CSNI Report No. 49 ARCHIVES NUCLEAR SAFETY DIVISION TWO-PHASE CRITICAL FLOW MODELS A technical addendum to the CSNI state of the art report on critical flow modelling F. D'AURIA - P. VIGNI Université degli Studi di Pisa Istituto di Impianti Nucleâri May 1980 Work sponsored by: COMITATO NAZIONALE PER L'ENERGIA NUCLEARE Roma NUCLEAR ENERGY AGENCY ORGANIZATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENT COMMITTEE ON THE SAFETY OF NUCLEAR INSTALLATIONS
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
CSNI Report No. 49
ARCHIVES
NUCLEAR SAFETY DIVISION
TWO-PHASE CRITICAL FLOW MODELS
A technical addendum to the CSNI state of the art report on critical flow modelling
F. D'AURIA - P. VIGNI
Un ivers i té degli Studi di Pisa Is t i tu to di Impianti Nucleâr i
May 1980
Work sponsored by:
COMITATO NAZIONALE PER L'ENERGIA NUCLEARE
Roma
NUCLEAR ENERGY AGENCY ORGANIZATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENT
COMMITTEE ON THE SAFETY OF NUCLEAR INSTALLATIONS
CSNI Report No. 49
TWO-PHASE CRITICAL FLOW MODELS
A technical addendum to the CSNI state of the art report
on critical flow modelling
F. D'AURIA - P. VIGNI
U n i v e r s i t é d e g l i S t u d i di P i s a
I s t i t u t o d i I m p i a n t i N u c l e a r i
May 1980
Work sponsored by:
COMITATO NAZIONALE PER L'ENERGIA NUCLEARE
Roma
NUCLEAR ENERGY AGENCY ORGANIZATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENT
COMMITTEE ON THE SAFETY OF NUCLEAR INSTALLATIONS
Riprodotto in offset presso il Laboratorio Tecnografico délia Direzione Centrale Relazioni Esterne del CNEN - Vialc Rcgina Margherita 125, Roma
The Organisation for Economic Co-operation and Development (OECD) was set up under a Convention signed in Paris on 14th December, 1960, which provides that the OECD shall promote policies designed:
— to achieve the highest sustainable economic growth and employment and a rising standard of living in Member countries, while maintaining financial stability, and thus to contribute to the development of the world economy ;
— to contribute to sound economic expansion in Member as well as non-member countries in the process of economic development;
— to contribute to the expansion of world trade on a multilateral, non-discriminatory basis in accordance with international obligations.
The Members of OECD are Australia, Austria, Belgium, Canada, Denmark, Finland, France, the Federal Republic of Germany, Greece, Iceland, Ireland, Italy, Japan, Luxembourg, the Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, Turkey, the United Kingdom and the United States.
The OECD Nuclear Energy Agency (NEA) was established on 20th April 1972, replacing OECD's European Nuclear Energy Agency (ENEA) on the adhesion of Japan as a full Member.
NEA now groups all the European Member countries of OECD and Australia, Canada, Japan, and the United States. The Commission of the European Communities takes part in the work of the Agency.
The primary objectives of NEA are to promote co-operation between its Member governments on the safety and regulatory aspects of nuclear development, and on assessing the future role of nuclear energy as a contributor to economic progress.
This is achieved by: — encouraging harmonisation of governments' regulatory policies and practices in
the nuclear field, with particular reference to the safety of nuclear installations, protection of man against ionising radiation and preservation of the environment, radioactive waste management, and nuclear third party liability and insurance;
— keeping under review the technical and economic characteristics of nuclear power growth and of the nuclear fuel cycle, and assessing demand and supply for the different phases of the nuclear fuel cycle and the potential future contribution of nuclear power to overall energy demand;
— developing exchanges of scientific and technical information on nuclear energy, particularly through participation in common services;
— setting up international research and development programmes and undertakings jointly organised and operated by OECD countries.
In these and related tasks, NEA works in close collaboration with the International Atomic Energy Agency in Vienna, with which it has concluded a Co-operation Agreement, as well as with other international organisations in the nuclear field.
LEGAL NOTICE
The opinion expressed and arguments employed in this publication are the responsibility of the Authors and do not necessarily represent those of the OECD.
Copyright OECD, 19^0
Queries concerning permissions or translation rights should be addressed to: Director of Information, OECD
2, rue André-Pascal, 75775 PARIS CEDEX 16. France.
Riprodocto in offset presso il Laboratorio Tecnografico della Direzione Cen-trale Relazioni Esterne del CNEN Viale Regina Margherita 125, Roma
- Ill -
ACKHOWLEVGEMENT
The. authors would like, to ac.knowle.dge. M. Mazzlnl o£ Visa Uni
ve.i<y and G. SantafioA&a o& CNEhl &on the.lt. hzlp dating the. de.-
ve.lopme.nt o£ the. wonk, and all the. pzK&on& who have contributed
to the. print and to the. translation.
- V -
INDEX
ABSTRACT Pag. IX
LIST OF SYMBOLS " XI
3.1 INTRODUCTION " 1
3.2 CLASSIFICATION OF THE MODELS " 3
3.2.1 Criteria " 3
3.2.2 Classification " 6
3.3 SCHEMATIC DESCRIPTION OF EXAMINED MODELS " 13
3.3.1 Generality " 13
3.3.2 Perfect fluid " 15
3.3.2.1 Perfect gas - Classical theory " 15
3.3.2.2 Perfect gas - Flow from cylindrical
ducts M 16
3.3.2.3 Incompressible liquid - Classical
theory " 18
3.3.3 Theories assuming thermodynamical equili
brium between liquid and vapor phases " 20
3.3.3.1 Homogeneous equilibrium theory " 20
3.3.3.2 'Fluid dynamic' approach in the fo_r
mulation of a homogeneous thermody
namical equilibrium model " 22
3.3.3.3 Babitskiy 1973 " 24
3.3.3.4 Non homogeneous equilibrium theo
ries - Introduction " 26
3.3.3.5 Moody 1965 " 27
3.3.3.6 Moody 1966 " 29
3.3.3.7 Fauske 1964 " 31
3.3.3.8 Levy 1965 " 33
- VI -
3.3.3.9 Cruver-Moulton 1967
3.3.3.10 Ogasawara 1969
3.3.3.11 Ogasawara 1969
3.3.3.12 Malnes 1977
3.3.3.13 Adachi 1973
3.3.3.14 Afechi 1974 (cylindrical duct)
3.3.3.15 Adachi 1974 (orifice)
3.3.3.16 Moody 1975
3.3.3.17 Castiglia-Oliveri-Vella 1979
3.3.3.18 Tentner-Weisman 1978
3.3.3.19 Wallis-Richter 1978
3.3.3.20 Ransom-Trapp 1978
3.3.4 Non equilibrium models
3.3.4.1 "Frozen" theories
3.3.4.2 Burnell 1947
3.3.4.3 Zaloudek 1963
3.3.4.4 Starkman-Schrock-Neusen-Maneely '64
3.3.4.5 Moody 1969
3.3.4.6 Henry-Fauske 1971
3.3.4.7 D'Arcy 1971
3.3.4.8 Ardron-Furness 1976
3.3.4.9 Ransom-Trapp 1978
3.3.4.10 Non equilibrium, general theories
3.3.4.11 Henry-Fauske 1970
3.3.4.12 Henry 1970
3.3.4.13 Klingelbiel-Moulton 1971
3.3.4.14 Klingelbiel-Moulton 1971
3.3.4.15 Malnes 1975
3.3.4.16 Porter 1975
Pag. 36
39
41
43
44
47
49
51
53
56
59
61
65
66
67
68
69
71
73
76
78
80
81
81
85
88
90
91
94
- VII -
3.3.4.17 Rivard-Torrey 1975 Pag. 97
3.3.4.18 Kroeger 1976 " 100
3.3.4.19 Bouré-Giot-Fritte-Réocreux 1976 " 105
3.3.4.20 Avdeev-Maidanik-Seleznev-Shanin *77 " 108
3.3.4.21 Travis-Hirt-Rivard 1978 " 111
3 .3 .4 .22 Tentner-Weismann 1978 " 114
3.3.4.23 Moesinger 1978 " 116
3.3.4.24 Winters - Merte 1979 " 118
3.3.4.25 Romanacci 1976 " 121
3.3.5 Theories briefly described by Giot " 124
3.3.5.1 Sudo-Katto 1974 " 124
3.3.5.2 Giot-Meunier 1968 " 124
3.3.5.3 Meunier 1969, Giot 1970, Giot-Frijt
te 1972 " 125
3.3.5.4 Flinta 1973, Flinta 1975 _ " 126
3.3.5.5 Bauer 1976 " 126
3.3.5.6 Stadtke 1977 " 126
3.3.5.7 Seynhaeve 1977 " 126
3.4 COMPARISON AMONG SOME MODELS " 129
3.4.1 Generalities " 129
3.4.2 Quantitative comparison among different m£
dels " 130
3.5 CONCLUSION " 133
3.5.1 Main conclusions of some authors " 133
3.5.2 Results and discussion " 138
BIBLIOGRAPHY " 141
FIGURES " 165
APPENDICES - APP. 3.1: About the calculation of perfect
gas sound velocity " Al
- VIII -
APP. 3.2: Definition of functions f , f_,
f of paragraph 3.3.3.6 Pag. A3
APP. 3.3: Definition of vaporization and
condensation rates (Par. 3.3.3.16)
per unit pressure reduction " A5
APP. 3.4: Definition of function q.. and q
relating to paragraph 3.3.3.17 " A7
APP. 3.5: Hughmark correlation (See Par.
3.3.3.18) " A9
APP. 3.6: Discussion of Eq. (6) of the Pa.
ragraph 3.3.3.20 " A l l
APP. 3.7: Definition of the parameters A ,
and X related to Par. 3.3.4.7 " A13
APP. 3.8: Formula to obtain slip ratio
(Par. 3.3.4.13) " A15
APP. 3.9: Definition of the state function
n and of the friction factors sg
(Par. 3.3.4.17.) " A17
APP. 3.10: Evaluation of the vaporization
rate for Adveev model " A19
APP. 3.11: Description of the vapor genera
tion rate for the model of Travis
et al. " A21
- IX -
ABSTRACT
This won.k was originally thz thi.ua chaptzn. o£ a Statz o£
thz Kut Rzpon.t (SOAR) which will bz publlshzd by the. Commit
tzz on the. Sa.6e.ty orf Nuclzan. Installation* (CSWIJ. ton. ptiac
tical Kzasons, aktzn.wan.ds, it has bzzn nzczssan.y to shoKtzn
the. a^onzsaid chaptzn using only a position o£ this contnibu
tion in the. SOAR. Howz\>zn, in oKde.fi to kzzp the. monz dzta.il
zd {,in.st dn.a^t intact too, it was dzcidzd to pKoducz this
as a CSWI technical addzndum to the SOAR itself.
Thz puHposz o£ this wofik is to obtain a compKzhznsivz
sunvzy on thz two-phasz ilow dynamics during accidzntal si
tuations in nuclzan. Kzactons.
About sixty thzofiizs n.zgan.di.ng thz two-phasz £low calcu
lation havz bzzn Kzvizwzd in this Kzpon.t with paKticulan. nz
fiznznez to thzin physical basis and assumptions ; thz aim is
to control thzin. applicability to nuclzan. safety pKoblzms .
Thz main conclusions may be dfiawn as follows:
- thz zxaminzd thzonJLzs anz vzn.y di^zn.znt both £on. faofimula
tion and KzsultS',
- gznzfial validity oi most thzon.izs is tJioublz'somz to chzck
&on thz usz o& zmpifiical cozh&iciznts.
Monzovzn, accon.di.ng to thz authons ' opinion, it is nzczs_
s any to szt up an onganic pn.ogA.am to obtain Kzliablz zxpzni
mzntal nzsults in this iizld and to dzvzlop a modzl considz
King thz wholz blowdown tKansiznt.
- XI -
LIST OF SYMBOLS
a sound velocity
A flow area
b hydraulic head
c specific heat at constant pressure
c„ specific heat at constant volume
C_ area reduction coefficient
D pipe diameter
D equivalent hydraulic diameter eq n
e specific internal energy
E total internal energy
f friction coefficient f„ Fanning friction coefficient F
F force
Fr Froude number
g gravity constant (exceptionally used in this work)
G flowrate = TA
h specific enthalpy
H total enthalpy
k slip ratio
kp slip ratio by Fauske
kM slip ratio by Moody
L pipe lenght or work
m mass
M total mass
n polytropic coefficient
p pressure
P wetted flowing duct perimeter
- XII -
Pr Prandtl number
q,Q heat exchanged by two phase mixture
R energy loss (Par. 3.3.3.10)
R bubble radius
Re Reynold number
s specific entropy
S thermal exchange surface
t time
t dimensional time
T temperature
u phase velocity when more than one dimension is considered
U vectorial velocity
v specific volume
V perturbation velocity
x quality or cartesian coordinate when more than one dimension is considered
w phase velocity
W specific volumetric flowrate
We Weber number
z cartesian coordinate
a void fraction
Y c /c 1 p' v
r specific flowrate
A(E) determinantal equation used when energy equation is adopted
A(s) determinantal equation used when entropy equation is adopted
e circumference
ç pressure loss n Pc/P0 X water conductivity y dynamic viscosity
- XIII -
v
ç
p
a
T
X
SI
kinematic viscosity
loss coefficient
mixture density
surface tension
wall shear
two phase flow multiplier
bubble wall interfacial heat flux
evaporation rate
volume
moreover: D/DN total derivative with respect to N
3/8N partial derivative with respect to N
<N> cross section average value
N vectorial quantity
(N is the generical variable)
SUBSCRIPTS INDEX
a
A
c
con
down
e
ex
E
f
fg
F
ambient
assigned
critical value
condensation
downstream
exit
exchanged
equilibrium
liquid phase
such that N-. =N -N-fg g f
frictional
- XIV -
g vapor phase
G vapor phase (only in Par. 3.3.4.?.9)
GSP gas single phase
h constant enthalpy
H homogeneous
HE homogeneous equilibrium
HF homogeneous frozen
i inlet
irr irreversible
L liquid phase (only in Par. 3.3.4.19)
LSP liquid single phase
m mixture
M momentum averaged
n phase index (may be f or g)
ns non isentropic
o initial value or reservoir value
p constant pressure
rev reversible
s constant entropy
sat saturation
st stationary
t throat
T total
up upstream
vise viscous
w wall
- XV -
SUPERSCRIPTS INDEX
x critical value
static value
derivative with respect to time
NOTE: When in the text we write "Mpody" we intend "Moody 1965'
•
3.1 INTRODUCTION
In the previous chapters the phenomenological aspects of the
Loss of Coolant Accident (LOCA) have been emphasized from a ther
mo-fluiddynamic point of view. In particular the concept of two
phase "maximum" flow has been analyzed, outlining the main para
meters it depends upon, and consequently the difficulties in
the mathematical models.
As far as the theoretical study is concerned, the difficul
ties are mainly due to "quantitative" evaluation of the follo
wing points :
- multidimensional effects and flow pattern (bubble flow, slug
flow, annular flow, etc.);
- phase change, generally under strong non-equilibrium conditions, 2
which take place when the high pressure fluid (70-150 kg/cm )
meets the ambient pressure (problems related to mass, momen
tum and energy exchange between liquid and vapour phases and
between each phase and the exterior);
- influence of the geometrical and thermodynamical situation on
critical conditions (internal flow paths, rupture position
with respect to liquid level, heat source, etc.).
In particular the last point makes troublesome the formula
tion of one theory valid for all plant situations.
In this chapter models presented so far are taken into ac
count, in the context of the SOAR concerning both the applicabi^
lity to any real situation and the future development of the
theoretical problem.
The following section deals with the criteria for the classji
fication of different models.
9
- 3 -
3.2 CLASSIFICATION OF THE MODELS
3.2.1 - Criteria
In the last 40 years, several tens of models have been publi
shed about calculation of the critical flow rate of a two phase
mixture.
In some models there is no theoretical support and they are
rather semi-empirical formulas, linking critical flow rate to
thermodynamical variables, generally representing the fluid sta
te in the pressure vessel. These models adopt adimensional empi
rical coefficients in order to fit the esperimental data.
Other models, instead, derive from the solution of a set of
two or more (up to 6) equilibrium equations, describing the con
servation of mass, momentum and energy, for each phase separate
ly or for homogeneous mixtures.
From a mathematical point of view, it ought to be possible to
draw, with suitable simplifying assumptions from complex theo
ries (e.g. based upon all of the six balance equations) models
with a smaller number of equations; this is hardly ever achieva
ble as in most models the result is obtained introducing evolu
tion laws, which are more or less particular or arbitrary.
Moreover, many theories become acceptable for the engineering
calculations, if these are related to a given experimental a£
paratus and/or to conditions assigned beforehand. On the contra
ry, if boundary conditions (that are usually simplifying assump
tions) are varied, the results sometimes diverge or become unusa
ble.
From this brief outline, one can foresee quite different for
mulations of the models and difficulty in group classification.
- 4 -
The following subdivision in four groups has usually been /10, 54/
adopted
1) Theories which assume thermodynamic equilibrium throughout the expansion
2) Non-equilibrium theories
1.a - homogeneous theories (k=l)
l.b - non-homogeneous theories (k ? 1)
2.a - "frozen"theories (k/ 1) but having given value (*)
2.b - non-homogeneous theories (k f 1)
This subdivision is maintained throughout this work.
Nevertheless, other characteristics allowing a deeper compari
son and giving an idea of the model applicability are referred
to; such characteristics are listed as following:
A) Model formulation
A.l) number of conservation equations, e.g. equations taking
account of mass, momentum and energy conservation of the
flowing system;
A.2) number of state and/or transformation equations: the sta
te equations are those describing the system state
through some variables (e.g. p, T, h, s, etc.); the tran
sformation equations outline the state change of the system,
according to given criteria (**)
(x) By "frozen" we intend that the composition at the inlet of the flowing pipe is the same as at the outlet.
(xx) The two types of equation have been matched in order to avoid sophi sticated definitions to distinguish them. Considering, e.g., the e-quation ds=0, at the same time it is a state equation and it indiyi duates an isoentropical transformation.
- 5 -
A.3) number of constitutive equations: they are usually empi
rical equations, derived either from adimensional analy
sis or from assumptions regarding the system behaviour ;
A.4) number of analytical conditions, added to the model, so
metimes without any physical meaning;
A.5) Necessity of semi-empirical parameters for problem solu
tion.
B) Assumptions (phenomenological aspects and parametersconside-
red)
B.l) transient phenomena;
B.2) multidimensional-effects;
B.3) non-homogeneity in pressure vessel;
B.4) heat exchange with exterior;
B.5) pipe length (essentially for friction);
B.6) orifice (I), nozzle (II), constant pipe area (III), any
kind of pipe (IV).
C) Output
C.l) diagrams or relations linking p0, ho and/or x0 to T;
C.2) diagrams or relations linking p and/or x to T;
C.3) diagrams giving exit thermodynamical variables vs reser
voir ones;
C.4) model applicability
In the following paragraph an answer to the aforesaid que
stion is given for each model taken into account. In the Par.3.3
a brief description of the considered models is reported.
(x) For example, T=f (h£,hg,p,v); %x~^2 (hf»hg»P»v) a r e intended as constitutive relationships,
(xx) For the definition of model applicability see paragraph 3.3.1
- 6 -
3.2.2 - Classification
The examined models are divided into the four main groups
mentioned above. Each model is located in Table 3.1 according
to the first author's name and to presentation date, with ref£
rence to literature available.
The models about maximum flowrate of a perfect fluid, both
throughout a De Laval nozzle and a cylindrical duct, are exa
mined in detail also owing to their use as reference point for
two phase fluid. Finally in Table 3.II for each model we give
the critical flowrate expression when it is written in expli
cit form by the authors.
We will note that Table 3.1 doesn't give an exact vision of
each model. Infact much more time would have been necessary to
analyze each theory in order to minimize the number of adopted
equations, to verify the consistency of the assumptions and to
understand its power. Rather the table (together with the Ta
ble 3.II) may be helpful to characterize each model and to
show the great differences among their analytical formulations.
[
1 1
1
!
f
1 i
!
!
i
1 1
1 f
1 I
5 i
MAM or TOI V I R S T AtrntoR
PERFECT CAS ( C l a e a i e a l Theory)
n V K T CAS (AaaiaaptLon i ) )
PEItPECT CAS (Aaauaetlon U »
INCOMPRESSIBLE LIQUID
•OMOClrtOUS BJOILIMItM HODCL
1ARST a t a l . 1977
SABITSKTT M 73
HOODT 19*5
HDOPT 19W
P A U U 1964
LETT 1965
a t o m « t a l . 1967
OGUAHAJtA 1969
OCASAVAJU 19*9
ABACRI 1975
ADACHI 1974
ADACEII 197*
HOODT 1975
CASTICLIA a t a l . 1979
mmmt at ai. »?s
«ALUS a t a l . 1 9 »
RAHSOH a t a l . 1979
ÉMMËt 1947
EAUXJDU 1943
STANDtt* a t a l . 19M
MOOT 1 H 9
KURT 1971
P'AICT 1971
ARDKM a t a l . 1976
RANSOM « t a l . 197S
RtJHR a t a l . 1970
HURT ( L / 0 >12) 1970
MEURT <UB >12) 1970
KUMXUIEL * t a l . 1971
ELÏRCEI1IEL a t a l . 1971
HALRES 197S
H i m 1973
RITA» a t a l . 197S
KMEGU 1976
MURE « t a l . 1976
ATOUT a t a l . 1977
TENTNER a t a l . I97S
HOUIRCU 1971
WIHTEM at a l . 1979
KATTO a t a l . 1974
«SOT a t a l . 1968
MEUNIER a t a l . 19M
CIOT a t a t . 1972
FLIRTA at a l . 197}
BADER a t a l . 1976
t T A M O 1977
m ^ m . 1 , , ,
HDOSL FOWOWÏlW
Al
i 1 1 i i
z
1
2
I
2
3
2
3
2
3
2
4
3
3
3
3
3
2
3
2
4
A2
j 1 j ! o
1 o
a
2
2
2
2
4
3
2
4
4
2
2
3
2
2
3
3
3
3
4
3
3
2
/ / 2
2
2
4
2
6
3
3
3
3
2
4
3
» 4
6
4
3
S
2
! 3
4
4
3
f
1
2
5
S
3
5
4
3
4
3
2
2
2
4
6
5
3
5
* 5
3
A3
a
I i
ï 1 / / / / / i
/ / i
/ / / / / / / / / / 3
/ CI
Î
1
/ / 1
1
1
<« / 1
1
/ / 3
1
4
« ' I
2
HR
3
A4
j 1
/
,
«
S
r
t
i
t
i
i
i
i
i
TES
TES
/. 1
TES
7
TES
TES
TES
1
1
1
1
TES
ASsïnrTICm (H ina.i.:le«.eal
Bl
|
j S
RO
NO
NO
HO
RO
NO
NO
NO
RO
RO
m
NO
HO
RO
NO
NO
RO
NO
NO
ru
RO
TES
TES
TES
/ 1
T U
/ /
TES
TES
TES
TES
TES
TES
TES
TES
TES
TES
t
TES
TES
TES
TES
RO
NO
m
RO
NO
RO
NO
TES
RO
NO
NO
NO
NO
TES
NO
TES
TES
TES
NO
TES
TES
RO
BI
S
1 ] 3
i NO
NO
RO
RO
RO
NO
NO
NO
RO
RO
RO
RO
RO
NO
RO
NO
NO
NO
RO
RO
rts
RO
RO
NO
NO
NO
RO
RO
RO
RO
RO
RO
HO
NO
RO
RO
RO
TE!
RO
NO
NO
NO
TM
NO
R3
i i
U II
II
II
II
II
T U
M
NO
T E S »
T E S »
RO
HO
HO
TES
T E S »
r t s " '
T E S »
T E S »
T E . "
T E . » '
RO
T E S " '
RO
II
II
TES
RO
TES
HO
T E S » )
NO
RO
T E S » '
T E S » '
HO
NO
HO
T U < "
TES
TES
TES
„,m HO
TES
TES
S4
3
•s
i
1 s RO
TES
HO
II
RO
NO
RO
RO
RO
RO
RO
RO
TES
II
RO
RO
II
NO
RO
TES
RO
TES
BJ
|
!
NO
NO
TES
HO
NO
TES
NO
RO
TES
TES
HO
RO
TES
// TES
TES
// HO
NO
TES
HO
TES
II
II
RO
RO
RO
NO
NO
TES
NO
NO
RO
HO
RO
TES
TES
TES
TES
TES
HO
TES
TES
NO
// // NO
RO
RO
RO
HO
TES
RO
HO
RO
HO
RO
TES
TES
TES
TES
TES
TES
TES
TES
NO
B6
! !
lli u
m
m
IT
IT
IT
IT
11,1?
I l l
m
IT
I I I
I »
I
I I
I I I
I
IT
IV
I I I
w
IV
I
I
I I
I I I
I I I
I I I .
I l l
IV
IV
IV
IV
IV
I I I
IV
IT
I I I
onrm
a I
| 3
l a
ù TES
' NO
RO
HO
TES
RO
NO
TES
TES
NO
NO
RO
HO
TES
TES
TES
TE»
RO
TES
NO
TES
NE
C2 es
I
3 a i
\ 1! 1! 3Z
8 s V.
ji 1| 1! II TES
TES
TBS
T U
TES
TES
HO
TES
TES
TES
TES
TES
TES
TES
TES
TES
II
TES
TES
TES
TES
RR
TES
HO
M
NO
HO
RO
NO
TES
TES
TES
RO
HO
tes
T U
T U
T U
RO
TES
RO
HO
T U
TES
C4
"J?
III
III
III
III
m
RO
HO
RO
TES
HO
RO
RO
RO
III
III
T U
III
NO
HO
RO
RO
III
NO
RO
TES
NO
TES
RO
TES
RR
HO
TES
TES
RO
RO
TES
TES
NR
NR
HR
RR
NR
TES
RR
TE*
TES
RO
TES
TES
TES
TES
RR
TES
TES
TES
TES
TES
TES
TES
RR
RR
RR
RR
TES
TES
RR
/// IN
RO
TES
T U
T U
RO
T U
HO
RO
HO
TES
HO
T U
HO
HO
HO
HO
RO
NO
T U
RR
III
III
III
HO
/// RO
HO
/// HO
HO
HO
HO
HO
T U
T U
T U
T U
TES
HO
RO
TES
HO
It « s
w S I
ïîl III
III
III
III
m
T U
RO
T U
T U
T U
RO
HO
T U
III
III
T U
III
T U
T U
T U
T U
l i t
"
1 a m
mi
IKI
mi
Ml
ni
mi
IUI
nv mi
IM
nu i-ni
tm
mi
/»/ /•/ ni
ni
im
nv mi
mi
l/l
VI
III
T U
l i t
T U
T U
III
T U
T U
T U
T U
T U
T U
T U
T U
T U
ru
T U
T U
T U
T U
EQUILIBRIUM ADIABATIC LIQUID-VAPOR FLOW
SOLVED IT THE CHARACTERISTIC DETERMINANT /
/
ARALTSIS OF ROCLEATtOH PROCESS /
RON EQUILURIim QUAUTT /
1MEVEHIBUE TRRIttODYKAMCS AHALTSIS f
•«COOLED LtQOlO TfflWOOl ORIFICU /
m lui 1191
mi
liai
iml
mi
nu mi
nu mi
mi
mi
lui
/Ml
IV»
ira
mi
mi
noi
nu
1X1
nu 100/
îos;
nu
1 M /
î a i
10»/
110/
mu " t J " U
3
1 i
i a f!
h 3.3.2.1
3.3.2.2
3.3.2.2
3.3.2.3
S.3.3.1
3.3.3.2
3.3.3.3
3.3.3.3
3.3.3.*
3.3.3.7
3.3.3.1
3.3.3.9
3.3.3.U
3.3.3.11
3.3.3.13
3.3.3.14
3.3.3.15
3.3.3.11
3.3.3.17
3.3.3.11
3.3.3.19
3.3.3.M
3.3.4.2
3.3.4.3
3.3.4.*
3.3.4.3
3.3.4.6
Î.J.4.7
3.3.4.»
3.3-4.9
3.3.4.11
3.3.4.12
3.3.4.12
9.3.4.13
3.3.4.1*
3.3.4.13
3.3.4.11
3.3.4.17
3.3.4.11
3.3.4.19
3.3.4.2C
3.3.4.22
3.1.4.23
3.3.4.21
3.3.3.1
3.3.5.2
3.3.5.3
3.3.3.3
3.3.5.4
3.3.3.5
3.3.5.»
3.3.5.7
/ - «a intéad " •» oaa"
/ / - t a l a t a * * %aat«caaaaer'1
/ / / - Ra intend "net intaraat lag l a th la analTata"
RR - «a lataad "act reported l a avai lable bibliography"
(1) » AeeLicabUtt* l a d e t l M d In ear. 3 .3 .1
'2> I t l a eeceeearr to e>>e ea ineut to tha calculat ion* the eathalpy evaluated a t tea height ot the piae-veaeal eoaaeetloe
Tafelt 3.1 - Suaaurr et the analysa*1 sod* I t .
9 -
NAME OF THE FIRST AUTHOR AND PUBLICATION DATE
MAIN MODEL CHARACTERISTIC
MAXIMUM FLOWRATE EXPRESSION (HHEN ACHIEVABLE)
PERFECT GAS CLASSICAL THEORY
- 1 I * • [ " . ' . ( ^ ) M ] PERFECT GAS (ASSUMPTION i))
max(p) p(pQ - p)
[»ho-h>] PERFECT GAS (ASSUMPTION ii))
m a x ( p ) p | 2 ( h Q - h )
INCOMPRESSIBLE LIQUID 2 P (P0 - P e)
IEMXENEOUS EQUJ_ LIBRIUM MODEL HEM T « max(p) ._
S o ( p o ) " S f ( p )
Vf •'
so{po)-sf(p)
— ^ 5 5 a w
LAHEY et al. 1977
FLUID-DYNAMIC APPROACH
-1
/ d v f v f g d \ [dp s f g dp
dv. v - d s . " f s . _l£ _ i i dp s f g dp^
)
J
f (p e )
BABITSKIY 1975
EQUILIBRIUM SCHEME
NOT EXPLICIT
MOODY 1965
SLIP EQUILIBRIUM MODEL (ENERGY MODEL)
r « max(p.k) <
nfg h - h , - (s - s , )
O f Sfg *" O f
k(s -s ) s - s , g o o f — s v,+ v s f g f s f f c fi
12 s - s r s -s„ o f + g o s f t k s f&
MOODY 1966
FIRST ANALYSIS OF THE KHOLE DEPRESSURIZATION PHENOMENON
NOT EXPLICIT
FAUSKE 1964
SLIP EQUILIBRIUM MODEL ^OMENTUM MODEL)
- k
a», 2 2 r U-x*k x)x f** [vg(l»2xk -2x) + vf(2xk -2k -2xk *k )] ^î
f(P„): x - f(p#); k - f(pe)
ï
LEVY 1965
LUMPED MODEL NOT EXPLICIT
CRUVER et a1.. 1967
DEFINITION AND ANALYSIS: Area average spe cific volume, momentum average specific volume, kinetic energy ave rage specific volume, velocity weighted spe cific volume
]!
U["<Ht*l3l-T''°MSl-HS)J j. _ A. w _ r. A. i 1/3 ds - O; k (VV
Table 3.II ./• continue
- 10 -
. / . Table 3 . II coi
OGASAKARA 1969
OGASAKARA 1969
ADACHI 1973
ADACHI 1974
ADACHI 1974
MOODY 1975 CASTIGLIA e t a l . 1979
TENTNER e t a l . 1978
KALLIS e t a l . 1978
RANSCM e t a l . 1978
BURNELL 1947
ZALOUDEK 1963
STARKMAN e t a l . 1964
MOODY 1969
HENRY e t a l . 1971
D'ARCY 1971
i i t inued
EIGENVALUE METHOD APPROACH TO CRITICAL TWO PHASE FLOW
AS ABOVE
TWO INDEPENDENT ENERGY EQUATIONS METHODS
AS ABOVE (FLOW FROM CYLINDRICAL DUCT)
AS ABOVE (FLOW FROM ORIFICES)
CONSISTENT SLIP MODEL
MAXIMUM ENTROPIC FLOW
METHOD OF CHARACTERISTICS TO SOLVE THE PROBLEM
ISENTROPIC STREAM TUBE MODEL
CHARACTERISTICS METHOD TO SOLVE THE PROBLEM (Four e q u a t i o n s )
SD1IB«»IRICAL CORRELATION
AS ABOVE
FROZEN COMPOSITION
PRESSURE PULSE MODEL
RELATED TO LOW TERMA NENCE TIME OF THE MIXTURE IN THE EXIT DUCT
PERTURBATION METHOD
NOT' EXPLICIT
AS ABOVE
NOT EXPLICIT
AS ABOVE
AS ABOVE
NOT EXPLICIT
AS MOODY 1 9 6 5 WITH CONDITIONS —- - 0 , I 5 - - 0 «fa a p
r •
r
r
r
r2
NOT EXPLICIT
• m a x ( p ) U yi . Y" ] i " l p . w. p , w i^n î .n fn n _
- 1
NOT EXPLICIT
- V2 - < 2 p f [ P u p - ( l - C ) p s a t J >
- C l { £ 2 > f ( P « p - P s a t > ï } / 2
i [ -, _ _y_( i * v • ( l - x ) v " ° *° o * ' 1
o g v o go L 1 0
fa ' ( l -o) ' 1
* «feal,kgtol ; x = x ° ;
VgUp ]s v f I 3 p j s
NOT EXPLICIT
NOT EXPLICIT
Table 3.II . / . continue
1* J
- 11 -
./. Table 3.II continued
ARDRON et al. 1976
RANSOM et al. 1978
HENRY et al. 1970
HENRY 1970
HENRY 1970
•
KLINGELBIEL et al. 1971
KLINGELBIEL et al. 1971
MALNES 197S
PORTER 197S
RIVARD et al. 1975
KROEGER 1976
BOURE et al. 1976
AVDEEV et al. 1977
TENTNER et al. 1978
MOESINGER 1978
WINTERS et al. 1979
UPPER BOUND FLOW
METHOD OF CHARACTERI STICS TO SOLVE THE PROBLEM (six equations)
LOW QUALITIES (x^ <0.02) MODEL
HIGH L/D RATIOS (L/D =12)
VERY HIGH L/D RATIOS (L/D >12)
ENTRAINED SEPARATED FLOW (ESF)
SEMI EMPIRICAL APPROACH
RELEASE OF DISSOLVED GASES
ANALYSIS PERFORMED OVER ALL MOLLIER DIAGRAM (from sub-cooled liquid to superheated vapor) TWO FIELD TWO PHASE MODEL
DRIFT FLUX APPROXIMATION
COMPLETE ONE DIMENSIO NAL TWO PHASE FLOW ANALYSIS ORIGINAL METHOD IN DETERMINING VAPOR FORMATION RATE CHARACTERISTICS ME THOD TO SOLVE THE PROBLEM DRIFT FLUX APPROXIMA TION ASSESSMENT LUMPED NON-EQUILI BRIUM MODEL; BUBBLE GROUT! 1 ANALYSIS
,»P l2fPcï 1 l x J P f
c |pgo goj IpJ X • K^ff^ff NOT EXPLICIT
r - FcHE
C ^
r 2 « -c
r 2 -c
lV Vfoj dp
x v ( 1
-f- - lv*>l '
- i
c
« ^ 1 ' dp
-1
c
NOT EXPLICIT
I" - LIKE MOODY 1965 KITH "k" OBTAINED FROM EXPERIMENT
NOT EXPLICIT
NOT EXPLICIT
NOT EXPLICIT
NOT EXPLICIT
NOT EXPLICIT
NOT EXPLICIT
NOT EXPLICIT
NOT EXPLICIT
NOT EXPLICIT
Table 3.II - Analysis of models with reference to: a) maximum flow rate analytical expression (when explicitly given);b) main model characteristic. All the models are referenced in preceding table.
- 13 -
3.3 SCHEMATIC DESCRIPTION OF EXAMINED MODELS
3.3.1 - Generality
In formulating any theory, a certain physical pattern, which
more or less reflects reality, is referred to.
In the present case , we take the situation shown in Fig.
3.1 as reference. In order to obtain flowrate it needs to con
sider at least:
a) initial thermodynamical conditions in the vessel: enthalpy
(H ) , mass (M ) and pressure (p );
b) geometrical data: length (L) and diameter (D) of the broken
pipe; position of the connection between broken pipe and pre£
sure vessel, with respect to initial liquid level (dimension
'b' in Fig. 1) .
Moreover, in the most general case, the theory should consi
der the following aspects:
- possible difference in pressure and temperature between liquid
and vapor phases while flowing in the duct, and consequently,
thermodynamical non-equilibrium;
- friction and heat exchange between each phase and the exte
rior;
- initial acceleration of the mixture as a function of breaking
time, initial thermodynamical conditions and rupture size;
- link between critical flowrate and pressure history in the vessel;
QO In this work we only examine models studying the flow of a two phase mix ture from a pressure vessel without internals. (The complex thermonuclear effects and heat exchange phenomena in a real vessel are not consi dered").
- 14 -
- mass, momentum and energy exchange between the two phases;
- multidimensional effects, expecially near geometrical discon
tinuities.
In the following we don't emphasize if the model matches or
not such aspects.
The flowrate and the exit thermodynamical variables have to
be the output of the calculation: they shall be expressed as a
function of one or more of the parameters given in the above
points a) and b) .
The description of the models will outline four aspects:
- model purpose;
- basic assumptions;
- essential equations;
- model applicability (we call a theory 'applicable' if it con
siders, at least, all the variables given in the above points
a) and b) as input, and if it shows, as result,a correlation
of the specific maximum mass flowrate (T) as a function of i-
nitial enthalpy (h ), pressure (p ) and/or temperature (T )
of the fluid in the vessel).
- 15 -
3.3.2 - Perfect fluid
3.3.2.1 - Perfect gas - Classical theory /66/
The purpose of this theory is the calculation of the maximum
flowrate of a gas flowing from a reservoir of infinite capacity.
The assumptions are:
- stationary and isentropic flow;
- thermally and calorically perfect gas.
The equations are as follows:
a) Mass conservation
G = pwA = const. (1)
b) Energy conservation
1 2 •£ w + c T - c T 2 p p o c) Transformation equation
d4 = (V
(2)
(3)
d) State equation
pV = RT (4)
By combining e q s . (1) through (4) we o b t a i n :
1
A =
p o Y
Gv (—) O P
Hi _ I 2c T (1 - - £ - ) Y
P o p o
(5 )
It can easily be seen that this expression has a minimum
when :
(6)
- 16 -
For this value of P , r results
/
= L Y Po Po Y+l |
r • h P„ p. ( T T T ) (7)
Moreover for A = A we have:
w* = (2 c T ) (^44) C8) p o Y + 1
It can be seen independently , that w is exactly the
sound propagation velocity (and, consequently, a small pertur
bation propagation velocity) in a perfect gas for the thermod^
namical condition existing in section A = A (Appendix 3.1).
Note that r depends only upon constant reservoir conditions.
In order to achieve such a thermodynamical transformation
the duct must have a convergent shape in the flow direction.
In Fig.3.2 the nondimensional mass flow T/T vs p/p is
represented; Fig. 3.3 shows T vs p .
3.3.2.2 - Perfect gas - Flow from cylindrical ducts
We shall describe this model in order to clarify some impor
tant differences from the preceding theory.
In the assumptions of perfect gas stationary flow, in a cy
lindrical duct, the balance equations may be written:
- mass conservation:
pw = const. (1)
- momentum conservation:
pw + p = pQ (2)
- energy conservation:
h + \ w2 = h (3) i o
- 17 -
Note that:
i) equation (2) is valid also with heat exchange with the ext£
rior, but in absence of friction;
ii) equation (3) is valid also with friction, but in adiabatic
conditions.
By combining equations (1) and (2)(after adding assumption
i)) we have:
1
r • [ P ( P 0 - P ) ] 5 (4)
By combining eqs (1) and (3) (after adding assumption ii))
we have:
r = P |^2(ho-h)j 5
Taking into account the perfect gas state equations
(5 )
= Rh p c P
(6)
and
h = const • c ' exp ( s / c ) p R / c v (7)
two diagrams in h/s plane may be drawn. The first, related to
formula (4), is the Rayleigh line and it is given qualitatively
in Fig. 3.4. The second, related to formula (5) is the Fanno line
and it is given in Fig. 3.5. It may be observed that in both
diagrams the entropy has a maximum; it is easily shown'65'
that in such maximum the gas velocity is the sound velocity
(M = 1).
The curve in Fig. 3.4 may be passed through in the direction
from B to C through A and from C to B through A since an entro
py decrease may be obtained working adequately on the quantity
of heat exchanged;as a consequence a gas subsonic at the entran
- 18 -
ce into a cylindrical duct (in assumption i)) may become super
sonic.
The case of Fig. 3.5 is different: in absence of shock wa
ves , the path can only be from B to A and from C to A. This
implies that a gas subsonic at the entrance in assumption ii)
cannot possibly become supersonic in a cylindrical duct.
3.3.2.3 - Incompressible liquid - Classical theory'65'
With the assumptions of: a) stationary flow and b) isentro-
pic flow (ds = 0), we can write the following equations:
- continuity equation:
p w A = const. (1)
- energy equation:
2 " h - h + "r- W
o 2
- state equation:
dh = Tds + vdp (3)
which, taking into account assumption b) becomes
p
From the above equations
dh = **• ( 4 ) P
r - [_2P ( P 0 - P e)J * (5)
i s e a s i l y o b t a i n e d .
Unlike the p e r f e c t gas t h e o r y , in t h i s c a s e , r depends only
upon p and i t i n c r e a s e s i n d e f i n i t e l y as p d e c r e a s e s . This
(x) These shocks can only transform a supersonic flow into a subsonic one and not viceversa.
- 19 -
fact is shown qualitatively in Fig. 3.6 where the trend of a
perfect gas specific flowrate is reported as comparison.
It is interesting to note that an incompressible liquid
doesn't possess internal energy; the motion is possible only
by an external energy source.
- 20 -
3.3.3 - Theories assuming thermodynamical equilibrium between
liquid and vapor phases
The basic assumption of these models is the existence of thermo_
dynamical equilibrium between liquid and vapor phases in each
section of the flow duct. By thermodynamical equilibrium we in
tend that:
- pressure and temperature of liquid and vapor phases are equal;
- pressure and temperature are linked by the Mollier diagram sa
turation curve.
The mixture quality change is allowed along the duct: strictly
speaking this faculty is in contrast with the thermodynamical
hypothesis, because the condensation and evaporation phenomena
depend upon differences of temperature and/or pressure through
the phases. Implicitly, quality change is supposed to take place
at infinite velocity.
In the preceding section we have distinguished the equili
brium models in two groups:
a) homogeneous theories: equal vapor and liquid velocity;
b) non homogeneous theories: different velocities for the two
phases.
/9/ 3.3.3.1 - Homogeneous equilibrium theory
This theory is known as HEM (Homogeneous Equilibrium Model).
It is the most simple equilibrium model (except for the semiemp^
rical formulas) that may be formulated in the analysis of two
phase flow.
It has been developed initially for the analysis of situations
in which a fluid contained in a pressure vessel outflows through
21
a pipe the diameter of which is much smaller than the vessel dia
meter.
The basic assumptions are:
- homogeneity (i.e.: the mixture is considered as one component
fluid with thermodynamical properties that are
the average between the two phases,and liquid and
vapor velocity are equal);
- thermodynamical equilibrium;
- isentropic and stationary flow.
The balance equations are:
- continuity equation:
p w A = const. m CD
- energy equation:
1 2 u u -=- p w + h = h 2 m o
(2)
state equations:
h =hf(p) + x h£g(p)
-- = v = v -(p) + x v (p) p m m i fp m
fg
(3)
(4)
s = s (p) + x s (p) f fg
transformation equation:
ds = 0
By combining the above equations we obtain: SoCpo)_Sf(p)
2 h (P ) - h (p) - , * — h„ (p) o^cr gViV s. (p) fg^'
r = IE!
vf + s0(P0)-sf(P)
sfg(p) V P )
(5)
(6)
(7)
In this expression, after giving the reservoir conditions,
- 22 -
the specific flow rate depends only upon the pressure' 'local*
value.
The critical flow rate is obtained in a manner similar to the
perfect gas theory.
In Figs. 3.7 and 3.8 the functions r vs h and p /p vs h 6 o e o o
are reported.
Moreover, having fixed the initial values:
p = 1000 psia
h = 552 BTU/lbm o
in Fig. 3.9 we plot equation (7);
In the same Fig. 3.9 the trend of x vs p is also reported;
in Fig. 3.10 r vs x again with the same reservoir data is shown.
It should be noted that in the above assumptions (particular
ly from eq. (1)), the duct area 'A' must decrease and then in
crease if the specific flow rate (T) has a trend similar to the
one shown in Fig. 3.9. This fact is also valid for other theo
ries described in the following.
3.3.3.2 - 'Fluid dynamic' approach in the formulation of a homo-/fi 7 /
geneous thermodynamical equilibrium model
The ternT'fluid dynamic"has been introduced by Lahey-Moody in
opposition to the 'thermodynamic* approach given in the prece
ding paragraph.
This theory may be obtained, with some simplifying assump
tions, by more complex models describing the two phase mixture e-
volutions through six independent equations (the three balance
equations written separately for vapor and liquid). In order to
obtain results from this model, the knowledge of at least one
- 23 -
thermodynamic variable at the exit section is necessary.
With the assumption of steady state let us now consider the
mass and momentum balance equations:
Tz (FA) - 0
A dz r2A p»
d£ dz
T P, W r
(1)
(2)
where p' is the momentum density, generally given by:
P' pf(l-a) p a (3)
If a critical condition exists, it follows that a maximum in
critical flow rate exists too and it is independent from the ab
, that is :
dr dz
= 0 (4)
Taking the latter equation into account, the left hand si
de of eq. (2) may be written as follows:
, f 1 1 dA
Ap' dz P'
dp dp dz = 0 (5)
Moreover from eqs. (2) and (5) it results 2
_ d A
dp = Up1
T„P1 ' T* dA Twrj ^p' dz A J
(6)
With the further assumption that the pressure gradient, at the
critical section, becomes infinite or indeterminate, we obtain:
- 24 -
If we limit ourselves to the analysis of the homogeneous mix
ture isentropic espansion, the following equations must be used:
- state equations:
1 P' = V_ + X V_
f fg
s = sf + x s
- transformation equation
ds = 0
fg
(8)
(9)
(10)
By combining eqs (7) through (10) the following expression
for the maximum flowrate results :
r
r* = dVj. Vr r t d s
dp s £ g dp
-1
+ X
"av. fg
dp fg fg >
s r dp fg J J
I (11)
From eq. (11) it may be observed that a value of the specific
flowrate corresponds to each thermodynamical situation in the
exit section; it results, that in a certain range the flowrate
doesn't depend upon the reservoir conditions.
By changing eqs. (8) - (10) appropriately, other models can
be obtained without the homogeneity assumption as well.
The abscissa where the choking condition establishes itself
may be obtained by vanishing the numerator of eq. (6). The trend of
r vs p for an assigned value of s is shown in Fig. 3.11; note
that s only is not sufficient to individuate the reservoir con-o '
ditions.
*J • O • -J • O BABITSKIY 19 73 /68/
The peculiarity of this model is that the two-phase mixture
- 25 -
homogeneous equilibrium expansion is theoretically treated in
two different manners:
1) Integration of continuity, energy and momentum equations ;
2) Resolution of continuity, energy and isentropic relations
(i.e. the momentum equation is substituted by the isentropic
one [_ds = Oj ).
The author points out the differences between these two ap
proaches .
Balance equations :
- mass continuity:
p w A = const. (1) m
- momentum equation:
1 2 p + — p w = const. (2)
- energy equation: 2
x h + (1-x) hc + -T- = const. (3) g f 2
This is a system of three equations in the three unknowns x,
p, w, when all the reservoir quantities are given.
Instead of eq. (2) the author uses:
ds = 0 (4)
In the two cases (non isentropic flow and isentropic flow)
the following espressions for (1-x) are obtained:
h -h - — (p -p) o g p *o
(l-x)= ^ • (5)
h„-h + (p -p j(p - p) £ g pogpof 0g °£ ° s - s
(1-x) =—2 L. (6) s,.-- s f g
- 26 -
The results are reported in Figs. 3.12 and 3.13. Particularly
in Fig. 3.12 the ratios of different quantities calculated by
the momentum equation with respect to the same quantities found
by the isentropic assumption are shown. In Fig. 3.13 the entro
py difference is shown for the two cases; in the same figure the
mixture velocity calculated from (1), (2) and (3) and from (1),
(2), (4) is also reported. Another curve shows the experimental
trend for w. The abscissa is the temperature (°K) .
The author's conclusions are:
- the fluid expansion calculated from (1), (2), (3) occurs with
entropy increase;
- the mixture velocity calculated according to eq. (2) is smal
ler than the one calculated from eq. (4);
- the velocity reduction in the non isentropic situation is com
pensated by additional thermal energy losses for formation of
a larger mass of steam than at s=const.;
- when the entire thermal energy released is spent only for va
porization (kinetic energy = 0) the entropy mixture variation
will be maximal (i.e. maximum vaporization = maximum entropy
production).
The opposite also is valid.
3.3.3.4 - Non homogeneous equilibrium theories - Introduction
With respect to the preceding models, in these theories at
least a new variable appears: the void fraction defined as:
a = T ^ vT- CD 1 + k (_L2L)_f_
X V„
- 27 -
Consequently, a further equation at least is necessary for
the problem solution.
The models described in the following have usually two common (*) assumptions :
- thermodynamic equilibrium;
- adiabatic flow.
On the contrary no common assumption is given with regard to
the slip ratio (k); indeed the calculation of this quantity di
stinguishes one model from the other.
/13/ 3.3.3.5 - MOODY 1965'
This is one of the theories (together with the Fauske and
Levy ones described in the following) that had the greatest dif
fusion. It has been considered by the US NRC as the best approach
in predicting the flowrate from long channels (L/D» 1). Moreover
it is also used in some large diffusion codes, *as RELAP 4 Code.
In addition to the assumptions given in the above paragraph,
Moody assumes frictionless and stationary flow. The balance equa
tions result:
Mass conservation:
X V 1-X V-
Energy conservation 2 .2
w w hQ = x(h + - i ) + (1-x) (h + -f) (2) g 2 J v ' v f 2
(*) These assumptions wi l l not be repeated each time.
- 28 -
The state equations and the trasformation equations are iden
tical, respectively, to those given in section 3.3.3.1.
From eqs (1) and (2) of this paragraph and from eqs.(3)
through (6) of par. 3.3.3.1 it results:
T=<f
kfg h -hu. (s -s_) O f Sfg V O fJ
k(s -s ) s -s,, g o o f s y + y
sfg f sfg g
s -s,, o f sfg
s -s g o
• " 2 — ks £ g
l
(3)
When the reservoir conditions are assigned the flowrate depends
y upon local pressure (p ) and slip
ximum flowrate is obtained by imposing:
only upon local pressure (p ) and slip (k). The mathematical ma-
9r 3k
3r 3p J
= 0 p
( * )
= 0 k
(4)
(5)
Particularly from eq (4) it results:
k~ = r v £.
X/3 (6)
By evaluating reservoir enthalpy (h ) at the pipe-vessel con
nection height, the gravitational head "b" of Fig. 1 may be ta
ken into account.
The author observes that both liquid and vapor velocities are
slower than sonic velocity in the critical section; it follows
that the ambient pressure change has a certain influence (even
if minimal) on the critical flowrate.
(x) This condition is not present in the perfect gas theory.
- 29 -
In Fig.3.14r vs h is given, the parameter is p ; in Fig.3.15
p vs h is given, the parameter is p . Finally in Fig.3.16 r vs
p , adopting x as parameter, is reported.
3.3.3.6 - MOODY 1966/69/
Unlike the preceding model, the global study of blowdown is
the aim of this theory: it may be summarized in three points:
1) to apply existing two phase flow models in order to predict
maximum steam water pipe flowrate in terms of upstream stagna
tion properties and pipe flowrate in terms of upstream proper
ties and pipe resistance;
2) to obtain theoretical blowdown transients (dp /dt) with va
rious pipe diameter to length ratios and flow areas;
3) to compare maximum flowrates and estimated blowdown transients
with experimental data.
We shall analyze point l)only.
With reference to Fig.3.17 one obtains:
- mass balance equation:
r = const. (1)
- momentum balance equation:
dQ = Adp + T P dz (2) c w w
- energy balance equation: w w
hQ = x(hg + -£-) + (1-x) (h£ + -j-) (3)
In equation (2), fi is defined as:
Q = X W + (1-X) W-_ g £
TA (4)
- 30 -
The basic assumptions are".
a) thermodynamic equilibrium;
b) isenthalpic flow;
c) minimum kinetic energy.
In particular from the last assumption (as Zivi
lows :
/ll/
k = ' v I3* °°
v , !
) it fol-
(5)
The wall shear stress "T " is found from the constitutive re-w lationship:
w F — Vf
1-x 1-a
(6)
where "f "is the local Fanning friction factor, related to the b
liquid Reynold number, which in turn is
Re = D r •
1-1
L
1 - X
-a J
(7)
From equa t ions ( 1 ) , (2) and (3) i t may be w r i t t e n :
f F cp, ho, n - D dz (8 )
where:
F(p,ho,r)=-
3f,
3x
e q r ~ î
-1 3P
+r 2 ^ 1 3P_L 3fi
3fr P 3
-"x -"x 2^4
3x +r - I 3xJp
8P -*x
-1
g 1+ k v f
12 (9)
and f i s an average Darcy f r i c t i o n f a c t o r eva lua ted for the ave
rage l i q u i d Reynold number ( e q . ( 7 ) ) . The func t ions f , f , f4
a re def ined in Appendix 3 . 2 .
(x) Obviously this is the same result for k obtained in the preceding se£ tion with a different assumption.
- 31 -
If the mixture expands isentropically between the reservoir
condition and section '1' (see Fig.3.17) , it results:
fP2 fT
\ eq (10)
s = s (11)
o l
Equations (9) through (11) were programmed for machine calcu
lations;some results are shown in Fig.3.18 and 3.19, related to
the two extreme situations: fL/D =0 and fL/D =100 respective-eq eq
iy-
Figures 3.20 a), 3.20 b), 3.20 c) show pressure transients p =p (t)
obtained with such model, for different values of parameter
fL/D, when initial flow is water,steam-water mixture and steam
respectively.
3.3.3.7 - FAUSKE 1964/30/
The basic equations are the same as the ones referred in con
nection with the homogeneous theory of paragraph 3.3.2.2. The
main differences are:
- the friction is not taken into consideration (also in the ab£
ve theory, however, friction doesn't affect the critical flow
rate calculation);
- the mixture non homogeneity is considered through the quantity
slip that is assumed to be different from one.
The main differences from the Moody classical model (Par.
3.3.2.4) are the use of momentum conservation equation in the
- 32 -
place of energy conservation equation and the assumption of dif
ferent criticality conditions.
The balance equations are:
- mass balance
r = const CD
- momentum balance
.2 dv + 4 E - = o dz dz
where
v = P*
(2)
(3)
and p' is defined by equation (3) of paragraph 3.3.3.2,
From equation (2) one obtains:
T = -dv dp
(4)
By substituting the expression of a (eq.(l) of section 3.3.3.4)
in equation (3) of section 3.3.3.2, it results:
v = f (p,k) (5)
A firstly imposed criticality condition is:
9v
from which
is obtained.
Finally we
„*2
9k
k* -
have:
r v ï g
r~ -- k
(6)
(7)
{ (l-x+k*x)x -r-£+ [vg(l+2xk*-2x) + vf(2xk*-2k*-2xk* +k* )J -j*}
(8)
- 33 -
(in which the term dvf/dp has been neglected) and then
* ., dx , r = f(x , — j — , p )
^ c dp rc (9)
dx The values of x and of —7—
c dp "I are made functions of p through an assumption (second assumption) with respect to the
mixture thermodynamic transformation:
lo = hf ( pc ) + x hfg ( pc) = c o n s t
from which
x =
h -h^ o f hf S
(10)
(11)
and
dx dp
xfC o dp
u
r dh. <*V hf g dp" " hf dp (12)
By substituting these last two equations in eq. 8 we obtain
r*=f(Pc).
The same author suggests the values to be used for the ratio
p /p as a function of L/D (see Fig.3.21) . In Fig.3.22 r vs h0 is
reported, pQ being the parameter.
It may be observed that the same expression for r given in
(8) can be obtained once again starting from an equation similar
to eq. (9) and following a different procedure
3.3.3.8 - LEVY 1965 /12/
This is a "lumped model": each phase is represented by a sin
gle mean velocity.
Besides assuming thermodynamical equilibrium, the basic as
sumptions are:
34 -
- no frictional or head losses;
- static pressure drop is the same for the two phases;
- the expansion is supposed to be isentropic.
Balance equations:
- mass conservation:
A. w- A w = D = —*- o —*- =
A pf 1-x A Mg x const (1)
momentum conservation equations:
- liquid phase
dp*p f w £ dw f - - ^ LTP
dz
vapor phase
1 2 wf dp+ — d(Ag pg wg) + j±- d(A£ p£ w£) •dz
(2 )
dz (3) CTP
From equa t ions (2) and ( 3 ) , one o b t a i n s :
r = -dp
dv m
- 1
-1 s dv m dpj s
where 2 (^ ^
x (1-x) V = V + V - - ^ -m g a f 1 - o
By s u b t r a c t i n g (2) from (3) i t r e s u l t s :
(4)
(5)
a(l-2cQ +a/(l-2a)2+ar(2pf/pg)(l-a)2-Hq(l-2al
X ~ 2 pf/pg (l-a)2 + d(l-2a) (6)
This is like imposing a constraint to the slip. For the calcu
lation of critical flowrate eq. (4) is directly employed, diffe
rentiating vm with respect to p. By taking into account the as
- 35 -
sumption:
i t r e s u l t s
ds = ds 3PJ X
+ 3s 3x
dx = 0 (7)
dvm
dp
3v, m 3p
9vm dsg dsf
3xJ |_ dp v dp_
In this expression the derivatives 3v m and
/Sfg
3v m_
3x
(8)
are obtained 3Pjx WA Jp
by differentiating eq. (5) e (6) respectively. After calculating
r it is possible to obtain the reservoir enthalpy from the eq.
(2) of section 3.3.3.5.
It should be noted that if we don't want the flow to be isen-
tropic, for instance an isenthalpic process, all the above equa
tions apply except for the replacement of entropy "s" with entalphy
"h" in equations (7) and (8).
Once neglected the frictional and head losses of the present
model, no further assumptions about slip are needed.
The functional link between "x" and "a" is obtained from a
momentum balance equation; it doesn't depend upon other adjoined
conditions that are necessary in the preceding theories (a pecu
liarity of the Levy model) .
In Fig.3.23 the maximum flowrate is given versus the exit qua
lity, the local static pressure is the parameter. In this figure
the dotted lines represent critical flowrate when assuming isen
thalpic process, the unbroken lines represent isentropic proces
ses. In Fig.5.24 the pressure is reported versus the critical
flowrate. With regard to these two figures, one may observe:
- the very little difference between isentropic and isenthalpic
processes ;
- 36 -
- that the solution exhibits a maximum in terms of flow quality (this
maximum in the pressure range reported stands in the range
0.01 <x < 0. 2) .
The author also gives a well known formula in order to calculate the
reservoir enthalpy when the exit quantities are found.
3.3.3.9 - tRUVER-MOULTON 1967
As the authors say, this is a unified theory of one dimensio
nal, isentropic, equilibrium, separated two phase flow.
In the initial part of the work, four conservation equations
are reported: they are related to mass, momentum, "mechanical"
energy and energy respectively. Only two of these equations are
used in the critical flowrate evaluation.
In the assumptions of constant flow area and steady state, n£
glecting gravity effects, the mass and mechanical energy conser
vation equations become:
- mass continuity
r = const. (1)
- mechanical energy conservation
r2 - - 2vH 3P
^ K E ^ (2)
-J s
The quantities v u and v„„ will be defined later. H Kb
Moreover the transformation equation is
ds = 0 (3)
at critical conditions. By solving equation (2), taking into ac
count equations (3) and (1), it results:
- 37 -
r= < -v.
2
H 3 v ^ - i [-HtriSl/iv^ia— S
(4)
From assumption (3) it still easily follows:
dx dp
1 Sfg
dsf dsfg' +x (5)
dp dp
The slip ratio value still has to be established in order to
obtain r.
Therefore the authors have written the following expressions
respectively for:
- velocity weighted specific volume:
1 f v = H TA I «
/A dA = (1-x) v,. + x v
g (6)
area average specific volume:
VA = X f P A JA
-1 (l-x)v£k + xVg
(1-x) k + x
- momentum average specific volume
VM = Jl f 2
pw dA = r A -A
XVg
TT t f l-x ) vf l+x(k-l)
kinetic energy average specific volume:
2 VkE =
1 f 3 AK - = - | p w dA I A } J IL
I <
- \ -£* + (1-x) vf l+x(k -1)
_ )
11 >
( 7 )
( 8 )
(9)
when k=l it follows that v. =v__ - vw = v. _. From the above espres A H M kE —
sions it may be clearly seen that the further degree of freedom
(k) appears when considering non-homogeneous flow instead of ho
mogeneous flow.
- 38 -
In order to find an appropriate value for k the authors prcj
ceed as follows: by differentiating the mechanical energy conse_r
vation equation (where eq. (3) has not yet been employed) with
respect to k, they obtain:
?AS r2 3(A vkE)2
cas i Kt _ ç 1 0 ) 3k 2T 3k avg
2 (x) It is easily shown that the function 3(v,p) /3k is negati
ve in the region l<k<(v /v-J™, it equals zero for k=(vg/vf)"^5 g 1/ & T
and it is greater than zero for k>(Vg/vf)'3 . it follows that the
entropy of the system increases with slip ratio, reaches a maxi
mum at k=(Vg/vf)/3 and decreases for k>(vg/v£) . Therefore as
sumption (3) implies:
f v
k = g Vf
%
/13/ /ll/
like Moody and Zivi
Therefore equations (4), (5), and (11) solve the problem.
In Fig. 3.25 r vs h is shown; another interesting result is
shown in Fig.3.26, where v., v , v , v are plotted as a function A H M kt
of k, when quality and pressure are assigned. It is shown that v., doesn't depend from k (obvious), v. continuously decreases M r Aj . with k, v has a minimum for k=(vg/v£)
2 and v, has a minimum 1/5 for k=(vg/v£)
2 2 (x) The authors show that in the above assumptions 3/3k(Av,p) = 3/3k(v )
- 3*1
5.3.5.10 - OGASAWARA 1969 772/
Another theoretical approach is used by Ogasawara in order to
determine critical flowrate when thermodynamical equilibrium be;t
ween phases is present.
In this model the criticality condition is obtained by em
ploying separate momentum equilibrium in order to adequately des_
cribe the momentum between the two phases.
With the assumption of axial steady flow where radial varia
tion of the thermodynamical quantities is neglected, the below
equations result:
- mass conservation:
_d_ dz
pa w + Pf (1 - a) wf =0 (1)
- momentum conservation in vapor phase:
dz p aw + ap g g
] = -= -F. -F fg g
(2)
- momentum conservation in liquid phase
à_ dz
p£(l - a) wf + (1 -a)p = F -F fg f
(3)
energy conservation:
d_ dz pgawg(hg + i) + pf(l-a) W T M = "R C4
The author points out that the momentum equations are written
separately so that the force F acting between the phases may
not be cancelled as internal force, as in other models.
Indeed the same separation is not necessary for energy equation
because of the assumption of saturated flow.
- 40 -
Rearranging the set (1) t- (4) appropriately, it may be written
as :
A(E)
dw.
dz
dp_ dz
dk dz
=
R
-F + F fg £
F . - F fg g
(5)
Since the right side member includes only finite terms and d w £ dp dk
the left side terms f—;—, -r> T~) are assumed as divergent in a dz dz dz
critical situation, it follows that the determinant A(E) must be
zero, that is the criticality condition is
A(E) = 0 (6)
This leads to a cubic equation in w , the roots of which indi
viduate the critical velocities. Since there is more than one
root, the author takes the root that better approximates the ex
perimental results as the real root. The slip ratio "k" is itéra
tively calculated when p and x are given. Therefore these two ' c c
quantities are necessary as input to the calculations. By inter
polating the theoretical results the author also gives a mathema_ % — — V tical expression of k , vs the ratio I p_(p )/p (p ) I 2,
•— r c g c —'
The results are given in Fig. 3.27 in which r vs x is repor-,
ted, p is a parameter; in Fig.3.28 r vs p , adopting the parame
ter x , is shown. In the same work Ogasawara presents another
theory obtained by substituting the energy equation (1) with an
entropy conservation equation. He concludes that critical condi
tions with energy conservation give a more accurate solution
than the ones with entropy conservation equation. In the latter case (en
- 41 -
tropy conservation) only few values of T are presented in a ta
ble. The values of r so obtained are slightly inferior to the
ones calculated when total energy is conserved (obviously p and
x are the same in the two cases). c
3.3.3.11 OGASAWARA 1969 774/
This other model formulated by Ogasawara is valid for flow
through orifices in which the flowing fluid friction and heat ex
change may obviously be neglected; moreover the fluid velocity
upstream the orifice is assumed as zero. The other explicit
assumption consists in neglected kinetic energy in comparison
with enthalpy at the exit. The equations are as follows:
- mass equation:
dz _ xpf+k(l-x)p J (1)
momentum equation:
-^ I r(l-x+kx) wf + p = F (2)
- energy equation
dz
v 2 2 2 — K W W
x C - y ^ + h ^ + C l - x H y + h£) = R (3 )
By integrating these equations one obtains f h. h. -h.
, fgo. fo fc x = x (r-*-) + r r— c oh, h -h
fgc go gc p = p - k p p_(l-x+kx)w_ /M *c o c gc fc c c c fc
(4)
(5)
where
42 -
M = xpr + k(l - x) p £ g
These are two equations containing four unkowns (i.e. p , x ,
Two further equations are then necessary to achieve solution.
One is obtained by the "eigenvalue method" (see also the prece
ding paragraph) setting the determinant of equations (1), (2),
(3) as equal to zero (following the same process of the prece
ding paragraph). The other comes from the FAUSKE's result, i.e. V2
k = (v /v ) g (6)
Substituting determinantal equation (determinantal equation is
the equation (6) of the preceding paragraph) and equation (6) in equa
tions (4) and (5) the problem is resolved. Unlike the preceding
model, in this theory Ogasawara gives r vs p (see Fig. 3.29);
with x as parameter. In Fig. 3.30 p /p vs x is shown; p is the o c " *c *o o o
parameter. The following result as:
- the smaller the fluid compressibility the larger the pressure
ratio becomes (Fig. 3.30); /123 13/
- since it has been experimentally confirmed ' that p /p
becomes smaller in the case of small reservoir quality, owing
to thermodynamical metastability, this model isn't worth using
at small qualities;
- the results differ from Moody's for about 201.
In the same report the author makes a very interesting analy_
sis (comparing theoretical and experimental results) regarding
two phase flow through orifices;in particular , he observes a drastic r£
duction in specific flowrate when orifice diameter increases
(all other variables being constant).
43 -
/HT,/ 3.3.3.12 MALNKS 1977
This theory is taken into consideration only to point out the
quantitative and qualitative differences arising in the two pha
se mixture maximum velocity calculation when the following two
assumptions are separately applied (the basic equations are the
same in the two cases):
a) homogeneous frozen flow (k=l, ip=0) ;
b) homogeneous equilibrium flow (k-1, sufficient to mantain
thermal equilibrium or equal temperatures of the two phases).
The steady state flow equations are:
- vapor-mass conservation:
T - (ap„ w ) - ty = 0 9z g g (1)
liquid-mass conservation:
9 9z
(1-a) pf wf i|> = 0 (2)
mixture momentum balance:
3p + 3
9z 9z 2 2
a p w + (1-a) p. w_ g g f f
= 0 (3)
In assumption a) we obtain:
w HF a
< 9p ^
9p 1-a
9p£
~3p~ ) s
In assumption b) we obtain:
a 2 1 w = - < HE p p
f 9 p * l 9p
t " J
f l - a '
T P f
' 3 P f
9p +
T
a
LV
f*«l 9T
I J
+ l - a ' 3 p f '
9T P.
f ï 9T 9p
I )
( ^2 3T
3P apg CPg+(1"°° pf CPf
(4)
(5)
- 44 -
The large differences between these two extremes may be seen
in Fig. 3.31.
The above equations are part of a more general non equilibrium
theory presented by the same author and to be considered later
in this work.
IM 3.3.3.13 - ADACHI 1973
The formulation of this theory is strongly different from th£
se shown so far.
The basic assumptions are the usual ones (apart from the im
plicit assumptions we have referred to in Par. 3.3.1). We shall
repeat them only for convenience:
a) one dimensional two-phase flow (separated flow);
b) steady flow;
c) adiabatic flow;
d) state change follows the saturation curve.
The author takes into consideration two energy balance equa
tions: the one related to the "flowing fluid", the other related
to the "existing fluid"; in the case of non homogeneous mixture
he shows that the two espressions are independent.
Let us consider the fluid flowing in a duct between sec
tions (1) (upstream) and (2) (downstream). The vapor and liquid
(if slip exists) which have simultaneously passed through control
surface (1) cannot reach section (2) at the same time. Therefore
there must be mass and energy exchange between the two phases;
as long as the "existing fluid" is considered the flow between
sections (1) and (2) is not adiabatic.
Viceversa, for the above assumption c) the "flowing fluid" e-
nergy equation is of adiabatic type.
45 -
Equations adopted:
mass conservation:
TA = T A = A c c
ap w + (1-a) p. w. g g £ £
= const. (1)
- energy of the flowing fluid conservation:
2 2 2
x w + (1-x) w.
L g i
+ d xh + (1-x) h^ g J f
= 0 (2)
- energy of the existing fluid conservation:
2 2 6w + (1-8) w£
+ (1-Ç ) JL+idL dp = 0 (3) w p p- x v 7
g £
where:
a p 0 = g
ap + (1-a) p g f x + (1-x) k
(4)
From the state equations (section 3.3.3.1) and from the assump
tions :
ds = 0
dh = 0
we obtain respectively:
d x ] s - •
and
dx
where Ç is defined by:
x ds + (1-x) ds,. g £
fg (1-ÇT)
x dh + (1-x) dh^ g . L r h. 4 fg
dp = (1-Ç_) dp « dp s ^T v Krev
(5)
(6)
(7)
(8)
(9)
- 46
and dp, = Krr, dp = dp. *h T r i r r
By s e t t i n g dx=dxj +dx_| and so lv ing in the unkn
(10)
awn x , i t
r e s u l t s :
x = x -o
p _ _ p [_x ds +(l-x)ds f J (1-&J.) r
g
J, ' fg J,
x dh + (l-x) dh^ g f
fg
( * )
(11)
The following procedure i s shown in the fol lowing flow diagram:
input po'VÇT'Çw
rp\ppi
| x from (11) 1
k for the new state ( ini t ia l guess)
w„ and W£ from (2)
| kj from (3) and (4)| (iainteraction index, in this case)
-GE£Q NO
r from (1) and a from (1) of s e c . 3 .3 .3 .4
output p »x
e»wge»wf e , a e» ' ' c ^ e
YES •<£^C>" NO TËNDI
S«-S o->f . (x) Note that i f ^=0 i t follows x= i . e . isentropic flow Sfg
47 -
Note :
- the iterative calculation of k and its dependence from £ and r w
- the absence of criticality assumptions in the calculation of
the exit thermodynamic quantities.
In Fig. 3.32 the variables r, w , w_ are shown as function of g f
fluid instantaneous pressure. The left side of the curve (with
respect to their maximum) can be achieved only in the case of
expansion occurring in a constant area duct (this means that "supe£
critical" flow cannot be observed in a constant area duct).
In Fig. 3.33 we report r vs p: x is the parameter for the a£ 2
signed initial value p = 60 kg/cm .
In Fig. 3.34, finally, k vs p is shown (the initial value is 2
still p = 60 kg/cm ).
It is still to be noted that no idea is given about the calcu
lation of two quantities K and Ç (we remind that they repre-W X
sent the losses).
On the basis of this formulation the authors present two other
theories applicable to cylindrical ducts and orifices respectively.
3.3.3.14 - ADACHI 1974 (cylindrical duct) /6/
The equations and the solution method are those reported in
the preceding paragraph. The author admits that in the case of
a cylindrical duct the friction losses have the major weight in
the total loss evaluation, whence it results:
T w
Moreover this hypothesis requires specific flowrate to be
constant.
- 48 -
The integral pressure drop and the integral frictional pressu
re drop respectively can be defined as follows:
fP C - -I dp = (p -p) (1)
i O
'Pp çF - - j
P dpF - (po-p)ç (2) *Po
The solution of the equations of the preceding paragraph is
shown in the following figures;
- Fig. 3.35 represents ç„ vs ç with r as parameter;
- Fig. 3.36 represents r vs p with parameter E, •
In both diagrams the pressure and the vapour quality of the
reservoir are supposed to be constant.
We observe that:
-. the maximum in Fig. 3.35 represents the critical flow: in fact
assuming one is not in a maximum point (point 2 of Fig..3.37)
at an exit pressure equal to p~, lowering this value the flow
rate increases and the representative point shifts to a curve
with higher flowrate. The transformation line on this diagram
will have a positive slope as it has to increase with ç . r
The right-hand points of the maximum values are not real ones
since the frictional pressure drop would decrease along the
flow;
-. the straight line 9 of Fig. 3.35 represents a limit case (r=0)
since the losses are only due to friction and not to fluid a£
celeration;
-. the dotted line in Fig. 3.36 corresponds to the dotted line
in Fig. 3.35: it represents the relationship between the cri
tical pressure and the critical weight velocity for given re
servoir quality and pressure;
49 -
-. the change of the flow in the constant area channel (r=const.)
yields a straight line parallel to the horizontal axis (in
Fig. 3.36, i.e., the line a-c-d represents the flow behaviour).
The interaction of lines c-d with the lines having constant Ç
means that Ç varies along the channel axis;
-. ç must be deducted separately if we want to obtain the criti
cal flowrate; from value of ç we may obtain both ç=p -p and
r from Fig. 3.35.
In the same work ADACHI also presents other curves showing the
relationship between the channel pressure and other parameters
(ct,w ,w ,k,x).
A semiempirical relationship is finally drawn between the pi
pe current adscissa and the pressure. The comparison with the ex
perimental results is also performed.
in 3.3.3.15 - ADACHI 1974 (orifice)
In this work information about the values to be given to the
efflux coefficient C area provided when the efflux from orifi-IM ces is studied and when the theory presented in (Par.3.3.2.13)
may be applied.
The author observes that:
- the coefficient is considered so as to include the effects of contrac
tion, friction, compressibility, and thermal non equilibrium of
the two phase discharge flow through the break. Fig. 3.38 (C
vs x ) and Fig. 3.39 (C vs p ) result from an interpolation of
experimental values.
However, when the orifice is attached to the pressure vessel,
the functional relationships can be written as (for a given
break geometry):
50
S = £(D>VV (1)
It may be observed that:
-. the relationship between the discharge coefficient C_ and the
high pressure reservoir tank quality x can be separated into
two regions:
a) region of x j*0 where x has little effect and o o
bl region of x =0 where x has a significant effect;
-. in the region of x 9*0, C is almost independent from p and
x and it can be determined from the orifice diameter. Cn is o D
larger for smaller diameter values; for 25<D<70 mm C is in
the range of approximately 0.8-0.52. The reason may be due to
the fact that the critical flowrate decreases because of the
contraction immediately downstream from the orifice and, in
addition, the extreme supercooling (50°C) phenomenon of the
vapor phase which occurs at the orifice; -. in the region x =0, C„ sharply increases as x decreases and & o D o
it reaches 1.5 from the 25 mm orifice. However for the same
value of x ,C becomes larger for smaller D. For higher p ,
C slightly decreases.
The reason for C exceeding 1 may be as follows: because of
the lack of nuclei, vaporization cannot follow the fast depressu
rization at the orifice and thus the discharge flow behaves more
like single phase fluid. The theoretical maximum value of C for
x =0 is given in Fig. 3.39 which is obtained from o
1 2Pf CP0-Pe) CD - °-« f(po,0) <2'
f(p ,0) being equation (1).
- 51 -
3.3.3.16 MOODY 1975 79/
The author recognizes that Fauske,Levy and his own preceding
models are based on either momentum or energy conservation but not
upon both and therefore they are not consistent. However he says
that the assumptions of the above models do not necessarily des
cribe the physical behaviour. On the contrary this study uses all
the conservation laws.
Besides the thermodynamic equation assumption, the author im
poses:
- monodimensional flow;
- any dissipative loss equal to zero;
- constant pressure in any section.
The equations are:
- continuity equation
d(TA) + d
- momentum equation
AV(- a v g
vf J CD
r2 Av m
+ d vrA + Adp = 0 (2)
- energy equation
fA h + d f A£ Ag
V — * W + — - h o L [ Vf £ Va 8-U g
- AVdp = 0 (3)
Second thermodynamic law:
rAs + d [ A £ Ag
V —i- s,-+ —s- s _lv f f vg 8J
50 (4)
in which
m x v + (1-x) kv,.
g t x +
1-x (5)
- 52 -
The critical conditions are expressed as follows:
V = 0 (6)
dV dp
dr dp
= 0 (7)
= 0 (8)
(i.e. perturbation velocity and its derivative with respect to
pressure are supposed as equal to zero) .
By substituting equations (6) and (7) in the set (1), (2),
(3), (4), we obtain a new system of three equations in the four
unknowns A, x, r, k if p is considered an independent variable. /7 ?/ 128/
The author says Ogasawara and Giot showed that mo
mentum or energy conservation for either phase is sufficient for
one more independent equation; however an approach which invol
ves entropy production is equivalent and it is employed in this
study.
The following state equations for either phase are used:
- Gibbs equation
T ds = dh - v dp (9)
- Clapeyron equation
T sfg ' hfg (10>
- state equation
h = hf + x h f p (11)
(x) For the author these conditions are representative of the physical fact consisting in wave blockage at the critical section.
- 53 -
Equation (9), (10) and (11) together with equations (1), (2)
and (3) produce a new relationship among dependent variables,
namely:
ds (p,x) = — mr + m„ _ fg gf
w - w _ _g L
2
; i
dp = £(p,x,k,r) (12)
where the terms m and mpf represent the vaporization and con
densation flowrates the expression of which is given in App. 3.3
Equations (1), (2), (3) with conditions (6), (7) and (8)
constitute a closed system. In order to obtain a solution it is
further necessary to find consistent values for k.
The author assumes
k * 1 (13)
and that k is maximum consistently with the condition ds=0.
The results are shown in Figs 3.40 and 3.41.
One may observe that:
- these results imply the thermodynamic quantities at the exit
knowledge ;
- the flowrate variation is too great with respect to k which
hasn't been fixed clearly and moreover it is difficult to ob
tain from blowdown experiments.
3.3.3.17 CASTIGLIA-OLIVERI-VELLA 1979 775/
This model is based upon the assumptions of a onedimensional
stationary motion and thermodynamic equilibrium.
The expansion is supposed to take place in a duct with fric
tion and adiabatic wall.
The balance equations are written as following:
- 54
- mass continuity
Ar = A
- momentum continuity
aw wf
_g_ + (i_a) — g f
= cons t (1)
dp _ r2 d dz dz
2 n ,2 V + — V-
a g (1-a) f + F (2)
- energy continuity
h =h +x h + — o g fg 2
3 n ^ 3
x ..(l-x) —;r v +- — v_ 2 g f1 .2 £
a (1-a) (3)
In order to evaluate ?, the assumption of maximum entropie
flowrate is used; according to such assumption , when reservoir
enthalpy and flowrate are assigned, the effect of irreversible
phenomena is such that entropie flowrate is maximum in each point
of the expansion and this maximum is obtained with respect to
void fraction, which is assumed to be the only restrained quant^
ty.
Analytically we have:
3s 3a
= 0 (4)
Taking the classical expression for "a" into account we ob
tain:
r k =
v ïV3
_g_ (5)
which is nothing more than the result obtained with different
assumptions by Moody and Zivi
It is to be observed that in this case equation (5) is valid
throughout the transformation, whereas in Moody's cases it was
- 55
obtained as a criticality condition and therefore valid only at
the critical section.
By substituting (5) in (3) and solving for x we obtain:
vf
x(p)=-
I , 2 Z.nys , 2 3.f]l/3
2/5 2/5 vg " vf
(6)
where q and q are functions of static pressure and of flow
rate (see App. 3.4).
By substituting equation (6) in the entropy espression we ob
tain mixture entropy through all the transformation as a func
tion of "p" only.
In Fig. 3.42 we report, in a p/s plane, the transformation
trends related to different r values. The line below point c) is
not admissible, since the fluid system is isolated (the entropy
cannot decrease); it follows that the other criticality condition
results: *
d S = 0 (7) dp
Moreover, in order to obtain the critical point only it is a.L
so necessary that:
d 2 s < 0 (8) dp2
From eq. (3) and taking into account conditions (4) and (7),
the diagrams of figures 3.43 and 3.44 are obtained.
It can be observed that in the region of very low qualities s£
me curves (having h as parameter) show a maximum: this means
that when reservoir enthalpy and flowrate are fixed, two points
of validity of equation (7) are present. The authors say that on
ly the point with greater quality satisfies both conditions (7)
- 56 -
and (8). Therefore in Fig. 3.43 only the region on the right of
the line connecting the maximum points is the real region. /13/
The analogy of this model with Moody's is evident, regar
ding numerical results too; in particular it refers to the pro-/9/
cedure later adopted by Moody , which consists in assuming
equation (5) as valid throughout the transformation.
At the conclusion of their work, the authors observe that
this model too allows to connect inlet thermodynamic condi
tions and flowrate to the pipe physical and geometrical charac
teristics: it is sufficient to integrate equation (2).
3.3.3.18 - TENTNER-WEISMANN 1978/10/
This model distinguishes itself from those presented above
especially for the solution method adopted, that is the charac
teristic method. Therefore the variable time appears in its e-
quations;it follows that the critical flowrate should be deter
mined, as a function of time, until it reaches the critical va
lue.
The prediction of critical flow is based upon the magnitude
of the characteristic slopes. Critical flow is predicted when
the smallest characteristic slope becomes equal to zero. This
corresponds to the physical fact that a pressure pulse cannot
propagate upstream.
Balance equations:
- mass conservation:
^[apg+(l-oOp£ >^-[apgkw£+(l-a)p£wf J =0 (1)
- momentum conservation:
- 57 -
3 r~ , ,-. •> ~î 3 r i 2 2 ., . 2-7 8p Tw . . . -Lap kwf+(l-a)pfw,J+-Lapkwf + Q - a)pfw J + - f = - - (2) 9 t L _ - g — £ ' f f - l 3z L.-"g" "f
- energy conservation:
L«P_B • (l-a)PfBf J + ^ L « P w£kE + (1-oOp f J -£-£ (3)
9 r ïïLVg
where i 2 2
K W f E = h + — -g g 2
Wf Ef " h£ + T
These are three equations in the three unknowns (a, p, w ) ,
since the other quantity (k) is found separately.
The authors note that a term accounting for form losses may
be added to the right side of equation (2) and that this term
will not change the nature of the characteristic directions be
cause it doesn't contain derivatives of w and a. For the same
reason terms T and q too don't influence the characteristic w nw slopes.
Besides the assumption of thermodynamical equilibrium and
the physically justifiable hypothesis:
3p£
~3P" = 0 (4)
the authors assume that"the slip ratio variation is low e-
nough to allow terms containing partial derivatives of k with
respect to p, w and a to be neglected"; i.e.
58 -
As we have already said, in order to solve the system it is
necessary to give the functional form of the slip ratio as input.
For void fraction below %0.7 the following Hughmark correla
t i o n i s used:
l f p f ï ( l - Z ">
T-Hifjhr-J (6)
where Z is a non linear function of an assigned parameter w
(see App. 3.5).
For void fraction greater than 0.7, values of k are obtain
ed using the classical expression:
k = H l-o ^ pf
l 1-x (7)
g ft ?
At mass velocities above ^2«10 lb /hr ft homogeneous flow m
is assumed (k=l).
Moreover, in their calculation, the authors use the Moody ex-
k=(vg/vff3
pression of the slip too .With regard to this last
assumption they conclude that Moody slip values are too high on
two counts :
- first,they are above experimental results;
- second,they cause a mathematically ill-posed problem.
In Fig. 3.45 r vs x is reported in two cases: e
a) i n i t i a l model-
b) k from Moody's c o r r e l a t i o n .
In order to so lve the problem complete ly terms T and q (*) w w
in equa t ions (2) and (3) need to be eva lua ted . (x) For this reason we have written in Table 3.1 that three constitutive re
lationships are necessary to solve the model: 1) related to k (given); (2) related to T (not given); 3) related to q (not given).
- 59 -
In the second part of the same report a simple approach to
non equilibrium problems is presented: it will be described in
the following paragraphs.
/76/ 3.3.3.19 - WALLIS-RICHTER 1978
The authors conclude that this model doesn't represent the de_
tails of choking realistically (or better, more realistically
than other models), but it may be considered as a certain limit;
its purpose may rather be to help in providing standards for com
parison with other theories and with actual phenomena and it may
be considered as a starting point for the development of more
elaborate theories.
The basis of this model can be understood from Fig. 3.46.
Saturated water is assumed at the entrance of a tube; a cer
tain amount of pressure decrease will cause the first evapora
tion (first stream tube); a further Ap will cause a further eva
poration (second stream tube); this second Ap also causes expan
sion of the vapor in the first stream tube; and so on.
In Fig.3.46bis the process in a Mollier diagram is shown.
After changing phase the espansion in each stream tube conti
nues independently from the other stream tubes.
If the water is not saturated at the inlet, the authors sug
gest that the conditions up to the saturation curve may be ob
tained from Bernoulli's formula giving for the initial velocity
to be considered as input: 1 2
(1) w = o
p -p ^ - o sat
The basic assumptions are:
- 60 -
- isentropic flow and liquid in equilibrium with the vapor in
each stream tube;
- pressure uniform in each flow section;
- the velocity components perpendicular to the main flow direc
tion negligible.
The equations for each streamtube are:
- mass balance
Y- , = Y. + y. l-l l i
energy balance
f w. ">
l-l| fi-1 2 = Y.
)
entropy conservation
2 i w J v + 41
fi 2 I
2 <, w .
h +-Si gi 2
(2)
(3)
Y. .. s_. n = Y. s_. + y. s . l-l fi-1 l fi 7i gi (4)
Besides the well known symbols we have
i-1 liquid flowrate from which the stream tube "i" follows
(see eq (2)) ;
Y. normalized liquid mass flowrate in stream tube "i"; l
y. fraction of total mass flowrate in i-th stream tube;
i,n number of Ap steps.
The authors also show that by considering equation (2), (3)
and (4) and thermodynamic identity:
hjr = T sr (5)
one obtains Bernoulli's equation applied to the liquid phase.
From equations (2), (3), (4) it follows
r = X—-±— 1=1 p. W.
i,n i,n
n pfn Wn
-1 (6)
- 61 -
3.3.3.20 - RANSOM-TRAPP 1978 778/
This model is only a part of the recently developed code
RELAP 5 /M0D"0M.
In order to obtain chocking, the flow is described by the ove
rail mass continuity equation, by two momentum equations and by
a mixture energy equation as follows:
- mixture mass equation
-k |_a Y ( 1"a ) pf|+ ~t La pg V e 1-0 0 pf wfJ= ° (1)
- vapor momentum equation
aw aw
g I 9 z S 9 x a p + (l-a)g+caCl-o)p
3w 3w 3w. 3w_ _g+w —S L w _ J at fak az g ax
= o (2 )
- l i q u i d momentum equa t ion
(1-cOP, aw aw
+w
at fax +(l-a)|?+co(l-a)p
3wf 3wf 3w 3w +W - * - W i r — *
at sax at fax = o (3 )
- mixture energy equa t ion
a at
a pg s g + ( 1 " a ) pf s f Hi* pg sg wg + ( 1" a ) pf sf wf = 0 (4)
This system may easily be written in terms of four dependent
variables a,p,w ,w . It appears as
A(U) au at
+ B(U) au 3x
+ C(U) = 0 (5)
where A(U), B(U), C(U) are square matrix (4*4 in this case) and
[u~| is the vector of the dependent variables.
By defining the characteristic directions of the system as
the roots A. of the characteristic polinomial: l
AX - B | = 0 (6)
62 -
where
Pi,n= 1-x.
pfn
(7)
gn
and w. i,n
1 7 2|h.-h. + wT _ { i i,nj i_
(8)
In the above expressions M. is the generical thermodynami-
cal quantity risen in the i-th Ap step and evaluated in the n-th
Ap step.
Expression (6) for r (relating to a unit cross sectional area)
is the reciprocal of the sum of the areas of all stream tubes per
unit normalized flow. It results function only of local pressure.
The criticality condition is expressed as:
dr dp
= 0 (9)
The values of r vs Ap (for given p and T ) are shown in r *o o
Fig. 3.47; in Fig. 3.48 r vs x (with p as a parameter) is pre
sented.
With regard to«this model it may be further observed that:
- it bypasses the slip evaluation difficulties found in other mo
dels ;
- it admits differences in velocity along the flow radius, so
that, within certain limits, it may be considered a bidimensio
nal theory;
- it doesn't consider momentum transfer between the streamtubes;
- by adopting very small Ap steps a continuous expansion may be
analytically simulated, even if the decrease of step size may
cause certain instabilities in calculation depending (according
to the authors) upon accuracy of steam tables.
- 63 -
It has been shown that the mathematical condition for cho
ked flow is
A. £ 0 for all j * n (7)
where n is the nunber of equations. This corresponds to the
physical fact that reduction in downstream pressure ceases to r£
suit in increased flowrate.
With reference to this model the following observations can
be made:
- the momentum equation includes the force terms due to relative
motion between the phases;
- the product ca (l-a)p is the coefficient of virtual mass: "c"
has to be fixed each time, although it has a theoretical value
of 0.5 for dispersed flow and for separated flow the value may
approach to zero;
- the energy equation is written in terms of entropy which is
constant for adiabatic equilibrium flow;
- the term C(U) is not given by the authors in equations (1),
(2), (3), (4) since it is assumed that it doesn't affect the
characteristic equation (6) because it doesn't contain deriva
tives of the dependent variables;
- the term C(U) represents the constitutive relations (not given
here): these relations include interphase momentum and mass
transfer, wall heat transfer and friction.
In this model thermodynamic equilibrium is assumed, however
the authors have presented also a similar model valid when
frozen flow is assumed: it is shown in following sections.
The eigenvalues resulting from equation (6) and the mathema
tical discussion of equation (6) are given in App. 3.6.
Fig. 3.49 shows the trend of equilibrium Mach number versus
- 64 -
vapor fraction. The choked conditions are defined as the inter
sections of the lines having c= const, with Mach number unit li*-
ne.
In the bibliography available no other interesting results for
this work are given.
- 65 -
3.3.4 - Non equilibrium models
Non equilibrium models include equilibrium models as a parti
cular case. Non equilibrium effects derive from the physical
fact that two-phase depressurization velocity may be greater
than thermal exchange velocity between liquid and vapor. When
the mixture at the inlet of the broken pipe is subcooled or satura
ted water, non equilibrium consists of a delay in vaporization
and of superheating when there is vapor at the inlet. /26,124,etc/
According to different experimental works the time during which the mixture remains in a metastable state is about
/9/ one millisecond. Starting from such a value, Moody calculated that non equilibrium degree for pipe length greater than 12
centimeters is negligible. /31/
According to other researchers , instead, the non equili
brium value depends on the L/D ratio: i.e. non equilibrium pheno
mena must be considered for L/D£3*-12.
However also in long channels it probably plays an important
role on thermodynamical conditions at the exit section.
Besides, all researchers seem to agree that non thermal equi
librium can appreciably increase both critical flowrate and pro
pagation velocity of rarefaction wave normally generated at the
ruptured section.
Many theories treating non equilibrium have been developed:
some of them do not help more than the empirical formulas to
quickly calculate maximum flow of initially subcooled water; o-
thers make use of very sophisticated models.
Most of these theories adopt empirically determined coeffi
cients: the latter are the number of vaporization nuclei per
- 66 -
unit mass of flowing mixture, a number relating non equilibrium
quantities to equilibrium ones, a number that is needed to take
into account a certain assumed flow contraction, experimentally
observed in single phase flow from orifices.
The feature which is common to all these numbers is that they
are obtained by matching experimental data to theoretical results.
As already said (par. 3.2), we have distinguished these
theories in two groups:
- frozen theories (which admit no variation in mixture composi
tion while flowing);
- non homogeneous non equilibrium theories (in which no assump
tion is given referring to the link between pressure and tem-(*) perature of the two phases) .
With regard to non equilibrium models, the greatest part has
been developed in the last years; in this report we want only to
show the qualitative and quantitative differences existing bet
ween mixture velocities in a homogeneous equilibrium theory and
in I
a frozen theory (See for example Fig.3.50)
3 . 3 . 4 . 1 - "Frozen" t h e o r i e s
The assumptions g e n e r a l l y common to these models a r e :
- the v e l o c i t y r a t i o between the two phases i s g iven;
- no hea t or mass t r a n s f e r t a k e s p l ace between the p h a s e s : thus
the q u a l i t y remains c o n s t a n t throughout the expansion ( t h i s
assumption c h a r a c t e r i z e s "frozen f low") ;
- the vapor expands accord ing to an ass igned law: for example
(X) We actually intend a theory of non equilibrium type when at least one thermodynamical quantity doesn't follow saturation l ine .
- 67 -
according to gasdynamic principles.
All vapor and liquid transformations are then assumed as
independent from one another, as far as the exit section at
least.
3.3.4.2 - BURNELL 1947 78/
It is fairly well noted that maximum flowrate consistently
increases when decreasing L/D ratio and an even greater flowrate takes pla_
ce when subcooled conditions are present just upstream the break,
as in the case of orifices.
The experimental results in this case significantly exceed
the prediction of most of the models presented so far.
In the fourtie s many works have been performed with regard
to this situation and are still valid (see for example ' ' ').
Burnell's work is one of the studies more frequently mentio
ned in bibliography.
He relates the non equilibrium in two-phase flow to the sur
face tension of the liquid droplets that delays the vapor bubble
formation.
In his model the critical flowrate is proportional to the
square root of (p -p 1: in particular he obtains: n VIup *sat r
r * = < 2 P f L P u p - C i - c ) P s a t ] > / 2 CD
The coefficient C is (Fig. 3.51) a function of saturation
pressure. As it may easily be seen from equation (1) the coeffi_
cient "C" decreases in a fictitious manner the value of the satu
ration pressure itself.
Physically speaking this means a certain liquid superheating
- 68 -
is to be considered.
From the above formula a diagram showing r vs p is quickly
obtained when p is known. up
3.3.4.3 - ZALOUDEK 1963
A similar approach has been employed by ZALOUDEK who analy
zed an initially subcooled two-phase flow from short pipes. He
observed two possibilities of choking phenomena:
1) upstream choking with respect to exit section which can be fo_r
med at a so called "vena contracta";
2) downstream choking forming after the exit section, due to prej;
sure built up in turn and to water flashing.
In the first case he gives £he following correlation in order
to calculate V. '
r * - c i { L 2 p f c p U p - p s a t r j > / 2 a )
where the values of empyrical coefficient "C" may vary from about
0.6 to 0.64 .
It is to be noted that physically this coefficient (unlike Bur
nell's coefficient) is equal to a flow area reduction.
The last two theories are assumed to be "frozen" because they
do not consider any change in mixture composition during the
flow. Moreover the formulas expressing maximum flowrate by Bur-
nell and Zaloudek are very similar to those obtained from the pei:
feet gas theory flowing through a cylindrical duct when the as
sumption i) of paragraph 3.3.2.2 is adopted; for convenience we
shall remind this assumption:
- absence of dissipative losses, but admittance for heat exchan
ge with the exterior.
- 69 -
/19,64/ 3.3.4.4 - STARKMAN-SCHROCK-NEUSEN-MANEELY 1964
This study too, is based upon an experimental work; all the
results are related to a convergent-divergent exit pipe.
The authors' assumptions are (apart from those valid for frc)
zen flow given in section 3.3.4.1):
- adiabatic expansion of vapor (Y= 1.3);
- no-slip;
- all the kinetic energy in the steam evolves from the vapor ex
pansion;
- flow conditions are controlled by a throat defined according
to gasdynamic principles.
The employed balance equations are:
- mass continuity:
r - - . (i) m
- vapor energy conservation: Y - l
h = h +x v p -Z- ( 1 - r ' ) (2)
o e o g o o y - 1
where "r" is the pressure ratio (p/p ) and v is defined as:
v = x v* + (1 -x )v. (3) m o g ^ o fo
From another form of equation (2) and from the perfect gas
theory (par. 3.3.2.1) it follows:
- V 2 w = L2(VVJ (4)
_x__
* o Y-l
ro
70 -
Moreover from the adiabatic vapor expansion assumption it fol
lows:
v* = v r* Y (6) g go
x and r is given by
* 1 Y *
r* è 2x v p — ^ ( 1 - r 0 (7) X V + fl -X )V 6 1
o g o go *- ->
In Fig. 3.52 the maximum flowrate versus reservoir critical
quality is shown: the reservoir pressure is the parameter.
The strongly marked maximum corresponding to low reservoir-
throat quality can easily be noted; they cannot obviously be ob_
served experimentally.
Moreover the authors say that the assumption of frozen flow
is not as restrictive as one may think, because the time needed -4
to travel through the nozzle is about 10 second. /64/
In the report later presented the authors make an interesting experimental data analysis from which it particularly results that:
- the pressure ratios (p/p ) in the diverging part of the noz
zle greatly decrease when subcooling increases while the flow
rate is constant;
- in the case of initial low quality the pressure ratio increa
ses as the stagnation pressure is reached.
- 71 -
3.3.4.5 MOODY 1969 /79/
The transmission of a pressure pulse in a two phase mixture
is the basis for this two-phase critical flowrate frozen theo
ry.
In this work the formulas are developed for evaluating both
critical flow and sonic velocities in the cases of separated and
homogeneous phases.
Here we shall only describe the critical flowrate frozen thec
ry and illustrate the main results of homogeneous assumption.
The author considers a pressure pulse travelling counter-flow
at velocity "v". The balance equations are written for a moving
control volume which is so thin that neither mass, nor momentum,
nor energy storage occur; neglecting body and potential forces
we have :
- mass conservation:
dr + Vd (—) = 0 (1)
- momentum conservation:
d(r v j + Vd r = -dp M
(2)
- energy conservation:
r(h
2 2 r y
KK + V d(— +r -y)-dp - H
= 0 (3)
In these equations v , v , v«F and v.. are defined in par.
3.3.3.9 and h is given by:
h = xh + (1 -x)kh, g J (4)
- 72
The critical flow in the case of separated phases is obtained
through the subsequent assumptions:
V=0 ; dx=0 ; ds =ds. g f
(5)
from the first of which we have c*) .
M
By solving equations (1), (2) and (3) it follows:
.2 'LlF pg+(T^Ep pfJ
(6)
r = vg tap J v£
(7)
L 3p js
with:,, a=a(k,x) ;, :y • =v -<p) ; vf=y£(p) .
Mpreover x is given through assumption (5), instead k is
assumed to be equal unity or given from the relationship obtai-/l3 9/ ned in the author's reports ' previously described:
k = JL v,
(8)
Taking the above into account, the critical flowrate results
to be a function of both x and p.
In figures 3.53 and 3.54 T is reported versus x ,p being the
parameter. The two figures are related respectively to k from
equation (8) and to k=l, note the very little difference between
these two results.
As already said, in this report Moody also follows a similar
procedure in order to obtain the critical flowrate with homoge
neity assumption. Fig. 3.55 shows the flowrate r in a diagram
(x) One can easily note that this assumption is the same as neglecting ener gy equation.
- 7;
with the same variables as in Figs. 3.53 and 3.54. It may be in-
teresting to note that the results for k=(v /v ) in Fig. 3.54
(separated flow), are quite different from those related to hom£
geneity assumption particularly in low quality region.
3.3.4.6 - HENRY-FAUSKE 1971 720/
The purpose of this model is to solve the problem of two-pha
se critical flow requiring only the knowledge of the stagnation
conditions and, at the same time, accounting for the non-equili
brium nature of the flow.
The duct considered is a DeLaval nozzle.
The conservation equations are:
- mass continuity
G v + G_ v. = A_ w +A w g g f f f . f g g
(1)
momentum conservation
- A dp = d (G w + G _ w J + d F g g f £ w
(2)
In a converging nozzle the wall shear forces are assumed to
be negligible by comparison with the forces due to momentum va
riation and pressure gradient.
With the assumption that at the throat
dr = 0 (3) dp
it results
T2-c d
dp X k + (1-X) f
I (1-x) k v£ + x vg
(4)
* - 74 -
The authors make further assumptions as following and justify
them through experimental arguments (the whole formulation is r£
lated expecially to low quality region):
(5)
— - --^JT"
X =
c
T ' f t
w = g
s gc
fc
X 0
fo
w f =
= s go
= S fo
w
n n (x) p v = p v 7
*o go *c gc
= 0 -1 c
= N i_ - -J c
~ d x E ~ dp
(6)
(7)
(8)
(9)
(10)
(ID
(12)
where x_ is the equilibrium quality given by:
XE = o fE
'fgE
and N is an empirically determined function of x Ec
Moreover from wel l known thermodynamic r e l a t i o n s h i p s we ha-
(X) From this condition (polytropic expansion of vapor) i t follows: dv -
g dp
v —
np
where n is given by the expression:
(1-x) cf/cpg * 1
(1-x) Cf/c' + 1/Y n =
- 75 -
ve ds g
dp M.
n Y (13)
and from the above assumptions the critical flowrate can be wri t
ten as :
x v r=<-^-£+(v - v - )
np g fo
(l-xo)N d s f E xQ c p g ( l / n - l / Y )
S g E - S £E d P ps £gE > (14)
c
This is an equation containing two unknowns (r and p ): in
order to obtain the solution, this equation is coupled with the
momentum equation (2), integrated between the stagnation and
throat conditions:
2 x Y
(1-x )v. (p -p )+-^r v o fo ' o *c Y-l
p v -p v o go c gc
r(i-x^)v. +x v ~i t-—° fo0 °
gd r2 (is)
Equations (14) and (15) solve the problem.
Some results of this theory are presented in Fig. 3.56 and
3.57 where the trends of r and of p /p vs x respectively are 1 c *o o v
shown.
It shall be noted that:
- if N is taken equal unity the solution approximates the
homogeneous equilibrium model;
- if N equals zero the solution approximates the homogeneous
frozen model;
- the function N is:
xT N = <
(TÎT £or ° « xo * °'14
1 for x £ 0.14 o
- 76 -
- when analyzing results from orifices or short tubes the authors
suggest to use value 0.84 for the actual flowrate reduction
coefficient;
- the model assumes neither completely frozen, nor complete equi
librium heat and mass transfer processes.
3.3.4.7 - D'ARCY 1971 780/
The starting point of this theory is the pressure wave propa
gation velocity in a two-phase system. The conditions in which
the wave velocity is zero are assumed as critical conditions.
The fundamental assumptions are:
a) no mass exchange between liquid and vapor phases;
b) the values of x , a , p are assumed to be known: the author G C C
suggests an empirical relationship between x and a so
that only one of these two quantities must be known.
Directly from mass and momentum conservation equations, writ
ten separately for the two phases, the author obtains the follo
wing four equations in the four unknowns (6w , 6w , 6m, <5p) :
p -(l-a) 6w +6m+ fw _-V) -r— f f f dp P£d-a) 6p = 0
p a<5w -6m+(w -V)^—(p a) ôp = 0 g g g dp g
p£(l-a)(wf-V)ôwf+(w-w£)ôm+-rd- p(l-a)
p afw -V) ôw -(w-w ) ôm+-;—(pa)ôp = 0 g g g g dp F
ôp = 0
(1)
(2)
(3)
(4)
where w is an unknown mean velocity so that w 6m is the net
momentum transfer from the liquid to the vapor. Once assumed
6m=0 the value of w becomes "immaterial".
(x) With the symbol 6 the author refers to small perturbations.
- 77 -
The compatibility conditions for these homogeneous linear equai
tions is the vanishing of the coefficient determinant; i.e.:
A = 0 (5)
When equation (5) is satisfied, equations (1) through (4) are
not independent and any of them can be solved, for example in
terms of op.
Taking into account assumption a) it follows:
(w -V)2-^-(p o)--r- (pa) 6m. g ^ dp VMg dp ^ _ Q ( 6 )
6 p -(w- 2w + V) g
By the definition of w it follows that the denominator of
equation (6) is always different from zero. Then: d r ^
7 -rr (p «
dF cpg"> and by substituting this equation in the determinantal equation
(5) we have:
d/dp [p (1-coJ
d/dp [Pg(l-a)] (wf-V)
2= = ~- (8)
Equations (7) and (8) form the compatibility conditions for
equations (1) - (4) and must be satisfied simultaneously.
In order to achieve the solution the author introduces two pa
rameters A and X whose expressions are given in App. 3.7. He
gives in a diagram (Fig. 3.58) X vs A satisfying equations (7)
and (8). The four intersections of the two curves (an hyperbola
and a parabola) so obtained are the two solutions for the sonic
velocities, i.e. (with symbols of Fig. 3.58):
- 78 -
a i = ( V l + V ï ) / 2 C9)
V = CV V2 ) / 2 (10)
Two-phase critical flow is obtained when it is no longer pos
sible for disturbance waves to propagate upstream, i.e. analiti-
cally: r _ v = o
(ii) Y" = 0
These conditions can be found by adjusting T until the V=0
point of the parabola in Fig. 3.58 falls either on one branch of
the hyperbola or another; as a conseguence two admissible values
for r are obtained. They are compared in Fig. 3.59, for a given
reference pressure.
/55/ 3.3.4.8 - ARDRON-FURNESS 1976
A simple frozen theory on the basis of Moody's model is deve
loped for the calculation of what the authors call "upper bound
mass flowrates". /13/ They write all Moody's 1965 equations in order to obtain
(see paragraph 3.3.3-5, eq. (2) that is written here in a dif
ferent form):
Ixw2+i(l-x)w2=x (h -h )+(l-x H h £ -hr )-h. (x-x ) (1) 2 g 2 f oK go ge' oJ K fo fe' fgev o v J
Consideration of these equations shows that since the enthal
py of vaporization is always greater than zero, the kinetic ener
gy flux and hence the flowrate increases with the decrease of
outlet quality. In order to achieve maximum flowrate they assume:
- 79 -
x = x e o
(2)
and isentropic vapor expansion, i.e.
( T) ) VY
P =
P . g go I P0 )
(3)
Moreover the authors assume:
p = const
and, like Moody, the criticality conditions are (*)
9k
3p = 0
(4)
(5)
(6)
Taking into account the thermodynamic identities valid in con
stant quality isentropic flow:
and
8h 3p J s
l - . i
1-x
h -h=x w { ° o go (. p0j i. x0 j
1 -x ^ p Y - l
P£oY Y - l
f iî ï Y
i-l-H the maximum flowrate is easily obtained:
r2-C
P W •go go
Y+ l
7 Y I X J P f
2 " f P . ^ f P o l ^ f l - x P g ^ L lpgoJ IPCJ I X P f
(7)
(8)
(9)
In the expression (9) p i s calculated by applying equations
(5) and (6) to equation (1) , taking into account equations (3)
(x) Condensation (see also PORTER) is forbidden for simplicity.
Jf
>.
> i
}
- 80 -
and (4). In Figg. 3.60 and 3.61 some results are shown; it may be
observed that when:
x 0—:-o
the theory predicts,. Berhouilli's flow.
/78/ 3.3.4.9 - RANSOM-TRAPP 1978
We have already presented an equilibrium model by the same au
thors (Par. 3.3.3.20). This frozen theory and the preceding (by
the authors) actually bound two-phase flow.
In developing this model two balance equations are adjoined to
those given in Par. 3.3.3.20,namely:
- vapor mass continuity
3 Cap) + -4- Cap. wj = 0 (1) 3t 'g' 9x Kg g
- vapor entropy conservation
9 3 -r— (ap s ) + -r— (ap s w ) = 0 (2) 3t g g' 3x Kg g gJ
These two equations with equations (1) through (4) of para
graph 3.3.3.20 forma closed system of six equations whose depen
dent variables are: a.p.w ,w_,s_,s . Two state equations charac-g £ f g
terizing the frozen transformation need to be adjoined to the ba
lance equations; these are the following functional relationships: Pg = f (P.sg) (3)
pf = f (p,s£) (4)
Finally two other transformation equations completely indivi
duate the frozen flow:
81 -
ds g
dp
d s £
= n
0 dp
(5)
(6)
Equations (1) , (2) (of this paragraph) , (1) through (4) of pa.
ragraph 3.3.3.20 and the conditions (3) through (6) solve the pro
blem. The solution is obtained through the characteristics method,
as in the case of equilibrium model. The only mathematical diffe_
rence is that this model produces a sixth order determinantal e-
quation.
In Fig. 3.62 we present the same variables of Fig. 3.49, in
order to compare the quantitative and qualitative differences bet
ween these two diagrams and the different value of velocity rela
ted to the two analytical models.
Apart from the observations made in paragraph 3.3.3.20, it may
also be noted that:
- in the choking flow criterion the only empirical data is the
thermal non-equilibrium degree (in fact the constant c can be
evaluated theorically);
- partial non-equilibrium actually has not be included in calcu
lations, since the lack of reliable vapor generation models:
- the frozen theory applied to EDWARDS' data predicts blow-
down to occur in about half the actual time, while equilibrium
theory well agrees with the experimental results.
3.3.4.10 - Non equilibrium, general theories
/82/ 3.3.4.11 - HENRY-FAUSKE 1970
• These authors started from the consideration that the greater
•
- 82 -
part of the theories developed before their study were equili
brium ones, whose main déficiences were the discrepancies from
experimental data at low qualities and the too high slip ratio
provided.
Stating that the assumption of frozen flow (dx/dp=0) is far
too restrictive, they propose a partially non-equilibrium model
suitable to predict critical flow for equilibrium quality less
than 0.02 approximately. The basic equations of this model are:
- vapor continuity equation
dz
A w g g X V
g = 0
- liquid continuity equation
d dz
Af Wf (l-x)v.
= 0
CD
(2)
- mixture momentum equation
dz = r dz x w +(l-x) L g •<] (3)
in which the wall shear and the hydrostatic head are neglected
in comparison with other pressure terms.
A first criticality condition is:
d r T 0
-"c
(4)
therefore equation (3) may be written
r * - < _d_ dp
xk+(l-x) ' k
(l-x)kv-+xv f gj_
-1 (5)
- 83 -
As in the examined region of the Mollier diagram
V- « V £ g
dv,. dv
dp dp
(6)
with the assumption
k £ 3
and making the possible simplifications consequent to the hypo
thesis 0<x<0.02, equation (5) may be written as:
" -1 > r = - < N
dv dx XC-T-£ + V -r-
E dp g dp
dN g E dp (7)
where N is a non equilibrium parameter defined as
N = kx.
(8)
The term dx /dp can easily be obtained as a function of p
from equation:
ds = 0 (9)
with the further assumptions:
- polytropic expansion process (the authors show tha t the variat ion of n
be tween 1 and 1.3 i n t h e i n v e s t i g a t e d f i e l d ( p * 50 p s i a ) af
f e c t s t h e c r i t i c a l f l o w r a t e of l e s s t h a n 1 $ ) ;
- dv / d p determined from s a t u r a t i o n p r o p e r t i e s ( n = l ) .
E q u a t i o n (7) becomes:
r2 -C •N
cHE
dN - V X^ —;
g E dp
(10)
- 84 -
where T Tt_ is the critical flowrate resulting from hoiaoge-cHE
neous equilibrium theory (see paragraphs 3.3.3.2 and 3.3.3.3).
Moreover in equation (10) the influence of the non equilibrium
parameter N may be clearly seen (infact for N=l, it results
r =r ) . c cHEJ
Regarding to N, it is easily shown that equation (8) can be reduced to
N = f1 n, (11) xE (1-a) vg
However the value of N is obtained esperimentally and, as
a first approximation, it results only as a function of x :
N = 20x c (12) e be
Moreover, the d e r i v a t i v e dN/dp may be w r i t t e n a s
dN dN , fc dp dz dz
and then as dp/dz at the exit is very large and dN/dz appears
to be nearly zero, it follows that dN/dp can be neglected.
Finally, from equations (10) and (12) and from the above con
clusion, it follows:
r = r / / N . c cHE ' e
Some results are shown in figures 3.63 and 3.64; in Fig. 3.65
the ratio r /T TTT, is plotted vs x_ . c cHE Ee
The simplicity of this theory that takes into account slip
and metastable effects is to be observed; the simplicity is coun
terbalanced by the too narrow field of application.
In the same work the authors present also a two component cri_
85 -
tical flow theory which is based on the frozen flow assumption
(dx/dp=0) .
3.3.4.12 HENRY 1970 731/
This other theory developed by Henry applies non-equilibrium
effects to nozzles with high (L/D=12) and very high L/D (L/D >
12) ratios, smooth or sharp edged at the entrance. Now we shall refer
to sharp entrance and initially saturated or subcooled liquid.
The author starts from the following phenomenological observa,
tions:
a) a great pressure drop is present at the inlet;
b) the pressure remains about constant up to L/D-12;
c) for L/D>12 flashing of the mixture generates a momentum pressu
re drop that cannot be neglected.
By further neglecting the quantity d vf/dp and assuming slip
ratio equal unity, from equations (1)T(3) of the preceding sec
tion it easily follows:
r = i / dv
X —r-^- + (V -V ) dp
dx g fJ dp (1)
In order to evaluate the mass transfer term (dx/dp) the au
thor obtains the following correlation from experimental results:
d XE • N — £ - (2)
dx dp dp
(x) In particular, from more or less empirical considerations, the authors reach the formula rc= IcHE
k n o t unexpected because for x=xE, N=l/k (see eq. (8)).
- 86 -
where
N = < 20 x for x < 0,05
and dx /dp, being ds=0, is
dp
(3) for x * 0,05
ds( ds (1-aO- dp dp
'fg (4)
At this point it is necessary to distinguish two situations:
A) L/D =12;
B) L/D > 12.
In the first situation, from the phenomenological assumptions
a) and b) it follows that we have a pressure decay only at the in
let and it may be obtained from:
p - p = Ap. n . o c "inlet
r 2 v f
C fO 2 C 2
(5)
where C is a constant (the orifice contraction coefficient),
whose value is 0.61.
Moreover the last assumption to solve the problem is:
x = 0 c (6)
Equations (1) and (5) which take into account equations (2),
(3), (4) and (6) constitute a set of two implicit equations in the
unknowns p and r.
In situation B), two are the differences from the case A): the
first is related to the fact that equation (5) is no more valid
and, in accordance with the phenomenological aspect c), it is nece£
sary to write a further term in order to obtain p -p ; for homo
geneous flow flashing in a constant area duct the infinitesimal
- 87 -
/128/ frictionless momentum pressure drop (following WALLIS 1969 )
can be written as:
dp= -r d (l-x)v +x v £ g
(7)
and integrating between the inlet (x=0) and the exit section (x=x ),
taking into account equation (5), we obtain:
v, p -p = r *o *c c
2C -=-+x (v -v_ ) 2 e ge fo (8)
The second difference is related to the fact that not even e_
quation (6) is valid any more. Equation (6) is substituted by:
x =Nx <q e E 1-exp
-B(L/D-12) (9)
in which the constant B is obtained from experimental data and
results :
B = 0.0523 (10)
Analogously to situation a) it may be concluded that equations
(1) and (8), with equations (2), (3), (4), (9) and (10), consti
tute a set of two implicit equations in the unknowns r and p .
Some results are shown in figures 3.66 and 3.67. Namely in
Fig. 3.66 r is plotted vs p , L/D being the parameter; in
Fig. 3.67 r vs L/D is shown, p being the parameter. In Fig.
3.68 p /p is shown vs L/D, p being the parameter. Finally, in
figures 3.69 and 3.70 the influence on the flowrate of subcoo-
ling degree is clearly shown.
Up to this point we have treated the sharp entrance: in the
smooth edge case, the author notes that there should be no signi
ficant separation of the main flow from the duct walls; thus the
- 88 -
coefficient C of eq. (5) results equal to one.
Moreover the assumption x=0 at L/D=0 produces a flow pattern
identical to the one at L/D=12 in the sharp entrance configura
tion and therefore the exit quality may be expressed as:
xe=NxEk-exp(-BL/D)j (11)
where B has the same value given in equation (10). Equation
(11) substitutes equation (9) in the solution of the problem for
smooth edged nozzle; some results are shown in Fig. 3.71.
The simplicity and the completeness of this model justify its
diffusion.
/83/ 3.3.4.13 - KLINGELBIEL-MOULTON 1971
The Klingelbiel-Moulton work is based on experimental data in
which, apart from the classical quantities, the two-phase jet
thrust is also measured.
The authors note that condensation rather than vaporization
is the result of high quality isentropic flow. Moreover no model
is really very accurate in the range of quality below about 201.
Therefore two new phenomenologically based models for critical
flow of two phase mixture were formulated.
The first one accounts for the above mentioned condensation,
the second takes into account the fact that some mass of liquid
is entrained like droplets moving at velocity of vapor stream.
We shall describe the latter, because it seems more analytically
developed. It was called Entrained-Separated-Flow (ESF) model.
The conservation equations are:
- 89 -
- mass continuity
r = p w = const, m m CD
- vapor momentum balance
dp A =< g 7
r A +d(r A ) g g g g
(w +dw )-T w A -Dw d(T A )-(l-D)w^d(r A ) g g g g g g g g f g g
> (2)
- momentum equation for the liquid phase
(w£+dw£) (l-D) + (r A,)+d(r Af) (w +dw )D-
-wfAfrf(l-D)-WgrfA£D-Wgd(rfA£)D-wfd(r£Af)(l-D)
dpA£= < r£A£+d(r£A£)
(3)
where D is the constant weight fraction of liquid moving at va.
por velocity and (1-D) is the fraction moving at liquid velocity.
We assumes that evaporation takes place from the entrained
and non-entrained liquid in the same ratio as the weight distribu
tion of the liquid itself. From the above equations it results:
r = I" -dvT (4)
where v is the volume of an entrained-separated system given
by:
|_x2+Dx(l-x)J v ]x +Dx g
a
v£(l-x)2(l-D)2
The expression for the slip ratio (k) is obtained by differen
tiating the kinetic energy expression with respect to void frac
tion or slip ratio, with the assumption that choking involves mi.
nimization of entropy production.
- 90 -
It results
x = D+(l-D) V
g
h* J
J / J
v f x
1 + D 1-x (6)
The formula of D is obtained from experimental data:
f1+0,14 In x D = <
for 0.19<x<l (7)
0.94-0.204 In x for 0.01<x<0.25
Equations (5), (6) and (7) permit r to be a function of p
and x only, e
Some results are presented in Fig. 3.72.
The authors conclude that neither of the two models offer sub
stantial improvements over existent models.
3.3.4.14 - KLINGELBIEL-MOULTON 1971 /83/
Taking into account what has been said in the preceding para
graph, this new method involves calculating critical flowrate
from the energy balance written in the same form as Moody (see
Par. 3.3.3.5). Similarly from mass and energy balance they ob
tain:
r =< 2(h -h.-xh.
•L — xv 12 ,-- ^ + (l-x)v£l ll + x(k2-l)
(1)
The slip ratio and the values of "D" are obtained from ex
pression (6) and (7) of the preceding paragraph respectively.
- 91 -
By substituting the expression of "k" in equation (1), r remains
a function of p and x only and x values may be calculated e e e
from an empirical equation related to the authors' experimental
results; the expression for k is reDorted in appendix 3.8.
Therefore for each value of p (once given h ) a value of r *e 6 o
is obtained.
This formula well agrees with experimental data, with an ave
rage error less than 2% in the quality range 0.01 <x<0.99 and
pressure range 20<p <75 psia. /84/
3.3.4.15 - MALNES 1975 '
The author employs a set of four one-dimensional two-phase
conservation equations, including thermodynamic non-equilibrium.
By simplifying this model with the assumption of steady state
conditions, the pressure gradient goes to infinity at the exit
section when critical two-phase flow occurs: however,in a real
system,irreversible phenomena and two-dimensional effects cause
exit pressure gradient limited, exit pressure higher and exit v£
locities lower than those predicted by one-dimensional theory.
In this synthesis we shall not show the analytical develop
ment of the above but we shall only expose the main model.
The characteristic of this theory is the consideration that:
- the release of dissolved gases is the main mechanism for bub
ble formation during expansion of two-phase mixture.
In order to achieve this a conduction controlled flashing co£
relation is developed.
The balance equations are:
- vapor mass conservation:
- 92 -
9 3re — ap = - — 4 + 3t 3z
*
liquid mass conservation:
3 r i arf *
- mixture momentum conservation:
(1)
(2)
_3_ 3t apgW£ + (1~a)pfWf
= _ JE . JL 3z 3z g g f £ J
- mixture energy conservation:
3E 3t y 3t V^v^i^vptyv
(3)
(4)
where "F" i s a gene r i c term t ak ing f r i c t i o n i n t o account and "E'
( t o t a l i n t e r n a l energy) i s given by the e q u a t i o n :
1 2 1 2 E = ap (e + - wJ + (1 - a) p f (e f + - wf)
(5)
The above system may be solved when the following quantities
are known independently:
- slip ratio
- friction (F)
- flashing O ) .
The author solves equations (1) through (4) by a finite dif
ference technique from upstream reservoir conditions up to the
inlet of a nozzle diffuser. He assumes:
1) slip ratio equal to one ;
2) two-phase friction multiplier given by Becker ;
3) evaporation rate ij; calculated by an original expression
which will be briefly discussed.
- 93 -
The author says that flashing starts essentially from:
a) wall boiling;
b) release of gases;
c) impurities;
d) statistical fluctuations of the vaporization in a superheated
liquid.
Referring to the last point he assumes that "dissolved gases
are the main mechanism of flashing in reactor conditions, surfa
ce boiling is insignificant expecially for large diameter pipes",
justifying this hypothesis with the flashing delay experimental
ly observed in a gas free liquid .
Taking into account the Zuber results for the calculation
of the diameter of a single bubble as a time function, the evapo
ration rate \l> may be calculated by the following equation:
P c \ p_ - - V. <j; = (R -S B + RJ P-g *
° pf l hi fg
(1-cO '3.„2 AT (6)
where R and R„ are dimensionless empirical constants, whose va-° 7 5
lues are 7x10' and 2x10 respectively (for a particular case ) and
1 - 2a for a < 0.5 B = < (7)
0 for a > 0.5
The equa t ions (1) through (4) and ( 6 ) , wi th the assumptions
(1) and ( 2 ) , mathemat ica l ly so lve the problem. Some r e s u l t s a r e
shown in F i g . 3.73 (T vs p , h i s t he p a r a m e t e r ) ; in F i g . 3.74
(x) The author says that typical values of dissolved gases in a LWR are : 0.5 ml/1 in a BWR at 1. bar and 25 °C and 10 ml/1 in a PWR at l .bar and 25 °C.
- 94 -
r versus exit pressure is reported*, finally irt Fig. 3.75 pressu
re and void fraction versus pipe length are shown, for r = 11100
kg/m sec.
/89/ 3.3.4.16 - PORTER 1975
A very complete analysis is thé dnë developed by Porter. He
considers both equilibrium and nônJequiiibrium thermodynamic ef
fects, allowing also for a change in slip ratio and calcula
ting in the two cases, (equilibrium and non-equilibrium) the cri
tical flowrate, in the range from subcooled to superheated sta
tes through saturated state. ' •'•
Moody's theory is taken as à reference ; the equations adopted
are the same as those derived by Moody in treating saturated mix
ture in thermodynamical equilibrium.
In this work we consider the hpn-equilibrium theory only but
we shall also present some results related to the equilibrium
one. ,
I) Mixture in the saturated_field
Fundamental equations:
- mass conservation:
r = const. (1)
- momentum equation:
Td Lk(l-x*)v f+x*v gJ l x * + * -x k
= -dp+2f f L/Dvér 2 d z + E * + k ( 1 ~ * y dz g k(l-x>,nv
(2)
fx) Two values of slip ratio are used: k=l and k= (-=•)
- 95 -
- energy equation:
2 2 * Vt = H + T&(l-x*)v f • x*v J j V + f - J (3)
6 k
where "Q" is the heat exchanged with the exterior per unit length
of exchange surface, "f _" is the Fanning factor for water (f =0.046 -0 2
Re» ' for a smooth pipe) and "<J>" is the two phase friction imU
tipl ier .
In the case of non-equilibrium the author assumes:
- no water flashing;
- isentropic expansion of steam.
When considering the first assumption the initial water flow
is:
r. = (l -x )r fo v o
The vapor expansion causes a partial condensation that may
quantitatively be expressed as:
s - s£
x = -&—£- (4) con s- v J
fg
from which the total water flow results:
r. = (i-x )r + x r(i-x ) = (i-xx )r (5)
fe v o o v con' v o con v J
and the total steam flow results:
r = x x r (6)
ge o con
In this case, therefore, we have:
x* = x x (7) o con
Equations (2) and (3) are derived with respect to "z" and
- 96 -
dp are solved simultaneously to give -p-
field, the results are obtained (as
ming:
k = 1
or v V k = (-*) 3
II) Subcooled water
The only equation is the momentuir
dp = 2£f L/D vf r2 dz
where we have neglected inertial and
III) Superheated steam
- momentum conservation:
2 v
r d(-f) =-dP + 2f£
- enthalpy conservation:
2
H •Sf. = H + ^ r 2
o TA g 2
and —.— . dz
mentioned
i equation
gravity
L/D v r2d g
These equations are differentiated with re dp
may be solved in order to obtain -j*-
The results related to situations
iteratively, for assigned values of
and -r— . dz
I), II),
In the saturation
above) by assu-
(8)
(9)
given by:
(10)
forces.
z
spect
III)
total pressure
and by calculating the inlet pressure through pressure
at each step the two values coincide
In this way the three sets: (1),
', the cal
(2), (3),
culat:
(11)
(12)
to "z" and
are obtained
at the outlet
drops. When
Ion stops.
(7) and (8) or
- 97 -
(9) for saturation non-equilibrium flow; (1) and (10) for sub-
cooled non-equilibrium flow; and (1), (11) and (12) for super
heated steam flow are solved.
Figs. 3.76 and 3.77 show the critical flow versus stagnation
enthalpy in a form similar to Moody's. In Fig. 3.78 and 3.79 the
static pressure is plotted versus stagnation enthalpy. All the a_
bove figures are related to non-equilibrium flow.
EqiT^Librium flow results are presented in Fig. 3.80 and 3.81
in which r vs h and p vs h have been calculated respectively 0 e 1/3°
for k = 1 and k = (v /v_)
By the examination of above figures one can draw the follo
wing basic conclusions:
- in non-equilibrium case, there is not a great difference if <s
quation (8) instead of eq. (9) is used;
- in equilibrium case, a strong dependence of r from subcooling
is observed on the left of the saturated liquid line; in case
of non-equilibrium the strong dependence is observed on the
right of the same curve;
- in every case curves converge towards the same values at the
saturated vapor line, since the assumptions in the saturated
field become identical.
/90/ 3.3.4.17 - RIVARD-TORREY 1975
This work analyzes the transient depressurization phenomenon
entirely.
A two-dimensional time dependent theory is used and the au
thors emphasize some relationships describing:
a) compressibility of the liquid phase;
b) phase transition;
- 98 -
c) heat transfer and friction between the two phases;
d) friction with the duct wall.
The balance equations are written as:
- vapor continuity:
—: (cxp J + V ' (ap w ) = i|> - ty 3t V - v *g-gJ Ye rc
- liquid mass continuity:
Testo aervenuto nel settembre 1980 - vapor-momentum conservation: g
^-(otp w ) + V(ap w w ) = - a V p + F(w_-wJ +\\> w_-4> w + f
- liquid-momentum conservation:
CD
(2)
( 3 ) , (3a)
az Q l - a ) p ^ J + V • £ ( l -a) P£w£wf2=- (1-a)V p + F (w -v^) «l> • w - * ^ + f£
(4),(4a)
- vapor-energy conservation:
ap g
3e —£ + ap w • v —f- + ap w • y e = - p v * I aw + (l-a)w_ |+F(wJ.-w ) + 3t ^ g - g - g ~ J g ^C-1 f g
• (*e-tc)P(vg-vf) • Q(If-Tg) • I ^ H . ^ (5)
- liquid-energy conservation:
3e (l-a) pf-r*+ d-a) p A . Ve. - -(*,-*) L.P(v -v ) + e - e J + R(T -T ) + Jf 3t ' ^ ^ *& * -~f "' "re V L r v ' g "f • ' ~g w£-" " v g f
fvix Ticond
The authors note that some transient phenomena linked to the
Riprodotto in offrt pntêo il Laboratorio Tecnografico délia Direzione Centrale Relazioni Externe <kl CNEN • Viale Rcgina Margherita 125, Roma
- 99 -
microphysics of phase change are necessarily neglected.
Two implicit state equations relating vapor pressure and tern
perature are written as:
P = pg(nsg(P) -1) eg(p,Tg) (7)
T =T + fh (p,T ) -h fp)J /c (8) g s L g^' gJ sg^'-1 ps
and the state function n is given in Appendix 3.9. These re
lations are valid for a > a (a being a fixed value) and for o o &
T < T . g s
The authors also present other forms of equation (7) and (8)
that are not presented here because they yield no quite diffe
rent results.
The transformation equations describing the mass exchange
are:
V2 T f " T s \\> = X B ( l - a ) p , a ( T R) — - — - v a l i d f o r T r > T (9)
e e f ^ s T f - s
\b = 0 o t h e r w i s e e
1/ T - T ij> = A B ( l - a ) a p (T R) - 5 - — & v a l i d f o r T <T (10)
c c g s T g - s
^ = 0 o t h e r w i s e
In these equations "B" is proportional to the contact area
between the two-phases and depends upon an empirical constant
"N" (number of bubbles per unit of mixture volume). The fric
tion between phases is:
- 100 -
F = f <j_p|_cdlwg - wf I + ^ L y + (i - «)v£)jj> B
in which "r" is another function of "N" and "a".
The pipe wall friction is written as:
— 2 2 — 2 T = I -af p u a / 2D I <f> wg L g g g -1 g
(ID
and
T .= -(l-a)Lf.P-:4(l-a)2/ 2DJ4»? wf LfHf "F
(12)
(13)
in which the friction factors f and f _ depend upon the Reynolds
number and the pipe roughness (see App. 3.9). The heat exchange
is analyzed through the function MQM which has not a definite
functional dependence.
After investigating, amongst other things (KACHINA code), the
effects of large relative velocities and large temperature difftî
rences between the phases, the authors conclude that they are ne_
gligible. / ? f\ I
The authors analyzed Edwards experiments and find a good
agreement between experimental and theoretical results over all
the transient.
In conclusion this is one of the most evolute models adopta-
ble also in analyzing depressurization transients in complex ge£
metries. No result is given about the flowrate.
3.3.4.18 - KROEGER 1976 Ml
A complete and analytically consistent analysis is the one
performed by Kroeger. In reviewing the preceding models he points
out two interesting aspects that are not always appropriately con
- 101 -
sidered:
- the vapor generation rate analytical description;
- mathematically self-consistency of a model.
The primary interest of this work is the analysis of expan
sion of initially subcooled water, also considering the flashing.
The main explicit assumption is to neglect dissipation terms in
energy equation; moreover, axial heat conduction, surface ten
sion and uneven distribution of the phases in a plane normal
to flow direction have not been taken into consideration.
This model can be characterized by the following quantity,
called vapor drift velocity:
w = w - w (1) gm g m
where w is the mixture centre of mass velocity, m
The balance equations are:
- mixture mass conservation:
A£(pJ + Ptn~ (AWJ =0 (2) Dt m m 3 z m
- vapor-mass conservation:
D 3 Ap - x =Ai^-— (Ax o w ) (3) m Dt st 3z st m gnr
- mixture momentum conservation:
Ap £• w = -A P- - AP -ET -•£• (A r-^- p w2 ) (4) m Dt m dz m w dz 1-x m gm
st **
- mixture thermal energy:
AP ^ n = A S " + A x * ~ w 7 + sq -'— l~Ax p w hc ~\ -Mm Dt m Dt st p gm dz Hex 3z «- stMm gm fg-1
- Ar ^ 1,- st.,
w + -(- ) w m 2 1 -x gm
w gm
(5)
1-x . st
- 102 -
V J^ _ p m
pgCiyp}
1 " X 4- X 4.
St ( St
f g
with the state equations:
Pf = P£(hf,p) (6)
(7)
(8)
h = (1-x Jh_ + x Ji (9) m v stJ f st g K J
All the quantities are generally not taken along a satura
tion line. In order to solve the set constituted by equation (2)
through (5) four constitutive relationships are further neces
sary; these are written by the author as:
* - vvv^st'V (io)
wgm = g'V^st'V ^
Tw = VVV^st'V (12)
q = f. (h ,h,.,p,x .,w ) (13) ex 4 g f r st m v J
where on the right hand side the derivatives with respect to
"z" and "t" of h,,, h , p, x . can also appear. Since the field ± g st
of study of these equations is essentially referred to low qua
lities (see above) it is not a very restrictive assumption to
write:
h = h . (14) g gsat v -*
In the following we shall briefly present the author's con
clusions about the function f and f , describing the vapor gen£
ration rate (ip) and the vapor drift velocity (w ) . The vapor
- 103 -
generation rate, for homogeneous equilibrium conditions, may be
evaluated by the relationship:
Eq rax. Dp_ _1_ nex
TΠ" pm l3pJc Dt h. A s fg
(15)
where the first term on the right hand side represents the gene
ration rate due to pressure variation and the second term ac
counts for exterior heat addition.
To take into account non-equilibrium phenomenon the author aj5
sûmes:
0 for Xp £ x.
£ r^ (16) 9x
m 9t> Dp. Dt
for > x.
Also another model of vapor generation rate is presented by
the author, through the evaluation of vapor drift velocity.
In this model, the function f? is related to a so called di£
tribution parameter "C ", defined as: o
< Y >
< a >
const
Fr n (17)
where <Y > is the flowing volume concentration. The author shows
that in the examined range of values, Fr >> 1 and then the se
cond term in the right hand side is negligible. Moreover the
distribution parameter C is obtained by interpolating some
[mental data of Zuber' ': expern
c = i + o i9f 3 2 0 6 "F) Lo 1 + u-iyt 3206 J
2.4 (18)
where "p" must be expressed in psia.
- 104 -
The vapor drift velocity equation is then written as
(l-x)+x - pf
w = (C -1) s w (19) gm o pf pf m
(1-x) + x — + x C (—-1) g g
Also another expression for w is obtained by the author gm
and is not presented here.
In conclusion the problem is defined by:
- the constitutive equations (16) and (19) characterizing this
model, and (12) and (13), of which the former has the classi
cal form, and the latter has not been expressed;
- state equations (6) through (9) and eq. (14);
- balance equations (2) through (5).
The set is solved by the method of characteristics in which
the dependent variables are assumed to be: h_, p, x, w and the
independent variables are obviously "z" and "t". The observations
are as following:
- the vapor drift relationships characterize the momentum dis
tribution between the two phases (as the slip ratio in other
models);
- the assumption (14) corresponds to a statement on the energy
distribution between the phases;
- the two relationships for drift flux and for vaporization ra
te fairly increase the free degree of the solution;
- all the chosen constitutive relationships lead to real roots
of the determinantal equation;
- in the available reference only a comparison of theoretical
results with Edwards depressurization experiments is presen-
- 105 -
ted; this comparison is quite good when constitutive relation
ships are opportunely chosen.
/29/ 3.3.4.19 - BOURE -GIOT-FRITTE-REOCREUX 1976
From a mathematical point of view this is undoubtedly the most
developed among the theories presented in this review.
The authors' aim is to present a general model from which any
other consistent model should be obtained; moreover they wish
to point out the effects of some constitutive terms in the balan
ce equations on the critical flowrate evaluation. Regarding to
this, the authors say that in analyzing any thermodynamical tran
sformation, a rational procedure intends to postulate the trans
fer laws (cause) rather than the nature of the transformation ijt
self (depending upon the transfer laws).
The assumptions adopted are:
- two dimensional effects negligible;
- pressure uniform in any cross section;
- diffusive and turbolence effects negligible.
Moreover the importance that these three assumptions may ha
ve is recognized by the authors.
A complete system of six balance equations and seven consti
tutive relations is written as:
- mass cont inui ty equation:
+ 9aG+ f9pKl j n [3pKl 8 A Y 3aG f3pK] 3D
±o — + a — —*-+a — ••' ± o w — + a w —— -*-+ MK9t K[3pj 9t K[3hKJ 3t 1 K3z K K(3pJ 3z AhK p AhK
+ VK irlVGirJ V K ^ J IT±M=-VKWKÂ
C1>la)
only i f K=G p
- 106 -
- momentum conservation equation:
9 aG ±PK "ôTVlC
(dp ^
3p I )
3OL 3p L 3t T K 3t
29aG , 2 1 + \
r3p AhK
3AW Ot 0 -z—
GPG 3 t *Vfc
K
VÊ fdpJ 3Ah
3hK.
K. 3z
3P
r i m
K )
9 w L r
¥z+2aA\1T+
; o n l y i f K=G 3AK
3P,
31V
3Ah I
3t
Ah f only i f K=G
(2,2a)
- e n e r g y c o n s e r v a t i o n e q u a t i o n ;
K h +-r-K 2
I J
8 a G f +a
3 t K
Y 2 ï
h +—-K 2
.1 J
Kl 3p
.
±p„ h +— —
\
^ ^ C s a t . K p -*-+a.j3„w„-—+
3Aw
only i f K=G
+a K
w.,
Vf f ^
*K JV _
3AhK. 3t_±PKWK
2
Vf 3aG W,
Vf
Vi 3W(
V T
2 3wT
3z aGPG 'H 3 W ] 3Aw
3z
r 1
MI
t. ) • Q K = - a K P K \ g C O S 0 - a K P K \
only i f K=G
K A» h +—
K 2
+a w K K
3 P
r 2,
+p a*
sa t
V / Ah K p
K
ap 3z
W_,
Vf 3p
K 3z
(3,3a)
Moreover:
- pressure drop per uni t length ( * )
3x . 3x.
h = T k , o + f ( T T ' T T 3 (4 )
(x) This relationship takes into account friction between the two phases of fluid and between each phase and the duct wall.
- 107 -
- heat transfer to each phase from the exterior per unit volume:
' 3x. 3x.
Qk = «k.o^TT'T^ C5)
- mass transfer from the liquid to the gas phase:
9x. 3x.
M = M + M(-rr f ~ ) (6) o 3t 3z
- momentum transfer from the liquid to the gas phase:
3x. ax. MV = (MV)o + (MV) (-~ , -~) (7)
- energy transfer from the liquid to the gas phase:
3oc. 3x. MH = (MH)o + (MH) (- , ~ ) (8)
In all above equations the subscript K refers both to liquid
(L) and to vapor phase (G), the term A' means dA/dz, and by x.
we intend all the dependent variables (see below). Substituting
equations (4) through (8) in equations (1) - (3) yields a set of
six equations in which the dependent variables are ar, p, w. ,
Aw = w - wT , AhT = hT - hT . Ah0 = hn - h_ . G L L L Lsat G G Gsat
The critical condition results from the formulation of the
model. At the same time it implies the vanishing of the deternu
nant of the coefficients A of equations (1) - (3) and of the de
terminants N. which are obtained from the matrix of A, by subs-
l '
tituting ith column with the column on the right hand side mem
bers of equations (1) through (6J.
In particular the condition:
A = 0 (9)
- 108 -
is a necessary flow criterion and:
N. = 0 (10)
is the "compatibility condition" involving the nozzle geometry
among other terms.
The flow may be considered critical if we verify that, accor
ding to physical experience, the flowrate cannot increase when
the external conditions are varied.
The authors analyze some particular cases, studying the in
fluence of the constitutive laws. The conclusions may be drawn
as following:
- by varying the constitutive laws any consistent model should
be obtained from this theory; in particular, the authors af
firm that this is achievable only when in the transfer terms
the presence of the partial derivatives of the dependent va
riables is allowed;
- the gradients of the dependent variables are generally not in
finite in the critical section;
- in the case of single phase flow investigated by the authors
together with two-phase flow, critical velocities different
from sonic velocity are found: this too is a consequence of
the presence of differential terms in the external constitutif
ve laws.
/92/ 3.3.4.20 - AVDEEV-MAIDANIK-SELEZNEV-SHANIN 1977
This theory is related to cylindrical ducts and takes non-£
quilibrium into account by imposing the vapor temperature to
follow the saturation line and by calculating the liquid tempe
rature independently.
- 109 -
The main assumptions a r e :
- monodimensional flow;
- p_ = const.
- slip ratio equal to one.
The balance equations are:
- continuity equation for vapor phase:
-T- (x p w ) = i|> dz m (1)
- continuity equation for liquid phase:
£[_wpm(l-x)J = -• (2)
- mixture momentum equation:
dw dp J.X. p w-r-=--rtL-f Hm dz dz (3)
- liquid energy conservation equation:
CpfPfW 1 -P x
ë)
f 2 dw , ,_JL\ <3> -r—= - p w -r--Tl>hr. - w p x ( - T ~ ) -? 1
dz Km dz y fg nm dp dz
( 4 )
in which "f " (frictional resistance) is a known function of
Reynold number:
f* = f(Re)
and, as usually, p is the mean fluid density, weighted between
vapor and liquid volume fractions.
With regard to the vapor rate formation (if/) the authors deve_
lop a complicated formula in order to obtain it. In particular,
it is a function of empirical parameters of the current abscis
sa and of some thermodynamical properties of the mixture; its
- 110 -
expression is given in appendix 3.10
The other assumptions are:
T = T ^ g sat
dh
dz
dp/ dz
° - 0
(6)
(7)
(8) z=L
the boundary conditions (at z=0) are:
w p _ o K£
P o " 2C DO
x. = 0 l
T. = T l o
w = w
The equations (1) through (4) with the constitutive relation
ship for "£" and the one for "ij>" are solved with respect to
T , w, a, p, taking into account assumptions (6), (7), (8) and
of the above boundary conditions.
In particular "w " is found by a trial and error method, its
value being varied so that the critical cross section (in which
dp/dz = oo) corresponds to the outlet cross section.
This method has been successfully compared with experimental
data for L/D > 8. No graphical results interesting for this
work are furnished.
- Ill -
/39/ 3.3.4.21 - TRAVIS-HIRT-RIVARD 1978
In this work different questions inquired at present and invo^
ved in critical two-phase flowrate investigation have been poin
ted out ^ .
In the first part of the work, five of the best known and pr£
viously described models have been discussed and compared with
experimental results obtained from SEMISCALE apparatus. The con
clusion is: "none of these models accurately predicts both the
throat pressure and mass flowrate over the entire blowdown tran
sient".
In this report the authors want to show that the discharge
coefficients, necessarily used in order to accord experimental
results with theoretical ones, are due partly to non-equilibrium,
but expecially to multidimensional effects.
In order to achieve this from a theoretical point of view, they
have produced numerical solutions of two-dimensional time-depen
dent equations of motion for a two-phase fluid mixture. In parti
cular they use two recently developed codes:
- K-FIX the formulation of which is based upon a complete
two-fluid description (8 equations); /137/
- SOLA-DF the formula t ion of which i s based upon mixture con
s e r v a t i o n equa t ions (4 e q s . ) , wi th a d d i t i o n a l equa t ions t ak ing
i n t o account : a) vapor mass c o n s e r v a t i o n ; b) d r i f t v e l o c i t y ;
c) tempera ture d i f f e r e n c e between the two p h a s e s .
(x) This i s the reason for which we describe this work, although i t i s not a two-phase flow theory.
(XX) These models are HEM (Par. 3.3.3.1), HENRY-FAUSKE (Par. 3.3.4.6), BURNELL (Par. 3.3.4.2) and a modified BURNELL model.
- 112 -
A reduced set of equations only is here reported, namely:
- mixture mass continuity equation :
3p + _J_ f 3Spu 3t + S [ 3x
3Spw = 0
3t S { 3x 3z .
- mixture momentum continuity equations :
3u 1 3p
CD
3z
3w 3z
3x
3w + U 3x
-r w — - — -
3z
3w + W _ .
3z
3x
iE. 3z
(2)
(3)
mixture energy conservation equation:
3pe _1_ 3t +S
— (Speu) + —(Spew) p f 3Su 3Sw'
3x 3z (4)
(*) in which S is a length in the third direction and coincides
with the duct area in the case of monodimensional equations. Mor£
over, in this case, x is the second cartesian coordinate.
To solve the problem a further equation considering the vapor
mass balance must be written
80ipg . i 3t S
-2— (aSp u) + -^— (aSp w) 3x v g 3z Kg
= * (5)
The term |ip| (evaporation rate) is very important and its ex
pression is reported in App. 3.11.
For the calculations the authors assume:
- no-slip;
- equal phase tempera ture ( a l so i f t he code K-FIX could o p e r a t e
(X) I t is obtained by integrating the complete tridimensional equations in this direction.
- 113 -
with T ^ T and k f 1).
Although these last two points appear as restrictive approxima.
tions, the elimination of their effects on computations permits
to better investigate phase change and multidimensional effects.
With reference to a SEMISCALE experiment (S--02-4), maximum
flow and throat pressure as a function of exit pressure (varied
between 4.5 and l.MPa) were calculated. The results are shown in
figures 3.82 and 3.83 for bidimensional and monodimensional cases
respectively.
The authors point out the following observations:
- the calculations results are in very good agreement with the ex
perimental ones both regarding critical flowrate and throat
pressure;
- the monodimensional model overestimates critical flowrate and
throat pressure and then it is necessary to multiply the dichaj_
ge area by reduction coefficient (0.84) in order to adjust the
calculated results;
- applying the same procedure to subcooled inlet conditions, the
models yield an error of about 401 in both critical flowrate
and throat pressure evaluation: this disagreement was initial
ly ascribed to the neglecting of slip or of non-equilibrium fla
shing; however by changing 4> numerically, a good accordance
between computed and measured values was found; this last re
sult suggests that slip effects must be small in that situa
tion;
- the theoretically evaluated two-dimensional effects vanish for
L/D>5.
- 114 -
3 . 3 . 4 . 2 2 - TENTNER-WEISMAN 1978 no/
These authors (of whom we have already presented an equili
brium theory, in Par. 3.3.3.18) having experimentally observed
sonic velocities conditions, state that non equilibrium effects
cannot be neglected at low qualities.
In particular they show that a very little fluid enthalpy
change is required (by freon) to go through saturated liquid to
a void fraction of 0.1. This means that "during a severe tran
sient any given control volume will pass through the low void r£
gion very quickly".
In order to take into account these non-equilibrium phenome
na, the authors suggest to select (independently) a velocity va
rying between equilibrium velocity for a=0.1 and single phase s£
nic velocity (a=0). This new equation would be used in conjunc
tion with conservation equations of a single phase fluid.
The balance equations are:
- mass conservation :
3p
i_3P_ih
JE. 3t
+w 3p 3p -Jh
JP_+pJïL+ 3z y 3z
3p 3h 3t +w
3p
3h - T
3h 3z
= 0 (1)
- momentum c o n s e r v a t i o n
w 3p
3p +<w
+ w
3p
- 9 p - h
3p
i 3p 3w ^ . 3w _,_ + 1 ^ + p i r + 2 p w -3T + w
3h
)
3h 3z
3p
3h -«p
3h 3t
w
( 2 )
- e n e r g y c o n s e r v a t i o n
- 115 -
K - S - ^ «*fëijh~ïh«»s» aw 3z
ie. , , 3h V L JP '
J c 3p A 1 9h %r _ n + WE™-+Wp — r— = 0
{ dh J 3z A
(3)
where E i s the same as i n paragraph 3 . 3 . 3 . 1 8 .
As shown in another
t h i s model i s given by
/ 3 8 / As shown in another work, (HANCOX ) the son ic v e l o c i t y for
a = < 3p 9p .-K-3hJ n
PJ
(4)
By imposing a=£(h) in an assigned field of variation of quali
ty, it easily results:
2 (5) 3p
3P " i x ± _f(h)J + p
_3p p 3h
and by substituting equation (5) in equations (1), (2) and (3)
the problem may be solved. The solution is:
2 2 2 r = a p c
A result of this approach is shown in Fig.3.84. It is compared
with MOODY (1975) slip theory. Referring to this comparison the
author concluded that Moody is wrong when he explains that the
experimental increase in critical flowrate at low qualities
(x<0.1) is due to slip reduction. In fact these calculations
(executed for k=l) show exactly the contrary: i.e. by increasing
the slip ratio the critical flowrate actually increases.
- 116 -
/93/ 3.3.4.23 - MOESINGER 1978
As the model of RIVARD-TORREY (Par. 3.3.4.17) this is another
theory calculating the whole depressurization phenomenon.
The author establishes seven unknowns (nine considering that
two of them are vectorial quantities) to be calculated: p . p . m g
w, wr, e-, e , p (that is assumed to be equal for both phases).
He writes a set of five balance equations to which he adds a
state equation and one vectorial equation for the relative velocity
of one phase with respect to the other. By considering this equji
tion entirely he obtains a "six equations model"; by neglecting
the inertia force he obtains an "ordinary drift flux approxima
tion"; by neglecting only the spatial derivative of w he obtains
a "refined drift flux approximation".
The balance equations are written as:
- conservation of mixture mass 9p + div (pw) = 0 (1) 3t
conservation of vapor mass
3p_ a ( l - a ) p p £
£- + div (p w+ £• w r ) = \\> ( 2 ) 3z g - p
conservation of mixture momentum
a(l-a)p_p 3 (pw)+div(pwx w+ •—^—wrxwr)=-grad p+ f . (3) , (3a)
3z
- conservation of mixture energy
<x(l-a)p£p a(l-a)p p£ 2
— (pe)+div(wpe+ £ e wj =- p div(w+ ^— v£g ifr +|wr| +
(4) F + -L2-I interfacial 2
+ L . vise
- 117 -
In order to determine the internal energy of the mixture, the
author imposes
T - T-g f
(5)
The pressure field is obtained by means of a state equation
of the type:
v = <
£1L«Pg»(
1-a)P£»e£»
egJ for a<ac
f2l«'V
egU for a*a
c (6)
The expression for the relative velocity characterizing this
model is (in vectorial form):
dw —r dt a(l-ct)p p..
S f
(dp -apf) (l-o) 1
* grad p - wr — • P ~ P
(l-o)pf ap +F interfacial
(7), (7a)
in which:
dw
dt "inertia force";
a(l-a)p • 8 grad p = force due to pressure gradient;
g
w
o(l-o)p£p • Y = force due to momentum transition during phase
change.
The "ordinary" drift flux approximation is written as
B grad p wr = ~~ „ n _r y + F (8),(8a)
interfacial
The "refined" drift flux approximation is written as:
- 118
3w
8t
where
= y 3gradp- w r ( y + F i n t e r f a c i a l ) ] (9), (9a)
B =
Y =
a(l-a)(Pf-Pg)
Pf(l-a) •(|*|+*D + op„(|*|-*)/p Î
g
y = a(l-a)p p£
The non equilibrium degree is varied through a parameter de
scribing the phase transition rate.
Equations (1)T(6) and (7) or (8) or (9) solve the problem.
The author makes a comparison among the three types of ap
proach and concludes that, apart from initial blowdown transient
in which the simpler approximation (eqs.8,8a) shows some physi
cal inconsistencies, the drift phase approximation is advantageous
because it saves time in calculations (see Fig. 3.85 and 3.86).
3.3.4.24 - WINTERS-MERTE 1979 /50/
This is another theory recently developed, which refers to the
whole blowdown process. It is based on a bubble growth model.
The assumptions presented by the authors are the following:
- all bubbles begin to grow simultaneously and their number re
mains constant during the depressurization;
- the growth of the bubbles is conduction controlled; the effects
of surface tension and liquid inertia are neglected;
- uniform pressure dependent vapor properties are assumed with
each bubble;
- 119 -
- the motion of the thermal boundary layer surrounding each bub
ble is neglected for purposes of calculating the heat transfer
rate per unit area at the bubble wall (x);
- plane geometry is assumed in calculating "xM; a symmetry factor
"K " is used to account for non plane geometry effects;
- constant liquid properties are assumed; also T. outside ther
mal boundary layer, is assumed as constant; this latter assum£
tion remains valid as long as there is no interference between
thermal boundary layers surrounding adjacent bubbles;
- the flow through the opening in the blowdown vessel can be pre
dieted in terms of upstream fluid properties, using a suitable
choke flow model;
- the heat exchanged with the exterior (q ) is supposed equal
to zero.
Starting from the above assumptions, the bubble growth model
is defined by the following equations set:
dR
dm 6 af +
3
m
R
Pg
d
d d t dt h. p 3 p dt g
fg g " 6 P. CD
T (2) dt m T--T dt g
f g *» 2 K K_ (T.-T ) r .
x C t) - 5 f
m f ? W (3)
where m is the thermal boundary layer thickness and af is the
fluid thermal diffusivity.
The model is coupled with two balance equations:
dp/dt = - pw (4)
(*) The authors also write another more general expression for "x" which is not taken into consideration here.
- 120 -
and
( 3p ^
The authors call this last equation a "choked flow condition",
but apart from coefficient C , it can be obtained directly from
balance equations.
In the bubble model, the void fraction is defined as:
a = j t R3 N P (6)
where p is simply given by
p = p + a p (7)
g fg
By further imposing to h_ , p and T to follow the satura-fg g g
tion line, the equations (1) through (7) constitute a closed set;
the unknowns are:p, w, p, R, x>ni, a.
The energy is implicitly conserved through equations (1)*(3),
which require the energy equation to be solved across the wall
surrounding the steam bubbles.
Only two empirical parameters are needed in this formulation:
N (number of bubble per unit mass of mixture assumed to be of the
order of 10 /kgm) and C (discharge coefficient assumed to be
about 0.75) *" J .
In the same work the authors present a single lumped phase e-
quilibrium model and compare the outputs obtained using each mo
del with some experimental pressure data. They show that it is
possible to have a good agreement between calculated and experi-
(x) In fact the authors show that the other empirical parameter "k " is a function of "N".
- m
mental results by appropriately adjusting C^ and N in the non e-
quilibrium model; equilibrium model yields unacceptable results.
The differences between non equilibrium model and experimental da
ta are attributed mainly to two dimensional effects, liquid iner
tia influence, bubble growth and limitations in the choked expression.
No result, concerning the flowrate values is given.
3.3.4.25 - ROMANACCI 1976 753/
We present, finally, this simple model describing the medium
trend of depressurization from a vessel of volume Œ, in order to
show some analytical conclusions that fairly well agree with ex
perimental results.
The main assumptions are:
- pressure is uniform in each point of the volume ft;
- liquid and vapor phase are in thermodynamical equilibrium.
The balance equations are:
- l i q u i d mass conse rva t ion
dM„ CD IT- " G i ( 1 -V G. - G (1-x )
fg e v eJ
vapor mass conse rva t i on
dM g
d t = G. x . + G . - G x
i l fg e e (2)
- t o t a l energy c o n s e r v a t i o n
-^—(M-h+M h ) - ft J[Ç—G.h.-G h +Q d t f f g gJ d t l l e e x (3)
- volume conse rva t i on
- 122 -
-r—-(M, v. + M v ) = 0 dt £ £ g g
By combining these equations it results
h .v -h Vr + G
fg g£-h - Q G.
dp_=.J_J; dt M
h.v -h v.
Lg_fiLi-h.. Vj. 1 )
(1-x) hr dv. dhT £g £ £ _L v£ dp dp g
'fg
+ x h. dv
v dp I fg
dh — £ + v dp g
C4)
(5)
where the subscript "i" means inlet quantity and the subscript
"e" means outlet quantity.
From eq (5) the following can be observed:
1) let us assume Q =0, G.=0, keeping all other variables con-
stant: '
if x =0 it follows dp/dt proportional to Gf v_
if x =1 it follows dp/dt proportional to G v e v r r g e g
Since v » v_, even if G <Gr, it generally results (see for g f g f
example Moody flowrates for x=l and for x=0) that the depres-
surization rate is greater in the case of flow from the vapor
zone than in the case from the liquid zone;
2) If Q>0 (heat introduction into the system) the depressurization
rate decreases, if Q value is such that numerator of eq. (5)
is greater than zero; the pressure increases notwithstanding
the mass loss ;
3) let us now consider the global pressure transient taking pla
ce in a vessel initially containing a mass M with quality x
o ^ o.
Since M decreases more quickly when the rupture is in the wa
ter zone with respect to the blowdown from the vapor-zone and
since it appears in the denominator of eq. (5), it follows
(still adopting Moody's results) that the pressure transient
- 123 -
is quicker when the break is in the liquid zone. This is in appa.
rent contrast with what said at point (1), but it must be remem
bered that we then studied the influence of break position only
in a short interval of time.
- 124 -
3.3.5 - Theories briefly described by Giot
/96/ 3.3.5.1 - SUDO-KATTO 1974
Katto (1968) and Sudo and Katto (1974) consider an equili
brium adiabatic liquid-vapor flow (Ah,, = Ah = 0) described by
a set of equations involving a mixture mass balance equation,
two one-phase momentum equations, a mixture energy balance equ<i
tion and a vapor energy balance equation. According to this au
thor the critical condition is produced as the limiting condi
tion of the variation of state which satisfies the fundamental
equations mentioned above, and the limit is determined as put
ting the cause of the variation of state to be zero, i.e. consi
dering that all the terms of the right side members of the equa
tion are zero. Then the necessary conditions for the coexisten
ce of the five homogeneous equations with four unknowns (p, x,
wf, w ) are the vanishing conditions of two determinants of the
fourth order. They can be solved to give w and w as functions
of p and x at the critical section. Comparing the results with
various data for 0.02 _< x < 1, they find a rather good agree
ment .
/100/ 3.3.5.2 - GIOT-MEUNIER 1968
A method analogous tb that of Fauske and Moody has been
tried by Giot and Meunier (1968) . It consists on writing the va
nishing condition of the set of the mass, momentum and energy
mixture balance equations, the four variables being p, x, r and
k. If one takes Fauske's critical slip-ratio, the resulting
- 125 -
flow-rates are very similar to those obtained by Fauske. On the
contrary, when Moody's critical slip-ratio is taken, the resul
ting flow-rates are lower.
/103,101,15/ 3.3.5.3 - MEUNIER 1969, GIOT 1970, GIOT-FRITTE 1972
Meunier and Fritte (1969), Giot (1970) and Giot and Fritte
(1972) have studied a model taking into account a mass mixture
balance equation, two one-phase momentum equations and an ener
gy mixture equation. The critical flow condition deduced from
this set of four equations enables to calculate the critical
mass velocity as a function of the slip-ratio for given pressu
re and quality. The result is illustrated in Fig. 3.87 for
freon 12 (curve l).0ne sees that the critical mass velocity
increases with decreasing slip-ratio; this result is in accor
dance with the experimental data showing an increase of mass ye
locity with a decrease of the L/D ratio. Fig. 3.87 also pre
sents results obtained by replacing, like Fauske, the energy e
quation by a simplified entropy equation (curve 2), and by re
placing, like Moody, the momentum mixture equation by a simply
fied entropy equation (curve 3). For the value of the critical
slip-ratio recommended by Fauske, curve 2 intersects the curve
given by Fauske's model, when k is allowed to vary. This also
applies to curve 3 and to Moody's model. The results given by
Fauske's and Moody's model are not very different because the
curve r-k is rather flat for the high values of the slip-ratio.
- 126 -
/104/ 3.3.5.4 - FLINTA 1973, FLINTA 1975' '
The studies o£ Flinta and al. (1973) and Flinta (1975) deal
with the analysis of the nucleation process which plays an im
portant role in this problem and significantly affects the crji
tical flow-rate.
/105/ 3.3.5.5 - BAUER 1976
A non-equilibrium axial two-phase flow model is developed at
EDF by Bauer and al. (1976) who make use of the three mixture
balance equations and an equation for the non-equilibrium quali
ty: x - x
3x ax eq 3t 3z 0
where 0 is a change of state delay. They find good agreement with
Reocreux's data, except that they find a so-called pseudo-critic
cal flow (see above) instead of a critical flow.
3.3.5.6 - STADTKE 1977
We shall now. mention the study carried out by Stadtke (1S77)
using the formulation of the theory of irreversible thermodyna
mics and analyzing the influence of the interfacial transfer
terms on the critical flow.
3.3.5.7 - SEYNHAEVE 1977/110/
Some aspects of the flow of subcooled liquid through orifi
ces have been discussed in preceding sections. Recent experi-
- 127 -
mental and theoretical studies have been made by Seynhaeve
(1977) who has shown that the critical section is located in
the tube downstream the orifice at a position below that of the
rupture of the jet. In this case, the homogeneous model predicts
the experimental values of the critical flow-rate with a preci
sion better than 10$. A consequence of this result is that, in
these conditions, the critical flow-rate depends upon the diam£
ter of the tube downstream the orifice. This particularity is /74/
also a conclusion of Ogasawara's study
- 129 -
3.4 COMPARISON AMONG SOME MODELS
3.4.1 - Generalities
Before drawing conclusions we wish to point out that not all
models presented in the bibliography have been taken into consi
deration throughout this work , on the other hand the most g<e
neral conclusions can be obtained from the above described thee)
ries.
A first comparison may be derived from the analysis of tables
3.1 and 3.II presented in paragraph 3.3.1. In particular it may
be observed:
1) probably in no engineering field we have such a discrepancy
of number and type of equations used to solve the same pro
blem (precisely the critical two-phase flowrate);
2) the importance attribued to the physical aspects, mentioned
in part B of table 3.1, is different from author to author;
3) very few authors give as output of their theories quickly
comparable results: i.e. there is not a generally accepted
format to show the results. Such format could be, for exam
ple, a diagram showing r vs p with h as parameter or r
vs p with x as parameter; ' e e ^ '
4) the p h i l o s o p h i e s of approach to the problem and the methods
of mathemat ical s o l u t i o n a re in number more or l e s s equal to
the number of models;
5) most models p r e s e n t e d d o n ' t have an e x p l i c i t expres s ion for
the f l o w r a t e : from the few cases in which the a n a l y t i c a l ex
(x) With regard to this we think that about two tenths of models are lacking in this analysis.
- 130 -
pression for r is given the difference in results among the
models easily follows from table 3.II; such a fact also re
sults from figures 3.2 to 3.86 reported in this work;
6) the uniqueness in development of each physical model leads
to the incompatibility of the various existing two phase thee)
ries: i.e. it is not possible starting from a general theory
and making appropriate assumptions to arrive at all existing
theories.
From the above points,among other things, the difficulties
when comparing the models easily appear. Moreover we think it is
useless to show that a model predicts a flowrate greater than
another at low reservoir quality and, in case, it predicts an op
posite behaviour at high quality, since each model is generally
considered valid in a certain zone of the Mollier diagram and
since such a comparison doesn't reach any concrete conclusion
with regard to the absolute validity of the model itself. Not
withstanding this statement we will show some differences among
the theories in the next paragraph,
A better procedural method for the comparison of the models
could be to take as reference one or more experimental results
in which all the measured quantities (especially flowrate) are
clearly given with the respective uncertainties, and then to a£
ply the models in question. Unfortunately it has not been poss^
ble to perform such work also for lack of time.
3.4.2 - Quantitative comparison among different models
We have already reported some figures showing the comparison
among two or more theories (see for example Fig. 3.60 and 3.61).
- 131 -
/115,27,28,55,etc./ , - , , Many researchers have performed such a
work; some results are shown in figures 3.87 through 3.93. The
main conclusions drawn are as following:
- the quantitative differences among the results obtained by s^
me models (see Figs. 3.60, 3.61, 3.89) are unacceptable from an
engineering point of view;
- the worst discrepancies are related to low quality region and
to low values of L/D ratios (L/D <_ 8);
- sometimes theories give very different results when varying a
parameter which is not clearly chosen by the authors; the slip-
ratio is one of such quantities: figures 3.88 and 3.90 show
its influence on critical conditions.
Many other observations analogous to the above may be obta_i
ned by analyzing all the figures presented in the work, but
this is useless without the support of experimental data; more
over it is difficult to individuate the causes or the quantities
leading to such discrepancies.
When studying the most evolute models (paragraphs 3.3.4.17
and following) the difficulties of a comparative analysis are e_
ven greater both for the lack of standard results in terms of
flowrate, and for the great number of variables involved in ca.1
culations.
Finally it may be interesting to mention the type of compare
son adopted by Wallis and by Hall who studied the be
haviour of different choked theories over a whole blowdown tran
sient. In particular Hall studied the output of calculations
performed with five different models (see paragraph 3.3.4.21)
both in terms of flowrate and of exit pressure and he concluded
that none of them was satisfactory from both points of view. In
- 132 -
Fig. 3.92 and 3.93 we report the results obtained by Wallis com /122/
pared to experimental values measured by Hutcherson
- 133
3.5 CONCLUSION
The general conclusions of this SOAR will be drawn later in
time, together with other subgroup members, taking also into ac:
count of the other chapters. Either the proposal for the deve
lopment of a new model either the recommendation for the use of
an existent model too, is task of all subgroup members. In the
two next paragraphs we will show the main conclusions of some
authors and a brief discussion about the results of this chap
ter.
3.5.1 - Main conclusions of some authors
OGASAWARA_1969/72/
- "Eigenvalue method defines the criticality as the disconti-
nous condition to study flowdifferential equations system and
can treat easily such a complicated system composed of sepa
rate momentum between two phases, energy and mass conserva
tion equation".
- "Critical condition with energy conservation gives a more a£
curate solution than the one with entropy conservation".
MOODY_1975/9/
With reference to theoretical predictions of the two-phase
equilibrium discharge rate from nozzles of a pressure vessel he
says that:
- "Theory which predicts critical flow data in terms of pipe exit
pressure and quality severely over predicts flowrate in terms
of vessel fluid properties. This study shows that the discre-
134 -
pancy is explained by the flow pattern".
In his conclusion, Moody discusses of many phenomenological
aspects influencing flowrate.
MALNES 1975 784/
- "Critical conditions in a one-dimensional sense are not obtai_
ned in two phase flow, due to two dimensional effects".
- "Critical mass flow may, however, be calculated from the one
dimensional continuity equation utilizing a flashing correla
tion as mass flows are not sensitive to outlet conditions".
- "Gas content seems to be an important parameter in critical
two phase flow".
ARDRON-FURNESS 1976 755/
After comparing different theories (see figures 3.60, 3,61,
3.89) with experimental data these authors conclude as:
- "There is an absence of a general critical flow theory which
satisfactorily describes observed effects of outlet geometry
and discharge flowrates, and can thus be confidently applied to
reactor blowdown calculations".
KROEGER 1976 732/
- "Treatment of the phase change front as a discontinuity simi
lar to the treatment of shocks in simple phase gas dynamics,
permits very accurate solutions".
BOURE -FRITTE-G10T-RE0CREUX 1976 729/
- "In studying two-phase flow, since the evolution of the mixtu
- 135 -
re is, in fact, a consequence of the transfers at the wall and
at the interface, it is more rational to postulate transfer laws
than to assume mixture evolution".
- "The mathematical form of the above transfer laws is of a pri
mary importance and it is proposed to allow for the presence
in the transfer terms of partial derivatives of dependent va
riables" .
TENTNER-WEISMAN 1978 /10,27/
- "A simple fluid model incorporating slip and coupled to the
method of characteristics appears to provide a useful techni
que for analysis of two-phase flow behavior";
- "Choked flow in long pipe lines can be predicted from a homo
geneous equilibrium model" (many other authors agree with this
observation); "discrepancies are observed when this model is
used to compute pressure at pipe exit".
- "Thermodynamic non equilibrium at the exit, rather than high
slip ratios,>may cause inconsistencies arising in the vicini
ty of a break".
- Finally they require for carefully controlled experiments in
order to check proposed models.
MOESINGER 1978 793/
- "The essential blowdown quantities, namely flowrate and pressu
re history are described enough well by drift flux approxima
tion".
- "Phenomena which are only poorly described by the drift flux
model, like phase separation, and the behaviour of the sepa-
- 136 -
rated phases during the very transient stage of the blowdown,
play an inferior role because they do not influence the beha
viour of the overall blowdown process".
- "For other problems where phase separation and the flow fields
of the simple phases are really of interest, a six equation mo
del, which takes the inertia of the droplets or bubbles into
consideration should be used".
TRAVIS-RIVARD-TORREY 197 9 7130,116/
- "The success in comparison experimental and theoretical data
has istilled confidenc
vapor production model
has istilled confidence in the predictive ability of the new /116/„
WINTERS-MERTE 197 9 750/
- "The non equilibrium model predictions were considerably more
accurate than those of the phase equilibrium model, although
consistent discrepancies were observed in all model data com
parisons".
- "It is believed that two dimensional effects near the pipe
exit, liquid inertia influences on bubble growth, and limita
tions in the choked flow model are primarily responsable for
the discrepancies".
RANSOM-TRAPP 1980 777/
- "Comparison of the RELAP5/M0D "0" calculations with the Mar-
viken III Test 4 provided a good evaluation of the ability of
a two-phase thermal-hydraulic model to correctly predict mass
discharge rates under choked flow conditions at large scale;
this both in the subcooled and low quality flow regime".
- 137 -
WALLIS_1979' '
In this recently published work Wallis compares different
theories with experimental data. His main conclusions are:
- "The HEM is not a bad way of predicting critical flow in long
pipes where there is sufficient time for equilibrium to be a-
chieved and when the flow pattern is conducive to interphase
forces that are adeguate to repress relative motion. Errors
can be large for short pipes (a factor of 5) and significatj^
ve in longer pipes (factor of 2J if the flow regime, such as
annular flow, allows large differences in phase velocities".
After having analyzed theories considering nucleation process
Wallis concludes:
- "Although various authors make various assumptions about the
source of bubbles, none is able to escape entirely from empi-
rism: it is either blatant or hidden somewhere in the alge
braic derivations ; the numbers are chosen to correlate the re
sulting critical flow behaviour rather than to represent mea
sured nucleation characteristics. Thus the increase in rea
lism gained by incorporating this process into the analysis
is largely offset by uncertainties about the quantitative m£
chanism involved".
He achieved a similar conclusion by analyzing vapor genera
tion models. With regard to "two fluid"models (six equation mo
dels] he says that this is undoubtly the best approach to the
problem even if some questions (as the transfer laws) remain to be
established. Moreover he points out that attempts to reduce the
full two fluid model to a smaller number of equation are not, g£
nerally, a good job .
(x) On this subject he says that drift flux approximation approach is ill-advised.
- 138 -
Finally Wallis concludes:
- "I believe it is a good rule that the sophistication of a the£
retical analysis should match the degree to which the physical
phenomena can be specified".
3.5.2 - Results and discussion
Before concluding we confirm our agreement with Wallis's con
clusions, reported in the preceding paragraph.
Two aspects immediately result from the review of about sixty
different theories and many other works concerning the two-phase
flow:
- the great difference in formulation and results we have alrea
dy spoken about;
- the fact that nearly all the authors well compare their theo
retical results with the given experimental data.
With regard to the differences about the theories formulation
and the results obtained the main cause is due to the complexi
ty of the phenomenon studied and the lack of reliable experimen
tal results. With particular reference to this aspect,we can net
te that almost all researchers feel the need for a deeper experi
mental understanding of the variables involved in critical two-
-phase flow.
In these last four years many relatively advanced theories
have been published (see, for example the references 10, 32,29,
90,93 and 131 of the bibliography) but their formulation is far
from being unique.
A problem common to most of them concerns the analytical mo
del used to describe vaporization rate.
- 139 -
At this point it may be interesting to distinguish two possi
bilities of approach to critical two phase flowrate evaluation:
a) an engineeristic-computational approach;
b) a physical-mathematical approach.
In case a) the aims of each models should be the following:
to fit the experimental data as much as possible (eventually by
adopting also empirical coefficients and analytical conditions
as in par. 3.3.1.)and to achieve maximum analytical simplicity.
In case b) instead it is necessary to strictly analyze the
phenomena and to develop equations systems which enable the en
gineer to evaluate the main aspects of two phase flow dynamics,
without any further reference to experimental results. The cri-
ticality condition must be implicit in these systems.
In the bulk of the examined theories, the two approaches are
considered simultaneously.
Moreover we believe that from a fluid-dynamic point of view a
synthesis of the whole LOCA in a LWR, also considering the ef
fects due to the vessel's internal geometry on critical two
phase flowrate,cannot at present be handled from a mathematical
point of view.
Therefore approach a) seems useful.
On the contrary, a more simple situation, at least like the
one shown in Fig. 3.1, should be the aim of a theory adopting
approach b), With regard to this we think that for a complete
knowledge of the whole blowdown phenomenon it is necessary to
calculate both the istantaneous value of the pressure in the ves
sel and the flowrate. Such a theory must consider (apart from
the aspects we have spoken about in the introduction):
1) dynamics of rarefaction wave generated at the break;
- 140 -
2) acceleration of the two-phase mixture;
3) flashing phenomena in the pressure vessel subsequent to the
arrival of the depressurization wave.
BIBLIOGRAPHY
.
- 143 -
LIST OF ABBREVIATIONS USED IN THE BIBLIOGRAPHY
AIChE American Institute of Chemical Engineer
ANS American Nuclear Society
BNES British Nuclear Energy Society
E Engineering
EN Energia Nucleare
EP Energie Primaire
FMsr Fluid Mechanic soviet research
HT Heat transfer
HTjr Heat transfer Japanese research
U N Istituto Impianti Nucleari - Université di Pisa
IJHMT International Journal of Heat and Mass Transfer
IJMF International Journal of Multiphase Flow
JBE Journal of Basic Engineering
JFE Journal of Fluid Engineering
JHT Journal of Heat Transfer
JSME Japan Society of Mechanical Engineering
LT La Termotecnica
NED Nuclear Engineering and Design
NS Nuclear Safety
NSE Nuclear Scientific Engineering
NT Nuclear Technology
PHT Progress in Heat Transfer
PNE Progress in Nuclear Energy
T Teploenergetika
TANS Transactions of American Nuclear Society
- 144 -
/l/ SIHWEIL I.S.
"General structural design criteria for PWR contain
ment-interior structures"
NS, 1977, n. 13
IH ARDRON K.H.,BAUM M.R.,LEE M.H.
"On the evaluation of dynamic forces and missile ener
gies arising in a water reactor LOCA"
BNES, 1977, n.l
/3/ MOODY F.J.
"Time dependent pipe forces caused by blowdown and flow
stoppage"
JFE, 1973
IM ADACHI H.
"Theory of axial change of flow parameters and critical
flows" (I)
HTjr, 1973
/5/ ADACHI H.
"Flashing flow through a converging-diverging nozzle" (II)
HTjr, 1973
/6/ ADACHI H.
"Theory on critical flow in constant flow area channel"
(III)
HTjr, 1974
- 145 -
PI ADACHI H.
"Two-phase discharge coefficient of large, sharp-edged
orifice" (IV)
HTjr, 1974
/8/ BURNELL J.G.
"Flow of boiling water, through nozzles, orifices and
pipes"
E, 1947, n. 164
/9/ MOODY F.J.
"Maximum discharge rate of liquid-vapor mixtures from
vessels"
'Non equilibrium two-phase flows', Ed. Lahey-Wallis, ASME
New York, 1975
/10/ TENTNER A., WEISMANN J.
"The use of the method of characteristics for examination
of two phase flow behaviour"
NT, 1978, n. 37
/11/ ZIVI S. M.
"Estimation of steady state steam void fraction by means
of the principle of minimum' entropy production"
JHT, 1964, n, 86
/12/ LEVY S.
"Prediction of two phase critical flow-rate"
JHT, 1965, n.87
/13/ MOODY F. J.
"Maximum flow-rate of a single component two-phase mixture"
JHT, 1965, n. 87
•
- 146 -
/14/ ALLEMANN R.T. et. al.
"Experimental high enthalpy water blowdown from a vessel
through bottom outlet"
BNWL-1411, Battelle North West Lab., Richland, Washington
/15/ GIOT M., FRITTE A.
"Two phase two and one component critical flows with the
variable slip model"
PHT, 1970, n. 6
/16/ TREMBLE G. D., TURNER W.J.
"Characteristics of two-phase one component flow with slip"
NED, 1977, n. 42
/I7/ TENTNER A.M., WEISMANN
"Characteristic equation for a single fluid model incor
porating slip"
TANS, 1976, -ii. 23
/18/ ZALOUDEK F.R.
"The critical flow of hot water through short tubes"
Hanford Atomics Products Operations, Interim Report, n.
HW 77594, 1963
/19/ SCHROCK V.E., STARKMANN E.S., BROWN R.A.
"Flashing flow of initially subcooled water in convergent-
divergent nozzles"
JHT, 1977, n. 99
/20/ HENRY, FAUSKE H.K.
"The two phase critical flow of one component mixtures in
nozzles, orifices and short tubes"
JHT, 1971, n. 93
- 147 -
/21/ VIGNI P.,GAROFALO C , ROSA U.
"Esperienze preliminari sull'efflusso rapido di miscele
acqua-vapore inizialmente alio stato saturo"
UN, 1973, RL 149(73)
/22/ VIGNI P.
"Contratto per ricerche sull'incidente di perdita del ré
frigérante, -relazione intermedia-"
UN, Atti dell'Istituto
/23/ VIGNI P., D'AURIA F., ROSA U.
"Esperienze di efflusso rapido da un recipiente in pres-
sione con strutture interne"
UN, 1978, RL 317(78)
/24/ VIGNI P., D'AURIA F.
"Forze di reazione prodotte da getti di fluido bifase in
condizioni di efflusso non stazionario"
UN, 1979, RP 344(79)
/2S/ TONG L.S., BENNETT G.L.
"NRC Water reactor safety research progress"
NS, 1977, n. 18
/26/ EDWARDS A.R., 0' BRIEN P.
"Studies of phenomenon connected with the depressurization
of water reactors"
BNES, 1970, n. 9
/27/ WEISMANN J., TENTNER A.
"Models for estimation of critical flow in two-phase systems"
PNE, 1978, n.2
- 148 -
/28/ GIOT M.
"Two phase flow in nuclear reactors'*
'Von Kàrmann Institute for fluid dynamics' -Lecture series
1978-5
/29/ BOURE J.A., FRITTE A.A., GIOT M.M. , REOCREUX M.L.
"Highlights of two phase critical flow: on the links be
tween maximum flowrates, sonic velocities, propagation and
transfer phenomena in single and two phase flows"
IJMF, 1976, n. 3
/30/ FAUSKE H.K.
"Contribution to the theory of two phase, one component
critical flow"
USAEC Report ANL 6633, 1962
/31/ HENRY R.E. 1970, NSE, n. 41
/32/ KROEGER P.G.
"Application of non equilibrium drift flux model to two
phase blowdown experiments"
BNL-NUREG-21506-R 1977
/33/ ZIMMERMANN M., STEIN K,, RUDIGER B.
"Der Grossbehalterprufstand zur Simulation der Drucken-
tlastung wassergekuhlter Reaktoren"
'Forschungsauftrag Nr. RS16. Battelle Institut E.V. Frank
furt am Main
/34/ EL WAKIL
"Nuclear heat transport"
International Textbook Company
- 149 -
/35/ SCHUMANN U. et al.
"Fluid structure interactions in PWR vessels during blow-
down-code development at Karlsruhe and results"
SMiRT 1979, B6/1
/36/ DIENES J.K. et al.
"Hydroelastic effects of a LOCA in a PWR"
SMiRT, 1979, B6/2
/37/ VERBIESE S. et al.
"MEL finite element analysis of water shells interactions
in the context of a PWR LOCA"
SMiRT, 1979
/38/ VEAH.W., LAHEY R.T, Jr
"An exact analytical solution of pool swell dynamics du
ring depressurization by the method of characteristics"
NED, 1978, n. 45
/39/ TRAVIS J. R., HIRT C.W., RIVARD W.C.
"Multidimensional effects in critical two phase flow"
NSE, 1978, n. 68
/40/ TAKEUCHI K.
"Hydraulic force calculation with idrostructural interac
tions"
NT, 1978, n. 39
/41/ SMITH A.V.
"A fast response multibeam X-ray absorption technique for
identifying phase distributions during steam-water blow-
down"
BNES, 1975, n. 3
- 150 -
/42/ LYNCH CF., SEGEL S.L.
"Direct measurement of the void fraction of a two-phase
fluid by nuclear magnetic resonance"
IJHMT, 1977, n. 20
/43/ PANA P., MULLER M.
"Pressure wave propagation in a LWR downcomer and resul
tant load on the core shroud due to rupture of a primary
coolant loop"
NED, 1976, n. 36
/44/ HARAYAMA Y.
"Calculated effect of radially asymmetric heat generation
on temperature and heat flux distribution in a fuel rod"
NED, 1974, n. 31
/45/ MOORE C.V.
"The design of barricades for hazardous pressure systems"
NED, 1967, n. 5
/46/ DITTMAR, MUNCACIU D,, THIES J,
"Steam hammer in main steam piping system"
SMiRT, 1977, F6/9
/47/ SHIMUZU T., SATO Y. , YOSHINAGU T.
"Non linear dynamic behavior of pipe restraint system"
SMiRT, 1977, F2/5
/48/ KASAWARA R.P., BUSHMAM F.H.
"Dynamic analysis of reactor coolant system under LOCA
conditions"
SMiRT, 1977, F4/1
- 151 -
/49/ REIMER R.M.
"Computation of the critical flow function, pressure ratio,
and temperature ratio for real air"
JBE, 1964
/50/ WINTERS W.S., MERTE H. Jr.
"Experiments and non equilibrium analysis of pipe blowdown"
NSE, 1979, n. 69
/51/ MARTIN C.S., PEDMANANABHAN M.
"Pressure pulse propagation in two component slug flow"
JFE, 1979, n.101
/52/ HENRIKSSON T., FREDELL J.
"A Theoretical and experimental study of the structural
responses to safety/relief valves discharge loads"
SMiRT, 1979, J2/4
/53/ ROMANACCI R.
"Incidente di perdita del réfrigérante: decompressione e
svuotamento del reattore"
'Lezione al Corso di Perfezionamento in Ingegneria Nucleare
dell'Università di Pisa', (1977)
/54/ BURNETTE G.W., SOZZI G.L.
"Synopsis of the BWR blowdown heat transfer program"
NS, 1979, n. 20
/55/ ARDRON K.H., FURNESS R.A.
"A Study of the critical flow model used in reactor blow-
down analysis"
NED, 1976, n. 39
- 152 -
/56/ LINNING D.
"The adiabatic flow of evaporating fluids in pipes of
uniform bore"
Proc. Inst. Mech. Eng., 1952, 18(2)64
/57/ KLINGEBIEL W.J.
"Critical flow slip ratios of steam-water mixtures"
Ph. Thesis, University of Washington, 1964
/58/ MANEELY D.J.
"A study of the expansion process of low quality steam
through a De Laval nozzle"
University of California, Livermore Report UCRL 6230, 1962
/59/ FALETTI D.W.
"Two-phase critical flow of steam water mixtures"
Ph. D. Thesis, University of Washington, 1959
/60/ FRIEDRICH H.
"Flow through single stage nozzles with different thermo
dynamic states"
Energie, 1960, n. 3
/61/ UCHIDA H., NARIAI H.
' Discharge of saturated water through pipes and orifices"
Proc. of Third Int. Heat Transfer Conf., 1966, n. 5
/62/ SOZZI G. L., SUTHERLAND W.A.
"Critical flow of saturated and subcooled flow at high
pressures"
GE Report, NEDO-13418, 1975
- 153 -
/63/ FAUSKE H.K.
"The discharge of saturated water through tubes"
Chem. Eng. Progr. Symp. 1965, Ser. 61
/64/ STARKMAN S., SCHROCK V.E., NEUSEN K.F.
"Expansion of a very low quality two-phase fluid
through a convergent-divergent nozzle"
JBE, 1964, n. 86
/65/ LIEPMANN H.W., ROSHKO A.
"Elements of gasdynamics"
John Wiley and Sons Inc., 1957
/66/ POGGI L.
"Termotecnica con element! di fluidodinamica"
Vallerini Editore, Pisa, 1971
/67/ LAHEY K.T. Jr., MOODY F.J.
"The Thermal-hydraulics of a boiling water nuclear
reactor"
ANS Monograph, 197 7
/68/ BABITSKIY A.F.
"Concerning the discharge of boiling fluids"
FMsr n. 4, 1975
/69/ MOODY F.J.
"Maximum two-phase vessel blowdown from pipes"
JHT, august 1966
- 154 -
/70/ CRUVER J.E., MOULTON R.W.
"Critical flow of liquid-vapor mixtures"
AIChE, 1967
/71/ OGASAWARA H.
"A theoretical prediction of two-phase critical flow"
JSME Bull. 1967
/7 2/ OGASAWARA H.
"A theoretical approach to two-phase critical flow
(3rd report, the critical condition including inter-
phasic slip)"
JSME Bull. 1969
/73/ OGASAWARA H.
"A theoretical approach to two-phase critical flow
(4th report, experiments on saturated water dischar
ging through long tubes)"
JSME Bull. 1969
/74/ OGASAWARA H.
"A theoretical approach to two-phase critical flow
(5th report, several problems on discharging of satu
rated water through orifices)"
JSME Bull. 1969
/75/ CASTIGLIA F., OLIVERI E., VELLA G.
"Sull'efflusso critico di miscele bifasi monocompo-
nente"
EN, 1979
- 155 -
/76/ WALLIS G.B., RICHTER H.J.
"An isentropic stream tube model for flashing two-
-phase vapor liquid flow"
JTH, 1978
111 I RANSOM V.H., TRAPP J.A.
"The Relap 5 choked flow model and application to a
large scale flow test"
EG&G Idaho Inc., Paper presented at the ASME Nu.Rea£
tor thermohydr. Topic Meeting 1980 Saratoga
/78/ RANSOM V.H., TRAPP J.A.
"Relap 5 progress summary analytical choking crite
rion for two-phase flow"
Idaho Report, 1978 n. CDAP-TR-013
179/ MOODY F.J.
"A pressure pulse model for two-phase critical flow
and sonic velocity"
JHT, 1969
/80/ D'ARCY D.F.
"On acoustic wave propagation and critical mass flux
in two-phase flow"
JHT, 1971
/81/ HENRY R.E., FAUSKE H.K., Mc COMAS S.T.
"Two-phase critical flow at low qualities. Part I:
experimental"
NSE, 1970
- 156 -
/8 2/ HENRY R.E., FAUSKE H.K., Mc COMAS S.T.
"Two-phase critical flow at low qualities. Part II:
Analysis"
NSE, 1970
/83/ KLINGELBIEL W.J., MOULTON R.W.
"Analysis of flow choking of two-phases - one compo
nent mixtures"
AIChE, 1971
/84/ MALNES D.
"Critical two-phase flow based on non equilibrium ef
fects"
Winter Meeting of ASME, dec. 1975, Houston
/85/ MALNES D.
"Discharge flow rates from breaks in tanks and tubes"
Inst, for Atomenergy n. SD-223, 1977
/86/ PORTER W.H.L.
"Rapvoid - A computer code for deriving the release of
two phase single component mixture through a complex
array of pipes and the resulting depressurization of
the discharge vessel"
UKAEA Report n. AEEW-M-1512, 1977
/87/ PORTER W.H.L.
"The thermodynamical properties and critical release
rate for Freon 12"
UKAEA Report n. AEEW-M-1478, 1977
- 157 -
/88/ PORTER W.H.L.
"A method for analyzing critical flow through vents
with complex ducting"
IME C271/79, 1979
/89/ PORTER W.H.L.
"A method for analyzing critical flow of steam-water
mixtures"
UKAEA Report n. AEEW-M1364, 1976
/90/ RIVARD W.C.
"Numerical calculation of flashing from long pipes u
sing a two-field model"
Las Alamos Report n. LA-6104-MS, 1975
/91/ ANDEEN G.B., GRIFFITH P.
"Momentum flux in two-phase flow"
JHT, 1968
/92/ AVDEEV A.A., MAIDAMIK V.N., SELEZNEV L.I., SHANIN V.K.
"Calculating the critical flowrate with saturated and
subcooled water flowing in a cylindrical duct"
. T, 1977
/93/ MOESINGER H.
"Assessment of a drift flow approximation for a strong
ly transient two phase flow"
Spec. Meet, on Trans, two-phase flow, Paris, 1978
- 158 -
/94/ MOESINGER H.
"Zweidimensionale numerische expérimente zur instati£
nâren zweiphasen-Wasser Str5mung am Beispiel der HDR
Blowdown-versuche mit DRIX 2D"
KfK Report, n. 2853, 1979
/95/ KATTO Y.
"Dynamics of compressible saturated two-phase flow"
JSME Bull. 1968
/96/ KATTO Y., SUDO Y.
"Study of critical flow (completely separated gas li
quid two-phase flow)"
JSME Bull. 1973
/97/ IVANDAEV A.I., NIGMATULIN R.I.
"Elementary theory of critical flow rate of two-phase
mixtures"
H.T., 1972
/98/ SUDO Y., KATTO Y.
"Mechanics of critical flow"
Theor. and Applied mech., Univ. of Tokio Press, 22,
67-78, 1974.
/99/ VIGNI P., D'AURIA F.
"Problemi relativi alla valutazione délie reazioni vin
colari durante un incidente di perdita di réfrigérante
in un impianto nucleare"
UN, RP 359(79), 1979.
- 159 -
/100/ GIOT M., MEUNIER D.
"Méthodes de détermination du débit critique en écou
lement monophasique et biphasique à un constituant"
E.P., 1968
/101/ GIOT M.
"Débits critiques des écoulements biphasiques"
Thèse Université Catholique de Louvain, 1970.
/102/ ALAD'YEV S.E.
"Two-phase flow with coalescence and break up of dro
plets"
FMSr. 1975
/103/ MEUNIER D., FRITTE A.
"Modèle à glissement variable pour l'étude des débits
critiques en double phase"
IIF, Com. II et VI,' 1969
/104/ FLINTA J., HERNBERG G., SIDEN L., SKINSTAD A.
"Blowdown through nozzles of different types"
Europ. two-phase flow group meeting, Brussel, 1973
,'105/ BAUER E.G., HONDAYER G.R., SUREAN H.M.
"A non equilibrium axial flow model and application
to loss of coolant accident analysis: the CLYSTERE
System code"
OECD Spec. Meet, on transient two-phase flow, Toron
to, 1976
- 160 -
/106/ CHEN P.C., ISBIN H.S.
"Two-phase flow through apertures"
EN. 1966
/107/ WALLIS G.B., SULLIVAN D.A.
"Two-phase air water nozzle flow"
JBE, 1972
/108/ TREMBLAY P.E., ANDREWS D.G.
"A physical basis for two phase pressure gradient
and critical flow calculation"
NSE 1971
/109/ STADTKE H.
"Two-phase critical flow of initially subcooled wa
ter through orifices"
Europ. two-phase flow group Meet., Grenoble, 1977
/l10/ SEYNHAEVE J.M.
"Critical flow through orifices"
Europ. two-phase flow group Meet., Grenoble, 1977
/111/ BANKOFF S.G.
"A variable density single fluid model for two-phase
flow with particular reference to steam water flow"
JHT, 1960
/112/ CULOTTA S., VOLPES R.
"Espansioni instazionarie in miscele bifasiche omog£
nee gas-liquido"
L.T. 1977
- 161 -
/113/ JONES O.C., SAHA P.
"Non equilibrium aspects of water reactor safety"
BNL-NUREG-23143, 1977
/114/ HUGHMARK G.A.
"Hold-up in gas liquid flow"
Chem. Eng. Proc. 62, 1962
/115/ WALLIS G.B.
"Critical two-phase flow"
EPRI work shop, 1979
/116/ RIVARD W.C., TRAVIS J.R.
"A non equilibrium vapor production model for cri
tical flow"
Los Alamos Report - LA-UR-79, 1058, 1979
/117/ REOCREUX M.L.
"Experimental study of steam water choked flow"
Proc. of CSNI Spec. Meet. 1976, Vol. 2
/118/ KARWAT H.
"Analytical simulations in the field of two-phase
flow: a promising scaling law for the interpretations
of experiments"
Brux. Meeting 1978
/119/ THEOFANOUS T.G., BOHRER T., CHEN M., PATEL P.D.
"Universal solutions for bubble growth and the influen
ce of Microlayers"
15th Nat. Heat Transf. Conf. San Francisco, 1975.
- 162 -
/120/ MOALEM D., SIDEMAN S. *
IJ HMT, 1973
/121/ HALL D.G.
"A study for critical flow prediction for SEMISCALE
Mod-1 loss of coolant accident Experiments"
Tree-Nureg-1006, EG&G, Idaho Inc., 1976
/122/ HUTCHERSON M.N.
"Contribution to the theory of the two phase blow-
-down phenomena"
AWL-75-82, 1975
/123/ OGASAWARA
"Ph. D. Dissertation"
University of Tokyo, 1967-5
/124/ GALLAGHER E.V.
"Water decompression experiments and analyses for
blowdown of nuclear reactors"
IITRI-578, p. 21-39, 1970, Illinois Institute of
Technology, Chicago, 111.
/125/ BENJAMIN M.W., GAND-MILLER J.
Transactions ASME, 63, 419, 1941.
/126/ HODKINSON B.
Engineering, London, 43, 629, 1937.
/127/ SILVER R.S., MITCHELL J.A•
"Discharge of saturated water through nozzles"
Trans.WE. Cst. Instr. Engrs, Shipbldrs, 62,51, 1945.
- 163 -
/128/ WALLIS G.B.
"One dimensional two phase flow"
Mc Graw-Hill Book Company, New York, 1969.
/129/ DURACK D., WENDROFF B.
"Relaxation and choked two-phase flow"
LA-UR 78-1212, 1978.
/130/ TRAVIS J.R., RIVARD W.C., TORREY M.D.
"Critical flow studies"
LA-UR-79-3066, 1979.
/131/ ARDRON K.A.
"A two-fluid model for critical vapour liquid flow"
IJMF, 1978.
/132/ ARDRON K.A., ACKERMAN M.C.
"Studies of the critical flow of subcooled water in
a pipe"
CSNI Spec. Meet., on Trans, two-phase flow, Paris,
1978.
/133/ WOLFERT K.
"The simulation of blowdown processes with conside
ration of thermodynamic non equilibrium phenomena"
Proc. of CSNI Spec. Meet., Toronto, 1976.
/134/ BECKER K.M. et al.
"An experimental study of pressure gradients for flow
of boiling water in vertical ducts"
AE - 86, 1962.
- 164 -
/135/ ZUBER N.
"Recent trends in the boiling heat transfer research"
AMR, Vol. 17 n° 9, 1964.
/136/ RIVARD W.C., TORREY M.D.
"K-FIX a computer program for transient, two dimensio_
nal, two fluid flow"
LA-NUREG-6623, 1977.
/137/ HIRT C.W., ROMERO N.C.
"Application of a drift flux model to flashing in
straight pipes"
LA-6005-MS, 1975.
/138/ HANCOX et al.
"Analysis of transient flow boiling"
Proc. 15 Conf. Heat Transf. - S. Francisco, AECL 1973.
F I G U R E S
m
o
Po , m 0,T 0
l
k-k
Fig. 3.1 - Reference scheme for this work
M
0.5-
1.0-
1.5-
2.0-
3.0-u0 0.2 0.4 0.6 0.8 1.0
gw Q*\N*
F i g . 3.2 - Liepman e t a l . (Par . 3 . 3 . 2 . 1 )
08
06
04
02
M<1
M=1
M>1
i ï 8 p (ARBITRARY UNITS )
Fig . 3.3 - (Par . 3 . 3 . 2 . 1 )
>! \ i
E
CO i _
•4-»
IB CD
RAYLEIGH LINE
arbitrary units
Fig. 3.4 - Perfect gas, flow from cylindrical duct. Ass. i) (Par .3.3.2.2.).
s
4->
ry
un
arb
itra
B-
FANNO LINE
/ /
arbitrary units
A
\
h
f * 0
qex=o
^ c
Fig. 3.5 - Perfect gas, flow from cylindrical duct. Ass. ii) (Par. 3.3.2.2.) .
r ( 2 g c P o p o )
Incompressible liquid
F i g . 3 .6 - (Par . 3 . 3 . 2 . 3 ) .
100.0 p
0.01
1689.5 kN/m , 1100 Btu/lbm
r e f 1(2.326) 105J/kg m
r „ 11000 tbm/sec-ft2
r e f (4882kgm/»ec-«n2
PRESSURE, p / p = 0.25 o ref
X 2.0 4.0 6.0 8.0 10.0 12.0
h / h , o r e i
Fig. 3.7 - Results from HEM (Par. 3.3.3.1
ioao —
o.i
p MOOIbf/in.2 I P r e £ * |689.5 kN/m 2 j
, f 100 Bto/ibm | \ e £ ° 1(2.326) 105J/kgm|
STAGNATION PRESSURE, p Q / p ref/ 0.25
2.0 4.0 6.0 8.0 10.0 12.0
h /h o r e t
F i g . 3 .8 - R e s u l t s from HEM ( P a r . 3 . 3 . 3 . 1 ) .
5000
4000
£ 3000
E -Q
2000
1000
'0 0.2 04 0.6
Fig. 3.9 - Results from HEM (Par. 3.3.3.1).
5000
- 4000
o CD
£ -Q
3000
2000
1000,
po=1000 p s
X0 = 1.52-10" hD=552 BT
s0 =0.753 Bl
ia 2
U/lbm
U/lbm'F
'0 0.1 Xe 0.2
Fig. 3.10 - Results from HEM (Par. 3.3.3.1).
7000
200
Fig. 3.11 - Resul
400 600 800 1000 Pe (psia)
;sults from Lahey-Moody, s =0,753 BTU/lbm °F 'ar. 3.3.3.2").
(1-X)
0.50
0.25
0
y y^y
sly AS
y ^^«--""^ 2
— — —
5.--" ^ ^ ^
3 " * » ^ ^ ^
I--
0 50 100 150 •è/p
AS,rr ,w r
1.0
0.5
0 200
F i g . 3 .12 - B a b i t s k i y 1973 ( P a r . 3 . 3 . 3 1 ) ( s - s ) / s =AS when x i s 2 ) w n s / w s = w r ; 3 ) r r = r n s / r s ; 4 ) ( 1 - x ) from e q . ( 6 ) ; 5 ) ( 1 - x ) from e q . ( 5 ) .
3) . c;i 1 c u l a ted from eq . ( 5) ;
240 w m
200
160
120
80
40
0,
3/
2 /
1
295 31
AS
0X)4
333 353 373 oK
Fig. 3.13 - Babitskiy 1973 (Par. 3.3.3.3). 1)(s0-s) when is calculated from eq.(5),
(kJ/kg°C); 2) w from eq. (2); _ _ 3) w from the equilibrium scheme [w(m/sec)J
c lbm
sec f t 2 jw» -
a 100 TOO 300 400 MO «C ?» KB WO 1000 11» SÎCC 1300 « « ^ ( B T U / l b m )
F i g . 3 .14 - Moody 1965 f P a r . 3 . 3 . 3 . S) .
P c(psia)
KKS
h (BTU/lbm)
F i g . 3 .15 - Moody 1965 ( P a r . 3 . 3 . 3 . 5 )
ri I M
c UJK»
lbm sec f t 2
| l i N
u.n
10 K*
-
-
f s
w ^
ï^ 1
*
- . . . 1 „ . . . . , 1 i
\\\
i • n « i u n i n i n i n IUO I U ?.« 22m 2:00 KM ax jot» L-JO
P c(psia)
F i g . 3 .16 - Moody 1965 ( P a r . 3 . 3 . 3 . 5 )
STAGNATION PROPERTIES
v s
ENTRANCE PROPERTIES
< x , >
r
• ; • /
'J ISENTROPIC u / ENTRANCE
STATION 1
LIQUID
VAPOR
EXIT PROPERTIES
< x 2 > r = r
STATION 2
F i g . 3 .17 - Moody 1966 ( P a r . 3 . 3 . 3 . 6 ) .
1000 3000
Po(psia)
F i g . 3 .18 - Moody 1966 ( P a r . 3 . 3 . 3 . 6 )
p .Cpsia)
F i g . 3 .19 - Moody 1966 ( P a r . 3 . 3 . 3 . 6 )
i i - 1 T
at Z)
VA
PO
— i 1 1 — i —
i i i i o o o
(eisd) od o o
< Z g CO
Z LU
O O tn
m
Kl
1
1 X
TU
RE
i
S
A
ii
i i — i — i
^ s ''S
Y jy
1 1 1 . . . .
—71—
jp^ 3-""""
0
i i i
v— -
-
-
1
§ (eisd)Od
o
o o m
r-LLI
o*;-\— _ l
< Z o ^ CO Z LU
2 O
S j Q
—~ CD
- * - » *— CT (/)
K I I 4->
vO O C7> i—1
O O
<« 1
o r^i
to
ci •H
- I —
o
1
LIQ
M i i
"T - f"
/ *2-
i i
1 1 1 1
^J^^^ ^ 5 _ —
i i j i
o o o (eisd) od
o o
§i=
g CO
I
Oatg Initial Press,, psiq o lo» (100-500) a lnte»med. (700-1200) A High (1200 1800)
-CÔ-L/D=40-
I Reqion
m
Fig ,
? 4 6 8 10 12 14 16
3 . 2 1 - Fauske 1964 ( P a r . 3 . 3 . 3 . 7 )
18 20 L/D
w4 r
eg •P
O 10 >\r
x> 10* r
10
. '
-
r
• ' r-'•
•
1 ' 1 '
X/ «V / ^ v
° \ / / J \ /
£^
1 i 1 •
r •,T T—» r -r-ri
*O~~*-~L//
1 . 1 . 1 . 1
«5"'T~T'"r
5 0 6 0 ^
. » i . >
i
70
• ! ' !
SO 90 i 1 . 1
' I
100,
1
• :
1
: •
-•
» 10 11 12 13
BTU/lbm-10" 2
F i g . 3 .22 - Fauske 1964 ( P a r . 3 . 3 . 3 . 7 )
.001 0.01 0.1 1.0 STEAM QUALITY, WEIGHT FRACTION
F i g . 3 . 2 3 - Levy 1965 ( P a r . 3 . 3 . 3 . 8 )
1000
P fpsia)
100
10
. I I I
"
-
• r
/ f
I i I
' ,
/ /
/
I
X=0.9
•
I
—r-r -
1 1
-rr
JLL
l i
i i
f ""
X-0 .4
...1
,1 l
,l.,
-
1 100 200 400 700 1000 2000 r, 4000
( l b m / s e c f t 2 )
F i g . 3 . 2 4 - L e v y . 1 9 6 5 ( P a r . 3 . 3 . 3 . 8 ) .
I3UU
ho
btu | Ibm
1000
700
400
100 1 01
I00r~
9or
80^
70
•p e
u.
ioV "
50V^
4oY
3(
THE ENDPOIN I 5 1
\ ~ ~ \ .
2oNr**^\s^
1 *sT\ T IS X=104 -^*-
02 !
* \ ^ ^ 2 0 0
/V&OC
1000
' 5 0 0 , *
« * , #
^ 50 <?*
^ = - 1 0 2 0
I 5 1'03 - 5 1 0 4 '
T f i bm/sec f t 2 )
CRITICAL / / POINT 3206 psia
F i g . 3.25 - Cruver e t a l . 1967 (Par . 3 . 3 . 3 . 9 ) .
2.8
-O 2.4
CO
Z 2.0
| 1 . 6
§ 1.-2 U-
Ô
CO
0.8
0.4
0 0
V H -
STEAM=WATER MIXTURE QUALITY = 0.1 PRESSURE =14.7 psia
8 Ï4 24 32 40 48 56 SL IP RATIO k
Fig. 3.26 - Cruver et a l . 1967 (Par . 3 . 3 . 3 . 9 ) .
jy< r
tO
o o T—
^ O c ifi E
o ^ D)
-* "—" n°
o
i n
to
• to m
u rt a, v—' CTl VO CTi i—1
cd !-. c3 S rt in rt bo O
CO C J
to
bo
to
to
to
*-< a s a.
en CT)
?-<
tn nj b û O i
t-~
m
bû •H Ci,
W\-
100 Po» a t a
F i g . 3 .29 - Ogasawara 1969 ( P a r . 3 . 3 , 3 . 9 )
1.0
_Pc Po 0.8
0.6
0.4
0.2
S— Pc W - 2
70
> -2 ata
0
'0 0.2 0.4 0.6 0.8 yn 1.0
F i g . 3 . 3 0 - Ogasawara 1969 ( P a r . 3 . 3 . 3 . 9 )
10.000
1000
HOM EQUIL PRESSURE
~ p s i a
~ 100 o
o _1 LU
> O t-
Z> o o <
10
10
0.1
i r
HOM FROZEN PRESSURE ps ia
0
1 J L 0 0.2 0.4 0.6 0.8
VOID FRACTION, a 1.0
F i g . 3 . 3 1 - Malnes 1977 ( P a r . 3 . 3 . 3 . 1 2 )
CO
s o <d-
o no
O C\J
O
O
CM
E o • v .
05 -SE
O.
3.3
. r.
3.
rt
»o 1 ^ en f - t
•H ^
Ad
ac
i
to #o
Fig
. 3
.
t . * s 'h O
CD CM
< 0
CM
E o
o *-; co o
— n il
o ©. o o Il II .
I l ' 1 •• ^ o O .H- S
5
c \
?i
|o > £ CD
00
o CO
o © CNJ
o CO
o un
o *r
o c*>
o CNJ
© T -
o >
•"N CM
fc U • " > •
0) . * w
Q.
<~>
3.1
2
3.3
(H
rt G, < w ^
to
rH
•H J3
<
Fig
. 3
.32 -
0
^
\<£
• ^ 5 0
7n
^ 9 0 _ _
.95
2 0
8» i
= § T= 0 .0
1 1 CRITICAL FL
A"
i i i i
4 0 p(kg-
OW
'cm 2 ) 6 0
Fig . 3.34 - Adachi 1973 (Par . 3 . 3 .3 .13 )
80
(kgfcmfjl
60
40
20
Po = 60. Xo=0.10
1 G =2 72 104kg/mZsec
V0 20 40 f. 60 .. . 2 btotal ( k g / c m J
Fig . 3.35 - Adachi 1974 (Par . 3 .3 .3 .14 )
20 4 0 P (kg/cm2)
Fig . 3.36 - Adachi 1974 (Par . 3 .3 .3 .14 )
t. Po = const
Xo = const
Fig . 3.37 - Adachi 1974 (Par . 3 .3 .3 .14 )
CD
1.5
1.0
0.5 bj D = 25mm
\ 70 mm 30mm
0 0.2 0.4 0.6 0.8 Xo 1.0
F i g . 3 . 3 8 - A d a c h i 1974 ( P a r . 3 . 3 . 3 . 1 5 )
CD; o
2.0
1.5
1.0
05
SHAR P EDGE D ORIF ICE
50 100 150 p 0 (kg/cm2 )
F i g . 3 . 3 9 - Adachi 1974 ( P a r . 3 . 3 . 3 . 1 5 )
10000-
u
e
lOOOr-
100 0.001 0 .01 0.1 1.0
F i g . 3 . 4 0 - Moody 1975 ( P a r . 3 . 3 . 3 . 1 b
0.001 0.01
Fig. 3.41 - Moody 1975 (Par. 3.3.3.16)
0.1 1.0
20
P (bar)
15
10-
R = 0
2.4465 2.4470 2.4475
S (kJ/kg °K)
F i g . 3 . 4 2 - C a s t i g l i n e t a l . ( l ' a i . 3 /) . .i . i i
i
60000
40000
20000
0-(.
, r
N
&
)
(kg,
i
/
2 Ti s e c J
' ! ' — i —
0.02
Ho = 1300 kl/kg
1200
-1100
~ 1000
— — 9 0 0
— — 8 0 0 - 7 0 0
600 - Ï — 5 0 0 ^ 0 a
0.04 X
Fig. 3.43 - Castiçlia et al. (Par. 3.3.3.1
p (kgm/m s e c J
80000 -
H0=1000
•1500
H0=2000 kJ/kg
F i g . 3 . 4 4 - C a s t i g l i a e t a l . ( P a r . 3 . 3 . 3 . 1
'cr. Tref
10
0.1
_ _ — — —
^ - - -
- k = 1
• k = (vg /v
«• •> * — ^ — —
rref = K p=1C
f ) l /3
— — ,
)00 Ibm/f )0 Ibf/
- H
~*~"».
t2sec
in2
P/pref-20
^.Jû^r
• ^
^ \
^ V
0.001 0.01 0.1
F i g . 3.45 - Tentner e t a l . 1978 (Par . 3 . 3 . 3 . 1 8 ) .
l iquid -vapor interface
vapor streamtubes ^> (including droplets)
Fig. 3.46 - Wallis et al. 1978 (Par. 3.3.3.19
liquid streamtu be
•1st streamtu be
2nd streamtube
3rd streamtube
I I
constant pressure lines
Fig. 3.46 bis) -Wallis et al. 1978 (3.3.3.19)
1.6
F i g . 3 .47 -
0.4 ' 0.8 1.2 Ap (MP a)
W a l l i s e t a l . 1978 ( P a r . 3 . 3 . 3 . 1 9 )
- - - : • : * * * % • • -
r ( k
29 )
nrrsec
30
20
10
°, ) 0.( )5 0.
A ^
1 o:
pj5^
5 Xo 0 2
- 3 . 5
- 2 . 1
- 0 . 7
0.2
Fig. 3.48 - Wallis et al. 1978 (Par. 3.3.3.19)
r
LO EQUILIBRIUM MACH NUMBER
C = 0 — C - 0 . 5
C - INFINITY
M = l . G
d
d ».
»».
to
0.0 0.2 0.4 0.6 0.8 a
1.0
F i g . 3 .49 - Ransom e t a l . 1978 ( P a r . 3 . 3 . 3 . 2 0 )
500
wc
(m/sec)
400
300
200
100
FROZEI
HEM
si MODE
* ^ ^
L
/
/
h / / / / / /
/ / / /
/ / / /
/ / / /
/ /
/ f
0 0.2 0.4 0.6 0.8 a 1.0
F i g . 3 . 5 0 - ( P a r . 3 .3 .4 . )
0.32
C
0.28
0.24
0.20
0.16
*
200 4Q0 600 800 1000 1200
Psat fpsia)
F i g . 3 . 5 1 - B u r n e l l 1974 ( P a r . 3 . 3 . 4 . 2 )
'0 5 10 10 CHAMBER QUALITY , %
20
Fig . 3.52 - Starkman e t a l . 1964 (Par . 3 . 3 . 4 . 4 )
i c \
STEAM WATER NO PHASE CHANGE,NO HEAT TRANSFER
Fig. 3.53 - Moody 1969 (Par. 3.3.4.57
Fig. 3.54 - Moody 1969 (Par. 3.3.4.5)
m
0.8 x 1-0
Fig. 3.55 - Moody 1969 (Par. 3.3.4.5).
Homogeneous phases.
a<vs
8000
^r 6000 u eu
E -D 4000
1000
1 — I i | i i.iil 1—I I I 11 H
PRESSURE po psia
0.001 0.01 Xo
F i g . 3 .56 - Henry e t a l . 1971 ( P a r . 3 . 3 . 4 . 6 ) .
% 0.9
0.7
0.5
—1
* • « • ,
1 1 — 1 — 1 PROPOSED HOMOGENE
p 0 =17.6 ps
MO OUS
>i
ia
r •
DEL
EQL
FRC
ILIBF
)ZEN
ÎIUM
- " —
0.1 0.5 Xo 1.0
F i g . 3.57 - Henry e t a l . 1971 (Par . 3 . 3 . 4 . 6 )
"-».
vi
V: ^-hyperbola-*
p a r a b o l a - ^ , / ^
Mlys ' M l /
Af
Fig. 3.58 - D'Arcy 1971 (Par. 3.3.4.7)
10'
r Ibm
ft2sec
4 10
10
sH
s5
10 0.001
Fig. 3.59
0.01 0.1 1.0
D^Arcy 1971 (Par. 3.3.4.7). r is referred to solution on" branchlof the hyperbola and Y2 to solution on branch 2 of the hyperbola (see Fig. 3.58).
F i g . 3.60 - Ardron e t a l . 1976 ( P a r . 3 . 3 . 4 . 8 )
fpo,62bar]
120-10 r
nrsec 100
80
60
40
20
0 15 30 I 45 ' 6 0 75 90 ! Po^barj
Fig. 3.61 - Ardi-on et a l . 1976 (Par . 3 . 3 . 4 . 8 )
Fig. 3.62 - Ransom et al. 1978 (Par. 3.3.4.9)
0.01 XEe
Fig . 3.63 - Henry e t a l . 1970 (Par . 3 . 3 . 4 . 1 1 ) .
•
* 4000
E xi
3000
.2000
Q "• 1000
C/>
< 5
1 ' 1 Pe, psia
^ 1 0 0
^ 5 0
— 7^»^ r
0.2 0.4 0.6 0.8 VOID FRACTION , CLe
1.0
F i g . 3 . 6 4 - H e n r y e t ;; 1 . 1D"0 ( P a r . 3 . 3 . 4 . 1 1
2 l 0.010 0.020 QUALITY, XEe
F i g . 3 . 6 5 - H e n r y e t n i . 1 9 " 0 ' a r 3 . 3 . 4 . 1 1
10-10'
ë 8 (/î
E
rj-> 6
a> (O
§ 4 o
<0
^ 2
O
SATURATED WATER DATA L/D=12,
4 0 ^ *
400 800 1200 1600 P0 (psia)
2000
F i g . 3 .66 - Henry e t a l . 1970 ( P a r . 3 . 3 . 4 . 1 2 )
10-10v
? 8
5«5L ™ - 6 *- o j , 0) s «
_ -O
o w 9
O
SATURATED WATER DATA
28.4
Po=1500psia
450
113
20 40 60 80 L / D
F i g . 3 .67 - Henry e t a l . 1970 ( P a r . 3 . 3 . 4 . 1 2 )
t—
< œ. LU Q£ S
LU
a: Q
_ i < O I—
Q£
O
0.9
0.8
0.7
O.b
0.5
0.4
1 ^-i 1 1— SATURATED WATER
I l Po ,= 450
r
psia
>000 psia
20 40 60 L / D 80
F i g . 3 . 6 8 - Henry 1970 ( P a r . 3 . 3 . 4 . 1 2 )
c<T 3500
O
8 ^.
E
£ 2500 •s.
LU
OC
<: 9 1500 < O
oc O 500
7 o=271c F v V
^~28S °F
SATU ÎATED
20 60 100 140 STAGNATION PRESSURE pQ(psia)
F i g . 3 .69 - Henry 1970 ( P a r . 3 . 3 . 4 . 1 2 )
LU 8. < Où
< O
oc O
103
7
6
5
4 ? 3.
I I I I I "I 1 T 1
_ SATURATED AND SUBCOOLED DATA.
_ P o = 4 4 1 p s i a diam = 0.157 in. L/D = 15
50 370 390
P0=441 psia
4' 0 430 450 STAGNATION TEMP,T0(°F)
F i g . 3 . 70 - Henry 1970 ( 3 . 3 . 4 . 1 2 )
400 800 1000 1400 1800 STAGNATION PRESSURE , P0(psia)
F i g . 3 . 7 1 - Henry 1970 ( P a r . 3 . 3 . 4 . 1 2 )
2000 r Ibm
sec ft2
1000
400
200
100
-D=0. (30 psia).
.0 2 .0 4 .1 .2 .4 x 1.0
Fig. 3.72 - Klingelbiel et al. 1971 (Par. 3.3.4.14).
103-100
r •c—g-j
m*sec
OU
•
0
ZT1 ! INF
'
ho=
1 I 1 I I
LUENCE OF hç ; a =
421 btu Ibm
A ' r
' /
f
463^
y
? r
~7
'A r 30
5-19 m3/m3
52E
50 100 po ( bar}
F i g . 3 . 7 3 - Malnes 1975 ( Par. 3 . 3 . 4 . 1 5 ) . (a=gas content at 25°C and lbar)
6000
r
rr^sec
4000
2000
0 J > t 6 Pe (bar)
F i g . 3 . 7 4 - Malaes 1975 ( P a r . 3 . 3 . 4 . 1 5 )
p (bar)
-0.5-
Tm= 11100 kg/m2sec
rc= 10350 » » a = 0.02 m3/m3
01 02 03 L (m)
Fig . 3.75 - Malnes 1975 (Par . 3 . 3 . 4 . 1 5 ) . (a = gas con ten t a t 25°C and 1 bar)
1 0 0 , 0 0 0
SLIP RATIO ' I O
SHORT PIPS
IO.OOO
o
< X
<
ipoo
STAGNATION PRESSURE a i p n a i
l i O O
IOOO
7 0 0
5 0 0
3 OO
2 0 0
I O O
IOO 2 0 0 4 0 0 6 0 0 BOO lOOO I 2 0 0
STAGNATION ENTHALPY ( B f u / l b )
F i g . 3 .76 - P o r t e r 1975 ( P a r . 3 . 3 . 4 . 1 6 )
_L HOO
1 0 0 , 0 0 0
IO.OOO
s o
V
ipoo —
too
/ 1
l\\
1
' 1 \
1 1
SLIP RATIO =
SHORT PIPE
a.' < v >
\ »—' X . <
^ N ^ I
1
1 1
M
1
STAGNATION
PRESSURE
fo fp. ro. ;
ISOO
" -IOOO
^ 7 0 0
" * " - 5 0 0
' 300
"*"-• 2 0 0
IOO
! 2 0 0 4 0 0 bOO 8 0 0 IOOO 1200
STAGNATION ENTHALPY ( B t u / l b J
H O O
F i g . 3 .77 - P o r t e r 1975 ( P a r . 3 . 3 . 4 . 1 6 )
1 0 , 0 0 0
1 , 0 0 0
3
<
too
IO
SLIP RATIO * I O
SHORT PIPE
< >
< STAGNATION
PRESSURE fo(rti.a)
ENVIRONMENTAL PRESSURE (p.i.i.a)
I 2 0 0 4 0 0 6 0 0 BOO IOOO 1200
STAGNATION ENTHALPY ( B f u / l b J
ISOO
I O O O
TOO
5 0 0
3 0 0
2 0 0
IOO
,JOO
F i g . 3 .78 - P o r t e r 1975 ( P a r . 3 . 3 . 4 . 1 6 ) .
io,ooo|
NOTE STATIC PRESSURE - 14-7 pj.i.a.
IMMEDIATELY SATURATED
LIQUID 60UN0ARY IS «ACHED
SLIP RATIO = k M
SHORT PIPE
l,ooo^
u <
IOO
STAGNATION
PRESSURE
p o (p i i . o )
iSOO
l O O O
7 0 0
5 0 0
J O O
2 0 0
I O O
ENVIRONMENTAL PRESSURE: I 4 - 7 p . i . i . o .
I O _L _L 2 0 0 4 0 0 6OO SOO IOOO 12 OO
STAGNATION ENTHALPY I B f u / l b l
F i g . 3.79 - P o r t e r 1975 (Par . 3 . 3 . 4 . 1 6 )
1400
100,000
SLIP RATIO "IO
LONG HPI
10,000 —
9 O
Î
5
1,000 —
STAGNATION
PRESSURE
P0 ( P l i a )
IJOO
IOOO
7 0 0
5 0 0
3 0 0
2 0 0
IOO
IOO ± 200 400 tOO BOO IOOO I200
STAGNATION ENTHALPY (Btu/lbJ MOO
Fig . 3,80 - P o r t e r 1975 (Par , 3 .3 ,4 ,16 )
1 0 , 0 0 0
1 ,000
I 0 O
2 0 0 4 0 0 6 0 0 BOO IOÔO IÎOO
STAGNATION ENTHALPY (Btu / lb)
MOO
Fig . 3.81 - P o r t e r 1975 (Par . 3 .3 .4 .16 )
(A
ID
<o
(D -t-> CO
o I
CM
o I
•
è
ir
J CO
o
CVJ
CO
£
i
J? u. u.
< O 10
S O O u-C0 </) O ) ^
i _
I
CM
o (0
a. in CO'
o CO'
«VI CO
•p to
to t/>
•H • > U rt at
^ a. H <--'
i
t o 00 to
oo r-» • fr> /—* r-H r-l
. « ^ •-I • ft Tf
• +-» to d) •
to t/>
•i-i • > u «3 d >-. a, I
rvi oo
• to
00 •H
.
10
r re f
r r e f = 1000 lbm/ft2sec
Pref= 100 psia
0.001 0.01 0.1
Fig. 3.84 - Tentner et al. 1978 (Par. 3.3.4.22)
SIX-EQUATION SYSTEM ORDINARY DRIFT-FLUF REFINED DRIFT-FLUX
30 40 time (ms)
10 20
F i g . 3 . 8 5 - M o e s i n g e r 1978 ( P a r . 3 . 3 . 4 . 2 3 )
50
120 Pe
(bar) 80
40 f**-I
S-Pi s
;at ( i n i t i a l )
10 20 30 40 50 60 time (ms)
F i g . 3 . 8 6 - Moes inger 1978 ( P a r . 3 . 3 . 4 . 2 3 )
O Il —1 X
CM
E z
p =
9810
0
-
FR
EO
N 1
2
A
i
/
CM/
MO
OD
Y
lOi
, ^_
JSK
E
/ / \ <
^ M _ M _ • > —
5
o CM
_ v
00 r—
to ~
^3" T_
CM T—
o
0 0
CD
CM o o o m
o C\J
O o o <3-
o o o 00
o o o CM
S-
O H
<v +J o
•H C3
I
00
dû •H
K
50
30
10
*jt
« $ ) *
10 20 30 40 50 D 60
F i g . 3 . 8 8 - A r d r o n e t a l . 1976 ( P a r . 3 . 4 . 2 )
- A r d r o n e t . a l . 1976 ( P a r . 3 . 4 . 2 )
1.0
p = 6 . 2 b a r ) ~x o
F i g . 3 . 9 0 - W a l l i s 1979 ( P a r . 3 . 4 . 2 )
i
Fig . 3.91 - w a l l i s 1979 (Par . 3 .4 .2 )
1 sec f t 2
4000
3000
2000
1000
M 11
HEM
L - MOODY
\ HENRY-FAUSKE \ y | |
m i N E THROUGH DA
H^-38^o ("initial rTidbs^ .1 remaining ' I I 1 a —so-
'A
1.0 2-°t(sec) 3"° F i g . 3 .92 - W a l l i s 1979 ( P a r . 3 . 4 . 2 )
Ic 8000
( l b m , ) secft*
6000
4000
2000
• N
LIN
• ^ *
ETH
I ' l l ' -HENRY-FAUSKE
ROUGH DATA
- H E M -"°i—
MOODY
^ *
* »N i »
12., 16 20 t(sec|
F i g . 3 .93 - W a l l i s 1979 ( P a r . 3 . 4 . 2 )
A P P E N D I C E S
- Al -
APPENDIX 3.1 : ABOUT THE CALCULATION OF PERFECT GAS SOUND VELOCITY.
We will show here that the sound velocity is the same as the
throat gas velocity obtained in the minimum section of a De La
val nozzle in par. 3.3.2.1.
In any matter the sound velocity is:
a - C^) * = (-v2 ) V2
dp v dv = A V 2 (Al.l)
where the derivative has to be executed along the supposed trans
formation line .
Particularly in an isentropic transformation we have:
a = (- v dp_ dv
V. ) CM. 2)
Let us consider now the continuity equation (1) of the para
graph 3.3.2.1 written as:
A = Gv Gv w L 2 c h
0 - h ) J 1/2 (A1.3)
which when s= const, is function of only "v". The minimum sec
tion will be found by:
dA dv
= 0 (A1.4)
this leads to
dh dv
2(h -h ) c o V
w c V
(A1.5)
- A2
1/ 2 , v d h , 2 v . ,
w = ( - — ) i A i . ( |
c dv
But when s= const, from a known thermodynamica 1 relationship
it results:
dh = vdp ( Al.7j
and substituting into (A1.6) we have:
2 - r 2 dn ^2 w = ( - v -jM c d v '
which is the same as equation (A1.2); then
w = a c
- A3
APPENDIX 3.2 : DEFINITION OF FUNCTIONS fx, f3, f4 OF PARAGRAPH
3.3.3.6.
£x = |_k(l - x ) v f + k v g J (x + - i - ^ ) (A2.i;
£_ = h + x h r (A2.2) 3 g fg
f4 = \ [ k ( l - x ) v f + x v g J 2 ( x + ^ — ) (A2.3)
A5
APPENDIX 3.3 : DEFINITION OF VAPORIZATION AND CONDENSATION RATES
(PAR. 3.3.3.16) PER UNIT PRESSURE REDUCTION.
[ m fg
( i s - ds —I (1-x) I T * Z x i^J
m gf
fg 1 + Z
p ds ds
T L'-dif + Z ^ d 7
fg 1 + Z
TA (A3.1)
(A3.2)
where (w - w )
Z = — K — 2h
fg (A3.3)
'•:i'vr;
- A7 -
APPENDIX 3.4 : DEFINITION OF FUNCTION q AND q2 RELATING TO PA
RAGRAPH 3.3.3.17.
? n f f T
«i = - T -5—ZZ 2 7 - (A4.1) r (w ° - w.-5) g f
H - h . _ 2/_ o x 3 / 3 , . . 0 .
q 2 = — - 2 2 q i Wf ( A 4 , 2 )
- A9 -
APPENDIX 3.5 : HUGHMARK CORRELATION (SEE PAR. 3.3.3.18).
The quality has the following expression:
'- = 1 - (1 - *) pg
(A5.1)
in which "w" is a non linear function of "Z", where:
Z = D r e i_ y_(l - a) + y a
C _2 ^ 8 (1 - x)p +
PgD iL__f
(1 - x)p g
(A5.2)
- All -
APPENDIX 3,6 : DISCUSSION OF EQ. (6) OF THE PARAGRAPH 3.3.3.20
With reference to the characteristic polynominal
AA - B = O (A6.1),
R The real part of any root A. gives the velocity of signal
propagation along the corresponding characteristic path in the I
space-time plane. The imaginary part of any complex root, X.,
gives the rate of growth or decay of the signal propagating
along the respective path.
In the case of equilibrium flow the first two roots are
2 y2 U l - c O P + f - L C ¥ - ) - a ( l - a ) p f p J } v
X = g ^ g ( A 6 . 2 ) K
2 ' V2
{ ( l - a ) p + ^ - L<-,1 - a ( l - c O p , p J } v (A6.3) i = & L i r & £
2
K = ( l - a ) p + pc + a p f CA6.4^
It is shown that these two roots have values between "w " g
and "w ". The remaining two roots are:
A, . = w + D(w - w _) - a (A6.5) 3,4 g f
where:
w = I apw + (1-alPj.w l/p (A6.6) u g g £ g^
- A12 -
— 2 — 2 / 2 a = a
H E { U C P + p C a p £
+ ( l - a ) p ) J / ( c p + p f p )} (A6.7)
__ — _ ^s ^s~_ (ap f - (l-oOpg) P g P f L U - a ) P f - a p g J 2 P ^ P ^ (I-OOP^J
pc+ ( l - a ) p + a p f P(PgPf+ CP ) P P f s
f
( A 6 . 8 )
The quantity a is the homogeneous equilibrium speed of
sound and is defined in paragraph 3.3.3.12.
- A13 -
APPENDIX 3.7 : DEFINITION OF THE PARAMETERS A AND X RELATED TO g f
PARAGRAPH 3.3.4.7.
Xf = 7 ^ L P C 1 - « ) J / -^LPf(l-a)J (A7.1)
p d d
x = _S ( 3 / _ (- ) CA7.2) g p dp ^ dp VKg
- A15
APPENDIX 3.8 : FORMULA TO OBTAIN SLIP RATIO (PAR. 3.3.4.13)
From the expression of the thrust (F) one obtains
k 2 - k
2 2 -F-(l-x) v„ - x v
f g x (1 - x) vr
V
+ -£ = 0 (A8.1)
where:
F = F + VPo-Pe)
Ar2 (A8.2)
The thrust is measured experimentally
*
- A17 -
APPENDIX 3.9 : DEFINITION OF THE STATE FUNCTION n AND OF THE sg
FRICTION FACTORS (PAR. 3,3.4.17)
n = 1,0666 +1 .02-10 _ 9 p-2 .548-10 _ 1 7 p 2
'sg F F
val id for p £ 2-107 Pa (A9.1)
-1] -19 ? n = 1.0764 +3.625-10 p - 9 . 0 6 3 ' 1 0 p^ 'sg v v
7 val id for p > 2'10 Pa
(A9.2)
vapor friction factor
- V 2 _ _V2
f = 1 .74-2 log I 2k /D+18.7 f / aw D/v I g 6 L s g g g J (A9.3)
liquid friction factor
J/2 ff = 1.74 - 21og [_2ks/V + 18.7 ff / (1 - a)w f D/v£ J (A9.4)
In this expressions "k " is related to the roughness of the
pipe so that k /D is the relative roughness.
A19 -
APPENDIX 3.10 : EVALUATION OF THE VAPORIZATION RATE FOR AVDEEV
MODEL.
The expression for vapour formation rate is:
dm fç.z) <J>
= l ^P- V dS+I(S)mfg,z) i D dz n (AlO.l)
The development of the author leads t o :
4 s ~2S rz & ^ =-TT{I(Z) r (z) p (z) + 3p (z) F(z). | I ( ç ) p \ 0 g 8
- V 3 _ [Z „ , . -%
r(0 +
+ P U (O I F(n) P^ ' (n) dn
2
dU
(A10.2)
In this formulas, other than the already known symbols:
1(C) = number of active vapour formation sites
Ç = cross section of incipience of vapour bubble
m = mass of bubbles in cross section z n
= 2a T / p (T.-T ) g g f g
= surface tension
P r C c T
f - y 8 Mf p h
fg
- A21 -
APPENDIX 3.11 : DESCRIPTION OF THE VAPOR GENERATION RATE FOR
THE MODEL OF TRAVIS ET AL. (PAR. 3.3.4.21).
After defining "S" as the interfacial area per unit volume
and "q" as the interfacial heat flux,a simple energy balance
shows that:
_ as * (All.l)
where "A" is the latent heat of vaporization.
By expliciting "q", this may be written as:
i> =
k(T.-T. J'S f fsat (All.2)
where "k" is the thermal conductivity and "£" is a length chara£
terizing the thickness of the thermal boundary layer over which
the liquid temperature changes from its interior, bulk value ( T ) ' ,
toT, . fsat
The authors give two expressions for I; one relating to non-
traslating bubble growing in an infinite fluid region (£ ) and
the other relating to bubble traslating with respect to the sur
rounding liquid with speed U (£ ). These expressions are:
I = r, c b
, p r c - T _ - T £ _f pf f f s a t IT p g
-1
V. / = r r ^
u bLRe Pr J
( A l l . 3 ) '
( A l l . 4 )
where r , i s t h e i n s t a n t a n e o u s b u b b l e r a d i u s u
„ • • . * i x /H9 ,120 / . . . x These expressions are taken from respect ively.
- A22 -
Moreover the authors define:
(All.5) 1 . =
I 1 1
= _, _ _ . _ l _ • - .
t I c u
The model still requires the definition of r, and il. b
It is shown that r, is a function of Weber number (assumed b
equal 4), and of the number of bubbles per unit volume "N" (as
sumed as parameter); it is also function of other dependent va
riables already appearing in the calculation or known quantities.
The authors define r, as the minimum of r and r where: b o u
r = i.-f—') ( A l l . 6) O 4 TT N
a W r = ^ ( A l l . 7)
u ? H 2 2 pf U
Finally u is assumed as:
u = | U I + 0 I U-. I ( A l l . 8)
where U is the relative speed between phases and U is a mass r ^ r M
average mixture velocity and 6 is a parameter to take into ac
count primarily for turbulence. :
About the question of determining ^, see also ref.
Related Documents
![[INIS] Earthquake resistance of nuclear power plantsf-archive.jaea.go.jp/Reference_information/files/file/...[INIS] Earthquake resistance of nuclear power plants No. INIS Ref. No.](https://static.cupdf.com/doc/110x72/5eaf9c39fd2b5e73cc08868b/inis-earthquake-resistance-of-nuclear-power-plantsf-inis-earthquake-resistance.jpg)