numbers for laminar and transitional to turbulence regimes of fluid ..... Fluid Dynamics, Hermosa Pub- lishers, Albuquerque, New Mexico 1979). 1 0-13-3 ...
a THE4TH INTERNATIONAL TOPICAL MEETING ON NUCLEAR THERMAL HYDRAULICS, OPERATIONS AND SAFETY April 68,1994, Taipei, Taiwan
XA04NO538
NUMERICAL STUDY OF NATURAL CONVECTION OF A HEAT-GENERATING FLUID IN NUCLEAR REACTOR SAFETY PROBLEMS
L.A. Bolshov, R.V. Arutyunyan, A.G. Popkov, V.V. Chudanov Institute of Nuclear Safety Russian Ac.Sci. 52 B.Tulskaya, Moscow 113191, Russia Fax: 095) 230-20-23 and P.N. Vabishchevich, A.G. Curbanov Institute for Mathematical Modeling Russian Ac.Sci. 4 Miusskaya Sq., Moscow 125047, Russia Fax: 095) 972-07223
ABSTRACT
USA developed approximately in a similar way.
Unsteady natural convection of a heat-generating fluid in axisymmetric enclosures (cylindrical and downward-facing hemispherical) with isothermal walls is investigated numerically in the present work. This problem is considered from the stand point of solving the problem of molten corium retention at the vessel bottom of a Pressurized Water Reactor (PWR). The peculiarities of convective heat transfer are studied i a wide range of Prandtl and Rayleigh numbers for laminar and transitional to turbulence regimes of fluid inotion. Te turbulent regime is not considered in this study. The predictions are compared with the numerical and experimental results of other scientists.
11.
GOVERNING EQUATIONS
Laminar natural convection of a heat-generating fluid is governed by the Navier-Stokes equations in the Boussinesq approximation for buoyancy, which can be written with the account of axial symmetry in the stream function-vorticity formulation as: 00 1 49(ruO) 49(0) T + --- jr- + r r OZ - I -_ 9 r-a9 +-D20 +I (1) Pr [r Or Or) aZ2
1. INTRODUCTION
49W a(uw)
The peculiarities of convective beat transfer in a heat-generating fluid layer for the problem of molten coriurn retention at the vessel bottom of PWR are examined. This problem has various formulations as from the viewpoint of geometry of the domain under the consideration (cylindrical, emispherical, horizontal layer, etc.), and also from the point of view of imposing the boundary conditions (isothermal or adiabatic) (see a comprehensive reviewl).
49(vw)
= [a (I
rW) ) + 02w]
Ra 0 (2)
Or a
r
r
(I
00
OZ2 I
Pr r
0
(3) r Or r O2 - latp = I 010 (4) r az' r Or Here 0, w, io, u and v stand for the dimensionless temperature, vorticity, stream function, radial and axial velocities, respectively; r is the dimensionless radial coordinate, z is the dimensionless axial coordinate and r is the dimensionless time. Next, Pr = llet is the Prandtl number; Ra = j6sD')1(avk) is the modified Rayleigh number, derived using te rate of volurnetric heat generation instead of a reference temperature difference (in this problem a tempersture difference is not known a priori). Uniformly distributed internal heat sources are considered in te problem. Normalization is done here on the basis of the cavity diameter D, the kinematic viscosity v and the value aDl/k, proportional to a temperature difference. Or U
In the present work this problem is considered for the closed cylindrical and downward-facing hemispherical cavities with isothermal rigid walls. Such a formulation is coincident with the conditions of pysical experiments on the BAFOND device 2,3 and umerical experiments,1,4 that allows to use some results of these works for the verification of the accuracy of the mathematical model being used i our calculations and for the validation of prediction reliability. The urgency of the problem is conditioned by the necessity to predict correctly the behaviour of a molten heat-gcnerating orium for various scenarios of hypothetical accidents at PWR. The situation, where a molten coriurn flowed down to the bottom of a PWR reactor vessel and a buoyancy-driven corium flow was occurred there, can be considered as an example of such scenarios. The scenario of the accident at the Three Mile Island NPP in the
A flow domain is a closed cylindrical or downward-facing hemispherical cavity with isothermal rigid walls. With the account of axial symmetry a calculation domain forms a half of te
10-13-1
vertical cavity section with the following boundary conditions:
Unfortunately, in the work a model problem for cylinder or
no slip, no permeability conditions on the rigid walls;
spherical cavities filled with a gas has been studied. But we are interested in regimes corresponding to real situations of evere accidents. So the regimes with higher Rayleigh and Prandtl numbers have been investigated. With the increase of Rayleigh nmber up to 108 the structure of thermal and hydrodynamic fields becomes essentially complicated.
fixed temperature value on the walls; 9 zero radial derivatives of temperature and fixed zero values of vorticity and stream function on the symmetry axis. 111.
SOLUTION METHOD A secondary vortex i formed at the upper part of the cavity Convective heat transfer equations (1(4)
were solved nu-
near the axis, this results in descending flow at the symmetry axis
merically with the above mentioned boundary conditions using an
in this region. Such change of the vertical velocity sign near the
efficient finite-difference method.
The ADI scheme was used for
axis at the top part of the cavity at high Rayleigh numbers is con-
1),(2) and evaluation of the temper-
firmed by experimental measurements on the BAFOND device. 2.3
attire and vorticity at a new time-level. The conservative upwind
At the same time, the temperature local maximum, existing due
scheme-' has been employed for convective terms. The vorticity on
to the volumetric heat sources, shifts up and nearer to the side
solid boundaries was calculated via explicit three-point formulas
wall. Pronounced vertical stratification of the temperature as well
of a second order of accuracy (the Jensen formula5). The Poisson
as increasing the heat transfer rate through the top surface of a
equation
cylinder is occurred in these regimes.
solving unsteady equations
3) for the stream function was calculated by means of the
AD] method with the Jordan optimal see of iterative parameters.6 Steady-state solutions (if they exist) have been obtained as a limit of a time-evolution process. was used in
The quiescent state
=
=
The
=
flow
becomes oscillating
beginning
with Ra sts 10".
Transient splitting and redistribution of secondary vortices is ob-
ll calculations as the initial conditions for the time
served in these regimes, whereas the integral characteristics of heat
integration. More details of the numerical method are given in the
transfer through the cylinder bases and the side surface are
work.'
practically changed, i.e. the heat transfer regime is almost quasisteady state. Calculations at higher Rayleigh numbers require an
The uniform grid of 20 x 100 steps for cylindrical and the
ot
introduction of some turbulence model.
nonuniform grid of 50 x 50 for hemispherical cavities have been used in our calculations. The sufficiency of these grids was validated via omparative predictions on more fine grids. It should be
Streamlines
Isotherms
Nulbp = 0086
noted, that odinary cylindrical coordinates have been employed not only for the cylindrical cavity but for the hemispherical
avity
calculation, too. A nonuniform rectangular mesh in the cylindrical coordinates with gridpoints falling onto the curvilimear problem
NuT,,p
boundary has been constructed for the hemispherical cavity. Doing o, we can without problem consider the cavity with any fluid fillings.
W.
.0 N
0
RESULTS AND DISCUSSIONS
Z .1
Parametrical study of convective heat transfer in
Pt 0
a closed
V NUSide
cylindrical and hemispherical cavities was carried out in the following range of parameter values: Prandtl number: 0.
< Pr < 10;
NuBotm
Raleigh number: 103 < Ra < 10'0;
0
aspect ratio: 025 < WID < 3
R
1q.6, Ca' TA 9.8 a.$' 1. R
The regimes considered for a cylinder are close by the pa-
RZ YuBot.
rameters to the investigated ones in the physical experiments on the BAFOND device. 1,3 These experiments have been conducted in
a cylinder, filled with salted water and heating via the elec-
tric current. Further, some are available from the work'
easurements for cylindrical cavities at low ayleigh numbers. As for
Figure I -
Oscillating flow regime, Ra = 10".
downward-facing hemispherical cavity convection, only numerical resultsl have been used for a comparison. Good agreement between or pedimental date HID
= 13, Pr
The characteristic results for the oscillating regime at Ra
steady-state results and ex-
10"
has been obtained for a cylindrical cavity at = 07 and Rayleigh number up to 10'.
circulates in the cavity as a single convective
are shown in Fig. 1 in the following way. On the left side there
is an instantaneous flow pattern (equidistant streamlines). In the
A liquid
middle of the picture a thermal field is depicted via the equidis-
ell, flowing upward
tant isotherms. This allows to judge not only the distribution of
at the center of the cavity and downward near its side walls. At
temperature itself, but the behaviour of its gradient (heat flux).
high Ra the vortex center shifts up closely to the cylinder top.
And finally, on the right there are distributions of the local modi-
The temperature maximum is also up-shifted and is- located at the
fied Nusselt number (dimensionless heat flux) Nu =
symmetry axis in this case.
0)..11 on the top, bottom and side walls of the cavity. The abscissa for these plots is nothing but dimensionless distance along the surface of the corresponding wall, i.e.,
I 0-13-2
"I
=
(grad
NuTp and NuI3.t.,
are depicted in dependence o the radial coordinate r, whereas Nusid, is presented versus the axial distance along the side wall. Due to the fact, that the aspect ratio in this figure is equal to 6 the plots for top and bottom walls terminate at the second trick of the abscissa with the distance value of 1. The total Nu for the corresponding part of the surface are also given here. This aows
In conclusion it should be noted, that three various regime have been investigated numerically in our study for Cylindrical and hemispherical enclosures with isothermal walls, containing a uid with the uniform distribution of volumetric thermal sources. Namely, conduction dominated regimes, thermal boundary layer regimes and oscillating multicellular ones have been obtained and
to judge the nonunifortnity of heat transfer through the various
analyzed.
parts of the cylinder surface. NOMENCLATURE Similar numerical results are obtained for downward-facing hemispherical cavities. Fine agreement between our steady-state and oscillating periodic results and predictions I Rayleigh number up to 109 and Pr
= 1.
D =
As the strength of con-
vective flow increases with increasing Rayleigh tendencies as in
cp - specific heat, JI(kg - K);
are obtained at umber, the same
the cylindrical case are observed.
- diameter, m;
g - gravitational acceleration, m/s2 H - cavity height, m;
A boundary
k - thermal conductivity, W/(m - K);
08, as shown
in Fig.2. Beginning with Ra Sze 109 the flow becomes oscillating,
Nu = I = -(grad 0)..u - local modified Nusselt number; 'D - Prandtl number; Pr = vla
as it is illustrated in Fig.3. Some small discrepancies are observed
q. = -k(grad
in our results in comparison with the predictions
r = RID - dimensionless radial coordinate;
layer regime occurred in the thermal field at Ra =
Flow pattern
at Pr
10.
T) - local heat flux on the boundary surface, W/M2;
R - radial coordinate, rn; Ra = O'Da - modified Rayleigh number; aVk s - rate of volumetric heat generation, Wlyn3
Isotherms
T - temperature, K; T.
- fixed temperature on the walls, K;
u = UDIv - dimensionless radial velocity; U - radial velocity, m/s; v = VDIP - dimensionless axial velocity; V - axial velocity, m/s; z = ZID
- dimensionless axial coordinate;
Z - axial coordinate, rn; Cr = 0 Figure 2 -
Steady-state flow regime, Ra
1013.
k - thermal diffusivity, m2/s; pep thermal expansion coefficient, K T-T - dimensionless temperature;
P - kinematic viscosity, m2/s; p - density, kglm3 ; T
Flow pattern
WID2- dimensionless time; * - dimensionless stream function; VD stream function;
Isotherms w
OD21V - dimensionless vorticity;
11 - vorticity. REFERENCES I. X.M. KELKAR, R.C. SCHMIDT and S-V. PATANKAR,'Numerical Analysis of Laminar Natural Convection of an Internally Heated Fluid in a Hemispherical Cavity," Proceed.
9th Int.
Transfer Conference, pp.355-364. San Diego, USA
1991).
2. D.
ALVAREZ, P.
MALTERRE and J.M. SEILER, 'Natural
Convection in Volume Heated Liquid Pools periments:
Heat
the BAFOND Ex-
Proposals for New Correlations," Science and Tech-
nology of Fast Reactor Safety, pp.331-336.
BNES, London, UK
(1986). 3. J.M. SEILER, "Cooling of Molten Material Liquid Pool Submitted to Volumetric Heating: New Correlations for Various Cooling Conditions,
Proceed. Int.
ENSIANS Conference on Thermal
Reactor Safety NUCSAFE 88, pp.885--895
1988).
4. R.J. KEE, C.S. LANDRAM and J.C. MILES, "Natural Convection of a Heat-Generating Fluid Within Closed Vertical Cylinders and Spheres,'
Figure 3 -
A half of a period for oscillating regime, Ra
10'.
ASME J.Heat
ransfer 98, 55-61 (1976).
5 P. ROACHE, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, New Mexico 1979).
1 0-13-3
6. A. SAMARSKII and E. NIKOLAEV, Numerical Methods for Grid Equations, Vol.1,2. Birkhauser Verlag, Basel 1989). 7. B.P. GERASIMOV, T.G. ELIZAROVA, I.S. KALACHINSKAYA, A.V. LESUNOVSKII and A.G. CHURBANOV, "Program package NEPTUNE for the numerical simulation of incompressible viscous fluid flows," Packages of Applied Programs (Eds. A.A. Samarskii et. at.), pp.3-17. Nauka Publishers, Moscow 1986). in Russian
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