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Nuclear Engineering and Design 240 (2010) 489–499

Contents lists available at ScienceDirect

Nuclear Engineering and Design

journa l homepage: www.e lsev ier .com/ locate /nucengdes

asic investigation of particle swarm optimization performance inreduced scale PWR passive safety system design

oão J. da Cunhaa, Celso Marcelo F. Lapab,e,∗, Antonio Carlos M. Alvimc,e,arlos A. Souza Lima Jr. b,d, Cláudio Márcio do N.A. Pereirab,e

Eletronuclear Eletrobras Termonuclear, Gerência de Análise de Seguranca Nuclear, Rua da Candelária, 65, 7◦ andar. Centro, Rio de Janeiro 20091-906, BrazilInstituto de Engenharia Nuclear, Divisão de Reatores/PPGIEN, P.O. Box 68550, Rua Hélio de Almeida 75 Cidade Universitária, Ilha do Fundão, Rio de Janeiro 21941-972, BrazilUniversidade Federal do Rio de Janeiro, COPPE/Nuclear, P.O. Box 68509, Cidade Universitária, Ilha do Fundão s/n, Rio de Janeiro 21945-970, BrazilInstituto Politécnico, Universidade do Estado do Rio de Janeiro, Pós-Graduacão em Modelagem Computacional, Rua Alberto Rangel, s/n, Vila Nova, Nova Friburgo 28630-050, BrazilInstituto Nacional de Ciência e Tecnologia de Reatores Nucleares Inovadores, Brazil

r t i c l e i n f o

rticle history:eceived 2 April 2009eceived in revised form

a b s t r a c t

This work presents a methodology to investigate the viability of using particle swarm optimization tech-nique to obtain the best combination of physical and operational parameters that lead to the best adjusted

3 November 2009ccepted 3 December 2009

dimensionless groups, calculated by similarity laws, that are able to simulate the most relevant phys-ical phenomena in single-phase flow under natural circulation and to offer an appropriate alternativereduced scale design for reactor primary loops with this flow characteristics.

A PWR reactor core, under natural circulation, based on LOFT test facility, was used as the case study.The particle swarm optimization technique was applied to a problem with these thermo-hydraulicsconditions and results demonstrated the viability and adequacy of the method to design similar systems

.

with these characteristics

. Introduction

The study presented in this paper combines a new methodol-gy to design experimental systems and/or facilities in reducedcale (Lapa, 2004; Cunha, 2007) with the particle swarm optimiza-ion technique that has been successfully used recently in classicaluclear engineering problems.

.1. The similarity problem

The use of reduced scale test sections to simulate and under-tand physical phenomena is an approach widely used whenesigning and constructing industrial facilities or facilities of anyther nature, whose proportions and magnitude involve largemounts of money or where some analyses are needed that cannote conducted in real scale, due to operational or physical reasons

e.g., simulation of severe accident in a nuclear station or the anal-sis of the behavior of an estuary). However, those experimentalacilities, whose operation and construction costs are significantlyower than their similar in real scale, should be able to qualitatively

∗ Corresponding author at: Instituto de Engenharia Nuclear, Divisão deeatores/PPGIEN, P.O. Box 68550, Rua Hélio de Almeida 75 Cidade Universitária,

lha do Fundão, Rio de Janeiro 21941-972, Brazil. Tel.: +55 21 21733907.E-mail address: lapa@ien.gov.br (C.M.F. Lapa).

029-5493/$ – see front matter © 2009 Published by Elsevier B.V.oi:10.1016/j.nucengdes.2009.12.003

© 2009 Published by Elsevier B.V.

and quantitatively simulate the physical phenomena that mayoccur in real facility situations. Needless to say that design of newengineering systems is supported by the construction of similardevices in reduced scale, mainly in the aerospace, petrochemi-cal, nuclear, naval, metal-mechanics and metallurgical industries.Unfortunately, however, it is not possible to obtain a facility of moremodest dimensions and operational conditions which truly repre-sents the phenomena that occur in real scale. Absolute similarityonly exists in a one-to-one scale and under the same operationalconditions as the original system (Ishii, 1984). Thus, the project ofa reduced scale facility should take into account the best param-eter combination (dimensions, operational conditions, etc.) whichbetter simulates the most important phenomena of the experimentunder study, considering also practical and economical constraints.Recently, a new methodology was proposed to solve this problem(Lapa, 2004, Cunha, 2007). This methodology formulates the prob-lem of designing an experimental test section as an optimizationproblem with constraints and tests a large number of combinationsof reduced scale facilities until the best combination, consideringproject constraints, is obtained.

1.2. Intrinsically safe reactors

Decay heat removal of first and second generation nuclearpower stations is done by an active system, based on forced

490 J.J. da Cunha et al. / Nuclear Engineering and Design 240 (2010) 489–499

Nomenclature

A accelerationa flow areaa0, ai heated flow area, i-th section flow areaas solid cross sectional areaBi Biot numberc1 self-confidence factorc2 swarm confidence factorCp fluid heat capacityCps solid heat capacityd hydraulic diameterf friction factorFi friction numberFr Froude numberg gravityGi expected value of a given dimensionless group i

obtained with the operating conditions of the pro-totype

Gi value corresponding to the dimensionless group i,obtained with the operating conditions of the can-didate solution (design scale);

Gr Grashof numberh heat transfer coefficientK conductivity of fluidKi i-th section loss coefficientks conductivity of solidl, L axial lengthlh distance between centers of core and steam gener-

atorN number of full flow channels in each directionNg total number of significant dimensionless groupsNc total number of channels flowNV total number of fuel pinNu Nusselt numberPr Prandtl numberP pressurep pitch�Pi

k-best,j best position of each particle over time

�pik-gbest,j best global value in the current swarm

q′′′si

heat generation in solidQsi heat source numberR Richardson numberRe Reynolds numbersi, si operational and structural parameters that make up

the groups Gi, Gi

St modified Stanton numbert timeT fluid temperatureTs solid temperatureTsat saturation temperatureT* time ratio numberu, U fluid velocityur, u0 representative velocity, reference velocityV volumewi, w weights for the relative importance of each non-

dimensional group; to the problem, inertial factor

Greek letters˛s solid thermal diffusivityˇ thermal expansion coefficientı conduction thickness�T characteristic temperature rise�w wetted perimeter

�h heated perimeter� scale factor� density of fluid�s density of solid

� viscosity of fluid� dynamic viscosity of the fluid

circulation of coolant through the core, when reactor is shutdown,should an accident occur or during fuel reload.

Even considering the already known self-control characteristicsof a PWR (moderator and fuel feedback effects, etc.), TMI accidentanalysis have indicated that decay heat removal poses a big chal-lenge and is a basic issue, in order to reduce the risk of accidentalradiation release, mainly for those long-lasting accidents (loss ofcooling accidents, for example). To solve this problem, third gen-eration nuclear reactor concepts adopted residual heat removalsystems which make use of natural circulation, besides coping witha set of objectives, including radioactive waste reduction, economiccompetitiveness, proliferation resistance, etc. Natural circulationshould occur as a consequence of the temperature differencebetween cold sink (heat exchanger) and hot source (reactor), whichshould have sufficiently difference in height. Considering the largecoolant density difference in the core and at the heat exchanger,gravitational force promotes circulation. The existence of this nat-ural device is a remarkable example of compliance with anotherimportant safety concept of this new generation of reactors, calledintrinsic safety.

2. Prominence

Considering the forthcoming of large scale construction ofthird and fourth generation nuclear stations in the next decadesand that countless tests and experiments in similar circuits willhave to be conducted, to ensure functional and safety require-ments, both in the design and license steps, Lapa et al. (2004)and Cunha et al. (2007) developed a method that consists informulating the design problem of an experimental test section,similar to a reference prototype (real system), as an optimiza-tion problem with constraints, to be solved by using advancedglobal optimization techniques. These studies used as case exam-ple the reduced scale design of a typical PWR core, under forcedcirculation regime and at 100% nominal power. Genetic Algo-rithms (Holland, 1975) were adopted as the optimization techniquein that work. Although the results were very auspicious andshowed that the new methodology could be applied to designsimilar experiments, some problems and improvement sugges-tions were established. Among them, the following two should beemphasized:

• The generated search space showed, according to qualitativeevaluation and sensitivity analysis of the obtained results, greatmultimodality and a highly irregular topology, with very highderivatives and extremely acute “peaks” and “valleys”. As a conse-quence, the solutions obtained with Genetic Algorithms (GAs), inseveral occasions, clearly showed premature convergence char-acteristics, besides requiring to much computer simulation time.Still, this analysis indicated that these problems may get worse,when the problem under study increases in size. Presuming that

complexity exponentially increases with the size of the problem,we conclude that an optimization technique that preserves GAgood characteristics, though faster and more efficient, should betested.

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The second aspect refers to the need of finding a relevant caseexample for the new generations of nuclear reactors. Thus, a nat-ural convection residual heat removal system, usually found inthird generation PWR’s, was adopted as a reference prototype.

. Theoretical basis

This section explains the concepts involved in this work. Itescribes the reasons behind the concepts of similarity and scal-

ng laws. A quick approach places the optimization problem in theontext to be studied, and finally emphasizes the particle swarmptimization.

PSO was chosen as an optimization tool because of excellentesults obtained in recent nuclear engineering problems, wheret proved to be more efficient and to require less computer timehan GA. Its applicability to nuclear engineering was particularlyxtended by Domingos et al. (2006). They used it as an alternativeethod to solve optimization problems in typical PWR cells involv-

ng moderator, cladding and fuel, minimizing the average powereaking factor, for a given average thermal power, considering con-traints due to criticality and sub-moderation.

Medeiros and Schirru (2008) has also applied PSO to a nuclearransient classification system, aiming at helping the operator toerform a quick and correct diagnosis of current events. This systemllows an increase of the time available for the operator to takeorrective measures in order to keep the plant in a safe condition.

.1. Similarity and scaling laws

According to Szirtes (1998), the objective of dimensional model-ng is to perform scale experiments in a similar construction, called

odel, of the original facility, called prototype, aiming at applyinghe results obtained with the model to the prototype.

This shows that the main aspect of the similarity concepts the dimensional analysis ability to reduce, or abbreviate theunctional form of physical relationships. Such relationships arestablished by dimensionless numbers or groups, which repre-ent the physical phenomena involved in a particular problem.hus, it seems reasonable to state that, in physical terms, similarityefers to some equivalence between two things or phenomena thatre really different. Since phenomena are represented by dimen-ionless numbers or groups (Barenblatt, 2003), it makes sense totudy a phenomenon using a reduced scale model and to infer onhe prototype performance from this model. For example, underery particular conditions, a direct relationship exists between theorces that act in a real size airplane and those in a reduced scale

odel.Although geometric and kinematic similarities (Duncan, 1955)

re necessary for an experimental model to portray the prototypeow, they are not enough. Both flow fields are considered simi-

ar when dynamic similarity also occurs, that is, when their forceoefficients are equal. Dynamic and kinematic similarities will bensured if the dimensionless groups or numbers found, for theodel and the prototype, have the same value (Szirtes, 1998).

.2. Optimization problem

Optimization problems are present in several branches ofuman activity. Then, it is important to use techniques that areble to present a set of solutions for the problem to support deci-

ion making. The formulation of a practical optimization problemsually contains two parts:

A fitness to be reached;Constraints that should be satisfied.

and Design 240 (2010) 489–499 491

A solution should be found (sometimes there are many) whichminimizes or maximizes the fitness and also simultaneously sat-isfies the constraints. That is, the solution found should belong tothe search space or viable region that includes the set of possi-ble or viable solutions of the problem to be optimized. In orderto attain optimal conditions, the system should be free to handledecision variables of the problem under study, conveniently calleddesign variables or also, search variables. Thus, some operationaland structural conditions will be modified in order to attain theoptimum viable point. The fitness that represents the problem mayhave one or several minima (or maxima) points, being respectivelydefined as unimodal or multimodal.

Optimization is the search for the best solution for a given prob-lem within a finite or infinite set of possible solutions. The searchprocess may start from an initial solution or a set of them, with pro-gressive improvements until another set is attained, containing oneor all of the best possible solutions within the search space. Conven-tional methods are conceived to solve generic cases that belong tosubclasses where linearity, differentiability or non-differentiabilityrule in isolated points. They are classified, according the character-istics of the fitness and its constraints, into subclasses as LinearProgramming, Non-Linear Programming or Quadratic Program-ming.

When the fitness and the constraints are linear functions of thedesign variables, the optimization problem is linear. In the sameway, if the fitness or at least one of the constraints is a non-linearfunction of the design variables, the optimization problem is non-linear and may be a problem with constraints or not

Finally, when the fitness is quadratic and the constraints arelinear functions of the design variables, the optimization problemis quadratic.

Most of the modern computational methods of global opti-mization are based on the observation of biological processes,which interpret natural evolution, competition and collaborationas an intelligent way of adaptation, self-organization and opti-mization. In general, these methods are conceived to solve genericproblems in generic environments, where non-linearity or evennon-differentiability is irrelevant and they operate with a popula-tion of points that evolve through probabilistic rules. Usually, theyensure a good approximation to the solution, for problems whereconventional methods do not ensure the same performance.

The optimization process of the dimensional design of reducedscale residual heat removal system under natural convection,includes non-linearity, discontinuities and, multimodalities, andmay involve the optimization of parameters whose representationis continuous, together with other parameters having discrete rep-resentations. In this case, conventional methods fail to determinea global optimum. Considering the previous argument, the particleswarm optimization technique (PSO) was adopted in this study, asa working tool for the problem of optimization of dimensionlessparameters, calculated for a natural circulation flow.

3.3. Particle swarm optimization (PSO)

PSO is a computational technique based on optimization of indi-vidual exploration, combined with the ideas of cooperation andcompetition of the population. It was derived considering simpli-fied social models, as flock of birds, school of fish and the theoryof cluster (Kennedy and Eberhart, 1995). Observation of simplifiedsocial models identified the metaphor-generating algorithm: Thekey idea of the swarm, flock or school of fish is that, searching for

food, any agent of the group can profit from the discoveries and/orprevious experiences of all members of the swarm. This advantagecan become decisive where food sources are distributed in unpre-dictable ways, as opposed to the inconvenience of competition forfood items “inter-bands.” This means that there is an evolutionary

4 eering and Design 240 (2010) 489–499

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92 J.J. da Cunha et al. / Nuclear Engin

dvantage when information is socially distributed by agents, inhe school of fish or flock of birds. Because optimization is basedn the metaphor of foraging by a swarm of agents, where the richocal food is the best overall solution to be considered in the searchpace of the real problem, then each particle has its position deter-ined by a combination of parameters. One of them gives the best

ocal position, and the other represents position of the best particlehosen at each generation step. Moreover, the trajectory of eachndividual in search space is dynamically adjusted, by changing thepeed of each particle according to its own search experience andearch experience of other particles in space.

To mathematically illustrate the qualitative description pre-ented, it can be stated that the basic algorithm of PSO is itself ahree-step process: generation of cluster of particles (with deter-

ination of position and velocity of the particles), updating theosition (movement of particles to a new position) and, finally,valuation of the new position (Hassan et al., 2005).

.4. Methodology

The methodology to be described makes use of the theoreticalackground described previously to formulate the problem of scal-

ng a section of an experimental facility as an optimization problemith constraints; using the particle swarm methodology as the

ptimization tool.The proposed methodology can be summarized in the following

teps:

. Definition of physical model suitable for the problem to be stud-ied.

. Dimensional analysis of the physical model and determinationof the relevant dimensionless groups.

. Calculation of dimensionless numbers of the real system thatwill serve as reference for the search.

. Formulation of optimization problem with constraints.

. Definition of the search variables of the problem as the minimumset and sufficient for the calculation of all the dimensionlessgroups of interest.

. Design of a test section that has the same dimensionless numbersas the original system (prototype), but with design parameterslimited by economic, structural or operational considerations.

. Use the method of particle swarm to solve the optimization prob-lem.

. Case study

This section presents several case studies, using the methodol-gy described in the previous section, aiming at obtaining a reducedcale thermo-hydraulic loop under natural circulation, similar tohe reference prototype. Two different case studies were devel-ped:

Methodology Performance and Validation Test;Reduced Scale Project.

A description of the steps for both case studies follows.

.1. Definition of physical model suitable for the problem

This step consists of establishing mass, energy and momentumonservation equations, with initial and boundary conditions, forhe problem at hand.

Fig. 1. Schematics of natural circulation in LWR.Source: Ishii and Kataoka (1984).

4.2. Dimensionless physical model and determination of relevantdimensionless numbers

Step 2 focus on the work of Ishii and Kataoka (1984). A typi-cal single-phase flow PWR primary loop under natural circulation(Fig. 1) was analyzed, aiming at establishing new similarity criteriafor the existing model, based on physical laws applied to an averageloop configuration.

When the governing equations representing the physical modelare made dimensionless and typical or reference values are used forthe independent and dependent system variables, dimensionlessgroups or numbers are produced as coefficients of the several termsof the differential equations.

However, it is more difficult to establish such similarity criteriain a closed loop. Thus, the most relevant dimensionless groups mustbe established, based on average values (Ransom et al., 1998).

Ishii and Kataoka (1984) used two energy equations and aone-dimensional boundary condition, besides the continuity andintegral momentum equations around a loop composed of severalsections. A brief description follows:

Continuity equation for the i-th section:

ui = a0

ai· ur (1)

Integral momentum equation for the i-th section:

�dur

dt

∑i

a0

aili = ˇg�Tlh − �u2

r

2

∑i

(fl

d+ K

)(a0

ai

)2(2)

Fluid energy equation for the i-th section:

�Cp

(∂Ti

∂t+ ui

∂Ti

∂z

)= 4h

di(Tsi − Ti) (3)

Solid energy equation for the i-th section:

�sCps∂Tsi

∂t+ ks∇2Tsi − q′′′

s = 0 (4)

The coupling between both energy equations is obtainedthrough the boundary condition:

−ks∂Tsi

∂y= h(Tsi − Ti) (5)

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here ui = velocity through the i-th section, ur = velocity throughhe core, a0 = flow area of the core (used as reference area for otheri areas) and lh = vertical separation between hot and cold thermalenters of the loop.

Note that the ur value will be, in general, a function of time.The value under steady state natural circulation is named u0, and

ndicates a reference velocity used in the non-dimensional process.n the same way, l0, the active height of the rods is taken as referenceength. The equation system will be made non-dimensional if theollowing dimensionless parameters are used:

∗i = ui/u0, u∗

r = ur/u0, a∗i = ai/a0

∗i = li/l0, l∗h = lh/l0, z∗ = z/l0, y∗ = y/l0

∗ = tu0/l0, T∗ = T/�T0, �T∗ = �T/�T0

∗2 = ı2∇2 (6)

The value �T0 is the maximum temperature difference in theoop during steady state natural circulation. The following simpli-ying premises for a steady state flow should be considered:

Viscous dissipation and axial conduction effects are negligible;Heat losses are negligible;Since the analysis is performed for liquid conditions in singlephase regime, the fluid is incompressible;Boussinesq approximation is valid, that is, the properties of thefluid may be considered constant in the conservation equations,except for density as a driving force, which is supposed to varylinearly with temperature.

Considering the dimensionless parameters described in Eq. (4),nd substituting them into the conservation equations and estab-ishing the necessary simplifications, those equations may turn outimensionless, we get the following non-dimensional equations:

Continuity equation for the i-th section:

∗i = u∗

r /a∗i (7)

Integral momentum equation for the i-th section:

du∗r

dt∗∑

i

l∗i

a∗i

= ˇg�T0l0u2

0

�T∗lh − u∗2r

2

∑i

((fl/d) + K)i

a∗2i

(8)

Fluid energy equation for the i-th section:

∂T∗i

∂t∗ + u∗i

∂T∗i

∂z∗ = 4hl0�Cpu0di

(T∗si − T∗

i ) (9)

Solid energy equation for the i-th section:

∂T∗si

∂t∗ + T∗∇∗2i Tsi − q′′′

s l0�sCpsu0�T0

= 0 (10)

Boundary conditions that couple fluid and solid energy equa-ions for the i-th section:

∂T∗si

∂y∗ =(

hı

ks

)i

(Tsi − Ti) (11)

here di = hydraulic diameter for the i-th section; ıi = conductionepth for the i-th section, that is, the ratio between the volume ofhe solid and its surface area at the i-th section.

The hydraulic diameter is defined by:

i = 4ai

(�w)i(12)

here (�w)i = wetted perimeter of the i-th section

and Design 240 (2010) 489–499 493

The conduction thickness is defined by:

ıi = asi

(�w)i(13)

where asi = straight transversal section of the solid of the i-th sec-tion.

The hydraulic diameter and conduction thickness are relatedthrough the expression:

di = 4ıiai

asi(14)

When the system of dimensionless equations is observed fromEqs. (7) to (11), the similarity dimensionless groups or numbers(coefficients of those equation terms) defined below, turn up. Theclose relationship between similarity and dimensional analysisshould be enhanced, because the last was responsible for estab-lishing the dimensionless numbers which represent the physicalphenomena to be simulated.

Richardson number

R = gˇ�T0l0u2

0

(15)

Friction number

Fi =(

fl

d+ K

)i

(16)

Stanton modified number

St =(

4hl0�Cpu0d

)i

(17)

Time ratio number

T∗i =

(˛s

ı2· l0

u0

)(18)

Biot number

Bi =(

h · ı

Ks

)i

(19)

Heat source number

Qsi = q′′′s l0

�Cpu0�To(20)

Solving the system of dimensionless conservation equations forthe steady state, the temperature difference between the entranceand the exit of the core, and the flow velocity in the core, respec-tively �T0 and u0 are determined.

�T0 = q′′′0 l0

�Cpu0.(

as0

a0

)(21)

u0 =[

ˇ · (q′′′0 l0/�Cp) · lh · (as0/a0)

1/2g ·∑

iFi/a2i

]1/3

(22)

In addition to the previous development, the coupling of tem-perature and velocity distributions can be confirmed from the twoabove equations (Eqs. (20) and (21)), performing the followingmanipulation:

u30 =

[ˇ · (q′′′

0 l0/�Cp) · lh · (as0/a0)

1/2g ·∑

iFi/a2i

]⇒ u2

0

=[

(ˇ · (q′′′0 l0/�Cp) · lh · (as0/a0))/u0

1/2g · ∑iFi/a2i

]

The u20 expression may be rewritten as follows:

u20 =

[ˇ · (q

′′′0 l0/�Cpu0) · lh · (as0/a0)

1/2g · ∑iFi/a2i

](23)

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94 J.J. da Cunha et al. / Nuclear Engin

The numerator of Eq. (23) may be substituted according to Eq.21), leading to:

20 =

[ˇ · �T0 · lh

1/2g · ∑iFi/a2i

]

0 =[

ˇ · �T0 · lh1/2g ·

∑iFi/a2

i

]1/2

(24)

The complexity of the similarity analysis for natural circulationystems is due to the coupling that exists between the dynamic sys-ems (gravitational force and momentum) and the thermodynamicystems (heat transfer processes), which prevents solving the flowelocity scaling independently of the temperature distribution, ashown by Eqs. (21) and (24).

Based on the concepts of the physical meaning of each of theimensionless groups, during their analysis and selection, Biot,ime ratio and heat source numbers are concluded to be irrelevantor the problem at hand, due to the following reasons:

Biot number – this group relates resistance to thermal conduc-tance inside the solid with heat convection resistance through theboundary layer. There is no need to consider it, since there is nosolid fuel producing heat in the experiment;Time ratio number – since this group provides a measure of therelative effectiveness of conduction in solid and stores thermalenergy, its use is also irrelevant for the same reason mentionedfor Biot number;Heat source number – this group relates the heat generation ratein the solid to the heat transfer rate by conduction. Since thereis no solid fuel in the experiment, this relation has no meaningfor this problem. Thus, numbers mentioned below are the onesrepresentative for the phenomenon to be simulated:• Richardson number• Friction number• Modified Stanton number

In addition to those physical similarity groups, the followingeometrical similarity groups must be considered.

Thermal length scale:

h = lhl0

(25)

Flow area scale:

i = ai

a0(26)

Conduction thickness:

i = di

4

(as

a

)i

(27)

Thus, the experiment is characterized by a set of six dimension-ess numbers or groups.

The determination of the best combinations of design param-ters and operational conditions constitutes a series of objectiveso be maximized or minimized, since a trade-off exists and tryingo improve one or more objectives may cause the deterioration ofthers.

.3. Calculation of dimensionless numbers

This step will be detailed along with the description of the cal-ulations.

and Design 240 (2010) 489–499

4.4. The formulation of the optimization problem

The design of a thermo-hydraulic experiment, similar to a natu-ral scale one, is characterized by determining the best combinationsof design parameters and operating, construction and safety con-ditions, within practical constraints, leading to a set dimensionlessreference numbers that are as close as possible to the respec-tive dimensionless numbers of the real system (prototype). Theseobjectives (several dimensionless numbers to be obtained), as evi-denced in Cunha et al. (2007), compete with each other, because ofthe interdependence between the dimensionless numbers and thestrong non-linearity of the problem. Therefore, trying to improvesome number may cause deterioration of others. This fact indi-cates the need for a balance between these dimensionless numbers,in order to obtain the best combinations of structural parametersand operating conditions that lead to an optimal solution of theproblem.

Thus, the goal is to get values or optimal solutions, i.e. a maxi-mum or minimum of a function that represents the phenomenonbeing studied in a search space, penalizing candidate solutions thatviolate safety conditions or lead to undesirable designs.

Mathematically, the problem in question is to approach thedimensionless numbers using a weighted least squares process,which can be formalized as follows.

Let Gi be the dimensionless groups in the scale system and Gi

be the dimensionless groups in the design scale. Consider also pi

and pi the structural and operational parameters that belong tothe dimensionless groups for the prototype and small-scale systemrespectively. Then the problem can be stated as:

min f [Gi(si), Gi(si)] =

√√√√ Ng∑i=1

wi

(Gi − Gi

Gi

)2

(28)

Subjected to:

s1 mín ≤ s1 < s1 máxs2 mín ≤ s2 < s2 máx

......

...sM mín ≤ sM < sM máx

where Ng is the total number of significant dimensionless groups;Gi the expected value of a particular group dimensionless i to pro-totype expected value of a given dimensionless group i, obtainedwith the operating conditions of the prototype; G the value corre-

Fig. 2. Geometric representation of the optimization problem.

eering and Design 240 (2010) 489–499 495

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Restrictions: solutions with boiling fluid conditions are penal-zed because they do not represent the physical situation to beimulated.

Geometrically, the problem can be interpreted in two dimen-ions, i.e., two design parameters (two search variables), as follows.

It can be inferred from Fig. 2, that optimization is to obtain, in theearch space, a certain value X(s1, s2) that minimizes the distance detween the reference value and the value suggested by the searchonducted by the PSO algorithm.

.5. Definition of search variables of the problem

This step will be detailed along with the description of the cal-ulations.

.6. Design of a test section

This step will be detailed along with the description of the cal-ulations.

.7. Using the PSO method to solve the problem

This step will be detailed along with the description of the cal-ulations.

.8. Methodology Performance and Validation Test

The objective of this case study is to test the performance ofhe proposed method, in a problem usually difficult to be solvedy conventional optimization methods, aiming at determining itsdequacy.

This problem presents good similarity with the original problemo be solved; differing only in the number of search variables, theange of physical properties and the search intervals.

From the geometric data and operational test and the descrip-ion of the LOFT (Reeder, 1978) and specified in Table 1, wealculated the dimensionless numbers representing the prototypehat will serve as reference for subsequent searches of step 3 of the

ethodology (calculation of dimensionless numbers).The problem thought to test PSO performance and consistency

nvolves the search for the best possible design, considering searchntervals for the project variables that contain the real operationalalues of the reference system (LOFT). Thus, the searches are ini-ially thought to systematically converge to the search space wherehe optimum solution is located, that is, to the proper system in realcale (prototype).

Considering that a schematic square array of rods was adopted,f the number of complete flow channels in each direction is N,hen the total number of rods is NV = (N + 1)2. If the contributionf incomplete flow channels is taken into account, the total numberf flow channels will also be Nc = (N + 1)2, equal to 36 (362 = 1296),

able 1perational and geometric data of (LOFT) test facility.

Operational and geometric data Specification

w0 Mass flow ratel0 Rod active heightq0 Rod powerp0 Operating pressure�h Wetted perimeter�q Heated perimeterA Flow areaT0 Inlet temperature�0 Density at the inlet�0 Viscosity at the inletk0 Thermal conductivity at the inletCp Specific heat at constant pressure

Fig. 3. Cross section of the square array of thermal rods with three complete chan-nels.

which allows the best possible approximation to 1300, the numberof rods of the original facility (Fig. 3).

Considering a square array, where p is the pitch, d is the roddiameter, the geometrical data of the test section are defined below.

Channel flow area:

ai = p2 − d2

4(29)

Total flow area:

a0 = (N + 1)2 · ai (30)

Wetted perimeter:

�w = 4(N + 1)p + (N + 1)2d (31)

Heated perimeter:

�h = (N + 1)2d (32)

Considering the model previously described, under natural cir-culation regime, Table 2 shows the physical variables chosen asparameters or search variables that are directly or indirectly suf-ficient to calculate all dimensionless numbers. This procedurecharacterizes step 5 (definition of the search variables of the prob-lem).

From the definition of values specified for the search parametersor LOFT test facility, listed in the second column of Table 3, thevalues of the reference dimensionless numbers that represent the

flow conditions as a function of those parameters were determined.This procedure characterizes step 3.

Aiming to find the best PSO configuration, several configura-tions were tested, combining multiple conditions and values fortheir parameters. At least 138 simulations were performed in each

Table 2Search parameters.

Search parameters Specification

�H (m) Height between thermal centersd (m) Rod outer diameterq0 Rod powerp0 Operation pressurel0 Rod active length

496 J.J. da Cunha et al. / Nuclear Engineering

Table 3Design parameters and dimensionless numbers of the standardproject.

Search parameters Specification

�H (m) 4.6725d (m) 0.0107p0 (Mpa) 15.5q0 (w) 390l0 (m) 1.68Richardson (E−2) 2.08103Stanton (E−2) 8.50364Friction (E+2) 3.51633

os

ee

aa

cb

ua

gt

ap

TS

Thermal length ratio 2.778125Conduction depth ratio 1.60613Flow area ratio 1.0000

f these configurations and PSO started up from different randomeeds for each one, that is, from different initial conditions.

As previously mentioned, the condition supplied so that PSO doach simulation search, established search intervals that consid-red the previously known values of the global best.

Those search intervals were obtained by taking a variation ofbout fifty percent around the best global values of each variables,s shown in Table 4.

About 2000 tests were performed, proposing to PSO differentonfigurations for the search parameters, aiming at obtaining theest results in the search process.

At the end of each process, a statistical analysis was conducted,sing a 2.5% (more rigid) or 5% (more flexible) tolerance criteriaround each value of the real design parameters.

Thus, if all the dimensionless group values of the solutions sug-ested by PSO were included in these phenotypic proximity criteria,hen the solution would be admitted to be around the global best.

Five conditions and configurations, with different initial char-cteristics, were tested. Table 5 presents each condition of the PSOarameters used in each of the five configurations.

Description of table parameters is presented below:

Objective PSO – this parameter has values 0 or 1, dependingwhether local maximum or minimum is desired;PSO c1 constant – coefficient of individual learning of the particles;PSO c2 constant – coefficient of collective learning;Number of cycles – total number of particle generations;Number of iterations – number of evaluations in each cycle;Stop condition (% Num Iter.) – iterations stop after this number ofiterations, if no evolution is achieved;Number of particles – population size.Initial and final weights – inertia factors of the particle, w, varyingfrom 0.8 to 0.1;Multip factor (Dw = 1, 2, 3, 4, 5) – acts as an angular coefficientfor the line joining initial and final weights, allowing a balancebetween local and global searches;

Particle maximum velocity ([0] = self) – design criterion estab-lished by the algorithm for the maximum velocity of particle;Number of dimensions (20 max.) – defines the dimension of thesearch space by the number of search variables.

able 4earch interval of the design parameters.

Search variables Search Interval

Minimum Maximum

�H 2.4650 7.3950d 0.05350 0.016050p0 11.0000 17.0000q0 195.000 585.0000l0 0.8400 2.5200

and Design 240 (2010) 489–499

Several simulations were performed with different randomseeds, based on the information in Table 5 and configuration 1attained the global best in 30.44% and 20.29% of the time, with5% and 2.5% tolerance criteria respectively, while configuration 2attained global best in 29.71% and 10.15% of the time. The valuesobtained 3 were 26.09% and 13.04% for configuration, 26.19% and11.91% for configuration 4 and 26.87% and 14.93% for configuration5.

From 138 simulations performed with configuration 1 that hasproved to be the most adequate for this kind of problems, the 3best results, represented by the smallest values of the fitness, wereselected. The last line of Table 6 lists fitness values and presents acomparison between the dimensionless values calculated from thedesign parameters, for each one of the fitness, with the dimension-less numbers of the reference system (prototype). This procedurecharacterizes step 7 (use of the PSO method to solve the problem).

From the best simulations, selected from the best tested config-urations under several operational conditions and comparing thedimensionless numbers found with the design standard dimen-sionless numbers, that is, with the global best, a general view ofthe method behavior can be obtained and also preliminary conclu-sions can be taken on the applicability of PSO to thermal-hydraulicsystem design with the characteristics of the studied problem.

Although the values of the project parameters are transparentfor PSO, a brief analysis of them in the three simulations presentedin Table 6 is interesting. The distance between the thermal centersshowed regularity among simulations 1, 2 and 3, with a relativeerror corresponding to 0.0542%, 0.0526% and 0.0548%, respectively.

The fuel diameter deviated just 0.112% in simulation 3, while insimulations 1 and 2 deviation obtained were 0.0654% and 0.0936%,respectively. The pressure attained the following deviation in rela-tion to the reference: 0.179%, 0.251% and 0.300%. Thermal powershowed a little more irregular behavior than pressure, results forsimulations 1 and 2 were 0.202% and 0.285%. However, the figurefor simulation 3 was 0.340%. Finally, the active height of the rodpresented a high regularity, with very low deviations of 0.0007%,0.0001% and 0.00013% for simulations 1, 2 and 3, respectively.

The main focus of the simulation analysis should be directed tothe dimensionless numbers. Since they are representative of thethermo-hydraulic experiments obtained in the three simulations,they indicate how accurately the simulated phenomena correspondto the phenomena occurring in the prototype. Considering simu-lations 1–3, the deviations for Richardson number were 0.148%,0.153% and 0.153%. As for Stanton number, they were 0.005%, for allthree simulations. The Friction number presented 0.0003%, 0.0009%and 0.0009% errors. The three simulations presented deviations of0.0536% for the thermal length ratio number. The deviation indi-cated for the conduction depth ratio Number was 0.006% only forsimulation 3. There was no percent deviation for the flow area rationumber. Finally, Table 6 shows that the fitness function attainederrors of 10−7 order in simulation 1 and 10−6 in simulations 2 and3, which may be considered excellent, if the high degree of com-plexity due to the multimodal nature and non-linearity conditionsof the search space are taken into account.

The fact that PSO attained virtually identical values as the realdesign parameters, based on LOFT test facility, in more than 30%of the simulations should be enhanced. Thus, the methodologyattained the global best, which in this case was deliberately known,or produced a result close to it. The solutions obtained in othersimulations were not considered in the neighborhood of the globalbest, according to the adopted tolerance criteria. However, all of

them showed fitnesses on the order of 10−5, maximum. Then, evenif these candidates were not phenotypic identical to the prototype,for all search parameters, fitnesses were extremely low.

These results provide indications of the multimodal nature ofthe search space, since phenotypic candidates, different from the

J.J. da Cunha et al. / Nuclear Engineering and Design 240 (2010) 489–499 497

Table 5PSO operational conditions of the best configurations.

Operational conditions Configuration

1 2 3 4 5

Objective PSO [0] = min or [1] = max 0 0 0 0 0PSO c1 (constant) 3.0 2.5 1.4 1.5 4.0PSO c2 (constant) 1.0 1.5 1.4 1.5 1.0Number of cycles (max. = 100). 100 100 100 100 100Number of iterations 2500 2500 2500 2500 2500Stop condition (% Num Iter.) 12 12 12 12 12Number of particles (max = 500) 500 500 500 500 500Initial weight (particles) 0.8 0.8 0.8 0.8 0.8Final weight (particles) 0.1 0.1 0.1 0.1 0.1Multip factor. (Dw = 1, 2, 3, 4, 5) 2 2 2 2 2Particle maximum velocity ([0] = self) 0.9 0.9 0.9 0.9 0.9Number of dimensions (20 max.) 5 5 5 5 5

Table 6Design parameters and expected dimensionless numbers (known global best) and its respective values for 3 simulations selected among those that better approximates theglobal best.

Search parameters and dimensionless numbers Global best Simulation

1 2 3

�H (m) 4.67250 4.66997 4.67005 4.66994d (m) 0.010700 0.010707 0.010690 0.010712p0 (Mpa) 15.50000 15.47228 15.53900 15.45344q0 (w) 390.000 389.209 391.114 388.673l0 (m) 1.680000 1.679988 1.680017 1.679979Richardson (E−2) 2.08103 2.08411 2.08421 2.08421Stanton (E−2) 8.50364 8.50410 8.50402 8.50402Friction (E+2) 3.51633 3.51634 3.51636 3.51636Thermal length ratio 2.78125 2.77976 2.77976 2.77976

utatio

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a

as number of sets of thermal rods (NAQ), and the number thatindirectly calculates the lattice pitch (DELTAx) and a temperaturerelated number at the core inlet, NSUB. This number (NSUB) iscalculated at a given saturation pressure, down to an inlet temper-ature corresponding to about 290 ◦C. This was necessary in order to

Table 7Search variables and search intervals for reduced scale experiments.

Search variables Search interval

Minimum Maximum

�H 0.5 4.0d 0.00535 0.01605NAQ 1 16

Conduction depth ratio 1.60613Flow area ratio 1.00000Fitness Zero (considering comp

lobal best, presented excellent fitnesses. From the technologicaloint of view, simulations that converged to local bests, due to opti-um fitnesses, would be considered excellent simulations of the

hysical phenomena occurring in the real system.A slight flexibility on the tolerance criteria of what was admit-

ed as being in the neighborhood of the global best, would lead tomuch larger percent of simulations that would consistently be

cceptable.It is worth noting that, these criteria should be established con-

idering not only precision and accuracy of the measurements to bebtained in the future test section, but considering also the uncer-ainties and inaccuracies typical of a mechanical mock up.

The analysis of the three previous conclusions: good index forlobal optimum determination; excellent fitnesses, even for solu-ions that converged to local maxima, and the fact that a slightexibility on tolerance criteria would certainly conduct to increase

n the number of designs in the neighborhood of the global opti-um, allows us to conclude that PSO presents good performance

n this kind of problems. This conclusion leads us to investigate PSOerformance for the problem of designing a reduced scale experi-ent.

.9. Reduced scale design

Although the case study previously presented, rigorously speak-ng, differs from the problem under study, both in the number

f dimensions and in the range of search of the design variables,ts results showed that PSO is adequate to consistently find goodolutions for this kind of problem.

So, we were encouraged to investigate the PSO performance inreduced scale experiment.

1.60613 1.60613 1.606141.00000 1.00000 1.00000

nal precision) 9.91090E−7 13.0740E−7 14.0175E−7

Thus, these results were used to support that would follow theresearch directions on the performance of PSO for experimentalscale.

All simulations had economical, physical and operational con-straints, which influenced the search intervals for the followingdesign variables:

• Number of thermal rod arrays;• Rod thermal power;• Operational pressure.

Table 7 establishes new conditions for the search intervals nec-essary to conduct the reduced scale experiments, according to thementioned constraints and introduces new search variables such

NSUB 128 255p0 1 4�x 0.0018 0.0054q0 10 200l0 0.5 1.5

498 J.J. da Cunha et al. / Nuclear Engineering and Design 240 (2010) 489–499

Table 8Design parameters and dimensionless numbers of the best reduced scale cases.

Search parameters and dimensionless numbers Global best Case

1 2 3 4 5

�H 4.67250 4.00000 4.00000 4.00000 3.99984 4.00000d 0.01070 0.01598 0.01580 0.01555 0.015850 0.01594NAQ 36 15 15 15 15 15NSUB 214 204 210 206 215 201p0 15.5 2.169 2.003 2.144 1.794 2.311�x (E−03) 3.6 2.2231 2.1352 2.0566 2.1040 2.2472q0 390 163.764 134.42 138.662 112.223 182.391l0 1.68 1.50 1.50 1.50 1.50 1.50Richardson (E−2) 2.08103 2.08420 2.08423 2.08423 2.08420 2.08422Stanton (E−2) 8.50364 8.50402 8.50403 8.50398 8.50403 8.50399FRICTION (E+2) 3.51633 3.51633 3.52958 3.53050 3.53022 3.52791Thermal length ratio 2.78125 2.77976 2.66667 2.66667 2.66656 2.66667Conduction depth 1.60613 1.37573 1.37602 1.37558 1.37597 1.37514

1.00002.1362

cts

stdb

spfi

4

bip2at(

P

stwitoa

pwiri

tln

t

Flow area ratio 1.00000Fitness (E−1) Zero

ircumvent the interdependence between pressure and saturationemperature. This procedure characterizes step 5 (definition of theearch variables).

The constraints and the increase in search space dimensionhould naturally pose a greater degree of difficulty for the methodo adjust design parameters and also to obtain the best adjustedimensionless numbers that represent the physical phenomenaeing simulated.

Thus, in this case study, the main objective is to obtain dimen-ionless numbers and groups which are the closest to the valuesreviously indicated in Table 3, with no reference to the expectedtness reference numbers.

.10. Results

With the information of configuration 1, which presented theest performance in the preliminary case study, and the search

ntervals established in Table 7, a total of 700 simulations wereerformed, where the fitness values presented a mean value of.17129 × 10−1 and a standard deviation of 5.00603 × 10−3. A quicknalysis of the data variability around the mean indicates a rela-ive dispersion, represented by the Pearson correlation coefficientPCC), of 2.31%. The Pearson correlation coefficient is defined as:

CC (%) =

x× 100 (33)

This shows the homogeneity of results obtained with reducedcale designs, PSO obtained fitness values of 2.13 × 10−1, 5.29% ofhe time, corresponding to 37 results of all simulations. There is noay to formally prove that these are global optima, but consider-

ng the extremely low phenotypic dispersion among the 37 results,hese are certainly bests results to be considered and the best resultbtained in 700 PSO simulations. The best five results were selectedmong them, and are listed in Table 8.

In this case study, there is no sense in comparing the designarameters with those of the prototype, since the search intervalsere strongly restricted to simulate real constraints of econom-

cal, constructive or operational character, that would occur in aeal engineering design of a reduced scale thermo-hydraulic exper-ment.

Considering similarity theory, those constraints are supposed

o impede the optimization problem, making impossible the abso-ute adjustment between the obtained and expected dimensionlessumbers.

In short, the search in this case is for a combination of parame-ers which originate a set of design dimensionless numbers which

0 1.00000 1.00000 1.00000 1.000003 2.13644 2.13655 2.13664 2.13673

approximate as much as possible (according the metrics estab-lished in Eq. (31)) the prototype numbers.

The analysis of the best obtained cases shows that the relativeerror for the Richardson number were 0.1521%, 0.1535%, 0.1535%,0.1521% and 0.1531% respectively, for experiments 1, 2, 3, 4 and5, evidencing an optimal regularity in the representation of thisphysical phenomenon.

The Stanton number showed identical regularity. Its relativeerrors were 0.0045%, 0.0046%, 0.0040%, 0.0046%, and 0.0041% forthe respective experiments.

The Friction number made the adjustment in the first case andshowed large regularity, with relative error of 0.0085%, 0.3669%,0.3929%, 0.3850% and 0.3197%, considering the sequence of exper-iments 1, 2, 3, 4 and 5.

The variations exhibited by the thermal length ratio numberwere 0.0536% for case 1, 4.2968% for cases 2 and 3, 4.3015% for 4and 4.2967% for case 5.

The conduction depth ratio number accused percent correla-tions of 16.7444%, 16.7223%, 16.7603%, 16.7271% and 16.7976%,respectively.

Although the relative error of the other physical phenomenawere larger, an excellent regularity was attained in the representa-tion of this physical phenomenon, probably indicating a consistentsearch of the method towards the global best.

Finally, the flow area ratio number was not difficult to beadjusted and it did not present significant errors.

PSO performance along the search is shown in simulation 1, withconvergence attained after 752 generations. Since it did not evolveto a different value after 300 consecutive generations, the searchwas interrupted (stop criterion established for all simulations).

Considering not only the practical experiences with reducedscale systems previously constructed (Botelho and Faccini, 2002),but also the objectives of the fitness function (dimensionlessnumbers), the analysis of simulations showed that PSO attainedsatisfactory results, from the point of view of an engineering designand also because of the consistency and low variability of theobtained results

5. Final evaluations and conclusions

The investigations conducted in the present work showed that

the methodology proposed for reduced scale thermo-hydraulicdesigns, to simulate natural circulation regime is viable.

This methodology offers the designers the support for decisionmaking in design stages, allowing to obtain more efficient and eco-nomic designs.

eering

spepit

ott3gns

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itdt

Ransom, V.H., Wang, W., Ishii, M., 1998. Use of an ideal scaled model for scalingevaluation. Nuclear Engineering and Design 186 (1–2), 135–148.

J.J. da Cunha et al. / Nuclear Engin

The first case study, which was formed by a set of 138imulations of thermo-hydraulic experiments, indicated that theresented methodology was able to find combinations of param-ters that lead to a large percent of solutions with fitnessesractically equal to zero (always smaller than 10−5). This is an

mportant indication of the consistency of the method, its abilityo systematically obtain solutions having excellent fitnesses.

In addition, its robustness, that is, the capacity of the methodol-gy to consistently attain phenotypic similar solutions, was showno be reasonable enough, particularly when the characteristics ofhe optimization problem are considered. The solutions found in0% of the simulations were in the neighborhood of the knownlobal best. These two analyses points out the potential of the tech-ique, but also the difficulty to work in a multimodal and equimodalearch space.

Considering PSO good performance in the test problem (firstase study) and the small dispersion obtained among the results inhe reduced scale problem (second case study) the results of theecond case study may be supposed to be among the best possi-le combinations of parameters (good designs, similar in scale),onsidering the severe constraints imposed on the limits of theearch variables. In a real engineering problem, when designingreduced scale thermo-hydraulic system, those limits would have

o be defined as a function of the pre-existent economical, physicalnd operational constraints.

Besides the academic importance of this work, better simulationxperiments could have been obtained, had experimental devices,uch as reduction valves, mixers, etc. been used. Those devicesould be added to the physical modeling of the loop and wouldave allowed a larger degree of freedom during the dimensionalnalysis of the problem, thus favoring the PSO search. So, even bet-er results could have been obtained with the insertion of theseevices.

The placement of a valve, for example, would have been verymportant to correct the friction factor and consequently the Fric-ion number. This procedure would not undermine the otherimensionless numbers of the experiment, but would have reducedhe interdependency of some dimensionless groups.

and Design 240 (2010) 489–499 499

Acknowledgements

The authors would like to thank Fundacão de Amparo à Pesquisado Estado do Rio de Janeiro - FAPERJ and Conselho Nacional deDesenvolvimento Científico e Tecnológico – CNPq for their supportto this work.

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