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R73-22

DSR 73801

DSR 80004

RALPH M. PARSOLIJS LABORATORY

FOR WATER RESOURCES

AND HYDRODYNAMICS

Department of C i v i l Engineer ing

M a s s a c h u s e t t s I n s t i t u t e of Technology

THE

MECHAiJICS

OF SUBMERGED MULTIPORT DIFFUSERS

FOR BUOYANT

DISCHARGES I N SHALLOW

WATER

P

Gerha r d

J

r k a

and

Donald

R. F.

Harleman

Repor t N o . 169

March, 1 9 7 3

red mder

t h

suppor t

€

Stone and Webster Eng inee r ing Corporatio n

Boston, Massachusetts

' and

Long Island Lighting Company

H i c k s v i l l e ,

New

York

and

Nat iona l Sc ien ce Founda tion

Engineer ing E ne rg et ic s Program

Grant No.

GK-32472



ABSTRACT

A

submerged mul t ipor t d i f fuser

i s

a n e f f e c t i v e d e vi ce f o r d i s p o s a l

of

water

co n t iPn in g h ea t o r o th e r d eg rad ab l e

wastes

i n t o

a

n a tu ra l b o d y

of

water .

t a l

impact o f concen t ra ted

waste

c a n b e c o n s t r a i n e d t o

a

small

a rea .

A

high degree of d i lu t i on can be ob ta in ed and the env ironmen-

An a n a l y t i c a l a nd e x pe r im e n ta l i n v e s t i g a t i o n

i s

c on du ct ed f o r t h e

purpose o f develop ing p re d ic t iv e methods f o r buoyant d i scha rges f rom sub-

merged rn u l t i p o r t d i f fu s e r s .

A

m u l t i p o r t d i f f u s e r w it h g i ve n l e n gt h ,

n o z z l e s p ac i n g and v e r t i c a l a n g l e

of nozz les i s loc a te d on the bo ttom of a l a r g e body of

water

of uniform

depth. The ambient

water i s

u n s t r a t i f i ed and may b e s t ag n an t o r h av e

a

uni fo rm cu rr en t which runs

a t

a n a r b i t r a r y a n g l e t o t h e a x i s of t h e d i f -

fu ser . The ge ner a l case of

a

d i f f u s e r i n a r b i t r a r y d e pth of water and

a r b i t r a r y b uo ya nc y

i s

tr ea te d. However, emphasis

i s

p u t on t h e d i f f u s e r

i n s h al l ow

r e c e i v i n g water with low buoyancy, th e type used f o r d i sc har ge

of condenser

c o o l i n g w a t e r from thermal power plants .

The f o l l o w i n g p h y s i c a l s i t u a t i o n

i s

co n s id e red :

A

m u l t i p o r t d i f f u s e r

w i l l

produce

a

genera l th ree-d imens ional

f l o w f i e l d .

u l a t e d t o e x i s t i n t h e c e n t e r p o r ti o n o f t h e t h re e- di me ns io na l d i f f u s e r

cad

b e an a ly zed

as

a two-dimensional " cha nne l model", th a t

i s a

d i f f u s e r

sect ion bounded by w a l l s o f f i n i t e l e n g t h a nd op e ni ng s

a t

both ends

i n t o

a

l a r g e r e s e r v o i r .

Matching of t h e s o l u t i o n s f o r t h e f o u r d i s t i n c t

f low re g io ns which can be d i s cerne d i n th e c hannel model ,

namely,

a

buoyant j e t r e g i o n , a s u r f ace im pin gement r eg io n , an i n t e r n a l h y d r au l i c

jump region and

a

s t r a t i f i e d co u nt e rf l ow r e g io n , y i e l d s t h e s e r e s u l t s :

The ne ar - f ie ld zone is

s t a b l e o nl y f o r a l im i t ed r an g e o f

j e t

d e n s i m e t r i c

Froude numbers and r e l a t iv e dep ths .

The s t a b i l i t y i s a lso dependen t on

t h e

j e t

d i s ch a rg e an g l e .

buoyant j e t models assuming an unbounded re ce iv in g

water a re

a p p l i c a b l e

t o p r e d i c t d i l u t i o n s . O u t si d e of t h e p a ra m e te r r a n ge which y i e l d s

s t a b l e n e a r - f i e l d c o n d it i o n s, t h e d i f f us e r - i nd u c e d d i l u t i o n s a re essen-

t i a l l y d e t e r m in e d by t h e i n t e r p l a y o f

two f a c t o r s : f r i c t i o n a l e f f e c t s

i n t h e f a r - f i e l d an d t h e h o r i zo n t a l momentum in p u t of t h e j e t d i s ch a rg e .

T hr ee f a r - f i e l d f l ow c o n f i g u r a t i o n s

are p o s s i b l e , a coun ter f low sys tem,

a

s tagnant wedge system and

a

v e r t i c a l l y f u l l y m ixed f lo w, which

i s

t h e

ex tr eme cas e o f s u r f ace and b ot to m in t e ra c t i o n .

Y e t th e predominant ly two-dimensional f low which

i s

p o s t -

I t

i s o n ly i n t h i s l i m i t e d ra ng e t h a t p r e v io u s

A

t h ree -dim en s ion a l m odel fo r t h e d i f fu se r - i n du ced f lo w f i e l d i s

d eve lo p ed . B as ed on eq u iv a l en cy o f f a r - f i e l d e f f e c t s , t h e p re d i c t i o n s

of th e two-dimensional channel model can be l ink ed t o th e three-dimen-

s i o n a l d i f f u s e r c h a r a c t e r i s t i c s . D i f f u s e r s w i t h a n u n s ta b l e n e a r - f i e l d

p ro du ce t h r e e- d im e n si o na l c i r c u l a t i o n s w hi ch l e a d t o r e c i r c u l a t i o n

a t

t h e d i f f u s e r l i n e : e f f e c t i v e c o n t r o l

of

t h e s e c i r c u l a t i o n s

i s

p o s s i b l e

t h r o u g h h o r i z o n t a l n o z z l e o r i e n t a t i o n .

2



The di f f user i n an ambi ent cr oss- cur r ent i s st udi ed exper i ment al l y.

Di f f erent ext r eme r egi mes of di f f user behavi our can be descr i bed. Per -

f or mance

i s dependent on t he ar r angement of t he di f f user axi s wi t h res-

pect to t he cr ossf l ow di rect i on.

Exper i ment s ar e perf ormed i n t wo set - ups, i nvest i gat i ng bot h t wo-

di mensi onal sl ot s and thr ee- di mensi onal di f f user s. Good agr eement bet ween

t heor et i cal pr edi ct i ons and exper i ment al r esul t s

i s

f ound.

The r esul t s of t hi s st udy ar e pr esent ed i n f or m of di l ut i on gr aphs

whi ch can be used f or t hr ee- di mensi onal di f f user desi gn or prel i m nar y

desi gn i f pr oper schemat i zat i on of t he ambi ent geomet r y i s possi bl e.

Desi gn consi derat i ons ar e di scussed and exampl es ar e gi ven. For more

compl i cat ed ambi ent condi t i ons, hydr aul i c scal e model s are necessary.

The r esul t s of t hi s st udy i ndi cat e t hat onl y

undi st ort ed scal e model s

si mul ate

t he cor r ect ar eal ext ent of t he t emper at ur e f i el d and t he i nt er -

act i on wi t h cur r ent s, but ar e al ways somewhat conservat i ve i n di l ut i on

pr edi ct i on. The degree of conservati sm can be est i mat ed. Di st or t ed

model s are l ess conser vat i ve i n pr edi cti ng near - f i el d di l ut i ons, but

exagger at e t he ext ent of t he near - f i el d m xi ng zone.

3



ACKNOWLEDGEMENTS

This s tudy w a s supp orte d by Sto ne and Webster Engineering Corpora-

t io n, Boston, M assa chus et ts , and Long Is la nd Li gh t i ng Compnay, Hic ks vi l le ,

New

Y ork , i n c o n j u n c t i o n w i t h a n i n v e s t i g a t i o n of t h e d i f f u s e r d i s c h a r g e

s ys tem fo r t h e N o r th p o r t Gen e ra t i ng S t a t i o n l o ca t e d o n Long I s l an d

(DSR 73801). The a ut ho r s wish t o thank Messrs. B. Brodfeld and D.

H.

Matchet t

of

Stone and Webs ter fo r th e i r coop era t ion th roughou t the s tudy .

Addi t iona l funds

f o r p ub l ic a t io n of t h i s r e p o r t w e r e provided by t he

Na t io n a l

Science

Founda t ion, Engineering En er ge t i cs Program, Grant

No.

GK-32472

(DSR

8 0 0 0 4 ) .

T h e p r o j e c t o f f i c e r i s D r . Royal E. Rostenbach.

The ass is tance of g r ad u at e s t u d e n t s a nd t h e s t a f f

of

the

R . M .

Pars o n s L ab o ra to ry fo r

Water

Resources and Hydrodynamics i n car ryi ng o ut

the experimental program

i s

gr a t ef u l ly acknowledged . Pa r t ic u l ar men t ion

i s made of M. Watanabe, D. H. Evans, E. E. A d a m s , Graduate Research A s s i s -

t an t s ; E. McCaffrey, DSR St af f (In str um en tat ion ); R. Mil ley , Mach in is t

and S. E l l i s , undergra duate s tuden t . The manuscr i p t

w a s

typed by

K.

Emperor, S.

W i l l i a m s

and

S . Demeris.

The r es ea rc h work w a s con d uc ted u n de r t h e t ech n i ca l s u p e rv i s i o n

of D. R. F. Harleman, Pr ofe sso r of C iv i l Engi neer ing. The material

c on ta in ed i n t h i s r e p o r t w a s submit t ed by G. J i r k a , R e s e a r c h A s s i s t a n t ,

i n p a r t i a l f u l f i l l m e n t of t h e r eq ui r em en t f o r t h e d eg ree

of

Doctor of

Phi losophy a t M. I.

T.

4



FOREWORD

The r e s ea rch co n t a in ed i n t h i s r e p o r t

i s

p a r t o f a cont inuing

re s ea rch e f fo r t by t h e R a lp h M . Parsons Labora to ry fo r Water Resources

and Hydrodynamics on th e en gi ne er in g as pe ct s of w a s t e heat d i sposa l f rom

e l e c t r i c power ge nera t ion by means of submerged mu l t i por t d i f fus er s .

gu id ing ob jec t ive of the res ear ch p rogram i s th e development of pr edic-

The

t i v e m o d e l s f o r d i f fu s e r d i s ch a rg e which form th e b a s i s o f so und en gin ee r -

in g des ign compat ible wi t h env i ronmenta l requ i rements .

s i t e - r e l a t e d s t u d i e s co n cern ed wi th o p tim ized d i f f u s e r d es ig n u nd er

s p ec i f i c amb ient c o n d i t i o n s a re conducted.

I n a d d i t i o n ,

P r ev i ou s r e p o r t s r e l a t e d t o submerged d i f f u s e r s t u d i e s

are:

Thermal Diffus ion o f condenser Water i n a River During Steady and

1

Unsteady Flows" by Harleman, D. R. F.,

H a l l ,

L. C . a n d C u r t i s , T . G.,

M.I.T.

Hydrodynamics La bo ra tor y Techn ical R eport No.

111,

September 1968.

A

Study

of

Submerged Mult i -por t Di ff use rs f o r Condenser Water Discharge

1

wi th App lica tion t o t he Shoreham Nuclear Power St at io n" by Harleman,

D.

R. F., J i rka ,

G.

and Stolzenbach, K. D., M.I .T. Parsons Laboratory

f o r Water Resources and Hydrodynamics Tec hni ca l Repo rt No. 139, August

1971.

In v es t i g a t i o n o f a Submerged,

S l o t t ed P ip e Di f fus er f o r Condenser Water

1

Di sch arg e from t he Canal P l a n t , Cape Cod Canal" by

H a r l e m a n ,

D .

R.

F.,

J i r k a , G., Adams, E.

E.

and Watanabe, M . , M.I.T. Parsons Labora to ry fo r

Water Resources and Hydrodynamics Te ch ni ca l Rep ort

N o .

141, October 1971.

"Experimental In ve st ig at io n of Submerged Mult i port Di ff us er s f or Conden-

s e r Water Disch arg e wi th A p p l i ca t i o n t o t h e No r th p o rt E l ec t r i c Generat ion

S ta t i o n "

by

Harleman, D.

R.

F.,

Jirka, G.

and Evans, D. H.,

M.I.T.

Parsons

L ab o ra to ry fo r Water Resourc es and Hydrodynamics, Te ch ni ca l Report

No.

165,

February 1973.

5



TABLE OF CONTENTS

ABSTRACT

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

I.

I NTRODUCTI ON

Page

2

4

5

11

1.1

H stori cal Perspect i ve

1 .2

Basi c Features of Mul t i port D f f users

f or Buoyant Di scharges

1.3 Obj ecti ves of thi s Study

1.4 Summary of the Pr esent Work

11. CRI TI CAL REVI EWOF PREVI OUS PRED CTI VE MODELS FOR

SUBMERGED MULTI PORT DI FFUSERS

2.1

I nvest i gat i ons of Buoyant J ets

2.1.1

General Character i st i cs

2.1.2

Round Buoyant J ets

2.1.3 Sl ot Buoyant J et s

2.1.4

2.1.5

Ef f ect

of

t he Free Surf ace

2.1.6

2.1.7

Ef f ect of Crossf l ow

Lateral I nter f erence of Round Buoyant

J ets

Ef f ect of Ambi ent Densi ty St rat i f i cat i on

2.2

One- D mensi onal Average Model s f or Hori zontal

D f f user D scharge i nto Shal l owWater

2.2.1

The Two-D mensi onal Channel Case

2.2.2 The Three- D mensi onal Case

2.3

Apprai sal of Previ ous Know edge About the

Character i sti cs of a Mul t i port D f f user

111.

THEORETI CAL FRAMEWORK: TWO-DD ENSI ONAL CHANNEL MODEL

3.1

Basi c Approach

1 2

13

15

17

1 9

1 9

1 9

2 1

26

29

37 I

38

J'

38

39

39

42

46

50

50

3.2 Probl emDef i ni t i on; Two- D mensi onal Channel Model 53

3 . 3

Sol ut i on Method 55

3.4

Domnant Fl owRegi ons 59

. 6



Page

3.4.1 Buoyant J et Regi on

59

3.4.1.1 Approxi mati ons and Governi ng

Equati ons

3.4.1.2 Dependence of the Entrai nment

on Local J et Characteri sti cs

3.4.1.3

I ni t i al Condi t i ons: Zone of

Fl owEstabl i shment

3414 Sol ut i on of the Equati ons

3.4.2 Surface I mpi ngement Regi on

3.4.2.1 General Sol uti on

3422 Speci al Cases

3.4.2.3 Vert i cal Fl owD stri but i on

Pri or

t o

the Hydraul i c J ump

3.4.3 Hydraul i c J ump Regi on

3.4.3.1 General Sol uti on

3.4.3.2

Sol ut i on for J umps w th

Low

Vel oci t i es and Weak Buoyancy

344 Strati f i ed Counterf l owRegi on

3441 Approxi mati ons and Governi ng

3442 Si mpl i f i ed Equati ons, Negl ect-

Equati ons

i ng Surf ace Heat Loss and

I nterf aci al Mxi ng

3443 Head

Loss

i n Strati f i ed Fl ow

3.4.4.4 Speci al Cases

35

Mafchfng of

Sol ut i ons

3.5.1 Governi ng Mon-D mensi onal Parameters

36 Theoreti cal Predi ct i ons: D f fusers w th

no

Net

Hori zontal

MomcDtum

361

The

Near-Fi el d Zone

3.6.2 The Far-Fi el d Zone

3621 I nteracti on w th Near-Fi el d

3.6.2.2 I nt erf acbl Fri cti on Factor

3623 Sol uti on Graphs

3.7 Theoreti cal Predi cti ons: D f f users w th Net

Hori zontal Momentum

7

59

67

72

75

81

81

89

93

95

95

98

102

102

109

117

120

124

124

127

127

131

131

134

134

140



3.7.1 The Near - Fi el d Zone

3. 7. 2 The Far - Fi el d

Zone

141

141

3. 7. 2. 1 Possi bl e Fl ow Condi t i ons 141

3. 7.2.2 Sol ut i on Bet hod 145

3. 7. 2. 3 Sol ut i on Gr aphs 147

3.8

Summar y 148

I V. THREE-DI MENSI ONAL ASPECTS OF

THE

DI FFUSER I NDUCED

FLOW

FI ELD 157

v.

4.1

Rel at i ng t he Two- Di mensi onal Channel Model

t o t he Thr ee- Di mensi onal Fl ow Fi el d 15 8

4.1.1

Di f f user s wi t h No Net Hor i zont al

Moment um 15a

4.1.1.1 Equi val ency Requi r ement s 158

4.1.1.2

Model f or t he Thr ee- Di mensi onal 162

Fl ow Di str i but i on

4.1.2 Di f f user s wi t h Net Hor i zont al Moment um 169

4. 2 Di f f user I nduced Hor i zont al Ci r cul at i ons 169

4. 2. 1 Di f f users wi t h No Net Hori zont al Moment um 170

4. 2.1.1 Generat i ng Mechani sm

4.2.1.2 Cont r ol Met hods

170

163

4.2.2 Di f f user s wi t h Net Hor i zont al Moment um 177

4.2. 2.1 Generat i ng Mechani sm

4.2.2. 2 Cont r ol Methods

177

179

EXPERI MENTAL EQUI PMENT AND PROCEDURES 180

5. 1 Basi c Consi derat i ons f or Di f f user Exper i ment s 180

5. 1.1 Exper i ment al Pr ogram

5. 1. 2 Exper i ment al Li m t at i ons

5. 2 The Fl ume Set - Up

5.2.1 Equi pment

5.2.2 Experi ment al Pr ocedur e

5. 2. 3 Exper i ment al Runs

5. 2. 4 Data Reduct i on

8

180

183

185

185

19

0

190

19

1



VI.

V I I .

5.3 The Basi n Set-Up

5.3.1 Equi pment

5.3.2 Experi ment al Pr ocedure

5.3.3

Experi ment al Runs

5.3.4

Dat a Reduct i on

5 . 4

Exper i ment s

by

Ot her I nvest i gat or s

DI FFUSERS

WITHOUT

AMBIENT CROSSFLOW COMPARI SON

OF THEORY AM, FXPERIMENTS

6.1 Di f f user s w t h No Net Hor i zont al Moment um

6.1.1

Two- Di mensi onal Fl ume Exper i ment s

6.1.2 Thr ee- Di mensi onal Basi n Experi ment s

6.2

Di f f users w t h Net Hor i zont al Moment um

6.2.1 Two- Di mensi onal Fl ume Exper i ment s

6.2.2

Thr ee- Di mensi onal Basi n Experi ment s

6.3 Concl usi ons: Di f f user s W t hout Ambi ent Cr ossf l ow

DI FFUSERS

I N

AMBIENT CROSSFLOW

EXPERIMENTAL

RESULTS

7.1 Basi c Consi der at i ons

7.2 Method of Anal ysi s

7.3 Fl ume Exper i ment s: Per pendi cul ar Di f f user

7 . 4 Three- Di mensi onal Basi n Exper i ment s

7.4.1 Di f f users wi t h No Net Hori zont al Moment um

7.4.2 Di f f user s w th Net Hor i zontal Moment um

7.5 Concl usi ons: Di f f users wi t h Ambi ent Cr ossf l ow

V I I I .

APPLI CATI ON OF RESULTS TO DESI GN ANTI HYDRAULIC

SCALE MODELI NG

OF SUBMERGED MULTIPORT

DI FFUSERS

8.1

Si t e Char acter i st i cs

8.2

Di f f user

Desi gn

f or Di l ut i on Requi r ement

8.2.1 Gl ossary of Desi gn Parameters

8.2.2 Desi gn Obj ect i ves

Page

192

192

198

199

199

200

202

202

202

205

2 6

216

2 8

229

232

232

233

237

243

243

248

256

260

261

263

263

265

9



8 . 2 . 3 Desi gn Procedure

8 . 2 . 3 . 1 Exampl e: D f f user i n a

Reversi ng Ti dal Current System

8 . 2 . 3 . 2 Exampl e: D f f user i n a Steady

Uni f ormCurrent

8 . 3 The Use of Hydraul i c Scal e Model s

8 . 3 . 1 Model i ng Requi rements

8 . 3 . 2 Undi storted Model s

8 . 3 . 3 D storted Model s

8 . 3 . 4

Boundary Cont rol

I X

.

SUMMARY AND CONCLUSI ONS

9. 1

Background

9 . 2 Previ ous Predi ct i ve Techni ques

9 . 3 Summary

9 . 3 . 1 D f f users Wt hout Ambi ent Crossf l ow

9 . 3 . 2 D f f users Wth Ambi ent Crossf l ow

9 . 4 Concl usi ons

9 . 5 Recommendat i ons f or Fut ure Research

LI ST OF REFERENCES

LI ST OF F I GWS

LI ST OF TABLES

GLOSSARY OF SYMBOLS

Page

267

2 6 8

2 7 6

27 9

2 8 0

2 8 3

2 8 4

287

2 8 8

2 8 8

2 8 9

2 8 9

2 8 9

2 9 4

2 9 5

2 9 8

300

3 0 3

3 0 8

3 0 9

10



I.

INTRODUCTION

I n managi ng t he wast e wat er whi ch accr ues as a r esul t of man' s

domest i c and i ndust r i al act i vi t i es di f f er ent met hods of t r eat ment , re-

cycl i ng and di sposal are used.

wat er management i s det erm ned by economc and engi neer i ng consi der at i ons,

such as cost s and avai l abl e t echnol ogy,

and by consi der at i ons

of

envi r on-

ment al qual i t y, each scheme havi ng a cert ai n i mpact on t he natur al envi r -

onment.

The choi ce of a speci f i c scheme of wast e

I n

many i nst ances t he di schar ge of water cont ai ni ng heat or other

degr adabl e wast es i nt o a natur al body of wat er

i s

a vi abl e economc and

engi neer i ng sol ut i on.

t o

r egul at e t he adver se ef f ect s of such di schar ges on t he recei vi ng

wat er .

t he bi ol ogi cal , chem cal and physi cal pr ocesses whi ch occur i n r esponse

t o t he wast e water di scharge. The st andards have the obj ect i ve of pre-

ser vi ng or enhanci ng t he use

of

t he natur al water body f or a var i et y

of

human needs.

Water qual i t y st andards' ' have been est abl i shed

These st andar ds are based on exi st i ng sci ent i f i c know edge of

A common f eat ure of al l wat er qual i t y st andar ds,

as set f ort h by

var i ous l egal aut hor i t i es, i s a hi gh di l ut i on requi rement : W t hi n a

l i m t ed m xi ng zone the wast e water has to be thor oughl y m xed w t h t he

r ecei vi ng water. The purpose of t hi s requi r ement i s t o const r ai n the

i mpact of concent r ated wast e water t o a smal l area.

I t

i s agai nst t hi s backgr ound t hat t he i ncreasi ng appl i cat i on of

submer ged mul t i por t di f f user s as an ef f ect i ve devi ce f or di sposal of

wast e wat er must be underst ood. A submerged mul t i port di f f user i s

essent i al l y a pi pel i ne l ai d on the bot t om of t he r ecei vi ng wat er .

The

11



waste water i s d i s ch a rg ed i n t h e form o f rou nd t u rb u l en t j e t s th rough

p o r t s o r n o z z l e s w hi ch are s pa ce d a l on g t h e p i p e l i n e .

t r i b u t i o n

of

c o n c e n t r a t i o n of t h e d i s c h a r g e d

w a s t e

mater ia ls w i t h i n t h e

r e c e i v i n g

w a t e r

depends on

a

v a r i e t y of p h y s i c a l p r o c e ss e s .

unders tand ing of t h e s e p r o ce s se s

i s

needed s o t h a t p r e d i c t i v e mo de ls c a n

be developed which

form the ba s i s o f

a

sound eng ineer ing des ign .

1.1 H i s t o r i c a l P e r s pe c t iv e

T h e r e s u l t i n g d i s -

A c l e a r

F o r severa l decades many co a st al

c i t i e s

have u t i l i zed submerged

m u l t i p o r t d i f f u s e r s f o r t h e d i s c h a r g e of m u n i c i p a l sewage w a t e r . Note-

wor thy as pe c t s o f th es e "sewage d i f fus er s ' '

are :

1 )

Water

q u a l i t y s t a n -

d a rd s d i c t a t e d i l u t i o n r e q ui re m en ts i n t h e o r d e r o f 100 and h i g h e r when

sewage w a t e r

i s

d i s ch a rg ed .

t o f a i r l y d ee p w a t e r (more than 100 f ee t deep) .

d i s ch a rg ed water i s s i g n i f i c a n t .

sewage

water

and ocean

water

i s

ab o u t 2.5%.

A s a

c o n s e q u e n c e t h e s e d i f f u s e r s

are

l i m i t e d

2 ) The buoyancy of t h e

The

re la t ive

d e n s i t y d i f f e r e n c e b et we en

O nly i n v e r y r e c e n t y e a r s h av e m u l t i p o r t d i f f u s e r s f ou nd a p p l i -

c a t i o n f o r t h e d i s c h a r g e of h e a te d c o n de n se r c o o l i n g

water

from thermal

power plants .

The main impe tus ha s come from t h e im plem entat ion of

s t r i n g e n t t e m p e r a t u r e s t a n da r d s .

f i c a t i o n of t h e r e c e i v in g

w a t e r

a nd o n t h e c o o l i n g

water

t em p era tu re

r i s e

d i lu t i o n s b e tween ab o u t 5 and

20 a re

r e q u i r e d w i t h i n a s p e c i f i e d m i xi ng

area.

T h i s d i l u t i o n r eq ui re me nt f r e q u e n t l y r u l e s o u t r e l a t i v e l y s i m p le

d is po sa l schemes , such as discharge by means of a s u r fa c e c a n a l o r

a

Depending on

the water

q u a l i t y

c lass i -

s ing le submerged p ipe .

p la ce d i n r e l a t i v e l y s ha ll ow water

( c o n s i d e r a b l y

l e s s

t h a n 1 0 0 f t . d e ep )

and s t i l l a t t a i n t h e r e q u ir e d di l u t i o ns .

On t h e o t h e r ha nd , m u l t i p o r t d i f f u s e r s c a n b e

The economic advantage i n

12



keep ing t he conveyance d i s tan ce f rom th e sh or e l in e sh or t might be sub-

s t a n t i a l ,

i n p a r t i c u l a r i n la k es , e s t u a r i e s o r c o a s t a l w at er s w i t h ex-

tended sha l lo w nea rsh ore zones. "Thermal d if fu se rs " have the se char-

a c t e r i s t i c s :

buoyancy of th e dis cha rge d water i s low.

are i n

t h e

o rd e r o f 0.3% correspond ing to a t e mp e ra t ur e d i f f e r e n t i a l o f

about 20"F, an av e rag e v a lu e fo r t h e rm a l power p l an t s .

1)

They may b e l o ca t ed i n r e l a t i v e l y s h a ll o w

water.

2)

The

Rela t ive d e n s i ty d i f f e r e n c e s

Due t o t h e s e e s s e n t i a l d i f f e r e n c e s , r e g a r d i n g d ep th of t h e

re-

ce iv in g w a t e r and buoyancy o f t he d i schar ge , th er e i s a pronounced

d i f f e r en ce i n the mechanics o f "sewage d i f fu se rs " and "thermal d i f fu ser s" .

Consequen tly , p r ed ic t i ve models which have been es ta b l i she d and ve r i f i ed

f o r t h e class o f "sewage d i f fu s e r s " f a i l t o g i v e c o r r e c t p r e d i c t i o n s

when ap p l i ed f o r t h e class of " thermal d i f fusers" .

1 . 2

Basic

Featu res o f Mul t ipor t Di f fu se rs f o r Buoyant Discharges

The p e rfo rm an ce ch a ra c t e r i s t i c s o f

a

m u l t i po r t d i f f u s e r , t h a t i s

t h e d i s t r i b u t i o n s of v e l o c i t i e s ,

d e n s i t i e s an d co n cen t r a t i o n s wh ich

r e s u l t

when

t h e d i f f u s e r i s o p e ra t i n g ,

are

inf lue nce d by many ph ys ic al

p rocesses .

-- d i v i d e d i n t o

two

groups.

These pro ces ses may be co nve nie nt l y -- y e t somewhat l oo se ly

I1

Near- f ie ld" p rocesses are d i re c t l y g ov erned by t h e g eo m e t ri c ,

dynamic and b uo ya nt c h a r a c t e r i s t i c s o f t h e d i f f u s e r i t s e l f a nd of t h e

ambient

w a t e r

i n t h e immediate d i f f u s e r v i c i n i t y . S i g n i fi c a n t f e a t u r e s

are: T u rb u l en t j e t d i f fu s i o n p ro du ces a g r a d u a l i n c r e a s e i n j e t t h i ck -

ness (" je t sp read ing") and

a

s im u lt an eou s r ed u c t i o n of v e lo c i t i e s and

c o n c e n t r a t i o n s w i t h i n t h e j e t s through entrainment of ambient w a t e r . The

t r a j e c t o r y o f

t h e

j e t s

i s d e te rmin ed by t h e i n i t i a l an g le and b y i n f l u en ce

13



of buoyancy causing

a

r i s e t oward s t h e s u r f ace . B e fo re s u r fac in g t h e

j e t

spreading becomes s o l a r g e t h a t l a t e r a l i n t e r f e re n c e b etween ad j a -

cent j e t s forms a two-dimensional j e t a l o n g t h e d i f f u s e r l i n e . Upon

impingement on the surface

of

t h e r e c e i v i n g

w a t e r

t h e

j e t

i s

t rans fo rmed

i n t o

a

h o r i z o n t a l l y moving b uo ya nt l a y e r . S t a b i l i t y o f t h i s l a y e r

i s

of

c r u c i a l i mp or ta nc e. I n s t a b i l i t i e s r e s u l t i n r e- e nt ra in m en t o f a l r e ad y

mixed

water

i n t o t h e j e t d i f f u s i o n p r o c es s . I n a d d i t i o n t o t he s e b a s i c

processes , ambien t con d i t io ns such as c r o s s - c u r r e n ts and e x i s t i n g n a t u r a l

d en si ty s t r a t i f i c a t i o n c an ha ve

a

s t r o n g e f f e c t o n t h e n e a r- f ie l d.

F a r - f ie l d ' ' p r o c e s s e s i n f l u e n c e t h e m o ti on an d d i s t r i b u t i o n o f

1

mixed

w a t e r

away from th e ne ar -f ie ld zone. The mixed water i s d r iv en

by i t s buoyancy a g a i n s t i n t e r f a c i a l f r i c t i o n a l r e s i s t a n c e as d e n s i t y

c u r r e n t s , t h u s a f low away from th e d i f f us er i s g en e ra t ed .

a flow t ow ar d t h e d i f f u s e r a g a i n s t i n t e r f a c i a l a nd b ot to m f r i c t i o n i s

s e t

up as t h e t u r b u l e n t e n tr a in m e nt i n t o t h e j e t s

ac t s

l i k e a s i nk f o r

ambient water .

cu r ren t s an d t h e d i f f u s io n b y am bien t t u rb u l en ce an d t h e co n cen t r a t i o n

red uct ion through t ime-dependent decay proc ess es may be impo rtant pro-

cesses .

Conversely ,

Fur thermore th e convect ion of th e mixed water by ambient

The e f f i c i e n c y o f t h e n e a r - f i e l d p r o c e s s e s ( n o t ab l y j e t mixing)

i n r ed u c in g t h e c o n c e n t r a t i o n s of t h e d i s c h a r ge d

water

i s

dominant over

f a r - f i e l d p r o c e s s e s w hi ch u s u a l l y a c t o v e r

a

l o n g e r t i m e sca le . How-

e v e r , t h e r e

i s a

co u p l in g b etween n ea r and f a r - f i e l d p ro ces s e s , n ea r-

f i e l d p r oc e ss e s a f f e c t i n g t h e f a r - f i e l d a nd v i c e

versa.

a t o t a l p r e d i c t i o n o f t h e pe rf or ma nc e c h a r a c t e r i s t i c s of

a

rn u l t i p o r t

d i f f u s e r must i n c l u d e t h i s c o u p l in g .

Thus i n g en e ra l ,

14



Y e t

i n s p e c i a l cases th e coup l ing ef fe c t may be so weak th a t th e

nea r- f ie l d p roce sses may be assumed no t t o be in f lu ence d by the fa r - f ie ld .

D i f f u s e r s i n d e ep water wi th high buoyancy of th e dis ch ar ge ("sewage

d i f f us e r s " ) f a l l i n t o t h i s c at eg o ry . Th es e d i f f u s e r s p ro duc e

a

s t a b l e

sur fa ce la ye r which moves away from the d if f us e r

as

a d e n s i t y c u r r e n t .

N e a r - f i e l d d i l u t i o n s

a r e then p r imar i ly caused by j e t entrainment and

t h e d i f f u s e r c a n b e a n al y ze d

as a se r ies

o f r o u n d i n t e r a c t i n g

j e t s

i n

i n f i n i t e water.

T h i s

a n a l y s i s

i s

t h e b a s i s o f most e x i s t i n g p r e d i c t i v e

models f o r d i f f u s e r d i s ch a rg es .

On th e o t he r hand, d i f f us er s

i n s h a l l ow

water

with low buoyancy

(" thermal d i f fus ers ") may no t c r ea te a s t a b l e s u r f a ce l a y e r.

ly , a l r ead y mixed

w a t e r

g e t s re - en tr a in e d i n t o t h e

j e t s

t o s uc h

a

d eg ree

t h a t t h e i n c r e a s e d buoyancy f o r c e o f t h e s u r f a c e l a y e r i s s u f f ic i e n t t o

overcome t h e f r i c t i o n a l e f f e c t s i n t h e f a r - f i e l d . Hence i n t h i s c as e a

com po si te an a ly s i s o f n ea r - f i e l d an d f a r - f i e l d m us t b e u nd e rt aken i n

develop ing p red ic t ive models .

Subsequent-

T h i s co n t r a s t i n g d i f f e r en c e b etween t h es e two t y p es o f d i f fu s e r s

i s

q u a l i t a t iv e l y i l l u s t r a t e d i n F ig ur e 1-1.

t i c a l a nd n o n - ve r t i c a l d i s c h a rg e s wi t ho u t a m bi en t c u r r e n t s . A s a n ex-

treme case

of

t h e n o n - v er t i c a l d i s c h a r ge i n s ha l l ow water a uni-direc-

t i o n a l f l o w o f am bient water toward the diffuser and of mixed water away

f ro m th e d i f fu s e r i s e s t a b l i s h e d

( s ee F ig u re 1 - l d ) .

1 . 3 Ob jec t i v es of t h i s Stud y

Examples are shown f o r ver-

T h i s i n v e s t i g a t i o n i s concerned wi t h t he development o f p r ed ic t i ve

methods fo r buoyant d is cha rge s from submerged mu lt i por t d i ff us er s . The

f o ll ow i ng p h y s i c a l s i t u a t i o n i s cons idered :

A

m u l t i p o r t d i f f u s e r w i t h

15



C

L

0

-e

L-

Q)

4

3 .

d

16



g iv e n l e n g t h , n o z z l e s p a c i n g and v e r t i c a l a n g l e o f n o z z l e s i s loca ted on

the bot tom of a l a rg e body o f water of uniform dep th. The ambient

water

i s u n s t r a t i f i e d and may b e s t ag n an t o r h av e

a

uniform current which runs

a t

an a r b i t r a r y a n g l e

t o

the

axis

o f t h e d i f f u s e r .

A l l o f t h e n e a r - f i e l d p r o ce s se s b u t o nl y p a r t

of

t h e f a r - f i e l d

processes

(excluding e f f e c t s of ambient turbulence and decay processes)

a r e t ak en i n t o acco u nt .

.

T h i s s t u d y ad d re s s e s t h e g en e ra l case of a d i f f u s e r i n a r b i tr a r y

depth of water and a r b i t r a ry buoyancy.

on t h e d i f f u s e r i n s h a l l o w r e c e i v i n g

water

wi th low buoyancy, the type

frequ en t ly used fo r d i sc har ge of condenser coo l ing

water

from thermal

power plants.

o f the d i f f us er p ipe (mani fo ld des ign problem) .

However, s p e c i a l emphasis i s p u t

The study

i s

not concerned w i t h t h e i n t e r n a l h y dr a u li c s

A p pl ic a ti on of t h e r e s u l t s o f t h i s i n v e s t i g a t i o n i s a n t i c i p a t e d

f o r va r i o u s a s p e c t s :

-- E co nom ical d e s ig n o f t h e d i f fu s e r s t r u c tu r e .

--

Design to

m e e t

s p e c i f i c

water

q u a l i t y r eq u i r em en t s .

-- E v a lu a t io n o f t h e im p ac t i n r eg io n s away f ro m th e d i f f u s e r ,

such as t h e p o s s i b i l i t y of r e c i r c u l a t i o n i n t o t h e c o ol in g

water i n t a k e of thermal power plants .

-- Design and ope ra t io n o f hydr au l i c scale models.

1 .4

Summary

of the Present Work

An a n a l y t i c a l a n d e x p er i m en t a l i n v e s t i g a t i o n

i s

conducted.

I n Chap ter 2 a

c r i t i c a l

review o f e x i s t i n g p r e d i c t i o n t e c h n i q u e s

f o r m u lt i po r t d i f f u s e r s i s given.

Chapter

3

p r e s e n t s t h e t h e o r e t i c a l f ra mew ork f o r t h e s t u d y

of

17



di f f users wi t hout ambi ent cr ossf l ow: Recogni zi ng t he pr edom nant l y two-

di mensi onal f l ow pat t er n whi ch pr evai l s i n the cent er por t i on of a di f -

f user , pr edi ct i ve model s ar e devel oped f or a t wo- di mensi onal ' khannel

model , i . e. a di f f user sect i on i s bounded l at er al l y by wal l s

of

f i ni t e

l ength.

Thi s concept ual i zat i on al l ows t he anal ysi s of ver t i cal and

l ongi t udi nal var i at i ons of t he di f f user- i nduced f l ow f i el d.

Chapter 4 di scusses t hr ee- di mensi onal aspect s of di f f user di s-

char ge. Thr ough a quant i t at i ve anal ysi s r egar di ng f ar - f i el d ef f ect s

( f r i cti onal r esi st ance i n t he f l ow away zone) t he l engt h of t he two-

di mensi onal channel model i s l i nked

t o

t he t hr ee- di mensi onal di f f user

l engt h. Thus t he t heor et i cal pr edi ct i ons devel oped f or t he t wo- di men-

si onal channel model become appl i cabl e t o

t he general t hr ee- di mensi onal

case. Chapt er

4

al so di scusses the cont r ol of hor i zont al ci r cul at i ons

i nduced by t he di f f user act i on.

The experi ment al f aci l i t i es and pr ocedur es are descr i bed i n

Chapt er

5.

( channel model s ) and three- di mensi onal model s.

Exper i ment s wer e per f ormed bot h on t wo- di mensi onal model s

I n Chapt er

6

exper i ment al r esul t s f or di f f users wi t hout ambi ent

cr ossf l ow ar e r eport ed and compared t o t heoret i cal pr edi ct i ons.

The ef f ect of a uni f orm ambi ent cr ossf l ow on di f f user perf or mance

i s

st udi ed i n Chapt er

7.

Thi s par t i s mai nl y experi ment al ; however ,

l i m t i ng cases of crossf l ow i nf l uence ar e di scussed theoreti cal l y.

Di f f user arr angement s

w th the

di f f user axi s ei t her per pendi cul ar or

par al l el t o the crossf l ow di r ect i on wer e tested.

The appl i cat i on

of the

r esul t s t o pr act i cal pr obl ems

of

di f f user

desi gn and oper ati on of hydraul i c scal e model s i s di scussed i n Chapt er

8.

18



11-

CRITIC& REVIEW O F PREVIOUS PREDICTIVE MODELS FOR

SUBMERGED MULTIPORT DIFFUSERS

E x i s t i n g p re d i c t i v e t ech niq ues fo r

the

a n a l y s i s o f submerged

m u l t ip o r t d i f f u s e r s f a l l i n t o two r e s t r i c t e d g ro up s:

models descr ibe the p hys i ca l p rocesses govern ing buoyan t j e t s i n a n i n -

f i n i t e body o f water.

(without proof) assumed t h a t t h e e f f e c t of t h e f i n i t e w a te r d e pt h ca n b e

neglected and a s t a b l e f low away from th e l i n e of su rf a ce impingement

ex is t s .

Only

t h e most s i g n i f i c a n t c o n t r i b u t i o n s

are

reviewed.

s i o n a l a v er a ge models f o r h o r i z o n t a l d i f f u s e r d i s c h a r g e

assume full

ver-

t i c a l mixing downstream of th e di f f u s e r and

a re

v a l i d o n ly fo r s h a l l o w

r e c e i v i n g

water.

2.1 In v es t i g a t i o n s

of

Buoyant

J e t s

F i r s t , b uo yant

j e t

I n ap p ly in g t h es e m od el s

it

i s u s u a l l y t a c i t l y

There i s a large amount of l i t e r a t u r e on b uo ya n t j e t s .

Secondly, one-dimen-

2 . 1 . 1 General

Charac ter is t ics

Turbulent buoyant

j e t s

( a l s o ca l l ed fo rced p lu mes) are examples of

f l u i d motion wi th shear-genera ted

f r ee

tu rbu lence .

turbulent buoyant j e t

a re

the simple non-buoyant j e t , dr iven by t he

momentum of fluid discharged into a homogeneous medium, and th e

simple

plume, emanating from

a

co n cen t r a t ed s o u rce o f dens i ty def ic iency and

dr ive n by buoyancy fo rc es .

S p e c i a l

cases

of

t h e

Dominat ing t ran spo r t p rocesses govern ing th e d i s t r ib u t io n of f low

q u a n t i t i e s

a re

convect ion by the mean v e l o c i t i e s , a c c e l e r a t i o n i n the

d i r e c t i o n o f

the

buoyancy fo r ce and t u rb u l en t d i f fu s io n b y t h e i r r eg u l a r

eddy motion within the

j e t .

19



Main p roper t ies

of the je t

f l o w f i e l d and their impor tan t impl i -

ca t i o n s o n p o s s ib l e me th ods

o f

a n a l y s i s are (Abramoyich ( 1 9 6 3 ) , Schl ich-

t i n g (1968)

) :

1)

Gradual sp read ing o f

the j e t

width. The

j e t

w i d t h i s

s m a l l

compared t o t h e d i s t an ce f rom

the

s o u r c e a l o n g

the axis

of

the

j e t . T h i s a l l ow s t o make t h e t y p i c a l

boundary

l a y e r

assumptions:

verse v e l o c i t i e s can be neg lec t ed compared t o con vect ion

by mean

axia l

v e l o c i t i e s . D i ff u si on i n t h e

a x i a l

d i r e c -

t i o n

i s

small com pared t o d i f f u s i o n i n t h e t r a n s v e r s e

d i r e c t i o n .

Convection by mean t r a n s -

2) S e l f - s i m i l a r i t y o f t h e

f l o w .

The t r a n s v e r s e p r o f i l e s

of v e l o c i t y , m a s s and

heat

a t d i f f e r e n t a x i a l d i s t a n c e s

a l o n g t h e j e t

a r e

s i m i l a r t o e a c h o t h e r .

q u a n t i t i e s c a n b e e x p r e s s e d as a f u n c t i o n of c e n t e r l i n e

q u a n t i t i e s a n d j e t wid th .

L o ca l j e t

3)

F l u ct u a ti n g t u r b u l e n t q u a n t i t i e s

are

s m a l l compared

t o mean c e n t e r l i n e q u a n t i t i e s .

4 ) For j e t s i s s u i n g i n t o u nc on f in e d r e gi o n s p r e s s u r e

g r a d i e n t s are n e g l i g i b l e .

I f s em i- em pi ri ca l r e l a t i o n s h i p s r e l a t i n g

the

t u r b u l e n t s t r u c t u r e

O f the j e t t o i t s mean p r op er t i es ( such as the m ixin g l en g th h y p o th es i s )

a r e invoked,

a

s i m i l a r i t y s o l u t i o n t 9 the s i m p l i f i e d g o v e rn i n g e q u a t i o n s

wi th s p ec i f i e d b ou nd ary co n d i t i o n s i s p o s s ib l e .

T h i s

is shown by

Sch l i ch t i n g (1968) f o r s im p le

j e t s .

The s o l u t i o n r e q u i r e s

the

s p e c i f i -

c a t i o n of one exper imenta l ly de termined cons tan t and y ie lds

t h e f u n c t i o n

20



of t he si ml ari ty prof i l e and gross j et characteri sti cs as a f uncti on

of l ongi tudi nal di stance.

An al ternate approach, somewhat mor e conveni ent to use, i s the a

pr i or i speci f i cati on

of

si ml ari ty functi ons.

can then be i ntegrated i n t he t ransverse di recti on.

of

equati ons shows onl y dependence on the axi al coordi nate.

sol ut i on requi res an experi mental l y determned coef f i ci ent .

The governi ng equati ons

The resul t i ng set

Agai n, f ul l

The coef f i -

ci ent ei ther refers t o the rate of spreadi ng (method f i rst descri bed by

Al bertson et.al . (1950) ) or to the rate of entrai nment (f i rst descri bed

by Morton et.al .

(1956) ) , both coef f i ci ents bei ng rel ated to each other.

I n general , these coef f i ci ents are not constants, bei ng di f f erent f or

si ngl e j et and pl umes,

Usage of

the

i ntegral techni que for buoyant j et

predi ct i on i s common to model s

descri bed i n the f ol l ow ng secti ons.

2.1.2 Round Buctyant J e t s

The schemati cs of a round buoyant j et are shown i n Fi gure 2-1.

Af t er an i ni t i al zone

of

f l ow establ i shment

the j et moti on becomes sel f -

si ml ar.

be wel l f i tted t o

the observed di stri but i on of vel oci ty, densi ty def i -

ci ency and mass:

Experi mental data show that a Gaussi an

pr of i l e

can usual l y

2

- ( , * I

c(s,r)

=

cc(s)

e

2 1



Fig. 2-1: Schematics of a

Round Buoyant

Jet

22



where

s r r = axi al and transverse coordi nates

ii = axi al vel oci ty

m

U

= centerl i ne axi al vel oci ty

C

b = nomnal j et w dth

= ambi ent densi ty

p = densi ty i n the j et

Pa

= densi ty i n j et centerl i ne

P C

A = spreadi ng rati o between vel oci ty and mass

c

=

concentrat i on of some di scharged materi al

C

= centerl i ne concent rat i on

C

Th e

spreadi ng rat i o

X

accounts f or the fact that exper i mental observa-

t i ons show i n general stronger l at eral di f f usi on

( ~ > 1 )

or BcaIai quan-

t i t i es such as mass or heat than f or vel oci t i es. Wth the speci f i cat i on

of the vel oci ty prof i l e a vol ume f lux i s determned as

The entrai nment concept as f ormul ated by Morton et. al . assumes a

t ransverse entrai nment vel oci ty v

at the nomnal j et boundary b to be

rel ated

to

the centerl i ne vel oci ty as

e

v = a u

e C

(2- 51

where a i s a coef f i ci ent

of

proport i onal i ty (entrai nment coef f i ci ent).

Wth thi s assumpti on the change i n vol ume f l ux f ol l ows as

23



dQ

=

2% a;c (2-6)

d s

Using the p r o f i l e assumptions (2-1)? (2-2) and (2-31, in te gr at e d cons er-

v a t i o n

c an b e w r i t t e n .

e q u a t i o n s f o r

the ve r t i c a l

and hor iz on ta l momentum and f o r

m a s s

S o l u t io n t o the s ys te m o f o r d i n a r y d i f f e r e n t i a l e qu a t i o n s w i t h

i n i t i a l d i sc h ar g e c o n di ti o ns y i e l d s

th e sh ap e o f t h e j e t t r a j e c t o r y and

v a lu es o f

G

’ p c , cc and b alo ng the t r a j e c t o r y .

Th is approach forms the ba s i s of many buoyant j e t t h e o r i e s s i n c e

Morton

e t .

a l . (1956) . The the or ie s , however , d i f f e r on sp ec i f ic assump-

t i o n s r e g a r d i n g t h e e n t ra in m en t c o e f f i c i e n t a .

menta l da t a shows th a t a

i s

c l e a r l y a fu n c t i o n of t h e l o ca l bu o yan t

Examinat ion of experi-

c h a r a c t e r i s t i c s o f t h e

j e t

which can b e ex p re s s ed i n an av e rag e f a s h io n

by a l o c a l F rou de number FL

The va lue o f a i s dependent on the f o r m of t h e s i m i l a r i t y p r o f i l e .

For G a u s s i a n p r o f i l e s as s p e c i f i ed ab o ve , d a t a by Alb e r t s o n e t . a l . (1950)

s u g g e s t f o r

the

s h p l e j e t

(FL -f =)

Q = 0.057

and d a t a by Rouse et .a l . (1952) f o r t h e plume (FL s m a l l )

= 0.082

24



Buoyant j e t s t e nd to the c o n d i t i o n o f a s i m p l e plume f a r away from

the

so ur ce when

the

i n i t i a l momentum becomes

small

i n c o mp ar is on t o t h e

buoyancy ind uc ed momentum. Ac co rd in gl y, Morton (1959) and Fan and

Brooks (1966) assumed

oc

=

0.082 co ns ta nt throughout th e

j e t .

For

j e t s

w i t h s u b s t a n t i a l i n i t i a l momentum

a cer ta in

e r r o r

i s

i n h e r e n t .

Using the i n t e g r a t e d e ner gy c onse r va t ion e qua t ion Fox (1970)

showed that the dependence

of

t h e e n t r a inm e n t c oe f f i c i e n t on

FL as

a p

a = + -

1

FL2

(2-10)

f o r t h e

case

of

ve r t i c a l

d i sc ha r ge

charge

H i r s t

(1971) ex tended Fox's argument t o show

(0,

=

90").

For non-v er t ica l d is -

a2 A )

FL

a +-

s i n

8

1

2

=

(2-11)

where 8 i s t h e l o c a l a ng l e of t h e j e t t r a j e c t o r y . I n b o t h E q ua ti o ns

(2-10) an d (2-11)

a1 can

be de te rmined

from

t he s im ple

j e t ( F L - + w )

from

Equat ion

(2-8)

and

a 2

i s found

as

a un ique f unc t ion of t h e s p r e a d in g

r a t i o

A.

Buoyant j e t models based

on

t he

i n t e g r a l t e ch n iq u e b u t

w i t h

spec-

i f i c a t i o n of a c oe f f i c i e n t of th e sp r e a d ing r a t e

were

developed by

Abraham (1963).

r a t e

i s f ou nd t o b e v a r i a b l e i n b uo y an t j e t s , approaching

a

c ons ta n t

v a l u e f o r

the

l i m i t i n g

cases

of s im ple

j e t

and plume.

o f th e ve r t i c a l ly d i e cha r ge d buoya nt

j e t

Abraham assumed

a

c o n s t a n t

r a t e

of spread ing. For

the

c a se o f a h o r i z o n t a l l y d i sc h ar ge d

j e t

t h e sp r e a d -

i n g

r a t e

w a s p o s t u l at e d t o be r e l a t e d t o t h e l o c a l j e t a n g le , a nd n o t t o

S i m i l a r t o

the

e n t r a inm e n t c oe f f i c i e n t

the

spreading

I n t h e a n a l y s i s

25



t h e l o c a l b u oy an t c h a r a c t e r i s t i c s

as

is p h y s i c a l l y more r ea s o n ab l e .

So l u t i on g raphs f o r round huoyant j e t s d i s ch a rg ed a t Various

a n g l e s 9 have been pub l i shed by Fan and Broo ks (19 691 , Abraham (1963)

0

and o thers .

I n

a l l

th es e models some adju stme nt

i s

made for the i n i t i a l

zone of f low es tab l is hm en t . From the p r a c t i c a l s ta n dp o in t t h e r e

i s

l i t t l e

v a r i a t i o n b etw ee n t h e p r e d i c t i o n s o f v a r i o u s mo de ls , t y p i c a l

v a r i a t i o n s i n c e n t e r l i n e d i l u t i o n s f o r exam ple be i n g

l e ss

than 20%,

w e l l

below t h e s c a t t e r

of

usua l expe r imenta l da t a (se e Fan (1967)

).

The

choice of a p a r t i c u l a r m o d e l i s t h u s d e t e rm ined by t h e co r rec tn es s o f

t h e p r e s e n t a t i o n o f t h e p h y s i c a l p r o c e s s e s a n d by t h e a p p l i c a b i l i t y t o

vary ing des ig n p rob lems, such as d i s ch a r g e i n t o s t r a t i f i e d a mb ie nt s. I n

t h i s

respect

a n i n t e g r a l model w i t h e n t r a in m e n t c o e f f i c i e n t s

as

g i v e n

by Fox seems t o b e most s a t i s f y i n g .

2.1.3 S l o t Buoyant

Je ts

Fig ure 2-2 shows th e two-dimensional f low pa t t er n fo r

a

buoyan t

A f t e r t h e

e t i s s u i n g from

a

s l o t w i th w i dt h B and v e r t i c a l a n g l e 8

i n i t i a l z on e of f l ow es t a b li s h m e nt the f o ll o wi ng s i m i l a r i t y p r o f i l e s

f i t

w e l l

t o e x p e r i m e n t a l d a t a :

(2-12)

(2-13)

(2-14)

where

s , n

= a x i a l a nd t r a n s v e r s e c o o r d i n a t e s .

26



Fig. 2-2: Schematics

of a

Slot

Buoyant J e t

27



The volume flux i n t h e a x i a l d i r e c t i o n

i s

t h e n

The e n t r a in m e n t v e l o c i t y

a t

the j e t b o u n d a r i e s i s assumed as

e =

similar

t o t h e r o un d j e t .

Thus t h e c o n t i n u i t y e q u a t i o n

i s

(2-15)

(2-16)

(2-17)

A f t e r f o r m u l a t io n of t h e o t h e r c o n s e r v a t i o n eq u a ti o n s o l u t i o n s

proceed ana logous ly t o t he round buoyant j e t . The dependence of the

e n tr a in m en t c o e f f i c i e n t o n t h e b uo ya nt c h a r a c t e r i s t i c s o f t h e j e t i s

i n d i c a t e d b y e x p e r i m e n t a l d a t a .

For th e s imple

j e t ~1

i s

found

as

~1 = 0.069

(A lbe r t son e t . a l .

(1950)

)

and for the plume

(2-18)

a = 0.16 (2-19)

(Rouse

e t . a l .

(1952)

).

An an a ly s i s w i th

a

c o n s t a n t

a w a s f i r s t

c a r r i e d o u t by

L e e

and

Emmons (1962) and l a t e r by Fan and

Brooks

(1969).

A n an improvement,dependence on th e l o c a l Froude number w a s proposed

f o r t h e

ve r t i c a l

buoyant

j e t

by F a x (1970) i n a r e l a t i o n s h i p a n a l o g o u s t o

Equation (2-10) but

w i t h

d i f f e r e n t v a lu e s f o r a

and

a

1 2'

Abraham (1963) t r e a t e d t h e s l o t bu oy an t

j e t

( v e r t i c a l and hor i -

28



z on ta l d i sc ha r ge ) i n

a

f a s h i o n

similar

to the round buoyant j e t as des-

cribed above.

Less

e xpe r im e n ta l da ta

i s

a v a i l a b l e on s l o t b uo ya nt

j e t s .

Ceder-

w a l l

(1971) gives

a

comparison

02

e xpe r im e n ta l va lue s wi th

the

t h e o r i e s

by Abraham and Fan and Brooks. Rea son abl e agr eem ent i s found.

2.1.4 Latera l In te rf er en ce of Round Buoyant

Je t s

I n

a

submerged mul t ipor t d i f fuse r the round buoyant j e t s i s s u i n g

w i t h v e l o c i t y

U

L g r a du a l ly b eg i n t o

i n t e r a c t w i t h ea ch o t h e r

a

c e r t a i n d i s t a nc e away

f rom nozz les wi th d iamete r

D

and spaced

a t

a d i s t a n c e

0

from t he d i sc ha r ge . I n

a

t r a n s i t i o n zone t h e t y p i c a l s i m i l a r i t y pro-

f i l e s

of t h e ser ies of round j e t s are mod ified t o two-dimensional j e t

pr o f i l e s . From the n on th e d i sc ha r ge be ha ves l i k e

a

s l o t buoyant

j e t .

This process i s i nd ic a t e d i n F igur e 2-3. M a them a t ic a l a na lys i s a long

the above out l in ed procedures i s

im poss ib le as th e assumpt ion of se l f -

s i m i l a r i t y i s n o t v a l i d i n t h e t r a n s i t i o n zone.

assumptions

are

u s u a l l y m ade i n t h e a n a l y t i c a l t r e a t me n t .

Hence some approximate

The f lo w f i e l d o f

the

m ul t ipo r t d i f f us e r c a n be compared t o th a t

of

a n " e q ui v a le n t s l o t d i f f u s e r" .

u n i t d i f f u s e r l e n g t h an d t h e

same

momentum f l u x pe r u n i t le ng th

a

wid th

B of

the

e q u iv a le n t s l o t d i f f u s e r c an b e r e l a t e d t o the dimensions of

t h e mu l t i p or t d i f f u s e r by

By re qu ir in g th e same d i sc ha r ge pe r

2

B = h

4R

( 2 - 2 0 )

A

common c r i t e r i o n re gardi ng th e merging between round j e t s t o

two-dimensional j e t s i s t o assume t r a ns i t ion whe n

29



TOP V I E W , /

Transition

Fully

developed

L

zone

J p o

- dimensional

j e t

ound jets

SIDE

V I E W

2

Fig. 2-3: Jet Interference for

a

Submerged Multiport Diffuser

X



b

=

R/2 (2-21)

The nominal

j e t

w i d t h b (as def ined by Eq. 2-1) i s only a char-

a c t e r i s t i c

measure

of

the t a a n s v e r s e j e t dimension, namely the d i s t a n c e

where

the j e t

veloci ty becomes l / e of the c e n t e r l i n e v e l o ci t y . Thus t h e

assumption of Equat ion (2-21) cannot be su pport ed by ph ysi ca l arguments ,

b u t o n l y on i n t u i t i v e g r ou n ds ,

r e a so n in g t h a t w hen t h e v e l o c i t y p r o f i l e s

o v e r l ap t o s u ch

a

d e g r e e t h e l a t e r a l en t ra inment i s l a r g e l y i n h i bi t e d .

Using the assumptions of Equa tion 2-19 Cederw all

(1971) ca r r ie d ou t

a

comparison between th e aver age d i l u t io ns produced by

a

m u l t ip o r t d i f f u s e r

and by

i t s

e q ui va le n t s l o t d i f f u s e r a t t h e d i s t a n c e o f i n t e r f e r e n c e of

t h e m u l t ip o rt d i f f u s e r .

f o r t he volume f lu x and r a t e of sp read ing pub l i shed by Alber t son e t , a l .

on the s imple j e t and by Rouse et.al. on th e plume. Ced erw all found

H e u sed ex p e r im en ta l l y d e t e rm in ed r e l a t i o n s h ip s

f o r th e s imple non-buoyant j e t :

=

0.95

(2-22)i l u t i o n o f

the


d i l u t i o n o f t h e e qu iv al e nt s l o t d i f f u s e r

=

and f o r t he buoyant plume:

R = 0.78 (2-23)

I n v i e w of the u n ce r t a in ty i n v o ly ed these v a l u e s of R should only

b e i n t e r p r e t e d on

an

o r d e r of m ag ni tu de b a s i s , i n d i c a t i n g p rac t i ca l ly

similar

d i l u t i o n

character is t ics

f o r s l o t

j e t s

an d i n t e r f e r i n g ro u n d

j e t s .

Another comparison can be made

as

fo l lows .

Koh and Fan (1970)

proposed

a

t r a ns i t io n c r i t e r i o n as when th e entra inme nt

ra te

i n t o t he

31



round

j e t s

becomes equ al t o tha t

the

e q ui v al e nt s l o t

j e t .

They remarked

t ha t

this a ss um pt io n y i e l d e d e s s e n t i a l l y

the

same r e s u l t as

the assump-

t i o n of E q ua ti o n

(2-21) .

Davis

(1972) t o com pute m u l t i p o r t d i f f u s e r

cha rac te r i s t i c s

f o r

a

Y a r i e t y

of

co n d i t i o n s r eg a rd in g

j e t

a n g l e e

namics

of th e disc har ge gi ve n by t he Froude number

Their c r i t e r i o n w a s ap p l i ed b y Sh i r az i and

r e l a t i ve

s p ac in g a / D and the dy-

0’

U

(2-24)

as

a

f u n c t i o n

of

th e d im en s io n l e s s

ve r t i c a l

d i s t a n c e

z / D . The

d i l u t i o n s

/-,@a

s l o t j e t with discharge Froude number

(2-25)

a r e c a l c u l a t e d by t h e same

numerical

method as u s ed by Sh i r az i an d Davis

u s i n g t h e i r v a l ue s f o r t h e c o e f f i c i e n t s and A C e n t e r l i n e d i l u t i o n s

S

and Fs i n F ig u re 2-4 fo r t he case of h o r i z o n t a l d i s c h a r g e .

compared t o S h i ra z i and

Davis’

r e s u l t s by u si n g t h e d e f i n i t i o n of t h e

equivalent s l o t w id th

(Eq.

2-40) namely

are p l o t t e d

as

a

f u n c t i o n o f t h e dimens ion less

v e r t i c a l

d i s t a n c e z/B

T h i s can b e

C

and

(2-26)

32

(2-27)



2000,

I I

50

4 0

"""F

-

. \

\ \ \ -

1 \ 1 , \ I I I I

X

17

7

= sc

-

w -

\

-

\

\

\

\

-

\

\

I \ I I ),

I

\

\ 5

\ \

\ \

\

-

-

\

\

\

0, -

\

\ \ \ \

Values computed

by

Shirazi and

Dav is

(1972)

fo r t he mu l t i po r t d i f f user

- -

-

omputed values

(*=.16, A = 1 . 0 ) + = 30

f o r

the s lot buoyant j e t

0

Fig.

2-4:

Centerline Dilutions

S

for

a

Slot Buoyant

J e t

C

With

Horizontal Discharge

33



Valu es co n v er t ed i n this f a s h i o n f o r p o in t s a f t e r

the

t r a n s i t i o n

zon e an d fo r t h e r an g e of L/D

=

10 ,

20

and 30 and o f Fn

=

1 0 , 30 and 100

a re

shown i n F ig u re 2-4.

I n g e n e r a l , S h i r a z i and Davis

resul ts

show no

v a r i a b i l i t y w it h 2/D and have the same.funct iona1 dependence on z/B and

F as the r e s u l t s f o r t he s l o t j e t . However, there i s a sys temat ic under-

e s t i m a t i o n

of

d i l u t i o n , t h i s o f co u r s e b e in g a s p ec i f i c co ns equ ence o f

S

t h e a do pt ed c r i t e r i o n f o r t r a n s i t i o n .

Thus, u n t i l exper imenta l ev idence t o th e con t r ary becomes

avai l -

a b l e

--

and t h i s q u es t i o n can o n ly b e s e t t l ed ex p e r im en ta l l y

-- i t

s u f f i c e s f o r a l l p r a c t i c a l p u r p o s e s

t o

assume t h a t t h e f l ow f i e l d c h ar -

a c t e r i s t i c s of a m u l t i p o r t d i f f u s e r

are

eq u a l l y p re s en t ed b y

i t s

equiv-

a l e n t s l o t d i f f us e r .

A f r eq u en t l y u sed d i f f u s e r g eo me try i s discharge

t h r o u g h p o r t s

o r n o zz l e s i s s u in g i n to a l t e rn a t i n g d i r ec t i o n s from th e common m an i fo ld

p i p e (see Figure 2-5).

With t h i s a r rangement a complicated flow p a t t e r n

TOP

V I E W

Fig. 2-5;

M u l t i p o r t D i f fu s e r

w i , t h

A l t e r n a ti n g P o r t s

i n Deep

Water

3 4



evolves. T h e j e t s a t b o t h s i d e s i n t e r f e r e l a t e r a l l y and r i s e upward

under the in f luence o f buoyancy.

can p en e t r a t e in to

the

region b e t w e e n t h e r i s ing two-dbensional j e t s

as

the

are a between th e

j e t s

b e fo r e l a t e r a l i n t er f e re n c e

is

r e s t r i c t e d .

As t h e tu rb u len t ent ra in men t a t the i n n e r j e t b ou nd ar ie s a c t s l i k e a

su ct io n mechanism, a low pres sur e ar ea i s created between the j e t s . Con-

seq u en t ly , th e

j e t s

ar e gradually bent over un ti l merging over t h e d i f f u s e r

O n l y a limited amount

o f

amhient water

l i ne . Th is case was ex tensive ly s tud ied in a series of

Li se th (1971). Averaged values of cen te r l in e di l ut i on s

d i f f e r e n t l e v e ls z/E w e r e presented i n g raph ica l form.

a l s o y i e l d ed

an

approximate expression for the location

above the d i f fuser

zm/E Fn

exper men

t

s by

measured a t

L i s e t h ' s s t u d y

z of merging

m

(2-28)

As th e discharge by means of al te rn at in g horizon tal buoyant j e t s does not

intro duc e any i n i t i a l momentum i n

the

v e r t i c a l d i r ec t io n , t h e f lo w abov e

the l i n e of merging can b e compared to the flow in the buoyant plume.

The r e l a t io n sh ip fo r th e cen t e r l in e d en s i ty d e fi c ien cy + c i n the buoyant

plume i s given by Rouse et.al . (1952) as

(2-29)

i n

which

w

i s the fl ux of weight d efic ienc y emanating from

the

l i n e s ou rc e.

For the discharge from the s l o t w can b e expressed as

w =

A p o

g UoB

35

(2-30)



100

50

SC

10

F

-

I

1

I I I 1

I

I I

/ '

/Buoyant plume ,

' +

f r om Rouse et a l .

(1952

x

Z/k

=

8 B e s t - f i t d a t a f r o m

Mu1 iport

0 =20

1

L i s e t h

(1971)

diffuser

I

+ =80

Fig. 2-6: Comparison Between the Centerline Dilutions Above the

Point of Merging for a Buoyant Plume

and

a Multiport

Diffuser with Alternating Nozzles

36



where Ap =

-

i s the i n i t i a l d e n s i t y d e fi ci e nc y . U si ng

t he

d e f i n i -

o

P a Po

t i o n of the s l o t Froude number J? (Eq, (2-25)) Eq. (2-29) ca n be tr an s-

formed t o g iv e the c e n t e r l i n e d i l u t i o n Sc

s

-2 /3

.39

z/B

Fs

p

0

*Q

C

s

= - =

C

(2-31)

I n F i g u r e 2-6 t h i s r e l a t i o n s h i p i s compared t o a se r ies o f d a t a p o i n ts

f o r z/Q

=

8, 20 and 80 t ak en f ro m L i s e t h ' s b e s t - f i t cu rv es . The d a t a

p o i n t s

w e r e

c o nv e rt e d us i n g t h e r e l a t i o n s h i p s b etw een m u l t i p o r t d i f f u s e r s

and e q u i v a l e n t s l o t d i f f u s e r s .

c luded .

Data po in ts f o r which zm/l l< Fn w e r e ex-

There i s good a gr ee me nt , a g a i n i n d i c a t i n g t h a t l o c a l d e t a i l s

of

th e d i sc har ge geomet ry , such

as

noz zle spac ing , have indeed a n e g l i g i b l e

i n f l u e n c e o n t h e o v e r a l l C h a r a c t e r i s t i c s o f m u l ti p or t d i f f u s e r s .

2.1.5 Ef fe c t o f the Free Surf ace

The d e n s i t y d i s c o n t i n u i t y

a t

t he air-water i n t e r f a c e

ac ts as

an

e f fe c t i v e b a r r i e r t o t h e u pward mo t ion o f t h e bu oy an t

j e t .

on

t h e k i n e t i c e ne rg y o f t h e

j e t

o n l y

a

s m a l l s u r f a c e

r i se

w i l l occur .

As

a

consequence

t he

j e t

w i l l

s p re a d l a t e r a l l y a lo ng

the

s u rf a ce i n

a

Depending

l a y e r o f a c e r t a i n t h i c k n e s s .

As a l l

the buoyant j e t t h e o r i e s d i s cu ss e d i n the preceding presume

discharge in to an unconf ined env i ronment ,

t h e p re s ence o f

the f ree

s u r -

f ace

i s

u s u a l l y a cc ou nt ed f o r by a s su mi ng e f f e c t i v e e n t r a i nm e n t i n t o t h e

j e t s

up t o

the

lower edge of

the

s u r f a c e l a y e r . I n

the

ab s en ce o f a n

a n a l y t i c a l m odel f o r

the

e s t i m a t i o n o f the s u r f a c e l a y e r t h i c k n e s s , e x pe r -

im en ta l va lu es r ep or te d by Abraham (1963) are of ten used .

buoyant j e t

( a f t e r

l a t e r a l

in te rac t ion) Abraham g ives the l a y e r thickness

F or t h e s l o t

37



t o b e e q u a l t o a b ou t

114

of th e l e ng th o f

the

j e t t r a j e c t o r y .

J' 2.1,6

Ef f e c t o f Am bie n t De ns i ty S t r a t i f i c a t ion

S t a b le de ns it y s t r a t i i i c a t i o n --

tha t

i s d e c re a s in g d e n s i t y w i t h

e l e v a ti o n d ue t o y a r i a t i o n s i n t e mp e ra t ur e an d s a l i n i t y -- i s a common

o c c ur r en c e , i n p a r t i c u l a r f o r deep

water

o u t f a l l s i n o ce an s and l a k e s .

Under such con di t ion s th e

j e t

c a n r e a c h a n e qu i l ib r ium l eve l and spreads

l a t e r a l l y i n t h e form of a n i n t e r n a l c u r r e n t when i t s de ns it y becomes

equa l t o th e ambient dens i ty . Pre di c t i on of t h i s phenomenon i s im por ta n t .

J e t t h e o r i e s f o r d i sc h ar g e i n t o l i n e a r l y s t r a t i f i e d a mbien ts

are

a l l

based on th e ent ra inm ent concept (Morton

e t . a l . (1956),

Fan (1967),

Fox

(1970) ).

s t a b le e nv i r onm e n t s ( D i t m a r s (1969), Sh i ra zi and Davis (1972) ).

T he se methods ha ve a l s o b e e n ad a pt e d f o r a r b i t r a r i l y s t r a t i f i e d

2.1.7 Ef fe ct of Crossf low

A

s in gl e round buoyant

j e t

d i s c h a r g e d i n t o

a

crossf low u g et s de-

a

f l e c t e d i n t o t h e d i r e c t i o n o f t h e c r os s f lo w . The d e f l e c t i o n i s a f f e c t e d

by two f o r c e mechanisms

a c t i n g on t h e

j e t ,

a p r e s s u r e d r a g f o r c e

3

L

2b

aua

2

D

= CD

(2-32)

where

C

l o s s of ambient momentum due to entra inme nt of ambient f l u i d i n t o t he j e t

i s a

dr a g c oe f f i c i e n t a nd

a

f o r c e Fe r e s u l t in g from

t h e r a t e of

D

(2-33)

C h a r a c t e r i s t i c f e a t u r e

of

j e t s

i n c r o s s f lo w

i s

a

s i g n if i c an t d i s t o r t i o n

of th e us ua l ly symmetric j e t p r o f i l e s t o h or se -s ho e l i k e sh a pe s

w i t h

a

s t ron g wake regin n.

The entra inment concept w a s modif ied by Fan

(1967)

38



to

(2-34)

CI

wher e the t erm

between ambi ent vel oci t y and j et vel oci t y t o account f or t he ef f ect of

-

ucI denot es t he magni t ude of t he vect or di f f erence

crossf l ow vel oci t y on t he ent rai nment mechani sm ( shear i ng acti on) . Val ues

for

si der abl y l ar ger t han i n t he st agnant case, i ndi cat i ng the i ncreased

a when st i l l r et ai ni ng t he assumpt i on of Gaussi an pr of i l es ar e con-

di l ut i on ef f i ci ency i n t he presence of a cr ossf l ow.

No anal yt i cal model s have yet been advanced f or t he def l ect i on of

a ser i es

of

i nt er act i ng r ound j et s as i n t he mul t i por t di f f user or f or

a sl ot di f f user.

cases w t h eddyi ng and re-ent r ai nment i n t he wake zone behi nd t he j et as

has been observed experi ment al l y by Ceder wal l (1971). The assumpt i on

of

sel f - si m l ar i t y i s not val i d any l onger.

2.2 One- Di mensi onal Aver age Model s f or Hor i zont al Di f f user Di schar ge

The def l ect i ng mechani sm i s hi ghl y compl i cat ed i n these

i nt o Shal l ow Water

A

sever e exampl e of t he i nadequacy of buoyant j et model s devel oped

on t he assumpt i on of an unbounded r ecei vi ng wat er i s gi ven by t he hori -

zont al (or near hor i zont al ) di f f user di schar ge i nt o shal l ow wat er . I n

thi s case st r ong sur f ace and bot t om i nt er act i on causes a ver t i cal l y f ul l y

m xed concent r at i on f i el d downst r eam of t he di schar ge as i l l ust r at ed i n

Fi gur e 1- l d.

Maki ng use of t hi s f ul l y ani xed condi t i on t he pr obl em can

be anal yzed i n a gr oss f ashi on.

2. 2. 1 The Two- Di mensi onal Channel Case

W t h the r at i onal e to examne t he appr oxi matel y t wo- di mensi onal

39



f low f i e l d which

p e r s i s t s

i n

the

c e n t e r p o r t i o n

of a

d i f f us e r l i n e

Harleman e t , a l . (1971) s tudied the i o l l o win g co n f ig u ra t i o n : An a r r a y o f

d i f f u s e r n o z z l e s i s p u t b e tw ee n v e r t i c a l

w a l l s

of f i n i t e l e ng t h 2L. The

channel thus formed

i s

p l ac e d i n

a

l a r g e b a s i n

as

shown i n F igure

2-7.

The j e t d i s c h a r g e se t s up a cu r ren t o f m ag ni tud e

u

th rough the

m

entrainm ent pro ces s . Ambient water

i s

a c c e l e r a t e d from z e r o v e l o c i t y

o u t si d e i n t h e b a s i n ( f a r f i e l d ) t o v e l o c i t y urn i n t h e c h an ne l.

t h e c h a nn e l t h e c u r r e n t e x p e r i e nc e s a head loss ex p res s ed as usua l as

2

Ck-

co e f f i c i e n t s d e s c r i b in g t h e ch an n e l g eo me try.

a t the downstream end th e ve lo ci ty head um2/2g is d i s s i p a t e d . Hence t h e

t o t a l head l o s s i s

I n s i d e

U

m

a g a i n s t f r i c t i o n a l r e s i s t a n c e , w here

Ck i s

t h e sum o f h ead l o s s

Upon leav ing the channel

2g

2

(2-35)

I n s t e a dy s t a t e

the

p res s u re fo r ce cau s ed b y t h i s head loss i s b a l an ced

by th e momentum f l u x of t h e j e t d i s ch a rg e .

tum e q u a t i o n ( p e r u n i t c h an n el w i dt h ) c a n b e w r i t t e n betw ee n s e c t i o n s 1

and 2 of F igure 2-7 as

Thus a one-dimensional momen-

( 2 - 36 )

The d i l u t i o n S

i n the

fu l l y m ix ed f l o w away i s s imply g iven by t h e volume

f l ux r a t i o

U,Hk

DZa

=

0 4

40

(2-37)



/ hannel

W a l l s

-

2 L

P L A N

VIEW

J e t E n

rai

n r n e n t

Z o n e S p e c if i c H e a d L i n e

C h a n n e l E n d

h a n n e l E n d

ELEVATION

V I E W

Fig. 2-7:

Schematics

of

Channel Model for

a

Multiport Diffuser with

Horizontal Discharge in Shallow Water

41



I f the

ap p ro x im a t io n s po/p, ::

1,

S/S-1

d e f i n i t i o n f o r t h e e q ui v al e nt s l o t d i i f u s e r , Eq. (2-20) are i n t ro d u ced ,

the d i l u t i o n

can

b e ex p re ss ed

as

1 ( l a r g e d i l u t i o n s ) a n d

t he

(2-38)

The s t r i k i n g f e a t u r e s of t h i s e q u a ti o n are t h e independence on t h e Froude

number o f t h e d i s c h a rg e and on t h e l o c a l d i f f u s e r g eo me try,

i . e .

n o z z l e

spac ing .

t ion .

The

v a l i d i t y

i s

o f c ou rs e r e s t r i c t e d t o the f u l l y mixed cond i -

S a t i s f a c t o r y a gr ee me nt w i t h e x p e r i m e n ta l r e s u l t s

w a s

found.

2.2.2 The Three-D imensiona l Case

The three-dimensional aspects o f h o r i z o n t a l d i f f u s e r di s c h a r g e

i n s h a l l ow water wi th o r w i th o u t t h e p re s e n ce o f an am bien t cu r re n t u a

were s t u d ie d by Adams (1972).

H e

o b se r ve d i n t h e a b se n c e of c o n f i n i n g

walls (as i n t h e p r ev io u s case) a co n t ra c t io n o f t he f low downs tream of

t h e d i f f u s e r

l i n e

as ind ica ted i n F igu re 2-8. Using a sca l ing argument

Adams n eg l ec t ed l o ca l f r i c t i o n a l d i s s i p a t i o n an d made a one-dimensional

i n v i s c i d a n a l y s i s s i m i l a r t o t h a t u se d f o r i d e a l p r o p el l e r t h eo r y ( P r an d t l

(1952)

)

t o

a r r i v e

a t

a n e s t i m a t io n o f t h e i n d u ce d average d i l u t i o n S

downstream of the d i f f u s e r .

e ra t ed f ro m

a

s e c t i o n

1

f a r b eh in d the d i f f u s e r t o u2

f u s e r l i n e .

R efe r r i n g

t o

F i g u r e

2-8

the flow i s a c c e l -

u3 a t t h e d i f -

A f t e r p a s s i n g o v e r

the

d i f f u s e r

l i n e

and being mixed

w i t h t he

j e t d i s c h a r g e

the

f lo w f i e l d c o nt in u es t o c o n t r a c t d ue t o i t s i n e r t i a

u n t i l s e c t i o n 4 w he re t h e s p e c i f i c h ea d r e t u r n s t o

i t s

o r i g i n a l v a l u e H.

42



TOP

V I E W

I

IDE V I E W

f

ill

1

c

Dividing

Stream

Line

*

Fig. 2-8:

Schematics faor One-Dimensional Analysis of a Multiport

Diffuser with Horizontal Discharge in Shallow Water

43



T h e theory does no t

t r e a t

the reg ion beyond s ec t i on

4

i n which t he f low

g rad u a l l y

w i l l

r e t u r n

t o i t s

p r i g i n a l v e l o c i ty ua t h ro u gh y is c o u s d i s s i -

pat ion of the excess

v e l o c i t y

head (u

2 2

c.

u

, ) / 2 g .

4

Apply ing Be rn ou l l i ' s theorem between sec t i on s

1

and 2 and

sec-

t i o n s 3 and

4

y i e l d s the head change ac ros s

the

d i f f u s e r

1 2 2

AJI = 2 g ( u 4 - u

)

a

(2-39)

The p ressure fo rce thus p roduced i s balanced

by

t he momentum f l u x of t h e

d i f f u s e r w hic h c o n s i s t s o f n n o z z l e s w i t h s p a c i n g R,

s o

tha t , us ing

Pa=Po=Pm

Y

(2-40)

as i n the two-d imensional channel case. Another momentum equation can

b e w r i t t e n for t h e c o n t ro l v olume b etween s e c t i o n s

1

and 4

For l a r g e d i l u t i o n s

a 1 u4R2H

so that

U,QIH

2

D T

u0

-

4

s : :

(2-41)

( 2 -4 2 )

(2-43)

44



di l uti on i s

( 2 - 4 4 )

For st agnant recei vi ng water, ua

=

0, E q . ( 2 - 4 4 ) reduces t o

1 1 2

( 2 - 4 5 )

s

=

T ( 2 E )

H

For the case when the crossf l ow i s very l arge E q . ( 2 - 4 4 ) becomes

U

a H

s -

5 :

( 2 - 4 6 )

i ndi cat i ng proport i onal mxi ng w th the oncomng f l ow

The contract i on

of the f l ow between the di f f user and sect i on 4 i s f ound

t o

be

€C

whi ch reduces t o

1

c 2

c

= -

( 2 - 4 7 )

( 2 - 4 8 )

i n the case of zero ambi ent f l ow

I t i s i l l umnat i ng to compare E q . ( 2 - 4 2 ) w th Eq. ( 2 - 3 8 ) for

the two-di mensi onal channel case maki ng t he si ml ar assumpti on

o f

neg-

l ecti ng f ri cti on i nsi de the channel (Ck = 01, namel y

1 1 2

H

s = ( 2 : ) (2-4 9

Thus the predi cted di l ut i on capaci ty of the three-di mensi onal case i s

one hal f of the two- di mensi onal channel . The di f f erence i s att r i butabl e

45



t o t he cont r act i on whi ch occurs i n t he f or mer case whi ch causes mor e

vel oci t y head

to

be di ssi pat ed i n t he regi on beyond.

Despi t e

the

appr oxi mat i ons i nvol ved -- no bot t om f r i c t i on, no

di f f usi on at t he boundar y of the current -- Adams f ound sat i sf actory

agr eement

w th

exper i ment al l y det er m ned aver age di l ut i ons

i n

a sect i on

downst r eam f r om t he di schar ge ( see Chapt er

7 ) .

2 . 3

Appr ai sal of Previ ous Knowl edge About t he Char act er i st i cs

of a Mul t i port Di f f user

The obj ect i ve of pr edi ct i ve model s f or mul t i port di f f user s i s t he

det er m nat i on of vel oci t i es and concent r at i on di st r i but i ons i nduced by

t he di f f user di schar ge.

shown two const r ast i ng l i m t i ng cases of di f f user di schar ge:

i n pr act i cal l y unconf i ned deep wat er i n f or m of buoyant j et s and di schar ge

i nt o f ai r l y shal l ow wat er

w th

extr eme boundar y i nt er act i on r esul t i ng i n

a uni f or m y m xed cur r ent .

behavi or i mmedi atel y suggest s quest i ons r egardi ng t he di f f user per f ormance

i n

t he i nt er medi at e r ange ( conf i ned r ecei vi ng wat er ) and t he appl i cabi l i t y

of s uch si mpl e model s as di scussed i n t he r evi ew

--

si mpl e i n t he sense

t hat t hey consi der onl y one dom nat i ng physi cal pr ocess.

The r evi ew of exi st i ng pr edi ct i on t echni ques has

di schar ge

Thi s st r i ki ng di f f er ence

i n

t he r esul t ant

Det ai l ed obser vat i ons r egar di ng the degr ee of est abl i shed phys-

i cal under st andi ng can be summar i zed:

. Ar eas of Adequat e Under st andi nq

I n

t hese pr obl em ar eas underst andi ng

has

been achi eyed to

such a poi nt t hat f ai r l y rel i abl e pr edi ct i ons c an be made.

4 6



1)

Buoyant

J e t s i n Deep

Water

The

d i f f e r e n t t h e o r i e s f o r z w n d and s l o t

buoyant

j e t s

have

b ee n l a r g e l y v e r i f i e d i n l ab o ra t or y e x p e r a e n t s .

t i on s between models do not vary app rec i abl y a l though

var ious assumpt ions regarding the

j e t charac te r i s t i c s

have

been made.

on the phys ica l "cor rec tness" of t he se assumpt ions and

on

t h e a p p l i c a b i l i t y t o d i f f e r e n t s i t u a t i o n s . I n t h i s

r e spe c t an i n t e g r a l a n a l y s i s w i t h v a r i a b l e e n tr ai n me n t

c o e f f i c i e n t s d ep en di ng on t h e l o c a l b u oy an t c h a r a c t e r i s t i c s

seems

t o b e p r e f e r a b l e .

2)

In te rf er en ce o f Round Buoyant

Je t s

The

l a t e r a l

in te r f e r en ce of t he round buoyant

j e t s

i s s u i n g

from

a

m u l t i p o r t d i f f u s e r t o form a two-dimensional

j e t

has not ye t been st ud ie d experimental ly. However, reason-

ab le assumptions regardin g a t r a n s i t i o n c r i t e r i o n i n t he

a n a l y t i c a l

treatment can

b e made. Comparisons show t h a t

the f low f i e l d p roduc ed by

a


i s similar

t o tha t one p roduc e d by a n " e qu iva le n t s l o t d i f f use r " .

Hence f o r mathem atical convenience

t h i s

concept should be

r e t a ine d . The

same

argument per ta in s t o th e merging

of

j e t s

above

the

d i f f u s e r l i ne i n t h e

case

o f a l t e r n a t i n g l y

d i sc ha r g ing noz z le s

as

s tudi ed by Lise t h .

3) Hor iz on ta l D i f f us e r s i n S hal low

Water

H o ri zo n ta l d i f f u s e r s d is c ha rg i ng i n t o f a i r l y s h a ll ow water

p ro du ce f u l l

v e r t i c a l

mixing due t o s tr on g boundary in ter-

Predic-

Choice of

a

pa r t i c u la r model

should be based

47



a c t i o n s . P r e d i c t i o n s

o f

average d i lu t ions downst ream

o f

t h e d isc har ge l i n e can be made us ing Adams' experimen-

t a l l y v a l i d a te d r e l a t i o n s hi p .

B. Areas o f

Lnsuff ic ient Knowledge

1 ) E f f e c t

o f a Vertical ly Confined Flow Region

T h i s i s the

ge ne r a l

case

of

a

d i f f u s e r d i s ch a r ge . S o l u t i o n

of

t h i s p roblem r e qu i r e s th e a s se s sm e n t o f :

a )

The e f f e c t of t h e f r e e s u r f a c e . P r e d i c t i o n o f t h e

th ickness of th e sur fa ce impingement la ye r as

a

func t ion of

j e t

pa r a m e te r s . Th i s de f in e s th e upper

l e v e l up t o wh ich e f f e c t i v e e n t ra i nm e nt t a k e s p l a c e

i n t o t h e j e t .

b ) The s t a b i l i t y of t h e s u r f a c e l a y e r .

ing f rom the l i n e of impingement can be uns t abl e .

Hence

water

c a n b e r e -e n t ra i n ed i n t o t h e

j e t

r e g ion .

c)

The

f low away f rom the d i f f u se r l i n e i n th e f or m

of

a

d e n s i t y c u r r e n t .

The f low spread-

d) The e f f e c t of b o t t o m i n t e r a c t i o n . Je ts d i s c h a r g e d

h o r i z o n t al l y a nd c l o s e t o the bottom can become

a t t a c h e d t o the bottom.

Solut ion of t h i s genera l problem

w i l l

encompass the

l i m i t i n g

cases

of d i s c h a r g e i n f a i r l y deep an d f a i r l y

sha l low

water.

Hence, c r i t e r i a o f a p p l i c a b i l i t y

of

the

"simple" models rep or t ed above can be pre sent ed .

2 ) Three-Dimensional Behavior

As

exempl i f ied by th e

case

o f h o r i z o n t a l d i f f u s e r d i s c h a r g e

4 8



i nto shal l ow recei vi ng water whi ch produces a f l ow away

w t h si gni f i cant cont racti on . thus reduci ng di l ut i on,

three- di mensi onal aspects

o f

t he

di f f user f l ow f i el d

are extremel y i mportant .

3)

Ef f ect of Crossf l ow

The ef f ect of crossf l ow has onl y been i nvest i gated f or

the si ngl e round j et .

i nteracti ng di f f user j ets or sl ot j ets i s avai l abl e.

No quant i tat i ve i nf ormat i on on

I n general the three- di mensi onal di f f user i nduced f l ow

f i el d i s superposed on but al so modi f i ed by, the

ambi ent f l ow f i el d. The overal l l ayout of the di f f user

axi s w th respect to the ambi ent current di rect i on i s

an

i mportant f actor , as shown by exper i mental i nvest i ga-

t i ons reported by Harl eman, et.al . (1971).

A l l

these probl emareas are addressed i n the fol l ow ng

chapters of thi s study.

49



111. THEORETICAL

FRAMEWORK:

TWO-DIMENSIONAL CHANNEL MODEL

3.1 Basic Approach

T h e

review of the preceding chapter showed

the

l i m i t a t i o n s o f

e x i s t i n g t h e o r i e s f o r the p r e d i c t i o n o f m u l t ip o r t d i f f u s e r b e h a v io r .

A n a l y t i c a l m od els

are

a v a i l a b l e o nl y f o r the ex treme ca se s o f (1 ) buoyant

j e t s i n d ee p

water,

n e g l e c t i n g the d yn am ic e f f e c t s c a u se d b y t h e f r e e

su r f a c e , a n d

( 2 )

d i s c h a r g e i n t o s h a l l o w w a t e r w i t h s t r o n g b ou nd ar y i n t e r -

a c t io n r e s u l t i n g i n a v e r t i c a l l y mixed c u r r e n t .

have been developed for t h e i n t e r m e d i a t e r a n ge i n w hich b ou nd ar y e f f e c t s

a r e

i m p o rt a n t and no c r i t e r i a of a p p l i c a b i l i t y f o r t h e e x i s t i n g mo de ls

have been de r ived .

No math em ati cal models

The p r es e n t s t u dy a t t e m p ts t o f u l f i l l t h i s n ee d.

The complex i ty of th e gen e ra l th ree -d imens iona l p rob lem of mul t i -

p o r t d i f f u s e r d is c h ar g e

i s

s u c h t h a t

no

s i n g l e a n a l y t i c a l d e s c r i p t i o n

( a s i n g l e s e t of govern ing equa t ions w i th th e appro pr i a t e boundary con-

d i t i o n s ) o f t h e f l u i d f lo w c a n b e s o l v e d by a v a i l a b l e me th od s.

t h e f o l l o w i n g a p p r o a c h i s u n d e r t a k e n i n t h e d ev el op me nt

of

a p r e d i c t i v e

model :

Hence,

1) T h e t h e o r e t i c a l treatment

i s

l i m i te d t o t h e d i f f u s e r -

i n du c ed s t e a d y - s t a t e f l o w f i e l d w i t ho u t t h e p r e s e n c e

of an ambien t c ro ss f low.

2)

A two-dimensional "cha nne l model" sim ul at es

t he

pre-

dominantly two-dimensional

f l o w

f i e l d w hich i s E Q s t u l a t e d

t o

exist

i n t h e c e n t e r p o r t i o n of a t h ree -d imens iona l

d i f f u s e r . T h i s

is

i l l u s t r a t e d

in

F i g u r e

3-1

f o r the

case

of a s ta b le f low away zone.

The two-dimensional

50



PLAN

V I E W

SURFACE

FLOW PAT7

t

\

I -

-

-

Y

E R N

ELEVATION ALONG

X - y PLANE

t z

Fig. 3-1: Three-Dimensional Flow Field for a Submerged Diffuser

Two-Dimensional Behavior in Center Portion

(Stable Flow Away Zone)

51



channel model assumes a di f f user secti on bounded l ater-

al l y by wal l s of f i ni te l engt h, 2L. Thi s conceptual i za-

t i on al l ows t he anal ysi s

of

the yer t i cal and l o, n, gi tudi nal

var i at i on of the di f f user- i nduced f l ow f i el d.

3) Through a quanti tat i ve anal ysi s

regardi ng f ar- f i el d ef f ects

(f r i cti onal resi stance of the f l ow away zone) the l engt h,

ZL, of the two-di mensi onal channel model i s l i nked to

the l ength of the three- di mensi onal di f f user, 2LD

thi s manner theoret i cal predi ct i ons of the two- di mensi onal

I n

channel model become appl i cabl e

t o

the general three- di men-

si onal case.

4 ) The i nteracti on of the di f f user i nduced f l ow w th a cross

f l ow i n the recei vi ng water body i s st udi ed exper i mental l y.

I n thi s chapter the theoret i cal f ramework f or the t reatment of

the f l ow di st r i but i on i n the two- di mensi onal channel model i s devel oped.

The di f f user di scharge exhi bi ts several di st i nct f l ow regi ons. Anal yt i -

cal t reatment of each of these regi mes becomes possi bl e by i ntroduci ng

approxi mati ons to the governi ng equati ons of f l ui d moti on.

Matchi ng of the sol ut i ons at t he boundar i es o f the var i ous

regi ons

resul ts i n an overal l predi cti on

of

the two- di mensi onal channel f l ow

fi el d. I n Chapter

4

the quant i tati ve compari son between the

f l ow

f i el ds

i n the two-di mensi onal channel model and the general t hree- di mensi onal

case i s made.

I n Chapter

6 ,

the theoret i cal model predi cti ons are compared w t h

exper i mental resul ts.

Experi ments were perf ormed both f or t he two-di men-

52



s i on a l channe l mode l and th e three-dimensiona l

case.

I n C h ap t er 7 , t he m o d i f i c a t i o n o f t he d i f fuse r- induc e d f low f i e l d

through

the

ef fe c t s of ambient c ro ss f low i s s tud ie d e xpe r im e n ta l ly .

For

the

pur pose

o f

e s t a b l i s h i n g s c a l i n g r e l a t i o n s h i p s the

ana ly t i c a l

t r e a tm e n t i s direc ted toward the d i sc ha r ge o f he a ted

water. T h i s

i s moti -

vated by

the

f a c t t h a t t he rm al d i f f u s e r s l o c a t e d i n s ha ll ow w a t e r , w i t h

low buoya nc y of th e d i sc ha r ge, t y p i c a l l y

are

s t r ong ly in f lue nc e d by

t h e

f i n i t e d ep th

of

t h e r e c e iv i n g w a t e r body.

3.2

Problem De fi ni ti on : Two-Dimensional Chan nel Model

R e f e r r i n g t o F i g ur e

3-2

th e fo l low ing problem

i s

considered: The

s te a dy- s t a t e d i sc ha r ge of he a te d w a t e r with tempera ture To a n d v e l o c i t y

u

t h r ough

a

s l o t w i t h w i dt h

B

and

v e r t i c a l

o r i e n t a t i o n

e0

i n t o

a

c ha nne l

0

of uni form depth

H,

un i t w id th a nd l e ng th

2L.

opening above the bottom

i s small

compared t o t h e t o t a l d e p t h ,

The he igh t hs o f th e s l o t

The cha nn el opens

a t

b o t h e nd s i n t o a l a r g e r e s er v o i r .

The

r a t i o n a l e f o r s t ud y in g

t h i s

model

i s

provided by:

a)

In

the

mathemat ica l t rea tment a m u l t i p o r t d i f f u s e r c a n

b e represented by an e q u i v al e n t s l o t d i f f u s e r a s d i s -

c us se d i n

the

previous chapter (Eq.

(2-20)).

b)

T h e channel

model approximates

the

predominantly two-

d im e ns ional f low f i e l d wh ich

i s

p o s t u l a t e d t o

exist

i n the c e n t e r p o r t i o n

of a

t h r e e - d im e ns iona l d i f f use r

as shown i n Fig ure 3-1.

It w i l l

be shown

l a t e r

t h a t

53



Channel Walls

TOP

V I E W

*

Ta

ambient

t em pera t u re

x

4

Ibt

l4 2L

I

heat loss t o

t h e a t m osphere

E L E V A T I O N V I E W

To discharge t emP

erst

Urc

Fig. 3-2:

Problem

Definition: Two-Dimensional Channel

Model

54



under cer t ai n condi t i ons, namel y i nst abi l i t y o f t he

f l ow away, t he di f f uaer di schar ge does not exhi hi t

t hi s pr edom nant l y t wo- di mensi onal regi on. Howeyer ,

t hr ough var i at i on

o f

the

br i zont al nozzl e or i enta-

t i on i t i s possi bl e

t o

cont r ol t he t hr ee- di mensi onal

f l ow

s o

as

t o

appr oxi mate t he t wo- di mensi onal behavi or.

I n t he i nt er est of achi evi ng hi gh di l ut i ons t hi s con-

t r ol i s desi r abl e. These t hr ee- di mensi onal aspect s of

di f f user di schar ge ar e tr eat ed i n mor e det ai l i n

Chapt ers 4 and 6 .

3.3 Sol ut i on Method

For t he pr obl em def i ned, t he gover ni ng equat i ons of f l ui d mot i on

and heat conservat i on ar e wr i t t en under t he f ol l ow ng assumpt i ons:

1) The f l ow f i el d i s t wo- di mensi onal i n t he ver t i cal

No

l at eral var i at i ons w t h

y

occur .

r xz- pl ane.

2)

The f l ow i s tur bul ent , but st eady i n t he mean. Local

f l ow quant i t i es are composed of a mean and a f l uctuat i ng

component .

3) Mol ecul ar t r anspor t pr ocesses f or moment um

mass

and

heat ar e negl ect ed i n compar i son t o t r anspor t by t he

f l uctuat i ng eddy vel oci t i es.

.-

4 ) The Boussi nesq approxi mat i on i s appl i ed. Densi t y devi a-

t i ons

Ap

f r om t he ambi ent densi t y

pa

i ntr oduced by

t he

di f f user di schar ge are smal l compar ed

t o

t he l ocal den-

si ty P (x,z)

& < <

1 (3-2)

P

55



Hence p i s approxi mated by p i n al l terms except the

gravi tat i onal (buoyant) term. Fur t hermore the

mass

conservat i on equat i on i s repl aced

by

the equati on o f

i nc omp res

s

i b

l

y

.

a

5 ) I n the heat conservat i on equat i on, the heat producti on

due to vi scous di ssi pat i on is negl ected i n compari son

w th the heat added

by

the heated di scharge.

Wth these approxi mati ons,

the t i me-averaged equati ons

of

moti on

and heat conservat i on are

aw

u

ax a z

- = o

3-3)

i n whi ch

x,z = Cartesi an coordi nates, w t h

z

upwards agai nst

the gravi ty f orce

u, w = mean vel oci t i es i n

x,z

di recti ons

U,W = vel oci ty f l uctuat i ons

= mean l ocal densi ty

=

const ant ambi ent densi ty

'a

56



p

= mean pressure

T =

m e a n

temperature

T ' =

temperature f l uctuat i on

and the bar denotes t he t i me-averaged turbul ent transf er term.

A

l i neari zed equat i on of state rel ates densi ty and temperature

P = Pa

11

- B(T -

Tal l

(3-7)

where

B

i s the coef f i ci ent of thermal expansi on. The si mul taneous sol u-

t i on of Equat i ons

3-3) t o (3-7)

w th gi ven boundary condi t i ons deter-

mnes the f l ow and temperature f i el d. No such general sol ut i on i s poss-

i bl e by present anal yt i cal techni ques.

However, i nspect i on of actual di f f user perf ormance -- as can be

made i n a l aboratory experi ment

--

i ndi cates that the f l ow f i el d i s

actual l y made up of several regi ons w th di st i nct hydrodynamc proper-

t i es. By maki ng use of these propert i es, addi t i onal approxi mat i ons to

the governi ng equati ons can be i ntroduced. Thi s enabl es sol ut i ons to be

obtai ned by anal yti cal or shpl e numeri cal methods w thi n these regi ons.

By matchi ng these sol ut i ons, an overal l descr i pt i on

of

the f l ow f i el d

can be gi ven.

The obseryed vert i cal st ructure

of

t he f l ow f i el d f o r a di f f user

di scharge w thi n the two-di mensi onal channel

i s

i ndi cated i n Fi gure

3-3

f or the case of a stabl e near- f i el d zone w thout re-entrai nment.

Four

f l ow regi ons can be di scerned i n thi s general case:

1)

Buoyant J et Regi on:

Forced by i ts i ni t i al momentumand

under t he acti on of gravi ty, the two- di mensi onal sl ot

57



a

rl

aJ

d

Frc

r

kl

a

Q

c)

7

a

C

H

k

Q

I

w

-4

a

rH

0

Q

U

0

3

k

U

m

rl

a

0

7(

U

k

z

$

..

m

I

m

M

d

F

58



j e t

r i ses towards

t h e

s u r f a c e e n t r a i n i n g a mb ie nt water.

2) Su rf ac e Lmpingement Region: The pr e se nc e o f

the’free

s u r f a c e , w i t h

i t s

d e n s i ty d i s c o n t i n u i t y , d i v e r t s t h e

impinging j e t i n t h e h o ri z on t al d i r e c t i o n s .

3) Hy dra uli c Jump Region:

An

a br up t t r a ns i t io n between the

h igh ve lo c i t y f low i n the su r f a c e impingem en t r e g ion t o

lower ve l oc i t i e s i n the f low away z one i s provided by

an in te rna l hydraul ic jump.

4) S t ra t i f i ed Counte r flow Region:

A

counte r f low sys tem i s

se t

up

as a

buoya nc y- dr iven c u r r e n t i n t he upper l a ye r

and an ent ra inment-induced cur ren t i n th e lower laye r .

Region 1, 2 and 3 c o ns t i tu t e the ne a r - f i e ld z one; r e g ion 4 and the water

body out s id e the channe l , th e fa r - f i e l d zone . Sur face hea t

l o s s

t o t h e

atmosphere

is

only impor tant

i n

t h e f a r - f i e l d

zone

d ue t o t h e a rea l

res-

t r i c t i o n of t h e n e a r- f i e ld .

The a n a l y t i c a l t r e a tm e n t s for these four dominant f low regions

are

given

i n

t h e f o l l o w in g

s e c t i o n s .

3 .4 Dominant

F l o w

Regions

3.4.1 Buoyant

Je t

Region

3.4.1.1 Approximations and GoverninP Equations

T h e d e f i n i t i o n o f a l o c a l c o o r d i n a t e s y s t e m

s ,

n with velo-

c i t y

components

G,

i s c onve n ie n t f o r the upward curved j e t t r a j e c t o r y ,

as

shown i n Fi gur e

3-4.

I n

terms of t h i s

coo rdi nat e system th e gov erning Equations (3-3)

-

59



t

Fig. 3-4: Definition Diagram for

Buoyant

Jet Region

g

60



to

(3-6)

can

be

t ransf ormed

-

-

2

(3-10)

ai i '

an Pa as Pa an

av - aG

- Jii'3'

P u -

+ pa an

-

-

+

,,gcosg -

a as

(3-11)

w th

0 =

angl e between

s

and

x

axes.

Fromj et observati ons the f ol l ow ng approxi mat i ons are i nf erred:

1) The f l ow phenomenon i s predomnantl y i n the l ongi tudi nal

di rect i on,

b

-

< < 1

S

(3-12)

By conti nui ty i t f ol l ows that i nduced vel oci t i es i n the

l ateral di recti on are smal l

*

* v

- <<1

i3

Thi s boundary-l ayer approxi mat i on al l ows the negl ect

of

certai n term i n the governi ng equat i ons whi ch are f ound

t o

be o f

secondary i mportance

by

a strai ght f orward scal i ng

procedure usi ng Equat i ons (3-12) and

(3-13).

61



2)

The pressure ou ts ide the j e t proper i s h y d r o s t a t i c , ph,

a n d s a t i s f i e s

= o

'h

P a

8

- -

a Z

(

3-14)

Consequently, the p res sur e i n the j e t i t s e l f can be

w r i t t e n

as

P = Ph

+ P r

where pr

is a

reduced pre ss ure , namely t he pres su re

d is tu rbance due t o

j e t

motion.

The two components of

Eq.

(3-14) i n

s,

r are

(3- 15)

-h

-

p g s i n 8

- 0

as

a

(3-16)

p g c o s e = o

ph

an a

- -

Noting tha t

P = Pa + AP (3-17)

where Ap i s the densi ty def i c ien cy due t o the tempera ture anomaly

AT

=

T

-

Ta

, p r e s su r e and g r a v i t a t i o n a l terms i n th e momentum

e q ua ti on s (3-9) and (3-10) can be s i m p l i f i e d

as

- *

p g s i n e

=

- -

+ Ap

g

sin

e (3-18)

as 3s

and

r e s p e c t i v e l y .

22

+ p g c o s e =

- -

+ AP GOS e

an

an

(3-19)

62



Experimental observations have shown

t h a t

the

p r e s s u r e

d e v i a t i o n pr

i s p ra c t i ca l ly n eg l ig ib le i n th e ab sence of b o u nd a ri e s.

diffuser problem boundaries

are

gi ve n by t he bottom and

t h e

free sur face .

Thus

i n making

t h e

approximation

I n t h e

- 0

(3-20)

P r

i t i s

impl ied tha t

t h e

j e t s t a y s

clear

of t he bottom (as j e t a t tach-

ment) and

t h a t

t h e e f f ec t of th e f r ee su r face where t h e p r e ssu re

d e v i a t i o n

i s

documented by t h e s u r f a c e hump ( se e Fig . 3-3)

i s

l i m i t e d

t o t h e j e t impingement region.

co n s id e ra t io n of th e

j e t

impingement region.

This

i s

d iscussed i n

more

d e t a i l i n t he

The s impl i f ie d governing equat ions fo r the buoyant s l o t j e t are,

using th e con t inu i ty equat ion

i n

the transformation

of

the convective

terms,

(3-21)

(3-22)

(3-23)

(3-26)

The

l a t e ra l

momentum equation (3-23)

i s

rep laced by t he hor izo n ta l

momentum equation,

a

l i n ea r combination of (3-22) and (3-231,



which relates d i r e c t ly t o t h e boundary condition of h o r i z o n t a l momentum

f l u x

a t

t h e s l o t o pen ing as i s shown below.

The fu r th e r so lu t i o n

procedure

uses

the entrainment concept proposed by Morton

e t

a l . (1956).

A va r i ab le en t rainment co ef f ic i en t dependent on the l oc a l buoyant

c h a r a c t e r i s t i c s of t h e

j e t

i s assumed of the form Eq. (2-11).

method of obta inin g t h i s dependence foll aws the procedure us ed by

Fox

(1970)

f o r v e r t i c a l round and s l o t j e t s and by

H i r s t

(1971) f o r non-

v e r t i ca l r o u n d j e t s .

&sc r ib in g th e f l u x of mech an ical en ergy i n th e lo n g i tu d in a l d i r e c t io n

wi th v e lo c i ty

3

i s found from EQ. (3-22)

as

The

For

th is purpose a s impl i f ied energy equat ion

o r by v i r t u e o f th e co n t in u i ty eq u a tio n

(3-26)

(3-27)

Based on experimental evidence the fol lowing similar i ty functions

are s p e c i f i e d f o r

6

and

T

L

n

-(

T&

1

AT(s,>n)

=

ATc(s)e

6 4

(3-28)

(3-29)



i n which X

I s

a

dispers ion r a t i o be tween momentum an d h ea t or mass.

T h e p r o f i l e f o r Ap is re la ted to (3 -29)

by t h e

equation of s t a t e ,

thus

(3-

30)

.

uc , Tc , ApC

are

c e n t e r l i n e v a lu e s .

These functions

are

in t roduced i n t o Eqs. (3-211, (3-2 2), (3-24)

and (3-25) and

t h e

equations are i n t e g r a t e d i n t h e nor ma l d i r e c t i o n

n

.

I f th e boundary condit ions

-

.,

V

+ + 5

e

are n ot e d, t h e r e s u l t i n g or d in a ry d i f f e r e n t i a l e qu a ti on s

are,

2

d ..,

x [

u

e

d n l

-

2 Ge

= 0

n

-( g 1

a

(3-32)

65



2

Q)

-2 (

2

dn]

= 0

h 2

- - [

u

c o s e e

ds - 0 0

C

3-34)

*

The normal ve loc i t i es a t

t h e j e t

boundary,

v , i .e .

the "entrainment

veloci ty" i s assumed to be r e l at e d t o t h e c h a r a c t e r i s t i c j e t v e l o c i t y

-

U

C

lGe1

a iic

3-36)

This c on st it u te s th e entrainment concept by Morton

e t a l .

After

eva lua t ion o f the in te gra l s

t h e

equations become

continuity:

C

a t

(Gc

b) =

-

d i r

a x i a l momentum:

g

hb

s i n 8

- 2 %

(uC

b) =

fi

-

pa

horizontal momentum:

heat conservat ion

:

d

ds

Gc ATc b) 0

3-37)

3-38)

3-39)

3-40)

o r a l t e r n a t i v e l y i n s t ead of

( 3 - 4 0 )

66



conservat ion

of

dens i ty def ic iency :

3.4.1.2

Dependence of t h e Entrainment on

Local Jet

Characteristics

Experimentally deduced values for a

as

r epo r ted i n t h e

pre vio us ch ap te rs , Eqs. (2-18) and (2-19)

show

t h e dependence

of

a

with

the buoyant j e t characteristics.

This dependence can be examined usi ng

th e energy equat ion (3-27) following Pox's app ro ach f o r t h e v e r t i c a l j e t .

To

i n t eg r a t e t h i s eq u a t i o n i t is assumed that t h e stress

G G

i s

d i s tr i b ut e d i n a s i m i l a r i t y p r o f i l e f (n / b ) and r e l a t e d t o t h e c en te r-

l i n e v e l oc i t y

iic

There

is

exper imental support to t h i s assumption (Mih

(1972)) bu t t h e accuracy of the data does not a l low spe cif ica t io n of

With (3-42) and th e othe r s i mi la r i ty pr of i l es t h e energy equat ion (3-27)

and Hoopes

f .

is i n t e g r a t e d t o

2

1

- ( 1 + r H

g ii s-n 8

e

3

L r ~ u e dn]

=

2 I

-

d s

~

c

dn

-3 (n/b)

* p a c

W

4 -(n/b)2 d

- 2

-OD

1

u c e

-n f (nib) dn

a n d a f t e r

evaluation



where

a0

2

- ( d b )

d

(n/b) dn

dn

-

$ e

ioD

Noting th e ide nt i t y

t o

1

1 + X

( G C b ) = 2 - g b - [ f i X -

"C

U

C

ds Pa

+ 2 . \ 1 +

Gc

I

( 3 - 4 5 )

( 3 - 4 6 )

Equation ( 3 - 4 4 ) can be transformed using th e axia l momentum Eq. ( 3 - 3 8 )

( 3 - 4 7 )

When t h i s eq uat ion

i s

compared with the continuity Eq. ( 3 . 3 7 ) t h e

ent ra inment coef f ic ient

a i s

expressed as

i f a local densimetric Froude number i s defined

(3-49

)

Equation

( 3 - 4 8 ) i s

similar t o

Fox's

equat ion except for the fac tor

s i n

8

.

The expression

( 3 - 4 8 )

c o n s i s t s

of t w o

p a r t s

( 3 - 5 0 )

68



showing a co n s t an t

a

modified by a fu nc ti on which depends on t h e buoyant j e t c h a r a c t e r i s t i c s ,

t ,

,

and the j e t geometry, sin 8 .

explored.

,

as

f i r s t pos tu l a ted by Morton e t a1 (1956),

1

This dependence

i s

f u r t h e r

For t h e simp le non-buoyant momentum

j e t

a reduces to

a

= al = & I as F ~ - c -

(3-51)

As t h e l a ck of s p ec i f i c a t i o n o f a s i m i l a r i t y p r o f il e f o r

c’?’

does

no t a l low eva lua t ion of th e in teg ra l

I ,

a,

i s

determined from

.I.

experimental data f o r t h e momentum j e t

= 0.069

al

by Albertson e t a1

(1950) which value has been

o t h e r i n v es t i g a t i o n s .

(3-52)

w e l l

subs tan t i a ted by

For d i s tances

far from

th e d isc har ge po,nt a l l buoyant j e t s

tend t o the condi t ion of a pure ve r t i c a l ly r i s i n g plume.

asymptotic case

i s

characterized by

a

constant densimetric Froude

This

n

FL

nrnnber

as

i s shown below.

The change in t h e l o c a l densim etric Froude number FL alo ng

t h e j e t t r a j e c t o r y s is from Eq. (3-49)

(3-53)

69



a f t e r some manipulation. Using Eqs. (3-37), (3-38) and (3-40) t h i s

becomes

2 N

g X b s i n 8 - - uC]

(3-54)

1 1

- = F L z ~ [ - T -

Pa

6

.

FL

ds

uc

or

i n

a

simpler, im pl ic i t form

I

X

s i n 8

dFL

d s

For the ve r t i ca l ly r i s in g plume s in 8

= 1 .

The necessary condi t ion s fo r an asymptotic valu e FL

are

n

h

as

FL

FL

dFL

- = o

ds

and

if

FL FL

- > o

da

- < oFL

i f FL > FL

d s

These conditions are s a t i s f i e d f o r

(3-55)

(3-56)

(3-5 7)

n

where i s the entrainment coe ff i cie nt and X t he sp read ing r a t io

of th e sim ple plume.

than

FL ,

then the j e t

w i l l

be accelerated and

w i l l

monotonically

approach the plume condition.

belong i n th is group.

case

i s

discharge with large Froude numbers

If th e Froude number of th e discha rge is less

A

Convection phenomena over l i n e f i r e s

Of d i re c t in te re s t in the submerged diffuser

A

FL

> FL

where the flow

70



becomes gradually decelerated until the balance between buoyancy and

shear forces which i s ty pi ca l of the plume i s reached.

and th e ex istence of th e asymptotic value

was

f i r s t described by Lee

This behavior

-

and Emmons (19621, re fe rr in g t o the two cases as restrained and impelled

sources , respect ively .

In the v er t i ca l l y r i s in g buoyant plume condi t ion the equat ion

n A

A

fo r th e entrainment (3-50) should be s at is f i e d by a

, X ,

and

FL

from Eq. (3-57) with sin 8

-

1

, hus

n

a

-

0.069 +

[a

qy

(3-58)

1 + X

The only extensive experimental investigation regarding plume

behavior was made by

Rome

e t a1

(1952) on buoyancy sour ces from l i n e

f i r e s .

Measurements of veloci ty and temperature

w e r e

taken a t d i s t ances

above the source such that the inf luence of an i n i t i a l d e vi at io n

FL #

tL can be negl ect ed and tr u e plume values are approached. However,

th ei r da ta f i t t i n g procedure which y ie lded t he f requent ly used values

h

n

a

-

0.160 and X

=

0.89 (3-59)

implied l inear spreading from the source,

a Con diti on which is

vi ol at ed when th e

ini t ia l

FL

#

GL .

s a t i s f y Eq. (3-58). Abraham (1963) re-examined Rouse e t

al . 's

da ta

The

values (3-59) do not

and found tha t i n t he region above the source where ther e i s e f f e c t i v e

plume behavior th e val ues

h

n

a

= 0.130 and X

= 1.24

(3-60)

h

d e sc r ib e t h e d i s t r i b u t i o n s much b e t t e r . A s p re a d in g r a t i o X >

1 i s

.

71

__



a l so c ons i s t e n t w i th obse rva t i ons on o the r f r ee t u rb u l e n t phenomena.

In t roducing h

=

1.24 i n t o Eq.

(3-58)

y i e l d s a

=

0.128. This good

agreement for

A

h

6

m u s t

be judged somewhat fo r t u i to us s i nc e i t

is

based only on

a

v a l i d i t y of t h e

c h a r a c t e r i s

i c s

of lo c a l Froude

s i n g l e

s e t

of experiments , but seems t o s u p p o r t t h e

fu nc t io na l dependence

of 01

on the buoyant

j e t

exp res sed by Eq. (3-48). With th es e va lu es th e magnitude

number f o r

a

s imple plume

i s

A

FL

= 3.48.

The

s p r e a d i n g r a t i o X also shows some v a r i a t i o n w i th j e t

buoyancy as ind ica ted by exper iments .

f o r the s i m p l e momentum j e t (Re ich ard t (1942)) and = 1 .24 fo r t he

plume (3-60)

i s

observed. However,

as

the second

term

i n t h e e n t r a in -

m en t r e l a t i onsh ip

i s

impor tan t on ly f or p lume- like behavior a cons tan t

A

var i a t i on be tween

X =

1.41

X =

h

=

1.24 is assumed throughout th e j e t domain.

With t hese da t a , t he fo l l owing qu an t i t a t i ve form

i s

proposed for

t h e e nt ra in m en t r e l a t i o n s h i p (3-50) i n a s l o t b u o ya nt j e t

(3-61a)

a = 0.069 + [3.11 - 2.39 s i n 01

FL

and

a

c o n s t a n t s p r ea d i ng r a t i o

3.4.1.3 I n i t i a l Conditions: Zone of Flow Establishment

The governing eq ua tio ns (3-37) t o (3-41) do no t adequa tely

descr ibe the zone of f low es t ab l i sh men t i n which a t r an s i t i on be tween

t h e uniform s l o t e x i t v e l o c i t y

Uo

the g ener a l flow reg io n tak es

place as

see n i n Fig. 3-5. Therefore

and t h e v e l o c i t y d i s t r i b u t i o n i n

f o r t he s p e c i f i c a t io n of i n i t i a l c on d it io n s some approximate s t e p s are

taken:

7 2



Z

-

Velocity u0

Fig.

3-5: Zone

of Flow

Establishment

73



1) The i n i t i a l c o n d i ti o n s

are

s p e c i f i e d a t t h e e nd of

t h e

zone

of flow establ ishm ent and are r e l a t e d t o t h e s l o t d is c h ar g e c o nd i ti o ns

by cons ervation equations.

2) The e ff e c t of buoyancy i n th e zone of estab lishm ent i s

neglec ted

.

Requir ing constan t momentum f l u x i n the ax ia l d i re c t io n one ob ta ins

n

OD -2(n/b)

U ~ B / iii

e dn = f i U : bo (3-62)

C

00

0

(3-63)

(3-64)

which i s i n c los e agreement on exper imenta l ly der ived re la t io ns h i ps

f o r th e zone of establ ishment by Albertson e t

a1

(1950)

thus

b0

= $ B

The volume flux

a t

the end of th e zone follows th en

e v a l u a t e d f o r t h e length of th e zone s

=

5.2 B as

=

1.42 UoB

i n which

4e

= volume

f l u x .

Fi n a l ly , co n serv a tio n of d en s i ty d e f i c i en cy r eq u i re s

(3-65)

(3-66)

Q) - ( 1 + 1/X2)(n/b) 2

uo8

Apo = / Go Apc e dn (3-67)

C

0

OD

74



and with

(3-62)

(

3-6

8)

as t h e

i n i t i a l

condition

a t

th e end of th e zone.

In

the fu r ther t r ea tment ,

t h e

e x t e n t s / B = 5.2 of the zone

of es tabl ishment

w i l l

be neglected as

t h e r egi o n of i n t e r e s t

s/B

is considerably larger.

3.4.1.4 Sol uti on of th e Equations

The complete statement of the heated buoyant j e t problem is

summarized

as

follaws:

(3-69)

g A

b s i n

0

-2

(uC b) = A -

Pa

d

- 6:

b

cos

6 ) =

0

ds

0

--iclud,ng geometric re la ti on s descri bing the j e t t r a j e c t o r y

w i t h

and

dx

-

-

cos 0

ds

dz

- -

s i n 0

ds

1

a

-

0.069 + [3.11 - 2.39 s i n 01

FL

A = 1.24

75

(3 - 70)

(3-71)

(3-72)

(3-73)

(3-74)

(3-75)

(3 - 76)



The center ine t emperature

r i se

r e l a t e d t o

Apc

by th e equat ion

of s t a t e

ATc

above the ambient temp erat ure i s

1

ATc =

p

ApC

The i n i t i a l c o nd it io n s a r e

b = O

I

(3 -7 7 )

I

( 3 - 7 8 )

x - 0

y - 0

which

are

g iven a t the end

of

the zone

of

f low es tab l i shment and

are

r e l a t e d t o t h e s l o t e x i t c o n d i t i o n s by E qs. (3 -63) and

(3-68) .

S o l u t i o n

of

t he equa t i ons

is

n o t p o s s i b le i n c l o s e d a n a l y t i c a l

form.

However,

t h e i n i t i a l value problem

is

r e a d i l y i n t e g r a t e d by

num eric al methods.

is

used.

A

four t h or de r Runge-Kutta in te gr a t io n techn ique

The buoyant j e t d i scharge is governed by t he f ol lo win g

dimensionless parameters

densimetric Froude

determined f rom the s l o t e x i t c o n d i t i o n s :

(3-79)

a n g l e

of

d i scharge

76

( 3 - 8 0 )



I f i n a d d i t i o n t h e d i s ta n c e

s

i s s c a l e d b y B

S

dimension less d is tance

( 3 - 8 1 )

the buoyant j e t p ro p e r t i e s , n amely v e lo c i ty , d en s i ty d e f i c i en cy , w id th

a n d t r a j e c t o r y , are funct iona l ly dependent

as

( 3 - 8 2 )

An important r e s u l t i n g parameter is t h e j e t c e n t e r li n e d i l u t i o n

S

which determines th e decr eas e

i n

d en s i ty d e f i c i en cy (o r t emp era tu r e )

C

w i t h r e s p e c t t o t h e d i s c h a r g e

bTO

4% %

s I =

C

( 3 - 8 3 )

and wi th respec , t o t h e end of

zone

of flow estaAishment

( 3 - 8 4 )

by

virtue of Eq.

( 3 - 6 8 ) .

-

An av e rag e d i lu t io n S

i s

d e f in ed

as

t h e

r a t i o

of

f l o w

a t any

d i s t a n c e

s t o

the d ischarge

a t

t h e

s l o t .

m

(3-85)

Mass

co n se rv a t io n g iv es

the

r e l a t i o n

- = s + y -

x

( 3 - 8 6 )



A

com parison betw een t h e o r e t i c a l p r e d i c t i o n s f o r c e n t e r l i n e d i l u t i o n

S

p ro po sed i n

E q . (3-75)

versus using a c o n s t a n t e nt ra in m en t c o e f f i c i e n t

wi th o u t su r f ace e f f ec t s u s in g a v a r i a b l e e nt ra in m en t c o e f f i c i e n t as

C

(Fan and Brooks (1969),

c1

=

0.16)

i s

g iv e n i n

F i g .

3-6

for

a

v e r t i c a l

and i n Fig . 3-7 f o r a h o r i z o n t a l

j e t

discharge .

d a t a

are

a l s o i nc l ud e d.

Limited experimental

F or t h e v e r t i c a l d i sc ha rg e b o th p r e d i c t i o n s are about e q u a l i n

the l o w Froude number range.

Fan and Brooks' pre di ct io n i s , however,

(due to the chosen value of a), w h i l e t h e

FS

too high

f o r

l a r g e

pre d ic t ion s of t h i s s tudy approach Alber tsen e t a l .

's

(1950) experimental ly

-

v e r i fi e d r e s u l t for t h e a v e r a g e d i l u t i o n S i n non-buoyant

j e t s

-

S = 0.62

which can be converted t o

(3-87)

S =

0 . 6 4 B

1 + X

(3-88)

b y v i r t u e

of

Eq.

(3-86)

and taking account of th e proper

i n i t i a l

condit ion

a t

t h e end of the zone of flow es tab lis hm ent . Some measure-

ments of

S

w e r e made i n t h e ex p e r imen ta l p a r t of t h i s s t u d y ( s e e

Chapter 5 ) and the agreement is b e t t e r w it h t h e t h e o r e t i c a l p r e di c t i on

of t h i s s t u d y .

C

For th e h o r i z o n ta l d i sch a rg e t h i s s tu d y p r ed ic t s somewhat h ig h e r

d i lu t ion s fo r low Fs

.

fo r h igh F i s again due t o the con sta n t va lue o f

c1 .

Experimental

d a t a are re po rt ed by Cederwell (1971) and

are

l imi ted to the very low

The s t ro n g s l o p e of Fan and Brook's s o lu t i on

S

78



oC=

0.16

A

=

0

89

A

- 1 2 4

-. - -

Non-buoyant

j e t %+OO

Fan and Brooks

This s tud y

CC

Eq (3-75)

- - -

Eq (3 8 8 )

X E x p e r i m e n t s

-

th is S tudy

Fig. 3-6: Centerline Dilutions

S

for Buoyant Slot Jets Without

C

Surface Interaction. Vertical Discharge. Comparison

of theory

and

experiments.



20c

OC =0.16

- - - an

and

Brooks

=0.89

oC

E q . ( 3 - 7 5 )

T h i s s t u d y

Experiments by Cederwal l

(1971)

Fig.

3-7:

Cen te r l in e Di lu t io n s

S c

f o r Buoyant S lot Je t s Without

Sur face Int era cti on . Hor izon tal Discharge. Comparison

of Theory and Experiments.

80



*S

range. A l l d a t a l i e between th e p red i c t io ns o f both inves t iga t ion s ,

th e agreement bei ng somewhat cl os er wi th Fan and Brooks' so lu ti on .

3 . 4 . 2 Surf ace Impingement Region

3.4.2.1) General So lu ti on

The buoyant

j e t

region

i s

bounded by t h e s u r f ace

impingement regio n i n which an abrupt t r a n s i t i o n between the j e t flow,

with

a

s t ro ng v e r t i c a l component, to a hor izontal spreading motion

occurs. Discussion

of

t h e j e t impingement

a t

the su r face a l so has

some rele vanc e t o t he buoyant

j e t

r e gi o n i n p a r t i c u l a r w it h r eg ar d t o

P r

he previous assumption of neg li gi ble reduced pressure

The main features of

t h e impingement are i n d i ca t ed i n F i g. 3-8.

The momentum

of

t h e j e t

sets

up a surface hump

TI

i n con junc tion wi th

a p res sure dev ia t ion

Pr

d ec reas in g i n t h e v e r t i c a l d i r ec t i o n . T hi s

__

pres sure g rad ien t

causes

a s tagna t ion of the

ver t i ca l

flow accompanied

by a h o r i zo n t a l sp readi n g i n b ot h d i r ec t i o n s .

The buoyant

j e t

impingement

i s a

complex fl ow phenomenon.

Thus

no attempt i s

made

t o s o l v e f o r f low p r o p e r t i e s i n s i d e t h e r eg io n.

Rather, a c o n tr o l volume approach i s taken which yields a d e s c r i p t i o n

of

t h e ho r iz on ta l spreading lay ers af t e r impingement. The th ickness

of

t h e

spread ing lay er s de te rmines t he e le va t io n (H

=

hi)

e f f e c t i v e

j e t

entrainment occurs and th e dynamic c ha rac te r is t ic s

are

t o which

d e c i si v e f o r t h e s t a b i l i t y of t h e f l o w away zone.

Refe rring t o Fig. 3-8,

a

co nt ro l volume i s def ined by se ct io n

i

with th e incoming j e t f low, by sect ions a , b wi th the hor izon ta l

spreading motion and by

t h e f r e e s u r f a c e .

The f l a w

i s described by a

se t

of s im pl i f i ed sp a t ia l l y averaged equa t ions:

-

81



Fig.

3-8:

Schematics

of

Surface Impingement Region

82



Energy equations

:

2

a

1

-2

i

APi =

'a

2g APa

= + Pa

hL (3-89)

-

a

- -

pa

2g

-

2

(3-90)

-

Lb

i Api z

=

ulb

'Pb

+

Pa

-2

P a 2g

U

- -

pa 2g

Ho r izo n ta l momentum equat ion :

2

)

(3-9

1)

Pa

$

cos

8

dn

+

$

Pa

u% dz =

pa

uf dz

H-hl

a

a

-m

Continuity

:

H

H-hl

a

a

02

$

Gi dn -

la

us, dz

+ I

u1 dz (3-92)

-03

= energy lo ss es i n the impingement process.

This

La h \

where h

formula t ion inc ludes

some

of th e assumptions made

in

the treatment of

the buoyant j e t r eg io n , n o tab ly th e

Boussinesq

approximation.

more, th es e assumptions are i n h e ren t :

Further-

a )

T h e d i s t r i b u t i o n

of

t h e p r e s s u r e d e v i a t i o n

pr : The

ver t i ca l

e x t e n t

of

t h e p r e s s u re i n f l u e n c e

is

l im i t ed .

by

Cola (1966) and Murota and Muraoka (1967) on v e r t i c a l non-

buoyant

j e t s

show th a t t h e p r e ssu re in f lu en ce is n e g l i g i b l e

below

z/H <

0.75.

Thus

t h e p r e ssu re p r

i s

neglec ted a t

Experiments

_-

s e c t i o n i as

a

f i r s t app ro xima tion . The h o r i z o n ta l ex ten t

83

.--



of t h e p r e s s u r e i n f l u e n c e

i s

r e l a t e d t o t h e j e t width

S ec t i ons a and b

are

assum ed t o be l oc a t ed ou t s i d e t h i s

zone.

bi

b) Entra inment i n the spreading process i s n e g l i g i b l e .

S o l u t i o n of t h e s e e q u a t i on s r e q u i r e s t h e s p e c i f i c a t i o n

of

v e l o c i t y and d e n si t y p r o f i l e s f o r a l l s e c t i o n s as w e l l

as

express ions f o r the energy los se s . Hence th e fo l lowing

addi t ional assumpt ions

are

in t roduced:

c)

V e l o ci t y an d d e n s i t y d i s t r i b u t i o n s a t s e c t i o n i

are

repre sente d by t he Gaussian pr of i l es Eqs . (3-28) and

(3-30)

t yp i ca l of t he buoyan t

j e t

r e g i o n . D i s t r i b u t i o n s

a t

s e c t i o n s a and b are dependent on the buoyant c ha ra cte ris -

t i c s of th e impinging j e t .

Cola and Murota and Nuraoka found

the ve loc i t y d i s t r i bu t i on fo r non -buoyan t j e t s t o b e

es s en t i a l l y j e t - l i ke . For buoyant j e t s , however, the

buoyancy

in

t h e s p r e a d i n g l a y e r exerts a s t a b i l i z i n g e f f e c t

on the f low which s upp res ses th e j e t d i f fu s i on i n t he

v e r t i c a l d i r e c t i o n . V e l oc i ty a nd d e n s i t y p r o f i l e s ca n b e

reasona bly approxim ated by a r e c t a n g u l a r p r o f i l e . T hi s c an

b e se e n i n

Fig.

3-9. Photog raph (Fig . 3-9a) shows t h e dye

traces obtained from an ins tan taneo us dye in je c t i o n by means

of a

probe

which had holes spaced a t

1”

nte rva l s . Due t o

the d i f f u s io n of t he i n j e c t e d dye, t he p robe canno t be used

fo r qu an t i t a t i ve de t e rm ina t i on o f ve lo c i t i e s . However, t he

pho tograph shows t he qua l i t a t i v e ve loc i t y d i s t r i b u t i o n ,

namely a s t ro ng approximate ly uni form f low i n t h e upper

5

i n .

of th e depth and a weak c o un te rf lo w t o t h e l e f t i n t h e lower

84



a ) Dye Traces From Instantaneous Inject ion Showing

V e l o c i t y D i s t r i b u t i o n

in.

X = + 6 in.

T= 65.8 F

b ) T em p era tu re P ro f i l e s

a t

Sec t i o n s a and

b

Fig. 3-9:

A

Observed Veloci ty and Temperature (Densi ty)

D i s t r i b u t i o n s f o r Ver t ica l Buoyant J e t

(Fs = 31, H / B = 416)

85

G r i d Size

2

i n .

x

4 i n .



por t ion caused by the j e t e n t ra i n m en t . S i m i l a r l y , t h e

obse rve d t e mpe ra tur e d i s t r ib u t io ns a f t e r im pingem en t a re

a pp ro xi ma te ly u ni fo rm e x c e p t f o r t y p i c a l t u r b u l e n t f l u c t u a -

t i o n s .

i n

Chapter

5.)

Thus t h e v e l o c i t y a nd d e n s i t y d i s t r i b u t i o n s

(The

e xpe r im e n ta l s e t - up

i s

d es cr ib ed i n d e t a i l

are

i n s e ct i on

a

a

1

(4

=

U

a

H - h l <

z < H

a

Apa(z>

=

Apa

and

i n

s e c t i o n b

= u1

a

(3-9 3)

(3-94)

A Consequence of

t h e u n if o rm l y mixed d i s t r i b u t i o n s

i n t h e s p r e ad i ng l a y e r s

i s

d)

The energy losses due

t o

s ec on da ry c i r c u l a t i o n s i n

the impingement process a r e r e l a t e d t o t h e v e l o c i t y h ea d

a t

s e c t i o n

i as

(3-96)

86



are head

loss

coef f i c i en t s wh ich are dependent

where

5

kLb

a

on the ang le and curv at ur e of th e f low bending and can be

approximated by experimentally determined head

loss

c o e f f i c i e n t s

f o r

f lows i n smooth p i pe bends ( f or example,

Ito

(1960)).

With t h e assumed s i m i l a r i t y p r o f i l e s t he s e c t i o na l l y ave raged

q u a n t i t i e s i n t h e en er gy e q u a t i o n

can

be de f i ned

as:

00

- 2

U

C

-2

ci

dn

i 1 - 0

1

i

3,

dn

(3-9 7)

- -

U

- n -

2g 45

g

2g

OD

-Q)

and

b

l + hl

a

P

hi 2

87

(3-99)

(3-100)

(3-101)



The

Eqs.

(3-89) t o (3-92) become, a f t e r con sid era tio n of

Eq.

(3-95)

2

2

cos €Ii + ulb hlb = u1

hl

a a

6 bi fi

i

(3-105)

a a

l a a

These 4 equat ions de termine the 4 unknowns u

as a

f u n c t i o n of j e t condi t ions

a t

s e c t i o n

i ,

h a t i s

.

h i )

,

ApC , bi = f(z H

-

i

C

i

(3-106)

as

obta ined from the so lu t i on

of

th e buoyant j e t equa tion s (3-69) t o

(3-76).

As no

closed form so l u t io n s to th e b uoyant j e t equat ions

are p o s s i b l e , t h e a l g e b r a i c equ ati ons (3-102) t o (3-105) of the su rf ac e

impingement can be s im pl i f ied by s ub st i t u t in g a l i n e a r ap pro ximat ion

of Eqs. (3-106) i n th e range of z where the layer depth i s expected.

The a lg eb ra equations are then ea s i ly so lvab l e by s imple numer ica l

techniques, such as the Newton-Raphson method.

8 8

(3-104)



3.4.2.2 Special Cases

Valuable insight on the relat ive importance of the terms

i n Eqs. (3-102) to (3-104)

i s

gained by considering the ver t i ca l

d i s -

charge case for both the simple momentum j e t and t h e simple plume.

The f low pattern i s then

symmetric u1

= ulb

=

u1 , l

hlb

=

hi ,

a a

and the equations reduce t o

-

u bi fi = 2 u1 hi

i

(3-107)

(3-108)

and by subst i tu t ion

a ) Vertical Momentum Jet (Apc = 0): Eq. (3-109) becomes

i

(3-110)

The j e t width bi

equations.

From

Eq. (3-70) one ob ta in s

i s

obtained from th e so lut i on of the buoyant

j e t

by conside ring the i n i t i a l cond i tion (3-63). Subs t i tu t ion in t o

Eq. (3-69) yields ( 8 = z)

(3-111)

4

- a

b

- =

dz

J;;

(3 112)

89



and in tegrated

4

b = -

3 L z

J;;

i f t he

i n i t i a l

width bo

i s

n egl ect ed f o r l a r g e

z ,

b >>

bo

.

hi

ence a t z =

H

-

4

- u(H

-

hi)

bi

=

and

a f t e r

s u b s t i t u t i n g i n t o

Eq.

(3-110)

L

- =

H

(3-113)

(3-114)

(3-115)

For t h e momentum j e t

a

= 0.069, and % is estimated from I t o ' s

(1960) data as kL-0.2

curv atur es. The va ri at io n of hi/H with

% i s

se en i n t h e

fol lowing table:

for a 90" bend and a wide range of

0.2

-

0.4

0.167 0.188

i

- = 0.152

H

b) Plume: The constancy 6f the local dens imetr ic Froude

h

number F,, = FL ha s been shown i n S ec ti on 3.4.1.2.

T h i s f a c t , in

combination with th e constant buoyancy fl ux

Eq.

(3-72), implies

t h a t t h e j e t cen t e r l i n e v e l o ci t y i s a l so cons tan t i n t h e

co nv ec tiv e pl an e plume. Thus

Eq.

(3-69)

can

be s impl i f i ed to

d

uc

- ab 2 ;

J;;

s

90

(3-116)



hi

nd in t eg ra t ed fo r

z

=

H

-

-

a(H

-

hi)

J;;

i

=

(3-117)

aga in neg lec t ing the i n i t i a l width.

ing using th e value of

After

su bs t i tu t io n and rear rang-

A

FL

(Eq.

(3-57)) Eq. (3-109) becomes

(3-118)

n

h

With

% =

0.2,

a

= 0.128 and X

-

1.24 th e equation is evaluated

as

(3-119)

i

-

p

0.149

H

The second terra on

effect of buoyancy

the r igh t s ide o f Eq. (3-118) exp res ses th e

on the th ickness . I f t h i s term i s neglected

(3-120)

i

-

-

0.159

€I

exhibi t in g only weak sen si t iv i ty .

Thus ca lcu la t ions fo r bo th the vertical momentum

j e t

and the

plume show

that

the th ickness of the spreading laye r is about 1/6 of

the t o t a l depth

H and the

j e t

entrainment region can be assumed to

extend t o th i s e l eva t ion.

seen

i n

Fig. 3.9.

neglect ing th e pressure devia t ion

pr

preceding se cti on s. However, th e error introduced appears t o be

small.

Prel iminary evidence of th is r es ul t can be

The

result

somewhat v io la te s t h e assumption of

a t s e c t i o n i made i n t h e

91



3.4.2.3

Vert ical Flow D i s t r i b u t i o n P r i o r t o t h e

Hydraulic

Jump

A f t e r t h e combined evaluation of th e buoyant j e t and the

impingement r egi ons , th e ver t i ca l f low dis t r ib ut io n can be determined.

With re fer en ce t o Fig. 3-10, a t both Sect ions

a

and b, a counterflow

system

i s

pr ese nt with f low in the upper la ye r away from the l i n e of

impingement and flow i n th e

lower

layer towards

j e t

entrainment

I

Fig. 3-10: Schematic of

Vert ical

Flow Distr ibution

P r i o r t o

In te rna l

Hydraulic Jump.

I n i t i a l an d s u r f ac e l ay e r volume f l u x es

per

u ni t channel width are

=

Uo B

4 0

91

-

u1 hl

a a 2

qlb =

r e s p ec t i v e l y .

The average di lu t ion

i s by

d e f i n i t i o n

-

q1

+ q l b

s =

QO

(3-121)

(3-122)

92



- and the t o t a l entrainment f low q fo l lows as

e

( 3 -

i s obtained by considering a

and *%

he magnitude of

q2

horizontal momentum equation between a and b . Writ ing th is

equation notice can be made of t he f a c t that the hor i zon ta l j e t d i s -

charge momentum i s ac tua l ly conserved within th e buoyant

j e t

region

and th e impingement re gi on (Eq. (3-104)). Thus

a

momentum equation

a

can be

writ ten

for the lower layer alone excluding these regions.

With th e assumptions of hyd rost atic pr es su re and uniform ve loc ity

d i s tr i b ut i on t h i s

i s

simply

2

q2b

Subs t i tu t ion of Eq. (3-123) gives

(3-

(3-

Gross

densimetric Froude numbers for upper and

lower

l a y e r s

are i mp or ta nt v a r ia b l e s a f f e c t i n g t h e s t a b i l i t y in the subsequent

in te rn al h ydra ulic jump region and

are

defined as (omit t ing the seca

- subsc r ip t )

93



(3-126)

(3-127)

‘ a

-

i n which Ap - bpi

th e uniform dens i ty de f ic ie ncy i n the upper layer .

In di ca ti on s of th e magnitude of th es e Froude numbers are drawn

from the special cases considered earlier.

For th e v e r t i c a l momentum

F1 9 F2

=

(3-128)

For

t h e plume,

(3-129)

h, 312

following

a

similar

procedure to that used

i n

the der iva t ion

of

h

Eq. (3-118), with A = 1.24, a = 0.128 and hi/H = 0.149 (Eq. (3-119)),

one gets

= 2.63

=

0.19

F1

F2

(3-130)

For ve rt ic al buoyant j e t s Eq.

(3-130) gi ve s t h e

lower

bounds on the

R o d e

numbers.

9 4



3.4.3 In te rn al Hydraulic Jump Region

3.4.3.1) General Sol uti on

The in te rn a l hydraulic jump region provid es t h e t r a n s i t i o n

between the flow conditions

a t t he

end of t h e s ur fa c e impingement

region and the f low away in t o the fa r f i e ld .

The region

is

analyzed

as an

in te rn a l hydrau l ic jump i n

a

two-layered counterf l o w .

The de fi ni t i on diagram,

Fig.

3-11, shows t h e upstream

q 1

q 2

J-

and

down-

Fig. 3-11:

Definition Diagram: Internal Hydraulic Jump.

stream

condi t ions of the i n t er na l hydraul ic jump.

are r e f e r r e d t o as conjuga te to each o ther , t h a t is they represent

These conditions

two flow states which are dynamical ly po ss i ble with regard t o the

governing momentum equation.

The abrupt change of flow states i n the

jump is asso cia ted with cons iderable energy l os ses and poss ible

entrainment

a t

t h e i n t e r f a c e .

An

approximate a na ly si s of the jump

can

be made by apply ing t h e

momentum conserva t ion p r inc ip le to both flaw la ye rs between th e up-

and downstream sections.

The analysis assumes hydros t a t i c p ressure

95



d i s t r i b u t i on and un iform ve lo c i t i e s

a t

both se ct io ns . Fur thermore the

entrainment i n the jump

is

neglected

as are

t he i n t e r f ac i a l and bottom

shear .

an al ys is of simple one-layered jumps.

All

these assumptions are c o n s i s t e n t

w i t h

those made i n the

The

momentum

e q u a t i o n

is

f i r s t g iven f o r

the

control volume

comprising the

lower

l a y e r as

i n

which the mean head ac t i ng over t he j um p sec t i on

is

approximated by

1

2

(hl

+

h i )

.

For th e upper l a ye r i t fo l lows

I f f r e e su rf ac e Froude numbers are def ined in the us ua l form

96

(3-133)



Equat ions (3-131) and (3-132) can

b e

r e w r i t t e n

as

h i

-

hl

2 2

2F1 h (h ' - hl)

1 1

+ 1 -

a

h 2

h2 h i (h1

+

h i )

h 2

(

3-134)

This

d e r i v a t i o n

was

f i r s t give n by Yih and Guha (1955). These au tho r s

a l s o d i s cus sed t he pos s ib l e so lu t i ons fo r h i / h l and h;/h2 . I n

genera l , t he re are 9 r o o t s t o

t h e

above equat ions, out of

which

only

4 are

pos i t iv e and thus phys ica l ly meaningfu l.

These

4

roo t s are

given by t he in te rs ec t io n of t w o parabol ic branches of the equat ions

as

i l l u s t r a t e d

in

Fig. 3-12a. The energy co nte nt (s pe ci f i c head) fo r

each of the

4

conjugate s t a t e s can

b e

eva lua t ed .

I t

i s found (Yih

(1965)) th a t the

4

s o l u t i o n s ha ve d i f f e r e n t s p e c i f i c he ads

as i nd i ca t ed

in

Fig. 3-12a with

1

fo r t he cond i t i on wi th h ighe s t ene rgy.

A

jump

can only occur from

a

s t a t e of higher energy

to

one of laser energy.

For the

i n t e r n a l

jump

s t a t e 3 i s the given upstream s t a t e and s t a t e 4

is

the conjugate s t a t e w i t h

lower

energy,

t h a t

in

t he i n t e rn a l jump the

lower

l a y e r d e c r e a s e s ,

and the upper

This fol lows from the fa c t

l a y e r

increases

i t s thickness. Only states

3

and 4 s a t i s f y t h i s

requirement

.

Under cer ta in c o n d i t i o n s t h e r e i s no conjugate s t a t e t o t h e

given

s t a t e

(3)

as shown

i n Fig. 3-12b. The ph ys ic al im pli ca tio n

of

t h i s f a c t

i s

discussed

i n the

sequel .

9 7



Fig. 3-12: So lu ti on graph for the momentum equations

a )

General case

with

4

conjugate s t a t e s ,

b) C a s e with only 2 conjugate s ta tes , no

i n t e r n a l j ump possib le .

3.4.3.2 Sol uti on f o r Jumps with Low Velocit ies and

Low

Buoyancy

Yih 's so lu t ion , Eqs. (3-134) and (3-1351, g ive th e jump

conditions

as

a fun cti on of 4 parameters

hl

*

* 'P -

A P

i h i

hl

' h2 h2 Pa

f [

- F1

9 F2

Y

-

P

(3-136)

I n the problem of i n te r e s t , namely, di ffu se r discharge, both f r e e

surface Froude numbers and relative density differences

are

s m a l l .

Hence an asymptotic solution

is

attempted of the form

98



while f i n i t e (3-137)

so that the jump conditions are

a

function of 3 parameters

only

hl

hl

'

h2 h2

f [ - y 2 1

i

P

-i

(3-138)

Equations (3-134) and (3-135) are t ransformed in to l ine ar ly

dependent equations i n which t h is l im iti ng process can be made val idly .

For t h i s purpose

Ahl

=

h i - hl , Ah2 h i

-

hl (3-139)

are defin ed and t h e ex pr es sio ns f o r Ah2/Ahl obt ain ed from Eq.

(3-134) and (3-135)

are equated t o g ive

h i h i 2 h i h i 2

P a - &

2

2

lF1 F2

hl hl h2 h 2

pa

[

-

- +

1 ) - 2F1][

- - +

1 )

-

2 F2]

4(

(3-140)

A second equation

i s

obt ain ed by tak ing th e in ve rs e of Eq. (3-134) and

again forming t he expre ssion fo r

t h e Ah2/Ah1 from Eq. (3-1351, thus

Ah2/hhl

which i s then equated

with

2

2 F

[ - - + 1)

-

2 F1]

= -

i h i

hl hl

h i h i

-

- +

1 )

h i h i

- - + 1)

hl hl

h2 h 2

( - - - 1)

h2

hl

( - - -i 1)

-2

hl

(3-141

9 9



The

express ion (pa

-

&)/pa

u n i t y .

A

s imple equat ion f o r h i / h l can then be formed by su bs t i tu t i ng

t h e

value

o f h v 2 /h 2

i n Eq. (3-140) can b e approximated

t o

from

Eq

. (3-140) i n t o (3-141), namely

2 2

F i

2Fi

hi

[ (

-

hl -

h h

(3-142)

I I

-(-+l)

hl hl

The d e v i a t i o n in t ro d uc e d by s u b s t i t u t i n g t h e a s ym p t ot i c s o l u t i o n f o r

h i / h l

n e g l i g i b l e . F o r Ap/pa =

0.05

which i s co n s ide rab ly h ig h e r th an

d e n s i t y d i f f e re n c e s i n p r a c t i c a l d i f f u s e r a p p l i c a t i o n s (e.g.,

see

S e c t i o n 1 .1 ) t h e e r r o r

i s

about

1 .Thus Eq.

(3-142) d es cr ib es th e

Eq. (3-142) t o t he true s o l u t i o n

Eqs.

(3-134) an d (3-135) i s

dynamics of an i n t e r n a l h y d r a u l i c jump wi th low v e l o c i t i e s a n d l o w

d e n s i t y d i f f e r e n c e s .

Impor tan t in format ion regard ing t he existence

of

a h y d r a u l i c

jump

can be d er iv ed from Eq. (3-142).

one which i s c onjug ate t o i t s e l f , t h a t i s

v a lu e th e eq u a t io n g iv es th e c r i t i c a l condi t ion

A c r i t i c a l s t a t e i s

def ined

as

h;/hl

= 1 .

With th is

2

2

F1

+

F2 -

1

(3-143)

In ana lo gy to f r ee su r f a ce flow su p e rc r i t i ca l flow i s c h a r a c t e r i z e d

by

F1

+ F i > 1

and

s u b c r i t i c a l f l o w

by

3-144)

2 2

F1

+ F2

<

1

100

(3-145)



-

A hydrau l i c

jump

c an on ly o c c ur from s u p e r c r i t i c a l t o s u b c r i t i c a l

flow

as

the energy con ten t i s lower f o r t h e

l a t t e r

and energy i s

d i s s i p a t e d i n the jump.

That th e upstream con di t ion i n

the

problem

considered

is

i n d e e d s u p e r c r i t i c a l

is

demons t ra ted by su bs t i tu t in g

the values

of densimetric Froude numbers

Eqs.

(3-130), which

as

plume condi t io ns repr es en t

lower

bounds f o r buoyant

j e t s :

Condition

(3-144)

is

s a t i s f i e d .

c r i t i c a l

s ta te t o t h e s u b c r i t i c a l s t a t e p r ev a il in g i n t h e s t r a t i f i e d

counterflow region.

A jump w i l l occur which changes t h i s super-

For ce rt ai n combinat ions of Fl

, FZ

and hl/h2

,

however,

Eq.

(3-142) does

not

g i v e

a

conjugate su bc r i t i c a l downstream sec t io n

wi th

lower

energy (s ee Fig. 3-12b).

The phy sica l im pli ca t io n of t he non-existence of a s o l u t i o n is

a

hydrodynamical ly unstable condi t ion.

The excess energy

i s

d i s s i p a t e d

by tur bulen t d i f fu s io n over the whole reg ion , l ead ing t o re-en tra inment

of

already mixed

water,

i n t o t he j e t reg ion ,

as

de pic ted i n Fig. 3-13.

A

ver t ica l

region i s thus formed i n the

near

f i e l d zone.

I n

steady

s t a t e

t h i s d i f f u s i o n a nd

re-entrainment

process

will

act

t o

such

a

degree th at ou t s ide th e mixing reg ion t he

c r i t i c a l

relat ion (3-143)

i s obta ined. I n o th er

w o r d s ,

through d ecrea se i n ki ne t i c energy and

i n c r e a s e i n buoyancy t h e s u p e r c r i t i c a l s ta te , a f te r

j e t

impingement,

i s

t rans fo rm ed i n to t h e l im i t i n g

case

of

a

s u b c r i t i c a l

s t a t e , a

c r i t i c a l

s t a t e .

s ub se qu en t s t r a t i f i e d c o u n te r f

law

region which is cha rac ter iz ed by

s u b c r i t i c a l flow

as

t h e

next

s e c t i o n w i l l show.

T h e c r i t i c a l

s ta te

is then a s t a r t i n g c o n d it i o n f o r t h e

101



. C r i t i c a l Section

Fig. 3-13: Non-existence of an i n t e r n a l hy dr a u li c jump:

Turbulent di ff us io n and re-entrainment

.

3.4.4 S tr a t i f i e d Counterflow Region

3.4.4.1 Approximations and Governing Equ atio ns

The governing equations for s lowly va ry ing s t r a t i f i e d coun te r-

flow i n t h e f a r - f i e l d , w i t h h e a t d i s s i p a t i o n

t o

the atmosphere, are

developed.

f low with

a

f a i r l y d i s t i n c t d e n s i t y change Ap =

B

AT ac ros s the

Figure 3-14 defines the problem of

a

two-layered f luid

in t e r face de f i ned

as

t h e e l e v a t i o n o f t h e z e ro h o r i z o n t a l v e l o c i t y

poin t .

The flow is predominant ly hor izon ta l , th e

thickness

of t he

upper and

lower

layer is hl and

h2 ,

r e s p e c t i v e l y .

Under these

circumstances the

governing equat ions (3-3) t o

(3-6)

can

b e s i m p l i f i e d , n e g l e c t i n g

ver t i ca l

acce l e ra t i ons and

l o n g i t u d in a l t u r b u l e n t t r a n s p o r t

terms,

t o

au/ax + a w i a z - o

(3-146)

102



Z

HEAT LOSS

'FACE

VELOC

IT Y

DENSI T Y

x

Fig.

3-14

: Stra tifi ed flow definiti ons.

(3-147)

0 = - *

pg

(3- 148)

az

a T

a

a T

+

W - z K Z z )

a T

a z

-

ax

(3-149)

where

E

=

Vertical eddy dif fu sion coef f ic ient for momentum

2

KZ * Vertical eddy diffu sion co eff ici en t for

mass

or

heat

and the turbulent

transfer

term

have been written i n diffus ion

analogy as

10

-



The fu r th e r an a ly s i s makes u se o f th e d i s t i n c t l y l aye red flow s t ru c tu r e :

The flow-field

i s

divided i nt o two l ay er s which are coupled through

kin em at i c , dynamic and hea t f l ux con di t ions a t t h e i n t e r f a c e .

S i m i l a r i t y p r o f i l e s f o r h o r i z o n t a l v e l o c i t y and t em pe ra tu re

a r e

defined

fo r each l ay e r so tha t t h e eq u a ti o ns c an b e i n t e g r a t e d v e r t i c a l l y .

a )

Kinematic boundary conditions

are

s p e c i f i e d as:

a(hl + h2)

= w a t z = hl + h2

ax S

s u r f a c e U

S

a t

z = h (3-151)

i

1

= w

h2

ulax

n t e r f a c e

w = o

a t 2 - 0

b

ottom

The in t e r fa ce condi t ion (3 -15lb)assumes th a t t here is

no

mean flow .

ac r o s s th e in t e r f ac e , h en ce t h e volume f l u x i n each l ay e r

i s

constan t .

This assumption does not al low for entrainment.

o f t h e en tra inment p rocess as discussed earlier are t h e e x i s t e n c e o f a

zone of f l u i d f l ow

wi th

h i g h t u r b u l e n t i n t e n s i t y

as

compared t o t he

surrounding f l ui d . Turbulence a t th e zone boundary le ad s t o incorpora-

The s a l i e n t f e a t u r e s

t i o n o f a mb ie nt f l u i d i n t o t h e

act ive

zone.

No

such

ac t ive

zo n e ex i s t s

i n t h e

case

of s t r a t i f ie d counter flow: bo th laye rs move

a s

d e n s i t y

c u r r e n t s and

have

ap p ro x ima te ly eq u a l tu rb u len t ch a rac te r i s t i c s .

Furthermore, even i f such an

act ive

l ay e r would p r ev a i l , t h e

existence

of

a

s t a b l e de n s it y s t r a t i f i c a t i o n g r e a t l y re du ce s t h e e nt ra in me nt a t

th e boundary. Experimental s tu di e s by El li so n and Turner (1959) show

that for bulk Richardson numbers

>

0 . 8 5

Ri -

104

(3-152)



ver t ica l en t ra inment

p r a c t i c a l l y

ceases, where Ri

i s

defined and

r e l a t e d t o

a

F’roude number

F

by

(3-153)

with

h =

depth of turbul ent lay er

V = layer veloci ty

The s t ra t i f i ed counter flow regime of in ter es t i n th is s tud y

i s

s u b c r i t i c a l

as

w i l l be shown below, hence c on di tion (3-145)

i s

app lic abl e and simultaneously condit ion (3 ~1 52 )

s

s a t i s f i e d .

Zero entrainment, however, does not ru l e out t ra ns po rt of mass

by f luc tua t in g ve loc i t i e s ac ross the in t e r f ace , t h a t

i s ,

i n t e r f a c i a l

mixing.

b)

Dynamic boundary conditions

are

given as:

-

a U

fs

= pa €z

aZ

=

Ti = pa €2 aZ

surface:

’} z = h l + h 2 (3-154)

P ‘ O

z

* h2

U

i n t e r a c e:

bottom: z = o

where

‘ tS , T~ and

T~

are shear

stresses

which

are

later r e l a t e d

t o t he mean f low qua nti t ie s.

c)

Heat

fl ux conditions : The fl ux of heat,

qH

,

from

t he

S

f re e su rfac e t o th e atmosphere, which

i s

induced by the e levated

water

tempera ture, can be considered

in

a sim ple concept by (Edinger and Geyer

(1965) ) :

-

105



i n

which

% =

-

p

c

K ( T ~ Te)

P

s

(3-155)

c

- s p e c i f i c h e a t

P

K

= heat exchange coeff ic ient

=

water surface temperature

-

T(z = h l + h 2)

TS

= equilibrium temperature

Te

Both fac to r s Te and K

are

dependent on meteorological parameters,

which determine the t r ans fer processes across the water sur face .

i s def ined a s t h e

water

temperature a t which there would be no net heat

f lux acr os s the su r face . In genera l , Te i s not equal to the ambient

temperature T

Te

(without heat in pu t) , only on

a

long

term

average.

a

The remaining heat flux cond i t ions

are:

z =

h2

T

= p c K -

p

z

az

qHi

i n t e r f ace :

(3-156)

z = o

T

bottom:

*l$)

- cp

KZ

aZ

d)

Both velocity and temperature are approximated

to

be

uniform i n each l ay er and are averaged by

layer

:

u1

p1

*1

-

hl

1

-

hl

n -

hl*2

I

P

dz

u dz

h2

(3-157)

CONT

'D.

106



h2

1

u d z

2

u 2 = - = -

ower l a y e r :

h 2 h 2 0

(3-15 7)

p2

1

-

h2

L

I P dz

0

Using th es e pr o f il e assumptions and boundary con diti ons , Eqs. (3-146)

t o (3-149) can be transformed in to ordinary d i f f e r e n t i a l e q u a t i o n s for

each la ye r. The hy dro st ati c equat ion (3-148)

is

i n t eg ra t ed to

P P 1 g(hl

+

h2 -

Z )

(3-158)

which can be su bs ti tu te d in to t he momentum equati on (3-147) t o gi ve

(3-159)

a aU

+ az

(Ez

az

a l s o

making

use

of th e co nt in ui ty equ atio n (3-146). Equation (3-159)

is i n t e gr a t ed i n t h e z dire ct io n over th e upper layer .

(3-160)

The l e f t hand s id e i s in tegra te d , us ing Leibniz 's rule on t he f i r s t

i n t e g r a l , t o

+

h2) ah2

+ h

2 2 2 + usws - uiwi

ax

+ u i

x

~ Z - U

a

8

L

10



and a f t e r u s e

of

th e k inematic condi t ions (3-151) only th e f i r s t t e r m ,

a (u h ) , rema ins. F i n a l l y , Eq. (3-160) becomes, usi ng q

=

u h

x

1 1 1 11

(3-161)

2 dPl

P 1

dhl

T

( ax

dx

pa

hl

h

dx

'a dx Pa

+ - )

-

1 dhl

- -

-

- -

- - -

The hea t c onservat ion equat ion for t he upper l ay er i s i n t e g r a t e d , a f t e r

s u b s t i t u t i o n of t h e co n t i n u i t y eq u at i on , t o

hl+h,

aT

-

l+h2

dz 5

(Kz

)

az

uT dz + 1

h2

ax

h2 h2

'a 'p

Again making use of th e Leibmlz r u l e and t h e kinematic

t o

In an analogous

manner

and

equations

for

the lower

dP2

+ -

dx

d p l

dhl

dx l p 1

d x

- Tb>

(3-162)

L

cond i t ions l eads

l a y e r

are

found

(3-163)

as

(3-164)

(3-165)



Eq ua tion s (3-161), (3-163), (3-164) an d (3-165) are t h e g e n e ra l

equations des cr i bi ng the motion and temperature d is tr ib ut io n of non-

e n t r a i n i n g s t r a t i f i e d f lo w w it h h e a t d i s s i p a t i o n t o t h e a tm os ph ere .

Dependent variables

are

th e mean f low pr op er t i es

which re qu ir e statement of boundary c on dit io ns

a t some

p o s i t i o n

hl,h2 and

T1,

T2

,

x

.

‘Ii ,

rb and q

can

be re la ted to t h e mean f low proper t ies by empirical

re1at ons.

Hi

The so lu t ion of t h i s set of equations i s not r ea d i l y ach ieved .

Thus i n t he fo l lowing sec t io n the equat ions are s imp l i f i ed by ap p ro p r i a t e

sca l ing procedures f o r the f low area i n t h e d i f f u s e r v i c i n i t y .

3.4.4.2)

Simp lified Equations, Neglecting Surface

Heat

Loss

a n d I n t e r f a c i a l

Mixing

I t

can be shown

that

under ty p i ca l d i f fus er des ign condi-

t i o n s s u r f a c e

heat

l o s s an d in t e r f a c i a l m ix in g

can

be neglected

f o r t h e

s t r a t i f i e d

f lo w re gi on i n t h e d i f f u s e r v i c i n i t y ,

as t h e

time

scale f o r

these p rocesses

is

considerab ly la rger than

that

fo r th e co nv ec tiv e

tr an sp or t. For t h i s purpose, Equation (3-163)

w i l l

be sc ale d and

typ ica l p ro to type magnitudes fo r the sc a l in g parameters are i n s e r t e d

to compare the

r e l a t ive

importance of

the terms.

Dimen sion le ss v a r i ab le s (with a s t e r i sk s ) are defined by

x*

=

x/L

h* = h,/H

u*

=

u p

AT*

=

AT/ATo

(3-166)



i n which

L

=

channel leng th

H = water depth

U = t y p i c a l i n du ce d v e l o c i t y

ATo =

t y p ica l su r f ace t emp era tu r e d i f f e r en ce

The

i n t e r f a c i a l h e at f l u x

term i s

s u b s t i t u t e d from Eq. (3-156a)

and approximated as

-..

T1

- T2

qHi cpKz H

(3-167)

Su b s t i t u t in g th e v a r i ab le s (3-166) in to th e h e a t t r an sp o r t eq ua t io n

one obtain s, d iv idi ng by

U A T ~ / L

u*-

dAT* -.. - K

- )AT* - ( 2 )AT*

dx*

Pa

cp H u

Ch arac te r i s t i c mag ni tu des fo r th e sca l in g

parameters

i n p rototy pe

condi t ions fo r

a

t h e rma l d i f fu se r

are:

(3-168)

L =

1000 f t . ( I n

Chap te r 4

t h e chan ne l l en g th

L i s

shown t o b e r e l a t e d t o t h e c h a r a c t e r i s t i c h o r i z o n t a l

d i f fuser d imension ,

i t s

h a l f l e n g t h

LD as

L

$1'

H

- 30 f t

u = 0.1 f t

ATs = S°F

2

K

= 150

BTU/OF,

f t , day

= f t2 / sec . Th e v e r t i c a l ed d y d i f fu s io n i s s t rongly

Koh and Fan

KZ

i n h i b i t e d by t h e d e n s it y s t r a t i f i c a t i o n .

(1970) compiled available

data

and f i t t e d an emp ir i ca l

110



r e l a t i o n which

is

an i n v e r se fu n c t io n of th e d en s i ty

g rad ien t . The d en s i ty g r ad ien t a t t h e i n t e r f a c e

w a s

approximated as i n

Eq.

(3-167), but is i n f a c t much

higher .

Further:

pa 'p - 62 BTIJ/ft3

With the se va lues t h e express ions i n paren theses are ca lcu la t ed to b e

of the magni tude -0.01 fo r t he sur fa ce hea t lo s s and ~0.03 o r i n t e r -

f a c i a l mixing. Consequently , th es e tr an sf er processes become only

imp o r tan t f o r l a rg e r d i s t an ces L

and can be neg lec ted i n th e channel

problem considered. Thus, th e he at equa tio n

i s

s i m p l i f i e d

t o

dT1

- 5 0

dx

and by similar arguments it i s found

dT2

- = o

dx

(3-169)

(3-170)

Hence t h e problem reduces t o t h e w e l l - k n o w n e qu at io ns f o r t h e s t r a t i f i e d

f lo w wi th co n s tan t d en s i t i e s , f i r s t d e riv ed

by

Schijf and Schonfeld

(1953) :

(3-171)

hl dh2

=i

1

c'nx dx pa g hl

d x =

+ - ) +

hl

with

111

(3-172)



With the de fi ni t i on of f r e e sur fac e Froude numbers (3-133) t h e s e

equations ca n be transformed t o

-

hl

dx

*2

1 - F2

+ q

h 2

*b

‘i

Pa

g h2

Pa

g

- -P

’a

*2 *2

-

F Y F1 F 2

(3-173)

Adding up t he se equat ion s g ives t he change i n t o t a l depth

(3-175)

Considering Fig. 3-15 t h e t o t a l d ep th i s w r i t t e n a s

= H + n

hl h2

where

H i s

t h e s t i l l

water

depth and

n

th e dis tu rbanc e. From

hydros ta t i c cons idera t ions and a l s o

by

i n s p ec t i o n

of

Eqs.

(3-175)

and

(3-174)

(3-176)

11



TERFACE

Fig.

3-15:

Depth re la t ionships in s t r a t i f i e d flow

Thus for

Ap/p

<< 1, F*2 << 1

the change i n su rfa ce elevat ion is small

a

compared to th e in te r fa c ia l depth change dh2/dx . Consequently, as

a f i r s t approximation dh2/dx can be cal cul ate d by assuming

hl

+ h2

= const .

= H

(3-1 77)

The solut ion fo r hl(x) , h2(x)

evaluate

the surface disturbance T)(x) . Knowledge of 0 which

i s

caused by f r i c t io na l forces

i s

important

in the

analy sis of d i f fuser

discharge s which in tro duc e h or iz on ta l mmentum which is balanced by

f r i c t i o n a l f o r ce s

.

thus obtained can then be used to

Then s u b s t i tu t i n g Eq. (3-177) i n t o Eq. (3-173) Which

i s

subtracted from

Eq.

(3-174) one ob ta in s

-

dx

(3-178)

1

-

Pi - P;

113



where F1,F2

are

densimetric Froude numbers and A p /p

,

F*2 have been

neglected with respect to un i ty . A t th i s po in t bo ttom and in te r fac ia l

stresses

a r e expressed i n terms of mean flow

parameters

as

(3-179

o r

L

(3-180)

2

f i

Ti

=

- p . ( - - -

8 a

hl

h2

S u b s t it u t i o n i n t o

Eq.

(3-178) gives

dx 1

-

Pi - F;

This equation i s put i n t o non-dimensional form by

X, H1,H2

=

(x, hl,h2)/H (3-182)

Constant densimetric Froude numbers based on the t o t a l depth

are

de-

f i n ed

as

2 2 3

F l H = Fl H1

(3-183)

2

2 3

F2H

=

F2

H2

114



A

f low r a t i o

is

given by

Q T

Q = -

42

Hence

For counterflow, always

(3-184)

(3-185)

(3-186)

Using these definit ions the counterflow equation

is

w r i t t e n

and can be integrated in inverted form

x2-x1

-

8

2

F2H

-

.\ .l

(3-188)

Equation (3-188) has been giv en i n

a

s l i gh t ly d i f fe r en t form by

R ig te r

(1970).

(q2)

=

- 1 and consists of 3 branches which are div ided by c r i t i ca l

sect ion s, where the slope of th e in ter fac e goes to in fi n it y. The

c r i t i c a l r el a ti o n

is

obtained by

s e t t i n g

the denominator

in

Eq. (3-187)

e qual t o

0,

The general form of th e sol ut io n

is

given in Fig. 3-16 for sign

(3-189)

3

-

H~

( I - H ~ ) ~

o

iH[Q H2 i - ( 1 - H 2 ) 1

115



a

I

i

I

I

I

1

i

,

I

i

U

2 0 1

I

x

E

-I

u

ll

00

a3

u

w

W

u

..

\o

4

i

m



The so lu t ion t o th i s c r i t i c a l equation i s plot ted i n Fig . 3-17 a s

a function of

F2H

and

1Q1

. Subcr i t ical f low is only poss ib le i f

(3-190)

In

t h i s

case

there exist two

c r i t i c a l

sec t ion Hi and €I;

A

c r i t i c a l sec t ion de l inea tes the t r an s i t io n between s ub cr i t i c a l and super-

C C

c r i t i c a l flow

states

and can be explaine d by energy consi der ati ons .

In

general , a c r i t i c a l sec t ion is give n by abrupt changes i n flo w geometry.

The channel ending In

a

l a r g e r e s e r v o i r (see Fig. 3-21 i s such

a

s i t u a t i o n . I n ad d it i on ,

a

c r i t i c a l se cti on can form th e end of a l oc al

mixing region

in

case of -a n uns tab le in te rna l hydrau l ic jump fo rce , as

discu ssed i n Section 3.4.3.2.

L* between two c r i t i c a l s ec t i o n s i s found by evaluating the

in tegra l

The length of the su bc r i t i ca l f low zone

C

(3-188) with

limits

H; and HZ . This in tegrat ion w a s car r ied ou t

C C

by Rigte r (1970) numerically f or th e parameter range.

Analyt ical

s o l u t i o n s f o r s p ec i a l cases of

Q

are

given i n S ec tio n 3.4.4.4.

3.4.4.3 Head Loss i n S t r a t i f i e d

Flow

The change i n t o t a l dept h H-tq

relates

t o t he

head

l o s s

incurr ed by th e motion aga ins t shea r fo rces a t the bottom and i nte rfa ce.

As commented in the preceding section

t h e

depth change

i s

ca lcu la ted by

sol vi ng Eq. (3-175) with the flow fi e l d obtained through the constant

de pth approxim ation (3-177). One can

write

11

(3-19 1 )



U

01

I

118



a n d s u b s t i t u t i n g

Eq.

(3-175) and t h e inver ted Eq. (3-174)

dn

- =

dh2

F;2

+

- 1

*2

1

+ F1

h 2

h2

+ T i ( - -

hl

l +

i

( -

T -

*2

h2

1

-Tb

h2

hl

(3-192)

Div id ing bo th s id es by

Ap/p,

and neglect ing

P I 2 ,

F;2

<< 1

i n t h e

denominator yields the equation for t h e normalized dep th change

as

d

-

dh2

Using t h e sh ea r

b e fo re , w i th

rl

i n a d d i t i o n , on e ob t ai n s

1 1

T ( -

+ - )

"b

h2

hl

h2

stress de f i n i t i on s and non-dimensibnal izing

*

= n / H

-

2 2

1 )

H2

2

-tF2H

(3-193)

(3-194)

(3 19 5 )

1 3 I 3

1

+ 1 ] + ) + 1

-H2

H2 2 H2

H2

[Q 7)

1 1

119



This equat ion can be i n tegr a ted between the same l i m i t s as Eq. (3-188)

t o give th e head change

and i n par t i cu la r the head change

9 can be found. In te gr a-ver the c r i t i c a l length

t io n h as t o be n umer ical , except i n s p e c i a l

cases,

as

d is cu ss ed i n t h e

fo l lowing sec t ion .

*

l

Lg ,

A P I P ,

1

APIPa

3.4.4.4 Special

Cases

Mrec t in te gr at io ns of t he in te rf ac e Eq. (3-188)

and

t h e

head l o s s Eq. (3-195) can b e g iven f o r sp ec ia l v alues of

Q

.

A l l

t h e s e cases have st ron g sign if ic an ce i n buoyant discharge problems.

a) Q 5 O : sta gna nt s urf ace wedge

Equation (3-188) reduce s t o

(3-196)

I n t e g r a t i o n g i v e s

(Bata

(1957)

)

3

i

[A(1+Aj2 -

F&]H2

+ A[(I+A)

i n which

fi

fo

A = -

(3-197)

120



The i n tr u si o n lengt hof the wedge, Lss , upstream from a c r i t i c a l

s e c t i o n is found

by

taking

t h e

l i m i t s

( 3-19

8 )

There is zero depth change over the leng th of the in tr us io n

as

is

ev i d en t from Eq. (3-195)

= o

H*

q

(3-199)

b)

P

:

stagnant bottom wedge

In t h i s case F

= 0

as

w e l l , so t h a t i t

is

advantageous

?I

t o r e w r i t e Eq. (3-188) in terms of F

:

t i

(3-200)

v i t h Q

-t

QD

(3-201)

or Integrated

(3-202)

8

2

2 -

x1

= -

p$ fi

121



*

The

length of th e wedge intru sio n,

Ls

s ec t i o n

i s

found by taking the l i m i t s

, upstream from

c r i t i c a l

b

H

(L* ) =

1- F Z 3

'

I

(3-203)

This expres s ion fo r the in t rus ion leng th w a s given by Schijf and

Schonfeld (1953).

i s

a f t e r similar modification of Eq. (3-195)

The t o t a l head change over t he i n t r us i on length

2 ) dH2 (3-204)

1

h p l p O

I

(-

'b

(1-H2) 3

F%

(1-H2) 3

and

(3-205)

c) Q - -1 : equal counterflaw

I n t h i s case Eq. (3-188)

i s

given by

(F

(3-206)

1 2 2



which

can be integrated to

4

3 3

2

x2

-

x1 =

1-

$ (1-H2) + ( 1 - H 2 ) - ( 1 ~ ~ )

f o

FH

i n which

2

(a+(1-H 1

2

2a

6a >FH)Rn (a+l-€I2) 2 - 3a(l-H2)

1 1

+ ( - - -

2 3 3

-

(a3

-

F ~ )

n

[a

+ (I-H~)

I

+

2 4

2 4 3 2

Fa +a H

a + 3a +

a

- -

(

43 a3fi

(3-20

7)

The l eng th of the sub cr i t i c a l f l av sec t ion ,

Lc

, i s found by sub st it u-

t ing the so lu t ions

of

t h e c r i t i c a l e q u a t i o n

€I2 ( 1 - €I2 ) 3 - P2(H3 + ( 1 -

H2

13) -

0

C C a

2c C

(3-208)

as

limits

i n t o

Eq. (3-207).

change is also possib le , but i s omit ted here , as

i t

is not needed in

the f u r t h e r a n a l y s i s .

Inspection of

E q .

(3-208) which i s p lo t t ed as

the curve 1Q1

=

1

i n Fig.

3-16 shaws t h a t

equal

counterflow can only

Closed

form

i n t e g r a t i o n

of

the depth

1 2 3



e x is t i f

FH 5 0.25

For the l im i t i ng

value

FH = 0.25 i t

i s

fo u n d th a t

HI

- H

C 2C

L c = L

3.5

Matching of Solutions

I n the p reced ing se c t io n the f our f low reg ions which c

discerned i n th e f low f i e l d induced by

a

multiport d i f fu s e r disc

were analyzed. Governing equ ati ons

w e r e

deve lope d which took ac

of th e di s t i n c t hydrodynamic pro pe rt i es of each region and solul

were given o r ou t l ined . The so l u t i on s f o r each reg ion can be mi

t o provide an o v e r a l l p r ed ic t i o n of th e d i f fu se r in du ced fl aw f :

In

th is way important c r i t e r i a regard ing t he near - f ie ld s a b i l i l

d iffuser discharge

w i l l

be deduced. Furthermore, i n case of i n s t i

ties

w i t h r e s u l t i n g j e t

re-entrainment,predictions

o n d i f fu se r

performance w i l l be developed.

Th i s sy n th es i s

i s

done i n Se c t io n 3.6 for t h e case of

discharges with no

net

h o r i z o n t a l momentum

(symmetric

f low f ielc

i n

Section

3.7 f o r t he more complex case of

discharges with hor:

mamenturn.

3.5.1 Governing Non-dimensional Par am ete rs

By dimensional ana lys is of th e problem va ri ab le s as dc

i n F ig .

3-2

and by in sp ec ti on of th e governing equ ati ons and thc

boundary condit ions der ive d f o r each f law regi on the steady-sta

and tempera ture f i e l d i s w r i t t e n as



and

ATO

S = -

AT

(3-211)

is

t h e d i lu t ion

a t

any point. This formulat ion

assumes

f u l l y

turbulent j e t flaw, hence independence of the j e t Reynolds number

of t h e s l o t

U 2B

IR = - +

1

(3-212)

where v = kinemat ic v iscosi ty

and

neglects he at lo s s to the atmosphere. Furthermore, the bottom

shear

stress

c o e f f ic i e n t i n t h e

flow

away region

i s

assumed

t o

be i n

the turbulent flow range of the form

(3-213)

(White-Colebrook re la ti o n) i n which

=

lower

l a y e r

Reynolds

number

2 Rn

I R 2 -

\ =

hydraul ic radius

ks

= abs olute bottom roughness

and the in t e r f ac i a l shea r stress coef f i c i en t i s related by

E q .

(3-19%)

fi

= A fo (3-214)

The

r a t i o

f l a w condit ions, as

i s shown la te r .

i t is found th a t f o and

L/H

can be combined t o a f a r f i e l d

A can be assmed constant over a wide range of pra ct ic al

Through in sp ec ti on of

4 .

(3-188)

parameter

Q =

f o L/H (3-215)

125



So

tha t , f in al ly th e problem of the diff us er induced

flow

and

temperature field (without ambient cross flow) i s defined by these

dimensionless parameters :

Near-field parameters

:

Fs =

s l o t

densimetric Froude

number

8 = v e r t i c a l a ng le of discharge

0

H/B

=

relat ive

water depth

Far-f i e l d parameter: H

and Eq. (3-210) i s reduced to:

(3-216)

The discuss ion of th eo re t i ca l and experimental re su l t s

i s

given

f o r

the se parameters. The range of i n t e r e s t

i s

Fs =

10

t o

1,000

8 =

90° (Section 3.6)

<

90° (Section 3.7)

0

H/B

=

50 t o 5,000

=

0 .1 t o 1 .0

which conforms t o pra ct ic a l d i f fu ser appl i ca t io ns , as i s shown i n

examples given i n C hapter

5.

In the p resen ta t ion

of

resul ts, emphasis w i l l be laid on

the gross proper t ies of d i f fus e r d i scharges . I n pa r t i c u la r , t he

average non-dimensional su rf ac e temperature

r ise ,

or inver se ly the

average sur face d i lu t ion

are

chosen

as

the

main

des cri pt i ve parameters

of the temperature f i el d. The equivale nt s l o t concept

(F H / B )

i s

S’

126



used throughout i n describ ing t he multiport d if fu se r mechanics, based

on the discuss ion i n Chapter 2.

3.6 Theoret ical Predic tions: Diffusers With

No N e t

Horizontal Momentum

Vertical discharge o r d i scharge wi th nozz les po in t ing i n a l t e rna-

ting directions have no net horizontal momentum and thus w i l l produce

a f low fi el d which is symmetrical t o t h e d i f f u s e r axis .

3.6.1 The Near-Fie ld Zone

The numerical int eg rat ion of th e buoyant

j e t

equations

(3-69) to (3-76) with i n i t i a l con diti ons (3-78)

is

ca r r i ed o ut f o r

Bo

-

90

and various values of

Fs .

proper t i es

as a

func t ion

of

v e r t i c a l d i s t a n c e z/B . As an example

t h e c e n t e r l i ne d i l u t i o n

The eff ec t of the presence

of

t h e f r e e s u r f a c e is considered by solving

The in teg ra t ion y ie ld s

j e t

Sc

along the j e t pa th

was

shown i n Fig. 3-6.

the impingement equation (3-109). In the so lu t ion the energy los s

co e f f i c i en t

5 is

taken

as

0.2 throughout.

The

solut ion gives values

for th e thicknes s of th e impingement lay er

The densimetric Froude number

F1

ment i s calculated from Eq.

(3-126) and

a lso

i n c l u d ed i n

Fig.

3-18.

hi/H

as

shown

i n Fig.

3-18.

of t he spreading lay er af t e r impinge-

The graph indi ca tes th e asymptotic values fo r th e plume in t he high

H/B,

low Fs rang e, namely hi/H

=

0.149 and F1

=

2.63

(Eqs.

(3-120) and

(3-129)).

hi/H

-

0.167

(Eq. (3-115)) is approached, although

a t

low H/B t h e r e

is a

s l i g h t

For the less buoyant range (large

FS )

th e va lue

-

1 / 6 obt ain ed from t h e sol u ti o n of t h e momentum

j e t

d evi a ti o n t o t h e f ac t t h a t

Eq.

(3-115) assumes ne gl ig ib le s l o t width B

compared t o th e de pth H

.

For low

H/B

, high Fs th e spreading lay er

.-

1 2 7

-



Fig. 3-18:

Thfckness hi/H and Densimetric Froude Number

F1

of the Surface Impingement Layer, Vertical Discharge

128



Fro ude number F1 becomes ve ry l a r g e .

With given impingement la ye r thickness, th e average su rfa ce

is by vir tu e of th e uniform mixing i n

sS

d i l u t i o n

Ss is

ca l cu l a t ed .

the impingement process equal to the average

j e t

d i l u t i o n

a t

the lower

edge of th e impingement re gi on

-

Ss

=

S ( Z H

- hi)

Values of Ss are shown i n Fig . 3-19.

(3-217)

With th e g iven cond i t ions p r io r to the in te rn a l hydrau l ic jump

t h e jump eq uat ion (3-142) is eva lua ted g iv ing so lu t ions fo r the

conjugate depth hi/H

,

as

in di ca te d i n Fig. 3-19. For high

H/B

,

low FS

h;/H

0.5.

Proceeding toward th e low H / B , high FS range h;/H

increases

u n t i l

so lu ti on t o Eq. (3-142) po ss ib le (see Fig. 3-12b). This in di ca te s

t h e absence of a s t ab le su bc r i t i ca l con juga te cond i tion and re -ent ra in -

ment i n to th e

j e t

region w i l l occur, forming a l o c a l mixing zone. The

t r a n s i t i o n

i s

described by

the conjugate depth approaches one-half of t he t o t a l depth ,

hi/H

=

1 - hl/H , beyond which there is no p o s i t i v e

FH

0.20

(3-218)

where FH i s th e average of (Eq. (3-183)) and

character izes the dynamic

characteristics of

th e counterflow system.

H/B

t can be deduced from the

FS'

I n

terms

of discharge parameters

g raph tha t

4/3

H/B

=

1.84 Fs

(3-219)

g ives the c r i t e r i on between s ta b l e and uns tab le near - f ie l d cond i t ions .

(See Fig. 3-20.) In consequence, di lu ti on pr ed ic ti on s

Ss

f o r t h e

129



- - - -

- e

I I

I

1 1 1 1

I I I

I

I I L

5 50 100 500 1000

FS

10

Q=W'

Fi g. 3-19: Average Surface D i l u t i o n

S

Accounting f o r Thickness

S

1

of Impingement Layer and Conjugate Depth h

/H

f o r

I n t e r n a l H y d r a u l i c Jump.

1

C r i t e r i o n L i n e D e l i n e a te s

St ab le and Unsta ble Near-Field Zone

130



unstable parameter domain

(drawn a s

dashed l i n e s i n Fig. 3-19) which

are obtained as direct buoyant j e t so lu t ions , on ly accoun t ing fo r

th e su rf ac e impingement,

are

i n f a c t n o t ap p li c abl e . R a th e r, d i l u t i o n

-

i n th e uns tab le parameter domain

a r e

dependent on t h e dynamic co nd it io ns

i n t h e f a r f i e l d zone.

3.6.2 The Far- Fiel d Zone

3.6.2.1 In t era ct i on wit h near- f ield

The flow away from t h e near f i e l d zone forms a s u b c r i t i c a l

s t r a t i f i e d cou nt er fl ow r egi on . The p o s s i b i l i t i e s of i n t e r ac t i o n w i t h

the near - f ie ld zone

are

indic ated i n Fig. 3-20.

A) S t ab l e n ear -f ie l d: I n d i r ec t ana lo gy t o f r e e s u r f ace h y d r au l i c

jumps, two con dit ion s

are

poss ib le depend ing on th e backwater e f fe c t s i n

the f a r - f i e ld which

are

determined by the co n t ro l s ec t ion ( c r i t i c a l flow)

of t h e channel end.

a )

A normal i n t e r n a l jump i s give n when con ju ga te

d ep th h i i s la rg er than t h e depth hl determined from

th e coun ter flow so lu t io n .

8

b)

A submerged internal jump. is g iv en i n t h e o t h e r case.

Some re-entrainment of al re ad y mixed water i n t o t h e j e t r e g i o n

w i l l

o ccu r u n t i l a co n d i t i o n is es tab l is h ed i n s t e ad y - s t a t e

such that the increased buoyancy in t h e f a r - f i e l d w i l l decrease

t h e d ep th hl u n t i l h i = hl .

B )

Unstable near-f ield:

As

no s u b c r i t i c a l con j u ga t e

s t a t e

e x i s t s

8 S

i n t h i s case, lo ca l mixing and re-entrainment in to th e j e t r eg ion takes

p l a c e t o a degree such that a c r i t i c a l s ec t i o n i s es t ab l i s h ed a t t h e end

-

of t h e l o ca l mixing zone as t h e l i m i t i n g

case

of a s u b c r i t i c a l

flow

131

_ _



A ) STABLE NEAR-FIELD

1.

Normal Internal Jump

c

v

I

2.

Submerged Internal Jump

CHANEL

END

r

B

)

UNSTABLE NEAR-FIELD

Local Mixing and Reentrainment

Fig .

3-20:

I n t e r ac t i o n

of

Near-Field and Far-F ield Zones

(Ver t ic al Dif fuser Discharge)

i32



. -

condit ion. S t ra t i f i e d counterf low

w i l l

then occur over the length,

Lc , bounded by two c r i t i c a l sec t ions .

i s est imated as

one,

The length of th e l o c a l mixing

Lm 9

2

2.5

H

(3-2 20)

Lm

from experiments made i n t h i s s tudy which

are

i n good agreement with

Iamandi and Rouse's (1969) observations on c i r c u l a t i o n p a t t e r n s

induced by nonbuoyant j e t s i n

narrow

channels. Consideration of

Lm

may be s i gn i f i c an t i n the so lu t ion f o r shor t channel s, f o r l ong

channels, however, i t may be neglected i n comparison to

Lc

.

For t h e v e r t i c a l d i f f u s e r d i s ch a rg e t h e e f f e c t of t h e f a r - f i e l d

i s considered by assuming an equal counterflow

system.

approximation for

large

dil ut i ons , fo r which the entrainment f low

i s

This i s

a

good

about e qu al t o t h e flow-away of mixed water.

f o r t h e p o s i t i o n of the in te rf ac e, Eq. (3-207), i s appl icable . The

v a l i d i t y of Eq. (3-207)

i s

r e s t r i c t e d t o s u b c r i t i c a l f lo w,

FH

5 0.25 .

For the case of s t ab le near - f ie ld

The closed form solu t ion

FH

i s

known

from the solut ion of

3 3

the near - f ie ld

(FH

Fh =

F1

h

/H

)

and the depth of the i n t er fa ce

hl a t t he d i s t ance L from the channel end i s of in t e r e s t . For the

case of an unstable near - f ie ld the value of FH = F

length of t h e s u b c r i t i c a l s e c t i o n ,

Lc

,

i s

equa l t o

i s of in t e r es t . The overa l l d i lu t io n Ss (outs ide the local mixing

zone) i s then determined from

8

fo r which t he

HC

L (neglect ing L)

133



as

(3-222)

The f a r - f i e l d e f f e c t

( f r i c t io n and channe l l eng th )

i s

given by th e

s ingle parameter

@

=

f o

L/H i f

A

=

f i /fo i s assumed constant.

3.6.2.2 I n t e r f a c i a l f r i c t i o n f a c t o r

While t h e boundary f r i c t i o n f a c t o r fo can u s u a l l y b e

es timated wi th good re l i a b i l i t y as

a

f u n c t i o n

of

Reynolds number and

boundary roughness, th er e

is

a s c a r c i t y of data on t he magnitude of

t he i n t e r f a c i a l f r i c t i o n c o e f f i c i e n t f i n t u r b u le n t fl ow s, i n

p a r t i c u l a r , i n t h e high Reynolds number range. In ad di ti on , inve sti ga -

t i o n s i n d i c a t e a certain dependence on the densimetric Froude numbers

(F1,

F2)

of th e la ye r f low (Lofquis t (1960)) .

such

as

under

a

s t ag n an t s u r f ace wedge, t h e r a t i o

For density underflows,

is about

- f i /fo

0.43 (Harleman (1961)). For lo ck exchange flow s, which resemble t h e

equa l coun terfl ow si tu a t io n , Abraham and Eysink (1971) gave f i as

drawn i n Fig . 3-21. For la rg e val ues of t h e lower la ye r Reynolds number

a cons tan t va lue o f f is approached. For comparison t h e smooth

w a l l

f r i c t i o n r e l a t i o n f o r fo

is

included in Fig . 3-21. In t h e

i

2

Reynolds number range 5

X

10 to l o 5

A =

0 .4 to

0.5

can be assumed.

Based on t h i s l im ite d evidence

A

is taken as

(3-223)

i n a l l t h e o r e t i c a l p r e d i c ti o n s g iv en i n t h i s s t ud y.

3.6.2.3 Sol uti on graphs

-

S u r f ac e d i l u t i o n

Solut ion graphs giving the value of t h e s u r f ace

d i l u t i o n Ss ( o u t si d e t h e l o c a l mixing r e gi o n i n case of an uns tab le

134



4 -

2 -

1

0

Fig. 3-21: Var iat ion of i nt er f ac ia l stress c oe ff ic ie nt f i

wi th Reynolds numbers (Abraham and Eysink (1971)).

fi - Abraham and Eysink (19711

-

-

I I I I I I

I

I I I I

I 1

near - f ie ld ) as a fu nc ti on of Fs , H/B and

Q

= f o are developed

as

follows:

a)

Fo r th e s t a b le n ea r - fi e ld co n d i t io n

(Fig. 3-20 A-1)

Ss

the impingement layer thickness.

is t ak en d i r ec t ly f rom th e j e t so lu t io n tak ing account

of

Thus t h e p o s s i b i l i t y of a

submerged

internal

jump (Fig. 3-20 A-2) i s not considered i n

t h e t h e o r e t i c a l s o l u t i o n s ,

as

t h i s c o n d i ti o n o n ly

becomes

imp or tant f o r l a rg e v alu es

of

0

.

b )

For the unstab le near - f ie ld condi t ion

(Fig.

3-20B)

the Froude number

FH

of t he equ al counterf low system f o r

which the length of t h e s u b c r i t i c a l s e c t i o n i s eq u a l to the

C



channel leng th

i s

c a l c u l a t e d f i r s t . The v a l u e i s FH

as

a

funct ion

of 4

(with

A

=

0 . 5 )

i s

p l o t t e d i n F i g.

3-22.

Th e d i lu t io n s Ss

then obtained from

Eq.

(3-222).

Examples of so lu t i on graphs wi th in th e p ra c t i c a l range

are

C

f o r any combination of F

,

H/B are

S

g iv en fo r weak f a r - f i e ld e f f e c t s , P = 0 .1 i n F ig .

3-23,

and strong

f a r - f i e l d e f f e c t s , P

=

1 .0 i n F ig .

3-24.

Comparison shows t he dec rea se

i n d i l u t i on s f o r l a r g e r P . Furthermore, while f o r 4 - 0 .1 the

l i n e s f o r e q u a l Ss

each other a t t h e c r i t e r i o n l i n e , t h e re

i s a

n o t i ceab le l ack o f

matching for

= 1.0.

This ind i ca t es the submerged in te rn a l

jump

condit ion which is n e g l i g i b le f o r

f o r bo th near- and far - f ie ld approximately meet

Q = 0 .1

but which would provide for

a smooth t r an s i t i o n of eq u a l d i l u t i o n l i n e s fo r

0 =

1.0. This behavior

can be seen by inspecting Fig.

3-22.

l i n e is cha rac ter ize d by FH

0.20

which i s

smaller

than FH (0

=

0.1)

=

0.22

so

t h a t no

back

water e f f e c t s l ea di ng t o

a

submerged jump should

As mentioned above the cr i t er io n

-

C

be expected.

The important inf luence of n e ar - fi e ld i n s t a b i l i t i e s i n co mb ina ti on

w ith f a r - f i e ld e f f e c t s

i s

obvious when comparing di lu t i on pre dic t io ns

of Fig. 3-19 and Figs. 3-23 and

3-24 .

S c h e m a t i c i l l u s t r a t i o n of t h e d i f f u se r d i sch a rg e , such as

Fig. 3-20, showed a

ve r t i ca l

d i f fu se r d i sch a rg e eo = 90").

th e ap p l i ca b i l i t y o f th e two -d imen sio nal s l o t r ep re sen t a t io n wi th

8

d i r e c t i o n s a l o n g t h e d i f f u s e r l i n e .

can b e in f e r r ed d i r ec t l y from th e d i scuss io n of Liseth 's (1970)

However,

=

90"

is

p oi nt ed o u t a l s o f o r d i f f u s e r s w i t h n oz z le s i n a l t e r n a t i n g

0

For t h e s t a b l e nea r - fi e ld case t h i s

13



a

0

a

137



Fig.

3-23:

Surface Dilution S as a Function of

F

H/B.

S S’

Vertical Diffuser, Weak Far-Field Effects



10 50 100 500 1000

e,=

goo

@ =1.0 Fs

Fig.

3 - 2 4 :

Surface Dilutions

S as a

Function of F H/b.

s

s ’

Vertical

Diffuser, Strong

Far-Field

Effects

139



ex p er imen tal r e su l t ,

as

shown i n F ig. 2-6. For t he uns tab le near-

f i e l d

a

l o c a l mixing zone

is

c rea ted which su p presses d e t a i l s of

t h e

j e t d i sch a rg e f lo w p a t t e rn ( se e F ig. 3-25). D i l u t i o n is determined

Fig.

3-25: Local

b ehav ior of d i f fu s e r d i sch a rg e wi th

a l t e rn a t i n g n o zzle s , u n s t ab le n ea r - f ie ld .

merely by th e in t e rp lay of fa r- f i el d e ff ec ts and buoyancy supply from

th e near - f ie ld zone, exac t ly

as

i n t he

case

of v e r t i c a l ly di sch a rgin g

j e t s .

The equ iva len t s l o t concep t

is

thus app l icab le .

3.7

Th eo re ti ca l P r ed ic t io n s : D i f fu se r s w i th N e t Horizontal Momentum

Dif fusers wi th non-ver t ica l nozz les po in t ing

in

t h e same

di re c t io n (un i -d i rec t iona l d ischarge) can produce

a

v a r i e t y o f f lo w

condi t ions

w h i c h

are somewhat more complicated to analyze than the

previously discussed

symmetric

flow

f i e l d g e ne ra te d b y d i f f u s e r s wi th

no n e t ho riz ont al momentum. Pr es ent at ion of re s u l t s

is

l i m i te d t o

0 = 45" and Bo = 0" (h o r i zo n ta l d i sch a rg e ).

0

140



3.7.1 Near-Field Zone

The analysis

of

the buoyant j e t reg ion and the su r fa ce

impingement i s analogous t o th e p rev ious se c t i on .

A

h y d rau l i c

jump

occurs on bo th s id es of t h e

l i ne

of

impingement, however, t h e jump

a t

t h e r i g h t ( s e ct i o n a

i n

Fig.

3-8) becomes unstable

ear l ier

w i t h

o r d ec reas in g H / B . Furthermore, the s t a b i l i t y

is

FS

i n c r e a s i n g

st ro ng ly dependent on th e angle of discharge, e0 .

t h e c r i t e r i o n l i ne d e l i n e a t i n g t h e s able and uns t ab le n ea r - f i e ld

Figure 3-26 shows

co n d it io ns fo r v a r io u s

B o .

ment layer hi/H

w i t h i n t h e s t a b l e ra nge

is

again found t o be about

1 / 6 f o r a l l B o

.

Outside the s t a b l e range hi/H i s r ap id ly in c r eas in g

fo r in c r eas ing

Fs

and decreasing H/B . Y e t t h e t h i ck n es s i n t h i a

range

is

not as important ,

as the near-field zone w i l l be engulfed i n a

The average thickness of the impinge-

lo ca l mixing region due t o t he uns table jump condit ion.

3.7.2 Far-Field Zone

3.7.2 .1 Po ss ibl e f low condit ion s

The pos s ib le flow condi t ions i n the f a r - f ie ld and the

i n t e r a c t i o n w i th

t h e n e a r - f i e ld

are

i n d i c a t e d i n F ig .

3-27.

. A) S t a b l e n e a r - f i e ld

a )

A

normal

internal

lump and

b)

a submerged internal jump

are similar

t o t h e

ver t ica l

discharge case.

B)

Unstab le near - f ie ld

I n s t a b i l i t i e s a n d j e t re-entrainment le ad t o th e formation

of

th e lo ca l mixing zone,

However, t h e co nd iti on s of t h e flow-away from

141



I

I

I

I

~

1

I

I

j

I

I

j

~

1

1

I

I

j

1

I

1

001

I I I I I 1 1 1 1 1

UNS A BLE

NEAR-FIELD

J U

10

50

100

500 1000

F s '

Fig . 3-26: E f f e c t

of

Angle of Dis ch a rg e

O0

on t h e S t a b i l i t y

of t he Near-Field Zone

14



A ) S T A B L E N E A R - F I E L D

1 )

Normal Internal Jump

PCHANNEL

ND

YI

HANNEL END

I

X

-L ' 0

* L

0

2 L

*

2 ) Submerged Internal Jump

8 ) UNSTABLE NEAR- FIELD

1 )

Counterflow System

X

* L

- L 0

2 ) Stagnant Wedge System

3 ) Supercritical System

Fig.

3-27:

Possible Flow Conditions f o r Discharges with Net

Horizontal Momentum ( C Denotes

a

Critical Section)

143



th is mixing

zone are a f f e c t e d

by

t h e s t r e n g t h

of

t he h o r i z o n t a l

j e t

momentum, which

causes a

n e t f l ow i n one d i r ec t i o n .

h o r i z o n t a l momentum 3 cases can be d i s t ingu ished .

Depending on the

a )

A

Su bc ri t i ca l Counterflaw System is given for weak

h o r i z o n t a l momentum, somewhat s i m i l a r t o t h e v e r t i c a l d is ch a rg e

case bu t wi th non-equal flaws (Q #

-

e x i s te n c e c r i t e r i o n for t h i s t y p e o f

1) i n each lay er . The

f law i s ( E q . ( 3 - 1 9 0 ) )

( 3 - 2 2 4 )

b)

A

Stagnant Wedge System

is

s e t up f o r l a r g e r h o r i z o n t a l

I n t e r f ac i a l f r i c t i o n p rev ent s m otion of w a t e r a g a i n s t

omentum.

t h e d i r e c t i o n of t h e

j e t

discharge. S tagnant sur fa ce and bottom

wedges

are

formed. For the ups tream flaw sec t io n th e c r i t e r i on

i s

(Q

=

0)

( 3 - 2 2 5 )

and s i mi la r l y fo r the downstream sec t i on

c)

A

S u p e r c r i t i c a l System r e s u l t s f o r

s t i l l

l a r g e r

ho r iz o n ta l momentum, ex pe ll in g th e st ag na nt wedges. The densi-

metric

Froude number of t h e fl aw

is

simply

FH >

1

( 3 - 2 2 6 )

This

extreme

f la w system r e s u l t i n g i n f u l l v e r t i c a l m ixi ng

i s t h e

case

stu di ed by

Harleman

e t

a l .

(1971) f o r hor i zo n ta l

discharge (cos e0 =

1)

and i s described by

Eq.

( 2 - 3 8 ) which for

144



th e purpose of in t roduc ing the f a r - f i e l d parameter

Q

and

t h e an g le e0 r ew r i t t en as

(i

g

cos u )

( 1 + Q/2)1'2 0

(3-227)

For

th e ana ly t ic a l t r ea tment of un s tab le near - fie ld cond i t ions

it i s necessary t o cons ider th e t o t a l head changes,

r)

, d u e t o

f r i c t i o n i n t h e f a r - f i e l d .

r) i s

ind ica ted schematical ly i n F ig . 3-27.

The head change AH causes a p r e s su r e d i f f e r e n t i a l a c ro s s t h e d i f f u s e r

mixing zone which i n stead y

s t a t e

i s balanced by th e ho r iz on tal

momentum

of t he d if fus er . The equat ion fo r the head change

r)

i n

s t r a t i f i e d f l o w

w a s

developed earl ier

(Eq.

3-195) and so lu ti on s f o r

the wedge cases were given

i n

paragraph

3.4.4.4.

3.7.2.2 So lu ti on method

The cou nte rfl ow system (Fig. 3-2 B1) i s th e most gen era l form

of t h e f lo w d i s t r i b u t i o n

i n

case of an uns t abl e near- f ie ld . The f low

system i s made up of two c o u n t e r f l w regi on s bounded by c r i t i c a l

s ec t i o n s . I t s p r o p e r t i e s are des cr ib ed by th e fo l lowing 8 v a r i a b l e s

( s u b s c r i p t s

a

and b re fe r t o co nd it ion s down- and upstream,

r e s p ec t i v e l y ) :

H/B ,

Q and

B 0

.

S

,

iven parameters

are

The var iables are determined by t he s imultaneous s ol ut io n of t h e

following system of non-linear al ge br ai c equations: (The equat ions are

/

14

5



a l l w r i t t e n i n e x p l i c i t , sym bolic

form

t o s u gg es t t h e i t e r a t i v e

so lu ti on method)

where Qa(F , @) d en ot es t h e e x p l i c i t form

of

t h e

fI

-

a

i n t e r f a c e

Eq.

(3-188) ev al ua te d between 0 and

L .

a

42

q% = (ss

- l )

where Qb(F2,, , a) is t h e e x p l i c i t form of Eq. (3-188)

"b

evaluated between

-L

and 0

.

1

:,

2

-

- F\ H - h2

a

C

a

C

a a

which

i s a

h o r i z o n t a l

momentum

equation obtained by taking

a

c on t ro l volume between t h e two

c r i t i c a l

sect ions bounding

t h e loca l mixing zone. The

c r i t i c a l

depths h2

implici t ly known

from

t h e s o l u t i o n

of

t h e

i n t e r f a c e e q ua t io ns

are

' h2

'b

a

Y

146



T h e f i r s t

term is

the head lo ss equat ion

(3-195)

evalua ted

between -L and 0 and th e second term acco u n t s f o r th e

lo ss of th e v e lo c i ty h ead i n t h e ch an ne l and due t o

d i s s i p a t i o n o u t s i d e t h e c ha nn el .

oa/H

where Qa(

hplp

is the impl ic i t fo rm of

an

e q u a t i o n f o r t h e

&region

similar

t o

6 ) .

A

Gauss-Seidel i t e ra t i o n method

w a s

used.

of equations

1)

t o

8),

s p e c i f i c a t i o n

of

i n i t i a l va lu es f o r

Q,

is needed. Convergence t o a s t a b l e v a lu e f o r

only 5 t o 10 i te ra t ions depending on the i n i t i a l guess.

I n t he given arrangement

Ss

and

i s

f a s t , t a k i n g

s S

a

I n t h e above formulat ion t he len gth of the mixing zone w a s

Derivation of th e

eg lec ted wi th r e sp ec t t o t h e channel leng th .

h o r i z o n t a l momentum equation assumed hy dr os ta t i c condi t ions a t t h e

end

of

t h e mixing zone which is co ns is te n t wi th assumptions

made

f o r

t h e s t r a t i f i e d f low re gi on s.

The st ag na nt wedge system (Fi g.

3-27 B2)

is

described by a

similar,

but

s i m p l e r

system of equations.

3.7.2.3

Solution Graphs - S u r f a c e D i l u t i o n

So lu t io n g raph s g iv ing su r f a ce d i lu t io n s

Ss

f o r b ot h t h e

s tab le and unstab le near - f ie ld range are given fo r weak and st ro ng far-

147



f i e l d e f f e c t s f o r

0

= 1.0) and f o r

0 = 1.0).

1)

t h e c r i t e r io n b etween s t a b l e and u n s tab le n ea r - f i e ld co n d it io n s ,

2 ) th e t ra ns i t io n be tween t he counter flow system and

t h e

s tagnant

wedge system is giv en when t h e wedge l en gt h i s j u s t e qu al t o t h e

channel len gth, and 3 ) t h e t ra ns i t io n between t h e wedge system and

s u p e r c r i t i c a l flow, given when

Based on the so l u t i on

i n

€Io

45"

(Fig.

3-28 ,

9 =

0 . 1

and Fig.

3-29 ,

€Io

0

(Fig. 3 - 3 0 ,

0 = 0 . 1

and Fi g. 3-31,

I n

a l l

g ra ph s t h r e e t r a n s i t i o n l i n e s a r e included:

FH

=

1 .

t h e

s u p e r c r i t i c a l case E q .

( 3 - 2 2 7 )

t h i s t r a n s i t i o n c an be g iv en

as

(3-228)

In

t h e s u p e r c r i t i c a l case t h e d i l u t i o n

Ss i s

independent of

FS

a t th e poi nt of maximum wedge in tr us io n le ng th .

ver t i ca l d i s c h a r g e case, t h e matching a t t h e t r a n s i t i o n between s t a b l e

and u n s t a b l e n e a r - f i e l d d i l u t i o n p r e d i c t i o n s

w i l l

be smoothed out

and t h e f low is f u l l y mixed v e r t i c a l l y .

The

d i l u t i o n

i s a

minimum

S i m il a r t o t h e

through t he submerged in te rn a l j u m p regime which

i s

n o t co ns ide red i n

t h e pred ic t ion .

3 . 8 Stnnmary

The mechanics of

a

submerged diffuser

w e r e

an alyzed i n

a two-

dimensional "channel model".

se ct io n bounded by channel

w a l l s of

f i n i t e l en g th and o penin g a t both

The channel model consists of a d i f f u s e r

ends in to a l a rg e r e se rv o i r . The r a t i o n a l e fo r th e ch an n el model

i s

t o sim ulat e t h e predominantly two-dimensional flow which

i s

postu la ted

148



JV

10

50 m

a=45O

=

0.1

500

1000

Fs

Fig . 3-28:

S u r f a c e D i l u t i o n S

as a F unc t ion of F ,H/B.

S S

45 Discharg e, Weak Far-F ield Ef fe ct s

149



n .

rig.

3-29:

Surface Dilution

S s

as a Function of Fs, H/B.

45

Discharge, Strong Far-Field Effects

150



Fig.

3-30:

Surface Dilution

S

as a Function of F

H/B.

S

S’

Horizontal Discharge, Weak Far-Field Effects

151



5 0 0 0

1000

x

50C

1oc

5c

ss=lo -

-

-

I

I

I I I I I I

I

I I I 1 1 1

10

50

100 500

1000

Q = 0 g = 1.0

F,

Fig .

3-31:

S u r f a c e D i l u t i o n S

as

a

Func t ion of

F H/B.

S S '

Hor i z on ta l D i sc ha r ge , S t r ong Far F i e l d E f f e c t s

152



t o e x i s t i n t h e c e nt e r

char ge. The mul tip ort arrangement w a s r epresen ted by t he equ ivalen t s l o t

concept , p rese rv in g th e dynamic ch ar ac te r i s t i c s o f the

j e t

discharge.

p o r t i o n o f

a

three-dimensional d i f f us er d is -

In analyzing th e two-dimensional f low f i e l d emphasis

w a s

l a i d

o n

a

de ta i le d treatment of t h e four d is t i nc t f low reg ion s which can be

d i sce r ned i n t h e g en er a l case. The object ive

w a s

t o o b t a i n a n o v e r a l l

desc r ip t ion of the f low f i e l d by matching o f the so lu t ion s fo r the

ind iv i dua l r eg ions .

th e s urf ace impingement reg ion , th e i n t er na l hydraul ic jump region and

t h e s t r a t i f i e d cou nt er fl ow r eg io n .

The four f low regions are: the buoyant j e t region,

The buoyant

j e t

r e g i o n w a s analyzed using t h e entrainment concept

proposed by Morton e t a l . (1956).

c oe f fi c ie n t (Eq. (3-48)), which depends on t h e l o c a l buoyant

characteris-

t i c s of the

j e t

was deduced for j e t s w it h a r b i t r a r y d i s c ha r g e a n gl e i n

a

f a s h io n s i m il a r t o

FOX'S

(1970) so lu t io n fo r th e

ver t i ca l

j e t . I t was

shown t h a t

a l l

buoyant j e t s tend to a n asymptotic

case,

t h e plume, which

i s characterized by

a

constant local densimetric Froude number,

Eq. (3-57).

The value of th e entrainment coe ff ic ie nt (Eq. (3-48)) i n t h i s a sy mp to ti c

case agrees w e l l wit h Abraham's (1963) eva lu at io n of Rouse

e t a l . ' s

(1952)

exper imental da ta .

A r e la t ion sh ip f o r th e en trainment

Impor tant a spe ct s of th e an al ys i s of t he sur fa ce impingement reg ion

w e r e the inc lus ion of

an

energy l o ss and

a

buoyancy term i n the energy

equation.

I t

w a s

found

that

t h e v e r t i c a l

flow

d i s t r i b u t i o n i n t h e s e c t io n

a f t e r impingement

is

a lw ay s d i s t i n g u i s h ed b y d en s i m e t r i c a l l y s u p e r c r i t i c a l

f low cond i t ions l ead ing t o a subsequent in te rn al hyd rau l ic jump.

153



The in t e r na l hydrau l ic

jump

r e g i o n i s descr ibed by a se t

of

equa tion s f i r s t derived by Yih and Guha (19551, which i n gen er al i s

dependent on the densi ty r a t i o be tween th e two laye rs and th e f r e e

su rf ac e Froude numbers.

As

th e p roblem of in te re s t

is

ch a rac te r i zed

by

small

d en s ity d i f f e r en ce s and small f re e sur fac e Froude numbers, an

asymptot ic so l u t i on fo r f i n i t e densimetr ic Froude numbers w a s derived

giv ing

a

simple equatio n (3-142) f o r t h e conjugate jump co nd it io n.

The equat ions fo r a non-ent rain ing s t ra t i f i e d counter f low reg ion

w i th s u r f a c e h e at l o s s an d i n t e r f a c i a l mixing were developed. Scaling

showed

that

fo r p r a c t i c a l co n d i t io n s o f chann el l en g th scales t h e

su r f ac e h ea t t r an s f e r p ro cesses can be n eg lected ,

so t h a t t h e eq ua t io n s

r e du c e t o t h e class ical Schi j f and Sc hh fe ld (1953) equat ions f o r

s t r a t i f i e d f l o w .

The

s o l u t i o n f o r t h e p o s i t i o n of t h e i n t e r f a c e , Eq.

(3-188), is obtained assuming constant

t o t a l d ep th as a f i r s t approxima-

t ion, which again implies

small

d e n s i t y d i f f e r e n c e s and f r e e s u r f a c e

Froude numbers.

t o t a l h ead caused b y f r i c t io n a l e f f e c t s i n th e f lo w sy st em

was

der ived ,

Eq. (3 -195). Analy t ica l so lu t ion s fo r the in te r fa ce and t o t a l head

e q u a t i o n s were given.

With the g iven in te r f ac e posi t ion , th e change of t he

The diffuser problem

is

governed by four dimensionless parameters :

H/B and

B o

are near-f ield parameters, 4J

=

f o

L/H i s a

f a r - f i e l d

F*’

parameter .

Matching of the so l u t i ons f o r t he f low reg ions y ie ld s th e fol lowing

i mp or ta nt r e s u l t s : s t a b i l i t y of t h e n e a r- f i e l d zone

is

given on ly fo r a

li mi te d range of low Fs , high H/B . Furthermore, th e range decrease s

fo r decreas ing (more hor izon t a l )

B o

(Fig. 3-25).

I t is only i n t h i s

15



l im i t e d r ang e that th e buoyant j e t models i n an unbounded re ce iv in g

water

(discussed i n Chapter

2)

are a p p li c a bl e t o p r e d i c t d i l u t i o n s .

i s found that i n t h i s range t he th ick ness of the sur fa ce impingement

l ay e r which h as to b e acco un ted f o r in d i l u t i o n p r ed i c t io n s

i s

about

1 /6 of t h e water depth.

e f f e c t on n e ar - fi e ld d i l u t i o n ( e x ce pt f o r l a r g e (3 l e a d i n g t o a sub-

merged in te rn a l jump) and thus can be neglec ted f o r d i l u t io n pred ic t ions .

I t

The flow-away i n the far -f ie ld has l i t t l e

Ou ts id e th e s t ab le n ea r - f i e ld r an ge th e d i lu t ed

w a t e r

i s

cont inuously re -ent ra ined in to the j e t region forming a lo ca l mix ing

zone.

f i e l d w a t e r u n t i l i n s te ad y s t a t e an equ i l ib r ium is reached which i s

es se n t i a l l y d etermined by th e in t e rp lay o f

two

f a c to r s : f r i c t i o n a l e f f e c t s

i n th e f a r - f i e ld , r epre sen ted by (3

,

and t h e ho ri zo n ta l momentum input

This re-entrainment leads t o a build-up of buoyancy of t h e near-

- of the j e t discharge , represen ted by 8 .

0

For di ff u se rs with no ne t ho ri zo nt al momentum

( e o

= 90 ' ) t h e f a r -

f i e l d f l ow i s given by an equal counterflow system between two

c r i t i c a l

s e c t i o n s , one

a t

the edge

of

th e l o c a l mixing zone, one

a t

th e channel

end.

-

For d i ff us er s with n et ho ri zo nt al momentum

( e o

< 90') t h e r e are

3

p o ss ib le f a r - f i e ld f lo w co n f ig u ra tio n s : a counterflow system, a

s tag nant wedge system or su pe rc r i t i ca l f low.

w it h r e s u l t i n g f u l l v e r t i c a l mixing downstream

i s

t h e

extreme case

of

sur fac e and bot tom in te ra c t io n descr ibed by Harleman e t a l .

(1971) and

reviewed i n Chapter

2 .

ho ri zo nt al momentum of t h e dis cha rge i s balanced by the depth change

ac ro s s th e mixing zone r e s u l t i n g fro m f a r - f i e ld e f f ec t s .

The su pe rc r i t i ca l flow case

In th e an a ly s i s o f each o f th es e co n d i tio n s th e

-

15

-



Composi te su rfa ce d i l u t io n graphs descr ib ing both t he near- and

fa r- f i e l d r ange were presented.

The u t i l i t y of th e two-dimens ional channel model i n th e s tudy

of t he d i f fu se r induced f low f i e l d

i s

obvious:

(1) I t provided necessary c r i t e r i a g iv ing t h e r ange

of

a p p l i c a b i l i t y f o r b uo ya nt j e t models t o p r e d i c t d i l u t i o n s

f o r d i f f u s e r d i sc h ar g e i n f i n i t e d ep th .

(2) I t

gave i n s i g h t i n t o t h e

ver t i ca l

and l ong i t ud ina l

v a r i a t i o n s of t h e f lo w f i e l d .

(3)

I t

demonstrated that

i t i s

neces sa ry t o desc r i be

t h e

dynamics of the t o t a l flow f ie ld , and not merely th e j e t

r e g i o n ,

t o give

d i l u t i o n p r e d i c t i o n s o u t s id e t h e s t a b l e ne ar -

f i e l d ra ng e.

However,

as

t he underlying ob j ec t i ve of t h i s s t udy i s p r e d i c t i o n

of t h e th ree -d im ens iona l d i f f u se r fl ow f i e l d it w i l l b e ne c e ss a r y t o

p rovide some l inkage between t he f a r - f i e l d e f fe c t s p resen t

i n

t h e

two-

d imensiona l channe l model and t h e f a r - f i e l d e f fe c t s p resen t i n t h e t h ree -

dimensional

case.

In par t i cu la r , based on the requirement of equivalency

of t h e f a r - f i e l d e f f e c t s a r e l a t i on sh i p be tween t he charac ter i s t ic

h o r i z o n t a l l e n g t h scales, th e two-dimensional channel le ng th and th e three -

d im ens iona l d i f fu se r l eng th , has t o be developed .

p rovided i n ana ly t i c a l f a sh ion

i n

Chapter

4

and comparisons with

This

l i n k a g e is

e x pe r im e nt al r e s u l t s

are

given i n Chapter

6 .

15



..-

I V. THREE- DI MENSI ONAL ASPECTS

OF

THE DI FFUSER I NDUCED FL OW FI ELD

A

mul t i por t di f f user l i ne of l engt h, 2LD, pl aced i n a l ar ge body

of water of uni f or m dept h,

H, wi l l gener at e a thr ee- di mensi onal f l ow

pat t er n.

f or mul ated si mul ati ng t he pr edom nant l y t wo- di mensi onal f l ow f i el d whi ch

i s post ul at ed to exi st i n t he cent er por t i on of t he di f f user l i ne (Fi g.

3- 1 ) .

unst abl e near - f i el d condi t i on per si sts. For t he st abl e near - f i el d, t he

di f f user - i nduced di l ut i on i s pr i mar i l y gover ned by j et ent r ai nment .

ever , f or unst abl e near- f i el d condi t i ons, t he di l ut i on i s i nf l uenced by

f ar - f i el d ef f ects r el at i ng t o t he t ot al r esi st ance i n t he f l owaway of

m xed water f r om t he near - f i el d.

t i on pr edi ct i on i n t he gener al t hr ee- di mensi onal case, t hi s chapt er di s-

cusses t he i mpor t ant t hr ee- di mensi onal aspect s of t he di f f user- i nduced

f l ow f i el d and thei r r el at i on

t o

t he t wo- di mensi onal channel model :

i )

A si mpl i f i ed model

of

t he t hr ee- di mensi onal f l ow f i el d i s

I n t he pr ecedi ng chapt er a t wo- di mensi onal channel model

was

Dependi ng on di schar ge condi t i ons i t was f ound t hat a st abl e or

How-

A s t he obj ect i ve of t hi s study i s di l u-

devel oped and the f ar - f i el d ef f ect s ar e eval uat ed.

on equi val ency of f ar - f i el d ef f ect s, t he channel l engt h

of t he cor r espondi ng t wo- di mensi onal model i s

r el at ed

t o

t he l engt h of t he t hr ee- di mensi onal di f f user. I n t hi s

way pr edi ct i ons

f or t he t hr ee- di mensi onal appl i cat i on may

be gi ven thr ough the cor r espondi ng t wo- di mensi onal channel

model .

Based

i i ) I n addi t i on to ver t i cal c i r cul at i ons (such as i n t he

st r at i f i ed count er f l ow system,

the di f f user i s capabl e

of pr oduci ng ci r cul at i ons i n t he hor i zont al pl ane. The

157



e x i s t e n c e o f t h e s e c i r c u l a t i o n s w hi ch u l t im a t e l y l e a d

t o re -en t ra inment o f mixed w a t e r

i s

r e l a t e d t o n e a r - f i el d

i n s t a b i l i t i e s . The c o n t r o l o f t h e s e c i r c u l a t i o n s th ro ug h

o r i e n t a t i o n of t h e d i f f u s e r n o zz l e s i n t h e h o r i z o n t a l

p l a n e i s d i s cu s s ed .

4 . 1

Re la t in g t he Two-Dimensional Channel Model t o t he Three-

Dimensional Flow Field

A q u a n t i t a t i v e r e l a t i o n s hi p

i s

g i v e n o n ly f o r d i f f u s e r d i s c h a r ge

w i t h

no

n e t h o r i z o n t a l momentum an d ze r o c ro s s fl ow. D i f fu s e r s w i t h n e t

horizontal momentum are d i sc u s se d q u a l i t a t i v e l y and e xp e r im e n t al r e s u l t s

a re given .

4 . 1 . 1 Di ff us er s w i th No N e t H or iz on ta l Momentum

4.1.1.1 Equiva lency Requi reme nts

F i g u r e

4-1

shows the

v e r t i c a l

c i r c u l a t i o n a nd t h e ho r iz o n-

t a l f lo w p a t t e r n i n t h e l ow er l a y e r p o s t ul a t e d f o r a d i f f u s e r w i t h no n e t

h o r i zo n t a l momentum. The h o r i zo n t a l f l o w p a t t e r n i n t h e u p pe r l a y e r i s

s i m i l a r w i t h r e v e r s e d d i r e c t i o n s . The t h re e -d i me n si o na l s i t u a t i o n a nd

th e two-d imens ional channel model co nc ep tu a l iz a t io n

a re

shown. I n bo th

cases

t h e

f low i s s e t up by th e ent rain men t demand and th e buoyancy sup pl y

w i t h i n t h e n e a r - f i e l d z on e. F or co m pa r is on o f t h e d i l u t i o n c h a r a c t e r i s -

t i c s of b o th s y s t em s i t

i s

r e q u i r e d t h a t t h e n e a r - f i e l d p a r a m et e r s F

H/B a re the s a m e . Furthermore,

the

s a m e f r i c t i o n c o e f f i c i e n t s f a nd

S’

0

=

Af

are

given .f i

0

The ob

j

ect v e

L

s o t h a t t h e

same d

of

th e compar ison i s t o d e t e r m i n e t h e c h a n n e l l e n g t h

l u t i o n i s obt a in ed f rom t h e two-d imensional channel

model as f rom the th ree-d imens ional case.

T o

o b t a i n t h e s a m e d i l u t i o n s

158



X

f

5

\

159



two requirements h ave t o hold :

1 )

Kinematic Requirement

T he d i l u t i o n f o r t h e t wo -d im en si on al case i s

= -q2

i s

t h e flow i n e ac h l a y e r ( l a r g e d i l u t i o n s )

4 1

here 141‘

-

u = l a y e r v e l o c i t y

u

a ve ra ge d o v e r t o t a l d e p t h

H.

-

u i s c ons ta n t th r oughou t th e c ha nnel ,

- -

U ’ U

.

C

The d i l u t i o n f o r t h e t h re e -d i m en s io n al

case

i s

(4-2)

1

f

where

e va lua te d

a t


x

=O,

-

L

<y<LD

i s

the normal

(x)

component of t h e f low vec to r q

=

(qx,qy

‘d

D

-

u i s t h e no rm a l component o f th e ve lo c i ty ve c to r a ve r a ge d ove r

d

t h e t o t a l d ep th

3 =

u;) a t t h e d i f f u s e r .

=&

The magnitude

of

u de c r e a ses away f r om t h e d i f f u se r l i n e .

The k i n e m at i c r e qu i re m en t f o r e q u i v a l e n t d i l u t i o n f o l lo w s

2 ) Dynamic Requirement

D i sc h ar g e w i t h s t a b l e n e a r- f i e l d c o n d i t i o n s

i s l i t t l e

a f f e c t e d by

f a r - f i e l d e f f e c t s and t h u s w i l l b eh av e s i m i l a r l y i n b o th cases.

D i s -

c ha rge s wi th uns ta b le ne a r - f i e ld c ond i t ion s depend s t r on g ly on t h e f a r -

160



_-

f i e l d .

For th e two-dimensional channel model d i lu t io n

i s

uniquely con-

t r o l l e d by t h e b a l a n c e o f f a r - f i e l d f r i c t i o n a nd b uoy anc y

i n

t he nea r -

f i e l d r e g i o n , e x p r e s se d by t h e e q u a l c o u n te r f l o w E q u a ti o n ( 3 2 0 7 ) which

i s p l o t t e d i n F i g ur e 3-22 (A

=

0 . 5 ) .

as

The f u n c t i o n a l r e l a t i o n i s g i v e n

L

@ = fo = f l (FH 1

C

( 4 -5 )

where F

(Eq. 3-221). By v i r t u e o f t h e d i l u t i o n r e l a t i o n sh i p ,

E q .

( 3 - 2 2 2 ) ,

t h i s

equa t ion i s

w r i t t e n as

i s

the densimetr ic Froude number of t he counte r f low sys tem

HC

a nd f u r t h e r m o r e, m u l t i p l y i n g b o t h s i d e s

by

u 2 /@

- 2

f o L Uc

3

(’2-D)

- - -

4

H 2g

2-D

f

( 4 - 6 )

( 4 - 7 )

where h

i s a

head l o s s expr essi on f o r th e two-dimensional channel

f low. By v i r t u e of th e constancy of

( Eq .

( 4 - 2 ) ) , h

,

i s equiv-

a l e n t t o t h e i n t e g r a t i o n o v e r t h e f l o w do main

2-D

2-D

’

The st ro ng ly two-dimensional ch ar ac te r of t h e three-dimensional

flo w f i e l d i n t h e v i c i n i t y o f t h e d i f f u s e r c e nt e rp o rt i on i s noted .

Ex-

c e p t f o r d i s t a n c e s f a r from t h e d i f f u s e r l i n e , t h e v e r t i c a l c ou nt er fl ow

1 6 1



resembl es that of the two- di mensi onal model . Therefore a f unct i onal

rel at i onshi p f o r the bal ance

of

f ar- f i el d f r i ct i on and buoyancy i s hypo-

thesi zed, si ml ar to Eq. (4-71,

=

f3

(S3-*)

3-D

f

( 4 - 9

1

where hf

i nfi ni ty.

i s the i ntegrat i on over the f l ow path f romt he di f f user to

3-D

I n part i cul ar, by i ntegrat i on al ong the x- axi s

(4-10)

To eval uate the i ntegral a descri pt i on of the f l ow f i el d

u,;)

i s neces-

sary.

The dynamc requi rement f or equi val ent di l ut i on (S

=

S3D i s

2-D

the equal i ty of f ar- f i el d f ri cti onal ef f ects

= hf (4-11)

2-D 3-D

f

4.1.1.2 Model f or the Three- D mensi onal Fl ow D stri but i on

Eval uat i on

of

the three- di mensi onal f ar- f i el d ef f ects, Eq.

(4-l o), requi res speci f i cat i on of the f l ow di str i but i on.

A si mpl i f i ed

f l owmodel i s gi ven descri bi ng the hor i zontal mot i on i n each f l ui d l ayer.

Densi ty changes due t o heat di ssi pat i on or i nterf aci al mxi ng are neg-

l ected i n vi ew of the f act that the model appl i cat i on i s pri mari l y

focused on the two- di mensi onal behavi or i n the di f f user vi ci ni ty; the

heat l oss scal i ng perf ormed i n paragraph 3.4.4.2 appl i es then as wel l .

The f l ow i s gradual l y varyi ng and two- di mensi onal i n the hori zon-

tal pl ane.

The equat i ons of moti on are wr i t t en f or the l ower l ayer.

The

162



- -

flo w v e l o c i t i e s u ,

v a re

a v er a ge d o v er t h e t o t a l d e p th , H .

(4-12)

(4-13)

(4-14)

-

where

7

f r i c t i o n , w r i t t e n as

T

a re

t o t a l

s t ress terms

f o r bo t h i n t e r f a c i a l and b ot to m

zx' zy

(4-15)

where

5

i s t he t o t a l f r i c t i o n co e f f ic i e nt .

The

r e l a t ive

importance

of

t h e

terms i n

the momentum equations

i s

de te rmined by sca l i ng :

(u*,v*)

=

(U,V)/Gd

The d i f f u s e r l e n g t h i s chosen as t h e c h a r a c t e r i s t i c l en g th

s c a l e

of t h e problem as t h e l o c a l f lo w f i e l d i n t h e d i f f u s e r v i c i n i t y i s o f

i n t e r e s t .

s u b s t i t u t i n g T from E q . (4-15) as

The x-momentum equ ati on may be w r i t t e n i n dimen sionles s form,

XZ

a

u*

+ v * - -

*-

u*

ax* a Y * ax*

8 H

-

163

( 4 - 1 7 )



Typi cal prototype val ues are chosen as i n paragraph 3.4 .4 .2 .

H

=

30

f t . and 7 i s t aken as 0.03. The bracketed term i s eval uated as

LD=lOOO

f t ,

- 0. l i ndi cat i ng that the f l ow f i el d i s governed by pressure and i nert i al

f orces.

the

f l ow f i el d can be determned assumng i nvi sci d condi t i ons (potenti al

Thus negl ecti ng the f r i cti onal termas a f i rst approxi mat i on,

fl ow. Wt h the cal cul ated f l ow f i el d the f r i cti onal t erm h , can

then be eval uated.

3-D

The potent i al f l ow f ormul at i on

i s

v20

= 0

w th

and the boundary condi t i on at the di f f user l i ne

-

= - u x ++o

($

ax d

(4-18)

(4-19)

-

= + u X - t - 0

d

The probl emi s sol ved i n the compl ex pl ane (Fi gure 4-2).

A

con-

pl ex potent i al W i s def i ned as

w = b + i $

The compl ex potent i al due

t o

a poi nt si nk at

5

i s

164

( 4 - 2 0 )

( 4 - 2 1 )



Fig.

4-2: Complex.

o l u t i o n

Domain

-

S u p e r p os i t i on of p o i n t s i n k s a l on g t h e d i f f u s e r l i n e l e a d s t o

a

l i n e

s i n k

I iLD

a nd a f t e r i n t e g r a t i o n

(4-22)

( 4 - 2 3 )

The

s t r e a m l i n e s $

a r e

given by th e imag inary

p a r t

of

W

and

a r e

p l o t t e d

i n

Figure 4-3 fo r h a l f t h e f l o w f i e l d ex h ib i t i n g t h e two -d im en sion al

- -

c h a r a c t e r i n t h e c e n t e r p o r t i o n .

b y d i f f e r e n t i a t i o n

The complex ve loc i t i es

u,

v

are

obta ined

165

_-

, ( 4 - 2 4 )



r

L

\Diffuser

Line

x

Fig. 4 - 3 : Streamlines for One Quadrant

of

the Flow Field

166



-

The f lo w d i s t r i b u t i o n a t t h e d i f f u s e r l i n e i s g iv en as

Us in g t h e b ou nd ary co n d i t i o n f o r t h e normal v e l o c i t y E

g i v e s

d

2Ud

m--

0-

L

D

T h e v e lo c i t y a lo n g t h e x-axis i s now given

as

(4-25)

(4-26)

(4-27)

The head loss i n t e g r a l ,

Eq.

(5 - lo ) , f o r t h e t h ree -d im en sio n al

f l o w f i e l d c an b e e v a l u a t e d as

CLD

o r w i t h x

=

(4-28)

(4-29)

The value

of

t h e d ef i n i t e i n t e g r a l i n Eq. (4-29)

i s

gi ve n by Gradshteyn

and Ryshik (1965) as -T l o g 2

s o

t h a t

2

- 2

-

fo Ud

h - - -

€ 3 4 4H 2g

(0.884)

LD

(4-30)



-1

The i nt egr al

0

pl ot t ed i n Fi gur e

4-4.

The shape

o f

dE

i s comput ed numer i cal l y and

t he f uncti on i ndi cat es t hat t he

us'

4 H

29

.5

1

2

5

x/LD

Fi g. 4 - 4 :

, Al ong t heFl ow Pat h, x / L

D

umul at i ye . Head Loss ,

maj or i nf l uence of t he f ar- f i el d ef f ects

i s

i ndeed r est r i ct ed t o t he

di f f user vi ci ni t y wher e the st r ongl y t wo- di mensi onal f l ow char act er

wi t h r easonabl y hi gh vel oci t i es per si sts.

t he asymptot i c val ue i s approached

t o

wi t hi n 10%

At 5 di f f user hal f - l engt hs

I nvoki ng t he ki nemat i c and dynam c r equi r ement s f or equi val ency

of di f f user - i nduced di l ut i ons, by equat i ng Equat i ons

(4-8)

and

( 4 - 30 )

and subst i t ut i ng Eq. (4-4), i t i s f ound t hat

L

= 0. 884 LD

(4-31)

gi ves t he l engt h of t he cor r espondi ng t wo- di mensi onal channel model .

For pract i cal pur poses Eq.

(4-31)

can be approxi mat ed as

L

=

LD ( 4 -

32)

For a di f f user w th no net hor i zont al moment um t he channel l engt h of

t he cor r espondi ng t wo- di mensi onal channel shoul d be t aken about equal t o

168



t h e l en g th o f

the

t h ree -d im en s io n a l d i f fu s e r .

4 .1 .2 Dif fu s e r s w i t h

N e t

Horizontal Momentum

Kinematic and dynamic requir ement s f or equivalency of d i ff us er -

i nd uc e d d i l u t i o n s c an b e d e r iv e d i n a manner s im i l a r t o Equat ions (4 -4 )

and (4-11). The evalua t ion o f the three-dimensional head l o s s eq u a t i o n ,

(Eq.

(4-10) ) ,

r e q ui r e s s p e c i f i c a t i o n of t h e h o r i z o n t a l v e l o c i t y

hf3-D

-

d i s t r i b u t i o n u;).

In develop ing a model fo r (t,;) t h e h o r i z o n t a l

mo-

mentum inpu t o f t he d i f f us er

( e < 9 0 " )

h a s t o b e i n c l ud e d

as

a boundary

0

co n d i t i o n .

I n the complex p lan e t h i s can be accompli shed by i n t eg ra t i ng

a d i p o l e d i s t r i b u t i o n a l on g the d i f f u s e r l i n e .

w i l l be t h a t o f two v o r t i ce s c en t e red a t b o t h d i f f u s e r ends.

The r e s u l t i n g fl ow f i e l d

However,

t h e

t o t a l

discharge over the d i f f us e r l i n e i s found t o b e i n f i n i t e f o r

t h i s s i m p l e model. This i s n o n r e a l i s t i c i n

terms of the

k in em a t i c

re-

quirement . Therefore,

a

more in t r i c a t e model must be developed to des-

c r i b e t he t hree-d im ens ion a l f l o w f i e l d .

I n t h i s s t ud y i t

i s assumed

t h a t t h e r e l a t i o n , L = LD, developed

f o r

the

di f f us er w i th no n e t ho r i zo n t a l momentum

i s

a l s o ap p ro x im a te ly

c o r r e c t f o r t h e d i f f u se r w i th

net

ho ri zo nt al momentum. Th is assum ption

i s a p p r o p r i a te f o r t h e

case

of a d i f f u s e r w i t h a r e s u l t i n g c o u n t e r f l o w

sys tem (Figure 3-27B1) which s t r on g l y resembles the eq ual coun ter f low

sys tem of th e d i f fu se r wi th no

ne t

h or iz o nt a l momentum.

The assumption

i s more h y p o t h e ti c a l f o r d i s c ha r g e s w i t h f u l l v e r t i c a l mixing (Fig .

3-27B3)

a nd e s s e n t i a l l y h as t o be sub s ta n t i a te d by exper im enta l ev idence.

4.2 Dif f u s e r Ind uced Ho r i zo n t a l C i rc u l a t i o n s

Diffuser d ischarges , even with no net horizontal momentum, a re

u n de r ce r t a in co n d i t i o n s , c ap ab l e

of

p ro du ci ng s i g n i f i c a n t h o r i z o n t a l

1 6 9



ci rcul at i ons. These ci rcul at i ons are def i ned as currents whi ch ul t i m

atel y l ead to reci rcul at i on of al ready mxed water i nto the di f f user

l i ne. D f f users w th stabl e near- f i el d condi t i ons do not produce such

ci rcul at i ons, as a stabl e vert i cal l y str at i f i ed f l ow systemi s set up.

The exi st ence

of

hori zontal ci rcul at i ons i s i nt i matel y rel ated to near-

f i el d i nstabi l i t i es. The generat i on mechani sm for these ci rcul at i ons

i s di scussed i n a qual i tat i ve f ashi on. Cont rol of the ci rcul at i ons can

be achi eved through speci f i c or i entat i on of the di f f user nozzl es i n the

hor i zontal pl ane.

Such control i s desi rabl e f rompracti cal consi dera-

t i ons to prevent reci rcul at i on and maxi mze di f f user eff i ci ency.

4.2.1 . D f f user w th No Net Hori zontal Momentum

4.2.1.1 Generati ng Mechani sm

The vel oci ty di str i but i on at the di f f user l i ne i s gi ven

f romthe hor i zont al f l owmodel f or the l ower l ayer by Eq. ( 4 - 2 5 ) w th

Eq.

( 4 - 2 6 )

subst i tuted

-

The vel oci ty i s thus made up of a normal component, ud, account i ng for

the ent rai nment i nto the di f f user l i ne and a tangent i al component , vd

-

-

- Ud W I L D

= -

Vd IT l og l -y/ LD

( 4 - 3 4 )

The hor i zontal component i s zero at the di f f user center and i nf i ni tel y

hi gh at t he di f f user ends. Such a sweepi ng mot i an f romthe di f f user

end

t o

i t s center i s present as can be

seen

i n Fi gure 4-3.

The

170



e x i s t e n c e o f

t h i s cu r re n t i n

t h e

lower l ay er has been observed by

L i s e t h

(1970)

and ca use s some inward bending of th e j e t s i s s u in g f ro m

t h e n o z z l e s a t th e d i f f us er end . The mot ion

i s

o f no fu r t h e r co n ce rn

f o r s t a b l e n ea r - fi e ld c o n di t io n s.

F o r u n s t a b l e n e a r - f i e l d c o n d i t i o n s , how ever, t h e m ot io n r e s u l t s

i n

the

g e n e r a t i o n o f a s t ro n g h o r i zo n t a l ed d y in g m o t io n u n l e s s

some cow

t r o l i s invoked. The mechanism for t h i s somewhat unexpected phenomenon

( t h e r e

i s

no ne t ho ri zo nt al momentum )

i s

ex p l a in ed as f o l l o w s , r e f e r -

r i n g t o F i gu r e 4-5:

A s s u m e a f lo w p a r t i c l e i s e n tr a i ne d i n t o t h e vcr-

t i c a l

j e t

near

t h e d i f f u s e r e nd . The p a r t i c l e

i s

c a r r i e d upward i n

t h e j e t and def lec ted due to su r face impingement . As

t he

n e a r - f i e l d

zone is d yn am ic a ll y u n s t a b l e t h e f l o w p a r t i c l e i s

c a r r i e d a g a in i n t o

th e l o wer l a y e r wh i l e b e in g s wept

i nw ar d a l on g t h e d i f f u s e r l i n e .

p a r t i c l e g e t s r e- en tr ai ne d i n t o t h e j e t s .

The

L ag rang i an p a th o f t h e

p a r t i c l e i s i n d ic a t ed . The t o t a l e f f e c t of t h e b e h a vi o r i s t h a t a l l

the r e p e a t e d ly e n t r a i n e d d i l u t e d f lo w i s t r a n s p o r t e d t o w ar ds t h e d i f -

The

fu se r c en te r f rom which i t d e p a r t s

i n the form of

a v e r t i c a l l y f u l l y

m ixed s t ro n g cu r ren t a l o n g t h e

x-axis.

f low a long t he y -ax is i s a

s i m i l a r

s t r o n g c u r r e n t of ambient

water.

A

By c o n t i n u i t y t h e t o t a l a pp ro ac h

h o r i z o n t a l c i r c u l a t i o n i s g e n e r a t e d w hic h u l t i m a t e l y l e a d s t o r e c i r c u l a -

t i o n an d u n st ead y co n cen t r a t i o n b u il d -u p e f f e c t s .

The c l o s e c o n ne c ti o n of t h e g e n e r a t i n g mechanism t o t h e s t a b i l i t y

of t h e n ea r - f i e l d zon e

i s

im p o r t an t . The v e r t i c a l l y f u l l y m ixed flow-

away a t S e c t i o n

B-B

(F ig .

4-5)

d oe s n o t a g r e e w i th t h e v e r t i c a l l y s t r a t -

i f i e d f lo w i n t h e d i f f u s e r c e n t e r p o rt i on as i n d i c a t e d o n Fig. 4-1.

In

- f a c t , t h i s p o s t u la t e d v e r t i c a l l y s t r a t i f i e d flow c on d i ti o n ( which a l s o

171

-



PLAN

V I E W

A - A

I

reentrainment

in

unstable

near field zone

I

-B

ve r t callv

fully mixed.

f low - away

Fig. 4-5: Three-Dimensional Flow Field f o r Diffuser with

Unstable Near-Field Zone (No Control)

172



f orms t he basi s of the two-di mensi onal channel model conceptual i zat i on)

requi res cont rol of

t he

three-di mensi onal f l ow f i el d. Thi s cont rol can

he

achi eved through or i entat i on

of

the di f f user nozzl es i n the hor i zon-

tal pl ane and i s desi rabl e i n vi ew

of

the prevent i on of t he repeated

re- entrai nment.

4.2.1.2 Control Methods

The obj ecti ve of the cont rol i s the prevent i on of repeat -

ed ent rai nment w thi n the unstabl e near- f i el d zone ( l ocal mxi ng zone).

Thi s i s achi eved by opposi ng the i nward current w thi n the l ocal mxi ng

zone through or i entat i on of the j et nozzl es i n the hori zont al pl ane.

The momentum

of

the i nward f l ow i s bal anced by the momentum

of

the j et

di scharge. Thi s stagnat i on of the i nward f l oww thi n the l ocal mxi ng

\

zone, however, does not i mpl y zero tangent i al f l ow at the edge of t he

l ocal mxi ng zone.

predi cted by t he si mpl e l ayer model and i ndi cated i n Fi gure 4-3.

Con-

t rol through nozzl e or i entat i on merel y guarantees zero i nward vel oci ty

w thi n

the l ocal mxi ng zone and provi des the proper st art i ng condi -

t i ons at the edge of the mxi ng zone f or the establ i shment of a verti -

The f l ow outsi de the mxi ng zone w l l behave as

cal l y str at i f i ed f l ow systemoutsi de.

The l ocal angl e @ ( y ) (see Fi g. 4-3) under whi ch the hori zont al

ent rai nment f l ow i s enter i ng t he di f f user l i ne i s

d

(4-35)

To

counteract thi s ent rai nment f l oww t hi n the l ocal mxi ng zone the

i ndi vi dual nozzl es of the al ternat i ng di f f user are di rected agai nst

t he

-

173



ent r ai nment f l ow.

The var i at i on of t he hor i zont al nozzl e or i ent at i on

al ong the di f f user l i ne i s then

(4-36

Local det ai l s of a di f f user sect i on wi t h al t er nat i ng nozzl es are shown

i n Fi gur e 4-6.

i ng of t he ent r ai nment f l ow wi t hi n t he l ocal m xi ng zone i s onl y poss-

i bl e i f t he moment um f l ux of a nozzl e di schar ge act i ng over t he wi dt h

The ver t i cal nozzl e angl e i s e0.

Ef f i c i ent count er act-

Fi g.

4-6:

Pl an

V i e w

of

Di f f user Sect i on; Al t er nat i ng Nozzl es wi t h

Ver t i cal Angl e

0

2Rsi nB

no cont r ol i s possi bl e i f

eo = 90"

( ver t i cal di scharge).

maxi mum angl e

8

i s hi gher t han t he moment um

of

t he ent r ai nment f l ow.

Obvi ousl y,

Ther e i s a

The moment um f l ux

p

t o

whi ch cont r ol i s possi bl e.

0

max

of t he ent r ai nment f l ow over a di f f user l engt h Ay i s

( 4 - 3 7 )

The moment um f l u x

of

t he nozzl e di schar ge i s, over t he di f f user l engt h

2 a , ( see Fi g. 4-6

174



2 u

DTr

0

cos

e

_ c

po '

4

s i n g 0

m =

n

( 4 - 3 8 )

and u s i ng t h e e q u i v a l en t s l o t d e f i n i t i o n ,

Eq,

( 2 - 2 0 )

and a l so

p,po

9 1

m = U ~ ' B c o s

e

n P a 0

I

R

s ing

( 4 - 3 9 )

The spacing R

i s

cons ide red

a

d i f f e r e n t i a l l e n g t h of t h e t o t a l d i f f u s e r ,

thus

Ay-R

and Am-m

n

Am

I

u

L ~

cos e

P a

o

o

s i n g

To make

a c o n t r o l p o s s i b l e

3

AY

With

the

d e f i n i t i o n (Eq.

( 4 - 3 )

and qo =

UoB

o n e can w r i t e

4 H

-

c o s

8 >1

0

s S

( 4 - 4 0 )

( 4 - 4 1 )

( 4 - 4 2 )

I n t h e p a r am e te r r an g e f o r t h e u n s t a b l e n e a r - f ie l d

S

by Eq.

( 3 - 2 2 2 )

which

i s

s u b s t i t u t e d i n t o E q .

( 4 - 4 2 )

t o g i ve

i s g i v e n

S

( 4 - 4 3 )

w he re F a c c ou n t s f o r t h e f a r - f i e l d

e f f e c t s

@ by v i r t u e o f t h e i n t e r -

f a c i a l e q u a t i o n (see Fig.

3 - 2 2 ) .

e s t i m a t e d :

C

H

The maximum value of eo =

O 0

H

i s

m a x

N e g l i g i b l e f a r - f i e l d e f f e c t s mean F

=

0 . 2 5

and

a t

t h e

C

175



c r i t e r i o n which d e s c r i b e s t h e t r a n s i t i o n b et we en t h e s t a b l e and u n s t a b l e

parameter range

w a s

e s t a b l i s h e d

as

413

H/B

=

1.84

Fs

so t h a t

= cos- l P-lI3 (1.84) (0.25)4']

79

max

o

(3-219)

(4-44)

F o r l a r g e r F

d i f f u s e r d i s c h a rg e o n t h e r e c e i v in g

w a t e r ,

t h e a n g l e O

ger .

f i e l d , eo i s l a r g e r . However, u s i n g Eq. (4-44) as the lower es t i -

mate,

i t

can b e s t a t e d

i n

o r d e r t o e n a b l e c o n t r o l o f t h e th re e- di me n-

o r smaller H/B, i . e . r e l a t i v e l y s t r o n g e r im pact of t h e

S

i s even

l a r -

max

S i m i l a r l y , f o r smaller FH

,

i.e. l a r g e r r e s i s t a n c e i n t h e f a r -

C

max

s i o n a l f l o w f i e l d t h r o ug h h o r i z o n t a l o r i e n t a t i o n o f t h e n o z zl e s, B ( y )

given by

E q .

(4-36) t h e v e r t i c a l a n g l e o f t h e d i s c h a rg e ha s t o b e l e ss

t han

~ 7 9 .

horizontal nozzle momentum

i s

d i f f u s e d i n t h e l o c a l m ix in g z o ne , much

It i s immaterial how much l e s s , s i n c e

i f 0

< emax t h e e x c e s s

0

l i k e i n t h e f a sh i o n i n d i c a t e d i n Fi g . 3-25. As mentioned above, t h e

o b j e c t i v e of t h e c o n t r o l i s t o c o u n te r ac t t h e t a n g e n t i a l v e l o c i t y ,

w i t h i n

t he

l o ca l mix ing zone .

-

Vd'

N o h o r i z o n t a l m o ti o n c an b e

induced by

the nozzle discharge momentum outside

th e mix ing zone, as can be shown

by drawing a c o n t r o l volume a ro un d t h e d i f f u se r area:

t a l momentum i n any d i re c t io n

i s

ze ro

The net hor izon-

Thus i t

i s

emphasized,

t h a t

t h e

h o r i z o n t a l c i r c u l a t i o n s w hich

are

se t

up by

a

d i f f u s e r wi th v e r t i c a l

and a l t e r n a t i n g n o z z l es

a re

by no means a r e s u l t o f h o r i z o n t a l d i s c h ar g e

momentum, b u t ra t h e r

a

c o m p l e x i n t e r a c t i o n of v e r t i c a l n ea r- fi el d

1 7 6



i nstabi l i t i es whi ch are ampl i f i ed by the entrai nment f l ow sweepi ng

al ong the di f f user l i ne.

I n summary,

through control

by

l ocal hori zontal ori entati on, B(y)

i n

Eq.

(4-36) of the nozzl es

of

a mul t i port di f f user the dexel opment of

hori zontal ci rcul at i ons can be prevented.

the l ocal mxi ng zone.

be characteri zed by the two- l ayered systemwhi ch i s i n equi l i br i umbe-

tween t he buoyancy f orce of the near- f i el d and the resi stance i n the

The control consol i dates

The f l ow outsi de the l ocal mxi ng zone w l l

then

far- f i el d (Fi g. 4-1). The two-di mensi onal channel model conceptual i za-

t i on i s appl i cabl e.

No

control i s requi red

f o r

di f f users w th a stabl e

near- f i el d zone:

the strat i f i ed two- l ayered type.

The f l ow f i el d outsi de the near- f i el d w l l al ways be

Eval uat i on of the sensi t i vi ty of the f ormof the f l ow f i el d f or

di f f erent or i entat i ons, ~(y) , can onl y be made experi mental l y. I n the

experi mental programvari ous di str i but i ons, ~ ( y ) ,were tested and

changes i n f l ow f i el d behavi or and resul t i ng overal l di l ut i on are

re-

ported (Chapter 6 ) .

4.2.2 D f f users w th Net Hori zontal Momentum -

4.2.2.1 Generati ng Mechani sm

D f f users w th uni di recti onal nozzl es

produce hori zontal

ci rcul at i ons due to two f actors: i ) hor i zontal momentumof the di s-

charge, and i i ) i nstabi l i t i es i n the near- f i el d zone, si ml ar t o the

di scharge w th no hori zontal momentum The rel ati ve st rength

of

the

two mechani sms i s i mportant .

As no anal yti cal model f o r the descri pt i on of the far- f i el d has

been devel oped, the fol l ow ng i s hypothet i cal , partl y i n anal ogy t o the

177



dis cha rge wi th no ho ri z o n t a l momentum.

1)

D i f f u s e r s w i t h

a

s t a b l e n e a r- f ie l d z on e e x h i b i t a two-

layered sys tem i n t h e f a r - f i e l d . The h o r i z o n t a l o r ie n -

t a t i o n o f t h e n o z zl e s ,

B(y), i s

n o t d e c i s i v e.

2 )

D i f f u s e r s w i t h a n u n s t a b l e n e a r - f i e l d w i l l produce

ci rcula-

t i on s wh ic h

are

de pe nde nt on the noz z l e o r i e n t a t i on ,

~ ( y ) .

The d i s c u s s i o n i s g iv en f o r p a r a l l e l n o z zl e s as shown i n

Fig. 4-7.

k G d

LAN VIEW

A - A

-

Fig. 4-7;

U n i d i r e c t i o n a l D i sc h ar g e

w i t h P a r a l l e l

Noz z le Or ie n ta t ion

@(y) = c o n s t

=

90 )

a ) For weak ho ri zo n ta l momentum (cou nter flow s ystem) th e set-up

o f h o r i z o n t a l c i r c u l a t i o n s i s p ri m ar il y due t o i n s t a b i l i t i e s

i n the ne a r - f i e ld z one i n con junc t ion wi t h a n inwar d f low

a l o n g t h e d i f f u s e r l i n e .

A f l ow f i e l d much l i k e t h e o ne

f o r the d i f fu se r wi th no n e t ho r iz on ta l momentum (F ig .

4-5)

' w i l l

result only somewhat modif ied

by the

momentum of t h e

d i sc ha r ge

s o

that the flow-away

w i l l b e

m a in ly a long the

+x-axis.

178



b) For st r ong hor i zont al moment um ( super cr i t i cal syst emw t h

f ul l ver t i cal mx i ng)

the hor i zont al c i r cul at i on i s pr i m

ari l y caused by

the

moment um

of

t he di schar ge.

essent i al l y t he si t uat i on whi ch has been descri bed anal y-

t i cal l y by Adams (1971) as di scussed i n Chapter

2.

Adams'

model , when appl i ed t o zero cr ossf l ow condi t i ons, Eq. ( 2 - 4 5 1 ,

pr edi cts a si gni f i cant cont r acti on ( cc =

1 / 2 )

of t he f l ow

f i el d downst r eam of t he di f f user l i ne. Thi s cont r acti on

i s accompani ed by a di l ut i on decr ease

of

l / 2 as compar ed

to t he cor r espondi ng t wo- di mensi onal channel model w t hout

f ar - f i el d f r i ct i on

( @

= 0).

4.2.2.2 Cont r ol Met hods

Cont r ol

of

t he hor i zont al di f f user- i nduced c i r cul at i ons i s

Thi s i s

ai med pr i mar i l y at t he pr event i on of t he f l ow cont r act i on downst r eam of

t he di f f user l i ne. Such cont r ol i s desi r abl e as i s shown by compar i son

of

t he di l ut i on pr edi ct i ons f r omAdams' model and t he t wo- di mensi onal

channel model .

Prevent i on of t he cont r act i on i s agai n achi eved by count er act i ng

t he y- component of t he ent r ai nment f l ow as i t ent er s t he di f f user l i ne

t hr ough hor i zont al or i ent at i on

of

t he di f f user nozzl es, The var i at i on

of nozzl e or i ent at i on, B(y), cannot be comput ed anal yt i cal l y due to

l ack of a model f or t he t hr ee- di mensi onal f l ow f i el d. I n t he exper i men-

t al program uni di r ect i onal di f f users w th par al l el nozzl es ( B

=

90")

and nozzl es wi t h

~ ( y )

s gi ven by E q . 4-36 wer e st udi ed.

of

t hi s By) may be wel l j ust i f i ed at l east i n t he case o f a s t rat i f i ed

The assumpt i on

count er f l ow system i n vi ew of the f l ow- f i el d si m l ar i t i es .

-

1 7 9

-



V. EXPERIMENTAL

EQUIPMENT

AND

PROCEDURES

This chapter descr ibes the laboratory equipment and procedures

used i n t h e exp e rimen ta l p a r t of t h e in v es t ig a t io n .

set-ups w e r e used:

Two d i f f e r e n t

(1) a flume set-up di re ct ed toward the study of

t h e two-dimensional cha nne l model, and

(2) a

ba si n set-up mainly

di re ct ed toward the study of three-dimensional di ff us e r behavior ,

wi th few addi t iona l tests on t he two-dimensional chann el concept.

I n b oth se t-up s th e e f f e c t of cross-flow w a s invest iga ted . The

buoyancy of th e dis cha rge w a s in troduced by using heated water.

The

set-ups and proced ures

are

descr ibed

i n

d e t a i l below.

However,

f i r s t b a s i c co n s id e ra t io n s r ep o r t in g th e ex p e rimen tal p rogram

are

discussed . Experimenta l re su l t s are p resen ted i n Chapter s

6

and

7 , r e sp ec t iv e ly .

5 . 1 Basic Considerations on Di ff us er Experiments

5.1.1 Expe rimen tal Program

The ob je ct iv es of th e experimental program w e r e :

a )

to g a in in s i g h t i n t o t h e h ydrodynamic p ro p e r t i e s o f th e d i f fu se r -

induced flow f i e l d and examine assumptions made

i n

t h e t h e o r e t i c a l

development

;

b) t o measure the gross behavior of the diffuser- induced f low f i e l d

and compare i t wi th t h eo re t i ca l so lu t io n s (Chapter 6);

c) t o determine the e f f e c t of ambient cross-flow on di ff us er behavior

and der iv e empir ica l re la t ion sh i ps (Chapter

7) .

The experimental program

w a s

planned t o encompass

a

broad range

of

the govern ing d imensionless parameters fo r mu l t ip ar t d i f fu se r

dicharges . For t he purel y di f fuser-induced behavio r the se parameters

180



are

given i n Eq. (3-215), namely,

the densimetric Froude number of the

( e qu i va l en t ) s l o t d i f f u s e r , FS, t h e

ver t i ca l

discharge ang le , B o , t h e

re la t ive depth, H/B, and the f a r - f i e l d parameter, @ .

Two

more

parameters

are

i n tro du ced fo r th e

case

of

a

d i f f u s e r i n ambient c ro ss-

flaw, namely,

u H

a

U B

V E - X volume f l ux r a t i o

0

and

y =

an g le o f d i f f u se r axis w i t h d i r e c t i o n

of cros s-cu rren t, ua

.

The

parameter

range was i n ten ded to in clu de p r a c t i c a l p ro to ty pe

ap pl ic at io ns , the ord er of magnitude of which are o u t li n e d i n t h e

following examples

wi th

r e f er en ce t o th e d i scu ss io n i n Chapter 1:

(1)

p p ca l sewage d i f f us er wi th h igh d i lu t io n requ irement:

Liseth (1970) gave a su rv ey of sewage mu l t ip o r t d i f f u s e r o u t f a l l s

on

t h e U.S. Pa ci f i c Coast. Se le c ted average dimensions

are

shown in

Table 5.1 and non-dimensional parameters f o r the equi va le nt s l o t

d i f f u s e r

are

ca lcu la t ed .

(2)

Typi c a l t h ermal d i f fu s e r w i th lower di lu t i on requi rement :

The discharge conditions from a

600

MW nuclear power plant with a

d i f f us e r i n a shallaw near-shore area are i n d i c a t e d i n Table 5 . 1 f o r

comparison (see, f o r example, Harleman e t .

a l .

(19 71)

) .

Equivalent

s lo t parameters are ca lcu la t ed .

A

bottom f r i c t i o n c o e f f i c i e n t f o =

0.02

w a s

assumed i n both

cases,

corresponding t o

an

average Chezy C 110 (ft)’”/sec. I n

Chapter

4

the r e la t i on between channel length, L ,

and d i f fu ser leng th ,

LD,

is

derived as L

z

LD

,

hence

f4 fo

LD/H

.

Comparison

shows

-

181



Table 5.1: Comparison of Relevant Parameters for

Typ ical Sewage and Thermal Dif fu se r

9 p l i c a t i o n s

Sewage Di ff us er Thermal Di ffu sera r i a b l e s:

Water

depth, H ( f t ) 200

20

Tota l d i s cha rge , Qo (cfs) 400 1000

APf

Pa 0.025 0.003

Tota l D i f fu s e r Leng th ,

2% ( f t )

Nozzle

Diameter, D

( f t )

Nozzle Spacing,

11

( f t )

( f r e s h - s a l t water)

(ATo - 20°F)

3000

3000

0.5

10

Discharge ve lo c i t y , U,,

(fps)

6.8

Ambient cu rr en t velo ci ty ,

u Ups)

0

t o 0 . 1

0

Bottom f r i c t i o n c o e f f i c i e n t , f o 0.02

Equ iva l en t s l o t w id th , B ( f t ) 0.02

Dimensionless

parameters:

H/B

v

Y

70

10,000

v a r a b

e

0.1

0 t o 100

v a r i a b l e

1 o

20

8 . 5

0 t o 0.5

0.02

0.04

140

500

v a r i a b l e

1.5

0 t o 50

v a r i a b l e

18



some

o f th e s t r i k i n g d i f f e r en ces be tween th ese g en e ra l d i f fu s e r ty p es:

The

re la t ive

depth

is

considerab ly l a r ge r fo r the sewage d i f fu se r ,

w h i l e i t s densimetric Froude number

is

somewhat smaller. The ef fec t

of t h e f a r f i e l d

is

more pronounced f o r t he thermal d i f fuser .

5.1.2 Experimental Li mi ta tio ns

While laboratory experiments on submerged mult iport d iffusers

have

the d is t i nc t advantage of a loca l ized observa t ion

area

and a

control lable environment,

they have

some

l im i t a t io n s wh ich h av e d i r e c t

repercus sions on the range of dimensionless parameters which can be

s t u d i e d .

l a b o r a t o r y f a c i l i t i e s:

These

l i m i t a t i o n s

are

r e l a t e d t o t h e s i z e of u s u a l l y a v a i l a b l e

(1) Boundary ef fe ct s: The th eo re t i ca l t reatment

w a s

d i r e c t e d

toward the steady-s tate di ff us er performance i n a ( i n f i n i t e l y ) l a r g e

body of

water

with uniform depth.

b a s i n of f i n i t e e xt e n t,

i t

is clear that, depending on t h e s i z e of t h e

b as in r e l a t i v e t o th e d i f fu se r l en g th , t h e f lo w ind uced by th e

d i f f u s e r

w i l l

be in f luenced t o

a

s t r on g e r o r

lesser

degree through

boundary ef fe ct s . There is no i n f l u e nc e i n case of t h e two-dimensional

channe l model, i n which c r i t i c a l s e c t i o n s

a t

the channel ends def ine a

clear

s

teady-s

ta te

boundary cond itio n, th e only requirement being

t h a t

th e r es er vo ir outside t he channel be somewhat lar ge r than th e channel

i t s e l f .

There is n e g l i g ib l e i n f lu e n ce i n t h e case of a s t ro n g c ross -

flow.

I f t h e d i f f u s e r i s l o c a t e d i n a

There

are cases

which have some quasi-steady

s t a t e

condi t ion ,

such

as

discharg es with no n e t ho ri zo nt al momentum and co nt ro l, which

create

v e r t i c a l s t r a t i f i c a t i o n and a n uppe r l a y e r which

f a r

from the

183



d i f fu se r ( n ea r t o th e b as in b o un d ar i es ) i s o nly s lo wly in c r eas in g i n

depth and thus has l i t t l e i n f lu en ce on th e behav io r i n r eg io n s c l o s e r

t o t h e d i f f u s e r.

There may be considerable boundary in f lu enc e fo r di sch arg es

with strong horizontal momentum:

experimental

t i m e s

long enough t o overcome t he

i n e r t i a of

t h e i n i t i a l l y

undisturbed receiving water. On t h e oth er hand, f o r long er t i m e s

The s t ead y - s t a t e co n d i t io n r eq u i r e s

re c i rc u l a t io n of the mixed water i n t o t h e d i ff u s e r l i n e w i l l occur due

t o b ou nd ary e f f ec t s . A rea l dilemma may r e s u l t i n ce r t a i n ins tan ces .

Therefore , i n genera l, min imiza t ion of boundary ef fe c t s re qu i res

d i f fuser d imensions ,

LD/H,

which are s m a l l compared t o t h e m i n i m u m

s i z e

of t h e a v a i l a b l e b a s i n Ldn/H,

(2) Turbulence problems: I n th e ana lys is a f u l l y t u r b u l e n t

j e t

This requires maintenance of

a j e t

Reynolds numbereg ion

i s

assumed,

which i s

l a r g e r than some c r i t i c a l

value,

and poses a considerab le

co ns t r a i n t i n s tudy ing the parameter range wi th low

F

and high H / B

i n view of th e requirement (5-1).

S

The change of the tu rb u le n t s t ru c t ur e

of a round non-buoyant j e t

w i t h

a Reynolds number defined as

UOD

w

= -

j

V

(5-2)

a s vi su a l ly observed by

Pearce

(1968).

Pearce

found a f u l l y t u rb u l en t

j e t s t r u c t u r e w i t h

constant angle

of

spreading t o be g iven fo r

IR

> 3000

.

For IR < 500 the j e t e f f l u e n t

i s

laminar with on ly

small

i n s t a b i l i t i e s . The t r a n s i t i o n r eg io n 500 <

IR

< 3000 i s

j

j

1

184



d i s t i ngu i shed by an i nc reas ing ly t u rbu l e n t j e t s t r u c t u r e . Pearce

d i d n o t i n v e s t i g a t e t h e e n tr a in m en t e f f i c i e n c y

of

j e t s i n

this

t r an s i t i on reg ion , bu t presumably i t will be

lower

and only approach-

i n g t h a t of

the

f u l l y t u r b ul e n t j e t ,

IR >3000

.

Hence,

for a

j e t

Reynolds number

IR

based on th e hydra ul ic rad ius

of

t h e d i s c h a r g e

opening, R , a c r i t e r i o n

j

> 3000

4 %

V

R =

j

(5-3)

should be

m e t

i n e x pe ri me nt al s t u d i e s

.

3)

Measurement technique: The ve lo ci ty and temperatu re f i e l d s

are

of

in te res t .

p o i n t

of

d is c ha r ge and h o r i z o n t a l l a y e r v e l o c i t i e s

are

very

l o w .

i s

Y e t

the

j e t

ve lo c i t i e s dec rease r ap id ly f rom the

There

as

of

now

no prac t ic a l l y f ea s i b l e technique which

allows

v e l o c i t y

measurement of t h i s order

of

magnitude.

Only

approximate v i sual

or

photograph ic obse rvati ons can be made

in certain

i n s t a n c e s .

Thus

t h e

g ros s behav io r o f t he d i f fu se r d i s cha rge

has

t o be p r im ar i l y de te rmined

by measuring th e temperature f i e ld .

5 . 2

The Flume Set-Up

P r i m a r y

purpose of

t h e

flume set-up

w a s

the observat ion of

t h e v e r t i c a l s t r u c t u r e of the temperature f i e l d produced by

a

h e a t e d

water

d i f fu se r d i s cha rge .

The m u l t i po r t d i f f u se r d i s cha rge

was

simu-

la ted as a cont inuous s lo t .

5.2.1

E<1

uipmen

t

Figures 5-1 and 5-2 show the experimental set-up.

A

1 f t . wide channel t e s t s e c t i o n of v a r i a b l e l e n gt h 2L

was

i n s t a l l e d

i n a

metal

flume 50' X 4.5'

x

3' . The

water

volume ou ts id e th e t e s t

185



Fig.

5-1:

Photograph of Flume

Set-Up



partition wall 1" plywood for tests with crossflow

TOP

V I E W

at.

etal flume

U

A

4.5'

b

2L4 4

I I

t

i f '

1 .

\ hannel wall 1/2" plywood

I-

S I D E V I E W

W

-1-

H

I .

K J

A

1

\

A - A

point-gage

mounted

themistor

pr b e

f i xe d

thermistor

Drobes

slot jet

7

44

'

alse floor 2-14 plywood

injection device

DETAIL :

1

t

flume

;racing

. J E T

'

sliding plate

perforated

damping plate

3

ID

plexiglass pipe

c

1

3 .5 l

INFLOW

Fig. 5-2:

Flume

Set-Up



se ct io n formed a re se rv oi r . The channel

w a l l

and fa l s e - f l oo r

w e r e

of ma rin e plywood.

The length of the channel could be var ied by

adding o r removing 8 f t . long plywood pan els .

10 f t . l on g g l a s s w a l l se ct ion a l lowing obse rvat i on of the f low f rom

The flume has a

t h e s i d e .

The s l o t

j e t

i n j e c t i o n dev i ce had a

1

f t . lo ng , 3 I D p l e x i g l a s s

p ipe

a s i t s

b a si c component

and was

equ ipped wi th an ad jus t ab l e s l i d i ng

p l a t e f o r c h an gi ng t h t s l o t w i dt h

B.

A pe rf or Bed damping p la te pro-

vided f o r uniform

j e t exit

v e l o c i t i e s .

The

i n j e c t i o n d e vi c e

w a s

mounted on U-shaped su pp or ts i n an open ing o f t h e f a l s e channe l f loo r

and cou ld be ro t a t ed t o g ive t he des i r ed d i s cha rge ang l e

t e s t s

were made using a s i ng le round nozzle p laced

in

the channel

B o

.

A few

center as

t h e j e t source .

copper f i t t i n g s wi th va r i ous d i am ete r s and ang l e s .

The nozzles used w e r e commercial ly a va ia bl e

Heated discharge

water was

supp l ied f rom

a s t e a m

heat exchanger capable of d el i ver ing up

t o

60

g a l l o n s p e r

minute

of

water

a t

a

co ns ta nt temperature up t o 150OF.

An

amount eq ual t o

the j e t

disc harg e could b e withdrawn from t he

reservoir

i n t h e back o f

the

channel t o guarant ee cons t a n t water depth.

For t e s t s with cross-f low a plywood par t i t ion w a l l w a s i n s e r t e d

i n t h e r e se r vo i r beh ind t he channe l and a f i v e HP ce nt ri fu ga l pump w a s

used

t o

gene ra te cross - f low i n th e channel by pumping a cro ss

the

p a r t i t i o n .

A l l

f lows

are

monitored by

Brooks

rotameter flowmeters.

Temperature measurements were t aken i n t w o components: (1)

Fixed probe system:

50

Yellow Springs Instruments

No.

401 thermis tor

probes were mounted i n f ixed pos i t io ns

i n the

set-up.

Th i r t y - s i x (36)

probes were mounted through the channel

w a l l s

forming 4 v e r t i c a l

188



t r an sec t s o f 9 probes each (s ee Fig.

5-2)

and the remainder w e r e

i n s t a l l e d i n t h e r e s e r vo i r and i n

in-

and ou t f low l ines .

w e r e

connected to a Dif i t ec (United Systems) sca nnin g and pri nt in g

un i t . The tempera ture i n degrees Fahrenhei t

w a s

p r in ted

on

paper

t ap e a t

a rate

of 1.5 seconds

p e r

probe.

The

probes have an accuracy

of - 0.2S0F and

were

i n d i v i du a l l y c a l i b r a t e d .

7

sec. which f i l t e r s o ut t u rb u l en t f l u c t u a t i o n s .

The probes

The t i m e co n s tan t is

(2) Moveable probe: A Fenwal GA51SM2 th ermis to r pro be

w a s

i n s t a l l e d on a point gage which

i s

drive n by a

s m a l l

DC

motor capable

of v e r t i c a l l y t r a v e r s i n g

1.5

f e e t p e r m in ute.

The

v e r t i c a l p o s i t io n

of t h e probe w a s recorded via a poten t iometer c i r cu i t on t h e v e r t i c a l

axis

of a Valtec x-y p l o t t e r . The thermis to r was connected to

a

w he at st on e br i d g e c i r c u i t t o t h e h o r i z o n t a l axis of the

x-y

p l o t t e r .

As

the

moveable probe made a v e r t i c a l

traverse a

d i r ec t p lo t of d ep th

versus t h e mi l l iv o l t o u tpu t of th e th e rmis to r c i r c u i t r e su l t ed . The

therm is to r had a s m a l l time cons tan t of 0.07 sec, imp or tant i n view of

the trav er si ng speed. The probe

w a s

ca l ib r a t ed ag a in s t t h e r ead in g

of

a

mercury thermometer.

instrument c a r r i a g e running on t h e r a i l s of t h e

m e t a l

flume and could

be moved i n t o any posi t ion.

The p o in t g ag e i t s e l f w a s mounted on a n

The glass

w a l l

se c t io n of the f lume a l lowed v i su a l observa t ion

of

the f low f i e l d through dye

(FDC-Blue)

i n j e c t e d i n t o t h e j e t d i s -

charge. Limite d ve lo ci ty measurements could be made through th e

time-lapse phot ogra phic reco rding s of dye

traces

formed by falling dye

c r y s t a l s.

189



by th e in s t a l l a t i o n of h o rse -h ai r mats

w a s

provided.

monitored with Brooks flow meters.

A l l f lo ws w e r e

Somewhat dif fe re nt te mper atur e measurement in st ru me nt at io n w a s

used in

t h e

ba si n set-up: (1) Fixed probe system: 60 Yellow Spri ngs

Ins t ruments No. 701 l inear thermis to r p robes w e r e mounted within the

b as in .

and were ind iv i dua l ly ca l ib r a te d . The probe arrangement i n th e bas in

The probes have an accu rac y of 0.25'F, a t i m e co n s tan t of 9 s e

i s

shown i n Fi gs . 5-5a and 5-5b f o r

the

two d i f fu se r p ip e l ay o u t s . The

probe mounting can be seen i n Fi g. 5-3

.

Thirty-seven

(37)

probes w e r e

mounted on prob e ho ld er s which

w e r e

at ta ch ed t o th e wooden probe frame.

The

probe f rame could be leve led wi th ja cks so th a t th e p robes were

accu ra te ly p o s i t io n ed a t 1 /4 i n be lo w th e

water

su r f ace

(30 probes) and

3

i n be lo w th e w a t e r sur fac e (7 probes) . Sixt een (16) probes

w e r e

mounted

1

n a bo ve t h e b a s i n f l o o r .

The

remainder of th e probes (7) w e r e

used t o measure temperatures

of

t h e h ea ted water and of t h e c ro ss- f l aw

water.

i n s i d e

the

d i f f u s e r p i pe .

The

probes

w e r e

automatical ly scanned by

a

Digi tec scanner -pr in ter system connected to a IIP computer f o r on-l ine

d a t a a c q u i s i t i o n .

probes:

d e s i r e d l o c a t i o n s w it h i n t h e b a s i n .

as desc r ibed i n the f lume se t-up .

A

switching mechanism al lowed s eq ue nt ia l co nt ro l of th e 4 probe s and

The hea ted d ischarge tempera ture

w a s

measured with

3

probes

A complete scan took about 1.5 minutes.

(2) Moveable

Four moveable probe units w e r e b u i l t and cou ld b e p o s i t io n ed a t

The basic components w e r e t h e s a m e

The poi n t gages

w e r e

mounted on stands.

recording on a Houston Instrumen ts x-y p lo t te r .

a l l

4 probes took about

2

minutes.

A complete record of

196



X

f t

X

X

X

Near

field

zone

-

To

-

+

t

alse wall

fcr

2 - 0 channel

model

I x

I

-15 -

10

-5 0 + 5 +

10 +I5

t

a) Fixe d prob e arrangement, Basin Set -Up A

X Pr ob es 1/4 in below surface

0 Probes 3 in

below

surface

+ R o b e s

1

in above bottom

X

X

X

x +

+

+

. x

X

-I

r---;;----

A N

ear ti.4d zone

-

-

X

UI

Diffuser l ine

mc

I

-15 -10 -5 0 +5

+10

+I5 t

b ) Fix ed probe arrangem ent, Basin Set - Up

8

’ + 5

’ t

’

-5

. -10

F i g . 5-5: Arrangement of Temperature Measurement System

197



5.3.2 Experimental Procedure

An i n i t i a l t emperature scan served t o check uni formi ty and

to def ine th e ambient bas in t emperature . For runs wi th cross -f low

the des i r e d cross -f l ow ve loc i t y w a s es t a b l i s hed . B efore t he

s t a r t

t h e d i f f u s e r p i p e w a s "primed", i . e . , h e a t e d water was d i scharged

i n t o an en d o f t h e d i f f u s e r p i p e w h i l e a t t h e o t h e r e nd a n e q u a l

amount w a s withdrawn. This procedu re (ab out 1 t o

2

minutes dura t ion)

e s t a b l i s h e d t h e d e s i r e d d i sc h a rg e t e m pe r a tu r e i n t h e

p i p e .

A t

the run

s t a r t

t he wi thd rawal f low f rom the d i f f u s e r p ipe

w a s

s topped and t he hea t ed water disch arged through t he nozzles .

A n

amount equ a l t o t he d i f fu se r f low w a s withdrawn f rom the ba s i n s i d e

channels t o keep th e

water

l e ve l con s tan t dur in g the run . Dye

w a s

i n j e c t e d t o o b se r ve t h e c i r c u l a t i o n

p a t t e r n .

Automatic temperature

scans were t aken

a t 5

minute in te rv a ls throughout th e run. The

moveable probes

were

o p er a te d i n similar i n t e r v a l s .

The

s t e a d y s t a t e or quas i - s t eady

s t a t e

p o r t i o n

w a s

l i m i t e d

t o

a

vary ing deg ree

f o r each experiment and depended s t r on gl y on

d i f f u s e r c h a r a c t e r i s t i c s . The main f a c t o r w a s t h e n e t h o r i z o n t a l

momentum of

the

d i f f u s e r d i sc h a r ge . I n g e n e r a l , e s t a b l is h m e n t o f a

s t eady s t a t e c i r c u l a t i o n p a t te r n

i n

t he bas in r equ i red abou t

5 t o

10

minutes . For runs wit h

no

ne t ho ri zo nt a l momentum th i s w a s fo l lowed

by a n e s s e n t i a l l y s t e a d y p o r t i o n of 10 t o

30

minutes , i . e . d i l u t i o n s

i n t he d i f f u s e r v i c in i t y

were

not changing.

ho ri zo nt al momentum th e st ead y

s t a t e

p o r t i o n w it h r e s p e c t

t o

d i l u t i o n s

For

runs wi th ne t

i n t h e d i f f u s e r v i c i n i t y w a s i n

some

i n s t a n c e s s tr o n g l y l i m i t e d , t h a t

i s ,

r e c i r c u l a t i o n of mixed water r e su l te d i n an uns teady d i l u t i on dec rease .

138



5.3.3 Experimental

Runs

The

run parameters

are

g iv en i n

Chapters 6

and

7 ,

t o g e t h e r

wi th exper imenta l re su l t s . Three

ser ies

of experiments

were

performed

i n t h e b a s i n:

S e r i e s

BN - No ne t

h o r i z o n t a l momentum (S ectio n 6.1)

S e r i e s BH - N e t h o r i z o n t a l momentum (Section 6.2)

S e r i e s

BC

- With ambient cross -flaw (Chapter

7)

5.3.4

Data Reduction

The pr in c i p l e of the da t a reduct ion

i s

t h e same as f o r t h e

flume experiments. Normalized s ur fa ce tem perature

rises

are

p l o t t e d

in th e form of isotherms i n Chapters 6 and

7

fo r se l e c t ed ex pe rimen ts .

In addit ion the non-dimensional

ve r t i ca l

temperature rise p r o f i l e s

given a t f o u r p o i n t s

i n t h e ba s in are i n d i c a t e d .

For the purpose o f comparing wi t h t he t he or e t ic a l su r f ac e

d i l u t i o n

Ss

d e f in ed arou nd th e d i f f u s e r l i n e as i n d i c a t e d i n Fig. 5-5. The ex ten t

o f t h i s

area

is comparable t o

Eq.

(3-220) describing t h e leng th o f th e

and area, 4 f t w id e, d e s c r i b i n g t h e n e a r f i e l d z one , i s

lo ca l mixing zone. The ex p e r imen ta l d i lu t io n

is

then def ined as t h e

in v e r se o f th e maximum non-dimensional su rf ac e temperature observed a t

the edge of t h i s near - f i e ld zone.

The bo tt om f r i c t i o n f a c t o r f o r t h e b a s i n is c a l c u l a t e d by t h e

ks

=

0.03 f t f o r

hite-Colebrook re la t i o n with an a bs olu te roughness

grou ted concre te sur faces .

@

i s

c a l c u l a t e d

as

4

f o

LD/H

.

For the

channel

tes ts

a c o r r e c t i o n f a c t o r ,

Eq.

( 5 - 5 ) , w a s applied.

199



5.4 Experiments by Oth e r In v es t ig a to r s

Basic

exper imenta l inv est iga t io n of submerged mul t ipor t d i f fu se rs

,

i n pa r t i cu la r , three-dimensional phenomena,

are

scarce. Most d i f fuser

s t u d ie s r e po r te d i n t h e l i t e r a t u r e

are

ap p l i ed model s tu d ie s of sp ec i f i c

design problems and thus have pec ul ia r fe at ur es of topography and cur re nt

d i s t r i b u t i o n . Few inv es t i ga t i on s addres s th e gen era l problem of a

d if f us e r i n

water

of uniform depth with o r withou t an ambient cu rr en t ,

as considered

i n

t h i s s t u d y .

The two-dimensional experiments by Liseth (19 70) o n d i f fu se r s

in deep

water

( s t ab le n ea r - f i e ld )

w e r e

discussed i n Chapter

2.

L i s e t h

o nl y r e p o r t s d i l u t i o n s i n the buoyant j e t reg ion and does not consi der

t h e e f f e c t of t h e f r e e su r f ac e .

mo-dimension al chann el model exper iment s were reported by

Harleman

e t a l .

(1971).

i n

a

21 f t l on g,

4.5 f t wide c ha nn el p l a ce d i n a l a r g e r b as in .

nozzle discharge w a s u n d i r ec t io n a l w i th

a

z e ro o r n e ar z e ro v e r t i c a l

angle.

v e r t i c a l m ix in g ( su p e r c r i t i c a l flow) and o nly a few ru n s r e su l t ed i n

The bulk of

th a t exper iment

w a s

performed

The

Most of t h e i r runs

w e r e

made

i n

t h e p a rameter r ang e wi th f u l l

a st ag na nt wedge system.

the presen t theory .

In Chapter 6 t h e i r r e s u l ts

are

compared w it h

Two-dimensional experiments were a l s o made by

Larsen

and Hecker

(1972).

However,

i n t h e i r e xp er im en ts

the

nozzles

were

i n s t a l l e d a t

a considerable hei ght above the bottom ( i n some cases a t hal f dep th)

which h a s l i m i t e d p r a c t i c a l s i g n i f i c a n c e .

compared t o t he pre se nt theo ry.

Th e i r r e su l t s can n o t b e

200



T h r e e - d i m e n s i o n a l diffuser tests were a l s o performed by

Harleman e t a l . ( 1 9 7 1 ) a n d are summ arized by Adams (1 9 7 2 ). A l l t h e s e

tests

were made

i n

a b a s i n 46

X

28 X 1 . 5

f t and

had ambient

cross-

flow.

In

Chapter

7

t h e i r r e s u l t s

are

c o m p a r e d w i t h t h e c r o s s - f l o w

e x p e r i m e n t s of this

study .

201



V I .

DIFFUSERS WITHOUT

AMBIENT

CROSSFLOW:

COMPARISON OF

THEORY

AND EXPERIMENTS

6 . 1 Di ff us er s w it h No N e t Horizontal Momentum

6 . 1 . 1

TwoLDirnensioaal Flume EXDerimentS

The two-dimensional f lume experim ents w it h

v e r t i c a l

d i sc ha r ge

a ng le ,

Ser i e s

FN, a r e summarized i n Table 6 . 1 . The r e l e va n t phys ic a l

va r i ab le s and non-dimensional param eters a re gi ve n. Most of th e

experiments used a c o nt i nu o us s l o t d i s c h a r g e , a few experiments

were

done wi th a s i n g l e n o z z l e . The

j e t

Reynolds number d id n ot ,

i n

a l l

cases, s a t i s f y t h e r e qu i re m en t , E q ua ti on (5-3). Although

v i s u a l i n s p e c t i o n showed t h a t t h e

j e t

s t r u c t u r e w a s a l w a y s c l e a r l y

t u r b u l e n t , n e v e r t h e l e s s

i t

i s p o s s i b l e t h a t t h e t u r b u l e n t e n tr a in -

ment capa c i t y of t h e j e t i s w eak er t h a n p r e d i c t e d b y t h e t h e o r e t i c a l

ana lys is assuming a f u l l y t u rb u l en t j e t . I n t h e e x p er i m e nt a l r an g e

t h e b ot to m f r i c t i o n f a c t o r , f o , i s both dependent on channel Reynolds

number and r e l a t i v e rough ness. The chan nel Reynolds number,

Equation

( 3 - 2 1 2 ) ,

c a n b e e s t i ma t e d u s i n g t h e t h e o r e t i c a l p r e d i c t i o n

o f d i l u t i o n a s a good approximation.

The v e r t i c a l s t r u c t u r e

of

t he f low f i e l d i n t he two-d im ensional

channel model can be in fe rr ed from temp erat ure measurements and

v i s u a l o b s e r v a t i o n . F i g u re

6-1

g i v e s

a

comparison between t h e

t y p i c a l f lo w f i e l d s f o r ( a) a s t a b l e and ( b) a n u n s t a b l e n e a r - f i e l d

zone.

The

t h e o r e t i c a l l o c a t i o n of t h e i n t e r f a c e , E q ua ti on

(3-207),

shows good agreement wi t h t he observed tempera ture pr o f i le .

in t e r n a l hyd r a u l i c jump r e g ion i s c l e a r l y v i s i b l e i n F i gu re 6 - l a,

The

202



P;,

O

W

H

Table 6 . 1 :

Flume

Diffuser Experiments,

Series

FN

Governing

Physical Variables Secondary Parameters Parameters

(ft) (10%) (ft/sec) (ft) (OF) (OF) m L/H

fo c 2 fo

Fs

H/B

( d w ) Ss

@bserved Predicted

eO

L

Ta ATo

0 *

1

FN-1 2.00 10.4

2 2.00 10.4

3 2.00 10.4

4 2.00

3.3

5

2.00

3.3

6 2.00

3.3

7 2.00

3.3

8

1.33 3.3

9 1.00 5.2

10 1.00 5.2

11 2.17 5.2

12 2.00 10.4

13 1.25 20.8

14

2.17 1.7

15 2.17 1.7

16 2.17 1.7

17 2.00

10.4

18

2.00 3.3

19 1.25 20.8

20n

2.00 3.4n

21n 2.00 3.4n

22n 2.17 1.7n

23n 1.33

3.4n

FN-24n 2.17 5.8n

0.64

0.43

4.27

1.72

0.88

1.51

4.71

2.69

1.86

5.34

0.64

2.13

1.07

1.31

3.92

1.30

1.86

4.71

1.07

2.61

1.63

3.92

2.61

0.57

20 67.5 15.3 1,610 10

0.039 1.7 0.066

25 192

9n 0.7 11.8

20 66.4 29.0 1,070 10

0.039 1.7 0.066 13

192 90 0.7 17.0

20 64.7 16.3 10,700 10

0.032 1.7 0.054

160 192

90 0.5 4.5

20 67.4 24.1 1,370 10

0.038 1.7 0.065

87

600 90 0.6

15.0

20 68.0 10.9 700 10

0.042 1.7 0.071

71

600 90 0.7

15.5

20 68.2 29.5 1,200 10

0.035 1.7 0.060 67

600 90 0.6

14.7

20 66.1 17.5 3,750 10

0.036 1.7 0.061 291

600

90 0.6 6.7

20 67.3 22.0 2,140 15

0.041 1.4 0.058

144

400

90 0.9 7.3

20 66.9 17.9 2,330 20

0.047 1.3 0.061

91 192 9r) 1.2

5.1

20 67.6 16.9 6,700 20 0.042 1.3 0.055

270 192

90

1.1 2.7

20 65.4 24.8 810 9

0.040 1.7 0.068

31 416 90 n.6

17.7

20 65.8 16.1 5,350 10

0.036 1.3 0.047

80 192

90 0.5 5.5

20 67.5 21.0 5,360 16

0.037 1.4 0.052

23 50

90 0.8 4.6

20 69.0 14.4 540 9

0.041 1.7 0.070

128 1300 90 0.6

20.6

20 69.0 23.0 1,610 9

0.037 1.7 0.063 293 i3no ?n

0.6 17.7

20 69.4 28.4 540 9 0.040 1.7 0.068 85 1300 90 0.6 25.6

12 67.5 15.5

4,760 6

0.035 1.7 0.060 70 192

90 0.4 6.7

12 68.2 15.1

3,750 6

0.036 1.7 0.061 312 600 90 0.4

7.5

12 68.2 21.4 5,360 10 0.037 1.4 0.052

23 60 90 0.5

4.1

20 73.0 22.2 2,140 10 0.037 1.7 0.063

131 585

90 0.6 11.1

20 73.2 28.3 1,340 10

0.038 1.7 0.065

71 585

90 0.6 13.17

20 71.2 23.7 1,610 9

0.037 1.7 0.063

231 1240 90 0.6

15.0

20 67.5 21.8 2,140 15

0.041 1.4 0.058

140

390

90 0.9

7.2

20 67.5 24.6 810 9 0.041 1.7 0.070 22 377 90 0.6 12.3

Note:"n" denotes experiments with

a

single nozzle, equivalent slot width (spacing 1.0

ft)

s s

12.2

17.0

3.5

16.5

19

o

19.2

7.3

7.0

4.9

3.2

21 0

5.6

4.4

28.n

16.4

36

.O

6.2

7

.0

4

.0

12.1

18.0

18.2

7.0

23.0



- - -

I

- -

m

-5 0

c

D

*

f

C

0

N

n

V

.-

-

‘a

z

0

1

a

d

aJ

.rl

Fr

I

&I

ul

Q)

2

P)

rl

U

cn

L D

0

II

f

m

f i

a

5

0 0

a d

204



as

i s t h e l o c a l m i xi ng z on e w i t h a c r i t i c a l se c t ion ( r a p id c ha nge

i n t h e l o c a t i o n of t h e i n t e r f a c e ) i n F ig u r e 6 -l b.

Table 6-1 a l so p rovid es th e comparison be tween observed

AT

's

AT

- (where

AT

i s t h e s u r f a c e t e mp e ra t ur e

u r f a c e d i l u t i o n ,

r i se

a t t h e e dg e o f t h e n e a r f i e l d zo ne

as

shown i n Fi gu re 6-11, and

t h e t h e o r e t i c a l d i l u t i o n as deve loped i n Chapte r

3 .

This comparison

i s a l s o g i v en i n F i g u re s 6-6 @ 0.5) and

6-7 (@

1.0) a long wi th

th e r e s u l t s f rom th e th r e e- d im ens ional ba s in t e s t s . ( F i l l e d o r un -

f i l l e d s ymbols i n d i c a t e a n u n s t a b l e o r s t a b l e n e a r - f i e l d , r e s pe c ti v e -

ly,

a s

de te r mined by v i su a l obse r va t ion a nd by t e m p e r a tu r e p r o f i l e s ) .

I n g e n e r a l, t h e r e i s ex ce l l en t agreement be tween the ory and exper i -

m en ts , b o t h w i th r e g a r d t o t h e c r i t e r i o n be tw ee n s t a b l e and u n s t a b l e

domain and wi th r e ga r d t o the d i lu t i on s . The obse rve d d i lu t i on s

are somewhat lower tha n pre di c t ed fo r some exper iments i n th e

s t ab l e domain, which i s a t t r ib u t ed t o low Reynolds numbers of th e

j e t d i sc ha r ge .

The r e s u l t s f o r t h e s i n g l e n oz z le t e s t s a gr e e w e l l

w it h t h e s l o t t es t s f o r t h e u n s t a b l e d omain, b u t are somewhat

l ow er i n t h e

s t a b l e

domain indica t ing incomple te l a t e r a l merging

f o r a spa c ing ,

R,

e q u a l t o h a l f t h e d e pt h , H.

6.1.2

Three-Dimensional Bas in Experi ments

Table 6.2

i s

t h e summary

of

t h e b a s i n e x p er i m en t s of m ul t i -

po r t d i f f u se r s w i th no ne t ho r i z o n ta l momentum, S e r i e s

BN.

Most

o f th e r uns

were

t h r e e - d im e ns iona l ,

a

few

were

done on the two-

dimensio nal channel model by int rod uci ng

a

f a l s e

w a l l

p a r a l l e l

t o th e x - ax i s. Again i n some t e s t s th e n ozz le Reynolds number

i s

lower than the g iven requirement .

205



BN-1 0. 67

2

0.67

3

0.67

4 0. 67

5 0. 67

6 0.67

7 0. 67

8

0.43

9 0. 67

10 0. 67

11 0.67

12 0.92

13 0. 67

14 0.67

15 0.67

16 0. 67

17c 0.67

18c 0.66

19c 0. 67

20c 0. 67

21c 0.67

22c 0.67

BN- 23c 0.67

Table 6.2: Basin Diffuser Experiments, Series

BN

Physi cal Vari abl es Secondary Parameters Governi ng Parameters

H . D

uo LD

Ta eO

( f t )

(10- 3f t>

( f t )

(f t/ sec) ( f t ) (OF)

(OF)

IRj L/ H LD/H

f o Fs

H/ B

(dpr) 4

B(v)

10. 4

0.08

2. 42 6. 4 58.8

27. 0

2, 870 0. 12 10 0.09 158 628 90 0.9 NOR

10. 4

0.08

2.42 6. 4

64. 0 25.3 3,000 0. 12 10 0. 09 160

628 45-A 0.9 LIN

10. 4

0.08

2.42 6. 4 65. 3 26. 4 3,090 0.12 10 0.09 157 628 45-A 0.9 SIN

10.4 0.08

2.42 6.4 67.9 27. 3 3,210 0.12

10 0.09 154 628 45-A 0.9 N-S

10. 4 0.08

2.42 6. 4

65. 6 27. 4 3,120 0.12 10 0.09 154

628 45-A 0.9

LOG

10. 4 0.08 3.60 6.4 67. 2 26. 6 4,700 1-1.1210 0.08 231 628 454 0.9 LOG

10. 4 0.16

4.84 6. 4 68. 5 24. 6 6, 260 0.24 10 0.09 458 1256

45-A 0.9 LOG

10. 4

0.08 1.21

6. 4 63. 2 24. 9 1,480 0.19 15 a08 81 402

45-A 1.2 LOC

15.6 0.16

2.15 6. 4

63.7 26. 0 4, 010 0.24 10 0.09 131 558

45-A 0.9

LOG

21. 3

0.32 2. 31

6.4 64.A 24. 7 5,A10 0.48 10 0. 11 150 600 45-A 1.1 LOG

10. 4

0.16 1.50

6.4 64. 2 23. 0 1,820 0.24

10 0. 10 148 1256

90 1.0

NOR

10. 4

0.16

1.50 6.4

64. 0 23.0 1, 810 0. 17

7

0.08 148 1727

90 0. 6 NOR

10.4

0.08

2. 42 6. 4

64. 7 26. 1 3,040 0. 12 10

o m

158 628 0-A

0.9

LOG

10. 4

0.16

4.84

6. 4 65. 2 25. 3 6, 070 0.24 10 0.09 454 1256 0-A 0.9 LOG

15. 6

0.16

2. 15 6.4 65. 9 25. 9 4, 110 n.24

10 0.09 133 558

0-A

0.9 LOG

15. 6

0.16

0.87

6.4 66.6 22. 5 1, 610 0.24 10 0.10 58 558 0-A

1.0

LOG

10. 4

O a O 8

2.42 6 . 0 ~3.9 29. 3 3,130 0.12 9c 009 149 628 90 0.8

-

15. 6

0.16

2.15

6 . 0 ~4.4 22. 7 3, 890 0.24 9c 0.09 142 552 45-A 0. 8

-

10. 4

0.08

2.42

6 . 0 ~4. 7 27.1 3, 080 0.12 9~ 0. 09 155 628

45-A 0. 8 -

10. 4

0.16

2.43

6 . 0~3. 4 26. 4 3, 020 0.24 9c 0.09 223 1256 45-A

O . R -

10. 4

0.16

150

6 . 0 ~3.4 26. 5 1, 860 0.24 9~ 0.09 137 1256

45-A 0.8

-

15. 6

0.16

2.15

6 . 0 ~4.3 28. 3 4, 140 0.24 9~ 0.09 127 558

0-A

0.8 -

10. 4

0.16

4.84

6 . 0 ~3.4 28. 4 6, 150 0.24 9~ 0.09 429 1256

0-A O R -

4

ote: c denotes channel experi ments, w th

the

channel l ength, L, gi ven under

I1A

t

denotes al ternati ng nozzles

Observed Predi cted

sS

10.0

10.0

5.9

11. 8

9. 5

8. 3

13. 2

9.1

10.0

10. 5

17. 3

27 0

10. 4

10. 0

10. 5

15. 1

8. 6

9.1

9. 4

12. 2

15.0

9. 2

9. 3

s s

N. A.

N. A.

N. A.

N. A.

10. 6

8 . 0

10. 4

10. 5

i n. 4

22

n

10. 5

37 . 0

10. 5

10. 4

i n.4

19. 5

10. 4

10. 4

i n. 5

16. 7

22. 5

10. 4

10. 7



The f i r s t f i v e e x p er im en ts s e r v ed t o e s t a b l i s h t h e u s e o f t h e h or -

i z o n t a l n o zz le o r i e n t a t i o n as

a

c o n t r o l of t h e h o r i z o n t a l c i r c u la -

t i o n s

induced by

a

d i f f u s e r d i s c h a rg e w i th a n u n s t a b l e n e a r - f i el d .

I n C h a p t e r 4 t h e t h e o r e t i c a l s o l u t i o n f o r t h e h o r i z o n t a l n o zz le

o r i e n t a t i o n , B ( y ) ,

a long

t h e d i f f u s e r l i n e h a s be en d e r i v e d ,

E q u a t i o n ( 4 - 3 6 ) , b a sed on t h e m od el of t h e t h r e e -d i m e n si o n a l s t r a f i -

f i e d fl ow f i e l d .

To

c he ck t h e v a l i d i t y o f t h i s e q u a t i o n and a l s o

t h e s e n s i t i v i t y d u e

t o

d e v i a t io n s i n t h e n o z z le o r i e n t a t i o n t h e

f o l l o w i n g f? (y) were checked:

1) N o r m a l d i s t r i b u t i o n (NOR):

All

n o z z l e s

are

n or ma l t o t h e

d i f f u s e r a x i s o r v e r t i c a l l y u pw ard , t h u s p r o v i di n g no d i sc h a r g e

momentum i n the y -d i re c t io n (a long the axis o f t h e d i f f u s e r ) .

B (y) = 90

=

c o n s t . (6-1)

2) L i n ea r d i s t r i b u t i o n (LIN) :

B(Y)

=

goo

(1- )

LD

B(y)

=

cos-l(Y-)

LD

3 )

S i n u s o i d a l d i s t r i b u t i o n ( SIN ):

4 )

Normal

-

n u s

o

i d a l

d i s t r i b u t i o n

(N-

S

(900

B(Y) =

The d i s t r i b u t i o n g i v e n b y E q ua t i on ( 4 - 3 6 ) i s c a l le d f o r s i m p l i c i ty

t h e :

5 ) " L o g a r i t h i c " d i s t r i b u t i o n (LOG)

( 4 - 3 6 )

207



T h e d i f f e r e n t B(y) a re p l o t t ed i n F ig u re 6-2 . Runs

BN - 1

t o 5 have

c l o s e l y

s i m i l a r

d i s c h a r g e c o n d i t i o n s . The r e s u l t i n g te m p e r a t u re

f i e l d s f o r r u ns 1 , 2 and

5

a re shown i n Fi gu re s 6-3a,b and 6-4a.

V e r t i c a l t e mp e ra t ur e p r o f i l e s

a t

f o u r p o i n t s a r e g i v e n a nd s u r f a c e

c ur r e n t d i r e c t io ns and m a gn i tude s

a re

i n d i c a te d q u a l i t a t i v e l y .

For t h e no rm al d i s t r i b u t i o n (NOR), F igur e 6-3a, w i thou t any d i f f us e r

momentum opposing th e inward f low a lon g th e d i f f us e r l i n e , a s t r o n g ,

v e r t i c a l l y f u l l y mixed flow-away a long th e x - a x is r e s u l t s . The

s i t u a t i o n

i s

r ev e rs e d f o r th e l i n e a r d i s t r i b u t i o n

(LIN),

Figure 6-3b;

t o o

much n oz zl e o r i e n t a t i o n i n t h e y d i r e c t i o n r e s u l t s i n

a

dominant

flow-away a long the y -a x is ( d i f f u se r l i n e ) which

is

a g a i n r a t h e r

w e l l

m ixed . F or the " loga r i thm ic d i s t r i bu t i on "

(LOG)

, F ig u re 6-4a, how-

ever ,

t h e f low f i e l d i s a ppr ox im a te ly un if o rm i n

a l l

d i r e c t i o n s a nd t h e

v e r t i c a l t em p er at ur e p r o f i l e s i n d i c a t e t h e s t r a t i f i e d c ou n te rf lo w

system

a t a l l

p o i n t s i n t h e d i f f u s e r v i c i n i t y . I n g e n er a l , i t

i s

found (see F igur e

6-2)

t h a t n o z z l e o r i e n t a t i o n s p r ov i d i ng

l e s s

d i s -

charge momentum i n th e y-di r ec t ion tha n in dic a te d by Eq ua t ion (4-36)

t en d t o

a

flow-away alo ng t h e

x-axis

a nd v ic e

versa.

The flow con-

d i t i o n n e a r t o t h e LOG d i s t r i b u t i o n is n o t v e r y s e n s i t i v e . The

t h e o r e t i c a l d i l u t i o n , a ss um in g c o n t r o l , i s 1 0. 6 f o r t h e s e cases.

I t i s seen t h a t t h e

NOR

d i s t r i b u t i o n h a s a s t r o n g l y r ed uc ed d i l u t i o n ,

w h i l e f o r t h e o t h e r n o zz l e d i s t r i b u t i o n s , t h e o bs er ve d and t he o-

r e t i c a l

d i l u t i o n s

are

i n good agreement.

It

sho uld be n ot ed , how-

e v er , t h a t

a l l

d i s t r i b u t i o n s , e x c ep t t h e

LOG

d i s t r i b u t i o n ,

s e t

up

h o r i z o n t a l c i r c u l a t i o n s which u l t i m a t e l y l e a d t o r e c i r c u l a t i o n

208



9c

80

Y>

60'

40"

20'

OC

0

NOR

-4

6

-8

y/

1 - 0

LD

Example:

LOG, 10

unidirectional nozzles

I

I

E n d

Fig.

6-2:

Nozzle Orientation, @(y),

Along

the Diffuser Line

209



'/H

'

I

I I I

I I

I

1

I I '

30 Surface lsother ms -

a)

Run BN-l,Fs = 158,

H / B

= 628,

Bo

=

go", @

=

0.9, B(y)

=

NOR

Surface Isotherms

fH

3 0

2 0

a5

0 1 0 2

1 0

-4

*To

0

b) Run BN-2, Fs = 160, H/B

=

628,

e 0 =

45 -A , @

= 0.9, B(y)

= LIN

Fig . 6-3: Observed Temperature and Flow Fi el d, S e r i e s

BN

210



a) Run BN-5,

FS = 154, H/B

=

628, g o = 45"-A, 0 =

0.9, B(y) = LOG

1 I I I I I k I I I I I

1 1 I

50

Surface

Isotherms

AT

-0.075

4

x-

40

-

-

-

30

20 -

-

10 -

I

I I 1 I I

-20 -10

'

-

4 0 -30

b) Run

BN-8, Fs = 81,

H / B

= 402,

e

= 45"-A,

@

= 1.2,

B(y)

=

LOG

Fig . 6-4: Observed Temperature and Flow F i e l d , Series BN

211



i n t o t he d i f f u s e r l i n e . The d i l u t i o n s g i ve n do n o t a cc ou nt f o r t h i s

e f f e c t . I n t h i s r e sp e ct t h e LOG d i s t r i b u t i o n i s the only one which

c r e a t e s

a

s t a b l y s t r a t i f i e d f lo w sy st em i n t h e f a r - f i e l d w hich i s

n o t p ro n e t o r e c i r c u l a t i o n .

Two more examples of th e te mper a tur e f i e l d produced by

a

d i sc ha r ge wi th LOG d i s t r ib u t io n

are

g i v e n

i n

Figure 6-4b (8

more sh all ow wa te r) and Figure 6-5a

(0

have a r e l a t i ve ly h ighe r va lue f o r th e y-momentum of the d i sc ha r ge

= 4 5 O ,

0

= 0') . These exper iments

0

as compared t o F igu re 6-4a. The re s u l t i n g f low f i e l d , however ,

i s s i m i l a r ,

g i v i n g s u p p o rt t o t h e d i s c u s s i o n i n C ha pt er 4 , t h a t

i t :

i s no t t h e abso lu te va lue of the d is ch ar ge y-momentum t h a t i s

i m p o r t a n t , b u t r a t h e r

i t s

d i s t r i b u t i o n as l o n g

as

t h e

v e r t i c a l

a n g l e

1 i s below Bo

,

Equation (4-43)

max

A d i f f u s e r w i t h a s t a b l e n e a r - f i e l d ( t h us re q u i r i n g no c o n t r o l

B(y) = NOR) i s

shown

i n F igu re 6-5b.

The f low f ie ld can be compared

t o F ig ure 6-3a. With a s t a b le ne a r - f i e ld th e flow-away i s uniform

i n a l l d i r e c t i on s . The e xper ime n t i n F igu r e 6-5b i s ne a r t he e dge

o f t h e s t a b l e z on e

(see

Figure 6-61

and may be somewhat affected

by i n s t a b i l i t i e s e x p l a i n i n g t h e w ea ke r flow-away i n t h e y - d i r e c t i o n .

O bs erv ed and t h e o r e t i c a l s u r f a c e d i l u t i o n s

are

compared i n

Table 6.2 and Figures 6-6 (@ 0.5) and 6-7

(@

1.0). Good

agreement i s found f o r both three-dimens ional and two-dimensional

b a s i n e x p e ri m e n ts . Due t o e x p e r i m e n t al l i m i t a t i o n s ( j e t t u rb u -

l e nc e ) t he e xper im e n t s were m ai nl y r e s t r i c t e d t o t h e u n s t a b l e

nea r - f ie l d domain. Var ious exper imen ts wi th

s imi l a r

F

H / B

b u t

d i f f e r e n t

v e r t i c a l

a n g l e

€lo

nd spa c ing k / H , were made t o assess

S'

212



fH

30

I- i

I

I I

I

I 1 I 1 I

1

Surface Isotherms

-

0 . 1 s

a2

0 1

AT-

a) Run BN-13, Fs

=

158, H/B = 628, 8

= 0'-A, @ = 0.9, B(y) = LOG

0

0.03

* I

.

n

0.5zF-

0 -

A le

b)

Run

BN-12,

Fs =

148, H/B

=

1727,

eo = go ' ,

@

= 0.6, B(y) =

NOR, Stable

Fig.

6-5:

Observed Temperature

and

Velocity Field, Series BN

2L3



10000

50 100 5

00

~

8,

=

go c) Series F N

-

stable near-f ie ld

Series F N

-

unstable near-f ie ld

@ Series BN - stable near-field

=

0.5

ss

i g . 6 - 6 :

P r e d i c t e d

V s ,

O b s e r v e d

Dilutions,

214



Fig. 6-7 ;

P r e d i c t e d

Vs.

Observed Dilut ions ,

' ss.

215



t h e e q u i v a l e nt s l o t c on ce p t f o r d i f f u s e r s w i t h a l t e r n a t i n g n o z z le s .

A s c an b e s e e n t he d i l u t i o n s a r e , i n de e d, i n de p en d en t of v e r t i c a l

a n g l e a nd sp a ci n g, s o t h a t t h e e q u i v a l e n t s l o t r e p r e s e n t a t i o n i s

v a l i d .

6 . 2 D i f f u s e r s w i t h N e t H or iz o nt a l Momentum

6 . 2 . 1 Two-Dimensional Flume Exp eri men ts

Run p a ram e te r s f o r t h e f l um e ex p e r im en t s w i t h u n id i r ec t i o n a l

d i s c h a r g e , S e r i e s FH, are g iv en i n Table 6 . 3 . The runs

were

l i m i t e d

t o t h e lo w Fs, h ig h H / B r a n g e f o r t h e f o l l ow i n g r ea s o n s :

( i )

The f u l l y mixed range (high Fs , low H/B, eo

= 0)

has been

ex t en s iv e l y i n v e s t i g a t e d i n t h e ch an n e l ex p e r im en t s by Harleman

e t a l . ( 1 9 7 1 ) . T h e i r e x p e r i m e n t a l r e s u l t s w e r e an a ly s ed u s in g

k = 0.002 f t (planed-wood f lum e) ,a nd

are

compared i n F igu res

6 - 1 2

(@

Z 0.5) and 6-13 (a z 1) t o t h e o r e t i c a l p r e d i ct i o n s .

S

(ii)Pre l im in a ry ex p e r im en t s

of

t h e fl um e s e t-u p i n t h e f u l l y mixed

ran g e e s t ab l i s h ed two s h o r t co m in g s:

(a) The " re s e rv o i r " o u t s i d e

the channel t e s t s e c t i o n

w a s

n o t l a r g e e no ug h, t h u s v e l o c i t i e s

and i n cu r red h ead l o s s e s w e r e n o t n e g l i g i b l e ; ( b) F o r s t r o n g

j e t d i sc h ar g es t h e s l o t j e t became at tached

t o

t h e ch an n e l

b o tt o m r e s u l t i n g i n s i g n i f i c a n t momentum d i s s i p a t i o n .

It may

b e c o nc lu de d t h a t w h i l e t h e e q u i v a l e n t s l o t c o n c e p t c an a lw ay s

b e u s e d i n m at h e m a ti c a l terms t o r e p re s e nt a m u l t i p o r t d i f f u s e r

i t

i s

n o t

a

d e s i r a b l e e x p e r im e n t a l s c h e m a t i z a t io n u n de r a l l

c o nd it i on s . I n p a r t i c u l a r , i t d oes n o t a l l o w p en e t r a t i o n of

ambien t en t ra inment w a t e r coming f rom the ups t ream d i r ec t i on

216



Table

6.3: Flume Diffuser Experiments, Series FH

Physica l Variab le s Secondary Parameters

Governing

Parameters

FH-1

2.00

3.3

1.68 20 71.4 31.3

2

2.00 10.4

0.64 20 65.7 16.9

3n

2.00

3.4n

1.63 20 69.1 29.9

4

2.17

1 . 7

1.3 1 20 66.6 14.9

5 2.00

3.3

1.68 20 67.0 29.0

6

2.00

10.4 0.64

20 70.2 13.6

FH-7n

2.00

3.4n

1.64 20 68.9 29.1

*

fO

R L/H

f o

c 2

1

1,340 10 0.036 1 . 7 0.061

1,61 0 10 0.038 1 . 7 0.065

1,340 10 0.037 1. 7 0.063

540 9 0.038 1. 7 0.065

1,340 10 0.037 1.7 0.063

1,610 10 0.037 1.7 0.063

1,340 10 0.037 1.7 0.063

Observed Predicted

0

8

Fs

H / B

(dgr)

@

Ss

64 600 45 0.6 17.5 23. 0

23 192 45 0.7 10. 1 13.2

7 1 585 45 0.6 13.6 20.5

129 1300 0 0.6 29.5 37 . O

86 600

0

0.6 16.1 26.0

20 192 0 0.6 13.6 15.0

72 585

0

0.6 19.5 25 .0

s S

Note: "nl ' denote s si ng le nozzle experiments, spacing 1.0 f t ,



and t h u s l ead s t o

a

l o w pressure zone and

j e t

a t t ach m en t

downstream of th e disc har ge .

O bserved d i l u t i o n s i n t h e e x pe r im e nt a l r a n g e ( s t r a t i f i e d

c o u n t e r f l o w c o n d i t i o n s )

a r e

somewhat lower than p r ed ic te d , p robab ly

a consequence

of

t h e l o w Reynolds number

of

t h e d i s c h a r g e . T h e r e

i s

good co r re sp o n d ence b e tween s l o t an d i n d iv id u a l n o zz l e ex p e r im en t s .

All, but th re e , o f th e channe l exper iments by Harleman e t a l .

( 197 1) r e s u l t e d i n a f u l l y mixed r eg im e. I n g e n e r a l , t h e i r r e s u l t s

a g r e e w e l l

w i t h

t h e t h e o r e t i c a l p r e d i c t i o n s . The t h r e e e x pe r im e n ts

r e s u l t i n g i n

a

s ta gna n t wedge reg ime ex h i b i t somewhat lower d i l u t io ns

tha n p re d ic ted (F igure 6-13)

6 . 2 . 2 Three-Dimensional

Basin

Experiments

The d i f f u s e r e x p e ri m en t s w i t h u n d i r e c t i o n a l d i s c h a r g e i n t h e

b a s i n , S e r i e s

BH,

are summarized

i n

Table

6.4. Runs 1 t o 9 w e r e

made with a 4 v e r t i c a l a n g le , t h e r e s t w i t h h o r i z o n t a l d i s c h a r g e

(0').

co n cep t fo cu s in g on the "deep

water"

regime (high H / B , low Fs).

The bu lk o f th e exper iments

w a s

concen t ra ted on the th ree-d imens ion-

a l b eh av io u r :

w e r e

a l l

i n t h e d ee p water r an g e , wh i l e ru n s 16 t o 2 4 (wi th 0

)

w e r e i n t h e ' 's ha ll ow

w a t e r "

range ( low

H/B ,

high F ) r e s u l t i n g i n

f u l l v e r t i c a l mixing.

A few experiments w e r e made on th e two-dimensional cha nne l

Runs 1 t o 6 (w i th 45') and ru n s 1 0 t o 15 (wi th 0)

0

S

The ques t ion

of

t h e h o r i z o n t a l n o z z le o r i e n t a t i o n , B (y ),

f o r d i f f u s e r c o n tr o l w a s a d d r e s s e d b y i n v e s t i g a t i n g t h e n or ma l d i s -

t r i b u t i o n

(NOR)

and t h e " lo g a r it h m i c " d i s t r i b u t i o n

(LOG)

i n p a i r s

218



T abl e

6.4:

Basin

Diffuser

Experiments,

Series BB

Predicted

hysical Variables

Secondary Parameters Governing Parameters

H D Observed This studv Adams (1972)

BH-1 0.92

18.7 0.16

0.58 6.4 58.2 24.1 1,190 0.18 7

0.090 31 534

2 0.92 15.6 0.16 0.83 6.4

57.5

24.8 1,430 0.18

7

0.085 53 768

3

0.65

15.6

0.16

2.15 6.4 61.5 25.6 3,900 0.24

10

0.090 137 546

4

0.65 15.6

0.16

1.08 6.4 61.5 23.9 1,910 0.24 10

0.095 69 546

5

0.67

15.6 0.32

1.66

6.4 62.6 22.4 2,950 0.48 10 0.095 156 1116

6 0.67

15.6

0.32

1.66

6.4 62.9 22.4 2,950 0.48

10

0.095 156 1116

7c

0.92

21.3 0.32

0.89 6 . 0 ~ 3.5 24.6 2,230 0.35 7c

0.090 59 824

8c

0.92

21.3 0.16

0.45

6 . 0 ~ 3.5 24.6 1,110 0.18

7~

0.090 21 412

9c

0.92 15.6

0.16

0.83

6 . 0 ~ 3.6 21.8 1,480 0.18 7~ 0.090 56 768

10

0.67 15.6 0.16 1.08

6.4

60.7 25.4 1,930 0.24 10 0.095 67 558

11 0.67

15.6 0.16

1.08 6.4

59.5 26.4 1,930 0.24 10 0.100 66 558

12 0.92 21.3 0.32

0.89

6.4

59.5 24.0 2,120 0.35 7 0.090 59 824

13 0.92

21.3 0.32

0.89 6.4

59.9 23.8 2,120

0.35

7

0.090

60

824

14 0.67 15.6 0.32 1.66

6.4

61.6 23.5 2,950 0.48 10 0.090 152 1116

15 0.92 18.7

0.16 0.58

6.4 58.8 23.7 1,190 0.18 7 0.090 31 534

16 0.67 15.6 0.16 2.15

6.4 60.7 27.2 3,930 0.24

10

0.100 130 558

17 0.67 15.6 0.16 2.15 6.4 59.5 28.2 3,930 0.24 10 0.085 128 558

18 0.49 21.3 0.32

1.56 6.4

59.8 27.2 3,850 0.64 13

0.100

97 436

19 0.49 21.3 0.32

1.56

6.4

60.6 26.6 3,850 0.64 13 0.100 99 440

20 0.50 15.6 0.32 2.91

6.4

61.9 25.1 5,250 0.64 13 0.090 258 837

21 0.50 15.6 0.32 2.91

6.4

62.2 25.9 5,320 0.64 13 0.090 254 837

22 0.50 10.4 0.08 2.42 6.4 62.5 26.5 2,980

0.16 13

0.085 157 471

24 0.50 21.3

0.32

2.31 6.4 60.3

25.0

4,420 0.64

1 3

0.085

150

449

23 0.50 10.4 0.08 2.42 6.4 62.8 26.1 2,980

0.15

13 0.085 158 471

25c 0.92

15.6 0.16

0.83 6 . 0 ~ 3.1 22.5 1,480

0.18

7c 0.080 55 768

26c 0.92

21.3 0.32

0.89 6 . 0 ~ 3.2 23.6 2,190 0.35

7c

0.080

60 824

BH-27c 0.92 21.3

0.16

0.45 6. 0 ~ 3.3 22.6 1,090 0.18 7c 0.085 22 412

45 0.6

LOG

17.9

45 0.6

LOG

18.5

45 0.9

LOG 13.3

45 1.0

LOG 11.8

45

1.0 LOG 18.2

45

1.0

NOR

16.7

45

0.6

-

21.8

45 0.6

-

18.9

45 0.6 - 18.2

0 1.0

LOG 14.3

0

1.0

NOR

13.2

0 0.6 LOG

20.0

0

0.6 NOR 16.1

0

0.9

LOC-

21.3

0

0.6

LOG

19.6

0

1.0 LOG 18.2

0

0.9

NOR

13.5

n

1.3 LOG 18.2

0

1.3

NOR

16.7

0

1.2

NOR

17.2

0

1.2 LOG 19.2

0 1.1 LOG 13.9

0

1.1 NOR 14.3

0

1.1 NOR

16.7

0

0.6

-

18.2

0 0 . 6

-

19.6

0

0.6

-

18.2

sS

28.0

30.0

21.0

18.5

27

.O

N . A .

30.0

28.0

28.0

20.5

N . A .

32.0

N . A .

34.5

27.2

25.6

N . A .

22.5

N . A .

31.8

N . A .

23.6

N . A .

.23.4

28.7

29.5

26.0

sS

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

N . A .

16.7

N . A .

14.7

N . A .

21-1.5

N . A .

15.3

15.0

N . A .

N . A .

N . A .

LD

ote:

“c”

denotes channel experiments, with

the

channel length, L , given under



wi th equa l

H / B , Fs,

€lo (Runs

5

and 6, 10 and 11,

1 6

and 17,

18

and 19, 20 and 21,

22

and

2 3 ) .

The r e s u l t s of t h e i n v e s t i g a t i o n w i t h r e s p e c t t o t h e ob se r ve d

f lo w f i e l d c o nf ir me d ( a ) t h e c r i t e r i a f o r t h e o c c u r r e n c e o f

a

s t r a t i f i e d o r fu l l y mixed flow-away, e s t a b l i sh ed i n C hapt e r 3,

and ( b ) t h e e f f e c t of d i f f u s e r c o n t r o l ,

B(y).

However, observed

d i l u t i o n s ,

i n c e r t a in cases up t o 40% lower. This

i s

a t t r ib u t a b l e t o

e x p e r i m e n t a l l i m i t a t i o n s as discussed below.

were

i n g e ne r a l l ow e r t h a n pr e d i c t e d by t h i s s t u d y ,

s S ,

D i f f u s e r f lo w f i e l d s i n " deep

water"

f o r

a

LOG


(Run

5 )

a n d f o r

a

NOR d i s t r i bu t i on (R un

6 )

a r e shown i n F igu re

6-8. The d i f fu se r w ith con t ro l (LOG) t ends t o e s t a b l i sh

a

s t r o n g e r

l a t e r a l sp read ing of t he d i s cha r ge wi th

a

d i s t i n c t ly s t r a t i f i e d

condi t ion , which

i s

pre dict ed from Figu re 6-11. On t h e o t h e r h an d,

t h e d i f f u s e r w i th o ut c o n t r o l

(NOR) h a s a concen t ra t ed f u l l y m ixed

flow-away al ong t h e

x-axis.

Both d i f f us e rs show some su rf ac e

eddying

a t

t h e d i f f u s e r e n d .

I n t h e " s h a l l o w

water"

range the compar i son

i s

g iven

i n

Fi gu re 6-9 (Run

18, LOG,

and Run 19,

NOR).

Both diffusers show

f u l l

v e r t i c a l

mixing as p r e d i c t e d .

co n t r o l t ends t o e s t ab l i sh t he concen t ra t ed flow-away wi th a con-

t r a c t i o n , as pr ed ic te d by th e the or y of Adams (1972). A t t h e

edge of t h e f low-away zone d e n s i t y e f f e c t s l e a d t o s t r a t i f i c a t i o n

and l a t e r a l s p r e a d i n g , t o

a

s t r o n g e r d e g r e e s o f o r t he LOG di s -

t r i b u t i o n .

This

phenomenon makes t h e obs er vat ion of t h e co nt ra ct io n

A gain t h e d i f f u s e r w i t h o u t

220



5

2 0 -

10

0

'4

I

I I I

1

-

-

-

-

. I

0 5

T ,' 4

-

,

I I I

1

I

0

AT

O 1

m

'3

y

-

b ) Run BH-6,

Fs

=

156, H/B

=

1116,

8

=

45O,@

=

1.0,

B(y)

=

NOR

0

Fig .

6-8:

Observed Temperature

and

V e l oc i t y F i e l d , S e r i e s

BH

I

(Nozzles Di rected to

the L e f t )

221

1

- IU

U

10

'/H

0.5

'

0

zp-.5

a)

Run BH-5, Fs

=

156, H/B

=

1116, e0 = 45",

@

= 1.0, B(y)

= LOG

\

k

Surface lsot

herms

0.5

I

.-

0

0.1



I I

I I

I

I

I

1

40

'/H

30

2 0

1 0 -

-

-

-

-

z

' 2\

-

' ; H I F ,

AT

0.'

ATe

- 0 5

2 ' 3

4

I

1

I I

I

1

I 1

0

O'

-310

'

-

2 0

-

10

0

10

20

30

/H

-

a) Run BH-18, Fs

=

9 7 , H/B

= 436,

Oo = 0", Q

=

1.3, B(y) = LOG

0.5

0.1

0

b ) Run BH-19,

Fs

= 99, H/B =

440,

e 0

= O ,

@

= 1.3,

@(y) =

NOR

Fig.

6-9:

Observed Tempera ture and Veloc i ty F ie ld , Se r i es

BH

(Nozzles D i r ec t e d t o t h e

L e f t )

222



of th e f low f i e l d somewhat

o b t a in ed f rom dy e o b s e rv a t

i n a c c u r a t e , b u t

ons, a c o n t r a c t

as

an approx imate measure ,

on of 0.7 i s t y p i c a l

f o r d i f f u s e r s i n t h e s ha ll ow

water

rang e (a s compared t o

0.5

from

Adams t h eo r y , n e g l ec t i n g f r i c t i o n i n t h e flow-away)

.

I n T a b le 6.4 o b s e r v e d d i l u t i o n s ,

S s ,

are compared with the pre-

d i c t i o n s of t h i s s t ud y ( a p p l i c a b l e o n l y t o d i f f u s e r s w i t h c o n t r o l ,

h av in g no co n t r ac t i o n ) and t h e p re d i c t i o n s by Adams (ap p l i c ab l e o n ly

( a )

t o

d i f f u s e r s w i th o ut c o n t r o l , w i t h c o n t r a ct i o n , a n d ( b ) i n

t h e

s h a l l o w water r an ge w i th f u l l v e r t i c a l mixing) . As m en t io n ed , t h e

o b se r ve d d i l u t i o n s i n t h e b a s i n e x pe r im e nt s are i n g e n e r a l l ow er

t h a t p r e d i c te d by t h i s s tu dy , i n c e r t a i n

cases

up t o

40%

lower

(see F i g u r e s 6-10 t o 6-13) .

a gr ee me nt s o l e l y r e f l e c t s t h e e x p er i m en t a l l i m i t a t i o n s

of a re l -

I t i s a rg ue d t h a t t h i s l a c k o f

a t i v e l y s m a l l b a s i n

as

d i s c u s s e d i n C h a pt er 4:

(i)

In the "deep

water"

r a n g e t h e

j e t

t u rb u l en ce i s too low,

as

i n

t h e case of t h e ex p e r im en t s o n d i f f u s e r s w i th no n e t h o r i zo n -

t a l momentum.

( i i ) I n t h e " s ha ll ow

water"

r a ng e t h e r e l a t i v e s t r e n g t h o f t h e

h o r i z o n t a l c i r c u l a t i o n s as i n f l u en ced b y t h e b as i n b o u n d a r i e s

l e ad s t o a dilemma: ( a ) To o b t a in s t ead y - s t a t e co n d i t i o n s t h e

ex p e rim en t h as t o l a s t l o n g en ou gh , w h i l e (b) w i th i n c re as i n g

t i m e a n i n cr e a si n g r e c i r c u l a t i o n i n t o t h e d i f f u s e r l i n e t a k e s

p l a c e .

g i v e n i n T a b le

6.4

do not c o n t a i n

s S

*

l l m eas u red d i l u t i o n s ,

2 2 3



8,=45"

@ = 0 5

Fig. 6-10: P r e d i c t e d

V s .

O b s e r v e d Dilu t ions , Ss

0

Series

F H

x Series BH

2 2 4

?



1ooc

500

100

5 0

10

r

I I

I

F i g . 6-11;

Predicted Vs. Obseryed

Dilut ions ,

S S

225



r

e = oo Q =

Series

FH

x = Series BH

@

0.5

A = Harleman

et. al.

(1971)

- Fully

mixed

-Stagnant wedge

. .226



?

€3 =oo x Series BH

@ = I A Harleman et al (1971)-

fully

mixed

I

I I

Fig. 6-13:

Predicted

Vs.

Obs e r ve d D i l u t i o n s ,

S s

227



any e f f e c t of r e c i r c u l a t i o n , i . e . , c a r e

w a s

take n t o make a measure-

ment as l a t e

as

p o s s i b l e w h il e n o t h av in g r e c i r c u l a t i o n . F i g u re s

6-8

and

6-9

are

c h a r a c t e r i s t i c p l o t s

of

t h i s s i t u a t i o n . On t h e

o the r ha nd , th e e s t a b l i shm e n t

of

an approaching "s te ady- s ta te " could

n o t b e i n su r e d i n t h i s f a s h i on .

I n t h i s c o n t e x t i t i s emphasized that

a l l

d i f f u s e r s w it h

s i gn i f i c a n t ho r i z on ta l m omentum, e ven i n a t r u l y unbounded s i t u a t i o n

(absence

o f

b a s i n w a l l s )

w i l l

e x h i b i t s ome r e c i r c u l a t i o n ,

as

can be

shown by th e Rankine-theorem of t h e p o t e n ti a l f lo w analog y (vo rtex

f low).

I t

i s ,

however, th e degree and th e t i m e

scale

o f t h e

re-

c i r c u l a t i o n w hich

i s

s t r o n g l y a f f e c t e d by t h e b a s i n b o u nd a r ie s .

Another fa c t or which probably has some e f f e c t on th e low va lu es of

t h e o b s er ve d d i l u t i o n s i s j e t a tt ac he me nt f o r t h e h o r i z o n t a l

d i f f u s e r d i s c h a r g e

(eo =

0 l e ad i n g t o d i s s i p a t i o n o f h o r i z o n ta l

momentum. Ob ser vat ion thro ugh th e mi rr or flume showed a pronounced

a t tachement of t h e i n d i v i d u a l r o u n d

j e t s

bef ore merging and of t he

s l o t j e t a f t e r m er gi ng .

The obser ved d i lu t i o ns (see Table 6 . 4 ) f o r " s h a l l o w

water"

d i f f u s e r s w i th o ut c o n t r o l a g r e e

w e l l

wi th Adams' pr ed ic t io n , but have

t o b e t ak e n w i t h r e s e r v a t i o n i n t h e l i g h t of t h e a bo ve d i sc u s s i o n

as

t he se e xpe r im e n t s

are

s i m i l a r l y i n f l u e n c e d by u n st e a d y e f f e c t s

and boundary e f fe c t s . Fur thermore , compar ison shows t h a t , i n

g e n er a l, f o r t h e case of ne t ho r i z on t a l momentum d i f f us e r s w i th

c o n tr o l p er fo rm o n l y s l i g h t l y b e t t e r (%

10%

h i g h e r d i l u t i o n ) t h an

d i f f u s e r s w i t h o ut c o n t r o l . The d i f f e r e n c e

l i e s

m ai nl y i n t h e

228



s t r e n g t h of t h e h o r i z o n t a l c u r r e n t g e n er a t e d .

The e f f e c t o f s p ac in g ,

R/H, w a s

s t ud i ed i n c e r t a i n

t e s t

combina t ions (keep ing

Fs

and H / B abou t equa l ) , f o r example

( 2 3 , 2 4 ) .

(7c,9c) and (25c,26c). No i n f l u en ce o f n o zz l e s p ac in g on t h e

d i f fu s e r p e r fo rm an ce i s ev id en t .

6 . 3

Conclusions: Di ff us er s with out Ambient Crossflow

The t h e o r e t i c a l a n a l y s i s o f

two -d im en s io n a l ch a rac t e r i s t i c s

(Chapter 3 ) and th ree-d imens ional as pec ts (Chap ter 4 ) o f d i f f u s e r s

wi thou t ambien t c ross f low

i n

combina t ion wi th th e exper imenta l

program ( t h i s ch ap t e r ) en ab l e s t h e fo l l o win g co n c lu s io n s t o b e

d r a w n :

1) E q u iv a l ent s l o t co ncep t

Any m u l t i p o r t d i f f u s e r c a n , f o r a n a l y t i c a l p ur po se s. b e r e p r e s e n-

k d .

by

an eq u iv a l en t s l o t d i f f u s e r . T h i s i n c l u d e s d i f f u s e r s w i t h

a l t e r n a t i n g n o z zl e s

as

has been shown by t h e a na ly si s of

L i s e t h ' s

experime nts of deep water d i f f u s e r s ( C h a p t e r 2 ) an d by t h e r e s u l t s

of t h i s s tu d y f o r s h a ll o w

water

d i f f u s e r s w i t h a n u n s t a b l e ne ar -

f i e l d . U t i l i z i n g t h e e qu i v al e n t s l o t co n ce p t i n tr o d uc e s a convenient

t o o l f o r com paring d i f fu s e r s t u d i e s and p ro to ty p e ap p l i ca t i o n s :

t h e v a r i a b l e s D and R are r e p la c e d by t h e s i n g l e v a r i a b l e B. A

d i f f u s e r i s ch a ra c t e r i z ed by

F

= 90' f o r

d i f f u s e r s w it h a l t e r n a t i n g n o z zl e s.

2 ) T wo-dim en sion al d i f fu s e r ch a r ac t e r i s t i c s

H / B and

e0,

where

0

s ' 0

A s t a b l e n e a r - f i e l d z on e, w i th

a

dominant buoyant

j e t

reg ion ,

e x i s t s o nl y f o r a l imi ted range o f Fs and

H/B.

The range i s

229



dependent on 0

.

O u t si d e t h i s r a n g e t h e d i f f u s e r p er fo rm an ce

i s

gove rne d by the f r i c t io na l c h a r a c te r i s t i c s of the flow-away i n the

0

fa r - f i e l d , governed by

0

= f L / H . D i f fe r en t v e r t i c a l f lo w s tr u c -

t u r e s

w i l l

r e s u l t as a f unc t ion o f 8 and 0.

3)

T h re e- di me ns io n al d i f f u s e r a s p e c t s

0

0

The f l ow d i s t r i b u t i o n i n t h e h o r i z o n t a l p l a n e i s i n t i m a t e l y

r e l a t e d t o n ea r -f i el d i n s t a b i l i t i e s a lo ng th e d i f f u s e r l i n e and t o

th e s t ren gt h of the ho r i zo nt a l momentum input of

t h e d i f f u s e r d i s -

c h ar g e. H o r i z o n t a l c i r c u l a t i o n s w i l l b e g e n e ra t e d by t h i s i n t e r a c -

t i o n , e x ce p t f o r t h e

case

of

a

s t a b l e n e a r - f i e l d ("deep

water"

d i f f u s e r s ) w h e r e

a

v e r t i c a l l y s t r a t i f i e d t wo -l ay er ed s ys te m i s

b u i l t up.

4 ) C o n t ro l of h o r i z o n t a l c i r c u l a t i o n s

H o r i z o n t a l c i r c u l a t i o n s c a n b e c o n t r o l l e d t h r o u g h h o r i z o n t a l

o r i e n t a t i o n , B (y ), o f t h e i n d i v i d u a l n o z z le s a l o n g t h e d i f f u s e r .

For d i f f us e r s wi th no ne t h or iz on ta l momentum, B(y) g iven by

Equa t ion ( 4 - 4 3 ) ( " l o g a r i t h m i c " d i s t r i b u t i o n

-

LOG), insures a

s t a b l y s t r a t i f i e d c ou nt er -f lo w s ys te m i n t h e f a r - f i e l d . Fo r

d i f f us e r s w i th ne t ho r i z on t a l momentum, th e LOG d i s t r i b u t i o n

pr e ve n t s c on t r a c t io n of th e f low downs tre am of th e d i f f u se r and

promotes v e r t i c a l s t r a t i f i c a t i o n .

5)

D i l u t i o n p r e d i c t i o n s

The o r e t i c a l d i lu t io ns ob ta ine d from th e two- dim ensiona l

"channel model'' can be ap pl ie d t o t h e t h re e -d i me n si o na l d i f f u s e r

wi th c on tr o l by means of t h e requirement of equiva le ncy of f a r - f ie ld

230



e f f e c t s , E q u a t i o n s

( 4 - 4 )

and (4 -11) . In mathematical

terms,

e q u iv a l en t f a r - f i e l d e f f e c t s r e q u i r e L

2

t h e two-dimensional "chann el model"

i s

a bo ut e q u a l t o t h e d i f f u s e r

l e n g t h . E xp er im en ta ll y, d i f f u s e r s t u d i e s i n a two-dimensional

channel model re qu ir e a s u f f i c i e n t l y l a r g e r e s e r v o i r o u t s id e th e

channel and accoun t has t o be t aken fo r t h e i nc reased f r i c t i on due

t o t h e s i dewal l s of t h e channe l .

The im p l i ca t i ons of t hes e f i nd ings on p r ac t i ca l d i f fu se r

i . e . ,

the l ength of

LD ,

desig n and on the op era t io n of hydraul ic

scale

models of d i f fu se rs

are

discussed i n Chapter

8 .

The in t er ac t io n of t h - d i f fuser- induced flows and temperatur

f i e l d wi th an ambient cur ren t sys tem

is

t r e a t ed i n t he subsequent

chap t e r . .

231



V I I .

DI FFUSERS I N AMBI ENT CROSSFLOU: EXPERI MENTAL RESULTS

7. 1

Basi c Consi der at i ons

I n t he pr ecedi ng chapter s

t he f l ow and t emper at ure i nduced by t he

act i on of a mul t i por t di f f user i n an ot her wi se st agnant unst r at i f i ed

body of wat er of uni f or m dept h was anal yzed.

di f f user

perf ormance i s basi cal l y determ ned by f our di mensi onl ess par a-

t he rel at i ve dept h,

eters , namel y t he equi val ent sl ot Fr oude number,

H / B , t he ver t i cal angl e of di schar ge, and t he f ar - f i el d ef f ect,

@

= f L/H.

I n addi t i on, f or cont r ol of hor i zont al c i rcul at i ons t he

hori zont al nozzl e or i ent at i on, By), i s i mpor t ant .

I t

was f ound t hat t he

FS

,

0

The compl exi t y

of

t he pr obl em i s appr eci abl y i ncr eased i f t he

i nt er ac t i on of t he

di f f user di schar ge wi t h an ambi ent cur r ent syst em i s

consi der ed, as shown i n Fi gur e

7-1.

-

“ a

Two

mor e di mensi onl ess par amet ers

F i g . 7-1:

Mul t i por t Di f f user i n Ambi ent Cr ossf l oy

232

x



c a n be in t r oduc ed ,

u H

u n i t w i d t h d i s c h a rg e of c r o s a f l o w

u n i t w i d t h d i s h c a r g e

of

d i f f u s e r

= volume f lux

= - -

OB r a t i o

y =

a n g l e o f d i f f u s e r axis w i t h d i r e c t i o n o f c r o ss f lo w

i n wh ic h u

t h e d i f f u s e r fl ow f i e l d i n a mb ie nt c r o ss fl o w

i s

depending on 7 dimen-

s i o n l e s s

parameters.

=

und i s tu r bed c r oss f low ve l oc i t y ups t re a m.

Thus i n ge ne r a l ,

a

The e f f e c t s o f c r oss f low may be c l a s s i f i e d i n t o two g r oups:

1)

Ef fec t on th e Near-Field Zone: The cro ss f lo w cause s a

d e f l e c t i o n o f the buoyant j e t t r a j e c t o r s

as

d i sc usse d

i n Paragraph 2.1.7. Depending on V and

Y,

such e f f ec t s

as

b en di ng i n t o a d j a c e n t j e t s , j e t a t t a c hm en t t o t he bo ttom

and i n s t a b i l i t i e s o f t h e s u r f a c e l a y e r ca n en su e.

I n t h e

case

of an un s t abl e near - f i e ld zone

a

cu r re nt sweeping a long

t h e d i f f u s e r l i n e

(v

=

0)

can cause repea ted re -ent ra inment ,

somewhat s i m i l a r t o d i f f u s e r in du ced c i r c u l a t i o n s w i t ho u t

c o n t r o l by n o z z l e o r i e n t a t i o n .

2) Ef fec t on th e Far-Field Zone:

dr iv en by buoyancy and by t h e di sc ha rg e momentum,

are

modi-

The d i f f us e r induce d c u r r e n t s ,

f i ed through th e ambient cur r ent sys tem.

Spreading of t he

d i f f u s e r f l ow a g a i n s t t h e c u r r e n t i s i n h i b i t e d t o

a

s t r o n g e r

o r

lesser

degree.

7.2 Method

of

Ana lys i s

The d i f f u s e r i n a uniform ambient cross-current w a s i n v e s t i g a t e d

exper imenta l ly .

No

c omplete the o r y de sc r ib ing the f u l l ra nge o f the

233



governi ng parameters has been devel oped. However, f or

the

purpose

of

qual i tat i ve di scussi on and quant i tat i ve data presentat i on, extreme

cases of di f f user behavi or (regi mes) can be i sol ated and descr i bed

by

mathemati cal model s.

The advantage of doi ng

so

i s that

the number

of

si gni f i cant parameters i s consi derabl y reduced i n these regi mes. To

i sol ate the regi mes, the tr eatment i s restri cted

t o

di f f users w th an

unstabl e near- f i el d zone (shal l owwater type). D f f users w th a stabl e

near- f i el d zone are i n general l i t t l e af f ected by the crossf l ow

ex-

cept f or the def l ecti on of the buoyant j ets.

The crossf l oww l l merel y

t ransl ate the stabl e surf ace l ayer w thout causi ng re-entrai nment .

For di f f users w th an unstabl e near- f i el d zone i n a crossf l ow

three regi mes can

be

i sol ated: A ) buoyancy i nduced counterf l ow B) mo-

mentumi nduced currents, and C) crossf l ow mxi ng. The two f i rst

regi mes

have

been di scussed i n the theoret i cal anal ysi s

of Chapter 3

and are summari zed here:

A ) Buoyancy I nduced Counterf l ow For weak crossf l owand

l i t t l e hori zontal di f f user momentum

the f l ow f i el d

w l l approach that of

the di f f user w th no net hori -

zontal momentum The theoret i cal near- f i el d di l ut i on

i s then

(3-222)

i n whi ch

F

i s the densi metr i c Froude number of the

counterf l ow systemdependent on the f r i ct i onal ef f ects

i n

the far- f i el d, (counterf l ow Equat i on

(3-207) and

Fi g. 3-23).

HC

234



Def i ni ng the di f f user l oad , F,

,

Equat i on (3-222) i s wr i t t en as

(7-2)

The di f fuser l oad' ' i s a si gni f i cant parameter f or

di f f users w th an unstabl e near- f i el d zone (shal l ow

water) and i s a densi metr i c Froude number descr i bi ng the

i mpact ( di scharge,

qo

the di scharge on the total depth.

on di f f user desi gn are further di scussed i n Chapter 8 .

B) Momentum I nduced Currents: For weak crossf l owand st rong

U B y and buoyancy, Apo/ pa)

of

0

I ts i mpl i cat i ons

hori zontal di f f user momentum hori zontal cur rent s w th

f ul l vert i cal mxi ng w l l be set up.

these currents i s dependent on the di f f user cont rol through

hori zontal nozzl e or i entat i on, By):

i ) For the LOG di str i but i on (f3(y) f rom

Eq.

( 4 - 3 6 ) , ) , w th a

predomnant l y two- di mensi onal f l owpattern i n the di f -

f user vi ci ni ty, t he two- di mensi onal channel model con-

The behavi or of

cept can be used:

(3-227)

i i ) For the

NOR

di str i but i on (B(y) =

go",

no control ), w i t h

235



t yp i ca l c on t r ac t i on of t h e f low f i e l d downstream o f

t h e

disch arge, Adams' model can he used ne gl ec t i ng

f r i c t i o n a l f a r - f i e ld e f f e c t s .

m od if ie d, i n cl u d in g t h e e f f e c t of v e r t i c a l a n g l e

Equat ion (2-45)

i s

t o

0

C ) Cross-Flow Mixing:

F or s t r on g c ros s f l ow and weak e f f ec t s

of buoyancy and h or iz on ta l d i f f us er momentum th e d i l u t i o n

i s e q ua l t o t h e r a t i o of the crossflow sweeping over the

d i f f u s e r t o t h e d i f f u s e r d is ch a rg e .

ss

= sin

(7-41

Essen t i a l l y , r eg im es

B

and C , b o t h w i t h

f d l

v e r t i c a l

mixing, can be combined by s imple sup er po si t ion :

i ) F or t h e

LOG

d i s t r i b u t i o n :

I n th e two-dimensional channel

model de r iv at i on (Sect ion 2.2.1 wi t h F igure

2-7)

t h e

t o t a l head l o s s (Eq.

(2-35)) i s

i n presence of an

ambient cu r r en t , t aken t o

be

r e l a t e d t o t h e

excess

ve lo ci ty head, (urn

-

u

s in y ) / 2g .

The d i l u t i on pre-

a

d i c t i o n fo l l ows i n s im i l a r i t y t o Eq. (3-227)

2

H/B

cos

eo

' sS

= v*+ 2 2i n y

+

14-

@/2

(7-5)

The e x p re s s io n s i n t h e b r a c k e t show

the re la t ive

con-

t r i b u t i o n s o f d i l u t i o n by the cross - f low and by d i f f us er

induced momentum.

236



ii) For the NOR d i s t r i b u t i o n : The ef fe c t o f c ross-f low

i s

includ ed i n Adams' E q . ( 2 - 44 ) w r i t t e n as

1 1 2

(7-6)

= y* =

+

v s i n y +

L

(V2sin2y +

2

H cos go>

S 2

I n t h e g e n e r a l case of a mixed regime, d i l ut io n s can'

be assumed

t o

be somewhat dependent on t he se l i m i t i ng

cases. A s a consequence

i t

f ol lo w s t h a t t h e 7 parameter

problem

may

be reasonably approximated by th e

4

parameters:

FT,

8 ,

V*,

@(y) (Note: ~ ( y ) e t ermines whe th er

V*

i s

taken from

Eq.

(7-5) o r ( 7 - 6 ) ) . These parameters

are

us ed i n t h e d a t a p r e s e n t a t i o n .

I n th e experimental program the dif fuser a r rangement

w a s l im i te d t o two va lues o f y:

a )

y

=

0 P a r a l l e l D i ff u se r ,

i . e .


axis

i s

p a r a l l e l t o t h e c ro s s- fl ow d i r e c t i o n , and

b) y = 90": Perp en d icu la r D i f fu se r ,

i . e .


a x i s

i s

p e rp end icu lar to t h e c ro ss flo w d i r ec t io n .

Two-dimensional chann el model s tu di e s on t h e perpen-

d ic u la r d i f fu se r pe rformed i n th e f lu me se t -u p a re

d es cr ib ed f i r s t ( Se c ti on

7 . 3 ) .

Both perpendicular

and p a r a l l e l d i f f u s e r s w e r e t e s t e d r i n t h e b a s i n se t -

ups

A

and B , and

are

discussed

i n

S e c t i o n

7 .4 .

7 . 3

Flume

Experiments: Perpe ndicul ar Di ff us er

Table 7 . 1 summarizes t h e r un c h a r a c t e r i s t i c s f o r t h e d i f f u s e r

experiments with ambient crossf low

i n

the f lume set-up, Se r i es FC. The

2 3 7



1

Table 7.1 : Flume Diffuser Experiments w i t h Crossflow, Series FC

Physical Variables

Secondarv

Pa rme ter s Governing Parameters

FC-1 2- 00 0.100

2 2.00

0.044

3 2.00 0.022

4 2.00 0.011

5 2.00 0.049

6

2.00 0.067

7

2.00 0.100

8 2.00 0.033

9 2.00 0.067

10 1.33 0.100

11

2.17

0.049

12 2.17 0.020

13

2 . 1 7

0.061

14

1.00

0.133

15 1.00 0.067

16 1.00 0.278

1 7 2.00 0.033

18 1.25 0.153

19 1.25 0.062

20 2.00 0.967

21 2.00 0.044

22 2.00 0.078

23 1.33 0.033

24 2.17 0.041

25 2.00 0.100

26 2.00 0.044

FC-27 1. 33 0.033

3.3

3 .3

3 .3

3.3

3 .3

3.3

3.3

3.3

3.3

3.3

1 . 7

1 .7

5.2

5.2

5.2

5.2

10.4

20.8

20.8

3 .3

3.3

3.3

3.3

1 . 7

3 .3

3.3

3.3

2.69 69.9 21.9 2,140

10

0.066

1 4 1

600 90 0.7 22.5 9.5

2.69 70.3 22.5 2,140 10 0.073 122 600 90 0.7 10.0 A.2

2.69 71.0 22.0 2,140 10 0.080 122 600 90 0. 8 5. 0 8.2

2.69 71.8 21.7 2,140 10 0.076 139 600 90

0.8

2.5 9.4

1.6 8 72.3 31.5 1,34 0 10 0.080 69 600 90 0.8 16. 0 4.6

1.68 72.9 30.8 1,3 40 10 0.066 70 600 90 0.7 24.0

4 . 7

4 . 7 1

67.5 15. 1 3,750 10 0.066 314 600 90 0.7 12.9 21.0

4.71 68.5 15.5 3.750 10 0.076 306

600

90 0.8 4.3 20.7

0.88 69.5 9.0 700 10 0.082 77 600 90 0. 8 45.8 5.2

2.69 69. 8 23.6 2,140 15 0.053 135 400

90 0.8

15.0 16.7

3.92 68.6 23.1 1,610 9 0.078 292 1300 90 0.7 13.3 6.4

3.92 69.4 22.0 1,610 9 0.082 298 1300 90 0.7 6.7 6.6

0.55 67.1 24. 1 700 9 0.085 27 416 90 0.8 46.2 3.2

5.34 68.7 15.9 6,700 LO 0.044 276 192 90 0.9 4.8 104.0

5.34 68.7 15.7 6,700 20 0.051 278 192 90 1.0

2 .4

104.4

1.87 70.3 16.8 2,350 20 0.059 78 192 90 1.2 5.7 29.2

0.64 66.9 15.4 1,610 10 0.085 24 192 90 0.9 10. 0 9.0

1.07 67.8 20.4 5,350 16 0.048 24 60 90 0. 8 8.6 51. 5

1.07 67.8 20.6 5,350 16 0.056 24 60 90 0.9 3.5 51.5

2.69 75.2 22.3 2,140 10 0.065 133 600 0 0.7 21.8 8.9

2.69 75.2 21.9 2,140 10 0.075 134 600 0 0. 8 10. 0 9.0

1.68 73.6 30.8 1,340 10 0.072 70 600 0 0.7 28.0

4 . 7

2.69 69.3 22.0 2,140 15 0.064

1 4 1

400

0

1.0 5.0 17.5

3.92 68.8 2 1 . 1 1,610 9 0.078 309 1300 0 0.7 13.3 6.8

2.69 66.5 24.1 2,140 10 0.065 139 600 180

0 . 7

22.5 9.3

1.68 67.5 20.5 1,340 10

0,080

73 600 180 0.8 16. 0 4.9

2.69 68.3 24.6 2,140 15 0.066 133 400 180

1 .0

5.0 16.5

s s

2 1 . 7

14.1

9.6

12.8

16.7

31.3

11.6

7.4

30.3

14.7

15.4

13.7

27.8

5.0

3.8

7.0

11.0

8 .5

5 . 1

20.4

13.7

25.6

8.5

15.2

21.3

1 7

.O

7.6

Note: 1) Channel length, L

=

20 f t .

2)

Bo

=

180' indica tes horizonta l discharge against the current



c r o s s f l o w

w a s

g en e ra t ed b y pumping across a p a r t i t i o n

w a l l

i n t h e reser-

v o i r b e h i nd t h e c h an n el as w a s skown i n F ig ur e 5-1.

d i s ch a rg e w a s from

a

c o n t i n u o u s s l o t .

p r i m a r i l y

90"

w i t h a d d i t i o n a l

t e s t s w i t h 0" ( d i sc h a r ge i n t h e c u r r e n t

d i r e c t i o n ) a n d 180" ( a g a in s t t h e c u r r e n t ) . R e s ul t a nt s u r f a c e d i l u t i o n s ,

S s ,

a r e included on Table

7 . 1 .

F i g u r e 7-2.

I n

a l l

t e s t s t he

The

d i s ch a rg e an g l e ,

O0, w a s

T y p i c a l fl ow d i s t r i b u t i o n s

a r e

shown i n

The data

are

a na ly ze d co n s i d e r in g t h e s p e c i f i c f e a t u r e of t h e

flume set-up: A cu r ren t h av in g a u n i t w i d t h d i s c h a r g e u H i s f o r c e d

th rough the channel .

canno t

a l t e r

t h i s d i sc ha rg e; t h e e f f e c t

of

the horizontal momentum

i s

m erely t o b u i l d up

a

high and

a

low p r e s s u r e r e g i o n i n t h e c h a n ne l .

con seq uen ce o f t h i s fo rced cu r ren t i s t h e i nd ep end ence o f t h e d i f f u s e r

performance (o ut s i de some lo ca l zone) on the ho r iz on ta l momentum (th us

0 =

90"

as with no net horizontal momentum).

a

The hor iz on ta l d i f f u s e r momentum

O o

=

0" o r

180")

The

0

The two l i m i t i n g re gim es f o r t h i s d i s c h a r ge s i t u a t i o n are t h e

coun ter f lo w sys tem (F a ) and the

T '

These

@ 1, FH i s

c r o s s f l o w i s

mixed regime

C

l im i t i n g r eg im es a re

t aken as

0.165

(Fig .

assumed .a t V = 1. A

c ro s s f l o w m ixin g (V, as s i n Y =

1).

shown i n Fig . 7-3. A s most ru ns had

3 - 2 4 ) .

c r i t e r i o n of a p p l i c a bi l i ty o f t h e f u l l y

The l im i t i n g v a lu e f o r low

i s

g iv en by co n s id e r i n g

t ha t

below a crossflow Froude number

FH =

1

s t ag na n t wedges w i l l form.

Thus

and using S s = V and t h e a p p ro p r ia t e d e f i n i t i o n s t h e c r i t e r i o n i s

239



a 4

a) R u ~

C-3,

Fs = 69, H/B = 558, 8

= 45"-A,9 = 0.4,

@(y)

= LOG,

0

V

=

14.0,

y =

90

b)

Run

BC-B,

Fs

= 66, H/B

= 558, B 0

=

45"-A,@

=

0.2, B(y) = LOG

V

=

12.4, y =

Oo

Fig. 7-4: Observed Temperature and Velocity Field, Series BC,

Alternating Nozzles

245



e x p e ri m e n ta l s et -u p, t h e t o t a l l e n g t h of t h e p e r p e n d i c u l a r d i f f u s e r

w a s

t w ic e t h e l e n g th of t h e p a r a l l e l d i f f u s e r . T h i s d i f f e r e n c e

i s

expressed

by

the

v a l u e of t h e f a r - f i e l d p a ra m et e r

t i o n s . I n b ot h c a s es t h e f l ow f i e l d i s w e l l s t r a t i f i e d and u pst re am

wedges a r e p res en t .

as

shown

i n

t h e f i g u r e c a p -

I n F i g u r e 7-5a t h e e x p e r i m e n ta l d i l u t i o n s f o r

the

p e rp en d i cu l a r

d i f f u s e r t a k e n a t t h e ed g e o f t h e n ea r - f i e l d zon e a r e compared with the

p re d i c t i o n s fo r t h e l i m i t i n g r egim es. F ig u re 7-5a w a s d ev el op ed i n t h e

same

manner as Figur e 7-3, bu t wi th

cp

= 0.5 (FH = 0.187) correspond ing

t o m ost ru n s .

t o l i m i t a t i o n s i n t h e c r o s s f lo w g e n e r at i o n s ys te m. )

C

(No ex p e r im en t s w e r e made i n t h e f u l l y mixed reg ime due

Figure 7-5a a l s o

shows f o r

the g e n e r a l

case V*

= Y s i n y t h e

c r i t e r i a f o r t h e d i f f e r e n t r e g im es , w i t h h y p o t h e t i c a l i s o- d i l u ti o n l i n e s

i n t h e c o u nt e r f lo w r e g i o n . O nly t h e r e s u l t s o f t h e p e r pe n d i c u l a r d i f -

f u s e r e x pe r im e nt s

( y

= 90") are shown i n Fig ure 7-5a, w it h very

good

agreement. This would mean that

t h e d i l u t i o n i s dependent o n l y on th e c h a r a c t e r i s t i c s o f th e buoyant

coun ter f low reg ime. Th is

i s

n o t c o n s i d e r e d a p p r o p r i a t e a n d i n d i c a t e s

t h a t due t o th e " lumping" o f t he parameters , such as i n E q ua t io n s

4-6

and 4-7, some of th e dependency of th e or ig in a l 7 para met er problem i s

l o s t . It c an b e ex pe ct ed t h a t f o r t h e p a r a l l e l d i f f u s e r , t h e c ur r e nt

s weep in g a lo n g t h e d i f f u s e r l i n e ( s ee F ig u re 7-4b)

w i l l

cau s e r ep ea t ed

re -en tr a in men t and some d ec reas ed d i l u t i o n s , e s p e c i a l l y f o r

a

s t r o n g

c u r r e n t .

d i f f u s e r v e r s u s t h o s e o f t h e p e r p e n d i c u la r o n e i s

given i n F igu re 7-5b

(see also Table 7-2a) as a f u n c t i o n o f

Y

(i .e. t h e p o t e n t i a l d i l u t i o n ).

F or t h e p a r a l l e l d i f f u s e r , V* = 0.

A comparison between

the

d i l u t i o n s o b t a i ne d w i t h

the

p a r a l l e l

2 4 6



Series B C

L

30 = S s

I I I I I I I I I 1

1

IXI

N

\

5

10 50 100

$=0.5 v*= v SINT

a) Dilutions of Perpendicular Diffuser

o . 2 k l L l5 l o 50

100

b) Dilutions of Parallel V s . V Perpendicular Diffuser

Fig.

7-5:

Three-Dimensional Di ffusers i n Crossflow,

(Series B C )

No

Net

Horizontal Momentum, LOG Distribution

247



I n general , the paral l el di f f user has about a 20%l ower di l ut i on

capaci ty t han the correspondi ng perpendi cul ar one whi ch as has been

noted i s tw ce as l ong i n the present exper i mental set -up (resul t i ng i n

a di f f erence

of Q)

.

7.4.2 D f f users w th Net Hor i zontal Momentum

Tabl e 7.2b summari zes the three- di mensi onal di f f user experi ments

i n crossf l ow Seri es

BC, w th uni di recti onal nozzl es. Runs 21 to

34

rel ate to the perpendi cul ar, Runs

3 5

t o 48 t o the paral l el di f f user. I n

addi t i on to the di f f user arrangement , Y, emphasi s was put on the cont rol

of the hor i zontal ci rcul at i ons through nozzl e or i ent at i on ~(y) : LOG and

NOR di st r i but i ons were tested. The vert i cal angl e

eo

was vari ed between

45 and 0 .

The tests on di f f users w t h net hor i zontal momentumi n the pres-

ence of an ambi ent current exhi bi t some of the exper i mental di f f i cul t i es

and l i mtat i ons as was di scussed i n Sect i on 6.2.2 f or the no crossf l ow

tests (Seri es

BH),

namel y unst eady ef f ects and boundary ef f ects. Agai n,

i n taki ng temperature measurements car e was taken not t o i ncl ude reci rcu-

l at i on ef f ects due

t o

basi n boundar i es.

Character i sti c f l ow f i el ds f or correspondi ng paral l el and perpen-

di cul ar di f f users w th net hori zontal momentum are shown i n Fi gures 7- 6,

7-7 and 7-8. Fi gure 7-6 ( LOG) shoul d be compared w th the al ternat i ng

nozzl e case (Fi gure 7-4) as t he exper i ments have si ml ar parameters. The

f ul l mxi ng downstreamof the di scharge f or the perpendi cul ar di f f user

i s

character i st i c and predi cted f romtheoret i cal consi derat i ons (see

Fi gure 7-9). The f ul l mxi ng f or

t he

paral l el di f f user i s l ess pro-

nounced i ndi cat i ng penet rati on and t ravel

of

ambi ent crossf l owwater

2 4 8



Table 7.2b:

Basin Diffuser Experiments with Crossflow, Series BC

b) Discharge with N e t Horizontal Momentum

Secondary

Governing P u m a -

Phyrical Variables Parameters

Y ?T Obrerved

H Ua Uo LD Ta ATo 80

(ft) (ftlsec) (ft/rec)

(it)

(OF) (OF) E/H fo

PI

H/B

(dgr) 0

B(y)

V

(dgr)

v

BC-21 0.67 0.024

22 0.67 0.012

23 0.67 0.024

24 0.67 0.009

25 0.67 0.024

26 0.67 0.009

27 0.67 0.024

28 0.67 0.009

29 0.50 0.024

30 0.50

0.012

31 0.50 0.024

32 0.50 0.012

33 0.50 0.024

34 0.50 0.012

35 0.67 0.024

36 0.67 0.012

37

0.67 0.024

38 0.67 0.012

39 0.67 0.024

40 0.67 0.012

41 0.67 0.024

42 0.67 0.009

43 0.67 0.024

44 0.67 0.009

45 0.50 0.024

46 0.50 0.012

47 0.50 0.024

BC-48 0.50 0.012

2.12

2.12

1.08

1.08

1.08

1.08

1.08

1.08

1.45

1.45

1.45

1.45

1.45

1.45

2.15

2.15

1.08

1.08

1.08

1.08

1.08

1.08

1.08

1.08

1.45

1.45

1.45

1.45

6.4 64.8 26.3 4.060 0.24 0.043 132 558 45

6.4 64.8 26.2 4,060 0.24 0.050 132 358 45

6.4 65.4 23.5 1,990 0.24 0.043 70 558 45

6.4 65.4 23.5 1.990 0.24 0.053 80 558 45

6.4 66.2 24.3 2,020 0.24 0.043 69 558 0

6.4 66.2 24.2 2,020 0.24 0.053 69 558 0

6.4 66.5 23.8 2,929 0.24 0.043 69 558 0

6.4 66.5 23.9 2,929 0.24 0.053 69 558 0

6.4 67.3 27.6 2,880 0.32 0.045 87 419 0

6.4 67.3 26.6 2,840 0.32 0.053

88

419 0

3.2 67.4 25.0 2,800 0.32 0.045 91 419 0

3.2

67.4 25.2 2,800 0.32 0.053 91 419 0

6.4 61.7 26.9 2,870 0.32 0.045

88

419

0

6.4 67.7 26.9 2,870 0.32 0.053

88

419 0

3.2 60.5 26.0 3,870 0.24 0.043 133 558 45

3.2 60.5 25.9 3,870 0.24 0.050 133 558 45

3.2 61.2 23.2 1,890 0.24 0.043 70 558 45

3.2

61.2 23.6 1,900 0.24 0.050 70 558 45

3.2 61.6 23.1 1.900 0.24 0.043 70 558 45

3.2 61.6 23.3 1.900 0.24 0.050 70 558 45

3.2

58.8

21.4 1,810 0.24 0.043 73 558 0

3.2 58.8 21.0 1,810 0.24 0.053 73 558 0

3.2 59.7 26.1 1,920 0.24 0.043 66 558 0

3.2 59.7 26.1 1,920 0.24 0.053 66 558 0

3.2 61.0 25.7 2,620 0.32 0.045 90 419 0

3.2 61.0 25.7 2,630 0.32 0.053 90 419 0

3.2 60.2 26.8 2,630 0.32 0.045 88 419

0

3.2 60.2 26.8 2.630 0.32 0.053 88 419 0

0.4

LOG

0.5 LOG

0.4 LOG

0.5 LOG

0.4 LOG

6.2 90

3.1 90

4.4 90

4.7 90

2.4 90

10.0 25.90

10.0 25.38

5.3

25.6.

5.3

29.0a

5.3

32.4s

0.5 LOG 4.7 90

0.4 NOR 12.4 90

0.5 NOR 4.7 90

0.4 LOG 6.7 90

0.6 LOG 3.5 90

0 . 3

LOG 6.9 90

0 . 3

UX;

3.5 90

0.5 NOR 6.9

90

0.6 NOR

3.5

90

0.2 LOG 6.2 0

0 . 3 LOG 3 . 1 0

0.2 LOG 12.4 0

0 .3

LOG

6.2

0

0.2 NOR 12.4 0

0.3 NOR 6.2 0

0.2 LOG 12.4 0

0.3

LOG

4.7

0

0.2 NOR 12.4 0

0.3 NOR 4.7 0

0.2 LOG 6.7 0

0.3 LOG 3.5 0

0.2 NOR 6.7 0

0.3 NOR 3.5 0

5.3

30.2a

5.3 24.0b

5.3

19.2b

10.3 26.6a

10.3 26.1.

10.7

26.8a

10.6 26.la

10.3 18.3b

10.2 16.3b

10.0 25.la

10.0 25.10

5.3 25.1.9

5.3 25.10

5.3 14.0b

5.3 14.0b

5.5 29.9a

5.5 29.9a

5.0 16.7b

5.0 16.7b

10.5 25.9a

10.5 25.9a

10.3 14.5b

10.3 14.5b

Note:

1) All runs with D = 0.0156

ft

and

E -

0.16 f t .

. - - .

2)

In column V*:

a calculated

f rm

Equation (7-6),

b

calculated

from

Equation (7-7)

sr

15.4

13.8

16.4

14.3

17.6

15.4

13.4

12.5

18.2

15.4

19.2

16.7

13.3

12.2

13.0

11.1

12.5

12.5

12

o

11.8

12.7

10.6

12.5

12.0

12.0

10.8

9.5

10.2



r

1 I

I 1

I I I I 1

SURFACE

ISOT HERMS

-

-

04

10 -

-

-

- .I

1 I

1 I

1

-

2 0

-10 0 10 2o "/H

Run

BC =

23,

Fs =

70,

H/B

-

5 5 8 ,

= 45",

@

= 0.4,

B(y)

=

V

= 6 . 2 , y

= 90' ( No zz le s D i r e c t e d t o t h e L e f t )

0.5

0.1

0

.5p

0

AT 01

AT

-

LOG,

1

I I I

I

I

I 1

a5

c

0.1

0.5

0

b )

Run BC-35,

Fs

= 7 0 ,

H/B

=

5 5 8 ,

eo =

4 5 ' ,

cp

= 0.2,B(y) =

LOG,

F i g .

7-6:

Observed Tempera ture and Velo c i ty F ie ld wi th Crossflow,

V

= 6 .2 , y

=

0"

(Nozzles Directed Downward)

S e r i e s BC, U n i d i r e c t i o n a l N oz zl es

250



*

1 I

1

I I I I I I

1

I

.

SURFACE

ISOTHERMS -

0-

-

-

-

- :

0

c

20

-

-

-

10 -

-

I

1 I I I

0

*4

0.5

0.1

lz

zim

0.5 .T

0

-T 0.1

201

I I I

I I I I

1

I I

I

1

SURFACE

ISOTHERMS

-

I

I

t

/

/ -

1 1

I I

2 0

3 0

X

I I

1 I I 1

I I

//I 1

-

30

-2

0

- 1 0

0

10

0.5 i - p

0

AT 0.1

A L

-

b)

Run

BC-45,

Fs

=

9 0 , H / B

- 4 1 9 , 8 = O o , @ = 0.2, B(y)

=

LOG

V

=

6.7, y =

0"


Fig. 7-7: Observed Temperature and Velocity Field, Series B C,

Unidirectional Nozzles

2 5 1



I

1 I I I

1

I I I I

I

I I

1

i

SURFACE ISOTHERMS

-

10-

-

YH -

h

0

2 0

1c

n I

I

1 I

I

0

- 3 0 - 2 0 - 1 0

SURFACE

ISOTHERMS

- 1

1 I 1 1 1

I

10 2 0 3 0 X A

1

z/H

0.5

I

10 2 0 3 0 xA.

0.5

0

0.1

'H

a) Run BC-33, Fs = 88,

H/B

= 4 1 9 , B0

=

O ,

@ = 0.5,

B(y) = NOR,

V

=

6 .9 , y = 90"

(Nozzles Directed to the Left)

0

-1 0

-

2 0

b) Run BC = 4 7 ,

F

= 88, H/B = 4 1 9 ,

0 = 0",

@ = 0.2, B(y)

=

NOR

S

0

V = 6.7,

y = 0


Fig .

7-8: Observed Temperature and Velocity Field, Series BC,

Unidirectional Nozzles

252



u n d e rn e a th t h e s u r f a c e l a y e r .

t u a t e d b y bo un da ry e f f e c t s .

T h i s phenomenon may b e somewhat accen-

F i g u r e

7-7

i n compari son

w i t h

F i g u r e 7-6 shows

t h e

e f f e c t o f

de-

c reased ambien t dep th ,

H,

and increased diffuser momentum

( 8

= 0" vs.

4 5 ) . Both f igures show LOG d i s t r i b u t i o n s . The i n c r e a s e d s t r e n g t h o f

t h e d i f f u s e r i nd uc ed c u r r e n t s

i s

apparent; boundary e f f e c t s

a r e

enhanced.

0

The e f f e c t o f LOG v e r s u s

NOR

d i s t r i b u t i o n i s shown by comparing

F i g u r e s 7-8

(NOR)

and 7-7 (LOG). The furt :her incr eas ed s tr e n g t h of t h e

c i r c u l a t i o n a n d t h e t e n de nc y t o c o n t r a c t d own stream o f

t h e d i f f u s e r l i n e

c a n b e s e e n .

r e gi m e s a nd c r i t e r i a . F i g u r e 7-9 a p p l i e s t o t h e LOG d i s t r i b u t i o n s w i t h

a predominant ly two-dimensional f low f i e l d :

V*,

of cros sf lo w and ho ri zo nt a l d i f f u s e r momentum

i s

given by Equa t ion

7-5. The ra ng e of a p p l i c a b i l i t y of the f u l l y mixed and s t a gn ant wedge

domains i s give n by arguments, Equ atio ns (7-7) and ( 7 - 9 ) , o n l y s u b s t i -

t u t i n g

S

conc en t r a t ed i n F igure 7-9, i n d i ca t i ng t he dominance of t h e h o r i z o n t a l

momentum induced mixing over the l i m i t e d e x p e r i m e n t a l r a n g e .

a r e

c o n s i d e r a b l y

(30 t o 5 0% ) l ow er t h a n p r e d i c t e d b ot h f o r p a r a l l e l a n d .

p e r p e n d i c u l a r d i f f u se r s , w h i c h i s c o n s i s t e n t w i th t h e o b s e r v at i o n f o r

t h e b a s i n e x p er i me n ts w i t h z e r o c r o s s f l o w

( S e r i e s

BH)

and

may

b e a t t r i -

b u t e d t o e x p e ri m en t al l i m i t a t i o n s .

Obse rved d i lu t ions are co mp ared w it h p r e d i c t e d l i m i t i n g

The combined mixing e f f e c t ,

= V* (FH = 0 .52 for @ = 0.5).

The run pa ramete rs are s t r o n g l y

S

D i l u t i o n s

F igure 7-10 app l i e s t o t h e NOR d i s t r i b u t i o n w i t h a c o n t r a c t e d

f l o w f i e l d : Adams' Eq. (7-6) i s used and c r i t e r i a l i n e s are developed

i n s i m i l a r f a s h i o n

a s

fo r F i gure 7-9.

V*+O,

i . e .

i n t h e c o un t e r fl o w r a n g e . I n t h e f u l l y mixed r a n g e o b se r v ed

2 5 3

There i s no p r e d i c t i v e m od el f o r



-

-

-

12.:

12.5

5 -

-

0

4IX

t

f

5=

.5

1

j

.4

q2

12.0

10.8

perpendicular

d=90°

- d= 0

Y

Fig. 7-9:

D i f f u s e r s

i n

Crossf low,

N e t

H o r i z o n t a l

Momentum,

w i t h Cont ro l (LOG D i s t r i b u t i o n )

254



12.0@

11.8 +

12.5

12.0

-

2 0

1x1

5 . 8

3.5

L0.6

\

F

30

-

\

\

-r

5 0

= S

-

\

I

I

I I I I I 1 1 1

I

I I I

5

10

50

'/2

* = ~ V S I N $ + + ( V

2I N

g + 2 +cos e m )

2

x

Go

= d

+ =o

Adams

(1972)

perpendicular

8

=

90

0 =

4 5 O

para1

el

r = 0

Fig. 7-10:

Di f fu se r s i n C ros sf low,

N e t

Horizontal Momentum,

no Control

(NOR

D i s t r i b u t i o n )

255



d i l u t i o n s a g r e e r e a s o n a b l y w e l l (abou t 20% lower) w i th th e p red ic te d

v a lu es . A l s o i n c lu d ed

a re

the da ta by Adams (1972) f o r perp end icul ar

d i f fu s e r s s ho wing b e t t e r ag reemen t, a l t h o u g h i t

i s

p o i n te d o u t t h a t

Adams' d i l u t io n da t a a r e averaged va lues

taken

a

c o n s i d e r a b l e d i s t a n c e

downstream from the d if fu se r

as

opposed t o t h i s s t u d y

where

d i l u t i o n

i s

def ined as th e

maximum

t empera tu re

a t

th e edge o f t he near- f ie ld zone.

A comparison of

r e l a t i v e

d i f f u s e r p e rfo rm anced

i s

g i ve n i n F i g .

7-11 f o r e x pe r im e n ts w i t h s imi la r d i f f u s e r p a r a m e t e r s b u t v a ry i n g

y

and

B(y). Fig u re 7 - l l a com pares co r re s p on d in g p a r a l l e l and p e rp en d i cu l a r

d i f f us er s , showing abou t

20%

r ed uc ed d i l u t i o n c a p a c i t y f o r t h e

f i r s t .

Sim i l a r l y ab o u t

20%

reduced performance i s seen f o r t h e NOR d i s t r i b u t i o n

(no control) compared t o the

LOG

d i s t r i b u t i o n ( c o n t r o l ) . T hi s

I s

con-

s i s t e n t w i t h o b s e r v a t i o n o f t h e z e r o c r o s s f l o w

t e s t s

(Series

BH).

7.5

Conclusions : Di f f us er s wi th Ambient Cross f low

R e s u l t s o f t h e e x p e r i m en t a l i n v e s t i g a t i o n a n a ly z e d w i t h i n t h e

framework of t h e l i m i t i n g f l o w reg im es en ab l e t h e fo l l owin g co n c lu s ion s

t o b e drawn on t h e e f f e c t

of

an

ambien t c ross f low on d i f fuser per fo rm-

ance:

1) Diffuser Arrangement

I n t h e p r e s e nc e

of

am bient cu r ren t s t h e s i n g l e m ost c r i t i c a l para-

meters of d i f f u s e r b e h a vi o r i s t h e a r r an g emen t , y , of t h e d i f -

f u s e r w i th r e s p e c t t o t h e c u r r e n t d i r e c t i o n . Depending on t h e

arrangement ,

t h e e f f e c t o f c ro s s f l o w may

increase

o r d ec reas e

o b s e r v e d d i l u t i o n s w i t h r e f e r e n c e t o

t he

s i t u a t i o n of z e r o

cross f low:

256



0

NOR

.4 -

-

-

-

0.2

-

-

-

-

0 -

I I I I I I l l l I I I I I I I I

1 5 10 50 100

V

- .

a) Parallel Diffuser

V s .

Perpendicular Diffuser

0%

1.0 I I I

I I I I I

I i s 9 I

-

I

048

I

I #?

3T

z x 2 8

-

0.8 34 0% .27

0.6

-

run

umber

-

-

-

-

x

perpendicular

9

32

-

0.4

- 0 parallel

02

0

OG

-

-

-

-

-

I

I I

I

I I I I I I I I I I I I I

1

5 1 0 50

100

V

b)

NOR

Distribution V s .

LOG

Distribution

Fig . 7-11: Comparison of Relative Diffuser Performance in Crossflow

(S er ies BC) with Unidi rect ion al Discharge

257



a) Perpendi cul ar D f f user:

The presence of a crossf l ow

i ncreases

the

di l ut i on obseryed downst reamof t he

di f f user i n the case of al ternat i ng nozzl es and i n

i n the case of nozzl es di schargi ng i n t he di recti on

of the current .

The case of nozzl es di schargi ng

agai nst the current was not studi ed i n thi s i nvesti -

gat i on, however, other studi es (Harl eman et. al . (1971))

i ndi cate

qual i tat i vel y a marked decrease i n di l ut i on

due t o st agnat i on ef f ects and unsteady r eci rcul at i on.

b) Paral l el D f f user:

The

ef f ect of cross currents causes

i n al l condi t i ons (al ternat i ng or uni di recti onal di s-

charge)

a decrease i n di l ut i ons as compared t o the

zero crossf l ow si tuat i on. Thi s f act i s due to (i ) the

current sweepi ng al ong the di f f user l i ne causi ng repeated

re- entrai nment and ( i i ) f or the case of uni di recti onal

di scharge w t h ful l ver t i cal mxi ng, an ef f ecti ve bl ocki ng

of the oncomng ambi ent f l owi s gi ven causi ng eddyi ng

and unsteady reci rcul ati on of the mxed water at the down-

streamdi f f user end.

2) D f f user Cont rol

Control of the i nduced hori zontal ci rcul at i ons t hrough hori zontal

nozzl e or i entat i on, ~ ( y ) , s equal l y i mportant as i n the zero

crossf l ow case.

3)

Two-D mensi onal Experi ments

The i nterpl ay between the di f f user i nduced and the ambi ent f l ow

f i el d f or

t he

three- di mensi onal di f f user cannot be si mul ated

258



i n a two-dimensional chan nel modal wi th fo rc ed ambient

crossflow.

t o l a t e r a l l y s t r o n g l y c on fi ne d s i t u a t i o n s .

The a p p l i c a t i o n

of

such exper iments

i s

l i m i t e d

259



V I I I .

APPLICATION OF RESULTS

TO

D E S I G N A N D

HYDRAULIC

SCALE

MODELING OF SUBMERGED MULTIPORT DIFFUSERS

--

The t heo re t ica l and exper imenta l an a ly s i s t r ea te d th e mechanics

of a submerged mult ipor t d if fu se r f o r buoyant di schar ges in to an un-

s t r a t i f i e d , l a r g e body o f

water

of cons tant depth wi t h

a

uniform

current system running a t a n a r b i t r a r y a n gl e t o t h e d i f f u s e r a x i s . The

r e s u l t s o f t h i s st ud y,

presented

i n the f orm of so lu t ion g r aphs ,

can be

use d f o r d i f f use r de s ign f o r th e c onf igu r a t ion c onside re d a nd - with

proper schematization

-

f o r t h e p r e li m i na r y d e s i g n e s t i m a t e f o r

a

va r i e t y of o ther , more complicated d ischarge conf ig ura t ion s .

For

compl ica ted d ischarge con di t ion s hyd raul icsc a lem ode ls have t o be used

to eva lua te d i f fuse r per formance .

I n t h i s case the theory developed

i n th is s tudy enables th e formula t ion of model ing requirements f or the

opera t ion of sca le models.

The design of d i f f u s e r s t r u c t u r e s

is

an i n t e g r a l p a r t of a

waste

management system and t h u s

i s

governed by a h o s t of f a c t o r s r e l a t i n g

t o economical, engineer ing and environmental obj ect iv es. A complete

d i f f use r de s ign i s c a r r i e d ou t i n th r e e br oa d s t e p s , e ac h wi th im pli ca -

t ion s on the above ob je c t iv e s :

( i ) S i t e S e l e c t i o n i s th e choice of a g e n e r a l area

f o r t h e l o c a t i o n

of

t h e d if f u se r o u t f a l l .

( i i )

Di ffu ser Design f o r Dilu tion Requirement nvolves

th e hydrodynamic design of th e disc ha rge s t ruc tu re

w i t hi n t h e s e l e c t e d s i t e with t h e purpose of reducing

t h e c o n c e n t r at i o n of t h e e f f l u e n t

material

so

as

t o

conform t o environmental c r i t e r i a .

260



(iii)

I n t e r n a l Hydr a u l i c Des ign o f the Di f f use r PQe

i s the computation of th e manifold problem wit h

th e goa l of achieving reasonably uni form nozz le

e x i t v e l o c i t i e s a lo ng t h e d i f f u s e r l i n e .

--

-

This s tudy

w a s

p r i m a r i l y a d dr e ss e d t o d e s i g n s t e p

(ii)

However,

c e r t a i n c o n s id e r at i o n s r e g a r d i ng s t e p

i)

a n b e d i s c us s e d i n t h e

c o n t e x t

of

t h i s s tu dy . The i n t e r n a l hy d ra u li c de s ig n , s t e p ( i i i ) , h as

been t rea ted e lsewhere

(see

Rmm e t a l . (1960)) and computer programs

have

been developed.

summarized here, as

i t

d i r e c t l y re la tes t o s t e p

( i i ) : I n o rd e r to i n s u re

reasonable uni formi ty of t h e d i sc h a rg e v e l o c i t i e s f rom the d i f f us e r

noz z le s i t i s necessary that

the

c r oss - se c t iona l

area of the

d i f f u s e r

p i p e

( fe ed e r l i n e ) b e l a r g e r t h a n t h e t o t a l area of the nozz les . ( In o ther

words, no uniformi ty

is

p o s s i b l e i f t h e f l ow

has

to be de c e le r a t e d

be f o r e l e a v ing the p ipe .)

The

de s ign c ons ide r a t ions p r e se n te d he r e in

a re p r i m a r i l y d i r e c t e d a t " thermal" d i f f us e r s fo r d isc harge of hea ted

condenser water from steam-electric power plants.

f o r a the rmal d i f f us e r i n comparison wi th a t yp ic a l s e wa ge d i f f use r

ha s bee n g ive n i n Ta ble

5.1.

Sewage d i f f us er s w i th

a

h i g h d i l u t i o n

requirement (more than

100) a r e

g e n e r a l ly l i m i t e d

t o

deep

water

and thus

do

not e x h i b i t t h e i n t r i c a c i e s i n tr o du c ed by

a

c o n s t r a i n i n g

water

depth,

namely an uns tab le near - f ie l d

zone

with loc a l re -ent ra inment and the

g e n e r a t io n o f h o r i z o n t a l c i r c u l a t i o n s .

8.1

S i t e

C h a r a c t e r i s t i c s

An important re s u lt of R a w n e t a l . ' s work i s

0

A

typical example

S e le c t ion

of

a s i t e f o r t h e l o c at i o n of a t he r m a l d i f f use r

o u t f a l l i nv ol ve s a v a r i e t y o f c o n s i d e r a t i o ns ,

such

a s d i s t a n c e f r o n t h e

261



s h o r e l i n e ,

av a i l ab le

o f f s h o r e area, prevent i on of r e c i r c u l a t i on i n to

th e power pla nt i nta ke, requirements f o r providing a zone of passage

etc..

i nd iv idua l l y i n each

case,

a

desc r i p t i on o f pos s ib l e

s i t e

character is t ics

i s given here:

Rather than discussing th es e

aspec ts

which

have

t o be evaluat ed

(i)

S i t e topography. The ve r t i c a l exte n t and t he l a t e r a l confine-

ment

of

the receiv ing water are important .

l a t e r a l ly unconfined ( d i f fus er located f a r away f r o m sho re, o r o f f sho re

power plant),

l a t e ra l l y semi-conf ined (d i f fu se r l oca t ed c lo se t o shore-

l i n e ) o r l a t e r a l l y c on fin ed

( d if f u se r i n r i v e r s ,

canals

or other con-

s t ra in ing geomet r ies ) .

fo r di ff us er s located i n embayments o r re gions which a re somehow

sepa ra te d from the main body of w a t e r and do not

have

a st r ong through-

The water body can be

A

pa r t ic u l ar ly s t rong boundary ef fe c t may occur

flow.

The v e r t i c a l e x t e n t of

t h e d i f f u s e r s i t e can be uniformly shallow

(such

as

on a c o a s t a l s h e lf o r i n r i v e r s )

o r

have var iab le s lop ing off -

shore depths.

ii) Current S y s t e m . The ambient current a t t h e d i f f u s e r s i t e may

b e weak and unpronounced

(in

i n t e rm ed ia t e s i ze lakes and

r e s e r v o i r s

pri mar i ly dependent on unsteady wind ac t io n) .

ac t i on may occur i n r i ve r app l ica t io ns and

i n

c o a s t a l o r l a r g e l a k e

S i g n i f i c a n t c u r r e n t

s i t e s

w ith s t ro n g t i d a l o r i n e r t i a l c u r r e n t s .

predominantly i n one d i r ec t io n ( r i ve r , l ake , o r ocean longshore curren ts )

o r may b e o s c i l l a t i n g ( t i d a l c u r r e n t s ) . For d i f f u s e r

s i t e s

i n

estuaries

o r t i d a l bays the ambient f low may pass rep eatedly (during several t i d a l

cyc l es ) over t h e s i t e .

The c u rr e nt system may be

262



( i i i ) Ambient Densi ty S t r a t i f i c a t io n . Occur rence of tempera ture

s t r a t i f i c a t i o n d uring

s u m m e r

months

i s

t y p i c a l f o r l a k e s and r e s e r v o i r s

i n th e mod era te c lima te zone . S imi la r ly , t emperatur e s t r a t i f i ca t i o n

has a l s o been observed i n sha l low coa st a l zones a l though l imi t ed to

p e r io ds of mod era te wave ac t io n . Sa l i n i ty s t r a t i f i ca t i o n

i s

t y p ic a l i n

e s t u a r i n e

s i t e s .

( i v ) Ambient Turbulence. The ambient t u r bu l e n t i n t e n s i t y h a s

imp l ica t ion s on th e s t a b i l i ty o f bo th ambient and d i f fuser - induced

s t r a t i f i ca t i o n . I nc reased tu rb u len ce, as through s ig n i f i ca n t wave ac t i on ,

may lea d t o e ros ion and breakdown of dens i ty s t r a t i f ic a t i o n .

(v) Wind S t r e ss . I n ad d i t i o n t o c r ea t in g co nv ec tiv e cu r r e n t s

and ambient turbulence, th e wind ac t i ng on th e water s u r f a c e e x e r t s

a

stress

on th e di ff us er induced f low f i e l d and may in f lu en ce th e f low-a..-y,

or

l e ad t o r e c i r c u l a t i o n a t t h e i n t a k e .

8.2

Di ffu se r Design fo r D il ut io n Requirement

Once the genera l s i t e f o r t h e d i f f u s e r l o c a t i o n

has

been se lec ted ,

t he d i f f u s e r s t r u c t u r e i s designed with th e primary purpose of achievi ng

a

c e r t a i n mix ing of the d ischarged hea ted

water

with the ambient water

(n ea r f i e l d d i l u t i o n r equ i remen t ).

I n a d d i t i o n , t h e d i f f u s e r s ho ul d meet

other environmental and economical object ives.

8.2.1 Glossary of design parameters

The following nomenclature i s sug gested fo r u se i n d es ig n

of submerged mult iport d iffusers:

Equiva len t S lo t Dif fuser :

Any

mu l t ipo r t d i f fu se r wi th n ozz le

diameter , D, and spac in g , & , c a n i n p h ys ica l terms be descr ibed by

i t s

e q ui v al e nt s l o t d i f f u s e r .

Th i s co n cep t r e t a in s th e

relat ive

dynamic

26 3



ch ar ac te r i s t i c s of the d i f fu ser d i scharge and should be used f o r

mathemat ica l ana lys i s , c l as s i f i ca t i on of d i f fu ser s , compar ison of

d i f f e r en t d i f fu ser s and des ign . The s l o t w id th , B , o f t h e eq u i v a l en t

s l o t d i f f u s e r

is

n

D L R

B z -

4R

(2-20)

Important design parameters

are

t he n t h e s l o t d e ns im et ri c k o u d e

number

U

and the re la t ive water depth, H/B

.

Diffuser Load:

FT , is a de ns im e tr ic Froude number

(7-1)

where qo

= di f fu ser d i scharge per u n i t l eng th which descr ibes th e

impact (d ischarge and buoyancy e f f ec ts ) of th e d ischarge on th e t o t a l

depth.

Vertical Angle of Discharge, 8 : Varying between horizontal

( e o = 0 ) and ve r t i ca l ( e o

=

90").

Di re ct io n of Discharge:

The nozzles may

a l l

b e d i r e c t e d t o o ne

s i d e of t h e d i f f u s e r axis ( u n i d i r ec t i o n a l di sch ar g e w i th n e t h o r i zo n t a l

d i f f u s e r momentum)

or

may p o i n t i n t o b o th d i r ec t i o n s ( a l t e r n a t i n g d i s -

charge with no n et ho ri zo nt al d i f f u s e r momentum).

A

l i m i t i n g

case

of

264



a l t e r n a t i n g d i s c h a r g e is t h e v e r t i c a l d i s c ha r ge .

eq u iva len t s l o t con cept a l l a l t e rn a t i n g d i scha rg es can b e r sp r e sen ted

by e0

=

90" .

In terms

of

t h e

Control through Horizontal Nozzle Or ien tat ion , B(y): The ho ri zo nt al

ci rc ul at io ns produced by

a

dif fus er d ischarge (wi th unstab le near -f ie ld )

can be co n t r o l l ed th ro ug h in d iv id u a l h o r i z o n ta l o r i en ta t io n of th e

d i f f u se r no zz les a lo ng th e d i f fu s e r l i n e of h a l f l eng th,

Pa r t i c u la r ly imp or tant d i s t r i b u t io n s fo r B(y) are t h e

"LOG"


(see

Fig. 6-2) .

( 4 - 3 6 )

which i n t he

case

of a l t e r n a t i n g n oz zl es r e s u l t s i n a three-dimensional

s t r a t i f i e d c o un te rf lo w s i t u a t i o n and t h e

"NOR"


B(y) = 90"

i n which a l l nozzles

are

o r i e n t e d norm al t o t h e d i f f u s e r

axis

t y p i c a l l y

r e s ul t in g i n a

contracted flow-away from th e di ff us er .

Arrangement of th e Diff user

Axis

with Respect to the Ambient

Curren t Direc t ion , Y

:

Refe rr in g t o F ig . 7-1, l im i t i ng va lues o f

y

a re

t h e p a r a l l e l d i f f u s e r

(y

-

0 )

and th e p e rpen d icu la r d i f fu s e r

(y =

90").

8.2.2

Design object ives

Typica l thermal d i f f us er s wi th a low dilut ion requirement

and located

i n

shallow water

can

have

a

s ig ni fi c an t dynamic impact on

th e r ece iv in g

water.

r e l a t e d t o wh ich

i s

the genera t ion

of

h o r i zo n ta l c i r cu la t io n s in d u ced

i n t h e r e c e i v in g w a t e r as d i scu ssed i n Chapter

4.

c i r c u l a t i o n s are caused by the

ne t

disc harg e momentum ( un id ir ec ti on al

Vert ical

i n s t a b i l i t i e s

ar i se

i n t h e n e ar - fi e ld

I n a d d i t i o n h o r l z o n t a l

265



nozzles ) . Nonetheless , through ef fe c t iv e d i f f u se r design t h i s dynamic

impact

can

be con t ro l l ed t o such

a

degree

as

t o ach ieve g iven des ign

o b j e c t i v e s .

Genera l ob ject ives

re la te

to environmental and engineering require-

ments. Most impor tan t fo r des ign

i s

the conformi ty wi th l egal t emperature

standards : di f f user-induced temperatures s h a l l not exceed prescr ibed

l i m i t s d ur i ng c e r t a i n

t i m e

per iods and ou t s i de a given

area

("mixing

zone"). Typica l ly th es e

l i m i t s

app ly t o su r fa ce t em peratures. In

add i t i on ,

environmenta l goals may incl ude th e d es i r ab i l i t y of

ver t i ca l

s t ra t i f i c a t i o n to minimize thermal impact

on

t h e bottom sub s t r a t e ,

and t he sp ec i f i c a t i o n of

a

passageway i n l a t e r a l l y bounded

s i t e s ,

t o

a m i d

a

"thermal blockage" fo r aq ua t i c organisms. Also i t might be

required t o keep t h e heated discharge away from the shore l ine.

ing requirements are concerned wi th t he con t ro l of t he d i f f u se r f low f i e l d

Engineer-

for proper loc at i on of th e cool ing w a t e r i n t a ke and t o p revent r ec i r cu l a -

t i o n i n t o t h e i n t a k e .

Obviously, from the design poi nt

of v i e w

t hese genera l ob j ec t i ves

can always be m e t (assuming the s i t e geometry

i s

no t cons t r a in ing ) ,

f o r

example, by bu il di ng a " long d i f f use r" f a r away from the sh ore l ine .

A t

t h i s po i n t economic requirements e n t er th e d i scuss ion:


should be designed

t o

m e e t t he above genera l ob j ec t i ves a t a minimum

cos t .

a c t i o n o b ta i ne d i n t h i s s tu d y t h e f o l lo w in g s p e c i f i c ob j e c t i v e s c a n b e

formulated:

Thus, based on th e understanding of t h e mechanics of th e dif fu se r

a )

The dif fu se r induced temperature

rises

should be uniform

along t he whole d i f fu se r

l i n e ,

or

else

t h e d i f f u s e r c a p a c i t y i s not

266



f u l l y u t i l i z ed . This

re la tes

t o ( i ) t h e fl ow d i s t r i b u t i o n

("d i f fuser load") i n case of

a

v a r i a b l e d ep t h,

H,

and ( i i ) t o

t h e

maximum

nozzle spac ing (p reven t ion of i s o la ted % h o t ' '

s p o t s )

.

b)

No cu r r en t s , e i th e r in d u ced

by

t h e d i f f u s e r a c t i o n

i t s e l f o r given by th e ambient crossf low, should sweep along

t h e d i f f u s e r a x i s r e s u l t i n g i n r e p ea t ed e nt ra in me nt .

q u es t io n s of d i f fu s e r co n t ro l t h ro ug h h o r i zo n t a l n o zz le

or ie nt at io n, B(y), and th e arrangement with r e s p e c t t o t h e

crossf low,

y , are

important.

The

c)

In o rd e r th a t d i f fu s e r d es ig n s p er fo rm

w e l l

under

a

va ri et y of ambient condit ions, such as t i d a l or wind-dr iven cur ren ts

wi th vary ing d i r ec t i on and magnitude, i t is d e s i r a b l e t o a c h ie v e

a s t r a t i f i e d f low f i e l d .

d i f fu s e r l i n e un de r ad v e rse flow co n d i t io n s is minimized f o r such

d i f f u s e r s .

8.2.3

Design

procedure

The d eg ree o f r ec i r cu la t io n i n t o th e

By v i r t u e of th e number of impor tan t va r ia b le s invo lved , d i f f us er

design

has

t o b e a n i t e ra t ive procedure evaluating

a set

o f a l t e r n a t i v e s

( a l so d i f f e r e n t s i t es ) .

schematized

a s

t o resemble t he geometry considered i n t h is b as ic s tudy ,

a d i f f u s e r d es i gn

or

pre l iminary des ign estirnate ( e.g. f o r f u r t h e r

te s t ing i n a

h y d r a u l i c

scale

model) can be developed. The l o g i c a l

st ep s and de cis io ns which have t o be made i n t h e design procedure

are

i l l u s t ra t e d i n two specif ic examples:

a) Design

of

a

d i f fu s e r i n

a

r e ve r s in g t i d a l c u r r e n t s ys tem ( t h i s i n c l u d e s t h e l i m i t i n g caseof zero

Whenever the discharge geometry can be

26 7



cross f low) , and b ) Design of a d i f fu s e r i n a uniform (approximately)

steady

crossflow (such as a lake

o r

ocean

longshore cu r r en t ) .

The following

i s

given

i n

t h e d es ig n

problem:

1)

S i t e charac ter i s t ics

(depth,

geometry, cur rents) .

2)

Environmental

and engineering requirements

(allowable

m a x i m u m

temperature

r ise,

area of "mixing zone").

3)

Plan t charac ter i s t ics ,

namely:

- t o t a l

condenser water

f l o w ra te

To

= temperature of condenser

water. For

QO

s i m p l i c i t y , t h e ambient w a t e r temperature,

is

assumed

homogeneous,

so

t h a tTa

,

ATo To - Ta = discharge

temperature

r ise .

The

effect

of a s t r a t i f i e d

rece iving

water i s

evaluated as

the l a s t

design

s t e p .

8.2.3.1

E x a m p l e :

Diffuser

i n a

reversing t i d a l current

system

Figure 8-1 shows the hypothet ica l

s i t u a t i o n

w i t h a s loping

revers ing

cu r r en t s

-30 f t = t i d a l

-40

t

:I-

+ -

* 0.5 f p s

Scale

-20 f t

-

-

f t

-10 f t

T

= 70°F

a

/ /

ATo

= 20°F

Fig.

8-1:

Diffuser Des ign i n T i d a l

Current

System

268



offshore topography and

t i d a l

cu r r en t s r u n n in g p a ra l l e l t o th e sh o re -

line. The

ambient temperature

is T =

70°F, th e pl an t f low

is

1,000

cfs wi th

a

temperature r i se

The bo t tom coef f ic ien t , fo

,

i n t h e o f f s ho r e

area is

est imated

as

" 0.015

(average val ue f o r sandy bottom).

AT = 2OF,

t h u s g iv in g

a

minimum nea r-fi eld d i l u t i o n requirement

a

ATo =

20°F (t h e corresponding

Apo/pa

= 0.03).

The governing temperature standard

is

maX

=

AT

/AT = 10

.

0

max

min

S

The fol lowing design steps are considered:

A) Choice of Arrangement and D ir ec ti on of Disch arge

(Al terna t ing

vs.

Unidirect ional Nozzles) :

A d i f fu se r ar rang emen t p e rp en d icu la r t o th e p r ev a i l in g

t i d a l c u r r e nt d i r e c t i on i s chosen, s i n c e t h i s r e s u l t s as

a

g e ne r al r u l e

i n hi g he r d i l u t i o n s as compared t o th e corresponding p ar a l le l d i f fu se r

( t h i s

is

t r u e f o r b ot h a l t e r n a t i n g no zz l es ,

see

Fig. 7-5b, and

for uni-

d i r e c t i o n a l n oz z le s,

see

Fig.

7 - l l a ) .

(Note: A p o ss ib le excep tio n t o

t h i s

rule

may

be a

p a r a l l e l d i f f u s e r l o c a t e d on a sha l low she l f d ischarg-

i ng u n i d i r e c t i o n a ll y i n t o

an

offshore deep

w a t e r

region with strong

ad vec t iv e cu r r en t s , p r even tin g r ec i r cu la t io n ) .

A d i f f u s e r d i sc h a rg e w i th a l t e r n a t i n g n o z z l es

( g o = 9 0 ) i s chosen.

A

di f f us er wi th un id i re c t i ona l nozz les would perform

w e l l

only during

one ha lf of t he

t i d a l

cycle (when the cur ren t d i rec t ion is t h e

same as

t h e no z zl e d i r e c t i o n ) .

n o zz le s p o in t a g a i n s t t h e c u r r e n t , s t a g n a t i o n e f f e c t s o c cu r, r e s u l t i n g

i n unsteady temperature

increase

and st ro ng ly reduced performance (as

an example

for

t h i s phenomenon, see Harleman

e t

a l . (1971).

Dur in g th e o th e r h a l f o f t h e

t i d a l

cycle when the

269



B) Control through Horizon

a1 Nozzl

O r e n

a l t e r n a t i n g d i f f u s e r s

(no

ne t h or iz on ta l momentum) c on tr ol through

hor i zon t a l nozzl e o r i en t a t i on acco rding t o t he "LOG" d i s t r i b u t i o n

( E q .

(4-36), see

a l s o F i g.

6-2)

should always be used. This di s t r i bu -

t i o n i n s u r e s a s t r a t i f i e d f low f i e l d i n t h e f a r - f i e l d ( o u t s i d e t h e

unstable nea r- f ie ld zone) which

i s

des i r ab l e f rom t he s t andpo in t

of

prevent ing re c i rc u l a t io n wi th unsteady temperature

r ises. I f t h e

LOG


i s

not used (e.g. NOR d i s t r i b u t i o n ) , c u r r e n t s w i l l

s w e e p a long t he d i f fu se r ax i s r e su l t i n g i n r epea t ed r e-en trainm en t and

strongly reduced performance

(see

Fig. 4-5 and F ig ,

6-3a as

compared to

Fig. 6-4a).

Applicat ion

of

the three-dimensional

LOG

d i s t r i bu t i on , wh ich

produces

a

predominant ly two-dimensional f low f i e l d i n

the

d i f f u s e r

cen t e rp os i t i on , a l l ows usage of t he nea r- f i e ld d i l u t i o n g raphs fo r

design (e.g. Figs .

3-23, 3-24 or 6-6).

C)

I n i t i a l Design: The of fsh ore topography

i s

s loping . A n i n i t i a l

design can be made using an average depth.

subsequently).

(The design

i s

r e f i n e d

Assume the di f fu se r i s centered around t he

30

f t depth

contour, hence

Have = 30

f t

i s

taken

as

the des ign depth .

The ambient

cur ren t s are

unsteady and reve rs in g . The i n i t i a l

design

is

made for the

worst

cond i t i on , namely s l ac k t i de

(zero

curren t ) .

The performance under current conditions

i s

improved (Chapter 7 ) and

can be est imated l a te r .

I n o r d e r t o u s e

a

d i l u t i o n gr aph f o r a l t e r n a t i n g d i sc h ar ge

i t i s

necessary t o ob ta in

a

va lue fo r t he f a r - f i e l d pa ram eter ,

where

LD i s

t h e ha l f - leng th of t h e d i f fu se r .

($

=

f o

$/Have

,

I f

i t

is assumed that

270



t he t o t a l d i ff u s e r l e n gt h

is

t o

be ad jus ted i f th e computed

2LD

from t he or ig in a l assumption). Thus graph, Fig.

6 - 6 , is

a p p l i c a b l e

( i f

@

is

d i f f e r e n t f ro m t h e v a lu e s p l o t t e d i n t h i s s t ud y, gra ph s

2LD

z

2000

f t , P = 0 . 5

( t h i s

w i l l

have

is

more than 10 t o

20%

d i f f e r e n t

can be c ons tru cte d follow ing t he procedure of Paragraph 3.6.2.3).

Figure

6-6

shows t h a t t he d i l u t io n requ irement

(Ss =

10) can be

m e t

f o r di ff er en t combinations of Fs and H / B a nd t h a t t h e d i l u t i o n

requiremen t can be m e t w i t h e i t h e r a s t a b l e or u n s tab le n ea r - f i e ld

regime.

In

terms of th ree-d imensional d i f f us er ch ar ac te r i s t ic s , the

parameters

Fs

, H/B

can be expressed as

(Qo = Uo

B (2%))

Qo 9

APo/Pa and Have (assumed location)

are

f i x e d

a t

t h i s p o in t.

Only

Uo

and

(2 )

can be var ied .

I n

s e l e c t i n g a n i n i t i a l c om bi na ti on of

Fs , H/B

on the graph,

it i s

d e s i ra b l e t o b e as c l o s e as p o s s i b l e t o t h e s t a b l e n ea r -f ie l d

reg ion

because

d i f f u s e r s w it h

a

s t a b l e n e a r -f i e l d

are less

prone to t he

o n se t o f u n des i rab le h o r i z o n ta l c i r c u la t io n s .

co n t ro l t hrou gh h o r i zo n ta l n o zz le o r i en ta t io n . )

is taken a t t h e c r i t e r i o n l i n e between t h e s t a b l e a nd u n s t ab l e n ea r-

f i e l d

( i .e . ,

Fs

=

32,

H / B =

200).

(They do not require

An

i n i t i a l combin at ion

With t h e s e val ue s t h e two unknowns

2 7 1



i n Eqs. (8-1) and (8-2) are evalua ted

2LD

uO

P

312

H

Qo

12

H3/2

F s ( - g )

'a

-

B

(8-3)

(8-4)

as 2LD

=

1 ,700 f t and Uo

=

3.7 fps .

A t t h i s po in t economic considera t ions of in te rn a l d i f fus er hydrau l ics

e n t e r t h e a n a l y s i s .

Low

j e t

d i sc h ar g e v e l o c i t i e s r e q u i r e a l a r g e

d iame te r d i f fu s e r p ip e i n o rd e r t o have uniformity of discharge along

t h e d i f f u s e r .

o f sed imenta t ion i n th e p ipe.

vary between 10 t o

15

fps , wi th th e upper l im i t g iven by considera t ions

This r e s u l t s i n e x ce s si v el y hi gh c o s t s and t h e p o s s i b i l i t y

In p r a c t i c a l p ro blems, j e t e x i t v e l o c i t ie s

of pump

s i z e

and pumping co st s. Incr eas ing Uo

also

i n c r e a s e s H/B

as

seen

from Eq.

(8-2) .

Thus i n o r d e r t o i n c r e a s e t h e j e t e x i t

v e l o c i t y i t i s necessary t o move t o th e r ig h t a long t he

i n Fig. 6-6 ( i . e . move fur th er in to the unstab le reg io n) .

Ss

=

1 0 l i n e

It should be

n oted th a t t y p ic a l t he rma l d i f fu s e r d es ign s a r ea lmo s t a lway s loca ted i n

the unstable parameter range,

pure ly fo r economic reasons (keeping

shor t , and Uo reasonably high) .

2LD

I n moving upward along t h e l i n e

Ss = 1 0, t h e r e q u i r e d d i f f u s e r

This i s a consequence of the

ength, 2LD

=

1,700 f t does not change.

s lo p e of

the l i n e s and can al so be in fe rr ed from Eq. (3-222)

o r

Eq.

(7-21, which

s t a t e

t h a t t h e ne a r -f i e ld d i l u t i o n

Ss

is

dependent

272



FT

nly on th e d i f fu ser load

Thus,

which combination of Fs

, H/B is

chosen. The ch oi ce

i s

dependent on

the des ign

j e t

v e l o c i t y an d on t h e o b j ec t i v e t o b e c l o s e as p o s s ib l e t o

2LD =

1 , 7 0 0 f t , i s t h e r eq u i r ed d i f f u s e r l en g t h , r eg a r d l e s s of

t h e s t ab le range. (These two requ irem ents are i n o p p o s it i o n ) . I f

Uo

=

1 0 f p s

i s

chosen , th en from Eq. (8-2)

Fs =

150.

which

is

c lo se enough t o th e i n i t i a l assumpt ion

( 0 . 5 ) , so

t h a t no

H/B

=

51.0 and from Fig.

6-6

i s c a l c u l a t e d a g a i n ,

(D = 0.43,

f o

he value of

i t e ra t ion i s

necessary.

I f t h e d i l u t i o n r eq ui rement w e r e higher, S > 10, only

two

S

measures could be taken: in cr ea sin g th e di ff us er len gt h, 2LD

,

o r moving

t h e d i f f u s e r f u r t h e r o f f sh o r e , i n c r ea s i n g t h e d ep t h, H .

D) Design Refinement: The i n i t i a l design with an average depth,

= 30

f t , y i e l d e d a l en g t h

Have

p e r u n i t length i s

is cente red around th e

30

f t depth contour and roughly extends between

2LD

=

1700 f t

;

the average discharge

qo = 0.59 c f s / f t .

Refe r r ing to F ig .

8-1


t h e 20 f t a nd 40 f t c on to ur s.

un i t l eng th , q

uni fo rm temperature f i e l d a long th e d i f fuse r .

Obvious ly , the d if fu se r d ischarge per

, shou ld be var i ed wi th the o b je c t iv e o f ach iev ing

a

0

By v i r t u e of Eq. (7-2)

th e requirement of con s tan t

t o th e requirement of a co n s t an t d i f f u s e r o ad,

FT

,

Ss i n

case

of var i ab le dep th is tantamount

273



I f H

i s

a

v a r i ab l e , th en q s h o u ld ch an ge as

0

274

T h us , i n t h e examp le , t h e d i s c h a rg e a t t h e o f f s h o r e en d ( t w i c e

as

deep) shou ld be incr ease d by

a

f a c t o r

of

2

= 2.8 compared t o th e

/ 2

d i s c h a r g e a t th e near-shore end .

E ) Deta i l s of M u l t i p o r t D i f f u s e r

I t r e ma in s t o c h oo se t h e s e co n da r y v a r i a b l e s of t h e - m u l t i p o r t

d i f f u s e r

R ,

D and kl0 ( v e r t i c a l a n g le ) .

be chosen

s o

t h a t t h e r e

i s

good l o ca l u n i fo rm i ty of t em p era tu re r i s e

a l o n g t h e d i f f u s e r l i n e . B as ed

on

j e t s p read in g

l a w s ,

complete j e t

The nozz le spac ing ,

R ,

shou ld

in t e r f e r en ce b e fo re s u r f ace im pin gement r eq u i r e s R/H <, 0.5.

T h i s

might be somewhat r e s t r i c t i v e as

a d d i t i o n a l i n t e r a c t i o n t a k e s p l a c e

l a t e r a l l y ev en a f t e r imp in gement . B as ed on ex p e r im en ta l o b s e rv a t i o n s

R 5

H i s recommended i n o r d e r t o p rev en t i s o l a t e d "h ot s p o t s " and j e t

" b o i l areas". The nozz le d iameter D ,

i s

t h e n D =

,

where

B = Qo/(Uo2LD>.

IT

The v e r t i c a l a n gl e of t h e j e t s

s h o u ld b e l a rg e eno ug h t o p rev en t

bo t tom a t tachement and re su l t an t bo t tom scou r . Aminimum ang l e o f

0 % 20

Fo r t h e

same

r ea s o ns t h e h e ig h t , h s , o f t h e n o zz l e s ab ov e b o tt o m

should be h >, 3 D . For t h e co n t ro l (B (y ) g iv en by t h e LOG d i s t r i b u t i o n )

t h e maximum v e r t i c a l n o z z l e a n g l e

0

0

i s s u gg e st e d f o r b o t h a l t e r n a t i n g a nd u n i d i r e c t i o n a l d is c h a r g e s .

0

S

<

79O,

Equat ion

( 4 - 4 4 ) .

No

0



v e r t i c a l d i s c h ar g e s ho ul d b e us ed f o r u ns t a b l e n e a r- f i e l d c o n d i t i o n s .

F) Ef f ec t of Cross Cur r ents

The i n i t i a l d es ig n w a s c a r r i e d o u t f o r t h e s l ac k

( w o r s t ) c o n d i t i o n .

For p e r p en d i c u l a r d i f f u s e r s w i t h a l t e r n a t i n g no z zl e s t h e e f f e c t of c r o s s

f low i s a lwa ys be ne f i c i a l on d i lu t i on pe rf o rm a nce . Assuming the t i d a l

c u r r e n t var ies between u = k 0 .5 f p s , t h e pe rf or m ance

a t

non-slack

co nd it io ns may be checked by u si ng a crossf low graph, F igure 7-5

(@=

0.5). The

e f f e c t of c r oss f low i s i n t r o d u c e d by V* = Vsiny

(y

= 90

, e r p e n d i c u l a r d i f f u s e r an d V = u

H

l oa d ,

FT,

i s

f rom Equation

(8-5)

w i t h t h e a p p r o p r i a t e v a l u e s , FT

=

0.012.

The value of V

i s

f p s , V =

(0.5)(30)/(0.50) = 25.4 a nd th e co r r e spond ing d i lu t i on i s

a

0

/qo). T h e d i f f u s e r

o ave

changing wi th th e c ur re nt magni tude,

f o r u

=

0.5

a

maX

= 25 .4 . Under t h i s c ond i t ion the d i f f u s e r

i s

o p e ra t i ng i n t h e

sS

fu l l y mixed regime.

1 2. 7, t h e d i l u t i o n Ss = 1 2.0 a nd t h e d i f f u s e r i s o p e r a t i n g

a t

t h e

edge of th e sta gn an t wedge regime.

For ua < 0 .2 5 f p s t h e r e i s a

s t r a t i f i e d c ou nt er fl ow u p st re am o f the d i s ch a r ge and t h e d i l u t i o n s

a pproa ch those f o r :he s l a c k t i d e c ond i t ion .

For

an

a ve r a ge c u r r e n t ua = 0.25 fps, and

V =

G rap hs o f n e a r -f i e l d d i l u t i o n s ( n o t a cc o un t in g f o r any p o s s i b l e

h e a t r e t u r n d u e t o t i d a l mo ti on )

as

a f u n c t i o n of t i d a l t i m e can be

c o n s t r u c t e d .

G)

Ef f e c t of Ambient S t r a t i f i c a t i on

I n t h i s e xam ple the a m bie nt

water

has been assumed t o be

u n s t r a t i f i e d .

as

f o l l o w s :

The e f f e c t o f a m b ie n t s t r a t i f i c a t i o n may b e e v a l u at e d

I f a s t e p c ha nge i n a m bie nt de ns i ty ( t e m pe r a tu r e) oc c ur s

275



a t

h a l f d e p t h and p r ov i de s an e f f e c t i v e " c e i l i n g " t o t h e f l ow f i e l d ,

then t he des ign dep th would be reduced t o H / 2 .

t h e d i l u t i o n would d ec reas e by

a

f a c t o r

o r

2 (Equat ion

(8-3))

and the

A s a conseq-uence,

t em p era tu re

r i s e

of th e mixed

w a t e r

would be doubled . Unless the

ambient s t r a t i f i c a t i o n i s v e ry pro no un ced (o f t h e o rd e r o f ab o u t h a l f

t h e d i s c h a r g e t e m p e r a t u r e

r i se)

, t h e d i f f u s e r d is ch ar ge

w i l l

b r e a k

up t h e n a t u r a l s t r a t i f i c a t i o n and " s u bs t i tu t e1 ' an a r t i f i c i a l

s t r a t i f i c a t i o n . The a mbi en t t e m p e ra t u r e c a n t h e n be d e f i n e d as t h e

average v e r t i c a l t em p era tu re ,

T

9 and

a

ave

AT

=

To

-

T

0

a

ave

i s

t a k e n as

t h e d e s i gn v a r i a b l e a nd a l l induced tempera tu re

r i s e s

a re

s i m i l a r l y r e f e r r e d t o T .

a

ave

8 . 2 . 3 . 2 E X a m D le : D i f f u s e r

i n

Stead y Uniform Crossf low

The s i t e c h a r a c t e r i s t i c s

are

shown i n Fig u re 8-2

-

=

a

0.5 fps.

H

=

20 f t ( un if or m)

S c a l e

I

0

10od f t

T

=

70°F

a

F i g .

8-2:

Di ffu ser Des ign i n S teady Uniform Cross f low

276



The of fshore topography i s un i f o r m ly sha l low, H =

20

f t , t h e

cur rent magni tude i s 0.5

fp s and does not have s t r ong uns teady

f l u c t u a t i o n s . T h e

same

p l a n t c h a r a c t e r i s t i c s and t h e same d i l u t i o n

requirement (10) as i n t he p r e v ious e xa mple are assumed.

A)

A n arrangement y = 60

Choice of Arrangement and Di re ct io n

of

Nozzle Discharge

0

and

a

u n i d i r e c t i o n a l d i s ch a r ge

i s

chosen.

The ob je c t ive i s t o in t e r c e p t s ub s t a n t i a l amount o f c r oss f low ( he nce

a vo id ing

y % 0)

whil e a ls o prov id in g some momentum i n t o th e of f sho re

d i r e c t i o n t o " push" t h e t e mp e ra t ur e f i e l d i n t o d e ep e r o f f s h o r e

areas .

The

v e r t i c a l

noz z le a ng le

i s

chosen

as

B o = 20°, t o provide m a x i m u m

ho r i zo nt a l momentum whi le preve nt in g bot tom scou r .

B)

Hor izo nta l Nozzle Or ie nta t io n , B(y)

A l l u n i d i r e c t i o n a l d i f f u s e r s w h ic h p ro v i de e nough h o r i z o n t a l

momentum t o r e su l t i n f u l l mixing ( c r i t e r i a l i n e s i n F i g u r es 3-28

t o 3-31 and

6-10

t o 6-13) are s u b j e c t t o r e c i r c u la t i o n .

of

re c i rc ul a t io n depends pr im ar i ly on an ambient advec t ive mechanism

(such

as

th e presen t cu r ren t svs tem) ,which car r ie s t he t e m pe r a tu r e

f i e l d away f ro m t h e d i f f u s e r l i n e , a n d o n t h e n o z z l e o r i e n t a t i o n ,

f 3( y) . I n ge ne r a l , t he

LOG

d i s t r i b u t i o n w h i l e pr o v id i n g somewhat

h i g he r n e a r - f i e l d d i l u t i o n s i s somewhat more l i a b l e t o r e c i r c u la t io n .

The

NOR d i s t r i b u t i o n (B(y)

= 9

r e s u l t s i n a c o n t r ac t e d f lo w f i e l d

and

i s l es s

l i a b l e t o r e c i r c ul a t i on . A NOR d i s t r i b u t i o n i s chosen.

The

degree

C)

I n i t i a l D esign

F or

a

u n i d i r e c t i o n a l d i f f u s e r w i t h NOR d i s t r i b u t io n , f u l l

v e r t i c a l m ix in g a nd u nd er t h e i n f l u e n c e o f

a

crossf low,Figure 7-10

277



( A d a s ' model) i s a p p l i c ab l e . F i g ur e 7-10 a l s o i n c l u de s c r i t e r i a

of a p p l i c a b i l i t y f o r t h e t h e o r e t i c a l p r e d i c t i o n . In t h e fu l l y mixed

r e g i m e t h e d i l u t i o n , S i s e q u a l t o

V*

S

(7-6)

1 1 2 2

H

1 / 2

2 2

0

V*

= -

Vsiny

+

-

(V

s i n y

+

2

co s

8

)

t h a t i s , the combined e f f e c t of cr os s f l ow and d if f us e r momentum, and

t h e d i l u t i o n i s i n d ep en d en t of t h e d i f f u s e r l o ad , FT.

i n d i c a t e t he d es i r ab i l i t y o f m ak in g FT

as

l a r g e

as

p o s s i b l e t o r e d uc e

t h e d i f f u s e r l e n g th , 2% (s ee Equa tion (8-5 )). By doing s o , however,

th e s t r en gt h of t h e momentum-induced cu rr en t w i l l i n c r e a s e and t h e

p o t e n t i a l f o r r e c i r c u l a t i o n

a t


w i l l

a l s o i n c r e a s e .

Usual ly , the c o n s t r a i n t on d i f f u s e r l e n g t h

i s

give n by the range o f

p o s s i b l e j e t e x i t v e l o c i t i e s ( s e e c o n s i de r a t i o ns i n t he f i r s t ex am pl e) .

A j e t e x i t v e l o c i t y

(7-6) can

be

m o di f ie d t o gi v e t h e r e q u i r e d d i f f u s e r l e n g t h

This would

Uo = 10 f p s i s chosen. With V* = Ss, Equat ion

1

(8-8)

1

0

2 0

2% =

s s

H

S u s i n y + Uocos 8

s a 0

Using the above va lue s , Equat ion(8-8) g i ve s

t o check

whether

i t

i s l i a b l e t o r e c i r cu l at i on e f f e c t s .

2 5 = 5 5 0 f t .

I t remains

a) w h e th e r t h i s d e s ig n i s i n t h e f u l l y mixed r a n g e and b)

T

)

The d i f fuser load , FT , Equat ion (8-5)

i s

c a l c u l a t e d as F

=

0 .06 , which f rom Figure 7-10 in d i ca te s tha t i t

i s

w e l l

i n s i d e t h e f u l l y mixed r a ng e . N o upstream wedge

w i l l

o ccu r .

The c r o s s f l o w m ix in g r a t i o i s Vsiny

=

u Hsiny

/q

The r em ai ni ng c o n t r ib u t i o n t o t h e t o t a l d i l u t i o n

( S

= 1 0 )

i s 5 . 1 from momentum-induced c u r r e n t s . A s a r u l e

of

thumb

b)

=

4 . 9 .

a 0

S

278



i t

i s h y p ot h es i ze d t h a t t h e d e g r e e o f r e c i r c u l a t i o n i s

n e g l i g i b l e o nl y i f t h e c o n t r i b u t i o n f ro m t h e c r o s sf l ow

mixing

i s

a t

l e a s t Q 50%

of t h e t o t a l d i l u t i o n . I n o t h e r

cas es , th e s t re ng th of th e ambient advec t ive mechanism

w i l l

n o t b e s u f f i c i e n t t o p re v en t c i r c u l a t i o n

Thus i n case of u n s te ad y c u r r e n t s ( l o w er v e l o c i t i e s ) t h e d i f f u s e r

per formance may be dr as t i ca l l y decreased .

depends on the dur a t i on o f th e low c u r r e n t ve loc i t i e s :

have t o be determined from a h y d r a u l i c scale model.

The o ve ra l l pe r formance

S uc h e f f e c t s

E)

Deta i l s

of M ul t ipo r t D i f f use r

D i f fu s e r d e t a i l s are determined as i n th e p r e v ious e xa mple .

8 . 3

The Use of Hydraul ic Scale Models

C e r t a i n s i t e s having complex topographic features and non-

u n if or m o r u n st e a d y a mb ie nt c u r r e n t s may p r e s e n t d i f f i c u l t i e s i n

sc he m a t i z a t ion and a p p l i c a t i o n of t he t h e o r e t i c a l p r e d i c t io n s d ev el op ed

i n t h i s s t ud y . N e v er t h el e s s,

i t w i l l

u s u a l l y b e p o s s i b l e

t o

u s e t h e

d e si g n c h a r t s t o ar r ive a t a n i n i t i a l d e s i g n wh ic h c an b e t e s te d . an d

mo dif ied by means of a h y d r a u l i c sca le model.

t h a t

c an b e i n cl u d ed i n

a

model study

are

t h e p o s s i b i l i t y of re-

c i r c u l a t i o n

a t

the condenser water i n l e t and the unsteady build-up

of heat due t o r e c i r c u l a t i o n a t t h e d i f f u s e r . The l a t t e r i s par -

t i c u l a r l y i m po r ta n t i n t h e

case

of d i f f u s e r s w i t h s i g n i f i c a n t

horizontal momentum input.

A d d i ti o n a l f e a t u r e s

T y p i c a l p ro bl em s a s s o c i a t e d w i t h t h e o p e r a t i o n of d i f f u s e r

s c a l e

models are those which were d i s c us s e d f o r t h e p r e s e n t b a s i c e x p er i m en t a l

279



s t u d i e s ( S e c t i o n

6.1.21,

n am e ly , b ou nd ary e f f e c t s and t u rb u l e n ce

requir ements . Both problems r e l a t e t o t h e l i m i t e d s i z e of a v a i l a bl e

t e s t i ng f a c i l i t i e s .

I n t h i s s e c t i o n mo de lin g re qu ir em e nt s f o r d i f f u s e r

s ca l e

models

are

d i sc u ss e d, i n p a r t i c u l a r t h e u s e of d i s t o r t e d

sca le

models i s

in ve s t iga te d , and methods of model boundary co n t ro l , t o min imize

b o u n d a ry e f f ec t s , are ex p lo red .

8.3.1

Modeling Req,uirements

The d i f f u s e r f lo w f i e l d i n v o lv e s seve ra l d i f f e r e n t

regions.

Modeling l a w s re qu i r e the eq ua l i t y of geom et r i c and dynamic non-

d im en sio n a l p a ram e te r s f o r m od el ( s u b s c r i p t

m)

an d p ro to ty p e ( s u b s c r i p t

p ) . In g en e ra l , i n any m o de l in g p ro b lem , a l l parameters canno t be

k ep t eq u a l , and

a

c h oi c e r e l a t i n g t o t h e

r e l a t i v e

impor tance o f ce r t a i n

p a r a m et e r s h a s t o be made.

The

o b j e c t i v e o f sca le models i s t h e d e t e r m i n a ti o n of t h e

t h re e -d i m en s io n al t e m p e ra t u r e ( d i l u t i o n s ) a nd v e l o c i t y f i e l d .

( o u t s i d e t h e l o c a l

s S

a)

The n e a r - f i e l d s u r f a c e d i l u t i o n ,

m i xi ng z on e i n c a s e of a n u n s t a b l e n e a r - f i e l d ) i n

a

g e n e r a l

c r o s s f l o w s i t u a t i o n ( C ha pt er

7 )

i s

a

f u n c t i o n

of

t h e

d i m e n s i o n l e s s v a r i a b l e s .

(8-9)

b) I n a d d i t i o n , d i l u t i o n s w i t h i n t h e b u oy an t j e t r e g i o n are

d epen den t on t h e d i s t an ce

s

B

(3-81)

and the j e t Reynolds number

280



2

3,000

-R

= -

4Rh

j

V

(5-3)

t o i n s u r e f u l l y t u r b u le n t

j e t

behaviour .

c ) The t h r ee - d im e ns iona l t e m pe r a tu r e f i e l d ou t s i de of th e

immediate ne ar- f ie ld zone i s dependent on the r a t i o of t h e

d i f f u s e r h a l f l e n g t h t o d ep th

(8-10)

D

H

d)

d i s t r i b u t i o n i s a f f e c t e d b y s u r f a c e

heat

l o s s .

p a ra m et e r f o r t h i s r e g io n i s ( f rom Equation 3-168)

A t

l a r ge r d is t anc es away f rom

the

n e a r - f i e l d , t h e t e m pe r at u r e

The governing

ta k ing LD as t h e cha rac te r i s t i c h o r i z o n t a l l e n g t h and

U

v e l o c i t y scale.

I t

i s p o i nt e d ou t a ga in t h a t l o c a l d i f f u s e r d e t a i l s , R / H , hs/H,

as t h e

0

d o n o t i n f l u e n c e t h e d i f f u s e r p er fo rm an ce as l ong as reasonable j e t

i n t e r f e r e n c e i s obtained (R/H

< 1). The

n o z z l e o r i e n t a t i o n , p ( y ) ,

however, i s im por ta n t .

The r a t i o o f th e gove rn ing parameters f o r mode l and proto typ e

i s i n v e s t i g a t e d

f o r

bo th und i s to r t e d a nd d i s t o r t e d m odel s. F or t h i s

p u r p o s e t h e f o l l o w i n g r a t i o s

are

def ined:

L =

h o r i z o n t a l l en g th r a t i o

z

= v e r t i c a l

l en gt h r a t i o

u = v e l oc i t y r a t i o

r

r

r

281



*PO

P a

(-

g),

=

r ed uc ed g r av i t y r a t i o

z

r

L r

T h e d i s t o r t i o n i s given bv

-.

For an un di s t or ted model z

= L

.

A l l models have t o be den s im etr ic Froude models

as

a minimum require-

r

r

ment. Thus

Fsr

= 1

Heated discharge

same temperature

(8-12)

g I r

: , i . e . ,

odels

are

u s u a l l y o p e r a te d w i t h

(-

A P O

P a

d i f f e r e n t i a l s

as

i n the p r o to type , due t o measurem en t

..

I n

and c on tr o l problems,

s o

t h a t u

:

r IL t y p i c a l l y

as

i n

a

f r e e

r

surface Froude model.

The bottom f r i c t i o n f a c t o r s , f o , i n t h e f a r - f i e l d

parameter,

@ =

f o

L,,/H

i n Equa t ion ( 8 - 9 ), i s d i f f e r e n t f o r m odel and p r o to type .

The

param eter f depends on t he r e l a t i ve roughness of th e bot tom

0

ks/\.

P r o t o t y p e v a l u e s

are f 'L

0 . 0 1 t o

0.02

independent of the

Reynolds

number, wh il e model val ue s

are

s t r o n g l y

scale

(Reynolds

number) dep end ent and may va ry bet ween f 'L 0 .03 t o 0 .08. F ur the r -

more, Equation

(8-9)

assumes

f i

=

Afo

where

A

%

0.4 t o

0.5

( se e

Paragraph 3.6.2.2)

ove r

a

wide range of Reynolds numbers.

0

P

0

m

T y p ic a l v a l u e s f o r t h e s u r f a c e h e a t e x ch an ge c o e f f i c i e n t ,

K ,

F and f o r the

0

are f o r t h e p r o t ot y p e K

model K~

=

100 to

150 B T U / f t ,

day,

fl ux phenomena in vol ved .

=

1 50 t o

200 B T U / f t ,

da y ,

P

2

0

F r e f l e c t i n g t h e d i f f e r e n t h ea t

282



8.3.2

Undistor ted Models

(zr =

Lr)

F o r u n di s t o rt e d d i f f u s e r mo del s, t h e r a t i o o f a l l governing

paramete rs (a ) t o d) above)

i s

u n i t y ex c ep t f o r

f

(8-13)

and

The e f f e c t of Equations (8-13)and (8-P4)is s t ro ng ly

sca le

dependent.

F or d i f f us e r m odel s

i t

i s

d e s i r a b l e t o r ep ro d uc e

as

l a r ge a n

area as

pos s ib le ( to minimize boundar y e f f e c t s ) .

however, i s given by t he d i scharg e turbu lence requirement , Equa tion

The minimum scale ra t io,

L r ,

( 5- 3) , f o r u s u a l co n d i t i o n s

Lr

equa ls 1/100 o 1/120.

scale r a t i o h a s i m p l i c a t i o n s on t h e v a l u e s of f o

The

s m a l l

(and hence @ ) and

r

Kr/Lr1’2.

Typic a l va lue s

are

m

’L 2 t o 5

(8-15)

1,2

‘L 4 t o

15

r

Lr

(8-16)

w i t h

K

between 0.5 and 0.75.

r

The e f fec t of

the

e xa gge r ate d f a r - f i e ld e f f e c t s , Equa t ion (8-101,

c an b e es t i m a te d f o r a n a l t e r n a t i n g d i f f u s e r

ss

n n e a r- f i e ld d i l u t i o n ,

from Equation

(7 -2 ) :

t h e d i lu t i o n r a t i o

S

=:

0.83 (using Fig ure 3-23). For la rg er

@

t h e

e r r o r i n p r e di c ti o n

i s

decreased . In gen era l , an und is t or t ed model

g i v e s a somewhat c on se r va t iv e p r e d ic t ion o f t he ne a r f i e l d d i l u t i on

A s s u m e @

=

0.5 and

a m = 2.0 ( i . e . , Q r = 4 )

S P

P

r

283



( i n t h e

extreme

about

20%

l o w e r ) .

The e f f e c t of t h e i n c r e a s e d h e a t l o s s t o t h e a t m o sp h er e ,

Equat ion (8-16), becomes important only

a t

d i s t a n c e s f ro m t h e d i f f u s e r ;

i t

h a s a n e g l i g i b l e i n f l u e n c e on n e a r - f i e l d d i l u t i o n s ( s e e p ar ag r ap h

3 . 4 . 4 . 2 ) .

A s

u n d i s to r t e d d i f f u s e r m od els r e p l i c a t e

a

l i m i t e d f a r - f i e l d

r an ge , t h i s e f f e c t

i s

n o t i m p o r t a n t .

U n d i st o r te d d i f f u s e r m o de ls r e p r od u ce c o r r e c t l y t h e g e om e tr i c

p r o p e r t i e s of t h e d i f f u s e r f lo w f i e l d , s u ch as e x t e n t of t h e

tempera-

t u r e f i e l d , l o c a l m ix in g z on e, i n t e r a c t i o n w i t h a n am bi en t c u r r e n t ,

b u t a r e somewhat co n s e rv a t i v e i n d i l u t i o n p red i c t i o n . The co n s erv a t iv e -

n es s can b e e s t im a t ed i n any

case.

T h e i r m a jo r d i s ad v an t a g e i s t h e

l i m i t e d f a r - f i e l d r a ng e w hi ch c a n b e s i m u l a te d .

8 . 3 . 3 D i s t o r t e d Models z > Lr)

I t

i s

r e q u i r e d t o r e pr o d u ce t h e same n e a r - f i e l d c h a r a c t e r i s t i c s

r

r e l a t i n g t o t h e s t a b i l i t y c r i t e r i o n a n d t o t h e h o r i z o n t a l momentum i n p ut ,

s o t h a t i n E q ua ti on (8-9)

(8-17

T h i s

i s

t an ta mo un t t o s a y i n g t h a t t h e d e t a i l e d d i f f u s e r g eom et ry

s h ou ld b e l o c a l l y u n d i s t o r t e d ( n o t e0

=

zr/Lr ) , t h a t

i s ,


r

s h o u ld b e m o de l led wi th

t h e

v e r t i c a l

sca le r a t i o , z - B r . Then

r

r

r

-

2

(8-18

)

a nd i n p a r t i c u l a r i f t he d i s t o r t i o n r a t i o i s ch o s en s u ch t h a t

z /L

f

=

r r

, Q r = 1, t h e same f a r - f i e l d ef fec t s .

0

r

284



However,

t h e r e l a t ive d i f f u s e r ha l f l e n g t h, (8-lo), -is a f f e c t e d

b y d i s t o r t i o n

(8-19)

i n s t e a d o f 1, and the ext en t of t he near - f ie ld zone , governed by

j e t

dynamics, (3-81),

i s

e x a gg e r at e d s i n c e

(3,

= I i n s t e a d o f Lr / zr .

The

h e at l o s s r a t i o i s

(8- 20)

3/2

L r

=

1.

r

h ic h c an b e e q ua l t o u n i t y o n l y i f

z

As i n t h e u n d i s t o r t e d mo de l, t h e

ver t i ca l sca le

r a t i o

z

ha s a

r

m i n i m u m of 1/100 t o 1 /1 20 f o r

j e t

turbulence requirements .

The d i s t o r t i o n f a c t o r may b e c h os en p r i m a r i l y i n c o n s i d e ra t i o n

of

Eq uat io n (8-13), th us z,/Lr = f o .

r

r e a l i s t i c f a r - f i e l d he a t lo s s , Equa t ion ( 8 - 20 , wh ic h may be im por tan t

Th i s w i l l a l s o i n s u r e

a

more

i n d i s t o r t e d models r e p l i c a t i n g l a r g e r areas.

However, the geometr ic e f fe c ts , Equa t ions

(8-19)

and

($1; = 1,

may

b e c r i t i c a l i n c e rt a in cases: Dis to r t e d m odel s do no t r e p r e se n t t he

c o r r e c t t h r e e -d im e ns ional t e m pe r a tu r e f i e l d induc e d by

a

d i f f u s e r .

The e x t e n t o f the ne a r - f i e ld z one i s exaggerated by

z

/ L r , as

shown

above, which can have re per cus s io ns when des ign ing f o r

a

"mixing

r

zone" d e fi n ed i n l e g a l s t a n d a rd s . I n a d d i t i o n , t h e r e l a t i v e

s h or t en i ng ' ' o f t h e d i f f u s e r h a l f l e n g t h ,

b / H ,

h a s t w o e f f e c t s :

1

(i) Whenever currents are swe ep ing a long the d i f f us e r axis ( p a r a l l e l

d i f f u s e r , o r d i f f u s e r w i t h a l t e r n a t i n g n o z z l es and no c o n t r o l ) t h e

285



d eg ree o f r ep ea t ed en t r a in m en t

i s

s t ro ng ly dependen t on L

/H.

( i i ) F or u n i d i r e c t i o n a l d i s c h a r g e w i t h a v e r t i c a l l y f u l l y mixed

flow-away, l a t e r a l d e n s i t y c u r r e n t s a c t a t the end of the f low away

zone ( l a t e r a l wedge i n t r u s i o n ) . T h is e f f e c t i s ex ag g e ra t ed d u e t o

D

t h e r e l a t i v e s h or t en i ng of t h e d i f f u s e r a nd r e s u l t s i n e a s i e r

s t r a t i f i c a t i o n and l e s s t en de nc y t o r e c i r c u l a t e . I n t h e s e c a s e s

p r e d i c t i o n s from d i s t o r t e d m od el s are non-conserva t ive .

To c ou n t e rb a l a nc e t h e s e e f f e c t s ,

a

d i s t o r t i on r a t i o

z /L

l e ss

r r

than f

t h e n somewhat c o n s e r v a ti v e , y e t t h e f a r - f i e l d e f f e c t s are modeled i n

a

more r e l i a b le manner.

may b e cho sen: The n ea r - f i e l d d i l u t i o n p r ed i c t i o n ,

S s , i s

0

r

It

i s

c o nc lu de d, t h a t d i s t o r t e d mo de ls w h i l e b e i n g less con-

servat ive

i n t h e p r e d i c t i o n of t h e n e a r -f i e l d d i l u t i o n , do n o t

r e p r e s e n t t h e t r u e t hr e e -d i me n si o na l f l ow f i e l d : The e x t e n t of t h e

mixing zone i s ex ag g era t ed ( s o n et im es g ro s s ly co n s e rv a t i v e ) an d f o r

ce r t a i n d i f f u s e r s chemes, t h e i n t e r a c t i o n w i th c u r r e n t s i s d ec reas ed

(n o n -co n s e rv a t i v e ) . Y e t i n

a

va r i e ty of s i t u a t i o n s , s u c h

as

a n a l t e r -

n a t i n g , p e r p en d i c ul a r d i f f u s e r , a d i s t o r t i o n can a lways b e u sed and

may even be de s i ra b le

( a c c o un t i ng f o r t h e e x a g ge r a t io n o f a l o c a l

mixing zone). The major advanta ge of d is to rt e d models

i s

t h e l a r g e

h o r i z o n t a l

area

which can be reproduced .

The c h oi c e of n o z z l e d e t a i l s , R, hs,

i s

u ni mp or ta nt i n t h e

h y d r a u l i c scale m od el , and n eed n o t a g ree wi th t h e p ro to ty p e v a lu es ,

p ro v id ed , h owev er , t h a t t h e eq u iv a l en t s l o t co ncep t i s m ain t a in ed

an d t h a t

R/H

5 1 , p rev en t i n g no n -u n ifo rm i ty o f t em p era tu re s i n t h e

n ea r - f i e l d .

286



8 . 3 . 4

Boundary Control

The ob je c t ive

of

boundary co nt ro l methods

i s

t he minimiza t ion

These

boundary e f fec ts

a re

p a r t i c u l a r l y

f

model boundary ef fe ct s .

d i f f i c u l t i n s tu dy in g d i f f u s e r i n du ce d c i r c u l a ti o n s . I t i s e s s e n t i a l l y

d e s i r e d t o s i mu l at e

a

c o n d i t i o n as i f t he boundar y wou ld no t be p r e se n t ,

t h a t i s mixed or ambient

w a t e r i s

withdrawn o r sup pl ied a t t h e

bounda ry wi thou t a c c e le r a t i ng o r de c e le r a t ing the c u r r e n t s . However,

th e se c on t r o l methods r e qu i r e th e

a

pr ior i knowledge of t h e c o r r e c t

wi thd rawal and supply

ra te ,

and are t hu s r a t he r su b j e c t ive . An

i t e r a t i v e method observing th e behav iour of th e curr en t system could

be sugge s te d , bu t i s a g ai n d i f f i c u l t t o e v al u at e . I n g e n e r a l , em phasis

shou ld be pu t on d i f f us e r de s igns which do no t inp u t c ons id e r a b le

ho r i zo nt a l momentum and thus do no t r eq ui re ex ten s iv e boundary c on tro l .

287



IX

SUMMARY AND

CONCLUSIONS

9.1 Background

A s u b m erg ed m u l t i p o r t d i f fu s e r i s a n e f f e c t i v e d e vi c e f o r d i s -

posa l o f

w a t e r

c o n t a i n in g h e a t o r o t h e r d e g r ad a b le

wastes

i n t o

a

na tu ra l body of

w a t e r .

A h i g h d e g re e of d i l u t i o n c a n b e o b t a i n e d

and th e env i ronmenta l impact of conc en t ra ted waste c an b e c o n s t r a i n e d

t o a s m a l l area.

The s u b m erg ed m u l t i p o r t d i f fu s e r i s e s s e n t i a l l y a p i p e l i n e l a i d

on t h e b o t t om o f t h e r ece iv in g

w a t e r .

The

w a s t e w a t e r

i s d i s ch a rg ed

i n t h e fo rm o f ro un d t u rb u l en t j e t s t h ro u g h p o r t s o r n o zz l e s wh ich are

s pa ce d a l o n g t h e p i p e l i n e .

The

r e s u l t i n g d i s t r i b u t i o n of c on ce nt ra -

t i o n o f t h e d i s c ha r g e d

waste

mate r ia l s w i t h i n t h e r e c e iv i ng water de-

pends

on

a v a r i e t y of p h y s i c a l p r o c e s s e s .

T h i s i n v e s t i g a t i o n i s concerned wi th th e development of pre -

d i c t iv e methods fo r buoyan t d i sch arg es f rom submerged mu l t i po r t

d i f f u s e r s .

The

f o ll o wi n g p h y s i ca l s i t u a t i o n i s cons idered : A m u l t i -

p o r t d i f f u s e r w i t h g i ve n l e n gt h , n o z z l e s pa c in g and v e r t i c a l a n g l e of

n o zz l e s

i s

lo ca te d on th e bot tom of

a

l a rg e body of water of uniform

dep th .

The

ambient water i s u n s t r a t i f i e d

and may be stagnant

o r have

a uni form c ur re n t which runs a t a n a r b i t r a r y a n gl e t o t h e axis of

t h e d i f f u s e r .

The genera l case o f

a

d i f f u s e r i n a r b i t r a r y d ep th of w a t e r and

a r b i t r a r y b uo yancy i s t r ea te d. However, emphasis i s p u t o n t h e

d i f f u s e r i n s h a l lo w r e c e iv i n g water wi th low buoyancy, th e ty pe used

f o r d i s ch a rg e of co nd ens e r coo l i n g water f rom thermal power p lan ts .

288



The devel opment

of

predi cti ve methods i s i mportant i n vi ew of :

--

Economcal desi gn of t he di f f user structure.

--

Desi gn

t o

meet speci f i c water qual i ty requi rements.

--

Eval uat i on

of

the i mpact

i n regi ons away f romt he di f f user ,

such as the possi bi l i ty of reci rcul at i on i nto the cool i ng

water i ntake of thermal power pl ants.

--

Desi gn and operati on

of hydraul i c scal e model s.

9 . 2

Previ ous Predi ct i ve Techni ques

Previ ous predi cti ve model s f or submerged di f f user di scharge

have been devel oped for two l i mt i ng cases of di f f user di scharge:

di scharge i n unconf i ned deep water i n the form

of

ri si ng buoyant j ets

(j et model s),

and di scharge i nto shal l owwater w th extreme boundarv

i nteracti on resul t i ng i n a uni f ormy mxed current .

These model s are not appl i cabl e for submerged di f f users i n water

of i ntermedi ate depth and no cri ter i a of appl i cabi l i ty have been

establ i shed.

9.3 summary

An

anal yt i cal

and

experi mental i nvest i gat i on of the di f f user

probl emi s conducted.

I n the anal ysi s the equi val ent sl ot di f f user

concept was used throughout as the sal i ent representat i on of a mul t i -

port di f f user w th l ateral l y i nterf er i ng or mergi ng j ets.

9.3.1 D f f users w thout Ambi ent Crossf l ows

a) Two-D mensi onal Channel Model

A

mul t i port di f f user w l l produce a general three-di mensi onal

f l ow fi el d.

Yet the predomnant l y two-di mensi onal f l owwhi ch i s

289



al ong the di f f user axi s. I n part i cul ar, the nozzl e ori entat i on

( LOG di str i buti on)

( 4 - 3 6 )

i nsures a stabl y strat i f i ed counterf l owregi me outsi de the unstabl e

near- f i el d zone.

The maxi mumvert i cal angl e

8

whi ch can be ut i l i zed t o control

0

the hori zontal ci rcul at i on i s found to

be %

79' (Equat i on

( 4 - 4 4 ) ) .

control i s needed for di f f users w th a stabl e near- f i el d zone.

N o

No

three- di mensi onal f l owmodel has been devel oped f or t he case

of

di f f user di scharge

w i t h

net hori zontal momentum

that L

5 5

al so provi des equi val ency of f ar- f i el d ef f ects f or thi s

case.

I t

i s hypothesi zed

c) Experi ments

Experi ments were perf ormed i n two set - ups, i nvest i gati ng both

a two-di mensi onal sl ot di f f user and a three- di mensi onal di f f user.

D f f user i nduced temperature ri ses are measured and compared w th

theoret i cal predi cti ons.

9 . 3 . 2

D f f users w th Ambi ent Crossf l ows

The di f f user i n a uni f ormambi ent cross- current i s i nvesti gated

experi mental l y.

governi ng parameters i s devel oped. However, f or the purpose

of

qual i tat i ve di scussi ons and quant i tat i ve data presentati on, extreme

cases of di f f user behavi or (regi mes) can be- i sol ated and descri bed

anal yti cal l y. The advantage i s that the number of si gni f i cant

No compl ete theory descri bi ng the ful l range of the

29

4



parameters i s cons iderab ly reduced in these reg imes .

regimes,

the

treatment i s

r e s t r i c t e d t o d i f f u s e r s w i t h a n u ns ta bl e

n ea r - f i e l d zone . D i f fu s e r s w i th a s t ab l e n ea r - f i e l d zone are i n

g e n e r a l l i t t l e a f fec t ed by t h e c ro s s fl o w ex cep t fo r t h e d e f l ec t i o n of

th e buoyant

j e t s .

fac e l aye r wi thou t caus ing re -en t ra inment .

T o

i s o l a t e t h e

The crossflow w i l l m erely t r an s l a t e t h e s t a b l e s ur -

Fo r d i f f u s e r s wi th an u n s t ab l e nea r - f i e l d zon e i n

a

crossflow

th ree

regimes

are

important :

induced curr ent s , and crossflow mixing.

d e l i n e a t i n g

the

f low regimes

are

presented. The di f fu se r performance

i s found to be dependent on th e arrangement of t h e d i f f u s e r axis with

r e s p e c t t o the c ro s s f l o w d i r ec t i o n .

buoyancy induced counterflow, momentum

Di lu t i o n g rap h s wi th c r i t e r i a

A

p e rp en d i cu l a r d i f fu s e r gen-

e r a l l y y i e ld s b e t t e r p er fo rm an ce t han a p a r a l l e l d i f fu s e r .

9 .4 Conclusions

The agreement between theo re t i ca l p red ic t ion s o f near- f ie ld

d i l u t i o n s and e x pe r im e nt a l r e s u l t s

i s

s a t i s f a c t o r y f o r d i f f u s e r s

wi th no ho ri zo nt al momentum. Di ff use rs w i t h h o r i z o n t a l momentum

ex h ib i t u n st eady c i r c u l a t i o n p a t t e rn s which

are

s t ro n g ly a f f ec t ed by

t h e b ou nd ar ie s of t h e e x p er i me n ta l f a c i l i t i e s .

p red ic te d and observed d i lu t i on s i s f a i r .

T h e

agreement between

The

fo l lowing genera l conclus ions are made:

1)

Any m u l t ip o r t d i f f u s e r

can,

f o r a n a l y t i c a l p ur p os e s, b e

rep re s ent ed b y an eq u iv al en t s l o t d i f fu s e r .

d i f f u s e r s w i t h a l t e r n a t i n g n o z zl e s.

nea r-f iel d geometry , such

as

nozzle spacings and height

This includes

Thus d e t a i l s

of

t h e

29 5



above t he bo t tom, have on ly

a

s eco n da ry i n f l u en c e on t h e

d i f f u s e r b e ha v io u r. The e q u i v a l e n t s l o t co n c e pt i n t r o d u c e s

a co nv eni ent t o o l fo r com paring d i f fu s e r s t u d i e s and pro to -

t y p e a p p l i c a t i o n s .

2 ) The mechanics o f mu l t i po r t d i f fu se rs a re s u b s t a n t i a l l y

d i f f e r e n t f o r d i f f u s e r s i n deep water with high buoyancy o f

t h e d is c h a rg e ( t y p i c a l se wa ge d i f f u s e r s ) a nd f o r d i f f u s e r s

i n s ha ll ow

water

w i t h low bu oy ancy ( t y p i c a l t h e rm a l d i f fu -

sers , T able 5 . 1 ) . Sewage d i f fu s e r s ex h i b i t

a

s t a b l e near-

f i e l d w hich i s a) n o t p ro ne t o v e r t i c a l i n s t a b i l i t i e s and

r e- en tr ai nm e nt , a nd b ) n o t l i a b l e t o g e n e r a t e h o r i z o n t a l

c i r c u l a t i o n s . On t h e c o n t r ar y , t y p i c a l t h e rm a l d i f f u s e r s

h av e an u n s t a b l e n e a r - f i e l d w i t h c o n s i d e r a b l e r e - e n tr a i n -

ment and

are

i n t h e a b s en c e of c o n t r o l t h ro u gh n o z z l e

o r i e n t a t i o n l i a b l e t o g e ne r a t e h o r i z on t a l c i r c u l a t i o n s

l e a d i n g t o u n s te a d y t e m p e r a tu r e i n c r e a s e s .

3) A

s u c c e s s fu l d i f f u s e r d e s i g n sh ou ld

m e e t

t h e f o l lo w i ng

ob

ec

ives :

i ) The i n du ced t em p era ture f i e l d sh o u ld b e u n ifo rm a lon g

t h e d i f f u s e r l i n e , t h us

i n

case of v a r i a b l e d e p t h t h e d i s -

charge

p e r

u n i t d i f f u s e r l e n g t h sh ou ld b e v a r i e d a c c or d in g

t o E q ua t io n

( 8 -

7 ) , i . e . , prov id ing

a

c o n s t a n t d i f f u s e r

l o ad , and t h e n o z z l e s p ac in g s h ou ld b e

less

t h a n t h e w a t e r

dep th .

29 6



i i ) The d i f f u s e r s h ou l d b e a rr a n ge d p e r p e n d i c u l a r l y t o t h e

p r e v a i l i n g cr o s s - c u rr e n t ( i f p o s s i b le ) and

t h e

d i f f u s e r i n -

du ced f lo w f i e l d s h o u ld b e co n t ro l l ed t h ro u g h h o r i zo n t a l

n o z zl e o r i e n t a t i o n a l o n g the d i f f u s e r axis

(LOG

d i s t r i b u t i o n ) .

Both t h e s e f e a t u re s p rev en t cu r ren t s f ro m sweep in g a lo n g t h e

d i f f u s e r l i n e wh ic h r e s u l t i n r e p e a t e d re - e n tr a i nm e n t and

thus i n decreased perfo rmance .

iii) For

a

d i f f u s e r

t o

p er fo rm i n

a

c o n s i s t e n t a n d

sa t i s -

fa ct or y manner under a v a r i e t y of am b ient co n d i t i o n s ( e . g ,

r e v er s i ng t i d a l c u r r e n t s ) i t

i s

d e s i r a b l e t o a c hi e ve a

s t r a t -

i f i e d fl ow i n t h e f a r - f i e l d , t h u s m in im iz in g t h e d e g re e of

r e c i r c u l a t i o n i n t o t h e n e a r - f ie l d . D i f f u s e r s w i t h al ter-

n a t i n g n o zz l e s

ach i ev e s u ch a s t r a t i f i e d f low f i e l d .

With p roper sche mat iz a t ion o f th e s i t e geometry the theo-

)

r e t i c a l

p r e d i c t i o n s o f t h i s s t u d y ca n b e u s ed t o pr o v id e

a d i f f u s e r d e s i g n o r p r e l i m i n a r y d e s ig n est imate f o r t h e

s c r ee n i ng of a l t e r n a t i v e d i s c ha r g e schemes a n d / o r f o r f u r t h e r

i n v e st i g a ti o n i n a h y d r a u l i c scale model.

5 ) U n d i s t o r t e d d i f f u s e r sca le m od el s b o r re c t l y r ep ro d u ce t h e

area l e x t e n t o f t h e t e m p er a t ur e f i e l d a nd t h e c u r r e n t i n t e r -

a c t i o n , b u t are a lways somewhat con serv a t ive i n p r ed ic t in g

n e a r - f i e l d d i l u t i o n . The degree o f c onser va t iv eness can be

e s t i m a t e d . D i s t o r t e d

sca le

m od el s, w i t h p r o p e r d i s t o r t i o n ,

g iv e a l e s s c o n s e r va t i v e p r e d i c t i o n of n e a r - f i e l d d i l u t i o n

and can also s i mu l at e t h e he a t l o s s i n t h e f a r - f i e l d .

297



However, th e ex ten t of t he near - f ie ld zone ( impor tant fo r

c onfo rm ing t o l e ga l t e m pe r a tu r e s t a nda r ds )

i s

exaggera ted

and t he d i s t o r t i o n o f t h e r a t i o of d i f f u s e r l e n g t h t o w a t e r

depth can have non-conservat ive e f fe c t s on the per formance

p r e d i c t i o n .

9.5

Recommendations f o r Fut ur e Res earch

The t h e o r e t i c a l a p pr oa ch s h o u ld b e a d ap t ed t o i n c l u d e t h e i n t e r -

a c t i on wi th an a mbien t c r oss f low. I n a dd i t io n , t he e f f e c t o f w ind

s t re ss on t h e f r e e s u r f a c e c o ul d b e i n c o rp o r a t ed t o d et e rm i n e t h e

t r a n s l a t i o n of t h e te mp e ra tu re f i e l d .

For d i f f us e r s wi th n e t ho r i zo nt a l momentum th e need

i s

f o r

the development of a s imple c i rc ul a t io n model which provid es a ) t h e

l i n k ag e t o t h e

two-dimensional cha nne l model and b ) i n f o r m a t ion on th e

d e g re e o f p o t e n t i a l c i r c u l a t i o n .

d i f f u s e r s ( o n l y

a

f e w

m u l t i p l e s

of

t h e

water

depth long) which do

no t ha ve a c l e a r l y two- dime ns iona l c e n te r po r t ion shou ld be inv e s t i -

g a t e d .

Also the behaviour of " shor t"

The t o t a l pe rformance of a d i f f u s e r o u t f a l l i n t h e

w a t e r

environ-

ment

i s

c l o s e l y l i n k e d t o t h e i n t e r a c t i o n b etween n e a r - f i e l d and f a r -

f i e l d e f f ec t s .

d im e ns ional , v e r t i c a l l y two-laye re d m a the m a ti c a l m odel , i nc o r po r a t iug

e f f e c t s of f a r - f i e l d b o u n d a ri e s , a d v e c ti o n , h e a t l o s s an d t u r b u l en c e ,

a nd c o u pl i n g t o t h e n e a r - f i e l d p r o p e r t i e s

as

de ve lope d i n t h i s work,

s h o u l d b e i n v e s t i g a t e d .

To t h i s end th e development of a hor iz on ta l ly two-

Some ca re fu l exper imenta l work sh ould be d i re c te d t o t he buoyant

298



j e t r e g i o n t o v a l i d a t e en t ra i nm e nt r e l a t i o n s h i p s and t o s t u d y t h e e f f e c t

of

j e t

i n t e r a c t i o n . I n a d d i t i on ,

there i s

im me dia te ne ed f o r l a r ge

scale exper iments on d i f f us e r behaviour , which

are w e l l

i n t h e t u r b u le n t

domain and reasonably f r ee of boundary e f f ec ts .

299



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302



Fi gur e

1-1

2-1

2-2

2-3

2-4

2-5

2-6

2-

7

2-8

3-1

3- 2

3-3

3-4

3-5

3-6

3 7

L I S T OF

FIGURES

Page

Qual i tat i ye I l l ustrat i ons

of

Resul t i ng D str i hut i ons

of M xed Uater (no ambi ent currents)

Schemati cs

of

a Round Buoyant J et

Schemati cs of a Sl ot Buoyant J et

J et I nterf erence f or a Submerged Mul t i port D f f user

Centerl i ne D l ut i ons Sc for a Sl ot Buoyant J et Wth

Hori zontal D scharge

Mul t i port D f f user w th Al ternat i ng Ports

i n

Deep Water

Compari son Between the Centerl i ne D l uti ons Above

the Poi nt

of

Mergi ng f or a Buoyant Pl ume and a

Mul t i port D f f user w th Al ternat i ng Nozzl es

Schemati cs

of

Channel Model f or a Mul t i port

D f f user w th Hor i zontal D scharge i n Shal l ow Water

Schemati cs f or One- D mensi onal Anal ysi s of a Mul t i -

port D f f user w th Hor i zontal D scharge i n Shal l ow

Water

Three-D mensi onal Fl ow Fi el d f or a Submerged

D f f user Two- D mensi onal Behavi or i n Center

Port i on (Stabl e Fl owAway Zone)

Probl emDef i ni t i on: Two-D mensi onal Channel Model

Vert i cal St ructure of D f f user I nduced Fl ow Fi el d

Def i ni t i on D agramfor Buoyant J et Regi on

Zone

of

Fl ow Establ i shment

Centerl i ne D l ut i ons Sc for Buoyant Sl ot J et s

Wthout Surf ace I nteracti on. Vert i cal D scharge.

Compari son

of

Theory and Experi ments

Centerl i ne D l ut i ons Sc f or Buoyant Sl ot J et s

Wthout Surf ace I nteracti on. Hori zontal D scharge.

Compari son of Theory and Experi ments

16

22

27

30

3 3

34

36

41

4 3

51

54

5 8

60

7 3

7 9

80

3-

8

Schemati cs

of

Surf ace I mpi ngement Regi on

3 0 3

82



Page

ig u re

3-9

3-10

3-11

3-12

3-13

3-14

315

3-16

3-17

3-18

3-19

3-20

3-21

3-22

3-23

3-24

3-25

Observed Yeloci ty and Temperature (Densi ty)

D i s t r i b u t i o n s f o r Yer t ica l Buoyant

Jet

(Fs

= 31,

H / B =

416)

85

Schematic of

V e r t i c a l

F low D i s t r i b u t i o n P r i o r 92

to In t e rn a l Hy d rau l i c J u m p

Def i n i t i o n Diagram: In t e rn a l Hy d raul i c Jump 95

So lu ti on Graph f o r t h e Momentum Equa tion s 98

Non-Existence of an I n t e r n a l H y d r a u l i c Jump;

102

Turbulent Diffusion and Re-Entrainment

S t r a t i f i e d Flow D e f i n i t i o n s 1 0 3

D epth R e l a t i o n s h i p s i n S t r a t i f i e d Flow 113

In t e r f ac i a l He ig h t i n C ou n te r fl o w from Eq. (3-188) 116

C r i t i c a l

Depth

H

Thic kne ss hi/H and De ns im et ri c Froude Number F

of t he Sur fac e Impingement Layer, Ver t i ca l Dis ch a rg e

as

a

Func tion of F2

, IQI 11

8

128

C H

1

Average Surface Di l u t i on S Accounting f o r Thick- 130

nes s of Impingenent Layer and Conjugate Depth

h;/H fo r In t e rn a l Hy d rau l i c J u m p .

Del i nea te s S ta b l e and Uns tab le Near-F ie ld Zone

S

C r i t e r i o n L i n e

In te ra ct io n of Near-Field and Far-Field Zones 132

V a ri a ti on of I n t e r f a c i a l St r e s s C o e ff i ci e nt f i

w i t h Reynolds numbers (Abraham and Eys ink (197 1))

13

5

Froude Number FH of t h e Su bc r i ti c a l Equal Counter -

1 3 7

flow System a s aC Fu n c t i o n 0 (Eq. (3-207) w i t h

A

=

0.5)

S u r f ac e D i l u t i o n S as

a

Fun ctio n of F ,H/B.

13

8

Vertical D i f f u s e r , Weak Far -F i e ld E f fez t s

S u r f a c e D i l u t i o n

S

V e r t i c a l

Dif fu s e r , S t ro n g Fa r -F ie ld E f fe c t s

as a Fu n c t i o n o f I?,, H / b .

S

L o ca l B eh avior o f D i f fu s e r D i s ch a rg e wi th

Al t ern a t i ng Nozzles , Un s tab l e Near-F ie ld

3 0 4

139

140



Fi gur e

3-26

3-27

3-28

3-29

3-30

3-31

4-

1

4- 2

4-3

4-4

4- 5

4-6

4-7

5- 1

5- 2

5- 3

5- 4

5- 5

Ef f ect of Angl e of Di schar ge Oo on t he

St ahLl i t y

of

t he Neax- Fi el d Zone

Possi bl e Fl ow Condi t i ons

f o x

Di scharges wi t h

Net Hor i zont al Mament um ( C Denotes a Cr i t i cal

Sect i on)

Sur f ace Di l ut i on S, as a Funct i on of F

H/ B.

45 Di schar ge, Weak Far - Fi el d Ef f ect s

S

Sur f ace Di l ut i on

S ,

as a Funct i on of F, H/ B.

45 Di schar ge, St r ong Far - Fi el d Ef f ect s


S

Hor i zont al Di schar i e, Weak Far - Fi el d Ef f ect s

as a Funct i on of F,, H/ b.


S,

as a Funct i on of F, H/ B.

Hor i zont al Di schar ge, St r ong Far Fi el d Ef f ect s

Compar i son of Fl ow Fi el ds f or a) Thr ee- Di men-

si onal Di f f user wi t h Cont r ol and b)

Two-

Di mensi onal Channel Model Concept ual i zat i on

Compl ex Sol ut i on Domai n

St r eam i nes f or One Quadr ant of t he Fl ow Fi el d

Cumul at i ve Head Loss hf

Thr ee- Di mensi onal Fl ow Fi el d f or Di f f user wi t h

Unst abl e Near - Fi el d Zone (No Cont rol )

Pl ane Vi ew of Di f f user Sect i on, Al t er nat i ng

Nozzl es wi t h Ver t i cal Angl e €lo

Uni di r ecti onal Di schar ge w t h Par al l el Nozzl e

Or i ent at i on (B(y) = const =

90")

Photogr aph of Fl ume Set - Up

Fl ume Set - Up

Photogr aphs of Basi n Set - Up

Basi n Set- Up

Ar r angement of Temper at ur e Measurement Syst em

,

Al ong t he Fl ow

Pat h,

x/L,,

3-D

305

Page

142

14

3

149

150

151

152

159

165

166

16 8

172

174

178

186

187

193

194

19

7



F i g u r e

6-1

6-2

6-3

6-4

6-5

6-6

6-

7

6-8

6-9

6-10

6-11

6-12

6-13

7-1

7- 2

7-3

7-4

7-5

7- 6

7-7

7-

8

Steady-Sta te

Yert ical

Flow St ruc tu re :

Ohserved

T em p era tu re P ro f i l e s

Ys.

Lntexface

Eq.

(3-2(17]

No zz l e Or i en t a t i o n ,

B y ) ,

Along th.e Di f f us er L ine

Observed Tanpe ra tu re and Flow Fie l d , Se r i es

BN

Observed Temperature and Flow Field , Series BN

Observed Temperature and Veloci ty Field ,

S e r i e s

BN

s S

s S

P r e d i c t e d V s . Observed Di lu t ions ,

P r e d i c t e d

V s .

Observed Dilut ions ,

Observed Temperature and Veloci ty Field ,

S e r i e s BH

Observed Temperatu re and Veloc i ty F ie ld , Se r i es

BH

P r e d i c t e d

Vs.

Observed Di lu t ions ,

S

P r e d i c t e d V s . Observed Di lu t ions , S

P r e d i c t e d V s . Observed D i lu t ions ,

P r e d i c t e d V s . Observed Di lu t ions , S

M u l t i p o r t D i f fu s e r i n Ambient C ro ss f l ow

S

S

s S

S

Typica l F low Regimes f o r Di f fuse rs w i th

Forced Ambient Cr oss flo w

Flume Experiments

Observed Tempera ture and Veloc i ty F ie ld , Se r i es BC

Al te rn a t i n g No zz l e s

Three-Dimensional Di f fuse rs i n Cross f low, (S er i es

BC) No N e t

Horizontal Momentum,

LOG

D i s t r i b u t i o n

Observed Temperature and Veloci ty Field

w i t h

Cross f low, Ser ies BC, Un id i r ec t i o n a l No zz l e s

Observed Temperature and Ve loci ty Fkeld , Se ri es

BC,

Un id i r ec t i o n a l No zz l e s

Observed Tempera tu re and Vel oc i t y F ie ld , Se r i es

BC,

Un id i r ec t i o n a l No zz l e s

306

Page

204

209

210

211

213

2 14

215

221

222

224

225

226

227

232

240

242

245

247

25d

251

252



F i g u r e

7-9

7-10

7-11

8- 1

8-2

Page

D i ff u se r s i n Crosszlov, N e t Hor i zon t a l M a m e n t u m ,

w i t h

C ont ro l CLOG D i s t r i b t i Q n )

254

Dif fuse r s i n C ros sf low, N e t Ho ri zo nt al Momentum,

No

C ont ro l

(NOR

D i s t r i h u t i o n )

255

Comparison of Relative Dif fuse r P e r fo rmance i n

Crossflow (S er ie s BC) with Un idi rec t io na l

Discharge

257

Dif fus er Design i n Tid al Curren t System

Dif fus er Design i n Steady Uniform Crossflow

.

268

276

307



Table

5.1

6 .1

6.2

6 .3

6.4

7 . 1

7.2a

7.2b

L I S T OF TABLES

Comparison of Releyant Parameters for Typical

Sewage and Thermal Diffuser Applications

Flume Di ffu se r Experiments, Se ri es FN

Basin Di ffu se r Experiments, S er ie s BN

Flume

Di ffu se r Experiments, Se ri es FH

Basin Di ffu se r Experiments, S er ie s

BH

Flume Diffuser Experiments with Crossflow,

Ser ies

FC

Basin Experiments w i t h Crossf loy, Ser ies BC,

a) Discharge with

No

Net Horizontal Momentum

Basin Experiments with Crossflow,

Ser ies

BC,

b)

Discharge with Net Horizontal Momentum

Page

182

203

206

217

219

2 38

244

2 49

308



GLOSSARY OF SYVBOLS

Subscri pts

upper , l ower l agers i n strat i f i ed f l ow

92

a, b vert i cal secti ons a, b

C

cri t i cal sect i on i n str ati f i ed f l ow

0

di scharge vari abl es

s, i , b sur f ace, i nter f ace, bottomboundary condi t i ons

Superscri pts

* di mensi onl ess vari abl es

A

rat i o of i nterf aci al to bottom shear stress coef f i ci ents

B equi val ent sl ot w dth

b nomnal j et w dth

nomnal j et w dth at i mpi ngement

bi

C concentrat i on

C

centerl i ne concentrat i on

C cont racti on coef f i ci ent

C

speci f i c heat

C

C

P

c2

correcti on f actor f or channel wal l ef f ects

D

nozzl e di ameter

F l aver densi metr i c Froude number

densi metr i c Froude number based on total depth

FH

FHc

cri t i cal F f or counterf l ow system

l ocal j et densi metr i c Froude nuxi bey

constant pl ume densi metri c Fr oude number

H

nozzl e densi metr i c Froude number

Fn

309



*S

FT

F*

f i

f O

f *

0

g

H

H1 H 2

hf

hi

hL

h

S

h

I

KZ

kL

kS

k

L

L

C

LD

Lm

LS

R

m

0

sl ot densi metr i c Froude number

di f f user l oad

f ree sur f ace Froude number

i nt er f aci al stress coef f i ci ent

bot tom st ress coef f i ci ent

bot tom stress coef f i ci ent incu,,ig channel wal l

e

gravi tat i onal accel erat i on

water depth

normal i zed l ayer depths

l ayer depth i n strat i f i ed f l ow

head

l o s s

i n l ayer moti on

thi ckness of j et i mpi ngement l aver

head

l oss

i n surf ace i mpi ngement

hei ght of nozzl es above bottom

i ntegral Equat i on ( 3 - 4 5 )

ver t i cal eddy di f f usi on coef f i ci ent f or mass, heat

f r i ct i onal head l oss coef f i ci ent

head

l oss

coef f i ci ent f or i mpi ngement

absol ute wal l roughness

l ength of two-di mensi onal channel model

l ength of subcri ti cal strat i f i ed f l ow secti on

di f f user hal f l ength

l ength of l ocal mxi ng zone

l ength of wedge i nt rusi on

nozzl e sDaci np

si nk or source st rength

310

f ects



n

P

P'

Pn

Pr

9

QO

q

Rh

r

S

sC

-

S

sS

S

T

T'

Ta

TC

Te

0

T

s

T

t ransverse coordi nate

pressure

pressure f l uctuat i on

hydrostat i c pressure

reduced pressure

f l ow rati o i n strati f i ed f l ow

total di f f user f l ow (pl ant di scharge)

vol ume f l ux per uni t w dth

heat f l ux

di scharge per uni t di f f user l ength

Reynol ds number

j et Reynol ds number

hydraul i c radi us

radi al coordi nate

d l ur on

j et centerl i ne di l ut i on

average j et di l ut i on

sur f ace di l ut i on

axi al coordi nate

tempera ur e

temperature f l uctuat i on

ambi ent temperature

j et centerl i ne temperature

equi l i br i umt emperature

di scharge temperature

surf ace temperature

311



0

U

U

a

-

U

C

C

Ud

' Vd

.

i

V

V*

V

e

w

i et di scharge vel oci ty

vel oci t i es i n Cartesi an coordi nate svstem

l ayer vel oci t i es averaged over total depth

j et vel oci t i es i n l ocal coordi nates

ambi ent cross f l ow vel oci ty

averaged l ayer vel oci ty i n channel

j et center l i ne vel oci ty

averaged l ayer vel oci t i es at di f f user

j et vel oci ty at i mpi ngement

vol ume f l ux rat i o

combi ned mxi ng by crossf l ow and di f f user momentum

Equati on (7-5) or

(7-6)

j et entrai nment vel oci ty

compl ex potent i al

normal i zed hori zontal di st ance

Cartesi an coordi nates

Y

AH

AT

ATC

buoyant j et entrai nment coef f i ci ent

constant pl ume ent rai nment coef f i ci ent

non- buoyant j et entrai nment coef f i ci ent

coef f i ci ent of thermal expansi on

hor i zontal nozzl e or i entat i on w th di f fuser axi s

arrangement

of

di f f user axi s w th crossf l ow

head change i n f ree sur f ace f l ow

temperature ri se above ambi ent

j et center l i ne temperature r i se

di scharge temperature r i se

312

difusores mecanica

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