ML20217G763

Draft Paper Entitled, Extracting Scaling Info from Fluid Conservation Equations, Submitted to Nuclear Science & Engineering &/Or Nuclear Engineering & Design

ML20217G763
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Issue date:	12/30/1996
From:	Shotkin L NRC
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PR-961230, NUDOCS 9804290206
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i I-l 1

December 30,1996 Extractina Scalina information from the Fluid Conservation Eauations l Louis M. Shotkin, NRC 1

Abstract Recently work has ' ,en conducted to extend pievious scalirg re Its to a more global or " top-down", view of the reactor system. The intent has been to develop new scaling criteria that can be used to determine the sufficiency and relevance of data from l integral test facilities of different scales and imperfect geometric similitude. These  !

several approaches to scaling, both " local" and " global", are phrased in different terminology, which often makes it difficult to discern original contributions. The current effort puts these scaling efforts into a common terminology so that differences and similarities can be easily noted and compared.

The commonality between " local" and " global" scaling approaches is demonstrated.

The approach is to start with the conservation equations of mass, energy and momentum from the TRAC-PWR computer code. These equations are then used to derive traditional scaling parameter ratios, or n groups. For scaling to be exact, these groups should be identical in both the model and prototype. In particular, the relation to the Ishii scaling groups is discussed.

in order to compare with the n groups of the global approaches, it is first shown how the .

conservation equations used in the global studies can be derived from the " local" TRAC-PWR equations. This is done in order to highlight the assumptions and limitations of the global approaches. The n groups of the global approaches are then related to the local n-groups to indicate what new scaling information can be obtained. i The commonality between the n-groups of the different approaches is shown. j Finally, the limitations of the use of n-groups for scaling analysis is discussed. i Submitted to Nuclear Science and Engineering and/or D\

l t

Nuclear Eng/neer/ng and Design. This preprint (draft) is of j l not to be cited or reproduced. c0 '

g)\

This is a preprint (draft) of a paper intended for publication in ajournal. Since changes may be made before publication,

.-2.  ; /g this preprint (draft) is made available with the understanding that it will not be cited or reproduced without the permission of the author.

9804290206 961230 PDR MISC 980429o206 PDR

2 I

1. Introduction to n-Grouos An acceptable technique for achieving similitude between model and prototype is to define dimensionless groups of important variables and require that these groups be equal in both model and prototype. These dimensionless groups are often called n-l

' groups, after the Buckingham n-theorem (1). They can be derived in several ways (2).

l If one does not know the mathematical equations describing the physical laws under I

investigation, then one can list important variables, such as length, density, velocity,

viscosity, etc., and then combine them into dimensionless ratios using purely l dimensional arguments (1,2). Altematively, one can define ratios of known forces or l ratios of known heat transfer mechanisms (2). In either case, experimental data is required to define the functional relationship between the various n-groups.

In the field of nuclear reactor thermal-hydraulics these n-groups are typically derived from the conservation equations of mass, energy and momentum (3,4). With choice of apnropriate correlations for heat kansfer and mass and momentum interchange, these conservation equations are felt to adequately represent the key phenomena important to reactor safety. The same equations and correlations are assumed to apply equally in both model and prototype. Some limitations of the conservation equations, as well as the need for experimental data to assess the assumptions, is discussed in Ref. (Q)

The scaling technique used starts by making the equations non-dimensional. Similitude is then assumed to be achieved if the coefficients of each of the terms is separately made equal in both model and prototype. Such a scaling analysis is called a fractional analysis (2) because one obtains only a fraction of the information that can be obtained by solving the equations in both model and prototype'and comparing results.

The thermal-hydraulic conservation equations can be applied either locally (3,4) or

. globally (Q,Z,8) and appropriate n-groups derived. The objective of the present study is to show the commonality between the " local" scaling approaches and the " global" scaling approaches. The approach is to start with the conservation equations of mass, ,

energy and momentum from the TRAC-PWR computer code (2). These equations are l i

l then used to derive traditional scaling parameter ratios, or n groups. Since these equations were derived for an arbitrary computational node, the n-groups derived from them can be considered as " local". On the_ other hand, " global" scaling is based on the same fluid conservation equations, but conducted on a larger spatial control volume or grid and, typically, on a temporal grid of larger time intervals. The present analysis will j focus on showing the common dimensionality of the n-groups obtained by the various  ;

y approaches. This commonality can be shown by demonstrating that the same n-group L can be applied to different parts of the system (spatial grid) and different times during l l ' the transient (temporal grid). j 1

3 l

L ll. Four Basic Assumotions of Analvtic Similitude. or Scaling l

l The first two assumptions form the common basis for scaling analysis:

l @ The same set of equations for the physical laws governing the phenomena apply to both the prototype and the model.

@ Scaling information can be derived by requiring the non-dimensionalized l coefficients of the individual terms in the conservation equations to be identical in the prototype and the model.

The third and fourth assumptions are implicit to scaling analysis.

@ There is geometric similitude between the prototype and the model.

@ The number and type of computational nodes, u tiie spatial and temporal grids, are identicalin the analytic simulation of the prototype and the model so that

! initial and boundary, as well as local, conditions are properly accounted for.

The effect of not having geometric similitude, as well as the grids chosen, must be carefully considered in scaling analyses. Geometric similitude includes having the same relative heights, volumes and aspect ratios (UD). Lack of geometric similitude occurs, for example, when the prototype has parallel pipes connecting two components and the model has only a single pipe connecting the same components.

Ill. Derivation of Basic ri-Grouos The approach is to start with the conservation equations of mass, energy and i momentum from the TRAC-PWR computer code . These equations are then used to ,

derive traditional scaling parameter ratios, or n-groups.

l For this discussion, it is sufficient to start with the 1-D single-phase conservation I equations, applicable to a fixed volume / node. They ate identical with the TRAC-PWR equations for the liquid phase when no gas phase is present. The extension to two-phase conditions is covered in Section Vill.

Mass: b+ =0 (1)

Bt Bx l

i

4 Momentum: 5 + V5 =

Bt Ox

?p 8x- FV +g {2)

Thermal Energy: 09' + 09v' = - P 5 + q,, (3)

Bt ex ax

)

where:

p = density V = velocity e = internal energy q, = wall heat source g = gravity acceleration P = pressure

F = friction = (2/Dn)[Co + C,(Re)) + K/L (4)

K is the form loss Dn is the hydraulic diameter Co is a constant, dependent on Reynold's number Ci (Re) is a constant times a function of Reynold's number; its form is also dependent on Reynold's number.

L is a length.

Use of this particular form of the friction term in the momentum equation ( FV2) assumes a relatively turbulent-type flow. If the flow were largely laminar, a viscous term of the form pd2V/dx2 ,instead of FV2, would have been appropriate. Such a term leads directly to the Reynold's number in the non-dimensional form of the equation.

The three conservation equations contain four unknowns:

l p, V, P and e The fourth equation is provided by an empirical equation of state:

p = p(P,e) (5)

- The first basic assumption is that these same set of equations apply to both the full-

5 scale plant, or " prototype", and smaller-scale test facilities, or "models".

To find the necessary conditions for this assumption to hold, non-dimensionalize the equations by substituting (subscripted variables are dimensioned constants)::

p=p'

• po ; e=e' e o V=V' +Vo ; t=t' *to ; x=x'.xo ;

P=P'

• Po ; AP = AP' .APo (momentum equation only) F=F'

• Fo ; q ,=q ,'+q o and obtain a set of equations that holds in both the model and the prototype i po 89' po oV 89'V'

. 9 to Ot' xn ex' I

i VO- l o

+ y,2 v' OV' APc 1 BP ' p.1 p y,2.

to at x,Bx, 9aox p 8x 90 0 0ap'e' PoVeo o Sp' V'e > PVoo , ayo ,

to St' xo Bx' x, ex' ,

This formulation recognizes that for a geometrically scaled system, the pressure can be equal in model and prototype while the spatial differential pressure, AP, can be different. The implications are discussed in Section IV.

Multiply through each equation by the constants in front of the time derivative:

ap' fXVo o 89[ = 0 at' o BX

\

BV' tV aptoo

. oo ,.BV' , _ 1 BP ' _ g y p p'y 2 +

6t' Xo BX' Po*oV o 9 O*

• 6 ap'e '

tV oo ap* y*e' _ _ P o tV oa ,3y* , toq,o i 8t' xo Bx' poe o x o 8x' poeo ,

These equations hold in both model and prototype if the following ratios are equal in both'model and prototype:

, tV o

(USCALE h l

AP*

Euler

~

(ngggggugg) poVo, l Fxoo Friction (nrniction) gx* (6) 11Froude (n,gouog) v=o P*

j Euler =Eckert (ngugggy) poeo x*

q,o Heat Source Number (ngg,7)

, poVe oa l

i l

l

These were obtained from the ratios in the conservation equations by using l

Vo

$oYo , $oYo N$o$model ,

\ 0 A prototype

0) prototype ( o prototype k 0) model \

\ 0) model That is, by eliminating to from the scaling relations, as shown in the following l

I 7

l transformations . It is of interest to note that Ref. (R) took a different approach and did not normalize the time-scale, but rather derived scaling information in terms of characteristic frequencies.

Using Eq. (7) one can perform the following transformations:

l r 3 r 3 r r 3 9*00 = 9" 0 in the energy equation becomes 9*

• 3 - 9*0*O

< P000 ),nacer e PoBo >parorype . < PoBoVo Ames < Po*oVo A m a oto ' , ' APo to ' in the momentum equation becomes: 0' = 0* i LPoNV o o Adel NV LPo o oimp r PoVo JMet V o > pW

< Po (toVo F o),,,'= (toVoFo) becomesoo (F x ),,, = (Fjo),,,

f i f $ f T f 3 becomes g A g- 10 = g 1*- = gi

< Vo > w er < Vo>moorye < Vo* > w er < Vo* ) n o w This leads to the non-dimensionalized equations with the n-groups explicitly displayed:

Sp' + n

,. SCAte OP ". =0 (8) et ex o

+

SCALE go ~ PRESSURE o

~

FRICTION

+

FRoUDE h i

ap'e' +

6p'V'e' "

, BV'

+

ENERGY SCALE HEAT A,W (10h ggr SCALE ga ga l

l

8 IV. Scalino Imolications of n-Grouos The n-groups in Eq. (6) contain all the scaling information that can be obtained from these general (single-phase) equations using the technique of non-dimensionalizing the equations, before applying them to specific spatial grids over specific time periods.

These scaling groups can lead to conflicting requirements.

The Euler number comes from the spatial derivative of the pressure term in the momentum equation. It is thus scaled to the differential pressure (4, fi). Ifit had been scaled 2

to the pressure itself it would not be possible to satisfy batt1 the Euler number 2

(1No ) and the Froude number (Vo /gxo) in a facility which is not full-height. -

l One should note that Nahavandi's time-reduction or " linear" scaling laws (3,1Q) sigt with preserving the Euler number, written with Po rather than APo, giving (Vo)a = 1. This gives (to)n = a(l )a and (q )a = 1/(l a )a, which differs from Ishii's relations (4) unless (lo)a=1=

full-height scaling.

1 Preserving Fo ox means that as one decreases the length in the model one must increase the friction. This is one reason why test facilities use orifices in their pipes.

This statement is true provided the form loss, K, does not dominate the friction (see Eq.

4). If the form loss dominates, one has to be sure to scale its effect with appropriate hardware, especially under natural circulation conditions.

Explicit equations for heat structures have not been included for simplicity; they are treated as a heat source boundary condition. Thus, the only place where the flow area, or diameter, appears is in the Friction number, as 1/Dn (Eq. 4). In practice as one decreases the piping diameter in a test facility the friction increases. If the friction is too high, one has to use a larger diameter.

From the above one can deduce that there are often conflicting requirements for scaling friction between model and prototype. It is thus one of the most difficult exercises in designing a scaled thermal-hydraulic test facility (11).

Note that the Froude number in Eq. 6 is written on the length scale, but it is usually invoked on a diameter scale. It is also important to note that it is impossible to satisfy both the Froude number and the Reynolds number, which appears in the friction number, in both model and prototype. This is because to satisfy the Froude number corresponding velocities in model and prototype must be proportional to the square root of the diameter, while the Reynolds number requires that velocities be inversely i

L

c ,

9 1

proportional to the diameter. The Reynolds number is used to calculate the friction, while the Froude number is typically used to set the minimum diameter to preserve flow

. regimes between model and prototype (11).

If one takes the operating pressure, temperature and working fluid to be the same between model and prototype, then Po, eo , and pa are the same in model and prototype. This is equivalent to " full-pressure" scaling.

Defining ()n to be the ratio of parameters between model and prototype, one obtains-from the Froude number the requirement that -

u2 (Vo )a = (xo )a and from the toVo /x o relation the requirement that (to )a = (x o )a" These "Ishii" reise , (4) state that for a " full-height" test facility, (xo )a = 1, the velocity and time are presuv 4 while for a reduced height facility they are both reduced by the square root of the length scale ratio,- xo.

The heat source number ratio then becomes:

i l (q,)n = (xo )a*

l l which states that the power density (power to volume ratio) in the model core has to be increased as the length scale is reduced. For full-height scaling the power density is preserved between model and prototype.

" Power to volume" scaling preserves the relation for the heat source number in Eq. 6. It is thus equivalent to full-height, full-pressure scaling.

l V. Derivation of " Global" Conservation Equations from the " Local" TRAC-PWR Equations

- The first step in comparing the above ri-groups with those of recent " global" scaling '

studies (Z, R) is to derive a time-dependent equation for the system pressure from the TRAC-PWR conservation of energy equation. This is done by performing the following:

1. Expand the time derivative in Eq. (3):

I l

r 1 .

10 8pe , ,8p pae e at at 8t

2. Substitute for 8p/8t using the conservation of mass equation (Eq. (1)):

i 8p , _8pV l 8t 8x l

3. Combine spatial derivatives on the right-hand side p*=e 0

09 - 09V* -Pv O

+ qw Bt 8x 8x 8x l

l 4. Assume the pressure, P, is constant spatially (in global analyses (Z, B) this ase'imption is only used in the energy equation) i 0

9*=e epV _ 8pv , p, ,q 8t 8x 8x

5. Define h=e+P/p i

l pBe =e apV _ 8pVh +q, Bt 8x 8x

6. Expand the time derivative using the equation of state (H):

I p 0* N + 0* ? =e 09V - 8pVh , qw '

.8p Bt BP 8t . Bx 8x l

7. Rearrange the left-hand side l

9 8e BP _ _ 8e Q _ ,BpV _ 8pVh , "

BP 8t 6p 8t Bx 8x l

8. Again use conservation of mass to eliminate 8p/8t on the right-hand side:

p Be BP Be 'BpV - 6pVh + qw BP Bt e + p 8p. Bx 8x l

L

11

9. Difference the equation spatially to obtain:

ApV p 6e SP ,

ae ApVh , q

,,p g 6F at 6p. Ax Ar l

! To compare with Ref. (B), multiply through by Ve = AAx and use pAV = W AApV = EW, AApVh = EW,h, l to get Eq. 6 of Ref. (B), the time-dependent pressure equation:

V,p ae BP , ,, g y, _ y,,ycq , (11b) 6P 6t 6p-l Thus, the pressure rate equation of Ref. (8) can be derived directly from the TRAC-PWR conservation equations of energy and mass, assuming only that the pressure is constant spatially.

l Note that one must also use Se _ Bh _ P_

i 6p 6p p2 l

l to obtain Eq. (7) of Ref. (H).

! Next, turn to the energy equation used in Ref. (Z) (Eq. A-13)

Is .y, (eoy,-9) - Is,,, (e a-9) + I4,y, - I ,,,

4 +M =0 dt This equation agrees with Eqs. (11) because (notation of (Z) on the left-hand side)

M = pVc l

l

12 Equ - Iq,n = gno = q, p=e e=h dm/dt = W = pAV l

l Note that Ref. (Z) also differentiates the equation of state, but in the form P = P (e,p)

I rather than the form e = e (P,p) used in Ref. (B), to get a pressure rate equation from l the energy equation. Also note that Ref. (Z) does not define " compliances" from the derivatives of the equation of state. Otherwise, the energy equation of Refs. (Z) and (H)  !

are identical and both are derivable from the TRAC-PWR conservation of energy and l mass equations.

Next, compare the momentum equation in Ref. (I), Eq. B.11, with that of TRAC-PWR pf = AP _ p y2 pg TRAC-PWR (12a) dt Ax where p5 + pVBV _ p _V d O! 8x di Eq. B.11 of Ref. (Z) is: j l

pt dQ P

=P 1

-F 2 + pg6Y,,,, - Q2 (12b)

A ,,,, dt 2A* l l

l with (notation of (Z) on the left-hand side)

Q=AV (V = velocity)

K/2 = F Ax AY = tcos0 = vertical component of L l Thus the two momentum equations, Eqs. (12), can be seen to be directly related.

I l

4

13 VI. Non-dimensionalize the " Global" Conservation Ecuations As the next step in comparing local and global n-groups, non-dimensionalize the energy equation, Eq. (11a) or (11b) using h = e + P/p and Ve =Axo . After dividing through by the area, A, obtain: 4 X90 o0 0 Be' BP' , , y Be' gp y,a to p, SP' St' ,,,p, dp' (13) l l

en poV o [p',V'p', - P oo V P'[V', + xa q ,o q',

Multiply through by the coefficient of the time derivative, t o/xopo eo:

Be' BP' fV o o ,Be' p,6P' Bt'

,, y y _

xn Sp' to V* [p',V'p ', - P*to V* P'[V', + t* q,o q ',

Xo 90X 0 0o 90 0 0 Using the definition of the n groups, (Eq. (6)), the " pressure" equation can be seen to include the same n groups as the TRAC-PWR energy equation (Eq. (10)):

l

= n SCALE O'

• 9' 9'iV'l ~

p' BP' Bt op,

~ D ENERGY SCALE +

DsCatE WIW I HEAT Ww This shows that the " global" form of the energy (plus mass) equation leads to the same form of the dimensionless n-groups as in the TRAC-PWR energy equation. It also l shows that differentiating the equation of state does not introduce any new n-groups.  ;

l Similarly, by nondimensionalizing the momentum equation (Eq. (12)), and multiplying through by the coefficient of the time derivative term, one can show that the resulting n-1 groups are identical with Eq. (6), since (notation of (Z) on LHS):

Lo = Axo = x, '

l

)

i.

14 K/2 = Fo ax, = Fox, (14)

Q, = A V.

P, = AP -

Vll. Comoarison of Local vs. Global n-Grouos. Sinate-Phase Having demonstrated that the " global" conservation equations (Z,8) are directly related l

to the " local" TRAC-PWR equations, it should come as no surprise that the rt-groups that can be derived from these equations are also directly related. In this section specific n-groups will be directly compared.

This is most easily done by returning to Eq. (13) and multiplying through by the area, A, using V,=Axo to get:

Vc9o"o Be' BP' , y ,,,p,Be' yp, y, _

l to p, BP' Bt' 09' e,p,V, [p'f V'p', - P V aaP'[A V', + V,q ,o q',

i The intent is to compare with the n's in Eq. 69 of Ref. (H):ue n , ng, ny, and nu .

i instead of dividing through by the coefficient of the time derivative, as was done above, follow Ref. (S) and divide through by the coefficient of the heat source, qd, defining (B) l HT,o caw 0 to get V,poeo de'SP' e,poVo

, ,, p,Be' pp,g y, _

p, BP' St' W to n7,a O H 7,,

'o9 0Yo y*p*p*, - Po V, p, ,

HT,o HT,o To get nue (multiplying the time derivative term) two more substitutions are required.

L.

15 First, rather than keeping the time-scale, to, general, Ref. (R) chose to explicitly define it as g , YMLOo HT,o where Vut = volume of main loop War, = mass flow rate through main loop Second, Ref. (8) chose to normalize the entire term pSe/SP together, rather than normalizing p, e and P separately. That is, 1

p ee , ' iBe'90' 0 0 ,'p ae ' p &e '

! BPio BP r BP ) r Po e r BP) r Solving for p'ae'/SP' and substituting that, and the definition for to into the time derivative term gives:

V,poco Be' 6P' Ve9a*oWu7,o Po  :

, Be ' ' p beBP' to6n7 ,o p, BP' 6t' y pf poea r p BP > o r BPi St' }

which gives directly (8, Eq. 87) l V Be' n,, = c #"?o p, p Vut po Q u7,n < cPio where Po is equivalent to the Po - P,in Ref. (R). i i

! Scaling the general variables in nuc, it follows that l i

i

i .

A I

16 n uc =

SCALE HEAT Op (and no. ) is the " normalized pumping power"(" normalized heat loss") which here have been included as part of qu. Therefore the form of the scaling relations are identical; that is, they are all normalized to Os7,o = V, q,o.

Instead of Osr.o Mr = AhoWo.o (B,Eq.91),

the terminology in this paper is pcVoea + VoP o = poVo (en + Po /po) = poVoho

which agrees with the form of Dr (E)-

l l It then follows that l

n, = 1

+

HEAT HEAT in other words, the " form" of Mr is the same as that derived from the TRAC-PWR l equations. The only difference is that Ref. (H) chooses to define some of the scaling i constants explicitly, while in Eq. (6) they have been left general.  ;

Note that Ref. (B) relates Wo.o to the " initial discharge rate", which is different than the definition for War,, used in n ue of the same equation..

Finally, n u(Eqs. 92a and 9 of Ref. (S)):

vv.o Oh n =

4 QHT.o ' OP'"

has the same dimensions as Dr and thus is equivalent: j i

(

I l i

i l

1

l 17 UENERay p =n, 1 HEAT U HEAT Thus, the forms of the n-groups in the pressure (energy + mass) equation of Ref. (B) i have been shown to be equivalent to the n groups derived from the TRAC-PWR energy equation.

t i Next, turn to the n-groups in the momentum equation. For illustratica use the i momentum equation, or surge-line equation of Ref. (Z), Eq. B.46. In that equation, the l

coefficient of the AP' term is Pt oo hPtao U1 * * *0 PRESSURE b hoYXoo poQo o 0

using the notation changes in Eq. (14).

The coefficient of the gravity term is U2 ~ gYtao ~ gte ~ gte ~U FROUDE b #

0, o V0a 0

0 0 since in the above formulation, based on the TRAC-PWR equations, the direction of the gravity vector (cos0) has not been explicitly included.

Finally, the coefficient of the friction term is Otoo 1

A*o Us = =FVt ooo=Fx oo =n FRicToon

(

2 "-

Ao Thus the form of the dimensionless n-groups from the line momentum equation in Ref.

(Z) agree with those from the TRAC-PWR momentum equation.

The vessel inventory equation of Ref. (Z), Eq. B.34, which is a conservation of mass i

18 equation, gives:

• oot poAVt ooo toV, n, = ,

cALE PoVolum e, poAn x, x, which agrees with TRAC-PWR. Another n-group derived from a conservation of mass equation by Ref. (Z) is:

Qtao AVto oo toV,

"' 5 " "'

ALoa ALoa x, Vill. Two-ohase Eauations and n-arouos To illustrate the extension of the above analysis from the single-phase to the two-phase situation, it is sufficient to consider the gas conservation equations of mass and energy.

The TRAC-PWR equations of mass and thermal energy for the gas phase when a liquid phase is present are (2):

I dap, aap,V as Mass St ax Sap,e, Sap,e,V, Et ex Gas Thermal Energy

-PSa - -P Ba y* + q,, + q,, + Vh ',

1 l where

.q,, + q ,,

V _- .

h, - h, i q, and q, are the interfacial heat transfer to the liquid and gas respectively; I a is the void fraction; h', is the vapor enthalpy if the vapor is condensing or the vapor saturation enthalpy if liquid is vaporizing.

l The power deposited directly into the fluid, and vapor generated by subcooled boiling l

1'

\ .

i 19 l'

are both neglected for simplicity.

Following the same procedure as for the single-phase equations, nondimensionalize the above gas-phase equations and divide through by the coefficient of the time derivative term to get the following n-groups which must be satisfied in both model and prototype:

to V,, j SCALE O l 1

00 MASS UP D g0 P,

EnlERG Y P,oe,o Nwpo 0 Nipo o g

HEAT O P0 g0 g0 0P0 0 g0 g0 O b'vo$0 HEA TIMASS i

nSCALE' UENERGYe OUd NHEAT are the same n-groups as in Eq. 6, except they are applied to the gas rather than to the liquid phase. A new n-group, n ASS, u is introduced to account for the scaling of the mass transfer between liquid and vapor, with the energy transferred by this mass scaled by nHEATMASS. These two-phase n-groups come from what is effectively a two-fluid formulation. n-groups derived from a drift-flux formulation have a different form (4).

IX. Comoarison of Local vs. Global n-arouos. Two-Phase As a final comparison of local vs. global n-groups, the n-groups in Eq. B.36 of Ref. 7.

This is the equation for the mass of steam in the vessel. It contains three n-groups.

The first two groups share commonality with those derived in Eq. (6) above:

p* , 00 _

00 _

p"' "

h,,om,o a op,o

l l .

1 o

' ~

20 l

l since l T=o

m ,o = aa p,o V , ; V, = volume h,,oV, mtoo a ap,oA V,o to taV,o g* , #*'

m,o a ap,oAx o x, ,

l The third n-group, no, measures the subcooling of the incoming liquid flow. There is no corresponding n-group in the TRAC-PWR formulation because TRAC-PWR treats this l information as a boundary condition for the individual cell and incorporates it into the local cell temperature and heat transfer coefficient.

Next, look at the Ref. I equation (Eq. B.40) for the bubble temperature and compare its four n-groups with those of the TRAC-PWR thermal energy equation.

YP eo Po i 11 ENERGY C,o oT m,o a op,oe,o i

g _ PO go o _ 0 go _ g C yo m yo xo These assume that the specific heats are equal in model and prototype.

Similarly, n,3 and ni, of Ref. (Z) can be easily shown to be common with n scue.

Thus the two-phase global n-groups in Ref. I are equivalent to the local n-groups derived from the TRAC-PWR conservation equations.

X Discussion l

Dimensionless groups of variables, such as those in the n-groups, are useful tools to perform scaling analyses between model and prototype. There are only a few basic n-groups that can be derived from the fluid conservation equations and each one has the j same dimensional form no matter where it is applied in the system and no matter during

I .

, 21 which time period it is applied. For example, the Reynolds number has the same form, pDV/p, but can have different numerical values depending on the part of the system and time during the transient when it is calculated. Each application of the basic n-groups (e.g., those in Eq. 6) can have specific choices for the characteristic time, to, and different definitions for the characteristic velocity while still maintaining their same dimensional form. Thus these applications of the same n-groups to different spatial and temporal grids introduce no new scaling groups but do introduce new applications of these groups.

In order to make proper use of these n-groups it is important to be aware of their limitations. They cannot directly account for lack of geometric similitude between model and prototype. One cannot account for different conditions in a model having one pipe in place of a prototype with two parallel pipes. Oscillatory flows that may exist among components within a system can also not be accounted for solely through use of the basic n-groups. Since there are conflicting scaling requirements among the parameters in the n-groups (e.g. Froude vs. Reynolds numbers for a geumetncal sca!e ratio <1, discussed above), it is not possible to satisfy all n-groups equally well in all parts of the system and during all time periods of a transient.

The two primary uses of scaling are: 1) to design a test facility (model); and 2) to interpret the spplicability of data from the test facility to the full-scale plant (prototype).

In the first use, one must design the facility hardware to simulate the entire time period, ,

and all spatiallocations, of the planned set of transients. In the second use, one can examine the data in a limited portion of the system and for a limited time period.

Because of the limitations of application of the fluid n-groups they are potentially more useful in analyzing data from test facilities over limited spatial and temporal grids than they are in designing test facilities over the full range of transient spatial and temporal conditions. In the former use, analyzing data, assumptions are typically required on phase distribution in specific parts of the system over chosen time periods (I, H). For l the latter use, designing test facilities, the fractional analysis (2) supplied by comparing the individual terms in the conservation equations can be supplemented with information obtained by solving the complete set of equations (H) on a much smaller spatial and temporal grid.

XI CONCLUSIONS 1

The scaling technique used in this study, and used in many other similar studies, is to require the nondimensional n-groups multiplying the terms in the thermal-hydraulic conservation equations to be equalin both model and prototype. We have shown that the type of conservation equations used in local scaling analyses are identical to those l used in global scaling ana!yses. It is thus not surprising that the form of the n-groups derived from either global or local analyses are also identical. The only difference between the n-groups is due to what variables were used to make the equations non-i

22 l~

dimensional. This of course relates to, or is determined by, the spatial and temporal grids used in the analyses. That is, for the spatial grid whether the equations are considered for the whole system (Ref. 8, energy equation); a major component within the system (Ref. Z, mass and energy equations) or a local computational cell within a component (TRAC-PWR). For the temporal grid, the normalizing variables used depend on which terms in the conservation equations are felt to be dominant over that time period.

Introducing the time derivative of the equation of state is usefulin formulating a pressure rate relation from the energy equation, but it does not introduce any new type of scaling relation.

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23

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2. S. Kline, Similitude and Approximation Theory, McGraw-Hill,1965.

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