MARINE STEAM CONDENSERDESIGN USING NUMERICAL

OPTIMIZATION

17

Charles Michael Johnson

Report Number--NPS69-77-002

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESISMARINE STEAM CONDENSERDESIGN USING NUMERICAL

OPTIMIZATION

by

Charles Michael Johnson

December 1977

Thesis Adv isors : Paul J. MartoG. N. Vanderp laats

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Marine Steam Condenser Design UsingNumerical Optimization

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Engineer's Thesis;December 1977

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Charles Michael Johnson

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19. KEY WORDS (Contlnua on ravaraa aida It nacaaaary and Idmnttty by block numbar)

Marine Condenser DesignCondensersAutomated Ship Design

20. ABSTRACT (Conttrma on ravaraa atda it nacaaaary and idantity by block numbar)

Two separate computer codes were coupled with a constrainedfunction minimization code to produce automated marinecondenser design and optimization programs of vastly differentcomplexity. The first program, 0PC0DE1, was developed fromthe Heat Exchange Institute's Standards for Steam SurfaceCondensers (HEI) . The second program, 0PC0DE2, was developedfrom the sophisticated 0RC0N1 , a computer code produced by the

DD FORMI JAN 73 1473 EDITION OF I NOV 68 IS OBSOLETE

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Oak Ridge National Laboratory. CONMIN, the optimizationprogram, was developed at the Ames Research Center.

0PC0DE1 was well verified using main condenser inputdata of an aircraft carrier and a destroyer escort.Verification of 0PC0DE2 , using main condenser data of anaircraft carrier, was less satisfactory due to theconservative nature of flooding effects on the outside filmheat transfer coefficient used in 0RC0N1

.

0PC0DE1 is an excellent design tool for the conceptualdesign of a marine condenser. Optimized test cases run with0PC0DE1 show that a condenser designed by the HEI method isnearly optimum with respect to volume.

Test cases with 0PC0DE2 show that enhancing the heattransfer on the shell-side by 80 percent yields a condenserwith ten percent less volume than the unenhanced case.

DDlJan^

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Approved for public release; distribution unlimited

Marine Steam CondenserDesign Using Numerical

Optimization

by

Charles Michael JohnsonLieutenant, United States Navy

B.S.E., University of South Florida, 1968

Submitted in partial fulfillment of therequirements for the degrees of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

and

MECHANICAL ENGINEER

from the

NAVAL POSTGRADUATE SCHOOL

December 1977

ABSTRACT

Two separate computer codes were coupled with a con-

strained function minimization code to produce automated

marine condenser design and optimization programs of vastly

different complexity. The first program, 0PC0DE1 , was

developed from the Heat Exchange Institute's Standards for

Steam Surface Condensers (HEI) . The second program, 0PC0DE2,

was developed from the sophisticated 0RC0N1 , a computer code

produced by the Oak Ridge National Laboratory. CONMIN, the

optimization program, was developed at the Ames Research

Center.

0PC0DE1 was well verified using main condenser input data

of an aircraft carrier and a destroyer escort. Verification

of 0PC0DE2, using main condenser data of an aircraft carrier,

was less satisfactory due to the conservative nature of

flooding effects on the outside film heat transfer coeffi-

cient used in 0RC0N1

.

0PC0DE1 is an excellent design tool for the conceptual

design of a marine condenser. Optimized test cases run with

0PC0DE1 show that a condenser designed by the HEI method is

nearly optimum with respect to volume.

Test cases with 0PC0DE2 show that enhancing the heat

transfer on the shell-side by 80 percent yields a condenser

with ten percent less volume than the unenhanced case.

TABLE OF CONTENTS

I. INTRODUCTION 15

A. BACKGROUND 15

B. METHODOLOGY 16

C. OBJECTIVES 18

II. NUMERICAL OPTIMIZATION 20

A. BACKGROUND 2

B. CONSTRAINED FUNCTION MINIMIZATION (CONMIN)-- 21

C. CONTROL PROGRAM FOR ENGINEERINGSYNTHESIS (COPES) 31

III. OPTIMIZED CONDENSER DESIGN, VERSION 1

(OPCODE1) 33

A. DEVELOPMENT 3 3

B. OPCODE1 VERIFICATION 36

C. LIMITATIONS OF OPCODE1 38

IV. OPTIMIZED CONDENSER DESIGN, VERSION 2

XOPCODE2) 41

A. BACKGROUND 41

B. MODIFICATIONS TO ORCON1 44

C. OPCODE2 VERIFICATION 48

D. LIMITATIONS OF OPCODE2 49

V. RESULTS 50

A. EXPLANATION OF THE CASE STUDIES 50

1. Constraint Framework for OPCODE1 50

2. Design Variable Framework for OPCODE1 -- 55

3. Constraint Framework for OPCODE2 54

4. Design Variable Framework for 0PC0DE2 -- 56

B. CASE STUDIES USING 0PC0DE1 57

1. Case One - Minimize Pumping Power;Rejected Heat Constant 57

2. Case Two - Minimize Volume;Rejected Heat Constant 58

5. Case Three - Maximize Rejected Heat;Pumping Power Constant 59

4. Case Four - Minimize Volume; PumpingPower and Rejected Heat Constant 61

5. Case Five - Minimize Condenser Length;Pumping Power and Rejected HeatConstant 63

6. Case Six - Minimize Pumping Power;Condenser Volume and Rejected HeatConstant 64

7. Case Seven - Maximize Rejected Heat;Pumping Power Increased by 50 Percentand Held Constant 65

C. CASE STUDIES USING OPCODE2 67

1. Case Eight - Minimize Volume UsingKorodense Relation and 14 PercentEnhancement 67

2. Case Nine - Minimize Volume UsingKorodense Relation and 80 PercentEnhancement 68

3. Case Ten - Minimize Volume UsingEissenberg Relation and 14 PercentEnhancement 69

VI. CONCLUSIONS 71

VII. RECOMMENDATIONS 7 3

VIII. FIGURES 75

IX. TABLES 89

APPENDIX A: DEVELOPMENT OF THE ANAL I Z SUBROUTINEFOR 0PC0DE1 104

APPENDIX B: DEVELOPMENT OF SUPPORTING SUBROUTINESFOR OPCODE1 111

APPENDIX C: USER'S MANUAL FOR OPCODE1 -- 114

APPENDIX D: SAMPLE OUTPUT FROM OPCODE1 136

APPENDIX E: OPCODE1 PROGRAM LISTING 147

BIBLIOGRAPHY 162

INITIAL DISTRIBUTION LIST 165

LIST OF FIGURES

FIGURE 1.

FIGURE 2.

FIGURE 3.

FIGURE 4.

FIGURE 5.

FIGURE 6.

FIGURE 7.

FIGURE 8.

FIGURE 9.

FIGURE 10.

FIGURE 11.

FIGURE 12.

FIGURE 13.

FIGURE 14.

Significance of the ConstraintThickness (CT) Parameter 75

Two-Variable Design Space Showingthe Initial Design at Point A 76

Two-Variable Design Space Showingthe First Design Iteration 77

Two-Variable Design Space Showingthe Second Design Iteration 78

Two-Variable Design Space Showingthe Third Design Iteration 79

Two-Variable Design Space Showingthe Fourth Design Iteration 80

Two-Variable Design Space Showingthe Fifth Design Iteration 81

Two-Variable Design Space Showingthe Need for Optimization Techniques 82

Two-Variable Design Space ShowingRelative Minima for a ConstrainedMinimization Problem 83

Circular Condenser Bundle Designedby 0RC0N1 84

CVA 67 Main Condenser as Modeledby ORCON1 and 0PC0DE2 85

CVA 67 Main Condenser Tube Sheet 86

Flowchart from ORCON1 87

Flowchart Showing Modification ofORCON1 to OPCODE2 88

TABLE I

TABLE II

TABLE III

TABLE IV

TABLE V

TABLE VI

TABLE VII

TABLE VIII

TABLE IX

TABLE X

TABLE XI

TABLE XII

TABLE XIII

TABLE XIV

TABLE XV

LIST OF TABLES

Verification of the CVA 67 Main CondenserDesign Using OPCODE1 89

Verification of the DE 1040 Class ofMain Condenser Using OPCODE1 90

Verification of the CVA 67 Main CondenserUsing OPCODE2 91

Input Parameters Used in OPCODE1 forCases One Through Seven 92

Initial Design by OPCODE1 for CasesOne Through Seven 93

Case One Optimization Results 94

Case Two Optimization Results 95

Case Three Optimization Results 96

Case Four Optimization Results 97

Case Five Optimization Results 98

Case Six Optimization Results 99

Case Seven Optimization Results 100

Case Eight Initial Design andOptimization Results 101

Case Nine Initial Design andOptimization Results 102

Case Ten Initial Design andOptimization Results 103

NOMENCLATURE

English Letter Symbols

?

At: - tube sheet area/number of tubes , ft /tube

2Atj - heat transfer area, ft

?A- - internal tube area per linear foot, ft "/ft

?A - external tube area per linear foot, ft"/fto

2A_p - flow area for a single tube, ft

C - coefficient for calculation of overall heattransfer coefficient

D - tube outside diameter, ft

d - tube inside diameter, ft

E. - internal heat transfer enhancement factor

E - external heat transfer enhancement factoro

F(X) - objective function

F, - average tube flooding factor

F - tube flooding factor for the n-th tuben &

F, - fouling factor

F-, - material correction factor

F, - temperature correction factor

f - tube-side friction factor

G - cooling water flow, gpm

G.(X) - inequality constraint

g - tube flow factor

2g L

- acceleration due to gravity, ft/hr

7

g - acceleration due to gravity, ft/sec"

H, (X) - equality constraint

10

h

h

h*o

fg

ICALC

k

kw

kv

L

LMTD

m

N

NAC

NCON

NDV

n*

P

PP

APext

AP

AP

AP

ent

enthalpy, BTU/lb

internal film heat transfer coefficient,BTU/(hr) (ft 2

) (°F)

external film heat transfer coefficient,BTU/(hr) (ft 2

) (°F)

external film heat transfer coefficientcorrected for inundation, BTU/ (hr) (f

t

2) (°F)

latent heat of condensation, BTU/lb

control flag for COPES

tube constant, k = s/g

fluid conductivity, BTU/ (hr) (f t) (°F)

wall conductivity, BTU/ (hr) (ft) (°F)

vapor conductivity, BTU/ (hr) (f t ) (°F)

tube length, ft

log mean temperature difference, °F

mass flow rate, lb/hr

number of tubes

number of active constraints

number of constraints

number of design variables

number of tubes in a vertical row abovethe n-th tube

absolute pressure, psia

pumping power, ft-lbf/sec

tube exit loss, ft w.c.

tube entrance loss, ft w.c.

sum of all tube-side pressure losses, ft w.c

internal tube friction loss, ft w.c.

11

Q- heat transferred, BTU/hr

q- iteration number

Rf

- fouling resistance, (f t) (hr) (°F) /BTU

r. - tube inside radius, ft1

r - tube outside radius, fto

S"- search direction

2s - outside area of tube per linear foot, ft~/ft

T - temperature, °R

AT\ M- log mean temperature difference, °F

t. - cooling water inlet temperature, °F

t - cooling water outlet temperature, °F

t - saturation temperature, °F

t - vapor temperature, °F

t - wall temperature, °Fw r

'

U - uncorrected overall heat transfer coefficient,BTU/(hr) (ft 2

) (°F)

U - corrected overall heat transfer coefficient,c BTU/(hr) (ft 2

) (°F)

U. - overall heat transfer coefficient based on1 inside tube area, BTU/ (hr) (f

t

2) (°F)

V - cooling water velocity, ft/sec

VLB. - lower side constraint on i-th design variablel °

VUB

.

- upper side constraint on i-th design variable

W - steam flow, lb/hr

w.c. - water column

w - weight of cooling water, lb/gal

X - vector of design variables

12

Dimensionless Groups

Pr - Prandtl number

Re - Reynolds number

Greek Letter Symbols

a* - move parameter in optimization problem

6 - ratio of Ap/A

f

3 - parameter in method of feasible directions

e - absolute roughness, ft

push off factor in method of feasible directions

?J

Pr - fluid absolute viscosity, lb sec/ft

p- condensate density, lb/ft"

5

3p

- sea water density, lb/ft^sw 7 '

p - vapor density, lb/ft"3

ACKNOWLEDGEMENTS

The author wishes to express his appreciation to

Professor Paul Marto , Doctor Gary Vanderplaats and

Professor Paul Pucci for their advice and guidance during

the development of this project. Without their valuable

assistance, this thesis could not have been completed.

The author wishes to thank Sharon Raney , Richard Donat

and especially Edwin Donnellan, all of the W.R. Church

Computer Center, for their time and patience in teaching

an engineer how to use all of the tools available.

Infinite thanks and grateful appreciation are due my

wife, Lari, for understanding the project, encouraging

its completion and making the difficult times bearable.

iMahalo

.

14

I. INTRODUCTION

A. BACKGROUND

In the last ten years a revolution has swept the marine

power plant industry that could result in the obsolescence

of the marine steam power plant. The revolution was caused

primarily by the use of the gas turbine as an alternative

to marine, and more recently, to naval propulsion. Gas

turbines brought about power plants which were more compact,

lighter, but less fuel efficient than the steam power plants

which they displaced.

When compared with the compact gas turbine, the massive

size and weight of the marine steam power plant, which

evolved in stride with the behemoth of the power industry,

the stationary steam power plant, made this means of propul-

sion less desirable for naval vessels. Therefore, it has

become imperative for the naval engineering community to

develop a more efficient, more compact, lighter weight steam

power plant.

As steam power plants are durable and they can burn a

variety of fuels — both essential qualities when considering

the Navy's combat readiness — they must not be allowed to

be overdesigned out of existence. To make steam propulsion

competitive with marine gas turbines, advanced concepts must

be explored in all areas of steam propulsion. Such concepts

as pressurized boilers, super critical cycles, enhanced

15

condenser tubes, and dropwise condensation must be developed

further. Above all, overdesign by the use of unnecessary

safety factors must be curtailed, and the minimum safe design

must be developed and identified.

B. METHODOLOGY

In the United States the most prevalent criterion for the

design and specification of surface condensers is based on

the "square root of V" relationship as developed by the Heat

Exchange Institute (HEI) [1]. Using this method, the overall

heat transfer coefficient is calculated as a function of the

square root of the cooling water velocity multiplied by

correction factors for inlet cooling water temperature, tube

wall thickness and material, and fouling.

The HEI method was adopted by the Department of the Navy,

Bureau of Ships (now Naval Sea Systems Command) for the

specification of U. S. Navy condensers by issuing Design

Data Sheet 4601-1 (DDS) [2] in 1953. Henceforth this thesis

will designate the preceding methodology as the HEI/DDS

method.

With the advent of the high speed digital computer,

numerical methods of solving complex engineering problems

are now possible. A computer code has been developed to

calculate the local heat transfer and thermodynamic properties

of a large surface condenser on a row by row basis. Known as

ORCON1, this code was developed by Oak Ridge National

Laboratory (ORNL) under contract to the Office of Saline

16

Water during the period from 1968 to 1970 [3] . The program

was based, in part, on the work performed by Eissenberg [4].

Eissenberg's experimental results led to correction factors

on the basic Nusselt equation to account for inundation

effects of tubes within a condenser tube bundle. Addi-

tionally, logic was developed to account for the pressure

loss caused by the steam's passage through the tube bank

with the accompanying reduction in saturation temperature;

the heat resistance due to the presence of a noncondensable

gas film; heat transfer enhancement factors on both sides of

the tubes; and other important factors to yield a program

which could calculate heat flux, overall heat transfer

coefficient, noncondensable gas concentration, and fifteen

additional parameters on a local, row by row basis.

Search [5] has used 0RC0N1 to perform parametric studies

of an actual naval condenser. Tube enhancement, the high

velocity flow allowed with titanium tubes, and dropwise

condensation were investigated. The penalties paid in

increased pumping power and increased cost were weighed

against the gains realized with a more compact and with a

lighter condenser.

In the open literature there is but one reference [6]

to coupling a condenser analysis and design program with

an optimization procedure that is capable of improving a

given design.

17

The case for utilizing an optimizing design scheme for

the condenser portion of a steam propulsion plant can easily

be made by re-emphasizing the fact that in order for steam

propulsion to remain a viable contender when the naval

vessels of the late 1980's are designed, it must compete

and succeed in an area that is rapidly being dominated by

gas turbines. All components of the naval ship's steam

propulsion plant will have to be critically designed to

ensure that the minimum design will still perform as and

when required.

C. OBJECTIVES

There were two primary objectives of this thesis.

The first objective was to develop a computer code based

on the HEI/DDS method of condenser design, as this method

is considered the industry standard. Coupling the HEI/DDS

code with a numerical optimization program yields a complete

design package. The design package can be used for trade-

off studies, first cut analysis, and conceptual design.

The second objective was to couple 0RC0N1 , with its

capability of varying both internal and external tube

enhancement factors, with a numerical optimization program

to produce a more detailed design program.

These design tools provide the Naval architect and the

Naval engineer with the means to optimize size, weight,

design, and cost of the marine steam propulsion plant for

ships of the 1980's; the enhanced design provides the optimum

18

streamlined design of a steam plant resulting in its

reinstatement and continuance as a viable alternative to

gas turbine propulsion.

19

II. NUMERICAL OPTIMIZATION

A. BACKGROUND

Nearly all design problems require either the minimiza-

tion or maximization of a parameter or function. This

parameter will be called the problem's objective function

or design objective [7]. For the design to be acceptable,

it must satisfy a set of design constraints. For example,

if an engineer was designing a piping system to achieve the

minimum in required pumping power, the minimum allowable

flow delivered would be a meaningful constraint. Likewise

a constraint that required the inside diameter of the pipe

to be less than the outside diameter of the pipe would be

a necessity.

If the problem could be formulated analytically with a

great deal of simplicity, the minima or maxima could be

found by using the methods of differential calculus or the

calculus of variations. However, these methods would fail

for all but the very simplest of problems.

A numerical method that would be satisfactory for rela-

tively small scale problems would be an iterative solution

technique. A simple computer program could be written

containing a series of nested iteration loops that would

vary the design parameters and solve the problem for a

variety of values for each of the parameters. For small,

easily formulated problems, the cost in central processor (CPU)

time would not be excessive, and this method would be

satisfactory

.

20

However, for a serious engineering problem, this method

quickly becomes too costly to pursue. For example, if the

engineer had ten design parameters for which he wanted to try

ten separate values, he would need to make ten billion

calculations. If each calculation took ten CPU seconds,

the solution would be available in approximately 3200 years!

Thus a rational approach to design automation and optimiza-

tion is obviously needed.

There are many optimization schemes available to the

engineer. The various methods fall into three broad cate-

gories based on the type of problem to be solved:

unconstrained minimization, solution of constrained problems

by unconstrained minimization, and direct methods for

solution of constrained problems [8]. An optimization

program based on the last method was chosen for this

research work.

B. CONSTRAINED FUNCTION MINIMIZATION (CONMIN)

Vanderplaats [9] developed an optimization program,

CONMIN, capable of optimizing a very wide class of

engineering problems. CONMIN is a FORTRAN program, in

subprogram form, that optimizes a multi-variable function

subject to a set of inequality constraints.

Three basic definitions are required [10]:

Design Variables - Those parameters which theoptimization program is permitted to change inorder to improve the design. Design variablesappear only on the right side of an equation,are continuous and have continuous first derivatives.

21

Design Constraints - Any parameter which mustnot exceed specified bounds for the design to beacceptable. Design constraints may be linearor non linear, implicit or explicit, but theymust be functions of the design variables.Design constraints appear only on the left sideof equations

.

Objective Function - The parameter which is goingto be minimized or maximized during the optimizationprocess. The objective function may also be linearor non linear, implicit or explicit, and must be a

function of the design variables. The objectivefunction usually appears on the left side of anequation. The only exception is if the objectivefunction is also a design variable.

As can readily be seen by the definitions above, design

constraints and objective functions are usually

interchangeable

.

The number of design variables being utilized for an

optimization is equivalent to the dimension of the design

space in which the design is being calculated. Thus, if

an optimization problem has four design variables specified,

the design will be carried out in four-space.

Assuming that the optimization process requires the

minimization of a particular objective function, the general

optimization problem can be stated as:

Find the vector of design variables, X, to

minimize F(X) subject to the constraints

G. (X) <_ 0.0, j = l,NCON (1)

VLB. < X < VUB. , i = 1,NDV . (2)

In the general problem statement, F (X) is the objective

function, there are NDV design variables, and NCON constraints

VLB. and VUB. are the lower bounds and upper bounds respec-

tively on the i-th design variable. If the inequality

22

condition of equation (1) is violated, (G.(X) > 0) , for any

constraint, that constraint is said to be violated. If the

equality condition is met, (G.(X) = 0), the constraint is

active. If the inequality condition is met, (G.(X) < 0),

the constraint is inactive. Because of the numerical

problems involved in representing exact zero on a computer

with a finite number of significant figures, the equality

condition is represented by a band around the value G. (X) =

The band is equal to twice the constraint thickness (CT) and

is shown in Figure 1. Figure 1 is a representation of a

two-variable design space with the values of G.(X) =

plotted. In this instance,

X.

X,

For n-space, G.(X) would appear as a hypersurface

whereas for n = 2, the hypersurface degenerates to a single

curve that can be easily represented.

Any design which satisfies the inequalities of

equations (1) and (2) is referred to as a feasible design.

If a design violates any of these constraints, it is an

infeasible design. The minimum feasible design is said to

be optimal. Note that if it is desired to maximize an

objective function, the process reduces to minimizing the

negative of the objective function.

CONMIN requires an initial vector of design variables,

X, which may or may not yield a feasible design. If the

23

initial design is infeasible, CONMIN moves towards a

feasible solution with a minimal increase in the objective

function [9]. The optimization process then proceeds in

an iterative fashion with the following recursive

relationship

:

where q is the iteration number and a* is the move parameter,

a scalar, which defines the distance of travel in the

direction of search, S.

The optimization process is divided into two parts.

The first is the determination of S which will reduce the

objective function without violating any constraints. The

second is the determination of the scalar a* so that F(X)

is minimized in this direction, a new constraint is

encountered, or a currently active constraint is encountered

again.

For the sake of discussion, consider a condenser design

problem with two design variables, X, and X , where

X, = condenser tube outside diameter, and

X~ = tube pitch to diameter ratio.

The objective function is condenser volume, VOL(X). Assume

that the tube bundle diameter must be greater than a

specified value, BD . , and that the cooling water pumping

power must be less than a specified value, HP . Figure 2r r' max °

24

is a graphical representation of the problem. Note the

lines of constant objective function, with VOL, (X) > VOL- (X)

,

and the initial design at point (a) .

It must be reiterated that, while this example assumes

a feasible initial design, this is not a requirement and

CONMIN will also optimize when given an infeasible initial

design.

The optimization begins by calculating the gradient of

the objective function by using the finite difference tech-

nique. A perturbation of 0.01 is applied to each of the

design variables in a single forward step. The gradient

vector is therefore

VF(X) VVOL

3 (VOL)

Because no constraints are active or violated, the greatest

improvement in the objective function will be realized by

moving in the direction of steepest descent so that

VVOL

as shown in Figure 3.

With the value of S now determined, it remains to find

the move parameter, a*, that will allow the greatest

25

improvement in the objective function. A one-dimensional

search is carried out in the S direction until the value for

a* is found. This is point (B) on Figure 3. The location

of point (B) terminates the first design iteration.

The second design iteration is begun by, once again,

perturbing X to find VVOL. Instead of moving in the

direction of steepest descent, however, a new S is found

by the method of conjugate directions, developed by

Fletcher and Reeves [11]. With this method, S is calculated

by the following relationship:

2

V F(X) q

' - V F(X) q +gqgq-1

V F(X)q-1

This is shown in Figure 4.

The advantage of using the Fletcher-Reeves method

instead of the steepest descent is that convergence to an

optimum is much faster. With the new search direction, S,

a search is performed in this direction until a constraint

is encountered. This occurs at point (c) on Figure 4 at

the pumping power constraint, thus terminating the second

design iteration.

The third design iteration begins with the HP& & max

constraint active at point (cT) on Figure 5. As VVOL is

found, the gradient of the active constraint is found using

the information from the same forward finite difference step

Not only is a new S required that will reduce the

objective function, but this new S must not violate the

26

active constraint. This problem may be solved by the method

of feasible directions developed by Zoutendijk [12] and

implemented by Vanderplaats and Moses [13]

.

The finding of a new S has now become a sub-problem

which is a linear programming problem with a single

quadratic constraint. This sub-problem [14] is stated as:

Find a vector S to maximize 3 subject to the constraints

V F(X) • S + 3 < (3)

V G. (X) • S + 9 .3 £ j = 1,NAC (4)

S < 1 (5)

where, for the case at hand, V F(X) E7V0L,VG.(X) = -VHP,

and G.(X) = HP - HP. NAC , the number of active constraints,

j* ' max ' '

is one in this instance.

If equation (3) is satisfied and 3 is positive, the

resulting search direction will reduce the objective function

and is defined as a usable direction. If equation (4) is

satisfied and 3 is positive, S is a feasible direction

because, for a small move in this direction, no constraints

will be violated. 9. is defined as the push-off factor for

G. (X) and has the effect of pushing the design away from

the active constraint. 9- must be zero or positive to

maintain a feasible design. For 9- = 0, S would be tangent

to the active constraint. For 9. >> 1.0, S would be3

27

pushed away from the active constraint and very nearly

tangent to a line of constant objective function. For

9. = 1.0, the angle between constant objective function and

the active constraint would be approximately bisected. If

the maximum value of 6 obtainable from equations (3) , (4)

and (5) is zero, then no direction exists which will both

reduce the objective function and satisfy the constraint,

and the current design is optimal or is at least a local

minimum. In this example, a usable and feasible direction

exists and a one dimensional search leads to point (D) in

Figure 5 where the minimum bundle diameter, BD . , constraints ' mm'is encountered. This ends the third iteration in the

optimization process.

The fourth iteration begins as before, with the calcula-

tion of the gradient of the objective function and the

active constraint. The sub-problem of equations (3) through

(5) is again solved for a new S.

It should be noted that the minimum bundle diameter

constraint is assumed to be linear; therefore in equation (4)

the push-off factor, 8., is set to zero allowing S to

follow the constraint as shown in Figure 6. A one

dimensional search is again carried out in the new S direction

until a new constraint is encountered or an active constraint

is re-encountered. Thus, the activated BD . constraint is' mm"ridden" until no further design improvement is realized.

This occurs at point (|) on Figure 6 and the fourth design

iteration is terminated.

28

For the fifth iteration, the same procedure is followed

and the solution to the sub-problem characterized by

equations (3) through (5) yields a value of zero for 8.

Thus, there is no direction that will both reduce the

objective function and satisfy the constraint and the

current design at point (E) in Figure 7 is optimal.

Figure 8 illustrates the value of using optimization

techniques to solve the design problem. Assume an initial

design at point (A) such that the minimum tube outside

diameter is active. A reduction in the tube pitch to

diameter ratio will yield a design improvement until the

minimum bundle diameter constraint is encountered at point

(B) . At this point, neither the pitch to diameter ratio

nor the outside tube diameter can be reduced independently

as required to reduce the objective function, without

violating the active constraints. Only by changing the

two design variables in a particular manner can the minimum

value of the objective function at point © be achieved.

This discussion of the methodology involved with CONMIN

would not be complete without citing the program's limita-

tions. The number of design variables (NDV) directly

affects the computational time required to reach an optimum.

Since the calculation of the gradient information required

for each design variable at the beginning of each design

iteration is found by using a single forward finite difference

step, requiring a complete pass through the analysis portion

of the program, there is a subsequent increase in CPU time

29

as NDV increases. Also, problems with many design variables

tend to converge more slowly due to the interaction between

the design variables and because of the numerical inaccuracy

(machine related) generated during the optimization process.

The number of design variables should therefore be kept

small in order to expedite the optimization process.

Vanderplaats [7] recommends a practical limit of twenty

design variables.

The number of design constraints used is not limited in

the same manner. Recall that the only time gradient

information is stored for a constraint is when that constraint

is active. Therefore, the sheer number of constraints will

not adversely affect the optimization.

As can be seen in Figure 9, it is quite possible that

the optimal design found is actually a relative optimum and

not a global optimum. This problem can be overcome by

starting the design with several different initial vectors,

X, until the same optimal design is repeated.

Equality constraints of the type

Hk(X) =

are very difficult to provide for in the general optimization

scheme [7]. By defining an objective function in which a

weighting factor multiplies the parameter to be held constant,

Y, and this product is summed with the parameter to be

optimized, X,

objective function = X + weighting factor x Y

30

the parameter to be held constant can be forced to the

appropriate bound. The weighting factor is generally one

that will keep both terms in the same order of magnitude

but trial and error is sometimes required. Parameters can

be held "constant" within to. 5 percent.

C. CONTROL PROGRAM FOR ENGINEERING SYNTHESIS (COPES)

Recall that the optimization program, CONMIN, was

written in subroutine form. Vanderplaats [15] has developed

a main program to simplify the use of CONMIN and to further

aid in the design optimization process.

The user must supply an analysis subroutine with the

name ANALIZ. ANALIZ, in keeping with general good program-

ming practice, must have three segments: input, analysis and

output. Based on the value of a flag from COPES

(ICALC = 1,2 or 3), ANALIZ performs the proper function.

Finally, CONMIN and ANALIZ do not communicate directly with

each other as COPES is the main program.

COPES is constantly being revised by Vanderplaats to

better meet the needs of the engineer.

The COPES program currently provides four specific

capabilities

:

1. Single analysis - just as if COPES/CONMIN were

not used.

2. Optimization - minimization or maximization of a

multivariable function with limits imposed on other

functions

.

31

3. Sensitivity analysis - used to investigate the

effect of changing one or more design variables

on one or more calculated functions.

4. Two-variable function space - provides analyses

of all specified combinations of two design

variables

.

This work is concerned only with the application of

COPES/CONMIN to the optimized design of a marine condenser,

therefore, items 3 and 4 are included only for completeness.

If there is no relative or absolute change in the value

of the objective function for three design iterations,

the optimum is found and COPES prints the appropriate

message and terminates the program. If no feasible design

can be found after ten design iterations, COPES prints the

appropriate message and terminates the program.

Experience has shown that most design problems can be

optimized within 20 design iterations and the maximum number

of design iterations permitted is defaulted to this value.

Thus, if the optimum value has not occurred in 20 design

iterations, COPES will print the appropriate message and

terminate the procedure.

COPES has simplified the procedures involved in using a

sophisticated program such as CONMIN. Thus the engineer is

freed from the unwanted role of systems analyst and may

devote his talents to engineering.

32

III. OPTIMIZED CONDENSER DESIGN, VERSION 1 (OPCODE1)

A. DEVELOPMENT

The Bureau of Ships adopted the Heat Exchange Institute's

Standards for Steam Surface Condensers [1] by issuing Design

Data Sheet DPS 4601-1 [2] in October, 1953. The DDS is still

being used to specify naval condensers.

The technique involved is based on calculating a value

of the overall heat transfer coefficient, U , based on the' c

'

cooling water velocity through the condenser tubes, the

condenser tube wall thickness and material, the tube fouling

factor, and the cooling water inlet (injection) temperature.

Knowledge of these parameters and their associated correction

factors leads to the simple formulation of U

U = F n F F, C /Vc 12 3

The correction factors F, , F- , and F, are tabulated in12 o

references [1] and [2]. Reference [2] specifies a value of

C = 270 for 0.625 and 0.750 inch outside diameter (o.d.),

18 Birmingham Wire Gauge (BWG) tubes. A value of C = 263 is

used for 0.875 and 1.00 inch o.d., 18 BWG tubes.

Because of the simplicity of the HEI/DDS method of

condenser design, and since the DDS is still the specifying

document for naval condensers, the author chose to implement

the HEI/DDS method for automated design. The combination of

33

COPES/CONMIN with the HEI/DDS method was named 0PC0DE1

(Optimized COndenser DEsign, Version 1 ) . The development of

0PC0DE1 is presented in Appendices A and B.

To add greater versatility to the HEI/DDS method,

algorithms were developed for 0PC0DE1 to calculate a

circular bundle geometry and the cooling water pumping

requirement. The circular bundle geometry was designed from

the long axis out. A central 12 inch diameter void to serve

as the collection header for noncondensable gasses was

provided along the bundle's longitudinal axis. Once the

number of tubes was calculated, the tubes were placed in

rows concentric to the central void and spaced with a 60

degree triangular pattern. To simplify the algorithm, and to

provide continuous functions to COPES/CONMIN, partial tubes

were permitted in a row and the outermost row was allowed

to be partially filled.

The calculation of the mass flow rate of cooling water

required began the calculation of the sea water (S-W) pressure

drop and pumping power requirement.

The S-W pressure drop was divided into a component based

on the sudden expansion and contraction losses caused by the

flow entering and exiting the waterboxes; a component based

on the sudden contraction and expansion losses caused by

flow from the inlet waterbox into the inlet tube sheet and

from the outlet tube sheet into the outlet waterbox; and a

component based on the normal frictional losses associated

with internal tube flow.

34

Reference 1 presents, in graphical form, the pressure

drops for all the components. The graphical data from

Figure SF-9 of reference 1 for the calculation of waterbox

losses was implemented using a systems library interpolating

subroutine. The S-W velocity in the waterboxes was taken

as 75 percent of the velocity in the condenser tubes as

specified in reference 16.

To simplify the tube-side pressure loss algorithm, the

friction factor, f, was first calculated by solving the

transcendental Colebrook equation [17]. Details of the

solution are provided in Appendix B. An absolute roughness

of 5 x 10 feet was assumed as representative of drawn

tubing. With the value of f calculated, the tube-side

pressure drop, AP , was calculated using the familiar

Darcy-Weisbach formula [17]:

APt

- £ & C^O

The tube sheet entrance and exit losses were calculated

by utilizing the area ratio technique developed in

reference 18. With this method, the flow area associated

with each tube was calculated

a - tube sheet areaF number of tubes

followed by the calculation of the internal flow area for

an individual tube, Ar and the ratio of the two areas:

35

A£

6 A '

With the value of 8 calculated, the entrance loss was

calculated with the relation

2

AP «. = 0. 5(1.0 - 62

) (-r—) [feet of water column]ent ^ ' ^2g J L J

& o

and the exit loss was calculated with the relation

2 22 V

AP . = (1.0- 6 ) f*— ) [feet of water column]ext 2g L

*o

Although the heat transfer calculations performed in

0PC0DE1 were for a single pass condenser, expansion of the

program to multiple pass condensers is feasible. To

simplify this future expansion, the waterbox pressure loss

was multiplied by the number of tube passes, read as an

input variable.

The pumping power was now easily calculated

PP = N • V • Ar * p • P j-

ft-lbfjf sw L sec J

The input variables required for the use of 0PC0DE1

are listed in the User's Manual provided as Appendix C.

B. 0PC0DE1 VERIFICATION

It was desirable to verify the performance of 0PC0DE1

as a predictor of condenser performance by comparison with

36

actual experimental data. Complete and accurate data was

not available, therefore, another method of model verification

was sought.

Using the design data from the condenser technical

manuals for two classes of ships, the CVA 6 7 and the DE 1040,

an attempt was made to repeat the design of the condensers.

The two condensers are vastly different in size; the CVA 67

has a heat transfer area of 16,011 square feet while the

DE 1040 class condenser has a heat transfer area of

6600 square feet.

The results of the design of the CVA 67 by 0PC0DE1

are tabulated in Table I with the data from the condenser

technical manual [19] included for comparison. Very close

agreement was achieved for all parameters except tube-side

pressure drop. This difference was attributed to the tube

sheet layout by 0PC0DE1 and the fact that no steam lanes

were accounted for. The addition of steam lanes would tend

to increase the tube-side pressure drop since more tube sheet

area would be unused as tube sites. Thus the entrance and

exit losses associated with the tube sheets would be greater

than predicted by 0PC0DE1.

The results of the design of the DE 1040 class condenser

by 0PC0DE1 are tabulated in Table II with the data from the

condenser technical manual [20] included for comparison.

Excellent correlation was achieved with all parameters within

three percent of the specifications except for bundle

diameter and tube-side pressure drop. Since no provision was

37

made for the calculation of circumferential steam lanes

in 0PC0DE1, the bundle diameter calculated was less than

the prototype.

Since the DE 1040 condenser has a basically circular

bundle geometry, close agreement was expected between the

calculated tube-side pressure drop and the value specified

in the technical manual. This was not the case. One

plausible explanation could be that the designers specified

a large factor of safety for this parameter.

These results confirm that the designers of the CVA 67

and the DE 1040 main condensers did, in fact, use the

HEI/DDS method for the calculation of the heat transfer

parameters and, with the notable exceptions of tube-side

pressure drop and bundle diameter, 0PC0DE1 accurately

predicts these parameters at the full power design point.

If the limitations imposed on tube-side pressure drop

and bundle diameter by the exclusion of steam lanes from

0PC0DE1 are acknowledged, the results received from 0PC0DE1

can be viewed as a "first cut" analysis. This was the

original purpose of developing this program.

C. LIMITATIONS OF 0PC0DE1

0PC0DE1 is insensitive to shell-side conditions. The

saturation steam pressure and the saturation steam temperature

are assumed to remain constant as the steam passes through

the bundle, whereas the steam flow actually experiences a

pressure drop with a resulting decrease in saturation

38

temperature as the steam passes through each row of tubes.

As a result, 0PC0DE1 makes no attempt to specify steam

lanes, either circumferential or radial, since the proper

design of steam lanes is a strong function of local steam

pressure

.

0PC0DE1 has no provision for the application of tube

enhancement factors due to the simple manner in which the

corrected overall heat transfer coefficient is computed.

The usual method for the calculation of U- [21] is:

U- = (6)

i+\ m Cy^j A.

,

R~ 2¥kT Rf A " h~

1 W

where the internal film heat transfer coefficient is given

by the Colburn form of the convective film heat transfer

correlation [21]

:

khi

= 0.023 (Re)°-8(Pr)

1/3(-^) (7)

The outside film heat transfer coefficient is given by

Nusselt's equation [21] corrected for the inundation, or

condensate rain, effects of upper tubes on lower tubes in

the bundle by including the factor n* in the denominator,

where n* is the number of tubes in a vertical row above the

i-th tube

39

0.725P (p - P ) g T

h r kc^ c v^ & L fg c

U n*D (tv V

1/4

(3)

With h. and h in the forms given by equations (7) and

(8), enhancement factors can be applied as multipliers of

these equations and equation (6) will yield a value for

the enhanced overall heat transfer coefficient, U.*.' i

None of the foregoing is possible with the HEI/DDS

based 0PC0DE1.

Finally, since 0PC0DE1 is insensitive to steam conditions,

the effect of noncondensable gasses on the heat transfer

characteristics of the condenser are unknown.

40

IV. OPTIMIZED CONDENSER DESIGN, VERSION 2_ (OPCODE2)

A. BACKGROUND

In the late 1960's, engineers at the Oak Ridge National

Laboratory developed a sophisticated computer code under

contract to the Office of Saline Water. This code, called

0RC0N1 [3], was generated to aid in the analysis and

parametric study of large, generally circular steam

condensers for use in large scale, multistage distillation

plants for the production of potable water from sea water

by the flash evaporation process.

Much of 0RC0N1 was dependent on Eissenberg's research

work [4] on the effects of condensate rain on the shell-

side convective heat transfer coefficient.

The program analyzes a single pass, circular or semi-

circular condenser, with steam flowing on the shell-side

of the tubes and variable salinity water flowing on the

tube-side. An optional, rectangular air cooler bundle is

provided for, as well as elementary, shell-side baffles.

The bundle is divided into 30 degree sectors and symmetry

about the central axis may be employed to reduce computa-

tional effort. The tubes are placed on a 60 degree equilat-

eral triangular pattern of concentric rows with the rows

added from the outermost row to an inner void provided

along the bundle's longitudinal axis. This serves as a

collection header for noncondensable gasses prior to passage

through the air cooler, if specified. The steam is assumed

to flow radially from the outside of the bundle to the

central void. Figure 10 shows the ORCON1 model of a steam

condenser with optional baffles and cooler.

0RC0N1 proceeds with sector by sector, row by row

calculations of the following quantities:

a) Saturation temperature of the steam entering therow.

b) Pressure of the steam plus noncondensable gasentering the row.

c) Steam flow entering the row.

d) Steam and noncondensable gas velocity at the minimumcross section in the row.

e) The fraction of noncondensable gas by weight.

f) The overall heat transfer coefficient for theaverage tube in the row.

g) The steam-side condensing coefficient.

h) The tube-side heat transfer coefficient.

i) The shell-side film heat transfer coefficientcomposed of the noncondensable gas film plus thecondensate film.

j) The shell-side Reynolds number based on the massflow at the minimum cross sectional area in the row.

k) The heat flux per square foot of condenser tube.

1) The shell-side friction factor.

m) The mass flow rate of steam plus noncondensable gasat the minimum cross section in the row.

n) The cooling water temperature at the inlet end ofthe condenser tube.

o) The cooling water temperature at the outlet end ofthe condenser tube.

p) The heat transfer coefficient for the noncondensablegas film.

42

q) The number of tubes per row.

r) The cumulative shell-side pressure drop from row 1

to row n.

In addition to the above parameters, the area weighted

overall heat transfer coefficients for the condenser section,

the cooler section, and the combined condenser are used to

calculate the "back calculated" log mean temperature

difference (LMTD) . This value of the LMTD , when compared

with the LMTD calculated by standard means using the

saturation steam temperature and the average cooling water

inlet and exit temperatures, represents the loss in average

thermal driving force due to pressure drops within the

bundle

.

The tube internal film heat transfer coefficient is

calculated by the Colburn equation multiplied by the

internal enhancement factor

0.023 (Re)' 8

(Pr)1/3

(-/) (Ei

)

The uncorrected external film heat transfer coefficient

is calculated using the basic Nusselt equation [21]:

h 0.725

7 3kc P,

-i 1/4hfg g L

THT (E ) (9)

The value of h calculated in equation (9) is for a

single tube. Eissenberg [4] corrects for inundation effects

43

by first calculating a tube flooding factor using the

following relation:

F = 0.6F, + (1 - 0.5647F,)n" - 20

n d & J

F_, is an input parameter and is a function of both tube

spacing to diameter ratio and tube orientation [3,22].

The condensate film coefficient for the typical tube in

the n-th row is then calculated by correcting the value of

h from equation (9) :

h *(n) = [nF - (n - 1)F - ]ho v J L n J n-l J o

To model the main condenser of the CVA 67, one quarter

of the full circular bundle was specified. With two bundle

quarters placed back to back as shown in Figure 11, the

steam lanes and tube arrangement closely approximated the

CVA 67 tube sheet as shown in Figure 12 from reference 19.

The combination of the COPES/CONMIN optimization package

with a suitably modified 0RC0N1 produced 0PC0DE2 (Optimized

COndenser DEsign , Version 2) .

B. MODIFICATIONS TO 0RC0N1

The original 0RC0N1 had neither tube-side pressure drop

calculations nor volumetric calculations. The subroutines

developed for 0PC0DE1 to calculate the tube-side pressure

drop and pumping power were installed in 0RC0N1 when

44

converting it to 0PC0DE2. An equation for the calculation

of condenser volume based strictly on the model as developed

for the CVA 67 main condenser and shown in Figure 10 was

added to 0RC0N1

.

The flowchart given in Figure 13 illustrates the

original program logic for 0RC0N1 . Once all calculations

were completed, a test for the exit steam fraction was made

against an input' target value. If the target value was not

met, and depending on the value of an input flag, either

the length of the condenser tubes or the quantity of inlet

steam was varied, and the calculations were repeated until

the target value and the calculated values agreed. The

adjustment of tube length or inlet steam was performed in

subroutine ADJUST.

It was felt that the quantity of inlet steam was generally

the value which should drive the entire design of the

condenser and should therefore remain a constant.

Similarly, the tube length was a good candidate for inclusion

in the optimization program as a design variable. For these

reasons, subroutine ADJUST was removed from 0RC0N1

.

The original ORCON1 provided logic for multiple data

runs. This capability was removed since COPES/CONMIN

requires a "once through" analysis program. Figure 14 shows

the sections of 0RC0N1 removed during the conversion to

0PC0DE2.

Search [5] had previously modified 0RC0N1 so as to

calculate the shell-side heat transfer coefficient either

45

with the original equation from the work of Eissenberg [4]

or by using the relation specified by the manufacturer of

Korodense condenser tubes [23]. This facility was left in

0PC0DE2.

0RC0N1 was not originally a machine independent program

as it had been tailored for operation on IBM (International

Business Machines) equipment. This was not a desirable

characteristic and programming changes were made to ensure

that 0RC0N1 was machine independent and that 0PC0DE2 would

remain machine independent.

0RC0N1 uses iterative techniques to solve for quantities

such as condensation rate, steam mass flow rate, and pressure

drop balance between sectors. The description of subroutine

ADJUST is an excellent example of such an application.

CONMIN uses perturbation techniques to calculate the

gradient information required for each design variable and

for each active design constraint during an optimization

iteration

.

Since 0RC0N1 has the capability to make design decisions,

a perturbation by CONMIN would cause an adjustment by 0RC0N1.

The two programs therefore worked at cross purposes.

It was not possible to remove all of these design

decision points from 0RC0N1 because of the strong coupling

between the subroutines. The decision was made to remove

only ADJUST and to leave the other design decision points

intact

.

46

This problem affected the choice of parameters that

could be used as design variables and the range of values

that the chosen design variable was permitted to assume.

Another problem area developed because many of the thermo-

dynamic properties were calculated in 0RC0N1 by subroutines

that use logarithmic functions to approximate the thermo-

dynamic curves. If the arguments for these functions take

on values less than or equal to zero, the function is

undefined. While constraints could be added as part of the

optimization process, the computation would stop during the

pass through the analysis.

Tests were put before all calculations that involved

logarithmic evaluation and before all function evaluations

where the denominator could take on values very close to

zero. These tests, when activated, would make a "design

decision" by setting the function to an approximate value

such as

:

if x <_ 0.0001 , then y = -10.0

otherwise, y = In x .

These approximate values would cascade due to the large

number of times the function was evaluated. Other mathematical

instabilities also occurred.

The design variables or the values of the design variables

that would trigger the original instability were found. The

tests were removed and the troublesome design variables and/or

the particular values that would cause the undesirable

response were avoided.

47

C. 0PC0DE2 VERIFICATION

Verification of 0PC0DE2 as a predictor of condenser

performance was attempted by inputing the design values

from the CVA 67 technical manual [19] and comparing the

condenser designed by 0PC0DE2 with the values given by the

condenser technical manual [19]. Both the Eissenberg [3]

and the Korodense [23] relations for tube inundation

effects were used.

Because only one half of the condenser is designed by

0RC0N1 , the quantity of inlet steam and the number of tubes

specified by reference 19 were halved.

The program would not accept an inlet steam rate of

greater than 200,000 pounds per hour (400,000 pounds per hour

for the entire bundle)

.

Holman [21] states that up to a 20 percent increase in

heat transfer rate may be realized by the ripples set up in

the condensate film by steam passing over the film. The

Korodense literature [23] indicates an enhancement due to

the same phenomena as less than the 20 percent reported by

Holman but also shows the enhancement to be a weak function

of tube location in the bundle.

An enhancement, due to film ripples, of 14 percent was

assumed for the current work.

The results of the verifications using both the Eissenberg

and the Korodense relations are tabulated in Table III with

the data from the CVA 67 technical manual [19] included for

48

comparison. Both relations yielded unsatisfactory design

verification and in both cases a very conservative condenser

was designed.

All parameters, as calculated by 0PC0DE2, were ten to

twenty percent less than those from the actual condenser.

With the removal of subroutine ADJUST, the condenser designed

by 0PC0DE2 vented in excess of 20 percent of the inlet steam

without condensing it.

It is believed that the effects of tube inundation give

a shell-side film heat transfer coefficient which is too

conservative as reflected in the reduced value of the overall

heat transfer coefficient.

D. LIMITATIONS OF 0PC0DE2

Because 0RC0N1 has the capability to make design decisions

and COPES/CONMIN requires a "once through" analysis, the

coupling of these two programs in 0PC0DE2 created a situation

where the programs were working against each other. This

placed a limitation on which parameters could be used as

design variables and on what range of values these design

variables could use.

The condenser designed by 0PC0DE2 was very conservative,

a condition caused primarily by the conservative value of

the overall heat transfer coefficient calculated.

Accepting these limitations, it was felt that 0PC0DE2

should be further developed and that case studies should be

performed

.

49

V. RESULTS

A. EXPLANATION OF THE CASE STUDIES

The case studies were devised to best exercise the

attributes of 0PC0DE1 and 0PC0DE2, and were made as realistic

as possible so as to simulate the problem of a condenser

design and specification during the early stages of power

plant design. Only input parameters that would normally

be available were used.

When comparing the results from the different cases, two

cautions must be kept in mind. First, since all the cases

involve four to six design variables, the design is taking

place in a four to six dimensional design space and intuition

on how an optimized design "should" turn out is not always

applicable. Secondly, the percentage change referred to in

each case is calculated based on the initial design for that

particular case.

1 . Constraint Framework for 0PC0DE1

In order to simulate an actual trade off study, the

constraints for each case study were kept the same, even

though an unimportant constraint could become active during

a particular case study. In this way, each case study could

be directly compared with all others.

The main condenser for the CVA 67 was to be designed

with a maximum bundle diameter of ten feet; a terminal

temperature difference (pinch point) of at least five degrees

50

Fahrenheit (°F) but not more than 35°F; a cooling water

temperature rise of at least five °F, but not more than 20°F;

and a ratio of tube sheet hole area to tube sheet area

without the drilled holes of less than 0.36.

The constraint on bundle diameter was chosen only

because it is a reasonable value. If more or less vertical

space was available in a proposed machinery arrangement, the

diameter constraint would be appropriately changed. The

lower constraint on pinch point came from reference 1

which called for a minimum terminal temperature difference

of five °F. No reference to an upper limit on pinch point

could be found, but several technical manuals specified

temperatures in the range of 20°F to 45°F. A value of 35°F

was therefore chosen as the upper bound on the terminal

difference

.

Reference 24 states that the difference in temperature

between the steam and cooling water streams entering the

condenser, or temperature range, is ordinarily kept to 20°F,

and that the cooling water temperature rise is usually made

about five °F less than the temperature range. Since the

CVA 67 exhausts steam with a saturation temperature of 125°F

and reference 3 calls for a cooling water injection tempera-

ture of 75°F, the temperature range used was 50°F and the

cooling water temperature rise upper limit was 45°F. As

the guidance given by reference 24 seemed inappropriate and

since neither reference 1 nor reference 2 specifies limits

on cooling water temperature rise, the lower limit was

51

arbitrarily set at five °F and the upper limit was

arbitrarily set at 20°F. The amount of tube sheet material

that can be removed by drilling for the installation of

tubes is specified at 24 percent of the blank tube sheet

area by reference 2. Since 0PC0DE1 does not allow for

steam lanes in the design of a condenser, and since these

steam lanes would provide blank tube sheet area, the

24 percent limit was raised to 56 percent.

To ensure that an unrealistic tube wall thickness

was not specified, a constraint on the wall thickness was

added. The wall thickness was calculated from the current

values of tube outside diameter and tube inside diameter,

and the constraint applied. Values of wall thickness in

the range from BWG 24 (0.022 inch) to BWG 12 (0.109 inch)

were used as the lower and upper constraints.

In summary, the general design constraints and the

associated upper and lower bounds were:

0.022 <_ tube wall thickness (inch) 0.109

1.0 <_ bundle diameter (feet) <_ 10.0

5.0 <_ terminal temperature difference (°F) <_ 55.0

5.0 <_ sea water temperature rise (°F) 20.0 .

These design constraints and the associated upper and lower

bounds were used in all of the OPCODE1 test cases except

where specifically modified.

52

2 . Design Variable Framework for 0PC0DE1

The condenser tube outside and inside diameters

were used as design variables. The side constraints were

set to correspond with the values of normally available

tubes [1] . The tube outside diameter was allowed to vary

in the range between 0.625 inch and 1.25 inch. The tube

inside diameter was allowed to vary from 0.407 inch to

1.206 inch.

The tube pitch is defined as the center to center

spacing between tubes. The pitch to diameter ratio (S/D)

is an accurate measure of how closely packed the tube

bundle is. Generally accepted S/D ratios lie in the range

of 1.3 to 1.7. However, to give greater latitude to this

design variable, and since 0PC0DE1 was insensitive to

shell-side conditions, the S/D ratio was allowed to vary

within the range from 1.1 to 3.0.

There was no guidance available on the range of tube

lengths that would be applicable to a condenser of this size

using 0.625 inch tubes. Reference 1 gave a range of

recommended tube lengths of eight to fourteen feet for

0.625 inch outside diameter tubes with an upper limit on

heat transfer surface area of 1000 square feet. In the

range of the expected heat transfer area of 12,000 to 18,000

square feet, the recommended tube outside diameters were

from 0.75 inch to 0.875 inch with tube lengths ranging

from 16 to 24 feet. Since the lower bound was not considered

to be as crucial as the upper bound, it was set at one foot.

The upper bound on tube length was set at 25 feet.

53

Cooling water velocity generally ranges from three

to nine feet per second for all common tube materials

except titanium which has an upper bound of 15 feet per

second.

In summary, the general design variables and the

associated side constraints were:

0.625 <_ tube outside diameter (inch) <_ 1.25

0.407 <_ tube inside diameter (inch) <_ 1.206

1.1 <_ pitch/diameter ratio 3.0

1.0 <_ tube length (feet) < 25.0

3.0 <_ cooling water velocity (feet/second) <_ 9.0

These design variables and the associated side

constraints were used in all of the 0PC0DE1 test cases

except where specifically modified.

3 . Constraint Framework for 0PC0DE2

The same constraints that were used for 0PC0DE1

were used for 0PC0DE2 with several additional constraints

required due to the physical calculations performed.

To ensure that the cooler was not designed to be

wider than the void inside diameter, the ratio of cooler

width to void diameter had an upper bound of 1.0 and a lower

bound of 0.1

As the initial design with 0PC0DE2 vented over

20 percent of the inlet steam without condensing it, an

54

upper bound on the exit steam fraction was set at five

percent. The lower bound was set to zero.

Since 0PC0DE2 was attempting to allow for a central

steam lane, the upper limit on bundle diameter was set at

12 feet and the lower limit was set at five feet.

Bundle volume had an upper limit of 1500 cubic

feet and a lower limit of 100 cubic feet. These limits

had been used satisfactorily with 0PC0DE1.

To ensure that the cooler height did not become

larger than the band of condenser tubes from the outside

diameter of the bundle to the inner void, the ratio of

cooler height to tube band width was calculated. This

ratio had an upper limit of 1.0 and a lower limit of 0.01.

The pinch point and the sea water temperature rise

had the same range as was used for 0PC0DE1.

The tube wall thickness was not used as a design

constraint in 0PC0DE2 as it had been in 0PC0DE1. Instead

the tube inside diameter was used, with the lower constraint

set at 0.407 inches and the upper constraint set at 1.206

inches to correspond with the values of normally available [1]

In summary, the general design constraints and the

associated upper and lower bounds were:

0.1 <_ cooler width/void diameter <_ 1.0

0.0 <_ exit steam percentage <_ 5.0

5.0 < bundle diameter (feet) < 12.0

55

100. <_ bundle volume (cubic feet) <_ 1500.

0.01 <_ cooler height/tube band width <_ 1.0

0.407 <_ tube inside diameter (inch) <_ 1.206

5 . <_ terminal temperature difference (°F) <_ 35.

5.0 <_ sea water temperature rise (°F) <_ 20.

4 . Design Variable Framework for OPCODE2

The condenser tube outside diameter and wall

thickness were used as design variables. The side con-

straints were set to correspond with the values of normally

specified tubes [1]. The tube outside diameter had a

lower side constraint of 0.625 inch and an upper side

constraint of 1.25 inch. The tube wall had a lower side

constraint of 0.022 inch and an upper side constraint of

0.109 inch.

Tube length had the same side constraints as were

used for 0PC0DE1.

The tube pitch to diameter ratio had a lower side

constraint of 1.4 and an upper side constraint of 2.0.

This band of allowable values was reduced from that used

with 0PC0DE1 because of instabilities that developed for

values of this variable outside of this band. More prob-

lems occurred at the lower end of the range than at the

upper end. These problems originated from the tubes coming

too close together and causing an excessively high steam

velocity with a subsequent large pressure drop through a

row of tubes. Since condenser pressure was already low,

56

passage through very few rows led to a negative steam

pressure and thus the computational problems associated with

the logarithmic calculation of thermodynamic properties

discussed in Chapter IV.

The sea water velocity was allowed to vary within

the range from three feet per second to ten feet per second.

The upper side constraint was raised from the nine feet

per second specified in 0PC0DE1 in an attempt to increase

the mass flow rate of the cooling water and, therefore, the

heat rejection rate.

In summary, the general design variables and the

associated side constraints were:

5.0 <_ tube length (feet) <_ 25.0

3.0 <_ sea water velocity (feet per second) 10.0

0.625 tube outside diameter (inch) <_ 1.25

0.022 <_ tube wall thickness (inch) <_ 0.109

1.4 <_ tube pitch/diameter ratio <_ 2.0

B. CASE STUDIES USING 0PC0DE1

1 . Case One

The objective of this test case was to minimize

the pumping power requirement while holding the heat load

to the condenser constant. The input parameters are pre-

sented in Table IV, the' initial design is presented in

Table V, and the results of the optimization are presented

in Table VI.

57

These results show an 84 percent decrease in pumping

power with a commensurate 37 percent increase in heat

transfer surface area, a 55 percent increase in condenser

volume, and a decrease of 27 percent in overall heat trans-

fer coefficient. The log mean temperature difference

remained constant.

The lower bound on tube wall thickness constraint

and the upper bound on sea water temperature rise constraint

were both active. No side constraints were active and no

constraints were violated.

The decrease in pumping power is impressive but

so are the increases in condenser dimensions with the implied

increases in weight and cost.

2 . Case Two

The objective of this test case was to minimize

condenser volume with the heat load to the condenser held

constant. The input parameters are presented in Table IV,

the initial design is presented in Table V, and the results

of the optimization are presented in Table VII.

These results show a 15 percent decrease in condenser

volume with an unexpected 5.0 percent decrease in pumping

power. This design can be understood by noting that the

tube wall thickness was reduced from 0.049 inch to 0.022

inch causing an increase in material correction factor

from 0.90 to 0.99. This increase was the primary factor

in increasing the overall heat transfer coefficient 9.9

percent

.

58

The lower constraint on tube wall thickness, and

the upper constraint on tube sheet area ratio were both

active. In addition, the lower side constraint on tube

outside diameter and the upper side constraint on sea water

velocity were both active. There were no violated constraints

3 . Case Three

The objective of this case was to maximize the heat

rejected while holding pumping power constant. Since the

pumping power is a calculated quantity, the method of com-

bining pumping power and the heat rejected with an appro-

priate weighting factor in the objective function was used.

The objective function was therefore:

OBJ = QREJ + A * POWER

and POWER was added as a design constraint with the target

value of 68836 foot pounds per second as the upper constraint.

With the weighting factor, A, set at 5600, the pumping

power was constant within one percent. Turbine exhaust

steam and auxiliary exhaust steam were both added as design

variables to provide the driving force to increase the heat

rejection rate.

The input parameters are presented in Table IV,

the initial design is presented in Table V and the results

of the optimization are presented in Table VIII.

59

These results show an 82 percent increase in the

heat rejection rate with an accompanying 88 percent increase

in surface area. The log mean temperature difference

remained constant and the overall heat transfer coefficient

decreased by 3.7 percent. The condenser volume increased

by 78 percent and the number of tubes increased by 96 percent

The upper constraint on pumping power was active, as

desired, and the upper constraint on the tube sheet area

ratio was active. There were no violated constraints and

no active side constraints.

It should be noted that the turbine exhaust steam

rate increased by 83 percent, whereas the auxiliary exhaust

steam rate increased only 1.0 percent. Inspection of the

component parts of QREJ explains the dominance by the main

steam rate. The rejected heat was calculated by:

Q . = m.. c Ah,, + m, c Ah, c + heat from other sourcesxrej MS MS AS AS

where the subscript MS corresponds to main steam and the

subscript AS corresponds to auxiliary steam. Heat from

other sources was assumed constant during the optimization.

The product nu,^ AhMC, was several orders of magnitude greater

than the product m.~ Ah.- and it therefore dominated the

optimization process as a change in the mMC, design variable

would yield a greater design improvement than would a like

change in the m.- design parameter.

60

In all probability, the m.~ design variable would

not begin to make an appreciable move until the niwg design

variable approached its upper side constraint. This could

not occur as the mMC, design variable had an unbounded upper

side constraint.

No attempt was made to balance the flow of steam

from the two sources to more evenly distribute the incoming

heat load. Balancing could be accomplished by appropriately

scaling the two mass flow rates so they would have approximately

the same values

.

4 . Case Four

The objective of this case was to minimize the

condenser volume while holding pumping power and the rejected

heat constant. POWER was again included in the objective

function

OBJ = VOL + A * POWER

and as a design constraint with a lower bound of 68836 foot

pounds per second. The lower bound was the target value

for pumping power. Trial and error solutions with various

values of the weighting factor showed that a value of

A = 0.0 gave a value of POWER which was constant within

0.5 percent. This was interpreted to mean that the optimi-

zation would have reduced the objective function further

if the pumping power constraint was not present and that

61

the pumping power lower constraint was therefore always

active. With the value of this constraint set at the

target value for constant pumping power, inclusion of POWER

in the objective function was unnecessary and A = 0.0 was

appropriate

.

The input parameters are presented in Table IV,

the initial deisgn is presented in Table V and the results

of the optimization are presented in Table IX.

These results show only a 2.0 percent decrease in

condenser volume. The tube pitch to tube diameter ratio

remained essentially constant as did the tube outside diameter

The tube wall thickness increased by 11 percent with an

accompanying decrease in material correction factor from

0.90 to 0.88.

The lower constraint on pumping power was active,

as discussed above. The upper constraint on sea water

temperature rise, as well as the upper constraint on tube

sheet area ratio were active. The upper constraint on sea

water velocity was the only active side constraint. There

were no violated constraints.

The results of this case study indicated that the

circular bundle prototype was very close to having an opti-

mum volume within the given design variable and design

constraint framework.

Comparison of Cases Two and Four shows that a

smaller condenser with essentially constant pumping power

62

was achieved with Case Two then with Case Four. This

anomaly can be understood by comparing the constraints

for the two cases.

Pumping power was not a constraint for Case Two

and it was an active constraint for Case Four. Thus,

the Case Four optimization had to contend with an additional

constraint whereas, Case Two had more latitude within the

design space and a better optimum was found.

5 . Case Five

The objective of this case was to minimize the

condenser overall length while holding pumping power and

the heat rejected constant. Pumping power was included

in the objective function:

OBJ = CLOA + A * POWER

and as a design constraint with a lower bound of 68856 foot

pounds per second as the target value. With a weighting

- 4factor of 3.526 x 10 , pumping power was held constant

within 0.5 percent.

The input parameters are presented in Tabe IV, the

initial design is presented in Table V and the results of

the optimization are presented in Table X.

The overall condenser length was reduced by 28 percent

The heat transfer area increased by 9.7 percent and the

overall heat transfer coefficient decreased by 15 percent,

63

while the log mean temperature difference increased by

7.1 percent. The number,'of tubes increased 96 percent and

the tube length decreased 44 percent. The tube wall thick-

ness increased causing a decrease of 14 percent in the

material correction factor thereby contributing to the

decrease in overall heat transfer coefficient.

Accompanying the 23 percent decrease in condenser

length was a compensating 37 percent increase in bundle

diameter and an 18 percent increase in condenser volume.

The lower constraint on pumping power was active,

as desired. The upper constraints on pinch point and on

tube sheet area ratio were active. There were no active

side constraints and no violated constraints.

6. Case Six

The objective of this case was to minimize pumping

power while holding condenser volume and the heat rejected

constant. For this case, volume was included in the

objective function:

OBJ = POWER + A * VOL

and it was also included as a design constraint with an

upper bound of 928.8 cubic feet as the target value. The

usual trial and error procedure with various weighting

functions was performed and the best results were obtained

for a weighting factor of -2.5. With this value, condenser

volume was held constant to within 1.3 percent.

64

The input parameters are presented in Table IV,

the initial design is presented in Table V and the results

of the optimization are presented in Table XI.

The pumping power was reduced by 35 percent. The

log mean temperature difference and the heat transfer area

remained essentially constant, while the tube wall thickness

decreased 52 percent causing a ten percent increase in the

tube material correction factor. The sea water velocity

was reduced by 16 percent, however, and the overall heat

transfer coefficient remained essentially constant. There

were no active constraints, no violated constraints and

no active side constraints.

Volume was not forced against its upper constraint

as planned because the two members of the objective function

were attempting to move in opposite directions. As power

was decreased, the volume tended to increase. The negative

weighting factor helped to turn this process around but

the optimum was reached before the constraint was activated.

7 . Case Seven

English has shown [25] that by placing boundary

layer fences in front of the injection scoop of a ship,

the pressure coefficient, and hence pumping power, will

increase by 68 percent. These boundary layer fences cause

shedding vortices to form behind them thus pumping water

from the boundary layer around the ship into the condenser

intake

.

65

This case takes advantage of the increase in

available pumping power by setting pumping power at a

value 50 percent greater than the normal 68836 foot pounds

per second, and holding this new value of pumping power

constant while maximizing the rejected heat.

The constraints and design variables were set up

exactly as they were for Case Three with the exception that

the upper constraint on pumping power was raised to 103,250

foot pounds per second. POWER was included in the objective

function as before and a weighting factor of 4095 yielded

pumping power constant within 0.5 percent.

The input parameters are presented in Table IV, the

initial design is presented in Table V and the results

of the optimization are presented in Table XII.

The heat rejected was increased by 107 percent. The

heat transfer area increased by 101 percent, and the log

mean temperature difference remained constant. The tube

wall thinned down by 53 percent causing an increase in the

tube wall material correction factor of ten percent. The

sea water velocity decreased 15 percent and the overall heat

transfer coefficient increased 2.5 percent.

The upper constraints on pumping power and the tube

sheet area ratio were active. There were no active side

constraints and no violated constraints.

66

C. CASE STUDIES USING 0PC0DE2

1. Case Eight

This case was run assuming plain copper-nickel

tubes with a 14 percent enhancement factor applied to the

outside film heat transfer coefficient as described in

Section IV(C) of this work. Steam baffles were specified

and the inlet steam flow rate was set at 200,000 pounds per

hour as only one half of a condenser bundle was being simu-

lated. The Korodense equation from reference [23] was used

to calculate the correction factor for tube inundation by

condensate rain.

The objective of this case was to minimize condenser

volume. The initial and optimum designs are presented in

Table XVI.

The volume was reduced 13 percent and the heat

transfer area increased 14 percent. The log mean tempera-

ture difference decreased 5.6 percent, the overall heat

transfer coefficient increased 11 percent and the heat

rejected increased 21 percent. The vented steam rate

decreased 76 percent to five percent of the inlet steam

flow rate. Finally, pumping power increased 65 percent

and the sea water flow rate increased 35 percent.

The upper side constraint on sea water velocity as

well as the lower side constraints on tube outside diameter,

tube pitch to diameter ratio and tube wall thickness were

all active. The upper constraint on vented steam rate

was active. There were no violated constraints.

67

2 . Case Nine

The objective of this case was to minimize the

condenser volume when enhanced tubes were specified.

The initial parameters were the same as those used in

Case Nine with the exception of the outside enhancement

factor which was set to 1.8. It was felt that this modest

enhancement was attainable with the augmented tubes

currently being manufactured.

The initial and optimum designs are presented in

Table XVII.

Use of enhanced tubes reduced the condenser volume

by 21 percent. The heat transfer area increased 2.6 per-

cent, the overall heat transfer coefficient increased 34

percent, the log mean temperature coefficient decreased

11 percent and the heat rejected increased 21 percent. The

vented steam rate decreased 76 percent to five percent of

the inlet steam flow rate.

Pumping power increased 52 percent and the cooling

water flow rate increased 35 percent. Tube length was

increased by 2.7 percent and the tube inside diameter

increased ten percent.

The pumping power calculated was based on smooth

tubes. The most common augmented condenser tubes use either

a rope design, a twist design, or internal flow promoters

to enhance heat transfer. Reilly [26] has found that the

pressure drop through enhanced tubes is 1.5 to 3.0 times the

pressure drop through smooth tubes for the same flow conditions

68

The upper side constraint on sea water velocity,

as well as the lower side constraints on tube outside

diameter, tube pitch to diameter ratio and tube wall

thickness were all active. The upper constraint on vented

steam rate was active, and there were no violated constraints.

3 . Case Ten

This case was run using the same conditions as

Case Eight with the exception that the original equation

from 0RC0N1 was used to correct for tube inundation. Reference

3 specified a value for the flooding factor, FDAVE , of

zero for pitch to diameter ratios greater than 1.4.

This value was used during this run of 0PC0DE2.

The objective of this case was to minimize the condenser

volume. The initial and optimum designs are presented in

Table XVIII.

A more conservative design was produced with the

Eissenberg equation than the design produced with the

Korodense equation. The volume was reduced 5.3 percent and

the heat transfer area increased 24 percent. The log mean

temperature difference remained constant, the overall heat

transfer coefficient increased by 8.9 percent, and the

heat rejected increased by 33 percent. The vented steam

rate decreased 83 percent to five percent of the inlet

steam flow rate. Pumping power increased 75 percent and the

sea water flow rate increased 35 percent.

The upper side constraint on sea water velocity,

as well as the lower side constraints on tube outside

69

diameter, tube pitch to diameter ratio and tube wall

thickness were all active. The upper constraint on vented

steam rate was active. There were no violated constraints

70

VI. CONCLUSIONS

The intent of this investigation was to couple a

numerical optimization code with both a simple computer

code for condenser design based on the HEI/DDS method

and the more sophisticated 0RC0N1 , and to test the programs

developed with a variety of test cases to prove their

viability. The results of these test cases were presented

in Section V; the resulting conclusions are summarized

here

.

A. A condenser designed by the HEI/DDS method is very

near the volumetric optimum in design, as only a

15 percent decrease in condenser volume was

realized when volume was optimized using 0PC0DE1.

B. 0PC0DE1 is an excellent design tool for the concep-

tual design of a condenser to meet specified con-

straints such as length, height, and weight.

C. 0PC0DE1 can optimize a variety of objective func-

tions, and with some trial and error application

of weighting factors, equality constraints can

be met.

D. 0PC0DE1 will yield an optimum design on which

experience-based safety factors may be applied.

E. The ability to enhance tubes and the sensitivity

of the program to shell-side conditions make the

ORCONl-based 0PC0DE2 very attractive.

71

F. The basic equations and the research applied to

the development of 0RC0N1 are sound, but a conser-

vatively-designed condenser results. More baffles

are required to more closely approximate an actual

condenser and to aid in reducing the inundation

effect on interior tubes.

G. The 0PC0DE2 designed condensers were more conserva-

tive than those designed by 0PC0DE1. This was

caused by the detrimental effect on outside film

coefficient by tube flooding. It is felt that this

effect could be reduced by the addition of more

baffles.

72

VII. RECOMMENDATIONS

In addition to the insight that this investigation has

given into the generation of automated design programs for

condenser design, it has also generated an awareness of

this investigation's shortcomings. Presented herein are

recommendations for improving upon and furthering develop-

ment of the OPCODE family of condenser design programs.

A. 0PC0DE1 should be expanded to include the capability

for multi-pass condenser calculations, geometrical

options, material costing equations, strength of

material considerations, and an algorithm to compute

steam velocity through the rows of tubes to ensure

that the velocity remains realistic.

B. Research should be performed to develop enhancement

factor data for various tube types that can be

applied to the basic equations used in 0PC0DE1.

C. 0RC0N1 should be rewritten with the intent of

applying optimization techniques to the new program,

and based on the equations and logic developed by

Eissenberg, Korodense, and other investigators and

manufactures, with the option to choose the

desired relationship to be used.

D. In an effort to reduce the computer time required

for a run using 0PC0DE2 (typically 55 CPU minutes)

,

an investigation into using 0PC0DE1 as a pre-

processor in front of a detailed analysis scheme is

attractive

.

E. Expand 0PC0DE2 to perform calculations for other

than a circular bundle.

74

VIII. FIGURES

h A

FEASIBLEREGIONG-<CT

INFEASIBLE REGION

Gj> [CT|

FIGURE 1. Significance of the Constraint Thickness (CT)Parameter

75

2.2 r-

2.0 -

© 1.8 -

cc

a:LUI—

<QIEC_3

CO

1.6 -

1.4 -

1.2 -

SIDE CONSTRAINT

CONTOUR OF CONSTANT VOLUME

1.00.0

=

0.2 0.4 0.6 0.8

TUBE OUTSIDE DIAMETER (INCH)

FIGURE 2. Two-Variable Design Space Showing the InitialDesign at Point (A) .

76

2.2

2.0

1.8

a:

£ 1.6

t—

I

Q

1.4

CD

1.2

1.0

SIDE CONSTRAINT

CONTOUR OF CONSTANT VOLUME

OD r=0.18> gLL"

0.0 0.2 0.4 0.6 0.8

TUBE OUTSIDE DIAMETER (INCH)

FIGURE 3. Two-Variable Design Space Showing the FirstDesign Iteration

77

2.2 i-

SIDE CONSTRAINT

CONTOUR OF CONSTANT VOLUME

2.0 -

1.8 -

0£

£ 1-6

O

1.4 -

CO13

1.2 -

1.00.0 0.2 0.4 0.6 0.8

TUBE OUTSIDE DIAMETER (INCH)

FIGURE 4. Two-Variable Design Space Showing the SecondDesign Iteration

2.2 »-

SIDE CONSTRAINT

CONTOUR OF CONSTANT VOLUME

2.0 -

1.8

<

!- 1.6 -UJ

<(—1

Q

~ 4

CO

1.2 -

1.00.0 0.2 0.4 0.6

TUBE OUTSIDE DIAMETER (INCH)

HP -HP =0max

0.8

FIGURE 5. Two-Variable Design Space Showing the ThirdDesign Iteration

79

2.2 r-

2.0 -

1.8 -

<

<1—1

Qa:o

CO

1.6 -

1.4 -

1.2 -

1.0

SIDE CONSTRAINT

CONTOUR OF CONSTANT VOLUME

0.0 0.2 0.4 0.6 0.8

TUBE OUTSIDE DIAMETER (INCH)

FIGURE 6. Two-Variable Design Space Showing the FourthDesign Iteration

80

2.2 »-

SIDE CONSTRAINT

CONTOUR OF CONSTANT VOLUME

2.0 -

1.8 -

<en

ccLU

h 1.6 -LU

<Q

I—""• 1.4

LUOQ

1.2

1.00.0 0.2 0.4 0.6 0.8

TUBE OUTSIDE DIAMETER (INCH)

FIGURE 7. Two-Variable Design Space Showing the FifthDesign Iteration

81

SIDE CONSTRAINT

2.2

2.0

1.8

<

UJ

<I—

t

Q

CD

1.6

1.4

1.2

1.0•OD r=0.18

J P n1 i

0.0 0.2 0.4 0.6 0.8

TUBE OUTSIDE DIAMETER (INCH)

FIGURE 8. Two-Variable Design Space Showing the Need forOptimization Techniques

82

g(X) = o

F(X)= CONSTANT

Y^^b \

\c \

\

^_p\ \^ sXsyysyv

FIGURE 9. Two-Variable Design Space Showing RelativeMinima for a Constrained Minimization Problem

83

STEAMFLOW

AIR COOLERBAFFLE

0.866-q X DBAFFLE

STEAMFLOW

FIGURE 10. Circular Condenser Bundle Designed by ORCON1

84

STEAM IN

SEA WATER OUT-

CONDENSATE OUT

STEAM IN

JO AIR EJECTOR

AIRCOOLER

TO AIR EJECTOR.

AIR COOLER

-TUBE SHEETS—'

SECTION 0-0

FIGURE 11. CVA 67 Main Condenser as Modeled by ORCON1and OPCODE2

85

£• V. © >j x ;> o_« •• »i a -4-0 *t-»o-&

K

9 la

cd I® 4 -

-4 -.

••> o a

-

o 3 • • 1 a £

I

~1x a

J

O

co

<D

-Q3HUCD

c

-a

o

ti

<>

CsJ

0BS

1—

1

S'JmO'a »3i.-i* 3Qi<;ki

— h-,«i •

86

f MAIN )

CAULINPUT

CALCULATE

INLET STEAM

FACTORS

CALCULATETUBE LENGTH

FACTORS

CALL

SECALC

CALCULATECONOITIONS

AFTER

CONOENSER

GEj

FIGURE 13. Flowchart from 0RC0N1

87

C MAIN J

REMOVED

REMOVED

FIGURE 14. Flowchart Showing Modification of 0RC0N1to 0PC0DE2

S8

IX. TABLES

TECHNICALMANUAL 0PC0DE1 DEVIATION

m2

Heat transfer area; ft 16,011 16,099 + 0.55

Number of tubes 6,612 6,648 + 0.54

Bundle diameter; ft N/A 7.26 -

Heat rejected; BTU/hr 4.035X10 8 4.035xl0 80.00

Overall heat transfercoefficient

•

BTU/(hr) (ft 2) C°F)

635 637 + 0.31

Log mean temperaturedifference; °F

39.7 39.3 -1.01

Sea water temperature rise;op

19.9 19.9 0.0

Terminal temperaturedifference; °F

30.53 30.2 -1.08

Sea water flow rate, gpm 40,565 40,645 + 0.20

Tube-side pressure drop;f t . w. c

.

16.3 12.4 -23.9

TABLE I. Verification of the CVA 67 MainCondenser Design Using 0PC0DE1

89

TECHNICALMANUAL 0PC0DE1

DEVIATIONm

2Heat transfer area; ft 6,600 6,527 -l.ii

Number of tubes 3,892 3,850 -1.08

Bundle diameter; ft 5.9 4.6 -22.1

Heat rejected; BTU/hr 2.05xl08

2.047xl08 -0.15

Overall heat transfercoefficient*BTU/(hr) (ftO (°F)

635 637 + 0.31

Log mean temperaturedifference; °F

50.0 49.2 -1.6

Sea water temperature rise;op

16.9 17.4 + 2.96

Terminal temperaturedifference; °F

41.9 41.0 -2.15

Sea water flow rate; gpm 23,900 23,539 -1.51

Tube-side pressure drop;ft. w. c.

11.5 8.21 -2S.6

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100

INITIALVALUE

VALUE ATOPTIMUM

CHANGE

2Heat transfer area; ft 16,008 18,290 + 14

Number of tubes 6,612 6,612 0.0

Bundle diameter; ft 10.23 8.95 -13

Heat rejected; BTU/hr 3.241xl08

3.916xl08

+ 21

Overall heat transfercoefficientBTU/Chr) (ft2) (°F)

562 625 + 11

Log mean temperaturedifference; °F

36 34 -5.6

Sea water temperaturerise; °F

15.6 13.7 -12

Terminal temperaturedifference; °F

25.6 24.6 -3.9

Sea water flow rate; gpm 40,341 54,479 + 35

Tube-side pressure drop;f t . w. c

.

12.3 - -

Tube length; ft 14.8 16.9 + 14

Tube outside diam. ; in. 0.625 0.625 0.0

Tube inside diam.; in. 0. 527 0. 581 + 10

Tube wall; in. 0.049 0.022 -55

S-W velocity; fps 9.0 10.0 + 11

Pitch/diameter 1.6 1.4 -13

Exit steam; % of input 21 5 -76

Pumping power; ft- lb/sec 55,409 58,290 + 65

Condenser volume; ft 775 678 -13

- - —

—

TABLE XIII. Case Eight Initial Design andOptimization Results

101

INITIALVALUE

VALUE ATOPTIMUM

CHANGEm2

Heat transfer area; ft 16,008 16.431 + 2.6

Number of tubes 6,612 6,612 0.0

Bundle diameter; ft 10.23 8.95 -13

Heat rejected; BTU/hr 3.241xl08

3.924xl08

+ 21

Overall heat transfercoefficient,-,BTU/Chr) (ft

z) (°F)

562 751 + 34

Log mean temperaturedifference; °F

36 32 -11

Sea water temperature 15.6 13.1 -16rise; °F

Terminal temperaturedifference; °F

25.6 21.6 -16

Sea water flow rate; gpm 40,341 54,479 + 35

Tube-side pressure drop;f t . w. c

.

12.5 - -

Tube length; ft 14.8 15.2 + 2.7

Tube outside diam. ; in. 0.625 0.625 0.0

Tube inside diam.; in. 0.527 0. 581 + 10

Tube wall; in. 0.049 0.022 -55

S-W velocity; fps 9 10 + 11

Pitch/diameter 1.6 1.4 -13

Exit steam; % of input 21 5 -76

Pumping power; ft- lb/sec 35,409 53,674 + 52

Condenser volume; ft-3

775 609 -21

TABLE XIV. Case Nine Initial Design andOptimization Results

10

INITIALVALUE

VALUE ATOPTIMUM

CHANGEm

2Heat transfer area; ft 16,008 19,808 + 24

Number of tubes 6,612 6,612 0.0

Bundle diameter; ft 10.23 8.95 -13

Heat rejected; BTU/hr 2.937xl08

3.910xl08

+ 33

Overall heat transfercoefficient

•

BTU/(hr) (ft*) (°F)

505 550 + 8.9

Log mean temperaturedifference; °F

56 36 0.0

Sea water temperaturerise; °F

14.4 14.2 -1.4

Terminal temperaturedifference; °F

25.2 26.3 + 4.4

Sea water flow rate; gpm 40.341 54,479 + 35

Tube-side pressure drop;f t . w. c

.

12.3 16.0 + 30

Tube length; ft 14.8 18.5 + 24

Tube outside diam. ; in. 0.625 0.625 0.0

Tube inside diam.; in. 0.527 0. 581 + 10

Tube wall; in. 0.049 0.022 + 11

S-W velocity; fps 9 10 + 11

Pitch/diameter 1.6 1.4 -13

Exit steam; % of input 29 5 -83

Pumping power; ft -lb/sec 35,409 62,059 + 75

Condenser volume ; ft 775 734 -5.3

TABLE XV. Case Ten Initial Design andOptimization Results

105

APPENDIX A

DEVELOPMENT OF THE ANAL I Z SUBROUTINE FOR OPCODE1

OPCODE1 was based on the HEI/DDS method of calculation

of heat transfer and thermodynamic parameters.

The following equations are used in the design of

a condenser using the HEI/DDS method.

U = F,F.,F-C/V (A-l)c 1 2 :>

Q = Uc

AH

ATLM

(A- 2)

Q = W Ah (A- 3)

Q = 500 G (tQ

- t^ (A-4)

ATLM k -Y (A - 5 )

in (^_^i)

S

AR

= L N s (A-6)

G = Ng V (A-7)

k = s/g (A-8)

The ANAL I Z subroutine of 0PC0DE1 was divided into three

sections: input, analysis and output. The correct section

104

of ANALIZ was entered by testing for the value of ICALC

that was passed from COPES to ANALIZ.

When ICALC = 1, the input section of ANALIZ was entered

and the input variables and problem identification were

read. The input section had the capability to default the

latent heat of vaporization to 950 BTU per pound as speci-

fied by reference [2] and to calculate either the saturation

temperature or the saturation pressure depending on which

value was not read in on the data card.

When COPES set ICALC = 2, the analysis section of

ANALIZ was entered and the calculation of the outside

area of a condenser tube per foot of tube length was made:

?ft

it D[ff-]

.

The rate of flow, in gallons per minute, through a

condenser tube at a velocity of one foot per second was

then made

:

Gl = (_j) (d2

) (60x7.481) = 448.46(j)d'

riL_][f t2 ]r£eC]rgal] = .gaK

L sec H J L min J L^T J L min J

The value of C in equation (A-l) was assumed to be a

constant 270 when in fact, C was weakly dependent on tube

outside diameter. With this assumption, the uncorrected

heat transfer coefficient was calculated as:

105

u = c/v [

BTU ] .

hr-ft .°F

A call was then made to subroutine MATFAC to find the

new value of the material correction factor, F2 , based on

the current value of tube wall thickness. The temperature

correction factor, F,, was found with the function sub-

routine TEMFAC.

The data from Figure 1 of reference [2] was implemented

in TEMFAC and a value of F„ as a function of injection3

J

temperature was retrieved using subroutine INTRPL, a systems

supplied interpolator.

The fouling factor, F, , was set to 0.85 in accordance

with references land 2 during the input section of ANALIZ.

The corrected value of the overall heat transfer

coefficient was calculated:

U = F1

F 9 F- Uc 1 2 j>

The original form of equation (A-4) was

Q = 60 W C G (t - t.)p ^ o i

J

For sea water, the following value of 60 W C was&p

calculated

:

60 (W C )s . w

= (60) (8.55) (0.94) = 482

106

while, for fresh water, 60 W C was calculated asP

60(W Cp ) F . w

= (60) (8.33) CI. 0) = 500

Reference [2] recommends the use of 60 W C = 500 inP

keeping with industry standards. This approximation will

induce an error of approximately one-half percent.

Be setting equation (A-2) equal to equation (A-4)

and substituting for values of G, AH and AT TM , the following

relation was obtained:

(A-9)

ts

t

.

1 U L K

ts

• V c

500 V

which can be written as

:

t - t

.

si a

V~^ = e

where

U L K_ _c

a " 500 V

Equation (A-9) was solved for the cooling water outlet

temperature

t - t.

t = tx

o s a

107

The sea water temperature rise and the pinch point were

calculated, as was the initial temperature difference. All

the factors were now available for the calculation of the

logarithmic mean temperature difference, ^Tjw, using

equation (A- 5)

.

The input heat load was divided into the auxiliary

steam flow, main steam flow, and heat from other sources.

The auxiliary and main steam flows were multiplied by

their respective changes in latent heat, Ah, and summed

with the heat from other sources to yield the total heat

input, Q.

If an input value for Ah is not read in for either

auxiliary or main steam flow, the default value of 950 BTU

per pound, specified by reference [2], is used.

With the value of Q, equation (A-4) was utilized to

calculate the required flow rate of cooling water, G. With

the value of G known, the heat transfer area, A„ , was

found using equations (A-6), (A-7) and (A-8):

A =L G KA

H V

The heat rejected to the cooling water was found by

using equation (A-2).

The next portion of ANALIZ was devoted to the calcu-

lation of bundle geometry. The call to subroutine GEOM

yielded a condenser design of circular bundle cross section

108

with a 12 inch diameter void along the longitudinal center-

line to serve as a header for the removal of nonconden-

sable gasses by an air ejector. The rows of tubes were

filled from the inner void and were concentric to the

center void. Partial tubes were permitted in order to

simplify the algorithm and the final, outermost row was

only partially filled. The bundle radius was calculated

with no allowance for circumferential steam lanes, since

it is beyond the capability of the HEI/DDS method to provide

for steam lanes

.

It was assumed that the waterboxes were hemispherical

caps on the ends of the cylindrical bundle. The waterboxes

may not be any deeper than 45 inches [2] and logic was

included to ensure that this specification was met.

The ratio of tube sheet material removed for tube

installation to original undrilled tube sheet area was

calculated and used as a design constraint during the

optimization process. Reference [2] requires that the

ratio calculated must be less than or equal to 0.24 to

allow for adequate tube sheet strength.

A hotwell capacity capable of receiving the condensate

from one minute of full power operation is specified by

reference [2]. This value was calculated as the product

of the specific volume of the incoming steam and the

steam flow rate.

The calls to subroutines FRIFLAC and PRSDRP found the

factors required for the calculation of tube side pressure

drop and pumping power.

109

Total condenser volume was the sum of the tube bundle

volume, the waterbox volume, and the hotwell volume with

the individual watervoxes treated as spherical caps.

The final step in the analysis section of subroutine

ANALIZ was to define the objective function with any

necessary weighting factors as described in Chapter II.

COPES repeatedly enters the ANALIZ subroutine with

ICALC = 2 during the optimization process — often on the

order of hundreds of times. Therefore, no WRITE statements

were included in the analysis section. All output was

performed within the output portion of ANALIZ, a region

that is entered only when the optimization process is

terminated and ICALC is set equal to three.

110

APPENDIX B

DEVELOPMENT OF SUPPORTING SUBROUTINES FOR OPCODE1

In this appendix, the subroutines that were utilized

during the execution of ANALIZ in OPCODE1 are briefly

described.

FRIFAC

This subroutine iteratively solved the transcendental

Colebrook equation [17]:

2.51+

/T " ^Re/f3

2 log C^= + t^)

for the value of the internal friction factor for tube

flow. FRIFAC was valid for the range

0.01 <_ f <_ 0.10 .

This method was chosen over the simpler Blasius equation

because of the inclusion of the desirable dependence on

the relative roughness, e/d, in the Colebrook equation.

GEOM

This subroutine calculated bundle geometry. The

condenser bundle was assumed to have a 12 inch central void

along the longitudinal centerline with the rows of tubes

in concentric rows about the void. The void served as a

111

header for the venting of noncondensable gasses. The rows

were filled from the inside out; partial tubes were permitted

and the outermost row was unfilled.

PRSDRP

This subroutine calculated the cooling water pressure

drop through the condenser bundle. The development of this

subroutine was thoroughly discussed in Chapter III and will

not be repeated here.

DENSE

This subroutine calculated the density of water as a

function of temperature with the relation from reference [27]:

p = 63.8 - 0.01781 t + 1.132 x 10" 5t2

- 6.786 x 10" 8t3

.

Based on the assumption made for determining the coeffi-

cient for equation (A-4) in Appendix A, the relation given

above was used for the sea water as well as the steam and

condensate densities.

For the calculation of cooling water density, the

numerical average of the cooling water injection temperature

and the cooling water overboard temperature was used.

PSATFN

This function subroutine calculated the saturation

pressure of steam as a function of the saturation temperature,

112

in degrees Rankine, using the relation from reference [3]:

In Ps

= 14.150119 -64f-

562-

S57Sl

3 - 21(B-l)

s Ts

TEMFAC

The retrieval of the temperature correction factor,

F,, was performed in this function subroutine. The data

presented in Figure SF-2 of reference [1] was implemented

in tabular form and was retrieved as a function of cooling

water injection temperature by a call to the library

supplied interpolation subroutine, INTRPL.

TSATFN

This function subroutine calculated the saturation

temperature as a function of saturation pressure by solving

equation (B-l) for T .

MATFAC

This subroutine retrieved the material correction

factor, Fy , as a function of wall thickness and tube material

The data presented in Figure ST-1 of reference [1] was

in tabular form and was retrieved using the library supplied

interpolation subroutine, INTRPL.

113

APPENDIX C

USER'S MANUAL FOR OPCODE 1

This Appendix describes the data cards required for

the use of 0PC0DE1. A complete optimization run is included

as Appendix D.

The data is divided into the COPES/CONMIN program section

and the HEI/DDS-based condenser design program section.

The COPES data is segmented into "blocks" for convenience.

All formats are alphanumeric for TITLE, END, and STOP cards,

F10 for real data and 110 for integer data. Comment cards

may be inserted anywhere in the data stack prior to the END

card and are identified by a dollar sign ($) in column 1.

The COPES data stack must terminate with an end card

containing the word "END" in columns 1-3.

The analysis data is also segmented into blocks for

convenience and must immediately follow the "END" card.

No comment cards are permitted and the analysis data stack

must terminate with the word "STOP" in columns 1-4.

It should be noted that only the information for using

OPCODE1 for either a single analysis or for optimization

is included in this User's Manual. Information pertaining

to the use of the sensitivity analysis and the two-variable

function space features of COPES can be found in reference [15]

114

DATA BLOCK A

DESCRIPTION: COPES Title Card

FORMAT: 20A4

TITLE CARD

REMARKS

1) Program is terminated by the word 'STOP' in columns 1-4

115

DATA BLOCK B

DESCRIPTION: COPES Program Control Parameters

FORMAT 7110

1 2 3 4 5 6 7 8

NCALC NDV IPNPUT

FIELD CONTENTS

NCALC: Calculation control- Read input and stop. Data of blocks

A-B is required. Remaining data isoptional

.

1 - One cycle through program. Data ofblocks A-B is required. Remaining datais optional.

2 - Optimization. Data of blocks A-I isrequired. Remaining data is optional.

NDV: Number of independent design variablesin optimization or optimum sensitivitystudy.

IPNPUT: Input print control- Print card images plus formated print

of input.1 - Formated print of input only.2 - No print of input.

REMARKS1) Field 1 determines program execution.

2) Fields 5, 4, 6, 7, and 8 to be left blank for the0PC0DE1 application of COPES/CONMIN.

116

DATA BLOCK C

DESCRIPTION: COPES Integer Optimization Control Parameters

FORMAT: 8110

1 2 3 4 5 6 7 8

IPRINT ITMAX ICNDIR NSCAL ITRM LINOBJ NACMX1 NFDG

FIELD CONTENTS

1

2

3

4

5

2 ITMAX:

3 ICNDIR

4 NSCAL:

IPRINT: Print control used in optimization program,CONMIN.No print during optimization.Print initial and final optimizationinformation

.

Print above plus function value anddesign variable values at each iteration.Print above plus constraint values

,

direction vector and move parameterat each iteration.Print above plust gradient information.Print above plus each proposed designvector, objective function and constraintsduring the one-dimensional search.Maximum number of optimization iterationsallowed. DEFAULT = 20.Conjugate direction restart parameter.DEFAULT = NDV+1.Scaling parameter. GT.O - Scale designvariables to order of magnitude oneevery NSCAL iterations. LT . - Scaledesign variables according to scalingvalues input. DEFAULT = No scaling.Number of subsequent iterations whichmust satisfy relative or absoluteconvergence criterion before optimizationprocess is terminated. DEFAULT = 3.

Linear objective function identifier.If the optimization objective is knownto be a linear function of the designvariables, set LINOBJ = 1.

DEFAULT = Non-Linear.NACMX1 : One plus the maximum number of active

constraints anticipated.DEFAULT = NDV+2.

ITRM:

LINOBJ

117

DATA BLOCK C (Continued)

FIELD CONTENTS

NFDG : Finite difference gradient identifier.- All gradient information is computed

by finite difference.1 - Gradient of objective is computed

analytically. Gradients of constraintsare computed by finite difference.

2 - All gradient information is computedanalytically.

REMARKS

1) The value of NSCAL = 5 is suggested andITRM = NACMX1 = should be used.

2) The value of IPRINT may be reduced when the user isfamiliar with the optimization output.

118

DATA BLOCK

DESCRIPTION COPES Floating Point Optimization ProgramParameters

FORMAT: 8F10

1 2 3 4 5 6 7 8

FDCH FDCHM CT CTMIN CTL CTLMIN THETA PHI

Note: Two cards of data are read here

FIELD CONTENTS

1 FDCH: Relative changecalculating finDEFAULT =0.01

2 FDCHM: Minimum absolutgradient calculDEFAULT = 0.001

3 CT: Constraint thicDEFAULT = -0.1.

4 CTMIN: Minimum absolutin the optimizaDEFAULT = 0.004

5 CTL: Constraint thiclinear and sideDEFAULT = -0.01

6 CTLMIN: Minimum absolutin the optimizaDEFAULT = 0.001

7 THETA: Mean value of pmethod of feasiDEFAULT =1.0.

8 PHI: Participation c

or more constraDEFAULT = 5.0.

in design variables inite difference gradients.

e step in finite differenceations

.

kness parameter.

e value of CT consideredtion process.

kness parameter forconstraints

.

e value of CTL consideredtion process

.

ush-off factor in theble directions.

oefficient, used if oneints are violated.

119

DATA BLOCK D (Continued)

FORMAT: 2F10

1 2T

4 5 6 7 8

DELFUN DABFUN

FIELD CONTENTS

1 DELFUN: Minimum relative change in objectivefunction to indicate convergence ofoptimization process.DEFAULT = 0.001.

2 DABFUN: Minimum absolute change in objectivefunction to indicate convergence ofthe optimization process.DEFAULT = 0.001 times the initial

objective value.

120

DATA BLOCK

DESCRIPTION Total Number of Design Variables, DesignObjective Identification and Sign on DesignObj ective

.

FORMAT: 2110, F10

1 9 3 4 5 6 7 8

NDVTOT IOBJ SGNOPT

FIELD CONTENTS

NDVTOT: Total number of variables linked to thedesign variables. NDVTOT must begreater than or equal to NDV. Thisoption allows two or more parametersto be assigned to a single designvariable. The value of each parameteris the value of the design variabletimes a multiplier which may bedifferent for each parameter.DEFAULT = NDV.

IOBJ: Global variable number associated withobjective function in optimizationor optimum sensitivity analysis.

SGNOPT: Sign used on objective of optimizationto identify whether function is to bemaximized or minimized. +1.0 indicatesmaximization. -1.0 indicates minimizationDEFAULT = -1.0.

121

DATA BLOCK

DESCRIPTION : Design variable bounds, initial values andscaling factors

.

FORMAT: 4F10

1 2 3 4 5 6 7 8

VLB VUB X SCAL

Note : Read one card for each of the NDV independent designvariables

.

FIELD CONTENTS

VLBVUB

X

SCAL

Lower bound on the design variable.Upper bound on the design variable.Initial value of the design variable.If X is non-zero, this will supercedethe value initialized by subroutineANAL I Z

.

Design variable scale factor. Not usedif NSCAL.GE.O in Block C.

122

DATA BLOCK G

DESCRIPTION : Design Variable Identification

FORMAT: 2110, F10

1 2 3 4 5 6 7 8

NDSGN IDSGN AMULT

Note : Read one card for each of the NDVTOT Design Variables

FIELD CONTENTS

1 NDSGN: Design variable number associated withthe variable.

2 IDSGN: Global variable number associated withthe variable.

3 AMULT: Constant multiplier on the variable.The value of the variable will be thevalue of the design variable, NDSGNtimes AMULT.DEFAULT =1.0.

123

DATA BLOCK H

DESCRIPTION : Number of constrained parameters

FORMAT: 110

12 3 4 5 6

NCONS

FIELD CONTENTS

NCONS: Number of constraint sets in theoptimization problem.

REMARKS

1) If two or more adjacent parameters in the Global commonblock have the same limits imposed, these are part ofthe same constraint set.

124

DATA BLOCK I

DESCRIPTION: Constraint Identification and Bounds

FORMAT: 3110

1 2 3 4 5 6 7 8

ICON JCON LCON

Note: Read two cards for each of the NCONS constraint sets

FIELD CONTENTS

1 ICON: First Global number corresponding tothe constraint set.

2 JCON: Last Global number corresponding tothe constraint set.DEFAULT = ICON.

3 LCON: Linear constraint identifier for thisset of constrained variables.LCON = 1 indicates linear constraintsDEFAULT = = Nonlinear constraint.

125

DATA BLOCK I (continued)

FORMAT: 4F10

1 2 3 4 5 6 7 8

BL SCAL1 BU SCAL2

FIELD CONTENTS

BL: Lower bound on the constrained variablesValue less than -l.OE+15 is assumedunbounded.

SCAL1 : Normalization factor on lower bound.DEFAULT = Max of ABS(BL), 0.1.

BU : Upper bound on the constrained variablesValue greater than l.OE+15 is assumedunbounded.

SCAL2 : Normalization factor on upper bound.DEFAULT = Max of ABS (BU) , 0.1.

REMARKS

1) The normalization factor should usually be defaulted.

126

DATA BLOCK

DESCRIPTION: COPES data 'END' card

FORMAT : 3A1

END CARD

END

FIELD CONTENTS

The word 'END' in columns 1-3

REMARKS

1) This card must appear at the end of the COPES data

2) This ends the COPES input data.

127

0PC0DE1 Analysis

Data for the condenser analysis follows the 'END'

card in the COPES data deck. If the general design

capability of COPES/CONMIN is not needed, the condenser

analysis portion of 0PC0DE1 can be run in a stand-alone

mode by using the following main program:

C MAIN PROGRAM FOR STAND-ALONE CONDENSERANALYSIS

C INPUT SECTIONICALC=1CALL ANAL I Z (ICALC)

C EXECUTION SECTIONICALC=2CALL ANAL I Z (ICALC)

C OUTPUT SECTIONICALC=3CALL ANAL I Z (ICALC)

STOPEND

128

DATA BLOCK AA

DESCRIPTION : Condenser Analysis Title Card

FORMAT: 20A4

TITLE CARD

129

DATA BLOCK BB

DESCRIPTION : Objective Function Weighting Factor

FORMAT: E12.5

130

DATA BLOCK CC

DESCRIPTION: Condenser Tube Input Data

FORMAT: 6F10, 12

1 2 3 4 5 6 7 8

SDO SDI XW SDD ALGTH ROUGH NPASS

FIELD CONTENTS

SDOSDIXW

SDDALGTHROUGHNPASS

Tube outside diameter; inch.Tube inside diameter; inch.Tube wall thickness; inch.Tube pitch to diameter ratio.Tube length, feet.Tube inside absolute roughnessNumber of tube passes.

foot

151

BLOCK DD

DESCRIPTION: Sea Water Information

FORMAT: 8F10

1 2 4 5 6 7 8

VELC Tl

FIELD CONTENTS

1 VELC: Sea water velocity in the condensertubes, feet per second.

2 Tl : Sea water injection temperature, °F

152

BLOCK EE

DESCRIPTION: Inlet Steam and Heat Load Information

FORMAT: 7F10

1 2 3 4 5 6 7 8

WMS HFGMS WAS HFGAS PSAT TSAT TRNSHT

FIELD CONTENTS

WMS: Incoming main steam rate, pound per hourHFGMS: Latent heat of condensation for main

steam, BTU per pound.WAS: Incoming auxiliary steam rate, pound

per hourHFGAS: Latent heat of condensation for

auxiliary steam, BTU per pound.PSAT: Saturation pressure of incoming steam,

pounds per square inch absolute.TSAT: Saturation temperature of incoming

steam, degrees Fahrenheit.TRNSHT: Heat load from other sources, BTU

per hour.

REMARKS

1) If no value for HFGMS or HFGAS is read in, theseparameters will default to 950 BTU per pound.

2) Either PSAT or TSAT is to be specified.

133

DATA BLOCK FF

DESCRIPTION: Tube Material Identification

FORMAT: 110

IDMATL

FIELD CONTENTS

IDMATL: Material identificationfollowing tabl e

.

TUBE MATERIAL CODE

ADMIRALTY METAL 1

ARSENICAL COPPER 1

ALUMINUM 1

ALUMINUM BRASS 2

ALUMINUM BRONZE 2

MUNTZ METAL 2

90-10 CU-NI 3

70-30 CU-NI 4

COLD ROLLED LOWCARBON STEEL 5

STAINLESS STEELSTYPE 410/430 6

TYPE 304/316 7

TYPE 329 8

TITANIUM 9

from the

134

DATA BLOCK GG

DESCRIPTION: STOP Card

FORMAT: 4A1

STOP CARD

STOP

FIELD CONTENTS

The word 'STOP' in columns 1-4

REMARKS

1) This card must appear at the end of the analysis data

2) This ends the analysis data.

135

APPENDIX D

SAMPLE OUTPUT FROM OPCODE

1

CCCCCCC CCCCOQO PPPPPPP EEEEEEE SSSSSSSC G P P E SC Q P P E SC QO PPFPPPP EEEE SSSSSSSC OOP E SC OOP E SCCCCCCC CCOGOOCJ P EEEEEEE SSSSSSS

C C NTRCL PRCGRAMFOR

ENGINEERING SYNTHESIS

TITLE

CVA / CASE ONE /0EJ=PGwER/r1IN: POKER WITH CONSTANT CREJ/

136

CARC IMAGES OF CONTROL DATA

CARO IMAGE

1) CVA / CASE LM: /CuJ=PC*ER/hI\: PCkER i»ITH CONSTANT CRFJ/2) 2 b31 1 40 5 154)5)6) 18 -1.07) S TUBE 0.0. . INCHES8J C.625 1.259) i TUBE I.O., INCHES

1C) C.4C7 1.2C611) S PITCh/OIAMETER RATIO12) 1.1 3.C12) i TUBE LENGTH, FEET14) l.C 25.

C

15) S S-W VELOCITY IN TUBES, FPS16) 2.0 9.C17) 1 1

18) 2 2i<=) 3 620) 4 421) 5 512) 522) $ TUBE WALL THICKNESS, INCHES24) 7 725) C.022 0.10926) S CCNOENSER DIA.METER, FEET27) S 328) 1.0 10.025) J PINCH POINT, F30) 12 1231) 5.C 35.022) * S-k TEMPERATURE RISE, F •

'

33) 13 1334) 5.0 20.035) $ RATIC OF TLBE HCL E AREA TO TUeE SHEET AREA WITHOUT ORILLEC HOLES26) 16 1637) C.l 0.3636) ENO

137

TITLE:CVA / CASE ONE /OBJ»PQWgR/MIN: POWER WITH CONSTANT CREJ/

CCNTFCL PARAMETERS :

CALCLLATICN CCNTRCL.NUM3ER OF GLOBAL DESIGN VARIABLES,NU>eER CF SENSITIVITY VARIABLES,NUMBER OF FUNCTIONS IN TwO-SPACE,INPUT INFORMATION PRINT COOE,SENSITIVITY ^RINT CCCE,TWO-SPACE PRINT CODE,CEeUG PRINT CCOE,

CALCULATICN CCNTROL, NCALCVALUE MEANING

1 SINGLE ANALYSIS2 OPTIMIZATION3 SENSITIVITY4 TWC-VARIAolc FLNCTION SPACE

GLCBAL VARIABLE NUMEtR CF OBJECTIVE = 18MULTIPLIER (NEGATIVE INOICATES MINIMIZATION) = -O.IOOOE 01

CCNMN PARAMETERS I IF ZERO, CQNMIN DEFAULT WILL OVER-RIOE)

NCALC 3 2NOV 3 5NSV 3

N2V AR 3

IPNPUT =IPSENS 3IP2VAR 3

IPOBG S

IPRINT1

ITMAX40

ICNCIR NSCAL5

ITRM LINOBJ NACMXl15

NFOG

FOCH0.0

FOCHM0.0

CT0.0

CTMIN0.0

CTL0.0

CTLMINO.C

THETA0.0

PHI0.0

OELFUN0.0

0A6FUNO.C

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ROW NR. RADIUS ( IN. J ,\R. CF TU

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30 31.11 195 .5031 31.56 200.9432 32.55 206 .5322 23.71 211.8234 34.56 217.2635 35.44 222.7036 st. 31 223. 1437 37.18 233.5638 28.04 239.0339 26.51 244.4740 35.77 245.9141 4C.a4 255.3542 41.51 260. 79*»3 42.37 266 . 2344 43.24 113.58

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146

APPENDIX E

0PC0DE1 PROGRAM LISTING

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22. United States Atomic Energy Commission ReportORNL-TM- 2972 , Computer Model and Correlations forPrediction of Horizontal Tube Condenser Performancein Seawater Distillation Plants , by D.M. Eissenbergand H.M. Noritake, April 1970.

23. Universal Oil Products Company Bulletin No. 4020,Wolverine Korodense Tube: General Information andHeat Transfer Design Notes~ 10 January 1973.

24. Fraas , A. P. and Ozisik, M.N., Heat Exchanger Design,

Wiley, 1965.

25. English, J.W., Hydrodynamic Considerations in the Designof Condenser Cooling Water Systems for Large Ships

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26. Reilly, D.J., An Experimental Investigation of EnhancedHeat Transfer on Horizontal Condenser Tubes , MSMEThesis, Naval Postgraduate School, March 1978

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27. Westinghouse Electric Corporation WAPD-TM-213,STDY-3 A Program for the Thermal Analysis of aPressurized Water Nuclear Reactor During Steady-StateOperation, by Pyle, R. S

., June 1960

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164

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Thesis 173370J574 Johnson

c.l Marine steam conden-ser design using nu-merical optimization.

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