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Scaling of two-phase flow transients using reduced pressure system and simulant fluid
Nuclear Engineering and Design 104 (1987) 121-132 North-Holland, Amsterdam
SCALING OF TWO-PHASE AND SIMULANT FLUID
121
FLOW TRANSIENTS
USING REDUCED
PRESSURE
SYSTEM
G. KOCAMUSTAFAOGULLARI University of Wisconsin- Milwaukee, Department of Mechanical Engineering, Milwaukee, WI 53201, USA and M. ISHII Argonne National Laboratory, Reactor Analysis and Safety Division, Argonne, IL 60439, USA Received May 1987
Scaling criteria for a natural circulation loop under single-phase and two-phase flow conditions are derived. Based on these criteria, practical applications for designing a scaled-down model are considered. Particular emphasis is placed on scaling a test model at reduced pressure levels compared to a prototype and on fluid-to-fluid scaling. The large number of similarity groups which are to be matched between model and prototype makes the design of a scale model a challenging task. The present study demonstrates a new approach to this classical problem using two-phase flow scaling parameters. It indicates that a real time scaling is not a practical solution and a scaled-down model should have an accelerated (shortened) time scale. An important result is the proposed new scaling methodology for simulating pressure transients. It is obtained by considering the changes of the fluid property groups which appear within the two-phase similarity pargmeters and the single-phase to two-phase flow transition parameters. Sample calculations are performed for modeling two-phase flow transients of a high-pressure water system by a low-pressure water system or a Freon system. It is shown that modeling is possible for both cases for simulating pressure transients. However, simulation of phase change transitions is not possible by a reduced pressure water system without distortion in either power or time.
1. Introduction Similarity laws and scaling criteria are important for designing, performing and analyzing simulation experiments using a scale model. They are particularly important for studying phenomena which can not be easily observed in a prototype model, such as occur in nuclear reactor transients and accidents, as well as in some petrochemical experiments. In view of the inherent difficulties associated with full-scale testing of such systems, it is necessary to use scale models and simulation experiments. However, for a justifiable extrapolation of experimental model results to full-scale prototype transients, it is necessary that the scaling rationale be soundly based. Scaling thus becomes the single most important issue in the conceptual design of experiments whose aim is to investigate transients in a prototype, Amrouni et at. [1].
In line with the central importance in establishing similarity relations between prototype and scale models, several scaling techniques have been developed. The simplest technique is linear scaling, in which all length ratios are preserved. While there are some single-phase flow situations where linear scaling is adequate, its use results in thermal and flow distribution distortions which are generally considered unacceptable for two-phase flow conditions, [2-5]. For these reasons, volumetric scaling techniques are frequently used for thermohydrautic experiments [4-7]. Model facilities using this approach are usually full-length (with the notable exception of LOFT), while area, volume and power are reduced proportionally. Both linear and volumetric scaling techniques described here are developed based on single-phase flow conservation equations. However, extending these techniques to investigate phenomena associated with two-phase flow, particularly for non-homo-
0 0 2 9 - 5 4 9 3 / 8 7 / $ 0 3 . 5 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. (North-Holland Physics Publishing Division)
122
G. Kocamustafaogullari, M. Ishii / Scaling of two-phase flow transients
geneous flows, is certainly incorrect, Zuber [8]. Recently, extensive investigation have been undertaken by the present authors to develop the similarity criteria for the natural circulation loops under singlephase and two-phase flow conditions [9-12]. The similarity criteria obtained from these investigations were, further, extended to practical applications for designing a scaled-down model. In these applications, however, emphasis was concentrated on exploring the modeling of a prototype by a scaled-down model using the same fluid at the same pressure as the prototype. Therefore, the working fluid property effects and fluid-to-fluid simulations were not studied in detail. Based on the rigorous similarity analysis of the recent studies, [9-12], the present study attempts to develop scaling criteria for a natural circulation loop under single a n d / o r two-phase flow conditions, and to apply the criteria for modeling a high pressure water system by a low pressure water system or a Freon system. One of the most important results of the present study is a new methodology for simulating pressure transients. It is obtained by studying the single-phase to two-phase flow transition and by considering changes of the fluid property groups arising from the two-phase flow similarity parameters.
STEAM GENERATOR (COOLED SECTION)
PUMP
HEATED SECTION
J - -
SINGLE--PHASE REGION
,',;'/,G'; TWO--PHASE REGION Fig. 1. Schematic natural circulation system under consideration. [9-12]. They are given as follows: Richardson number:
R =- gBa T0t0/u02, Friction number: F, -~ [ ( f l / d ) + K ],,
2. Single-phase flow similarity 2.1. Similarity parameters
(J)
(2)
Stanton number: Sti ~- (4hlo/prCpfuod),,
(3)
Time ratio number: Although the major effort in this study is directed at developing similarity criteria for simulating two-phase flow pressure transients, single-phase flow similarity requirements are needed first to simulate the singiephase to two-phase flow transition. A typical system under consideration is illustrated in fig. 1. This system consists of a thermal energy source, energy sink, and a connecting piping system between components. For a natural circulation loop under single-phase flow conditions, the similarity parameters are obtained from the integral effects of the local balance equations along the entire loop, [9-12]. Rather than duplicating the work of these references, the essential parts of the similarity analysis are summarized below. The fluid continuity, integral momentum, and energy equations in one-dimensional, area-averaged forms have been used together with the appropriate boundary conditions and the solid energy equation. From the one-dimensional form of these equations, important dimensionless groups characterizing geometric, kinematic, dynamic and energetic similarity parameters are derived,
Ti* =_ ( l o / u o ) / ( 3 2 / a s ) i ,
(4)
Biot number: B, ~_ ( h S / t , s ) , ,
(5)
Heat source number:
Qsi -~ ( i1~)"lo/psCpsuo A To ) i"
(6)
Here subscripts i, f and s identify the i th component of the loop, fluid and solid, whereas u o, A T o and l 0 are the prescribed reference velocity, temperature difference, and equivalent length, respectively. Most of the symbols appearing above conform to standard nomenclature. However, for the purpose of completeness, all symbols are listed in the nomenclature. In addition to the above physical similarity groups, two geometric similarity groups are obtained: Axial length scale:
L,-= l,/lo,
(7)
G. Kocamustafaogullari, M. Ishii /Scaling of two-phase flow transients Flow area scale:
123
plete similarity:
A i =- a i / a o.
(8)
2 uoR= [flqo.... lo/(OsCp~)] 1/3 R '
(14)
The hydraulic diameter d i and the conduction depth 8i are define by
aToR = [ 0o"to/(O, Cps) u0] R'
(15)
d, = 4a,/~,
3,R = 3R = (a,Io/Uo)R
1/2
and
3 i = asi/~i,
(9)
,
(16)
2,
(17)
where ai, a~i and ~, are the flow cross-sectional area, sohd structure cross-sectional area, and wetted perimeter of ith section. Hence, d i and 3 i are related by
d,R = dR =
d, = 4 ( a / a ~ ) , 3 , .
In addition to the above constraints, the following geometric and dynamic similarity requirements must be
(10)
The reference velocity u 0 and temperature difference ATo are obtained from the steady-state solution. By taking the heated section as the representative section, they are expressed as follows: u 0 = [(4flg(lo" 12/Of Cpr )( a~o/ao ) / ~
and A TO = ( qo" lo/P, Cpf Uo ) ( a~o/ao ),
(12)
where the subscript 0 denotes the heated section. 2.2. Similarity criteria Eqs. (1) through (8) represent relationships between the dimensionless parameters and the generalized variables of the system. The similarity criteria between two different systems can be obtained from a detailed consideration of the above similarity groups together with the necessary closure conditions. If the similarity is to be achieved between processes observed in a prototype and in a model, the following requirements must be satisfied: 1, FiR = 1, S t i R = 1,
Bia
=
1,
LiR = 1, R R = 1, T/~ = 1, QsiR
=
h iR = h R = ksR (uo/loa~)a
,
(18)
met.
( l J l o ) R = ( a i / a o ) R = 1,
(19)
( ~ F i / A ~ ) R = 1.
(20)
( F / / A 2 ) ] ,/3 (11)
AiR =
1/2
(13)
1,
where subscript R denotes the model to prototype ratio of the variables, i.e., ~bR --- ~m/~p" As discussed in detail in [9-12], the frictional similarity requirement, FiR, can be satisfied independently of the remaining scaling requirements. Hence, from the remaining scaling requirements, it can be shown that the following conditions should be satisfied for a c o m -
The conditions established in eqs. (14)-(20) assure that the effects of each term in the conservation equations are preserved in both model and prototype without any distortions. If they are not all satisfied, the effects of some of the processes observed in the model and prototype will be distorted. At this point, a few comments might be appropriate as to the practical implications of the similarity requirements: (1) The friction similarity requirement imposed by eq. (20) can be easily satisfied by introducing an appropriate flow resistance in the loop. However, as demonstrated by Kocamustafaogullari and Ishii [10,12], this requirement imposes a hmit on the size of a scaled-down model. (2) The conduction depth ratio and hydraulic diameter ratio as expressed by eqs. (16) and (17), respectively, will affect the design of the scaled-down model. Although it may be difficult to satisfy both requirements over the entire loop, they are likely to be important only at the major heat transfer components where they can be met easily. (3) In contrast to the design parameters such as 3 i and d i, the heat transfer coefficient ratio criterion, h R, given by eq. (18) may be difficult to satisfy in general since the heat transfer coefficient depends strongly on the flow field and fluid properties. Thus eq. (18) imposes an additional constraint on the flow field, i.e., laminar or turbulent. It is customary to represent a correlation for h in terms of the Nusselt number defined by Nu --- h d / k f .
(21)
G. KocamustafaogullarL M. Ishii /Scaring of two-phase flow transients
124
There are a number of correlations for Nu for a flow in a relatively long tube. However, the following two correlations typically represent a wide range of applications: Nu = 4.36 Nu
=
for laminar flow, ~" = const.,
C, Re" Pr n for turbulent flow,
(22) (23)
where Ct, rn and n are constant and their numerical values change slightly from one correlation to another. In view of eq. (21), the heat transfer correlation yields (hR)c
=
ktR/dR
for laminar flow,
(24)
and ( h R)c = ( k f a / d a ) ( R e ) a (Pr) R for turbulent flow. (25) Here, the subscript c refers to the heat transfer coefficient based on correlations. Furthermore, using eq. (17) for d R, eqs. (24) and (25) can be expressed, respectively, as follows: Laminar flow: (hR) c = kfR(pfCpf/psCps) ( uo/loas)-,/2 R ,
(26)
Turbulent flow: (hR) c = kfR(pfGf//PsCps) ( Uo/tOets)R _, ,,1/2(Re)Rm(Pr)Rn (27) It is evident that eqs. (26) and (27) impose quite different constraints on operational and design parameters than those imposed by eq. (18). Therefore, these different constitutive relations for h indicate that the similarity requirements resulting from the Stanton and Biot numbers are not easy to meet. These difficulties may be overcome by requiring that h R = (hR) c.
(28)
Combining eq. (18) with eq. (26) for laminar flow and with eq. (27) for turbulent flow, eq. (28) yields
(kfOfCpt/k~p~Cp~) = 1 for laminar flow,
(29)
and
( kfpfCpt/kspsCps)R ( R e R ) " (Pr)R = 1 (30) for turbulent flow. For a laminar-laminar flow simulation the constraint expressed by eq. (29) depends only on the fluid and solid structure property ratio, whereas for a turbulent-turbulent simulation additional terms such as Re-
ynolds and Prandtl number ratios appear in eq. (30). These numbers can change greatly from prototype to scaled model, especially if non-prototypical pressures are applied a n d / o r different fluids are used in the scaled model. Even for the same fluids at prototypical pressure level, eq. (30) requires that the ratio of the Reynolds number be close to one. For a scaled model this may result in higher model velocity and very high model power. Because of these facts, the similarity condition based on the Biot and Stanton number should be carefully evaluated. It is to be noted that the difficulties discussed above result from the laminar-laminar or turbulent-turbulent flow simulations only. It may be entirely possible to have turbulent flow in the prototype with associated heat transfer and hydraulic resistances and have laminar flow in the model. This makes the design of a scaled model problem much more complicated than the laminar-laminar and turbulent-turbulent flow simulation described above. An exact treatment of scaling should address these issues in a greater detail. For the problem addressed in this paper, i.e., simulating a natural circulation loop, the difficulties in meeting eq. (18) may be relaxed since the Biot and Stanton number similarity conditions with the constitutive relations for the heat transfer coefficient mainly simulate the thermal boundary layer temperature drop. When the heat transfer mechanism is not completely simulated, the system would adjust locally to a different temperature drop in the thermal boundary layer. However, the over-all flow and energy distribution in the complete loop will not be strongly affected during the slow transients typical of a natural circulation system. The violation of the Biot or Stanton number similarity within the liquid flow condition should not cause a major problem except at very rapid power transients. (4) It is important to note that the above set of requirements does not put a constraint on the power density scale, qoR- However, it puts a restriction on the time scale as follows: r
zrp 2
"
~1 1/3
~R =- (Io/Uo)R = 1OR~[BOo lo/(ps/Cp~)lR
. (31) Any additional requirement on the time scale fixes the power density scale. For example, for a real time scaling, e.g., ~'R = 1, the similarity requirements expressed by eqs. (15) through (18) yield the following set of equations: UOR = ]0R; [
8R = (t~sR)
AToR = /0R/fiR; .1/2
;
dR
=
[(p~Cp~)R/(PfCpt)R]:
~ OtsR )..,/2 ;
(32) ~o~ =
(PsGs)Rt0~R;
hR = (~sG~P~)R
G. Kocamustafaogullari, M. Ishii /Scaring of two-phase flow transients With these restrictions, and the additional requirements imposed by eqs. (19) and (20), real time scaling is preserved and complete scaling can be achieved.
125
Friction number:
i ( l + x A P / l l g ) °'2' 3. Two-phase flow similarity
Orifice number:
3.1. Similarity parameters
No -= r, [1 + x ,2(z p/p,)] (ao/a,) 2,
The similarity parameters based on the drift-flux model have been developed by Ishii and Zuber [13] and Ishii and Jones [14]. Two different methods have been used. The first method is based on the one dimensional drift-flux model by choosing proper scales for various parameters. Since it is obtained from the differential equations, it has the characteristics of local scales. The second method is based on the small perturbation technique and consideration of the whole system responses. The local responses of main variables are obtained by solving the differential equations; then the integral effects are found. The resulting transfer functions are nondimensionalized. From these, the governing similarity parameters are obtained. Recently, a combination of results obtained from the above two methods has been used by Ishii and Kataoka [9,11] to develop similarity criteria for two-phase flow systems. The important dimensionless groups that characterize the kinematic, dynamic, and energetic flow fields were given as follows: Phase change number:
where Vgj, Ahfg, Ahs, b and X are the drift velocity of the vapor phase, heat of evaporation, subcoohng, and vapor flow quality, respectively. In addition to the above-defined physical similarity groups, several geometric similarity groups such as (li/lo) and (ai/ao) are obtained. The Froude, friction and orifice numbers, together with the time ratio and thermal inertia groups, have their usual significance as in single-phase flow. The subcooling, phase change and drift-flux numbers are unique to two-phase flow systems. Their physical significance is discussed in detail elsewhere, [9-12].
' " ¢~lo/duopfAhfs)( Ap/pg ), Npeh-~ ( 4 qo
(33)
Subcooling number: Ns. b =- ( Ahs.b/Ahfg)(
Ap/pg),
(34)
Froude number:
S~r -- ( '&glo )( P,/"o'aP ),
(35)
(40)
3.2. Similarity criteria Eqs. (23) through (30) represent relationships between the dimensionless groups and the generalized variables of a two-phase flow system. The dimensionless groups must be the same in both the prototype and the model if similarity requirements are to be satisfied. This leads to the following conditions: (Nwh)a = 1,
(Ns.b) R = 1,
(NFr)R = 1 ,
(Na,)R=I,
(T,*)a = 1,
(Nth,) R = 1,
(Nf,)a = 1,
(No) a = 1.
(41)
It can be shown from the steady-state energy balance over the heated section that Np¢h and Nsub are related by
Nmh - Ns,,b = x e ( A p/p~ ),
(42)
Drift-flux number:
Ndi - ( Vgj/u o) , (or void-quality relation),
(36)
x~a(ap/pg)a= 1.
Time ratio number:
r,. - (10/.0)/(82/.s),,
(37)
Thermal inertia ratio:
N,h, " ( P,%~8/P, C~, d ) ,,
where x¢ is the quality at the exit of the heated section. Therefore, the similarity based on the phase change and subcooling numbers yields
(38)
(43)
This relation scales the quality in terms of the density ratio, and the similarity in vapor quality is satisfied. Eq. (43), when combined with eqs. (29) and (30), ensures that the similarity in the friction number, Nt, and orifice number, No, are approximately satisfied. By
G. Kocarnustafaogullari, M. Ishii / Scaling of two-phase flow transients
126
using the definition of drift-flux, = ug - [(1
Vg; = ug - j
- a) uf + aug],
(44)
it can be shown that the drift-flux number can be expressed as
Nd
(Apx/pg)[(pf/Apa)
=
-
1] - 1,
(45)
where j is the volumetric flux of the two-phase mixture. From eqs. (43) and (45), it can be observed that the similarity in the drift-flux number requires similarity in the void-quahty correlation. This implies that OteR(Ap/pf)R = 1.
(46)
Hence, excluding friction, orifice and drift-flux number similarities from the set of eq. (41), and solving the remaining equations, one obtains the following similarity requirements: _
11/2
/'/OR -- "0R ,
(47)
( Ah~,b )R = ( Ahfgog/AP )a,
(48)
OOR = ( OfosAhfJaO )( d/8 )Rlod/z,
(49)
It
8 R = (loR)
'~l/4z xl/2 [asR) ,
dR= [(0sCps)R//(pfCpt)R]
1*OR 1 / 4 "~sR 1/2
"
(50) (51)
As demonstrated by eq. (31), there remains flexibility in establishing the time scale in single-phase flow. However, in the case of two-phase flow, the time scale is uniquely determined from eq. (47) alone. Thus, "R -= ( l o / u o ) R
-- -oR"/2.
(52)
If the axial length in the model is reduced, real time simulation can not be achieved in model and prototype two-phase flow natural circulation loops. Instead time events are accelerated in the scaled-down model by a factor of tl/2 "0R over the prototype. When the two-phase flow velocity scale, eq. (47), is used in the single-phase flow geometric scale requirements, eqs. (16) and (17), the geometric similarity requirements in both phases become the same. However, the single-phase real time-scale can not be preserved.
4. Single-phase to two-phase scaling Single-phase and two-phase flows occur simultaneously in a natural circulation loop as shown in fig. 1. In general, single-phase and two-phase flow requirements are different from each other. This introduces
difficulties in meeting geometric, power and time scaling requirements when both similarity requirements have to be satisfied simultaneously. However, the problem can be resolved by using the flexibility of the singlephase flow similarity requirements. Depending on the choice of time or power scale simulations in single and two-phase flows, two alternatives exist to meet singlephase and two-phase flow requirements simultaneously as discussed below.
4.1. Time scale simulation This alternative is based on using the same time scale in the single-phase and two-phase flow. Since the time scale for two-phase flow similarity is fixed by eq. (52), the same time scaring in the single-phase flow can be used. Thus, ( TR )sp = ( TR )tp =/1/2 "OR"
(53)
The velocity scale then becomes ( U o . ) s p = 1,/2 -OR.
(54)
Power and geometric scaring parameters can be obtained from eqs. (14), (16) and (17). They are given by .... ( pscps i (qoR)sp = ~ flllo/2 JR'
(55)
( ~R )sp = "0R11/4--1/2UsR '
(56)
(aR)sp = [(osCps)/(PfCpe)] R'0RI'/4='/2'*sR•
(57)
By comparing eq. (56) with eq. (50) and eq. (57) with eq. (51), it is observed that the single-phase and twophase flow geometric scales are equal; thus the same geometric scale criterion can be used in both phases. However, eqs. (55) and (49) indicate that the power density in single-phase flow sections should be scaled differentially from two-phase flow sections by a factor of (Oo~)s#(od~),.
=
(/~Ogah,g/AoG,)~'
(58)
Although changing the power density from single-phase to two-phase flow transition is not impossible, it does not seem to be a practical solution. In order to avoid this difficulty, an additional constraint should be imposed by setting
(BogAhfs/aOCp,)R
= 1.
(59)
Similarity in the property group expressed by eq. (59) ensures that geometric, power and time scales are
G. Kocamustafaogullari, M. Ishii /Scaling of two-phaseflow transients the same during the single-phase to two-phase flow transition. Otherwise, the power density will be distorted by a factor given by eq. (58). This distortion can not be tolerated because it may lead to completely different two-phase flow phenomena associated with phase change in the model as compared to the prototype. As demonstrated later it is possible to meet this requirement by using different fluids in model and prototype. However, it is impossible to satisfy eq. (59) for modeling high pressure systems with low pressure systems using the same fluid in the prototype and model.
4.2. Power scale simulation The solution to the original problem of the singlephase to two-phase flow transition scaling can also be achieved by using the power scale given by eq. (49) for both phases:
= (pfpgAhfg/AP)R(PsCps)/(pfCpf)glo
1/2 .
(60) Using eq. (50) in eq. (14), the single-phase flow velocity scale then becomes 1/2 1/3 ( U0g )sp = 10g ( tiPs A h fg//ApCpf )R "
) -- 1/6
( 8 R )sp = 10R OtsR ( flA h fgps/A p Cpf /R
(62)
and
(d.)sp = (psC s). / ( p , cpf )RIoR 1/4~XsR 1/2 ( flAhfsPg//ApCpf )R
. (63)
Similarly, the single-phase time scale becomes (~'g)sp -= log/Uog = llo/2( flAhfgPg//ApCp, )R 1/3.
(64)
Comparing the single-phase flow scaling parameters given by eqs. (62), (63) and (64) with their two-phase flow counterparts, it can be shown that ( UOR)sp( U0R )tp = ( d R ) s p / ( dR
)tp
= (flpgAhfg//ApCpf)R
1/3
= (~pgAhfs/ApCpf)R
(~'R)sp/(~'R)W = (flpsAhfg/ApCpf)R
- 1/6 -- 1/3
,
(66)
.
(67)
As in the case of time scale simulation, for a power scale simulation the condition imposed by eq. (59) must be met in order to have a complete similarity during the single-phase to two-phase flow transition. This presents no problem for fluid-to-fluid simulation. However, distortions are unavoidable in simulating high pressure systems with low pressure systems by using the same fluid. If the same fluid has to be used for reduced pressure modeling, the second alternative, power scaling, is advisable for the following reasons: (1) Time simulation is not practical since it requires different powers for the single-phase and two-phase sections. Otherwise the power scale will be distorted by the factor given by eq. (59). This distortion can not be tolerated since it may lead to completely different two-phase flow phenomena in the model as compared to the prototype. (2) Although the power simulation causes different time scales in single-phase and two-phase flows, it can be accounted for when evaluating model data. (3) The geometric distortions caused by power scaling may not be significant because of the very small exponent appearing in eq. (66).
(61)
Furthermore, substituting this velocity scale in eqs. (16) and (17) yields the following single-phase geometric scaling parameters 1/4 1/2
(SR)sp//(~a)tp
127
,
(65)
5. Simulating pressure transients for two-phase flow
A characteristic pressure transient for a LWR under LOCA is illustrated in fig. 2 [15]. After a LOCA the system pressure drops rapidly from its normal operating pressure, Pi, to the saturation pressure, Pa, and boiling starts at this point. In region I, which is very short, the fluid is single-phase. Subsequently two-phase flow dominates the circulation, and the system pressure is identified as /2, which changes with time as shown in fig. 2. It is the objective of this section to investigate the possibility of simulating the pressure transients in region II. With the exception of state 1 shown on the figure where a phase transition occurs, two-phase flow dominates in the second region. To simulate the pressure transient in region II it is important to meet the two-phase flow scaling conditions, eqs. (47) through (51) where eq. (47) is based on the axial length scale and eqs. (50) and (51) are based on a given geometric scale. Hence, once the model is designed, the geometric parameters /OR, 8R and d R can not be altered while pressure transients occur. Since the power scale and the
G. Kocamustafaogullari, M. Ishii /Scaling of two-phaseflow transients
128
where subscripts 1 identifies state 1, and subscript 2 identifies any state in region II, as shown in fig. 2. The first condition simulates subcooling while the second one ensures that the phase change is simulated. Fortunately, these two requirements are not too different from each other since Of, the only difference in the two requirements, does not change appreciably within the practical limits of a pressure transient from Pt to "°2Since the phase change is more important than the subcooling for two-phase flow simulations, only the second group needs to be considered further. Recalling the original definition of the ratios, eq. (69) can also be expressed as
Pi
1
P1 I
P2
2
HI I
Time
I] - -
Iv.
Two-Phase Region
rmJ Ym, r.~/r., =
Single-Phase Region Fig. 2. A typical pressure transient for a LWR under LOCA.
subcooling are fixed by state 1, eqs. (48) and (49) require, respectively, that the following two conditions should be met during the transient:
(Ap/pgahfs)R , = (AP/PsAhfg)R:
(68)
and ( A
p/pfpgA h fg ) R1 = 102
1
( A p/pfpgA h fg ) R2'
I ~I
I
\
]
]
! l
(69)
[
i
1. Water, Ppl = 110 bar 2. R-11, Pm1=37.5 bar 3. R-11, Pro1= 20.0 bar 4. R-11, Pm1=9"5 bar R - 1 1 , Pro1 =3.8 bar
\ i
10o
10-1 10o
L Lli
I
102
10
where Y is defined as Y=-Ap/pfpgAhts. Once the prototype pressure, Ppa, and the model pressure, Pml, are specified at state 1, where the phase change starts, the problem of simulating pressure transients reduces to seeking solutions in the form of Pro2 = f(Pp2) which satisfy eq. (70). To investigate the possibilities of locating such solutions, two separate cases are considered. The first one simulates the pressure transients in a model with a different fluid than is used in the prototype, i.e., fluid-to-fluid simulation. The second simulates pressure transients with a model using the same fluid as the prototype at lower pressures relative to the prototype, i.e., reduced pressure simulation.
5.1. Fluid-to-fluid simulation
I
Y ~ Apl pfpgAhfg
I0
(70)
I 1
10 3
P2 (bar) Fig. 3. Variation of two-phase pressure transient similarity group ratio for water and R-11.
The use of refrigerants for scaling of water systems has been an accepted practice for modeling various two-phase flow phenomena, such as modeling the CHF and boiling, etc. Considerable facility cost savins are achieved by the lower absolute pressure, and temperatures realized at similar reduced pressures. It would, therefore, be of considerable interest to accomplish the same result for simulating transients encountered during a LWR and LOCA. Among the various refrigerants, R-11, 12, 22, 113 and 114 that could be considered, R-12, 22 and 114 are excluded from further study since they exhibit very low boiling temperatures at relatively low pressures. Consequently, the analysis was limited to R-11 and R-113. For these two model fluids, a series of solutions in the form of Pro2 = f(Pp2) are sought in three steps as follows: (a) Assuming that the typical prototype pressure at state 1 is Ppl = 110 bar, [15], (Y2/Y1) curves for the prototype and the model fluids are produced in fig. 3
G. Kocamustafaogullari, M. Ishii /Scaring of two-phaseflow transients 102 _
[
~
[
- - T
20 --
y_= Ap/pfpgAhfg 1. Water, Pp1=110 bar 2. R-113, Pml = 30.0 bar 3. R-113, Pro1=21.4 bar 4. R-113, Pml: 12.1 bar _ _ 5. R-113, Pm1=7"0 bar -
108 --
~.~
[
Water to
129
~
T
R-113 Simulation
1. Water, Pp2:80 bar 2. Water, Pp2=60 bar 3. Water. Pp2=40 bar 4. Water, Pp2:20 bar
T
/
/
/ 2 / /
/
//--/// / /
--
"~
, waterP,,=lobar/ / /
--
/
T
11
t
2
~
~
4
6
8 10
20
40
60
Pml (bar)
100
102
10 '
103
Fig. 6. Model pressure curves which satisfy two-phase flow pressure transient similarity requirement for water to R-113 simulation at Ppl = 110 bar.
P2 (bar)
Fig. 4. Variation of two-phase flow pressure transient similarity group ratio for water and R-113.
useful in locating Pm2 once Ppl, Pp2 and Pml are known. Hence, knowing P,m is crucial in obtaining Pm2 = f ( P p 2 ) .
for R-11 and fig. 4 for R-113 as function of the transient pressures, Pp2 o Since the model pressure at the point where boiling initiates, Pro1, is not yet known, it is taken as a parameter. (b) Utilizing figs. 3 and 4, Pro2 = f ( P m l ) curves are generated for given values of Ppl and Pv2 as illustrated in fig. 5 for R-11 and fig. 6 for R-113. These figures are
(c) As noted above, state 1 is characterized by the single-phase to two-phase flow transition. Simulation of this state by a model fluid was discussed in detail in the preceding section where it was concluded that the condition imposed by eq. (59) must be met between the 100
I
I
I I I
I
I
r
r
20 -- Water t° R'I' Slmulati°n Pp2:80 bar 2 . W a t e r , ,,2:60 bar 3. Water, Pp2:40 bar 4. Water, Pp2=20bar
"~'/
I
~
1. W a t e r ,
10 - .~ 8 - -
_
/ / /
/ /
/ /
''
I
'
'
I
I
i
I
I/
~
/
10"1-
I
"~ /
1. Water (Prototype fluid) 2. R-11 (Model Fluid) 3. R-113 (Model Fluid)
I
/
q
%
T
l
10.2
2
1
2
4
6
8 10
20
40
60
Pml (bar)
Fig. 5. Model pressure curves which satisfy two-phase flow pressure transient similarity requirement for water to R-11 simulation at Pp1 = 110 bar.
10"
I
10
I I
102 • P1 (bar)
103
Fig. 7. Variation of single-phase to two-phase flow transition similarity group for prototype and selected model fluids.
G. Kocamustafaogullari, M. Ishii /Scaling of two-phaseflow transients
130
(d) Using these Pm] values in figs. 5 and 6, pressure transient simulations curves in the form of Pm2 = f(Pp2) are generated in fig. 8 for both model fluids. Each Pmz =f(Ppz) curve appearing in fig. 8 simultaneously satisfies the two-phase flow similarity requirements and the single-phase to two-phase flow transition requirement. The latter is important for locating state 1. In addition, the pressures of both model fluids are within the practical range of applicability, indicating that R-11 and R-113 can be used for simulating the pressure transients occurring in a L W R under L O C A condition.
20--
10
C
8--
O.~ "~
6
2 ~
1, R-113
/ /
//11
Pp1=110 bar Pml : 23 bar
/ /
///
Pml : 26 bar
2 -
5.2. Reducing pressure scaling 1 10
L 20
L 40
L 60
I L 80 100
--~
Pp2 (bar)
200
Fig. 8. Pressure transient simulation for water to R-11 and to R-113 modeling at Pp] = 110 bar. prototype and model fluids. Variations in this property group appearing in eq. (59) are illustrated in fig. 7 for the prototype and model fluids. For Ppl = 110 bar, it is evident from the figure that Pm~ = 23 bar for R-113 and Pml = 26 bar for R-11. Both of these values are within the practical range of applicability for the selected model fluids.
It is also interesting to investigate the possibilities of simulating high pressure water transients with a low pressure water system. Corresponding to the prior example, figs. 9 and 10 are generated for this purpose. The final solution in the form of Pro2 = f(Pp2) is given in fig. 11. Unlike the fluid-to-fluid simulation investigated previously, a unique solution Pro2 = f ( P p 2 ) can no longer be found. As shown in fig. 11, Pml has to be taken as a parameter, and curves appearing in this figure satisfy only the two-phase flow scaling requirements over a wide range of pressure transients. However, these curves
100 -
46~ 2 10 s8
"~ >-
4
f i
2
J
i
ii
I
2
4
6
10
Water to Water Simulation 1. Water, Pp2 =10 bar 2. Water, Pp2 = 20 bar 3. Water, Pp2 = 40 bar 4 - - 4. Water, Pp2 = 60 bar 5. Water, Pp2 • 80 bar 6. Water, Pp2 = 110 bar
8-6 --
y~ Ap//pfpg,..~hfg
~ 1 \ ~ ~ ' ~ \ ~a~\ \ ~ " ~ ~ ~ 5 ~ ~
1. Water, 2. Water, 3. Water, 4. Water,
Ppl =Pml =110 bar P m l : 80 bar Pml : 60 bar Pml = 40 bar
S. W a t e r ,
Pro1 : 2 0 b a r
_
6. Water, Pm, : 10 bar
ito
\ \
8
10o 8
6 4 2 10-I 100
2
4
6 8 10
2
4
6 8 102
2
4
6 ; 103
P2 (bar)
Fig. 9. Variation of the two-phase flow pressure transient similarity group ratio for reduced pressure simulation.
2
t
4
i 8iJ10
6
Pml (bar)
Fig. 10. Model pressure curves which satisfy two-phase flow pressure transient similarity requirement for high-pressure water to reduced-pressure water simulation at Ppl = 110 bar.
G. Kocamustafaogullari, M. lshii /Scaling of two-phaseflow transients 10(
m ---
v
I
I
I
I I
[
Water to Water Simulation 1. Water, Pml =10 bar 2. Water, Pm2 : 20 bar 3. Water, Pm2 =40 bar 4. Water, Pm2 = 60 bar 5. Water, Pm2 " 8 0 bar 6. Water, Pm2 . 1 1 0 bar
I
I
I
4
6
8 100
65
o.
T
2
4
6
8 10
2
' Pp2 (bar)
Fig. 11. Pressure transient simulation for high-pressure water to reduced-pressure water modeling at Pp] = 110 bar.
131
was shown that the real time scaling is not possible for an axially scaled-down model. The scaling criteria are also applied for simulating pressure transients occurring in a LWR under LOCA conditions. Crucial to this study were the fluid property groups which appear within the two-phase similarity parameters and the single-phase to two-phase transition similarity. Important property groups to be considered w e r e ( Ap//Ahfgpfpg) and (flPsAhfg//ApCpf). The first group ensures the similarity in two-phase flows transients while the second one permits the similarity from single-phase to two-phase flow transition. Finally, computations were performed to investigate the feasibility of modeling a typical LWR system under LOCA by a low pressure water system and a Freon system, using R-11 or R-113. Modeling is possible in both cases for simulating pressure transients. However, simulation of phase change transitions is not possible with a low pressure water system without distortions in power or time.
Acknowledgements
no longer satisfy the single-phase to two-phase flow similarity requirement given by eq. (59). Therefore distortions at the power scale, as expressed by eq. (58), can not be avoided for the reduced pressure simulation.
The authors would like to thank Drs. N. Zuber and R. Lee of the U.S. Nuclear Regulatory Commission for valuable discussion on the subject. This work was carried out under the auspices of the U.S. Nuclear Regulatory Commission.
6. S u m m a r y and conclusions
Nomenclature
Based on the similarity analysis of [9-12], scaling criteria for a natural circulation loop under single-phase and two-phase flow conditions have been derived. The criteria include the effects of fluid properties to allow for modeling of a high pressure water system by a low pressure water system or by a different fluid system. In addition to the usual single-phase flow dimensionless groups, it is shown that the phase change, subcooling, drift-flux and Froude numbers are important for twophase flow systems. Combining the single-phase and two-phase flow criteria, single-phase to two-phase flow scaling was studied. Two possibilities, namely time simulation and power simulation, were investigated in detail. Particular emphasis was placed on scaling at reduced pressure levels relative to a prototype system and on fluid-to-fluid scaling. It was noted that power scaling is more practical than time scaling for modeling a high pressure water system by a low pressure water system. Furthermore, it
A a as B Cp d F f g h K k L l j Npch Nsub
NFr Nd
non-dimensional area, flow cross-sectional area, solid cross-sectional area, Blot number, specific heat capacity at constant pressure, hydraulic diameter, friction number, friction factor, gravity, heat transfer coefficient, orifice coefficient, conductivity, non-dimensional length, axial length, volumetric flux, phase change number, subcooling number, Froude number, drift-flux number,
132
N~ No Nu P Pr •
*tt
qo q o//
Qs R Re St T T* U U0
v~j X Xe ~t Ote Ots
8 ahf s A h sub
aT0 ao At, P
G. Kocamustafaogullari, M. Ishii /Scaling of two-phase flow transients friction nuber (two-phase), orifice number (two-phase), Nusselt number, pressure, Prandtl number, heat generation density in solid, heat flux, heat source number, Richardson number, Reynolds number, Stanton number, temperature, characteristic time ratio, fluid velocity, heated section representative velocity, drift velocity, vapor flow quality, exit quality, void fraction, exit void fraction, solid thermal diffusivity, thermal expansion coefficient, conduction thickness, latent heat of evaporation, subcooling, characteristic temperature rise, density difference, viscosity difference, wetted perimeter, mass density, non-dimensional time.
Subscripts e
f g i m
P R S
sp tp 0
exit, hquid, gas or vapor, i th section, model, prototype, model-to-prototype ratio, solid, single-phase flow, two-phase flow, reference constant (heated section).
References [1] J.C. Amrouni, S. Nijhawan, A.M.M. Aly, R.L. Ferah, and A.M. Omar, Sscaling requirements for an experimental facility to study the thermosyphoning behavior of a CANDU reactor, Presented at the Canadian Nuclear Society 6th Annual Conf., Ottawa, Canada (June 1985). [2] A. Carbiener and A. Chudnik, Similitude considerations for modeling nuclear reactor blowdowns, Trans. Am. Nucl. Soc. 12 (1969) 361-362. [3] A.N. Nahavandi, F.S. Castellana and E.N. Moradkhanian, Scaling laws for modeling nuclear reactor systems. Nucl. Sci. Engng. 72 (1982) 347-354. [4] L. Ybarrando, L.Y. Griffith, P. Fabic and G.D. McPherson, Examination of LOFT scaling, ASME, Annual Winter Meeting, New York (November 1974). [5] T.K. Larson, J.L. Anderson, and D.J. Shineck, Scaling criteria and assessment of semiscale Mod-3 scaling for small-break-loss-of-coolant transients, EGG-SEMI-5121 (March 1980). [6] E.R. Rosal, PWR FLECHT SEASET systems effects natural circulation and reflux condensation task plan report. NUREG/CR-2401 (February 1983). [7] R.P.Rose, Heat transfer problems associated with a LOFT (loss of fluid test), Proc. Conf. ASME-AIChE Heat Transfer, Los Angeles (August 8-11, 1965). [8] N. Zuber, Problems in modeling of small break LOCA, NUREG-0724 (October 1980). [9] M. Ishii and I. Kataoka, Similarity analysis and scaling criteria for LWR's under single-phase and two-phase natural circulation, NUREG/CR-3267, ANL-83-82, Argonne National Laboratory, Argonne, Illinois (March 1983). [10] G. Kocamustafaogullari and M. Ishii, Scaling criteria for two-phase flow natural and forced convection loop and their application to conceptual 2×2 simulation loop design. NUREG/CR-3420, ANL-83-61, Argonne National Laboratory, Argonne, Illinois (May 1983). [11] M. Ishii and I. Kataoka, Scaling laws for thermal-hydraulic system under single-phase and two-phase natural circulation, Nucl. Engrg. Des. 81 (1984) 411-425. [12] G. Kocamustafaognllari and M. Ishii, Scaling criteria for two-phase flow loops and their application to conceptual 2 × 4 simulation loop design, Nucl. Technology 65 (1984) 146-160. [13] M. Ishii and N. Zuber, Thermally induced flow instabilities in two-phase mixtures. Proc. 4th Conf. Heat Transfer. Paris, France (1970) Paper B5.11. [14] M. Ishii and D. Jones, Derivation and application of scaling criteria for two-phase flows. Proc. Conf. Two-Phase flows and Heat Transfer, NATO Advanced Study Institute, 1, Istanbul, Turkey (1976) 163-185. [15] Nuclear Safety Analysis Center Report, The analysis of Three-Mile Island-Unit 2 Accident, NSAC-1 (July 1979).
SCALING OF TWO-PHASE AND SIMULANT FLUID
121
FLOW TRANSIENTS
USING REDUCED
PRESSURE
SYSTEM
G. KOCAMUSTAFAOGULLARI University of Wisconsin- Milwaukee, Department of Mechanical Engineering, Milwaukee, WI 53201, USA and M. ISHII Argonne National Laboratory, Reactor Analysis and Safety Division, Argonne, IL 60439, USA Received May 1987
Scaling criteria for a natural circulation loop under single-phase and two-phase flow conditions are derived. Based on these criteria, practical applications for designing a scaled-down model are considered. Particular emphasis is placed on scaling a test model at reduced pressure levels compared to a prototype and on fluid-to-fluid scaling. The large number of similarity groups which are to be matched between model and prototype makes the design of a scale model a challenging task. The present study demonstrates a new approach to this classical problem using two-phase flow scaling parameters. It indicates that a real time scaling is not a practical solution and a scaled-down model should have an accelerated (shortened) time scale. An important result is the proposed new scaling methodology for simulating pressure transients. It is obtained by considering the changes of the fluid property groups which appear within the two-phase similarity pargmeters and the single-phase to two-phase flow transition parameters. Sample calculations are performed for modeling two-phase flow transients of a high-pressure water system by a low-pressure water system or a Freon system. It is shown that modeling is possible for both cases for simulating pressure transients. However, simulation of phase change transitions is not possible by a reduced pressure water system without distortion in either power or time.
1. Introduction Similarity laws and scaling criteria are important for designing, performing and analyzing simulation experiments using a scale model. They are particularly important for studying phenomena which can not be easily observed in a prototype model, such as occur in nuclear reactor transients and accidents, as well as in some petrochemical experiments. In view of the inherent difficulties associated with full-scale testing of such systems, it is necessary to use scale models and simulation experiments. However, for a justifiable extrapolation of experimental model results to full-scale prototype transients, it is necessary that the scaling rationale be soundly based. Scaling thus becomes the single most important issue in the conceptual design of experiments whose aim is to investigate transients in a prototype, Amrouni et at. [1].
In line with the central importance in establishing similarity relations between prototype and scale models, several scaling techniques have been developed. The simplest technique is linear scaling, in which all length ratios are preserved. While there are some single-phase flow situations where linear scaling is adequate, its use results in thermal and flow distribution distortions which are generally considered unacceptable for two-phase flow conditions, [2-5]. For these reasons, volumetric scaling techniques are frequently used for thermohydrautic experiments [4-7]. Model facilities using this approach are usually full-length (with the notable exception of LOFT), while area, volume and power are reduced proportionally. Both linear and volumetric scaling techniques described here are developed based on single-phase flow conservation equations. However, extending these techniques to investigate phenomena associated with two-phase flow, particularly for non-homo-
0 0 2 9 - 5 4 9 3 / 8 7 / $ 0 3 . 5 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. (North-Holland Physics Publishing Division)
122
G. Kocamustafaogullari, M. Ishii / Scaling of two-phase flow transients
geneous flows, is certainly incorrect, Zuber [8]. Recently, extensive investigation have been undertaken by the present authors to develop the similarity criteria for the natural circulation loops under singlephase and two-phase flow conditions [9-12]. The similarity criteria obtained from these investigations were, further, extended to practical applications for designing a scaled-down model. In these applications, however, emphasis was concentrated on exploring the modeling of a prototype by a scaled-down model using the same fluid at the same pressure as the prototype. Therefore, the working fluid property effects and fluid-to-fluid simulations were not studied in detail. Based on the rigorous similarity analysis of the recent studies, [9-12], the present study attempts to develop scaling criteria for a natural circulation loop under single a n d / o r two-phase flow conditions, and to apply the criteria for modeling a high pressure water system by a low pressure water system or a Freon system. One of the most important results of the present study is a new methodology for simulating pressure transients. It is obtained by studying the single-phase to two-phase flow transition and by considering changes of the fluid property groups arising from the two-phase flow similarity parameters.
STEAM GENERATOR (COOLED SECTION)
PUMP
HEATED SECTION
J - -
SINGLE--PHASE REGION
,',;'/,G'; TWO--PHASE REGION Fig. 1. Schematic natural circulation system under consideration. [9-12]. They are given as follows: Richardson number:
R =- gBa T0t0/u02, Friction number: F, -~ [ ( f l / d ) + K ],,
2. Single-phase flow similarity 2.1. Similarity parameters
(J)
(2)
Stanton number: Sti ~- (4hlo/prCpfuod),,
(3)
Time ratio number: Although the major effort in this study is directed at developing similarity criteria for simulating two-phase flow pressure transients, single-phase flow similarity requirements are needed first to simulate the singiephase to two-phase flow transition. A typical system under consideration is illustrated in fig. 1. This system consists of a thermal energy source, energy sink, and a connecting piping system between components. For a natural circulation loop under single-phase flow conditions, the similarity parameters are obtained from the integral effects of the local balance equations along the entire loop, [9-12]. Rather than duplicating the work of these references, the essential parts of the similarity analysis are summarized below. The fluid continuity, integral momentum, and energy equations in one-dimensional, area-averaged forms have been used together with the appropriate boundary conditions and the solid energy equation. From the one-dimensional form of these equations, important dimensionless groups characterizing geometric, kinematic, dynamic and energetic similarity parameters are derived,
Ti* =_ ( l o / u o ) / ( 3 2 / a s ) i ,
(4)
Biot number: B, ~_ ( h S / t , s ) , ,
(5)
Heat source number:
Qsi -~ ( i1~)"lo/psCpsuo A To ) i"
(6)
Here subscripts i, f and s identify the i th component of the loop, fluid and solid, whereas u o, A T o and l 0 are the prescribed reference velocity, temperature difference, and equivalent length, respectively. Most of the symbols appearing above conform to standard nomenclature. However, for the purpose of completeness, all symbols are listed in the nomenclature. In addition to the above physical similarity groups, two geometric similarity groups are obtained: Axial length scale:
L,-= l,/lo,
(7)
G. Kocamustafaogullari, M. Ishii /Scaling of two-phase flow transients Flow area scale:
123
plete similarity:
A i =- a i / a o.
(8)
2 uoR= [flqo.... lo/(OsCp~)] 1/3 R '
(14)
The hydraulic diameter d i and the conduction depth 8i are define by
aToR = [ 0o"to/(O, Cps) u0] R'
(15)
d, = 4a,/~,
3,R = 3R = (a,Io/Uo)R
1/2
and
3 i = asi/~i,
(9)
,
(16)
2,
(17)
where ai, a~i and ~, are the flow cross-sectional area, sohd structure cross-sectional area, and wetted perimeter of ith section. Hence, d i and 3 i are related by
d,R = dR =
d, = 4 ( a / a ~ ) , 3 , .
In addition to the above constraints, the following geometric and dynamic similarity requirements must be
(10)
The reference velocity u 0 and temperature difference ATo are obtained from the steady-state solution. By taking the heated section as the representative section, they are expressed as follows: u 0 = [(4flg(lo" 12/Of Cpr )( a~o/ao ) / ~
and A TO = ( qo" lo/P, Cpf Uo ) ( a~o/ao ),
(12)
where the subscript 0 denotes the heated section. 2.2. Similarity criteria Eqs. (1) through (8) represent relationships between the dimensionless parameters and the generalized variables of the system. The similarity criteria between two different systems can be obtained from a detailed consideration of the above similarity groups together with the necessary closure conditions. If the similarity is to be achieved between processes observed in a prototype and in a model, the following requirements must be satisfied: 1, FiR = 1, S t i R = 1,
Bia
=
1,
LiR = 1, R R = 1, T/~ = 1, QsiR
=
h iR = h R = ksR (uo/loa~)a
,
(18)
met.
( l J l o ) R = ( a i / a o ) R = 1,
(19)
( ~ F i / A ~ ) R = 1.
(20)
( F / / A 2 ) ] ,/3 (11)
AiR =
1/2
(13)
1,
where subscript R denotes the model to prototype ratio of the variables, i.e., ~bR --- ~m/~p" As discussed in detail in [9-12], the frictional similarity requirement, FiR, can be satisfied independently of the remaining scaling requirements. Hence, from the remaining scaling requirements, it can be shown that the following conditions should be satisfied for a c o m -
The conditions established in eqs. (14)-(20) assure that the effects of each term in the conservation equations are preserved in both model and prototype without any distortions. If they are not all satisfied, the effects of some of the processes observed in the model and prototype will be distorted. At this point, a few comments might be appropriate as to the practical implications of the similarity requirements: (1) The friction similarity requirement imposed by eq. (20) can be easily satisfied by introducing an appropriate flow resistance in the loop. However, as demonstrated by Kocamustafaogullari and Ishii [10,12], this requirement imposes a hmit on the size of a scaled-down model. (2) The conduction depth ratio and hydraulic diameter ratio as expressed by eqs. (16) and (17), respectively, will affect the design of the scaled-down model. Although it may be difficult to satisfy both requirements over the entire loop, they are likely to be important only at the major heat transfer components where they can be met easily. (3) In contrast to the design parameters such as 3 i and d i, the heat transfer coefficient ratio criterion, h R, given by eq. (18) may be difficult to satisfy in general since the heat transfer coefficient depends strongly on the flow field and fluid properties. Thus eq. (18) imposes an additional constraint on the flow field, i.e., laminar or turbulent. It is customary to represent a correlation for h in terms of the Nusselt number defined by Nu --- h d / k f .
(21)
G. KocamustafaogullarL M. Ishii /Scaring of two-phase flow transients
124
There are a number of correlations for Nu for a flow in a relatively long tube. However, the following two correlations typically represent a wide range of applications: Nu = 4.36 Nu
=
for laminar flow, ~" = const.,
C, Re" Pr n for turbulent flow,
(22) (23)
where Ct, rn and n are constant and their numerical values change slightly from one correlation to another. In view of eq. (21), the heat transfer correlation yields (hR)c
=
ktR/dR
for laminar flow,
(24)
and ( h R)c = ( k f a / d a ) ( R e ) a (Pr) R for turbulent flow. (25) Here, the subscript c refers to the heat transfer coefficient based on correlations. Furthermore, using eq. (17) for d R, eqs. (24) and (25) can be expressed, respectively, as follows: Laminar flow: (hR) c = kfR(pfCpf/psCps) ( uo/loas)-,/2 R ,
(26)
Turbulent flow: (hR) c = kfR(pfGf//PsCps) ( Uo/tOets)R _, ,,1/2(Re)Rm(Pr)Rn (27) It is evident that eqs. (26) and (27) impose quite different constraints on operational and design parameters than those imposed by eq. (18). Therefore, these different constitutive relations for h indicate that the similarity requirements resulting from the Stanton and Biot numbers are not easy to meet. These difficulties may be overcome by requiring that h R = (hR) c.
(28)
Combining eq. (18) with eq. (26) for laminar flow and with eq. (27) for turbulent flow, eq. (28) yields
(kfOfCpt/k~p~Cp~) = 1 for laminar flow,
(29)
and
( kfpfCpt/kspsCps)R ( R e R ) " (Pr)R = 1 (30) for turbulent flow. For a laminar-laminar flow simulation the constraint expressed by eq. (29) depends only on the fluid and solid structure property ratio, whereas for a turbulent-turbulent simulation additional terms such as Re-
ynolds and Prandtl number ratios appear in eq. (30). These numbers can change greatly from prototype to scaled model, especially if non-prototypical pressures are applied a n d / o r different fluids are used in the scaled model. Even for the same fluids at prototypical pressure level, eq. (30) requires that the ratio of the Reynolds number be close to one. For a scaled model this may result in higher model velocity and very high model power. Because of these facts, the similarity condition based on the Biot and Stanton number should be carefully evaluated. It is to be noted that the difficulties discussed above result from the laminar-laminar or turbulent-turbulent flow simulations only. It may be entirely possible to have turbulent flow in the prototype with associated heat transfer and hydraulic resistances and have laminar flow in the model. This makes the design of a scaled model problem much more complicated than the laminar-laminar and turbulent-turbulent flow simulation described above. An exact treatment of scaling should address these issues in a greater detail. For the problem addressed in this paper, i.e., simulating a natural circulation loop, the difficulties in meeting eq. (18) may be relaxed since the Biot and Stanton number similarity conditions with the constitutive relations for the heat transfer coefficient mainly simulate the thermal boundary layer temperature drop. When the heat transfer mechanism is not completely simulated, the system would adjust locally to a different temperature drop in the thermal boundary layer. However, the over-all flow and energy distribution in the complete loop will not be strongly affected during the slow transients typical of a natural circulation system. The violation of the Biot or Stanton number similarity within the liquid flow condition should not cause a major problem except at very rapid power transients. (4) It is important to note that the above set of requirements does not put a constraint on the power density scale, qoR- However, it puts a restriction on the time scale as follows: r
zrp 2
"
~1 1/3
~R =- (Io/Uo)R = 1OR~[BOo lo/(ps/Cp~)lR
. (31) Any additional requirement on the time scale fixes the power density scale. For example, for a real time scaling, e.g., ~'R = 1, the similarity requirements expressed by eqs. (15) through (18) yield the following set of equations: UOR = ]0R; [
8R = (t~sR)
AToR = /0R/fiR; .1/2
;
dR
=
[(p~Cp~)R/(PfCpt)R]:
~ OtsR )..,/2 ;
(32) ~o~ =
(PsGs)Rt0~R;
hR = (~sG~P~)R
G. Kocamustafaogullari, M. Ishii /Scaring of two-phase flow transients With these restrictions, and the additional requirements imposed by eqs. (19) and (20), real time scaling is preserved and complete scaling can be achieved.
125
Friction number:
i ( l + x A P / l l g ) °'2' 3. Two-phase flow similarity
Orifice number:
3.1. Similarity parameters
No -= r, [1 + x ,2(z p/p,)] (ao/a,) 2,
The similarity parameters based on the drift-flux model have been developed by Ishii and Zuber [13] and Ishii and Jones [14]. Two different methods have been used. The first method is based on the one dimensional drift-flux model by choosing proper scales for various parameters. Since it is obtained from the differential equations, it has the characteristics of local scales. The second method is based on the small perturbation technique and consideration of the whole system responses. The local responses of main variables are obtained by solving the differential equations; then the integral effects are found. The resulting transfer functions are nondimensionalized. From these, the governing similarity parameters are obtained. Recently, a combination of results obtained from the above two methods has been used by Ishii and Kataoka [9,11] to develop similarity criteria for two-phase flow systems. The important dimensionless groups that characterize the kinematic, dynamic, and energetic flow fields were given as follows: Phase change number:
where Vgj, Ahfg, Ahs, b and X are the drift velocity of the vapor phase, heat of evaporation, subcoohng, and vapor flow quality, respectively. In addition to the above-defined physical similarity groups, several geometric similarity groups such as (li/lo) and (ai/ao) are obtained. The Froude, friction and orifice numbers, together with the time ratio and thermal inertia groups, have their usual significance as in single-phase flow. The subcooling, phase change and drift-flux numbers are unique to two-phase flow systems. Their physical significance is discussed in detail elsewhere, [9-12].
' " ¢~lo/duopfAhfs)( Ap/pg ), Npeh-~ ( 4 qo
(33)
Subcooling number: Ns. b =- ( Ahs.b/Ahfg)(
Ap/pg),
(34)
Froude number:
S~r -- ( '&glo )( P,/"o'aP ),
(35)
(40)
3.2. Similarity criteria Eqs. (23) through (30) represent relationships between the dimensionless groups and the generalized variables of a two-phase flow system. The dimensionless groups must be the same in both the prototype and the model if similarity requirements are to be satisfied. This leads to the following conditions: (Nwh)a = 1,
(Ns.b) R = 1,
(NFr)R = 1 ,
(Na,)R=I,
(T,*)a = 1,
(Nth,) R = 1,
(Nf,)a = 1,
(No) a = 1.
(41)
It can be shown from the steady-state energy balance over the heated section that Np¢h and Nsub are related by
Nmh - Ns,,b = x e ( A p/p~ ),
(42)
Drift-flux number:
Ndi - ( Vgj/u o) , (or void-quality relation),
(36)
x~a(ap/pg)a= 1.
Time ratio number:
r,. - (10/.0)/(82/.s),,
(37)
Thermal inertia ratio:
N,h, " ( P,%~8/P, C~, d ) ,,
where x¢ is the quality at the exit of the heated section. Therefore, the similarity based on the phase change and subcooling numbers yields
(38)
(43)
This relation scales the quality in terms of the density ratio, and the similarity in vapor quality is satisfied. Eq. (43), when combined with eqs. (29) and (30), ensures that the similarity in the friction number, Nt, and orifice number, No, are approximately satisfied. By
G. Kocarnustafaogullari, M. Ishii / Scaling of two-phase flow transients
126
using the definition of drift-flux, = ug - [(1
Vg; = ug - j
- a) uf + aug],
(44)
it can be shown that the drift-flux number can be expressed as
Nd
(Apx/pg)[(pf/Apa)
=
-
1] - 1,
(45)
where j is the volumetric flux of the two-phase mixture. From eqs. (43) and (45), it can be observed that the similarity in the drift-flux number requires similarity in the void-quahty correlation. This implies that OteR(Ap/pf)R = 1.
(46)
Hence, excluding friction, orifice and drift-flux number similarities from the set of eq. (41), and solving the remaining equations, one obtains the following similarity requirements: _
11/2
/'/OR -- "0R ,
(47)
( Ah~,b )R = ( Ahfgog/AP )a,
(48)
OOR = ( OfosAhfJaO )( d/8 )Rlod/z,
(49)
It
8 R = (loR)
'~l/4z xl/2 [asR) ,
dR= [(0sCps)R//(pfCpt)R]
1*OR 1 / 4 "~sR 1/2
"
(50) (51)
As demonstrated by eq. (31), there remains flexibility in establishing the time scale in single-phase flow. However, in the case of two-phase flow, the time scale is uniquely determined from eq. (47) alone. Thus, "R -= ( l o / u o ) R
-- -oR"/2.
(52)
If the axial length in the model is reduced, real time simulation can not be achieved in model and prototype two-phase flow natural circulation loops. Instead time events are accelerated in the scaled-down model by a factor of tl/2 "0R over the prototype. When the two-phase flow velocity scale, eq. (47), is used in the single-phase flow geometric scale requirements, eqs. (16) and (17), the geometric similarity requirements in both phases become the same. However, the single-phase real time-scale can not be preserved.
4. Single-phase to two-phase scaling Single-phase and two-phase flows occur simultaneously in a natural circulation loop as shown in fig. 1. In general, single-phase and two-phase flow requirements are different from each other. This introduces
difficulties in meeting geometric, power and time scaling requirements when both similarity requirements have to be satisfied simultaneously. However, the problem can be resolved by using the flexibility of the singlephase flow similarity requirements. Depending on the choice of time or power scale simulations in single and two-phase flows, two alternatives exist to meet singlephase and two-phase flow requirements simultaneously as discussed below.
4.1. Time scale simulation This alternative is based on using the same time scale in the single-phase and two-phase flow. Since the time scale for two-phase flow similarity is fixed by eq. (52), the same time scaring in the single-phase flow can be used. Thus, ( TR )sp = ( TR )tp =/1/2 "OR"
(53)
The velocity scale then becomes ( U o . ) s p = 1,/2 -OR.
(54)
Power and geometric scaring parameters can be obtained from eqs. (14), (16) and (17). They are given by .... ( pscps i (qoR)sp = ~ flllo/2 JR'
(55)
( ~R )sp = "0R11/4--1/2UsR '
(56)
(aR)sp = [(osCps)/(PfCpe)] R'0RI'/4='/2'*sR•
(57)
By comparing eq. (56) with eq. (50) and eq. (57) with eq. (51), it is observed that the single-phase and twophase flow geometric scales are equal; thus the same geometric scale criterion can be used in both phases. However, eqs. (55) and (49) indicate that the power density in single-phase flow sections should be scaled differentially from two-phase flow sections by a factor of (Oo~)s#(od~),.
=
(/~Ogah,g/AoG,)~'
(58)
Although changing the power density from single-phase to two-phase flow transition is not impossible, it does not seem to be a practical solution. In order to avoid this difficulty, an additional constraint should be imposed by setting
(BogAhfs/aOCp,)R
= 1.
(59)
Similarity in the property group expressed by eq. (59) ensures that geometric, power and time scales are
G. Kocamustafaogullari, M. Ishii /Scaling of two-phaseflow transients the same during the single-phase to two-phase flow transition. Otherwise, the power density will be distorted by a factor given by eq. (58). This distortion can not be tolerated because it may lead to completely different two-phase flow phenomena associated with phase change in the model as compared to the prototype. As demonstrated later it is possible to meet this requirement by using different fluids in model and prototype. However, it is impossible to satisfy eq. (59) for modeling high pressure systems with low pressure systems using the same fluid in the prototype and model.
4.2. Power scale simulation The solution to the original problem of the singlephase to two-phase flow transition scaling can also be achieved by using the power scale given by eq. (49) for both phases:
= (pfpgAhfg/AP)R(PsCps)/(pfCpf)glo
1/2 .
(60) Using eq. (50) in eq. (14), the single-phase flow velocity scale then becomes 1/2 1/3 ( U0g )sp = 10g ( tiPs A h fg//ApCpf )R "
) -- 1/6
( 8 R )sp = 10R OtsR ( flA h fgps/A p Cpf /R
(62)
and
(d.)sp = (psC s). / ( p , cpf )RIoR 1/4~XsR 1/2 ( flAhfsPg//ApCpf )R
. (63)
Similarly, the single-phase time scale becomes (~'g)sp -= log/Uog = llo/2( flAhfgPg//ApCp, )R 1/3.
(64)
Comparing the single-phase flow scaling parameters given by eqs. (62), (63) and (64) with their two-phase flow counterparts, it can be shown that ( UOR)sp( U0R )tp = ( d R ) s p / ( dR
)tp
= (flpgAhfg//ApCpf)R
1/3
= (~pgAhfs/ApCpf)R
(~'R)sp/(~'R)W = (flpsAhfg/ApCpf)R
- 1/6 -- 1/3
,
(66)
.
(67)
As in the case of time scale simulation, for a power scale simulation the condition imposed by eq. (59) must be met in order to have a complete similarity during the single-phase to two-phase flow transition. This presents no problem for fluid-to-fluid simulation. However, distortions are unavoidable in simulating high pressure systems with low pressure systems by using the same fluid. If the same fluid has to be used for reduced pressure modeling, the second alternative, power scaling, is advisable for the following reasons: (1) Time simulation is not practical since it requires different powers for the single-phase and two-phase sections. Otherwise the power scale will be distorted by the factor given by eq. (59). This distortion can not be tolerated since it may lead to completely different two-phase flow phenomena in the model as compared to the prototype. (2) Although the power simulation causes different time scales in single-phase and two-phase flows, it can be accounted for when evaluating model data. (3) The geometric distortions caused by power scaling may not be significant because of the very small exponent appearing in eq. (66).
(61)
Furthermore, substituting this velocity scale in eqs. (16) and (17) yields the following single-phase geometric scaling parameters 1/4 1/2
(SR)sp//(~a)tp
127
,
(65)
5. Simulating pressure transients for two-phase flow
A characteristic pressure transient for a LWR under LOCA is illustrated in fig. 2 [15]. After a LOCA the system pressure drops rapidly from its normal operating pressure, Pi, to the saturation pressure, Pa, and boiling starts at this point. In region I, which is very short, the fluid is single-phase. Subsequently two-phase flow dominates the circulation, and the system pressure is identified as /2, which changes with time as shown in fig. 2. It is the objective of this section to investigate the possibility of simulating the pressure transients in region II. With the exception of state 1 shown on the figure where a phase transition occurs, two-phase flow dominates in the second region. To simulate the pressure transient in region II it is important to meet the two-phase flow scaling conditions, eqs. (47) through (51) where eq. (47) is based on the axial length scale and eqs. (50) and (51) are based on a given geometric scale. Hence, once the model is designed, the geometric parameters /OR, 8R and d R can not be altered while pressure transients occur. Since the power scale and the
G. Kocamustafaogullari, M. Ishii /Scaling of two-phaseflow transients
128
where subscripts 1 identifies state 1, and subscript 2 identifies any state in region II, as shown in fig. 2. The first condition simulates subcooling while the second one ensures that the phase change is simulated. Fortunately, these two requirements are not too different from each other since Of, the only difference in the two requirements, does not change appreciably within the practical limits of a pressure transient from Pt to "°2Since the phase change is more important than the subcooling for two-phase flow simulations, only the second group needs to be considered further. Recalling the original definition of the ratios, eq. (69) can also be expressed as
Pi
1
P1 I
P2
2
HI I
Time
I] - -
Iv.
Two-Phase Region
rmJ Ym, r.~/r., =
Single-Phase Region Fig. 2. A typical pressure transient for a LWR under LOCA.
subcooling are fixed by state 1, eqs. (48) and (49) require, respectively, that the following two conditions should be met during the transient:
(Ap/pgahfs)R , = (AP/PsAhfg)R:
(68)
and ( A
p/pfpgA h fg ) R1 = 102
1
( A p/pfpgA h fg ) R2'
I ~I
I
\
]
]
! l
(69)
[
i
1. Water, Ppl = 110 bar 2. R-11, Pm1=37.5 bar 3. R-11, Pro1= 20.0 bar 4. R-11, Pm1=9"5 bar R - 1 1 , Pro1 =3.8 bar
\ i
10o
10-1 10o
L Lli
I
102
10
where Y is defined as Y=-Ap/pfpgAhts. Once the prototype pressure, Ppa, and the model pressure, Pml, are specified at state 1, where the phase change starts, the problem of simulating pressure transients reduces to seeking solutions in the form of Pro2 = f(Pp2) which satisfy eq. (70). To investigate the possibilities of locating such solutions, two separate cases are considered. The first one simulates the pressure transients in a model with a different fluid than is used in the prototype, i.e., fluid-to-fluid simulation. The second simulates pressure transients with a model using the same fluid as the prototype at lower pressures relative to the prototype, i.e., reduced pressure simulation.
5.1. Fluid-to-fluid simulation
I
Y ~ Apl pfpgAhfg
I0
(70)
I 1
10 3
P2 (bar) Fig. 3. Variation of two-phase pressure transient similarity group ratio for water and R-11.
The use of refrigerants for scaling of water systems has been an accepted practice for modeling various two-phase flow phenomena, such as modeling the CHF and boiling, etc. Considerable facility cost savins are achieved by the lower absolute pressure, and temperatures realized at similar reduced pressures. It would, therefore, be of considerable interest to accomplish the same result for simulating transients encountered during a LWR and LOCA. Among the various refrigerants, R-11, 12, 22, 113 and 114 that could be considered, R-12, 22 and 114 are excluded from further study since they exhibit very low boiling temperatures at relatively low pressures. Consequently, the analysis was limited to R-11 and R-113. For these two model fluids, a series of solutions in the form of Pro2 = f(Pp2) are sought in three steps as follows: (a) Assuming that the typical prototype pressure at state 1 is Ppl = 110 bar, [15], (Y2/Y1) curves for the prototype and the model fluids are produced in fig. 3
G. Kocamustafaogullari, M. Ishii /Scaring of two-phaseflow transients 102 _
[
~
[
- - T
20 --
y_= Ap/pfpgAhfg 1. Water, Pp1=110 bar 2. R-113, Pml = 30.0 bar 3. R-113, Pro1=21.4 bar 4. R-113, Pml: 12.1 bar _ _ 5. R-113, Pm1=7"0 bar -
108 --
~.~
[
Water to
129
~
T
R-113 Simulation
1. Water, Pp2:80 bar 2. Water, Pp2=60 bar 3. Water. Pp2=40 bar 4. Water, Pp2:20 bar
T
/
/
/ 2 / /
/
//--/// / /
--
"~
, waterP,,=lobar/ / /
--
/
T
11
t
2
~
~
4
6
8 10
20
40
60
Pml (bar)
100
102
10 '
103
Fig. 6. Model pressure curves which satisfy two-phase flow pressure transient similarity requirement for water to R-113 simulation at Ppl = 110 bar.
P2 (bar)
Fig. 4. Variation of two-phase flow pressure transient similarity group ratio for water and R-113.
useful in locating Pm2 once Ppl, Pp2 and Pml are known. Hence, knowing P,m is crucial in obtaining Pm2 = f ( P p 2 ) .
for R-11 and fig. 4 for R-113 as function of the transient pressures, Pp2 o Since the model pressure at the point where boiling initiates, Pro1, is not yet known, it is taken as a parameter. (b) Utilizing figs. 3 and 4, Pro2 = f ( P m l ) curves are generated for given values of Ppl and Pv2 as illustrated in fig. 5 for R-11 and fig. 6 for R-113. These figures are
(c) As noted above, state 1 is characterized by the single-phase to two-phase flow transition. Simulation of this state by a model fluid was discussed in detail in the preceding section where it was concluded that the condition imposed by eq. (59) must be met between the 100
I
I
I I I
I
I
r
r
20 -- Water t° R'I' Slmulati°n Pp2:80 bar 2 . W a t e r , ,,2:60 bar 3. Water, Pp2:40 bar 4. Water, Pp2=20bar
"~'/
I
~
1. W a t e r ,
10 - .~ 8 - -
_
/ / /
/ /
/ /
''
I
'
'
I
I
i
I
I/
~
/
10"1-
I
"~ /
1. Water (Prototype fluid) 2. R-11 (Model Fluid) 3. R-113 (Model Fluid)
I
/
q
%
T
l
10.2
2
1
2
4
6
8 10
20
40
60
Pml (bar)
Fig. 5. Model pressure curves which satisfy two-phase flow pressure transient similarity requirement for water to R-11 simulation at Pp1 = 110 bar.
10"
I
10
I I
102 • P1 (bar)
103
Fig. 7. Variation of single-phase to two-phase flow transition similarity group for prototype and selected model fluids.
G. Kocamustafaogullari, M. Ishii /Scaling of two-phaseflow transients
130
(d) Using these Pm] values in figs. 5 and 6, pressure transient simulations curves in the form of Pm2 = f(Pp2) are generated in fig. 8 for both model fluids. Each Pmz =f(Ppz) curve appearing in fig. 8 simultaneously satisfies the two-phase flow similarity requirements and the single-phase to two-phase flow transition requirement. The latter is important for locating state 1. In addition, the pressures of both model fluids are within the practical range of applicability, indicating that R-11 and R-113 can be used for simulating the pressure transients occurring in a L W R under L O C A condition.
20--
10
C
8--
O.~ "~
6
2 ~
1, R-113
/ /
//11
Pp1=110 bar Pml : 23 bar
/ /
///
Pml : 26 bar
2 -
5.2. Reducing pressure scaling 1 10
L 20
L 40
L 60
I L 80 100
--~
Pp2 (bar)
200
Fig. 8. Pressure transient simulation for water to R-11 and to R-113 modeling at Pp] = 110 bar. prototype and model fluids. Variations in this property group appearing in eq. (59) are illustrated in fig. 7 for the prototype and model fluids. For Ppl = 110 bar, it is evident from the figure that Pm~ = 23 bar for R-113 and Pml = 26 bar for R-11. Both of these values are within the practical range of applicability for the selected model fluids.
It is also interesting to investigate the possibilities of simulating high pressure water transients with a low pressure water system. Corresponding to the prior example, figs. 9 and 10 are generated for this purpose. The final solution in the form of Pro2 = f(Pp2) is given in fig. 11. Unlike the fluid-to-fluid simulation investigated previously, a unique solution Pro2 = f ( P p 2 ) can no longer be found. As shown in fig. 11, Pml has to be taken as a parameter, and curves appearing in this figure satisfy only the two-phase flow scaling requirements over a wide range of pressure transients. However, these curves
100 -
46~ 2 10 s8
"~ >-
4
f i
2
J
i
ii
I
2
4
6
10
Water to Water Simulation 1. Water, Pp2 =10 bar 2. Water, Pp2 = 20 bar 3. Water, Pp2 = 40 bar 4 - - 4. Water, Pp2 = 60 bar 5. Water, Pp2 • 80 bar 6. Water, Pp2 = 110 bar
8-6 --
y~ Ap//pfpg,..~hfg
~ 1 \ ~ ~ ' ~ \ ~a~\ \ ~ " ~ ~ ~ 5 ~ ~
1. Water, 2. Water, 3. Water, 4. Water,
Ppl =Pml =110 bar P m l : 80 bar Pml : 60 bar Pml = 40 bar
S. W a t e r ,
Pro1 : 2 0 b a r
_
6. Water, Pm, : 10 bar
ito
\ \
8
10o 8
6 4 2 10-I 100
2
4
6 8 10
2
4
6 8 102
2
4
6 ; 103
P2 (bar)
Fig. 9. Variation of the two-phase flow pressure transient similarity group ratio for reduced pressure simulation.
2
t
4
i 8iJ10
6
Pml (bar)
Fig. 10. Model pressure curves which satisfy two-phase flow pressure transient similarity requirement for high-pressure water to reduced-pressure water simulation at Ppl = 110 bar.
G. Kocamustafaogullari, M. lshii /Scaling of two-phaseflow transients 10(
m ---
v
I
I
I
I I
[
Water to Water Simulation 1. Water, Pml =10 bar 2. Water, Pm2 : 20 bar 3. Water, Pm2 =40 bar 4. Water, Pm2 = 60 bar 5. Water, Pm2 " 8 0 bar 6. Water, Pm2 . 1 1 0 bar
I
I
I
4
6
8 100
65
o.
T
2
4
6
8 10
2
' Pp2 (bar)
Fig. 11. Pressure transient simulation for high-pressure water to reduced-pressure water modeling at Pp] = 110 bar.
131
was shown that the real time scaling is not possible for an axially scaled-down model. The scaling criteria are also applied for simulating pressure transients occurring in a LWR under LOCA conditions. Crucial to this study were the fluid property groups which appear within the two-phase similarity parameters and the single-phase to two-phase transition similarity. Important property groups to be considered w e r e ( Ap//Ahfgpfpg) and (flPsAhfg//ApCpf). The first group ensures the similarity in two-phase flows transients while the second one permits the similarity from single-phase to two-phase flow transition. Finally, computations were performed to investigate the feasibility of modeling a typical LWR system under LOCA by a low pressure water system and a Freon system, using R-11 or R-113. Modeling is possible in both cases for simulating pressure transients. However, simulation of phase change transitions is not possible with a low pressure water system without distortions in power or time.
Acknowledgements
no longer satisfy the single-phase to two-phase flow similarity requirement given by eq. (59). Therefore distortions at the power scale, as expressed by eq. (58), can not be avoided for the reduced pressure simulation.
The authors would like to thank Drs. N. Zuber and R. Lee of the U.S. Nuclear Regulatory Commission for valuable discussion on the subject. This work was carried out under the auspices of the U.S. Nuclear Regulatory Commission.
6. S u m m a r y and conclusions
Nomenclature
Based on the similarity analysis of [9-12], scaling criteria for a natural circulation loop under single-phase and two-phase flow conditions have been derived. The criteria include the effects of fluid properties to allow for modeling of a high pressure water system by a low pressure water system or by a different fluid system. In addition to the usual single-phase flow dimensionless groups, it is shown that the phase change, subcooling, drift-flux and Froude numbers are important for twophase flow systems. Combining the single-phase and two-phase flow criteria, single-phase to two-phase flow scaling was studied. Two possibilities, namely time simulation and power simulation, were investigated in detail. Particular emphasis was placed on scaling at reduced pressure levels relative to a prototype system and on fluid-to-fluid scaling. It was noted that power scaling is more practical than time scaling for modeling a high pressure water system by a low pressure water system. Furthermore, it
A a as B Cp d F f g h K k L l j Npch Nsub
NFr Nd
non-dimensional area, flow cross-sectional area, solid cross-sectional area, Blot number, specific heat capacity at constant pressure, hydraulic diameter, friction number, friction factor, gravity, heat transfer coefficient, orifice coefficient, conductivity, non-dimensional length, axial length, volumetric flux, phase change number, subcooling number, Froude number, drift-flux number,
132
N~ No Nu P Pr •
*tt
qo q o//
Qs R Re St T T* U U0
v~j X Xe ~t Ote Ots
8 ahf s A h sub
aT0 ao At, P
G. Kocamustafaogullari, M. Ishii /Scaling of two-phase flow transients friction nuber (two-phase), orifice number (two-phase), Nusselt number, pressure, Prandtl number, heat generation density in solid, heat flux, heat source number, Richardson number, Reynolds number, Stanton number, temperature, characteristic time ratio, fluid velocity, heated section representative velocity, drift velocity, vapor flow quality, exit quality, void fraction, exit void fraction, solid thermal diffusivity, thermal expansion coefficient, conduction thickness, latent heat of evaporation, subcooling, characteristic temperature rise, density difference, viscosity difference, wetted perimeter, mass density, non-dimensional time.
Subscripts e
f g i m
P R S
sp tp 0
exit, hquid, gas or vapor, i th section, model, prototype, model-to-prototype ratio, solid, single-phase flow, two-phase flow, reference constant (heated section).
References [1] J.C. Amrouni, S. Nijhawan, A.M.M. Aly, R.L. Ferah, and A.M. Omar, Sscaling requirements for an experimental facility to study the thermosyphoning behavior of a CANDU reactor, Presented at the Canadian Nuclear Society 6th Annual Conf., Ottawa, Canada (June 1985). [2] A. Carbiener and A. Chudnik, Similitude considerations for modeling nuclear reactor blowdowns, Trans. Am. Nucl. Soc. 12 (1969) 361-362. [3] A.N. Nahavandi, F.S. Castellana and E.N. Moradkhanian, Scaling laws for modeling nuclear reactor systems. Nucl. Sci. Engng. 72 (1982) 347-354. [4] L. Ybarrando, L.Y. Griffith, P. Fabic and G.D. McPherson, Examination of LOFT scaling, ASME, Annual Winter Meeting, New York (November 1974). [5] T.K. Larson, J.L. Anderson, and D.J. Shineck, Scaling criteria and assessment of semiscale Mod-3 scaling for small-break-loss-of-coolant transients, EGG-SEMI-5121 (March 1980). [6] E.R. Rosal, PWR FLECHT SEASET systems effects natural circulation and reflux condensation task plan report. NUREG/CR-2401 (February 1983). [7] R.P.Rose, Heat transfer problems associated with a LOFT (loss of fluid test), Proc. Conf. ASME-AIChE Heat Transfer, Los Angeles (August 8-11, 1965). [8] N. Zuber, Problems in modeling of small break LOCA, NUREG-0724 (October 1980). [9] M. Ishii and I. Kataoka, Similarity analysis and scaling criteria for LWR's under single-phase and two-phase natural circulation, NUREG/CR-3267, ANL-83-82, Argonne National Laboratory, Argonne, Illinois (March 1983). [10] G. Kocamustafaogullari and M. Ishii, Scaling criteria for two-phase flow natural and forced convection loop and their application to conceptual 2×2 simulation loop design. NUREG/CR-3420, ANL-83-61, Argonne National Laboratory, Argonne, Illinois (May 1983). [11] M. Ishii and I. Kataoka, Scaling laws for thermal-hydraulic system under single-phase and two-phase natural circulation, Nucl. Engrg. Des. 81 (1984) 411-425. [12] G. Kocamustafaognllari and M. Ishii, Scaling criteria for two-phase flow loops and their application to conceptual 2 × 4 simulation loop design, Nucl. Technology 65 (1984) 146-160. [13] M. Ishii and N. Zuber, Thermally induced flow instabilities in two-phase mixtures. Proc. 4th Conf. Heat Transfer. Paris, France (1970) Paper B5.11. [14] M. Ishii and D. Jones, Derivation and application of scaling criteria for two-phase flows. Proc. Conf. Two-Phase flows and Heat Transfer, NATO Advanced Study Institute, 1, Istanbul, Turkey (1976) 163-185. [15] Nuclear Safety Analysis Center Report, The analysis of Three-Mile Island-Unit 2 Accident, NSAC-1 (July 1979).