An experimental study of vortex flow

Item Type text; Thesis-Reproduction (electronic)

Authors Neuls, Allen Scott, 1943-

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.

Download date 12/04/2021 11:36:14

Link to Item http://hdl.handle.net/10150/557251

AM -EiPERIMM.TAL STUDY

. .. OF- VORTEX FLOW

by.

A llen S c o tt Meals

A Thesis Subm itted to th e F acu lty o f th e

DEPAKTMMT OF. CSEBG&L . ERtHNEEamiGy ;

In P a r t i a l F u lfillm e n t o f t h e ,Requirements ■ :For the Degree o f

:',MA5TMl-.0F SCIENCE .

a ^ . Ih: th e Graduate College

t : 1 ; '‘HIE UNIVERSITY OF ARIZONA

STATEMENT BY AUTHOR

This th e s is has been subm itted in p a r t i a l fu lf i l lm e n t o f re ­quirem ents fo r an advanced degree a t the U n iv e rs ity o f Arizona and i s deposited in the U n iv e rs ity L ib ra ry to be made a v a ila b le to borrowers under ru le s o f the L ib ra ry .

B rie f q u o ta tio n s from th is th e s is a re allow able w ithout sp e c ia l perm ission , provided th a t accu ra te acknowledgment o f source i s made. Requests fo r perm ission fo r extended q u o ta tio n from o r reproduction o f th i s m anuscrip t in whole o r in p a r t may be g ran ted by th e head o f the m ajor departm ent o r the Dean o f the Graduate College when in h is judg­ment th e proposed use o f the m a te r ia l i s in th e in te r e s t s o f sch o la r­sh ip . In a l l o th e r in s ta n c e s , however, perm ission must be ob ta ined from the au th o r.

SIGHED:

APPROVAL BY THESIS DIRECTOR

This th e s is has been approved on the da te shown below:

A ssociate P ro fe sso r o f Chemical Eh g ineering

A'GKNOWLEDGEBMT

The au th o r w ishes to express h is s in ce re a p p re c ia tio n to the

Chemical Ehgineering Depa.rtment o f th e U n iv e rs ity o f A rizona fo r con­

tin u ed support, during th is study* S p ec ia l appreciation- i s expressed

to th e d i r e c to r and o rg in a to r o f t h i s re sea rch , Dp« N*-D* Cox, who

provided h is in v a lu ab le ass is tan ce*

The au th o r a lso w ishes to acknowledge th e a s s is ta n c e o f h is

.wife during th i s extended p e rio d o f s tu d y »

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . v

LIST OF XABLiiiS . . . o o . o . . . . . . . . . . . . . . . . . vn ~i

ABbTRACT . . . . . . . . . . . . o . . . . . . . o o . o . . v m

INTRODUCTION . . . . . . . . . . . . . o . o o o o o o o o a . I

THEORETICAL CONSIDERATIONS . . . . . . . . . . . . . . . . . . 3

EQUIPMENT . . . . . . . . . . . . . . . o . o . . . . o . . . .7PROCEDURE . . . . . . . . . . o . o . . . . o . o . 0 . . . 0 I 3

RESULTS . . . . . . . . . . . . . . . . . o . o . . . . . . . 1 9

DISCUSSION OF RESULTS . . . . . . . . . . . . . . . . . . . . 33

CONCLUSION AND RECOMMENDATIONS . . . . . . . . . . . . . . . . 39

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . 0^-2

APPENDIX A

APPENDIX B

APPENDIX C

APPENDIX D

EQUIPMENT DESCRIPTION DATA . . . . . . . . . . . 44

PRESSURE PROFILE ORIGINAL DATA . . . . . . . . . 51

DIMEN SIGNLESS PROCESS DATA . . . . . . . . . . . 56

SAMPLE CALCULATIONS . . . . . . . . . . . . . . . 64

LITERATURE CITED . . o . . . . . . . . . . . . . . . . . . . . 65

iv

LIST OF ILLUSTRATIONS

Figure Page

1 D escrip tio n o f Flow and Nomenclature „ 0 „ <, o o <, » <> 4

2 Photograph Showing the Small Tube, Nozzleand I n l e t P ressu re F i t t in g . . . . . . . . . . . . 8

3 Photograph Showing th e S h ie ld , Small Tubeand P ressu re F i t t in g . . . . . . . . . . . . . . .10

4 Small Tube Dimensions and Tap L oca tio n s . . . . . . . . 1 2

5 Large Tube Dimensions and Tap L oca tio n s . . . . . . . .13

6 Measured H eights and T heir R ela tionsh ipsto Various P re s su re s . . . . . . . . . . . . . . .16

7 The E ffec t o f L ongitud inal P o s itio n on Yfo r th e Small Tube. . . . . . . . . . . . . . . . 20

8 The E ffe c t o f L ongitud inal P o s itio n on Yf o r the Large Tube. . . . o . . . . . . . . . . .21

9 The E ffe c t o f L ong itud inal P o s itio n on Y/Xfo i* the SnidZLiL xxfoe o o o o o o o o o o o o o o o o 22

10 The E ffe c t o f L ongitud inal P o s itio n on Y/Xfo r th e Large Tube. . . . . . . . . . a . . . . .23

11 The E ffe c t o f Flow Rate on Dim ensionless A irCore a t the Top o f the Large Tube . . . . . . . .24

12 The E ffe c t o f R adial P o s itio n on Y/X a t 2 1 .6gpm in the Large Tube . . . . . . . . . . . . . . 2 6

13 The E ffec t o f R adial P o s itio n on Y/X a t 7.93gpm in the Small Tube . . . . . . . o . . . . . . 27

14 The E ffec t o f R adial P o s itio n on Y/X a t 16.85gpm in the Small Tube . . . . . . . . . . . . . . 2 8

15 P o s itio n o f T an g en tia lly Located Taps and theR esu lting P ressu res . . . . . . . . . . . . . . . 2 9

v

LIST OF ILLUSTRATIONS—Continued

Figure

16 Photographs Comparing Nodes in th e A ir Core inth e Small Tube a t Moderate and High FlowRateS o o o o o o o o o o o o o o o o o o o

17 Photograph Showing Foam P iece AccompanyingFlow T ran s itio n in Small Tube 0 0 0 0 0 0 .

A-1 The E ffe c t o f I n le t P ressu re on Flow Ratefo r th e Large Tube. . . . . . . . . . . . o

A- 2 The E ffec t o f I n le t P ressu re on Flow Ratefo r th e Small Tube. . . . . . . . . o o o .

A-3 Small Rotam eter C a lib ra tio n Curve. » . . . . . .

A-4 Large Rotameter C a lib ra tio n Curve. . . . . . . .

Page

. .30

. .31

o .45

. o .46

. . .4?

o o .48

\

LIST OF TABLES

Table

A-1

A—2

B_1

B- 2

B-3

B_4

C-1

C- 2

c-3

C-4

0 - 5

C- 6

0-7

Flow R a te - Ih le t P ressu re Data

Rotameter C a lib ra tio n Data

e o o o o o o o o b o

o o o o o o o o o o o o

L o n g itu d in a lly Measured Values o f h fo r th e Small Tube,' o o o e o o o o o o o o o o o o o o

L o n g itu d in a lly Measured Values o f h fo r th eLarge Tube o o o o o o o o o o o o o o o o o o

R ad ia lly Measured Values o f h fo r th e Small Tube „

R ad ia lly Measured Values o f h fo r the Large Tube „

Y Values a t th e M all, Small Tube 0 0 0 = 0 0 0 0 0

Y Values a t th e M all, Large Tube = 0 0 0 0 0 = 0 0

Flow R ate, X, n ^ , Reynolds Number, FroudeNumber, and Weber Number. 0 0 0 0 0 0 0 0 0 =

"Wall Values o f Y/X, Small Tube 0 0 0 0 0 0 0 0 0 =

Wall Values o f Y/X, Large Tube

R adial Values o f Y/X fo r the Small Tube, =

R adial Values o f Y/X fo r th e Large Tube0 = = = = =

0 0 0 0 0 0 0 0 0 0

Page

, =49

o =50

O =52

o o53

o =54

- <>55

o =57

o 0.58

O *59

o 060

o 061

o 062

o 063

v i i

ABSTRACT

An ex p lo ra to ry in v e s tig a tio n o f th e fo llow ing system was com­

p le te d ; A co n stan t d e n s ity f lu id , w a te r, was in je c te d ta n g e n tia l ly in to

th e bottom o f a s ta t io n a ry p le x ig la s s tu b e , c re a tin g a sw irlin g vo rtex

flow . Two d i f f e r e n t 's i z e v o rtex tubes were used to in v e s t ig a te the

lo n g itu d in a l and r a d ia l p re ssu re p r o f i l e s . A ir core and a x ia l v e lo c ity

c h a r a c te r is t ic s were a lso considered . The flow regime was found to be.

tu rb u le n t .

L ong itud inal p re ssu re d is t r ib u t io n s were ob ta in ed a t Reynolds

Numbers between 4,500 and 73 ,000, The d is t r ib u t io n s were n o n - lin e a r

w ith re sp e c t to lo n g itu d in a l p o s it io n , Ehtrance and e x i t cond itions

a f fe c te d th e p re ssu re p r o f i l e s , p a r t ic u la r ly a t h igh flow r a t e s .

R adial p re ssu re d is t r ib u t io n s were found to be approxim ately

l i n e a r w ith re sp e c t to r a d ia l p o s itio n '.

The dim ensionless a i r core rad iu s a t th e . tube o u t l e t approached

an asymptote o f 0 ,7 2 , N egative a x ia l v e lo c i t ie s n ea r th e a i r core were

observed . These n eg a tiv e v e lo c i t ie s have been p re v io u s ly rep o rted ,

.In t h i s .in v e s t ig a t io n , they d isappeared a t h igh flow r a t e s ,

f, : '.' 'V:As the- flow r a te was in c re a sed in th e sm all tu b e , a t r a n s i t io n

in flow regime was observed . The h e l ic a l ly shaped a i r core suddenly

changed form to an o th er h e l ix o f many more tu rn s , \ This change in the

a i r core was th e r e s u l t o f a h y d rau lic jump.

INTEDDUCHON

In re c e n t years th e re has been considerab le experim ental and

th e o re t ic a l work on v ario u s v o rtex c o n fig u ra tio n s . These in c lu d e the

Borda f re e j e t in tank drainage s tu d ied by B innie (1964), th e ro ta tin g

tube in a ir -c o n d itio n in g in v e s tig a te d by White (1964), and numerous

problems d eriv ed from flow over a wing..as summarized by Kuchemann (1965)«

However, th e re has been very l i t t l e work pub lished on the problem o f a

eo n s ta n t-d e n s ity f lu id in je c te d ta n g e n tia l ly in to a s ta t io n a ry tube

thereby c re a tin g a sw irlin g flow w ith , under high flow c o n d itio n s , an

a i r core in th e center® F u rth e r knowledge o f th i s problem would a id in

th e design o f a wide v a r ie ty o f spray n o zz le s , such as those used to

atomize l iq u id s « Spray nozzles a re c u r re n tly designed u s in g on ly empir­

i c a l factors®

To th e a u th o r is knowledge, th e re e x is ts only one th e o r e t ic a l

so lu tio n to th i s v o rtex problem® N® D® Cox ( 1967) developed a two-

dim ensional lam inar model® Experim ental Work i s very lim ited® J» B,

N u tta l ( 1953) . made a sh o rt re p o r t on th e a x ia l v e lo c i ty c h a ra c te r is t ic s

o f th e above problem. M arshall (195^) mentioned a i r core d iam eters in .

sp ray n o zz les , which a re s im ila r t o , t h i s system® But, th e re have been

no experim ental r e s u l ts pub lished on th e p re ssu re o r v e lo c i ty p r o f i l e s .

In fo m a tio n r e la t in g to th ese p r o f i le s would a id trem endously in devel­

oping a complete model f o r the system® This p ro je c t was i n i t i a t e d on

th e b a s is th a t a low -cost ex p lo ra to ry .in v e s tig a tio n o f the p re ssu re

p r o f i le would fu rn ish considerab le in s ig h t in to the p h y s ic a l • behavior o f

th i s type .of vortex* The s p e c if ic o b je c tiv e s o .f th i s work were as

fo llow s: to measure lo n g itu d in a l and r a d ia l p re ssu re p r o f i l e s in two

d i f f e r e n t ly s ized tubes a t a v a r ie ty o f flow r a te s using w ater as the

f lu id ; to measure a i r core d iam eters a t the top o f each tube a t various

flow r a te s ; and to in v e s t ig a te th e a x ia l v e lo c i ty c h a r a c te r is t ic s o f

th e system.

THEORETICAL CONSIDERATIONS

The only ■ cu rren tly , a v a ila b le th e o re t ic a l development fo r th e ■

s ta te d problem i s an a n a ly t ic a l so lu tio n f o r lam inar, flow . A g rap h ica l

d e sc r ip tio n o f the flow w ith re la te d nom enclature i s 'shorn .in Figure f «

In o b ta in in g - th is so lu tio n , Cox ( 196?) assumed symmetry about th e z~.axis,

constan t d e n s ity and v is c o s ity , fu lly , developed lam inar flow , a n e g l ig i­

b ly sm all r a d ia l v e lo c ity component, and an a i r core along the c e n te r l in e , .

Hie a i r core along the c e n te r lin e i s assumed because, in a f re e v o rtex ,

t h e . ta n g e n tia l v e lo c ity in c re a se s from th e w all to th e c e n te r l in e , caus­

in g a decrease in. th e f lu id p re ssu re . Since the p re ssu re decreases to

atm ospheric a t some p o in t not, on the c e n te r l in e , an a i r core must e x is t

.in the fre e v o rte x . With th e a d d it io n a l cond itions, o f th e A xial and

ta n g e n tia l v e lo c i t ie s being zero a t th e w all, th e fo llow ing p re ssu re

p r o f i le fo r lam inar flow .was p resen ted by Cox:

Y % f W . X f(nco)

where the dim ensionless p re ssu re ,

v _ Pq. . r o ” r a •

th e dim ensionless w all p ressu re a t th e e x i t .

( 1)

( 2).

' X 0 < X < 1 = : . y ' ( 2 ) .

the. d im ensionless ra d iu s , . . .

w ,

4

p = Barom etric P ressure

a x ia l v e lo c ity

Nozzle I n le t P ressure Tap

o*

S te e l I n s e r

Cast Iron Sw irl Chamber

I n le t Pipe

v, ta n g e n tia l v e lo c ity

Figure 1

D escrip tion o f Flow and Nomenclature

5

. and ; •

f(n ) == ™2 -* n2 * 4- l n ( n ) «, ( 5 )

' n ■ ■ ' 'Also 9. .

f(nco) = 1— ^ . ■ (6 )

.The follox-ting d e f in it io n was made f o r the combined p re ssu re and poten­

t i a l head:

' P = p + egz ' (7)

V a.th ;th is d e f in i t io n , i t was necessary f o r c o n s is te n t dim ensionless p re s ­

su res, to d efine th e follow ing fo r Pa : •

' pa = Pa + ( 8 )

t-ihere pa i s the barom etric, p re s s u re . The n ecessary cond ition o f X being -

-un ity fo r no flow wass thereby ,, s a t i s f i e d . The a x ia l, v e lo c ity fo r th i s

case was,

-w = — - [ 1 - n 2 > g ln (n )] > . (9 )

■ where g i s c o n s ta n t.

The preceding development. WTas fo r lam inar flow . However, an

in k in je c t io n t e s t showed th a t th e flow was tu rb u le n t in th e a v a ila b le

experim ental apparatus a t a very low flow r a te , th a t i s , below Be' = 1500

(Reynolds Number based on tube d iam eter and average a x ia l v e lo c i ty ) .

The lam inar model was, th e re fo re , n o t a p p lic ab le to the p re se n t ex p eri­

m ental work. In tu rb u le n t flow, th e tu rb u le n t sh ear s t r e s s i s s tro n g ly

p o s itio n dependent. Because th is p o s it io n dependence was unknown, i t

was im possib le to solve f o r the p re ssu re o r v e lo c ity p r o f i l e , even in th e

much s im p lif ie d tw o-dim ensional case . However, the lam inar theo ry should

suggest how to c o r re la te experim entally determ ined p re ssu re p ro f i le s f o r

tu rb u le n t flow,, Hence, the d a ta ob ta ined in th i s in v e s tig a tio n were

c o r re la te d w ith th e use o f th e dim ensionless groups p resen ted above®

EQUIPMENT

Two d i f f e r e n t v o rtex tubes were c o n s tru c ted » The la r g e r one was

made o f a 2 13 / 16 - in ch in s id e diam eter, 1 / 8- inch w a ll, 31"inch long

p le x ig la s s tu b e . In o rd e r to provide sw ir l to the w ater, th i s tube was

connected to a nozzle which had a ta n g e n tia l i n l e t s im ila r to the one

shown in F igure 2 . The nozzle, was a 2 CRC I~55~iP5 Spraying System, Go,

W h irlje t type w ith the s te e l i n s e r t removed. The p le x ig la s s tube was

cemented to the o u ts id e rim o f th e c a s t iro n housing . The i n l e t to th e

nozzle was f i t t e d w ith a 2 -in ch n ip p le which had an i n l e t p ressu re tap

2 inches upstream from th e nozzle i n l e t . . The flow r a te in th is la r g e r

v o rtex tube was approxim ately p ro p o rtio n a l to th e square ro o t o f the in ­

l e t p re ssu re as in d ic a te d by th e 0=59 slope o f the i n l e t p ressu re -flo w

r a te p lo t seen in F igure A-1. A sm alle r v o rtex tube was made from a

1 1/ 8 - inch in s id e d iam eter, 1/ 16 - in c h w a ll, 33=-Inch long p le x ig la s s tu b e .

This tube was a lso connected to a n o zz le which had a ta n g e n tia l i n l e t .

This nozzle was a 1 1/h CRC 1-20-45 Whirl j e t ty p e . The p le x ig la s s tube

was f i t t e d and cemented to th e in s id e o f th e s te e l nozzle i n s e r t . The

nozzle i n l e t was f i t t e d w ith a 1 1/4 -in c h p ipe n ip p le which had a p re s ­

su re tap approxim ately 1-inch b efo re th e nozz le , as seen, in Figure 2 .

The flow r a te in th is tube was again approxim ately p ro p o rtio n a l to th e

square ro o t o f th e i n l e t p re ssu re as shown in F igure A-2 . Table A-1

con tains th e ta b u la te d va lues o f i n l e t p re ssu re and flow r a te s fo r both

tu b e s . . ; • " - -

8

Cast - Iron Nozzle

Figure 2

Photograph Showing the Small Tube, Nozzle and I n le t P ressure F i t t in g

A m odified fume hood was p laced on a support over the vortex

tube to a c t as a w ater s h ie ld in o rd e r to allow c lose in v e s tig a tio n under

flow c o n d itio n s» The arrangem ent i s shown, in F igure 3 * A 2 -inch w ater

l i n e was run in to the s h ie ld to a p o s it io n where th e v o rte x tube could

be a tta c h e d . The 2-inch i n l e t l in e was p a r t o f an e x is tin g f lu id flow

.experim ent in the U nit O perations L aboratory o f the Department o f Chem­

i c a l Engin.eer in g . The experim ent had a 2 f t . x 6 f t . feed tank , the

o u t l e t o f which supp lied w ater to a 14 -inch O liv e t i Acid Pump (O liv e r

U nited F i l t e r s I n c . , Spec. 4015). The.pump was run by a 1200 rpm, 1.

horsepower m otor. -Downstream from th e pump, the p ip e and valves on the .

e x is t in g experim ent were such th a t th e flow to th e v o rtex tube could be

run through one o f two ro ta m e te rs . These ro tam eters were m anufactured

by Brooks In strum en t Co., the la r g e r a type 13-1110* 70 gpm cap ac ity , and

th e sm alle r a type 10-1110$ 8 .8 gpm. c a p a c ity . The c a l ib ra t io n curves

fo r w ater a re shown in F igures A-3 and.A -4. Tables A- 2 con ta ins the

ta b u la te d c a l ib ra t io n d a ta .

In o rd e r to measure p ressu re d is t r ib u t io n s , tap s were d r ille d , in

th e w alls o f th e v o rtex tu b e s . H oles, 25 m ils in d iam eter, were p laced

1 /2 , .4, ? , 13, 19, and 25 inches below th e top o f th e la rg e t u b e . . The

low er fo u r holes were on a lo n g itu d in a l a x is which was about 30 degrees

counter-clockw ise from th e i n l e t as viewed from the to p . S im ila r ho les

were p laced 1/ 2 , 4 , 8 , 1 2 ,.1 6 , 2 0 , 24, and 28 inches below the to p ,o f

the sm all tu b e . The low er f iv e tap s were again on a lo n g itu d in a l ax is

which was about 30 degrees clockwise from th e i n l e t . The upper th re e

ho les were on an a x is 180 degrees from th e lower f iv e h o le s . A contour-:

f i t t e d rec ta n g u la r p iece o f p le x ig la s s , in to which a 1/ 4 - in c h b rass

10

Figure 3

Photograph Showing th e S h ie ld , Small Tube and P ressu re F i t t in g

/ ; V : ’ ■ ■ ■ . . " '■ . ■ 11 ■

f i t t i n g had been, cemented,, was glued over each h o le 3 ;F igures 4 and 5

. show the/ tap p o s itio n s and- the r e la te d tube dimensions e.

To f a c i l i t a t e lo n g itu d in a l w a ll p re s s u re . measurements, a 1 /4-inch.;.',

b rass ;s to p co ck . valve was connected to th e f i t t i n g over each . ta p e; From ,

each valve, 1/4 -in c h c le a r po lyethy lene tubing was run to a $0 tale b u re tte ,

secu re ly mounted to a w all bracket® A s h o rt p iece o f rubber tubing was

clamped to th e bottom o f the b u r e t te » Cemented a t th e o th e r end o f th e

rubber tubing was a 1/4 -inch- b ra ss f i t t i n g which could be qu ick ly a tta c h e d

to the p o ly e th y len e-tu b in g from the. p ressu re tap® - The b u re t te was .

' mounted w ith the bottom o f the s c a le approxim ately even w ith the le v e l

o f the v o rtex tube o u tle t* .. The b u re tte , was th e means by which th e w ater

h e ig h t, and th e re fo re th e p re ssu re , could be read® (A 25 ml p ip e tte was

used .in p lace o f th e b u re tte fo r the f i r s t th re e flow r a te s in the sm all

tube and th e f i r s t - two flow ra te s in th e .la rg e tube®) ■_

To o b ta in r a d ia l p ressu re measurements,- a 5 1 /4 -inch 20 gauge

s ta in le s s - s t e e l hypodermic needle was used® ; The n ee d le was cemented to ...■

■ a len g th o f 1 /4 -in c h c le a r po ly e th y len e tu b in g , ; which could be a tta ch e d ..

t o th e burette® In o rd e r to i n s e r t th e need le in to th e v o r te x .tu b e , i t :

was necessary', to remove the valves from th e tap s and en la rg e th e ho les , .

to 37 m ils . in diameter® New f i t t i n g s , c o n s is tin g o f p l a s t i c capped .

po lyethy lene f i t t i n g s , were p laced over th re e o f the; tap f i t t i n g s , the :

remaining were plugged w ith a 1 /4 -in ch nut® A ho le was punctured in the

p la s t i c cap to allow the- needle to go through® The cap a c te d as. a .p re s s - :.

■ fit s e a l and in conjunction w ith th e h o le in th e w a ll, a support fo r , th e

"n eed le .' 1 '

12

H 1 1/8"

12 in .

4 in ,

4 in .

4 in .

4 in .

4 in .

_ L

3-| i n .

#8

#7

4 in .

#6 I ...I,..

. J5

- # 4

#2

t 1/2 in .

r

L = 28 in .

8 iI I

F igure 4

Small Tube Dimensions and Tap Locations

13

7 in

6 in

L = 25 in6 in

. _#2

6 in .

6 in

F igure 5

Large Tube Dimensions and Tap Locations

' 14

The "pressure tap befo re the' nozzle i n l e t was connected by p la s ­

t i c tubing to a mercury monometer w ith th e o p p o site end open to th e ■

atmosphere* .This .manometer was used to o b ta in th e p rev io u s ly mentioned

i n l e t p re ssu re s a t the various flow r a t e s «,.

A fte r a m a jo rity o f . th e d a ta was c o lle c te d , i t was n o ticed th a t

th e e x i t v e lo c ity from th e tube was n o t uniform , b u t s l ig h t ly g re a te r •

•near th e p o rtio n d i r e c t ly o p posite th e in le t* This suggested a lack o f

f u l ly developed flow* To f a c i l i t a t e a check on th i s p o in t , two more w a ll

tap s were in s ta l l e d 25 inches below the to p , 120 degrees a p a r t on the

la rg e tube* The h o le d iam eter was 3? m ils * P la s t ic tub ing was f i t t e d .

to each tap as was done f o r th e lo n g itu d in a l measurements.

PROCEDURE

Several p re lim in ary fa m ilia r iz a tio n runs were, made using the

la rg e tube before p reparing fo r the lo n g itu d in a l p re ssu re measurements.

During these runs i t was. apparent., th a t th e a i r core was n o t s t r a ig h t ■

b u t had a h e l ic a l form. This h e l ic a l a i r core form was a r e s u l t o f the

c e n te r o f sw irl being o f f c en te r from the lo n g itu d in a l c e n te r l in e . • The

c e n te r o f sw irl follow ed a path o f a c y l in d r ic a l h e lix , a s i t progressed

up the tu b e . The h e l ic a l form a lso p e r s is te d even to flow r a te s where

th e a i r core d isappeared except a t th e to p . At low flow r a t e s , a i r bub­

b le s , c a r r ie d down th e tube by th e n eg a tiv e a x ia l v e lo c i ty n e a r the

c en te r , follow ed the id e n t ic a l h e l ic a l p a tte rn p re se n t a t h ig h er flow

r a te s . The h e l ic a l a i r core was a lso p re se n t in th e sm all tu b e . No

e f fo r ts were made to re so lv e th e problem o f the h e l ic a l a i r core .

The la rg e tube was d r i l l e d and se t.u p f o r th e lo n g itu d in a l p re s - .

su re measurements. S evera l u n su ccessfu l experim ents u sing various

d i f f e r e n t i a l manometers were conducted before the b u re tte system was

suggested . The b u re tte method proved slow a t f i r s t because th e tap h o le

d iam eter was only 13 m ils . The la r g e r 25 m il tap s proved su c c e ss fu l. The

w ater l e v e l in th e b u re tte , w ith the tube f u l l b u t no flow , was determ ined

fo r a re fe ren ce p o in t . The amount o f w a te r ' above th is , s t a t i c le v e l was

recorded f o r each tap a t a s p e c if ic flow r a t e . . Knowing the le v e l above

s t a t i c and the d is tan ce from th e tap to the f i t t i n g , th e gage p ressu re

in inches o f w ater was determ ined fo r each lo n g itu d in a l p o in t . See

F igure 6 f o r a v isu a l exp lanation o f th e measured value and i t s r e la t io n

16

Tube Top Level

P tap

pg(L + h)

F igure 6

Measured H eights and Their R e la tio n sh ip s to Various P ressu res

For r a d ia l runs, h can be neg a tiv e and equal to -(L -z) in the a i r co re . The h e ig h t h0 was measured a t the P0 tap : P0 - P g(L + h0) .

• V .. 1?

to o th e r q u a n tit ie s ,. The p ressu re d i f f e r e n t i a l between tap s was pro-

p 'o rtio n a l to th e le v e l d iffe re n c e e A p ressu re p r o f i le fo r th e given;

flow r a te could then be co n s tru c ted . The above procedure was c a rr ie d

o u t f o r 'e ig h t flow: ra te s i n .th e la rg e tu b e „ The sm all tube'w as then

s e t up fo r th e lo n g itu d in a l measurements and th e procedure c a rr ie d o u t

f o r n ine flow ra te s „ D up licate measurements were made f o r a l l flow

ra te s on both tu b es ,

The b u re tte measuring device proved adequate f o r moderate flow

ra te s but inadequate a t very low flow ra te s (lam inar re g io n ) , because

p ressu res were too low fo r accu ra te measurement o f h . At h igh flow

r a t e s 9 th e b u re tte had to be rep laced by a long g la ss column because h

was very la r g e . The le v e l above s t a t i c h 9 was determ ined w ith a measur­

ing tap e . Before each measurement, the p re ssu re l in e to be a ttach ed to

the b u re tte was purged w ith w ater and checked fo r a i r bubb les« The

c le a r p l a s t i c l in e proved q u ite u se fu l f o r t h i s .

For the r a d ia l p r o f i le s , th re e flow r a te s and th re e lo n g itu d in a l

p o s itio n s were used f o r e ach ;tube® ■ ■ The. p ro p er r a d ia l need le t i p posi-.

t io n fo r a given measurement was determ ined by pushing th e needle a l l .

. the way in u n t i l i t touched the o p p o site s id e , marking th e needle w ith

' p a in t n ex t to the bushing, then withdrawing th e need le th e tube rad iu s

p lus the d e s ire d ra d iu s fo r the measurement. This procedure o f measur­

ing the p ressu re a t a given ra d iu s , w ith th e need le on th e s id e n e a re r

the ta p , was considered su p e rio r to m easuring p re ssu re on th e same

given d iam eter with, th e need le on the s id e opposite th e ta p , because

the needle d id n o t v ib ra te as much and c rea te d l e s s flow in te r fe re n c e

n e a r the. c e n te r . The need le and p l a s t i c l in e was purged o f a i r before

' . 18

s e t t in g the r a d ia l p o s i t io n «

F o r .th e r a d ia l d is t r ib u t io n , gage, p ressu res were determ ined in

the manner describ ed above. The rad ia l- p o s itio n o f the t i p o f the

need le was checked a f t e r each run . Only one s e t o f d a ta was' d u p lica ted

because o f th e la rg e amount o f time req u ired to f in d th e equ ilib rium

le v e l o f the w ater in the b u re t te .

The w all p ressu re a,t th e bottom o f the la rg e tube.was measured

using th re e w all taps' eq u a lly spaced a t 120 degrees around th e tube,

in o rd e r to t e s t fo r asymmetry due to an en trance e f f e c t . Measurements

were taken a t th re e flow r a t e s „

The a i r core rad iu s a t the v o rtex tube o u t le t was measured w ith

the use o f a .greased 22 gauge needle which hs,d been cemented in to a

.notch a t the top o f the p le x ig la s s tu b e . The need le could be moved so

as to determ ine the r a d ia l p o s itio n o f the air-.w ater in te r f a c e o f the

a i r co re . Measurements were made in both tubes fo r a l l flow ra te s used

th a t had a w ell defin ed a i r co re . . . . . .

RESULTS

. For the lo n g itu d in a l, p re ssu re p r o f i le s , the average o f two

measured values o f dim ensionless p re ssu re , Y, were computed fo r the d a ta .

l i s t e d in Tables C~1. and C-2, th e Y values a t th e .w a ll fo r the la rg e and

sm all tube re sp e c tiv e ly , and p lo t te d a g a in s t d im ensionless d is ta n c e , z/L ,

as seen in F igures 7 and 8 , The X v a lu es , the dim ensionless w all pres™ .

su re a t th e tube e x i t , were then determ ined by e x tra p o ta tin g th e la s t-

two p o in ts to z/L = 1 , s in ce Y i s equal to X a t z/L =1« The X values

a re ta b u la te d 'in Table 0-3- Next, the experim ental values o f Y/X a t

the w all could be p lo t te d a g a in s t z/L according to Equation 1 fo r each

flow r a t e . F igures 9 and 10 show.some o f th ese r e s u l t s . W all values

o f Y/X a t th e various z/L values fo r a l l flow r a te s a re l i s t e d in Tables

C~4 and C-5.

In o rd e r to c o r re la te the. d a ta u sing Equations 1-8, the a i r

core diam eters a t the top must be known. - .. ,

./ Values o f 'n C2_s the dim ensionless a i r . core d iam eter a t th e .tube

top , a re l i s t e d in Table C-3 fo r various flow r a te s . F igure 11, measured

. values o f n c l a t various flow ra te s in th e la rg e tube , i l l u s t r a t e s th a t

. - th e .v a lu e o f n^p approaches an 3,symptdte ,o f approxim ately 0 .? 2 .

. For the r a d ia l p re ssu re p r o f i le s , Y/X values were computed a t

vario u s values o f dim ensionless ra d iu s , n , a t th re e lo n g itu d in a l p o s i-

. t io n s . Tables B~1 , 2 , 3 , and 4 con ta in a l l the measured d a ta . The Y/X

v alues fo r various r a d ia l p o s itio n s , c a lc u la te d from the measured d a ta ,

a re l i s t e d in .Tables 0-6- and 0-7« F igu res 12, 13, and 14 a re ty p ic a l

. . ' " 19 ' ' .

20

0 .8

0 .6

Y

0 .4- 20.2 gpm

0 .2

0.6 0.8 1 .00 . 20

• z/L

Figure 7

The E ffe c t o f L ongitud inal P o s itio n on Y fo r the Small Tube

21

0.8

0.6

Y

0.4

0 . 2

0.60 .40 . 2

Figure 8

The E ffec t o f L ongitud inal P o s itio n on Y fo r the Large Tube

22

0 .8

0.6

Y/X

Laminar Theory

0 . 2

0 .80 . 2

14.2 21.)10.7

Tube Diameters from Bottom o f Nozzle

Figure 9

The E ffe c t o f L ongitud inal P o sitio n on Y/X fo r th e Small Tube

23

0.8

0.6

Y/X

0 .4

Laminar Theory

0.6 0 .80 .4 1.00 . 20z/L

3.20 5.34 7 .4 7 9 .6 1 1 0 .7 11 .9

Tube Diameters from Bottom of Nozzle

f ig u re 10

The E ffec t o f Longitudinal P o s itio n on Y/X fo r the Large Tube

24

0.8

0.6

0 .4

4010 20

Flow Rate, gpm

Figure 11

The E ffec t o f Flow Rate on Dim ensionless A ir Core a t the Top o f th e Large Tube

v : ; ■■■'■ . ' : " - 25p lo ts i l l u s t r a t i n g . Iiow - th e values o f X/X vary w ith r a d ia l p o s itio n * The •

lam inar, theo ry l in e s were c a lc u la ted u s in g Equation 1 w ith 2 = 0 , Y/X =

f(n )/X f ( n ^ ) , and .Equation 2 , f ( n co) = f ( n 0]_)/l - X* The th e o re t ic a l

l i n e p rov ides a comparison between th e p o s tu la te d lam inar p r o f i l e , under

th e same co n d itio n s o f p re ssu re drop and a i r core d iam eter, and the meas­

u red tu rb u le n t p r o f i l e e

T h e .re su lts o f th e t e s t fo r en trance e f f e c t , using w all taps

120 degrees a p a r t n e a r th e base o f th e la rg e tu b e , a re shown in Figure

15 o The p re ssu re was c o n s is te n tly h ig h e s t a t tap P0-j 0 This tap co rre s ­

ponded to the p0 tap f o r lo n g itu d in a l measurements in th e la rg e tube* -

. .A sudden t r a n s i t io n in flow, regime was observed in the small ‘ .

tube as the flow r a te was increased* The frequency o f th e a i r core

h e l ix changed from one o f approxim ately th re e nodes in 8 inches to a

h e l ix o f approxim ately 7 nodes in 8 .inches (nodes as viewed hwo-dimension-

a l l y ) 8 a s .se e n in F igure l 6 e Accompanying th i s change.was what s h a ll be

c a lle d , fo r la ck o f a b e t t e r term , a foam piece ' as seen in F igure 1 7 *

.The foam p ie c e was a reg ion approxim ately, one -inch long th a t contained a

la rg e amount o f a i r en tra in ed in the c y lin d e r o f w ater n ex t to the wall* .

The w ater reg ions above and below th e foam p iece con tained l i t t l e o r no

a ir* A rio ticeab le change in the a i r core d iam eter was observed by com­

paring the a i r core d iam eters above and below th e foam piece* This foam

region f i r s t appeared a t th e nozzle o u t l e t and climbed, w ith in c reas in g .

flow r a te , to a p o in t about two inches below, the f i r s t p re ssu re tap*

Here the flow r a te was lim ite d by th e feed system c a p a b il i t ie s * The

' t r a n s i t io n s in a i r core h e l ix frequency took p lace a t a flow r a te o f

19 gpm* The phenomenon d id not. occur in the la rg e tube, even a t .the

26

Y/X

7

6

5

z/L = 0.840

3 z/L = 0.480

2

Laminar Theory, z/L = 01

00.6 0.80 .4 1.00 . 20

n

Figure 12

The E ffec t o f Radial P o s itio n o f Y/X a t 21.6 gpm in the Large Tubs

27

3.0

1.0

z/L = 0.428

Laminar Theory, z/L = 0

00 0 .2 0 .4 0 .6 0 .8 1.0

n

Figure 13

The E ffe c t o f Radial P o s itio n on Y/X. a t 7.93 gpm in th e Small Tube

28

2.0

•6

.2

0 .

z/L = 0

Laminar Theory, z/L = 0

00 0 .2 0 .4 0 .6 0 .8 1.0

n

F igure 14

The E ffe c t o f Radial P o s itio n on Y/X a t 16.85 gpm in the Small Tube

29

o3

Top View

P o sitio n po1 po2 p03

Flow(gpm)

12?g

(in ch es o f water)

21.6 27.32 27.20 27.24

3 0 .2 29.81 29.57 2 9 .6 6

41.1 34.43 33.68 33.81

Figure 15

P o sitio n o f T an g en tia lly Located Taps and th e .R esu lting P ressures

Flow Rate o f Approximately

12 gpm

Figure 16

Flow Rate o f Approximately

25 gpm

Photographs Comparing Nodes in the A ir Core in the Small Tube a t Moderate and High Flow Rates

31

Figure 1?

Photograph Showing Foam P iece Accompanying Flow T ran s itio n inSmall Tube

: ' ' • ■ . ; 32

maximum flow r a te o f 47.7 gpm. Some d im ensionless groups which may. be

re la te d to th is phenomenon a re l i s t e d in Table G»>3» These groups were,

computed u s in g the average a x ia l v e lo c ity and the tube d iam eter.

Also o f im portance were the- r e s u l t s o f ink in jec tio n - t e s t s .

These t e s t s showed th a t th e re was a n eg a tiv e a x ia l v e lo c i ty e i th e r n e a r

the c e n te r i f no a i r core was p re se n t, o r n e a r and along th e a i r core

when i t was p re se n t. The n eg a tiv e a x ia l v e lo c ity was observed in both

tu b es , bu t i t d id d isap p ear a t a flow r a te o f approxim ately 16 gpm in th e\ '

small tube and 45 gpm in the la r g e tube-. Ey in je c t in g in k in to th e top

o f th e la rg e tube a t low flow r a te s , i t was observed th a t th e re was a

region in which th e re were no n eg a tiv e a x ia l v e lo c ity components along

the a i r core in te r f a c e . The reg ion extended approxim ately l / 2 inch below

th e . l i p o f th e la r g e r v o rtex tu b e . As the f lo w .ra te was in c reased , th i s

reg ion became l a r g e r . Also, by dropping ink in to th e 'c e n te r o f the a i r

core in a ease when the. a i r core extended- approxim ately two inches down

from th e top , i t was p o ss ib le to observe th e in k t ra v e l in g a h e l ic a l

p a th down th e c e n te r p o rtio n o f th e .tube below th e a i r - c o r e . This r e s u l t

in d ic a te d th a t th e h e l ic a l flow p a tte rn was p re se n t even when the a i r

core was n o t . . • • -

DISCUSSION OF RESULTS

The l/X .m lJL . values fo r various Ion .g itud lnal p o s it io n s , l i s t e d

in Tables 0-4 and C-5 and i l l u s t r a t e d in F igures 9 and 10 > showed non­

l i n e a r behavioro The l in e a r lam inar th eo ry l i n e , p lo t te d in F igures 9

and 10 , i s Equation 1 w ith n = 1. The double a b sc issa in F igures 9 and

- 10 should be n o tic e d , . The steep i n i t i a l s lo p e , which i s p ro p o rtio n a l to

( . - d F / '<9z), ex h ib ited in the la rg e tube , as seen in F igure 10, was the

re su lt , o f incom pletely developed flow a t th e bottom o f th e tu b e . This

en trance e f f e c t was n o tic e a b le up to a p o in t approxim ately s ix tube diam­

e te r s from the bottom o f the n o zz le . Since th e i n i t i a l , p re ssu re top in

th e sm all tube was ?«,1 tube d iam eters from th e bottom o f the nozz le , th e

en trance e f f e c t was l e s s s ig n if ic a n t, in the sm all tu b e . The d iffe re n c e

in th e methods o f a tta c h in g th e tubes to th e nozzles m ig h t,. however,

account f o r some "of th e apparen t red u c tio n o f an en tran ce e f f e c t r e la t iv e

to th a t found in the la rg e tu b e ,.

The r e s u l t s o f th e t e s t using the th re e w all tap s a t the base

o f th e la rg e tube , as seen in F igure 15, d id in d ic a te incom pletely devel­

oped flow . The w all p re ssu re was c o n s is te n tly low er a t tap s and P0 q,

By using the B ern o u lli equation , i t can b e suggested th a t th e bulk v e l-

.: o c i ty .was h ig h e r in th e neighborhood o f P0 2 and Pq- than n ea r .. This

en trance e f f e c t was n e g lig ib le a t the low er flow r a te s : bu t i t became

more mariced w ith in c re a s in g flow r a te , r e s u l t in g i n . a 2 , 2^ d iffe re n c e

- between th e h ig h e s t and low est p re ssu re s i n the la rg e tube a t 41 *1gpm, h

. ' 3 ^ '

I t i s doubtful' t h a t th i s en trance cond ition would a f f e c t the lo n g itu d in a l

p r o f i le s a t the upper p o r tio n o f e i th e r tube "where th e p ressu re taps '

were p laced 180 degrees o p p o s ite ' the p rev ious taps® For the la rg e tube,

the f i r s t tap lo c a te d 180 degrees around the circum ference from the i n i ­

t i a l p ressu re tap was more than ? tube d iam eters from th e i n i t i a l tap®

In th e sm all tube, n o t only was the i n i t i a l p re ssu re tap 4 more tube

diam eters fu r th e r from the base o f the nozz le , bu t a lso th e f i r s t a l te r e d

p o s itio n tap was more- than 1? tube d iam eters from the i n i t i a l ta p , .

At th e high flow r a t e s ' i n both tu b es , the Y/X values n ea r the

top f e l l below the Y/X = z/L l in e and then climbed back to. the p o in t

Y/X = 1, z/L - 1, The s lo p e , .which i s p ro p o rtio n a l to (-c* P/ 5 z ) , changed

in th i s reg ion from le s s th an u n ity a t low flow ra te s to g re a te r than

u n ity a t high flow ra te s as seen in F igures 9 and 10 . This slope change

suggests an e x i t e f f e c t a t the h igh flow r a te s . The e x i t e f f e c t extends

approxim ately fo u r tube d iam eters down from the top in both tubes a t the

high flow r a te s . . .

. At low flow r a te s , the. p ressu re .p ro f i le r i s e s above the Y/X = z/L

l in e and in te r s e c ts th i s same l i n e ' a t . th e end p o in ts . This cu rv a tu re

was probably a c h a r a c te r is t ic o f th e tu rb u le n t v o r te x , . In the la rg e

tubeg. th e bottom p o rtio n o f the curve was d is to r te d by an en trance e f f e c t .

At h igh flow r a te s , the p r o f i l e s . f o r both tubes w e re .d is to r te d a t the

upper end by an e x i t e f f e c t . In th e la rg e tube a t h igh flow r a te s , the

curves were com pletely d is to r te d by th e en trance and e x i t e f fe c ts due to

the r e la t iv e ly sh o rt tube, in terms o f tube d iam eters . I t was i n t e r e s t ­

ing th a t th e re was l i t t l e change in the lo n g itu d in a l p re ssu re p r o f i le o f

. 35

the sm all tube a f t e r th e t r a n s i t io n o f the a i r core h e l ix . This can, be

seen by comparing the Y/X values a t 16,85 gpm and 20,2 gpm in Table C-4,

In F igure 11 , th e measured values o f n c-j a t various flow ra te s in

the la rg e tube in d ic a te th a t an asymptote i s approached. The dimension™

le s s a i r core rad iu s a t the top o f the tube approached a l im i t o f 0 , 7 2 *

The n c-j_ values in the sm all tube, l i s t e d in Table C-3, were n o t p lo t te d

because o f the la rg e e r ro r involved in the m easured.values* This e r ro r

was due to the sm a ll 'd is ta n c e s invo lved . The la rg e tube measurements

were considerab ly more a c c u ra te , M arshall (1954) rep o rted a l im itin g

dim ensionless a i r core in sp ray nozzles o f 0 ,6 1 ,. This rep o rted value

was measured by photographing flow co n d itio n s in sm all I n c i te spray

n o z z le s ,

' The Y/X values fo r various r a d ia l p o s it io n s , l i s t e d in Tables 0-6

and 0-7 and i l l u s t r a t e d in F igures 12, 13? and 14, showed some n o n - lin e a r

c h a r a c te r is t ic s . The degree o f n o n - lin e a r i ty was, however, n o t as g re a t

as the p o s tu la te d lam inar behavior f o r the same co n d itio n s o f lo n g itu ­

d in a l p ressu re and a i r core d iam eter a t the to p , I t should be no ted th a t

• the measured r a d ia l values a re known.to be in e r ro r . Measured p re ssu re s •

in the w ater p o rtio n o f th e v o rtex th a t corresponded to a vacuum o f 14 .

inches o f w ater were recorded in th e sm all tube a t th e h ig h e s t flow r a t e .

A lso, an apparen t vacuum cond ition was c rea ted by in je c t in g the same

need le p e rp en d icu la rly in to a s t r a ig h t w ater j e t . Flow sep ara tio n a t th e •

needle t ip caused "the d iscrepancy in both cases. The need le caused the

flow s tream lin es to d e f le c t and c re a te a low p re ssu re a rea r ig h t a t th e

t i p . At high flow r a te s , foam was v i s ib le in th is low p re ssu re a re a .

However, the e f f e c t o f th i s e r ro r was to induce more n o n - l in e a r i ty in to

the measured v a lu e s . The v e lo c ity n ea r th e tube c e n te r , n = 0, was

known to be g re a te r than n ea r the w all, n = 1 . The values n e a r th e tube

c e n te r were, th e re fo re , subject- to a h ig h e r magnitude o f e r ro r than th e -

measured values n e a r the tube w a ll . The e r ro r in measurement caused an

in c re a se in the c a lc u la te d X/X value over the tru e v a lu e . Since the

e f f e c t o f the e r ro r was to r a is e the Y/X value more n e a r th e c en te r than

n e a r th e w all, th e a c tu a l r a d ia l p r o f i le .should be approxim ately l i n e a r .

The in c re a se in the frequency o f the a i r core, h e l ix observed in

th e sm all tube, shown in Figure 16, occurred sim ultaheon-sly w ith the

appearance o f th e foam p iece , which i s shown in F igure 1?. The foam

p iece had the appearance o f a h y d rau lic jump. -S tre e te r ( 1962) d escrib es

a h y d rau lic jump.as fo llow s: "Under, p ro p er cond itions a ra p id ly flow ing

stream o f l iq u id in an open channel suddenly changes to a slow ly flow ing

stream, w ith a . la r g e r G ro ss-sec tio n a l a rea and a sudden r i s e in e le v a tio n

o f l iq u id su rfa c e . . . . the ra p id ly flow ing j e t expands and converts

k in e t ic energy in to p o te n t ia l energy and lo s s e s o r i r r e v e r s i b i l i t i e s . ' A

r o l l e r develops on the in c lin e d su rfac e o f the expanding l iq u id j e t and

draws a i r in to the l iq u id ." The foam p iece was an a rea o f a i r e n tra in ­

men t . There was a change in th e a i r core d iam eter o r w ater th ick n ess a t

the foam p ie c e . A d e f in i te change in v e lo c ity was ev id en t by d i r e c t

o b serv a tio n o f th e foam p iece and was suggested by Figure 1?. Thus, th e

foam p iece was probably a h y d rau lic jump.

A h y drau lic jump in an open channel i s , according to S tr e e te r ,

d i r e c t ly r e la te d to th e Froude lumber* the r a t io o f k in e t ic forces to

:g r a v i ta t io n a l fo rc e s . By r e f e r r in g 'to T able-0-3, i t can be seen th a t the

Froude Number could be re la te d to the jump observed in th e sm all tu b e .

■ ■ - 3?

The Froude Number reached a value o f 10 In th e small tube before th e ,

jump appeared* The Froude Number ■ reached a maximum o f o n ly 0*816 in th e

la rg e tube® According to S tr e e te r ( 1962) , A Froude Number range o f 20 -

t o 80 i s th e region o f w ell developed jumps in open re c ta n g u la r channels $

where the Froude Number i s based on th e l iq u id depth® The Froude Number

in the sm all tube when the jump appeared was, th e re fo re , th e p roper o rd e r

o f magnitude® I t should be mentioned th a t th e Reynolds.Number m ight a lso

be r e la te d since i t had a value o f approxim ately 6 1 ,0 0 0 when th e t r a n s i ­

tio n occured and the jump appeared® This was n e a r the same value as the

maximum in the la rg e tube® : The d iffe re n c e in . the method. o f .a ttach in g the

nozzles to the p le x ig la s s tubes m ight a lso have been a co n tr ib u tin g fa c ­

to r to th e appearance o f the jump in th e sm all tube bu t n o t in the la rg e

tu b e «

. As mentioned b e fo re , th e re was a change in a i r core d iam eter a t

th e h y d rau lic jump® This d is c o n tin u ity , when p re se n t, was the low er

. bound f o r th e a i r core helix® The appearance o f the h y d ra u lic jump,

■ th e re fo re ,-c a u se d a change in th e le n g th o f th e a i r core helix® Assume .

• th a t the a i r core su rfac e , S, could be d escrib ed by

iaxsl.S-j(r, z , 6 ) = f 2 r , z , ti, son ^ (to).where 1 i s the len g th o f . th e tube from the nozzle to th e e x i t , 8 i s the :

angu lar co o rd ina te , and n - is an in te g e r . Then, th e a i r core su rface

. a f t e r th e appearance o f th e h y d rau lic jump could be d esc rib ed by -

S2 ( r , .z, 8 ) = f 2 z , 8 , s in ( 11)

where A i s th e .change in le n g th o f th e h e l ix , and m, i s an in te g e r . The

’ observed t r a n s i t io n in th e frequency o f the a i r core h e l ix , th e re fo re ,,

38

could be a t t r ib u te d to : the appearance o f the h y d rau lic jump s ince A fo rces

a change in the fundamental mode o f th e a i r core s u rfa c e , a s seen b y '

comparing Equations (10) and (1 1 )e -

The n eg a tiv e a x ia l v e lo c i t ie s , n e a r the c e n te r o r along the a i r

core, , found in th is study v e r ify a re p o rt made by H u tta l (1-953) <> But,

i t should be noted th a t th i s n eg a tiv e a x ia l v e lo c ity d id d isappear a t

h ig h er flow ra te s in th is stu d y . The p resence o f n eg a tiv e a x ia l v e l­

o c i t ie s in d ic a te d th a t th e re was a re c irc u la t io n o f a p o r tio n o f the

.f lu id . . This means th a t flow in a f re e v o rtex i s d e f in i te ly th ree dimen- .

s io n a l . • ■ .. ..

CONCLUSION AND RECOMMENDATIONS

The lo n g itu d in a l p ressu re p r o f i le s were measured, and i t was.

found th a t the d im en sio n le ss 'p re ssu re , was a n o n - lin e a r function

o f z/Lc The r a d ia l p ressu re p r o f i le s were a lso m easured, b u t w ith more

d i f f i c u l ty than the lo n g itu d in a l p r o f i l e s e The r a d ia l r e s u l t s showed

th a t the d im ensionless p ressu re was approxim ately a l i n e a r function o f

d im ensionless ra d iu s , n . The dim ensionless a i r core rad iu s a t the tube

e x i t approached an asymptote .of 0 »?2o The n eg a tiv e a x ia l v e lo c i t ie s

rep o rted in th is s tudy v e r ify the re p o r t o f N u tta l (1953); b u t, the

re p o rt was extended by showing th a t the n eg a tiv e a x ia l v e lo c i ty d id

in d ic a te r a d ia l v e lo c ity components and the need fo r a th ree-d im ensional

model o f the system » .

During the period o f the in v e s t ig a t io n , some problems w ith the .

equipment were experienced .. and should be remedied b e fo re any f u r th e r

. in v e s tig a tio n s a re undertaken . A w ater s h ie ld th a t would cap ture a l l

: w ater and spray e x itin g from the v o rtex tube o u t le t should be designed

and c o n s tru c ted . This sh ie ld should in co rp o ra te a system fo r re tu rn in g

the f lu id to th e feed tan k . The feed 'system should have adequate capac­

i t y fo r a wide range o f f lo w :ra te s in th e la r g e s t tu b e to be used . A

wide, s e le c tio n o f ro tam eters should be a v a ila b le f o r acc u ra te flow r a te

d e te n a in a tib n s . . I f sev e ra l tube s iz e s a re to be used , a quick method

fo r changing tubes would be advantageous. .

• . ; -. . S everal d i f f i c u l t i e s , were encountered w hile measuring the

lo n g itu d in a l p ressu re p r o f i l e s . The p r o f i le s were d is to r te d in th e .

' ' 39/ . ' ■ ■ • , V ,

" w

la rg e tube by a lo n g itu d in a l en trance e f f e c t th a t extended approxim ately

s ix tube d iam eters from the bottom o f the n o zz le 0 An asymmetric en trance

e f f e c t was found experim en tally a t the bottom o f th e la rg e tube a t h igh

flow r a te s . This en trance e f f e c t c o n s is te d o f a h ig h e r bulk v e lo c ity on

the s id e o p p o site th e ta n g e n tia l: i n l e t . I t i s . l ik e ly th a t th i s asymmet­

r i c a l problem could be remedied by the use o f a sym m etrical en trance ,

such as tro ta n g e n tia l i n l e t s which a re opposite, one a n o th e r . An e x i t ■

e f f e c t was observed in both tubes a t th e h ig h e r flow r a t e s . The e x i t

. e f f e c t extended approxim ately fo u r tube diam eters down from the to p , - At

high flow r a te s , n e a r ly the e n t i re p r o f i l e o f the la rg e tube was b lanketed

by en trance and e x i t e f f e c t s . This in d ic a te d th a t lo n g e r tubes should be

used f o r any -fu rth e r s tu d ie s o f lo n g itu d in a l c h a ra c te r is t ic s =■ A very

long in c lin e d manometer tube w ith a f in e sc a le might be u se fu l fo r f u r th e r

s tu d ie s o f lo n g itu d in a l p r o f i le s , p a r t ic u la r ly a t the low er flow r a te s

where p ressu re d iffe re n c e s a re sm all.

R adial p re ssu re p r o f i le s proved to be considerab ly more d i f ­

f i c u l t to measure than the. lo n g itu d in a l p r o f i l e s . Known e r ro rs in th e

measured r a d ia l p re ssu re s in d ic a te d th a t an improved procedure fo r meas­

u rin g p re ssu re s i s needed, A method fo r p reven ting flow sep ara tio n a t

the measuring device i s necessary fo r a p roper r a d ia l p r o f i le measure­

ment, O btaining a r a d ia l .v e lo c i ty p r o f i l e by a method th a t would n o t

involve a probe might be e a s ie r and more in fo rm ative than ob ta in ing a

p ressu re p r o f i l e u sin g a probe. The la r g e r d iam eter tubes a re advanta­

geous fo r measuring a r a d ia l p r o f i le s in ce more d a ta p o in ts can be

ob ta ined w ith g re a te r accuracy than w ith a. sm all tu b e ,

41

The de term ina tion o f ' the a i r core rad iu s a t th e tube e x i t was

com plicated by the h ig h -f lo w -ra te t r a n s i t io n o f the asym atric a i r core

helix® The t r a n s i t io n in the frequency o f the a i r core h e l ix in the

sm all tube was caused by the appearance o f a h y d rau lic jump. The appear­

ance o f th i s jump was re la te d to the Fronde Number and p o ss ib ly the

Reynolds Number® A symm etrical en trance m ight e lim in a te th e h e l ic a l

flow pattern®

nomW clature

A change in a i r core len g th , f t ,

g g ra v i ta tio n a l a c c e le ra tio n , f t , / s e c , ^

h measured w ater le v e l , see F igure 6, f t ,

£ _ a i r core le n g th , f t ,

L . len g th between th e f i r s t p re s su re ' tap and. th e tube top,, f t ,

0 an g u lar .p o s itio n , rad ian s . ,

; p d e n s ity o f w ater, lb m ,/ f t ,3

p p re ssu re ,' l b m , / f t 0~sec»2

r r a d ia l p o s it io n co o rd ina te , f t , .

R tube ra d iu s , f t# : ■ , ■

S ' a i r core su rfa ce , Equation (10)

- v ta n g e n tia l v e lo c i ty , . f t , / s e c ,

w . a x ia l v e lo c ity , f t . / s e c , ../■.->■

z lo n g itu d in a l p o s it io n co o rd in a te , f t .

■ ' ' n = r/R . ‘

v - _■ • : P. = p + p g z • ■ 1 V ,:V ,V: ./

S u b scrip ts

a ' barom etric

c a i r core

1 z = L

O 2 = 0

APPENDIX A

EQUIPMENT DESCRIPTION DATA

44

45

100

FlowRate(gpm)

104 10 100

Nozzle I n le t P ressu re ( in . H20)

Figure A-1

Tlie E ffec t o f I n le t P ressure on Flow Rate fo r the Large Tube

Slope = 0 .5 9 1

46

FlowRate(gpm)

100

100 3 0 0Nozzle I n le t P ressu re

( in . HgP)

Figure A-2

The E ffe c t o f I n le t P ressu re on Flow Rate fo r th e Small Tube

4?

10

8

6Flow Rate ( gpm)

4

2

w ater a t 80° F

6040 8020 100 120

$ fo r Flow

Figure A-3

Small Rotameter C a lib ra tio n Curve

48

60

50

40

FlowRate 30 (gpm)

20

10

010 20 30 40 30 60 70

io o f Flow

Figure A-4

Large Rotameter C a lib ra tio n Curve

w ater a t 80° F

49

TABLE -A-t

F io w -In le t P ressu re Data

Small Tube . Large Tube

Flow 12 Pj . Flow - 12 P1Rate - • Pg Rate Pg(gpm) lino.,of.,Tja-terI I shb)_______ ( in . of Water).

2*02 4=99 12,55 5*77

3.41 7 = 50 ' 16=85 8=89

4*31 9=38 21=6 13*28

5=30 12=51 25=9 . 18=33

6,45 18=8 30=2 24*0

7=93 27*5 33*9. 32=0

9=08 37=6 41*1 44,6

16*85 116.0 47=7 « » 0 0 moot*

20 * 2 183=0

23=0 247=0 ,

50

TABLE A-2

Rotameter C a lib ra tio n Data . . '

Small Rotameter

;l_QTJ5-og Weight, .(lbs). T im e jm in ,! • ggmZ

Zk 167.75 10;00.0 ' 2.02

40 283.25 . 10:00.0 ■ 3.41

50 358.5 10:00.0 , 4,31

60 302.0 : 7 :00 .0 5 .3 0 ,

' 74 375.5 : . 7 :0 0 .0 . 6 .45

90 . 394.75 . 6 :0 0 .0 7.93

Large Rotameter

^ o f Flow Weight ( lb s ) Time (m in.) gpm*

19 523.5 . ■ 5 :00 .0 ' 12.55

25 560.0 4 :00 .0 16.85

32 . 597.5 . 3 :20 .0 21.6

38 ■ 609 .0 2 : 5 0 .0 , 2 5 .9 .

• 44 ■ ■■ 503.0 2:00.0 ' 30 .2

^■ 50; ' : /564.0 ' ; 2 :00.0 .. ; 33.9

60 655.5 ' 1 :55.0 . 41.1

70 - - - 596.5 (V' : 1 0 0 .0 . 47.7

* gpm f ig u re i s based on a d e n s ity o f 6 2 .2 I b / f t ^ a t " 80° F

APPEMHEX B

PRESSU RE P R O F IL E O R IG IN AL DATA

51

TABLE 3 .1

L o n g itu d in a lly Measured Values o f h fo r th e Small Tube

12h ( in . )

FlowRate(gpm) 0 : *4 ■ 8

2 ( in . )

. I 2 16 . ' . 2 0 _ . 24 87 9.5

2*02 0.4750.432

0 .3 6 70=367

0.3020.281

0 . 25?0.238

0.1940.194

0.1730.173

0.1300.151

0.0860.108

3.41 1 .5 1 21.340

1/123 1.100

0.863O.8 6 3

0.7550.734

J3"0 0

\q\o0

0 O O 0=410

0.4320.3020.323

4.31 2.10 2 .12 .

1.771=75

1*321.34

1.171.16?

0 .9 3 00 .9 5 0

. 0.842 0.842

0.5800.604

0.433 0.388 •

6 .45 5.104.70

4.402 .8 9

3.472 .9 6

2.662.59

2.1-22.10

1=791=77

1.121.10

0 .65 . 0 .6 5

7.93 7.282.40

6.186 .02

4 .844.66

4 .154.10

3=433=39

2.852 .8 9

1 .921 =90

1.151.12

16.85 30.6332=19

2 6 .44 26125

20.7020.56

18.7118.84

15=9316.59

13=4513=20

10.0010.20

6 .0 36 .1 5

20.2 43=7542=75

36.5036.00

28.63.28.06

26.5026.38

22.5122.42

18.50 18.57

13=9514.05

8 .9 58.20

23.0 • 55=50 56 = 50

46.0045.13

36 = 25 3 6 .6 3

34.5634.75

30.0630.00

23.4324.12

19=78.1 9 .2 2

10.7510.97

53

TABLE B-2

L o n g itu d in a lly M easured.Values o f h fo r the Large Tube

12b ( i n e)

2 ( in . )

SlowRateI s s s I 0 6 12 18 21 ' -24.JL

12*55 0.995 0 .692* 0 .6 9 2 0 .4 9 7 0.432 0.3890 .9 0 7 • 0 .7 1 3 0,648 0.475 • 0.432 0.389

16.85 1.51 0 .9 5 0* • 0.950 0.713 0.649 0 .4 9 71.40 1 .0 7 0.973 . 0,886 0 .6 2 6 . • 0 .4 7 5

21.6 2.46 . 1 . 47* 1.51- 1,12 0.995 0.7772 .2 5 1 .5 8 1.49 1 .0 6 0.950 0.735

25 »9 3.50 2.12* 2,24 1.68 1.40 1,103.50 2.31 2,22 1.58 1.42 1.10

3 0 .2 4.86 2.91* 3.19 2.38 . 2,05 ; 1.514.84 3 .2 6 3*1? •2.38 2.01 1.49

33.9 6.39 2.38* 4.23 3.17 2,68 1.996 .3 8 4.32 4,23 3.22 2.70 1.96

41.1 9 .45 . 5,69* 6 .24 ' 4 .84 4.01 1.949 .34 , 6 .57: 6 ,2 6 • 4.86 ■ 4.01 2,94

47.7 12.90 ' 9 .00 8 .82 : 6 .7 6 5.41 3.9713.05 . 9 .04 . 8.57 \ , 6 .75 - 5.49 3.98

* bad point# le a k in tap sea l

12 y (iru ) .

0*00*1850*31250.4375

0*0.0*1875 0*3125

• 0..4375

0*00*18750*31250*4375

o.o0*18750.31250*4375

: ; ■, . ■ , 54

TABLE B»3

R ad ia lly Measured Values o f h fo r t h e 'Small Tube

= 0 12z = 12 i n . 12a = 20 in*

12h ( in . ) 12r ( i n . ) . 12h .( jr i8) 12r ( i n . ) . 12h ( in . )

Elow. Rate = 4*31 gpK ' •

-2*95 ' 0 .0 -2*07, -2 ,2 8 0 .0 -1*31-0*971 0 .1 8 7 5 -0*42, -0 .3 8 - 0*1975 - 0 .? 6+0*253 0*3125 +0.21, +0*17 0*3125 -0 .1 3+0 *928 0.4375 +0*63, +0*59 0*4375 +0*25

Flow Rate = 6 .45 gpai

- 7*30 0*0 - 5«79 0*0 -3 .1 8- 2*95 0 *1875 - 1*98 0 .1 2 5 . - I .90- 0*169 0 *3125 +0*04 0 .3 1 2 5 +0 .1 3+1*35 0 .4 3 7 5 +1*98 0 .4 3 7 5 +0.34

Flow Rate = 7*43 gpm

,10.80 0*0 -9 *24 0*0 ~5«50-4*39 0.1875 -2 .7 4 0*1875 -2 .2 4-0*80 2 0*3125 -0 .21 0*3125. -0*30+2.00 ■ 0.4375 +1.09 ■ 0*4375 - +0.55

• Flow f e te = 16.85 gpm

,28.00* 0*0 - 1 6 . 0 0 * 0*0 .-8 .0*,23*20 0.1875 -16.00** 0 . 1 8 7 5 - 8 . 0 * *-8 .80 O.3 1 2 5 -8*40 0.3125 -2*55-h6 .5 0 0*4375 *2.54 0.4375 +’2*41

* need le in a i r core

**.need le a t edge o f a i r core

55

TABLE BJ+

R a d i a l l y Measured Values o f h . f o r th e 'L arg e Tube

12a = 0 12a = '12 in , 12a = 21 in®

12r .( in .) 12r ( i n . ) . , 12h ( in . ) • I 2 r ( in .,)., 12h (In

ELow Rate = 2 1 .6 gpm

0 .0 ' - 8.96 0 .0 - 8 ,9 0 0 ,0 -5 .0* .0.375 -3 .90 ■ 0.375 —3 ®71• 0.356 -5.0**0 .6 2 5 -2 .1 5 0 .6 2 5 -1 ,71 0 .5 0 0 -3 ,500.875 -0 .7 7 , 0.875 . 0 .5 6 0.750 . -1 .501 .1 2 5 40.05 1.125 40.25 1 .0 0 . 0 .2 1

Flow Rate r= 30.2 gpm

0 ,0 • -16.77 0 .0 .13.0* 0 .0 -5 ,0*0.375 - 8 .6 1 0.375 - 8 .0 2 0 .5 0 0 : -5 .0**0.625 -5 .7 7 0 .6 2 5 - 3 <.56 0 .7 5 0 -3 .830.875 -2 .50 0.875 -1 .3 3 1.00 -1 .601.125 0.0 1.125 40.17 «ss» ms cs*ts»ee»

. Flow R ate 33.9 gpm

0.0 - 2 5 . 0 * : . 0 ,0 4 3 ,0 * 0 .0 ■ -5 .0*0.375 -13.56 0.375 -11,52 0 .6 2 5 ..-5,0**0.625 -6 .1 5 0.625 - 5 .0 2 : 0.875 -3 .500.875 -3 .50 0 .8 7 5 - 1 ,7 7 I . I 25 - 0 .7 71.125 — 1.50 1.125 >1.11 esa cBOisato tea B20 «k> esa <317 TO

: * needle in a i r core

** need le a t edge o f a i r core

A PPEN D IX C

D IM M S IO N L E S S PROCESS DATA

56

57

TABLE C»1

Y Values a t the Wall, Small Tube

•z/L •

FlowRatei s e a l '' 0.143 0.286 0.428

2.02 0.20?.0.150

0 0 0.455

0.449

3A1 0.2570.179

0 .4 2 90.358

• 0.501 0,455

4.31 0 . 16?0.165

0 .3 6 20.377

0.4430.448

5.30. 0 .1 5 10.174

O O 0.450

0.428

6.45 0.1370.189

0 .3 2 00.370

0.4780.449

7.93 0.1510.187

0.3350.371

0.4440.446

16.85 0.1370.184-

0.3240.361

0.3900.415

20.2 • 0.165 0.158

0.3450.344 0

.0

00 \o

-£*

-P*

2 3 .0 0.2010.171

0 <r\

0 0 0.385

0.377

O ts n 0=714 . M S - 0.981

W Cvj

33O O 0 .6 3 6

0 .6 0 00 .7 2 60 .6 5 0

0.8020.750

O O 0 .644

.0 .5 9 70 .7 2 8

■ 0 ,6 7 80.8020 .7 6 2

O.5470 .5 6 2

0 .6 0 00.604 O

O

~<5 -<} 0 .814

0.798

0.557 0 .5 6 6

0 .6 1 20.625

0 .7 4 5 0 ,7 6 0

0.8620.843

0 0 0.649

0.6230.781O.767

0.8730 .8 6 2

SI

0 0 0,609

0 ,6 0 90 ,7 3 60.744

0.830 0.848

0.4810.484

O.5 6 20.590

0.6740.684

0.8040.809

0.4860.474

0.578 0 ,5 6 6 .

0 .682 O.6 7 4 .

0,8120.808

O O

43 0.5740.578

0.6610.644

0.806 0.806

58

TABLE. 0-2

T Values a t the Wall, Large Tube

z/L111

QeZkO 0.480 0.720 0.840 o .2Z2 __

12,55 0,305* 0.305 0.497 0 .5 6 6 ' 0 .6 0 80,214 0.286 0.476 0.513 0 .5 7 1

16,85 0,371* O.3 7 I 0.530 0.569 0 .6 6 90,236 0 ,3 0 8 0,364 0 ,4 7 8 0 .6 5 7

21,6 . 0.402* 0.386 0.544 0.594. 0,6830.298 0.338 0,529 0.578 0 .6 7 2

25.9 0.394* 0 .3 6 0 0.520 0 .6 0 0 0.6860.340 0 .3 6 6 0.548 0 .5 9 4 0.686

30.2 0.402* 0.344 0 .5 1 0 0 .0 5 7 8 0.6900 .3 2 6 0 b 345 0,508 0.584 0 ,6 9 2

33.9 • 0.393* 0.338 . 0.504 0 .5 8 0 0.6880 .3 2 2 0.337 0 .4 9 5 . 0.577 . O.6 9 2

41,1 0.398* . 0.340 0,488 0.58? 0 .6 8 9.0 .2 9 6 *■ 0.330 0.480 0.565 0 .6 7 8

47,7 0.302 0.332 . 0.476 0.581 O.6 9 20.307 0.344 A 0 "475 0.579 O.6 9 4 ...,

*■bad p o in ts , le ak in tap sea l

; • 59

; ' TABLE.C~3:

Flow Rate, I , nci». Reynolds Ntmber, Fronde Rumber and Weber Number

Flow Rate .. ( sm ) ■ X

n c l ReynoldsNumber

FroudeNumber

WeberNumber

Small Tube

2,02 0 .?85 4,510 0.139 188

3.41 0.790 1 0 ,9 0 0 0.401 555

4.31 0.818 ■ 13,800 0.640 885

5.30 0.875 16,600 0.923 1,280

6 .4 5 0.887 0.459 2 0 ,7 0 0 • 1.43 1,980

7.93 . 0.857 0.556 2 5 ,2 0 0 2.17 3,010

9.80 0.542 2 9 ,1 0 0 2.84 3,930

16.85 0.820 0 .6 9 4 54,000 . 9*77 13,500

20.2 0.825 64,800 • 14.0 19,400

2 3 ,0 0.825 72,900 18.3 25,400

Large Tube -

12.55 0.598 16,200 0.055 177

16.85 0.675 - 0.419 21,700 0 .1 0 1 312

21.6 0.682 0.618 27,900 0.166 , ' .512;

25.9 0.697 0.674 . 32,700 0.235 ... : 726

3 0 .2 0.701 0.695 38,900 0 .3 2 6 1,010

33.9 0.700 0.693 43,700 0,411 1,270

41,1 ■ 0 .6 9 6 : • 0.713 53,000 ; 0.603 1,860

4?.7 0 .7 0 0 . 0.713 61,600 0,816 2 ,5 2 0

60

TABLE. 0-4

Wall Values o f Y/X*, Small Tube

Flow . RateI eeeI- M M 0.286 0.428

2.02 0.22? 0 .4 5 5 0.575

3-41 0.262 0.497 0 .6 0 2

4.31 0.203 0.452 0.545

5-30 0.186 0.429 0.502

.6=45 0.189 . 0.389 0.524

7-93 0.197 0.412 0.519

16.85 0.195 0.417 0.491

20.2 0.196 0 .4 1 8 ' 0.472

23=0 0 .2 2 5 0.423 0.462

'/L ...

WZ1 M B M S ■ MM0 .7 3 0 0 .7 8 7 0 .8 7 6 0 .9 8 9

0 .7 2 9 0 .7 8 7 0 .8 9 0 0 .9 9 0

0.677 0.737 0.880 0.986

0.641 0 .7 0 6 0 .8 5 9 - O.9 7 4

•0.642 0.716 0.872 0.978

0.626 0.711 0.863 O.9 8 I

0.589 0.70 2 0.840 0=985

0.582 0.695 ■ 0.832 0.982

0.561 O.699 O.79O . 0.978

* average Y values used

6i

TABLE C~5

Wall. Values o f Y/X*, Large Tube

z/L

FlowRate.(.Spm) 0.240 0.480 O.720 0.840 M 2 2 .

12=55 0.358 ' 0.495 0.794 0.903 0 .9 8 6

16.85 . 0 .350 ' 0,504 0 .6 6 8 ' 0 .774 0.983

2 1 .6 0.437 0.531 O.786. 0 .8 6 0 0.992

25.9 0 .489 o .521 0 .7 6 9 • 0.857 0.971

30.2 0.465 ' 0 .4 9 2 O.7 2 6 0 .8 2 9 0 .9 8 5

33.9 . 0.461 0.482 0.715 0.827 0 .9 8 6

^1.1 0.425 0.481 O.6 9 4 0.827 0,983

47.7 .0 .437 0.483 0.681 0.828 0 ,9 8 8

* average Y values used

62

TABLE: 0-6

R adial Values o f X/X fo r the Small Tube

z/Ij ~ Q z/L = 0' =428 z/L f 0=714

n I/X n Y/X , . n X/X«f**MWct*tirss=5ses

Flow Rate ~: 4=31.gpm

0,0 . 2 .9 3 0=0 2.46 0 .0 1=980=333 1=79 0=333 1 =46 0=333 1 =66o°55& , 1=08 0=556 1=11 0=556 1=300=778 0 =684 0=778 • 0 =870 0=778 1=081=00 0=0 1=00 0 0 544 1=00 0=734-

Flow Rate = 6 =45 gpm.

0 .0 2=81 0=0 2=67 . 0=0 1 .8 60=333 1 =80 . ' 0=333 1=58 0=333 1=560=556 1 =67 0=556 , . 1=12 . 0=556 1=100=778 0=817 . 0=778 0 = 596 0=778 1=051=00 0=0 1=00 0=522 1 =00 0=718

- - Flow Rate = 7 =43 gpm

0=0 2=89 0=0 2=64 .0=0 ■ 2.040=333 1 =87 0=333 1=60 • 0=333 1=520=556 1=30 0=556 1 =20 0=556 . 1.220=778 . 0=850 0 =778 0 =994- 0=778 1 =001=00 0=0 1=00 0 = 512 1 =00 0.712

Flow R a te '= 16=85 gpm

0=0 . 2=31* 0.0 1=84* ■ 0=0 1=53*0=333 2=12 0=333 1 =84* 0=333 1=53*0=556 . 1 =56 0=556 1=54 0=556 1=320=778 0 =968 0 .7 7 8 1=12 0=778 1=131=00 0=0 1=00 0=494 1 =00 0=705

* denotes needle In a i r core o r a t in te r f a c e

63

. TABLE C~?

Radial' Values o f X/X fo r the Large' Tube

z /l , = 0 . ■ z/L = 0,480 ■ z/L = 0.840

n . n . Y/X n Y/X

' Flow Rate = 2 1 ,6 gpm

0.0 ' 7 .0 6 ' 0 .0 7.04. ; 0 .0 . 3»96*0 , 26? ■ 3.90 0 .2 6 7 ■ 3.78 O.2 6 7 3.96*0 e l'4'4' 2.81 0,444 2 .5 3 0.356 - 3.590.623 1 *95 0.623 1.82 0.533 2.340,801 1.22 0.801 ‘ 1,32 0.713 1.60■1.00 0,0 1,00 0.524 1.00 0.863

F3.ow Rate = 30.2 gpm

0.0 6 .3 6 0,0 5.25* 0 .0 2 . 6 0 *0 .2 6 7 3.96 0.267 3.79 0 .2 6 7 . 2 . 6 0 *0 . 4 # 2.83 0,444 2.45 0 .3 5 6 2 .60*0.623 2.13 0.623 1.82 0 .5 3 3 2.550.801 1,00 0.801 1.38 0 .7 1 3 1.901.00 0 .0 1,00 0.488 1 .0 0 . 0.832

Flow Rate =33*9 gpm

0 .0 7,04 0 ,0 4 .34* . 0 .0 2 . 31*0 .2 6 7 4.44 0 .2 6 7 4,01 0 .2 6 7 2 . 31*0 .6 2 3 2 .1 9 0 .6 2 3 1.83 0.623 2,190,801 1.74 0 .8 0 1 1.18 0 .8 0 1 1 ,6 01,00 0.0 - ' 1.00 0.481 1.00 0.82?

> denotes need le in a i r core o r a t in te r f a c e

APPMDIX D

SAMPLE CALCULATIONS

C alcu la tio n o f Y/X fo r the small tube a t 12z = 24 i n , , It

z 24.i n ,IT = 2 5 -15 : = 0-857 .

From Table B-1

12h0 = 30,63 in<

12h = 10,00 i n .

But, P0 ~ P <*Ll2pg(h0 - h) = 12pg(20,63 in ,)

And, P0 - P a ^ e g h 0 ,

| i § = 0.674

From Table 3, X = 0 ,820 ,

Thea r = § iS§ = °»823-

,85 gpm;

64

LITERATURE CITED

B in n ie , A. M. A nnular Borda Flow, J o u rn a l o f F lu id M echanics, 19,2: 18? (1964)

Cox, Weil D. An A n a ly tica l So lu tion o f Laminar Vortex Flows (unpublished)( 1967) .

Kuchemann, D, R eport on the I . U. T. A. M. Symposium on C oncen tra ted V ortex Motion in F lu id s , Jo u rn a l o f F lu id M echanics, 21, 1: 1(1965).

M arsh a ll, V. R. A. I . Ch. E. Monograph S e r ie s , 50, 2 (195^)*

N u tta l , J . B. N atu re , 1?2: 582 (1953).

S tre e te r , V. L. F lu id M echanics. McGraw-Hill, New York, pp. 125-126, 493 ( 1962) .

W hite, A. Flow o f F lu id in A x ia lly R o ta tin g P ip e , J o u rn a l o f M echanical E ngineering S c ie n ce , 6 , 1: 47 (1964).

65

I 3 U