ML19340A532

Stability of Two-Phase Flow Loops & Response to Ship Motion, Revision 1

ML19340A532
Person / Time
Site:	Dresden
Issue date:	09/26/1960
From:	Beckjord E GENERAL ELECTRIC CO.
To:	NRC
References
CON-AT(04-3)189-PA-5, CON-AT(4-3)189-PA-5 GEAP-3493, GEAP-3493-R1, NUDOCS 8008250805
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CZAP 3k93 Ecvision I AEC Research and Development Report 9

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THE STABILITY OF TWC-PHASE FLO'd LOOPS

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8 --

s

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c

= =. -

-a AUD RESPONSE TO SHIP'S IMION

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....,,,.., : =a Q\\

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E. S. Beckjord s N. -

u September 26, 1960 Prepared under AEC CONTRACT AT(ok-3)l69,PA #5 GENERAL ELECTRIC ATOMIC POWER EQUIPMENT DEPARTMENT SAN JOSE, CAllFORNIA l

.J811 TO REGULiiURY DE, SIR,il FILES "0M 018 RESULATO.lY 00CXET fil.E COPY

.80082sopo5 p

n CEAP 3493 Eevision I AEC Eesearch and Development Eeport 21e Stability of Two-Phase Flow Loops and Ecoponse to Ship's ibtion by E. S. reckjord September 26, 1560 f

Approved by:

,S li < I 8.a /-[

D. II. Imhoff, Ih:tager enc neerinC I:evelopment i

Prepared under AEC CONTRACT AT(04-3)189, PA #5

i LEGAL NOTICE This report was prepared as an account of Govemment sponsored work. ?!cither the United States, nor the Co:rmis-cion, nor any person acting on behalf of the Corr.ission:

A.

Fhkes any warranty or representation, expressed or implied, with respect to the accuracy, ccepleter. css, or uccfulness of the information contained in this report, or that the use of any infomation, apparatus, method, or pro-cess disclosed in this report may not infringe privately owned rights; or B. -Assumes any liabilities with respect to the use of, or for anmnges resulting from the use of any infoma-tion, apparatus, method, or process disclosed in this report.

As used in the above, " person acting on behalf of the Commission" includes any employee or contractor of the Coc-mission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or.e=ployee of such contractor prepares, disseminates, or provides access to, any information pursuant to his er.:ployment or contract with the Com=1ssion, or his employment with such contractor.

1

i -

e SU:2EY lle investigate.the dynamics or stationary two-phase flow loops and of those accelerated by ship's notion,.and determine the controlling parameters. Qc i

results apply to boiling water rec.ctors, and one.ble prediction and desi n ol C

stable two-phase flov in reactors. An analogue computor circuit is developed for. calculating loop transients. Analytical predictions of stability and-transient response are compared with A.P.S.D. Heat Tmsfer Loop test results:

1.

Analysis correctly predicted the trend of four tests-progressing from very stable flow to a case on the verge of instability.

1 1

2.

The analogue predicts the-temporary decrease in inlet velocity which occurs when the loop heating power is increased. Se quantitative comparison for a heating rate increase of 7% is:

Cbserved Predicted inlet velocity trancient

-326

-30 5%

oscillation period 3 0 sec.

2.4 acc.

damping factor.

0.066 0.1 inalogue computation of the effect on the T-7 core of a 20% of norcal gravita-tional acceleration increase Sives these results:

pressure drop 20%

chimney water velocity

+27%

inlet water velocity

+ 8.5%

average stecr. void

-11 7%

Inalo6ue computation of the effect of a 20% heating increase in channel 1.o' 2 coupled parallel cham ols similar to T-7 core channels gives these results:

Channel 1 inlet water velocity i 5%

average steam void

+17%

Channel 2 inlet water velocity

- 0. 4'S average steam void t 0 9%

2e following conclusions are drawn on the factors which determine loop dynamics:

1.

':he natural period of the loop is governed by the transit time of fluid across the two-phase vertical section.

If oscillatory, the period vill be between 4/3 T and 2T, where T is the transit time.

2.

'Ihe primary cause of loon instability is subcooling with high steam voids, because subcooling makes the natural circulation drivin5 head decrease i

vhen the inlet vater velocity increases, and vice-versa. A design line of subcooling lirit versus operating pressure is given.

i 3

unstable flow loops can be stabilized by velocity head losses, such as l-at an orifice in the douncemer, and by the use long downcomer pipes and I

consequent high single phase fluid inertia.

-v

-r y-

-~

-,e,--

,w-,

,iier

---

ivrrr

- w

- + - =

m

)

4.

Friction pressure drops and velocity head losses in the riser can help to stabilize, provided inlet subcooling is not excessive.

If it is excessive, the riser friction pressure drop and head losses actually decrease when inlet water velocity increases. The result is a negative increnental pressure drop '.hich destabilizes. This point is important to the design of reactors with internal steam separation.

2

?

i The work reported here was perforced under contract !.T(04-3)-18') Pit 5,':S Lith the San Francisco Operations Office of the U. S. Atomic Energy Cet=1 scion.

INTRODUCTICU 2e purpose of thic work is to explain the dynamics of two-phase caturated atcan and water flov loops in general, and particular the effectc of chip's motion accelerations on such loops. Se dynamics of both stationary loops and acceler-ated loops have obvious bearing on the decign of stable boiling water reactors.

The bacic questions of two-phase flow in pipes or channels of boiling water reactorc arc (a) what is the pressure drop?

(b) what is the phase volume fraction or phase relative velocity?, and (c) is the flov steady or pulcatinC.'

In the pact empirical correlations (1)

(2) f prcccure dros and of volute fractionn o

with quality have been uced to deal with the firct tro questionc.

T.ccently a cethod vac introduced

, which relates preccure drop and vclure fraction by the force between the phasco. 210 acthod ic baced upon the flow cyctea ccorc-try, the precsure, the heat input, and the cincle-phace friction ractor of b)

Calculations of Cook'c (5) and pipes.

It cricinatec vith vork by levy thrchaterrc's (6) enperimental data conciating of void fraction and prescure drop were made by thic method, and the calculated valucs were within the bcnd of experimental error of the measured values, for test runs between 275 and 61!+ poia.

Cne of the calculated cases from reference (3) is chovn in Figure 1.

The calculation is made frca operating pressure, ialet velocity, heating rate, and channel hydraulic diameter, but without the uce of any correlatica of void fraction with quality. Iecauce the equations of this method are derived from hydrodyncaic forces, they can be applied to the third questica of fiev l

2.4 0 - OSO PRESSURE DROP

./

APXIg3 v

• \\

./

KG-M F

TION 2

2

[c l

STEAM VELOClTY SEC -M

\\

0

./

20 050

.x s

/.#*

N 9

"*p' STEAM VOID

.. + #

.'/',

.s

\\

FRACTION O

81.6

-4 0.40 of N

\\

W o./

• N 1.2

/

-6 0.30 O /

Ei N

g

• g IPSI-w 0

N, 9

] O.8

,e j.

N o

-8

- 020 O[. #

WATER VELOCITY VW

.*of MEASURED AP

/

I O.4 CALCULATED- --

-10 0.10 1

-IFT/SEC.*

COOKS DATA O

.O O

1 r

i

.O O

O O2 0.4 0.6 QB 10 (2

X, METERS VERTICAL DIMENSION l

COMPARISON OF MEASURED AND EXPERIMENTALVOID FRACTION OF COOKS RUN 445 FIGURE I

a, a

stability and transients. The correlation tethods, en the other hand, are not based en dynamical flow relations of force and inertia, and hence are not descrip-tive 'of transient problems, without some patching up to account for the forces.

In this report, the equations and assumptions of the dynamical method are des-4 cribed and applied to a ceneral two-phase, natural circulation flow loop. The

[

conditions of stability.for the linearized equations are derived, cnd the predic-tions are compared with the results of flow stability experiments conducted en the A.P.Z.D. Heat Transfer Loop. An cnalo6ue cc=puter circuit for solution of i

the equations is developed, and is applied to the ship's pitching motion problem, 1

and to a parallel flow channel problem as ucll. Finally the main cause of tuo-phase flow pulsations, which is the dependence of the natural circulation driving 4

head on inlet water velocity and subcooling, is discussed. The ccthods of i

stabilizing loops, as indicated by the physical parameters in the equations, are pointed out.

i AUALYSIG OF TUO-PHASE FLOW LOOPS Se analysis of _ two-phase flow loops is divided into three topics: - the mechan-i ics of two-phase flow, the application of the mechanics to a flow loop, and the solution for small disturbances.

1.

TI!E MECHARICS OF TUO-PHASE FLOW Ge essence of the two-phase mechanics described in reference (3) is the statccent of Ucuton's third law for each phase:

j 2c net force actin 6 cn the phase, plus the inertia of the phase, plus the phase to pipe shear force, plus the phase to phase shearing force, plus the phase weight equal zero.

2e assumptions to give =athematical e::pression to the problem are as follows:

l 1.'

Steam and unter flowing concurrently are in saturated equilibrium.

2.

2c flow process is adiabatic.

[ +W-e

--e W

w w

t w

4 m

tr*fyy

'7--~vW v

w-v-yw-e

+

vyr'

k

~

JA 3 A

i'- - - -

3 Se transverse pressure gradient is nc611 ible compared to the axial C

pressure gradient.

k. - he shear stress between the phases and between each phanc and the channel vall are analytic functions of the phase velocities, the phase volume fractions, and flow rate ratios.

5 Be flow system pressure drop is small enough compared to the absolute pressure that the saturation enthalpies and densities are essentially constant.

6.

He fluid mechanical energy is negligible compared to the thermal energy.

Be two-phase flov equations follov from application of Gauss' law to momentum flow, and mass and energy conservation.

Let P be pressure, u, steam void fraction, W and S vater and steam velocities, pdensity, P shear stress per unit 3

length, g gravity, 3 heat rate per unit volume, and h_ enthalpy per unit mass.

He equations take the fom:

I for water pressure Cradient, (1}

(1-u)

(1-u) p W2+

(1-u)p.JT (1-u)pg = 0

+

+ P P.,3

+

y y

for steam pressure cradient, BP D

u g + g upaG2 + - up S + Pg + Pj3 + up,g = 0 (2) g 3 for conservation of nnas t

g (1-u) p v + g (1-u) o + Bg up,s + g up, = 0 (3)

B B

B y

y and for conservation of energy b

b b

b g (1-u) og'ahy + g (1-u) @ + g up,Sh E."9 h = q(x,t)

(4) r s

ss Uor nlly-the fluid pressure drop around a loop is small compared to the operating pressure in cases of interest, and the interchange betueen the phases that results from flashing or condensation is negligible outside of the heated sec-d tion of the loop, from assumption (3). Bus, above the heated section, the. -. -

i 1

i l

l water terms and the steam tems of equation (3) can be cet ceparately equal to Zero.

It is necessary to define the shear ctreco per unit length terms, T, P, and r y

3 y3 For vater to pipe ctress per unit length, the follcuinc relation is assumed.

2 Tw = f(Re)

(l p W ),

y f = D'Arcy Wiesbach friction factor (5) pv = perimeter consnon to water and pipe A = pipe section area

%c friction factor' depends on Reyr. olds number, which is taken to be that of water flow only. We ccamon water perincter is in cencral difficult to deter-minc without measurement. It is equal to or lecc than the pipe perimeter.

A vay of calculating it suCGested by S. Levy is described in Appendi:c A.

Inalogously for cteam, the relation is Ps=f(Re)g(}psS)

(6) 2 where Reynolds number is calculated for stec= only.

Le. rater to steam chear streca must be zero when the fluid is all water or all steam, and it surely depends on the number and size of the bubblcc, cr chnrne-toristic radii of curvature of the interface between the phases when' flov ic not in bubble form. For vator to steam chcar stress per unit length, the re-l lation ic taken to be 2

us" f(U )

rp ]P(0-U) "(l'")

(7)

P r-bubble radiuc.

e

'The Beynolds number used is based on bubble flov, and the density is tha. of the predominant phase. The hydraulic diameter of bubble flov is baced on 7

the volenetric definition of Cunter and Chav ('/).

For bubble radiuc the ces::.ctric recan or.aluca nucccated by Zuber and Tribuc, Ec f. (C), is used:

i

- (3 r/ Cow - ps) c (c) r3 surface tension of water where v =

Cubotitution of the au::iliary equationc (S) thrcach (8) in the principle equations (1) throuch (h) cives a cyctem of four equaticno c.nd four unknounc, p, u, u, and 2.

If the boundary co::ditiona cre known, the colution is detettined.

2.

Application to the Flow Loop 3e ci:.. ale, cincle channel natural circulation loop of Figure 2 io chocen for the cake of clarity in presentation of the eccentiala in loop dyncnicc. Two accu =ptionc help to simplify algebra considerably. The first is that feed-water flow is adjusted to hecp inlet unter cubcooling constant:

Ric ascunp-tien clininntec douncener trancit tice effects. The general case is discucced in Appendix 3.

Tac accend accu =ption, which clininates encrcy storace effectc in the heated cection, is that heat is injected at a point rather than along a finite cea!.on.

Se equations can, of cource, be ceneralized to acccunt for a finite heated lencth.

The equationc of the pmcedinC cection can be applied directly to the tuo-phase portion of the loop. One additional equation is nececoury uhich relatcc pressure drop in the downcomer to cincle phase water velocity. ' lith five equationc and the five unhncvnc, p, u, u, c, and v, the problem colution is Cne way of sinpliS ing the solution is determined, but ic not very tractable.

f to wor! uith at.cruce valucc of the variaolec in the two-phase ccation. Tac price of this ci:r;lification is loac of accuracy throuch ignorf n_, velocity

' lith this accunption, ve vill not deceribe and void frac tic'1 cradiento.

_8

,a a

i

?oO L

.(

• a si_

POINT o

y V

HEAT j

SOURCE Ay j

LO SINGLE CHANNEL i

FW

_w_wu f

6 T

!.'s 6

f4

) 0 V

2 -*.( [3

$v 1 i

i 2

COUPLED PARALLEL CHANNELS NATURAL CIRCULATION LOOPS FIGURE 2 9_

4 sluccing flow where, for example, water is rushing in at the bottou and falling back from the top of'the cection.

The average pressure gradient across the two-phase portion of the loop is a j

useful quantity to work with, because it is necessary to calculate the average steam void in the channel in order to ascertain the driving force. We vill 1

integrate and average equations (1) and (2) over the vertical length, L.

The 3

average void fraction is defined to bc R, L

f u dx (9)

R

=

0 Integration of the acceleration terms is done in the following ny.

We make use of the identity d(1.- u)W B(1 - u)W B(1 - u)W B(1 - u) ) +

~

,y dx 8t dx 8t j

(

(10) 1 (1 - u)W Ng + (1 - u) y N

3 and the fact that the vater and steam portions of equation (3) are separ-ately equal to zero away from the heating point, and finally that at the heating point, equations (3) and (4) nay be conbined, with the result:

d(1 - u)W E(1 - u)

)

ex 6t (11) 1 q(x,t)

~

p (hs - hv) y J

j Use of the nean value theorem gives for equation (1) 2 2

+ pv (1 - R) k - k + (1 - R)

~

+

y

}.

L f

(P - P,) dx (1 - R)g

=0

+

y o.

and for equation (2) 4

~

2 Q

1 R OP RS R dS-S s

+ ' Ps 3+

_g+E APs (hs - hy)

+ 6 1

t I

L (13) f (Ps + Pvs)dx

= 0

+

'o j.

Where Qs is the stencing heat rate. 'Ihe steaming rate itself is derived from conservation of enerCy:

L

~S

(

}

kotal*Sctecning subcooling Gaubcooling p.Av(h

- h.7)

(15) 3e where igc c hiet cubcooled water enthalpy i

'Io continue with reduction of the pressure drop equations, we add equations (12) 1 f

i and (13), divide by water density, and clininate the z.teaminc rate variable with equations (14) and (lC

' e result is 2

+ (1 - R) h - k + (1 - R) h +

h+8{

+

Ag(hs~h) v 2

R-W S

+ (1 - R)6 + 3 g + Gy,y- + Gj 7 =

0.

Q6)

E where S

=

ps At qualitico lecc than 10,i, and with vater to atcan density ratioc creater than 10, the steau precouro drop ter=3 in equation (16) may be ncClocted without seriouc crror. The reactor conditions of current interest arc within thece l

l limits. Outside of the limits, account should be taken of the steam acceleration and head teins. Accordingly we vill neglect the term with sos, and with (S 'd)Q,

3 whic14 ic the acceleration of vnter that has beccme steam at the heated point.

'2e _ time derivative of ' oteam velocity till alco be dropped, because the steam reaches its equilibriun velocity relative to vater in tinco much shorter than the fluid trancit time of ricers in cases of current interest.

• _~

Equation (16) thuc becomes 2

g + (1 - R) V M + (1 - R) g Gy 2 2

dW W

1 (1 - R)g = 0.

+

Pw 2L 2L (17) i In similar fachion equation (13) becoccc g + G, (1 - R) (S - W)2 1

(18) y 2

4xrB f (R )

where G

=

vs e

4 3

g* y x r3 l

i for the cacc of vator fraction predominent.

1 Dic cum of preocure dropa around the cloced loop ic zero. 01erefore the preocure drop acroco the two phase section is the static head of the section heicht lecc the velocity head locccc in the dovnconor. 21uc b dy l a K y g+g g 7 7 g =

0.

(19) where K cinC c phace head loss cccfficient l

and Lp total cincle phase lenCth.

In the ricer above the point of heat.1njection, the =acc cnd encrcy concervation equationc reduce to conservation of water and of steam ceparately. As a recult, 4

the volume flov throuch all coas sections of the ricer is the care at cny instant of time. Thuc (1 - R) W + PS 1-u(x,0)

'?(x = 0) u(x, 0) S(x = 0)

(20)

=

+

y q

h

-b Pw oc y

h

-h p

oc

--l

-v 1+

=

P Apy(hs - h )

h

-h As h

-h c

y 3

y s

y A boundary condition at the two-phace cocticn is required. 7110 is obtained frca equations (14) and (15), and from nacs cencervation' at the heating point.

4,!

4

.e m-o

The equation in Q - Ap y(h h,g) y y

(1 - u) U(x = 0)

~

=

v-Apy(h - hy)

(21) 3 inlet subcooled water endialpy.

where b :

cc Finally we need the average steam void as a Ibnetion of time:

L t

R b

u(x, t) dx ~ b u(o, 7) vdr.

(22)

=

L o=_L L

=

Equation (21) is an approximation which assumes that a change in void fraction is transport.d across the section at the velocity of water.

Equations (17) throu6h (22) are a set of non-linear differential and algebraic equations in the six variables, P, R, u(x,0 ), S, W, and v which approximate the simple loop dynamical system. The set has no solution in closed fona for even simple initial disturbances, and is most accurately solved numerically.

3 Solution for C=all Disturbances For anall disturbances the loop dynamics can be investigated by linearization of the equations: that is to say, by takinC small variations of the variables about a steady-state solution, and ignoring second and higher order terms and c ross products in the equatiens wh1ch relate the variables. The expanded form of the variables is e

s AP = /6

.AP ox (Ax

\\S~

a S-W = (s -v ) + 6(S-W) o o W=W

+ AW g

u=u

+ 441 o

R=ua+ AR v=v

+ av g

i Che set of variables with the subscript zero ecmprise the steady state solution, which is found by setting the increncntal variables equal to zero, and by solvin6 equations (17) throu6h (21) simultaneously. Equation (22) is not required for the steady-state solution, because the average value of void fraction is u. With linear differential equations it is also convenient to o

apply the Laplace 'Iransfomation, and write the equations in matrix-operator form. The transformation is straightforward except in the case of the average void, 3, which i t

L a-f u(t)vdt h

1-e

[(u)

(23)

Xferm (R) h

=

=

L t

,7 complex frequency RAD /SEC.

vhere e =

'21e equations are normalized by the substitution of a dimensionless frequency Ia variable

= 7, t11ch converts the time scale from seconds to transit time units. The set of equations in matrix form is 61ven in equation (24).

The disturbances we are interested in are normally heat input, and gravity chances caused by ship's motion. These disturbances are, therefore, placed on the ri ht-hand side of equation (24).

C 1

fl

^

[

)

4 2

(

y A

)

g 1

y o

o 6

o o

t p

a u

(

- Eox

)

y R

u R

y A

t o

A g

(

y

- 1p 3

)

K*

p a

p

)

a a

w[

o o

+

+

Lp y0L D

1 1

(

(

0 g

W-2

)

L W

2 0

W 3

O O

1

(

s w

9 c.

S

(

9 0

o O

O w

d" o

U v

G

+

p

+

)

9 O.

O O

1 u

0 h

W L L

)

)

o o

1 8

v u

u h

h h

(

1 1

(

(

Q y

h" Wp a

S p

h A

)

m o

u

=

=

=

1

(

c D

r

)

O o

O u

O 0

W-o e

3 r

(

e a

h y

w G

1 l

1 O

O 0

r

. U. '

m

)

J

~D 0

We are interested in both the transient responce of the flow loop and in its stability. First let us consider stability. The system is stable if the transient response to a disturbance decreases exponentially. The theory of ordinary linear differential equations with constant coefficients shows this to be the case $f the real parts of the roots of the characteristic equation are negative.

In the problem at hand the characteristic equation is si= ply the deteminant of the natrix operator of aquation (24) set equal to zero.

Reduction of this determinant gives

~

o (25)

B

+

B2+B3 M +%+35

=

y L /L D

o o where B

=

1 1+c IGy3(So - W )(1 - u )

o g

Y "o

a "o a8 o

1~

+

B

+

~

2 1+c L

G,( S -U)(1-u)

(1+a)G,(S -W )1-u y

g g

k/L "o

G 11 y 0 1

+

+

1-a u, G,(S, - W )(1 - u,)

1 - u, -

y 2

o o

1 b/L "o(8+

)

1 + b /L 1+

B

=

3 2(1-u,)

1+c W G,(S,-V )1-u, 1-a N W,

oy o

1-

+

B e

+

g 1-u 1+c 1+c Gus(B ~w )(1-u )-

1+a o o y

u G il +WM g

y 0

y g g

+

"o u 1-u Gyg(S,-W )(1-u ) -

g g

m

+1 p0 -W v

g g

y o

y B

=

• -+

+

+

o -(

y, c, y

iu W

2(1 - u )

o g

1 "o

a(3 b

u Ky k+2 o

o

+

1-u 1+c IG,(S - W )(1 - u ) -

U y

g g

To 'ind the roots of equation (25) is a time censuming task.

'1he question of stability, however, can be answered by an indirect =ethod.

Let us call the left hand side of equation (25) a function F of the conglex varichic p To fir.d out 1: Z has zeroes in the riCht half of the p plene, uc draw a contour enclosing the right half plane, as shown in Fic. 3(a) and cap the contour on the F plane.

Ey inspec-tion of equation (25), we see that F, haa r.o poles in the right half plane excluding the origin.

If F has zeroes in the right half u plane, then frc= Cauchy's Index

'Iheorem the contour on the F - plane must encircle the origin once for each zero in the clockwise direction.

In Fig. 3(b) a representative F contour is shown for the case of no such zeroes. Fig. 3(c) shows a contour indicating two zeroes in the right half plane.

In order to detemine whether er not a two phase loop is stable, ve can calculate the coefficients of equatica (25) from the steady-state solution, and plot the F_ contour for the positive imaginary values or frequency beCinninc with 0, uatil the imaC nary anis. As a practical catter, the sycten is the contour crocco i

stable if the crossing is above the origin, and unctable if below.

The transient solution of equation (2' ) for arbitrary disturbances is readily 4

done by rteans of an analogue computor.

'de shall return to this topic after diccuccion of some experimental results.

COUPARISCU OF PREDICTICU AFD EXPERIIST Hydraulic instability tests on the A.P.E.D. large heat transfer loop were performed and reported in Reference (9). At the time of test, the analysis had not been finished.

Mien the analysis was finished, it became apparent that the test data was incomplete and could not be adequately checked vitb predictioa.

Several of tre tests were therefore repeated, and the missing information, which was pres-sure drop acrocs the two phase portion of the loop, was obtained. The experimental -

8 d A

F PLANE t k A

8 A

g D

C D

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(A) CLOSED CONTOUR ENCIRCLING (B) F CONTOUR-NO ENCIRCLEMENT-NO ROOTS RIGHT-HALF COMPLEX FREQUENCY fN RIGHT-HALF PLANE.

PLANE d

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(C) F CONTOUR-TWO ENCIRCLEMENT-TWO ROOTS IN RIGHT-HAND PLANE INSPECTION OF CHARACTERISTIC EQUATION TO LEARN WHEREABOUTS OF ROOTS FIGURE 3 s

values of heat input, subcooling, inlet velocity and two-phase total pressure drop vere used to solve the analytical equations for steam void fraction (which could not at the time be =casured). "ihis value for void fraction was trustworthy to require accuracy because it checked closely with Larson's void fraction versus quality data (Ref. 2).

2e coefficients of equation (25) were calculated from the data, and the partial F contours for four cases are shovn in Fig. 4.

The four cases proceed from Case 1 - very stable, to Case 4 - marginally stable. From the discussion in the preceeding paragrapu, it is clear that the predictions of the four cases vould be qualitatively the same, because the contour of the first case crosses far chove the origin, and the fourth almost at the orf gin.

Let us now look at the inlet flow and power traces for the first three cases, in Fig. 5, 6, and 7 The disturbances vore imposed on the loop by c=all changes of the loop heating rate. We first case is strongly damped, the second sif htly less damped, and 6

the thi:1 is noisy and overshoots. The fourth case - the trace has been mis-placed - vas observed to show lightly damped oscillations after the power change. The degree of damping of response in the different cases was controlled by a valve setting in the do.;necter line.

TIIE TRA'ISIEIIT RESPOI!SE OF TWO-PHASE FLOW LOOPS Transient traces of flow loop operation are important to developing quantita-tive knowledge of two-phase flov mechanics.

For exc=ple, in developing a model for SWR dynamics, it was hypothesized that a sudden increase of heat input in the reactor would cause the inlet water velocity to decrease temporarily as a result of the impulse of steam formation increase. Eccently analogue computer l

tmrh on the two-phase flow equations showed the same velocity effect, and it a

l ( F('u.= ]v))

l CBSERVAT!ON OF PRESSURE DROP 8 LOW TRACES 300 n

CASE I VERY STABLE CASE 2 QUITE STABLE CASE 3 STABLE BUT UNDERDAMPED CASE 4 MARGINALLY STABLE 2

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l 1 was obvious that a simple experiment could bc performed tc reasure it. 2e heat increase and decrease experiment was performed first at 30 psia on the visual burnout loop designed by E. Janssen. Two typical charts of power input to the loop and inlet water velocity are shown in Figures 8 and 9 The i experiment was repeated twelve times and in each case a flov decrease pip appeared after a power increase and vice versa. A similar experiment was carried out on the heat transfer loop at 1000 psia under more carefully controlled conditions than are feasible on the atmo:- pheric loop. Typical test results are shown in the chart of Figure 10. Se same effect is also evident in Figures j, 6, and 7 In every test run, the inlet vater velocity initially decreased after the heat rate increased and settled to a new equilibrium value which was higher than the initial value if the loop was operating on the rising portion of the natural circulation flow curve, or lover if on the descending part of the curve. In Figure 10 the power was increased twice and decaying oscillations ensued. 'Ihe loop conditions which produced the flow trace in Fig.10 were used to obtain coefficients for an analogue ecmputor solutien of equation (24), which is shown in Fig.11. The actual flow trace and the computer trace are quite similar, and compare quantitatively as follows for a heating rate increase of 7is; Observed Computed Inlet velocity i transient - 32ii - 30 5 Oscillation period 3 0 sec.

.4 sec.

I Damping factor 0.066 0.1 -2k-l l l r

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.. 4. At.41 r: re- " g/ i x., w.;., d. - -r- >.N-i i i i.> - i L, i p:7 7,4 - p. 1 -.-j L,. , z fx rrC "(. .i,.: ~j. p.p; -~i". .t l. ', L.. j e i - wm -.e7 c j ,I-e' &,L 9 ;: 3 y g ]_,._, m et-i ,e.. 'T.L, t-t - l _. - i T' 7 " r,, 1 s -4! -J _ -1_ 4 t l ,. ;,,jf,'3 -. _i .-~----,.~.y,- 1._ n ' j_.: g +l-i-_ . g.j._ u,,... j'*T-Q i 7.4d - ' ^4 L. !!,,4 .t< p f- -q.,,,,,., 6- _i-- .. h;,,, 4 ! 7,, m-. w.;....... u

g. 3> :., a_..a

~w w a3 w= = n u,, o =. -oo => .oW w a55 s e -2 Wwp og 38 W Io w a = J5 - W a .E-5>= l'j $ ally - WzN z n. D * * ~D) 90 T - 3 u,, M e M.1.- = l 4 -, - ~ ,,y y

b The error in period and danping factor is attributable at least in part to the approximation of riser transit time used'in the analogue simulation. He approximation errs at hich frequencies. The pressure gradient trace in Figure 11 deserves con: ment. Se input step disturbance causes an i= pulse to appear initially. This impulse is a math-ematical consequence of the assumption that the fluids in the two-phase mixture are incompressible. It is therefore not physical, as is clearly demonstrated in the test results. ANALOGUE C0!?tffATION OF TRANSIENTS The analogue computor circuit which was used to calculate the traces in Fig.11 is shown in F16 12. The circuit is composed of linear networks and follows directly from equation (24). We analogue computation of flow transients was developed to detemine ship's motion effects on reactor dynamics. We principal effects arc (1) the accel-eration of pitching and rolling which affects all channels symmetrically and (2) the effects of closhing which may affect channels across the core assynetrically. To conclude the discussion, therefore, analogue si=ulation of varying acceleration and of hydraulic coupling of parallel two-phace flow channels, which is necessary for the study of assynetric effects, is presented. Uatural circulation in the two-phase flow loop is caused by the difference in veight between two fluid colu=ns: the mean density of the riser column is less than the density of the downcomer, because part of the riser volume i contains steam. Weicht or Gravity force appears in'the loop pressure drop equations with the tem containing the factor "g", which is the acceleration of gravity. Mien the flow loop is accelerated, the weight of the colu.ns r-,

'e e I.aP A(S-w)= gg )g,, R n A3 : V A9 : &O+ 07; ! }--- OIS-W) ^ Al : M -Aw I +A w AR n A7, M v A2 /V\\r-- A Av ^ A8 : M v 0 /vv--- -Ag = t n R(t) = f U(r) d r L_ t-r m \\ AW -Au ANAL UE DELAY APPROXIMATION : A3 ; O /vv- -AV A4 A9 ; O /vv-daa ,a

• T2 R = p,a uW O

d ez ' T de +Ay -1l \\ - Au /Vb / --C /VV /% A5 A6 A4 : AR / 2 A7 m /\\r-l ,,, At s -w) n \\ +6v \\ ~# / a -Ag = 0 JV%-- JV\\-- O-- ANALOG COMPUTER CIRCUIT FOR TRANSIENT SOLUTION OF LOOP DYNAMICS FIGURE 12 l l

- changes by the component of acceleration force which parallels the gravitational vector. In ship's motion the accelerations are transverse and rotational, as well as along the riser flow direction. 'Ihe flow loop is confined by channels j and thus the transverse accelerations affect the mass distrubution of the i two-phase mixture across the channel. The change in fluid distribution a across the channel vill in turn. affect the pressure drops along the flow direction. In a first approximation, however, the transverse effects are 1 assumed to be negligible. Under this assu: ption, the transients that follow accclcration changes can be computed by variation of the weight term in equations (17), (18), and (19) by the amount of the axial component of f acceleration. Se solution following a step change of 20% of gravitational acceleration along the vertical axis is shown in Fig.13 for conditions of the T-7 Tanker 4 Tcforence Core. S e traces show the inlet water velocity to reach a higher equilibrium value and the average channel steam void a lower value, as is expected. 'Ihe small dip in average channel water velocity, W, immediately followini; the step is intriguing. Since the velocities are measured from the channel frame of reference, the water velocity in the heated section vill initially decrease because the channel is accelerated upward. Shortly, how-ever, the heavier downcomer leg predominates and increases both inlet and channel-velocities. The equilibrium values after the 20% acceleration i increase are: i pressure drop + 20 % chimney water velocity 27% inlet water velocity + 8.$$ l averace void fraction - 11.'(% 4 4, . ~.

.) l 1 1 I i_;,iq..,:.N.u.Wdn t,L.,p aq_.P._! r. r W_ p".. .... hu!. cio,Mi .m .n a y, s+3._,._..". L. ;..L _. L.,a.F. l.'s : s. a.r N '.

• y".J-m... ?.j. 2,3., G $.

m Hi - J Iw I M

1. L

...J=+a.. 2 .. a % ~.,.. a PRESSURE P " 4J.g" ic h_.o :. L.,' s' ". :_1 WP 1 : F... W. pi P.4' R.u...,. , - - -4.: a4 ep - : e ._.L. ' i.. i s- ..",_ad d7 GRADIENT a i j! et . a '3-- l. - -.'-.- ~ ' p.,. 4 1.2 a p i,,,j d '.Ml.-.c.1 i,,,M N -a.' Fl T. M .I. c.. q..j _.. M: y. .. p;4 3x .i ~ +:i m: t mj - 4 q.b_. ! m. .. u._4...4.7 a. E o. y a i..1 4I i l i..'.,'l }. ^ Il: i q'j/.j !, l, j t'.l-l .,'l, .., a f [ l l. ','. [._(' t t I 1 ' h.d ).. q".' '.. _.p/.C._,,_.,l.W.; ' j.6 . u' j.,.. ; .. ( 4_ ..[ l " ' "p D 4.7..J'..C [ I..IG WQ n e4 : n 14i A P, ....b.+ L _._.I y %(;KCC. 7._r 4.._f ; hhy' p +.

4..p

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• [:

I e h e ,_.p_.- ... _' L'. ' +.i 0 - J..>< 7 d WATER VELOCITY ,_ A., q'... E_.'... a : y. i ,_.ici.._i !'._+'..!_ E,_m. IN HEATED 0 : F.T o ? i i 7. . '.,'!. '..t. '.i .,J..y' 's CHANNEL 1.04 w. '.': ? L-M _.! .y 7' 9.,;. 3 L......~ m l

. 3,.

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14. _ _. _ _

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4.. - ', ! ' l l

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• e.

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7.. -,-_.+...!_'

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1. :, ; ' I =,

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• U i - - - IP R

n-

.'**'"-?'"-**',~~~ '.p'*-"*~

O i 4 ..c'~,.. C,..;P 2W 2 ~ - ai 4:, AVERAGE .'):. e s 's.y m..

b...y G y 0.

t-s STEAM VOID ... m,,.: %.; ; -, r, -i-C .L.. 'H "i y-W. r. -"p... 'u;.y r. 1 - r%. .J rp c-i 1-;-, ",... ' ' l.. f. - y- "h...,'. - l l#

. k"., #--i. v.,..- l- + -,--,:.',.-

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a<'

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: L

%;.a. 7_-v1 c.j ; gc .., y ty wa. r, 74,, p &.,v. ~.,i Std, 4..t. h.; M @'. 0 a.

.-@L u.a.

g l.' A .i FIGURE 13 FLOW TRANSIENT CAUSED BY ACCELERATION si - D**D " DW d* 4

M

.; t. The damped natural period is 2.6 seconds. Se results given in this paragit'h c are purely theoretical. A co=parison of theory and experiments on an accelerated, rockin6 two-phase flow loop at 1000 psia are reported in Ref. (10). Se simulation of parallel channel flow problems is a simple extension of the i mechanics of single channel problems. Se parallel channels are coupled by a common downcomer velocity head loss, v3, as shown in Figure 2(b). Che set of equations is written for each heated channel, including the individual head losses. An additional equation relates the common head loss to the driving pressure. Se effect of an increase in heat. rate in one channel on the other channel of a coupled, inicially symmetrical system is shown in Figure 14. The parm:eters re-present two channels s'imilar to the T-7 Reference Core Channels. In this problem the velocity head loss in the cocmon leg was chosen to be equal to the separate leg losses. We result is to decrease the flov in channel whose input of heat is unchanged, by decreasing the driving pressure. Also, it is noted, the sving of variables in one channel is cut of phase with the sving of corresponding i variables in the other channel. Be equilibrium values followinG a 20% increase of heat rate in channel 1 arc-Channel 1 inlet vcLer velocity v 5% average steam void fraction +17% chnnnel 2 inlet water velocity -0.4% avert.go steam void fraction +0 9% i ne effects of parallel channel hydraulic coupling are explored more fully in Ref. (11). A discussion of downco=er transit time effects is given in Appendix B. '

s i 4 I i n.. ~.'!'i, '.'I '.{. ! l T',e 4', l . l ' < +. 3.!.T ' ~ 'l ~ ~ ~ ~ 7.7:.. v...; >. c.l'- -.'.. [;.4 !! },. ' I,; !,. l, 'I, .e!.!l!il...hl l l f, l l ^. r l. . l' I' ' h], '. "I, t , - g gr,:,t. q'.g' '.tF "4@#..tt/..n- ..M r- *4 * ~1 T;,9Fp4

q q--l 1

la. .;, ] -.. m] n ', i.,3 S i n CHANNEL l '~l M.,. ~ ~;x LL. g m,, r q.,p I i. . F] yi-1, i s J j'".' '4, %. t y(...p,,.,p.g a e INLET WATER v M I, t.3 gy m' .g. g'7 9 7 i "7 h,.pg,. f l j+ +-j.. p %ggg VELOCITY 10 u.+,.1-%y n v.a bg. g.

e. a... r. 4...W,7. -._) -e.Ls.

__.T..:a.2a ..; -m, ~.-. y_e . c a.. 6.l,- , h.3 + uww . r,. +t. y . _x....;_ . L. _.7, ; ;.f M.{.g. q..,a..,1 . t. j l.. );.,,.,... ~. ' l ;._4,.,.._y'. ' i.;.,7. l_ .i . a. ,p, ' h. .l. j. l o po,,1 c. l [' LIi, L..u... h i '.. !I l.. ' -- _i..'{ t 2- +- 2 L i-o IE' iy*,, ! ',i. a t .- i ' '.-4.' u ,( },,

j.. '. " '.

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m.. { t

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J. L.

ej. STEAM VOID [ 7, - . ;c 7..;/ ! i _..' .,i.. ,f,- q. z[.. 7...~- r_, 6 i r ,p.,i 10 :+fl -t-i- +.. r 1.,.._.a... 1.1 R l ,. b. r!. -l 1 i LJ.. g., ,t 1 * ._ 4 5 . ~ _ _ _ ,-i . i,.,, ,..p. 1

'y.,.

f.V_t ' j l. I. 7 --.! .y. 1 i y R ,.q ,i [,,.. '. i ,6 ) io . i. . +- 2 . r. ; y 3... p.. j.... 7!.,. i l ~p,...', . ;~ .i .i . l ..l q. u,,,i ; o +, - t , i i.. t ( a j '. ~ j i 20 b H M, y n. b.. ._,.t u._..; 4a. i_! 2 - N. : - b+ 5. - M, - b..,.. E 1.04 v ~ a a CHANNEL 2 DQ,- + q c-M : Tr p7', by i m bii ~ 1 b ' [' + 1 -1 " N ;J ^ !"'~ r 1NLET WATER ' ~ ~ "tp VELOCITY -I -1+ Y- - 'l.. i -! -~ V M c-

~ '

20 r i .i..1.,. is i W.g,.'Ii D,(! j-i -[ i ~..... r + -,. i~ ...,_.1..

.,,..q a.i..'g'(.. '.

3, -}..~ 5 6 t. ).t. .i ^ a -.i. : .3.. g.. ., i t. ...._u i... 1

s..

u. l.i l l i ! i ) i q.. , I (' ',..'.] i i [. i. a-.,, i .m p...4. { i .i g.y. ._i 3. _. . g... 1 ei .e i .a c..x i.. _g ,..t i..,_.," p 1.02 R 20 @J.c s..l '. '.C I ~' 7 ". Aq..%'". 1 .[ t. _. _1:. t.1. _. L' L.:. i CHANNEL 2 .i.. .t a i K. 4,. t .J.. AVERAGE a q p y...s.._, p. ...p. 1

.n, ;

i2 ~. . c. Pt -iW W " - n"I t

t.. ' "rr

- M; ;"-t 9 STEAM VOID -R i, i 20 i _ a. p.. y.a., L e..- * - e -, - - -- - - -,--_.,;_..'..,.,;.,....;._'..r,----*---- -:--e 1 - -"- .-+ 4 e .c a 1-i .-. j _t, _...g _;... 9 ,,, i 3 ,.c . *I.. L.... D, . 3. 1.. g _. "I.L......,,. - A ', ;. .HI.: _c ,1 s. ,[ .r - i-p.-h. +y.- i [-h 1.j 4 5

. 4..y

{ J L L $_.'..,a.L'w J ..r. U1.'..1.1.: d... . g.....!. i ..a.J ) 9.4. , p '.;. - i 7 [-( y. 6?.a. .. __' u._._y.._L),y.$ Q !.. l i ]4 '.! j I ni g.: ; a. FIGURE I4 PARALLEL CHANNEL TRANSIENT . u. s D Y 9 D4 g. JL.}L L u

-. -.... ~. - t l l l C0!iCLU3IOUS s he two-phase flow mechanics presented in this report closely check the steady i state void measurements of W. H. Cook. She mechanics check his results a priori, usin6 only the input conditions of open. ting pressure, section ecoactry, inlet velocity, and heating rate. h e loop analysis and the two-phase mechanics.together t i i successfully explain the pubations and transients observed on the A.P.E.D.1000 psia Heat Transfer Loop. he parameters governing two-phase flow loop dynamics are revealed by studying the loop equations. First we consider fluid transit time of the two-phase i section. Because the loop driving force is veight difference of the columns, j and because changes in the average void fraction are transported across the I riser at the average two-phase fluid velocity, it is cicar that the loop dynam-i 1 1 ics in the time domain vill be dominated by the transit time. /.nother way of l stating this fact is that a change in steaming conditions at the inlet builds 1- ~up an effect on the driving head during one complete transit time of the riser. I' } The mathematical consequences are evident in the time delay tezu in equation (22). 1 Experimental'_y it has been noted that loop oscillations when they occur, do so { at cyclic periods a bit longer than the transit time. Analysis predicts the period of an oscillatory system h/3 to 2 times the +.vo-phase section transit 1 j time. 1 1 j Re second point is the dependence of steaming rate on inlet water velocity l and subcoolin6 Bis dependence is the primary cause of flow pulsations in the A.P.E.D. loop, and it is of concern in the range of interest to boiling vater reactor design. Paradoxically this cause is therredynamic snd not hydrodynamic. Consider the energy equation: stenning heat rate - total heat rate - o (subcooling per unit mass) (mass flow rate) (area). 1 3h.

_~ - i 10 i I -If ve suppose inlet velocity increases in a loop that has been in a steady state of operation, the equation states that steaming rate vill decreac.c The riser j steam fraction vill then decrease, and with it the drivin6 head during the l passage of a transit time. H e loop velocity vill decelerate as a consequence, j and as velocity decreases, the steamin6 rate vill rise. If certain conditions are met, the process vill be oscillatory and undamped. W e main conditions are 1 that steaminE be strongly dependent on inlet velocity, and that the natural i circulation driving head be substantial. The conditions can be cet with high subcooling, and high heating rate. Sey can also be met at low heatin6 rates with low inlet velocities, but the latter conditiens are of less practical interest. Se degree of subcooling is related to the operating pressure and the specific volume ratio of saturated water and steam, as will be discussed. I It is useful to examine the structure of the characteristic equation (25) to find out what the stabilizing and de-stabilizin6 influences are. ',ic deduced " rom the complex variable plots in Fig. 3 that the flow loop is stable if the plot of F (p) = F (iv) passes from the first to the second quadrant without i crossin6 under the origin as v is increased from zero to high positive frequency, and vice-versa. If the coefficients of the delay function in equation (25), 3 ""d 3, are id nti ally z r, the fu eti n is a imple quadratic. which are B 5 Furthemore, if the renainin6 coefficients B, B, and B are positive, it is 1 2 obvious that F passes from quadrant 1 to quadrant 2 above the origin. 'le ~" will now see how destabilization is caused by the del'ay function u i which is plotted in Fig. 15. The numerator is sinusoidal at periods of 2 H, f and the function proceeds in a spiral-like form in a clockwise direction i under the abaciosa for positive and increasing values of v, and its =agnitude l approaches zero. If the coefficients B3 ""E 25 "#* " " "11 **d P Siti ""1"*8' it' is apparent that they displace the F contour at each point by a vector

IMAGINARY AXIS a REAL Axis 97 24 7 =O O 22/*~ 28 20,/ 19 [ 9=l -0. 2 - 18 t 2 z 17 ' y_ 24 j 16 *\\ / -0.4 i I 15*\\ / I4 \\ 5 f 13 A 6 \\ 12 \\ 7 /- ' 'O. 6 Il 8 p'

• g "0%.

+#, l 9 -0.8 _ c _iv iv - 0. 2 o 0.2 o4 0.6 o.g 1.0 PLOT OF DELAY FUNCTION FIGURE IS -

sving the direction of the delay fbnetion for each frequency. In the case of B, the direction, usin6 compass terms, repeats itself pointing first South-East and then South-West. It is the South-West direction, with H > v > 2 H that destabilizes, for a sufficiently large coefficient B will cause the F Contour to pass under the origin. As a practical matter, the frequency of 3R oscillation vill be H<v< 7 ' hich corresponds to period c between 4/3T and 2 T. A condition of instability is that B must exceed B by about a factor of 4, 4 because the delay function magnitude is attenuated by roughly that factor in its destabilizing direction. The system can nevertheless be stable if the coefficient B is large n uch and B, is sufficiently smaller than B. 2 .) The parameters which comprise the coefficients show how to determine system characteristics, and also give considerable physical insight into the problem. The destabilizing coefficient, B, is effectively the velocity dependent portion of the static head pressure drop in the riser. It is increased by long risers and by subecoling. It is decreased by hi h fluid velocity. 'he 6 stabilizing coefficients are B and B. B represents the velocity head 2 g losses in both the single phase and two-phase sections, and large head losses vill always stabilize the system. 3 is a mbination of head loss terr.s 2 and fluid inertia ter=s by virtue of the downecmer to riser length ratio L / L, nd will also tend to stabilize. The coefficient g is essentially an d inertia quantity. As g which is proportional to L 1' is increased from d/ zero (by supplyin6 vater from an infinite reservoir at the section inlet) its effect on stability proceeda from no effect to possibly adverse, and fi.lly to an absolutely stabilizing effect, when the dovnecmer inertia is so large that the velocity cannot change appreciably during one riser transit time. The effect of subcooling on the velo < it,y dependent static head has 'xen diccuoced, and is evident in the tor OS in coefficient B. Subcooling has another impor-g 5 ter.t and curprisinC cffect. It reduces the stab 111:inC influence of riuer velocity head loss by means of the tern os in 3 and D. To understand sty, g g uc vill examine equation (20). At cufficiently h1Ch subcooling, a S 1+a the riuer atear' +'r volute flow rate decreaccc for an increase of inlet velocity. . is caused by subcooling. I;ormally the stean 311,3 velocity 1. sensitive to inlet velocity chances, and therefore the void fraction decrease: with the steam volume rate. Ous the water acquires a lar;;er flov area. Althouch water volume rate increases, its velocity may decreace. I" the vater velocity decreases, so will the velocity head loss. Friction forces that increase with velocity tend to stabilize the loop, stether they are in the sincio phase or two phace part of the loop. High subcooling causes riser frietten to decreacc with inlet velocity inercaces, and therefore cahes the loop less stable. A useful des 1 n curve of subcooling limit versus operating G y aure is given in Appendix C. The effect of riser velocity head loss is 1:portant in the design of reactors with internal stoc= acparatica, and the stability of cuch designs chould be checked. Qis report extends prior work on two-phace flov technnics in Ecf. (3), and a simple flov loop model in Fef. (9). The conditions of stability of flow arc derived under cinpli;yinc ussumptions. The work does not deal with trancient velocity gradients, such as in sluc flow. Two other topico, as well as slug flow, need further exploration. The first is the trcnsient behavior of two-phace flow at an crifice. The second is the ctcan-vater chcar stress. The possibility is cuccested in the equations of this report, of a decreare of the shearing stress between water and stcam at hich void fractions which vculd lead to a hi her clip velocity and a lever void fraction. Such behavior could recult t in an oscillation ci:nilar to the one caused by subcoolinc. ACICO'JLEDGD SES 'Lhe author wishes to acknowledce the painstaking work of J. M. Case in comparing test results and analysis, the assistance of J. M. !brlock for test loop operation, and the supervision of C. L. Svan for tests and instrunentation. 4 1 1* REFERENCES .l. Ihrtinelli and relcon, ' Prediction of Precoure Drop Durins Forced Circulation .of Doiling Water", ASIG Transactionc Vol. 70, p. 695, Aug.1948. l 2. Iarson, H. C., " Void Fractions of Two-Phace Steau-Water Mixtures", :!.S. Shecic University of Minnesota,1957 . 3. Deckjord and Harker, " Hercules I - Steady State Calculation of Two-Phace Flow", GEAP 3261. I h.

Levy, S., " Steam Slip - Theoretical Prediction from Ibmentum tbdel", AS!I Paper, 59-HT-15 i

5 Cook, W. H., " Boiling Density in Vertical Rectangular f.blti-Channel Sections l with Untural Circulation", ACL-5621. 6. thrchaterre, J. F., "7he Effect of Pressure on Boiling Density in Shlti-Channel RectanEular Channels", AHL-5522. 7 Cunter, A. Y and Shaw, W. A., "A General Correlation of Friction for Various Types of Surfaces in Crossflow", ASLE Transactions, Nov.19%, ~ Vol. 67, ro. 8. 6. Zuber, H., and Tribu s, M., "Further Remarks on the Stability of Eoilin6

r. eat Transfer", AECU 3631.

9 Levy, S. and Eeckjord, E. S., GEAP 3215, July 15,1959, " Hydraulic Instability in a Natural Circulation Loop vith Ect Stca: Generation at 1000 pcia" 10. Quinn, E. P., Part I, "Untural' Circulation Loop Perfor=ance at 1000 psia Under Periodic Accelerations", GEAP 3397, Rev.1; Case, J. M., Part II, 11. Case, J. M., " Neutron and Parallel Flow Channel Coupling Effects on the T-7 Flux Trap Reactor",.GEAP 3508. 12. Eben, R. H., Pressure Drop in Two Phase Flow", thiversity of Minnesota Thesis, Au6ust 1956. i r i 40

l 3 /TF4.' DIX A - Cl.IEULATIO ' OF FEIC'iIO:.' PI2030EF. D00P FAC'ZCE S. Invy succested a way of calculating the effective water to Vall perimeter to obtain two-phase friction pressure drop hva established nothods. From equation (5) J ve asse.c pv - p F(u), where p is the Ccometrical perinctor, and F(u) is to bc detcInined. Uc have now P 2 f (Re) gh W ) F (u), P = y y cad from the literature G G G G TP (1 - x) 6 13 ry = = 1 o using : ben's (Ref.12) parameter 4 ' and is shown in Fig. A. In this fashion y Dy can be determined from the literature or from experimental data, i l 4 1 1 -i-f ~ - - -, r,,r

g 6 r db I l.O GW (u) G N (u:0) O USING MOEN'S DATA o.8 N O O.6 n \\ O g - a 0.4 U: G W (u 2 N 0.2 1 0 0 0.2 0.4 0.6 0.8 1.0 STEAM VOID FRACTION DEPENDENCE OF WATER-PIPE FRICTION f'RESSURE DROP ON VolD FRACTION FIG A 5-

t = APPEITDIX B - D0'WI!CCIGR TRAUSIT TII2 EFFECTS ' In the body of the report, the heated section inlet subcooling was assumed to be fixed. In practice, feedvator is controlled to maintain wat' r level, and subcoolin5 varies with feedvater flow and with recirculation flow. The effect of subcooling chances is dcicyed by the downcomer transit time. Usually the dovncomer transit time is many times the riser transit time. For purposes of loop stability, therefore, the inlet subcooling is effectively constant over the period of several riser transit times. For purposes of calculating transit response, subcooling variations and downcomer transit time should, on the contrary,' be included. Two of the loop equations are affected by subcooling changes: the volume flow, equation (20), and the inlet vater flow, equation (21). Instead of constant subcooling, let us assume that feedvater flov, C, is constant and equal to the averace utcacinc rate. She averaSe water enthalpy at the feedvater mixing point is therefore v-c) hoc h v y 3e(t-T ), where T The average water enthalpy at the inlet is h p D transit tice. Incorporation of these relations results in the following changes to equations (20) and (21). RS + (1-R)W = v(t)-(E-1)h..-hc C *(t-T ) + (3-1) y h,-h v D v (1-u)W - v(t) + hu-h C 7(t) h -hv v(t-Tp) 3 For chances that occur in periods less than the downcomer transit time, the - two formulations are equivalent. -111-i -n.

For transicuts of longer duration, the ratio v(t)/v(t-T ) tends to citigate D the destab11 icing tendency of subcooling. Increasing loop velocity tends to incroace t'ic proportion of recirculating water to feedvater, and thereby to decreace nubcoolin6 Pulsatin6 flow traces of two-phase flov loops often show a basic frequency of oscillation detemined by riser transit time, and a lon6 envelope modulation of the basic frequency of about the downcomer transit time. The pulsation is continuous, but its magnitude increases and decrec:,en cyclically. -iv-

=__ - l APPElJDLt C As yet a convenient fomula for predicting two-phase flow loop stability has not been devised, and it is necessary to do the detailed calculations. Eccause of the prime importance of subcooling to stability, a graph of operating pressure P h -( l ':rsus subcooling is useful. Two curves of the parameter a S y se = are P h -h 8 8 V plotted. in Fig. C. A loop is probably stable if it operates above the curve

1. If the riser void fraction is high, the region below a S

=1 vill ap = 4 be unstable. At low void fractions the region between a p 1 and a p 2 = = vill be margirial. 4 i f d i t b l 4 1 1 mYe t i w y y - +- s- ,y v.

1OOO 7OO aO =I 400 aO=2 2OO 5 E N h -hw 00= 1OO h 'h" '~ s E E _$ 70 5 + 40 20 l0 0 10 20 30 INLET SUBCOOLING BTU /L B OPERATING PRESSURE VS. SUBCOOLING FIGURE C .g-

I!CflE:iCLATURE Cross scetional area A Constants of characteristic equation 3 Condenser feedwater flov C Pipe friction factor f Characteristic function F Acceleration of gravity C h Water caturation enthalpy y h Steam saturation enthalpy 3 h Fcedvater enthalpy e Subcooled water enthalpy he 3 Velocity head loss coefficient K L Lencth of two-phase vertical portion of loop L Lcncth of downcocer D Pressure P p Perimeter Heat input rate Q q I: cat input rato per unit coolant volume R Average steam void S Steam velocity Surface tension v Trar ~ ' t time T Steam void u Inlet water velocity v v ?ro-phase fluid water velocity Channel vertical dimension x .;1-

3D a

h.. - 12cc/h -h 3 y PP y3 Y

Q/AD (h -h ) y 3 y P Water-wall shear stress / unit length y P Steam-wall shear stress / unit length 3 P Steam-water shear stress / unit length vs p L/v + 7 dimensionless complex frequency Co= plex frequency (sec-1) 7 Steam density 9 3 p Water density y -viii-J .}}