The three-level scaling approach with application to the Purdue University Multi-Dimensional Integral Test Assembly (PUMA)

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1 Nuclear Engineering and Design 186 (1998) The three-level scaling approach with application to the Purdue University Multi-Dimensional Integral Test Assembly (PUMA) M. Ishii a, *, S.T. Revankar a, T. Leonardi a, R. Dowlati a, M.L. Bertodano a, I. Babelli 1,a, W. Wang a, H. Pokharna a, V.H. Ransom a, R. Viskanta a, J.T. Han b a School of Nuclear Engineering, Purdue Uniersity, West Lafayette, IN , USA b Diision of Systems Technology, Office of Nuclear Regulatory Research, U.S. Nuclear Regulatory Commission, Washington, DC , USA Received 16 December 1996; received in revised form 1 November 1997; accepted 16 December 1997 Abstract The three-level scaling approach was developed for the scientific design of an integral test facility and then it was applied to the design of the scaled facility known as the Purdue University Multi-Dimensional Integral Test Assembly (PUMA). The NRC Technical Program Group for severe accident scaling developed the conceptual framework for this scaling methodology. The present scaling method consists of the integral system scaling, whose components comprise the first two levels, and the phenomenological scaling constitutes the third level of scaling. More specifically, the scaling is considered as follows: (1) the integral response function scaling, () control volume and boundary flow scaling, and (3) local phenomena scaling. The first two levels are termed the top-down approach while the third level is the bottom-up approach. This scheme provides a scaling methodology that is practical and yields technically justifiable results. It ensures that both the steady state and dynamic conditions are simulated within each component, and also scales the inter-component mass and energy flows as well as the mass and energy inventories within each component. Published by Elsevier Science S.A. 1. Introduction The importance of similarity laws and scaling criteria has been shown extensively in industrial modeling and research. This is especially true for * Corresponding author. Tel.: ; fax: ; ishii@ecn.purdue.deu 1 Present address: King Abduhlaziz City for Science and Technology, Riyadh, Saudi Arabia. the design, operation and analysis of simulation experiments using a scaled model. More importantly, the scaling methods are necessary when the study of phenomena cannot be accomplished using a prototypic model. This is often the case for large systems where full-scale experiments would be dangerous and costly. The scaling laws for forced convection single phase flow are well established and modeling with these criteria is common practice. The natural circulation single phase problem has been investi /98/$ - see front matter Published by Elsevier Science S.A. PII S (98)00-

2 178 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) gated by Heisler and Singer (1981), Heisler (198). The similarity analysis in this case is very complex since there is a coupling of the driving force and heat transfer process. Twophase flow similarity criteria were developed by Ishii and Zuber (1970), Ishii (1971), Ishii and Jones (1976), Ishii and Kataoka (1983), Kocamustafaogullari and Ishii (1983, 1986). The method used was a perturbation technique along with the steady state solution using the drift flux model and constitutive relations Oerall scaling philosophy The present research presents the development of a systematic scaling method for the design of a thermal hydraulic integral test facility and for the analysis of experimental data relative to the prototypic conditions. The scaling method is described in Fig. 1. The basis of the scaling criteria is the conservation principles and the constitutive laws. The actual scaling method consists of three levels of scaling analysis, namely the integral system scaling, control volume scaling, and local phenomena scaling. The first two levels correspond to the top-down scaling and the third level represents the bottom-up scaling. The prototypic system consists of multiple inter-connected components. The first level of the scaling method focuses on the development of scaling criteria for each component using the response function scaling. It is assumed that each component can be mathematically described as a one-dimensional system. A combination of the single phase formulation and the drift-flux twophase flow formulation is used to express the conservation principles of mass, momentum and energy. First, the system conservation equations are solved under a transient condition using a linear small-perturbation analysis. The solution yields various transfer functions between variables, for example between the inlet flow, void fraction, enthalpy, and pressure. It is noted that these transfer functions describe the system dynamic responses. The similarity criteria are developed by non-dimensionalizing the transfer functions and then by identifying the conditions to make the non-dimensional transfer functions to be identical between the prototype and an integral test facility. Therefore, it can be said that the scaling method gives the dynamic scaling of a whole component. The second level of the scaling analysis focuses on the mass and energy inventory of each component and the inter-component mass and energy transfers. This is accomplished by introducing a scaling method based on the control volume balance equations of mass, momentum and energy. The first and second level scaling analyses are, therefore, based on the conservation principles of mass, momentum and energy. Together they ensure that the dynamic responses of each component as well as the dynamic responses of the inter-component transfers are simulated. These two levels of scaling analyses yield the bulk of the information necessary to develop the scientific design of a test facility. However, these two levels of analyses are not sufficient to guarantee the development of a well-scaled facility design. This is due to the fact that, in two-phase flow various local phenomena have their own internal length scales and the micro-scale physical phenomena that affect various transfer mechanisms may not be fully represented by a simple one-dimensional drift-flux formulation. Therefore, it is necessary to evaluate the key local phenomena and various constitutive relations in terms of scaling. The third level of scaling analyses focuses on the various local phenomena, constitutive laws, and their impact on the overall scaling strategy. The scaling analysis at this level is typically carried out for each phenomenon separately by considering it as a simple rate process or as a transition criterion for a bifurcation. It is noted, however, that the first and second level scaling criteria are the backbone of the present method. The third level of scaling analysis gives the potential scaling distortions and a possible way to minimize such distortions. The PUMA design report authored by Ishii et al. (1996) documents the details of this scaling method and its application to the facility design for PUMA.

3 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) Fig. 1. Integral test facility scaling methodology flow chart (as applied to PUMA).

4 180 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) SBWR General Electric (GE) Nuclear Energy developed a boiling reactor designated as the simplified boiling reactor (SBWR). The differences between this reactor and current boiling reactors (BWRs) are the simplification of the coolant circulation system and the implementation of a passive emergency cooling system. The engineered safety systems include the automatic depressurization system (ADS), the gravity driven cooling system (GDCS), the passive containment cooling system (PCCS), the isolation condenser systems (ICS), and the pressure suppression pool (SP). The GDCS and PCCS are designs that are unique to the SBWR while the ICS is similar to those already seen in some operating BWRs. It is necessary to study the performance and interactions of these unique safety systems to assess the response of the SBWR under postulated accident conditions. Since it is not feasible to build and test a full power prototypical system, a scaled integral system is required PUMA As a result of the scaling that will be discussed below, the PUMA facility was constructed based on the detailed scaling analysis. The PUMA facility thus contains all of the thermal-hydraulic components of the SBWR. It can be applied over the range of conditions required for the assessment of the response of the passive safety systems. It also addresses the integral system response by simulating the interactions between the different components. The facility was constructed as a distributed system. This means that the internal tanks of the system were removed from the containment and were then connected to the containment using artificial lines. This distributed system allowed for the facilitation of instrumentation for the different components. To assure the scaled system s frictional resistance similarity, additional calculations were performed for the connecting lines. The objectives of the PUMA program are: provide integral test data to the NRC for the assessment of the RELAP5 code for SBWR application. assess the integral performance of the GDCS and PCCS, assess the relevant SBWR phenomena important to LOCAs and other transients, perform various separate effects tests. The scaling method addresses the following phenomena and issues: 1. in-vessel natural circulation and two-phase flow instability. flashing in the chimney 3. inflow or outflow from various components 4. inter-component flow 5. initial and boundary conditions 6. important containment phenomena 7. single phase and two-phase natural circulation 8. condensation phenomena in the presence of noncondensable gases, and 9. system stored energy and decay heat. The scaling methodology for designing PUMA is mapped out in Fig. 1 and discussed in detail in the PUMA design report by Ishii et al. (1996).. Integral system scaling (1st level) The first level of scaling analysis is carried out by using the scaling criteria developed from the response function scaling by Ishii (1971), Ishii and Jones (1976), Ishii and Kataoka (1983). The concept is described in Fig.. First, the one-dimensional conservation equations are integrated for either the single phase flow model or the drift flux model. This is done using the linear small-perturbation analysis. Since the equations are integrated over time and space, the resulting solutions represent the dynamic system responses. These responses are expressed in terms of the transfer functions, for example, the system pressure drop, p, can be expressed in terms of the inlet velocity perturbation,, as p=q, (1) where Q is the transfer function. Since during the system transients both the single phase and twophase flow can exist, the scaling criteria for single phase and two-phase flow are imposed simultaneously. This approach makes the result quite general, see details in Ishii and Kataoka (1983). It

5 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) Fig.. Response function scaling. is noted that the two-phase flow scaling criteria are more restrictive, however they can satisfy the requirements of the single phase flow scaling criteria..1. Single phase similarity parameters For single phase flow, the one-dimensional area averaged forms of the continuity, integral momentum, and energy equations are used. Relevant scales for the basic parameters are determined, then the similarity groups are obtained from the conservation equations and boundary conditions. The heat transfer between the fluid and structure can be included in the analysis using the energy equation for the structure. From these equations, the important dimensionless groups characterizing geometric, kinematic, dynamic and energetic similarity parameters are derived. They are as follows: Richardson number, R gt ol o = Buoyancy u o Inertia force () Friction number, F i fl d +Ki = Friction Inertia force Modified Stanton number, St i 4hl o f c pf u o d i Wall convection = Axial convection Time ratio number, T* l o/u o Transport time i = / s i Conduction time Heat source number, Q si q s l o s c ps u o T o Biot number, B ii =(h/k s ) i = Heat source = Axial energy change i (3) (4) (5) (6) Wall convection, (7) conduction where the subscripts i, f and s represent the ith component of the loop, the fluid component and the solid component, respectively. The parameters u o, T o and l o are the reference velocity, reference temperature difference and reference

6 18 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) length, respectively. The hydraulic diameter of the ith section, d i, and the conduction depth, i, are defined by: d i 4a i / i (8) and i a si / i, (9) where a i, a si and i are the flow cross sectional area, solid structure cross-sectional area and wetted perimeter of the ith section. Hence, d i and i are related by: d i =4(a i /a si ) i. (10) In addition to the physical similarity groups defined above, several geometric similarity groups are necessary as well. These are: Axial length scale: L i l i /l o (11) Flow area scale: A i a i /a o, (1) where a o is the cross-sectional flow area at the reference component. The reference velocity, u o, and the reference temperature difference, T o are obtained from the steady-state solution. If the heated section is taken as the reference section, these characteristic parameters are expressed as follows: a so a o lo 4g q o l o u o = f c pf (F i /A i ) i 1/3 (13) for the natural circulation loop or = ) gh d u o (14) (F i /A i i for the forced convection case and T o = q o l o a so f c pf u o a o, (15) where the subscript o denotes the heated section and a so is the reference heated surface area. Therefore, u o and T o are the natural circulation reference velocity and temperature rise over the heated section that can be obtained if the system is operated under steady state conditions... Single phase similarity laws The similarity criteria between different systems can be obtained through detailed consideration of the similarity groups listed above, together with the necessary closure conditions. If similarity is to be achieved between the processes observed between the prototype and the model, the scaled property should be defined as follows: R m for model = p for prototype. (16) Similarity is then achieved if the following requirements are satisfied: A ir =(a i /a o ) R =1 (17) L ir =(l i /l o ) R =1 (18) F i /A i l i R= fi +K i i d i i /(ai /a o ) nr =1 (19) R R =(T o l o /u o ) R =1 (0) St ir =(hl o / f c pf u o d i ) R =1 (1) T* ir =[(l o /u o )/( / s ) i ] R =1 () B ir =(h/k s ) ir =1 (3) Q sir =(q s l o / s c ps u o T o ) ir =1, (4) where the subscript i designates the particular component and R denotes the ratio of the property of the model to that of the prototype. From the above set of equations, it is apparent that the complete transverse area similarity is required as expressed by Eq. (17) and the complete axial geometrical similarity is required by Eq. (18). In view of the complete transverse area similarity, the dynamic similarity of Eq. (19) can be reduced to i (F i ) R =1. (5) This illustrates that the pipe friction loss and the minor losses associated with the loss coefficient can be interchanged without changing the overall value of the pressure loss term. So, with the addition of orifices, which provide flow restriction and increased frictional losses, a wide range of

7 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) scaling conditions can be simulated. The pressure loss term can then be satisfied independently of the remaining scaling requirements. Physically, the time ratio scales the speed of the transport processes within the fluid and solid. The Biot number is the ratio of the thermal conductance of the fluid and solid, so it is the scale for the boundary layer temperature drop and thermal gradient in the solid. The Stanton number is the ratio of the convection at the wall and the axial convection. The Stanton and Biot numbers are related to the fluid solid interfacial boundary condition. The similarity requirement from these two quantities contribute to the simulation of the temperature drop in the boundary layer and interface for natural circulation conditions. The heat source number is important for the temperature in the solid and also for the overall energy balance of the system. From the steady state requirements outlined above, the ratio of the reference velocities becomes: (u o ) R = q 1/3 o l o, (6) s c ps R then the reference temperature rise ratio becomes: (T o ) R = q o l o. (7) s c ps u o R The requirements from the time constant ratio and heat source number also lead to some additional geometric constraints. The conduction depth, which can be related to the wall thickness in most cases, becomes: 1/ ( i ) R =() R = sl o (8) u o R while the hydraulic diameter is given by: (d i ) R =(d) R = 1/ sc ps s l o. (9) f c pf R u o R From the Biot number similarity, it is required that the following be satisfied: (h i ) R =(h) R =(k s ) R u o l o s 1/ R, (30) where, for the above equations, the parameters without the component subscript, i, denote universal values that must be satisfied in all components. In addition to the above single phase scaling requirements, the geometric similarity requirements dictate that l i =1 (31) l o R and a i =1 (3) a o R must also be met. With these conditions, the effects of each term in the conservation equations are preserved in the model and prototype without any distortions. If some of these requirements are not satisfied, then the effects of some of the processes observed in the model and prototype will be distorted. With regard to the practical implications of the similarity requirements, the friction similarity may be difficult to satisfy individually for each component, except in components having a subchannel geometry. Also, the conduction depth ratio and hydraulic diameter ratio should satisfy certain criteria. However, satisfying the criteria over the entire loop may be difficult. They are important mainly at the major heat transfer components where these conditions can be easily satisfied. However, the distortions in the criteria over a loop may lead to an overall scale distortion in terms of structural heat losses. In addition, the heat transfer coefficient cannot be independent of the flow field. Therefore, there may be some difficulties in meeting the constraint imposed by Eq. (30). However, relaxation of this similarity requirement influences only the boundary layer temperature drop. When the heat transfer mechanism is not completely simulated, the system will adjust to a different temperature drop in the boundary layer. The overall flow and energy distribution will not be strongly affected during the slow transients typical of a natural circulation system. It is important to note that the above set of requirements does not put constraints on the power density ratio, q or, however, there is a restriction on the time scale as follows: R l o u o R l or =. (33) 1/3 [(q o l o ) R /( s c ps ) R ]

8 184 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) Two-phase similarity parameters For a two-phase natural circulation system, similarity groups have been developed from a perturbation analysis based on the one-dimensional drift flux model. The set of mass, momentum and energy equations are integrated along the loop, and the transfer functions between the inlet perturbation and various variables are obtained. The scaling parameters which are developed from the integral transfer functions, represent the whole-system similarity conditions, and are applicable to transient thermal hydraulic phenomena (Ishii, 1971). Ishii and Jones (1976) obtained, in a study that extended the study by Ishii (1971), the integral system scaling criteria for two-phase flow systems from the application of the small perturbation technique to the one-dimensional drift flux model. The dynamic behavior and stability of a boiling flow system can be analyzed by using a onedimensional drift flux model and a small perturbation method. A perturbation of inlet flow is introduced as: (s, t)=e st, (34) where s=a+j. Thus, s is a complex number; the real part gives the amplification coefficient and the imaginary part represents the angular frequency,. By formally integrating the four differential balance equations in the one-dimensional drift flux model, various transfer functions between major variables, such as the velocity, void fraction, density, enthalpy, pressure drop and inlet velocity, can be obtained. These can be expressed symbolically as: f(s, z, t)=q(s, z) (s, t), (35) where Q represents the transfer function and f is the perturbed part of the variable, f, at location z. It has been shown that both the dynamic and transient responses of the system are governed by the transfer function between the internal pressure drop over the system and the inlet flow. Thus, the most important relation is given by p(s, t)=q(s) (s, t). (36) In a real physical system, the perturbation comes from the boundary condition on the pressure, which induces flow change. The disappearance of z in Q is due to the formal integration of the momentum equation over the entire system length, thus: = 1 p (37) Q(s) The dynamic response depends on the form of the transfer function, 1/Q(s), whereas the linear stability of the system depends on the root of the characteristic equation given by: Q(s) =0. (38) The characteristic function, Q(s) can be nondimensionalized by introducing proper scales for various variables. Thus Q*(s*)=Q*(s*, N 1, N, N m ), (39) where N 1,, N m are the same dimensionless groups listed in the subsequent section. This indicates that the dynamics of the system can be simulated if the scaling parameters, N 1,, N m, are identical between the two systems. The nondimensionalization of these response functions yields the key integral scaling parameters. From these, the scaling criteria for dynamic simulation can be obtained. The important dimensionless groups that characterize the kinematic, dynamic and energy similarities are given as follows (Ishii, 1976): Phase change no. N pch 4q o l o du o f i fg g =NZu, (40) where this phase change number has been recently renamed the Zuber number, N zu, in recognition of Novak Zuber s significant contribution to the field of two-phase flow. Also defined are: Subcooling no. N sub i sub i fg g (41) Froude no. N Fr u o f (4) gl o o

9 Drift-flux no. N di V gj u o M. Ishii et al. / Nuclear Engineering and Design 186 (1998) (void-quality relation) (43) Time ratio no. Thermal inertia ratio T i * l o/u o / s i i N thi sc ps f c pf d Friction no. N fi fl di 1+x(/ g ) (1+x/ g ) 0.5na o Orifice no. N oi K i [1+x 3/ (/ g )] a o, a i i a i (44) (45) (46) (47) where o, the reference void fraction in Eq. (4), is given by: o = f 1 1+(N d +1)(N Zu N sub ). (48) V gj, i fg, i sub, and x are the drift velocity of the vapor phase, heat of evaporation, subcooling and equilibrium quality, respectively. In addition to the above-defined physical similarity groups, several geometric similarity groups, such as (l i /l o ) and (a i /a o ), are obtained. The Froude, friction, and orifice numbers, together with the time ratio and thermal inertia groups, have the standard significance. The subcooling, Zuber, and drift-flux numbers are associated with two-phase flow systems. For a natural circulation dominated flow, the Froude number becomes important. Thus the void fraction plays a very important role for flow with low quality. However, at a high quality flow, the void fraction is close to unity and the driving force becomes only dependent on the length of the two-phase region. Physically, the phase change number is the scale for the amount of heating and vapor flow generation by phase change. The subcooling number is the scale for the cooling in the condensation section or the pressurization of liquid relative to the saturation condition. The excess cooling in this section or pressurization in the downcomer mainly determines the subcooled liquid temperature. The similarity analysis becomes much more complicated when there is not sufficient cooling to condense all of the steam or if the subcooling cannot be well controlled by the condensation. In this case, the detailed modeling of the condensation process and the analysis of the secondary loop may become necessary in order to determine the exit quality or the subcooling. The phase change number is one of the decisive parameters for the kinematic similarity. Together with the subcooling number, the phase change number is significant for the energy balance and dynamics of a system and also for the description of the steady state operational conditions. From the steady state energy equation balanced over the heated section using a control volume analysis, N Zu and N sub are related by: g xe =N Zu N sub, (49) where x e is the vapor equilibrium quality at the exit of the heated section. Therefore, the similarity of the Zuber and subcooling numbers yields: (x e ) R =1. (50) g R This indicates that the vapor quality should be scaled by the density ratio. If this condition is satisfied, the friction similarity in terms of N fi and N oi can be approximated by dropping the terms related to the two-phase friction multiplier. Furthermore, by definition it can be shown that N d = g x e f o 1 1. (51) Therefore, similarity of the drift flux number requires void fraction similarity. ( e ) R =1 or ( e ) R 1 (5) f R The drift flux number takes into account the drift effects due to the relative motion of the fluid. Thus, it plays an important role in the two-phase flow which is similar to diffusion processes. Also, since V gj depends on the flow regime, this group parameter also characterizes the flow pattern. The density ratio group, given by the (/ g ) term, scales the dynamic effect of the system pressure. This also appears in the groups N sub, N Zu, N f, and N o.

10 186 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) n The CHF number replaces the groups related to the thermal boundary layer in single phase flow. These groups include the Biot number and the Stanton number. In two-phase flow systems with heating, the boiling heat transfer is rather efficient and the value of the two-phase flow convective heat transfer coefficient is generally very high. In normal conditions, the wall superheat, T s T sat,is relatively small. However, the occurrence of the critical heat flux is significant since the heat transfer coefficient is drastically reduced at CHF. So the two-phase flow simulation of the CHF condition is more important than that of the thermal boundary layer. The occurrence of CHF can be considered as a flow regime transition due to change in heat transfer mechanisms. For the above similarity groups, Eqs. (41) (47), the constitutive relations for the relative motion between two-phases and the critical heat flux should be specified. These include the vapor drift velocity, V gj and the critical heat flux, q. c Several different forms (Ishii, 1977; Kataoka and Ishii, 1987) can specify the relative motion. The representative constitutive equation for the relative motion based on the drift velocity correlation is given by (Ishii, 1977): V gj =C o j+1.4 g f 1 4, (53) where the volumetric flux, j, in the heated section is given by: j= 1+x. (54) g uo The first term in Eq. (53) represents the contribution of the transverse velocity and void profiles. The second term is the contribution due to the local slip. If the relative motion, on the other hand, is based on the classical void-quality correlation, it may be expressed using = x, g, g f f, etc.. (55) Mathematically, this is equivalent to Eq. (53). With regard to Eqs. (53) and (54), the relative motion similarity based on the drift velocity correlation becomes: N d =C o 1+x g u o g f 1 4 (56) or it should have the same void quality relation given by Eq. (55). The CHF condition at low flow conditions has been reviewed by Leung (1977), Katto (1978), Mishima and Ishii (198). The modified Zuber (1980) correlation for low flow is given by: 1 q c =0.14(1 ) g i fgg4. (57) g Based on the blowdown experiments, this pool boiling CHF based correlation is recommended for the mass velocity range of 40 to 100 kg m s 1. It is only applicable to transients that involve flow reversal. For slow transient situations at low flow rate ranges, Katto s correlation may be used: q c = 1 4 i fgg d o l o p i sub G l o i fgn. (58) This equation implies that the critical quality is x c =(p/g l o ) 0.043, where G is the mass velocity. The typical values are between 0.5 and 0.8 implying that the underlying mechanism should be the annular flow film dryout. However, there is a possibility that the critical heat flux may occur at much lower exit quality than that given above due to a change in two-phase flow regimes. In a natural circulation system with very small flow fluctuations, the occurrences of CHF have been observed at the transition between churn turbulent and annular flow. Beyond this transition, the lack of large disturbance waves eliminate the rewetting of dry patches. This leads to the formation of permanent dry patches and CHF. The criteria developed by Mishima and Ishii (198) for this case is given by: 1 q c = d o 1 i 4l o C fg g gd o +Gi o subn (59) where C o is the distribution parameter for the drift flux model and is given by: C o =1. 0. g f. (60) The above CHF criteria should be used to develop a similarity criterion for the fluid solid

11 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) boundary. This ensures that the critical heat flux occurs under similar conditions in the simulated system..4. Two-phase similarity laws Eqs. (41) (47) represent the relationships for the dimensionless groups and the generalized variables of a two-phase flow system. For similarity to be achieved, these groups must be equal in the prototype and the model. In general, the scaling of the two-phase flow dynamics with complete similarity is impossible when there are so many groups which must be satisfied simultaneously. However, a scale model with the same fluid under the same pressure simplifies the conditions significantly such that close similarity may be attained. In this case, the fluid properties between the model and prototype are considered to be the same: R = gr = R =C pr =k R = R = gr =i fgr =1. (61) The resulting similarity laws, combined with the above condition, are: (N Zu ) R =1 (6) (N sub ) R =1 (63) (N Fr ) R =1 (64) (N di ) R =1 (65) (T*) i R =1 (66) (N thi ) R =1 (67) (N fi ) R =1 (68) and (N o ) R =1. (69) The drift flux number can be automatically satisfied if the contribution of the local slip is small in comparison to the slip due to the transverse velocity and void profiles. When the local slip is the dominant factor, the similarity requirement becomes u R =1, which is very restrictive. For most cases, the former applies. From the requirements of the similarity laws for the Zuber number, Eq. (6), the fluid properties, Eq. (61), and the quality relation given by Eq. (50), the similarity in terms of the vapor quality is satisfied, thus x R =1. With the local slip controlled, the void similarity is also obtained, as R =1. Hence, excluding the friction similarity conditions, Eq. (68), it is required that: R q R l R =1 (70) d R u R (i sub ) R =1 (71) u =1 (7) l R and in addition, the critical heat flux similarity requires: q c =1. (73) q 50 R Using the Katto CHF relation, Eq. (58), and the phase change similarity requirement, Eq. (6), the above equation becomes: (x c ) R = =1. (74) u l R Excluding the friction, orifice and drift-flux number similarities from the set of similarity requirements, the following similarity requirements can be obtained: (u o ) R =(l o ) R 1/ (75) (i sub ) R = i fg g R (76) (q o ) R = f g i fg d 1/ (l o ) R (77) R R 1/4 1/ R =(l o ) R ( s ) R (78) d R = sc ps 1/4 1/ (l o ) R ( s ) R. (79) f c pf R The velocity scale shows that, in contrast to the case of single phase flow scaling, the time scale for a two-phase flow is not an independent parameter. From Eq. (75), the time scale in two-phase flow is uniquely established to be: R = l o u o R =(l o ) R 1/. (80)

12 188 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) This implies that if the axial length is reduced in the model, then the time scale is shifted in the two-phase flow natural circulation loops. In such a case, the events are accelerated in the scaleddown model by a factor of (l or ) 1/ over the prototype. It is important to note that when the two-phase flow velocity scale is used in the single phase flow geometric scale requirements, the geometric similarity requirements in both cases become the same. Hence, the same geometric scale can be used for single phase and two-phase flows. However, using the time scale indicated by the two-phase flow scaling, namely R =l or, the single phase time events are also scaled by the same criterion. This leads to the very important conclusion that for systems involving both single and two-phase flow in a reduced length model, real-time scaling is not appropriate. 3. Mass and energy inventory and boundary flow scaling (nd level) A nuclear reactor system such as the SBWR consists of several inter-connected components. Therefore, it is essential to simulate the thermalhydraulic interactions between these components. The physical processes involved in the system are governed by the conservation principles of mass, momentum and energy. Among them, both the mass and energy balances are key to the proper scaling of the inter-component relations. The conservation of momentum is important for the forces acting on the structure, however, it is not essential to the scaling of the inter-component thermal-hydraulics. For a system consisting of several components, the scaled mass and energy inventory histories must be preserved for the integral similarity of the thermodynamic state of each component. The conservation of momentum becomes important in determining the boundary mass flow. The scaling criteria can be obtained from the control volume balance equations for mass and energy. In particular, important scaling criteria are obtained for the boundary flow of mass and energy at the interface between two connected components. As is the case of many types of transients, both choked and non-choked flow can occur at the same junction. If the non-choked flow is governed by the frictional resistance, it can be scaled by the integral scaling criteria based on the response function. However, during the blowdown phase, non-frictional momentum effects dominate the choked flow. At such discharge points, the fluid velocity depends upon the local pressure ratio across the device, which is preserved in a full-pressure scaled system. In nonfrictional momentum-dominated choked flows, the fluid velocity is the same in the model as in the prototype. Therefore, the flow area at such discharge points must be scaled to preserve mass and energy inventory rather than loop kinematics. The purpose of this section is to develop the appropriate scaling relations to be applied at such points. An overall criterion for similar behavior between the prototype and the model is that the depressurization histories be the same when compared in the respective (scaled) time frames, i.e. p m (t m )=p p (t p ). (81) This integral condition will be satisfied if the differential pressure change is the same at corresponding times, i.e. dp m = 1 dp p. (8) dt m R dt p The scaling criteria for similarity of the frictiondominated natural circulation flows, yields the result that the time scale of the model or laboratory time, is related to the prototype time by: 1/ t m =(l o ) R t p = R t p (83) and the depressurization rates of the model and the prototype are related by: 1 dp m = 1 dp p. (84) dt m l o R dt p This condition will be satisfied if the corresponding component vessel inventories are similar, i.e. M m = m t M p (85) m p t p where M p and M m are the prototype and model vessel inventory masses, and p and m are the respective prototype and model vessel volumes.

13 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) This relation must hold for each component as well as for the overall system if complete similarity is to be ensured Mass inentory and mass flow scaling For integral experiments, accurate simulation of the mass and energy inventory is essential. This requires a separate scaling criteria for the system boundary flows such as the break flow and various ECCS injection flows. The scaling criteria, stated in Eq. (85), are obtained from the overall control volume balance equations. For the coolant mass inventory, the total mass for a particular component is given by: d dt M= m in m out (86) By denoting the total volume by V and the mean density by, the mass conservation equation can be written in a dimensionless form that applies to both the model and the prototype system as: d dt* *= m * in m * out, (87) where t*t/(l o /u o ) and */ o (88) and a in u in a o u o, (89) m * in m in o in o where o =(l o /u o ) for either the prototype or the model. The definition for m * out can be given similarly. For model and prototype having the same pressure (* out ) R =( out /) R is simply unity. Hence, the simulation of the boundary flow requires: a in u in a o u o R and a out a o u out u o =1 (90) R =1. (91) This is a similarity condition for the flow area and velocity combined. Therefore, it is not necessary at discharge points to satisfy the independent conditions for area and flow given by Eqs. (31), (3) and (75), which must be satisfied by the other components of the loop. The forms of the discharge scaling criteria given by Eqs. (90) and (91) are very convenient from the standpoint of practical implementation. For example, the break flow velocity, u out, can not be independently controlled if choking occurs. In the case of choking, Mach number similarity is maintained. Thus, for equal-pressure system the break flow velocity is prototypical. However, for 1/ the basic scaling (u o ) R =(l o ) R and the criteria given in Eqs. (90) and (91) predict that the break flow area should be scaled according to: a in =(l a or ) 1/ (9) o R which would result in a reduction of the break flow area beyond the geometrical scale used for the loop flows. For the case of ECCS injection flows, the breakflow scaling criterion is also very useful. If the injection lines are scaled according to the geometrical scaling condition, Eqs. (31) and (3), the line diameters become very small and the frictional resistance can be very large. This will result in mismatched ECCS injection flow, which is unacceptable. Fortunately, the boundary flow scaling criteria, Eqs. (84) and (85), permit an enlarged flow area to obtain the correct volumetric or mass flow rate. 3.. Energy inentory and energy flow scaling For the energy inventory, E, the control volume balance is given by: de dt =q w+ m ini in m out i out. (93) By non-dimensionalizing the above equation, it can be shown that the scaling criteria obtained for the natural circulation satisfies the similarity requirement for the heat input, q. The dimensionless form of the above equation is given by: de* dt* =q* w*+ m *i in * in m * out i* out (94)

14 190 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) where E*= E E o, q*= q q o, w*= w w o and i*= i i o(95) and m *i in * in =m in i o in in i o a in a u in u i in. (96) i It can be shown that the integral system scaling criteria automatically satisfies the requirement for the power q*, hence no additional constraints are imposed beyond the similarity of the enthalpy at the boundary. In view of Eqs. (84) and (85), for a full pressure simulation, i.e. (i o ) R =1, it is necessary to require: (i in ) R =1 and (i out ) R =1. (97) This physically implies that the inflow or outflow should have a prototypic enthalpy. The above dimensionless energy equation also shows that the initial energy inventory should be scaled by the volume ratio Basis for reduced height scaling Under a prototypic pressure simulation, the system geometry can be determined from the integral system scaling and the boundary flow scaling discussed above. The dynamic scaling requirements for a two-phase flow system are given by Eqs. (40) (47). In general it is difficult to match all these similarity criteria for a scaled down system, so a careful evaluation of each of these requirements should be made. Based on the original scaling study by Ishii and Kataoka (1984) it is evident that the Froude number scales the gravitational driving head and the Friction number scales the frictional resistance against the inertia term. The Zuber (phase change) number and subcooling number scale the energy transfer for a boiling process. It is essential that these latter numbers are satisfied for the energy and kinematic similarities. As indicated by Eqs. (49), (50) and (5), these two similarity criteria give the simulation of the void fraction and the steam quality under the prototypic pressure simulation. In considering the dynamics of the system, two conditions should be considered separately. The first is the quasi-steady flow simulation and the second is the dynamic response of the system, including the inertia effect. It is clear that the Froude number and friction number scale the dynamic response. When the inertia forces are not important, only the balance between the frictional resistance and gravitational force should be considered. This can be achieved by taking the product of these two numbers. Thus, the natural circulation number is defined as: N nc =N f N Fr = friction inertia (98) inertia gravity head This equation can be extended to include the minor loss coefficient as: N nc =(N f +N o )N Fr. (99) In general, the requirement of: (N nc ) R =1 or [(N f +N o )N Fr ] R =1 (100) is less restrictive than (N f ) R =(N o ) R =(N Fr ) R =1. However, the energy and kinematic similarities require that the velocity be scaled by Eq. (75) and the void fraction by Eq. (5). Under these conditions, it can be shown that: f (N Fr ) R = u =1. (101) gl R Hence Eq. (100) can be reduced to: (N f +N o ) R =1. (10) Combining the above equation with Froude number similarity, it is seen that these two also constitute an approximate dynamic similarity between the inertia term, gravitational term and flow resistance. The advantage of Eq. (10) relative to the two independent requirements of (N f ) R =1 and (N o ) R =1 is significant. Under a homogeneous flow assumption, the requirement given by Eq. (10) can be approximated by: (N f +N o ) R fl +KRa o =1. (103) d a i R By using the geometrical similarity criteria,

15 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) fl d +KR =1. (104) These two scaling criteria apply to the gravity driven flow. For break flow, a different criterion applies. A careful analysis of Eq. (104) clearly indicates, for a given volume scale, the advantage of using the reduced-height system in satisfying the dynamic similarity criteria. By reducing the flow area, the hydraulic diameter is reduced by d R = a R, except at bundled sections, such as the core. For most small integral test facilities, it is necessary to have l R d R in order to maintain a reasonably large axial height so that the naturally existing two-phase level fluctuations do not adversely affect various transient phenomena. In general, the ratio of the first friction term itself is always larger than unity. However, by reducing the height of a facility, this ratio can be made closer to unity by increasing d R for a fixed value of R. The second significant point is that the minor loss coefficient is an easy parameter to adjust through small design modifications in such a way that K R 1 to compensate for increased friction. Hence, by properly modifying the K value, Eq. (104) can be satisfied PUMA geometrical scaling For the scaling study performed for the PUMA facility, the following comparison of various parameters between the prototype and PUMA was made in Table 1. The values presented are used in the following section for the analysis of the local phenomena scaling which is the 3rd level of scaling. A more extensive discussion of these values is presented in Section 5. Table 1 Parameter comparison between prototype and PUMA Description Parameter Pressure ratio p R 1 Area ratio a R 1/100 1 Length ratio l R 4 Velocity ratio u R 1 Power density ratio q R Time ratio R Numerical value 1 4. Local phenomena scaling (3rd level) with application to PUMA Although the global scaling criteria satisfy the system response similarity, the local phenomena may not be satisfied. Hence, it is important to study the local phenomena scaling in detail for the specific system. In the following sections, the detailed scaling of the relevant phenomena for the scaled model of the SBWR (PUMA) is discussed Break and ADS flow scaling The RPV is depressurized by the discharge of steam or water from a break and by the SRV and DPV flows when these systems are activated. In the early phase of the depressurization, the upstream pressure is sufficient to cause sonic velocity at the minimum-area section of the steam line venturi, and at the throat of the SRV or DPV, or at the break location. A bottom drain line break (BDLB) also results in choked flow when cavitation occurs at the throat or minimum area in the break line. For all of these cases, the velocity at the break or the throat will be the same in the model as in the prototype since the pressures and thermodynamic conditions are the same. The prototype-to-model ratio of velocity multiplied by the throat area should satisfy the conditions for similarity of the mass and energy inventories, as previously discussed. As the system depressurizes, a transition to unchoked flow will occur. For choked flow, no additional restrictions on the geometry or loss mechanisms are necessary, but additional restrictions are needed to preserve the pressure or pressure ratio at which the transition from choked flow to unchoked flow will occur. Two different nozzle geometries are found in the SBWR system. First, the steam line contains a converging diverging de Laval-type nozzle that is designed to limit the steam discharge rate in the case of a main steam line break, yet results in little pressure loss under normal operation. The low pressure loss under normal operation is achieved by use of a low-angle conical diffuser downstream of the nozzle throat so that flow separation is avoided and good pressure recovery is achieved. This type of nozzle will become choked whenever

16 19 M. Ishii et al. / Nuclear Engineering and Design 186 (1998) a modest drop in the discharge pressure occurs as a result of a break or other decrease in the downstream flow resistance (decrease in discharge pressure). The flow will remain choked until the upstream pressure drops to near the downstream value. The flow in short nozzles and valve contraction sections can be considered to be nearly frictionless and adiabatic so that an isentropic model of the flow process is a good approximation. The SRV and DPV systems have a different nozzle geometry that consists of a smooth contraction down to the throat followed by an abrupt increase in the flow area. This type of nozzle has different flow characteristics. The abrupt increase in flow area downstream of the throat results in large pressure losses due to the irreversibility associated with turbulence downstream of the abrupt increase in area. The behavior of this type of nozzle for low downstream pressure (choked) is even more complex than for the converging diverging nozzle and consists of separated flow regions, possible oblique shocks combined with a normal shock, and reattachment of the flow to the cylindrical downstream passage. In order to maintain similarity in the scaled mass and energy inventories, it is necessary to scale the throat area between the model and the prototype by the ratio given by Eq. (91) for choked flows. However, in order to assure correct transition to subsonic or unchoked flow, it will also be necessary to preserve the diffusion characteristics of the downstream section of the nozzle. This requires that geometric similarity be maintained and, to a lesser degree, that Reynolds similarity be maintained. Thus, the nozzle contour and especially the diverging section cone angle must be geometrically similar in the model and the prototype. For nozzles having abrupt area change, it is simply necessary for the cylindrical section area ratio to be similar to the prototype. For a break flow area, a t, with break flow velocity, u t, the boundary flow scaling requirement is given by: (a t u t ) R =(a line u line ) R =(a line ) R (u line ) R =(a o u o ) R =1/00. (105) For critical flow, the ratio of velocity at the throat is given by (u t ) R =1. From boundary flow scaling, (a t u t ) R =(a o u o ) R =1/00. Since the model has prototypic pressure, the density ratio is R =1. Thus, the area ratio is (a t ) R =1/00. (106) This shows that the throat area where choking occurs should be scaled differently from loop sections in which (a line ) R =1/100 is used. One additional case needs to be considered, and that is the case of a cavitating venturi such as that which would occur for a bottom drain line break (BDLB). This case is more complex than the ideal gas case just discussed. However, the considerations are very similar, and the resulting conditions for similarity are the same. The reason for this is that even though the choking phenomena is due to the onset of vaporization caused by the lowering of the static pressure to less than the saturation pressure, the pressure behavior is primarily governed by the same inviscid flow mechanism. The pressure recovery/loss mechanism that occurs downstream of an abrupt area change is more complex than the single phase flow case, but it is also mainly governed by the geometry and is thus, approximately simulated by preserving geometric similarity. As the pressure downstream of a nozzle or break approaches the upstream pressure, a transition to unchoked flow will occur. Each of these nozzles has different unchoked flow characteristics that need to be considered, relative to the flow area scaling that is used to preserve volumetric similarity of loop components. First, consider the steam line venturi, which is a converging diverging nozzle and is designed to have negligible head loss under normal unchoked operating conditions. The pressure drop for this component is negligibly small, and so is the associated minor loss coefficient. Thus, other losses in the interconnecting components, such as line friction and minor flow losses, will dominate the resultant inter-component flow. For this case, no additional scaling considerations are needed for unchoked flow and the nozzle throat can be scaled by 1/00, as needed to maintain choked flow similarity. The second case of the