Experimental modelling of hypothetical core disruptive accidents

Nuclear Engineering and Design 55 (1979) 225-233 © North-Holland Publishing Company

EXPERIMENTAL MODELLING OF HYPOTHETICAL CORE DISRUPTIVE ACCIDENTS R. STANIFORTH United Kingdom Atomic Energy Authority, AEE Winfrith, Dorchester, Dorset, UK Received 20 June 1979

A common feature to reactor containment programmes is the use of detailed models to furnish data for design and safety assessment purposes. Despite the great strides which have been made in computational methods it is expected that the experimental approach will have a continuing role. It is therefore still pertinent to review the basis of such experiments, to see how they could be improved, and to see how well model experiments describe other processes occurring during an hypothetical core disruptive accident (HCDA). Numerous papers have described experiments on detailed models of a fast reactor scheme, and in all these, the sodium coolant of the reactor is replaced by water in the model for obvious practical reasons, but the scaling consequences of this change seem to have been given little attention. Therefore the object of this paper is to review the fundamentals of the scaling process, and then to discuss in more detail the effects of changing the working fluid in HCDA experiments. It is shown that the usual practice of using a geometrically scaled model, water as the working fluid, and a charge of the same characteristics as expected in the reactor excursion results in an inexact simulation, requiring somewhat uncertain corrections before the data can be used for the reactor case. An alternative possibility which is discussed in this paper would be to model the compressible characteristics of the sodium and the results could then be applied directly to the reactor scale using well defined scaling factors. This proposal, however, does require detailed changes to the experimental model and to the charge, but neither of these is expected to give undue difficulty. Modelling of an HCDA normally refers to modelling of the compressible fluid/structure interaction but in recent years interest has grown in other processes, such as heat and mass transfer. By looking at the appropriate dimensionless numbers in the model and reactor, the possibilities of using scale experiments to investigate certain features can be gauged. It is concluded that with experiments using water as the working fluid many processes associated with heat and mass transfer will not be modelled correctly and therefore special experiments have to be devised. For the same reason, caution should be used in extrapolating to the reactor heat and mass transfer data from experiments designed to reproduce structure deformation and loading. Although the modelling of compressible fluid/structure interactions is without doubt the main interest at the present time, other processes can be modelled without difficulty. In the example given, it is shown that buoyancy effects can be modelled provided an incompressible fluid simulation is sufficient. This simulation requires a low pressure charge such as might be provided by the evaporation of FREON released from a frangible container.

1. Introduction

been noted that replacing sodium in an experiment by water reduces the structural deformation, and the pressures and impulses on impact of the coolant on the roof are markedly altered [6]. Corrections for these differences are not straightforward because they affect the detailed processes in a complex way. This difficulty can be avoided if the model correctly scales the full size reactor not only in the geometrical sense, b u t also in the other important processes. It is with this problem of correct scaling in reactor containment experiments that this paper is mainly concerned.

The testing of detailed models of fast breeder reactors is a significant feature of the containment design and assessment programmes [ 1 - 5 ] , providing data directly on the structural response and loading, including much detail which cannot yet be obtained using computational methods. However, because in these models the reactor coolant is simulated by water to avoid difficulties associated with sodium, problems are introduced in relating the data to the reactor scale. Thus it has 225

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R. Stanffbrth / Core disruptive accidents

2. Theory of modelling A model of a situation is one in which all the processes being studied are reproduced despite differences in size and properties of the materials involved. Certain conditions have to be met for this to be so and these are sometimes termed the Scaling Laws. These require that certain groups of dimensionless variables are equal in both the real and model situations. In general simple processes are easily modelled, but with increasing complexity of the process, the modelling conditions become increasingly restrictive until in the end, only full-size testing is adequate. Later it will be shown that with containment experiments reasonably accurate modelling should be achievable. Commonly two methods are used to identify the dimensionless groups which make up the Scaling Law - dimensional analysis (given the variables involved) or non-dimensionalizing the equations describing the process. The latter procedure is used in this paper to derive the dimensionless variables for the more important processes relevant to reactor containment models. Because dimensionless variables can be changed by addition, subtraction, multiplication and division amongst themselves (as long as the number of independent groups remains the same), there are many possible forms for each. However certain dimensionless variable arrangements are well established and therefore the following analysis is arranged to give the variables in these forms. An advantage of these well-known forms of dimensionless variable is that each can be associated with a particular physical process. To demonstrate the derivation of the dimensionless variables from the equations describing the process under investigation, the two-dimensional form of equations of fluid flow will be used although it will be obvious that the arguments are also applicable to either one- or three-dimensional forms of the equations. Initially only steady-state inviscid incompressible flow equations will be employed and this case will be analysed in detail. The dimensionless variables arising from the addition of non-steady flow and fluid compressibility will be derived. Other dimensionless variables of possible interest in containment modelling are listed but not derived in table 1.

The equations describing two-dimensional incompressible, inviscid steady state fluid flow without body forces are:

p

=--~X

+0

Momentum equations (U~O't))

~p

(1) ~u by --+ --=0 Ox ay

Continuity equation.

(2) Substituting the dimensionless variables p* = Cp - po)/poo~ , u* = u/v 0 , x* = X/lo,

V* = V/VO , y* = y / t o ,

(where Po, vo, 1o are conveniently defined values of fluid density, fluid velocity and length respectively) in eqs. (1) and (2) gives ,au* _

_

,~u*~ ht -

u ox* V~y,] ( ,~v*

,~v*~

u ~ x * + V ~y* ] __

au* +

~)v*

0p*

Ox*

(3)

ap* ~y*

= 0

(4)

ax* ~ " Thus an experiment which describes u, v and p as functions o f x and y is a particular solution of eqs. (1) and (2) and also a solution of eqs. (3) and (4) in terms of the variables u*, v*, p*, x* and y*. This latter solution will also apply to other flow situations with different po, v0 or lo (but with the same boundary conditions on the variables) and hence other values of u, v, p as functions o f x andy can be calculated. This is the basis of the procedure of expressing pressure differences as a "pressure coefficient" and velocities as "relative velocities" in steady state fluid dynamics. If the flow is unsteady, additional terms must be added to the LHS of eq. (1) giving p

+u--+o ~x

(or +u~+ ou,

- - ~x

Op

(s)

R. Staniforth / Core disruptive accidents

227

Table 1 List of dimensionless variables of interest in the modelling of HCDAs (Ideally the mechanical properties of the materials used in the model and reactor should be similar so that at all strains the stresses are in constant proportion. The tr in the below expressions would be taken at a convenient strain level e.)

Aa

Phenomenon being modelled

Dimensionless variable

Usual name

Incompressible and inviscid fluid flow (steady state)

P - PO povo2

Pressure coefficient (usually with ~ added to denominator)

Additional for unsteady flow

tuo/lo

Additionally for compressible flow

oo/a 0

Mach number

Additionally for viscous flow

volo00/~

Reynolds number

Additionally for gravitational effects

v2/glo

Froude number

Bubble and droplet formation: (here o is the liquid surface tension)

v~Polo/o

Weber number

Convective heat transfer

hlo/k uCplk

Nusselt number

Unsteady heat condition Free convection heat transfer: where here (only)/~ is the coefficient of expansion Radiant and conduction heat Transfer: where here (only) tr = Stephan-Boltzmann constant 5.6697 X 10 - 8 Wm - 2 K-4 Perfect gas (e.g., cover gas)

Distortion of thin shells

C Distortion of thick shells (scaled thickness)

1

Prandtl number

(zt t

Fourier number Grashof number

g2 oloT3/k

Stephan number

= %1c,

Ratio of specific heat capacities

pa6/polo

a~/poo~/0 Ps/Po vo/as

olpoo~

a A, B and C in this column denote: A, fluid flow; B, heat transfer; C, structural distortion.

S u b s t i t u t i n g t h e d i m e n s i o n l e s s variable t* -results i n

au* 0t" + u

,au*

,au*~

tUofl o

/'av* +u ,av* a-~ . .av'~ \at*

.v

ap*

a-f~) -- _ a y . .

(6)

ap* -

ax"

T h u s i f u n s t e a d y p h e n o m e n a are t o b e m o d e l l e d , t h e a d d i t i o n a l d i m e n s i o n l e s s variable is

R. Staniforth / Core disruptive accidents

228

t* = too/lo .

temperature i.e.,

If the fluid density is not constant and the flow unsteady then the continuity equation, eq. (2) becomes

p* = P/Po = F(p, T)/F(Po, To).

0p + afpu) ...... + 0(pv) = 0 . at ax ay

(7)

It is convenient to consider the density distribution to be composed o f two parts: that due to pressure fluctuations and that due to general variations in density of the fluid. The former propagates with the pressure field whereas the latter is convected by the fluid motion. Since the fluid density changes resulting from the pressures occurring during an HCDA are small, the acoustic approximation is appropriate. Thus the compressibility of the fluid can be described in terms of/3 = - ( 1 / V) OV/Op (where commonly this process is adiabatic), so the substitution of/3 ° =/3PoOo2 and P* = P/Po into eq. (7) gives ,.,Op*

Ofp*u*)

o ~ a-~- + - a x ,

Ofp*v*)

+--ay* =0.

(8)

Since ao2 = 1/~po the dimensionless fluid compressibility/3* = {3poo~ can be expressed as (vo/ao) 2. This is simply the square of the Mach number Mao2 so that for scaling compressibility effects the Mach number has to be correct. Clearly for correct modelling of density transport, the general density distribution described by p* must be scaled and therefore this should have the same relationship to pressure and

However, in practice the effect of general density variations on the pressure field (and hence on the containment response) is likely to be quite negligible.

3. Modelling in practice Ideally one would use the correct working fluid in a model, since this obviates one source of error. However practical considerations make it essential to use a more convenient fluid than sodium, except perhaps for a small number of check tests. Invariably water is used because it is convenient, and has many physical properties not too dissimilar from those of sodium. Table 2 compares the properties of sodium at 560°C with those of water at room temperature (15°C). These physical properties will be used to see which situations can be modelled exactly and what compromises are necessary because of practical constraints.

3.1. Incompressible inviscid flow The simplest modelling would be that assuming the fluid flow within the reactor vessel is both incompressible and inviscid. From table 1, it will be seen that the appropriate dimensionless variables

Table 2 Properties of sodium and water Physical Property

Sodium

Water

Units

Temperature Density Viscosity Specific Heat Thermal conductivity Surface tension Vapour pressure Velocity of sound Prandtl number Dynamic velocity Thermal diffusivity

560 818 2.174 1257 64.1 0.1507 1.651 2284 4.262 2.66 6.236

15 999 11.5 × 10 --4 4188 0.595 0.0735 1.704 X 10-2 1465 8.1 1.15 × 10 - 6 0.0142× 10 -5

°C

× 10~ '

× 10 -2 X 10 - 3 x 10 -7 × 10-5

Kg m -3 Kgm -1 s-1

J Kg-1 K -1 Wm -1 K-1 Kg s-2 Bar ms- I m 2 s-1 m2s -1

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R. Staniforth / Core disruptive accidents

Thus the pressure and time scaling factors are

are (p - po)/Oo og

and too~to.

These dimensionless variables are to be the same in the model and reactor and therefore the first term provides a relationship between pressure, density and velocity in the model, and the second term a relationship between time, velocity and length. In a modelling situation, Po and lo will be specified leaving (p - Po), Oo and t to be determined by the scaling variables. Clearly in this case, one of the three variables can be chosen quite arbitrarily. This freedom in choosing the model parameters is lost when additional features are modelled. The commonly used technique of using a geometrically scaled model and the same pressures as expected in the reactor HCDA fits within this scaling scheme. Thus if the subscripts m and r stand for the model and reactor respectively, then the velocity and time scaling factors are (VO)r/(VO) m = [(,O0)m/(PO)r I 1/2

and tr _ (lo)r (O0)m _ (lo)r ( (,O0)r 11/2 tm

(lo)m

(VO)~r

(~O)m\(~O)m]

"

Substituting the properties for sodium and water from table 2 gives (Vo)r/(Vo)m = 1.105

and tr/tm = [(lo)r/(lo)m] 0.905

1 (equals 18.1 for a ~ scale model). However, processes controlled by compressibility such as pressure wave arrival times, or pressures during fluid impact on the roof would not be correctly modelled. 3.2. Compressible inviscid f l o w

If in addition to the inviscid flow, compressibility effects are also to be modelled then from table 1 it will be seen that the list of dimensionless variables is now (p - po)/Po v2 ;

too~to and oo/ao .

Now the value of Oo is constrained by the velocity of sound of the modelling fluid and therefore there is no freedom of choice, other than the scale of the experiment.

~p- po)r _ ~ooabr and t r _ (lo/ao)~ (P - PO)m (POa2)m tm (lo/ao)m " Substituting for the properties of sodium at 560°C and water at 15°C; the numerical values of the scaling factors become 1.99 and [(lo)r/qo)m] × 0.64 1 (equals 12.8 for ~-6scale model) for pressure and time respectively. Note that as the charge pressure is lower than that of the expected HCDA, the charge energy will be lower than in current experiments, i.e., E r I E m = (poaolo)r/(Poaolo) 2 3 2 3 m •

If the model includes a compressible gas volume such as the cover gas then we have p V n = constant

or p - Po = P o [ ( V o / F ) n - 1 ]

(taking the subscript zero to denote as the conditions before the explosion). Since for a geometric model V o / V must be scaled 1 : 1, scaling p - Po will require Po, the initial cover gas pressure, to be scaled by the pressure scale factor. Further, if the model contains deformable structures, then these structures must produce the same relative deformation with the scaled pressure differences at the rather different rates of strain experienced by the model. For thin shells, this can be achieved by either a change in material or material thickness, or by using a composite material. Additionally, to simulate the dynamic characteristics correctly, the dimensionless variable psSs/polo must be the same in both model and full-size so that the choice of materials has to be made with some care. The choice of materials to simulate thick shells is more restricted (see table 1). It will be noted that the vapour pressures of sodium and water at the appropriate temperatures are quite close so that some aspects of cavitation might be modelled. However this is a complex phenomenon and others [7] have suggested that Weber number (which does not scale correctly) is the controlling parameter. In summary, a model can be constructed using water instead of sodium which reproduces the compressible unsteady flow of fluid and associated structure deformation. This model requires that all pressures be about ½that of that in the full-scale and the structure weaker in the same proportion.

R. Staniforth / Core disruptive accidents

230

Table 3 Test for scaling of various phenomena with a model that simulates compressible fluid flow (model = ~0 reactor size) Phenomenon

Dimensionless variable (DV)

Ratio a (Reactor DV t \ Model DV !

Viscous flow Gravitational effects (surface motion, buoyancy) Bubble and droplet formation Convective heat transfer Unsteady heat conduction

Reynolds number Froude number

134.8 0.1215

Weber number Prandtl number Fourier number

sitate the velocities in the model being some 86 times that of the reactor, and dynamic pressures 9100 times! The other possibilities are listed in table 4 together with the required scaling factors for velocity, dynamic pressure and time. The first and second of these are entirely practical but some difficulties would be experienced with the third due to the fact that water boiling would occur at the required initial pressure. In this paper, only the first of the above will be investigated further because it is believed that gravitational effects significantly affect the fluid motion in the later stages o f an HCDA.

19.37 0.000526 14.08

3.4. Incompressible inviscid flow with gravitational effects

a This should be 1.0 for correct scaling of phenomenon.

Other processes are not reproduced and a measure of the error involved can be gauged from table 3 in 1 which the model is assumed to be ~ reactor size. Therefore care must be taken not to assume that the above phenomena are correctly scaled when analysing the results of experiments in which only the compressible fluid dynamics are correctly modelled.

3. 3. Other possibilities Since the unsteady incompressible flow scaling laws allow one further phenomenon to be scaled, many other possibilities exist apart from the one just discussed. It is impractical to scale viscous effects (and hence fluid heat transfer) by making Reynolds number the same since this would neces-

From table 4 it will be seen that an important characteristic of a model scaling unsteady incompressible flow and gravitational effects is that pressures are all reduced by a factor of 16. It follows that if structural distortion is also to be modelled, then the effective material strength has also to be reduced by this same ratio. Although not impossible, this may be difficult to attain in practice and therefore it is expected that this scaling may be of more value with overstrong structures. Indeed, due to the low pressures involved, models destined eventually for conventional testing could be used for this type of scaling without risk of deformation. Table 5 lists the ratios between the dimensionless variables in the reactor and in the model for those phenomena which are not explicitly modelled. It will be seen that none o f the listed phenomena scale exactly although compressibility and unsteady heat

Table 4 1 Scaling factors for incompressible fluid flow plus an extra phenomenon (model = ~ reactor size) Phenomenon modelled

Dimensionless variable

Gravitational effects Bubble and droplet formation Unsteady heat conduction

Froude number

4.47

Weber number

0.355

Fourier number

Velocity (reactor) Velocity (model)

21.9

Dyn p(reactor) Dyn p(model) 16.32 0.103 391.5

Time (reactor) Time (model) 4.47 56.3 0.913

R. Staniforth / Core disruptive accidents

Table 5 Test for scaling of various phenomena with a model that simulates incompressiblefluid flow and gravitational effects (model = 2-160reactor size) Phenomenon

Dimensionless wariable (DV)

Ratio a Reactor DV Model DV

Compressible flow Viscous flow Bubble and droplet Formation Convectiveheat transfer Unsteady heat conduction

Mach number 2.87 Reynolds number 391.0 Webernumber 159.0 Prandtl number

0.000526

Fourier number

4.90

a This should be 1.0 for correct scaling of phenomenon. conduction are fairly close and therefore will be qualitatively scaled in a model of this type.

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spurious effects. In these circumstances, there is no alternative to experimental validation. A useful validation exercise for the scaling of compressible fluid flow would be a comparison between two experiments, one using sodium and the other designed to simulate the first but using water as the working fluid. This would require two different charges, one having half the pressure and half the energy of the other. Possible means of achieving this would be the use of a compressed gas or a low density explosive. [8]. A comparison between scaled pressures and deformations of the vessels would go some way towards the validation of the scaling technique. A partial validation of the scaling techniques which correctly models gravitational effects can be achieved by testing two different sized models with appropriately designed charges. Observation of the behaviour of the bubble in the two cases would provide suitable data on which the validity of the scaling could be assessed.

4. Practical problems

4.2. Core power excursion simulated in a model

4.1. Validation o f the scaling techniques

Prediction of the characteristics of the HCDA explosion is fraught with difficulties and uncertainties so that it is unprofitable in the current paper to consider anything other than the broadest principles of simulation. It is generally assumed that an HCDA will result in pressures of the order of 300 bar so that experiments modelling compressible fluid motion and incompressible fluid motion with gravitational effects will require charges having initial pressures of 152.0 and 18.4 bars respectively. Currently used methods of producing explosions in containment modelling include slow burning charges [2--4] and detonating explosives, undiluted [2,4] or with an inert diluent [5]. The characteristics of any of these can be modified by the use of a suitable ventilated container [1 ]. Table 6 lists the initial pressures produced by certain examples of the above techniques. Development to give lower initial pressures is possible but the scope for this is not large. It is clear from a comparison between table 6 and the required initial pressures that compressible flow simulation could use a chemical charge of the slow burning or vented container types, but high explosives are not suitable and low density explosives

Although there is an abundance of evidence from different branches of technology that the dimensionless variable approach is valid in practice, there are no validation experiments which encompass the group of phenomena relevant to the modelling of HCDAs. Thus underwater explosion experiments and containment experiments [1] have shown that in a scale experiment, the relative velocities and pressures are the same and that the timescale is equal to the geometric scaling factor. This is as predicted for a model using the same working fluid. Aircraft acre-dynamics has demonstrated the validity of the Mach Number and Reynolds Number, and ship hull dynamics has demonstrated the validity of the Froude Number and so on. There is no point in using computer codes to "validate" a scaling technique because these codes are based on the same flow equations as used in the scaling theory and therefore will certainly "confirm" them. Clearly if validation is necessary it is because it is not sufficiently certain that the important parameters can be correctly modelled without introducing

232

R. Staniforth / Core disruptive accidents

Table 6 Initial pressures produced by Charges used to simulate an HCDA Charge type

Initial pressure bar

Slow burning charge Normal density high explosive Low density high explosive Low density high explosive in vented container

300 105 1600 200-300

still produce pressures which are significantly too high. The pressures required to simulate incompressible fluid motion and gravitational effects are too low to be generated by chemical explosives, and it is suggested that an attractive solution is the use of evaporating liquid FREON. The FREON could be released using a frangible container or by a quick release mechanism [9] and the requisite pressures could be generated by quite modest temperatures (50-100°C for FREON 12). Although such an evaporating fluid would tend to model the pressure characteristics of an HCDA more closely than a chemical explosive, difficulty may be experienced in characterizing the charge due to the effects of non-equilibrium and heat transfer [9]. Heat and mass transfer from an HCDA bubble clearly affects the course of an accident, but scaling this is expected to be difficult because of the large number of possible processes which could be involved. However, if experimental studies isolated one or perhaps two dominant mechanisms, then a scale model would then become more feasible.

5. Conclusions The current practice in the experimental simulation of the effects of an HCDA does not take full advantage of the technique in that only incompressible flow of the simulated coolant is correctly modelled. By making quite practical changes to the model and the imposed pressures, correct modelling of either compressible flow, or incompressible flow

with gravitational effects can be achieved. Compressible flow modelling is eminently suitable for the study of the flows and structural distortion close to the core in the time interval immediately after the explosion and impact loads on the reactor roof, whereas the incompressible flow with gravity model can be used to study the long term effects including such features as the rise of the core bubble into the cover gas region. These types of experiment require quite different charges to reproduce the effects of a core power excursion. The first could be similar to that in current use, i.e., a low density explosive in a vented container and the second could consist of a suitable liquid such as FREON held in a frangible container. Many heat and mass transfer processes are incorrectly simulated in tests on models in which water replaces the sodium coolant of the reactor, so caution should be exercised in relating data on heat and mass transfer from this source to the reactor situation.

Nomenclature a Cp Co E g h k l p t T u, o V x, y t3 6 7 e 0 /a p o

= Velocity of sound = Specific heat at constant pressure = Specific heat at constant volume = Charge energy = Gravitational acceleration = Heat transfer coefficient = Thermal conductivity = Representative length = Pressure = Time = Absolute temperature = Velocities in x and y directions respectively = Specific volume = Co-ordinates of a point in the fluid = Compressibility of fluid = Thickness of thin shell = Ratio of specific heat capacities = Material strain = Temperature difference = Viscosity of fluid = Density of fluid or structural material = Surface tension (also material stress)

R. Staniforth / Core disruptive accidents Subscripts 0 s

= Value at a c o n v e n i e n t p o i n t for use as a datum = Shell

Superscripts *

= Non-dimensional variable

References [ 1 ] G.R. Abrahamson and D.J. Cagliostro, Nucl. Eng. Des. 42 (1977) 89. [2] Y. Ando et al., Nucl. Eng. Des. 42 (1977) 105.

233

[3] M. Egleme et al., Nucl. Eng. Des. 42 (1977) 115. [4] R. Cenerini et al., 2nd Int. Conf. Structural Mechanics in Reactor Technology, Berlin (September 1973) Paper E2/9. [5] K.M. Leigh et al., 5th Int. Conf. Structural Mechanics in Reactor Technology, Berlin (August 1979) Paper E1/7. [6] G.A.V. Drevon et al., Proc. Conf. Safety, Fuels and Core Design in Large Fast Power Reactors, ANL-7120, (October 1965) 720. [7] A.V. Jones, 4th Int. Conf. Structural Mechanics in Reactor Technology, San Francisco (August 1977) Paper E1/4. [8] I.G. Cameron et al., 4th Int. Conf. Structural Mechanics in Reactor Technology, San Francisco (August 1977) Paper E2/1. 9 ] D.W. Ploeger et al., 4th Int. Conf. Structural Mechanics in Reactor Technology, San Francisco (August 1977) Paper E4/2.