analysis ultimate heat sink · cl specific heat of water cv specific heat of water vapor cg...

Departrr.ent of Mechanical and Industrial Engineering

J

Method for Analysis of Ultimate Heat Sink

. Cooling Tower Performance

University of Illinois at Urbana-Champaign

. by

S. M. Sullivan and

. W. E. Dunn

Performed for

Mr. Rex Wescott

U. S. Nuclear Regulat2!.Y C,o.rnmis.sjo_n •• _ • c_ .�.. -- .�

Washington, DC

April, 1986

METHOD FOR ANALYSIS OF ULTIMATE HEAT SINK COOLING TOWER PERFORMANCE

by

S. M. Sullivan

and

W. E. Dunn

of the

Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign

1206 W. Green Street Urbana, Illinois 61801

217-333-1176

prepared for

Mr. Rex Wescott U. S. Nuclear Regulatory Commission

Washington, DC 20555

April, 1986

iii

ABSTRACT

This report developes computer models which can be used in the

design and analysis of the cooling tower/basin systems of ultimate heat

sinks within nuclear power plants, and demonstrates the ways in which

these models are employed to determine the design basis required by the

U . S. Nuclear Regulatory Commission Regulatory Guide 1 . 2 7 .

The tower characteristic value is generated and the tower cold

water temperature is predicted at the generated tower characteristic

using a mathematical tower submodel. The predicted cold water

temperature values are then used �o evaluate the manufacturer' s

performance cold water temperature values.

The time constant of the tower/basin system is approximated by

minimizing the sum of squares of a predicted basin temperature and a

corresponding temperature value determined assuming a first-order system

response. The time constant is used in a data scanning model which

scans a long-term weather record from a representative meteorological

station to determine the periods of most adverse meteorology for cooling

or evaporation. The identified periods are used in an ultimate heat

sink tower/basin simulation model to estimate design-basis basin

temperature, basin mass, and basin salinity.

iv

ACKNowLEDGMENTS

The authors wish to express their sincere gratitude to Mr. Rex Wescott

of the Nuclear Regulatory Commission for his contribution to the completion

of this work. The assistance he provided during the progress of the study

as well as the time he took to review the draft report are deeply appreciated.

The authors also wish to thank Dr. Anthony J. Policastro of the Environ

mental Research Division of Argonne National Laboratory for reviewing the

draft report. His many helpful suggestions greatly improved the quality of

the finished document.

1 .

2 .

3 .

v

TABLE OF CONT E NT S

Page

I NTRODUCT I ON ••••••••••••••••••••••••••••••••••••••• � ••••••••• 1

1 . 1 U l timate Heat S i n k Cool i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . 2 Obj ect i ves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

"t=::� L IT ERATURE REV I EW •••••••••••••••••••••••••••••••• �.� ••••••••• 5

2 .1 U l t i mate Heat S i n k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 . 1 . 1 2 . 1 . 2 2 . 1 . 3

Coo l i n g Ponds ••••••••••••••••••••••••••••••••••••

Sp ray Ponds . , � ••••••••••••••••••••••••••••••••••••

Mode l V a l i dat i on •••••••••••••••••••••••••••••••••

6 8 9

2 . 2 E v a p o rat i ve Coo l i n g T owe rs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

FORMULAT I ON OF MODE L ••••••••••••••••••••••••••••••••••••••••• 1 5

3 . 1 T ower Submodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

3 . 1 . 1 3 . 1 . 2 3 . 1 . 3

Ma rcroscop i c An a l y s i s ••••••••••••••••••••••••••••

M i c ro s c op i c Ana l ys i s - Me rke l Met hod •••••••••••••

E n t h a l py D r i v i n g Force I mp l ement at i on ••••••••••••

1 7 1 8 27

3 . 2 Tow e r/B as i n Mode l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 3 . 3 ' Wo r st-Case Met eo rol o g i c a l Se l ect i on A l g o r i t hm . . . . . . . . . .. 3 7

3. 3 . 1 3 . 3 . 2

R u nn i ng Wet - B ulb Temp e ratu re •••••••••••• � ••••••• •

T i me Con s t ant ••••••••••••••••••••••••••••••••••••

37 4 1

3 . 4 Performance Data A n a l ys i s . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 42

4 . DESCR I PT I ON OF COMPUTE R PROGRAMS AND SAMP LE PROBLEM . . . . . . . . . . 46

4 .1 4 . 2 4 . 3 4 . 4 4 . 5

Desc r i pt i on of the S amp l e P rob l em •••••••••••••••••••••••

Pe rfo rma nce Dat a An a l y s i s P rog r am •••••••••••••••••••••••

T i me Con st ant E va l u at i on ••••••••••••••••••••••••••••••••

P rog ram NWS - - Dat a Sca n n i n g P rocedu re ••••••••••••••••••

Nume r i ca l Simu l at i on of UHS Coo l i ng T owe r

49 5 3 7 1 8 1

P e rformance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7 4 . 6 S e n s i t i v i ty St u dy of Met e o ro l o g i c a l D a t a S c a n ni n g

Procedu re . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4 . 7 F i n a l S i mu l at i on Re s u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 . CONCLU S I ONS AND RECOMME NDAT I ON S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 2

vi

REFERENCES • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

APPENDIX A • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

1 0 5

107

vii

L I ST OF TABLE S

page

Tab l e 4 . 1 Desi gn P a ramet ers for Test C a s e UHS C o o l i ng T ower 51

Tab l e 4 . 2 Heat Load a nd Cool i n g Wate r F l ow Rate Du r i n g LOCA . ( Adapted F rom F i na l Safety Ana l y s i s R e p o rt . ) • • • • • • • • • 52

Tab l e 4 . 3 Des c ri pt i on of I n put V a r i a b l e s for P ro g ram PDAP • • • • • • 54

Tab l e 4 . 4 Pe rforman ce Data for Test C a s e • • • • • • • • • • • • • • • • • . • • • • • 56

Tabl e 4 . 5 Des c r i pt i on of I n put Va r i ables for P rog ram UHS S IM • • • • 74

Tab l e 4 . 6 Out put L i st i n g f rom P ro g ram UHSS IM for C o n stant Wet -Bu l b Temp e ratu re • • • • • • • • � • • • • • • • • • • • • • • • • • • • • • • • • 80

Tab l e 4 . 7 Des c r i pt i on of I n put V a r i a b l es f o r P ro g ram NWS • • • • • • • 8 3

Tab l e 4 .8 Va r i at i on o f I mp o rtant Meteoro l og i c a l , Heat Loa d , a nd B as i n V a r i ab l es w i t h T i me Du r i n g LOCA • • • • • • • • • • • • 89

Tab l e 4 .9 C o r rel at i on Between Ma ximum P red i ct ed Ba s i n Temp e r at u re a n d Maxi mum Ru nn i n g Wet - B u l b Temp e ratu re fo r F i ve Met eoro l o g i ca l W i n dows • • • • • • • • • • 92

Tab l e 4 . 10 V a r i ation of I mp o rtant Met e o ro l o g i c a l , Heat L oad , a n d Ba s i n V a r i ab l es wi t h T i me Du r ing LOCA • • • • • • • • • • • • 96

v i i i

L IST OF F I GURES

page

F i gu re 3 . 1 Heat Exch a n g e r Dev i ces • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 16

Fi g u re 3 . 2 I n fi n it es i ma l Vo l ume of Cool i n g Tower Pac k i n g • • • • • • • 19

Fi gu re 3 .3 Nume rica l F o rmu l at i on of Coo l i ng Tow e r P rob l em • • • • • • 28

Fi gu re 3 . 4 S c h emat i c of P l a nt /Towe r /Basi n Simulati o n • • • • • • • • • • • 34

Fi g u re 3 . 5 F l ow C h a rt of P rog ram UHSS I M • • • • • • • • • • • • • • • • • • • • • • • • 35

F i gu re 3 . 6 F l ow Ch a rt of P e rforma n c e Dat a Analy s i s P rog ram • • • • • 43

F i gu re 3 . 7 Met h o ds of O bt a i ni n g Coo l i n g Tower C o l d W at e r Temp e ratu re • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 44

Fi gu re 3 .8 F u n cti on FM I N - M i nimum KaV/ L C a l cu l ati o n • • • • • • • • • • • 44

F i g u re 4 . 1 F l ow Ch a rt of UHS Cool i n g Towe r Perfo rmance Ana l y si s • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 48

Fi gu re 4 . 2 Schemati c V i ew of UHS Cool i n g Towe r for Test Ca s e • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 5 0

Fi gu re 4 . 3 I n put L i stin g fo r P rogr am PDAP • • • • • • • • • • • • • • • • • • • • • • 57

F i gu re 4 . 4 Out put Li st i ng for P ro g ram PDAP • • • • • • • • • • • • • • • • • • • • • 60

Fi gu re 4 . 5 Compa r i s o n of Man u fact u re r ' s Pe rfo rma n c e Dat a with P re d i ct i on s of Mat hemat i ca l Model for 50% of Design F l ow Rat e • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 64

F i gu re 4 .6 C omp a r i s on of Ma n u fact u re r ' s Pe rfo rman c e D at a w i t h P red i ct i on s of Math emat i c a l Mode l for 60% of D es i gn F l ow Rat e • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 65

Fi gu re 4.7 Comp a r i s o n of Ma nu fact u re r ' s Pe rforma n c e Data w i t h P red i cti ons of Mathemat i ca l Model for 70% o f Des i gn F l ow Rate • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 66

Fi gu re 4 .8 C omp a r i son of M a n u fa ct u re r ' s P er f o rma nce Dat a wit h P redi ct i on s of Mat h emat i ca l Mode l for 80% of D es i gn F l ow Rate • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 67

Fi gu re 4 .9 Compa ri s on of Man u fact u re r ' s Pe rfo rma nce Dat a w i th P red i ct i o n s of Mathemat i ca l Mode l for 90% of Desi g n F l ow Rate • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 68

i x

F i g u re 4 . 10 Compa r i s o n of Ma nuf a ctu re r1 s Pe rfo rma nce Dat a w i t h P red i ct i ons of Mathemat i ca l Mode l f o r 100% of Des i gn F l ow Rate • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 69

F i gu re 4 . 1 1 Compa r i son o f M a n u fa ctu rerlS P e rforman ce Data wi t h P red i cti o n s of Mathemat i ca l Model fo r 1 10% of D es i g n F l ow Rate • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 70

F i gu re 4 . 1 2 V a r i at i on of Re l at i ve Towe r Pa ramet e r KaV (R at i o of KaV at t h e G i ven F l ow R ate to KaV at D es i gn F l ow Rate ) t o Re l at i ve F l ow Rate ( R at i o of G i ve n F l ow R ate to D es i gn F l ow R at e ) • • • • • • • • • • • • • • • • • • • • • • • • • • 7 2

F i g u re 4 . 1 3 I nput Fo rmat f o r P rog ram UHSS I M • • • • • • • • • • • • • • • • • • • • 7 5

F i gu re 4 . 14 I n put L i st i n g f o r P ro g ram UHSS I M • • • • • • • • • • • • • • • • • • • 76

F i g u re 4 . 1 5 St ep C h a nge i n L , KaV /L , a nd LjG Va l u e s • • • • • • • • • • • • 78

F i g u re 4 . 16 Compa r i son of Tem p e r at u re H i s t o r i es P re d i cted by F u l l Towe r/B a s i n S i mu l at i on ( UHSS IM ) a n d S i mp l i f i ed Model f o r Con st ant Wet-Bu l b Tem p e rat u re • • • • • • • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • 82

F i gu re 4 . 1 7 Out put L i st i n g · f rom P rog ram NWS - Th ree W i n dow Fo rmat • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 86

F i g u re 4 . 18 Max i mum B as i n Temp e r at u re as F u n ct i on of D i s p l acement of Acc i dent I n i t i at i on f rom T i me of Max i mum R u n n i n g W et -B u l b Temperatu re • • • • • • • • • • • • 90

F i gu re 4 . 19 Ba s i n Temp e ratu re Du r i n g LOCA as a F u n ct i on of T i me f rom B e g i n n i n g of Acci dent f o r Va r i a b l e I n i t i a l B a s i n Temperatu re • • • • • • • • • • • • • • • • • 93

F i gu re 4.20 C omp a r i s on of Temp e r at u re H i s t o r i es P red i cte d by F u l l Towe r / Ba s i n S i mu l at i on ( UHSS I M ) a n d S i mp l i f i ed Model fo r P ri ma ry Met e o ro l o g i c a l W i n dow . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 9 5

F i gu re 4 . 2 1 B a s i n T empe ratu re Du r i n g LOCA a s F u n ct i on o f T i me f rom Beg i n n i n g o f Acc i dent • • • • • • • • • • • • • • • • • • • • 97

F i gu re 4 . 2 2 B a s i n Mas s D u r i n g LOCA a s F u n ct i on of T i me •

F rom Beg i n n i n g of Ac c i dent • • • • • • • • • • • • • • • • • • • • • • • • • 98

F i gu re 4.23 Heat Lo ad t o UHS Sys t em Du r i n g LOCA a s F u n ct i o n o f Ti me F r om Be g i n n i n g of Acc i dent • • • • • • • • • • • • • • • • • 99

x

Fi gu re 4 . 24 B a si n S a l i nity Du ri n g LOCA as F u n ct i o n of T i me f rom B egi n ni n g o f Acci dent for 30 Day s • • • • • • • 101

x i

NOMENCLATURE

E NGL I SH SY MBOLS

a Su rface a rea per u n i t vol ume of t ower f i l l

A Pond s u rface a rea ( u sed i n d i s cu s s i on of UHS coo l i n g

ponds )

c a Spec i f i c heat of d ry a i r

cL Spec i f i c h eat of wat e r

C v Spe c i f i c h eat of wat e r vapor

cG Spec i f i c h eat at con st a nt p re s s u re of a i r a n d wat e r

v a por mi x t u re

e �t /t

E Eq u i l i b r i um t empe rat u re ( u s e d i n di cu s s i on o f UHS

cool i ng p o n d s )

f F ract i on of wate r evapo rat ed

G Ma s s f l ow rat e of a i r

h Heat t ra n s f e r coef f i c i ent

h a E nt h a l py of d ry a i r

h fg Spec i f i c heat of vap o r i zat i on at act u a l wat e r

t emp e ratu re o

h fg . Spec i f i c heat of vap o r i z at i o n at wate r t emp e rat u re To

h v E nt h a l py of wat e r vapor

h O Ma s s t ra n s fe r coeff i c i e n t

hG E nt h a l py of mo i st a i r

hG* Nond i me n s i o na l e nt h a l py of mo i s t a i r ( u s e d i n f i n i t e

x i i

d i ffe rence app roximat ion )

hL E nt h a l py of l i q u i d wat e r

hL* Nond i men s i o n a l ent h a l py of wat e r ( u s e d i n f i n i t e

d i ffe rence a ppr ox i mat i on )

K

KaV / L

E n t h a l py ( at numbered ent ra n ce a n d ex i t s t ate s )

R at i o of d i ffe rentia l e nt h a l py t o d i ffe rent i a l

E ffect i ve heat t ra n s fer coef f i c i ent

Towe r perfo rma n ce c ha ract e r i s t i c

E ffect i ve overa l l ( po n d ) s u rface heat tr a n s fe r

coeff i cient (u s e d i n di s cu s s i o n o f U H S cool i ng ponds )

L M a s s f l ow rate of wat e r

L e Lewi s numbe r

LHF L at ent Heat F a ct o r

L. Ma s s fl ow rate of wat e r at s tep "; II ,

L/G L i qu; d over g a s fl ow rate rati o

m Ba s; n ma s s

m; Ba s i n mas s at s tep " i ll

M P o n d ma s s ( u s e d i n di s cu s s i o n of UHS coo l i n g ponds )

Nu N u s s e l t numbe r

P Tot a l p re s s u re

Pa P a rt i a l p res s u re of a i r

Pv P a rt i a l p re s s u re of wat e r v a p o r

P r P ra n dl t n umbe r

q Heat t ra n s fe r rate

xii i

qi Heat load at step "i II

qp Heat transfer rate of the plant

qs Rate of surface heat transfer (used in discussion of

UHS cooling ponds)

qt Heat load of the tower

qB Heat load of the basin

ql Plant. heat load (used in discussion of UHS cooling

ponds)

Q Cooling tower heat load (plant heat load)

R hL(n)/cL (used in trldiagonal matrix solution to

finite difference approximation)

Ra Perfect gas law constant for air

Re Reynolds number

Rv Perfect gas law constant for water vapor

s Basin salinity

S Sum of squares (used in calculating the system time

constant)

Sc Schmidt number

Sh Sherwood number

t Time

tl Time (used in running wet-bulb temperature

integration)

T Pond temperature (used in discussion of UHS cooling

ponds)

Tact Tower cold water temperatures from manufacturerer1s

xiv

performance curves

Ti Basin temperature predicted by program UHSSIM at step "i II

Tin Tower inlet temperature

Tm Mixed-mean temperature of dry air and water-vapor

mixture

To Enthalpy reference temperature

TL Temperature of water

TL . Temperature of water at step "i II ,1

Tout Tpred

Twb,i

TB,wb TB,Q T'

Tower outlet temperature

Predicted tower cold water temperature using

mathematical tower submodel

Wet-bulb temperature

Wet-bulb temperature at step "i "

Basin temperature

Basin temperature due to wet-bulb temperature effects

Basin temperature due to plant heat loaa effects

Pond temperature without heat load (used in dic'ussion

of UHS cooling ponds ) T i Basin temperature predicted by first-order response

....

at step "ill

TWb Running wet-bulb temperature (used in program NWS to

follow changing wet-bulb temperature ) V Volume of tower fill

V* Nondimens;onal volume (used in finite difference

xv

approximation)

w Mass flow rate

GREEK SYMBOLS

e:

T

a

w

Change in heat load

Change in time

Efficiency ( (Tin-Tout)/ (Tin-Twb))

Density of water vapor

Density of saturated water vapor

Density of moist air

Summation

Time constant of tower/basin system

Excess temperature (used in discussion of UHS cooling

ponds)

Absolute humidity

Absolute saturation humidity

1

1 . INTRODUCTION

1 . 1 Ultimate Heat Sink Cooling

This work centers on predicting the thermal performance of cooling

towers used as the ultimate heat sink ( UHS ) of a proposed nuclear power

plant. The purpose of the ultimate heat sink is to provide for the safe

shutdown and cool down of the nuclear reactor ( s ) in the event of a de-

sign-basis accident. Since issues of riuclear safety are involved, the

analysis set forth herein is conservative in nature in that parameters

are selected to provide the worst performance which may be reasonably

expected. It is intended that this work can form the basis of a stan

dardized analysis of UHS cooling tower performance.

Stringent standards for the design of the UHS have been set forth

by the U. S. Nuclear Regulatory Commission ( NRC ) in Regulatory Guide

1 . 27 (1976) . The three basic requirements set forth in that document

are: (a ) the UHS system should be able to dissipate the heat of a de

sign-basis accident, such as a loss-of-coolant accident, of one unit

while assuring the concurrent safe shutdown and cool down of all remain

ing units; ( b ) the UHS system must provide a 30-day supply of cooling

water at or below the design basis temperature for all safety related

equipment; and ( c ) the UHS system must be capable of operating under two

sets of meteorological conditions, the first resulting in the greatest

water loss and the second resulting in the worst thermal performance.

The UHS cooling system differs from a conventional cooling system

in several key respects, and, thus, the analysis must recognize and pro

perly account for these differences. First, a UHS system must be imper-

2

vious to natural and man-made disasters. As a result, unusual designs

are frequently employed. For example, UHS cooling tower structures must

often be built to withstand earthquakes and tornadoes leading to complex

air and water flow patterns unlike those of conventional cooling towers.

Second, UHS cooling systems are heavily loaded relative to conventional

cooling systems since they are intended to operate only under emergency

conditions for which plant efficiency is not a consideration. Indeed,

the thermal performance of the UHS system is measured in terms of its

ability to maintain the temperature of cooling water entering the plant

below the design basis limit of safety related equipment. This value is

typically far above the temperature of cooling water required for effi

cient plant operation under normal conditions. Third, the UHS system

must be able to operate for a period of at least 30 days without the ad

dition of make-up water. Thus, the ability of the system to maintain an

adequate water inventory is an additional constraint not normally em

ployed in the design of conventional cooling systems. Fourth, the ana

lysis must be able to handle off-design conditions such as a reduced

number of fans, a reduced water flow rate ( loss of a pump ), or the ef

fect of unusually high dissolved solids content resulting from the sig

nificant evaporative losses which may be expected. Lastly, since the

UHS cooling system is intended for emergency use only, the UHS system

must be designed to operate reliability on demand after a period of dor

mancy.

The NRC has found many types of UHS systems to be acceptable. Some

examples of these types are: (a ) a large body of water such as a river,

3

a lake, or an ocean, (b) a spray pond with a reservoir or other large

body of water, ( c) a submerged pond within a larger cooling lake which

will remain filled even in the event of a major dam rupture, and ( d ) a

specially designed mechanical-draft cooling tower. Cooling ponds, spray

ponds, and cooling towers are the three major types of UHS systems in

use today.

1 . 2 Objectives

The first of several objectives for this work is to establish a

standardized method for analyzing the performance of UHS mechanical

draft cooling towers. Standardized methods already exist for UHS cool

ing ponds and UHS spray ponds. The development and use of these latter

methods is presented by Cadell and Nuttle ( 1 980 ) for the UHS cooling

pond and by Codell ( 1 981 ) for the UHS spray pond as is more fully dis

cussed in Chapter 2 . Standardized methods are valuable because unifor

mity can be established in determining the limits of system performance

across the broad spectrum of possible UHS cooling tower designs. Like

the methods developed for UHS cooling ponds and UHS spray ponds, the

present method utilizes the entire set of data recorded by the National

Weather Service at the site nearest to ( and most representative of) the

proposed plant. A major obstacle here is to identify, from within this

typically massive amount of meteorological data, the periods of worst

case system performance. Development of a reliable selection algorithm

is a major focus of the present work as is more fully described later.

The second objective of this work is to develop a realistically

conservative analysis which is flexible enough to handle any proposed

4

UHS cooling tower design. The model must couple an analysis of tower

heat and mass transfer characteristics with a prediction of basin tem

perature, basin mass, and basin dissolved solids content as functions of

time. The model must incorporate the time-varying heat rejection rate

of the plant and must allow for towers or tower cells to be configured

in complex loop arrangements made necessary by the need for redundency.

The model must allow for variable air and water flow rates as well as

operating conditions greatly different from the design point.

A third objective of this work is to apply the method to a real

world example. Application of a real-world example will verify the

workability of the method and will also provide ill ustrative examples

for other potential users.'

The final objective of this work is to make the method easy to

apply to many different sites. To accomplish this task, the input to

the computer model must be structured in such a way that site dependent

data can be used in a convenient manner. Moreover, the form of this

data should be that traditionally available to the design �ngineer.

5

2. L I TERATURE REV I EW

Becau s e of the spec i a l cons i de rat i o n s att endant wi t h the UHS app l i

cat i o n , t h e s cope of l i t e ratu re s peci f i c a l l y dea l i n g wit h the des i gn of

UHS cool i n g systems i s l i mi ted. Howeve r , t h e U. S. Nu c l e a r Regu l at o ry

C omm i s s i on h a s p u b l i s h ed methods for t he a na l ys i s of UHS cool i n g p o n d s

( Code11 a n d Nutt l e , 1980 ) a n d met h ods fo r t h e a na l y s i s of U H S s p ray

p o n ds ( C o de1 1 , 1 9 8 1 ) . T hese m et h o ds a d d ress n ot o n l y t h e p roper mode l

i n g of heat t ra n s f e r cha ract e r i s t i cs u n de r acc i dent cond i t i o n s , but al so

p ro v i de p rocedu res for s elect i n g worst - c a s e met e o ro l o g i ca l cond i t i o n s.

I n addit i on , Batte l l e Pac i f i c No rt hwes t Labo rat o ry h a s col l ect e d dat a by

wh i ch t o c h a ract e r i ze t he t h e rma l p e rforma n ce a n d wat e r l os s of coo l i n g

p o n ds ( H a dl ock et a1. , 1 978 ) a n d s p ray pon d s ( Had l o c k et a 1 . , 1 9 8 1 )

u n de r hea v i l y l oaded c o n d i t i o n s s u ch a s t h o s e wh i ch e x i st du ri n g UHS

ope rat i o n. Codel l ( 1 98 2 ) has com p a red t he s e dat a w i t h t he NRC coo l i n g

pond a n d s p ray pond mode l s. The m ethods for a n a l y s i s of UHS coo l i n g

p on ds a re rel e va nt t o t h e p re s e nt wo rk not on l y i n t e rms of gene ra l

b a c k g round for t he UHS a p p l i catio n , but mo re s p ec i fi ca l l y beca u s e t h ey

i l l u st rat e t he l eve l o f con s e rvat i sm a n d s op h i s ti c at i o n i n he rent i n t he

a n a l yses app l i ed t o UHS systems ot h e r t h a n coo l i ng t owe r s.

The l i t e rat u re dea l i n g wi t h t h e t h e rm a l p e r f o rm a n c e of cool i n g

towe rs i n gene ral i s , on t he ot h e r h an d , ext e ns i v e . As was p rev i o u s l y

n ot e d , h oweve r , t h e des i gn s o f UHS coo l i n g t owe r sys t ems oft en d i f f e r

ma rked ly from con vent i ona l des i g n s and t he t h e rma l p e r f o rman ce a n a l y s e s

mu st be modi f i e d acco r di ng l y . A b r i ef su rvey of t h e gene ra l l i t e ratu re

b a se w i l l be p resented h e re i n so as to p rov i de a context for v i ew i ng t h e

p resent wo rk.

2.1 U l t i mate Heat S i n k s

2 .1.1 C o o l i n g P o n ds

6

Ca del l a nd Nutt l e ( 1980) des c r ibe an ana l y s i s of UHS coo l i n g ponds

w i th fou r majo r component s :

( a) a met hod for t reat i n g su rface heat t ran s f e r ba s e d o n t h e equ i

l i b r i um tempe ratu re concept of B ra dy , G ra ves , a n d Gey e r ( 1969) ,

( b) t h ree l i mi t i n g -case mode l s o f pon d hyd rody n ami c s [f u l l y mi xed

pond ( ze ro di men s i o n a l ) , p l u g f l ow pond ( one d i men s i o na l h o r i

zont a l l y) , and st rat i f i e d pon d model ( o ne di men s i o n a l ve rt i c a l )J .

( c) a method for s c a n n i n g a l ong record of meteoro l o g i ca l data t o

det e rmi n e t he amb i ent con di t i o n s g i v i n g wo rst pon d p e r f o rma nce ,

a n d

(d) a met h o d f o r compa r i n g the typ i c a l l y l on g reco r d o f offs i t e dat a

w i th the s h o rter but mo re det a i l ed record of o n s ite d at a .

O f t h e s e seve ra l component s , the meth o d u sed to s e l ect t h e c o n d i t i o n s

of wo rst- case po nd perfo rmance i s of g reatest re l e v a n ce to the p re s e n t

wo r k . Th i s met h o d h i n ge s o n t h e u s e of a l i nea r i zed fo rm o f t h e s u rface

heat t ra n sfer re l at i on s h i p a s fo l l ows .

q s = K' A ( T - E), (2 . 1)

7

whe re q s i s t he rat e of su rface heat t ra n s fe r , T i s pon d tempe rat u re , A i s

pond s u rface a re a , KI i s t he " effect i ve o v e r a l l ( po n d ) su r f a ce heat t ra n s

f e r coeff i c i e nt " , a n d E i s a p a ra meter commo n l y k nown a s t h e " eq u i l i b r i u m

t empe rat u re " . I f t he l i nea r i zed f o rm we re e xact a n d K were a co n stant ,

t h e " eq u i l i b r i u m t emp e rat u re " wou l d be t h e t emp e ratu re th at t he pond wou l d

reach u n de r s teady state en v i ronme nt a l c o n d i t i o n s w i t h o ut e xt e rna l heat

i n p u t s o r remo va l s . S i nce t he dependence of heat t ra n s fer rate on pon d

t empe rat u re i s i n fact not l i n e a r as s u p p os e d above , t h e eq u i l i b r i um t emp

erat u re i s me rel y a dev i ce fo r sep a rat i n g t h e effect s o f heat l oa d f rom

the e f fects of c h a n g i n g amb i ent condi t i o n s . U n de r t h e a s s u mpt i on s of t h i s

a n a l y s i s , t he pon d t emp e ratu re at any g i ven i n s t a nt o f t i me i s simp l y t h e

t emp e ratu re t he p o n d wou l d h a ve atta i ned w i t h out h eat i nput ( gove rned by

env i ronme nt a l cond i t i o n s) p l u s t h e exces s temp e ra t u re p rodu ced by the heat

rej ect i on rate of the p l a nt . To s ee t hat t h i s i s i n deed t h e ca s e , con

s i de r the eq uat i on of a fu l l y mi xed pond wi t h ma s s M , s pe c i f i c heat cl'

a nd p l a nt h eat l oa d ql :

M cl ( dT / dt ) = KI A ( E - T) + ql . ( 2 . 2 )

S i nce the equ at i on i s l i nea r i n T ( a s s umi n g K , A a n d q ar e c o n st a nt s i n

depen dent of T) , t h e s o l u t i o n c a n wri t ten as t h e s um

T = TI + a , ( 2 . 3 )

8

where TI i s t he pond temp e rat u re wi t hout h eat l oad g i ven by t h e so l u t i on

of t he eq u at i on

M cL ( d TI / d t ) = KI A ( E - TI ) ( 2 . 4 )

a n d a i s t he exces s t empe ratu re g i ven by t h e s o l u t i on of

M cL ( da/ dt ) = - KI A a + ql • ( 2 . 5 )

The fo rme r equ at i on can be s o l ved to det e rmi ne a v a l u e of TI du r i n g each

hou r fo r wh i c h a met eoro l o g i c a l o b s e rvat i on i s ava i l ab l e . The st a rt i n g

t i me of t he acc i d e nt i s t hen s e l ected so t h at t he max i mum v a l ue of TI a n d

t h e ma x i mum va l u e of 6 o ccu r s i mu l taneou s l y . Th u s , TI can b e u s e d t o

i de nt i fy t h e per i od o f wo r st-case t h e rmal p e r fo rma n ce f rom w i t h i n t he l o n g

reco rd o f met eo ro l o g i ca l cond i t i o n s .

S i nce t he above ana l y s i s i s o n l y app rox i mat e , Code l l recomme nds t hat

a s e r i es of ru n s be made t o s t u dy the s e n s i t i v i ty of t h e ma x i mum pond

t emp e ratu re t o t h e s t a rt i n g t i me of t he acc i den t .

2 . 1 .2 Sp ray Ponds

The a na l ys i s of UHS s p ray pon ds p a r a l l e l s t hat o f the UHS coo l i n g

p o n d d i s c u s sed above except t h at t he e n h a n ced heat t ra n sfer a s s o c i at ed

w i t h t h e d ro p l et sp ray mu st be adde d . I n add i t i on , t he wat e r l os s du e to

d r i ft ( remova l of s p ray drop l ets by t h e w i n d ) mu st be accou nted f o r . T h e

ba s i c concept of u s i n g a l i nea r app roxi mat i on t o s e p a rate t h e effect s o f

the c ha n g i n g e n v i ronmenta l c o n d i t i o n s f rom those of t h e t i me - v a ry i n g h eat

. rej ect i on rat e i s ret a i n e d .

9

I n t h e case of the sp ray pon d , t he cont r i but i on to t he ove ra l l heat

and wat e r l oss a r i s i n g f rom t he pond s u rface i s smal l re l at i ve t o t h at

f rom t he s p ray . Al s o , du e to t h e enh a n ce d heat an d mas s t ra n s fe r a s soc i

ated w i th t he s p ray , t he UHS s p ray pond i s much smal l e r i n s i ze t h a n a

convent i o n a l UHS cool i n g pon d . T h e s p ray pond i s u s u a l l y rega r ded a s

fu l l y m i xed .

2 . 1 . 3 Mo d e l V a l i d at i on

C ode l l has comp a red the p re d i ct i o n s of the NRC cool i n g pon d a n d s p ray

pond w i t h data t a ken by B atte l l e P a c i f i c N o rt hwest La b o r at o ry at two sma l l

geot h e rm a l 1 y heat ed ponds [1 982J . These sma l l heav i l y l oa de d ponds we re

con s i de red to be a p p rop ri ate a n a l o gs of UHS coo l i ng p o n ds a n d s p ray

p o n ds .

B atte l l e con s i de red many s i te s as cand i d at es fo r t h e i r st u dy . Geo

t h e rma l s i t es we re c ho sen aft e r c a refu l con s i derat i on becau s e of t h e

a v a i l ab i l i ty of l a rge quant i t i es of heat e d wat e r . P on d s we re con s t ru ct e d

a t two geot h e rmal sit e s , s uit a b l e f o r s u rface heat t ra n s fe r mea s u rement s .

A sp ray sy stem was f i tted t o o ne of t h e ponds to a l l ow s p r�y dat a co l

l ect i on . A comp l ete des c r i pt i on of t h e data co l l ected i s g i ven by

Had l ock a n d Abbey [1978 , 1981J . The comp a r i s o n s p re s e nt e d by C o de 1 1

demon st rate the re l i ab i l i1 i ty of t h e s u rface re l a t i o n s h i p s u sed t o p re

d i ct su rface heat t ra n s fe r i n the NRC model s .

2 . 2 E v a p o r at i ve Coo l i ng Towe rs

Fo r ove r 50 yea rs , the p e rfo r ma n c e of eva p o r at i ve cool i n g t owe r s

h a s been est i mated by t he meth o d o f Me rke l [ 1 92 5J . M e r k e l ' s met hod i s

b a s e d on a cont ro l vo l ume ene rgy b a l a n c e at an a rb i t ra ry h e i g h t wi t h i n

10

t h e towe r . Me rke l' s met h od comb i nes the eq uat i on s of h eat a n d ma s s

t ra n s fe r i nto a n e nt h a l py- d i ffe rence d r i v i n g f o rce to a l l ow for both

s e n s i b l e a n d l atent heat t ra n s fe r . Th i s a s s umpt i o n may be exp res sed

mat hemat i ca l l y a s fo l l ows . ��JY �� f /

q = K a V ( h L - hG ) , ( 2 . 6 )

w h e re q i s t he heat t ra n s fe r r at e , K i s a n effect i ve t ra n s f e r coeff i -

c i ent , a i s su rface a re a per u n i t vol ume of towe r f i l l , V i s the vol ume

of t ower f i l l , hL i s t h e e nt h a l py of l i qu id wate r , a n d hG i s the e n

t h a l py of mo i st a i r . Th i s rel at i o n s h i p i s comb i n e d w i t h a s t atement of

e n e r gy con se rvat i on ( F i rst L aw of The rmody n am i c s )

( 2 . 7 )

w here G i s t he mas s f l ow rate of a i r , L i s t h e ma s s f l ow rate of wate r ,

cL i s t he s pec i f i c heat of wate r ( o fte n omi tted s i n c e cL = 1 Bt u / l bm-F ) ,

a n d TL i s the t emp e ratu re of wate r .

N a hava n di a n d D e l l i n ge r [ 1 9 7 7J po i nt out t h at t h e theo ry of Me r k e l

req u i res two maj o r a n d fou r m i n o r a s s umpt i on s . The most i mp o rt a nt of

these a s s umpt i o n s i s t h at the redu ct i on i n the wat e r fl ow rate du e to

e v a p o rat i on i s neg l i g i b l e . The s econd maj o r a s s umpt i on i s t h at the

L ew i s Numbe r fo r a n a i r -wat e r - va p o r system i s o ne . T h e fou r mi n o r a s -

s umpt i ons a re : ( a ) the humi d i ty rat i o i s n eg l i g i b l e compa red w i th t h e

rat i o of t h e a i r-t o - v a p o r ga s con s t ant s i n t he exp re s s i on f o r t he p a r -

1 1

ti a l p res s u re of vap o r i n t h e mo i st a i r , (b ) t he s atu rat i on humi d i ty

rat i o i s n e g l i g i b l e comp a red w i th the r at i o of the a i r-to-vapor g a s co n

s t a n t s i n t he exp res s i on for t he p a rt i a l p res s u re of vapo r co r re s p o n di n g

to th e mo i st a i r a t t h e i nt e rface t empe rat u re , ( c ) t h e mo i st a i r h umi d

h eat , wh i c h i s t h e sum of the dry a i r s pec i f i c heat a nd the p roduct of

t h e v apor s pe c i f i c h eat and hum i d i ty r at i o , i s c on s id e red to be co n

s t ant , a n d ( d ) t h e ca l cu l at io n of the mo i s t a i r entha l py co r re s p o n d i n g

to t h e i nt e rface t empe ratu re i s b a s ed o n t h e b u l k wat e r t em p e rat u re

rat h e r t h a n t he i nt e rface tem p e rat u re .

T he ent h a l py t h e o ry of Me rke l h a s bee n a l most u n i ve r s a l l y accepted

due to t he fact t h at dea l i n g r i g o r ou s l y w i t h h eat and ma s s t ra n s fe r re

q u i res a d et a i l ed nume r i ca l so l ut i on . Mo reove r , the a na ly s i s of coo l i n g

t owe r heat a nd mas s t ra n s fe r rests on t h e u se of emp i r i ca l data conce r n

i n g f i l l perfo rma nce . Me rke l' s met h od p ro v i des a s i mp l e mea n s fo r i n

t e rp ret i n g t h ese data i n te rms of two n o n - d i men s i ona l p a ramet ers K a V /L

a n d L/ G . W i t h t he a v a i l ab i l i ty of t he h i g h speed comput e r , acc u rate

p red i ct i on s of t ower p e r f o rmance may be ma de a n d the effects of t h e a s

s umpt i o n s may b e acce s s e d .

N a h a v a n di , et a le [ 1 9 7 5 ] con s i de r e d t h e effect of eva po rat i ve

l o s s es on the pe r f o rma nce a n a l y s i s of t h e counte rfl ow coo l i n g t owe r .

N a havand i comp a red KaV / L ca l c u l at e d u s i n g Me r ke l ' s met hod wi th a res u l t

i n c l u d i n g evapo rat i ve l os s e s . Ove r t h e r a n ge of p a ramete rs con s i de red

i n the i r s t u dy , a ma x i mum rel at i ve error g reat e r t h a n 1 2 pe rcent wa s

found between the methods w i t h Me r k e l ' s meth o d g i v i ng the g reat e r

v a l u e . Wh at i s n ot i mmedi atel y c l e a r f rom th i s s t udy h oweve r i s t h e

12

ma g n i t u de of the e r r o r ma d e i n u s i ng Me r k e l ' s met hod bot h for t h e i nt e r

p ret at i on of emp i r i ca l pe rfo rma nce d at a a n d t hen i n a p red i ct i ve s e n s e

f o r oth e r operat i n g con d i t i o n s . I t may be a rgued t hat t he effect wou l d

i n deed be mu ch l es s t han i n d i cated by t he a bo ve s t u dy .

I n a s econd pape r , Na h a v a n d i and Oe l l i n g e r [1977J cons i d e r t h e ef

fects of a l l s i x a s s umpt i on s . T h ey f i n d t h at the fou r mi n o r a s s umpt i o n s

a n d t h e a s s umpt i on of t h e Lewi s Numb e r eq u a l t o u n i ty h a ve l es s th a n 0 . 5

pe rcent effect on the outcome of t h e an a l y s i s . Naha v a n di a n d O e l l i n ge r

conc l u de t h at the ma i n cons i d e rat i on i n i mp ro v i n g Me r k e l ' s met h o d i s to

i n c l u d e t h e effects of evaporat i ve l os s e s on the ene r gy b a l a n c e .

I n t h e above stud i es by Nah a va n d i et a l . , a f i n i t e d i ffe r n ce met h o d

i s u sed to s o l ve t h e h eat a nd mass t ra n s fe r eq u at i o n s . The t ower i s

d i v i ded i nt o N cont ro l vol umes . A set o f f i n i t e d i ffe rence eq u at i o n s

a re deve l oped l i n k i n g s u cces s i ve a d j a cent v o l umes . T h e f i n i te d i f fe r

e n c e eq uat i on s a re then so l ved i t e rat i ve l y .

Y a di ga rog l u a n d P a s t o r [1974J a l s o i n vest i gated t h e effect of t h e

a pp rox i mat i on s of t h e Me rke l met h od o n coo l i n g tower p e rforma n c e . A n

e r ro r of l e s s t h a n 1 0 pe rcent was fou n d when compa r i n g t h e Me rke l s o l u

t i on t o a n "exact" formu l at i on a s des c r i bed i n th e i r p a pe r . Y a di g a rog l u

a n d P a s t o r fou nd t h at t h ree a s s umpt i o n s , ( i ) Le = 1, ( i i ) the r at i o o f

t h e gas con s t a n t s be i n g mu ch g reat e r t h a n t h e spec i f i c humi d i ty , a n d

( i i i ) no e v ap o rat i ve l os s es i n t h e watm'" heat b a l a n c e , were t h e m a i n

cont r i b ut o r s to t h e e r r o r . The Le = 1 a s s u mp t i on t e n d s to u n de res t i mate

wat e r coo l i n g r a n ge whe reas t he o t h e r two a p p rox imat i o n s t e n d t o o v e r

est i mat e wate r coo l i n g r ange . T h e net o v e r e s t i mat e of wat e r coo l i n g

13

range cou l d l ea d t o a non-con s e r v a t i ve d es i g n , depen d in g o n how t h e

met hod i s i mp l emented . Y a d e g a ro g l u a n d P a s t o r a l s o u s ed a f i n i te d i f

ference met ho d to i nt e g rate the gove rn i n g equ at i on ac ros s th e towe r

p ack i n g .

Th e effect s o f t h e app rox i mat i ons o f Me r k e l hq ve been s tu d i e d by

ot h e r i n vest i gat o rs as wel l . S uther l a n d [1983J f o u n d t h at t ower vo l ume

cou l d be u n d e re s t i mat e d by 5 t o 15 pe rcent by not con s i deri n g the ef

fect s of e vaporat i on w i t h i n the t owe r . Poppe [1973J con s i d e red t h e

d e v i at i on s f rom Me r k e l ' s resu l t s i n ca s e s w h e n eq u i l i b r i um i s a l most

reached w i t h i n t he sys t em . Poppe ext e n ded t he d i ffe rent ia l eq u at i ons to

t a k e i nt o accou nt con d en s at i o n when the a i r s tates p a s s e s t h rou g h t he

s at u rat i on l i ne i nto t he fog reg i o n . S uth e r l an d u sed a fou rt h - o r d e r

R u n ge-Kutta ba se d s hoot i n g met h o d to s o l ve t h e gove rn i n g eq u at i on s .

E l ect r i c i t e d e F ra n ce ( s ee t ra n s l at ed rep o rt p u b l i s he d by t h e

E l ect r i c Powe r Re sea rch I n s t i t u t e [1983J ) d eve l o pe d a d i t i g a l comp u t e r

c o d e t h at s o l ves t h e fo rmu l at i on devel oped by P o p p e . T h i s comp u t e r

co de , ent i t l e d TEF ER I , u s e s t h e fou rt h - o r d e r R u n ge - K u kt a m et h o d o f

nume r i ca l i nt e r at i on . T h e mod e l i s const r u ct e d to c a l cu l at e b oth t h e

P o p p e fo rmu l at i on a n d t h e s i mp l i f i ed fo rmu l a t i on of Me r k e l . T he mod e l

was fou nd t o p re d i ct evapo rat ed water f l ow rate we l l a n d t h e c o n de n s a t e

content of the wa rm a i r wi t h good accu racy i n a va l i d at i on s tu dy d o n e by

E l ect r i c i te de F ra n c e . The p re d i ct i on s we re comp a red w i th fu l l - s c a l e

test resu l t s f rom a coo l i n g t ow e r at t h e Neu rat h P owe r Stat i on i n t h e

F ed e ra l Repu b l i c of Ge rmany .

Maj umda r et a l e [ 1983J deve l oped a mat h emat i ca l model wh i c h com-

1 4

p u t e s two - d i me n s i on a l di st r i but i o n s of a i r ve l oc i ty , a i r t empe ratu re ,

p re s s u re , wat e r v a p o r ma s s f ra ct i on and l i q u i d wat e r ma s s f ract i on . The

l oc a l i nt e rface heat a nd ma s s t ra n s fe r rates a re ca l c u l ated f rom emp i r i

ca l c o r re l at i on s u s i n g one of two methods . The f i rst method i n c l u de s

o n e f ree p a ramet e r a n d fol l ow s t h e t reatment of M e r k e l i n wh i c h heat a nd

mas s t ra n s fe r a re comb i ned; t h e second method emp l oys two f ree p a rame

t e rs , t reat i n g heat and ma s s t ra n s fe r sepa rate l y . Maj umda r et a l e u s e a

mu l t i - d i me n s i ona l f i n i te d i ffe rence met h o d t o s o l v e t h e f l ow a n d ene rgy

equat i on s i n t he t owe r . Th i s code i s l i c e n s e d by the E l ect r i c Power

R e s ea rch I n s t i t ute a s t he VERA2D s o ftwa re p roduct .

15

3. FORMU LAT I ON OF MOD EL

As g i ven by the object i ves of the f i r st ch apt e r , t he maj o r pu r p o s e

of t h i s wo rk i s to deve l op a st a n da rd i zed met hod for a na l y z i ng t he p e r

f o rma nce of t h e UHS coo l i n g towe r sys t em t h at i s a l s o f l ex i b l e enou gh to

a l l ow a rea l i st i ca l l y con s e r v at i ve a n a l y s i s of a ny p roposed UHS coo l i n g

t owe r des i g n . Th i s sect i on wi l l d i s cu s s t he met hodo l o gy back i n g the

va r i o u s compute r p rog rams wh i ch comb i ne to e va l u ate t he p e r f o rmance of a

UHS coo l i n g towe r sys t em . The des c r i p t i o n s that fol l ow a re i nt ended

o n l y to g i ve the u n de r l y i ng rea s o n i n g s of t he computer codes , a ctua l a p

p l i c at i o n w i l l be d i s cu s s ed i n det a i l i n C hapter 4 .

The pe rformance a na l ys i s c a n be d i v i ded i nto four d i s t i n ct g rou p s :

t he t owe r s ubmodel , TOW E R; t h e t ow e r / ba s i n mode l , UHSS I M; th e wo r s t - c a s e

se l e ct i on a l go r i t h m , NWS; a nd t h e P e rf o rman ce Data An a l y s i s P rog ram ,

P DAP . Mo r e s peci f i c a l l y , fo rmu l at i on of the tower ana l y s i s , t h e s u b rou

t i ne TOW E R , wi l l be de ve l oped r i g o rou s l y f rom the ent h a l py d r i v i n g fo rce

concept of the Me rke l Met h o d dis c u s s e d i n the seco n d chapte r . T he a s

s umpt i o n s a n d app roxi mat i ons d i s cu s sed ea r l i e r i n t h e deve l o pment of t h e

Me r ke l eq u at i on wi l l be p resent e d a s we l l a s a di s c u s s i on of t h e f i n i t e

d i ffe rence s o l ut i on tech n i que u se d i n t h e c a l c u l at i on of the t ower o u t

l et temp e rat u re . A comp l ete l i s t i n g of the comput e r p rog rams ca n b e

fou n d i n A P P E ND I X A .

3.1 Towe r Su bmode l

T he di s cu s s i o n of the coo l i n g towe r model wi l l beg i n at i t s s i m

p l est f o rm , as a b a s i c h eat exch a n ge r . Th ree types of heat exc h a n ge r

dev i ces a re s h own i n F i gu re 3.1. Any one of these t h ree cou l d rep re s e n t

. �

L

G+fL

16

1 • .. • ..

.... • ... I

4

a) COL,lnterflow

...... � •

.... • ..... ·

4

b) Paral lei Flow

.� 3 1,

2 .... • ... •

.� 4 �,

c) Crossf low

Figure 3.1 Heat Exchanger Devices

2 I •

I..&. --L .... ,

3

1 ..- � .... ,

t.... --L ... I

3

1 ...... • ... .

.. - L( 1-1)

G

1 7

s i mp l i f i ed cool i n g towe r ope rat i on of two fl u i ds t ran s f e r r i n g heat and

ma s s . T hou gh t h ree d i ffe rent h e at exch a n ge r devi ces a re shown , t h e

t owe r an a l y si s formu l at e d h e re wi l l b e for cou nt e rf l ow di rect contact

water coo l i ng by a i r a s s h own i n Fi gu re 3 .1 a .

3 .1 .1 Macros copi c An a l y si s

T o b e g i n t hi s tow e r ana l y si s , con si de r t h e coo l i n g t ower cont ro l

vo l ume of F i gu re 3 .1a . App l yin g the con s e r v at i on of ma s s y i e l ds

( 3 . 1 )

w h e re w i s t he mas s f l ow rate of t h e f l ui d at t he r e s p e cti ve ent ran ce

a n d exi t stat e s . The app l i cati o n of the f i r st l aw of t he rmody n ami c s ,

t he c o n s e rvati on of e n e r gy , to t h e cont ro l vo l ume , a s s umi ng t h at ad i aba

t i c cond i ti o n s exi st at t h e wa l l s an d s u rfac e s an d t h at n o wor k i s be i n g

done o n t he system , g i ves

( 3 .2 )

where h i s the ent h a l py of t h e f l u i d at t he res pect i v e ent rance and ex i t

s t at e s . The i n l et cond i t i ons at s t ates 1 a n d 3 a re t h e k n own v a l u es a n d

t h e exi t con d i t i o n s a t states 2 a n d 4 a re t h e va l ue s wh i ch mu s t b e de

t e rmi ned . C o n s eq u e nt l y wI' w3' hI and h 3 a re k n own q u a nt i ties and w2 ,

w4' h2 a n d h4 a re u n k n ow n va l u e s . W i t h two equat i on s a n d fou r u n kn ow n s ,

two more const ra i nts a re needed t o s o l ve t h i s p ro b l em . T h e se con

s t r ai n t s a re fou n d u s i n g a mi c ro s cop i c a na l y s i s of t h e heat a nd mas s

t ra n s fe r wi t h i n t he t owe r .

18

3.1.2 M i c ro s copic An a l y s i s - Me rkel Meth o d

The m i c roscop i c a n a l y s i s of t h i s s ect i on wi l l c o n s i st of deve l o p i n g

t he di f f e rent i a l equ at i o n s wh i ch des c r i b e t h e heat a n d ma s s t ra n s fe r

p henome n a w i t h i n an e vaporat i ve coo l i ng t owe r . A segment o f i n f i n i t es -

i ma l vol ume wi t h i n t h e t owe r i s s h own i n F i gu re 3 . 2 . A heat ba l a n ce

equ at i on f o r the wate r , a s s umi n g l oca l l y u n i form water temp e ratu re TL ,

i s writt e n i n di ff erent i a l form a s

( 3 . 3 )

whe re q i s the heat t ra n s fe r rat e , L i s t h e mas s f l ow rate of wat e r , cL

i s t h e s p e c i f i c heat of wat e r , TL i s t h e t empe ratu re of wat e r , a n d To i s

t he e n t h a l py refe rence t emperatu r e . By exp a n di ng Equ ation (3.3) a n d

neg l e ct i n g p rodu ct s of d iffe rent i a l q u a ntit i es , the wat e r heat -b a l a n c e

equ at i on becomes

( 3 . 4 )

A heat b a l a n ce eq u at i on fo r t h e a i r st ream i s gi ven i n di ffe r e nt i a l fo rm

as

( 3 . 5 )

w a ter stream

19

air stream

Figure 3.2 I'nfinitesimal Volume of Cooling Tower Packing

dV

20

where G i s the ma s s f l ow rat e of a i r , ca i s t h e s pec i fic heat of dry

a i r , Tm i s t he m i xed-mean tempe rat u re of d ry a i r a n d wat er- vapor mi x-

t u re , w i s abs o l u te humi di ty , Cv i s the s pec i f i c heat of wat e r vapo r , 0

and h f g i s t he s p e c i f i c h eat of va po r i z at i on at t emp e r at u re To ' Ag a i n

expand i n g a n d neg l ect i n g p rodu ct s of di fferent i a l q u a nt i t i es , t h e a i r

h eat -ba l a nce equat i on becomes

dq = G (c + w C ) dT + G [c (T - T ) + hOf

] dw. (3.6) a v m v m 0 9

The def i n i t i on of humi d a i r e ntha l py ,

= ca (T - T ) + w [c (T - T ) + hO

f ] , m 0 v m ° 9 (3.7)

where hG i s t he e n t ha l py of mo i st a i r , h a t h e ent h a l py of d ry a i r , a n d

h v t h e ent h a l py o f wa t e r va p o r , c a n b e di f f e r e nt i a t e d a s

(3.8)

and s u bs t i t uted i nt o Equat i on ( 3 .6 ) to compa ct t h e a i r st ream h eat b a l -

a nce equ at i on t o

dq = G dhG • (3.9)

Equat i on s (3.4) a n d ( 3 . 9 ) a re t he h eat ba l a nce equat i on s for t h e wat e r

a nd a i r st reams res pect i vel y .

21

Heat i s t ra n s fo rmed at the i nte rface s h own i n F i g u re 3 . 2 by two

mec h a n i sms . The f i rst mecha n i sm i s s e n s i b l e h eat c o n v e ct i on

( 3 .10 )

when h i s the heat t ra n s f e r coeff i c i e nt , a i s the s u rface a i r p e r u n i t

vo l u me of towe r f i l l , a nd V i s the vol ume o f tower f i l l . The s econ d

mech a n i s m i s l atent heat t ra n s p o rted by eva p o r at i on

( 3 .11 )

where hf g i s the spec i f i c heat of vapo r i z at i o n at the a ct u a l wat e r temp

e r atu re . E q uat i on ( 3 .11) rep resents the amo u nt of e n e r gy t ra n s fe r re d

dur i n g the evaporat i on of a mas s o f wate r dL . The s u mmi n g o f Eq u at i o n s

( 3 .1 0 ) a n d ( 3 .11 ) y i el ds

( 3 .1 2 )

the tota l heat t ra n s fe r .

The mas s tran s fe r rate, o r rate of eva p o r at i on, at the a i r wat e r

i nt e r face, a s suming t h at a t t h e i n t e r fa ce t h e water v a p o r i s s at u rat e d

at the l i qu i d temp e ratu re, may be wr itten as

( 3 .1 3 )

2 2

where hO i s the ma s s t r a n s f e r coef f i c i ent , P v , s at ( TL ) i s the den s i ty of

s atu rated w ater v apor at t emp e rat u re TL , a n d pv ( Tm ) i s the den s i ty of

wat e r vap o r at temp e rat u re Tm .

I n Eq u at i on ( 3 .1 3 ) , t h e dr i v i n g fo rce for mas s t ra n s f e r i s the den -

s i ty d i ffe rence b etween water v apor at t h e i nte rface a n d wat er v a p o r i n

the humi d a i r st ream . I n Eq uat i o n ( 3 .10 ) the dr i v i n g force for s e n s i b l e

heat t ra n s fe r i s the temp e r atu re d i ffe rence b etween the wat e r st ream a n d

t h e a i r st ream . Th e s i mp l i f i ed f o rmu l at i on of Merke l comb i n e s t h e s e n -

s i b l e h eat t ra n sfer a n d the heat t ra n s fe r d u e to evapo rat i on to o bta i n a

s i n g l e dr i v i n g force for tot a l heat t ra n s fe r . T he s i n g l e d r i v i n g fo rce

p roposed by Me rke l i s the d i ffe ren ce between t he ent h a l py of the s at u r-

ated a i r at the i nte rface and t h e e nt h a l py of the h um i d a i r st ream .

To b eg i n the f o rmu l at i on of t h e s i n g l e d r i v i n g fo rce concept , con-

s i de r that s i nce humi d a i r i s a mi xtu re of d ry a i r a nd wat e r va�or bot h

components a re a s s u med t o obey the i dea l gas l aw

a n d

P = P R T v v v m

( 3 .1 4 )

( 3 .15)

whe re Pv a n d P a a re t h e p a rt i a l p re s s u res of wat e r vap o r a n d a i r res pec

t i ve l y , P v a n d P a a re t he den s i t i e s of water vapor a n d a i r r e s p e ct i v e l y ,

a n d R i s t he perfect gas l aw con s t ant . T h e s e two equ at i on s a re re l ate d

by O a l ton ' s L aw between t ot al a n d p a rt i a l p res s u res

P = P + P , a v

23

a n d by the def i n i t i o n of abso l ute humi d i ty

(3.16)

(3.17)

U s i ng Equat i on s (3.14) t h rough (3.17) , vapo r den s i ty can be expre s s e d

a s

P (

W P v

= � W + R /R v m a v ) . (3.18)

E q uat i on (3.18) can t he n be s ub st i t uted i nt o t h e equ at i on for ma s s

t ra n s fe r , (3.l3), to g i ve

w ( T ) (T ) +

mR /R J a dV , ( 3 .1 9 )

w m a v

where w s at ( TL ) i s t he abso l ute s atu rat i o n humi d i ty at the wat e r temp e r a

t u re TL•

To cont i nu e , heat a n d mas s t ra n s fe r a re re l at e d by the Ch i l t o n -

Co l b u rn a n a l o gy wh i ch u s es fami l i a r di men s i on l e s s g roup s to rel ate t h e

heat a n d ma s s t ran s fe r coeff i c i ent s ,

Nu Sh -�� = -""--�-:::-

Re Pr1 /3 Re Sc

1 / 3 ' ( 3 . 20 )

2 4

whe re Nu i s the Nu s s e l t n umbe r , Re t h e Rey no l ds numbe r , P r t he P r a n dl t

n u mbe r , Sc the Schmi dt n u mbe r , a n d Sh the S h e rwood n u mbe r . C ance l i n g

t he Reyno l ds number f rom each s i de of Eq uat i on ( 3 . 20 ) a n d man i p u l at i n g

a l l ows t h e rel at i o n s h i p b etween t h e heat t ra n s fer coeff i c i e nt a n d mas s

t ra n s f e r coeff i c i ent t o be g i ven a s

( 3 . 21 )

where PG i s the den s i ty of the mo i s t a i r a n d cG i s the s pec i f i c heat of

mo i st a i r . Def i n i ng the Lewis n umbe r , Le , as the P ra n dl t n umber o ve r

t h e Sc hmi dt numbe r a l l ows Eq uat i on ( 3 . 21 ) t o b e rew r i t t e n as

h Le 2 / 3 h D = -

P G-c-G

S u b s t i tut i n g t h i s exp res s i on i nto Equat i on ( 3 .19 ) g i ves

( 3 . 2 2 )

( 3 . 2 3 )

To s impl i fy t h i s rat h e r fo rmi dab l e equ at i on fou r app r o x i mat i o n s a re

made . The f i rst a s s umpt i on i s t h at for an a i r -wat e r - va p o r sys t em t h e

Lewi s numbe r i s equ a l t o u n i ty . T h e sec o n d i s t hat t h e temp e ratu re

rat i o TL/Tm i s a s s umed to be equ a l to u n i ty . Th i rd , s i n ce the rat i o

2 5

R a /R v = 0 . 6 2 2 a n d a typ i c a l va l ue f o r w i s 0 . 0 5 , w s at a n d w a re el i mi n

ated f rom t h e denomi n ator s o f E q uat i on ( 3 . 2 3 ) . A n d l as t l y , the g roup

P / (P GR vTL ) i s co n s i de red to be eq u a l to u n i ty . W i th t h i s a p p rox i

mat i o n , Eq u at i on ( 3 . 2 3 ) reduces t o

( 3 . 2 4 )

Th i s equat i on can now b e s u b st i t u t e d i nto t h e equ at i on f o r t ot al i nt e r -

face h eat t ra n s fe r , E q u at i on ( 3 .1 2 ) , to y i el d

( 3 . 2 5 )

wh i ch cont a i n s on l y the h eat t ra n s fe r coeff i c i ent h .

S i nce

( 3 . 26 )

E q u at i on ( 3 . 2 5 ) can be rew r i tten a s

h 0 dq = � {CG ( TL - Tm ) + [ hf g + Cv ( TL - To

)][ws at( TL ) - w ( Tm ) ]} a dV .

( 3 . 2 7 )

2 6

Th e v a r i a b l e cG ' t he spec i f i c heat at con s t a nt p res s u re of a i r a n d wat e r

v a p o r mi xt u re , can b e w r i tten

p aca + p c v v

cG =

Pa + P v ( 3 . 28 )

o r

c + w C a v cG

= 1 + W ( 3 . 29 )

T he f i fth app rox i mat i on i s to l et 1 + W _ 1 so t hat

By s ub st i t ut i n g t h i s exp res s i o n i nt o the b r a c k et e d t e rm i n Eq u a -

t i on ( 3 . 2 7) a n d u s i n g the def i n i t io n of s p e c i f i c e nt h a l py of t h e a i r

wat e r - v a p o r mi xtu re p e r u n i t ma s s of d ry a i r , E q u at i o n s ( 3 . 7 ) a nd ( 3 . 2 7 )

can be w ritten as

( 3 . 31 )

where h/ cG rep resents a ma ss t ra n s fer coeff i c i ent wh i ch i s u s u a l l y de

n oted by K .

The si xt h a n d f i n a l a s s umpt i on t o be made cons i s t s of negl ect i n g

t h e t e rm cL ( TL - To ) dL , the change i n wat er f l ow due t o evapo rat i on i n

t h e wat e r heat - ba l a n ce i n Eq uat i on ( 3 . 4 ) . Once t h i s t e rm i s neg l ect e d ,

2 7

t h e th ree equ at i o n s of d i ffe rent i a l heat t ra n s fe r can be equ at ed , E q u a

t i o n s ( 3 . 4 ) , ( 3 . 9 ) , a nd ( 3 . 3 1 ) , t o y i e l d

( 3 . 32 )

wh i ch i s t he Merke l equ at i on f o r t ot a l h eat t ra n sf e r i n te rms of t he e n

t h a l py d r i v i n g force .

3 . 1 . 3 E n t h a l py D r i v i n g Fo rce I mp l eme n t at i on

The ent h a l py d r i v i n g fo rce concept de r i ved i n the p re v i ous s e ct i o n

i s u s e d t o so l ve for t h e ou t l et co l d wate r t emp e ra t u re of t h e coo l i n g

t owe r i n s u b rout i ne TOWER . T h i s c o n cept i s i mp l emented u s i n g a fo rwa r d

t i me , cente re d - s p ace f i n i te d iff e re n c e app rox i mat i on .

The i mp l ement at i on b eg i n s w i th t h e gove rn i n g equ ati}ns of Me rke l ,

E q u at i o n ( 3 .32 ) , g i ven above . F i g u re 3 . 3 a i l l u st rates t he e s s e nt i a l

f eat u res o f t h e cou n t e rf l ow coo l i n g t owe r . T h e e n t h a l py d r i v i n g fo rce

for the f i n i t e d iff e rence a n a l y s i s i s con s i d e re d t o be vapor s atu rat e d

at t emp e ratu re TL a nd v a p o r s at u rated at t h e wet - bu l b t emperatu re .

E q u at i on ( 3 . 32 ) can be rew r i tten a s

( 3 . 33 )

Nond i men s i o n a l va ri a b l es i n t e rms of e n t h a l p i es a re i nt rodu ced a s

( 3 . 3 4)

2 8

TL,o T wb,1

--Vo=V

W A A I T R E R

V=O

T L I T wb,o ,

a) Simpl ified Cool i ng Tower

ha,o hL,1 1

1 2

2 3

• •

I n [1

b) Discrete Control Volumes

n

c) Generic Cell

Figure 3.3 Numerical Formulation of Cool ing Tower Problem.

29

f o r t he ga s s i de a nd

(3.35)

f o r t he l i q u i d s i de , a l l ow i ng t h e e n t h a l py d i f f e rence o f E q u a t i on (3.33) t o be exp re s s e d as

(3.36)

The ex p re s s io n dTL i s rew r i tten i n t e rms of e n t h a l py a s

o r

(3.37)

The d i f ferent i a l entha l p i es a re computed i n t e rms of n o n d i me n s i onal

p a r amete r s a s

(3.38)

a n d

(3.39)

3 0

a n d a l on g w i t h Eq u at i ons ( 3 . 36 ) a n d ( 3 . 3 7 ) a re s u b st i t uted i nt o Equ at i on

( 3 . 3 3 ) to y i e l d

= ( K a ) / G ( hL* + hG* - 1 ) dV* , ( 3 . 40 )

w h e re V* = V olVo

Equ at i on ( 3 . 40 ) i s n ow app rox i mat e d nume r i c a l l y u s i n g d i s c rete con

t rol v o l u mes as s hown i n F i gu re 3 . 3 b . Equ at i on ( 3 . 3 3 ) become s , u s i n g

f i n i t e di ffe rence app rox i mat i on ,

( 3 . 4 1 )

whe re hL ( n ) i s equa l to hL at ( TL ( n ) + TL ( n+ 1 ) ) / 2 .

Nond i men s i on a l i z i n g a n d a l geb ra i c a l l y s i mp l i fy i n g Equat i o n ( 3 . 4 1 )

y i e l ds a t r i d i agona l set of equ at i o n s of t h e form

(3 . 4 2 )

for 1 � n � N+1 , wh e re N i s equ a l t o t h e n umber of d i s c rete ce l ls o r

3 1

d i s c ret e cont ro l vo l ume s . The va l u e s of the coef f i c i ent s A , B , C , a n d D

of E q u at i on ( 3 . 42 ) va ry depen d i ng on wh i ch ce l l i s be i ng c o n s i dered .

The bou n d ry cel l s mu st be cons i de red i n d i v i du a l l y becau s e o f t he i r i n i

t i a l cond i t i o n s w h e reas t h e i nt e r i o r ce l l s may be con s i dered g e n e r i c a l

l y , a s s how n i n F i g u re 3 . 3 c . F o r i nt e r i o r ce l l s 2 t h rou gh N t h e coef f i

c i ents for equ at i on ( 3 . 42 ) a re

A = ( K a �V* ) / ( 2 L ) [R ( n ) - ( L R ( n ) ) / ( G R ( n - 1 ) ) ] - R ( n ) / R ( n - 1 ) ,

B = ( K a �V* ) / ( 2 L ) [ R ( n ) /R ( n- l ) - L / G] + R ( n ) /R ( n - 1 ) + 1 ,

C = ( K a �V* ) / ( 2 L ) [ L/ G - R ( n ) ] - 1 ,

a n d

D = 0 , ( 3 . 4 3 )

whe re R ( n ) = h L ( n ) / cL a n d R ( n - l ) = hL ( n- 1 ) / cL • F o r bou n da ry ce l l s , w h e n

n i s equ a l t o N+1 t h e a i r i n l et i s k now n , t h e refore hG = hG , i a n d

hG* ( N+ 1 ) = O . The coeff i c i ent s f o r t h i s c a se a re

A = 0 ,

B = ( K a �V* ) / ( 2 L ) [R ( N ) -L/ G] - 1 ,

3 2

c = ( K a �V* ) / ( 2 L ) [R ( N ) - L/G ] - 1 ,

a n d

o = ( K a �V* R ( N ) ) / L .

When n = 1 , the wat e r i n l et t emp e r atu re i s k n ow n , t h e refore h L =

a n d hL* ( 1 ) = O. The coeff i c i e nt s for t h i s ca se a re

A = 0 ,

B = 2 ,

c = 1 ,

a n d 0 = O.

(3. 44 )

hL . , ,

(3. 45 )

The t r i di agona l s et of equ at i ons gene rated u s i n g t h e f i n i te d i fference

a p p rox i mat i on i s so l ved u s i n g the t ri d i agon a l mat r i x so l ve r TOMS . A

comp l ete l i st i n g of t he s u b rout i ne i s l i sted i n AP P E ND I X A for fu rth e r

refe rence .

3 . 2 Towe r / Ba s i n Model

The second maj o r code t o be d i s cu s s ed i s the t owe r /ba s i n model .

T h i s model , wh i ch i nc l udes sub rout i ne TOWER , a l l ows the ent i re coo l i n g

sys t em to be a n a l yzed . A s i mp l i f i ed schemat i c of the p l ant / t owe r / ba s i n

3 3

l ayout i s s h own i n F i gu re 3 . 4 . As can be seen , wat e r i s drawn f rom t h e

b a s i n , h eated t o a t empe ratu re T i n wh i ch depen ds on the heat l oad of t h e

p l a nt , a n d p a s s e d th rou g h t h e cool i n g tow e rs whe re a i r i s u s e d i n di rect

cont act t o remo ve heat f rom the wate r . The i nt e g r at i on of each of t h e s e

p a rt s i nt o a system f o r ana l y s i s i s done i n p rog ram UHSS I M . A fl ow

c h a rt of p ro g ram UHSS IM s h ow i ng t h i s i nt e g r at i on i s g i ven i n F i gu re

3 . 5 . As can be seen f rom the f l ow cha rt of F i gu re 3 . 5 , prog ram UHSS I M

can b e di v i ded i nto t h ree g rou p s .

T h e f i rst g rou p con s i s t s of rea di n g dat a a n d f i tt i n g s ome of t he

data to c u r ves to a l l ow d i s c rete p o i nts to be r e p r e sented as cont i n uou s

c u r ves . T i me - v a ry i n g va l u es of t h e pa ramet e r s KaV / L and L /G a l ong wi t h

t i me- v a ry i n g v a l u es of l i q u i d f l ow rate a n d h eat l oad a re read and f i t

t o l i ne a r cu rves to a l l ow i nt e rpo l at i o n t o be p e r f o rmed . T i me- va ry i n g

v a l ues of wet - bu l b t emp e ratu re a re r e a d a n d f i t t o a cu rve u s i n g a He r

mi t e-Cub i c Sp l i ne f i t a l s o f o r i nt e rpo l at i on to be eas i l y pe rforme d .

Ot h e r s et v a l u es s u ch a s t h e numbe r o f coo l i n g t owe r s , numb e r o f t i me

step s , a n d step du rat i on a re a l s o rea d i n t he i n i t i a l port i on of p ro g ram

UHSS I M . The s econd g roup i n p ro g r am UHSS IM i s t h e man a ge r . A t i me l oo p

i s sta rt e d t o s i mu l ate acc i dent i n i t i at i o n a n d c a l l s a re made to s u b r o u

t i nes as req u i re d .

T h e t h i rd g rou p con s i s t s of s u b rout i n e s BAS I N a n d TOWER . S u b rou

t i ne TOWER , as ment i oned e a r l i e r , i s needed t o c a l c u l ate the t emp e ratu re

of the wat e r l ea v i n g t he cool i n g t owe r s . S u b rout i n e BAS I N s i mu l ates t h e

c ha n g i n g bas i n c o n d i t i o n s w i th t i me . Two o r d i n a ry d i ffe rent i a l eq u a -

L ( t ) T I N

P L A N T

t L ( t ) _

TS

a i r f l o w >

+ L 1 ( t )

.:t. .... ... a: w 3: o ...

+ L 2( t )

� (II ... a: w 3: o ... -.

I- L3 ( t )

..t. C? ... a: w 3: o ...

+1B 1 +1B 2 tTS 3 L 1 ( 1 - f1 ) I L 2 ( 1 - f2 ) L 3( 1 - f3)

.. .. ..

B A S I N

T B ( t ), M B( t ) , S ( t )

I

F i g u re 3 . 4 S c h ema t i c o f P l a n t / Towe r / B a s i n S i mu l a t i o n .

., 'lit ... a: UJ 3: o ...

L4( t )

+TS 4

(4 ( 1 - 1 ) � � 4

3 5

F i gu re 3 . 5 F l ow Cha rt o f P rogram UHSS I M .

I N I T I A L

I N F O R M A T I O N

T S T E P , N S T E P , e c t

I

S E T

P A R A M E T E R S

� , -- �, T I M E L O O P

T I M E - V A R Y I N G

K a V / L , L / G ,

L, Q

'0

L I N E A R

F I T

O F D A T A

� ,

I N T E R P O L A T E L / G , K a V / L , L

a t e a c h t i m e s t e p

O D E

B A S I N

I n t e r p o l a t e Tw B ' Q a t e a c h t i m e s t e p

C a l c u l a t e d M / d t , d T / d t

T O W E R

C a l c u l a t e t o we r e x i t

t e m p e r a t u r e s

....


W E T - B U L B

T E M P E R A T U R E

'0

H E R M I T E

C U B I C S P L I N E

F I T

�,

3 6

t i o n s a re u s e d t o rep resent t he c hang i n g ba s i n temp e rat u re a n d cha n g i n g

b a s i n ma s s at each t i me s tep . The c h a n g i n g b a s i n t empe ratu re i s g i ven

by t h e f i r st l aw of the rmody nami c s a s

( 3 . 46 )

where m i s bas i n ma s s , TB i s ba s i n temp e rat u re , Lj i s the l i q u i d f l ow

rates of each t owe r , j , Tout , j i s the o ut l et temp e ratu re for each t owe r ,

j , a n d L i s the i n i t i a l l i q u i d f l ow rat e . The chan g i n g ba s i n mas s i s

g i ven a s

d m _ .JL dt - h LHF ,

fg ( 3 . 47 )

whe re q i s t he heat t ra n s fe r , hfg i s t h e s pec i f i c heat of vapo r i z at i o n ,

a nd LHF i s k n own as the Latent Heat F a c t o r . Th i s equat i on can be re-

wr i t t e n as

L cL

( T . - T ) LHF l n out

( 3 . 48 )

whe re Ti n and Tout a re towe r i n l et and out l et t emp e rat u res of t he

t owe r .

A n o rd i n a ry di fferent i a l equ at i on s o l v e r ca l l e d ODE i s u s e d t o

so l ve E q u at i ons ( 3 . 46 ) a n d ( 3 . 48 ) . I n t h i s c a s e ODE i nteg rates f rom T i n

t o Tout . Once t h e di fferent i a l eq u at i o n s a re s o l ved , u p dat e d v a l ve s o f

bas i n t empe ratu re a nd ba s i n mas s a re o bt a i ned a n d the t i me l oop c o nt i n -

3 7

ues . The c h a n g i n g s a l i n i ty a n d d i s s o l ved so l i ds concent rat i o n i s det e r

m i ned d i rect l y f rom t he c h a n g i n g ba s i n ma s s s i n c e a s ba s i n mas s de

c reases , di s s o l ved so l i d or s a l i n i ty cont ent i n c re a s e s . Th e refore t he

c h a n g i ng s a l i n i ty concent r at i on i s fou n d by di v i d i n g t he u p dated ba s i n

ma s s va l ve by t h e o r i g i n a l s a l i n i ty and d i s s o l v e d so l i d cont ent .

3 . 3 W o rst-Case Meteoro l o g i ca l Se l ect i on Al go r i t hm

T he p rog ram NWS (�at i o n a l Wea t h e r �e r v i c e ) reads met eo ro l o g i ca l

d ata f rom Nat i o na l Weat h e r S e r v i ce Tape Data F am i l y- 14 ( TDF - 1 4 ) ma g n et i c

t apes . Hou r l y o r t h ree- h ou r l y va l u e s of u p t o 48 met eoro l o g i c a l va r i

a b l es a re s t o red on t h e s e t apes i n an a l p h a n u me r i c fo rmat . T h i s p ro g ram

i nt e rp ret s on l y t he wet -b u l b t emp e rat u re va l u e s f rom the t ape . P rog ram

NWS s cans the t a p e , f i n d i n g w i n dows of wors t - c a s e met eoro l o g i ca l c o nd i

t i on s . A u n i q u e feat u re t o t h i s p rocedu re i s t h e fact that t he w i n dow s

fo l d t o p revent w i n dows f rom o verl a p p i n g a n d l i m i t i ng the data ra n g e .

Two te rms u sed i n t h e p rog ram NWS wi l l be de r i ved h e re a n d t h e re

ma i n de r of the code w i l l be d i s c u s s ed by way of a p p l i cat i on i n C h a p t e r

4 .

3 . 3 . 1 Ru n n i n g Wet -B u l b Temp e r at u r e

The occu r re n c e s o f wo r st -c a s e met eoro l o g i c a l cond i t i on s a re fou n d

by s e g regat i n g p e r i od s , o r w i n dows , o f ma x i mum ru n n i ng wet - b u l b t emp e r a

t u re s o n t h e t a p e . Th e ru n n i n g wet -b u l b temp e rat u re , a wei g hted a v e r a g e

o f t h e p resent wet -b u l b t emp e r at u re w i th the p re v i ous ru n n i n g wet -bu l b

t emp e r at u re , rep res e n t s t h e t i me l ag of the sys t em t o con s t a nt l y cha n g

i n g wet -bu l b t emp e r at u res .

3 8

The de r i vat i o n o f the ru n n i n g wet -bu l b temp e rat u re beg i n s w i th an

e n e r gy b a l a n ce

( 3 . 49 )

of t h e p l a nt /towe r / b as i n system where qp ( t ) i s t he heat t ra n s fer of the

p l a nt . The heat l oa d of t he bas i n , qS ( t ) , a nd t h e heat l oa d of the

t owe r , qt ( t ) , can be def i n e d a s

( 3 . 5 0 )

a n d

( 3 . 5 1 )

res pect i ve l y , whe re

( 3 . 5 2 )

T i n i s t he t owe r i n l et temp e rat u re , TWb i s the wet - b u l b temp e r at u re , T8

i s t h e bas i n t emp e rat u re , m i s t he bas i n mas s , cL i s t h e s p e c i f i c heat

of wate r , L i s the l i qu i d f l ow rat e , a n d € i s the e f f i c i ency of t he

towe r g i ven by ( T i n - Tout ) / ( T i n - Twb ) where Tout i s t he tow e r out l et

t emp e r at u re . S u b st i t ut i n g equ at i on s ( 3 . 5 0 ) , ( 3 . 5 1 ) , a n d ( 3 . 5 2 ) i nt o

equat i on ( 3 . 49 ) y i e l ds

3 9

( 3 . 5 3 )

The t i me- depen dent b a s i n tempe rat u re can be rep res e nted by comb i n i n g

p l a nt heat l oa d t e rms a n d di v i di n g by m cL

( 3 . 5 4 )

whe re L i s equ a l to T = m/ ( L € cL ) .

E q u at i o n ( 3 . 54 ) s hows that t he b a s i n t empe rat u re wi l l b e i n f l u e n c e d

by the c h an g i n g amb i ent wet -bu l b t emp e rat u res a nd the temperatu re d u e t o

p l a nt heat rej ect i on rep resented by t h e l as t t e rm i n t he equ a t i on .

T herefo re , t he b a s i n t emp e rat u re may be w r i tten a s

( 3 . 5 5 )

t h e sum of the two i nf l u ences , whe re TS ,wb i s t he ba s i n temp e ratu re due

t o wet - b u l b t emp e ratu re effect s a n d Ts , Q i s t he b a s i n temperat u re due t o

p l a nt heat l oa d effect s . The re fo r e , by t h e p r i n c i p l e o f s u p e r p os i t i on ,

t he b as i n t emp e ratu re may be rep resented by

( 3 . 56 )

a n d

4 0

( 3 . 5 7 )

for wet - bu l b tempe ratu re ef fect and for heat l oa d e f fec t , respect i ve l y .

To i nt e rp ret mo re f u l l y t he effect s of t he va ryi n g amb i e n t wet - b u l b

temp e r atu re , Equat i on ( 3 . 56 ) , w r i t ten i n o r d i n a ry d i f fe rent i a l equat i on

f o rm ,

( 3 . 58 )

i s s o l ved to y i e l d

( 3 . 5 9 )

I nt e g r at i ng f rom t i me t to t i me ze ro , a nd s o l v i n g fo r the b a s i n te rnp e ra-

t u re du e to the wet -bu l b t emp e rat u re effect s g i v e s

t T

B ,wb = TB ,wb (O ) exp ( -t /T ) + exp ( -t / T ) fo

exp ( t l / T ) T (� I ) dt l .

F rom Equat i on ( 3 . 60 ) , the ru n n i n g wet -bu l b temp e ratu re

t = f

-T T ( t l )

e xp [ ( t l -t ) / t J wb dt l T

i s obt a i ned . The ru n n i n g wet -bu l b temp e ratu re i s w r i tten a s

( 3 . 60 )

( 3 . 6 1 )

A

dTB , wb dt

A A

T T B , wb + wb T T

4 1

( 3 . 6 2 )

f o r eva l u at i on . E q uat i on ( 3 . 6 2 ) i s exp res s e d i n f i n i t e di f f e rence f rom

a s

( 3 . 6 3 )

by a pp l y i n g the i mp l i c i t Backwa r d E u l e r s cheme . The n+ 1 v a l u e s a re

t hose b e i ng c a l c u l ated , a nd t h e n v a l ues a re k n own q u a nt i t i e s . E q u a t i on

( 3 . 6 3 ) i s rew ri tten a s

( 3 . 6 4 )

t o ca l c u l a te the ru n n i n g wet - b u l b temp e r a t u re at each t i me s t ep w i t h i n

p ro g ram NWS .

3 . 3 . 2 T i me Con s t a nt

The t i me c on s t ant of t he t owe r / ba s i n system rep r e s e nt s the res p o n s e

of the b a s i n temp e rat u re t o e xt e r n a l i n f l uences , n a me l y wet - b u l b t emp e r

a t u re a n d p l a nt h e a t l oa d . Th i s syst em res p o n s e i s a s s umed t o b e o n e o f

f i r s t - o rde r .

T he ca l c u l at i on of the t i me con s t ant begi n s f rom a n e n e r gy b a l a n ce

of the sy stem as s h own p re v i o u s l y i n E q u at i on s ( 3 . 49 ) t h rou gh ( 3 . 54 ) .

F rom Equat i on ( 3 . 54 ) a recu r s i ve rel a t i o n s h i p c a n be w r i tt e n a s

4 2

( 3 . 6 5 )

where i rep resents t he cu r rent t i me s t ep a n d i - 1 the p re v i ous t i me

s t e p . To det e rmi n e t he t i me con s t a n t . t he sum of the sq u a res of t h e

b a s i n temp e r a t u re TB • i a nd a t emp e ra t u re p re d i cted u s i ng p rog ram UHSS I M i s mi n i mi z e d . Th i s mi n i m i z i n g p rocedu re i s exp l a i n ed mo re fu l l y i n Sect i o n s 3 . 4 a nd d i s c u s s ed v i a examp l e i n C h a p t e r 4 .

3 . 4 - P e r f o rma nce Da ta An a l y s i s

T h e Perfo rman c e Data An a l ys i s P rog ram ( P DAP ) c a l c u l a t e s the t ow e r

cha racte r i s t i c KaV /l v a l ues a n d . u s i n g t hese generated KaV/l v a l u e s .

p re d i c t s t he tower col d wat e r temp e ra t u re . Wet -bu l b t emp e ra tu re va l ue s .

r a n ge v a l u e s . a n d co l d water t emp e rat u re v a l u es a re obta i ned f rom ma n u

f a ct u re r s u p p l i ed pe rfo rma nce cu r v e s , a n d a compa r i s o n of the ma n u fa c t u re r s u p p l i ed co l d wat e r tempe ratu r e va l ues w i th p re d i cted co l d wa t e r

t emp e ra t u re v a l ue s a l l ow s ver i f i c a t i o n of t h e man u fact u re r ' s da t a . A

fl ow d i a g ram for p rogram POAP i s g i ven i n F i g u re 3 . 6 .

As s h ow n o n the f l ow cha rt o f F i gu re 3 . 6 the t ow e r s u bmode l i s u s e d

i n p ro g ram PDAP t o p re d i ct t h e t ow e r out l et t emperat u re , rep res e n t e d a s

Tp re d ' Fo l l ow i ng t h e f l ow cha rt s s hows that t h e a ct u a l col d wat e r temp

e rat u re v a l u e s , Ta ct ' a re s u bt ra ct e d f rom the p red i ct ed co l d w a t e r t emp

e rat u re va l u e s an d t h i s di f fe re n c e i s squ a red . Th i s d i fference p roce du re i s repe a t ed fo r each co l d wat e r temperatu re v a l u e a v a i l a b l e f r om

the man u f a ct u r e r at a g i ven des i gn f l ow ra t e . A s i mp l e schemat i c g i ven

i n F i gu re 3 . 7 s hows the two meth ods of obt a i n i n g the co l d wat e r t emp e r a -

4 3

F i g u re 3 . 6 Fl ow C ha rt o f Pe rfo rma n c e Da ta Ana l ys i s P ro Q ram .

I N P U T

M a n u f a c t u r e r ' s D s t a

R a n g e , W e t - B u l b T e m p e r a t u r e ,

C o l d W a t e r T e m p e r a t u r e - Ta c t

F M I N ·

..

I s S T 2 m i n i m u m ?

N O

�, N E W

K a V / L

� --. T O W E R

S U B M O O E L

� , Tp r e d j

� O = ( Tp r e d - Ta c t ) 2

I ..

Y E S ... . R e s u l t s

K a V / L � f o r P l o t t i n g

fJJtt�d T �,. ()dU -7 (JAAJV'--' 1 lr'-, .c.- !J,.#J".lJy-1f'�JJ:

K a V / L

4 4

W E T - B U L B P E R F O R M A N C E

T E M P E R A T U R E ,� I---� Ta c t C U R V E S

R A N G E

T O W E R

S U B M O D E L 1-----------»>- T p r e d

F i g u re 3 . 7 Me t h o d s o f O b t a i n i n g C o o l i n g Towe r C o l d Wa t e r Tempe ra t u re .

I I I I

S T . I I I I I I I I I I I I

K a V / L l o w K a V / L h i g h

K a V / L

F i g u re 3 . 8 Fun c t i o n FM I N - M i n i mum Ka V/ L C a l c u l a t i o n .

4 5

t u re va l u es . Th e act u a l co l d wat er t emp e ratu re i s obt a i ned d i rect l y

f rom man u factu rers pe rfo rmance c u rves a n d the p re d i cted col d water temp

e ratu re f rom t h e towe r submode l a s stat ed e a r l i e r .

The t ower cha racte r i s t i c KaV/L v a l ve u s ed i n the t ower s u bmodel i s

s h ow n i n F i gu re 3 . 6 . The fu n ct i o n rout i ne FM I N f i n d s a n app rox i mate

KaV /L v a l ve when a second f u n ct i o n , S T 2 , att a i n s a m i n i mum w i t h i n a

c h o s e n i nt e r v a l . The funct i on ST2 i s the summat i on of the d i ffe rence of

t h e act u a l and p red i cted col d water t emp e r at u re v a l ue s as des c r i bed

above . When t he mi n i mum s ummed va l u e i s obt a i ned , t he erro r i n t h e p re

d i cted co l d w a t e r t empe ratu re i s mi n i mi zed . The KaV / L v a l ves u sed t o

c a l cu l a t e t h e p redi ct ed co l d wat e r t emp e ra t u re when t h e mi n i mu m summa

t i on i s o bt a i ned i s def i ned a s t he KaV/L v a l ue for t h e tower system .

F i g u re 3 . 8 i l l u st rates t h i s mi n i m i z i n g p roce s s • . The i nt e r v a l o f KaV / L

v a l ue to b e sea rched by fu n ct i on FM I N i s det e rmi ned by t h e u s e r as a

rea s o n a b l e range for the KaV / L v a l u e to l i e .

4 6

4 . DESCR I PT I O N O F COMPUTER PROGRAMS AND SAMP LE PROBLEM

The p u rpose of t h i s sect i o n i s two - fo l d . F i rst , the va r i o u s comp u

t e r p ro g r ams wr i tten t o i mp l ement t he met h odol ogy s e t forth i n C h a p t e r 3 a re des c r i bed . Seco n d , the a pp l i c at i on of the methodo l ogy i s i l l u s t ra

t e d by way of a t est case o r examp l e . T h e paramet e r s for t h i s test ca s e

we re s e l ected so as to be rep res e n t at i ve of an act u a l UHS des i g n e v a l u a

t i on .

T h e fo l l ow i n g i n fo rmat i on i s req u i red to beg i n a n eva l u at i o n of a

p roposed UHS coo l i ng tower sy stem .

1 ) T he con f i gu rat i on of t he UHS cool i n g tower sy stem . Th i s i n

c l u des i n fo rmat i on on t h e number and s i ze of t owe rs to be em

p l oyed , t he l i qu i d a n d a i r f l ow p at h s , an d t he ba s i n vo l ume a n d

i n i t i a l s a l i n i ty .

2 ) T he des i g n - b as i s ope rat i n g s cena r i o fo r t he towe r s . T h i s i n

c l u des a s u mma ry of t h e p o s t u l ated ope rat i n g cond i t i o n s s uch a s

t h e numbe r o f pump s a n d f a n s a s s u med t o b e ope rab l e , t h e t i me

v a ry i n g heat l oa ds t o t h e sy stem , a n d t he t i me - va ry i n g wat e r

a n d a i r f l ow ra t es to b e emp l oyed .

3 ) The ma x i mum a l l owab l e temp e r at u re o f t he coo l i ng wat e r ret u r n e d

t o t h e p l a n t . Th i s v a l u e i s det e rmi n e d b a s e d o n t h e o p e r a t i on

a l l i m i t s o f s a fety- rel ated equ i pment w i t h i n the p l a nt a n d , fo r

t he pu rpos e s of t he UHS sys t em p e r f o rmance eva l u at i on , may be

reg a rded a s a g i ven .

4 ) T he fu l l h i s t o ry o f met eo ro l o g i c a l cond i t i o n s rep re s e n t at i ve of

s i t e . Typ i c a l l y , t h i s c o n s i s t s o f 3 0 or mo re yea rs o f o b s e r v a -

4 7

t i on s at the nea rest Nat i o n a l Weat h e r Se rv i c e repo rt i n g s t a

t i on . Th i s d a t a i s a va i l ab l e on computer readab l e ma gnet i c

tape f rom t he Nat i o n a l C l i mat i c Ce nte r . I n choos i n g the NWS

s t at i o n , t he p r i ma ry con s i de rat i on i s u s u a l l y p ro x i mi ty to t h e

p roposed p l a nt ; h owe ve r , the s i mi l a r i ty of amb i ent temp e rat u re

a n d humi d i t i es i s act u a l l y the i mp o rt a nt con s i derat i on . F o r

examp l e , coa s t a l s i tes may b e typ i c a l l y cool e r and mo re hum i d

t h a n a n i n l a n d s tat i on fo r wh i ch NWS data a re a va i l a b l e .

The app l i ca t i o n of the U H S system p e r f o rma nce a n a l y s i s i s a step

by- st ep p rocess w i t h severa l i mp o rtant eva l u a t i on p o i n t s a l on g t h e

way . C r i t i c a l eva l u at i on of the resu l t s at e a c h o f t he s e j u nct u re s i s

e s s e nt i a l to p roper a p p l i c at i on of t he met h o d o l ogy . I n deed , a l t h ou g h

t h e a na l y s i s p resented he re i n i s syst emat i c a n d t h e compute r codes a re

i ntended t o be e a sy t o a p p l y , t h e i mporta n c e o f e n g i nee r i ng j u dgment i n

e va l u at i n g t he out p u t of the va r i o u s p r o g r ams c a n n ot be ove r s t a t e d .

A fl owc h a rt of t he ent i re p rocess i s g i ven i n F i gu re 4 . 1 . T h e

l e ft - h a n d s i de of F i gu re 4 . 1 �hows t h e es s e nt i a l i np ut s des c r i bed

above . The v a r i ous other boxes rep resent p ro g ram modu l e s , key i nt e rmed

i at e a n d f i n a l res u l t s , and maj o r eva l u at i o n p o i nt s . The d i s cu s s i o n of

each modu l e w i l l focus f i rst o n descr i b i n g t he i n puts a nd o utputs to t h e

modu l e . Th i s d i s cu s s i o n w i l l be fo l l owed i n t u rn by t h e a p p l i c at i on o f

the correspon d i ng p rog ram to t he test c a s e . T hu s , t h e t est c a s e w i l l

s e rve to c l a r i fy a ny pa rt i cu l a r l y confu s i n g po i nt s wh i ch a r i s e . T h e re

s u l t s of the test case w i l l a l so be u s ed t o c o n f i rm the va l i d i ty of t h e

methodo l o gy u se d t o det e rmi n e the per i o d o f wo rst - c a s e tow e r p e r f o r -

ma n ce .

M A N U F A C T U R E R ' S D A T A P R O G R A M P R O G R A M

.. r-----. I--D e . l g n S c e n a r i o

po

P D A P K A V L D e . l g n L / G

D E S I G N P A R A M E T E R S

B a . l n m a • • , . a l l n l t y , a n d t e m p e r a t u r e

N u m b e r o f t o w e , . --"' .. P R O G R A M

E v a p o r a t i o n -� ..

.� .. .. U H S S I M I -.


E v a l u a t i o n H e a t L o a d ,

i , Ir L i q u i d F l o w R a t e ,

O U T P U T L / G , K a V I L O U T P U T c o n . t a n t

m e t e o r o l o g i c a l W B - Tb • • l n w i n d o w .

i N A T I O N A L W E A T H E R·

P R O G R A M S E R V I C E

� � E v a l u a ti o n � M e t e o r o l o g i c a l D a t a N W S

T a p e - T D F - 1 4

F i q u re 4 . 1 F l o� C h a rt o f U H S Coo l i n q Towe r P e r fo rma n c e An a l y s i s .

I

o u t p u t �

T A U

• P R O G R A M

T M C O N

) ,

),

� (X)

4 9

4 . 1 Desc r i pt i o n o f the Samp l e P rob l em

T h e goa l of a U H S perfo rma nce a n a l y s i s i s to det e rmi ne whet h e r t h e

p roposed c o o l i ng system may be rea s o n a b l y expected t o p rov i de ( a ) s u f f i

c i ent cool i n g that the temp e rat u re l i mi t of sa fety- re l at ed equ i pment i n

, t h e p l a nt i s n ot exceeded a n d ( b ) s u f f i c i ent ba s i n c a p a c i ty t h at o p e r a

t i on of the U H S sy stem fo r a per i o d of 30 days wi t h out mak e - u p wat e r i s

a s s u red .

T h e hypot het i ca l p l a nt to be a n a l yzed a s Ou r test case con s i s t s of

a s i n g le n u c l ea r u n i t of 1 000 MW c a p ac i ty . The U H S sy stem c o n s i s t s o f a

s i n g l e coo l i n g t owe r of the des i g n s hown i n F i gu re 4 . 2 . Th e u nu s u a l

p hy s i ca l s t ruct u re o f t he t ower ref l ec ts t h e fact t h at i t i s de s i g n e d t o

meet s e i sm i c category 1 c r i t e r i on . T he t ow e r con s i s t s o f fou r ce l l s ,

each of wh i ch emp l oys f i ve fa n s . The des i gn a i r f l ow rate of each f a n

i s 1 1 6 , 440 cub i c feet per mi n ute . T h e fou r cel l s a re o rgan i ze d i nt o two

i n depen dent water l oops , each l oop w i t h a des i gn wat e r f l ow rat e o f

1 6 , 500 g a l l o n s p e r mi nute . Al t hough bot h l oo p s can be ope r a t e d s i mu l

t a neou s l y , the des i g n -bas i s s cena r i o p o s t u a l ted for the tes� c a s e a s

sumes that o n l y two o f the fou r ce l l s a re u t i l i zed . T h e des i g n - b a s i s

a c c i dent s cena r i o fu rther a s sumes a wate r f l ow rate of 1 2 , 1 50 g a l l o n s

p e r mi n u t e o r 7 3 . 6 percent of des i g n f l ow r a t e f o r t h e f i rs t 2 4 h o u r s

fo l l ow i ng an a cc i dent and a f l ow r a t e of 1 1 , 6 1 0 g a l l ons per m i n ute o r

7 0 . 4 pe rcent of des i gn fl ow rat e t he rea ft e r . The e s s ent i a l des i g n p a r a

met ers o f the U H S cool i n g-t owe r /b a s i n sy stem a re s u mma r i zed i n T a b l e

4 . 1 . Th e t i me- dependent heat l oa d s a nd coo l i n g wat e r fl ow r at e s p os t u

l at ed fo r t h e a c c i dent a re p resent ed i n Tab l e 4 . 2 .

A i r I n l e t

S I D E V I E W

F a n a n d F i l l H a n d l i n g M o n o r a i l

P u m p R o o m

F a n ( t y p )

T O P V I E W ( C o v e r R e m o v e d )

P u m p ( t y p )

O p e n C o o l i n g B a s i n

C o o l i n g B a s i n

N O T E : D r a w i n g N o t T o S c a l e

A i r O u t l e t

T O P V I E W ( C o v e r e d )

F i g u re 4 . 2 S c hema t i c V i ew o f U H S C o o l i n g Towe r fo r Te s t Ca s e .

(jl o

5 1

T a b l e 4 . 1 Des i gn Pa ramet e r s for Tes t C a s e UHS Coo l i n g Tow e r

Des i gn C i r c u l a t i n g �Ia t e r F l ow R a t e

L /G

KAV / L Oes i gn W e t Bu l b Temp e rat u re

n e s i g n Ap p roa c h

Des i gn R a n ge

N umbe r of Ce l l s

N umber of F an s p e r C e l l

A i r F l ow Rate p e r F a n

�� a t e r Vo l u me i n B a s i n ( i n i t i a l ) W at e r Ma s s i n B a s i n ( i n i t ; a 1 ) D i s s o l ved So l i ds C o n t ent of B a s i n Wat e r

3 3 , 0 00 g pm t ot a l i n two l oo p s 1 6 , 50 0 g p m i n e a c h l oo p

1 . 5 9 *

* 1 . 3 0

4

5

1 1 6 , 44 0 c flll

6 , 5 0 0 , 000 ga l l o n s

5 4 , 200 , 000 l brn

M i s s i s s i p p i R i ve r W at e r

Ma x i mum Al l owab l e Ret u rn Temp e rat u re t o P l a nt

* Det e rm i ned f rom ma n u f a ct u re r ' s perfo rma n c e c u rves .

----- - -- --- --

5 2

Ta b l e 4 . 2 Hea t Lo a d a n d C o o l i n g Wa t e r Fl ow Ra te Du r i n q LOCA . ( Ad a pte d F rom F i na l � a fe t y A n a l y s i s R e po r t . )

T i me H e a t L o a d S erv i c e (·Ia t e r T i me i � ��2 j ��[�!'L _ E!2�_ i 9 !?�lL _

0 0 6 . 6 2 X 1 0 7 1 2 1 5 0 0 . 5 1 . 8 x 1 0 3 1 . 9 3 x l O B 1 2 1 5 0 1 . 0 3 . 6 X 1 0 3 1 . 9 3 X 1 0 8 1 2 1 5 0 1 . 5 5 . 4 x 1 0 3 1 . 9 3 X 1 0 8 1 2 1 50 2 . 0 7 . 2 x 1 0 3 1 . 9 3 X l O B 1 2 1 5 0 2 . 5 9 . 0 x 1 0 3 1 . 9 3 X l O B 1 2 1 5 0 3 1 . 08 x 1 0 4 1 . 9 3 X l O B 1 2 1 5 0 4 1 . 44 x 1 0 4 1 . 9 3 X 1 0 8 1 2 1 5 0 5 1 . 80 x 1 0 4 1 . 9 3 X 1 0 8 1 2 1 5 0 6 2 . 1 6 x 1 0 4 1 . 9 3 X l O B 1 2 1 5 0 8 2 . 88 x 1 0 4 1 . 9 3 X 1 0 8 1 2 1 5 0 1 0 3 . 60 x 1 0 4 1 . 9 3 X l O B 1 2 1 5 0 1 2 4 . 32 x 1 0 " 1 . 9 0 X 1 0 8 1 2 1 50 1 6 5 . 7 6 x 1 0 " 1 . 8 2 x 1 0 8 1 2 1 5 0 2 0 7 . 20 x 1 0 " 1 . 78 x 1 0 8 1 2 1 5 0 2 4 8 . 64 x 1 0 " 1 . 3 1 X l O B 1 2 1 5 0

2 days 1 . 73 x 1 0 5 9 . 5 4 x 1 0 7 1 1 6 1 0 3 2 . 5 9 X 1 0 5 8 . 86 X 1 0 7 1 1 6 1 0 4 3 . 4 6 X 1 0 5 8 . 4 1 X 1 0 7 1 1 6 1 0 5 4 . 3 2 x 1 0 5 8 . 1 1 X 1 0 7 1 1 6 1 0 6 5 . 1 8 X 1 0 5 7 . 7 6 x 1 0 7 1 1 6 1 0 7 6 . 0 5 X 1 0 5 7 . 6 2 X 1 0 7 1 1 6 1 0 8 6 . 9 1 X 1 0 5 7 . 4 6 X 1 0 7 1 1 6 1 0 9 7 . 7 8 x l 0 5 7 . 3 1 X 1 0 7 1 1 6 1 0

1 0 8 . 64 x 1 0 5 7 . 1 2 X 1 0 7 1 1 6 1 0 1 5 1 . 2 9 x 1 0 6 6 . 70 X 1 0 7 1 16 1 0 2 0 1 . 7 3 X 1 0 6 6 . 4 0 X 1 0 7 1 1 6 1 0 2 5 2 . 1 6 x 1 0 6 6 . 1 6 X 1 0 7 1 1 6 1 0 30 2 . 5 9 X 1 0 6 6 . 0 4 x 1 0 7 1 1 6 1 0

5 3

4 . 2 P e r f o rma nce Dat a An a l y s i s P rog ram

The p u rpose of the p e r f o rma n ce dat a a n a l y s i s p rog ram ( P DAP ) i s to

dete rmi ne f rom the man u fact u re r ' s data the c h a ract e r i s t i c pa ramet e r

v a l ues f o r t h e tow e r ( s ) to b e u se d . The p ro g ram a l s o a l l ows a comp a r i

son between ma n u fa ct u re r ' s p e rf o rmance p red i ct i ons for off-des i gn ope ra

t i on w i t h t he p red i ct i on of the nume r i c a l tow e r performance s i mu l at i on

u sed i n t he UHS system a n a l ys i s p rocedu re . The p rog ram det e rmi nes K a V / L

v a l ue s f o r e a c h of the op e r at i n g cond i t i o n s for wh i ch perfo rmance dat a

a re a v a i l ab l e . I dea l l y , s uch data s h ou l d be a v a i l ab l e for a fu l l s p e c

t rum o f wate r a n d a i r f l ow rat es but t h e deg ree to wh i c h s u c h i n f o rma

t i on i s a va i l ab l e va r i e s f rom case to ca s e . I n some c a s e s , t he i n fo rma

t i on p res ented i s me re l y p red i ct i on s by t h e man u fact u re r u s i n g p rop r i e

t a ry data on f i l l c ha ract e ri s t i cs a n d , i n ma ny i n s ta nces , p r op r i et a ry

methods o f ana l y s i s . F o r t h i s rea son , i t i s i mp o rt ant that the p e r f o r

mance o f t h e sy stem be re-eva l u a t e d when coo l i ng t ower accepta nce dat a

a re a va i l a b l e . When on l y KaV / L va l ue s o r s i n g l e " des i g n p o i nt " dat a a re

p resent e d , i t may be gen e ra l l y a s s umed t h at K aV rema i n s re l at i ve l y con

s t a nt for a rea s o n ab l e r ange o f nea r-des i gn - p o i nt con d i t i o n s . I f f l ow

rates (wat e r a nd / o r a i r ) du r i ng a cc i dent o p e r at i on f a l l be l ow des i g n

condi t i o n s , a pena l ty on KaV / L i s app rop ri ate . T h e mag n i t u de of t h i s

p e n a l ty mu s t , u n fo rt u n ate l y , be b a s ed on best e n g i nee r i n g j u dgme n t .

I np u t s to the performance dat a a na l y s i s p rog ram a re l i s t e d i n T a b l e

4 . 3 i n t e rms of v a r i a b l e name , v a r i ab l e def i n i t i on , a n d v a r i a b l e

u n i t s . T he f i rst l i ne of t he i np u t f i l e i s a a l p h a nu me r i c " c a s e " l abe l

u sed to i d ent i fy the out p u t . The rema i n de r of t he i n put data a re b roken

5 4

Tab l e 4 . 3 De s c r i p t i o n o f I n p u t V a r i a b l e s fo r P ro g r a m P OAP

VAR I AB L E NAME D E S C R I PT I O N AND U N I TS

CAS E BO - Co l umn H e a de r U s e d t o I d e n t i fy Ou t p u t F i l e

FL0\4 BO - Co l umn Hea d e r U s e d t o I d e n t i fy L i qu i d F1 0 \-J Ra te s

W i t h i n t he Ou t p u t

LOVERG

PERC

S

N RA N

R LAB E L

RA N

N\�B

WB D

TCOLD

L i q u i d to Ga s F l ow Rate Rat i o

Da t a L i q u i d Fl ow Ra t e Pe rc e n t

I n i t i a l Ba s i n S a l i n i ty

Numbe r of Ra n ge V a l u e s W i th i n a F l ow Ra t e

BO - Co l umn Hea de r U s e d to I de n t i fy t h e Temp e ra t u re

V a l u e s O b t a i n e d a t a S p ec i f i c Ra n g e V a l u e

Array C o n t a i n i n g t h e Ra n g e V a l u e s ( F )

A rray C o n t a i n i n g t h e N umb er o f We t - B u l b Tempe ra tu re

V a l u e s Wi t h i n E a c h Ra n g e Va l u e

We t - B u l b Tempe r a t u re D a t a ( F )

Col d Wa t e r Temp e r a t u re ( F )

5 5

i n t o g rou p s acco rd i n g t o percent fl ow rat e . W i t h i n e ach " f l ow rat e "

g rou p i n g , t h e f i rst ent ry i s t h e a l p h a n ume r i c " f l ow r at e " l a be l u s e d t o

i de nt i fy t h e fl ow rate o n out p u t . The " f l ow rat e " l ab e l i s t hen fo l

l owed by t h ree nume r i ca l e nt r i e s : t he L/G v a l u e , the p e rcent f l ow rat e ,

a n d t h e s a l i n i ty of t he cool i ng wat e r u n de r test con d i t i o n s . F o l l ow i n g

t n e c o n vent i on a l p ra ct i ce i n g i v i n g towe r pe rforma nce dat a , the v a l u e s

a re fu rt h e r subd i v i ded i nt o g rou p s acco rd i n g t o cool i n g ran ge , w i t h c o l d

w a t e r t emp e r at u re t abu l ated a g a i n st amb i e nt wet-bu l b t empe ratu re w i t h i n

e a c h g i ven range . P rog ram PDAP u s es the tot a l body o f dat a for al l

r a n ge s at a g i ven f l ow rate to det e rmi ne a c o r respon d i n g KaV/L v a l ue .

T h i s v a l u e of KaV/ L i s u se d , i n t u rn , to p red i ct col d wat e r temp e r a t u re

at each of the operat i ng p o i n t s ( r a n ge a n d amb i ent wet - bu l b temp e r a t u re )

g i v e n i n the i np u t . A comp a r i s o n between rep o rt e d a n d p red i c t e d co l d

water t emp e ratu res p ro v i des a t e s t of the a b i l i ty of t h e t ower model t o

rep roduce t h e performance c ha ract e ri s t i c s of t h e towe r .

The p e rforman ce data a v a i l ab l e for the hypothet i c a l t e st c a se a re

g i ven i n Tab l e 4 . 4 wi t h t he corres pondi n g i nput f i l e t o p rog ram PDAP

s h own i n F i gu re 4 . 3 . The i nput f i l e s h own i n F i gu re 4 . 3 i s di v i ded

a c r o s s th ree pages for t h e pu rpose of p resentat i on h e re . As i n d i c a t e d ,

dat a a re a v a i l ab l e for wat e r f l ow rates ra n g i ng f rom 50 p e rcent to 1 1 0

p e rcent o f the des i gn va l u e .

F i gu re 4 . 4 s h ow s the p r i n t e d out put of the PDAP ru n fo r t h e t e s t

ca s e ; compa r i sons o f p red i cted a n d repo rted co l d wat e r temp e ratu res a re

s h ow n i n F i gu re 4 . 5 - 4 . 1 1 . As was don e for the i np u t f i l e , t h e o ut p u t

i s cont i n u ed over fou r pages f o r t h e s ak e o f p resentat i on . T he n umer i -

5 6

T a b l e 4 . 4 Pe rfo rma n c e Da t a fo r Te s t Ca s e . ( VJB - Wet - Bu l b Temp e ra t u re ; CW - C o l d Wa t e r Tempe r a t u re )

P E R C E N T L I Q U I D F L O W R A T E

5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0

L ( g p m ) 8 2 5 0 9 9 0 0 1 1 5 5 0 1 3 2 0 0 1 4 8 5 0 1 6 5 0 0 1 8 1 5 0

L / G 0 . 7 9 3 0 . 9 5 2 1 . 1 1 1 " 1 . 2 7 0 1 . 4 2 8 1 . 5 7 8 1 . 7 4 5

R a n g e ( F ) W C W W C W W W C W W W W B B B

C W B

C W B C W C W B B

4 0 5 6 . 0 4 0 5 8 . 6 4 0 6 1 . 2 4 0 6 3 . 5 4 0 6 5 . 8 4 0 6 7 . 9 4 0 6 9 . 9 4 5 5 9 . 1 4 5 6 1 . 4 4 5 6 ) . 9 4 5 6 6 . 0 4 5 6 8 . 1 4 5 7 0 . 1 4 5 7 2 . 0 5 0 6 2 . 1 5 0 6 4 . 4 5 0 6 6 . 8 5 0 6 8 . 8 5 0 7 0 . 8 5 0 7 2 . 5 5 0 1 4 . 4 5 5 6 5 . 7 5 5 5 6 7 . 7 5 5 6 9 . 7 5 5 7 1 . 6 5 5 7 3 . ) 5 5 7 5 . 0 5 5 1 6 . 7

1 5 6 0 6 9 . a 6 0 7 1 . 0 6 0 7 2 . 8 6 0 1 4 . 5 6 0 7 6 .; 1 6 0 7 7 . 8 6 0 7 9 . ) 6 5 7 2 . 7 6 5 1 4 . 5 " 6 5 7 6 . 1 6 5 7 7 . 8 6 5 7 9 . 0 6 5 8 0 . 4 6 5 8 2 . 1 7 0 7 6 . 3 7 0 7 1 . 9 7 0 7 9 . 4 7 0 8 0 . 8 7 0 8 2 . 0 7 0 8 3 . 2 7 0 8 4 . 9 7 5 8 0 . 0 5 7 5 8 1 . 7 7 5 8 ) . 0 7 5 1 4 . ) 7 5 8 5 . 3 7 5 8 6 . 4 7 5 8 7 . 9 8 0 1 4 . 1 8 0 8 5 . 5 8 0 8 6 . 6 8 0 8 7 . 7 8 0 8 8 . 7 8 0 8 9 . 9 8 0 9 1 . 0 8 5 8 8 . 5 8 5 8 9 . 5 8 5 9 0 . 4 8 5 9 1 . 6 8 5 9 2 . 2 a 5 9 3 . 2 8 5 H . )

4 0 5 9 . 1 4 0 6 2 . 2 4 0 6 5 . 0 4 0 6 7 . 7 4 0 7 0 . 1 4 0 7 2 . 2 4 0 H . S 4 5 6 1 . 8 4 5 6 4 . 7 4 5 6 7 . 4 4 5 U . 8 4 5 7 2 . 1 4 5 H . 2 4 5 7 6 . 4 5 0 6 4 . 7 5 0 6 7 . 4 5 0 6 9 . 9 5 0 7 2 . 2 5 0 7 4 . 3 5 0 7 6 . 2 5 0 7 8 . 3 5 5 6 7 . 9 5 5 7 0 . 4 5 5 7 2 . 5 5 5 1 4 . 7 5 5 7 6 . 7 5 5 7 8 . 8 5 5 8 0 . 3

2 0 6 0 7 0 . 9 6 0 7 3 . 2 6 0 7 5 . 3 6 0 7 7 . 2 6 0 7 9 . 2 6 0 8 1 . 0 6 0 8 2 . 7 6 5 H . 2 6 5 7 6 . 3 6 5 7 8 . 3 6 5 8 0 . 0 6 5 8 1 . 8 6 5 8 3 . 2 6 5 8 4 . 8 7 0 7 1 . 7 7 0 7 9 . 4 7 0 8 1 . 3 7 0 8 2 . 9 7 0 U . S 7 0 8 5 . 9 7 0 8 7 . 5 7 5 8 1 . 2 7 5 8 3 . 0 7 5 U . S 7 5 8 5 . 9 7 5 8 7 . 5 7 5 8 8 . 5 7 5 9 0 . 1 8 0 1 4 . 9 5 8 0 8 6 . 5 8 0 8 7 . 9 8 0 8 9 . 1 8 0 9 0 . 7 8 0 9 1 . 4 8 0 9 ) . 0 8 5 8 8 . 9 8 5 9 0 . 4 , 8 5 9 1 . 7 8 5 9 2 . 8 8 5 9 3 . 7 8 5 9 4 . 8 8 5 9 6 . 0

4 0 U . S 4 0 " . 9 4 0 6 8 . 0 4 0 7 0 . 8 4 0 7 3 . 3 " 4 0 7 5 . 6 4 O 7 8 . 0 4 5 6 4 . 0 4 5 6 7 . 2 4 5 7 0 . 2 4 5 7 2 . 1 4 5 7 5 . 2 4 5 1 7 . 5 4 5 7 9 . 7 5 0 6 6 . 1 5 0 6 9 . 6 · 5 0 7 2 . 3 5 0 H . 9 5 0 7 1 . 2 5 0 7 9 . 2 5 0 8 1 . 2 5 5 6 9 . 7 5 5 7 2 . 3 5 5 7 4 . 9 5 5 7 1 . 2 5 5 7 9 . 4 5 5 8 1 . 1 5 5 8 3 . 2 6 0 7 2 . 4 6 0 7 5 . 0 6 0 7 1 . 4 6 0 7 9 . 4 6 0 8 1 . 4 6 0 8 ) . 2 6 0 8 5 . 2 2 5 6 5 7 5 . 5 6 5 7 7 . 8 6 5 8 0 . 1 6 5 8 2 . 1 6 5 8 ) . 9 6 5 8 5 . 3 6 5 8 7 . 3 7 0 7 8 . 8 7 0 8 0 . 1 1 0 8 2 . 9 7 0 8 4 . 7 7 0 8 6 . ) 7 0 8 7 . 8 1 0 8 9 . 7 7 5 8 2 . 1 7 5 8 ) . 9 7 5 8 6 . 0 7 5 8 7 . 3 7 5 8 9 . 1 1 5 9 0 . 2 7 5 9 1 . 9 8 0 8 5 . 9 8 0 8 7 . 1 8 0 8 9 . 0 8 0 9 0 . 3 8 0 9 1 . 8 8 0 9 2 . 9 8 0 9 4 . 6 8 5 8 9 . 7 8 5 9 1 . 0 8 5 9 2 . 4 8 5 9 3 . 9 8 5 9 4 . 7 8 5 9 6 . 1 8 5 9 1 . 2

4 0 6 3 . 5 4 0 6 7 . 2 4 0 7 0 . 3 4 0 7 3 . 3 4 0 7 5 . 9 4 0 7 8 . 3 4 0 8 0 . 7 4 5 6 5 . 9 4 5 6 9 . 2 4 5 7 2 . 2 4 5 7 5 . 1 4 5 7 7 . 5 4 5 7 9 . 9 " 5 8 2 . 2 S O 6 8 . 4 5 0 1 1 . 5 5 0 1 4 . 4 5 0 7 7 . 0 5 0 7 9 . 4 5 0 8 1 . 5 5 0 8 3 . 7

5 5 1 1 . 1 5 5 7 3 . 9 5 5 7 6 . 7 5 5 7 9 . 0 5 5 8 1 . 3 5 5 8 3 . 1 5 5 8 5 . )

6 0 7 3 . 7 6 0 7 6 . 3 6 0 7 8 . 9 6 0 8 1 . 0 6 0 8 ) . 3 6 0 8 5 . 0 5 6 0 8 7 . 2 3 0 6 5 7 6 . 7 6 5 7 9 . 1 6 5 8 1 . 5 6 5 8 3 . 4 6 5 8 5 . 4 6 5 8 7 . 0 6 5 8 8 . 9

7 0 7 9 . 8 7 0 8 1 . 8 7 0 1 4 . 0 7 0 8 5 . 9 7 0 8 7 . 8 7 0 8 9 . 1 7 0 9 1 . 2 7 5 8 ) . 0 7 5 8 4 . 9 7 5 8 6 . 8 7 5 8 8 . 6 7 5 9 0 . 3 7 5 9 1 . 5 7 5 9 3 . " 8 0 8 6 . 3 8 0 8 8 . 1 8 0 8 9 . 9 8 0 9 1 . 3 8 0 9 2 . 9 8 0 9 4 . 0 8 0 9 5 . 9 8 5 9 0 . 2 8 5 9 1 . 6 8 5 9 3 . 2 8 5 9 4 . 5 8 5 9 5 . 7 8 5 9 7 . 0 8 5 9 8 . 3

TEST CAS E 5 0 \ O F D E S I G N FLOW RATE ( 8 2 5 0 GPH ) 0 . 7 93 5 , 5 0 , 0 . 0 4 RANGE = 1 5 1 5 1 0 4 0 56 . 0 4 5 5 9 . 1 5 0 6 2 . 1 5 5 6 5 . 7 5 6 0 6 9 . 0 6 5 7 2 . 7 7 0 7 6 . 3 7 5 8 0 . 0 5 8 0 8 4 . 1 8 5 8 8 . 5 RANG E = 2 0 2 0 1 0 4 0 5 9 . 1 4 5 6 1 . 8 5 0 6 4 . 7 5 5 6 7 . 9 6 0 7 0 . 9 6 5 7 4 . 2 7 0 7 7 . 7 7 5 8 1 . 2 B O 8 4 . 9 5 8 5 8 8 . 9 RANG E = 2 5 2 5 1 0 4 0 6 1 . 5 4 5 6 4 . 0 5 0 6 6 . 7 5 5 6 9 . 7 6 0 7 2 . 4 6 5 7 5 . 5 7 0 7 B . 8 7 5 8 2 . 1 B O 85 . 9 B 5 8 9 . 7 RANG E " 3 0 3 0 1 0 4 0 6 3 . 5 4 5 6 5 . 9 5 0 6 B . 4

5 5 7 1 . 1 8 0 8 7 . 3

6 0 7 3 . 7 8 5 9 1 . 0

6 5 7 6 . 7 RANG E " 3 0

7 0 7 9 . 8 3 0 1 0

7 5 8 3 . 0 4 0 6 7 . 2

8 0 8 6 . 3 4 5 6 9 . 2 8 5 9 0 . 2 5 0 7 1 . 5 6 0 , OF D E S I G N FLOW RATE ( 9 9 0 0 GPM ) 5 5 7 3 . 9 0 . 9 5 2 2 , 6 0 , 0 . 0 6 0 7 6 . 3 4 6 5 7 9 . 1 RANG E .. 1 5 7 0 8 1 . 8 1 5 1 0 7 5 8 4 . 9 4 0 5 8 . 6 8 0 8 8 . 1 4 5 6 1 . 4 8 5 9 1 . 6 5 0 6 4 . 4 7 0 , OF D E S I G N 5 5 6 7 . 7 1 . 1 1 1 , 7 0 , O . 0 6 0 7 1 . 0 4 6 5 7 4 . 5 RANG E = 1 5 7 0 7 7 . 9 1 5 1 0

7 5 8 1 . 7 4 0 6 1 . 2

8 0 8 5 . 5 4 5 6 3 . 9

8 5 B 9 . 5 5 0 6 6 . B

RANG E = 2 0 5 5 6 9 . 7 2 0 1 0 6 0 7 2 . 8 4 0 6 2 . 2 6 5 7 6 . 1 4 5 6 4 . 7 7 0 7 9 . 4 5 0 6 7 . 4 7 5 B 3 . 0 5 5 7 0 . 4 8 0 8 6 . 6 6 0 7 3 . 2 8 5 9 0 . 4 6 5 7 6 . 3 RANG E .. 2 0 7 0 7 9 . 4 2 0 1 0

7 5 8 3 . 0 4 0 6 5 . 0 8 0 8 6 . 5 4 5 6 7 . 4 8 5 9 0 . 4 5 0 6 9 . 9 RANG E .. 2 5 5 5 7 2 . 5 2 5 1 0 6 0 7 5 . 3 4 0 6 4 . 9 6 5 7 8 . 3

4 5 6 7 . 2 7 0 B 1 . 3

5 0 6 9 . 6 7 5 B 4 . 5

5 5 7 2 . 3 8 0 87. 9 6 0 7 5 . 0 8 5 9 1 . 7

6 5 7 7 . 8 RANG E " 2 5

7 0 8 0 . 7 2 5 1 0 7 5 8 3 . 9 4 0 6 8 . 0

F i g u re 4 . 3 I n pu t L i s t i n g fo r P ro g ra m P OAP .

FLOW RATE ( 1 1 5 5 0 GPM )

(Jl -.,J

4 5 7 0 . 2 5 0 7 2 . 3 5 5 7 4 . 9 6 0 7 7 . 4 6 5 8 0 . 1 7 0 8 2 . 9 7 5 8 6 . 0 8 0 8 9 . 0 8 5 9 2 . 4 RANGE '" 3 0 3 0 1 0 4 0 7 0 . 3 4 5 7 2 . 2 5 0 7 4 . 4 5 5 7 6 . 7 6 0 7 8 . 9 6 5 8 1 . 5 7 0 8 4 . 0 7 5 8 6 . 8 8 0 8 9 : 9 8 5 9 3 . 2 8 0 , OF D E S I GN FLOW RATE ( 1 3 2 0 0 GPH ) 1 . 27 0 , 8 0 , 0 . 0 4 RANG E '" 1 5 1 5 1 0 4 0 6 3 . 5 4 5 6 6 . 0 5 0 6 B . B 5 5 7 1 . 6 6 0 7 4 . 5 6 5 7 7 . B 7 0 B O . 8 7 5 B 4 . 3 8 0 87 . 7 8 5 9 1 . 6 RANG E '" 2 0 2 0 1 0 4 0 6 7 . 7 4 5 6 9 . B 5 0 7 2 . 2 5 5 7 4 . 7 6 0 7 7 . 2 6 5 8 0 . 0 7 0 8 2 . 9

7 5 B 5 . 9 8 0 8 9 . 1 8 5 9 2 . 8 RANG E .. 2 5 2 5 1 0 4 0 7 0 . 8 4 5 7 2 . 7 5 0 7 4 . 9 5 5 7 7 . 2 6 0 7 9 . 4 6 5 8 2 . 1 7 0 8 4 . 7 7 5 87 . 3 8 0 9 0 . 3 8 5 9 3 . 9 RANGE ... 3 0 3 0 1 0 4 0 7 3 . 3 4 5 7 5 . 1 5 0 7 7 . 0 5 5 7 9 . 0 6 0 8 1 . 0 6 5 8 3 . 4 7 0 8 5 . 9 7 5 8 8 . 6 8 0 9 1 . 3 8 5 9 4 . 5 9 0 , OF D E S I GN FLOW RATE ( 1 4 8 5 0 G PM ) 1 . 4 28 , 9 0 , 0 . 0 4 RANG E '" 1 5 1 5 1 0 4 0 6 5 . 8 4 5 6 8 . 1 5 0 7 0 . 8 5 5 7 3 . 3 6 0 7 6 . 1 6 5 7 9 . 0 7 0 B 2 . 0 7 5 8 5 . 3 B O 8 B . 7

8 5 9 2 . 2 RANG E .. 2 0 2 0 1 0

F i g u re 4 . 3 Co n t i n u e d

4 0 7 0 . 1 4 5 7 2 . 1 5 0 7 4 . 3 5 5 7 6 . 7 6 0 7 9 . 2 6 5 8 1 . 8 7 0 8 4 . 5 7 5 8 7 . 5 8 0 9 0 . 7 8 5 9 3 . 7 RANG E ... 2 5 2 5 1 0 4 0 7 3 . 3 4 5 7 5 . 2 5 0 7 7 . 2 5 5 7 9 . 4 6 0 8 1 . 4 6 5 8 3 . 9 7 0 8 6 . 3 7 5 8 9 . 1 B O 9 1 . 8 8 5 9 4 . 7 RANG E '" 3 0 3 0 1 0 4 0 7 5 . 9 4 5 7 7 . 5 5 0 7 9 . 4 5 5 8 1 . 3 6 0 8 3 . 3 6 5 8 5 . 4 7 0 B7 . 8 7 5 9 0 . 3 8 0 9 2 . 9 8 5 9 5 . 7 1 0 0 , OF D E S I GN 1 . 5 7 B , 1 0 0 , 0 . 0 4 RANG E ... 1 5 1 5 1 0 4 0 6 7 . 9 4 5 7 0 . 1 5 0 7 2 . 5 5 5 7 5 . 0 6 0 7 7 . B 6 5 6 0 . 4

(J1 ex:;

FLOW RATE ( 1 6 5 0 0 GPM )

7 0 8 3 . 2 7 5 8 6 . 4 8 0 8 9 . 9 8 5 9 3 . 2 RANGE .. 2 0 2 0 1 0 4 0 7 2 . 2 4 5 7 4 . 2 5 0 7 6 . 2 5 5 7 8 . 8 6 0 8 1 . 0 6 5 8 3 . 2 7 0 8 5 . 9 7 5 8 8 . 5 8 0 9 1 . 4 8 5 9 4 . 8 RANG E .. 2 5 2 5 1 0 4 0 7 5 . 6 4 5 7 7 . 5 5 0 7 9 . 2 5 5 8 1 . 2 6 0 8 3 . 2 6 5 8 5 . 3 7 0 8 7 . 8 7 5 9 0 . 2 8 0 9 2 . 9 8 5 9 6 . 1 RANG E " 3 0 3 0 1 0 4 0 7 8 . 3 4 5 7 9 . 9 5 0 8 1 . 5 5 5 8 3 . 1 6 0 8 5 . 0 5 6 5 87 . 0 7 0 8 9 . 1 7 5 9 1 . 5 8 0 9 ,," 0 8 5 97 . 0 1 1 0 , OF DE S I GN FLOW RATE ( 1 8 1 5 0 G PM )

1 . 7 4 5 , 1 1 0 , 0 . 0 4 RANG E .. 1 5

1 5 1 0 4 0 6 9 . 9 4 5 7 2 . 0 5 0 7 4 . 4 5 5 7 6 . 7 6 0 7 9 . 3 6 5 8 2 . 1 7 0 8 4 . 9 7 5 87 . 9 8 0 9 1 . 0 8 5 9 4 . 3 RANG E • 2 0 2 0 1 0 4 0 7 4 . 5 4 5 7 6 . 4 5 0 7 8 . 3 5 5 8 0 . 3 6 0 8 2 . 7 6 5 8 4 . 8 7 0 8 7 . 5 7 5 9 0 . 1 8 0 9 3 . 0 8 5 9 6 . 0 RANG E .. 2 5 2 5 1 0 4 0 7 8 . 0 4 5 7 9 . 7 5 0 81 . 2 5 5 83 . 2 6 0 8 5 . 2 6 5 87 . 3 7 0 8 9 . 7 7 5 9 1 . 9 8 0 9 4 . 6 8 5 97 . 2 RANG E " 3 0 3 0 1 0 4 0 8 0 . 7 4 5 8 2 . 2 5 0 8 3 . 7 5 5 8 5 . 3 6 0 87 . 2 6 5 8 8 . 9 7 0 9 1 . 2

F i � u re 4 . 3 C o n t i n u e d

7 5 93 . 4 8 0 9 5 . 9 8 5 9 8 . 3

U1 1.0

TEST CAS E 5 5 . 0 0 0 0 7 1 . 1 0 0 0 7 1 . 07 4 1 - . 0 2 5 9 S O , O F DES I GN FLOW RATE ( 8 2 5 0 G PM ) 6 0 . 0 0 0 0 7 3 . 7 0 0 0 7 3 . 7 5 4 4 . 0 5 4 4 1 . 9 0 4 0 . 7 9 3 5 5 0 . 0 0 . 0 0 0 0

4 RANG E .. I S

1 5 . 0 1 0 . 0 4 0 . 0 0 0 0 5 6 . 0 0 0 0 5 6 . 0 1 5 1 4 5 . 0 0 0 0 5 9 . 1 0 0 0 5 9 . 1 4 5 5 5 0 . 0 0 0 0 6 2 . 1 0 0 0 6 2 . 2 9 4 8 5 5 . 0 0 0 0 6 5 . 7 5 0 0 6 5 . 7 3 4 8 6 0 . 0 0 0 0 6 9 . 0 0 0 0 6 9 . 1 07 5 6 5 . 0 0 0 0 7 2 . 7 0 0 0 7 2 . 6 9 3 3 7 0 . 0 0 0 0 7 6 . 3 0 0 0 7 6 . 3 3 8 3 7 5 . 0 0 0 0 8 0 . 0 5 0 0 8 0 . 1 1 9 2 8 0 . 0 0 0 0 8 4 . 1 0 0 0 8 4 . 0 6 2 2 8 5 . 0 0 0 0 8 8 . 5 0 0 0 8 8 . 1 6 0 4

RANG E .. 2 0 2 0 . 0 1 0 . 0

4 0 . 0 0 0 0 5 9 . 1 0 0 0 5 9 . 1 6 7 1 4 5 . 0 0 0 0 6 1 . 8 0 0 0 6 1 . 9 3 9 4 5 0 . 0 0 0 0 6 4 . 7 0 0 0 6 4 . 8 43 1 5 5 . 0 0 0 0 6 7 . 9 0 0 0 6 7 . 9 0 8 0 6 0 . 0 0 0 0 7 0 . 9 0 0 0 7 1 . 0 0 0 4 6 5 . 0 0 0 0 7 4 . 2 0 0 0 7 4 . 2 6 7 9 7 0 . 0 0 0 0 7 7 . 7 0 0 0 7 7 . 6 8 4 0 7 5 . 0 0 0 0 8 1 . 2 0 0 0 8 1 . 2 1 8 2 8 0 . 0 0 0 0 8 4 . 9 5 0 0 8 4 . 9 2 3 2 8 5 . 0 0 0 0 8 8 . 9 0 0 0 8 8 . 7 9 0 8

RANG E .. 2 5 2 5 . 0 1 0 . 0

4 0 . 0 0 0 0 6 1 . 5 0 0 0 6 1 . 6 5 6 9 4 5 . 0 0 0 0 6 4 . 0 0 0 0 6 4 . 1 9 0 5 5 0 . 0 0 0 0 6 6 . 7 0 0 0 6 6 . 8 5 2 7 5 5 . 0 0 0 0 6 9 . 7 0 0 0 6 9 . 6 7 2 6 6 0 . 0 0 0 0 7 2 . 4 0 0 0 7 2 . 5 1 8 0 6 5 . 0 0 0 0 7 5 . 5 0 0 0 7 5 . 5 6 3 6 7 0 . 0 0 0 0 7 8 . 8 0 0 0 7 8 . 7 6 6 6 7 5 . 0 0 0 0 8 2 . 1 0 0 0 8 2 . 1 0 5 0 8 0 . 0 0 0 0 8 5 . 9 0 0 0 8 5 . 6 5 5 4 85 . 0 0 0 0 8 9 . 7 0 0 0 8 9 . 3 5 6 8

RANG E .. 3 0 3 0 . 0 1 0 . 0

4 0 . 0 0 0 0 6 3 . 5 0 0 0 6 3 . 6 7 6 4 4 5 . 0 0 0 0 6 5 . 9 0 0 0 6 6 . 0 2 6 9 5 0 . 0 0 0 0 6 8 . 4 0 0 0 6 8 . 4 8 2 4

6 5 . 0 0 0 0 7 6 . 7 0 0 0 7 6 . 6 2 8 9 - . 07 1 1 7 0 . 0 0 0 0 7 9 . 8 0 0 0 7 9 . 6 5 1 9 - . 1 4 8 1 7 5 . 0 0 0 0 8 3 . 0 0 0 0 8 2 . 8 3 8 2 - . 1 6 1 8 8 0 . 0 0 0 0 8 6 . 3 0 0 0 8 6 . 2 0 2 5 - . 0 9 7 5 . 0 1 5 1 8 5 . 0 0 0 0 9 0 . 2 0 0 0 8 9 . 7 9 4 4 - . 4 0 5 6 . 0 4 5 5 6 0 \ O F DES I GN F LOW RATE ( 9 9 0 0 GPM )

. 1 9 4 8 1 . 7 1 4 0 . 9 5 2 2 6 0 . 0 0 . 0 0 0 0 - . 0 1 5 2 4

. 1 07 5 RANG E .. 1 5 - . 0 0 6 7 1 5 . 0 1 0 . 0

. 0 3 83 4 0 . 0 0 0 0 5 8 . 6 0 0 0 5 8 . 7 1 2 1 . 1 1 2 1 . 06 9 2 4 5 . 0 0 0 0 6 1 . 4 0 0 0 6 1 . 5 97 8 . 1 9 7 8 - . 0 3 7 8 5 0 . 0 0 0 0 6 4 . 4 0 0 0 6 4 . 6 2 0 4 . 2 2 0 4 - . 3 3 96 5 5 . 0 0 0 0 6 7 . 7 0 0 0 6 7 . 81 4 8 . 1 1 4 8 6 0 . 0 0 0 0 7 1 . 0 0 0 0 7 1 . 06 9 4 . 06 9 4 6 5 . 0 0 0 0 7 4 . 5 0 0 0 7 4 . 46 0 2 - . 0 3 9 8

. 06 7 1 7 0 . 0 0 0 0 7 7 . 9 0 0 0 7 7 . 9 0 5 6 . 0 0 5 6

. 1 3 9 4 7 5 . 0 0 0 0 8 1 . 7 0 0 0 8 1 . 5 4 87 - . 1 5 1 3

. 1 4 3 1 8 0 . 0 0 0 0 8 5 . 5 0 0 0 8 5 . 2 8 8 5 - . 2 1 1 5 . 0 0 8 0 8 5 . 0 0 0 0 8 9 . 5 0 0 0 8 9 . 1 7 5 5 - . 3 2 4 5 . 1 0 0 4 RANG E .. 2 0

. 06 7 9 2 0 . 0 1 0 . 0 - . 0 1 6 0 4 0 . 0 0 0 0 6 2 . 2 0 0 0 6 2 . 2 9 9 2 . 0 9 9 2

. 0 1 8 2 4 5 . 0 0 0 0 6 4 . 7 0 0 0 6 4 . 8 5 2 0 . 1 5 2 0 - . 0 26 8 5 0 . 0 0 0 0 6 7 . 4 0 0 0 6 7 . 5 4 1 2 . 1 4 1 2 - . 1 0 9 2 5 5 . 0 0 0 0 7 0 . 4 0 0 0 7 0 . 3 9 6 7 - . 0 0 3 3

6 0 . 0 0 0 0 7 3 . 2 0 0 0 7 3 . 2 7 07 . 07 07 6 5 . 0 0 0 0 7 6 . 3 0 0 0 7 6 . 3 2 4 3 . 0 2 4 3

. 1 5 6 9 7 0 . 0 0 0 0 7 9 . 4 0 0 0 7 9 . 4 8 4 2 . 0 8 4 2

. 1 9 0 5 7 5 . 0 0 0 0 8 3 . 0 0 0 0 8 2 . 8 6 6 5 - . 1 3 3 5

. 1 5 27 8 0 . 0 0 0 0 8 6 . 5 0 0 0 86 . 3 4 97 - . 1 5 0 3 - . 0 27 4 85 . 0 0 0 0 9 0 . 4 0 0 0 9 0 . 0 2 8 6 - . 3 7 1 4

. 1 1 8 0 RANGE .. 2 5

. 06 3 6 2 5 . 0 1 0 . 0 - . 0 3 3 4 4 0 . 0 0 0 0 6 4 . 9 0 0 0 6 5 . 05 47 . 1 5 4 7 . 0 0 5 0 4 5 . 0 0 0 0 6 7 . 2 0 0 0 6 7 . 3 7 0 1 . 1 7 0 1 - . 2 4 4 6 5 0 . 0 0 0 0 6 9 . 6 0 0 0 6 9 . 7 8 4 8 . 1 8 4 8 - . 3 4 3 2 5 5 . 0 0 0 0 7 2 . 3 0 0 0 7 2 . 3 6 4 4 . 0 6 4 4

6 0 . 0 0 0 0 7 5 . 0 0 0 0 7 5 . 0 3 3 7 . 0 3 3 7 6 5 . 0 0 0 0 7 7 . 8 0 0 0 7 7 . 8 3 3 1 . 0 3 3 1

. 1 7 6 4 7 0 . 0 0 0 0 8 0 . 7 0 0 0 8 0 . 7 7 5 5 . 0 7 5 5

. 1 26 9 7 5 . 0 0 0 0 83 . 9 0 0 0 8 3 . 9 0 6 4 . 0 06 4

. 0 8 2 4

F i g u re 4 . 4 Ou t p u t L i s t i n g f o r P r o g ram P DA P .

0'1 0

8 0 . 0 0 0 0 8 1 . 3 0 0 0 81 . 2 1 0 5 - . 0 8 9 5 8 5 . 0 0 0 0 9 1 . 0 0 0 0 9 0 . 1 0 4 8 - . 2 9 5 2

RANGE " 3 0 3 0 . 0 1 0 . 0

4 0 . 0 0 0 0 6 1 . 2 0 0 0 6 1 . 2 8 2 5 . 0 8 2 5 4 5 . 0 0 0 0 6 9 . 2 0 0 0 6 9 . 3 6 5 9 . 1 6 5 9 5 0 . 0 0 0 0 1 1 . 5 0 0 0 1 1 . 6 0 3 3 . 1 0 3 3 5 5 . 0 0 0 0 1 3 . 9 0 0 0 1 3 . 9 5 2 8 . 0 5 2 8 6 0 . 0 0 0 0 1 6 . 3 0 0 0 1 6 . 4 0 6 6 . 1 06 6 6 5 . 0 0 0 0 1 9 . 1 0 0 0 7 9 . 0 5 5 1 - . 0 4 4 3 7 0 . 0 0 0 0 8 1 . 8 0 0 0 8 1 . 8 1 6 5 . 0 1 6 5 1 5 . 0 0 0 0 8 4 . 9 0 0 0 8 4 . 1 8 4 2 - . 1 1 5 8 8 0 . 0 0 0 0 8 8 . 1 0 0 0 81 . 9 2 09 - . 1 1 9 1 8 5 . 0 0 0 0 9 1 . 6 0 0 0 9 1 . 2 6 0 2 - . 3 3 9 8

1 0 , OF DESIGN FLOW RATE ( 1 1 5 5 0 GPM ) 1 . 5 5 9 5 1 . 1 1 1 0 1 0 . 0 0 . 0 0 0 0

4 RANGE .. 1 5

1 5 . 0 1 0 . 0 4 0 . 0 0 0 0 6 1 . 2 0 0 0 6 1 . 3 3 4 6 . 1 3 4 6 4 5 . 0 0 0 0 6 3 . 9 0 0 0 6 4 . 0 6 8 8 . 1 6 8 8 5 0 . 0 0 0 0 6 6 . 8 0 0 0 6 6 . 93 9 1 . 1 3 9 1 5 5 . 0 0 0 0 6 9 . 7 0 0 0 6 9 . 8 6 5 8 . 1 6 5 8 6 0 . 0 0 0 0 7 2 . 8 0 0 0 7 2 . 9 3 6 1 . 1 3 6 1 6 5 . 0 0 0 0 7 6 . 1 0 0 0 7 6 . 1 4 9 9 . 0 4 9 9 7 0 . 0 0 0 0 7 9 . 4 0 0 0 1 9 . 4 43 6 • 0 4 3 6 7 5 . 0 0 0 0 8 3 . 0 0 0 0 8 2 . 9 1 4 6 - . 0 8 5 4 8 0 . 0 0 0 0 8 6 . 6 0 0 0 8 6 . 4 8 2 4 - . 1 1 7 6 85 . 0 0 0 0 9 0 . 4 0 0 0 9 0 . 2 0 4 2 - . 1 9 5 8

RANG E .. 2 0 2 0 . 0 1 0 . 0

4 0 . 0 0 0 0 6 5 . 0 0 0 0 6 5 . 1 9 8 5 . 1 9 85 4 5 . 0 0 0 0 6 1 . 4 0 0 0 6 1 . 5 9 46 . 1 9 46 5 0 . 0 0 0 0 6 9 . 9 0 0 0 1 0 . 0 B 96 . 1 ,8 96 5 5 . 0 0 0 0 1 2 . 5 0 0 0 1 2 . 6 8 91 . 1 8 91 6 0 . 0 0 0 0 7 5 . 3 0 0 0 1 5 . 4 3 4 0 . 1 3 4 0 6 5 . 0 0 0 0 7 8 . :i 0 0 0 1 8 . 3 2 4 0 . 0 2 4 0 1 0 . 0 0 0 0 8 1 . 3 0 0 0 81 . 3 1 1 8 . 0 1 1 8 7 5 . 0 0 0 0 8 4 . 5 0 0 0 8 4 . 4 6 07 - . 0 3 9 3 8 0 . 0 0 0 0 8 7 . 9 0 0 0 81 . 1 1 3 3 - . 1 26 1

8 5 . 0 0 0 0 9 1 . 7 0 0 0 9 1 . 2 8 3 5 - . 4 1 6 5

RANG E '" 2 5 2 5 . 0 1 0 . 0

4 0 . 0 0 0 0 6 8 . 0 0 0 0 6 8 . 2 03 0 . 2 03 0

F i g u re 4 . 4

4 5 . 0 0 0 0 1 0 . 2 0 0 0 5 0 . 0 0 0 0 1 2 . 3 0 0 0 5 5 . 0 0 0 0 1 4 . 9 0 0 0 6 0 . 0 0 0 0 1 1 . 4 0 0 0 6 5 . 0 0 0 0 8 0 . 1 0 0 0 1 0 . 0 0 0 0 8 2 . 9 0 0 0 1 5 . 0 0 0 0 8 6 . 0 0 0 0 8 0 . 0 0 0 0 8 9 . 0 0 0 0 8 5 . 0 0 0 0 9 2 . 4 0 0 0

RANG E '" 3 0 3 0 . 0 1 0 . 0

4 0 . 0 0 0 0 7 0 . 3 0 0 0 4 5 . 0 0 0 0 7 2 . 2 0 0 0 5 0 . 0 0 0 0 1 4 . 4 0 0 0 5 5 . 0 0 0 0 7 6 . 7 0 0 0 6 0 . 0 0 0 0 7 8 . 9 0 0 0 6 5 . 0 0 0 0 8 1 . 5 0 0 0 1 0 . 0 0 0 0 8 4 . 0 0 0 0 7 5 . 0 0 0 0 8 6 . 8 0 0 0 8 0 . 0 0 0 0 8 9 . 9 0 0 0 8 5 . 0 0 0 0 9 3 . 2 0 0 0

1 0 . 3 5 8 1 . 1 5 8 1 1 2 . 5 4 5 3 . 2 4 5 3 1 4 . 9 6 0 8 . 06 0 8 1 1 . 4 2 8 2 . 0 2 8 2 6 0 . 0 5 0 2 - . 0 4 9 8 8 2 . 8 0 1 6 - . 0 9 2 4 8 5 . 1 4 9 5 - . 2 5 05 8 8 . 8 1 3 0 - . 1 8 1 0 9 2 . 0 9 5 4 - . 3 0 4 6

1 0 . 5 3 9 5 • 23 9 5 7 2 . 4 51 8 • 2 51 8 1 4 . 5 3 01 . 1 3 01 7 6 . 7 0 9 0 . 0 0 9 0 1 8 . 9 6 1 1 . 06 1 1 8 1 . 4 0 9 3 - . 0 9 07 8 3 . 9 6 1 5 - . 0 3 8 5 8 6 . 7 0 6 1 - . 0 9 3 9 8 9 . 6 4 5 2 - . 2 5 4 8 9 2 . 7 7 1 1 - . 4 2 8 9

8 0 , OF DES I GN FLOW RATE ( 1 3 2 0 0 G PM ) 1 . 4 5 4 9 1 . 27 0 0 8 0 . 0 0 . 0 0 0 0

4 RANGE " 1 5

1 5 . 0 1 0 . 0 4 0 . 0 0 0 0 6 3 . 5 0 0 0 6 3 . 6 2 6 0 4 5 . 0 0 0 0 6 6 . 0 0 0 0 6 6 . 1 81 6 5 0 . 0 0 0 0 6 8 . 8 0 0 0 6 8 . 9 2 2 3 5 5 . 0 0 0 0 1 1 . 6 0 0 0 7 1 . 1 1 5 5 6 0 . 0 0 0 0 1 4 . 5 0 0 0 7 4 . 6 1 4 4 6 5 . 0 0 0 0 1 7 . B O O O 7 1 . 1 3 1 7 1 0 . 0 0 0 0 8 0 . 8 0 0 0 8 0 . 8 2 2 9 1 5 . 0 0 0 0 8 4 . 3 0 0 0 8 4 . 1 6 3 8 8 0 . 0 0 0 0 8 7 . 7 0 0 0 8 1 . 5 1 2 0 8 5 . 0 0 0 0 9 1 . 6 0 0 0 9 1 . 2 03 5

RANG E .. 2 0 2 0 . 0 1 0 . 0

4 0 . 0 0 0 0 6 1 . 1 0 0 0 6 1 . 7 9 5 6 4 5 . 0 0 0 0 6 9 . 8 0 0 0 6 9 . 9 1'2 7 5 0 . 0 0 0 0 7 2 . 2 0 0 0 1 2 . 3 3 0 2 5 5 . 0 0 0 0 7 4 . 7 0 0 0 7 4 . 1 8 9 6 6 0 . 0 0 0 0 1 1 . 2 0 0 0 7 7 . 3 2 6 8 6 5 . 0 0 0 0 8 0 . 0 0 0 0 8 0 . 0 4 7 9 1 0 . 0 0 0 0 B 2 . 9 0 0 0 8 2 . 8 93 1

C o n t i n u e d

. 1 26 0

. 1 8 1 6

. 1 2 2 3

. 1 1 5 5

. 1 1 4 4 - . 0 6 8 3

. 0 2 2 9 - . 1 3 6 2 - . 1 2 8 0 - . 3 9 6 5

. 0 9 5 6

. 1 7 2 1

. 1 3 0 2

. 0 8 9 6

. 1 26 B

. 0 4 7 9 - . 0 06 9

0"> .......

7 5 . 0 0 0 0 8 5 . 9 0 0 0 8 5 . 87 3 5 - . 0 26 5 8 0 . 0 0 0 0 8 9 . 1 0 0 0 8 9 . 0 2 1 4 - . 0 7 86 8 5 . 0 0 0 0 9 2 . 8 0 0 0 9 2 . 3 9 5 3 - . 4 0 4 7

RANG E .. 2 5 2 5 . 0 1 0 . 0

4 0 . 0 0 0 0 7 0 . 8 0 0 0 7 0 . 9 2 2 8 . 1 2 2 8 4 5 . 0 0 0 0 7 2 . 7 0 0 0 7 2 . 8 6 9 8 . 1 6 9 B 5 0 . 0 0 0 0 7 4 . 9 0 0 0 7 4 . 97 6 8 . 07 6 8 5 5 . 0 0 0 0 7 7 . 2 0 0 0 7 7 . 1 8 9 9 - . 0 1 0 1 6 0 . 0 0 0 0 7 9 . 4 0 0 0 7 9 . 4 6 1 3 . 0 6 1 3 6 5 . 0 0 0 0 8 2 . 1 0 0 0 8 1 . 9 6 3 2 - . 1 3 6 8 7 0 . 0 0 0 0 8 4 . 7 0 0 0 8 4 . 5 46 4 - . 1 5 3 6 7 5 . 0 0 0 0 87 . 3 0 0 0 8 7 . 2 57 9 - . 0 4 2 1 8 0 . 0 0 0 0 9 0 . 3 0 0 0 9 0 . 1 8 7 5 - . 1 1 2 5 85 . 0 0 0 0 9 3 . 9 0 0 0 9 3 . 3 5 5 1 - . 5 4 4 9

RANG E .. 3 0 3 0 . 0 1 0 . 0

4 0 . 0 0 0 0 7 3 . 3 0 0 0 7 3 . 3 7 2 5 . 07 2 5 4 5 . 0 0 0 0 7 5 . 1 0 0 0 7 5 . 1 5 1 5 . 0 5 1 5 5 0 . 0 0 0 0 7 7 . 0 0 0 0 7 7 . 0 2 97 . 0 297

5 5 . 0 0 0 0 7 9 . 0 0 0 0 7 9 . 0 1 1 7 . 0 117 6 0 . 0 0 0 0 8 1 . 0 0 0 0 8 1 . 0 9 5 5 . 09 5 5 6 5 . 0 0 0 0 8 3 . 4 0 0 0 83 . 3 7 6 5 - . 0 23 5 7 0 . 0 0 0 0 8 5 . 9 0 0 0 8 5 . 7 9 46 - . 1 05 4 7 5 . 0 0 0 0 8 8 . 6 0 0 0 8 8 . 3 8 0 2 - . 2 1 9 8 8 0 . 0 0 0 0 9 1 . 3 0 0 0 9 1 . 1 1 4 5 - . 1 8 5 5 8 5 . 0 0 0 0 9 4 . 5 0 0 0 9 4 . 07 8 9 - . 4 21 1

90 , O F DESIGN FLOW RA T E ( 1 4 8 5 0 GPH) 1 . 3 6 3 3 1 . 4 2 8 0 9 0 . 0 0 . 0 0 0 0

4 RANG E .. 1 5

1 5 . 0 4 0 . 0 0 0 0 4 5 . 0 0 0 0 5 0 . 0 0 0 0

5 5 . 0 0 0 0 6 0 . 0 0 0 0 6 5 . 0 0 0 0 7 0 . 0 0 0 0 7 5 . 0 0 0 0 8 0 . 0 0 0 0 8 5 . 0 0 0 0

RANG E .. 2 0 2 0 . 0

4 0 . 0 0 0 0

1 0 . 0 6 5 . 8 0 0 0 6 5 . 8 9 0 5 6 8 . 1 0 0 0 6 8 . 27 2 0 7 0 . 8 0 0 0 7 0 . 8 9 03 7 3 . 3 0 0 0 7 3 . 47 21 7 6 . 1 0 0 0 7 6 . 2 4 9 1 7 9 . 0 0 0 0 7 9 . 1 3 7 0 8 2 . 0 0 0 0 8 2 . 1 4 3 6 8 5 . 3 0 0 0 8 5 . 3 3 6 7 8 8 . 7 0 0 0 8 8 . 6 5 4 1 9 2 . 2 0 0 0 9 2 . 1 0 2 7

1 0 . 0 7 0 . 1 0 0 0 7 0 . 2 2 2 4

. 0 9 0 5

. 17 2 0

. 0 9 0 3

. 17 2 1

. 1 4 9 1

. 1 3 7 0

. 1 4 3 6

. 0 3 6 7 - . 0 4 5 9

- . 0 9 7 3

. 1 2 2 4

4 5 . 0 0 0 0 7 2 . 1 0 0 0 7 2 . 27 3 2 . 1 7 3 2 5 0 . 0 0 0 0 7 4 . 3 0 0 0 7 4 . 4 6 6 3 . 1 6 6 3 5 5 . 0 0 0 0 7 6 . 7 0 0 0 7 6 . 7 9 9 6 . 0 9 9 6 6 0 . 0 0 0 0 7 9 . 2 0 0 0 7 9 . 2 4 0 0 . 0 4 0 0 6 5 . 0 0 0 0 8 1 . 8 0 0 0 8 1 . 7 9 9 5 - . 0 0 0 5 7 0 . 0 0 0 0 8 4 . 5 0 0 0 8 4 . 4 8 1 9 - . 0 1 81 7 5 . 0 0 0 0 8 7 . 5 0 0 0 8 7 . 3 5 1 8 - . 1 4 8 2 8 0 . 0 0 0 0 9 0 . 7 0 0 0 9 0 . 3 8 0 7 - . 3 1 9 3 8 5 . 0 0 0 0 9 3 . 7 0 0 0 9 3 . 4 9 47 - . 2 05 3

RANG E " 2 5 2 5 . 0 1 0 . 0

4 0 . 0 0 0 0 7 3 . 3 0 0 0 7 3 . 47 0 7 . 1 7 07 4 5 . 0 0 0 0 7 5 . 2 0 0 0 7 5 . 3 2 4 3 . 1 2 4 3 5 0 . 0 0 0 0 7 7 . 2 0 0 0 7 7 . 27 5 8 . 0 7 5 8 5 5 . 0 0 0 0 7 9 . 4 0 0 0 7 9 . 3 6 0 9 - . 0 3 9 1 6 0 . 0 0 0 0 8 1 . 4 0 0 0 81 . 47 0 2 . 07 0 2 6 5 . 0 0 0 0 8 3 . 9 0 0 0 8 3 . 8 1 4 8 - . 0 8 5 2 7 0 . 0 0 0 0 8 6 . 3 0 0 0 8 6 . 2 3 8 1 - . e6 l 9 7 5 . 0 0 0 0 8 9 . 1 0 0 0 8 8 . 8 7 1 6 - . 2 2 8 4 8 0 . 0 0 0 0 9 1 . 8 0 0 0 9 1 . 6 1 5 0 - . 1 8 5 0 8 5 . 0 0 0 0 9 4 . 7 0 0 0 9 4 . 5 4 5 5 - . 1 5 4 5

RANGE co 3 0 3 0 . 0 1 0 . 0

4 0 . 0 0 0 0 7 5 . 9 0 0 0 7 6 . 0 0 4 2 . 1 0 4 2 en N

4 5 . 0 0 0 0 7 7 . 5 0 0 0 7 7 . 6 3 3 6 . 1 3 3 6 5 0 . 0 0 0 0 7 9 . 4 0 0 0 7 9 . U 5 8 . 0 1 5 8 5 5 . 0 0 0 0 8 1 . 3 0 0 0 81 . 27 4 1 - . 0 2 5 9 6 0 . 0 0 0 0 8 3 . 3 0 0 0 83 . 2 4 7 3 - . 0 527 6 5 . 0 0 0 0 8 5 . 4 0 0 0 85 . 3 47 2 - . 0 5 2 8 7 0 . 0 0 0 0 87 . B O O O 87 . 6 2 4 9 - . 1 7 5 1 7 5 . 0 0 0 0 9 0 . 3 0 0 0 9 0 . 0 4 8 0 - . 2 5 2 0 8 0 . 0 0 0 0 9 2 . 9 0 0 0 9 2 . 6 3 1 6 - . 2 6 8 4 8 5 . 0 0 0 0 9 5 . 7 0 0 0 9 5 . 4 0 4 8 - . 2 9 5 2

1 00 , O F DES IGN FLOW RATE ( 1 6 5 0 0 GPH ) 1 . 2 9 7 9 1 . 57 8 0 1 0 0 . 0 0 . 0 0 0 0 0 0 0 0

4 RANG E - 1 5

1 5 . 0 1 0 . 0 4 0 . 0 0 0 0 6 7 . 9 0 0 0 6 7 . 9 1 0 1 . 0 1 0 1 4 5 . 0 0 0 0 7 0 . 1 0 0 0 7 0 . 1 7 6 1 . 07 6 1 5 0 . 0 0 0 0 7 2 . 5 0 0 0 7 2 . 5 8 7 5 . 0 8 7 5 5 5 . 0 0 0 0 7 5 . 0 0 0 0 7 5 . 1 0 0 6 . 1 0 0 6 6 0 . 0 0 0 0 7 7 . 8 0 0 0 7 7 . 8 0 5 1 . 0 051 6 5 . 0 0 0 0 8 0 . 4 0 0 0 8 0 . 4 97 4 . 0 9 7 " 7 0 . 0 0 0 0 8 3 . 2 0 0 0 83 . 3 5 2 5 . 1 5 2 5

F i g u re 4 . 4 C o n t i n u e d

7 5 . 0 0 0 0 8 6 . 4 0 0 0 8 6 . 4 3 9 4 . 0 3 9 4 4 5 . 0 0 0 0 7 2 . 0 0 0 0 7 2 . 1 8 5 0 . 1 8 5 0 8 0 . 0 0 0 0 8 9 . 9 0 0 0 8 9 . 7 0 5 3 - . 1 9 47 5 0 . 0 0 0 0 7 4 . 4 0 0 0 7 4 . 5 3 3 6 . 1 3 3 6 8 5 . 0 0 0 0 9 3 . 2 0 0 0 9 3 . 0 1 1 6 - . 1 8 8 4 5 5 . 0 0 0 0 7 6 . 7 0 0 0 7 6 . 8 86 1 . 1 8 6 1

RANGE .. 2 0 6 0 . 0 0 0 0 7 9 . 3 0 0 0 7 9 . 4 3 5 1 . 1 3 5 1 2 0 . 0 1 0 . 0 6 5 . 0 0 0 0 8 2 . 1 0 0 0 8 2 . 1 3 4 6 . 0 3 4 6

4 0 . 0 0 0 0 7 2 . 2 0 0 0 7 2 . 3 2 2 6 . 1 2 26 7 0 . 0 0 0 0 8 4 . 9 0 0 0 8 4 . 9 06 0 . 0 06 0 4 5 . 0 0 0 0 7 4 . 2 0 0 0 7 4 . 3 0 3 6 . 1 03 6 7 5 . 0 0 0 0 87 . 9 0 0 0 87 . 8 3 4 9 - . 06 5 1 5 0 . 0 0 0 0 7 6 . 2 0 0 0 7 6 . 3 4 3 7 . 1 4 3 7 8 0 . 0 0 0 0 9 1 . 0 0 0 0 9 0 . 8 8 9 3 - . 1 1 07 5 5 . 0 0 0 0 7 8 . 8 0 0 0 7 8 . 6 7 97 - . 1 2 0 3 85 . 0 0 0 0 9 4 . 3 0 0 0 9 4 . 1 06 0 - . 1 9 4 0 6 0 . 0 0 0 0 8 1 . 0 0 0 0 8 0 . 9 2 9 8 - . 07 0 2 RANGE .. 2 0 6 5 . 0 0 0 0 8 3 . 2 0 0 0 8 3 . 27 1 1 . 07 1 1 2 0 . 0 1 0 . 0 7 0 . 0 0 0 0 8 5 . 9 0 0 0 8 5 . 87 0 3 - . 0 2 9 7 4 0 . 0 0 0 0 H . 5 0 0 0 7 4 . 6 97 9 . 1 97 9 7 5 . 0 0 0 0 8 8 . 5 0 0 0 8 8 . 5 4 4 2 . 0 4 4 2 4 5 . 0 0 0 0 7 6 . 4 0 0 0 7 6 . 5 6 7 1 . 1 6 1 1 8 0 . 0 0 0 0 9 1 . 4 0 0 0 9 1 . 4 1 7 7 . 0 17 7 5 0 . 0 0 0 0 7 8 . 3 0 0 0 7 8 . 4 96 0 . 1 9 6 0 85 . 0 0 0 0 9 4 . 8 0 0 0 9 4 . 5 27 1 - . 2 7 2 9 5 5 . 0 0 0 0 8 0 . 3 0 0 0 8 0 . 5 26 4 . 2 26 4

RANG E • 2 5 6 0 . 0 0 0 0 8 2 . 7 0 0 0 8 2 . 7 7 8 1 . 07 8 1 2 5 . 0 1 0 . 0 6 5 . 0 0 0 0 8 4 . 8 0 0 0 8 5 . 0 0 0 4 . 2 0 0 4

4 0 . 0 0 0 0 7 5 . 6 0 0 0 7 5 . 7 0 3 0 . 1 03 0 7 0 . 0 0 0 0 8 7 . 5 0 0 0 8 7 . 5 1 4 9 . 0 1 4 9 4 5 . 0 0 q O 7 7 . 5 0 0 0 7 7 . 4 81 6 - . 0 1 8 4 7 5 . 0 0 0 0 9 0 . 1 0 0 0 9 0 . 0 9 1 6 - . 0 0 8 4 5 0 . 0 0 0 0 7 9 . 2 0 0 0 7 9 . 2 5 4 4 . 05 4 4 8 0 . 0 0 0 0 9 3 . 0 0 0 0 9 2 . 8 6 1 1 - . 1 3 8 9 5 5 . 0 0 0 0 8 1 . 2 0 0 0 8 1 . 1 9 8 2 - . 0 0 1 8 8 5 . 0 0 0 0 9 6 . 0 0 0 0 9 5 . 7 7 2 5 - . 2 27 5

6 0 . 0 0 0 0 8 3 . 2 0 0 0 8 3 . 2 2 3 2 . 0 23 2 RANGE .. 2 5

6 5 . 0 0 0 0 8 5 . 3 0 0 0 8 5 . 3 7 2 5 . 07 2 5 2 5 . 0 1 0 . 0

7 0 . 0 0 0 0 8 7 . 8 0 0 0 87 . 7 3 1 4 - . 0 6 8 6 4 0 . 0 0 0 0 7 8 . 0 0 0 0 7 8 . 1 9 3 5 . 1 9 3 5 0'\

7 5 . 0 0 0 0 9 0 . 2 0 0 0 9 0 . 1 8 0 7 - . 0 1 9 3 4 5 . 0 0 0 0 7 9 . 7 0 0 0 7 9 . 8 2 8 4 . 1 2 8 4 w

8 0 . 0 0 0 0 9 2 . 9 0 0 0 9 2 . 8 2 6 6 - . 07 3 4 5 0 . 0 0 0 0 8 1 . 2 0 0 0 81 . 4 5 6 7 . 2 56 7

85 . 0 0 0 0 9 6 . 1 0 0 0 9 5 . 7 0 2 2 - . 3 97 8 5 5 . 0 0 0 0 8 3 . 2 0 0 0 83 . 3 2 3 3 . 1 23 3

RANG E " 3 0 6 0 . 0 0 0 0 8 5 . 2 0 0 0 85 . 26 2 9 • 0 6 2 9

3 0 . 0 1 0 . 0 6 5 . 0 0 0 0 8 7 . 3 0 0 0 87 . 3 1 8 8 • 01 8 8

4 0 . 0 0 0 0 7 8 . 3 0 0 0 7 8 . 2 9 47 - . 0 05 3 7 0 . 0 0 0 0 8 9 . 7 0 0 0 8 9 . 5 5 2 6 - . 1 4 7 4 4 5 . 0 0 0 0 7 9 . 9 0 0 0 7 9 . 8 47 4 - . 0 5 26 7 5 . 0 0 0 0 9 1 . 9 0 0 0 91 . 8 4 6 4 - . 0 5 3 6 5 0 . 0 0 0 0 8 1 . 5 0 0 0 8 1 . 4 6 5 4 - . 0 3 4 6 8 0 . 0 0 0 0 9 4 . 6 0 0 0 9 4 . 3 8 4 1 - . 2 1 5 9 5 5 . 0 0 0 0 8 3 . 1 0 0 0 83 . 1 6 2 9 . 06 2 9 8 5 . 0 0 0 0 97. 2 0 0 0 9 7 . 0 2 87 - . 1 7 1 3

6 0 . 0 0 0 0 8 5 . 0 5 0 0 85 . 0 3 9 1 - . 01 09 RANGE .. 3 0

6 5 . 0 0 0 0 87 . 0 0 0 0 87 . 0 1 4 4 . 0 1 4 4 3 0 . 0 1 0 . 0

7 0 . 0 0 0 0 8 9 . 1 0 0 0 8 9 . 1 3 6 9 . 0 3 6 9 4 0 . 0 0 0 0 8 0 . 7 0 0 0 8 0 . 8 4 2 4 . 1 4 2 4

7 5 . 0 0 0 0 9 1 . 5 0 0 0 9 1 . 4 4 47 - . 0 5 5 3 4 5 . 0 0 0 0 8 2 . 2 0 0 0 8 2 . 2 8 5 9 . 0 8 5 9

8 0 . 0 0 0 0 9 4 . 0 0 0 0 9 3 . 9 0 8 3 - . 0 9 1 7 5 0 . 0 0 0 0 8 3 . 7 0 0 0 83 . 7 96 7 . 0 9 6 7

8 5 . 0 0 0 0 97 . 0 0 0 0 9 6 . 6 0 3 8 - . 3 9 6 2 5 5 . 0 0 0 0 8 5 . 3 0 0 0 8 5 . 4 1 4 5 . 1 1 4 5

1 1 0 , OF D E S I GN F LOW RATE ( 1 8 1 5 0 GPM) 6 0 . 0 0 0 0 87 . 2 0 0 0 8 7 . 1 8 8 6 - . 0 1 1 4

1 . 2 1 87 1 . 7 4 5 0 1 1 0 . 0 0 . 0 0 0 0 6 5 . 0 0 0 0 8 8 . 9 0 0 0 8 9 . 0 0 7 6 . 1 07 6

4 7 0 . 0 0 0 0 9 1 . 2 0 0 0 9 1 . 07 03 - . 1 2 9 7

RANGE .. 1 5 7 5 . 0 0 0 0 9 3 . 4 0 0 0 93 . 2 2 5 5 - . 1 7 4 5

1 5 . 0 1 0 . 0 8 0 . 0 0 0 0 9 5 . 9 0 0 0 9 5 . 5 6 8 0 - . 3 3 2 0

4 0 . 0 0 0 0 6 9 . 9 0 0 0 7 0 . 0 3 0 5 . 1 3 0 5 8 5 . 0 0 0 0 9 8 . 3 0 0 0 9 8 . 0 4 1 4 - . 2 5 8 6

F i g u re 4 . 4 C o n t i n u e d

1 00 P E R F O R M A N C E C U R V E C O M P A R I S O N Ra n g e C · F)

E 90 � 30 SY M B O L S- DATA F RO M M A N U FACT U R E R 2 5 C U RV E S- M O D E L P R E D I C T I O N S A 20

1 5 4Jb'-// 0...

:2 W 80 l-n: W l- 70 � <.{ ./ ... // � -1 � 0 5 60 r �/ K a V / l = 1 . 9040

L = 8250 g p m (50% of d es i gn) 0

50 I� ________ � ________ � ________ � ________ �� ________ � ________ � ________ �

30

F i g u re 4 . 5

40 50 60 7 0 8 0 I N L E T W E T - B U L B T E M P ( O F )

90 1 0 0

Comp a r i s o n o f Ma n u fa c t u re r ' s P e r f o rma n c e Da t a w i t h P re d i c t i o n s o f Ma t h ema t i c a l Mo d e l fo r 5 0 % o f De s i g n F l ow Ra t e .

O'l .po

1 00

E 90 � �

80 f � W r- 70 « �

� 60 � U

P E R F O R M A N C E C U RV E CO M P A R I S O N Ra n g e ( - f)

SY M B O L S- DA T A F R O M M A N U FACT U R E R � CU RV ES- tv1 0 D E l P R E D I CT I O N S �� . "" /'

h-� ' ' C'Y

�-7 '�' / ,�. /� �. /'

�.::::::.� 'l ..",AI" / ./ / �' . .er .� ./ It""" /

/' (!J' KaV/l = 1 .7 1 40

30 2 5 20 1 5

� l = 9 9 0 0 g p m (60% of d es i g n )

5 0 IL-______ � ________ � ________ � ________ � ________ � ________ � ______ �

30 4 0 5 0 6 0 7 0 8 0 I N L ET W ET - B U L B T E M P ( O F )

9 0 1 0 0

F i g u re 4 . 6 Compa r i s o n o f Ma n u fa c t u re r ' s Pe rfo rma n c e Da t a w i t h P re d i c t i o n s o f Ma t hema t i c a l Mo d e l fo r 6 0 % o f De s i g n F l ow Ra te .

0"1 <.TI

100 P E R F O R M A N C E C U RV E C O M PA R I SO N Ra n g e C · F)

30 r"" LL S Y M BO L S- DATA F R O M M A N U F A CT U R E R .t 25 � C U R V E S- M O D E L P R E D I CT I O N S �/:f2 20 90 �/'// 1 5 � �/ 2 £/ W /�� ,//. / / � � er:: 80 :% / / W �,/. .0'

� /�� ........ ��I!I'"/ � ./' /' o 70 r ..... /" � --1 If"'" /' K a V/ l = 1 .5595 o / / '/ L = 1 1 550 g p m (70:1: of d es i g n ) o (!f

60 '� ______ � ________ � ______ � ________ � ______ � ________ � ______ -J

30 40 50 6 0 70 80 I N L E T W ET - B U L B T E M P ( O F )

9 0 1 0 0

F i g u re 4 . 7 Compa r i s o n o f Ma n u f a c t u re r ' s Pe rfo rma n c e Da t a wi t h P red i c t i o n s o f Ma t h ema t i c a l Mo d e l -for 7 0 % o f De s i g n F l o w Ra t e .

1 00

r-. LL 0 '-' 90 0... 2 W r-

P E R F O R M A N C E C U RV E CO M P A R I S O N Ra n g e ( e F)

30 SY M B O L S- DA T A F R O M M A N U F AC T U R E R � 25 C U R V E S- M O D E L P R E D I CT I O N S �� 20

��/ 1 5 �� / . ./

�;0 �. /. n::: 80 .,.K. fII' '/ / / ./ W .,......... /' .,., ,/' r- _ / /' .,., A

-< .,.�.,./ .,.,eT � ..... �. AI /

� 70 r 0

¥"". / � fIf -" / K e Vi l - 1 . 4 5 4 9

L = 1 3 2 0 0 g p m (80% o f d es I g n )

6 0 LI ________ � ________ � ________ � ________ � ________ � ________ �------�

3 0 40 90 1 00

F i g u re 4 . 8 Compa r i s o n o f Ma n u fa c t u r e r ' s Pe rfo rma n c e Da t a w i t h P re d i c t i o n s o f Ma t h ema t i c a l Mo d e l fo r 8 0 % o f Des i g n F l ow Ra t e .

'" .......

1 00 P E R FO R M A N C E C U RV E C O M PA R I S O N Ra n g e C - F)

A 30 G:" SYM B O L S - D A T A F RO M M A N U FACT U R E R �:;?; 25 o C U R V E S- M O D E L P R E D I CT I O N S �X-./ � 20

90 /� .// 1 5 � �" ./ 2 :% ", It' W ,�" � ./ L- ,.,Y / ./ .- � /" � er::: 80 �� /" ./ ./ � " AI W ,.A"' � ./ ./ 0\

r-- � .. / / 00

« " �" ./ S If'. ./ -t!f / ./

-.J K a V I L ::I 1 .3633 o 70 r t!f

o / L � 1 4 850 g p m (90% of d es i g n ) o

60 �, ________ � ______ � ________ � ______ � ________ � ______ � ______ �

30 4 0 50 6 0 7 0 80 90 1 00 I N L ET W E T - B U L B T E M P ( O F )

F i g u re 4 . 9 Compa r i s o r. o f t la n u fa c t u re r ' s Pe rf o rma n c e Da t a w i t h P red i c t i o n s o f Ma t hema t i c a l �lo d e l fo r 9 0 � o f De s i g n F l o w Ra t e .

1 0 5

r---. LL 0 '-'

95 0... :2 W r--

0:: 85 w r--« 3

� 75 r

0

P E R F O R M A N C E C U RV E C O M PA R I S O N Ra n g e ( - F )

SY M B O LS- DATA F R O M M A N U FACT U R E R A C U RV E S- M O D E L P R E D I CT I O N S �� :::;-:: / m

"/0/ /.% �%1!1" ",..... ./ �/' �

---/�/. / / � �/ JiY /' ./ ................ �� /'

.� � /' II""" /'

K a V I L = 1 .2 9 7 9

30 2 5 20 15

ff / / / L - 1 6500 g p m ( 1 00% o r d e s l o n )

65 ,� ________ � ______ � ______ �� ______ � ________ � ______ � ______ �

30 40 50 6 0 70 80 I N L ET W ET - B U L B T E M P ( O F)

9 0 1 0 0

F i g u re 4 . 1 0 Compa r i s o n o f t1a n u fa c t u re r ' s Pe r fo rma n c e Da t a w i t h P red i c t i o n s o f Ma t h ema t i c a l Mo d e l . for 1 00 % o f De s i g n F l ow Ra t e .

Q) \.0

1 05

,....... LL 0 '-/

95 n.. 2 w t-eL 85 W t-oe::( � o 75 --1 0 0

P E R F O R M A N C E C U RV E C O M PA R I S O N Ra n g e C e F) 30

SY M BO LS- DATA F R O M M A N U FA C T U R E R ;! 25 CU RVES- M O D E L P R E D I CT I O N S � /' (i) 20 �� /' � 1 5 ... /� /' � ... �.// .... � /" ,/' � �. Ilf' //' ,/' / .............. /. A

. -� .,/' eI' ./ -� � ,/' ./ .,,- .� .A> ,� . ff' ./ r ./

,...-e/ ./

(!f Ka V/ l = 1 .2 1 87 l = 1 8 1 50 g p m ( 1 1 0% of d es i g n )

65 �I __________ � __________ � ________ � ________ � _________ � _________ � ________ �

30 40 50 6 0 7 0 80 I N L ET W E T - B U L B T E �v1 P ( O F)

9 0 1 00

F i g u re 4 . 1 1 Compa r i s o n o f Ma n u fa c t u re r ' s Pe rfo rma n c e Da t a w i th P r ed i c t i o n s o f Ma t hema t i c a l Mo d e l fo r 1 1 0 : o f De s i g n F l ow Ra t e .

--.J a

71

c a l d i ffe rence between the p red i cted a n d repo rt ed tempe rat u res i s s hown

i n t he fou rth co l umn of F i gu re 4 . 4 . As i s r e a d i l y ob s e r ved f rom F i g u re s

4 . 5 - 4 . 1 1 a s wel l a s the t abu l at e d ou t p u t g i ve n i n F i gu re 4 . 4 , t h e

a g reement i s exce l l en t , i n d i c at i n g t h at t he t ower . s i mu l a t i on i s i n deed

ab l e to rep roduce t he tower pe r fo rma n ce c h a ract e r i s t i cs . F i gu re 4 . 1 2

comp a res re l at i ve KaV ( i . e . , t he r at i o of KaV at t he g i ven f l ow rate t o

K a V a t t h e des i gn f l ow rate ) t o re l at i ve f l ow rat e ( i . e . , t h e rat i o o f

t he g i ven f l ow rate to t h e des i gn f l ow rate ) . T h i s p l ot i l l u s t r a t e s

t h at KaV dec l i n e s w i t h dec l i n i n g f l ow rate a s may b e expect ed .

S i nce the des i g n - b a s i s o p e r at i n g p o i nts ( 7 3 . 6 p e rcent f l ow rate a n d

7 0 . 4 percent f l ow rat e ) d o not c o i n c i de wi t h a ny of t h e f l ow r a t e s f o r

wh i ch p e r f o rmance data a re a v a i l ab l e , a s i mp l e p ro g ram ent i t l ed KAVL i s

p ro v i ded to i nt e rp o l ate by He rmi t e cub i c s p l i ne the va l ues o f KaV / L at

the act u a l o pe rat i n g p o i nt s . T he i np u t s to p ro g ram KAVL a re t a ken d i

rect l y f rom t h e out p u t of p rog ram POAP .

4 . 3 T i me C o n s t a nt E v a l u at i o n

P rog rams UHSS I M a n d TMC O N (�IME C O N STANT ) a re needed to det e rmi n e

the a pp rox i mate t i me c o n s t a nt of the t owe r / b a s i n system for u s e i n t h e

dat a s c a n n i n g p rocedu re to be des c r i bed l at e r . Reca l l t h at , for t h e

p u rpose o f det e rmi n i ng t h e p e r i o d o f worst - c a s e t ower p e rforma nce , t h e

res p o n s e of t h e bas i n temp e ratu r e i s app rox i mat e d a s be i n g exponent i a l

i n t i me .. A l t hou gh the a s s umpt i on of a f i r st - o r der t i me r e s p o n s e i s

b o r ne out wel l i n p ract i ce , t h e va l u e o f the t i me con s t a n t may v a ry ove r

the range of p o s s i b l e o perat i n g temp e r at u res for t h e t owe r / b a s i n sy s

tem . Thu s , t he v a l u e o f t he t i me con s t ant emp l oyed must be rep r e s e n t a -

0 0 r4

> 0

� "-> 0 �

1 . 2 K a V P e rf o r m a n c e - T E ST C A S E

e P o i n t s f r o m f i t t o D a t a f r o m M a n u f a ct u r e r 1 . 1 I- B e st L i n e a r F i t

1 .0

0.9 �

0 . 8

0.7

0 .6 1� __________ �� ________ �� __________ � ____________ � 0 . 4 0 . 6 1 . 2 0 . 8

L/ L 1 00 1 . 0

F i g u re 4 . 1 2 Va r i a t i o n o f Re l a t i v e Towe r P a r a me t e r Ka V ( Ra t i o o f Ka V a t t he G i v e n F l ow Ra t e t o Ka V a t De s i g n F l ow Ra t e ) to Re l a t i v e F l ow Ra t e ( Ra t i o o f G i v e n F l ow Ra t e t o De s i g n Fl ow Ra t e ) .

........ N

7 3

t i ve o f the s pec i f i c acc i dent con d i t i o n s hyp ot h es i zed . F o r t he pu r po se

o f det ermi n i n g t he t i me c o n s t a nt o f the t owe r / b a s i n sy stem , a c o n s t a n t

amb i ent wet - b u l b temp e r at u re i s u s e d . Al s o , a rea s o n a b l e cho i ce f o r

i n i t i a l b a s i n t empe rat u re i s nece s s a ry . The exact va l ues s e l ected f o r

t h e s e i np u t s a re n o t c ru c i a l t o t h e t i me-con stant det e rmi n at i o n ; r athe r ,

i t i s me re l y neces s a ry t h at t he o pe rat i ng t emp e rat u re of the t owe r ( s ) be

i n the p rop e r ran ge .

T h e req u i red i n put f o r p rog ram UHSS I M i s s et fort h i n Ta b l e 4 . 5 a n d

F i gu res 4 . 1 3 a n d 4 . l 4 � Spec i f i c a l l y , Tab l e 4 . 5 de f i nes e a c h F ORTRAN

v a r i a b l e n ame , F i gu re 4 . 1 3 i l l u s t rate s the format i n wh i ch the i n p ut i s

expect e d , a n d F i g u re 4 . 1 4 s h ows t he actua l i n put f i l e u sed i n the t e s t

c a s e s i mu l at i on . As s h own i n F i gu re 4 . 1 3 , t h e i n p ut i s di v i ded i nt o s i x

s e ct i o n s . T he f i rst sect i on g i ves the number of t owe r s , t h e i n i t i a l

ma s s o f wate r i n the bas i n a n d the i n i t i a l s a l i n i ty o f the b a s i n . F o r

the t est c a s e , on l y a s i n g l e t owe r i s p re s e n t , t he mas s o f water i n i t i

a l l y i n the bas i n i s 5 . 42 x 1 0 ' l bm , a nd t h e i n i t i a l s a l i n i ty i s t a k e n

t o b e 5 p a rt s p e r t h o u s and ( ppt ) .

T h e s econ d s ect i on l i s t s the cont ro l v a r i a b l e s f o r t he s i mu l a

t i o n : t he t i me i nc rement t o be u s e d i n p r i nt i ng t he re s u l t s , t he n u mbe r

o f s u c h t i me step s to be s i mu l ated , a l og i c a l v a l u e i nd i cat i n g whet h e r

e vapo rat i on i s to b e a l l owed , a n d a l og i ca l v a l u e i n d i c at i n g whether a

c o n s t a nt wet - bu l b t emp e rat u re i s t o be emp l oye d . I n t h e s a mp l e c a s e ,

t h e t i me step i s set t o one h ou r , t he n umber of steps i s s et t o 1 0 0 , a n d

evapo rat i on i s a l l owed . As noted e a r l i e r , a u s e r- s p ec i f i e d con s t a nt

wet - bu l b t emperat u re i s u s ed fo r t h e pu rpose of det e rm i n i n g t he t i me

7 4 Ta b l e 4 . 5 De s c r i p t i o n o f I n p u t V a r i a b l e s fo r P ro g ram U H S S I M

VA R I AB L E NAME

NTOWE R

BASMA S

S

TSTEP

NSTEP

NO EVAP

CONWB

NL

TE

LOV RGS

KAV L S

L S

NH

Tf1E

WB

BASTMP

NW I N ml

N S TART

D ES C R I PT I O N AND UN I TS

N umbe r o f Towe r s

I n i t i a l B a s i n Ma s s ( l bm )

I n i t i a l B a s i n S a l i n i ty

P r i n t i n g T i me S t e p S i z e ( h r )

Number o f S t e p s t o b e Tal e n

Log i c a l V a r i a b l e S e t . Tr u e . fo r No Ev a po ra t i o n

. Fa l s e . O t h e rw i s e

L o g i c a l Va r i a b l e S e t . Tr u e . fo r Con s t a n t We t - B u l b

Temre ra t u re . Fa l s e . Ot h e rw i s e

Numb e r o f F l ow Ra t e V a l u e s P e r Towe r

Array C o n t a i n i n g T i m e s o f C o r r e s po n d i n g Fl ow Ra t e

Va l u e s ( h r )

Array Co n t a i n i n g L / G V a l u e s

Array C o n ta i n i n g Ka V/ L V a l u e s

A r ray C o n t a i n i n g L i q u i d F l ow R a t e V a l u e s ( l bm/ h r )

N um b e r o f H e a t L o a d V a l u e s P e r TO\'/e r

A rray Co n t a i n i n g T i me s o f Co r r e s p on d i n g He a t L o a d

Va l u e s ( B t u / h r )

Con s t a n t Wet - Bu l b Tempe r a t u re V a l u e ( F )

I n i t i a l Ba s i n Temp e r a t u re ( F )

Numbe r o f Me t e o ro l o g i c a l W i n dow t o be Re a d

Po s i t i o n W i t h i n Me t e o ro l o g i c a l W i n dow t o B e g i n

Rea d i n g We t - Bu l b Tempe ra t u re V a l u e s

7 5

F i g u re 4 . 1 3 I n pu t Fo rma t fo r P ro g ram UH S S I M

Sec t i on 1

Sec t i on 2

Sec t i on 3

Sec t i on 4

Sect i on 5

Sect i on 6

NTOWE R . BAS�1AS • S

TSTEP . NS T E P . NO EVAP . CONWB

NL ( l ) . NL ( 2 ) . . . . TE ( 1 . 1 ) , LOVRG S ( l , 1 ) , KA VLS ( 1 . 1 ) • LS ( 1 , 1 ) } �E ( 2 , 1 ) , lOVRGS ( 2 , 1 ) , KAVl S ( 2 , l ) . l S ( 2 , 1 ) T o w e r + 1

TE { 1 . 2 } . LOVRGS ( 1 . 2 ) , KAVL S ( l . 2 ) , L S ( l . 2 ) } TE ( 2 . 2 ) . LOV RGS ( 2 . 2 ) . KAV L S ( 2 . 2 ) . L S ( 2 . 2 )

NH ( l ) . NH ( 2 ) • . • •

TME ( l , 1 ) , Q ( 1 , l ) } TME ( 2 , 1 ) . Q ( 2 , l ) T o w e r + 1

TME ( 1 , 2 ) • Q ( 1 , 2 )

} TME ( 2 , 2 ) . Q ( 2 . 2 ) T o w e r + 2

e t c . . .

WB , BASTMP

NW I NDW , NSTART

T o w e r + 2

e t c . . .

7 6

F i g u re 4 . 1 4 I n pu t L i s t i n g fo r Prog ram U H S S I M .

1 , S . 4 2 E 7 , O . 0 0 5 1 . O , l O O , F , T 4 O . O , 1 . 1 6 8 7 , 1 . 5 1 8 9 , 6 . 0 6 6 1 E 6 2 5 . 0 , 1 . 1 6 8 7 , 1 . 5 1 8 9 , 6 . 0 6 6 1 E 6 2 5 . 0 , 1 . 1 1 6 7 , 1 . 5 5 5 1 , 5 . 7 9 6 E 6 1 0 0 . 0 , 1 . 1 1 6 7 , 1 . 5 5 5 1 , 5 . 7 9 6 E 6 2 9 O . O , 6 6 . 2 E 6 O . 5 , 1 9 3 . 0 E 6 1 . 0 , 1 9 3 . 0 E 6 1 . 5 , 1 9 3 . 0 E 6 2 . 0 , 1 9 3 . 0 E 6 2 . 5 , 1 9 3 . 0 E 6 3 . 0 , 1 9 3 . 0 E 6 4 . 0 , 1 9 3 . 0 E 6 5 . 0 , 1 9 3 . 0 E 6 6 . 0 , 1 9 3 . 0 E 6 8 . 0 , 1 9 3 . 0 E 6 1 0 . 0 , 1 9 3 . 0 E 6 1 2 . 0 , 1 9 0 . 0 E 6 1 6 . 0 , 1 8 2 . 0 E 6 2 0 . 0 , 1 7 8 . 0 E 6 2 4 . 0 , 1 3 1 . 0 E 6 4 8 . 0 , 9 5 . 4 E 6 7 2 . 0 , 8 8 . 6 E 6 · 9 6 . 0 , 8 4 . 1 E 6 1 2 0 . 0 , 8 1 . 1 E 6 1 4 4 . 0 , 7 7 . 6 E 6 1 6 8 . 0 , 7 6 . 2 E 6 1 9 2 . 0 , 7 4 . 6 E 6 2 1 6 . 0 , 7 3 . 1 E 6 2 4 0 . 0 , 7 1 . 2 E 6 3 6 0 . 0 , 6 7 . 0 E 6 4 8 0 . 0 , 6 4 . 0 E 6 6 0 0 . 0 , 6 1 . 6 E 6 7 2 0 . 0 , 6 0 . 4 E 6 8 1 . , 8 1 . 1 , 7 9

7 7

con s t a nt o f the t owe r/b a s i n sy stem , so the va r i a b l e CONWB i s set to be

t ru e .

The t h i rd s ect i o n p ro v i des i n f o rmat i on o n how the va r i o u s tow e r

p e r f o rma nce p a ramet e rs c h a n ge wi th t i me for e a c h towe r du r i n g the s i mu

l at e d acc i dent . The s pec i f i c pa ramet e r s req u i red i nc l u de t h e two cha r

a ct e r i s t i c v a l ues L /G a n d KaV/L as wel l as t h e act u a l water f l ow rat e ,

a l l as funct i o n s of t i me for each t ower sep a rate l y . The f i r s t l i ne i n

Sect i on 3 g i ves the number o f l i n es of i n put requ i red for e ach t owe r .

Th u s , i n t he ca se of two towers , t h e va l ue of NTOWER set i n Sect i o n 1

w i l l be 2 , a nd two va l ues of NL ( f i rst l i ne of Sect i on 3 ) a re expect

ed . I f t h e s e va l ues a re 1 0 a n d 1 5 , res p ect i ve l y , t h e n 1 0 l i n es o f i np u t

a re c o r re s p on d i n g l y expected for t owe r 1 fo l l owed by 1 5 l i nes o f i n put

f o r towe r 2 .

F o r the test cas e , o n l y one towe r i s p re s e nt ; t h u s , o n l y one va l ue

of NL , n ame l y 4 , i s rea d , i n d i c at i n g t h at fou r l i nes of i n put a re to b e

g i ven for the t owe r . A c l o s e exami n at i o n of the i np u t s hows t h a t two

c o n s ecut i ve l i nes a re g i ven for a t i me of 25 hou rs a fter st a rt of t h e

a c c i de nt . Th i s i s done t o cor rect l y rep res ent t he s t e p cha n ge i n t h e

f l ow rate a t t h at t i me as F i g u re 4 . 1 5 i l l u st rates .

Sect i on 4 p rov i des the t i me- dependent acc i dent heat l oa d s for each

t owe r i n a man n e r pa ra l l e l i n g that u sed to s pec i fy t ower p e r f o rma nce

dat a i n Sect i o n 3 . F o r t h e te st c a s e , 29 ent r i e s a re ma de a s s h ow n i n

F i g u re 4 . 1 4 .

Sect i on 5 p ro v i des the va l u e s of the con s t a n t wet - bu l b temp e r at u re

a nd i n i t i a l ba s i n tempe rat u re u s ed i n t he t i me c o n s t a nt det e rmi n at i o n .

7 8

F i g u re 4 . 1 5 S t e p Cha n g e i n L , Ka V/ L , a n d L/ G V a l u e s .

L / G .

L

K a V / L

T I M E ( h r )

T I M E ( h r )

B o t h v a l u e s e n t e r e d o n c o n s e c u t i v e l i n e s o f I n p u t

7 9

S i nce Sect ; on 6 i s req u i red o n l y for the f i n a l s i mu l at i on u s i n g t he

met e o ro l o g i ca l data w i n dows e xt racted by the data s c a n n i n g p rocedu re ,

d i s c u s s i o n of i t wi l l be defe r red . F o r the test ca s e , t he wet -bu l b tem

p e r at u re and i n i t i a l b a s i n t empe rat u re were both s et to 81 ° F .

The a ct u a l t i me con s t a nt i s n ow ca l c u l ated u s i n g p ro g ram TMCON .

P ro g ram UHSS I M c reates an out put f i l e wh i ch i s d i rect l y i n put to p ro g ram

TMCON as we l l as an i nt e rp ret e d output f i l e fo r p r i n t i n g a s s hown i n

T a b l e 4 . 6 . The t i me c o n s t ant i s det e rm i ned by mi n i m i z i ng t h e s um of

s qu a res S g i ven by

( 4 . 1 )

whe re T i i s t h e ba s i n temp e r at u re at step lI i l l p re d i cted by UHSS IM a nd T i i s t he corre s p o n d i n g va l ue det e rmi ned a s s umi n g f i r st -o rde r system re

s p o n s e . The v a l ues of T i a re g i ven by t he recu r s i ve re l at i o n s h i p

w h e re e i s At /T: , Twb , i i s t h e wet - bu l b t empe ratu re at step , " i " , q i i s

t h e heat l oa d at step " i " , L i i s t h e fl ow rate at step " i ll , mi i s the

b a s i n ma ss a t s tep lI i ll , e: i i s ( T i n , i - Tout , i ) / ( T i n , i - Twb , i ) ' T i n , i i s

t h e towe r ; n l et temp e ratu re at step 1 1 ; 1 1, Tout , i i s t he t owe r exi t temp

e rat u re at step lI i l l , a nd cL i s t h e spec i f i c heat of wat e r .

The t i me con s t a nt va l u e fou n d by p rog ram TMCON s h ou l d b e eva l u at e d

a s t o whether i t a p p e a rs rea s on ab l e f o r t h e sys t em u n de r c on s i de r a

t i on . Typ i c a l va l u e s l i e i n t he range of seve r a l hou r s t o s e v e r a l t e n s

of hou rs . For the t e s t c a s e , the t i me c o n s t a nt wa s found t o be 1 1 . 4 6

8 0

Ta b l e 4 . 6 Ou tpu t L i s t i n g from P rog ram UH S S I M fo r Co n s ta n t Wet - Bu l b Tempera t u re .

T I M E TBAS I N BMAS S S TWB Q L ( H R ) ( F ) ( LB M ) ( P PT ) ( F ) ( BTU/ H R ) ( LB M / H R ) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 . 0 8 1 . 0 0 5 . 4 2 E + 0 7 5 . 0 8 1 . 0 6 . 6 2 E + 0 7 6 . 0 7 E + 0 6 1 . 0 8 1 . 8 0 5 . 4 1 E + 0 7 5 . 0 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 2 . 0 8 2 . 6 3 5 . 3 9 E + 0 7 5 . 0 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 3 . 0 8 3 . 3 9 5 . 3 8 E + 0 7 5 . 0 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 4 . 0 8 4 . 0 9 5 . 3 7 E + 0 7 5 . 1 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 5 . 0 8 4 . 7 2 5 . 3 5 E + 0 7 5 . 1 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 6 . 0 8 5 . 3 0 5 . 3 4 E + 0 7 5 . 1 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 7 . 0 8 5 . 8 3 5 . 3 2 E + 0 7 5 . 1 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 8 . 0 8 6 . 3 1 5 . 3 0 E + 0 7 5 . 1 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 9 . 0 8 6 . 7 5 5 . 2 9 E + 0 7 5 . 1 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6

1 0 . 0 8 7 . 1 5 5 . 2 7 E + 0 7 5 . 1 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 1 1 . 0 8 7 . 5 1 5 . 2 5 E + 0 7 5 . 2 8 1 . 0 1 . 9 2 E + 0 8 6 . 0 7 E + 0 6 1 2 . 0 8 7 . 8 4 5 . 2 4 E + 0 7 5 . 2 8 1 . 0 1 . 9 0 E + 0 8 6 . 0 7 E + 0 6 1 3 . 0 8 8 . 1 3 5 . 2 2 E + 0 7 5 . 2 8 1 . 0 1 . 8 8 E + 0 8 6 . 0 7 E + 0 6 1 4 . 0 8 8 . 3 9 5 . 2 0 E + 0 7 5 . 2 8 1 . 0 1 . 8 6 E + 0 8 6 . 0 7 E + 0 6 1 5 . 0 8 8 . 6 3 5 . 1 9 E + 0 7 5 . 2 8 1 . 0 1 . 8 4 E + 0 8 6 . 0 7 £ + 0 6 1 6 . 0 8 8 . 8 4 5 . 1 7 E + 0 7 5 . 2 8 1 . 0 1 . 8 2 E + 0 8 6 . 0 7 E + 0 6 1 7 . 0 8 9 . 0 2 5 . 1 6 E + 0 7 5 . 3 8 1 . 0 1 . 8 1 E + 0 8 6 . 0 7 E + 0 6 1 8 . 0 8 9 . 1 8 5 . 1 4 E + 0 7 5 . 3 8 1 . 0 1 . 8 0 E + 0 8 6 . 0 7 E + 0 6 1 9 . 0 8 9 . 3 3 5 . 1 2 E + 0 7 5 . 3 8 1 . 0 1 . 7 9 E + 0 8 6 . 0 7 E + 0 6 2 0 . 0 8 9 . 4 5 5 . 1 1 E + 0 7 5 . 3 8 1 . 0 1 . 7 8 E + 0 8 6 . 0 7 E + 0 6 2 1 . 0 8 9 . 5 5 5 . 0 9 E + 0 7 5 . 3 8 1 . 0 1 . 6 6 E + 0 8 6 . 0 7 E + 0 6 2 2 . 0 8 9. 6 2 . 5 . 0 8 E + 0 7 5 . 3 8 1 . 0 1 . 5 5 E + 0 8 6 . 0 7 E + 0 6 2 3 . 0 8 9 . 6 5 5 . 0 6 E + 0 7 5 . 4 8 1 . 0 1 . 4 3 E + 0 8 6 . 0 7 E + 0 6

,---� ....

2 4 . 0 8 9 . 6 4 5 . 0 5 E + 0 7 5 . 4 8 1 . 0 1 . 3 1 E + 0 8 6 . 0 7 E + 0 6 2 5 . 0 8 9 . 6 0 5 . 0 4 E + 0 7 5 . 4 8 1 . 0 1 . 3 0 E + 0 8 5 . 8 0 E + 0 6 2 6 . 0 8 9 . 5 5 5 . 0 2 E + 0 7 5 . 4 8 1 . 0 1 . 2 8 E + 0 8 5 . 8 0 E + 0 6

2 7 . 0 8 9 . 4 9 5 . 0 1 E + 0 7 5 . 4 8 1 . 0 1 . 2 7 E + 0 8 5 . 8 0 E + 0 6 2 8 . 0 8 9 . 4 3 5 . 0 0 E + 0 7 5 . 4 8 1 . 0 1 . 2 5 E + 0 8 5 . 8 0 E + 0 6 2 9 . 0 8 9 . 3 8 4 . 9 9 E + 0 7 5 . 4 8 1 . 0 1 . 2 4 E + 0 8 5 . 8 0 E + 0 6 3 0 . 0 8 9 . 3 2 4 . 9 7 E + 0 7 5 . 4 8 1 . 0 1 . 2 2 E + 0 8 5 . 8 0 E + 0 6 3 1 . 0 8 9 . 2 7 4 . 9 6 E + 0 7 5 . 5 8 1 . 0 1 . 2 1 E + 0 8 5 . 8 0 E + 0 6 3 2 . 0 8 9 . 2 1 4 . 9 5 E + 0 7 5 . 5 8 1 . 0 1 . 1 9 E + 0 8 5 . 8 0 E + 0 6 3 3 . 0 8 9 . 1 6 4 . 9 4 E + 0 7 5 . 5 8 1 . 0 1 . 1 8 E + 0 8 5 . 8 0 E + 0 6 3 4 . 0 8 9 . 1 0 4 . 9 3 E + 0 7 5 . 5 8 1 . 0 1 . 1 6 E + 0 8 5 . 8 0 E + 0 6 3 5 . 0 8 9 . 0 4 4 . 9 2 E + 0 7 5 . 5 8 1 . 0 1 . 1 5 E + 0 8 5 . 8 0 E + 0 6 3 6 . 0 8 8 . 9 9 4 . 9 1 E + 0 7 5 . 5 8 1 . 0 1 . 1 3 E + 0 8 5 . 8 0 E + 0 6

3 7 . 0 8 8 . 9 3 4 . 8 9 E + 0 7 5 . 5 8 1 . 0 1 . 1 2 E + 0 8 5 . 8 0 E + 0 6

3 8 . 0 8 8 . 8 7 4 . 8 8 E + 0 7 5 . 5 8 1 . 0 1 . 1 0 E + 0 8 5 . 8 0 E + 0 6

3 9 . 0 8 8 . 8 1 4 . 8 7 E + 0 7 5 . 6 8 1 . 0 1 . 0 9 E + 0 8 5 . 8 0 E + 0 6

4 0 . 0 8 8 . 7 6 4 . 8 6 E + 0 7 5 . 6 8 1 . 0 1 . 0 7 E + 0 8 5 . 8 0 E + 0 6

4 1 . 0 8 8 . 7 0 4 . 8 5 E + 0 7 5 . 6 8 1 . 0 1 . 0 6 E + 0 8 5 . 8 0 E + 0 6

4 2 . 0 8 8 . 6 4 4 . 8 4 E + 0 7 5 . 6 8 1 . 0 1 . 0 4 E + 0 8 5 . 8 0 E + 0 6

4 3 . 0 8 8 . 5 8 4 . 8 3 E + 0 7 5 . 6 8 1 . 0 · 1 . 0 3 E + 0 8 5 . 8 0 E + 0 6

4 4 . 0 8 8 . 5 2 4 . 8 2 E + 0 7 5 . 6 8 1 . 0 1 . 0 1 E + 0 8 5 . 8 0 E + 0 6

4 5 . 0 8 8 . 4 7 4 . 8 1 E + 0 7 5 . 6 8 1 . 0 9 . 9 9 E + 0 7 5 . 8 0 E + 0 6

4 6 . 0 8 8 . 4 2 4 . 8 0 E + 0 7 5 . 6 8 1 . 0 9 . 8 4 E + 0 7 5 . 8 0 E + 0 6

4 7 . 0 8 8 . 3 6 4 . 7 9 E + 0 7 5 . 7 8 1 . 0 9 . 6 9 E + 0 7 5 . 8 0 E + 0 6

4 8 . 0 8 8 . 3 0 4 . 7 8 E + 0 7 5 . 7 8 1 . 0 9 . 5 4 E + 0 7 5 . 8 0 1:: + 0 6

8 1

h ou rs . A l s o , t h e effect s o f i n i t i a l ba s i n temp e ratu re a nd nomi n a l am

b i ent wet -bu l b t empe ratu re c a n be s t u d i ed by ru n n i ng severa l c a s es w i t h

i n t h e rea s o n a b l e range .

F i gu re 4 . 1 6 p ro v i de s a comp a r i s o n of the temp e ratu re h i s t o r i e s p re

d i cted by the fu l l t ower/ bas i n s i mu l at i on ( UHSS I M ) a nd the s i mp l i f i e d

mode l . As may be rea di l y obs e rved , t he app rox i mat i on as a f i r s t o rde r

sys t em i s a good one .

4 . 4 P rog ram NWS -- Dat a Sca n n i n g P rocedu re

The fu l l p e r i od of met eoro l o g i ca l data a v a i l a b l e from the N at i o n a l

Weat h e r Se rv i ce for t h e nea rest rep o rt i n g stat i on i s ana l yzed u s i n g p ro

g ram NWS to o b t a i n the wo r s t -ca se met e o ro l o g i ca l c o nd i t i o n s for t h e

a rea . The i np u t " va r i a b l es for p ro g ram N W S a re s et u s i n g pa ramet e r

statements w i t h i n the code . The v a r i a b l e n ames , des c r i pt i on s , a n d u n i t s

a re g i ven i n Ta b l e 4 . 7 .

P rog ram NWS a l l ows the w i n dow s i ze , t h e numbe r of wi n dows t o be

s a ved s i mu l t a ne ou s l y , and t he numb e r of reco rds to be read i n e ach r u n

t o be se l ected by t h e u s e r . T h e term "w i n dow " refe rs to a s egme n t of

consecut i ve rec o r ds f rom the met eoro l o g i c a l data b a s e . I n the dat a

s c a n n i n g p rocedu re , the wi n dow i s moved c h rono l o g i c a l l y t h ro u g h t h e dat a

b a s e by a dd i n g t he next reco r d f rom the met eoro l o g i ca l data b a s e to t h e

b ot t om of t h e wi n dow a n d remo v i n g t h e o l des t rec o r d i n t h e wi n dow f rom

t he top of t he w i n dow . P rog r am NWS i s s et u p to s a ve the w i n dows wh i ch

h a ve the h i g h e s t ru n n i n g wet -b u l b temp e r a t u res . Rat h e r t h a n s a v i n g j u st

t he s i n g l e w i n dow w i th t he h i ghest va l ue seen t hu s fa r i n s c a n n i ng t h e

dat a b a se , p rog ram NWS a l l ows t h e u s e r t o s pec i fy t h e numb e r o f w i n dows

r--..

LL o "-"

CD

I I 1 00 I

S Y S T E M R ES PO N S E -- S i m u l a t i o n B a s i n T e m pe ra t u re - - - S i m p l i f i e d M od e l B a s i n Te m pe r a t u re

L 90 .... .:J

of-/" a

yo - - - - - j ? - -....; - - - - - - - - === /'"

/'

� V 0.. (/ E (l)

l-c. (f) o

CO

W a t e r f l ow R a t e, L = 7 3 . 6 4 % a f t e r 2 4 h o u rs. L = 7 0 . 3 6 % C o n st a n t T W B = 8 1 .0 e f I n i t i a l T BASI N = 8 1 . 0 e f

T = 1 1 . 4 6 h r

7 0 I I I I , o 20 4 0 60 80

F i g u re 4 . 1 6

T i m e ( h r ) Compa r i s o n o f Tempe ra t u re H i s t o r i e s P red i c t e d by Fu l l Towe r/ B a s i n S i mu l a t i o n ( UHSS I M ) a n d S i mp l i f i ed Mo de l fo r Con s t a n t We t - B u l b Tempe ra t u re .

-

1 00

00 N

\\ ,� 0 �

'3 )'Oc:'\\ 1-C;O

\0\

-7 � V ·

8 3

Tab l e 4 . 7 De s c r i pt i on of I n put V a r i a b l e s fo r P r o g r am NWS

VAR I ARL E NAt1E

H E AD E R

TAi l

DELTAT

NRE C S

L E NGTH ([)') d�) I P E AK

N SAVE

DE SCR I PT I ON AND U N I TS

40- C o l umn Heade r U s e d t o I de n t i fy Ou t p ut F i l e

T i me C o n st a nt of Tower Sy s t em ( h r )

T i me Between Succes s i ve Reco rds on Met e o r o l og i ca l D ata Tape ( h r )

Numbe r of Reco rds t o be P roces s e d i n Ru n

Numb e r of Data Records i n E a c h W i n dow

Des i re d Locat i o n of Peak Ru n n i n g Wet - B u l b Temp e r at u re w i th i n W i n dow

Numbe r of D i s t i n ct W i n dows t o b e S a v e d

8 4

t o be s a v e d at a ny t i me . S a v i n g mo re t h a n j u st the s i n g l e h i g h e s t wi n

dow c a n p re vent the mi s l ead i n g re s u l t s wh i ch may b e cau sed by sect i on s

o f t h e dat a b a s e wi th h i gh f ract i on s o f mi s s i n g data .

I n a d d i t i o n , p rog ram NWS i s s et u p t o be executed i n s e ve ra l ru n s

rathe r t h a n j u st a s i n g l e ru n . T h i s dev i ce serves two pu rpos e s . F i r st ,

by a n a l y z i n g the data base i n c h u n k s , a l ower i n vestment i n comp u t at i on

a l res ou rces i s made i n each ru n . Thu s , i f a u s e r - p roduced o r comp u t e r

rel ated p rob l em s h ou l d devel op , on l y a s ma l l amo u nt of comput e r re sou r

c e s w i l l h a v e been wa sted . The n umbe r of records may b e se l e c t e d for

u s e r c o n ven i en c e , a l t hough 3 5 , 064 i s a p a rt i c u l a r l y att ract i ve c h o i ce

s i nce i t co r re spond s to exact l y 4 yea r s of data ( i n c l u d i n g l e a p yea r ) .

The second p u rpose i n a l l owi n g s ma l l er c h u n k s of data to be a n a l yzed i n

s epa rate comp u t e r ru n s stems f rom t h e opp o rt u n i ty affo rded t he u s e r t o

revi ew t he s a ved w i n dows aft e r e a c h run a n d t hu s bett e r t rack t h e dat a

s c a n n i n g p rocedu re .

F o r t h e test ca s e , t h i rty- fou r - a n d - a - h a l f yea r s of meteo ro l o g i c a l

data a re a s s u med to b e ava i l ab l e co ve r i n g t he p e r i od � rom 06 : 00 hou rs on

J u l y 1 , 1948 , t h rough 2 3 : 00 hou r s o n Decemb e r 3 1 , 1982 . The met e o ro l o g

i cal data b a s e conta i ns h ou r l y reco rds except f o r a few st ret c h e s of

mi s s i n g data a nd a pe r i od of seve ra l yea r s for wh i ch o n l y t r i - h ou r l y ob

se rvat i on s we re made . T hese data w e re s c a n ned u s i n g the t i me c o n s t a nt

o f 1 1 . 46 h o u r s di s c u s sed ea r l i e r i n 4-yea r ( 3 5 , 064 reco rds ) b l oc k s . T he

number of w i n dows to be sa ved c o n c u r re nt l y was s et at 3 and t h e w i n dow

l en gt h was set at 200 a n d t he t i me of max i mum ru n n i n g wet -b u l b t emp e r a

t u re w a s s et to b e reco rd 1 0 1 .

8 5

The p e r i ods o f wo rst - c a s e meteo ro l o g i c a l' cond i t i on s i de nt i f i ed by

o u r p rocedu re a re , i n o rde r of dec reas i n g s e ve r i ty :

a ) 1 4 : 00 hou rs on Au g u s t 1 3 , 1981 , t h rou gh 2 1 : 00 hou rs on Au gu st

2 1 , wh e re i n t he ru n n i n g wet-bu l b t emp e rat u re reached a ma x i mum

o f 81 . 2 7 ° F at 18 : 00 hou r s on Au g u s t 1 7 , 1 98 1 .

b ) 1 9 : 00 h o u rs on Au g u st 6 , 1 969 , t h rough 2 : 00 hou rs on Au g u st 1 5 ,

1 969 , where i n t he ru n n i n g wet -bu l b temp e ratu re reached a ma x i

mum of 80 . 70°F at 2 3 : 00 h ou rs on Au g u s t 1 0 , 1 969 .

c ) 1 3 : 00 hou rs on J u ne 2 3 , 1 980 , t h rou gh 20 : 00 hou rs on J u l y 1 ,

1 980 , where i n t he ru n n i n g wet - b u l b t emp e rat u re reached a ma x i

mum of 80 . 68 ° F at 1 7 : 00 h o u r s on J u ne 2 7 , 1 980 .

F i gu re 4 . 1 7 g i ves an a b b re v i ated l i s t i ng of a f i l e cont a i n i n g t h e

met eo ro l o g i ca l dat a w i n dows . Th e heade r i n fo rmat i on g i ves t h e a l p h a

nume r i c i de nt i f i e r f o r t h e c a s e ( TEST DATA 0 7 3084 ) , t h e v a l ues o f t h e

t i me con s tant ( 1 1 . 46 h ou rs ) a n d t i me i nc rement ( 1 h ou r ) , t he numb e r o f

rec o r ds p roce s s ed i n t h i s ru n ( 3 5 , 064 ) , t h e wi n dow l en gth ( 200 ) , t h e

ma x i mum number o f wi n dows t o b e s a ved ( 3 ) , t h e act u a l numbe r o f wi n dows

s a ved thus far ( 3 ) , t he n umbe r of w i ndow s a ve req u e s t s p roce s s ed ( 24 ) ,

a n d t he t ot a l numbe r o f reco rds s c a nned i n a l l ru n s ma de t h u s f a r

( 302 , 41 8 ) .

The heade r i n fo rmat i on i s fol l owed by t h e t h ree wi n dows . Th e f i r s t

l i ne of each wi n dow g i ves t h e max i mum ru n n i n g wet - bu l b temp e ratu re f o r

t h e w i n dow ( e . g . , 81 . 2 7 0 1 7 0 7 203 1 ) a n d t h e pos i t i on numb e r w i t h i n t h e

wi ndow a t wh i ch t he max i mum occu rs ( e . g . , 1 0 1 ) . The data f o r e a c h w i n

dow a re g i ven i n t h ree co l umn s co r res p on d i n g t o ( i ) t h e yea r , mon t h ,

8 6

F i g u re 4 . 1 7 O u t p u t L i s t i n g f rom P �o g ra m NWS - Three W i n dow Fo rma t .

" < �:.\

. . :J ' Ify "

\� '<J, , . ')/. \ I..

TEST DATA 0 7 3 0 8 4 1 1 . 4 6 1 . 3 5 0 6 4 2 0 0 3 2 4 3 0 2 4 1 8 8 1 . 2 7 0 1 7 0 7 2 0 3 1

3 �;....... /' \l �-// \" . ',M .l,) \t 1 0 1 / / '" \",!\, i � « 8 1 0 8 1 3 1 4

� 0 1 0 8 1 3 1 5 8 1 0 8 1 3 1 6

�1 ·

8 1 0 8 1 7 1 7 'i·- 8 1 0 8 1 7 1 8

8 1 0 8 1 7 1 9

8 1 0 8 2 1 1 9 8 1 0 8 2 1 2 0 8 1 0 8 2 1 2 1

8 0 . 6 9 9 2 4 2 3 1 8 1 2 ( 6 9 0 8 0 6 1 9

6 9 0 8 0 6 2 0 6 9 0 8 0 6 2 1

, 6 9 0 8 1 0 2 2 � 9 0 8 1 0 2 3

6 9 0 8 1 1 0 0

6 9 0 8 1 5 0 0 6 9 0 8 1 5 0 1 6 9 0 8 1 5 0 2

8 0 . 6 7 5 3 8 0 2 9 2 9 7 " 8 0 0 6 2 3 1 3 \ a 0 0 6 2 3 1 4

8 0 0 6 2 3 1 5 .

l a O O: 2 7 1 6 8 0 0 6 2 7 1 7 8 0 0 6 2 7 1 8

8 0 0 7 0 1 1 8 8 0 0 7 0 1 1 9 8 0 0 7 0 1 2 0

7 6 . 8 1 7 9 . 0 0 7 7 . 0 5 8 0 . 0 0 7 7 . 1 9 7 9 . 0 0

•

8 1 . 2 1 8 1 . 2 7 8 1 . 2 5

7 4 . 3 1 7 4 . 2 1 7 4 . 1 2

1 0 1 7 4 . 6 7 7 4 . 6 2 7 4 . 5 0

8 0 . 6 8 8 0 . 7 0 8 0 . 6 5

•

7 4 . 8 7 7 4 . 7 3 7 4 . 6 1

1 0 1 7 3 . 5 5 7 3 . 5 8 7 3 . 8 3

8 0 . 4 9 8 0 . 6 8 8 0 . 6 3

7 8 . 5 8 7 8 . 6 1 7 8 . 6 4

8 2 . 0 0 8 2 . 0 0 8 1 . 0 0

7 4 . 0 0 7 3 . 0 0 7 3 . 0 0

9 9 9 .. 0 0 9 9 9 . 0 0

7 3 . 0 0

9 9 9 . 0 0 9 9 9 . 0 0

8 0 . 0 0

7 3 . 0 0 9 9 9 . 0 0 9 9 9 . 0 0

9 9 9 . 0 0 9 9 9 . 0 0

7 7 . 0 0

•

9 9 9 . 0 0 9 9 9 . 0 0

8 0 . 0 0

•

7 9 . 0 0 9 9 9 . 0 0 9 9 9 . 0 0

8 7

day , a n d hou r ( Y YMMDDh h format ) , ( i i ) t he ru n n i n g wet - b u l b t emp e ratu re ,

a n d ( i i i ) the actua l wet -b u l b t emp e ratu re for t hat hou r . T he v a l ue of

" 999 . 0 " i n the wet - bu l b temp e ratu re col umn i nd i cates a hou r for wh i c h no

o b s e rvat i on i s a v a i l ah l e . As can be obs e rved f rom F i g u re 4 . 1 7 , b oth t he

s ec o n d a n d t h i rd w i ndows fa l l i n per i ods du r i n g wh i c h o n l y t r i -h o u rl y

ob se rvat i on s a re a v a i l ab l e .

4 . 5 Nume r i ca l S i mu l at i on of UHS Cool i n g Tow e r Performa nce

The f i na l step i n the UHS coo l i ng t ower p e rfo rma n ce a n a l ys i s i s the

a c c i dent s i mu l at i on . Th i s s tep i s car r i ed out u s i n g p rog ram U H S S I M

wh i ch h a s a l rea dy been d i s c u s sed i n Sect i on 4 . 3 i n t he context o f the

t i me - c o n s t ant eva l u at i on . For t h a t pu r p o s e , t h e p ro g r am wa s ru n wi t h a

u s e r s p ec i f i ed i n i t i a l ba s i n t emp e rat u re a n d a u s e r s p e c i f i ed c o n s t a nt

wet - b u l b temp e rat u re . I n cont r a s t , the t i me - v a ry i n g wet - b u l b t empera

t u re w i n dows s e l ected by t he data s c a nn i n g p rocedu re a re u s ed f o r t he

a ct u a l acc i dent s i mu l at i on .

F o r a noncon s t a n t wet - b u l b t empe ratu re ca s e , t h e u s e r mu s t set the

cont rol v a r i a b l e CONWB fa l se a n d add Sect i on 6 to the e nd o f t h e i n put

dat a f i l e whe re i n the w i n dow numb e r a n d st a rt i n g t i me of t h e acc i dent

a re s p ec i f i ed . Norma l l y w i n dow n umbe r 1 , i . e . , t he w i ndow w i th t h e

g reatest ru n n i n g wet -bu l b t emp e r a t u re , i s u sed a n d t h e st a rt i n g t i me i s

adj u sted so t h at t he peak ru n n i n g wet - b u l b t emp e r at u re a n d t h e peak a c

c i dent-heat - l oad i nc rement co i n c i de . Howeve r , t h e f reedom to exami ne

a l t e rn at i ve w i n dows and t o a l t e r the s t a rt i n g t i me of t h e a c c i dent a l l ow

the s e n s i t i v i ty of p red i ct i on s t o be s tud i e d . A l ow sen s i t i v i ty to t h e

ma n n e r i n wh i ch t he a n a l y s i s i s c a r r i ed o u t e n h a n ces t he s i g n i f i c a nce o f

t h e e nd res u l t t h u s obt a i n e d .

8 8

F o r t h e test ca s e , a s i mu l at i on wa s ma de f o r 1 0 0 hou r s a f t e r s ta rt

o f the a c c i dent w i th resu l t s p r i nted at the end of each hou r . The i n p u t

f i l e i s t h e s ame a s t h a t s hown i n F i gu re 4 . 1 4 except that t h e cont r o l

v a r i a b l e CONWB i s s et fa l se a nd t h e w i n dow n umber ( NW I NDW ) a n d s t a rt i n g

t i me of t he acc i dent wi t h i n t h e wi n dow ( NSTART ) a re a dde d as t h e l a s t

l i ne of t he f i l e . NW I NDW i s i n i t i a l l y s et t o 1 to fo rce u s e of t h e p r i

ma ry wi n dow obt a i n e d by the dat a s c a n n i n g p rocedu re . NSTART i s i n i t i a l

l y s et to 78 . T h i s v a l ue i s t he d i f ference between the l ocat i on of t h e

ma x i mum ru n n i n g wet - b u l b t emp e rat u re wi t h i n the wi n dow ( 1 0 1 ) a n d t h e

l oc at i on of the max i mum heat - l oad i nc rement ( 2 3 ) det e rmi ned f rom t h e

res u l t s of the con s t a n t -wet-bu 1 b -t empe r at u re s i mu l at i on p re s e n t e d i n

F i gu re 4 . 1 7 a n d Tab l e 4 .6 . F o r t h i s c ho i c e , t he ma x i mum ba s i n t emp e ra

tu re does i n deed rea c h a max i mum of 89 . 90 o f , 2 3 hou rs a ft e r i n i t i at i o n

of the a c c i dent as s h own i n T a b l e 4 . 8 .

4 . 6 Sen s i t i v i ty St u dy o f Met eoro l og i c a l D ata Sca n n i n g P rocedu re

F i gu re 4 . 1 8 i l l u st rates the s e n s i v i ty o f the ma x i mum ba s i n t emp e ra

t u re p red i cted by p ro g ram UHSS I M t o t h e st a rt i n g t i me of t he a c c i dent .

He re , ma x i mum p red i cted ba s i n temp e ratu re i s p l otted v e r s u s the t i me

d i s p l acement between the sta rt of t he acc i dent a nd the l ocat i o n of t h e

max i mum ru n n i n g wet - h u l b t emp e ratu re w i t h i n t h e p ri ma ry data w i n dow .

Th u s , a n off s et of 0 hou rs mea n s t hat the acc i dent was i n i t i at e d at h ou r

1 0 1 , i . e . , the t i me of max i mum ru n n i ng wet - bu l b t empe ratu re w i t h i n t h e

w i ndow . S i mi l a r l y , a n offset of 6 0 hou rs mea n s t h a t the acc i dent wa s

s t a rted at hou r 4 1 , s i xty hou rs b e fore t he t i me of ma x i mum ru n n i n g wet

b u l b temp e rat u re . Not ab l y , each hou r on t h i s p l ot rep res e nt s a s e p a r a t e

r u n of p rog ram l I HSS I M .

8 9 Ta b l e 4 . 8 Va r i a t i o n o f I mpo r t a n t Me t e o r o l o g i c a l , Hea t L o a d , a n d Ba s i n

Va r i a b l e s w i t h T i me Du r i n g LOCA .

T I M E TBAS I N BMAS S S TWB Q L ( H R ) ( F ) ( LB M ) ( PPT ) ( F ) ( BTU/H R ) ( LBM/H R ) - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - -0 . 0 8 1 . 0 4 5 . 4 2 E + 0 7 5 . 0 8 1 . 0 6 . 6 2 E + 0 7 6 . 0 7 E + 0 6 1 . 0 8 1 . 8 3 5 . 4 1 E + 0 7 5 . 0 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 2 . 0 8 2 . 6 6 5 . 3 9 E + 0 7 5 . 0 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 3 . 0 8 3 . 4 0 5 . 3 8 E + 0 7 5 . 0 8 0 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 4 . 0 8 4 . 0 4 5 . 3 7 E + 0 7 5 . 1 8 0 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 5 . 0 8 4 . 6 2 5 . 3 5 E + 0 7 5 . 1 8 0 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 6 . 0 8 5 . 1 5 5 . 3 3 E + 0 7 5 . 1 8 0 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 7 . 0 8 5 . 6 0 5 . 3 2 E + 0 7 5 . 1 7 9 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 8 . 0 8 6 . 0 0 5 . 3 0 E + 0 7 5 . 1 7 9 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 9 . 0 8 6 . 3 6 5 . 2 9 E + 0 7 5 . 1 7 9 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6

1 0 . 0 8 6 . 6 8 5 . 2 7 E + 0 7 5 . 1 7 9 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 1 1 . 0 8 6 . 9 7 5 . 2 5 E + 0 7 5 . 2 7 8 . 0 1 . 9 2 E + 0 8 6 . 0 7 E + 0 6 1 2 . 0 8 7 . 2 3 5 . 2 3 E + 0 7 5 . 2 8 0 . 0 1 . 9 0 E + 0 8 6 . 0 7 E + 0 6 1 3 . 0 8 7 . 5 3 5 . 2 2 E + 0 7 5 . 2 8 0 . 0 1 . 8 8 E + 0 8 6 . 0 7 E + 0 6 1 4 . 0 8 7 . 7 9 5 . 2 0 E + 0 7 5 . 2 8 0 . 0 1 . 8 6 E + 0 8 6 . 0 7 E + 0 6 1 5 . 0 8 8 . 0 7 5 . 1 9 E + 0 7 5 . 2 8 2 . 0 1 . 8 4 E + 0 8 6 . 0 7 E + 0 6 1 6 . 0 8 8 . 4 0 5 . 1 7 E + 0 7 5 . 2 8 2 . 0 1 . 8 2 E + 0 8 6 . 0 7 E + 0 6 1 7 . 0 8 8 . 6 7 5 . 1 5 E + 0 7 5 . 3 8 2 . 0 1 . 8 1 E + 0 8 6 . 0 7 E + 0 6

. 1 8 . 0 8 8 . 9 6 5 . 1 4 E + 0 7 5 . 3 8 3 . 0 1 . B O E + 0 8 6 . 0 7 E + 0 6 1 9 . 0 8 9 . 2 5 5 . 1 2 E + 0 7 5 . 3 .8 3 . 0 1 . 7 9 E + 0 8 6 . 0 7 E + 0 6 2 0 . 0 8 9 . 5 0 5 . 1 1 E + 0 7 5 . 3 8 3 . 0 1 . 7 8 E + 0 8 6 . 0 7 E + 0 6 2 1 . 0 8 9 . 7 1 5 . 0 9 E + 0 7 5 . 3 8 3 . 0 1 . 6 6 E + 0 8 6 . 0 7 E + 0 6 2 2 . 0 8 9 . 8 5 5 . 0 8 E + 0 7 5 . 3 8 2 . 0 1 . 5 5 E + 0 8 6 . 0 7 E + 0 6 2 3 . 0 8 9 . 9 0 5 . 0 6 E + 0 7 5 . 4 8 2 . 0 1 . 4 3 E + 0 8 6 . 0 7 E + 0 6 2 4 . 0 8 9 . 8 9 5 . 0 5 E + 0 7 5 . 4 8 1 . 0 1 . 3 1 E + 0 8 6 . 0 7 E + 0 6 2 5 . 0 8 9 . 8 4 5 . 0 4 E + 0 7 5 . 4 8 1 . 0 1 . 3 0 E + O B 5 . 8 0 E + 0 6 2 6 . 0 8 9 . 7 4 5 . 0 2 E + 0 7 5 . 4 8 0 . 0 . 1 . 2 8 E + 0 8 5 . B O E + 0 6 2 7 . 0 8 9 . 5 5 5 . 0 1 E + 0 7 5 . 4 7 8 . 0 1 . 2 7 E + 0 8 5 . 8 0 E + 0 6 2 8 . 0 8 9 . 2 6 5 . 0 0 E + 0 7 5 . 4 7 6 . 0 1 . 2 5 E + 0 8 5 . 8 0 E + 0 6 2 9 . 0 8 8 . 9 2 4 . 9 8 E + 0 7 5 . 4 7 6 . 0 1 . 2 4 E + 0 8 5 . B O E + 0 6 3 0 . 0 8 8 . 6 5 4 . 9 7 E + 0 7 5 . 5 7 7 . 0 1 . 2 .2 E + 0 8 5 . 8 0 E + 0 6 3 1 . 0 8 8 . 4 0 4 . 9 6 E + 0 7 5 . 5 7 6 . 0 1 . 2 1 E + 0 8 5 . 8 0 E + 0 6

3 2 . 0 8 8 . 1 2 4 . 9 5 E + 0 7 5 . 5 7 6 . 0 1 . 1 9 E + 0 8 5 . B O E + 0 6

3 3 . 0 B 7 . 8 8 4 . 9 3 E + 07 5 . 5 7 6 . 0 1 . 1 8 E + O B 5 . B O E + 0 6

3 4 . 0 8 7 . 6 5 4 . 9 2 E + 0 7 5 . 5 7 6 . 0 1 . 1 6 E + O B 5 . B O E + 0 6

3 5 . 0 B 7 . 4 7 4 . 9 1 E + 0 7 5 . 5 7 8 . 0 1 . 1 5 E + O B 5 . B O E + 0 6

3 6 . 0 B 7 . 4 2 4 . 9 0 E + 0 7 5 . 5 7 9 . 0 1 . 1 3 E + O B 5 . B O E + 0 6

3 7 . 0 B 7 . 3 8 4 . 8 9 E + 0 7 5 . 5 7 9 . 0 1 . 1 2 E + O B 5 . 8 0 E + 0 6

3 8 . 0 B 7 . 4 0 4 . 8 B E + 0 7 5 . 6 8 1 . 0 1 . 1 0 E + O B 5 . B O E + 0 6

3 9 . 0 B 7 . 5 1 4 . B 7 E + 0 7 5 . 6 8 2 . 0 1 . 0 9 E + 0 8 5 . B O E + 0 6

4 0 . 0 B 7 . 6 4 4 . B 6,E + 0 7 5 . 6 8 2 . 0 1 . 0 7 E + O B 5 . B O E + 0 6

4 1 . 0 B 7 . 7 8 4 . 8 5 E + 0 7 5 . 6 B 3 . 0 1 . 0 6 E + O B 5 . B O E + 0 6

4 2 . 0 8 7 . B 7 4 . B 4 E + 0 7 5 . 6 B 1 . 0 1 . 0 4 E + O B 5 . 8 0 E + 0 6

4 3 . 0 8 7 . 9 2 4 . B 3 E + 0 7 5 . 6 B 2 . 0 1 . 0 3 E + O B 5 . 8 0 E + 0 6

4 4 . 0 8 7 . 9 5 4 . B 2 E + 0 7 5 . 6 B 1 . 0 1 . 0 1 E + 0 8 5 . 8 0 E + 0 6

4 5 . 0 8 7 . 9 1 4 . B 1 E + 0 7 5 . 6 8 0 . 0 9 . 9 9 E + 0 7 5 . B O E + 0 6

4 6 . 0 B 7 . B 6 4 . B O E + 0 7 5 . 6 8 1 . 0 9 . 8 4 E + 0 7 5 . 8 0 E + 0 6

4 7 . 0 B 7 . B 6 4 . 7 9 E + 0 7 5 . 7 B 1 . 0 9 . 6 9 E + 0 7 5 . B O E + 0 6

4 B . 0 8 7 . B 4 4 . 7 8 E + 0 7 5 . 7 . 8 1 . 0 9 . 5 4 E + 0 7 5 . B O E + 0 6

8 0 I I I I I I o 1 0 2 0 3 0 4 0· 5 0

P o s i t i o n of T RW B A p p l i c a t i o n m a x

F i g u re 4 . 1 8 Ma x i mum Ba s i n Tempe ra t u re a s Fu n c t i o n o f D i s p l a c eme n t o f Ac c i d e n t I n i t i a t i o n f rom T i me of Ma x i mum Ru n n i n q We t - Bu l b Tempe ra t u re .

5 0

91

F i g u re 4 . 1 8 s hows f i r st that t h e max i mu m p redi cted ba s i n temp e ra

t u re i s n ot h i gh l y s e n s i t i ve t o c h a n ges i n a c c i dent sta rt i n g t i me s i nce

t h e di ffe rence hetween the sma l l e s t ( 88 . 6 5 O F ) a nd l a rgest ( 89 . 9 4 O F )

v a l ues p l otted i s l es s t h a n one a n d a h a l f deg rees F a h r e n h e i t . Secon d ,

t h e g reat e s t va l ue occu rs at a n offset of 2 2 hou rs , on l y one hou r d i f

fe rent f rom the i n i t i a l c ho i ce b a sed on t he resu l t s of the c o n s t a nt -wet

b u l b-temp e rat u re s i mu l at i on . V a ry i n g t h e sta rt i n g t i me o f t he a c c i dent

by 3 hou r s on e i t her s i de of t he ma x i mum y i e l ds a c h a n ge i n ma x i mum p re

d i cted ba s i n temp e ratu re of l es s than 0 . 1 o f . F i n a l l y , t he p l ot re

f l ects t h e d i u rna l c h a racter of t h e amb i ent wet -bu l b t empe ratu re dat a .

T he corre l at i on between max i mu m p red i cted ba s i n t emp e ra t u re a n d

ma x i mum ru n n i n g wet - bu l b t emp e rat u re w a s s t u d i e d by exami n i n g f i ve d i f

fe rent w i n dows that h a d been s a ve d du r i n g the data sca n n i n g p rocedu re . The resu l t s of t h i s s t u dy a re s umma r i zed i n T a b l e 4 . 9 , wh e re i n the t i me

o f the wi n dow , the correspon d i n g ma x i mum ru n n i n g wet -bu l b temp e ra t u re ,

the max i mum p red i cted ba s i n t empe ra t u re , the i n i t i a l ba s i n temp e r at u re ,

a n d t he heat l oa d i nc rement ( def i n e d a s the di f ference betwee n t h e ma x i

mum p re d i cted b a s i n t emp e ratu re a nd the max i mum ru n n i n g wet - bu l b t emp e r

atu re ) a re l i s ted . T h e res u l t s s how t h at the max i mum ru n n i n g wet -b u l b

temp e rat u re i s a ve ry goo d , a l t hough not neces s a r i l y p e r fect , p re d i ct o r

o f ma x i mum bas i n tempe ratu re . T he heat l oa d i nc rement appea r s t o de

c l i ne w i t h i n c reas i ng max i mum ru n n i ng wet - bu l b tempe ratu re , a l t h o u g h the

u n de r l y i n g phYS i ca l ba s i s of t h i s t re n d i s not c l ea r .

F i g u re 4 . 1 9 s h ows the ba s i n t emperat u re response u n de r a c c i de n t

cond i t i on s for f i ve d i fferent i n i t i a l b a s i n t emp e ratu res . A l t h o u g h a

Tabl e 4 . 9 Co rre l a t i o n B e twee n Ma x i mum P re d i c t e d Ba s i n Tempe r a t u re a n d Ma x i mum Ru n n i n g We t - B u l b Tempe ra tu re fo r F i ve Met e o ro l o g i ca l W i ndows .

M a x i m u m H e a l L o a d W i n d o w R u n n i n g M a x i m u m B a s i n I n i t i a l B a s i n

I n c r e m e n t ( F ) , W a t - B u l b D a t e T e m p e r a t u r e ( F ) T e m p e r a t u r e ( F )

T e m p e r a l u r e ( F ) T - T B m a x R W Bm a x

1 4 : 0 0 h o u r s ,

8 / 1 3 / 8 1 t o 8 1 . 2 7 8 9 . 9 4 8 1 . 0 4 8 . 6 7 2 1 : 0 0 h o u r s ,

8 / 2 1 / 8 1

1 9 : 0 0 h o u r s ,

8 / 0 6 / 6 9 t o 8 0 . 7 0 8 9 . 5 7 7 9 . 3 8 8 . 8 7 o 2 : 0 0 h 0 u r s,

8 / 1 5 / 6 9

1 0 : 0 0 h o u r s ,

8 / 0 4 / 5 9 t o 7 9 . 7 4 8 8 . 8 8 7 8 . 7 5 9 . 1 4 1 7 : 0 0 h o u r s ,

8 / 1 2 (5 9

1 9 : 0 0 h o u r s , 7 / 0 1 / 5 7 t o

7 9 . 5 6 8 8 . 7 4 7 8 . 8 4 9 . 1 8 0 2 : 0 0 h o u r s , I 7 / 1 0 / 5 7

1 4 : 0 0 h o u r s , I 8 / 2 7 / 5 6 t o 7 9 . 3 2 S 8 . 7 2 7 8 . 5 9 9 . 4 0 2 1 : 0 0 h o u r . ,

I 9 / 0 4 / 5 6

� N

100 r' --------11--------,---------.---------.---------

,,-... lL. 0

'-../ 90 Q) L :J

+-0 L CD 8 0 0-E Q)

b-e 70 (J) 0

CO

2 0 4 0

T i m e

T EST C A S E I n i t i a l T 8 = 8 1 .0 4 · F

- - - I n i t i a l T & = 7 8 .00 · F -.- I n i t i a l T & = 75.00 · F --- I n I t I a l T & = 70 . 00 · F · · · · · · · · · · · I n l t l a l T8 = 60 .00 · F

6 0 8 0

( h r ) F i Q u re 4 . 1 9 B a s i n Tempe ra t u re Du r i n g L OCA a s F u n c t i o n o f T i me f rom B e g i n n i n g o f Ac c i d e n t fo r -

V a r i a b l e I n i t i a l B a s i n Tempe ra t u re .

\.C W I

1 0 0

9 4

h i ghe r i n i t i a l b a s i n t emperat u re does con s i s tent l y l ea d t o a h i g h e r

b a s i n temp e rat u re a t a ny g i ven t i me , t h e e f fect o f t h e i n i t i a l c o n d i t i on

d i e s away rou g h l y exponent i a l l y i n t i me wi t h a t i me c o n s t a n t equ a l t o

t h a t of t he t owe r/ba s i n system . T h i s beh a v i o r may b e expected due t o

t he a pp rox i mat e f i r s t - o rde r res p o n s e of t h e system a s d i s cu s sed ea rl i e r .

F i g u re 4 . 20 comp a res the ba s i n tempe ratu re h i s t o ry p red i cted by t he

s i mu l at i on p ro g ram UHSS I M wi t h t hat g i ven us i n g the s i mp l i f i ed model f o r

the p r i ma ry w i ndow of t i me-va ry i n g wet - bu l b tempe ratu res s e l ected by

p ro g ram NWS . U s i n g t h i s set o f met eo ro l o g i c a l cond i t i o n s , p rog ram TMCO N

dete rm i ned a t i me con st ant of 1 1 . 7 3 hou rs wh i ch i s o n l y neg l i g i b l y d i f

fe rent f rom the va l u e o f 1 1 . 46 h ou r s fou n d u s i n g a c o n s t a n t wet -bu l b

t emp e ratu re o f 81 ° F . T h i s compa r i son p ro v i des an i mp o rt a nt test of t h e

v a l i d i ty of the t i me con s t a n t as wel l a s on t he val i d i ty o f the f i rs t

o rde r a p p r o x i mat i on i t s e l f .

4 . 7 F i n a l S i mu l at i on Res u l t s

T h e res u l t s for t h e p r i ma ry s et o f meteoro l o g i c a l cond i t i o n s a re

summa r i zed i n Tab l e 4 . 1 0 a nd F i gu res 4 . 2 1 th rough 4 . 2 3 . T�e ba s i n t em

p e rat u re ( a s s umed equ a l to the co l d wat e r ret u r n temp e ratu re ) , p l ot t ed

i n F i g u re 4 . 2 1 , reaches a ma x i mum o f 89 . 94 o f 2 3 hou rs a ft e r i n i t i at i on

o f t he hyp ot h et i c a l a c c i dent , a s n ot e d ea r l i e r . T he va r i at i o n of ba s i n

ma s s w i th t i me i s s h own i n F i gu re 4 . 2 2 . As can be seen h e re , b a s i n mas s

dec l i ne s s l ow l y du e t o evapo rat i ve l os s es . F i gu re 4 . 2 3 s h ow s t h e va r i a

t i on of h eat l oad to t he t owe r / ba s i n sy stem.

S i n ce t h e ba s i n t emp e rat u re ma x i mum of 89 . 94 ° F i s we l l be l ow the

des i gn c r i t e r i o n of 9 3 ° F , the sy st em i s j u dged to meet t he ma x i mum t em-

1 00 ,' --------,---------.---------.-�----�---------

S Y S T E M R E S PO N S E � LL o .........,

Q) 90 L ::::J

+-o L Q) 0.. E Q)

l- 80 c: (/) o

())

-- S i m u l a t i o n Ba s i n Te m pe ra t u re - - - S i m p l i f i e d M o d e l B a s i n T e m pe r a t u r e

W a t e r F l ow R a t e , L = 7 3 . 6 4 % a f t e r 2 4 h o u rs , L = 7 0 .36% I n i t i a l T B A S I N = 8 1 .0 - F

L = 1 1 .7 3 h r

7 0 L' ______________ � ____________ � ______________ � ____________ � ____________ �

o 2 0 4 0 T i m e

6 0 8 0 ( h r )

F i g u re 4 . 20 Comp a ri s o n o f Tempe ra t u re H i s t o r i e s P r e d i c t e d by Fu l l Towe r/ Ba s i n S i mu l a t i o n ( U H S S H1 ) a n d S i mp l i f i e d �1o d e l f o r P r i ma ry t1e t e o ro l o q i c a l �J i n d ow .

1 0 0

� U1

9 6 Ta b l e 4 . 1 0 Va r i a t i o n o f I mpo rt a n t Me t e o r o l o g i c a l , Hea t Lo a d , a n d Ba s i n

V a r i a b l e s w i t h T i me Du r i n g LOCA .

T I M E TBAS I N BMAS S S TWB Q L ( H R ) ( F ) ( LB M ) ( P PT ) ( F ) ( BTU/ H R ) ( LBM/H R ) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 . 0 8 1 . 0 4 5 . 4 2 E + 0 7 5 . 0 8 1 . 0 6 . 6 2 E + 0 7 6 . 0 7 E + 0 6 1 . 0 8 1 . 8 4 5 . 4 1 E + 0 7 5 . 0 8 1 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 2 . 0 8 2 . 6 4 5 . 3 9 E + 0 7 5 . 0 8 0 , 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 3 . 0 8 3 . 3 4 5 . 3 8 E + 0 7 5 . 0 8 0 . 0 1 . 9 3 £ + 0 8 6 . 0 7 E + 0 6 4 . 0 8 3 . 9 9 5 . 3 6 E + 0 7 5 . 1 8 0 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 5 . 0 8 4 . 5 8 5 . 3 5 E + 0 7 5 . 1 8 0 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 6 . 0 8 5 . 0 9 5 . 3 3 E + 0 7 5 . 1 7 9 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 7 . 0 8 5 . 5 2 5 . 3 2 E + 0 7 5 . 1 7 9 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 8 . 0 8 5 . 9 2 5 . 3 0 E + 0 7 5 . 1 7 9 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 9 . 0 8 6 . 2 9 5 . 2 8 E + 0 7 5 . 1 7 9 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6

1 0 . 0 8 6 . 6 2 5 . 2 7 E + 0 7 5 . 1 7 8 . 0 1 . 9 3 E + 0 8 6 . 0 7 E + 0 6 1 1 . 0 8 6 . 9 6 5 . 2 5 E + 0 7 5 . 2 8 0 ·. 0 1 . 9 2 E + 0 8 6 . 0 7 E + 0 6 1 2 . 0 8 7 . 2 8 5 . 2 3 E + 0 7 5 . 2 8 0 . 0 1 . 9 0 E + 0 8 6 . 0 7 E + 0 6 1 3 . 0 8 7 . 5 7 5 . 2 2 E + 0 7 5 . 2 8 0 . 0 1 . 8 8 E + 0 8 6 . 0 7 E + 0 6 1 4 . 0 8 7 . 8 8 5 . 2 0 E + 0 7 5 . 2 8 2 . 0 1 . 8 6 E + 0 8 6 . 0 7 E + 0 6 1 5 . 0 8 8 . 2 2 5 . 1 9 E + 0 7 5 . 2 8 2 . 0 1 . 8 4 E + 0 8 6 . 0 7 E + 0 6 1 6 . 0 8 8 . 5 2 5 . 1 7 E + 0 7 5 . 2 8 2 . 0 1 . 8 2 E + 0 8 6 . 0 7 E + 0 6 1 7 . 0 8 8 . 8 2 5 . 1 5 E + 0 7 5 . 3 8 3 . 0 1 . 8 1 E + 0 8 6 . 0 7 E + 0 6 1 8 . 0 8 9 . 1 2 5 . 1 4 E + 0 7 5 . 3 8 3 . 0 1 . 8 0 E + 0 8 6 . 0 7 E + 0 6 1 9 . 0 8 9 . 3 9 5 . 1 2 E + 0 7 5 . 3 8 3 . 0 1 . 7 9 E + 0 8 6 . 0 7 E + 0 6 2 0 . 0 8 9 . 6 3 5 . 1 1 E + 0 7 5 . 3 8 3 . 0 1 . 7 8 E + 0 8 6 . 0 7 E + 0 6 2 1 . 0 8 9 . 8 1 5 . 0 9 E + 0 7 5 . 3 8 2 . 0 1 . 6 6 E + 0 8 6 . 0 7 E + 0 6 2 2 . 0 8 9 . 9 0 5 . 0 8 E + 0 7 5 . 3 8 2 . 0 1 . 5 5 E + 0 8 6 . 0 7 E +- 0 6 2 3 . 0 8 9 . 9 4 5 . 0 6 E + 0 7 5 . 4 8 1 . 0 1 . 4 3 E + 0 8 6 . 0 7 £ + 0 6 2 4 . 0 8 9 . 8 9 5 . 0 5 E + 0 7 5 . 4 8 1 . 0 1 . 3 1 E + 0 8 6 . 0 7 E + 0 6 2 5 . 0 8 9 . 8 1 5 . 0 4 E + 0 7 5 . 4 8 0 . 0 1 . 3 0 E + 0 8 5 . 8 0 E + 0 6 2 6 . 0 8 9 . 6 2 5 . 0 2 E + 0 7 5 . 4 7 8 . 0 1 . 2 8 E + 0 8 5 . 8 0 E + 0 6 2 7 . 0 8 9 . 3 3 5 . 0 1 E + 0 7 5 . 4 7 6 . 0 1 . 2 7 E + 0 8 5 . 8 0 E + 0 6 2 8 . 0 8 9 . 0 0 5 . 0 0 E + 0 7 5 . 4 7 6 . 0 1 . 2 5 £ + 0 8 5 . 8 0 £ + 0 6 2 9 . 0 8 8 . 7 2 4 . 9 8 E + 0 7 5 . 4 7 7 . 0 1 . 2 4 £ + 0 8 5 . 8 0 £ + 0 6 3 O . 0 8 8 . 4 7 4 . 9 7 £ + 0 7 5 . 5 7 6 . 0 1 . 2 2 E + 0 8 5 . 8 0 £ + 0 6

3 1 . 0 8 8 . 2 0 4 . 9 6 £ + 0 7 5 . 5 7 6 . 0 1 . 2 1 £ + 0 8 5 . 8 0 E + 0 6

3 2 . 0 8 7 . 9 5 4 . 9 4 E + 0 7 5 . 5 7 6 . 0 1 . 1 9 E + 0 8 5 . 8 0 E + 0 6

3 3 . 0 8 7 . 7 2 4 . 9 3 E + 0 7 5 . 5 7 6 . 0 1 . 1 8 E + 0 8 5 . 8 0 £ + 0 6

3 4 . 0 8 7 . 5 5 4 . 9 2 E + 0 7 5 . 5 7 8 . 0 1 . 1 6 £ + 0 8 5 . 8 0 E + 0 6

3 5 . 0 8 7 . 4 9 4 . 9 1 E + 0 7 5 . 5 7 9 . 0 1 . 1 5 £ + 0 8 5 . 8 0 f'; + 0 6

3 6 . 0 8 7 . 4 5 4 . 9 0 E + 0 7 5 . 5 7 9 . 0 1 . 1 3 E + 0 8 5 . 8 0 E + 0 6

3 7 . 0 8 7 . 4 7 4 . 8 9 £ + 0 7 5 . 5 8 1" . 0 1 . 1 2 E + 0 8 5 . 8 0 £ + 0 6

3 8 . 0 8 7 . 5 8 4 . 8 8 E + 0 7 5 . 6 8 2 . 0 1 . 1 0 £ + 0 8 5 . 8 0 £ + 0 6

3 9 . 0 8 7 . 7 1 4 . 8 7 £ + 0 7 5 . 6 8 2 . 0 1 . 0 9 £ + 0 8 5 . 8 0 E + 0 6

4 0 . 0 8 7 . 8 4 4 . 8 6 £ + 0 7 5 . 6 8 3 . 0 1 . 0 7 £ + 0 8 5 . 8 0 £ + 0 6

4 1 . 0 8 7 . 9 4 4 . 8 5 E + 0 7 5 . 6 8 1 . 0 1 . 0 6 £ + 0 8 5 . 8 0 E + 0 6

4 2 . 0 8 7 . 9 7 4 . 8 4 E + 0 7 5 . 6 8 2 . 0 1 . 0 4 E + 0 8 5 . 8 0 E + 0 6

4 3 . 0 8 8 . 0 1 4 . 8 3 E + 0 7 5 . 6 8 1 . 0 1 . 0 3 E + 0 8 5 . 8 0 E + 0 6

4 4 . 0 8 7 . 9 7 4 . 8 2 E + 0 7 5 . 6 8 0 . 0 1 . 0 1 E + 0 8 5 . 8 0 E + 0 6

4 5 . 0 8 7 . 9 3 4 . 8 1 £ + 0 7 5 . 6 8 1 . 0 9 . 9 9 £ + 0 7 5 . 8 0 E + 0 6

4 6 . 0 8 7 . 9 2 4 . 8 0 E + 0 7 5 . 6 8 1 . 0 9 . 8 4 E + 0 7 5 . 8 0 E + 0 6 4 7 . 0 8 7 . 9 0 4 . 7 9 E + 0 7 5 . 7 B l . 0 9 . 6 9 E + 0 7 5 . 8 0 E + 0 6 4 8 . 0 8 7 . 8 5 4 . 7 8 £ + 0 7 5 . 7 8 0 . 0 9 . 5 4 E + 0 7 5 . 8 0 £ + 0 6

1 00 � LL 0 '-..-/

<D L :J 90

+-0 L <D 0..

E <D 80 l-e U1 0

CO 7 0

T E ST CAS E B a s i n T e m p e r a t u r e a f t e r L O C A

I I

0 2 0 4 0 60 80 T i m e ( h r )

F i g u re 4 . 2 1 B a s i n Tempe ra t u re Du r i n g L OCA a s F u n c t i o n o f T i me f rom Be g i n n i n g o f Ac c i d e n t .

I �

1 0 0

6X l 07�-------.--------�---------.--------�---------T E ST CAS E

,...-... E

...0 v 5x l 07 (f) (f) o 2 C 4x l 07 - -(f) o

en

--- B a s i n M a ss a f t e r L O CA

� co

3x 1 07 �1 ________ �--------�--------��--------L-------� o 20 4 0

T i m e 6 0 8 0 1 00

( h r ) F i g u re 4 . 2 2 Ba s i n Ma s s Du r i n q LOCA a s Fu n c t i o n o f T i me f rom Be q i n n i n g o f Ac c i dent .

2.2x l 08 r' ---------r--------.---------.---------�---------

,-..... L

..c. " 1 . 8x l 08 :J

+-...0 '-.-/

D 1 . 4 x l 08 o o

--.J +-o 1 . 0 x l 08 Q)

I

T EST C A S E --- H e a t L o a d a f t e r L O C A

2 0 40 T i m e

6 . 0X l 07 �, ________ ��------�--------�--------L-------� o 60 1 0 0 80

( h r ) F i g u re 4 . 2 3 H e a t L o a d t o U H S Sy s t em Du r i n 9 L O C A a s F u n c t i o n o f T i me f rom Be g i n n i n g o f Ac c i d e n t .

� '"

1 0 0

p e ra t u re l i m i t i mp osed . An d s i n c e the ba s i n vol ume i s rou g h l y 443 , 000

g a l l o n s of wat e r aft e r 30 day s , the sy stem meets the req u i rements o f 30

day s of operat i on wi thout ma k e - u p wat e r s u p p l y . The s a l i n i ty content of

the b a s i n a ft e r the 30- day p e r i od was fou n d t o be 7 6 . 8 p pt a s s h own i n

F i gu re 4 . 2 4 , i n di cat i n g t h at the effects of d i s s o l ved s o l i ds wa s neg l i

g i b l y smal l for th i s c a s e .

A +-

C

0 (J) C (J) 0

CO

0 . 1 0 I

T EST CAS E 0 . 0 8 I

B a s i n S a l i n i t y a f t e r L O C A

0 . 06

0 . 04

0 . 02

0 . 0 0 �, ________ � ______ � ______ � ________ � ______ � ______ �

o 1 20 2 4 0 3 6 0 4 8 0 600 T i m e ( h r )

F i g u re 4 . 24 B a s i n S a l i n i ty Du r i n 9 LOCA a s F u n c t i o n o f T i me f ro� B e g i n n i n g o f Ac c i d e n t fo r 30 d a y s .

7 2 0

1 0 2

5 . CONCLU S I ONS AND RECOMME NDAT I ON S

Th e a na l ys i s o f u l t i mate heat s i n k coo l i n g t owe r perfo rma n c e du r i n g

a des i g n - b as i s acc i dent i s of i mp o rt a n ce i n det e rmi n i ng whet her t h e p ro

posed sys t em i s adeq uate to a l l ow safe s h u t down a n d coo l down of t he nu

c l e a r react o r ( s } . The U . S . Nuc l e a r R e g u l at o ry C ommi s s i on h a s p re v i ou s

l y p ub l i s hed met hods o f a n a l y s i s f o r U H S cool i n g ponds a n d s p ray ponds .

I n t he a p p l i cat i on of t h es e methods , p a ramet e rs a re s e l ected to p ro v i de

the wo r s t - c a s e performa nce wh i c h may be rea s o n a b l y expected fo r the p ro

posed des i gn a n d g i ven the h i st o r i ca l met eoro l o g i ca l dat a f o r t he s i t e .

Thu s , a h i gh deg ree of con s e rvat i s m i s bu i l t i nt o the ana l y s i s ow i n g t o

the s e v e re c o n s equences a s s o c i ated w i t h t h e f a i l u re of t he U H S coo l i n g

sys t em .

To ext e n d t h i s type of ana l y s i s t o UHS cool i n g t owe rs , we h a v e de

ve l oped a s et of computer codes wh i ch p re d i ct the t h e rma l p e rfo rma n ce o f

t he U H S coo l i n g t owe r . L i ke the met hods deve l oped fo r the U H S coo l i n g

pond a nd t he UHS s p ray p on d , t h i s met h o d u t i l i zes the fu l l p e r i od of

met eo ro l o g i c a l dat a reco r de d by the Nat i o n a l Weat h e r Se rvi c e fo r the

s t at i on most rep resentat i ve of t he p ro p o s ed p l a nt . The pro g ram NWS was

devel oped to eas i l y h a n dl e the l a r ge amo u n t s of data on the Nat i o n a l

Weat her S e r v i ce t apes a n d to i de nt i fy t h e p e r i ods o f worst - c ase sy s t em

performance . T h rough examp l e , t he val i di ty of th e NWS met h o do l o gy i s

demon st rate d .

B ased o n a theo ret i c a l towe r/ heat exc h a n ge r ana l y s i s , a met h o d i s

devel oped for c a l cu l at i n g t he t owe r p e r f o rma n ce c h a racte ri s t i c v a l u e s .

P rog ram PDAP , wh i c h ca l c u l ates t he towe r pe rfo rma nce ch a rac t e r i s t i c

1 0 3

va l u e s a nd the p redi cted t ow e r col d wat e r temp e ratu re , a l l ow s con v en i e n t

u s e of ma n u fact u re r ' s pe rforma nce dat a as i n p u t . T h e p red i ct i on of t owe r

c o l d wat e r temp e ratu re by p rog ram PDAP a l l ow s c r i t i c a l eva l u at i o n of t h e

ma n u factu re r ' s des i gn c u rves , a s s hown by t he comp a r i sons g i ven i n

C h ap t e r 4 .

O n ce a rel i ab l e s i mu l at i on of coo l i n g towe r performa n c e i s obt a i n

e d , a s i mp l i f i ed model o f t h e towe r/ba s i n sys t em i s con st ructed a s s umi n g

a s i mp l e f i r s t - o rde r ( e xp o nent i a l rel axat i on ) res p o n s e of ba s i n t emp e r a

t u re w i th t i me . T h e t i me c o n stant o f t h i s s i mp l i f i ed model ; s det er

mi n e d u s i n g a wet - bu l b temp e ratu re rep resentat i ve of the acc i den t sce n

a r i o . T h i s t i me con s t a nt i s t hen . u sed i n t h e d a t a s c a n n i n g p rocedu re

( i n p rog ram NWS ) whe re i n a " ru n n i n g ave rage wet - b u l b tempe rat u re " i s

c a l c u l ated for each hou r f o r wh i ch a met � o ro l og i c a l o b s e rvat i on i s

a v a i l a b l e . The per i od of wo r s t - c a s e t h e rma l p e r f o rma nce then c o r r e s

p o n ds to the p e r i od of ma x i mum r u n n i n g wet - bu l b t empe ratu re . The de

t a i l s of t h i s met hod as wel l a s the u n de r l y i n g t h eo ret i c a l fou n dat i o n i s

fu l l y d i s c u s sed i n C h a p t e r 3 . The va l i d i ty of t h e app roach i s fu rt h e r

demo n s t rated by way o f examp l e i n Chapt e r 4 .

U s i ng t he wo rs t -case meteoro l og i ca l con d i t i o n s det e rmi ned by p ro

g ram NWS , the des i g n-bas i s acc i dent i s s i mu l ated and the ba s i n cond i

t i o n s of tempe ratu re , mas s , a n d s a l i n i ty a re det e rmi ned as funct i ons o f

t i me f o r up to 3 0 day s fo l l ow i ng t he acc i dent . T h e sys t em can t hen be

j u dged as to whet her t he des i gn i s s u f f i c i ent to s a fe l y s h u t down a n d

c o o l down t h e n uc l ea r react o r ( s ) a s requ i red by U . S . Nu c l ea r Regu l ato ry

C ommi s i on Regu l a tory G u i de 1 . 2 7 . I n s u mma ry , t he p rocedu re p re s e nted i n

1 0 4

t h i s wo rk p ro v i des a n obj e ct i v e mea n s of eva l u at i n g the adeq u a cy of t he

UHS coo l i ng t ower sy stem .

To comp l et e l y i n s u re t h e rel i ab i l i ty of t he p red i c t i on s o f UHS sy s

t em p e r f o rma nce made u s i n g i n fo rma t i on s u p p l i ed i n the F i na l S a f ety

An a l y s i s Report , we recommen d that a l l c a ses be re- e v a 1 uat e d u s i n g t he

coo l i n g t ower accept ance dat a o nce i t i s a v a i l a b l e .

1 0 5

REFER E NCE S

Beckw i t h , T . G . , a n d N . L . B u c k , 1 9 7 3 : Mech a n i ca l Mea s u reme n t s ( Mas s a c h u sett s : Addi s on-We s l ey P u b l i s h i n g Comp a ny ) .

B ra dy , D . K . , W . L . G raves , a n d J . C . Gey e r , 1 969 : " S u rface Heat E xc h a n g e at P l a nt Coo l i n g La k es , " Report No . 5 , E d i s o n E l ect r i c I n st i t ut e , E E l P u b l i c at i on 69- 401 .

Code l 1 , R . , a n d W . K . Nu tt l e , 1 980 : "An a l y s i s of U l t i ma t e Heat S i n k C oo l i n g P o n d s , " U . S . N RC R e p o rt NUREG- 069 3 .

Code l l , R . , 1 98 1 : " An a l y s i s o f U l t i mate-Heat - S i n k Sp ray Pon ds , " U . S . NRC R e p o rt NUREG- 0 7 3 3 .

Code l l , R . , 1 982 : "Comp a r i s o n Between F i e l d Dat a and U l t i ma t e Heat S i n k C o o l i n g P o n d a n d S p ray Pond Model s , " U . S . NRC R e p o rt NUREG- 0858 .

Du n n , W . E . , 1 98 1 : " P l ot l i b Sub rout i ne L i b ra ry U s e r ' s Ma n u a l , " Depa rt ment of Mech a n i Ca l a nd I n du st r i a l E n g i nee r i n g , U n i v e rs i ty o f I l l i n o i s , U rb an a , I L .

D u n n , W . E . , 1 984 : P e r s o n a l Commu n i c at i o n , U n i vers i ty o f I l l i no i s at U rb a n a - C h ampa i g n .

E l ect r i c P ow e r Res e a rch I n s t i t u t e , Spec i a l Repo rt CS- 3 2 1 2-SR , 1 983 : " TE F E R I , Nume r i ca l Model for Ca l cu l at i n g t he Pe rformance o f a n E va p o rat i ve Coo l i n g Towe r , " E l ect r i c P owe r Res e a rch I n s t i t u t e , P a l o A l t o , CA .

Had l ock , R . K . , a n d O . B . Abbey , 1978 : " T h e rma l P e r f o rmance Mea s u rements on U l t i mate Heat S i n k s - Coo l i ng Ponds , " U . S . NRC R e p o rt NUREG/CR- 0008 , P NL- 2463 .

Had l o c k , R . K . , a n d O . B . Abb ey , 1981 : " T h e rma l P e r fo rma n c e a n d Wat e r Ut i l i zat i o n Mea s u rements o n U l t i mate Heat S i n k s - Coo l i n g P o n ds a n d Sp ray Ponds , " U . S . NRC Report NUREG / C R - 1 886 , P NL- 3689 .

Ho l l , J . H . , 1 9 5 1 : " Heat a n d Ma s s T r a n s f e r i n E v a p o r at i ve C o o l i n g , " Ma s t e r s The s i s , U n i vers i ty of I l l i no i s at U rba n a- Ch amp a i g n .

Maj u mda r , A . K . , A . K . S i n g h a l , a n d D . B . S p a l d i n g , 1 983 : " N u me r i c a l Mo del i n g o f Wet Cool i n g Towe rs - Pa rt 1 : Mathemat i ca l a n d P hys i ca l Mode l s , " J . Heat T ra n s fe r , 1 0 5 , 7 2 8- 7 3 5 .

Maj umda r , A . K . , A . K . S i n g h a l , H . E . Re i l l y , a nd J . A . Ba rt z , 1 983 : " Nu me r i c a l Model i ng of Wet C o o l i n g T owe rs - P a rt 2 : App l i c at i on t o Nat u ra l D ra ft Towe rs , " J . Heat Tra n s fe r , 1 0 5 , 7 36- 7 4 3 .

1 0 6

Maj umda r , A . K . , A . K . S i n gh a l , a n d O . B . Sp a l d i n g , 1 983 : " V ERA20 - A C ompu t e r P rog ram for Two- D i men s i o n a l Ana l y s i s of F l ow , Heat , a n d Ma s s T ra n s f e r i n E v a p o rat i ve Coo l i n g Towe rs : V o l . 1 - Ma t h emat i c a l , F o rmu l at i on , S o l u t i on P rocedu r e , a n d A p p l i c at i o n , " EPR I R e p o rt N o . C S - 29 2 3 .

Maj u mda r , A . K . , a n d A . K . S i n g h a l , 1983 : for Two- D i me n s i ona l A n a l y s i s o f F l ow , E v a p o rat i ve Coo l i n g Towe rs : V o l . 2 -No . C S - 2 9 2 3 .

" VERA2D - A Comp u t e r P ro g r a m Heat a n d M a s s T ra n s fe r i n Us e r ' s Ma n u a l , " E P R I Report

Merke l , F . , 1 9 2 5 : " Ve rdu n s l u n g s h� h l u n g , " VD I - F o r�ch u n gs a rbe i t e n , 2 7 5 .

N a havand i , A . N . , R . M . K e rs h a h , a n d B . J . S e r i c o , 1 9 7 5 : "The E ffect s o f E v a p o rat i ve Lo s ses i n the An a l y s i s of Cou n t e r f l ow Coo l i n g Towe r s , " N uc . E n g . Des . , 32 , 2 9 - 36 .

Na h a v a n d i , A . N . , a nd J . J . Oel l i n ge r , 1 9 7 7 : " An I mp roved Model for t h e Ana l ys i s of E va p o rat i ve Counte rf l ow C o o l i ng Towe rs , " N u c . E n g . De s . , 40 , 3 2 7 - 3 3 6 .

Poppe , M . , 1 9 7 3 : " W� rme- u n d S l offubert r a g u n g be i de r Verdums l u n g s k i i h l u n g i n Ge rgen u n d K reu z s t rom , " VD I - F o r s c h u n g s heft , 560 , 3 9-44 .

Reyno l ds , W . C . , a n d H . C . Pe rk i n s , 1 9 7 7 : The rmody n ami c s ( New Yo r k : McG r aw-H i l l B o ok Comp a ny ) .

S k e l l a n d , A . H . P . , 1 9 74 : D i ffu s i o n a l Ma s s Tra n s f e r ( New Yo r k : J o h n W i l ey a n d S o n s ) .

St e i n , H . L . , 1982 : " Coo l i n g Towe rs , " Spec i fy i n g E n g i nee r , 47 , 1 2 1 - 1 26 .

Su l l i va n , S . M . , a nd W . E . Du n n , 1 984 : " U s e r ' s Ma n u a l for UHS The rma l P e rfo rmance A n a l y s i s Codes , " Depa rtment of Mecha n i c a l a n d I n d u s t r i a l E n g i neer i n g , U n i ve r s i ty o f I l l i no i s , U r b a n a , I L .

S uther l a n d , J . W . , 1 983 : "Ana l ys i s of Mech a n i c a l - D rau ght C o u n t e rf l ow Ai r/Wa t e r Cool i n g Towe rs , " J . Heat T r a n s fe r , 1 0 5 , 576- 586 .

U . S . Nu c l e a r Regu l atory C ommi s s i o n , Regu l at o ry Gu i de 1 . 2 7 , 1 9 7 6 : " U l t i mate Heat S i n k for Nu c l ea r Pow e r P l ant s . "

Y a d i ga rog l u , G . , a n d E . J . P a s t o r , 1 9 7 4 : " An I n ve st i gat i on of t h e Accu racy of t h e Me rke l Equ at i on f o r E v a p o r at i ve Coo l i n g Tow e r Ca l c u l at i o n s , " ASME P aper 7 4-HT- 5 9 .

1 0 7

A P P E N D I X A S I MU LA T I ON COMPU T E R C OD E S

F i g u r e A . l U s t i ng o f P rog r am rOAP

f i g u r e A . 2 U s t i ng o f Prog r am KAV L

f i g u re A . 3 L i s t ; ng o f Program TMeON

F i g u r e A . 4 L i s t i ng o f P r o g r am U W S

f 1 g u rc A . S L i s t i ng o f P ro g r am U H S S I M

C

1 0 8 PROGRAM PDAP ( TAPE 5 , TAPE 6 , TAPE 7 , OUTPUT )

c • • • TH I S PROG RAM CALCU LATE S , G IV E N TH E I N LET WET- B U L B C • • • TEMPE RATURE , TH E RANG E , AND TH E COLD WATER TEM P E RATU RE C • • • F ROM P E RFORMAN CE CURV E S , A B E ST F I T L E AST SQUARES VAL U E C • • • OF TH E TOWER CHARACTER I S T I C , KAV/ L , FOR A U H S COOL I NG TOW E R . C . . . TH I S PROG RAM ALSO CAL ClJ LATES , U S I NG TH I S PRED I C'fED KAV/ L , A C • • • CO L D WAT E R TEM P E RATU RE O F TH E TOW E R . C c • • • THE DATA I S ENTERED FROM LOWEST TO H I GH E ST RANG E VAL U E S C • • • I N ORD E R OF I NCREAS I NG WET- BULB TEMPERATU R E W I TH I N EACH C • • • RANG E . E NT E R I NG DATA I N TH I S MAN N E R PERM I TS TH E OUTPUT C • • • TO B E USED D I R ECTLY FOR PLOTT I NG IN P ROG RAM PDAPLT . C

C

CHARACT E R * B O CAS E , FLOW , RLAB EL ( 6 ) PARAMETER ( MAXDAT=5 0 ) EXTERNAL ST 2 REAL RAN ( MAXDAT ) , NWB ( MAXDAT) COMMON WBD ( MAXDAT ) , RAN G E ( MAXDAT ) , TCOLD ( MAXDAT ) , TPRED ( MAXDAT ) COMMON D I F F ( MAXDAT ) , N PT COMMON / ,rOWE RC/ S , LOV E RG REAL KAVL , KLOI� , K HG H , LOV E RG DATA K LOW/ 0 . 5 / , KHGH/3 . 0/ , TOL/ O . O l/

C . " . READ I N P E RFO RMAN CE DATA C C C C C C C C C C C C C C

S LOV E RG P E RC NWB

RAN N RAN CAS E F LOW

RLABEL

- SAL I N I TY - L I QU I D TO GAS F LOW RATE RAT IO - L I Q U I D F LOW RATE PERCE NTAG E B E I NG EVAL UAT ED - ARRAY CONTA I N I NG T H E NUMBER OF WET-BU LB T E M P E RATU R E S

W I TH I N EACH RANG E VAL U E - ARRAY CONTAI N I NG TH E RANG E VALU E S - NUMBER OF RANG E VAL U E S W I TH I N A F LOW RATE - B O -CHARACTER LABEL U S ED TO I D E NT I FY OUT PUT - B O -CHARACTER LABEL U S ED TO I D E NT I FY L I QU I D F LOW

RAT ES W I TH I N T H E OUTPUT - 8 0 -CIIARACTER LABEL U S ED TO I DE NT I FY T EM P E RATU R E

VAULUES READ W n'H I N S P EC I F I C RANG E VAL U E S

R E AD ( 5 , 1 0 0 , END = 9 9 ) CAS E W R I T E ( 6 , 1 0 0 ) CAS E

5 NDATAc1 N PT = O

2

1 C C • • •

C

C C • • •

C • • •

C

READ ( 5 , 1 0 0 , END- 9 9 ) F LOW READ ( 5 , . ) LOV E RG , PE RC , S READ ( 5 , · ) NRAN DO 1 J = 1 , NRAN

READ ( 5 , 1 0 0 ) RLAB EL ( J ) READ ( 5 , · ) RAN ( J ) , NWB ( J ) DO 2 I =NDATA , NDATA+ NW B ( J ) - l

READ ( 5 , · ) WBD ( I ) , TCOLD ( I ) RANG E ( I ) -RAN ( J ) N PT = N PT+l

CONT I N U E NDATA =NDATA+NWB ( J ) CONT I N U E

F I ND L EAST SQUARES B E S T F I T VALU E OF KAV/ L

KAVL=FM I N ( KLOW , KHGH , ST 2 , TOL )

WRI T E RESU LTS TO TAPE 6 FOR PLOTT I NG I N PROG RAM PDAPLT , AND TO TAPE 7 FOR PLOTT I NG I N PROGRAM KAVL .

W R I T E ( 6 , 1 0 0 ) FLOW WRI TE ( 6 , 1 0 1 ) KAV L , LOVE RG , PE RC , S WRI TE ( 6 , 1 0 5 ) NRAN

F i g u re A . l L i s t i ng o f P ro q ram P DAP

C

NDATA= l D O 4 J '"' l , NRAN WRI T E ( 6 , 1 0 0 ) RLAB EL ( J ) WRI T E ( 6 , 1 0 2 ) RAN ( J ) , NWB ( J ) DO 3 I = NDATA , NDATA+NW B ( J ) - l

1 0 9

WR I TE ( 6 , 1 0 3 ) WBD ( l ) , TCOLD ( I ) , TPRED ( I ) , DI F F ( I ) 3 CONT I N U E

NDATA�NDATA+NWB ( J ) 4 CONT I N U E

WR I TE ( 7 , 1 0 4 ) KAVL , PE RC GO TO 5

9 9 STOP

c . . . FORHAT STATEMENTS C

C

1 0 0 FO RHAT ( A ) 1 0 1 FORHAT ( 2F 8 . 4 , F 8 . 1 , F B . 4 ) 1 0 2 FO RHAT ( 2 F 1 0 . 1 ) 1 0 3 FO RHAT ( 4 F I O . 4 ) 1 0 4 FO RMAT ( 2F 1 0 . 4 ) 1 0 5 FO RMAT ( 1 1 )

END REAL FUNCT I ON ST2 ( KAVL )

c . . . TH I S FUNCTION I S U S ED TO CALC U LATE T H E SUM OF TH E SQUARES C • • • O F TH E D I F F E RENCE OF THE PE RFO RMANC E CURV E TOUT AND TH E C • • • TOWER PREDI CTED TOUT . C

C

R EAL KAVL PARAMETER ( MAXDAT= 5 0 ) COMMON WBD ( MAXDAT ) , RANG E ( MAXDAT ) , TCOLD ( MAXDAT ) , TPRED ( MAXDA T ) COMMO N D I F F ( MAXDAT ) , N PT

C • • • LOOP TO CALCULATE SUM OF SQUARES C

ST 2 = 0 . 0 DO 1 I = l , N PT

C C . . • CALL S U B ROUTI NE TOW E R U S ED TO CALCU LATE TOUT O F TH E TO\'I E R . C

CALL TOW E R ( TCOLD ( I ) +RANG E ( I ) , WBD ( I ) , KAVL , TPRED ( I » S T 2 = ST 2 + ( TPRED ( I ) -TCOLD ( I » * * 2 D I F F ( I ) =TPRED ( I ) -TCOLD ( I )

1 CONT I N U E RETURN END S U B ROU T I N E TOWER . ( T I N , TWB , KAVL , TOUT ) SAVE TOLD , A , B , C , D , T , H HAT R EAL LOVERG , KAVL COMMO N /TOWERC/ S , LOVERG D I ME N S ION A ( l O ) , B ( 1 0 ) , C ( l O ) , D ( l O ) , T ( l 2 ) , B HAT ( 1 2 ) , WORK ( 2 0 ) DATA A/ 0 . , 9 * l . / , B/ l O * - 2 . / , C/ l O * l . / , D/ 1 0 * O . / DATA N/ I O/ , N P 1 / 1 1/ , N P 2/ 1 2/ , AN/ 0 . 0 5/ DATA A O/ 1 . 0/ , Al/ - 0 . 6 6 1 7 7 / , A 2/ 0 . 3 6 5 0 8/ , A3 / - 5 . 2 5 9 2 6 / DATA C O/ 1 3 . 3 1 B 5/ , C l/ 1 . 9 7 6/ , C 2/ 0 . 6 4 4 5/ , C 3/ 0 . 1 2 9 9/ DATA TT1/ 2 1 2 . / , TT 2/ 4 5 9 . 6 7 / , WRAT I O/ 0 . 6 2 2/ DATA H V 1/ 0 . 4 27 5 5 5/ , HV 2/ l 06 2 . 3/ DATA HA1/ O . 2 4 0 3 / DATA S MALL/ O . O l/ , TOLD/ O . O/ E ( X ) = « ( C3 * X -C2 ) * X+Cl ) * X + C O ) * X F ( X ) =A O +X * ( A l +X * ( A 2 +X * A3 » T ( N P 1 ) =TI N T ( N P 2 ) =TWB AK =AN * KAVL

1 D ( l ) = ( B ( l ) +C ( l » *TWB D ( N ) =-C ( N ) * T I N CALL TDMS ( A , B , C , T , D , N , I E R , WORK ) I F ( AB S ( TOLD-T ( l » . LT . SMAL L ) GO TO 4

F i � u re A . l C o n t i n u e d

C C C C C C C C C C C C C C C C C C C C

TOLD =T ( l ) DO 2 I - l , N P 2 TT= ( TTI -T ( I » / ( T ( I ) +TT 2 ) P P = E X P ( - E ( TT » * F ( S ) W=WRAT I O * PP/ ( 1 . 0 - P P ) HV = HV l * T ( I ) + I I V 2 H A = I I A I * '1' ( I )

2 HHAT ( I ) =HA+W * H V

1 1 0

DO 3 I = 2 , N D H = ( H H AT ( I + l ) -HHAT ( I - l » / ( T ( I + l ) -T ( I - l » Q = AK * ( DI I - LOVERG ) A ( I ) = 1 . 0 + Q

3 C ( I ) = 1 . O - Q B ( 1 ) = - 1 . O -AK * ( ( !I liA'!' ( 1 ) - 1I 1 1AT ( NP 2 ) ) I ( T ( 1 ) -T ( N P 2 » +LOV E RG ) C ( 1 ) = 1 . 0 -AK * « H HAT ( 2 ) - H HAT ( N P 2 » / ( T ( 2 ) -T ( N P 2 » - LOV E RG ) GO TO 1

4 TOU T =T ( l ) TOL D =T ( l ) RE'fU RN EN D R EAL F U NCT I ON FM I N ( AX , BX , F , TOL ) REAL AX , BX , F , TOL

AN AP PROX I MAT I O N TH E I NT E RVAL ( AX , BX )

I NPUT • •

X TO Til E PO INT WHERE IS D E T E RM I N ED .

L EFT ENDPO I NT OF I N I T I AL I NTERVAL R I GHT ENDPO I NT OF I N I T I AL I NTERVAL

F ATTA I N S A M I N I MU M

AX BX F F U NCT I O N S U B PROGRAM WII I CH EVAL UATES F ( X ) FOR ANY X

I N TH E I NTERVAL ( AX , aX ) TOL D E S I RED L E NG TH OF TH E I NTERVAL OF UNCERTA I NTY OF TH E F I NAL

RESULT ( . G E . 0 . 0 0 0 )

OUTPUT • •

FM I N ABC I S SA APPROX I MAT I NG T H E PO INT WHERE p ATTA I N S A M I N I MUM

atl

C TH E METHOD U S E D I S A COM B I NAT I ON OF GOLDE N SECT I O N SEARCH AN D C SUCCE S S I VE PARABOL I C I NT E RPOLAT I O N . CONVERG E NCE I S N EV E R MUCH S LOW E R

C THAN THAT F O R A F I BONACC I S EARCH . I F F HAS A CONT I N UOU S S E COND C D E R I VAT I V E WH I CH I S POS I T I V E AT TH E M I N I MUM ( WH I CH I S NOT AT AX OR C BX ) , TH EN CONV ERG EN C E IS S U P E RL I N EAR , AND U S UALLY OF TH E O RD E R OF C ABOUT 1 . 3 2 4 • • • •

C TH E F U NCT I O N F I S NEVER EVAL UATED AT TWO PO I NT S CLOS ER TOG ETH E R C THAN E PS * ABS ( FM I N ) + ( TOL/3 ) , WHERE E P S I S AP PROX I MATEL Y TH E SQUARE C ROOT O F TH E RELAT IVE MACH I N E PREC I S I ON . IF F IS A U N I MODAL C F U NCT I O N AND THE COM PUTED VAL U E S O F F ARE ALWAYS U NI MODAL WHEN C S EPARATED B Y AT LEAST E P S * AB S ( X ) + ( TOL/3 ) , TH E N FM I N AP PROX I MAT E S C T H E ABC I S SA O F THE G LOBA� M I N I MU M O F F ON TH E I NT E RVAL AX , a X W I TH C AN ERROR L E S S THAN 3 * E P S *AB S ( FM I N ) + TOL . I F F I S NOT U N I MODAL , C THEN FM I N MAY APPROX I MATE A LOCAL , BUT P E RHAPS NON-GLO BAL , M I N I MU M TO C THE SAME ACCURACY . C TH I S F U NCT ION SUBPROGRAM I S A S L I GHTLY MOD I F I ED V ERS I ON OF TH E C ALGOL 6 0 PROCEDURE LOCALM I N G I VEN I N RI CHARD B R EN T , ALGORI THMS FOR C M I N I M I Z ATION W I THOUT D E R I VATIVES , PRENT I CE - H AL L , I N C . ( 1 9 7 3 ) . C C

C

REAL A , B , C , D , E , E P S , XM , P , Q , R , TOL1 , TOL 2 , U , V , W REAL FU , FV , FW , FX , X

C C I S TH E SQUARED I NVERSE O F TH E GOLD E N RAT I O C

C = 0 . 5 * ( 3 . -SQRT ( S . 0 »

F i g u re A . l Con t i n u e d

C 1 1 1

C E P S I S APPROX I MATELY TH E SOUARE ROOT OF TH E RELAT I VE MACH I N E C PRECI S I ON . C

C

E P S = I . 0 1 0 E P S = EPS/ 2 . 0

TOL l = I . 0 + EPS IF ( TOL l . GT . l . 0 ) GO TO 1 0 E P S = SORT ( E PS )

C I N I T I AL I Z AT ION C

C

A =AX a - a x V"A+ C * ( B-A ) W=V X =V E = O . O F X =F ( X ) FV"FX FW:o:FX

C MA I N LOOP STARTS HERE C

C

2 0 XM = O . S * ( A+B ) TOL I - EPS *ABS ( X ) +TOL/3 . 0 TOL 2 = 2 . 0 * TOL I

C CHECK STO P P I NG CRI TERION C

C I F ( ABS ( X -XM ) . LE . ( TOL 2 - 0 . S * ( B-A» ) GO TO 9 0

C I S GOLDEN-SECT ION NECESSARY C

I F ( AaS ( E ) . LE . TOLl ) GO TO 4 0 C C F I T PARABOLLLLA C

C

R= ( X-W) * ( FX-FV) o- e X-V ) * ( FX -FW ) p - e x -v ) *o- e X -W ) * R 0= 2 . 0 * ( O- R ) I F ( O . GT . O . O ) P--P O =AB S ( O ) R = E E = O

C IS �ARABOLA ACCEPTABLE C

C

I F (ABS l PJ . GE . ABS ( O . 5 *0 * R » GO TO 4 0 I F ( P . LE . O* ( A-X » GO TO 4 Q I F ( P . GE . O* ( B-X » GO TO 4 6

C A PARABOL I C INTERPOLATION STEP C

C

D=P/O U"X+D

C F MUST NOT BE EVALUATED TOO CLOSE TO AX 0& ax C

c

I F « U-A ) . LT . TOL 2 ) D-S IGN ( TOL 1 , XH-X ) I F « B-U ) . LT . TOL2 ) D = S I G N ( TOL l , XM-X ) GO TO 5 0

C A GOLDEN-SEC'1'IOH STEP C

4 0 IF ( X . GB . XM ) B-A-X

. F i g u r e A . l Con t i n u e d

C

I F ( X . LT . XH ) E - B-X D "'C * E

1 1 2

C F MUST NOT B E EVAL UATED TOO CLO S E TO X C

C

5 0 I F ( AB S ( D ) . G E . TOLl ) U=X +D I F ( AB S ( D ) . LT . TOLl ) U = X + S I G N ( TOL l , D ) F U = F ( U )

C UPDATE A , B , V , W , AND X C

C

I F ( F U . GT . FX ) GO TO 6 0 I F ( U . G E . X ) A=X I F ( U . LT . X ) B=X V=W FV= FW W=X FW=FX X =U F X = F U GO T O 2 0

6 0 I F ( U . LT . X ) A=U IF ( U . G E . X ) B=U I F ( F U . L E . FW ) GO TO 7 0 I F ( W . EQ . X ) GO TO 7 0 I F ( F U . L E . FV ) GO TO 8 0 I F ( V . EQ . X ) G O TO 8 0 I F ( V . EQ . W ) G O TO 8 0 G O T O 2 0

7 0 V =W ' FV=FW W=U FW=FU G O TO 2 0

8 0 V=U FV "' F U GO TO 2 0

C END O F MAI N LOOP C

C C C C C C C C C C C C C C C

9 0 FM I N =X RETU RN END S U B ROUTI N E TDMS ( A , B , C , X , D , N , I E R , WO RK ) D I MEN S I ON A ( N ) , B ( N ) , C ( N ) , X ( N ) , D ( N ) , WORK ( l )

TDMS SOLVES A S ET O F ALG EBRA I C EQUAT I O N S OF THE FORM :

A ( I ) * X ( I - l ) +B ( I ) * X ( I ) +C ( I ) * X ( I + l ) =D ( I )

FOR I s l , 2 , • • • • , N , U S I NG T H E TRI D I AGONAL MATRI X SOLUT I ON .

OTH E R VAR I ABLES :

I ER .. 0 NORMAL RETU RN - 1 - - ERROR B E CAU S E N I S L E S S THAN 1 a 2 -- EQUAT I ON S ARE S I NG U LAR

WORK A WORK ARRAY D I MENSI ONED AT L EAST 2 * ( N-l )

I F ( N . LE . 1 ) GO TO 4 I ER = 2 I F ( B ( l ) . EO . O . O ) RETU RN N M 1 "N-l BOL D =C ( l ) /B ( l ) GOLD-D ( l ) /B ( l ) WORK ( l ) -GOLD

Fi g u re A . l Co n t i nued

WORK ( 1 + NM1 ) -BOLD I F ( N . EQ . 2 ) GO TO 2 DO 1 I " 2 , NM 1 D I V- B ( I ) -A ( I ) * BOLD I F ( D I V . EQ . O . O ) RETU RN GOLD = ( D ( I ) -A ( I ) *GOLD ) /D I V WORK ( I ) "GOLD BOLD=C ( I ) /D I V

1 WORK ( I + NM1 ) =BOLD 2 D I V= B ( N ) -A ( N ) * BOLD

I F ( D I V . EQ . O . O ) RETURN GOLD- ( D ( N ) -A ( N ) *GOLD ) /D I V X ( N ) -GOL D L = N D O 3 I = 2 , N

11. 3

L = L - 1 GOLD=WORK ( L ) -WORK ( L +NM1 ) *GOLD

3 X ( L ) -GOLD I E R=O RETURN

4 I E R= l I F ( N . LE . O ) RETURN I E R" 2 I F ( B ( l ) . EQ . O . O ) RETURN X ( l ) =0 ( 1 ) / 8 ( 1 ) I E R"O RETURN END

F i g u re A . l C o n t i n u e d

C

1 1 4 PROGRAM KAVL ( TAPE7 , TAPE 8 , OUTPUT )

c . • . TH I S PROGRAM U S ES H E RM I TE CU B I C SPL I N E CUVRE C . . . F I TT I NG TO EVAL UATE KAV/ L VAL U E S AT D E S I RED FLOW C • • • RATES ALONG TH E CURV E G E N E RATED BY P E RFORMAN CE C • • • KAV/ L VAL UES AND TH E I R RESP ECTIVE L I QU I D F LOW RA T E S . C C . • • I N PUT TAP E 7 I S ODTAINED D I R ECTLY FROM PROGRAM PDA P OUTPU'r . C

C

PARAMETER ( NPTS = 2 0 ) LOG I CAL H E RM I T REAL KAVLS ( N PTS ) , KAVL S 8 REAL PERC ( N PTS ) , WO RK ( l O O ) DI MENSION A ( NPTS ) , B ( N PTS ) , C ( N PTS ) DATA H E RM I T/ . TRU E . /

C • • • READ CALCULATED VAL U E S OF KAV/L AND TH E I R RESPECT I V E L I QU I D FLOW RA T E P E RC E NTAG E S c . . .

C

1 C

DO 1 I " l , N PTS READ ( 7 , 1 0 0 , END = 9 9 ) KAVLS ( I ) , PE RC ( I ) NPT=I

CONT I N U E

c . . . EVAL UATE F LOW RATES POSTU LATED I N THE D E S I GN AC C I D E NT FO R WH I CH NO MAN U FACTU RER DATA I S AVA ILABLE C • • •

C

C

9 9 PR I NT 6 0 0 READ * , EVAL I F ( EVAL . EQ . O ) STOP I F ( EVAL . EQ . l ) THEN

PRINT * , ' ENTE R PERCENTAGE ' READ * , PERCI N CALL SPLF I T ( P E RC , KAVLS , NPT , A , I E R ) KAVLS 8 =SPLFCN ( PE RC I N , PERC , KAVLS , N PT , A , I E R ) PRINT * , ' AT ' , PERC I N , ' KAV/ L = ' , KAVLS 8 WRI TE ( 8 , 1 0 2 ) P E RC I N , KAVLS 8

ENDI F GO TO 9 9

C • • • FO RMAT STATEMENTS C

C

1 0 0 FORMAT 1 0 2 FO RMAT 6 0 0 FORMAT

1 2 3

END

( 2 F l O . 4 ) ( S X , ' AT ' ' , F 6 . 2 , SX , ' KAV/L . ' , F6 . 4 , / ) ( ' I F KAV/L EVAL UAT I ON I S D E S I RED ALO NG T H E CURV E ' ,

, OF VAL UE S , ENTER 1 1 ' , / , ' I F NOT , , ENTER 0 ' )

SUBROU T I N E SPLF I T ( X , Y , N , HE RM I TE , A , B , C , I NT ER , I ER ) LOG I CAL HERMITE D I MENSION X ( N ) , Y I N ) , A ( N ) , B ( N ) , C ( N )

C SUBROU T I N E SPLFIT CALCULATES TH E CO EF F I C I ENTS O F AN I NTERPOLAT I NG C NATURAL OR H E RM I TE CUB I C S P L I N E . SPLFCN CAN BE U S ED TO EVAU LAT E C THE I NTERPOLAT I NG POLYNOM I AL . C C C C C C C C C C C

TH E 1 -0 ARRAYS X AND Y CONTAIN N DATA PA IRS WH I CH ARE TO BE TH E KNOTS OF TH E I NT ERPOLATI O N . TH E I NTERPOLATI NG FU NCT I ON I S D E F I NED BY

WHERE

AND

Y =Y ( I ) + « C ( I ) *T+B ( I » *T+A ( I » * T

TaX-X ( I ) X ( I ) . LE . X . LT . X ( I +1 ) A , B , AND C ARE ARRAYS OF L E NGTH N CONTA I N I NG T H E I NTE RPOLAT I NG CO EF F I C I ENT S .

F i g u re A . 2 L i s t i n g o f Prog ram KAV L

• ,

C C C C C C C C C C C C C C C C C C C

..

1 1 5 I I E RM I T E I S A LOG I CAL VAR I AB L E ( . T RU E . OR . FALSE . ) WH I CH D E T E RM I N E S WH ETH E R A H E RM I T E O R NATURAL S PL I N E I S U S ED AS FOLLOW S .

H E RM I TE = . TRU E . .. = > H E RM I TE S P L I N E U S ED I F N > 3 lI E RM I T E = . F A L S E . = = > NATU RAL S P L I N E U S ED I F N > 3 N = 2 O R 3 = = > POLYNOM I AL I NT ERPOLAT ION U S ED

I N'l' E R I S AN I NTEGER VAR I A B L E D E S I GNAT I NG THE I N'l' E HVAL OF I NT E H PO LA'l' I O N . S PL F I 'l' I N I TAL I Z ES I NT E l{ '1'0 1

I E R I S AN I NTEG E R ERROR F LAG D E F I N ED A S FOLLOWS .

I ER = O I E R = 1 I E R= 2

= � > N O E RROR - - NORMAL RETU RN = = > N . LE . 1

= = > X A RRAY NOT I N STRI C'l'LY ASCENDI NG O R D E S CEND I NG O RD E R

C • • • • • I N I T I AL I Z E I NT E R = l

I NTER

C C • • • • • CH E C K I N PU T DATA

I E R = 1 I F ( N . LE . l ) RETURN I E R = 2 I F ( X ( 2 ) . EO . X ( I » RETU RN I F ( N . GT . 2 ) GO TO 1

C • • • • • F I T A STRA IGHT L I N E TO DATA A ( l ) = ( Y ( 2 ) -Y ( I » / ( X ( 2 ) -X ( 1 » B ( I ) = O . O C ( I ) = O . O RETURN

1 H' ( X ( 2 ) . LT . X ( l » GO TO 3 DO 2 I =3 , N I F ( X ( I ) . L E . X ( I - l » RETU RN

2 CONT I N U E G O T O 5

3 DO 4 I = 3 , N I F ( X ( I ) . G E . X ( I - l » RETU RN

4 CONT I NU E 5 I ER = O

I F ( N . GT . 3 ) G O TO 6 C • • • • • F I T A PARABOLA TO DATA

X 2X l =X ( 2 ) -X ( 1 ) X 3 X l =X ( 3 ) -X ( 1 ) Y 2 Y l =Y ( 2 ) -Y ( 1 ) y 3 Y l =Y ( 3 ) -y ( 1 ) D =X 2 X l * X 3 X l D =D * X 2 X I -D * X 3 X l B B = ( Y 2 Y l * X 3 X l -Y 3Y l * X 2X l ) /D AA = ( X 2 X l * X 2X l * Y 3 Y l -X 3 X l * X3 X 1 * Y 2 Y l ) /D A ( 1 ) =AA 8 ( 1 ) =B13 C ( I ) = O . O 8 ( 2 ) =B 8 A ( 2 ) =AA + 2 . *8B * X 2X 1 C ( 2 ) = O . 0 RETU RN

6 IF ( HE RM I T E ) GO TO 1 1 C • • • • • NATU RAL S P L I N E I N T E RPOLAT I ON

l'1= N - l C I =X ( 2 ) -X ( 1 ) C ( I ) =CI B I = ( Y ( 2 ) -Y ( I » / C I 6 ( 2 ) = B I DO 7 I = 2 , M CC=X ( I + 1 ) -X ( I )

F i g u re A . 2 Co n t i nued

C ( I ) =CC B B = ( Y ( I + l ) -Y ( I » / CC B ( I ) = B B - B I B I = B B AA =CC + C I A ( I ) =AA+AA

7 C I "'CC DO 8 I = 3 , M AA=C ( I - l ) /A ( I - l ) A ( I ) =A ( I ) -AA * C ( I - l )

8 B ( I ) = B ( I ) -AA * B ( I - l ) S ( 1 ) = O . O B ( N ) = O . O Il B = B ( M ) /A ( foI ) B ( M ) = B B JJ = M D O 9 I =3 , M J J =J J - l B B = ( B ( J J ) -C ( J J ) * B B ) /A ( J J )

9 B ( J J ) = B B B I = O . O J J =N DO 1 0 I = l , M J J =JJ - l B B= B ( J J )

1 1 6

CC=C ( J J ) A ( JJ ) = ( Y ( J J + l ) -Y ( JJ » /CC-CC * ( B I +B B + B B ) C ( J J ) = ( B I - BB ) /CC B ( J J ) = B B + B B + B B

1 0 ' B I = B B RETURN

C • • • • • HE RM I T E SPL I N E

C

1 1 RM 3 - ( Y ( 2 ) -Y ( 1 ) ) / ( X ( 2 ) -X ( l » Tl = RM 3 - ( Y ( 2 ) -Y ( 3 » / ( X ( 2 ) -X ( 3 » RM 2 =RM3 +Tl RM l =RM 2 +T l N O = N - 2 D O 1 2 I = l , N RM 4 = RM 3 - RM 2 +RM 3 I F ( I . L E . N O ) RM 4 = ( Y ( I + 2 ) -Y ( I + l » / ( X ( I + 2 ) - X ( I + l » T l =ABS ( RM 4 - RM 3 ) 'l' 2 "ABS ( RM 2 -RtH ) B B ='l' l + T 2 A ( I ) = 0 . 5 * ( RM 2 +RM 3 ) I F ( B B . N E . O . O ) A ( I ) = ( T l * RM 2 +T 2 * RM 3 ) / B B RM l = RM 2 RM 2 .. RM 3

1 2 RM 3 =RM 4 N O " N-l DO 1 3 I "' l , N O Tl = l . O/ ( X ( I + l ) -X ( I » T 2 - ( Y ( I + l ) -Y ( I » *Tl B B = ( A ( I ) +A ( I +l ) -T2-T 2 ) *Tl B ( I ) = - BB + ( T 2-A ( I » *Tl

13 C ( I ) &BB *Tl RETURN END REAL FUNCT I ON S PLFCN ( X I N , X , Y , N , A , B , C , I NT ER , I ER ) D I M E N S I O N X ( N ) , y e N ) , A ( N ) , B ( N ) , C ( N )

C FUNCT I ON S PLFCN EVALUATES A S P L I N E I N T ERPOLATION AT X I N

C C ARGU MENTS X , Y , N , A , B AND C ARE DE F I N ED AS I N SPLF I T .

C C I NTER I S AN I NT EG E R VAR I ABLE I N D I CAT ING THE INTERVAL U S ED

C I N TH E I NT ERPOLAT ION ( EXTRAPOLAT ION ) C

F i q u re A . 2 Co n t i n u e d

c c C C C C

1 1 7 I E R I S AN I NT EG E R ERROR FLAG D E F I NED AS FOLLOWS .

I E R= O I E R= l I E R = 2

= = ) N O ERRO R - - NORMAL RETURN ==) X I N . I.T . XM I N - - EXTRAPOLAT I ON = = ) X I N . GT . XMAX -- EXTRAPOLAT ION

C W H E R E XMAX AND XM I N A R E TH E MAX I MUM AND M I N I MU M VAL U E S C O F TH E X -ARRAY , R E S PEC'U V E L Y . C

I F ( I NTE R . LE . O . O R �I NTER . GT . N ) I NTER=l I E R = O T;::X I N - X ( I N T E R ) U =X I N -X ( I NTER+ l ) I F ( U *T . GT . O . O ) GO TO 1 SPLFCN =Y ( I N TER ) + « C ( I NTER ) *T+B ( I NT E R » * T+A ( I NT E R » * T RETURN

1 K = l I F ( T . LT . O . O ) K =-K I F « T-U ) . LT . O . 0 ) K =- K

2 I NT E R = I N T E R+ K I F « N- I N T E R ) * I NTER . LE . O ) GO T O 3 T =-X I N -X ( I NTER ) U = X I N-X ( I N T E R + l ) I F ( U *T . GT . O . O ) GO TO 2 S PLFCN =Y ( I NT E R ) + « ( C ( I N T ER ) * T+B ( I NTER » *T+A ( I NT E R » * T RETURN

3 I E R = l

I F ( T . LT . O . O ) I E R= 2 I NTER=INTER-K S PL F C N =Y ( I NT E R ) + « C ( I NTER ) * T + B ( I NT E R » * T+A ( I NT E R » * T RETURN END

F i g u re A . 2 Co n t i n u e d

1 1 8

C PROGRAM TMCON ( TAPE 7 , TAPE 6 , TAPE 9 , OUTPUT )

c . . . PROGRAM TMCON DETE RM I N E S TH E F I RST ORDER SYSTEM T I M E CONSTANT OF C AN U H S COOL I NG TOW E R BAS I N S Y S TEM F ROM TH E T I M E -D E P EN D E NT B AS I N C TEM P E RA TU R E G I V I::N 'l'H E T I ME H I STORY OF H EAT I N P U T AND WI::T- BU LB C TEM P E RAT U R E

C C . . . S E T PARAMETERS -- S E E AL S O F U NCTION ST2 C C MSTEP "" MAX I MUM NUMBER OF T I M E STEPS ( USED I N D I MENS I ONI NG ) C MTOW E R .. MAX I MUM NUMBER O F TOWERS ALLOWE D ( US ED I N D I M E NS I ON I NG ) C

PARAMETER ( MSTEP-1 0 1 , MTOW E R= 5 ) C C . . . D ECLARE VAR I ABLES C

C c . . . C

1 C c • • •

C C C

2 C c . . • C C C

C C • • • C

C

EXTERNAL ST2 L OG I CAL NO EVAP , CONWB COMMO N NTOWE R , TSTE P , NSTEP , TI M E ( MSTEP ) , T ( MSTEP ) , BM ( MSTEP ) ,

1 TWB ( MSTEP ) , Q ( MSTEP , MTOWE R ) , FLOW ( MSTEP , MTOWE R ) , 2 HL ( MSTEP ) , HM ( MSTEP ) , TF I T ( MSTEP )

DATA TOL/ 1 . 0 E-5/ , ALOW/ 0 . 0 5/ , AHGH/ 0 . 5/

R EAD I NPUT DATA

READ ( 7 , 1 01 ) NTOWER , BASMAS , S , TSTEP , NSTP , NOEVAP , CONWB NSTEP=NSTP+1 DO 2 I =l , NSTEP READ ( 7 , 1 0 2 ) T I ME ( I ) , T ( I ) , BM ( I ) , SS , TWB ( I ) , ( O ( I , J ) , FLOW ( I , J ) ,

1 J -l , NTOWER) S U M = O . O S U ML - O . O DO 1 J=l , NTOWER SUM=SUM+Q ( I , J ) SUML = S U ML+Q ( I , J ) /FLOW ( I , J ) CONT I N U E

TH E NEXT L I NE I S TST E P * OTOTAL / « BAS I N MASS ) * ( SPECI F I C H EAT » , B U T I N ENG L I S H U N I TS TH E S P EC I F I C HEAT OF WATER I S AP PROX I MATELY 1 BTU/ LBM-F

H M ( I ) ""TSTEP * SUM/ BM ( I ) HL ( I ) "SUML/ FLOAT ( NTOWER ) CONT I N U E

D E TE RM I N E T H E VALUE OF ' TAU ' WH I CH M I N I M I Z ES THE S U M OF TH E SQUARES OF TH E D I F FERENCES BETWEEN TH E FI RST O RD E R A PPROX I MAT I ON AND T H E PRED I CT IONS OF U H SS I M

A-FM I N ( ALOW , AHGH , ST 2 , TOL ) TAU =TSTEP/A

WRI TE OUT TH E RESULTS

WRI TE ( 6 , 1 03 ) NTOWER , BASMAS , S , TSTEP , NSTP , NOEVAP , CONWB , TAU WRITE ( 6 , 1 0 4 ) WRITE ( 6 , 1 05 ) ( TIME ( I ) , T ( I ) , TF I T ( I ) , TWB ( I ) , I - l , NSTEP ) WR I TE ( 9 , 1 06 ) NSTEP , TAU WRITE ( 9 , 1 07 ) ( T I ME ( I ) , T ( I ) , TF I T ( I ) , TWB ( I ) , I - 1 , NSTEP ) STOP

1 0 1 FORMAT 1 0 2 FORMAT 1 0 3 FO RMAT

( I I 0 , 3 E 1 0 . 3 , I 1 0 , 2L 1 0 ) ( 8 E 1 0 . 3 ) ( ' 1 T I M E CONSTAN T ANALYS I S PROGRAM ' , // , ' I N P U T DATA ' , // ,

' , 1 1 0 , / , ' , l PE 1 0 . 3 , / ,

1 2 3 4

, NUMBER OF TOWERS BAS I N MAS S ( LBM ) SAL I N I TY OF BAS I N ( PPM ) T I ME STEP S I Z E ( H R )

, , 3 PF 1 0 . 2 , / , ' , O PF I O . 2 , / ,

F i g u re A , 3 L i s t i n g o f P ro g ram TMCOM

C

5 6 7 8

1 0 4 FORMAT

1 0 5 1 06 1 07

1 FORMAT FO RMAT FORMAT E N D

1 1 9 NUMB E R OF S T E PS TO B E TAK E N ' , 1 1 0 , / , NO EVAPO RAT I ON ' , L I O , / , CONSTANT WET-BULB T EM P E RATURE ' , L I O , // ,

, T I M E CON STANT ( H R ) ' , F I 0 . 2 , // ) ( 6 X , ' TI ME ' , 4 X , • TBAS I N ' , 6 X , ' TF I T ' , 7 X , ' TW B ' , / ,

3 X , ' ( HOURS ) " 7 X , ' ( F ) " 7 X , , ( F ) ' , 7 X , ' ( F ) • , I ) ( 4 F I O . l ) ( I 1 0 , I P E l O . 3 ) ( l P 4 E I O . 3 )

F U NCT I ON S T 2 ( A )

C • • • S ET PARAMETERS - - S E E AL S O PROG RAM TMCON C C C C

C

M S T EP MTml E R

MAX I MUM NUMBE R OF T I ME STEPS ( USED I N D I MENS I ON I NG ) MAX I MU M N U M B E R OF TOWERS ALLOW ED ( US E D I N D I M E N S I ON I NG )

PARAMETER ( MSTEP=1 0 1 , MTOW E R= 5 )

c . . . D ECLARE VAR I ABLES C

C

COMMON NTOWE R , TSTE P , NSTEP , T I M E ( MSTEP ) , T ( MSTEP ) , BM ( MSTEP ) , 1 TWB ( MSTEP ) , Q ( MS'l'EP , MTOW E R ) , F LOW ( MSTEP , MTOW E R ) , 2 HL ( MSTEP ) , HM ( MS T E P ) , TF I T ( MS T E P )

c . . . D E TE RM I N E TH E F I RST ORD E R R ESPONS E FOR ' A ' WH E R E A = T S T E P/TAU C

C

T F I T ( l ) =T ( l ) DO 1 1 = 2 , N STEP T F I T ( I ) = ( TF I T ( I -l ) +A* ( TWB ( I ) -HL ( I » +HM ( I » / ( l . O +A )

1 CONT I N U E

c . . . CALCU LATE SUM OF S QUARE OF D I F F EREN C E S C

C C C C C C C C C C C C C C C C C C C C C C C C C

ST 2 = 0 . O DO 2 I = 2 , NSTEP S T 2 = ST 2 + ( T ( I ) -TF I T ( I » * * 2

2 CONT I N U E RETU RN END REAL FUNCT ION FM I N ( AX , B X , F , TOI. ) REAL AX , BX , F , TOL E XT E RNAL F

AN APPROX I MAT I ON X TO TH E PO I NT WHERE F ATTA I N S A M I N I MU M ON TH E I NT E RVAL ( AX , BX ) IS D E T E RM I N ED .

I N PUT • •

AX L E FT ENDPO I NT OF I N I TIAL I NTERVAL ax R I G H T ENDPO I NT OF I N I T I AL I N T E RVAL F F U NCT ION SUBPROG RAM W H I CH EVALUAT E S F ( X ) FOR ANY X

I N THE I NTERVAL ( AX , BX ) TOL D E S I RED L E NGTH OF TH E I NTERVAL OF U NCERTAI NTY OF TH E F I NAL

R E S U LT ( . G E . 0 . 00 0 )

OUTPUT • •

F M I N ABC I S SA APPROX I MAT ING THE PO'I NT W H E R E F ATTA I N S A M I N I MUM

THE METHOD U S ED IS A COM B I NAT I ON OF GOLDEN S ECT I O N S EARCH AND SUCCE S S IVE PARABOL I C I NT E RPOLAT I ON . CONVE RG ENCE IS N EV E R MUCH SLOW E R THAN THAT FOR A F I BONAC C I S EARCH . I F F HAS A CONT I N UOU S S ECOND D E R I VATI V E WH ICH IS PO S I T IV E AT TH E M I N I MUM ( WH I CH I S NOT AT AX OR BX ) , T H E N CONVERG ENCE IS SUPERL I N EAR, AND USUALLY OF TH E O RD E R OF


1 2 0 C ABOUT 1 . 3 2 4 • • • •

C T H E F U NCT ION F I S NEVER EVAL UATED AT TWO PO INTS CLO S ER TOG E TH E R C THAN E P S *ABS ( FM I N ) + ( TOL/ 3 ) , WHERE EPS I S APPROX I MATELY TH E SQUARE C ROOT O F THE RELATI VE MACH I N E P R EC I S I ON . IF F I S A U N I MODAL C F U NCT I O N AND TH E COM PUT ED VAL U E S OF F ARE ALWAY S U N I MODAI. WH E N C S E PARAT ED B Y A T LEAST E P S *ABS ( X ) + ( TOL/ 3 ) , TH E N FM I N A P PROX I MAT E S C T H E A BC I S SA O F TH E GLOBAL M I N I MU M OF F ON TH E I NT E RVAL AX , BX W I TH C AN ERROR L E S S THAN 3 * E PS *ABS ( FM I N ) + TOL . I F F I S NOT U N I MODAL , C TH E N FM I N �t AY AP PROX I MA'l' E A LOCAL , BUT P E RHAPS NON-G LOBAL , M I N I MU M TO C TH E SAME ACCURACY . C Til l S F U NCT ION SUB PROGRAM I S A S L I GHTLY MOD I F I ED VERS I ON Or' TH E C ALGOr. 6 0 PROCEDURE LOCAL M I N G I VEN I N RI CHARD B RE NT , ALG O R I T H l1S F O R C M I N I M I Z AT ION W I TH O U T D E R I VAT I V E S , PR ENT I CE - HALL , I N C . ( 1 9 7 3 ) . C C

C

REAL A , B , C , D , E , E PS , XM , P , Q , R , TOLl , TOL 2 , U , V , W REAL FU , FV , FW , FX , X

C C I S TH E SQUARED I NVERSE OF TH E GOLDEN RAT I O C

C = 0 . 5 * ( 3 . -SQRT ( 5 . 0 » C C EPS I S APPROX I MATELY TH E SQUARE ROOT OF TH E RELATIVE MACH I N E: C PREC I S I ON . C

C

E P S = I . 0 1 0 E P S = EP S/ 2 . 0

TOL l =l . O+EPS IF ( TOLl . GT . l . 0 ) GO TO 1 0 E P S = SQRT ( EPS )

C I N I T I AL I Z ATION C

C

A=AX B = BX V =A+C * ( B-A ) W=V X =-V E = O . O F X = F ( X ) FV = F X FW= F X

C MA I N LOOP STARTS H E RE C

C

2 0 XM= 0 . 5 * ( A+B ) TOL l =EPS *ABS ( X ) +TOL/3 . 0 TOL 2 = 2 . 0 *TOL I

C CHECK STOPP I NG CR I TERION C

I F ( AB S ( X -XM ) . LE . ( TOL 2- 0 . 5 * ( B-A» ) GO TO 9 0 C C I S GOLDEN-SECT ION NECESSARY C

I F ( ABS ( E ) . LE . TOL l ) GO TO 4 0 C C F I T PARABOLA C

C

R= ( X-W ) * ( FX-FV ) Q= ( X -V ) * ( FX -r'W ) P = ( X-V ) * Q- ( X -W ) *R Q= 2 . 0 * ( Q- R ) I F ( Q . GT . 0 . 0 ) P = - P Q=ABS ( Q ) R=E E=D

F i g u re A . 3 Cont i n ued

1 2 1 C I S PARABOLA ACCE PTABL E C

C

I F ( ABS ( P ) . GE . ABS ( 0 . 5 * O * R » GO TO 4 0 I F ( P . LE . O * ( A- X » GO TO 4 0 I F ( P . GE . O * ( B-X » G O TO 4 0

C A PARABOL I C I NTERPOLAT ION STEP C

C

D = P/ O U = X +D

C F MUST NOT B E EVALUATED TOO CLO S E TO AX OR BX C

C

I F « U-A ) . LT . TOL 2 ) D - S I G N ( TOL l , XM-X ) I F « B- U ) . LT . TOL 2 ) D - S I G N ( TOLl , XM-X ) G O TO S O

C A GOLDEN- SECT ION STEP C

C

4 0 I F ( X . G E . XM ) E -A-X IF ( X . LT . XM ) E = B-X D = C * E

C F MUST NOT B E EVALUATED TOO CLOS E TO X C

C

S O I F ( ABS ( D ) . G E . TOLl ) U=X+D IF ( AB S ( D ) . LT . TOLl ) U=X + S I G N c 'rOLl , D ) F U = F ( U )

C U PDATE A , B , V , W , AND X C

C C C

I F ( FU . GT . FX ) GO TO 6 0 I F ( U . GE . X ) A=X IF ( U . LT . X ) BaX V=W FV= FW W=X FW=FX X "U FX =FU GO TO 2 0

6 0 I F ( U . LT . X ) A-U IF ( U . GE . X ) B=U IF ( F U . LE . PW ) GO TO 7 0 I F ( W . EO . X ) GO TO 7 0 I F ( FU . LE . FV ) GO TO 8 0 I F ( V . EO . X ) GO TO 8 0 I F ( V . EO . W ) GO T O 8 0 G O TO 2 0

7 0 V=W FV=FW W=U FW=FU GO TO 2 0

8 0 V=U FV=FU GO TO 2 0

END O F MAIN LOOP

9 0 FM I N = X RETURN END


C C • • •

c • • •

c • • •

c • • •

c • • •

c . . . c . . . C

1 2 2 PROGRAM UHS S I M ( TAPE 5 , TAPE 8 , TAPE 6 , TAPE 7 , OUTPUT )

PROGRAM U H S S I M ( U LT I MATE H E AT S I N K S I MU LAT I ON ) S I MULATES TH E PERFORMANCE OF AN U LT I MATE H EAT S I N K TOWER/ BAS I N SYSTEM . TH E BAS I N TEMPERATU R E , BAS I N MAS S , AND BAS I N SAL I N I TY ARE CALCU LATED W I TH RESPECT TO T I M E FOL LOW I NG A LOCA ( LOSS OF COOLANT ACC I DENT ) . WORST-CAS E METEOROLOG I CAL CON D I TIONS OF WET- B U LB TEMPERATURE ALONG W I TH D E S I G N - BAS I S ACC I DENT FLOW RATE VAL U E S ARE U S ED I N TH E S I MULAT I ON .

C • • • COD E NWS SUPPL I ES TH E WORST-CAS E WET-BULB TEM PERATU RES . C C C • • • D ECLARE EXTERNALS C C BAS I N - A S UBROUT I N E U S ED TO COM PUTE DERIVAT IVES OF C BAS I N TEMPERATURE AND MASS ( NAME PAS S ED TO C SUBROU T I N E ODE AS AN ARGUMENT ) C

EXTERNAL BAS I N C C • • • D ECLARE M ISCELLANEOU S VARI ABLE TYP E S C C I TOWER - MAX I MUM NUMBER OF L I QU I D FLOW RATE , KAV/ L , C AND L/G VAL U E S FOR EACH TOWER C J TOWER - MAX I MUM NUMBER O F TOWERS C I H EAT - MAX I MUM NUMB E R OF HEAT LOAD VAL UES FOR EACH C TOWER C I WNDL - MAX I MUM L E NGTH OF WET-BULB TEMPERATU RE W I N DOW C

C

PARAMETER ( I TOWE R= 5 0 , JTOWE R= 5 , I H EAT= 5 0 , IWNDL= 2 0 0 ) LOG I CAL HERM I T , NO EVAP , CONWB CHARACTER* 4 0 HEAD E R CHARACT ER* 8 YMDHH , YMDH ( I WND L )

C • • • D ECLARE REAL ARRAYS C C C C C C C C C

C

LOVERG L KAVL LOVRG S KAVLS L S LOLD

- L I QU I D OVER GAS F LOW RATE RAT I O - L I QU I D FLOW RATE - TOWER CHARACTER I S T I C - ARRAY CONTA I N I NG T I M E-VARY I NG L/G VAL U E S - AR RAY CONTA I N I NG T I ME-VARY I NG KAV/L VAL U E S - ARRAY CONTA I N I NG T I ME-VARY I NG L VAL U E S - DUMMY VAR I ABLE U S ED FOR L VAL UE COMPARI SONS

REAL LOVERG , L , KAVL , LOLD ( JTOWER) REAL LOVRG S ( I TOWE R , JTOWER) , KAVLS ( I TOWER , JTOWER ) , LS ( I TOWE R , JTOW E R )

C • • • DECLARE ARRAYS C C C C C

C

Y Y P WORK

- ARRAY STORI NG I NT EGRAT ION VAR I ABLES ( T AND M ) - ARRAY STORI NG DER I VAT IVES - WORK ARRAY FOR SUBROU T I N E ODE

D I MENS I ON Y ( 2 ) , YP ( 2 ) , WORK ( 4 2 6 )

C • • • COMMON BLOCKS C

C

COMMON /TOWERC/ LOVERG ( JTOWE R ) , L ( JTOWER) , KAVL ( J TOWE R ) , NH ( J TOWER) , 1 NL ( JTOWER) , NTOW E R , SALT

COMMON /TOWERC2/ TE ( I TOWER , JTOWER ) , ALG ( I TOWER , JTOW E R ) , 1 AL ( I TOWE R , JTOWE R ) , AK ( I TOWER , JTOWE R )

COMMON /HEATC/ Q ( I H EAT , J TOW E R ) , TME ( I H EAT , JTOWER ) , AH ( I H E AT , JTOWER ) COMMON /AI RC/ WB ( IWNDL ) , TM ( I WNDL ) , AWB ( IWNDL ) , BWB ( IWNDL ) ,

1 CWB ( IWNDL ) , NAMB , RWB ( IWNDL ) COMMON /OUT/ QQ ( JTOWER) , TWB , S COMMON /EVAPC/ NOEVAP

F i q u re A . 5 L i s t i n q o f Prog ram UHSS I M

1 2 3 c . . . SET M I SCELLANEOUS DATA C C C C C C C C C

C c . . . C C C C C C C C C C C C

C

N EOU RE L E RR A S B ERR H E RM I T

D I NTER START

- NUMB ER OF EQUAT IONS TO BE I NTEGRATED - RELAT IVE E RROR CR I TERION FOR I NTEGRATION ROUT I N E - ABSOLUTE E RROR CR I TERI ON F O R I NTEGRAT I ON ROUT I N E - LOG I CAL VAR I ABLE U S ED TO S P E C I F Y H E RM I TE S PL I N E

I NTERPOLAT ION - DATA I NTERVAL - I N I TIAL STARTI NG LOCAT ION

DATA N EON/ 2/ , RELE RR/ l . E- 4/ , AB S ERR/ l . E- 4/ , H E RH I T/ . TRUE . / DATA D I NTER/ l . O/ , START/ O . O/

READ I N I T I AL TOWER I NFORMATION

TSTEP N STEP BAS MAS S NTOWER NO EVAP

CONWB

BASTMP

- TI ME STEP - NUMBER OF T I ME STEPS TO B E TAK E N - I N I TIAL BAS I N MAS S - SAL I N I TY - NUMBER OF 'rOWE RS - LOG I CAL VAR I ABLE S ET . TRU E . FOR NO EVAPORAT I O N

. FALS E . OTH ERW I S E - LOG I CAL VAR I ABLE S ET . TRUE . FOR CONSTANT WET- BULB

TEMPE RATU RE . FALS E . OTH E RW I S E - I N I TI AL BAS I N TEMPERATU R E

READ ( 5 , * ) NTOW E R , BASMAS , S READ ( 5 , * ) TSTE P , N STEP , NO EVAP , CONWB

C . . . WRI T E I N I T I AL VAL UES TO TAP E 6 AND 7 C

C c . • •

C C C

WRITE ( 6 , 2 0 3 ) NTOWER, BASMAS , S , Ts'rEP , NSTE P , NO EVAP , CONWB WRITE ( 7 , 1 0 3 ) NTOWER , BASMAS , S , TSTEP , NSTEP , NO EVAP , CONWB WRI T E ( 6 , 2 0 2 )

READ T I ME-VARY I NG L I Q U I D FLOW RATE , L , KAV/ L , AND L/G VAL U E S

N L - NUMBE R OF FLOW RA T E VALUES P E R TOWER

READ ( 5 , * ) ( NL ( I ) , 1 =1 , NTOWER) D O 1 I = l , NTOWER NP=NL ( I ) I F ( N P . GT . O ) READ ( 5 , * ) ( T E ( J , I ) , LOVRGS ( J , I ) , KAVLS ( J , I ) , LS ( J , I ) ,

1 J '"'l , NP ) 1 CONT I N U E

C C • • • READ TIME -VARYI NG H EAT LOAD VALUES FOR EACH TOWER C C TME - T I ME OF CORESPOND I NG H EAT LOAD C 0 - H EAT LOAD VALUES C NH - NUMBER OF H EAT LOAD VALUES P E R TOWER C

C

READ ( 5 , * ) ( NH ( I ) , 1 =1 , NTOW E R ) D O 2 I = l , NTOWER NN=NH ( I ) I F ( NN . GT . O ) READ ( 5 , * ) ( TM E ( J , I ) , O ( J , I ) , J - l , NN )

2 CON T I N U E

C . . . I F WET-BULB TEMPE RATURE I S CONSTANT , READ WET-BULB

C • • • TEMPERATURE AND I N I T I AL BAS I N TEMPERATU RE

C

C c . . .

I F ( CONWB ) THEN NAMB:l READ ( 5 , * ) WB ( l ) , BASTMP GO TO 2 1

ELSE

READ W I N DOW TO BE READ PROM TH E WORST-CAS E METEOROLOG I CAL

F i g u re A . 5 Cont i n u e d

1 2 4 c • • • WI NDOWS DETERM I N ED BY PROG RAM NWS AND TH E N START POS I T I ON ; C • • • READ H EAD E R I N FORMAT ION ABOUT WINDOWS LOCATED AT TH E BEG I N N I NG C • • • OF TH E TAP E . ADVANCE TO WINDOW S ELECTED U S I NG DUMM I NG C • • • VAR I ABLES YMDH H , RWBB , WBB . VARY TH E ACC I D E NT START I NG C • • • HOUR U S I NG DUMMY VAR I ABLES YMDH H , RWBB , WBB . C C C C C C C C

C

C

C

NAMB TAPE 8 NWI NDW

N START

- NUMBE R OF AMB I E NT WET-BULB TEMPERATU RES - TAPE OF W I NDOWS DETERM I N ED FROM CODE NWS - I NTEG E R NUMBER COORESPONDI NG TO METEOROLOG I CAL

W I N DOW TO BE READ - PO S I T I ON AT WH I CH TO START READ I NG TWB ;

S E T = 0 I MPL I E S CONSTANT TWB

READ ( 5 , * ) WB B , BAST READ ( 5 , * ) NW INDW , N START READ ( 8 , 1 0 5 ) H EADER R EAD ( 8 , * ) TAU , DELTAT , NRECS , L ENGTH , N SAVE READ ( 8 , * ) MSAVE , NPAS S , MREAD

DO 2 0 I =NSAV E , 2 , -1 I F ( NW I NDW . NE . I ) GO TO 2 0 N= ( FLOAT ( I ) -l ) *LENGTH + ( FLOAT ( I ) -l ) READ ( 8 , 1 0 0 ) ( YMDHH , RWBB , WB B , J = l , N ) GO TO 3

2 0 CONT I N U E

3 READ ( 8 , * ) TMX , I l READ ( 8 , 1 0 0 , EN D = 1 4 ) ( YMDH H , RWBB , WBB , I = 2 , N START ) NAMB=FLOAT ( N STE P ) *TSTEP/DI NTER+l . O BASTMP"'RWBB READ ( 8 , 1 0 0 , END"'1 4 ) ( YMDH ( I ) , RWB ( I ) , WB ( I ) , I =l , NAMB )

END I F

C . . . I N I T I AL I Z E VARI ABLES FOR INTEGRATION C C C C C C C C

C

T I ME I FLAG TOUT Y ( 1 ) Y ( 2 ) S ALT

- I NTEGRAT ION TIME ( HR ) - CONTROL FLAG - CONT I N UO U S TIME ( HR ) - I N I T IAL BAS I N TEMPERATURE ( F ) - I N I T I AL BAS I N MAS S ( LBM ) - I N I TIAL MAS S OF S ALT I N BAS I N

2 1 KOU NT=- l I F LAG = l T I M E = START TOUT=TI ME Y ( l ) =BASTMP Y ( 2 ) =BASMAS SALT=Y ( 2 ) * S XXX=START

C • • • I F THE WET-BULB TEMPERATURE I S NOT CONSTANT , C . . . F I T H E RM I T E CUB I C S PL I N E TO T I ME VB . TEMPERATURE C C • • • CH ECK FOR ' BAD ' WB VALUES W I TH I N WINDOW C

I F ( NAMB . GT . l ) TH EN YYY= 9 9 9 . 0 DO 4 I =l , NAMB I F ( WB ( I ) . GT . 9 9 8 . 0 ) GO TO 4 YYY =WB ( I ) GO TO 5

4 CONT I N U E WRI TE ( 6 , * ) ' NO VAL U E WB FOU ND I N REQ U E STED W I NDOW ' STOP

5 DO 6 I =l , NAMB I F ( WB ( I ) . GT . 9 9 8 . 0 ) WB ( I ) "'YYY YYY =WB ( I )

F i gu re A . S Con t i n u e d

C

TM ( I ) cXXX 6 XXX =XXX+DINTER

1 2 5

CALL SPLF I T ( TM , WB , NAMB , H E RM I T , AWB , BWB , CWB , I NTER , I E R ) I F ( I ER. NE . O ) G O T O 1 5

'

END I F

C . . . H' TH E HEAT LOAD I S T I M E D E P ENDENT , F I T C • • • DATA O F T I ME VS . H EAT LOAD B Y L I NEAR I NTERPOLAT ION C

C

DO 7 I = I , NTOWER IF ( NH ( I ) . L E . l ) GO TO 7 CALL L F I T ( TME ( l , I ) , O ( I , I ) , NH ( I ) , AH ( I , I ) , I E R ) I F ( I E R . NE . O ) GO T O 1 6

7 CONT I N U E

C . . . I F TH E LIOU I D FLOW RATE I S T IME DEPENDENT , F I T DATA O F C . . . T I ME VB . L , T I ME VS . L/G , AND TIME VS . KAV/L C . . . U S I NG LI NEAR I NTERPOLAT ION C

C

DO 8 I - l , NTOWER I F ( NL ( I ) . LE . l ) GO TO 8 CALL L F I T ( TE ( l , I ) , LOVRG S ( l , I ) , NL ( I ) , ALG ( l , I ) , I E R )

I F ( I E R . NE . O ) G O T O 1 7 CALL L F I T ( TE ( l , I ) , KAVLS ( l , I ) , NL ( I ) , AK ( l , I ) , I ER)

I F ( I ER . NE . O ) GO TO 1 8 CALL L F I T ( TE ( I , I ) , LS ( I , I ) , NL ( I ) , AL ( l , I ) , I E R )

I F ( I ER . NE . O ) GO T O 1 9 8 CONT I N U E

C . . . B EG I N ACCI DENT ANALYS I S C C . . . S ET L/G , KAV/ L , L AT EACH TIME STEP C

C

1 2 DO 9 I = I , NTOWER I F ( NL ( I ) . GT . l ) TH EN

LOV ERG ( I ) =SL FCN ( T I HE , TE ( l , I ) , LOVRG S ( l , I ) , NL ( I ) , ALG ( l , I ) , I E R ) KAVL ( I ) -SLFCN ( T I ME , TE ( I , I ) , KAVLS ( l , I ) , NL ( I ) , AK ( l , I ) , I ER ) L ( I ) =SLFCN ( T I ME , TE ( l , I ) , LS ( l , I ) , NL ( I ) , AL ( l , I ) , I E R )

ELS E I F ( NL ( I ) . EO . l ) TH EN LOV ERG ( I ) -LOVRG S ( I , I ) KAVL ( I ) =KAVLS ( l , I ) L ( I ) =LS ( I , I )

END I F 9 CONT INUE

C . . . I F L I O U I D FLOW RATE CHANGES , REI N I T I AL I Z E ODE C

C

c

I F ( T I ME . GT . l ) THEN DO 1 0 I = I , NTOWER I F ( L ( I ) . NE . LOLD ( I » I FLAG - l

1 0 CONT INUE END I F

KOU NT=KOUNT+l I F ( NAMB . EO . l ) Tll EN

TWB"WB ( I ) END I F CALL BAS IN ( T I ME , y , y P )

c . . . WRITE RESU LTS TO TAPE 6 C

C c • • •

C

C

WRITE ( 6 , 1 0 2 ) T I ME , Y , S , TWB , ( 00 ( 1 ) , L ( I ) , I = I , NTOW E R )

WRITE RESU LTS F O R PLOTT I NG T O TAPE 7

WRITE ( 7 , 1 0 4 ) T I ME , Y , S , TWB , ( OO ( I ) , L ( I ) , I � l , NTOW E R ) I F ( KOUNT . G E . NSTEP ) STOP

F i g u re A . S Co n t i n u e d

C • • • BEG I N I N T EGRATION LOOP C

1 2 6

C

TOU'f=TOUT+TSTEP . 1 3 CAL L OD E ( BAS I N , NEQN , Y , T I ME , TOUT , RELERR , ABSERR , I FLAG , WORK )

I F ( I FLAG . EQ . 2 ) TH EN DO 11 1 = 1 , N'rOWER

1 1 LOLD ( I ) =L ( I ) G O TO 1 2

END I F I F ( I FLAG . EQ . 3 ) GO TO 1 3 WR I TE ( 6 , 1 0 1 ) I FLAG , Y , REL ERR , ABS ERR STOP

C • • • FORMAT STATEMENTS C

C

1 0 0 FORMAT ( SX , A , 2 F I 0 . 2 ) 1 01 FORMAT ( I , ' * * * ERROR* * * E RROR NUMBER ' , 1 5 ,

1 • I N BAS I N I NTEG RAT ION ' , 1 , 1 4X , ' DTDT= ' , l P E 1 5 . 3 , ' F/HR ' , I , 2 1 4X , ' DMDT= ' , l PE l S . 3 , ' LBM/HR ' , 1 , 1 4 X , ' REL ERR= ' , l PE 1 4 . 3 , / , 3 1 4 X , ' ABS ERR= ' , l P E 1 4 . 3 , 1/ )

1 0 2 FORMAT ( F I 0 . 1 , F I 0 . 2 , l P EI O . 2 , 3 PF I 0 . l , O P F l O . 1 , l P 2 E1 0 . 2 , : , I , ( 5 0X , 1 I P 2 E I 0 . 2 »

1 0 3 FORMAT ( I I O , l PE I O . 3 , 3 PF I O . 2 , O P F I 0 . 2 , I I O , 2L I 0 ) 1 0 4 FO RMAT ( l P 8 E I O . 3 ) 1 0 5 FORMAT ( A I 2 0 1 FORMAT ( l P 2EI O . 3 ) 2 0 2 FO RMAT ( 6 X , ' TI ME ' , 4X , ' TBAS I N ' , 3 X , ' BMAS S ' , 1 0X , ' S ' , 7 X , ' TWB ' ,

1 6 X , ' Q ' , 9 X , ' L ' , / , 6 X , ' ( HR ) I , 6 X , ' ( F ) ' , 4X , ' ( LBM ) " 2 8X , I ( PPT ) ' , S X , , ( F ) I , 3 X , , ( B'rU/ H R ) ' , 2X , , ( LBM/ H R ) , , / , 3 6 X , ' ---- · , 4X , ' ----- - · , 3 X , ' ----- ' , 8X , · ----- ' , S X , · -- - ' , 4 3 X , ' ------- - ' , 2X , · --- ----- ' )

2 03 FORMAT ( J / , I NUMBER m' TOWERS 1 ' BAS I N MAS S ( LBM ) 2 I SAL I N I TY OF BAS I N ( PPM ) 3 ' TI ME STEP S I Z E ( HR ) 4 I NUMBER OF STEPS TO B E TAKE N 5 ' NO EVAPORAT ION 6 ' CONSTANT WET- BULB TEMP E RATURE

14 STOP , I N SUFF I C I ENT AMB I ENT DATA ' 1 5 STOP , ERROR I N WB ' 1 6 STOP I ERROR I N H EAT ' 1 7 STOP I ERROR I N L/G ' 1 8 STOP I ERROR I N KAV/L ' 1 9 STOP I ERROR I N L '

END SUBROUT I N E BAS I N ( T I ME , Y , YP )

1 , 1 1 0 , 1 , ' , l PE I O . 3 , 1 , ' , 3 PF I O . 2 , 1 , ' , O P F I 0 . 2 , 1 , 1 , 1 1 0 , 1 , I , L l O , I , ' , L l O ' ;/ )

C • • • TH I S S U B ROUTINE I S U S ED TO EVALUATE DERIVAT IVES OF TH E C • • • FO RM Y P ( I ) =DY ( I ) /DT . BAS I N H AS TWO EQUATIONS ; BAS I N C • • • T E M PERATURE AND BAS I N MAS S W RT T I M E . C

C

PARAMETER ( I TOWER= 5 0 , JTOWER= 5 , I H E AT= 5 0 , IWNDL- 2 0 0 ) REAL LOVERG , L , KAVL D I ME N S ION Y ( 2 ) , YP ( 2 ) COMMON ITOWERCI LOVERG ( J TOWE R ) , L ( JTOWER ) , KAVL ( JTOWER ) , NH ( JTOW E R ) ,

1 NL ( JTOWERI , NTOWER, SALT COMMON ITOWERC2/ TE ( I TOWER , J TOWE R ) , ALG ( I TOWER , JTOWE R ) ,

1 AL ( I TOWE R , J TOWE R ) , AK ( I TOWER , JTOWER ) COMMO N IHEATCI Q ( I H EAT , JTOW E R ) , TME ( I H EAT , JTOWER) , AH ( I H EAT , JTOW E R ) COMMO N I A I RCI WB ( IWNDL ) , TH ( I WNDL ) , AWB ( IWNDL ) , BWB ( IWNDL ) ,

1 CWB ( IWNDL ) , NAMB , RWB ( IWNDL ) COMMON 10UTI QQ ( JTOWER) , TWB , S

C . . . S ET WET-BULB TEMPE RATU RE AT EACH T I ME STEP C

I F ( NAMB . GT . l ) TH EN , TWB=SPLFCN ( TI ME , TM , WB , NAMB , AWB , BWB , CWB , I NTER , I ER )

END I F

F i g u r e A . 5 Con t i n u e d

1 2 7 C c . . . CALCULATE SAL I N I TY C

S a SALT/Y ( 2 ) C c • • • B EG I N LOOP OV E R TOWERS C

C

000"' 0 . 0 DTDT " O . O DMDT = O . O

c . . . SET H EAT LOAD AT EACH TIME STEP C

C c . . . c . . . C C C

C

DO 1 I " l , NTOWER I F ( NH ( I ) . GT . l ) TH EN

OOO=SLFCN ( T I ME , TME ( l , I ) , O ( l , I ) , NH ( I ) , AH ( l , I ) , I E R ) EL S E I F ( NH ( I ) . EO . l ) TH EN

000=0 ( 1 , 1 ) END I F 00 ( 1 ) =000 T I N =Y ( l ) +OOO/L ( I )

CALCULATE OUTLET TEM PERATUR E OF TOWER ( TOUT ) AND TH E FRACT ION OF WATER EVAPORA'l'ED , G IVEN

T I N - TOWER I NLET TEMPERATU R E , AND S , TWB , KAV/ L , AND L/G •

CALL TOWER ( TIN , S , TWB , KAVL ( I ) , LOVERG ( I ) , TOUT , F ) DTDTcDTDT+ L ( I ) * ( 1 . 0 -F ) * ( TOUT-Y ( 1 » / ( Y ( 2 ) -F * L ( I » DMDT=DMDT-F * L ( I )

1 CONT I N U E YP ( l ) =DTDT Y P ( 2 ) =DMDT RETURN END SUBROUTI N E TOWER ( T I N , S , TWB , KAVL , LOVERG , TOUT , FEVAP )

C . . . TH I S S U B ROUTI N E I S U S ED TO CALCULATE TOUT-TH E OUTLET C • • • TEMPE RATU R E OR COLD WATER TEM PE RATU RE-OF TH E TOWER, C • • • AND TH E FRACT ION OF WATER EVAPO RATED W I TH I N EACH TOWER . C C

C

SAVE TOLD , A , B , C , D , T , HHAT REAL LOVERG , KAVL LOG I CAL NOEVAP COMMON / EVAPC/ NOEVAP D I ME N S I O N A ( 1 0 ) , B ( l O ) , C ( l O ) , D ( l O ) , T ( 1 2 ) , HHAT ( 1 2 ) , WORK ( 2 0 )

C . . . SET M I S CELLAN EOU S DATA C

C

C

DATA A/ O . , 9 * 1 . / , B/ I O * - 2 . / , C/ 1 0 * 1 . / , D/ I O * O . / DATA N/ I O/ , NP1/ l l/ , NP2/ l 2/ , AN/ O . 0 5/ DATA A O/ l . O/ , Al/ - O . 6 6 1 7 7 / , A2/ 0 . 3 6 5 0 8/ , A3 / - 5 . 2 5 9 26/ DATA C O/ 1 3 . 3 1 8 5/ , C l / l . 97 6/ , C2/ 0 . 6 4 4 5/ , C3/ 0 . l 2 9 9/ DATA TT1/ 2 1 2 . / , TT 2/ 4 5 9 . 6 7/ , WRATI O/ 0 . 6 2 2/ DATA HV1/ 0 . 4 27 5 5 5/ , HV 2/ 1 06 2 . 3/ DATA HA1/ O . 2 4 0 3/ DATA H FG/I 0 4 8 . 0/ , LHF/l . O/ DATA SMALL/ O . O l/ , TOLD/ O . O/

E ( X ) a « ( C3 * X-C 2 ) *X +Cl ) *X+C O ) * X F ( X ) =A O +X * ( Al +X * ( A 2 +X *A3 » T ( N P 1 ) =T I N T ( N P 2 ) =TWB

C • • • AN=1/ 2 * 1/N WHE RE l/N-DELTA A * C

AK 1 -AN * KAVL

F i gu re A . S Co n t i n u e d

C 1 2 8 c . . . SET-UP CO EPF I CI ENTS FOR TH E TRI D I AGONAL MATRI X SOLUT I ON C

C

1 D ( l ) = ( B ( l ) +C ( l » * TWB D ( N ) " - C ( N ) *T I N CAL L TDMS ( A , B , C , T , D , N , I E R , WORK ) I F ( AB S ( TOLD-T ( l » . LT . SMALL ) GO TO 4 TOLD"T ( l )

c . . . CALCULATE TH E TOTAL EN'l'HALPY , HHAT , FOR EACH CELL BY CALCULATI NG TH E RELAT I V E H UM I D I TY , W , AT EACH TEM P E RATU RE , MU LT I PLYI NG TH E ENTHAL PY D U E TO SATU RATED VAPOR BY W ;

c . . . c . . . C • • •

c . . . AND ADD I NG TH I S TO TH E ENTHAPLY DU E TO SATU RATED A I R ALL FOUND A T TH E CELL TEMPERATURE .

C DO 2 I = 1 , NP 2 TT= ( TT I -T ( I » / ( T ( I ) +TT 2 ) PP�EXP ( -E ( TT » *AHAX 1 ( 0 . 7 7 5 , F ( S » W=WRAT I O * P P/ ( I . O-PP ) HV" HV 1 *T ( I ) +HV 2 HA=HA1 *T ( I )

2 H H AT ( I ) -HA+W*HV C C . . . CALCULATE DH/DT POR EACH CALL TO CALCULATE MATRI X COEFF I CI ENTS C

C

DO 3 I = 2 , N DH= ( H HAT ( I + I ) -HHAT { I -I » / ( T ( I +l ) -T ( I -l » O=AK I * ( DH- LOVERG ) A ( I ) =1 . O +Q

3 C ( I ) -1 . 0-0 B ( 1 ) --1 . 0 -AK 1 * « HHAT ( 1 ) -HHAT ( NP 2 » / ( T ( I ) -T ( N P 2 » +LOVERG ) C ( I ) -1 . O-AK I * ( ( H HAT ( 2 ) -HHAT ( N P 2 ) ) / ( T ( 2 ) -T ( NP 2 ) ) -LOVERG ) GO TO 1

4 TOUT=T ( I ) TOLD-T ( l ) I F ( NO EVAP ) TH EN

FEVAP = O . O

C . . . CALCU LATE TH E EFFECT S OF EVAPORATION C

C

ELS E F EVAP '" ( T I N-TOUT ) / H FG * LlIF

END I F RETU RN END SUBROUTINE SPLF I T ( X , Y , N , HE RM I TE , A , B , C , INTER , I E R ) LOG I CAL HERMITE D I MEN S I ON X ( N ) , Y I N ) , A ( N ) , B ( N ) , C ( N )

C S UBROUT I N E SPLF I T CALCULATES TH E CO EFF I C I ENTS O F AN I NTERPOLATI NG C NATU RAL OR H ERMI T E CUB I C S P L I N E . SPLFCN CAN B E U S E D TO EVAU LATE C TH E I NTERPOLATING POL YNOM I AL . C C C C C C C C C C C C C C C C C C

TH E I -D ARRAYS X AND Y CONTAI N N DATA PAIRS WH I CH ARE TO BE TH E KNOTS OF TH E I NT E RPOLAT ION . TH E I NTERPOLATI NG FUNCT I O N I S D E F I NED B Y

WHERE

AND

Y=Y ( I ) + « C ( I ) *T+B ( I » *T+A ( I » *T

T=X-X ( I ) X ( I ) . LE . X . LT . X ( I + l ) A , B , AND C ARE ARRAYS OF L ENGTH N CONTA I N I NG THE I NTERPOLATI NG CO EF F I C I ENTS .

H E RM I T E I S A LOG I CAL VAR I ABLE ( . �RU E . OR . FA L S E . ) WH I CH D ETERM I N E S WHETH E R A H E RM I T E OR NATU RAL S PL I N E I S U S ED AS FOLLOW S .

H ERM I TE - . TRU E . - - > H E RM I TE S PL I N E U S ED I F N > 3 H E RM ITE- . FALS E . _ a > NATU RAL SPL I N E U S ED I F N > 3


C C

N - 2 OR 3 1 2 9

- - > POLYNOM I AL I N TERPOLAT I ON U S ED

C I NTER I S AN I NTEG ER VAR I ABLE D E S I GNAT I NG TH E I NTERVAL OF I NT ERC POLAT I O N . SPLF I T I N I TAL I Z ES I NTER TO 1 C C C C C C C C

I ER I S AN I N T EG E R E RROR FLAG D E F I N ED AS FOLLOWS .

I E R"' O I E R = l I E R = 2

= - > NO E RROR -- NORMAL R ETU RN = = > N . L E . 1 = = > X ARRAY NOT I N STRI CTLY ASCEND I NG

O R D E SCEND I NG ORD E R

C • • • • • I N I T I AL I Z E INTER I NT E R .. l

C C • • • • • CHECK I N PUT DATA

I E R = l I F ( N . L E . l ) RETU RN I E R= 2 I F ( X ( 2 ) . EQ . X ( l » RETU R N I F ( N . GT . 2 ) G O T O 1

C • • • • • F I T A STRA IGHT L I N E TO DATA A ( l ) - ( Y ( 2 ) -Y ( 1 » / ( X ( 2 ) -X ( l » B ( l ) = O . O C ( l ) = O . O I E R= O RETU RN

1 I F ( X ( 2 ) . LT . X ( 1 » GO TO 3 DO 2 I = 3 , N I F ( X ( I ) . LE . X ( I - l » RETU R N

2 CONT I N U E GO TO 5

3 DO 4 I = 3 , N I F ( X ( I ) . G E . X ( I - l » RETU RN

4 CONT I N U E 5 I E R = O

I F ( N . GT . 3 ) GO T O 6 C • • • • • F I T A PARABOLA TO DATA

X 2 X 1 "X ( 2 ) -X ( 1 ) X 3 X l =X ( 3 ) -X ( 1 ) Y 2Y l =Y ( 2 ) -Y ( 1 ) Y 3 Y l =Y ( 3 ) -Y ( 1 ) D=X 2X 1 * X 3 X l D=D * X 2 X l -D * X 3 X 1 B B = ( Y 2Y 1 * X 3 X 1 -Y 3 Y l * X 2X l ) /D AA= ( X 2X 1 * X 2X l * Y 3 Y I -X 3 X 1 * X 3 X l *Y 2Y 1 ) /D A ( l ) =AA B ( l ) =BB C ( l ) = O . O B ( 2 ) = B B A ( 2 ) =AA + 2 . * B B * X 2X l C ( 2 ) =0 . 0 RETU RN

6 I F ( H E RM I TE ) GO TO 1 1 C • • • • • NATU RAL S P L I N E I NTERPOLAT I O N

M:N-1 . C I :X ( 2 ) -X ( 1 )

C ( l ) =C I B I = ( Y ( 2 ) -Y ( 1 » /C I B ( 2 ) : B I _

DO 7 I = 2 , M CC=X ( I + l ) -X ( I ) C ( I ) =CC B B = ( Y ( I + 1 ) -Y ( I » /CC B ( I ) =B B - B I B I =BB AA-CC + C I A U ) "AA+AA

F i qu re A . S Cont i n u e d

7 C I =CC DO 8 I - 3 , M AA= C ( I - 1 ) /A ( I - 1 ) A ( I ) =A ( I ) -AA * C ( I - 1 )

8 B ( I ) = B ( I ) -AA * B ( I - l ) B ( 1 ) = 0 . 0 B ( N ) = O . O B B = B ( M ) /A ( M ) B ( M ) = B B J J = M D O 9 I = 3 , M J J = J J - l B B= ( B ( JJ ) -C ( JJ ) * B B ) /A ( JJ )

9 B ( J J ) =BB B I = O . O JJ"N DO 1 0 I - I , M JJ ·J J - l BB:oB ( JJ )

1 3 0

C C = C ( J J ) A ( JJ ) " ( Y ( J J +l ) -Y ( JJ » /CC-CC * ( B I +BB+B B ) C ( JJ ) = ( B I - B B ) /CC B ( J J ) = B B + B B + B B

1 0 B I = B B RETURN

C • • • • • H E RM I TE S PL I N E

C C C C C C C

· C C C C C C C

1 1 RM 3 = ( Y ( 2 ) -Y ( I » / ( X ( 2 ) -X ( I » T1 =RM 3 - ( Y ( 2 ) -Y ( 3 » / ( X ( 2 ) -X ( 3 » RM 2 =RM 3 +Tl RM1 = RM 2 +T1 N O " N- 2 DO 1 2 I = l , N RM 4 = RM 3 - RM 2 +RM 3 I F ( I . LE . N O ) RM 4 - ( Y ( I + 2 ) -Y ( I +l » / ( X ( I + 2 ) -X ( I + l » T1 =AB S ( RM 4 - RM 3 ) T 2 =ABS ( RM 2 - RM l ) B B=T1 +T2 A ( I ) = 0 . 5 * ( RM 2 + RM 3 ) I F ( B B . NE . O . O ) A ( I ) = ( Tl * RM 2 +T 2 * RM 3 ) /B B RM 1 =RM 2 RM 2 = RM 3

1 2 RM 3 =RM 4 N O "N - l D O 1 3 I = 1 , N O Tl =1 . 0/ ( X ( I +1 ) -X ( I » T 2 = ( Y ( I + l ) -Y ( I » *Tl B B= ( A ( I ) +A ( I + 1 ) -T2-T2 ) *T 1 B ( I ) = - B B + ( T2 �A ( I » *Tl

13 C ( I ) = B B * T l RETURN END REAL FU NCT ION SPLFCN ( X I N , X , Y , N , A , B , C , I N T ER , I ER ) D I MEN S I ON X ( N ) , y e N ) , A ( N ) , B ( N ) , C ( N )

FUNCT ION S PLFCN EVAL UATES A S PL I N E INTERPOLATION AT X I N

ARGU MENTS X , Y , N , A , B AND C ARE D E F I NED A S I N S PL F I T .

I NT ER I S AN I NTEG ER VARI ABLE I ND I CATI NG THE I NTERVAL U S ED I N THE I N T ERPOLAT ION ( EXTRAPOLAT I ON )

I ER I S A N I NT EG E R ERROR FLAG D E F I N ED A S FOLLOW S .

I ER = O I ER - 1 I ER :0 2

�=> NO ERROR -- NORMAL RETU RN &=> X I N . LT . XM I N EXTRAPOLAT ION s_> XIN . GT . XMAX -- EXTRAPOLATION


1 3 1 C WHE RE XMAX AND XM I N ARE TH E MAX I MUM AND M I N I MU M VAL U E S C OF TH E X -ARRAY , RES PECT IVELY . C

C

I F ( I NTER . L E . O . OR . INTER . GT . N ) INTER=1 I ER = O T = X I N-X ( I NTER ) U = X I N-X ( I NTER+ 1 ) I F ( U *T . GT . O . O ) GO TO 1 S PL FCN=Y ( I NTER) + « C ( I NTER) *T+B ( I NT E R » *T+A ( I NT E R » *T RETURN

1 K = 1 I F ( T . LT . O . O ) K =-K IF « T-U ) . LT . O . O ) K=-K

2 I NT ER= I NTER+K IF « N- I NT E R ) * I N T E R . L E . O ) GO TO 3 T = X I N-X ( I NTER) U = X I N - X ( I NTER+1 ) I F ( U * T . GT . O . O ) GO TO 2 SPLFCN=Y ( I NTER) + « C ( I NTER) * T + B ( I NTER» * T +A ( I NTER» * T RETURN

3 I E R = 1 I F ( T . LT . 0 . 0 ) I E R = 2 I NTER= I NTER-K S PL F CN=Y ( I NTER) + « C ( I NTER) *T+B ( I NTER» *T+A ( I NTER » *T RETURN END S U B ROU T I N E LF I T ( XTABLE , YTAB L E , NDATA , AAA , I E R ) D I MEN S I ON XTABL E ( NDATA ) , YTABLE ( NDATA ) , AAA ( NDATA )

c . . . SUBROU T I N E LF I T CALCULATE S TH E S LOPE FOR L I N EAR C . . . I NTERPOLAT I ON OVE R AN INTERVAL . C . . . SLFCN CAN B E U S ED TO EVAL UATE A L I N EAR F U N CT I ON C • • • W I TH I N AN INTERVAL . C C . . . XTABLE AND YTAB L E ARE 1 -D ARRAYS CONTA I N I NG NDATA C . . . PO I NTS WH I CH FORM NDATA- 1 INTERVAL S FOR I NT E RPOLAT I ON . C c . . . C C C C C C C C C C c • • •

C C C C C C C

TH E L I N EAR FUNCT ION I S

Y =YTABLE ( I ) + ( X -XTAB L E ( I » * « YTAB L E ( I + 1 ) -YTAB L E ( I »

( XTAB L E ( I + 1 ) -XTAB L E ( I » )

WHERE X L I ES B ETWE EN XTAB L E ( I ) AND XTABLE ( I + 1 )

AAA I S AN ARRAY CONTA I N I NG TH E S LOPE VAL U E S FOR EACH I N T E RVAL

I E R I S AN E RROR FLAG D E F I N ED AS FOL LOWS

I E R"' O I E R"'1 I E R = 2

= = > NO E RROR ==> N DATA . LE . 1 = = > X ARRAY I S NOT I N STRI CTL Y

ASC E N D I NG O R D E SCEND I NG ORDER

C . . . CH ECK I N PU T C

C

I ER=1 I F ( NDATA . L E . 1 ) RETURN I E R= O

.

I F ( NDATA . EQ . 2 ) GO TO 5 I E R=2

C . . . CHECK TO MAK E SURE DATA I S I N STRI CTLY ASCEND I NG OR C . . . DESCEND I NG X ORD E R C

I F ( XTAB L E ( 2 ) . LT . XTAB L E ( 1 » GO TO 2

F i gu re A . 5 Co n t i n u e d

C C • • •

C

C

1

2

3 4

5

DO 1 I = 3 , NDATA 1 3 2 I F ( XTABLE ( NDATA ) . LT . XTABLE ( NDATA-l » RETURN

CONT I N U E _

GO TO 4 DO 3 I = 3 , NDATA

I F ( XTAB L E ( NDATA ) . G T . XTABL E ( NDATA- l » RETU RN CONT I N U E I E R = O

CALCU LATE SLOPE F O R EACH I NTERVAL

DO 6 I = l , NDATA- l AA=XTABLE ( I + l ) -XTABL E ( I ) I F ( AA . EQ . O . O ) TH EN

AAA ( I ) =O . O GO TO 6

END I F AAA ( I ) = ( YTABLE ( I +l ) -YTAB L E ( I » /AA

6 CONT I N U E RETU RN END REAL FUNCT ION SLFCN ( X I N , XTABL E , YTABLE , NDATA , AAA , I E R ) D I ME N S I ON XTABLE ( NDATA ) , YTABL E ( NDATA ) , AAA ( NDATA ) SAV E I NTER , K

C • • • TH I S FUNCT I O N EVAL UATES A L I NEAR I NTERPOLAT ION AT X I N . C C C • • • ARG UMENTS XTABLE , YTABLE , NDATA , AND AAA ARE AS D E F I NED C I N L F I T . C C . . . I ER I S AN I NTERGER ERROR FLAG DE F I N ED AS FOLLOWS . C C C C C C

I ER-O I ER=1 I ER=2

� = > NO ERROR --> X I N . LT . M I N I MUM VALUE O F X -ARRAY • • > X I N . GT . MAX I MUM VAL U E OF X -ARRAY

C • • • TH E I NTER ( I NTERVAL ) VALU E IS SAVED EACH T I M E S LFCN C . . . I S CALL TO PR EVENT REPEATATI V E S EARCH I NG . C

C

DATA I NTER/1/ I E R= O I NTER�MI N O ( I NTER , NDATA- l ) XX I =X I N -XTABLE ( I NT E R ) XXI P l =X I N-XTABLE ( I NTER+ l ) I F ( XX I * XX I P 1 . GT . O . O ) GO TO 1 SL FCN =YTABLE ( I NTER ) +XX I *AAA ( I NTER) RETURN

C . . . S ET K • 1 , - 1 DEPENDI NG ON PRES ENT X LOCATION , TO F I ND C • • • TH E INTERVAL CONTAIN I NG X C

C

1 X - I I F ( XX I . LT . O . O ) K --K I F « X X I -XX I P l ) . LT . O . O ) K--K

2 I NTER= I NTER+ K I F « NDATA- I NTER) * I NT E R . L E . O ) GO TO 3 XX I =X I N-XTABLE ( I NTER ) X X I P 1 =X I N-XTABLE ( I NTER+1 ) I F ( XX I * X X I P l . GT . O . O ) GO TO 2 SLFCN =YTABL E ( I NTER) +XX I *AAA ( I NTER) RETURN

C • • • EXTRAPOLATION C

3 I ER=l I F ( XX I . LT . O . O ) I E R- 2


C c . . . C C C c . . . C c . . . C C C C C

I NTER= I NTER-K 1 3 3

SLFCN -YTAB L E ( I NTER ) +XX I *AAA ( I NTER) RETURN END S U B ROU T I N E TOMS ( A , B , C , X , D , N , I ER , WORK ) D I ME N S ION A ( N ) , B ( N ) , C ( N ) , X ( N ) , D ( N ) , WORK ( l )

TOMS S OLV ES A S ET O F ALG EBRA I C EOUATIONS O F TH E FORM :

A ( I ) * X ( I - l ) +B ( I ) * X ( I ) +C ( I ) * X ( I + l ) =D ( I )

FOR I =l , 2 , • • • • , N , U S I NG TH E TR I D I AG ONAL MATR I X SOLUT ION .

OTH ER VAR I ABL ES :

I ER - 0 NORMAL RETU RN • 1 -- ERROR BECAU S E N IS L E S S THAN 1 - 2 -- EQUAT IONS ARE S I NG U LAR

C . . . WORK A WORK ARRAY D I MENS I O N ED AT L EAST 2 * ( N- l ) C

I F ( N . LE . l ) GO TO 4 I ERe 2 I F ( B ( l ) . EO . O . O ) RETURN NHI -N- l BOLD"C ( l ) /B ( l ) GOLD=D ( 1 ) / B ( 1 ) WORK ( l ) =GOLD WORK ( l +NMl ) =BOLD I F ( N . EO . 2 ) GO TO 2 DO 1 I = 2 , NHI D I V = B ( I ) -A ( I ) * BOLD I F ( D I V . EO . O . O ) RETURN GOLD= ( D ( I ) -A ( I ) *GOLD ) /DIV HORK ( I ) =GOLD BOLD=C ( I ) /D I V

1 WORK ( I +NHl ) =BOLD 2 D IV=B ( N ) -A ( N ) * BOLD

I F ( O I V . EO . O . O ) RETURN GOLD= ( D ( N ) -A ( N ) *GOLD ) /D I V X ( N ) "GOLD L=N DO 3 I = 2 , N L-L- l GOLD-WORK ( L ) -WORK ( L+NM1 ) *GOLD

3 X ( L ) "'GOLD I E R= O RETURN

4 I E R .. 1 I F ( N . LE . O ) RETURN I ER- 2 I F ( B ( l } . EO . O . O ) RETURN X ( l ) =D ( l ) /B ( l ) I ER= O RETU RN END S U B ROUTI N E ODE ( P , NEON , Y , T , TOUT , REL E RR , ABSERR , I FLAG , WORK )

C C SUBROU T I N E ODE I NTEG RATES A SYSTEM OF N EON F I RST ORD E R C ORD I NARY D I F FERENT IAL EOUATION S OF TH E FORM C DY ( I ) /DT .. F ( T , Y ( 1 ) , Y ( 2 ) , • • • , Y ( NEON ) } C Y ( I ) G IVEN AT T . C TH E S U B ROU T I N E I NTEGRATES FROM T TO TOUT . ON RETU RN TH E C PARAMETERS I N TH E CALL L I ST ARE S ET FOR CONT I NU I NG TH E I NT EG RAT I ON . C TH E U S E R HAS ONLY TO DE F I N E A N EW VALU E TOUT AND CALL ODE AGA I N . C C THE D I FFERENT I AL EOUATIONS ARE ACTUALLY SOLVED BY A S U I T E OF COD E S C DE , STEP , AN D I NTRP . ODE ALLOCATES V I RTUAL STORAG E I N TH E

F i g u re A . 5 C o n t i n u e d

1 3 4 C ARRAYS WORK AND IWORK AND CALL S DE . D E I S A S U P E RV I S OR WH I CH C D I R ECTS TH E SOL UT I O N . I T CALLS ON TH E ROUT I N E S STEP AND I NTRP C TO ADVANCE THE I N TEG RAT ION AND TO I NTERPOLATE AT OUTPUT PO I NTS . C STEP U S ES A MOD I F I ED D I V I DED D I F FERENCE FORM OF TH E ADAMS PECE C FO RMULAS AND LOCAL EXTRAPOLAT ION . I T ADJ U STS TH E O RDER AND STEP C S I Z E TO CONTROL TH E LOCAL ERROR PER U N I T STEP I N A GENERAL I Z ED C S E N S E . NORMAL LY EACH CALL TO STEP ADVAN C E S 'l'H E S OL U'l' I O N O N E S TEP C I N TH E D I RECT I ON OF TOUT . FOR REASON S OF E F F I C I ENCY D E C I NTEG RATES B EYOND TOUT I NTERNALLY , THOUGH NEVER B EYOND C T+ I O * ( TOUT-T ) , AND CALLS I NTRP TO INTERPOLATE TH E SOL UT I ON AT C TOUT . AN OPT I ON I S PROV I DED TO STOP TH E I NTEGRAT ION AT TOUT BUT C I T SHOULD B E U S ED ONLY I F I T I S I MPOS S I B L E TO CONT I N U E TH E C I NTEG RAT ION B EYOND TOU T . C C TH I S CODE I S COM PLETELY EX PLAI N ED AND DOCUMENTED I N TH E TEXT , C COMPUTER SOLUT I ON OF O RD I NARY D I FFERENT IAL EQUAT I ON S : TH E I N I T I AL C VALUE PROBLEM BY L . F . SHAMP I N E AND M . K . GORDON . C C TH E PARAMETERS REPRES ENT : C F -- SUBROU T I N E F ( T , Y , YP ) TO EVALUATE DERIVATIVES Y P ( I ) =DY ( I ) !DT C NEQN -- NUMB ER OF EQUATIONS TO BE I NTEGRATED C Y ( * ) -- SOLUT ION VECTOR AT T C T -- INDE PENDENT VAR I ABLE C TOUT -- PO INT AT WH I CH SOL UT ION I S D E S I RED C RELE RR , ABS ERR -- RELATIVE AND ABSOLUTE ERROR TOL ERANCES FOR LOCAL C E RROR TEST . AT EACH STEP TH E CODE REQU I RE S C ABS ( LOCAL E RROR ) . LE . AB S ( Y ) * RELERR + AB S ERR C FOR EACH COM PONENT OF TH E LOCAL ERROR AND SOL UT ION V ECTORS C I FLAG -- IND I CATE S STATU S OF I NTEG RAT ION C WORK ( * ) -- ARRAY TO H OLD I N FORMAT I ON I N TERNAL TO CODE C WH I C H I S NECESSARY E'OR S U B S EQU ENT CALLS C C F I RST CALL TO ODE -

C C TH E U S E R MU ST PROVIDE STORAG E I N H I S CAL L I NG PROG RAM FOR TH E ARRAY S C I N TH E CALL L I S T , C Y ( NEQN ) , WORK ( 1 0 0 +2 1 * N EQN ) , C DECLARE F I N AN EXTERNAL STATEMENT , S U PPLY TH E S UBROU T IN E C F ( T , Y , YP ) TO EVALUATE C DY ( I ) !DT = Y P ( I ) = F ( T , Y ( 1 ) , Y ( 2 ) , • • • , Y ( NEQN » C AND I N I T I AL I Z E TH E PARAMETERS : C NEQN -- NUMB E R OF EQUAT I ONS TO B E I NTEG RATED C Y ( * ) -- VECTOR OF I N I T I AL COND I T I ON S C T -- START I NG PO INT OF I NTEGRAT ION C TOUT -- PO I NT AT WH I CH SOL UT I ON I S D E S I RED C REL ERR , ABS E RR -- RELAT IVE AND ABSOLUT E LOCAL E RROR TOL ERAN C E S C I FLAG -- +1 , -1 . I ND I CATOR TO I N I T IAL I Z E TH E COD E . NORMAL I N PUT C I S + 1 . TH E U S ER S HOULD S ET I FLAG - - l ONLY I F I T I S '

C I MPOS S I BLE TO CONT I N U E TH E I NTEGRATION B EYOND TOUT . C ALL PARAMETERS EXCEPT F , NEQN AND TOUT MAY B E ALT ERED BY TH E C CODE ON OUTPUT SO MU ST B E VAR IABLES I N TH E CAL L I NG PROGRAM . C C OUTPUT FROM ODE C C NEQN -- UNCHANG ED C Y ( * ) -- SOL UT I ON AT T C T -- LAST PO I N T R EACH ED I N I NT EGRAT I ON . NORMAL RETU RN HAS C T = TOUT •

C TOUT -- UNCHANG ED C REL ERR , AB S E RR -- NORMAL RETURN HAS TOL ERAN C E S U NCHANG ED . I FLAG = 3 C S I G NAL S TOL ERAN C ES I N CR EAS ED C I FLAG c 2 NORMAL RETURN . I NTEGRAT I ON REACH ED TOUT C = 3 I NTEG RAT ION D I D N OT REACH TOU T B E CAU S E E RROR

C TOLERANCES TOO SMALL . RELERR , AB S ERR INCREAS ED C APPROPRI ATELY FOR CONT INU I NG

C • 4 INTEG RAT ION D I D N OT REACH TOU T B E CAU S E MORE THAN

C 5 0 0 STEPS N E EDED C = 5 I NTEGRAT ION D I D NOT REACH TOU T BECAU S E EQUAT IONS


1 3 5 C APPEAR TO B E S T I F F C • 6 -- I NVAL I D I NPUT PARAMETERS ( FATAL ERROR) C TH E VAL U E OF I FLAG I S RETURNED NEG AT I V E WHEN TH E I N PUT C VAL UE I S NEGAT I V E AND TH E INTEGRATION DOES NOT R EACH TOUT , C I . E . , - 3 , - 4 , - 5 . C WORK ( * ) , IWORK ( * ) -- I N FORMAT ION G EN E RALLY OF NO I NTEREST TO Til E C USER BUT NECESSARY FOR S U B S EQ U E NT CALLS . C C S U B S EQUENT CALLS TO ODE C C S U B ROU T I N E ODE RETU RNS W I TH AL L I N FO RMAT ION N E EDED TO CONT I N U E C TH E I NTEG RATION . I F TH E I NT EG RATION REACH ED 'I'OUT , TH E U S ER N E ED C ONLY D E F I N E A N EW TOU T AND CAL L AGA I N . I F TH E I N T EGRAT ION D I D NOT C REACH TOU T AND TH E U S ER WANTS TO CONT IN U E , H E J U ST CAL L S AGA I N . C TH E OUTPUT VAL U E OF U'LAG I S THE APPROPRIATE I NPUT VAL U E FOR C S U B S EQUENT CALLS . TH E ONLY S I TUATION I N WHI CH I T S HOU LD B E ALTERED C I S TO STOP TH E I NT EG RAT ION I NTERNAL LY AT TH E NEW TOUT , I . E . , C CHANG E OUTPUT I FLAG = 2 TO I N PUT I FLAG = - 2 . ERROR TOL ERANC E S MAY C B E CHANG ED BY TH E U S ER BEFORE CONT I N U I NG . ALL OTH E R PARAME'I'ERS MUST C REMA I N UNCH ANG ED . C

C * * * * * * * * * * * * * * * · * · * * · * · * · · · · · * · · * * · * · · * * · * * · · · * * * * · · * * . * * * * * * * * * * * * * * * *

C * S U B ROUTINES D E AND STEP CONTAI N MACH I N E D E P E ND E NT CONSTAN T S . *

C * B E SURE THEY ARE S ET B E FORE U S I NG ODE . *

C * * * * * * * * * * * * * * * * * * * * * * * * * * * · · · * * * · * * * · * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

C

C C C C C C e C C

LOG I CAL START , PHAS E l , NORND D I MEN S I ON Y ( NEQN ) , WORK ( 1 ) , IWORK ( 5 ) DATA I AL PHA , I B ETA , I S I G , I V , IW , IG , I PHAS E , I PS I , I X , I H , I HOLD , I START ,

1 I TOLD , I DELSN/ l , 1 3 , 2 5 , 3 8 , 5 0 , 6 2 , 7 5 , 7 6 , 8 8 , 8 9 , 9 0 , 9 1 , 9 2 , 9 3 / I Y Y • 1 0 0 IWT • IYY + NEQN IP • IWT · + NEQN I Y P - I P + NEQN IY POUT • IYP + NEQN I PH I = I Y POU T + NEQN I F ( I ABS ( I FLAG ) . EQ . 1 ) GO TO 1 START = WORK ( I START ) . GT . 0 . 0 PHAS E l = WO RK ( I PHAS E ) . GT . 0 . 0 NORND = IWORK ( 2 ) . NE . - 1

1 CALL DE ( F , NEQN , Y , T , TOUT , REL E RR , AB S ERR , I FLAG , WORK ( I Y Y ) , 1 WORK ( I WT ) , WORK ( I P ) , WORK ( I Y P ) , WORK ( I Y POUT ) , WORK ( I PH I ) , 2 WORK ( I AL PHA ) , WORK ( I B ETA ) , WO RK ( I S I G ) , WORK ( I V ) , WORK ( I W ) , WORK ( I G ) , 3 PHAS E l , WORK ( I PS I ) , WORK ( I X ) , WORK ( I H ) , WORK ( I H OLD ) , START , 4 WORK ( I TOLD ) , WORK ( I D ELSN ) , I WORK ( 1 ) , NORND , IWORK ( 3 ) , I WORK ( 4 ) , 5 IWORK ( S »

WORK ( I START ) - -1 . 0 I F ( START ) WORK ( I START ) = 1 . 0 WORK ( I PHAS E ) a - 1 . 0 I F ( PHAS E 1 ) WORK ( I PHAS E ) = 1 . 0 IWORK ( 2 ) ... - 1 I F ( NORND ) IWORK ( 2 ) - 1 RETURN END SUBROU T I N E DE ( F , NEQN , Y , T , TOUT , RELERR , ABSERR , I FLAG ,

1 YY , WT , P , Y P , YPOUT , PH I , ALPHA , BETA , S IG , V , W , G , P H AS E 1 , PS I , X , H , HOLD , 2 START , TOLD , DEL SGN , NS , NORND , K , KOLD , I S NOLD )

ODE MERELY ALLOCATES STORAG E FOR DE TO R EL I EV E TH E U S E R OF TH E I N CONV EN I ENCE OF A LONG CALL L I ST . CONSEQU ENTLY DE I S U S ED AS DESCRIBED I N TH E COMMENTS FOR ODE .

TH I S CODE I S COMPL ETELY EXPLA I N ED AND DOCUMENTED I N TH E TEXT , COM PUTER SOLUT ION OF ORDI NARY D I FFERENTIAL EQUAT I ONS : TH E I N I T I AL

' VAL U E PROBL EM BY L . F . SHAMP I N E AND M . K . GORDO N .

LOG I CAL STI F F , CRAS H , START , PHASE1 , NORND D I MENSION Y ( NEQN ) , YY ( NEQN ) , WT ( N EQN ) , PH I ( NEQN , 1 6 ) , P ( NEQN ) , Y P ( NEQN ) ,

F i � u re A . S Con t i n ued

1 3 6 1 Y POUT ( N EON ) , PS I ( 1 2 ) , AL PH A ( 1 2 ) , B ETA ( 1 2 ) , S I G ( 1 3 ) , V ( 1 2 ) , W ( 1 2 ) , G ( 1 3 )

EXTE RNAL F C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

C * TH E ONLY MACH I N E D E P ENDENT CONSTANT I S BAS ED ON TH E MACH I N E U N I T *

C * ROUNDO F F E RROR U WH I CH I S TH E SMAL L E S T PO S I T IV E NUMBER S U C H THAT *

C * 1 . 0 + U . GT . 1 . 0 . U M U S T B E CAL CULATED AND FOU RU = 4 . 0 * U I N S E RTED *

C * I N TH E FOLLOW I NG DATA STATEMENT BEFORE U S I NG D E . TH E ROUT I N E *

C * MACH I N CAL CULATES U . FOURU AND TWOU = 2 . 0 * U MUST AL SO B E *

C * I N SE RT ED I N SUBROUT I N E STEP B EFORE CAL L I NG D E . *

DATA FOURU/ 2 8 . E- 1 5/ C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

C C C C

C C C C

C

TH E CONSTANT MAXNUM I S TH E MAX I MU M NUMBER OF STEPS ALLOW E D I N CAL L T O DE . TH E U S E R MAY CHANG E TH I S L I M I T BY ALT ER I NG T H E FOLLOW I NG STATE�IENT

DATA MAX NUM/ 5 0 0/

* * * * * *

TEST FOR I MPROPER PARAMETERS

I F ( NEON . LT . 1 ) GO TO 1 0 I F ( T . EO . TOU T ) GO TO 1 0

* * *

I F ( RELERR . LT . 0 . 0 . OR . ABS ERR . LT . 0 . 0 ) GO TO 1 0 E P S = AMAX l ( RELERR , ABSE RR ) I F ( E PS . L E . 0 . 0 ) GO TO 1 0 I F ( I FLAG . EO . 0 ) GO TO 1 0 I S N = I S I G N ( l , I FLAG ) I FLAG = IABS ( I FLAG ) I F ( I FLAG . EO . 1 ) GO TO 2 0 I F ( T . NE . TOLD ) GO TO 1 0 I F ( I FLAG . G E . 2 . AN D . I FLAG . LE . 5 ) GO TO 2 0

1 0 I FLAG = 6 RETU RN

ONE

C ON EACH CALL SET I NTERVAL OF INTEG RAT ION AND COU NT E R FOR N U M B E R O F C S T E PS . ADJ UST I N PUT E RROR TOLE RAN C E S TO DEF I N E WE I G H T VECTOR FOR C S U B ROU T I N E STEP C

C

2 0 D E L = TOUT - T ABSDEL = AB S ( D EL ) TEND = T + 1 0 . 0 *D E L I F ( I S N . LT . 0 ) T E N D = TOUT NOST EP = 0 KL E 4 = 0 ST I F F = . FALS E . REL E P S .. RELERR/EPS ABS E P S = ABS E RR/ E P S I F ( I FLAG . EO . 1 ) GO T O 3 0 I F ( I S NOLD . LT . 0 ) GO TO 3 0 I F ( DELSGN * DEL . GT . 0 . 0 ) GO TO 5 0

C ON START AND RESTART ALS O S ET WORK VAR I ABLES X AND YY ( * ) , STORE TH E C D I R ECT ION OF I NTEGRATION AND I N I T IAL I Z E TH E STEP S I Z E C

3 0 START = . TRUE . X = T DO 4 0 L = I , NEON

4 0 Y Y ( L ) = y e L l DELSGN = S IGN ( l . O , DEL ) H = S I G N ( AMAX l ( AB S ( TOUT-X ) , FOURU *ABS ( X » , TOUT-X )

C C I F ALREADY PAST OUTPUT PO I N T , I NTERPOLATE AND R ETU RN C

5 0 I F ( AB S ( X -T ) . LT . ABSDEL ) GO TO 6 0 CALL I NTRP ( X , YY , TOUT , Y , YPOU T , NEON , KOLD , PH I , PS I ) I FLAG .. 2 T .. TOU T

F i �u re A . 5 C o n t i n u e d

C

TOLD .. T I S NOLD • I S N RETURN

1 3 7

C I F CANNOT G O PAS T OUTPUT PO I NT AND S U F F I C I ENTLY CLOS E , C E XTRAPOLATE AND R ETU RN C

C

6 0 I F ( I S N . GT . 0 . OR . ABS ( TOUT-X ) . G E . FOURU *AB S ( X » GO TO 8 0

H .. TOUT - X CALL F ( X , YY , yP ) DO 7 0 L .. 1 , NEQN

7 0 Y ( L ) .. YY ( L ) + H *Y P ( L ) I FLAG = 2 T .. TOUT TOLD .. T I S NOLD .. I S N RETU RN

C TEST FOR TOO MAN Y STEPS C

C

8 0 I F ( NOSTEP . LT . MAXNUM ) GO TO 1 0 0 I FLAG .. I S N * 4 I F ( ST I F F ) I FLAG .. I S N * 5 DO 9 0 L .. 1 , NEQN

9 0 Y ( L ) " Y'Y ( L ) T .. X TOLD '" T I S NOLD .. 1 RETURN

C L I M I T STEP S I Z E , SET WE IGHT V ECTOR AND TAK E A STEP C

C

1 0 0 H .. S I GN ( AM I N l ( ABS ( H ) , AB S ( TEND-X » , H ) DO 1 1 0 L z 1 , NEQN

1 1 0 WT ( L ) " RELE P S *ABS ( YY ( L » + ABSEPS CALL STEP ( X , YY , F , NEQN , H , E PS , WT , START ,

1 HOLD , K , KOLD , CRAS H , PH I , P , Y P , PS I , 2 AL PHA , B ETA , S IG , V , W, G , PHAS E l , NS , NORND )

C TEST FOR TOL ERAN C E S TOO SMAL L C

C

I F ( . NOT . CRAS H ) GO TO 1 3 0 I F LAG .. I S N * 3 REL E RR '" EPS * RELEPS ABS ERR .. EPS *AB S E PS DO 1 2 0 L .. 1 , NEQN

1 2 0 Y ( L ) " YY ( L ) T .. X TOLD .. T I S NOLD 1 RETU RN

C AUGMENT COU NTER ON NUMB ER OF STEPS AND TEST FOR STI F FNESS C

C

1 3 0 NOSTEP .. NOSTEP + 1 KLE 4 .. KLE 4 + 1 I F ( KOLD . GT . 4 ) KLE 4 = 0 I F ( KL E 4 . GE . 5 0 ) ST I P F a . TRU E . GO TO 5 0 END SUBROUTI N E STEP ( X , Y , P , NEQN , H , EPS , WT , START ,

1 HOLD , K , KOLD , CRASH , PH I , P , Y P , PS I , 2 AL PHA , B ETA , S IG , V , W , G , PHAS E l , N S , NORN D )

C SUB ROU T I N E S T E P I NTEG RATES A S Y STEM O F F I RST O RD E R ORD I N ARY

C D I FFEREN T I AL EQUATIONS ONE STEP , NORMALLY FROM X TO X + H , U S I NG A

C MOD I F I ED D I V I D E D D I F FERENCE FO RM OF TH E ADAMS PEeE FORMULA S . LOCAL

F i g u re A . S C o n t i n u e d

1 3 8 C EXTRA POLAT ION I S U S ED TO I M PROV E ABS OL UTE STAB I L I TY AND ACCU RACY . C TH E COD E ADJ U STS I TS ORD E R AND STEP S I Z E TO CO NTROL TH E LOCAL E RROR C PER U N I T STEP I N A G E N E RAL I Z ED S EN S E . SPECIAL DEV I C E S ARE I N C L U D ED C TO CONTROL ROU NDOF F ERROR AND TO DETECT WHEN TH E U S E R I S R EQ U E S T I NG C TOO MUCH ACCURACY . C C TH I S CODE I S COM PL ETELY EXPLA I N ED AND DOCUMENTED I N TH E TEXT , C COM PU TER SOLUTION OF O RD I NARY D I FFERENT I AL EQUAT I ONS : TH E I N I T I AL C VAL U E PROBL EM BY L . F . SHAMP I N E AND M . K . GORDO N . C C C TH E PARAMETERS REPRES ENT : C X -- I N D E P E NDENT VAR I ABLE C Y ( * ) -- SOLUT I ON VECTOR AT X C Y P ( * ) -- D E R I VAT I V E OF SOLUT ION VECTOR AT X AFTER SUCCE S S F U L C STEP C N EQN -- NUMB E R OF EQUAT I ON S TO B E I NTEGRATED C H -- AP PROPR I ATE STEP S I Z E FOR N EXT STEP . NORMALLY DETERM I N ED B Y C CODE C E P S �- LOCAL ERROR TOLE RANCE . MUST B E VAR I ABLE C WT ( * ) -- VECTOR OF W E I G HTS �'OR ERROR CR ITERION C START -- LOG I CAL VAR I ABLE S ET . TRU E . FOR F I RST STEP , . FAL S E . C OTH ERW I S E C HOLD -- STEP S I Z E U S ED FOR LAST SUCCESSFUL S T E P C K -- APPROPR I AT E ORD E R FOR N EXT STEP ( D ETERM I N ED BY COD E ) C KOLD -- ORDE R U S ED FOR LAST SUCCE SSFUL STEP C CRAS H -- LOG I CAL VAR IABLE S ET . TRU E . WHE N N O STEP CAN B E TAK E N , C . FALS E . OTH ERW I S E . C TH E ARRAYS PH I , PS I AR E REQU I R ED FOR TH E INTERPOLAT ION S U B ROUT I N E C I NTRP . TH E ARRAY P I S I NT E RNAL TO THE COD E . C C I N PU T TO STEP C C F I RST CALL --C C TH E U S E R MUST PROV I D E STORAG E I N H I S DRIVER PROG RAM FOR AL L ARRAYS C I N T H E CAL L L I ST , NAMELY C C D I MEN S I ON Y ( N EQN ) , WT ( NEQN ) , PH I ( NEQN , 1 6 ) , P ( NEQ N ) , Y P ( NEQN ) , PS I ( 1 2 ) C C TH E U SER MUST ALSO D E CLAR E START AND CRASH LOG I CAL VAR I AB L E S C AND F AN EXTERNAL SUB ROUT I N E , SUPPLY TH E S UB ROUT I N E F ( X , Y , Y P ) C TO EVALUATE C DY ( I ) /DX .. YP ( I ) • F ( X , Y ( l ) , Y ( 2 ) , • • • , Y ( NEQN » C AN D I N I T IAL I Z E ONLY TH E FOL LOW I NG PARAMETERS : C X -- I N I T IAL VALU E OF TH E I NDEPENDENT VAR I AB L E C Y ( * ) -- VECTOR OF I N I T I AL VAL UES OF DEPENDENT VAR I ABLES C NEQN -- NUMBER OF EQUAT IONS TO B E I NTEGRATED C H -- NOM I NAL STEP S I Z E I N D I CAT I NG D I RECT I ON O F I NTEGRAT I ON C AND MAX I MU M S I Z E OF STEP . MU ST B E VAR I ABLE C EPS -- LOCAL ERROR TOL E RANC E PER STEP . MU ST B E VARI AB L E C WT ( * ) -- VECTOR OF NON - Z ERO WE IGHTS FOR ERROR CRI TERION C START -- . TRU E . C C STEP REQU I RES TH E L2 NORM OF THE VECTOR W I TH COM PO N ENTS C LOCAL E RROR ( L ) /WT ( L ) B E LESS THAN EPS FOR A S U C CE S S F U L STEP . TH E C ARRAY WT ALLOWS TH E U S E R TO S P EC I FY AN ERROR TEST A PPROPR IATE C FOR H I S PROBLEM . FOR EXAMPL E , C WT ( L ) = 1 . 0 S P E C I F I ES ABSOLUTE E RROR , C = ABS ( Y ( L » ERROR R ELAT IVE TO TH E MOST RECENT VAL U E OF TH E C L-TH COMPON ENT O F TH E SOL UT ION , C = AB S ( Y P ( L » E RROR R ELATIVE TO THE MOS T RECENT VAL U E OF C TH E L-TH COMPONENT OF TH E DERIVAT IV E , . C • AMAX 1 ( WT ( L ) , AB S ( Y ( L » ) E RROR R ELAT I V E TO TH E LARG E S T C MAG N I TUDE OF L - TH COMPONENT OBTA I N ED S O FAR , C • ABS ( Y ( L » * RELERR/ E P S + AB SERR/EPS S PEC I F I ES A M I X E D C RELATIVE-ABSOLUTE TEST WHERE R EL E RR I S R ELAT I V E C ERROR , ABS ERR I S ABSOLUTE E RROR AND EPS ·

F i g u re A . 5 Co n t i nued

1 3 9 C AMAX 1 ( RELERR , ABSERR) •

C C S U B S EQU ENT CALLS --C C S U B ROUT I N E STEP I S DES I G N ED S O THAT ALL I N FORMATI ON N E EDED TO C CONT I N U E TH E I NTEGRAT I ON , I N CL UD I NG TH E STEP S I Z E H AND TH E ORD E R C K , I S RETURNED W I TH EACH STE P . WITH TH E EXCE PTION OF TH E S T E P C S I Z E , TH E ERROR TOL ERANCE , AND TH E WE IGHTS , NON E OF TH E PARAMETERS C S HOU LD BE ALTERED . TH E ARRAY WT MUST B E U P DATED AFT ER EACH STEP C TO MA I NTA I N RELATIVE ERROR T ESTS L I K E THOS E ABOVE . NORMAL LY TH E C I NTEGRATION I S CONT I N U ED J U ST B EYOND TH E D E S I RED ENDPO I NT AN D TH E C SOLUT ION I N T ERPOLATED TH ERE W I TH SUBROUT I N E I NTRP . I F I T I S C I MPO SS I BL E TO I NTEG RATE B EYOND TH E ENDPO I N T , TH E STEP S I Z E MAY B E C REDUCED TO H I T TH E END PO I N T S I NC E TH E COD E W I L L NOT TAK E A STEP C LARG ER THAN TH E H I N PUT . CHANG I NG TH E D I RECT ION OF I NTEG RAT I O N , C I . E . , TH E S I G N OF H , REQU I R ES TH E U S ER S ET START = . TRU E . BEFORE C CAL L I NG STEP AGA I N . TH I S I S THE ONLY S I TUAT I ON I N W H I CH START C S HOU LD B E ALTERED . C C OU TPUT FROM STEP C C SU CCESSFUL STEP C C TH E SUBROU T I N E RETURNS AFTER EACH SUCCESSFUL STEP W I TH START AND C CRAS H S ET . FALS E . • , X REPRES ENTS TH E I N D EPENDENT VAR I ABLE C ADVANCED ONE STEP OF L ENGTH HOLD FROM I TS VAL U E ON I N PUT AND Y C TH E SOLUT ION VECTOR AT TH E N EW VALUE OF X . AL L OTH E R PARAMETERS C REPRESENT I N FORMAT ION CORRESPOND I NG TO TH E NEW X N E EDED TO C CONT I N U E TH E INTEGRAT I ON . C C UNSU CCE SSFUL STEP --C C WHEN TH E E RROR TOL E RANCE I S TOO SMALL POR TH E MACH I N E PREC I S I O N , C TH E S U B ROU T I N E RETU RN S W I THOUT TAK I NG A STEP AND CRASH = . TRU E • •

C AN APPROPRIATE STEP S I Z E AND ERROR TOLERANCE FOR CONT I N U ING ARE C EST I MATED AND ALL OTH ER I N FO RMATION I S RESTORED AS UPON I N PUT C B EFORE RETU R N I NG . TO CONT I N U E W I TH TH E LARGER TOL E RAN C E , TH E U S ER C J UST CALLS TH E CODE AGA I N . A R ESTART I S N E I THER REQU I RED NOR C DES IRABL E . C

EXTERNAL P LOG I CAL START , CRAS H , PHAS E 1 , NORND D I MEN S I ON Y ( NEQN ) , WT ( NEQN ) , PH I ( NEQN , 1 6 ) , P ( NEQN ) , Y P ( NEQN ) , PS I ( 1 2 ) D I MEN S ION AL PHA ( 1 2 ) , B ETA ( 1 2 ) , S IG ( 1 3 ) , W ( 1 2 ) , V ( 1 2 ) , G ( 1 3 ) ,

1 G S TR ( 1 3 ) , TWO ( 1 3 ) C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C * TH E ONLY MACH I N E DEPENDENT CONSTANTS ARE BAS ED ON TH E MACH I N E U N I T * C * ROU NDO F F E RROR U WH I CH I S TH E SMALLEST POS I TIVE NUMB E R ' SUCH THAT * C * 1 . 0 + U . GT . 1 . 0 TH E U S ER MU ST CALCU LATE U AND I N S ERT *

C * TWOU = 2 . 0 * U AND FOU RU = 4 . 0 * U I N TH E DATA STATEMENT B E FORE CAL L I NG * C * TH E CODE . TH E ROUT I N E MACH I N CALCULATES U . *

DATA TWOU , FOURU/ 1 4 . 2 E- 1 5 , 2 8 . E- 1 5/ C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

DATA TWO/ 2 . 0 , 4 . 0 , 8 . 0 , 1 6 . 0 , 3 2 . 0 , 6 4 . 0 , 1 2 8 . 0 , 2 56 . 0 , 5 1 2 . 0 , 1 0 2 4 . 0 ,

c C C C C C C C C C C

1 2 0 4 8 . 0 , 4 0 96 . 0 , 8 1 9 2 . 0/ DATA G STR/ 0 . 5 0 0 , 0 . 0 8 3 3 , 0 . 0 4 1 7 , 0 . 0 2 6 4 , 0 . 0 1 8 8 , O . 0 1 4 3 , O . 0 1 1 4 , O . 0 0 93 6 ,

1 0 . 0 07 8 9 , 0 . 0 06 7 9 , 0 . 0 0 5 9 2 , 0 . 0 0 5 2 4 , 0 . 0 0 46 8/ 'DATA G ( 1 ) , G ( 2 ) /1 . 0 , 0 . 5/ , S I G ( 1 ) / 1 . 0/

* * * B EG I N BLOCK 0 * * * CHECK I F STEP S I Z E O R ERROR TOL ERANCE PREC I S ION . IF F I RST STEP , I N I T I AL I Z E START I NG STEP S I Z E .

* * *

I S TOO SMALL FOR MACH I N E PH I ARRAY AND E S T I MATE A

I F STEP S I Z E I S TOO SMAL L , DETERMINE AN ACCEPTABLE ONE

F i qu re A . S Con t i n u e d

C

CRAS H " . TRUE . I F ( AB S ( H ) . G E . FOURU *ABS ( X » H = S I G N ( FOU RU *ABS ( X ) , H ) RETURN

5 P S E P S .. 0 . 5 * E PS

1 4 0

GO TO 5

C I F E RROR TOL E RANCE I S TOO SHAL L , I N CR EAS E I T TO AN ACCE PTAO LE VAL U E C

C

ROU N D a 0 . 0 DO 1 0 L a 1 , NEQN

1 0 ROU ND .. ROUND + ( Y ( L ) /WT ( L » * * 2 ROU ND a TWOU * SQRT ( ROU ND ) I F ( P 5 E PS . G E . ROU ND ) GO TO 1 5 E P S = 2 . 0 * ROUND * ( 1 . 0 + E'OURU ) RETURN

1 5 CRAS H = . FAL S E . G ( l ) a1 . 0 G ( 2 ) a O . S S I G ( 1 ) = 1 . 0 I F ( . NOT . START ) GO TO 9 9

C I N I T I AL I Z E . COMPUTE APPROPR I AT E STEP S I Z E FOR F I RST STEP C

C C C C C C C

C

CALL F ( X , Y , yP ) SUM " 0 . 0 DO 2 0 L • 1 , N EQN

PH I ( L , 1 ) - YP ( L ) P H I ( L , 2 ) .. 0 . 0

2 0 SU M " SUM + ( Y P { L ) /WT ( L » * * 2 SUM " SQRT ( SU M ) ABSH • ABS ( H ) I F ( E PS . LT . 1 6 . 0 *SUM* H * H ) ABS H .. 0 . 2 5 * SQRT ( EPS/SU M ) H .. S I G N ( AMAX 1 { ABSH , FOU RU *ABS ( X » , H ) HOLD = 0 . 0 K .. 1 KOLD .. 0 STAR'r .. . FAL S E . PHAS E 1 .. . TRU E . NORND = . TRU E . I F ( P S E PS . GT . 1 0 0 . 0 * ROUND ) GO TO 9 9 NORND .. . FAL S E . DO 2 5 L '" 1 , NEQN

2 5 P H I ( L , 1 5 ) .. O . 0 9 9 I FA I L .. 0

* * * END BLOCK 0 * * *

* * * B EG I N BLOCK 1 * * * COM PUTE CO E F F I CI ENTS OF FORMULAS FOR TH I S STEP . AVOI D COM PU T I NG THO S E QUANT I T I E S NOT CHANG ED W H E N STEP S I Z E I S NOT CHANG ED .

1 0 0 K P l = K+ l K P 2 • K + 2 , KM1 .. K - 1 K M 2 '" K - 2

* * *

C N S I S TH E NUMBER OF STEPS TAK EN W I TH S I Z E H , I N C L UD I NG TH E CU RRENT C ONE . WHEN K . LT . NS , NO CO EF F I C I ENTS CHANG E C

C

I F ( H . NE . HOLD ) NS - 0 I F ( N S . LE . KOLD ) NS-NS+1 NSP1 .. NS + 1 I F ( K . LT . N S ) GO TO 1 9 9

C COMPUTE THOS E COMPON E NTS OF AL PHA ( * ) , B ETA ( * ) , PS I { * ) , S IG ( * ) WH I CH C ARE CHANG ED C

B ETA ( NS ) - 1 . 0


REAL N S - NS AL PHA ( N S ) - 1 . 0/REAL N S TEM P 1 = H * REALNS S IG ( N SP 1 ) .. 1 . 0 I F ( K . LT . N SP l ) GO TO 1 1 0 DO 1 0 5 I .. N S P l , K

I M I = I - I T EM P 2 = PS I ( I Ml ) PS I ( I Ml ) a TEM P I

1 4 1

U �TA ( I ) - 8 � TA ( I M l ) * PS I ( I M l ) /T EM P 2 TEM P I a TE M P 2 + H

C

AL P H A ( I ) .. H/TEM P I REAL I = I

1 0 5 S I G ( I + l ) . REAL I *AL PHA ( I ) * S I G ( I ) 1 1 0 PS I ( K ) - TEM P 1

C COM PUTE COEF F I CI ENTS G ( * ) C C I N I T I AL I Z E V ( * ) AND S ET W ( * ) . G ( 2 ) I S S ET I N DATA STATEMENT C

C

I F ( N S . GT . 1 ) GO TO 1 2 0 DO 1 1 5 1 0 .. 1 , K

TEM P 3 = 1 0 * ( 1 0+ 1 ) V ( I O ) = 1 . 0/TEM P 3

1 1 5 W ( I O ) " V ( I O ) G O TO 1 4 0

C I F ORD E R WAS RA I S ED , UPDATE D I ACONAL PART OF V ( * ) C

C

1 2 0 I F ( K . LE . KOLD ) GO TO 1 3 0 TE M P 4 .. K * K P I V ( K ) .. 1 . 0/TEM P 4 NSM2 .. N S - 2

.

I F ( N S M 2 . LT . 1 ) GO TO 1 3 0 DO 1 2 5 J .. 1 , NSM2

I .. K -J 1 2 5 V ( I ) " V ( I ) - AL PHA ( J + 1 ) *V ( I + l )

C U PDATE V ( * ) AND S ET W ( * ) C

C

1 3 0 L I M I T1 • KP 1 - NS TEMP5 .. ALPHA ( N S ) D O 1 3 5 1 0 a 1 , L I M I TI

V ( I O ) .. V ( I O ) - TEMP5 *V ( I O + l ) 1 3 5 W ( I O ) " V ( I O )

G ( NSP 1 ) .. W ( l )

C COM PUT E TH E G ( * ) I N TH E WO RK VECTOR W e * ) C

1 4 0 N S P 2 c NS + 2 I F ( KP 1 . LT . N S P 2 ) GO TO 1 9 9 DO 1 5 0 I .. N S P 2 , KP 1

L I M I T 2 .. K P 2 - I TEMP6 • ALPHA ( I - 1 ) DO 1 4 5 10 .. 1 , L I M I T2

1 4 5 W ( I O ) " W ( I O ) - TEMP 6 *W ( I O+l ) 1 5 0 G ( I ) = W ( l ) 1 9 9 CONT I N U E

C * * * E N D BLOCK 1 * * * C C * * * B EG I N BLOCK 2 * * * C PREDI CT A SOLUTION P ( * ) , EVALUATE DERIVATIVES U S I NG PREDI CTED C SOLUTION , EST I MATE LOCAL ERROR AT ORD E R K AND ERRORS AT ORD E RS X , C K - 1 , K - 2 AS I F CONSTANT STEP S I Z E WERE U S ED . C * * * C C CHANG E PH I TO PH I STAR C


C

2 0 5 2 1 0

I F ( K . LT . NS P 1 ) G O TO 2 1 5 DO 2 1 0 I .. N S P 1 , K

T EM P 1 = BETA ( I ) DO 2 0 5 L � 1 , NEQN

pll l e L , I ) = TEM P 1 * pH I ( L , I ) CONT I N U E

1 4 2

C PR ED I CT SOLUTION AND D I F FE R E N C E S C

C

2 1 5 DO 2 2 0 L = 1 , NEQN P H I ( L , K p 2 ) '" 1'11 1 ( L , Kp 1 ) PH I ( L , K P 1 ) = 0 . 0

2 2 0 p e L ) '" 0 . 0 DO 2 3 0 J '" 1 , K

I '" K P I - J I P I '" 1 + 1 TEM P 2 = G ( I ) DO 2 2 5 L '" I , NEQN

p e L ) .. p e L ) + TEM P 2 * PH I ( L , I ) 2 2 5 PH I ( L , I ) = PH I ( L , I } + PH I ( L , I P 1 ) 2 3 0 CONT I N U E

I F ( NORND ) G O T O 2 4 0 DO 2 3 5 L = 1 , NEQN

TAU = H * P ( L ) - PII I ( L , 1 5 ) p e L } a Y ( L ) + TAU

2 3 5 PH I ( L , 1 6 ) = ( P ( L ) - Y e L l ) - TAU GO TO 2 5 0

2 4 0 DO 2 4 5 L = 1 , NEQN 2 4 5 p e L ) '" Y e L l T H * P ( L ) 2 5 0 XOLD = X

X = X + II ABS H = ABS ( H ) CALL F ( X , P , Y P )

C EST I MATE E RRORS AT ORDERS K , K -l , K -2 C

C C C

C C C

C C

ERKM 2 = 0 . 0 ERKMI = 0 . 0 E R K = 0 . 0 DO 2 6 5 L = 1 , NEQN

T EM P 3 '" 1 . 0/WT ( L ) T EM P 4 .. Y P ( L ) - PII I ( L , I ) I F ( KM 2 ) 2 6 5 , 2 6 0 , 2 5 5

2 5 5 ERKM 2 .. ERKM 2 + « PII I ( L , KM 1 ) +TEM P 4 ) * T EM P3 ) * * 2 2 6 0 E RKM l '" ERKM l + « PH I ( L , K ) +TEMP 4 ) * T E M P 3 ) * * 2 2 6 5 ERK '" ERK + ( TEMP 4 *TEM P 3 ) * * 2

I F ( KM 2 ) 2 8 0 , 2 7 5 , 27 0 2 7 0 ERKM 2 .. ABSH * S I G ( KM 1 ) * G STR ( KM 2 ) *SQR'r ( E RKM 2 ) 2 1 5 ERKM1 = ABSH * S I G ( K ) *GST R ( KM 1 ) * SQRT ( E RKM 1 ) 2 8 0 TEM P S .. AB SH *SQRT ( ERK )

ERR " TEM P5 * ( G ( K ) -G ( K P 1 » E R K .. TEM P5 * S I G ( K P 1 ) *GSTR ( K ) K N EW '" K

TEST I F ORDER S HOULD B E LOWE RED

I F ( KM 2 ) 2 9 9 , 2 9 0 , 2 8 5 2 8 5 I F ( AMAX l ( E RKM1 , ERKM 2 ) . LE . ERK )

GO TO 2 9 9 KNEW

2 9 0 I F ( ERKM I . LE . 0 . 5 * E RK ) K N EW '" KMI

TEST I F STEP SUCCESSFUL

2 9 9 I F ( ERR . LE . EPS } GO TO 4 0 0 * * * END BLOCK 2 * * *

=


KM I

1 4 3 C * * * B EG I N B LOCK 3 * * * C TH E S T E P I S U N SUCCESS FUL . RESTORE X , PH I ( * , * ) , PS I ( * ) •

C I F TH I R D CON S E C U T I V E FA ILURE , S E T ORD E R TO O N E . I F STEP FA I L S MOR E C THAN TH R E E T I ME S , CON S I D E R AN OPT I MAL STEP S I Z E . DOU B L E E R R O R C TOL E HA N C E AND H ETU HN H' E S 'r I MA'l'ED s'n;p S I Z E I S 'roo SMAL L Fon HAClI I N E C P R EC I S I ON . C * * * C C R ESTORE x , PII I ( * , ' ) AND PS I ( * ) C

C

PHAS E I '" . FA L S E . X '" XOLD 0 0 3 1 0 I '" I , K

'rE M 1' 1 '" 1 . 0/BETA ( I ) I P I '" 1 + 1 D O 3 0 5 L '" 1 , NEQN

3 0 5 PH I ( L , I ) '" TEM P l * ( PH I ( L , I ) - PH I ( L , I P l » 3 1 0 CONT I N U E

I F ( K . LT . 2 ) G O TO 3 2 0 D O 3 1 5 1 '" 2 , K

3 1 5 PS I ( I - I ) '" PS I ( I ) - H

C ON TH I RD FA I L U R E , S ET ORD E R TO O N E . 'l'H E R EAFT E R , U S E O PT I MAL S T E P C S I Z!:: C

C C C C C C C

C

3 2 0 I FA I L '" I FA I L + 1 T E M P 2 '" 0 . 5 I F ( I FA I L - 3 ) 3 3 5 , 3 3 0 , 3 2 5

3 2 5 I F ( P 5 E P S . LT . 0 . 2 5 * E RK ) TEM P 2 '" SQRT ( P 5 EPS/ E RK ) 3 3 0 K N EW '" 1 3 3 5 H '" TEM P 2 *H

K = K N EW I F ( AB S ( II ) . G E . FOURU *ABS ( X » ) GO TO 3 4 0 C RAS H '" . TRU E . H '" S I G N ( FOU RU * ABS ( X ) , H ) E P S '" E PS + EPS R ETURN

3 4 0 GO TO 1 0 0 * * * END BLOCK 3 * * *

* * * B EG IN B LOCK 4 * * * TH E S T E P I S S U CC E S S FU L . CORR ECT TH E PRED I CTED SOL UT I O N , EVAL UATE TH E D E R I VAT IV E S U S I NG TH E CORRECT ED S OL UT I O N AND U PDATE TH E D I F FERENCE S . DETERM I N E BEST O RD E R AND S T E P S I Z E FOR N EXT STEP .

4 0 0 KOLD '" K HOLD '" H

* * *

C CORRECT AND EVAL UATE C

C

T E M P I • H *G ( K P 1 ) I F ( NORND ) GO TO 4 1 0 DO 4 0 5 L - 1 , NEQN

RHO '" TEMP 1 * ( Y P ( L ) - PH I ( L , I » - PH I ( L , 1 6 ) y e L l = p e L ) + RHO

4 0 5 P il I ( L , I S ) = ( Y ( L ) - P ( L » - RIIO GO TO 4 2 0

4 1 0 DO 4 1 5 L '" I , NEQN 4 1 5 Y ( L ) " p e L ) + TEMP l * ( Y P ( L ) - pa I ( L , I ) ) 4 2 0 CALL F ( X , Y , Y P )

C U PDATE D I F F ERENCES FOR NEXT S T E P C

DO 4 2 5 L '" I , NEQN PH I ( L , K P l ) '" YP ( L ) - PH I ( L , I )

4 2 5 PH I ( L , KP 2 ) - PH I ( L , K P l ) - PH I ( L , KP 2 ) DO 4 3 5 I - I , K

F i qu re A . 5 C o n t i n u e d

C C C C C C

C C C C

C

4 3 0 4 3 5

DO 4 3 0 L • 1 , NEQN 1 4 4 PH I ( L , I ) '" PU I ( L , I ) + PH I ( L , KP l )

CON T I N U E

E S T I MAT E E RROR A T O RD E R K + l U NLESS : I N F I RST PHAS E WHEN ALWAYS RA I S E ORDER , AL R EADY DE C I D ED TO LOW E R ORD E R , S T E P S I Z E N O T CONSTANT S O ES'l' I MA'l'E U N REL I ABf. E

E RK P I .. 0 . 0 I F ( K N EW . EQ . KM I . OR . K . EQ . 1 2 ) PHAS E I .. . FAL S E . I F ( PH AS E l ) GO TO 4 5 0 I F ( K N EW . EQ . KM l ) GO TO 4 5 5 I F ( K P I . GT . N S ) GO TO 4 6 0 DO 4 4 0 L .. 1 , N EQN

4 4 0 E R K P l .. ERK P I + ( PH I ( L , K P 2 ) /WT ( L » * * 2 E RK P l .. ABS H * G STR ( KP l ) * S Q RT ( ERKP 1 )

U S I NG E S T I MATED ERROR AT O RDE R K + l , DETERM I N E APPROPR I ATE O RD E R FO R N E XT S T E P

I F ( K . GT . 1 ) GO T O 4 4 5 I F ( E R K P 1 . G E . 0 . 5 * E RK ) GO TO 4 6 0 GO TO 4 5 0

4 4 5 I F ( ERKMI . L E . AM I N 1 ( E RK , E R K P l » GO TO 4 5 5 I F ( ER K P 1 . G E . ERK . OR . K . EQ . 1 2 ) G O TO 4 6 0

C H E RE E R K P I . LT . ERK . LT . AMAX 1 ( E RKM 1 , ERKM 2 ) EL S E ORD E R WOULD HAV E C B E E N LOWE RED I N B LOCK 2 . TH U S ORDER I S TO B E RA I S ED C C RA I S E O RD E R C

C

4 5 0 K .. K P I ERK '" ERK P 1 GO T O 4 6 0

C LOW E R O RD E R C

4 5 5 K .. KMI ERK .. ERKM1

C C W I TH N EW ORD E R DETERM I N E AP PROPR I ATE S T E P S I Z E FOR NEXT STEP C

c

4 6 0 H N EW = H + II I F ( PH AS E l ) GO TO 4 6 5 I F ( P 5 EPS . G E . ERK * TWO ( K + 1 » GO TO 4 6 5 H N EW = H I F ( P 5 EPS . G E . ERK ) GO TO 4 6 5 T E M P 2 .. K + l R .. ( P 5 E PS/ E RK ) * * ( 1 . 0/TEM P 2 ) H N EW '" ABSH * AMAX l ( 0 . 5 , AM I N 1 ( 0 . 9 , R » H N EW " S I G N ( AMAX 1 ( H N EW , FOU RU *ABS ( X » , H )

4 6 5 H = H N EW RETU RN

* * *

END END BLOC K 4 * * *

SUBROU T I N E I NTRP ( X , Y , XOUT , YOUT , Y PO U T , NEQN , KOLD , PH I , PS I ) C C TH E METHODS I N SUBROU T I N E STEP AP PROX I MATE TH E SOLUT I ON N EAR X C BY A POLY NOM I AL . SUBROU T I N E I NTRP AP PROX I MAT E S TH E SOLUT I O N AT C XOU T BY EVAL UAT I NG THE POLYNOM I AL TH E R E . I N FO RMAT I O N D E E-' I N I NG TH I S C POL YNOM IAL I S PAS S ED F ROM ST E P SO I N TRP CAN NOT B E U S ED ALON E .

C C TH I S COD E I S COMPL ETELY EXPLA I N ED AND POCUM E NTED I N TH E TEXT , C COM PUTER SOLUT I O N OF ORD I NARY D I F F E R E NT I AL EQUAT I O N S : 'rH E I N I 'l' I AL

C VAL U E PIlOB L EM BY L . F . S I IAM P I N E AND M . K . GOHDO N . C C I N PU T TO I NTRP --


1 4 5 C C T H E U S E R PROV I DES S'l'ORAG E I N TH E CALL I NG PROG RAM FOR TH E ARRAY S I N C THE CALL L I ST

' DI MEN S ION Y ( NEQN ) , YOUT ( NEQN ) , YPOUT ( NEQN ) , PH I ( NEQN , 1 6 ) , PS I ( 1 2 ) C AND D E F I N E S C XO UT -- PO I NT A T W H I CH SOL UT I ON I S D E S I RED . C Til E REMA I N I NG PARAME'l'I:: RS ARE D E F I N ED I N S1'E ? AND PAS S ED 'ro I NTRl' C F ROM THAT S U B ROUT I N E C C OUTPUT F ROM I NTRP -C C YOUT ( * ) -- SOL U T I O N AT XOUT C Y POU T ( * ) -- D E R I VATI V E OF S OL UT ION AT XOUT C TH E REMA I N I NG PARAMETERS ARE R ETU RNED U NALT E R ED FROM TH E I R I N P U T C VAL U E S . I N T EG RAT I ON W I TH S'rEP MAY B E CONT I N U ED . C

C

C

D I ME N S ION G ( 1 3 ) , W ( 1 3 ) , RH O ( 1 3 ) DATA G ( l ) / l . O/ , RHO ( l ) / l . O/

H I .. XOU T - X K I ;; KOLD + 1 K I P I ;; K I + 1

C I N I TI AL I Z E W ( * ) FOR COMPUT I NG G ( * ) C

C

DO 5 I .. 1 , K I T EMPI ;; I

5 W ( l ) " 1 . 0/TEM PI TERM ;; 0 . 0

C COM PUTE G ( * ) C

C

DO 1 5 J ;; 2 , K I J M1 .. J - 1 PS I J M 1 - PS I ( J M 1 ) GAMMA ;; ( H I + TE RM ) /PS I J MI ETA .. H I/ PS IJ M 1

L I MI T1 = K I P 1 - J DO 1 0 I .. I , L I M I T1

1 0 W ( I ) " GAMMA *W ( I ) - ETA *W ( I + 1 } G ( J ) .. W ( l ) RHO ( J ) .. GAMMA * RHO ( J M1 )

1 5 T E RM ;; PS I J M 1

C I NT E RPOLATE C

DO 2 0 L = I , NEQN Y POU'l' ( L ) .. 0 . 0

2 0 YOUT ( L ) ;; 0 . 0 DO 3 0 J .. I , K I

I .. K I P 1 - J T EM P 2 .. G ( I ) T EM P 3 .. RHO ( I ) DO 2 5 L .. I , NEQN

YOUT ( L ) .. YOUT ( L ) + TEM P 2 * PH I ( L , I ) 2 5 YPOUT ( L ) " YPOUT ( L ) + TEMP3 * P H I ( L , I ) 3 0 CONT I N U E

DO 3 5 L .. I , N EQN 3 5 YOUT ( L ) = Y ( L ) + H I *YOUT ( L )

RETU RN EN D

F i q u re A . 5 Co n t i n u e d

analysis ultimate heat sink · cl specific heat of water cv specific heat of water vapor cg...

Documents