DOTTORATO DI RICERCA IN INGEGNERIA INDUSTRIALE
Curriculum in Ingegneria Nucleare e della Sicurezza Industriale Ciclo XXIX
Analysis of fluiddynamic and heat transfer phenomena with
supercritical fluids
Aprile 2017
Author
Andrea Pucciarelli
Supervisors
Prof. Walter Ambrosini
Prof. Shuisheng He
Dr. Ing. Medhat Sharabi
Coordinator of the PhD program
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The objective of this work is obtaining a better understanding of the heat transfer phenomena occurring when dealing with fluids at supercritical pressures, with the aim of paving the way for the development of the Generation IV Supercritical Water Cooled Reactor (SCWR) nuclear power plant. At the beginning of the present work, no reliable technique for predicting heat transfer phenomena in these conditions was available, including both CFD and heat transfer correlations.
The phenomena occurring in heat transfer to supercritical fluids are in fact much more complex than the ones occurring in fluids in standard conditions. In particular, this is due to the strong variations of the thermodynamic properties occurring in the vicinity of the so called “pseudo-critical temperature”, which marks the single-phase transition from the liquid-like to the gas-like conditions. In addition, buoyancy phenomena imply both impairments and improvements of heat transfer conditions depending on the flow direction and thermal conditions. In fact, in upward flows, the buoyancy forces may imply a relaminarization of the flow inducing a heat transfer deterioration phenomenon. Further along the heated length the same phenomena may induce new velocity distributions (M-shaped) which result in a recovery of the turbulence conditions and, as a consequence, in a new heat transfer improvement. In downward flow cases, instead, buoyancy forces always have a positive effect since they increase the shear stresses in the vicinity of the wall and, as a consequence, improve the heat transfer conditions.
Heat transfer deterioration and heat transfer recovery occurring in upward flows are the hardest conditions to deal with and a better prediction of these phenomena adopting CFD analysis is the topic of the present research. The RANS techniques adopted in the study do not require large computational effort and allow studying even complicated geometries; on the other hand, they are less accurate and reliable than LES and DNS. As a consequence, some particular phenomena may be neglected or modelled in a too simple way for dealing with supercritical fluids, making the results inaccurate. Different paths were considered in the present research project in order to find out which could be the lacking ingredient in the CFD models that are now providing us with better results.
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The University of Sheffield is acknowledged for hosting me in the frame of my six months period of study in the UK. In particular, I wish to thank Prof. Shuisheng He and the “Heat, Flow and Turbulence Research Group” for the support and kind help during my staying in Sheffield.
Dr. Medhat Sharabi is acknowledged for his help and suggestions in the development of the present work.
Ing. Irene Borroni and Ing. Vera Papp are acknowledged for our mutual cooperation in some part of the present work.
The IAEA is acknowledged for including the present research in the Coordinated Research Project on “Understanding and Prediction of Thermal-Hydraulics Phenomena Relevant to SCWRs” through the IAEA Research Agreement No: 18425/R0.
The support to this work by the European Commission through the THINS Project (Grant agreement no.: 249337) is gratefully acknowledged.
CD-Adapco is also acknowledged for making possible this work.
Prof. Walter Ambrosini is, finally, acknowledged for his continuous support and the good cooperative relationship developed during these years.
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Abstract
Acknowledgements
Table of Content
Nomenclature
I
II
III
VII
Chapter 1 -Introduction
1
Chapter 2 -SCWR and Heat Transfer to Supercritical
Fluids
2.1 SCWR: Concept and International Background
2.2 Fluid Properties and Deterioration Mechanisms
2.3 Heat Transfer prediction capabilities
2.4 Considered experimental data sets
Data by Ornatskiy et al. (1971) Data by Fewster (1976) Data by Watts (1980)
Data by Pis’menny et al. (2005a and 2005b) Data provided by Delft University (2014)
5
5
8
15
22
22 22 24 26 27Chapter 3 –Considered modelling techniques
3.1 RANS techniques: considered turbulence models
AKN Low-Re κ-ε model (Abe et al., 1994)
Lien Low-Re κ-ε model (Lien et al., 1996)
Wilcox κ-ω model (Wilcox, 1998 version)
SST κ-ω model (Menter, 1994)
3.2 Modelling the turbulent heat flux
3.3 Favre averaging
29
29
31 32 35 3739
43
Chapter 4 –Application of available models to rod bundles
analysis
4.1 International Benchmark – Supercritical flow in a
7-rod bundle
4.1.1 Test facility and operating conditions 4.1.2 Adopted strategy for the blind simulations
Discretization of the domain The computational effort problem Further assumptions
4.1.3 Results
Case A1: P =25.00 MPa, Tin 297.35 K, G = 2283.44 kg/m2s
Case B1: P =24.98 MPa, Tin 353.38 K, G = 1447.56 kg/m2s Case B2: P =25.03 MPa, Tin 519.58 K, G = 1432.97 kg/m2s
46
47
47 51 52 56 57 59 59 61 66IV
4.1.4 Comparison of the obtained results with the experimental data
Case A1: P =25.00 MPa, Tin 297.35 K, G = 2283.44 kg/m2s
Case B1: P =24.98 MPa, Tin 353.38 K, G = 1447.56 kg/m2s
Case B2: P =25.03 MPa, Tin 519.58 K, G = 1432.97 kg/m2s 4.1.5 Further analyses on non-homogenous distributions of the
supplied power
Simulations performed adopting a helical electric wire as heater inside the rods
4.1.6 Concluding remarks
4.2 Experimental data by Zhao et al. (2013)
4.2.1 Test facility and operating conditions 4.2.2 Results Discretization of the domain Case 1: obtained results Case 2: obtained results
4.2.3 Comments on the obtained results
4.3 Concluding remarks
72 72 73 75 78 78 8384
84 88 88 92 97 102102
Chapter 5 –Inclusion of wall roughness effects in Low-Re
turbulence models
5.1 Introduction
5.2 Basis of the methodology
5.3 First results and validation
5.4 Further analyses and developments
5.5 Results obtained after the improvement process
5.6 Concluding remarks
105
106
107
111
118
122
133
Chapter 6 –Implementation of AHFM in available CFD
codes: procedure and results
6.1 Implementation of AHFM in STAR-CCM+
6.1.1 A first attempt: implementation of GGDH 6.1.2 Implementation of AHFM6.1.3 Selecting suitable coefficients for AHFM
6.1.4 Set AHFM-Ht: first results on the influence of the selected AHFM parameters
6.1.5 Set AHFM-Ht: Results for the Pis’menny (2005b) data in upward flow conditions
6.1.6 Set AHFM-Ht: Results for the Watts (1980) data set in upward flow conditions
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136
136 140 145 147 150 156V
6.1.7 Set AHFM-Ht: Results for the Ornatskiy et al. (1980) data set in upward flow conditions
6.1.8 Set AHFM-Ht: Results for Carbon Dioxide in upward flow conditions
6.1.9 Set AHFM-Ht: Results for R23 in upward flow conditions 6.1.10 Set AHFM-Ht: Results downward flow conditions
6.1.11 Set AHFM-Sc: A first updating attempt 6.1.12 Final remarks
6.2 Implementation of AHFM in THEMAT
6.2.1 Adopted implementation strategy 6.2.2 First results6.2.3 A systematic application of the obtained model to Pis’menny et al. (2005a and 2005b) case series in the BSc work by Vera Papp
6.2.4 Further results: experimental conditions with supercritical carbon dioxide
6.3 Further improvements in predicting heat transfer to
supercritical fluids
6.3.1 Updated model 6.3.2 Obtained results
6.3.3 Sensitivity to boundary conditions
6.4 Concluding remarks
161 163 169 171 173 176176
177 178 180 184 185 185 190 195200
Chapter 7 –Fluid-to-fluid scaling to fluids at supercritical
pressure
7.1 Introductory remarks
7.2 Dimensionless parameters: previous results and
motivations for the present work
7.3 Updated theoretical considerations
7.4 Results obtained from RANS analyses
Reference Case 1: Water Data by Pis’menny et al. (2005a) Reference Case 2: CO2 Data by Fewster (1976)
Reference Case 3: Water Data by Watts (1980)
Reference Case 4: CO2 DNS Data by Bae et al. (2005)
Sensitivity analyses on Pressure selection
Effect of the inlet temperature on observed phenomena An approximate relation for the identification of a suitable
heat flux scaling factor Varying the pipe diameter
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203
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211
217
220 222 224 226 228 229 231 235VI
7.5 Results obtained from LES analyses
7.6 Results obtained from DNS analyses
7.7 Concluding remarks
236
241
245
Chapter 9 –Conclusions and future perspectives
Study on heat transfer to supercritical water in rod bundles Study on roughness effects on heat transfer to supercritical
fluids
Implementation of the AHFM in the energy equation of the Lien κ-ε (1996) turbulence model
Fluid-to-fluid scaling Future perspectives Final remark
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246 247 248 249 251 251References
252
VII
Roman Letters
A Flow area [m2]
A, A2, A2θ A3, aij Parameters in the R relation according to Kenjeres et al. (2005)
Bo* Buoyancy parameter CP1, CP2, CD1,
CD2
Parameters of the temperature variance equation Ct, Ct1, Ct2, Ct3,
Ct4
Constants of the AHFM model
𝐶𝑡𝑃𝑟𝑡𝑢𝑟 Substitute of the Ct parameter of AHFM when calculating the
modified Prt
Cε1, Cε2,Cε3 Parameters in the turbulence dissipation rate equation
𝐶𝑅𝑜𝑢𝑔ℎ Roughness Emprical Coefficient
cp Specific Heat [J/kgK]
D Pipe diameter [m]
Dh Hydraulic diameter [m]
f Friction factor fε1, fε2 Damping functions
fq(z) Heating distribution factor
𝑓𝑠𝑚𝑜𝑜𝑡ℎ Roughness smoothing function
Fr Froude number
𝐹𝑟𝐷 Froude number adopting the diameter as reference length
g Gravity [m/s2]
G Mass flux [kg/m2s]
Gk Production term of turbulence due to buoyancy [m2/s3]
Gr Grashof number
ID Internal Diameter [m] H Convective heat transfer coeffiient [W/m2K]
h Specific Enthalpy [J/kg] ht Auxiliary temperature squared fluctuation [K∙J/kg]
ℎ∗ Dimensionless Enthalpy
Kin , Kout Singular pressure drop coefficient at channel inlet and outlet
Krough Form drag coefficient of the roughness element
VIII L Channel length [m] NSPC Sub-pseudocritical number NTPC , NTPC′ Trans-pseudocritical numbers Nu Nusselt number P,p Pressure [Pa]
P/D Pitch over Diameter Ratio
Pk Production term of turbulence due to shearing [m2/s3]
Pr Prandtld number
Prt Turbulent Prandtl number
q’’ Heat flux [W/m2]
R Time scale ratio Re Reynolds number 𝑅𝑖𝐷 Richardson number
𝑆𝑘,𝑟𝑜𝑢𝑔ℎ Roughness additional production term of turbulence [m2/s3]
𝑆𝜀,𝑟𝑜𝑢𝑔ℎ Roughness additional production term of dissipation [m2/s4]
St Stanton number t Time [s] 𝑡′2 ̅̅̅̅, κ𝑡 Temperature variance [K2] T Temperature [K] Tin Inlet temperature [K] Tpc Pseudo-critical temperature [K] u, U Velocity [m/s] ux, w Axial velocity [m/s] u+ Dimensionless velocity v Radial velocity [m/s]
W, 𝑚̇ Mass flow rate [kg/s]
x Axial position [m]
y+ Dimensionless distance from the wall
Greek Letters
αt Eddy diffusivity [m2/s]
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δij Kronecker delta
ε Turbulent dissipation rate [m2/s3]
εt Dissipation rate of t'̅̅̅̅ 2 [K2/s]
𝜀𝑟𝑜𝑢𝑔ℎ Roughness height [m]
ε/D Roughness height over diameter Ratio 𝜀+ Dimensonless Roughness height
κ Turbulent kinetic energy [m2/s2]
Λ Friction parameter
λ Thermal conductivity [W/mK] λt Eddy thermal conductivity [W/mK]
μ Dynamic viscosity [Pa s] μt Eddy Dynamic viscosity [Pa s]
ν Kinematic viscosity [m2/s]
νt Eddy viscosity [m2/s]
𝛱ℎ Heated perimeter [m]
ρ Density [kg/m3]
σκ σε σh Model constants
τw Wall shear stress [Pa]
ω Specific dissipation rate [1/s]
Subscripts
0 Reference value b, bulk Bulk
in Inlet out Outlet
pc Calculated at the pseudo-critical temperature w wall
Superscripts
𝜑̅ Time Averaged Value 𝜑̃ Favre Averaged Value
X
𝜑′′ Favre Averaging Fluctuations
* starred quantities are dimensionless
Abbreviations
AHFM Algebraic Heat Flux Model AKN Abe Kondoh Nagano BWR Boiling Water Reactor CANDU Canadian Deuterium Uranium
DC Direct Current
CFD Computational fluid dynamics CRP Coordinated Research Project DNS Direct Numerical Simulation
GIF Generation IV International Forum
GGDH Generalized Gradient Diffusion Hypothesis HPLWR High Performance Light Water Reactor
IAEA International Atomic Energy Agency JAEA Japanese Atomic Energy Agency
JL Jones and Launder LES Large Eddy Simulation
LS Launder and Sharma LWR Light Water Reactor PWR Pressurised Water Reactor RANS Reynolds Averaged Navier Stokes SCWR Supercritical Water-Cooled Reactor SGDH Simple Gradient Diffusion Hypothesis
THINS Thermal Hydraulics of Innovative Nuclear Systems YS Yang-Shih
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The study of the complex phenomena occurring when dealing with fluids at supercritical pressures currently represents a relevant topic for several research and industrial sectors ranging from chemistry to power engineering. Following the path identified by previous studies performed at the University of Pisa (Sharabi, 2008; De Rosa, 2010; Badiali, 2011; Pucciarelli, 2013; Borroni, 2014), the present work aims at contributing to pave the way for the development of the SuperCritical Water-cooled Reactor (SCWR), one of the presently proposed concepts for the Generation IV of Nuclear Power Plants (see e.g., Pioro and Duffey, 2007; Oka et al., 2010; Schulenberg and Starflinger, 2012).
SCWR adopts supercritical water as primary coolant in the reactor core; inter alia, the forecasted improvements include both net thermodynamic efficiency increases and capital costs reduction in comparison to the present Pressurised Water Reactors (PWRs) and Boiling Water Reactos (BWRs). On the other hand, in order to achieve the feasibility of a prototype reactor, researchers have to cope with several issues, in particular related to material science and thermal-hydraulics, which make the development of all new Generation IV reactors a challenging endeavour. It must be also mentioned that supercritical pressure fluids are also proposed as working fluids for the secondary side of Generation IV reactors, while their importance in the case of fossil-fuelled power station is also well known.
The specific objective of present work is the improvement of the state-of-the-art in the field of CFD predictions of heat transfer to supercritical pressure fluids, which presently faces several issues in reproducing the involved phenomena, in particular in operating conditions close to the so-called pseudo-critical temperature. In fact, in correspondence of this temperature threshold, the fluids undergo strong thermodynamic property variations which give rise to complex phenomena, both involving heat transfer deterioration and enhancement; this situation is definitely challenging for the presently available prediction techniques, which often provide strong wall temperature overestimates when compared with the corresponding experimental data. Previous theoretical and experimental works, performed in past decades at different laboratories in several countries in the world, allowed to accumulate a considerable knowledge about phenomena and prediction tools,
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preparing for next steps that are anyway needed to improve present prediction capabilities (Bringer and Smith, 1957; Krasnoshchekov and Protopopov, 1959; Swenson et al., 1965; Ornatskiy et al., 1971; Jackson and Fewster, 1975; Fewster, 1976; Jackson and Hall, 1979a and 1979b; Watts, 1980; Jiang et al., 1995; Jackson, 2002; Jiang et al., 2004; McEligot and Jackson 2004; Bae et al., 2005; He et al., 2005; Pis’menny et al., 2005a; Pis’menny et al., 2005b; Cheng et al., 2007; Pioro and Duffey, 2007; Ambrosini and Sharabi, 2008; He et al., 2008; Kim et al., 2008; Loewenberg et al., 2008; Palko and Anglart 2008; Sharabi, 2008; Sharabi and Ambrosini, 2009; De Rosa, 2010; Zahlan et al., 2010; Zhang et al., 2010; Badiali, 2011; Mokry, 2011; Jaromin, 2012, Laurien, 2012; Jaromin and Anglart 2013; Zhao et al., 2013; Jackson, 2014).
Taking advantage from results obtained in the frame of previous studies performed at the University of Pisa, this work represents their natural follow-up, trying to achieve some of the improvements considered necessary in their recommendations. Attention is paid to the analysis of the effects of phenomena whose role in heat transfer to supercritical fluids is currently disregarded or not properly accounted for, such as roughness and buoyancy; heat transfer in 3D geometries relevant for industrial applications, such as rod bundles, is considered as well. A fluid-to-fluid scaling methodology is also proposed with the aim of trying to suggest a new similarity theory and providing interesting results which may inspire further improvements of the presently available predicting techniques.
The outcomes of the present work are contributing to the second IAEA Coordinated Research Project on SCWR, started in 2014 and entitled “Understanding and Prediction of Thermal-Hydraulics Phenomena Relevant to SCWRs”.
Thesis Outline
The thesis mainly compares the obtained computational results with a varied set of experimental data in terms of wall temperature; further analyses are reported as well, in order to try understanding the causes of the observed macroscopic heat transfer behaviour.
The present work is mainly divided into 8 chapters, including this Introduction, whose content is summarised below.
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Chapter 2 introduces the subject of the thesis. The main features of SCWR are briefly summarised together with the description of the main phenomena occurring when dealing with heat transfer to supercritical fluids. The experimental data sets adopted for comparison and validation of the proposed models are reported as well.
Chapter 3 summarises the predictive tools adopted in the present work. In particular, attention is paid to the description of the addressed RANS turbulence models and the considered Algebraic Heat Flux Model (AHFM) relation for the calculation of the turbulent heat flux. A brief summary about the adoption of the Favre averaging techniques is reported as well.
Chapter 4 presents the results obtained in the analysis of heat transfer to supercritical fluids flowing in rod-bundles. Together with the comparison of the calculated and measured wall temperature trends, further analyses are performed in order to try understanding the effect of the spacer grids on the fluid flow and heat transfer phenomena.
Chapter 5 proposes a new methodology for including wall roughness effects when dealing with Low-Reynolds κ-ε turbulence models. Starting from simple physical considerations, the methodology manages to reliably reproduce a significant part of the Moody’s diagram, both in transition and fully turbulent flow conditions. Its application to heat transfer to supercritical fluids provided limited results, though roughness is definitely to be considered one of the uncertain aspects having a role in the observed heat transfer behaviour. Chapter 6 describes the assumptions at the basis of the introduction of the
AHFM as a valid tool for the calculation of the turbulent heat fluxes in the RANS energy equation. The implementation was performed in different ways both in the commercial code STAR-CCM+ (CD-adapco, 2015) and in the in-house code THEMAT (Sharabi, 2008). The comparison of the obtained results with the experimental data is performed as well, showing promising capabilities. The latest improvements, obtained thanks to the development of the fluid-to-fluid scaling methodology presented in Chapter 7 are presented as well. A new set of parameters for AHFM is proposed: reliable results are finally obtained for a wide enough range of operating conditions including water and CO2 as supercritical fluids.
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Chapter 7 reports on the proposal of a new fluid-to-fluid scaling methodology to be applied to heat transfer to supercritical fluids. Adopting previously proposed dimensionless parameters and updated considerations, the new methodology finally manages to provide very good results, revealing the capability of reproducing similarities both for axial and radial trends as simulated by RANS, LES and DNS techniques, which were adopted in support to the reached conclusions.
Chapter 8 summarises the outcomes of the present work and provides recommendations for future developments.
Supercritical Fluids
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The present Chapter briefly summarizes the main features of SCWR, together with the latest proposed concepts and the related international background. The challenges to cope with for the development of the reactor will be presented, in particular focusing on the present understanding of the main phenomena involved in heat transfer to supercritical fluids, with main reference to the three main known heat transfer deterioration mechanisms. The present predicting capabilities of the available turbulence model will be discussed and the considered experimental data set will be briefly summarized.
2.1 SCWR: concepts and international background
In 2002, the Generation IV International Forum (GIF) selected six promising reactor concepts to be developed for the new generation of nuclear power plants. Among the objectives of these new designs, the increase of the plant safety, economical competitiveness, efficiency and proliferation resistance were particularly stressed. The six proposed concepts included both fast and thermal reactors and considered various fluids to be used as coolants in the primary loop; in particular, the Super Critical Water cooled Reactor (SCWR) is considered in the present work.
The SCWR represents the natural evolution of the presently available LWRs, sharing some features characterising both PWRs and BWRs; among the other proposals, it is the only one that could be achieved through an evolutionary path, by progressively increasing the operating pressure of the plants to be built, thus somehow compensating its research and development costs. This plant concept, in fact, takes advantage from two well experienced power engineering sectors, the nuclear LWRs and the conventional supercritical power plants, in order to develop a new reactor concept that could increase the present 30-35% efficiency of LWRs to the 40-45%, forecasted by several designers (see e.g., Pioro and Duffey, 2007; Oka et al., 2010; Schulenberg and Starflinger, 2012). Figure 2.1 reports the proposed overall plant concept.
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Figure 2.1 SCWR overall plant concept, adapted from Squarer et al. (2002)
Together with the mentioned efficiency increase, implying lower operational costs, the concept aims also at decreasing the capital costs since, owing to the absence of the two-phase transition between liquid and vapour granted by supercritical fluids, a simplified plant layout may be obtained with respect to PWRs; in particular, steam generators and separators will be no more required (Pioro and Duffey, 2007) and a single loop layout will be adopted, in similarity with the presently available BWRs. The development of SCWR obtained support in the past years through European funding projects; in particular, the FP5 HPLWR, FP6 HPLWR Phase 2 and the FP7 THINS projects have to be mentioned. In particular, in the frame of the latter project works were carried out at the University of Pisa on the several challenges posed by the prediction of heat transfer to supercritical fluids (e.g. Badiali, 2012; Pucciarelli, 2013; Borroni, 2014). Though nowadays the European Union is focusing mainly on the development of other Generation IV reactor concepts, in particular the Sodium-Cooled Fast Reactor, the Lead Cooled Fast Reactor and the Gas Cooled Fast Reactor, there is still interest in Europe and worldwide for supercritical water as a coolant and several research groups are still actively participating in developing the SCWR technology. Together with the European HPLWR concept, described by Schulenberg and
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Starflinger (2012), also Canada, China and Russian Federation are currently developing their own concepts.
The International Atomic Energy Agency coordinates part of the international efforts, through Coordinated Research Projects (CRP) on SCWR. In particular, a first CRP entitled “Heat Transfer Behaviour and Thermo-hydraulics Code Testing for SCWRs” started in 2008 and was completed in 2012; a second one, to which the present work gives contribution, started in 2014 and is entitled “Understanding and Prediction of Thermal-Hydraulics Phenomena Relevant to SCWRs”.
The different proposed designs follow different technological paths. The Canadian concept represents a supercritical water evolution of the present CANDU reactors, maintaining the calandria and heavy water as a moderator. The electrical net power is 1200 MWe, with a target operational life of about 75 years; concerning the operating conditions, instead, supercritical water at 25 MPa being heated throughout the core from 350 to 625 °C will be adopted as primary coolant. The development of a smaller version of the Canadian SCWR is also envisaged, in order to deal with both the large distances and the presence of small isolated centres characterizing the Canadian population distribution (Leung, 2016).
On the European side, the proposed HPLWR is more similar to the presently operating PWRs, though considering a single direct loop as in the BWRs. Among the various challenges to deal with, the possibility of considering a “three-pass” core has to be mentioned. In fact, by increasing the core outlet temperature, dangerous situations may be possibly faced in hot channels; as suggested by Starflinger and Schulenberg (2016), by considering a single pass core, temperatures up to 1200 °C may be obtained in the hot channel and material failure would be likely implied. The introduction of three passages in the core, (two upwards and one downward), with the interposition of a mixing chamber between each one, would help at maintaining a lower temperature peak, thus resulting in more feasible plant operating conditions; this way, a maximum cladding temperature of 625 °C, close to the expected 500 °C average outlet temperature, may be obtained. Figure 2.2 shows the three-pass-core design concept and a possible example of the core arrangement.
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Figure 2.2 Three-pass-core: design concept and possible core arrangement, adapted from Schulenberg et al. (2008)
As a consequence, the main problems for the development of SCWR seem to lay in the material limitations and in thermal-hydraulics; in the former field, together with the mentioned high temperature problems, a very important challenge is represented by the aggressive nature of supercritical water, which implies corrosion rates higher than the ones reported by fluids in standard conditions, in particular in the near critical region.
However, since the present work deals with fluiddynamic analyses, the problems encountered in modelling the heat transfer to supercritical fluids will be mainly considered; the next sections summarize both the present understanding of the involved heat transfer phenomena and the predicting capabilities of the currently available models.
2.2 Fluid Properties and Deterioration Mechanisms
Unlike fluids in standard conditions, fluids at supercritical pressure may undergo strong thermodynamic property variations, localised in the vicinity of the so called “pseudo-critical temperature”. This temperature represents a sort of threshold marking the transition between a high density fluid, in “liquid-like” conditions, and a lighter one, in “gas-like” conditions. In fact, being at a pressure above the critical one, no
two-9
phase transition may be observed, being the phase change phenomenon replaced by a single-phase process, in which the fluid properties may change sharply but continuously. Such changes really depend on the considered fluid pressure; in particular, as reported by Figures 2.3 to 2.6, which refer to water, they become milder as the pressure is increased above the critical value.
Figure 2.3 Water: Density trends at some supercritical pressures.
Figure 2.4 Water: Specific Heat trends at some supercritical pressures.
Figure 2.5 Water: Thermal Conductivity trends at some supercritical pressures.
Figure 2.6 Water: Dynamic viscosity trends at some supercritical pressures.
In particular, it is interesting highlighting the behaviour of the specific heat, which shows progressively lower peaks as the operating pressure is increased. In addition, its trend is relevant for the definition of the pseudo-critical temperature, being conventionally the temperature value at which the specific heat achieves its maximum, a temperature value which is shown to increase with pressure. It can be argued that the observed trends mimic, in a continuous and single-phase fashion, the changes that the fluid undergoes in the frame of a two-phase transition between liquid and vapour. In fact, in such conditions, being isothermal and isobaric, the specific heat may be considered infinite, something similar to the very high value reported in the
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supercritical fluid case; the other properties, though in a simpler way, also seem to mimic in a smoother way the changes observed for the two-phase transition.
The reported Figures are related to supercritical water; different fluids show anyway similar behaviour in qualitative terms.
Among the consequences of the fluid properties changes, the deterioration and
enhancement phenomena shown by heat transfer definitely are the most relevant ones.
Heat transfer deterioration is often defined as the decrease of the heat transfer capabilities in respect to the ones obtained in the correspondent fully developed turbulent conditions. In particular, firstly a reference correlation for the calculation of the Nusselt number in fully developed turbulence conditions has to be selected (in example the Krasnoschekov and Protopopov relation as in Jackson and Hall, 1979a); consequently, a threshold value for the ratio between the experimentally calculated Nusselt number and the one returned by the selected correlation has to be defined. If the obtained 𝑁𝑢𝐸𝑥𝑝
𝑁𝑢𝐶𝑜𝑟𝑟 is lower than the selected threshold, heat transfer deterioration is obtained; on the contrary, with 𝑁𝑢𝐸𝑥𝑝
𝑁𝑢𝐶𝑜𝑟𝑟 values higher than unity, we talk about heat transfer enhancement.
In example, the large increase of the specific heat in the temperature region around the pseudo-critical threshold implies a very peaked distribution of the Prandtl number, which consequently improves the heat transfer capabilities (enhancing them) in the same region (see e.g., Figure 2.7). As a consequence of the specific heat decrease in the gas-like region, instead, heat transfer gets impaired, making the fluid prone to possible strong temperature increases.
In addition, heat transfer deterioration and enhancement phenomena may occur even in regions far from the pseudo-critical temperature as an effect of buoyancy forces. This mechanism was observed by Jackson and Hall (1979a and 1979b) and is observed to possibly occur in vertical upward flow, when dealing with relatively low mass flux values. As a further complication, the buoyancy effects act differently for the upward and downward flow cases.
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Figure 2.7 Water: Prandtl number trend at some supercritical pressures.
In particular, in downward flow, heat transfer is mainly enhanced as the buoyancy effects improve the turbulence conditions. In fact, the hotter fluid in the vicinity of the wall is decelerated because of adverse buoyancy effects, thus increasing the wall shear stress; this in turn causes an increase in the production of turbulence due to shearing. As a global effect, heat transfer is then enhanced.
When considering the upward flow case instead, the situation becomes more complex; buoyancy may have in fact both positive and negative consequences. Figure 2.8 and Figure 2.9 describe the observed phenomena, which are commonly called “laminarization” and “heat transfer recovery”, for the resulting impairment and enhancement of heat transfer respectively.
Figure 2.8 Data by Watts (1980). Water, 25 MPa, 25.4 mm ID, Tin = 200°C, G=340 kg/m2s, q’’=250 kW/m2.Axial Velocity
Trends. Adapted from De Rosa (2010)
Figure 2.9 Data by Watts (1980). Water, 25 MPa, 25.4 mm ID, Tin = 200°C, G=340 kg/m2s, q’’=250 kW/m2.Turbulent Kinetic
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During the very first phase of the heating process, the classical mean axial velocity profile becomes flatter in the vicinity of the wall because of the acceleration caused by buoyancy effects. As a consequence, the shear stresses become smaller, thus resulting in a reduction of turbulence production due to shearing. As a consequence, the turbulent kinetic energy decreases, thus impairing the heat transfer. Proceeding downstream, buoyancy effects keep affecting the velocity field and, because of the acceleration close to the wall, the axial velocity profile becomes “M-shaped” (Jackson and Hall, 1979a and 1979b). With respect to the previous flatter shape, the “M-shaped” profile induces larger velocity gradients and stresses, thus causing the recovery of the turbulence production and of the heat transfer. The effect on the observed wall temperature trend is obviously an increase because of laminarization and a subsequent decrease, when instead recovery occurs. Figure 2.10 reports the calculated wall temperature trends for the same operating conditions considered in Figures 2.8 and 2.9.
Figure 2.10 Data by Watts (1980). Water, 25 MPa, 25.4 mm ID, Tin = 200°C, G=340 kg/m2s,
q’’=250 kW/m2.Adapted from De Rosa (2010)
Several authors tried to identify the flow and fluid conditions leading to a buoyancy induced heat transfer deterioration; in particular the definition of the buoyancy parameter, called Bo*, proposed by Jackson and Hall (1979b) is often adopted in the available literature. Bo* was later defined by (McEligot and Jackson, 2004) as
𝐵𝑜∗ = 𝐺𝑟
∗
𝑅𝑒3.425𝑃𝑟0.8 (2.1)
where 𝐺𝑟∗ = 𝑔𝛽𝑑4𝑞′′
13
According to this definition of buoyancy parameter, buoyancy induced heat transfer deterioration phenomena may be observed for 𝐵𝑜∗ > 6 ∙ 10−7; however, buoyancy effects already start affecting the heat transfer phenomenon for 𝐵𝑜∗ > 2 ∙ 10−7. The third heat transfer deterioration mechanism is caused by flow acceleration and usually occurs when dealing with operating conditions characterised by a very high mass flux, which makes the buoyancy effects negligible. In particular, this may especially occur when the bulk temperature, and not only the one in the vicinity of the wall, gets close to the pseudo-critical threshold thus implying strong density variations, along all the considered test section. Concerning the effects of flow acceleration on the axial velocity distribution, as it can be noted from Figure 2.11, it must be remarked that the profile remains turbulent, though progressively achieving higher velocity values as the density increases. Nevertheless, the changes in the velocity distribution may imply an impairment of the turbulent conditions thus inducing a deterioration of the heat transfer; this is clear when observing the changes occurring between the curves at x = 0.8 m and x = 1.2 m in Figure 2.12, which reports the turbulent kinetic energy trends calculated for the same operating conditions considered in Figure 2.11. Similarly to the definition of a buoyancy parameter, an acceleration parameter may be defined in order to check if acceleration may imply whether or not heat transfer deterioration is occurring. McEligot et al., (1970) introduced the parameter Kv defined
as
𝐾𝑣 =
4q’’dβ 𝑅𝑒2𝜇
𝑏𝐶𝑝,𝑏 (2.2) According to the authors, the value 𝐾𝑣 = 3 ∙ 10−6 represents the threshold which, once exceeded, implies the occurrence of heat transfer deterioration due to acceleration phenomena.
In addition, all the considered deterioration mechanisms seem to be very sensitive to changes in the operating conditions; as highlighted by several experimental results, in fact, regions exist in which even small changes in the operating conditions may imply very large variations in the observed measured trends.
14
Figure 2.11 Predictions for data by Ornatskiy (1971).Water, 25.5 MPa, 3 mm ID, Tin = 273 K, G=1500 kg/m2s, q’’=1810 kW/m2.Axial Velocity
Trends. Adapted from Pucciarelli, 2013
Figure 2.12 Predictions for data by Ornatskiy (1971). Water, 25.5 MPa, 3 mm ID, Tin = 273 K, G=1500 kg/m2s, q’’=1810 kW/m2.Turbulent
Kinetic energy Trends. Adapted from Pucciarelli, 2013
Figures 2.13 and 2.14 report two cases showing this feature. In particular Figure 2.13 reports the effect of mass flux variation for one of the operating conditions considered by Watts (1980) in his experimental campaign. As it can be noted, as the mass flux is increased, the heat transfer deterioration moves downstream, while the measured wall temperature peak is maintained almost constant. Nevertheless, when passing from 367 to 382 kg/m2s, with a relative mass flux increase of about 4%; the situation changes
abruptly; deterioration becomes milder and the measured trend changes both qualitatively and quantitatively. With further mass flux increases, deterioration almost disappears and normal-like heat transfer conditions are obtained.
Cases involving near critical conditions may be even more prone to changes. Figure 2.14 reports a sensitivity analysis concerning the inlet temperature for one of the experimental conditions considered by Fewster (1976) in his experimental campaign. Strong heat transfer deterioration phenomena are shown when imposing low inlet temperatures, while when approaching the pseudo-critical temperature, deterioration disappears. Nevertheless, this variation does not occur progressively and an abrupt change occurs when moving from 24.5 to 25 °C passing in practice from deteriorated to normal heat transfer conditions.
All the presented phenomena are very challenging to deal with by predictive techniques, whose main past results, capabilities and weaknesses are summarised in the next section.
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Figure 2.13 Data by Watts (1980). Water, 25 MPa, 25.4 mm ID, Tin = 150°C, q’’=250 kW/m2.Measured trends for different mass flux
values.
Figure 2.14 Data by Fewster (1976). CO2, 7.584 MPa, 7.88 mm ID,𝑚̇ = 0.02 𝑘𝑔/𝑠 , q’’=33.6
kW/m2.
2.3 Heat transfer prediction capabilities
During the last years, several modelling techniques have been proposed in order to deal with heat transfer to supercritical fluids.
Concerning the classical heat transfer correlations, many proposals are currently available in literature, though none of them has general application (see, e.g., Pioro and Duffey, 2007). Large part of these correlations were in fact developed focusing on specific and limited data sets; in particular, they are often related to a specific fluid (e.g., Bishop et al., 1964). Other attempts, such as the correlations developed by Bringer and Smith (1957), Swenson et al. (1965), Krasnoshchekov et al. (1967) try to broaden the applicability range by considering both supercritical water and carbon dioxide, though maintaining well defined validity ranges in terms of Reynolds number, temperature and mass flux values.
In this frame, the works performed by Jackson (see e.g. Jackson and Fewster, 1975; Jackson, 2002; Jackson, 2014) have to be mentioned. With the introduction of a buoyancy parameter, Jackson tried to identify the different flow patterns observed in the study of supercritical fluids, thus modifying the predicted heat transfer coefficient accordingly; in particular, attention has been paid to the different behaviours obtained for upward and downward flow conditions. Though in the last years improvements
16
have been obtained, the usual ± 20% uncertainty range on heat transfer coefficient was not yet achieved. In this regard, the work by Mokry (2011) can be considered remarkable; in fact, by taking into account a set of data considering operating conditions relevant for SCWR applications, the Author managed to obtain a ± 25% uncertainty in the prediction of the heat transfer coefficient. Operating conditions both reporting deterioration and enhancement were considered; promising results were obtained for normal and improved heat transfer conditions, while larger discrepancies were observed for cases reporting large heat transfer deterioration phenomena.
Among the various adopted predictive techniques, during the last decade CFD analyses started to become a valuable alternative both because of their increasing accuracy and decreasing computational cost. In particular, since in the particular field of heat transfer to supercritical fluids no relevant success could be claimed in previous literature by CFD models, this field has been selected for the main topic of the present work with the aim of improving the state of the art.
In the frame of previous works, both performed at the University of Pisa (Sharabi, 2008; De Rosa, 2010; Badiali, 2012; Pucciarelli, 2013; Borroni, 2014) and in other institutions (see e.g. He et al., 2005; Cheng et al., 2007; He et al., 2008; Kim et al., 2008; Palko and Anglart 2008; Jaromin and Anglart, 2013 ), weaknesses and capabilities of the currently available modelling techniques have been evaluated; a review of the obtained results is reported in the present section.
In the frame of the first studies performed in Pisa, the use of traditional RANS two-equation turbulence models was considered; some interesting features of these models were pointed out, though relevant difficulties in dealing with near critical conditions were highlighted as well. Figures 2.15 to 2.18 are chosen to summarise some typical obtained results.
In particular, Figures 2.15 and 2.16 refer to cases in which the pseudo-critical temperature is exceeded after a certain axial position. As it can be noted, unlike the measured trends, the predictions provided by the considered models tend to report too early and strong temperature increases; in particular, deterioration often occurs as the pseudo-critical temperature is exceeded at the wall revealing a certain unsuitability of the models in dealing with such a situation. Nevertheless, it must be noted that this
17
behaviour is usually reported by κ-ε models, while the κ-ω ones tend to report trends showing lower temperature excursions.
On the other hand, when dealing with conditions far from the pseudo-critical temperature, κ-ε models really seem being able to provide sufficiently reliable predictions. In fact, as Figures 2.17 and 2.18 clearly show, the Lien κ-ε model (Lien et al., 1996) manages to reproduce quite correctly the transition from deteriorated to normal heat transfer obtained when varying the mass flux and shows even better capabilities in the cases at lower temperature reported in Figure 2.18. In such conditions, the selected κ-ω model cannot reproduce the observed deterioration and returns instead normal heat transfer conditions.
Figure 2.15 Data by Pis’menny (2005a). Water, 23.5 MPa, 6.28 mm ID, Tin = 300°C,
G=509 kg/m2s, q’’=390 kW/ m2.
Adapted from Sharabi, 2008
Figure 2.16 Data by Fewster (1976):CO2, 7.584 MPa, 5.1 mm ID, Tin = 20°C, G = 631.48 kg/(m2s), q’’ = 68 kW/m2.
Adapted from Badiali, 2011
Figure 2.17 Data by Watts (1980). Water, 25 MPa, 25.4 mm ID, Tin = 200°C, q’’=250 kW/m2.Adapted from De Rosa, 2010
Figure 2.18 Data by Pis’menny (2005b). Water, 23.5 MPa, 6.28 mm ID, Tin = 17 °C,
G=248 kg/m2s, q’’=181 kW/ m2.
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After this very first studies, it was clear that, despite the observed inaccuracies, κ-ε models seemed having the correct ingredients needed for predicting the occurrence of heat transfer deterioration; nevertheless, further efforts were necessary in order to cope with the problems encountered with these models when exceeding the pseudo-critical temperature.
With the aim of obtaining better results, Zhang et al., (2010) proposed a four-equation turbulence model specifically developed for dealing with heat transfer to supercritical fluids. The very feature of their proposal was the adoption of the algebraic heat flux model (AHFM), a formulation used to evaluate the turbulent heat flux described in Chapter 3, which is proposed as a valuable tool for improving the prediction of the production of turbulence kinetic energy due to buoyancy. Notwithstanding the improved evaluation of the turbulent heat flux, the usual simple gradient diffusion hypothesis was instead maintained for the purpose of calculating the turbulent heat flux contribution in the energy balance equation. The AHFM relation as proposed by Zhang et al. (2010) is u' it' ̅̅̅̅= -Ct κ ε[Ct1u̅̅̅̅̅̅ ∂T'iu'j∂xj+(1-Ct2)u ' jt' ̅̅̅̅ ∂𝑢̅i ∂xj+(1-Ct3)βgi t '2 ̅ ] (2.3)
As it can be noted, it mainly consists of three contributions and the definition of four coefficients is required for its application. By performing a tuning of the model coefficients, Zhang et al. (2010) managed to obtain promising results for some considered experimental conditions by Alekseev et al. (1976) for water at a supercritical pressure of 24.0 MPa, thus suggesting a possible route for model development.
On the basis of such results, similar analyses were performed at the University of Pisa (Pucciarelli, 2013; Borroni, 2014; Pucciarelli et al. 2015) by considering the use of the AHFM for the purpose of calculating the production term of turbulence due to buoyancy to be adopted in association with four different “four-equation” turbulence models (Abe et al., 1995; Hwang and Lin, 1999; Deng et al., 2000; Zhang et al., 2010) which were implemented in the in-house code THEMAT (Sharabi, 2008). Promising results were obtained; in particular, the Zhang et al. (2010) model confirmed some interesting capabilities in the near critical region though often reporting strong wall temperature underestimations for operating conditions far from the pseudo-critical
19
threshold. On the other hand, the Abe et al. (1995) model showed interesting capabilities in predicting buoyancy induced heat transfer deterioration phenomena in the region far from the pseudo-critical temperature, where instead reported a strong sensitivity when exceeding this threshold.
Figures 2.19 to 2.21 report some of the obtained results, highlighting both some interesting features and weaknesses of the adopted methodology. In particular, Figure 2.19 reports the same experimental conditions considered in Figure 2.16; by comparing the results reported in the two Figures, it is clear that by adopting the Zhang et al. (2010) model relevant improvements were obtained. Figure 2.20 reports instead the results of some sensitivity analyses performed with the aim of understanding the actual relevance of each of the AHFM contributions in defining the obtained trends.
Figure 2.19 Data by Pis’menny (2005a). Water, 23.5 MPa, 6.28 mm ID, Tin = 300°C, G=509 kg/m2s, q’’=390 kW/ m2.Adapted from
Pucciarelli (2013)
Figure 2.20 Data by Pis’menny (2005a). Water, 23.5 MPa, 6.28 mm ID, Tin = 300°C, G=508 kg/m2s, q’’=390 kW/ m2.Effect of
various AHFM contributions. Adapted from Pucciarelli (2013)
Figure 2.21 Data by Pis’menny (2005b). Water, 23.5 MPa, 6.28 mm ID, Tin = 17 °C, G=248 kg/m2s, q’’=181 kW/ m2.Adapted from Pucciarelli (2013)
20
As highlighted by Figure 2.20, the first and the second contributions in Eq (2.3) are required for predicting the heat transfer impairment due to buoyancy effects, the third one is instead necessary for predicting the heat transfer recovery phase which is instead absent when this is neglected. This can be simply deduced by observing the corresponding temperature trends, which report strong temperature increases when considering the first two components, and return a definitely lower temperature trend when instead adopting the last contribution only. In particular, this last contribution is connected with the temperature variance, a quantity that is not calculated by commonly used two-equation models; the introduction of the AHFM thus requires adopting more complex turbulence models, at least considering three equations, which increases the computational effort. Figure 2.21 again reports further results for the same operating conditions considered in Figure 2.18; as it can be easily noted, the Zhang et al. (2010) model cannot reproduce the observed heat transfer deterioration, while the Abe et al. (1995) model instead manages to provide better results, in similarity with the previously considered Lien κ-ε turbulence model (Lien et al., 1996).
Improvements could then be obtained in the near critical region by adopting the Zhang et al. (2010) model; nevertheless, the same technique seems unsuitable for lower temperature applications. However, the introduction of AHFM was recognised as having a positive effect on the quality of predictions. In addition, it helped at highlighting some threshold phenomena that the considered models seem reporting. Figure 2.22 shows the results of further sensitivity analyses, performed for the same case of the Fewster (1976) data set considered in Figure 2.16, adopting the Abe et al., (1995) model, for different values of one of the coefficients appearing in the dissipation rate balance equation.
As it can be noted, in this case the model seems being very sensitive to changes in the coefficients when approaching a certain threshold, 𝐶𝜀1= 1.4464, which defines the boundary for the occurrence of the heat transfer deterioration phenomena. Similar behaviours were recognised also for variations in the boundary conditions, somehow reproducing the trends highlighted in Figures 2.13 and 2.14. In this respect, the AHFM seemed to be a useful ingredient that, with further efforts, may grant the possibility of obtaining better predictions.
21
Figure 2.22 Data by Fewster (1976):CO2, 7.584 MPa, 5.1 mm ID, Tin = 20°C, G = 631.48 kg/(m2s), q’’ = 68 kW/m2.
Sensitivity analysis for Cε1 parameter in AKN, 1994 model. Adapted from Pucciarelli (2013)
As a natural following step, the introduction of AHFM for the calculation of the turbulent heat fluxes even in the energy equation, thus replacing the usual simple gradient hypothesis, was considered in the present work. The results of a recent work performed by Xiong and Cheng (2014) provided as a further motivation for this development. In fact, in their work, Xiong and Cheng considered the application of a modified form of the AHFM relation, called EB-AHFM, as a useful tool for predicting heat transfer to supercritical fluids, in association with the κ-ε-ζ-f model. In particular, the EB-AHFM is obtained through an elliptical blending approach (Dehoux et al., 2012) of the AHFM relation and one of its features is to account for variable coefficients for the AHFM relation, which are calculated on the basis of an additional parameter, obtained solving a dedicated balance equation. Xiong and Cheng (2014) adopted the mentioned model combination for reproducing the results obtained by Bae et al. (2005) in their DNS, obtaining promising results; unfortunately, no further comparison with other data was reported.
Following a simpler approach, in order to understand the consequences of the adopted modelling techniques, in the present work AHFM has been initially used by assuming constant coefficients. The analysis of a wider range of experimental data is also carried out, with the objective of trying to obtain improvements in a large enough number of operating conditions, being convinced that showing good result in front of single experimental cases is not sufficient to claim success in the solution of the challenging problems involved in the prediction of heat transfer to supercritical fluids.
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2.4 Considered experimental data sets
All the considered experimental conditions, except for a few cases, lay in the buoyancy affected region. Heat transfer deterioration is consequently often implied by buoyancy phenomena; the exceeding of the pseudo-critical threshold also helps at obtaining heat transfer deterioration for the cases reporting borderline Bo* values.
Data by Ornatskiy et al. (1971)
Ornatskiy et al. (1971) performed experiments considering water flowing in circular pipes at the supercritical pressure of 25.5 MPa. In comparison to other data sets considered in the present work, these data show high mass flux values, which make buoyancy effects almost negligible, while more likely reporting an acceleration induced heat transfer deterioration.
Unfortunately, it was not possible finding the original paper by the Auhors; the information was consequently taken from a secondary source (Yu Zhu, 2010). Concerning the geometry, 3 mm diameters pipes were considered; the total heated length was 1.5 m long, with the usual unheated entrance region being 0.5 m long. Just one of the operating conditions investigated by Ornatskiy et al. (1971) is considered in the present work; Table 2.1 summarises its boundary conditions.
For this case, heat transfer deterioration is not implied by buoyancy effects as the buoyancy parameter is well below the 𝐵𝑜∗ = 2 ∙ 10−7 threshold; deterioration is consequently mainly due to the exceeding of the pseudo-critical temperature.
Inlet Temperature [K] Heat flux [kW/ m2] G [kg/m2s] Flow direction
300 1810 1500 Upward
Table 2.1 Operating conditions considered in present work for the data by Ornatskiy (1971)
Data by Fewster (1976)
These data were collected at the University of Manchester, in the frame of Fewster’s PhD thesis and recently released in a report provided in the frame of the first IAEA CRP on SCWR (Jackson, 2009b). In particular, heat transfer to supercritical carbon dioxide was investigated; several fluid pressures were considered; nevertheless, cases
23
involving an operating pressure of 7.584 MPa only will be considered in the present work. Experiments were performed with different pipe diameters (5.1, 7.88 and 19 mm ID) both in upward and downward flow conditions.
Reporting a low pseudo-critical temperature (roughly at 32 °C) a pre-cooler was included before the inlet section in order to reach bulk inlet temperatures close to 0°C thus broadening the investigated flow regime range. A preheater is also present in order to reach near critical inlet temperatures as well. In particular, as Figure 2.23 clearly shows, the two devices are arranged in parallel, in order to be able to select the desired tool.
Figure 2.23 Scheme of the test facility adopted for the experiments by Fewster (1976), from which the Figure is adapted.
Concerning the geometrical data, a heated length of 150 diameters was considered for the cases adopting the pipes with the smaller diameter; the heated length is instead 129 diameters for the 19 mm ID pipe. An unheated entrance region was considered for all the experimental conditions, being 75 diameters for the smaller pipes and 64 diameter for the larger one; in the former case, a final unheated length of 75 diameters was adopted as well.
Inlet and outlet bulk temperatures were measured at the corresponding sections, while wall temperature was instead measured by adopting cromel-alumel thermocouples localized along the heated sections.
24
Though a very large number of experimental conditions were investigated, only a representative part of them has been selected in the present work; Table 2.2 summarises the considered boundary conditions.
Inlet Temperature [°C] G [kg/m2s] Heat flux [kW/ m2] D [mm] Upward Downward 20 631.48 68 5.1 ✔ ✔ 13 3157.4 318 5.1 ✔ ✘ 11.5 283.0 17.7 5.1 ✔ ✘ 8.5 184.54 6.64 7.88 ✔ ✘ 7.5 to 28 410.1 33.6 7.88 ✔ ✘ 8 205.2 6.95 19 ✔ ✘
Table 2.2 Operating conditions considered in present work for the data by Fewster (1976)
Concerning the deterioration mechanisms, buoyancy and exceeding the pseudo-critical temperature in the vicinity of the wall are the leading phenomena for all the considered cases, except for the one reporting the highest mass flux in which no buoyancy effects were observed.
Data by Watts (1980)
These data were collected at the University of Manchester; nevertheless, unlike Fewster (1976), Watts used water at the supercritical pressure of 25 MPa. In comparison with the other presently considered data sets, the operating conditions by Watts (1980) report the largest pipe diameter, 25.4 mm, being selected for reproducing the size of pipes usually adopted in power plant steam generators. The 2 m adopted heated length is very short if compared to the ones adopted in power plants, which may even be 40 m long; nevertheless, considering longer pipes would have not been feasible because of obvious space and power input limitations in the laboratory. Figure 2.24 shows the test facility adopted for the mentioned experiments.
Pressures up to 30 MPa and temperatures up to 450 °C could be considered as upper safety bounds; a maximum heat flux of 750 kW/m2, obtained by direct heating, could
25
for the bulk conditions, while the wall temperature was measured along the heated length with the help of 50 thermocouples. In particular, the temperature on the external side of the pipe was measured; the internal wall temperature was subsequently deduced by performing a data reduction, considering the conductive effects occurring in the pipe wall. Table 2.3 reports the operating conditions selected for assessing the capabilities of the models considered in the present work. In particular, the nominal values are here reported; measurements inaccuracies were however experienced and sensitivity analyses on heat and mass flux variations are reported in Chapter 6.
26 Nominal inlet Temperature [°C] Heat flux [kW/ m2] G [kg/m2s] Flow direction 150 250 273-367 Upward 150 340 364 Upward 200 250 269-318-340-356 Upward 200 400 390-562 Upward 250 340 392 Upward 250 400 477 Upward 310 400 361-505 Upward
Table 2.3 Operating conditions considered in present work for the data by Watts (1980)
Data by Pis’menny et al. (2005a and 2005b)
These data were collected at the National Technological University of Ukraine; a sketch of the adopted test facility is reported in Figure 2.25. The selected operating conditions concerned water at a supercritical pressure of 23.5 MPa; both upward and downward flows were considered. A maximum temperature of 600 °C, then in the range of the SCWR possible outlet temperature, could be afforded, while the pressure could reach 28 MPa. Circular pipes with an internal diameter of 6.28 and 9.50 mm were considered; in particular, with the aim of simulating the bare subchannels of SCWR fuel bundles.
27
The inlet temperature could be set with the help of a pre-heater, power was instead supplied along the test section by Joule effect, with direct electrical current passing through the pipe walls. The wall temperature was measured along the test section with the help of 13 thermocouples; inlet and outlet bulk temperature were measured as well. Several experiments with different mass and heat flux values were performed; the operating conditions considered in the present work are reported in Table 2.4
Inlet Temperature [°C] G [kg/m2 s] Heat flux [kW/ m2] D [mm] Upward Downward 17 248 113-370 6.28 ✔ ✘ 200 249 76-289-361 6.28 ✔ ✘ 300 509 390 6.28 ✔ ✔ 100 248 118-275-396 9.5 ✔ ✘
Table 2.4 Operating conditions considered in present work for the data by Pis’menny (2005)
Data provided by Delft University (2014)
These data were collected in the frame of the EU THINS project; the University of Pisa participated by performing predictions of the considered operating conditions and the first results were presented in the frame of the THINS project closure meeting held in Mol in February 2015. Further analyses were subsequently performed by adopting the updated modelling techniques presented in Chapter 6.
Unlike the other data sets, an annular flow was selected in this case; R23 at a supercritical pressure of 5.7 MPa was considered as working fluid. The test section was 6.18 m long; nevertheless, only five wall temperature measurements were available, all located in the very final part of the heated length, which was 1.069 m long. Concerning the annular section, the outer pipe diameter was 3.5 mm while the inner one, from which power was provided, was instead 2.54 mm.
For this set of data, exceeding the pseudo-critical temperature in the vicinity of the wall is the leading phenomenon for the occurrence of heat transfer deterioration. Table 2.5 summarises the investigated operating conditions.
28
Inlet Temperature [°C ] Mass Flow Rate [g/s] Supplied Power [W]
24.90 31.76 1882
24.83 32.22 4391
29
In the present Chapter, the modelling techniques adopted in the following sections are described; in particular, the turbulence model equations and the related parameter definitions are reported.
In addition, a brief description of the Algebraic Heat Flux Model (AHFM) is included as well since, in the present work, it has been widely used as a valid tool for the calculation of the turbulent heat fluxes.
Finally, some information about the Favre-averaging technique is reported, being the considered approach for the calculation of the time averaged quantities in the frame of the LES calculations reported in Chapter 7. In fact, since they account for density fluctuations, they seem more suitable than classical Reynolds-averaging techniques, when dealing with fluids at supercritical pressure.
3.1 RANS techniques: considered turbulence models
The very feature of performing RANS calculations is the resolution of time-averaged balance equations; the intrinsic instantaneous flow fluctuations, both in the thermal and velocity fields thus are not described. In terms of computational time, this leads to less expensive calculations, thus implying a lower accuracy with respect to more complicated techniques such as LES and DNS.
Nevertheless, the instantaneous fluctuations do have an important role in defining flow patterns and heat transfer and cannot be disregarded. In fact, as well known, the balance equations time-averaging process implies the presence of additional terms, namely the double and triple correlations, which essentially represent the time-averaged values of the product of the fluctuations of some relevant quantities. These terms need to be explicitly modelled and different techniques are currently available in literature.
The common approach assumes that the turbulent contribution may act in similarity with the molecular diffusion. The introduced two equalities, reported below, are the
30
Boussinesq’s approssimation for the velocity field (Eq. 3.1) and the Simple Gradient Hypothesis (SGDH) for the thermal one (Eq. 3.2).
−𝜌𝑢̅̅̅̅̅̅ = 𝜇𝑖′𝑢𝑗′ 𝑡( 𝜕𝑢𝑖 𝜕𝑥𝑗 +𝜕𝑢𝑗 𝜕𝑥𝑖 ) − 2 3𝜌𝛿𝑖𝑗𝜅 (3.1)
where τij = − 𝑢̅̅̅̅̅̅𝑖′𝑢𝑗′, is also known as the (specific) Reynolds Stress tensor and
−𝜌𝑢̅̅̅̅̅ = 𝑖′𝑡′ 𝜇𝑡 𝑃𝑟𝑡 𝜕𝑇 𝜕𝑥𝑖 = 𝛼𝑡 𝜕𝑇 𝜕𝑥𝑖 (3.2) expressing the turbulent heat flux in its simplest form (Simple Gradient Diffusion Hypothesis, SGDH).
As it can be noticed, their introduction requires the knowledge of three further parameters: the turbulent kinetic energy κ, the turbulent viscosity μt and the turbulent
Prandtl number Prt. In particular, the turbulent kinetic energy is defined as
𝜅 = 1 2 𝑢𝑖 ′𝑢 𝑖 ′ ̅̅̅̅̅̅ =1 2 ( 𝑢1 ′ 2 ̅̅̅̅̅ + 𝑢̅̅̅̅̅ + 𝑢2′ 2 3 ′ 2 ̅̅̅̅̅) [𝑚2 𝑠2] (3.3)
These quantities may be obtained in several ways, nevertheless the most common path is represented by two-equation turbulence models. In particular, they solve two further transported scalar equations, one of which always regarding the turbulent kinetic energy, with the aim of obtaining an estimate of the turbulent viscosity and diffusivity to be adopted in the momentum and energy equations respectively. The turbulent Prandtl number is instead usually considered as a constant input value, commonly set to 0.85-0.90.
