2CX2/(1078.-0.6*X(4 ) ) IFIVC2.LE.0.) CVODX2=0. DERY(2)=-LAMDA(2)»(Wl2*( DH1DX2-1. ) +DV2DX2* ( X ( 2 )-Hl V) 1V21*(1.-CFVDX2) + 0G1DX2)/X( 1 ) -LAMDA U ) * ( DQ2DX2 + DVODX2 2H2V))/X(3)+ (LAMDAl 1)»0V2DX2+L *(X(4)AMOA(3) 3DVCDX2) DETERMINE RIGHT HAND SIDE OF D(LAMDA4)/DT 155 CONTINUE XXil)=X(4) K0UNT=0 160 CALL PCERIV(XX,FF,2, 2..K0UNT, DELX4,DFDX,W ) IF(KCUNT)5000,180,165 165 Y(1)=XX(KCUNT) IP0SIT(1)=4 CALL MAPl(X,8 f .Y f IPOSIT, ltD) CALL FUNCS(C,8,F, 8) FF(KCUNT,1)=F(5) FF(KGUNT,2)=F(6) GO TC 160 180 DG1DX4=CFCX(1 ) DG2DX4=CFCX(2) IFU21.LE.0. )CQ2DX4=0. DH1CX4=0. IF(W12.GT.0. )CH1DX4=X(6)/W12 DV2DX4 = CG2DXW( 1078.-0. 6*X( 2) ) DENR=1078.-0.6*D( 4) DV0DX4=( CENR*( CQ2CX4-FEEC ) +0 . 6* ( Q2-F EED* ( X( 4 )-HF ) ) ) / (DENR+D ENR)

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178 IF(VC2.LE.0.)CVODX4=C. CHVDX4=0.4 DERY{4) = -LAI"DA(2)»(W12*DHlCX4*(X(2)-HlV)»DV2DX4-«-.DQlDX4)/X(l )lLA^OAK) »(-X(5)+DC2CX4*(X(4)-H2V ) »DV0DX4 + V02» ( 1.-DHVDX4) )/X (3) + 2 (LAMDA( l)*CV2DXA + LAMDA(3)*DVODX4) IF( I (M.£C.1)CERYK)=C. IF( I (2) .EQ.O)DERYi4)=0. GC TC 112 XI, X2 AND X4 AT STEACY STATE 150 CCNTINUE DERY(1)=0. OERY(A) . GC TC 109 XI ANC X2 AT STEACY STATE 170 CONTINUE DERY( 1)=0. GC TC 1G9 DETERMINE RIGHT HAND SICE OF C(LAMDA3)/DT 112 XX(1)=X(3) KOUNT=0 113 CALL PCERIV(XX,FF,.2, 2 , KOUNT, CELX 3 , DFDX ,W ) IF(K0UNT)5000,115,114 114 Y(1)=XX(KCUNT ) IPCSIT(1)=3 CALL NAP1 (X,S,Y,IPOSIT, 1,C) CALL FUNCS(C,8,F,8) FF(KCUNT,1 )=F(6) FF(KCUNT,2)=F(5) GO TC 113 115 CC2DX3=CFCX( 1 ) IF(V21.LE.0.)CC2DX3=C. D^1DX3=DFCX«2 ) CV2DX 3= CC2CX 3/(1078. -Q.6*X(2>) DVODX3=DQ20X3/I1078.-0.6*X(4) ) 0ERY(3)=-LAMCA(2)»( ( X ( 2 )-HlV ) *DV2DX3 + DQ 10X3 ) /X ( 1 ) +LAMDA ( 3 ) * 1DV0DX3 + LACDA(4)*( FEEC*(HF-X(4 ) J+Q2 + V0 2M X(4)-H2V))/(X(3)+X( 3) )2LAMDA(4)MCU2CX3-HX(4 )-H2V )* C V0DX3 ) / X ( 3 ) + (LAMDA( 1 )*0V2D X3) IF(X(3) .LT.l. )DERY(3 )=0. GC TC 200 ALL VARIABLES AT STEACY STATE

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179 125 5C0O 5010 200 DERY ( 1 ) =0. 0ERY(2)=0. DERY(3) = '3. DERY(A)=0. C" D TC 200 WRITE (6,5C10)K0UNT FORMATtSX^KOUNT*' , 14) RETURN CONTINUE RETURN ERROR EXIT. I OUT OF RANGE. STOP EXECUTION 50 WRITE(6 f 2C50) ( I(J)t J = lf A) STCP 2050 F0RMAT(5X,«LAMD0T. I OUT OF RANGE%AI3) END

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180 C CONVAR C PURPCSE C DETERMINES THE OPTIMAL CONTROL LAW BY MINIMIZATION OF C THE HAMILTONIAN C USAGE C CALL CCNVAR(TOUT,LAMDA) C PARAMETERS C TCUT TIME, MINUTES, AT WHICH CONTROLS ARE TO BE C FCUND, INPUT C LAMDA INPUT VECTOR CONTAINING ADJOINT VARIABLES AT C TIME TOUT C SUBPROGRAMS REQUIRED C SUBROUTINE SEARCH C SUBROUTINE MAPI C SUeRCUTINE FUNCS C THE FOLLOWING SUBPROGRAMS ARE USED FOR THE MI NI MI ZAR I ON. C THEY ARE NCT LISTEC SEPARATELY IN APPENDIX B C SUBROUTINE PARTI C SUERCUTINE PART2 C SUBROUTINE MAPCWN C SUBROUTINE MAPUP C SUBROUTINE VA04A C SUBROUTINE CONVAR ( TOUT ,L AMDA ) REAL LAMCA(A) CIMENSICN UK),E(4),VU),F(8),Dm,X(8),W(60),IPOSIT(4), 1ZU),XL0U),XUP(4),IL0(4),IUPU),ICLS(4),DX(4) C0MM0N/DELT/DELX1,DELX2,CELX3,.DELX4,DELU1,DELU2,DELU3,DELU4 COMMCN/CCNTRL/UK 120 ) , U2 ( 120 ) ,.U3 ( 120 ) , U4 ( 120 ) C0MMCN/CATA1/NDATA1,TEMP(22),THALP(22),THALPV(22) , 1ALAMCA(22),PRESS(22) ,RH0V(22 ) C OMMCK/ STATES/ XI { 120), X2( 120) ,-X 3(120 ),X4( 120) COMMCN/STEACY/XHAT( A ),XINIT( A ) , T F r HF , NORDER , U1MAX , U1MI N , 1U2MAX,U2MIN,U3MAX, U3M IN, U4MAX ,U4M IN COMMCN/I \CEX/ I (4) COMMON/ CNEW/U1NEW( 120),U2NEW( 120),U3NEW( 120) , U4NEWU20) COMMCN/TI^ER/TIME(120),ITIME, I MAX ,. I COUNT CAT A ZERC,FRChG,?ISCALE,SC,MAXIT,. IP, ICON/ 1 .E5 , 0. 5 , 1 . E + 70 , 11. E-3, 103,1,1/ C C INTERPCLATE FOR VALUES OF THE STATE AND ADJOINT C VARIABLES AT TIME TOUT C CALL SEARCH(TIME,X1,T0UT, I MAX , NORDER , X ( 1) ) CALL SEARCH (T IME,X2,T0UT, I MAX ^NORDER , X ( 2 ) ) CALL SEARCh(TIME,X3,T0UT, I MAX , NORDER , X ( 3 ) ) CALL SEARCH (TIME,X A, TOUT, I MAX .NORDER , X ( 4 ) ) CALL SEARCH (TIME,U2, TOUT, I MAX ,NORDER , X ( 6 ) ) TT=TCUT

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181 C C c cc IF(X( 1J.LE.0. )X( l)=O.0Ol IF(X(3).LE.O. )X{ 3)=0.07 DO 1 J=5,8 IFCXl J).LT.O. )X( J )=0. DC 3 J=l,4 I ( J)=0 IF(X( J) .LT.XHATt J ) )GG TO 3 I { J)=l CGNTINUE THE FGLLOWING VALUES OF THE CONTROL VARIABLES WERE SET BECAUSE OF THE FEEDBACK LAW X(8)=U4KAX X(5)=U1MAX IF ( I (3).EC.1)X(5)=U1MIN X(7)=U3MAX I F ( I (1).EG.0)X(7)=U3MIN FEED=X(5) 00 2 J = l,4 ILC( J)=l IUP( J)=l IGOTG=I( 1)+2»I(2)+4*I(3)+8*I( 4) + l GO TC (5!?0, 5, 200, 5, 2C%.5, 200, 5, 200,200, 200, 5, 200, 200, 200, 15) ,IGCTC THIS SECTION OF THE SUBPROGRAM IS EXECUTED IN ALL STAGES EXCEPT WHEN NO STATE VARIABLE IS AT STEADY STATE Tl T TF PUT CONTROL VARIABLES THAT HAVE TO BE SEARCHED OVER IN VECTOR Z Z(1)=X(7) Z(2)=X(8) Z(3)=X(5) N=3 SPECIFY LOWER (XLO) CCNTRCL VARIABLES AND UPPER (XUP) BOUNDS ON THE XL0(l)=U3f IN XL0(2)=U4MIN XL0(3)=U1MIN XUP( 3)=U1MAX XUP( 1 )=U3MAX XUP(2)=U4MAX SET UP PARAMETERS FOR MINIMIZATION ROUTINE VA04A

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182 DC 7 J=1,N E( J)=SC*XUPl J ) RANGE=XUP( J)-XLC( J ) ESC=RANGE*FRCFG/E(J) IF(ESCALE.GT.ESC)ESCALE=ESC 7 CONTINUE JTEST=1 10 IT=3 SUERCUTINE PARTI PARTITIONS THE VARIABLES IN Z INTO THCSE WHICH CANNOT BE SEARCHEC OVER AND INTO THOSE WHICH CAN BE SEARCHEC OVER. IF A CONSTRAINT IS ENCOUNTERED IT NUMERICALLY EVALUATES THE KUHN-TUCKER MULTIPLIER ASSOCIATED WITH ThE PARTICULAR CONTROL VARIABLE TO DETERMINE WHETHER THE CONSTRAINT IS TO BE HELD OR RELEASEC. IT ALSC DETERMINES WHEN THE SEARCH IS COMPLETE AFTER ACCOUNTING FOR THE CONSTRAINTS CALL PARTI (Z,E,XLC,XUP,ILC, I UP,1 CL S, N , H, IF( JTEST-A)80,20,90 JTEST,DX) SUBROUTINE MAPDWN MAPS THE UNCONSTRAINED VARIABLES IN Z AS DETERMINED IN PARTI INTO A VECTOR U 20 CALL MAPCWN(Z,E, ICLS ,N,U,V,MU) IFCMU.EQ.01G0 TO 10 GC TC (30,40,50,60), IT SUBROUTINE VA04A IS THE CONJUGATE GRADIENT SEARCH PROGRAM WHICH SEARCHES OVER THE VARIABLES IN U TO MINIMIZE THE HAMILTONIAN H 30 CALL VAG4A1 GC TC 70 40 CALL VA04A2 GC TC 70 50 CALL VA04A(U,V,MU,H, ESCALE, IP, ICON,MAXIT, IT,W) GC TC 70 60 CALL VA04A4 SUERCUTINE MAPUP RE-MAPS THE VARIABLES IN VECTOR U INTO VECTC3 Z 70 CALL MAPUPIZ, ICLS ,N,U,MU) SUBROUTINE PART2 CHECKS WHETHER THE SEARCH CAUSED ANY CF THE VARIABLES TO VIOLATE CONSTRAINTS CALL FART2(Z,XL0,XUP, ILO, IUP, ICLS ,N,IT) 80 CONTINUE

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183 THIS SECTION EVALUATES THE HAMILTONIAN H. IPOS IPOS IPOS CALL CALL 00 8 ' X(J) H12= V21 = V02 = H12 = G1 = F Q2 = F HIV= H2V = H=LA 1»(X( 2X(6) IF( J GO T IT ( 1 IT(2 IT(3 PAP FUN 7 J = = C( J F(l) F(2) F(3) F K) (5) (6) F(7) F(8) NCA( 5)»< -V02 TEST 10 )=7 )=8 )=5 1 (X, 8f.Z f IPOSITtNiD) CS(C,B,F,8) 1,8 ) 2)*(W12»(H12-X(2) )+V21»(X(2)-HlV)+Ql)/X(l)+LAMDA(4) HF-X14) )+Q2+VC2*(X(4)-H2V) ) /X ( 3 ) +LAMDA ( 3 ) * ( X { 5 ) ) .EQ.4.AND.IT.NE.5)G0 TO 20 CONVERGENCE IS OBTAINED. THE OPTIMAL VALUES OF THE CONTROL VARIABLES ARE STORED IN ARRAYS IN COMMON BLOCK CNEW 90 U3NEW(ITIPE)=Z(1) U4NEH(ITIME)=Z(2) UlNEW(ITiyc)=Z(3) U2NEW( ITINE)=X(6) RETURN THIS SECTION OF THE SUBPROGRAM IS EXECUTED MEN NO STATE VARIABLES ARE AT STEADY STATE. T Tl 500 Z(1)=X(6) Z(2)=X(8) N=2 XL0(1)=U2MIN XL0(2)=UAVIN XUP( 1)=U2PAX XUP(2)=U4^AX DO 505 J = 1,N EU) = SC*XUP(J) RANGE = XUP( J)-XLO( J) ESC=RANGE»FRCHG/E(J ) IF(ESCALE.GT.ESC)ESCALE=ESC 505 CONTIKUl JTEST=1 510 IT=3

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184 CALL PARTKZ, E,XLO,XLP, ILC, IIP, ICLS, N,H, JTEST.DX) 511 CONTINUE IF (JTEST-4) 580,520, 600 520 CALL PAPCWMZ,.E, ICLS ,N,U,V,MU) IFIPU.EC.OGO TO 510 GO TC (530,540,550,560), IT 530 CALL VA04A1 GO TC 570 540 CALL VA04A2 GO TC 570 550 CALL VAG4MU,V,MU,H, ESCALE, IP, ICON,MAXIT, IT,W) GO TC 570 560 CALL VA04A4 570 CALL yAPUP(Z,ICLS ,N,U,MU) CALL PART2(Z,XL0, XUP, ILO, IUP, ICLS ,.N,IT) 571 CONTINUE 580 IPCSIT(1)=6 IP0SIT(2)=8 CALL N"AP1 (X,8,Z, IPCSIT,N,D) CALL FUNCS(C,8,F,8) OC 590 J=l,8 590 XI J) = C( J) 591 CONTINUE W12=F(1) V21=F(2) V02=F(3> H12=F(4) Q1=F(5) C2=F(6) H1V=F(7) H2V=F(8) H=LAVCA(1 )*(W12-V21-X(7) ) +L AMDA ( 2 ) * ( W12* ( H12-X ( 2 ) ) + V21 *(X(2) 1-H1V)+Q1)/X(1)+LAMCA(3)*(X(5)-XI 6 )-V0 2 ) + LAMDA ( 4 ) * ( X ( 5 ) * ( HFX(4) ) 2+Q2+VC2«(X{4)-H2V))/X(3) 592 CCNTINUE IF( JTEST.EC.4.AND.IT.NE.5) GO TO 520 GO TG 51? 600 U2NEW(ITIPE)=Z(1) U3NEW( ITINE )=C. U4NE*(ITI^E)=Z(2) U1NEW(ITIME) = X.I5) -tETURN ERROR EXIT. I OUT OF RANGE. STOP EXECUTION 200 WRITE(6,2050) ( K J ), J = l,4) 2050 F0RVAT(2X,« IN CONVAR. I = • , 4 ( 2X, I 2 ) ) STOP END

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185 C INPUT c***« *****»«*«*******»»««*«*************** *********************** PURPOSE TC INPUT DATA FOR COMMON BLOCKS DATA1, DATA2, STEADY AND CELT USAGE CALL INPUT SUBPROGRAMS REQUIRED SUBRCUTINE SEARCH SUBROUTINE INPUT C0MMCN/CATA1/NCATA1,TEMP( 2 2),THALP( 22 ) , THALP V ( 22 ) , 1ALAMCA(22),PRESS( 22),RHOV(22) COMMON/ ST EACY/XHAT( 4 ),XINIT( 4 ) , TF , HF , NOROER , U1MA X , Ul Ml N, 1U2MAX,U2MIN,U3MAX,U3MIN,U4MAX,UAMIN C0MMCN/DELT/DELX1,DELX2,0ELX3,DELX4,DELU1,DELU2,DELU3,DELU4 COMMCN/DATA2/NDATA2»T( 8) , RHO( 8 ) ,CP ( 8 ) , VI SC I 8 ) ,THCOND(8) t 1PRANTL(8),BETA READ NUMBER OF POINTS IN ARRAYS IN DATAU AND DATA2, CRCER OF INTERPOLATING POLYNOMIAL, FEED TEMPERATURE ANC EXPANSION COEFFICIENT OF WATER READ(5,10CO)NCATA1,NCATA2,NORCER,TF,.BETA READ PROPERTIES OF WATER AND STEAM VS TEMPERATURE, I.E. ENTHALPY, VAPOR ENTHALPY, LATENT HEAT,. SATURATION PRESSURE AND VAPOR DENSITY READ (5, 1010) (TEMPI I) ,THALP( I ), THALP V( I ),ALAMDA
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186 CALL SEARCH < T EMP , THAL P , T 1 SS , N DATA 1 , NORDER , XHAT ( 2 ) ) CALL SEARCMTEMP,THALP,T2SS,NDATA1,N0RDER,XHAT(4) ) C C READ INITIAL VALUES OF STATE VARIABLES C READ(5,103C) (XINITl I ), I = U4) C C READ MAXIMUM AND MINIMUM VALUES OF CONTROL VARIABLES C READ(5,1040)U1MAX,U1MIN,U2MAX,U2MIN,U3MAX,U3MIN,U4MAX,U4MIN C C READ INITIAL PERTURBATIONS IN STATE AND CONTROLS FOR C USE IN SUBROUTINE PCERIV C REACC5,1040)DELX1,CELX2, CELX 3 ,DELX4, DELU 1 ,.DELU2, DELU3 , DE LU4 C C CUTPUT DATA READ IN C WRITE(6,2000) WRITE(6,2010)(TEMP(I),THALP(I),THALPV(I) .ALAMDAd ) , 1 PRESS ( I ) , RHOV( I),I=1,NCATA1) WRITt (6,2020) WRITE(6,20 30)(T(I),RI-C(I),CP(I ),THCOND( I ) , V I SC ( I ) , 1PRANTH I ) , I = 1,NDATA2 ) WRITE(6,2040)TF, HF,BETA WRITE (6,2050) ( I, X INIT( I ) ,XHAT( I), 1=1,4) WRITE(6,2080) WRITE(6,2C9U)U1MAX,U1MIN,U2MAX,U2MIN,U3MAX,U3MIN,U4MAX, LU4MIN WRITE(6,3040)CELXl,CELX2,DELX3,DELX4,DELUl,DELU2,DELU3, 1DELU4 RETURN 1000 FORMAT(3I2,3F10.0) 1C1C F0RMAT(6F10.0) 1020 F0RMATU2) 1C30 FCRMAT(5F10.0 ) 1040 F0RMAT(8F10. ) 1C60 FURMAT( IX , I 2 , 2X , F6 . 3 , F 10 . 7,.F 10 . 6 , F10. 7, F 10.6 , 3F8 . 5, F 5 . 1 ) 2000 FORMAT( lhl, 1CX,« PROPERTIES OF WATER AND STEAM •,//, 5X ,• TE MP • 1,2X,'LIC ENTHALPY' ,2X, «VAP ENTHALPY •, 2X, ' LATENT HE A T •, 3X , 2« PRESSURE* ,2X, «VAP D ENS I T Y • , / ,.4X , • ( DEG F ) • , 3X , • ( B TU/LB ) • , 5X B.MBTU/LBPfTXtM 8TU/LB) • , 5X , • ( PS I ) • , 4X , • (LBS/CU FT)' J 2010 F0RMAT(4X,F5.1,4X,F6.2,7X,F7.2,.8X,F7.2,5X,F7.4,4X,F7.5) 2 020 FCRMAT(//5X,'TEMP«,3X,'DENSITY , ,4X,«SP.HT. • , 2 X , • TH.COND. ' ,• 13X,'VISCCSITY' ,2X, 'PRANDTL NO • , / , 4X , • ( DEG F)',1X, 2 , (LBS/CUFT) , ,1X,MB/LB.F)«,1X,MBF/HSQFF)«,1X, MLBS/FTHR) • ) 2030 F0RMAT(4X,F5. 1,4X,F5.2,5X,F6.4,.5X,F5.3,4X,F6.4,5X,F5.3) 2040 FORMAT! lhl,5X, 'STEADY STATE AND INITIAL VALUE S ',/, 5X , l'FEED TEKPERATURESFS.lt/iSX, 'FEED ENTHALPY = • , F 7. 2 , / , 5X , 2'BETA',12X,'=',F9.7,//8X,'XINIT',6X,'XHAT') 2 50 FORMAT(5X,Il,2X,F5.2,5X,F6.2)

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187 2C80 F0RMAT(1H1,5X, 'MAX AND MIN VALUES OF CONTROL VARIABLES',//, 117X,'FAX VALUE', 5X, 'fIN VALUE') 2C90 FORMAT(18X,F7.3,7X,F7.3) 3040 FORMAT(1H1,10X,' INITIAL INCREMENT IN INDEPENDENT VARIABLES' 1,//,5X,'CELX1 = ',F5.2,/,.5X, • DELX2= • ,F 5. 2 , / , 5X , • DELX3= • , F5 . 2 , 2/,5X,'DELX4=',F5.2,/,5X,'DELUl=',F5.2,/,5X,'DELU2=',F5.2,/, 3 5X,' CELU3=',F5.2,/,5X,'DELU4=',F5.2) END

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C SEARCH PURPCSE TO INTERPOLATE USING A POLYNOMIAL OF DESIRED ORDER USAGE CALL SEARCH! X,Y,XG I VEN , NDATA , NORDER , YFOUND > DESCRIPTION OF PARAMETERS X -INPUT VECTOR OF INDEPENDENT VARIABLE Y -INPUT VECTOR OF DEPENDENT VARIABLE XGIVEN-INPUT POINT AT WHICH VALUE OF Y IS REQUIRED NCATA -INPUT NUMBER OF CATA POINTS NORCER-INPUT ORCER OF INTERPOLATING POLYNOMIAL YFCUND-OUTPUT VALUE OF CEPENDENT VARIABLE REMARKS MAX CIMENSION OF INPUT VECTOR IS 20 AND MAX ORDER OF INTERPCLATING PCLYNOMIAL IS 4 (IF GREATER , CHANGE CIMENSION STATEMENTS ACCORDINGLY SUBPRCGRAMS REQUIRED FUN CT I CM AITKEN(X,Y,NDATA,XGIVEN) CcTHCC TAKES NORCER+1 POINTS AND INTERPOLATES USING AITKENLAGRANGE INTERPOLATION SUBROUTINE SE ARCH ( X, Y, XG IV EN , NDATA, NORDER , YFOUND ) DIMENSION X(20),Y(20),A{5),B(5) SEARCH FOR NORDER POINTS YFCUNC=0. DC 30 M=l, NCATA IF(XGIVEN-X(M) >10,20,30 10 JJ=N GO TC 40 20 YFCU.\C = Y(M) RETURN 30 CONTINUE 40 LL=JJ-NCRCER/2+l IF(LL.LE.0)LL=1 LU=LL+NCRCER-1 IFILU.LT. NCATA) GO TC 50 LU=NCATA LL=LU-NCRCER+1 50 DO 60 M=LLtLU K=M-LL+1 A(K)=X(M) BKK)=Y(M) 60 CONTINUE IF(YFCUNC.EC.0)YFOUNC = AITKEN( A, B , NORDER, XGI VEN ) RETURN

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189 END

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190 £•«•««««»»•«»«•«*«««*•*•«*#»•*«•*##••*•**»«••*»»****«*•»**•*•*»*» C AITKEN c purpc.se C INTERPOLATES USING POLYNOMIAL OF DESIRED ORDER C USAGE C FUNCTION AITKEN(X t Y,NCATA,XGIVEN) C DESCRIPTION OF PARAMETERS C X -I!\PUT VECTOR OF INDEPENDENT VARIABLE C Y -INPUT VECTOR OF CEPENDENT VARIABLE C NCATA -NUMBER OF DATA POINTS C XGIVEN-INPUT POINT AT WHICH VALUE OF Y IS REQUIRED C REMARKS C XGIVEN MUST LIE IN THE RANGE OF THE TABLE. IF NDATA IS C GREATER THAN 5, ADJUST CIMENSION STATEMENT C METHCC C AITKEN-LAGRANGE INTERPOLATION C FUNCTION A I TK EN (X,Y, NCATA, XGIVEN ) DIMENSION X(5),Y(5),Z<5,5) DC 10 1=1, NCATA 10 Z(I,1)=Y( I) NN=NCATA-1 DC 20 L=1,NN LL=L+1 DC 30 K=LL, NDATA 30 UK, L+l)=( (X{K)-XGIVEN)»Z(L,L)-(X(L)-XGIVEN)*Z(K,L) ) /(X(K)X(L) ) 20 CONTINUE AITKEN=Z(NDATA, NDATA) RETURN END

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191 C#**«# ***»»*«*«*»*»««************* ******************************* C FUNCS c**»**«»»«******************************************************* C PURPOSE C CALCULATES VALUES OF ALGEBRAIC VARIABLES FROM C VALUES OF CONTROL ANC DIFFERENTIATED VARIABLES. C USAGE C CALL FUNCS(X,N,.F,M) C DESCRIPTICN OF PARAMETERSO C X N DIMENSIONAL VECTOR OF DIFFERENTIATED AND C CONTROL VARIABLES AS DESCRIBED IN SUBROUTINE XDOT C F M CIMENSIONAL VECTOR CONTAINING VALUES OF C ALGEBRAIC VAR IABLESt WHERE, C F(l) FEED TO FIRST EFFECT, W12 (LBS/MIN) C F(2) VAPOR FROM FIRST EFFECT, V21 (LBS/MIN) C F(3) VAPOR FROM SECOND EFFECT, V02 (LBS/MIN) C F(4) ENTHALPY OF FEED TO FIRST EFFECT, H12,. C BTU/LB C F(5) HEAT TRANSFER IN FIRST EFFECT, Ql, C OTU/MIN C F(6) FFAT TRANSFER IN SECOND EFFECT, Q2 C BTU/MIN C F(7) VAPOR ENTHALPY IN FIRST EFFECT, HIV, C BTU/LB C F(8) VAPOR ENTHALPY IN SECOND EFFECT, H2V, C BTU/LB C SUBPROGRAMS REGUIRED C SUBROUTINE H^AT C SUBROUTINE SEARCH C C IN THE RANGE T = 212 DEC F TO 220 DEG F (I.E. ENTHALPY C 180.07 BTU/LB TO 188.13 BTU/LB) C RELATIONSHIP BETWEEN VAPOR ENTHALPY AND TEMPERATUREO C HV = 1G66.0 + O.A*T C RELATIONSHIP BETWEEN TEMPERATURE AND LIQUID ENTHALPYO C T = HL + 32.0 C WHERE, C T = TEMPERATURE (CEG F) C HL = LIQUIC ENTHALPY (BTU/LB) C whERE hV = VAPOR ENTHALPY (BTU/LB) C RHO = VAPOR DENSITY (LBS/CU.FT) C DIMENSION X(l ) ,F( 1) C0MMCN/CATA1/NDATA1,TEMP( 22 ) , THALP ( 22 ) , THALP V( 22 ) , 1ALAMDA(22 ), PRESS ( 22) ,RHOV( 22) COMMON/ST EADY/XHAT( 4 ),X I NITU),TF,HF,N0RDER,U1MAX,U1MIN, 1U2MAX,U2MIN,U3MAX,U3MIN,U4MAX,U4MIN CONMCN/INCEX/ I (4) C C FIND TEMPERATURES AND VAPOR ENTHALPIES IN EFFECTS C

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192 5 Tl=X(2)+32.0 T2=X(4)+32.0 H1V=1066.+0.4*T1 H2V=1066.+0.4*T2 I F ( I (2).EUi)G0 TO 10 C C X2.NE.XFAT(2), FIRST EFFECT IS BEING HEATED. C W2=0. V21 = . VC2=0. M2=X(6)+X(7) IF(W12.NE.0.)G0 TO 1 T12=T1 GC TC 2 1 H12=(X(7)«X(2 )+X(6)*X(4) J/W12 T12=H12+32. 2 CALL HEAT(CltX(l),W12iV21.X(8)i.Tl f T12»l,i) GC TC 70 C C X2.EU.XHAT(2) , FIRST EFFECT IS BOILING. C 10 CONTINUE CALL FEAT(C2,X(3),0.,0.,T1,T2,TF,3,2) CALL SEARCH { T EMP, ALAMnAt Tlf NDATA It NORDERtAL AMI) V21=G2/ALAM1 IF(I (1).EC.1)X(6)=V21 W12=X(6)+X(7) IF(W12.NE.C.)G0 TO 190 T12=T1 GC TC 20 190 T12=(X(7)*TH-X(6)*T2)/W12 H12=T12-32. 200 CALL FEAT(G1,X(1 ) ,W12,V21,X< 8)„Tl,T12, 2, 1) I F { I (4).EC.0)G0 TO 70 C C X4 = XHATU), SECONC EFFECT IS ALSO BOILING. C CALL SEARCH (TEMP, AL ANTA , T 2, NDATA 1 , NORDER , ALAM2 > V02=(C2-X(5)*(X(4)-HF))/ALAM2 GO TO 80 70 VC2. 30 F(1)=U12 F (2)=V21 F(3)=V02 F(4)=H12 F(5)-tl F(6)=G2 F17)=H1V F(8)=F2V kETURN

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193 END

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194 C*** c c» *« c c c c c c c c c c c c c c c c c c c c c c c c c c ••••••••••••A******************************************** HEAT THE HEAT TRANSFER RATE PURPCSE EVALUATES USAGE CALL HEAT(C«H,W,V,TO,TI,TIN, IPHASE.IEFECT) PARAMETERS Q H W V TO TI TIN IPHASE IEFECT SUBPROGRAMS FUNCTION FUNCTION FUNCTION FUNCTICN FUNCTION HE HO LI VA OU IN IN PA PA Rl FC FC FC FC HE SUBROUTINE AT TRAN LO-UP, GUIC FL POR FLO TSICE ( SIDE (L LET TEM RAMETER It INCI 2, INCI 3, INCI RAMETER ANSFER GUI RED INB IN IF IGHT ALPHA BTU/MIN, OUTPUT SFER RATE, LBS, INPUT OW RATE, LBS/HR, INPUT W RATE, LBS/HR, INPUT STEAM SICE) TEMPERATURE, DEG F, INPUT IQUID SIDE) TEMPERATURE, DEG F, INPUT PERATURE OF LIQUID, DEG F, INPUT DESCRIPTIVE OF TYPE OF BOILING CATES SINGLE PHASE FLOW CATES TWO PHASE FLOW CATES NATURAL CONVECTION INDICATING EFFECT FOR WHICH HEAT RATE IS TO SE CALCULATED, INPUT SUBROUTINE HE AT ( Q, H, W, V, TO, T I ,.T IN, IPHASE, IEFECT) REAL LIS, LI, LIB DIMENSION THETA{ 3) COMMON/ C AT A 1/ N DAT A 1, TEMP (22) , THALP ( 22 ) ,.THALP V ( 22 ) , 1ALAMDA(22) , PRESS ( 22 ) , RHOV ( 22 ) COMMCN/DATA2/NCATA2,T(8),RHO( 8),CP(8),VISC{8) ,THC0ND(8) , 1PRANTH8 ) ,BETA COMMCN/INCEX/ I (4) DATA NCRCER/4/ IFUC.LE.TI )GO TO 25 LI IS THE MAXIMUM LENGTH OF THE TUBES IN FEET. A IS THE MAXIMUM HEAT TRANSFER AREA IN SQ.FT. Ll=9.5 IF(IEFECT.EC2)L1 = 1.917 -=7.46128 IFCW.EQ.O. )TIN=TI TBULK=(TII\ + TI )*0.5 IFtV.LE.}. )IPFASE=1 IF( IEFECT. EQ. 2) A= 9. 08116

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195 10 11 12 THE CUTER WALL TEMPERATURE, TW2 AND INNER WALL TEMPERATURE TWl ARE GUESSED INITIALLY AND THEN ITERATED ON BY EVALUATION CF THE FILM COEFFICIENTS AND THEN RE-EVALUATING THE WALL TEMPERATURES TW2=G.5»(TC+TBULK) TW1=TW2 ITIME=ITIME+1 THE PARAMETERS THETA ARE CORRECTION PARAMETERS FOR THE TWC EFFECTS THETA1 = 0.1395 = 0.0928 THETA2 = 0.1359 = 0.1763 WHEN FIRST EFFECT IS NOT BOILING OTHERWISE WHEN SECOND EFFECT IS NOT BOILING OTHERWISE THETA(1)=0.1395 THETA(2)=1. THETA(3)=1. I F ( I (2) .EG.OJGO TO 5 THETA(1)=0.0928 THETA(2)=0.1359 THETA(3)=0.1763 CONTINUE KCUNT=1 TH1=TFETA(1) IFUEFECT.EQ.UGO TO 11 TH1=THETA(2) IF(I(4).EC.1)TH1=THETA(3) GC TC 12 IF(W.NE.O. )G0 TO 15 IF ( I (4).NE.1.0R.IEFECT.NE.2)G0 TO 13 HIS IS THE INSIDE FILM COEFFICIENT HCS IS THE OUTSICE FILM COEFFICIENT TWIN AND TW2N ARE THE NEW INNER AND OUTER WALL TEMPERATURES HIS=FCING(TWlt.TI ) GO TC 17 13 HIS=FCIN(TBULK,TW1»H, IEFECT) GC TC 17 15 HIS=FCIF(TEULK,W,TW1) 17 H0S=FC0(TC,TW2, IEFECT) HW=849.62A TW1N=(TBULK*TH1 *H IS * ( HOS + HW ) +HW»HOS* TO ) / ( TH1 *HIS» 1 (HOS+HW)+hOS»HW) TW2N=(H0S«T0 + HW*TW1N )/( HOS+HW) IF(ABS(TW1N-TW1).LT.1..AND.ABS(TW2N-TW2).LT.1. )G0 TO 20 IF{KCUNT.GT.15)WRITE(6,21CO)HIS,HOS,TW1,TW2,TW1N,TW2N IF(IU) .NE.1.CR.IEFECT.NE.2.0R.KOUNT.GT.10)GO TO 18

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196 TWl=Twl+J.2*(TWlN TW2=TW2+0.2*( TW2N GC TC 19 18 TW1=TW1N TW2 = TW2i\ 19 KOUNT = KCUi\T + l IFIKCUNT.LT.20 )GO WRITE(6,2000)KOUN STOP 20 IF(ABS(HIS).LT..1E-70)G0 TO 25 l-TWl ) I-TW2) TO LO T,Tl.,TI,TIN,IPHASE,IEFECT CALCULATE OVER SENSIBLE HEATI LIS IS THE LEM DELT1S IS THE SENSIBLE HEATI CIS IS THE HEA ZCNE ALL HEAT TRANSFER COEFFICIENT UlS IN NG ZONE GTH OF THE SENSIBLE HEATING ZONE MEAN TEMPERATURE DIFFERENCE IN THE NG ZONE T TRANSFER RATE IN THE SENSIBLE HEATING 25 U1S=(1./(1. 25357/ l1s=height( iphas5 deltis=tc-(ti+;tin q1s=u1s*a «l1s»de if( ipfase.eq.2)g0 C = G1S RETURN Q=0RETURN (TH1 *HIS)+1./HOS+0.001315+0.002) )/60. ,H f TBULK,TW1,TIN,.W,IEFECT) )*0.5 LT1S/L1 TO 30 THIS PART OF T CALCULATION OF EFFECT WHEN BO THE EQUATIONS HE SUBROUTINE IS USED ONLY FOR THE THE HEAT TRANSFER RATE IN THE FIRST RING OCCURS. REFER TO APPENDIX A FOR 30 TW2 = u.5»(TC+TI ) T In 1 = T V. 2 I TI ME= IT INE + 1 31 CONTINUE TB = TI KCUNT=1 32 IF( (W-V) .GT.O. )GQ TO 35 X12=l. HIB=22.15727*V«*0.8 GC TC AO 35 X12=V/W HIB=FCIF(TB,W ,TW1 ) 40 HCB=FC0(TC,TW2, IEFECT ) HW=849.624 TW1N = (T8*THETA(1 ) 1HIB*(H0B+HW)+H0B* TW2N = (HCB»TC+HW#T *HIE«(HOB+HW )+Hfc«HOB*TO )/( THETA(l)* HU ) WIN )/(HOB + HW )

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197 IF( ABS(TwlN-TWl) .LT. 1 . . AND. ABS( TW2N-TW2 ) .LT.l. )G0 TO 50 TW1=TW1N TW2 = TVv2N IF(TW1.LT.TI)TW1=TI IF(TW2.GT.TO)TW2=TO KCUNT=KCUNT+1 IF(KCUNT.LT.10)G0 TO 32 wRITE(6,2C00)K0UNT,TC,TI,TIN, I PHASE, I EFECT STOP 50 IF( (W-V) .LE.O. )G0 TO 65 X12PR=0.4«X12 CALL SEARCH(T,RHO,TB , N0ATA2, NORDER ,RHOL ) CALL SEARCH(T, VISCtTB , NDATA2, NORDER, VI SCL ) CALL SEARCh(TEMP,RHOV,TI,NDATAl,NORDER,RHOG) VISCG=0.0195 3 29+5.10 7E-0 5*TI XTT=( (l.-X12)/X12 )**0.9*(RHOG/RHOL)**0.5*(VISCL/VISCG)**G.l XTTPR=( ( l.-X12PR)/X12PR)**O.9*(RHOG/RH0L)**0.5*( VISCL/VISCG 1 > **01 HITP = 3.5*HIB/SGRTUTT) GT=4844.673*W CALL ALPHA(XTT,GT,ALFA12) CALL ALPI-A{XTTPR,GT, ALFAPR) ALFA^=0.5*( ALFA12+ALFAPR) HINB=0.074*(TW1-TI )*«2.86 HIB=ALFAM*HINE+HITP 65 U1B=( l./( 1.25357/ (ThETA( 1 ) *H I B )+ 1 . /H0B+0.C01 315+0. 002 ) )/60. LlS=HEIGhT( IPhASE,H,TBULK,TWl,TlN,W,.IEFECT) L1B=L1-L1S DELT1S=TC-(TI+TIN)*0.5 Q1S=U1S*A «L1S*DELT1S/L1 DELT1E=TC-TI Q1B=U1B*A *L1B*DELT1B/L1 G=Q1S+Q1B RETURN 2000 F0RMAT12X,' AFTER' , I4,2X, • ITERATIONS IN HEAT THERE WAS NO ', 1' CONVERGENCE' ,/,2X, ' ARGUMENTS WERE',/,5X, • TO = • ,F 10. 3 , 2X , 2'TI=' ,F1J.3,2X,'TIN=',F10.3, 'PHASE= f , I4,2X, • EFFECT= • , 14 ) END

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198 C HEIGHT C»**« •«««*«****««**«««**•*«*«********* *************************** C PURPOSE C DETERMINES LIQUIC HEIGHT IN EACH EFFECT C USAGE FUNCTION HEIGHT ( I PHAS E» H, TX , T 1BW , T IN, FLOW , I EFEC T ) C PARAMETERS C IPFASE =1, SINGLE PHASE FLOW C = 2, TWO PHASE FLOW C H HOLD UP OF LIQUIC (LBS) C TX MEAN LIQUID TEMPERATURE (DEG F) C T1BW INNER WALL TEMPERATURE. LIQUID SIDE. (DEG F) C TIN ENTERING LIQUID TEMPERATURE (DEG F) C FLOW FLOW RATE OF ENTERING LIQUID (LBS/MIN) C IEFECT =1, FIRST EFFECT C = 2, SECCNC EFFECT C SUBPROGRAMS RECUIREC C SUBROUTINE SEARCH C SUBROUTINE DET5 (FROM THE IBM SSP ) C FUNCTION HEIGHTl I PHA S E , H , TX, T 1BW , T IN , FLOW ,1 E FECT ) DINENSIGN Z(20) COMMON/ C AT Al / N CAT A 1, TEMP (22) ,THALP( 2 2 ) , THALP V ( 22 ) » 1ALAMCM22),PRESS(22) ,RHOV(22) COMMGN/CATA2/NCATA2,T(8),RHO(8),CP(8),VISC(8) ,THCOND(8) , 1PRANTH8) ,BETA CATA NORDERM/ AL=2.3125 C C TUBE CIAMETER IN BOTH EFFECTS = 0.0725 FT. C C=0.0725 C C NUMBER OF TUBES. FIRST EFFECT = 3. SECOND EFFECT = 15. C NT = 3 IF( IEFECT. EC. 2)NT = 15 TBULK=TX CALL SEARCH(T,RH0,TBULK,NDATA2,N0RDER,RH0F) IF( IPHASE. EQ.2JG0 TO 20 AT=3.14159*C»CM. C C DEAD HCLDUPS (DO NOT CONTRIBUTE TO HEIGHT). C FIRST EFFECT = 1.5315 LBS. SECOND EFFECT = 16.336 LBS. C HCLC = H-1.5315 IF(IEFECT.EQ.2)H0LD=H-16.3 36 HEIGHT =HCLC/( FLOAT ( NT ) »RHOF* AT ) IFlHEIGHT.Lr. 0. )HEIGFT=0. IF( IEFECT. EQ.2JG0 TO 10

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199 LENGTH CF TUBES. (ALSO MAXIMUM HEIGHT) FIRST EFFECT =9.5 FT. SECOND EFFECT = 1.917 FT. IF(HEIGHT.GE.9.5)HEIGFT=9.5 RETURN THIS SECTION OF THE SUBROUTINE DETERMINES THE LENGTH OF THE SENSIBLE HEATING ZONE IN TWO-PHASE FLOW. REFER TO APPENDIX A FOR THE EQUATIONS AND REFERENCES 10 IF(HEIGHT.GE.1.917)HEIGHT=1.917 RETURN 20 HL=FCIF(TBULK,FLOW) CALL SEARCH(T,CP,TBULK,NDATA2,N0RDER,CPB ) IF(FLCW.EC.O. )G0 TO 30 DTDL=(3.14159*D*FL0AT(NT)*HL*(T1BW-TBULK) ) / ( 60.*FLOW»CPB ) 30 CPDL=-RH0F/144. CALL CET5(10.»PRESS,Z,NDATAl t IER ) CALL SEARCH (TEMP,Z,T BULK, NDATA1 , NORDER, DPDT ) DTDP=1./DPDT IF(FLCW.EC.O. )G0 TO 40 HEIGHT=AL«CTDP/ (-( CTCL/CPDL MDTCP) GC TC 50 40 DELT=2.*(TX-TIN) HEIGHT=1.+(CELT/(DPDL*DTDP) ) 50 IF(HEIGHT.GT.9.5 )HEIGHT=9.5 RETURN END

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200 FCINB PURPCSE CALC CCNV USAGE FUNCTICN FCINB(TW,TI) PARANET TI SUBPRCG SUER ULATES THE INSIDE FILM COEFFICIENT FOR NATURAL ECTION AND POOL BOILING USING THE RHOSENOW EQUATION ERS INNER WALL TEMPERATURE, DEG F, INPUT INNER BULK TEMPERATURE* DEG F, INPUT RAMS REQUIRED CUTINE SEARCH FUNCTION FCINB(TW,TI ) COMMON/CATA2/NDATA2,T<8) , RHO < 8 ) , CP < 8 ) , VI SC ( 8 ) , THCOND ( 8 ) , 1PRANTH8) ,8ETA COMMON/ DATA 1/ NDAT Al, TEMP ( 22) , THALP ( 2 2 ) , THALP V ( 22 ) , 1ALAMDA(22),PRESS(22),RH0V(22) DATA NCRCERM/ DELTAT = TVi-TI IFIDELTAT.LE.O. ) GO TC 10 TFILM=TW-0.75*(TW-TI ) CALL SEARCF(T,VISC,TI,NCATA2,NORCER,VISCF) CALL SEARCF(TEMP,THALPV,TFILM,NDATA1,N0RDER,HFG) CALL SEARCH(T,RH0,TI,NDATA2,N0RDER,RH0L) CALL SEARCF(T,CP,TI,NCATA2,NCRDER,CL ) CALL SEARCh(T,PRANTL,TI,NCATA2,NORDER,PRANTF ) CALL SEARCH (TEMP, ALAMC A, TF ILM , NDATA1 , NORDER , ALAMF ) CCNVERT ALAMF TO CALS/G-MOLE ALAMF=ALAMF*10. CCNVERT RHCL TC GMS/CC RHCLB=RHCL*0.016 CALCULATE SURFACE TENSION FROM WALDEN'S EQUATION (PERRY) IN DYNES/CM ST=( ALAMF»RH0LB)/364. CCNVERT SURFACE TENSION TO LBS/FT ST=ST*68.5218E-06 A=CL»DELTAT/HFG B=ST/RHCL FCIN3=(VISCF»FFG*A*A«A)/( 2 . 1 <5 7E-C6*DELTAT»SQRT ( B ) »PRANTF «»5 .1) RETURN

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201 DELTAT IS NEGATIVE 10 FCINe=0. RETURN END

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202 C FCIF PURPOSE CALCU CCNVE USAGE FUNCT PARAKETE TBULK W TW1 FUNCTION COI*NCN/C 1PRANTLI8 DATA NOR CALL SEA CALL SEA CALL SEA CALL SEA CALL SEA CALL SEA FL0W=60. RE=4.856 IF(RE.LE 1 ( VISCF/V IF(RE.LE LPRANTF** IF(RE.GT KVISCF/V RETURN END LATES THE INSIDE FILM COEFFICIENT DUE TO FORCED CTION USING THE C I TTUS-BOELTER EQUATION ICN FCIF(TBULK,W,TW1 ) RS BULK TEMPERATURE OF LIQUID, DEG F, INPUT FLOW RATE OF LIQUID,. LBS/HR, INPUT INNER WALL TEMPERATURE,. DEG F, INPUT FCIF(TBULK, W, TW1 ) ATA2/NCATA2,T(8),RHO(8),CP(8) ,VISC(8) ,THCOND(8), ) ,EETA CER/4/ RCMT.RHO, TBULK, NDAT A2,NORDER , RHOF ) RCMT,VISC,TBULK,NDATA2,N0RDER,VISCF ) RCH(T»CP, TBULK,NCATA2,N0RDER,CPF ) RCH ( T, PR ANTLf TBULK, NDATA2 f NORDERt PRANTF) RCF(T, THCCND,TBULK,NCATA2,NORDER,THCF) RCHT,VISC,TW1,NCATA2,.N0RDER,VISCW) *W 7»FLOW/VISCF .2COO. )FCIF=2.C9 3 414«THCF*RE«*0.545*PRANTF**0.4* ISCW)«*0.14 . 5000..ANC.RE.GT.200C. ) FC IF = 0.8 8655* THCF*RE**0.66* :-.4*( VISCF/V ISCW)**0. 14 .5000. )FCIF = C.274 5000*THCF»RE**0.8*PRANTF**0.4* ISCW) »»0. 14

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203 C PURPCSE C CALCU C FCO *«##**«»»»»«****«»#«*«#**«»***##*»**#»»*»*»*»##*»# LATES THE OUTSIDE (VAPOR SIDE) FILM COEFFICIENT USI NG USSELT EQUATION ION FC0(T0,TW2, IEFECT) RS MEAN VAPCR TEMPERATURE,. DEG F, INPUT OUTER WALL TEMPERATURE, DEG F, INPUT T PARAMETER INDICATING THE EFFECT AMS REQUIRED UTINE SEARCH 10 THE N USAGE FUNCT PARAMET6 TO TW2 IEFEC SUBPRCGR SUERC FUNCTION COVMCK/C 1PRANTLU CONPCN/C 1ALAMDA(2 DATA NOR CALL SEA TFIL^=TC CALL SEA CALL SEA CALL SEA DELTAT=T AL=9.5 IFUEFEC G=4.17E0 IF(DELTA FC0=0.1E RETURN FC0=0.9A RETURN END FCO(T ATA2/N ),BETA ATA1/N 2), PRE CER/4/ RCF(TE -0.75* RCH(T, RCHtT, RCH(T, C-TW2 0,TW2, IEFECT) DATA2,T(8),RH0( 8 ) , CP ( 8 ) ,.VI SC ( 8 ) ,THC0ND(8) , CAT A 1, TEMP (22) , THALP ( 22 ) , THALP V ( 22 ) , SS(22),RHOV(22) MP, ALAMCA,T0,NCATA1,.N0RDER,ALAMF) (T0-TW2) THC0ND,TFILM,NCATA2,N0RDER,THCF) RH0,TFILM,NDATA2,N0RDER,RH0F) VISC,TFILM,NDATA2,N0RDER,VISCF ) T.EQ.2)AL=2.3125 8 T.GT.O.IGO TC 10 + 40 3*(THCF**3*RF0F*»2*G*ALAMF/(DELTAT*AL*VISCF))**0.25

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204 U c c * » »* » * C P c c c c c c c c c c c c c FCIN URPCSE CALCULATES THE INSIDE FILM COEFFICIENT FOR NATURAL CCNVECTICN USING THE NATURAL CONVECTION EQUATION SAGE FUNCTICN FCIN(TBULK,TW,H, IEFECT) ARAMETERS T8ULK BULK TEMPERATURE OF LIQUID, DEG F, INPUT TW INNER WALL TEMPERATURE, DEG F, INPUT H HOLD-UP, LBS, INPUT IEFECT PARAMETER INDICATING THE EFFECT UBPRCGRANS REQUIRED SUBROUTINE SEARCH FUNCTICN HEIGHT F C 1A D T I C c c c X I G Y I 1* I I R F R F R F E 10 2100 UNCTICN CNMCN/L LAMCA(2 ATA NOR FILM=0. F( (Tk-T ALL SEA ALL SEA ALL SEA ALL SEA L=HEIGh F(A3S(X R=(XL»* =GR*PRA F( (Y.GE (Y»*0.2 FtY.GE. FIY.LT. ETURN RITE(6, CIN=(IC ETURN CIN= . ETURN 0RMAT(5 ND FCIN( ATA1/N 2) ,PRE DERM/ 5MTBU BULK) . RCHIT, RCH(T, RCH(T, RCH(T, T(1,H, D.LT. 3*RH0F NTF .1.E4) 5) 3.5E7) 1.E4JG TBULK,TW,H, IEFECT) CAT Al, T EMP { 22 ),THALP( 22 ),THALPV< 22) , SSI 22),RH0V(22 ) LK+TW) LE.O.)GO TO 10 THC0NC,TFILM,NCATA2,NORDER,THCF) RH0,TFILM,NDATA2,N0RDER,RH0F) VISC,TFILM,NCATA2,N0RDER,VISCF) PRANTL,TFILM,NCATA2,N0RDER,PRANTF) TBULK,TW,0.,0., IEFECT) •1E-70JG0 TO 10 *RH0F*4.17E8*BETA*(TW-TBULK) ) / ( VI SCF»VI SCF ) .AMU.(Y.LT.3.5E7))FCIN=( (0.5 5»THCF )/XL) FCIN=( (0.13*THCF)/XL)»(Y»*C33 3) TO 2C00 2100) .55*THCF)/XL )*(Y»»C25) X.'Y IS OUT CF RANGE' )

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205 C*«««**«****«* £«#*«#«*»»**»* 10 20 THIS DESIR USAGE CALL PARAI^ETE X N Y IPCSI M Z SUBRCUTI CINENSIC DO 10 1= Z(I)=X( I CONTINUE IC0UNT=1 J=IPCSIT Z( J)=Y( I ICOUNT=I IFdCCUN RETURN END MAPI SUBROUTINE MAPS THE ELEMENTS OF A VECTOR INTO EC POSITIONS IN ANOTHER VECTOR MAPK X,N, Y f IPOSIT,M,Z) R S INTO VECTOR INTO WHICH ELEMENTS ARE MAPPED DIMENSION OF X AND Z, INPUT INPUT VECTOR TO REPLACE ELEMENTS IN X T VECTOR CONTAINING INTEGERS INDICATING WHICH ELEMENTS IN X ARE TO BE REPLACED BY THE ELEMENTS OF Y DIMENSION OF Y AND IPOSIT, INPUT OUTPUT VECTOR NE MAPI (X,N,Y, IPOSIT, M, Z ) N X(1),Y( 1), IPOSITU ),Z( 1) 1,N ) ( ICOUNT) COUNT ) COUNT 41 T.LE.M)GO TO 20

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LITERATURE CITED Andre, H., and R. A. Ritter, "Dynamic Response of a Double Effect Evaporator," Can. J. Chem. Eng., 46, 259 (1968). Bryson, A. E., W. F. Denham, and S. E. Dreyfus, "Optimal Programming Problems with Inequality Constraints. 1: Necessary Conditions for Extremal Solutions," AIAA J., 1, 2544 (1963). Bryson, A. E., and Y. C. Ho, Applied Optimal Control, Ginn, Waltham, Mass. (1969). Chant, V. G., and R. Luus, "Time Suboptimal Control of a Gas Absorber," Can. J. Chem. Eng., 46, 376 (1968). Denn, M. M., Optimization by Variational Methods, McGraw-Hill, N.Y. (1969). Eschenbacher, R. C, Experimental Studies in Optimal Control of a Continuous Distillation Column, Ph.D. thesis, University of Florida (1970). Fletcher, R., and M. J. D. Powell, "A Rapidly Convergent Descent Method for Minimization," Computer J., 6_, 163 (1963). Foss, A. S., "Critique of Chemical Process Theory," AIChE J., ^9, 209 (1973). Fair, J. R., "What You Need to Design Thermosiphon Reboilers," Petr. Ref., 39, 2, 105 (1960). Fair, J. R., "Vaporizer and Reboiler Design—Part I," Chem. Eng., July £ 119 (1963a). Fair, J. R. , "Vaporizer and Reboiler Design—Part II," Chem. Eng., Aug. 5, 101 (1963b). GIPSI Hardware Manual, Revised edition, Chem. Eng. Dept., University of Florida (1973). GIPSI Software Manual, Revised edition, Chem. Eng. Dept., University of Florida (1973). Hughmark, G. A., "Designing Thermosiphon Reboilers," CEP, 65, 7, 67 (1969). ~" Jackson, R., "Optimum Start-up Procedures for an Autothermic Reaction System," Chem. Eng. Sci., 21, 241 (1966). 206

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207 Jazwinski, A. H., Stochastic Processes and Filtering Theory, Academic Press, N.Y. (1970). Joffe, B. L., and R. W. H. Sargent, "The Design of an On-line Control Scheme for a Tubular Reactor," Imperial College, London (1971). McCabe, W. L., and J. C. Smith, Unit Operations of Chemical Engineering, McGraw-Hill, N.Y. (1967). Newell, R. B., Ph.D. thesis, University of Alberta (1970). Newell, R. B., and D. G. Fisher, "Optimal Multi-variable Computer Control of an Evaporator," University of Alberta (1971). Newell, R. B., D. G. Fisher and D. E. Seborg, "Computer Control using Optimal Multivariate Feedforward Feedback Algorithms," AIChE J., J8, 976 (1972). Nieman, R. E., and D. G. Fisher, "Computer Control Using Optimal State Driving Techniques," Nat'l Conf. on Auto Control, University of Waterloo, Ontario (1970). Nieman, R. E., D. G. Fisher, and D. E. Seborg, "A Review of Process Identification and Parameter Estimation Techniques," Int. J. of Control, 1_3, 2, 209 (1971). Padmanabhan, L., "A New Approach to Recursive Estimation in Continuous Dynamical Systems," Chem. Eng. J., 1, 232 (1970). Pollard, G. P., and R. W. H. Sargent, "Off-line Computation of Optimum Controls for a Plate Distillation Column," Imperial College, London (1 966] Powell, M. J. D., "An Efficient Method for Finding the Minimum of a Function of Several Variables Without Calculating Derivatives," Computer J., 7, 155 (1964). Siebenthal, C. D., and R. Aris, "Studies in 0ptimization--IV," Chem. Eng. Sci., ]9_, 729 (1964). Tong, L. S., Boiling Heat Transfer and Two-Phase Flow, Wiley, N.Y. (1965). Tsang, A. C. C, and R. Luus, "Approximation in Control of Nonlinear Dynamic Systems," AIChE J., j_9, 327 (1973). Vermeychuk, J. G., and L. Lapidus, "Suboptimal Control of Distributed Systems Parts I and II," AIChE J., 1_9, 123 (1973). Weber, A. P. J., and L. Lapidus, "Suboptimal Control of Nonlinear Systems Parts I and II," AIChE J., 17., 641 (1971). Westerberg, A. W., and R. C. Eschenbacher, "A Real Time Computer Control Facility," Chem. Eng. Education, 5, 7, 32 (1971). Westerberg, A. W., 'RMINSQ', Private Communication (1969).

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BIOGRAPHICAL SKETCH Santosh Nayak was born on March 5, 1946 at Mysore, India. He graduated from St. Joseph's High School, Bangalore in December 1961. He then attended the Indian Institute of Technology, Madras and received a Bachelor of Technology degree in chemical engineering in June 1967. He spent the next two years at the Indian Institute of Science, Bangalore and received a Master of Engineering degree in September 1969. In January 1970 he joined the Chemical Engineering Department at the University of Florida. He received a second Master of Engineering degree in June 1971 and continued work towards the degree Doctor of Philosophy. He was initially on a teaching assistantship and later on a research assistantship at the University of Florida. 208

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Arthur W . ""Wester berg ,\ Chairman Associate Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ^L**,?~ fib#~£Thomas E. Bullock Associate Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. \-V VV Y^vw-tw^y Ulrich H. Kurzweg Associate Professor of Engineering Science and Mechanics This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. March, 1974 <^t U/vi a n, Coll,ege~bf Engjwpeer i ng J u-