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Retrospective Theses and Dissertations

Summer 1982

Two-Phase Flow Pressure Drop Across Thick Restrictions of Two-Phase Flow Pressure Drop Across Thick Restrictions of

Annular Geometries Annular Geometries

Saeed Ghandeharioun University of Central Florida

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STARS Citation STARS Citation Ghandeharioun, Saeed, "Two-Phase Flow Pressure Drop Across Thick Restrictions of Annular Geometries" (1982). Retrospective Theses and Dissertations. 624. https://stars.library.ucf.edu/rtd/624

T 0-PHASE FLOW PRESSURE DROP ACROSS THICK RESTRICTIONS OF A NULAR GEOMETRIES

BY

SAEED GHANDEHARIOUN B.S.M.E., Uni ersity of Miami, 1979

RESEARCH REPORT

Submitte in partial fulfillment of the requirements for the Master of Science in Engineering in the

Graduate Studies Program of the College of Engineering un·versity of Central Florida

Orlando, Florida

Summer Term 1982

ABSTRACT

This paper presents the methods of predicting the

steady- state two - phase flow (steam and water) pressure

drop across the restrictions of annular geometries

formed when tubes extend through circular holes in tube

su port plates .

Two approaches are discussed and a detailed sample

calculation of the one selected is presented . The major

areas of discussion are the orientation of tubes - to ­

tube s pport plate holes and the thickness of tube

support plate .

Finally, the conclusion gives a comparison of the

methods and recommendations for future investigations.

ACK~OWLEDGEAENTS

The author wishes to express special thanks to

Dr . E . R. Hosler , the academic and research report

advisor , for his guidance and support throughout the

course of this study .

Thanks are also extended to the other members of

my com ittee , Dr . F . S . Gunnerson , for his helpful advice

and donation of his time and his books , and Dr . R. G.

Denning , for serving on my committee.

Finall , my sincere appreciation and special

thanks is given to iss Dian Brandstetter for her

complete cooperation and expert accomplishment of typing

this report.

iii

ACK OWLEDGEME TS

LIST OF TABLES •

LIST OF FIGURES

0 E CLATURE

SUBSCRIPTS .

Chapter

TABLE OF CO TENTS

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

1 . PRESSURE DROP OFT 0- PHASE FLOW ACROSS A ULAR ORIFICES . . . . • . . • . . .

2 . PRESSURE DROP OF T 0 - PHASE FLOW ACROSS

iii

v

vi

. . vii

ix

4

SHORT A D LO G LE GTH RESTRICTIO S . . . 18

3 . CO CLU IO S A D RECO E DATIO S .

APPE IX 1 . . . . . . . . . . . . . . . APPE DIX 2 • . . . . . . . . . . . . . . APPE DIX 3 . . . . . . . .

. . . . . . . . . . . REFERE CES CITED .

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . .

iv

23

26

41

45

48

49

LIST OF TABLES

1 . Results of calculations for saturated single - phase pressure drop for various flow conditions •••

2 . Values of ~ as a function of ~ . . . . . . . .

v

34

46

1 •

2 .

J.

4.

5.

6. ? .. 8 .

9 •

1 0 .

11 •

1 2 .

1 3 .

1 4 .

1 5 .

LIST OF FIGURES

Annular gap between tube and tube support plate ••

Kinetic term for viscous flow in annular orifice • • •••••

Friction factor for annuli of fine clearance and for parallel plates •

Annular orifice coefficient versus Re nolds number for Sharp - Edge orifice

Summar of concentric - orifice coefficients • • • • ...

•

Summary of tangent-orifice coefficients . Short and long length restrictions . Configuration of two - phase flow channel

and its tube support plate for the numerical example •

ariation of quality with height in a uniforml heated channel • • •

Pressure drop versus mixture quality across the support plate for both concentric and tangent orifices •

Pressure drop versus mixture quality across the support plate for various slip ratios (concentric orifices) •

Pressure drop versus mixture quality across the support plate for various slip ratios (tangent orifices) ••••••

Pressure drop versus mixture quality across the support plate for various flow rates (concentric orifices) .

Pressure drop versus mixture quality across the support plate for various flow rates (tangent orifices) ••.•••••

Void fraction versus mixture quality for the annular gap between the tube and the support plate at the system pressure of 1000 psia • .. • • •

v i

. . . .

.2

1 3

14

1 5

16 1 7 19

27

29

35

36

37

38

39

47

A

c

c c

D

d

Fr

f p

G

H

h

* K

K

L

~p

NOMENCLATURE

cross - sectional area , ft 2

void fraction

overall annular orifice coefficient

coefficient of stream contraction in an orifice

area vena contracta/area of restriction

outside diameter of annular orifice , ft

inside diameter of annular orifice or outside diameter, ft

u2 Froude number, gDh

friction factor for flow between parallel plates of infinite, Fig . 3

maRs flux, lbm/hr-ft 2

height, ft

fluid enthalpy, BTU/lbm

single - phase loss coefficient

kinetic term for viscous flow in annular orifice, Fig. 2

restriction (orifice) length or thickness of tube support plate, ft

mass flow rate , lbm/s

total pressure drop across restriction or orifice, psi

Q volume flow rate, ft 3 /s

vii

R

r

Re

s

u

x z µ

p

a

2

sensible heat added per pound mass of incoming coolant~ BTU/lbm

total heat added in channel per pound of mass of incoming coolant , BTU/lbm

outer radius of annular orifice, ft

inner radius of annular orifice or outer radius of tube, ft

DhG Re nolds number ,

u slip ra io

volumetric flux (superficial velocity) , ft/s

fluid specific volume, ft 3 /lbm

floN quality

annulus length - to - width ratio

fluid absolute viscosity , lbm/hr - ft

fluid densit , lbm/ft 3

two-phase multiplier

re triction(s) flow area/channel flow area

hydraulic diameter (for annular gap; Dh = D-d), ft

viii

SUBSCRIP TS

B

e

f

g

H

i

boiling region

exit of channel

saturated liquid

saturated vapor

homogeneous flow

inlet of channel

non - oiling region

o orifice

S separated flow

T total

TP two - phase

TSP tube support plate

ix

INTRODUC TI ON

In shell and tube type heat exchangers , support

plates are spaced pe r iodi cally a long the tube bundles

to maintain the proper geom e t r i c a rran gement among the

tubes . \hen the shell side flow is paral l el to the tube

axis , the flow must pass thr ough t he annular shaped

clearance between the outside di a meter of the tubes and

the hole in the support p l ate (F igure 1 ), increasing

the shell - side pressure d r op . Since the support plates

contribute a major portion of the pressure drop, pre ­

dicting shell - ide pressure drop f or t wo - p h ase flow is

an important design conside r ation .

A review of literatur e i ndicates t hat there has

been previous work to

(a) predict single - pha se pressure drop in compli ­

cated geometrie s (s uch as support plates) and

(b ) predict two - ph a s e pressure drop in simple

geometrie s (su ch as tubes and channels) .

Howe v e r, th er e h as been litt l e or no work to predict

t wo-pha se p r e ssure drop i n c omplica t ed geometries .

Th i s wor k is t o provi d e one or more rational

approaches of predicting two - phase pressure drop by

tube

tube support plate

r flow

A. CO CE TRIG B. TANGENT

Fig . 1. Annular gap between tube and tube support plate

2

tying together the previous work in (a) and (b) to be

able to predict shell - side pressure drop with two ­

phase flow .

3

CHAPTER 1

PRESSURE DROP OF TWO - PHASE FLOW

ACROSS A I ULAR ORIFICES

In general, orifices are used to measure flow

rates. However , many flow restrictions, such as

clearance between a tube and a tube support plate may

be analyzed b treating them as orifices . It is the

intent here to evaluate the pressure drop for two - phase

flow across annular orifices .

A re ie of literature indicates that in two -

phase systems , it has been observed experimentally

/ that fo a given mass flux , the pressure drop can be

much greater than for a corresponding single - phase

system (1). The classical approach , which has been

taken to predict two - phase pressure losses , is to

multiply the equivalent saturated single - phase pressure

loss b a multiplier, $ 2 , which is a function of (at

least) flow quality and system pressure . Thus

~p TP == ~pf • <P 2

( X ' p ' • • • )

or

( 1 -1 )

4

5

* where K is the single-phase loss coefficient, which will

be discussed in more detail for various geometries of

annular gaps between a tube and a tube support plate

later in this chapter .

Since the pressure drop in two-phase flow is closely

related to the flow pattern , two principle types of flow

models ill appear in the analysis of two - phase pressure

drop in this chapter. They are homogeneous flow model

whi h regards the t o - phase to flow as a single - phase

posse sing mean fluid properties , and the separated flow

odel which considers ~he phases to be artificially segre -

ated into t o streams ; one of liquid and one of vapor .

efore presenting the commonly accepted expressions

fa multiplier , 2 . h , in omogeneous flow model and sepa -

rated flo~ model, it is necessary to make the following

assumptions;

1 . One - dimensional flow

2. Steady- state flow

3 . Adiabatic flow across the tube support plate,

so that the quality , X, is constant .

4. Pressure drop across support plate is small

compared to the total pressure , so that the

densities , Pr and pg, do not change .

5. Flow properties are in terms of cross -sectional

averages taken across the annular gap flow

cross section .

6

6. The parameters affecting voids; i.e.; quality,

pressure, and mass velocity are nearly the same

upstream and downstream of the tube support

plate, so that void fraction, a, is nearly

constant , and is given by (4) 1

a = ( 1 - 2)

-here S is the slip ratio, and it is defined as

the ratio of the average velocity of the vapor

phase to that of the liquid. For homogeneous

flow the slip ratio is equal to 1.0. and for

nonhomogeneous flow is greater than 11.0 due to

the fact that the vapor , because its buoyancy,

has a tendency to slip past the liquid.

Therefore, in order to find the commonly accepted

expression for ~ 2 based on the foregoing assumptions for

homogeneous and separated flow models, it is suggested

(1) that the expected behavior of $ 2 is to be examined.

For this purpose , for two-phase flow it can be written

in which PTp is the appropriate two-phase density. For

the case of saturated homogeneous two-phase flow (PTP = pH),

equation (1-3) can be written

( ) * G2

(Pf) ~PTP H = K2gcPf PH

where PH is homogeneous density as

( 1 - 4)

1 v + v x f f g

7

( 1 - 5)

Thus , by comparing equations (1 - 5) , (1 - 4 ), and

(1-1) , it is found that

( 1 - 6)

or

( 1 - 7)

~hich is the expression for prediction of pressure drop

in two - phase homogeneous flow .

The appropriate two - phase density for separated flow

is not as ell defined (1) . It has been suggested

(Chisholm , 1973) that the momentum density should be

used . Thus equation (1 - 3) becomes

( 1 - 8)

-where p is the momentum density , and is given by

- 1 P = ( 1 - x ~ 2 + x2

Pr (1 - a pga

( 1 - 9)

Thus , by comparing equations (1 - 9) , (1-8) , and (1 - 1) , it

is found that

¢ 2 = ( 1 - x )2 + ~ x 2

s 1 .... a Vr ex ( 1 - 10)

or

( 1 -11 )

8

which is the expression for prediction of pressure drop

in two-phase separated flow.

For the clearance between a tube and a tube support

plate which may be regarded as an annular orifice , the

single - phase loss coefficient can be defined as

* 1 K = c2 (1-12)

here C is called the overall annular orifice coefficient.

The experimental st dy by Bell and Bergelin (2)

o the single - phase flow (water and oil) through various

single annular orifices indicates that the annular orifice

coefficient , C, is a function of the annular orifice di-

mensions , the annular orifice Reyno lds number , and the

orientation of annular orifice ; i . e ., concentric or

tangent position of tube to tube support plate hole

(Figure 1) . The equations defining orifice length-to-

width ratio , and orifice Reynolds number are given by

z = 2L D - d (1-13)

= ( D - d) G0 (1-14)

where G0 is mass flux through an annular orifice.

It should be noted that for the two-phase flow

through an annular orifice , the orifice coefficient may

be estimated at the Reynolds number of the entire flow

rate considered in the state of saturated liquid

Consequently, the results of Bell and

Bergelin ' s analysis (2) for prediction of the overall

annular orifice coefficient may be used.

9

Bell and Bergelin have developed suitable equations

expressing the overall annular orifice coefficient for

various geometries of thick annular orifices with single ­

phase flow , with the designation of the flow ranges as

viscous , turbulent , or transition , based on the following

Re nolds-nurnber ranges :

1 . The viscous-flow range refers to Reynolds num ­

bers less than 40, where the predominant

effect is energy loss by the viscous shear in

the fluid, and the kinetic effects are confined

to the fluid as it flows into the orifice .

2. The turbulent - flow range refers to Reynolds

numbers above 4000 , where the predominant

effects are the kinetic - energy losses associated

with stream acceleration , contraction, limited

expansion , and turbulent friction . It should

be noted that the stream expands from the vena

contracta to the full area of the annulus with

a partial recovery of the kinetic energy as

pressure . The expansion from the vena contracta

begins a finite distance downstream from the

orifice entrance, and no pressure recovery will

be obtained in an orifice whose thickness is

less than this distance (Figure 7) .

10

3 . The transition-flow range refers to Reynolds

numbers between 40 and 4000, where both kinetic

and viscous phenomena are important .

The equations of the overall annular thick orifice

coefficients are given by

(A) Viscous Range , Concentric Orifice .

11 _ 9J± + 48 Z + K 2 - Re Re

where K is taken from Figure 2 .

(B) Viscous Range, Tangent Orifice.

_1 = 128 + 96 Z + K c2 Re S Re

here K is taken from Figure 2 .

(C) Transition Range, Conce~tric Orifice .

1 c2 =

here F = 0 , for Z > 1 .1 5 , and

(1 -15 )

(1 - 16)"

F = 1 _ e - 0.95(Z - 1.15) , for z > 1 . 15 , where Cc

and fp are taken from Figures 4 and 3 at the

appropriate Reynolds number.

(D) Turbulent Range , Concentric Orifice .

c\ = c1 2 - [ c2 - 2) F + 2r P • z c c

where F = O, for Z < 1.15, and

F = 1 - e- o. 9 5(z- 1 • 1 5 ) , for Z > 1.15.

C0

and fp are taken from Figs . 4 and 3 ,

respectively.

(1-1s)

Orifice .

+L(2r - R) < [ i - - 1 -2

-_-~:_2~_-2-J

[_1 c 2

c

[; +

+

Cc Cc

2 )] ~]

2 2 - (- - 2)

Cc

1 ( C2c - 2 ) - z f ) ] sin _ 1(_c_~_2 __ ~_2 ___________ P

-:--TC - (~ - 2) + Zf P c Cc

H)} 2

~ - 2) Cc

1 1

for Z > 9, and Re > 10 , 000 . Cc and fp a r e

taken from Figures 4 and 3 , respectively .

12

The experimental analysis of Bell and Bergelin for

the flow through annular openings are presented for

wide range of flow rates and orifice dimensions in the

Figures 5 and 6 . From these curves , it was concluded

that for high Reynolds number (Re > 5000) in thick 0

orifices with a fixed mass flow rate, the pressure

recovery begins at a Z- value of about 1 . 0 up to a

Z-value of 6 . 0 . For longer orifice channels the wall

friction auses an increase in the pressure drop until

the channel length give Z- numbers in the range of 10

to 100 here the overall annular orifice coefficients

drop to a out 0 . 65, the value for sharped-edg e orifice

(Figure 4) . At hig er Z- values the friction resistance

lo~ers the value of overall annular orifice coefficient

still more .

A numerical example is presented in Appendi x 1,

which has been solved based on the foregoing discussion

of predicting the two - phase flow pressure drop across

the annular orifice .

K

1 J

1 ,. 8 I

-

1 • 6 L----~ l

,_,,...-----1 • 4 /

I~ I I

/ ! I I

1 • 2 I I

I I

I I

l

1 • 0 I .

0 0 . 005 0 . 01 0. 01 5

Z/Re

0 . 02 0 . 025 0 . 03

Fig . 2 . Kinetic term for viscous flow in annular orifice

SOURCE : K. J . Bell , and O. P . Bergelin , Flow Through Annular Orifices (Transactions of the ASME , 79 April ,, 195 7) , p. 59 7 .

0. 1

8 6 4 2

f 0

.01

p

8 6 4

0.0

00

2

I '

f 1

I "'

I ~

1 " i

"' i

I

·"~

I ' ~

I

"-I

I'. ~-

l i

"""'

I ~

' I '..

...

I I

" "' .....

....._

-.... "'

---....

.. r--..

...._

.

r----

--1--

-I

I --

--.

--,.._,

I """"

" 1'"-

r-.,

I

Re

= (

D-d

)G/µ

I I

I 11

I

l I

2 4

6 8

2 4

6 8

2 4

6 8

100

1000

1

0,0

00

1

00

,00

0

Fri

cti

on

fa

cto

r fo

r an

nu

li

of

fin

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ara

nce

and

fo

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ara

llel

pla

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Fig

. 3

•

~

SOU

RC

E:

K.

J.

Bell

, an

d O

. P

. B

erg

eli

n,

Flo

w

Thr

9ugh

A

nn

ula

r O

rifi

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~

(Tra

nsa

cti

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ME

79

A

pri

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195

7),

p

. 59

8.

c

1 • 0

0 . 8

o. 6

0. 4

0 . 3

0 . 2

0 . 1

1 5

I I I I i

I I I I ...... r • I I 11 I I I l • I

-~~ ~ ~I- - - -i-- - ____ .._. ___

// , I

'h / I

best fitted line ~/ ~q (1 - 16~ ~

I through data, in

1>-J Vi turbulent range

I c = c ~ o . 65 c /, K ' I

I

v Eq(1 -1 5)

1 2

Fig . 4.

SOURCE :

I I I I

4 6 8

1 0

2 4 6 8

100

2 4 6 8

1 000

2 4 6 8

10,000

Reynolds number, Re = (D- d)G/u

Annular orifice coefficient versus Reynolds ' number for Sharp-Edge Orifice

K. J. Bell , and O. P. Bergelin , Flow Through Annular Orifices (Transactions of the ASE 79 April, 1957), p. 596.

---

-

--

-

23

0.9

0.8

0.7

o.6

0.

5

c 0.4

0.3

0.2

0. 1

o.o

!

I --

~ 1

I I

---

./

,,.~-

~ -

-r--

...._

·~

·r--~ "

"""

. ..._ ~;r----

. /

---

..._

~ ~

-r-

........

... ......

....

~ v

-----r--

.. -..

.,_

'~

"-R

e =

20

00

0

--

T"T

-:7

-

--~

~

" ...

Re

=

10

00

0

-,_

__ -

-~

-..

--

-.....

.. ~",

Re

=

50

00

-~-----

..........

~--

--..

. " ....

........

v ..

----

---

-1

----..

........

r-,

' ~

' R

e =

20

00

~--

' .__

_ .

---

~

--.......

.. I

-.. ~

--

- i---

. ~

~ "

' '-.

.. R

e =

10

00

-..

.. ....

........

....

........

·"-l

I ""

.... ""-

--

-1

""

' R

e =

500

-

--"'

...... ...

--....

.....

--~

"" --

........

....

-.

"""

--

--....

.......

' ""

--~

....

....

... ......

'-I

-~

-.... ....

........

... ,

' R

e =

20

0

-..

-~

' -"

'-"""'

" ....

......

~

--

----~

~

"-.

--....

.. .....

...

' R

e 1

00

=

--~

. -

--~

'--~

--....

. .... ~

..........

.._ -

---

......

...

--....

. .....

--t-

-.....

.... R

e =

50

-....

-~

..........

. ---

......_

-....

r-.

-R

e 2

0

-.....

--....

-

=

-_

,

z =

21/

(D-d

) I

---

Re

= 1

0

----

---

Re

= 5

I

I I

I

0. 1

2

4 6

8 1

2 4

6 8

10

2

4 6

8 1

00

Fig

. 5.

Su

mm

ary

of

co

ncen

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-ori

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co

eff

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nts

SOU

RCE

: K

. J.

B

ell

, an

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erg

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Flo

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ME

79

Ap

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1

957

),

p.

596.

°'

0.9

0.8

0.7

0.6

0.5

c 0.

4

0.3

0.2

0. 1

o.o

I l

l I

I /-=

-"""-

- 1--_

R

e=

2

0,0

00

/

v---

--.....

r--.....

.. ;--

Re

= 1

0,0

00

I 7

--

~

,,. - -

--I

~_;.

... -

- --

-.....

--

Re

=

5000

-

----

... -

-~

-V

-?-

-r-

-..

...

-.... ~

-....

..._

--

.._

-.....

-......

Re

= 2

00

0

--

....._

-~ ~

~

---

--

----

-..

,... --

-.......

-...... ....

.... ~Re =

100

0 .....

.. -

,.._.

...

....._

. --

-~ _

/

......

..........

. .....

.

' -

~

-------

-....

.......

' ~

.-..

i--

----

,,_.,_

, .-

., "

""'-'

........

. "-

Re

' 5

00

..._

....

.....

' =

---

--

-....

........

....

--

........

.. ....

-'

.....

-.....

.. ....

......

-....

.......

' .....

. '

........_

Re

20

0

........

... ....

.......

........

.. =

---

---

----

~

--

' ~

..._

......

--....

. '-

J

........

.. .....

.....

"" ....

.......

........

...

' ......

Re

= 1

00

-..

'""

' "'

........

... -

..........

._ -

-_,

_

-....

.....

--

.........,

....

... ....

......

--..

........

"'-..

Re

50

-=

-..

........

' ...._

r-

-....

.......

-....

-,.

-~

.. .....

..........

' Re

=

2

0

......

..........

.....

"' ...._

_ ....

....

-..

Re

1

0

-....

-=

-

z =

21/

(D-d

) -..

..._

Re

=

5

I I

I I

0.

1 2

4 6

8 1

2 4

6 8

10

2 4

6 8

100

Fig

. 6.

Su

mm

ary

of

tan

gen

t-o

rifi

ce

co

eff

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nts

SOU

RC

E:

K.

J.

Bell

, an

d

O.

P.

Berg

eli

n,

Flo

w

Thr

ough

A

nn

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r O

rifi

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(Tra

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Ap

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19

57),

p

. 59

6.

~

-..J

CHAPTER 2

PRESSURE DROP OF TWO - PHASE FLOW ACROSS

SHORT A D LONG LENGTH RESTRICTIONS

This chapter primarily deals with the Janssen 1 s

pred·ction of steady state two-phase flow pressure

drop across short, and long length restrictions, based

on a one-dimensional momentum balance .

A review of literature indicates that Janssen (3)

obtained t o equations (2-1 and 2 - 2) for prediction of

pressure drop across restrictions of circular and

rec angular geometries depend on whether the vena

cont acta occurs in ide or outside of the restriction

Fig. 7). Thus, for short length restriction where the

vena contracta is o side of the restriction , the equation

of pressure drop is given by

p ( short ) G2 1 [~ 1 x2&3 = TP restriction 2g pf o2 c2 c

{_L - 0 2 2 } + ( 1 - X) 2 (1 - &3) { ( 1 1

ex a25 - a3)2 3

o2cs } - 2aC {~ x2 ( -1- _ ac) + ( 1 - X) 2

( 1 - a 5) 2 Ct 3 Ct 5

( 1 1 ~ca5)}] ( 2 - 1 )

- Ct 3 1

1 9

vena contracta

....... _,, ...... ~ -- --flow .. -- -

- -- - --- -r I .._

-... / ........

1 3 5

A. Short length restriction

/

vena contracta

- - ' J I, __ ....... <-_ r- ...... - -

-flow

> -

1 2 3 4 5

B. Long length restriction

Fig. 7 . Short and long length restrictions

SOURCE : B. Harshe , A. Hussain , J . Weisman . Two -Phase Pressure Dro Across Restrictions and Other Abrupt Area Changes . U.S. Nuclear Regulatory Commission , Report Number NUREG-0062, April, 1976), p . 17.

where

G1

= mass flux in channel

&3 = (et3 + a.5)/ 2

and the a ' s are void fractions at different locations

20

as shown in Figure ? with the assumption of slip flow at

all locations .

In long length restrictions where the vena

contracta occurs within the restriction , the equation

of pressure drop is given by

fiP ( long ) = TP restriction

( 1 - _ 1_ ) ( 1 + -

G z 1

X) 2 ( 1

cz er. 3 <i2- 4 1 \ 4)2 } c2<1 - ) 2 ( 1 -

3 1 1 a 02

<c a.3 - + - ) + ( 1 et4 f:J.4 Cl5

1 1 a c ( 1 a. 3)

- 1 + - - Cl 4 1 -

where

- &1)

2{~ o2 Vr

x2

- X) 2

02 J - +

0.4 1 - Cl5

(2 - 2)

21

and the a ' s a r e v oid fr a c ti on s at different locati ons as

shown i n Figur e 7 .

On the basis of h i s own steam- water data , Janssen

suggested that the void frac t i ons b e estimated by

assuming slip flow eve r ywher e except at the vena

contracta .

In view of the forego ing e qua t i ons , it can be seen

that he vena contracta rat i o is th e only major factor

to be depended on the geometry of the r e s t riction .

Consequentl , it would be possible to app ly t h e Janssen ' s

equation to an form of restrictions ; i . e ., the annular

gap between a tube and a tube supp ort p l ate , providing

that the proper value of vena con tracta ratio is used .

Sin e anssen did not addr es s t he ques t ion of wh en

a restriction ma be cons i der ed short or long, the experi ­

mental stud by Harshe , Hussa i n , a n d Weisman (3) with

freon and its vapor on both si ng le and multiple hole

circular cross - sectional area restric t ions suggests that

equation (2 - 1) may be us ed for s hort length restrict i on ,

the r atio of restric tion len g th t o restriction diameter

is less than o r equal to 2 . 0 , and equation (2 - 2) for

l o n g l ength restrictions wh e n the ratio is greater than

2 . 0 . Mo r eo v e r , the study recommends that in sho r t l ength

re s triction s (0 . 5 < L/D < 2 . 0) and long length res tr i ct i on

(with void fraction greater than 50 %) , the vo i d frac tion

at the vena contracta be based on part i al mixing at the

22

yena contracta which depends on void fraction and geome t ry

of restriction. It is also suggested that for multiple

hole restrictions the ratio of restriction length to

restriction diameter should be determined using diamete r

of a single hole.

However, it should be noted that the validity of

foregoing limitations on the ratio of restriction length

to restriction diameter remains to be tested for this

case study here there is an annular gap, not a circular

hole etween the tube and the tube support plate .

The restriction diameter should be taken as hydraulic

diameter of the annular gap.

hen there is significant vaporization across the

restrictio (large difference between the inlet and the

outlet voi fraction), it is suggested by Harshe ,

Hus in, an ei man (3) that the void fraction at the

vena contracta be based on the exit quality of restriction .

Furthermore, the study recommends the Hug ark correlation

(Appendix 2) for obtaining the relationship between void

fraction and quality for slip flow condition .

CHAPTER 3

CO CLUSIO S AND RECOMMENDATIONS

The goal of this paper was to present the methods

of predicting two - phase flow pressure drop across the

tube support plate due to the existence of the small

annular gap between tubes and tube support plate . This

was ac ieved b considering two approaches :

1 . The first method considers annular clearances

between the tubes and the tube support plate

as annular orifices , assuming that the entire

two - pha -e flow is considered in the state of

saturated liquid. The approach is found to

e ea y to use based on the available data

and equations for finding the overall annular

orifice coefficients , and simple calculation

of two - phase multiplier. However , the method

has lack of data in the area of the annular

orifice coefficient for the cases of multiple

annular orifices , and various eccentric orifices

other than tangent.

2 . The second method deals with the Janssen ' s

23

24

prediction of the two - phase flow pressure dr op

across short and long length restrictions .

Since the Janssen's approach deals with the

nature of the two - phase flow at every loca ­

tion throughout the restrictions, the method

should be more accurate . However , there are

many difficulties associated with this method ;

e . g. , accurate prediction of the void

fractions at various axial locations and the

appro riate values of the vena contracta

ratios .

Since the Janssen 1 s equations of two-phase pressure

drops require the extensive measurements of void fractions

at various axial locations , it would be difficult to

apply them for design purposes of large scale heat

exchangers . However, the first method is easier to use

since it does not require any measurements of flow con­

ditions throughout the shell and should provide adequate

accurac for design purposes .

Based on the findings of this study, it is

recommended that future investigations are needed to

1 . Examine the effect of geometrical parameters ;

i . e . , number of tubes and pitch- to - d i ameter

ratio of tubes , on loss coefficient .

2. Experimentally confirm Janssen's method on

restrictions with annular cross - section

area of flow.

25

APPENDIX 1

UMERICAL EXA PLE OF CALCULATING THE /

T~O - PHASE FLO PRESSURE DROP ACROSS

A ULAR ORIFICES

A 3 - ft . -high two - phase flow channel is in the shape

of a cylindrical shell with the inside diameter of 2 . 5

inche • The channel contains seven heating tubes ,

which pass through a 3/4 inch thick tube support plate

at the middle of the channel (Fig. 8) . Assuming the

channel receives heat uniformly, and operates at a pres -

sure of 1000 psia, an exit quality of 10 percent , and

inlet water temperature of 520°F with flow rate of

6 GP • Compute the pressure drop across the tube support

plate for the following two-phase flow models :

(a) homogeneous flow model

(b) separated flow model with the slip ratio of 2 .

Solution :

The thermodynamic and physical properties of

saturated steam and water are given by

At 520°F ; hi= 511 . 9 BTU/lbm

Vr = 0 . 0209 ft 3 /lbm

26

2 . 5 11

... ...

3/ 4 I

~ I

i I

If

18"

a

x == o. e

I

1

~ 0.53" ~

~o . 50 11tE-

T. = 520°F l

6 GPM

27

Fig . 8 . Configuration of two - phase flow channel and its tube support plate for the numerical example

28

At 1 000 psia ; h f = 542 . 4 BTU/lbm

hf g = 649 .. 4 BTU/lbm

Vr - • 0216 ft 3 /lbm

vg = .4456 ft 3 /lbm

Vrg = . 4240 ft 3 /lbm

µf = ,. 233 1 bm/hr . - ft .

First , it is necessary to determine if the tube

support is in the two - phase flow region of the channel .

For this purpose , we calculate the non - boiling he ight

for the uniformly heated channel (Fig . 9) • Thus

H qs hf - h . l

H = = (hf+ x hf ) qT - h« e g l

= 542 . 4 - 511 . 9 (542 . 4 + 0 . 1 x 649 . 4) - 511 . 9

:'.:: 0 . 32

therefore. the non boiling height is

H == 0 . 32 H = 0 . 32 (36 11)

= 11.52 in .

and the boiling height is

HB = H - HN = 36 - 11.52 = 24.48 in.

Since , HTSP > HN ' the two - phase flow condition exists at

the tube support plate .

H =

HTSP =

H =

HB =

x =

XTSP =

height of

I channe l

height of tube suppo r t plate

non - boiling height HB

boiling height H

quality at exit T quality at tube TUBE support plate HTSP SUPPORT

ll, PLATE

Fig . 9 . Variation of quality wi th h eight in a uniformly heated channel

29

Assuming the existence of the tube suppor t plate d oes n ot

change the linear ehavior of quality wi th h e i gh t , th e

qualit at the tube support plate can be obtained f r om

1 8 =

x e

- 11 . 52 I = 2 . 65% 24 . 48 '

Under the steady state condition , the mas s flow rate

remains constant throughou t the channe l (m T = mH = ~B) .

Therefore

1 = = vr 47 . 847 lbm/ft 3

an d

ifl T - 3 3/ = 6 g a 1 x 2 • 2 3 x 1 0 . ft s x 4 7 • 8 4 7 1 bm

min 1 g~l ft " 3

min

30

= 0 .640 lbm/s

Since the cross - sectional areas of the annular

gaps between the tubes and the tube support plate are

the same , we assume each gap carries the same mass flow

rate under the steady state condition regardless of

their annular geometries . Thus

1 1 0

= 7 = - x o . 640 - 0.0914 lbm/s T 7

and mass flux is

0 ::: = Ao

0 . 0914 I 2 n(

2 5 2 ) 1 = 542.33 lbm ft - s 4 0 . 53 - - o. 0 144

Regarding the annular gaps between the tu bes and

the tube support plate as annular orifices , the annular

orifice Reynolds number can be obtained from equation

(1--14) :

=

(0.53 - 0 . 50)ft (542 . 33 )lbm/ft 2 -s 1 2

0 .23 3 lbm/hr-ft

x (3600) s/hr = 20 , 948

Since Re > 4000 , the flow is turbulent. 0

Also , Z = 2L = 2(3/4) " D- d 0.53 - 0.50 = 50

To obtain the two -phase flow pressure drop across the

tube support plate, we examine the following cases:

31

CASE I

Assume the tubes are in the concentric position inside

the support plate holes . Therefore, from equation (1-18)

* 1 K = c2

where

z = 50

= 1

c 2 c [ 2 J - - 2 1 Cc I

F = 1 _8-. 95(50 - 1 . 15) ~ 1

f = 0 . 00?3 (from Fig. p

c ~ o . 65 (from Fig . 4) c

*

F + 2f Z p

3)

1 resulting K = = 2 . 020 c2 Thus , the pressure drops in two - phase flow are

(a) Homogeneous floN" model

( 1 7. ) ( P ) = K* G2

( 1 + ~ X) Eq • - : fJ TP H 2 V gcpf f

where X = XTSP = 2 . 65%

Thus

- (542 . 33) 2

(~PTP)H - 2 • 020 2(32 . 2)(46 . 296)144

I[ . 4240 . l 1 + • 021 6( .0 265 ~= 2 .1 04 psi

(b) Separated flow model (S = 2)

( Eq • 1 -· 1 1 ) : ~~ Gz [_(.1 X)2 V

(6 PTP) s = K 2gc Pr _,__1 ___ (), ___ + ~

where

(), = ~SP = 1

~2 J

32

1 =

1 + (1 - . 0265)2( • 0216) .0265 . 4456

= 21 .92 %

Therefore,

(5 ,42.33) 2 [(1 - .0265) 2

( l'.P TP ) S - 2 • O 2 O 2 (3 2 • 2 )( 4 6 • 2 96 ) 1 4 4 _ 1 - • 21 9 2

+ ( . 4456)( . 0265)2J = 1 771 .

. 0216 .2192 · psi

CASE II

Assume the tubes are in the tangent position inside the

support plate holes . Therefore , from equation (1-19)

with

z = 50

f = 0 . 00'73 (from Fig . 3) p

c :::: o . 65 (from Fig . 4) c

A = zco.53 2 0 . 502)1!.4 0

L = 3/4 in. = 0 . 0625 ft.

R = 0 . 0221 ft •.

r = 0 .0208 ft .

resulting C = 0.752

* 1 or K = 2 - 1.?68 c

= 1 • 68 x 1 o- 4

Thus, the pressure drop in two - phase flow are

ft 2

(a) Homogeneous flow model from equation (1-7)

(11.PTPlH = 1 • 768 2(32~~tf4t:~~6)144

[1 + : ~~t6( . 0265)J = 1 .841 psi

33 (b) Separated flow model from equation (1-11)

) (542 . 33) 2

(~PTP S = 1 • 768 2(32 . 2){46 . 296)144

[11 - . 0265) 2 + . 4456 ( . 0265) 2

]

1 - . 2192 .0216 . 2192

= 1.550 psi

The foregoing procedure have been applied to the

following flow conditions

Flow rate = 6 to 8 GPM

Exit quality = 0 . 05 to 0 . 25 percent

Slip ratio = 1 to 3

which the results are presented in Table 1 and the

graphs of pressure drop across tube support

plate versus the quality at the tube support plate .

FLOW

RA

TE

GPM

6 7 8

TAB

LE

1

RESU

LTS

OF

CALC

ULA

TIO

NS

FOR

SATU

RATE

D

SIN

GLE

-PH

ASE

PR

ESSU

RE

DROP

FO

R V

ARI

OU

S FL

OW

CON

DIT

ION

S

CONC

ENTR

IC

TANG

ENT

OR

IFIC

E O

RIF

ICE

mT

m

G

Re

f 1

6Pf

1 ll

Pf

0 0

0 p

~

CT

.640

• 0

914

542

.33

2094

8 .0

073

2.0

2 1

. 38

4 1

. 768

1

• 211

.747

.1

067

633

.11

2445

5 . 0

071

2.0

0 1

. 867

1

.75

7

1 • 6

40

.854

.1

220

723.

89

2796

1 .0

0707

1

. 997

2

.437

1

• 7 56

2

.143

I

\>.)

+'-

.. ~ 0

3 . 0

FLOW RATE = 6 GPM

SLIP RATIO = 2

2 . 5

2 . 0

1 • 5

1 • 0

0 . 5

o.o

./

TA GENT

0 2 4 6 8

Mixture quality, X percent

Fig. 10. Pressure drop versus mixture quality across the support plate for both concentric and tangent orifices

35

10

·rl Cf)

0..

Pot 0 H

'"d

Q)

H ;:1 en C/l (!)

f-c ~

5. 0

4.5

4. 0

3 . 5

3 . 0

2 . 5

2 . 0

1 • 5

1 • 0

0 . 5

o. o 0 2

GONCE TRIG ORIFICES

FLO~ RATE = 6 GPM

4 6 8

Mixture quality, X percent

Fig . 11. Pressure drop versus mixture quality across the support plate for various slip ratios

36

1 0

. ,.; Cf)

p.

. p. 0 H

'TI

().)

H :::::1 Cf.)

Cf.)

Q)

H ~

4. 0

3 . 5

3 . 0

TA GE T OR I F I CES

FLO RATE = 6 GPM

s = 1

~

37

/ /

/

/ /

/

/ / s 2 - / 2 . 5 / ~ /

/ / ---/ ../' ....--....--2 . 0 / / --,,,,..,.....

' ./ /' ----./" ....--/ ./" --/

....-- s 3 ...,,.,,.... __.....,... -

1 • 5 / /' --/

./ /" --/ _.. / ;::::::. ...-

, _::;::?

1 • 0

0 . 5

0 2 4 6 8

Mixture quality, X percent

Fig. 1 2 . Pressure drop versus mixture quality across the support plate for various slip ratios

_...

·rl Cl)

0...

... ~ 0 H

"d

Q)

H :::s Cl)

C/J Q)

~ P-1

38

6. o

5 . 0

4 . 0

3 . 0

2 . 0

1 • 0

CONCENTRIC ORIFICES

SLIP RATIO ::: 2

8 GPM

6 GPM

o. o .._ __________________ ..._ __ _,., ____ .._ __ _.. ________________ _

0 2 4 6 8

Mixture quality, X percent

Fig. 13. Pressure drop versus mixture quality across the support plate for various flow rates

1 0

·rl Ul p...

... p... 0 H rd

Q)

f..t ;:s Cl)

Cl)

Q)

H P-t

39

TANGENT ORIFICES 5 . 0

SLIP RATIO == 2

4.5 / 8 GPM /

j / 4.0

/

3 . 5 /

/ ,-7 GP

/ ,,,-

I /

/ /

3 . 0 /

/ ~

/ ,,..,....

/ .,,,....

6 GPM / .,,,,.,....

2 . 5 / /

.,,,...,, j _.,....-

/ ,,..,..... _.,,..,,,..

~ --2 ., 0 / ~

_..--

/""' _..-

.....-,,,,. .....,.-"

1 • 5 ......-...--_... __..

...,...

1 • 0

0 . 5

0 . 0 .._ ________ .__ __ _,_ __ -.l. ____ ._ __ ..... ____ _,_ __ _._ __ ....1~--..._

0 2 4 6 8

Mixture quality, X percent

Fig. 14. Pressure drop versus mixture quality across the support plate for various flow rates

1 0

40

DISCUSSION

From these curves (Figure 10 through Figure 14) ,

it can be seen that :

1 . The pressure drop across the support plate

increases linearly as the quality across the

support plate increases . This is found to be

true for the following cases:

(i) both homogeneous and separated flow models

(ii) both concentric and tangent orifices

(iii) for all flow rates .

2. The pressure drop in ornogeneous flow models

are found to be uch greater than the separated

flow models .

3 . It is found that the pressure drop across the

oncentric annular gaps is greater than the

tangent annular gaps .

APPE -DIX 2

SA PLE CALCULATIO OF THE TIO - PHASE

(AIR - ATER) FLO. PRESSURE DROP

ACROSS AJ ULAR ORIFICES

This Appendix gives the sample calculation of

t o-pha e flo (air - water) pressure drop across the tube

s pport plate si ilar to Appendix (1), and comparison

ith the expe i ental result from the University of

Central Florida to - phase flo apparatus (Fig . 8) , when

a ix ure of 6 GP of ~ater ~ith 20 ft 3 /hr of air at

room tern erature of 78°F and the atmospheric condition

flows in the hannel.

Solution:

The ph sical properties of air and water are given

by

For air ; pg = 0 . 0737 lbm/ft 3

v = 1 3 . 5 68 ft 3 /lbm g

For water ; Pr = 62 . 32 lbm/ft 3

vr - 0 . 016 ft 3 /lbm

µf = 2 . 08 lbm/hr - ft

41

42

Therefore , the mass flow rate of air and wa t e r c an

be found

ft 3

= mg = 20 hr x m . air

= 0 . 00041 lbm/s

0 .. 0737 lbm x 1 ft 3 1 h r

3 6 00 s

m = . = 6 gal x 2 . 23 x 10 - 3 ft 3 /s water f min 1 g~l

min

lbm x 62 . 32 ft 3 = 0 . 83384 lbrn/s

Consequentl the total mass flow rate is given by

= + f = 0.00041 + 0 . 83384 = 0 . 83425 lbm/s T g

Under the stead -state condition , ..re assume the

total as flo rate remains constant throughout the

channel and each of the annular gaps bet een the tubes

and the support late holes carries the same mass flow

rate. Thu

1 = 7 T = ~ x D. 83425 = 0 . 11918 lbm/s

and mass lux is

G 0

0 = = A

0

0 . 11918 = 707 . 16 lbrn/ f t 2 - s 1f 2 2 1 4(0 . 53 - 0 . 50 )144

Considering the annular gaps between the t u be s a nd

the tube support plate holes as annular or i fices , t h e

annular orifice Re nolds number can be found from

equation (1 - 14) :

Re 0

=

= (D - d) G0

JJf

c0

•53

1 ; 0

·50

)rt ( 707 .1 6 )l bm/ft 2 -s( 3600)hr / s

2 . 08 l brn/hr-:ft

43

~ 3060

Since 40 < Re < 4000 , the flow is transitional . 0

Assuming the tubes are in the concentric position

inside the support plate holes , from equation (1 -1 7) can

be found

* K =

where

z =

F ~

f = p

c ~

c

resulting * K =

1 c2

50

1

= 1

C2 c

0 . 01 3 5 (from

o . 65 (from

1 2 . 654 c2 =

Fig .

Fig .

3)

4)

2f z p

Since there is no state change from water to air,

no relationship can be found between the specific

volumes of air and water for defining Vfg in the

equation (1 - 7) of pressure drop in two - phase homogeneous

flow . Therefore , the equation (1 - 11) of pressure drop

in separated flow with slip ratio equal to 1 • 0 is being

used

where

instead

v xz J Eq . ~E- G2 [~~ - X) 2

( 1 - 11 ): (6PTP) = K + J. -2 gcpf - Ct v f Ct H

~ X = XTSP = rtiT = 0.00041 = 0 00049

0 . 83425 . and quality is assumed to remain cons~ant throughout the channel under the steady­sta te condition .

44

Cl = aH a TSP 1 = = 1 - XTSP Vf

1 + ( )-XTSP Vg

1 = - 0 . 00049)0.61605 1 + ( 1

0 . 00049 1,3.568

= 29 . 30%

Thus, the predicted value of two - phase pressure

drop can be found

(707.16) 2

= 2 · 654 2(32 . 2)(62.32)144

[( 1 - 0 • 0 0 0 4 9 ) 2 + ( 1 3 • 5 6 8 ) ( 0 • 0 0 0 4 9 ) 2 l

1 - 0 . 293 0.01605 0 . 293 J = J.246 psi

From the comparison between the predicted value

(3.246 psi) and the experimental value (4 . 729 psi) of

two-pha e pressure drop, it can be seen that the predicted

value underestimated the experimental value by 45 . 7% .

The difference might be caused by :

1. The assumptions which are stated in Chapter 1 .

2 . The positions of pressure taps before and after

the support plate .

Also , it should be noted that the two - phase multi -

plier, ¢ , are derived for two - phase flow with one component

~iquid and its vapor) , not two components (air and waten.

Since the two - phase pressure drop for separated flow

model is less than the homogeneous flow model , the

calculation of pressure drop for the separated f l ow model

will not be presented .

APPE DIX 3

THE HUGH1ARK CORRELATIO (4)

In a two-phase flow, the relationship between

quality and void fraction is given by Hughmark 1962 as

1 v R x = 1 - vf (1 ... a)

here ft is related to a parameter ~ (Table 2) which is

defined as follo s :

Z = (Re) 1 / 6 (Fr) 1 / 8 (1 - a) - 1 /4

or

here U is volumetric flux, and is given by

u = A

Also, Figure (15) is provided for comparison between

the Hughmark Correlation and equation (1 - 2) for void

fractions versus mixture qualities .

45

* z * R * z JI ,,. R

1 • 3

0 .1

85

8.0

0.7

67

TABL

E 2

* ~

VALU

ES

OF

R A

S A

FU

NCT

ION

OF

Z

1 • 5

2

.0

3.0

0.2

25

0,,3

25

0.49

10.0

15

.0

20.0

0.7

8 0

.808

0

. 83

4.0

5.0

6

.0

0.6

05

0.6

75

0.7

2

40.0

70

.0

13

0. 0

0.8

8 0

.93

0.9

8

SOU

RCE

: Jo

hn

G.

Co

llie

r,

Co

nv

ecti

ve

Bo

ilin

g

and

Co

nd

ensa

tio

n.

New

Yor

k:

McG

raw

H

ill

Boo

k C

ompa

ny,

1972

, p

. 68

.

.p-- 0'-

100

90

80 ..µ

s:::: 70 Q)

F-i Q)

60 0..

z 50 0 H E-i 40 0 c::t; 0:: Px-t 30 p H 0 20 >

1 0

0 0

Eq (1 - 2) for S = 2

-- ---

HUGH ARK CORRELATIO for slip flow with G

0 = 542.33 lbm/ft 2 - s

20 40 60 80

Quality, X percent

47

100

Fig . 15 . Void fraction versus mixture quality for the annular gap between the tube and the support plate (Appendix 1) at the system pressure of 1000 psia

REFERENCES CITED

1 . Lahey , R. T., Jr ., and Moody , F . J . The The r mal ­Hydraulics of Boiling Water Nuclear Reactor . La Grange Park , IL: American uclear Society , 1979 , pp . 173 - 245 . ~

2 . Bell , K. J ., and Bergelin , 0 . P . Flow Through Annular Orifices . AS E Transactions , 79 , April , 1957, pp. 593- 601 .

3 . Harshe, B.; Hussain, A.; and eisman , J . Two - Phase Pressure Drop Across Restrictions and Other Abrupt Area Changes . Universit of Cincinnati for U. S . uclear Regulator Commission, Report umber UREG - 0062 , April, 1976 , Springfield , VA : ational Technical Infor ation Service .

4. Collier , John G. Convective Boiling and Condensation . ew York : cGraw-Hill Book Company , 1972, p . 68 .

48

BIBLIOGRAPHY

Bell , K. J. , and Bergelin , O. P . "Flow Through Annular Orifices . 1 AS E Transactions 79 (April 1957): 593 - 601 .

Butterworth , D., and Hewitt , G. F . Two - Phase Flow and Heat Transfer . Oxford : Oxford University Press , 1977 .

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