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Scholarship at UWindsor

Scholarship at UWindsor

Electronic Theses and Dissertations Theses, Dissertations, and Major Papers

1-1-1971

Turbulent, single-phase velocity distributions for parallel flow in

Turbulent, single-phase velocity distributions for parallel flow in

rod bundle assemblies.

rod bundle assemblies.

Joseph F. Martyniuk

University of Windsor

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation Recommended Citation

Martyniuk, Joseph F., "Turbulent, single-phase velocity distributions for parallel flow in rod bundle assemblies." (1971). Electronic Theses and Dissertations. 6692.

https://scholar.uwindsor.ca/etd/6692

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TURBULENT, SINGLE-PHASE VELOCITY

DISTRIBUTIONS FOR PARALLEL FLOW IN

ROD BUNDLE ASSEMBLIES

by

Joseph F. Martyniuk

A Thesi s

Submitted to the F acu lty o f Graduate Studies through the Department o f Chemical Engineering in P a rtia l F u lfillm e n t of

the Requirements fo r the Degree o f Master o f Applied Science a t the

U n iv e rs ity o f Windsor

Windsor, Ontario

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INFORMATION TO USERS

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A knowledge of the velocity distribution in nuclear reactor fuel

rod bundles is essential for the accurate prediction of critical heat

flux. In this study, a mathematical computer model was developed based

upon Deissler and Taylor's [1] work for axial flow between banks of rods

and tubes. Flow split predictions for a triangular-square array were

determined over a Reynolds number range of 2 x 10^ to 10"*, and at pitch/

rod diameter ratios from 1 to 2. Variations of shear stress and

cross-sectional average velocities with circumferential direction were also pre­

dicted for an infinite triangular array. It was found that friction

factor and the variation of shear stress and cross-sectional average

velocities were in good agreement with earlier models. The mass flux

ratio for triangular/square subchannels increased with gap spacing and

was weakly dependent on Reynolds number. In conclusion, no satisfactory

mathematical model has yet been developed to predict flow and temperature

distributions in fuel rod clusters as all the previous mathematical

models ignored secondary flow effects.

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ACKNOWLEDGEMENTS

The author wishes to express his g ra titu d e to Dr. C. C. S t. P ie rre

fo r his guidance and advice throughout the study.

Thanks are also due Mr. K. Singh and Mr. K. Petrunik who gave con­

s tru c tiv e comments and c ritic is m s when confronted fo r advice. Special

thanks go to Mr. D. Candido fo r his many e f f o r t s , e s p e c ia lly the drawings

and graphical p lots appearing in th is th e s is .

The fin a n c ia l support provided by the National Research Council o f

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P iS i

ABSTRACT i i i

ACKNOWLEDGEMENTS iv

TABLE OF CONTENTS V

LIST OF FIGURES v i i

LIST OF TABLES ix

I INTRODUCTION 1

I I LITERATURE SURVEY 5

A. D eissler and Taylor 5

B. Rapier and Redman 8

C. Bender and S w itick 10

D. N ijs in g , Gargantini and E if l e r 14

E. E if l e r and N ijs in g 17

F. Launder and Spalding 19

I I I PROGRAM DOCUMENTATION 21

A. Basic D e fin itio n s 21

B. Evaluation o f y+ 23

C. F ric tio n Factor C alcu latio n s 25

++ ++

D. Determination o f u , r , dp/dx 25

E. C alculatio n o f Sectional Areas (An) 26

F. C alcu latio n o f Wall Shear Stress (t ) 26

G. C alculatio n o f Maximum Dimensionless

Distance from Wall (ym+ ) 28

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Page

H. C alcu latio n o f Maximum Dimensionless

-J.

V e lo c ity in each Sector (u^ ] 28

I . C alcu latio n o f Local V e lo c itie s (u) and

Distances from the Wall (y) 29

J . Subchannel Bulk V e lo c ity C alculatio n s 30

K. Conclusion o f Ite r a tio n Procedure 31

L. Determ ination o f Flow S p lit in a T ria n g u la

r-Square Array (Gt /G s ) 31

M. Output 31

IV RESULTS AND DISCUSSION 33

V CONCLUSIONS 47

NOMENCLATURE 48

REFERENCES 50

APPENDIX I : Program Input and Sample Output 53

APPENDIX I I : Flow S p lit P red ictio n Using F ric tio n 70

Factor C o rrelatio n s fo r Pipe Flow

VITA AUCTORIS

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F ig u re

Page

1. Cross-section o f ty p ic a l rod bundle 2

2. Triang ular-square arra y showing symmetrical sections

studied 4

3. Generalized v e lo c ity d is tr ib u tio n fo r a d ia b a tic

tu rb u le n t flo w . 7

4. Nodal representation o f Bender and S w itick 13

5. Lines o f symmetry and section studied 22

6. Generalized v e lo c ity d is tr ib u tio n fo r ad ia b a tic

tu rb u le n t flow 24

7. Determ ination o f areas in a tria n g u la r a rra y 27

8. Elemental area and determ ination o f cross-section al

average v e lo c ity 27

9. T ria n g u la r-a rra y Fanning f r ic t i o n fa c to r vs

Reynolds number 35

10. Square-array Fanning f r ic t io n fa c to r vs

Reynolds number 36

11. T ria n g u la r-a rra y Fanning f r ic t io n fa c to r vs

p itc h /ro d diameter r a tio 38

12. V a ria tio n o f shear stress w ith 0 39

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F ig u re

Page

14. Comparison o f shear stress v a ria tio n w ith 0 44

15. Flow s p li t p red ictio n s fo r varying p itc h /ro d diameter

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TABLE Page

I . Computer re s u lts a t various Re 3*f

I I . 1 Summary o f flow s p l i t p redictio ns using f r ic t io n fa c to r

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I. INTRODUCTION

For reasons of nuclear-physical and technological nature, the

fuel element geometry in nuclear reactors often differs greatly from

simple geometries like the circular duct. A fuel element geometry

often used is the rod bundle (Fig. 1). It consists of a number of

parallel rods from which the heat generated by nuclear fission is re­

moved by an axial flow of coolant. The design of such fuel elements

requires the solution of important problems pertaining to fluid flow

and heajt transfer. Only in some cases has reactor thermal analysis

quantitatively accounted for the mixing of coolant between the bundle

subchannels (Fig. 1).

For both heavy and light moderated water reactors the interest in

improved subchannel property definitions centers on the problem of

critical heat flux (CHF) predictions. Significant improvements in the

correlation of experimental CHF data from rod bundles should be possible

by considering the local coolant and heat flux conditions at the

ob-ser|j|d critical location rather than bundle averaged conditions. The

turbulent cooling flow in any subchannel mixes with that in neighbouring

channels on its journey through the core and transfers energy from the

hotter to the cooler subchannels thus affecting CHF predictions.

Ob-viously, improved definition mixing of parameters within and between

subchannels would provide an improvement in the prediction of fuel bundle

conditions. This relates to an increase in the net electrical output,

removal of heat for power generation purposes, and maintaining the tem­

perature within tolerable limits. It thus becomes

1

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f u e l r o d s

ROD BUNDLE

HOUSING

*

u

.TRIANGULAR

SUBCHANNEL

SQUARE „

SUBCHANNEL

TRAPEZOIDAL

SUBCHANNEL

(14)

obvious th a t a knowledge o f the v e lo c ity p r o file must be known before a

temperature p r o f ile may be estab lish ed .

The primary purpose o f th is study was to t h e o r e tic a lly p re d ic t the

natural flo w s p lits in various tria n g u la r-s q u a re arrays in rod bundles

(F ig . 2 ) ; th a t is determining the r a tio o f the mass flu x in a t r i ­

angular array to the mass flu x in a square arra y as a function o f the

Reynolds number ( Re) w ith a r a tio o f p itch to rod diameter (p /d ) as a

4 4 5

parameter. Reynolds numbers o f 2 x 10 , 5 x 10 and 10 were studied

over a p/d range o f 1 . 0 0 to 2 . 0 0 which corresponded to a gap width range

o f 0.000" to 0.750". Theo retical p redictio ns o f f r ic t i o n fa c to r , local

and cross-sectional average v e lo c it ie s , and the v e lo c ity and shear stress

v a ria tio n in the c irc u m fe re n tia l d ire c tio n were also studied . This re p o rt

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H LITERATURE SURVEY

A DEISSLER AND TAYLOR [1 ]

D e is sle r and Taylor [ 1 ] have in vestig ated a x ia l flow between rods

or tubes as configuration s which may have been advantageous fo r remov­

ing heat from nuclear re a c to rs . A basic assumption was th a t expressions

for fully developed velocity and temperature profiles in circular

tubes apply along lines normal to a wall. Further assumptions were:

1) constant f lu id prop erties

2) equal eddy d i f f u s i v i t ie s fo r momentum and heat

3) along lin e s normal to a w a ll, the v a ria tio n o f the

shear stress and heat tra n s fe r per u n it area have a

n e g lig ib le e f f e c t on the v e lo c ity and temperature

d is trib u tio n s

4) in the region away from the w a ll, molecular d i f f u s iv it ie s

are n e g lig ib le in comparison to the eddy v is c o s ity and

eddy c o n d u c tiv ity .

D e is s le r's [ 2 ] universal v e lo c ity d is tr ib u tio n curve fo r smooth

c ir c u la r tubes was used to c a lc u la te the v e lo c ity d is tr ib u tio n in each

sector o f the flow area. An it e r a t iv e procedure was used as v e lo c ity

grad ien t lin e s were unknown a t the o u tset. F i r s t ly , lin e s o f maximum

v e lo c ity and assumed v e lo c ity grad ien t lin e s were drawn. Using these

assumed v e lo c ity gradient lin e s , lin e s o f constant v e lo c ity were c a l­

c u lated . Then a more accurate set of v e lo c ity g rad ien t lin e s was drawn

normal to the constant v e lo c ity lin e s . This procedure was repeated u n til

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In order to c a lc u la te these constant v e lo c ity lin e s , the fo llo w in g

v e lo c ity parameters were defined:

u = ^ t 3/p ( 2 . 1 )

u+ = u /u * ( 2 . 2 )

u = u/ /-rg(dp/dx)/p

(2 .3 )

4*4*

where u was preferred fo r useage in c a lc u la tin g lin e s o f constant

-J*

v e lo c ity because the shear stress in the d e fin itio n o f u va ries w ith

p o s itio n .

++

A tube radius parameter r was defined as

r = /-rg(dp/dx)/p r /v ( 2 . 4 )

which is a kind o f Re.

The dimensionless distance from the wall

y+ = u* y /v (2 .5 )

was s u bstitu ted w ith ®d. 2 .4 in to eq. 2 .3 to give

u++

= u+ j r ym+ ( 2 . 6 )

+ +

r y

where r was the radius o f the rod

r ++ was assigned an a r b itr a r y value

y was the maximum distance from the w all and was calculated m

dependent on the geometry o f the subchannel a rra y .

Note th a t u+ and y+ were re la te d through D e is s le r's generalized v e lo c ity

d is tr ib u tio n fo r a d ia b a tic tu rb u le n t flow ( F ig .3 ).

(18)

7

in in

o

o

o

o

o

ID

OJ

3 >> II h

>>

OJ OJ

CO

CD

n/n = n

* +

G

e

ne

ra

liz

e

d

V

e

lo

c

it

y

D

is

tr

ib

u

ti

o

n

fo

r

A

d

ia

b

a

tic

Tu

rbu

len

t

(19)

T0 = - AA d£ (2 .7 ) Ai ax

where tq was the shear force acting on an arc length (a&) and (dp/dx) was the pressure grad ient acting over the faces o f the element. There

were no shear forces acting on the v e lo c ity gradient lin e s because the

normal v e lo c ity d e riv a tiv e s were zero along these lin e s .

S u b stitu tio n o f eg* 2.7 in to 2 .5 gave

ym+ =

AA/r

(A£) ym r ++/ r (2 .8 )

By specifying r ++ and considering a given lin e normal to the su rface,

the value o f y/ym fo r a given u++ was calcu lated by using eqs 2 . 8 , 2 . 6

and D e is s le r's universal v e lo c ity p r o f ile in th a t order. The same c a l­

c u la tio n a t constant u++ was c a rrie d out fo r various lin e s normal to

the surface and lin e s o f constant v e lo c ity (constant u++) were obtained.

New and more accurate v e lo c ity grad ien t lin e s were drawn and the tech­

nique was repeated u n til the v e lo c ity and grad ient lin e s converged.

D e is sle r and Taylor [1 ] determined v e lo c ity d is trib u tio n s fo r both

tria n g u la r and square a rra y subchannels a t p/d r a tio s o f 1 . 0 , 1 . 0 2 , 1 . 1 ,

and 2 . 0 , and found th a t the v e lo c ity lin e s near the center o f the pass­

age form closed lin e s about the po int o f maximum v e lo c ity , whereas those

close to the wall formed closed lin e s around the tubes.

B RAPIER AND REDMAN [31

A refinem ent to D eis sle r and T a y lo r's [1 ] method fo r c a lc u la tin g

v e lo c ity d is trib u tio n s in rod c lu s te r flow was made by Rapier and Redman

[ 3 ] . T h eir method was n o n -ite r a tiv e , a p p licab le to both rough and smooth

passages, and used a modified universal v e lo c ity p r o f ile which did not

(20)

9

The basic assumptions used were:

7) constant physical prop erties

2) A0 (angle o f increm ent) approached zero in the l i m i t

A

3) y was the average distance from the w all to the surface

o f no shear

4) pressure forces acted on a surface area estimated by the

d iffe re n c e between two sectors

5) the v e lo c ity p r o f ile fo r c ir c u la r pipes and between

p a r a lle l planes was used fo r n o n -c irc u la r passages

Since th e ir o rig in a l force balance on a element included shear

stress in the c irc u m fe re n tia l d ir e c tio n , fu rth e r s im p lify in g assumptions

were made:

6) the momentum d i f f u s i v i t y (and also the shear s tre s s ) in the

c irc u m fe re n tia l d ire c tio n was independent o f r . Rapier [4 ]

approximated e as

dropped, thus in cu rrin g a 12.5% e rro r in c a lc u la tio n s .

This assumption was made in conjunction w ith the former to

c a lc u la te a t 1 in the c irc u m fe re n tia l d ire c tio n independent

o f r .

The shear stress equation to be solved reduced to

(2 .9 ) e = u y/ 1 0

7 ) since u+ = 20, a constant value o f 2 .5 added to u+ was

U = 2Y (1 + Y) + | r : }

(

2

.

1 0

)

1 + Y 5 3G

where U = t q / r ( d p / d x ) ii r , + \ 2 or U = (u /u )

(2.11)

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which is D eissler and T a y lo r's [ 1 ] term inology, and Y = y /d (2 .1 3 )

This equation fo r U was num erically solved by computer fo r both a t r i ­

angular and square arra y a t p/d ra tio s o f 1 .0 0 , 1 .0 5 , 1 .1 2 5 , 1.250 and

1.5 0 . T h e ir re s u lts a t a constant u+ o f 10,20 and 30 compared favour­

ably with those o f D eissler and Taylor [1 ] and the shear stress d is ­

tr ib u tio n changed very l i t t l e w ith tube roughness.

Rapier and Redman fu rth e r proceeded to develop a v e lo c ity d is t r ib u ­

tio n based on a d if f u s i v i t y p r o f ile consistent w ith the universal v e lo c ity

A

p r o file up to y = y/ 2 and w ith non-zero d i f f u s iv it y a t a surface o f no

shear. Assuming a lin e a r v a ria tio n in shear stress fo r c lu s te r passages:

1iGa) *

t/ t q = 1 - y /y (2 .1 4 )

they proceeded to derive v e lo c ity p r o file s fo r two regions o f flow

o < y < y /2 (2 .1 5 )

and y/ 2 < y < y (2 .1 6 )

T h e ir re s u lts showed th a t agreement between experimental and th e o re tic a l

v e lo c ity d is trib u tio n s was f a i r l y good [ 1 , 4 , 5 ] .

C BENDER AND SWITICK T6l

Bender and S w itick [ 6] presented a model fo r determining the

two-dimensional v e lo c ity d is tr ib u tio n fo r tu rb u le n t, constant p ro p erty,

sin g le phase flow through an i n f i n i t e tria n g u la r rod bundle. The tim

e-average tu rb u le n t momentum equation was in teg rated on a nodal basis by

developing expressions fo r hydraulic resistance between nodes due to a

change in v e lo c ity caused by the molecular and eddy v is c o s itie s .

Eddy d i f f u s iv it ie s were defined in terms o f the lo cal v e lo c ity

(22)

-duced the c a p a b ility o f describing tu rb u le n t momentum tra n s fe r in both

the ra d ia l and circum ferential directions a t each po int in the flo w f i e l d .

By balancing pressure forces w ith viscous forces on each node, a set of

lin e a r , simultaneous alg eb raic equations resu lted fo r the v e lo c ity .

The follo w in g assumptions were made

1) constant physical pro p erties

2) f u l l y developed, sing le-phase, s te a d y -s ta te ,

tu rb u le n t flow '

3) no momentum flu x across the boundaries o f the channel

4) temperature d i f f e r e n t ia ls in the coolant cross-section

were not severe, so th a t m a terial p ro p erties could be

evaluated a t the bulk f lu id temperature and considered

constant across the flow cro ss-section .

Applying the c o n tin u ity equation and reducing the momentum equations,

the governing flow equation became the z-component o f the momentum

equation:

where u ( r ,e ) = uz ( r , e )

In th is equation,Emr and £m0 were the eddy d if f u s i v i t i e s o f momentum

in the ra d ia l and c irc u m fere n tia l d ire c tio n s re s p e c tiv e ly and were defined

by the shear stress and v e lo c ity gradients in the coolant: p

U + ( v + e ) 3U

- r r v mr'

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T = _£. ( v + e ) — (2 .1 8 )

r g mrJ 9r

_ p / * \ 9 U

t q - ( v + e m Q ) C 2 i l 9 )

g r 30

N ext, an a r b itr a r y segment o f f lu id was considered a t some point

in the flo w f ie l d to give the force per u n it length a t the face a t r as

F = (rA 0 )x r (2 .2 0 )

S u b stitu tin g eq. 2.18 in to 2 .2 0 , rearranging and in te g ra tin g , gave

A u

p/g m{>2 / r p/ *Q<v + Emr)

(2

.

21)

which i s analogous in c i r c u i t th e o ry to

I = E/R (2 .2 2 )

where au = a p o t e n t ia l f l u x

^ = a r e s u l t in g f l u x p/g

and the hydraulic resistance is

l n ( r 0/ r \ )

R -2, (2 .2 3 )

40 (v + cm r )

The same analysis was used in obtaining a hydraulic resistance to a c i r ­

cum ferential change in v e lo c ity as

R = ..AQ- .(r.)--- ( 2. 24 )

A r ( v + emQ)

A nodal admittance (Y) was defined as the recip ro cal o f the re sista n c e .

A force balance on the center node with fo ur adjacent nodes (F ig . 4)

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13

FIG. 4: Nodal Representation o f Bender and S w itick

to obtain an equation governing the v e lo c ity d is tr ib u tio n .

5

£_ I Y ?i (u , - u ?) = A (dp/dz)

g i - i

(2 .2 5 )

The pressure gradient was given as

d£ . - PUb2 f

(2 -2 6 )

(2 .2 7 )

dz 2g De

where f = 1 / (1.82 log Re - 1.64)^

and was determined from experimental data [ 8 , 9] .

The eddy d i f f u s iv it y o f momentum was determined by considering a

mixing length model based on the harmonic mean length o f Buleev [1 0 ]

and Hooper [ 7 ] .

emr ’ *-c 3u/3r

/ r ) 3 u / 3 0

(2 .2 8 )

(2 .2 9 ) "me ' c

where the mixing length was described in the form o f an e l l i p t i c

in te g r a l.

The authors developed a computer program "VELVET" fo r an 84-node

case which simultaneously solved a set o f lin e a r , alg eb raic equations

(25)

and 2.25 was continued u n til the eddy e f fe c t and the v e lo c ity p r o f ile

converged to w ith in a desired l i m i t .

Comparison o f t h e ir re s u lts w ith experimental data [3 ,1 1 ] in dicated

th a t the model overpredicted the c irc u m fe re n tia l v a ria tio n in v e lo c ity ,

suggesting th a t too low a value o f peripheral d i f f u s iv it y was used.

D NIJSING, GARGANTINI AND EIFLER [1 2 ]

N ijs in g , Gargantini and E if l e r [1 2 ] presented a th e o re tic a l

t r e a tis e to evaluate the d is tr ib u tio n o f lo cal coolant v e lo c ity , shear

stress and temperature in a tria n g u la r arra y o f p a r a lle l heat generating

rods. T h eir analysis was s im ila r to th a t o f D eis sle r and Taylor [1 ]

but n o n -ite ra tiv e and included the tu rb u le n t d iffu s io n o f momentum and

heat tra n s fe r around the rod periphery. The assumptions made were:

1) constant physical p ro p erties

2) f u l l y developed flow conditions

3) incompressible f lu id

4 ) Prandtl number g re a te r than u n ity

5) m olecular and eddy d i f f u s i v i t y were a d d itiv e

The momentum equation in c y lin d r ic a l co-ordinates became:

1 = ( v + j A + ( — I f r P ) M + ( U L f e r j A (2>30)

p ar^ r 3r r^ 30^

The s o lu tio n o f the energy equation required a knowledge o f the

f lu id v e lo c ity d is tr ib u tio n which must be found from the momentum

equation. Since no exact s o lu tio n was known from eq. 2 .3 0 when e was

a function or r and 0 , the authors looked fo r an approximate s o lu tio n .

The f i r s t two terms o f eq. 2.30 accounted fo r tra n s fe r o f momentum

(26)

distance away from the wall and soon became gre a ter than molecular

d i f f u s i v i t i e s . Consequently the ra te o f ra d ia l heat tra n s fe r was

m ainly determined by conditions in the immediate v ic in i t y o f the rod

surface.

The th ird term accounted fo r the tra n s fe r o f momentum around the

rod periphery. In the central p a rt o f the flo w , m olecular d if f u s iv it ie s

can be disregarded in comparison w ith the corresponding eddy d i f f u s i v i ­

t ie s . The ra te o f peripheral tra n s fe r then became a fun ctio n o f the

distance from the rod surface ( y ) . Since the ra d ia l v e lo c ity v a ria tio n

is decreased by periph eral momentum tra n s p o rt, a sin g le o v e ra ll e f f e c t ­

ive eddy d if f u s i v i t y % was assumed. T h eir an alysis employed a non­

isotropic nodel which allowed fo r periph eral and ra d ia l tra n s fe r o f momen

turn. The fo llo w in g assumptions were added:

6) the flow was f u l l y tu rb u le n t a t a l l peripheral

positions around the rod.

7) only the mean flow p ro p erties were considered in an

element dA by employing a lo cal v e lo c ity u^ Q

8) the process o f peripheral tu rb u le n t momentum d iffu s io n

was described in terms o f a sin g le distance parameter.

A momentum balance on the element dA and a lo cal eq u ivalen t hydraul

ic diameter defined as

4 dA

(2 .3 1 ) ye =

-r do

gave the expression fo r the lo cal shear stress TQas:

(2 .3 2 )

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terms o f local flow and geometry parameters.

Based on E ld e r's [1 3 ] work, the o v e ra ll e ffe c tiv e d i f f u s i v it y fo r

periph eral momentum tra n s fe r was re la te d to the lo cal Re as

Assuming also th a t the local shear stress was re la te d to the lo cal

mean v e lo c ity by the B la s iu s 1 equation

Equation 2 .3 6 , being an in te g r o -d iffe r e n tia l equatio n, was numer­

ic a ll y solved to provide the dimensionless v e lo c ity (n) and the periph eral

shear stress ( t 0/ t q av) d is tr ib u tio n s . Th eir re s u lts in dicated th a t the

periph eral momentum tra n s fe r diminished the v a ria tio n in shear stress and

v e lo c ity . A comparison with D eissler and T a y lo r's [ 1 ] re s u lts gave f a i r

agreement fo r an assumption o f zero peripheral momentum tr a n s fe r.

D eissler and Taylor [1 ] found n higher near 0= 0 and lower near 0 = n/6.

Eg/v = 0.0115n(ye/De)Re7 /8 (2 .3 3 )

(2 .3 4 ) where n = ua J u

(2 .3 5 )

gave:

(De/ye) - ( y e /D e ) ( 6/n ) 1' 7 5 (D e/ye)O ,25d0

= 0.287 Re°*1 25(D e/p r) d ^m ^ dr. (2 .3 6 )

de De r d0

subject to the boundary conditions

dn

= O at the gap and 6 = tt/6 (2 .3 7 )

(28)

For decreased rod spacing, the peripheral v e lo c ity v a ria tio n

increased but the momentum tra n s fe r became r e la t iv e ly less im portant.

The in fluence o f Re on the peripheral v e lo c ity d is tr ib u tio n was com­

p a ra tiv e ly small due to the fa c t th a t a l l three momentum tra n s fe r terms

increased p ro p o rtio n a lly w ith the periph eral momentum tra n s fe r s lig h t ly

dominant.

E EIFLER AND WIJSING [1 4 ]

This work was based on E i f l e r 's [1 5 ] e a r l ie r work on concentric .

annul i and extended the in v e s tig a tio n o f N ijs in g , G argantini and E if l e r

[ 12] so th a t the secondary e ffe c ts were included in terms o f an increased

c irc u m fe re n tia l eddy d i f f u s iv it y re la te d to th a t in the ra d ia l d ire c tio n .

A lo cal law fo r xQwas formulated in which t q was dependent on ra d ia l co­

ordinates so th a t:

The an isotro p ic fa c to r was assumed to be independent o f p o s itio n .

The ra d ia l components o f e and u from an inner concentric annuli

an alysis [1 5 ] were su b stitu ted in to eq. 2.39 assuming th a t the c i r ­

cum ferential momentum tran sp o rt had a n e g lig ib le e ff e c t on the ra d ia l

components. Equations fo r er and u as a function o f xr and K (a local

geometry parameter which was a function o f 0 ) were s ub stitu ted in to

2.39 and w ith the w all shear stress known, then t q was also known. An o rdinary second order d i f f e r e n t ia l equation o f the form

(2 .3 8 )

where n was an a n iso tro p ic fa c to r. Then

T

0

nper 3u (2 .3 9 )

r 30

L = 0 (— ,x , 0 )

d0 r

(29)

was then solved w ith the boundary c o n d itio n s :

a t 0 = o , t q = o

( 2 . 4 1 )

0 = A9 ; T q — o (2 .4 2 )

where A0 was the c i r c u m f e r e n t i a l l y symmetrical known angle f o r the rod

c l u s t e r ; i . e . ,

AO = n /6 f o r a t r i a n g u l a r a r r a y ,

A0 = n /4 f o r a square a r r a y

and w ith physical c o m p a t i b i l i t y c o n d itio n s :

a t 0 = o or A0 , ^ r = o (2 .4 3 )

Equations 2 .4 3 , 2.44 and 2.45 implied th a t the momentum flo w must

be zero normal to the lin e o f constant v e lo c ity and along lin e s of

symmetry, and th a t the average ra d ia l momentum flow on the wetted wall

equalled the a x ia l momentum flow defined by the pressure g ra d ie n t.

Equation 2.40 was solved num erically by the Runge-Kutta-Nijstrom

method to determine the v e lo c ity f i e l d . By comparing t h e ir th e o re tic a l

so lu tio n to experimental data [1 5 ] fo r a p/d ra tio , o f 1.08 and various

Re, i t was concluded th a t n should be equal to 10 and be independent o f

Re fo r a tria n g u la r a rra y . The wall shear stress and v e lo c ity v a ria ­

tio n in the c irc u m fe re n tia l d ire c tio n was g re a ter fo r the rectangular

array compared to the tria n g u la r a rra y and the d iffe re n c e increased as

p/d increased. They concluded th a t more experimental work was needed (2 .4 4 )

(30)

to v e r if y th a t n was indeed an a n tis tro p ic fa c to r.

F LAUNDER AND SPALDING [16]

Launder and Spalding [1 6 ] ch ro n o lo g ic ally reviewed various t u r ­

bulence models as they re la te d to the p re d ic tio n o f in te rn a l flow s.

The models were assessed from an engineer's point o f view to give the

turbulence model o f s u f f ic ie n t accuracy o f p re d ic tio n w ith the minimum

o f computational expense. Viewed from th e ir stan d po in t, the fo llo w in g

were the most important featu res a t the tim e:

1) "For a large number o f boundary layers where the turbulence

s tru c tu re changes slow ly in the d ire c tio n o f flow (in clu d in g flows in

j e t s , wakes, d iffu s io n flames and non-separated flows in pipes and

channels), a version o f the Prandtl mixing length model is most to be

recommended. The necessary em pirical constants w i ll be v a lid only fo r

a r e s tr ic te d class o f phenomena; but an engineer who is concerned w ith

a lim ite d range o f problems can endure th is r e s tr ic tio n w ith complacence;

2) For boundary la y e r flows where the boundary conditions enforce

rapid streamwise v a ria tio n s o f the turbulence p ro p erties and fo r almost

a l l separated flo w s , i t seems to be necessary to employ a model in which

a t le a s t two d if f e r e n t ia l equations are solved fo r prop erties o f the t u r ­

bulence; fo r convection and d iffu s io n o f turbulence energy are u su ally

s ig n ific a n t fo r such flow s; and the geometries are too various fo r the

length scale to be obtainable from g e n e ra lly - v a lid alg eb raic formulae.

A ll the two equation models employ the Boussinesq formula fo r e v a lu a t­

ing the shear stresses.

T ~ "ye f f — ( 2 - 46)

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3) There are some flows o f engineering importance fo r which the

Boussinesq formula is inadequate. For example, "the p re d ic tio n of

v e lo c ity d is tr ib u tio n fo r p a r a lle l flow on rod bundle geometry . . . " I f

these flows are to be computed r e a l i s t i c a l l y , a three equation model

must be employed. The v a riab le s might well be the turbulence energy,

the e nerg y-d issip atio n r a te , and the shear stress i t s e l f .

4) As two- and three equation models become re fin e d through

extensive t r i a l and development, the need fo r models having gre a ter

numbers o f equations w ill become apparent, demonstrated by the incom­

p le te u n iv e r s a lity o f the simple models. However, th is s itu a tio n has

not y e t a rriv e d ; a t present, none o f the more complex models has been

developed to the point a t which i t is superior in performance to the

(32)

I l l PROGRAM DOCUMENTATION

Before the energy equation could be solved to give a temperature

p r o f ile in any subchannel a rra y , the v e lo c ity p r o f ile had to be d e te r­

mined. To c a lc u la te the v e lo c ity d is tr ib u tio n in banks o f rods or tubes

under the conditions o f a x ia l, tu rb u le n t flo w , a modified version o f

D e is s le r and T a y lo r's [ 1 ] model was adopted to a computer s o lu tio n . The

it e r a t iv e procedure which was developed in a computer program LOOP

(Appendix I ) to solve the equations o f change is documented here. An

i n f i n i t e tria n g u la r arra y o f smooth rods was i n i t i a l l y analyzed. Due

to symmetry, only a 30° increment had to be considered and th is sector

was divided in to t h i r t y - 1° sections from the gap to the lin e o f

symmetry a t e = 30° ( F ig .5 ) . For an i n f in i t e square a rra y o f fu e l rods,

a 45° increment was analyzed. The gap width and Re were allowed to

v a ry , and the program was w ritte n so th a t any physical p ro p erties may

be va ried .

A BASIC DEFINITIONS

The p itch to rod diameter r a tio is the distance between rod centers

The equivalent diameter is defined as fou r times the flow area over

the wetted perim eter. For a tria n g u la r array

p/d = 1 + b /2 r ( 3 .1 )

De = 2r f t O L - (p /d )2 - 1] (3 .2 )

n

and fo r a square arra y

De = 2r [ — Cp/d}2 - 1] (3 .3 )

(33)

L IN E S OF M A X IM U M V E L O C IT Y

E L E M E N T A L A R E A

(34)

23

The to ta l area under consideration fo r a tr ia n g u la r arra y is

At = 0 / 1 2 ) l 4 r 2 Cp/d)2 sin(2nX) - n r2] (3 .4 )

where n = 30

X = ne/180 to convert degrees to radians (3 .5 )

For the square array

As - i-2[ ( p / d ) 2/2 - n /8 ] (3 .6 )

The arc length is given by

Ai, = rX (3 .7 )

Other basic dimensionless v a riab le s are the dimensionless v e lo c ity

u+ = u /u * = u / A Qg/p (3 .8 )

and the dimensionless distance from the wall

y+ = u *y /v (3 .9 )

B EVALUATION OF y *

From the universal v e lo c ity p r o f ile [2 ] ( F ig .6) i t is evident th a t

"f* 4* 4*

fo r y > 26, a s tra ig h t lin e re la tio n s h ip e x is ts between u and y . For

. +

any given u

y+ = e x p [0.36(u + - 3 .8 ) ] (3 .1 0 )

However, fo r y+ < 26 the fo llo w in g equation is used:

0.109u+

+ 2 +

exp(-0.0059 u ) du_______ (3 11)

+2 exp(-0 .0 0 5 9 u )

Equation 3.11 was developed by D eis sle r [1 7 ] and the in te g ra l is

(35)

o

O '

o

o

o

3 >>

<3

CVJ

io

cn

CVJ CVJ

n/n

Ge

ne

ra

liz

e

d

V

e

lo

c

it

y

D

is

tr

ib

u

ti

o

n

fo

r

A

d

ia

b

a

tic

Tu

rb

u

le

n

t

Fl

(36)

C FRICTION FACTOR CALCULATIONS

To s t a r t the it e r a t io n , a Fanning f r ic t io n fa c to r ( f ) was req uired.

This was calcu lated by assuming a Re and using the Colebrook equation

] _ = - 4 log + L 2 6 ) 5 Re > 2000 (3 .1 2 )

^ 3,7 R e / T

f = 16/Re Re < 2000 (3 .1 3 )

From Knudsen & Katz [1 8 ]

' - De (-g dp/dx) (3 .1 4 )

^pUb ,av

D DETERMINATION OF u++, r ++, dp/dx

I f a dimensionless v e lo c ity parameter is defined as:

u u /,/-rg (dp/dx )/p (3 .1 5 )

then, by using eq. 3 .1 4 , the dimensionless cro ss-section al average

v e lo c ity is

ub,av = 1/,72 f(r /D e ) ^3 ' 16^

A dimensionless Re is defined as

++

/-r = /

rg (d p /d x) / p r /v (3 .1 7 )

(37)

per unit length may be written:

E CALCULATION OF SECTIONAL AREAS (A )

The elemental areas comprising the to ta l subchannel flow area were

found as in Fig. 7:

A = area o f large sector ( r + ymax) area o f rod sector ( r )

-area o f segment - -area o f tr ia n g le - I other calcu lated areas

where n = number o f angle increments

0 = angle increment

F CALCULATION OF WALL SHEAR STRESS (tq )

The forces on an element (F ig .8 ) in the flow are the shear force

acting on an arc length (aji) and the pressure forces actin g on the faces

o f the element. There are no shear forces acting on the v e lo c ity grad­

ie n t lin e s (assumed to be the ra d ia l lin e s as a f i r s t assumption) because

the normal v e lo c ity d e riv a tiv e s are zero along these 1in e s .W ritin g a force

balance on the element gives

The r a tio o f shear stress to average shear stress is e a s ily c a l­

culated from

(.3*19)

dx rg r

(3 .2 0 )

360

T

(38)

27

^ A R E A OF S E C T I O N U N D E R C ONSI DERATI ON

A R E A O F A L L O T H E R S E C T I O N S

A R E A OF T R I A N G L E

A R E A OF S E G M E N T

F ig. 7: Determ ination o f Areas in a T ria n g u la r Array

F ig . 8: Elemental Area and Determ ination of

(39)

To _ AA A _ „ aA

t AA At At ^3 - 22)

0 3 V t t

6 CALCULATION OF MAXIMUM- DIMENSIONLESS DISTANCE FROM MALL (.ym* )

In order to c a lc u la te the maximum dimensionless distance from the

w all (ym+ ) ,e q s . 3.19 and 3.21 were s u bstitu ted in to eq . 3 .9 to give

ym " ^ 1 aA r ^ (3 .2 3 )

ym

r aa r

A lso, by trigonom etry

ym = r(l-c o s n e + b /2 ) (3 24)

cos rB

The average dimensionless distance from the wall was found by

averaging two adjacent distances:

1 , e , » ym £V tyn ,n + ym ,n - l ) / 2 (3 .2 5 )

+ / + +

, = (ym „ I I I j I I " I J / C * i

+ ,

H CALCULATION OF MAXIMUM DIMENSIONLESS VELOCITY IN EACH SECTOR (u )

To c a lc u la te the maximum dimensionless v e lo c ity in any one s e c to r,

i t was only necessary to reverse equations 3.10 and 3 .1 1 . This was re ­

la t iv e ly simple fo re q . 3 .1 0 :

um+ = 2 .7 8 ln y m+ + 3 .8 fo r y * > 26 (3 .2 6 )

(40)

29

dy+ +

fo r y < 26 (3 .2 7 )

o 1 + n 2 u+y+£ l-e x p (-n 2u+y+ ^}

where n = 0 .1 2 4

To solve eq. 3 .2 7 , two subroutines SIMPU and SIMPL, which in co r­

porated t h e N e w t o n - R a p h s o n technique, were w r itte n . I f we set the

expression a f t e r the in te g ra l sign in eq.3*27 equal to f ( u + ) so th a t

f ( u + ) = --- (3 .2 8 )

1 + n2u+y +[ l - e x p (- n 2u+y + ) ]

then the Newton-Raphson procedure f o r an i n i t i a l u j is

f ( u + ) (3 .3 0 )

u+ = u t - f ( u + ) / f (u+ ) ^ + (3 .2 9 )

where f (u+ ) = - f ( u + ) = 1 - —

+

3U 3U /

/ °

which was solved employing the L ie b n itz formula.

I CALCULATION OF LOCAL VELOCITIES (u) AND DISTANCES FROM THE WALL (y)

The lo cal v e lo c ity as a fun ctio n o f the distance from the w all was

found from equations 3 .8 and 3 .9 as

+ * (3 .3 1 )

u = u u J

y = y+v /u * (3 .3 2 )

Since lin e s o f constant v e lo c ity are to be c a lc u la te d , a v e lo c ity

(41)

values o f u and y are found fo r every p o sitio n J a t which a v e lo c ity

has been c a lc u la te d . This procedure is repeated fo r every section and

lin e s o f constant v e lo c ity may be found by comparing output in each o f

the sections.

■1 siirchannel bulk v e l o c i t y c al cu l at io ns

By d e f in it io n , the cro ss-sectio n al average v e lo c ity is calcu lated

by in te g ra tin g over the section

An incremental area ( F ig .8) was approxim ately the d iffe re n c e between two

sectors o f radius r + y (J ) and r + y ( J - l ) . The v e lo c ity o f th is element

is the average o f the predetermined v e lo c itie s a t (J ) and ( J - l ) .

By summing over the e n tir e incremental a rea , the average v e lo c ity is de­

termined. Then, by summing these cross-section al average v e lo c itie s in

each s e c tio n , the bulk v e lo c ity (u, ,,,) was determined fo r the sub-D j a v

channel.

r

AA

Jo ub dA

(3 .3 3 )

o

dA

u = u (J ) + u ( J - l )

2

(3 .3 4 )

dA = s a [ ( r + y ( j) ) 2 - ( r + y ( J - l ) ) 2] (3 .3 5 )

(42)

31

K CONCLUSION OF ITERATION PROCEDURE

Knowing the average bulk v e lo c ity , a new Fanning f r ic t io n fa c to r

( eq. 3 .1 4 ) is c a lc u la te d . With th is revised f , a new ub ay++ is d e te r­

mined ( e q. 3 .1 6 ) and consequently a revised Re is c a lcu lated by re ­

arranging eq. 3.18:

Re = r++ub,av++/ t r /D e ) (3 *36)

I f th is Re agreed w ith th a t desired the it e r a t io n was term inated. I f

n o t, the ite r a tio n continued from e q . 3.17 u n til a desired convergence

was obtained.

L DETERMINATION OF FLOW SPLIT IN A TRIANGULAR-SQUARE ARRAY (G./G 1

w 3

The flo w s p l i t was defined as the r a tio o f the mass flu x fo r a

tria n g u la r channel to the mass flu x fo r a square channel, where

G = pub,av (3 - 37)

For constant physical p ro p e rtie s , the flo w s p l i t became the r a tio o f

cro ss-sectio n al average v e lo c itie s . To determine th is r a t i o , the average

cro ss-section al v e lo c ity fo r a tria n g u la r arra y was found by the method

ju s t described a t a p a r tic u la r Re. Since the pressure drop must be

uniform in each channel, the dp/dx calcu lated from the tria n g u la r array

- must be the value fo r the square a rra y . This value was used as a con­

s ta n t.to determine the cross-sectional average v e lo c ity fo r the square

a rra y . No ite r a tio n s were required fo r th is case.

M OUTPUT

Enough data was prin ted to show the v a ria tio n o f f r i c t io n fa c to r

(43)

spacing parameter; and the v a ria tio n o f shear stress on the rod w ith

angle 0 fo r various values o f the spacing parameter fo r both arrays.

More im p o rta n tly, however, s u f fic ie n t values o f the v e lo c ity a t

a po int in each section were c a lcu lated so as to give a p re d ic tio n o f

the v e lo c ity d is tr ib u tio n between tubes fo r d if fe r e n t gap widths and Re.

The section studied was expanded (100 - 175 tim es) w ith a SCALE fa c to r

so th a t meaningful p lo ts could be drawn and studied.

These plots also served the purpose o f determining more accurate

v e lo c ity g rad ien t lin e s (as opposed to our i n i t i a l assumption o f using

the ra d ia l lin e s as a v e lo c ity g ra d ie n t). By using a planim eter to re ­

c a lc u la te the new area and subm itting these in to a s im ila r program as

the one developed, a second t r i a l w ith more accurate constant v e lo c ity

lin e s may be drawn. Once the v e lo c ity gradient lin e s and lin e s o f con­

s ta n t v e lo c ity have repeated themselves, no more t r i a l s need be done

(44)

iy RESULTS AND DISCUSSION

In the lit e r a t u r e survey, i t was made evident th a t the method of

D eis sle r and Taylor I I ] was a common model fo r determining v e lo c ity d is ­

trib u tio n s in banks o f rods or tubes. For th is reason, a basic computer

model was made of th e ir analysis to which changes could be made in the

fu tu re . A computer program LOOP was developed to determine the flo w

-s p l i t in an i n f i n i t e a rra y o f tria n g u la r--s q u a re -subchannel-s. In ad d itio n

th is program calcu lated local and cross-section al average v e lo c it ie s ,

f r ic t io n fa c to rs , and w all shear stress fo r both arrays. A re p resen tative

input and output o f the program, which corresponds to the program do­

cumentation, is given in Appendix I . In ta b le 1 are summarized the pre­

d icted f r ic t io n fa c to r , lo cal v e lo c ity , e tc . fo r a tria n g u la r a rra y a t

Re o f 2 x 10^, 5 x 10^ and 1 .0 x 10^ fo r a range o f p/d r a tio s .

Figure 9 illu s t r a te d th a t f r ic t io n fa c to r p redictio n s fo r a t r i

-angular a rra y o f rods were in good agreement w ith those o f D eis sle r and

T a y lo r. Experimental data o f E if l e r and N ijs in g [1 4 ] a t a p/d o f 1 .0

and 1.1 were higher than predicted f r ic t io n fa c to rs , but the experimental

data o f Walton e t al [2 0 ] a t a p/d o f 1.05 were in good agreement. F r ic ­

tio n fa c to rs fo r a square arra y are compared w ith those o f D eis sle r and

Taylor in Figure 10 where, a t a p/d o f 1 .0 , the present an alysis pre­

d icted f r ic t io n fa c to rs appreciably lower than those o f D e is sle r and

T aylo r. However, a t a p/d o f 1 .1 , the agreement was g re a tly improved,

and a t a p/d o f 2 .0 the comparison was e x c e lle n t. These discrepancies

are a ttrib u te d to : i ) the increased accuracy obtainable w ith a computer

as compared to the hand c a lc u la tio n o f D eis sle r and T a y lo r, and i i )

(45)

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o o o o o o o O CD O O O O O o o o o o o o

O N ^ T C O O O L D O CM <3- CO O O LO o CVJ CO o o LO O O O O I— c v ir^ O o o o r— CVJ Ds. o o o o r— CMD s .

a a • • a • • • • • • • • • • • • • a « a

(46)

R

ep

ro

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w

ith

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Fu

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production

p

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ib

it

e

d

w

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e

rm

is

s

io

n

.

DEISSLER AND TAYLOR

PRESENT ANALYSIS

WALTON ET AL & 0 ]

E IF L E R AND N IJ S IN i

P /D

O . O I O

0 .0 0 8

P / D

1.05 0.006

i . i o

0 .0 0 4

1.0

0.002

1 J I L

1 1

(47)

ro

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Fu

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production

p

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n

.

DEISSLER AND TAYLOR M

CIRCULAR PIPE

O.OIO PRESENT ANALYSIS

0.008

P/D

2 .0

0.000

f

s 0.004 1.05

1.0

0.002

0.001

(48)

37

only one ite r a tio n (ra d ia l lin e s as y e lo c ity gradien t lin e s ] was in ­

corporated in to the computer model w hile D eis sle r and Taylor used an

it e r a t iv e technique to update v e lo c ity gradient lin e s . The v a ria tio n

o f f r ic t io n fa c to r w ith p/d fo r a tria n g u la r arra y is shown in Figure 11

w ith Re as a parameter. This fig u re demonstrates the rapid increase o f

f r ic t io n fa c to r a t low p/d ra tio s and the excellent'agreem ent w ith the

analysis o f D eissler and Taylor £ 1 ].

Shear stress v a ria tio n s , (tq/ t0 ay) , in the c irc u m fe re n tia l d ire c tio n

fo r a tria n g u la r arra y are compared in Figure 12. The re s u lts o f D eissler

and Taylor were c o n s is te n tly higher a t any p/d r a t i o , except in the upper

c irc u m fe re n tia l sections where the present analysis co n stan tly overpre­

d icted the shear stress v a ria tio n . However, in both models, the shear

stress was lowest in the regions where the rods were clo sest together

and approached zero as p/d approached u n ity , i . e . where the rods touched.

A s im ila r comparison was made in Figure 13 w ith u^/u^ ay being the

dependent v a ria b le . Again the re s u lts o f D eissler and T aylo r were con­

s is te n tly higher except fo r angles g re a ter than 20° where the present

analysis d iffe re d markedly. Both fig u re s 12 and 13 demonstrate the in ­

crease in shear stress and v e lo c ity w ith rod spacing and w ith distance

from the gap region.

As previously mentioned, one attempt was assumed to be accurate

enough in the computer model to give a reasonably f a i r agreement with

the model o f D eissler and T aylo r. Thus lin e s normal to the rod surface

were the v e lo c ity gradient lin e s used in determining the v e lo c ity d is ­

(49)

0.010

0 .0 0 9

0 .0 0 8

0 .0 0 7

0 .0 0 6

R

e

= i.o x io

0 .0 0 5

0 .0 0 4

D E I S S L E R AND T A Y L OR

P R E S E N T ANALYSI S

0 .0 0 3

0.002

1.0 1.5 2.0

PITCH / ROD D I A M E T E R , P / D

Fig. 11: T ria n g u la r-A rra y Fanning F ric tio n Factor vs.

Figure

Figure 1. Cross-section of typical rod bundle.
Fig. 2:
FIG. 4:
Fig. 5:
+7

References

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