Document text
OPTIMIZATION STUDIES OF TOO
WATER PURIFICATION SYSTEMS
7^3
by
THOMAS DEAN DENCHFIELD
B. S.s KANSAS STATE UNIVERSITY, 1965
A MASTER'S REPORT
submitted in partial fulfillment of the
requirements for the degree
MASTER OF SCIENCE
Department of Chemical Engineering
KANSAS STATE UNIVERSITY
Manhattan, Kansas
1967
Approved by:
/Wtfr^-fj^ZUU
rJ^fe^
Major Professor
i-y
c ^ ACKNOWLEDGEMENTS
The author is indebted to many people who assisted with this
work.
The Chemical Engineering Faculty, under the direction of Professor
Wm. H. Honstead, Head, provided graduate course work which was most
useful in preparation for writing this work.
Dr. L. V. Sorg and Mr. J. C. Lamkin of the American Oil Company
supplied written material on the Sugar Creek aerated lagoon. Also,
discussions with these gentlemen about aerated lagoon processes were
invaluable.
The Kansas State University Computing Center Staff is thanked for
running many of the computer programs used in this work. Dr. E. S. Lee
is thanked for having the POP computer programs run.
The financial assistance provided by a National Science Foundation
Traineeship made the author!s graduate study possible, and it is greatly
appreciated.
The help of Dr. L. E, Erickson who offered many suggestions which appear
in the final copy of this work is gratefully acknowledged.
Finally, the enthusiasm, guidance, and encouragement imparted to the
author by Professor L. T. Fan, his major advisor, during the author's grad-
uate study and the writing of this work are sincerely appreciated.
iii
TABLE OF CONTENTS
PART I. THE USE OF THE NONLINEAR PROGRAMMING
TECHNIQUE, POP-II TO OPTIMIZE A
MULTI-EFFECT MULTI-STAGE (MEMS)
SEAWATER DISTILLATION PLANT
Page
1.0
INTRODUCTION
1.1 Some Basic Concepts and Definitions Useful in
1
Nonlinear Programming
1
1.2 The Kuhn-Tucker Conditions
6
2.0
NONLINEAR PROGRAMMING METHODS
9
2.1 The Classical Calculus
9
2.2 Separable Programming Problems
10
2.3 Quadratic Programming Problems
10
2.4 The Process Optimization Program (POP-II)
12
3.0
OPTIMIZATION OF A MEMS PLANT USING POP-II
23
3.1 Process Description
23
3.2 Explanation of the SUBROUTINE MODEL Used in the
POP-II Program to Optimize the MEMS Plant
27
3.3 Results, Discussions and Conclusions Concerning
the Usefulness of POP-II for Optimizing the
MEMS Plant
40
3.4 Suggestions for Further Work
49
4.0
REFERENCES
51
5.0
NOMENCLATURE
52
APPENDIX I. INPUT DATA FORMS FOR PROPOSED FUTURE WORK ON
THE MEMS PROCESS
60
APPENDIX II. POP-II SAMPLE OUTPUT
72
APPENDIX III. DESCRIPTION OF COMPUTER PROGRAM - MODEL
90
PART II. OPTIMIZATION OF A MULTI-STAGE
AERATED LAGOON BY THE DISCRETE
MAXIMUM PRINCIPLE
1,0 INTRODUCTION
2.0 DESCRIPTION OF LAGOON MODEL
2.1 Ideal Component Assumption and Kinetic Model
2.2 BOD Material Balance
2.3 Aerator Motor Size Equation
2.4 Economic Model
3.0 PROCESS OPTIMIZATION
3.1 Development of the Performance Equation
3.2 Statement of the Optimization Problem
3.3 Computational Procedure - The Discrete Maximum
Principle
4.0 RESULTS AND CONCLUSIONS
5.0 PROPOSED FUTURE WORK
6.0 REFERENCES
7.0 NOMENCLATURE
APPENDIX I. Explanation of Lagoon Optimization Computer
Program - OPT and Subroutine DMP
APPENDIX II. Explanation of Lagoon Simulation Computer
Program - SIM
APPENDIX III. Explanation of Computer Program - SEARCH
Page
102
104
104
109
109
110
114
114
117
113
120
135
140
141
145
161
165
PART I.
THE Uoji OF THE NONLINEAR PROGRAMMING TECHNIQUE POP-II TO
OPTIMIZE a MULTI-EFFECT MULTI-ST^SE (S4EMS)
SE&WATEH DISTILLATION PLANT
1.0 INTRODUCTION
In this part of the report a Multi-Effect Multi-Stage (MEMS)
seawater distillation plant was optimized by applying the nonlinear
programming technique known as POP-II. Several concepts basic to the
understanding of nonlinear programming will be presented first.
1.1 Some Basic Concepts and Definitions Useful in Nonlinear Programming
Unfortunately no single algorithm exists, with the exception of a
systematic exhaustive search, for solving the general nonlinear program-
ming problem which has an objective function of the form
S = f(x1, x2 , x ) m f(x) (la)
X — (.Xj , x~, ...., X )
and constraints of the form
,00
t>- (lb)
i = 1, 2, ...., m
It should be apparent that there is no single method which is best
for attacking the above problem. Many times the best method for attacking
a problem will depend a great deal upon the exact functional forms of the
objective function and the constraints. For example, if all the constraints
are equality constraints, the methods given by Hadley (1) can be applied.
A few special cases of the general nonlinear programming problem will
be given in the sections that follow. Fortunately, most problems of
practical interest can be handled by one or more of the special cases or
by more general techniques such as POP-II.
.
Before proceeding further, certain definitions which are important to
nonlinear programming will be introduced.
A bounded region of n dimensional space, sometimes called n-space or
hyperspace, is a portion of hyperspace which has been closed off or sur-
rounded. Hypersurfaces are used to form the bounded region. The concept
of a bounded region is best visualized in terms of 2 or 3 dimensional
space. Figure la shows a portion of the first quadrant which is bounded
by plane curves. Figure lb shows a portion of the first octant which is
bounded by a spherical surface.
A point which is inside a bounded region is called an interior point.
A point on the boundary is a surface point. A point outside of the
bounded region is called an exterior point. Interior and surface points
are feasible points whereas an exterior point is said to be infeasible.
The quadratic form is useful in nonlinear programming in that by
examining the quadratic form of a function it is possible to make important
statements about the nature of the function. A function is said to be
strictly concave if its quadratic form is negative definite over all of
Euclidean space. For a function with one variable this corresponds to the
case in which there is a unimodal maximum. On the other hand, if the
quadratic form of the function is positive definite, the function is said
to be strictly convex. This corresponds to a function with a unimodal
minimum in the one variable case. Convex and concave functions of a single
variable are shown in Figures 2a and 2b, respectively.
Recall that if the quadratic form of a function, h(x), is positive
definite, then the quadratic form of -h(x) is negative definite. This
implies that if h(x) is convex, then -h(x) is concave and conversely. Or,
3
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if h(x) is unimodal with a maximum, then -h(x) is unimodal with a minimum.
That is, it is possible to change a minimization problem into a maximiza-
tion problem or a maximization problem into a minimization problem. There-
fore, one may talk exclusively about either, maximization or minimization
processes.
The following relations also hold, where
(Convex Function) f% | 1x^(1- X )x2 j £ \f ^x, )+(l- A ) f^) (2a)
(Concave Function) i% ("Ax^I- X )x2 ] > XfjfcjHU-X) i^x^ (2b)
One way of thinking about the significance of relation, Equation (2a), is
to consider the problem of approximating the convex function pictured in
Figure 2a by a straight line through any two points on the curve. Observe
that the straight line lies above the function it is approximating. That
is, it over estimates the function. Similarly, the straight line approxi-
mation to a concave function always under estimates the function it is
approximating. Relations given by Equations (2a) and (2b) can be genera-
lized to higher spaces by replacing x by the s-dimensional vector, x -
(x^> x9' -•••' xs)* *n t'le general case of s-space these relations are
orten used to define convex and concave functions.
The important use of the definitions of convexity and concavity is to
determine whether or not an extremum is a local or a global extremum. For
the case of minimization of an objective function, a local minimum which
occurs within a closed convex constraint set is a global minimum if the
objective function is convex. A local maximum is a global maximum if the
objective function is concave and the constraints are closed and convex.
Note in both cases the constraint set must be closed and convex.
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For many problems, especially those with large numbers of constraint
equations, it is quite a laborious task to determine if the objective
function is convex or concave and if the constraint equations are convex.
Therefore, what is generally done is to solve the problem without examining
the object function and constraints with respect to convexity and con-
cavity. Simulation and/or a search technique are then used to verify the
results. If this reveals a solution point which gives a better extremum
than the previous solution, the new point is used as a starting point for
applying the particular nonlinear programming method again. The danger of
a local extremum should be kept well in mind, but for many problems a
local optimum is better than none at all.
The careful reader will also note that it is possible for an objective
function to be convex over parts of the constraint region and concave over
other parts. This case is shown in Figure 3.
Furthermore, some functions, such as, the one given by equation (3),
may be neither convex nor concave.
S = C a.x. (3)
i-1 1 X
1.2 The Kuhn-Tucker Conditions
The Kuhn-Tucker conditions are useful because they give some insight
into nonlinear programming theory and they enable one to determine whether
or not a point which has been found by a computational procedure is an
optimum point.
Kuhn and Tucker (2) were the first to derive the conditions which bear
their names. Carr and Howe (3) also give a very readable derivation of the
conditions.
7
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The derivation of the Kuhti-Tucker conditions given here is due to
Wilde and Beightler (4).
Consider the problem:
min: S = f(x) (4a)
subject to the constraints
%i (x) i. b. (4b)
There is no loss in generality by using constraint equation (4b), in
place of constraint equation (lb) since constraint equation (4b) can be
obtained from constraint equation (lb) by introducing more slack variables
and constraints.
Constraint equation (4b) can be charged to an equality constraint by
introducing the slack variable, u, which is squared to insure that the
squared quantity remains positive.
g. (x) - b. - u.2 - 0 (5)
A Lagrangian function can be formed
L = f(x) - £ *i J g, (x) - b. - u.2 (6)
Then, three of the four Kuhn-Tucker conditions which are necessary
conditions for a minimum are:
in
J l - 0 ■ Jf(x) - t_. Ai J gi(x), j = 1, 2 s (7)
cJx.
h
= 0
1, 2, ..., m (S)
g,-(x)
cUi
,)l = 0 = 2/L a., i = 1, 2, ..., m (9)
dui
The fourth necessary Kuhn-Tucker condition for a minimization problem as
given by Wilde and Beightler (4) requires that the Lagrange Multipliers be
non-negative,
^ i - 0, I - 1, 2, ...., m (10)
If a point does satisfy the Kuhn-Tucker conditions, it may or may not
be a minimum. However, if it does not satisfy them, it can not possibly
be a minimum. If both the objective function and the constraints are
convex, the Kuhn-Tucker conditions are sufficient conditions for a global
minimum. The Kuhn-Tucker conditions in themselves do not provide a com-
putational procedure for finding optimal solutions to nonlinear programming
problems.
2.0 NONLINEAR PROGRAMMING METHODS
2.1 The Classical Calculus
For some nonlinear programming problems, classical calculus techniques
can be used to find the optimum extreme point. The steps for finding the
optimum extreme point given the objective function, equation (la), and the
constraints, equation (lb), are as follows: Ignoring the inequality con-
straints, find the stationary points of the objective function. Determine
which if any of the stationary points are within the feasible region. For
those points in the feasible region determine at which point the objective
function takes on an extreme value. Search the boundaries of the feasible
region to determine if the interior extreme point is a global extremum.
An example illustrating this procedure is:
Find the maximum value of the objective function
2
S ■ IOx-l + 20x2 + x.x. - 2xL - 2x2 (11)
which is subject to the constraints
0 - xx i 7 (12)
0 - x2 - 8 (13)
10
The stationary points and the values of the objective function at these
points are given in Table 1. For this problem the maximum is at the in-
terior point. Note however, that if the goal would have been to minimize
the objective function, the optimal point would have been on the boundary.
Several mathematical programming algorithms are extentions of the
simplex method of linear programming. The reluctance to abandon the sim-
plex method may be partially explained by noting that large sums of money
have been invested in the development of computer codes to solve large
scale linear programming problems since 1947 when the simplex method
started its rapid development. Furthermore, experienced people are avail-
able who can cast a nonlinear programming problem into a linear program-
ing format.
Two classes of nonlinear programming problems which can be solved by
modified simplex algorithms are separable and quadratic programming problems.
2.2 Separable Programming Problems
The separable programming technique can be applied when the constraints
are linear and it is possible to separate the objective function into a sum
of functions each of which is a function of only one independent variable.
The general separable programming problem has the following form.
Objective function:
f(x) f t. (x ) (14)
i=l
Constraint equations:
t aij xj (=) V i = l m <15)
j=l
2.3 Quadratic Programming Problems
When the objective function has the quadratic form
n
Table 1. Stationary Points and Values of the Objective
Function for the Classical Calculus Example
Location
of
Stationar
y
Point
X-i
x2
5
interior
4.0
6.0
80.0
Boundary
0.0
5.0
50.0
Boundary
7.0
6.8
63.1
Boundary
4.5
8.0
72.5
Boundary
2.5
0.0
12.5
12
f(x) - E a.x + T HC xixi (16)
1-1 x x 1-1 j-1 « l J
and the constraints are linear,
£ a., x. U / b.; i = 1 , m (17)
j-i 1J J UJ l
quadratic programming methods can be used to extremize the objective
function.
Separable programming, quadratic programming, and methods for solving
other types of nonlinear programming problems are discussed in detail by
several authors (1, 3, 4). In this work, little emphasis will be placed
on methods for solving problems which have specific objective and constraint
equation forms because more general algorithms exist for solving more general
types of mathematical programming problems.
2.4 The Process Optimization Program (POP-II)
For many of the nonlinear programming algorithms, no "general", easy to
use, computer program (computer code) exists for implementing the calculations
for large scale problems. Because this is a serious disadvantage to using
these methods, there has been a great impetus to develop a general computer
program to handle a wide variety of nonlinear optimization problems, which
would require a minimum of effort, knowledge, and time on the users part.
One computer code for doing this which has been quite successful is the
Process Optimization Program II (POP-II) which has been developed by Smith
(5), of IBM's System Research Institute.
POP-II uses a truncated Taylor series to obtain a sectionally linear-
ized linear programming problem from the nonlinear programming problem.
To start the computations, the processes' performance and objective equations
13
are linearized. A linear programming problem is formed from these
linear equations. The solution of the linear programming problem, hope-
fully, gives values of the independent variables that are closer to the
optimum values. The above procedure is repeated until the optimum values
of the independent variables have been determined.
POP-II is a nonlinear programming technique which is easy to use.
All that is necessary for using the technique is a mathematical model of
the process or system which the user wishes to optimize and a mathematical
statement of the optimization objective (objective function). The model
must be constructed so that the processes' dependent variables can be
calculated once the independent variables have been specified. If the
user can provide a model which will calculate the dependent variables given
the independent variables, then POP may well be a good technique to use.
Some of the terms discussed in this section may be easier to understand
if one refers to Appendix II where the computer output for a POP problem
is given.
The process model or simulation program which the user provides is
named SUBROUTINE MODEL. SUBROUTINE MODEL must be programmed in such a
way that each of the processes' dependent variables, the y. values, is
written as a function of one or more of the processes' independent variables,
the x values, and/or a function of one or more of the previously defined
dependent variables. In effect each dependent variable must be written
as a function of one or more of the independent variables.
The sectionally linearized linear programming technique used by POP
is described in the following paragraphs.
Consider the objective function and the equality constraints of the
14
generalized nonlinear programming problem given by equations (la) and
(lb).
S » f(xy, x, , Xg) (18)
gp (*l. *2 , xs) = bp, p = 1, 2, ..., n (19)
If the inequality constraints are ignored for the present, the number of
independent variables for the nonlinear programming problem posed by
equations (18) and (19) is equal to (s-n). Let the number of independent
variables be r, that is, r is equal to (s-n).
Let the i th dependent variable be given by ?.. However, y1, is
reserved for the objective function S. It is now possible to rewrite
equations (18) and (19) in the form
f(y,, y3, ..... yn+1, xv *2, *r) (2°)
and
where
gp (y2. y3, •••.. yn+1. V x2> Kr) " V
p - 1, 2 , n (21>
Equations (20) and (21) represent respectively, the general functional
forms of the objective function and the equality constraints. In most
instances, an individual g will not be a function of all of the y^,
i = 2, 3, n+1 and x , y = 1, 2, ...., r. For purposes of program-
ming a SUBROUTINE MODEL, each equality constraint is solved for a particular
y. value. The equality constraints, having been solved for a particular
y. value, are next arranged in an orderly fashion, such that each y. is a
function of the other dependent variables y2, y^, ...., Y^_i an& the
independent variables, that is,
y2 = h2 (x)
y ■ h3 (h2(x), x) = h^(x)
15
y. = h. (h,<x), h3(x) , h1_1 (x), x) = h. (x) (22)
= h ,, (h. (x), h,(x), , h (x), x) - h . , (x)
'n+1 "n+1 v"2
n+Iv
where the abbreviated notation x ' (x., x,m .... x ) is used. After the
values of y,, y,, ...., y ,, have been determined, y. is calculated using
equation (20).
The dependent variables given by equation (22) may be approximated in
a small region near a point x , denoted by x = (x , x , ...., x ), by
o o o
a truncated Taylor Series if a coordinate system is defined such that the
distance from x in the i th direction is denoted by S x. . Thus we can
, j=r o> hi
y. (xq+ Sx) ~ h.(x) + C Sx ^x
(23)
lj <-, ....5 n
Equation (23) is linear in Sx. and can be simplified to give
~ j=r r
y. (x + x) - e. + T3 a-' Jx. ,
j = l
where
i = 2, 3 , n+1
(24)
e.= h. (x)
(25)
and
lj
(26)
A similar approximation holds for y. in the small region near x .
16
The set of linear equations obtained by linearizing the nonlinear
equations may be used to form a linear programming (LP) problem. The
LP problem is solved in a small region near x • In most instances the
solution will give a new x point which gives an improved value of the
objective function.
When inequality constraints are imposed they are introduced into
the LP problem by the POP program after the user has inserted the maximum
and/or minimum limits of a variable on the input data cards. The POP
program handles the inequalities by automatically introducing the neces-
sary slack variables which are required.
The size of the small region near x in which the LP problem is
solved may be controlled by user-specified constants which are called
move limits. The j th move limit is the maximum step size that is
permitted for 5x..
Since the y. values obtained in the linear programming solution may
deviate from the true y. values, the new x values are used in SUBROUTINE
'i ' o
MODEL to calculate the true y. values. The user must also supply error
limits for the dependent variables. These error limits are used by POP
to minimize the deviations that occur from the true y. values.
J x
The procedure described up until now is called a loop and is repeated
until an extreme value of the objective function is found. In the event
that an extreme value is not found, the procedure is terminated when the
maximum number of optimization loops which the user specifies has been
exceeded.
Many nonlinear optimization techniques require the calculation of
derivatives of the same type. A significant feature of POP is that the
values of the first partial derivatives required, the a. . values, are
17
calculated automatically by a subroutine of POP. The central difference
technique which is used to calculate the partial derivatives is given as
h.
U
<)XJ
£X1 | jc I AXi (27)
2 Ax.
where Che Ax. values are specified by the user. This technique is
illustrated graphically in Figure 4 for the case where h. is a function
of one independent variable, x, . Note that h. is shown as a smooth con-
tinuous curve in Figure 4. Obviously, if the function whose first partial
derivative is to be computed by a numerical method has one or more discon-
tinuities for example, then the calculated derivative may be a very poor
approximation. This is a disadvantage inherent in the use of POP. How-
ever, on the other hand, techniques which require analytical expressions'
for first partial derivatives require logical statements in their computer
programs for handling piecewise smooth functions (see Appendix I of Part II
of this work for an example.)
As an optional feature, the subroutine which calculates the partial
derivatives can also calculate new move limit values as the optimization
progresses. The process is called making an adaptive move limit calculation.
This feature or operating mode of POP may be used with some SUBROUTINE
MODEL'S to reach the optimum in a fewer number of loops. However, with
some SUBROUTINE MODEL'S the adaptive move limit calculation reduces the
values of the move limits so much that POP shuts off before reaching the
optimum. Smith (5) gives a detailed description of the adaptive move
limit operating mode.
In writing the SUBROUTINE MODEL used to simulate the MEMS plant,
16
3h,
ax,
Slope of £S
h,(x)
h.
'lx, +AX,
'o
x^-Ax,
Fig. 4. Central difference -approximation to the slope of
the function h. (x.) at the point x. .
19
several procedures were found to help speed the programming and debugging
processes. Since it is thought that these procedures will be helpful in
general programming, they will be mentioned here.
In SUBROUTINE MODEL the j th independent variable is denoted as X(J).
The i th dependent variable is denoted as Y(l). The i th constant is
denoted as CONST(I). If large numbers of the symbols X(J), Y(I), and
CONST(I) are used in a SUBROUTINE MODEL, it may be difficult to remember'
what each symbol represents. Consequently, it may be easier to debug
SUBROUTINE MODEL by writing it in terms of easily recogni2able symbols.
The independent variables which are written in terms of X(J) variables
at the beginning of the program. At the end of SUBROUTINE MODEL the Y(I)
variable can be defined in terms of the easily recognizable symbols which
are used in order to make the programming easier.
From experience with the MEMS plant problem, it was also determined
that it is not necessary to define all dependent variables as Y(l) vari-
ables. For example, the horsepower requirements in the MEMS plant problem
were not defined by Y(I) variables. Since the program, without modification,
has a maximum limit of 50 Y(I) variables, this procedure may also help
spread out the Y(I) variables and thus increase the problem size that can
be handled.
The programming procedure described above will require more computer
cards. However, from knowledge of the problem studied, it is felt that the
additional computer cards in SUBROUTINE MODEL will not have a significant
effect on the time required for execution of the POP program. Furthermore,
the time from problem inception to completion may be greatly decreased.
One difficulty inherent in the use of POP, which also occurs in the
20
use of many other techniques, is the possibility of finding a relative or
local extreraum rather than a global extremum. If there are several local
extreme points in the feasible answer space, POP will probably find the
one which is nearest the starting point. This difficulty has been called
"nearsightedness" by Baumol (6, 3). The way that "nearsightedness" is
usually overcome is to start the program at several widely spaced feasible
points and then observe what happens. Another problem which may be en-
countered when SUBROUTINE MODEL is large and/or has complex performance
equations is that it may be difficult to find a feasible starting point.
However, to aid the user, POP will double the move limits on successive
loops until a feasible point has been found if one can be found at all.
A diagram giving the steps required to solve a nonlinear programming
problem using POP is given in Figure 4a.
One of the important features of POP is' the matrix which is printed
out after POP has reached an optimum point or optionally after each loop.
Each element of this matrix, which is called the DYDX MATRIX, is one of
the a., values given by equation (27). Specifically, the element a., is
the number used in the i th row of the LP tableau with the variable x;.
For the i th row the y variable is y..
Kith the DYDX MATRIX, it is a simple matter to perform an incremental
sensitivity analysis to determine where to concentrate effort to improve
the process being optimized. In addition, inspection of the first row of
the matrix, that is, the row for the objective variable y1 , will give a
good indication of the nature of the response surface near the x point
for which the matrix holds. This is especially true if the A x. values
used are small.
21
Derive process performance
equations and establish
objective function.
Equations (20) and (21)
Identify
dependent
variables :
and
independent
variables
x. > X(J)
Arrange performance equations
and write as PORTRAH state-
ments to obtain
SUBROUTIHE KODEL
Specify PCP input data such as
move limits, dependent
variable error limits, and
Initial values of X(J) *
Insert SUBROUTINE KODSL and
input data into the POP deck
?.un x^up starting with initial
X(j) values
Calculate partial derivatives:
Form and solve LP problem for
new X(j) values and Y(l)
values which may be incorrect
due to linearization errors
Insert new X(j) values into
SUBROUTINE MODEL to calculate
orrect Y(l) values
Fig. 'ta. The steps required to solve a nonlinear programming
problem using POP-II.
22
Each coefficient of the DYDX MATRIX can be examined with respect to
sign and size. Consider the case when the coefficient for the i th row
and the j th column is -a. This indicates that if the independent variable
of the j th column is increased by one unit then the dependent variable
of the i th row will be decreased by approximately a_ of its units. This
analysis holds true only for a small region around the x point for which
the matrix holds. Furthermore, the above statements are not strictly true
for variables that appear in equality constraints.
If each element, a. ., of the objective variable row of the matrix is
a very small number, then small charges in the independent variables will
not change the objective function significantly. If this situation occurs
at the optimum, this is called a flat optimum and the response surface is
said to be well behaved near the optimum.
However, if one of the elements, a 0f tne objective variable row
is very large, then only a small change in the independent variable of
that element, x , will cause a large change in the objective function,,
This is the most interesting type of problem as it may be a constraint which
causes this situation to happen. In the event that it is a constraint on a
variable, which causes the element of the objective variable row to be
large, one should look into the possibilities for removing or relaxing the
constraint to some extent.
As an example, consider a problem in which reactor temperature appears
as an independent variable. Furthermore, assume an upper limit is placed
on the reactor temperature because the reactor cannot withstand high tem-
peratures. Now if the system is optimized by POP and it is found that the
reactor temperature element of the y. row of the DYDX MATRIX is the only
large element in the row, and furthermore if this is due to the upper
temperature imposed, then obviously searching for construction materials
23
which can withstand higher temperatures should be seriously considered.
Admittedly this attack will necessitate resolving the problem with new
cost data.
3.0 OPTIMIZATION OF A MEMS PLANT USING POP-II
An example problem of a Multi-Effect Multi-Stage (MEMS) seawater
distillation plant which demonstrates the use of POP-II in process design
will be given next.
3.1 Process Description
Fan, et. al. (7) give an excellent description of this process which
is included here for convenience.
Figure 5 illustrates a three-effect multistage flash system. In
order to facilitate the discussion, some critical locations in the system
are denoted by letters, Z, A, B, C, D, etc. and the system is divided into
various sections which are denoted by HR-1, R-l, HR-2, etc. The first
effect consists of a brine heater H-l, a heat recovery section, HR-1, and
a cooling section, R-l. The second effect consists of a brine heater H-2,
a heat recovery section, HR-2, and a cooling section, R-2. The third
effect consists of a brine heater H-3, a heat recovery section, KR-3, and
a cooling section, R-3. The section between locations B and C serves a
double purpose; it is the cooling section for the first effect, R-l, and
the brine heater for the second effect H-2. Similarly, the section between
locations E and F serves as the cooling section for the second effect, R-2,
and the brine heater for the third effect, H-3. Sea water is used as a
coolant in R-3.
F and L represent the flow rate of feed brine and flashing brine,
respectively* The feed brine and recycle brine are referred to together as
24
£
CD
to
>-■
(/)
c
o
CO
'■o
CO
a
o
CO
=5
C
o
o
CD
25
the non-flashing brine stream. T,, T. and T represent respectively the
r j c
temperature of the flashing brine, non-flashing brine, and condensate,
respectively. Subscript notation will be used to indicate the location.
For example, (T-)„, (T.)„ and (T )„ represent the temperature of the
flashing brine at location F, temperature of the non-flashing brine at
location B and the temperature of the condensate at location H respectively.
R , R, and R, represent the recycle flow rate in the first, second and
third effect, respectively, and W., W, and W, represent the condensate
produced in the first, second, and third effects, respectively. R, repre-
sent the cooling water (sea water) used in the third effect.
The sea water feed is heated in R-3 and then acidified and degasified
to remove C0_ and other dissolved gases. After being heated successively
in KR-3, H-3, HR-2, H-2, HR-1 it is mixed with recycle brine R, to form a
brine stream which is heated in brine heater H-l and then introduced into
the first effect as the flashing brine (L) .
A
The flashing brine at location C is divided into two streams. One
stream, (L)„, is fed into the second effect and the other stream, R, , is
recirculated by a recycle pump, J., heated in HR-1 and then mixed with the
feed stream at the mixing point M-, As has been described, the. combined
stream is heated in brine heater H-l and introduced into the first effect
as flashing brine (L),. Similarly the flashing brine at locations F and K
are divided into (L)„ and R„ , and (L)„ and R„ , respectively. (L) is fed
into the third effect, and (L) is discharged from the third effect as the
reject brine from the system. Streams R„ and R„ are recirculated by pumps
J, and J, respectively, heated in sections HR-2 and H-2, and HR-3 and H-3,
respectively, mixed with (L)„ and (L) at mixing points M, and M, respective-
ly, and introduced to the second effect and the third effect as (L)B and
26
(L) , respectively.
The feed brine and the recycle brine are heated in each stage by the
water vapor evaporated from the flashing brine in that stage. It is possible
to arrange the flow-system so that the temperatures of the feed brine and the
recycle brine are equal at any location. In the following discussion, such
an arrangement is assumed. As has been described, the feed brine and the
recycle brine are referred to jointly as the non-flashing brine and its
temperature is denoted by T.. The recycle brine, R , which is a part of
the flashing brine at location C, is introduced into the condensing chamber
at location B where it becomes a part of the non-flashing brine stream.
Therefore, the following relation should hold.
<Vb"(Vc <28>
Similarly, we can write
(T.)E = (Tf)r (29)
and
(Vh=(Vk' . (30)
Mixing is thermodynamically irreversible when two solutions, which
differ in temperature and/or composition, are mixed together. The solutions
mixed at the mixing points, M^, M, and M , have differences in composition;
however, by suitably locating point B, the temperature of the recycle
solution R,, (T.)fi, can be adjusted to (Tf) , the temperature of stream '
(L)c, and in this way the thermodynamic irreversibility due to mixing can
be minimized. In this study isothermal mixing is at M. , M , and M and the
heat of mixing of the sodium chloride-water system is neglected. Thus,
27
(Vb= (Vc= (TfV (3«
Similarly,
(T.)E- (Tf)F= (Tf)G. (32)
The unit enthalpy of dilute salt solutions is assumed to be a function
of temperature but independent of composition. This assumption is justified
because of the small heat of mixing and the rather limited concentration
range of approximately from 3.57. to 77. encountered in this process.
A stage within each effect consists of a flashing chamber and a con-
denser chamber and a demister which separates the two chambers. When the
flashing brine leaving one stage (the (n-1) th stage) is released into the
next stage (the n-th stage), water vapor flashed out of the solution. The
water vapor then passes through the demister on the way to the condenser
chamber, where it is condensed to heat the nonflashing brine, i.e., the
feed and recycle streams.
3.2 Explanation of the SUBROUTINE MODEL Used in the POP-II Program ;
Optimize the MEMS Plant
Fan, et. al. (7) have given the theoretical background and the
derivation of the equations that are given in this section. The POP-II
SUBROUTINE MODEL used in this example is listed in Table 2. The equations
used in the SUBROUTINE MODEL are given immediately after the listing of
SUBROUTINE MODEL in a one-to-one correspondence with the FORTRAN statements
of SUBROUTINE MODEL. Also, some of the equation numbers are given beside
the corresponding SUBROUTINE MODEL FORTRAN statements. For example, the
first FORTRAN statement of SUBROUTINE MODEL under the PERFORMANCE EQUATIONS
heading, is the same as Equation (33), the first equation in the following
explanation of SUBROUTINE MODEL. It is hoped that this method of presentation
2a
will give a clearer understanding of how a SUBROUTINE MODEL may be programed.
The symbols used in SUBROUTINE MODEL are given in Table III-l of
Appendix III. Because of the simplicity of SUBROUTINE MODEL, no logic
diagram is given for it.
29
fable 2. SUBROUTINE MODEL .-'or The MEMS' Plant
SUE ROUTINE MODEL
COMMON P(5244)
DIMFNSICN X(50), Y(5C), CONST(400)
EQUIVALENCE (X(1).P(5120),(Y(1),P(5180)) ,( CONST ( 1) ,P(451) )
C SIMULATION PROGRAM MEMS SEAWATER DISTILLATION PLANT
C ******************** ****** ******** ********************
C CONSTANTS USED IN PERFORMANCE EQUATIONS
AL=1000.
3=17.01723
AN1=23.
AN2=23.
AN3=22.
CF=0.035
CP-1.
XA=230.
XB=250.
XC=270.
XD=290.
YA=958.8
YS=945.5
YC=931-8
YD=917.5
U0=510.
ui=sm.
■ u ? = ■> J 0 .
U3=510.
C **** I-*************************************************
C PERFORMANCE EQUATIONS
Y(2)=X(1)+X( 2)+X(3) (33)
Y(3)=Y(2)/( l.-CF/X(3) )
AL10 = Y(3)-X( 1) (35)
AL2O=AL10-X(2)
AL3C=AL20-X(3)
ALl!=Y(3)+X(4>
AL2I=Y(3)+X( 5)-X(l )
AL3I=Y(3)+X(6)-X!1)-X(2)
CSF10 = CF*Y!3J/AL10 (41)
CSF^O=CF*Y(3)/AL20
CSF1I = CSF10*CA:.1C+X(4) J/AL1I
CSF2I=CSF2 0*(AL20+X( 5) 1/AL2I
CSF3I=X( 3)*( AL30+XI6 ) 1/AL3I
Y{4)=XP.O )-< AL/CP)*AL0G(CSF1C/CSF1I ) (46)
Y(5)=Y(4)-<AL/CP)*ALGGCCSF20/CSF2I )
Y(6)*Yt5)-(AL/CP)*ALCG(X<8 I/CSF3I )
A:=:.OIOO+(CSF1I+CSF10)/(2.*0.0300) (49)
A2=1.0C75+(CSF2I+CSF2G)/(2.*0.0347>
A3=0.320l+!CSF3I+X(8) 1/(2. *C. 0315)
Y(7)=Y(4)-A1 (52)
30
.... 2. (tion't)
Y(8)=Y(5)-A2
Y <9)=Y(6)-A3
2 = X(-->)
LAGRANGIAN POLYNOMIAL
AA=YA*(Z-XB)*(Z-XC)*(Z-XD)/( ( XA-XB ) * ( XA-XC ) * I X A-XD ) )
AB=YR*(Z-XA)*CZ-XC)*(Z-XD)/I ! XB-XA )* ( XB-XC) *< XB-XD) )
AC=YC*(Z-XA)*(Z-XB)*(Z-XD)/< 1 XC-XA >* ( XC-XB)*(XC-XD> )
AD=YD*(Z-XA>#(Z-XB)*(Z-XC)/( ( XO-XA )* ( XD-XB ) *( XD-XC ) )
ALS=AA+AB+AC+AD
******************************** ** ************** ******
YC0)=X(7)*ALS (55)
TLF0I=Xt 1C]-Y< 10)/(CP*(Y!3)+XC4> ) ) (56)
TLF1I=(CP*( ( AL10+X(5 ) ) *Y (4 )+X t 1 ) *Y ( 7) )-Y( 10 ) )
1 /iC?iiV(3]+X(S))l
TLF2!=(CP*( [AL20+X(6 ) ) *Y ( 5 ) + ( X ! 1 ) +X ( 2 ) ) *Y ( 8 ) ) -Y ( 10 ) )
1 /(C?*(Y(3)+X(6) ) )
Y(15)=(Y( 10)+Y(3)*CP*X(ll)-(CP*(AL3O*Y(6) +Y( 2 ) *Y( 9) ) ) )
1 /(CP*(Y<6) -X(ll) ) ) (59)
DT0=X(9)-0.5*(X( 101+TLF0I ) (60)
DT1=X( 10)-TLFCI-Ai-(X(10)-Y(4) )/(2.*ANl )
DT2=Y(4)-TLF1I-A2-(Y(4)-Y( 5 ) ) / I 2.*AN2 )
DT3=Y(5)-TLr2I-A3-(Y(5)-Y(6) )/(2.*AN3)
Y( 11 )=Y( 10)/[DT0*U0) (64)
' Y(12)=X( l)*AL/OTl*Ul)
Y!13)=X(2)*AL/(DT2*U2)
Y tl4)=X(3)#AL/(DT 3*113)
HP1=X(4)*3*(EXP t-AL/(0.1104*(X(lC 1+460. )) ) (68)
1 - EX? (-AL/(C.1104*(Y<4) +46C.)!))
HP2=XC5)*B*(EXP (-AL/(0.1104*(Y(4) +460.)))
1 - EXP (-AL/(C1104*(Y(5) +460.))!)
H?3=X(6)*8*(EXP (-AL/(0.1104*( Y(5) +460.)))
1 - EXP 1-AL/(0.1104*(Y(6) +460.))))
******************************************************
C0S7 EQUATIONS
Y(16)=C0NST( 1)*Y(3)/1.E+10 (71)
Y(17)=CCNST(2)*X(7)/1.E+10
Y(18)=CO\ST(3)*HP1/1.E+10 (73)
Y(19)=CCNST(3)*HP2/1.E+10
Y(20)=COMSTC3)*HP3/l.E+10
Y ( 2 1 ) =CCN ST { 4 ) * Y ; 1 5 ) / 1 . F+ 1 0
Y{22)=CCNST(6)*Y(11)/1.E+10
Y(23)»CCNST(5)«YI 12J/1.E+10 (76)
Y(24)=CONST(5)*Y< 13 1/1.E+10
Y(25)=CCNST( 5)*Y(14) /l.E+10
Y(1)=Y< 16 1+Y1 17i+Y( 18)+Y(19)+Y;20)+Y!21 )
1 +Y<22)+Y(23)+YI24)+Y(25)+CONST(7)/l.E-t-10 (77)
RETURN
31
POP-II PERFORMANCE EQUATIONS FOR A MEMS PLANT
The total production rate of fresh water, Cw is equal to the sua
of the individual distilled water streams from each effect
Z\ ■ Wx + W2 + W, (33)
A salt material balance" around the entire plant gives the seaw2ter
feed rate
i - (34)
<cf)k
The flow rates of the effluent flashing brine streams for the individual
effects are given, respectively, by
(L)c = F - WL (35)
(L)p - (L)c - W2 (36)
(L)K = (L)p - W3 (37)
The flow rates of the influent flashing brine streams for the individual
effects are given, respectively, by
(L)A = F + Rj_ (38)
(L)D = F + R2 - Wx (39)
(L)G = F + R3 - Wj_ - W2 (40)
In the following discussion, the concentration of a brine solution refers
to the salinity as defined by Badger and Associates (8).
32
Salinity material balances give the concentrations of the flashing
brine solutions lea
ving effects one and two respectively:
C0 F/(L)C (41)
CQ F/(L)p (42)
The concentrations for the flashing brine solutions entering effects
one, two, and three
, respectively, are given by the following material
balance equations :
(C ) =
A
(Cf) ( (L)c + RL )/(L)A (43)
(C-) ( (L)„ + R, )/(L) (44)
t F n J. D
(Cf) ( (L)K + R3 )/(L)G (45)
K
As the flashin
g brine proceeds through the effect, it is cooled due
to flashing. Fan,
et. al. (7) derived an equation which gives the tempera-
ture drop of the fl
ashing brine solution across an effect as a function of
the ratio of the effect's outlet to inlet concentration. The equation can
be rearranged to give the temperatures of the effluent flashing brine
solutions for the f
irst, second, and third effects respectively:
(Tf) -
C
(T.) - A/C ln( (C ) /(C) ) (46)
A P C A
(T.) - A/C ln( (C.) /(CJ ) (47)
1 C P f F £ D
(Tf) =
K
(T,) - A/C ln( (C.) /(C,) ) (45)
f F p r K £ G
where A is the latent heat of evaporation of steam in the effect and C
P
is the heat capacit
y of the flashing brine.
'
33
The boiling point elevation in an effect is taken as a linear function
of the average flashing brine concentration in the effect. Furthermore, it
is assumed that a one degree Fahrenheit temperature drop occurs across the
demister which is placed in each effect to prevent the flashing brine
solution from splashing onto the overhead heat transfer area and into the
distilled water collection troughs.
Badger and Associates (8) give graphs which can be linearized about a
point to give the boiling point elevation in an effect as a linear function
of the average flashing brine concentration.
Equations (49), (50), and (51) were formulated to calculate the com-
bined effects of the average boiling point elevation and the demister temper-
ature drop for effects one, two, and three respectively
( (Cf) + (Cf) )
«1»" * l'Ol0O+fos55ff "\ * C <4S>
( (c ) + (c ) )
<*V - whof ^2 " (50)
( (C ) + (C-) )
W) ,av = 0.3201 + " G r K (51)
0.0315 ,
where (°0 ,av is the combined effect of the average boiling point elevation
and the demister temperature drop for the n-th effect.
The temperature of the distilled water leaving a stage is given by the
flashing brine temperature leaving the stage minus the corresponding (°c) ,av
for that stage.
(T ) = (T ) - (cC) ,av (52)
C C X C L
(T ) ■> (T ) - (ct) ay (53)
F F
(T ) - (T ) - (<*),,av (54)
K r K J
34
The latent heat of vaporization of the saturated brine heater steam
is written as a function of the steam temperature by using a Lagrangian
Polynomial. However, in this study the brine heater steam temperature is
allowed to vary only in one case study and then by only a small amount,
because increasing the temperature will bring about a significant increase
in the pressure within the brine heater. Higher pressures require that the
brine heater be built out of sturdier, more expensive material. The
effect of brine heater pressure upon brine heater cost has not been taken
into account except at the brine heater temperature used for this study.
The rate of heat addition to the brine heater by the brine heater
steam is given as
q = m* (55)
^s s s
where m is the steam flow rate and X is the latent heat of vaporization
of the brine heater steam.
A heat balance around the brine heater, including all effects up to
the (n + l)th effect, will give the temperature of the seawater and recycle
brine entering the n-th effect. Equations (56), (57), and (58) give the
temperatures of the combined non-flashing brine streams entering the brine
heater, effect one, and effect two, respectively. The seawater coming
into the plant is assumed to be at an ambient seawater temperature of 85° F.
(T. ) = (T.)
J 'A fA Cp(F+V (56)
C (((L ) +R ) (T ) + W (T ) ) - q
(T.) = P t C Z r C X C C (57)
2 c CP <F + V
C (((L ) + R ) (T ) + (W +W,)(T ) -a
(T.) - P t F J r F l - I F ' (53)
F C (F + R )
P -1
35
A heat balance around the entire plant will give the required cooling
water flow rate
<!„ + FC„ <T
) - C f (L) (T,) + ( W ) (T ) ]
J K p I K t n c RJ
R4 = I J K '■"I. 2—1 - :; <59)
(TJ - (Ts) I
f k j k;
The temperature driving force available for heat transfer in the
brine heater is taken as the arithmetic mean of the approach temperature
differences.
(At)Q = T= -(1/2)((T.) + (T ) ) (60)
A •'A
Fan, et.al. (7) derive equations for calculating the effective heat
transfer driving force for each effect. The equations take into account
the boiling point elevation of the flashing brine, the demister temperature
drop, and the fact that the effect has a finite number of stages rather than
an infinite number. The effective heat transfer driving forces for effects
one, two, and three are given by Equations (61), (62), and (63) respectively
(Tf) - (Tf)
(At) = (T.) - (T ) - (oC) av -(1/2)— i— — L (61)
A J A '1
(Tf) - (Tf)
(At), - (Tf) - (T.) -(<*),, av -(1/2) 2 L. (62)
» JB N2
<Tf) - (Tf)
(At) - (T ) - (T ) - (cC) av -(l/2)__£ _L (63)
where, N is the number of stages in effect n.
The heat transfer areas required for the brine heater and each effect
are given as follows
A0 - qs / (At)Q UQ (64)
Ax = Vx / (flt)L UL (65)
36
A2 = W2 / (At)2 U2 (66)
A3 » W3 / (At)3 U3 (67)
where U is the overall heat transfer coefficient for the brine heater at
n th effect. Equations (65), (66), and (67) give a somewhat approximate
evaluation of the heat transfer areas of the heat rejecting sections of
the respective effects. This approximation is thought to be justified
since the sizes and thus the costs of the heat rejection areas are a small
part of the total heat transfer area of each effect.
The power requirement of a recycle brine pump is taken to be propor-
tional to the product of the pressure head which is developed by the pump
and the recycle brine mass flow rate. The pressure head developed by the
pump must be enough to overcome the frictional loss in the recirculation
line and the pressure increase caused by different vapor pressures at the
inlet and outlet of the stage. In order to determine the power requirements,
the following assumptions are made:
1. The vapor pressure of the flashing brine solution is given
by an integrated Clausius-Clapeyron equation.
2. The friction loss is proportional to the pressure drop
between the outlet and inlet of the effect.
3. The density of the flashing brine solutions are constant
throughout the plant.
4. The horsepower calculated has been corrected for the
mechanical inefficiency of the pump.
The power requirements for effects one, two, and three, respectively, are
given oy
37
(HP),
"1 "1
Exp
0.1104 ( (Tf) + 460.)
A
(HP)2 = Bx R2
(HP)3 - B]_ R3
-Exp
Exp
-Exp
Exp
-Exp
-\
0.1104 ( (T ) + 460.)
C
0.1104 ( (T ) + 460.)
C
0.1104 ( (Tj + 460.)
0.1104 ( (T.) + 460.)
F
0.1104 ( (T ) + 460.)
K
(68)
(69)
(70)
where (HP) is the power requirement for effect n and B. is a constant
which takes into account the friction loss, the recirculating brine density,
the mechanical efficiency of the pump, and the integration constant of the
integrated Clausius-Clapeyron equation.
POP-II ECONOMIC EQUATIONS FOR A MEMS PLANT
Both initial equipment costs and operating costs for the life of the
equipment are taken into account in the objective function. The following
costs are considered to be significant in the selection of equipment size:
I. Initial equipment costs
a. Heat transfer area costs
1. Brine heater
2. Each Effect
b. Pumps - 3rine recirculation
c. Outer shell of each effect
II. Operating costs
a. Feed brine pretreatment and pumping
b. Cooling water pumping
c. Recirculation brine pumping
d. Brine heater steam
In the following cost equations, the cost coefficients have been
divided by the total production rate of fresh water to obtain the total
cost on the basis of a unit of production.
The initial cost of the brine feed pump, the brine pretreatment cost,'
and the brine feed pump operating cost are all considered to be proportional
to the brine feed rate. Therefore,
Ei = V CD
where C_, is the unit feed water cost.
w
The brine heater steam cost is proportional to the steam consumption
rate
E2 - Csms <72>
where C is the unit steam cost.
39
The
energy cost associated with the operation of the
recycle purr.p in
the n-th
effect and the initial cost of each pump can be
calculated by the
following equation:
(ce + YCj) (hp)b = cHp (IIP) n
(73)
where C
e
is the unit power cost, r is the capitalization
charge, C is the
capital cost per horsepower, and C = (C + ^ C ).
The
cooling water cost is directly proportional to the quantity used
E, = Cr R.
3 u 4
(74)
where C„
is the unit cooling water cost.
The
brine heater capital cost per hour, E«, is given
by
E2 - VCBAQ = CktAq
(75)
where C_
is the capital cost per unit heat transfer area,
andCHT=yV
The
heat transfer area cost for the n-th effect can be calculated as
follows:
<EV =YVn = CeA
(76)
where A
n
is the heat transfer area of the n-th effect, C„
is the capital
cost per
unit heat transfer area, and C__ = j C . There should be a
different cost coefficient for the brine heater area cost
than for the
effect areas.
The
total outer shell cost for the plant based on a unit of production
is taken
into account by adding a constant, E, , the sum o]
the previous
costs. The total water cost per unit of production is given by the sum
of the cost terms previously stated.
40
The objective cost to be minimized is:
C„F + C.m_ + g CHp(KP)n + CcR4 + C^ +
n+1
N=3
C c A + E„ (77)
n=l EF n u
3.3 Results, Discussion and Conclusions Concerning the Usefulness of
POP-II for Optimizing the MEMS Plant
The results of three representative runs made with the POP-II
optimization program are summarized in Table 3. Runs 1, 2, and 3 were
started at different feasible points. The initial and final feasible
X(J), values and the total cost obtained for each run are given in the
table. Three independent variables, the brine heater steam temperature,
X(9), the temperature of the flashing brine entering effect one, X(10),
and the temperature of the fresh seawater entering the plant, X(ll), were
held constant in this work so tha.. the' results could be compared with the
work by Fan, et. al. (7). The complete computer output for Run 1, includ-
ing the loop-to-loop output is given in Appendix II. The loop-to-loop
outputs for Runs 2 and 3 are also given in Appendix II.
The starting values of X(J) used in Run 1 were essentially the same
as the overall optimum for the MEMS process obtained by Fan, et. al. (7)
using the discrete version of the maximum principle. The total cost for
this set of initial input values was ?0.2867 per thousand gallons of potable
water, which is again very close to the optimal value obtained by Fan,
et. al. As shown in Table 3, an appreciable change occurred between the
initial and final values of only two independent variables, X(7)= m and
X(8)=(Cf)k. The brine heater steam consumption rate, m , decreased by
41
1.9 percent. The concentration of the discharge brine solution, (C_) ,
increased by 14.1 percent. The final total cost was $0.2866 per thousand
gallons of potable water produced which is again very close but slightly
higher than that obtained by Fan, et. al. Since Run 1 was the lowest cost
case obtained with POP, the final values of the dependent variables are
given in Table 4.
The total cost essentially stayed constant throughout Run 1. No
infeasible intermediate loops were encountered during the run as can be
seen by noting that the production equality constraint was always satisfied
and the independent and dependent variables remained positive throughout
the run. The adaptive move limit option was used in this run. The run
terminated at the end of twenty-five loops because the calculated move
limits became too small.
Run 2 was started with all X(J) values except X(3) less than the
corresponding starting X(J) values of Run 1. The initial total cost was
$0.2926 per thousand gallons of potable water produced. As was the case
in Run 1, only the brine heater steam consumption rate and the concentration
of the discharge brine solution changed appreciably during the run. The
brine heater steam consumption rate increased by 10.8 percent. The con-
centration of the discharge brine solution increased by 14.6 percent. The
final total cost, $0.2890 par thousand gallons of potable water produced,
was only 0.84 percent larger than the optimum cost found in Run 1.
During the course of Run 2, several loops ended at infeasible points.
For these loops, the process required a negative amount of cooling water in
effect three, R4# The next loop after an infeasible loop was always feasible
and the last ten loops were all feasible. Also, small oscillations in the
total cost in going from one loop to the next were noted at the beginning
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47
of the run. This run also terminated at the end of twenty-five loops
because the calculated move limits became too small.
Run 3 was started with all X(J) values except X(3) greater than the
corresponding X(J) values of Run 1. The initial cost was $0.3340 per thous-
and gallons of potable water produced. In this run all of the independent
variables changed appreciably during the run. The significant changes
in the values of the independent variables can perhaps be explained by the
fact that the error limits on some of the dependent variables were larger
in this run than in Runs 1 and 2. Furthermore, the initial move limits of
the independent variables were larger in this run than in Runs 1 and 2.
Loop number six was the best loop obtained in this run. The total cost
at the end of loop six was $0,306 per thousand gallons of potable water
produced. Even though this run ended infeasible, the total cost at the
end of loop six was only 6.S percent higher than the optimum cost obtained
in Run 1.
At the end of loop one of Run 3, the total cost was nearly three times
the starting total cost. Thereafter, the total cost progressively decreased
until the end of loop six. The infeasible loops of this run had negative
cooling water requirements for effect three. This run terminated at the
end of eleven loops because the calculated move limits became too small.
Preliminary simulation of the operation of the MEMS plant was carried
out using computer program MODEL. The main portion of MODEL consists of
the FORTRAN statements used in SUBROUTINE MODEL. MODEL'S main function
was to debug the FORTRAN statements used in SUBROUTINE MODEL. Since remote
computer facilities were used to run the POP program, it was decided to
debug SUBROUTINE MODEL'S FORTRAN statements on available computer facilities
48
in order to speed up the debugging process. Normally, the POP program
may be used to do the debugging. The starting points of Runs 2 and 3
were also determined by using MODEL. MODEL is explained in Appendix III.
The loop-to-loop output is a very useful aid for studying a particular
SUBROUTINE MODEL. Three general observations about the MEMS SUBROUTINE
MODEL were made by inspecting the loop-to-loop outputs for the three runs.
First, there appeared to be a strong interaction between the brine heater
steam consumption rate, m , and the discharge brine concentration, (C-), .
This was readily seen by observing the loop-to-loop output of Run 2.
Secondly, small charges in m and (C_). in m and (C„). caused rather
large changes in the cooling water requirement, R, . Most of the runs
which became infeasible did so because R, became negative. Many tines
R, became negative after only a small change in either m or (Cf), .
The DYDX matrix is also very useful for studying a particular SUB-
ROUTINE MODEL. For the MEMS SUBROUTINE MODEL most of the a., elements of
the m and (C.), columns of the DYDX matrices of all three runs were large
in comparison to the a,, elements of the other independent variables.
Consequently, as was observed in the loop-to-loop outputs, small changes in
these independent variables caused relatively large changes in the other
dependent variables in addition to the total cost. Most of the a., elercnts
ij
of the cooling water, R, , row of the DYDX matrices were larger than the
elements of the other rows. This explains why small changes in most of the
independent variables had a large effect on the cooling water requirements.
Several conclusions regarding the usefulness of POP have been formed
in working with the MEMS plant problem. It is not a routine matter to
apply POP to optimize complex processes. Experienced programmers may even
49
be required to use POP on simple models. It appears that some experimenta-
tion with the modes of operation and the input data may be required on
simple models.
In the next section, future work is proposed in which changes in the
mode of operation and the input data cards can be made now that more infor-
mation is available on the SUBROUTINE MODEL used in this work.
3.4 Suggestions for Further Work
There are several possible changes that can be made in the POP input
data forms and in the manner of operating POP which may help POP to con-
verge to the optimum when it is started at points which are far away from
the optimum.
A minimum limit can be put on the dependent variable R, which hopefully
will prevent it from becoming negative. The optimizer may be run without
using the adaptive move limit calculation feature. This mode of operation
should enable the optimizer to get closer to the minimum cost during a run.
Since the model is sensitive to small changes in the independent variable
(C„), , this variable can be held at a different constant value for a series
of runs as was done by Fan et. al. (7). Furthermore, a reformulation of
SUBROUTINE MODEL using different independent variables may be advisable.
Oscillations in the total cost may possibly be reduced by making the
following adjustments: Move limits on the less sensitive independent vari-
ables may be increased. It may be necessary to individually adjust each
move limit during a series of runs. In combination with the above steps,
changes in the values of the constants used in the optimizer's convergence
logic may help.
A general procedure for conducting POP runs is outlined next. Analyze
the loop-to-loop output and the DYDX matrix. Make necessary adjustments such
50
as those previously described. Restart the next run from the best point
obtained in the previous run. Many runs which do not end at a feasible
point contain useful information and often one of their loops is an ex-
cellent starting point for the next run. See loop six of Run 3. Sample
input data forms, which contain the above changes are given in Appendix I.
A more elaborate process simulation program or SUBROUTINE MODEL can
be investigated in the future. The number of stages per effect can easily
be made an independent variable as is suggested by Fan, et. al. (7).
Information giving equipment cost as a function of equipment size and steam
cost as a function of the steam's physical properties (9) is becoming more
readily available as time progresses.
Certain advantages may be realized by combining POP with other optimi-
zation techniques such as the discrete maximum principle.
51
4.0 REFERENCES
1. Hadley, C, "Nonlinear and Dynamic Programming," Addison-Wesley,
Reading, Mass. (1964).
2. Kuhn, H. W. , and Tucker, A. W. , "Nonlinear Programming", Proceedings
of the Second Berkley Symposium on Mathematical Statistics and
Probability, J. Neyman, Ed., U. of California Press, Berkeley,
Calif. (1951).
3. Carr, C. R., and C. W. Howe, "Quantitative Decision Procedures in
Management and Economics", McGraw-Hill, New York (1964).
4. Wilds, D. J., and C. S. Beightler, "Optimization Theory", Prentice-
Hall, Englewood Cliffs, New Jersey (1964).
5. Smith, H. V., "A Process Optimization Program for Nonlinear Systems:
POP-II", POP-II 7090 H9IBM0021, IBM' Share General Program Library,
IBM Corp., Houston, Texas 77025.
6. Baumol, W. J., "Economic Theory and Operations Analysis", Prentice-
Hall, Englewood Cliffs, N. J. (1961).
7. Fan, L. T. , and Associates, "Analysis and Optimization of a Multi-
effect, Multistage Flash Evaporation System," Special Report No. 74,
Kansas Engineering Experiment Station, Manhattan, Kansas.
8. Badger, W. L. and Associates, "Critical Review of Literature on
Formation and Prevention of Scale", Office of Saline Water Research
and Development Progress Report No. 25, U. S. Department of the
Interior, Washington 25, D.C. (1959).
9. Detman, R. F. , Chem. Eng. Progr. 63, 80-89 (1967).
52
5.0 NOMENCLATURE
ai A constant in several equations.
eij As used in equations (15) and (17), a constant
which denotes the coefficient of the i th row
and the j th column of a system of linear
equations. As used in equation (26) it repre-
sents the partial derivative of the i th
equality constraint with respect to the j th
independent variable.
^0 Keat transfer area of the brine heater, sq. ft.
Al Heat transfer area of the first effect, sq. ft.
A2 --eat transfer area of the second effect, sq. ft.
A3 H*at transfer area of the third effect, sq. ft.
bi A constant which is first introduced in the
equality constraint, equation (lb).
Bl A constant used in the equations which calculate
the horsepower requirements of the recycle pumps,
H.?.-hr./lb.
ce Unit power cost, $/H.P.
(Cf)A Concentration of the flashing brine stream flowing
into the first effect, wt, %.
(-f )c Concentration of the flashing brine stream flowing
out of the first effect, wt, %.
<Cf>D
<cf)F
<Cf>G
<cf>K
Cij
co
S
cs
CB
CC '
C^,- :
GH ,
-
■ CHT I
CJ c
°w l
-
Goncentratior. of
the flashing
brine stream
flowing into the
second effect, wt. %,
Concentration of
the flashing
brine stream
flowing from the
second effect, wt. %,
Concentration of
the flashing
brine stream
flowing into the
third effect,
wt. %.
Concentration of
the f lashing
brine stream
flowing from the
third effect,
wt. %.
The coefficient of the quadratic tern of
equation (16)
Concentration of
seawater fed
to the MEMS
Plant, wt. %.
Heat capacity of
flashing brine solution, 3tu/
lb. °P.
Unit steam cost,
?/lb.
Unit cost of brir
e heater heat
transfer area,
S/sq. ft.
Unit cooling water cost, $/lb.
Equal to Y CH
Unit cost of heat
transfer are
a in an effect,
?/sq. ft.
Equal to (Ce+fCj
)
Equal to YC-,
Capital cost per
horsepower, $
/:-:.?.
Unit feed water cost, $/lb.
54
CONST(I) The i th constant in the ^rray of constants
which can be used with POP-II
J*- - rrie name of the matrix printed out at the end of
a POP-II run which can be used to make an
incremental sensativity analysis near the point
at which the run ends.
ei Tke value of the i th equality constraint when It
evaluated at a feasible point
El Total cost associated with the seawater fed to
the MEMS plant, 5 /1000 gal. fresh water
E2 Total cost of brine heater steam, S/1000 gal.
fresh water
E3 Total cost of cooling water, S/1000 gal. fresh
water
u3 ; Total cost of heat transfer area for the a th effect
S/1000 gal. fresh water
Total cost of the outer shell of the MEMS plant
based on 1000 gal. fresh water produced,
3/1000 gal. fresh water
^x' The objective function
-Jeed rate to the MEMS plant, lbs/hr.
S±(.x) The i th equality or inequality constraint
h^ A function used in the discussion about the
quadratic form
\(x' The *• th equality constraint (used in the
explanation of POP-II)
-6
-
55
(HP),
Horsepower required for the recycle pump of the
first effect, H.?.
(HP)2
Horsepower required for the recycle pump of the
second effect, H.P.
(H?)3
Horsepower required for the recycle pump of the
third effect, H.P.
L
The Lagrangian function used in the development
of the Kuhn- Tucker conditions.
(L)
A
Flow rate of lashing brine stream into the
first effect, Ibs./hr.
U)c
Flow rate of lashing brine stream from the
first effect, lbs./hr.
(L)D
Flow rate of lashing brine stream into the
second effect, lbs./hr.
(L)?
Flow rate of flashing brine stream from the second
effect, lbs./hr.
a>G
Flow rate of flashing brine stream into the
third effect, lbs./hr.
<L)K
Flow rate of flashing brine stream from the
third effect, lbs/hr.
m
s
Brine heater steam flow rate, lbs/hr
^s
Heat transfer rate ir. the brine heater, BTU/hr.
-_
Recycle brine flow rate in the first effect,
lbs. /hr .
56
R2
Recycle brine flow rate in the second effect,
lbs./hr.
Recycle brine flow rate in the third effect,
lbs. /hr .
H
Cooling water flow rate in the third effect,
lbs./hr.
s
Value of the objective function of an
optimization problem
(Xc)r
Temperature of distillate water from effect one,
°P
Cfc),
Temperature of distillate water from effect two,
Temperature of distillate water from effect
(Tc)
K
three, °F
* A
Temperature of flashing brine stream into
effect one, °F
C«£>q
Temperature of flashing brine stream from effect
one, °F
(Tf)F
Temperature of flashing brine stream from
TOO, °?
<*f>K
Temperature of flashing brine stream from
effect three, °'S
<Va
Temperature of combined recycle and feed streams
into the brine heater, °F
(Tj>c
Temperature of combined recycle and feed streams
into effect one, °? ■
„
(tj),
Temperature of combined recycle and feed streams
into effect two, P
<VK
Temperature of combined recycle and feed streams
into effect three, F
-s
Temperature of brine heater steam, °F
(At)0
Temperature differences used for heat transfer
in brine heater, °?
CAt)x
Temperature difference used for heat transfer
in effect one, °?
(At)2
Temperature difference used for heat transfer in
effect two, °?
(At)3
Temperature difference used for heat transfer in
effect three, °f
ui
Slack variable used in development of Kuhn-
Tucker Conditions
U0
Overall heat transfer coefficient for the brine
heater, Btu/hr-f t2_°j?
°1
Overall heat transfer coefficient for effect one,
Btu/hr-ft -°F
U2
Overall heat transfer coefficient for effect two,
Btu/hr-ft2-°y
s
Overall heat transfer coefficient for effect
three, Btu/hr-f t2-°x''
Ul
Equal to ^
*1
Kate of distillate water production in effect one,
Ibs./hr.
£*n
Rate of distillate water production in effect
two, los./.'.r.
... te of cistillate water production in effect
three, lbs./hr.
Total rate of distillate water production in
the MEMS plant, lbs./hr.
x An abbreviated notation which stands for the
point (x,, X2 , ••••, xs) in s-space.
xQ A feasible point (xLq, x?q, , xgQ)
X(I) The i th independent variable (used in POP
programming)
y« The i th dependent variable
yiCx.-r §x) The i th dependent variable approximated as a
linear function of £x
Y(l) The i th dependent variable (used in POP
programming )
Y(l) The objective function (used in POP programming)
Greek Letters
( d. } The average boiling point elevation in the n th
r.,av
effect
Latent heat of vaporization of flashing brine
stream in an effect, otu/lb.
Lagrange multiplier
X s Latent heat of steam used in brine heater,
Btu/lb.
Capitalization charge
X
r
60
... .■ NQIX I
IKPUX' -J-.:-. POEMS P08 PROPOSED PUiUHE ....... O.N r&E -■....- VHQC ...
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L . Smith, K. V. , , , ! near
as: PUP-II," POP-II 7090 H9Iba0021, laii Shs ieneral
Pro; >rary, IBM Corp., Houston, rexas 770.25.
72
Ar INDIX 11
POI -XI 3 ... __ OUTPUT
Ihe computer o t for Jun 1 Is in this Ape
.... .. _. ..:. optimum cas - for - — ork. Also, the loop-to-
loop computer outputs for Suns 2 and 3 are given in tnis
..dix.
73
RUN Ml FR 1 PROBLEM NUMBER ! I OOP NUMBER 0
INPUT r y* I /» FOR PLANT IPTIKIZATION PROGRAM, PAGE I
12/12/66
***** KU
- o u; c - o - o - o - o 3
-0 I -0 1 -0 -i) -0
12 12 66 -0 I -0 -0 -0 9 1 I I -0
-0 5 _-0 -0 -o -o -0 -0 -0. -0 I -0 -0
«.** CS1 - CPTIMIZER CCNSTANJS *****_**J*
O.i- 994E-04 0.099999994E-04 0.5G0000000E 03
- 0 . 0 - 1999 9 9 9 961 C L - 0 .
0.9999999R5E-0 2
-0.
0-499999993E-0 I
0.50Q000000E 00
0.499999993C-02
o. n ooooooo or no
****** c. c h s r - mxcbl 0:1. i r ic ii.-k rs****
0.17 64 ' I' II 0 5 0.25 00 000 00E 0 7 0.575899996E 08
0 .20 /'.''• )9 n 09
0.598399997E 04 0.239699997E 06
0.375999995F 06
74
IvLN MNEI R l
INPU r I. Al /< FOR
: I ' ' ■ Bl k
P L ^ i\ I OPTIMZATIGN
3 LOOP NUMBER
PROGRAM, PAGI 2
12/12/66
: ■ ■ ■ [NCI 'l I LEJ •■ * *
V AR I A E L I I • ■ II
. I
W2
V- !
R I
R2
MS
CSI 1C
TS
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>**** CI It iv Dl N I VAP I ^ 0 L E S
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CW
AO
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204
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10 3
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152
101
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NUK8FR '5
UN MARY
2 LOOP
12/12/66
OF INDIVIDUAL
5 LOOP 4
I OPS
I OP
LOOP
1 nop
LOOP 8
!
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.02,
21
907
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02. J2 0M
2829..8S0
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NUMBER 18
SUMMARY
11 LOOP
12/12/66
OF INDIVIDUAL
12 LGGP 13
LOCI'S
LOOP
14
32 0 26 02.
8.9 Q 282SL-
79 0 29 0 7.
0 0 0 5 9 8 0 0 .
0 0 0 4 2 4 0 0 .
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TCTAL CCST
PRODUCT ION
FEED
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TSF2C
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T C 1 G
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80
RUN ISLfEfR 1
I
INDEPENCEM V.ARJLMLES
:.a2iJ
2 9 0 7
R 1 3 9 8 0 0
R 2 ^2400
R 3 4 6 0 0 0
MS 531
C S F 3 C 0
rs 274
T S F 1 I 2 5 0
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PROBLEM NUMBER 3
NPUT LOUP 19
LOOP NUMBER 25
SUMMARY OF
LOOP 2 0 LOOP 21
12/12/66
INDIVIDUAL
LCOP 22
I I OPS
LCOP
23
LOOP 24
LOOP 25
.32 0
. o' 0
. 79 0
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. 0 C 0 4
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LOOP 1
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JUM 3ER 9
SUMMARY
2 LOOP
12/ 12/66
OF INDIVIDUAL
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LOOP /
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85
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uuk NUMBER 2 I LEN NUMBER 1 LOOP NUMBER LB 12/12/66
RUIV fvLMMv * SUMMARY OF INDIVIDUAL LOOPS
INPUT M:no l0 LOOP LI LOOP 12 LOOP 13 LOOP 14 LOOP 15 LOOP 16 LOOP 17 LOl !'
INCEPENC1KT.-V./UUAL1LFS ■ , ' - ^ 2231.164 2231.164 2231.164 2231.164 2231.164 223l7l64 2?3l7lM
2426 274:242^276 2426.276 .-,26.276 2.426.276 2426.276 2426.27.6 2426,276
3682J55'3 -682.530 682.530 J682.530 3682.310 3682.3 50 3682.530 3682.530 0 >682.
; d] %2Q U23 520 34123.520 34123.520 34123.520 34123.520 34123.520 34123.520 34123.520 ^123.520
*\ 5^'70036352 700 6352.700 36332.700 36332.700 36352.700 36332.700 36332.700 36352.700 '6352.700
p' M43S.25 0 39439.230 3-e39.230 3.9.435*. 25.0 39439.230 39_43Su25il I I '439.2
1*^6515 lltl* 510.266M 515.266M 510.266M 515.266M 510.266M 512.766M il2.766 510.2;
0 0 0 033 0.053 0.055 0.053 0.055 0.053 0.054 1 0.055M
274 390U 274.390U 274.390U 274.390U 274.390U 274.390U 274.390U 274.390U
250.010U 250.010U 250.010U 250.010U 250.010U 25Q.0.10U 250.010U 25Q...01.0U
5 000 .590U 84.990U 84.990U 84.990U 84.990U 84.990U 84.990U >.990U 84.990U
K3
IA S i 55 . 266
I , il
,S 274.4 00
, | i 25il._Q0.Q
i
,:,:' s" 0 25,3 0.285 0.290 0.289 0.290 0.289 0.290 0.289 CL.2S.9 0.2
' ' c 833^999 833^9 7 CU 8339.970U 3339.970U 8339. 970U 8339.970U 8339.970U 8339. 970U 8339. 970U 8339. 97 0U
l\ 1U,LN ,7 ,23 22 663 24/^9.554 23174.004 24675.825 23121,176 24605.854 23546.623 2Z6JL6. 92J
710 652 210 '26 211.369 210.292 211.326 210.254 211.779 210.553 209.894 .553
' '78 !o \\\.lll 167.022 169.185 166.-44 160. OH 1*7.570 I '•;/
]\\ \ 10 2 752 1 Clio 2 08.409 101.373 105.249 101.233 105.073 102.354 99.882 102.354
:i 2oJ:c3o 111:111 W.o^ ^.^ 207.973 209.010 20a. 279 ^...^ ^
,/,:, s',1 \tU 832 167.050 164.779 166.960 164.700 166.861 165.331 163.939 163.331
; : I^Jj \^\lll 59.373 103.302 99.230 103.1,4 100.368 < > 10.
0 47E 06 0.48F 06 0.47E 06 0.48F06 0.47E06 0.48H06 0.47E06 0.48E06 0.48E 06 0.47E06
fA n 23 5CQ 32.863 32.712 32.853 32,707 32,354 32.702 32.7'>3 L6 /,'
104.;: , 840!270 K85.101 839.290 883.760 833.227 882.289 853.464
ARE* 1202.444 363.537 1012.147 362.923 1010.752 061.872 1009.224 97&.4S7 *
AREA - 2 22.0971673.786 1740.128 1672.670 1738,610 1671.457 1736.737 I • ' 6
?CCL J 145.224 6767.176 -736.742 7053.236 -482.606 7367.931 199.560 ^32.65110493.423
CCS >42 0.041 0.044 0.041 0.044 0.041 0.043 0.0
0.11', 0.175 0.128 0,179 0,173 0.129 0.12.8 0 .J
; 0.005 0,003 0.005 0,003 0.005 0.005 0.005
CCS1 0.003 0,003 0.003 0.003 0.003 0.00 5 0.003 0.003
LST o.oi 0,007 0.002 ).002 0.007 0,00, 0,002 0.002
, | | ,. -O.00Q o,o,4 -0.000 0.004
, CCS! 0. 0.001 ' 101 0.001 0.001 0,0)1 0.001
0.021 0.020 0.1
A?" CCS1 0.1] 0.023 0.074 0.07 3 0.024 0.03 3
A3 CCST 0.051 0.040 0.042 0.040 0.042 0.040
0.04 7 )41 '.04 0 0.04.1
87
iujn m;
IBLEN NUMBER 1
! !
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INPUT LC.0 P 1
P.I m:i NT VJR [AE1 l S
2231.164 2 2
242 6*276 24
3'6 82.5 5 9 3 6
14123.52.0341
1 6 3 5 2.700363
3343.9,. 23:iO_3£4
LOOP
LOOP
NUMBER 25
SUMMARY
2 0 I. COP
12/ 17/66
n INDIVIDUAL
2 1 LOUP 22
LOOPS
I i OP
2 3
LOOP 24
I HOP 25
)C
1 1
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'74. 40 0U
i . 0 0 0 U
! /-'• . 400U
>5 0.oooy
l1 .01 ou
85. 0] OU
I
fl > ;.l .000 P '''•?
i ) i •>■. .;,/. .'.; rust*.
194.909
1 S2.256
I _ ,.. ,
l ',1 . ? 1
0. ?4P 06 0 . 7'i
! ! 0 ft . 4 2 0
' i
S3,
'1 .
b.
9.,
o.
Q
n
90
APPENDIX III
0ESC3IPTION OF CQMPU'IEH PHQ3IUM - riQDEL
Computer program MODEL may be used for simulation studies
of the MEMS process. Furthermore, the program may be used to
obtain feasible starting points for SUBROUTINE MQOEL which are
near the boundary of the feasible constraint region*. ■
Table III-l describes the variables used in MODEL and the
SOBrlQUTIJJS MODEL used in the optimization study. | a logic dia-
gram for program MODEL is given in Figure III-l. Program
MODEL is listed in Table III-2. *
91
Table III.
Explanation of ComDuted Program Variables Used
In Program MODEL and subroutine; MODEL
Symbol
Explanation
A
Constant used as multiplier to change size
of independent variables
AA
The first term of the Lagnangian Polynomial
AS
The second term of the Lagnangian Polynomial
AC
The third term of the Lagnangian Polynomial
AD
The fourth term of the Lagnangian
Polynomial
AL
Latent heat of vaporization of distillate
water in each effect
ALS •
Latent heat of brine heater steam
ALII
Flashing brine stream flow rate into effect
one
AL2I
Flashing brine stream flow rate into effect
two
AL3I
Flashing brine stream flow rate into
effect three
ALIO
Flashing brine stream flow rate out of
effect one
AL20
Flashing brine stream flow rate out of
effect two
AL30
Flashing brine stream flow rate out of
effect three
AN1
Number of stages in effect one
92
Table 3.
(Continued )
Symbol
Explanation
AN2
Number of stages in effect two
AN3
Number of stages in effect three
Al
Average boiling point elevation in effect
one
A2
Average boiling point elevation in effect
two
A3
Average boiling point elevation in effect
three
3
A constant used in the equations which
calculate horsepower
CF
Salinity of feed stream
CONST(l)
through
co;nst(7)
Cost coefficients used in the terms of the
process objective function
CF
Heat capacity of flashing brine solution
CSF1I
Salinity of flashing brine stream fed to
effect one
CSF2I
Salinity of flashing brine stream fed to
.
effect two
CSF3I
Salinity of flashing brine stream fed to
effect three
CSF10
Salinity of flashing brine stream discharged
from effect one
CSF10
Salinity of flashing brine stream discharged
from effect two
CSF10
Salinity of flashing brine stream discharged
from effect three
93
Table 3.
(Continued)
Symbol
Explanation
DTO
Temperature
difference available for heat
transfer in
brine
DTI
Temperature
difference available for heat
transfer in
effect one
DT2
Temperature
difference available for heat
transfer in
effect two
DT3
Temperature
difference available for heat
transfer in
effect three
HP1
Horsepower of the recycle pump of effect one
HP2
Horsepower of the recycle pump of effect two
H?3 •
Horsepower of the recycle pump of effect
three
l.J.K
Subscripts and counters used in program MODEL
TLFOI
Temperature
indicated by (Tj ), on Figure 5
TLF1I
Temperature
indicated by (Tj )q on Figure 5
TLF21
Temperature
indicated by (Tj )« on Figure 5
00, Ul
Overall heat transfer coefficients used in
the brine heater and effects one, two and
U2, U3
three respectively
XA
230° F
XS
250°F
XC
270°F
XD
290°F
X(l)
Distillate produced in effect one
94
Table 3.
(Continued)
Symbol
Explanation
X(2)
Distillate produced in effect two
X(3)
Distillate produced in effect three
X(4)
Recycle used in effect one
X(5)
Recycle used in effect two
X(6)
Recycle used in effect three
X(7)
Srine heater steam consumption rate
X(8)
Salinity of flashing brine stream dis-
charged from the MEMS plant
X(9)
Brine heater steam temperature
X(10)
Temperature of flashing brine stream fed
to effect one
X(ll)
Temperature of seawater fed to MEMS plant
YA
Enthalpy of saturated steam at temperature
XA
YB
Enthalpy of saturated steam at temperature
XB
YC
Enthalpy of saturated steam at temperature
XC
YD
Enthalpy of saturated steam at temperature
XD
Y(l)
Total cost per 1000 gal. distillate water
produced
Y(2)
Total distillate water production rate
Y(3)
Seawater feed rate
Y(4)
_
Temperature of flashing brine stream leav-
ing effect one
95
Table 3.
(Continued)
Symbol
Explanation
Y(5)
Temperature of flashing brine stream
leaving effect two
Y(6)
Temperature of flashing brine stream
leaving effect three
Y(7)
Temperature of distillate stream flowing
from effect one
Y(8)
Temperature of distillate stream flowing
from effect two
Y(9)
Temperature of distillate stream flowing
from effect three
Y(10)
Brine heater heat transfer rate
Y(U)
Brine heater heat transfer area
Y(12)
Effect one heat transfer area
Y(13)
Effect two heat transfer area
Y(14)
Effect three heat transfer area
Y(15)
Cooling water flow rate
Y(16)
Cost for feed pretreatment and pumping
Y(17)
Cost of brine heater steam
Y(18)
Cost of recycling brine in effect one
Y(19)
Cost of recycling brine in effect two
Y(20)
Cost of recycling brine in effect three
Y(21)
Cooling water cost
Y(22)
Cost of heat transfer area in brine heater
Y(23)
Cost of heat transfer area in effect one
96
T.nble 3. (Continued)
Symbol Explanation
Y(24) Cost of heat transfer area in effect two
Y(25) Cost of heat transfer area in effect
three
Z Dummy variable used in Lagrange polynamial
97
98
Table III-2. Coraouter program MODEL
HCNSS JOS
MCNSS COMT 8 MINUTES. 9 PAGES.
MCNSS ASGN MJR.12
HCNSS ASGN MGC16
MCNSS MODE GC
MCNSS EXEO FORTRAN. ,.,., , MODEL
DIMENSICNX(12) .YI25 1 .CCNSTI7)
C
SIMULATION PROGRAM MEMS SEAWATER DISTILLATION PLANT
1 FCRMATi2CX,F 13.6)
2 FCRMAT(20X,E13.6)
3 FORMAT (5E13. 6)
4 FCRMAT16F1G.0)
READ(1.4) (CONST (I ) .1=1,7)
6 CONTINUE
RFADd .DA
WRITFI 3,2) A
K = l
IFiA.NE.l. 1GOTC13
J=26
GCTC14
13 J-l
14 READ( 1 ,1) (X( I ) .1 = 1 .11)
15 CONTINUE
*w
SIMULATION PROGRAM MEMS SEAWATER DISTILLATION PLANT
C
i *#*•*** *•»•****#**#*******■*#**#**#*******♦# ******♦***#**•**»**
c
CONSTANTS USED IN PERFORMANCE EQUATIONS
■ AL=100C.
8=17. 01723
ANI=?3.
AN2=>?3.
AN3=22«
CF=0.035
CP=1.
XA = 23C
X8=250.
XC=27C.
XD=290.
YA=953.8
YB=945.5
YC=931.8
YD=9]7.5
U0=510.
Ul=510.
U2=510.
U 3 = 5 1 C .
C
ft**************************************** **«#*******#*****
C
PERFORMANCE EOUATIONS
Y(2)=X(1)+X(2)+X(3)
Y!3)=Y
ALlCiY
Ai_2C = A
A13C=A
A|_II=Y
AL2I*Y
AL3I=Y
csfic=
CSF2C=
CSF1I"
CSF2I=
CSF3I
Y(4)=X
Y(5)=Y
Y(6)=Y
Ai=1.0
A2=1.0
A3=0.3
Y(7)=Y
Y (8)=Y
Y (9)=Y
2=X(9)
LAGRAN
AA-YA*
AS=YB*
AC=YC*
AD«YD*
ALS=AA
*■#*#**
YQC} =
TLFOI«
T|_F1I =
1/ iCP*{
TLF2I=
1/ [CP*<
Y(15)=
1/<C?*(
DTC=X(
DT1=X(
DT2=Yt
DT3=YC
Y<11)=
Y(12)=
Y<13)=
Y(14)=
HP1»X<
1-EXP1-
HP2=X!
1-EXPi-
f 2) /( 1
(3)-X(
L10-XI
L2C-X(
(3) +X(
(3)+X<
(3) +X<
CF*Y(3
CF*Y(3
CSF1C*
CSF2C*
X( 8)*(
( 10 )-(
!4)-<A
(5)-< A
100+ (C
:;75 + <C
201+IC
( 4 ) -A 1
(51-A2
(6) -A3
GIAN P
(Z-XB)
(2-XA)
(Z-XA)
(Z-XA)
+ A 3t AC
***■*-**
X(7)*A
XQO)-
(CP*( <
V(3)-rX
<CP*( (
YOJ+X
tYC 10)
Y(6 )-X
91-C.5
ICJ-TL
41-TLF
5 1-TLF
Y( 10)/
X! 1 )#A
X( 2)*A
X(3)*A
4)*B*(
L/(0.
5)*B*(
AL/IO.
-CF/X1S) )
X(
X(
AL1
AL2
L1C
L2
30+
/CP
CP)
CP)
11+
21 +
31
X(2)
+ X<4) 1/AL1I
+X( 5) 1/AL2I
X(6) 1/AL3I
)*ALCG(CSF1C/CSF1I )
*ALCG(CSF2C/CSF2I 1
*ALCG(X(8 1/CSF3I )
CSF1C)/(2.*0.0300 )
CSF2C>/(2.*0.0347)
X<8) 1/(2. *0. 0315)
LYNCMIAL
*(Z-XC)*(Z-XD) /( (XA-XB)*(XA-XC)*(XA-XD) )
*(Z-XC)*(Z-X0) /( (XB-XA)*(X3-XC)*(XB-XD) )
*<Z-XS)* (Z-XD) /( (XC-XA)*(XC-XB)*(XC-XD) )
*(Z-X3)* (Z-XC)/( <XD-XA)*(XD-X3)*(X[>-XC) )
D
*#**#*#*######* ######**Hf •&■«■■«■*#*■*■****■***■*#■*****
LS
Y( 10>/(CP*(Y(3)+X(4> ) )
AL1C+XI5 ) )*Y(4)+X(1)*Y<7) ) -Y ( 10) )
(5) ! )
AL2C+XI6 ) )*Y!5)+tX( D+XI2 ) ) *Y < 8 ) ) -Y ( 10) )
! 6 ) ) )
+Y!3)*CP*X( 11)-(CP*(AL3C*Y(6)+Y(2)*Y(9) ) ) )
(11 ) ) )
*(X(10)+TLF0I )
F0I-A1-(X(10)-Y(4) >/<2.*ANl.)
1I-A2-(Y(4)-Y< 5)1/(2. *AN2)
2I-A3-(Y(5 1-Y(6) )/(2.*AN3l
(D70*L>0)
L/(DT1*U1)
L/(DT2*U2)
L/(DT3*U3)
HXP1-AL/ (0.1104*(X(1C)+460.11)
1104*(Y( 4J+460. ) ) ) 1
EXP (-AL/I0. 1104* < Y(4)+460.) ) 1
1104*(Y( 51+460. 1 1 1 1
100
Sable
III-2. (Con't)
hP3=X(6)*B*(EXP(-AL/(0.1104*(Y<5)+460.)))
l-EXPl-AL/ (0.1104*1 Yt 6 1+460. ) ) ) )
c
*#■»#»*» it* ********************************** *************** 1
c
COST EQUATIONS
Y( 16)=C0NST( 1)*Y(3) /l.E+10
Yil7)=CONST{2)*X(7)/l.E+10
Y ( 18 >=CONST( 3 )*HP1/ l.E+10
Y! 19)=CONST(3)*HP2/1.E+10
Y(20)=CONST(3)*HP3/1.E+10
Y(21 )=C0NST(4)*Y( 15) /l.E+10
Y(22)=CONJST(6)*Y( 11 ) /l.E+10
Y(23)=CONST( 5)*Y(12) /l.E+10
Y(2t)=CCNST(5)*Y(13)/l.E+10
Y(25)=CONST( 5 1*Y( 14) /l.E+10
Yd >=Y( 16 )+Y( 17)+Y( 18)+Y(19)+Y(20)+Y(21 )
1 + Y ( 22 ) +Y ( 2 3 ) +Y ( 24 ) +Y ( 25 1 +CONST ( 71 /l.E+10
IF(K-2)91,110,91
91
DC93I=1,3
IF(X( I ) ) 104,104,94
93
CONTINUE
04
DC96I=1,15 .
IF! Y( I ) ) 104,104,96
96
CONTINUE
97
DC99I=1 ,8
XI I )=A*X( I )
99
CONTINUE
X (31=8340. -X(2)-X( 1)
IFI25-J) 104,102,102
102
J=J + 1
GOT015
104
D0106I=1 ,8
X ( I )=X( I ) /A
106
CONTINUE
X(3)=8340.-X(2)-X( 1)
K = 2
G0T015
no
WRITE (3, 2) (CONST! I ) , 1 = 1 ,7)
WRITE (3,1) <X( I 1 ,1 = 1 ,11 )
WRITEI3.2) (Y( I ) ,1=1,25)
WRITF(3,3)AL10,AL20,AL30
WRIT-:(3,3)AL1I ,AL2I ,AL3I
WRITE(3,3)CSF10,CSF20
WRITE(3,3)CSF1I .CSF2I .CSF3I
WR:TE(3,3)A1,A2.A3,ALS
WRITE(3,3)TLF0I ,TLF1 I ,TLF2I
WPITF(3.3)DTO,OT1,OT2,DT3
WRITE(3»3)HPl,HP2,HP3
G0T06
END
101
Table III-2. (Con't)
MONSS
EXEQ
CALL
LiNKLCAD
MODEL
MONSS
EXEQ
MODEL. MJB
DATA
•
17650.0
208000000.
2500000.
57590000.0
1.
2602.32
2829.86
2907.79
36961.
39376.
42719.
510.
.065
274. A
250.
85.
5984.0
239700.0
376000.0
PART II.
OPTIMIZATION OP A MULTI-STAG 3 AERATED LAGOON BY THE
DISCRETE MAXIMUM PBINCIPLE
102
1.0 INTRODUCTION
Pollution abatement has become a subject of increasing concern both
to the technical expert in this area and to the common citizen. Sources
of pollution range from plant life to large manufacturing complexes. The
tools for combatting pollution range from plant life to man made pollution
control devices.
In most industries it is not economically feasible to prevent waste
formation. As a result these industries must concentrate on destroying
or disposing of wastes once they have been formed. The chemical process
industry and the petroleum industry serve as examples of this type of
industry.
One means of measuring the strength of a pollutant is to determine
the total amount of oxygen that is required to reduce the pollutant to a
harmless state. Some pollutants react directly with oxygen. The rates of
reaction of these pollutants are usually quite rapid. Other pollutants are
degraded by bacteria or microorganisms in an oxygen enriched environment.
Microorganism feeding processes usually occur at a considerably slower
rate than direct oxidation reactions.
This work deals with the modeling and optimization of a system for
treating petroleum refinery waste water. The system or process considered
is an aerated lagoon. In general the process consists of introducing
waste water solutions into a large body of water wherein they are degraded
sufficiently to allow the effluent stream to be discharged from the refinery.
The lagoon model used in this paper is partially patterned after the
aeration basins of American Oil Company's Sugar Creek Refinery which is
located in Sugar Creek Missouri. Several of the constants used in the model
103
were calculated from operating data from this particular aerated lagoon.
A detailed description of the Sugar Creek aerated lagoon, its operating
characteristics, and the nature of the wastes it treats is given by
Stroud, Sorg, and Lamkin (1) and Burkhead (2). A short description of
the Sugar Creek Lagoon is given here. Waste water is first introduced into
a pond which has an oil skimmer trough at its outlet for removing any
surface oil slick which forms. The effluent from the oil skimming pond
is introduced by gravity flow evenly across the inlet of the first aeration
basin. The first aeration basin is approximately 713 feet in length by 120
feet in width with a depth of 10 feet.
The effluent from the first basin is introduced by gravity flow into
the second aeration basin. This basin is approximately 700 feet in length
by 120 feet in width with a depth of 10 feet. The effluent from this
basin is introduced by gravity flow into the second aeration basin. This
basin is approximately 700 feet in length by 120 feet in width with a depth
of 10 feet. The effluent from this basin is introduced into a settling
basin before it is discharged from the refinery.
In the first aeration basin, three mechanical surface aerators which
are driven by 60 horsepower electric motors are mounted on steel platforms.
The platforms are positioned down the basin center line which is parallel
to the over all direction of waste water flow. The second aeration basin
has three 15 horsepower surface aerators positioned down its center line.
Recently four more 20 horsepower aerators have been added to the first
basin. These aerators are currently located along a line which is adjacent
to and parallel to the inlet baffle. However, these aerators are not on
immobile platforms as are the other aerators. The aerators oxygenate and
keep the waste water in the basins mixed.
104
The optimization goal is to determine the size of lagoon and the sizes
and positions of the aerators required to achieve a specified waste water
conversion such that the total cost of the aerated lagoon is minimized.
The lagoon is considered to be a stagewise process. The optimum
aerator horsepower and the lagoon volume required at each stage are de-
termined by a discrete version of Pontryagin's maximum principle as
elucidated by Fan and Wang (3).
2.0 DESCRIPTION OF THE LAGOON MODEL
In this section the kinetic, flow, and economic models of the process
are developed.
2.1 Ideal Component Assumption and Kinetic Model.
In the lagoon model a real waste solution which may have many types
of impurities is assumed to be composed of wastes which fit into one of
three categories. The impurities in each category are further assumed to
act as a single idealized component.
The first idealized impurity is a mixture of organic and inorganic
impurities which can be degraded to harmless products if it remains in the
presence of degrading aerobic bacteria in an oxygenated environment for
a sufficient length of time. This component is commonly measured in terms
of biological oxygen demand (BOD) which is defined by Eckenfelder and
O'Connor (4) as "... that quantity of oxygen required during the stabilization
of decomposable organic matter and oxidizable inorganic matter by aerobic
biological action."
The kinetic expression which will be used to describe the rate of
destruction of the BOD component will now be derived.
105
Inspection of equations given by Grieves, Milbury, and Pipes (5) shows
that the rate of formation of aerobic microorganisms, r„ , can be written as
B
dCB kMCBx
'■B dt (K + x) (1)
where,
C = aerobic microorganism concentration
M
k « growth rate constant
x » BOD concentration
K = Miachaelis-Menton constant
If it is assumed that the organism population increase is proportional
to the increase in BOD concentration, that is
dC - -Y dx (2)
or
^=-Y^ (3)
dt dt w;
or
rB " "Y rx («
where
Y = a yield constant (lbs. microorganisms formed per lb.
BOD consumed)
-r ■ BOD reduction rate
106
the BOD reduction rate can be written by combining Equations (1) and
(4) as
M
. k s *
rx = Y(K+x)
(5)
To make the eventual optimization problem easier to solve, it is
assumed that Equation (5) may be approximated by a pseudo first-order
kinetic expression in terms of BOD. The approximation requires that
M
the quotient, k C /(Y(K-toc)), remain nearly constant throughout the portion
of the lagoon which decomposes BOD, that is
-rx = k x (6)
where
M
1 k C
k " B ~* constant
Y(K-hx)
The validity of the assumption made to obtain Equation (6) will be
verified after the optimization study has been completed.
The second idealized waste component is assumed to be a mixture of
organic and inorganic compounds which pass through the lagoon without
being directly oxidized or acted upon by microorganisms. This component
does have an oxygen demand; however, neither direct oxidation nor attack
by microorganisms while the component is in the lagoon will reduce the oxygen
demand. Consequently, since the component's oxygen demand is not reduced
in passing through the lagoon, it is called the nondegradeable component.
The kinetics of decomposition of certain detergents are such that they are
effectively nondegradeable in an aerated lagoon.
The third idealized component is assumed to be a fast reacting in-
organic component with the following characteristics: It is oxidized
107
directly by oxygen in a relatively fast reaction. The reaction rate is
assumed to be so fast that it is limited by the rate of mass transfer
and the rate of addition of oxygen by the oxygenation equipment. The
rate of oxidation is primarily limited by the rate of diffusion of
oxygen into the waste solution. The primary example of this type of
component in refinery waste water is hydrogen sulfide.
The theoretical oxygen demand (TOD) exhibited by the three components,
that is, the sum of the oxygen demands of all three components, is of
primary interest in this work. The chemical oxygen demand (COD) includes
the biological oxygen demand and the oxygen demand of the fast reacting
inorganic component.
Further description of the lagoon model can be facilitated by referring
to Figure 1. The values of BOD, COD, and TOD are shown at various points
along the lagoon aeration basin in the figure. At the inlet of the aeration
section of the lagoon the TOD will include the oxygen demands of the BOD
component, the nondegradeable component, and the fast reacting inorganic
component. At the outlet of the inorganic reduction section, the TOD will
be equal to the sum of the oxygen demands of the BOD and nondegradeable
components, that is, in the lagoon model all of the directly oxidizable
inorganics are removed in a section at the beginning of the lagoon whose
volume should theoretically be determined by a diffusion limiting rate
equation. In practice the aerators for this section are set as close as
practically possible to the inlet.
In the lagoon model the inorganic reduction section is located at the
inlet because it is desired to have as large a portion of the lagoon as
possible for BOD reduction. This method of aeration is chosen because it
is assumed that no aerobic bacteria can survive in the oxygen deficient
108
_Q
Q Q
T-,<~>
O O
rUJ
(J H-
aj
Ol
c
o
u
Gj
q' <g
q ,<g
<D
cn
a Z
C/)
a
T"
-< —
JL
ai
cn
a —
c
o
o
■o
cc
a
o
m
(A
J.
c
a
cn
O
c
a
O r- u
Q
■o
cu
K
'ii 'no / 20 -sqj
PUOlUBQ. U36XXQ
109
waste water which contains even a small amount of fast reacting inorganic
component.
At the outlet of the aeration sections of the lagoon, the TOD will
be the sum of the oxygen demands of the nondegradeable component plus the
BOD of the outlet water.
2.2 BOD Material Balance
The sequential arrangement of aerators in the Sugar Creek lagoon
suggested that the BOD reduction section of the lagoon could be modeled
by a series of ideal backmix reactors,
A steady state BOD material balance about the nth ideal backmix
(completely mixed) reactor for the biodegradable component can be written
Qx11"1 - Qxu + r. Vn = 0 (7)
where Q is the volumetric feed rate and V is the reactor or stage volume.
The volume of the nth stage of the BOD reduction section can be calculated
by combining Equations (6) and (7) to obtain
., n-1 a,
Q(x - x ) , .
,1 n C8)
k x
2.3 Aerator Motor Size Equation
It is assumed that Equation (8) is valid as long as a certain minimum
oxygen concentration is maintained in each stage. The size of the electric
motor required for maintaining this minimum oxygen concentration is directly
proportional to the product of the waste water flow rate and the difference
in BOD between the stage's inlet and outlet streams, that is,
., n-1 n.
pn . 9<* ■»)
no
where
n
p « aerator electric motor size at stage n
Q ■ waste water flow rate through the lagoon
n-1
x - oxygen demand of waste solution flowing into stage n
x - oxygen demand of waste solution flowing out of stage n
Ro " oxy§en transfer rate constant
E = mechanical efficiency of aeration unit
2.4 Economic Model
In connection with designing an aerated lagoon, the sizes and locations
of the aerators, the number of aeration stages to be used, and the lagoon
volume should be determined by economic considerations.
The objective cost function used in this study takes into account both
the initial equipment costs and the operating costs for the life of the
equipment. A present worth objective function of the form used by Hwa (6)
is used to do this. The initial costs considered are the aerator motor
costs as a function of horsepower, the lagoon volume, and the costs of the
aerators and supporting platforms. Only one operating cost, the electric
power cost for aerator operation, is considered to have a significant effect
on selecting the equipment size and lagoon volume.
Bauman (7) has proposed the following relation which can be used to
obtain the initial cost of an explosion proof induction motor, (C n)., as
a function of the motor size, Pn,
(c,a), = /S(Pn)
"I '1 " /•"•* ' (10)
where
°^ - aL for l^p'i 20
^ = a[ for 20 < Pn±200
»
fi
« C, for
1 i Pn & 20
ft
i
» C, for
20 <■ Pn £-200
The symb
3 Is a.
i
t
and C1 are constants.
The
initi
ll cost of
the aeration basin is taken
as
the
sum of
land
real
estate cost, the cost of digging the
basin, and
the
cost of 1
aying
rock
sid
mg to
prevent erosion. The unit
land, digging,
anc
rock
lining
costs
are represented by
C_, C , and C respectively
The aeration basin
cost
for
each stage or aerator, (CT);>> is
calculated
by
Wj*J
2 - C2j2w
+ C3J2wh + C4(£w + 2h^ )
(11)
where
■
St •
length of aerator's basin
w =
width of
aerator's basin
h =
depth of
aerator's basin
To re
late
lagoon geometry
to aerator requirements, the ratio
of
J2:w
:h = 23:
12:1
(12)
is us
ed.
This
is approximately the ratio
of the dimensions
of the
Sugar
Creek
aeration
basis.
Equation
(12) y
ields
w -
11 0
23
(12a)
and
h =
Si
23
(12b)
By de
finition
Vn =
iwh
(12c)
112
Substitution of
Equations (12a) and (12b) into (12c) gives
vn =
-, i3 (124)
23"
Solving Equation (12d) for J£ gives
SL3 =
232 Vn
12 (12e)
2
S0lving Equation (12e) for £ gives
J22 -
f232"
2/3 n 5/<!
(Vn)2/3 (12f)
12
Substitution of
Equations (12a) and (12b) into Equation (11) gives
«&
" C2 I*' + C3 ~2 ^ + C4 M^ (U)
23
Substitution of
Equations (12e) and (12f) into Equation,' (13) and
simplifing gives
the aeration basin cost as a function of the basin
volume
(cn)
- C3Vn + C5 (vV2 (13a)
where
a2 "
2/3
and
s -
2I (^]2/3(6C2+ V
The aerator
turbine and supporting stand is assumed to have a constant
initial cost, C,
irregardless of the aerator motor size.
Since the e
.ectrical power cost is paid over a period of years, it is
necessary to estimate the present worth of this money. The operating cost
(CQ), is given by
113
(C*) -
i=M
C7 T YZ (1 + r)"1
PU (14)
where
C- = electrical power cost (assumed constant)
T ■ operating hours per year
M ■ life of system
r = annual interest rate
Equation (14) reduces to
(C») - C8 Pn (15)
where
M
i=l
The total cost for each stage of the organic degradation section of
the aeration basin is then written as the sum of the costs given previous
ly:
Gn - (c°)1 + (c»)2 + (eg) + c5
or
Gn = Cfi + C5(Vn) 2 + C3Vn + Cg Pn + /9(Fn)U (16)
The objective function for minimizing the cost of the organic de-
gradation section of the aeration basin is then defined as
n=N
S " T. a" (17)
n=l
where
S ■ total cost of BOD degradation section
114
total number of aerators or stages
3.0 PROCESS OPTIMIZATION
3.1 Development of the Performance Equations
To restate our objective, the purpose of this study is to determine
the volume per stage, the aerator motor size, and the number of stages
required for the BOD degradation section of an aeration basin which is
used to achieve a specified waste reduction while at the same time mini-
mizing the cost of the total system.
The maximum principle is used to perform the optimization because
it provides a systematic method for optimizing multi-stage processes.
In anticipation of the forms of the equations that may be used in
the maximum principle solution, some of the previous equations will be
combined and rearranged. Equation (8) can be rearranged to give
n-1
Xl , ,
*1 ■ ? (18)
1 + vn
Q
Substitution of Equation (18) into Equation (9) and rearranging gives
n-1
xl
R E ( * + J_ ) (19)
° k' V" Q
The electric motor size can be eliminated from the stage cost equation
by substituting Equation (19) into Equation (16)
n-1
Gn = C, + C. (Vn) 2 + C, Vn + C0 1
to j Jo — - — ~~
RuE t + )
k'V11 Q
115
ft
oL
n-l
*1
R E (-
k-vn + Q )
(20)
A new variable, x2, which is equal to the sum of the costs of all
the stages up to and including stage n is defined next as
where
x"-1 + Gn, x° = 0
(21)
n-l
the sum of the costs of all stages up to stage n
G - the cost of stage n
By defining the stage volume of the nth stage as, 0n, a change in
notation to conform with the notation given by Fan and Wang (3) can be
performed. It is hoped that the change in notation will make it easier
to follow through the algorithm which is used to solve this problem. In
terms of the new notation, the dependent variables which are called state
variables are denoted by the letter x. The superscript n on a state
variable indicates that it is the result of a decision made in the nth
stage. Independent or decision variables are denoted by the Greek letter
8. The superscript n on a decision variable indicates that it is a
decision which is made at the nth stage. The BOD reduction section of
the aerated lagoon process can be visualized in terms of the discrete
maximum principle by referring to Figure 2.
Given the values of the state variables entering a stage and the
values of the decision variables at that stage, the value of a state
variable leaving the stage is calculated by using its transformation
116
A
"
2
Id
SO
■H
1
a)
,
2
en
s
■H
>
1 «
>>
i-H
C3
A
:
o
■H
z
-P
i
I
ft
CD
O
o
.
I
o
c
o
c
o
X
CS
O
-P
1 °
c0 »J o> c
^
1 "#*I
o
vy
c
ai
A
T
<H
c
O
xl
i
i
i
c
o
-H
~> 03
o to
1
a> Q)
wx|
CO o
o
O ft
j
-p a
1 o.
o w
W „ 1 ! .
3 -H
O W .V (\i
•o >
j o w^
CD OJ
1 -*"~
fc t*Q
(A
as
o w
P3
1
03
.
Q)
"x1
,G CO
I
1 O
CNJ
O H a —
U
•H
t5
&4
1
o
X
■
1
equation (Equation (18) for x. and Equation (21) for x,). In general,
the transformation equations are the performance or constraint equations
of the process.
3.2 Statement of the Optimization Problem
The transformation and economic equations which are needed for the
optimization study can be rewritten in terms of the new notation and
summarized as follows. The first number to the right of each equation
gives the original equation number from which the equation is obtained.
The BOD material balance equation which is the transformation equation is
n-1
Xl
i + k en
T(x^S 6n)
(18) - (22)
where T(x. ; 8 ) is the symbolic notation used for the transformation
equation. The cost at each stage is
G(x^-1; 9n)
= C6 + C5 (8n) 2 + C3
+ C0
+ $
^
n-1
Xl
R E (-
•-JJ
1-1
R E (
k'8
-Li
Q
(20) - (23)
where
esi-
al
for
«U
»i
for
ft =
ci
for
fi-
<
for
1 < P < 20 horsepower
20 < Pn < 200 horsepower
I <_ P < 20 horsepower
20 < 20 _< 200 horsepower
lis
and the accumulated cost is
x^ = x*"1 + G(xf S 9n), 4 - 0 (21) - (24)
The objective function which is to be minimized is
(17) - (25)
The transformation equation, Equation (22), may be used to calculate
the BOD concentration at the outlet of the nth stage given the inlet BOD
concentration and the stage volume (the decision variable) at that stage.
Equation (23) gives the cost of the nth stage in terms of the inlet BOD
concentration and the stage volume. Equation (24) gives the total cost
of all stages up to and including the nth stage. The total cost of the
BOD reduction section of the lagoon is given by Equation (25) as the sum
of the costs for each stage of the BOD reduction section.
Equations (22) through (25) are the performance equations for a one-
dimensional multistage decision process which is defined by Fan and Wang
(3) as a " ... process which can be completely characterized for the purpose
of optimization by a single-state variable..." The stage volume is the
decision or state variable for the lagoon process.
3.3 Computational Procedure - The Discrete Maximum Principle
Fan and Wang (3) have derived a necessary but not sufficient recur-
rence relation which can be used to calculate the optimal values of state
and decision variables for many one-dimensional processes.
-„/ n-1 „n, n„. n .n+1.
3G(x ; e ) 3G(x ; 6 )
„n+lN „„, n n+lN
3
,n+l ST(xj; e"Ti) 300^; 8"r*)
„, n-1 , n. ,„, n ,n+l, . n . n
r(x ; -j ) 3i(x1; 6 ) 3x1 3x1
~ . „n+l
36 38
(26)
119
Equation (26) has the general form
gCx?"1; 9n) - f(x"; 9n+1) (26a)
where
, n-1 niiv , -/ n An+1*
g(x ; 9 ) and ffcCjl 9 )
are the left and right hand sides, respectively, of equation (26).
One solution procedure for a problem with fixed end points, i.e.,
known inlet and outlet BOD concentrations, is to start at the first
stage of the process and assume a value of the decision variable for
this stage, 8 . With Equation (22), the transformation equation, x, is
calculated. Next, Equation (26a) is used to calculate the optimum value
2
of the decision variable for the next stage, 8 , for the assumed value of
8 . Repeated application of the transformation equation and the optimum
recurrence equation for all of the stages of the process will yield a
value of the outlet BOD concentration, x,. If this value of x^ is equal
to the required value, the problem has been solved. If not, another
value of 8 is assumed and the above procedure is repeated.
The optimization calculation was carried out using the algorithm
described above. However, in carrying out the computations, the electric
motor size at each stage of the process, P , was used as the decision
variable, 8n, in place of the stage volume in order to simplify the
computations. The performance equations which were used to perform the
optimization are given in Appendix I. Theoretically, either the set of
performance equations with stage volume as the decision variable or the
set of performance equations given in Appendix I which have the stage
120
electric motor size as the decision variable will work equally well when
performing the optimization calculations. Computer program OPT which was
programmed to perform the optimization is explained in Appendix I.
4.0 RESULTS AND CONCLUSIONS
The constants which were employed in the numerical solution of the
problem are given below
a. = 0.53 for 1 < Pn < 20 horsepower
«j = 1.08 for 20 < Pn < 200 horsepower
a2 = 0.6667
(BOD). = 0.0109025 lb. oxygen/ cu. ft. solution
(BOD) = 0.0026166 lb. oxygen/cu. ft. solution
(COD). " 0.0290941 lb. oxygen/cu. ft. solution
(COD) = 0.0090958 lb. oxygen/cu. ft. solution
Subscripts :
i = quantity in parenthesis is measured at the inlet of
aeration basin
o = quantity in parenthesis is measured at the outlet of
aeration basin
C. = $100.00 for 1 < P < 20 horsepower
C = $ 21.57 for 20 < P < 200 horsepower
1.00 $/sq. ft.
C3 = 9.259 x 10"3 $/cu. ft.
121
4
3.333 x 10"2
s -
6.764
C6 "
5000.
C7 ■
0.02
C8 -
1492.
E =
0.75
I
k =
0.75
M «
20
Q -
45,120
r =
0.10
R ■
o
3.2
T -
8760
$/sq. ft.
S/sq. ft.
$
$/horsepower-hour
$/horsepower
(hour)
years
cu. ft. /hour
lb. oxygen/horsepower
hours
x: m (BOD). = 0.0109025 lb. oxygen/cu. ft. solutic
Determination of the aerator motor size of the inlet section of the
aeration basin (that portion of the lagoon used for directly oxidizing the
idealized inorganic component) was not considered to be part of the opti-
mization problem since the size is fixed once the waste water flow rate
and the BOD's and COD's of the aeration basin inlet and outlet streams
are given. The size of the inlet aerator motor, P , was calculated using
Equation (27) and the numerical values of the constants given previously.
R E
l(C0D)i - (BOD).] - f (C0D)q - (B0D)o ]] (27)
A value of 220 horsepower was obtained for P .
122
The outlet waste solution BOD was used as a parameter in the optimiza-
tion study. Three cases were considered in which it was required to obtain
reductions of the inlet BOD of 76, 90, and 99 percent respectively. For
the 76 percent inlet BOD reduction case, optimum policies were calculated
for one, two, and three stage processes. The least expensive process is
the one stage process. Optimum policies were calculated for processes
composed of from one to four stages for the 90% BOD reduction case. For
this case, the two stage process is the least expensive process. Similarly,
optimum policies for processes with stages numbering from one to six were
calculated for the 99 percent BOD reduction case. For this case, the four
stage process is least expensive. The results for the cases of 76, 90, and
99 percent inlet BOD reduction are summarized in Tables 1, 2, and 3, respective
ly. The optimum process for each case is given in Table 4.
Since the maximum principle does not guarantee a minimum, it is necessary
to do simulation studies with a mathematical model of the lagoon to insure
that a minimum has been found. Simulation studies were performed with com-
puter program SIM which was programmed to simulate the process taking place
in the lagoon aeration basin based upon performance Equations (22) through
(25).
The results of the simulation studies for the 99 percent BOD reduction
case are given in Table 5. Three cases given in Table 5 are slightly less
expensive than the optimum process found by the maximum principle solution.
These less expensive processes resulted because of discretization error
which was introduced by computer program OPT during the optimization
calculations. The optimum process can be computed more accurately, however,
the improvement obtained in the total process cost will not be significant
enough to justify additional computations. In the fourth simulation study
123
given in Table 5, the aerator motor size of each stage was obtained by
rounding off the optimum aerator motor sizes of the first three stages
to integer sizes as would be required should it be desired to purchase
the aerator motors. The total process cost of the fourth simulation
study is not significantly more than the total cost of the optimum
process .
For the model used in this study, several trends in the relationships
among the variables are apparent upon inspection of Tables 1, 2, 3, and 4.
At a given conversion, the optimum total volume required to achieve this
conversion decreases as the number of stages increases. Both the total
cost and the optimum number of stages for the BOD reduction section of the
lagoon increases as the percentage BOD conversion increases. For all of
the multi-stage processes, tapered aeration as discussed by Sawyer (8),
is the best policy. This is due to the fact that the rate of BOD reduc-
tion decreases as the treatment progresses.
The discrete maximum principle shows promise as a useful technique
for the optimization of aerated lagoons which are operated in a stage-
wise fashion. The use of this technique with the aerated lagoon model
used in this work predicts, as was expected, that tapered aeration is the
optimal operating policy for multi-stage lagoons.
For all of the optimal cases, the total aeration volume in each case
is an order of magnitude smaller than the Sugar Creek aeration basin volume
(about 1,700,000 cu. ft.). This may be partially due to the value of the
first order reaction rate constant, k , which was used in the numerical
computations. Too large a value of k would account for the smaller
optimal aeration volume calculated. An improved model with calculations
to support this belief is given below.
124
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•
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m
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f-l r-1 CO
m
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in
r-s in co
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