A translation from

Atomic Energy of Canada Limited

CALCULATION OF HEAT-TRANSFER CRISIS FOR ANNULAR

TWO-PHASE FLOW OF A STEAM-LIQUID MIXTURE

THROUGH AN ANNULAR CHANNEL

by P.L. KIRILLOV and I.P. SMOGALEV

Translated from report FEI-297 of the Physics and

Power-Engineering Institute, Obninsk, USSR, 1972.

Translated by GEOFFREY PHILLIPS

Atomic Energy of Canada LimitedChalk River, OntarioFebruary 1974

AECL-4752, - ^

Calculation of heat-transfer crisis for annular two-phase flowof a steam-liquid mixture through an annular channel*

P.L. Kiriilov and I.P. SmogalevPhysics and Power-Engineering Institute

Obninsk, USSR

Translated by Geoffrey Phillips

TRANSLATOR'S RESUME

In 1969, the Physics and Power-Engineering Institute,Obninsk, issued Report No. FEI-181, by the same authors,entitled "Calculating the heat-exchange crisis for a vapour-liquid mixture on the basis of a droplet-diffusion model".

The present report extends that work from simple circulargeometry to channels of annular cross section.

The authors claim that annular channels, while of fairlysimple configuration, permit verification of a number of effectscharacteristic of more complex forms (bundles, etc.), and arealso of interest per se in view of their practical application.

Two cases are considered, namely: i) heat transfer takingplace at the inner surface only; ii) heat transfer taking placeat both surfaces.

EDITOR'S NOTE - In view of apparent inconsistencies in themathematics of the original article, videTranslator's Critique, Atomic Energy ofCanada Limited does not bear responsibilityfor the contents of this paper.

Chalk River Nuclear LaboratoriesChalk River, Ontario

February 1974

AECL-4752

* Translated for Atomic Energy of Canada Limited from reportFEI-297 of the Physics and Power-Engineering Institute,Obninsk, USSR, 1972.

Calcul des crises de transfert thermiquedans l'écoulement annulaire biphasé d'un mélangevapeur-liquide passant dans un canal annulaire*

par

P.L. Kirillov et I.P. Smogalev

Institut de Physique et d'Ingénierie énergétiqueObninsk, URSS

*Traduction du russe en anglais par M. GeoffreyPhillips, pour l'EACL, du Rapport FEI-297 del'Institut de Physique et d'Ingénierie énergé-tique, Obninsk, URSS, 1972.

Résumé du Traducteur

En 1969, l'Institut de Physique et d'Ingénie-rie énergétique (Obninsk, URSS) a publié le RapportN° FEI-181 rédigé par les mêmes auteurs et intituléCalcul de la crise d'échange thermique pour un mélangeliquide-vapeur sur la base d'un modèle de diffusionde gouttelettes".

Le présent Rapport fait passer cette étudede la géométrie circulaire simple a des canaux desection annulaire.

Les auteurs affirment que les canaux annulaires,bien que de configuration assez simple, permettent devérifier un certain nombre d'effets caractéristiquesde formes plus complexes (grappes, etc.) et qu'ils sontégalement d'un intérêt en soi, par suite de leurapplication pratique.

Deux cas sont considérés, à savoir: i)transfert thermique n'ayant lieu que sur la surfaceinterne; ii) transfert thermique ayant lieu sur lesdeux surfaces.

No_te de l'éditeur - Par suite des contradictions semblantexister dans les mathématiques durapport original (cf. la Critique duTraducteur) l'EACL ne peut assumeraucune responsabilité quant aucontenu de ce rapport.

L'Energie Atomique du Canada, LimitéeLaboratoires Nucléaires de Chalk River

Chalk River, Ontario

Février 1974AECL-4752

- 1 -

TABLE OF CONTENTS

Notation: 2

Introduction: 9

1. Crisis with unilateral heat supply: 9

2. Crisis with bilateral heat supply: 15

3. Comparison with experimental data and discussionof the results: 17

Conclusions: 19

Translator's Critique: 20

Bibliography: 23

Figures: 25

Table: M

Postscript: 35

- 2 -

0 T A T I 0 N

symbol d e f ined in [ 7 J , p . 2 . (24) a

subscript adiabatic

atm unit atmosphere (absolute)

symbol defined in [7J, p. 2.

symbol coefficient of term in squar*brackets in (15), vide (15J-)

$62)

(24+)

(16)

ata

symbol defined in [7], p. 2.

term

term

= K, , vide [3 ] , (19)

--K. 3 vide [3] , (23)

(24)

(12b)

(126)

am urn t centimetre CM

cv subscript critical KP

differential operator

d symbol diameter

d, termhe

heated equivalent diameter,defined in [7], p. 1.

§19

(251)

u

Key to columns:

19

34

Translator's symbol;category;meaning and remarks;dimensions;where enaountered t;Authors' symJiol.

Order of notation:

Latin letters;Greek letters;other symbols.

.j. § para.; $ figure;1i table; ( ) equation,

- 3 -

d term hydraulic equivalent diameter, L §20 ddefined in [7], p. 1. (mm) r

(Cf. vh J vide [4], p. 81)

dep /subscript deposition (fall-out) Bblfl

E symbol fraction of liquid entrained — (9a) £

in core stream

•f function designator r

f subscript film (8) FIJI

G symbol specific mass flowrate, —^— (Mb) ywhereinafter called: ^ "mass velocity". (kg/m2secin Fig. 5; kg/m2hr in 4 & 6a)

g symbol acceleration due to gravity —j (12a) g(symbol ubiquitous in [4], (134-)in which mass occurs as W/g )

L

h symbol specific enthalpy, vide A?z^ —

he subscript heated equivalent, vide d^

hr unit hour

hy subscript hydraulic equivalent, vide d^

K symbol mass transfer coefficient — (5)

(m/hr in Fig. 6b) NB: £20.

kg unit kilogram

kgf unit kilogram-force

L symbol length as a dimension

- 4 -

1 2 3 4

I symbol heated length L

In natural logarithm designator (13a) In

M symbol mass as a dimension

m unit metre M

max subscript maximum max

min subscript minimum \ntn

mm unit millimetre MM

N unit newton §2M H

n subscript vide xn & f' §4 n

n} subscript vide zK (1) ni

nq compound vide n & a (7) nq

subscript

V symbol pressure {kgf/cm2 in Fig. 5; —g- paim in Fig. 6b: in former casenot clear whether pressure isgauge or absolute)

q symbol heat flux (W/^rr.2 in Fig. 5) ^y qT X

q subscript non-adiabatic (5) n

r1 symbol radius (mm in Table 1) L (11) p

5 symbol slip — (14) 4

6- subscript stable (12&) CT

- 5 -

sec umv second CEK

symbol time as a dimension

tpf subscript in V'^ , benchmark for theonset of annular two-phase flow

§5VK

symbol velocity

term friction velocity, vide [3],p. 4 & [4], p. 189.

(11)

(12M)

symbol weight as a concept, vide a

W unit watt

x symbol steam quality, dryness fractionor gravimetric steam content,vide the following terms;

xX

x term defined, within the frameworkof the present report, by (10)and (20+) for unilateral andbilateral heat supply,respectively. However, for thepurpose of (25) and (?6), thedefinition of P.G. Barnett [7]applies, vide (254-).

limx~ benchmark used in lieu of T \p (vide

Fig. 1 of present report) inanalogous Fig, 1 and Fig. 11of [l] and [17], respectively.

[16]

x term "Extreme value of steam content"ffor G < 2000 kg/mzsec)3 videTranslator's Critique.

§4(1)

x term "Extreme value of steam contentfor G 2 2000 kg/mzsec"

(1)(17)

x

- 6 -

x

x

a

r'

Ky

teim true steam qualitycorresponding to xn ,vide Translator's Critique.

term counterpart of the above formixture in core stream, videTranslator's Critique.

symbol defined in [7], p. 2.

symbol defined in [7], p. 2.

symbol mass flowrate

term critical mass flowrate of theliquid phase for:

unilateral heat supply:bilateral heat supply:

benchmark used improperly to mark end ofmicrofilm, i.e. point of filmdryout, observing that (9)indicates outer film extant at

t crisis. (Cf. x°lim )

symbol defined in [7], p. 3.

(31)

(33)

MT

MT

(25)

(25)

§5

(9)(20)

a

6

G

G'KP

(25)

Aftsub

Ap

prefix increment designator, simply

or in the compounds Aj & A2

term inlet sub-cooling (vide h)

subscript „ designator of section wherebenchmark microfilm begins, this

corresponding to hydraulic--resistance crisis, vide $2[16], in which x is theabscissa of the maximumvalue of Ap/Apo (q.v.).

(6)

(19)

(244-)(25+)

§5$1(1)

A

ApBX*

Used in (2) and (9a) in the sense of entry into the microfilm span<f E>- " Tr.

For insight into meaning of subscript n see italicizedtext between equations (5) and (6) concerning f^ . Tr.

- 7 -

1

Apterm

symboI

symbol

ratio of the pressure dropwith a steam-water mixtureto the pressure drop withwater at saturation temp.

(formulae for Ap and ApQ

given in [2], equations (1)and (2), respectively)

film thickness

defined in [7], p. 3.

§7

(25)

subscript „ in F , entry to test sectionbenchmark

§5

nBX

symbol defined in [7], p. 3. (25)

symbol hydraulic-resistance factor forsmooth tubes, in the formulafor the denominator of Ap/Ap(q.v.)y vide [14].

symbol fraction of total heatingsurface:

(3)(19)

symbol

symbol

symboI

symboI

latent heat of vaporization —2

dynamic viscosity

coined by authors of presentreport to facilitatesimplification of (264-) to (26)

kinematic viscosity

MLT

T

(10) ft

(12a) y

(26) u

symbol defined in [7], p. 4, (7). (26)

constant 3.14159...

symbol density

symbol surface tension

symbol shear stress, defined in (13)

MUM

MLT2

§9

(122?)

symbol true volumetric steam content,wide [15], needed to evaluatethe numerator of Ap/Aj?o (q.v.)3using [2], (1).

symbol fraction of liquid flowing asa film along outer surface ofannular channel when unheated,vide §22.

(15)

term fraction of liquid flowing asa film along outer surface ofannular channel when heated

(22)

il 10 defined in (14a) & --19 (14)§19

subscript inner surface of annular channel

subscript outer surface of annular channel

superscript liquid phase, in film or dropletform, i.e. the liquid content ofthe mixture as a substance,regardless of geometry.

superscript steam phase, i.e. the dry-steamcontent of the mixture as asubstance, regardless of geometry,

ovcrdot designator of terms which, in thetranslator's opinion, apply tothe core stream only, i.e. whichexclude the film or films.

- 9 -

I N T R O D U C T I O N

§1. Latterly, definite progress in analytical description of heat

transfer crisis phenomena for two-phase-mixture flow in tubes has been

achieved with the aid of the droplet-diffusion model. Detailed research on

droplet-mass-transfer processes will apparently soon permit the physical

basis of this theory to be so well developed t*hat it will soon become

possible to proceed from a semi-empirical approach to more rigorous

analytical calculations.

§2. Annular channels, while of fairly simple configuration, at the

same time permit verification of a number of effects characteristic of

more complex shapes (bundles, eta.). Therefore, calculations pertaining

to such channels will help to elucidate the nature of these effects, for

example the diverse character of the results of heat flux on surfaces in

the vicinity of the one under consideration.

§3. On the other hand, annular channels may be used in engineering,

so that formulae obtained are of direct interest.

1. CRISIS WITH UNILATERAL HEAT SUPPLY

§4. In bibliographical reference [l] we introduced the parameter xn,

namely the steam quality at which droplet deposition from the core stream

onto the channel wall deteriorates markedly or ceases completely, for

adiabatic flow of a steam-liquid mixture. According to contemporary ideas,

development of the flow throughout the length of a heated channel takes

place in the following manner (vxde Figure 1).

§5. Formation of the annular two-phase flow regime, with a wavy film

on the channel wall and liquid droplets in the core stream, takes place

from TE to 1 . From the point V^f to the point T'^ , droplets

break away from the wavy surface of the film and are entrained in the core

stream, partly as a result of interaction between the film and the

core stream and partly as a result of boiling and evaporation of the film.

- 10 -

Upon attainment of a certain film thickness, breakaway of droplets from the

film ceases, which corresponds, in V.E. Doroshchuk's notation, to the point

T' . The region from T' to T' is characterized by an annular tvo-phaseAp o Ap <-r

flow regime with a fairly smooth film on the channel wall and a core stream

consisting of steam with droplets of liquid in it. Thus, if the (normal.^

steam flow hindering droplet deposition on the channel wall be neglected,

then such deposition in the microfilm region will take place until the

(liquidj mass flowrate in the core stream reaches its extreme* value Tn.

It may be hypothesized that in a heated channel, when evaporation occurs at

the wall and the steam departing from the wall hinders droplet deposition, the

value of the droplet mass flowrate fr , at which deterioration of mass

transfer begins, will increase by a certain amount, depending on the intensity

of evaporation.

§6. In contrast to a plain circular tube, in an annular channel the

droplets from the core stream may be deposited on two surfaces. For the sake

of definiteness, let us first consider an annular channel in which heat is

liberated only at the inner surface.

§7. The character of the function Ap/Apo = f(x~) is similar for tubes

and annular channels, and confirmation of this will be found in [2] (vide

Figure 2 of the present report). This being so, it is apparently permissible

to assume that for the two types of channel the hydrodynamic flow conditions

for a steam-water mixture are similar and that heat-transfer crisis is

determined by physically identical processes.

§8. As may be seen from Figure 2b, the difference ("interval of abscissaj

between the maximum and minimum of these curves does not depend on the: mass

velocity (xn — x does not depend on G), and for high values of G, tends

to x :

x — x = x . u (i)n up ni " \ x /

Let us proceed to consider the flow of a steam-liquid mixture through an

annular channel in the region where the liquid mass flowrate changes from T '

Italic brackets ( ) enclosetranslator's parentheses. Tr.

Minimum in this context. Tr,

Vide Translator's Critique.

- 11 -

to Tcr . If the (normal,) steam velocity hindering droplet deposition on the

heated surface be neglected, then the mass flowrate of liquid deposited in

this region will be:

l&r ~ [^f ~ ln • (2)

Moreover, deposition on the (inner) heated surface will be:

rU = \ rlp , (3)

and on the (outer) unheated surface:

F 2 i : = "2 TL • (4)

As there is no evaporation at the outer surface, it may be- hypothesized

that if the region is long enough a stable film will form on this surface,

and that the quantity of liquid deposited will equal the quantity removed

by entrainment. Let us denote the liquid mass flowrate in this film by TJ.

§9. Taking cognizance of the fact that when evaporation takes place

at the heated surface there is a normal component of the steam velocity

q/\p" which hinders deposition, the quantity of liquid deposited on the

surface diminishes in the ratio K./Ka , and:

Accordingly, the mass flowrate of droplets in the core stream at which

their mass transfer1 toward the wall deteriorates will increase by the

amount:

Then the droplet mass flowrate in the core stream at which mass transfer

deteriorates will be:

In this case, the liquid mass flowrate at any section in the region will

- 12 -

be determined by the relation:

r' = f. + CT'E-T^ ) + r^ + Y'zf

For the onset of crisis it is necessary that:

Y'E = ?l-

and:

so that:

= 0

T-. ' £ ' Ks

Or, substituting (2) and (7) into (9), taking cognizance of the fact that:

fA', = r.; . /<;,.

and dividing by the total mass flowrate, we obtain:

*, = xH - n, [Ci - x^ ) £;.. - i + xn ]

Before this formula can be used, the value of F, .. must be computed. The

liquid mass flowrate in the (outer,' film may be estimated with the

expression:

blO. Thus, in order to determine Y2 . it is necessary to know the

stable film thickness <S. A large number of empirical formulae are now

available for this purpose, but they give little insight into the effect

of the various stream parameters on film formation.

What is actually obtained as a result of the prescribedsubstitutions is:

f,: i(i - * , „ ) .*•, - i + llL

(8)

(8a)

(8b)

(9)

(9a)

(11)

Ka\p"

and since nx is arbitrary, equation (10) must be based onthe assumption that xn = I — fn'/T, which will not bearscrutiny, x'fdc Translator's Critique. Tr.

- 13 -

§11. Of existing theoretical formulae for film thickness, two in

particular seem to us to be in best agreement with reality, namely those

of Ch'eng She-Fu* and Tippets. The former, vide [l3]#, i s :

= 1.22Cp')

§12. It may be seen from the above formula that * is inveraelv

proportional to the mass velocity of the steam phase (Gx) . As will be

shown below, the same result emerges from the formula of Tippets [3]. The

shear stress occurs in the latter formula explicitly (sic), which is very

important for calculation in the case of flow in annular channels, where

the values of the shear stress at the various surfaces differ. According

to the formula proposed in [3] ({22i)) 3 the stable film thickness is:

(12/:)

Knudsen and Katz [4] give formulae for the shear stress at each surface of

an annular channel (taking into account the shift of the velocity profile)

for single-phase flow:4 u" y" (V - rz )

T = , " (13)^ y> (j>2 -|- r

2 — 2r )2 2 1 max

where: r .„ = + / — zr- . (13a)

* # Formula (12a) not found in [13], and spelling of Author's nameconjectural (Romanization given here corresponds to M3H U]3-*Y) . Tr.

? Notation of (12a) not verifiable, but if correctly construed

there is a dimensional discrepancy of V"", which suggests the

variant:

8 . = 1.22 • 10s 5l u" CvO1" g{ •mzn GX

•'• Equation (13) is based on (4-65) of [4], namely:

4 \i v O 2 - r 2 )for r > v

g v iv\ + r\ -

- 14 -

In [ll] it is shown that the shear stress at the wall is approximately

equal to the shear stress at the phase interface. As is done in [5], the

mean velocity in the film may be related to the stream velocity by the

expression:

S - ^ = ^b v' p"

where:

(14)

(14a)

The theoretically highest possible value of the slip occurs when £2 = ^. (14/0

Using this value, substituting (12i>) , (13) and (14) into (11), and dividing

by the total mass flowrate, we obtain the fraction of liquid in the (outerJ

film, namely:

f o"

r'1 2/

rI \ 2

* (15)

The coefficient of the term in squared brackets evidently depends on the

pressure, and is inversely proportional to the mass velocity. Let us call

it B. Then:

~ B*\~ ~)

#

The fraction of liquid in the core stream at high velocities is not yet

known. However, it is quite evident that the fraction of liquid in the core

stream will increase constantly with increase in steam velocity, vide [12],

Equation (15) is dimensionally inconsistent, and should read:

' I P" I 2 f 1 P'T\ -

/p 7 2

1 + /—V P"

2 G y"

The value of B corresponding to (15+) is:

B =

1 + / ^{ V P"

and differs from the coefficient of the squa:in (15) by the factor cr/2y".

-bracketed termTr.

(150

(16 + )

- 15 -

Moreover, fit is quite evidentj that: g = \ir (16,0

Under this condition, the function q^ Or)

will describe a sloping straight line in accordance with equation (10).

Later (vide Section 3) it will be shown that many formulae, describing the

experimental data of various investigators, give just such a linear function.

In virtue of the foregoing, equation (10) transforms into:

, +

2. CRISIS WITH BILATERAL HEAT SUPPLY

§13. The process whereby crisis sets on when both surfaces of an

annular channel are heated is complicated by the fact that crisis does not

occur simultaneously at both surfaces. The film flowing along the second

(outerj heated surface affects the quantity of liquid at the crisis section

in two ways:

§14. On the one hard, the thickness of the outer film diminishes in the

direction of flow, vide [6], thus:

u _ • 2

dl ~ ~ A p ' v~

This diminution depends on the intensity of evaporation along the channel.

§15. On the other hand, in consequence of evaporation of the film, the

fnormalj component of the steam velocity q/Xp" hinders droplet deposition,• f

which increases the liquid nass flowrate in the core stream, T'n , by the

amount:

A f' = n Cf' - N ) - q? . *

Using its unilateral counterpart as an analogue, we write the equation for

(18)

(19) is analogous to (6) in vir tue of: ' Y., 1 .•

(deducible from: Kq = Ka - j ^ ) Jr { ^a I - \ - ' c "

- 16 -

liquid balance at the crisis section with bilateral heat supply as follows:

The counterpart of (10) for the case of b i l a t e r a l heat supply,impl ic i t in the Authors' t r an s i t i on from (20) to (23), i s :

The text's version of (20) contains q, in lieu of a . Tr.

The factor IT which appears in the numerator of the fractionalterm of the text's version of (22) and (36) has been omitted inthe belief that it is redundant. Tr.

The film thickness at the crisis section is determined from the relation:

6q = 6 - A6 . (21)

Taking cognizance of the expression (18), the fraction of liquid flowing

along the (second, outer) heated surface may be written in the form:

2 r (S, ~ S-to) q\\) = l|; g . 11 (22)

Dividing r ' in (20) by the t o t a l mass flowrate, taking cognizance of (22),

we obtain:v + v K Xp" p

q = ~L-~Z " „ ° ~ (-x. ~^> ~~ x™ ^ ~ T" "?, • (2 3)

Here if) characterizes the fraction of liquid in the ("outer and only

rxtanU film at the crisis section. The final term in (23) appears as a

result of the counter-current of steam on the flux of settling liquid,

this counter-current hindering the deposition of droplets.

§16. Equation (23) clearly shows the dual effect of q on q

For one and the same value of q , its effect on the quantity of liquid

in the crisis section is various, depending on q at the inner surface,

- 17 -

vide Figure 3. As follows from the heat balance, the difference £ — £up

must diminish with an increase in the total quantity of heat supplied, i.e.

for high values of q^ the fraction of liquid at the second ("outer) surface

in the crisis section is greater than for low values of q . Thus, for high

values of q a the third term inside the brackets in (23) is small, whereas

the final term of the equation remains unchanged, so that the negative effect

of q on q is more telling. For low values of q , when the second

term inside the brackets in (23) is fairly large, the converse is true, i.e.

the effect is positive, as the fraction of liquid remaining in the core

stream as a result of the counter effect of the fnormalj steam flow from the

second fouter) surface is approximately the same in both cases.

3. COMPARISON WITH EXPERIMENTAL DATAAND DISCUSSION OF THE RESULTS

§17. In order to determine the value of B in (16) (vide Table lj^ we

used the well known empirical formulae of:

Barnett [7]

Janssen [8]

and Bertoletti [9]

%, '

q

q

' 10- b

• io-6

• 10ar

I _ e - * j (25)

, (26)

References [8] and [9] need not be consulted, as the authors of thepresent report do not go beyond [7], which ^ives (244), (25) and(264-). Barnett's formula for X^ is included here as (25+). Tr.

io-6A + B • ATz^

C + % '

1 f 4 M (X (254)

hd (G x 10-5) + h dh CG x l(Tb)

Bllb

^r

- 18 -

where A, H, C, a, B, Y» &*> e> V* a r e parameters depending on the pressure,

mass velocity and section parameters. Comparison of (24-26) with (17) revealsA Y

their structural similarity. Moreover, j^ , j* - £ and E, correspond to

x ~i>, so that the value of B may readily be determined.

118. Calculations {vide Table 1) show that B diminishes with increase

in mass velocity. However, this functional relationship is not quite

accurately described by the formula (15), and the discrepancy is probably due

to the inaccuracy of the chosen expression for slip (14) , as well as to the

assumption concerning the equivalence of the shear stresses at the channel

wall and at the. phase interface. The dependence of B on mass velocity,

based on the three empirical formulae (24-26), is depicted in Figure 4.

Presented in Figure 5 is the function qap = £(xjr ) according to [13] for

various values of q at the outer surface. As may be seen from this figure,

the lower q , the greater the positive effect of q on q , which is

in good agreement with the formula (23) . With increase in mass velocity (.of.

Figure 5a, b), the effect of the heat flux q becomes weaker.

§19. It is natural to hypothesize that with a reduction in the hydraulic

equivalent diameter of the channel, the velocity profiles and concentrations

will change, which in the final analysis must affect the mass-transfer

velocity of droplets toward the wall. In [lO] there is an analysis of a

number of works of various authors, and it is shown that:

qcf - ~ , (26a)d

where: 0 ^ u) $ 0.5 . (26i)

§20. I t follows from (24-26) that to = 0.4. Inasmuch as we assume the

(critical,) heat flux q^ to be proportional to the mass transfer coefficient,

i t must be concluded that:

Ka " -5T7 •

Then, taking this correction into account: K = K *'~* , (26a")

where K is the mass transfer coefficient d hy

from [l] not corrected for the effect of diameter, and d is in mm.

- 19 -

Calculations done on the basis of equations (24-26) show that the mass transfer

coefficient is a fairly conservative quantity, and is approximately the same

for tubes and annular channels.

§21. Figure 6a shows the dependence of xn on mass velocity and pressure,

ani Figure 6b depicts K as a function of pressure.

§22. Thus, calculation of heat-transfer crisis in annular channels requires

a knowledge of i» the fraction of liquid flowing along the unheated surface,

as well as a knowledge of xn and Ka.

C O N C L U S I O N S

§23. The formula for calculating crisis in annular channels with unilateral

finternal,) heat supply is:

r + r K Ap"q = -* ^ • — Gc - * - x ) , (27)^ at- T\ X " •-''

1 -I

where xn and xn may be found in Figure 6a, and A' in Figure 61..

, (28)* Qr2 + r\ -

= B>2 _ 2 y 2 _ 2N

2 ' re A i 2 z 1 ^

where B may be found in Table 1 or Figure 4.

Formula (27) is valid for the following parameter ranges:

I > 1 m , (28a)

70 $ p i 200 atm t (28i)

2000 $ G 4 5500 kg/m2sec ,0.17 . {lid)

§24. The formula for calculating crisis in annular channels with bilateral

heat supply is:

a:

where xn and xn may be found in Figure ha, and # c in Figure 62?.

- 20 -

The counterpart of I|J for the case of bilateral heat supply is:

2 r a - i .,) ~ - (30)

G X iv\ -

§25. The authors are indebted to R.F. Masagitov for helping with the

computation, and to G.M. Gorbach and M.A. Demidova for processing the data.

TRANSLATOR'S CRITIQUE

§26. With the Lransition from tubular to annular geometry, a translator

of pertinent Russian texts into English meets in the target language a

semantic problem having no counterpart in the source language. In Russian,

the outside of an annular channel is the tube (TPY5A) and the inside is the

rod CCTERPKEHb), so that the locus midway between them may be called the core

(:WO) without ambiguity. In English, however, the inside of an annular

channel is itself a core in the foundry-parlance sense, so that another word

is needed for the above-mentioned locus. For lack of such a word, the term

core stream has been used throughout this translation to denote the domain

between the inner and outer films, which in the present report are numbered

1 and ,':, respectively, in text and translation alike.

§27. The central concept of the present report is the x^ of [l], in

which it is defined by equation (.20) as: xn = Ajj:o, where Aj is defined

by equation (24) of [l] and x is the quality at entry. The physical

meaning of this concept is stated on page 20 of [l] as follows: " x is the

steam content at which the deposition of droplets ceases under conditions of

adiabatic flow of the dispersed mixture". It is noteworthy that the pr^- nt

report uses this statement, rvutatrs mutandis, as its crisis criteric... Tn

fact, cessation of droplet deposition under non-adiabatic conditions is the

criterion used, vide §9, equation (9) in particular. This is at variance

with reference [l], in which ths criterion for the onset of crisis is that

the droplet concentration at the wall should vanish, vide the foot of p. 10.

In [l] the role of xn is merely that of an incidental parameter in terms

of which q^r and x ^ are expressed.

- 21 -

§28. There is also a discrepancy between reference [l] and the present

report concerning the mass-velocity range to which the parameter xt. applies.

The critical formulae derived in [l] are stated to be valid, vide page 26 of

that reference, for: xjl < x^ < xn, and C > 2000 kg/m2sec. The xn

of the present report, on the other hand, presumably applies to the mass-

-velocity range: G < 2000 kg/m2see, observing that the alternative xn

is expressly defined for: G >- 2000 kgM2sec.

§29. Quite apart from the questions of relevancy and range of validity,

there are grave doubts as to whether the xr of the present report is the

xn of reference [l]. These doubts arise as follows: The Authors' notation

makes no distinction between terms which apply to the entire channel and

those which apply to the core stream only. However, the text is not entirely

meaningful, and equations (2) and (9a) not intelligible, until such a

distinction is made. In some cases, e.g. in §5 and Figure 1, this distinction

is neither clear nor important. Elsewhere, however, both in the wording of

the text and in the logic of the equations, there is abundant evidence that

wherever the Authors' symbol for mass flowrate is accompanied by the liquid-

phase superscript and the subscript n or nq3 the term so formed applies

to the core stream only. Such terms are identified in the translation by an

overdot, thus: f „ and f . In the light of this symbology, it will now be

apparent, vide the footnote on page 12 of this typescript, that the general

expression for steam quality in terms of mass flowrate has been extended to a

fallacious particular instance, thus:

II r - r ' = j _ Lr. /

but: xn = 1 - •— x

because the domain of Tn is not the

same as the domain of T. It may be verified that the true steam quality for

the entire channel in terms of the text's spurious xn is given by:

x\ = — — . <31>

1 - Y

On page 11 of [17], the following expression is given for the quality of the

core stream in terms of the quality of the channel's entire content:

x1 + (1 -

(32)

- 22 -

Substituting the ^ of (31) for the x in (32), the true quality of the

core stream in terms of the text's spurious xn may be obtained, namely:

x - . (33)

Accordingly, if it is desired to use quality data from the post-film-dryout

region in the subscript n category in equation (17) of the present report,

these data should be entered as xn in the inverse of (33) for conversion

to the text's spurious xn3 thus:(2TJJ — 1) x — xjj

x. = v . (34)

However, the above-described remedial procedure is valid only for unilateral

heat supply, as in the bilateral case, equation (23) contains a spurious x

as well as a spurious xn.

§30. Since the final term in (10) is F /T, whereas by definition

fy = V ' /Y ', the result (17) tacitly depends on the approximation: V' - T.

Analogously, equation (20) depends on the same approximation.

§31. Observing that equation (13) expressly applies to single-phase

flow, its relevancy to the context is questionable.

§32. As equation (16a), namely: E^ = 1, implies film dryout at the

benchmark for the onset of microfilm, it would appear to be a contradiction

in terms. Moreover, it is at variance with Figure 3 and §16, which indicate

that the outer film is extant at crisis.

533. Finally, equation (1), namely: xn - x^ = x , is enigmatic. In

view of what has been said in 529 above, both terms on the left-hand side are

suspect, and the equation's meaning is not intuitively obvious from Figure 2b,

nor from its prototype, Figure 2/?+. The possibility of a typographical error

is precluded by the fact that the equivalence of (23) and (29) depends on (1) .

Further enquiry into this question would entail mastery of references [2] and

[16], which are beyond the scope of this critique.

- 23 -

B I B L I O G R A P H Y

P.L. Kirillov £ I.P. Smogalev, Physics and Power EngineeringInstitute, Obninsk, preprint of Report H°. FEI-181 (1969) .

V.E. Klyushnev & N.V. Tarasova, "Hydraulic resistance tosteam-water flow in narrow annular channels", TeploeneraetikaN° 11, pp. 65-68 (1966).

(English translation in Thermal Engineering, H°. 11, p. 87 (1966))

F.E. Tippets, "Analysis of the critical heat-flux condition inhigh-pressure boiling water flows", ASME Paper N? 62-WA-161.*

James G. Knudsen & Donald L. Katz, Fluid Dynamics and HeatTransfer, McGraw-Hill Book Company, Inc., New York (1958).(Library of Congress Catalog Card Number 57-10224J

Heat Transfer Problems ("in Russian,1 (symposium of articles underthe editorship of P.L. Kirillov), Atomizdat (1967).

Achievements in the Field of Heat Transfer Tin RussianJ (symposiumof articles under the editorship of B.M. Borishanskii), PublishingHouse "Mir" (1970) .

7 P.G. Barnett, "A comparison of the accuracy of some correlationsfor burnout in annuli and rod bundles", AEEW - R 558 (1968).

8 E. Janssen & J.A. Kervinen, "Burnout conditions for single rodin annular geometry, water at 600 to 1400 psia", General ElectricCo. Atomic Power Equipment Dept., San Jose, Calif., Report N?GEAP-3899 (1963).(Vide Nuclear Science Abstracts, Vol. 18, N° 4, p. 740, N? 5464J

9 S. Bertoletti, et at., "Heat transfer crisis with steam-watermixtures" (in EnglisbJ, Energia Nuoleare, Vol. 12, N° 3,pp. 121-172 (1965).

10 V.E. Doroshchuk, Heat Transfer Crises for Water Boiling in Tubes,Publishing House "Energiya" (1970).

For paper and discussion, =ee Transacti;--•• /' th: AWF.. Joof Heat Transfer, Vol. 86, Series C, N° ! J f b . 1964), pp. 23-38. Tr.

- 24 -

11 Pogson et. al., TepLoperedaoha*3 Series C (Nov., 1970).

12 B.A. Zenkevich, P.L. Kirillov, et al., "Heat transfer burnoutin water flow through round tubes and annuii". Paper H°. B 6.13,presented at the Fourth International Heat Transfer Conference,Paris -Versailles, 1970, vide "Heat Transfer 1970", Vol. VI,Elsevier Publishing Company, Amsterdam (1970).

13 N. Adorni, At a!., "Heat transfer crisis with steam-watermixtures in complex geometries: experimental data on annuii andclusters", Report W. EURAEC-1359 fin English; (=CISE-R-123),Milan (1964) .

'Hde Nuclear Science Abstracts, Vol. 21, N° 3, p. 366, H°. 3208J

14 G.K. Filonenko, Tn U^ievjetikaM, N? 4 (1954).

15 N.I, Semenov & A.A. Tochigin, Tnzhenermo-fizicheskii shurnal,h_, N? 7, pp. 30-34 (1961) .

16 V.E. Doroshchuk, "Heat transfer crisis in an evaporating pipe",Teplofizika vysokikh temperatur, Vol. 4, N? 4, pp. 552-561 (1966)

('English translation in tiioh Temperature,, Vol. 4, N° 4, pp. 522--529 (1966))

17 P.L. Kirillov, Physics and Power-Engineering Institute, Obninsk,Report N? /•>T-224 (1970).

Published by the Energeticheskii Institut, Moscow. The onlylibrary code Msted for this journal in North America (OCoB)pertains to the Battelle Columbus Laboratories. Tr.

Issues prior to 1957 not held in National Science Library. Tr,

- 25 -

FIGURE 1.

Caption: Development of flow throughout the length of a heated channel.

Film dryout -*•

Microfilm span

Hydraulic-resistancecrisis •*

Beginning of annulartwo-phase flow -*•

Entry to test section

r'

Cf. x° J vide Notation, page 5.** Vim

- 26 -

FIGURE 2 a .

Ap r -,Caption: The function: — = f(aO, according to |_2J

Ap

/

TJOO

1A0.5 x

Legend:o

tube,annular channel.

Prototype:

Fig. 2. Experimental data on resistance during steam-water flow in a tube and in an annular chamiel, at

p = 147 x 10* N'/W and pui = 2000 kg/m's.

- 27 -

FIGURE 2 b .

Caption: The funct ion: -^ = fCaO for an annular channel, according to [2],

Legend:

Ap

4.2

3.4

2 .6

1.8

4

frJr>

—-i

Mass

•

0

in:

velocity

1030

1450

2063

kg/m2sec

Prototype:

0.5 FIGURE 2b+.

4-

0*^0.7 02 0.3 - 3.9 0.5 0.6 OJ O.i 0.1 10

Fig. 6. Comparison of calculated dependence ofAp/iV0 on x fox tests on an annular channel

(8= 1.82mm).p = 147 X ICf N/ma; I - p» •* 2063 kg/ro's; 2 -1450k«/mJs;3 - 1030 kg/m lSif.= 196x Iff

4 - pu> =* 2045 kg/'m's.

FIGURE 3.

Caption: The effect of q on the fraction of liquid on the second surface at the crisis section.1 or

/ / / / / / / / y / / / / / / / / to00

/ / s / , /A / / / / / / / . / / ,-l / /" / / •r7~;~7~,

- 29 -

FIGURE 4.

Caption: The function: B = f(G).

B

0.9

0.7

n s

\

8 G * 10~6

Abscissa:

Ordinate:

Mass velocity in kg/mzhr3

Dimensionless coefficient, vide (16), (161) & Table 1.

- 30 -

FIGURE 5a. G = 1100 kg/mzsea

Caption: The function q = fGEcp ) for bilateral heating,crisis occurring at the inner surface.

400

200

Parameters: r} - 7.06 mm,

P2 = 12.52 mm;

p = 5 1 kgf/cm2.

Legend:

a

q2 = 0 W/am2,

q = 37 V/cm2.,

? . = 63 W/am2.

Abscissa:

Ordinate:

3 vide Notation and Translator's Critique;

cr i t i ca l heat flux at inner surface in W/am2,

Whether gauge or absolute, not clear. Tr.

- 31 -

FIGURE 5b. G = 2200 kg/m2sec

Caption: The function qm = f(.xor ) for bilateral heating,crisis occurring at the inner surface.

400

200

Parameters:

Legend:

A

D

0.1 0.2

v = 7.06 mm3

v = 12.52 mm;

p = 51 kgf/cm2. *

0 W/em2!

37 ftycmzJ

63 W/em2.

x

Abscissa: a; , vide Notation and Translator's Critique;

Ordinate: = q 3 i.e. critical heat flux at inner surface in W/cm'lor

Whether gauge or absolute, not clear, Tr.

- 32 -

FIGURE 6a.

Caption: The function xn = f(GO.

1.0

x

0.8

0.6

0.4

G x 10"

Abscissa:

Ordinate:

Legend:

mass velocity in kg/m2hr,

vide Notation and Translator's Critique.

Key to dotted lines not given in text. Tr.

Judging by Figure &b,• the text's caption which is common to bothfigures 6a and 6b (on separate pages) is a blanket caption, butthe possibility is not excluded that it should have been givenhere in extenso, viz.: "The dependence of xn and K on massvelocity and pressure". Tr.

- 33 -

FIGURE 6b.

Caption: The function: K = fCp).

K

200

150

100

50

\

-i

50 100 150 200

Abscissa: pressure in atir (absolute),

Ordinate: mass transfer coefficient* rn/hr.

It is not clear whether this is the >: defined io §20, orwhether it is K . in view of §24. Tr •

TABLE 1.

Caption: Values of the parameter B3 designated B? and E according to the data of [7], [8] and [9], respectively.

mm

2

2

2

2

2

2

3

3

3

3

3

3

3

5

5

5

5

5

10

10

10

15

15

P2

mm

2 . 5

3

4

5

8

10

4

5

8

10

15

20

25

8

10

15

20

25

15

20

25

20

25

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1.

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

G

S 7

.083

.940

.821

.749

.784

.608

.970

.890

.730

678

.608

572

551

911

000

701

643

608

941

821

749

969

890

1

1

1

1

1

1

1

1

1

1

1

1

1

1.

1

1.

0 .

1.

1 .

1 .

1.

1.

1.

500

* .

.654

.528

.453

.404

.617

.297

.515

.495

.389

352

.297

266

247

508

769

369

326

297

529

455

404

483

495

kg/m2sea

B9

0.428

0.512

0.628

0.728

1.091

0.966

0.519

1.012

0.927

1.016

1.136

1.202

1.242

0.720

1.269

1.223

1.289

1.337

0.708

1.434

1.514

1.231

1.555

1

0

0

0

1

0

1

1

1

1

1

1

1

1

1

1.

1.

1 .

1 .

1 .

1 .

1 .

1 .

B

.06

.99

.97

.96

.16

.96

.00

.13

02

02

.01

01

01

05

35

10

09

08

06

24

22

23

31

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 .

0 .

0 .

0 .

0 .

0 .

0 .

0 .

c

B 7

.015

.882

.769

.702

.735

.570

.909

.834

684

635

570

536

516

854

937

657

603

570

882

769

702

909

834

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1

1

1

1

1

1.

1.

1.

1.

1000

Ba

.332

.232

.169

.129

.297

.040

.212

.201

.115

.084

.040

016

994

219

424

099

064

040

233

032

129

207

201

ku

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

I

1

1

1

1

1

1

/m2sea

B,

.350

.229

.153

.112

.297

.054

.268

.257

.177

.150

.121

.107

.096

.449

.512

.264

.226

.198

.524

.479

.335

.571

.604

1

1

1

0

1

0

1

1

0

0

0

0

0

1

1

1

0

0

1

1

1

1

1

fl

.23

.11

.03

.98

.11

.89

.13

.10

.99

.96

.91

.89

.87

.17

.29

.01

.96

.94

.21

.09

06

.23

.21

G

B 7

0.7520.6540.5700.5200.5450.4220.6740.6180.5070.4710.4220.3980.3830.6331.695

0.4870.4470.4220.6540.5700.512

0.6740.618

= 1500

fls

0.9600.8980.8680.8460.9860.796

0.8850.8830.8380.8220.7960.7810.7710.8911.057

0.8290.8080.7960.8990.8680.846

0.8850.883

kg/m2sec

0.157

0.226

0.316

0.373

0.578

0.516

0.233

0.343

0.495

0.545

0.542

O.64.V

0.66.1

0.46:

0.682

0.659

0.697

0.715

0.630

0.776

0.942

0.669

0.836

B

0.623

0.593

0.585

0.580

0.703

0.578

0.597

0.615

0.613

0.613

0.587

0.608

0.605

0.661

0.811

0.6580.6510.6440.7280.7380.7690.743

0.779

G

B 7

0.5930.5170.4510.4120.4310.3340.533

0.4890.4020.3730.3340.3150.3030.5010.5500.3860.3540.3340.5180.4510.412

0.5330.489

= 2000

Be

0.7440.7030.6880.6800.8030.6500.6880.696

0.6750.6660.6500.6420.6350.7020.8400.6700.6590.6500.7030.6890.680

0.6880.696

kg/m2sea

Bs

0.2760.3130.5500.4140.5840.5070.3300.411

0.5060.5400.5820.6020.6130.5030.700

0.6360.6570.6650.6660.7530.769

0.6910.812

B

0.5380.5110.5630.5020.6060.4970.5170.5320.5280.5260.5220.5200.5170.5690.697

0.5640.5570.5500.6290.6310.620

0.6370.666

- 35 -

P O S T S C R I P T

§34. Referring to §15, it is not clear why the Authors mention the

effect of the outer film only, vide (19), observing that the result (20)

also implies:

and, of course:

§35. The location of inner film dryout is not indicated explicitly

in the text, and there is no a priori reason why it should correspond

to T'n However, in view of the preceding paragraph, the result (20),

and its tacit counterpart (9) in extenso (incorporating (2) and (7)) for

that matter, presupposes such a correspondence.

(19a)

(19b)

23.10.73G.P.