Mathematics and North-Holland

Computers in Simulation XXVI (1984) 475-484 475

TWO DIMENSIONAL MODELLING OF THE HOMOGENIZING EFFECTS OF FLOW OBSTRUCTIONS IN TWO FLUID FLOW *

M.B. CARVER

Applied Mathematics Branch, Aromic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, KOJ lJ0, Canada

M.A. SALCUDEAN

Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, Canada

It is well known that obstructions in two fluid flow have the effect of homogenizing the two iIuid mixture, resulting not only in a more uniform phase distribution, but also in a significant reduction of the ratio of fluid velocities. This paper shows that these effects can be modelled using a two fIuid computer program, and compares results to those obtained in a rcc:cnt experiment.

1. Introduction

As a nuclear reactor system relies entirely on fluid circuits for energy transport, mathematical modelling of thermalhydraulic phenomena plays an important role in reactor design and develop- * .tent, and methods of improving the accuracy and efficiency of thermalhydraulic computations are sought continually. A simplified fluid circuit dia- gram of a CANDU reactor is shown in Fig. 1. Throughout most of the piping network the fluid behaviour may be adequately described by one-di- mensional models such as those described in [8,9]. However, in the reactor fuel channel, flow must distribute itself amongs the intricate flow passages of the fuel bundle. In the secondary side of the steam generator, and in the calandria, the flow distribution is also complex. One-dimensional analysis is adequate to simulate overall or bulk energy transfer, but multi-dimensional analysis is necessary to model detailed local distribution of flows and temperatures in any of these geometri- cally complex components. The secondary side of

* This paper was presented at the IMACS International Sym- posium on Computer Methods for PDE’s, Bethlehem., PA, U.S.A., June 1984.

the steam generator develops two-phase flow, but this flows vertically, so the homogeneous or single

HEAVY WATER REACTOR (CANDU-PH’JV)

(PRESSURISED HEAVY

HEAVY WATER MODER:TOR 6 COOLANT

Fig. 1. Simplified view of CANDU reactor showing flow paths.

0378-4754/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland)

b . M. B. Carver. M-A. Salcudean Homogenizing in IWO fluid jlow

!,*R: . tel al two-phase flow is reasonable [S]. ln / i” &Pm . fuel bundle, however, which is shown

I g#f ) FL2 2. the flow passages are horizontally ori- :3; y b “det boiling conditions and lower flows

‘B* I*#‘, i,- li, therefore, a tendency for the steam and ;= it+T w flow in opposite directions with respect to T: .!;I: u I A model of two-phase flow, which permits &C ~~~~t~r and steam phases to flow wit’h different

ttc~ is required to simulate this gravity in- aration tendency. The Chalk Ri*ler

,:tboratory (CRNL) is involved in an program of research and development of ibrium two-phase flow modelling and

nsbnal computational analysis of flow I hundIes and related geometries. Develop- of prototype codes incorporating advanced

cl;% is underway at CRNL and the Univer- tt;rwa [6,10,11~, and models and numerical

s wCpCsd are discussed in a number of publi- ~~~ti~~~~ ~3.15.16]. ParaPlel phenomenological inves-

tm IS required to ensure that the computer s incorporate the mechanism necessary to late experimentally observed trends. The de-

v&Iprncnt program, therefore. contains a number carrdinated experimental and analytical pro- ts. each of which acts as an auxiliary building

1t~L providing information essential for the n~rtl project. ‘This paper concentrates on one of csia’ funtiamcntal projects, which examines the

of flow blockages on two fluid flow in ntal channels. This study is relevant to the

effect of the structural endplates in a CANDU bundle, which, as clearly seen in Fig. 2, introduce a blockage in some of the flow channels. It is well known that such a flow blockage tends to re- distribute flow and introduce a homogenization effect. Currently, in a subchannel code, such as ASSERT [6], this effect is modelled by introducing local loss and diffusion parameters and an empiri- cal modification to the drift flux model [6]. How- ever, studies are needed to quantify this effect in further detail. A recent experiment reported iby Salcudean et al. [ 12,131 studied the effect of flow obstructions on pressure drop and void distribu- tion in horizontal air-water flow, and confirmed that a flow obstruction causes several separately identifiable but related local phenomena; a pres- sure drop, a mixing of the two fluids, a reduction of the ratio of air to water velocities, and an eventual return toward stratification. It is of inter- est to know whether a two fluid model contains the basic mechanisms necessary to simulate these phenomena, and the current study investigates this question. Obviously, a one-dimensional two fluid mode1 cannot simulate either the tendency towards stratification or the homogenizing effects of the blockage. The effect is three dimensional in na- ture, but can be sufficiently characterized in two dimensions to provide an adequate proof-of-con- cept study. The two-dimensional two fluid code TOFFEA [3] has been used for this purpose.

M.B. Carver, M.A. Sakudean / Homogenizing in two fluid f/m

Plan Tube

7) I.95

p * 153.3 We0

Plain Tuba

d I6 5 %

7 I95

p 1533 KPo

Tap I

..I_ . --

__.--* i”” />A =- /

.-A-.

.

.

,-/

) . . * a 22.1 Y.

7 : I.15

P: 153.7 KPP

---_)

Dwoction of Flow

tap 2

5 ! 26.6 Y.

7: II9

P : ISI. KPO

too 3

Flow obrtructlon outho

ti. 21.0 % ,,*..I. .*..a

9 I 64

P 1506 KRY

(a). Change of void distribution. Peripheral ob&sction. Bubbly flow. W:ltw I.1 kg/s. Air, 0.000775 kg/s. Quality: 0 0704%.

Top I

t- ‘\

tj 25 6 %

7. I 10

[>:I557 KPo

6 54 0 %

7’ 034

p 148.3 KPa

47,

Tap 3

ii 17 4 */a

‘7 066

P 1494 KPa

- -- 6@

--. 50

_. _ 40

_-__ _ ___ a0

-_ _ -_._ 20

---- -. ‘G

flow Obstf”Ct~om ouwm@

(b). Change of void distributron. Central otwtructwn. Bubbly flw. Wdlsr. I.1 kg/\ Atr

0.077.5 kg/s. Q&y 0.0704q.

Fig. 3. Results from the flow blockage experiments.

M. B. Caruer, M. A. Ssrlcudean / Homogenizing in two fluid flow

2. IBe flaw l&&age experiment

test section was a 25.4 mm id. 3 m long ho tat plexiglass tube. Air and water flow at w?ected rates through a mixer and a 3 nn long

h to establish flow prior to the test section. action was measured at three stations using

an q&al probe. Typical results for peripheral and central flow obstructions blocking 25% of the flow area are shown in Fig. 3 and illustrate the

h~n~rnen~ mentioned above.

f d. +IKMFEA computer program

EA (Two Fluid Flow Equation Anal- 3 a two dimensional computer program that

I?C ha-;c8 on the same principles as the well known ‘fW’K3i (7] single phase code, and was developed

niversity of Ottawa version ill]. The aflv dmclopment of TOFFEA is described in [ 1]

ad (41. $hc main features of the numerical scheme ;WC now di,wusscd,

“hr conservation equations governing adiabatic two fluid flow can be written in Cartesian coordi- natcs for fluid 1 3s follows:

C *onscrt~ation of max.5 :

cp dcnatcs volume fraction, p is density, u are ~‘Axif~ components, g is gravitational

acceleration, and + and r are functional relation- ships for interphase drag, and friction and viscous stresses respectively.

For brevity, these may be generalized in vector notation for each fluid k, k = 1, 2:

(3)

(4)

The velocity component along coordinate X, is denoted by toi and aUJ/aXj E ill~/iLx + au/ay: S, contains the remaining terms from (2), and (4) comprises both (2a) and (2b).

3.2. Numerical solution

The solution scheme used in TOFFEA is a novel extension of the TEACH scheme for solu- tion of the single fluid equations. The equati#ons are first discretised by integrating over the control volumes for each fluid k, using the usual staggered grid concept:

c [(QPu,uJA),+ - ( apu,u,A),_] k

+A%( P,+ - PJ = s,,v, (6)

A and V denote cross sectional area, and volume of the appropriate control volume and i - or i + denote upstream and downstream values.

If an initial pressure field is assumed, (6) can now be reduced to a form which gives a first estimate of mass flux in terms of the neighbouring mass fluxes and the relevant pressure gradient:

[ a(apu), +C(hapu), + e( P,+ - P,__) = SJ.

n 1 k

The coefficients Q, h and e contain the appropriart, expressions reduced from (6), Fut (7) is now lin- earized, and can be solved in point or matrix form for a new matrix of mass fluxes, However, as the pressure field was not necessarily guessed COP

M. B. Carver, M.A. Salcudean / Homogenizing in IWO fluid/low

rectly, the new mass fluxes will not satisfy the continuity equation (5) but yield a mass imbalance D:

c bwd,+ -(we),-] k = D,. (8) ._

As there are two continuity equations, the question of how they may be used most efficiently to ex- tract the remaining two variables, pressure and volume fraction, must be addressed. Shah et al. [14] have suggested combining the two squations (8) to form a total mixture mass balance:

Ii c bP~),+ -(ap~&]~==D,+D,=q,. k-l i

( 1 8

A pressure equation can then be derived in the classical ‘SIMPLE’ [3] manner. It is necessary to find a new pressure filed that will drive 9, to zero. This can be done most efficiently by the New ton- Raphson technique, which yields

dP= - D,/(dDs/dP). (10)

Thus the required equation for the change in the pressure field, dP, is obtained from rearranging (10):

= -Ds.

As expression for d D, is readily obtained by dif- ferentiating (8):

k-l i

=z -o,,

and from (7)

-Adl(cupuA)i..] k

(12)

+e(dP,.+ - aPi_) (13)

Equation (13) is handled mar-r;! readily by ne- gkcting the variations in neighbowing masL: fluxes, thus relating changes in each mass flux component at each point to the changes in pressure upstream

and downstream, as in SIMPLE. Substituting tflc simplified equation (13) in;o ( 1 B ) now yicitls :i definitive matrix equation fc<:r the changes in the pressure field required to drive I?, towards rtcr~. ttt the form

(A, +A,)dP=A dF -D,p= .-{D, -+ r,,). (14

This is a Poisson equation and rellatcs. for both fluids, the pressure correction required at ~clr node to reduce the mass flux in\baliulccs cxistin between nodes.

The pressure correction resulting from (14) will not precisely satisfy continuity because neighh(:ur- ing flux variations were neglected in (13) and void fraction variations were also neglected. Shah et al. [14] obtain a further expression for void fraction by solving (8) for cyl with D, irnposcd as zero.

Numerical experiments at CXVL using this a~- preach for kr-water flow have show11 thnt the lighter fluid is poorly conserved. This can bo sx- plained by noting that (14) taxis to dictate i\ pressure equation based on D, -t- D, = 0, not m

D, = D2 = G. When p, GC pZ, as for air end w;itQ”r, the lighter fluid hardly influcncos the ~rcsstw field at all, 3s D, e D,.

Thl.: problem is clrcumventcd ilrl the ‘l’Of+‘f~:~ code by normalizing each continuity equirtic)tl (8) by the reference density 9,) to give 11 vcrlum:‘tr~~ conservation equation eqilivalent to (9):

k-l I

where r = pk/pukq 0; = Dk/p,,c. In this case as /‘A z 1, the phnses illIt thclt

contributions D’ to the pressure cq\li\ti(lll :\r~ equally wei,yhtcd, and a pressure field thnt impcp\t‘h continuity in each is coil>puied.

.+

uQLUPX FRFXTION - RIR

--- ---------_.._ _ ii _ _ __ _ _ _--I__ _..-- ._c- -_--._ _ “_ ---_----- _ - - - - - - - _ _ ___ - - - - - _ - _ _ _ _. -

Ii -_ - - - - _ - - - _ -. _I - _ _ - _ - .- _ I _ - -

- I - 1 - - - _ - - _ _- - __- - - - I-- . _

- - - - _ - - ___-_-______-_^

_ _ _

- ___ _. ~________ - _ _

much fuiih3 downstream than for tht: peripheral different velocity L-~W~~~I~LW~~. 111 III{- I\\(’ ph.t~“\ obstruction, clgain as observed in the experiment, The variatron in slip iilonb ) :eilt.i i\r<f”(h\ I~hB’ illil~~ll~‘~

In these, the vertical dimension has been aug- can be dedur~! qu:llitativcl~ fr crm tl’tc+i* k f.9, !a\’ mented in scaie to accentuate clearly the distinctly fields, quantitative figures arc pit cn hIon

3. M.iR Carver, MA. Sakudean / hmogenizing in two fluid frow

E-“~-------“-=---- :- s _ _ s 9 - - - - - _ _ - - -_ --__-_-___A

t * - . _ i i _ - 1

I _ _ _

c

.:of~.ametric fiow Felds and void fraction profile for the central blockage.

M. B. Carver, M.A. Salcudean / Homogenizing in two fluid flow

PLANE OF

BLoCKtGE

OE 0 avq

AXIAL SLIP

VV"

AXIAL POSlTlON

Fig. 6. Computed void fraction and slip ratio, average and centre line values (peripheral).

4.2. Particular analysis of velocity and phase distri- bution

An overall impression of the nature of velocity and phase distribution can be gleaned from the flow fields depicted, particular features are now discussed. Graphs of computed cross sectional average void fraction and axial velocity ratio (air/wz:ter) are given for the peripheral and central obstruc.:tion cases in Figs. 6 and 7. Note that the trends in void fraction and slip are again in agree- ment with the experiments of Figs. 3 and 4. In particular, the slip can be seen to increase in the plane of the obstruction and decrease markedly just downstream, with the overall void fractions varying inversely as expected. The increase in slip and slight decrease in void in the plane of the constriction are due to the greater acceleration induced in the lighter phase. The subsequent de- celeration causes an opposite tendency, but the mixing effects cause comparably larger variations, the average void increasing above upstream values, and the average slip decreasing proportionally. This is also compatible with the experiment. The computed centre line values do not show strict proportionality, but are not necessarily con-

AYIAL POSITION

Fig. 7. Computed void fraction and 41p r;w~. rtv~~qw .IIIJ centre line values (central).

strained to, as this is a local measurement 111 ,t varying field.

Again as in the experiment, the cent;,al obhtruc- tion causes a significantly greater rcbistributicm than does the peripheral one. The flow around ihr central obstruction shows the liquid pha,w hc- having in a similar way as single phase flow past an obstruction. A zone of recirculation is fornrvc~ in the wake, and air flowing into this zone C~UWS ai slow moving air pocket, giving a veloci t! riltt*.l smaller than one. This accounts for the h’ph WIT region observed in both experitncnts and Calculi)- tion. The effect is less pronounced in the ~orrrpPat;t- tion due to the two-dimensional trcatmcl;t of thy flow field.

Further downstream, pressure recover> (KLUD with an associated increase in slip and decre;lsc 111 void. The recovery is computecl to be mnuch fa\tcr for the peripheral obstruction than for the it:nir:~1 one, again as observed in expcrriment.

5. Conclusion

As stated, the intent of the study was tcr de- termine whether the two fluid1 model contain\ the

M. B. Carver, M. A. Sakudeaj

ry flexibitiry to simulate the several physi- nsmcna which occur in two phase flow

constriction. Because this was a feasibil- no attempt was made to simulate the

c:nsionhif nature of the flow, and as this is a major influence no attempt was made to pre-

reduce experimental conditions. tn spite pproximations it is clear that the numeri- tion reproduces the main phenomena of

~nt~r~t and therefore contains the necessary rn~~~~~i~rn~. It would be counterproductive to at-

mpt to tune a two dimensional model to agree ’ h the experiment in more detail, but the agree- nt is sufficient to promise reasonable success

for a three dimensional simulation. One feature on which a discrepancy exists between the computed and experimental results is the precise amount of mixing_ Jn the central blockage experiment, the mixing appears to temporarily invert the void frac- tion p~dile. As the blockage is symmetric, this is quite surprising and may be due to the inducement sf swift. a phenomenon that would require multi- dimcrtsiomrl simulation.

$1 B Caner. Conservatiou s.nd the pressure continuity rc:lstlonkhio In multidimensionz! two fluid computation, on R Vlchnmetsky. Ed.. Advances in Compurer Methods for Parrtal biferennrtal Equartons, J V (IMACS Press, 1981) 16% 172.

[21 M.B. Carver. A method of limiting intermediate values of volume fraction in iterative two fluid computations, 2.

M& &qg Sci. 24 (4) (1982) 221-224. [3J M.B. Carver, Numerical computation in two fluid flow

using the two dimensional TOFFEA code, AECL-8066 (1983).

4) M.B. Carver, Numerical computation of phase separation

/ Homogbzing in two fluid flow

[51

161

[71

[81

891

IlO1

[Ill

1121

1131

I141

1151

1161

in two fluid flow, J. Fluiak Eng. 106 (1984) 147-153. M.B. Carver, L.N. Carlucci and W.W.R. Inch, Ther- malhydrauhcs in recirculating stream generators, THIRST Code User’s Manual, Atomic Energy of Canada Limited report AECL-7254 (1981). M.B. Carver, A. Tahir, D.S. Rowe, A. Tapucu and S.Y. Ahmad, Computational analyses of two-phase flow in horizontal rod bundles, Nucl. Eng. & Des. 66 (1984) 101-123. A.D. Gosman and W.M. Pun, Lecture notes for course in calculation of recirculating flows, Imperial College Report HTS /74/2. W.T. Hancox and S. Banejee, Numerical standards for flow boiling analysis, Nucl. Sci. & Eng. 64 (1977) 106-123. W.G. Mathers, W.W. Zuzak, B.H. McDonald and W.T. Hancox, On finite difference solutions to the transient flow-boiling equations, CSNJ Specialisis Meeting on Tran- sient Two-Phase Flow, Toronto (1976). D.S. Rowe, C.A. McMonagle and S.Y. Ahmad, Numerical solutions for two-phase flow in horizontal nuclear fuel assembles, in: R. Vichnevetsky, Ed., Advances in Computer Methodr for Partial Di//erenrial Equations, IV (IMACS Press, 1981) 351-358. M. Salcudean and Z. Abdelrehim, Heat transfer in turbu- lent recirculation flows affected by buoyancy in rectangu- lar cavities, Chem. Eng. Comm. 4 (1980) 249-268. M. Salcudean, J.H. Chen, D.C. Groeneveld and L. Leung. Effect of flow obstruction geometry in pressure drop in horizontal air water flow, Internar. J. Multiphase Flow 9 (1) (1983) 73-85. M. Salcudean, D.C. Groeneveld and L. Leung, Effect of flow obstruction on void distribution in horizontal air- water flow, Internat. J. Multiphase F!ow 9 (1) (1983) 91-96. V.L. Shah et al., Numerical procedures for calculating steady/unsteady single/two phase three dimensional fluid flow, NUREG-CR-0789 (1979). M. Shoulcri, A. Tahir and M.B. Carver, Numerical simula- tion of two phase flow in horizontal interconnected sub- channels, CNS/ANS International Conference on Numeri- cal Methods .+I Nuclear Engineering, Montreal (1983). A. Tahir and M.B. Carver, ASSERT & COBRA prediction of flow distribution in vertical bundles, Proc. CNS/ANS lnternarional Conference on Numerical Methods in Nuclear Engrnpering, Montreal (1983) 371-387.