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Forschungszentrum Karlsruhe holtzGemeinschaft in der Helm Wissenschaftliche Berichte FZKA 7520
Fluid Dynamic Design of Complex Mixing Chambers
Alexander Wank
ut für Kern und Energietechnik Instit
schinenbau der Universität Karlsruhe (TH) Von der Fakultät für Ma
igte Dissertation genehm
bH, Karlsruhe Forschungszentrum Karlsruhe Gm
2009
Für diesen Bericht behalten wir uns alle Rechte vor
ForschunPostfach 364gszentrum Ka0, 76021 Karlrlsruhe Gmbsruhe H
Mitglied der HerDeutscher Fomann von Helmhrschungszentren (oHGFltzGemeinschaft )
ISSN 09478 620
075206 e:0005urn:nbn:d
Fluid Dynamic Design of Complex Mixing Chambers
Zur Erlangung des akademischen Grades eines
Doktors der Ingenieurwissenschaften
hinenbau der Universität Karlsruhe (TH) der Fakultät für Masc
igte genehm
Dissertation
von
Dipl. Ing. Alexander Wank
aus Reutlingen
Prüfung: 12.10.2009 Tag der mündlichen
Vorsitzender: Prof.
Hauptreferent: Prof.
Korreferent:
.Ing. H.J. Bauer Dr
Dr.Ing. T. Schulenberg
H. Oertel . Dr.Ing. habilo. Prof. Prof. e.h.
Vorwort:
Die vorliegende Arbeit entstand während meiner Tätigkeit als wissenschaftlicher Mitarbeiter
am Institut für Kern und Energietechnik (IKET) des Forschungszentrums Karlsruhe GmbH.
Instituts für Kern und Energietechnik (IKET) Herrn Prof. Dr.Ing. T. Schulenberg, Leiter des des Forschungszentrums Karlsruhe GmbH und Hauptreferent dieser Arbeit gilt mein herz
e stets vorhandene Unterstützung und Diskussilichster Dank. Sein persönlicher Einsatz, di Maß zumrordentlichemBetreuung haben in außeonsbereitschaft sowie die sehr persönliche Gelingen dieser Arbeit beigetragen.
Dem Leiter des Instituts für Strömungslehre (ISL) der Universität Karlsruhe (TH), Herrn Prof.
referats dieser Arbeit, sein e des Kosonders für die ÜbernahmDr.Ing. H. Oertel danke ich begroßes Interesse und die verschiedenen anregenden Diskussionen.
Für die fachliche Unterstützung sowie die persönliche Förderung danke ich meinem Gruppen
leiter Dr.Ing. J. Starflinger. Außerdem möchte ich mich bei den Mitarbeitern des Instituts,
Class und Herrn Dr.Ing. Grötzbach für die wertbesonders bei Herrn Prof. Dr.Ing. habil. A. rte und die fachliche Betreuung bedanken. vollen Ratschläge, die ermutigenden Wo
ereitschaft, das her ich für ihre Hilfsbarbeitern dankeAllen Kollegen, Studien und Diplomvorragende Arbeitsklima und ihren Einsatz. Besonderer Dank gilt Herrn Dr.Ing. B. Vogt,
Dipl.Ing. C. Bruzzese ng. C. Pfeifer und Herrn g. K. Fischer sowie Herrn Dipl.IHerrn Dr.Inien Diskussionen. Speziell danke ich auch mfür ihre Unterstützung und die vielen anregendeten Freund Herrn Dipl.Ing. C. Haasi. langjährigen Mitbewohner, Kollegen und gunem
Ganz besonders herzlich danke ich meiner Familie für ihre uneingeschränkte Unterstützung,
Hilfe und Motivation während meiner gesamten Ausbildungszeit sowie meinen ehemaligen
ste Version probegelesen haben. Bob Decker, die geduldig die er Carol undGasteltern Frau Tati für die tatkräftige Unterstützung, inere Herzen mSchließlich danke ich von ganzemebe. iarbeit in allen Bereichen und ihre Ldie gute Team
Karlsruhe, im Oktober 2009
Alexander Wank
Kurzfassung:
Strömungsmechanische Auslegung komplexer Mischkammern
Ein Ansatz für die Simulation von komplexen Mischkammern mit vergleichbar geringem
numerischem Aufwand ist ein interessantes Werkzeug für Auslegungsprozesse von Misch
kammern sowie von andern komplexen Geometrien in Strömungen. Vor allem in Designpro
zessen müssen oft mehrere Simulationen einer Geometrie durchgeführt werden um optimale
nd Methoden, die zwar alle Einflüsse einer Grund siAnordnungen zu entwickeln. Aus diesemkomplexen Geometrie auf die Strömung berücksichtigen, den numerischen Aufwand aber
drastisch reduzieren von großem Interesse. In dieser Arbeit wird ein Ansatz vorgestellt, in
nicht detailliert aufgelöst wird, ihr Einfluss etrie ung beeinflussende Geom die, die Strömdemauf die Strömung aber erhalten bleibt. In der so genannten Vereinfachungsmethode kommen
Quellterme in den Strömungsmechanischen Gleichungen zum Einsatz. Diese Quellterme rep
zutage standardisierten CFD ente in heutetrieelemtigten Geomräsentieren die nicht berücksichSimulationen. Als Anwendung für die Vereinfachungsmethode wurde die obere Mischkam
mer des High Performance Light Water Reactor (HPLWR) ausgewählt. Anhand der komple
eitet. In der oberen ndete Methode ausgearbetrie der Mischkammer wurde die verwexen GeomMischkammer werden verschiedene Einlassströme unterschiedlicher Temperatur vermischt.
Ziel ist dabei eine gute Homogenisierung der Temperatur um eine Weitergabe heißer Sträh
eiden. Eine weitere Herausforderung bei der rmnen in eine folgende Aufheizungsstufe zu veBerechnung der Vermischung resultiert daraus, dass überkritisches Wasser nahe dem Pseudo
ischt werden soll. Die besonders starken Änderungen in den relevanten unkts vermkritischen PStoffeigenschaften müssen in das Modell integriert werden um mögliche temperaturabhängi
ge Effekte, wie z.B. Auftrieb zu berücksichtigen. Beide Herausforderungen, das Abbilden der
komplexen Geometrie sowie die Berücksichtigung temperaturabhängiger Effekte werden un
thode auf die obere en der Vereinfachungsmabhängig voneinander angegangen. Das Anwendengsstruku vereinfachten Modell, das jedoch die globale StrömMischkammer führt zu einemtur in der Mischkammer wiedergibt. Das vereinfachte Modell wird dann optimiert und erst im
Anschluss werden die Vereinfachungen mit Quelltermen ins Modell integriert. Zur Bestim
pulsme wurden einfache Handrechnungen auf Basis des I Quelltermmung der erforderlichenlationen verwendet. Die erhaltenen Ergebnisse wurden dann satzes sowie Druckverlustkorremit einer Simulation eines Detailausschnitts der oberen Mischkammer, in dem alle geometri
peraturabhängiger Für die Berücksichtigung temschen Details aufgelöst wurden, validiert. Effekte wurden Funktionen für die relevanten Stoffeigenschaften in das Modell aufgenom
men. Obwohl Auftriebseffekte keine dominante Rolle spielen, hat die Berücksichtigung der
variablen Stoffeigenschaften doch einen Einfluss auf die Strömung und die Vermischung.
Gesamtergebnisse, die sowohl die abgebildete komplexe Geometrie als auch temperaturab
iertes Design renzdesign sowie für ein optimhängige Effekte enthalten wurden für das Refe. terzeug
wurden, sind grundsätzlich für jede andere Die vorgestellten Methoden, die für die OpMischkammer oder komtimierung der oberen Mischkammplexe Geomer verwendet etrie in einer
Strömnwendbar. ung a
i
Abstract:
Complex Mixing Chambers Fluid Dynamic Design of
introduAn approach to simced, which mulate comakes this appplex mroiach an inxing chamberste with compresting tool for design processes of arably little numerical efmfioxing rt is
chambers and also to some extent for other systems with complex geometries. Especially in
traints are necessary to find consesimulations under the samdesign processes, where various optimal configurations, methods reducing the numerical effort, while still capturing the domi
thod has been developed, with e this work, a mnant influences on the flow are interesting. Inwhich complex geometrical structures in a flow are not resolved in detail, while still including
ing the flowtheir effects. This socalled sim and thus accounts for disregarded geplification metomhod uses source termetric details in state of the art CFD sims in the equations describu
milations. As an exemxing chamber of the High Performplary application of thance Light We simplification mater Reacetor (HPLWRthod, the very complex upper ) has been used and
the method has been developed with the given geometry of this rather complex mixing cham
ber. In the upper mixing chamber, different inlet flows with different temperatures are mixed
to achieve a good temperature homogenization of all inlet flows. Thus, the propagation of hot
mix water, at supercritical conditionstreaks to a following heatup section is avoides close to thd. The fact that the upper me pseudo critical point leads to an additionixing chamber shall al
ties. Thchallenge, wey hhave to bich is the ce inonsideratroducedtion of into the app the verylied m large gradients thodels to be able to analyzeat occur in the fluid proper temperature
ometry and the introduction of thdependant effects, such as buoyancy. Both che temperature dependant effectallenges, the reproduction of the complex ges into the model are tackled
independently. Applying the simplification method to the upper mixing chamber leads to a
much more simplified model, which to some extent still resembles the global flow structure in
the mixing chamber. This simplified model is then optimized, before the effects of the simpli
calculations based on the integralfications are introduced by source term balance of ms. To deriomentum as well ve the necessary source termas on pressure drop correlas, simple hand
tions are used. The obtained results are validated with simulations of a detailed cutout model
tempof the upper mierature dependant effects inxing chamber, in which all geom the flow, functions describing the stronetric entities are included. To account for g changes in the dif
odel. Even though buoyancy effects do not play mferent fluid properties are included into the a dominant role, temperature dependant effects still have some visible influences on the flow
field in the upper mixing chamber and therefore on the mixing. Integrated results containing
nd the consideration of temperature dependant etry alex geompboth, the influences of the comeffects are presented for the reference case and the optimized case of the upper mixing cham
ber.
The mother meixing chamthods to optimber or flow inize the upper mi or around complex geomxing chambeetries. r should be generally applicable to any
ii
CONTENTSE OF TABL
1 .........................................................................................................................1 Introduction1.1 Fluid Mixing in Complex Systems Concepts and Modeling......................................1
1.2 High Performance Light Water Reactor HPLWR.......................................................3
4 Three Pass Core Design..................................................................................................1.3 11 ........................................................................................The Upper Mixing Chamber1.4 13 ........................................................................................The Lower Mixing Chamber1.5 1.6 Aim and Outline of this Work......................................................................................14
16 .........................................................................................................Equations2 Governing 2.1 Characterization of the Flow in the Upper Mixing Chamber.......................................16
19 ................................................................................................Equations2.2 Conservation 24 ....................................................................Characterization of Buoyancy Influences2.3 26 ows............................................................................................................lF2.4 Turbulent 36 ..........................................................................................................Strategy2.5 Analysis 38 .................................................. Complex Mixing ChambersMethod for the Analysis of3 38 ..................................................................................................Methodplification 3.1 Sim41 ..........................................................................................................Modelplified 3.2 Sim48 .....................................................................................................................3.3 Validation55 ......................................................................Verification Grid Sensitivity Analysis3.4 4 Design optimization using the simplified model..............................................................57
57 .......................................................................................................Headpieces4.1 Turned 4.3 4.2 Outlets Shielded fromIn and Outlets at Different Heights Inlet Side...............................................................................................................................................................60 59
62 ........................................................Collection and Redistribution of the Inlet Flows4.4 4.5 Collection and Redistribution of the Inlet Flows Meander Alignment.....................63
64 ...................................................................Evaluation of the Different Modifications4.6 5 Including the Effects of the Omitted Structures................................................................67
5.1 Detailed Analysis of the Flow in the Headpiece Structures.........................................67
76 of the Headpiece Influences.....................................................................Introduction5.2 6 6.1 Analysis of TemCharacteristic Flow Patternsperature Depending Effects.............................................................................................................................................................97 94
6.2 Case with a Specified Inlet Temperature Distribution...............................................102
7 Integrated Results for the Upper Mixing Chamber.........................................................105
111 ..............................................................................................ary and ConclusionsSumm8 Nomenclature.........................................................................................................................113
117 .........................................................................................................................Abbreviations118 ..............................................................................................................................ReferencesAnnex A Lower Mixing Chamber......................................................................................125
Annex B Dimensions of the Upper Mixing Chamber and Headpiece Geometry...............129
iii
TABLE OF CONTENTS  DETAILED
1 .........................................................................................................................1 Introduction1.1 Fluid Mixing in Complex Systems Concepts and Modeling......................................1
1.2 High Performance Light Water Reactor HPLWR.......................................................3
4 Three Pass Core Design..................................................................................................1.3 11 ........................................................................................The Upper Mixing Chamber1.4 13 ........................................................................................The Lower Mixing Chamber1.5 1.6 Aim and Outline of this Work......................................................................................14
16 .........................................................................................................Equations2 Governing 2.1 Characterization of the Flow in the Upper Mixing Chamber.......................................16
2.2.1 2.2 Conservation Conservation of MassEquations............................................................................................................................................................................................19 19
2.2.2 Conservation of Momentum..................................................................................19
22 .........................................................................................Conservation of Energy2.2.3 24 ....................................................................Characterization of Buoyancy Influences2.3 26 ows............................................................................................................lF2.4 Turbulent 2.4.1 Reynolds Equations for Turbulent Flows..............................................................26
2.4.3 2.4.2 Equations for Turbulent FlEnergy Equation for Turbulent Flowsows with Variable Density............................................................................................................29 28
2.4.4.1 2.4.4 Turbulence Eddy Viscosity ApproachModeling.............................................................................................................................................................................31 30
33 ...............................................................................Applied Boundary Conditions2.4.5 34 Boundary..............................................................................................etry 2.4.6 Symm2.4.7 Numerical Methods...............................................................................................34
36 ..........................................................................................................Strategy2.5 Analysis 38 .................................................. Complex Mixing ChambersMethod for the Analysis of3 3.1 Sim3.2 Simplified plification ModelMethod............................................................................................................................................................................................................41 38
3.2.1 Numerical Model for the Simplified Geometry....................................................41
44 ..............................................................Reference Case of the Simplified Model3.2.2 48 .....................................................................................................................3.3 Validation55 ......................................................................Verification Grid Sensitivity Analysis3.4 4 Design optimization using the simplified model..............................................................57
57 .......................................................................................................Headpieces4.1 Turned 59 .................................................................................. Inlet SideOutlets Shielded from4.2 60 .............................................................................In and Outlets at Different Heights4.3 62 ........................................................Collection and Redistribution of the Inlet Flows4.4 4.5 Collection and Redistribution of the Inlet Flows Meander Alignment.....................63
64 ...................................................................Evaluation of the Different Modifications4.6 5 Including the Effects of the Omitted Structures................................................................67
5.1 Detailed Analysis of the Flow in the Headpiece Structures.........................................67
5.1.1 Detailed Headpiece Model....................................................................................67
5.1.2 Verification Grid Sensitivity Analysis of the Detailed Headpiece Model.........70
5.1.3 Simplified Headpiece Model.................................................................................72
iv
5.1.4 Comparison of the Detailed and Simplified Headpiece Model.............................73
76 of the Headpiece Influences.....................................................................Introduction5.2 76 .......................................................................................Insertion of Local Forces5.2.1 5.2.2 5.2.3 Insertion of Global ForcesInsertion of Global Forces Accelerating the Flow.....................................................................................................................................85 82
93 ..............................troduced Headpiece InfluencesConclusions Regarding the In5.2.4
94 ....................................................................perature Depending EffectsAnalysis of Tem6 97 .........................................................................................Characteristic Flow Patterns6.1 6.1.1 Constant Volume Flow..........................................................................................98
100 ............................................................................................Constant Mass Flow6.1.2 6.2 Case with a Specified Inlet Temperature Distribution...............................................102
7 Integrated Results for the Upper Mixing Chamber.........................................................105
111 ..............................................................................................ary and ConclusionsSumm8
Nomenclature.........................................................................................................................113
117 .........................................................................................................................Abbreviations
118 ..............................................................................................................................References
Annex A
Annex B
mLower Mixing Cha125 ......................................................................................ber
Dimensions of the Upper Mixing Chamber and Headpiece Geometry...............129
v
oduction Intr 1
nIntroductio
Complex mixing chambers with different functions are found in many industries. Examples
include the mixing of chemical components in the chemical or petrol industry, processes in
the metallurgy, and many others. In the sector of power production, mixing to reduce tem
eponents and surfaces somcient cooling of comperature peaks in flows, e.g. to ensure suffitimes also requires very complex mixing chambers.
compIn this work, a mlex upper miexing chamthod to optimize the mber of the High Peixing of rformance Light Wawater at supercritical pressure in the very ter Reactor (HPLWR),
d. The basic principle is developed and appliewhich is a new concept of light water reactors, of this meternal flows. In the upper mthod should be generally applicableixing chamber, diffe for comprent inlet flowlex mis with different temxing chambers or complex inperatures
are mixed to achieve a good tempropagation of hot streaks to the following hperature homogenization of all inlet eatup section is avoided. This tempflows and thus the erature ho
ing chammogenizatiober an imn is a crucial aspect of the feasibportant part within the reactor design. ility of the HPLWR, which makes the upper mix
Two main aspects are discussed and ways to model them are presented in this work. The chal
and their influences on the flow and thereplex structureslenge of reproducing the very comfore on the mixing is the leading subject. Also, it is expected that the presented solution of
this problem will find broader application in different fields. Another challenge is the consid
eration of the distinctive aspects occurring when analyzing water at supercritical pressures.
Here, very large gradients occur in the fluid properties that have to be introduced into the ap
odels. plied m
1.1Fluid Mixing in Complex Systems Concepts and Modeling
Mixing systems can be classified in different ways. According to the book of Kraume [38]
and the book of Schubert [67], mixers can be divided in continuous and discontinuous mixing
systems. While in continuous mixers a constant stream from the inlet to the outlet side is
found, discontinuous mixers are loaded with batches. Here the mixing process can be divided
into different steps like filling, mixing, and emptying. Another classification for mixing appli
cations is the division in static and dynamic mixers. While either internals or the whole cham
ber is moving in dynamic mixers, no moving parts are found in static or motionless mixers
and the mixing is a result of the energy introduced into the system by the flow field. All static
mixers are by definition continuous mixers. Here the flow usually passes through the mixing
volume and the mixing is enhanced with included passive / non moving internals. Also, mixer
systems can be characterized by the components which have to be mixed. Mixer systems dif
fer strongly weather solid substances or fluids are mixed. In this work, only static mixers for
one single phase fluid are analyzed.
For static or motionless mixers, different methods of operation can be found. The ones work
ing with, what sometimes is referred to as free turbulence, consist of alignments where jets
interact with each other or a global swirl is imposed. A typical alignment is presented in the
1
Introductio n
catalog by Pfaudler in [59]. Other mixers mainly use the effects of flow separation and
me passive ny of the saaeatures include mchanges in the flow direction. Common design felements introduced into the flow one after the other to change the flow direction significantly
in each step. An overview of different motionless mixers is delivered in [56]. The evaluation
of the achieved mixing quality in a mixture is often described with the statistical deviation of
samples around a mean value as described in detail in the book of Oldshue [55].
Difble smfiall miculties arise in the designing anxing systems or mixers with a limd dimensioning ofited comp complexity, numlex miericaxer systeml simulations. For comsp are theara
state of the art in industrial [1]. Essential advances in the simulation of stdesign processes as described inatic mi the paper by Arimxers are presented by Hobbs and Muzzio ond and Ervin
in [28] and another, more recent example for the numerical simulation of the flow in mixing
applications is given by Visser et al. in [80]. Here it is shown that the periodically repeating
structure of the mixer is used and only a small section of the complex geometry is modeled,
approaches using the periodically repeating ilar lifying the task significantly. Simpthus simstructure of the analyzed mixing systems are found in [37] and a more recent analysis in [61].
results to experimeGood results have been obtained by Bertolottonts carried out for a double Tjunction in et al., who have comp[6]. In their setup, it can be seen ared their simulation
that already for a small and rather simple geometry, a very large number of numerical meshes
promare necessary and therefore the numising approach that delivers good results,erical effort is very high. Anothe while reducing the numer recently published, rical effort, is pre
all structures only been successful for sm[70]. However, all presented studies have sented in or structures for which it is possible to extract a small, representative geometry.
On the other hand, for many large and highly complex mixers, numerical simulations often
play an imlead to a very largportane numt role, especially inerical effort and ar design pre not feasibocesses. Therefore, thle within satisfactore design of my time limaitsny mi, which xing
systems in industrial use today is based on empirical data as described for example by [59], in
type. An overview of the applwhich the design process of the Pfaudler Wication and the design erke Gmprocess of static mbH is briefly summarized for one mixers is given in i[57]. xer
Here, also the emplarily. The interest in efficipirical design process and the dient simulation mefferent desigthods describing comn criteria areplex mi summxing elemed up exements or 
[10] and Pfaudler just any complex flow structures w[58]). Especially intereas also affirmsting are med by the industry (amethods applicable in design processes, ong others by Bürkert
where many simulations are necessary to develop optimized structures.
In this work, an approach to simulate complex mixing chambers with comparably little nu
merical effort is introduced, which makes this approach an useful tool for design processes of
miapplication of the mxing chambers and also to somethod, the upper mei extent othexing chamr systems with comber of the HPLWR, whplex flow geometries. As ich is explained in
ity ofdetail in the f the mioxing chamllowing chapter, hber at all times is cas been defriucneial, therefd. Being a noreu moving parts are noclear reactor systemt perm, the reliabilitted.
This avoids additional maintenance and control of the system. The upper mixing chamber,
which is explained in more detail in chapter 1.4, is an example for a very complex static mix
ber. ing cham
2
1.2 High Performance Light Water Reactor HPLWR
nIntroductio
The High Performcooled and moderated with supercance Light Writical steater Reactor HPLWam. It is R currently developed asis a new concept of light water reacto one of the Generars,
tion IV advanced nuclear systems according to the generation IV technology roadmap [78].
(LWRThe idea was to develop) concepts, but is mo a systemre efficient and le, which is baads to further imsed on well establishedprove Light Wment in an economater Reactoical r
ter cooled reactors with supercritical prespoint of view. Oka has presented a review of wain sures in [16]. An improved core design has b[54]. A first conceptual design of such a een presented by Yamaji et al. in reactor has been proposed by Dobas[87] and in [88]. hi et al.
the use of a single phase fluid in the core, R concept are vantages of the HPLWThe key adby the long termwhich avoids steam experience of supercritical stea separators and dryers, a direct once through steamm cycles for fossil fired power plants. Also, a cycle, and the support
arized by Squarer et al in enabled, as summal efficiency of around 44% shall be high therm[72].
HPLWR, can be reduced significanThe capital costs of a SuperCritical Lightly in comt parWater Reactor (SCWR)ison with todays nuclear power plants. Rea system, such as the
a smsons for the comaller pressure vesspetitive economel leading to a smical advantage oallef r containmthese systems are the vent and therefore a smery compaller reactor act reactor,
steambuilding. Other com generators can be omitted inponents used in todays L this design. BittermWRs, like steamann et al. [7] estim separators, steamated the very low dryers, and
construction costs in the vicinity of 1000 per kWe and also expected very low electricity
production costs in the range of 3 to 4 cents per kWh.
ter shall enter the core at a itions of around 25 MPa, liquid waAt supercritical pressure condtemHPLWR steamperature of 280°C and exit as superhea cycle as proposed by Bittermted steamann et al. in of around 500°C. For com[8] and a reference LWR steamparison, the cy
temcle are shown in the temperatperature in the reactor of the HPLWR stureentropy (Ts) diagramays above the twophase in Fig. 11. It can be seen that the region throughout the
than for the LWentire heat up and the area suR reference design, explrrounded by the curve describing thaining the higher thermae steaml efficiency. cycle is much larger
temUsing superperatures. Imcritical steam in the reaportant aspects are the testing ctor leads toand developm challenges due to the very high pressures and ent of eligible materials, e.g.
eat transfer deterioration under well as the investigation of hconcerning corrosion behavior, as the HPLWR steamsupercritical conditions. In addition cycle have to be designed a a plant dend dimesign has to be developednsioned, safety system, components withins have to be
planed and analyzed, and a reactor pressure vessel with all its internals has to be designed.
[69], theribed different core design concepts in While Schulenberg and Starflinger have descmore detail, since all analdesign of the core for the HPLWR yses presented in this wpresented by Fischer et al. in ork are based on this design. [20] shall be explained in
3
nIntroductio
HPRLWRLW
linpseuedocritical
aaPaPM258MP.7MPa
M31
aPM500.0
Fig. 11 The HPLWR steam cycle and a reference LWR steam cycle in the Ts diagram
Water above the critical pressure is called supercritical. It is a single phase fluid with liquid
like properties at low temperatures and steam like properties at high temperatures. The critical
point of water is found at the critical temperature TC=374°C and the critical pres
surepC=22.1MPa. A pseudocritical line can be defined by the peak values of the specific
ssure. Along this line, the gradients of the points, at a given preheat, called the pseudocritical tivity and density are very strong. fluid properties, the viscosity, conduc
miThe mxing chamber of the HPLWR is presented inodeling of the changes in the fluid prope chapter rties due to tem6 (pAnalysis of Temperature changes in the upper erature Depend
ore on the mixing of such effects ow and therefing Effects). Here, also the influences on the flare discussed.
Three Pass Core Design 1.3
ost 2000kJ/kg, thus ex core inlet to outlet is almThe enthalpy rise of the cooling water fromceeding the enthalpyrise leads to concerns for the hot rise of pressutest srized water ubchannel in the core.reactors by a factor of 8. It is assumed that the worst cas This large enthalpy e
hot channel experiences a largcalled hot channel factors, as described by Strauß in er enthalpy rise than a nom[73], containing certaiinal subchannel due to several, so n statistical uncer
Schulenberg et al. tainties instead of absolute to[68] were assuming the follolerances. First predictions withwing hot channel factors: thermalhydraulic analyses by
4
nIntroductio
1) A radial form factor, considering differences in the power distribution and in the mass
flux distribution in the core.
2) each fuel assemA local peaking factor, regardibly cluster. ng similar differences not in the entire core, but within
due to differences in channel factors arise The differences taken into account by these two hotthe fuel composition and distribution, in the wbution of subchannels, in neutron leakage, and atedifferences due to burn r density distribution, in the size and distriup effects. They also
take into account differences in the positions of the control rods and differences arising due to
. Additional factors are: the usage of burnable poisons
s regarding the core design. ng all statistical uncertaintieAn uncertainty factor includi 3)
rties, neutron physical terials, fluid propeang mThis factor includes all uncertainties concerniinlet temand thermperature distribution, asohydraulical modeling, ma well as measuremnufacturing tolerances, deforent uncertainmties of the installed mations, differences in the easure
nt systems. em
4) control, as well as smA factor regards concessions in the planall transients during operation. t operation, like power, flow, and pressure
l factor of 2.0 is obtained. Theultiplied, a total hot channee mIf these hot channel factors artotal hot channel factor should be multiplied with the enthalpy rise of a nominal subchannel
to mtoday suggest that a model the hottest subchaaximum cladding temnnel under worst case perconditions. Cladding mature of 620°C should not be exceeded. Conaterials available
sidering this hot channel factor and the maximum cladding temperature, the high core outlet
mtemepdiate mierature can only be achieved wxing to avoid hot streaks. ith a stepwise heat up, e.g. in three stages, and with inter
Each of the three heating stages of the coolant includes 52 assembly clusters. In the central
part of the core, the first stage is situated, the socalled evaporator, which has upward flow.
Another 52 assembly clusters with downward flow acting as a first superheater are arranged
around the evaporator. The second and final superheater, again with upward flow, also made
of 52 assembly clusters is situated at the outer periphery of the core. This way, superheater II,
the periphery of the core, where the neutron eratures, is atpwhich has the highest coolant temflux and therefore the pin power are lowest. Between the evaporator and the superheater I, the
flow is mixed in the upper mixing chamber and between the two superheaters, in the lower
mixing chamber. The concept of this three pass core and the actual cluster arrangement in the
Fig. 12. cted in i the core are depcross section of a quarter of
235with energyCertain fuels in nuclear reactors, lik below 1keV to sustaine uranium the chain r with a certain ameaction, since the fissionount of 92U probability for these, require neutrons
l neutrons, because their kinetic acalled thermneutrons is much higher. These neutrons are energy is in the range of the energy due to the thermal molecular motion within the material
which mof the reactor. The neutrons produced by fissieans that these fast neutrons have to on have an average kibe decelerated for the continuous chain reaction netic energy of 2MeV,
5
nIntroductio
in the reactor as explained by Smidt in [71] and in the books of Todreas [75] and [76]. The
deceleration is accomplished with socalled moderators, in which the neutron dispenses its
which is desienergy due to elastic collisiongned as a thermal reactor, water shs with the nucleuses of the mould be used as moderator moderator, which haterial. In thae HPLWs the adR, 
density is reduced with increasing temperature. vantage that the chain reaction decreases if the This leads to additional safety, since the core switches off automatically if the temperature
s too high. ebecom
C0°50
C5°43
upper mixing chamber
°039C
evaporator
r IteahesuperpusIr Ieteahre
310°C435°C
435°C310°C435°Ceevavappororatatoror
ssuperheuperheaater Iter I
lower mixing chamberssuuppeerhrheaeatterer I III
with three different heat up steps as proConcept of the HPLWR three pass core Fig. 12 ][68posed in
In the HPLWR, the density ratio between the core outlet steam and the inlet feed water is very
tion in the core. Therefore, which leads to an uneven moderaing a factor of eight, high exceedHofmeister et al. proposed a fuel element design in [30] and [31] according to first analyses of
Cheng et al. [15] in which the moderator water should flow downwards through moderator
blies, while the coolant flows upwards in mboxes and through the gaps between the fuel assethe subchannhave to be enclosed by a fuel assembly box aels, which are the various channels between the fuel rods. The fuel assemnd in the center of the assemblies, a moderator blies
bly box of the HPLWR hasof the fuel assemechanical analysis box has to be introduced. A mbeen carried out by Himmel et al. in [26]. Another aspect, when designing fuel elements is the
neutron absorption by structure material is in the core. Hofmeister et al. [30] identified that
small fuel assemblies need less structural material than large ones, which leads to a design
lies have been designed, because Cheng et al. bents. Square fuel assemall fuel elemwith sm[15] expected that they have a more uniform heat up than hexagonal arrangements.
Since the selected fuel assembly only has an outer width of 70mm, it is not possible to equip
each fuel assembly with a control rod. Moreover, because so many small fuel assemblies are
ing, and shuffling. To le during revision, refuel very difficult to handenecessary, they becom
6
nIntroductio
overcomand housed fuel assemble these difficulties, a fuel elemies in a 3x3 arrangement clusent, with 40 fuelter has been proposed consisting of nine sm rods and a moderator rod each. all
Fig. 13. t out of one cluster are depicted in ents and a cuThe design of the fuel elem
d rolfue
subchannels
moderator box
fuel assembly box
moderator gaps
moderator and gap water
tancool
Fig. 13 Fuel elem ent of the HPLWR [31]
thus to control the power output and to shut dowEach cluster has a common headpiece and footpiecn the reactor, control rods, me. To actively control the reactivity andade out of neu
by Hofmtron absorbing meister et al. a[31] with a comterials, are used. A cross shapmon drive from the top into 5 of the 9 med design, the control rod spider, is proposed oderator boxes of
access throuone cluster. For the cogh the headrner mpiece geometry is possoderator boxes, no control rodsible. The headpiece and are foreseen, since no straight footpiece of a fuel
element clubeen proposed by Hister are depimmel et al. in cted in [25]. WiFig. 14. As spacers between the re wraps around each fuel rofuel rods, wire wraps have d provide good mix
xing between the subchannels of a iand on the ming of the coolant. Their effect on the flow fuel assembly has also been analyzed by Himmel et al. in [24].
The design of the reactor and its internals has been carried out and presented by Fischer et al.
in [19] and [20]. A stress analysis of the reactor pressure vessel (RPV) has been presented by
Fischer et al. in [18]. In Fig. 15, a general overview of the RPV and its internals is given and
rranged in the core. The footpieces are inserted ent clusters are ait is shown how the fuel eleminto the lower plenum and the headpieces are inside the steam plenum above the core. In these
two plena, the steam is collected before it is passed on to the next heat up stage. The steam
plenum is divided into two regions. The upper mixing chamber in the center contains all the
o heat up stages takes xing between these twie mevaporator and superheater I clusters. Here thplace. An outer region, where the coolant is collected before it leaves the core, is built around
this mixing chamber. The lower plenum is also separated in an inner and outer region. In the
7
nIntroductio
inner region, the coolant is mixed before it enters the evaporator, whereas in the outer region
of the lower plenum, the socalled lower mixing chamber, the coolant is mixed between the
two superheaters.
control rod spider
d roroltcon
window element
erator boxdomheadpiece struts
zzle notransitionetom plabottentemel elfu
piecefoot
fuel element
eatler plow
eadpiecehFig. 14 headpiece and footpiece of the HPLWR fuel assembly cluster [19]
ector is introduced to reduce neutron leaklfent clusters, a steel reSurrounding the fuel elemages from the core. This way, a more uniform power distribution is achieved and the RPV is
neutron flux. All internals in the RPV are contained and fix aging due to high protected fromated in the core barrel. The core barrel is a cylindrical structure, which hangs in the RPV to
allow thermal expansion due to different temperatures of the different parts in the RPV. The
d guiding tubes are placed above om the top, therefore control rocontrol rods enter the core frontrol rods/the control rod spiders for each containing and guiding the c plenumthe steamcluster. On the top of the RPV, the vessel is closed with the closure head. Two redundant o
of the vessel. On top of the vessel the penes to assure the leaktightneseals are foreseenring strations for the control rods need to be foreseen. The height of the RPV is determined by the
active length of the core, which is 4.2m (including fission gas plena, the fuel rods have a
amlength of 4.7meter is found, when adding up the diam) and the height of the extended eter of control rods on top. Its smthe core, the thickness of the steel reflector allest possible di
and the core barrel, the downcomer, and the smallest possible thickness of the RPV. Thus, the
height of the RPV is 14.3m and the inner diameter of the vessel is 4.47m with a maximum
of the HPLWR three pass core is thoroughly shell thickness of 0.56m. The detailed design of the HPLWR, the coolant [19]. To understand the concept explained by Fischer et al. in flow pass will be explained in more detail.
8
closure head
reactor pressure vessel
steam outlet
steam plenum
core barrel
steel reflector
nIntroductio
control rod guiding tubes
t inletncoola
fuel element cluster
loweumnr ple
Fig. 15 The HPLWR Core and its internals according to [19] and [20]
The coolant entering the core is split up; 25% of the cooling water entering the reactor pres
oderator rods inside the lies and to mbe assemsure vessel shall be supplied to gaps between thassemblies. The flow pass of this gap and moderator water is indicated on the left side in Fig.
vessel, and reaches the core from the top. T16. It flows upwards in the pressure vessel, pahus, the region in the RPV assing the closure head of the reactor pressurebove the steam ple
num is entirely filled with the gap and moderator water. From here, 2/3 of this flow is lead
lies as gap bbe kept outside the fuel assem, to through connection tubes in the steam plenumwater. The remaining moderator water flows downwards in the moderator boxes inside the
fuel assemblies. The lower plate of each footpiece, which is indicated in Fig. 14, has two
levels. The moderator boxes are extended to the lower level, where the moderator water is
collected and then released to the outside of the footpiece, where it is mixed with the gap wa
holes in the central partter surrounding the assemblies. The com of the lower plenumbined . gap and moderator water is then led through
in the gap beThe other 75% of the cooling watween the RPV and the coreter entering bar therel core,and en the doters the cenwncomtral per water,art of th flow doe lower plewnwards 
num from below via holes. The flow pass of this downcomer water is shown in the center of
Fig. 16.
9
Introductio n
, which is indicated with the black rectangle xing chamber of the lower plenumiIn the inner min Fig. 16, the gap and moderator water is mixed with the downcomer water. At the core
inlet, above the central part of the lower mixing plenum, the temperature of the mixed coolant
is expected to be around 310°C. The mixing of the moderator water by jets of the downcomer
[29] and appeared to be very eister et al. in alyzed by Hofmas been anwater in the footpieces heffective.
pass core is depicted. Each heat up Fig. 16, the flow pass inside the three On the right side of zone, the evaporator and the two superheaters are indicated with only one fuel element cluster
exemplarily. The colors of the arrows indicating the flow direction stand for the temperature
oderator water, as well ure of the gap and mperatof the coolant in a qualitative way. The temas of the downcomer water is close to the core inlet temperature. In the evaporator and the
two superheaters, the coolant is heated up significantly before it exits the core with a core
tely 500°C. aperature of approximoutlet tem
Coolant flow pass in the HPLWR Fig. 16 left: gap and moderator water, center: downcomer water right: heat up in the three pass core
The coolant enters the reactor through four inlets circumferentially positioned at the cylindri
cal part of the vessel, well above the four outlets. Inlets and outlets are positioned every 90°,
while thhigh tempe inletserature gr have a 45° offadients have to be avoided. set to the outlets. To avoid high thermTo achieve this, the insiade surface ol stresses in the RPV, f the RPV
the inlet temperature of the core. Therefore, should only be in contact with coolant close to coaxial pipes. The downcomthe hot steam outlet pipes are surrounded by a thermer water flows around al sleeve, realized withthe hot outlet pipe, shielding the RPV fro an alignment of to m
[22]. alyzed by Foulon et al. in es as anrthe high temperatu
and [20]. All analyses presented in this work are based on the design proposed by Fischer et al. in [19]
10
The Upper Mixing Chamber 1.4
nIntroductio
The inner part of the steam plenum is called the upper mixing chamber. Considering the flow
perheapass ofter I. It is sepa the coolant in thrated fre reacomtor, it is loca the outer part btey a wall welded into the plenumd above the core between the evaporator and su. The steam
plenumcluded, the upper m with the upper miixing chamxiber ing chamber is ds an exameplpicted in e for a static miFig. 17. Since no mxing chamber with the aimoving parts are in to
achieve good mixing in order to accomplish homogenization of the temperature.
opening for coolant exit
upper mixing chamber
connection tube
zoneexit
separation wall
opening for headpiece
m m048
m m9653Fig. 17 HPLWR Steam plenum upper mixing chamber [19]
The outer part has a rather simpercritical coolant is collected before leaviple geometry wng the reactor thith only few obstacles in the flow. Herough the connected steam outlet re the su
in and outlets to the upper mpipes. The inner part has a very complex geomixing chamber. Inetry due to the ma addition to the headpieces, a high numny headpieces formbing the er of
transporting the gap water to ber, in the upper mixing chamconnections tubes are encounteredwith indications of the cthe gaps between the fuel assemoolant flow and of bly clusters. Athe flow of the gap a cross section of the upper mnd mioderator water through xing chamber
Again, one exemthe connection tubes and the mplary cluster is shown oderator boxesrepresenting each heat up stage. within the headpieces is shown in Fig. 18.
plenum, thus separating the inner and outer region, All headpieces are captured by the steamwhile miheater II to the reactoxing the flow comri outlets. The diamng from the evaporateter of the steam plenumor and guiding the flow com is 3965mm, alming fromost 4m super; its
height is 480mmof each connection tube is 82mm. , while the outer diameter of one headpiece is 218mm and the outer diameter
11
nIntroductio
Fig. 18 Coolant flow in / moderator and gap water flow through the upper mixing chamber
The complexity of the upper mixing chamber becomes more evident when looking at Fig. 19.
ent clusters. The ith all introduced fuel elem plenum is shown wHere, a cut through the steamspace in the upper mixing chamber is filled with either connection tubes or headpieces with
their struts and moderator boxes to a significant amount. The mixing of the flows coming
ount ters is strongly influenced by the large ament clus the different evaporator fuel elemfromof obstacles in the flow field. Challenges arise when modeling the flow in the mixing chamber
due to this complicated structure and due to the fact that the mean coolant temperature in the
upper mixing chamber is 390°C, thus being close to the pseudocritical point. High gradients
of the fluid properties are expected.
12
Fig. 19 Cut through the steam plenum with introduced fuel element clusters [19]
er Mixing Chamber The Low 1.5
nIntroductio
The lower mixing chamber is the outside ring of the mixing plenum. Unlike the upper mixing
chamchamber, which is aber can be characterized as rather full s antru emcturepty volum (connectione with no flow obstacles tubes, headpieces) th inside to infe lower mliuencexing
mixing. In ber and the lower mixing chamFig. 110 the lower plenum with its two mber between superhixing chameater I and II is depicted. The footpieces obers, the inner mixing chamf
the fuel element clusters are inThus, the core support plate, which is supported troduced in the by the core barrel, is carrying the wforeseen openings of the core support plate. eight of
sters. the clu
core support plate
reirlsw
separation wall
lower mixing chamber
opening for footpiece
inlet gap and moderator water
inlets downcomer water
inner mixing chamber
Fig. 110 HPLWR lower plenum lower mixing chamber [19]
Sufficient mixing in the lower mixing chamber is accomplished with socalled swirlers, indi
cated in Fig. 110. These swirlers are welded to the bottom of the core support plate at the
tacles are found in the lower r I footpieces. Since no flow obsoutlet openings of the superheatemiof the lower mixing chamber, a global swirl is created, exxing chamber. The analysis of the mtending the miixing in the lower chamxing length between in and outlet ber with and with
n that a global swirl [85]. It has been showesented by Wank et al. in out swirlers has been praccording tocan be created leading to good m the introduced classiixing. Thus, thfication as a static me lower miixer usxing chaming free turbulencber can be characterized e. A more
detailed analysis of the mixing in the lower mixing chamber is presented in Annex A.
13
nIntroductio
Aim and Outline of this Work 1.6
The aim of this work is to develop a method for the fluid dynamic design of complex mixing
chambers, using the example of the complex upper mixing chamber of the HPLWR. When
ber, two challenges are encountered: xing chamianalyzing the flow in the upper m
1) stacles, suchThe consideration of the com as headpieces and connection tubes plex structure with the very high number of complex ob
2) let tempThe consideration of the very large gradients eratures near the pseudocritical point in the upper min the fluid propertiesixing cham due to different inber as a result
of differences in the heat up of the evaporator clusters
Since more than a single analyslations need to be carried out. Therefore the nis is necessary for a design optimeed for an analysis method arization, many different simises, using Compuu
tational Fluid Dynamwhile leading to results icscapturing the dom (CFD), which requireinating s only little comeffects of the mixing. puting and preparation time
xiplex upper m to develop a design of the comUsing such analyses, the technical objective ising chamture and thus to avoid the propaber that ensures good mixing of the diffegation of hot streaks into thrent inlet flow to home superheater I fuel elemogenize the tement cluperas
outlinters. In ed. Fig. 111 the process of the analysis and optimization of the upper mixing chamber is
upper mixing chamber
¾complex geometry
¾temperature close to pseudocritical point
include temperature
dependant effects
)(chapter 6
optimize complex
mixi(chapter ng ge3, 4om, 5)etry
integrated results for the optimized upper mixing chamber
)(chapter 7 Fig. 111 Process of the analysis and optimization of the upper mixing chamber
ent steps. TThe challenges for the analysis which are explhus, the task at hand is split up in several independent mained above are tackled independently in differodules. In chapter 3, 4,
miand xing chamber is carried out. 5 the complex structures are included in thIn chapter 6 the teme simuperature dependlation and a design optimant effects are handled,ization of the
7. fferent aspects is presented in chapter before an integrated result including all di
14
e thesis h tfOutline o
Introductio n
hnical tasks as well as the HPLWR Introduction), the scientific and tecIn the first chapter (necessary mixing chamhave been described. The focus lies hereby on thbers. The technical designe fl of the HPLWow pass in the thrR is presented, leading to the ee pass core and on the
ber. xing chamidesign of the upper m
chamChapter 2 (ber in the context of fluid dynamGoverning Equations) gives an overview ofics. The equations describing the flow are derived. the analyzed flow in the upper mixing
is introduced. Also, the applied turbulence models are explained and the evaluation strategy for the mixing
In chapter 3 (Method for the Analysis of Complex Mixing Chambers), the method for the
analysis is explained. It is shown how it is intended to deal with complex structures in mixing
chambers. In addition, the numerical model is described and a verification of the numerical
the applied CFD model is ion oftso the Validamodel, a grid sensitivity analysis, is shown. Alpresented. Therefore, an experiment of a similar mixing chamber is described and compared
settings. elations for the samuwith the carried out sim
the OmIn chapters 4 (itted StructurDesign optimes), the actual dization using the esign optimisimplified mzation is carodel) and 5 (ried out and the approacIncluding the Effects of h, intro
duced in chapter 3 is applied.
The effects on the flow due to the large changes in the fluid properties with temperature are
perature Depending Effects). meAnalysis of Tdiscussed in chapter 6 (
the fluidThe consid properation of both analyerties, combined in one zed aspects,analysis th set ise com shown inplex geom chapteetry and r 7 (the laIntegrargeted Re gradsultsients f inor
the Upper Mixing Chamber). The conclusions are presented in chapter 8 (Summary and Con
clusion).
15
quationsGoverning E
Governing Equations 2
In this chapter, the analyzed problem will be concerted into the context of the field of fluid
mechanics. Therefore, the analyzed flow will be described with characteristic dimensionless
numbers. Then, the governing equations necessary for the performed analyses in this work are
introduced. In the case of the adiabatic approach, these are the mass conservation equation
and the momentum conservation equations. For further analyses, also the energy conservation
gnitude are introduced. at of a scalar mequation and an equation for the transpor
2.1in the Upper Mixing Chamber Characterization of the Flow
Based on a one dimeaverage temperature of 390°C and a pressure ofnsional thermalhydraulic analysis, Schulenberg et al. around 25MPa at the evaporator outlet. The [68] predicted an
bly cluster. The fluid prop22.3kg/s per assemss flow was 1160kg/s, which yields acoolant mof the upper merties of water at this poixing chamber the int were defined accocharacteristic dimrding to ensionless [81]. Due to the cnumompbers, defined by Oertel in licated structure
geom[50], [52] aetrical paramend Schlichting in ters used fo[66] vary strongly accordr their definition are shown in ingFig. 21. to where they are defined. The
A
A
dls
:Across section A
D
Fig. 21 Definition of the lengths for the dimensionless numbers describing the flow
With: D=199.6mm, d=23.4mm, s=26.88mm.
is defined according to: deter The hydraulic diamhdH=4 ⋅A
Pee wetted perimss section area A and thwith the croter P, defined as:
16
quationsGoverning E22A=πD−4⋅πd−5⋅s2=25957.5mm2,
44P=πD+4⋅πd+5⋅4⋅s=1458.7mm.
eter: hydraulic diamThis leads to the AdH=4⋅P=71.2mm.
In Tab. 21, the fluid properties at the evaporator outlet are listed.
Pressure: p=25MPa
Temperature: T=390°C
kgDensity: ρ=215.189m3
Dynamic viscosity: μ=31.704⋅10−6kg
msKinematic viscosity: ν=0.147⋅10−6m2
sJSpecific heat: cp=28461kgK
WConductivity: λ=0.2398mK
Thermal diffusivity: a=0.553⋅10−6m2
sSound velocity: c=403.34m
sVolumetric thermal expansion coefficient: α=0.042K−1
Tab. 21 Fluid properties at the evaporator outlet
The analysis of Schulenberg et al. [68] results in the following average inlet velocity into to
the mixing chamber, defined with the given density, mass flow, and inlet cross section area,
of a headpiece: ection at the inlet which corresponds to the inner cross su=&m=1160kg/s⋅1=3.997m
Inlet velocity: ρA52(215.2kg/m3)⋅(0.2596m2)s
17
quationsGoverning E
duced. The different characteristic dimIn order to characterize the flow, different ensionlecharacterisss numbtic, dimensionless numers describing the flow are defined as: bers are intro
ber: MachnumReynoldsnumber section: ossrlet cat in
Reynoldsnumber r hwith inneeight:
Prandtlnumber:
M=flowvelocityu=0.0099
csoundvelocityRein=inertia=ρudh=udh=1.9⋅106
frictionμν
Rel=inertia=ρul=ul=1.3⋅107
frictionμν
Pr=viscousdiffusivity=ν=3.76
athermaldiffusivity
Pecletnumber: Pe=RePr=convectiveheattransfer=udh=7.3⋅106
adiffusiveheattransferGrashofnumber Gr=buoyancyforce=αgl3(T−T0)=1.1⋅1014
inside the mixing chamber: viscousforceν2
city u and the sound velocity c. In the Reyber is defined with the flow veloThe Machnumnoldsnumber, the kinematic viscosity ν and the hydraulic diameter dh/the inner height l are
used. And in the Prandtlnumber, a is the thermal diffusivity. In the Grashofnumber, α
represents the volumetric thermal expansion coefficient, g the acceleration due to gravity, and
red into the Grashoffor buoyancy effects enteperature. The characteristic length T the temnumber, is the height of the mixing chamber.
mixing chaTwo Reynoldsnummbers are defined: ber and another one with the inner heone with the hydraulic diamight of the mieter xing chamdhber. The at the inlet of very high the
Reynoldsnumber at the inlet of more than 106 implies that the flow is highly turbulent, since
geomthe critical Reynolds number foetry with wire wrap spacers to enhance thr a pipe flow is approxime miaxing between the dtely 2300. Ailfferent subso, the very comchannels of plex
the fuel element clusters, through which the flow has to pass before entering the upper mixing
ber inlet. The other bulent at the mixing chamber, suggests that the flow is highly turchamReynoldsnumterization of buoyancy influencesber, built with the height of the as presented in chapter mi2.3 and applied in chapter xing chamber, is necessary for the charac6.
(Pr=0.7The Prandtlnumb) at a pressure of 1bar and a temer has a value of 3.8. perature of 20°C. Thus, it lies between the one of water (Pr=7) and air
e flow can be regarded as incompressible, ber of 0.0099, thDue to the very low Machnumwhich mvariation with temeans that the dperature. ensity variation with pressure is negligible compared with the density
18
quationsGoverning E
ansfer due to diffusion can be disregarded and ber shows that the heat trThe large Pecletnum has an influence on the analysis strategy exthe convective heat transfer predominates. Thisderived in chapter plained in chapter 2.3 (2.5. A criterion if buoyancy effects haveCharacterization of Buoyancy Influences). The Grashofnum a significant effect on the flow is ber,
of 1014which is defined with the inner height. A discussion, weather buoyancy effects have l of the upper mixing chaman influence on the flow can be found in ber, is very high at a value
6. chapter
2.2Conservation Equations
The governing equations for the mass, momentum, and energy conservation are derived from
an infinitesimal small volume at an arbitrary position within the analyzed flow according to
[43]. All of the conservation equations have change of the conserved quantity within the analthe same structure, where the timyzed volume equals the flow of the conserved e dependant
ported quanquantity in and out of the volumtity. The noe plus corresptation according to the book of Oertel and Laurien onding sources and sinks influencing the trans[43] is generally
used.
2.2.1Conservation of Mass
Applying a mass balance to the analyzed fluid volume in which the time dependant changes
of the mass equal the difference of the mass flow entering and exiting the fluid volume results
in the mass conservation equation:
∂ρ+∂()ρu+∂(ρv)+∂(ρw)=0. (2.1)
∂t∂x∂y∂z
In this equation ρ represents the density and u, v, w are the velocities in the different spatial
directions. The flow in all analyzed applications in this work can be regarded as incompressi
ble and the density therefore is only a function of the temperature (ρ=ρ(T)). In cases,
where temperature dependant effects are disregarded and the density in the flow is constant,
ss conservation equation is simplified to: athe m
∂u+∂v+∂w=0. (2.2)
∂x∂y∂z
Applying vector notation, the continuity equation for flows with constant density is written as:
∇⋅rv=0. (2.3)
Conservation of Momentum 2.2.2
The NavierStokes Equations, which describe the conservation of momentum, can be derived
when looking at the time dependant changes of the momentum in an analyzed fluid volume,
which are the sum of the momentum fluxes in and out of the analyzed fluid volume, the shear
and normal stresses, and the forces acting on the volume. For the change in momentum in x
quation can be derived: direction, the following e
19
quationsGoverning E
∂(ρu)+∂(ρuu)+∂(ρuv)+∂(ρuw)=F+∂τxx+∂τyx+∂τzx. (2.4)
∂t∂x∂y∂zx∂x∂y∂z
The pressure acts as a negative normal stress, thus it can be written as:
τxx+τyy+τzz
(2.5) . =−p3The three normal stressesτxx, τyy, and τzzcan be written as a composite of the pressure p and
the contributions due to the friction of the fluid σxx, σyy, and σzz:
τxx=σxx−p, τyy=σyy−p, τzz=σzz−p.
: (2.6) leads toApplying the relationship found in
(2.6)
τ∂ρ⎛⎜⎜∂u+u∂u+v∂u+w∂u⎞⎟⎟=Fx−∂p+∂σxx+yx+∂τzx, (2.7)
⎝∂t∂x∂y∂z⎠∂x∂x∂y∂z
ρ⎛⎜∂v+u∂v+v∂v+w∂v⎞⎟=F−∂p+∂τxy+∂σyy+∂τzy, (2.8)
⎝⎜∂t∂x∂y∂z⎠⎟y∂y∂x∂y∂z
ρ⎛⎜⎜∂w+u∂w+v∂w+w∂w⎞⎟⎟=Fz−∂p+∂τxz+∂τyz+∂σzz. (2.9)
⎝∂t∂x∂y∂z⎠∂z∂x∂y∂z
Further relations for Newtonian fluids according to [89] lead to the NavierStokes equations
for flows with variable density.
ρ⎛⎝⎜⎜∂∂ut+u∂∂ux+v∂∂uy+w∂∂uz⎞⎠⎟⎟=Fx−∂∂px+∂∂x⎢⎡⎣μ⎛⎝⎜2∂∂ux−23()∇⋅rv⎞⎠⎟⎤⎦⎥
(2.10) ⎡⎤+∂∂y⎣⎢μ⎛⎝⎜⎜∂∂uy+∂∂vx⎞⎠⎟⎟⎦⎥+∂∂z⎡⎣⎢μ⎛⎝⎜∂∂wx+∂∂uz⎞⎠⎟⎤⎦⎥,
ρ⎛⎝⎜⎜∂∂vt+u∂∂vx+v∂∂yv+w∂∂vz⎞⎠⎟⎟=Fy−∂∂py+∂∂y⎡⎣⎢μ⎛⎝⎜⎜2∂∂vy−23()∇⋅vr⎞⎠⎟⎟⎤⎦⎥
(2.11) +∂∂z⎡⎣⎢μ⎛⎝⎜⎜∂∂vz+∂∂wy⎞⎠⎟⎟⎤⎦⎥+∂∂z⎡⎣⎢μ⎛⎝⎜⎜∂∂uy+∂∂vx⎞⎠⎟⎟⎤⎦⎥,
ρ⎛⎝⎜⎜∂∂wt+u∂∂wx+v∂∂wy+w∂∂wz⎞⎠⎟⎟=Fz−∂∂pz+∂∂z⎡⎣⎢μ⎛⎝⎜2∂∂wz−23()∇⋅rv⎞⎠⎟⎤⎦⎥
(2.12) ⎡⎤+∂∂x⎡⎣⎢μ⎛⎝⎜∂∂wx+∂∂uz⎞⎠⎟⎤⎦⎥+∂∂z⎣⎢μ⎛⎝⎜⎜∂∂vz+∂∂wy⎞⎠⎟⎟⎦⎥.
constant density introducing the continuity eqThese can be further simplified for flows with constantuation and assum visciosity can be written asng constant viscosity. The Navi: erStokes equations for constant density and
20
Governing E quations
222ρ⎛⎜⎜∂u+u∂u+v∂u+w∂u⎞⎟⎟=Fx−∂p+μ⎛⎜∂u+∂u+∂u⎞⎟,
⎝∂t∂x∂y∂z⎠∂x⎝⎜∂x2∂y2∂z2⎠⎟
222⎛⎞ρ⎛⎜⎜∂v+u∂v+v∂v+w∂v⎞⎟⎟=Fy−∂p+μ⎜∂v+∂v+∂v⎟,
⎝∂t∂x∂y∂z⎠∂y⎝⎜∂x2∂y2∂z2⎠⎟
222ρ⎛⎜∂w+u∂w+v∂w+w∂w⎞⎟=Fz−∂p+μ⎛⎜∂w+∂w+∂w⎞⎟.
⎝⎜∂t∂x∂y∂z⎠⎟∂z⎝⎜∂x2∂y2∂z2⎠⎟
Using vector notation, these equations can be summed up as:
(2.13)
(2.14)
(2.15)
(2.16)
rρ⎛⎝⎜∂∂vt+()rv⋅∇rv⎞⎠⎟=Fr−∇p+μΔrv. (2.16)
ity equation For flows with constant density(2.3) describe the analy, the NavierStokes equations zed flow. These equations for(2.16) togem a systemther with the continu of four non
boundary conditions as exemlinear, secondorder partial differential equationsplarily described in chapter , which has to be solved2.4.7. The four unknowns are: u, v, for given initial and
w, and p.
If temanalyzedp flowerature dependan is describted by the N effects are regardavier Stoked aes equations according tond the density is not regarded as constant, the (2.10), (2.11), (2.12),
and the continuity equation accordinsary, since the density is also an unknown. g the (2.1). In this case, an additional equation is neces
21
Governing E quations
Conservation of Energy 2.2.3
e conservation of energy is , the energy equation describing thFor flows with constant densityerwise it is necessanot a constitutive equation for thry to close the equation systeme density, but me rely a transpordescribing the flow. The energy equation t equation for the energy; oth
r and kinetic energy in an analyzed fluid vole dependant change of the innedescribes the timume due to the energy fluxes in and out of the volume, the energy fluxes in and out of the
volumand norme by mal stresses, the energy from outside, eans of conduction, the work done on and the work duthe volumee to volum due to pressure forces, shear etric forces.
⎛⎡V2⎤⎞⎛⎛⎡V2⎤⎞⎛⎡V2⎤⎞⎛⎡V2⎤⎞⎞
∂⎝⎜⎜ρ⎣⎢⎢e+2⎦⎥⎥⎠⎟⎟⎜⎜∂⎝⎜⎜ρ⎣⎢⎢e+2⎦⎥⎥u⎠⎟⎟∂⎝⎜⎜ρ⎣⎢⎢e+2⎦⎥⎥v⎠⎟⎟∂⎝⎜⎜ρ⎣⎢⎢e+2⎦⎥⎥w⎠⎟⎟⎟⎟
1424∂t434=−⎜⎜∂x+∂y+∂z⎟⎟
⎜⎟Changeoftotalenergy1⎝⎜424444444444434444444444⎠⎟
Convectiveterms
(2.17)
+⎛⎜∂⎡⎢λ∂T⎥⎤+∂⎡⎢λ∂T⎤⎥+∂⎡⎢λ∂T⎤⎥⎞⎟
⎝⎜1∂x⎣424∂4x⎦44∂4y4⎣∂y434⎦4∂4z⎣44∂z4⎦⎠⎟
Changebymeansofheatconduction(2.17)
⎛⎜−∂()pu+∂()σxxu+∂()τxyv+∂()τxzw⎞⎟+⎛⎜−∂()pv+∂()τyxu+∂()σyyv+∂()τyzw⎞⎟
⎜⎝∂x∂x∂x∂x⎠⎟⎝⎜∂y∂y∂y∂y⎠⎟
++⎛⎜−∂()pw+∂()τzxu+∂()τzyv+∂()σzzw⎞⎟
⎜⎟1⎝424∂4z444∂4z444∂4z444∂4z4⎠43444444444444444
pressure,normal,andshearstressforces
r+F1⋅4vr2+4ρq&3s.
workbyvolumeforcesandoutsideenergy
Here e is the internal energy, the kinetic energy is [ρ V2/2], and λ is the thermal conductivity.
the different spatial e velocity components ingnitude is calculated with thaThe velocity mdirections, as: V2=u2+v2+w2. When excluding the term [e+V2/2] from the term describ
ing the change of total energy and from the convective terms, it is found that one multiplier is
l and shear aand the approach used for the normthe continuity equation, which equals 0. This stresses for Newtonian fluids analogue as for the conservation of momentum, leads to:
22
quationsGoverning E
⎛∂e∂e∂e∂e⎞⎛∂⎡∂T⎤∂⎡∂T⎤∂⎡∂T⎤⎞
ρ⎝⎜⎜∂t+u∂x+v∂y+w∂z⎠⎟⎟=⎜⎜∂x⎣⎢λ∂x⎦⎥+∂y⎣⎢λ∂y⎦⎥+∂z⎣⎢λ∂z⎦⎥⎟⎟
⎝⎠r−p()∇⋅rv+μΦ+F⋅rv+ρ&qs,
: Φwith the dissipation function
⎤222⎡Φ=2⎢⎢⎛⎝⎜∂∂ux⎞⎠⎟+⎛⎝⎜⎜∂∂vy⎞⎠⎟⎟+⎛⎝⎜∂∂wz⎞⎠⎟⎥⎥
⎣22⎦22
+⎛⎝⎜⎜∂∂vx+∂∂uy⎞⎠⎟⎟+⎛⎝⎜⎜∂∂wy+∂∂vz⎞⎠⎟⎟+⎛⎝⎜∂∂uz+∂∂wx⎞⎠⎟−23⎛⎝⎜⎜∂∂ux+∂∂vy+∂∂wz⎞⎠⎟⎟.
Rewriting the energy equation for the enthalpy, defined as:
with:
h=e+p,
ρ
dh=⎛⎝⎜∂∂hT⎞⎠⎟pdT+⎛⎝⎜⎜∂∂hp⎞⎠⎟⎟Vdp, with cp=⎛⎝⎜∂∂hT⎞⎠⎟p,
Assuming that the pressure is constant, the change in enthalpy can be written as:
(2.18)
(2.19)
dh=cpdT.
th the enthalpy, when disregarding radiation, y equation expressed wiThis leads to the energwork imposed by outside forces, and the usually very small dissipation:
⎛∂h∂h∂h∂h⎞∂⎛⎜λ∂h⎞⎟∂⎛⎜λ∂h⎞⎟∂⎛⎜λ∂h⎞⎟
ρ⎝⎜⎜∂t+u∂x+v∂y+w∂z⎠⎟⎟=∂x⎜cp∂x⎟+∂y⎜cp∂y⎟+∂z⎜cp∂z⎟. (2.20)
⎝⎠⎝⎠⎝⎠
For all the introduced conservation equations no restrictions were made. They are valid in
wtonian fluids. ous flows of Neogenegeneral, describing all hom
23
quationsGoverning E
Characterization of Buoyancy Influences 2.3
As the flble. The change in the dow is incompresesible, the chnsity with tempange in deraturensity duee due to therm to chaanges in the pressurl expansion, however, is very e is negligi
Turner in significant. In convective flows this[77] and in the book of Kakac [35]. is the origin of buoyancy effects as described in detail by
The Boussinesq approximation holds for flows, in which the change in density can be disre
garded in all equations except the buoyancy term in the momentum equation. Temperature
dependant effects are only taken into account by this gravity term. This lift term accelerates
the flow in the opposite direction of the gravity:
g(ρ−ρ∞). (2.21)
This termthe volum is approximetric buoyancy forceated in the mρ()Tgom in which entumρ(T) equation in th is linearized as: e direction of gravity describing
ρ()T=ρ∞[1−α(T−T∞)]. (2.22)
Here, ρ∞ is the reference density, α the thermal expansion coefficient, and T∞ a reference
temperature. ρ=ρ()T is only introduced in this term, in all other terms the density change
will be neglected. The Boussinesq approximation also implies that all other fluid properties
tant.are cons
neve a significant effect on the flow, the dimTo obtain a criterion if buoyancy effects hasionless Boussinesq equations are needed. The following dimensionless quantities are intro
: duced to derive them
2xi*=xi, withi=1,2,3; rv*=lrv; T*=T−T∞; p*=()p+ρ∞gx3l. (2.23)
la∞TW−T∞ρ∞ν∞a∞
Whereas the thermal diffusivitya∞, the acceleration due to gravityg, the kinematic viscos
ityν∝, and a characteristic length scale l are defined for the bulk. Introducing these dimen
sionless quantities lead to the dimensionless Boussinesq equations, however formulated for
the temperature in steady state flow (Oertel [52]):
∇⋅rv*=0, (2.24)
0⎛⎞1()rv*⋅∇rv*=−∇p*+Δrv*+RaT*⎜⎜0⎟⎟, (2.25)
Pr⎜⎝1⎠⎟
rv*⋅∇T*=ΔT*. (2.26)
Rais the dimensionless Rayleighnumber, which describes the ratio of the buoyancy acceler
ating the flow against the gravity to the retaining friction. The Rayleighnumber is defined as:
24
3glRa∞=α()T−T∞.
νa∞∞
(2.27)
ers: bnsionless numeIntroducing the following dim
Governing E quations
u∞lν∞u∞lν∞u∝l
Rel,∞=, Pr∞=, Pe∞=Rel,∞Pr∞==, (2.28)
ν∞a∞ν∞a∞a∞
together with another definition for a dimensionless velocity:
rrrrvrlvv′rlrrr
v′=r, v*=∞, v*=v=Re∞Pr∞v′=Pe∞v′, (2.29)
v∞a∞a∞
and then inserting this dimensionless velocity into the Boussinesq equations, the following
equation for the momentum in the direction of gravity is obtained:
Pe2⎛⎜u′∂w′+v′∂w′+w′∂w′⎞⎟=−∂p*+Peμ⎛⎜∂2w′+∂2w′+∂2w′⎞⎟+RaT*. (2.30)
Pr⎝⎜∂x∂y∂z⎠⎟∂z⎝⎜∂x2∂y2∂z2⎠⎟
Dividing equation (2.30) with ()RePe leads to:
⎛⎜u′∂w′+v′∂w′+w′∂w′⎞⎟=+1μ⎛⎜∂2w′+∂2w′+∂2w′⎞⎟−1∂p*+RaT*
⎝⎜1∂42x444∂y4344∂4z⎠⎟1Re42⎝⎜4∂4x244∂y24344∂4z24⎠⎟1RePe424∂z31RePe4243(2.31)
pressurebuoyancyconvectionfriction flows with the coefficient ofificance of buoyancy effects in it can be defined for the signA limthe buoyancy term in (2.31). The introduction of the Grashofnum2ber, defined asGr=Ra/Pr,
leads to the rewritten coefficient in the buoyancy term: Gr/Re.
(2.31) that buoyancy eff equation We conclude fromects are negligible if:
ReGr2<<1. (2.32)
If the changes in the fluid properties are larger, the linear approximation (2.22) will not be
valid and the specific hwell. In such cases, this critereat, viscosity, and thermion can only lead to an approxal conductivityim will vary withate evaluation of buoyancy ef temperature as
fects, since is derived with the simplified Boussinesq approximation.
25
Governing E quations
s Turbulent Flow 2.4
In comthe flow mpaarison with laminar flows, turbulent flowgnitudes leading to additional exchange of momes are characterized byntum local fluctu and of energy as described ations of
aminstance, becomeong others in the book by Pope s turbulent at hi[85]. The sgher velocities, thus atmooth and straight higher Reynolds numlamibnar flow in a pipe, for ers. It is highly
irregular and is full ofthe flow lead to a higher pressure eddy modrop and to better mtions. The velocity fluctuaixing of the flow. tions, which are superimposed on
The transition from laminar to turbulent flows is characterized by the critical Reynolds num
ber Rec for each flow type. The laminar flow is influence by small perturbations, which arent
damped away at high enough Reynolds numbers. If the critical Reynolds number is obtained
in the flow, the laminar flow is superimposed with two dimensional perturbating waves
(TollmienSchlichting waves). Further downstream, three dimensional perturbations lead to
flowso called . The onset ofΛ turbuvortices decaying intolence/the lami lonacal turbulenrturbulent transition t spots, whis described in ich lead to the fu[52] and lly tur[51]. bulent
The most accurate way of simulating turbulent flow is the socalled Direct Numerical Simula
tokes equa[17]. Here, the NavierSe book of Ferziger and Peric tion DNS as described in th. The only errors arise due to tionsaing or the use of approximtions are solved without averagthe numerical discretization, which can be estimated and controlled. The disadvantage of this
approach is the high numerical effort, which makes it not applicable as a design tool. Also,
the very detailed information often exceeds the demand.
An approach in which the large scales, which are in general much more energetic and thus
all scales, usually conlated and the smutransport most of the conserved properties, are simtaining much less energy are modeled, is called Large Eddy Simulation LES. The numerical
effort is decreased significantly in comparison with the DNS, but it is still very high when
thods for fluid dynamics. etional mconsidering the use of computa
Stokes (RANAn effective approach to mS) approach. Here, odel complex turbthe turbulent fluctuations arulent flows is the ReynoldsAveraged Navier e regarded, without resolving
them. All simulations presented in this work use this approach, which is explained in more
. detail below
s Reynolds Equations for Turbulent Flow 2.4.1
Even though the derived equations in chapter 2.2 are valid for all flows, solving them for tur
bulent flows, which are encountered in many technical applications, is linked to an enormous
effort and thus, to enable the analysis ofputational putational effort. To reduce the comcomvery comdescribe the signifipcant aspectslex, technical flows, these equations of the flow according to are modified to sim[43]. The mpler eoquations, wdified NavierStokes hich still
constant density (equations for turbulent flows arρ = const.), the Reynolds ansatz, which spe called Reynolds equations. To derive themlits all the velocity com for fluids with ponents: u,
denoted with a dash. v, w, and the pressure p into timeaveraged values according to (2.34) and a fluctuating value
26
quationsGoverning E
tions. This way, time aDoing so, it is possible to separate the global veraging the equations unsteady flow fromincluding the separated the local turbuflow malgnitudes leads ent fluctua
to the timeucts of the local fluctuations. Th averaged continuity equation aese then have to be mnd the Reynolds equation, odeled. without losing the prod
u=u+u′, v=v+v′, w=w+w′, p=p+p′
e Reynolds averaged: gnitudes araThe flow m
T1u=T∫()udt.
0This ansatz is now introduced into (2.4):
(2.33)
(2.34)
(2.35)
2∂ρ()u+u′+∂ρ()u+u′+∂ρ(u+u′)(v+v′)+∂ρ(u+u′)(w+w′)=
∂t∂x∂y∂z (2.35)
F+∂τxx+∂τyx+∂τzx.
X∂x∂y∂z
whereas the turbulent e global unsteady flow, gnitudes describe thaThe Reynolds averaged mfluctuations are present in the additional fluctuation terms on the right side of (2.38), (2.39),
e called Reynolds stresses. The separation of s ar(2.40). The additional fluctuation termand Fig. 22. plarily in ction is shown exemponent in xdirethe velocity com
ponent eraged and turbulent fluctuation comeavSeparation of u velocity in timFig. 22
Whaveraged anen time averagd fluctuatinging term(2.35), in which all flow param, it has to be considered that: eters have been separated into a time
f=f f′=0 ff′′≠0. (2.36)
to the definition of the Reynolds averaging gnitude. Due aHere, f represents any flow mf′=0, as shown exemplarily for the u velocity:
1T1T1T1T
u=∫()udt=u=∫()u+u′dt=∫()udt+∫()u′dt⇒u=u+u′;⇒u′=0.
T0T0T0T0
27
quationsGoverning E
Thus, the time averaged continuity equation for flows with constant density is written as:
∂u+∂v+∂w=0.
∂x∂y∂z
And the Reynolds equations for flows with constant density are: ∂()ρu+∂()ρuu+∂()ρuv+∂(ρuw)=Fx−∂p+∂σxx+∂τyx+∂τzx
∂t∂x∂y∂z∂x∂x∂y∂z
−⎛⎜∂()ρu′u′+∂()ρu′v′+∂()ρu′w′⎞⎟,
⎝⎜∂x∂y∂z⎠⎟
∂()ρv+∂()ρvu+∂()ρvv+∂(ρvw)=F−∂p+∂τxy+∂σyy+∂τzy
∂t∂x∂y∂zy∂y∂x∂y∂z
−⎛⎜∂()ρv′u′+∂()ρv′v′+∂()ρw′v′⎞⎟,
⎜⎝∂x∂y∂z⎠⎟
∂()ρw+∂()ρwu+∂()ρwv+∂(ρww)=Fz−∂p+∂τxz+∂τyz+∂σzz
∂t∂x∂y∂z∂z∂x∂y∂z
−⎛⎜∂()ρw′u′+∂()ρw′v′+∂()ρw′w′⎞⎟.
⎝⎜∂x∂y∂z⎠⎟
s Energy Equation for Turbulent Flow 2.4.2
(2.37)
(2.38)
(2.39)
(2.40)
duced to the energy equation Analogue to the approach leadifor incompressible flows as: ng to the Reynolds equations, the Reynolds ansatz is intro
Thus, the energy equation describih=h+h′. ng turbulent flows with constant density can be written as: (2.41)
ρ⎛⎜∂h+u∂h+v∂h+w∂h⎞⎟=∂⎛⎜λ∂h⎞⎟+∂⎛⎜λ∂h⎞⎟+∂⎛⎜λ∂h⎞⎟+Fr⋅rv+ρ&qs
⎝⎜∂t∂x∂y∂z⎠⎟∂x⎝⎜cp∂x⎠⎟∂y⎝⎜cp∂y⎠⎟∂z⎝⎜cp∂z⎠⎟
−⎛⎜∂()ρh′u′+∂()ρh′v′+∂()ρh′w′⎞⎟.(2.42)
⎝⎜∂x∂y∂z⎠⎟
28
2.4.3 Equations for Turbulent Flows with Variable Density
Governing E quations
For flows in which the density changes, as analyzed in chapters 6 (Analysis of Temperature
Depending Effects) and 7 (Integrated Results for the Upper Mixing Chamber), another ap
proach is used. Massaveraged quantities are introduced as:
~u=ρu, ~v=ρv, ~w=ρw, ~h=ρh.
ρρρρgnitudes are Favre averaged: amThe flow
(2.43)
Tρu=1∫()ρudt. (2.44)
T0fferent quantities are density changes, the diAccording to the approach for flows without made up of time averaged and fluctuating quantities. For the mass averaged quantities the
fluctuations will be denoted by two dashes. The density and the pressure do not have to be
satz for flows with variable density: ss averaged. This leads to the Reynolds anamu=~u+u′′, v=~v+v′′, w=~w+w′′, h=~h+h′′,
p=p+p′, ρ=~ρ+ρ′′. (2.45)
In addition it has to be considered that: f′′≠0, ρf′′=0, ρ′u~=0, ρu′′=0. (2.46)
the continuity equation, the flows with variable density inIntroducing the Reynolds ansatz for describing all flows with variable density Navier Stokes equations, and the energy equationvariable density: quation for flows with ity ee averaged continuleads to the tim∂ρ+∂(ρ~u)+∂(ρ~v)+∂(ρ~w)=0, (2.47)
∂t∂x∂y∂z
the Reynolds equations for flows with variable density: ∂()ρ~u+∂()ρ~u~u+∂()ρ~u~v+∂(ρ~u~w)=F−∂p+∂σxx+∂τyx+∂τzx
∂t∂x∂y∂zx∂x∂x∂y∂z
(2.48) −⎛⎜∂()ρu′′u′′+∂()ρu′′v′′+∂()ρu′′w′′⎞⎟,
⎝⎜∂x∂y∂z⎠⎟
∂()ρ~v+∂()ρ~v~u+∂()ρ~v~v+∂(ρ~v~w)=F−∂p+∂τxy+∂σyy+∂τzy
∂t∂x∂y∂zy∂y∂x∂y∂z
(2.49) −⎛⎜∂()ρv′′u′′+∂()ρv′′v′′+∂()ρw′′v′′⎞⎟,
⎝⎜∂x∂y∂z⎠⎟
∂()ρ~w+∂()ρ~w~u+∂()ρ~w~v+∂(ρ~w~w)=F−∂p+∂τxz+∂τyz+∂σzz
∂t∂x∂y∂zz∂z∂x∂y∂z
(2.50) −⎛⎜∂()ρw′′u′′+∂()ρw′′v′′+∂()ρw′′w′′⎞⎟,
⎝⎜∂x∂y∂z⎠⎟
(2.48)
(2.49)
(2.50)
29
Governing E quations
ith: w
(2.51)
σii=μ⎛⎜2∂~ui−2()∇⋅r~v⎞⎟+μ⎛⎜2∂uii′′−2()∇⋅rv′′⎞⎟,
⎝⎜∂xi3⎠⎟⎝⎜∂xi3⎠⎟
(2.51) ~~τii=μ⎛⎜⎜∂ui+∂uj⎞⎟⎟+μ⎛⎜⎜∂ui′′+∂uj′′⎞⎟⎟,
⎝∂xj∂xi⎠⎝∂xj∂xi⎠
and the time averaged energy equation for flows with variable density:
⎛⎜∂(ρh~)+∂(ρh~u~)+∂(ρh~v~)+∂(ρh~w~)⎞⎟=∂⎛⎜λ∂h~⎞⎟+∂⎛⎜λ∂h~⎟⎞+∂⎛⎜λ∂h~⎞⎟
⎝⎜∂t∂x∂y∂z⎠⎟∂x⎝⎜cp∂x⎠⎟∂y⎝⎜cp∂y⎠⎟∂z⎝⎜cp∂z⎠⎟
(2.52) −⎛⎜⎜∂()ρh′′u′′+∂()ρh′′v′′+∂()ρh′′w′′⎞⎟⎟.
⎝∂x∂y∂z⎠
ergy allows accounting for the local fluctuationsThis Reynolds Averaged NavierStokes approach for the conservation of m in turbulent flows without resolving themomentum and en
e and space. detailed in tim
Turbulence Modeling 2.4.4
equations, as well as in the energy equation s in the ReynoldsThe additional fluctuation termfor turbulent flows have to be mthe equations describing flows wodeled using ith constant density. The msocalled turbulence models for flows with variable denodels. Starting point are
the Reynolds equations and in a vectorsity are derived in analogy. The additional fluctuation term for the turbulent energy equation. s can be summarized in a tensor for
⎛u′u′u′v′u′w′⎞⎛⎜u′h′⎞⎟
⎜⎟τt=−ρ⎜v′u′v′v′v′w′⎟ qt=−ρ⎜v′h′⎟ (2.53)
⎜⎟⎝⎜w′u′w′v′w′w′⎠⎟⎝w′h′⎠
Due to symmetry of the matrix, 6 Reynolds stresses and 3 turbulent heat fluxes have to be
itten as follows: rbulent energy equation are wmodeled. The Reynolds equations and the turr⎛∂vrr⎞r
ρ⎜⎜+()v⋅∇v⎟⎟=F−∇p+∇τ+∇τt, (2.54)
∂t⎝⎠
(2.54)
ρ⎛⎜⎜∂h+vr⋅∇h⎞⎟⎟=∂⎛⎜λ∂h⎞⎟+∂⎛⎜λ∂h⎞⎟+∂⎛⎜λ∂h⎞⎟−qt. (2.55)
⎝∂t⎠∂x⎝⎜cp∂x⎠⎟∂y⎝⎜cp∂y⎠⎟∂z⎝⎜cp∂z⎠⎟
The eddies in turbulent flows lead to additional mixing and transport of mass, momentum,
transfer at walls. Considering the fact, that and energy, thus leading to higher forces and heatin a microscopic scale, momentum and energy are transported by diffusion, the turbulent
transport will be mReynolds stresses. odeled in analogy. This approach leads to the eddyviscosity model for the
30
2.4.4.1Eddy Viscosity Approach
quationsGoverning E
To enable a more compact way of writing the necessary equations, the velocity components in
x, y, and zdirection will be written as u1, u2, and u3 or ui, uj, respectively. The Reynolds
(2.13), approach for Newtonian fluids applied in odeled in analogy to the stresses will be m(2.15), as: (2.14),
⎛∂u∂u⎞2
−ρui′u′j=μt⎜⎜∂xij+∂xij⎟⎟−3ρKδij, δij=1, for i=jand ,δij=0 for i≠j. (2.56)
⎠⎝In this equation μt is the eddy viscosity. In laminar flows, the average velocity can be de
scribed as the mean velocity of all molecules. The specific velocity of the molecules is aver
aged over a certain, very sflows is the averaged velocity mall length scale. of the specific, localThe Reynolds averaged velocity in turbulent turbulent eddy structures, also averaged
over an adequate turbulent length scale.
The numerical effort of these, so called eddy viscosity models is smaller compared to models,
in which all the additional terms are modeled separately. The turbulent kinetic energy per
mass K, which is used in (2.56), is defined as:
11K=ui′ui′=(u1′+u2′+u3′). (2.57)
22222The turbulent heat fluxes are assumed to be proportional to the gradient of the mean enthalpy
odeled as: and are m
−ρui′h=λt∂∂xh or −ui′h=at∂∂xh. (2.58)
iiEitherλt, which is the turbulent thermal conductivity orat, the turbulent thermal diffusivity
has to be modeled. Considering fusion are transported by the same turbulent fluctuthe fact that the eddy viscositations, it can be seen that they are not indey and the turbulent thermal dif
pendent from each other and are linked by the turbulent Prandtlnumber, defined as:
νPrt=t. (2.59)
atFor fluids with small heat conductivity as used in the analyses presented in this workνt≈at,
is constant and close to one. Prandt
In the kε model defined by Launder and Spalding in [42], the eddy viscosity μt is expressed
as:
2Kμt=ρCμε. (2.60)
For the turbulent kinetic energy K and the dissipationε, transport equations are solved, to
identify the local distribution of the eddy viscosity μt in the flow. The kε model is only
applicable to completely turbulent flows. K is describing the energy of the turbulent fluctua
31
quationsGoverning E
(2.61)
tion. Both magnitudes are produced, trans is describing their decay due to fricεtions, while ported, and dissipated in the flow. ∂()ρK+∂()ρujK=μ∂ui⎛⎜∂ui+∂uj⎞⎟+∂⎛⎜μ∂K+μt∂K⎞⎟+
t⎜⎟⎜⎟∂t∂xj1∂42xj4⎝4∂xj434∂4xi⎠∂1xj42⎝44∂x4j43σ4k4∂xi4⎠
ProductionDiffusion (2.61) μtgi1∂ρ−ρε
1Pr442tρ43∂4xiDestructio{n
Buoyancy∂()ρε+∂()ρujε=Cεμ∂ui⎛⎜∂ui+∂uj⎞⎟+∂⎛⎜μ∂ε+μt∂ε⎞⎟
ε1t⎜⎟⎜⎟
∂t∂xj142K44∂4x4j⎝∂x43j44∂4xi4⎠∂1xj42⎝4∂4x4j43σ4ε∂4xi4⎠
ProductionDiffusion (2.62) 2+Cρεμgi1∂ρ−Cρε
1ε3424K4tPr43t4ρ4∂xi1ε2424K3
DestructionBuoyancyWith the following empirical model parameters defined in the model: Cμ=0.09, Cε1=1.44,
Cε2=1.92, σK=1.0, σε=1.3. The left side of the K and ε equations is composed of the time
on the right side of both equations represents s. The first term dependant and convective termthe third one describes buoyancy influences, , the production, the second one diffusive termwith gi being the vector of the acceleration due to gravity, and the last term in the Kequation
is the sink term.Cε3 in the buoyancy term of the ε equation describes the ration between the
l to the direction of gravity. flow velocity parallel and orthogona
(2.62)
Turbulent structures emerge due to instabilities. First, large structures or eddies are produced
which eventually decay into small structures. The large structures carry most of the energy
and can be associated with the turbulent kinetic energy K. The turbulent kinetic energy is
mostly dissipated in the smaller structures, which therefore can be associated withε. For in
dustrial applications, where shear stresses or heat fluxes at the wall do not play the dominant
role, the kε model has become the most widely used turbulence model. The flow patterns
within the flow field are described well in the kε model.
Other turbulence mbest practice guidelines odels have been developed [11] a general overview is given ovfor different applicatioer the performance of tuns. In the ERCOFTAC rbulence
Wilcox models for different flows. The k[86], for instance, performϖs very well for model, among other sources described in the book ofboundary layer flows close to the wall. Be
compromise between thsides the kequation, a transport equatiese two most widely used models (kon for the frequency of the large eddies ε and kϖ) is rϖealized with is solved. A the
kϖ SST (shear stress transport) model proposed by Menter in [48]. This model maintains
suited kε mothe good solutions obtained by the kdel away from the wall. ϖmodel close to the wall and blending into the better
32
Reynolds Stress Models and NonLinear Models
quationsGoverning E
To capture anisotropic effects of the flow, e.g.been conducted. These are nonlinear eddy viscosity m strong swirls, two diffodels and the Reyerent approaches have nolds stress models
RSM. In Reynolds stress mseparately, thus, not only two aodels, each of thdditional equations have to be e 6 independent Reynolds solved to accountstresses are modeled for the turbu
which requires malent fluctuations, but six. Since the goal of ny simulations and the numerical effort of ththis work is a method for design optime Reynolds stress models is ization,
odels, however, work. Nonlinear eddy viscosity mmuch higher, they arent applied in this might be a promand the rate of strain are adopted as ising approach. Here, nonlineardescribed in the methodology of StarCD relationships between the Reynolds stresses [14].
xing iodel for the analysis of the flow in the upper mbulence mThe choice of an adequate turchamber is based on the comparison between an experiment and the obtained results for this
3.3. nt as shown in chapter echosen experim
nditions Applied Boundary Co 2.4.5
Inlet Boundary
so, the densityAt the inlets to the fluid doρ, for nonisothermmain, the velocity componentsal simulations, the temuiperature T and values for the turbu in all directions are defined. Al
have to be given. εlent kinetic energy K and the dissipation
Outlet Boundary
At the outlets, specifications for the velocity componentsui, the temperature T, and the turbu
boundary condition at the outlet should only have alent variables are required. In general, the weak influence on the upstream flow and therefore it is defined far away from the analyzed
lations. At the uries are applied for all simetry. In this work, so called pressure boundageomoutlet, the pressure is defined at a certain value, whereas the gradients of all other variables
in flow. ajust to the mables can adare set to zero. Thus, the flow vari
Wall Boundary
Directly at the wall, the fluid velocity is equal to the velocity of the wall, since a viscous fluid
sticks to the wall. Here, all walls are motionless, which means that the velocity at the wall is
zero for all simulations carried out within this work. Thus, the gradients in the flow variables
zero. To resolve the e wall distance reduces toely large as thclose to the wall become extrember of cells would be necessary in large numflow in the boundary layer close to the wall, a this region. Due to the fact that the viscous effects close to the wall become dominant, since
the turbulent fluctuations are damped, the standard turbulence models are not valid in this
region and have to be adapted. For the here applied turbulence models, the nearwall region is
erical effort reducing the numalled wall functions are applied,not explicitly resolved, but socsignificantly.
33
quationsGoverning E
The stationary turbulent flow close to a solid surface can be described for the dimensionless
velocity u+ with the dimensionless wall distance y+, defined with the friction velocity uτac
cording to Versteeg and Malalasekera [79] as:
u+=u, y+=ρΔyuτ, uτ=τw.
ρμuτHere, Δy is the wall distance and τwis the wall shear stress.
(2.63)
The near wall region can be divided into three parts. In the viscous sublayer, which is in con
tact with the wall, it is assumed that they are no turbulent eddying motions due to the wall and
thus, in the absence of turbulence, the shear stress is constant and equal to the wall shear
stress τw throughout the layer. The viscous sublayer layer is very thin, it starts at y+=0 and
ends at y+=5. Due to u+=y+, it is also called linear sublayer.
The loglaw layer is the region just outside of the viscous sublayer. Here, viscous and turbu
ribing the functional relationship and the loglaw is valid desclent effects influence the flowbetween u+ and y+:
u+=κ1lny++C, (2.64)
with κ=0.41 and C=5.5. For the applied models in this work, wall functions are used at
following relationship: ording to thethe wall acc
+⎧⎪y+;y+≤y+m
u=⎨⎪1lny++C;y+≥y+m. (2.65)
κ⎩The two equations meet atym+≈12. To obtain good results when using highReynolds turbu
lent models with wall functions, the criteria 30≤y+≥100should be applied for the cell cen
ters, according to the best practice guidelines ([11]).
Symmetry Boundary 2.4.6
Symmetry boundaries are applied to the planes, where the analyzed geometry model has been
cut. They are defined that all flow variables in one side of the surface are a mirror image to
the ones on the other side. This means that the velocities orthogonal to the symmetry plane
are called Dirichlet boundary, etry boundaries wall, and symmdisappear. The applied inlet, which means that the values of the flow variables are defined directly, whereas the pressure
ables are specifboundary at the outlet is a Neumied. ann boundary condition. Here, the gradients of the flow vari
Numerical Methods 2.4.7
The equations describing general flows usually arent solvable analytically. The approach is
to discretize themdescribed by Oertel and Laurien and then solve the coupled sy[43] and Ferziger and Peric stem of discretized eq[17]. uations iteratively as
34
Discretization
quationsGoverning E
In this work, only steady state simulations are carried out, which is why the discretization in
timchosen, a discretization me is not regarded. After the governing set of ethod has to be selected to approximequations for the analyzed problemate the differential equations has been
in this work, uses the finite e used programleading to algebraic equations. Star CD, being thvolume method (FVM). The fluid domain is divided into a number of contiguous volumes or
cells connected by nodes, which represent the limitation of each cell. In the center of the cells,
the fluid variables are calculated applying the conservation equations; therefore, the Gauss
e is equal to the agnitude in a volumnge of a m is applied, which states that the chatheoremfluxes over the regarded volumes surface area. Since the position of the cell centers, where
all fluid variables are defined differs from the position where the fluxes are calculated, an
interpolation method has to be selected to approximate the surface integrals.
and the Quadratic UpstreamThe applied interpolation me Interpolthods in this woation of Convective Kinemrk are the Upwind Differencing Schematics (QUICK). For the UDe (UD)
Schemthe flow comes frome, the value for each fluid variab. For the QUICK mle is appethod, roximated fromthe value of the neighboring cell is not ap the value of the cell, where
proximated by a straight line, bucell values are needed, not only one or two neigt by a quadratic function. Therefhboring cells. To constrore, three points, respectively uct the quadratic func
cell on the upstreamtion, the two neighboring cell valu side. es are used and the third point is taken from an additional
Solution Method
ing system of nonlinear algedescribing the flow, the remainAfter discretizing the equations braic equations has to be solved. For the analyses presented in this work, the SemiImplicit
Method for Pressure Linked Equations (SIMPLE), which suitable for steady state conditions
equations are calculated with mentumo the mhas been used. Here, the velocity components ofa guessed pressure field. The solution of this calculation is the velocity fieldu*i, which holds
eld has to also hold for the continuity equa equations. Since the velocity fimentumofor the muity constraint is solved with the the contintion, a pressure correction equation that includes velocity componentsu*i. With the corrected pressure field, corrected velocity components
u*iare calculated. This process is carried out iteratively until the obtained velocity field con
verges, respectively until the changes for the velocity components from one iteration to the
it. tain lim a cere stay withinprevious on
Numerical Errors
There are different sources for errors in numerical simulations. The model error is the differ
eans the com reality. A validation, which mparison toodels show in comence the applied mparison of the simulation with an experiment, needs to be carried out to quantify the model
error. Another error is the discretization erroresolution. Generally speaking, regions with large gradr due to the discretizatients in the flowion me variables need to be thod and the grid
resolved much higher. Thus, it needs to be checked if the solution is not dependent on the
thod. ediscretization mdiscretization and the
35
quationsGoverning E
2.5Analysis Strategy
The exact temperature dbe unknown. As outlined in chapter istribution at the in1.3, it will dlet of the upper mepend on power aind mxing chamass flber will continue toow distribution in
ith statistical effects e, but also wup and timblies, which vary with burnthe evaporator assemlike tolerancThe hot channel factor of 2, eses, uncertainties, locatimal petrturbationsed by Schulenberg et al. in [68] ca, and fluctuations of the operating conditions. n in general occur in
foany assemre to limit the influence ofbly of the evaporator. The strategy fo a temperature pertr optimizatiourbation in any inlet to the mn of the miixing chamxing chamber on ber is there
ts. tle its ouany of
An effective method to study the mone single CFD analysis is the use of passive scalars as mixing of several inlet flows with diffearkers. Applyingrent tem this strategpyeratures in , each
inlethis idt is charea with the cologed with a passive scring of the difalar ffoerent inler whict flh a transporows. The inlet scalat equation is solved; pr distributionictured can is then
thermal flow is assumevaluated at each outlet. This med, as well ase to cases wthod is applicableith a certain tem to first, prelimperature inary andistribution at the inlet alyses in which iso
side. The transport equations for the energy, here expressed with the temperature T (2.66) and
(2.67) for turbulent flows are defined as follows: Yfor an exemplary scalar m
∂T∂∂⎡⎛ν⎞∂T⎤
∂t+∂xj()ujT=∂xj⎢⎢⎝⎜⎜a+Prtt⎠⎟⎟∂xj⎥⎥, (2.66)
⎣⎦∂Ym∂∂⎡⎛νt⎞∂Ym⎤
∂t+∂xj()ujYm=∂xj⎣⎢⎢⎝⎜⎜D+Sct⎠⎟⎟∂xj⎦⎥⎥. (2.67)
s a, describing the therr in the diffusion termIt can be seen that the two equations only diffemal diffusivity and D, describing the diffusion coefficient of the transported scalar and in the
turbulent PrandtlnumberPrt and turbulent SchmidtnumberSct.
The very high Pecletnumber with a value of Pe=7.3⋅106calculated in chapter 2.1 suggests
that the thermal diffusivity a is very small compared to the turbulent term and can be ne
glected. Also, since the transport of the scalar should reflect the energy transport in the flow,
Sctusing passiv is set equal toe scalarsPrt for the ev, thus leading toaluation of the m thei possibility oxing and thisf applying the descri way of the temperature hombed strategy of og
enization, if the thermal diffusivity a and the diffusion coefficient D is set to equal zero.
ese scalars are: using then vantages whThe great ad
36
1) the inlet from whFor a potential hot outlet, not only the increasich the transported hot stred temeak is derived. perature can be identified, but also
2) Predictions for the mixing performance of a mixing chamber used for temperature
homogenization can be made in one single simulation, rather than evaluating various
given inlet temperature distributions and the enthalpy distribution at the outlets.
quationsGoverning E
A vital requirement for this applied evaluation strategy is that the used scalars are passive,
which means that they do not influence the flow and each other.
The analyzonly steady state analyed flow has bses are peeen characterizedrforme as incompd and mixing shall be evalressible (ρ=uaρ(T)ted w) and turbuith passliveent. scala Alrso,s.
In case of small temperature perturbations, we can describe the flow with constant density
with the following equations:
inuity equation: te averaged conThe tim
∂ui=0,
x∂i, describing turbulent flows: the steadystate Reynolds equations
(2.68)
(2.69)
⎛∂ui⎞∂pi∂⎡⎛∂ui∂uj⎞⎤
ρ⎝⎜⎜uj∂xi⎠⎟⎟=−∂xi+∂xi⎣⎢⎢()μ+μt⎝⎜⎜∂xj+∂xi⎠⎟⎟⎦⎥⎥, (2.69)
the steadystate transport equations for the turbulent quantities Kandε:
∂()ρujK∂ui⎛∂ui∂uj⎞∂⎛∂Kμt∂K⎞gi1∂ρ
∂xj=μt∂xj⎝⎜⎜∂xj+∂xi⎠⎟⎟+∂xj⎝⎜⎜μ∂xj+σk∂xi⎠⎟⎟+μtPrtρ∂xi−ρε, (2.70)
∂()ρujεε∂u⎛∂u∂uj⎞∂⎛∂εμ∂ε⎞
∂xj=Cε1Kμt∂xij⎜⎜∂xij+∂xi⎟⎟+∂xj⎜⎜μ∂xj+σεt∂xi⎟⎟
⎝⎠⎝⎠ (2.71)
2+Cε3ρεKμtPrgi1ρ∂∂xρ−Cε2ρεK,
which are linked with the equations ti(2.69) by the equations: (2.53), (2.56), and (2.60).
For the transport of a scalarYi, including energy, a transport equation is solved according to:
ρ⎛⎜⎜uj∂∂Yxi⎞⎟⎟=∂∂x⎛⎜⎜()D+Dt∂∂Yxi⎞⎟⎟. (2.72)
⎝i⎠i⎝i⎠
ogether with a variable amount for the differA system of 6 nonlinear differential equations, tent scalars transported, has to be soinfluence on the flow field, the equations desclved, wherribing flows with variable density have to beeas, in cases where the energy equation has an
solved.
37
Method for the Analysis of Complex Mixing Chambers
3Analysis of Complex Mixing Chambers the Method for
tion Method Simplifica 3.1
xing chamber that should be applicable to imethod for the analysis of the upper The applied mother complex mixing chambers is based on the simplification of the geometry. If complex
are not taken into account as egarded, their effects on the flowstructures in the flow are disrnumwell. For flows in or around comerical effort since a large numplex geometries, simber of cells is necessary to resolvulations are often e the geomlinked to a very high etric details.
puting and thods, this leads to very long comization using CFD meor design optimEspecially fpreparhere. The difation timfes ferent stepsor the sim ofu the simlationplifs, in parication mticuolar ifdel proc medurany sime are aulso oulations are nectlined in [84essa]r. y, like
The simplification method describes the approach in which a simplified grid is used to calcu
late flows in and around complex geometries by applying source terms in the equations de
scribing the flow.
simplify geometry
quantify influence of the omitted structures
include effects of the omitted structures as source terms thod eplification mProcedure of the simFig. 31
The idea is to economize the numerical effort significantly, while still taking into account the
effects, complex geometries have on the flow. However, the complex structures shall not be
resolved in a detailed way. For the highly anisotropic flow in the upper mixing chamber with
a rather distinctive global flow field, source terms in the Reynolds equations seem to be the
most promising approach. Therefore, volumetric forces are introduced to the right side of
g to: adin(2.69) le
⎛∂ui⎞∂pi∂⎡⎛∂ui∂uj⎞⎤
ρ⎝⎜⎜uj∂xi⎠⎟⎟=−∂xi+∂xi⎢⎢()μ+μt⎜⎜∂xj+∂xi⎟⎟⎥⎥+F{iS. (3.1)
⎣⎝⎠⎦volumetricforces
Since these volumetric forcesFiS represent complex geometries, their definition is the crucial
aspect to successfully apply them instead of resolving the detailed geometry.
38
Method for the Analysis of Complex Mixing Chambers
The definition of the adequate forces differs from mixing chamber to mixing chamber. For the
upper mixing chamber, different approaches were pursued as described in chapter 5
(Including the Effects of the Omitted Structures).
thod is: eThe first step of the simplification m
1) Simplifying the geometry.
Applying this first step of the simplification method to the upper mixing chamber of the
HPLWR leads to the simplified model. In this model the geometry is simplified by disregard
headpiece struts are oming the complex headpiece struitted and only the outer cture. Thus, the mcontour of the headpieces struts, the windowoderator boxes and the thickness of the
elements with 2dimensional, impermeable baffle cells as shown in Fig. 32 are introduced.
Fig. 32 Headpiece geometry in the simplified model reduced to window element
The simplified model is shown on the right side of Fig. 33, while the actual geometry of the
. Even though the structure of the simplified ber is shown on the left sidexing chamiupper mmodel is still rather complicated, it is much simpler than the actual geometry. Adequate re
sults for the mithxing and the pressure drop now are achieved with less then one million cells,
etry. of the geomwhen regarding 1/8
Fig. 33 Upper mixing chamber: actual geometry vs. simplified model
39
Method for the Analysis of Complex Mixing Chambers
The next two steps, when applying the simplification method to the upper mixing chamber
are: 2) Quantifying the influences of the disregarded headpiece structures on the mixing and
3) Reynolds equations) into Introducing these influences via source termthe simplified model. s (in this case via volumetric forces in the
These two steps are presented in chapter 5, while in this chapter the numerical model for the
on of the used code StarCD and the applied plified model is described. Also, a validatisimsimnumpelifrical mied meodel. In the fothods is presented and a grid sensllowing Fig. 34, it is outlined hitivity stuow the sidy as verificationmplification m is sheown for the thod is
ber. xing chamicess of the upper mapplied within the design proupper mixing chamber
¾complex geometry
¾temperature close to pseudocritical point
include temperature
dependa(chanpter 6t e)ffects
simplify mixing
chamber design
)(chapter 3
perform design
optimization
)(chapter 4
reproduce complex
headpiece geometry
)(chapter 5
simplifmethoicatidon
integrated results for the optimized upper mixing chamber
)(chapter 7 Fig. 34 Outline of the design process of the HPLWR upper mixing chamber with emphasis
thod elification mpon the included sim
40
Simplified Model 3.2
Method for the Analysis of Complex Mixing Chambers
3.2.1 Numerical Model for the Simplified Geometry
All calculations have been carried out with the CFD software package STARCD version
ing of3.26. Moder the inlet fator boxlows is mes and the thickness ofodeled with passive scalars fo the headpiece str which thruts have bee transen omport equatioitted. The mn (2.72), ix
(2.70), the Reynolds equations (2.71) of the introduced (2.69), the mk−εass balance equation model were solved. Only steady state analyses have (2.68), and the K and ε equations
been performed. For this first analysis step, the flow is assumed to be isothermal so that fluid
odel, the standard high RANSturbulence mproperties were assumed to be constant. AsReynolds k−ε together with the standard wall function has been chosen.
odeled and the sidewalls of the fluid doeen mxing chamber has biOnly 1/8th of the total mmain, where the mixing chamber has been cut, were modeled as symmetry boundary condi
chamtions, since no circumber model is formferential effects are expeed by the separation wall.cted. The radial lim As only one eight itation of the uppeof the upper mr miixing xing
chamber is modeled, there are eigh35. These include also half in and outlet regit inlets aons as a resund seven outlets as sketched from below in lt of the cutting plane that runFig. s
through several headpieces.
485337562124617
bering Inlet (green) and outlet (red) numFig. 35
To minimize errors due to numerical diffusion in the mesh, a block structured mesh has been
created for the analyses as can be seen in Fig. 36. The mesh consists of approximately one
million cells, of which about 21,000 are ba+ffle cells. Cells near the walls have been arranged
to adjust the dimcells for the application of highReynoldsturbulensionless wall distance yence models in combto a value in between 30 and 100 for most of the ination with lawsofthe
wall. The headpiece structure is created with a socalled ogrid, which is a standard procedure
dimto meensional baffle cells are created as shown in sh round tubes with hexahedral cells. At the outer periphery of these o grids, the two Fig. 32. The created ogrids are extended to
the bottom to enlarge the distance between the mixing chamber and the applied boundaries.
iects of the boundaries on the flow shall be mThe outlet extensions are longer, since the effnimized. A pressure drop of around 105 Pa in first superheater assemblies, between the upper
mixing chamber and the lower mixing chamber, has been estimated by Schulenberg et al.
[68]. The pressure difference in the fuel assembly clusters of the first superheater has been
implemented by three layers of porous media cells at the end of the outlet pipes as shown in
41
Method for the Analysis of Complex Mixing Chambers
timated presFig. 36. The resistsure drop of about 100kPa will be obance coefficients of this porous mtained. edia have been defined such that the es
Since the connection tubes are included in the simplified model, their geometry is omitted in
the mesh. Four blocks are created around them as shown on the lower left side in the picture
the walls of the connection tubes to capture below. The introduced cells are refined towards equately. the effects at the walls ad
connection tubes
headpiece ogrid
porous media
Fig. 36 Numerical mesh of the upper mixing chamber
All boundaries are indicated in Fig. 37. The two cutting planes are defined as symmetry
r surfaces are defined as noslip walls. As the boundaries, as indicated in blue, whereas all othemoderator boxes are omitted, a smaller and uniform inlet velocity in vertical direction
(w=2.78m/s) is defined according to the average mass flow in each cluster for all eight inlet
areas indicated with the green circles. The values for the turbulent energy K and the turbulent
dissipation ε at the inlet are taken from the analysis by Himmel et al. [24]. At the outlets,
the pressure boundaries the red circles. Since pressure boundaries are defined indicated with drop of the superheaow field and the pressure fl the analyzedare located very far away fromter I elements is realized with the introduced porous media cells in the model, the value for
uence the result. A constant pressure oflthe pressure at the boundaries does not inf is applied. ≈25pMPa
differentiateHere, each evaporator o between thue contrtlet shall ibe mbutions of diffearked with a differenrent inlet flowt passs into the miive scalar which allows toxing chamber to
42
Method for the Analysis of Complex Mixing Chambers
analysindividual ois. Assumuitlet flows into each supng a certain inlet temperature erheater clustedistribution instead of these mr. Buoyancy forces will be neglected in thisarkers, on the
studied exemplarily. other hand, would require a separate CFD analysis for each case, so that mixing could only be
try IIemsym
Ietrymmsy
tlein
erupress
Fig. 37 Boundaries applied to the upper m ixing chamber
All model variables are summarized in:
Fluid: water at a pressure of p=25MPa
perature of 390°C and a tem
3Fluid properties: DynaDensity: miρ=c viscosity: 215kg/μm= 31.704⋅10−6kg/ms
Specific heat:Conductivity: cpλ==028..2398461WkJ//mKkgK
Boundaries: Inlet: u=2.78m/s, K=0.002m2/s2, ε=0.0007m2/s3 ([24])
WaOutlet: p≈ll: no slip conditions250MPa
etry conditions Cutting planes: symm
Tab. 31 Boundary conditions for the numerical model of the simplified model
ects), buoyancy effects are analyzed and perature Depending EffAnalysis of Tem6 (In chapter a distinctive, defined temperature distribution will be shown in chapter 6.2 (Case with a
tion). erature Distribuplet TemSpecified In
43
Method for the Analysis of Complex Mixing Chambers
Reference Case of the Simplified Model 3.2.2ber by Fischer xing chamie design of the upper mThe first analysis has been performed for thet al. [20]. Here, no extra measures have been introduced to enhance mixing. For this refer
ence case, the procedure to analyze the mixing of passive scalars will be shown exemplarily
results will be presented. and the evaluation of the obtained The quality of m38. The colors repiresent the concenxing is expressed by the mtrations of arker conceneach of the 8trations in th scalars use outleed as mts, showarkers in thn in Fig. e
analysis, in the arbitrary scale from 0 to 1. A red inlet color indicates the origin of the particu
lar marker, whereas blue regions are occupied only by the other seven inlet scalars in this
case. For the visual evaluation, the distribution of each scalar is plotted from underneath. The
pression of the flow distriads to a general imsummary of all the distributions in one graph lebution in the upper mixing plenum. Fig. 38 shows the marker distribution for the base case
xing. ieasures to enhance mwithout m
inlet_1
3t_lein
inlet_5
inlet_7
inlet_2
4t_lein
t_6inle
inlet_8
Fig. 38 Scalar/ marker concentrations in the upper mixing chamber, seen from underneath
44
Method for the Analysis of Complex Mixing Chambers
It is noticeable that the inlet flows 1 to 4, which enter in the center of the upper mixing cham
stributing their scalars already rather uniber and do not have a neighboring outlet, are diformly to the outlets in the outer part of the chamber. Inlets 6 to 8, however, which are close
to the outlets 1, 2, and 3, are disposing their scalars primarily there, such that the outlet con
centrations are exceeding even 70% locally. The scalar distribution of the inlet 6 scalar is
Fig. 39. It describes the worst case. shown in the following
Fig. 39 Distribution of the inlet 6 scalar
neighboring in and outlets. In This result can be explained by a horizontally Fig. 310 this layered flow strulayered flow structure with short cuts forcture is demonstrated. It shows
boundary II, shown in a vertical cut through the upper mixing chamFig. 37. The cut runs through the inleber, parallel to thts 1, 2, 4, 6, and through the oute cutting plane with symmetry 
lets 1 and 7, marked on the right side in Fig. 310. It can be seen that the central inlet flows
e concentration of the ected, thts. As expcloser to the outleare obstructed by the inlet flows inlet markers at the outlets are higher the closer the in and outlets come together.
1
2
46Fig. 310 Vertical cut through let_1, inlethe upper mt_i2, inlexing chamt_4, and inleber and distt_6 ribution of the scalars: in
45
Method for the Analysis of Complex Mixing Chambers
For quantitative evaluation of the mixing performance, the concentration of each of the eight
edia cells. st above the porous me cell layer of each outlet jue oninlet scalars is evaluated in thaveraged coThe value of the concentration foncentration, as well as the local mar each scalar is readout for exima for the concentration are calculated inach cell and then both, a volume
each outlet region. The volume averaged value is obtained by multiplying the value for each
scalar in one cell with its volume, building the sum of all the cell values and then dividing the
obtained value by the volume of all evaluated cells. The index j stands for the inlet scalar; i
ber. e cell numents threpres
1cj=V∑ciVi,j=1,...,8
totalcj is the volume averaged concentration of each scalar at the analyzed outlet,
ci is the concentration of each scalar at the analyzed cell,
Vi is the volume of the analyzed cell,
Vtotal is the sum of all the cell volumes per outlet
(3.2)
Thus, an average value for each scalar is received at each outlet. For later analyses, not only
the averaged values will be of interest, but also local hot spots, so that the maximum values
for each scalar (cj,max) were analyzed as well.
The results for the reference case of the upper mixing chamber without any additional mixing
Fig. 311. The outlets are displayed on the xcoordinate and an inlet devices are shown in m(marker distribution is sarker), according to the color scale situathown for each outlet; each bar ined underneath the diagramdicates the value for one scalar . The diagram in Fig.
311 shows the marker concentration of each inlet averaged over the cross section of each
outlet, whereas the diagram in Fig. 312 shows the peak values for each inlet marker at the
ts. outle
To evaluate the mixing the standard deviation σ can be introduced. This way each configura
ber. The can be characterized by only one numzation ition for the later presented design optimean value, can be analyzed variable around its mstandard deviation, as the dispersion of the used to characterize the mixing quality. The standard deviation applied for the evaluation of
xing according to e.g. ithe m[23] is defined as:
(3.3)
n1σ=n−1∑()xi−x2 (3.3)
1i=lue of each inlet scalar averaged over each outlet, ais the vxixis the mean value of all inlet scalars at all outlets,
n is the number of evaluated values for the inlet scalars. Since 8 inlet scalars are evaluated at
7 outlets, n=56.
xing quality serves well, since the iof the mUsing the standard deviation for the evaluation mean value of the inlet scalars at the outlet corresponds to the ideal value for perfect mixing
sion, is an adequate ed deviation, or disperxing chamber. Therefore, the indicatiin the upper mspecification for the mixing quality in the upper mixing chamber, since it quantifies the dif
ference between the achieved mixing and the perfect mixing. The obtained standard deviation
46
Method for the Analysis of Complex Mixing Chambers
for the reference case of the simplified model is: σ=12.1%. For comparison, Schulenberg et
al. [68] assumed in their analysis a standard deviation of σ=5% only. Thus, the mixing can
not yet be considered sufficient.
0,80,70,60,5Marker Fraction0,40,30,20,10
miidexial ng
1234567etOutlInlet_1Inlet_2Inlet_3Inlet_4
Inlet_5Inlet_6Inlet_7Inlet_8
Fig. 311 Scalar concentrations, av eraged over each outlet cross section
0,80,70,60,5er Maximum0,40,3k0,2Mar0,10
123
4567123tOutleInlet_4let_3InInlet_2Inlet_1Inlet_5Inlet_6Inlet_7Inlet_8
Fig. 312 Peak values of the scalar c oncentrations in each outlet cross section
As an examneighboring inlet 6, while the local mple, outlet 1 receives a high maximaum of inrker fraction of mlet scalar 6 is alore thmoan 50% in average fromst 70% at outlet 1. Out the
the inlet 3, for exlet 8 scamalar at outleple, gets a hight 3 is just abo peak fractionve 30%. of 65% from inlet 8, while the average value of
47
Method for the Analysis of Complex Mixing Chambers
The red line in the top graph shows the ideal marker distribution. For ideal mixing, one
eighth,ously, the p or 1resent distri2.5% of each inlet shoubution is far fromld leav ideal. e the upper mThey highest mixing plenum arker concentration, avat each outlet. Obvieraged
over each outlet cross section, is 4 times the ideal homogeneous concentration.
Another important aspect for the design of the upper mixing chamber is the created pressure
drop, which should be minimized, since the pressure drop in the core of the HPLWR is di
rectly linked to the thermal efficiency of the entire power plant. Analyses for the steam cycle
haufer et al.of the HPLW in [64] and in R and the influence of the core pr[65], as well as by Brandauer et al. in essure drop have been presented by Schlagen[9]. The obtained pressure
drop for the reference case of the simplified model is:
Δp≈4⋅103Pa.
It is measured as the pressure difference between the inlets and the outlets of the mixing
ber. cham
presented in chapter Different ways to enhance m4 (iDesign optimxing and an optimization using the simpized design of the upper mlified model). ixing chamber are
Validation 3.3
lations are gained and an adequate turuTo ensure that adequate results for the carried out simbulence model is used, a simulation applying the same numerical methods for an existing ex
periment has to be compared to the obtained experimental results in the experiment. Valida
tion therefore means the procedure of testing the extent to which the model accurately repre
sents the reality. The chosen validation experiment should be comparable to the analyzed ge
ometry, the upper mixing chamber, and should especially exhibit the same flow features as
etry. expected in the analyzed geom
found: When looking at the flow in the upper mixing chamber, two characteristic flow features are
A strong redirection of the flow of 180° 1)
2) Flow separation at a large number of obstacles in the upper mixing chamber such as
ctures connection tubes and headpiece stru
In an adequate validation experiment, the same flow features should be contained.
mThe model is thae selejor function of the comction of the best suitabplarison between e turbulence man experiment and thodel. Therefore different turbulence e applied numerical
models, which according to the best practice guidelines [11] seem promising to capture the
ared. pave been comain flow feature, hm
48
Method for the Analysis of Complex Mixing Chambers
An appropriate experiment was carried out by Inagaki et al. [33] in 1990. Here, the core bot
tom structure (CBS) of the gas cooled high temperature engineering test reactor (HTTR) as
presented in [34], developed by the Japan Atomic Energy Research Institute (JAERI), has
been analyzed. Other tests of the CBS have also been presented in [41].The CBS is a complex
passive mixing chamber, used to mix gas at different inlet temperatures to achieve a rather
homogeneous temperature distribution at the outlet of the mixing chamber to avoid hot spots
in following high temperature components. Thus, the task of this mixing chamber is somehow
comparable to the upper mixing chamber of the HPLWR. For the presented experimental
fluid and a oneseventh scale of 0.3MPa was used as teststudy, however, water at a pressure zed.odel of the CBS was analytest m
The temperature and flow rates were measured with thermocouples and electromagnetic flow
meters, respectively. For the temperature the measurement errors were within ±0.6°C and
for the flow rate within±0.5%. The analyzed test section is shown in Fig. 313. It is made of
let nozzles above the plenumacrylic resin and consists of a plenum, an ou. Inlet nozzle number tlet nozzle underneath the 1 is placed in the center of the test mplenum, and seven inodel
cated by red circles in and is surrounded by the others conFig. 313. centrically, which are numbered 2 to 7. All inlets are indi
cross section A A:
Fig. 313 CBS mixing chamber of the HTTR test section analyzed in [33]
In the plenum 15 cylinders, the socalled support posts, as well as a disc to promote the mix
ing of the coolant are inserted. The flow rate of the water in each inlet nozzle was controlled
by control valves to maintain equality. Hot water at a temperature of 55°C enters through inlet
through all perature of 25°C enters the plenummnozzle 1 in the center, while cold water at a teother inlets. Water passing through the inlet nozzles, is then mixed in the plenum before it
exits the mixing chamber through the outlet nozzle. Thus, the temperature difference analyzed
in this setting is 30°C and the Reynolds number in the outlet nozzle defined with the diameter
of the nozzle isRe=45000. The positions of the thermocouples in the plenum and in the out
Fig. 314. Fig. 313 and in let nozzle are indicated in 49
Method for the Analysis of Complex Mixing Chambers
Fig. 314 Cut through the CBS mixing chamber with indicated positions of thermocouples
In Fig. 314 the measurement positions that are compared to the simulation are highlighted.
Inside the plenum, the temperature is measured within cross section AA at a radius of
r=60mm between the bottom of the plenum and the indicated mixing promoter. In the outlet
nozzle, the measurements at two positions behind the plenum, at 100mm and 750mm, are
compared to the simulation. The characteristics of the numerical model for the CBS mixing
chamber are summarized in Tab. 32. At the inlets, the velocity profile of a fully developed
pipe flow with a mean velocity of u=0.11m/s and the corresponding values for the turbulent
ped pipe flow guaranteed in the fully develognitudes has been defined, according to the amexperiment (by Inagaki [32]). At the outlet an arbitrary pressure has been defined at a satisfac
with zero gradient conditions for all other flow the analyzed flow field, tory distance fromvariables and all walls are assumed to be adiabatic. The high Reynolds k−ε model has been
used for the first simulation, thus the flow at the wall is calculated with the standard wall
function. When building the Reynolds number with the velocityu=0.11m/s, the kinematic
viscosityν()T=25°C=0.89⋅10−6m2/s, and the diameterdh=50mm, all at an inlet nozzle,
Re=udh/ν≈6000 is obtained. To use the Reynolds number in the applied criteria (2.32)
for the consideration of buoyancy effects, it is built with the height of the mixing chamber as
characteristic lengthH=57mm. However, only small changes in the value of the Reynolds
number are obtained (Re≈6180). The Grashof number, built with the volumetric thermal
expansion coefficientα=1.9K−1, the acceleration due to gravityg=9.81m/s2, the kine
matic viscosityν()T=25°C=0.89⋅10−6m2/s, the temperature difference between the hot
inlet 1 and the cold inlets 2 to 7()T−T0=30K, and the height of the plenum as characteristic
lengthH=57mm, isGr=αgH3(T−T0)/ν2=1.3⋅1011. Applying the criteria (2.32) concern
ing the consideration of buoyancy:
GrRe2≈3400>>1
Buoyancy effects need to be taken account for the simulation of the CBS mixing chamber.
Approximation is applieTo account for buoyancy effects the Boussinesqd to the model as Characterization of Buoyancy Influences. 2.3 described in
50
Fluid:
Method for the Analysis of Complex Mixing Chambers
water at a pressure of p=0.3MPa
perature between 25°C and55°C and a tem
−6Fluid properties: DynamConductivity: ic viscosity: λ=0.μ6073=0.W/8900mK⋅ 10kg/ms
(at p=0.3MPa and 25°C) Specific heat:cp=4.18kJ/kgK
tion aDensity: BoussinesqApproxim
Boundaries:
Inlet: u=0.11m/s, K=0.00013m2/s2, ε=0.0002m2/s3
Outlet: p≈0Pa, else zero gradient
tic all: no slip, adiabaW
Tab. 32 Boundary conditions for the numerical model of the CBS mixing chamber
The grid used in the analysis of the CBS mixing chamber consists of 400,000 tetrahedral
cells. Different grids have been tested and no significant influence on the result has been ob
served.
The temperature distribution in the CBS mixing chamber is depicted in Fig. 315. On the
left side of the figure, the plenumis shown. It can be seen how the hot streak with all inentering in the center islet nozzles and the upper end of the outlet nozzles flowing around the mixing
promright side of oter, being miFig. 315 the AA cut through thxed with the cold streaks ene geomtering the plenumetry, as indicated in from the periphery. On the Fig. 313, can be
ogeneous right t nozzle is already fairly hom the water in the outlefseen. The temperature omafter theasureme plenum. Two ment position z, fairly larger greasurement positionsadient ars of the teme indicapted with erature are noticeable. the letters y and z. At the
Fig. 315 Result for the temperature distribution in the CBS mixing chamber
51
Method for the Analysis of Complex Mixing Chambers
In Fig. 316 the results of the experiment and the numerical simulation are compared across
the vertical direction in the plenum. Inagaki et al [33] have introduced the dimensionless tem
peratureΘ, which is plotted on the yaxis as follows:
Θ=T−Tcold (3.4)
ΔTIn which ΔT represents the difference between the inlet temperatures of the hot and the cold
inlets. The dimensionless temperatureΘ=1 represents the hot and Θ=0 the cold water in
the inlet nozzles, respectively. The error for the temperature, which can be included in the
presented diagrams, is indicated as error margin for each blue point showing the result of the
measurement (Experiment Ex. z/H), at a certain height. The xaxis shows the dimensionless
distance from the bottom of the plenumz/H. The red line represents the results for the simu
lation.
0,60,50,4Θ0,30,20,10
H/ z.Ex
z/H
00,10,20,30,40,5
Hz / Fig. 316 Temperature distribution across the vertical direction in the plenum experiment
(blue) and simulation (red)
It can be seen that the results for the experiment and the simulations agree well close to the
bottom of the plenum, while the discrepancy close to the mixing promoter is significant. Rea
sons for this difference in temperature could be within the applied numerical model. Also, the
influence of the thermocouple on the local flow field seems likely to have a large impact. Due
to the high gradients in the temperature in the region of the measurement position indicated in
Fig. 315 with z, vary small variations of the measurement position lead to noticeable changes
in the temperature distribution. In Fig. 317 the view from the top onto a horizontal cross sec
tion through the CBS mixing chamber is shown. The extension of the mixing promoter is
outlined by the black circle. The measurement position, indicated with the black arrow, is
located right in one of the hot streaks underneath the mixing promoter. Small influences on
region significantly. e distributions in thisperaturthe flow field are bound to change the tem
52
Method for the Analysis of Complex Mixing Chambers
Fig. 317 Temperature distribution at the surface underneath the mixing promoter
The results for the temperature distribution in the outlet nozzle according to the experimental
data and the numerical results are shown in Fig. 318. The xaxis shows the dimensionless
tance of 2 nozzle diamdistance diagonal in the outlet noeters and on the right side zzle. On the left side, the teat a distanmpce of 5 nozzle diamerature distribution at a diseters is shown.
It can be seen that the dimensionless temperature Θ obtained in the simulation stays well
within the range of the measurement errors in the experiment.
0,4EX_L/D0 = 2L/D0 = 20,4EX_L/D0 = 5L/D0 = 5
30,Θ20,
10,
30,Θ20,
0,1
0000,5y / D0100,5y / D01
Fig. 318 Temperature distribution in the outlet nozzle experiment (blue) and simulation
(red)
Using the same setup, other turbulence models have also been tested. Several recommenda
tions for the appliance of different turbulence models are suggested. Literature examples are
ERCROFTAC [11], MARNETCFD [2], and QNETCFD [12]. According to the best practice
guidelines [11], the k−ω−ShearStressTransport(SST) model proposed by Menter in
na. For highly anisotropic flows eon phenom[47] is very suitable for flow separati[48] and and turbulence driven secondary flows non−lineark−εmodels are recommended. The in
this work tested non−lineark−εmodel has been presented by Baglietto et al in [13], [4]
and [5]. It has been developed for fuel bundle simulations, but is not limited to nuclear appli
cations, since it is expected to deliver improved results for anisotropic flows and improved
sensitivity to secondary strains.
53
Method for the Analysis of Complex Mixing Chambers
In Fig. 319 the results for the temperature distribution obtained when applying the different
turbulence models are compared. Again, the experimental data is plotted and is indicated with
the blue dots marked Ex. z/H, Ex. y/D0, respectively. The red line shows the results obtained
for the standard k−ε model marked as KE. The non−lineark−ε model from Baglietto is
rked as quadKEBag, while the aue line mdescribed by the light blk−ω−ShearStressTransport()SST model is depicted in the diagram by the orange line,
arked as SST. mComparing the results obtained for the different turbulence models, no significant changes in
the results are noticeable. According to Baglietto [3], this can be explained by the fact that the
global flow structure dominates compared to local turbulent effects. Also, no differences in
the results for the different turbulence models are obtained, when applying them to finer grids.
Ex. z/Hz/H_KE
z/H_quadKEBag.z/H_SST
EX_L/D0 = 2L/D0 = 2 KE
L/D0 = 2 quad.KEBagL/D0 = 2 SST
0,4Ex. z/Hz/H_KE0,3
z/H_quadKEBag.z/H_SSTΘ
0,260,0,150,0Θ0,400,5y / D01
0,3EX_L/D0 = 5L/D0 = 5 KE
0,2L/D0 = 5 quad.KEBagL/D0 = 5 SST
0,410,0,30Θ0,200,10,20,30,40,5
H z /0,1000,5y / D01
Fig. 319 Temperature distribution in the plenum and in the outlet nozzle, obtained in the ex
periment (Ex. z/H, Ex. y/D0) and by numerical results with different turbulence models
the results are found, when applying the sameerences in fAs expected, again no noticeable difdifferent turbulence models to the simplified model of the upper mixing chamber. Therefore,
the standard high Reynolds k−ε turbulence model, which in comparison generates the least
numerical effort, is chosen for the analyses and design optimization of the upper mixing
chamber. It can be stated that the chosen numerical model is well applicable to the physical
problem and the numerical model can be regarded as validated, especially in the region be
hind the mixing chamber, which corresponds to the results of interest in the analysis of the
ber. xing chamiupper m 54
Method for the Analysis of Complex Mixing Chambers
3.4 Verification Grid Sensitivity Analysis
To evaluate the influence of the numerical resolution on the mixing, a grid sensitivity study
has been carried out in which the same simulation has been performed with five different gr
ids. To satisfy the validity of the applied wall functions for each model, the cell layer next to
the wall is kept inside the limit for the y+value, while the progressing cell layers are adapted
ting of around 300.000 cells has been refined in nt. The coarsest grid, consisefor the refinemseveral steps to the finest grid with around 1 million cells. In the grid sensitivity study, no
baffles have been applied which means that no additional internals, like the window elements
to account for the headpieces are regarded. For the grid sensitivity analysis, the regions where
the largest gradients were expected have been refined. In vertical direction the number of cell
layers has been increased, to provide a better resolution of the in and outlet regions, also the
regions around the connection tubes in the upper mixing chamber have been refined. This
separation where flow ection as well as regionsway, zones with strong changes in the flow diris expected, have been refined. The two evaluated qualities in the grid sensitivity study are the
maximum of the volume averaged scalar concentration for all outlets and the total pressure.
For the pressure, the difference between the inlet and outlet pressure is evaluated. In Fig.
320, the difference in total pressure for each grid is plotted over 1/N, where N represents the
number of cells in the model. The difference between the pressure drop obtained for the
coarsest and the fines grid is around 100Pa, which is almost two orders of magnitude smaller
than the pressure drop of interest for the performed analyses. The first order interpolation
scheme UD and the third order interpolation scheme QUICK for the spatial discretization
nt models. eve different refinemihave been tested for the f
When extrapolating the values obtained with the first order interpolation scheme UD and im
e, the anner, as expected for the first order scheme in a linear mplying that the results decreasnumextrapoelatiorical error for the different grids cn as defined by Roache in [63]. Thean be approxim values of the pressure drop for the different ated using the Richardson
grids are extrapolated to the yaxis which represents an infinite fine mesh. For the finest grid,
paring it to the r of 3.5% is found, when comted erroawith just over 1.000.000 cells, an estimextrapolated value from the Richardson extrapolation.
The other quantity evaluated in the grid sensitivity study is the maximum inlet scalar concen
tration, which is the maximum value of all averaged inlet scalars at all outlets, or the maxi
mum averaged value, which is detected at any outlet. It is the maximum percentage of any
inlet detected at an outlet, thus representing the least mixed streak. In Fig. 321 it is shown
that the maximum inlet scalar is not grid dependent.
55
Method for the Analysis of Complex Mixing Chambers
18001600p [Pa]1400Δ
1200
UDQuick (UD)Lineary = 5E+07x + 1253.7
1/N10000,E0+001,E062,E063,E064,E06
nt grids using the UD and QUICK udy pressure drop for differeGrid sensitivity stFig. 320 interpolation scheme, with the Richardson extrapolation applied to the UD scheme
UDQuick
UD1Quick80,60, inlet scalar40,max.20,1/N00,E0+001,E062,E063,E064,E06
Fig. 321 Grid sensitivity study maximum inlet scalar for different grids using the UD and
eQUICK interpolation schem
It can be seen that the vdiffer insignificantly, leading to the conclusion thalues obtained with both interpolatiat the first oron schemder interpoelation schems, UD and Quick, only e UD is
msufficient anodel, the finest grd will be uid shased inll be used to m future analyinimsesi. Foze the estimr further inated error. vestigations with the simplified
56
Design optimization using the simplified model
4Design optimization using the simplified model
Due to the results for the reference case of the simplified model presented in chapter 3.2.2
(Reference Case of the Simplified Model), a design optimization leading to improved mixing,
while limiting the pressure drop to an acceptable value, becomes necessary. According to the
simplification method proposed in chapter 3, the design optimization is carried out for the
simplified model. After an adequate design is found, the omitted geometry elements will be
introduced into the relevant models, applying the simplification method.
had originally been designed such that the ber xing chamiThe reference design of the upper mpressure losdirection in a horizontal cut thses in the mixing chamrough the upper mber are smallest. Ini xing chamber at Fig. 41, the velocitihalf its height are plotted. es in x, y, and z
obstacles. The depicted distinctive flow field consists of several pronounced streaks, facing almost no
[]m
/s Fig. 41 u, v, w velocity plot at a horizontal cut through the upper mixing chamber at half
its height
Several modifications to break up this distinctive flow structure and to introduce additional
turbulence to enhance the mixing have been analyzed. The different design modifications
xing and the established i for the mσhave been evaluated according to the standard deviation pressure drop of the upper mixing chamber. Regarding mixing, also the volume averaged inlet
and the peak values for each inlet scalar at the outlets)arker fractionscalar distribution (m(marker maximum) have been evaluated.
Turned Headpieces 4.1
As described above, the reference design had been designed to minimize the pressure drop.
Accordingly, windows in the headpieces of different clusters are facing each other and the
struts are placed close to the connection tubes. An idea to increase mixing caused by addi
tional obstacles in the flow could be to turn the head piece by 45°. The change of the distinct
flow structure and the introduction of additional turbulence have been expected by this modi
fication. As the fuel element clusters with its headpieces need to be interchangeable among
each other for burnup optimization, all head pieces need to be turned simultaneously. Now,
the inlet and outlet window elements do not face each other any more, and direct short cuts, as
57
Design optimization using the simplified model
scalar concentration e avoided. The averagedxing devices, ariin the reference case without m45° are shown in and peak values of the scalar concentration foFig. 42. Comparing the results of thisr this m modifiodification tocation with headpieces tu the reference case of rned by
the simplified model, we see that the situation became even worse.
0,80,80,70,70,60,60,50,50,40,4tionMarker Frac0,20,1mumier MaxMark0,10,2
0,30,3001234567Outlet1234567Outlet
Inlet_5Inlet_1InlInleet_6t_2InlInleet_7t_3InlInleet_8t_4InleInlet_5t_1InlInleet_2t_6Inlet_3Inlet_7InletInlet__48
and peak (right) in the upper , cross section averaged (left)Scalar concentrationsFig. 42 dpieces turned by 45° xing chamber with heaim
Outlet 1 receives even 60% of the markers from inlet 6. Slightly better results are obtained for
the local peak values of outlet 3, which receives only 50% from inlet 4 now. The reason for
at the u, v, w velocity plot in a horizonxing deterioration can be seen, when looking ithis mtal cross section at half its height as shown in Fig. 43.
[]/ms
6inlet_
inlet_7
let_8in
Fig. 43 u, v, w velocity plot at a horizontal cut at half its height with turned headpieces
and the results for the inlet scalars close to the outlet side
58
Design optimization using the simplified model
the reference case, when looking e even worse in comparison to The results for inlet 6 and 7 arat the cross section averaged values, while the peak values are slightly better, as a result of the
enhanced local turbulence. The flows coming from inlet 6 and 7 are now directed towards the
connection tubes framing the headpieces of the inlet headpieces towards the outlet side. When
the way of the inlet flows, these inlet flows colliding with the connection tubes which block are directed even more towards its neighboring outlets. The standard deviation for the mixing
for this configuration has a value of σ=12.9%(σ=12.1%), so it is slightly increased in
ber with essure drop of the upper mixing chamarison to the reference case. Also the prpcomthe turned headpieces is slightly increased in comparison to the reference case; for this con
figuration its value isΔp≈7⋅103Pa (Δp≈4⋅103Pa).
In conclusion, the configuration with the headpieces turned by 45° leads to worse results for
xing as well as for the pressure drop. iboth the m
Outlets Shielded from Inlet Side 4.2
Another measure to improve mixing is to close the window elements of the outlet headpieces
on the side facing the inlets, as shown in Fig. 44. This measure is similar to the turned head
pieces that it also focuses on influencing the in/ outlet geometry in the mixing chamber al
though it is more effective. This can be accomplished using plates that are welded into the
upper mixing chamber so that assembly clusters still remain exchangeable.
Fig. 44 Positions of the closed windows for the relevant window elements of the outlet
headpieces
odification is a Fig. 45. The advantage of this modification are shown in The results for this mreduction oflar 6 at outlet 1. Even though local m the local peak value from the mixing has baxieen immum of alproved, the average mmost 70% to just over 50% for scaarker fraction is
still high; with 3.7 times the ideal homogeneous mixing we are still far from optimum.
These additional obstacles, which have been welded in, extend the mixing length locally, thus
extent. The value of the standard deviation for e of the short cuts to somreducing the problemthe mixing has been reduced with this configuration to a value of σ=11.0%(σ=12.1%).
xing chamber, beiessure drop in the upper mAlong with the better mixing, however, the prtween the inlets and outlets, has been increased to Δp≈0.18⋅105Pa (Δp≈4⋅103Pa), which
it is still small compared to the pressure drop of the fuel assembly clusters.
59
Design optimization using the simplified model
0,80,80,60,60,40,4Mn Fractioerrka0,2mmurker MaxiMa0,2
001234567Outlet1234567Outlet
Inlet_5Inlet_1IInlet_6nlet_2InlInleet_t_73Inlet_Inlet_84Inlet_5Inlet_1Inlet_6Inlet_2InInlelet_7t_3IInnlelet_t_84
and peak (right) in the upper , cross section averaged (left)Scalar concentrationsFig. 45 ents in headpieces of outlet clusters emxing chamber with closed windows of window elim
Shielding the outlet side from the inlets is a very promising approach. It serves well to break
up a flow stposed walls, welded intoructure with any s the mixing chamhort circuits betweber, when certaich act as min ini and outlets. However, the proxing promoters for this configu
xing. iration, do not yet lead to sufficient m
4.3 In and Outlets at Different Heights
Since the reference case showedreduced inlet and outlet openings at different he a horizontally layered flow stights has been analyzed as a further modificaructure, a modification with
tion. The idea is to break up the horizontal layers, to extend the mixing length, and to separate
Fig. 46. be realized with a design depicted in the neighboring inlets and outlets. This shall into the upper mThe windows of the in and outleixing chamber so that all headt headpieces are partly closed pieces are still the sam with cae and thus exchangens which are welded
able. In thvertical positions of cans ae model, these cans are realized with baffle cells, like the headnd openings are also shown in Fig. 46. L stands for low, whereas piece geometry. The
H stands foror at the top of the upper mi a high position of the opening, mxing chamber, respeanectively. The dashed lining that the opening is located at the bottome indicates the cans,
openings are realized pression of how the imwhich are displayed underneath to give a betterat different heights.
HHH
HHHLLLHHH
LLLHHHHHHLLLHHH
LLLLLLLLLHHHHHHLLL
Fig. 46 and bottomPositions of high and low in and ou openings; right side: vertlet openings; left sidetical positions of the openings : distribution of top
The results for the constellation with in and outlets at different heights indicate that the dis
tribution has shifted drastically, as shown in Fig. 47. However, even though the short cuts
60
Design optimization using the simplified model
elimbetween neighboring in and outlets have been concentrations, volumeinated, very high averaged as well as for the local peak values, are obtained. As an example, outlet 7 receives
66% of the scalar released from inlet 6 and outlet 4 receives 60% from inlet 8. It appears that
they are not neighboring. A veryinlet and outlet head pieces, having their openings efficient mixing appears only if two head pieces have openat the same height, find a short cut even if
ings at different heights.
IInnlet_let_51InlInleet_t_62InlInleet_t_73InInlleet_t_84Inlet_1Inlet_2Inlet_3Inlet_4
Inlet_5Inlet_6Inlet_7Inlet_8
0,80,80,60,60,40,4Marknactio Frer0 MaximumrMarke0
0,20,21234567Outlet1234567Outlet
Fig. 47 Average and peak scalar concentration in the upper mixing chamber in case that in
and outlets are at different height
the sameIn general it can be seen height. This effect even outranges that the scalar transport takes plathe short cuts found betwce mainly between in and outleeen neighboring in and ts at
outleand 6 exemts for the referenceplarily, in the for case. In merly Fig. 48, this efpresented vertical cross section. fect is shown by looking at the inlet scalars 4
inlet_4:
inlet_6:
Fig. 48 Exemplary results for the configuration with in and outlet openings at different
heights
While the pictures on the leftscalar distribution in a more general way, the s, showing the view fromhort cut eff underneaect between theth, give an overview of the in and outlets at the
Fig. 48. Inlet scalar ections depicted on the right side of height is captured in the cross sesam61
Design optimization using the simplified model
4 hardly interacts with its neighboring inlet 6 and is transported mainly to outlet1, while inlet
is transported to the neighboring outlet 1 and xing chamber via itsiscalar 6 hardly leaves the mnext outlet in radial direction, outlet 7. The pressure difference between the inlet and outlet
side, on the other hand, is found to be aroundΔp≈0.29⋅105Pa (Δp≈4⋅103Pa). Thus, it is
still small compared with the pressure drop in the fuel assemblies.
in and outlets of the mixing chamThe proposed configuration shows that the inlet ber are locatedistribution is influend at different heights.ced strongly when the However, other and
modeteriorated mre severei short cuts axing and a re establishstandard deviation of eσd between in and outle=15.1%(σ=12.ts at the sa1%). me height leading to
4.4 Collection and Redistribution of the Inlet Flows
cally.Up to now, all m Approaches were focused on changes toodifications studied to enhance m thixing focused on infle direction of the inlet flows, or enhancing uencing the flow lo
mixing with introduced obstacles. A different approach is to collect the inlet flows, before
the different outlets. again todistributing them
Constructively, this can be realbetween the inlet and outlet side of the upper mized by simpily welding a verticalxing chamber. This way, the inlet flows are wall with a slot at the top
small cross section. In collected on the evaporator side of the wall before being forced Fig. 49 the position of the wall between evaporator and superheater to pass through a defined
assemtotal chambber height of 48 cmly clusters is depicted. For the presented first analy, leaving a slot of 2 cm underneath the top cover of the msis, the wall covers 46 cm of the ixing
ber. cham
t flows; separation wall with gap between inCollecting and redistribution of inleFig. 49 let and outlet regions
The results for the scalar distribution of this case are shown in Fig. 410. A much better scalar
distribution is obtained. Outlet 1 receives only 35% of inlet 6 and outlet 7 receives 34% of
let scalars. Pg the worst cases of the averaged ininlet 7, bein eak values for the concentrationare below 38% in worst cases. This mixing is much better than in the reference case. Aver
aged inlet scalar concentrations are less than 2.8 times the ideal, homogenized distribution,
62
Design optimization using the simplified model
and local peak concentrations achieve only 3 times the values of the case with ideal mixing.
The obtained standard deviation for this case is σ=9.3%(σ=12.1%).
hand, a very high pressure drop of xing, on the other iAlong with the good maboutΔp≈106Pa (Δp≈4⋅103Pa) is obtained for the rather small slot. Compared with the
in the fuel elemppressure drore drop is very high and can hardly be accepted. ents, this pressu
0,80,60,4onitcaMarker Fr0,2
0
0,880,60,0,6Marker Maximum40,onitcaMarker Fr0,20,2
0,4001234567Outlet1234567Outlet
InInlleet_t_51InInletlet__62InInlleet_t_73InInlelet_t_84InInlelet_5t_1InInlelet_t_62InInlelet_t_73InInlelet_4t_8
Fig. 410 Average and peak scalar concentration of the configuration with a separation wall
t side tlebetween the in and ou
4.5 Collection and Redistribution of the Inlet Flows Meander Alignment
After evaluating several cases, which separate the inlet from the outlet side, an alignment with
xing while keeping the pressure ito enhance mthree stages has been found as the best way drop within acceptable limits. In Fig. 411 such a meander structure and the results for exem
timplary inlet sized to a certain limcalars are dit. Furtheepicted. The size of thr optimization e gaps between the mican be achieved with a finer grid. The gap for xing stages has been op
the third gap is 14cm.the first introduced wall is defined as 18cm , the second gap is 18cm as well, and the value for
1
2
46Fig. 411 Meander structure dividing th e upper mixing chamber in three stages
63
Design optimization using the simplified model
When comparing the results for the mixing obtained for this meander structure, similar results
as for the case with only one separation wall and a very small gap are obtained. The obtained
standard deviation is σ=8.8%(σ=12.1%). Outlet 7 receives 37% percent of inlet 6. The
peak values are also small and stay below 39% as can be seen in Fig. 412. The pressure drop
for this constellation is, however, acceptable with a value ofΔp≈0.5⋅105Pa (0.5 bar).
0,880,0,660,0,440,nioer FractkMar0Marker Maximmu
0,220,01234567Outlet1234567Outlet
IInnlet_let_15InInlelet_t_26InInlleet_t_73IInnlet_let_84Inlet_1Inlet_2Inlet_3Inlet_4
Inlet_5Inlet_6Inlet_7Inlet_8
Fig. 412 Average and peak scalar concentration of the scalar distributions in the upper mixing
chamber with a meander structure, separating the upper mixing chamber into three stages
In Fig. 413, the histogram for the results shown in Fig. 412 is depicted. It confirms that the
standardized normal distribution (Gaussian distribution) has been reached approximately but
lly. not idea
30,0,25y20,oPritilabb0,1
0,150,0500,010,060,110,160,210,260,310,360,41
marker concentration at outlets
outlet arker reaches a certainlet mProbability that an inFig. 413
Evaluation of the Different Modifications 4.6
xing and for the pressure drop have iribing the mTarget values for the standard deviation descbeen given by Schulenberg in [68]. For the mixing a standard deviation of σ=5% and for
e0.15bar have been assumthe pressure drop a value of d.
Without any measures to enhance mixing (reference case), the scalar distribution at the outlets
is exceeding accep4.1 does not lead to an improvemtable limeits by far. Simnt as well; thply turning the he results for the scalar diseadpieces, by 4t5ribution ov °, as presented inerall are
achieved by introducing obstacles in the flow. even worse. Better results for the mixing are When the outlet headpieces are shielded from the inlet side, presented in 4.2, an improvement
for the volume averaged mixing is achieved and local peak values are decreased. In compari
64
Design optimization using the simplified model
son to this configuration, the configurations without additional obstacles do experience a neg
nds on how much resis drop, of course, depeligible additional pressure drop. The pressurer by blockages, obstacles, etc. For the evalubexing chamitance is introduced into the upper mated case with baffles cells at the windows of the outlet headpiece window elements facing
the inlet flow, a small and thus acceptable, additional pressure drop is obtained. Influencing
l heights, discussed at different, individuaxing by arranging the in and outlet openingsithe min 4.3, does have a strong influence on the mixing. Due to more severe short cuts between in
and outlets at the same height, this configuration provides even worse results than for the ref
erence case, the mixing reacts very sensitive to the introduced changes and the pressure drop
all. latively smis still re
achieveConcerning the m a collecition of the inlet xing, a very effective approach flows before distributing themis to separ toate th the outlets,e inlet and outle like pret resented in gions to
ributing the different inlet flows leads to best 4.4. This approach of collecting and then distbetween the separating wall and the top of the all gap xing. Due to the smiresults for the mmixing chamber, however, a very high pressure drop of around 1MPa is obtained. Thus, the
pressure drop in the upper mixing chamber exceeds the pressure drop in the fuel assemblies
pressure droby a factor of mpo is achieved with the alignmre than 5. The best resultent presented in ch concerning both miapter xing and the created 4.5, the presented madditional eander
structure, with which the upper mixing chamber is divided into three stages. In this configura
tion the gaps between the different zones of the mixing chamber can be enlarged, achieving
even better results for the mixing but decreasing the pressure drop significantly. A compari
son of the assumed values with the achieved values for the mixing and the pressure drop is
presented in shown in the HPLWR is shown in Annex A (Lower Mixing ChamTab. 41. Tber)he results f, for the upper moir the lower mxing chamiber, the mxing chamber are eander
4.5 is used. ent proposed in alignm
5% t mixingstandard dev. of coolan
pressure drop 0.15bar
5% t mixingstandard dev. of coolan
1bar pressure drop
upper mixing chamber
assumed
achieved
8.8%
0.5bar
chamberger mixinlow
assumed
achieved
5.5%
0.2bar
Tab. 41 Comparison of assumed and achieved values for the mixing chamber in the
R WHPL
65
Design optimization using the simplified model
Comparing the cases with enhanced mixing of the simplified model and the assumed values
for the mixing chambers in the three pass core proposal by Schulenberg et al [68], it is found
that the mixing is approximately within the assumed limits. The assumed standard deviation
for the mixing in both mixing chambers has been 5%. For the upper mixing chamber a value
e of 5.5% has been achieved, which is still ber a valuxing chamiof 8.8% and for the lower macceptable. The pressure drop of 0.5bar for the upper mixing chamber exceeds the assumed
value of 0.15bar, whereas the obtained value of 0.2bar for the lower mixing chamber is sig
nificantly lower compared to the assumed value of 1bar. The proposed solution for the opti
mized mixing chamber is shown in Fig. 414. For this picture, the top of the upper mixing
chamber has been removed to allow an insight. The introduced vertical walls forming the me
ander structure are shown in grey.
Fig. 414 Design of the optimized upper mixing chamber
According to the simplification method presented in this work, so far all analyses for the up
per mstep whenixing cham applying the mber of the HPLWethod for the analysR have been performis of comeplex md for the simiplxing chamified mbers outlinedodel. The next in
chapter 3.1 (Simplification Method) is described in the following chapter.
66
itted StructuresEffects of the OmIncluding the
es tructurIncluding the Effects of the Omitted S 5
The first step of the method to analyze complex mixing chambers, which has been introduced
in chapter 3.1, has been performed already by building and optimizing the simplified model as
r boxes, as well as the thickness of the headoderatodescribed in the chapters before. The mpiece struts however, will have an influence on the mixing. In the following sections, their
influences will be quantified and added to the simplified model by applying the simplification
method to the upper mixing chamber of the HPLWR. The required steps are listed in the be
3.1, they are: ginning of chapter
1) Simplifying the geometry
2) Quantifying the influences of the disregarded headpiece structures on the mixing
3) Introducing these influences via source terms into the simplified model.
To carry out the second step, a model of the headpiece geometry in all relevant detail has to
be built.
5.1 Detailed Analysis of the Flow in the Headpiece Structures
5.1.1del Detailed Headpiece Mo
Modeling the upper mixing chamber without all the rather complicated headpiece structures
served as a first step of the analysis. It has to be checked how much the omitted structures
influence the mixing. An analysis of a single headpiece in the transverse flow field of the up
per mixing chamber and an analysis of two neighboring headpieces without a transverse flow
xi analysis show good m[49]. The results of thisfield have been performed by Möbius et al. ing between the neighboring in and outlet headpiece, as well as great influence of the head
ber. For further analysis, xing chamithe upper mpieces inner obstacles on the flow field of xie flow field of the upper mces have been evaluated in thneighboring in and outlet headpiedpiece and the extended grid with two Fig. 51, the grid of one single heaber. In ing chames to apply the necessary boundary conditions used in d volumheadpieces and additional fluiis are shown. ented analysthe pres
The configuration applied to the detailed headpiece model is a combination of the inlet 6 and
the outlet 1 cluster, thus representing the worst case detected in the reference case using the
odel. ied mlifpsim
All presented results are based on the grid shown in the center of Fig. 51, which will be re
s are added to the sides of eece model. Additional fluid volumferred to as the detailed headpithe model to minimize the influence of the boundaries on the evaluated flow.
67
itted StructuresEffects of the OmIncluding the
16sMöbiu Fig. 51 Numerical mesh of a single headpiece proposed by Möbius [49] and of two
a transverse flow field neighboring headpieces in
The applied boundary conditions are depicted in Fig. 52; here all inlets are depicted with
yellowred color. At the bottom of the headpieces there are either nine inlets or nine outlets
for each headpiece representing the fuel elements united in one cluster. A transverse flow is
applied with the extracted boundary conditions from the solution of the simplified model. The
side inlet of the model introducing the transverse flow of the upper mixing chamber is ap
proximated with a block profile with an inlet velocity of 2m/s.
tseinl
iesrasymmetry bound
Fig. 52 Boundaries applied to the detailed headpiece model
outlets
For the headpiece representing the evaporator cluster, block profiles in vertical direction are
applied to the inlets on the bottom of the model with a velocity of 5.7m/s for the inlets in the
corners of the cluster and 4.6m/s for the other inlets. The inlet velocity for each inlet is esti
68
itted StructuresEffects of the OmIncluding the
mated using the mass flow and the inlet cross section of the nine inlets. No inlet velocity com
osed by opr even though the wire wrap spacers pponent in tangential direction has been added,Hiinlemmel in t swirls applied to e[25] are likely to introduach of the nine inlece a swirl to each fuel elemts have been analyzed by Möbius et al. in ent. Effects of thes[49], bute global
e pressure boundaries lied to the outlets arare found to be negligible. The boundaries app(brown). Thand to the outlet side of the mey are applied to the bottomodel in radial of the model for the mdirection. On the sideodeled s orthogonal to the msuperheater I headpiece ain
lue) are applied as a first guess. flow directions, symmetry boundaries (b
In this detailed headpiece model, all geometry features of the headpieces design are included
as demonstrated in Fig. 53. The transition nozzle of the headpiece geometry has been rebuilt
dow elemand all connection tubents are includesed by adding are now included in the m baffle cells to the headpiece outer diamodel. As in the simplified meter, thus framodel, the win
ing the windows by twodimensional, impermeable cells. It can be seen that the grid is refined
oderator boxes since here the largest gradients and towards the mtowards the window elemected in the flow field. ents are exp
moderator boxes
moderator boxes
transition nozzle
fuel assmbliese
moderator boxes
Fig. 53 Level of detail fort he detailed headpiece model of the headpiece geometry
The inlet boundary condition for the side inlet, with which the transverse flow is introduced,
tal cutting phas been extracted fromlane of the m ithe simplifxing chamber are sied model. In Fig. 54, the u,v,w hown, the red line shows the position where the velocities in a horizon
69
itted StructuresEffects of the OmIncluding the
cross flow velocities have been extracted. They have been averaged, to apply them to the de
odel. tailed m
Fig. 54 Velocities at a horizontal cross section of the simplified model
5.1.2 Verification Grid Sensitivity Analysis of the Detailed Headpiece Model
][/ms
Another grid sensitivity analysis has been performed for the detailed headpiece model to de
termine the influence of the numerical resolution on the evaluated magnitudes. Again, both
the pressure drop and the mixing will be evaluated and plotted over 1/N, where N represents
the number of cells in the model. Also, for all analyzed grids, the y+values are kept inside
the predetermined limits.
Five grids in a range between around 120.000 cells for the coarsest and 1.300.000 cells for the finest grid have been analyzed. Again, any baffles are disregarded for the grid sensitivity
analysis. Tober of cell layers ar refine especially the regions wheround the introduced me the largoderator boxes and est grthe numadients were expected, the number of cell layers in ver
tical dvolumirection has be averaged scalar concentratioeen increased. The two evn for all oualtluated quets and thalities are age total presain,su the mre. The fiaximumrst order i of the n
tization havterpolation scheme been teeste UD ad. nd the third order interpolation scheme QUICK for the spatial discre
This time, the difference between the inlet and outlet pressure is evaluated, between the eva
porator and superheater I fuel elements, respectively, that is the in/ outlets from the bottom of
the detailed headpiece model. The maximum inlet scalar concentration, or the maximum per
centage of any inlet detected at an outlet, is obtained in the same way as for the simplified
model, explained in 3.4 (Verification Grid Sensitivity Analysis). Now, each of the nine
inlets is charged with a separate scalar and the distribution is then evaluated at the outlets. The
result of the grid sensitivity analysis for the detailed headpiece model is shown in Fig. 55.
70
itted StructuresEffects of the OmIncluding the e finest grid is around 480Pa, re drop of the coarsest and thThe difference between the pressuwhich is much smaller (2 orders of magnitude) than the pressure drop of interest. For the
maximum inlet scalar almost no significant change is visible, when comparing the obtained
values of the different grids. Also, for both quantities no differences are detected between the
UD and the QUICK scheme. For further analysis, the UD scheme is chosen in combination
ffort, while ensuring rical eereduce the numwith a grid consisting of around 250.000 cells to lts. uadequate res
80007000]Pa6000p [Δ5000400000+E0,0
06E2,
UDkciQuLinear (UD)
N1/4,E066,E06
loc. conc. UD
kciQu
0,6loc. conc. UD
0,5Quick
40,30,aralc steln iax.m0,1
20,N1/00,E+0002,E064,E066,E06
odel is of the detailed headpiece m Grid sensitivity analysFig. 55
71
itted StructuresEffects of the OmIncluding the
Simplified Headpiece Model 5.1.3
When comparing the results obtained for this detailed model with the local result for the sim
plified model, it has to be taken into account that the flow field in the upper mixing chamber
s. A better comparison can be achieved, when is strongly influenced by the other inlet flowcomparing the detailed model with a cutout model of the simplified model, the socalled sim
plified headpiece model. Again, the same neighboring in and outlet headpieces are modeled,
but without any details like waterboxes or the reducing diameter in the lower part of the mod
el.
Also, a simple transverse flow as applied in the detailed headpiece model can not lead to a
tailed headpquantitative comiece mparison for the model. However, it can serve asixing and pressu a basis to qre drop of tuhe simplified mantify the influences of the disodel and the de
etry. meoed headpiece gregard
let as in theThe nine in simlets represplified menting the fuel elemodel. This is possiblents e,are com since the inbined in one inlelets on tht, re bottomespectively one out of the model,
representing any cluster of the mcharged with only one passive scalar for all the ixing chamber neighboring an outlet nine inlets, to evaluate the micluster have to be xing in the de
tailed headpiece model as well. This way, the mixing of this scalar can be compared to the
simmodel is shown in plified model, where the scalars are applieFig. 56; it consists of a much lower nd to each cluster. The simumber of cells thpan the detailelified headpiece d
model as do the dimeheadpiece model. However, its boundaries ansions of the added fluid volumes. pplied correspond to the ones of the detailed
Fig. 56 Grid of the simplified headpiece model
e model only towards the connection tubes and plified headpiecRefining the grid of the simtreating all geometry elements exactly as treated in the simplified model leads to a grid for the
simplified headpiece model consisting of less than 45.000 cells. Quantifying the influences of
the detailed headpiece geometry omitted in the simplified model, can be done by comparing
the results for the two headpiece models: the detailed headpiece model, based on the actual
geometry and the simplified headpiece geometry, based on the simplified model according to
ethod. lification mpthe sim
72
itted StructuresEffects of the OmIncluding the
5.1.4 Comparison of the Detailed and Simplified Headpiece Model
When comparing the detailed and the simplified headpiece model, large differences in the
Fig. 57; odels are displayed in The two different mmodels as well as in the results are found. on the bottom of the picture, a top view of each model is shown. Obviously, the numerical
effort for the simplified headpiece model consisting of around 45.000 cells is significantly
. Also, the analyzed ece model with 230.000 cellsparison to the detailed headpireduced in comch simpler. ue is mrflow structu
Fig. 57 Comparison of the simplified (left) and detailed (right) headpiece model
The results for the mixing of the simplified headpiece and of the detailed headpiece model can
the inlet scalar 6 is transported to out underneath. Whereas 53.5% of Fig. 58 frombe seen in let 1 in the simplified model, only 40.5% of the inlet scalar reaches the neighboring outlet in
the simplified headpiece model. When comparing the models where only two headpieces are
regarded, the difference due to the disregarded details is higher than due to the simplified
plified boundary conditions. on with simboundaries applied to the cutout secti
In the simplified headpiece model, the difference between the highest and lowest value for the
oreodel, a m between 29.8% and 62.6%. For the detailed mneighboring inlet scalar varieshomogenized flow is found for the outlets on the bottom of the model; the scalar concentra
ount also a large difference in the overall amtion varies between 25.4% and 40.0%. There is of the inlet scalar reaching the neighboring outlet. For the detailed headpiece model, only
73
itted StructuresEffects of the OmIncluding the
35.7% reaches the neighboring outlet, while in the simplified headpiece model this number is
higher at a value of 42.3%.
Fig. 58 Distribution of the inlet scalar in the neighboring outlet top: detailed headpiece
model, bottom simplified headpiece model
The large differences in the mixing of the fluid are caused by the disregarded moderator
pass througboxes. These boxes act as obstaclesh the headpieces. In these sm on the flow, leaving only smaller gaps, local jets are formall gaps where the water can ed resulting in a better
ng outlet headpieces. Also, the velocities in transport of the inlet scalar beyond the neighborie flow obstacles leading to a better mixing of the lateral direction are much higher due to ththe inlet f59 the velolow charged with the sccity fields of the two headpieces malar and thodels at a crose transverse flow in the mixing chams section at half the heigber. In ht of the Fig.
de of the figure, are displayed. model, as indicated on the right si
shaped by the waterboxes in the detailed headpiece mTwo very distinctive jets are odel. Here troduced by the obstacles in the rbulence is inl tuathe flow is accelerated strongly and additionflow. Also, the transverse flow hitting the headpieces is in some extent deflected to the sides.
higher transport of the inlet scalar past the uch This distinct flow structure leads to a mneighboring headpiece, representing outlet 1 of the simplified model. In addition, the mixing
74
itted StructuresEffects of the OmIncluding the
of the inlet 6 scalar is enhanced significantly, which is noticeable by the more homogeneous
distribution at outlet 1.
Fig. 59
[]/ms
Velocities in a horizontal cross section of the simplified (top) and detailed headpiece
)model (bottom
75
itted StructuresEffects of the OmIncluding the
of the Headpiece Influences nIntroductio 5.2
As described in chapter 3.1, the effect of the omitted structures on the mixing is introduced
into the Reynolds equations according to (3.1). The idea is to introduce source terms resem
bling the effects of the omitted structures.
Local Forces fInsertion o 5.2.1
boxes in the model, disregarded moderator An idea to introduce the effects of the initiallywithout actually introducing their geometry, is to insert local forces at the positions of the
cated exemplarily for the moderator boxes. This idea is outlined in central moderator boxes. Fig. 510 and the forces to be introduced are indi
[]/ms
Introduction of local forces to replace waterboxes Fig. 510
rally successful, the detailed headpiece model has proach can be geneMerely to see if this apthe mserved as a basis and adodel. Thus, the holes in the model (mditional cells are intrododuced where the merator boxes) are filled and a detailed moderator boxes had been inodel
e created model is shown on the left side of hTts as flow obstacles has been built. without struFig. 511 from above. While the cells of the detailed headpiece model are depicted in red, the
oderator boxes can be seen in a light brown color. On the madditional cells representing the right side of the figure below, the additional almost 50.000 cells for the moderator box struc
ze has been increased to 300.000 cells. odel situres are shown. Overall the m
Fig. 511 Detailed model with cells filling the positions of the moderator boxes, th us creating
odel without distinctive flow obstacles a m76
itted StructuresEffects of the OmIncluding the
For this created mheadpiece model, analogue resultodel without distinctive fls are obtained. To add the effow obstacles, which resembles the simects of the detailed headpiece plified
the mmodel to this model, source termoderator boxes. Since in reality no water s have been added to the newly introduccan pass through these cells, the applied source ed cells representing
terms are calculated with the pressure drop imposed by the omitted obstacles. For the steady
state analyses presented, the source terms are forces divided by unit volume with the
unitN/m3. The geometrical entities are taken from the quadratic shaped moderator boxes
that have a width of approximately a=0.03m and a height ofh=0.48m and are applied for
ssure drop, the pressure drop coefficient To calculate the preall additionally introduced cells. of a rectangular tube in a transverse flow ς=1.05 is applied according to [50] (page 149).
lated as: e calcuThe pressure drop can b
Δp=ς1ρ()u2. (5.1)
2For the source term and a constant value for the density ρ=215kg/m3 this leads to:
FR=AΔp=Aςρ()u2.
VVV1232(5.2)
COn the left side of (5.2) FR/V is the force per unit volume to be introduced, while FR is the
resting force.A is the area of the flow obstacle represented by the introduced force (width a
es height h of the rectangular mtim. eoderator box) and V is its volum
Since all values are known, a general prefactor can be introduced that defines the introduced
source term as a function of velocity. For the definition of the local forces, this leads to a pre
factor ofC=4000kg/m4. The source term as function of the velocity is plotted in Fig. 512.
FR/V [N/m3]
200000
150000100000
500000
m=2800
=ς1.05
]u [m/s
46802 Fig. 512 Source term as a function of the velocity for local forces
77
itted StructuresEffects of the OmIncluding the
StarCD requires a linear function at this point, therefore the function is approximated with a
straight line that has slope of m=2800 and starts at the origin. The introduced force repre
sents the rectangular pipe in thadditional error. In newer versions of the codee transverse flow field. Applyi, it should be possible tong a linear function leads to an apply quadratic func
tions, thus reducing the error. When looking at the results obtained for this detailed model
mwith source termodel. In a horizontal cross sectis, analogue results are obtaineon at half of the height, the flowd comp structure is veared to the results for the detailed ry similar to
the deta(top) is comiled mpared to the detailed model. This is shown in odel (bottom). Fig. 513, in which the detailed model with source terms
[]/ms Fig. 513 Velocity plotted for the detailed model, top: waterboxes introduced with source
terms, bottom: waterboxes geometrically resolved
When compjets are found. For the detailed maring the two velocityodel with th plots in the figure above, the e resolved moderator box geomsame two noticeable velocity etry, these jets
erated water in beadients between the accelare slightly more pronounced and the velocity grtween the waterboxes and the regions with small velocities are slightly higher.
and )is shown for the detailed model (bottomFig. 514 the scalar distribution at the outlet In the detailed model with introduced cells charged with source terms representing the modera
ilar and shows the same structure, but the simtor boxes (top). The scalar distribution is very values are generally higher for the model with applied source terms. In the detailed model,
35.5% of the neighboring inlet scalar is detected at the analyzed outlet with a local maximum
of 44% and a local minimum of 29.7%. For the model with the applied source terms, 38.5%
of the neighboring inlet scalar is detected at the outlet and the local maximum and minimum
vary between 35.1% and 44.6%.
78
With source terms: Sc1 averaged: 38.5% Max: 44.6%Min: 35.1%
Effects of the OmIncluding the itted Structures
: lly resolvedGeometrica Sc1 averaged: 35.5% Max: 44.0%Min: 29.7%
Min: 35.1% Max: 44.6% Min: 29.7% Max: 44.0%
oderator boxes introduced odel, top: mplotted for the detailed mScalar distribution Fig. 514 with source terms, bottom: moderator boxes geometrically resolved
The agreement between the two models is significantly better than the predictions of the sim
plified model as depicted in Fig. 514. Not only the results for the scalar distribution are al
most the same, but also the flow structure is very similar.
Reasons for the smthe quadratic function describing all discrepancy between the twthe applied source termo approaches m by a linear fight be the approximation of unction or the coarse
resolution, especially ofrator box structures. ode the cells filling the m
Even though these results are promising, adding additional cells that are then charged with
rt as intended. The additional erical effoa reduction in the nums does not lead to sources termdded source terms deteriorate the convergence of erical effort and the acells increase the numthe simmodel in order to combulation. Therefore, the derived sourceine the good results for the m termis are axing and the mdded to theu simplifch decreased numied headpiecerical e
effort. Due to the very coarse local discretization of the simplified headpiece model, an exact
the position of the moderator terms could not be achieved and of the source local applicationboxes had to be approxFig. 515 with the red cells. imated. The positions of the introduced source terms are indicated in
79
itted StructuresEffects of the OmIncluding the
Fig. 515 Position of cells with local source terms representing the initially omitted moderator
box structures
The results of this model, however, vary significantly from the results obtained for the de
tailed headpiece model. In Fig. 516 the velocities and the scalar distribution for the two mod
els are plotted. For the simplified headpiece model with local source terms, almost twice as
zed outlet and the range between ar is detected at the analymuch of the neighboring inlet scalthe minimum and maximum value for the scalar is much higher.
80
][/ms
With source terms: Geometrically resolved:
Sc1 averaged: 35.5% Sc1 averaged: 53.1% Min: 35.5% Max: 79.8% Min: 29.7% Max: 44.0%
Fig. 516 Velocities and scalar distribution left: simplified headpiece model with local
odel s, right: detailed msource term
itted StructuresEffects of the OmIncluding the
The reason for this very significant discrepancy between these models lies within the coarse
discretization of the simplified headpiece model, rather than in the rough approximation of
This can be seen s.e applied source termoderator boxes represented by thposition of the mwhen looking at the velocity plots for each model for a cross section at half its height as de
picted above. The flow structure stays the same, again two jets can be seen in the simplified
posed on the flow by the strong forces nts imheadpiece model, but since the very high gradieproduction of these achieve a better rey two cell layers. Tocannot be reproduced between onlgradients, local refinement of the grid (as performed for the detailed headpiece model) be
es necessary. com
The result of this failure to reproduce strong gradients leads to a general deceleration of the
flow within the entire headpiece not just locally and thus to a much higher scalar concentra
ated, since e sides is overestim, the scalar distribution to thtion in the neighboring outlet. Alsothe entire headpiece acts now almost as a blockage. Using the higher order QUICK discretiza
delivers analogue results. etion schem
ture the effect of the headpiece geomTo overcome these described disadvantages of thetry as a whole rathe mer thodel, global forces an locally. are introduced to cap
81
itted StructuresEffects of the OmIncluding the
5.2.2 Insertion of Global Forces
When introducing the effect of the headpiece geometry as a whole, claims to reproduce local
to the presented approach for the insertion of ilar et. The approach is simeffects cannot be mthe headplocal forces.iece geometry. Again, the applied source terms are derived with the pressure drop estimated for
to the porous mThe two approaches useing eitherdia approach, e.g. used by Kunik et al. local or global forces to out[40]. In the porous mweigh the momeentum are simdia approach ilar
omitted strucontrast to the general porous mctures are generaelly regdia approach arded by anit is poss additionible to intral pressure resistanoduce source termce. However, in s only to
specific regions of the grid, since no specifications for the boundary conditions between fluid
dia cells are necessary. ecells and porous m
the simpTherefore, all cells framed by the window elemlified headpiece model are charged with ents, reprsource terms. The resulting simesenting the headpiece geopmlified headetry in
piece model with introdcating the region where the forces are introduuced global forces isced. depicted in Fig. 517 outlining the idea by indi
Fig. 517 Insertion of global forces representing an entire headpiece
try is approximTo derive the forces which have to be intrated with a hand calculation usoduced, the pressure drop ing the pressure drop coefficient according to of the headpiece geome
Kays 518, the flow comi[36]. When looking at a crosng from the side is canalizeds section of the headp in two disitece geominctive jets frametry as depicted in ed by the centeFig. r
as shown in the figure on the right side of and outer waterboxes. If only the upper (or lower) Fig. 518, the effect of half of the headpiece the geomgeometry on the flowetry is regarded
rupt change in the flow cross section. ple and abcan be reproduced by a very sim
82
A1A2
Effects of the OmIncluding the itted Structures
A1A2
eadpiece with global forces e hodel thApproach to mFig. 518 This approach is not only applicable if the flow approaches the headpiece from one distinctive
side, but also if it approaches the headpiece with an angle. Due to its structure, certain chan
nels can always be distinguished, framed by the central and outer moderator boxes.
For the simplified structure displayed on the right side of Fig. 518 a pressure drop coefficient
coefficient for flows across an[36]. The pressure droping to Kays has been defined accordpressure droabrupt contraction is given as a functip coefficient of ς=0.3on of the ratio of th is found. Applying the same two cross sectionse hand calculations A1/A2(5.1) and. Here, a
factor C=160kg(5.2) as for the insertion of local/m4, leading to the depicted function in forces, a global resistant force FR,GlobalFig. 519 for the source term. To is found with a pre
slopeapproximm=750ate the quadratic f, has been chosen. unction, a linear function starting in the origin with the
FRGlobal/V [N/m3]A1A2
8000
6000
40002000
m=750
A1A2
00246u [m/s]
Fig. 519 Source term as a function of the veloc ity for global forces representing an entire
headpiece 83
itted StructuresEffects of the OmIncluding the
Wlarge differences in comhen applying the derived forces to each hparison with the detaeadpiece gliled headpiece geomobally, the maetry. The results for both rker distribution shows
s show that the flow is generally Fig. 520. The depicted velocity plotcases are depicted in decelerated, which leads to an even higher marker concentration at the neighboring outlet.
[]/ms
[]s/m Fig. 520 Compheadpiece model with glarison of the results for the detailed hobal forces (bottomeadpiece m) odel (top) and simplified
In this case, 43.7% of the inlet marker is transported directly to its neighboring outlet. The
local minimum for this model with global forces is 33.2% and the local maximum is 67.9%,
headpiece model and also for the simplified ing the values found for the detailed thus exceedheadpiece model.
ces are introduced, the pressure drop imposed by In the here analyzed case, where global forthe additional structure within the flow is captured. However, the forced deviation imposed on
the flow by the waterboxes within the headpiece geometry accelerating the flow locally is
neglected. This acceleration is responsible for the better mixing by transporting a larger frac
tion of the inlet scalar past its neighboring outlet and has to be regarded.
84
itted StructuresEffects of the OmIncluding the
5.2.3 Insertion of Global Forces Accelerating the Flow
The distinctive velocity jets formed in the gaps between the moderator boxes lead to an accel
piece geomeration of the flow. Sincetry, a force shall be introe this effect is responsduced into the simpible for the better mlified headpiece geomixing in the detailed headetry to generate
this acceleration.
between the mThe local acceleration within thoderator boxes by the forced deviate headpiece is based on contiion as a result of the reduction in the cross nuity. The flow is accelerated
ssipated. An approach to capture i jets are dsection. Further behind the headpieces, the velocitythis effect is to add forces directed in the flow direction.
Himmel has used an analogue approach to accelerate the flow in [27], which is based on the
ometriesbalance of mom inentum the flow are introduced. In comp as resisarison to the portanceous ms, here voelumetric forces are introdia approach, where the effects of geduced as
the direction of the flow. sources pointing in ntumemom
In the previous chapter 5.2.2 (Insertion of Global Forces) only the resistance of the water
viation has not been regboxes has been introduced into the farded. To adld ow, while these effects, an acceleration force the accelerating effects due to the forced deFA has to be intro
the volumduced, which can be derivee, where the forces will be appliedd by integrating the stead: ystate Reynolds equations (2.69) over
⎛⎛∂u⎞⎞∂p⎛∂⎡⎛∂u∂uj⎞⎤⎞
V∫1⎝⎜⎜ρ42⎝⎜⎜4u4j∂xii43⎠⎟⎟4⎠⎟⎟d4V=−1V∫42∂xii4d3V+V∫1⎝⎜⎜∂x42i4⎣⎢⎢4()μ4+4μt4⎝⎜⎜∂xij434+4∂xi4⎠⎟⎟4⎦⎥⎥⎠⎟⎟4dV.
ConvectionPressureThe following simplifications are introduced in equation (5.Resistance3):
a.ensional. The flow is regarded as onedim
(5.3)
b. The velocities are constant over the in and outlet cross section of the analyzed vol
e. um
c. The pressures are constant over the in and outlet cross section of the analyzed vol
e. um
The different terms are indicated in the equation (5.3). Since the resistance of the omitted
can be restance termiious chapter, the resstructure has already been included in the prevplaced by the global resistance forceFR. Thus, the integration leads to the following momen
balance: tum
(1ρ1A4214u124−ρ2434A24u22)=−1()A2424p2−43A14p1+F{R. (5.4)
AccelerationPressureResistanceHere the indices 1 and 2 represent the values at the in and outlet cross section of the regarded
e. volum
85
itted StructuresEffects of the OmIncluding the
(5.5)
The acceleration can be introduced with the acceleration force according to:
(ρ1A1u12−ρ2A2u22)=FA. (5.5)
Assuming constant density, this leads to:
ρ(A1u12−A2u22)=FA. (5.6)
FA can be calculated with the same cross sections as shown in Fig. 518. The velocity u2 can
be calculated applying the continuity equation:
AρA1u1=ρA2u2, u2=A1u1.
2unction of the velocity: can be written as a fFThus, the acceleration force A
(5.7)
22FA=ρ⎛⎜⎜⎜A1u12−A2⎛⎜⎜AA1u1⎞⎟⎟⎞⎟⎟⎟=ρ⎛⎜⎜⎜A1−A2⎛⎜⎜AA1⎞⎟⎟⎞⎟⎟⎟u12. (5.8)
⎝⎝2⎠⎠⎝⎝2⎠⎠
Since a volumwhich the force has been derived:etric force has to be applied, equation (5.8) has to be divided by the volume, for
22FVA=Vρ⎛⎜⎜⎜A1u12−A2⎛⎜⎜AA1u1⎞⎟⎟⎞⎟⎟⎟=Vρ⎛⎜⎜⎜A1−A2⎛⎜⎜AA1⎞⎟⎟⎞⎟⎟⎟u12=CA⋅u12. (5.9)
⎝2⎠⎝2⎠
The constant prefactor⎝CA=−480kg/⎠m4⎝ is found. It is negative, ⎠since the force is applied
and points in flow direction. on the right side of the equation
To combine the source terms for the resistance and the acceleration in one simulation, a re
scalculated. Therefore the two source termetric force has to be sulting global volumFSR=FR/V and FSA=FA/V have to be added as shown in Fig. 521. The resulting source
term for the entire headpiece is FS=FSA+FSR.
The acceleration force is directed into the direction of the flow. Adding the resistant force
ow reduces the acceleration force in flow diposite direction slowing down the flacting in oprection.
s leads to the following equation: termtric source eApplying the derived volum
⎡⎤ρ⎛⎜uj∂ui⎞⎟=−∂pi+∂⎢()μ+μt⎛⎜∂ui+∂uj⎞⎟⎥+FiSR+FiSA. (5.10)
⎝⎜∂xi⎠⎟∂xi∂xi⎢⎣⎝⎜∂xj∂xi⎠⎟⎦⎥
in which FiSR=FiR/V and FiSA=FiA/V are the volumetric source terms for the resistance
. sfferent directioniand acceleration in d
86
3]F/V [N/m120006000
0024600012000
Including the itted StructuresEffects of the Om
FSR
u [m/s]86F+ FSASR
18000FSA24000 s of the introduced structure describing the different effects Resulting source termFig. 521 In the following Fig. 522 only the resulting source term FS=FSA+FSR is plotted. In the
igin with a negative slope of a straight line staring in the orted with amodel it will be approxim. m=−1500
F/V [N/m3]
00
40000008
246
m= 1500
u [m/s]
FS= FSAFSR
00120 Fig. 522 Resulting source term accelerating the flow in the detailed headpiece structure
In the model, the resulting source term FiS in the different directions is entered according to:
⎡⎤ρ⎛⎜⎜uj∂ui⎞⎟⎟=−∂pi+∂⎢()μ+μt⎛⎜∂ui+∂uj⎞⎟⎥+FiS. (5.11)
⎝∂xi⎠∂xi∂xi⎣⎢⎝⎜∂xj∂xi⎠⎟⎦⎥
87
itted StructuresEffects of the OmIncluding the
the resuThe results achieved wlts for the simpith this mlified headpiece method leadodel with to good a prediction introduof the mced global forces in floixing. In w directioFig. 523, n
and the results for the detailed headpiece model are compared. The results for the model with
xing than the i even better results for the mglobal forces in direction of the flow reproducescture is entirely differough the local flow strudel with added local forces. Even thodetailed ment, the scalar distribution at the analyzed outlet is very similar when comparing the distribu
spots with a very small scalar fraction are tion displayed in the figure below. In the center,rners and especially tion are found on the cofound, while regions with a higher scalar fracrather on the right. For the model with the introduced global forces accelerating the flow,
r I outlet, for the e neighboring superheateodel through the the m37.1% of the inlet scalar leavdetailed model this value is only slightly smaller with 35.5%. For the differences of the peak
values at the analyzed outlet, good agreement is achieved as well. The local minimum for the
model with global forces in flow direction pared with values in between 29.7% and 44.0% fois 28.4% and the local mr the detailed maximumodel, the agreem is 46.6%. Coment is more 
ory. tactisfthan sa
[]/ms
[]/msWith source terms: Geometrica lly resolved:
Min: 28.4% Sc1 averaged: 37.1% Max: 46.6% Min: 29.7% Sc1 averaged: 35.5% Max: 44.0%
Fig. 523 Compheadpiece model with garison of the results for the detailed hlobal forces (bottom)eadpiece m accelerating the flow odel (top) and simplified
88
Including the itted StructuresEffects of the Om
chamSince only the effects of the headpiece geomber are analyzed and local differences canet ry on the global flow field in the upper mbe disregarded, the distribution of the inlet ixing
ent is well. Again, good agreemodel has to be analyzed as scalar to the sides of the machieved. While 9.6% of the inlet scalar exit the model on either side in the detailed model,
12.0% exit the model on the right and 10.7% on the left for the simplified model with global
below). odel fromooking at the mforces in flow direction (when l
Due to the very good agreement of the mixing, the comparison of the two models can be in
obal forces in flow direction to represent the terpreted as a validation of the approach using glheadpiece geomwhich have been derived exemetries in the upper mplarily for the ixing chamcombination ofber. Therefore, the defined source term the inlet 6 and outlet 1 heads,
pieces, now have to be introduced in the simplified model of the upper mixing chamber. Since
the source termheadpiece position in the upper mixing chams are defined as a function of the velocity, their introduber. ction is valid for each
The results for the pressure drop, however, can not be further regarded, since the pressure
drop of the additional structure is locally included as a resistance source term. This resistance
source term then is summed up with the acceleration source term locally to capture the local
acceleration effect. The resulting volumetric force pointing in the direction of the flow then
d headpiece structure. ssure drop of the detaileartificially revokes the pre
The defined global source terms are applied to the reference case of the simplified model as
presented in chapter 3.2.2 and to the optimized case with meander alignment, presented in
s, a model with only 300.000 cells is term4.5. For analyzes with introduced source chapter the presented results also hold for this grid. used. As shown with the grid sensitivity study, The effects for the mixing are shown in the following diagrams depicted in Fig. 524.
0,880,0,70,60,50,0,76
50,0,4tioner FrackMar0,1er MaximumMark0,0,21
40,0,330,0,2001234567Outlet1234567
Inlet_1Inlet_2Inlet_3Inlet_4Outlet
Inlet_5Inlet_6Inlet_7Inlet_8IInlet_nlet_51IInlet_nlet_62InInlet_7let_3IInlet_nlet_84
Fig. 524 Result for the scalar distributions for the simplified model with introduced forces
eces i the headpentingrepres
A velocity plot for the simplified model with introduced global source terms in flow direction
and for the simplified model without any introduced source terms at a horizontal cross section
Fig. 525. epicted in e model are dat half the height of th
ort cuts between the directly neighboring in and Due to the much larger flow velocities, the shoutlets are slightly decreased and especially the peak values of the different inlet scalars at the
89
itted StructuresEffects of the OmIncluding the
outlets are reduced significantly. For the simplified model without any included source terms,
peak values reach almost 70 %, whereas here, the maximum peaks values stay just below
60%. For the standard deviation a value of σ=11.5%is obtained.
[]/ms
Fig. 525 Comparison of the results fort he u,v, w velocity components at a horizontal cross
section for the reference case of the simrepresenting the headpplified iece geommodel without (top) and wetries (bottom) ith global forces
ble for the reduction of direct short cuts besied by the headpieces are responThe jets formtween neighboring in and Fig. 526. Here a cut through the moutleots. The efdel parallefects ofl to symm the jets can be visualized as canetry boundary II (as indicated on the be seen in
right side ofpast its neighboring outlet the figure) is shown. T 1 and therefore, a smhe plotted inlet scalar aller fraction is transporte6 forms a mod directly tore pronounced streak outlet 1.
plied source termIn the case without source terms, this value is reduced to s, 53.4% are detected at outlet 1,45.2%. For the neighboring whereas in the case with apoutlet 7, the inlet sca
31.4% to 40.4 %. increased fromlar 6 fraction, however, has been
90
Including the itted StructuresEffects of the Om
66Reference case: Reference case: without source terms with source terms
6 distribution in a vertical cross section for parison of the results for the scalarComFig. 526 s odel without and with source termplified mthe sim
Another distinctive influence of the headpieces is the better mixing of each inlet scalar itself.
The improvement can be shown, by analyzing the differences between the minimum and
maximum values for each scalar at each outlet. For the case with no introduced source terms,
the mean difference between minimum and maximum at each outlet measured for each inlet
outlescalar is 13.6%. The mt 3. Here, the maximauximmum diffe local concentrration ofence in the peak values inlet scalar 8 is 70.9are found for inlet scalar 8 at % and the minimum
s, the ced source termr the case with introduis 7.6% resulting in a difference of 63.2%. Fommumean difference between m difference of 48.0%, again found inimum and mfoar inlet sximumcala r 8 at outlet 3. is found to be only 8.5% with the maxi
1_tlein
5_tlein
inlet_2
inlet_6
inlet_3
lein7_t
4_tlein
inlet_8
Fig. 527 Scalar concentration for the simplified model seen from underneath with introduced
s global source term
To round up the application of the simplification method to the upper mixing chamber, the
last step is to apply the determined source terms to the optimized case of the simplified model
presented in chapter 4.5 (Collection and Redistribution of the Inlet Flows Meander Align
91
Effects of the OmIncluding the itted Structures
ized case with e not as significant for the optim forces arent). The effects due to introducedmmeander alignment. Very similar results are obtained for the optimized case without intro
ca for the different stween the peak valuess concerning the differences beduced source termlars at each outlet. In addition, almost no difference for the scalar distribution to the outlets is
obtained. The overall mixing is slightly better at a standard deviation of σ=8.4%compared
to σ=8.8% for the optimized case without source terms.
0,8n0,6er Fractio0,4rk0,2Ma0
0,80,6Marker Maximum0,40,20
12
Outlet67534Inlet_5Inlet_1Inlet_6Inlet_2Inlet_7Inlet_3InInlelet_t_84
123
4567123OutletIInnlet_5let_1Inlet_6Inlet_2Inlet_7Inlet_3Inlet_8Inlet_4
Fig. 528 Result for the scalar distributions for the optimized simplified model with introduced
eadpieces forces representing the hThe increased effect of the headpieces in the reference case compared to the optimized case
with meander alignment is explained by the significant change in the flow structure between
d by the introduced dpieces are suppresseintroduced by the heaents. The jets the two alignmmixing stages. In vertical walls in the optimFig. 529 the u, v, and wvelocities arized case and different jets are introduced in the gaps between the e plotted in the defined vertical cross
section. Almost the same jets between the mixing stages are obtained, only for the case with
source terms, shown on the bottom; the jets are slightly more pronounced showing less curva
the outlets. ture towards
92
itted StructuresEffects of the OmIncluding the
[m/s]
Fig. 529 Velocity plots for the optimsource termized cass at a vertical cross sectioe without (top) and withn (bottom) introduced
ces uced Headpiece InfluenConclusions Regarding the Introd 5.2.4
The better midue to the distinctive velocity xing and the better glojets formbal scalar died in the gaps betweenstribution in the m the detailedoderator boxes These jets headpiece model are
lead to a local acceleration of the flow due to the forced deviation as a result of the reduction
in the cross section.
e moderator boxes loduced resistance by thoSince the acceleration effects outweigh the intrtermcally, a resulting volumetr has been calculated and delivers good resuic force points in the direction of the flowlts concerning the mi. The resultingxing. However, the re source
pointsults for the pressure drop can not be further evaluated, since the resulting source termtailed headpiece structure. s the pressure drop of the deing in flow direction revoke
forces applied in chapter If the pressure drop introduced by 5.2.2 (Insertion ofthe headpiece stru Global Forces)ctures is of must be useinteresdt. In this, the glob caal resisse, thet efant
capturfect on the fled, while the veloow further behind thcity jets are dissipae hted. eadpiece structure is of interest. Here the pressure drop is
93
Analysis of Temperature Depending Effects
6 Analysis of Temperature Depending Effects
operties and buoyancy influt for constant fluid prSo far all the analyses have been carried ouCharacterization of 2.3 (ences have been disregarded. However, applying the in chapter play a significant role. Buoyancy Influences) derived criteria leads to the assumption that buoyancy effects might
A Grashofnumber of Gr=1.1⋅1014 is found when calculating it with the inner height of the
upper maverage cooilxing chamant tempber as the characteristic length erature and the peak coolant temlp, the temeraturperature of a hot channel accoe difference between the rding to
[68], which is ΔT=50K, the acceleration due to gravity g, and the volumetric thermal ex
pansion coefficient α=0.008K−1, as well as the kinematic viscosity ν=0.147⋅10−6m2/s
2.1 nd 390°C as given in the beginning of chapter defined for water at a pressure of 25MPa a(Characterization of the Flow in the Upper Mixing Chamber). The Reynoldsnumber also has
to be defined with the characteristic length l, the height of the upper mixing chamber and the
mean inlet velocityu=3.997m/s.
3Grashofnumber: Gr=buoyancyforce=αgl(T2−T0)=1.1⋅1014
viscousforceνReynoldsnumber: Re=inertia=ρul=ul=1.3⋅107
frictionμνGrRe2≈0.7
ation, where a vos, the Boussinesq approximDue to the strong changes in the fluid propertielumetric lift termflow. The strong changes in the fl is added as described in chapuid properties have to be regardter 2.3, does not hold fed as well. At supercritical or the here analyzed
temperatures close to thpressure conditions the densitye pseudo critical point. , conductivity, specific heat and viscosity vary strongly with
At a pressure of 25MPa, the density ratio changes by more than a factor of 8 in the tempera
heat capacity exhibits a very pronounced local R and the ture range of interest in the HPLWmaximum at the pseudocritical temperature.
ore ffects in the flow field could be muggests that buoyancy eThe strong change in density sxing and on the pressure drop iffects on the mperature dependent epronounced. Therefore, tementhalpy distributions at the outlet of the upper erature and pneed to be analyzed and the temmixing chamber need to be calculated. The curve progressions according to the IAPWSIF97
water and steam table [81] for the mentioned fluid properties are depicted in Fig. 61.
94
900800700600500]3m/g [kρ300
4002001000020
040
Analysis of Temperature Depending Effects
070060050040]Km/[W300
λ 0200100600T [°C]200
004
600T [°C]
8001270010608050cp [kgK]J/k20/6kg01 [μ]sm40
604030201000200400600T [°C]200400600T [°C]
Fig. 61 Fluid properties for water at a pressure of 25 MPa
Laurien et al. have included these changes in the fluid properties in their model for a pressure
et al. have introduced functions for a ogue to their approach, Kunik l[44]. Anaof 24.5MPa in [40]. To include these functions in [39] and odel as presented in pressure of 25MPa in their m for the enthalpy as a function of temperature nl, an additional functioodethe here presented mhad to be included according to the relationship: dh = cp(T) dt (6.1)
The included function is shown in Fig. 62, where the dotted line represents the values taken
from the IAPWSIF97 water and steam table [81] and the straight lines g1 and g2 are used for
the curve fit. The enthalpy h is approximated by the superposition:
h = g1f + g2 (1 f) (6.2)
of two linear functions: g1 (T) = h1 + (h2h1) / (T2T1) (TT1) (6.3)
g2 (T) = h3 + (h4 h3) / (T4T3) (TT3) (6.4)
by the blending function f: f = 1 / 1 + exp ((TTps) / w) (6.5)
Here Tps is the pseudocritical temperature and the enthalpies h1 to h4 as well as the tem
peratures T1 to T4 represent selected data points. The characteristic parameter w determines
curve fit possiieve the best tion and can be adjusted to achthe width of the exponential transible.
(6.3) (6.4)
95
Analysis of Temperature Depending Effects
Fig. 62 Curve fit for the enthalpy at 25 MPa
at a pressure of 25MPa is found to be: The retrieved function describing the enthalpy
g1g2==249643127436++(1496284(404400524964)/(603127436)/(31073273)*(T763)*273)(T763)
f1=1/(1+exp((T663)/10))
h=g1*f1+g2*(1f1)
(6.6)
To check the implemented functions, a test case has been defined in which a volumetric heat
up has been applied to a horizontaties correspond to the physical ones. A very thil, rectangular tube to verifn, 2m long pipe has been chosen as test gey that the predicted fluid proper
om0.5m/etry. For the simus and arbitrary values for K and lation a block profile has been ε in the sameapplied at range as for the sim its inlet with a mulations of the upper ean velocity of
miresults for all analyzed fluid xing chamber. At the outlet, a regulproperties over the temar outlet boundary has perature albeen applied, in ong the length of the test pipe Fig. 63, the
table and the values obtained in the test case are plotted. A very good correspondence between with the applied functithe values given for the water and steamons for the density, con
ductivity, heat capacityderived for the enthalpy as a function of tem, and viscosity according to Kunik perature plotted in [40] and the additional function Fig. 62 as curve fit. The val
ues given by the IAPWthe values according to the introduced fSIF97 water and steamunctions are depicted table by the pink lines. [81] are depicted with the blue lines and
96
0010800600m/g [kρ]3400
2000200
400
Analysis of Temperature Depending Effects
070600050040]K/m [Wλ300
0200100600T [°C]200
040
C]°T [006
12080701006080504060]Kgk/J [kcp20]smg/k601 [μ40
30201000200400600T [°C]200400600T [°C]
Fig. 63 Results for the test case with the introduced functions for the variable fluid proper
ties introduced the flow, the functions describing the variable perature dependant effects in To regard the temfluid properties as well as the additional volumetric lift term g(ρ−ρ∝) to regard the effects
e model for further analyzes. of buoyancy are introduced into th
Characteristic Flow 6.1 Patterns
For the analysis of temperature dependant effects, the simplified model as presented in chap
bout the influences of temperature dependant principal conclusions a3.2 is used. To drawter effects in the upper mixing chamber, one of the 8 inlets is charged with a higher temperature
the presented evaluatemperature of 390°C. Fore whereas the others are charged with the saming 100°C above the temperature perature of 490°C, thus betemtion, inlet 6 is charged with a of all the other inlets. Inlet 6 is chosen, since it represents the worst mixed inlet cluster. If a
hot inleinlet tempt streak occurs ferature should moake r this cluster, thsure that possible buoyancy effects are detected. Two extreme worst case is obtained. A significantly overstated e
cases can arise for this setting; either the mass flow rate or the volume flow rate in the hot
cluster stays constant. In this subchapter the analysis of these two cases is presented to dem
xed ie, where the least mtheoretical worst casonstrate and draw principal conclusions for the the miinlet flow is also the hottest, xing in the upper mixing chamber. and to evaluate the principal influences of buoyancy effects on
97
Analysis of Temperature Depending Effects
6.1.1 Constant Volume Flow
In the case with constant volume flow rate, the inlet velocity of 2.78m/s is kept for all inlets
despite their temcur. Two different cases are simperatures. This way, the mulated, one wherost significane the temt temperature perature of inlet 6 is 100K adependant effects ocbove the
0K below. erature and one where it is 10paverage inlet tem
the domThe results for these two cases, depinant influence as expected. In corresicted in Fig. 64, show that buoyancy effects do not have pondence to the very high Grashof number and
the criteria derived in direction has been expected due 2.3, a noticeable acceleration of the hot to buoyancy effects. However, foflow entering inlet 6 in vertical r the case in which inlet 6 is
hotter than the other inlets, the much lighter inlet flow with a much lower density is over
blown by the other inlets entering in the cenFig. 64. On the top, the reference case for the simplified mter of the upper moixing chamdel with constant fluid properties ber, as shown in
is depicted. On the bottomcan be seen for this configuration. Analogue resu left, the case with a hot inlet 6 lts are obtained when inlet 6 is mis depicted. No buoyancy effects uch colder
er mthan theo other inlets, dementum when entering the upper mpicted on the bottomi rixing chghamt side. The colder inleber due to its higher density. Tt flow has a much highhis way it
low density in comparison. ral inlets which have a veryis blocking all the other cent
nt fluid properties)reference case (constant fluid properties)reference case (consta
T = 390°C, T = 390°C, = 215kg/m3ρ = 215kg/m3ρ
ariable fluid properties and buoyancy)change in temperature at inlet 6 (variable fluid properties and buoyancy)change in temperature at inlet 6 (vhothotccoldold
(T=490°C, (T=490°C, 66(T=(T=290290°C, °C,
ρρ=93=93kkg/g/mm33))ρρ=76=761kg1kg/m/m33) ) 66
Fig. 64 Different temperatures applied to inlet 6 for the reference case of the simplified
e flow model with const. volum
hen looking xing is significant. Wie has on the mThe influence this change of the flow structurat the results for the hot inlet 6, a much better mixing of the hot inlet streak is obtained, whe
reas the overall mixing evaluated with the standard deviation stay pretty much the same. The
standard deviation for the case with a hot inlet 6 is σ=12.2%, while σ=12.1% is obtained
98
Analysis of Temperature Depending Effects
scalar fraction of the hotfor the reference case. In inlet scalar 6 detectFig. 65 the results for the scalared at the neighboring outlet 1 is decreased, higher distribution are shown. While the
fractions and peak values as in the reference case are obtained for other inlet scalars at the
ts. outle
80,0,860,0,6Marker Maximum40,Mation Fracrrke0,20,2
0,4001234567Outlet1234567Outlet
Inlet_5Inlet_1InlInleet_6t_2InletInlet__73InInlelet_8t_4InInlelet_t_51InInlet_let_62InInlet_let_73InlInlet_8et_4
Fig. 65 Result for the scalar distributions for the simplified model for the case with const.
volume flow and an inlet 6 temperature 100K above the mean temperature
the effect described before as overblowing. xing of the hot streak is the result of iBetter mThe fluid with the much smaller density is pushed aside by the fluid with the much higher
density leading to a better distribution of the hot fluid to all outlets. However, this effect also
ally for inlet 4, which neighboring inlets, especixing of the ileads to negative effects for the mng inlet 6, can reach troduced by its neighborinow, due to the decreased obstructing effect inthe inlets 1, 2, 4, and 6 and the Fig. 66, a vertical cut through outlet 7 in a more direct way. In outlets 1 and 7 of the upper mixing chamber illustrates the more direct short cut between inlet
4 and outlet 7.
11
22
6464Fig. 66 Scalar distribution ininle a vertical cut through the upt 6 per mixing chamber with a hot
99
Analysis of Temperature Depending Effects
As expected, a much worse result for the mixing is obtained in the case with an inlet 6 tem
perature 100K below the average inlet temperature. The obtained results are depicted in Fig.
67. A blockage of the colder fluid entering the upper mixing chamber through inlet 6 with a
much higher density is established and suspends the flow from the other more central inlets.
inlet 6 scalar at the outlets 1 and 7. The stanThis can be seen by the very high values for the dard deviation is much higher at a value of σ=15.8%, compared to σ=12.1% obtained for
the reference case.
0,80,80,60,60,40,4ekMarnotiac Frrm MaximuerMark0,2
0,2001234567Outlet1234567Outlet
InletInlet__51InleInlet_6t_2InleInlet_7t_3InleInlet_8t_4InInlet_let_51InInlleet_t_62InInlet_let_37InlInlet_4et_8
Fig. 67 Result for the scalar distributions for the simplified model for the case with const.
volume flow and an inlet 6 temperature 100K below the mean temperature
inertial effects clearly At this point it can be concoutweigh the buoyancy effluded that buoyancy effects and the hot inlet flects are not as dominant as expected. The ow entering at inlet 6
is actually overblown rather than showing an additional acceleration in vertical direction.
Constant Mass Flow 6.1.2
To analyze the case with constant mass flow through all inlets, the inlet velocity of inlet 6 has
to be adjusted according the density change. For the case with an inlet temperature 100K
above the average temperature, an inlet velocity of 6.31m/s is obtained, for the case with an
inlet temperature 100K below, the inlet velocity is much smaller at a value of0.79m/s. In
comparison, the inlet velocity for the mean temperature of T=390°Cand the corresponding
density of ρ=215kg/m3 an inlet velocity of 2.78m/s is obtained.
For this case, almost the same result is obtained as for the reference case without regarding
the temperature dependant effects. Also, the difference due to the change in inlet temperature
is smhigher inleall as can be seen in t velocity for the casFig. 68. The sme with thall differences in the e hot inlet than of buoyancy effects. flow are rather the result of the
The effects for the mixing are small, but also the same effect for the mixing of the hot leg can
be observed. Due to the smaller density, the lighter fluid is distributed much better to the dif
ferent outlets. The standard deviation for all three cases, the reference case, the case with an
inlet 6 temperature 100K above the mean temperature, and the case with the inlet temperature
100K below the mean temperature stays the same at σ=12.1%.
100
nt fluid properties)reference case (constant fluid properties)reference case (consta
Analysis of Temperature Depending Effects
T = 390°C, T = 390°C, = 215kg/m3ρ= 215kg/m3ρ
ariable fluid properties and buoyancy)change in temperature at inlet 6 (variable fluid properties and buoyancy)change in temperature at inlet 6 (vhothotccoldold
(T=490°C, (T=490°C, 66(T=(T=290290°C, °C,
ρρ=93=93kkg/g/mm33))ρρ=76=761kg1kg/m/m33) ) 66
Fig. 68 Different temperatures applied to inlet 6 for the reference case of the simplified
ss flow amodel with const. m
the case with the inlet ound that cases, it is fWhen evaluating the results for the two analyzed6 temperature 100K below the mean temperature delivers almost the same results as obtained
for the reference case. Fperature, analogue results are obtaor the case with the ined. The only significainlet 6 temperature 100K above the mean temnt difference is that the lighter inlet
6 fluid is now distributed better to the outlets, while the distribution of the other inlet scalars
stays almost the same. The result for this case can be seen in Fig. 69.
0,80,80,60,60,4ntioc FraerrkaMmumxir MaMarke
0,40,20,20012345671234567
OutletInlet_1Inlet_2Inlet_3Inlet_4Outlet
InlInleet_t_15Inlet_2Inlet_6InInlelett__37InInlelett__84Inlet_5Inlet_6Inlet_7Inlet_8
Fig. 69 Result for the scalar distributions for the simplified model for the case with const.
mass flow and an inlet 6 temperature 100K above the mean temperature
Summarizing the observed results for these extreme cases with a temperature change imposed
xierature dependent effects do influence the mponly at inlet 6, it can be concluded that teming, but buoyancy effects do not become as dominant as expected.
101
Analysis of Temperature Depending Effects
th a Specified Inlet Temperature Distribution iCase w 6.2
In addition to the presented cases, where extreme assumptions have been made for inlet 6
cases have been analyzed. The inlet boundaries ore realistic representing the worst case, mhave been defined according to the results for the thermal core analysis presented by Maráczy
et al. from[46]. The therm the Hungarian Academal core analyses have been cary of Sciences KFKI Atomic Energy Rried out with the KARATE Code as presentedesearch Institute in
in [45]. The given temperatures of the assemblies in each cluster have been averaged arith
mcertainties, etc. have noetically to define the inlet temt yet been conperature for sidered in these analyses. each cluster. However, tolerances, burnup, un
The inlet temperatures, shown on the right side of Fig. 610, and the inlet enthalpies for the
different inlets are listed on the left side of the figure. Here, inlet 7 is the inlet with the lowest
inlet temperature marked in grey and inlet 4 has the highest inlet temperature marked in blue.
InletInletT [T [°°C]C]T [T [KK]]h [kJ/kg]h [kJ/kg]44
11385,4385,4658,6658,62188,52188,588
22389,4389,4662,6662,62377,72377,7333355
33384,5384,5657,6657,62125,02125,011224545767621216767
44389,4389,4662,6662,62377,72377,7
55388,3388,3661,5661,52340,82340,8
66387,7387,7660,8660,82316,02316,0
77383,4383,4656,5656,52055,42055,4
88386,9386,9660,1660,12278,72278,7 [K]
Fig. 610 Realistic inlet temperature and enthalpy distribution according to [46]
Due to a relatively homogeneous neutron flux in the center of the core, the temperature and
ator only vary between 383.4°C and 389.4°C for the the enthalpy at the outlets of the evaporperature and between 2055.4kJ/kg and 2377.7kJ/kg for the enthalpy. This equals a temtemheperature difference of 6K and an enthalpy difference of 322.1kJ/kg. Wparing the ren comsults for the mixing and for the temperature and enthalpy distribution at the outlet for the ref
erence case, no significant differences are obtained for the assumption of constant volume
assumpflow or constant mtion of constant maass flow rss flow. For the case where thate and the corresponde inlet veloing inlet dencities have been defsity for eachined with the inlet tem
city is calculated with the density fraction perature has been looked up, the new inlet velotimes the reference inlet velocity. When assuming a constant volume flow rate, all inlets are
charged with the sament cases are analyzede: th inlet velocity and only e reference case of ththe simplife density anied mod temdel as prperatesented in chure varies. Two differapter 3.2.2
and the optimized design of the upper mixing chamber with meander structure to enhance
mixing as presented in 4.5. The results for the mixing are not significantly affected when
compwhere only constant fluid propearing the cases in which temprties are apperaturlied and no tempere depending affects are considature dependent efered with thfects are e cases
regarded.
102
Analysis of Temperature Depending Effects
For the standard deviation the same values are obtained for the case with constant mass flow
rate: σ=12.1% for the reference case and σ=8.8% for the optimized case. In addition, al
most no influence on the scalar distribution is noticeable for both cases. The achieved en
thalpy and temperature distributions are depicted in Fig. 611 for both analyzed cases. Here,
the upper mixing chamber is shown from the bottom and a dotted line is introduced to sepa
rate the inlet from the outlet side. Also, only the values on the outlet side are depicted to ad
just the scale to smaller values for the enthalpy, for the temperature the scale remains un
changed. While the result for the reference case, which is depicted on the left side, shows a
large variation in enthalpy and temperature, it can be seen that these differences in tempera
ture and enthalpy largely disappear.
[]/Jkg
[]K
Fig. 611 for the reference casEnthalpy (top) and teme (left) and for thperature (bottoe optimm)ized case (righ distribution in the upper mt) const. miass flow xing chamber
When looking at the outlet side of the optimized upper mixing chamber, the temperature var
py only varies between 2247kJ/kg and ies between 386.8°C and 388.4°C and the enthalpy of 322.3kJ/kg at the of 6.1K and in enthalerature p2208kJ/kg. Thus, the variations in temmixing chamber inlets have been reduced to 1.1K and 73 kJ/kg at the outlet side, as summa
Tab. 61. rized in
Tin= 6.1KΔ
hin= 322.3 kJ/kgΔ
Tout= 1.1KΔ
hout= 39 kJ/kgΔ
Tab. 61 Comparison of inlet and outlet temperature/enthalpy differences in the upper
ss flow) axing chamber (constant mim
103
Analysis of Temperature Depending Effects
For the case with constant volume flow, slightly higher differences in the outlet temperature
Fig. 612. The standard deviation for the and enthalpy distribution are found. As depicted in reference case is a little bit higher for the case with constant volume flow at a value of
σ=12.6%as for the case with constant fluid propertiesσ=12.1%. The same is true for the
optimized case where the value for the standard deviation has been increased toσ=9.1%,
compared toσ=8.8%. The scalar distribution has changed almost insignificantly.
[]/Jkg
][K
Fig. 612 for the reference casEnthalpy (top) and teme (left) and for thperature (bottoe optimm)ized case (right distribution in the upper m) constant voilumxing chame flow ber
For this case, the temperaturtween 2249kJ/kg and 2183kJ/kg. The differences in e varies between 659.1K and 661.6Ktemperature and enthalpy are summ and the enthalpy varies bearized
flow. eTab. 62 for the optimized case with constant volumin
Tin= 6.1KΔ
hin= 322.3 kJ/kgΔ
Δ Tout= 2.5 K
hout= 66 kJ/kgΔ
Tab. 62 Comparison of inlet and outlet temperature/enthalpy differences in the upper
e flow) xing chamber (constant volumimhen influences as expected. Wuch y effects do not have as mIt can be concluded that buoyancregarding the temperature dependant effects in the upper mixing chamber and applying them
ogenization of temperature is found that the homerature distribution, it pto a realistic inlet temof constant maand enthalpy is successful and only smss and volume flow. all differences are calculated when comparing the cases
104
berUpper Mixing Chamsults for the Integrated Re
Mixing Chamber the UpperIntegrated Results for 7
In this chapter, the results for the upper mixing chamber will be presented including both,
temperature dependant effects and the complex geometries, which had been disregarded in the
simplified model. Thus, all necessary approaches to capture the effects on the mixing are in
tegrated in one simulation only. However, the influence of the change in density has not been
included for the derivation of the introduced source terms. In this chapter, the integrated solu
tion for the mixing in the upper mixing chamber of the HPLWR is presented and the methods
ether. ltogpplied aters are aderived in the previous chap
Two models are analyzed: the reference case with no means to enhance mixing and the opti
mized case with the introduced meander structure as derived for the simplified model in chap
ter 4. The same calculated inlet temperature distribution delivered by AEKI as used in the
d, since slightly worse values ee flow is assumConstant volumprevious chapter is also used. perature distribution. more heterogeneous temxing, leading to a slightlyiare obtained for the m
Since the influences of the complex geometry are much more significant for the mixing than
the inclusion of the temperature dependent effects, the integrated results are very similar to
the results presented in chapter 5 (Including the Effects of the Omitted Structures). In Fig.
71, the results for the reference case with reproduced headpiece geometry and inlet tempera
ture distribution are depicted.
inlet_1
inlet_5
_letin2
inlet_6
inlet_3
7_letin
inlet_4
_letin8
Fig. 71 Integrated results for the scalar concentration with introduced source terms repre
senting the headpieces and a calculated inlet temperature distribution reference case
The global flow structure is comparable to the results of the simplified model reference case.
in the center are distributed to the outer peberxing chamiInlet flows entering the upper mriphery, while the inlet flows entering the mixing chamber close to the outlet side are prefer
sed due to the cuts are decreaoutlets. However, these short entially led to their neighboring
105
berUpper Mixing Chamsults for the Integrated Re
acceleration of the flow in radial direction by the simulated jets, which are generated by the
headpiece geomand 7 due to the acceleration of the flow in lateetries. Inlet scalar 6 for example is now better distriral direction. For one exbuted between outlet 1 emplary vertical cross
Fig. 72. section this is depicted in
6 erence case n for the refbutioInlet scalar 6 distriFig. 72 The better distribution of inlet scalar 6 can also bee seen in Fig. 73, where all inlet scalar
distributions are depicted.
80,70,60,50,40,noactier FrkMar0,1
30,20,0
0,80,70,6er Maximum0,50,40,3k0,2Mar0,10
6712345etlOutInlet_Inlet_51Inlet_Inlet_62Inlet_7Inlet_3Inlet_Inlet_84
6753412OutletInlInleet_5t_1IInnlet_6let_2InleInlett__73InlInleett__48
) and peak ection averaged (tope scalar concentrations, cross sIntegrated results for thFig. 73 (bottom) in the upper mixing chamber reference case
106
berUpper Mixing Chamsults for the Integrated Re
xing is better results for the m distribution, in general, showThe cross section averaged scalarthan in the simplified case. However, the high peaks are still not acceptable. Especially the
local maxima for the inlet scalars are almost not affected when comparing the simplified case
in chapter 3.2 with the integrated case. With a value for the standard deviation of σ=11.8%,
the result for the mixing is noticeably better than for the previous analyses.
When comparing the inlet temperature and enthalpy distribution given in Fig. 610 to the out
let side as shown in Fig. 74, a more homogeneous temperature and enthalpy distribution is
Fig. 74 the temperature distribution 6. In analyzed cases in chapter found than in the previous t eenthalpy, only the outlde are shown. For the and the enthalpy distribution on the outlet siside is depicted to adapt the scale in order to visualize smaller differences.
[]/Jkg
][K Fig. 74 Integrated results for outlet side of the upper mthe enthalpy (toixip) and temperatung chamber reference case re (bottom) distribution at the
ber thus vary betweenThe enthalpy and temperature at the outlets values of 2251kJ/kg and 2166kJ/kof the reference case for the upper mg for the enthalpy and between ixing cham
661.4K and 656.8K for the temperature. Differences in the enthalpy and temperature distribu
tion at the outlet are 85kJ/kg and 4.8K.
by the headSince the lopiece geomcal acceleration effects introduced inetries is weakened by the introduced mto the flow field of the upper meander structure for the optiixing chamber
mienthalpy distribution are expectedzed case developed in chapter to be sim4.5, the results for the milar to the resultis found for the simxing and temperature as well as plified model.
107
Upper Mixing Chamsults for the Integrated Re ber
The results for the scalar distribuare avoided by the wall welded into the upper mition are plotted from underneath in xing chamber between the inlet and outlet Fig. 75. Direct short cuts
side, only leaving a determduction of the other walls forming the foined gap at the top. rmerly explained mAdditional mieander structuxing is reare. lized with the intro
1inlet_
5inlet_
let_2in
let_6in
et_3lni
inlet_7
inlet_4
8inlet_
Fig. 75 Integrated results for the scalar concentration with introduced source terms repre
senting the headpieces and a calculated inlet temperature distribution optimized case
A vertical cross section shown in Fig. 76 illustrates the flow in the optimized mixing cham
eander structure. ber with the m
1
2
46 Fig. 76 Integrated results for the optimmiized xing chamber in three stages case with meander structure dividing the upper
108
berUpper Mixing Chamsults for the Integrated Re
The flow structure in this optimized upper mixing chamber is almost the same, when simu
lated with headpiece geometries and temperature dependant effect as in the simplified model,
to the vertical walls decelerating the flow and t regarded. This is due owhere these effects are nforming defined jets in the horizontal gaps between the introduced mixing walls and the top
or bottom wall of the upper mixing chamber. This is depicted in Fig. 77 exemplarily for one
vertical cross section.
[m/s]
Fig. 77 Velocity plot in a vertical cross section of the optimized upper mixing chamber
The value for the standard deviation of σ=8.8% remains unchanged in comparison to the
optimized case analyzed for the simplified model. As shown in Fig. 78, the scalar distribu
tion is almost identical to the scalar distribution presented for the simplified model.
0,80,70,60,50,4Marker Fraction0,30,20,10
0,80,70,60,50,4mumxirker MaMa0,1
0,30,20
4567321Inlet_1Inlet_2Inlet_3Inlet_4
Inlet_5Inlet_6Inlet_7Inlet_8
ltOute
6723451tletOuInInlelet_t_51InleInlet_6t_2InleInlet_7t_3Inlet_Inlet_84
Fig. 78 Integrated results for the scalar concentrations, cross section averaged (top ) and peak
(bottom) in the upper mixing chamber optimized case
109
berUpper Mixing Chamsults for the Integrated Re
Thus, the mixing achieved with the introduced meander structure is very successful and is
ature distribution. plied inlet temperalso only slightly affected by the ap
xing chamber, satiting the proposed technical solution for the upper mFor this case, represenisfactory resFig. 79. The enthalpy at the outlet lies beults are achieved for the enthalpy and temperature distributween 2238kJ/kg and 2191kJ/kg, while the temtion, which is shown in pera
nd 659.5K. ture lies between values of 661.0K a
][/Jkg
][K Fig. 79 Integrated results fort he enthalpy (top) and temperature (bottom) distribution at the
ized case xing chamber optimioutlet side of the upper m
The differences in temperature and enthalpy are summarized in Tab. 62 for the optimized
ber. xing chamicase of the upper m
Tin= 6.1KΔ
hin= 322.3 kJ/kgΔ
Tout= 1.5 KΔ
hout= 47 kJ/kgΔ
Tab. 71 Comparison of inlet and outlet temperature/enthalpy differences in the upper
e flow) xing chamber (constant volumim
ized case are the results proposed for the The here presented integrated results for the optimtechnical problem at hand, which is to enhance the mixing in the upper mixing chamber of the
HPLWR to achieve an acceptable temperature distribution at its outlets.
Due to the value for the achieved standard deviation and a satisfactory scalar distribution at
the outlets, with accepbe proposed as technical solution. Considering table peak values, the meander structure as presthe presented values for the enthalpy and temented in chapter 4.5 can 
that the good mperature distribution at the outixing leads to a very let of the optimsatisfactory temized upper mperature homixing chamogenization. ber, it can be concluded
110
Summary and Conclusions 8
Conclusions andSummary
A new method to analyze the mixing of fluids in complex mixing chambers has been pre
sented in this work. The simplification method allows the numerical analysis of mixing
etry, while still plex detail of the geomresolving every combers without geometrically chamcapturing its effects. This leads to numerical analyses with much less necessary computation
time. For this reason, the simplification method is an innovative approach in design processes,
ization. lations are necessary for optimuany simwhere m
HPLWThe proposed simR as an examplification mple for a very comepthod has been lex passive miderived for the upper mxing chamber necessaryixing cham for a tember of the pera
chamture homber with different temogenization. In the upper mperaturesi at the sxing chambeupercritical pressure of 25MPa. The temr, different water flows enter the miperature xing
of these different inlet flows hasection to avoid the propagation of hot streaks s to be homogenized before the in the core of the HPLWfluid enters the next heat up R. This task led to
cal stanother imeamportant part in the an in the used CFDcode StarCD to capture temalysis, which was to include the perature dependant effects. fluid properties of supercriti
Both challenges were tackled independently from each other, before they were combined in
one integrated simulation. The simplification method has been applied to model the complex
geometry of the upper mmoderator boxes. To apply the isimpxing chamber consistilification method, three stepng of a large nums are necessary, which are the ber of connection tubes and
simplification of the analyzed model, the quantification of the effects of the omitted geome
model. This strategy has been applied to the these effects in the ofctiontry, and the introduupper mixing chamber by omitting the complex geometry of the moderator boxes in the head
pieces through which the fluid enters and exits the chamber. For the resulting simplified
model, which consists of a much smaller number of cells, a design optimization has been per
formed which required much less computational time and effort. After finding the optimized
design, the influences of the omitted geometry on the mixing have been determined. As last
step in applying the simplification method, the influences of the omitted moderator boxes
the upper mihave been included into the mxing chamber. odel for the reference case as well as for the optimized case of
The influence of the detailed headpiece geometry with the moderator boxes on the flow has
been added in the model with source terms in the momentum equations. Since only steady
state analysbeen derived with an integrales have been perform balance of med, these omsouerntumce term and pres are volumssure drop correlations. The best etric forces, which have
performance of the method has been achieved, when regarding one headpiece as a whole and
adding the effect globally. For the pressure drop sen, describing the abrupt change of the flow cross section. coefficient, a simple approach has been cho
Hereby, an alignmAdditionally, a detailed CFD anent with neighboralysis of two neighboring heaing in and outlet headpieces in dpieces has been performa transverse flow field has ed.
simpbeen analyzed. Then thlified upper mixing chame introducedber without the mode forces have been applied to an exrator boxes inside the headpieces and the tracted cutout of the
ith the detailed CFD analysis. results have been validated w111
Conclusions andSummary
An effective method to study a large number of different inlet temperature distributions with a
single CFD analysis is the use of passive scalars as markers in an isothermal flow. Here, each
inlet has been marked with a different passive scalar, allowing differentiation between the
thod, endividual outlets. A disadvantage of this mcontributions of different inlet flows to the ihowever, is that buoyancy forces will be neglected. Assuming a certain inlet temperature dis
parate CFD analysis r hand, would require a serkers, on the otheatribution instead of these mfor each case, so that mixing could only be studied exemplarily.
Characterizing the flow and by buoyancy effects cannot be neglected a priori and temthe Grashof and the squared Reynolds numperature dependant effects also need to ber suggests that
be included in the siming approaches have been extended with a diulation. To include functirect relationship between the enthalpy and temons for the significant fluid properties, exist
perature. However, after has been found that buoyancy effeincluding the strong changects do not occus in the fr and only smluid propertieall influences on the ms into the model, it ixing
ature distribution. in the inlet temperhave been detected due to changes
For the integrated results of the upper mixing chamber, both temperature dependent effects
ve been introduced. The results are presentedetries haand the effects of the headpiece geomfor a reference case without means to enhance mixing as well as for an optimized case, repre
senting the technical solution for the task at hand, which was to improve the mixing in the
upper mixing chamenization are achieved in the optimber. Good mixing and thus an ized upper mixing chameffective tember. A mpeandeerature and enthalpy homr structure is introog
chamber. This way, the inlet flows duced in the flow by including vertical walls leare collected and redistribaving gaps at the top or bottomuted. of the mixing
The two components necessary to perform the analysis of this complex mixing chamber are
the introduction of the temciple it should be possible to peratapply these appure dependant effects and the roaches to other tesimpchnicalification ml tasks withouethod. In print major
difficulties. In particular the simplex flow structure. Although quantifplification mying the adeethod is designquate source terms med to be applicabight be a challenge. le to any com
112
eNomenclaturols Latin symb2m[a ]at ][m2
m[/c ]cp )][J/(
 ci cjcj,max 
kg[C ]m[d ]2m[D ]m[ ]dhJ[/e ]f  Fr ][N/
 f′  ff*, f′ 
Fx, Fy, Fz ][N/
N[ ]FSFiS ][N/
nclatureemNo
[m2/s Thermal diffusivity
[m2/s Turbulent thermal diffusivity
[m/s Sound velocity
[J/(kgK Specific heat
zed cell alyalue in the anScalar v  Volume averaged inlet scalar at each outlet
value of analyzed scalar per outlet muMaxim [kg/m4 Prefactor for definition of local forces as source terms
eter Diamm[[m2/s Diffusion coefficient of transported scalar
[m Hydraulic diameter
[J/m3 Inner energy
gnitude amany [N/m3 Volumetric forces vector
Magnitude is Reynolds averaged  Fluctuation of a magnitude
 Star and prime usually represent dimensionless magni
tude [N/m3 Volumetric forces in x, y, z  direction
[N Resulting force
[N/m3 Resulting volumetric force inserted as source term
113
nclatureemNo
]FR ]FiSR ]FA ]FiSAg ]h ]K ] nN p ] ]pc ]q&st ]q&s )]T ] ]Tc ]Tcoldu, v, w +  u ]uτui, uj ]
u′i, u′j ]
~ui, ~uj ]
114
force t ResistanN[[N/m3 Volumetric resistant force inserted as source term
force AccelerationN[[N/m3 Volumetric acceleration force inserted as source term
[m/s2 Acceleration due to gravity
Enthalpy Jkg[/[m2/s2 Turbulent kinetic energy
 Number of evaluated values for inlet scalars (8 inlet
scalars are evaluated at 7 outlets, n=56)
 Number of cells in model
pressure MPa[[MPa Critical pressure
[W/m2 Specific heat flux (per unit area), s denotes a source
[W/m2 turbulent heat flux vector
[kJ/(kgK Specific entropy
[K[°C] Temperature
[°C Critical temperature
[°C Temperature of cold inlet in validation experiment
[m/s] Velocity component in x, y, z direction
velocity nsionless emDi[m/s Friction velocity
[m/s Reynolds averaged velocity components
[m/s Turbulent fluctuations of velocity components
[m/s Favre averaged velocity components
u′i′, u′j′ ]
V ]r ]v ]Vi ]Vtotal xi x+ yYm, Ym 
y ]z ] Greek symbols α ςε λ λ t μ μ tν
nclatureemNo
[m/s Turbulent fluctuations of mass averaged velocity com
ponents [m/s Velocity magnitude (V2=u2+v2+w2)
[m/s Velocity vector
[m3 Volume of the analyzed cell
[m3 Sum of analyzed cell volumes per outlet
tlet Value of each inlet scalar at each ou  Mean value of all inlet scalars at all outlets
nsionless wall distance emDi scalar Transported [mm Horizontal coordinate for measurements in outlet pipe
nt eof validation experim[mm Vertical coordinate for measurements inside mixing
ent ber of validation experimcham
[1/K] Volumetric thermal expansion coefficient
p coefficient orPressure d [m3/s3] Turbulent dissipation
[W/(mK)] Conductivity
[W/(mK)] Turbulent conductivity
[kg/(ms)] Dynamic viscosity
[kg/(ms)] Eddy viscosity
[m2/s] Kinematic viscosity
115
emNo nclature
νt
ρ
τw
[%]σ
σxx, σyy, σzz ]
Θ
]τij
]τxy
]τxz ]τyzt ]τ
116
[m2/s] Turbulent kinematic viscosity
[kg/m3] Density
[N/m2] Wall
[N/m2

[N/m2 Shear
[N/m2
[N/m2
[N/m2
[N/m2
stress shear
deviation Standard
al stress in x, y, z  direction mNor
Dimensionless temperature (Θ=1 is hot and Θ=0
cold water in inlet nozzles of validation experiment)
stress tensor
Shear stress in xy  direction
Shear stress in xz  direction
Shear stress in yz  direction
Turbulent shear stress tensor
eviations Abbr
R BW
CBS
CFD
DNS
ERCOFTAC
HPLWR High
HTTR
JAERI
KFKI
LES Large
R LW
R Pressurized PW
QUICK
RANS
RPV
RSCW
SIMPLE
SST
StarCD
UD Upwind
rater ReactoBoiling W
Core Bottom Structure
cs iputational Fluid DynamCom
ulation Direct Numerical Sim
Abbreviations
bustion On Flow Turbulence And ComEuropean Research Community
Performance Light Water Reactor
eering Test Reactor nginperature eHigh Tem
ic Energy Research Institute Japan Atom
ic Energy Research Institute Atom
ulation SimEddy
ater Reactor Light W
ater WReactor
tics apolation of Convective Kinem InterQuadratic Upstream
Reynolds Averaged Navier Stokes
Reactor Pressure Vessel
ater Reactor Supercritical Light W
ImimSeplicit Method for Pressure Linked Equations
Shear Stress Transport
ulation of Turbulence in Arbitrary Regions Sim
Differencing
117
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uflage, Berlin, SpringerVerlag, 1993. e, 4. AungslehrZIEREP, J.:Grundzüge der Ström
Annex ALower Mixing Chamber
Numerical model and reference case A.1.
ber ChamgLower Mixin
The simulation settings for the lower mixing plenum are identical to the simulations carried
out for the upper mixing chamber. Only here, 1/4th of the geometry has been modeled. The
applied which consistsgrid consists of around 330000 cel of tetrahedral cells fls. For the lowor er mimost of the geomxing chamber a hybrid metry and hexahedral cells for esh has been
ulated. In analogy to the simced passive scalars are evaluathe outlet side, where the introdutions for the upper mixing chamber, porous media cells are introduced to the outlet geometry
llowing superheater II as ent clusters in the foto account for the pressure drop in the fuel elemshown in Fig. A1.
porous media
tetrahedral cells
hexahedral cells
Fig. A1 elemTetrahedral grid for the lower ments for the outlet geomixing chametry and introduced porous meber with block structured, hexahedral dia cells
xing iFig. A2. Since 1/4th of the lower mThe boundary conditions applied are shown in chamber is modeled, there are 13 inlets and 13 outlets defined in the model. The applied inlet
velocity is higher, since the density of the supercritical steam is decreased. For each inlet a
uniform inlet velocity of 11.7 m/s in vertical direction has been defined and the outlets are
etrically etry leads to geomomThe modeling of 1/4th of the gedefined as pressure boundaries. identical cutting planes on both sifor these surfaces. Cyclic or periodic boundaries des of the model, which allowsallow circumferential flow in the geom the use of cyclic boundaries etry,
r cutting plane. To enters through the otheodel through one, resince the fluid leaving the msimulate a global swirl in the lower mixing chamber, cyclic boundary conditions are applied
eich is why 1/4th of the geomodel, what both sides of the mrequiring identical cutting planes odeled. try is m
125
ber ChamgLower Mixin
esurespr
inlet
ccyicl
Fig. A2 Boundary conditions applied to the lower mixing chamber
The scalar dithe reference case, the inlet flows enter the stribution is evaluated at each outlowlet,er mi analogue toxing cham the upper mixing chamber without any tubes at theber. For
the used cyclic boundaries for uperheater I clusters. In spite of the sootpieces ofend of the fcumferential direction asthe cutting planes and the em can be seenpty volum in figue of rethe lower mi Fig. A3. The results foxing chamber, the flow has no cirr only four exemplary
ed in radial direction that the scalars are bundlscalars are shown in this picture. It can be seenpictuand almroe on thst no mie lower left xing takes place. Tside of figure Fig. A3. he different in and outlets are numbered according to the
scalar 3
ar 10scal
scalar 4scalar 11
Fig. A3 Lower mixing chamber: Results for the reference case and in and outlet numbering
When comparing this case to the reference case of the upper mixing chamber, it is found that
xing chamiand outlets, whereas for the upper mshort circuits are obtained between all in is in general higher, at a valuber, critical in and outlet pairs could be idene of around 40%, since eachtified. The volum inlet sce averagedalar is distributed scalar conc to onentration ly
of the upper the worst caset case here doesnt quite reachabout three outlets, but the wors126
ber ChamgLower Mixin
mixing chamber, which is found to be between inlet 6 and outlet 1, where over 50% of the
scalar is transmitted directly. Also the local peak values for the scalars, with about 50% arent
as high as in the upper mixing chamber where they reach local peak values over 70%.
A.2. Optimized design for the lower mixing chamber
Since the geometry of the lower mixing chamber is an empty volume, the mixing strategy
differs significantly from the one chosen for the upper mixing chamber. To improve mixing,
the choal swirl is esen aasy to generpproach is to adjusate. This wat the inlet fly it is possible to incows, since the mreaise the mxing volumie is emxing length signifpty and a globicantly
plished is demnd outlets. How this can be accomand to avoid short circuits between all in aonstrated in Fig. A4 on the left. On the right side it is shown how this approach can be simu
lated.
Fig. A4 Design improvement for the lower mixing plenum
The inlet flow can be influenced by curved tubes or swirlers that can be welded to the holes in
the core support plate (green) in which the footpieces of the fuel element clusters are entered.
For the simulation of a case which is optimized in this way, round pockets are introduced into
be ponditions canis way, the inlet boundary coere the inlet tubes are located. Ththe grid, whsitioned inside the lower mixing chamber and the flow angle can be varied additionally, as
shown on the right side of Fig. A4.
The scalar dithe reference case, the inlet flows enter the stribution is evaluated at each outlowlet,er mi analogue toxing cham the upper mixing chamber without any tubes at theber. For
the used cyclic boundaries for uperheater I clusters. In spite of the sootpieces ofend of the fthe cutting planes and the empty volume of the lower mixing chamber, the flow has no cir
mixing takes place. Wcumferential direction It is found hen compthat the scalars are bundled in aring this case to the reference case of the upper mradial direction and almiost no xing
chamthe upper mixing chamber, it is found that short circuits are obtber, certain critical in and outlet pairs could beained between all in a identified. The volumend outlets, whereas for
e of around 40%, since each inlet er, at a valuhighaveraged scalar concentration is in general worst case oscalar is distributed to of the upper mnily about thxing chamber, where overee outlets, but the worst case here dr 50% of the scalar is transmoesnt quitted directly.ite reach the
ing chamber where theyAlso the local peak values for the scalars, wi reach local peak values over 70%. Tth about 50% arent as high as in the upper mhe result for the optimized casixe
the imfor the lower mixing champroved case are 45° in vertical direction. Thber is shown in Fig. Ae radial direction for th5. The chosen angles for the inlet flows of e other inlet flows is
127
ChamgLower Mixin ber
orthogonal to the particular cutting surface, for the inlets 5 to 9 an angle of 45° between the
inflow direction and the cutting planes is chosen in circumferential direction, according to the
bering shown in Fig. A5 on the lower left side. num
scalar 3scalar 3
scalar 10scalar 10
scalar 4scalar 11scalar 4scalar 11 Fig. A5 Results for the improved case of the lower mixing chamber
With this constellation, a global inlet swirl, which is superimposed with the local swirl be
tween the in and outlet side, is achieved. Thus, the local swirl parallel to the cutting surfaces
is transported by the induced global swirl in circumferential direction and the mixing is im
plary scalar be seen when looking at exemxing can iproved significantly. The effect on the mdistributions. The inlet scalars are now distributed to at least five outlets and the mixing
length has been increased visibly. The inlets in Fig. A5 are accentuated with red windows.
Comparing the volume averaged scalar concentration, it is found that the global swirl in the
lower mixing chamber leads to an improvement from 40% to just over 20% for most inlet
scalars at the outlets. Peak values are also decreased by almost 30% from 50% to 20% in most
cases. For the lower mixing chamber with enhanced mixing, a standard deviation of
σ=5.5% has been achieved. In general it can be stated that that the mixing improvement
oposed swirlers is successful. rusing the p
er mixing chamber re and enthalpy distribution in the lowTemperatu A.3.
The inlet temperatures and enthalpies for the lower mixing chamber have been applied in
uperheater I outlets a higher discrepancy beber. For the sxing chamianalogy to the upper mclusters has been found by Mand enthalpies of the different eratures aptween the inlet temraczy et al. in [46]. The influence of the introduced temperature dependant effects into the
lower mixing chamber is insignificant. For the case with improved mixing, the variations in
temperature of 26.4K and in enthalpy of 151.1kJ/kg at the mixing chamber inlets have been
reduced to 7.1K and 40 kJ/kg at its outlets.
128
Dimensions of the Upper Mixing Chamber and Headpiece Geometry
Dimensions of the Upper Mixing Chamber and Annex B
Headpiece Geometry
Fig. B1 Dimensions of the upper mixing chamber reference case (without any measures to
xing) ienhance m
Fig. B2 Dimensions of the upper mixing chamber optimized case (with meander alignment)
ber: xing chamiInner height of upper m
ized case: Gap width for optim
480mm
1st: 180mm, 2nd: 180mm, 3rd: 140mm
129
Di
nsions of the Upper Mixing Chaem
130
Fig. B3
Di
madpiece Geor and Heeb
etrym
mensions of one cluster within the headpiece geometry
A
Dimensions of the Upper Mixing Chamber and Headpiece Geo
Fig. B4
Di
A
cross section
mensions of the headpiece geometry
A:A
etrym
131