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1951

A Visual Investigation of the Appearance ofTurbulence in Annular Spaces.Thomas Anderson FeazelLouisiana State University and Agricultural & Mechanical College

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Recommended CitationFeazel, Thomas Anderson, "A Visual Investigation of the Appearance of Turbulence in Annular Spaces." (1951). LSU HistoricalDissertations and Theses. 7982.https://digitalcommons.lsu.edu/gradschool_disstheses/7982

A VISUAL INVESTIGATION OF THE APPEARANCE OF TURBULENCE IN ANNULAR SPACES

A D is s e r ta t io n

Su b a i t ta d to th e G raduate Facu lty o f th e L ou isiana S ta te U n iv e rs ity and

A g ric u ltu ra l and Mechanical C ollage in p a r t i a l fu l f i l lm e n t o f th e requirem ents fo r th e degree o f

Doctor o f Philosophy

inThe Department o f Chemical Engineering

fcyThomas Anderson Feaasel

B*8«# L ouisiana Polytechnio In s t i tu te * 19^7 L ouisiana S ta te U niversity* 19^9

June* 1951

UMI Number: DP69360

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MANUSCRIPT THESES

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LOUISIANA STATE UNIVERSITY LIBRARY

ACKNOWLEDGMENT

The au tho r w ishes to express h is s in c e re a p p re c ia tio n

to Dr* B* S* Preseburgt under whoee d ir e c t io n t h i s work was

c a r r ie d out* fo r h ie advice and guidance* He a leo wishes

to thank Mr* E* E. Snyder whoee mechanioal s k i l l wae respon­

s ib le fo r th e e e n s tru e tio n o f th e apparatus*

He wishes to thank Dr* P» M. Horton o f th e Chemical

Engineering Department fo r supplying th e m a te ria l needed

in th ie in v e s tig a t io n and th e s t a f f o f th e Audubon Sugar

Factory fo r making th e space a v a ila b le fo r th e c o n s tru c tio n

o f th e apparatus*

i i

116573

TABLE OF CONTENTS

PAGE

acknowledgment............................................................. 11

LIST OF TABLES 9 . . lv

LIST OF FIG URES............................* ........................ V

ABSTRACT . . • . * v t l

CHAPTER

I INTRODUCTION . . .................................... X

I I THEORY OF FLOW MECHANISMS................... 4

A* C irc u la r Pipe • « * . . . . « « 4

B* Annular flection* 12

O* Conclusions 9 . . * .................. SO

I I I SUMMARY OF EXPERIMENTAL WORK « . * 22

A* Apparatus • • • • « » • « * « 22

B# Experim ental Procedure . * . • 27

0* L im ita tio n s o f Apparatus

and Data » * .................................... 26

D* T es ts o f Data . « • * « » • • * *>1

17 EXPERIMENTAL RESULTS............................

A* G e n e r a l ...................... • * « « * «

B. V elocity Measurements * . . . * 40

C# C r i t ie a l Values * * « » 56

7 CONCLUSIONS AID RECOMMENDATIONS . * 66

i l l

TABLE OF CONTENTS

Continued

APPENDIX

I B1BLIOQBAPHX............................... 69

I I NOMENCLATURE...........................................* . . . 95

I I I BATA * ........................... 95

IT CALIBRATION CURVES .............................. 1X5

V ITA ............................................... 115

i v

LIST OF TABLES

I* C r i t i c a l Reynolds Numbers fo r C irc u la r Pipe* • • 6

II* R e su lts o f In v e s tig a tio n * on Annular Spaces a t

Reported by Wiegand and Bakor • • » * * • • • * IT

I I I* Dimensions o f P ip es Usod in th e Experim ental

wark • * « • • • • * * • • • • • • • * • • • • 26

17* T h e o re tic a l P ro p e r tie s o f th e S ec tions T ested • 40

V-a L ast Observed S tream line Plow C onditions fo r

Annulus No* I * * * * * * * * * * * * * * * * * 59

v-b F i r s t Observed T urbu len t Flow C onditions fo r

Annulus No* l * * * * * * * * * * * * * * * * * 61

V I-a L as t Observed S tream line Flow C onditions fo r

Annulus No* £ • « • • • • • • * * • • • * * • • 65

VI-b F i r s t Observed T urbulen t Flow C onditions fo r

Annulus No* 2 « » • • « • • * » • * 64

V II-a L ast Observed S tream line Flow C onditions fo r

Annulus No* ^ * ........................ * * 6J

V II-b F i r s t Observed T urbulent Flow C onditions fo r

Annulus ^7

VIII* Summary o f C r i t i c a l Reynolds Numbers * * * * * 67

v

LiZST OF FIGURES

1 V isual O bservation Apparatus fu r Flow In Annull * * * 25

£ T ypical Wave Motions Observed « • * . * * * • * » • • J6

3 Pye S epara tion P a t te rn * • * • * * « * • • • • • * • 3®

4 V e lee ity D is tr ib u tio n fo r Annulus Wo• 1 * * • * • • • 4 l

3 V e lee ity D is tr ib u tio n fo r Anm lue No* 2 • » ................42

6 V e lo c ity D is tr ib u tio n fo r Annulus No* 5 * * * . . . * 45

7 V e lo c ity D is tr ib u tio n fo r Annulus N o * 4 « * * * * * * 44

8 R atio o f Average V eloc ity to Maximum V eloc ity in

Annulus No* 1 a s a Funotion o f Reynolds Number • • • 47

9 R atio o f Average V eloo ity to Maximum V e lo c ity in

Annul! No* 2 and 3 ** Functions o f Reynolds Number • 48

10 E ffe c t o f In c re ase in Reynolds Number on P o in t

V e lo o itie s Anm lue No* 1 • * * « « • .................................

11 E ffe e t o f In e rease In Reynolds Number on P o in t

V e lo o itie s in Annulus No. 5 52

12 V e lo c ity G rad ien ts a t Walls as Functions o f

Reynolds Number * * • • • • * • • • * • « 5?

13 von {Carman Reduced V elo c ity D is tr ib u tio n fo r

Annulus No* l * * * * * * * * * * * * * * * * * * * * 55

14 von Kerman Reduced V elo c ity D is tr ib u tio n fo r

Anm lue No. 3 ...................... * • • • • • * ♦ 57

15 C r i t i c a l Average Reynolds Numbers a t P o in ts in

Annulus No* • • • * 69

v i

LIST OF FIGUB&S

Continued

16 C r i t i c a l Average Reynolds Numbers a t P o in ta In

Annulus No* 2 * * * * « . * ♦ * * * • • « . » « . * 70

17 C r i t i c a l Average Reynolds Numbers a t P o in ts in

Annulus No* ^ » 71

16 C r i t i c a l Values o f u / ^ a t P o in ts In Annulus No* 1 74

19 C r i t i c a l Values o f u / 7 a t P o in ts in Annulus No* 2 75

20 C r i t i c a l Values o f u /V a t P o in ts in Annulus No* J 76

21 C r i t i c a l V alues o f u / ^ a t P o in ts in Annull No* 1

•nd 3 * * .................................................................................. 77

22 C r i t i c a l Values o f v j a t P o in ts In Annul! No. 1

and 3 .................................................................... 76

25 C r i t i c a l P o in t Reynolds Number as a Function o f

P o s itio n in Anmlue No* 1 . . • • * * * . . * * # ♦ 60

24 C r i t i c a l P o in t Reynolds Number as a Function o f

P o s itio n in Annulus No* 2 * * * # » * » « . » . . * 61

23 C r i t i c a l P o in t Reynolds Number as a Function o f

P o s itio n in Anmlue No* 3 * ♦ • • * * * • * • • • 62

26 C r i t ic a l Values o f ^ a t P o in ts in Annulus No* 1 65

27 C a lib ra tio n o f Rotameter * • • • « • • * • « * • • 113

26 P o s itio n o f Dye Tubing • • • • * • # . * • * * • • 114

vi l

ABSTRACT

Previous in v e s t ig a to r s have s tu d ied th e mechanism o f

s tre am lin e and tu rb u le n t flow* but no t th e c o n d itio n s under

which th e mechanism changes* This re se a rc h was concerned w ith

t h i s t r a n s i t i o n reg io n in whioh i t i s p o ss ib le to determ ine

what p o r tio n o f th e f lu id I s In s tream lin e and what p o rtio n

i s in tu rb u le n t motion*

Most p revious work has used th e o ire u la r se c tio n ; th e

an m lu e was s e le c te d fo r t e s t s here* avoiding th e symmetry

o f th e e l r o le and g iv ing d is s im ila r eu rfeces fo r f r ic t io n *

Dye in je c t io n tech n iq u es were used to observe th e be*

h av io r o f w ater flowing a t room tem peratu re In th re e annular

sections* formed using a 5- inch o u te r g la s s p ipe and cores

&*# 5/^* and l4* s tandard ga lvan ised iro n pipes* V eloc ity

d is tr ib u tio n s * ob ta ined a t a c o n stan t flow r a t e in ttfe stream ­

l in e re g io n were compared w ith accep ted th e o r e t ic a l r e la t io n s

and found to agree favorably* ju s t i fy in g th e technique*

V isual o b se rv a tio n o f th e dye behavior as th e mechanism

©hanged from s tream lin e to tu rb u le n t flow* in d ic a te d th a t

t r a n s i t i o n in flow f i r s t m anifested I t s e l f as a slow sinuous

motion in th e flu id * As th e flow r a te increased* th e wave

len g th decreased u n t i l eddying and d if fu s io n re su lte d * In a l l

v i l i

cases tu rb u le n ce was observed f i r s t in th e c e n tra l p o rtio n s o f

th e se c tio n and spread toward th e w a lls w ith in c re a s in g flow

ra te s*

C r i t i c a l flow c o n d itio n s were ob ta ined a t v a rio u s p o in ts

in a c ro s s -e e o tio n o f th e annull* The minimum c r i t i c a l Reynolds

Number, — t wa8 occur a t th e p o in t o f maximum

v e lo c ity fo r two sections* occu rring approxim ately .0 2 -.0 4 in ch es

( th e exac t v a lu e being a fu n c tio n o f th e r a t i o o f d iam eters) to

th e c e re s id e o f th e m id-poin t o f th e annulus* For th e se c tio n

w ith th e core* t h i s p o in t o f minimum va lue was d isp laced ap­

proxim ately 0 .2 5 inch toward th e core from th e p o in t o f maximum

v e lo c ity * The c r i t i c a l Reynolds Number a t v a rio u s p o in ts In th e

w idest an m lu e was found to in c re a se from 14^0 a t r 8 0*65 inches

to 55OO a t r 8 1*40 in ch e s3 a minimum o f 11^0 was in d ic a te d a t

r = 1*1*5 * max* 'In BUf one annulus p o in t c r i t i c a l v e lo c i t ie s v a ried w ith

p osition* in c re a s in g sharp ly and con tinuously from minimum v a lu es

a t each wall* Rectangular* lo g arith m ic and sem i-logarithm ic

p lo ts o f th e c r i t i c a l v e lo c ity a g a in s t d is ta n c e from a wall* gave

curved lin es* These curves were concave downward near th e walls*

bu t f l a t in th e middle o f th e section*

A p o in t Reynolds Number* y u / -7/ * y being th e d is ta n c e

from a su rfa ce and u th e p o in t ve lo c ity * was adopted to express

th e c r i t i c a l va lu es a t d i f f e r e n t position© in th e sections* The

i x

fo llow ing re la t io n s h ip s were o b ta in ed *

fo r th e l i* core* ^

and fo r th e cor a , ^0l)er> 'i. ' ^ 7e>Q

These expreBaione were found a p p lic a b le to a l l point© i n th e

a ee tlo n e p ro rlded y wae measured from th e core w all fo r p o in ts

between th a t su rface and r •max*F urther work was recommended to determ ine th e e f f e c t o f

w a ll roughness on th e se conditions*

x

c s m m i

i m m m n m

Rven b e fo re th e tim e of R eynolds' o r ig in a l dye experim ents

In 1883 w ith w ater flow ing in p ip es (2g ) , i t had been known th a t

f lu id s flow ing e x h ib ite d d if f e r e n t behav io r f o r d i f f e r e n t flow

r a t e s , t h i s was evidenced by th e f a c t th a t flow a t low v e lo o i t ie s

in sm all p ip es shoved p ressu re drops which were p ro p o rtio n a l to

th e f i r s t power of th e v e lo c ity * t o r flow a t h igh v e lo o i t ie s in

la r g e p ip e s , th ese p ressu re drops were found to be alm ost pro­

p o r t io n a l to the square o f the v e lo c ity * The knowledge of th ese1

f a c t s was th e m o tiv a tio n behind R eynolds' experim ent to determ ine

when th e f lu id m otion would y ie ld th ese d i f f e r e n t r e s u lts *

In h is work Reynolds found , by In je c tin g a dye in to th e f l u id ,

th a t a t low flow r a te s tiie in je c te d dye was c a r r ie d in a s t r a ig h t

l i n e by th e f lu id * T his type o f flow he d esc rib e d a s d irfeet flow ,

b u t i t i s now mere commonly r e f e r r e d to as s tream lin e or lam inar

flow* When th is type o f flow was observed. I t was found th a t th e

p re s su re lo s s doe to f r i c t io n was dependent on the f i r s t power o f

th e v e lo c ity * As th e flow r a t e was in c reased , however, he observed

th a t above c e r ta in c r i t i c a l r a t e s , the dye th rea d was to rn up and

the c o lo r was d isp e rsed throughout the p ip e , f o r th is type o f flow ,

i t was found th a t th e f r i c t i o n lo o s was dependent upon th e v e lo c ity

r a is e d to a power g re a te r than one. By using p ipes o f d i f f e r e n t

1

d iam eter and v a ry in g th e tem pera tu re o f th e w ater flowing*

Beyaolds found th a t th e grouping* » W ^ * • had th e same va lue

f o r th e p o in t o f change in flow behav io r fo r o i l systems te s te d .

T h is grouping has o inco been c a l le d th e Hsynold* Number.

S ince th a t t in e th e Reynold* Slumber hoe Veen w idely need to

o o r r e la te th e b eh av io r o f f lu id s In m otion. I t hoe been found to

bo a convenien t oenoept on which f r i c t i o n drop* h e a t t r a n s f e r and

ness t r a n s f e r s a l e o la t io n s o f system s may be b ased . In a l l such

trea tm en ts t however. th e re a r is e s a c e r ta in range of Reynolds Jfumbers

in which th e behav io r o f the f l u id I s no t c le a r ly d e fin e d . T his

c o n s t i tu te s th e s o -c a l le d t r a n s i t io n reg ion* a reg io n o f change

from flow obeying th e th e o re t ic a l law s of s tre am lin e m otion to

flow obeying the em p irica l r e la t io n s h ip s f o r tu rb u le n t m otion.

The manner in which th is t r a n s i t io n takes p la c e has n o t been

s a t i s f a c to r i l y ex p la in ed . A q u e s tio n vh loh has n o t been s e t t l e d

y e t is* does th e flow change from s tre am lin e to tu rb u le n t through­

ou t th e e ro s s -s e o t io n a t the same time* o r does more and more of

th e f l u id e x h ib it tu rb u le n t c h a r a c te r i s t i c s a s th e f lo v r a t e i s

g ra d u a lly increased?

T his re se a rc h i s concerned w ith the t r a n s i t io n reg io n o f flow

and has a s i t s purpose th e d e te rm in a tio n of th e co n d itio n s which

e x i s t when th e f lo v mechanism changes from stream lin e to tu rb u le n t

a t d i f f e r e n t p o in ts In the c ro s s -s e c tio n o f an annu lu s. V isual

* See Appendix I I fo r Nomenclature

5

o b se rv a tio n o f flow behavior i s p a r t ic u la r ly su ite d to thi©

in v e s tig a tio n s by o b se rv a tio n o f a p roperly in troduced dye stream*

th e ex ac t behavior o f th e f lu id flow ing can be observed* In th e

ease o f th e flow o f w ater i t has been found th a t a very d i lu te

s o lu t io n o f a w ater so lu b le dye can be used e ffe c tiv e ly * By t h i s

method i t l a n o t on ly p o ss ib le to observe th e type o f flow e x is tin g

b u t fo r s tream lin e flow i t i s p o ss ib le to o b ta in th e v e lo c ity o f th e

w ater a t th e o b se rv a tio n p o in t and a lso th e maximum v e lo c ity i n th e

section*

In th e flow o f a f lu id p a s t a su rfa ce a t flow r a te s w ell above

th e accepted maximum va lue fo r s tre am lin e flow* P re n d tl 'e boundary

la y e r th eo ry In d ic a te s t h a t th e re i s a s tream lin e su b -lay er o f th e

f lu id near th e su rface ( 52)» b u t i t s th ic k n e ss I s so sm all t h a t

measurement o f I t i s d i f f ic u l t* i f no t im possible* Presuming* how­

ever* th a t in th e t r a n s i t io n reg io n th e p o rtio n o f th e flow whloh

e x is ts i n th ie s tream lin e flow I s f in i te * t h i s can be determ ined by

t h i s dye in je c t io n method and perm it such e v a lu a tio n o f th e behavior

as w il l apply eq u a lly w ell to tu rb u le n t flow*

In th e past* in v e s tig a to r s o f flow behavior have used th e

c i r c u la r s e c tio n as th e flow standard* beeause o f th e symmetry o f

t h i s se c tio n end a lso beoaute o f th e wide a p p lic a tio n o f I t in

flow in s ta l la t io n s * The anzaulue i s se lec te d in t h i s work to avoid

th e symmetry o f th e o i r o le and a lso to a ffo rd two d is s im ila r sur­

fa c e s fo r f r ic t io n *

CHAPTER XI

THEORY O F FhOW MKGKAHZSIMI

A* C irc u la r P ico*

Since th e middle o f th e seven teen th cen tu ry , when T o r r ic e l l i

s tu d ied flow through an o r i f i c e , th e flow o f flu id® has hold coir-

s id e ra b le i n t e r e s t fo r many e e le n t ie t s ( 2) . I n th e m iddle o f th e

n in e tee n th cen tu ry , Darcy and one o f h ia co-w orkers wore th e f i r s t

to recogn ize th e e x is te n c e o f two flow mechanism, as dem onstrated

by t h e i r experim ents on p re ssu re drop In pipes* However, th e law

fo r viecoue flow had been s ta te d about te n y ea rs e a r l i e r by Hagen

and again by F o le e u l l le , who a lso v e r i f ie d th e law w ith s tu d ie s o f

flew in c a p i l la r ie s * This i s th e fa m ilia r H agenrF o leeu llle equa­

t io n fo r c i r c u la r p ip es ,

AF L

T his equation was ob tained fo r th e flow ease In which th e only

fo rc e s which must be considered a re th e v iscous fo rc e s o f one

la y e r e f f lu id s lip p in g p a s t another* I t has been g e n e ra lly ac­

cepted th a t a f lu id w il l obey Equation (1 ) fo r v a lu es o f Reynolds

Humber le s s th a n 2100, a lthough t h i s value i s em p irica l and excep­

t io n s have been reported* This i s c a lle d th e c r i t i c a l Reynolds

Number and flew fo r any value le e s th an t h i s le com pletely s i r earn-

lin e*

4

( i )

5

tli* fo o t t h a t f lu id s flow ing in s tream lin e motion obey t h i s

th e o r e t i c a l law fo r p re ssu re drop makes p o ss ib le tho trea tm e n t o f

ouch ooooo by sim ple m athem atical procedure* However# i f th o

f l u id flow* in tu rb u le n t motion# th en em pirica l r e la t io n s h ip s have

t o t o ob ta ined *o th a t th o trea tm e n t o f th la **•* may bo c a r r ie d

out* Ono o f th o oar H o s t formula* to bo u**d to compute th o flow

i n o i l * o rt* o f oondOit* wa* th a t o f Ohesy (H)# g iven below*

t t y .

L C a # H (a)Since th o appearance o f th ie form ula, sev era l o th e r expression*

have boon developed* two o f which a re g iven below*

Fanning Equation, — - ( 5)L J j D

Hasea-W illiam s Formula* /-?( f o r w ater on ly ) / * / S / & C r<n \ L / (b )

Avj

The Panning Equation 1* th e moot w idely used o f th e se expressions

and may be expressed in many ways, as*

Af_ _ 4 f G * _ 4 f^ v , 34-fiH1L ' 2 9 d ' ? s D f ' D n y g £ * ( 5 .J

In a f lu id system , th e re fo re , i t I s necessary i s know tho typo

o f flow whioh e x is ts* Although Reynolds' in v e s tig a tio n ind ica ted

t h a t th e type o f flow whleh e x is te d could be determined by th e vain#

o f th e Reynolds Number# i t remained fo r S tanton and Pannell ( 54) ^

6

apply t h i s group to tho oo rro l& tion o f f r l o t i o n lo s se s no measured

by numerous in v e s tig a to rs * They found th o t tho f r i c t i o n f a c to r woo

o unlquo fu n c tio n o f th o Reynolds Number fo r smooth c i r c u la r p ip es

c o n ta in in g a v a r ie ty o f f lu id s* They s to to d th e re fo re t h a t

* s im ila r i ty o f n o tio n i n f lu id s a t eono tan t v a lu es o f th o variab le#

nv # w i l l ex is t# provided th e s u r f uses r e l a t i v e to whleh th o

f lu id s aovo o re g e e m str is a lly s im ila r# whleh s im ila r ity # as herd

R ayleigh has po in ted out# o u s t extend to th o se i r r e g u l a r i t i e s in

th o su rfaoo whleh c o n s t i tu te roughness** from th e p lo t o f th e s e

f r i c t i o n f e s t e r s a g a in s t Reynolds Number* th e c r i t i c a l Reynolds

Number was in d ic a te d to he approxim ately 2100.

Table I g iv es some o f th e v a lu es o f c r i t i c a l Reynolds Numbers

ob ta ined by v a rio u s in v e s t ig a to r s u sin g c i r c u la r pipe*

TABLE I

C r i t i c a l Reynolds Numbers fo r C ircu la r P ipe

In v e s t ig a to r % ^( C r i t i c a l ) Method

S tan ton and P enne ll (24 ) 2100

Page and Tewnend (14) 2000*2700 Motion o f c o llo id a l p a r t ic le s*

F r lo tio n

Bend (7 )

B innie and Bowen ( 6 )

B innle ( 2 )

C lbson ( I d )

Reynolds (29 )

1690-2150

1760

1970

2700-4175

2300

Luminescence

P o la r is e d l i g h t

Color Band

Color Band

Stethoscope

T

I n th e development o f P o iB eu ille* * law fo r th e f r l o t i o n lo o s

i n c i r c u la r pipe* th o p o in t v e lo c ity o f th o f lu id l a ob ta ined as

an l a t e r m odiate fo ra* Thie p o in t v e lo c ity may be expressed as*

A ll o f theoe equations a re a p p lic a b le as long as th e flow through-*

o u t th e p ipe l e e e a p le te ly stream line*

When th e e r i t i e e l v a lu e o f th e Reynold* Humber l e exceeded*

th e r e la t io n s h ip s above no longer apply* S tan ton ($4 ) found th a t

th e value o f th e r a t i o o f average v e lo e ity to th e maximum v e lo c ity

in a re ae ee g ra d u a lly th roughou t th o t r a n s i t io n reg io n and reach es a

v a lu e o f approxim ately 0 .60 a t NR# * 10*000* Above th ie v a lue o f

th e Reynolde number# th e r a t i o ren a ln o approxim ately constant*

Throughout t h i s range th e v e lo e ity d i s t r ib u t io n le changing* bo any

ex p ress io n whleh re p re se n ts th e v e lo e ity d i s t r ib u t io n in t h i s reg io n

o u s t eozxt&ln fu n c tio n s o f th e Reynolds KUmber. Therefore* von

Kerman (55) has developed a u n iv e rsa l v e lo e ity d is tr ib u tio n * assuming#

among e th e r th ings* th a t a lanubtar eub -layer o f f lu id e x is ts* In th i s

(5 )

i s ob ta ined from Equation (1)* then*I f th e v a lu e o f

M )

from whleh*

' / 77<J»X

(7 )

a

ho BBk«» u s* o f a reduced v e lo e ity param tor, u% and a reduced

d le ta n s e y +# f b t l t v a lu e s a re defined act#

!

y * - ju

I n abi«h» Tb » tIboobb *h«ar «tr«B* « (d jy )g „ 0

<6)

(9)

( » >Z

For c ircu la r pipes la which 72 - fpAXg- m j

u * =(12)

* S Ir /2 (15 )

B ring th e s e r»UU oA<» N lk a ra d fi (58) determ ined th a t th o flow in

smooth p i poo oouXd ho d iv ided In to th ro o reg io n s a* follow s*

( 1 ) th o U a l s a t lay e r o f f lu id i n which*

u + * - t / + <1*>

(# ) t t * t r m l U r a ra g io n i n whish

U + = /°?K y + - 3 - a s - ( 15)

and (5 ) th o f u l ly tu rb u le n t flow , fo r which

u >* & 7 S * & Btf*+ ^ (W )

M ille r (29) h a s brought some se rio u s doubt in to th o aoooptanoo o f

th o se va lues as c a lc u la te d by H ikuradse, e sp e c ia l ly fo r th o v a lu es

Which wore in tended to prove th e e x is to n e s o f a lam inar lay e r near

th o surface* However, e lnee th a t tim e, o th e rs have ob tained r e s u l t s

9

whleh agree with the re suite of Nikuredee (IJ)#

Rothfua (2 ? ) used a m odified f r i c t i o n fa c to r p lo t to

c o r r e la te p o in t v e lo c i t i e s I n a c ir c u la r pipe* In th ie p lo t

and i t wee found th a t th e experim ental p o in ts f e l l upon th e

l in e s fo r th e f r l o t i o n f a s t e r p lo t in both th e s tre am lin e and

tu rb u le n t flew region* For th e p o rtio n o f th e f lu id in stream*

t 9 3j5/Nr ^ . The c r i t i c a l v a lu e o f $ y u /V was In d ic a te d to be

approxim ately 1000 f o r * 6# 210 and about 600 fo r

Throughout th e m a te r ia l p resen ted th u s far* freq u en t m ention

i s made o f th e Reynolds dumber a s th e b a s is fo r determ ining th e

flow c h a r a c te r i s t i c s o f a f lu id * M th e o r e t ic a l j u s t i f i c a t i o n

i s p resen ted fo r such use* According to Kllnfeenberg and Mooy (17)

th e Reynolds Humber "se rv es as a measure o f th e r e l a t i v e im portance

o f th e t r a n s f e r o f momentum by convection and by d if fu s io n reaper*

t iv e l y ff i . e .# o f th e fo rc e s o f i n e r t i a i n r e l a t io n to fo rce s on

account o f In te rn a l f r ic t io n * " As such i t re p re se n ts th e q u o tie n t

ob ta ined by d iv id in g one term o f th e Navier**Stokes equations by

one o f th e e th e r terms* Since a l l f lu id s obey th e s e eq u atio n s o f

motion# th en I t i s to be expected th a t suoh a r e la t io n s h ip should

se rv e a s a b a s is fo r flew behavior*

T his I s a ls o shown by Brown ( 6 )* In t h i s in stance# two system s

which a re g eo m e trica lly s im ila r a re considered and by assuming

2 TZfpvP* i s p lo tte d a g a in s t a p o in t Reynolds Number* © yu/y’

l in e motion# %T^Jp u 2 * 16 ^/©yu# which i s analogous to

*Rm s fct#30Q#

dynamic s im i la r i ty i t l a found th a t a Reynolda Number group l a

ob ta ined eueh t h a t th e v a lu e o f i t In one system l e equal to th a t

i n th e o th e r system* The fo llow ing sta tem en t le made*

*— *# when dynamic s im ila r i ty e x is ts i n two system s, th e product o f any c h a r a c te r i s t ic dim ension, any v e lo c ity , th e d e n s ity and th e re e ip ro o a l o f th e v ls e o e l ty i s th e seme fo r both system s when th e se v a r ia b le s a re chosen a t eorresponding lo ca tio n s* T here fo re , th e se v a r ia b le s (L , V, f> # / / ) in them selves w ill determ ine th e flow p a tte rn in

g eo m etrica lly s im ila r systems**

T h ere fo re , th e good c o r re la t io n s ob tained when using Reynolds

Numbers fo r c i r c u la r p ipes o f equal roughness a re to he expected

from t h i s statem ent* Although t h i s Reynolds Number i s u se fu l in

p re d ie tin g th e type o f flow which e x is t s i n th e se c i r c u la r p ip e s ,

i t does no t g iv e any in d ic a tio n o f th e mechanism by which th e

Change from s tre am lin e to tu rb u le n t flow ta k e s p isee*

By a purely m echanical a n a ly s is o f th e f a c to r s which would

a f f e c t th e growth o f a d is tu rb an c e in a flow pa tte rn # Reuse (21)

p re se n ts a so -c a lle d s t a b i l i t y parameter# V ,

-y =

^ (X7)

The v a lu e o f t h i s param eter w il l n e e e e sa r lly be se re a t a boundary

( y * 0 ) and again a t th e p o in t o f maximum v e lo e ity (du/dy s 0)*

T herefo re I t w i l l a t t a i n i t s maximum value a t some p o in t between

th e s e two* B ines a h igh value o f t h i s param eter in d ic a te d in ­

s t a b i l i t y o f flow , i t fo llow s th a t th e po in t o f g r e a te s t i n s t a b i l i t y

w i l l be a t a p o in t between th e su rfa ee and th e p o in t o f maximum

v e lo e ity } t h i s should be th e p o in t a t which tu rb u len ce w ill f i r s t

XI

occur* Hathen® t io e 1 ly th ie p o in t i s o n e -th ird o f th e ra d iu s

removed from th e se n io r o f a o lro u la r pipe* th e use o f t h i s

param eter does not seem c o rre o t when i t I s considered th a t i t s

v a lu e i s always aero a t th o p o in t o f maximum v e lo c ity * In d io a tr

th a t flow a t th a t p o in t i s very s ta b le to d istu rbance* Also

in aome p relim inary in v e s tig a tio n s by tho author ( 1^)» i t was

In d ica te d th a t tu rb u len ce appears f i r s t a t th e c e n te r o f a

c i r c u la r pipe* House does not p re sen t any experim ental d a ta

to in d ic a te th e r e l i a b i l i t y o f t h i s parameter*

(Jibson (16) i s one o f th e few workers to t r y to determ ine

th e p o in t a t whleh tu rb u le n ce w ill f i r s t appear in a c ir c u la r

pipe* 3y in je c t in g a dye in to w ater flowing in th e p ipe and by

o b se rv a tio n o f th e dye# he d e te r ad nod th a t tu rbu lonoe f i r s t

appeared a t a p o in t approxim ately s ix - te n th s o f th e ra d iu s from

th e c en te r o f th e pipe* He oonoluded th a t t h i s change occurs

f i r s t a t th e p o in t o f maximum change o f k in e t ic energy# whleh

has th e va lue o f 0*2773 f*or s tream lin e flow o f a f lu id in a

c ir c u la r seetlon* T h is again i s no t in agreement w ith th e pro*

▼ions f in d in g s o f th e author*

I t I s in d ic a te d above th a t th e v iscous ahear s t r e s s in a

th e in te g ra t io n o f t h i s s t r e s s over th e e n t i r e area o f flow y ie ld s

th e p ressu re drop due to th e f r ic t io n * However* when th e flow

becomes tu rb u len t* t h i s v iscous s t r e s s i s no t th e only c o n tr ib u to r

to th e f r i c t io n a l lo sses* Since th e flow o f th e f lu id pa rtic le®

f lu id may be expressed as and* fo r s tream lin e flow*

12

In tu rb u le n t flow has boon found to b© eddying and in d ire c t#

th « v e lo c ity a t a p o in t i© not oonetan t w ith time* Therefore#

a tem poral mean v e lo c ity # tf# must be used* I t i e a lso neoes-

aary to in d io a to th e e f f e c t iv e shear s t r o t s as#

In th is# rj i s o a lled th e eddy v iso o a ity and has been defined

by PrandtX ( 52) as#

In Equation (19) \ I s known as th e mixing len g th and s ig n i f ie s

th e d ie tsn o e in th e t ra n s v e rse d ir e c t io n which a f lu id p a r t i c le

a s s t t r a v e l s t th e mean v e lo c ity o f i t s o r ig in a l la y e r so t h a t

th e d if fe re n c e between i t s v e lo c ity end th e v e lo c ity o f th e new

lo c a tio n eq u a ls th e mean v e lo c ity f lu c tu a t io n o f th e tu rb u le n t

flow* I f v e lo c ity d is t r ib u t io n s and p ressu re drop measurements

a re made# i t ie p o ss ib le to c a lc u la te th e d is t r ib u t io n o f t h i s

mixing leng th in th e flow section*

B* Annular S ec tio n s*

The c o n s id e ra tio n s above have d e a l t alm ost e x c lu siv e ly w ith

c ir c u la r sections* Binoe flow in th e se flections her been more

thoroughly in v e s tig a te d th en th a t in any o f th e o th e r shapes*

This ia understandable* s in ce c ir c u la r p ipes a re th e most w idely

used flow seotienfl* Another widely used flow se c tio n ie th e

(18)

(19)

15

animlua* For etre&m line flow in a co n cen tric annulus, tho

p re ssu re drop may be calcu lated . by a m o d ifica tio n o f P o ise u ill© 1©

law fo r c i r c u la r pipes* Thin y ie ld s , (S I )

( 20)

The ra d iu s o f th o p o in t o f maximum v e lo c ity fo r t h i s type o f flow

in an annulus i s g iven by th e expression#

V - J 7 ^

( 21)

T his po in t i s always toward th e eore from th e m id-poin t o f th e

an m lu e , th e re fo re th e in te n s i ty o f shear a t th e oor© w all i s

g re a te r th a n t h a t a t th e o u te r w a ll, s in ce th e v e lo e ity g ra d ie n t

i s s te e p e r In th e in n e r p o rtio n o f th e annulus* The v e lo e ity

d i s t r ib u t io n fo r s tream lin e flow in an annulus i s g iven as#

* -a - G 1- * 1' / -£ 7( 22)

o r

U * ____ _ /7 j> 1 , / t ? - * ? / xfp>ls 7 ,,, -% f * f 7

(®5)

From t h i s i t can be seen th a t th e r a t i o o f th e average v e lo c ity

to th e maximum v e lo c ity in an anm lue depends upon th e r a d i i o f

th e annulua, to th th e a b so lu te and r e l a t i v e valued o f thee*

r a d i i and t h i s r a t i o may be expressed as#

The r a t i o o f th e shear s t r e s s e s a t th e two su rfa ce s in th e

annalue i s g iven by#

These th e o r e t i c a l r e la t io n s h ip s fo r th e s tream lin e flow

reg io n have been v e r i f ie d experim en ta lly re c e n tly by Rothfu© ( 28#

29) i n a study w ith smooth co n cen tric annuli# However$ to r t u r ­

b u len t flow c o n d itio n s th e knowledge o f flow In annul! i s n o t as

w ell c la r if ie d * Since th e Reynolds Humber has proved so s a t i s ­

fa c to ry fo r u se w ith c i r c u la r pipe# i t has a lso been used fo r

s m a ll# b u t th e r e s u l t s have not been as s a t is f a c to ry in t h i s

ap p lica tio n * The Reynolds Number should be as s ig n i f ic a n t In

d efin in g flew c h a r a c te r i s t ic s i n an annulue as in a c ir c u la r pipe#

provided th a t th e proper v a r ia b le s may be found and used* The

tro u b le which has been met In t h i s a p p lic a tio n seems to a r i s e

from th e a ttem pts to extend th e law o f s im ilitu d e from c i r c le s

t o annul 1* When Brown's sta tem ent on page 10 concerning th e use

o f t h i s Reynolds Humber l a considered , then i t i s seen th a t an

azmulus i s n o t g eo m etrica lly s im ila r to a c irc le *

(2 4 )

^ 2 ( r i - R ' )

(25)

Therefor** s in ce geom etrica l s im lle r l ty I s a p re re q u is i te ,

dynamic s im ila r i ty w il l no t e x is t a t th e same va lue o f th e

Reynold• Number In th e two shapes*

Thl* lock o f s im ila r i ty has been neg lec ted In th e p a s t,

however, and th e Reynolds Number has been c a lc u la te d using th e

e q u iv a le n t d iam eter concept* One o f th e e a r l i e s t form ulae to

be used to compute th e flew in a l l s o r t s o f condu its was th a t

o f Chesy, in which a h y d rau lic ra d iu s term appeared (2)* T his

h y d rau lic ra d iu s i s evaluated by d iv id in g th e c ro ss* e eo tlen a l

area o f flow by th e w etted perim ete r o f th e conduit* The Ohezy

eq u a tio n i s ,

I f t h i s i s compared to th e Fanning Squatlon , Squat ion (2 ) , i t i s

t ie n a to g ive th e same f r i c t i o n lo sses* Thus, th e eq u iv a len t

d iam eter, De , la th e va lue o f fou r tim es th e h y d rau lic rad ius*

For an a nonius, * (Dg-D^)/4 and th© eq u iv a len t d iam eter i s

u su a lly tak en to be &2 -

By using t h i s va lue o f eq u iv a len t diam eter to c a lc u la te th e

Reynolds 'umber fo r flow in an annulus and th e n by using t h i s

v a lue to determ ine th e f r i c t i o n fa c to r from c ir c u la r pip© d a ta ,

i t has been p o ss ib le to o b ta in a good approxim ation o f th e f r i c ­

t io n drop fo r tu rb u le n t flow in an annulue. Although t h i s method

o f extending date i s not s a t is f a c to ry because th e re i s no

2.

( 26)

seen th a t f must equal 2g/0^ and D must equal 4r^ , fo r th e ©qua-

16

s im ila r i ty between th e two shapes* th e re have boon no o u ts tan d in g ly

su c ce ss fu l a l t e r n a te Tcethodc proposed* T herefore! moot m a te r ia l

p resen ted on th e study o f th e behavior o f f lu id s in annul! has

used t h i s e q u iv a le n t diam eter*

A ctually th e amount o f work which has been done on th e flow

o f f lu id s in annu lar spaces i s no t very ex tensive* In a review

VIegand and Baker ( 36) p resen ted a com pila tion o f th e work done

b e fo re 19^*2 on f t lo t io n lo se and h e a t t r a n s f e r in annull* Much

o f th e work p resen ted d e a l t w ith h e a t t r a n s fe r * bu t a comprehensive

l i s t i n g o f th e f r i c t i o n drop s tu d ie s was a lso given* The r e s u l t s

o f th e se a re p resen ted In Table II* In t h i s tab le* j i s de fined as*

These va lu es o f j p resen ted In Table I I w il l no t be d iscussed &t

t h i s point* but w il l be covered la te r*

From Table II* i t can be seen th a t sev e ra l o f th e investlga?-

t o r s worked in both th e s tream lin e and tu rb u le n t reg io n s o f flow*

The rep o rte d c r i t i c a l valuee o f th e Reynolds Numbers cover q u i te

a range and th e re seems to be a marked e f f e c t o f th e width o f th e

annulue on th e se values* The d a ta o f P icrcy in d ic a te t h a t th e

c r i t i c a l va lue o f decreases as th e width o f th e

anratlue decreases* w ith & c o n stan t ja c k e t diam eter* Another i n t e r ­

e s t in g fo o t i s t h a t th e c r i t i c a l va lue fo r th e ja c k e t alone I s 1620

whereas t h i s va lue fo r th e w idest annulu® i s 1020*

2 j

(S7)

17

TABUS I I

R e su lts of In v e s tig a tio n s on Annular Spaces a s Reported toy

Viegand and Raker ( 36 )

V » « 2f e e t

C r i t i c a l(Dg-Dj

V

J A o f3250 10,000

F lu id In v e s t ig a to r

0 .9 :0 .!

0.0100 .5150 .M 90 .513O.&MO0.6830 .7200.75*0.SO3

.3^30 L 0 . ^ o.soo

o.sUto

0.975o.9«5

0 .000o.jito0.6280.805

0.00000.00170 .00580 .01550.0202

.0723

.072320002000

. 00UU

. 00*0.001*0 Water

.09a •0051 .0039 Water

.0622

.0827

.0622

.0827

.1303

.0622

.1303

.0827

.1303

279029202710263019401980241+02080

.0

*888

8

88

88

8 Water

.192

.192

.192

.007

.006

.007

Air* Water Fuel Oil

.0617 .0036 W a te r

.266

.266

.266,266

. 00**0

. 00H2

. 003U

.0030Water

.172

.172

.172

.172 ll+oo

.0056

.00U7

.OO56

.0050

.00U4

.0038

. 00UU

.00*40

Water andCaClgtorlne

. 01+25

.01+25

. 01+25

. 01+25

. 01*25

16201020

920

2 8

.00§7

.006a

. 0060U

.00599

.00582

A ir

Becker ( 3 )

Winkel (37)

Lonsdale ( 23 )

Caldw ell ( 9 )

Sohneokentoerg (33)

e t . a l . (20)

P le rc y # e t . a l . ( 27 )

(a£) treats

XB *^0 'JCOJ0J£

*IV

Jiv

GhOO*8£00*

6£z6£2

moo*s£oo*

06910691

StiOO* It) 00*

£S*iT£Snt

iCoo*xnoo*

£i\2X&«x

6200*g£oo*

2*100* inoo*

S£ox££ox

Si.900* x£loo* TB900*iti900*89900*8iS00*29600*

xtji

±88

8lt£90S899

SznoS2*t0£21t0S2tioSztjOSsnoS21|0

996*0000*0

8X6*0000*0

209*0000*0

Soi*o000*0

sna’o000*0

2X9*0£6tt*0062*0£81*0o£x*o

ioi.o*oxx£o*o

°°0'°T 0fi2£ “i(Vza) *•** . PWi JO hK • ( T»OT»XJIO 2j *0/^

(«>o) xx rcm

81

19

When Pierey* et* s i .* ( 2 7 )* ob ta ined th e low c r i t i c a l

v a lu es o f th e Reynolds Humber as mentioned above* an appara tus

was co n stru c ted fo r th e v isu a l o b se rv a tio n o f flow in an

annulue. To do th is* a dye was in troduced in to th e flowing

water* I t was rep o rted t h a t t h i s dye th read was s t r a ig h t a t

lew v e lo c it ie s * b u t developed a wave o f email am plitude a t

s l ig h t ly above th e l im i t fo r r e c t i l i n e a r flow* A a l ig h t in ­

c re ase o f v e lo c ity produced an e a s i ly seen slow sinuous motion*

approxim ately a c irc u m fe re n tia l swaying* which f u r th e r developed

through a wide range o f in c re a se o f flow through th e an m lu s

b e fo re break ing up in to in co h eren t eddying* Waving th rea d s

remained approxim ately a t a co n stan t ra d iu s so f a r as could

be seen . The a x is o f o s c i l l a t i o n o f th e th read wandered m ostly

in a c irc u m fe re n tia l d irec tio n * bu t sometimes ra d ia lly * I t never

c irc u la te d around th e core*

When Lea and Tadros (22) were working w ith wide annul!* a

low c r i t i c a l va lue was obtained* They a lso found th a t th e ex­

perim ental p ressu re drop fo r a wide annulue was not as g re a t as

th a t p red ic ted by theory* th u s i t was concluded th a t th e re was

s l ip o ccu rring a t one o f th e surfaces* probably th e su rface o f

th e core*

Rothfue ob ta ined c r i t i c a l v a lu es o f th e Reynolds tfumber on

two annular s e c tio n s . These were* 2100 fo r th e wider a n m lu s

and 2400 fo r th e o th er one* Carpenter* e t . a l (10) determ ined

th e p ressu re drops in an annulue fo r both s tream lin e end tu rb u le n t

80

flow conditions* Prom th o se d a ta fo r t h i s anm lus* th e c r i t i c a l

▼slue o f th e Reynolds Number was found to be approxim ately 12^0#

0* C onclusions*

Although co n sid erab le work has been done in th e general

f i e ld o f f lu id flow* moot o f t h i s has been confined to flow in

e i th e r th e s tream lin e re g io n o r th e tu rb u le n t region* There

seems to be a tendency to n e g le c t th e reg io n o f t r a n s i t i o n flow*

e sp e c ia lly th e manner i n which t h i s t r a n s i t i o n ta k e s p lace a t d i f -

f e r e n t p o in ts In th e flow section* Even when th e c r i t i c a l flow

cond itions have been reported* th e method by which th e se va lu es

wears ob tained I s no t u su a lly one o f d i r e c t o b se rv a tio n o f th e

flow mechanism, bu t r a th e r an in d i r e c t observation* most f r e ­

quen tly based upon p ressu re drop measurements* When t h i s s o r t

o f measurement le made* th e only way In which th e t r a n s i t io n may

be de tec ted i s by d e v ia tio n froar th e th e o re t ic a l l in e fo r stream*

l in e flew* T his d e v ia tio n probably ta k e s p lace in such a g radual

manner th a t th e p ro b a b il ity o f d e te c tin g i t i s low*

I t le a lso ev iden t th a t most o f th e work in t h i s f i e ld has

been confined to p ipes o f c ir c u la r c ro ss-sec tio n * From th e stand ­

p o in t o f s im ila r i ty th e o l r c le i s probably th e b e s t se c tio n fo r

study* because any l in e a r dimension in a c ir c u la r c ro ss se c tio n

may be expressed ae a fa c to r o f th e rad ius* However* t h i s

s im p lic ity o f measurement may be & m isleading fa c to r in th e study

o f f lu id flow* I f flow ic etud ied in an annulue* th en th e re a re

two su rfa ce s a t whlob f r i c t i o n occur© and a t l e a s t two l in e a r

a

dimension* a re req u ired to d e fin e th e se c tio n and point© in

th e ©action* A leo, due to th ee e two surf&eee o f d i f f e r e n t a rea

fo r f r ic t io n * th e v e lo c ity d is t r ib u t io n i s no t eyrametrioal about

a c e n tra l aids*

QHAP7KR I I I

SUMMRY OF EXPERIMENTAL WORK

A. A pparatus.

The ap p ara tu s ueed in t h i s work was a m o d ifica tio n o f th e

u su a l la b o ra to ry Reynolds Apparatus and i s shown in F igure Ho. !•

The major f e a tu re s o f i t were described in an e a r l i e r work {1^).

Two changes h a re been made einoe th a t tim e, however. The number

o f dye in ja e t io n tubes has been increased to three* two o f which

e re d ia m e tr ic a lly opposed and th e th i r d placed approxim ately th re e -

fo u r th e o f an inch o f f th e perpendioul& r b ise c to r o f th® ax is o f

th e o th e r two. In t h i s way t h i s tube does no t e n te r th e tube

r a d i a l l y . T his arrangem ent i e shown in th e d e ta i l o f Figure Ho* 1.

The dye tu b es could be moved by th e r o ta t io n o f th e notched wheel*

A* and th u s th e p o s it io n o f o b se rv a tio n changed. The dye tube* B*

was so ldered in to th e d r i l l e d end o f th e th readed b rass rod* C«

The dye was in troduced from above th® head tan k through th e dye

feed l in e and rubber tu b in g connected to a sh o rt tube* D* which

was so ldered in to th e b ra ss rod v e r t ic a l ly * These dye tubes were

made o f copper c a p i l la ry tu b in g , 0 .095 inch O.D. ©nd 0.045 inch

I .D . This dye in je c t io n se e tio n le a leo shown in T la tee I and II*

Im m ediately below th e g la s s o b se rv a tio n section* a one foot

le n g th o f s tandard iro n pipe was added so th a t th e e x i t e f f e c ts

would be minimized. This p ipe was connected to th e o u t le t se c tio n

which remained unchanged.

23

CONSTANT LEVEL HEAD TANK

Dye Injection Section, See Detail

'not -• K

itvorizod Iran Pip* Cara

ELEVATIONS

FRONT

*

i t

LEFT SIDE

Flo

w

DETAIL OF DYE INJECTION SECTION-Pip* to tubing connection

Dye tub*Dy# food tubing

Hond wheely g “ Brow rod

Iron pipe wall

ELEVATIONO I* t

Scale: I—i- I—i— I

TOP VIEW

LO U IS IA N A ST A TE U N I V E R S I T Y f y )

D E P A R T M E N T O F C H E M I C A L E N G I N E E R I N G Q J

VISUAL OBSERVATION APPARATUS FOR

FLOW IN ANNUL I__________DR. BY: T.A.FEAZEL IOATE . MAY, I SSI

SCALE: AS SHOWN I FIGURE NO. I

PLATE I

Photograph o f Dye In je c tio n S e c tio n , L e f t Side

PLATE I I

rhotogxaph o f Dye In je c tio n S e c tio n , F ron t View

26

Four cere© were te s te d i n t h i s apparatus* Th© diam eter

o f each core was ob ta in ed e t sev e ra l point® ©long th e len g th

o f th e c e re and th e average va lue o f thee# measurements wa©

ueed in th e e a le u la tio n a * Theee value© a re p resen ted in

Table I I I*

TABLE 111

Dimension© o f Pi pee Used in th e Experim ental Work

Anm luehttaber

Ja ck e tDiameter#Inches

C.D. o fCore#Inohee

Core D esc rip tio n

4

i

2

5.000

5.000

5.000

5.000

1.646 1 -i* s td . Galv. I ro n P ipe

1.047 5/4* S td . Gtelv. I ro n P ipe

0.6457 1/2* Bid. Oalv. I ro n Pipe

0.8457 1/2* S td . Copper P ipe

27

gw #rl»w nt« t p r o a t d w .

For each M o tio n te s ted * th e procedure follow ed was th e

same* The eo re wee plaoed i n th e ap p ara tu s and a d ju s te d u n til*

v is u a l ly i t seemed to be se n te re d in th e o b se rv a tio n se c tio n .

Flow was th en s ta r te d and by observ ing th e maximum v e lo c ity w ith

each o f th e th r e e dye j e t s t h i s c o n c e n tr ic i ty was checked* I f

th e maximum v e lo c i t i e s were n o t equal* th e n ad justm ents were made

w ith th e eore u n t i l e q u a lity was obtained* For s tream lin e flow

th e re i s no tra n s v e rs e p ressu re g ra d ie n t in a c ro s s -s e c tio n o f

flow* For t h i s to be t r u e in an e e e e n tr ie annulus* i t i s neces­

sary th a t th e boundary shear s t r e s s be th e same a t a l l p o in ts on

th e eore* T his shear s t r e s s l e th e produot o f th e v is c o s i ty o f

th e f lu id and th e v e lo c ity g ra d ie n t a t th e wall* I f th e se v e lo o lty

g ra d ie n ts a re to be equal* th e n th e v e lo c ity d i s t r ib u t io n i n th e

w idest p a r t o f th e annulus must be such th a t th e maximum v e lo o lty

i s g re a te r th a n th e m axim a v e lo c ity In th e narrow est p a r t o f th e

ansulue* These maximum v e lo c i t i e s a re equal only when th e oore I s

concen tric*

When i t was determ ined th a t th e oore was concentric* v e lo o lty

d is t r ib u t io n s were ob tained a t a co n stan t flow r a t e In th e Bireaar-

l in e flow region* The p o in t v e lo c i t i e s were ob ta ined by t h i s dye

in je c t io n method by in iro d u o tin g th e dye slowly so th a t i t emerged

from th e t i p o f th e dye tube a t approxim ately th e came v e lo c ity as

th e w ater and th en i t was allowed to t r a v e l a ab o rt d is tan c e b efo re

th e movement was timed* This time* was t h a t req u ire d fo r th© dye to

20

move between two measured po in ts* I t was neoee&ary to r e s t r i c t

t h i s v e lo c ity d e te rm in a tio n to th e s tream lin e flow co n d itio n

because th e dye w ill n o t rem ain a t th e same ra d iu s when th e

flew I s no longer stream line* th e se v e lo c ity d is t r ib u t io n s were

ob ta ined on bo th s id e s o f th e oore to g ive an in d ic a t io n o f th e

symmetry about th e sore*

The maximum v e lo c ity In th e se c tio n was ob tained by i n j e c t -

lug a la rg e mass o f dye in to th e moving stream and tim ing th e

t r a v e l o f th e f a s t e s t portion*

A fter sym m etrical v e lo c ity d i s t r ib u t io n s were obtained# one

o f th e dye tu b es was plaeed a t a d e s ired p o in t and th e flow o f

th e dye and w ater both s ta r te d a t a low ra te* The behavior o f

th e dye stream was observed and d escribed and th e v e lo c ity a t

th e o b se rv a tio n p o in t and th e maximum v e lo c ity in th e se c tio n

were determined* When t h i s was done a t t h i s flow ra te# th e r a t e

was in c reased s l ig h t ly and th e o b se rv a tio n repeated* This was

continued u n t i l tu rb u len ce appeared a t th e p o in t o f observation*

When th e se o b se rv a tio n s had been made e t one point# th e dye

j e t was moved to ano ther lo c a tio n and th e o b se rv a tio n rep ea ted

there*

0* L im ita tio n s o f Apparatus and P a ts *

The appara tu s has c e r ta in in h e re n t lim ita tio n s* Due to th e

s iz e o f th e dye tube# th e c lo s e s t th a t th e c en te r-iin © o f th e dye

j e t oould be brought to th e w all was 0 *0*W>9 inches* However# t h i s

v a lu e could not always be obtained* In bending and forming th e

29

dye tubes* c e r t a in im p e rfec tio n s arose* One o f ih ee e was th e

f e e t t h a t th e len g th o f dye tub ing p a r a l le l to th e w ater flow

wee s o t e x a c tly p e rp en d icu lar to th e tra n s v e rs e p o rtio n o f

th e dye tube* I f th e ang le between th e two wee lean th an 90°,

i t wee n e t p e ee lb le to p lace th e dye tube a g a in s t th e core w alls

and I f th e ang le was g re a te r th an 90°$ I t wae n o t p o ss ib le to

p la c e th e tube a g a in s t th e g lace wall* Also th e bend In th e

tu b in g wae curved and no t rig h t-an g led * so t h a t th e tu b e d id

n e t peso through th e s tu f f in g box and l i e f l a t a g a in s t th e p ipe

w all*

lie measurements o f proa aura drop were made* because th e

measurement o f p re ssu re d if fe re n c e s o f th e o rder o f magnitude

e f th o se which e x is te d h e re (approxim ately *00^ in ch o f w ater fo r

th e f i r s t f o o t o b se rv a tio n se c tio n ) would have req u ire d th e develop*

cen t and c a l ib r a t io n o f an e x ce p tio n a lly s e n s i t iv e ml©r©manometer•

As was sa id e a r l ie r* th e d e te rm in a tio n o f p o in t v e lo c i t i e s

by th e method e f dye in je c t io n tree l im ite d to th e s tream lin e flow

region* These v e lo c i t i e s were determ ined by tim ing th e t r a v e l o f

th e dye th rough len g th s o f 1* 1^ o r 2 fee t* The longer le n g th s

were used to detarm ine th e h igh v e lo c it ie s * When checks were

made on th e v e lo c ity d e te rm in a tio n s a t a point* i t was found th a t

th e maximum d e v ia tio n between th e two was 0*4 seconds i n th e

minimum observed tim e o f approxim ately 10 second a* T his gave a

maximum d e v ia tio n o f approxim ately

The major source o f e r ro r e n te r in g in to th e da ta o f t h i s

re se a rc h was th e doubt involved in th e p o s it io n o f th e dye tube*

30

A c a l ib r a t io n curve was made fo r th e p o e itio n o f each tube ae

a fu n c tio n o f th e number o f tu rn s o f th e handwheel* Theee

tu rn s were measured from th e n e a re s t p o s i t io n o f th e dye tube

to th e g la s s wall* The c a l ib r a t io n was ob ta ined by suspending

th e dye i n l e t s e c tio n above a p iece o f paper and determ ining

th e d ir e c t io n e f th e dye tube movement* Then th e assembly was

connected to th e g la s s p ip e se c tio n and read ings o f p o e itio n

made using a p iece o f s t e e l tap e placed aeroas th e section* I t

was p o ss ib le to read th e g rad u a tio n s on th e ta p e to 1/ 52% so

t h a t th e maximum probable e r ro r o f th e s e read ings was 1/ 52%

which was eq u iv a le n t to k tu rn o f th e handwheels*

In th e o b se rv a tio n s o f th e dye behavior* care had to be

tak e n th a t th e dye did not e n te r th e w ater a t a h ig h er v e lo c ity

th a n th a t o f th e water* I f t h i s was done* th e dye *piled**up*

and spread o u t in th e flow* g iv ing th e appearance o f non-stream*

l in e flow* A ctu a lly I t was not d i f f i c u l t to c o n tro l th e flow o f

dye so t h a t t h i s did not happen and th e c r i t e r io n used was th a t

th e diam eter o f th e dye stream leav in g th e dye tu b e should always

be sm aller th a n th e diam eter o f th e opening In th e tube* When

observing t h i s precaution* th e dye seemed to be drawn ou t o f th e

tu b e and picked up by th e w ater to be o&rriod downstream*

The lo c a tio n o f t h i s appara tus was suoh t h a t a t tim es i t was

sub jec ted to a g re e t deal o f ex traneous v ib ra tio n s and t h i s has

caused some o b se rv ers o f th e work to have q u estio n s regard ing th e

e f f e c t o f th e s e v ib ra t io n s on th e flow behavior* During th e

sugar season a c e n tr ifu g e nearby was operated q u ite fre q u e n tly a t

%

th© ©aw© tim e th a t experim ental work was being c a r r ie d out*

I t wee observed, however, t h a t air©amiin© flow was m aintained

under th e e e ©ever© v ib ra t io n s , and i t wae a le e found th a t

very good s tream lin e flow condition© e x is te d when th e eo re

could be seen to v ib ra te* I t i s to be concluded from th e s e

o b se rv a tio n s t h a t , ( l ) below some c r i t i c a l flow r a t e , stream*

l in e flew I s com pletely s ta b le to any v ib ra t io n , and ( 2 ) th e r e

was no p o s s ib i l i ty o f m e ta s ta t ic condition© g iv ing s tream lin e

flow a t h igh r a t e s o f flow*

D* T ests o f D ata,

The d a te ob tained were te s te d in sev e ra l ways* The v e lo c ity

d is t r ib u t io n s ob ta ined fo r s tream lin e flow co n d itio n s were com*

pared w ith th e th e o r e t ic a l d is t r ib u t io n s ob tained by Equations

(25) and (84)* These were combined to g ive a th e o re t ic a l va lue

o f th e r e t i e e f th e p o in t v e lo c ity to th e maximum v e lo o lty in th e

section* These th e o r e t ic a l r a t i o s app lied fo r any flow r a te as

long as th e flow was s tream lin e th roughout th© section*

The p o in t v e lo c i t ie s ob tained under a l l flow co n d itio n s were

compared to th e se th e o r e t ic a l r a t i o s to determ ine th e e f f e c t o f

th e In c re a se ir< flow r a t e upon th e v e lo c ity d is tr ib u tio n * With

th ee e p o in t v e lo c i t i e s i t was p o ss ib le to o b ta in th e v e lo o lty

d is t r ib u t io n s a t a number o f d i f f e r e n t Reynolde Numbers* Xt was

a ls o p o ss ib le to o b ta in th e v e lo c ity g ra d ie n t, du/dy, a t th e

v a rio u s p o in ts in th© c ro n e -se c tio n as a fu n c tio n o f th e Reynolds

Chamber* By p lo tt in g th ee e v&lues o f v e lo c ity g ra d ie n ts a t th e

5®

same Reynolds Number a g a in s t th e p o s i t io n In th® se c tio n , and

by e x tra p o la tin g th ee e p lo ts to th e w a lls , th e v e lo o lty

g ra d ie n ts a t each w all were obtained* The v a lu es o f th e se w all

g ra d ie n ts were determ ined a t se v e ra l Reynolds iumbere and th e

v a lu e s were p lo tte d a g a in s t Reynolds Number*

As a cheek on th e observed v e lo c ity d is t r ib u t io n s , th e

In te g ra l o f udA was o b ta in ed g ra p h ic a lly and compared w ith th e

flow r a t e in d io a te d by th e ro tam eter*

The von Kerman reduced fu n c tio n s , u * and y + , Equations

( a ) and (9 )# were c a lc u la te d , assuming th a t th e shear s t r e s s a t

a s Ion? as th e r e e x is t s a lam inar la y e r o f f lu id near th e w all*

The va lu es ob ta ined fo r th e s e ex p ress io n s were p lo tte d and com*

pared w ith E quations (1 4 ) , (12) and (16) p resen ted by Mlkuradse*

C r i t i c a l c o n d itio n s a t v a rio u s p o in ts i n th e se e tlo n e were

ob tained by observ ing th e l a s t s tream lin e behavior and th e f i r s t

tu rb u le n t behavior a t each point* The o r l t i e a l va lue a t th e

p o in t was th e n between these* Theee c r i t i c a l co n d itio n s were

expressed in se v e ra l ways* Zn th e p a s t th e o r l t i o a l flow r a t e

fo r an anisilue hr e been expressed as a c r i t i c a l va lue o f th e

Reynolds dumber, . T herefo re , th e value o f t h i s

term was ob tained fo r every d e te rm in a tio n made* The o r l t i e a l

v a lu es o f t h i s exp ress ion fo r each p o in t were p lo tte d a g a in s t

p o s it io n in th e s e c tio n , expressed as th e ra d iu s o f th e p o in t,

g iv in g on in d ic a tio n o f th e amount o f th e f lu id which was flowing

i n each o f th e mechanisms*

a v a lid assum ption

35

Since th e p o in t v e lo c i t i e s wore observed when determ ining

th e s e o r l t i e a l c o n d itio n s , th e p o in t o r l t i e a l v e lo c i t ie s wore

a ls o obtained* Since a l l de te rm ina tion* were n e t made a t

e x ac tly th e came tem pera tu re , i t was necessary to b ring them

to a common b a s ic ! accord ing ly th e v e lo o lty was d iv ided by th e

k inem atic v is c o s i ty o f th e f lu id * T his r a t io u/V was p lo tte d

a g a in s t p o s it io n in th e annulus*

A p o in t o r l t i e a l Reynolds Rumber was a lso ob ta ined fo r each

p o s it io n by m u ltip ly in g u /V by y th e d is ta n c e o f th e p o in t from

a wall* T his d is ta n c e was measured from th e g la s s w all fo r th o se

p o in ts between th e g la s s w all and th e p o in t o f maximum v e lo o lty

and from th e eore w all fo r th o se p o in ts between th e eore and th e

p o in t o f m axim a v e lo c ity * P lo ts were made o f th e s e c r i t i c a l

v a lu es a g a in s t y fo r each annulus*

R ouse's s t a b i l i t y param eter*%$ Equation (17)* was a lso

ob tained fo r th e o r l t i e a l flow c o n d itio n s a t each p o in t and th e

v a lu es o f t h i s p lo tte d a g a in s t th e p o s i t io n o f observation* This

wae done to determ ine th e u se fu ln ess o f t h i s parameter* A ll d a ta

necessary fo r th e d e te r mi n a tio n o f "3 * have been p resen ted above,

y , du/dy and V* •

CHAPTER X?

KXFSRIMBJflVJi RESULTS

A* G enera l*

Whan th e flow o f th e w ater was s ta r te d In th e se c tio n s end

m aintained a t a low r a t e , s tream lin e flow wae obtained* In th le

flow th e dye emerged from th e dye tu b e and tra v e le d th e len g th

o f th e o b se rv a tio n se c tio n w ithout a break o r iu te ru p tio n * As

th e flow r a t e was in c reased however* t h i s behavior d id n o t p e rs is t*

Instead* a slow« sinuous motion was im parted to th e dye* T his

motion was a wave o f long w ave-length and high amplitude* At

tim es i t s w ave-length was so long t h a t only two cy c les were p re sen t

in th e f iv e - f o o t o b se rv a tio n length* F igure 2 (a ) p re sen ts a

sketch o f t h i s motion* I n S ec tion A-A o f t h i s sketch# th e dye

th re a d I s p ic tu red as being d is t r ib u te d e iro u m fe re n tia lly a t th e

ra d iu s o f th e p o in t o f observation* T his view o f th e se c tio n was

n o t v is ib le # but i s based upon th e f a c t th a t th e dye th read remained

a t a p o s it io n o f c o n stan t v e lo c ity w hile in t h i s motion* For con­

c e n tr ic annuity p o in ts which have th e same v e lo c ity a re a t th e

same p o s it io n r e l a t i v e to th e cen te r o f th e system* This wave was

sym m etrical about th e im aginary l in e extending from th e dye tube

to th e bottom of th e o b se rv a tio n section* While e x h ib itin g t h i s

sinuous motion# th e dye was not broken in any way# th e re fo re th e

flow was described as stream line*

&

55

In F igu re 2 (b ) a wav© a c tio n I s shown o f much sh o rte r wove

le n g th th a n th a t In (a )# This motion was observed when th© flew

r a t e was in c re a sed above th a t a t which th e sinuous motion appeared*

This s h o r te r wave had a sm aller am plitude th an th e slow sinuous

motion* b u t was s t i l l seen to e x is t a s a continuous stream through**

o u t th e o b se rv a tio n length* When th e flow r a t e was increased

slowly* t h i s type o f wave a c tio n continued and m aintained approx­

im ately th e same wave length* b u t th e number o f waves passing a

p o in t in c reased w ith th e flow ra te * S t i l l th e dye e x h ib ited sym­

metry about th e p o s i t io n o f th e dye tube end gave no signs o f a

r a d ia l movement* This ty p e o f flow was a lso considered stream line*

s in ce th e dye stream remained continuous*

As th e flow r a t e was in creased more* th e ra p id wave a c tio n

was rep laced by a snapping n o tio n or tu rb u le n t bu rsts* as shown In

F igu re 1 (o)* In t h i s no tion th e dye th read was com pletely broken

by th e s e snaps and th e re g u la r wave p a tte rn was rep laced by an i r ­

re g u la r p a tte rn * This was considered th e f i r s t appearance o f

tu rbu lence* S lig h tly above t h i s flow ra te* edd ies were formed and

th e dye was d is t r ib u te d over a p o r tio n o f th e annulus*

The o b se rv a tio n s p resen ted In F igure 2 were obtained a t a l l

p o s it io n s in th e sections* b u t th e flow ra te s a t whloh each ty p e

e x is te d a t v a rio u s p o in ts in any one se c tio n were n o t th e same*

Thus th e d isc u ss io n above a p p lie s to th e general behavior o f th e

dye fix ed a t a p o in t as th e flow r a t e was increased*

Another type o f flow p a tte rn was observed when th e flow a t

a p o in t was no longer stream line* T his i s shown In f ig u re No* 5*

36

G l o s s J a c k e t

O

D y e T u b e

C o r e

D y e T u b e

SECTION A-A

- D y e S t r e a m

( b )( a )

D y eS t r e a m

( c )

FIGURE NO. 2 - TYPICAL WAVE MOTIONS OBSERVED

5 7

When viewed from th e Bide* th e wave p a tte rn was broken, h u t th e re

wee no in d ic a t io n o f mixing* From th e f ro n t o f th e apparatus*

however* i t wae eeen th a t th e dye had been p a r t i a l l y d is t r ib u te d

aereee th e e e e tio n eo th a t a l l o f i t wae no t in a p o e itio n o f

c o n s ta n t v e lo o lty * When th e dye wae d is t r ib u te d in t h i s manner

th e r e wae a d if fu s in g e f f e c t ev id en t in th e decreased concentra­

t io n o f c o lo r and a la o in th e widening o f th e co lo r band* When

th ie happened to th e flow pa tte rn* th e flow wae no longer eon*

e ldered stream line*

To e e ta b l is h s tre am lin e flow in th e se c tio n s te s ted * i t was

necessary to o p e ra te w ith very low flow ra te s* such th a t th e

average v e lo o lty in th e annulus wae approxim ately 0*10 f e e t per

second* These v a lu es were eo sm all t h a t i t was im possib le to

o b ta in s tream lin e flow when th e d if fe re n c e between th e tem perature

o f th e w ater and th e a i r wae g re a te r th a n 15° This was thought

to be caused by th e convective c u rre n ts e s ta b lish e d by h e a t t r a n s ­

fer* During th e w in ter months th e surroundings o f th e appara tus

were not heated so th a t th e tem pera tu re o f th e w ater was h ig h er

th an th e tem pera tu re o f th e a ir* As th e water flowed through th e

system i t l o s t some o f i t s h e a t and in doing so th e lay e r o f w ater

n ex t to th e ja c k e t became coo ler th a n th e r e s t o f th e water* This

d if fe re n c e in tem pera tu re in a c ro s s -s e c tio n caused a d if fe re n c e

in d e n s ity to e x is t so th a t motion in th e tra n s v e rse se c tio n wae

induced* T his e f f e c t wae g re a te r in a wide anm lue* Ho* 2* th an

in a narrow annulus* Ho* ;$* s in ce th© maximum v e lo c ity fo r e t roam-

l in e flow was about h ig h er in th e narrow section*

38

G l a s s J a c k e t

C o r e

D y e T u b e

SECTION A -A SECTION B-B

- D y e - S t r e a m

SIDE VIEW FRONT VIEW

FIGURE NO. 3 - DYE SEPARATION PATTERN

359

l a a lm ost any work i n th e flaw o f f lu id s* th e re l a u su a lly

• p m l& t lo n concerning th e e x is te n c e o f a lam inar * film M o f f lu id

a t a * u rf i# i« In t h i s ap p ara tu s t h i s *film * was on do v is u a l by

d i r e c t in g a stream o f dya toward th e ao ra w all* I f t h i s was dona

w h ile th e f lu id waa moving* a vary g ro a t v e lo c ity g ra d ie n t became

v i s i b le and th e dya wae " s tre tc h e d " in to a th in line* w ith one

end rem aining alm ost s ta t io n a ry a t th e p o in t where I t f i r s t ap­

proached th e eore* Whan viewed from d i f f e r e n t p o sitio n s* t h i s l in e

seemed to stand away from th e wall« However* t h i s d ls ta n o e could

n e t be measured as such due to m ag n ifica tio n a t t r ib u ta b le to th e

d i s to r t io n o f th e w ater and (o r ) th e cu rv a tu re o f th e g la s s Ja c k e t.

I f th e dye tu b e was p laced so th a t th e dye emerged from th e tube

and clung to th e wall* i t s p resence on th e w all a t th e p o in t o f

emergence was observed even a f t e r th e flow o f dye had been stopped

fo r approxim ately two m inu tes. A ll o f th e dye d id n o t remain

m otionless* b u t t h a t p o rtio n on th e eore did* T his was p a r t i a l

v e r i f i c a t io n o f a fundam ental assum ption in th e development o f

th e th e o r e t ic a l laws fo r s tream lin e flow t t h a t th e v e lo c ity o f

th e f lu id a t a su rfa ce i s zero*

I f one o f th e dye tu b es was p laced in th e c e n tra l p o rtio n o f

th e anoulue and ano ther plaoed so t h a t th e dye was near th e core*

i t was seen th a t th e flow became tu rb u le n t In th e c e n tra l p o r tio n

w h ile th e flow was s t i l l lam inar a t th e core w a ll. The same

o b se rv a tio n was ob tained by p lac ing th e dye tube so t h a t th e dye

was near th e g la s s w a ll. For v a lu es o f g re a te r th a n

55OO in Anoulue Do* 5 s tre am lin e flow was no longer in d ic a te d ©i

th e g la s s w ell* Howevert t h i s d id n o t preclude th e e x is te n c e

o f th e * fil» * a t t h i s ra te# because th e th ic k n e ss o f th e "m m *

was probably le e e th a n th e w idth o f th e dye stream# thus# th e

behav io r o f th e dye stream was no longer in d ic a t iv e o f th e f lu id

behavior*

B* T e lo c ity Measurements*

The th e o r e t i c a l v a lu es fo r th e ra d iu s o f maximum v e lo c ity

and th e r a t i o o f th e average to th e maximum v e lo c ity fo r stream*

l in e flew in th e an n u li te s te d a re p resen ted In Table IT* These

va lues were ob ta ined by u se o f fiqu&tlons ( 21 ) and (24)*

TABLS IV

T h e o re tic a l P ro p e r t ie s o f th e S ee tlo n s Tested

Annolus No* ®2in*

£ to r m*x.In* Vmax.

1 1.55* •5*9 1.144 0.669

2 1.955 •549 O.969 0 .6 6 2

5 2 .156 .282 0.904 O.657

4 2 .156 .282 0 .9 0 4 0 .6 5 7

In F igu res 4# 5# 6# and 7# th e v e lo c ity d is t r ib u t io n s as

ob ta ined in th e s tre am lin e flow reg io n a re presented* In thee© d is ­

t r ib u t io n s # th e p o in t v e lo c i t i e s were expressed a s p ercen t o f th e

rtvm

,-«w

yvM

y-

. »

ton

Mgxi

mjm

Vel

ocit

y

.

-<j

> •

o

Q C

,

FIGURE NO. 5 * Velocity Distribution for Anniiilus No. 2

Reynolds N u m b e r - 8 5 0

R t . T r a v e r s e L e f t T r a v e r j s

C l

>

<40=2

Q_

I . f3 07105 2 3

R a d iu s o f p o i n t , in

43

6 - Velocity Distribution for Reynolds fslumber -:9 70 i j

FIGURE

O <*

3

: i I : .^O i I ;R a d iu s cjf Po n t^ i|i.

; 70

44

f j g Or e NO. 7 “ V e lo c i ty D isftri l i j ;

Reynolds N um l

bufion lfoi[ A n r tu lu s N b . 4

b e | r - 1255 P P

100

. 9 0 j |

ladiuis cf Poibtjln,l.-O l . ' iO

4?

observed maximum v e lo c ity * The th e o r e t ic a l d is t r ib u t io n s an

g iv en by com bination o f aquations (2 1 ), ( 2 J ) , and ( 24) e re

in d ie a te d by th e s o lid curves*

From th e se f ig u re s i t seems th e b e s t agreement w ith th e

th e o r e t ic a l v a lu es was ob ta ined w ith Annultts Ho# 1* tthon each

o f th e s e o b se rv a tio n s was made th e flow was seen to be stream ­

l in e th roughou t th e c ro ss se e tio n j th e maximum v e lo c i t ie s ob­

ta in e d w ith th e th re e dye j e t s were very nearly th e same, and

th e r a t i o s o f th e average v e lo c ity to th e maximum v e lo c ity were

approxim ately equal to th e th e o re t io a l values* In a l l v e lo c ity

d is t r ib u t io n s o b ta in ed , th e maximum v e lo c ity found by in je c t in g

th e ty e blob was u su a lly h ig h er th an th e maximum o f th e p o in t

v e lo c it ie s * The v e lo c i t i e s in th e o u te r p o rtio n o f th e se c tio n

a g ree more e le s e ly w ith th e th e o re t io a l curve th an th o se in th e

in n e r portion# The measured v e lo c ity probably was th a t o f th e

p o s i t io n o f th e o u ts id e o f th e dye tube and no t th a t o f th e

e en te r* lln e* A c o n s tan t s h i f t i n p o s i t io n o f 0*04 inches toward

th e co re w all would g ive much c lo s e r agreement w ith th e th e o r e t ic a l

carve*

The b e s t smooth curves were drawn through th e se experim ental

v e lo c i t i e s and th e r a t e o f flow ob ta ined by g rap h ica l in te g ra t io n

fo r comparison w ith th e t r u e flow in th e section* These va lues

a r e p resen ted below#

m

A nm lusNo* -r'

Average V elocity f t* /s e c *

In te g ra te d Actual

% Error o f In te g ra te d V elocity

1-R ig h t 820 0*0669 0.0661 1*21- L e f t 820 0*0607 0,0661 - 8*1

2 -R ight 850 0,0*05 0.04*9 - 9*02 -L e ft 85O 0*040$ 0*0445 - 9*0

5 970 0,0*4* 0,0551 —16*4

A 1255 0,059* 0*0670 - 17 .5

The d is t r ib u t io n s ob tained in a n m li 5 and A showed th e l a r g e s t

e r ro r In in te g ra te d flow va lue* • Theee annul! were formed w ith

th e same e ls e core* but Ho* J wee made w ith ga lvan ised iron# a

rough core and Ho* A w ith Copper pipe# a smooth core* The f a c t

t h a t both o f th e s e gave th e same g en era l v e lo c ity d is t r ib u t io n

showed th a t th e r e p ro d u c ib il i ty o f th e d a ta was good*

The d e v ia tio n s o f th e p o in t v e lo c i t ie s in th e in n e r p o rtio n

o f th e a n m li could r e s u l t from th e f a c t th a t th e two su rfaces

o f th e a n m li a re o f d is s im ila r m ateria ls* This i s no t sup­

ported* however* by th e f a c t th a t th e v e lo c ity d is t r ib u t io n s in

F igures 6 and 7 a re approxim ately th e setae w ith a smooth oore

rep resen ted in F igure 7 end a rough one in Figure 6*

I n F igures 8 and 9 th e r a t io o f th e average v e lo c ity in th e

se c tio n s o f th e maximum observed v e lo c ity I s p lo tte d with va lues

o f w - f a # <rhe p o in ts p resen ted were determ ined in -Re -7^

d e p e n d e n t ly a t th e corresponding Reynolds Number and do not

Aye

rpqq

V

eloc

ity

Mox

imum

V

eloc

ity

Ratio of Annulus

. FIGURE NO. 8 - city . to notion of

imum Veto Reynolds Ni

erageVelc I as a

Q Q Q Q Q c Pr \ O (

Streorfime -Flow

T~ " T

rI-

0 : 8 0

— - *

oO <?o 0 \o o 0° oS5 -' 1 u a • t?p- o -

I o °o O . 0~JTV& p 5 ~n Stream line flow"

o

©-4SM - <5-

— i

■ i—

8 0 0

o o00 1 o1O f o

i—Q-

O o jAnnulus No. 2

lOpO W 1 4 0 0"D .jV

1000r ~

1 8 0 0

m

FTGlifRE NO. 9 | Ratio of Average Velocity to MaxFmum Velocity In Annulji No. 2: S^3Reynolds Numb

A nnulusan ju^S tream line Flow

1000

CD

S5 5 C 2600 m ) 0

Fdprecen t check d e te rm in a tio n s made a t 0. constan t flow ra te*

For Annulus No* 2# thi© r a t i o waa approxim ately equal to th e

v a lu e fo r pu rely s tream lin e flow to ®R * 1100* Above thl®

va lu e o f Reynold© tim ber th e r a t io in creased ©lowly a© th e

flow r a t e wbb increased* The f i r s t appearance o f tu rb u len ce

i n th e ©action was th e cause fo r th ie break from th e s tream lin e

flow v a lu e and th e r a t i o in c reased as th e amount o f th e f lu id

i n tu rb u le n t motion increased* The maximum value which th ie

r a t i o w il l a t t a i n wae not determined# but from Rothfu®‘e d a ta

fo r an annulus o f C^/Dg * 0*162 th e va lue a t * 21 #600 was

0*826 and fo r Dj/B2 * 0*650 th e va lue a t * 14#400 was

0 *807* Theee va lues in d ic a te th a t th e l im itin g value o f th e

r a t i o In an an ru lue ie g re a te r th an th e value fo r c ir c u la r pipe*

For Annulus No* 5# th e p lo t o f t h i s r a t i o shows a g radual

in c re a se ever th e e n t i r e range in v es tig a ted # from * 1000 to

*R . * *700* Although s tream lin e flow was ob tained throughout

th e s e c tio n a t v a lu es o f NRe le s s th an l 400t t h i s r a t io was

always found to be h igher th an th e th e o re t io a l value* For th e

range o f Reynolds Numbers between 1000 and 1400 th e average ex­

perim enta l va lue was approxim ately 8$ h igher th an th e th e o re t ic a l

va lue fo r t h i s anrulue* The r a te o f in c re a se o f t h i s r a t i o w ith

th e Reynolds Number was not so g re a t as th a t fo r Annulus No* 2#

b u t was approxim ately th e same as th a t fo r Annulus No* 1# th e

v a lu es fo r whieh a re p resen ted in F igure No. 8 * For t h i s annulus

th e experim ental r a t i o o f th e v e lo c i t ie s i e again always h igher

50

th an th e th e o r e t ic a l v a lu es by approxim ately 5%*

I n F igu res 10 and 11 th e v a r ia t io n In p o in t v a lo c i t ie s

to r Annuli No* 2 and 5 w ith tho Reynolds Number based on th e

average v e lo o ity i s shown* These v e lo c i t i e s a re expressed as

percen tage o f th e average v e lo c ity , For a p o s it io n w ith a

v e lo c ity l e s s th an th e average ve lo c ity * t h i s r a t i o in creased

as th e flew r a t e Increased* and th e op p o site was t ru e fo r th e

p o s it io n s w ith v e lo c i t i e s g re a te r th an th e average ve loc ity*

Using th e s e v a lu es o f p o in t v e lo c it ie s * v e lo c ity d i s t r i ­

b u tio n curves w e e ob ta ined a t d i f f e r e n t Reynolds Numbers and

from these* th e v e lo c ity g ra d ie n ts a t d i f f e r e n t p o in ts in th e

s e c t io n were obtained* P lo ts were th e n made o f v e lo c ity g ra d ie n t

v e rsu s p o s i t io n in th e s e c tio n and th e se p lo ts ex tra p o la ted to th e

wall* o b ta in in g an approxim ation o f th e g ra d ie n t a t th e wall* The

v a lu e s ob ta ined by t h i s method fo r Annul! 2 and 5 a r © p resen ted

i n F igure No* 12* In t h i s f ig u re th e v e lo c ity g ra d ie n t a t th e

Reynolds Number.

The w all shearing s tre s s* *72 * i s g iven by th e product o f th e

v is c o s i ty and t h i s v e lo c ity g ra d ie n t and fo r s tream lin e flow* th e

r a t i o o f th e shear a t th e in n er w all to th e shear a t th e o u te r

w a ll in an annulus i s g iven by Equation ( 25)* For Annulus No* 5»

th e value o f t h i s r a t i o in s tream lin e flow i s 1*5&* The experi­

m ental r a t i o found from th e e x tra p o la tio n o f tho v e lo o ity g ra d ie n t

p lo ts v a ried from 1*60 a t s 1600 to 1*665 s 2^00*

was p lo tte d a g a in s t th e

Avg

V

teJa

clty

IS

*r = r.049

OrTO

r~h 4 0 6

poo

FIGURE W U H E ffe

22 0 0 n v m > u

Iip Reyjnoltis Numof tncrea on Point [Velocities. . L . ... .

ulus No. 3

53

FIGURE NO. 12“ Velocity Gradients at Walls as Functionsof Reynolds Number

0 . Annulus No.t,. Wbll

Li O ^ i j l u ^ N p . . j j

(Og “ D| ) Vjjyq 7 ^ —

5*

T his lncr«ae« in th e r a t i o in d ic a te d th a t th e shear s t r e s s a t

th e in n e r w all was in c re a s in g more ra p id ly th an th a t a t th e

e n te r w all in th e range o f flow covered* The only way th a t

t h i s could occur so th a t both o f th e shear s t r e s s e s would g iv e

th e s a mo p re ssu re drop was fo r th e p o in t o f maximum v e lo o ity

to s h i f t toward th e o u te r w all 00 th a t th e value o f Equation (25)

could be increased* I f t h i s p o in t o f maximum v e lo c ity s h if te d

toward th e o u te r w a ll, and i f th e v e lo c ity g ra d ie n t a t th e core

w a ll In c reased f a s te r th an th a t a t th e o u te r wall* i t was

necessary th a t tu rb u le n ce appear between th e co re w all and th e

p o in t o f maximum v e lo c ity be fo re i t appeared In th e o u ter p o rtio n

o f th e annulus*

Prom th e v a lu es o f th e w all shearing s t r e s s ob tained from

th e v e lo o ity g ra d ie n ts a t th e walls* th e von Karatam reduced

v e lo o ity and p o s it io n term s g iven in Equations (6 ) and (9 ) were

evaluated fo r Annuli Ho* 1 and 5* In f ig u re 15 th e values o f u +

and y a re p lo tte d * Some o f th e se va lues were ob tained from

th e v e lo c ity d is t r ib u t io n de te rm ina tion p resen ted in F igure Ho* 4 .

In th e se th e v e lo c ity g ra d ie n t a t th e w all was assumed to be th a t

which would apply in th e s tream lin e flow condition* Tho o th e r

p o in ts in F igure 15 were ob tained from th e l a s t observed stream r

l in e flow c o n d itio n s and th e f i r s t observed tu rb u le n t flow con­

d i t io n s in th e section* These experim ental po in ts do no t fo llow

th e equations o f Kikuradee ae g iven In Equations ( 14}» ( 15) and

(1 6 )# b u t in d ic a te l in e s p a ra l le l to th ese expressions* The

2 0

10-

FIGURE NO. 13- 1. —

r '^’ 1 ^7 TI' r*fI .1 >i • I '" i r ‘’iM'w't’ 4(Vftr|», i ,1.1 I

J .I ' '■ \ - l I ,f* ... JT* IrT

i d Velocity Distribution for Annulua No.(

j ■ -l-t U*[m |

■ UlU

.* i**--* i.1. -„,u4

: t

f f

|-4 !:j ;" I 'o 'L o i t Observed Stream line Flow • ■

4 First Observed Turbulent Fiow ^

J+ Velocity Dtptrfbotieniot+ N*. *820 ~- NIkuradse*e 1 V6lu#s, : . ' * i

* *— i Experijiseh foi

t

•' t-t - * ■ f", . J . . . . . 4.444'■ i !■ 1 ■!- j . . . . • — - »+»tT+.

4 ,*4 : ..i ^ .j..,r L ^ • *-j........ • U> - 4 .-+

. . i

i-i- 4-

~l____

4, - I ---4------

;-H - <~H....-i - . - , 4 u_. . . . . ^T , . 4- . 4 : .

4-^...^4!- j•j-’j

j —J - --+ -f—; ji. . u J . 4 -^ .4

. . , . . - 4- . . .

.4 4 i'..!.-4 fc~ n .7 : ± ~ 4 ± :

. l - U

• t-

It* -i ..--- t - . i—........................

g h a r r i

■’ r

t i• - - • - -- • '4 rf 4*. *■ -*{"f**+-r*,*7**4 -ft""!

. 44-44 !4

i •

4 -

* — |_i LI j. J! 0"rjliZ4 4 — 1 , ! I , . I’1:’!''!!* t , '■r*-**' I" - '*-+ - t* **“rt*

* -|—* - - 4 - ! ^ — t - . . ' i 1 « i ■ »f i >u.—|*. +*it. u* _ |u .^4- . ..|.a i p. 4 -

t r~l- ■t- 4

Jrtr4:

4 •H ■

’T■ * TTT i t -

4 -4- .44i> Ur j— i)<i

-■-’— +- .- I ■ * r-y * r* - * r * - • T { • ■ - - • , ■ - - 4 - - -+ —,-4-+ - —.|- ‘ t r--.— j-+- *'4 yU4-.-. ■■-- +-U."*-T"' •-*

v tTTtrt t" LiLitii*:Mi

3 ^ . ; 4 0

ui_

aio»

96

r a m i t e in F ig u re No. 1J e re rep re sen ted by th e follow ing

equation©:

u f - / / .& /o j/a y f - 4 - 2

~ 7 *> S - ' ■ * (a6J

and

e /+ * 4 . 4 U jto y + / 3 - 9

g + '5 -5' (gp)

I n F igu re No* 14, th e Base e o r l o f p lo t was mad® fo r Annulus

No* % and th® .equation expressing th o se r e s u l t s la s

u * - / £ / < y / o -<*• 7

(50 )

Moat o f theco de te rm ina tlone war® mad® w hile th® flow a t th®

p o in t o f o b se rv a tio n was i n s tream lin e motion, b u t non® o f th®

experim ental va lues corresponded to th e lam inar reg io n in whioh

u ^ * y ^ aeeordlng to Niteur&dse* On c lo se r exam ination o f t h i s4~ 4r e la t io n s h ip , however, i t i s seen th a t fo r u to equal y # I t

i s zteoeseary th a t th e lam inar la y e r be so th in th a t th e v e lo c ity

a t y i e oqual to th e d is ta n c e from th e boundary m u ltip lied by th e

v o lo e ity g ra d ie n t a t th e wall* For th e flow co n d itio n s In t h i s

work, th e th ic k n e ss o f th e lam inar lay e r was g re a te r th an t h i s

a t a l l tim es*

.

' + * * * t t

— rFIGURE NO. 141 von Kormnn

* ............ * I 1 1 ' “ * ■ i t- t I: ...............

> in sj a> © 0- r—i*t—i—t • t" ! mi "| '■ • iH"i|T"fr:vri:ri»!iTM

- - 4i ,

• •" ■ ' • • ■ ^ 4 - r i -4.. k . j .*4*4 t, - i ■ < , ♦ , - t . ■ « » 1 ‘

,..) . . u , . . . .

j . . 4 • - I .................

. . . , j . . . .

. 1 1 . . , . ll

1 1 1 . 1 1

- * • ' T - t

■ 1 ............ - ]•>•■■■ ' 1 ........ 1 •■■!■■ '

. . . ' 4 4 : i

. . . . . . . . . .. . .. i . .. .. : j

• 1 • •*•• •• •••.. ~ L,--.j---- .... . 4.

Velocity Distributibn far:An rruluis :i* f - + r- 1 * t I *-*-1 ' *■■■ • ‘ ■ H*-*.

, , . L. j.«a. . , . 4 i-i -.4 44 .■F«.*.-v

4_..; . - i .

m Tir*<tr

-r - 4*fv* 4*4

a.-T

■fr

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I . . . i . ,

-20-—

*' UoM Observed*' :....................... ’ ^ ..J . 4 ►.

; Velocity D istribution o t ;N ^ -9 6 jO

rwlkurodse’s Valued Experim ental

■ t

f . . . . . 4- ^ - : j . . . i f . . . , -----

3 E H H

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- - 1 ’ -$► -40 20 sp :::ir^ (c u 3

N lkuradse’e ex p ress io n fo r th e b u ffe r layer ie*

U f - / / ^ % /0 <j 3 . o,j5"

( 15)

aad f a r th e tu rb u le n t p o rtio n

U r ~ —>• /&).\ o ^(1<S)

S ince h ie v a lu es wore ob tained a t extrem ely high Reynolds Numbers

fo r flow in o iro u la r pipes* i t was expected th a t th e re would be

b o me d if fe re n o e In N ik u rad se 's va lues and th o b o ob tained in t h le

work w ith flew o f low Reynold® Humbere in a n m li*

C* O r i t le a l V alues.

The c r i t i c a l flow co n d itio n s were ob tained a t varioue p o e itio n e

w ith in each aaaulus* and th e r e e u l te o f th e se f in d in g s a re p resen ted

i n T ables V, VI* end V II. In T ables V-a* V I-a, and VII~a th e va lues

a re p resen ted fo r th e l a s t observed s tream lin e flow conditions* and

in T ables V-b* VI-b* and V ll-b th e va lues fo r th e f i r s t observed

tu rb u le n t flow c o n d itio n s a re presented*

When th e c r i t i c a l v a lu es o f (pg ~ wer e p lo tte d a t th e

a p p ro p r ia te p o s it io n s in th e annul!* expressed as th e rad iu s o f th e

p o in t from th e c en te r o f th e core* th e curves In F igures 15* 16 and

17 re su lte d * There was co n sid erab le sc a tte r in g o f p o in ts In th e se

f ig u re s and no r e a l c o r re la t io n was obtained* Several g en era l ob­

se rv a tio n s could be made on th e b a s is o f th ese curves* however*

Basalts for Aanulns Bo. 1 Points lnslds r„ , last obsorTtd streamline flow conditions

59

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Point s out old# of t9§x • la s t ©bssrrsd s tr s s a lla s flow ooudltlouo

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i ’ d i j l a s *«• f t l l i T d M ltf f t./M O . f t " 1

1.1*22 .0065 •7*1 .0979 .930 1180 .0468 5.170 33.6

1.1*00 .00*33 .720 .104 1.027 11*5 .0935 9.230 76.8.0*75 .940 1050 .05*5 6.180 51.5

1. 3*1 .00991 .666 .1016 1.027 1U 5 .082 2.070 80.0

1.360 .01166 .607 .112 .990 1275 .103 10,400 121.5.108 1.038 1170 .099 10,900 122.2.0*75 .9$* 10*0 .0794 8,250 96.4

1.325 .0146 .50* 4 0 6 1.004 1190 .125 12.490 1*7

1.320 .0150 .*195 .1265 1.003 1*20 .1292 12,620 189

1.295 .01708 .1*21* .1265 1.045 13«5 .139 13.360 228a o s 1.023 1190 .131 12.730 217.5

1.232 .0223 .250 .11*5 1.045 1274 .140 13.220 295.104 .97* 1200 .125 12,760 2*4.5.1205 1.092 1360 •13*5 13.5®J 302.1205 1.030 1320 .147 14,080 5 4

1.169 .0276 .0703 .1160 1.03 1295 .1342 13,000 359.114 1310 .139 14,199.108 1.01b 1200 .1351 13,280 366.5

1. 1*5 .0296 0 .0935 1.002 1050 .1 % 12.790 37*.0935 1.002 1050 .119 11,880 352.108 1.000 1210 .1328 13,260 392

ttS&S vo. ?-»

ft# s ti lts fo r tiw fln s Vo. 1 P o in ts in sld o » f i r s t ofc#«rr#4 tn fb n lo n t flow co n d itio n s

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.917 .00723 .707 .108 .990 1160 .089 8.800 78.9

.951 .01067 .601 .102 .990 1160 .090 8.900 9M

.973 .0125 .533 .1015 .935 1225 .llU 12,200 152.5

. 9** .0 l3 te .*•99 .1185 1.0>*5 1280 .102 10,500 if c

l.O lH .0159 .*•05 .1185 .978 1365 .13S 13.900 221

1.029 .01718 .358 .0975 .935 U 75 .119 12,690 a s

l.O l^ .0185 .308 .116.1185

1.03.990

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1.096 .02275 .1*195 .106 1.03 1160 .li(2 13,200 300.5

1.110 .0239 .1059 .1225.1285.l l6.1185

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S.dlU8 of »int* Is*

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Averse*▼eleoityf t . / s e e .

V * i o 5f t* /* # * . y

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l . t2 2 .0069 j n -------

i . to o . 008J3 .7» .110.0975

1.027. 9*10

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.0715 J ’pO79.563

1 . 3& .00991 .666 .106 1.02 1172 .081) 8.300 82. 1)

1.3*0 .01166 .607 .1189 . 9*3 1360 .109 11,080 129.5

.0915 . 9** 1090 .082 8,630 100.3

1.325 .011)6 .50* .112 1.003 1260 .132 13,000 195

1.320 .0150 . 1)95 .1305 1.003 1*70 .13* 13,200 19*

1.295 .01708 .1)21) .1305ail*1.0381.023

11)201255

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21)6 .5226

1.232 .0223 .250 .1205.110.121)5.1225

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1.169 .0276 .0703 .1275.1185.111)

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TABUS KO. TI-B

Raaulta fo r Aumli» Vo* 2 Value* for f i r td oBoorrod turMLaat flow condition*

Hadlne a t F o la t, la .

yf t .

r - r_

*2-r»toreraceV elocityf t . /e e o .

V * io 5f*2/e e e .

*b Z“V Tm *

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U t Pwlftt V elocity f t./» e c * f t " 1

S t y

1.390 .00917 .79* .0820 .1(5 15*0

l .» 5 . 0179* .595 .0965 .886 1770 .0SB7 9.580 172

1.22 .0233* .*73 .093 .886 1710 .07*6 8 ,*20 19^.5

1.165 .0279 .370 .090 .880 1665 .0909 10,320 288

1.105 .03295 .258 .090 .875 1675 .096 10,980 362

1 . 0* .0383 .135 .0605 .835 1180

.9« .0*33 .0226 .0785 .865 1*80 .102 11,800 511

.97 .oWa .00

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1205103012551280

.0775

.0807

.0926

9.0709.570

10,320

*00*22*55

.92 .0331 .108 .0625 .863 1175

.8 6 .02805 .2*3 .0870.0950

.875

.875l €201770

.0925

.10010,380 11,*20

296.5320

.78 .o a * .*22 .0655 .897 1190 .0833 9.280 198.5

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TAKJ VO, TXX-n

I n tu its fo r Aa&ului Vo* 3

Points outoido *Bax,« Inst oboorrod B trtw a lla t f lo v condition*

Badiu* of Point* in*

y« .

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M N l | i

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f t 8/*#*.< V V \ „ «, Point

T *loeitjr ft./*** .

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i.Hoo .00833 .822 .1525 . 9ta 2910 .0595 6.320 52.6

1.560 .01166 .765 .lW *.1135

.966

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.05507,5005,800 67.6

1.320 .015 .698 .1523.1601

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28803120

.118•139

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186.5226.5

1.262 .01962 .601 .1H>*5 .930 2790 .123** 13,280 260

1.192 .0257 .*83 . 9*»«.958

25002705

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.12111,60012,620

298325

1.110 .0325 . 3*6 .1275. i *&5 :JS

21*302630

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1*1(9525

1 . 0!»5 .0379 .2365 .l lH . 9*a 2175 .139 1U.7S0 560

.96U .0^3 .0738 .0900.0985

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tooolto for Aonalms Yo« 3 Potato inoldo r„ $ first ofeoorrod tor talent flow condition*

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Results for Annul us Ho. 3 Points outsld# r _ • f i r s t observed turbulsnt flow conditionsBOX.

Radius of Point* in.

yf t .

Average Velocity f t./se c .

r 1 io 3f t /••« .

{®2"®l^Vf(j n , Point Velocity ft./ooc . f t"1 ¥

1.1(00 .00*33 ,822 .1605 . 9W 3000 .0626 6,600 55

1.3*0 .01166 .765 .1523 .966 2800 .077»* 8,010 93.1290 . 9<* 21150 .0595 6,290 73,

1.320 .015 .698 .16s •9W 3160 .11138 13,200 228.168 .920 321*0 .11(2 15 , 1(1(0 232

1.262 .01962 .601 .1525 .930 2900 .1312 ll(,100 277

1.192 .0257 M l .1366 . 9>(* 2565 .1150 12,11(0 312.1525 . 95* 2810 .128 13.300 3U2

1.110 .0365 .31*6 .13*5 •9& 2285 .11(7 15,600 507.1H55 .966 2685 .160 lb ,500 537

1.0U5 .0379 .2365 .1 3 5 . 9*1 2285 . 1>*7 15,600 591

. 98U .0U3 . 073* .0950 . 9la 1800 .122 12,960 If7.1072 .881 2160 .137 15.550 669

.915 .01*88 .0185 .1305 .95*1 2U25 . i p 17,800 S69.121(5 , 97« 2260 .161 l6,l*00 801

CNOo

tical Average Reynold s Numbers I in Annulus Nal

al PointsNO. 15-OrFIGURE

O La9t Observed Streamlinie Flow

• First Oosurved Turbulenl Flow

°\>a

!3 08 2 3 1.20- m ax

RadLusLai RointT in

O L ost Observed Sitreomlioej Flow

• F irs t Observed Turbulent Flow

2 0

R o d iu a _ o f Pointy u l_

Numbers ot Pointsticol Averope Reynold in Annulus No.2

FIGURE m Jfir_Cr.

71

:

: i rlGU1 RE NO tiqal Pbints

A v q ra je Anhu

lo w s N u n

-irstT 3403

8 0 €>

L4 0 0

Rad us

72

In F igu re No* 15# th e s e c r i t i c a l value*® fo r Anm lua ft0* 1 a rc

p resen ted* In t h i s f ig u re th e re i s no c le a r d i f f e r e n t ia t io n between

th e c r i t i e a l v a lu es fo r each point# t h a t i e th e flow became tu rb u -

l e n t th roughou t th e p o r t io n o f th e annulus s tu d ied a t approxim ately

the same v a lu e o f (Dg * There was some In d ic a tio n

t h a t th e minimum c r i t i c a l v a lu e was ob ta ined a t th e p o in t o f maxi-

tau Telocity# a v a lu e o f ll^O* The r^nge o f c r i t i c a l v a lu es o f

th e Reynolds Number fo r th e 70$ o f th e annulus w idth covered in

t h i s in v e s t ig a t io n was from ll^O to 1J50*

In F igu re No* 16# th e c r i t i e a l v a lu es a t v a rio u s p o s it io n s in

Arsaulua No* 2 a re presented* In t h i s s e c tio n th e minimum c r i t i c a l

Reynolds Number was observed approxim ately a t th e p o in t o f maximum

v e lo c ity * C r i t i c a l v a lu es in t h i s s e c tio n ranged from 14^0 to

1700 fo r th e co n te r y?% o f th e annulus in v e s tig a te d * Turbulence

was f i r s t observed a t th e p o in t o f maximum v e lo c ity and as th e

flew r a t e was increased# t h i s tu rb u le n ce spread toward th e w a lls

u n t i l th e flow a t p o in ts near th e w a lls became tu rb u len t*

The most thorough In v e s t ig a tio n was made o f Annulus No» 5

and th e r e s u l t s ob ta ined w ith t h i s se c tio n o re p resen ted in

F ig u re No* 17* These r e s u l t s in d ic a te d th a t th e ;ninlmum c r i t i c a l

v a lu e o f th e Reynolds Number d id not occur a t th e p o in t o f maximum

v e lo c ity # b u t occurred a t a p o in t approxim ately 0 .25 inch relieved

toward th e co re w all from th e p o in t o f oaxinttm ve lo c ity * Approx?*

ia a te ly of th e annulus w idth was covered in t h i s in v e s tig a tio n

and th e rang© o f th e c r i t i c a l Reynolds Numbers over th ic w idth was

79

f3roa to 9900* The c r i t i c a l va lu es in d ic a te d on th e g la s s

w e ll wore o b ta in ed by tu rn in g th e dye tube so t h a t th e dye w a s

in je c te d d i r e c t ly on th e g la s s and th e flow was i nor eased u n t i l

s tre a m lin e flow no longer ex is ted* The p o s it io n o f th e observed

stream wee Indeterm inate# b u t i t was approxim ately a t th e

w ell*

V alues o f were ob ta ined fo r each p o in t by

assuming t h a t th e c r i t i c a l va lue lay halfway between th e va lu es

f a r th e l a s t observed s tre am lin e flow and th e f i r s t observed

tu rb u len t flew cond itions# and th e s e were th en p lo tte d a g a in s t

position In th e axxnuli* F igu res 16# 19# and 20 re su lte d * There

did n e t seem to be a very good c o r re la t io n in these* I f th e

Indicated curves were e x tra p o la te d to th e wall# t h i s should

give th e v a lu e o f C u / V J ^ j ^ a t which th e com plete se c tio n

would be In tu rb u le n t flew* A ctua lly th e re w il l always be a

la y e r on th e w all which does n o t move# u n le s s s l ip occurs# so

t h i s v a lu e w i l l have to apply to a p o in t a d i f f e r e n t i a l d is ta n c e

away from th e boundary* From th e p lo ts o f th e se c r i t i c a l v a lu es

a g a in s t th e d is ta n c e from a w all on re c ta n g u la r coordinates# i t

was found th a t th e v a lu es ob ta ined by e x tra p o la tio n o f th e curves

to y ® 0 were p o s i t iv e fo r Annuli 2 and 9# bu t t h i s value fo r

Anoulus no* 1 was negative* P lo ttin g o f th e c r i t i c a l va lues fo r

A nm lus Ko* 3 was t r i e d on rec tan g u lar# lo g arith m ic and semi-

lo g a rith m ic co o rd in a te systems# b u t in a l l systems th© values

e s ta b lis h e d a curve and e x tra p o la tio n o f th ese curves could no t

be done w ithou t in tro d u c in g erro r*

74

VgluoSqf M/if at; Poirits Annulus No. I :

P o in ts Oiutside ' r (max

p o in t s In s l iM % 0J(

75

EKUBE MO. LSi (irliicc I PointsAnnuls

10,00 0

4>_ _ .P o ih K Qwt£ixLe_ I max—# P o in t s ' lnsiideN rmax

76

FIGURE NQ Vqlufes o f in :Annulus No. 3

10,000

O Points • P o in ts

TSS

► ^ i -r + ~ pII I , I !

+ n rrtt“- i m ;

& «m m ties i of

i~; u i i t ; :I 4 4-+ -* !■

f- H • f-

W

I t+i H -

**--4 -*—4—t- «~f *

k i m i i t i s

, | |_pi4

m & m• lArinO lus A . Annuiui

— IrCfcOOO rvnul u s

IO:OQG

5 ;Q0 0

1

FIGURE N 0 .22 - Critical Values of 4^ of Pointsin Annuli I S 3

19

To o o rre c t fo r p o tltlo A i th e c r i t i c a l r a t io s o f u /y wero

n w lt ip l ie d by y# converting th e s e v a lu es in to c r i t i c a l va lues

o f a Reynolds Number* in t h i a ease a p o in t Reynolds Number*

L ogarithm ic p lo ta ware made o f th e se p o in t c r i t i c a l Reynolds

a g a in e t th e d ia ta n e e from a w all and a re shown in

25* 24 and 25* R e su lts o f th e se p lo ts showed th a t p o in t

o r i t i e a l Reynolds Numbers could be expressed as fu n c tio n s o f

th e d is ta n c e from a boundary a s i

fo r Annulus No* 1* (yu/V )e r^ # = <59#000 y**^5 ( 51)

f o r A m ilu s No* 2, ( ^ w V ^ e r i t £ 15*710 y 1*12 (52)

aa4 fop Anmlaa So. J , (y w /V ) e r l t = 57.700 y 1* ^ (5 5 )

The c o n s ta n ts in th e se equations were ob tained by th e method o f

l e a s t squares* At f i r s t se p a ra te equa tions were determ ined fo r

th e p o in ts In th e se c tio n which lay between th e p o in t o f maximum

v e lo c ity and th e g la s s w all and fo r th e p o in ts which lay between

th e p o in t o f maximum v e lo c ity and th e so re wall* bu t In S ec tions

1 and 5» i t was found th a t th e se equations were very n early th e

same* Therefore* a l l o f th e va lues were used to o b ta in th e b e s t

l i n e through th e values*

These f in d in g s a re s ig n i f ic a n t In th a t they may be used to

determ ine th e th io k n esc o f th e lam inar lay e r fo r oond itions in

which t h i s la y e r i s aliaoet d i f f e r e n t i a l in th ickness* I f th e se

equations can be app lied to th e flow cond itions in which t h i s

d i f f e r e n t i a l th ic k n e ss i s p resen t, th en th e th ick n ess can be

c a lc u la te d i f th e p ressu re drop due to f r i c t io n i s known* This

80

FIGURE NO. 2 3 ~ Critical Point Reynolds Number as a Functionof Position in Annulus No.I

3 4 5 8 7 8 3 'I©y , f t .

81FIGURE NO. 2 4 “ Critical Point Reynolds Number as a Function

of Position in Annulus No.2

FIGURE NO. 25-Critical Point Reynolds Number as a Functionof Position in Annulus No.3

85

may be i l lu s tr a te d as fellows*

l a th is laminar lay e r, i t la assumed th a t the velocity a t

a point la given by u » y(dn/dy)y g For the genera l ease# th e o r i t l e a l value o f the point Reynolds Kumber w ill be g iven

( y « / y * A y n ( 54)

te b a ti tu tin g and sim plifying,

ya-n s A ^ / ( i n / i , )y • 0 ( 59)

Aa long as a lan inar layer ex la ta , the shear s tre ss a t the outer

v a i l i s given aa,

■a - rJ )^ ( W J r L * A p (" z~ y ^ 2 L Ra (5«)

Solving fo r (4u/djr)gj ,

(du/dy) _ s ^(57)

S u b s t itu t io n o f E quation 57 in Equation 555 g iv e s ,

_2~n _ g ' V R2“ 4 r e (

s /) /c f ig y

A | - E ( V “ ? ) ( 56)

Bquetion (58} w il l epply only i f Equation® ( 51) * (55 ) apply to

th e s e a t ions when th e flow reechos th e condition® in which th e

la y e r in s tre am lin e flow i s d i f f e r e n t i a l in th idcnoss*

Rouse* s s t a b i l i t y param eter, , was evaluated fo r th e

c r i t i e a l flow c o n d itio n s in Annulus ^o* 1 and th ese r e s u l t s a re

84

p resen ted I n F igure No* 26* In th ie f ig u re th e value o f jL was

p lo tte d a g a in s t p o s i t io n in th e pipe expressed as d is ta n c e from

a w all* The c r i t i c a l v a lu e o f t h i s param eter was not co n stan t

i n a s e c t io n and th e h ig h e s t va lues o f i t were not ob tained a t

th e p o s it io n s in which tu rb u len ce was f i r s t d e te s ted ) th e re fo re ,

i t s u se as a s t a b i l i t y param eter does no t seem appropria te* I t

was n o t p o s s ib le to e v a lu a te t h i s param eter a t e i th e r th e p o in t

o f "tffyiisum v e lo c ity o r a t th e w a lls , s in ce th e v a lu e a t both o f

th e s e p o in ts was n e o e ssa r ily z e ro . The maximum value o f ^ was

found to occur a t approxim ately th e same p o in t a s was p red ic ted

by th e u se o f th e s tream lin e flow re la tio n sh ip s* d l l in a l l ,

i s n o t considered a p p ro p ria te fo r u se in t h i s app lloa tion*

85

d \jaliie$ cf X of Poir in Anqulps No.f j ]

FIGURElNO. |26 " |Cifiti

4 0 0

1----50]

OKAiTKR V

CONCLUSIONS k m BKOOMI SJSiliATIOi S

Based on t h i s work* th e fo llow ing conclusions worn drawn*

(1 ) fho t r a n s i t i o n from s tre am lin e to tu rb u le n t flow In

on a n n ilu i occurred In euoh a manner th a t p o rt o f th e f lu id

rem ained in e t r e a a l ln e m otion w hile th e rem ainder o f th e f lu id

became tu rb u le n t* Thie tu rb u len ce o r ig in a te d a t th e p o in t o f

maxims* v e lo c ity f o r two o f th e e ee tlo n e tooted# b u t fo r

Annulus No* 5* i t o r ig in a te d a t a p o in t removed approxim ately

0*25 in ch toward th e eo re from th e p o in t o f maximum ve lo c ity *

The amount o f f lu id which remain* s tre am lin e was decreased as

th e flew r a t e was ino reased and f i n a l ly a flow r a t e was reached

a t which th e flow i n th e e n t i r e section# except fo r a very em ail

amount near th e w alls# was tu rb u len t*

( 2 ) T h is t r a n s i t i o n f i r s t m anifested i t s e l f by im parting

a wavy m otion to th e f lu id * At f i r s t t h i s motion was a slow

s im o u e motion# bu t i t s wave len g th decreased as th e flow r a t e

was increased u n t i l a fu r th e r in c re a se in flow r a t e r e s u l te d in

irregu l® 1* w iggles and tu rb u le n t bu rsts*

(5 ) The minimum c r i t i c a l Reynold* Number# (Dg - D1 ) Vavfi>»

v a rie d w ith th e width o f th e annulus# In t h i s InveeS ga tiO tt I t

was found th a t t h i s va lue wae lower fo r th e wider an rn li*

Table V III p re se n ts a summary o f th e se findings*

66

87

TABUS n x I

Summary o f C r i t i c a l Reynold* Numbera

AnzulueNumber

V » iin ch es

Minimum O r i t lc a l Reynolde Number

Range o f O r i t le a l Reynolds Numbers

% o f Width o f Annulus S tudied

1 1. 55* 1150 U 50-1250 70

2 1 .955 1450 1450-1700 55

5 2.156 1*50 1450-5500 85

The le n g th o f th e t r a n s i t i o n reg io n was eeen to vary w ith th e w idth

o f th e ammlus* also* The wider annulus had th e la rg e r range o f

Reynolds lumbers b e fo re a l l o f th e flew in th e s e c tio n was tu rb u len t*

(A) The c r i t i c a l v a lu e o f th e p o in t Reynolds Number* y u /-!^ *

was found to vary w ith p o s it io n in a g iven annulue and t h i s v a r ia ­

t io n was expressed as*

fo r Anmilus No* 1 ^ ^ ^ ^ e r i t ~

fo r Annulus No* 2 (y u /V ) oy^ ~ 15# 710

fo r Annulus No* 5 ^ T ^ ^ J e r i t S 57*7^0

The s ig n if io a n e e o f th e se f in d in g s l i e s in t h e i r a p p lic a tio n

to o b ta in th e th ic k n e ss o f th e lam inar "file** a t high r a te s o f

flow in th e s e c tio n s . I t w ill be necessary to o b ta in p re ssu re

drop measurements however* before th e s e th ick n esses can be oal—

culated*

67

TABLE V III

^amraary o f C r i t i c a l Reynolds Numbers

AnnulusNumber

V ® iinches

Minimum Oritical Reynolds Humber

Range of C ritical Reynolds Numbers

% of Width of Annulus Studied

1 1. 55* 1150 1150-1550 70

2 1 .955 1*50 1*50-1700 55

5 2*156 1^50 1*50-5300 85

The le n g th o f th e t r a n e i t io n reg io n was seen to vary w ith th e w idth

o f th e e n m lu i i a lso* The wider annulus had th e la rg e r range o f

Reynolds lumbers b e fo re a l l o f th e flow in th e aeo tio n was tu rb u len t*

(* ) The c r i t i c a l v a lu e o f th e p o in t Reynolds Number, yWV^ $

vac fbund to vary w ith p o s it io n in a g iven annulue and t h i s v a r ia ­

t io n was expressed a s ,

fo r Annulus No# 1 ^ ^ ^ ^ c r l t * JT*#^

fo r Anmlue No# 2 ^ o r i i * y l* l2

fo r Annulus No# 3 ^ jn a /^ ^ c r i t s 57#7G°

The e ig n if ie a n c e o f th e se f in d in g s l i e s in t h e i r a p p lic a tio n

to o b ta in th e th ic k n e ss o f th e lam inar "film* a t h igh r a te s o f

flow in th e s e c t io n s . I t w il l be necessary to o b ta in p re ssu re

drop measurements however, befo re th e se th ick n esses can be Cal­

culated#

88

( 5 ) T his method o f v is u a l o b se rv a tio n has many advantages

ever e th e r methods fo r determ ining c r i t i c a l flow co n d itio n s In

any ty p e o f flow emotion* I t p re se n ts th e only d i r e c t method

o f d e te rmi ni ng th e amount o f a f lu id stream which e x is ts in

tu rb u le n t flow* The method o f d e te rm ln ir^ v e lo c i t i e s l a e n tire *

ly s u i ta b le as long as th e f lu id rem ains I n s tre am lin e flow* but

can only be used to e b ta la maximum v e lo c i t i e s fo r th e p o rtio n

e f th e f lu id whieh i s i n tu rb u le n t flow*

(6 ) I t was concluded t h a t th e s t a b i l i t y param eter p resen ted

by House was n o t s u i ta b le to determ ine th e p o s it io n a t which

tu rb u le n c e would f i r s t appear in a section*

I t i s recommended t h a t fu r th e r study be made e f t h i s problem*

w ith p a r t ic u la r eiaphaela on th e e f f e c t e f roughness on th e c r i t i c a l

flow c o n d itio n s . I f a s u i ta b le sderoaanem eter were constructed*

i t would be p o s s ib le to o b ta in th e p re ssu re drop along w ith th e

v is u a l d e te rm in a tio n s and thus* more comprehensive d a ta would be

o b ta in e d . T his would a id In comparison o f th e w all shearing

s t r e s s e s as c a lc u la te d from th e v e lo c ity g ra d ie n ts and as ob ta ined

from th e p re s su re drop m easurements.

I f i t i s possib le* i t would be in te r e s t in g to arrange th e

system fo r th e use o f a more v iscous f lu id than w ater. This could

be done by i n s t a l l a t i o n o f a pump to r e tu rn th e f lu id to th e head

tsxdc. Blnee th e re a re now a v a ila b le commercial products which

can be d isso lv ed In sm all q u a n t i t ie s in water to in c re ase th e

v is c o s i ty g re a t ly , t h i s should be p o s s ib le .

APPENDIX I

BIBLIOGRAPHY

R eferenceNumber

1 A therton , D* H*f F lu id Flow in P ipes o f AnnularCross S ec tio n , T ran sac tio n s o f th e Amor loan 8«ol« iy « f M>ch»nleal Engln<>«rBT*xtV III.1*5-162 U 92ST.

2 B ardsley , C larence Edward, H is to r ic a l Resume o f th eDevelopment o f th e Science o f ^ y d rau liee , Oklahoma A H College* P i v is io n o f Engineer" ing P u b lic a tio n s , P u b lica tio n Ho* 59* IX* l A p r i l l ? ^ ) *

5 Becker, E ra s t , Stromungsrorgange in ringform lgenA palten ( la b y rin th d lo h tu n g en ), S e i te c h r l f t dee Y orelnee deuteoher Ingonlaura* £ l ,11 l y i i h i (1907)*

4 B lae iue , H» * Phyalkallaohe Z e ita c h rif t* XXI, 1175 ( 1911)<

5 B inn ie , A. M * , A D ouble-R efractive Method o f DeterminingTurbulence in L iqu id s, Proceedings o f ifo P hysica l Society ( London)* Lvil* wO^HSz

6 B innie, A* M«, and Bowen, E* J* , A Method fo r MakingS tream lines Momentarily V is ib le , Proceedings o f th e Cambridge P h ilo soph ica l Society*XXXVII, 4 5^457 (lp jfl)*

7 Bond, W. N ., T urbulen t Flow through P ip es, Proceedingeo f th e Physical Society (London), XLIII-1S3- ? ? (1951).

8 Brown, George G*, and A sso c ia te s , U nit O perations,John Wiley and Sons, Ino*, Now York, 19?0 *Pp* x i i / 611*

9 C aldw ell, J* , Jou rna l o f th e Royal Technical C ollege( G lasgow ),~ I I , 265 ( 1950)•

10 C arpen ter, F* G*, Colburn, A* P*, Schoenborn, E. M«,and W urster, A*, Heet T ransfer and th eF r ic t io n o f Water in an Annular Spaoe, T ran sac tio n s o f th e American I n s t i tu t e of Chemical 'E ngineers, XLII, 1$5“ 1®7 ( 1940/7

89

90

Refer©*©#^Washer

11

12

1?

14

15

16

1?

16

19

20

Cornish# R. j»# R trbuU nt Flow through Fine Eccentric

* s o a a u m w R M i wDelssler# Robert <$«# Analytical and Experimental

Investigation of Adiabatic Turbulent Flowin hwtk jgaugaU.ableac asmiuss

Fage# R* and Townend# H# 0* H## An Examination of Turbulent Flow with an Ullra^Mlefoeoope#

Soeletar at £ojgefi,

Fea*el# Thomoo A«t Visual Observation©f C ritical Reynold© Somber* in Shapes other than Oirouler, M.8.Tb»»l». LwrtgUn* Stoti OntT»r«lty, l w *

Gibson# A. H*# The Breakdown of Streamline Motion a t the Higher C rltloal Velocity in Pipes of Circular Croce Section# Philosophicals£ Ig M te'OT»

Rllnkenberg# A. and Mbey# H* H«» Dlmenslonles* Groups In Fluid Friction# Heat and Material Tramfwr, OhwAoal E nglaw lm Pr.SS.ttta*3 JY , 17- j i t ( 19W ).

Knudsen# Jernes G.# Heat Transfer# Friction and Velocity Gradients in Anmll Containing Plain and f lu Tuba** Pht-Bt tha»l»* »nl<ror«ity of Michigan# 1949*

Knud sen# J* G.# and Kate# Donald L.# Heat Transfer and Pressure Drop In Annul!# Chemical Engineering Progress# XLVI# tyQ^^dt}, (1950) •

Kratz# A. p## Maclntir©# H* J.# and Gould# R* Flew ©f Liquids In Pipes of Circular and Annular Croee-Sectiona# University of I llin o is#Engineeri m Experiment S ta tio n Bulletin8oT 222 ( 193l 7 o

91

R®f®r dumber

*»«**# Rwaw, Hydrodynamics, Sixth Edition, Cambridge University Prose, London, wr ^ 7Jft (1932)*

28 Lea, F# 0* and Tadroe, A* 0*, Flow o f Water through ftCircular Tub* with ft Control Cor® and through Rectangular Tubes, Philosophical Mftgasjn® (7)o XI#

25 Lonodal®, Thoaftl# Th® Flow of Water 1ft th® AnnularSpeoe between two Coaxial Cylindrical5 K fc JW ? ‘ SSl m $*&*£ <*)• XLVI,

24 Maurer, Edvard# Dio fcrltisch® Roynoldssehe Zahl furKrelarefere naoh dem Entropiepr insip,Zoltoohrlft fu r Physlk, o x m , 5^2- J2 ( 19*9 )*

25 Millar, Benjamin, 9ho Lead nftr~Film Hypothesis, Trane*ftg^loag &g J&2 jmgrleag ^olgtg o£ jfega&a&2&£B t i s s m t t h m i W J l p W h

26 Palmer, Louie Qftlvin, M« J g BSSSli MaUI&gS .SfeSlgPmftvorolty, 1939*

27 Pi«roy# H* A, Y*, Hooper# M* ft*, and dinar, H* F«,Viscous Flow through Pipes w ith Ceres,Philosophical dftgaaino and Journal of MfiAmse* i? , 7th s«rl®«# 769-7®3Tl933)*

28 RothfUe, Robert R«, V e loc ity D is tr ib u tio n and F lu idF r ic t io n In C oncentric Annul!, fte* D* th e C arnegie I n o t 1 tu t® o f Technology, JUn® ly

29 R othfue, Robert R*, Menrad, 0 . C«, and B®n®oal, V* E*,V e lo c ity D is tr ib u tio n and F lu id F r ic t io n In Smooth C oncentric Annul1,and E ngineering Chanda t r y , X t l l , a jjll*

Reynold#, Osborn®, An Experimental In v e s tig a tio n o f th e Circumatancee Which Determine Whether th e Motion o f Water Shell Be D irec t o r S inuous, and o f th e Law o f R esistance in P a r a l le l Channels, ihl.loftOPhl.fial pranftaotio n j o f th e Royal Society o f London, CLXHY, ( I f

92

H eferea&ee N eater

Jk Route, Hunter, glementary Mechanics o f Fluids* JohnWiley and Sen#, Wow fo rk , W T jT? / x li*

32 B o h lio h tin g , R«* Boundary ta y a r Theory*Adeleery Oottffldtioo foi n ie a l

Of Apr!

35 SahMokanborg, S ., Z e lta a h r in t e m w i f t i Mrfchamatlksad aaahw&k, h i . a ^ ir d g i; ! .

3* S taatoa* T»S*» and Pannell* J . P ., S im ila r ity o f MotionIn R e la tio n o f th e Surfaeo P r lo t io n o f P lu id o t FhU oeophleal T ran aae tlo n t o f th e Royal Sooloty o f hondotu A CiCXIV* 1 ^ *

55 you Karaan, Th+, Turbulence and Skin Fr lo t io n , Jou rnalo f A eronau tica l Splonoo, I# 1-20 ( lp jS K

56 tfleg&nd* J* H« and Baker* S . M*, T ransfe r Preeeeeeain Annul!* T ranaaotlona o f th e AmericanIn a tita te o f 'p fg & a a l ' f x m i l *‘j & J j M C jP b T T ^

37 Winkal, S ., ZattMfcrlft f ^ angwndte K>tfeaa»t.ife £*&Moohanlk, I I I* 2pl

56 Zerban* Alexander H*, Fh» D* Theolt» U n iv e rs ity o j

APPENDIX I I

NOMENCLATURE

Sy®hol D e fin itio n Dimension*

A C onstent

C C onstant —-

D Diameter o f Pipe L

f F r ic t io n f a c to r ——

g G ra v ita tio n a l constan t L/T^

C Maoe flow r a t e M/LT^

H V eloc ity head, V^/gg L

J T ranefor f a c to r

L Length, p a r a l le l to flow L

1 Mixing le n g th L

r Radius o f p o in t, v a r ia b le L

Vaax Radius o f p o in t o f maximum L▼ eloeity

R Radius o f w a ll, c o n stan t L

R^ H ydraulic ra d iu s L

u P o in t T e lo c ity L/T

V V e lo c ity , u su a lly average L/T

w Weight r a t e o f flow M/T

y D istance from a boundary L

F r ic t io n lo s s per fo o t o f p ipe L/LL

95

94

Symbol D e f in itio n Dimension!

A pL

P ressu re l e s s per fo o t o f p ipe f/\?

P D ensity o f f lu id m/ i ?

/* A bsolute v is so c i ty o f f lu id m/ lt

V Kinem atic v is c o s i ty o f f lu id Ls/T

n Sddy v is c o s i ty M/LT

'T Shear S tre s s M/LS

S ubscrip t*

1 Refer* to eore w all

2 R efers to ja c k e t w all

9 o r wbjl* maximum value

avg* average value

o r i t* c r i t i c a l va lue

APPENDIX I I I

Bey to Colson Barbers in tab les of Beta

Colson ItemM i r

1 Badius o f o b se rv a tio n p o in t . Inches fro o c e n te r­l i n e of

2 T e lo o ity a t p o in t of observation* f t . / s e c .

3 Average v e lo c ity in th e section* f t . / s e c .

h Bemarks* flev descriptionSL — streataHne to * b .- tnrbalent

5 Beynolds Vhaber* based on average v e lo c ity andV ° i

6 B a tlo o f p o in t v e lo c ity to average v e lo c ity

95

96

112—

0.917

0.951

0.973

0 .9 * *

1 .0 1 *

1 .089

l .o *(5

1 .0*5

1.077

DASA

m m a a no. 1

1 f t ............0 ? ................ B*___________ <1> <s>.0872.0975. lo t

.07*17 .0878

.o n o .0975— .102

.0839

.0973

.106

.107

.11*

A.0895-0955.1015

.0781 .0995

.09** .110 .1185

.125 .110

.198 .1185

.10*0 .0

.112 .0912

.11** .0975

.113 .0975

.1282 .10*1

.1332 .11*5

.135 .1185

.1153 .108

.102 .112

.130 .11*

.139 .11®

.098 .079

.100 .023

.103 .0895

.112 .0955

.119 .102

.1 0 .10$

.1*78 .1185

.1*9 .12*5

SISID iffusing

97211101192

.682

.832

SI91D iffusing

9871U01138

.868

.8*2

SI8181810 •IttUBtt* D iffusing

1080U H1230

I .0681.1151.119

1,12

8181D iffusing

106211901283

.786

.858tm

SIfus%«

12$01370

1.115i .lw

SISISIfurb*

8*8102*1100H75

1,2221.221.2291.222

8L» slxmous91SITurb.

11081180127513*5

l.lS1.831.181.1*

SLSL, sinuous 91Turb.

1180123$12511277

1.071.1**1.1*1.198

11 is

ii

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VITA

Thomas Anderson Fe&ael wee born Juno 17# 19&% i n D allas#

Teams* Ho moved to West knroe* Louisiana* and a ttended p u b lic

schoo ls in Monroo and West Monroof g raduating from Ou&ehlla

P a r i oh High School in 1940* Ho a tten d od Harvard Oollege on a

sc h o la rsh ip fo r th e 19401941 school year and then en te red

L ouisiana ro ly toohn io I n s t i t u t e in 1941. In 1944 ho wee d ra f te d

In to th e Amy and spen t to o y e a rs in th e serv ice* fo r so re than

a yoar o f t h i s serv ice* ho was assigned to th e ^ n h a t ta n D is tr ic t*

Corps o f Snglneers In Oak ttidge* Tennessee and Los Alamos* 2?ew

Mexico* Following h is discharge* he re -e n te re d L ouisiana Poly**

te e h n ie I n s t i t u t e in 1946 and was graduated in May* 1947 w ith

th e degree o f Bachelor o f Science in Chemical Engineering*

Following g rad u a tio n , he was employed in th e Heseareh Department

o f P h i l l ip s Te tre leua: Company in B & rtlSeville* Oklahoma* u n t i l

February* 1946* a t vhieh tim e he en te red th e Graduate School o f

L ou isiana S ta te M nirersity* Since th a t tim e he has been employed

a s a Graduate A ss is ta n t in th e Department o f Chemical Engineering

and rece ived th e degree o f Master o f Science In August* 1949* At

th e p resen t tim e he I s a cand idate fo r th e degree o f Doctor o f

Philosophy*

1 15

EXAMINATION AND THESIS REPORT

Candidate: Thomas Anderson F eaze l

Major Field: Chemical E ngineering

Title of Thesis: A V isual In v e s tig a tio n of th e Appearance o f Turbulence in AnnularSpaces

Approved:

Major Professor and Chainrfan

Dean ofjttee Graduate School

EXAMINING COMMITTEE:

/ j/

Date of Exam ination:

May lU j 1951